text
stringlengths 1.79k
370k
| summary
stringlengths 34
23.5k
|
|---|---|
You are an expert at summarizing long articles. Proceed to summarize the following text:
observations of black hole demographics locally is increasingly providing a strong constraint on models that explain the assembly and growth of black holes in the universe . the existence of a tight relation between the velocity dispersion of bulges and the mass of the central black hole has been reported by several authors ( merritt & ferrarese 2001 ; tremaine et al . 2002 ; gebhardt et al . this correlation is tighter than that between the luminosity of the bulge and the mass of the central black hole ( magorrian et al . 1998 ) . the physical processes that set up this correlation are not fully understood at the present time , although there are several proposed explanations that involve the regulation of star formation with black hole growth and assembly in galactic nuclei ( haehnelt , natarajan & rees 1998 ; natarajan & sigurdsson 1998 ; silk & rees 1999 ; murray , quataert and thompson 2004 ; king 2005 ) . recent work by several authors has suggested that umbhs are hereafter referred to as umbhs . ] ought to exist : bernardi et al . ( 2006 ) show that the high velocity dispersion tail of the velocity distribution function of early - type galaxies constructed from the sloan digital sky survey ( sdss ) had been under - estimated in earlier work suggestive of a corresponding high mass tail for the central black hole masses hosted in these nuclei . as first argued by lauer et al . ( 2007a ) and subequently by bernardi et al . ( 2007 ) and tundo et al . ( 2007 ) , even when the scatter in the observed @xmath4 correlation is taken into account it predicts fewer massive black holes compared to the @xmath5 relation . while bernardi et al . ( 2007 ) argue that this is due to the fact that the @xmath6 relation in currently available samples is inconsistent with the sdss sample from which the distributions of @xmath7 or @xmath3 are based . from an early - type galaxy sample observed by hst , lauer et al . ( 2007b ) argue that the relation between @xmath5 is likely the preferred one for bcgs ( brightest cluster galaxies ) consistent with the harboring of umbhs as evidenced by their large core sizes . the fact that the high mass end of the observed local black hole mass function is likely biased is a proposal that derives from optical data . deriving the mass functions of accreting black holes from optical quasars in the sloan digital sky survey data release 3 ( sdss dr3 ) , vestergaard et al . ( 2008 ) also find evidence for umbhs in the redshift range @xmath8 . in this paper , we show that umbhs exist using x - ray and bolometric agn luminosity functions and for consistency with local observations of the bh mass density , an upper limit to their masses is required . to probe the high mass end of the bh mass function , in earlier works the agn luminosity functions were simply extrapolated . this turns out to be inconsistent with local estimates of the bh mass function . here we focus on the high mass end of the predicted local black hole mass function , i.e. extrapolation of the @xmath4 relation to higher velocity dispersions and demonstrate that a self - limiting cut - off in the masses to which bhs grow at every epoch reconciles the x - ray and optical views . the outline of this paper is as follows : in section 2 , we briefly summarise the current observational census of black holes at high and low redshift including constraints from x - ray agn . the pathways to grow umbhs are described in section 3 . derivation of the local black hole mass function from the x - ray luminosity functions of agn is presented in section 4 . the arguement for the existence of an upper limit to black hole masses from various lines of evidence is presented in section 5 ; the prospects for detection of this population is presented in section 6 followed by conclusions and discussion . we adopt a cosmological model that is spatially flat with @xmath9 ; @xmath10 . the demography of local galaxies suggests that every galaxy hosts a quiescent supermassive black hole ( smbh ) at the present time and the properties of the black hole are correlated with those of the host . in particular , observational evidence points to the existence of a strong correlation between the mass of the central black hole and the velocity dispersion of the host spheroid ( tremaine et al . 2002 ; merritt & ferrarese 2001 , gebhardt et al . 2002 ) in nearby galaxies . this correlation strongly suggests coeval growth of the black hole and the stellar component via likely regulation of the gas supply in galactic nuclei ( silk & rees 1999 ; kauffmann & haehnelt 2000 ; cattaneo 2001 ; bromley , somerville & fabian 2004 ; king 2003 ; murray , quataert & thompson 2005 ; sazonov et al . 2005 ; begelman & nath 2005 ; alexander et al . 2005 ) . black hole growth is primarily powered by gas accretion ( lynden - bell 1969 ) and accreting black holes that are optically bright are detected as quasars . the build - up of smbhs is likely to have commenced at extremely high redshifts . indeed , optically bright quasars have now been detected at @xmath11 ( e.g. , fan et al . 2001a , 2003 ) in the sdss . there are also indications that high redshift quasar hosts are strong sources of dust emission ( omont et al . 2001 ; cox et al . 2002 ; carilli et al . 2002 ; walter et al . 2003 ; reuland et al . 2004 ) , suggesting that quasars were common in massive galaxies at a time when galaxies were undergoing copious star formation . the growth spurts of smbhs are also detected in the x - ray waveband . the summed emission from these agn generates the cosmic x - ray background ( xrb ) , and its spectrum suggests that most black - hole growth is optically obscured ( fabian 1999 ; di matteo et al . 1999 ; mushotzky et al . 2000 ; hasinger et al . 2001 ; barger et al . 2003 ; barger et al . 2005 ; worsley et al . there are clear examples of obscured black - hole growth in the form of ` type-2 ' quasars , and the detected numbers are in agreement with some recent xrb models ( treister & urry 2005 ; gilli et al . 2007 ) and have the expected luminosity dependence of the obscured fraction . additionally , there is tantalizing recent evidence from infra - red ( ir ) studies that dust - obscured accretion is ubiquitous ( martinez - sansigre et al . 2005 , 2007 ) . at present it is unknown what fraction of the total mass growth occurs in such an optically dim phase as a function of redshift . the build - up of bh mass in the universe has been traced using optical quasar activity . the current phenomenological approach to understanding the assembly of smbhs involves optical data from both high and low redshifts . these data are used to construct a consistent picture that fits within the larger framework of the growth and evolution of structure in the universe ( haehnelt , natarajan & rees 1998 ; haiman & loeb 1998 ; kauffmann & haehnelt 2000 ; 2002 ; wyithe & loeb 2002 ; volonteri et al . 2003 ; di matteo et al . 2003 ; steed & weinberg 2004 ) . black hole accretion histories derived from the quasar luminosity function ( e.g. soltan 1982 ; haehnelt , natarajan & rees 1998 ; salucci et al . 1999 ; yu & tremaine 2002 ; marconi et al . 2004 ; shankar et al . 2004 ; merloni et al . 2004 ) , synthesis models of the xrb ( e.g. comastri et al . 1995 ; gilli et al . 1999 ; elvis et al . 2002 ; ueda et al . 2003 ; barger et al . 2005 ; treister & urry 2005 ; gilli et al . 2007 ) , and observations of accretion rates in quasars at different redshifts ( vestergaard 2004 ; mclure & dunlop 2004 ) and composite models ( hopkins et al . 2005b ; 2006a ; 2006b ) suggest that supermassive black holes spend most of their lives in a low efficiency , low accretion rate state . in fact , only a small fraction of the smbhs lifetime is spent in the optically bright quasar phase , although the bulk of the mass growth occurs during these epochs . in this paper , we examine the consequences of such an accretion history for the high mass end of the local black hole mass function . surveys at x - ray energies allow us to obtain a more complete view of the agn population , as they cover a broader range in luminosity and are simultaneously less affected by baises due to obscuration . while optical surveys of quasars , like the sdss or 2df , are used to obtain a large sample of unobscured and high - luminosity sources , it is with x - ray surveys that the obscured low - luminosity population can be well traced . in particular , surveys at hard x - ray energies , 210 kev , are almost free of selection effects up to columns of @xmath12 . in the work of ueda et al . ( 2003 ) the agn x - ray luminosity function is computed based on a sample of @xmath13250 sources observed with various x - ray satellites . one of the important conclusions of this paper is the confirmation of a luminosity - dependent density evolution , in the sense that lower luminosity sources peak at lower redshifts , @xmath14 , while only the high luminosity sources are significantly more abundant at @xmath15 , as observed in optical quasar surveys ( e.g. , boyle et al . additionally , using this x - ray luminosity function and evolution it was possible for ueda et al . ( 2003 ) to convincingly account for the observed properties of the extragalactic xrb . extending the argument presented by soltan ( 1982 ) to the x - ray wave - band , agn activity can be used to trace the history of mass accretion onto supermassive black holes ( fabian & iwasawa 1999 ) . marconi et al . ( 2004 ) and shankar et al . ( 2004 ) used the luminosity function of ueda et al . ( 2003 ) to calculate the spatial density of supermassive black holes inferred from agn activity and compared that with observations . these authors reported in general a good agreement between observations and the density inferred from agn relics , suggesting that there is little or no room for further obscured accretion , once compton thick agn are properly accounted for . a similar conclusion was also obtained by barger et al . ( 2005 ) from an independently determination of the luminosity function , thus confirming this result . below we discuss plausible scenarios for forming these umbhs at low redshift . there are 2 feasible channels for doing so : ( i ) expect extremely rare umbhs to form from the merging of black holes due to the merging of galaxies via the picture suggested by volonteri et al . ( 2003 ) ; ( ii ) form from accretion onto high redshift ` seeds ' with perhaps a brief period of super - eddington accretion , the descendants of the smbhs that power the most luminous quasars at @xmath16 as proposed recently by volonteri & rees ( 2005 ) ; begelman , volonteri & rees ( 2006 ) ; lodato & natarajan ( 2007 ) and volonteri , lodato & natarajan ( 2007 ) . we discuss these two possible channels for growing umbhs in more detail below . following the merging dm hierarchy of halos starting with seed bhs at @xmath17 , populating the @xmath18 peaks , volonteri et al . ( 2003 ) are able to reproduce the mass function of local bhs as well as the abundance of the rare @xmath19 bhs that power the @xmath20 sdss quasars . proceeding to rarer peaks say , @xmath21 at @xmath17 in this scheme yields the rarer @xmath22 local umbhs . and in fact , the formation of a very small number density of umbhs at @xmath1 is inevitable in the standard hierarchical merging @xmath23cdm paradigm . a massive dm halo with mass , @xmath24 at @xmath1 which is the likely host to an umbh , is likely to have experienced about 100 mergers between @xmath16 and @xmath1 , starting with @xmath25 at @xmath16 . recently a numerical calculation of the merger scenario mentioned above has been performed in simulations by yoo et al . focusing on the merger history of high mass cluster - scale halos ( @xmath26 ) . they find that in ten realizations of halos on this mass scale , starting with the highest initial bh masses at @xmath27 of @xmath13 few times @xmath28 , 4 clusters contain umbhs at @xmath1 . therefore , rare umbhs are expected in the local universe . yoo et al . ( 2007 ) argue that black hole mergers can significantly augment the high end tail of the local bh mass function . similarly , using a model for quasar activity based on mergers of gas - rich galaxies , hopkins et al . ( 2006a ) showed that they could explain the observed local bh mass at low to intermediate bh masses ( 10@xmath29 - 10@xmath30@xmath31 ) . however , at higher bh masses , their calculations overpredict the observed values even considering a possible change in the eddington fraction at higher masses . conventional models of black hole formation and growth start with initial conditions at high redshift with seed bhs that are remnants of the first generation of stars in the universe . propagating these seeds via merger accompanied accretion events leading to mass growth for the bhs ( volonteri , haardt & madau 2003 ) it has been argued that in order to explain the masses of bhs powering the bright @xmath32 quasars by the sdss survey ( fan et al . 2004 ; 2006 ) that either a brief period of super - eddington accretion ( volonteri & rees 2005 ) or more massive seeds are needed ( begelman , volonteri & rees 2006 ; lodato & natarajan 2006 ; lodato & natarajan 2007 ) . massive seeds can alleviate the problem of assembling @xmath33 bhs by @xmath16 which is roughly 1 gyr after the big bang in the concordance @xmath23cdm model . the local relics of such super - grown black holes are expected to result in umbhs . we note here that following the evolution of the massive black holes that power the @xmath34 quasars , in a cosmological simulation , di matteo et al . ( 2008 ) find that these do not necessarily remain the most massive black holes at subsequent times . therefore , while umbhs might not be direct descendants of the smbhs that power the @xmath35 quasars , there is ample room for umbhs to form and grow . below , we briefly present scenarios that provide the massive bh seeds in the first place that will eventually result in a small population of umbhs by @xmath1 . these physically plausible mechanisms are critical to our prediction of umbhs at low redshift . two models have been proposed , one that involves starting from the remnants of population iii stars with brief episodes of accretion onto them exceeding the eddington rate to bump up their masses ( volonteri & rees 2006 ) and the other that explains direct formation of massive bh seeds prior to the formation of the first stars ( lodato & natarajan 2006 ; 2007 ) . volonteri & rees ( 2005 ) have proposed a scenario to explain the high bh masses @xmath36 needed to power the luminous quasars detected @xmath16 in the sdss . this is accomplished they argue by populating the @xmath37 peaks in the dark matter density field at @xmath38 with seed bhs which arise from the remnants of population iii stars in the mass ranges @xmath39 and @xmath40 . these remnant bhs then undergo an episode of super - eddington accretion from @xmath41 . they argue that in these high redshift , metal - free dark matter halos @xmath42 gas can cool in the absence of @xmath43 via atomic hydrogen lines to about @xmath44 . as shown by oh & haiman ( 2002 ) the gas at this temperature settles into a rotationally supported ` fat ' disk at the center of the halo under the assumption that the dm and the baryons have the same specific angular momentum . further , these disks are stable to fragmentation and therefore do not form stars and exclusively fuel the bh instead . the accretion is via stable super - critical accretion at rates well in excess of the eddington rate due to the formation of a thin , inner feeding disk . the accretion radius is comparable to the radiation trapping radius which implies that all the gas is likely to end up in the bh . any further cooling down to temperatures of @xmath45 for instance , halts the accretion , causes fragmentation of the disk which occurs when these regions of the universe have been enriched by metals . this process enables the comfortable formation of @xmath19 bhs by @xmath16 or so to explain the observed sdss quasars . in a @xmath23cdm universe , the time available from @xmath16 to @xmath1 is @xmath46 gyr . to grow by an order of magnitude during this epoch requires an accretion rate of @xmath47 which is well below the eddington rate ; however , it requires a gas rich environment . in recent work , lodato & natarajan ( 2007 ) have shown that an ab - initio prediction for the mass function of seed black holes at high redshift can be obtained in the context of the standard @xmath23cdm paradigm for structure formation combined with careful modeling of the formation , evolution and stability of pre - galactic disks . they show that in dark matter halos at high redshifts @xmath48 , where zero metallicity pre - galactic disks assemble ( prior to the formation of the first stars ) , gravitational instabilities in these disks transfer angular momentum out and mass inwards efficiently . note that the only coolants available to the gas at this epoch are either atomic or molecular hydrogen . taking into account the stability of these disks , in particular the possibility of fragmentation , the distribution of accumulated central masses in these halos can be computed . the central mass concentrations are expected to form seed black holes . the application of stability criteria to these disks leads to distinct regimes demarcated by the value of the @xmath49 where @xmath50 is the temperature of the gas and @xmath51 is the virial temperature of the halo . the three regimes and consequences are as follows : ( i ) when @xmath52 the disk fragments and forms stars instead of a central mass concentration ; ( ii ) @xmath53 , when both central mass concentrations and stars form ; ( iii)@xmath54 , when only central mass concentrations form and the disks are stable against fragmentation . using the predicted mass function of seed black holes at @xmath48 , and propagating their growth in a merger driven accretion scenario we find that the masses of black holes powering the @xmath35 optical quasars can be comfortably accommodated and consequently a small fraction of umbhs is predicted at @xmath1 . evolving and growing these seeds to @xmath1 , the abundance of umbhs can be estimated ( volonteri , lodato & natarajan 2008 ) . the new evidence that we present in this work for the existence of a rare population of umbhs stems from using x - ray luminosity functions of agn and the implied accretion history of black holes . hard x - rays have the advantage of tracing both obscured and unobscured agn , as the effects of obscuration are less important at these energies . in particular , we use the hard x - ray luminosity function and luminosity - dependent density evolution presented by ueda et al . ( 2003 ) defined from @xmath55 out to @xmath56 . we further assume that these agn are powered by bhs accreting at the eddington limit . in order to calculate bolometric luminosities starting from the hard x - ray luminosity the bolometric corrections derived from the agn spectral energy distribution library presented by treister et al . ( 2006 ) are used . these are based mainly on observations of local agn and quasars and depend only on the intrinsic x - ray luminosity of the source , as they are based on the x - ray to optical ratios reported by steffen et al . ( 2006 ) . to account for the contribution of compton - thick agn to the black hole mass density missed in x - ray luminosity functions , we use the column density distribution of treister & urry ( 2005 ) with the relative number of compton - thick agn adapted to match the spatial density of these sources observed by integral , obtained from the agn catalog of beckmann et al . ( 2006 ) . in order to account for sources with column densities @xmath57=10@xmath58 - 10@xmath59 @xmath60 which do not contribute much to the x - ray background , but can make a significant contribution to the bh mass density ( e.g. , marconi et al . 2004 ) , we multiply the bh mass density due to compton - thick agn by a factor of 2 , i.e. , we assume that they exist in the same numbers as in the @xmath57=10@xmath61 - 10@xmath58 @xmath60 range , in agreement with the assumption of marconi et al . ( 2004 ) and consistent with the @xmath57 distribution derived from a sample of nearby agn by risaliti et al . ( 1999 ) . under this assumption , the contribution of sources with @xmath57@xmath6210@xmath58 @xmath60 to the total population of smbhs is @xmath137% . we then convert these x - ray lf s to an equivalent bh mass function , and evolve these mass functions by assuming that accretion continues at the eddington rate down to @xmath55 . the results of this procedure are shown in fig . 2 for three different values of the accretion efficiency @xmath63 . note that we do not consider models in which the efficiency parameter varies with redshift or bh mass since such models merely add more unconstrained parameters . as can be seen clearly in fig . 2 , these simple models do not reproduce the observed local black hole mass function at the high mass end . the functional form adopted for the x - ray luminosity function is a double power - law as proposed by ueda et al . ( 2003 ) : @xmath64^{-1}.\ ] ] and the evolution is best described by the luminosity dependent density evolution model ( ldde model ) , where the cut - off redshift @xmath65 is expressed by a power law of @xmath66 , consistent with observational constraints ( see ueda et al . 2003 for more details ) : @xmath67 where @xmath68 @xmath69^{p2}\,\,\,\,\,(z \geq z_{\rm c}(l_{\rm x})).\end{aligned}\ ] ] this simple and conservative analysis predicts a population of umbhs with a local abundance of @xmath133@xmath7010@xmath71 ! this is fairly robust as this population is predicted for a large range of efficiencies . these lf s shown in fig . 2 also simultaneously account for the cosmic xrb , as shown by several authors ( for instance see treister & urry ( 2005 ) and gilli et al . ( 2007 ) and references therein ) , suggesting that the x - ray view presents a fairly complete picture of the accretion and growth of bhs . note that our estimates of the black hole mass function are in general agreement with those of marconi et al . ( 2004 ) [ for a direct comparison see their fig . 2 , right - hand panel ] , the very slight difference arises due to an alternate choice of bolometric correction factors and our prescription for including compton thick agn . estimates by other authors are also in agreement with our treatment here out to masses of a few times @xmath72 . for bh masses @xmath73 , there appears to be consistency between the optical and x - ray views of black hole growth . however for @xmath74 , all models that assume eddington accretion with varying efficiencies systematically over - estimate the local abundance of high mass black holes . as can be seen in fig . 2 , for a reasonable value of the efficiency , @xmath75 , there is a good agreement between the bh mass density at @xmath55 , as obtained from the velocity dispersion of bulges , and the density inferred from agn relics , for bh masses smaller than @xmath76 . however , for higher masses , in particular the umbh mass range , independent of the value of @xmath63 assumed , the bh mass density from agn relics is significantly higher than the observed value , indicating that umbhs should be more abundant than current observations suggest . if there is a mass dependent efficiency factor for accretion such that higher mass bhs tend to accrete at higher efficiency and hence at lower rates , then our estimate of the high mass tail would be an over - estimate . there is however no evidence for such a mass dependence at lower masses ( hopkins , narayan & hernquist 2006 ) . the sdss first data release covers approximately 2000 square degrees ( abazajian et al . 2003 ) , yielding a comoving volume of a cone on the sky out to @xmath77 of @xmath78 mpc@xmath79 . given our predicted abundance above , we expect @xmath13 1000 umbhs in the sdss volume , however only a few are detected . * no combination of assumed accretion efficiency and eddington ratio coupled with the x - ray agn lf can reproduce the observed local abundance at the high mass end . * however , we find that modifying one of the key assumptions made above brings the predicted abundance of local umbhs into better agreement with current observations . in the modeling we have extrapolated the observed x - ray agn lf slope to brighter luminosities . we find that if this slope is steepened at the bright end , we can reproduce the observed umbh mass function at @xmath1 for @xmath8010@xmath81 as well . in order to reconcile the observationally derived local black hole mass function at the high mass end , the slope @xmath82 in eqn . ( 1 ) needs to be modified . we find that the slope @xmath82 for black hole masses @xmath83 is @xmath84 , which however , does not provide a good - fit for higher masses . a slope steeper than @xmath85 is required to fit bh masses in excess of @xmath86 , we find that formally the best - fit is found in reduced-@xmath87 terms for the value of @xmath88 . * such a steepening simulates the cut - off of a self - regulation mechanism that limits black hole masses and sets in at every epoch*. in fig . 3 , the results of such a self - limiting growth model are plotted . the predicted abundance of umbhs is now in much better agreement with observations at @xmath1 and is consistent with the number of umbhs detected by sdss . in a self - regulated mass growth model we predict the abundance of umbhs at @xmath1 to be @xmath89 . therefore , requiring consistency between the x - ray and optical views of black hole growth and assembly with the observed number of umbhs at @xmath1 , points to the existence of a self - regulation mechanism that limits bh masses . the self - regulation is implemented as a steepening of the x - ray agn lf at the luminous end ( the value of @xmath82 needed is plotted in fig . 4 ) and does not have an important effect on the x - ray background , since this change in slope only affects sources with x - ray luminosities @xmath90 greater than 10@xmath91 erg s@xmath92 , while most of the x - ray background emission is produced by sources with luminosities of 10@xmath93 erg s@xmath92 , as found by treister & urry ( 2005 ) . note that the use of a bolometric luminosity function , as reported by hopkins , richards & hernquist ( 2007 ) does not match the slope at the high mass end ( shown as the dashed curve in fig . one key consequence of our model is that the slope of the @xmath94 relation at the high mass end likely evolves with redshift . recent cosmological simulations find evidence for such a trend ( di matteo et al 2008 ) . the functional form of this expected variation will depend on the specific model of self - regulation employed . below , we explore physical processes that are likely to regulate the growth of bhs in galactic nuclei . m@xmath95 , as shown by the _ black dashed line_.,width=340,height=340 ] converting the high end of the local black hole mass function into the equivalent velocity dispersions of the host spheroids we find values in excess of 350 kms@xmath92 . in the context of the currently popular hierarchical model for the assembly of structure , the most massive galaxies in the universe are expected to be the central galaxies in clusters . high-@xmath3 peaks in the density fluctuation field at early times seed clusters that assemble at later times , and hence these are the preferred locations for the formation of the most massive galaxies in a cold dark matter dominated universe . for the index @xmath82 in the x - ray agn lf required to match the high mass end of the local black hole mass density . in order to match the observed relation , the slope of the hard x - ray luminosity function was modified for masses higher than @xmath86m@xmath95.,width=340,height=340 ] while we predict above that a few , rare umbhs are likely to exist at the centers of the brightest central galaxies in clusters , we further argue that there likely exists an upper limit to black hole masses . evidence for this is presented using several plausible physical scenarios that attempt to explain the coeval formation of the black hole and the stellar component in galactic nuclei . clearly the existence of umbhs is intricately related to the highest mass galaxies that can form in the universe . given that star formation and black hole fueling appear to be coupled ( e.g. di matteo et al . 2005 and references therein ; silk & rees 1998 ) , it is likely that there is a self - limiting growth cycle for bhs and therefore a physical upper limit to their masses . here we present several distinct arguments that can be used to estimate the final masses of bhs ( haehnelt , natarajan & rees 1998 ; silk & rees 1998 ; murray , quataert & thompson 2004 and king 2005 ) . these involve self - limiting growth due to a momentum - driven wind , self - limiting growth due to the radiation pressure of a momentum - driven wind , and from an energy - driven superwind model . murray , quataert & thompson ( 2004 ) argue that the feedback from momentum driven winds , limits the stellar luminosity , which in turn regulates the bh mass . they argue for eddington limited star formation with a maximum stellar luminosity , @xmath96 where , @xmath97 is the gas fraction in the halo and @xmath3 the velocity dispersion of the host galaxy . star formation in this scheme is unlikely to evacuate the gas at small radius in the galactic nucleus , therefore , all the gas in the inner - most regions fuel the bh . the growing bh itself clears out this nuclear region with its accretion luminosity approaches @xmath98 . at this point the fuel supply to the bh is shut - off and this may shut off the star formation as well . the final bh mass is then given by , @xmath99 where @xmath100 is the electron scattering opacity . for the most massive , nearby early - type galaxies at the very tail of the measured sdss velocity dispersion function with velocity dispersions of @xmath13 350 - 400 kms@xmath92 ( bernardi et al . 2005 ) this gives a final bh mass of @xmath101 . therefore , normal galaxies with large velocity dispersions are the presumptive hosts for umbhs . furthermore , there appears to be a strong indication of the existence of an upper mass limit for accreting black holes derived from sdss dr3 by vestergard et al . ( 2008 ) in every redshift bin from @xmath102 . an alternative upper limit can be obtained when the emitted energy from the accreting bh back reacts with the accretion flow itself ( haehnelt , natarajan & rees 1998 ) . the final shut - down of accretion will depend on whether the emitted energy can back - react on the accretion flow prior to fuel exhaustion . this arguement provides a limit , @xmath103 where @xmath104 is the fraction of the accretion luminosity which is deposited as kinetic energy into the accretion flow ( cf . silk & rees 1998 ) , @xmath105 is the spin parameter of the dm halo , @xmath106 is the specific angular momentum of the disk , @xmath107 is the disk mass fraction . the back - reaction timescale will be related to the dynamical timescale of the outer parts of the disk and/or the core of the dm halo and should set the duration of the optically bright phase . it is interesting to note here that the accretion rate will change from super - eddington to sub - eddington without much gain in mass if the back - reaction timescale is shorter than the salpeter time . the overall emission efficiency is then determined by the value of @xmath108 when the back - reaction sets in and is reduced by a factor @xmath109 compared to accretion at below the eddington rate . by substituting the value of the velocity dispersion of nearby cd s @xmath110 , we obtain a limiting value of the mass , if we assume that the bulk of the mass growth occurs in the optically bright quasar phase . due to the dependence on the spin parameter @xmath105 of the dm halo , the desired umbh mass range can arise preferentially in high velocity dispersion halos with low spin . king ( 2005 ) presents a model that exploits the observed agn - starburst connection to couple black hole growth and star formation . as the black hole grows , an outflow drives a shell into the surrounding gas which stalls after a dynamical time - scale at a radius determined by the bh mass . the gas trapped inside this bubble cools , forms stars and is recycled as accretion and outflow . once the bh reaches a critical mass , this region attains a size such that the gas can no longer cool efficiently . the resulting energy - driven flow expels the remaining gas as a superwind , thereby fixing the observed @xmath4 relation as well as the total stellar mass of the bulge at values in good agreement with current observations . the limiting bh mass is given by : @xmath111 where @xmath97 is the gas fraction ( @xmath112 , @xmath113 the electron scattering opacity and @xmath3 the velocity dispersion . this model argues that black hole growth inevitably produces starburts and ultimately a superwind . note that both the murray , quataert & thompson ( 2004 ) model and the king ( 2005 ) model predict @xmath114 while the haehnelt et al . ( 1998 ) and silk & rees ( 1998 ) predict a @xmath115 dependence . the current error bars on the observational mass estimates for black holes preclude discrimination between these two possibilities . shutdown of star formation above a critical halo mass effected by the growing agn has also been proposed as a self - limiting mechanism to cap bh growth and simultaneously explain the dichotomy in galaxy properties ( croton et al . 2006 ; cattaneo et al . umbhs are expected to be rare in the local universe , from our analysis of the x - ray luminosity function of agn , we predict an abundance ranging from @xmath13 few times @xmath116 . these estimates are in good agreement with those obtained from optical quasars in the sdss dr3 by vestergard et al . the results of the first attempts to detect and measure masses for umbhs is promising . dalla bonta et al . ( 2007 ) selected 3 brightest cluster galaxies ( bcgs ) in abell 1836 , abell 2052 and abell 3565 . using acs ( advanced camera for surveys ) aboard the hubble space telescope and the imaging spectrograph ( stis ) , they obtained high resolution spectroscopy of the @xmath117 and @xmath118ii emission lines to measure the kinematics of the central ionized gas . they present bh mass estimates for 2 of these bcgs , @xmath119 and @xmath120 and an upper limit for the bh mass on the third candidate of @xmath121 . it is interesting to note that bernardi et al . ( 2005 ) in a census of the most massive galaxies in the sdss survey do find candidates with large velocity dispersions @xmath122 . the largest systems they find are claimed to be extremes of the early - type galaxy population , as they have the largest velocity dispersions . these @xmath123 systems ( see table 1 of bernardi et al . ( 2006 ) for details on these candidates ) are not distant outliers from the fundamental plane and the mass - to - light scaling relations defined by the bulk of the early - type galaxy population . clear outliers from these scaling relations tend to be objects in superposition for which they have evidence from spectra and images . we argue that these extreme early - type galaxies might harbour umbhs and likely their abundance offers key constraints on the physics of galaxy formation . although the observations are challenging , a more comprehensive and systematic survey of nearby bcgs is likely to yield our first local umbh before long . as discussed above , candidates from the sdss are promising targets for observational follow - up as they are extremely luminous . utilizing the hubble space telescope , the light profile might show evidence for the existence of an umbh in the center ( e.g. lauer et al . in fact , for @xmath124 and @xmath125 , it may be possible to measure spatially resolved velocity dispersion profiles even from ground - based facilities . the interplay between the evolution of bhs and the hierarchical build - up of galaxies appears as scaling relations between the masses of bhs and global properties of their hosts such as the bh mass vs. bulge velocity dispersion - the @xmath126 relation and the bh mass vs. bulge luminosity @xmath127 relation . the low bh mass end of this relation has recently been probed by ferrarese et al . ( 2006 ) in an acs survey of the virgo cluster galaxies . they find that galaxies brighter than @xmath128 host a supermassive central bh whereas fainter galaxies host a central nucleus , referred to as a central massive object ( cmo ) . ferrarese et al . report that a common @xmath129 relation leads smoothly down from the scaling relations observed for more more massive galaxies . extrapolating observed scaling relations to higher bh masses to the umbh range , we predict that these are likely hosted by the massive , high luminosity , central galaxies in clusters with large velocity dispersions . the velocity dispersion function of early - type galaxies measured from the sdss points to the existence of a high velocity dispersion tail with @xmath130 ( bernardi et al . if the observed scaling relations extend to the higher mass end as well , these early - types are the most likely hosts for umbhs . recent simulation work that follows the merger history of cluster scale dark matter halos and the growth of bhs hosted in them by yoo et al.(2007 ) also predict the existence of a rare population of local umbhs . however , theoretical arguments suggest that there may be an upper limit to the mass of a bh that can grow in a given galactic nucleus hosted in a dark matter halo of a given spin . clearly the issue of the existence of umbhs is intimately linked to the efficiency of galaxy formation and the formation of the largest , most luminous and massive galaxies in the universe . possible explanations for the tight correlation observed between the velocity dispersion of the spheroid and black hole mass involve a range of self - regulated feedback prescriptions . an estimate of the upper limits on the black hole mass that can assemble in the most massive spheroids can be derived for all these models and they all point to the existence of umbhs . in this paper , we have argued that while rare umbhs likely exist , there is nevertheless an upper limit of @xmath131 for the mass of bhs that inhabit galactic nuclei in the universe . we first show that our current understanding of the accretion history and mass build up of black holes allows and implies the existence of umbhs locally . this is primarily driven by new work that predicts the formation of massive black hole seeds at high redshift ( lodato & natarajan 2007 ) and their subsequent evolution ( volonteri , lodato & natarajan 2008 ) . starting with massive seeds and following their build - up through hierarchical merging in the context of structure formation in a cold dark matter dominated universe , we show that a viable pathway to the formation of umbhs exists . there is also compelling evidence from the observed evolution of x - ray agn for the existence of a local umbh population . convolving the observed x - ray lf s of agn , with a simple accretion model , the mass function of black holes at @xmath1 is estimated . mimic - ing the effect of self - regulation processes that impose an upper limit to bh masses and incorporating this into the x - ray agn lf we find that the observed umbh mass function at @xmath1 is reproduced . this self - regulation limited growth is implemented by steepening the high luminosity end of the agn lf at the bright end . we estimate the abundance of umbhs to be @xmath132 at @xmath1 . the key prediction of our model is that the slope of the @xmath4 relation likely evolves with redshift at the high mass end . probing this is observationally challenging at the present time but there are several bright , massive early - type galaxies that are promising host candidates from the sdss survey as well as a survey of bright central galaxies of nearby clusters . observational detection of umbhs will provide key insights into the physics of galaxy formation and black hole assembly in the universe . we thank steinn sigurdsson and meg urry for useful discussions . abazajian , k. , et al . , 2003 , aj , 126 , 2081 alexander , d. , smail , i. , bauer , f. , chapman , s. , blain , a. , brandt , w. , & ivison , r. , 2005 , nature , 434 , 738 barger , a. j. , et al . 2003 , aj , 126 , 632 barger , a. j. , cowie , l. l. , mushotzky , r. f. , yang , y. , wang , w .- h . , steffen , a. t. , & capak , p. 2005 , aj , 129 , 578 beckmann , v. , gehrels , n. , shrader , c. r. , & soldi , s. 2006 , apj , 638 , 642 begelman , m. , & meier , d. l. , 1982 , apj , 253 , 873 begelman , m. , & nath , b. , 2005 , mnras , 361 , 1387 begelman , m. , volonteri , m. & rees , m. j. , 2006 , mnras , 370 , 289 bernardi , m. , et al . , 2005 , aj , 129 , 61 bernardi , m. , et al . , 2006 , aj , 131 , 2018 bernardi , m. , sheth , r. , tundo , e. , & hyde , j. , 2007 , apj , 660 , 267 bland hawthorn , j. , wilson . & tully , r.b . , 1991 , apjl , 371 , l19 boyle , b. j. , shanks , t. , croom , s. m. , smith , r. j. , miller , l. , loaring , n. , & heymans , c. 2000 , mnras , 317 , 1014 carilli , c. , et am . 2002 , aj , 123 , 1838 cattaneo , a. , dekel , a. , devriendt , j. , guiderdoni , b. & blaizot , j. , 2006 , mnras , 370 , 1651 cox , p. , et al . 2002 , a & , 387 , 406 . corsini , e. m. , beifiori , a. , dalla bonta , e. , pizzella , a. , coccato , l. , sarzi , m. , bertola , f. , 2006 , proceedings of `` black holes : from stars to galaxies '' , iau symp . 238 , eds . v. karas & g. matt , cambridge univ . press , astro - ph/0610624 croton , d. , et al . , 2006 , mnras , 365 , 11 dalla bonta , e. , ferrarese , a. , corsini , e. , miralda - escude , j. , coccato , l. , & pizella , a. , 2007 , mmsai , 78 , 745 , astro - ph/07061959 di matteo , t. , esin , a. , fabian , a. c. , & narayan , r. , 1999 , mnras , 305 , l1 di matteo , t. , springel , v. , & hernquist , l. , 2005 , nature , 433 , 604 di matteo , t. , colberg , j. , springel , v. , hernquist , l. , sijacki , d. , 2008 , apj , 676 , 33 fan , x. , et al . 2001 , aj , 122 , 2833 fan , x. , et al . 2003 , aj , 125 , 1649 fan , x. , et al . 2004 , aj , 128 , 515 fan , x. , et al . 2006 , aj , 132 , 117 fabian , a. c. 1999 , mnras , 308 , l39 fabian , a. c. , & iwasawa , k. , 1999 , mnras , 303 , l34 merrit , d. & ferrarese , l. , 2001 , apj , 547 , 140 ferrarese , l. , et al . , 2006 , apj , 644 , 21 gebhardt , k. , et al . 2000 , apj , 539 , l13 gilli , r. , risaliti , g. , & salvati , m. , 1999 , a&a , 347 , 424 gilli , r. , comastri , a. , & hasinger , g. 2007 , a&a , 463 , 79 haehnelt , m.g . & rees , m.j . , 1993 , mnras , 263 , 168 haehnelt , m.g . , natarajan , p. & rees , m.j . , 1998 , mnras,300 , 817 hopkins , p. , hernquist , l. , cox , t. , di matteo , t. , martini , p. , robertson , b. , & springel , v. , 2005 , apj , 630 , 705 hopkins , p. , hernquist , l. , cox , t. , di matteo , t. , robertson , b. , & springel , v. , 2006a , apjs , 163 , 1 hopkins , p. , robertson , b. , krause , e. , hernquist , l. , & cox , t. , 2006b , apj , 652 , 107 hopkins , p. , narayan , r. , & hernquist , l. , 2006 , apj , 643 , 641 hopkins , p. ; richards , g. t. ; hernquist , l. , 2007 , apj , 654 , 731 kauffmann , g. & haehnelt , m. , 2000 , mnras , 311 , 576 king , a. , 2003 , apj , 596 , l27 king , a. , 2005 , apj , 635 , l121 lehto , h.j . & valtonen , m.j . , 1996 , apj , 460 , 207 lauer , t. , gebhardt , k. , & richstone , d. , et al . , 2002 , aj , 124 , 1975 lauer , t. , tremaine , s. , richstone , d. , & faber , s , m. , 2007a , apj , 670 , 249 lauer , t. , et al . , 2007b , apj , 662 , 808 lodato , g. , & natarajan , p. , 2006 , mnras , 371 , l1813 lodato , g. , & natarajan , p. , 2007 , mnras , 377 , l84 magorrian , j. , et al . 1998 , aj , 115 , 2285 marconi , a. , risaliti , g. , gilli , r. , hunt , l. , maiolino , r. & salvati , m. , 2004 , mnras , 351 , 169 martinez - sansigre , a. , et al . , 2005 , nature , 436 , 666 martnez - sansigre , a. , et al . 2007 , mnras , 379 , l6 mclure , r. , & dunlop , j. , 2004 , mnras , 352 , 1390 merloni , a. , rudnick , g. & di matteo , t. , 2004 , mnras , 354 , l37 murray , n. , quataert , e. , & thompson , t. , 2004 , apj , 618 , 569 mushotzky , r. f. , cowie , l. l. , barger , a. j. , & arnaud , k. a. 2000 , nature , 404 , 459 natarajan , p. & sigurdsson , s. , 1998 , mnras , 302 , 288 oh , p. , & haiman , z. , 2002 , apj , 569 , 558 omont , a. , cox , p. , bertoldi , f. , mcmahon , r. g. , carilli , c. , & isaak , k. g. 2001 , a&a , 374 , 371 reuland , m. , rottgering , h. , van breugel , w. , & de breuck , c. , 2004,mnras , 353 , 377 risaliti , g. , maiolino , r. , salvati , m. , 1999 , apj , 522 , 157 salucci , p. , szuszkiewicz , e. , monaco , p. , & danese , l. , 1999 , mnras , 307 , 637 sazonov , s. yu . , ostriker , j. p. , ciotti , l. , & sunyaev , r. , 2005 , mnras , 358 , 168 shankar , f. , salucci , p. , granato , g. l. , de zotti , g. , & danese , l. 2004 , mnras , 354 , 1020 silk , j. & rees , m.j . , 1998 , a & a , 331 , l1 sillanp , a. , haarala , s. , valtonen , m.j . , sundelius , b. & byrd , g.g . , 1988 , apj , 325 , 628 sotan , a. , 1982 , mnras , 200 , 115 steffen , a. t. , strateva , i. , brandt , w. n. , alexander , d. m. , koekemoer , a. m. , lehmer , b. d. , schneider , d. p. , & vignali , c. 2006 , aj , 131 , 2826 treister , e. , & urry , c. m. 2005 , apj , 630 , 115 treister , e. , et al . 2006 , apj , 640 , 603 tremaine , s. , et al . 2002 , apj , 574 , 740 tundo , e. , bernardi , m. , hyde , r. , sheth , r. , & pizzella , a. , 2007 , apj , 663 , 53 ueda , y. , akiyama , m. , ohta , k. , & miyaji , t. 2003 , apj , 598 , 886 valtonen , m.j . , lehto , h. & pietil , h. , 1999 , a & a , 342 , l29 van der marel , r.p . , 1994 , mnras , 270 , 271 vestergaard , m. , fan , x. , tremonti , c. r. , osmer , p. & richards , g. t. , 2008 , apj , 674 , l1 volonteri , m. , haardt , f. & madau , p. , 2003 , apj , 582 , 559 volonteri , m. & rees , m. j. , 2005 , apj , 633 , 624 volonteri , m. & rees , m. j. , 2006 , apj , 650 , 669 volonteri , m. , lodato , g. & natarajan , p. , 2008 , mnras , 383 , 1079 walter , f. , et al . 2003 , nature , 424 , 406 worsley , m. a. , et al . 2005 , mnras , 357 , 1281 wyithe , s. , & loeb , a. , 2002 , apj , 581 , 886 yoo , j. , miralda - escude , j. , weinberg , d. , zheng , z. , & morgan , c. , 2007 , apj , 667 , 813 , astro - ph/0702199 yu , q. , & tremaine , s. , 2002 , 335 , 965
|
we make a case for the existence for ultra - massive black holes ( umbhs ) in the universe , but argue that there exists a likely upper limit to black hole masses of the order of @xmath0 .
we show that there are three strong lines of argument that predicate the existence of umbhs : ( i ) expected as a natural extension of the observed black hole mass bulge luminosity relation , when extrapolated to the bulge luminosities of bright central galaxies in clusters ; ( ii ) new predictions for the mass function of seed black holes at high redshifts predict that growth via accretion or merger - induced accretion inevitably leads to the existence of rare umbhs at late times ; ( iii ) the local mass function of black holes computed from the observed x - ray luminosity functions of active galactic nuclei predict the existence of a high mass tail in the black hole mass function at @xmath1 .
consistency between the optical and x - ray census of the local black hole mass function requires an upper limit to black hole masses .
this consistent picture also predicts that the slope of the @xmath2-@xmath3 relation will evolve with redshift at the high mass end .
models of self - regulation that explain the co - evolution of the stellar component and nuclear black holes naturally provide such an upper limit .
the combination of multi - wavelength constraints predicts the existence of umbhs and simultaneously provides an upper limit to their masses .
the typical hosts for these local umbhs are likely the bright , central cluster galaxies in the nearby universe .
galaxies : evolution , active , nuclei .
x - rays : galaxies .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
it is known that one of the properties of linear diffusion equation solution @xmath3 is the infinite velocity of perturbations propagation . it is seen , for example , from the fundamental solution of this equation @xmath4 here @xmath5 is the fundamental solution presented in the gaussian form , @xmath6 is concentration , @xmath7 is the diffusion coefficient , @xmath8 defines the unit step heaviside function , @xmath9 is associated with current coordinate , @xmath10 is the current time . this solution contradicts to intuitive speculations that diffusive particles and any information about their movement are propagated with finite velocities . we should mark that equation ( [ eq:1 ] ) is a consequence of supposition that principles associated with locality and local thermodynamic equilibrium are valid . in the frame of this supposition one can use the diffusion fick s law in standard form @xmath11 connecting the current particle function and the concentration gradient . by analogy , for conductive heat transfer one can write the fourier law connecting the heat flux with temperature gradient and for liquids and gases involved in the filtration phenomenon in porous medium the darcy s law connecting the stream of liquid with pressure gradient . one of the approaches related to resolution of the problem connected to corrections for the infinite velocity of perturbations propagation lies in the usage of linear nonlocal models , for example , suggested by the extended of nonequilibrium thermodynamics @xcite , @xcite . in this case in expression ( 3 ) some additional relaxation terms are introduced . in the simplest case , the fick s law in the nonlocal interpretation is written as @xmath12 and equation ( [ eq:1 ] ) is reduced to the telegraph equation of the type @xmath13 the fundamental solution of the last one for @xmath14 case is expressed as @xmath15 where @xmath16 is the modified bessel s function of the first kind having zeroth order . the paradox of the infinite velocity is eliminated and the limiting velocity describing the perturbations propagation in a nonequilibrium medium becomes finite and is equaled to @xmath17 . in the given paper the proper selection of the fundamental solutions plays a key role . their knowledge determines the specific solutions of linear equations of mathematical physics as well , because the corresponding green s functions can be expressed in the form of linear combinations of fundamental solutions including their derivatives and integrals . equations of type ( [ eq:1 ] ) for many real situations of macroscopic physics are good approximation . but together with this statement the consideration of the nonlocal effects is an actual problem for relatively small intervals of observation times and distances . it is interesting to consider the evaluation of the limits of applicability of the local thermodynamic equilibrium hypothesis and the finding of approximate approaches related to determination of the current of the value studied and gradient of potential of the corresponding scalar field . taking into account the fact that approach ( 4 ) is also approximate , it would be interesting to evaluate errors which arise in the usage of similar expressions in comparison with solution of the `` exact '' transfer equation which is remained unknown . the certain complexity in experimental testing in laboratory conditions the adequacy of equations ( [ eq:1 ] ) and ( 4 ) lies in nonlinear dependencies of the key parameters involved . it creates some difficulties in interpretation of the results obtained . so , in the given paper the dynamics of hypothetical linear systems modelling the transfer phenomenon is considered . it is necessary to remind here about the inertia heat phenomenon when nonlinearity leads also to the finite velocities of perturbations @xcite but similar and relatively exotic models are out of the scope of this research . if one come back to micro - level the diffusion phenomenon , conductive heat transfer and filtration of liquids and gases in porous media and similar phenomena can be described with the help of random behavior of microparticles . for diffusion and heat transfer phenomenon one can introduce concepts as the characteristic length @xmath18 and time @xmath19 of free particles path or quasiparticles of phonon type ; for filtration we consider the characteristic distances and times associated with the hopping process of liquid droplets from one pore to another one . hence , from microscopic point of view of the phenomena considered it is important to establish the correlations between phenomenological parameters of the corresponding theories similar to the extended irreversible thermodynamics with characteristic relaxation times and microparameters of the transfer process studied . in random walk model the hopping length @xmath18 and hopping time @xmath19 characterize the dynamics of the system considered on microlevel . for various media and processes these microparameters are varied in wide limits . the length of free path for phonons in crystals is changed in the range @xmath20 m that corresponds to the free path time located in the range @xmath21 s for frequencies of the phonon spectrum from the interval @xmath22 hz . we should note that in paper @xcite with the usage of the molecular dynamics methods it was shown that characteristic relaxation times in nonlocal heat transfer model written in the form of telegraph equation ( 5 ) for solid argon crystal are very small ( @xmath23 s ) that comparable with times of the phonons free path in a crystal . the characteristic microscopic hopping length in diffusion phenomenon in the most cases is located in the range @xmath24 m , for filtration @xmath25 m and the corresponding characteristic times are located in the intervals @xmath26 s and @xmath27 s , accordingly . the random walk model at present time is widely used for description of many processes ; in particular , the fractal dynamic models are developed in @xcite . supposing that on validity not only nice equations can pretend but also the nice algorithms , as well , we will be based on supposition that true description of diffusion phenomenon is determined by random walk model and there is unknown linear equation which correctly describes of the transfer phenomenon of diffusion type . in this paper the simplest variant of the random walk model for @xmath14 system is considered . in this model the hoppings of a particle ( or quasiparticles ) are realized with equal probabilities during the interval of time @xmath19 for distances @xmath28 or @xmath29 . then for the discrete variant of the problem formulated one can obtain the solution for the probability @xmath30 of a particle location in coordinate @xmath31 in the moment @xmath32 in the form @xcite @xmath33 it is known that expression ( 2 ) presents one of the partial cases of solution of the random walk problem describing the probability density distribution of a random value . really , if divide ( 7 ) on @xmath34 and put @xmath35 at @xmath36 , then one can obtain expression for the probability density function describing a probability of location of diffusive particle in the vicinity of the point @xcite @xmath9 in the form ( 2 ) . in this case , simple calculations show that the velocity @xmath37 of perturbations propagation accepts the infinite value ( @xmath38 at @xmath36 ) but the solution itself has automodel character with respect to variable @xmath39 . we should note that in @xcite for simplification of expression ( 7 ) containing factorials with the usage of the stirling s decomposition formula @xmath40 only the first term in the braces is kept . it allows to introduce a criterion for correct evaluation of this approach in the form of the following inequalities [ eq : whole ] @xmath41 @xmath42 the fundamental solution for telegraph type equation ( 4 ) represents another limiting case of the random walk problem @xcite . in this case @xmath43 at @xmath36 . the selection of fundamental solutions as a ground for analysis of transfer processes is justified by the fact that for linear equations of mathematical physics the knowledge of a fundamental solution determines specific solutions of these equations as well , if the corresponding initial and boundary conditions are known . it is related to the fact that the corresponding green functions can be expressed in the form of linear combinations of fundamental solutions including their derivatives and integrals . then at transition to continuous variables it is convenient to measure a current length in the unit of elementary length @xmath18 and the current time in the units of @xmath19 . we put also @xmath44 , @xmath45 , @xmath46 . then the fundamental solution of equation ( [ eq:1 ] ) is written as @xmath47 and fundamental solution for ( 4 ) as @xmath48 and the probability density distribution function of a particle located in the vicinity of the coordinate @xmath9 and in the moment time @xmath10 for model ( 5 ) and considered in the given paper as the fundamental solution of unknown transfer equations is written as @xmath49 or , being presented in the form of the euler s gamma - functions it can be presented as @xmath50 here and below the symbol @xmath51 ( gauss ) indicates on equation eq . ( [ eq:1 ] ) , the symbol @xmath52 ( telegrapher equation ) on equation ( 5 ) , and symbol @xmath53 ( random walk ) on initial equation ( 7 ) . then we will use the following hypothesis : the true solution of evolution equation of the diffusion type presents the random walk solution in the form of fundamental solution ( 14 ) but presentations ( 2 , 11 ) and ( 6 , 12 ) present approximate approaches . solution ( 14 ) corresponds to the unknown linear evolution equation . let us consider the evolution of equations ( 11 , 12 and 14 ) . on figure1 we demonstrate the values of the function @xmath54 end its gradient @xmath55 for the fixed times t= 30 and 100 . on figure2 we show the differences of the function @xmath54 ( 14 ) from the functions @xmath56 ( 11 ) and @xmath57 ( 12 ) calculated at the same values of time . comparison of the values and for the fixed values of time t=30 and t=100 . ] the differences of solutions ( 11 ) , ( 12 ) and ( 14 ) for the values t=30 and t=100 . ] with increasing of the time the functions considered not changing their forms are decreased in their values and are stretched in the space . the differences between the functions @xmath56 , @xmath57 , @xmath54 become small and are decreased with the time growing . taking into account the influence of the second decomposition term figuring in the stirling s decomposition expression ( 11 ) for factorials and the asymptotic behavior of the bessel function at ( @xmath58 ) @xmath59 it is easy to calculate the values of the functions considered at the point @xmath60 in different values of time at @xmath61 . these values are equaled approximately [ eq : whole ] @xmath62 @xmath63 @xmath64 from these relationships it follows that the differences of the functions ( 16 ) are decreased with time as @xmath0 , their relative differences as @xmath1 and their absolute values as @xmath65 . the corresponding calculations show that for other values of the current coordinate @xmath9 the differences of the functions ( 11 , 12 and 14 ) values are decreased with time as @xmath0 . incidentally , the spatial stretching of the functions themselves at @xmath66 is proportional to @xmath2 . let us consider the gradient concentrations dynamics corresponding to solutions ( 11 ) , ( 12 ) and ( 14 ) . for the function @xmath54 in notations used the corresponding gradient will look as @xmath67 where @xmath68 , and for the functions @xmath69 , @xmath57 , as @xmath70 and @xmath71 for all functions considered the gradient concentrations distributions at the fixed moments of time coincide with the functions that tend to zero at @xmath72 and @xmath73 and having one maximum . for equation ( 18 ) at the fixed value of time the maximum value should be observed at @xmath74 and the value of function ( 18 ) in this point is @xmath75 , the velocity of the movement of the maximum point @xmath76 and its halfwidth @xmath77 will be equaled to @xmath78 and @xmath79 , correspondingly . the calculations show that the differences between maximum locations points of functions @xmath80 , @xmath81 @xmath82 and their relative differences are decayed as @xmath83 and the relative changes of the maximum values of the corresponding gradients @xmath84 , @xmath85 and @xmath86 for large values of @xmath10 are equaled [ eq : whole ] @xmath87 @xmath88 let us consider now the relationship between flux and concentration gradient . for the gaussian ( 2 ) the following relationship between flux and concentration gradient is valid @xmath89 for the functions @xmath90 and @xmath91 the fluxes in the points @xmath92 ( @xmath93 ) are calculated as [ eq : whole ] @xmath94 @xmath95 the calculations lead to the following dependencies of the square differences of the functions @xmath96,@xmath90 , @xmath91 , their gradients and the corresponding fluxes at @xmath97 : [ eq : whole ] @xmath98 @xmath99 @xmath100 @xmath101 @xmath102 @xmath103 @xmath104 @xmath105 @xmath106 one can notice that deviations of the values considered from each other are decreased with time faster than @xmath1 . so , for the temporal intervals @xmath107 calculating errors in replacement of the `` accurate '' solution by the functions , at calculations of the desired fluxes will be less then @xmath108 and for the times @xmath109 will be less then @xmath110 . we should remark also that absolute values of the fluxes and their concentration functions are relatively large only for the times @xmath111 . for the times @xmath112 the flux value is becoming less then @xmath108 at @xmath113 , for @xmath114 is less then @xmath108 at @xmath115 , for @xmath116 is less then @xmath108 at @xmath117 . let us consider the changings of deviations from @xmath118 for the function @xmath91 . these deviations are defined as @xmath119 it was proved that function ( 24 ) is decreased quickly with growth of the time as @xmath120 we introduce for equation ( 4 ) the corrected function @xmath121 by means of relationship @xmath122 here the index @xmath53 stresses the origin of this relationship with random walk problem . really , relationship ( 25 ) can present in the form of ( 4 ) for the random walk problem for @xmath14 case with replacement of the coefficient @xmath7 for the parameter @xmath123 . the analysis of ( 25 ) shows that with the increasing of the observation time @xmath124 , but for the various values of the coordinate @xmath9 the times of approaching of the function @xmath125 to the unit value @xmath126are different as it is shown on fig.3 . the values of the corrected function f at various values of the current coordinate x. ] we want to note here that simple and good enough analytical approximation for the function @xmath125 at the values @xmath127 and @xmath66 serves the following expression @xmath128 in this expression the function @xmath125 is presented as the approximation to the heaviside function @xmath8 for each fixed value of the coordinate @xmath9 and depends only on the characteristic velocity @xmath37 associated with random walk step . from this observation it is easy to conclude that for each moment of time @xmath129 if we define deviation of the function f from the unit value as @xmath130 the value of the corresponding coordinate corresponding to the interval @xmath66 can be written as @xmath131 . if we put , for example , @xmath132 , @xmath97 then @xmath133 . it signifies that the relative velocity of approaching of @xmath125 to the unit value ( @xmath134 ) is considerably higher in comparison with the propagation velocity of the fixed density distribution value proportional to @xmath2 . then equation ( 25 ) can be rewritten in the form @xmath135 where we use the notations introduced above @xmath136 , @xmath137 , @xmath138 , i.e. they are expressed by means of microparameters associated with random walk problem . we should note also the influence of fluctuations . really , it is necessary to take into account that in derivation of equation ( [ eq:1 ] ) the locality principle is used , as well . it imposes certain limitations of the spatial scales where the diffusion type process takes place . for the equilibrium values of concentration associated with number of particles and temperature one can use evaluations of their fluctuations in correspondence with formulas @xmath139 the following question can be posed , what is the size of a system should be to take into account the thermodynamic fluctuations of different macroscopic parameters as particles , temperature and etc . ? it is obvious that we can not show the accurate limits and it has a relative character . for example , for subsystem having @xmath140 particles the relative fluctuations give the value @xmath108 . but for small values of time and distances the thermodynamic fluctuations become very large and it is impossible to describe propagation perturbations front using only deterministic models . in other words , for the times @xmath141 and distances @xmath142 the usage of mesoscopic approaches becomes necessary . in this case we go out from the principle locality limits and continuous medium approach . this statement coincides with the conclusions given above associated with comparison of the functions ( [ eq:1 ] ) and ( 5 ) and `` true '' function ( 14 ) . in this paper we consider local - equilibrium and non - equilibrium models of transfer phenomena of the diffusion type . these phenomena are considered in the framework of the supposition that true solutions correspond to the solutions of the random walk problem ( 14 ) while the models ( 1 - 3 ) and ( 4 - 5 ) are considered as approximations to the `` true '' solution . the comparisons given above and analysis of the differences in fundamental solutions for equations ( [ eq:1 ] ) , ( 5 ) and continuous representation of the random walk function as the density distribution function for random value ( 14 ) allow to make the following conclusions : ( a ) the differences between the values of the functions considered are decreased with time as @xmath0 ; ( b ) the propagation velocity of the fixed values of the functions considered is proportional to @xmath2 ; ( c ) on relatively large values of observation times and distances the fundamental solutions becomes very close to each other . taking into account these conclusions one can formulate the criterion of applicability of the simplest equations as the telegraph equation ( 5 ) or heat transfer ( [ eq:1 ] ) . their criterion of applicability can be expressed in the form of inequalities : @xmath143 , @xmath144 and @xmath145 where @xmath18 , @xmath19 , @xmath146 are defined as microparameters . we should note that for the most problems of macrophysics that conditions are satisfied for relatively small values of the observation time @xmath147 . for more accurate calculations of the desired relationships between fluxes and concentration gradients in the framework of random walk problem one can use relationships ( 25 - 27 ) . in these relationships the additional parameters figuring in diffusion equations are expressed in the form of microparameters related to the random walk problem .
|
the fundamental solutions of diffusion equation for the local - equilibrium and nonlocal models are considered as the limiting cases of the solution of a problem related to consideration of the brownian particles random walks .
the differences between fundamental solutions were studied .
it was shown that on the period of observation time and distances exceeding the time and space associated with one step of random walk the fundamental solutions of diffusion and telegraph equations are very close to each other .
in particular , these fundamental solutions are very close to the values of the probability density distribution of diffusive particles that is obtained from the accurate solution of the random walk problem .
the difference between the probability density values for the models considered is decreased with time as @xmath0 , the relative difference as @xmath1 , and velocity of propagation of these fixed distribution densities is proportional to @xmath2 . the new modified non - local diffusion equation is suggested .
it contains only microparameters of the random walk problem .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the promising optoelectronic properties of spontaneously formed gan nanowires ( nws ) reported in the pioneering works of @xcite and @xcite have triggered world - wide research activities that have led to the demonstration of light - emitting @xcite and light - harvesting devices @xcite based on group - iii - nitride nws . despite this progress , several open questions still exist regarding the spontaneous formation of gan nws and their structural and optical properties . in particular , a prominent band at 3.45 ev has been widely reported in the low - temperature photoluminescence ( pl ) spectra of gan nws grown by plasma - assisted molecular beam epitaxy ( pambe ) on si(111 ) @xcite . the origin of this band in gan nws has been a subject of a lively debate for almost two decades . in bulk gan , two different recombination mechanisms are known to manifest themselves by luminescence lines at about 3.45 ev : first , the two - electron satellite ( tes ) of the donor - bound exciton transition [ @xmath0 @xcite and , second , excitons bound to inversion domain boundaries @xcite . the intensity of the tes transitions in bulk gan is about two orders of magnitude lower than that of the related @xmath1line @xcite . in contrast , the 3.45-ev band in gan nws is often prominent and sometimes even dominates the near band - edge pl spectrum . nevertheless , this band was ascribed to the tes by @xcite , who proposed that the distortion of the @xmath1wave function near the nw surface would lead to a strong enhancement of this transition . the same group substantiated this hypothesis by investigating the evolution of the 3.45-ev band with nw diameter and nw density @xcite . however , investigations of the fermi level pinning in gan nws @xcite as well as polarization - resolved pl and magneto - optical experiments @xcite later refuted the interpretation of the 3.45-ev band as an enhanced tes transition . inversion domain boundaries ( idbs ) may give rise to intense pl lines in gan films at 3.45 ev @xcite . an idb denotes a boundary between ga- and n - polar gan for which two different stacking sequences have been proposed by @xcite . the idb@xmath2notation refers to the specific atomic structure at which each atom remains fourfold coordinated by exclusively forming ga - n bonds across the boundary . in contrast , the unstarred idb indicates a structure where the formation of ga - ga or n - n bonds would occur . the idb@xmath2has an exceptionally low formation energy and does not induce electronic states in the band gap , thus facilitating the radiative recombination of excitons bound to these defects @xcite ( again in contrast to the unstarred idb , which entails electronic states in the band gap ) . @xcite suggested that the 3.45-ev band observed in pl spectra of gan nws is related to the presence of idb@xmath2sin gan nws just as in the bulk . this suggestion , however , was not supported by investigations of the microstructure of gan nws at that time . in fact , transmission electron microscopy ( tem ) performed on isolated gan nws invariably demonstrated the absence of extended defects in the nw volume @xcite . several groups therefore favored point defects as the origin for the 3.45-ev emission . in early work , in which gan nws were assumed to elongate axially via a ga - induced vapor - liquid - solid mechanism , ga interstitials were proposed as likely candidates for these point defects @xcite . later , gan nw growth was understood to proceed under n - rich conditions , which were suggested to result in an enhanced formation of ga vacancies near the nw surface @xcite . @xcite discussed the dependence of the 3.45-ev band on excitation density in terms of both planar defects such as the idb@xmath2and abundant point defects and eventually favored the latter for being most consistent with the whole set of available data . however , the observation of strong nw - to - nw variations in the intensity of the 3.45-ev emission from single nws @xcite contradicted this conclusion as well . recently , @xcite presented definitive experimental evidence for the presence of idb@xmath2sin gan nws grown on aln - buffered si(111 ) . subsequently , the same authors correlated -pl experiments with high - resolution scanning tem performed on single gan nws grown on aln - buffered si @xcite . they observed a systematic correlation between the presence of idb@xmath2sin the gan nw and transitions at 3.45 ev in its -pl spectrum and thus concluded that these transitions are caused by exciton recombination at idb@xmath2s . finally , they proposed that the 3.45-ev band is indicative for the presence of idb@xmath2sin gan nws also for other substrates . it is not uncommon to observe mixed polarities in aln films grown on si(111 ) @xcite , and the coexistence of n- and ga - polar nws on such a film is therefore not an actual surprise . the spontaneous formation of gan nws directly on si(111 ) , however , is largely believed to occur on an amorphous sin@xmath3 interlayer formed during the ( unintentional or intentionally promoted ) nitridation of the si substrate by the n plasma @xcite . the polarity of gan nws formed directly on such a nitridated si(111 ) surface has been debated for a long time . in earlier work , the gan nws were mostly reported to be ga polar @xcite , while recent studies ( which also include ensemble investigations with far better statitics ) indicate that gan nws grow predominantly or even exclusively n polar @xcite . in any case , whether idb@xmath2sform for gan nws on nitridated si(111 ) , and if so , at which density , are open questions . in this paper , we report a comprehensive investigation of the structural and optical properties of gan nws on si(111 ) fabricated with or without intentional substrate nitridation and within a wide range of substrate temperatures . this investigation focuses on the nature of the 3.45-ev band that is observed for all samples but with varying intensity . in sec . [ sec2:experimental ] , we briefly describe the samples under investigation and the experimental setups used for our study . low - temperature cathodoluminescence ( cl ) spectroscopy and tem are employed in sec . [ sec3:cl ] to investigate the origin of the 3.45-ev band . in accordance with the findings of @xcite , we attribute this band to the presence of idb@xmath2sin our nws . in sec . [ sec4:pl ] , we present an in - depth investigation of the 3.45-ev band comprising time - resolved and polarization - resolved pl spectroscopy accompanied by theoretical considerations based on data published by @xcite . the observed doublet structure of the 3.45-ev band is identified to be due to the recombination of localized and delocalized states at the idb@xmath2 . the paper closes with a summary and conclusion in sec . [ sec5:summary ] . the gan nw ensembles studied in this work ( see tab . [ tab : samples ] for an overview ) were grown by pambe on aln - buffered sic(0001 ) ( sample a ) or si(111 ) ( samples b1b7 , c , and d ) . they were fabricated over the course of six years and in four different pambe systems ( as indicated by the letters in their names ) . the si(111 ) substrates were intentionally nitridated prior to nw growth for all samples except for sample d. we have intentionally chosen nw ensembles synthesized over a wide range of substrate temperatures @xmath4between 720 and 875 @xmath5c . all samples were grown under n - rich conditions with the ga and n fluxes adjusted accordingly to account for the increased ga desorption at elevated temperatures @xcite . due to the wide range of conditions , the different nw samples exhibited incubation times between a few minutes and hours as well as different nw densities , average nw diameters , and coalescence degrees . further details on the growth conditions can be found elsewhere ( sample a @xcite , b1b4 @xcite , b5b7 @xcite , c @xcite , and d @xcite ) . .list of the investigated samples . the substrate temperature @xmath4during growth , the year of fabrication , and the reference containing growth details are also given . [ cols="^,^,^,^,^ " , ] the morphological and structural properties of the nws were investigated by scanning electron microscopy ( sem ) and tem . for tem , cross - sectional specimens were prepared by mechanical grinding , polishing , and subsequent ar - ion polishing . the tem images were recorded using an acceleration voltage of 300 kv . the polarity of the nws was determined using convergent beam electron diffraction ( cbed ) complemented by subsequent dark - field tem imaging . sem and cl spectroscopy were carried out in a field - emission instrument equipped with a cl system and a he - cooling stage . a photomultiplier tube was used for the acquisition of monochromatic images and a charge - coupled device ( ccd ) camera for recording cl spectra . throughout the experiments , the acceleration voltage and the probe current of the electron beam were set to 5 kv and 0.75 na , respectively . the spectral resolution amounted to 8 mev . due to the strong quenching of the cl intensity of gan nws under the electron beam @xcite , the irradiation time was kept to a minimum . since the @xmath1transition is more prone to quenching than the 3.45-ev band @xcite , the bichromatic cl images shown in this work consist of two superimposed monochromatic false - color images recorded first at 3.47 and then at 3.45 ev . control experiments performed in reverse order confirmed the validity of our results . all pl experiments were performed in backscatter geometry with the samples mounted in liquid he - cooled cryostats offering continuous temperature control from 5 to 300 k. all measurements were conducted with nw ensembles except for the polarization - resolved pl experiments , where single nws had been dispersed onto bare si(111 ) wafers prior to the measurement . for continuous - wave pl spectroscopy , the 325-nm line of a hecd laser was used to excite the samples . the laser was focused to a spot with a diameter of 1 m by a near - ultraviolet microscope objective with a numerical aperture of 0.65 . the photoluminescence signal was collected by the same objective and dispersed by a spectrometer with an energy resolution of 0.25 to 1 mev . the dispersed signal was detected by a ccd camera . polarization - resolved measurements were carried out using a half - wave plate followed by a linear polarizer @xcite . time - resolved pl measurements were performed by focusing the second harmonic ( 325 nm ) of an optical parametric oscillator pumped by a femtosecond ti : sapphire laser ( pulse width and repetition rate of 200 fs and 76 mhz , respectively ) . the pl signal was dispersed by a monochromator and detected by a streak camera operating in synchroscan mode . the energy and time resolutions are 2 mev and 20 ps , respectively . transition at 3.469 ev , several very narrow lines around 3.45 ev are observed . the high - resolution tem image in the inset shows the central part of a gan nw of sample a revealing a ga-/n - polar core / shell structure . the arrows denote the idb@xmath2sbetween the ga - polar core and the n - polar shell . ] prior to a systematic investigation of our gan nw ensembles on si(111 ) , we examine sample a which we already know to contain a very high density of idbs or , most likely , idb@xmath2s . this sample was part of our study devoted to the role of substrate polarity in the formation of gan nws @xcite . for this study , we attempted to induce the formation of gan nws on substrates with well - defined polarity , namely , aln / sic@xmath7 and aln / sic@xmath8 . in the present work , we discuss further aspects of the sample synthesized at a substrate temperature of 825@xmath5con sic@xmath7 , which is here referred to as sample a. the polar nature of the sic substrate is known to determine the polarity of group - iii - nitride layers deposited on it @xcite , and the aln buffer was consequently found to be al polar . initiating the deposition of gan at conditions typical for the growth of nws resulted in a highly faceted ga - polar gan layer interspersed with sparse vertical nws [ sem images of sample a can be found in figs . 2(c ) and 2(d ) in ref . we have found the majority of these nws to be n polar due to a si - induced polarity flip at the interface between aln and gan . consequently , idb@xmath2sform upon coalescence between the ga - polar matrix and the n - polar nws @xcite , and the sample is thus characterized by an exceptionally high fraction of nws containing an idb@xmath2 . transition . the substrate temperature of each sample is specified in the figure . ] the pl spectrum of sample a is shown in fig . [ fig01-pl - tem ] . the @xmath1transition is observed at 3.469 ev , but the spectrum also exhibits several intense and narrow lines around 3.45 ev . the slight redshift of the @xmath1transition with respect to the one usually observed for gan nws ( 3.471 ev ) as well as its comparatively large line width suggests that it mainly originates from the ga - polar gan layer . the strong transitions observed around 3.45 ev are consistent with the high fraction of nws with an idb@xmath2found in our previous analysis [ see fig . 4(b ) in ref . for an example ] . moreover , further investigations of sample a by tem conducted in the course of the present work revealed in addition the existence of ga-/n - polar core / shell nws analogous to those reported by @xcite ( see the high - resolution tem image displayed in the inset of fig . [ fig01-pl - tem ] ) . all these findings are in agreement with those reported in ref . and strongly suggest that the lines observed around 3.45 ev in fig . [ fig01-pl - tem ] are in fact due to excitons bound to idb@xmath2s . we will therefore label these transitions in all what follows as @xmath9 . an interesting finding in this context is the fine structure of the 3.45-ev band visible in fig . [ fig01-pl - tem ] . the origin of the distinct narrow lines in the pl spectrum will be addressed in sec . [ sec4:pl ] . a prominent band at 3.45 ev is commonly also observed for gan nws grown on si(111 ) @xcite . figure [ fig02-t - series - norm ] shows the evolution of the low - temperature ( 10 k ) pl spectrum from gan nw ensembles formed on si(111 ) ( samples b1b4 ) under identical conditions except for the substrate temperature . the linewidth of the @xmath1transition decreases with @xmath4increasing from 720 to 820@xmath5c , indicating a progressive reduction of micro - strain induced by nw coalescence @xcite . in parallel , the intensity of the 3.45-ev band with respect to that of the @xmath1line also decreases with increasing @xmath4 . if this band is also related to the presence of idb@xmath2sin gan nws grown directly on si(111 ) , its evolution seems to suggest that nws fabricated at a higher @xmath4exhibit a lower density of idb@xmath2s . in the following , we investigate whether this decrease in the relative @xmath9intensity with increasing @xmath4is correlated with changes in the morphology of the nw ensemble . furthermore , we examine the spatial distribution of the different spectral components for various nw ensembles . figures [ fig03-sem](a ) and [ fig03-sem](b ) display sem bird s eye view images of samples b4 and b1 , which were grown at @xmath10 and 720@xmath5c , respectively . the morphology of these samples is representative for nw ensembles grown at high and low @xmath4 . for substrate temperatures higher than 750@xmath5c , the nw ensemble usually exhibits a nw density of about @xmath11 @xmath12 together with a uniform height distribution as shown in fig . [ fig03-sem](a ) @xcite . in contrast , for temperatures significantly lower than this value , the ensemble morphology is characterized by a dense ( @xmath13 @xmath12 ) matrix of short and highly coalesced nws interspersed by long nws with a much lower density of @xmath14 @xmath12 [ cf . [ fig03-sem](b ) ] . top - view sem images ( not shown here ) indicate that most of the long nws are attached to short nws . to identify a possible correlation between the peculiar morphology of sample b1 and its intense 3.45-ev band , we record the spatial intensity distribution of the two distinct pl bands by cl spectroscopy . figure [ fig04-cl](a ) shows the superposition of an sem image with a bichromatic cl map recorded at 3.47 and 3.45 ev from sample b1 at 10 k. clearly , the 3.45-ev band ( color coded in green ) originates almost exclusively from long nws , whereas the @xmath1line ( color - coded in magenta ) stems mostly from the top part of short nws . figure [ fig04-cl](b ) shows the same measurement for sample c , which is another typical representative for a gan nw ensemble grown at comparatively low @xmath4 . the spatial distribution of the emission at 3.47 and 3.45 ev is the same as for sample b1 , as indeed for all samples exhibiting the morphology characteristic for growth at low substrate temperatures . to clarify the reason for this spatial distribution , we have performed cbed as well as tem imaging on several nw samples grown at low @xmath4 . using cbed , it is straightforward to determine the polarity of the short nws constituting the columnar matrix in these nw ensembles , since their effective diameter is large due to the high degree of coalescence . the polarity of this matrix was found to be exclusively n polar for both samples b1 and c ( not shown here ) . for the long nws with diameters below 50 nm , the polarity can not be reliably determined by cbed . however , since we know the polarity of the columnar matrix , we can instead employ dark - field tem and exploit the fact that opposite polarities induce a contrast inversion in images recorded by this technique . in addition , inverting the diffraction vector @xmath15 should also invert the contrast for both polarities . representative dark - field micrographs are displayed in fig . [ fig05-tem ] . the opposite contrast between the long nw and its short neighbor in fig . [ fig05-tem](a ) recorded with @xmath16 as well as the contrast inversion in fig . [ fig05-tem](b ) recorded with @xmath17 demonstrate that the long nw is ga polar . the situation is more complex in figs . [ fig05-tem](c ) and [ fig05-tem](d ) . here , the shell of the long nw has the same polarity as the adjacent material , but it clearly has a ga - polar core . these peculiar polarity core / shell nw structures seem to be identical to those observed in fig . [ fig01-pl - tem ] and in ref . , suggesting that the mechanism giving rise to the formation of such structures is a general one and does not depend on the substrate . the long nws are either ga polar [ figs . [ fig05-tem](a ) and [ fig05-tem](b ) ] or have a core / shell structure with a ga - polar core surrounded by a n - polar shell [ figs . [ fig05-tem](c ) and [ fig05-tem](d ) ] . in the former case , planar idb@xmath2sare formed at the coalescence boundaries between the long ga - polar nws and the n - polar columnar matrix . in the latter case , the ga-/n - polar core / shell structure results in the formation of an idb@xmath2tube . idb@xmath2sthus exist either at the junctions between the long nws and the surrounding columnar matrix or directly within the long nws themselves . it is thus very plausible that the 3.45-ev emission , which arises almost exclusively from the long nws [ see fig . [ fig04-cl ] ] , originates from the @xmath9complex . ga polar or [ ( c ) and ( d ) ] to exhibit an ga-/n - polar core / shell structure . an inverted diffraction vector @xmath15 also inverts the contrast between the ga- and n - polar material . ] we have shown so far that nw ensembles grown at a lower @xmath4exhibit both short and long nws and that the optical transition at 3.45 ev is due to exciton recombination at idb@xmath2sin long thin nws . however , with increasing @xmath4 , the nw ensembles are getting more homogeneous in diameter and length [ fig . [ fig03-sem](a ) ] , and the intensity of the transition at 3.45 ev strongly decreases in comparison to that of the @xmath1as depicted in fig . [ fig02-t - series - norm ] . figure [ fig06_pl - cl](a ) shows for sample b5 grown at @xmath18@xmath5cthat the intensity of the line at 3.45 ev for nw ensembles grown at high substrate temperatures can be two orders of magnitude smaller than that of the @xmath1line . in the bulk , the tes of the @xmath1is also centered at 3.45 ev and its intensity is about two orders of magnitude smaller than that of the @xmath1transition @xcite . consequently , the question arises whether the weak 3.45-ev band observed in fig . [ fig06_pl - cl](a ) is related to idb@xmath2sat all . figure [ fig06_pl - cl](b ) shows a bichromatic cl map of sample b5 and fig . [ fig06_pl - cl](c ) depicts cl spectra taken at 10 k on individual nws . clearly , the @xmath1and the 3.45-ev emission lines do not coincide spatially , ruling out the standard two - electron satellites as a possible origin of the 3.45-ev band and suggesting instead that the 3.45-ev band also arises from the presence of idb@xmath2sin these high-@xmath4nw ensembles . remarkably , the density of nws with dominant @xmath9transitions is comparable to that observed for low-@xmath4ensembles ( cf . [ fig04-cl ] ) , demonstrating that the density of idb@xmath2sdoes not change significantly with substrate temperature . this finding seems to contradict the evolution of the relative intensity of the @xmath9band with @xmath4as depicted in fig . [ fig02-t - series - norm ] . however , plotting the same data on an absolute intensity scale as done in fig . [ fig07-t - series - abs ] reveals that the intensity of the @xmath9band is actually not significantly reduced with increasing @xmath4 . the decrease of the relative intensity of the @xmath9band with increasing @xmath4is instead caused by the drastic increase in the @xmath1emission intensity , reflecting the reduced concentration of nonradiative point defects at high @xmath4@xcite . we have established in the previous section that the 3.45-ev transition in gan nws on si(111 ) is related to idb@xmath2sregardless of the substrate temperature . in this section , we investigate the optical properties of excitons bound to idb@xmath2sin more detail . in particular , we critically examine the consistence of our experimental results with the properties expected theoretically for this particular bound exciton state . we focus on samples which exhibit a detailed fine structure in their pl spectra . one basic property of the @xmath9transition can be deduced directly from the fact that the idb@xmath2is a planar defect , which laterally extends over several tens of nm and vertically spans the entire nw length . in analogy to the result in ref . for basal - plane stacking faults , we hence expect the density of states of the @xmath9to be two - dimensional . as a result of the large number of states available , the @xmath9transition should be difficult to saturate even for high excitation conditions . indeed , the @xmath9transition has been observed to scale linearly with excitation density even after the higher - energy @xmath1transition has started to saturate @xcite . using density - functional theory , @xcite calculated the electronic potential in the vicinity of an idb@xmath2based on the stacking sequence proposed by @xcite . this potential , depicted in fig . [ fig08_eff - mass ] , acts as a barrier for electrons and as a quantum well for holes : an idb@xmath2can therefore be seen as a type - ii quantum well . to get quantitative information on the properties of the electronic state associated with the idb@xmath2 , we calculated the wavefunction and the energy of an exciton in the potential profile obtained by @xcite using the variational approach described in ref . . the result of our calculations is shown in fig . [ fig08_eff - mass ] . the calculations yield an electron - hole overlap , defined as the absolute square of the overlap integral between the electron ( @xmath20 ) and hole ( @xmath21 ) wavefunctions , of @xmath22 . therefore , despite the type - ii band alignment across the idb@xmath2plane , the @xmath9transition possesses a large oscillator strength in agreement with the high intensity observed experimentally [ cf . figs . [ fig04-cl ] and [ fig06_pl - cl](b ) ] . for isotropic electron and hole masses of @xmath23 and @xmath24 , with @xmath25 denoting the mass of the free electron , the energy of this transition @xmath26 is found to be 3.445 ev , i.e. , close to the experimental value . @xcite also predicted a significant mixing between the @xmath27 and @xmath28 valence bands due to the idb@xmath2 . to verify this prediction , we have performed polarization - resolved pl experiments at 40 k on single nws from sample b7 dispersed on a si substrate . a typical polarization - resolved pl map taken on a single nw is shown in fig . [ fig09_polarization ] . in agreement with previous reports @xcite , the free a exciton ( @xmath29 ) and the @xmath1transitions are polarized perpendicular to the nw axis ( @xmath30 ) , whereas the @xmath9band is polarized parallel to the nw axis ( @xmath31 ) . the three lowest optical transitions in strain - free bulk gan obey selection rules such that the @xmath32 transition is allowed only with in - plane ( @xmath30 ) polarization , the @xmath33 transition with both in- and out - of - plane ( @xmath31 ) polarization , and the @xmath34 transition mostly with out - of - plane polarization . the out - of - plane polarization of the @xmath9band evident in fig . [ fig09_polarization ] thus seems to suggest that it originates from a pure @xmath34 transition , i.e. , the c exciton @xcite . for a nw with sub - wavelength diameter , however , we have to bear in mind that the dielectric contrast strongly suppresses in - plane polarized emission and thereby artificially enhances the out - of - plane component @xcite . the strong polarization of the @xmath9band along the nw axis ( @xmath31 ) thus provides clear evidence for the mixing of the @xmath27 and @xmath28 valence bands in the idb@xmath2 , in agreement with the theoretical result of fiorentini @xcite . note also the pure in - plane polarization for the @xmath35transition , demonstrating that the opposite behavior observed for the @xmath9transition is a consequence of the peculiar potential induced by the idb@xmath2and not a characteristic of excitons bound to planar defects in general . as derived from effective mass calculations applied to a planar idb@xmath2structure using the potentials given in ref . . for the electron , the dotted line reflects the self - consistent effective potential . the inset shows an enlarged section of the conduction band edge around the idb@xmath2 , clearly showing the reduction of the conduction band edge in its immediate vicinity due to the accumulation of holes . the values for the bandgap @xmath36 and the transition energy @xmath26 are also given . ] axis of gan , is oriented along 90@xmath5as indicated by the dashed line . ] c , respectively . the spectral windows containing the @xmath9and the @xmath1have been normalized independently to the strongest transition . ] the mixing between @xmath37 and @xmath28 states proposed in ref . and observed in fig . [ fig09_polarization ] is also consistent with the magneto - optical behavior of the @xmath9transition reported in ref . . the land factor @xmath38 was found to be close to zero for the @xmath9 , while for the @xmath1transition in gan nws it was observed to be 1.75 , a value similar to that reported for the bulk @xcite . as both the electron and hole land factors are extremely sensitive to valence band mixing @xcite , it is in fact not surprising to measure very different values of @xmath38 for the @xmath1and the @xmath9lines . all properties of the @xmath9transition discussed so far are consistently described by the electronic potential computed by @xcite . however , this model does not provide any explanation for the fact that the @xmath9band often exhibits a doublet structure . as shown first by @xcite , the band at 3.45 ev actually consists of two lines centered at about 3.449 and 3.455 ev @xcite . this finding is confirmed by the pl spectra of samples b6 , b7 , and d at 40 k as shown in fig . [ fig10-doublets ] . for these three samples , the lower and higher energy component of the doublet is centered at about 3.452 and 3.458 ev , respectively . note that the splitting between these two lines is almost identical to that between the @xmath1and the @xmath29transitions . figure [ fig11_t - p - dependencies ] shows the evolution of the @xmath9doublet with increasing excitation density and temperature for sample b6 and b7 , respectively . at low temperatures and excitation densities , the @xmath9doublet is dominated by the lower energy transition [ @xmath39 at 3.452 ev . however , with increasing excitation density [ fig . [ fig11_t - p - dependencies](a ) ] or temperature [ fig . [ fig11_t - p - dependencies](b ) ] , the higher energy transition [ @xmath40 takes over and eventually dominates the pl spectrum . at around 40 k , carriers start to escape from the idb@xmath2 , which manifests itself in a quenching of the @xmath9doublet @xcite . as noted in ref . , the excitation power and temperature dependences of the @xmath9doublet are similar to the ones observed for the @xmath1and @xmath29transitions . in view of the fact that idb@xmath2sact as quantum wells , we thus attribute the high - energy line of the @xmath9doublet to the recombination of excitons free to move along the idb@xmath2plane and the low - energy one to excitons localized within this plane . localization within the plane of a quantum well usually occurs at well width fluctuations or due to alloy disorder @xcite , which clearly can not be the origin of the intra - idb@xmath2localization observed here . following the results reported for the localization of excitons in i@xmath41 basal - plane stacking faults @xcite , we propose that the short - range potential of shallow donors such as si and o distributed in the vicinity of the idb@xmath2induce the localization of excitons within the idb@xmath2plane . to confirm this idea , we compare the excitation dependence of the intensity ratio at 10 k between the @xmath42and @xmath43lines with that between the @xmath29and the @xmath1 in fig . [ fig11_t - p - dependencies](c ) . both ratios remain nearly constant for low excitation powers , but increase together for powers higher than 18 w. this finding indicates that the density of localized states within the idb@xmath2sis comparable to the equivalent density of donors in nws and thus confirms that intra - idb@xmath2exciton localization occurs due to donors . note that we computed the characteristic extent of the exciton wavefunction perpendicular to the idb@xmath2plane to be 5 nm . assuming that the diffusion length of excitons in an idb@xmath2is 100 nm @xcite , a donor density of 10@xmath44 @xmath45 is found to be sufficient to localize excitons within the idb@xmath2plane , a value close to those reported for unintentionally @xmath46-doped gan nws @xcite . in ensemble measurements , the @xmath43band typically has a line width of several mev , whereas single nws exhibit numerous sharp lines in this region @xcite . the small differences in transition energies originate from the varying distances between the involved donor and the idb@xmath2plane @xcite . this effect can also be seen very clearly for sample a in fig . [ fig01-pl - tem ] . due to the low nw density of @xmath47 @xmath12 of this sample , only a small number of nws is probed simultaneously , and the individual narrow lines can be resolved even in an ensemble measurement . the line width of the sharp @xmath43lines in fig . [ fig01-pl - tem ] and also in fig . 3 of ref . is resolution limited , demonstrating that these lines stem from the radiative decay of bound excitons . in contrast , ensemble spectra such as the ones shown in figs . [ fig10-doublets ] and [ fig11_t - p - dependencies ] contain contributions from about @xmath48 nws , and the individual transitions can no longer be resolved , but blend together to an @xmath43band with a line width of several mev @xcite . note that @xcite observed a fine structure of the @xmath9band in bulk gan already in 2001 . the @xmath42band exhibits a linewidth of a few mev as expected for delocalized states . finally , we have performed time - resolved pl experiments to obtain quantitative information on the capture efficiency of excitons by idb@xmath2sand on the intra - idb@xmath2localization process . figure [ fig12_trpl ] shows pl transients of sample b6 measured at 10 k for the @xmath29and @xmath1transitions as well as for the @xmath42and @xmath43lines . while the initial increase of the @xmath29pl intensity is almost instantaneous , it takes 90 ps for the @xmath1pl intensity to reach a maximum . the latter time corresponds roughly to the characteristic time for the capture of excitons by donors with a density of about @xmath49 @xmath45 @xcite . in agreement with the results reported in ref . , the @xmath29and @xmath1pl decay in parallel at longer times as a result of the quasi - thermalization between these exciton states . using an exponential fit , the effective decay time for the @xmath29and @xmath1is found to be 195 ps , very similar to values obtained in previous reports @xcite . this decay is much faster than expected for the radiative decay of the @xmath1complex and is due to the nonradiative decay of the @xmath29at point defects @xcite . analogously to the @xmath1and @xmath29transitions , the initial increase of the @xmath43intensity is delayed compared to that of the @xmath42 . interestingly , the rise time of the @xmath42intensity is only 47 ps . therefore , the capture of excitons by idb@xmath2sis more efficient than their trapping by neutral donors due to the idb@xmath2s large capture cross - section . this situation differs from observations for the rise time of the @xmath35transition at 10 k @xcite , which is limited by the inefficient transport of excitons from one donor to the next @xcite . for time delays longer than 600 ps , the @xmath43and @xmath42decay in parallel , again demonstrating quasi - thermalization between these states . finally , for time delays longer than 1 ns , the pl decay of the coupled @xmath29and @xmath1states slows down and asymptotically approaches that of the @xmath9doublet ( not shown ) , indicating full thermalization between the @xmath29 , @xmath1and @xmath9states . this process is the origin of the biexponential pl decay reported for the @xmath1 in refs . . and @xmath29transitions as well as of the @xmath43and @xmath42lines of sample b6 the arrows denote the respective rise times , and the solid lines are exponential fits . ] the 3.45-ev band observed in low - temperature pl and cl spectra of spontaneously formed gan nws on si(111 ) arises from planar or tubular idb@xmath2s . while the former are due to the coalescence of adjacent ga- and n - polar nws , the latter form at the interface of ga-/n - polar core / shell nws . the intensity ratio between the @xmath9and the @xmath1transitions decreases with increasing @xmath4 . this decrease is a consequence of the reduction in the density of nonradiative point defects with increasing @xmath4 , leading to an increase in the absolute intensity of the @xmath1line . in contrast , the absolute intensity of the @xmath9band is neither directly governed by @xmath4nor by the presence or absence of an intentional nitridation step . the same applies to the abundance of idb@xmath2sobserved in spatially resolved , bichromatic cl maps . these results confirm the idea of @xcite that the 3.45-ev luminescence band in gan nws signifies the presence of idb@xmath2sregardless of the substrate . in fact , we now know with certainty that idb@xmath2soccur in gan nw ensembles synthesized by pambe on aln - buffered si(111 ) @xcite , on nitridated si(111 ) , and on sic(0001 ) . the exceptionally low formation energy of idb@xmath2sis obviously an important factor promoting their frequent occurrence . however , a prerequisite for the formation of an idb@xmath2is the simultaneous presence of both ga- and n - polar material . at present , it remains entirely unclear why an apparently constant fraction of the gan nuclei on all of these different substrates are ga polar . we also do not understand how ga - polar nws can evolve from these nuclei despite our inability to synthesize them intentionally on cation - polar substrates @xcite and at substrate temperatures at which gan(0001 ) usually decomposes . finally , it is unclear how the peculiar ga-/n - polar core / shell nws form , which have been observed by different groups and on different substrates . the lack of knowledge regarding these apparently universal and basic phenomena demonstrates that the nucleation and formation of gan nws in pambe are still far from being completely understood . concerning the electronic and optical properties of idb@xmath2s , it is helpful to imagine them as a thin type - ii quantum well that binds holes . the coulomb attraction exerted by these holes is strong enough to bind electrons , and the resulting @xmath9state decays radiatively and thus gives rise to intense light emission . the change in the symmetry of the fundamental hole state in idb@xmath2sstrongly modifies the polarization and the behavior of the @xmath9when subjected to magnetic fields . donor atoms distributed in the vicinity of the idb@xmath2plane localize the exciton within the idb@xmath2plane . as a result of this localization , the @xmath9band resolves into a doublet with the lines at low and high energy being associated with localized and free @xmath9states , respectively . their excitonic nature is manifested by the observation of a fine structure of the low energy line consisting of sharp , resolution - limited peaks . finally , while the capture of excitons by idb@xmath2stakes less than 50 ps , quasi - thermalization between the near - band edge excitons and the @xmath9takes much longer , resulting in the nonexponential decay usually observed for the @xmath1transition in gan nws at low temperatures . analogously to stacking faults @xcite , idb@xmath2sdo not suffer from fluctuations in layer thickness or composition as conventional quantum wells and thus offer unique possibilities for the study of low - dimensional excitons @xcite . particularly interesting in this context are the tubular idb@xmath2sformed in ga-/n - polar core / shell nws and intersections of these tubular idb@xmath2swith stacking faults forming perfect crystal - phase quantum rings . since the electronic states associated with all of these quantum structures are shallow @xcite , they should be of little practical relevance for conventional devices . however , we envisage that their exceptionally well - defined properties make them ideal model systems for an understanding of quantum effects important for a future generation of optoelectronic devices . the authors thank pierre lefebvre for fruitful discussions , vincent consonni and caroline chze for providing additional samples , and uwe jahn for a critical reading of the manuscript . c. acknowledges funding from the fonds national suisse de la recherche scientifique through project 161032 . this work was partly supported by the german bmbf joint research project monalisa ( contract no . 01bl0810 ) , by the deutsche forschungsgemeinschaft within sfb 951 , and by marie curie rtn parsem ( grant no . mrtn - ct-2004 - 005583 ) . 60ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty link:\doibase 10.1143/jjap.36.l459 [ * * , ( ) ] link:\doibase 10.1016/s0022 - 0248(97)00386 - 2 [ * * , ( ) ] link:\doibase 10.1143/jjap.43.l1524 [ * * , ( ) ] link:\doibase 10.1021/ja404043k [ * * , ( ) ] link:\doibase 10.7567/apex.8.042302 [ * * , ( ) ] link:\doibase 10.1103/physrevb.62.16826 [ * * , ( ) ] http://link.aip.org/link/?japiau/101/113506/1 [ * * , ( ) ] link:\doibase 10.1063/1.2980341 [ * * , ( ) ] link:\doibase 10.1063/1.3062742 [ * * , ( ) ] link:\doibase 10.1103/physrevb.81.045302 [ * * , ( ) ] link:\doibase 10.1063/1.4775492 [ * * , ( ) ] link:\doibase 10.1002/pssc.200674901 [ * * , ( ) ] link:\doibase 10.1063/1.1609632 [ * * , ( ) ] link:\doibase 10.1063/1.3556643 [ * * , ( ) ] link:\doibase 10.1103/physrevb.82.045320 [ * * , ( ) ] link:\doibase 10.1063/1.1390486 [ * * , ( ) ] link:\doibase 10.1063/1.3656987 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.77.103 [ * * , ( ) ] in http://scholar.google.com/scholar?hl=en{&}btng=search{&}q=intitle:tem+study+of+(+ga+,+al+)+n+nanocolumns+and+embedded+gan+nanodiscs{#}0[__ ] , , ( , ) p. link:\doibase 10.1007/s11664 - 006 - 0102 - 4 [ * * , ( ) ] link:\doibase 10.1063/1.2204836 [ * * , ( ) ] link:\doibase 10.1016/j.jcrysgro.2006.11.036 [ * * , ( ) ] link:\doibase 10.1007/s12274 - 010 - 0061 - 1 [ * * , ( ) ] link:\doibase 10.1063/1.4923024 [ * * , ( ) ] link:\doibase 10.1063/1.4927826 [ * * , ( ) ] link:\doibase 10.1063/1.3633522 [ * * , ( ) ] link:\doibase 10.1103/physrevb.83.035310 [ * * , ( ) ] link:\doibase 10.1063/1.2899944 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.2953087 [ * * , ( ) ] link:\doibase 10.1088/0957 - 4484/21/19/195202 [ * * , ( ) ] link:\doibase 10.1109/jstqe.2010.2098396 [ * * , ( ) ] http://prb.aps.org/abstract/prb/v84/i24/e245302 [ * * , ( ) ] link:\doibase 10.1021/nl203872q [ * * , ( ) ] link:\doibase 10.1021/nl302664q [ * * , ( ) ] link:\doibase 10.1021/nl400200 g [ * * , ( ) ] link:\doibase 10.1063/1.4905651 [ * * , ( ) ] link:\doibase 10.1063/1.1554776 [ * * , ( ) ] link:\doibase 10.1063/1.3267151 [ * * , ( ) ] link:\doibase 10.1016/j.jcrysgro.2010.12.081 [ * * , ( ) ] link:\doibase 10.1021/acs.cgd.5b00690 [ * * , ( ) ] link:\doibase 10.1063/1.3464956 [ * * , ( ) ] http://www.mrs.org/s{_}mrs/doc.asp?cid=26415{&}did=321215 [ * * , ( ) ] link:\doibase 10.1088/1367 - 2630/17/3/033040 [ * * , ( ) ] link:\doibase 10.1088/0957 - 4484/22/29/295714 [ * * , ( ) ] link:\doibase 10.1088/0957 - 4484/25/45/455702 [ * * , ( ) ] link:\doibase 10.1063/1.4822110 [ * * , ( ) ] link:\doibase 10.1063/1.4935522 [ * * , ( ) ] link:\doibase 10.1103/physrevb.93.115305 [ * * , ( ) ] link:\doibase 10.1063/1.4749789 [ * * , ( ) ] link:\doibase 10.1063/1.2794402 [ * * , ( ) ] link:\doibase 10.1103/physrev.114.90 [ * * , ( ) ] `` , '' in link:\doibase 10.1002/9781118984321.ch3 [ _ _ ] , ( , ) pp . link:\doibase 10.1103/physrevb.90.195309 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/0038-1098(81)90401-4 [ * * , ( ) ] link:\doibase 10.1103/physrevb.80.153309 [ * * , ( ) ] link:\doibase 10.1103/physrevb.83.245326 [ * * , ( ) ] link:\doibase 10.1063/1.4868131 [ * * , ( ) ] link:\doibase 10.1103/physrevb.88.075312 [ * * , ( ) ] link:\doibase 10.1063/1.3142396 [ * * , ( ) ] link:\doibase 10.1103/physrevb.90.165304 [ * * , ( ) ]
|
we investigate the 3.45-ev luminescence band of spontaneously formed gan nanowires on si(111 ) by photoluminescence and cathodoluminescence spectroscopy .
this band is found to be particularly prominent for samples synthesized at comparatively low temperatures . at the same time , these samples exhibit a peculiar morphology , namely , isolated long nanowires intersecting a dense matrix of short ones .
cathodoluminescence intensity maps reveal the 3.45-ev band to originate primarily from the long nanowires .
transmission electron microscopy shows that these long nanowires are either ga polar and are joined by an inversion domain boundary with their short n - polar neighbors , or exhibit a ga - polar core surrounded by a n - polar shell with a tubular inversion domain boundary at the core / shell interface . for samples grown at high temperatures , which exhibit a uniform nanowire morphology ,
the 3.45-ev band is also found to originate from particular nanowires in the ensemble and thus presumably from inversion domain boundaries stemming from the coexistence of n- and ga - polar nanowires . for several of the investigated samples , the 3.45-ev band splits into a doublet .
we demonstrate that the higher - energy component of this doublet arises from the recombination of two - dimensional excitons free to move in the plane of the inversion domain boundary .
in contrast , the lower - energy component of the doublet originates from excitons localized in the plane of the inversion domain boundary . we propose that this in - plane localization is due to shallow donors in the vicinity of the inversion domain boundaries .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
following the unification scheme the central nucleus of an active galaxy ( agn ) consists of a black hole ( bh ) , an accretion disk , line - emitting clouds , a dust torus , and emanates prominent jets when classified as radio - loud . the properties of radio - loud agn viewed at a small angle to the line - of - sight are in general agreement with the common blazar properties ( e.g. , * ? ? ? * ; * ? ? ? their broadband emission covers the complete electromagnetic band , from the radio up to the @xmath0-ray band , in some cases even reaching tev - energies , and is widely dominated by beamed non - thermal emission from a relativistic jet . the blazar class subdivides into bl lac objects and flat - spectrum radio quasars ( fsrqs ) . the difference dividing both subclasses is generally considered in the detection of strong emission lines in the case of fsrqs , while in bl lac objects the equivalent width of emission lines is depressed , or lines are absent at all . the physical reason is thought to lie predominantly in the weak accretion disk radiation field of bl lac objects , whereas bh in the nuclei of fsrqs accrete with high rates leading to luminous accretion disk photon fields . the present work deals with @xmath0-ray loud agn observed during an epoche of a bright accretion disk and accompagnied with the apprearance of strong emission lines . for simplicity i will refer to them as quasars in the following , although some radio - loud agn classified conventionally as bl lac objects may occasionally fall into this category as well ( e.g. , * ? ? ? * ; * ? ? ? * ) , and vice versa . gamma - ray production mechanisms , leptonic as well as hadronic ones , in agn jets often involve either the external radiation fields associated with the immediate agn environment or internal jet photons . for example , contributions from interactions in the accretion disk or broad - line region ( blr ) radiation fields are often mandatory to explain the overall @xmath0-ray spectral energy distribution ( sed ) from fsrqs ( e.g. , * ? ? ? * ; * ? ? ? * ) , while the sed of low - luminosity bl lac objects is often fitted with either synchrotron - self - compton models ( e.g. , * ? ? ? * ) or their hadronic equivalent , the synchrotron - proton blazar models ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the former scenario places the @xmath0-ray emission region rather close to the blr clouds . this has immediate consequences : if external radiation fields in quasar environments play a non - negligible role for @xmath0-ray production , then , at the same time they are also significant for the quasi - resonant process of photon - photon pair production with its peak cross section possessing a comparable value to the thomson cross section . it should be noted though , that the position of the high - energy emission region is still a matter of debate , with locations proposed to be also far away from the blr ( e.g. , * ? ? ? * ; * ? ? ? if this is the case , external compton emission ( e.g. , * ? ? ? * ) or pair cascade radiation initiated by ultra - high energy cosmic ray interactions on external photon fields ( e.g. , * ? ? ? * ; * ? ? ? * ) in those sources does not provide an appreciable contribution to the observed high - energy emission . the present work concerns specifically opacity features in the @xmath1 gev regime by photon absorption through e@xmath2-e@xmath3 pair production in radiation fields in the vicinity of the bh , but external to the jets in @xmath0-ray loud quasars . past works on this subject ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) indicate the importance of this process for constraining agn properties like the location of the @xmath0-ray region above the disk , disk radiation fields , torus temperature , etc . in contrast to these works , i will be focusing on the evolution of external radiation fields in quasar environments , and its consequences for the resulting opacity features in the @xmath0-ray band covered by current and near future instruments . this is primarily motivated by the anticipated studies of the evolution of the extragalactic background light ( ebl ) through the detection of absorption features in a large sample of high - redshift sources using the large area telescope ( lat ) onboard glast . dedicated methods have been developed here to probe the evolution of the ebl via detecting the horizon of @xmath0-rays emitted from extragalactic sources like agn and grbs while propagating through the ebl to earth ( e.g. , * ? ? ? * ; * ? ? ? they involve either the determination of the ratio of absorbed to unabsorbed flux versus redshift @xmath4 , or the detection of the e - folding cutoff energy @xmath5 versus redshift ( fazio - stecker relation ; * ? ? ? * ) in a large number of sources at various redshifts in order to disentangle intrinsic blazar features from absorption caused ones during propagation in the ebl . the common underlying reasoning for this procedure is that the observation of any redshift - dependent attenuation in @xmath0-ray agn can only be attributed to absorption in the ebl , and no other sources of redshift - dependent opacity exist in those sources . here i will demonstrate that opacity due to photon absorption from @xmath6-pair production caused in external radiation fields within the agn system ( in the following called local absorption , to be distinguished from `` self - absorption '' in the internal jet radiation fields ) will most likely result in optical depth values that increase with the source redshift , and coincidentally mimic redshift - dependent ebl - caused absorption . the outline of this paper is organized as follows : sect . 2 and sect . 3 describe the considered external target photon fields ( accretion disk and blr radiation field ) for photon - photon pair production , and the corresponding optical depth calculation , respectively . in sect . 5 i will apply various models of supermassive bh growth and accretion rate evolution , that are described in sect . 4 , to the @xmath0-ray attenuation calculations . the results with particular emphasis on the possibility of a redshift - dependence of the local opacity are presented in sect . the paper closes with conclusions and a discussion in sect . the most relevant target radiation fields in agn environments for photon absorption in the lat energy range , @xmath70.02 - 300 gev , are the optical / uv bands of the accretion disk photon field and the radiation field of the blr . these will be considered in the following . i assume the radiation fields to be located azimuthally symmetric with respect to the jet axis , and to radiate persistently during @xmath0-ray emission - ray domain , and thus allows the disk emission to be approximated as a constant photon field for the purpose of the present work . ] . the accretion disk spectrum in fsrqs is assumed to follow the cool , optically thick blackbody solution of shakura & sunyaev ( 1973 ) with a given accretion rate @xmath8 , suitable for agn that show strong emission lines ( `` strong - line agn '' ) . the differential photon density @xmath9 into a solid angle @xmath10 , @xmath11 reads @xmath12^{-1}\ ] ] with @xmath13 , @xmath14 the distance of the emission region above the bh , @xmath15 the photon energy , and @xmath16 with @xmath17 for a schwarzschild metric , @xmath18 and @xmath19 the boltzmann constant . for the opacity calculations the disk is considered to extend from 6 to @xmath20 . typical accretion rates for strong quasars approach the eddington accretion rate @xmath21 , @xmath22 , respectively the eddington luminosity @xmath23 . the accretion disk is considered as the photo - ionizing source of the blr material , and emission lines are produced through recombination . the blr geometry is approximated as a spherical shell with radius @xmath24 filled with clouds , extending from @xmath25 to @xmath26 . in this picture the accretion disk as located at @xmath27 . in the following the shell size is fixed to @xmath28pc to @xmath29pc , which are typical values for quasars ( e.g. , * ? ? ? a geometrically extended shell is supported from the observations by the agn watch campaigns . the opacity calculations require spectral as well as spatial knowledge of the emissivity of the blr radiation field . information about the cloud sizes and their spatial distribution can be deduced in general from reverberation mapping . although most details on blr physics and geometry have been derived by the study of radio - quiet agn , no major differences in the broad - line flux between radio - quiet and -loud sources have been found so far ( e.g. , * ? ? ? * ; * ? ? ? for the present calculations the number density @xmath30 and cross - section @xmath31 of the clouds are assumed to follow a power law with exponents @xmath32 and @xmath33 as derived by kaspi & netzer ( 1999 ) . a fraction @xmath34 of the central source luminosity @xmath35 is re - processed into line radiation such that the total blr luminosity , assumed to be optically thin , is @xmath36 , and these lines are assumed to radiate isotropically . observational support for a very narrow range of @xmath37 , i.e. the ratio of emitted to ionizing continuum photons , is provided by the observation of a linear correlation between balmer line and optical disk luminosity in agn over several orders of magnitude ( e.g. , * ? ? ? * ; * ? ? ? this is compatible with photoionization / recombination theory which expects the h - line brightness to be correlated with the ionizing continuum flux as the line luminosity is driven by this ionizing disk continuum . a statistical blazar study of celotti et al . ( 1997 ) implies @xmath38 , which is used in the following . changing the here fixed parameters ( e.g. @xmath37 , @xmath25 , @xmath26 , etc . ) will alter the absolute values of the @xmath0-ray attenuation , however , any redshift - dependence remains unaffected . the calculation of the @xmath0-ray opacity follows the procedure outlined in donea & protheroe ( 2003 ) for the geometrically thick shell case , but uses a more refined blr line spectrum here . average blr spectrum of francis et al . ( 1991 ) together with the h@xmath39 line strength reported by gaskell et al . ( 1981 ) sums up to 35 lines , with h@xmath39 and ly@xmath39 being the strongest lines . for the present work the blr spectrum is approximated as a two - component spectrum , @xmath40 , with the total luminosity of all lines at @xmath41 ( @xmath42 ) to be radiated at the strongest optical line ( h@xmath39 ) wavelength , and the total line luminosity at @xmath43 ( @xmath44 ) emitted at the strongest uv - line , ly@xmath39 . a refined treatment of the blr spectrum is straight forward , but will not alter the results on the redshift - dependence of the local opacity , nor add qualitatively new insights to the subject of the present work . the calculation of the optical depth @xmath45 with @xmath46 the primary @xmath0-ray photon energy , @xmath11 , @xmath47 , @xmath48 , @xmath49 , @xmath50 , @xmath51 and @xmath52 takes into account the full angle - dependent cross section ( e.g. , * ? ? ? * ; * ? ? ? @xmath53 , @xmath54 are determined by the respective geometry of the target photon field . the total cross section maximizes at @xmath55 , where @xmath56 ( @xmath57=photon interaction angle ) is the threshold condition of the pair production process . the very prominent peak of the cross section near threshold reaches roughly @xmath58 , where @xmath59 is the thomson cross section . the narrowness of the pair production cross section forces over half the interactions to occur in a small target photon energy interval @xmath60 centered on @xmath61ev for a smooth broadband spectrum @xcite . [ fig1 ] shows the resulting opacity from different distances @xmath62 of the @xmath0-ray production region above the black hole to @xmath63 , in the accretion disk ( red curves ) and blr radiation field ( blue curves ) for typical quasar accretion rates and bh masses . the two `` bumps '' in the blue opacity curves are the result of absorption in the h@xmath39 and ly@xmath39 lines of the blr , and smooth when a detailed multi(@xmath64)-line spectrum is used . in typical quasar environments the strength of local absorption is strongly dependent on the location of the emission region with respect to the target photon field . if the @xmath0-ray emission region is located not well beyond the blr , which is mandatory for @xmath0-ray production that involve external photon fields , local @xmath0-ray absorption features in fsrq spectra have to be expected at @xmath65 several tens of gev . the key step of this work is the application of cosmological black hole and quasar evolution to the expected pair production opacity of @xmath0-ray photons in agn , with direct implications for studies of the evolution of the ebl . with the availability of large agn data archives enormous advances in bh demographics were made . the cosmic evolution of the bh mass accretion rate has recently been studied by netzer & trakhtenbrot ( 2007 ) on the basis of @xmath66 sdss type - i radio - loud and radio - quiet agn in a large redshift range ( @xmath67 ) . significantly higher accretion rates at larger redshifts were derived with an eddington ratio of the accretion luminosity @xmath68 with @xmath69 . the intriguing similarity observed of the time history of star formation ( sf ) and bh accretion rate density ( e.g. , * ? ? ? * ) as well as established relations between bh mass and some bulge properties of the host galaxy lead to the widely accepted picture of a joint evolution of qsos / bhs and their host galaxies ( see also * ? ? ? * ; * ? ? ? moreover , the observed bh mass function is consistent with that inferred from quasar luminosities for simple assumptions of the accretion efficiency ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? the evolution of their luminosity functions at both , soft and hard x - rays , shows the number density of fainter agn to peak at lower redshifts than that of the brighter ones ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? this evidence of downsizing directly leads to the picture of an anti - hierarchical bh growth ( i.e. high - mass bhs grow faster and low - mass bhs grow preferably at lower redshift ) , and has meanwhile been confirmed by multiple studies : an analysis based on the fundamental plane of accreting bhs @xcite ; the phenomenological approach of marconi et al . ( 2004 ) for determining the evolution of the bh mass function using observational constraints from the local bh mass function , the evolving x - ray luminosity functions and energetics from the x - ray background ; determinations of low - luminosity high - redshift quasar luminosity functions by the goods collaboration ( e.g. , * ? ? ? * ) ; the combo-17 survey at higher luminosities @xcite , just to name a few . the combination of these observational findings with the theoretically advocated hierarchical clustering paradigm ( based on cold dark matter ) led to the awareness of feedback processes working during the process of bh mass growth in agn systems ( see e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? while the parameter space is still large , the advocated evolution of bh growth and thus accretion rates has severe implications for the @xmath0-ray quasar population where local @xmath0-ray absorption in accretion disk and blr radiation fields is potentially important . in the following i use three exemplary models for the evolution of the cosmic bh accretion rate ( fig . [ fig2 ] ) : a ) the netzer & trakhtenbrot ( 2007 ) analysis , complemented by the lapi et al . ( 2006 ) model for redshifts @xmath70 where only modest evolution is proposed there , b ) the anti - hierarchical bh growth picture of marconi et al . ( 2004 ) , and c ) a non - evolution scenario for comparison . the evolution of the accretion rate of model ( b ) has been derived by using the average bh growth history as published in marconi et al . ( 2004 ) from redshift @xmath71 ( where @xmath72 source activity has been assumed as an initial condition ) to @xmath73 together with their supported accretion radiation efficiency of 0.1 . model ( a ) and ( b ) both show strongly redshift - dependent bh accretion rates with higher rates at larger redshifts for bh masses of @xmath74m@xmath75 , typical for quasars . for the present work the chosen models are used plainly as means and agencies to demonstrate how any evolution of accretion rates transforms into a redshift - dependence of the local optical pair production depth in strong - line quasars . when studying the evolution of the ebl by means of a statistical analysis of signatures of @xmath0-ray attenuation with effective opacity @xmath76 from extragalactic objects , recognizing and disentangling absorption taking place within the source system ( local opacity ; @xmath77 ) and in the ebl during photon propagation to earth ( ebl - caused opacity ; @xmath78 ) is a crucial task . if both opacities each depend on redshift in the same direction , but with the exact value of each being unknown , the redshift parameter alone will not be sufficient to extract the evolution of the ebl - caused opacity . in order to assess the probability of this to take place in quasar - like agn , in the following i will apply realistic evolving , as well as non - evolving , cosmic accretion rate curves ( see fig . [ fig2 ] ) to the calculation of the expected opacity from @xmath0-ray absorption in their accretion disk and blr radiation fields , using the procedure outlined in sect . 2 . any evolution of accretion rates translates into an evolution of the target photon fields under consideration for photon - photon interactions , and thus into a redshift - dependence of the local opacity . for this study the position of the @xmath0-ray production is fixed to @xmath79pc . a location of the @xmath0-ray emission region close to the blr is particularly preferred by leptonic as well as hadronic blazar emission models that require external photon fields like accretion disk and blr radiation as an ingredient for @xmath0-ray production through particle - photon interactions . if @xmath0-ray production is located well beyond the blr , @xmath6-pair production on external photon fields ceases to be an important process , and so will external compton scattering above pair production threshold owing to their comparable cross section values there , or photopion production on accretion disk and blr photons . the goal of this excercise is to verify a possible _ redshift - dependence _ of the local opacity . the absolute @xmath80 values depend on the details of the target radiation fields and location of the @xmath0-ray production ( see e.g. * ? ? ? * ) , and thus come with uncertainties that reflect the dimension of the free parameter space . [ fig3 ] shows the resulting opacity curves in the observer frame when using the evolutionary behaviour of accretion rates following netzer & trakhtenbrot ( 2007 ) and lapi et al . ( 2006 ) for the target photon fields for @xmath6-interactions . the fast evolution of accretion rates at low redshifts is apparent . the corresponding curves for a m@xmath81m@xmath75 are inflated towards higher @xmath82 by more than an order of magnitude , and for @xmath83 gev the photon - photon collisions occur predominantly in the accretion disk radiation field . energy redshifting is the reason for the threshold and energy of maximum interaction probability , @xmath84tev , to decrease with redshift . note that for most energies @xmath85tev @xmath0-ray absorption occurs preferentially near the increasing part of the @xmath82 function near pair production threshold . if the distance @xmath62 of the @xmath0-ray production site from the disk rises beyond the blr , @xmath0-ray attenuation tails off as indicated in fig.[fig1 ] . an increase in blr size leads to an opacity decrease for an unchanged blr luminosity by an amount that corresponds to the decrease in target photon density with increasing blr volume . a direct view on the redshift - dependence of the local opacity opens up by slicing fig . [ fig3 ] at the energies of interest . [ fig4 ] shows the resulting @xmath86 curves at observer energy 100 gev ( all solid lines ) and 300 gev ( all dashed lines ) for all three evolution pictures for the accretion rate as shown in fig . [ fig2 ] and for typical quasar bh masses ( m@xmath87m@xmath75 : all red curves , m@xmath81m@xmath75 : all blue curves ) . the black curves represent the optical depths for the situation of a non - evolving high - accretion rate disk with a high - mass bh , and of a non - evolving low - accretion rate system with a lower - mass bh . a strong redshift - dependence is apparent in almost all cases . even for the case of non - evolving accretion rates the local opacities show redshift - dependence , if @xmath6 pair production occurs predominantly close to threshold , with an increasing slope of @xmath86 with redshift . the reason lies basically in the prominent peak of the pair production cross section , together with cosmological energy red - shifting : the prominent peak in the cross section leads to most photon - photon collisions occuring in a rather narrow energy range @xmath88 near threshold for smooth broadband target photon spectra . if those source spectra are located at cosmological distances , the energy red - shifting into the observer frame leads to the presented redshift - dependence of @xmath80 . it is straight forward to then determine the evolution of the e - folding cutoff energy @xmath89 for the series of accretion rate evolution models used so far . [ fig5 ] shows a fazio - stecker - like presentation for the local absorption and compares with the typically expected behaviour for absorption of @xmath0-rays in the evolving ebl ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? potential ebl probes of glast s lat are located at a redshift @xmath90 . in all cases , and this includes also non - evolving accretion rates , decreases the e - folding cutoff energy with redshift , similar to the fazio - stecker relation for ebl - caused absorption . if local absorption in the external radiation fields of agn leaves measurable imprints in the @xmath0-ray spectra , these will be almost unavoidably redshift - dependent , and remain to be distinguished from the @xmath0-ray opacity of the ebl . this work is devoted to investigate whether any redshift - dependence of the opacity @xmath80 of @xmath0-rays produced in systems of strong - line blazars can be uniquely attributed to photon absorption in the ebl . the possible existence of further sources of redshift - dependence of @xmath80 other than caused during propagation in the ebl will lead to ambiguities in estimating the ebl evolution . strong - line quasars are in general considered as high - luminosity sources in the @xmath0-ray domain , which have a marked probability of providing sufficient photon statistics to search for absorption breaks in a spectral analysis . their bright external photon fields ( i.e. accretion disk , blr ) are used in many high - energy emission models as a necessary ingredient for @xmath0-ray production , e.g. via inverse compton scattering . if this , however , takes place in the klein - nishina regime , and therefore above the pair production threshold , @xmath0-ray absorption due to photon - photon interactions is of comparable importance . in some hadronic blazar models the observed @xmath0-ray output is the result of re - distributing the injected nucleon energy via pair cascades that develop in external photon fields , and thus inevitably involves photon - photon pair production there . the focus of this study is therefore directed to @xmath6-pair production in photon environments ( here specifically accretion disk and blr radiation field ) of @xmath0-ray loud blazars where the interactions of relativistic particles in external photon fields potentially contributes significantly to the @xmath0-ray output above @xmath91 gev . * if the @xmath0-ray emission region is located not well beyond the blr , mandatory for @xmath0-ray production that involve external photon fields , local @xmath0-ray absorption features in strong - line quasar spectra have to be expected at @xmath92 several tens of gev . local @xmath0-ray absorption in external agn radiation fields ceases importance if the @xmath0-ray production site is sufficiently distant from the blr . while then evolutionary studies of the ebl by means of @xmath0-ray absorption signatures are not affected , it will lead to important implications for the high - energy blazar emission models . * following recent progress in bh demographics , bh growth and corresponding accretion rates turn out to show a redshift - dependence with higher rates at larger redshifts . correspondingly , the critical energy @xmath5 due to local absorption in quasar disk and blr radiation fields decreases with redshift , similar to fazio - stecker s relation for ebl absorption . * even for the case of no evolution of quasar disk accretion rates , @xmath5 decreases with redshift for sources of a given bh mass , at @xmath93 very similar to the fazio - stecker presentation of ebl absorption with the current knowledge of those systems . it results from the interplay of local absorption near pair production threshold and cosmological energy red - shifting . * any observed redshift - dependence of absorption features in blazars , that are prone to local @xmath0-ray absorption , can therefore _ not _ serve as a unique signature for absorption occurring in the ebl radiation field . this complicates approaches for estimating the evolution of the ebl using gev - sensitive instruments , that utilize the fazio - stecker relation or similar methods , and @xmath0-ray agn whose external photon fields are considered important in @xmath0-ray production . as a result , it seems that only naked @xmath0-ray jet sources ( i.e. agn without noticable optical / uv radiation fields close to the @xmath0-ray emission region ; `` true type-2 agn '' ) are unbiased probes for studies of the evolution of the ebl on the basis of the fazio - stecker relation and using gev instruments like glast , etc . consequently , an obvious choice for suitable candidate sources for this task would be blazars with particular weak or absent emission lines , generally classified as bl lac objects ( although exceptions exist , see sect . 1 ) . predictions for the expected number of gev bl lacs range from several hundred @xcite to a few thousand @xcite above the lat sensitivity . it remains to be seen whether the near future gev instruments will be detecting a sufficient number of suitable sources at @xmath90 to allow a sensible analysis on the basis of naked @xmath0-ray jet sources only . though the finding of this work adds fundamentally to already recognized complications ( e.g. flux and spectral source variability ) in analysis aiming to probe ebl evolution through the @xmath0-ray horizon , it also offers new options for constraining agn physics . the relevance of @xmath0-ray absorption for cutting off the sed at the high energy end in the different blazar types , possibly as a consequence of the location of @xmath0-ray production , could be probed by means of a statistical study of @xmath94 as a function of source type or object parameters . if simultaneously measured emission lines and/or accretion disk signatures indicate the presence of luminous photon fields external to the jet , the position of the @xmath0-ray production site could be constrained . monitoring both , the time history of the external target photon fields in agn and the jet @xmath0-ray flux , may offer independent verification of the importance of local absorption and extern inverse compton scattering there , and allows conclusions on the @xmath0-ray production location and on some properties of the blr material @xcite . at the same time , any non - detection of absorption features with sensitive gev instruments puts limits on the external radiation fields in those agn . if local opacity shapes part of the @xmath0-ray loud agn population , its evolution may influence luminosity function and extragalactic @xmath0-ray background contribution of agn above the 50 - 100 gev energy range . in both cases a redshift - dependence of the local opacity of @xmath0-ray loud quasars will have far - reaching implications . i d like to thank for valuable feedback from the glast - lat agn science working group , in particular for very useful comments from c. dermer , g. madejski and b. lott . i also thank f. stecker for providing his model curves and for interesting discussions . this work is supported by the national aeronautics and space administration under contract nas5 - 00147 with stanford university .
|
the case of @xmath0-ray absorption due to photon - photon pair production of jet photons in the external photon environment like accretion disk and broad - line region radiation field of @xmath0-ray loud active galactic nuclei ( agn ) that exhibit strong emission lines is considered .
i demonstrate that this local opacity , if detected , will almost unavoidably be redshift - dependent in the sub - tev range .
this introduces non - negligible biases , and complicates approaches for studying the evolution of the extragalactic background light with contemporary gev instruments like e.g. the gamma - ray large area space telescope ( glast ) , etc . , where the @xmath0-ray horizon is probed by means of statistical analysis of absorption features ( e.g. fazio - stecker relation , etc .
) in agn spectra at various redshifts .
it particularly applies to strong - line quasars where external photon fields are potentially involved in @xmath0-ray production .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
strange stars are astronomical compact objects which are entirely made of deconfined quarks ( for a review , see glendenning 1996 ; weber 1999 ) . the possible existence of strange stars is a direct consequence of the conjecture that strange quark matter ( sqm ) may be the absolute ground state of the strong interaction , i.e. , absolutely stable with respect to @xmath1fe ( bodmer 1971 ; witten 1984 ) . sqm with a density of @xmath2 g @xmath3 might exist up to the surface of a strange star . recently , the thermal emission from bare sqm surfaces of strange stars was considered ( usov 1998 , 2001a ) . it was shown that the surface emissivity of sqm in both equilibrium photons and @xmath0 pairs created by the coulomb barrier at the sqm surface is @xmath4% of the black body surface emissivity at the surface temperature @xmath5 k. below this temperature , @xmath6 k , the sqm surface emissivity decreases rapidly with decrease of @xmath7 . at the moment of formation of a strange star the surface temperature may be as high as @xmath8 k ( e.g. , haensel , paczyski , & amsterdamski 1991 ) . since sqm at the surface of a bare strange star is bound via strong interaction rather than gravity , such a star can radiate at the luminosity greatly exceeding the eddington limit , up to @xmath9 ergs s@xmath10 at @xmath11 k. ( alcock , farhi , & olinto 1986 ; chmaj , haensel , & slomiski 1991 ; usov 1998 , 2001a ) . a young strange star cools rapidly , and within about a month after its formation the surface temperature is less than @xmath12 k ( e.g. , pizzochero 1991 ) . in this case , the thermal luminosity from the stellar surface in both equilibrium photons and @xmath0 pairs is negligibly small , @xmath13 ergs s@xmath10 . a bare strange star with such a low surface temperature may be a strong source of radiation only if its surface is reheated . recently , response of a bare strange star to accretion of a massive comet - like object with the mass @xmath14 g onto the stellar surface was considered ( usov 2001b ) . it was shown that the light curves of the two giant bursts observed from the soft @xmath15-ray repeaters sgr 0526 - 66 and sgr 1900 + 14 may be easily explained in this model . in this paper we report on our numerical simulations of the response of a bare strange star to energy input onto its surface . we consider a wide range of the rate of the energy input . both normal and colour superconducting sqm are discussed . the model to be studied is the following . the energy input onto the surface of a bare strange star starts at the moment @xmath16 , and it is spherical and steady at @xmath17 . since in our simulations the surface temperature is not higher than @xmath18 k , @xmath0 pairs created by the coulomb barrier are the main component of the thermal emission from the stellar surface ( usov 2001a ) . we assume that the process of the energy input has no effect on the outflow of both created pairs and photons which form due to annihilation of some of these pairs ( cf . usov 2001b ) . the equation of heat transfer that describes the temperature distribution at the surface layers of a strange star is @xmath19 where @xmath20 is the specific heat for sqm per unit volume , @xmath21 is the thermal conductivity , and @xmath22 is the neutrino emissivity . the heat flux due to thermal conductivity is @xmath23 at the stellar surface , @xmath24 , the heat flux directed into the strange star is equal to ( usov 2001b ) @xmath25 where @xmath26 is the rate of the energy input onto the stellar surface , @xmath27 cm is the radius of the star , @xmath28 is the energy flux in @xmath0 pairs emitted from the sqm surface , @xmath29 is the mean energy of created pairs , @xmath30 is the flux of pairs from the unit sqm surface , @xmath31 @xmath32 , and @xmath33 is the surface temperature in units of @xmath34 k. eqs . @xmath35 give a boundary condition on @xmath36 at the stellar surface . we assume that at the initial moment , @xmath16 , the temperature in the surface layers is constant , @xmath37 k. it has been suggested ( bailin & love 1979 , 1984 ) that the quarks may eventually form cooper pairs . recently , superconductivity of sqm was considered in detail ( for a review , see rajagopal , & wilczek 2000 ; alford , bowers , & rajagopal 2001 ) , and it was shown that sqm is plausibly a colour superconductor if its temperature not too high . below , we consider both normal and superconducting sqm . for non - superconducting sqm , the contribution of the quarks to both the specific heat and the thermal conductivity prevails over the contributions of the electrons , photons and gluons . in this case we have ( iwamoto 1982 ; heiselberg & pethick 1993 ; benvenuto & althaus 1996 ) @xmath38 @xmath39 @xmath40 where @xmath41 @xmath3 is normal nuclear matter density , @xmath42 is the baryon number density of sqm , @xmath43 is the qcd fine structure constant , @xmath44 is the quark - gluon coupling constant , @xmath45 is the number of electrons per baryon , and @xmath46 is the temperature in units of @xmath34 k. sqm may be a colour superconductor if its temperature is below some critical value . in the classic model of bardeen , cooper , and schrieffer the critical temperature is @xmath47 , where @xmath48 is the energy gap at zero temperature and @xmath49 is the boltzmann constant ( e.g. , carter & reddy 2000 ) . the value of @xmath50 is very uncertain and lies in the range from @xmath51 mev ( bailin & love 1984 ) to @xmath52 mev ( alford , rajagopal , & wilczek 1998 ; alford , berges , & rajagopal 1999 ; pisarski & rischke 2000 ) . we use the following interpolation formula for the temperature dependence of the energy gap(e.g . , carter & reddy 2000 ) : @xmath53^{1/2}.\ ] ] superconductivity modifies the properties of sqm significantly . the specific heat of superconducting sqm increases discontinuously as the temperature falls below the critical temperature , and then decreases exponentially at lower temperatures . for the quark specific heat at @xmath54 we adopt the theoretical results of m " uhlschlegel ( 1959 ) for superconducting nucleons , i.e. , ( horvath , benvenuto , & vucetich 1991 ; blaschke , kl " ahn , & voskresensky 2000 ) @xmath55 @xmath56\,,\ ] ] here and below tilde signifies that this value relates to superconducting sqm . even at @xmath57 the suppression of the specific heat of sqm is never complete because the electrons remain unpaired . the specific heat of superconducting sqm which is used in our simulations is @xmath58 , where @xmath59 is the specific heat of the electron subsystem of sqm ( blaschke , grigorian , & voskresensky 2001 ) . at @xmath60 both the thermal conductivity of sqm and its neutrino emissivity are suppressed by a factor of @xmath61 ) . in our simulations , we adopt @xmath62+k_e\,,\ ] ] @xmath63\,,\ ] ] where @xmath64 and @xmath22 are given by equations ( 7 ) and ( 8) , respectively , and @xmath65 is the thermal conductivity of the electrons ( e.g. , blaschke et al . the set of equations @xmath66 was solved numerically for both normal @xmath67 and superconducting @xmath68 states of sqm . we assumed the typical values of @xmath69 , @xmath70 , and @xmath71 . figure 1 shows a typical temporal behaviour of the strange star luminosity , @xmath72 , in @xmath0 pairs . from this figure , we can see that @xmath73 increases eventually to its maximum value @xmath74 . the rate of this increase may be characterized by the rise time @xmath75 that is determined as a time interval from the initial moment , @xmath16 , to the moment when @xmath73 is equal to @xmath76 . the results of our simulations are presented in tables 1 and 2 . there is a critical value , @xmath77 , of the input luminosity at which the dependence of @xmath74 on @xmath26 changes qualitatively ( see table 1 ) . @xmath77 is @xmath78 ergs s@xmath10 for normal sqm and @xmath79 ergs s@xmath10 for superconducting sqm with @xmath80 mev . at @xmath81 , @xmath74 is about @xmath26 while when @xmath26 is a few times below than @xmath77 the thermal emission of @xmath0 pairs from the stellar surface is negligible . in our simulations , the rise time @xmath75 varies in a very wide range from @xmath82 s at @xmath83 ergs s@xmath10 and @xmath84 mev to @xmath85 s at @xmath86 ergs s@xmath10 and @xmath87 mev ( see table 2 ) . at @xmath87 mev the thermal emission of @xmath0 pairs does not depend on @xmath48 because in this case both the specific heat of the quarks and their thermal conductivity are strongly suppressed , and the heat transport is mostly determined by the electron subsystem of sqm . the rise time of the luminosity in neutrinos is many orders of magnitude larger that @xmath75 , especially when @xmath75 is small . in our model the neutrino luminosity may increase up to @xmath88 when @xmath89 goes to infinity . this is because all energy which is delivered onto the stellar surface is radiated either from the surface by @xmath0 pairs or from the stellar interior by neutrinos . since bare strange stars can radiate at the luminosities greatly exceeding the eddington limit ( alcock et al . 1986 ; chmaj et al . 1991 ; usov 1998 , 2001a ) , these stars are reasonable candidates for soft @xmath15-ray repeaters ( sgrs ) that are the sources of brief ( @xmath90 s ) bursts with super - eddington luminosities , up to @xmath91 . the bursting activity of a sgr may be explained by fast heating of the bare , rather cold ( @xmath92 k ) surface of a strange star up to the temperature of @xmath93 k and its subsequent thermal emission ( usov 2001a ) . the heating mechanism may be either fast decay of superstrong ( @xmath94 g ) magnetic fields ( usov 1984 ; duncan & thompson 1992 ; paczy ' nski 1992 ; thompson & duncan 1995 ; cheng & dai 1998 ; heyl & kulkarni 1998 ) or impacts of comet - like objects onto the stellar surface ( harwitt & salpeter 1973 ; newman & cox 1980 ; zhang , xu , & qiao 2000 ; usov 2001b ) . the magnetar model of sgrs which is based on the first mechanism is most popular now . in this model , the magnetic energy of a strongly magnetized strange star ( magnetar ) may be released from time to time due to mhd instabilities . a violent release of energy inside a magnetar excites surface oscillations ( e.g. , thompson & duncan 1995 ) . in turn , this shaking may generate strong electric fields in the magnetar magnetosphere which accelerate particles to high energies . these high energy particles bombard the surface of the strange star and heat it . at the input luminosities of @xmath95 ergs s@xmath10 , which are typical for sgrs , the efficiency of reradiation of the partical energy by the stellar surface to @xmath0 pairs is very high , @xmath96 , especially for superconducting sqm ( see table 1 ) . since for sgrs the burst luminosities are at least a few orders of magnitude higher than @xmath97 , the outflowing @xmath0 pairs mostly annihilate in the vicinity of the strange star ( e.g. , beloborodov 1999 ) . therefore , at @xmath98 ergs s@xmath10 far from the star the luminosity in x - ray and @xmath15-ray photons practically coincides with @xmath99 , @xmath100 . two giant bursts were observed on 5 march 1979 and 27 august 1998 from sgr @xmath101 and sgr @xmath102 , respectively . the peak luminosities of these bursts were @xmath103 ergs s@xmath10 ( fenimore , klebesadel , & laros 1996 ; hurley et al . 1999 ) . in this case , from table 2 the rise time expected in our model is @xmath104 s that is consistent with available data on the two giant bursts ( mazets et al . 1999 ) . this is valid irrespective of that sqm is a colour superconductor or not . for typical bursts of sgrs the luminosities are @xmath105 ergs s@xmath10 ( e.g. , kouveliotou 1995 ) , and the observed rise times ( @xmath106 s ) may be explained in our model only if sqm is a superconductor with the energy gap @xmath107 mev ( see table 2 ) . alcock , c. , farhi , e. , & olinto , a. 1986 , apj , 310 , 261 alford , m. , berges , j. , & rajagopal , k. 1999 , nucl . phys . b , 558 , 219 alford , m. , bowers , j.a . , & rajagopal , k. 2001 , j. phys . g , 27 , 541 alford , rajagopal , k , & wilczek , f. 1998 , pjys . lett . b , 422 , 247 bailin , d. , & love , a. 1979 , j. phys . a , 12 , l283 bailin , d. , & love , a. 1984 , phys . , 107 , 325 beloborodov , a.m. 1999 , mnras , 305 , 181 benvenuto , o.g . , & althaus , l.g . 1996 , apj , 462 , 364 blaschke , d. , grigorian , h. , & voskresensky , d.n . 2001 , a&a , 368 , 561 blaschke , d. , kl " ahn , t. , & voskresensky , d.n . 2000 , apj , 533 , 406 bodmer , d. 1971 , phys . d. , 4 , 1601 carter , g.w . , & reddy , s. 2000 , phys . d , 62 , 103002 cheng , k.s . , & dai , z.g . 1998 , phys . , 80 , 18 chmaj , t. , haensel , p. , & slomiski , w. 1991 , nucl . phys . b , 24 , 40 duncan , r.c . , & thompson , c. 1992 , apj , 392 , l9 fenimore , e.e . , klebesadel , r.w . , & laros , j.g . 1996 , apj , 460 , 964 glendenning , n.k . 1996 , compact stars : nuclear physics , particle physics , and general relativity ( verlag new york : springer ) haensel , p. , paczyski , b. , & amsterdamski , p. 1991 , apj , 375 , 209 harwitt , m. , & salpeter , e.e . 1973 , apj , 186 , l37 heiselberg , h. , & pethick , c.j . 1993 , phys . d , 48 , 2916 heyl , j.s . & kulkarni , s.r . 1998 , apj , 506 , l61 horvath , j.e . , benvenuto , o.g . , & vucetich , h. 1991 , phys . rev . d , 44 , 3797 hurley , k. , et al . 1999 , nature , 397 , 41 iwamoto , n. 1982 , ann . , 141 , 1 kouveliotou , c. 1995 , ap&ss , 231 , 49 mazets , e.p . , et al . 1999 , lett . , 25 , 635 m " uhlschlegel , b. 1959 , z. phys . , 155 , 313 newman , m.j . , & cox , a.n . 1980 , apj , 242 , 319 paczy ' nski , b. 1992 , acta astron . , 42 , 145 pisarski , r.d . , & rischke , d.h . 2000 , phys . d , 61 , 051501 pizzochero , p.m. 1991 , phys . rev . lett . , 66 , 2425 rajagopal , k. & wilczek , f. 2000 , preprint ( hep - ph/0011333 ) thompson , c. & duncan , r.c . 1995 , mnras , 275 , 255 usov , v.v . 1984 , ap&ss , 107 , 191 usov , v.v . 1998 , phys . , 80 , 230 usov , v.v . 2001a , apj , 550 , l179 usov , v.v . 2001b , phys . lett . , 87 , 021101 weber , f. 1999 , j. phys . g : nucl . part . phys . , 25 , 195 witten , e. 1984 , phys . d , 30 , 272 zhang , b. , xu , r.x . , & qiao , g.j . 2000 , apj , 545 , l127
|
we study numerically the thermal emission of @xmath0 pairs from a bare strange star heated by energy input onto its surface ; heating starts at some moment , and is steady afterwards .
the thermal luminosity in @xmath0 pairs increases to some constant value .
the rise time and the steady thermal luminosity are evaluated .
both normal and colour superconducting states of strange quark matter are considered .
the results are used to test the magnetar model of soft gamma - ray repeaters where the bursting activity is explained by fast decay of superstrong magnetic fields and heating of the strange star surface .
it is shown that the rise times observed in typical bursts may be explained in this model only if strange quark matter is a superconductor with an energy gap of more that 1 mev .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the quantum information topics are the subject of active research @xcite . the impressive progress have been reached in the quantum cryptography @xcite and study of quantum algorithms @xcite . due to an impact on security in quantum cryptography , the quantum cloning is still a significant topic . at the same time , a cloning itself is hardly sufficient for an eavesdropping @xcite . no - copying results have been established for pure states @xcite as well as for mixed states @xcite . in view of such evidences , the question arose how well quantum cloning machines could work . in effect , the basic importance of the no - cloning theorem is expressed much better in more detailed results , which also give explicit bounds on an amount of the noise . after the seminal work by buek and hillery @xcite , many approaches to approximate quantum cloning have been developed . in view of existing reviews @xcite , we cite only the literature that is directly connected to our results . an approximate cloning of two prescribed pure states was first considered in ref . this kind of cloning operation is usually referred to as _ state - dependent cloning _ @xcite . in general , various types of state - dependent cloners may be needed with respect to the question of interest @xcite . errors inevitably occur already in a cloning of two nonorthogonal states @xcite . how close to perfection can a cloning be ? of course , any explicit answer must utilize some optimality criterion . we will refer criterion used in ref . @xcite to as the _ absolute error _ @xcite . chefles and barnett @xcite derived the least upper bound on the global fidelity for cloning of two pure states with arbitrary prior probabilities . the quantum circuit that reaches this upper bound was also constructed @xcite . the global fidelity of cloning of several equiprobable pure states was examined in ref . @xcite . although cloning problems were mostly analyzed with respect to the fidelity criteria , other measures of closeness of quantum states are relevant . for example , the partial quantum cloning is easier to analyze with respect to the squared hilbert - schmidt distance @xcite . one of criteria , _ relative error _ @xcite , has been shown to be useful within the b92 protocol emerged in ref . @xcite . deriving bounds on the relative error was based on the spherical triangle inequality@xcite and the notion of the angle @xcite sometimes called the _ bures length _ @xcite . using this new method , a cloning of two equiprobable mixed states was studied with respect to the global fidelity @xcite . the results of ref . @xcite were partially extended to mixed - state cloning @xcite . in a traditional approach , the ancilla does not contain _ a priori _ information of state to be cloned just now . a more general case is the scope of the stronger no - cloning theorem @xcite . namely , a perfect cloning is achievable , if and only if the full information of the clone has already been provided in the ancilla state alone . in ref . @xcite we examined a cloning of finite set of states when the ancilla contains a partial information of the input state . so , the previous result of ref . @xcite was extended to both the mixed states and _ a priori _ information . in this paper , we study the relative error of cloning of several mixed states , having arbitrary prior probabilities . _ a priori _ information in the ancilla is also assumed . in section ii , the relative error criterion introduced in refs . @xcite is extended to the general cloning scenario . we derive the lower bounds on the relative error for cloning of two - state set ( see section iii ) and multi - state set ( see section iv ) . in section v , the relative error is compared with other optimality criteria . we also build the quantum circuit for cloning of two pure states ( see section vi ) . this circuit reaches the lower bound on the relative error for arbitrary prior probabilities and _ a priori _ knowledge about the input . section vii concludes the paper . the main problem posed formally is this . we have @xmath0 indistinguishable @xmath1-level systems that which are all prepared in the same state @xmath2 from the known set of density operators on the space @xmath3 . these @xmath0 systems form the register @xmath4 . its initial state is a density operator @xmath5 on the input hilbert space @xmath6 . the prior probabilities @xmath7 of states @xmath8 obey the normalization condition @xmath9 . we aim to get a larger number @xmath10 of copies of the given @xmath0 originals by means of the ancilla whose initial state is @xmath11 according to the input @xmath8 . here we mean a system @xmath12 composed of extra register @xmath13 and environment @xmath14 . the extra register @xmath13 contains @xmath15 additional @xmath1-level systems , each is to receive the clone of @xmath16 . if we include an environment space then any deterministic physical operation may be expressed as a unitary evolution . thus , the final state of two registers @xmath17 is the partial trace over environment space @xmath18 \ . \label{bd2}\ ] ] the output @xmath19 is a density operator on the output hilbert space @xmath20 . the actual output @xmath19 must be compared with the ideal output @xmath21 . many measures of distinguishability between mixed quantum states are based on the fidelity @xcite . we shall employ the angles and the sine metric @xcite . let @xmath22 denote a unique positive square root of @xmath23 . the fidelity between the two density operators @xmath24 and @xmath25 is equal to @xmath26 @xcite . in terms of this measure , the angle @xmath27 $ ] between @xmath24 and @xmath25 is defined by the equality @xmath28 @xcite . it is also referred to as the _ bures length _ @xcite , because of its close relation to the standard bures metric @xmath29 . due to the spherical triangle inequality @xcite , @xmath30 we introduce the _ sine distance _ @xcite between @xmath25 and @xmath24 as @xmath31 . this metric on the space of quantum states has a close relation to the trace distance @xcite and enjoys the following @xcite . for any povm measurement @xmath32 , there holds @xmath33 here @xmath34 is the probability of obtaining outcome @xmath35 , if the state right before measurement was @xmath25 . a more detailed characterization of such a kind can be posed via majorization relations @xcite . we also have @xmath36 . since the fidelity function can not decrease under any deterministic quantum operation @xcite , the last inequality can be extended to @xmath37 using the sine distance is reasonable approach due to the inequalities ( [ pr11 ] ) and ( [ pr14 ] ) . for brevity , let us denote @xmath38 when two inputs @xmath39 and @xmath40 are equiprobable , the relative error is defined by @xmath41 @xcite . meaning @xmath42 , it can be rewritten as @xmath43 the right - hand side of eq . ( [ bd6 ] ) is quite relevant to the case of arbitrary prior probabilities . since the distance @xmath44 estimates the difference between two probability distributions ( see eq . ( [ pr11 ] ) ) , a reliable identification of original input via measurement over clones may be provided only when @xmath45 so we see a reason for using a ratio of @xmath44 just to the half of @xmath46 . in addition , this choice implies that the tight lower bound on relative error generally recovers the range @xmath47 $ ] . we shall now extend the notion of relative error for the set @xmath48 with @xmath49 , when the number of different pairs is equal to @xmath50 . the probability of taking the pair @xmath51 is equal to @xmath52 where @xmath53 . we clearly have @xmath54 and @xmath55 for the set of @xmath56 equiprobable states . to each pair @xmath57 assign the quantity @xmath58 which takes into account that , perhaps , @xmath59 . it is natural to put the weighted average of the @xmath50 quantities ( [ aveid ] ) . * definition 1 . * _ the relative error of @xmath60 cloning of the set @xmath48 is defined by _ let the prior probability be value of order @xmath62 for all the states except @xmath63 and @xmath64 . that is , we take @xmath65 for @xmath66 , whence @xmath67 , @xmath68 for the rest pairs . the expression ( [ bd7 ] ) for relative error is simply reduced to @xmath69 . in the same manner , we can find @xmath70 , when @xmath71 and probabilities @xmath65 except for the states @xmath72 solely . we are interested in a nontrivial lower bound on the relative error ( [ bd7 ] ) . our approach to obtaining the limits utilizes triangle inequalities @xcite . following the method , we shall derive the angle relation from which bound on the relative error is simply obtained . it is handy to introduce the angle @xmath73 $ ] as @xmath74 the laws of quantum theory impose some restrictions on acceptable values of angles @xmath75 , whence nontrivial bounds for different figures of cloning merit follow . in the case of the two - state set @xmath76 , the initial state of ancilla is @xmath77 or @xmath78 according to the input which is @xmath79 or @xmath80 . we further assume that @xmath81 and , by the multiplicativity of fidelity , @xmath82 . the motivation is as follows . if the inequality ( [ upsres ] ) is not satisfied then there are states sufficient for perfect cloning @xcite . that is , there exist states @xmath77 and @xmath78 such that @xmath83 . hence we can mention a trivial bound @xmath84 only . so we presuppose that the inequality ( [ upsres ] ) is valid . as result , we have @xmath85 with no loos of generality , we assume that @xmath86 . * theorem 2 * _ the relative error @xmath87 of cloning of the set @xmath76 satisfies _ @xmath88 * proof * applying the inequality ( [ pr10 ] ) twice , we obtain @xmath89 recall that the fidelity function is multiplicative , preserved by unitary evolution and non - decreasing under the operation of partial trace @xcite . so we obtain @xmath90 whence @xmath91 . combining this with eq . ( [ tilde1 ] ) provides @xmath92 consider the function @xmath93 to be minimized . we want to minimize @xmath94 under the constraint ( [ star1 ] ) , @xmath95 and @xmath96 . this task is solved in appendix a. by substitutions , we then have @xmath97 and further the statement of theorem 2 . @xmath98 for equiprobable states , the bound ( [ theor3 ] ) is reduced to the lower bound deduced in ref . @xcite . in terms of @xmath99 and @xmath100 , we rewrite ( [ upsres ] ) as @xmath101 . by @xmath102 and @xmath103 , the bound ( [ theor3 ] ) becomes @xmath104 at fixed @xmath105 and @xmath106 , the right - hand side of eq . ( [ theor33 ] ) is an increasing function of probability @xmath107 . that is , it decreases as the prior probabilities differ . this is analog of that the upper bound on the global fidelity increases in such a situation @xcite . we are rather interested in dependence of the bound on @xmath106 . this parameter marks a top amount of an _ a priori _ information , which can initially be laid in the ancilla . the more a value of @xmath106 , the less this amount . the angle @xmath108 is a decreasing function of @xmath106 . in the range ( [ alpdel ] ) , the lower bound by theorem 3 is a decreasing function of @xmath108 . so the right - hand side of ( [ theor33 ] ) increases as the marker @xmath106 of additional information increases . for @xmath109 the perfect cloning can be reached @xcite . in line with this fact , we have @xmath110 and the vanishing bound on @xmath87 . on the contrary , in the usual cloning there is no _ a priori _ information , i.e. @xmath111 and @xmath112 . then the bound by theorem 2 reaches its maximum as a function of @xmath106 . the above points reproduce the observations of ref . @xcite in more general setting . if @xmath113 at fixed @xmath114 then the right - hand side of the inequality ( [ theor33 ] ) goes to zero . this is natural because infinite number @xmath0 of originals can provide almost perfect cloning . if @xmath115 at fixed @xmath0 then the right - hand side of ( [ theor33 ] ) recovers the value @xmath116 . in the standard cloning of equiprobable states ( @xmath112 , @xmath117 ) , this value can be arbitrarily close to 1 , since @xmath118 . it is not insignificant that the value @xmath116 gives the minimal size of probability of inconclusive answer for unambiguous discrimination at @xmath117 . namely , the success discrimination of the equiprobable pure states @xmath119 occurs with the optimal probability @xmath120 @xcite . the value @xmath116 is obtained for @xmath121 and @xmath122 . note that the upper bound on the global fidelity in the limit @xmath115 at fixed @xmath0 goes to well - known helstrom bound @xcite . it is the probability of correctly distinguishing between two pure states @xmath123 by the optimal strategy @xcite . we now obtain a lower bound on the relative error of cloning of the set @xmath48 . as before , the prior probabilities are arbitrary and constrained only by the normalization condition . like ( [ alpdel ] ) , we have the acceptable range @xmath124 according to theorem 2 , each term of sum in the right - hand side of ( [ bd7 ] ) obeys @xmath125 hence the desired bound is established as follows . * theorem 3 * _ the relative error of @xmath60 cloning of the set @xmath48 satisfies _ @xmath126 as a straightforward extension , the bound ( [ theor4 ] ) succeeds many features of the bound ( [ theor3 ] ) . if two probabilities , say , @xmath127 and @xmath128 are variable and the rest of parameters is fixed , then the bound ( [ theor4 ] ) decreases as these probabilities differ . if some one probability is close to 1 and other probabilities are small , then the bound is close to zero . this behavior is expected , because single known state can be cloned perfectly . for equal _ a priori _ probabilities @xmath54 , the bound by theorem 3 becomes @xmath129 it is natural that both the bounds given by ( [ theor4 ] ) and ( [ prieq ] ) decrease as @xmath130 increases . indeed , the parameter @xmath130 characterizes an amount of prior information . if the upper limit of eq . ( [ alpdel2 ] ) is saturated for some pair @xmath57 then corresponding summands in the right - hand sides of eqs . ( [ theor4 ] ) and ( [ prieq ] ) vanish . this is the case of potentially perfect cloning . on the whole , these conclusions on a role of _ a priori _ information in the ancilla add to the stronger no - cloning theorem . a question is , whether the lower bounds ( [ theor3 ] ) and ( [ theor4 ] ) can be reached ? in general , it is not the case , though the bound by theorem 3 is least for two pure states . the quantum circuit for optimal cloning will be built in the next section . the subject matter changes for @xmath49 . from the viewpoint of minimization the bound of theorem 3 is approximate . as reasons of appendix a show , saturating the inequality ( [ rjk1 ] ) holds if and only if @xmath131 , @xmath132 for @xmath133 ( for @xmath134 the angles @xmath135 and @xmath136 should be swapped in the two equalities ) . these two equalities per each of @xmath50 pairs totally give @xmath137 conditions . for saturating eq . ( [ theor4 ] ) , @xmath56 variables @xmath135 must satisfy all these @xmath137 conditions . except for some special cases , this is not possible . thus , the presented limit is somewhat rough . more rigorous way may be as follows . similar to ( [ star1 ] ) , we have arrived at the @xmath50 inequalities of a kind @xmath138 . together with the @xmath56 conditions @xmath139 , these relations specify some simplex in @xmath56-dimensional real space . the relative error ( [ bd7 ] ) can be rewritten in the form @xmath140 where @xmath141 . the task is to minimize the function ( [ bdel7 ] ) in the above simplex . so we come across a difficult problem of nonlinear programming ( the simple case @xmath142 of this problem is considered in appendix a ) . for @xmath139 , the minimized function is concave . so the problem of minimization is reduced to finding extremal points of the simplex . if the values of parameters are prescribed , the wanted minimum can be found numerically . at the same time , it is complicated to obtain an explicit formula for general case . but even if we should find it , we still would not have a complete solution to the problem of mixed - state cloning . indeed , it is not necessary that bound given by such a formula be least . so we have restricted our consideration to obtaining of the bound by theorem 3 . rough though this bound is , it has straightforward form and allows to estimate how a merit of state - dependent cloning is limited . we shall now expose the relative error in comparison with other optimality criteria . for the sake of simplicity , we restrict to the @xmath60 cloning of two equiprobable pure states @xmath143 without _ a priori _ information in the ancilla . how able to good cloning is the pair ? this question is central to applications of quantum cloning . in principle , we may assume both the deterministic cloning and probabilistic cloning @xcite . a merit of deterministic cloning may be viewed with respect to the global fidelity , the absolute error and the relative error . for equiprobable inputs , the global fidelity is expressed by @xmath144 @xcite . hillery and buek @xcite used the measure @xmath145 . this measure will be referred to as _ absolute error _ @xcite . the relative error is defined by eq . ( [ bd6 ] ) . in probabilistic cloning , the exact clone of an input successfully generated with the maximal probability @xcite @xmath146 where @xmath105 denote the overlap @xmath147 between states @xmath143 . as it is shown in refs . @xcite , the maximum of the global fidelity is equal to @xmath148 according to ( [ theor33 ] ) , the minimum of the relative error is reduced to @xmath149 for @xmath112 and @xmath42 . for the absolute error we have @xcite @xmath150 let us consider the two cases : ( i ) the states are @xmath143 are almost orthogonal , i.e. @xmath151 ; ( ii ) the states are @xmath143 are almost identical , i.e. @xmath152 with @xmath153 . a behaviour of each of the criteria is shown in table 1 ( @xmath154 ) . .an asymptotic behaviour of the four criteria . [ cols="^,^,^",options="header " , ] as it is clear from the second column , for the case ( i ) all the measures endorse a good merit of both the deterministic and probabilistic cloning . in effect , the optimum of global fidelity is close to one , the optimum of absolute and relative error is close to zero . the probability of success is close to one . it is natural because orthogonal states can perfectly be cloned . the principal distinction of the relative error is revealed in the case ( ii ) . it seems offhand that two almost identical states can be cloned very well . both the global fidelity and absolute error approve the conclusion ( @xmath155 and @xmath156 ) . it would be rash to accept this . in effect , the optimal probability @xmath157 is generally not close to one . the first term @xmath158 is almost one only if the number @xmath15 of actual clone is negligible in comparison with the number @xmath0 of originals . in line with this , the optimum of relative error is close to zero for @xmath159 . but the probability is close to zero and the relative error is close to one when the number @xmath15 of actual clone is large . we see that both the global fidelity and absolute error lose sight of the important aspect of deterministic cloning . even for the primary @xmath160 cloning , we have @xmath161 and @xmath162 , that is both the probabilistic and deterministic strategies are restricted enough . in contrast with the global fidelity and the absolute error , for the case ( ii ) a behaviour of relative error is crucially dependent on numbers @xmath0 and @xmath10 . similar to the optimal probability of success , the criterion of relative error emphasizes that any cloning is not isolated stage in quantum information processing . as a rule , the outputs of cloning machine are subjects of further operations , say , a discrimination . for example , in the cryptographic b92 scheme alice encodes the bits into two non - orthogonal pure states @xcite . so bob can apply the unambiguous discrimination @xcite . but the closer used states are to each other the larger number of discarded bits is in the total sequence . on the other hand , a sufficiently great closeness of the used states will prevent the eavesdropping . unlike both the global fidelity and absolute error , the notion of relative error allows to take such aspects into account . we shall now build quantum circuits for the optimal relative - error cloning of two pure states @xmath143 with arbitrary prior probabilities @xmath163 ( @xmath86 ) . _ a priori _ information about actually input state is contained in the state of ancilla which is either @xmath164 or @xmath165 . without loss of generality , we take the product @xmath166 to be positive real . these states are parametrized as @xmath167 , @xmath168 . the overlap is @xmath169 with @xmath170 . so , we have the register of @xmath171 qubits , where @xmath114 qubits are initially in the blank state @xmath172 , @xmath0 qubits are in the state to be cloned , and one qubit is ancillary . our aim is to transform these states according to the specification . the strategy is an extension of the known one @xcite and uses the _ distinguishability transfer gate _ ( see appendix b ) . first , the information about the input @xmath0 originals is transferred into one qubit . we mark the ancillary qubit by 0 , the @xmath0 original qubits by @xmath173 , and the @xmath114 additional qubits by @xmath174 . the just left gate acts on the qubits @xmath175 and @xmath0 as @xmath176 where @xmath177 . then an operation is applied to qubits @xmath178 and @xmath175 , and so on . in the first stage , the gate @xmath179 transfers the distinguishability from @xmath180th qubit to @xmath181th ( @xmath180 runs from @xmath0 to @xmath182 ) , i.e. @xmath183 where @xmath184 , @xmath185 . within the first stage , the state changes as @xmath186 where the gates @xmath187 are put from right to left with decreasing @xmath180 . this part transfers a total distinguishability of the @xmath0 originals @xmath188 into the one - qubit state @xmath189 . an example for @xmath190 cloning is shown on fig . 1 . cloning with the _ a priori _ information . for brevity , the gate @xmath191 in the left box and the gate @xmath192 the right box are both denoted as @xmath193 . ] for using an _ a priori _ information , we now include the _ turned _ gate @xmath194 . this gate transfers distinguishability of ancilla s states @xmath195 to those of qubit 1 , namely @xmath196 after the action of gate @xmath194 , the ancilla contains no information about distinguishability . all the distinguishability of inputs are now concentrated on two possible states @xmath197 of qubit 1 , where @xmath198 . now the scheme acts on the qubit 1 by the unitary operator @xmath199 specified as @xmath200 the values of angle @xmath201 and complex numbers @xmath202 and @xmath203 will be found below . so the second stage results in the final state @xmath204 of the @xmath10 qubits . in fig . 1 , the gates @xmath194 and @xmath199 between the dash boxes perform the second stage . its structure is independent of numbers @xmath0 and @xmath10 . note that this stage and an _ a priori _ information are not considered in ref.@xcite . we put two linear combinations of the ideal outputs @xmath205 and @xmath206 as @xmath207 the final stage of cloning is posed as @xmath208 . let us continue the sequence @xmath209 with respect to the above recurrence , that is @xmath210 where @xmath211 $ ] . hence we obtain @xmath212 . due to the property ( [ dp3012 ] ) of distinguishability transfer gate , we have @xmath213 in the third stage , the label @xmath214 in ( [ gdll ] ) runs from @xmath215 to @xmath216 . so , the gate @xmath217 acts on the qubits 1 and 2 , the gate @xmath218 acts on the qubits 2 and 3 , and so on . the total action is described by @xmath219 where the gates @xmath220 are put from right to left with increasing @xmath214 . in ( [ stag3 ] ) , the accumulated distinguishability is distributed among the @xmath10 qubits of interest . on fig . 1 , the four gates @xmath221 , @xmath222 , @xmath223 and @xmath224 of the third stage are grouped in the right dash box . using the linearity , we see that @xmath208 too . due to @xmath225 , @xmath226 that is actually correct . specifying concrete values of @xmath202 and @xmath203 and herewith the single - qubit gate @xmath199 in eq . ( [ datet ] ) , we can optimize either the relative error or the global fidelity . in each case , we superpose the @xmath227 onto the @xmath228 . then after the second stage the @xmath10 qubits of interest lie in the states @xmath229 . for the optimality with respect to the relative error , we demand that @xmath230 , whence we get @xmath231 and @xmath232 , @xmath233 from ( [ tildd3 ] ) . the angle between @xmath234 and @xmath235 is equal to @xmath236 , the angle between @xmath237 and @xmath238 is equal to @xmath239 . because unitary transformations preserve angles , the angle between @xmath240 and @xmath238 is equal to @xmath241 within the third stage , the state @xmath242 maps to @xmath243 . by definition , the value @xmath244 is angle between @xmath243 and @xmath245 . since @xmath246 and @xmath247 , we find the needed value @xmath248 . thus , the inequality ( [ theor3 ] ) is saturated too , and the built scheme is really optimal with respect to the relative error . note that @xmath249 and @xmath250 are found as @xmath251 and @xmath252 . but the described geometrical picture is quite sufficient for all the purposes . in the same manner , the optimization of cloning with respect to the global fidelity would be considered . as result , the generalization of the deterministic cloner of ref . @xcite to prior ancillary information can be obtained . we have analyzed a new optimality criterion for the state - dependent cloning of several states with arbitrary prior probabilities and an ancillary information . the notion of the relative error has been extended to the general cloning scenario . the lower bounds on the relative error have been obtained for both the two - state and multi - state cases . the attainability of the derived bounds has been discussed . the quantum circuit for optimal cloning of two pure states with respect to the relative error has been built . our approach is based on the simple geometrical description , which generally clarifies origins of a bound for one or another figure of merit . in principle , the described scheme allows to develop cloning circuit that is optimal with respect to any non - local figure of merit . the scenario with an _ a priori _ information in the ancilla was inspired by the stronger no - cloning theorem . the obtained conclusions on a possible merit of the cloning contribute to this subject . unequal prior probabilities of inputs are usual in communication systems . the examination of mixed - state cloning is needed because all the real devices are inevitably exposed to noise . analysis with respect to the relative error may have potential applications to the problem of eavesdropping in quantum cryptography . let us consider the function @xmath253 , where positive @xmath254 and @xmath255 obey @xmath256 . let @xmath257\,$ ] be a fixed parameter . the range of variables is stated by conditions @xmath258 , @xmath259 and @xmath260 . this domain @xmath261 is a square whose left - lower corner is cut off by line @xmath262 . * lemma 4 * _ the global minimum of the function @xmath263 in the domain @xmath261 is equal to _ @xmath264 . * proof * inside of the domain @xmath261 , we have @xmath265 and @xmath266 . so the extreme values are reached on the boundary @xmath267 . consider those segments that are parallel to either axis @xmath268 or axis @xmath269 . the minimum value on these segments is equal to either @xmath270 or @xmath271 , i.e. @xmath272 . on the segment @xmath273 , we put @xmath274 and @xmath275 with @xmath276 $ ] , whence @xmath277 . by calculus , we obtain the extreme value @xmath278 for @xmath276 $ ] . this value is not less than both the @xmath279 and @xmath280 . @xmath98 by this operation , a distiguishability of the possible states of second qubit is translated to those of the first . it is convenient to introduce a family of states @xmath281 with the inner product @xmath282 , where @xmath283 $ ] . as is well - known , one- and two - qubit gates are sufficient to implement universal computation . in the context of cloning , the writers of ref . @xcite note that only one type of pair - wise interaction is needed . the _ distiguishability transfer gate _ is described by @xcite @xmath284 where by the unitarity @xmath285 . it follows from eqs . ( [ dp1230 ] ) and ( [ dp3012 ] ) that the operation @xmath286 is hermitian @xcite . the action of distiguishability transfer gate on two - qubit register is shown on figure 2 . the corresponding circuit of @xmath287 elements and one - qubit operations is given in ref .
|
the relative error of cloning of quantum states with arbitrary prior probabilities is considered .
it is assumed that the ancilla may contain some _ a priori _ information about the input state to be cloned .
the lower bound on the relative error for general cloning scenario is derived .
both the case of two - state set and case of multi - state set are analyzed in details .
the treated figure of merit is compared with other optimality criteria .
the quantum circuit for optimal cloning of a pair of pure states is constructed .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the measurement of the cabibbo - kobayashi - maskawa ( ckm ) matrix @xcite element @xmath0 is currently one of the important goals of @xmath1 physics . standard model predictions employ the fundamental parameter @xmath0 as input . as the experiments at the @xmath1 factories are starting to take data , a precise determination of @xmath0 is increasingly vital for testing the standard model . for example , the standard model prediction for the cp - violating asymmetry in @xmath5 decays depends on the value of @xmath0 . evidence for the cp - violating asymmetry has already been provided by the cdf collaboration @xcite . as the dedicated experiments are expected to achieve much greater precision , an increase in the accuracy of @xmath0 is highly desirable in order to test whether the complex phase of the ckm matrix is the only source of cp violation . on the other hand , despite empirical successes , the standard model is definitely not the final theory of particle physics . there remain many outstanding issues : electroweak symmetry breaking , fermion masses and mixing , cp violation , replication of families , etc . a precise determination of @xmath0 is also an important link in pursuit of deeper principles to extend the standard model . @xmath0 can be determined from both inclusive and exclusive charmless semileptonic @xmath1 meson decays . the precision of the determination is limited by experimental and theoretical difficulties . charmless semileptonic @xmath1 decays are a rare process . the main experimental difficulty in observing signals from @xmath6 processes is the very large background due to @xmath7 whose branching fraction is two orders of magnitude larger . on the theoretical side , calculations are needed in order to relate the measured quantity to @xmath0 to extract a value from the data . strong interaction effects complicate the calculations , causing theoretical uncertainties . the uncertainties result not just from nonperturbative qcd which is notoriously difficult to calculate , but also from perturbative qcd although in priciple it is calculable in perturbation theory . with the developments in nonperturbative techniques , the uncertainty from perturbative qcd due to the truncation of perturbation series in practical calculations can be comparable to that from nonperturbative qcd , indicating that these two sources of uncertainty can be equally important . two approaches have been developed for a qcd treatment of inclusive decays of heavy hadrons . one called the heavy quark expansion approach @xcite is based on the operator product expansion . one called the light - cone approach @xcite is based on the light - cone expansion , using the notion and method of deep inelastic lepton - hadron scattering . in both approaches the heavy quark effective theory @xcite is exploited as a limit of qcd . there is important distinction between the two approaches . the former is a short - distance expansion in local operators of increasing dimension , while the latter is a non - local light - cone expansion in matrix elements of increasing twist . moreover , the heavy quark expansion approach invokes quark - hadron duality , which assumes rates evaluated at the parton level to be equal to observable rates summed over a sufficient number of hadronic channels . quark - hadron duality can not be exact @xcite . the heavy quark expansion approach has to use quark kinematics and can not account for the rate due to the extension of phase space from the parton level to the hadron level . the rate missing @xcite is just a manifestation of duality violation . in contrast , the light - cone approach does not rely on the assumption of quark - hadron duality . rates are evaluated in the light - cone approach using physical phase space at the hadron level . the light - cone approach was criticized in refs . however , it has been shown @xcite that that criticism is mistaken . the validity of the light - cone approach has been tested @xcite experimentally on the lepton energy spectrum in inclusive semileptonic decays of @xmath1 mesons . it is important to test theoretical approaches experimentally in various ways . to overcome the aforementioned difficulties in the determination of @xmath0 , we proposed @xcite a method of extracting @xmath0 from inclusive charmless semileptonic decays of @xmath1 mesons . we proposed to measure the lepton pair spectrum , i.e. , the decay distribution of the kinematic variable @xmath8 in the @xmath1-meson rest frame , where @xmath9 is the momentum transfer to the lepton pair and @xmath10 denotes the @xmath1 meson mass . because most of @xmath3 ( @xmath11 or @xmath12 ) events have a value of @xmath13 beyond the threshold allowed for @xmath4 decay , @xmath14 with @xmath15 being the @xmath16 meson mass , this kinematic requirement provides a powerful tool for background suppression . if nothing else , the decay distribution of @xmath13 is of direct interest from an experimental standpoint . the sum rule can then be used to extract @xmath0 from the weighted integral of the measured @xmath13 spectrum . the sum rule follows from the light - cone expansion for inclusive charmless semileptonic decays of @xmath1 mesons and @xmath2-flavored quantum number conservation . the sum rule is thus independent of phenomenological models . moreover , the sum rule does not receive any perturbative qcd correction . therefore , this method is not only experimentally very efficient , but theoretically also quite clean , allowing a precise determination of @xmath0 with a minimal overall ( experimental and theoretical ) error . the sum rule requires measuring the @xmath13 spectrum in inclusive charmless semileptonic @xmath1 meson decays , weighted with @xmath17 . the purpose of the present paper is to refine the analysis of @xcite . we calculate the perturbative qcd correction to the @xmath13 spectrum to order @xmath18 . we investigate in detail how gluon radiation and hadronic bound state effects affect the shape of the @xmath13 spectrum . the rest of the paper is organized as follows . in sec . ii the sum rule for inclusive charmless semileptonic decays of @xmath1 mesons is briefly described . we discuss how to extract @xmath0 from experiment exploiting the sum rule . in sec . iii we study the @xmath13 spectrum , including both the leading perturbative and nonperturbative qcd corrections . the exploration is extended to the weighted @xmath13 spectrum in sec . iv . finally in sec . v we present our conclusions . the sum rule for the decay distribution of @xmath13 in inclusive charmless semileptonic decays of @xmath1 mesons has been derived @xcite from a nonperturbative treatment of qcd based on light - cone expansion . the large @xmath1 meson mass sets a hard energy scale in the reactions , so that the light - cone expansion is applicable @xcite to inclusive @xmath1 decays that are dominated by light - cone singularities , just as deep inelastic scattering . for inclusive charmless semileptonic decays of @xmath1 mesons , @xmath19 of phase space has @xmath20 gev@xmath21 . this dominance of the high-@xmath22 region in the whole phase space @xmath23 renders the light - cone expansion especially reasonable . the theoretical expression for the weighted integral of the distribution of @xmath13 @xmath24 is very clean . in the leading - twist approximation of qcd , it was found @xcite that @xmath25 where @xmath26 is the momentum of the decaying @xmath1 meson . the masses of leptons , the @xmath27 quark and the @xmath28 meson are neglected . because @xmath2-flavored quantum number conservation under strong interactions implies @xmath29 the leading - twist contribution of nonperturbative qcd to the observable @xmath30 in eq . ( [ eq : leading ] ) is precisely calculable from first principles . the result is the sum rule @xcite @xmath31 the conserved current @xmath32 is not subjected to renormalization and the corresponding anomalous dimension vanishes thanks to the ward - takahashi identity . hence we recognize that the sum rule ( [ eq : sumrule ] ) does not receive any perturbative qcd correction . it might seem surprising that one is able to calculate @xmath30 so precisely in qcd when the experimentalist measures hadrons , and it is well known that hadrons can not be understood completely within perturbation theory . there is a physical way to make clear that we really do not need to know anything about how hadrons are formed in order to predict @xmath30 in inclusive @xmath1 decays . the observable @xmath30 measures something fundamental namely the @xmath2-flavored quantum number carried by the @xmath1 meson , and is basically determined by the underlying global symmetry of qcd . it is therefore insensitive to hadronic bound state effects . a measurement of the observable quantity @xmath30 in inclusive charmless semileptonic decays of @xmath1 mesons will lead to a determination of @xmath0 by use of the sum rule ( [ eq : sumrule ] ) which contains no additional parameter . this determination of @xmath0 is free of perturbative qcd uncertainties . meanwhile , the dominant hadronic uncertainty is avoided . higher - twist corrections of nonperturbative qcd to the sum rule ( [ eq : sumrule ] ) are suppressed by a power of @xmath33 . once the observable @xmath30 is measured , then eq . ( [ eq : sumrule ] ) yields @xmath0 with small theoretical uncertainty . moreover , this method is not only theoretically quite clean , but experimentally also very efficient in the discrimination between @xmath34 signal and @xmath35 background , as we shall discuss . in this section we describe the theoretical prediction for the @xmath13 spectrum . we explore step by step perturbative and nonperturbative qcd effects on the @xmath13 spectrum . we take into account first the perturbative qcd correction to order @xmath18 and then the leading nonperturbative qcd correction . ignoring qcd corrections , the tree - level @xmath13 spectrum in the free quark decay @xmath6 in the @xmath2-quark rest frame takes the form @xmath37 with @xmath38 where @xmath39 represents the @xmath2-quark mass . the resulting spectrum is a discrete line at @xmath40 . this is simply a consequence of kinematics that fixes @xmath13 to the single value @xmath41 , no other values of @xmath13 are kinematically allowed in @xmath6 decays . this is also the case for @xmath6 processes with virtual gluon emission . therefore , the @xmath13 spectrum in free quark decays is still a discrete line at @xmath40 , shown in fig . 1 , even if virtual gluon emission occurs . however , the above kinematic relation no longer holds for free @xmath2-quark decays with gluon bremsstrahlung , and hence the spectrum expands downward below the parton - level end point @xmath40 . = 9truecm including both virtual gluon emission and gluon bremsstrahlung , the differential decay rate as a function of @xmath13 in the @xmath2-quark rest frame is calculated to order @xmath18 to be @xmath42 \nonumber\\ \nonumber\\ & + & \frac{2\alpha_s}{3\pi}\frac{m_b}{m_b } \bigg [ -2\left(\frac{{\rm ln}r}{r}\right)_+ -\frac{13}{3}\left(\frac{1}{r}\right)_+ \nonumber\\ \nonumber\\ & + & \frac{79}{9}+\frac{407}{36}r-\frac{367}{12}r^2+\frac{59}{3}r^3 -\frac{50}{9}r^4 + \frac{11}{12}r^5-\frac{7}{36}r^6 \nonumber\\ \nonumber\\ & + & \left ( -\frac{2}{3}+\frac{23}{3}r+3r^2-\frac{8}{3}r^3\right ) { \rm ln}r + 2r^2(-3 + 2r){\rm ln}^2 r \bigg ] , \label{eq : free}\end{aligned}\ ] ] where @xmath43 varying in the range @xmath44 , corresponding to the kinematic range @xmath45 at the parton level of quarks and gluons . the distribution @xmath46_+$ ] is defined to coincide with the function @xmath47 for all values of @xmath48 greater than 0 , and to have a singularity at @xmath49 such that the integral of this distribution with any smooth function @xmath50 gives @xmath51_+ t(r ) = \int_0 ^ 1 dr\ , g(r)[t(r)-t(0 ) ] . \label{eq : def1}\ ] ] the result for the perturbative @xmath13 spectrum with the qcd radiative correction ) does not receive any perturbative qcd correction because it results from the conserved current . the numerical calculation by integrating eq . ( [ eq : free ] ) with the factor @xmath17 has not reached the absolute zero but yielded a tiny correction ( about @xmath52 ) . we have checked that this is only a numerical artifact due to the limited accuracy of numerical integration by computer , especially given the rapid oscillation of the integrand near the end point @xmath53 . ] to order @xmath18 is shown in fig . 2 . the perturbative @xmath13 spectrum is singular at its end point @xmath54 due to infrared divergences . we observe that gluon bremsstrahlung generates a small tail below @xmath54 . = 9truecm integrating eq . ( [ eq : free ] ) over @xmath13 yields the total perturbative decay rate @xmath55 . \label{eq : total}\ ] ] this agrees with the well - known result obtained in @xcite . to calculate the real physical decay distribution in inclusive decays @xmath3 , perturbative qcd alone is not sufficient . we must also account for hadronic bound state effects due to the confinement of the @xmath2 quark inside the @xmath1 meson . we now consider nonperturbative qcd effects on the @xmath13 spectrum . in the framework of the light - cone expansion , the leading nonperturbative qcd effect is incorporated in the @xmath2-quark distribution function @xcite @xmath56b(y ) \label{eq : def}\ ] ] where @xmath57 denotes path ordering . the distribution function @xmath58 has a simple physical interpretation : it is the probability of finding a @xmath2-quark with momentum @xmath59 inside the @xmath1 meson with momentum @xmath26 . the real physical spectrum is then obtained from a convolution of the hard perturbative spectrum with the soft nonperturbative distribution function : @xmath60 where the @xmath2-quark momentum @xmath61 in the perturbative spectrum is replaced by @xmath59 . the analytic result for the perturbative spectrum to order @xmath18 is given in eq . ( [ eq : free ] ) . the interplay between nonperturbative and perturbative qcd effects has been accounted for since confinement implies that free quarks are not asymptotic states of the theory and the separation of perturbative and nonperturbative effects can not be done in a clear - cut way . equation ( [ eq : convol ] ) demonstrates that the physical @xmath13 spectrum depends on the distribution function @xmath58 . the definition of it , eq . ( [ eq : def ] ) , involving the @xmath1-meson matrix element of the non - local @xmath2-quark operators separated along the light cone , makes clear that the distribution function is a nonperturbative quantity . although a complete calculation of the distribution function in qcd is impossible at present due to our ignorance of nonperturbative qcd , some basic properties of it are known @xcite . the distribution function is universal in the sense that the same distribution function also summarizes the leading nonperturbative qcd contribution in inclusive radiative @xmath1 decays @xmath62 . it is gauge invariant and obeys positivity . it has a support between 0 and 1 and is exactly normalized to unity because of @xmath2-flavored quantum number conservation ) . ] . it contains the free quark decay as a limiting case with @xmath63 . in the free quark limit , eq . ( [ eq : convol ] ) consistently reproduces the free quark spectrum . in addition , the mean @xmath12 and the variance @xmath64 of the distribution function were deduced @xcite using operator product expansion and heavy quark effective theory ( hqet ) @xcite : @xmath65 @xmath66 , \label{eq : variance}\ ] ] where @xmath67 and @xmath68 and @xmath69 are the dimensionless hqet parameters of order @xmath70 , which are often referred to by the alternate names @xmath71 and @xmath72 . the parameter @xmath73 can be extracted from the @xmath74 mass splitting : @xmath75 gev@xmath21 . the parameter @xmath76 suffers from large uncertainty . the mean value and variance of the distribution function characterize the location of the `` center of mass '' of the distribution function and the square of its width , respectively . they specify the primary shape of the distribution function . from eqs . ( [ eq : mean ] ) and ( [ eq : variance ] ) we know that the distribution function is sharply peaked around @xmath77 close to 1 and its width of order @xmath78 is narrow , suggesting that the distribution function is close to the delta function form in the free quark limit . nonperturbative qcd methods such as lattice simulation and qcd sum rules could help determine further the functional form of the distribution function . the distribution function can also be extracted directly from experiments of inclusive semileptonic @xcite or radiative @xcite decays of @xmath1 mesons . the universality of the distribution function implies great predictive power : once the distribution function is measured from one process , it can be used to make predictions in all other processes in a model - independent manner . since these are as yet not done , we perform the calculations using the parametrization @xcite of the distribution function @xmath79^\beta}\theta(\xi ) \theta(1-\xi ) , \label{eq : para}\ ] ] where @xmath80 , @xmath81 , @xmath82 , and @xmath2 are four parameters and @xmath83 is the normalization constant . the parametrization ( [ eq : para ] ) respects all the known properties of the distribution function , in particular the strong constraints of the sum rules ( [ eq : mean ] ) and ( [ eq : variance ] ) . the @xmath13 spectrum can be calculated using eqs . ( [ eq : convol ] ) , ( [ eq : free ] ) and ( [ eq : para ] ) . including both the leading nonperturbative and perturbative qcd corrections , we obtain the @xmath13 spectrum shown in fig . 3 , using @xmath84 , @xmath85 , @xmath86 and @xmath87 . here both values of @xmath80 and @xmath81 for the four - parameter distribution function ( [ eq : para ] ) are preset to be 1 , but in general they need not be integers . the values of @xmath82 and @xmath2 are then inferred from the sum rules ( [ eq : mean ] ) and ( [ eq : variance ] ) using @xmath88 gev and @xmath89 gev@xmath21 , giving the mean @xmath90 and the variance @xmath91 for the distribution function . bound - state effects lead to the extension of phase space from the parton level to the hadron level , also stretch the spectrum downward below @xmath41 , and are solely responsible for populating the spectrum upward in the gap between the parton - level end point @xmath40 and the hadron - level end point @xmath92 . the interplay between nonperturbative and perturbative qcd effects eliminates the singularity at the end point of the perturbative spectrum , so that the physical spectrum shows a smooth behavior over the entire range of @xmath13 , @xmath93 , as in fig . 3 . therefore , nonperturbative qcd effects play a crucial role in shaping the @xmath13 spectrum . = 9truecm by integrating eq . ( [ eq : convol ] ) over @xmath13 , we obtain the fraction of @xmath3 events above the charm threshold allowed for the predominant @xmath4 decays , defined as @xmath94 we find that @xmath95 for the spectrum shown in fig . 3 . this result refines our previous result @xcite . the earlier calculation without qcd radiative corrections found @xmath96 . gluon bremsstrahlung streches the @xmath13 spectrum downward , and hence gives rise to a decrease of the fraction of @xmath3 events above the charm threshold . nevertheless , most of @xmath3 events remain above the charm threshold . thus the kinematic cut on the observable quantity @xmath13 is very efficient in disentangling @xmath3 signal from @xmath4 background . this efficiency can be explained by the uniqueness of the @xmath13 spectrum : the @xmath13 spectrum stemming from the quark - level and virtual gluon exchange processes would only concentrate at @xmath40 , shown in fig . 1 , solely on kinematic grounds , and gluon bremsstrahlung and hadronic bound state effects smear the spectrum about this point , but most of the decay rate remains at large values of @xmath13 , as revealed by figs . 2 and 3 . the fraction @xmath97 of course depends on forms of the distribution function . however , we find that @xmath97 is relatively insensitive to forms of the distribution function once the mean and variance of it , which are known from hqet , given by eqs . ( [ eq : mean ] ) and ( [ eq : variance ] ) , are kept fixed . therefore , the above calculation of @xmath97 can be considered as a typical estimate of the fraction of @xmath3 events above the charm threshold . the weighted spectrum @xmath98 is more directly relevant for the measurement of the observable @xmath30 in inclusive charmless semileptonic decays of @xmath1 mesons . measurements of @xmath30 can be obtained from an extrapolation of the weighted spectrum measured above the charm threshold to the full phase space available in @xmath3 decays . while the normalization of the weighted spectrum given by the sum rule ( [ eq : sumrule ] ) does not depend on the @xmath2-quark distribution function @xmath58 , thus being model - independent , the shape of the weighted spectrum does . in this section we perform a detailed analysis of the shape of the weighted spectrum , using eqs . ( [ eq : convol ] ) , ( [ eq : free ] ) and ( [ eq : para ] ) . we first explore the impact of gluon radiation on the shape of the weighted @xmath13 spectrum . we show in fig . 4 the weighted @xmath13 spectram without and with the qcd radiative correction . it is evident from fig . 4 that the shape of the weighted @xmath13 spectrum receives a significant correction due to gluon radiation . however , the shape of the spectrum appears to be insensitive to the value of the strong coupling @xmath18 , varied within a reasonable range . = 9truecm we investigate next the sensitivity of the spectrum to the form of the distribution function , which would reflect the impact of hadronic bound state effects . for this purpose we choose to use two very different forms @xcite for the distribution function , albeit having the same mean value and variance . the calculated weighted spectra are shown in fig . 5 , taking into account the qcd radiative correction to order @xmath18 . the weighted spectrum exhibits a strong dependence on the form of the distribution function , even though the mean value and variance of the distribution functions are the same . = 9truecm generally , since the quark - level processes , exclusive of gluon bremsstrahlung , generate a discrete line , the shape of the @xmath13 spectrum directly reflects the inner long - distance dynamics of the reaction . this argument elucidates the strong variation of the weighted spectrum with the form of the distribution function , illustrated in fig . 5 . in fact , ignoring qcd radiative corrections , one obtains from eqs . ( [ eq : convol ] ) and ( [ eq : tree ] ) the weighted spectrum in the @xmath1 rest frame @xmath99 it makes clear that the resulting weighted spectrum is directly proportional to the distribution function . in other words , the weighted @xmath13 spectrum is most sensitive to the distribution function . this salient feature renders the weighted spectrum idealy suited for the direct extraction of the distribution function from experiment . after an observed spectrum @xmath98 is radiatively corrected , it is nothing but the @xmath2-quark distribution function if higher - twist terms are neglected . extrapolation of the weighted @xmath13 spectrum measured above the charm threshold to the full @xmath13 range requires a theoretical calculation of the rate ratio @xmath100 where @xmath30 is defined in eq . ( [ eq : integral ] ) . we calculate the ratio @xmath101 using eqs . ( [ eq : sumrule ] ) , ( [ eq : convol ] ) , ( [ eq : free ] ) and ( [ eq : para ] ) , and the theoretical uncertainties are estimated as follows : we study the variation of @xmath101 with respect to the mean value and the variance of the distribution function setting @xmath85 in eq . ( [ eq : para ] ) . actually , this amounts to the study of the ratio @xmath101 as functions of @xmath39 and @xmath76 , since , essentially , the mean value of the distribution function is determined by the @xmath2-quark mass and its variance is determined by @xmath76 according to the sum rules in eqs . ( [ eq : mean ] ) and ( [ eq : variance ] ) . at present , the estimated values of the @xmath2-quark mass and @xmath76 vary in the ranges @xmath102 the variation of @xmath39 leads to an uncertainty of @xmath103 in @xmath101 if other parameters are kept fixed . a small uncertainty of @xmath104 in @xmath101 results from the variation of @xmath76 . in other words , the ratio @xmath101 displays a strong dependence on the mean value of the distribution function of the @xmath2 quark inside the @xmath1 meson , but is insensitive to the variance of the distribution function . we examine the further sensitivity of the rate ratio @xmath101 to the form of the distribution function when keeping the mean value and variance of it fixed , by varying the values of the two additional parameters @xmath80 and @xmath81 in the parametrization ( [ eq : para ] ) . we estimate that the variation of @xmath101 is @xmath104 if the form of the distribution function is changed but with the same mean value and variance . we estimate the uncertainty due to the truncation of the perturbative series in eq . ( [ eq : free ] ) by varying the renormalization scale between @xmath105 and @xmath106 . we find that an uncertainty of @xmath103 in the ratio @xmath101 stems from the renormalization scale dependence . this analysis implies that at present the theoretical error in the calculation of @xmath101 has two main sources : the value of @xmath39 ( or equivalently , the mean value of the distribution function ) and the renormalization scale dependence . finally , adding all the uncertainties in quadrature we arrive at @xmath107 with an error of @xmath108 we have studied the @xmath13 spectrum in inclusive charmless semileptonic decays of @xmath1 mesons . the perturbative qcd correction to the spectrum is calculated to order @xmath18 . the leading nonperturbative qcd effect is calculated using light - cone expansion and heavy quark effective theory . the @xmath13 spectrum is unique in that the tree - level and virtual gluon exchange processes @xmath6 at the parton level generate a trivial @xmath13 spectrum a discrete line at @xmath109 in the @xmath2-quark rest frame , which is well above the charm threshold @xmath110 . gluon bremsstrahlung results in a small tail in the lepton pair spectrum below @xmath41 . bound - state effects lead to the extension of phase space from the parton level to the hadron level , also stretch the spectrum downward below @xmath41 , and are solely responsible for populating the spectrum upward in the gap between the parton - level endpoint @xmath54 and the hadron - level endpoint @xmath111 , smoothing out the singularity in the perturbative spectrum . as a result of these two distinct effects , the lepton pair spectrum spreads over the entire physical range @xmath93 . still , we find that about @xmath112 of the lepton pair spectrum in @xmath3 lies above the charm threshold , @xmath113 . this kinematic cut is most efficient in the suppression of the background of @xmath1 semileptonic decays into charmed particles . the observable @xmath30 can be measured by the extrapolation of the weighted spectrum @xmath98 measured above the charm threshold to the entire phase space . gluon bremsstrahlung and hadronic bound state effects strongly affect the shape of the weighted @xmath13 spectrum . however , the shape of the weighted @xmath13 spectrum is insensitive to the value of the strong coupling @xmath18 , varied in a reasonable range . the overall picture appears to be that the weighted @xmath13 spectrum is peaked towards larger values of @xmath13 with a narrow width . the contribution below @xmath114 is moderate and relatively insensitive to forms of the distribution function . this suggests that extrapolating the weighted @xmath13 spectrum down to low @xmath13 would not introduce a considerable uncertainty in the value of @xmath30 . quantitatively , our analysis determines the rate ratio for the extrapolation of the weighted @xmath13 spectrum to be @xmath115 with an error of @xmath108 at present . another interesting use of the weighted @xmath13 spectrum is its utility for directly extracting the @xmath2-quark distribution function . the universality of the distribution function implies that the distribution function extracted from inclusive semileptonic @xmath1 decays can be used to make predictions in inclusive radiative decays @xmath62 in a model - independent manner and vice versa . in particular , a measurement of the distribution function from the photon energy spectrum in @xmath62 @xcite would be very useful to improve the measurement of @xmath30 from extrapolation of the weighted @xmath13 spectrum . an improved knowledge of the form of the distribution function is important for an error reduction in extrapolation . more precisely , if by direct measurements or theoretical studies the uncertainty in the mean value of the distribution function can be improved from currently @xmath104 to @xmath116 in the future , the resulting uncertainty in the ratio @xmath101 for the extrapolation would decrease from @xmath103 to @xmath104 . a calculation of the @xmath117 perturbative qcd correction to the @xmath13 spectrum is also important for improving the accuracy of the extrapolation . the physical observable quantity @xmath30 is connected with @xmath0 via the sum rule ( [ eq : sumrule ] ) for semileptonic @xmath1 decay in the leading - twist approximation of qcd . the sum rule is a fundamental , model - independent prediction of qcd . note that the sum rule does not rely on the heavy quark effective theory . no arbitrary parameter other than @xmath0 enters the sum rule . the dominant hadronic uncertainty is avoided in the sum rule . as important , there is no perturbative qcd modification of this sum rule , so that the potential source of theoretical uncertainty associated with perturbative qcd calculations is totally averted . the kinematic cut on @xmath13 , @xmath118 , and the semileptonic @xmath1 decay sum rule , eq . ( [ eq : sumrule ] ) , offer an outstanding opportunity for the precise determination of @xmath0 from the observable @xmath30 . we wish to emphasize that this method is both exceptionally clean theoretically and very efficient experimentally in background suppression . there remain two types of theoretical uncertainties in the determination of @xmath0 . first , higher - twist ( or power suppressed ) corrections to the sum rule cause a theoretical error of order @xmath119 on @xmath0 . a quantitative study of higher - twist effects could further reduce this small theoretical uncertainty associated with simply extracting the value of @xmath0 from the measured value of @xmath30 . second , the extrapolation of the weighted @xmath13 spectrum to low @xmath13 gives rise to a systematic error for the measurement of @xmath30 . the status of this additional theoretical uncertainty associated with extrapolation , as well as how to improve it , have been discussed above . eventually , the error on @xmath0 determined by this method would mainly depend on how well the observable @xmath30 can be measured . to measure @xmath30 experimentally one needs to be able to reconstruct the neutrino . this poses a challenge to experiment . given the unique potential of determining @xmath0 , we would urge our experimental colleagues to examine the feasibility of the method . j. chay , h. georgi , and b. grinstein , phys . b * 247 * , 399 ( 1990 ) ; + i.i . bigi , n.g . uraltsev , and a.i . vainshtein , phys . b * 293 * , 430 ( 1992 ) ; * 297 * , 477(e ) ( 1993 ) ; + i.i . bigi , m.a . shifman , n.g . uraltsev , and a.i . vainshtein , phys . rev . lett . * 71 * , 496 ( 1993 ) ; int . phys . a * 9 * , 2467 ( 1994 ) ; + a.v . manohar and m.b . wise , phys . d * 49 * , 1310 ( 1994 ) ; + b. blok , l. koyrakh , m.a . shifman , and a.i . vainshtein , phys . d * 49 * , 3356 ( 1994 ) ; * 50 * , 3572(e ) ( 1994 ) ; + m. luke and m.j . savage , phys . b * 321 * , 88 ( 1994 ) ; + a.f . falk , m. luke , and m.j . savage , phys . d * 49 * , 3367 ( 1994 ) ; + t. mannel , nucl . b * 413 * , 396 ( 1994 ) ; + a.f . falk , z. ligeti , m. neubert , and y. nir , phys . b * 326 * , 145 ( 1994 ) ; + m. neubert , phys . d * 49 * , 3392 and 4623 ( 1994 ) ; + t. mannel and m. neubert , phys . d * 50 * , 2037 ( 1994 ) . jin and e.a . paschos , in _ proceedings of the international symposium on heavy flavor and electroweak theory _ , beijing , china , 1995 , edited by c.h . chang and c.s . huang ( world scientific , singapore , 1996 ) , p. 132 ; hep - ph/9504375 .
|
a precise determination of @xmath0 can be obtained exploiting the sum rule for inclusive charmless semileptonic @xmath1-meson decays .
the sum rule is derived on the basis of light - cone expansion and @xmath2-flavored quantum number conservation .
the sum rule does not receive any perturbative qcd correction . in this determination of @xmath0
, there is no perturbative qcd uncertainty , while the dominant hadronic uncertainty is avoided .
moreover , this method is not only theoretically quite clean , but experimentally also very efficient in the discrimination between @xmath3 signal and @xmath4 background .
the sum rule requires measuring the lepton pair spectrum .
we analyze the lepton pair spectrum , including the leading perturbative and nonperturbative qcd corrections .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
future electron - positrons colliders will require a flux of positrons much higher than what can be produced with current methods . one of the scheme that has been proposed to achieve such flux is to use an intense flux of polarized high energy gamma rays @xcite which can be converted into polarized electron - positron pairs . although promising this method requires the demonstration that such high intensity flux of gamma rays can be produced . polarized positrons can be produced by compton interaction between an electron beam and a laser . however given the low cross - section of this process , it is desirable to recycle both the photons and the electrons by , for instance , using an electron ring and a fabry - perot cavity ( fpc ) . we decided to investigate such a scheme by installing a four - mirror fpc on the accelerator test facility ( atf ) @xcite damping ring at kek in japan . the laser seed includes a modified commercial oscillator emmiting at 1031 nm with a repetition rate of 178.5 mhz . we use a standard chirp pulse amplification ( cpa ) architecture in order to limit the non - linearities in the amplifier . the amplifier is built around a commercial microstructure yb - doped double clad fiber . the system delivers recompressed pulses of around 60 ps and up to 50 w average power . the complete optical layout is shown on figure [ fig : optical_layout ] . the fpc is built using the experience gained at lal in previous experiments @xcite . the fpc is formed by 2 concave mirrors with a radius of curvature of 0.5 m and 2 flat mirrors arranged in a non - planar tetrahedron geometry ( see figure [ fig : cavity_geo ] ) . the two concave mirrors form a @xmath2 waist at which the laser beam crosses the electron beam with a collision angle of 8 degrees . the mirrors have a very high reflectivity ( 1 - 1060 ppm for one of them and 1 - 330 ppm for the others ) leading to a cavity finesse of the order of 3000 ( that is a power enhancement of about 1000 ) . the duration of a round trip in the fpc is 5.6 ns . great care had to be taken while designing the cavity to ensure that it is compatible with the ultra - high vacuum requirements of the atf ( @xmath3 mbar ) . all the mirrors are mounted on actuators carts or tables that allow a fine adjustment of the cavity length . furthermore one of the mirrors is mounted on a piezoelectric transducer ( pzt ) that allows to synchronize the length of the fpc with the repetition frequency of the atf by using a digital phase lock loop as shown on figure [ fig : digital_pll ] . in order to constructively stack the pulses in the fpc , it is mandatory to control the pulse to pulse phasing with an interferometric accuracy . this is done using the pound - drever - hall method @xcite implemented in a virtex - ii fpga board as shown on figure [ fig : digital_pdh ] . the error signal generated by this method is used to adjust the length of the laser oscillator . using this double digital feedback system we have been able to keep the fpc locked on the atf and the laser oscillator locked on the fpc for several hours . a detailed description of the atf at kek can be found in the literature @xcite . at the atf the 1.28 gev electron bunches are separated by 2.8 ns . although various fill patterns can be achieved in the atf damping ring ( dr ) , the data presented here were taken in single bunch single train mode in which the same electron bunch returns to the same position after 165 rf buckets ( one complete revolution of the atf dr equals to 462 ns ) . given that a round trip in the fpc is 5.6 ns this means that electrons collide with the laser pulse stored in the fpc every other turn . during the collisions the gamma rays are detected using a fast scintillation detector made of barium fluoride ( @xmath4 ) coupled with a photomultiplier tube ( pmt ) . to eliminate the slow component of scintillation an optical filter is installed in front of the pmt . the data acquisition is performed using a lecroy ws454 oscilloscope ( 1gs / s , 500 mhz bandwidth ) . geant4 simulations have been used to study this calorimeter . the fpc was commissioned in october 2010 and compton collisions were recorded on the first attempt . we present here a brief analysis of some of the data taken . during data taking we record the waveforms from the pmt as well as the 357 mhz atf clock and laser power transmitted by fpc measured by a photodiode . a full waveform contains approximately 200 000 samples spaced by 1 ns . on the day where the data presented here were taken the average power stored in the cavity was 160 w. the 357 mhz clock is used to define the beginning of the periods in time during which the collisions occurred . so , the length of the period is naturally 924 ns . for each period we define the gate which contains the compton peak to calculate consequently its height and integral . different quality cuts such as a cut on the shape of the electronic pulse , a cut on the arrival time of the signal etc . were applied to restrict the analysis to a high purity sample . the shape of the detector s response creates a linear relation between the total charge and the maximum charge measured . this can be seen as a correlation between the calculated peak height and peak integral as shown on figure [ fig : slope ] . the spectrum of the gamma rays is shown on figure [ fig : spectrum ] . it corresponds to the distribution of the energy deposited in the calorimeter expressed by the peak heights normalized by the laser power . since we took data over a wide range of power stored in the fpc , we can study the gamma ray production for different ranges of that power . on figure [ fig : laserbins ] , the average of the peak height distribution goes towards higher energy with increasing fpc power as expected . using cosmic rays for calibration , we can estimate the average number of scattered gammas per bunch crossing . according to this calibration , a peak height of 1 mv is equivalent to 34 mev of energy deposited in the calorimeter . assuming the mean energy of scattered gammas to be 24 mev , approximately 4 gammas are produced in average per bunch crossing ( for an average laser power stored in the cavity of 160 w ) . as the repetition frequency of the collisions is about 1 mhz the flux of gamma rays achieved is about @xmath0 @xmath1/s ( this does not take into account the 0.75 duty cycle of the atf ) . our experiment has been suspended due to the earthquake that struck japan in march 2011 however we plan to resume soon . to increase the compton flux we intend to replace the mirrors of the cavity by mirrors with a higher reflectivity , giving a finesse of about 30 000 . we also plan to improve the laser system and to replace our gamma rays data acquisition chain with a faster one . numerical simulations have shown that with the flux achieved so far we do not expect significant effects on the dynamics of the stored beam @xcite . however , with these future improvements , we intend to study the impact of compton collisions on the beam dynamics in the dr .
|
the next generation of e+/e- colliders will require a very intense flux of gamma rays to allow high current polarized positrons to be produced .
this can be achieved by converting polarized high energy photons in polarized pairs into a target . in that context
, an optical system consisting of a laser and a four - mirror passive fabry - perot cavity has recently been installed at the accelerator test facility ( atf ) at kek to produce a high flux of polarized gamma rays by inverse compton scattering . in this contribution , we describe the experimental system and present preliminary results . an ultra - stable four - mirror non planar geometry has been implemented to ensure the polarization of the gamma rays produced . a fiber amplifier is used to inject about 10w in the high finesse cavity with a gain of 1000 . a digital feedback system is used to keep the cavity at the length required for the optimal power enhancement .
preliminary measurements show that a flux of about @xmath0 @xmath1/s with an average energy of about 24 mev was generated .
several upgrades currently in progress are also described .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
mab problems @xcite are a class of resource allocation problems in which a decision - maker allocates a single resource by sequentially choosing one among a set of competing alternative options called arms . in the so - called stationary mab problem , a decision - maker at each discrete time instant chooses an arm and collects a reward drawn from an unknown stationary probability distribution associated with the selected arm . the objective of the decision - maker is to maximize the total expected reward aggregated over the sequential allocation process . these problems capture the fundamental trade - off between exploration ( collecting more information to reduce uncertainty ) and exploitation ( using the current information to maximize the immediate reward ) , and they model a variety of robotic missions including search and surveillance . recently , there has been significant interest in bayesian algorithms for the mab problem @xcite . bayesian methods are attractive because they allow for incorporating prior knowledge and spatial structure of the problem through the prior in the inference process . in this paper , we investigate the influence of the prior on the performance of a bayesian algorithm for the mab problem with gaussian rewards . mab problems became popular following the seminal paper by robbins @xcite and gathered interest in diverse areas including controls @xcite , robotics @xcite , machine learning @xcite , economics @xcite , ecology @xcite , and neuroscience @xcite . much recent work on mab problems focuses on a quantity termed _ cumulative expected regret_. the cumulative expected regret of a sequence of decisions is the cumulative difference between the expected reward of the options chosen and the maximum possible expected reward . in a ground - breaking work , lai and robbins @xcite established a logarithmic lower bound on the expected number of times a sub - optimal arm needs to be sampled by an optimal policy in a frequentist setting , thereby showing that cumulative expected regret is bounded below by a logarithmic function of time . their work established the best possible performance of any solution to the standard mab problem . they also developed an algorithm based on an upper confidence bound on estimated reward and showed that this algorithm achieves the performance bound asymptotically . in the following , we use the phrase _ logarithmic regret _ to refer to cumulative expected regret being bounded above by a logarithmic function of time , i.e. , having the same order of growth rate as the optimal solution . in the context of the bounded mab problem , i.e. , the mab problem in which the reward is sampled from a distribution with a bounded support , auer @xcite developed upper confidence bound - based algorithms that achieve logarithmic regret uniformly in time ; see @xcite for an extensive survey of upper confidence bound - based algorithms . bayesian approaches to the mab problem have also been considered . srinivas @xcite developed asymptotically optimal upper confidence bound - based algorithms for gaussian process optimization . agrawal and goyal @xcite showed that a bayesian algorithm known as thompson sampling @xcite is near - optimal for binary bandits with a uniform prior . liu and li @xcite characterize the sensitivity of the performance of thompson sampling to the assumptions on prior . kaufman @xcite developed a generic bayesian upper confidence bound - based algorithm and established its optimality for binary bandits with a uniform prior . reverdy @xcite studied the bayesian algorithm proposed in @xcite in the case of correlated gaussian rewards and analyzed its performance for uninformative priors . they called this algorithm the upper credible limit ( ucl ) algorithm and showed that the ucl algorithm models human decision - making in the spatially - embedded mab problem . we define a spatially - embedded mab problem as an mab problem in which the arms are embedded in a metric space and the correlation coefficient between arms is a function of distance between them . for example , in the problem of spatial search over an uncertain distributed resource field , patches in the environment can be modeled as spatially located alternatives and the spatial structure of the resource distribution as a prior on the spatially correlated reward . this is an example of a _ spatially - embedded mab problem . _ it was observed in @xcite that good assumptions on the correlation structure result in significant improvement of the performance of the ucl algorithm , and these assumptions can successfully account for the better performance of human subjects . in this note we rigorously study the influence of the assumptions in the prior on the performance of the ucl algorithm for a mab problem with gaussian rewards . since the ucl algorithm models human decision - making well , the results in this paper help us identify the set of parameters in the prior that explain the individual differences in performance of human subjects . the major contributions of this work are twofold : first , we study the ucl algorithm with uncorrelated informative prior and characterize its performance . we illuminate the opposing influences of the degree of confidence of a prior and the magnitude of its inaccuracy , i.e. , the gap between its mean prediction and the true mean reward value , on the decision - making performance . second , we propose and study a new correlated ucl algorithm with correlated informative prior and characterize its performance . we show that large correlation scales reduce the number of steps required to explore the surface . we then show that incorrectly assumed large correlation scales may lead to a much higher number of selections of suboptimal arms than suggested by the lai - robbins bound . this analysis provides insight into the structure of good priors in the context of explore - exploit problems . the remainder of the paper is organized in the following way . in section [ sec : review ] , we recall the mab problem and an associated bayesian algorithm , ucl . we analyze the ucl algorithm for uncorrelated informative prior and correlated informative prior in section [ sec : uncorr - ucl ] and [ sec : corr - ucl ] , respectively . we illustrate our results with some numerical examples in section [ sec : numerics ] , and we conclude in section [ sec : conclusions ] . in this section we recall the mab problem and the bayes - ucb algorithm proposed in @xcite . the @xmath0-armed bandit problem refers to the choice among @xmath0 options that a decision - making agent should make to maximize the cumulative expected reward . the agent collects reward @xmath1 by choosing arm @xmath2 at each time @xmath3 , where @xmath4 is the horizon length for the sequential decision process . in the so - called stationary mab problem , the reward from option @xmath5 is sampled from a stationary distribution @xmath6 and has an unknown mean @xmath7 . the decision - maker s objective is to maximize the cumulative expected reward @xmath8 by selecting a sequence of arms @xmath9 . equivalently , defining @xmath10 and @xmath11 as the expected _ regret _ at time @xmath12 , the objective can be formulated as minimizing the cumulative expected regret defined by @xmath13}}= \sum_{i=1}^n \delta_i { \ensuremath{\mathbb{e}\left [ n_{i}(t ) \right]}},\end{aligned}\ ] ] where @xmath14 is the total number of times option @xmath15 has been chosen until time @xmath16 and @xmath17 is the expected regret due to picking arm @xmath15 instead of arm @xmath18 . the bayes - ucb algorithm for the stationary @xmath0-armed bandit problem was proposed in @xcite . the bayes - ucb algorithm at each time 1 . computes the posterior distribution of the mean reward at each arm ; 2 . computes a @xmath19 upper credible limit for each arm ; 3 . selects the arm with highest upper credible limit . in step ( ii ) , the upper credible limit is defined as the least upper bound to the upper credible set , and the function @xmath20 is tuned to achieve efficient performance . in the context of bernoulli rewards , kaufmann @xcite set @xmath21 , for some @xmath22 , and show that for @xmath23 and uninformative priors , the bayes - ucb algorithm achieves the optimal performance . reverdy @xcite studied the bayes - ucb algorithm in the context of gaussian rewards with known variances . for simplicity the algorithm in @xcite is called the ucl ( upper credible limit ) algorithm . it is shown that for an uninformative prior , the ucl algorithm is order - optimal , i.e. , it achieves cumulative expected regret that is within a constant factor of that suggested by the lai - robbins bound . it is also shown that a variation of the ucl algorithm models human decision - making in an mab task . in this paper , we focus on the gaussian mab problem , i.e. , the reward distribution @xmath6 is gaussian with mean @xmath24 and variance @xmath25 . the variance @xmath25 is assumed known , e.g. , from previous observations or known characteristics of the reward generation process . we now recall the ucl algorithm and analyze its performance for a general prior . suppose the prior on the mean rewards at each arm is a gaussian random variable with mean vector @xmath26 and variance @xmath27 . for the above mab problem , let the number of times arm @xmath15 has been selected until time @xmath12 be denoted by @xmath28 . let the empirical mean of the rewards from arm @xmath15 until time @xmath12 be @xmath29 . then , the posterior distribution at time @xmath12 of the mean reward at arm @xmath15 has mean and variance @xmath30 respectively , where @xmath31 . moreover , @xmath32 & = \frac{\delta^2 \mu_{i}^0 + n_i(t ) m_i}{\delta^2+n_{i}(t ) } \ ; \text{and}\ ; \text{var}[\mu_{i}(t ) ] = \frac { n_i(t ) \sigma_s^2}{(\delta^2+n_{i}(t))^2}.\end{aligned}\ ] ] the ucl algorithm for the gaussian mab problem , at each decision instance @xmath3 , selects an arm with the maximum @xmath33-upper credible limit , i.e. , it selects an arm @xmath34 , where @xmath35 @xmath36 is the inverse cumulative distribution function for the standard gaussian random variable , @xmath37 , and @xmath38 and @xmath39 are tunable parameters . in the context of gaussian rewards , the function @xmath40 decomposes into two terms corresponding to the estimate of the mean reward and the associated variance . this makes the ucl algorithm amenable to an analysis akin to the analysis for ucb1 @xcite . using such an analysis , it was shown in @xcite that the ucl algorithm with an uninformative prior and parameter values @xmath41 and @xmath42 achieves an order - optimal performance . in the following , we investigate the performance of the ucl algorithm for general priors . to analyze the regret of the ucl algorithm , we require some inequalities that we recall in the following lemma . [ lem : ineq ] for the standard normal random variable @xmath43 and the associated inverse cumulative distribution function @xmath44 , the following statements hold : 1 . for any @xmath45 @xmath46 2 . for any @xmath47 $ ] , @xmath48 and @xmath49 , @xmath50 statement ( i ) in lemma [ lem : ineq ] can be found in @xcite . the first inequality in ( ii ) follows from ( i ) . the second inequality in ( ii ) was established in @xcite , and the last inequality can be easily verified using the second inequality in ( ii ) . [ lem : diff - of - squares ] for any @xmath51 such that @xmath52 , @xmath53 the inequality follows trivially using a completing the square argument . let @xmath54 , for each @xmath55 . set @xmath56 , @xmath57 , and @xmath58 , for some @xmath59 . [ thm : uncorr - regret ] for the gaussian mab problem , and the ucl algorithm with uncorrelated prior , the expected number of times a suboptimal arm @xmath15 is selected satisfies @xmath60 \le \eta_i + \hat n_i(t),\ ] ] where @xmath61 , and @xmath62 is defined in . @xmath63 ' '' '' see [ proof - uncorr - regret ] . [ rem : uncor - regret ] in this section , we study a new correlated ucl algorithm for the correlated mab problem . we first propose a modified ucl algorithm , and then analyze its performance . the modification is designed to leverage prior information on correlation structure . suppose the prior on the mean rewards at each arm is a multivariate gaussian random variable with mean vector @xmath64 and covariance matrix @xmath65 . for the above mab problem , the posterior distribution of the mean rewards at each arm at time @xmath12 is a gaussian distribution with mean @xmath66 and covariance @xmath67 defined by @xmath68 where @xmath69 is the column @xmath0-vector with @xmath2-th entry equal to one , and every other entry zero . in the following , we denote entries of @xmath70 and the diagonal entries of @xmath67 by @xmath71 and @xmath72 , respectively . as in section [ subsec : deterministic - ucl ] , let @xmath28 be the number of times arm @xmath15 has been selected until time @xmath12 , and @xmath29 be the empirical mean of the rewards from arm @xmath15 until time @xmath12 . then , it is easy to verify that @xmath73 where @xmath74 , @xmath75 is the diagonal matrix with entries @xmath76 , and @xmath77 is the vector of @xmath78 . the correlated ucl algorithm for the gaussian mab problem , at each decision instance @xmath3 , selects an arm with the maximum upper credible limit , i.e. , it selects an arm @xmath34 , where @xmath79 @xmath36 is the inverse cumulative distribution function for the standard gaussian random variable , @xmath37 , @xmath80 is the correlation coefficient between arm @xmath15 and arm @xmath81 at time @xmath12 and @xmath38 and @xmath39 are tunable parameters . note that for uncorrelated priors , @xmath82 and the correlated ucl algorithm reduces to the ucl algorithm . in the context of uninformative priors , @xmath83 for each @xmath55 , and the ucl algorithm selects each arm once in first @xmath0 steps . in a similar vein , we introduce an initialization phase for the correlated ucl algorithm . _ * initialization : * _ in the initialization phase , an arm @xmath2 defined by @xmath84 is selected at time @xmath12 . here , @xmath85 is a pre - specified positive constant . let @xmath86 be the number of steps in the initialization phase . [ lem : initialization - corr - stat ] for the correlated mab problem and the inference process , the initialization phase ends in at most @xmath0 steps and the variance following the initialization phase @xmath87 , for each @xmath5 . note that to prove the lemma , it suffices to show that no arm will be selected twice in the initialization phase . it follows from the sherman - morrison formula for the rank-@xmath88 update for the covariance in that @xmath89 where @xmath90 is the @xmath91 component of @xmath67 , for each @xmath5 . if @xmath92 , then @xmath93 . thus , arm @xmath81 will not be selected again in the initialization phase which establishes our claim . [ rem : corr - init ] for correlated priors , the inference equations yield the following expressions for the bias @xmath94 and covariance @xmath95 of the estimate @xmath70 @xmath96 - \bs m = ( \lambda_0 + p(t)^{-1})^{-1 } \lambda_0 ( \bs \mu_0 - \bs m)\\ \bar \sigma(t ) & : = \text{cov}(\bs \mu_t ) = ( \lambda_0 + p(t)^{-1})^{-1 } p(t)^{-1}(\lambda_0 + p(t)^{-1})^{-1},\end{aligned}\ ] ] where @xmath97 is the vector of mean reward . let @xmath98 and @xmath99 , @xmath100 be the diagonal and off - diagonal entries of @xmath67 , and @xmath101 be the diagonal entries of @xmath102 . we now analyze the properties of covariance matrices @xmath67 and @xmath102 . let @xmath103 be the submatrix of @xmath104 obtained after excluding the @xmath15-th row and @xmath15-th column . let @xmath105 be the row vector obtained after excluding the @xmath15-th entry from the @xmath15-th row of @xmath104 . we define the variance of arm @xmath15 conditioned on the mean reward at every other arm by @xmath106 let @xmath107 . with a slight abuse of notation , we refer to @xmath28 as the number of times arm @xmath15 is selected after the initialization phase . we also define for each @xmath5 @xmath108 where @xmath109 is the @xmath110 component of @xmath111 . [ lem : bounds - variances ] the following statements hold for the inference process : 1 . the variance @xmath98 satisfies @xmath112 2 . the variance @xmath113 satisfies @xmath114 we start by establishing the first statement . the covariance update in can be simplified using the sherman - morrison formula to obtain @xmath115 it follows that @xmath116 it follows that after the initialization phase @xmath117 . moreover , at each future round , if @xmath118 , then @xmath119 ; otherwise , @xmath120 . the upper bound on @xmath98 immediately follows from this observation and the induction argument . we now establish the lower bound on @xmath98 . since the inference process involves a stationary environment , the sequence in which arms are played is of no significance and the inference only depends on the number of times an arm has been played . consequently , the inference is the same if arms are played in blocks . in particular , each arm @xmath121 can be played in a block of size @xmath122 . further , any order in which these blocks are played leads to the same inference . suppose for such a modified allocation of arms , @xmath123 is the time when the block associated with arm @xmath81 begins . suppose that arm @xmath15 is played the last . then , from and for the modified allocation process , it follows that @xmath124 i.e. , the posterior variance @xmath125 is lower bounded by the conditional variance of arm @xmath15 under a noise free reward from arm @xmath81 . it follows that , for the modified allocation sequence , @xmath126 . now , the lower bound follows from the variance update after the last block . to establish the second statement , we note that @xmath127 . it follows that @xmath128 where the second inequality follows from the fact @xmath129 . similarly , @xmath130 establishing the lower bound . [ thm : corr - regret ] for the gaussian mab problem , and the correlated ucl algorithm , the expected number of times a suboptimal arm @xmath15 is selected after the initialization phase satisfies @xmath60 \le \eta_i + \hat n_i(t),\ ] ] where @xmath131 , and @xmath132 see [ proof : corr - regret ] . [ rem : corr - regret ] in this section , we illustrate the results of the preceding two sections with data from numerical simulations . the theoretical results pertain to different quality priors defined by how rich is the information they can capture about the rewards associated with the bandit . uninformative priors capture no information , while uncorrelated informative priors capture beliefs about individual arms . correlated ( informative ) priors add to uncorrelated informative priors the ability to capture beliefs about the relationship between different arms , which we leverage in our new correlated ucl algorithm . when an informative prior models the environment well , we refer to it as a _ well - informed _ prior ; conversely , if the prior models the environment poorly , we refer to it as _ ill - informed_. as in @xcite , our simulations focus on the case of a spatially - embedded bandit problem , for which @xcite showed that correlated priors can lead to higher performance . the simulations show that , among well - informed priors , those with richer information content result in higher performance . theorems [ thm : uncorr - regret ] and [ thm : corr - regret ] allow us to quantify the extent to which a prior is well - informed . we consider here the spatially - embedded bandit problem studied in @xcite . the reward surface is relatively smooth with regions of both high and low rewards . this means that a correlated prior capturing length scale information can improve performance . the mean reward value is equal to 30 , and the sampling variance for each arm is @xmath133 . figure [ fig : goodpriors ] shows simulations from cases where the informative priors are well - informed . mean cumulative regret computed from an ensemble of 100 simulations is shown for three priors : an uninformative prior , an informative uncorrelated prior , and an informative correlated prior . for all the simulations , the parameter @xmath134 was set equal to @xmath135 , and for correlated priors the parameter @xmath136 was set equal to 1 . the informative priors have an initial mean belief @xmath137 with a higher value ( equal to 100 ) in regions with high rewards , and a lower value of zero elsewhere . the uncorrelated prior sets @xmath138 , meaning the prior represents the equivalent of a single prior observation . the correlated prior sets @xmath139 as in the uncorrelated case , and uses a correlation structure representing an exponential kernel as in @xcite . this kernel encodes the information that the closer two arms are in the embedding space , the more correlated are their rewards . the richer information provided by the informative priors results in better performance in this case where the priors are well - informed : the informative correlated prior results in less regret than the informative uncorrelated prior , which in turn results in less regret than the uninformative prior . for short horizons , the informative priors result in cumulative regret which is less than the lai - robbins lower bound . the ucl algorithm and the correlated ucl algorithm can violate the lower bound because of the additional information provided by the priors , which effectively shifts the regret curve leftwards . asymptotically , however , the algorithms will tend to match the lai - robbins regret rate for any prior . in contrast , figure [ fig : badpriors ] shows simulations from cases where the informative priors are variously ill - informed . mean cumulative regret computed from an ensemble of @xmath140 simulations is shown for three increasingly informative priors , as in figure [ fig : goodpriors ] . the informative priors have an initial mean belief @xmath137 that is uniform with each element @xmath141 . as in figure [ fig : goodpriors ] , the uncorrelated prior sets @xmath138 , meaning the prior represents the equivalent of a single prior observation . the correlated prior sets @xmath139 and uses a correlation structure that again represents an exponential kernel but with a longer length scale to represent a smoother reward surface . although the informative priors accurately represent the overall mean value of the reward surface , they fail to capture the spatial heterogeneity of the reward surface , in particular the fact that it has high- and low - value patches . therefore , both informative priors are ill - informed about the mean rewards and the informative uncorrelated prior results in much poorer performance than the uninformative prior for moderate task horizons . however , by adding the correlation structure to the ill - informed uncorrelated prior , we can recover much of the performance exhibited by the well - informed correlated prior of figure [ fig : goodpriors ] . in a spatially - embedded task like the one studied here , information about correlation structure among arms can be as valuable as accurate information about the value of individual arms . . because of the additional information provided by the informative priors , the algorithms can sample arms more selectively from the initial time @xmath142 , which results in better performance than the uninformative prior and allows the algorithms to outperform the lai - robbins bound on regret.,scaledwidth=50.0% ] , the traces show mean cumulative regret from 100 simulations for each of three different priors . again the algorithms exhibit an initialization phase behavior for the uninformative and informative correlated priors , whose end can be seen in the bends in the regret curves near @xmath143 . the ill - informed correlated prior improves performance relative to the uninformative prior although not quite as much as the well - informed correlated prior does in figure [ fig : goodpriors ] . in contrast , the ill - informed uncorrelated prior significantly decreases performance relative to all other priors . by encoding a strong incorrect belief about the rewards , this prior requires multiple samples of suboptimal arms to learn that they are suboptimal . this appears in the regret curve as an initialization phase that lasts until @xmath144 4,500 , at which point the mean cumulative regret is approximately 35,000.,scaledwidth=50.0% ] in this note we studied and modified the ucl algorithm for the correlated mab problem with gaussian rewards . we investigated the influence of the assumptions in the prior on the performance of the ucl algorithm and the new correlated ucl algorithm . we characterized scenarios in which the informative priors perform better than the uninformative prior and characterized the improvement in the performance in terms of cumulative regret . in particular , we showed conditions in which an informative correlated prior can be leveraged to significantly reduce cumulative regret . there are several possible avenues of future research . first , we considered that the environment is stationary . an interesting future direction is to consider non - stationary environments in which the reward at each arm may be time - varying and the autocorrelation scale may be known . second , we considered these problems for a single player . many application scenarios involve a group of individuals and it is of interest to study collaborative and competitive multiplayer versions of these problems . 10 j. gittins , k. glazebrook , and r. weber . . wiley , second edition , 2011 . e. kaufmann , o. capp , and a. garivier . on bayesian upper confidence bounds for bandit problems . in _ international conference on artificial intelligence and statistics _ , pages 592600 , la palma , canary islands , spain , april 2012 . n. srinivas , a. krause , s. m. kakade , and m. seeger . information - theoretic regret bounds for gaussian process optimization in the bandit setting . , 58(5):32503265 , 2012 . s. agrawal and n. goyal . analysis of thompson sampling for the multi - armed bandit problem . in s. mannor , n. srebro , and r. c. williamson , editors , _ jmlr : workshop and conference proceedings _ , volume 23 : colt 2012 , pages 39.139.26 , 2012 . p. reverdy , v. srivastava , and n. e. leonard . modeling human decision making in generalized gaussian multiarmed bandits . , 102(4):544571 , 2014 . h. robbins . some aspects of the sequential design of experiments . , 58:527535 , 1952 . r. agrawal , m. v. hedge , and d. teneketzis . asymptotically efficient adaptive allocation rules for the multi - armed bandit problem with switching cost . , 33(10):899906 , 1988 . v. anantharam , p. varaiya , and j. walrand . . , 32(11):968976 , 1987 . j. l. ny , m. dahleh , and e. feron . multi - uav dynamic routing with partial observations using restless bandit allocation indices . in _ proceedings of american controls conference _ , pages 42204225 , seattle , washington , usa , june 2008 . m. y. cheung , j. leighton , and f. s. hover . autonomous mobile acoustic relay positioning as a multi - armed bandit with switching costs . in _ ieee / rsj international conference on intelligent robots and systems _ , pages 33683373 , tokyo , japan , november 2013 . v. srivastava , p. reverdy , and n. e. leonard . surveillance in an abruptly changing world via multiarmed bandits . in _ ieee conf . on decision and control _ , pages 692697 , los angeles , ca , december 2014 . m. babaioff , y. sharma , and a. slivkins . characterizing truthful multi - armed bandit mechanisms . in _ proceedings of the 10th acm conference on electronic commerce _ , pages 7988 , stanford , ca , usa , july 2009 . f. radlinski , r. kleinberg , and t. joachims . learning diverse rankings with multi - armed bandits . in _ proceedings of the 25th international conference on machine learning _ , pages 784791 , helsinki , finland , july 2008 . mccall and j. j. mccall . a sequential study of migration and job search . , 5(4):452476 , 1987 . j. r. krebs , a. kacelnik , and p. taylor . test of optimal sampling by foraging great tits . , 275(5675):2731 , 1978 . v. srivastava , p. reverdy , and n. e. leonard . optimal foraging and multi - armed bandits . in _ allerton conf . on communications , control and computing _ , pages 494499 , monticello , il , usa , october 2013 . r. c. wilson , a. geana , j. m. white , e. a. ludvig , and j. d. cohen . humans use directed and random exploration to solve the explore - exploit dilemma . , 143(6):20742081 , 2014 . p. reverdy , v. srivastava , r. c. wilson , and n. e. leonard . human decision making and the explore - exploit tradeoff : algorithmic models for multi - armed bandit problems . in simon haykin , editor , _ cognitive dynamic systems_. wiley series on adaptive and cognitive dynamic systems . ieee press / wiley , 2015 . t. l. lai and h. robbins . asymptotically efficient adaptive allocation rules . , 6(1):422 , 1985 . p. auer , n. cesa - bianchi , and p. fischer . finite - time analysis of the multiarmed bandit problem . , 47(2):235256 , 2002 . s. bubeck and n. cesa - bianchi . regret analysis of stochastic and nonstochastic multi - armed bandit problems . , 5(1):1122 , 2012 . e. kaufmann , n. korda , and r. munos . thompson sampling : an asymptotically optimal finite - time analysis . in _ algorithmic learning theory _ , pages 199213 . springer , 2012 . w. r. thompson . on the likelihood that one unknown probability exceeds another in view of the evidence of two samples . , 25(3/4):285294 , 1933 . c. y. liu and l. li . on the prior sensitivity of thompson sampling . , 2015 . m. abramowitz and i. a. stegun , editors . . dover publications , 1964 . in the spirit of @xcite , we bound @xmath14 as follows : @xmath145 where @xmath146 is some positive integer and @xmath147 is the indicator function , with @xmath148 if @xmath149 is a true statement and @xmath150 otherwise . at time @xmath12 , the agent picks option @xmath15 over @xmath18 only if @xmath151 this is true when at least one of the following equations holds : @xmath152 where @xmath153 and @xmath37 . otherwise , if none of the equations ( [ eq : qi*])-([eq : micomp ] ) holds , @xmath154 and option @xmath18 is picked over option @xmath15 at time t. as noted earlier , the posterior mean @xmath71 is a gaussian random variable : @xmath155 we will now analyze the events , , and . let @xmath156 be the probability of the event . [ lem - first - event - uncorr ] the following statements hold for event : 1 . if @xmath157 , then @xmath158 2 . if @xmath159 , then @xmath160 for @xmath161 , event is true if @xmath162 where @xmath163 is a standard normal random variable . similarly , for @xmath164 , event is not true if ( i ) @xmath165 , or ( ii ) @xmath166 and @xmath167 . we now establish the first statement . if @xmath165 and @xmath168 , then @xmath169 . if @xmath165 and @xmath170 , then @xmath171 therefore , @xmath172 to establish the second statement , we note that if @xmath173 and @xmath168 , then event does not hold if @xmath174 if @xmath173 and @xmath170 , then @xmath175 , where @xmath176 . note that @xmath177 , if @xmath178 . define @xmath179 it follows that for @xmath180 , @xmath181 where the second last inequality follows from lemma [ lem : diff - of - squares ] and @xmath182 and @xmath183 are as defined in section [ sec : uncorr - ucl - regret ] . therefore , @xmath184 let @xmath185 be the joint probability of the event and the event @xmath186 , for some @xmath187 . [ lem - sec - event - uncorr ] the following statements hold for event : 1 . if @xmath188 , then @xmath189 2 . if @xmath190 , then @xmath191 the event ( [ eq : qi ] ) holds if @xmath192 where @xmath163 is a standard normal random variable . we start with establishing the first statement . if @xmath193 and @xmath194 , then @xmath195 where @xmath196 . it follows that @xmath177 , if @xmath197 . it follows that for @xmath198 @xmath199 where the second last inequality follows from lemma [ lem : diff - of - squares ] . therefore , @xmath200 the second statement follows similarly to the first statement in lemma [ lem - first - event - uncorr ] . we now analyze the probability of event . @xmath201 where @xmath17 , the inequality follows from lemma [ lem : ineq ] , and the inequality follows from the monotonicity of the logarithmic function . therefore , the event ( [ eq : micomp ] ) is not true if @xmath202 setting @xmath61 , we get @xmath203 } } & \leq \eta_i + \sum_{t = 1}^t \prob(q_{i}^t > q_{i^*}^t , n_{i}(t-1 ) \geq \eta_i ) \\ & = \eta_i + \sum_{t = 1}^t \big(\prob_1(t ) + \prob_2(t)\big)\\ & < \eta_i + \hat n_i(t ) . \end{aligned}\ ] ] this completes the proof of the theorem . similar to the proof of theorem [ thm : uncorr - regret ] , at time @xmath12 , the agent picks option @xmath15 over @xmath18 only if @xmath204 . this is true when at least one of the following equations holds : @xmath205 where @xmath206 , @xmath37 . it follows using the same argument as in theorem [ thm : uncorr - regret ] that @xmath214 similarly , @xmath215 also , event is not true if @xmath216 adding the probabilities of the events - , we obtain the desired expression .
|
we consider the correlated multiarmed bandit ( mab ) problem in which the rewards associated with each arm are modeled by a multivariate gaussian random variable , and we investigate the influence of the assumptions in the bayesian prior on the performance of the upper credible limit ( ucl ) algorithm and a new correlated ucl algorithm .
we rigorously characterize the influence of accuracy , confidence , and correlation scale in the prior on the decision - making performance of the algorithms .
our results show how priors and correlation structure can be leveraged to improve performance . multiarmed bandit problem , bayesian algorithms , decision - making , spatial search , upper credible limit algorithm , influence of priors
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
let @xmath0 be a closed orientable surface of genus @xmath1 . according to the nielsen - thurston classification @xcite , every non - periodic irreducible automorphism of @xmath0 is isotopic to a pseudo - anosov diffeomorphism . thurston proved that the topological entropy of any pseudo - anosov diffeomorphism @xmath2 coincides with @xmath3 where @xmath4 is the dilatation of @xmath2 ( c.f . * expos 10 ) ) . also by the work of bers @xcite , @xmath3 is known to be equal to the translation distance of @xmath2 on the teichmller space with respect to the teichmller metric . the purpose of this paper is to demonstrate a `` random version '' of the work of bers and thurston . let @xmath5 denote the mapping class group of @xmath0 . we consider the random walk on @xmath5 which is determined by a probability measure @xmath6 on @xmath5 . this @xmath6 induces a probability measure @xmath7 on @xmath8 . throughout the paper we assume that @xmath6 has finite first moment with respect to the teichmller metric and the support of @xmath6 generates a non - elementary subgroup of @xmath5 ( see condition [ cond ] ) . before stating the main theorem , we prepare several terminologies briefly . formal definitions are given in [ sec.pre ] . first , the topological entropy @xmath9 of a sample path @xmath10 is defined using open coverings of @xmath0 , similarly to the one for surface diffeomorphisms . this measures growth rate of the number of distinguishable orbits of the random walk . next , karlsson @xcite proved that for @xmath7-a.e @xmath11 , the exponential growth rate of the length of the image @xmath12 of any simple closed curve @xmath13 with respect to any metric always gives the same quantity , which is called the `` lyapunov exponent '' @xmath14 of @xmath15 . moreover , it is also proved that @xmath16 almost surely coincides with the drift @xmath17 with respect to thurston s asymmetric lipschitz metric on the teichmller space @xmath18 of @xmath0 . roughly speaking , the drift is the translation distance of @xmath15 . the goal of this paper is to show that those quantities are the same almost surely . [ thm.main ] let @xmath6 be a probability measure on @xmath5 which satisfies condition [ cond ] and @xmath7 the probability measure on @xmath8 induced by @xmath6 . for @xmath7-a.e . @xmath19 , we have the following equality . @xmath20 where @xmath21 is the drift with respect to the teichmller metric . these quantities are independent of @xmath15 and they are invariants of the random walk . the strategy of the proof is similar to the one for pseudo - anosov diffeomorphisms in ( * ? ? ? * expos 9 - 10 ) . indeed , @xmath22 can be proved almost in the same way as the case of pseudo - anosov diffeomorphisms . to prove the opposite inequality for pseudo - anosovs , in @xcite , a subshift of finite type is associated to the dynamics of a pseudo - anosov iteration by constructing so called a markov partition . we will define a random subshift of finite type as a `` random version '' of a subshift of finite type ( see [ sec.random subshift ] ) . then we will construct a semi - markov partition of @xmath0 which respects the dynamics of @xmath15 ( see definition [ defi.semi-markov ] ) and associate to it a random subshift of finite type . the main difficulty , unlike pseudo - anosov diffeomorphisms , is that for @xmath7-a.e @xmath23 , some part of @xmath15 can be arbitrarily `` bad '' . for example , the orbit @xmath24 of any point @xmath25 may have large backtrack . on the other hand , for a pseudo - anosov @xmath2 case , the fact that @xmath2 acts as a translation on a teichmller geodesic is implicitly used in @xcite . to overcome the difficulty , in [ sec.markov]-[sec.main ] , we show that it suffices to observe only `` good '' elements in an orbit . the existence of such `` good '' elements follows from ergodic theorems . to consider dynamics of @xmath15 on the surface , we need to take representatives of mapping classes . let @xmath26 denote the space of orientation preserving diffeomorphisms on @xmath0 . let @xmath27 be a representative of @xmath28 and @xmath29 . another difficulty occurs after taking representatives , that is , we can not use ergodic theorems . this is because we can not take representatives so that they are compatible with the shift maps on @xmath8 , denoted @xmath30 . notations @xmath31 are used instead of @xmath32 to warn readers this issue . our goal in [ sec.markov ] is to prove the following theorem . [ thm.symbolic ] there exist a random subshift of finite type @xmath33 such that the following diagram commutes for any @xmath34 . @xmath35 where @xmath36 is a continuous surjective map and @xmath37 is the shift map . the topological entropy of @xmath37 in theorem [ thm.symbolic ] can be defined as the growth rate of the number of cylinder sets of length @xmath38 in @xmath39 . let @xmath40 denote the topological entropy of @xmath37 . the fact @xmath41 can be easily observed ( lemma [ lem.p - entropy ] ) . for pseudo - anosovs , the facts of type theorem [ thm.symbolic ] and @xmath41 suffice to prove a theorem of type theorem [ thm.main ] . however , we need to vary structures of @xmath0 to construct a semi - markov partition . hence we need to discuss how structures , especially the lebesgue number of a fixed open covering , vary as the steps . the lebesgue numbers are discussed in [ sec.lebesgue ] , and the rest of [ sec.main ] is devoted for a proof of @xmath42 . in this section , we prepare terminologies and basic facts which we need to prove theorem [ thm.main ] . we briefly recall the teichmller spaces and related facts . readers should refer @xcite for more details . let @xmath0 be a closed orientable surface of genus @xmath43 . a _ marked riemann surface _ is a pair of a riemann surface @xmath44 and a homeomorphism , called a _ marking _ , two marked riemann surfaces @xmath46 are said to be _ teichmller equivalent _ if there is a biholomorphic map @xmath47 such that @xmath48 is homotopic to @xmath49 . the teichmller space @xmath18 of @xmath0 is the space of marked riemann surfaces modulo teichmller equivalence . since each marking defines a complex structure on @xmath0 by pullback , we often confuse a point @xmath25 with a complex structure on @xmath0 . the mapping class group @xmath5 acts on @xmath18 so that for @xmath50 with a representative @xmath51 , @xmath52 . a holomorphic quadratic differential on @xmath25 is a family of holomorphic maps @xmath53 each defined on @xmath54 of a complex chart @xmath55 so that if @xmath56 then @xmath57 where @xmath58 . for @xmath25 , let @xmath59 denote the space of quadratic differentials . the vertical ( resp . horizontal ) trajectories of a quadratic differential @xmath60 are curves @xmath61 such that @xmath62 ( resp . @xmath63 ) . for each smooth arc @xmath64 , the transverse measures on the vertical and horizontal trajectories are defined by @xmath65 and @xmath66 respectively . thus each @xmath67 defines two measured foliations called _ vertical _ and _ horizontal _ foliations as the vertical and horizontal trajectories equipped with the transverse measures respectively . a theorem of teichmller says that given two points @xmath68 , there exists a quasi - conformal map @xmath69 and quadratic differentials @xmath70 and @xmath71 such that the map @xmath72 maps @xmath73 to @xmath74 so that it stretches ( resp . contracts ) the horizontal ( resp . vertical ) foliations . the logarithm of the stretch factor coincides with the teichmller distance @xmath75 . by integrating the square root of a quadratic differential @xmath60 , we have a singular euclidean metric on @xmath76 . with respect to the singular euclidean metric , the length of a smooth arc @xmath77 , denoted by @xmath78 , is equal to @xmath79 . for @xmath67 , we define the norm of @xmath60 by @xmath80 . let @xmath81 . we denote by @xmath82 the space of projective measured foliations . we consider the thurston compactification @xmath83 on which @xmath5 acts continuously . by the work of thurston , @xmath82 is homeomorphic to the sphere of dimension @xmath84 @xcite . we now recall the work of hubbard - masur . [ thm.hm ] the map @xmath85 associating the equivalence class of the horizontal foliation to each @xmath86 is a homeomorphism . let @xmath87 be transverse filling projective measured foliations . let @xmath88 denote the teichmller geodesic corresponding to a quadratic differential with horizontal and vertical foliation @xmath89 and @xmath90 respectively ( see @xcite for the existence of such geodesics ) . a projective measured foliation is called _ uniquely ergodic _ if its supporting foliation admits only one transverse measure up to scale . let @xmath91 denote the space of uniquely ergodic foliations . let @xmath90 be a countable group and @xmath92 $ ] a probability measure . by @xmath93 ( resp . @xmath94 ) , we denote the space of positive ( resp . negative ) integers . for group elements @xmath95 , the subset @xmath96:=\{\omega = ( \omega_{i})\in g^{\mathbb{z}_{+}}\mid \omega_{i } = x_{i } \text { for } 1\leq i\leq n\}\ ] ] is called a _ cylinder set_. the probability measure @xmath6 induces a probability measure @xmath7 on the space of sample paths @xmath97 so that @xmath98 ) = \mu(x_{0}^{-1}x_{1})\mu(x_{1}^{-1}x_{2})\cdots\mu(x_{n-1}^{-1}x_{n}),\ ] ] where @xmath99 is the initial element which is assumed to be the identity unless otherwise stated . we also consider the _ reflected measure _ @xmath100 . let @xmath101 be the probability measure on @xmath102 induced by @xmath103 . then by the map @xmath104 , the probability measure @xmath105 induces a probability measure on @xmath106 which we again denote by @xmath7 . we define the _ bernoulli shift _ , denoted by @xmath30 , as for any @xmath107 , @xmath108 recall that a subgroup of @xmath5 is called _ non - elementary _ if it contains two pseudo - anosov elements with disjoint fixed point sets in @xmath82 . from now on , we consider the random walk on @xmath5 which is determined by a probability measure @xmath6 which satisfies the following condition . [ cond ] the probability measure @xmath109 $ ] satisfies that * @xmath6 has finite first moment with respect to the teichmller metric on @xmath18 i.e. for any @xmath25 , @xmath110 , and * the support of @xmath6 generates a non - elementary subgroup of @xmath5 . let @xmath111 and @xmath112 be open coverings of @xmath0 . since @xmath0 is compact , each open covering has a finite subcover . let @xmath113 denote the number of sets in a subcover of @xmath114 with minimal cardinality . @xmath115 is said to be a _ refinement _ of a cover @xmath114 , denoted @xmath116 , if for any @xmath117 , there is @xmath118 such that @xmath119 . it can readily be seen that if @xmath116 then @xmath120 . we denote by @xmath121 the open cover @xmath122 . for an open covering @xmath114 of @xmath0 and a metric @xmath123 on s , the _ lebesgue number _ @xmath124 with respect to @xmath123 is defined to be @xmath125 where @xmath126 is the open ball centered at @xmath127 of radius @xmath128 with respect to @xmath123 . let @xmath129 be less than the lebesgue number of @xmath114 . then the covering consisting of all open balls of radius @xmath128 refines @xmath114 . let @xmath130 . we first choose an arbitrary representative @xmath131 of @xmath28 for each @xmath34 . let @xmath132 . for an open cover @xmath114 , let @xmath133 . we define @xmath134 note that if @xmath116 , then @xmath135 . the _ topological entropy _ of @xmath136 is @xmath137 where the supremum is taken over all open coverings of @xmath0 . finally we define @xmath138 where the infimum is taken over all representatives of @xmath15 . unlike the definition of topological entropy of surface automorphisms , we do not take inverses . this is natural because when we consider random walks , we multiply new elements from the right . we define the drift of random walks , which we may regard as a `` translation distance '' of the random walk . let @xmath139 be a metric space on which @xmath5 acts isometrically . suppose the probability measure @xmath6 has finite first moment with respect to @xmath140 , i.e. @xmath141 where @xmath142 is arbitrary . by kingman s subadditive ergodic theorem , the limit @xmath143 exists for @xmath7-a.e . @xmath15 and this limit is independent of @xmath127 and @xmath15 . this limit is called the _ drift _ of @xmath19 with respect to @xmath140 . let @xmath144 and @xmath145 denote the distance on @xmath18 by thurston s lipschitz metric and the teichmller metric respectively . we here recall the work of choi - rafi . [ thm.cr ] there is a constant @xmath146 depending on the surface @xmath0 and on @xmath147 such that for any @xmath148 in the @xmath147-thick part of @xmath18 , @xmath149 and @xmath150 differ from one another by at most @xmath146 . by theorem [ thm.cr ] , if the probability measure @xmath6 has finite first moment with respect to the teichmller metric , then it also has finite first moment with respect to thurston s lipschitz metric . let @xmath151 ( resp . @xmath152 ) denote the drift of @xmath15 with respect to @xmath144 ( resp . @xmath145 ) . since the drifts are also independent of the choice of base points , by taking a point in the thick part of @xmath18 , we have @xmath153 by theorem [ thm.cr ] . we let @xmath154 . in @xcite , karlsson proved the following . [ thm.kar ] there exists @xmath155 such that for @xmath7-a.e . @xmath156 , for any isotopy class @xmath13 of essential simple closed curves and riemannian metric @xmath157 of @xmath0 , @xmath158 where @xmath159 denotes the infimum of the length of curves in @xmath13 with respect to @xmath157 . moreover @xmath160 coincides with @xmath161 . note that for theorem [ thm.kar ] , we do not need to take a representative of @xmath15 . following @xcite , we call @xmath155 in theorem [ thm.kar ] the _ lyapunov exponent _ of the random walk . now we establish the following inequality . [ lem.lemma1 ] let @xmath155 be the lyapunov exponent of the random walk determined by @xmath6 . for @xmath7-a.e . @xmath15 , we have @xmath162 we first fix a hyperbolic metric @xmath157 on @xmath0 , a universal covering @xmath163 and a representative @xmath164 of @xmath15 . we also fix @xmath165 and @xmath166 in order to choose lifts @xmath167 of @xmath168 uniquely for all @xmath34 . in @xcite , a pseudo - anosov diffeomorphism @xmath169 is discussed . one can prove the following lemma by exchanging @xmath170 with @xmath171 , and following the same argument as in @xcite . [ lem.10.8 ] for any @xmath172 @xmath173 we may choose @xmath127 and @xmath174 in lemma [ lem.10.8 ] to be the endpoints of a lift of geodesic representative of a simple closed curve @xmath13 on @xmath0 . then since @xmath175 , we have @xmath176 for any representative @xmath136 of @xmath15 . in order to prove @xmath177 , we need a notion of random subshift of finite type . we define a random subshift of finite type which we use to prove theorem [ thm.main ] . our goal is to associate a random subshift of finite type to a sample path @xmath19 . since we have to overcome certain difficulty which is described briefly in the introduction , we need to slightly modify the definition from the standard one ( see e.g. ( * ? ? ? * definition 3.9 ) for the standard one ) . the main difference is that we can only associate a random subshift of finite type to a representative @xmath136 of @xmath178 . for later convenience , we use the notations with @xmath136 here . let @xmath179 be a function . suppose we have a family of @xmath180 matrices @xmath181 each of whose entry is @xmath182 or @xmath183 . for any @xmath34 , let @xmath184 we define the coordinate so that for each element @xmath185 , we have @xmath186 . a random subshift of finite type is a pair @xmath33 where @xmath187 and @xmath188 is the standard left shift . we consider the discrete topology on each @xmath189 and the product topology on @xmath190 . let @xmath191 for @xmath192 . the _ @xmath193-cylinder set _ in @xmath194 of @xmath195 is @xmath196 let @xmath197 denote the family of @xmath193-cylinder sets in @xmath194 . note that @xmath198 . a semi - markov partition with respect to @xmath19 is a sequence of partitions of the surface @xmath0 by _ birectangles _ with certain condition so that it respects the dynamics of @xmath15 ( see [ subsec.markov ] ) . we first construct a birectangle decomposition , denoted @xmath199 , from two transverse uniquely ergodic foliations @xmath200 , and a marked riemann surface @xmath201 which represents a point @xmath76 on the teichmller geodesic @xmath202 . the riemann surface structure @xmath44 lets us fix measured foliation representatives @xmath203 and @xmath204 of @xmath205 and @xmath206 respectively so that @xmath203 and @xmath204 are the horizontal and vertical foliation of a holomorphic quadratic differential on @xmath44 of norm 1 . their preimages by @xmath207 on @xmath0 are also denoted by the same notations . let @xmath208 denote the set of singular points of @xmath209 . note that with these representations , @xmath210 . a subset @xmath211 is called an _ @xmath212-rectangle _ , or a _ birectangle _ if @xmath213 is the image of some continuous map @xmath214\times[0,1]\rightarrow s$ ] such that * @xmath215 is an embedding , and * for all @xmath216 $ ] , @xmath217\times\{t\})$](resp . @xmath218)$ ] ) is a finite union of leaves and singularities of @xmath219 ( resp . @xmath220 ) , and in fact in one leaf if @xmath221 . we let @xmath222 , @xmath223\times \{0,1\})$ ] , @xmath224)$ ] , and @xmath225 . a family of birectangles @xmath226 is called a _ birectangle partition _ if 1 . @xmath227 , and 2 . @xmath228 for @xmath229 . for a singular measured foliation , we call a leaf which departs from a singularity a _ singular leaf_. any small neighborhood of a singular point is decomposed into several components by singular leaves . we call each component a _ sector_. a _ saddle connection _ is a singular leaf which connects two singular points . we will now construct a birectangle partition of @xmath0 . we imitate the construction in ( * ? ? ? * expos 9 ) . for each sector of @xmath220 of a singular point , we take a subarc of the singular leaf of @xmath219 in the sector , which starts from the singular point and have @xmath230 measure 1 . if @xmath219 has a saddle connection and we can not take a singular leaf of @xmath230 measure 1 , we instead take the whole saddle connection . let @xmath231 denote the family of such subarcs and saddle connections . then for each singular leaf of @xmath220 , we take the shortest subarc that starts from a singular point and intersects every element of @xmath77 which is not a saddle connection at least once . similarly to before , we take whole saddle connections if there are no such subarcs . let @xmath232 denote the family of such subarcs and saddle connections . then , for each @xmath233 , we truncate the component of @xmath234 which contains @xmath235 from @xmath236 , and denote by @xmath13 the resulting arc . note that saddle connections remain unchanged . let @xmath237 . then we extend each element of @xmath238 until it meets @xmath64 exactly once more . let @xmath239 denote the family of resulting subarcs . then we let @xmath240 @xmath199 is a birectangle partition . it suffices to prove that each element of @xmath241 is a birectangle . if @xmath242 contains @xmath64 , then by construction @xmath213 does not contain singular points in the interior . by the singular euclidean structure determined by @xmath243 , we see that two components of @xmath244 are parallel and in particular @xmath213 is a birectangle . if there were @xmath213 with @xmath245 , then @xmath242 must have contained a loop consisting of leaves of @xmath220 . however , since @xmath220 is uniquely ergodic , there are no such loops . let @xmath19 and @xmath164 be a representative of @xmath15 . our goal in this subsection is to construct a semi - markov partition from a birectangle partition obtained in the previous subsection , so that it respects the dynamics of @xmath136 . [ defi.semi-markov ] a sequence of birectangle partitions @xmath246 is a _ semi - markov partition _ with respect to @xmath136 if for every @xmath34 , 1 . @xmath247 , @xmath248 , and 2 . for each @xmath249 and @xmath250 , if @xmath251 and @xmath252 intersects , then the intersection is a single birectangle . we call it a _ semi_-markov partition because to have @xmath253 , we further need estimates for the size of birectangles . we here carefully construct a semi - markov partition so that it further satisfies certain estimates which we give in [ sec.main ] . recall that a markov partition for a pseudo - anosov diffeomorphism @xmath2 is constructed by using stable and unstable foliations @xmath254 and @xmath255 of @xmath2 . in terms of the thurston compactification @xmath256 , these foliations are characterized as limits @xmath257 where @xmath25 is an arbitrary point . kaimanovich - masur proved that for the case of random walks , we have similar limits . [ thm.km ] let @xmath6 be a probability measure which satisfies condition [ cond ] . then 1 . there exists a unique @xmath6-stationary probability measure @xmath258 on @xmath82 which is purely non - atomic and concentrated on @xmath259 . 2 . for @xmath7-a.e . @xmath260 and any @xmath25 , the sequence @xmath261 converges in @xmath82 to a limit @xmath262 and the distribution of the limits is given by the measure @xmath258 . we may apply theorem [ thm.km ] both to @xmath6 and @xmath103 . we denote by @xmath263 the @xmath103-stationary measure on @xmath82 . for @xmath7-a.e @xmath19 , let @xmath264 let @xmath265 denote the teichmller geodesic @xmath266 . note that by the definition of the bernoulli shift @xmath30 , we have @xmath267 and @xmath268 , and hence @xmath269 . we first fix @xmath270 on @xmath265 . by ( * lemma 1.4.3 ) , the function @xmath271 , @xmath272 is continuous where it is defined . we fix open neighborhoods @xmath273 of @xmath274 and @xmath275 of @xmath276 with the following condition . [ condi.u ] the neighborhoods @xmath277 satisfy * @xmath273 and @xmath275 have positive @xmath258 and @xmath263 measure respectively , and * for any @xmath278 and @xmath279 , there is the teichmller geodesic @xmath280 $ ] . * @xmath281 is bounded from above by some constant @xmath282 . the construction of semi - markov partition in this section works for any @xmath277 satisfying condition [ condi.u ] . we give @xmath277 which satisfy further condition that we need to prove theorem [ thm.main ] in [ sec.refine ] . finally let @xmath283 be small enough so that the @xmath284-neighborhood of @xmath270 is contained in the @xmath147-thick part of @xmath18 . we now choose points in @xmath18 to construct a semi - markov partition . see figure [ fig.position ] for a schematic picture . first , we choose @xmath285 to be a closest point to @xmath270 on @xmath286 . for @xmath38 positive , we define @xmath287 inductively by x_n : = x_n & if @xmath288 and @xmath289 + _ n^-1_n-1x_n-1 , & otherwise . we define @xmath290 for negative @xmath38 similarly . we then define @xmath291 as follows . for positive @xmath38 , we set @xmath292 if @xmath293 for all @xmath294 , and @xmath295 for otherwise . for negative @xmath38 , @xmath296 is defined similarly . we set @xmath297 . then for @xmath38 positive , we define @xmath298 inductively x_n : = x_n & if @xmath292 + _ n^-1_n-1x_n-1 , & if @xmath295 . we define @xmath298 for negative @xmath38 similarly . these @xmath298 are in the @xmath147-thick part and @xmath299 are located according to the order of @xmath38 on @xmath265 . even with this modification , the distance between @xmath300 and @xmath299 grows sublinearly . [ lem.tio ] for above @xmath15 and @xmath301 , @xmath302 since one can prove the statement for negative @xmath38 similarly , we assume that @xmath38 is positive . then since @xmath258 and @xmath263 are independent , @xmath303 hence by the ergodic theorem @xmath304 has positive density . we now recall the work of tiozzo . [ thm.tio ] for @xmath7-a.e . @xmath19 , let @xmath305 denote the teichmller geodesic @xmath265 with parametrization by arc length and @xmath306 . then we have @xmath307 where @xmath161 is the drift with respect to the teichmller metric . let @xmath308 since @xmath309 , @xmath310 is an integrable function . hence kingman s subadditive ergodic theorem implies for @xmath7-a.e . @xmath15 , @xmath311 then , we first estimate @xmath312 . let @xmath313 . by above observations , we may suppose that for large enough @xmath38 , we have @xmath314 where @xmath315 , and @xmath316 . then by definition , we have @xmath317 hence we have @xmath318 now we estimate @xmath319 . if @xmath320 , then there exists @xmath321 such that @xmath322 . suppose @xmath321 is chosen to be maximum with this property so that @xmath323 . let @xmath313 . we may suppose that @xmath38 is large enough so that @xmath324 for @xmath325 . then we have two cases ; @xmath326 or @xmath327 . since @xmath328 , in both cases we have , @xmath329 hence we have @xmath330 by theorem [ thm.tio ] and the triangle inequality , we have the conclusion . let @xmath331 be a representative of @xmath270 . since each @xmath299 is on @xmath265 , there is the teichmller map @xmath332 that stretches @xmath274 and contracts @xmath276 such that @xmath333 represents @xmath299 . then let @xmath334 , @xmath335 and @xmath336 . we denote the corresponding measured foliation representatives of @xmath274 and @xmath276 by @xmath337 and @xmath338 respectively . since @xmath299 are on @xmath265 , by the definition of @xmath296 , @xmath339 satisfies ( m1 ) . we need to decompose each birectangles in @xmath340 further to have a partition which satisfies ( m2 ) . given two birectangle partitions @xmath341 with @xmath342 and @xmath343 , let @xmath344 denote the birectangle partition we get by cutting @xmath0 by @xmath345 . let @xmath346 be indices which satisfy 1 . @xmath347 , and 2 . @xmath348 = 0 for all @xmath349 or @xmath350 , we define @xmath351 . we note that @xmath352 is equal to @xmath353 for @xmath354 with @xmath295 , let @xmath321 be the largest integer which is less than @xmath38 and @xmath355 . we define @xmath356 . for negative @xmath38 , @xmath357 is defined similarly . by the construction , @xmath358 still satisfies ( m1 ) . [ lem.markov ] @xmath358 is a semi - markov partition with respect to @xmath164 . if @xmath295 , the condition ( m2 ) is apparently satisfied . hence it suffices to prove for @xmath321 and @xmath38 with * @xmath359 and * @xmath360 for all @xmath361 , that for each @xmath362 and @xmath249 , the intersection @xmath363 is either empty or a single birectangle . since @xmath358 satisfies ( m1 ) , we see that if the intersection @xmath364 , it is a family of birectangles . note that each birectangle in @xmath365 or @xmath366 is a subset of a component of @xmath367 for some @xmath368 and @xmath369 . from each component @xmath370 of the intersection @xmath371 , a birectangle @xmath372 ( resp . @xmath373 ) is obtained by decomposing @xmath370 vertically by leaves of @xmath206 ( resp . horizontally by leaves of @xmath205 ) . hence each @xmath363 is connected . thus ( m2 ) follows . we now associate a random subshift of finite type to the representative @xmath136 of @xmath15 by using the semi - markov partition @xmath358 constructed in [ subsec.markov ] . let @xmath374 denote the number of birectangles in @xmath357 . we label birectangles in @xmath357 by @xmath375 . we define @xmath376 matrices @xmath377 by setting @xmath378 if @xmath379 and @xmath380 for otherwise . let @xmath33 be the random subshift of finite type with respect to @xmath381 . then each element in @xmath194 corresponds to a point in @xmath0 . [ lem.defi-p ] for any @xmath38 and @xmath382 , @xmath383 determines a single point in @xmath0 . let us fix @xmath382 . by the properties ( m1 ) and ( m2 ) of semi - markov partitions , we have that for each @xmath321 , @xmath384 is a birectangle with exactly one component . we consider the singular euclidean metric that determines the point @xmath299 on @xmath265 . let @xmath305 denote @xmath265 with parametrization by arc length so that @xmath385 . we will prove that the diameter of @xmath386 converges to @xmath182 as @xmath387 . by lemma [ lem.tio ] , we see that points @xmath388 for negative @xmath321 are close to @xmath389 . to construct @xmath390 , we considered arcs on @xmath391 of @xmath392 measure 1 which is @xmath393 measure almost equal to @xmath394 by lemma [ lem.tio ] and theorem [ thm.tio ] . hence the horizontal diameter of @xmath386 converges to @xmath182 as @xmath395 . on the other hand , for @xmath321 positive the arcs @xmath396 travel on singular leaves of @xmath205 longer as @xmath321 increases . since each infinite singular leaf is dense , it follows that the vertical diameter converges to @xmath182 as @xmath397 . thus we have a point on @xmath398 . finally by the marking @xmath399 , we fixed above , we have a point on @xmath0 . by lemma [ lem.defi-p ] , we define @xmath400 . [ lem.p - surj ] the map @xmath401 is continuous and surjective . since the image of a long cylinder set of @xmath194 is contained in a small birectangle , @xmath401 is continuous . let @xmath402 . for each @xmath403 , @xmath404 is an open dense set . then by the baire category theorem , @xmath405 is dense . each @xmath406 is contained in @xmath407 for some @xmath408 for every @xmath409 . let @xmath410 . we have @xmath411 , which implies @xmath412 . since @xmath413 is dense and @xmath194 is compact , @xmath401 is surjective . we are now ready to prove theorem [ thm.symbolic ] . note that @xmath414 changes only the metric and does not change the image . since @xmath400 is defined by using @xmath399 , we have @xmath415 . we now consider the topological entropy of the shift map @xmath188 . in order to prove theorem [ thm.main ] , it suffices to prove that growth ratio of the number of elements of cylinder sets . [ lem.p - entropy ] let @xmath33 be the random subshift of finite type defined above . then for any @xmath416 , @xmath417 where @xmath161 is the drift of @xmath15 with respect to the teichmller metric . note that by the property ( m1 ) and ( m2 ) of semi - markov partitions , the intersections @xmath418 is also a birectangle partition . for a given birectangle partition @xmath419 , let @xmath420 denotes the number of birectangles . by the map @xmath401 , we see that @xmath421 hence we will give a bound of @xmath422 . let @xmath423 be the shortest horizontal length of birectangles in @xmath424 measured by @xmath425 . let @xmath426 denote the maximum of @xmath425 measures of the arcs @xmath427 . each arc in @xmath428 cuts birectangles in @xmath424 at most @xmath429 times . the number of singular leaves of @xmath430 is bounded from above by some constant @xmath431 which depends only on @xmath0 . hence @xmath422 is at most @xmath432 . by lemma [ lem.tio ] and theorem [ thm.tio ] , @xmath433 which implies @xmath434 in [ sec.markov ] , we have constructed a semi - markov partition @xmath358 for any representative of @xmath7-a.e . @xmath19 . in this section , we will prove that for @xmath7-a.e . @xmath19 , we can find a representative @xmath435 of @xmath15 such that @xmath436 . in the case of pseudo - anosov diffeomorphisms , facts of type lemma [ lem.p - surj ] and [ lem.p - entropy ] suffice to prove that the topological entropy and the translation distance on the teichmller space agree . this is because to construct a markov partition for a pseudo - anosov diffeomorphism , we only need to use a single point in @xmath18 . on the other hand , for random walks , we need to use different @xmath298 s for each @xmath38 . one of the main difficulty caused for above reason is that the lebesgue number of a given open covering @xmath114 varies depending on the metric . in [ sec.lebesgue ] , we first give a suitable asymptotic bound for the lebesgue number . every argument so far works for any @xmath277 satisfying [ condi.u ] . in [ sec.positive-nu]-[sec.refine ] , the neighborhoods @xmath277 of @xmath437 which we need to prove theorem [ thm.main ] are given . to have a bound of the lebesgue number , we first observe how singular euclid structures may change in the @xmath147-thick part of @xmath18 . [ lem.bilipschiz ] there exists @xmath438 such that the following holds . let @xmath76 be in the @xmath147-thick part of @xmath18 , and @xmath439 . then the singular euclidean metric associated to @xmath440 and @xmath441 are @xmath442-bi - lipschitz . as pointed out in ( * lemma 9.22 ) , any two singular euclidean metrics are bi - lipschitz with some bi - lipschitz constant . since by theorem [ thm.hm ] , @xmath443 is homeomorphic to @xmath82 , a compact space , we see that for any @xmath25 , there exists @xmath444 such that any two singular euclidean metrics corresponding to elements in @xmath443 are @xmath445-bi - lipschitz . since @xmath445 varies continuously and the @xmath147-thick part of the moduli space of @xmath0 is compact , we have a desired bound . recall that the semi - markov partition @xmath358 is defined on @xmath446 with representative @xmath447 . then there is a quadratic differential @xmath448 that is the initial quadratic differential of the teichmller geodesic connecting @xmath270 and @xmath298 . let @xmath449 denote the complex structure we get by stretching ( resp . contracting ) horizontal ( resp . vertical ) foliation of @xmath450 so that it gives the same point as @xmath298 in @xmath18 . let @xmath451 be the corresponding teichmller map . since two markings @xmath399 and @xmath452 gives the same point in the teichmller space , there is a biholomorphic map @xmath453 so that @xmath454 is homotopic to @xmath207 . hence by homotopy , we may suppose @xmath455 . from now on , we use these representations and let @xmath164 . we now fix an open covering @xmath114 of @xmath0 . let @xmath456 be the quadratic differential on @xmath0 determined by @xmath457 and @xmath286 . for a quadratic differential @xmath60 , we denote by @xmath458 the lebesgue number of @xmath114 with respect to the singular euclidean metric defined by @xmath60 . by the choice of the representative @xmath459 , @xmath460 is equal to the lebesgue number of @xmath461 with respect to the quadratic differential determined by @xmath265 and @xmath398 that we used to construct @xmath462 . [ lem.lebesgue ] for @xmath7-a.e . @xmath15 , the @xmath463 defined above satisfies @xmath464 we first note that if two singular euclidean metrics determined by @xmath60 and @xmath465 are @xmath442-bi - lipschitz , then we have @xmath466 . since we have chosen @xmath298 so that they are in the @xmath147-thick part , we have @xmath467 and @xmath468 by lemma [ lem.bilipschiz ] . by the definition of @xmath469 , the ratio @xmath470 is bounded from above by @xmath471 . hence @xmath472 therefore , by lemma [ lem.tio ] , we have the conclusion . the goal of this subsection is the following proposition . the measure @xmath258 is from theorem [ thm.km ] . [ prop.positive-nu ] for @xmath7-a.e . @xmath15 , any open neighborhood @xmath413 of @xmath274 has positive @xmath258-measure . we first recall the curve graphs and shadows . the _ curve graph _ of @xmath0 , denoted @xmath473 , is the graph whose set of vertices are the set of isotopy classes of essential simple closed curves , and two vertices are connected by an edge of length @xmath183 if corresponding simple closed curves can be represented disjointly . for @xmath474 , the _ gromov product _ of @xmath174 and @xmath475 with respect to @xmath127 , denoted @xmath476 is defined by @xmath477 since @xmath473 is gromov hyperbolic by @xcite , it has the gromov boundary @xmath478 . let @xmath479 . a sequence of points @xmath480 converges to a point @xmath481 if @xmath482 . we define a _ shadow set _ by @xmath483 by ( * ? ? ? * theorem 1.2 and 1.4 ) and the fact that @xmath274 is uniquely ergodic , we see that @xmath484 as subsets of @xmath82 for sufficiently large @xmath213 . since @xmath485 converges to @xmath274 in @xmath486 , we see that for any @xmath431 , there exists @xmath487 such that for any @xmath488 , @xmath489 . hence by the work of maher ( * ? ? ? * proposition 2.13 ( 5 ) ) , for any @xmath313 , there is @xmath28 such that @xmath490 where @xmath491 . note that in @xcite , the measure @xmath6 is assumed to have a finite support , however , the finiteness is not used for the proof of results we need here . hence @xmath492 where @xmath493 is the @xmath38-fold convolution of @xmath6 . since @xmath494 , we have @xmath495 . we finally give neighborhoods @xmath273 and @xmath275 of @xmath274 and @xmath276 respectively . we would like to find @xmath277 so that the vertical lengths of birectangles in @xmath358 are bounded from above . recall that given two @xmath496 , we can construct a birectangle decomposition @xmath497 , where @xmath498 is a representative of a closest point projection @xmath499 on @xmath500 of @xmath270 . since the vertical length is independent of representatives of points of the teichmller space , we denote birectangle partitions by @xmath501 . we abuse notations similarly for @xmath64 and @xmath502 . let @xmath503 denote the maximum of the vertical lengths of birectangles in @xmath504 . since each @xmath357 is obtained from some @xmath501 by decomposing each birectangle in @xmath501 , it suffices to prove the following . [ lem.u ] there exist @xmath505 and open neighborhoods @xmath273 and @xmath275 of @xmath274 and @xmath276 respectively so that the following holds . suppose that * @xmath506 are written as @xmath507 respectively for some @xmath508 . then @xmath509 . suppose the contrary . then there are @xmath510 with @xmath511 and @xmath512 with @xmath513 such that @xmath514 . let @xmath515 be the rectangle with vertical length @xmath510 . note that since we assume that the total area is equal to @xmath183 , the horizontal length of @xmath516 converges to @xmath182 . hence by taking a subsequence if necessary , the vertical boundary @xmath517 converges in the hausdorff topology to an infinite subarc of a singular leaf of @xmath276 , which intersects @xmath518 only twice . however any infinite singular leaf of @xmath276 is dense , so such a singular leaf never exists . note that since @xmath273 and @xmath275 are open , we see that @xmath519 and @xmath520 by proposition [ prop.positive-nu ] . by taking smaller open neighborhoods if necessary , we may suppose that @xmath277 in lemma [ lem.u ] satisfy condition [ condi.u ] . from now on , we consider the semi - markov partition @xmath358 constructed with such @xmath277 and corresponding representation @xmath136 of @xmath15 whose construction is given in [ sec.lebesgue ] . we now consider images of cylinders in @xmath194 by @xmath401 . for notational simplicity , we call the image of cylinders by @xmath401 cylinders and omit to write @xmath401 . let @xmath521 be the number so that the set of cylinders @xmath522 refines @xmath114 . the existence of @xmath521 follows from lemma [ lem.defi-p ] . we now prove that @xmath521 grows sublinearly . [ lem.c(n ) ] for the above @xmath521 , we have @xmath523 it suffices to find @xmath521 so that the horizontal and the vertical lengths with respect to @xmath456 of @xmath524-cylinders are less than @xmath525 . let @xmath505 be the upper bound given by lemma [ lem.u ] . then the horizontal and vertical length of any @xmath524-cylinder is bounded from above by @xmath526 and @xmath527 respectively . since the bound for the horizontal length is given similarly , we only discuss the vertical lengths . let @xmath528 and let @xmath313 be arbitrary . by lemma [ lem.tio ] and theorem [ thm.tio ] , for sufficiently large @xmath38 , we have @xmath529 . we also have @xmath530 by lemma [ lem.lebesgue ] . let @xmath531 be the smallest integer with @xmath532 . then for large enough @xmath38 , @xmath533 refines @xmath114 . hence for large enough @xmath38 , @xmath534 . since @xmath313 is arbitrary , we have @xmath535 . we are now ready to prove the main theorem . since @xmath536 , there exists @xmath537 such that @xmath538 for all @xmath539 . therefore we have , @xmath540 by lemma [ lem.p - entropy ] , we have @xmath541 also by lemma [ lem.c(n ) ] , @xmath542 therefore , @xmath543 by putting all estimates together , we have @xmath544 hence the representative @xmath136 satisfy that for arbitrary open covering @xmath114 , @xmath545 . putting together with theorem [ thm.kar ] and lemma [ lem.lemma1 ] , we have @xmath546 . this work was partially supported by jsps research fellowship for young scientists . 99 r. l. adler , a. g. konheim , m. h. mcandrew . _ topological entropy_. transactions of the american mathematical society 114 ( 2 ) : 309 - 319 . l. bers , _ an extremal problem for quasiconformal mappings and a theorem by thurston _ , acta . math . , 141 ( 1978 ) , 73 - 98 . y. choi and k. rafi . _ comparison between teichmller and lipschitz metrics . _ j. lond . ( 2 ) , 76(3):739 - 756 , 2007 . math.gt/0510136 . a. fathi , f. laudenbach , v. ponaru et al . , travaux de thurston sur les surfaces , astrisque 66 - 67 , socit mathmatique de france , 1979 . b. farb and d. margalit , _ a primer on mapping class groups ( pms-49 ) _ , princeton university press , 2011 . f. gardiner and h. masur , _ extremal length geometry of teichmller space _ , complex variables and elliptic equations 16.2 - 3 ( 1991 ) : 209 - 237 . gundlach and y. kifer , _ random hyperbolic systems _ ( english summary ) stochastic dynamics ( bremen , 1997 ) , 17 - 145 , springer , new york , 1999 . j. hubbard and h. masur , _ quadratic differentials and foliations _ , acta . ( 1979 ) , 221 - 274 . v. kaimanovich and h. masur , _ the poisson boundary of the mapping class group _ , invent . ( 1996 ) 221 - 264 . a. karlsson , _ two extensions of thurston s spectral theorem for surface diffeomorphisms _ , bull . 46 ( 2014 ) , no . 2 , 217 - 226 . e. klarreich , _ the boundary at infinity of the curve complex and the relative teichmller space _ , unpublished manuscript , ann arbor 1999 . j. maher , _ exponential decay in the mapping class group _ , j. lond . ( 2 ) 86 ( 2012 ) , no . 2 , 366 - 386 . h. masur and y. minsky , _ geometry of the complex of curves . i. hyperbolicity _ , invent . , 138(1):103 - 149 , 1999 . w. p. thurston , _ on the geometry and dynamics of diffeomorphisms of surfaces _ , bulletin of the american mathematical society 19 , ( 1988 ) 417 - 431 . g. tiozzo , _ sublinear deviation between geodesics and sample paths _ , duke math . j. 164 ( 2015 ) , no . 3 , 511 - 539 .
|
for any pseudo - anosov diffeomorphism on a closed orientable surface @xmath0 of genus greater than one , it is known by the work of bers and thurston that the topological entropy agrees with the translation distance on the teichmller space with respect to the teichmller metric . in this paper
, we consider random walks on the mapping class group of @xmath0 .
the drift of a random walk is defined as the translation distance of the random walk .
we define the topological entropy of a random walk and prove that it almost surely agrees with the drift on the teichmller space with respect to the teichmller metric .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the 4f - electronic systems with spin and orbital degrees of freedom in rare earth compounds frequently reveal electric quadrupole orderings in addition to magnetic dipole orderings at low temperatures . the cubic compounds based on ce@xmath18 ion with a @xmath8 quartet ground state in particular have received much attention because the competitive phenomena associated with magnetic dipole , electric quadrupole and magnetic octupole degrees of freedom are expected . the direct product of @xmath19 is reduced to a direct sum @xmath20 . the magnetic dipole @xmath21 , @xmath22 , @xmath23 belonging to @xmath24 symmetry are order parameters for magnetic orderings . the quadrupole orderings of @xmath10 , @xmath25 with @xmath26 or @xmath27 , @xmath28 , @xmath29 with @xmath13 are interesting phenomena in the @xmath8 system . we refer to ceag exhibiting the ferroquadrupole ( fq ) ordering of @xmath10 at @xmath30 k.@xcite ceb@xmath2 is known as the antiferroquadrupole ( afq ) ordering of @xmath29-type with the propagation vector of = [ 111 ] at @xmath31 k.@xcite the octupole moments @xmath32 with @xmath33 symmetry , @xmath34 , @xmath35 , @xmath36 with @xmath24 and @xmath37 , @xmath38 , @xmath39 with @xmath13 may play a role in the @xmath8 system.@xcite cage of 4a site with o@xmath40 symmetry consisting of ge and pd2 atoms and cage of 8c site with t@xmath41 of pd1 and pd2 atoms in clathrate compound ce@xmath0pd@xmath1ge@xmath2 . the 4a site ce2 forms a simple cubic lattice , while 8c site ce1 makes a face centered cubic one . the ferroquadrupole ordering below @xmath11 is relevant for the 8c sites and the antiferromagnetic ordering below @xmath42 occurs at 4a sites . the @xmath13 rattling motion originates from the off - center ce1 atom in 4a - site cage . ] a cerium - based ternary compound ce@xmath0pd@xmath1ge@xmath2 with the @xmath8 ground state has received much attention because the competition between quadrupole and magnetic orderings is expected at low temperatures.@xcite ce@xmath0pd@xmath1ge@xmath2 crystallizes in a cubic cr@xmath3c@xmath2-type structure with a space group @xmath43 consisting of four molecular units with 116 atoms in a unit cell.@xcite the twelve ce sites located in cages are divided into two nonequivalent sites in crystallography . as shown in fig . [ eps01 ] the ce ion at 4a site in a cage consisting of twelve pd - atoms and six ge atoms possesses point group symmetry o@xmath40 , while the ce ion at 8c site in a cage of sixteen pd atoms has t@xmath41 . the 4a sites form a face - centered cubic lattice , while the 8c sites make a simple cubic lattice . inelastic neutron scattering on ce@xmath0pd@xmath1ge@xmath2 revealed overlapping two peaks for the crystalline electric field ( cef ) potentials , which correspond to magnetic dipole transitions from the @xmath8 ground quartet to the @xmath44 excited doublet at 60 k of the 4a site and from the @xmath8 ground quartet to the @xmath44 at 46 k of 8c site.@xcite the entropy obtained by low - temperature specific heat measurement on ce@xmath0pd@xmath1ge@xmath2 also indicates the ground state @xmath8 quartet at both 4a and 8c sites.@xcite the low - temperature specific heat of ce@xmath0pd@xmath1ge@xmath2 shows a rounded small peak at @xmath7 k and a sharp @xmath45-peak at @xmath12 k.@xcite magnetic susceptibility shows a clear cusp at @xmath42 , but exhibits no sign of anomaly at @xmath11.@xcite in addition to these experimental results , an elastic softening of @xmath4 in our preliminary paper suggests that the paramagnetic phase i transforms to the fq phase ii at @xmath11 and successively changes to the antiferromagnetic ( afm ) phase iii at @xmath42.@xcite the neutron scattering on ce@xmath0pd@xmath1ge@xmath2 reveals a paramagnetic state of ce ions at both 4a and 8c sites even in phase ii between @xmath11 and @xmath42 . the afm ordering in phase iii with a propagation vector @xmath46 $ ] for cerium ions at 4a site is observed below @xmath42.@xcite even in phase iii below @xmath42 , the 8c site still remains to be the paramagnetic state . the afm ordering with incommensurate structure at 8c site appears only below @xmath47 k. the clathrate compounds exhibiting the rattling motion or off - center motion in a cage have received attention because their remarkable reduction of thermal conductivity is favorable for application to thermoelectric device with a high figure of merit.@xcite the ultrasonic waves are scattered by the rattling motion in an over - sized cage of a semiconductor sr@xmath48ga@xmath49ge@xmath50 and a filled skutterudite compound pros@xmath51sb@xmath52.@xcite the off - center tunneling motion of oh ion doped in nacl gives rise to elastic softening at low temperatures.@xcite the rattling motion in the present compound ce@xmath0pd@xmath1ge@xmath2 with clathrate structure has not been reported so far . in the present paper we show ultrasonic measurements on ce@xmath0pd@xmath1ge@xmath2 in order to examine lattice effects associated with the quadrupole ordering and rattling motion in the system . the thermal expansion measurement is also employed to detect the spontaneous distortion below @xmath11 . in sec . ii , the experimental procedure and apparatus are described . the results of the elastic constant , magnetic phase diagram , thermal expansion are presented in sec . the ultrasonic dispersion due to rattling motion is also argued in sec . iii . in sec . iv , we present concluding remarks . single crystals of ce@xmath0pd@xmath1ge@xmath2 used in the present measurements were grown by a czochralski puling method . we have made the ultrasonic velocity measurements using an apparatus consisting of a phase difference detector . piezoelectric plates of linbo@xmath0 for the ultrasonic wave generation and detection are bonded on plane parallel surfaces of sample . x_-cut plate of linbo@xmath0 is available for transverse ultrasonic waves and the 36@xmath53_y_-cut plate is for longitudinal waves . the ultrasonic velocity _ v _ was measured by fundamental frequencies of 10 mhz and overtone excitations of 30 , 50 and 70 mhz . in the estimation of the elastic constant @xmath54 , we use the mass density @xmath55 g/@xmath56 for ce@xmath0pd@xmath1ge@xmath2 with a lattice parameter @xmath57 .@xcite a homemade @xmath58he - refrigerator equipped with a superconducting magnet was used for low - temperature measurements down to 450 mk in magnetic fields up to 12 t. a @xmath58he-@xmath59he dilution refrigerator with a top - loading probe was used for the ultrasonic measurements in low - temperature region down to 20 mk in fields up to 16 t. low input - power condition provides the low - temperature ultrasonic measurements free from a self - heating effect in the ultrasonic transducers . the sample length as a function of temperature or applied magnetic field was measured precisely by a capacitance dilatometer in the @xmath58he - refrigerator . the elastic constants of @xmath60 and @xmath61 of ce@xmath0pd@xmath1ge@xmath2 in fig . [ eps02 ] were measured by the longitudinal ultrasonic waves with frequencies 10 mhz propagating along the [ 100 ] and [ 110 ] directions , respectively . the elastic constant @xmath4 of ce@xmath0pd@xmath1ge@xmath2 in fig . [ eps03 ] was measured by the transverse ultrasonic wave of 10 mhz propagating along the [ 110 ] direction polarized to the @xmath62 $ ] one . the elastic constant @xmath5 of ce@xmath0pd@xmath1ge@xmath2 in fig . [ eps03 ] was determined by the transverse wave of 30 mhz propagating along [ 100 ] polarized to [ 010 ] . the bulk modulus @xmath63 in fig . [ eps02 ] was calculated by @xmath60 in fig . [ eps02 ] and @xmath4 in fig . [ eps03 ] . it is remarkable that @xmath4 exhibits a huge softening of 50% with decreasing temperature down to @xmath7 k. in phase ii below @xmath11 the ultrasonic echo signal of the @xmath4 mode completely disappears due to a marked ultrasonic attenuation . the softening of the longitudinal @xmath60 and @xmath64 modes in fig . 2 originates from the softening of @xmath4 , because @xmath60 and @xmath64 involve @xmath4 in part . the softening of @xmath4 above @xmath11 and the spontaneous tetragonal distortion below @xmath11 , that will be shown in sec . iii d , provide evidence for the fq ordering in phase ii . the @xmath5 in fig . [ eps03 ] also exhibits a softening of 2.5% down to @xmath11 . the low - temperature behavior of @xmath60 and @xmath5 shown in insets of figs . [ eps02 ] and [ eps03 ] indicates the transition to the fq phase ii at @xmath11 and successive transition to the afm phase iii at @xmath12 k. on the other hand , @xmath65 shows monotonic increase with decreasing temperature . neutron scattering on ce@xmath0pd@xmath1ge@xmath2 revealed the para - magnetic state for ce ions at both 4a and 8c sites in phase ii , which is consistent with the present scenario of the fq ordering at 8c site in phase ii below @xmath11.@xcite the afm ordering in phase iii at 4a site below @xmath42 has been detected by the neutron scattering . it has been proposed that the inter - site quadrupole interaction among 8c sites brings about the fq ordering at 8c sites in phase ii and ce ions at 4a sites still remain to be the para - state even in phase ii . the inter - site magnetic interaction among 4a sites gives rise to the afm ordering in phase iii below @xmath42 . the magnetic ordering at 8c sites appears only below 0.4 k. we discuss about this transition in the following sec . iii c. in order to analyze the elastic softening of @xmath4 and @xmath5 in ce@xmath0pd@xmath1ge@xmath2 of fig . [ eps03 ] , we introduce the coupling of the quadrupole @xmath66 of ce ions to the elastic strain @xmath67 as @xcite @xmath68 where the summation @xmath69 takes over ce ions in unit volume and @xmath70 is a coupling constant . the inter - site quadrupole interaction mediated by phonons and conduction electrons is written in a mean field approximation as @xmath71 where @xmath72 denotes a mean field of the quadrupole and @xmath73 means a coupling constant for the inter - site quadrupole interaction . by differentiating the total free energy consisting of 4f - electron and lattice parts with respect to the elastic strain @xmath67 , we obtain the temperature dependence of the elastic constant @xmath74 as @xcite temperature dependence of the elastic constants @xmath60 , @xmath64 and the bulk modulus @xmath65 of ce@xmath0pd@xmath1ge@xmath2 . inset shows the anomalies of @xmath60 around the ferroquadrupole transition @xmath7 k and the antiferromagnetic transition @xmath12 k. ] temperature dependence of the elastic constants @xmath4 and @xmath5 corresponding to shear waves in ce@xmath0pd@xmath1ge@xmath2 . inset shows the anomalies of @xmath5 around the ferroquadrupole transition @xmath7 k and the antiferromagnetic transition @xmath12 k. solid lines are the calculation by the quadrupole susceptibility for the @xmath8 ground state and @xmath44 state at 46 k of ce ions . the broken lines were back ground @xmath75 as shown in the text . a shoulder in @xmath5 around 30 k means the ultrasonic dispersion due to the @xmath13 rattling motion . ] @xmath76 here @xmath77 denotes a background elastic constant without the quadrupole - strain interaction and _ n _ is the number of ce ions in unit volume . the quadrupole susceptibility of @xmath78 in eq . ( 3 ) is written as @xmath79 where @xmath80 is a second - order perturbation energy with respect to @xmath67 for cef state . @xcite the first part in right hand of eq . ( 4 ) corresponds to the van vleck - term and the second part to the curie term . the ce ions at both 4a and 8c sites in ce@xmath0pd@xmath1ge@xmath2 have the @xmath8 ground state , while the @xmath44 state has excited energies of 46 k at 8c site and 60 k at the 4a site . as was already mentioned , the neutron scattering revealed that the fq ordering of 8c sites occurs at @xmath7 k and the afm ordering of 4a sites appears at @xmath12 k. these facts indicate that the inter - site quadrupole interaction of eq . ( 2 ) among the ce ions at 8c sites dominates the softening of @xmath4 as a precursor of the fq ordering at @xmath11 . in the following analysis we simply assume the quadrupole - strain interaction of eq . ( 1 ) and quadrupole interaction of eq . ( 2 ) for the 8c site with the cef splitting of @xmath8 ( 0 k ) and @xmath44 ( 46 k ) . the solid lines for @xmath4 and @xmath5 with eq . ( 3 ) in fig . [ eps03 ] reproduce well the softening in paramagnetic phase i above @xmath11 . it should be noted that the softening above @xmath11 proportional to the reciprocal temperature @xmath6 originates from the curie term of eq . the coupling constants were determined to be @xmath81 k , @xmath82 k for @xmath4 and @xmath83 k , @xmath84 k for @xmath5 . the back ground @xmath85 j / m@xmath58 and @xmath86 j / m@xmath58 indicated by broken lines in fig . [ eps03 ] was used . the positive value of @xmath87 are consistent with the fq ordering in ce@xmath0pd@xmath1ge@xmath2 . a shoulder like anomaly in @xmath5 around 10 k results from ultrasonic dispersion that is caused by a rattling motion of the rare - earth ion at 4a site in an oversized cage of fig . [ eps01 ] . we discuss about this remarkable behavior in sec . iii e. in order to examine the magnetic phase diagrams of the fq and afm orderings in ce@xmath0pd@xmath1ge@xmath2 , we have made the low - temperature ultrasonic measurements of @xmath60 , @xmath5 and @xmath4 under magnetic fields . the softening of @xmath60 of fig . [ eps04 ] reduces with increasing fields applied along the [ 001 ] direction parallel to the propagation direction of longitudinal wave . the fq transition points @xmath11 indicated by downward arrows in fig . [ eps04 ] shift to higher temperatures and become indistinct in high fields up to 6 t. in fig . [ eps05 ] , the fq transition points @xmath11 also shift to higher temperatures accompanied by reduction of the softening in @xmath5 with increasing applied fields along the [ 001 ] direction . low - temperature behavior of @xmath60 of ce@xmath0pd@xmath1ge@xmath2 under magnetic fields along the [ 001 ] direction . downward arrows indicate the ferroquadrupole ordering temperature @xmath11 and upward arrow means the antiferromagnetic ordering temperature @xmath42 . ] low - temperature behavior of @xmath5 of ce@xmath0pd@xmath1ge@xmath2 under magnetic fields along [ 001 ] . downward arrows indicate the ferroquadrupole ordering temperature @xmath11 and upward arrow means the antiferromagnetic ordering temperature @xmath42 . ] low temperature behavior of @xmath4 of ce@xmath0pd@xmath1ge@xmath2 under magnetic fields along the [ 001 ] direction . the successive phase transitions i - ii - iii are indicated by arrows . ] low temperature behavior of @xmath4 of ce@xmath0pd@xmath1ge@xmath2 under magnetic fields along the [ 110 ] direction . the successive phase transitions i - ii - ii@xmath88 are indicated by arrows . ] in figs . [ eps06 ] and [ eps07 ] , we show the field dependence of @xmath4 applying fields along the [ 001 ] and [ 110 ] directions , respectively . in zero magnetic field , the @xmath4 mode exhibits the considerable softening of 50% and the strong ultrasonic attenuation losing the echo signal in the vicinity of the fq transition @xmath7 k. the magnitude of the softening decreases abruptly with increasing fields along both [ 001 ] and [ 110 ] directions . in magnetic fields , clear minima of @xmath4 corresponding to the transitions @xmath11 from the paramagnetic phase i to the fq phase ii shift to higher temperatures . only broad round anomalies around @xmath11 have been observed in high fields of 5 t. this behavior of @xmath11 is similar to the results of the fq ordering accompanied by the soft @xmath5 mode in hob@xmath2.@xcite the neel temperature @xmath42 in fig . [ eps06 ] shifts slightly to lower temperatures with increasing magnetic fields . in fig . [ eps07 ] the anomalies associated with the transition between phase ii and ii@xmath88 below @xmath89 have been found . for the investigation of low - temperature and high - field behavior in fq ii and afm iii phases , we have measured field dependence of the @xmath5 and @xmath61 employing the dilution refrigerator . in fig . [ eps08 ] we show @xmath5 versus @xmath90 at 30 mk in fields up to 12 t applied along [ 001 ] . inset of fig . 8 is expanded view below 2.5 t. an anomaly of the phase ii - iii boundary at 2.1 t indicated by a vertical line has been observed . furthermore , several anomalies at 0.5 , 1.2 and 1.6 t associated with sub - phases of the phase iii have been found . it should be emphasized that appreciable hysteresis phenomena between increasing and decreasing field sequences have been found only in phase iii . in fig . [ eps09 ] we show the low - temperature field dependence of the @xmath64 in fields up to 16 t applied along [ 110 ] . low - field behavior below 2.5 t is shown in inset of fig . we have observed a new phase boundary around 8.2 t in phase ii , which is probably a sub - phase of the fq phase ii . however , this phase boundary is absent in fields along [ 001 ] as shown in fig . [ eps08 ] . at low field in phase iii , we have found several anomalies in @xmath64 of fig . [ eps09 ] showing a hysteresis behavior . as can been seen in inset of fig . [ eps09 ] , this hysteresis becomes pronounced with decreasing temperature . these sub - phases with hysteresis behavior in phase iii in magnetic fields along both [ 001 ] and [ 110 ] are well consistent with the results of neutron scattering experiments that detect weak incommensurate magnetic bragg peaks with a propagation vector = [ 0 0 1-@xmath91 , ( @xmath92 0.06 ) at 8c site.@xcite the magnetic phase diagrams of ce@xmath0pd@xmath1ge@xmath2 in figs . [ eps10 ] and [ eps11 ] are obtained in fields along the [ 001 ] and [ 110 ] directions , respectively . we present the results of the ultrasonic measurements together with the results of thermal expansions in sec . iii d. it is of importance that the fq phase ii is stabilized in fields for the [ 001 ] direction in fig . [ eps10 ] and the [ 110 ] direction in fig . [ eps11 ] . the fq sub - phase ii@xmath88 was added to the phase diagram and the upper limit at 8.2 t of the phase ii@xmath88 newly determined in fields along [ 110 ] of fig . [ eps11 ] . however , the fq sub - phase is absent in fields along [ 001 ] of fig . [ eps10 ] . this result indicates strong anisotropy of the quadrupole interaction of @xmath10 in ce@xmath0pd@xmath1ge@xmath2 . field dependence of the @xmath5 at 30 mk in fields along [ 001 ] up to 12 t. inset is expanded view below 2.5 t indicating the magnetic transitions . ] field dependence of the @xmath61 at various temperatures in fields along [ 110 ] up to 16 t. inset is expanded view below 2.5 t indicating the magnetic transitions ] magnetic phase diagram of ce@xmath0pd@xmath1ge@xmath2 under fields along the [ 001 ] direction . the boundary from paramagnetic phase i to the ferroquadrupole phase ii shifts to higher temperatures with increasing fields , while the boundary from phase ii to the antiferromagnetic phase iii shifts to lower temperatures in fields . ] magnetic phase diagram of ce@xmath0pd@xmath1ge@xmath2 under fields along the [ 110 ] direction . the sub - phase ii@xmath88 exists in the afq phase ii . ] the series of r@xmath0pd@xmath1x@xmath2 compounds usually show two successive afm orderings of 8c site at @xmath93 with a propagation vector @xmath94 $ ] and of 4a site at @xmath95 with @xmath96 $ ] , nd@xmath0pd@xmath1ge@xmath2 ( @xmath97 k , @xmath98 k ) @xcite , nd@xmath0pd@xmath1si@xmath2 ( @xmath99 k , @xmath100 k ) @xcite , tb@xmath0pd@xmath1si@xmath2 ( @xmath101 k , @xmath102 k ) @xcite , dy@xmath0pd@xmath1si@xmath2 ( @xmath103 k , @xmath104 k ) @xcite and so on . one can reasonably expect that the transition temperature @xmath93 at 8c site is always higher than @xmath42 at 4a site since the distance @xmath105 6.2 between rare - earth ion of 8c site is much shorter than the one @xmath105 8.8 of 4a site . in the present ce@xmath0pd@xmath1ge@xmath2 , at first the fq ordering at 8c site with a structural change from cubic lattice to tetragonal one occurs at @xmath7 k. therefore , the afm ordering at 8c site is hard to take place because the favorable propagation vector @xmath94 $ ] of 8c site does not match to the tetragonal lattice in phase ii . in other words , the afm ordering at 8c site is replaced by the fq ordering in ce@xmath0pd@xmath1ge@xmath2 . while , the afm ordering of 4a site with a propagation vector @xmath96 $ ] is easy to occur even in tetragonal structure below @xmath42 . neutron experiments detected large enough value of saturation cerium moments @xmath106/ce that is expected from the ground state quartet @xmath8 perpendicular to the @xmath46 $ ] in ce@xmath0pd@xmath1ge@xmath2 far below @xmath42 at 50 mk.@xcite in order to examine the structural change due to the fq ordering at @xmath7 k , we have measured the thermal expansion along the [ 001 ] direction in ce@xmath0pd@xmath1ge@xmath2 . the sample lengths along the [ 001 ] and [ 111 ] directions are written by the symmetry strains as @xmath107}=\varepsilon_{zz}=\varepsilon_{\rm b}/3+\varepsilon_u/\sqrt3 $ ] and @xmath108}=\varepsilon_{\rm b}/3 + 2(\varepsilon_{yz}+\varepsilon_{zx}+\varepsilon_{xy})/3 $ ] . here , @xmath109 is a volume strain with @xmath110 symmetry , @xmath111 is a tetragonal strain with @xmath26 and @xmath112 is a shear strain with @xmath13 . the length along [ 001 ] in fig . [ eps12 ] shows a monotonous decrease with decreasing temperature in paramagnetic phase i above @xmath11 and abruptly expands about @xmath9 below @xmath11 . the thermal expansion along [ 001 ] in phase ii and the huge softening of 50% in @xmath4 of fig . [ eps03 ] indicate the @xmath10-type fq ordering accompanied by the structural transition from cubic lattice to tetragonal one with the spontaneous strain @xmath113 in phase ii . this spontaneous strain is proportional to the order parameter as @xmath114 in mean - field approximation . below @xmath12 k , the @xmath115 along [ 001 ] slightly shrinks . inset of fig . [ eps12 ] is expanded view of @xmath115 and the coefficient of the thermal expansion @xmath116 at low temperatures . a sharp anomaly in the coefficient @xmath116 has been found at the fq transition @xmath89 . thermal expansion @xmath115 in ce@xmath0pd@xmath1ge@xmath2 . inset shows the thermal expansion coefficient @xmath116 and thermal expansion @xmath115 at low temperatures . ] thermal expansion @xmath115 along [ 001 ] in ce@xmath0pd@xmath1ge@xmath2 under fields up to 12 t along the [ 001 ] direction . ] measurements of @xmath115 versus @xmath117 in various magnetic fields parallel to [ 001 ] are shown in fig . [ eps13 ] . the magnitude of the expansion @xmath115 in fields exhibits noticeable increase up to @xmath118 compared with that in zero magnetic field . the sharp increase of @xmath115 at the transition point to the fq phase ii has been observed in low magnetic fields below 1 t. on the other hand , the gradual increase in the thermal expansion @xmath115 above 5 t up to 12 t indicates an obscure character of the i - ii phase boundary in high fields . this is consistent with the fact that the elastic constants in fields of figs . [ eps04 ] , [ eps05 ] , [ eps06 ] and [ eps07 ] show obscure transitions in fields . this behavior is similar to the liquid - gas transition near the critical end point under hydrostatic pressures . the thermal expansion along the [ 001 ] direction in fields parallel to [ 001 ] and the considerable softening of @xmath4 of 50% in ce@xmath0pd@xmath1ge@xmath2 strongly suggests that the order parameter of the fq ordering in phase ii is @xmath10 with @xmath26 symmetry . the relatively small softening of 2.5% in @xmath5 in fig . [ eps03 ] means that the quadrupole of @xmath29-type with @xmath13 is irrelevant for the transition at @xmath11 . the thermal expansion of ce@xmath0pd@xmath1ge@xmath2 along [ 111 ] is required to examine an interplay of the spontaneous strain @xmath112 for the phase ii . we refer to our recent study of the fq transition in hob@xmath2 and the phase iv in ce@xmath119la@xmath120b@xmath2 ( _ x_=0.75 , 0.70 ) , @xcite where the trigonal strain @xmath121 plays a significant role and the tetragonal strain @xmath122 is irrelevant . these facts are well consistent with the pronounced elastic softening in @xmath5 of 70% in hob@xmath2 and of 31% in ce@xmath119la@xmath120b@xmath2 ( _ x_=0.75 , 0.70 ) . the @xmath5 mode associated with the elastic strain @xmath124 of @xmath13 symmetry of ce@xmath0pd@xmath1ge@xmath2 in figs . [ eps03 ] and [ eps05 ] exhibits a shoulder like anomaly around 10 k in addition to the characteristic softening due to the quadrupolar coupling above @xmath7 k. it should be noted that this anomaly is absent for the @xmath4 mode associated with @xmath26 elastic strain @xmath125 and the bulk modulus @xmath65 with @xmath110 volume strain @xmath126 . in order to examine the origin of this anomaly , we have measured the frequency dependence of @xmath5 from 10 mhz up to 250 mhz . the elastic constant @xmath5 of fig . [ eps14](a ) exhibits shoulders showing remarkable frequency dependence . an increase in ultrasonic attenuation around shoulder has also been found , but not discussed here . we describe this frequency dependence of the elastic constant @xmath127 in terms of debye - type dispersion as @xmath128 where @xmath129 and @xmath130 are the elastic constants of high frequency limit and low frequency one , respectively . here @xmath131 is an angular frequency of the ultrasonic wave and @xmath132 means the relaxation time of the system . in fittings of fig . [ eps14](b ) , we take into account the superposition of two susceptibilities by the quadrupole one of eq.(3 ) and the debye - type dispersion of eq.(5 ) as @xmath133 . the inflection points around 10 k in @xmath5 indicated by arrows in fig . [ eps14](a ) mean the temperatures where the @xmath132 coincides with the @xmath131 as @xmath134 . the ultrasonic attenuation is expected to be maximum at the temperatures for @xmath134 . the solid lines of fig . [ eps14](b ) being the calculations with eq . ( 5 ) well reproduce the experimental results of fig . [ eps14](a ) . the relaxation time obeying the arrhenius - type temperature dependence @xmath135exp@xmath15 with the attempt time @xmath16 sec and the activation energy @xmath17 k has been determined . the parameter of @xmath136 j / m@xmath58 is used . frequency dependence of the elastic constant of the transverse @xmath5 mode in ce@xmath0pd@xmath1ge@xmath2 . ( a ) represents the experimental results with the frequencies from 10 mhz up to 250 mhz . ( b ) is the calculation in terms of the debye - type dispersion of eq . ( 5 ) in text . ] the ultrasonic dispersion due to electron thermal hopping has already found in the inhomogeneous valence fluctuation compounds of sm@xmath0x@xmath51 ( x = se , te ) , yb@xmath51(as@xmath137sb@xmath138)@xmath0 @xcite and sr@xmath52ca@xmath139cu@xmath140o@xmath141 @xcite . it is remarkable that the very slow relaxation time @xmath132 and extremely low activation energy @xmath142 for ce@xmath0pd@xmath1ge@xmath2 are exceptional as compared to those of charge fluctuation compounds sm@xmath0se@xmath51 @xcite and sm@xmath0te@xmath51 @xcite , @xmath143 sec and @xmath144 k , and sr@xmath52ca@xmath139cu@xmath140o@xmath141 , @xmath145 sec and @xmath146 k.@xcite this discrepancy of the order of @xmath147 and @xmath142 between the present ce@xmath0pd@xmath1ge@xmath2 and the charge fluctuation compounds indicates that thermally activated rattling motion of heavy mass particle , which is probably rare - earth ion in a cage , gives rise to the ultrasonic dispersion in ce@xmath0pd@xmath1ge@xmath2 . glass materials and charge fluctuation compounds exhibit frequently the ultrasonic dispersion , which results from a thermally activated motion in a double- or multi - well potential . the compounds such as sm@xmath0x@xmath51 ( x = s , se , te ) with different valence of sm@xmath148 and sm@xmath18 ions and sr@xmath52ca@xmath139cu@xmath140o@xmath141 with cu@xmath148 and cu@xmath18 ions cause the ultrasonic dispersions due to thermally assisted charge fluctuation in the temperature region between 100 - 200 k. the two - level system ( tls ) due to an atomic tunneling @xcite or electron tunneling @xcite manifests itself in glass materials at low temperatures , where the thermally activated motion dies out . the tls yields the decrease in the elastic constant proportional to ln@xmath117,@xcite the specific heat to @xmath117 and thermal conductivity to @xmath149.@xcite besides in the case of sm@xmath0te@xmath51 , remarkable logarithmic decrease in the elastic constant appears below about 15 k down to the spin glass transition at @xmath150 k , which suggests the existence of the 4f - electron tunneling motion between sm@xmath148 and sm@xmath18 ions situated charge glass state.@xcite [ cols="^,^,^",options="header " , ] the present clathrate compound ce@xmath0pd@xmath1ge@xmath2 is a crystal possessing an ideal periodic arrangement of cages in space . the stable trivalent ce ions in cages of ce@xmath0pd@xmath1ge@xmath2 are free from the valence fluctuation phenomena . as shown in fig . [ eps01 ] , the clathrate compound ce@xmath0pd@xmath1ge@xmath2 is made up of the cage at 4a site consisting of pd and ge with distances @xmath151=3.332 , @xmath152=3.067 , and the cage at 8c site of pd with @xmath153=2.868 , @xmath154=3.373 . the trivalent ce - ion with radii @xmath155 inside the 4a - site cage in particular is expected to show the rattling motion over the off - center positions being away from the center of the cage actually the neutron scattering on pr@xmath0pd@xmath1ge@xmath2 and nd@xmath0pd@xmath1ge@xmath2 revealed the sharp transition peaks indicating the stable cef splitting at 8c site and no indication for cef state at 4a site.@xcite these results may imply the obscure cef state due to the off - center ce ion at 4a site contrary to the well - defined cef splitting at 8c site being stable ce ion position . notable finding of fig . [ eps14 ] is that the ultrasonic dispersion in the @xmath5 mode associated with @xmath112-type strain indicates the rattling motion with specific @xmath13 symmetry in ce@xmath0pd@xmath1ge@xmath2 . it is expected that the ce ion in 4a cage with cubic symmetry o@xmath40 favors off - center positions along one of the three principle directions of [ 100 ] , [ 110 ] and [ 111 ] . as one can see the cage at 4a site in fig . [ eps01 ] , it is of particular interest that no atom exists along the three - fold [ 111 ] directions , while the ge atom occupies along the four - fold [ 100 ] and the pd atom along the two - fold [ 110 ] ones . this crystallographic character may promise a flat potential along the three - fold [ 111 ] directions and profound potentials along the four - fold [ 100 ] and two - fold [ 110 ] directions . presumably the ce ion at 4a site prefers the off - center eight positions along the [ 111 ] directions , which are defined as @xmath157 , @xmath158 , @xmath159 , @xmath160 , @xmath161 , @xmath162 , @xmath163 , @xmath164 . the atomic densities @xmath165 at the eight off - center positions are also defined . when 48 symmetry operators of o@xmath40 point group are acted on the atomic densities @xmath166 , one can derive the transfer representation matrices with @xmath167 elements . consequently , one obtains the characters @xmath168}$ ] for the rattling motion by tracing the diagonal elements of the representation matrices . using the characters @xmath168}$ ] and the character table for the irreducible representations of o@xmath40,@xcite the rattling motion over the eight off - center positions along the [ 111 ] direction is reduced to the direct sum of @xmath110(1d)@xmath169@xmath33(1d)@xmath169@xmath24(3d)@xmath169@xmath13(3d ) . employing projection operators on appropriate atomic density @xmath166 , we obtain the rattling modes for the irreducible representations as listed in table 1 together with the elastic strains @xmath67 . one can see the presence of @xmath13 rattling mode @xmath170 coupled to the strain @xmath171 , @xmath172 , @xmath112 contrary to the absence of @xmath26 rattling mode coupled to the strain @xmath122 , @xmath125 . this is consistent with the fact that the ultrasonic dispersion reveals in @xmath5 and is absent in @xmath173/2 . the ce atom in the cage of ce@xmath0pd@xmath1ge@xmath2 obeys a harmonic oscillation of @xmath174exp@xmath175 with a mean square displacement @xmath176 . the attempt time @xmath16 sec and the mass @xmath177 where @xmath178 is a proton mass , leads to the mean square displacement @xmath179 being approximately twice of off - center distances @xmath180 as @xmath181 .@xcite the full symmetry @xmath110 rattling mode @xmath182 means the uniform atomic distribution with fraction 1/8 at each eight off - center positions . while the @xmath13 rattling mode , for instance @xmath183 , represents anisotropic atomic distribution being quadrupole @xmath29 at the lowest order such as fraction 1/4 at @xmath184 , @xmath185 , @xmath186 , @xmath187 and zero at @xmath188 , @xmath189 , @xmath190 , @xmath191 as shown in fig . the present group theoretical analysis for the rattling mode is essentially the same treatment previously argued for the charge fluctuation mode.@xcite in the present system ce@xmath0pd@xmath1ge@xmath2 , the thermally activated @xmath13 rattling mode may be a ground state and the @xmath110 , @xmath33 , @xmath24 be excited states . our group has recently found similar ultrasonic dispersion around 30 k in a heavy fermion superconductor pros@xmath51sb@xmath52 with a filled skutterudite structure . it should be noted that ultrasonic dispersion in the @xmath4 mode of pros@xmath51sb@xmath52 indicates the @xmath26 rattling motion of pr atom over six fractionally occupied positions along [ 100 ] . the dispersion of @xmath4 in pros@xmath51sb@xmath52 is contrary to the one of @xmath5 in the present compound of ce@xmath0pd@xmath1ge@xmath2.@xcite the attempted time @xmath192 sec , activation energy @xmath193 k and mean square displacement @xmath194 in pros@xmath51sb@xmath52 are comparable to the present results of ce@xmath0pd@xmath1ge@xmath2 . the thermally activated @xmath13 rattling motion with fractional atomic state in ce@xmath0pd@xmath1ge@xmath2 dies out with decreasing temperature . at further low temperatures , the off - center tunneling state of ce ions in the 4a - site cages will appear , which means a quantum state being occupied four positions , for instance at @xmath184 , @xmath185 , @xmath186 , @xmath187 for the case of @xmath195 , at the same time . the charge glass state in the inhomogeneous mixed valence compound sm@xmath0te@xmath51 revealed the low - temperature softening in elastic constants proportional to ln@xmath117 being resemble the structural glass.@xcite the present scenario of the atomic tunneling of ce ions in 4a - site cages in ce@xmath0pd@xmath1ge@xmath2 is also expected to show the elastic softening at low temperatures . however , the low - temperature quadrupole and magnetic orderings in ce@xmath0pd@xmath1ge@xmath2 may mask the character of the tunneling state in the present case . in order to clarify the tunneling and rattling in the present clathrate compounds , the low - temperature thermodynamic and ultrasonic measurements on la@xmath0pd@xmath1ge@xmath2 without 4f - electron in particular is required . schematic view for the @xmath13 rattling mode @xmath195 due to the off - center ce - ion along the three - fold [ 111 ] direction in the 4a - site cage . the @xmath195 represents that fractional atomic density 1/4 is located at @xmath184 , @xmath185 , @xmath186 , @xmath187 and null at @xmath188 , @xmath189 , @xmath190 , @xmath191 . the freezing of the thermally activated motion of the @xmath13 rattling mode brings about the atomic tunneling state at low temperatures . ] in the present paper we have measured the elastic constants and thermal expansion of ce@xmath0pd@xmath1ge@xmath2 . the characteristic elastic softening in @xmath4 and @xmath5 is well described in terms of the quadrupole susceptibility for the @xmath8 ground state . the important finding is that the @xmath4 shows the huge softening of 50% towards @xmath7 k and the @xmath5 exhibits the softening of 2.5% only . this result strongly indicates the fq ordering with the order parameter of the @xmath26 symmetry in phase ii below @xmath11 . actually we have successfully observed the sharp increase of @xmath9 in length along the [ 001 ] direction below @xmath11 . this is the evidence for the @xmath10-type fq ordering accompanied by the structural change from cubic lattice to tetragonal one at @xmath11 in ce@xmath0pd@xmath1ge@xmath2 . for the investigation of the magnetic phase diagram concerning the fq phase ii and afm phase iii in ce@xmath0pd@xmath1ge@xmath2 , the elastic constants and thermal expansion in fields have been measured . we have found that the boundary from the paramagnetic phase i to the fq phase ii of @xmath10 shifts to higher temperatures with increasing magnetic fields . the result that the i - ii phase transition becomes obscure in fields is similar to the liquid - gas transition approaching to the critical end point under pressure . this result consistent with the fact that the fq order parameter @xmath10 in ce@xmath0pd@xmath1ge@xmath2 has the total symmetry under fields along the [ 001 ] direction.@xcite the boundary from the phase ii to the afm ordering shifts to lower temperatures as similar as the conventional afm ordering . we have found the ultrasonic dispersion in the @xmath5 mode indicating the rattling motion with @xmath13 symmetry . taking into account the absence of the atom along the [ 111 ] direction in 4a - site cage , we have successfully picked up the specific @xmath13 rattling mode @xmath196 , @xmath197 , @xmath195 with the fractional atomic density over the eight minimum positions of potential along the four - fold [ 111 ] , @xmath198 $ ] , @xmath199 $ ] and @xmath200 $ ] directions . the dispersion of the @xmath5 mode obeying the debye - formula revealed the thermally activated - type relaxation time @xmath135exp@xmath15 for the @xmath13 rattling mode with an attempt time @xmath16 sec and an activation energy @xmath17 k. the estimated mean square displacement @xmath201 for the harmonic oscillation of ce atom leads to the distance of the potential minima along the [ 111 ] direction as @xmath202 . in order to confirm the anisotropic atomic distribution in the 4a - site cage , the neutron or x - ray scattering is required . the freezing of the thermally activated motion due to the @xmath13 rattling mode with lowering temperature brings about the atomic tunneling state . by analogy of the charge glass compound sm@xmath0te@xmath51 , the ce - ion tunneling is expected at low temperatures . the ultrasonic investigation on la@xmath0pd@xmath1ge@xmath2 free from the long - range ordering due to 4f - electrons is now in progress to shed light on the tunneling and rattling in cages . 999 d. schmitt , p. morin , and j. pierre , j. magn . magn . matter . @xmath203 , 249 ( 1978 ) . r. takke , n. dolezal , w. assmus , and b. lthi , j. magn . magn . matter . @xmath204 , 247 ( 1981 ) . h. nakao , k. magishi , y. wakabayashi , y. murakami , k. koyama , k. hirota , y. endoh , and s. kunii , j. phys . @xmath205 , 1857 ( 2001 ) . s. nakamura , t. goto , s. kunii , k. iwashita , and a. tamaki , j. phys . @xmath206 , 623 ( 1994 ) . r. shiina , h. shiba , and p. thalmeier , j. phys . @xmath207 , 1741 ( 1997 ) ; r. shiina , o. sakai , h. shiba , and p. thalmeier , j. phys . @xmath208 , 941 ( 1998 ) . j. kitagawa , n. takeda , and m. ishikawa , phys . b @xmath209 , 5101 ( 1996 ) . a. v. grivanov , yu . d. seropegin , and o. i. bodak , j. alloys . @xmath210 , l9 ( 1994 ) . l. keller , a. dnni , m. zolliker , and t. komatsubara , physica b @xmath211-@xmath212 , 336 ( 1999 ) . j. kitagawa , n. takeda , m. ishikawa , t. yoshida , a. ishiguro , n. kimura , and t. komatsubara , phys . b @xmath213 , 7450 ( 1998 ) . o. suzuki , t. horino , y. nemoto , t. goto , a. dnni , t. komatsubara , and m. ishikawa , physica b @xmath211-@xmath212 , 334 ( 1999 ) ; t. goto , t. horino , y. nemoto , t. yamaguchi , a. dnni , o. suzuki , and t. komatsubara , physica b @xmath214-@xmath215 , 492 ( 2002 ) . a. dnni , t. herrmannsdrfer , p. fischer , l. keller , f. fauth , k a mcewen , t. goto , and t. komatsubara , j. phys . : matter @xmath216 , 9441 ( 2000 ) . l. mihaly , nature @xmath217 , 839 ( 1998 ) ; v. keppens , d. mandrus , b.c . sales , b.c . chakoumakos , p. dai , r. coldea , m.b . maple , d.a . gajewski , e.j . freeman , and s. bennington , nature @xmath217 , 876 ( 1998 ) . v. keppens , b.c . sales , d. mandrus , b.c . chakoumakos , and c. laermans , phil . mag . lett . , @xmath218 , 807 ( 2000 ) . t. goto , y. nemoto , k. sakai , t. yamaguchi , m. akatsu , t. yanagisawa , h. hazama , k. onuki , h. sugawara , and h. sato , to be submitted phys . rev . e. kanda , t. goto , h. yamada , s. suto , s. tanaka , t. fujita , and t. fujimura , j. phys . @xmath219 , 175 ( 1985 ) . , h. yamada , s. tanaka , y. kayanuma , t. kojima , j. phys . @xmath219 , 1180 ( 1985 ) . p. thalmeier and b. lthi , handbook on the physics and chemistry of rare earths , ed . k. a. gschneider jr . and l. eyring ( north - holland , 1991 ) vol . t. goto , y. nemoto , y. nakano , s. nakamura , t. kajitani , and s. kunii , physica b @xmath220-@xmath221 , 586 ( 2000 ) ; t. yamaguchi , m. akatsu , y. nakano , t. washizawa , y. nemoto , t. goto , a. dnni , s. nakamura , and s. kunii , physica b , @xmath222-@xmath223 , 622 ( 2003 ) . a. dnni , l. keller , p. fischer , y. aoki , h. sato , f. fauth , m. zolliker , t. komatsubara , and y. endoh , j. phys . : condens . matter @xmath224 , 7219 ( 1998 ) ; n. kimura , n. tateiwa , m. nakayama , h. aoki , t. komatsubara , t. sakon , m. motokawa , y. koike , and n. metoki , physica b @xmath211-@xmath212 , 338 ( 1999 ) . t. herrmannsdrfer , a. dnni , p. fischer , l. keller , e. clementyev , a. furrer , s. mango , b. van den brandt , and h. kitazawa , j. alloys . comp . @xmath225-@xmath226 , 509 ( 2001 ) . t. herrmannsdrfer , a. dnni , p. fischer , l. keller , g. bttger , m. gutmann , h. kitazawa , and j. tang , j. phys . : condens . matter @xmath227 , 2929 ( 1999 ) . t. herrmannsdrfer , a. dnni , p. fischer , l. keller , and h. kitazawa , physica b @xmath220-@xmath221 , 167 ( 2000 ) . o. suzuki , t. goto , s. nakamura , t. matsumura , and s. kunii , j. phys . @xmath208 , 4243 ( 1998 ) ; m. akatsu , t. goto , y. nemoto , o. suzuki , s. nakamura , and s. kunii , j. phys . @xmath228 , 205 ( 2003 ) . y. nemoto , h. aoki , t. goto , a. ochiai , and t. suzuki , physica b @xmath211-@xmath212 , 275 ( 1999 ) . t. goto and b. lthi , advances in phys . @xmath229 , 67 ( 2003 ) . a. tamaki , t , goto , s. kunii , t. suzuki , t. fujimura , and t. kasuya , j. phys . c : solid state phys . @xmath230 , 5849 ( 1985 ) . y. nemoto , t. goto , a. ochiai , and t. suzuki , phys . rev . b @xmath231 , 12050 ( 2000 ) . p. w. anderson , b. i. halperin , and c. m. varma , philos . @xmath232 , 1 ( 1972 ) . see for example ; s. hunklinger and w. arnold , physical acoustics ( academic , new york , 1976 ) , vol . xii , p. 155 . see for example ; amorphous solids : low - temperature properties , edited by w. a. phillips ( springer , berlin , 1981 ) . t. inui , y. tanabe , and y. onodera , group theory and its applications in physics ( springer , berlin , 1990 ) . cox and a. zawadowski , advances in phys . , @xmath233 , 599 ( 1998 ) . t. goto , y. nemoto , a. ochiai , and t. suzuki , phys . b @xmath234 , 269 ( 1999 ) .
|
lattice effects in a cerium based clathrate compound ce@xmath0pd@xmath1ge@xmath2 with a cubic cr@xmath3c@xmath2-type structure have been investigated by ultrasonic and thermal expansion measurements .
elastic softenings of @xmath4 and @xmath5 proportional to the reciprocal temperature @xmath6 above @xmath7 k are well described in terms of the quadrupole susceptibility for the ground state @xmath8 quartet .
a huge softening of 50 % in @xmath4 and a spontaneous expansion @xmath9 along the [ 001 ] direction in particular indicate the ferroquadrupole ordering of @xmath10 below @xmath11 .
the elastic anomalies associated with the antiferromagnetic ordering at @xmath12 k and the incommensurate antiferromagnetic ordering are also found .
notable frequency dependence of @xmath5 around 10 k is accounted for by the debye - type dispersion indicating a @xmath13 rattling motion of an off - center ce ion along the [ 111 ] direction with eight fractionally occupied positions around the 4a site in a cage .
the thermally activated @xmath13 rattling motion obeying a relaxation time @xmath14exp@xmath15 with an attempt time @xmath16 sec and an activation energy @xmath17 k dies out with decreasing temperature , and then the off - center tunneling state of ce ion in the 4a - site cage will appear at low temperatures .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
as presented in ref . @xcite , a periodically - dressed bose - einstein condensate ( pdbec ) is formed of atoms of mass @xmath3 with two internal states , @xmath4 and @xmath5 , separated by an energy difference @xmath6 . these atoms are exposed continuously to two laser beams of wavevectors @xmath7 and @xmath8 and frequencies @xmath9 and @xmath10 , which couple the states @xmath4 and @xmath5 via a raman transition ( see figure [ fig : scheme ] ) . in such a raman transition , an atom in internal state @xmath4 with wavevector @xmath11 absorbs a photon from beam @xmath12 and emits a photon into beam @xmath13 , arriving in internal state @xmath5 with wavevector @xmath14 gaining a momentum @xmath15 and a kinetic energy @xmath16 . and @xmath10 induce raman transitions between internal states @xmath17 and @xmath18 . the beams share a common large detuning @xmath19 from the excited atomic state , and this state is adiabatically eliminated in the theoretical treatment . the lasers are detuned by an amount @xmath20 from the two - photon raman transition . ( b ) such a raman transition imparts a momentum transfer of @xmath21 , where @xmath7 and @xmath8 are the wavevectors of the raman coupling lasers . , width=288 ] keeping in mind this coupling between atoms of different momenta , we let @xmath22 ( @xmath23 ) be the creation ( annihilation ) operator for an atom in internal state @xmath4 and wavevector @xmath11 , and similarly let @xmath24 ( @xmath25 ) be the creation ( annihilation ) operator for an atom in internal state @xmath5 and wavevector @xmath14 . we refer to these as the bare - state creation and annihilation operators . the justification for this particular notation lies in the dressed - state picture @xcite , in which one considers the driving optical fields to be part of the quantum system . in this picture , the raman transition connects the states @xmath26 and @xmath27 , where @xmath28 represents an atom in state @xmath4 and wavevector @xmath11 and an optical field with @xmath29 photons in beam @xmath12 and @xmath30 photons in beam @xmath13 , and similarly for @xmath31 . in the dressed - state picture , @xmath32 and @xmath31 are states for which the total momentum of the atom + photon system is equal by a proper choice of intertial frame , this total momentum is taken to be @xmath33 ) . the states are separated by an energy difference @xmath34 , and coupled by a static and ( in the dressed - state picture ) spatially uniform raman coupling . the system is then represented by a many - body hamiltonian of the form @xmath35\ ] ] here @xmath36 is the two - photon rabi frequency characterizing the strength of the raman transition , determined by the polarizations and intensities of the driving laser beams and by dipole matrix elements of the atoms . this hamiltonian is diagonalized by transforming to the dressed - state creation and annihilation operators @xmath37 , @xmath38 , @xmath39 , @xmath40 which are defined as linear combinations of the bare - state operators @xmath41 the mixing angle @xmath42 is defined by the relation @xmath43 . the dressed states have energies @xmath44 here , the minus sign refers to the lower dressed - state dispersion curve and to the operators @xmath37 and @xmath38 , while the plus sign refers to the upper dressed - state dispersion curve and to the operators @xmath39 and @xmath40 . a bose - einstein condensate formed from this periodically - dressed atomic gas has a macroscopic population of @xmath45 atoms in the lowest energy momentum state : this state lies on the lower dressed - state dispersion curve and is taken to have momentum @xmath46 @xcite . in ref.@xcite , a bogoliubov - approximation theory was developed to account for the effects of weak interatomic interactions on a pdbec . an interaction hamiltonian of the form @xmath47 was considered , where @xmath48 is the total number of atoms in the system and @xmath49 is the fourier transform of the density operator . quasi - particle energies and operators were found by diagonalizing a @xmath50 matrix @xmath51 where @xmath52 is a diagonal matrix with entries @xmath53 with @xmath54 , @xmath55 is the chemical potential , and @xmath56 , where @xmath57 . diagonalization yields @xmath58 where @xmath59 is a diagonal matrix with entries @xmath60 by which the lower ( minus sign ) and upper ( plus sign ) quasi - particle energies @xmath61 are defined . one also obtains the creation and annihilation operators for the lower ( @xmath62 , @xmath63 ) and upper ( @xmath64 , @xmath65 ) branch quasi - particles through the following relation : @xmath66 here @xmath67 contains trigonometric functions of @xmath68 and @xmath69 in accordance with eq.[eq : rotation ] . the bose commutation relations take the form @xmath70 = a_{i j } = \left\ { \begin{array}{c l } 0 & i \neq j \\ 1 & i = j = 1 \mbox { or } i = j = 3 \\ -1 & i = j = 2 \mbox { or } i = j = 4 \end{array } \right.\ ] ] where @xmath71 represents any of the four - component vectors in eq.[eq : baretoquasi ] . these relations are clearly preserved by the rotation matrix , i.e. @xmath72 , while the imposition of bose commutation relations for the quasi - particle operators enforces a normalization criterion @xmath73 . experimentally , values for the raman detuning @xmath20 , the rabi frequency @xmath36 , the chemical potential @xmath74 and the raman momentum transfer @xmath75 can be chosen over a wide range of parameters . for example , let us consider a particular realization of a pdbec using two hyperfine states of @xmath1rb . one may consider the case of raman excitation using two lasers both tuned near the @xmath76 optical transition at a wavelength @xmath77 nm . thus , the raman recoil energy @xmath78 can be readily tuned in the range @xmath79 khz . properties of a pdbec are governed by the ratios @xmath80 , @xmath81 and @xmath82 . the raman detuning @xmath20 and the rabi frequency @xmath36 can be chosen nearly arbitrarily , reaching maximum values which are orders of magnitude larger than @xmath83 . the chemical potential @xmath84 has a typical value of 1 khz for a density of @xmath85 and a scattering length @xmath86 bohr thus the ratio @xmath87 can be as small as @xmath88 for counter - propagating raman excitation lasers , and greatly increased by using raman beams intersecting at small angles . for illustration , we show in figure [ fig : quasidisprelations ] the quasiparticle dispersion relations for a pdbec with a chemical potential @xmath89 , a raman detuning of of @xmath90 and raman coupling strength @xmath91 varying between 0 and 2 . for this case of @xmath92 , a condensate is formed in the @xmath93 state of the lower dispersion relation . in the absence of raman coupling , this equality is exact , and the condensate is formed in the @xmath94 internal state . quasiparticle excitations in the @xmath94 state follow the standard bogoliubov result , while quasiparticles in state @xmath95 have a free - particle dispersion relation this can be though of as due to the lack of a spatial interference term between the condensate and excitations in the @xmath95 state at any wavevector . for small rabi coupling strengths , one speaks more properly of excitations in the lower and the upper quasiparticle dispersion relations , although the energies and internal - state compositions of these states remain relatively unchanged for wavevectors @xmath96 far from the doppler - shifted raman resonance condition @xmath97 . the lower quasiparticle dispersion relation has a local minimum near @xmath98 caused by the minimum of the dispersion relation for the bare @xmath95 state atoms . by analogy to the roton minimum in the excitation spectrum of superfluid @xmath0he @xcite , we denote this feature as an `` artificial roton , '' although we mean to suggest no similarity between the specific structure of a roton and of the plane - wave excitations of a pdbec . as the rabi frequency is increased further , a greater mixing of the internal state composition of the bose condensate and of the quasiparticle excitations occurs , and energy levels are shifted . in particular , the energy of the `` artificial roton '' is shifted upwards due to interactions as the internal state composition of the bose condensate and of these excitations becomes more similar . ) and upper ( created by @xmath99 ) dispersion curves with momentum @xmath100 with respect to the condensate . excitations parallel to the raman momentum transfer are considered , with the definition @xmath101 . black curves represent pdbec s with @xmath102 , @xmath103 and @xmath104 ( dashed ) , 1 ( dotted ) and 2 ( solid ) . the gray dot - dashed curve shows the bogoliubov excitation spectrum for a single component condensate with @xmath103 . , width=288 ] we now consider the scattering of impurities passing through a uniform pdbec . we consider impurities of mass @xmath105 which have a free - particle dispersion relation @xmath106 where @xmath107 is the impurity wavevector . such impurities interact with the two components of the periodically - dressed condensate through an interaction of the form @xmath108 here @xmath22 , @xmath23 , @xmath24 , and @xmath25 are defined as above , and @xmath109 and @xmath110 are the creation and annihilation operators , respectively , for the impurity particles . the coupling strength for the impurity scattering is @xmath111 , which we parameterize as @xmath112 . this parameterization correctly describes low - energy @xmath113-wave scattering with a scattering length of @xmath114 and the reduced mass @xmath115 . note that we have made the assumption that the scattering strengths between the impurity and atoms in states @xmath4 and @xmath5 are equal , and assume further that state changing collisions do not occur ( the impurity is considered `` non - magnetic '' ) . we may consider also the limit as the impurity mass becomes infinite while the impurity velocity @xmath116 remains constant ( @xmath117 is the impurity wavevector ) . we thereby obtain a description of the pdbec flowing past a rigid obstacle with a velocity @xmath118 , and the concept of impurity scattering is replaced by the dissipation of superfluid flow . the scattering rate of these impurities can be calculated using a perturbative approach as follows @xcite . we consider the system to be initially in the state latexmath:[$ |0\rangle_{ab } impurity particle of wavevector @xmath117 . the scattering hamiltonian of eq . [ eq : hscat ] may couple this state to the final state @xmath120 in which the periodically - dressed gas is in the state @xmath121 and the impurity wavevector changes from @xmath117 to @xmath122 . the matrix element for this transition is @xmath123 , where @xmath124 is described above , and where the @xmath125 subscript labeling the quantum states of the periodically - dressed gas has been dropped . we then obtain the rate of impurity scattering by summing over all final states @xmath121 and momentum transfers @xmath100 using fermi s golden rule . we now make use of the bogoliubov approximation . if we consider only weak scattering , we may restrict the final states to those containing a single quasiparticle , i.e. @xmath126 and @xmath127 . the rate of impurity scattering is then @xmath128 \label{eq : gammaeq}\ ] ] where @xmath129 is the impurity velocity . we have introduced the quantities @xmath130 and @xmath131 which relate to the static structure factor describing density - density correlations in the bose condensate . for a single - component bose - einstein condensate , the static structure factor @xmath132 has been evaluated and measured @xcite . for the pdbec , we separate the contributions of the lower and upper excitation spectra as @xmath133 physically , these quantities describe two effects . first , the magnitude of @xmath134 and @xmath135 describes the contributions of quasiparticles in the lower ( @xmath74 ) or upper ( @xmath136 ) quasiparticle state to density fluctuations ( as opposed to phase fluctuations , which dominate , for example , for phonon excitations of a scalar bose condensate @xcite ) . second , these quantities describe the degree to which the internal state composition of the condensate matches that of the quasiparticle . for instance , a condensate which is predominantly in the @xmath94 internal state will not scatter strongly into quasiparticle states in which the @xmath95 internal state is dominant , and thus the structure factor describing this scattering will be small . before proceeding to a calculation of @xmath134 and @xmath135 , it is useful to consider the scattering rate @xmath137 for an ideal - gas , single - component bose - einstein condensate as the target gas . in this case , we may conveniently take @xmath138 , @xmath139 and @xmath140 . replacing @xmath141 , and substituting for @xmath111 we evaluate eq . [ eq : gammaeq ] and obtain simply @xmath142 i.e. the conventional expression for the scattering rate where @xmath143 is the density of the target gas and @xmath144 is the collision cross section . we may then determine the impurity scattering rate in a pdbec relative to that in an ideal gas , @xmath145 : @xmath146 to calculate the structure factors @xmath134 and @xmath135 we start by expressing the density operator @xmath124 in terms of the quasiparticle operators @xmath147 acting on a pdbec , the only important terms in the sum are those which include the condensate operators @xmath148 and @xmath149 . applying the relations @xmath150 for @xmath151 , and the relation @xmath152 , we define matrices @xmath153 and @xmath154 and obtain @xmath155 so @xmath156 the static structure factors @xmath157 and @xmath131 for a pdbec are shown in figures [ fig : smuandspivaryrabi ] and [ fig : smuandspivarymu ] . the specific values for the raman detuning @xmath20 , the rabi frequency @xmath36 and the chemical potential @xmath74 are relevant for a particular realization of a pdbec using two hyperfine states of a @xmath1rb , as discussed above . the division of the scattering weight between the lower ( @xmath134 ) and upper ( @xmath135 ) quasiparticle branches can be understood by considering the composition of the dressed - state quasiparticles , which is related by eq . [ eq : rotation ] to the mixing angle @xmath158 . this angle ranges from @xmath159 for @xmath160 to @xmath161 for @xmath162 , reaching @xmath163 at the doppler - shifted raman resonance @xmath164 . for small rabi frequency , @xmath158 , and hence the internal state composition of the dressed states varies rapidly about the raman resonance condition , while for large rabi frequency , this variation occurs over a broader range of @xmath165 . this behaviour is reflected in the structure factors , as illustrated by figure [ fig : smuandspivaryrabi ] . by continuity , the internal state composition of the condensate in the @xmath166 state is nearly identical to that of excitations in the lower quasiparticle branch ( created by @xmath167 ) , and nearly orthogonal to that of excitations in the upper quasiparticle branch ( created by @xmath168 ) . thus , since elastic scattering from a `` non - magnetic '' impurity does not affect the internal state distribution , @xmath169 generally for small @xmath165 . away from @xmath170 , the division of the scattering weight between the upper and lower quasiparticle branches depends on the strength of the rabi coupling . for small rabi frequencies , the scattering strength shifts abruptly from the lower to the upper quasiparticle branch at the doppler - shifted raman resonance . for large rabi frequency , the changes in the scattering strengths are more gradual , with scattering into both branches showing a significant weight for a wide range about the raman resonance condition . this variation is particularly relevant for the scattering of impurities near the landau critical velocity , as discussed below . and ( b ) @xmath131 with the raman coupling strength @xmath36 . the wavevector @xmath100 is colinear with the raman momentum transfer @xmath117 with @xmath171 . three black curves describe a pdbec with @xmath172 , @xmath89 , and @xmath173 ( solid ) , @xmath174 ( dashed ) and @xmath12 ( dotted ) . scattering for @xmath175 and @xmath176 occurs only into the lower quasiparticle branch and for @xmath177 into the upper quasiparticle branch . in other regions , stronger raman coupling leads to eigenstates with mixed populations in the internal states @xmath95 and @xmath94 over a broader range of @xmath165 , and thus impurity scattering occurs into both the lower and upper quasiparticle branches . also shown is the structure factor @xmath178 for a scalar bose - einstein condensate ( gray dot - dashed curve ) with @xmath89 . , width=480 ] and ( b ) @xmath131 for a pdbec . the wavevector @xmath100 is colinear with the raman momentum transfer @xmath117 with @xmath171 . three black curves describe a pdbec with @xmath172 , @xmath179 , and @xmath180 ( solid ) , @xmath174 ( dashed ) and @xmath12 ( dotted ) . the condensate in the @xmath166 state is predominantly in the internal state @xmath5 ; thus scattering predominantly occurs into the lower quasiparticle branch for @xmath176 and into the upper quasi - particle branch for large @xmath181 . for @xmath182 both @xmath157 and @xmath131 are significant since quasiparticles contain large fractions of both @xmath95 and @xmath94 . interactions suppress scattering at small momentum transfer , an effect seen also in the structure factor @xmath178 for a scalar bose - einstein condensate ( gray dot - dashed curve ) with @xmath89 . , width=480 ] as illustrated in figure [ fig : smuandspivarymu ] , interatomic interactions suppress scattering at small momentum transfers . for a single - component homogeneous bose - einstein condensate , the structure factor @xmath183 is small for wavevectors of magnitude @xmath184 where @xmath185 is the healing length . similarly , for the pdbec , we find a suppression of the structure factors @xmath157 and @xmath131 for @xmath186 , as shown in the figure . this suppression of the structure factor in a pdbec could be probed experimentally as it was for a scalar bose - einstein condensate by studying shallow - angle optical bragg scattering @xcite . finally , let us use the formalism developed above to describe superfluid properties of a pdbec . in ref . @xcite , we applied the landau criterion to determine the critical velocity for superfluidity in a pdbec given the quasiparticle dispersion relations . this criterion gives the superfluid critical velocity along a direction @xmath187 as having a magnitude @xmath188 where @xmath189 are the energies of possible excitations with wavevector @xmath96 ; using the momentum definitions for a pdbec , these would be the quasiparticle energies @xmath190 . in an unperturbed , single component weakly - interacting condensate , this critical velocity @xmath191 is equal to the bogoliubov speed of sound @xmath192 in all directions . in a pdbec , the anisotropic quasiparticle dispersion relation leads directly to the prediction of an anisotropic superfluid critical velocity , and to the possibility of a critical velocity lower than that of the unperturbed condensate . one may determine this critical velocity graphically by finding the line of minimum absolute slope which connects the origin and the quasiparticle dispersion curve ( see figure [ fig : quasidisprelations ] ) . for the pdbec , the superfluid critical velocity may be suppressed by the presence of the `` artificial roton '' feature in the lower dispersion curve . this suppression occurs in the direction of the raman momentum transfer @xmath193 for the case of @xmath92 ( as shown in the figure ) , or in the opposite direction for @xmath194 . however , while the landau criterion determines the minimum velocity for the dissipation of superfluid flow , it does not quantify the onset of such dissipation as the velocity is increased beyond the critical value . indeed , for extremely weak raman coupling , even though the superfluid critical velocity is strictly different , one should expect the fluid properties of a pdbec to be nearly identical to that of a two - component bec in the complete absence of raman coupling . we can quantitatively assess this description using the formalism developed above for the determination of the matrix elements @xmath134 and @xmath134 , and therefrom the normalized scattering rates @xmath195 and @xmath196 . the results of such calculations for infinitely massive impurities ( i.e. superfluid flow about rigid obstacles ) are presented in figures [ fig : allpdplot ] and [ fig : onepdplot ] . values of @xmath195 and @xmath196 were determined by numerical integration in which the delta function in eqs . [ eq : fmuexpression ] and [ eq : fpiexpression ] was approximated by a narrow gaussian distribution whose width was adjusted to give reasonable numerical convergence . as we expect , for small rabi frequencies @xmath36 the properties of a periodically - dressed bec are quite similar to those of an uncoupled two component bose - einstein condensate . the scattering probability for impurity atoms ( related to the viscosity of the fluid ) is indeed non - zero at velocities lower than the superfluid critical velocity for the unperturbed gas , demonstrating that the critical velocity is suppressed along the direction of raman scattering . this scattering at low velocities occurs through the excitation of quasiparticles near the local minimum ( the `` artificial roton '' ) in the lower dispersion relation . however , this scattering channel is weak for small values of the rabi frequency @xmath36 , and thus the scattering strength @xmath197 remains weak until the curve joins that of the uncoupled condensate and the scattering strength is increased by phonon scattering . the effects of increasing the rabi frequency are twofold . first , as @xmath36 increases , the internal state composition of the `` artificial roton '' excitation becomes more similar to the internal state composition of the bose - einstein condensate . this causes an increase in the scattering strength ( as indicated also in the calculations of @xmath134 shown in figures [ fig : smuandspivarymu ] and [ fig : smuandspivaryrabi ] ) . simultaneously , because of this overlap of the internal state composition , the energy of these excitations is increased , approaching the energy given by the one - component bogoliubov spectrum , albeit with an appropriately modified effective mass . thus , the minimum velocity at which scattering occurs increases as the rabi frequency increases . at moderate values of the rabi frequency ( where @xmath198 ) , the scattering rate vs. velocity has a two - step shape as scattering is first enhanced by strong scattering into the `` artificial roton '' state , and then later enhanced by phonon scattering when the velocity is near @xmath191 . as shown in figure [ fig : onepdplot ] for one typical parameter setting , only @xmath195 contributes to the weak scattering at velocities smaller than @xmath191 . at velocities greater than @xmath191 , scattering into the upper dispersion curve is allowed and @xmath199 . for infinitely massive impurities traveling through the condensate , normalized to the scattering in an ideal gas bose - condensate at similar densities , is plotted vs. the velocity of the impurity . here we consider velocities parallel to the raman momentum transfer , with @xmath200 . black curves show results for pdbec s with varying strengths of the raman coupling ; for all curves , @xmath201 and @xmath89 while @xmath202 ( dashed ) , 1 ( solid ) or 2 ( dotted ) . the gray dot - dashed curve shows results for a one - component condensate with @xmath89 . the superfluid critical velocity of the pdbec in the direction of the raman momentum transfer is suppressed with respect to an uncoupled condensate . for weakly raman coupling , the dissipation at low velocities is weak . this dissipation grows stronger as the raman coupling strength is increased . , width=288 ] , dashed line ) and upper ( @xmath196 , dotted line ) quasiparticle excitations due to scattering off infinitely massive impurities moving at velocities @xmath203 in a pdbec . velocities parallel to the raman momentum transfer are considered , with @xmath200 . black curves show results for a pdbec with @xmath102 , @xmath89 , and @xmath204 , with the solid curve showing the total normalized scattering rate @xmath197 . the gray dot - dashed curve shows the normalized scattering rate for a one - component condensate with @xmath89 . for @xmath205 ( for this case where @xmath206 ) , impurity scattering at low velocities is enhanced as compared to the one - component condensate . this scattering leads to the creation of excitations in the lower dispersion curve ( @xmath207 ) around the `` artificial roton '' feature in the quasiparticle dispersion relations . at velocities near the critical velocity for the unperturbed condensate , @xmath191 , scattering is allowed into the upper dispersion relation @xmath208 , and the normalized scattering rate approaches that of a one - component condensate . for @xmath209 , energy and momentum conservation imply that only scattering into the lower quasiparticle excitations is allowed at the velocities considered . , width=288 ] this two - step curve gives a clear signature which should be visible in experimental probes of superfluid flow . indeed , experiments such as those performed by chikkatur _ _ e__t al @xcite , in which atoms of another ground state hyperfine level were used as scattering impurities in a dilute atomic bose condensate , could be carried out in a straightforward manner . it would be interesting to consider further how other aspects of superfluidity , such as the onset of turbulent flow @xcite or the metastability of quantized vortices @xcite , are affected by the anisotropic nature of a periodically - dressed bose - einstein condensate . in conclusion , dissipation of superfluid motion in a periodically - dressed bose - einstein condensate above the superfluid critical velocity was treated . a formalism was developed for the calculation of the static structure factors @xmath210 and @xmath211 describing density - density correlations from quasiparticles in the lower ( @xmath74 ) and upper ( @xmath136 ) excitation branches of the periodically coupled gas . following a perturbation theory approach , the scattering rate of massive impurities , or equivalently the damping rate of superfluid motion past microscopic obstacles , was calculated . raman coupling of a two - component condensed gas leads to a suppression of the superfluid critical velocity . the impurity scattering rate evolves smoothly as the raman coupling strength is increased . the the superfluid damping is weak for weak raman coupling , and becomes more significant as the raman coupling strength is increased . cornell , j.r . ensher , and c.e . wieman , in _ bose - einstein condensation in atomic gases _ , _ proceedings of the international school of physics `` enrico fermi , '' course cxl _ , edited by m. inguscio , s. stringari , and c.e . wieman ( ios press , amsterdam , 1999 ) . we do not consider here the possibility of a degenerate or near - degenerate ground state . proper treatment of a periodically - dressed bose - einstein condensate for such a situation requires a many - body treatment beyond the bogoliubov approximation , as will be discussed elsewhere .
|
the introduction of a steady - state spatially - periodic raman coupling between two components of an ultracold atomic gas produces a dressed - state gas with an anisotropic and tunable dispersion relation . a bose - einstein condensate formed in such a gas is consequently characterized by an anisotropic superfluid critical velocity . the anisotropic dissipation of superfluid flow is quantified by considering the scattering of impurities flowing through this superfluid
a gradual transition from the isotropic nature of an uncoupled bose - einstein condensate to the anisotropic periodically - dressed condensate is obtained as the strength of the raman coupling is varied .
these results present a clear signature for future experiemental realizations of this novel superfluid .
the experimental attainment of quantum degenerate dilute gases , composed of both bosonic and fermionic atoms , has created many new opportunities in the study of quantum fluids . in particular , recent years have seen a flurry of experimental studies of superfluidity in dilute , scalar bose - einstein condensates .
these include the observations of critical velocities for superfluid flow about microscopic @xcite and macroscopic @xcite obstacles , the onset of turbulent flow above this critical velocity @xcite , beautiful studies of quantized vortices and vortex lattices in rotating bose - einstein condensates @xcite , and other manifestations of irrotational superfluid flow @xcite .
these experiments are closely analogous to those performed on another scalar superfluid , liquid @xmath0he , but apply new experimental probes and allow new insights due to the vastly different parameter regime of the dilute atomic gases , and their amenability to a new set of tools for manipulation and probing .
dilute atomic gases also offer the opportunity to create novel quantum fluids by using these various tools to manipulate the internal and external states of the atoms comprising the superfluid .
for example , different types of multi - component condensates have been studied : externally - coupled two - component condensates of @xmath1rb at jila ( reviewed in @xcite ) , and @xmath2 spinor condensates of sodium at mit ( reviewed in @xcite ) .
one experiment at jila explored how a spatially - selective coupling between two trapped hyperfine levels of a rubidium condensate leads to a `` winding '' of the order parameter , akin to the phase winding of the order parameter which occurs due to rotation about a vortex core , and how this winding is recurrently `` undone '' due to the motional dynamics of a two - component condensate @xcite .
this observation illustrates that vortices in such a novel system should not be metastable ( similarly predicted for spinor condensates @xcite ) , a major modification from the superfluid behaviour of a single component gas . in a similar vein
, we have previously considered the novel superfluid properties introduced to a two - component bose condensed atomic gas which is coupled by a spatially periodic coupling field @xcite .
this spatially - periodic coupling results from intersecting , non - collinear laser beams which induce a raman coupling between two internal states of the ultracold gas .
the raman coupling introduces an anisotropy to this periodically - dressed fluid which should manifest itself in an anisotropic superfluid critical velocity .
controlling various parameters of the raman coupling laser beams allows one to dramatically alter this critical velocity , leading to the dissipation of superfluid flow at velocities much lower than the critical velocity for the uncoupled bose condensate . in this paper
, we develop further a theory describing the dissipation of superfluid flow in a periodically - dressed bose - einstein condensate . using the bogolibov approximation theory which was introduced in ref .
@xcite , the possible disspation of superfluid flow is treated by analyzing the scattering of massive impurities which flow through the superfluid . in analogy with similar calculations for a scalar bose - einstein condensate @xcite
, impurity scattering at low impurity velocities is found to be suppressed by the energetics of the quasi - particle spectrum ( related to the landau criterion for superfluidity @xcite ) , by the suppression of density fluctuations in the low - momentum phonon regime @xcite , and by variations of the internal - state compositions of quasiparticles at different wavevectors .
these effects combine to give a smooth evolution in the disspation rates for superfluid flow as the raman coupling strength is increased , gradually converting an uncoupled two component bose - einstein condensate ( essentially a scalar superfluid ) to a periodically - dressed condensate with dramatically different superfluid properties .
this superfluid dissipation rate gives an appealingly clear signature for future experimental studies .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
since its discovery the josephson effect @xcite has been studied for a variety of superconducting weak links @xcite . the research has recently entered a new phase with the experimental realization of quantum dot weak links exploiting electronic properties of finite - length carbon nanotubes coupled to superconducting leads @xcite . in particular , for the first time since its theoretical prediction @xcite resonant josephson tunneling through discrete electronic states has been observed in carbon nanotube quantum dots @xcite . as demonstrated in refs . , the novel type of weak links exhibits transistor - like functionalities , e.g. a periodic modulation of the critical current with a gate voltage tuning successive energy levels in the dot on- and off - resonance with the fermi energy in the leads . this property has already been implemented in a recently proposed carbon nanotube superconducting quantum interference device ( cnt - squid ) @xcite with possible applications in the field of molecular magnetism . motivated by the experiments on resonant josephson tunneling , in this paper we investigate theoretically how robust it is with respect to pair - breaking perturbations in the superconducting leads . cooper pair breaking can be induced by a number of factors , e.g. by paramagnetic impurities @xcite , an external magnetic field @xcite or by structural inhomogeneities producing spatial fluctuations of the superconducting coupling constant @xcite . it can cause a drastic distortion of the bardeen - cooper - schrieffer ( bcs ) superconducting state , which manifests itself in the smearing of the bsc density of states leading to gapless superconductivity @xcite . while the pair - breaking effect on bulk superconductivity is now well understood , its implications for quantum superconducting transport have been studied to a much lesser extent [ see , e.g. refs . ] which to our knowledge does not cover josephson tunneling through quantum dots . on the other hand , in low - dimensional systems pair - breaking effects may be observable in a common experimental situation when , for instance , a carbon nanotube weak link is subject to a magnetic field . since the orbital field effect in the quasi - one - dimensional channel is strongly suppressed , pair breaking in the superconducting leads can be the main source of the magnetic field dependence of the josephson current . this situation is addressed in our work . = 0.5 the influence of pair breaking on the josephson current can not , in general , be accounted for by mere suppression of the order parameter in the superconducting leads . as was pointed out in ref . , it is a more subtle effect involving the modification of the spectrum of current carrying states in the junction , in particular , the subgap states usually referred to as andreev bound states ( abs ) @xcite . we illustrate this idea for quantum dot junctions in the simple model of a short superconducting constriction with a scattering region containing a single breit - wigner resonance near the fermi energy . the josephson current is calculated using the normal - state scattering matrix of the system and the andreev reflection matrix @xcite . unlike refs . we focus on dirty superconductors for which the andreev matrix can be quite generally expressed in terms of the quasiclassical green functions @xcite , allowing us to treat pair breaking in the superconducting leads nonperturbatively . although we account for all energies ( below and above the abrikosov - gorkov gap @xmath2 ) , it turns out that the behavior of the josephson current can be well understood in terms of a pair - breaking - induced modification of the abs , which depends sensitively on the relation between the breit - wigner resonance width @xmath3 and the superconducting pairing energy @xmath4 . both the critical supercurrent and the josephson current - phase relation are analyzed under experimentally realizable conditions . we consider a junction between two superconductors @xmath5 and @xmath6 adiabatically narrowing into quasi - one - dimensional ballistic wires @xmath7 and @xmath8 coupled to a normal conductor @xmath9 [ fig . [ cons ] ] . the transformation from the superconducting electron spectrum to the normal - metal one is assumed to take place at the boundaries @xmath10 and @xmath11 , implying the pairing potential of the form @xcite : @xmath12 for @xmath13 , @xmath14 for @xmath15 and @xmath16 for @xmath17 with the order parameter phase difference @xmath18 and the junction length @xmath19 ( @xmath20 is the fermi velocity in @xmath21 ) . the josephson coupling can be interpreted in terms of the andreev process @xcite whereby an electron is retro - reflected as a fermi - sea hole from one of the superconductors with the subsequent hole - to - electron conversion in the other one . such an andreev reflection circle facilitates a cooper pair transfer between @xmath5 and @xmath6 . normal backscattering from disordered superconducting bulk into a single - channel junction is suppressed due to the smallness of the junction width compared to the elastic mean free path @xmath22 . the @xmath9 region in the middle of the junction is thus supposed to be the only source of normal scattering . in such type of weak links the josephson current is conveniently described by the scattering matrix expression of refs . that can be written at finite temperature @xmath23 as the following sum over the matsubara frequencies @xmath24 [ ref . ] : @xmath25_{e = i\omega_n . } \label{i}\end{aligned}\ ] ] here @xmath26 is a @xmath27 unitary matrix relating the incident electron and hole waves on the @xmath9 region to the outgoing ones [ fig . [ cons ] ] . it is diagonal in the electron - hole space : @xmath28 , \ , s_{ee}(e)= \left [ \begin{array}{cc } r_{11}(e ) & t_{12}(e)\\ t_{21}(e ) & r_{22}(e ) \end{array } \right ] . \nonumber\end{aligned}\ ] ] the matrix @xmath29 describes electron scattering in terms of the reflection and transmission amplitudes , @xmath30 and @xmath31 , for a transition from @xmath32 to @xmath33 ( @xmath34 ) . the hole scattering matrix is related to the electron one by @xmath35 . the andreev scattering matrix @xmath36 is off - diagonal in the electron - hole space : @xmath37 , \label{a}\end{aligned}\ ] ] where the @xmath38 matrices @xmath39 and @xmath40 govern the electron - to - hole and hole - to - electron scattering off the superconductors . equation ( [ i ] ) is valid for all energies as long as normal scattering from the superconductors is absent @xcite . in ref . the andreev matrix ( [ a ] ) was obtained by matching the solutions of the bogolubov - de gennes equations in the wires @xmath41 to the corresponding solutions in impurity - free leads . gorkov s green function formalism in combination with the quasiclassical theory @xcite allows one to generalize the results of ref . to dirty leads with a short mean free path @xmath42 . in the latter case the matrices @xmath39 and @xmath40 can be expressed in terms of the quasiclassical green functions of the superconductors as follows @xcite : @xmath43 , \ , s_{he}= \left [ \begin{array}{cc } \frac{-f^\dagger_1(e)}{g_1(e)+1 } & 0\\ 0 & \frac{-f^\dagger_2(e)}{g_2(e)+1 } \end{array } \right].\end{aligned}\ ] ] here @xmath44 and @xmath45 ( @xmath46 ) are , respectively , the normal and anomalous retarded green functions in @xmath21 . these matrices are diagonal in the electrode space due to a local character of andreev reflection in our geometry . neglecting the influence of the narrow weak link on the bulk superconductivity , we can use the green functions of the uncoupled superconductors @xmath21 described by the position - independent usadel equation @xcite , @xmath47=0 , \label{usadel}\end{aligned}\ ] ] with the normalization condition @xmath48 for the matrix green function @xmath49 , \,\ , \hat\delta_j=\left [ \begin{array}{cc } 0 & \delta { \rm e}^{i\varphi_j}\\ -\delta { \rm e}^{-i\varphi_j } & 0 \end{array } \right ] , \,\ , \end{aligned}\ ] ] here @xmath50 and @xmath51 are the unity and pauli matrices , respectively , and @xmath52 $ ] denotes a commutator . equation ( [ usadel ] ) accounts for a finite pair - breaking rate @xmath53 whose microscopic expression depends on the nature of the pair - breaking mechanism . for instance , for thin superconducting films in a parallel magnetic field , @xmath54 [ ref . ] where @xmath55 is the film thickness and @xmath56 is the flux quantum . for paramagnetic impurities , @xmath57 coincides with the spin - flip time @xcite . in the case of the spatial fluctuations of the superconducting coupling , @xmath53 is proportional to the variance of the fluctuations @xcite . from eq . ( [ usadel ] ) one obtains the green functions @xmath58 where , following refs . , we introduce a dimensionless pair - breaking parameter @xmath59 . the matrices @xmath60 and @xmath61 can be expressed using eqs . ( [ green ] ) as follows : @xmath62 , \,\ , s_{he}=\alpha \left [ \begin{array}{cc } { \rm e}^{-i\varphi_1 } & 0\\ 0 & { \rm e}^{-i\varphi_2 } \end{array } \right ] , & \label{sa}\\ & \alpha = u-\sqrt{u^2 - 1}. & \label{a}\end{aligned}\ ] ] we note that pair breaking modifies the energy dependence of the andreev reflection amplitude @xmath63 according to the non - bcs green functions ( [ green ] ) and ( [ u ] ) . a few words concerning the applicability of this result are due here . first of all , there is no restriction on energy @xmath64 , e.g. for @xmath65 , equations ( [ sa ] ) and ( [ a ] ) are valid both below and above the reduced ( abrikosov - gorkov ) quasiparticle gap @xmath66 . in particular , for @xmath67 one can show that @xmath68 is real and @xmath69 [ ref . ] , corresponding to perfect andreev reflection with @xmath70 . since in the usadel limit @xmath71 , normal scattering from the superconductors is suppressed due to the smallness of the junction width also in the presence of pair breaking . the absence of normal transmission at @xmath67 is consistent with the abrikosov - gorkov approach assuming no impurity states inside the gap and the validity of the born approximation @xcite . for @xmath72 the relevant solution of eq . ( [ u ] ) is complex and has positive @xmath73 related to the density of states of the superconductor @xcite . equations ( [ u ] ) and ( [ a ] ) are thus the generalization of the known result @xmath74 [ ref . ] for transparent point contact , where @xmath75 is the bcs gap . it is convenient to measure all energies in units of @xmath76 for which equations ( [ green])([a ] ) should be complemented with the self - consistency equation for @xmath4 . at @xmath77 , the case we are eventually interested in , this equation can be written as @xcite : @xmath78 with @xmath59 being now a function of a new pair - breaking parameter @xmath79 ranging from zero to the critical value @xmath80 at which @xmath81 and @xmath82 @xcite . inserting eqs . ( [ a ] ) and ( [ sa ] ) for @xmath36 into eq . ( [ i ] ) and taking the limit @xmath83 we obtain the josephson current for an arbitrary @xmath26 as @xmath84 \right\}_{e = i\omega . } \nonumber\end{aligned}\ ] ] = 0.8 let us assume that the @xmath9 region is a small quantum dot and electrons can only tunnel via one of its levels characterized by its position @xmath85 with respect to the fermi level and broadening @xmath3 . for the simplest breit - wigner scattering matrix with @xmath86 and @xmath87 , equation ( [ i0 ] ) reads @xmath88 where @xmath89 is the breit - wigner transmission probability at the fermi level . the parameter @xmath90 accounts for the energy dependence of the resonant superconducting tunneling . in eq . ( [ i1 ] ) the integrand has , in general , poles given by the equation @xmath91 along with eq . ( [ u ] ) they determine the energies of the andreev bound states ( abs ) localized below the abrikosov - gorkov gap @xmath66 . it is instructive to understand how the pair breaking modifies the abs spectrum since this is reflected on both the current - phase relation @xmath1 and the critical current @xmath92 . we start our analysis with an analytically accessible case of an infinitely broad resonant level , @xmath93 , where eq . ( [ eq ] ) reduces to @xmath94 , yielding the abs energies @xmath95 [ see , eq . ( [ u ] ) ] : @xmath96 . \label{broad}\end{aligned}\ ] ] requiring @xmath97 we find that the abs exist in the phase interval where @xmath98 and only if @xmath99 . the numerical solution of eqs . ( [ u ] ) , ( [ op1 ] ) and ( [ eq ] ) confirms that the interval of the existence of abs gradually shrinks from @xmath100 to a narrower one with increasing pair breaking [ see , fig . [ e15 ] ] . outside this interval the josephson current is carried by the continuum states ( @xmath101 ) alone , which is automatically accounted for by eq . ( [ i1 ] ) . an equation of the same form as eq . ( [ broad ] ) was derived earlier for a nonresonant system and by a different method @xcite . = 0.8 by contrast , the abs spectrum for a narrow resonant level turns out to be much less sensitive to pair breaking . indeed , under condition @xmath102 equations ( [ u ] ) and ( [ eq ] ) reproduce the known result , @xmath103 [ refs . ] . in particular , for @xmath104 the abs exist within the resonance width , @xmath105 and are separated from the continuum by a gap @xmath106 . solving eqs . ( [ u ] ) , ( [ op1 ] ) and ( [ eq ] ) numerically , we find that until this gap closes at a certain value of @xmath107 , the abs spectrum remains virtually intact [ see , fig . [ e03 ] ] . for bigger @xmath107 , the spectrum gets modified in a way similar to the previous case [ cf . , third panels in figs . [ e15 ] and [ e03 ] ] . in the case of a very narrow resonance , the characteristic value of @xmath107 is @xmath108 , corresponding to @xmath109 , i.e. to the onset of gapless superconductivity @xcite . for numerical evaluation of the josephson current ( [ i1 ] ) we first put @xmath110 in eq . ( [ u ] ) and then make the transformation @xmath111 , yielding @xmath112 . using this relation , in eq . ( [ i1 ] ) we change to the integration over @xmath113 with the jakobian @xmath114 $ ] : @xmath115 positiveness of @xmath116 in eq . ( [ i1 ] ) enforces the choice of the lower integration limit : @xmath117 for @xmath65 and @xmath118 for @xmath119 . using eqs . ( [ op1 ] ) , ( [ op2 ] ) and ( [ i2 ] ) we are able to analyze the critical current @xmath92 in the whole range of the pair - breaking parameter , @xmath120 [ see , fig . [ iz ] ] . in line with the discussed behavior of the andreev bound states , for a narrow resonance , @xmath121 , the critical current starts to drop significantly only upon entering the gapless superconductivity regime @xmath122 . on the other hand , for a broad resonance , @xmath123 , the suppression of @xmath0 is almost linear in the whole range . we note that in both cases the behavior of @xmath0 strongly deviates from that of the bulk order parameter ( red curve ) @xcite largely due to the pair - breaking effect on the abs . in practice , the @xmath124 dependence can be measured by applying a magnetic field [ the case where @xmath125 and @xmath126 in an experiment similar to ref . where a quantum dot , defined in a single - wall carbon nanotube , was strongly coupled to the leads with the ratio @xmath127 . carbon nanotube quantum dots with lower @xmath128 values are accessible experimentally , too @xcite . we also found that the crossover between the gapped and gapless regimes is accompanied by a qualitative change in the shape of the josephson current - phase relation @xmath1 as demonstrated in fig . [ i_f ] for the on - resonance case @xmath129 and @xmath130 . the @xmath1 relation is anharmonic as long as the junction with @xmath131 supports the abs ( black and blue curves ) . the vanishing of the abs upon entering the gapless regime leads to a nearly sinusoidal current - phase relation ( red curve ) . a closely related effect is demonstrated in fig . [ i_e ] showing the modification of the critical current resonance lineshape with the increasing pair - breaking strength . in the absence of pair breaking it is nonanalytic near @xmath129 ( black curves ) reflecting the anharmonic @xmath1 due to the abs in a transparent channel @xcite . on approaching the gapless regime this singularity is smeared out ( red curves ) , which is accompanied by the suppression of the @xmath0 amplitude . at finite temperatures @xmath132 the pair - breaking - induced smearing of the resonance peak will enhance the usual temperature effect . in conclusion , we have proposed a model describing resonant josephson tunneling through a quantum dot beyond the conventional bcs picture of the superconducting state in the leads . it allows for nonperturbative treatment of pair - breaking processes induced by a magnetic field or paramagnetic impurities in diffusive superconductors . we considered no coulomb blockade effects , assuming small charging energy in the dot @xmath133 , which was , for instance , the case in the experiment of ref . . our predictions , however , should be qualitatively correct also for weakly coupled dots with @xmath134 at least as far as the dependence of the ctitical supercurrent on the pair - breaking parameter is concerned . indeed , for a narrow resonance the andreev bound states begin to respond to pair breaking only when the gap @xmath2 becomes sufficiently small [ see , fig . [ e03 ] ] so that for a finite @xmath135 one can expect a sharp transition to the resistive state , too , similar to that shown in fig . [ iz ] for @xmath121 . we thank d. averin , c. bruder , p. fulde , a. golubov , m. hentschel , t. novotny , v. ryazanov and c. strunk for useful discussions . financial support by the deutsche forschungsgemeinschaft ( grk 638 at regensburg university ) is gratefully acknowledged .
|
we propose a model for resonant josephson tunneling through quantum dots that accounts for cooper pair - breaking processes in the superconducting leads caused by a magnetic field or spin - flip scattering .
the pair - breaking effect on the critical supercurrent @xmath0 and the josephson current - phase relation @xmath1 is largely due to the modification of the spectrum of andreev bound states below the reduced ( abrikosov - gorkov ) quasiparticle gap . for a quantum dot formed in a quasi - one - dimensional channel ,
both @xmath0 and @xmath1 can show a significant magnetic field dependence induced by pair breaking despite the suppression of the orbital magnetic field effect in the channel .
this case is relevant to recent experiments on quantum dot josephson junctions in carbon nanotubes .
pair - breaking processes are taken into account via the relation between the andreev scattering matrix and the quasiclassical green functions of the superconductors in the usadel limit .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
within the last few years _ relational database management systems _ ( rdbms or db in short ) have become essential for control system configuration . with availability of generic applications and hardware standards the interplay of components as well as the adaptation to site specific needs has become crucial . proper networking has been the central problem of the past . today s challenge is a central repository of reference and configuration data as well as an appropriate standard suite of db applications . target is a management system providing consistency in a programmable , comprehensible and automatic way for development , test and production phases of the control system . instead of careful bookkeeping of innumerable hand - edited files the ever needed modifications of the facility require ` only ' change of atomic and unique configuration data in the db and eventually adaptation of structures in the db ( applications ) . the update of tool configurations is then consistently accomplished by direct db connection , renewal of snapshot files generated by extraction scripts etc . very early a device oriented approach has been chosen for the description of the 3rd generation light source bessy ii . a naming convention was developed for easy identification and parsing of classifying properties like installation location , device family , type and instance . around the ` bootstrap ' information contained in the device names a first reference database has been set up @xcite , describing wiring , calibrations , geometries etc . utilisation of the epics control system toolkit implicates the network protocol called _ channel access _ for the i / o of process variables . at bessy the device model results in a scheme : < channel>. generation of the epics _ real time databases _ ( rtdb ) for device classes with high multiplicity and simple i / o ( power supplies , vacuum system , timings ) is based on two components : device class specific data are stored in the db . functionality and logic of is modelled with a graphical editor and stored in a template file . generating scripts merge both into the actual rtdb . for unique and complex systems ( rf , insertion devices ) [ rf ] structuring takes place in where no common naming convention has been defined so far . contrary to the previous approach of complex template / atomic substitution here all channels involved are assigned to sub - units and hierarchies . the structure is fully implemented in the db and for rtdb generation only connected with simple atomic templates . intermediate systems ( scraper , gp - ib devices ) are either adapted to one of these approaches or simply set up by ad - hoc created files . documentation of and transitions between different facility operation conditions are handled by a save / restore / compare tool that is working on a set of snapshot files . coarse configuration is provided by sql retrievals of name lists . hierarchies and partitioning are derived from device - name patterns and mapped into directories and files . this is feasible only because the relevant channels are restricted to setpoint , readback and status . deviations of this scheme are few and easily maintainable by hand . information required by modelling tools for conversion between engineering units of device i / o and the physics views ( magnet function , length , position , conversion factor etc . ) are fully available in the db @xcite . therefore the linear optics correction tools ( orbit , tune ) are instantaneously consistently configured provided the installed hardware matches the entries in the db . as long as the alarm handler has to monitor only hardware trips db retrieval of device name collections and script based interpretation of device name patterns are helpful at least for the simple devices with high multiplicity : typical channels are on / off status . again grouping and hierarchies can be derived from the class description embedded in the device name . for the complex devices ( rf ) control functionality and logic has already been mapped into db structures ( see [ rf ] ) . here genuine db calls produce alarm handler configuration files with sophisticated error reporting capabilities . in a similar fashion creation of configuration files for the data collector engine(s ) of the archiving system is simplified by db calls : for each device class a limited number of signals and associated frequencies are of interest for long term monitoring . the db basically serves as source for device name collections . high level software using the cdev api ( configured with a _ device description language _ file ) and ` foreign ' networks ( connected by the _ channel access gateway _ ) benefit also from the db : for cdev access definition and permission to selected i / o channels for each device class is defined by appropriate prototype descriptions ( in analogy to the rtdb templates ) . the associated lists of devices are compiled with db calls . _ ca _ gateway takes advantage of the naming convention by regular expression evaluation . facility modifications typically change the device inventory . the db supports consistent propagation of innovations @xcite : during an installation campaign devices are added or deleted in the db . in addition , new device classes are modelled within script logics and template ( prototype ) descriptions . running the configuration scripts on all control system levels updates the configurations within development environment , test system and production area . a number of restrictions imposed by the present device oriented db model are solvable by minor structural modifications and consequent introduction of channels necessary for the adequate description of essential device properties . dependent on the specific point of view , definition of device classes and assignment of equipment comprises ambiguities . e.g. the device class _ magnet _ ( m ) is tightly bound to the model aspect of the storage ring , whereas the the class _ power supply _ ( p ) covers the engineering aspects of the current converters . pulsed elements ( kicker , septa ) are very similar devices , but do not fit into this scheme - neither into the model nor the i / o aspects . the decision to assign a single dedicated device - class ( k ) introduces a new pattern ( fig . [ devnew ] ) and breaks the naming structure . : : + horizontal corrector ( windings ) in sext 4 of : : + @xmath0 power supply : : + i d ue56 as a complex device with lots of _ internal _ hardware units : : + @xmath0 power supply of + @xmath0 horizontal ue56bending magnet _ external access ! _ difficulties get worse for complex pieces of equipment : insertion devices have to be treated as ` units ' . they can be completely replaced by different entities with similar complexity . typically they consist of a number of devices partly belonging to already described device classes ( like bipolar power supplies ) . the straightforward solution here is the extension of the device naming convention : substructures and naming rules have to be introduced also for the ( up to now monolithic ) device property describing name element ( ` genome chart ' ) . corresponding db structures have to be implemented . similarly naming rules have to be developed where structuring takes place in ( fig . [ rfnew ] ) . in summary , the distinction and boundary between device and are dictated by specifics of the i / o connectivity and arbitrary with respect to the required db structure . it does not necessarily coincide with the most adequate classification aspects . several global states are well defined for the whole facility ( e.g. ` shutdown ' , ` machine development ' , ` user service ' etc . ) or for sub - sections or device collections : ` injection running ' , ` wave length shifter on ' etc . the role of devices depend on these states : in the context of ` shutdown ' insufficient liquid helium level at super - conducting devices has to generate a high severity alarm . for power supplies even off states are then no failure . but during ` user service ' alarms should notify the operator already when power supply readbacks are out of meaningful bounds . similar case distinctions have to be made for all conditioning applications ( reload constraints for save / restore , active / inactive elements for modelling ) , for data archiving ( active periods , frequency / monitor mode ) , for sequencing programs etc . today only the most general conditions are statically configured and available from the database / configuration scripts . exceptions are partly coded within the applications or have to be handled by the operators an error - prone situation . a clearly laid out man - machine interface provides an effective protection against accidental maloperation . presentation details and accessibility of devices should be tailored to the user groups addressed : operators , device experts , accelerator physicists . the required attributes attached to the device channels should be easily retrievable from the db . starting from the device model a very specific view of the control system structure has been mapped into db structure : device classes correspond to tailored set of tables . deviating aspects that do not fit into the scheme are modelled by relations , constraints , triggers and presented to the applications by pre - built views . the result is not very flexible , hard to extend and complicated to maintain in a consistent state free from redundancies . numerous device attributes are scattered over template / prototype files and configuration creating scripts in an implicit format without clear and explicit relation to the data within the db . configurations of archiver retrieval tools are much more demanding than those of data collector processes . in search of correlations arbitrary channel names have to be detected by their physical dimension : for a drift analysis the data sources for temperatures [ c ] , rf frequency [ mhz ] , bpm deviations [ mm ] and corrector kicks [ mrad ] have to be discovered . any non - static , not pre - configured retrieval interface to the most important performance analysis tool of the facility archived data requires clues to signal names , meaning , functionality , dimensions , attributes . with the rtdb template approach the channel attributes are hidden within these files and not available to the db . consequently , no db based data relations are available for correlations ( comparable dimensions ) , projections ( user , expert , device responsible ) or dependencies ( triggered by ) . there is no browsable common data source that allows to coordinate consistency of configurations requiring channel attributes ( that go beyond simple and clear assignments to well defined device classes ) . the device model is tailored for modelling applications . for the archiver retrieval it is useless continuously causing errors and inconsistencies . in a kind of clean - up effort the existing fragments used for rtdb generation will be put into a new scheme . sacrificing the feasibility to generate complex rtdb templates with a graphical editor , now device class tables , templates and i / o software modules ( _ records _ ) will be put together as a new db core . in view of the extendibility for high level applications a purely db based approach has been taken even though for rtdb generation the framework of xml seems to be an attractive and well suited alternative . a new entity _ gadget _ is introduced connecting the data spaces _ device _ , _ channel _ and _ iostruct_. these db sections have to be set up consequently in 3rd normal form , i.e. all device class properties , channel attributes and i / o specifics have to be modelled in data and relations , not in table structures . in blue - print the resulting data model looks abstract and ( intimidating ) complex , but promising with respect to db flexibility , extendibility and cleanness . expectation that today s complicated configuration scripts collapse to simple sequences of db queries seem to be justified . the most difficult problem of a comprehensive configuration management system is the determination of ` natural ' data structures and breaking them down into localised data areas , dependencies and relations . for a relatively small installation like bessy ii the expected reward in stability and configuration consistency does not clearly justify effort and risk of the envisaged re - engineering task . one could argue that the present mixture of db based and hand edited configuration is an efficient compromise . on the other hand the persistent and boring work - load due to manual configuration update requirements is a constant source of motivation to persue this reengineering project .
|
the reference rdbms for bessy ii has been set up with a device oriented data model .
this has proven adequate for e.g. template based rtdb generation , modelling etc .
but since assigned i / o channels have been stored outside the database ( a ) numerous specific conditions had to be maintained within the scripts generating configuration files and ( b ) several generic applications could not be set up automatically by scripts . in a larger re - design effort
the i / o channels are introduced into the rdbms . that modification allows to generate a larger set of rtdbs , map specific conditions into database relations and maintain application configurations by relatively simple extraction scripts .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the stereo mission provides us an unprecedented view of the solar corona , enabling us for the first time to fully constrain the three - dimensional ( 3-d ) geometry of the coronal magnetic field . stereoscopic triangulation of coronal loops has been conducted at small stereo spacecraft separation angles ( @xmath5 ) , for several active regions observed with stereo a(head ) and b(ehind ) in april and may 2007 ( aschwanden et al . 2008a , b ; 2009 ) . the reconstructed 3-d geometry of stereo - observed coronal loops has been compared with theoretical magnetic field models based on extrapolations from photospheric magnetograms , using nonlinear force - free field ( nlfff ) models ( derosa et al . 2009 ) , as well as potential and stretched potential field models ( sandman et al . 2009 ) , but surprisingly it turned out that the two types of magnetic field lines exhibited an average misalignment angle of @xmath6 , regardless of what type of theoretical magnetic field model was used . from this dilemma it was concluded that a more realistic physical model is needed to quantify the transition from the non - force - free photospheric boundary condition to the nearly force - free field at the base of the solar corona ( derosa et al . 2009 ) . at this junction , it is not clear what a viable method is to obtain a force - free boundary of the magnetic field at the coronal base , or how to correct the non - force - free magnetograms . however , the stereoscopic triangulation supposedly provides the correct 3-d directions of the magnetic field @xmath7 , which together with maxwell s equation of divergence - freeness ( @xmath8 , constrain also the absolute values of the field strengths . in this paper i we choose a magnetic field model that is defined in terms of multiple unipolar charges . an approach in terms of multiple dipoles is employed in paper ii ( sandman and aschwanden 2010 ) . since both unipolar or dipolar magnetic fields represent potential magnetic fields that fulfill the divergence - free condition , the superposition of multiple unipolar and dipolar magnetic field components fulfill the same condition . we develop a numerical code of such a parameterized divergence - free magnetic field that can be forward - fitted to the 3-d geometry of stereoscopically triangulated coronal loops . so , the simple goal of this study is to evaluate how closely the stereoscopically observed loops can be modeled in terms of potential fields , a goal that was already attempted with skylab observations ( sakurai and uchida 1977 ) . modeling with non - potential fields , such as nonlinear force - free field ( nlfff ) models , will be considered in future studies . this paper is organized as follows : the definition of a parameterized potential field is described in section 2 , the development and tests of a numeric magnetic field code and the results of forward - fitting to stereoscopically triangulated loops is presented in section 3 , and conclusions follow in section 4 . an alternative approach with dipolar magnetic fields is the subject of paper ii ( sandman and aschwanden 2010 ) . since the coronal plasma-@xmath9 parameter is generally less than unity ( gary 2001 ) , the magnetic pressure exceeds the thermal pressure , and thus all soft x - ray or euv - emitting plasma that fills or flows through coronal flux tubes traces out the coronal field . consequently , stereoscopic triangulation of euv loops provides the correct 3-d field directions along coronal loops . we denote the normalized 3-d field direction along a loop with the unity vector @xmath10 , which is parameterized as a function of a loop length coordinate @xmath11 , starting at the footpoint position @xmath12 at the base of the corona , @xmath13 \over \sqrt{b_x^2+b_y^2+b_z^2 } } \ , \ ] ] where the magnetic field is defined by the cartesian components @xmath14 $ ] . however , the absolute magnitude of the magnetic field strength , @xmath15 , is not known a priori . for a physical solution of the magnetic field , maxwell s equation of a divergence - free magnetic field has to be satisfied , @xmath16 which ( in its integral form ) corresponds to the theorem of magnetic flux conservation along a fluxtube , @xmath17 where @xmath18 is the integral over the cross - sectional area of a fluxtube defined in perpendicular direction to the magnetic field line ( at loop position @xmath19 ) . therefore , since the stereoscopic triangulation defines the magnetic field directions in adjacent flux tubes , it defines also the divergence of the field and the relative change of the magnetic field strength @xmath20 along the flux tubes , and this way implicitly defines also the isogauss surfaces perpendicular to each fluxtube , and therefore the full 3-d vector field @xmath21 , except for a scaling constant . the derivation of a 3-d magnetic field @xmath21 in an active region , @xmath22 requires also the knowledge of the scalar function @xmath23 in every 3-d location @xmath24 , which we constrain with a forward - fitting method of a divergene - free field model . a divergence - free 3-d magnetic field model @xmath7 can be parameterized by a superposition of divergence - free fields , because the divergence - free condition is linear ( or abelian ) , i.e. , if two components @xmath25 and @xmath21 fulfill @xmath26 and @xmath27 , then also their sum is divergence - free , @xmath28 . divergence - free magnetic field components are , for instance , a parallel field , unipolar fields ( a magnetic charge with spherical isogauss surfaces and a field that falls off with the square of the distance ) , dipole fields , quadrupolar fields , other multi - pole representations , or any potential field . the abelian property warrants that any superposition of divergence - free fields is also divergence - free . specifically , we will use divergence - free potential field models that consist of either _ i ) _ multiple unipolar charges ( in this paper i ) , or _ ii ) _ multiple dipoles ( in paper ii ) . our philosophy is the following : we will employ magnetic field models of the category of potential - field models , which are divergence - free by definition . we use particular potential field models that can be quantified with a finite number of free parameters . potential field models are not as general as non - potential and force - free field ( nlfff ) models . however , since both potential and nlllf models currently exhibit an equally poor misalignment with observed euv loops , we need first to investigate whether the misalignement between observations and _ any _ theoretical model can be mimized for the simplest class of magnetic field models , such as potential field models . if our approach proves to be successfull , refinements with non - potential or nlllf models can then be pursued in the future along the same avenue ( e.g. , see conlon and gallagher 2010 ) . unipolar potential fields often provide a good approximation to the magnetic field of sunspots , and thus can be used also for an active region that is composed of a finite number of spot - like magnetic polarities . conceptually , a unipolar field can be considered as an approximation to the upper half of a vertically positioned dipole . the simplest representation of a unipolar magnetic field that is a potential field , and hence fulfills maxwell s divergence - free condition , is a spherically symmetric field that drops off with an @xmath29-dependence with distance , so it has a potential function that drops off with @xmath30 , @xmath31 where @xmath32 is the potential field value at the solar surface vertically above the buried magnetic charge in depth @xmath33 . of course , the extrapolated magnetic field is only computed in the coronal domain @xmath34 , so that no `` magnetic monopole '' exists in the solar corona . the magnetic field model of a single magnetic charge requires 4 parameters : the maximum value of the potential field @xmath32 and the location @xmath35 of the buried charge , having a distance @xmath36 to any point @xmath37 in the solar corona , @xmath38^{1/2 } \ .\ ] ] the resulting magnetic field has then only a radial component @xmath39 in direction of @xmath40 , @xmath41 with the surface field strength @xmath42 . this unipolar potential field fulfills the divergence - free condition , as it can be calculated from the laplacian operator of the potential function , @xmath43 the fulfillment of the divergence - free condition can also be verified from the conservation of the magnetic flux theorem ( eq . 3 ) , if the envelope of a fluxtube is defined by radial field lines , so that the cross - sectional area @xmath44 remains constant for @xmath45 . in our first model we employ a superposition of @xmath46 multiple unipolar charges , @xmath47 in terms of the vector @xmath48 $ ] . for a single unipolar charge , the field lines will all be straight lines in radial direction away from the buried charge , which can approximate open - field regions . burying multiple magnetic charges of opposite magnetic polarity , however , can mimic closed - field regions . an example is given in fig . 1 , where we compare the magnetic field of a dipole with that of a combination of two unipolar charges with opposite magnetic polarity . actually , the two magnetic field models become identical when the two unipolar charges are moved close together at the location of the dipole moment , as it can be shown mathematically . although the two models are equivalent in the far - field approximation , a combination of two unipolar charges ( with @xmath49 free parameters ) allows more general solutions than a single dipole ( with 6 free parameters ) , especially in the case of strongly asymmetric fields ( sunspots ) or open - field regions , as they exist is most active regions . the forward - fitting of our analytical magnetic field model to a set of observed magnetic field vectors @xmath50 ( e.g. , using stereoscopically triangulated loop coordinates ) , is the task of optimizing the free parameters of the analytical model until the best match with the observed field lines is obtained . for the evaluation of the goodness or consistency of the analytical magnetic field models @xmath51 with the observed field line model @xmath52 , we define the 3-d misalignment angle @xmath53 , which is defined by the scalar product between the two field vectors @xmath54 and @xmath55 , @xmath56 or equivalently , between the unity field vectors @xmath57 and @xmath58 , @xmath59 this parameter is a single value at every spatial point @xmath24 , which can be averaged at the coronal base or over the lengths of the observed field lines , at @xmath60 positions , @xmath61^{1/2 } \ , \ ] ] which is similar to a @xmath62-criterion . since a unipolar magnetic charge can be parameterized with 4 free parameters ( @xmath63 ( eq . 9 ) , a model with @xmath60 components has @xmath64 free parameters . our strategy to bootstrap the coronal magnetic field with stereoscopic data consists of the following steps : ( 1 ) we create a model of subphotospheric magnetic charges by deconvolving an observed photospheric magnetogram into point charges , which defines the unit vectors of our parameterized theoretical magnetic field model @xmath65 ; ( 2 ) we perform stereoscopic triangulation for a set of coronal loops observed with stereo / euvi , which are quantified in terms of directional field vectors @xmath66 ; ( 3 ) we forward - fit the theoretical magnetic field model @xmath67 with a number of free parameters in the setup of unipolar magnetic charges by minimizing the mean misalignment angle @xmath68 , and which yields a best - fit solution @xmath51 . we can then compare the minimized misalignment of the bootstrapped best - fit model with those of standard methods based on extrapolation of the the photospheric boundaries using a magnetogram , e.g. , with the _ potential field source surface ( pfss ) _ model . the procedure is illustrated in fig . 2 , where a dipolar euv loop is observed and a magnetic field model is constructed from two unipolar charges . by adjusting the field strength of the second unipolar charge from @xmath69 to @xmath70 , the misalignment angle between the observed euv loop and the model field can be reduced from @xmath71 ( fig . 2 top ) to @xmath72 ( fig . 2 , bottom ) . note that the adjusted field is still a potential field and divergence - free , but represents a better match to the observed euv loop . we select four active regions observed with stereo / euvi and the _ michelson doppler imager ( mdi ) _ ( scherrer et al . 1995 ) onboard the _ solar and heliospheric observatory ( soho ) _ : 2007 april 30 , may 9 , may 19 , and dec 11 . the first ar 10953 ( 2007 apr 30 ) is identical to the case previously analyzed with stereo and hinode ( derosa et al . 2009 ; sandman et al . the second ar 10955 ( 2007 may 9 ) was subject of the first stereoscopically triangulated coronal loops , temperature and density measurements , and stereoscopic tomographic reconstruction ( aschwanden et al . 2008a , b , 2009 ; sandman et al . the third ar 10953 ( 2007 may 9 ) displayed a small flare ( during 12:40 - 13:20 ut ) as well as a partial filament eruption during the time of observations , and was featured in a few studies ( li et al . 2008 ; liewer et al . 2009 ; sandman et al . the fourth ar 10978 ( 2007 dec 11 ) is also subject of recent magnetic field modeling ( aad van ballegooijen and alec engell , private communication 2010 ) . some details of these four active regions are given in table 1 , such as the heliographic position of the ar center , the magnetic area for fluxes of @xmath73 g , the minimum and maximum field strengths , and the total unsigned magnetic flux . all four active regions were observed with the _ michelson doppler imager ( mdi ) _ onboard the _ solar and heliospheric observatory ( soho ) _ , which provides full - disk mdi magnetograms with a pixel size of 2 . subimages encompassing the active region of interest are shown in fig . 3 ( left column ) , with quadratic field - of - view sizes ranging from 145 to 339 pixels , or 0.3 to 0.7 solar radii , respectively . in order to create a realistic 3d magnetic field model we decompose the partial magnetograms into a number of @xmath74 positive and negative gaussian 2-d components , using an iterative 2-d gaussian fitting scheme that determines local maxima in decreasing order of field strengths . the composite magnetogram of these 200 decomposed sources is shown in fig . 2 ( middle column ) , and the difference to the original magnetogram is also shown ( right column ) . for each of the 200 magnetic source components we store the peak magnetic field value @xmath75 , the center position ( @xmath76 ) , and the half widths @xmath77 at full maximum . for a parameterization in terms of unipolar charges , however , we need to convert the half width @xmath77 into the corresponding depth @xmath78 at which the unipolar charge is buried . from the definition of the full width at half maximum ( fwhm ) at @xmath79 of a unipolar field ( eq . 9 ) , we have for the vertical magnetic field component @xmath80 ( see geometry in fig . 4 ) , @xmath81 with @xmath82 the angle between the vertical and a surface ring with radius @xmath83 ( fig . 4 ) , from which we can calculate the dipole depth @xmath78 , @xmath84 which corresponds approximately to the half width @xmath83 of the fitted gaussian component . since we have now all input parameters @xmath85 , @xmath86 , for a definition of a magnetic field model with multiple unipolar magnetic charges ( eq . 9 ) , we can calculate the full 3d magnetic field by superimposing the fields of all components . a particular field line is simply calculated by starting with the field at a footpoint position and by iterative stepping in the field direction , until the field line returns to the solar surface ( in case of closed field lines ) or to a selected boundary of the computation box ( for open field lines ) . for each of the four active regions we triangulate as many loops as can be discerned with a highpass filter in image pairs from stereo / euvi a and b. the method of stereoscopic triangulation is described in detail in aschwanden ( 2008a ) , from which we use identical loop coordinates for ar 10955 observed on 2007 may 9 . the stereoscopic triangulation requires accurate coalignment of both euvi / a and b images in an epipolar coordinate system . furthermore , we transform the 3-d loop coordinates measured in the epipolar coordinate system with the line - of - sight of the euvi / a spacecraft into the coordinate system of mdi , which sees the sun from the langrangian point l1 , almost in the same direction as seen from earth . the advantage of transforming the euvi loop coordinates into the mdi reference frame is the direct modeling of the longitudinal magnetic field component @xmath87 in the mdi reference frame , without requiring any knowledge of the absolute magnetic field strength @xmath88 , which is model - dependent , i.e. , it depends on the choice of the extrapolation method ( potential , force - free , or nlfff ) from photospheric magnetograms ( which measures only the longitudinal component @xmath87 ) . some parameters of the stereoscopic triangulation procedure are listed in table 1 , such as the heliographic coordinates of the active region , the spacecraft separation angle , and the number of triangulated euv loops ( varying between 70 and 200 per active region , combined from all three coronal wavelengths , 171 , 195 , and 284 ) . 3-d representations of the stereoscopically triangualted euv loops are shown in figs . 5 - 8 ( with blue color ) , seen along the line - of - sight of mdi ( in greyscale maps of figs . 5 - 8 ) and in the two orthogonal directions ( sideview and topview in figs . 5 - 8 ) . the height range of stereosopically triangulated loops generally does not exceed 0.1 solar radii , due to the drop of dynamic range in flux for altitudes in excess of one hydrostatic scale height . after we have parameterized the magnetic field with @xmath89 unipolar charges , each one defined by 4 parameters @xmath85 , using the procedure of iterative decomposition of a photospheric magnetogram as described in section 3.2 , we can vary these free model parameters to adjust it to the 3d geometry of the stereoscopically triangulated loops . however , since we typically represent an mdi magnetogram with @xmath90 components , we have @xmath91 free parameters , which are too many to optimize independently . standard optimization procedures , such as the powell algorithm ( press et al . 1986 ) , allow for optimization of @xmath92 free parameters , with good convergence behavior if the initial guesses are suitably chosen . therefore we choose 10 - 20 small circular zones ( with typical radii of @xmath93 solar radii ) containing the strongest field regions in the magnetogram of equal magnetic polarity and optimize the magnetic field strengths @xmath94 with a common correction factor @xmath95 in each zone . this is an empirical optimization of the lower boundary of the coronal magnetic field , optimized by varying the ( decomposed ) photospheric mdi magnetogram in such a way that the resulting potential field more closely matches the stereoscopically triangulated loops . the results of the best fits , based on the mimization of the median misalignment angle ( eqs . 11 - 12 ) , are shown in figs . 5 - 9 , in form of ( red ) model field lines that are extrapolated at the same locations as the footpoints of the stereoscopic loops ( blue in figs . 5 - 9 ) . the distribution of all misalignment angles , evaluated at about 80 locations of every loop is shown in form of histograms at the bottom of figs . 5 - 9 . each histogram is characterized with a gaussian peak , which are found to have a mean value and standard deviation of @xmath96 ( fig.5 : 2007 apr 30 ) , @xmath97 ( fig.6 : 2007 may 9 ) , @xmath98 ( fig.7 : 2007 may 19 ) , and @xmath99 ( fig.8 : 2007 dec 11 ) . if we compare these misalignment angles in the range of @xmath100 with the previously measured values using a potential field source surface ( pfss ) model ( @xmath101 ; sandman et al . 2009 ) , or using a nonlinear force - free field ( nlfff ) model ( @xmath102 ; derosa et al . 2009 ) , we see an improvement of the misalignment angle by about a factor of two . this is a remarkable result that demonstrates that the magnetic field at the lower boundary of the corona can be bootstrapped with stereoscopic measurements and with a suitable parameterization of a potential field model . although the extrapolation from a photospheric magnetogram should be unique , if there is no data noise present , an infinite number of potential field solutions can be obtained depending on how the boundary field is parameterized ( e.g. , by unipolar charges or dipolar magnetic moments ) . in our case , for every variation of the @xmath103 free parameters , a slightly different field with a different misalignment to the stereosocpic loops is obtained . the pfss code is designed to compute the potential field of the entire ( front and backside ) sun , and thus has a relatively coarse spatial resolution of typically @xmath104 ( one heliographic degree , i.e. , 12 mm ) for standard computations , but the small - scale magnetic field should not matter too much for our large - scale loops ( @xmath105 mm ) . hence , the best - fit potential - field bootstraps a force - free photospheric boundary field that is significantly different from the observed photospheric magnetograms which is observed in a non - force - free zone . the remaining misalignment between our best - fit potential field solution and the stereoscopic loops could be due to three sources of errors : ( 1 ) the bootstrapping method did not converge to the best solution ; ( 2 ) the 3d coordinates of the stereoscopically triangulated loops have some error ; or ( 3 ) the real coronal magnetic field that is represented by the stereoscopic loops could be non - potential . we consider the convergence of our code as satisfactory , because we ran many attempts for each case with different initial conditions and obtained about the same minimum misalignment . one possibility to vary the initial conditions is to vary the number of ( gaussian ) unipolar components . 9 shows the convergence behavior as a function of the number of ( decomposed ) unipolar components , where we computed a best - fit potential field solution for @xmath106 unipolar components for each of the 4 active regions . convergence to the minimum misalignment value typically recquires @xmath107 components for the simplest dipolar active regions ( 2007 may 9 or dec 11 ) , and @xmath108 for more complex active regions ( 2007 apr 30 or may 19 ) . we investigate the second possible source of errors that could contribute to the measured misalignment , i.e. , errors associated with the stereoscopic triangulation method . the errors of stereoscopic triangulation have been discussed in aschwanden et al . ( 2008a ) , which depend on ( 1 ) the ratio of the stereoscopic parallax angle to the spatial resolution of the instrument , and thus on the spacecraft separation angle , ( 2 ) the angle between the loop segment and east - west direction in the epipolar plane , being largest for loop segments parallel to the epipolar plane , and ( 3 ) the proper identification of a corresponding loop in image b to a selected loop in image a. while the first two sources of errors can be formally calculated , the correspondence problem is difficult to quantify . since stereoscopic triangulation can not be accomplished in an automated way at present time , the error of identifying corresponding loops could be estimated from the scatter obtained with different observers , but this is time - consuming . here we pursue another approach that is based on a self - consistency test . the procedure works as follows . if a sufficient large number of loops are triangulated in an active region , there should be for every triangulated loop ( which we call primary loop ) a neighbored ( secondary ) loop with an almost parallel direction . whatever the direction of the true magnetic field in the same neighborhood is , both the primary @xmath109 and the secondary stereo loop @xmath110 should have a similar misalignment angle with the local magnetic field , @xmath111 , or @xmath112 , in order to be self - consistent . therefore we can define a stereoscopic error ( se ) angle @xmath113 by averaging these differences in misalignements over all stereo loop positions with suitable weighting factors @xmath114 , @xmath115 where the index @xmath116 runs over all loop positions for each primary loop and the index @xmath117 runs over all loop positions of secondary loops , excluding the primary loop . for the weighting factor @xmath118 we should give less weight to more distant neighbors , because the true magnetic field and thus the local misalignment angle is likely to increase with distance . thus we should choose a negative power of the relative distance @xmath119 , @xmath120^{-p/2 } \ , \ ] ] where @xmath121 is a power index . if the index @xmath121 is small , say @xmath122 the weighting is reciprocal to the distance and the long - range neighbors are relatively strongly weighted . if the index @xmath121 is large , say @xmath123 , the short - range neighbors are relatively strongly weighted while the longe - range neighbors have almost no weight . so we expect that the so - defined stereoscopic error monotonously decreases with short - range weighting towards higher power indices @xmath124 . in fig . 10 we show the stereoscopic error calculated with eqs . ( 15 - 16 ) as a function of the power index from @xmath122 to @xmath125 , for all 4 active regions . indeed , the stereoscopic error monotonously decreases from @xmath122 to @xmath126 , because we give progressively less weight to the long - range misalignments . however , from @xmath126 towards @xmath125 the error increases again , probably because of the inhomogeneity of the closest neighbors at the shortest range and the excessive weighting to the very nearest neighbors . however , the plateau in the range of @xmath127 is a good indication that we measure a stable value at the minumum of @xmath128 . from this we obtain a misalignment contribution of @xmath129 ( 2007 apr 30 ) , @xmath130 ( 2007 may 9 ) , @xmath131 ( 2007 may 19 ) , and @xmath132 ( 2007 dec 11 ) , as listed in table 2 . a diagram of the various misalignment angles is shown in fig . 11 , which we plot as a function of the maximum goes soft x - ray flux during the observing interval . the goes flux was dominanted by the emission from the active regions analyzed here , since there was no other comparable active region present on the solar disk during the times of the observations . the case of 2007 may 9 with the lowest goes shows a single dipolar structure , while the case of 2007 may 19 with the highest goes flux exhibits multiple dipolar groups . the misalignment angles @xmath133 indicate those obtained with the _ potential field source surface ( pfss ) _ code , @xmath134 those with the _ nonlinear force - free field ( nlfff ) _ code ( only for 2007 apr 30 ) , @xmath135 those optimized with the _ potential field with unipolar charge ( pfu ) _ code , and @xmath113 indicates the stereosocpic errors . as a working hypothesis we might attribute the remaining residuals @xmath4 to the non - potentiality of the active regions , @xmath136 for which we measure the values : @xmath137 ( 2007 apr 30 ) , @xmath138 ( 2007 may 9 ) , @xmath139 ( 2007 may 19 ) , and @xmath140 ( 2007 dec 11 ) , listed also in table 2 . we consider these values as a new method to quantify the nonpotentialiy of active regions . we find that quiescent active regions that contain simple dipoles ( 2007 apr 30 and may 9 ) have small misalignments of @xmath141 with best - fit potential field models , while more complex active regions ( 2007 may 19 and dec 11 ) have somewhat larger non - potential misalignments in the order of @xmath142 compared with best - fit potential field models . while we express thge degree of nonpotentiality in terms of a mean misalignment angle here , other measures are the ratios of the total nonpotential to the potential energy in an active region , which amounts up to @xmath143 in one flaring active region ( schrijver et al . 2008 ) . since the four investigated active regions have quite different misalignment angles , and also the inferred nonpotentiality of the magnetic field varies significantly , we quantify the activity level of the active regions from the soft x - ray flux measured by goes . 12 shows the goes light curves for the 4 active regions during the times of stereoscopic triangulation . the lowest goes flux is measured for the ar of 2007 may 9 , with a level of @xmath144 w m@xmath145 ( goes class a4 ) , while the highest goes flux is measured for the ar of 2007 may 19 , which has a flare occurring during the observing period with a goes flux of @xmath146 w m@xmath145 ( goes class c0 ) . the three active regions with the low soft x - ray flux levels appear to be quiescent , judging from the goes flux profile , or may be subject to micro - flaring at a low level . there is a clear correlation between the soft x - ray level of the active region ( when it was on disk ) and the overall misalignment angle ( fig . 11 ) , as well as with the misalignment angle @xmath4 attributed to the non - potentiality , varying from @xmath147 for the lowest goes a - class levels to @xmath148 for an active region with a goes c - class flare . a higher soft x - ray flux generally means a higher heating rate with stronger impulsive heating or flaring . a higher degree of non - potentiality , on the other hand , indicates the presence of a higher level of electric currents ( which are non - potential ) . therefore , the observed correlation suggests a physical relationship between the electric currents in an active region and the amount of heating input . this is not surprising , since evidence for current - carrying emerging flux was demonstrated previously for h@xmath149 and soft x - ray structures that are non - potential ( e.g. , leka et al . 1996 ; jiao et al . 1997 ; schmieder et al . our measurement of the degree of non - potentiality with the magnetic field misalignment averaged over the entire active region , is a very coarse technique , but a more detailed investigation of the misalignment in separate parts of the active region that are quiescent or flaring will be pursued in paper ii . the agreement between theoretical magnetic field models of active regions in the solar corona with the true 3-d magnetic field as delineated from the stereoscopic triangulation of coronal loops in euv wavelengths has never been quantified until the recent advent of the stereo mission . to everybody s surprise , the average misalignment between the theoretical and observed magnetic field was quite substantial , in the amount of @xmath150 for both potential and nonlinear force - free field models ( derosa et al . 2009 ; sandman et al . 2009 ) . in this study we investigate the various contributions of this large misalignment for four different active regions observed with stereo and arrive at the following conclusions : 1 . the amount of misalignment can be reduced to about half of the values for potential - field models optimized by a bootstrapping method that minimizes the field directions with the stereoscopically triangulated loops . our potential - field model is parameterized with @xmath151 unipolar charges per active region , whose positions and field strengths are approximately derived from a gaussian decomposition of a photospheric magnetogram , and then varied until a best fit is obtained . the best - fit potential field model has an improved misalignment of @xmath152 . because the best - fit potential field model defines an improved magnetic field boundary condition at the bottom of the corona , the difference to the observed photospheric magnetogram contains information on the currents between the photosphere and the base of the force - free corona . 2 . we estimate the misalignment contriubtion caused by stereoscopic correlation errors from self - consistency measurements between the magnetic field misalignments of adjacent loops . we find contributions in the order of @xmath153 . we estimate the contributions to the field misalignment due to non - potentiality caused by electric currents from the residuals between the best - fit potential field and the stereoscopic triangulation errors and find misalignment contributions in the order of @xmath154 . 4 . the overall average misalignemnt angle between potential field models and stereoscopic loop directions , as well as the contribution to the misalignment due to non - potentiality , are found to correlate with the soft x - ray flux of the active region , which suggests a correlation between the amount of electric currents and the amount of energy dissipation in form of plasma heating in an active region . in this study we identify for the first time the contributions to the misalignment of the magnetic field , in terms of optimized potential field models , non - potentiality due to electric currents , and stereoscopic triangulation errors . these results open up a number of new avenues to improve theoretical modeling of the coronal magnetic field . first of all , optimized potential field models can be found that represent a suitable lower boundary condition at the base of the force - free corona , which provides a less computing - expensive method than nonlinear force - free codes . second , methods can be developed that allow us to localize electric currents in the non - force - free photophere and chromosphere . third , the misalignment angle can be used as a sensitive parameter to probe the evolution of current dissipation , energy build - up in form of non - potential magnetic energy in different quiescenct and flaring zones of active regions . the high - resolution magnetic field data from hinode and _ solar dynamics observatory _ provide excellent opportunities to obtain better theoretical models of the coronal magnetic field using our bootstrapping method , which is not restricted to stereoscopic data only , but can also be applied to single - spacecraft observations . acknowledgements : we are grateful to helpful discussions with marc derosa and allen gary . this work was partially supported by the nasa contract nas5 - 38099 of the trace mission and by nasa stereo under nrl contract n00173 - 02-c-2035 . the stereo / secchi data used here are produced by an international consortium of nrl , lmsal , ral , mpi , isas , and nasa . the mdi / soho data were produced by the mdi team at stanford university and nasa . # 1 [ aschwanden , m.j . , 2004 , _ physics of the solar corona - an introduction _ , praxis publishing ltd . , chichester uk , and springer , berlin . ] [ aschwanden , m.j . , wuelser , j.p . , nitta , n. , and lemen , j.r . : 2008a , , 827 . ] [ aschwanden , m.j . , nitta , n.v . , wuelser , j.p . , and lemen , j.r . : 2008b , , 1477 . ] [ aschwanden , m.j . , nitta , n.v . , wuelser , j.p . , lemen , j.r . , and sandman , a. : 2009 , , 12 . ] [ conlon , p.a . and gallagher , p.t . 2010 , apj ( submitted ) . ] [ derosa m.l . , schrijver , c.j . , barnes , g . , leka , k.d . , lites , b.w . , aschwanden , m.j . , amari , t . , canou , a . , mctiernan , j.m . , regnier , s . , thalmann , j . , valori , g . , wheatland , m.s . , wiegelmann , t . , cheung , m.c.m . , conlon , p.a . , fuhrmann , m . , inhester , b . , and tadesse , t . 2009 , , 1780 . ] [ gary , a. 2001 , solar phys . 203 , 71 . ] [ jiao , l. , mcclymont , a.n . , and mikic , z. 1997 , 174 , 311 . ] [ leka , k.d . , canfield , r.c . , mcclymont , a.n . , and van driel - gesztelyi , l. 1996 , 462 , 547 . ] [ li , y. , lynch , b.j . , stenborg , g. , luhmann , j.g . , huttunen , k.e.j . , welsch , b.t . , liewer , p.c . , and vourlidas , a. 2008 , , l37-l40 . ] [ liewer , p.c . , dejong , e.m . , hall , j.r . , howard , r.a . , thompson , w.t . , culhane , j.l . , bone , l . , and vandriel - gesztelyi , l . 2009 , , 57 - 72 . ] [ press , w.h . , flannery , b.p . , teukolsky , s.a . , and vetterling , w.t . 1986 , _ numerical recipes . the art of scientific computing _ , cambridge university press : new york . ] [ sakurai , t. and uchida , y. 1977 , , 397 . ] [ sandman , a.w . , aschwanden , m.j . , derosa , m.l . , wuelser , j.p . , and alexander , d. 2009 , , 1 . ] [ sandman , a.w . and aschwanden , m.j . , 2010 , ( in preparation ) , ( paper ii ) . ] [ scherrer , p.h . 1995 , solar phys . 162 , 129 . ] [ schmieder , b. , demoulin , p. , aulanier , g. , and golub , l . 1996 , 467 , 881 . ] [ schrijver , c.j . , derosa , t. metcalf , g. barnes , b. lites , t. tarbell , j. mctiernan , g. valori , t. wiegelmann , m.s . wheatland , t. amari , g. aulanier , p. demoulin , m. fuhrmann , k. kusano , s. regnier , and j.k . thalmann 2008 , 675 , 1637 . ] lllrrrrr 10953 ( s05e20 ) & 2007-apr-30 & 23:00 - 23:20 & 5.966 & 200 & 24.2 & [ -3134,+1425 ] & 8.7 + 10955 ( s09e24 ) & 2007-may-9 & 20:30 - 20:50 & 7.129 & 70 & 4.4 & [ -2396,+1926 ] & 1.6 + 10953 ( n03w03 ) & 2007-may-19 & 12:40 - 13:00 & 8.554 & 100 & 12.2 & [ -2056,+2307 ] & 4.0 + 10978 ( s09e06 ) & 2007-dec-11 & 16:30 - 16:50 & 42.698 & 87 & 8.2 & [ -2270,+2037 ] & 4.8 + lllll misalignment nlfff@xmath155 & 24@xmath15644 & & & + misalignment pfss@xmath157 & 25 @xmath158 8 & 19 @xmath1586 & 36 @xmath15813 & 32 @xmath15810 + misalignment pfu@xmath159 & [email protected] & [email protected] & [email protected] & [email protected] + median pfu@xmath159 & 20.0 & 16.2 & 25.8 & 15.7 + stereoscopy error@xmath160 & 9.4 & 7.6 & 11.5 & 8.9 + nonpotentiality@xmath161 & 4.9 & 5.7 & 8.8 & 7.2 + goes soft x - ray flux@xmath162&@xmath163 & @xmath144 & @xmath146 & @xmath164 + goes class & a7 & a4 & c0 & b1 +
|
we investigate the recently quantified misalignment of @xmath0 between the 3-d geometry of stereoscopically triangulated coronal loops observed with stereo / euvi ( in four active regions ) and theoretical ( potential or nonlinear force - free ) magnetic field models extrapolated from photospheric magnetograms .
we develop an efficient method of bootstrapping the coronal magnetic field by forward - fitting a parameterized potential field model to the stereo - observed loops .
the potential field model consists of a number of unipolar magnetic charges that are parameterized by decomposing a photospheric magnetogram from mdi .
the forward - fitting method yields a best - fit magnetic field model with a reduced misalignment of @xmath1 .
we evaluate also stereoscopic measurement errors and find a contribution of @xmath2 , which constrains the residual misalignment to @xmath3 , which is likely due to the nonpotentiality of the active regions .
the residual misalignment angle @xmath4 of the potential field due to nonpotentiality is found to correlate with the soft x - ray flux of the active region , which implies a relationship between electric currents and plasma heating .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the study of interfaces is often used to investigate the stability of ordered coexisting phases . low temperature ordered phases are spatially homogeneous , however , interfaces can be induced by forcing heterogeneous boundary conditions @xcite . for example , in the case of phases characterized by different values of an order parameter , one can fix the order parameter at the boundaries along one selected direction to two different values . for stable phases the induced interface costs an amount of free - energy proportional to a power of the interface size . at the lower critical dimension this power vanishes and long - range order becomes unstable . in spin glasses , due to the lack of a physical order parameter allowing to distinguish the different phases , this definition of an interface does not seem of practical use . different proposals have been put forward to define interfaces in spin glasses . a common procedure @xcite , involves the comparison of free - energies between systems with periodic and antiperiodic boundary conditions , respectively . this leads to a definition of an interface exponent , from the relation : @xmath0 which plays a fundamental rle in the droplet picture of spin glasses @xcite . measures of @xmath1 have been used to perform a high - precision estimate of the spin - glass lower critical dimension @xmath2 in ref . @xcite ( defined as the value of @xmath3 for which the interface exponent @xmath1 vanishes ) . an alternative definition @xcite uses the free - energy cost that is needed to impose a spatial heterogeneity in the order parameter , which in spin glasses is the overlap , i.e. a measure of the similarity between two equilibrated configurations of the same system @xcite . one needs to consider two identical copies of the system constrained in such a way to have their mutual overlaps on the boundaries fixed to some preassigned values . this definition is well suited to test an important property emerging in the replica symmetry breaking ( rsb ) of the spin - glass phase : while the probability distribution @xmath4 for the overlap is broad although having noticeable tails for finite systems reduces to two delta functions in the thermodynamic limit . ] , reflecting the presence of many different and almost degenerate equilibrium phases , one has that for any couple of equilibrated configurations their mutual overlap is spatially homogeneous on a large scale . two configurations with a given value of the overlap in a given macroscopic portion of space will have the same overlap everywhere . starting from this observation a pair of identical systems subject to `` twisted overlap '' boundary conditions was considered in @xcite . two different values @xmath5 and @xmath6 of the overlap were imposed on the two boundaries along a selected direction , chosen among the ones with non - zero probability density in the @xmath4 . it was then argued , that if rsb is present , the boundaries induce a smooth overlap profile in space that interpolates between the two values . above the lower critical dimension @xmath7 for spin - glass order this has a diverging free - energy cost , and one can expect the following scaling with @xmath8 and @xmath9 : @xmath10 where the exponent @xmath11 is positive for @xmath12 . a mean - field calculation of @xmath13 and the interface exponent @xmath14 gave the non - trivial values of @xmath15 and @xmath16 . remarkably , while measuring a quite different property , the resulting value of the lower critical dimension is in very good agreement with the one estimated by the periodic - antiperiodic boundary condition method @xcite . unfortunately , despite its potentially informative content on the nature of the spin - glass phase , the definition of @xcite was never used in numerical studies of the spin - glass phase . this is probably due to the difficulty of imposing values of the overlap at the boundaries . in this paper we start from the general observation , in fact not at all specific to spin glasses , that it is possible to induce interfaces with an alternative procedure . one can divide the physical system into two contiguous halves and impose fixed values to the bulk order parameters in each of the two halves . this procedure , theoretically equivalent to fixing heterogeneous boundary conditions , is better suited for numerical investigations , since the resulting free - energy cost can be related to the probability of a spontaneous fluctuation in an unconstrained system . in the case of spin glasses we have to consider two real replicas and impose values @xmath5 and @xmath6 of their mutual overlaps in two contiguous half spaces . this way of imposing heterogeneity in the system induces an overlap profile in space equal to the one induced by fixing the overlap on the boundaries to suitable values @xmath17 and @xmath18 . one then has a free - energy cost @xmath19 of the form ( [ uno ] ) with the same exponents @xmath14 and @xmath13 . the free - energy cost for imposing the different overlaps is related by boltzmann s relation to the probability that a spontaneous fluctuation of a couple of unconstrained systems produce the values @xmath5 and @xmath6 of the overlaps in the two half spaces : @xmath20 , which is a large deviation formula . this quantity has the advantage that it is easily accessible in numerical simulations and therefore we base our analysis on it . in this paper we focus our attention on overlap interfaces in dyson - like hierarchical spin - glass models . hierarchical models without disorder @xcite have played an important rle in the theoretical and mathematical understanding of critical points @xcite . in the ferromagnetic dyson model one can write exact renormalization group equations that involve the iteration of a function of a single variable ( for a review see @xcite ) . hierarchical spin - glass models of the dyson kind have not to our knowledge been considered in the literature and are in our opinion very attractive as they could allow for an analytical study of non - mean - field disordered critical points and nontrivial low temperature phases . these models provide a hierarchical version of spin glasses with power - law interactions introduced in @xcite . one - dimensional models with power - law interactions have recently received a great deal of attention as test grounds for theoretical ideas about finite - range spin - glass phases @xcite . here , following dyson , we propose to use the tree topology as a further simplification . we choose to start the study of these models focusing on interface properties , for which we can get particularly simple theoretical predictions in the rsb framework . we study the system through the replica method , deriving a recursion equation relating the replica partition functions at the different levels of the hierarchy . this relation , which in principle codes for all the thermodynamic properties of the system is analyzed in a self - consistent way to obtain the interface free - energy . we test our prediction in numerical simulations , finding good agreement . the organization of the paper is the following : in section [ sec : general ] we discuss the definition of the interfaces and how in principle interfaces can be evaluated and some expectations based on rsb theory . section [ sec : hierarchical ] is devoted to the definition of the models we use . in section [ sec : interfaces ] we discuss possible scenarios and expectations and in section [ sec : derivation ] we sketch the theoretical derivation of the interface free - energy cost in hierarchical models . section [ sec : numerics ] discusses the numerical simulations . finally , section [ sec : summary ] concludes the paper . in the appendix we discuss the numerical characterization of the critical point . in this section we define the probability distribution of overlaps in contiguous half spaces and discuss how it can be computed in principle in the large deviation regime . for definiteness we consider a spin - glass system with ising spins @xmath21 on a set of indices @xmath22 with @xmath23 elements , that we divide in two `` half spaces '' @xmath24 and @xmath25 with @xmath26 elements each . for two spin configurations @xmath27 and @xmath28 one can define the partial overlaps @xmath29 in each of the half spaces @xmath30 , @xmath31 as @xmath32 . the joint probability distribution function ( pdf ) of the two overlaps , for fixed value of the system size @xmath8 and quenched disorder @xmath33 , is given by @xmath34 where @xmath35 denotes the partition function . from this relation one gets the usual pdf of the total overlap by simple integration . one can consider the average over the disorder of ( [ p12 ] ) , however , here we will concentrate on the large deviation regime , where one can expect @xmath36 with a positive @xmath37 . the exponent @xmath38 represents an interface free - energy cost to maintain the constrained values . in this regime , the large deviation functional can be expected to be self - averaging and one needs to compute the average free - energy @xmath39 the rsb implies that if one chooses @xmath40 in the domain where @xmath4 is non - zero , then @xmath41 . the property of homogeneity of the overlap should then translate in a form for @xmath19 of the kind @xmath42 . from the complete knowledge of the joint distribution ( [ joint ] ) one can in principle extract the marginal distribution of the difference @xmath43 @xmath44 in the large deviation regime , where this analysis is supposed to be valid , we have @xmath45 for @xmath46 and hence the integral should be dominated by the value of @xmath47 that maximizes the function @xmath48 . as a consequence @xmath49 , the pdf of @xmath50 , should behave as @xmath51 in the tails . in order to understand the behavior for `` small '' values of @xmath50 we can suppose a smooth cross - over between the small and the large fluctuation regimes . in this case the form ( [ log ] ) suggests a finite probability for @xmath52 , i.e. , a scaling form @xmath53 and the marginal probability of the difference @xmath43 , @xmath54 . this quantity @xmath49 is particularly easily accessible in numerical simulations and contains the the information on the ratio between the interface exponents . the comparison of this method with the study of the overlap correlation function @xcite , as well as the comparison of our analytic results in section 5 with pertubative calculations @xcite will be not attempted in this paper . here we introduce hierarchical spin - glass models on dyson lattices . we will consider two families of models , a first one that is better suited for analytic studies and a second one that is more adapted to numerical simulations . in both families the spins are associated to the leaves of a binary tree , see fig . ( [ figuno ] ) . the distance between two spins that , rising up in the hierarchy , meet after @xmath55 branches is naturally defined as @xmath56 . the first family of models we would like to define , is the natural spin - glass generalization of the ferromagnetic dyson model ( @xcite ) . the hamiltonian can be constructed iteratively , connecting the two non - interacting systems of @xmath57 spins @xmath58 to form a composite system of @xmath59 spins in the following way : @xmath60 & = h_{k}^{j_1}[s_1, .... ,s_{2^{k}}]+h_{k}^{j_2}[s_{2^k+1}, .... ,s_{2^{k+1 } } ] - \nonumber\\ & - \frac{1}{2^{(k+1)\sigma}}\sum_{i < j}^{1,2^{k+1}}j_{ij}s_i s_j \label{i}\end{aligned}\ ] ] having defined the hamiltonian for a system at the first level of the hierarchy @xmath61 the couplings @xmath62 are independent and identically distributed gaussian random variables with zero mean and unit variance . the family is parametrized by the value of @xmath63 that tunes the decrease of the strength of the interaction with the distance . the interaction strenght decrease as a power of the distance , the model is therefore a hierarchical counterpart of the one - dimensional spin glass with power - law interactions @xcite which has received attention recently @xcite . as in this case , the model is defined for @xmath64 . exactly at the value @xmath65 it reduces to the infinite range model @xcite . notice that the sum of the squares of the interaction terms that couple the two subsystems @xmath66 scales as @xmath67 . this is on the order of the volume for the mean - field value @xmath68 and smaller than the volume for the nontrivial regime @xmath69 . the second family we consider is a dilute version of model ( [ i ] ) . we consider a hamiltonian for @xmath57 spins with a fixed number of terms @xmath70 @xmath71=-\sum_{\mu=1}^p j^\mu s_{i^\mu } s_{j^\mu } , \label{ii}\end{aligned}\ ] ] where the @xmath72 are taken as @xmath73 variables with equal probability , and we choose the interacting couples of sites @xmath74 independently term by term and in a way that , if the distance between @xmath75 and @xmath76 on the binary tree with @xmath57 branches is @xmath77 , we put the coupling with a probability given by @xmath78 where @xmath79 is a constant chosen such that it normalizes the probability . for @xmath65 the model reduces to the viana - bray spin - glass model @xcite on erds - rnyi random graph . we will refer to the two models defined above as model i ( [ i ] ) and model ii ( [ ii ] ) , respectively . both models should belong to the same universality class for the same value of @xmath63 , with respect to the critical and the low temperature properties . for @xmath80 $ ] the models have a finite temperature spin - glass transition . the critical point has a classical ( i.e. , gaussian ) character for @xmath81 $ ] , while it has a non - classical character for @xmath82 $ ] . as observed in @xcite , diluted models as the one in ( [ ii ] ) are convenient in numerical simulations since the number of interactions for each spin does not grow with the system size . spending an equal amount of computational effort we can therefore study much bigger sizes than for model i , hoping that the finite - size corrections are comparable in both models . let us discuss some scenarios for the behavior of the pdf @xmath83 corresponding to different possible physical situations . the simplest physical situation is the paramagnetic state where long - range order is absent . in this case we have a finite correlation length @xmath84 and for large @xmath85 one can expect @xmath5 and @xmath6 to be sums of @xmath86 independent terms . the resulting probability distribution of @xmath5 and @xmath6 is a product of two independent gaussians with variance proportional to @xmath87 . we can then consider a condensed phase with only two pure states . in this case , the space average value of the overlap can take two values @xmath88 . small fluctuations will still be gaussian as in the paramagnetic case . due to the symmetry of the model , where groups of spins at a given level of the hierarchy are on the same foot , one can expect that large fluctuations where @xmath89 and @xmath90 ( or vice versa ) will imply a free - energy cost @xmath91 and have a probability @xmath92 . the most interesting possibility is a spin - glass phase with rsb . in this case one has a zero mode in the free - energy associated to the existence of a couple of states with overlaps taking values in a finite interval @xcite . if one chooses @xmath93 one finds a broad distribution @xmath94 , which for large @xmath85 is close to the limiting distribution @xmath4 . consider first a system in absence of interactions at the @xmath95-th level . the two subsystems are independent and @xmath96 . if the interaction is switched on , we can expect a free - energy cost @xmath97 . the overlap interface exponent is just given by @xmath98 due to the fact that the total interaction strength _ squared _ between the two parts scales as @xmath99 . the value of the exponent @xmath13 as well as the function @xmath48 can be computed supposing rsb at the level @xmath85 . the detailed calculation is quite involved and is presented in the next section ; here we just give the net result , valid in the regime @xmath100 $ ] . neglecting prefactors we could not compute , it reads : @xmath101 notice the appearance of the @xmath102 function @xmath4 in the exponent of ( [ main ] ) . from this formula one can extract the conditional probability of the difference @xmath43 for a fixed value of the semi - sum @xmath47 : @xmath103 equation ( [ diff ] ) summarizes our prediction for the hierarchical model . for a fixed value of the sum , the difference of the overlap is distributed according to the exponential of the cube , which is different from a naive gaussian guess . the coefficient of the exponential is equal to the function @xmath4 which can be evaluated in independent measurements . in numerical simulations it might be complicated to collect sufficient statistics to condition @xmath50 to the value of @xmath104 . one can then turn to the unconditional distribution . in the large deviation regime , where @xmath105 , this should be dominated for large @xmath85 and finite @xmath50 by the value of @xmath104 which maximizes ( [ diff ] ) . for functions @xmath4 as the one commonly met in spin - glass systems this is the value @xmath106 which is the largest possible value of the overlap in the thermodynamic limit . the form ( [ diff ] ) suggests that the order of magnitude of the typical fluctuations is @xmath107 . in this regime , the unconditional distribution of @xmath50 involves the convolution of ( [ diff ] ) with a presently unknown prefactor and can not be computed . all the form ( [ diff ] ) tells us for the unconditioned distribution in this regime is that the finite volume distribution of @xmath50 admits the scaling form @xmath108 in order to compute the scaling function @xmath109 the knowledge of the prefactor in ( [ diff ] ) would be necessary . we remark that the exponent in this scaling is a strong consequence of rsb theory , a naive guess would have suggested a gaussian distribution with scaling variable @xmath110 . we analyze model i through the replica method . in order to compute the free - energy , it is natural to consider a recursion that relates the average partition function of @xmath111 replicas @xmath112 , @xmath113 , with fixed mutual overlaps @xmath114 . defining @xmath115 = e_j\left[\sum_{{\bf s}}\exp\left(-\sum_{a=1}^n h_k^{j}(s_1^a, ... ,s_{2^k}^a)\right)\prod_{a < b}^{1,n}\delta \left(q_{ab}-\frac{1}{2^k}\sum_{i=1}^{2^k } s_i^a s_i^b \right)\right]\!,\ ] ] where @xmath116 denotes the average over the disorder , we can write : @xmath115 = \exp\left ( \frac{\beta^2}{4}2^{2(1-\sigma)k } \tr { \bf q}^2 \right ) \int { \cal d } { \bf q}_1{\cal d } { \bf q}_2 z_{k-1 } [ { \bf q}_1]z_{k-1}[{\bf q}_2]\delta \left ( { \bf q}-\frac{{\bf q}_1+{\bf q}_2}{2 } \right)\!. \label{zeta}\ ] ] for integer @xmath111 this is an exact relation based on the independence of the hamiltonians of the sub - systems at level @xmath85 . in principle , the thermodynamical properties of the system are encoded in this recursion and in its analytic continuations for @xmath117 . for example it can be used to set up an epsilon expansion for the calculation of the critical indexes for @xmath118 $ ] @xcite . in this paper we just use ( [ zeta ] ) to study the probability distribution ( [ diff ] ) . this can be done with the technique of constrained replicas , introduced and discussed at length in @xcite . in order to consider constrained free - energies for two replicas with fixed overlaps one should fix some of the elements of the matrix @xmath119 @xmath31 to the values of the constraints . writing @xmath120 the constraint reads @xmath121 for @xmath31 and @xmath122 . introducing the replica free - energy at level @xmath85 for fixed @xmath5 and @xmath6 , @xmath123=e^{-\beta 2^k f_k[{\bf q}|q_1,q_2]}$ ] we see that one needs in principle to compute : @xmath124 } = & \int { \cal d } { \bf q}_1{\cal d } { \bf q}_2 e^{\left ( \frac{\beta^2}{4}2^{2(1-\sigma)k } \tr ( { \bf q}_1+{\bf q}_2)^2 \right ) } e^{-\beta 2^k ( f_{k}[{\bf q}_1]+f_k[{\bf q}_2 ] ) } \times \nonumber \\ & \times \prod_{a=1}^{n'}\delta(q^1_{a , a+n'}-q_1)\delta(q^2_{a , a+n'}-q_2 ) . \label{schif}\end{aligned}\ ] ] this form suggests that for large @xmath85 , supposing the knowledge about @xmath125 $ ] , the integral over @xmath126 and @xmath127 can be performed by the saddle - point approximation . notice that the interaction term is sub - extensive and scales as @xmath128 , while the partial free - energies scale as the volume @xmath57 . the interaction term does therefore not contribute to the saddle point , and the maximization with respect to the matrices @xmath129 can be performed separately in each of the two sub - systems . the value of the interaction term at the saddle point determines the free - energy difference @xmath19 . we now study the consequences of the hypothesis that there is rsb in the system . specifically , we suppose that in absence of any constraints rsb is described by a continuous parisi function @xmath130 taking values between the two extremes @xmath131 and @xmath132 . in the constrained problem , correspondingly , each of the matrices @xmath129 is parametrized by two functions @xmath133 and @xmath134 with @xmath135 $ ] . as implied by the analysis in @xcite the free - energy in each of the two sub - systems is then independent of @xmath136 , and the function @xmath133 and @xmath134 can be directly related to the function @xmath130 of the unconstrained system by the relations @xmath137 where @xmath138 is the value of @xmath139 such that @xmath140 . we then see that the free - energy difference from the unconstrained case is entirely due to the interaction term , which can be evaluated using the saddle - point value of the matrices @xmath141 and @xmath142 : @xmath143\!\!.\ ] ] substitution of ( [ p2 ] ) leads to @xmath144 where without loss of generality we have supposed @xmath145 . the first term is just the contribution that can be expected if @xmath40 . together with the two subsystems free - energy it just gives the free - energy of the system at the level @xmath95 . the second term is associated to the free - energy excess needed to impose @xmath146 . for small @xmath147 we can expand this last term and find @xmath148 where @xmath149 is the inverse function of @xmath130 . inserted in ( [ schif ] ) and identifying @xmath150 with @xmath4 as discussed in length in @xcite leads to ( [ diff ] ) @xmath151 in this section we discuss the results of numerical simulations used to test the behavior of the probability distribution of the overlap difference in the two subsystems . in order to deal with pdf s of a single variable we concentrated to the unconditional probability for which we have the theoretical prediction ( [ uncond ] ) . we did not try to test the more detailed prediction ( [ diff ] ) which would need the numerical determination joint pdf s of two variables . we have simulated model ii for @xmath152 , using parallel tempering to thermalize the system at low temperatures . we have tested our theoretical predictions concentrating on two values of @xmath63 : @xmath153 which lies in the classical region @xmath154 , where the spin - glass transition is well described by mean - field theory , and @xmath155 which lies in the non - classical region @xmath156 , where the exponents are nontrivial . we have characterized the critical point of the model following the procedure that we describe in the appendix . we estimate the critical temperatures to be equal to @xmath157 for @xmath153 and @xmath158 for @xmath155 . to test the predictions described in the previous sections we measured the distribution of the variables @xmath104 and @xmath50 as a function of temperature and system size . we considered systems sizes of @xmath159 and @xmath160 spins . averages were performed over 4000 samples for the smaller systems and 700 samples for the largest system . the configurations are thermalized during the first @xmath161 monte carlo ( mc ) sweeps of the runs and then data is collected for the subsequent @xmath161 mc steps . the first prediction we test is the validity of the cube - exponential form in the tails for large @xmath50 . this is well observed in all our simulations . a typical example of our findings is depicted in figure ( [ fig1 ] ) where we plot the function @xmath162 for @xmath163 for the two different values of @xmath63 in the classical and non - classical regime , respectively . the data are plotted together with functions of form @xmath164 which should be considered as a guide to the eye rather than the best fit . the parameters @xmath165 and @xmath166 were fixed by eye to be equal to @xmath167 , @xmath168 for @xmath155 and @xmath169 , @xmath170 for @xmath153 . a best fit procedure results to be sensitive to the chosen fitting interval and to the tails of the distribution that represent probabilities too small to be correctly estimated with our statistics . despite these caveats we believe that our data give an indication in favor of the cubic behavior for both values of @xmath63 . we then investigated the scaling of the pdf of @xmath50 with the system size , contrasting it with the behavior of the pdf of @xmath104 . in figures [ fig2 ] and [ fig4 ] we display the function @xmath171 for the two values of @xmath63 at low temperatures . the function @xmath171 has the characteristic appearance of the one of systems developing rsb for large volumes , with two symmetric peaks and a non - zero part for @xmath172 . figures [ fig3 ] and [ fig5 ] in contrast show that the distribution of @xmath50 , @xmath162 , is unimodal around zero . its width is as expected a decreasing function of @xmath85 . in the insets the unscaled data are presented , while the in the main panel the result of scaling the data using the variable @xmath173 is shown . we judge the scaling ( [ uncond ] ) to be very satisfactory , though not perfect for the values of @xmath85 we explored . indeed , while for @xmath153 we could not find a the value of the ratio @xmath174 producing a better data collapse than the theoretical value @xmath175 , for @xmath155 the value @xmath176 produces a better data collapse than the theoretical value @xmath177 . we believe that this discrepancy is due to finite size effects , but we can not exclude at present that the theory should be amended in the non - classical region . our data show that the best fitting exponent is largely temperature independent in the low temperature region . a direct scaling of the data in the high temperature region for the values of @xmath178 that we dispose , produces an effective exponent that crosses over slowly from the low temperature value towards the paramagnetic value @xmath179 . we believe that we are seeing a preasymptotic behavior due to the influence of the critical fixed point . this influence could be particularly marked due to the power - law interactions where the critical fixed point continues to attract the system on relatively large scales . larger and larger values of @xmath85 are necessary to observe the paramagnetic behavior closer and closer to the critical temperature . in this paper we have proposed a new method to determine numerically the overlap interface exponents first defined in @xcite . these can be obtained by looking at the finite - size scaling of the pdf of the differences of the overlaps between two replicas in two subsystems . we have applied the definition to hierarchical models , where we could give a theoretical prediction for the scaling of the overlap differences with size . though the interface exponent is naturally dictated by the model , the dependence in @xmath50 is found to be nontrivial in presence of rsb . we tested this dependence in numerical simulations finding very satisfactory agreement both in the classical and in the non - classical region . these results confirm the interest of hierarchical spin - glass models , that combine analytical tractability , nontrivial critical points and rsb low temperature phases . * acknowledgments * it is a pleasure to thank o. c. martin , m.mzard , f. ricci - tersenghi , p. contucci , c. giardin , c. giberti and c. vernia for interesting discussions . in this appendix we discuss the characterization of the critical point of the model for the values of the interaction parameter @xmath63 that we have considered in the text . the model has been simulated with the parallel tempering algorithm , with 10 values of the temperature , using @xmath161 thermalization steps before collecting data in the @xmath161 steps . we considered systems sizes of @xmath180 and @xmath160 spins . averages were performed over 4000 samples for the smaller systems and 700 samples for the largest system . simulating two replicas in parallel , we have studied the second and the fourth moment of the distribution of the mutual overlap @xmath181 and @xmath182 . we have identified the critical temperature and the exponents @xmath183 and @xmath184 using finite size scaling through the behavior of @xmath6 , @xmath185 and the corresponding binder parameter @xcite @xmath186 . in the non - classical region , @xmath187 where finite - size scaling should hold the various parameters exhibit the following dependence on the temperature and system size @xmath188 : @xmath189 the exponent @xmath183 should not renormalize in long range models , and analogously to the euclidian 1d model take the value @xmath190 both in the non - classical and in the classical regions@xcite . in the classical region , @xmath154 , the scaling implied by ( [ noncl ] ) does not hold @xcite . it is possible to show , using the fact that the critical theory is described by a cubic action analogous to the one for short - range spin glasses @xcite , that the various quantities scale according to the following : @xmath191 the exponent @xmath192 in the scaling functions can be derived from dimensional analysis from the cubic action . the exponent @xmath193 takes the value @xmath194 independently of @xmath63 as can be checked from the scaling relation @xmath195 with @xmath196 . let us now turn to the data considering the non - classical region first . to analyze the data we observe the following procedure : we first estimate the critical temperature from the crossing point of the binder parameter and the rescaled values , especially , @xmath197 and @xmath198 since these provide for a cleaner crossing than the binder parameter . we then fix the value of @xmath184 in order to collapse the curves . the result for @xmath155 is shown in figure [ fig6 ] , we present the data for @xmath6 , @xmath185 and @xmath199 scaled as in ( [ sc ] ) using a value of @xmath200 . if we try to use the scaling ( [ noncl ] ) in the classical regime we get inconsistent results : although we obtain an approximate crossing of the curves for the three quantities , the temperatures at which the curves cross do clearly not coincide . the crossing of the binder parameter indicates @xmath201 and as shown in figure [ fig6 ] for @xmath153 , we obtain consistent scaling assuming the form ( [ sc ] ) and not ( [ noncl ] ) .
|
we discuss interfaces in spin glasses .
we present new theoretical results and a numerical method to characterize overlap interfaces and the stability of the spin - glass phase in extended disordered systems .
we use this definition to characterize the low temperature phase of hierarchical spin - glass models .
we use the replica symmetry breaking theory to evaluate the cost for an overlap interface , which in these models is particularly simple .
a comparison of our results from numerical simulations with the theoretical predictions shows good agreement .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the non - keplerian rotation curves of spiral galaxies provided the first observational evidence that galaxies are embedded in extensive dark matter haloes ( @xcite ) . historically , the 21-cm emission from atomic neutral hydrogen gas ( hi ) has been the main tool to derive galaxy rotation curves , because of its capability to trace the gravitational field beyond the optical stellar disc . however , hi is less useful in constraining the rotation curve in the central parts of discs usually due to insufficient spatial resolution and the lack of hi gas in the inner parts of galaxies ( e.g. , @xcite ) . the interstellar medium in the centre is instead dominated by gas in the molecular and ionised phases ( e.g. , @xcite ) . unfortunately , the rotation curves derived through co emission , the common tracer for the molecular gas distribution , often show non - axisymmetric signatures such as wiggles ( @xcite ) . hot ionised gas has the additional disadvantage that the observed rotation alone is often insufficient to trace the total mass distribution and that its velocity dispersion needs to be included . this velocity dispersion is generally influenced by a typically unknown contribution from non - gravitational effects such as stellar winds and shocks ( e.g. , @xcite ) . some of these shortcomings of gas tracers can be overcome through careful correction and analysis . however , a broader shortcoming of gas tracers becomes apparent because of their dissipative nature . the gas is easily disturbed by perturbations in the plane from , for example , a bar or spiral arm ( @xcite ) . gas also settles in the galaxy disc plane ( or polar plane ) and is thus less sensitive to the mass distribution perpendicular to it . the typical velocity dispersions of the neutral and ionized gas , respectively , are @xmath410 kms@xmath5(@xcite ) and @xmath420 kms@xmath5(e.g . @xcite ) . instead of gas , stars could be used as a tracer of the underlying gravitational potential . stars are present in all galaxy types , are distributed in all three dimensions , and are collisionless making them less susceptible to perturbations . the random motion of the collisionless stars tipycally ranges from 20 kms@xmath5to above 300 kms@xmath5(@xcite ) . this implies a scale height difference of over an order of magnitude between the gaseous and stellar components ( e.g. , @xcite ) . however , we need to measure both the ordered and random motions of stars before they can be used as a dynamical tracer . their random motion can be different in all three directions , an effect known as the velocity anisotropy . this anisotropy requires more challenging observational and modelling techniques to uncover the total mass distribution . nonetheless , these techniques are becoming available . integral - field spectrographs like ` sauron ` ( @xcite ) , which is used in this study , allow us to extract high - quality stellar kinematic maps and enable us to perform dynamical models . a common approach for spiral galaxies is to apply the asymmetric drift correction ( adc ; @xcite ) to infer the circular velocity curve from the measured stellar mean velocity and velocity dispersion profiles . this approach is straightforward since the velocity and dispersion profiles can be obtained from long - slit spectroscopy , and no line - of - sight integration is required due to an underlying thin - disc assumption . using this method requires assuming both the disc inclination and the magnitude of the velocity anisotropy . as such , the adc approach is widely adopted in studies that use the stellar kinematics to infer the circular velocity curve . for example , when investigating the tully - fisher relation for earlier - type spirals ( e.g. , @xcite , @xcite ) , the speed of bars ( e.g. , @xcite ) , as well as the inner distribution of dark matter ( e.g. , @xcite ) . in this paper , we adopt an alternative approach , namely fitting a solution of the axisymmetric jeans equations to stellar mean velocity and velocity dispersion fields to infer mass distributions . we apply this approach to the data acquired with the integral - field spectrograph sauron from inner parts of 18 late - type sb sd spiral galaxies . these jeans models are less general than orbit - based models ( e.g. , schwarzschild s method ; @xcite ) but are far less computationally expensive while still providing a good description of galaxies dominated by stars on disc - like orbits ( e.g. , @xcite ) . the applicability of these fitted jeans solutions even applies to dynamically hot systems such as lenticular galaxies . the jeans models take into account the two - dimensional information in the stellar kinematic maps as well as integration along the line - of - sight . we also compare the resulting circular velocity curves with those obtained through adc to investigate the validity of the assumptions underlying the simpler adc approach . in section [ s : observations ] , we summarise the sample including the ` sauron ` observations and near - infrared imaging . the jeans and adc modelling approaches are described in section [ s : method ] and applied via markov chain monte carlo technique in section [ s : analysis ] . we then compare the circular velocity curves from both modelling approaches in section [ s : vcirc ] and discuss the possible reasons for the significant differences we find in section [ s : discussion ] . we draw our conclusions in section [ s : concl ] . = 2.0 mm * 12|l| ngc & type & d & pa & @xmath6 & _ i _ & @xmath7 & @xmath8 & @xmath9 & @xmath10 & @xmath11 & @xmath12 + ( 1 ) & ( 2 ) & ( 3 ) & ( 4 ) & ( 5 ) & ( 6 ) & ( 7 ) & ( 8) & ( 9 ) & ( 10 ) & ( 11 ) & ( 12 ) + 0488 & sa(r)b & 32.1 & 5 & 0.230 & 42 & 2299 & 53.6 & 19.2 & 42.7 & 9.9 & 0.4 + 0772 & sa(s)b & 35.6 & 126 & 0.340 & 51 & 2506 & 54.0 & 20.6 & 47.1 & 19.0 & 0.4 + 4102 & sab(s)b & 15.5 & 42 & 0.445 & 59 & 838 & 15.0 & 22.1 & 19.5 & 1.5 & 1.1 + 5678 & sab(rs)b & 31.3 & 5 & 0.475 & 61 & 1896 & 24.4 & 24.0 & 20.9 & 3.1 & 1.1 + 3949 & sa(s)bc & 14.6 & 122 & 0.360 & 53 & 808 & 20.4 & 22.1 & 16.8 & 4.6 & 1.3 + 4030 & sa(s)bc & 21.1 & 37 & 0.240 & 43 & 1443 & 30.2 & 19.2 & 26.5 & 15.3 & 0.7 + 2964 & sab(r)bc & 20.6 & 96 & 0.450 & 59 & 1324 & 20.8 & 21.0 & 16.9 & 1.0 & 1.2 + 0628 & sa(s)c & 9.8 & 25 & 0.190 & 38 & 703 & 90.7 & 22.1 & 70.7 & 12.3 & 0.3 + 0864 & sab(rs)c & 21.8 & 26 & 0.320 & 50 & 1606 & 38.3 & 15.1 & 28.1 & 1.6 & 0.5 + 4254 & sa(s)c & 19.4 & 50 & 0.270 & 46 & 2384 & 52.1 & 19.2 & 40.7 & 15.8 & 0.5 + 1042 & sab(rs)cd & 18.1 & 174 & 0.290 & 47 & 1404 & 73.4 & 15.1 & 52.5 & 4.4 & 0.3 + 3346 & sb(rs)cd & 18.9 & 100 & 0.160 & 42@xmath13 & 1257 & 55.3 & 17.8 & 35.4 & 3.2 & 0.5 + 3423 & sa(s)cd & 14.7 & 41 & 0.230 & 42 & 1001 & 52.6 & 20.1 & 40.1 & 9.7 & 0.5 + 4487 & sab(rs)cd & 14.7 & 77 & 0.370 & 53 & 1016 & 49.8 & 19.2 & 36.8 & 10.6 & 0.5 + 2805 & sab(rs)d & 28.2 & 125 & 0.240 & 43 & 1742 & 65.8 & 12.6 & 50.8 & 13.0 & 0.3 + 4775 & sa(s)d & 22.5 & 96 & 0.135 & 34@xmath13 & 1547 & 29.0 & 17.8 & 19.3 & 8.3 & 0.9 + 5585 & sab(s)d & 8.2 & 38 & 0.360 & 53 & 312 & 69.7 & 17.8 & 53.2 & 15.7 & 0.3 + 5668 & sa(s)d & 23.9 & 120 & 0.155 & 37@xmath13 & 1569 & 38.3 & 16.5 & 30.3 & 13.8 & 0.5 + [ tab : prop ] our sample consists of 18 nearby , late - type spiral galaxies with hubble types ranging from sb to sd . the sample selection , observations , and data reduction are presented in detail in ( * ? ? ? * ; * ? ? ? * hereafter g06 and g09 ) . the galaxies all have imaging data from the hubble space telescope ( hst ) including either wfpc2 and/or nicmos data @xcite . all targets were selected to be brighter than @xmath14 according to the values listed in the @xcite catalogue , where interacting and seyfert galaxies were discarded . the sample also excludes targets that are inaccessible to the 4.2-m william herschel telescope ( [ ss : sauronifs ] ) , so only galaxies with 0 @xmath15 ra @xmath16 and @xmath17@xmath18 have been selected . in table [ tab : prop ] we list the main properties of the 18 galaxies , combined from the literature and our own measurements . their morphological type ranging between sb and sd , together with the galactocentric distance ( d ) , are taken from g06 using nasa / ipac extragalactic database ( ned ) . the photometric position angle ( pa ) is measured by g09 using digital sky survey ( dss ) images , but the typical uncertainty of 5@xmath18 - 10@xmath18 comes from comparing the measured values of the pa in g06 to literature values compiled from the third reference catalog of bright galaxies ( rc3 ) . in the same way , we estimated the typical uncertainty of 15% for the ellipticity @xmath6 , which we also take from g06 to compare to the rc3 catalogue . from the axis ratio of the galaxies ( @xmath19 ) , we calculate the photometric inclination ( _ i _ ) of the galaxies using @xmath6 from table [ tab : prop ] in the following way ( @xcite ) : @xmath20 here @xmath21 is the intrinsic axis ratio of the galaxy ( e.g. @xcite ) and we adopt @xmath21 = 0.2 as commonly used ( e.g. , @xcite ) . the change in the value of @xmath21 slightly affects our estimation of the inclination ( 1@xmath183@xmath18 ) . the total uncertainty of the inclination , considering the errors of @xmath22 and @xmath21 can reach 5@xmath18 . in column 6 of table [ tab : prop ] , we present the inclinations we adopted throughout our entire analysis . note here that the listed inclinations of the galaxies ngc3346 , ngc4775 and ngc5668 are slightly larger than the corresponding photometric values calculated in eq . ( [ eq : incl ] ) by 7@xmath18 , 1@xmath18and 2@xmath18 , respectively . for these three galaxies , the estimated photometric inclination is below the minimum allowed mge inclination ( @xmath23 ) of the jam approach ( see sec . [ ss : jam ] ; eq.[eq : ilim ] ) , and thus we adopt @xmath24 . nevertheless , the difference between the adopted and photometric inclination for these three galaxies is generally within the uncertainty of the photometric inclination , and does not affect our results . we wish to keep a constant inclination through this analysis to focus on the differences between the jeans and adc approaches . further , we measure the systemic velocity ( @xmath7 ) using the enforced point - symmetry method with typical uncertainty of 1 km s@xmath5 ( see sec . [ ss : stellarkinematics ] ) and determine the galaxy effective radius @xmath25 ( half - light radius ) after applying a multi - gaussian expansion method ( sec . [ ss : surfacebrightness ] ) to the surface brightness profiles of the galaxies . using chi - squared values of the fit , we estimate that the typical @xmath26 uncertainty of the effective radius does not exceed 10% . the radial extent @xmath9 of the stellar kinematic data is estimated via @xmath27 package ( @xcite ) , yeilding a typical uncertainty for our galaxies of 1% . the disk scale length ( @xmath10 ) and effective radius of the bulge ( @xmath11 ) are taken from g09 including their typical uncertainties of 5% and 15% , respectively . we then calculate the radial extent ( @xmath28 ) of the stellar kinematic data in terms of disk scale lengths with an associated typical uncertainty of 5 % ( see table [ tab : prop ] ) . the sample galaxies were observed with the integral - field unit ( ifu ) spectrograph ` sauron ` at the 4.2-m william herschel telescope of the observatorio del roque de los muchachos on la palma , spain . the ` sauron ` ifu @xcite has a @xmath29 field - of - view ( fov ) , sampled by an array of @xmath30 square lenses . the fov corresponds to a typical radial extent of 1/5 to 1/3 of the galaxy s half - light radius ( @xmath25 ) . the spectral resolution is @xmath4 4.2 @xmath31 ( fwhm ) , corresponding to an instrumental velocity dispersion of @xmath32 kms@xmath5 in the observed spectral range 4800 - 5380 @xmath31 ( at 1.1 @xmath31 per pixel ) . this range includes a number of absorption features fe , mgb and h@xmath33 , which we use to measure the stellar kinematics . emission lines in this range , such as [ oiii ] , [ ni ] and h@xmath33 , can be used to probe the ionised gas properties ( g06 ) . the observations were reduced by g06 using the dedicated software @xmath34 @xcite . to obtain a sufficient signal - to - noise ratio ( s / n ) , we spatially binned the data cubes using the voronoi 2d binning algorithm of @xcite . we created compact bins with a minimum s / n @xmath35 per spectral resolution element . in the central regions many individual spectra have s / n @xmath36 60 and thus remained un - binned . we parametrize the light distribution of the galaxies with the surface brightness profiles obtained by g09 . they used archival ground - based @xmath37-band images from two - micron all sky survey ( hereafter , 2mass ) complemented with near - infrared hst nicmos / f160w images for 11 cases and optical hst wfpc2/f814w for the remaining 7 galaxies ( ngc1042 , ngc2805 , ngc3346 , ngc3423 , ngc4487 , ngc4775 , ngc5668 ) . dss data were used for the outer parts of the same galaxies to obtain an accurate determination of the sky level and the galaxy disc geometry . g09 derived the light distribution parameters using the ` ellipse ` task in imaging reduction and analysis facility ( iraf ) . they first fitted elliptical isophotes to galaxy images with the centre , position angle and ellipticity left as free parameters in order to obtain the centre coordinates . g09 then fit again ellipses to the images with the centre fixed , where the position angle and ellipticity were set as free parameters . in the end , there were three photometric profiles and their combination gave a single near - infrared @xmath37-band profile with the maximum extent and inner spatial resolution allowed by the data . the error introduced by this combination of optical and infrared images was negligible ( g09 , sec.3.3 ) and does not affect our analysis . for most of the 18 late - type spiral galaxies , the ` sauron ` stellar mean velocity and velocity dispersion maps are , within the observational uncertainties , consistent with axisymmetry . the non - axisymmetric features appear to be mainly due to dust obscuration . in half of the galaxies , the effect of bars seems to be weak or more dominant in the outer parts . these trends support the assumption of a stationary axisymmetric stellar system for the inner parts of the galaxies that we study here . the methodology of @xcite has proven to be powerful in building detailed models for spherical , axisymmetric ( e.g. * ? ? ? * ; * ? ? ? * ) , and triaxial nearby galaxies @xcite , as well as globular clusters ( e.g. * ? ? ? * ; * ? ? ? the method finds the set of weights of orbits computed in an arbitrary gravitational potential that best reproduces all available photometric and kinematic data at the same time . since higher - order stellar kinematic measurements are necessary to constrain the large freedom in this general modelling method , a less computationally intensive approach has been to construct simpler but still realistic dynamical models based on the solution of the axisymmetric jeans equations ( @xcite ) . we adopt the latter approach here . in the case of steady state axisymmetry both the potential @xmath38 and distribution function ( df ) are independent of azimuth @xmath39 and time . by jeans theorem @xcite the df depends only on the isolating integrals of motion : @xmath40 , with energy @xmath41 , angular momentum @xmath42 parallel to the symmetry @xmath43-axis , and a third integral @xmath44 for which in general no explicit expression is known . however , @xmath44 usually is invariant under the change @xmath45 , though @xmath44 may loose this symmetry if resonances are present . such symmetry implies that the mean velocity is in the azimuthal direction ( @xmath46 ) and that the velocity ellipsoid is aligned with the rotation direction ( @xmath47)20 degrees ( @xcite ) , and these vertex deviations may also correlate with morphological features ( @xcite ) ] . when we multiply the collisionless boltzmann equation in cylindrical coordinates by @xmath48 and @xmath49 respectively and integrate over all velocities , we obtain two of the jeans equations @xcite , @xmath50 where @xmath51 is the intrinsic luminosity density has the same meaning as the luminosity density @xmath52 in @xcite ] . due to the assumed axisymmetry , all terms in the third jeans equation , that follows from multiplying by @xmath53 , vanish . the _ jeans axisymmetric model _ ( jam ) is based on solving the above two jeans equations and . however , since there are four unknown second - order velocity moments @xmath54 , @xmath55 , @xmath56 and @xmath57 this is an underconstrained system , and we have to make two assumptions about the velocity anisotropy , or in other words about the shape and alignment of the velocity ellipsoid . first , we assume the velocity ellipsoid is aligned with the cylindrical @xmath58 coordinate system , which gives @xmath59 . we can then readily solve equation for @xmath55 . second , we also assume a constant flattening of the velocity ellipsoid in the meridional plane , so we can write @xmath60 and solve equation for @xmath56 . @xcite argue that this second assumption provides a good , general description for the kinematics of real disc galaxies . when @xmath61 , the velocity distribution is isotropic in the meridional plane , corresponding to the well - known case of a two - integral distribution function @xmath62 ( e.g. * ? ? * ; * ? ? ? . given the intrinsic second - order velocity moments , we can then calculate the observed second - order velocity moment by integrating along the line - of - sight through the stellar system viewed at an inclination @xmath63 away from the @xmath43-axis : @xmath64\ , { \mathrm{d}}z',\end{aligned}\ ] ] where @xmath65 is the ( observed ) surface brightness with the @xmath66-axis along the projected major axis . for each position @xmath67 on the sky - plane , @xmath68 yields a prediction of the ( luminosity weighted ) combination @xmath69 of the ( observed ) mean line - of - sight velocity @xmath70 and dispersion @xmath71 . under the above assumptions , the only unknown parameters are the anisotropy parameter @xmath72 , the inclination @xmath73 , and the gravitational potential @xmath38 , which is related to the total mass density @xmath74 through poisson s equation . to estimate @xmath74 , we derive the intrinsic luminosity density @xmath51 by deprojecting the observed surface brightness @xmath65 . we then assume a total mass - to - light ratio @xmath75 to derive @xmath76 . dynamical studies of the inner parts of galaxies typically consider @xmath75 as an additional parameter and further assume its value to be constant , i.e. , mass follows light ( e.g. * ? ? ? since @xmath75 may be larger than the stellar mass - to - light ratio @xmath77 , this assumption still allows for possible dark matter contribution , albeit with a constant fraction . a convenient way to arrive at @xmath38 is through the multi - gaussian expansion method ( mge ; * ? ? ? * ; * ? ? ? * ) , which models the observed surface brightness as a sum of @xmath78 gaussian components , @xmath79 \right\},\ ] ] each with three parameters : the central surface brightness @xmath80 , the dispersion @xmath81 along the major @xmath66-axis and the flattening @xmath82 . the mge approach has several advantages . even though gaussians do not form an orthogonal set of functions , surface density distributions are accurately reproduced . when the point spread function ( psf ) of the instrument is also represented as a sum of gaussians , the convolution with the psf becomes straightforward . given the viewing direction , the mge parametrization can be deprojected analytically into an intrinsic luminosity density @xmath51 . furthermore , the calculation of @xmath68 in equation reduces from the ( numerical ) evaluation of a triple integral to a straightforward single integral ( * ? ? ? * eq . similarly , the gravitational potential @xmath38 can be calculated by means of one - dimensional integral ( * ? ? ? * eq . 39 ) . given the latter , the circular velocity from the jam model in the equatorial plane then follows upon ( numerical ) evaluation of @xmath83 where @xmath84 and @xmath85 are the total luminosity and the mass - to - light ratio of the @xmath86th gaussian . the intrinsic dispersion and flattening , @xmath87 and @xmath88 , are related to their observed quantities , as @xmath89 with inclination @xmath73 , ranging from @xmath90 for face - on viewing to @xmath91 for edge - on viewing . note here that the oblate mge deprojection is valid if @xmath92 for all gaussians ( @xcite ) . therefore , the flattest gaussian in the mge fit defines the minimum possible inclination ( @xmath23 ) for which the mge model can be applied within the jam approach . in what follows , we assume mass follows light , so that the mass - to - light ratio is the same for each gaussian : @xmath93 . we fix the galaxies inclinations to the values shown in table [ tab : prop ] , and we are left with two free parameters in jam : the total or dynamical mass - to - light ratio @xmath75 and the velocity anisotropy @xmath94 . for each galaxy , we obtain the values of these two parameters by fitting the observed second - order velocity moment @xmath69 , after which the jam circular speed follows from equation . in this work , we compare the jam - derived circular speed curves to those from the commonly adopted adc method . here we relate the adc approach back to the jeans equations . instead of solving both jeans equations and , we can instead evaluate the first equation in the equatorial plane ( @xmath95 ) and use @xmath96 and @xmath97 ( by symmetry ) to rewrite eq . as @xmath98 .\end{gathered}\ ] ] here , @xmath99 is the intrinsic mean velocity , and @xmath100 , @xmath101 are ( the square of ) the intrinsic mean velocity dispersions . in case of a dynamically cold tracer such as neutral gas , observed through i and co emission at radio wavelengths , the mean velocity is typically much larger than the velocity dispersion ( @xmath102 ) , so that the circular velocity can be inferred from the deprojected observed rotation , @xmath103 ( e.g. , @xcite ) . however , the velocity dispersion can be non - negligible or even dominant for dynamically hot tracers like stars where it is possible that @xmath104 . in this case , the asymmetric drift must be taken into account since the observed rotation only captures part of the circular velocity ( e.g. , @xcite ) . from equation , this asymmetric drift correction depends on the two intrinsic velocity dispersions @xmath105 and @xmath106 , plus the cross term @xmath57 , which together define the velocity ellipsoid of the galaxy . to allow for a direct comparison with the jam model , we adopt the same assumption of alignment of the velocity ellipsoid with the cylindrical coordinate systems , so that @xmath107 . assuming that the velocity ellipsoid is furthermore symmetric around @xmath108 , it follows from appendix a from @xcite is efectively the epicycle approximation , which is applied only in adc and not in jam . ] of @xcite that @xmath109 with @xmath110 the radial logarithmic gradient of the intrinsic mean velocity : @xmath111 . under the assumption of a thin disc with @xmath112 , the circular velocity from the adc approach in the equatorial plane then reduces to @xmath113.\end{gathered}\ ] ] this final equation provides an estimate of @xmath114 that uses : ( 1 ) the surface brightness profile @xmath115 from the mge parametrization , ( 2 ) @xmath99 and @xmath110 from the observed mean line - of - sight velocity @xmath70 , and ( 3 ) @xmath105 from the observed line - of - sight velocity dispersion @xmath71 . the latter two inferences also involve the velocity anisotropy @xmath116 and inclination @xmath73 as described in sec . [ ss : radialprofiles ] below . here , we describe the application of our model analyses to the observational data . in sec . [ ss : surfacebrightness ] , we discuss the multi - gaussian expansion model adopted to the photometry of the ` sauron ` galaxies in order to derive a smooth , analytic representation of their surface brightness . we explain the extraction of the stellar kinematic maps in sec . [ ss : stellarkinematics ] . the performance of the markov chain monte carlo analysis to the jam and adc models is shown in sec . [ ss : mcmc ] . adc holds a thin disc assumption , whereas jam does a full line - of - sight integration through the luminosity distribution to model the observed velocity moments . to infer the mass distributions of the 18 ` sauron ` galaxies , we use the photometric bulge - disc decompositions of the total h - band surface brightness profile , as presented in @xcite and g09 ( see also sec . [ ss : nirimaging ] ) . these models represent each galaxy s profile as the superposition of an exponential disc and a srsic bulge . the galaxies in our sample can contain a significant amount of interstellar dust , which is mostly transparent at near - infrared wavelengths . the extinction @xmath117 is roughly a factor of 8 lower than the extinction @xmath118 @xcite , so we do not apply an internal extinction correction . additionally , the near - infrared is considered as a good tracer of the stellar mass in galaxies as the light is dominated by old stars @xcite , although there is some concern that this might be not true due to uncertain influence of intermediate age populations ( @xcite ) . @xcite and g09 use a one - dimensional mge decomposition @xcite of the bulge and disk profiles , using the implementation of @xcite . they adopt 10 gaussians to represent the discs and between 13 and 19 gaussians to represent the bulges . the fits to the one - dimensional profiles are excellent for all galaxies , but the mge is not applied to the two - dimensional images . most of the galaxies ( 16 out of 18 ) display a clear light excess above the srsic fit to the bulge , which can be attributed to a bright nuclear star cluster @xcite , where a luminous mass contributes to the kinematics in the inner regions . for these galaxies , we amend the mge decomposition of ( * ? ? ? * chapter 5 ; sec.5.4 . ) by including one additional , circular gaussian to accurately represent the light excess in the centre . figure [ fig : mge ] shows the mge light models of two representative galaxies in our sample . the green asterisks indicate their observed light profiles . the black thick curves represent the sum of the individual gaussians ( dotted lines ) of the three components : nucleus ( yellow ) , srsic bulge ( red ) , and exponential disc ( blue ) , which are marked in appendix [ a : mgetab ] with index of 0 , 1 and 2 , respectively . ngc628 ( left panels ) is an example of a very good fit to the data , which is typical for the majority of our galaxies . however , there are a few exceptions with mismatches due to bars or prominent spiral arms . these are ngc864 ( figure [ fig : mge ] , right panel ) , ngc772 , and ngc1042 . nevertheless , we considered the mge fits to these three galaxies to be satisfactory for our needs , because fitting non - symmetric features could lead to uneven representation of the galaxies surface brightness profiles , and hence , gravitational potentials . we converted the resulting peak surface brightnesses of the gaussians into physical units of @xmath119 , taking the absolute magnitude of the sun @xmath120 @xcite . we measured the stellar kinematics using the penalised pixel - fitting ( ppxf ) method of @xcite . for spectral templates , we used a sub - sample of the miles stellar library @xcite , containing @xmath4115 stars that span a large range in atmospheric parameters such as surface gravity , effective temperature and metallicity . we convolve the miles models of stars from their original spectral resolution to that of the data . this is done by convolving with a gaussian with dispersion equal to the difference in quadrature between the final and starting resolution . the ppxf method fits a non - negative linear combination of these stellar template spectra , convolved with the gaussian velocity distribution , to each galaxy spectrum by chi - square minimisation . the higher - order gauss - hermite moments h3 and h4 are not included in this fit as free parameters . the spectral regions affected by emission lines were masked out during this process . a low - order polynomial ( typically sixth degree ) is included in the fit to account for small differences in the flux calibration between the galaxy and the template spectra . this analysis yields the stellar mean velocity ( @xmath70 ) and stellar velocity dispersion ( @xmath71 ) for each bin . their errors are estimated through monte - carlo simulations with noise added to the galaxy s best fitting model spectrum . the ` sauron ` instrumental resolution is 105 kms@xmath5while the measured velocity dispersions of our galaxies can be significantly below that level . @xcite used monte - carlo simulations to show that the intrinsic velocity dispersion is still well recovered . for a spectrum with signal - to - noise of @xmath121 and @xmath122 kms@xmath5 , the ppxf method will give velocity dispersions differing from the intrinsic ones by @xmath123 kms@xmath5 , consistent with the measured errors . we might even expect larger uncertainties for some of our galaxies ( e.g. , ngc3346 , ngc4487 and ngc5585 ) , which own central velocity dispersions below this limit . on the other hand , recent papers put forward evidence that our adopted approach could accurately recover the intrinsic velocity dispersion to better than 10% precision given the simulations of @xcite and @xcite and our high s / n ( @xmath124 ) in each voronoi bin . in fig . [ fig : maps1 ] we show the stellar kinematic maps of our sample of 18 late - type spiral galaxies resulting from the ppxf fits . the first column shows the stellar flux in arbitrary units on a logarithmic scale . the next two columns display @xmath70 and @xmath71 maps in kms@xmath5 respectively . we over - plotted the surface brightness contours of the galaxies as derived from their intensity maps . for some of our galaxies , the centre position was not accurately determined during the data reduction due to foreground stars and dust lanes , which in some cases caused a significant offset in the measurement of the systemic velocity v@xmath125 . therefore , we use the velocity field symmetrisation method described in appendix a of @xcite to estimate a robust v@xmath125 . this method assumes that the velocity field ( 1 ) is symmetric with respect to the galaxy centre , ( 2 ) is uncorrelated and ( 3 ) varies linearly along the spatial coordinates . then , for each position that has a counterpart , it computes their weighted mean velocities and combined errors ( rejecting data with errors @xmath126 km s@xmath5 ) . in this way we obtain a robust estimate of v@xmath125 for each galaxy ( see table [ tab : prop ] ) . galaxies with more concentrated light distributions generally have larger central peaks in their velocity dispersion fields . elliptical galaxies usually have radially decreasing @xmath71 @xcite , and this is also the case for many early - type spirals , as a result of a centrally concentrated bulge . but for very late - type spirals , we expect the @xmath71 field to be either flat ( due to their lower bulge - to - disc ratios ) or with a central decrease , because of cold components or counter - rotating discs ( @xcite ) . in comparison with early - type galaxies , the bulges of the late - type spirals are smaller and have lower surface brightness ( * ? ? ? @xcite also find that @xmath71 generally decreases with radius with an @xmath127-folding length that is twice the photometric disc scale length . thus , the variety seen in @xmath71 profiles of ` sauron ` galaxies might be also due to their limited radial coverage , ranging between 0.4 and 1.2 scale lengths . to obtain reliable circular velocity curves and associated uncertainties , we assume these properties are random variables viewed in a bayesian framework . we then applied markov chain monte carlo ( mcmc ) to estimate these variables under both the adc and jam model approaches . we used the @xmath128 code of @xcite , an implementation of an affine invariant ensemble sampler for the mcmc method of parameter estimation , together with jam code ( @xcite ) and our own adc routines all of which are implemented in @xmath129 . this approach allows a robust understanding of the uncertainties in the modelling combined with their dependance on assumptions , which is particularly important since we ultimately compare adc and jam results . ) = 1.5 mm & + ngc & type & @xmath130 & @xmath131 & @xmath132 & @xmath133 & @xmath134 & @xmath135 & good fit + ( 1 ) & ( 2 ) & ( 3 ) & ( 4 ) & ( 5 ) & ( 6 ) & ( 7 ) & ( 8) & ( 9 ) + ngc0488 & sa(r)b & @xmath136 & @xmath137 & @xmath138 & @xmath139 & @xmath140 & @xmath141 & yes + ngc0772 & sa(s)b & @xmath142 & @xmath143 & @xmath138 & @xmath144 & @xmath145 & @xmath146 & yes + ngc4102 & sab(s)b & @xmath147 & @xmath148 & @xmath138 & @xmath149 & @xmath150 & @xmath151 & yes + ngc5678 & sab(rs)b & @xmath152 & @xmath153 & @xmath138 & @xmath154 & @xmath155 & @xmath156 & yes + ngc3949 & sa(s)bc & @xmath157 & @xmath158 & @xmath138 & @xmath159 & @xmath160 & @xmath161 & yes + ngc4030 & sa(s)bc & @xmath162 & @xmath163 & @xmath138 & @xmath164 & @xmath165 & @xmath166 & yes + ngc2964 & sab(r)bc & @xmath167 & @xmath168 & @xmath138 & @xmath169 & @xmath170 & @xmath171 & yes + ngc0628 & sa(s)c & @xmath138 & @xmath138 & @xmath172 & @xmath173 & @xmath174 & @xmath175 & no + ngc0864 & sab(rs)c & @xmath176 & @xmath177 & @xmath138 & @xmath178 & @xmath179 & @xmath180 & yes + ngc4254 & sa(s)c & @xmath181 & @xmath182 & @xmath138 & @xmath183 & @xmath184 & @xmath185 & yes + ngc1042 & sab(rs)cd & @xmath138 & @xmath138 & @xmath186 & @xmath187 & @xmath188 & @xmath189 & yes + ngc3346 & sb(rs)cd & @xmath138 & @xmath138 & @xmath190 & @xmath191 & @xmath192 & @xmath193 & yes + ngc3423 & sa(s)cd & @xmath194 & @xmath195 & @xmath138 & @xmath196 & @xmath197 & @xmath198 & yes + ngc4487 & sab(rs)cd & @xmath138 & @xmath138 & @xmath199 & @xmath200 & @xmath201 & @xmath202 & yes + ngc2805 & sab(rs)d & @xmath203 & @xmath204 & @xmath138 & @xmath205 & @xmath206 & @xmath207 & yes + ngc4775 & sa(s)d & @xmath138 & @xmath138 & @xmath208 & @xmath209 & @xmath210 & @xmath211 & no + ngc5585 & sab(s)d & @xmath138 & @xmath138 & @xmath212 & @xmath213 & @xmath214 & @xmath215 & no + ngc5668 & sa(s)d & @xmath138 & @xmath138 & @xmath216 & @xmath217 & @xmath218 & @xmath219 & no + [ tab : adcmc ] to infer circular velocity curves for our sample of galaxies using the adc method ( sect . [ ss : adc ] ) , we first need their kinematic profiles along the projected major axis . to this end , we use the ` kinemetry ` package of @xcite that is based on harmonic expansion of two - dimensional maps along ellipses . we extract the observed mean velocity @xmath220 and velocity dispersion @xmath221 profiles along ellipses with fixed axis ratio of the galaxies ( @xmath19 ) and photometric position angle , where @xmath6 and @xmath222 are shown in table [ tab : prop ] . however , for galaxies ngc3346 , ngc4775 and ngc5668 , we calculate the axis ratio @xmath22 from the adopted inclination @xmath223 using eq . [ eq : incl ] ( see sec . [ ss : sample ] ) . then under the thin - disc assumption we obtain the intrinsic mean velocity @xmath99 and radial velocity dispersion @xmath105 profiles from the observed profiles as : @xmath224 , \end{aligned}\ ] ] with systematic velocity @xmath225 and adopted inclination @xmath73 from table [ tab : prop ] . to get to equation , we have used @xmath109 under the assumptions that the velocity ellipsoid is aligned with the cylindrical coordinate system and symmetric around @xmath108 ( see section [ ss : adc ] ) . as before , @xmath226 is the velocity anisotropy in the meridional plane , and @xmath227 is the radial logarithmic gradient of the intrinsic mean velocity . + given the assumptions of the adc approach , we relate the model parameters to the observed data as follows : @xmath228 where @xmath229 and @xmath230 . to avoid numerical noise in the derivatives , we express @xmath231 and @xmath105 using a smooth , analytic functional forms and then compute the derivatives explicitly . as an analytical representation for @xmath232 , we use the power - law prescription of ( * ? ? ? * eq.2.11 ) , which in the equatorial plane ( @xmath233 ) and assuming flat rotation curve , becomes @xmath234 where @xmath235 is the asymptotic velocity and @xmath131 is the core radius . this model describes a rotation curve that increases linearly with radius @xmath236 when @xmath237 , and flattens to the value @xmath235 when @xmath238 . however , in several cases the velocity profile shows little indication of flattening the observed region . in such cases , the core radius @xmath131 is not well constrained and fitting eq . would lead to unphysical values for both @xmath131 and @xmath235 . such galaxies are in the regime of @xmath237 and we adopt a linear solid - body rotation instead : @xmath239 where @xmath132 is the linear coefficient . + for @xmath105 , we use a linear expression of the form @xmath240 where @xmath133 and @xmath134 are free parameters . we find the linear fits to the @xmath105 profiles are broadly representative , although the model does not reproduce the central part of the @xmath105 profiles for some of the galaxies ( e.g. , ngc772 , ngc5678 and ngc864 ) due to the marked dips in @xmath71 in the central parts of the galaxies ( @xcite ) . such a descrease in the observed stellar velocity dispersion might be the result of a small dynamically cold component or counter - rotating disc of high surface brightness that obscures the bulge , dominating the luminosity - derived kinematic measurements ( @xcite ) or due to @xmath127-folding length of our galaxies ( see @xcite ) . to apply a bayesian framework to the problem , we consider the free parameters of the model to be random variables and then calculate their posterior distributions using mcmc . we model 5 free parameters for the galaxies in the power - law model for @xmath232 ( @xmath130 , @xmath131 , @xmath133 , @xmath134 , @xmath135 ) and 4 free parameters ( @xmath132 , @xmath133 , @xmath134 , @xmath135 ) for the galaxies with linear models for @xmath232 . for these vectors of parameters @xmath241 , we compute model profiles for @xmath242 and @xmath243 using eq.([eq : obsvmc ] ) and ( [ eq : obssigmc ] ) . these profiles are compared to observed data using a chi - squared statistic , summing over radius positions with normalization set by the uncertainties in the observed data ( @xmath244 and @xmath245 ) : @xmath246 ^ 2/(\delta v_{\mathrm{obs},k}^2)\nonumber\\ & + & \sum_k [ \sigma_{\mathrm{mod},k}(\boldsymbol\theta ) - \sigma_{\mathrm{obs},k}]^2/(\delta \sigma_{\mathrm{obs},k}^2 ) . \label{eq : chisq}\end{aligned}\ ] ] we adopt a log - probability function for the parameters given the data as @xmath247 , where @xmath248 is a set of priors . we adopt uniform prior distributions for our parameters on fixed ranges : @xmath249 $ ] km s@xmath5 , @xmath250 $ ] , @xmath251 $ ] km s@xmath5 arcsec@xmath5 , @xmath252 $ ] km s@xmath5 , @xmath253 $ ] km s@xmath5 arcsec@xmath5 and @xmath254 $ ] . only for one galaxy ( ngc2964 ) we had to expand the range of the walkers to @xmath255 $ ] , since the distribution peaked at @xmath256 . we run the adc - mcmc code with 60 walkers ( i.e. , members of the ensemble sampler ) , which have initial position set by the prior information and physical information . we use a small random distribution around the parameters : @xmath130 , @xmath131 , @xmath132 , @xmath133 , @xmath134 , which are adopted from the least - squares minimisation routine ` mpfit ` in idl to find the values that minimize eq . [ eq : chisq ] . we then start the walkers around the optimized values in a normal distribution with scale determined by the uncertainties in the fit . the walkers of the parameter @xmath135 were allowed to move freely in the parameter space . on the other hand , the inclination of the galaxies is fixed to the value presented in column 6 of table [ tab : prop ] . the mcmc code samples the posterior distribution , using 400 steps for burn - in and 1000 steps for sampling . we find that all walkers converge after @xmath257 steps of burn - in to sample a similar distribution indicating a well - sampled unimodal posterior distribution . we summarize the posterior distributions for the parameters in table [ tab : adcmc ] of this paper . additionally , we assign a quality flag to each adc - mcmc model in table [ tab : adcmc ] depending on the residual of the v profile ( @xmath258 ) and @xmath71 profile ( @xmath259 ) , where the label `` yes / no '' correspond to `` good / bad '' fit , respectively . bad fit label is associated to galaxies ( i.e. , ngc0628 , ngc4775 , ngc5585 and ngc5668 ) if the median value of one of their residuals ( @xmath260 or @xmath261 ) is larger than 0.3 . the burn - in chains , posterior chains and corner plots of adc - mcmc method are shown in appendix [ c : chains ] . we present the results of the analysis in fig . [ fig : vc_c ] . in the first column , we show the observed mean line - of - sight velocity @xmath70 ( filled green circles ) and its fiting function ( eq . [ eq : obsvmc ] , filled magenta triangles ) . the galaxies are ordered by their morphological type from sb to sd . we observe that sb sc galaxies ( at @xmath262 ) have higher observed velocity in contrast to scd sd . we adopt a power - law model function to most of the @xmath232 radial profiles , except for the galaxies ngc0628 , ngc1042 , ngc3346 , ngc4487 , ngc4775 , ngc5585 and ngc5668 , which require a linear function . in the second column of fig . [ fig : vc_c ] , we present the observed line - of - sight velocity dispersion @xmath71 profiles ( filled green circles ) and its fiting function ( eq . [ eq : obssigmc ] , filled magenta triangles ) . the @xmath71 profiles are more varied than the @xmath70 profiles . some velocity dispersion profiles show an almost - linear decrease or increase with radius on the whole radial range available , some are flat , and others have a more complex behaviour that seems difficult to reproduce with a linear function . sbc galaxies ( at @xmath262 ) are characterised by high observed velocity dispersion @xmath71 with respect to sc most of our sb - sc galaxies have regular velocity fields with well - defined axisymmetric rotation and high amplitude , while sd velocity fields show more complex structure and lower amplitude of rotation . in our sample we have four types of @xmath71 behaviour : decreasing outward ( ngc4102 ) , increasing outward ( ngc3949 ) , a central @xmath71-dip ( ngc0772 ) or flat ( ngc4775 ) . we estimate the uncertainty of the velocity and velocity dispersion profiles at each radius ( see fig . [ fig : vc_c ] ) from the corresponding error maps since the provided uncertainties from ` kinemetry ` routine are unrealistically small ( @xmath4 1 km@xmath5 ) . thus , we took the absolute value of the median of the velocity bins and the median of the velocity dispersion bins within each annulus defined by ` kinemetry ` routine . as described in sect . [ s : method ] , we use a solution of the axisymmetric jeans equations based on a multi - gaussian expansion ( mge ) of the intrinsic luminosity density to predict the observed second velocity moment @xmath263 ( @xcite ) . the solution has a constant meridional plane velocity anisotropy @xmath264 and constant mass - to - light ratio @xmath265 within the galaxy . our analysis does not account for the variation of @xmath266 within each voronoi bin . such variations will cause the averaged @xmath267 in a bin to be larger than the intrinsic @xmath266 along a ray , as evaluated by the jam code . we do not anticipate this effect to bias our results significantly : since the surface brightness decreases with @xmath268 , the larger voronoi bins are found in the outer regions of the galaxies where the variation in the @xmath266 surface is low . at small @xmath268 , where the variation with position can be significant , the higher surface brightness means the bins are smaller . we symmetrize our observed @xmath266 fields in order to avoid outliers in the data ( see @xcite ) . in this way , the results will be not influenced by a few deviant values . this is especially important in the case of the mcmc method application , where the best fit @xmath266 model rely on @xmath269-statistics . moreover , the symmetrization renders the @xmath266 field axisymmetric which is the approximation consider by jam . at the same time we symmetrize the variance of the @xmath266 field : @xmath270 where the term @xmath271 indicates the symmetrized variance of @xmath266 field . + for this purpose we use the code presented in @xcite . the procedure interpolates the bin values over a new grid generated by permuting the @xmath86th - bin coordinates as follow : @xmath272 this operation generates three additional velocity fields ( the first permutation obtained by @xmath273 corresponds to the original velocity field ) . finally , the symmetrized velocity field is the mean of the four coordinate - permuted velocity fields . given this , equation ( [ eq : dvrms ] ) reduces to the common addition of the error in quadrature : @xmath274 where @xmath275 , and + @xmath276 of a certain permutation . + although the interpolation is precise , the code does not consider the systematic effect that a given bin at @xmath277 position not always have a corresponding bin atat the other positions , e.g. , ( -x , y ) . to account for this , the final @xmath266 field error is set by the maximum of either the values from equation ( [ eq : dvrms ] ) or half the deviation between neighbors @xmath266 field bin values ( m. cappellari private communication , see also @xcite ) . + we formulate a parallel mcmc approach for determining the posterior parameter distributions of the jam model . in this case , there are only 2 free parameters for all galaxies ( @xmath264 and @xmath75 ) . because of the large computational expense of evaluating a jam model , we run the jam - mcmc code with only 30 walkers , though the model still converges using 100 steps for a burn - in and 200 steps for sampling . the prior distributions are taken to be uniform with , e.g. , @xmath278 $ ] and @xmath279~m_{\odot}/l_{\odot}$ ] . in jam - mcmc method , we fix the inclination to the values in the column 6 of table [ tab : prop ] . for galaxies ngc3346 , ngc4775 , and ngc5668 , we adopt @xmath280 regarding to the minimum possible inclination condition of jam model ( see sec . [ ss : jam ] ; eq . [ eq : ilim ] ) . we use a similar log - likelihood statistic as we did for the adc , except in this case , only the model and observed @xmath266 are compared . summaries of the posterior distribution of parameters for jam - mcmc model are presented in table [ tab : jammc ] . the burn - in chains , posterior chains and corner plots are shown in appendix [ b : chains ] . in the fourth to fifth columns of fig . [ fig : maps1 ] , we show the observed and the best fitting @xmath266 maps with over - plotted mge contours . for all galaxies , the mge contours are aligned with the surface brightness contours in position angle and ellipticity . in the sixth colum , we calculate the residual maps @xmath281 between the observations ( @xmath282 ) and the models ( @xmath283 ) , where @xmath284=@xmath285 . the median values of these residual maps @xmath286 indicate that the uncertainties of jam fit ranges between 3 % and 10 % . note here that we do not count for systematics errors in these uncertainties . from the posterior distributions of jam - mcmc method , we define the goodness of the fit with the label `` yes / no '' for `` good / bad '' fit , respectively . bad fit labels are associated to galaxies ( i.e. , ngc4030 , ngc1042 , ngc3346 ) , which do not have well defined posterior distributions of the velocity anisotropy @xmath287 ( see appendix [ b : chains ] and table [ tab : jammc ] ) . further , we find that the jam produces the best fits to the second velocity moment @xmath267 for sb and sc spirals ( e.g. , ngc488 , ngc5678 , and ngc2964 , characterised by fast - rotating discs ) and the model gets worse for scd and sd galaxies . given the observed values of @xmath70 and @xmath71 show evidence for non - monotonic behaviour ( figure [ fig : maps1 ] ) , the model provides an average representation of the data in the late - type galaxies . we stress that the values derived for the parameters are the posterior distributions given the data , subject to the assumptions of the model . = 1.5 mm & + ngc & type & @xmath75 & @xmath287 & good fit + ( 1 ) & ( 2 ) & ( 3 ) & ( 4 ) & ( 5 ) + 0488 & sa(r)b & @xmath288 & @xmath289 & yes + 0772 & sa(s)b & @xmath290 & @xmath291 & yes + 4102 & sab(s)b & @xmath292 & @xmath293 & yes + 5678 & sab(rs)b & @xmath294 & @xmath295 & yes + 3949 & sa(s)bc & @xmath296 & @xmath297 & yes + 4030 & sa(s)bc & @xmath298 & @xmath299 & no + 2964 & sab(r)bc & @xmath300 & @xmath301 & yes + 0628 & sa(s)c & @xmath302 & @xmath303 & yes + 0864 & sab(rs)c & @xmath304 & @xmath305 & yes + 4254 & sa(s)c & @xmath306 & @xmath307 & yes + 1042 & sab(rs)cd & @xmath308 & @xmath309 & no + 3346 & sb(rs)cd & @xmath310 & @xmath311 & no + 3423 & sa(s)cd & @xmath312 & @xmath313 & yes + 4487 & sab(rs)cd & @xmath314 & @xmath315 & yes + 2805 & sab(rs)d & @xmath316 & @xmath317 & yes + 4775 & sa(s)d & @xmath318 & @xmath319 & yes + 5585 & sab(s)d & @xmath320 & @xmath321 & yes + 5668 & sa(s)d & @xmath322 & @xmath323 & yes + [ tab : jammc ] in order to compare the two modelling approaches adc and jam , we first combine the measured deprojected stellar velocity profile @xmath231 and radial dispersion profile @xmath105 . next , we derive the circular velocity curve @xmath324 using the asymmetric drift correction ( adc ) formula in equation ( [ eq : adcvcirc ] ) . these data are indicated by the blue - filled - pentagon curves with uncertainties derived from the transformed posterior distributions of adc - mcmc results . second , we construct axisymmetric jeans models ( jam ) that fit the combined observed stellar mean velocity and velocity dispersion fields , we use equation ( [ eq : mgevcirc ] ) to obtain the circular velocity curve for each galaxy . the resulting @xmath325 curves from the best - fit jam model are plotted as red - filled - squares curves in fig . [ fig : vc_a ] with their uncertainty from jam - mcmc code . for all galaxies , adc approach gives lower measurements of the circular velocity curve in comparison with the values obtained from jam models for at least part of the radial range . to quantify these differences , we plot in fig . [ fig : resratio ] the velocity ratio between @xmath326 and @xmath327 . overall , the velocity discrepancy for sb - sbc galaxies is larger in the inner parts ( @xmath328 ) and decreases in the outer parts ( @xmath329 ) , while for scd - sd galaxies the velocity discrepancy is constant or even increases in the outer parts . the differences we found between the circular velocity curves from the two modelling approaches can have a strong impact on the inferred total mass distribution and thus also on any follow - up inference like the amount of dark matter in the inner parts of these late - type spiral galaxies . we list the different assumptions adopted in both the axisymmetric jeans models ( jam ) and the asymmetric drift correction ( adc ) , and discuss the possible reasons for their discrepancy . _ mass - follows - light ( jam ) : _ within the small radial range covered by the stellar kinematics , typically at @xmath330 , we expect the variations in the mass - to - light ratio to be small . @xcite provides evidence that in a sample of 37 sb - sc galaxies , most of their rotation curves computes from the luminosity profiles assuming constant mass - to - light ratio provide a good match to the observed curve out to the radius where the predicted curve turns over . additionally , @xcite investigate the effects of realistic radially varying mass - to - light ratios and find the overall effect to only be @xmath4 10% variations on the derived kinematic properties within @xmath331 . _ constant anisotropy in the meridional plane ( jam ) : _ whereas the velocity anisotropy @xmath332 in the equatorial plane is inherent in the solution of the axisymmetric jeans equations ( section [ ss : axijeans ] ) , the velocity anisotropy @xmath333 in the meridional plane is a free parameter , given as @xmath334 . there is little evidence for strong variation of @xmath72 . for example , @xcite already argued that for spirals the @xmath333 is constant at approximately 0.6 , close to the measured value of @xmath335 in the solar neighbourhood @xcite . this ratio of the anisotropy is measured in a few spirals of type sa to sbc @xcite , yielding slightly larger constant values between 0.6 and 0.8 . based on long - slit spectra for a sample of 17 edge - on sb scd spirals , @xcite also adopt constant values , although slightly lower : 0.5 to 0.7 . while radial variation of the anisotropy is not excluded , a constant value of @xmath72 should be sufficient for our analysis . _ shape of the velocity ellipsoid ( jam , adc ) : _ the adc and jam models in our study assume that the velocity ellipsoid in the meridional plane is aligned with the cylindrical coordinate system so that @xmath107 . additionally , the adc approach may allow for a tilt of the velocity ellipsoid through the parameter @xmath336 having intermediate values between @xmath337 and @xmath338 , corresponding to the cylindrical and spherical coordinate system , respectively . however , we expect that the resulting circular velocity curve is only weakly dependent on this tilt , because most of the stellar mass is concentrated toward the equatorial plane , particularly for late - type spiral galaxies . in that case , the assumption of axisymmetry gives @xmath107 . _ dust ( jam , adc ) : _ the surface brightness distribution of the spiral galaxies that is used in both jam and adc modelling approaches can be strongly affected by extinction due to dust . we have tried to minimise the effects of dust in various ways ( g09 ) : ( i ) selecting galaxies with intermediate inclinations so that they are not edge - on ( where dust extinction is strongest ) or face - on ( where the stellar velocity dispersion would be significantly below the spectral resolution ) , ( ii ) inferring the surface brightness distribution from images in the near - infrared where the extinction is significantly smaller than in the optical , and ( iii ) fitting smooth , analytical mge profiles to the radial surface brightness profile after azimuthally averaging over annuli to suppress deviations caused by bars , spiral arms , and regions obscured by dust . the stellar kinematics are obtained from integral - field spectroscopy in the optical and thus could also be affected if the ( giant ) stars that contribute along the line - of - sight with different motions are affected by the dust in different ways . for example , if dynamically colder stars closer to the disc plane are relatively more obscured than dynamically hotter stars above the disc plane , the resulting combined ordered - over - random motion could be biased to lower values . the effects of dust are strongly reduced for lines of sight only a few degrees away from edge - on ( @xcite ) and the effects of extinction on the line - of - sight velocity are negligible : the change in @xmath339 is only 1.3% higher from the dustless case ( @xcite ) . given the selection in our sample , the effects of dust on the inferred circular velocity curves from both modelling approaches are expected to be minimal . _ thin - disc ( adc ) : _ circular velocity curves of spirals nearly always come from ( cold ) gas , which is naturally in a thin disc . the stellar discs of these late - type spiral galaxies are also believed to be thin , with inferred intrinsic flattening @xmath340 ( e.g. , * ? ? ? even the bulges in late - type spiral galaxies are different from the classical bulges in lenticular galaxies . srsic profile fits to their surface brightness ( @xmath341 ) show that towards later - types , the bulges are smaller in size , have profiles closer to exponential ( @xmath342 ) than de vaucouleurs ( @xmath343 ) , and are flatter ( e.g. , g09 ) . hence , the thin - disc assumption adopted in the adc approach also seems to be reasonable . independent of the nature of the dynamically hot stellar ( sub)system , it seems that the local value of the ordered - over - random motion is the key for understanding of the adc and jam discrepancy . to explore that , in fig . [ fig : resratio ] , we plot the radial velocity ratio @xmath344 of the galaxies , as well as the local luminosity fraction of the fitted exponential disc compared to the total luminosity as a thin solid curve . for the sb sbc galaxies , the discrepancy in the estimated velocity ratio between the jam and adc models seems to be larger in the inner parts ( @xmath328 ) , where the luminosity is dominated by the presence of the bulge , and decreases outwards . for scd sd galaxies the velocity ratio stays roughly constant and in a few cases increases towards larger radii , which could be due to the presence of a thick(er ) disc component . in the left panel of fig . [ fig : vsrel ] , we plot the velocity ratios @xmath344 obtained from fig.[fig : resratio ] for all galaxies as measured at different radii versus the ordered - over - random motion @xmath1 ( the deprojected rotation vs. radial velocity dispersion ) at the same radii . in the right panel of fig . [ fig : vsrel ] , we also show the ratio @xmath345 ( from the radial velocity dispersion profiles and deprojected rotation ) as measured at different radii versus the mass ratio @xmath346 at the same radii . the discrepancy between the adc and jam inferred enclosed masses becomes gradually larger for increasing @xmath345 . the impact of the models discrepancy is larger when is converted into mass ratio . it also has the same order of magnitude as the expected uncertainty of the epicycle approximation ( @xmath347 , @xcite ) , which is actually applied in adc ( but not in jam ) approach . we see that the circular velocity measurement of both modelling approaches are consistent for @xmath1 @xmath348 , corresponding to those radii in the sb sbc galaxies where the disc luminosity is dominating over the bulge luminosity , i.e. , where the thin solid curve in fig . [ fig : resratio ] approaches unity . since some of our galaxies do not have well fitting models , we mark their data with a different symbol in order to check for possible biases . however , even after their removal , the general trends of the velocity and mass ratios are preserved . the grey filled circles correspond to high - quality adc and jam model fits data , while the open pentagon and the open star correspond to low - quality adc and jam data , respectively . the black filled squares show the median of the distribution of the data per bin . the error bar corresponds to the uncertainties of the median value at the 25th and 75th percentiles of the distribution at each bin ( see also table [ tab : calc5 ] ) . = 1.2 mm @xmath1 & @xmath349 & @xmath350 & @xmath351 + ( 1 ) & ( 2 ) & ( 3 ) & ( 4 ) + 0.1 & @xmath352 & 0.4 & @xmath353 + 0.3 & @xmath354 & 0.8 & @xmath355 + 0.5 & @xmath356 & 1.3 & @xmath357 + 0.7 & @xmath358 & 2.4 & @xmath359 + 0.9 & @xmath360 & 4.2 & @xmath361 + 1.1 & @xmath362 & 8.3 & @xmath363 + 1.3 & @xmath364 & 14.1 & @xmath365 + 1.5 & @xmath366 & 24.8 & @xmath367 + 1.7 & @xmath368 & 49.6 & @xmath369 + 1.9 & @xmath370 & 85.5 & @xmath371 + 2.0 & @xmath372 & 150.1 & @xmath373 + [ tab : calc5 ] since we are probing the inner parts of these spiral galaxies it might well be the presence of bulges and/or thick stellar discs that causes a break - down of the thin - disc / epicycle approximation in adc . as can be seen from fig . [ fig : vc_a ] , the stellar rotation ( first column ) is of the same order and often even lower than the stellar dispersion ( second column ) . the small ordered - over - random motion values implies that the stars are far from dynamically cold . detailed photometric studies indicate that most disc galaxies contain a thick disc ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , and in low - mass galaxies with circular velocities @xmath15120kms@xmath5 , like the scd sd galaxies in this sample , thick disc stars can contribute nearly half the luminosity and dominate the stellar mass ( @xcite ) . moreover , recent hydrodynamical simulations that reproduce thick discs show that their typical scale lengths are around 35 kpc ( e.g. * ? ? ? * ) , i.e. , twice range typically covered by our analysis . additionally , various earlier studies have qualitatively indicated that the adc approach might not be suitable in the case of stellar systems that are ( locally ) not dynamically cold ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? we also find that the discrepancy between the two model is proportional to the error of the epicycle approximation in adc ( see @xcite ) , where @xmath374 . @xcite show that rotation curves of early - type galaxies based on the dynamically cold molecular gas are well traced by the @xmath325 , and that the @xmath375 rotation curves from their ` sauron ` data exhibit noticeable asymmetric drift . they also find the correction of the stellar rotation curve using adc approach is also consistent with @xmath325 . however , they note that there is some indication that this becomes less true as velocity dispersion approaches the rotation speed , as we find in this study . it is not excluded that there are strong streaming motions in the centers of the ` sauron ` galaxies due to presence of non - axisymmetric features , which prevent an accurate asymmetric - drft correction . nevertheless , some of the discrepancy might be arise from jam overestimation of the m / l ratio . @xcite showed that the overestimation can be severe in face - on barred system . @xcite also discussed the possibility of an overestimation of the @xmath376 ratio due to disk averaging over various stellar populations . we do not exclude the latter since the region probed by our data is in the central part of the galaxies , where multiple stellar populations can be present . the former case should not influence some of our measurements , since all of our galaxies have intermediate inclinations . in future studies , we will explore further this problem using large statistical sample of galaxies and morphologies . the rotation curves of spiral galaxies , traced by atomic gas , provide the most direct path to estimate the total galaxy masses outside the stellar disc . however , it is not straightforward to measure the rotation curve in the central parts of the galaxies , where the ionized and molecular gas dominate . gas settles in the equatorial or polar plane and is thus less sensitive to the mass distribution perpendicular to it . gas is also dissipative and easily disturbed by perturbations in the plane from features like bars or spiral arms . thus , stars appear to be a better tracer as they are distributed in all three dimensions and , being collisionless , they are less sensitive to perturbations . however , stars are not cold tracers as they move in orbits that are neither circular nor confined to a single plane . we therefore need to measure both velocity dispersions and line - of - sight velocities to recover the total mass distribution of galaxies . in this paper we compare two different approaches for inferring dynamical masses of spiral galaxies : the commonly used asymmetric drift correction ( adc ) and the axisymmetric jeans equations . we used the stellar kinematics derived by integral field spectroscopy of a sample of 18 sb sd galaxies , observed with the ` sauron ` spectrograph . we obtained stellar mean velocity and velocity dispersion maps and derived the galaxies circular velocity curves by fitting solutions to the jeans equations . using the same data we also derived the circular velocity curves via the adc technique . we use markov chain monte carlo methods to determine credible values for model parameters and their associated uncertainties . the adc approach gives systematically lower values than jam for the circular velocity curves , and hence the enclosed mass , when @xmath1 @xmath377 . the velocity ratio @xmath378 for sb sbc galaxies is larger in the inner parts ( @xmath328 ) and decreases outwards . however , for scd sd galaxies the mass ratio stays roughly constant and in a few cases increases towards larger radii . the complete python codes , implementing mcmc analysis on both jam and adc models , described in this paper , can be downloaded from ` https://github.com/kalinova/dyn_models ` . these codes provide a robust way to estimate the uncertainties in the derived mass distribution of galaxies through their circular velocity curves . we are extremely grateful to the anonymous referee for their valuable contribution towards the completeness of this paper . his / her reports were constructive and thorough and dramatically improved the quality of the result . thanks ronald lsker for the fruitful discussions . we thank michelle cappellari for the helpful suggestions about the calculation of the uncertainties of the symmetrized @xmath267 field . we acknowledge financial support to the dagal network from the people programme ( marie curie actions ) of the european union s seventh framework programme fp7/2007- 2013/ under rea grant agreement number pitn - ga-2011 - 289313 . j. f .- b . acknowledges support from grant aya2013 - 48226-c3 - 1-p from the spanish ministry of economy and competitiveness ( mineco ) . v.k , d.c . and e.r . are supported in part by a discovery grant from nserc of canada . [ lastpage ] [ s : appendix ] here we provide a table with the parameters of our mge models for each galaxy , where @xmath80 is the central surface brightness , @xmath379 is the the dispersion along the major @xmath66-axis , and @xmath82 is the flattening . additionally , @xmath380 indicates the gaussians of the central , bulge or disc component of the galaxies with respective values of 0,1 and 2 . [ cols="^,^,^,^,^,^,^,^,^,^,^,^,^ " , ]
|
we infer the central mass distributions within 0.41.2 disc scale lengths of 18 late - type spiral galaxies using two different dynamical modelling approaches the asymmetric drift correction ( adc ) and axisymmetric jeans anisotropic multi - gaussian expansion ( jam ) model .
adc adopts a thin disc assumption , whereas jam does a full line - of - sight velocity integration .
we use stellar kinematics maps obtained with the integral - field spectrograph @xmath0 to derive the corresponding circular velocity curves from the two models . to find their best - fit values , we apply markov chain monte carlo ( mcmc ) method .
adc and jam modelling approaches are consistent within 5% uncertainty when the ordered motions are significant comparable to the random motions , i.e , @xmath1 is locally greater than 1.5 . below this value ,
the ratio @xmath2 gradually increases with decreasing @xmath1 , reaching @xmath3 .
such conditions indicate that the stellar masses of the galaxies in our sample are not confined to their disk planes and likely have a non - negligible contribution from their bulges and thick disks .
[ firstpage ] galaxies : bulge galaxies : disc galaxies : kinematics and dynamics galaxies : structure
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
magnetic materials are essential for modern technology . up to date , most used magnetic materials involve elements with partially filled d or f - orbitals , like transitional metals , rare earth metals and their compounds . magnetic ordering in these compounds originates from the localization of the electrons so that the kinetic energy can be minimized . this is the result of competitions between the kinetic energy and the exchange splitting . the former favors nonmagnetic state while the latter is the source of spin polarization . magnetism from p - states rather than the above conventional d or f states has been a hot topic in recent years because of technical importance and theoretical interests as well . there are several ways to introduce spin polarization to the p - states . several alkali and alkaline earth compounds with elements from group iva ( carbides and silicides),@xcite va(nitrides , pnictides , and arsenides)@xcite , and via ( selenlides)@xcite are predicted to be magnetically ordered . oxides are not so usual , but doping with c and n , or introducing cation vacancies can be sources of magnetism.@xcite another large category comes from nanostructures of graphene , graphite and c@xmath2@xcite or their modification such as doping or absorption of other atoms . the origin of the spin polarization is explained mostly from the stoner s theory where localization of the p - orbitals triggers the instability of the nonmagnetic state to magnetic state@xcite . bulk carbon compounds are rarely found to be spin polarized . the most distinct feature of these compounds with p - orbital magnetism is that in most cases they are half metals , where in the one electron picture , one spin channel in the kohn - sham band structure has a finite density of states(dos ) , while the other is null . this is very important and promising for its possible applications in spintronics.@xcite monocarbides mc , where m may be alkali metal and alkaline earth metal show the above interesting magnetic properties , namely half metallic ferromagnetism , due to there unique electronic structures as predicted by several theoretical works.@xcite in our previous work,@xcite ferromagnetic ordering and its electronic origin were investigated in alkaline earth carbides with the rocksalt structure . it was found that the magnetism is from the polarization of the carbon p - orbitals , while the alkaline earth metals , due to their large atomic radii , guarantee the sufficient large distance between the carbons , which localizes the p - orbitals . at the same time , they are also electron provider because of the smaller electronegativity than that of c. in this case , we proposed that the electronic structure and the spin magnetic moment can be nicely tuned , which gives us freedom to tailor their properties when used as a spin filter , or spin injector . in this work , we show how the electronic structure can be tailored by doping ( a@xmath3b@xmath4c ) and by hydrostatic pressures . it turns out that the fundamental gap as well as the spin flip gap can be tuned and induce the transition from a semiconductor to metal . the calculations were performed by the full - potential local - orbital code@xcite in the version fplo9.00 - 33 with the default basis settings . all calculations were done within the scalar relativistic approximation . the general gradient approximation of exchange - correlation functional was chosen to be that parameterized by perdew , burke , and ernzerhof.@xcite the number of k - points in the whole brillouin zone was set to @xmath5 in order to ensure the convergency of the results . convergency of the total energy was set to be better than @xmath6 hartree together with that of the electron density better than @xmath7 in the internal unit of the codes . the fcc unit cell ( space group fm@xmath8 m ) was used to represent the rocksalt structure with wyckoff positions assigned to metal ( m ) and c as m(0,0,0 ) and c(0.5,0.5,0.5 ) , respectively . doping was modeled in a supercell with the total atom number up to 32 . as found in our previous work,@xcite the dos around the fermi level are mainly contributed by the 2p - electrons of c. we constructed 3 carbon centered wannier functions(wfs ) . the resulting wfs are p - like . in the basis of these wfs , if the small interactions below 0.001 ev are ignored , the elements of the tight - binding(tb ) non - magnetic hamiltonian @xmath9_{3\times3}$ ] can be written as , @xmath10 + 4t_4\cos(\frac{a}{2}k_j)\cos(\frac{a}{2}k_k)+2t_5\cos(ak_i)\ ] ] and @xmath11 where @xmath12 is the lattice constant , and subscripts @xmath13and@xmath14 and @xmath15 , respectively . the hoping constants @xmath16 is onsite contribution , @xmath17 , @xmath18 and @xmath19 are from the nearest neighbor interactions , and @xmath20 is from the next - nearest neighbor interactions as shown in fig . [ fig : tpp ] . the values are @xmath21 ev , @xmath22 ev , @xmath23 ev , @xmath24 ev and @xmath25 ev . we see that the strongest interactions come from neighboring carbons with orthogonal p - orbitals . the bands from this tb hamiltonian are shown in fig . [ fig : wfbands ] together with those from the full potential calculations . the agreement of the two results is quite satisfactory . ( color online ) cartoon representation of the interactions between different p - like wfs centered on the carbons . ] ( color online ) comparison of the bands from tb model ( tb dashed curves ) and full potential calculations(fp solid curves ) . ] in order to illustrate the binding in the real space , the charge density contour of the compounds is shown in fig . [ fig : edensity ] . spherical distribution of the electrons around the ions is a typical case of ionic bonding between the metals and carbon . in this scenario , the metallic ions determines the distances between the carbons . the characteristics of electrons around the fermi level are mostly determined by the interaction of the 2p - orbitals from c. exchange interaction is thus direct exchange between the carbons like in zinc - blende m@xmath26c@xcite . this specific electronic structure gives us a large degree of freedom to tune the properties of the material as will be discussed in the following sections . the contour lines of the electron charge density of kc at equilibrium lattice constant . ] as changes of the volume can result in changes of the bandwidth , the dos of these compounds are strongly dependent on the pressure which results in the dependence of magnetism on the volume ( pressure ) . the dependence of the spin magnetic moment on the volume of m@xmath27c is illustrated in fig . [ fig : ssia ] . dependence of the spin magnetic moment per primitive cell on the lattice constant . the corresponding pressure at the different lattice constant is also shown . the equilibrium lattice constant @xmath28 of the compounds are marked with arrows . ] it can be clearly seen that shrinking of the lattice constant can decrease the spin magnetic moment in all the compounds . this is the results of the increased kinetic energy of the electrons under pressure as expected . the lattice constant at which the spin polarization is set in is proportional to that at equilibrium , except that of nac which is close to the lattice constant at equilibrium . this is because nac is metallic at equilibrium as shown in our previous work @xcite . however , the processes of the magnetic moment approaching zero are different : the spin moment of kc , rbc and csc show an abrupt drop to zero at certain critical lattice constant while that of nac drops first to an intermediate value of 1.5 @xmath29 and then to zero . the difference comes from the different dos at the fermi level . by simple consideration from the rigid band model , the susceptibility of the magnetic moment to external parameters is proportional to the dos at the fermi level and its derivative . the dos of nac and kc in the vicinity of the transitional region are shown in fig . [ fig : doskc ] and fig . [ fig : dosnac ] in order to illustrate this point . comparing the dos of kc at lattice constant of 4.5 and 4.6 in fig . [ fig : doskc ] , we can see that a rigid shift of the states of spin up and spin down is obvious in the spin polarized state . in kc , the onset of spin polarization at 4.6 set the fermi level at the high dos about 2 ev@xmath30 in the spin up channel . the fermi level is between two saddle points . in nac , the fermi level is also between the two saddle points when the compound is at the intermediate magnetic moment state ( e.g. 1.2 @xmath29 when @xmath31 ) as shown in fig . [ fig : dosnac ] . the spin polarization at this lattice constant is because of the high dos at the fermi level in the nonmagnetic state ( not shown ) which triggers the magnetic instability according to the stoner s model . increase of the lattice constant enhances the exchange splitting , and the fermi level passes through the upper saddle point as shown in fig . [ fig : dosnac](b ) . in this case the magnetic moment increase quickly to a higher value . ( color online ) band of kc at lattice constant @xmath32 ( a ) , and dos of kc at lattice constant @xmath33 and 4.6 ( b ) . the minus sign in the dos indicates that it is of spin down channel , while the numbers without any signs indicate that they are of the spin up channel . the following dos figures follow this convention if not specified . ] ( color online ) band of nac at lattice constant @xmath31 ( a ) , and dos of nac at lattice constant @xmath33 , 5.0 and 5.5 ( b ) . ] the critical pressure of phase transition from the ferromagnetic ( fm ) state to the nonmagnetic ( nm ) state of csc and nac is approximately 28.2 gpa and 13.8 gpa . the fm state of kc is so stable and begins to decrease at about 30 gpa . under high pressure over 80 gpa we see the transition of the compound transforms into the nm state . not only the magnetic order , but also the band gap can be tuned by pressure . when the compounds are semiconductor with a finite gap , the bandwidth starts to broaden under pressure . in this case the bandgap decreases continuously . normally insulating states transforms into metallic states . an example from kc is shown in fig . [ fig : dos_kc ] . at zero pressure , corresponding to the lattice constant @xmath34 the gap is 1.0 ev formed between the two spin states as in fig . [ fig : dos_kc](a ) . it is called the spin flip gap as noted by capelle @xcite . the fundamental gaps are the same as the spin flip gap . from the band structure , we also find that the gap is an indirect gap between the @xmath35 and @xmath36point in the brillouin zone . under pressure , the gap comes to closure as shown in fig . [ fig : dos_kc](c ) . the compound transforms from magnetic insulator to magnetic metal . at the some critical pressure , the highest occupied spin up states just touches the lowest unoccupied spin down states as in fig.[fig : dos_kc](b ) . it reaches an interesting situation where no energy is required to flip the spin . this spin - gapless semiconductor has some important features including that the electron excitation from valence bands to conduction bands needs no energy and the carriers are fully spin - polarized . as the carriers are fully spin - polarized , they can be easily separated by hall effect . the features are useful in some novel applications as recently reviewed by wang@xcite _ et al_. dos of kc at different lattice constant : ( a ) 6.21 , ( b ) 5.18 and ( c ) 4.76 respectively . ] dos of k@xmath37ba@xmath38c at different doping levels indicated by @xmath1 . ] as was shown above , the electronic states near the fermi level are from the p - orbitals of carbon . the alkali metals serve as electron provider which determines the fermi level of the system . we already obtained that bac and kc have magnetic moment of 2.0 and 3.0 @xmath29 , respectively . in these two compounds , the spin up channel are fully occupied . in this case , the change of the electron number will only change the occupation of the spin down channel . if we substitute the ia elements by iia ones , forming compounds m@xmath3k@xmath4c , the magnetic moment @xmath39 varies as @xmath40 . the variation of the dos and the magnetic moment at different doping levels when k is substituted by ba in kc are shown in fig . [ fig : dos_kbac ] . starting from kc , the spin up states are fully occupied while the spin down states are empty , because there are 3 valence electrons and 3 p - states from c are available . doped with the iia element ba , for instance , the extra electrons can only go to the spin down channel . then the metallic state is produced and the electrons at the the fermi level is fully spin polarized . thus , a half metal is formed . the magnetic moment decreases according to @xmath40 . at the same time , the half metallicity preserves . thus a tunable half metal is obtained . the decrease of the magnetic moment can benefit devices where magnetic static interferences can be reduced . in summary , we performed electronic structure calculations of the mc compounds where @xmath39 is one of the ia elements ( na , k , rb , and cs ) based on dft . the results show that there is direct exchange of p - electrons of c which is the source of magnetism . the alkali metals serve as an electron source . the states near the fermi level are contributed mainly from carbon . all the compounds are ferromagnetic insulator at equilibrium with spin magnetic moment 3.0 @xmath29 , except that nac is metallic . nac can be insulator by slight expansion of the lattice constant . the electronic structure and magnetism of the compounds can be tuned by pressure and doping with alkaline earth metals . the variation of the physical parameters can be understood by the rigid band model . the compounds can have potential applications in spintronics and photoelectronics dure to its half metallicity and spin gapless feature . this work was financially supported by international science & technology cooperation program of china ( 2012dfa51430 ) and the fundamental research funds for the central universities ( zygx2011j021 ) . verma , mohini , p.s . bisht , p. jensen , semicond . sci . technol . * 25 * , 105002 ( 2010 ) . gao , k.l . yao , z.l . liu , y. min , j. zhang , s.w . fan , d.h . zhang , j. phys . : condens . matter * 21 * , 275502 ( 2009 ) . k. zdoan , e. aioiu , i. galanakis , j. appl . phys . , * 111 * , 113918 ( 2012 ) . m. sieberer , j. redinger , s. khmelevskyi , and p. mohn , phys . b , * 73 * , 024404(2006 ) . e. yan , physica b , * 407 * , 879 ( 2012 ) . gao , k.l . yao , m.h . song and z.l . liu , j. magn . , * 323 * , 2652(2011 ) . m. yogeswari , g. kalpana , computational materials science * 54 * , 219 ( 2012 ) . v. pardo , w.e . pickett , phys . b * 78 * , 134427 ( 2008 ) . yazyev , rep . * 73 * , 056501 ( 2010 ) . i. @xmath41uti , j. fabian , and s. das sarma , rev . * 76 * , 323 ( 2004 ) . o. volnianska , p. boguslawski , j. phys . : condens * 22 * , 073202 ( 2010 ) . gao , k.l . yao , appl . lett . * 91 * , 082512 ( 2007 ) . gao , k.l . yao , e. aiolu , l.m . sandratskii , z.l . liu , and j.l . jiang , phys . b , * 75 * , 174442 ( 2007 ) . ch.w . zhang , j. phys . phys . * 41 * , 085006 ( 2008 ) . w.x . zhang , z.d . song , b. peng , w.l . zhang , * 112 * , 043902 ( 2012 ) . s.j . dong , h. zhao , appl . . lett . * 98 * , 182501 ( 2011 ) . k. koepernik and h. eschrig , phys . b * 59 * , 1743 ( 1999 ) . perdew , k. burke , and m. ernzerhof , phys . lett . * 77 * , 3865 ( 1996 ) . k. capelle , g. vignale , c.a . ullrich , phys . b * 81 * , 125114 ( 2010 ) . wang , s.x . zhang , npg asia materials * 2 * , 31 ( 2010 ) .
|
electronic structures of carbides with the rocksalt structure were calculated by full potential electronic codes solving the kohn - sham equation .
bonding characters were analyzed by constructing tight - binding hamiltonian based on maximally - localized wannier functions .
it was found that the cations in these compounds act as an electron provider and the frame is formed by the carbon atoms .
the electronic states in the vicinity of the fermi level are mainly from the p - orbitals of c. pressure and doping are two efficient ways to tune the magnetic and electronic properties of these compounds .
it turns out that a spin gapless semiconductor can be obtained by applying hydrostatic pressure up to tens of gigapascal .
higher pressure induced an insulator to metal transition because of band broadening .
compounds of ia group ( na , k , rb , cs ) were magnetic semiconductor at ambient conditions .
alloying with iia elements decrease the magnetic moment according to the law of @xmath0 , where @xmath1 is the relative atomic ratio of the iia elements to the ia ones .
the behaviors of the compounds under the pressure and the doping effects can be understood by a rigid band model .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the international linear collider ( ilc ) will be a tev - scale lepton collider that will require non - invasive beam size monitors with micron and sub - micron resolution for beam phase space optimisation @xcite . laser - wire monitors operate by focussing a laser to a small spot size that can be scanned across the electron beam , producing compton - scattered photons ( and degraded electrons ) . these photons can then be detected further downstream using the total energy observed as a function of the laser spot position to infer the transverse profile of the electron bunch . the laser - wire system installed in the petra ring is part of an ongoing effort in the r&d of producing a feasible non - invasive beam size diagnostic tool . the petra accelerator was chosen for the installation of the laser - wire experiment because it is capable of producing bunch patterns similar to the ilc . laser - wire tests are run using a @xmath0 positron beam with a single bunch with a charge of typically @xmath1 . from the optics lattice the average beam size is @xmath2 for the horizontal and @xmath3 for the vertical dimension . preliminary simulations showed that the compton - scattered photons loose the majority of their energy in the material of the dipole magnet s beampipe due to hitting the wall with a shallow angle , resulting in an effective length of @xmath4 of aluminium . an exit window was therefore designed and installed ( by desy ) to allow these photons to reach the detector with little deterioration ( see fig . [ fig - petra_window ] ) . the laser pulses are created in a q - switched nd : yag laser operating at @xmath5 . the pulses are then transported via a matched gaussian relay made up of two lenses over a distance of @xmath6 from the laser hut via an access pipe into the tunnel housing the accelerator . the laser beam is then reflected off the scanning mirror before it reaches a focusing lens with @xmath7 back - focal length . the scanner is a piezo - driven platform with an attached high - reflectivity mirror which has a maximum scan range of @xmath8 . the peak power at the laser exit was measured to be @xmath9 . at the ip the peak power is reduced to @xmath10 as higher order modes carry some fraction of the beam power but these are focussed out of beam transport , which is only matched for the fundamental mode . the longitudinal profile was measured using a streak camera with @xmath11 time resolution . the data revealed a pulse length of @xmath12 fwhm with a sub - structure of roughly @xmath13 peak - to - peak and @xmath13 peak width at full contrast due to mode - beating . this causes the compton signal amplitude to vary between zero and full signal for different laser shots . in order to reduce the data taking time the current laser will be replaced with an injection seeded system enabling faster data taking . the laser - wire set up makes use of a calorimeter composed of 9 lead tungstate ( @xmath14 ) crystals arranged in a @xmath15 matrix fixed with optical grease to a square faced photomultiplier . the individual crystals have dimensions of @xmath16 . the complete detector set up was tested with a testbeam from the desy ii accelerator using electrons from @xmath17 to @xmath18 . energy resolution was shown to be better than @xmath19 for individual crystals and @xmath20 for the overall set up . simulations show that for the @xmath15 matrix , @xmath21 of the total energy deposit is collected for an incoming compton - scattered photon with @xmath22 energy @xcite . the laser - wire daq system has two main components : the hardware trigger which synchronises the laser and daq components to the electron ( positron ) bunch , and the software which controls the acquisition and collation of data from each sub - component of the system . the hardware trigger operates with two inputs from the petra integrated timing system ( pit ) and produces the necessary signals to fire the laser . the trigger card also produces a signal to trigger the ccd cameras and a signal to start the software data acquisition . when the signal from the trigger card is received a counter which runs for approximately @xmath23 is started . after this time a signal is sent to the integrator card , lasting around @xmath24 , to integrate the output from the calorimeter . the integrated signal is read by an adc channel . the daq software also produces a programmable signal , up to a peak of @xmath25 , which is amplified by a factor of 10 and this is used to drive the piezo - electric scanner . a scaled version of the scanner amplifier output is read by an adc channel . the other sub - components of the daq system : the bpm monitor , the petra data monitor and the ccd cameras are also read out . communication with each component is performed by a messaging system using tcp / ip . in order to determine the transverse size of the electron beam , it is necessary to know the properties of the laser that is being used to scan . particular attention is paid to the spot size at the laser waist , @xmath26 , and the rayleigh range , @xmath27 , ( the distance from the waist at which the beam size @xmath28 ) . these properties are related by eq . [ equ - laserprofile ] : @xmath29 where @xmath30 . + the laser is focused using the same final focus lens as described previously . a cmos camera is placed on a track rail so that it can be moved through the focal plane parallel to the beam direction . due to the high power of the laser , the beam was first passed through a @xmath31 reflective mirror , and then through a variable amount of neutral density filter in order to prevent saturation and damage to the camera pixels . the camera was moved along the track rail to a number of positions , and 100 images were taken in each location . the images taken by the camera are stored as 8-bit greyscale bitmap files . the pixel data is projected onto the y - axis , and fitted to a gaussian on a linear background in the region around the signal peak . the width at each location is then plotted , and fitted to eq . [ equ - laserprofile ] . from this we obtain @xmath32 , which is within the expected range , and @xmath33 , as shown in fig . [ fig - laserwaist ] . the laser is scanned across the electron beam by tilting a mirror on a piezo - electric stack to produce a deflection of @xmath34 . focusing through the lens produces a travel range for the focal spot at the ip of @xmath35 . the scanner voltage is applied in a stepped sinusoidal pattern ; @xmath36 triggers are taken at each of @xmath37 voltages over a whole @xmath38 . the trigger signal is taken from the laser trigger card running at @xmath39 , so a full scan takes approximately @xmath40 . the signal from the adc is expected to display two peaks ; one as the laser crosses the electron beam on a rising voltage to the scanner , and one on a falling voltage . the trigger number exactly half way between the peaks should correspond to a turning point in the scanner position . the mean of the background subracted adc counts at each voltage is then fitted to a gaussian whose width , @xmath41 is given by @xmath42 . [ fig - scanplots ] shows the typical results observed for a single scan and the results of multi - scan shifts are presented in table [ tbl - eresults ] . note that the large signal variation in fig . [ fig - scanplots]a is partly due to the sub - structure of each laser pulse and will be removed by a better laser . .,width=245 ] .data run results for extracted electron beam size , @xmath43 . the errors are the rms from several scans [ cols="^,^,>",options="header " , ] the entire petra laser - wire set up has been simulated using bdsim @xcite , which is a fast tracking code utilising the geant4 @xcite physics processes and framework . the simulation is a full model of the accelerator components including beampipe , magnets , and cooling water channel . for each simulated event a compton scattered photon is generated with an energy based upon the compton spectrum predicted for the petra laser - wire parameters . this photon is tracked to the detector whilst fully simulating any interactions with materials such as the beampipe wall . this process is repeated to create an effective single compton energy distribution and its corresponding distribution at the detector after passing through any matter along the photon path . the single photon distribution in the detector is extrapolated to the @xmath44 spectrum using poisson statistics whilst also accounting for the energy resolution of the calorimeter and the longitudinal sub - structure of a typical laser pulse . the simulated spectrum is compared directly to the experimental data ( see fig . [ fig - sim ] ) , where the laser and electron beam were well aligned . the expected number of compton - scattered photons , @xmath44 , per shot with the laser - wire setup parameters is approximately @xmath45 photons , which agrees with the theoretical value . the experimental data show an energy resolution of 34% which is dominated by the longitudinal fluctuations in the laser power . the simulation models these fluctuations using relatively old streak camera data as described above and so does not account for degradation in the quality of the laser since then . the calorimeter has also not been calibrated for the range of energy deposits now incident upon it and has been in the petra radiation environment for three years . this could explain why the simulation fails to completely model the experimental data in the lower energy region . the future strategy for the laser - wire project can be characterised in the short term to concentrate on non - laser issues like data acquisition , signal detection , vertical scanning , and implementation into a linac beamline . this aims at the development of a standard diagnostic tool to be placed at many locations along the accelerator beamline . in the long run r&d work is planned to develop a laser system producing pulses matching the ilc micro pulse structure . here the target is to have a beamsize monitor with full flexibilty . to meet the short - term targets it is planned to purchase an injection seeded q - switch laser with second harmonic generation having excellent longitudinal and transverse mode quality . a complimentary project concentrating on the achievement of micron - scale laser spot - sizes is underway at the accelerator test facility ( atf ) at kek .
|
the laser - wire will be an essential diagnostic tool at the international linear collider .
it uses a finely focussed laser beam to measure the transverse profile of electron bunches by detecting the compton - scattered photons ( or degraded electrons ) downstream of where the laser beam intersects the electron beam .
such a system has been installed at the petra storage ring at desy , which uses a piezo - driven mirror to scan the laser - light across the electron beam .
latest results of experimental data taking are presented and compared to detailed simulations using the geant4 based program bdsim .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the experimental discovery of bose - einstein condensation @xcite in dilute systems of trapped alkali - metal atoms , such as rubidium ( @xmath0 ) , lithium ( @xmath1 ) , sodium ( @xmath2 ) and ytterbium ( @xmath3 ) , has spurred a renewed interest into the investigation of macroscopic quantum phenomena and interference effects , allowing for a deeper understanding of the conceptual foundations of quantum mechanics @xcite . this fascinating research area has been growing up thanks to the high degree of experimental manipulation and control @xcite . interference between condensates released in a potential with a barrier was first observed in 1997 @xcite and that paved the way for further investigations on the problem of bose condensates in a double well potential . then josephson oscillations have been observed in one dimensional optical potential arrays @xcite . a single bosonic josephson junction was produced for the first time in 2005 with @xmath0 atoms and its dynamics was experimentally investigated both within tunneling as well as self - trapping regime @xcite@xcite@xcite . more recently , mixtures of @xmath4 and @xmath5 atoms have been produced and experimentally investigated @xcite as well , whose intraspecies scattering lengths could be tunable via magnetic and optical feshbach resonances . furthermore the realization of heteronuclear mixtures of @xmath5 and @xmath6 atoms with tunable interspecies interactions @xcite paved the way to the exploration of double species mott insulators and , in general , of the quantum phase diagram of two species bose - hubbard model @xcite . the interplay between the interspecies and intraspecies scattering produces deep consequences on the properties of the condensates , such as the density profile @xcite and the collective excitations @xcite . however , the wide tunability of such interactions makes a bec mixture a very interesting subject of investigation , both from experimental and theoretical side as a mean of studying new macroscopic quantum tunneling phenomena as well as the interplay between quantum coherence and nonlinearity . indeed novel and richer behaviours are expected in such a multicomponent bec . on the theoretical side , a bosonic josephson junction with a single species of bec has been widely investigated by means of a two - mode approximation @xcite@xcite@xcite , within the classical as well as the quantum regime . in the classical regime , characterized by large particle numbers and weak repulsive interactions , the gross - pitaevskii equation provides a reliable description . within the two mode approximation it reduces to two generalized josephson equations which describe the time evolution of the relative phase and the population imbalance between the wells @xcite and differ from their superconducting counterpart @xcite by the presence of a nonlinear term which couples the variables . because of such a term , a bosonic josephson junction exhibit a variety of novel phenomena which range from @xmath7-oscillations to macroscopic quantum self - trapping ( mqst ) @xcite . while the @xmath8-oscillations , as well the usual josephosn ones , deal with a symmetric oscillation of the condensate about the two wells , the mqst phenomenon is characterized by a broken symmetry phase with a population imbalance between the wells . in the quantum regime , characterized by smaller values of the particle number and strong interactions , an increasing of phase fluctuations is observed together with the suppression of number fluctuations . furthermore the time evolution is characterized by phase collapse and revival @xcite . the quantum behaviour of bosonic josephson junctions has been deeply investigated by means of the usual quantum phase model @xcite@xcite@xcite as well as by starting from a two - mode bose - hubbard hamiltonian @xcite@xcite@xcite . in this context the phase coherence of the junction has been characterized by studying the momentum distribution @xcite@xcite . the generation and detection of schroedinger cat states has been investigated as well ; indeed the presence of such a kind of states reflects in the strong reduction of the momentum - distribution contrast @xcite@xcite . more recently such a theoretical analysis has been successfully extended to a binary mixture of becs in a double well potential @xcite@xcite @xcite@xcite . the semiclassical regime in which the fluctuations around the mean values are small has been deeply investigated and found to be described by two coupled gross - pitaevskii equations . by means of a two - mode approximation such equations can be cast in the form of four coupled nonlinear ordinary differential equations for the population imbalance and the relative phase of each species . the solution results in a richer tunneling dynamics . in particular , two different mqst states with broken symmetry have been found @xcite , where the two species localize in the two different wells giving rise to a phase separation or coexist in the same well respectively . indeed , upon a variation of some parameters or initial conditions , the phase - separated mqst states evolve towards a symmetry - restoring phase where the two components swap places between the two wells , so avoiding each other . recently , the coherent dynamics of a two species bec in a double well has been analyzed as well focussing on the case where the two species are two hyperfine states of the same alkali metal @xcite . in this paper we study the quantum behaviour of a binary mixture of bose - einstein condensates ( bec ) in a double - well potential starting from a two - mode bose - hubbard hamiltonian . we analyze in detail the small tunneling amplitude regime where number fluctuations are suppressed and a mott - insulator behaviour is established . we perform a perturbative calculation up to second order in the tunneling amplitude and study the stationary states and the dynamics of the two species bosonic josephson junction . finally , the dynamical generation of schroedinger cat states is investigated starting from an initial coherent spin state and shown to affect the time - dependent population imbalance and momentum distribution@xcite@xcite . we focus on the contrast in the momentum distribution between the two wells and show how it vanishes for a two - component cat state . that could be interesting in view of the experimental realization of macroscopic superpositions of quantum states @xcite@xcite . the paper is organized as follows . in section 2 we introduce our model hamiltonian within the two - mode approximation and define the various parameters . then we adopt the angular momentum representation and focus on the small tunneling amplitude regime . in section 3 we apply perturbation theory in the tunneling amplitude to our hamiltonian and find analytical expressions for the energy eigenvalues and eigenstates up to second order . section 4 and 5 are devoted to the study of the quantum evolution of the number difference of bosons between the two wells in correspondence of two different initial conditions : completely localized states and coherent spin states . in the first case both the short and the long time dynamics is studied and a rich behaviour is evidenced , ranging from small amplitude oscillations and collapses and revivals to coherent tunneling . in the second case the short - time scale evolution of number difference is determined and a more irregular dynamics is evidenced , with suppression of the dominant frequency when the number of bosons increase . then , schroedinger cat states are shown to generate as a result of the time - evolution of an initial coherent state when the tunneling between the two wells is suppressed , and their influence on the contrast in the momentum distribution is studied . finally , in section 6 some conclusions and outlooks of this work are presented . a binary mixture of bose - einstein condensates @xcite@xcite loaded in a double - well potential is described by the general many - body hamiltonian : @xmath9 where @xmath10 @xmath11 are the hamiltonians for bosons of species @xmath12 and @xmath13 respectively and @xmath14 is the interaction term between bosons of different species . for dilute mixtures one can replace the interaction potentials @xmath15 , @xmath16 and @xmath17 with the effective contact interactions : @xmath18 where @xmath19 and @xmath20 are the intraspecies coupling constants of the species @xmath12 and @xmath13 respectively , @xmath21 and @xmath22 being the atomic masses and @xmath23 , @xmath24 the @xmath25-wave scattering lengths ; furthermore @xmath26 is the interspecies coupling constant , where @xmath27 is the reduced mass and @xmath28 is the associated @xmath25-wave scattering length . in this way the hamiltonian ( [ m1])-([m4 ] ) can be rewritten as : @xmath29 @xmath30 here @xmath31 is the double well trapping potential and , in the following , we assume @xmath32 ; @xmath33 @xmath34 , @xmath35 are the bosonic creation and annihilation operators for the two species , which satisfy the commutation rules : @xmath36 & = & \left [ \psi _ { i}^{+}\left ( \overrightarrow{r}\right ) , \psi _ { j}^{+}\left ( \overrightarrow{r}^{\prime } \right ) \right ] = 0 , \label{m8 } \\ \left [ \psi _ { i}\left ( \overrightarrow{r}\right ) , \psi _ { j}^{+}\left ( \overrightarrow{r}^{\prime } \right ) \right ] & = & \delta _ { ij}\delta \left ( \overrightarrow{r}-\overrightarrow{r}^{\prime } \right ) , \text { \ \ \ \ } % i , j = a , b , \label{m9}\end{aligned}\ ] ] and the normalization conditions : @xmath37 @xmath38 , @xmath35 being the number of atoms of species @xmath12 and @xmath13 respectively . the total number of atoms of the mixture is @xmath39 . now a weak link between the two wells produces a small energy splitting between the mean - field ground state and the first excited state of the double well potential and that allows us to reduce the dimension of the hilbert space of the initial many - body problem . indeed for low energy excitations and low temperatures it is possible to consider only such two states and neglect the contribution from the higher ones , the so called two - mode approximation @xcite @xcite@xcite . in this way , by taking into account for each of the two species @xmath12 and @xmath13 the mean - field ground states @xmath40 , @xmath41 and the mean - field excited states @xmath42 , @xmath43 , the wave functions @xmath44 , @xmath35 , can be rewritten as : @xmath45 where @xmath46 , @xmath35 , and @xmath47 , @xmath48 and @xmath49 , @xmath50 ( @xmath51 , @xmath52 and @xmath53 , @xmath54 ) are the creation ( annihilation ) operators for a particle of the species @xmath12 , @xmath13 in the ground and the excited state respectively . they satisfy the usual bosonic commutation relations @xmath55 = \left [ b_{i},b_{j}^{+}\right ] = \delta _ { ij}$ ] . furthermore @xmath56 , @xmath35 are assumed real for simplicity and such that @xmath57 , which simplifies the calculations . let us change the basis and switch to the atom number states in such a way that the expectation value of the population of the left and right well can be defined . the new annihilation operators are @xmath58 , @xmath59 and @xmath60 , @xmath61 for the species @xmath12 and @xmath13 respectively , so that the wave functions ( [ m11 ] ) become : @xmath62 by substituting equations ( [ m12 ] ) into the hamiltonian ( [ m6])-([m7 ] ) , after some algebra we obtain its second quantized version within the two - mode approximation : @xmath63 where @xmath64 , @xmath35 , is the number of atoms of species @xmath12 and @xmath13 respectively , expressed as a sum of numbers of atoms in the left and right well . the parameters are defined as follows : @xmath65 @xmath66 @xmath67 @xmath68 @xmath69 @xmath70 @xmath71 @xmath72 @xmath73 ; \text { \ \ \ } i , j = a , b;\text { \ \ \ } i\neq j. \label{m22}\ ] ] in eq . ( [ m13 ] ) , the terms proportional to @xmath74 , @xmath35 , describe tunneling of particles of species @xmath12 and @xmath13 from one to the other well while the terms proportional to @xmath75 , @xmath35 , deal with the local interaction within the two wells and the terms proportional to @xmath76 correspond to additional two - particle processes . finally the terms proportional to @xmath77 and @xmath78 couple the two species and then various constant terms follow , which we will drop for simplicity . in this paper we focus on the small tunneling amplitude regime where number fluctuations are suppressed and a mott - insulator behaviour is established , so it is convenient to introduce the angular momentum representation for the species @xmath12 and @xmath13 as follows : @xmath79 where the operators @xmath80 , @xmath81 , @xmath82 , obey to the usual angular momentum algebra and the following relations hold : @xmath83 in particular , the components @xmath84 , @xmath35 give the difference of the number of bosons of the species @xmath85 , @xmath86 and @xmath87 , occupying the two minima of the double well potential , i. e. the population imbalances , which are experimentally observable quantities . thus hamiltonian ( [ m13 ] ) can be cast in the following form : @xmath88 where the constant terms have been dropped . let us now simplify the notation by introducing the following parameters : @xmath89 and rewrite the hamiltonian ( [ m24 ] ) as : @xmath90 within the experimental parameters range it is possible to show that @xmath91 , @xmath35 , and @xmath92 @xcite@xcite , then in the following we put @xmath93 and @xmath94 , which corresponds to neglecting the spatial overlap integrals between the localized modes in the two wells . in this way the binary mixture of becs within two - mode approximation maps to two ising - type spin model in a transverse magnetic field . in the following we will focus on the symmetric case @xmath95 and @xmath96 because it allows us to perform analytical calculations while capturing many relevant phenomena characterizing the physics of the system . so the model hamiltonian ( [ m24 ] ) becomes : @xmath97 @xmath98 @xmath99 where , in the small tunneling amplitude regime , @xmath100 is considered as a perturbation . the total hamiltonian commutes with @xmath101 and @xmath102 , which leads to the conservation of total angular momentum with quantum numbers @xmath103 and @xmath104 respectively . so the whole hilbert space has finite dimension , equal to @xmath105 , thus it depends on the number of bosons of the species @xmath12 and @xmath13 respectively . the whole basis @xmath106 is given by the eigenvectors of @xmath107 ( @xmath108 ) and @xmath109 ( @xmath110 ) with @xmath111 and @xmath112 . as a first step we need to diagonalize the unperturbed hamiltonian ( [ m28 ] ) , which can be done by performing the following @xmath113 rotation on the operators @xmath107 , @xmath109 : @xmath114 while an analogous rotation needs to be carried out on @xmath115 , @xmath116 entering the perturbation ( [ m29 ] ) . as a result we get : @xmath117 which , by defining @xmath118 , @xmath119 , and @xmath120 , @xmath121 , @xmath122 , @xmath123 , can be cast in the final form : @xmath124 in the following section we will find analytical expressions for the eigenvalues and the eigenvectors up to second order by performing perturbation theory in the tunneling amplitude . in the present section we apply second - order perturbation theory to the hamiltonian of eq . ( [ m32 ] ) in the small tunneling amplitude limit , which allows us to derive analytical expressions for the stationary states of the system . in order to pursue this task let us rewrite eq . ( [ m32 ] ) in dimensionless form by assuming @xmath125 as unit of energy : @xmath126 where @xmath127 and @xmath128 , then take @xmath129 as unperturbed hamiltonian and @xmath130 as a small perturbation term . here @xmath131 , @xmath82 , obey the usual angular momentum algebra and the following relation holds : @xmath132 where : @xmath133 in principle , the rotated basis @xmath134 of the unperturbed hamiltonian ( [ m34 ] ) is given by the eigenvectors of @xmath135 ( @xmath136 ) and @xmath137 ( @xmath138 ) with @xmath139 and @xmath140 , whose corresponding eigenvalues are @xmath141 . the presence of the operator @xmath142 , which does not commute with the perturbation term @xmath143 , makes the problem of finding eigenvalues and eigenvectors of the full hamiltonian ( [ m33 ] ) within perturbation theory much more involved . in order to simplify the treatment and carry out analytical calculations while retaining the relevant phenomenology , we concentrate on the particular case of a binary mixture where the two species are equally populated , i. e. @xmath144 , and have the same population imbalance between the two wells , i. e. @xmath145 . this situation allows us to describe the quantum dynamics of the system in correspondence of the mqst regime , for which we need a completely localized initial state . that fixes @xmath146 while @xmath147 could be an even or odd integer depending on @xmath148 even or odd , and leads to the following zero - order eigenvalues : @xmath149 . each eigenvalue is two - fold degenerate , with the only exception of the ground state for @xmath150 even , @xmath151 , which is nondegenerate . the two - dimensional subspace of degeneracy is spanned by the states @xmath152 ( where @xmath153 ) and the corresponding zero - order eigenvectors are : @xmath154 by switching on the perturbation term ( [ m35 ] ) it is possible to show that the degeneration is lifted starting from the levels with smaller @xmath155 ; in general the double degeneracy of the zero - order eigenvalues @xmath156 will be lifted at the @xmath157-th order of perturbation theory @xcite . by applying perturbation theory @xcite up to order @xmath158 , we obtain the following corrected eigenvalues : @xmath159 @xmath160 @xmath161 where @xmath162 . furthermore , for @xmath163 even , the nondegenerate ground state @xmath164 belongs to the symmetry class of @xmath165 . the corresponding eigenvectors , up to order @xmath158 , are given in the appendix . in the following sections we use the analytical expressions of energy eigenvectors derived in the appendix , see eqs . ( [ m41])-([m48 ] ) , in order to study the quantum evolution of @xmath166 , that is the number difference of bosons of species @xmath12 and @xmath13 between the two wells of the potential . in this section we investigate the quantum evolution of the number difference of bosons of species @xmath12 and @xmath13 between the two wells assuming a completely localized state as initial condition . that could be interesting in order to elucidate the quantum behavior of the system in correspondence of the classical mqst regime and to put in evidence new phenomena including quantum coherence in a multicomponent system . in such a case we will study both the short and the long time dynamics : as a result a rich behaviour emerges , ranging from small amplitude oscillations and collapses and revivals to coherent tunneling . although such a physics is well known for the single component bose josephson junction , in our case the dynamics shows that the two species can coexist in the same potential well as if there would be an attractive interaction between them . as a first step let us recall the general formula which gives the time evolution of the mean value of @xmath167 @xcite : @xmath168 where @xmath169 is the dimensionless time , the sums are over all the eigenvectors @xmath170 , being @xmath171 or @xmath172 , and @xmath173 are the projections of the initial state @xmath174 on the basis @xmath175 : @xmath176 so it is clear how the knowledge of eigenvalues and eigenvectors is enough in order to study the quantum evolution of @xmath137 , the bohr frequencies involved , @xmath177 , and the corresponding weights @xmath178 . let us now study the dynamics of the system when all the bosons of species @xmath179 and @xmath13 are initially contained in one of the two wells of the potential , say the right one , and then the imbalances of the two species coincide , so that @xmath180 , @xmath181 , @xmath182 , @xmath183 ; furthermore the two species are equally populated , i.e. @xmath184 . that implies @xmath146 and @xmath185 in our _ center of mass _ rotated basis . the corresponding initial condition is : @xmath186 in order to investigate the short timescales evolution we need to keep terms up to second order in the tunneling amplitude @xmath187 when we compute the weights in eq . ( [ m49 ] ) . we find that : @xmath188 , \label{m52}\ ] ] where the frequency involved is : @xmath189 at short timescales small amplitude oscillations with frequency @xmath190 around the initial condition ( @xmath191 , @xmath192 ) are observed and that coincides with a strongly self - trapped regime . in order to investigate the dynamics at longer timescales we have to take into account also the small splittings @xmath193 and @xmath194 of the two higher pairs of quasidegenerate eigenvalues which provide two further frequencies ( see ref . @xcite for the derivation ) : @xmath195 the whole result is : @xmath196 \right . \nonumber \\ & & \left . + 2\cos ( \omega _ { \mu } \tau ) \cos ( \frac{\omega _ { 1}}{2}\tau ) -\cos ( \omega _ { 1}\tau ) -\cos ( \omega _ { 0}\tau ) \right ] , \label{m55}\end{aligned}\ ] ] and , by putting @xmath197 , the short timescale dynamics , eq . ( [ m52 ] ) , is recovered . summarizing , at longer timescales the two species bosons are still localized in the initial potential well but the quantum dynamics exhibits collapses and complete revivals . indeed the coefficient @xmath198 , which multiplies the higher frequency term @xmath199 , gives rise to the beat , which is responsible for the observed collapses and revivals at timescales fixed by @xmath200 , as shown in fig . [ fig : dynamics ] . finally , at very large timescales determined by the frequency @xmath201 all the bosons tunnel coherently back and forth between the two traps ; only the first term @xmath202 is responsible of such a coherent tunneling , since all harmonic functions containing the frequency @xmath200 and @xmath203 are small in amplitude and proportional to @xmath158 , thus they are unable to transfer bosons from one trap to the other . the tunneling dynamics within macroscopic quantum self - trapping regime described above is analogous to that of the @xmath204-mode fixed point obtained by the gross - pitaevski approach @xcite , where the two species localize in the same well despite the repulsive interaction between them . let us finally note that , despite the explicit dependence on @xmath205 of the frequencies ( [ m53])-([m54 ] ) , the different physics related to the three time scales described above is simply due to the energy splitting introduced by the renormalized tunneling for small @xmath77 . thus in the case of a mixture of becs with equal population the dynamics remains similar to that of a single component bec , apart the coexistence of the two species in the same well . as for the experimental detection of the long timescales phenomena ( collapses / revivals and coherent tunneling ) , since the time for their appearance is abruptly increased with @xmath163 , this implies a rapid decrease of the characteristic frequencies rendering more difficult the observation of the intermediate and long time behavior in current bec experiments . indeed pure condensates consisting of @xmath206 atoms of @xmath5 loaded in a double well have been recently realized @xcite@xcite thus rendering the detection of the intermediate time behavior possible . mixtures with a number of atoms ranging from @xmath207 and @xmath208 ( @xmath209 and @xmath6 @xcite ) to @xmath210 and @xmath211 ( @xmath4 and @xmath5 @xcite ) have also been recently realized , but in this case very small characteristic frequencies are implied . however , these phenomena may be relevant for molecular systems where the number of vibrational excited quanta is small . is @xmath212 and the boson number is @xmath213,title="fig : " ] is @xmath212 and the boson number is @xmath213,title="fig : " ] is @xmath212 and the boson number is @xmath213,title="fig : " ] in the next section we will further investigate the dynamics of the system by assuming as initial state a simple coherent state and then study the formation of a particular superposition of such coherent states , the so called schroedinger cat states . in this section we choose as initial condition a simple coherent spin state @xcite and study the short - time scale evolution of number difference ; in this way a more complex dynamics will appear . finally , we study the generation of schroedinger cat states ; in particular , we focus on the contrast in the momentum distribution and show how it vanishes for a two - component cat state . let us start by considering as initial condition the following coherent spin state @xcite : @xmath214 where the coefficient @xmath215 is : @xmath216 and @xmath217 and @xmath218 are two angles characterizing the superposition . the time evolution of the mean value of @xmath137 up to first order in the tunneling amplitude @xmath187 is given by : @xmath219 + c_{2}\sin \left ( \omega _ { e}\tau \right ) \right . \nonumber \\ & & \left . + \sum_{n=0or1/2}^{n_{2}/2 - 1}\frac{n_{2}!}{\left ( \frac{n_{2}}{2}% + n\right ) ! \left ( \frac{n_{2}}{2}-n\right ) ! } \frac{n_{2}-2n}{2\left ( 2n+1\right ) } a_{n}\right ] , \label{m58}\end{aligned}\ ] ] where @xmath220 -\frac{1}{\tan ^{2n+1}\left ( \frac{\theta } { 2}\right ) } \left [ \cos \left ( f_{n}\tau -\phi \right ) -\cos \left ( \phi \right ) \right ] \label{m59}\ ] ] with frequencies @xmath221 . furthermore the coefficients @xmath222 and @xmath223 are given by : @xmath224 -\left ( \frac{n_{2}}{2}% + 1\right ) \left [ \tan \left ( \frac{\theta } { 2}\right ) -\frac{1}{\tan \left ( \frac{\theta } { 2}\right ) } \right ] \right\ } , \label{m60}\ ] ] @xmath225 \left [ \left ( \frac{n_{2}}{6}-\frac{% 1}{3}\right ) \sin \left ( 3\phi \right ) -\left ( \frac{n_{2}}{2}+1\right ) \sin \left ( \phi \right ) \right ] , \label{m61}\ ] ] for @xmath163 even , and @xmath226 , \label{m62}\ ] ] @xmath227 , \label{m63}\ ] ] for @xmath163 odd , respectively . finally , for @xmath163 even , the zero - order mean value @xmath228 is given by : @xmath229 \left [ \cos ( \omega _ { e}\tau ) -1% \right ] + 2\sin \left ( 2\phi \right ) \sin \left ( \omega _ { e}\tau \right ) \right\ } , \label{m64}\end{aligned}\ ] ] where the dominant frequency @xmath230 is equal to @xmath231 . the corresponding expression for @xmath163 odd is : @xmath232 \left [ \cos ( \omega _ { e}\tau ) -1\right ] -2\sin \left ( \phi \right ) \sin \left ( \omega _ { e}\tau \right ) \right\ } , \label{m65}\end{aligned}\ ] ] where @xmath233 . as one can see , the dominant frequency is gradually suppressed with the number of bosons @xmath234 @xcite . this is clearly seen in fig . [ fig : st - dynamics ] where the boson number difference between the two traps is plotted for different @xmath235 ( even ) values as a function of the dimensionless time @xmath236 . one also notices a decrease of the oscillation amplitude at increasing @xmath235 . the effect of @xmath205 is instead shown in fig . [ fig : st - dynamics - lambda ] where the short - time dynamics of the boson number difference is analyzed for two values of the interspecies interaction . when @xmath205 increases the amplitude of the oscillations decreases . the detection of the mixture dynamics is thus more favorable for values of @xmath205 smaller than unity . is @xmath237 and @xmath213(black - dashed line ) , @xmath238(red - dotted line ) , @xmath239(blue - straight line ) , while @xmath240 and @xmath241 . ] . . the value of @xmath187 is @xmath242 while @xmath243(black - dashed line ) and @xmath244(green - straight line ) , while @xmath240 and @xmath245 . ] . let us consider the coherent spin state ( [ m56 ] ) ; the expectation value of the hamiltonian ( [ m33 ] ) on such state is given by : @xmath246 where @xmath247 and has the maximum value for @xmath248 . this result also corresponds to the mean - field result for the energy . now , starting from the coherent spin state ( [ m56 ] ) we are interested in looking for schroedinger cat states . such states are quantum superposition of macroscopic states and their realization has already been suggested for a single species bose - josephson junction in @xcite . also in the case of a bose - josephson junction with binary mixtures one might realize cat states from the time - evolution of an initially coherent state following a sudden rise of the barrier between the two wells . thus we consider at time @xmath249 a zero inter - well coupling @xmath187 , i.e. the time evolution is governed by the hamiltonian @xmath250 in eq . ( [ m34 ] ) . for each basis vector @xmath251 of the coherent state ( [ m56 ] ) , the time - evolution is given by @xmath252 , where @xmath253 is the so - called revival time such that @xmath254 . considering now the times @xmath255 , @xmath256 integer , the time evolution of the coherent state is governed by the factor @xmath257 which satisfies the property @xmath258 , depending on the parity of @xmath256 . for the choice of even @xmath256 , a discrete fourier transform leads to the cat state : @xmath259 i.e. a superposition of @xmath256 coherent states , where @xmath260 . in particular , the cat state affects the momentum distribution . this dependence could be important to probe experimentally their existence . in particular , when considering the two - component cat state , i.e. for the choice @xmath261 , one obtains that the contrast in the momentum distribution , i.e. the expectation value of @xmath262 on the unperturbed state , vanishes@xcite . furthermore , the amplitude of the intervals of time in which the contrast is zero increases with increasing @xmath163 as clearly shown in fig . [ fig : contrast ] . it should be noted that despite the close similarity in the behavior of the contrast between the single component bjj and the double one , the mixture will be a better candidate for the creation and detection of cat states . in fact their creation time is @xmath263 and since for repulsive interaction between the two species and @xmath264 we get @xmath265>1 $ ] , such time can be made short enough to render their detection more favorable . for example by fixing the ratio of @xmath266 interaction to @xmath267 interaction to be 2.13 , a parameter accessible in the jila setup@xcite , the detection time is twice smaller than the case of a single component bec . for @xmath268 and @xmath245 and even number of bosons . the red line is for @xmath213 , the blue one for @xmath269 and the black one for @xmath270 . the interval in which the contrast is zero increases with increasing @xmath235 . ] in this paper we investigated the quantum dynamics of a bose josephson junction made of a binary mixture of becs loaded in a double well potential within the two - mode approximation . we focused on the small tunneling amplitude limit and adopted the angular momentum representation for the bose - hubbard dimer hamiltonian . perturbation theory up to second order in the tunneling amplitude enabled us perform analytical calculations in the symmetric case where @xmath271 and @xmath96 . in this way we obtained the energy eigenvalues and eigenstates , whose knowledge is mandatory in order to investigate the quantum evolution of the number difference of bosons between the two potential wells . in order to study the quantum dynamics more easily and analitycally , we restricted to the case in which the two species are equally populated and imposed the condition of equal population imbalance of the species @xmath12 and @xmath13 between the two wells . we concentrated on the two following initial conditions : completely localized states and coherent spin states , and found a rich and complex behaviour , ranging from small amplitude oscillations and collapses and revivals to coherent tunneling . finally , we considered the generation of schroedinger cat states and pointed out their influence on the momentum distribution through the vanishing of the contrast . we showed that the creation time can be rendered short enough in the case of a mixture in order to render their detection more favorable . that could be crucial in order to build up an experimental protocol to produce and detect cat states within such systems . we stress that in this work we have chosen to study the symmetric case . this allowed us to obtain analytical results , while giving rise to the relevant phenomenology which characterizes the physics of the junction . the general case of different couplings between the two bosonic species and/or different populations needs to resort to numerical calculations and will be the subject of a future publication @xcite . another interesting issue which deserves further investigation is a careful analysis of the quantum manifestations of the self - trapping transition and in general of the mqst phenomenon in this more general context . the complex dynamics of the generalized bose josephson junctions investigated in the present paper could be experimentally testable within the current technology . for instance , the jila group recently @xcite succeeded in producing a mixture of @xmath4 and @xmath5 atoms , whose interactions are widely tunable via feshbach resonances . in particular it is possible to fix the scattering length of @xmath5 as well as the interspecies one and to tune the scattering length of @xmath4 . that allows one to explore the parameter space in a wide range and also to realize the symmetric regime @xmath271 . because of the high degree of experimental control , such a setup could be employed to reproduce the phenomenology described in this work . the authors would like to thank m. salerno for driving their attention on the topic of bose josephson junctions and e. orignac and a. minguzzi for discussions and for a critical reading of the manuscript . the eigenvectors of the full hamiltonian ( [ m33 ] ) , up to order @xmath158 , are : @xmath273 } { 2}}% \left| 0,2^{+}\right\rangle , \label{m41}\ ] ] @xmath274 \right ) \left| 0,1^{-}\right\rangle + \frac{k}{6\lambda } % \sqrt{\left [ j_{2}\left ( j_{2}+1\right ) -2\right ] } \left| 0,2^{-}\right\rangle \nonumber \\ & & + \frac{k^{2}}{96\lambda ^{2}}\sqrt{\frac{\left [ j_{2}\left ( j_{2}+1\right ) -2\right ] \left [ j_{2}\left ( j_{2}+1\right ) -6\right ] } { 2}}\left| 0,3^{-}\right\rangle , \label{m42}\end{aligned}\ ] ] @xmath275 \right ) \left| 0,1^{+}\right\rangle -\frac{k}{\lambda } % \sqrt{\frac{j_{2}\left ( j_{2}+1\right ) } { 2}}\left| 0,0\right\rangle + \frac{k}{6\lambda } \sqrt{\left [ j_{2}\left ( j_{2}+1\right ) -2\right ] } \left| 0,2^{+}\right\rangle \nonumber\\ & & + \frac{k^{2}}{96\lambda ^{2}}\sqrt{\frac{\left [ j_{2}\left ( j_{2}+1\right ) -2\right ] \left [ j_{2}\left ( j_{2}+1\right ) -6\right ] } { 2}}\left| 0,3^{+}\right\rangle , \label{m43}\end{aligned}\ ] ] @xmath276 \right ) \left| 0,\frac{1}{2}% ^{\pm } \right\rangle + \frac{k^{2}}{48\lambda ^{2}}\sqrt{\left [ j_{2}\left ( j_{2}+1\right ) -\frac{3}{4}\right ] \left [ j_{2}\left ( j_{2}+1\right ) -\frac{% 15}{4}\right ] } \left| 0,\frac{5}{2}^{\pm } \right\rangle \nonumber \\ & & + \left [ \frac{k}{4\lambda } \sqrt{j_{2}\left ( j_{2}+1\right ) -\frac{3}{4}}% \mp \frac{k^{2}}{16\lambda ^{2}}\sqrt{\left [ j_{2}\left ( j_{2}+1\right ) + % \frac{1}{4}\right ] \left [ j_{2}\left ( j_{2}+1\right ) -\frac{3}{4}\right ] } % \right ] \left| 0,\frac{3}{2}^{\pm } \right\rangle , \label{m44}\end{aligned}\ ] ] @xmath277 where the coefficients are defined as : @xmath278 @xmath279 @xmath280 99 m. h. anderson , j. r. ensher , m. r. matthews , c. e. wieman , e. a. cornell , _ science _ * 269 * ( 1995 ) 198 ; c. c. bradley , c. a. sackett , j. j. tollett , r. g. hulet , _ phys * 75 * ( 1995 ) 1687 ; k. b. davis , m. o. mewes , m. r. andrews , n. j. van druten , d. s. durfee , d. m. kurn , w. ketterle , _ phys . * 75 * ( 1995 ) 3969 . m. albiez , r. gati , j. folling , s. hunsmann , m. cristiani , m. k. oberthaler , _ phys . lett . _ * 95 * ( 2005 ) 010402 ; r. gati , m. albiez , j. folling , b. hemmerling , m. k. oberthaler , _ appl . b _ * 82 * ( 2006 ) 207 . j. catani , l. de sarlo , g. barontini , f. minardi , m. inguscio , _ phys . a _ * 77 * ( 2008 ) 011603(r ) ; p. buonsante , s. m. giampaolo , f. illuminati , v. penna , a. vezzani , _ phys . rev . * 100 * ( 2008 ) 240402 . a. smerzi , s. fantoni , s. giovanazzi , s. r. shenoy , _ phys . * 79 * ( 1997 ) 4950 ; s. raghavan , a. smerzi , s. fantoni , s. r. shenoy , _ phys . rev . a _ * 59 * ( 1999 ) 620 ; s. giovanazzi , a. smerzi , s. fantoni , _ phys . lett . _ * 84 * ( 2000 ) 4521 .
|
we study the quantum behaviour of a binary mixture of bose - einstein condensates ( bec ) in a double - well potential starting from a two - mode bose - hubbard hamiltonian .
we focus on the small tunneling amplitude regime and apply perturbation theory up to second order .
analytical expressions for the energy eigenvalues and eigenstates are obtained .
then the quantum evolution of the number difference of bosons between the two potential wells is fully investigated for two different initial conditions : completely localized states and coherent spin states . in the first case
both the short and the long time dynamics is studied and a rich behaviour is found , ranging from small amplitude oscillations and collapses and revivals to coherent tunneling . in the second case the short - time scale evolution of number difference
is determined and a more irregular dynamics is evidenced . finally , the formation of schroedinger cat states is considered and shown to affect the momentum distribution .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the jack symmetric polynomials @xcite are a system of orthogonal polynomials expressing the excited states of an integrable one - dimensional quantum many - body system with the trigonometric type potential called the calogero - sutherland model @xcite . to be precise , the jack polynomials are eigenfunctions of a hamiltonian @xmath5 which is obtained by a certain transformation of the calogero - sutherland hamiltonian . here @xmath6 is a parameter appearing in the calogero - sutherland model . the excited states can be constructed from the jack polynomials . ] these are one - parameter deformations of the schur symmetric polynomials . in general , being integrable means that the model has sufficiently many conserved quantities , and that system can be analytically solved . like the calogero - sutherland model , many of the integrable systems are not physical models of particles existing in the real world . however , the mathematical structure of the integrable models , e.g. , excellent solvability , can be used to advantage in many fields of mathematics . let us consider symmetric functions which are defined as a projective limit of symmetric polynomials with finite variables ( * ? ? ? * chap . 1 ) . in the case of the jack polynomials , the infinite - variable limit exists and is called the jack symmetric functions . the jack functions are parametrized by partitions or young diagrams , and has the complex parameter @xmath6 ( see also footnote [ footnote : hbeta ] ) . actually we can consider the parameter @xmath6 as an indeterminate , and then the jack functions are defined over the field @xmath7 . the surprising result due to mimachi and yamada is that the jack functions associated to rectangular young diagrams have a one - to - one correspondence with singular vectors of the virasoro algebra @xcite . the virasoro algebra is constructed by the infinitesimal conformal transformations in two dimensions , and is the lie algebra generated by @xmath8@xmath9 and the central element @xmath10 satisfying the relations @xmath11=(n - m)l_{n+m}+ c \ : \frac{n(n^2 - 1)}{12}\delta_{n+m,0 } , \quad n , m \in \mathbb{z},\ ] ] @xmath12=0,\quad n \in \mathbb{z}.\ ] ] this is an essential algebra to two - dimensional conformal field theories required for string theory and statistical mechanics . to obtain the irreducible representations of the virasoro algebra is important not only in representation theory but also in the conformal field theories . the irreducibility of highest weight representations can be determined by special vectors called singular vectors in the highest weight representation . although the singular vectors have an integral representation , the expression formula of the jack functions by the dunkl operator @xcite is more useful . further , various properties of jack functions are known . thus , the expression of the singular vectors by the jack functions is very convenient and beneficial . as a @xmath13-difference deformation of the jack polynomials , there is a system of orthogonal polynomials with rich theory called the macdonald polynomials @xcite . for later use let us introduce the notation for macdonald symmetric function , which is the infinite - variable version of the macdonald polynomial . we denote by @xmath14 the macdonald symmetric function associated to the partition @xmath15 . here @xmath13 and @xmath16 are free parameters , and they can be considered as complex numbers or indeterminates . in this paper , we regard the power sum symmetric functions @xmath17 as variables of the macdonald functions ( for more detail , see appendix [ sec : macdonald and hl ] ) . the macdonald polynomials are also simultaneous eigen - functions of commuting @xmath13-difference operators , now called macdonald difference operators . let us also mention that they are related to the ruijsenaars model @xcite which is a relativistic extension of the calogero - sutherland model . the @xmath13-deformation like the macdonald functions makes theory clearer and often mathematically easier to handle . for example , the jack functions can be characterized as the hamiltonian @xmath5 ( see footnote [ footnote : hbeta ] ) , but they have degenerate eigenvalues , and difficulties arise when we prove their orthogonality and coincidence with the singular vectors . in the theory of the macdonald functions , this degeneracy problem can be eliminated and the discussion is clearer . also , the hamiltonian @xmath5 has an infinite number of commuting operators . however , it is difficult to write down these operators explicitly @xcite , and in the macdonald theory we have an explicit formula for the commuting family of difference operators having @xmath14 as simultaneous eigenfunctions . for the above reason , it can be said that macdonald s theory is more beautiful . in the @xmath18 ( @xmath19 ) limit with @xmath6 fixed , the macdonald functions are reduced to the jack functions . on the other hand , in @xmath3 limit with @xmath16 fixed , they are reduced to the symmetric functions called the hall - littlewood functions . the hall - littlewood functions have a close connection to the character of the general linear group over finite fields , and they are also a generalization of the schur functions ( * ? ? ? * chap . it is one of the advantages that it is possible to unify and generalize the two generalizations of the schur functions . some applications in knot invariants @xcite and stochastic processes @xcite are also known . the macdonald functions are one of the important symmetric functions for modern mathematics . awata , kubo , odake and shiraishi introduced in @xcite a @xmath13-deformation of the virasoro algebra , which is named the deformed virasoro algebra . this deformed algebra is designed so that singular vectors of verma modules correspond to macdonald symmetric functions @xmath14 . the deformed virasoro algebra is an associative algebra defined over the base field @xmath20 , where @xmath13 and @xmath16 are the same parameters as in @xmath14 . the generators are denoted by @xmath21 ( @xmath22 ) , and the defining relation is @xmath23 = -\sum_{l=1}^{\infty}f_l(t_{n - l}t_{m+l}-t_{m - l}t_{n+l } ) -\frac{(1-q)(1-t^{-1})}{1-p}(p^n - p^{-n } ) \delta_{n+m,0},\ ] ] where @xmath24 and @xmath25 are the structure constants defined by @xmath26 it is shown that the singular vectors of the deformed virasoro algebra coincide with the macdonald functions associated with rectangular young diagrams . it is also possible to obtain the jack and macodnald functions associated with general partitions from the singular vectors of the @xmath27-algebra and the deformed @xmath27-algebra ( which is the ( deformed ) virasoro algebra when @xmath28 ) @xcite . to be exact , singular vectors of the ( deformed ) @xmath27-algebra can be realized by @xmath29 families of bosons under the free field representation . by a certain projection to one of these bosons , we can obtain the jack ( or macdonald ) functions associated with young diagrams with @xmath29 edges ( see figure [ fig : youngdiag_n-1edges ] ) . edges , width=226 ] * 1.2 . * the representation theory of the virasoro algebra plays an essential role in the two - dimensional conformal field theories . in 2009 , while studying the low energy effective theory of m5-branes , alday , gaiotto and tachikawa discovered the correspondence between the correlation functions of two - dimensional conformal field theories and the partition functions of four - dimensional supersymmetric gauge theories ( agt conjecture ) @xcite . gauge theory has a long history and is an attractive theory studied by a lot of mathematicians and physicists . although it is difficult to calculate the partition functions of gauge theories in general , nekrasov gave an explicit formula ( nekrasov formula ) for the instanton partition function of four - dimensional @xmath30 supersymmetric gauge theory in 2002 @xcite . the nekrasov formula @xmath31 is written by the summation of the terms @xmath32 parametrized by tuples of young diagrams : @xmath33 these terms are given in a factorized form , and as @xmath34 increases , the amount of calculation becomes enormous . however , it can be calculated by a simple combinatoric method . the discovery of @xcite is the following relation between two - dimensional and four - dimensional field theories . the nekrasov formula for the four - dimensional @xmath35 gauge theory with four matters in ( anti-)fundamental representation ( actually , it is the nekrasov formula of the @xmath36 gauge theory divided by the @xmath2 factor @xmath37 ) coincides with the four - point conformal block of the two - dimensional conformal field theory . basics of the conformal field theories were established by belavin , polyakov and zamolodchikov ( bpz ) in 1984 @xcite . they described the critical phenomenon of the two - dimensional ising model which is a model of the ferromagnet , and so on . the primary fields @xmath38 are operators on the representation space of the virasoro algebra such that @xmath39=z^n\left ( z\frac{\partial}{\partial z } + h(n+1 ) \right ) v(z ) , \quad z , h\in \mathbb{c}.\ ] ] the primary fields are the main research object in the conformal field theories . here @xmath40 is called the conformal dimension of the primary field . furthermore , in the conformal field theories , it is a fundamental problem to calculate the correlation functions of the primary fields . generally , in the quantum field theories , the calculations of correlation functions are difficult , and usually it is often solved by approximation . bpz succeeded in determining the exact forms of correlation functions in the conformal field theories . in particular , they derived differential equations with regular singularities for the correlation functions . however , the research by bpz was performed mainly for primary fields with the special conformal dimension , i.e. the minimal models , and they did not investigate the correlation functions in general forms . even if we derive the differential equations of the correlation functions , it is difficult to find their solutions . from the standpoint of conformal field theories , the agt conjecture that states the agreement between the nekrasov formulas and the conformal blocks ( originally in the liouville theory , that is the theory having the primary field with generic conformal dimensions ) is studied under the expectation that general formulas for the correlation functions can be obtained . various extensions were made immediately after the agt conjecture was discovered . first of all , the original agt conjecture deals with the four - dimensional gauge theory in the case that the number of ( anti-)fundamental matters @xmath41 is @xmath42 . immediately after this original conjecture @xcite , the cases with @xmath43 were studied in @xcite . these cases can be obtained from the case of @xmath44 by applying the same degenerate limits to the nekrasov formula and the conformal block . especially when @xmath45 , the conformal block degenerates to the inner product of the vector @xmath46 called the whittaker vector of the virasoro algebra . case , the degenerate conformal blocks can be realized by the inner product of certain vectors that are the general form of the vector @xmath46 . ] moreover , it is also expected that the four - dimensional gauge theories with the higher gauge group @xmath47 correspond with the @xmath27-algebra @xcite . the jack functions and the macdonald functions also play an important role in the agt conjecture . for example , the expansion coefficients of the whittaker vector @xmath46 by the jack functions are clarified @xcite . in addition , it is known that a good basis called aflt basis @xcite can be regarded as a sort of generalization of the jack functions . the aflt basis is a basis in the representation space of the algebra @xmath48 , which is first introduced by alba , fateev , litvinov and tarnopolskiy , and the conformal block can be combinatorially expanded by this basis . the aflt basis is an orthogonal basis which parametrized by pairs of young diagrams @xmath49 . in the @xmath27 algebra case , it is parametrized by @xmath0-tuples of young diagrams and exists in the representation space of the algebra @xmath50 . by inserting the identity @xmath51 with respect to the aflt basis @xmath52 , the calculation of correlation functions @xmath53 is attributed to that of the matrix element @xmath54 , where @xmath38 is a sort of the primary field defined by some relations with generators of the virasoro algebra and the heisenberg algebra . then the three - point functions @xmath55 are factorized and coincide with the significant factors called the nekrasov factors , which compose the nekrasov formula . namely , if we expand the correlation functions by using the aflt basis , then the form of its expansion is quite the same as that of the nekrasov formula ( [ eq : nek formula ] ) . further , the conformal block of the algebra @xmath56 coincides with the partition function @xmath31 of @xmath36 gauge theory . actually , such a good basis does not exist in the representation space of the virasoro algebra . since the @xmath2 factor contributes and complicates the agt conjecture , we need the adjustment by the heisenberg algebra . since the aflt basis correspond to the torus fixed points in the instanton moduli space , it is also called the fixed point basis . in @xcite , the original agt conjecture is `` proved '' with the help of the aflt basis ( the generalized jack functions ) and the free field representation . in @xcite by using zamolodchikov reccursion relation . some proofs from geometric representation theory are also given in @xcite . ] however , this `` proof '' is based on another conjecture . to explain it in more detail , recall that the free field representation of the conformal blocks can be written by the dotsenko - fateev integral @xmath57 , where @xmath58 means some integrals of the integrand @xmath59 . then @xmath59 can be expanded by a sum of the products of the generalized jack functions @xmath60 and their dual functions @xmath61 , which are parametrized by tuples of young diagrams . this expansion formula is called the cauchy formula . at that time , it was conjectured that the integral value of each term @xmath62 directly corresponds to @xmath32 in the nekrasov formula . this is the scenario of the `` proof . '' although this proof is straightforward without using recurrence formulas etc , since the integral value of the generalized jack functions is still a conjecture , it is necessary to prove it in order to complete this proof . for that , we need to investigate more properties of the generalized jack functions . @xmath13-deformed version of the agt conjecture is also provided . that is , the deformed virasoro/@xmath1-algebra is related to five - dimensional gauge theories ( 5d agt conjecture ) @xcite . in the simplest case , it is shown that the inner product of the whittaker vector of the deformed virasoro algebra coincides with the instanton partition function ( k - theoretical partition function ) of the five - dimensional @xmath63 pure @xmath36 gauge theory . also the same approach as @xcite is taken in the @xmath13-deformed case . in other words , it is conjectured that the @xmath13-deformed dotsenko - fateev integral corresponds to the partition function with @xmath44 matters , and this conjecture is checked by using the generalized macdonald functions @xcite . the @xmath13-deformed version of the aflt basis @xcite ( that is , the generalized macdonald functions ) exists in the representation space of the level @xmath0 representation of the ding - iohara - miki algebra ( dim algebra ) . the dim algebra ( explained in appendix [ sec : def of dim ] ) has the face of a @xmath13-deformation of the @xmath64 algebra as introduced by miki in @xcite , and the deformed virasoro/@xmath1-algebra appear in its representation @xcite . since the dim algebra has a lot of background , there are a lot of other names such as quantum toroidal @xmath65 algebra @xcite , quantum @xmath64 algebra @xcite , elliptic hall algebra @xcite and so on . the dim algebra has a hopf algebra structure which does not exist in the deformed virasoro/@xmath1-algebra , and the dim algebra is associated with the macdonald functions having rich theory . that is free field representation of the macdonald s difference operator , it is discovered that @xmath66 form a part of representation of the dim algebra @xcite . see also fact [ fact : lv . 1 rep of dim ] . ] unlike the case of the generalized jack functions , the generalized macdonald functions can be constructed by the coproduct of the dim algebra @xcite . it is a surprising phenomenon that the structure of the coproduct of the dim algebra has information on the partition functions of the five - dimensional gauge theories . furthermore , in the @xmath13-deformed case , awata - kanno s and iqbal - kozkaz - vafa s refined topological vertices @xcite are also reproduced by the matrix elements of some intertwining operator of the dim algebra , and the coincidence between the correlation function of the dim algebra and the 5d nekrasov formula is proved @xcite . the agt conjecture with respect to the @xmath13-deformed aflt basis @xcite ( recalled in section [ sec : reargument of di alg and agt ] ) is almost parallel to the undeformed case , and it suffices to consider the algebra @xmath67 , denoted by @xmath68 , which is generated by certain operators @xmath69 ( @xmath70 , @xmath9 ) obtained by the level @xmath0 representation of the dim algebra . the level @xmath0 representation is that on a fock module @xmath71 with the highest weight @xmath72 . the vertex operator @xmath73 on this fock module is defined by the relation ( definition [ df : dim vertex op ] ) @xmath74 where @xmath75 . @xmath76 can be regarded as an analog of the virasoro primary field . the generalized macdonald functions are defined to be the eigenfunctions of the generator @xmath77 constructed by the copoduct of the dim algebra . then , it is conjectured that the matrix elements of @xmath76 with respect to the generalized macdonald functions reproduce the five - dimensional nekrasov factors . under this conjecture , the four - point conformal block of @xmath76 corresponds to the 5d @xmath36 nekrasov formula with @xmath78 matters . * * the first main theorem in this thesis is the formula for the kac determinant of the algebra @xmath68 ( theorem [ thm : kacdet ] ) : @xmath79 where @xmath80 , @xmath81 , and @xmath82 denotes the number of the @xmath0-tuples of young diagrams of the size @xmath34 . for the definition of @xmath83 , see notations in the latter part of this section . this determinant can be proved by using the fact that the generators @xmath69 can be decomposed into the deformed @xmath1-algebra part and the @xmath2 part by a linear transformation of the bosons , and using the screening currents of the deformed @xmath1-algebra . by this formula , we can solve the conjecture ( * ? ? ? * conjecture 3.4 ) that the following pbw type vectors of the algebra @xmath68 ( definition [ df : ordinary pbw vct ] ) are a basis : @xmath84 we also discover that singular vectors of the algebra @xmath68 correspond to some generalized macdonald functions as the second main theorem . by this result , we can get singular vectors from generalized macdonald functions . can express all singular vectors of the algebra @xmath68 is incompletely understood . however , the kac determinant can be proved by the only vanishing points given by the singular vectors @xmath85 ( see ( [ eq : sing vct chi(i)rs ] ) ) corresponding to the simple roots , because the determinant has @xmath86 weyl group invariance . ] the singular vectors are intrinsically the same as those of the deformed @xmath1-algebra . however , as the projection of the bosons is necessary for the correspondence with the ordinary macdonald functions , the result of this thesis that does not need projections can be thought to be a generalization of @xcite . as a corollary of this fact , we can find a new relation of the ordinary macdonald functions and the generalized macdonald functions by the projection of the bosons . furthermore , since screening operators are written by integrals , we can also get an integral representation of generalized macdonald functions . concretely , the vector @xmath87 defined to be @xmath88 is a singular vector . here @xmath89 denotes the screening operator , the @xmath0-tuple of parameters @xmath90 is a function of @xmath91 , and for non - negative integers @xmath92 , @xmath93 , @xmath94 ( for more details , see section [ sec : sing vct and gn mac ] ) . the singular vector @xmath87 coincides with the generalized macdonald function with the @xmath0-tuple of young diagrams in figure [ fig : youngdiag_onlyrightside ] ( theorem [ thm : sing vct and gn mac 2 ] . ( main theorem ) ) . in fact , figure [ fig : youngdiag_n-1edges ] means the same young diagram being on the rightmost side in figure [ fig : youngdiag_onlyrightside ] . hence the projection of this generalized macdonald function corresponds to the ordinary macdonald functions associated with the rightmost young diagram with @xmath29 edges in figure [ fig : youngdiag_onlyrightside ] ( corollary [ cor : projection of gn mac ] ) . when the condition @xmath95 for the number of screening currents and parameter @xmath91 in @xmath87 is removed , the above figure is not a young diagram . however it turns out that the vector @xmath87 coincides with the generalized macdonald function obtained by cutting off the protruding part and moving boxes to the young diagrams on the left side . for example , if @xmath96 for all @xmath97 , the corresponding @xmath0-tuple of young diagrams of the generalized macdonald function is figure [ fig : youngdiag_somerectangle ] ( theorem [ thm : sing vct and gn mac 2 ] . ( main theorem ) ) . * 1.4 . * furthermore , we investigate behavior in the limit to the hall - littlewood functions , @xmath3 , of the deformed virasoro algebra and the algebra @xmath68 . also 5d agt conjecture is studied in this limit . the reason of considering such a limit is that the situation becomes simple and some problems are solved . in particular , the simplest 5d agt conjecture can be proved , and pbw type vectors can be expressed in terms of hall - littlewood functions . by virtue of the theory of hall - littlewood functions , we can obtain and prove an explicit formula ( theorem [ thm : main theorem ] ) for the four - point correlation function of a certain operator @xmath98 , which is the limit @xmath3 of the vertex operator @xmath76 associated with @xmath99 . here , @xmath71 is the fock module with the highest weight @xmath100 . @xmath101 here for a partition @xmath15 , @xmath102 , and @xmath103 is the same one in ( [ eq : kac det in intro ] ) . the function @xmath104 can be calculated by the generalized hall - littlewood functions in the same way as @xcite . however , we can obtain this formula by inserting the identity with respect to the pbw type vectors . we call this hall - littlewood limit @xmath3 ` crystallization ' after the use of the quantum groups @xcite , where the parameter @xmath13 represents the temperature in the rsos model @xcite which has symmetry of the deformed virasoro algebra , and the limit @xmath105 can be considered as the zero temperature limit . although our studies are mathematically different from the notion of the original crystal base of quantum groups , the physical meaning and the motivation to simplify phenomena are the same . to investigate their mathematical relationship is an interesting open problem . on the other hand , little is known about the physical meaning of the hall - littlewood limit in the gauge thoery at present . * in this thesis , the r - matrix of the dim algebra is also investigated . the result with respect to the r - matrix is based on the collaborative researches @xcite , and only works of the author is described . in general , a r - matrix is defined as a solution of the yang - baxter equation , and is closely related to the solvable lattice models , the knot invariants and so on . further , it is well - known that r - matrices can be constructed by hopf algebras such as the quantum groups . in general , a hopf algebra @xmath106 with the coproduct @xmath107 is called quasi - cocommutative if there exists an invertible element @xmath108 in the algebra @xmath109 such that @xmath110 this @xmath108 is called the universal r - matrix . if @xmath108 also satisfies the relations @xmath111 ( see definition of @xmath112 in section [ sec : r - matrix ] ) then @xmath106 is called quasi - triangular and @xmath108 satisfies the yang - baxter equation @xmath113 . the dim algebra is known to be quasi - triangular @xcite . in this thesis , the representation matrix of the universal r - matrix @xmath108 is explicitly calculated . in the tensor product of the level 1 representation of the dim algebra ( we denote it by @xmath114 ) , it is block - diagonalized at each level of the free boson fock space . also , it can be seen that the action of @xmath108 on the generalized macdonald functions corresponds to the exchange of spectral parameters , partitions , and variables in the generalized macdonald functions . moreover , by using the renormalized generalized macdonald functions ( the integral form @xmath115 ) , it can be conjectured that @xmath116 where @xmath117 is the vector obtained by exchanging partitions , variables and spectral parameters in @xmath118 ( see definitions in section [ sec : explicit cal of r ] ) . as a consequence , we have conjecture ( conjecture [ conj : r - matrix by int form ] ) of the explicit formula for the representation matrix @xmath119 of the universal r - matrix in the basis of @xmath118 : @xmath120 in @xcite , the rtt relation of the dim algebra is also studied using this r - matirx . this thesis is organized as follows . in section [ sec:5d agt ] , two examples of the 5d agt conjecture are reviewed . one is the correspondence between the whittaker vector of the deformed virasoro algebra and the partition function of the 5d pure gauge theory . the other is the conjecture on the aflt basis using the level @xmath0 representation of the dim algebra . in section [ sec : kac det and sing vct ] , we give a factorized formula for the kac determinant of the algebra @xmath68 . its proof depends on some results of the deformed @xmath1-algebra . the relationship between the singular vectors and the generalized macdonald functions is also revealed . in section [ sec : crystallization ] , we investigated the @xmath105 limit of the deformed virasoro algebra , the algebra @xmath68 and the 5d agt conjecture . in particular , the simplest 5d agt conjecture is proved in this limit . in section [ sec : r - matrix ] , the explicit form of the representation of the universal r - matrix of the dim algebra is calculated . its general form is also conjectured in terms of the generalized macdonald functions . in section [ sec : properties of gn mac ] , properties of the generalized macdonald functions are studied . first , to state the existence theorem of the generalized macdonald functions , we need partial orderings among @xmath0-tuples of partitions . in this thesis , by using the partial orderings @xmath121 ( see definition [ def : ordering1 ] ) and @xmath122 ( see definition [ df : ordering elaborated version ] ) , the existence theorem is proved . however , in @xcite , another ordering @xmath123 is used and the proof of existence theorem ( * ? ? ? * proposition 3.8 ) is omitted . we justify the theorem ( * ? ? ? * proposition 3.8 ) by comparing @xmath122 and @xmath123 in subsection [ sec : partial orderings ] . in subsection [ sec : realization of rank n rep ] , we also investigate the action of the generators @xmath124 and higher rank hamiltonians on the generalized macdonald functions . their actions are based on a conversion rule called spectral duality that exchanges the level @xmath0 representation and the rank @xmath0 representation of the dim algebra . furthermore , in subsection [ sec : limit to beta ] , the @xmath125 limit is also studied . since the generalized jack functions have degenerate eigenvalues , their cauchy formula used in the senario of proof of the agt conjecture @xcite is non - trivial . by taking the limit from the macdonald functions , we can justify the orthogonality of the generalized jack functions and show the cauchy formula . in appendix [ sec : macdonald and hl ] , the definition and basic facts of the ordinary macdonald functions and the hall - littlwood functions are briefly reviewed following @xcite . in appendix [ sec : def of dim ] , the definition of the dim algebra and the level @xmath0 representation are explained following mainly @xcite . moreover we also describe the definition of another representation of the dim algebra called level @xmath126 representation or the rank @xmath0 representation . in appendix [ sec : proofs and chekc of crystal ] , we present some proofs and checks of conjectures in section [ sec : crystallization ] . at last in appendix [ sec : ex of r - matrix ] , we give explicit examples of r - matrix at level 2 . * @xmath127 denote the set of positive integers , integers , rational numbers , real numbers , complex numbers , respectively . * @xmath128 denotes the set of non - negative integers . * @xmath129 denotes the set of integers except @xmath130 . * @xmath131 denotes the kronecker delta . * @xmath132 $ ] denotes the ring of polynomials in @xmath133 over a field @xmath134 . * @xmath135 denotes the cardinality of set . * functions @xmath136 depending on multiple variables @xmath137 ( @xmath138 ) are occasionally written as @xmath139 or @xmath140 for abbreviation . * for a partition @xmath15 , @xmath141 and @xmath142 denote the power sum symmetric function and the monomial symmetric function , respectively . * for @xmath143 , @xmath144 denotes the elementary symmetric function . let us explain the notation of partitions and young diagrams . a partition @xmath145 is a non - increasing sequence of integers @xmath146 . we write @xmath147 . the length of @xmath15 , denoted by @xmath148 , is the number of elements @xmath149 with @xmath150 . partitions are identified if all elements except @xmath130 are the same . for example , @xmath151 . @xmath83 denotes the number of elements that are equal to @xmath152 in @xmath15 , and we occasionally write partitions as @xmath153 . for example , @xmath154 . the partitions are identified with the young diagrams , which are the figures written by putting @xmath149 boxes on the @xmath152-th row and aligning the left side . for example , if @xmath155 , its young diagram is . the conjugate of a partition @xmath15 , denoted by @xmath156 , is the partition whose young diagram is the transpose of the diagram @xmath15 . for example , the conjugate of @xmath155 is @xmath157 . for a partition @xmath15 and a coordinate @xmath158 , define @xmath159 @xmath160 is called arm length and @xmath161 is called leg length . in the diagram , they mean the numbers of boxes in right side from or below the box being in the @xmath152-th row and @xmath162-th column . for example , if @xmath163 , then @xmath164 , @xmath165 . note that they can take negative values as @xmath166 , @xmath167 . for a partition @xmath15 , we define @xmath168 . this means the sum of the numbers obtained by attaching a zero to box in the top row of the young diagram of @xmath15 , a @xmath169 to each box in the second row , and so on . for @xmath0-tuple of partitions @xmath170 , define @xmath171 . if @xmath172 , we occasionally use the symbol `` @xmath173 '' as @xmath174 . the author would like to express his deepest gratitude to his supervisor hidetoshi awata for a great deal of advice . without his guidance and persistent help , this thesis would not have been possible . the author shows his greatest appreciation to hiroaki kanno for his insightful comments and suggestions , and h. fujino , t. matsumoto , a. mironov , al . morozov , and . morozov and y. zenkevich for the collaborative researches . some of the results in this thesis are based on the collaborations with them . the author also would like to thank m. hamanaka , k. iwaki , t. shiromizu , s. yanagida and friends for valuable discussions and supports . the author is supported in part by grant - in - aid for jsps fellow 26 - 10187 . we start with recapitulating the result of the whittaker vector of the deformed virasoro algebra and the simplest five - dimensional agt correspondence . let @xmath13 and @xmath16 be independent parameters and @xmath175 . the deformed virasoro algebra is the associative algebra over @xmath20 generated by @xmath21 ( @xmath22 ) with the commutation relation @xmath176 = -\sum_{l=1}^{\infty}f_l(t_{n - l}t_{m+l}-t_{m - l}t_{n+l } ) -\frac{(1-q)(1-t^{-1})}{1-p}(p^n - p^{-n } ) \delta_{n+m,0},\ ] ] where the structure constant @xmath177 is defined by @xmath26 the relation ( [ comm.rel.of qvir ] ) can be written in terms of the generating function @xmath178 as @xmath179,\ ] ] where @xmath180 . the deformed virasoro algebra is introduced in @xcite . let @xmath181 be the highest weight vector such that @xmath182 , @xmath183 ( @xmath184 ) , and @xmath185 be the verma module generated by @xmath181 . similarly , @xmath186 is the vector satisfying the condition that @xmath187 , @xmath188 ( @xmath189 ) . @xmath190 is the dual module generated by @xmath186 . the pbw type vectors @xmath191 for partitions @xmath15 form a basis over @xmath185 . also , @xmath192 form a basis over @xmath190 . here @xmath193 is a partition or a young diagram . the bilinear form @xmath194 is uniquely defined by @xmath195 . this bilinear form is called the shapovalov form . the whittaker vector @xmath196 is defined as follows . for a generic parameter @xmath197 , define the whittaker vector is also called the gaiotto state or the irregular vector . ] @xmath196 by the relations @xmath198 similarly , the dual whittaker vector @xmath199 is defined by the condition that @xmath200 this vector is in the form @xmath201 and its norm is calculated as @xmath202 , where @xmath203 denotes the inverse matrix element of the shapovalov matrix @xmath204 . it is useful to consider the free field representation of the deformed virasoro algebra . by the heisenberg algebra generated by @xmath137 ( @xmath9 ) and @xmath205 with the relations @xmath206=n\frac{1-q^{|n|}}{1-t^{|n| } } \delta_{n+m,0 } , \qquad [ a_n,{q}]=\delta_{n,0},\ ] ] the generating function @xmath207 can be represented as @xmath208 here @xmath209 . let @xmath210 be the highest weight vector in the fock module of the heisenberg algebra such that @xmath211 ( @xmath212 ) , and @xmath213 . then @xmath214 . furthermore , @xmath215 can be regarded as the highest weight vector @xmath181 of the deformed virasoro algebra with highest weight @xmath216 . in @xcite , awata and yamada conjectured an explicit formula for @xmath196 in terms of macdonald functions under the free field representation , and yanagida proved it in @xcite . the simplest five - dimensional agt conjecture is that the inner product @xmath217 coincides with the five - dimensional ( k - theoretic ) nekrasov formula for pure @xmath35 gauge theory @xcite : @xmath218 where @xmath160 and @xmath161 are the arm length and the leg length defined in introduction , and @xmath156 is the conjugate of @xmath15 . [ fact : simpleagt ] for @xmath219 , @xmath220 this fact is conjectured in @xcite and proved in @xcite when the parameter @xmath13 is generic . we now turn to the dim algebra @xcite . let us recall the aflt basis in the 5d agt correspondence of the @xmath47 gauge theory along @xcite . in this section , we use @xmath0 kinds of bosons @xmath221 ( @xmath222 , @xmath223 ) and @xmath224 with the relations @xmath225= n \frac{1-q^{|n|}}{1-t^{|n| } } \ , \delta_{i , j } \ , \delta_{n+m , 0},\ ] ] @xmath226=0 , \quad [ u_i , u_j]=0 , \qquad ( \forall i , j , n).\ ] ] here @xmath224 is the substitution of zero mode @xmath227 , which is realized in two different ways in sections [ sec : kac det and sing vct ] and [ sec : crystallization ] , respectively . let us define the vertex operators @xmath228 and @xmath229 . set @xmath230 define generators @xmath231 by @xmath232 where @xmath233 denotes the usual normal ordered product , and @xmath234 the generator @xmath235 arises from the level @xmath0 representation of ding - iohara - miki algebra @xcite , and is obtained by acting the coproduct of the dim algebra to the vertex operator @xmath66 @xmath0 times ( see appendix [ sec : def of dim ] ) . the other generators @xmath69 appear in the commutation relations of generators @xmath236 ( @xmath237 ) . when we just consider the agt conjecture , it suffices to deal with the subalgebra @xmath67 in some completion of the endomorphism of the algebra of @xmath0-tensored fock modules for our heisenberg algebra . we denote the algebra @xmath67 by @xmath68 . if @xmath238 , the commutation relations of the generators are @xmath239 where @xmath240 is the multiplicative delta function and the structure constant @xmath241 is defined by @xmath242 @xmath243 these relations are equivalent to @xmath244= -\sum_{l = 1}^{\infty } f^{(1)}_l ( x^{(1)}_{n - l } x^{(1)}_{m+l } - x^{(1)}_{m - l } x^{(1)}_{n+l } ) + \frac{(1-q)(1-t^{-1})}{1-p}(p^m - p^n ) x^{(2)}_{n+m } , \\ & [ x^{(2)}_n , x^{(2)}_m]= -\sum_{l = 1}^{\infty } f^{(2)}_l ( x^{(2)}_{n - l } x^{(2)}_{m+l } - x^{(2)}_{m - l } x^{(2)}_{n+l } ) , \\ & [ x^{(1)}_n , x^{(2)}_m]= -\sum_{l = 1}^{\infty } f^{(1)}_l ( p^{l } x^{(1)}_{n - l } x^{(2)}_{m+l } - x^{(2)}_{m - l } x^{(1)}_{n+l } ) .\end{aligned}\ ] ] the proof is similar to the calculation of the deformed virasoro algebra or the deformed @xmath1-algebra . in the formula ( [ eq : rel . of generator x^1 ] ) , we use @xmath245 for an @xmath0-tuple of parameters @xmath246 , define @xmath247 and @xmath248 to be the highest weight vectors such that @xmath249 ( @xmath250 , @xmath251 ) , @xmath252 and @xmath253 . @xmath254 is the highest weight module generated by @xmath255 , and @xmath256 is the dual module generated by @xmath257 . the bilinear form ( shapovalov form ) @xmath258 is uniquely determined by the condition @xmath259 . [ df : ordinary pbw vct ] for an @xmath0-tuple of partitions @xmath260 , set @xmath261 the pbw theorem can not be used because the algebra @xmath68 is not a lie algebra , but in @xcite it was conjectured that the pbw type vectors @xmath262 and @xmath263 are a basis over @xmath71 and @xmath256 , respectively . this conjecture can be solved by the kac determinant of the algebra @xmath68 , which is proved in section [ sec : kac det and sing vct ] . in this section , we consider another type of the pbw basis , since it has good expression in @xmath105 limit in terms of the hall - littlewood functions ( see section [ sec : crystallization of n=2 ] ) . [ df : another pbw vct ] for @xmath260 , set @xmath264 let us review the aflt basis in @xmath71 , which is also called generalized macdonald functions . in order to state its existence theorem , let us prepare the following ordering . [ def : ordering1 ] for @xmath0-tuple of partitions @xmath265 and @xmath266 , @xmath267 here @xmath268 . note that the second condition can be replaced with @xmath269 . we can state the existence theorem of generalized macdonald functions in the basis of products of macdonald functions @xmath270 , where @xmath271 are macdonald symmetric functions defined in appendix [ sec : macdonald and hl ] with substituting the bosons @xmath272 for the power sum symmetric functions @xmath17 . for each @xmath0-tuple of partitions @xmath265 , there exists a unique vector @xmath273 such that @xmath274 where @xmath275 is a constant , @xmath276 is the eigenvalue of @xmath277 . similarly , there exists a unique vector @xmath278 such that @xmath279 then the eigenvalues are @xmath280 although the ordering of definition [ def : ordering1 ] is different from the one in @xcite , the eigenfunctions @xmath281 are quite the same . the proof is similar to the one in section [ sec : partial orderings ] , which follows from triangulation of @xmath77 . by this proposition , it can be seen that @xmath281 is a basis over @xmath71 , and the eigenvalues of @xmath277 are non - degenerate . in section [ sec : partial orderings ] , a more elaborated ordering is introduced and a relationship between these orderings is explained . in section [ sec : limit to beta ] , it is shown that these vectors @xmath281 correspond to the generalized jack functions defined in @xcite in the @xmath18 limit . to use generalized macdonald functions in the agt correspondence , we need to consider its integral form . in this paper , we adopt the following renormalization , which is slightly different from that of @xcite . [ df : integral form of gn macdonald ] define the vectors @xmath282 and @xmath283 , called the integral forms , by the condition that @xmath284 the coefficients @xmath285 and @xmath286 are polynomials in @xmath287 , @xmath288 and @xmath289 with integer coefficients . if @xmath238 , the transition matrix @xmath290 is as follows : @xmath291 @xmath292 @xmath293 by using these integral forms , the five dimensional agt conjecture can be stated in the following form . * conjecture 3.11 and conjecture 3.13 ) ) the norm of @xmath118 reproduces the nekrasov factor : @xmath294 where @xmath295 . [ df : dim vertex op ] call the linear operator @xmath296 the vertex operator if it satisfies @xmath297 and @xmath298 . then the relations for the fourier components are @xmath299 for @xmath223 . if @xmath300 , it is known that @xmath76 exists and is given by @xmath301 where @xmath302 is the operator from @xmath303 to @xmath304 satisfying the relation @xmath305 . the matrix elements of @xmath306 with respect to generalized macdonald functions are @xmath307 under these conjectures , we can obtain a formula for multi - point correlation functions of @xmath76 by inserting the identity @xmath308 . in particular , the formula for the four - point functions agrees with the 5d @xmath309 nekrsov formula with @xmath78 matters . an m - theoretic derivation of this formula is also given by @xcite . in this section , we give the formula for the kac determinant of the algebra @xmath68 and prove it . moreover , it is shown that singular vectors correspond to the generalized macdonald functions . in order to prove the kac determinant , we need screening currents of the algebra @xmath68 . to construct them , it is necessary to realize the operator @xmath224 and the highest wight vector @xmath247 in terms of the charge operator @xmath310 and @xmath227 ( @xmath311 ) . let @xmath310 be the operator satisfying the relation @xmath312=\delta_{n,0}\delta_{i , j},\ ] ] @xmath210 be the highest weight vector in the fock module of the heisenberg algebra such that @xmath313 for @xmath314 . for an @xmath0-tuple of complex parameters @xmath315 with @xmath316 , we realize the highest wight vector @xmath247 and @xmath224 as @xmath317 where @xmath6 is defined by @xmath19 . then they satisfy the required relation @xmath318 . similarly , let @xmath319 be the dual highest weight vector , and @xmath320 . these highest wight vectors are normalized by @xmath321 , and satisfy the condition of the shapovalov form @xmath322 . and @xmath16 are assumed to be generic in this section . ] we obtain the formula for the kac determinant with respect to the pbw type vectors @xmath323.-algebra are proved in @xcite . ] [ thm : kacdet ] let @xmath324 . then @xmath325 where @xmath80 , @xmath81 , and @xmath326 is the number of entries in @xmath15 equal to @xmath152 . @xmath82 denotes the number of @xmath0-tuples of young diagrams of size @xmath34 , i.e. , @xmath327 . in particular , if @xmath300 , @xmath328 if @xmath329 and @xmath330 for any numbers @xmath152 , @xmath162 and integers @xmath331 , @xmath332 , then the pbw type vectors @xmath323 ( resp . @xmath333 ) are a basis over @xmath71 ( resp . @xmath334 ) . it can be seen that the representation of the algebra @xmath68 on the fock module @xmath71 is irreducible if and only if the parameters @xmath335 satisfy the condition that @xmath329 and @xmath330 . the proof of theorem [ thm : kacdet ] is given in the next section . it is known that the level @xmath0 representation of the dim algebra introduced in the last section can be regarded as the tensor product of the deformed @xmath27-algebra and the heisenberg algebra associated with the @xmath2 factor @xcite . this fact is obtained by a linear transformation of bosons . the point of proof of theorem [ thm : kacdet ] is to construct singular vectors by using screening currents of the deformed @xmath27-algebra under the decomposition of the generators @xmath336 into the deformed @xmath27-algebra part and the @xmath2 part . in general , a vector @xmath337 in the fock module @xmath71 is called the singular vector of the algebra @xmath68 if it satisfies @xmath338 for all @xmath152 and @xmath339 . the singular vectors obtained by the screening currents are intrinsically the same one of the deformed @xmath1-algebra . from this singular vector , we can get the vanishing line of the kac determinant in the similar way of the deformed @xmath27-algebra . at first , in the @xmath340 case , we introduce the following bosons . + * u(1 ) part boson * @xmath341 @xmath342 + * orthogonal component of @xmath343 for @xmath344 * @xmath345 + * fundamental boson of the deformed @xmath27-algebra part * @xmath346 @xmath347 then they satisfy the following relations @xmath348=-\frac{(1-q^{|n|})(1-t^{-|n|})(1-p^{|n|})}{n(1-p^{n|n| } ) } \delta_{n+m,0 } , \qquad [ { b^{(i)}}_n , b'_m]=[{h^{(i)}}_n , b'_m]=[{q^{(i)}}_h , b'_m]=0,\ ] ] @xmath349=\frac{(1-q^{|n|})(1-t^{-|n|})}{n}\delta_{i , j}\delta_{n+m,0 } -p^{(n-\frac{i+j}{2})|n|}\frac{(1-q^{|n|})(1-t^{-|n|})(1-p^{|n|})}{n(1-p^{n|n|})}\delta_{n+m,0},\ ] ] @xmath350= -\frac{(1-q^n)(1-t^{-n})(1-p^{(\delta_{i , j}n-1)n})}{n(1-p^{nn})}p^{nn \,\theta(i > j ) } \delta_{n+m,0},\ ] ] @xmath351=\delta_{i , j}-\frac{1}{n } , \qquad \sum_{i=1}^n p^{-in } { h^{(i)}}_n = 0 , \qquad { q^{(n)}}_{\lambda}=\sum_{i=1}^n q^{(i)}_h=0 .\ ] ] where @xmath352 is @xmath169 or @xmath130 if the proposition @xmath353 is true or false , respectively . correspond to the fundamental bosons @xmath354 in @xcite ] using these bosons , we can decompose the generator @xmath336 into the u(1 ) part and the deformed @xmath1-algebra part . that is to say , @xmath355 @xmath356 and @xmath357 @xmath358 @xmath359 is the generator of the deformed @xmath27-algebra . let us introduce the new parameters @xmath360 and @xmath361 defined by @xmath362 then the inner product of pbw type vectors can be written as @xmath363 hence , its determinant is also in the form @xmath364 where @xmath365 is a polynomial in @xmath360 ( @xmath366 ) which is independent of @xmath361 . now in @xcite , the screening currents of the deformed @xmath27-algebra are introduced : @xmath367 where @xmath368 is the root boson defined by @xmath369 and @xmath370 . the bosons @xmath368 and @xmath371 commute with @xmath372 , and it is known that the screening charge @xmath373 commutes with the generators @xmath374 . therefore , @xmath373 commutes with any generator @xmath375 , and it can be considered as the screening charges of the algebra @xmath68 . define parameters @xmath376 , @xmath377 by @xmath378 and @xmath379 , and set @xmath380 . then @xmath381 . for any number @xmath382 , the vector arising from the screening current @xmath89 , @xmath383 is a singular vector . @xmath85 is in the fock module @xmath384 with the parameter @xmath335 satisfying @xmath385 for @xmath386 and @xmath387 , @xmath388 . the @xmath389 vectors obtained by this singular vector @xmath390 contribute the vanishing point @xmath391 in the polynomial @xmath59 . similarly to the case of the deformed @xmath27-algebra ( see @xcite ) , by the @xmath86 weyl group invariance of the eigenvalues of @xmath392 , the polynomial @xmath59 has the factor @xmath393 ( @xmath394 ) . considering the degree of polynomials @xmath395 , we can see that when @xmath396 , the kac determinant is @xmath397 where @xmath398 is a rational function in parameters @xmath13 and @xmath16 and independent of the parameters @xmath289 . if @xmath300 , @xmath399 is clearly in the form @xmath400 next , the prefactor @xmath398 can be computed in general @xmath0 case by introducing another boson @xmath401 the commutation relation of the boson @xmath402 is @xmath403 = -\frac{(1-t^{-n})(1-q^n)}{n}\delta_{i , j } , \quad n>0.\ ] ] define the determinants @xmath404 and @xmath405 with @xmath406 given by the expansions @xmath407 where @xmath408 and @xmath409 are @xmath410 by using these determinants , the kac determinant can be written as @xmath411 here @xmath412 is the determinant of the diagonal matrix @xmath413 . this factor is independent of the parameters @xmath289 , and we have @xmath414 . in ( [ eq : kacdet1 ] ) , the factor depending on @xmath289 in @xmath399 was already clarified . hence , we can determine the prefactor @xmath398 by computing the leading term in @xmath415 . that is , the prefactor @xmath398 can be written as @xmath416 where we introduce the function @xmath417 which gives the leading term of @xmath418 as the polynomial in @xmath419 , and @xmath420 . to calculate this leading term , define the operators @xmath421 , @xmath422 by @xmath423 @xmath424 is arising from only the operator @xmath425 in @xmath426 . then @xmath427 let the matrices @xmath428 and @xmath429 be given by @xmath430 where @xmath431 and @xmath432 are defined in the usual way : @xmath433 then @xmath424 is expressed as @xmath434 since the matrix @xmath428 is lower triangular with respect to the partial ordering @xmath435 and @xmath436 are defined as follows : @xmath437 @xmath438 then we have @xmath439 unless @xmath440 . ] and its diagonal elements are @xmath441 we have @xmath442 the transition matrix @xmath443 is upper triangular with respect to the partial ordering @xmath436 , and all diagonal elements are @xmath169 . thus @xmath444 . similarly by considering the base transformation to @xmath445 , it can be seen that @xmath446 therefore the prefactor @xmath398 is @xmath447 hence theorem [ thm : kacdet ] is proved . in this subsection , the singular vectors of the algebra @xmath68 are discussed . trivially , when @xmath448 , the kac determinant ( [ eq : kacdet for dim ] ) degenerates , and it can be easily seen that the vectors @xmath449 are singular vectors . since the screening operator @xmath89 is the same one of the deformed @xmath27-algebra , the situation of the singular vectors of @xmath68 except contribution arising when @xmath448 is the same as the deformed @xmath27-algebra . we discover that singular vectors obtained by the screening currents @xmath89 correspond to generalized macdonald functions . algebra and the aflt basis is investigated in @xcite . ] first , we have the following simple theorem . [ thm : sing vct and gn mac 1 ] for a number @xmath450 , if @xmath451 and the other @xmath452 are generic , there exists a unique singular vector @xmath85 in @xmath71 , and it corresponds to the generalized macdonald function @xmath281 with @xmath453 that is , @xmath454 existence and uniqueness are understood by the formula for the kac determinant ( [ eq : kacdet for dim ] ) in the usual way . actually , the unique singular vector @xmath85 is the one of ( [ eq : sing vct chi(i)rs ] ) . since the screening charges commute with @xmath277 , the singular vector is an eigenfunction of @xmath77 of the eigenvalue @xmath455 . using the relations @xmath385 for @xmath386 and @xmath387 , @xmath388 , we have @xmath456 where @xmath457 is the eigenvalue of the generalized macdonald functions introduced in ( [ eq : eigenvalue of gn mac ] ) . thus , the singular vector @xmath85 and the generalized macdonald function @xmath458 are in the same eigenspace of @xmath459 . moreover , by comparing the eigenvalues @xmath460 , it can be shown that the dimension of the eigenspace of the eigenvalue @xmath461 is @xmath169 even when @xmath451 . therefore , this theorem follows . let us consider more complicated cases . for variables @xmath462 ( @xmath463 ) , define the function @xmath464 by @xmath465 then it satisfies @xmath466 . we focus on the following singular vectors @xmath467 where the parameter @xmath468 is @xmath469 , @xmath470 , and for non - negative integers @xmath92 and @xmath471 ( @xmath463 ) , @xmath472 then the singular vector @xmath473 is in the fock module @xmath71 of the highest weight @xmath474 defined by @xmath475 , @xmath476 , @xmath477 and @xmath478 now we obtain the following main theorem with respect to the generalized macdonald functions and the singular vectors of the dim algebra . this theorem can be regarded as a generalization of the result in @xcite . [ thm : sing vct and gn mac 2 ] let parameters @xmath289 satisfy @xmath479 for all @xmath152 . * ( a ) . * if @xmath93 for all @xmath97 , then the singular vector @xmath473 coincides with the generalized macdonald function @xmath480 with @xmath481 : @xmath482 see figure [ fig : youngdiag_onlyrightside ] in introduction . * if @xmath483 for all @xmath97 , the singular vector @xmath473 coincides with the generalized macdonald function associated with the tuple of young diagrams @xmath484 : @xmath485 see figure [ fig : youngdiag_somerectangle ] in introduction . the proof is quite similar to that of theorem [ thm : sing vct and gn mac 1 ] . the eigenvalue of this singular vector is @xmath486 on the other hand , the eigenvalue of the generalized macdonald function in the case * ( a ) * is calcurated as follows . firstly , @xmath487 and @xmath488 hence , by using the equation @xmath489 , we can see that @xmath490 this is equal to the eigenvalue of the singular vector . also , it can be seen that the dimension of the eigenspace of the eigenvalue @xmath491 is @xmath169 even when @xmath479 . if the condition @xmath95 does not hold , figure [ fig : youngdiag_onlyrightside ] is not a young diagram . in this case , the singular vector @xmath492 corresponds to the generalized macdonald function with the @xmath0-tuple of young diagram obtained by cutting off the protruding parts and moving the boxes to the young diagram in the left side . that is , the case * ( b)*. the proof in the case * ( b ) * is exactly the same as the case * ( a ) * , so it is omitted . it is known that projections of the singular vectors @xmath492 in the case * ( a ) * onto the diagonal components of the boson @xmath493 correspond to ordinary macdonald functions ( * ? ? ? * ( 35 ) ) . hence , ordinary macdonald functions are obtained by the projection of generalized macdonald functions . [ cor : projection of gn mac ] when @xmath479 for all @xmath152 , @xmath494 here , @xmath17 denotes the ordinary power sum symmetric functions . next , we consider a crystallization of the results of subsection [ sec : review of simplest 5d agt ] , namely the behavior in the @xmath3 limit of the deformed virasoro algebra and the simplest 5d agt correspondence . in this limit , the scaled generators @xmath495 satisfy the commutation relation @xmath496 = & -(1-t^{-1})\sum_{\ell = 1}^{n - m}{\tilde t } _ { n-\ell}{\tilde t } _ { m+\ell } \quad ( n > m > 0 \quad \mbox{or}\quad 0>n > m ) , \\ [ { \tilde t } _ n,{\tilde t } _ 0 ] = & -(1-t^{-1})\sum_{\ell = 1}^{n}{\tilde t } _ { n-\ell}{\tilde t } _ { \ell } -(t - t^{-1})\sum_{\ell = 1}^{\infty } t^{-\ell } { \tilde t } _ { -\ell}{\tilde t } _ { n+\ell } \quad ( n > 0 ) , \\ [ { \tilde t } _ 0,{\tilde t } _ m ] = & -(1-t^{-1})\sum_{\ell = 1}^{-m}{\tilde t } _ { -\ell}{\tilde t } _ { m+\ell } -(t - t^{-1})\sum_{\ell = 1}^{\infty } t^{-\ell } { \tilde t } _ { m-\ell}{\tilde t } _ { \ell } \quad ( 0 > m ) , \allowdisplaybreaks[4 ] \\ [ { \tilde t } _ n,{\tilde t } _ m ] = & -(1-t^{-1}){\tilde t } _ { m}{\tilde t } _ { n } -(t - t^{-1})\sum_{\ell = 1}^{\infty } t^{-\ell } { \tilde t } _ { m-\ell}{\tilde t } _ { n+\ell } \nonumber \\ & + ( 1-t^{-1})\delta_{n+m,0 } \quad ( n > 0 > m).\end{aligned}\ ] ] in @xcite , the above algebra is introduced and its free field representation is given . let the bosons @xmath497 ( @xmath22 ) satisfy the relations @xmath498=n \frac{1}{1-t^{|n|}}\delta_{n+m,0}$ ] , @xmath499 = \delta_{n,0}$ ] . these bosons can be regarded as the @xmath105 limit of the bosons @xmath137 and @xmath500 in ( [ eq : comm rel of qt - boson ] ) , i.e. , @xmath501 , @xmath502 . then @xmath503 is represented as @xmath504 \tilde{\lambda}^+(z ) + \theta[\,n\geq0\ , ] \tilde{\lambda}^-(z ) \right ) z^n,\ ] ] where @xmath505 and @xmath506 $ ] is 1 or 0 if the proposition @xmath353 is true or false , respectively . by this free field representation , we can write the pbw type vectors in terms of hall - littlewood functions @xmath507 defined in appendix [ sec : macdonald and hl ] : @xmath508 here @xmath509 is an abbreviation for @xmath510 , and @xmath215 and @xmath511 are the same highest weight vectors in section [ sec : review of simplest 5d agt ] such that @xmath512 and @xmath513 . these expressions are the consequences of jing s operators ( fact [ fact : jing s operator ] ) . because of ( [ eq : inner prod of hl poly ] ) , they are diagonalized as @xmath514 where @xmath515 is defined in appendix [ sec : macdonald and hl ] . since @xmath516 is non - degenerate , there is no singular vector in the limit @xmath3 . the disappearance of singular vectors can be understood by the fact that the highest weight which has singular vectors diverges at @xmath517 . the whittaker vector of this algebra is similarly defined . define the whittaker vector @xmath518 by the relation @xmath519 similarly , the dual whittaker vector @xmath520 is defined by @xmath521 then the crystallized whittaker vector is in the simple form @xmath522 and its inner product is @xmath523 on the other hand , recalling the nekrasov formula @xmath524 given in ( [ eq : nek formula for pure ] ) of subsection [ sec : review of simplest 5d agt ] , we can take the crystal limit with the following trick . the renormalization @xmath525 controls divergence in the @xmath3 limit ( @xmath526 , @xmath527 : fixed ) : @xmath528 @xmath529 removing parts which have singularity in the nekrasov factor , we have @xmath530 hence , @xmath531 @xmath532 if @xmath533 or @xmath534 for any integer @xmath34 , @xmath535 , then @xmath536 at @xmath3 . therefore , the sum with respect to partitions @xmath15 , @xmath537 can be rewritten as the sum with respect to integers @xmath34 , @xmath535 , i.e. , @xmath538 after some simple calculation , we get ( [ eq : tildez ] ) . using these calculations , we can get the following theorem which is an analog of the simplest 5d agt relation ( fact [ fact : simpleagt ] ) , and prove it more easily than the generic case . @xmath539 note that the left hand side is independent of @xmath97 . @xmath540 can be rewritten as @xmath541 which has simple poles at @xmath542 with @xmath543 , @xmath544 and @xmath545 . then @xmath546 note that @xmath547 thus @xmath548 is an odd function in @xmath331 . therefore @xmath549 residues at all singularities in @xmath500 of @xmath550 vanish , but @xmath551 . hence @xmath550 is independent of @xmath500 . therefore , @xmath552 in this paper , we discuss the crystallization only of the deformed virasoro algebra . it is expected that the limit can be taken for the general deformed @xmath27-algebra . however in the case of @xmath553 , an essential singularity seems to appear , and at present we do not know how to take an appropriate limit . to find an appropriate limit procedure and apply the agt conjecture for the deformed @xmath27-algebra @xcite we need further studies . in the crystallized case , the screening current diverges , which is one of the reasons why in this limit singular vectors disappear . hence it may be difficult to apply the agt correspondence studied by @xcite . next , we discuss a crystallization of the results of subsection [ sec : reargument of di alg and agt ] . in this subsection and the next subsection , unlike section [ sec : kac det and sing vct ] , the operators @xmath224 are assumed to be independent of the parameter @xmath13 in order to avoid difficulty in taking the @xmath105 limit . let us realize the operators @xmath224 and the vector @xmath247 as @xmath554 then they also satisfy relation @xmath252 . similarly @xmath555 . moreover , we consider the case that the parameters @xmath289 are independent of @xmath13 . the case where the parameters @xmath289 depend on @xmath13 is briefly described in section [ sec : another type of limit ] . at first , let us demonstrate the @xmath105 limit in the @xmath300 case . in this subsection , we use the same bosons @xmath497 and @xmath205 as subsection [ sec : crystal of qvir ] . since singularity in @xmath76 can be removed by normalization @xmath556 , define the vertex operator @xmath557 by @xmath558 if @xmath300 , @xmath559 are ordinary macdonald functions , and their integral forms @xmath560 have , at @xmath517 , the relation @xmath561 hence , the matrix elements @xmath562 can be written in terms of integrals by virtue of jing s operators @xmath563 and @xmath564 defined in ( [ eq : jing s op ] ) and ( [ eq : dual jing s op ] ) . using the usual normal ordered product @xmath233 with respect to the bosons @xmath497 , be the heisenberg algebra generated by the bosons @xmath497 ( @xmath9 ) , @xmath205 and @xmath169 . @xmath565 is the algebra obtained by making @xmath566 commutative . the normal ordered product @xmath233 is defined to be the linear map from @xmath565 to @xmath566 such that for @xmath567 , @xmath568 and @xmath569 . in the next subsection , the same symbol @xmath233 denotes the normal ordered product with respect to the bosons @xmath570 which is defined similarly . ] we have @xmath571 thus @xmath572 where @xmath573 , @xmath574 , @xmath575 , and the integration contour is @xmath576 . this integral reproduces the @xmath3 limit of the nekrasov factor . [ def : crystal nek factor ] set @xmath577 where @xmath578 is the set of boxes in @xmath537 whose arm length @xmath579 is not zero . for example , if @xmath580 , @xmath581 . this nekrasov factor has the property @xmath582 for any @xmath15 . therefore , the conjecture in the crystallized case of @xmath300 is @xmath583 the case of some particular partitions can be checked by calculating the contour integral ( appendix [ sec : check of n=1 conjecture ] ) . next , let us consider the @xmath105 limit in the case of @xmath238 . in this case , the generator @xmath584 of the heisenberg algebra is renormalized as @xmath585 and the generator @xmath586 is used as it is . by this normalization , it is possible to take the limit . also , the algebraic structure of @xmath587 and @xmath588 does not change . then @xmath587 and @xmath588 have the form @xmath589 moreover , let us use the bosons @xmath570 ( @xmath590 ) and @xmath591 with the relation @xmath592=n\frac{1}{1-t^{|n| } } \delta_{i , j } \ , \delta_{n+m , 0 } , \quad [ b_n^{(i ) } , { q}^{(j ) } ] = 0,\ ] ] and regard @xmath593 , @xmath594 . let us define the generator at @xmath3 . [ def : crystal first generator ] set @xmath595 [ prop : free field rep of x^1 ] definition [ def : crystal first generator ] is well - defined , i.e. , @xmath596 has no singularity at @xmath517 , and its free field representation is @xmath597 \tilde{\lambda}^{1}(z ) + \theta [ n\leq 0 ] \tilde{\lambda}^{2}(z ) \right\ } z^n,\ ] ] where @xmath598 is defined in section [ sec : crystal of qvir ] and @xmath599 define @xmath600 and @xmath601 by @xmath602 we can see @xmath603 is well - behaved in the limit @xmath3 by the form of @xmath604 . if @xmath339 , @xmath605 if @xmath606 , @xmath607 and if @xmath608 , @xmath609 thus @xmath610 is well - defined and ( [ eq : free field rep of txo ] ) is the natural free field representation . for the second generator , the following rescale is suitable . set @xmath611 the free field representation of @xmath612 is given by @xmath613 where @xmath614 this proposition is easily obtained by calculating @xmath615 . we can calculate the commutation relation of these generators as follows . the generators @xmath610 and @xmath612 satisfy the relations @xmath616 & = -(1-t^{-1 } ) \sum_{l=1}^{n - m } \tilde{x}^{(1)}_{n - l } \tilde{x}^{(1)}_{m+l } \qquad ( n > m>0 \;\ ; \mathrm{or } \;\ ; 0>n > m ) , \\ [ \tilde{x}^{(1)}_n , \tilde{x}^{(1)}_0 ] & = -(1-t^{-1 } ) \sum_{l=1}^{n-1 } \tilde{x}^{(1)}_{n - l } \tilde{x}^{(1)}_{l } -(1-t^{-1 } ) \sum_{l=1}^{\infty } \tilde{x}^{(1)}_{-l } \tilde{x}^{(1)}_{n+l } + ( 1-t^{-1 } ) \tilde{x}^{(2)}_{n } \quad ( n>0 ) , \\ [ \tilde{x}^{(1)}_n , \tilde{x}^{(1)}_m ] & = -(1-t^{-1})\sum_{l=0}^{\infty } \tilde{x}^{(1)}_{m - l } \tilde{x}^{(1)}_{n+l } + ( 1-t^{-1 } ) \tilde{x}^{(2)}_{n+m } \quad ( n>0>m ) , \\ [ \tilde{x}^{(1)}_0 , \tilde{x}^{(1)}_m ] & = -(1-t^{-1 } ) \sum_{l=1}^{-m-1 } \tilde{x}^{(1)}_{-l } \tilde{x}^{(1)}_{m+l } -(1-t^{-1 } ) \sum_{l=1}^{\infty } \tilde{x}^{(1)}_{m - l } \tilde{x}^{(1)}_{l } + ( 1-t^{-1 } ) \tilde{x}^{(2)}_{m } \quad ( 0>m ) , \end{aligned}\ ] ] @xmath617 & = ( 1-t^{-1 } ) \sum_{l=1}^{\infty } \tilde{x}^{(2)}_{m - l } \tilde{x}^{(1)}_{n+l } \qquad ( n>0,\ ; \forall m ) , \\ [ \tilde{x}^{(1)}_0 , \tilde{x}^{(2)}_m ] & = -(1-t^{-1 } ) \sum_{l=1}^{\infty } ( \tilde{x}^{(1)}_{-l } \tilde{x}^{(2)}_{m+l}- \tilde{x}^{(2)}_{m - l } \tilde{x}^{(1)}_{l } ) \quad ( \forall m ) , \\ [ \tilde{x}^{(1)}_n , \tilde{x}^{(2)}_m ] & = -(1-t^{-1 } ) \sum_{l=1}^{\infty } \tilde{x}^{(1)}_{n - l } \tilde{x}^{(2)}_{m+l } \qquad ( n<0,\ ; \forall m ) , \end{aligned}\ ] ] @xmath618 = -(1-t^{-1 } ) \sum_{l=1}^{\infty } ( \tilde{x}^{(2)}_{n - l } \tilde{x}^{(2)}_{m+l}- \tilde{x}^{(2)}_{m - l } \tilde{x}^{(2)}_{n+l } ) \qquad ( \forall n , m).\ ] ] these are obtained by the following relation of generating functions : @xmath619 where @xmath620 and for ( [ relation of tlo tlt ] ) we used the formula @xmath621 the algebra generated by @xmath622 and @xmath623 is closely related to the hall - littlewood functions . in particular , the pbw type vectors can be written as the product of two hall - littlewood functions . for a pair of partitions @xmath624 , set @xmath625 we have the expression of these vectors in terms of the hall - littlewood functions . [ prop : pbw vct rep by hall ] @xmath626 where @xmath627 the vectors @xmath628 do not have such a good expression . this proposition is proved by the theory of jing s operator . then the vectors @xmath629 are partially diagonalized as the following proposition . furthermore , with the help of hall - littlewood functions , we can calculate the shapovalov matrix @xmath630 and its inverse @xmath631 . [ prop : shapovalov in terms of hl poly ] we can express @xmath632 by the inner product @xmath633 of hall - littlewood functions defined in appendix [ sec : macdonald and hl ] : @xmath634 ( [ eq : shapovalov ] ) follows from proposition [ prop : pbw vct rep by hall ] . ( [ eq : inverse shapovalov ] ) can be obtained by the equation @xmath635 which is shown by inserting the complete system with respect to @xmath636 into the equation @xmath637 . existence of the inverse matrix @xmath638 leads linear independence of @xmath639 . since there are the same number of linear independent vectors as the dimension of each level of @xmath71 , we can see that @xmath639 forms a basis over @xmath71 . if @xmath16 is not a root of unity and @xmath640 , @xmath639 ( resp . @xmath641 ) is a basis of @xmath71 ( resp . @xmath642 ) . next , let us introduce generalized hall - littlewood functions which are specialization of generalized macdonald functions and give some crystallized versions of the agt conjecture . [ def : gn hall - littlewood ] define the vectors @xmath643 and @xmath644 as the @xmath3 limit of generalized macdonald functions , i.e. , @xmath645 we call the vectors @xmath643 generalized hall - littlewood functions . these are the eigenvectors of @xmath646 : @xmath647 moreover the eigenvalues are @xmath648 however there are too many degenerate eigenvalues to ensure the existence of generalized hall - littlewood functions . it is difficult to characterize @xmath649 as the eigenfunction of only @xmath650 . for example , @xmath651 and @xmath652 have the relation @xmath653 , but @xmath654 . is given under the hypothesis that the vector @xmath559 has no singulality in the limit @xmath3 . if we can show the existence therem of both generalized macdonald and generalized hall - littlewood functions by using the same partial ordering and the same basis , this hypothesis is guaranteed . ] let us define the transition matrices @xmath655 and @xmath656 by the expansions @xmath657 then up to the degree 2 the matrix elements @xmath658 are given by @xmath659 @xmath660 up to the degree 2 the matrix elements @xmath661 are given by @xmath662 @xmath663 similarly to the case of generic @xmath13 , we define the integral forms of generalized hall - littlewood functions and give a conjecture of their norms . the integral forms @xmath664 and @xmath665 are defined by @xmath666 note that the coefficients @xmath667 and @xmath668 can be zero at @xmath517 . [ conj : inner prod of gn hl ] @xmath669 next , let us define the vertex operator at crystal limit . the vertex operator @xmath670 is the linear operator satisfying the relations @xmath671 @xmath672 the existence of such an operator is shown by the renormalization @xmath673 . in the relation of @xmath76 and @xmath69 , it is understood that this renormalization is appropriate by considering the shift of @xmath674 such that @xmath557 and not all @xmath675 are commutative and the relation does not diverge . we give some simple properties of the vertex operator @xmath557 . @xmath676 for any @xmath250 , @xmath677 these follow from the commutation relations of @xmath557 . especially note that the three - point function which has generators @xmath675 on the left side , i.e. , @xmath678 , remains only in the case of special young diagrams @xmath679 with only one vertical column . [ conj : mat element of phi wrt gn hl ] the matrix elements of @xmath557 with respect to the integral form @xmath680 are @xmath681 under these conjectures [ conj : inner prod of gn hl ] and [ conj : mat element of phi wrt gn hl ] , we obtain the formula for correlation functions of the vertex operator @xmath557 . for example , the function corresponding to the four - point conformal block is @xmath682 ( [ eq : expansion by aflt ] ) is the agt conjecture in the limit @xmath3 with help of the aflt basis . however , in the crystallized case , we can prove another formula for this four - point correlation function by using the pbw type basis . at first , let us show the following two lemmas . [ lem : matrix el . of tphi wrt pbw ] the matrix elements with respect to pbw type vector @xmath683 and @xmath684 are @xmath685 for @xmath686 , by ( [ eq : comm rel of tphi and txt ] ) and the relation @xmath687 , @xmath688 repeating this calculation , we get @xmath689 where @xmath690 . when @xmath691 , by similar calculation @xmath692 by using above two formulas ( [ eq : preparation for i>1 ] ) and ( [ eq : preparation for i=1 ] ) , if we write @xmath693 , @xmath694 we have explicit form of formulas for some parts of the inverse shapovalov matrix . [ lem : exlicit form of inv . @xmath695 in this proof , we put @xmath696 . hall - littlewood function @xmath697 is the elementary symmetric function @xmath698 times @xmath699 . elementary symmetric functions have the generating function @xmath700 hence by the @xmath701 case of fact [ fact : spetialization of hl poly ] , @xmath702 therefore , the lemma follows from proposition [ prop : shapovalov in terms of hl poly ] . we give other proofs of this lemma in appendix [ seq : another proof of lemma ] , and the form of @xmath703 can be found in appendix [ sec : explicit form of < q , q > ] . by the property ( [ eq : simple property1 ] ) , proposition [ prop : shapovalov in terms of hl poly ] and lemmas [ lem : matrix el . of tphi wrt pbw ] and [ lem : exlicit form of inv . shapovalov ] , we can show the following theorem . [ thm : main theorem ] @xmath704 in this way , the explicit formula for the correlation function can be obtained , where we do nt use any conjecture . the formulas ( [ eq : expansion by aflt ] ) and ( [ eq : expansion formula by pbw ] ) are compared in appendix [ comparison of two formula ] . we expect that these works will be generalized to @xmath705 case . finally , we present other types of the crystal limit . in this paper , we investigated the crystal limit while the parameters @xmath289 , @xmath706 and @xmath707 are fixed . however , it is also important to study the cases when these parameters depend on @xmath13 . for example , let us consider the case that @xmath708 , @xmath709 , @xmath710 ( @xmath711 ) and @xmath712 , @xmath713 , @xmath714 are independent of @xmath13 or fixed in the limit @xmath3 . let @xmath715 for all @xmath716 and @xmath717 . then the nekrasov formula for generic @xmath13 case ( @xmath238 ) @xmath718 depends only on the partitions of the shape @xmath719 in the limit @xmath3 , where @xmath720 is fixed , and coincides with the partition function of the pure gauge theory ( [ eq : tildez ] ) : @xmath721 where @xmath722 . hence , we expect that the vector @xmath723 corresponds to the whittaker vector in the section [ sec : crystal of qvir ] in this limit , though we were not able to properly explain it . in this way , by considering the various other values of @xmath724 and @xmath725 , we can find special behavior of @xmath726 and the conformal block @xmath727 and may prove the relation . these are our future studies . in general , a bialgebra @xmath106 is called quasi - cocommutative if there exists an invertible element @xmath728 such that for all @xmath729 , @xmath730 this @xmath108 is called the universal r - matirx . the dim algebra is quasi - cocommutative @xcite . in this section , we explicitly calculate the representation of @xmath108 . moreover , the expression of its representation matrix is generally conjectured . the point of calculation is to make use of the condition that the generalized macdonald functions are the eigenfunctions of @xmath77 , to reduce the degree of freedom of the matrix in advance . in this section , we formally write the universal @xmath108-matrix as @xmath731 and set @xmath732 , @xmath733 , @xmath734 . occasionally , we explicitly write the variable of the generalized macdonald functions like @xmath735 and the pbw basis of the bosons is written as @xmath736 firstly , by definition of @xmath108 , we have @xmath737 here @xmath738 . the formula for the coproduct of the dim algebra and the definition of the representation @xmath739 are given in appendix [ sec : def of dim ] . hence , @xmath740 are eigenfunctions of @xmath741 . by comparing the forms of @xmath741 and @xmath742 , the eigenfunctions of @xmath741 are obtained by replacing @xmath743 with @xmath744 and @xmath745 with @xmath746 . moreover , by checking their eigenvalues , it can be seen that @xmath740 are proportional to @xmath747 : @xmath748 where @xmath749 are proportionality constants . by this property , the representation matrix of @xmath750 is block - diagonalized at each level of the boson @xmath343 . the proportionality constants @xmath751 can be calculated by using the generalized macdonald functions in the @xmath752 case . similarly to the @xmath238 case , from the relation @xmath753 we have @xmath754 where @xmath755 and @xmath756 are constants . since the generalized macdonald functions satisfy @xmath757 the proportionality constants have the relation @xmath758 . for example let us describe the calculation of the representation matrix at level 1 . the following are examples of @xmath759 at level 1 : @xmath760 } \rangle } \\ { | p_{\emptyset,[1],\emptyset } \rangle } \\ { | p_{[1],\emptyset,\emptyset } \rangle } \end{array } \right ) = a(u_1,u_2,u_3 ) \left ( \begin{array}{c } { | a_{\emptyset,\emptyset,[1 ] } \rangle } \\ { | a_{\emptyset,[1],\emptyset } \rangle } \\ { | a_{[1],\emptyset,\emptyset } \rangle } \end{array } \right),\ ] ] @xmath761 by the above discussion , if we set the matrix @xmath762 } & 0 & 0 \\ 0 & k^{(12)}_{\emptyset,[1],\emptyset } & 0 \\ 0 & 0 & k^{(12)}_{[1],\emptyset,\emptyset } \\ \end{array}\right ) a^{(12)}(u_1,u_2,u_3 ) a^{-1}(u_1,u_2,u_3),\ ] ] @xmath763 then the representation matrix of @xmath764 in the basis of the generalized macdonald functions is the transposed matrix of @xmath765 : @xmath766 } \right\rangle } & { \left| p_{\emptyset,[1],\emptyset } \right\rangle } & { \left| p_{[1]\emptyset,\emptyset } \right\rangle } \\ \end{array}\right ) = \left(\begin{array}{ccc } { \left| p_{\emptyset,\emptyset,[1 ] } \right\rangle } & { \left| p_{\emptyset,[1],\emptyset } \right\rangle } & { \left| p_{[1]\emptyset,\emptyset } \right\rangle } \\ \end{array}\right ) { } ^t b^{(12)}.\ ] ] the constants @xmath767 are determined as follows . at first , since scalar multiples of r - matrices are also r - matrices , we can normalize as @xmath768 . this means that @xmath769 . next , we consider the base change from @xmath770 to the bosons @xmath771 : @xmath772 then @xmath773 is in the form @xmath774 where @xmath775 is a function of @xmath776 . since for the action of @xmath777 to @xmath778 , variables @xmath779 and @xmath780 should not appear , we get equations @xmath781 . by solving these equations , we can see that @xmath782 substituting this value into the matrix ( [ eq : b12 containing k ] ) , we have @xmath783 in this way , we obtain the explicit expression @xmath773 of representation matrix of universal @xmath108 at level 1 . of course , it is possible to calculate the representation matrix of @xmath784 in the same way , but by using symmetry with respect to @xmath785 at different @xmath152 , we can easily understand the forms of @xmath784 and @xmath786 : @xmath787 indeed , we can check that they satisfy the yang - baxter equation @xmath788 incidentally , in the basis of the generalized macdonald functions , @xmath789 @xmath790 the representation matrix of @xmath791 is the @xmath792 matrix block at the lower right corner of @xmath765 or @xmath773 . for example , in the basis of generalized macdonald functions , its representation matrix is @xmath793 next , let us explain the case at level 2 . the generalized macdonald functions at level 2 in the @xmath752 case are expressed as @xmath794 } \rangle } & { | p_{\emptyset,\emptyset,[1,1 ] } \rangle } & { | p_{\emptyset,[1],[1 ] } \rangle } & { | p_{[1],\emptyset,[1 ] } \rangle } & { | p_{\emptyset,[2],\emptyset } \rangle } & { | p_{\emptyset,[1,1],\emptyset } \rangle } & { | p_{[1],[1],\emptyset } \rangle } & { | p_{[2],\emptyset,\emptyset } \rangle } & { | p_{[1,1],\emptyset,\emptyset } \rangle } \end{array } \right ) \nonumber \\ & = { } ^t\mathcal{a } \left ( \begin{array}{ccccccccc } { | p'_{\emptyset,\emptyset,[2 ] } \rangle } & { | p'_{\emptyset,\emptyset,[1,1 ] } \rangle } & { | p'_{\emptyset,[1],[1 ] } \rangle } & { | p'_{[1],\emptyset,[1 ] } \rangle } & { | p'_{\emptyset,[2],\emptyset } \rangle } & { | p'_{\emptyset,[1,1],\emptyset } \rangle } & { | p'_{[1],[1],\emptyset } \rangle } & { | p'_{[2],\emptyset,\emptyset } \rangle } & { | p'_{[1,1],\emptyset,\emptyset } \rangle } \end{array } \right),\end{aligned}\]]where @xmath795 denotes the product of ordinary macdonald functions @xmath796 , and the matrix @xmath797 is given in appendix [ sec : ex of r - matrix ] . in the same manner , we can get the representation matrix of @xmath108 . at first , @xmath765 at level 2 is in the form @xmath798 then we can find the proportionality constants such that all @xmath799 are zero just by solving equations @xmath800 ( @xmath801 ) . we have also checked that the representation matrix @xmath802 obtained in this way satisfies the yang - baxter equation up to level 3 . the explicit expressions of @xmath108 at level 2 are written in appendix [ sec : ex of r - matrix ] . the proportionality constants are in the form @xmath803 these proportionality constants can be simplified by using the integral forms of the generalized macdonald functions @xmath804 , which are defined in section [ sec : reargument of di alg and agt ] . define its opposite version by @xmath805 and the constants @xmath806 by @xmath807 by these renormalized functions , the relation ( [ eq : propo rel . ] ) can be written as @xmath808 then it is conjectured that @xmath809 this equation has been checked at @xmath810 . therefore , the representation matrix @xmath119 of @xmath108 in the basis of the integral forms can be expressed as the following conjecture . [ conj : r - matrix by int form ] @xmath811 since the formula ( [ eq : formula for r matrix ] ) means the expansion coefficients of @xmath117 in front of @xmath118 , by using the transition matrix defined by @xmath812 and its opposite version @xmath813 , the r - matrix can be calculated by the matrix operation @xmath814 this formula gives a much simpler way to get explicit expressions as compared with deducing them from the universal r - matrix @xcite . incidentally , the proportionality constants are conjectured to be @xmath815 where @xmath816 is the nekrasov factor defined in ( [ eq : def of nek factor ] ) , section [ sec : review of simplest 5d agt ] . the second equality follows from the formula ( eq.(2.34 ) in @xcite , eq.(102 ) in @xcite ) @xmath817 where @xmath818 is the framing factor @xcite . the equation ( [ eq : conj of prop const ] ) has been checked up to level 3 . the existence theorem of generalized macdonald functions can be stated by the ordering @xmath121 in definition [ def : ordering1 ] . in this subsection , we introduce a more elaborated ordering . using this ordering , we can find more elements which is @xmath130 in the transition matrix @xmath819 , where @xmath820 , and get more strict condition to existence theorem . [ df : ordering elaborated version ] for @xmath0-tuples of partitions @xmath265 and @xmath266 , @xmath821 for all @xmath822 . here @xmath823 denote that @xmath824 for all @xmath152 . if @xmath752 and the number of boxes is @xmath825 , then @xmath826 & & ( \emptyset , \emptyset , ( 2,1 ) ) \ar[ld]\ar[rd ] & & ( \emptyset , \emptyset , ( 1,1,1))\ar[ld ] & \\ & & ( \emptyset , ( 1 ) , ( 2))\ar[lld]\ar[d]\ar[rrrrd ] & & ( \emptyset , ( 1 ) , ( 1,1))\ar[lllld]\ar[d]\ar[rrd ] & & \\ ( \emptyset , ( 2 ) , ( 1))\ar[d]\ar[rrd]\ar[rrrrd ] & & ( ( 1 ) , \emptyset , ( 2))\ar[d ] & & ( ( 1 ) , \emptyset , ( 1,1))\ar[lld ] & & ( \emptyset , ( 1,1 ) , ( 1))\ar[d]\ar[lld]\ar[lllld ] \\ ( \emptyset , ( 3 ) , \emptyset)\ar[d ] & & ( ( 1 ) , ( 1 ) , ( 1))\ar[lld]\ar[d]\ar[rrd]\ar[rrrrd ] & & ( \emptyset , ( 2,1 ) , \emptyset)\ar[lllld]\ar[rrd ] & & ( \emptyset , ( 1,1,1 ) , \emptyset)\ar[d ] \\ ( ( 1 ) , ( 2 ) , \emptyset)\ar[rrd]\ar[rrrrd ] & & ( ( 2 ) , \emptyset , ( 1))\ar[d ] & & ( ( 1,1 ) , \emptyset , ( 1))\ar[d ] & & ( ( 1 ) , ( 1,1 ) , \emptyset)\ar[lllld]\ar[lld ] \\ & & ( ( 2 ) , ( 1 ) , \emptyset)\ar[ld]\ar[rd ] & & ( ( 1,1 ) , ( 1 ) , \emptyset)\ar[ld]\ar[rd ] & & \\ & ( ( 3 ) , \emptyset , \emptyset ) & & ( ( 2,1 ) , \emptyset , \emptyset ) & & ( ( 1,1,1 ) , \emptyset , \emptyset ) . & } \ ] ] here @xmath827 stands for @xmath828 . by using the following conjecture , we can state the existence theorem . [ conj : action of eta_n ] let @xmath829 . in the action of @xmath830 ( @xmath250 ) on macdonald functions @xmath831 , there only appear partitions @xmath537 contained in @xmath15 , i.e. , @xmath832 [ thm : another existence thm ] under the conjecture [ conj : action of eta_n ] , for an @xmath0-tuple of partitions @xmath265 , there exists an unique vector @xmath273 such that @xmath833 at first , @xmath830 satisfies @xmath834 if we act @xmath835 on the product of the macdonald functions , then @xmath836 where @xmath837 is a polynomial of degree @xmath838 of @xmath839 . such that @xmath840= n a^{(j)}_{-n}$ ] , the polynomial @xmath841 satisfies @xmath842 = ( |\lambda^{(i)}| - |\mu| ) c'_{\lambda^{(i ) } , \mu}$ ] . ] hence @xmath843 therefore one can easily diagonalize it and we have this theorem . in the basis of monomial symmetric functions @xmath844 , we have @xmath845 where @xmath846 ( @xmath847 ) . thus the partial ordering @xmath848 defined as follows also triangulates @xmath77 . @xmath849 it can be shown that the partial ordering @xmath848 is equivalent to the ordering @xmath850 introduced in @xcite . therefore theorem [ thm : another existence thm ] supports the existence theorem in ( * ? ? ? * proposition3.8 ) . a representation of the dim algebra called rank @xmath0 representation is provided in @xcite in terms of a basis @xmath851 called aflt basis . this rank @xmath0 representation corresponds to the @xmath0-fold tensor product of the level ( 0,1 ) representation described in appendix [ sec : def of dim ] . the level ( 0,1 ) representation can be considered as the spectral dual to the level ( 1,0 ) representation which is realized by the heisenberg algebra . in this subsection , based on this spectral duality we present conjectures for explicit expressions of the action of @xmath852 on the generalized macdonald functions , which are defined to be eigenfunctions of the hamiltonian @xmath77 . we can also conjecture the eigenvalues of higher rank hamiltonians on the generalized macdonald functions from those of the spectral dual generators provided in @xcite . our conjectures mean that the generalized macdonald functions concretely realize the spectral dual basis to @xmath851 in @xcite . although we already define the integral forms @xmath118 of the generalized macdonald functions , let us use another renormalization @xmath854 of them , which is defined by @xmath855 where @xmath856 is the nekrasov factor . this renormalization is the almost same as @xmath118 . their difference is conjectured to be the scalar multiplication of only monomials in parameter @xmath13 , @xmath16 and @xmath289 . it is expected that the basis @xmath857 corresponds to the aflt basis were already conjectured to be the aflt basis in this original sense . ] in @xcite and realizes the rank @xmath0 representation through the spectral duality @xmath858 . that is to say , for any generator @xmath859 in the dim algebra , the action of @xmath860 on the integral forms @xmath854 is in the same form as one of @xmath861 on the basis @xmath862 @xcite , where @xmath863 . indeed , we can check that the action of @xmath864 on the generalized macdonald functions is as the following conjecture . let us denote adding a box to or removing it from the young diagram @xmath265 through @xmath865 and @xmath866 respectively . we also use the notation @xmath867 for the triple @xmath868 , where @xmath869 is the coordinate of the box of the young diagram @xmath870 . [ conj : action of x ] @xmath871 where @xmath872 and for the triple @xmath873 , we put @xmath874 these actions of @xmath124 in this conjecture come from the corresponding actions of the generators @xmath875 and @xmath876 in @xcite respectively , i.e. , @xmath877 and @xmath878 in our notation , which are the spectral duals of @xmath879 and @xmath880 . incidentally , introducing the coefficients @xmath881 by @xmath882 i.e. , @xmath883 , we can further conjecture that @xmath884 conjecture [ conj : action of x ] with respect to @xmath885 and the formula ( [ eq : rel between cplus and cminus ] ) are checked on a computer for @xmath886 for @xmath300 , for @xmath887 for @xmath888 and for @xmath889 for @xmath890 . conjecture [ conj : action of x ] with respect to @xmath891 is also checked for the same size of @xmath266 . for each integer @xmath892 , the spectral dual of @xmath893 is @xmath894 defined by @xmath895 and @xmath896\cdots ] ] , \qquad k \geq 2.\ ] ] according to @xcite , @xmath894 are spectral dual to @xmath897 and consequently mutually commuting ; @xmath898=0 $ ] . thus the generalized macdonald functions @xmath281 are automatically eigenfunctions of all @xmath894 , i.e. , @xmath899 , and @xmath894 can be regarded as higher hamiltonians for the generalized macdonald functions . since @xmath894 are the spectral duals to @xmath897 ; @xmath900 , their eigenvalues are expected to be @xmath901 where @xmath902 is defined in ( [ eq : b+ ] ) . the eigenvalues @xmath903 correspond to those of the rank @xmath0 representation of the generators @xmath897 in @xcite . in the @xmath904 case , the conjecture ( [ eq : higher eigenvlue ] ) can be proved . we have checked it for @xmath886 for @xmath300 , for @xmath887 for @xmath238 , for @xmath889 for @xmath752 and for @xmath905 for @xmath890 in the @xmath906 case . in this subsection , we show that the generalized macdonald functions are reduced to the generalized jack functions introduced by morozov and smirnov in @xcite in the @xmath18 limit ( see also the sub - thesis @xcite ) . although the scenario of proof of agt correspondence is given in @xcite , the orthogonality of the generalized jack functions are non - trivial since there are degenerate eigenvalues . the cauchy formula used in the scenario of proof can not be proved without the orthogonality . however , the eigenvalues of generalized macdonald functions are non - degenerate . hence , we can prove the orthogonality of the generalized jack functions by using the limit in this section . when taking this limit , we set @xmath907 @xmath908 , @xmath19 , @xmath909 and take the limit @xmath910 with @xmath6 fixed . to expose the @xmath911 dependence of the generators @xmath343 in the heisenberg algebra , they are realized in terms of @xmath0 kinds of power sum symmetric functions @xmath912 with @xmath913 in the ring of symmetric functions @xmath914 in the variables @xmath915 ( @xmath70 , @xmath916 ) : @xmath917 since the operators @xmath224 become parameter @xmath289 in the representation space @xmath71 , they are transformed as @xmath918 from the beginning in this section . with the above transformation , the generator @xmath77 can be regarded as the operator over the ring of symmetric functions @xmath914 . define the isomorphism @xmath919 by @xmath920 where @xmath921 is defined in section [ sec : explicit cal of r ] . the inner product @xmath922 over @xmath914 is defined by @xmath923 this inner product naturally realize the one over the fock module @xmath71 , i.e. , @xmath924 let @xmath925 be the adjoint operator of @xmath77 with respect to the inner product . since the generalized macdonald functions @xmath926 have non - degenerate eigenvalues the orthogonality clearly follows : @xmath927 where @xmath928 is defined to be the eigenfunctions of the adjoint operator @xmath925 of the eigenvalue @xmath460 . now let us take the @xmath125 limit . at first , consider the @xmath911 expansion of @xmath929 . by @xmath930 and @xmath931 , @xmath932 , we have @xmath933 hence the @xmath911 expansion is @xmath934 thus we get @xmath935 where @xmath936 @xmath937 for @xmath938 , we define operators @xmath894 by @xmath939 with respect to @xmath940 , @xmath941 and @xmath942 , all homogeneous symmetric functions belong to the same eigenspace . hence , the eigenfunctions of @xmath943 are the same to those of @xmath944 . in addition , we have @xmath945 @xmath946 consequently the limit @xmath18 of the generalized macdonald functions are eigenfunctions of the differential operator @xmath947 . as a matter of fact , @xmath947 plus the momentum @xmath948 corresponds to the differential operator of @xcite , the eigenfunctions of which are called generalized jack symmetric functions . , @xmath949 . ] as in ( * ? ? * proposition 3.7 ) , we can triangulate @xmath947 similarly . moreover if @xmath950 ( resp . @xmath951 ) if and only if @xmath952 and @xmath953 @xmath954 for all @xmath955 and @xmath956 . ] and @xmath6 is generic , then @xmath957 . ( @xmath958 , @xmath959 are eigenvalues of @xmath947 . ) therefore we get the existence theorem of the generalized jack symmetric functions . there exists a unique symmetric function @xmath960 satisfying the following two conditions : @xmath961 where @xmath962 denotes the product of monomial symmetric functions @xmath963 . ( @xmath964 is the usual monomial symmetric function of variables @xmath965 . ) from the above argument and the uniqueness in this proposition we get the following important result . [ prop : limit of gnmac ] the limit of the generalized macdonald symmetric functions @xmath966 to @xmath6-deformation coincide with the generalized jack symmetric functions @xmath960 . that is @xmath967 for the dual functions @xmath968 and @xmath969 , a similar proposition holds . by the orthogonality ( [ eq : orthogonality of gn mac ] ) , proposition [ prop : limit of gnmac ] and the fact that the scalar product @xmath922 reduces to the scalar product @xmath970 which is defined by @xmath971 we obtain the orthogonality of the generalized jack symmetric functions . if @xmath972 , then @xmath973 by this proposition , we can prove the cauchy formula for generalized jack symmetric functions in the usual way . for example , in the @xmath238 case , we have @xmath974 where @xmath975 . this is the necessary formula in the scenario of proof of the agt conjecture @xcite . we give examples of proposition [ prop : limit of gnmac ] in the case @xmath238 . the generalized macdonald symmetric functions of level 1 and 2 have the forms : @xmath976 @xmath977 @xmath978 also the generalized jack symmetric functions have the forms : @xmath979 @xmath980 @xmath981 if we take the limit @xmath18 of @xmath982 , then @xmath983 appears . in this subsection , we briefly review some properties of hall - littlewood functions and macdonald functions following ( * ? ? ? * chap . iii , vi ) . let @xmath984^{s_n}$ ] be the ring of symmetric polynomials of @xmath0 variables and @xmath985 be the ring of symmetric functions . the inner product @xmath986 over @xmath197 is defined such that for power sum symmetric functions @xmath987 ( @xmath988 ) , @xmath989 where @xmath990 is the number of entries in @xmath15 equal to @xmath152 . for a partition @xmath15 , macdonald functions @xmath991 are uniquely determined by the following two conditions @xcite : @xmath992 here @xmath142 is the monomial symmetric function and @xmath993 is the ordinary dominance partial ordering , which is defined as follows : @xmath994 in this paper , we regard power sum symmetric functions @xmath17 ( @xmath143 ) as the variables of macdonald functions , i.e. , @xmath995 . here @xmath996 is an abbreviation for @xmath997 . in this paper , we often use the symbol @xmath998 , which is the polynomial of bosons @xmath999 obtained by replacing @xmath17 in macdonald functions with @xmath999 . next , let the hall - littlewood function @xmath1000 be given by @xmath1001 . if @xmath1002 , then for a partition @xmath15 of length @xmath1003 , the hall - littlewood polynomial @xmath1000 with @xmath1004 is expressed by @xmath1005 where @xmath1006 . note that @xmath1007 . the action of the symmetric group @xmath1008 of degree @xmath0 is defined by @xmath1009 for @xmath1010 . it is convenient to introduce functions @xmath1011 , which are defined by scalar multiples of @xmath1012 as follows : @xmath1013 where @xmath1014 . they are diagonalized as @xmath1015 these functions @xmath1011 can be constructed by using jing s operators @xmath1016 and @xmath1017 @xcite , which is defined by @xmath1018 where @xmath497 is the bosons realized by @xmath1019 [ fact : jing s operator ] let @xmath210 be the vector such that @xmath1020 ( @xmath339 ) . then for a partition @xmath15 , we have @xmath1021 furthermore , the following specialization formula is known . [ fact : spetialization of hl poly ] let @xmath331 be indeterminate . under the specialization @xmath1022 the hall - littlewood function @xmath1023 is specialized as @xmath1024 in this section , we recall the definition of the dim algebra and the level @xmath0 representation . for the notations , we follow @xcite . the dim algebra has two parameters @xmath13 and @xmath16 . let @xmath1025 be the formal series @xmath1026 then this series satisfies @xmath1027 . define the algebra @xmath1028 to be the unital associative algebra over @xmath20 generated by the currents @xmath1029 , @xmath1030 and the central element @xmath1031 satisfying the defining relations @xmath1032 = \dfrac{(1-q)(1 - 1/t)}{1-q / t } \big ( \delta(\gamma^{-1}z / w ) \psi^+(\gamma^{1/2}w)- \delta(\gamma z / w ) \psi^-(\gamma^{-1/2}w ) \big ) , \\ & g^{\mp}(z / w)x^\pm(z)x^\pm(w)=g^{\pm}(z / w)x^\pm(w)x^\pm(z ) . \end{aligned}\ ] ] note that @xmath1033 are central elements in @xmath1028 . let us contain the invertible elements @xmath1034 and @xmath1035 in the definition of @xmath1028 . further , set @xmath1036 . this algebra @xmath1028 is an example of the family topological hopf algebras introduced by ding and iohara @xcite . this family is a sort of generalization of the drinfeld realization of the quantum affine algebras . however , miki introduce a deformation of the @xmath64 algebra in @xcite , which is the quotient of the algebra @xmath1028 by the serre - type relation . hence we call the algebra @xmath1028 the ding - iohara - miki algebra ( dim algebra ) . since the algebra @xmath1028 has a lot of background , there are a lot of other names such as quantum toroidal @xmath65 algebra @xcite , quantum @xmath64 algebra @xcite , elliptic hall algebra @xcite and so on . this algebra has a hopf algebra structure . the formulas for its coproduct are @xmath1037 and @xmath1038 , where @xmath1039 and @xmath1040 . since we do not use the antipode and the counit in this thesis , we omit them . the dim algebra @xmath1028 can be realized by the heisenberg algebra defined in section [ sec : review of simplest 5d agt ] . [ fact : lv . 1 rep of dim ] the morphism @xmath1041 defined as follows is a representation of the dim algebra : @xmath1042 where @xmath1043 not that the zero mode @xmath1044 of @xmath1045 can be essentially identified with the macdonald difference operator @xcite . by using the coproduct of @xmath1028 , we can consider its tensor representations . for an @xmath0-tuple of parameters @xmath1046 , define the morphism @xmath1047 by @xmath1048 where @xmath1049 is inductively defined by @xmath1050 , @xmath1051 and @xmath1052 . the representation @xmath1047 is called the level @xmath0 representations . in section [ sec : reargument of di alg and agt ] , for simplicity , we write the @xmath152-th bosons as @xmath1053 the generator @xmath235 is defined by @xmath1054 the @xmath739 is also called the level @xmath1055 representation or the horizontal representation in @xcite in order to distinguish another representation of the dim algebra , which is called the level @xmath126 representation or the vertical representation . to define the level @xmath126 representation , we introduce some notations . for a partition @xmath193 and a number @xmath1056 , we set @xmath1057 here , @xmath1058 and @xmath1059 are considered as elements in @xmath1060 $ ] . let @xmath419 be an indeterminate parameter and @xmath1061 be the fock module generated by the highest weight vector @xmath210 . the morphism @xmath1062 defined as follows gives a representation of the dim algebra : @xmath1063 and @xmath1064 , where @xmath1065 denotes the ordinary macdonald function @xmath1066 associated with the partition @xmath15 . in this thesis , the vectors @xmath1065 are realized by the macdonald functions @xmath1067 along @xmath1068 . however , they can also be regarded as the abstract vectors labeled by young diagrams . note that the factors ( [ eq : a+])-([eq : b- ] ) can be written by the contribution from the edges of the young diagram @xmath15 , i.e. , the positions which we can add a box in or remove it from . the tensor representation of @xmath1069 is also called the rank @xmath0 representation , which is given in @xcite in terms of the edge contribution . this representation is connected to the level @xmath1055 representation under the change of basis and the automorphism of the dim algebra , that is defined as follows . this connection is called the spectral duality ( * ? ? ? * section 5 ) . there exists an automorphism @xmath1070 such that @xmath1071 @xmath1072 and @xmath1073 . although the algebra in @xcite slightly differs from the algebra @xmath1028 in the serre - type relation , this fact holds . this automorphism is of order four . by using this automorphism , we can check the correspondence between two representations of the dim algebra . in section [ sec : realization of rank n rep ] , we briefly explain the spectral duality and check it with respect to the generators @xmath852 and @xmath893 . in this subsection , let us explain other proofs of lemma [ lem : exlicit form of inv . shapovalov ] by the method of contour integrals . the generating function of elementary symmetric functions and jing s operator makes the equation @xmath1074 where we put @xmath696 , @xmath1075 , and @xmath1076 . it suffices to show that @xmath1077 , which is proved by a recursive relation of @xmath59 as follows . the contour integral @xmath1078 surrounding origin is represented as that surrounding @xmath1079 . since @xmath1080 , the residue of @xmath1081 at @xmath1082 is @xmath130 . hence , the only residue at @xmath1083 is left , and it is @xmath1084 by change of variable @xmath1085 , @xmath1086 therefore @xmath1087 thus the lemma [ lem : exlicit form of inv . shapovalov ] is proved . although it is slightly hard , one can also prove this lemma by reversing the order of integration , i.e. , first perform over a variable @xmath1088 surrounding origin . indeed @xmath1088 has the pole only at @xmath1089 , and its residue satisfies @xmath1090 where for @xmath1091 , @xmath1092 and @xmath1093 . by the assumption that @xmath1094 for @xmath1095 with @xmath1096 , we inductively get @xmath1097 by virtue of the equation @xmath1098 which is also proved by induction with respect to @xmath1099 , it can be seen that @xmath1100 . the formula for @xmath1102 is given in the lemma [ lem : exlicit form of inv . shapovalov ] and the last subsection . we also have an explicit form of @xmath1103 and @xmath1104 . by the proposition [ prop : shapovalov in terms of hl poly ] , it suffices to give the explicit form of @xmath1101 . @xmath1105 the proof is similar to the previous subsection . set @xmath1106 then @xmath1107 by jing s operator . integration of @xmath1088 around @xmath1079 makes recursive relation @xmath1108 and leads this proposition . for general partitions @xmath15 and @xmath537 , we can get the integral representation of @xmath1109 . however , it is very hard to give their explicit formula . the integral formula ( [ eq : n=1 conjecture ] ) can be checked by the similar way to subsections [ seq : another proof of lemma ] and [ sec : explicit form of < q , q > ] . let us set @xmath1110 then @xmath1111 , @xmath1112 . the integration of @xmath1081 around @xmath130 give the relation @xmath1113 on the other hand , the integration of @xmath1114 around @xmath1079 makes @xmath1115 thus @xmath1116 these agree with the right hand side of ( [ eq : n=1 conjecture ] ) . in this subsection , we compare two formulas ( [ eq : expansion by aflt ] ) and ( [ eq : expansion formula by pbw ] ) which are obtained by the other basis . comparing the coefficients of @xmath1117 , we have the equation @xmath1118 note that the left hand side is the summation with respect to pairs of partitions @xmath1119 and the right hand side is the summation with respect to single partitions @xmath15 . the right hand side depends only on the ratio @xmath1120 though the left hand side does nt look that way . for a single partition @xmath15 , let us define @xmath1121 to be the set of all pairs of partitions @xmath1122 such that a permutation of the sequence @xmath1123 coincides with @xmath15 . for example , if @xmath1124 , @xmath1125 then we obtain a strange factorization formula with respect to the partial summation of left hand side in ( [ eq : comparison of two formula ] ) @xmath1126 where @xmath1127 and @xmath1128 is introduced in definition [ def : crystal nek factor ] . if we prove that left hand side of ( [ eq : strange facorization formula ] ) depends only on @xmath1129 , ( [ eq : strange facorization formula ] ) is easily seen by checking the case of @xmath1130 . this equation almost reproduces each term of the right hand side in ( [ eq : comparison of two formula ] ) . hence ( [ eq : comparison of two formula ] ) may be proved by this equation . if ( [ eq : comparison of two formula ] ) holds , the agt conjecture at @xmath3 with the help of the aflt basis ( [ eq : expansion by aflt ] ) is completely proved . examples of the generalized macdonald functions at level 3 in the @xmath752 case : @xmath1131 @xmath1132 @xmath1133 b. sutherland , `` exact results for a quantum many body problem in one - dimension , '' http://dx.doi.org/10.1103/physreva.4.2019[_phys . rev . _ * a4 * ( 1971 ) 20192021 ] . b. sutherland , `` exact results for a quantum many body problem in one - dimension . 2 . , '' http://dx.doi.org/10.1103/physreva.5.1372[_phys . rev . _ * a5 * ( 1972 ) 13721376 ] . i. g. macdonald , _ symmetric functions and hall polynomials_. oxford classic texts in the physical sciences . the clarendon press , oxford university press , new york , second ed . , 2015 . m. aganagic and s. shakirov , `` knot homology and refined chern - simons index , '' http://dx.doi.org/10.1007/s00220-014-2197-4[_commun . phys . _ * 333 * no . 1 , ( 2015 ) 187228 ] , http://arxiv.org/abs/1105.5117[arxiv:1105.5117 [ hep - th ] ] . i. cherednik , `` jones polynomials of torus knots via daha , '' _ int . res . not . imrn _ no . 23 , ( 2013 ) 53665425 , http://arxiv.org/abs/1111.6195[arxiv:1111.6195 [ math.qa ] ] . e. gorsky and a. negut , `` refined knot invariants and hilbert schemes , '' _ j. math . * 104 * ( 2015 ) 403435 , http://arxiv.org/abs/1304.3328[arxiv:1304.3328 [ math.rt ] ] . a. borodin and i. corwin , `` macdonald processes , '' http://dx.doi.org/10.1007/s00440-013-0482-3[_probab . theory related fields _ * 158 * no . 1 - 2 , ( 2014 ) 225400 ] , http://arxiv.org/abs/1111.4408[arxiv:1111.4408 [ math.pr ] ] . j. shiraishi , h. kubo , h. awata , and s. odake , `` a quantum deformation of the virasoro algebra and the macdonald symmetric functions , '' http://dx.doi.org/10.1007/bf00398297[_lett . math . phys . _ * 38 * ( 1996 ) 3351 ] , http://arxiv.org/abs/q-alg/9507034[arxiv:q-alg/9507034 [ q - alg ] ] . k. mimachi and y. yamada , `` singular vectors of virasoro algebra in terms of jack symmetric polynomials , '' _ rims kokyuroku _ * 919 * ( 1995 ) 6878 . h. awata , y. matsuo , s. odake , and j. shiraishi , `` excited states of the calogero - sutherland model and singular vectors of the @xmath27 algebra , '' http://dx.doi.org/10.1016/0550-3213(95)00286-2[_nuclear phys . b _ * 449 * no . 1 - 2 , ( 1995 ) 347374 ] , http://arxiv.org/abs/hep-th/9503043[arxiv:hep-th/9503043 [ hep - th ] ] . h. awata , h. kubo , s. odake , and j. shiraishi , `` quantum @xmath1139 algebras and macdonald polynomials , '' http://dx.doi.org/10.1007/bf02102595[_commun . math . * 179 * ( 1996 ) 401416 ] , http://arxiv.org/abs/q-alg/9508011[arxiv:q-alg/9508011 [ q - alg ] ] . l. f. alday , d. gaiotto , and y. tachikawa , `` liouville correlation functions from four - dimensional gauge theories , '' http://dx.doi.org/10.1007/s11005-010-0369-5[_lett . * 91 * ( 2010 ) 167197 ] , http://arxiv.org/abs/0906.3219[arxiv:0906.3219 [ hep - th ] ] . n. a. nekrasov , `` seiberg - witten prepotential from instanton counting , '' http://dx.doi.org/10.4310/atmp.2003.v7.n5.a4[_adv . _ * 7 * no . 5 , ( 2003 ) 831864 ] , http://arxiv.org/abs/hep-th/0206161[arxiv:hep-th/0206161 [ hep - th ] ] . a. a. belavin , a. m. polyakov , and a. b. zamolodchikov , `` infinite conformal symmetry in two - dimensional quantum field theory , '' http://dx.doi.org/10.1016/0550-3213(84)90052-x[_nucl . * b241 * ( 1984 ) 333380 ] . m. bershtein and o. foda , `` agt , burge pairs and minimal models , '' http://dx.doi.org/10.1007/jhep06(2014)177[_jhep_ * 06 * ( 2014 ) 177 ] , http://arxiv.org/abs/1404.7075[arxiv:1404.7075 [ hep - th ] ] . d. gaiotto , `` asymptotically free @xmath1140 theories and irregular conformal blocks , '' http://dx.doi.org/10.1088/1742-6596/462/1/012014[_j . * 462 * no . 1 , ( 2013 ) 012014 ] , http://arxiv.org/abs/0908.0307[arxiv:0908.0307 [ hep - th ] ] . n. wyllard , `` a(n-1 ) conformal toda field theory correlation functions from conformal n = 2 su(n ) quiver gauge theories , '' http://dx.doi.org/10.1088/1126-6708/2009/11/002[_jhep_ * 11 * ( 2009 ) 002 ] , http://arxiv.org/abs/0907.2189[arxiv:0907.2189 [ hep - th ] ] . s. yanagida , `` whittaker vectors of the virasoro algebra in terms of jack symmetric polynomial , '' http://dx.doi.org/10.1016/j.jalgebra.2011.02.039[_j . algebra _ * 333 * ( 2011 ) 278294 ] , http://arxiv.org/abs/1003.1049[arxiv:1003.1049 [ math.qa ] ] . v. a. alba , v. a. fateev , a. v. litvinov , and g. m. tarnopolskiy , `` on combinatorial expansion of the conformal blocks arising from agt conjecture , '' http://dx.doi.org/10.1007/s11005-011-0503-z[_lett . * 98 * ( 2011 ) 3364 ] , http://arxiv.org/abs/1012.1312[arxiv:1012.1312 [ hep - th ] ] . v. a. fateev and a. v. litvinov , `` integrable structure , w - symmetry and agt relation , '' http://dx.doi.org/10.1007/jhep01(2012)051[_jhep_ * 01 * ( 2012 ) 051 ] , http://arxiv.org/abs/1109.4042[arxiv:1109.4042 [ hep - th ] ] . a. belavin and v. belavin , `` agt conjecture and integrable structure of conformal field theory for c=1 , '' http://dx.doi.org/10.1016/j.nuclphysb.2011.04.014[_nucl . phys . _ * b850 * ( 2011 ) 199213 ] , http://arxiv.org/abs/1102.0343[arxiv:1102.0343 [ hep - th ] ] . a. morozov and a. smirnov , `` towards the proof of agt relations with the help of the generalized jack polynomials , '' http://dx.doi.org/10.1007/s11005-014-0681-6[_lett . phys . _ * 104 * no . 5 , ( 2014 ) 585612 ] , http://arxiv.org/abs/1307.2576[arxiv:1307.2576 [ hep - th ] ] . l. hadasz , z. jaskolski , and p. suchanek , `` proving the agt relation for @xmath1141 antifundamentals , '' http://dx.doi.org/10.1007/jhep06(2010)046[_jhep_ * 06 * ( 2010 ) 046 ] , http://arxiv.org/abs/1004.1841[arxiv:1004.1841 [ hep - th ] ] . s. yanagida , `` norms of logarithmic primaries of virasoro algebra , '' http://dx.doi.org/10.1007/s11005-011-0502-0[_lett . * 98 * ( 2011 ) 133156 ] , http://arxiv.org/abs/1010.0528[arxiv:1010.0528 [ math.qa ] ] . d. maulik and a. okounkov , `` quantum groups and quantum cohomology , '' http://arxiv.org/abs/1211.1287[arxiv:1211.1287 [ math.ag ] ] . o. schiffmann and e. vasserot , `` cherednik algebras , w - algebras and the equivariant cohomology of the moduli space of instantons on @xmath1142 , '' http://dx.doi.org/10.1007/s10240-013-0052-3[_publ . hautes tudes sci . _ * 118 * ( 2013 ) 213342 ] , http://arxiv.org/abs/1202.2756[arxiv:1202.2756 [ math.qa ] ] . a. braverman , m. finkelberg , and h. nakajima , `` instanton moduli spaces and @xmath1143-algebras , '' http://arxiv.org/abs/1406.2381[arxiv:1406.2381 [ math.qa ] ] . f. nieri , `` an elliptic virasoro symmetry in 6d , '' http://arxiv.org/abs/1511.00574[arxiv:1511.00574 [ hep - th ] ] . a. iqbal , c. kozcaz , and s .- t . yau , `` elliptic virasoro conformal blocks , '' http://arxiv.org/abs/1511.00458[arxiv:1511.00458 [ hep - th ] ] . h. awata and y. yamada , `` five - dimensional agt conjecture and the deformed virasoro algebra , '' http://dx.doi.org/10.1007/jhep01(2010)125 [ _ jhep _ * 01 * ( 2010 ) 125 ] , http://arxiv.org/abs/0910.4431[arxiv:0910.4431 [ hep - th ] ] . h. awata and y. yamada , `` five - dimensional agt relation and the deformed beta - ensemble , '' http://dx.doi.org/10.1143/ptp.124.227[_prog . _ * 124 * ( 2010 ) 227262 ] , http://arxiv.org/abs/1004.5122[arxiv:1004.5122 [ hep - th ] ] . y. zenkevich , `` generalized macdonald polynomials , spectral duality for conformal blocks and agt correspondence in five dimensions , '' http://dx.doi.org/10.1007/jhep05(2015)131[_jhep_ * 05 * ( 2015 ) 131 ] , http://arxiv.org/abs/1412.8592[arxiv:1412.8592 [ hep - th ] ] . h. awata , b. feigin , a. hoshino , m. kanai , j. shiraishi , and s. yanagida , `` notes on ding - iohara algebra and agt conjecture , '' _ rims kokyuroku _ * 1765 * ( 2011 ) 1232 , http://arxiv.org/abs/1106.4088[arxiv:1106.4088 [ math - ph ] ] . k. miki , `` a @xmath1144 analog of the @xmath64 algebra , '' http://dx.doi.org/10.1063/1.2823979[_j . math _ * 48 * no . 12 , ( 2007 ) 123520 , b. feigin , a. hoshino , j. shibahara , j. shiraishi , and s. yanagida , `` kernel function and quantum algebra , '' _ rims kokyuroku _ * 1689 * ( 2010 ) 133152 , http://arxiv.org/abs/1002.2485[arxiv:1002.2485 [ math.qa ] ] . b. feigin , m. jimbo , t. miwa , and e. mukhin , `` quantum toroidal @xmath65 and bethe ansatz , '' http://dx.doi.org/10.1088/1751-8113/48/24/244001[_j . a _ * 48 * no . 24 , ( 2015 ) 244001 , 27 ] , http://arxiv.org/abs/1502.07194[arxiv:1502.07194 [ math.qa ] ] . bourgine , m. fukuda , y. matsuo , h. zhang , and r .- d . zhu , `` coherent states in quantum @xmath1145 algebra and qq - character for 5d super yang - mills , '' http://arxiv.org/abs/1606.08020[arxiv:1606.08020 [ hep - th ] ] . i. burban and o. schiffmann , `` on the hall algebra of an elliptic curve , i , '' http://dx.doi.org/10.1215/00127094-1593263[_duke math . j. _ * 161 * no . 7 , ( 2012 ) 11711231 ] , http://arxiv.org/abs/0505148[arxiv:0505148 [ math.ag ] ] . b. feigin , k. hashizume , a. hoshino , j. shiraishi , and s. yanagida , `` a commutative algebra on degenerate @xmath1146 and macdonald polynomials , '' _ j. math . _ * 50 * no . 9 , ( 2009 ) 095215 , 42 , http://arxiv.org/abs/0904.2291[arxiv:0904.2291 [ math.co ] ] . h. awata and h. kanno , `` refined bps state counting from nekrasov s formula and macdonald functions , '' http://dx.doi.org/10.1142/s0217751x09043006[_int . phys . _ * a24 * ( 2009 ) 22532306 ] , http://arxiv.org/abs/0805.0191[arxiv:0805.0191 [ hep - th ] ] . a. iqbal , c. kozcaz , and c. vafa , `` the refined topological vertex , '' http://dx.doi.org/10.1088/1126-6708/2009/10/069[_jhep_ * 10 * ( 2009 ) 069 ] , http://arxiv.org/abs/hep-th/0701156[arxiv:hep-th/0701156 [ hep - th ] ] . h. awata , b. feigin , and j. shiraishi , `` quantum algebraic approach to refined topological vertex , '' http://dx.doi.org/10.1007/jhep03(2012)041 [ _ jhep _ * 03 * ( 2012 ) 041 ] , http://arxiv.org/abs/1112.6074[arxiv:1112.6074 [ hep - th ] ] . m. kashiwara , `` crystalizing the @xmath500 analog of universal enveloping algebras , '' http://dx.doi.org/10.1007/bf02097367[_commun . math . * 133 * ( 1990 ) 249260 ] . s. l. lukyanov and y. pugai , `` multipoint local height probabilities in the integrable rsos model , '' http://dx.doi.org/10.1016/0550-3213(96)00221-0[_nucl . phys . _ * b473 * ( 1996 ) 631658 ] , http://arxiv.org/abs/hep-th/9602074[arxiv:hep-th/9602074 [ hep - th ] ] . h. awata , h. kanno , a. mironov , a. morozov , a. morozov , y. ohkubo , and y. zenkevich , `` toric calabi - yau threefolds as quantum integrable systems . @xmath1147 -matrix and @xmath1148 relations , '' http://dx.doi.org/10.1007/jhep10(2016)047[_jhep_ * 10 * ( 2016 ) 047 ] , http://arxiv.org/abs/1608.05351[arxiv:1608.05351 [ hep - th ] ] . h. awata , h. kanno , a. mironov , a. morozov , a. morozov , y. ohkubo , and y. zenkevich , `` anomaly in rtt relation for dim algebra and network matrix models , '' http://arxiv.org/abs/1611.07304[arxiv:1611.07304 [ hep - th ] ] . b. feigin , e. feigin , m. jimbo , t. miwa , and e. mukhin , `` quantum continuous @xmath1149 : semiinfinite construction of representations , '' http://dx.doi.org/10.1215/21562261-1214375[_kyoto j. math . _ * 51 * no . 2 , ( 2011 ) 337364 ] , http://arxiv.org/abs/1002.3100[arxiv:1002.3100 [ math.qa ] ] . b. l. feigin and a. i. tsymbaliuk , `` equivariant @xmath1150-theory of hilbert schemes via shuffle algebra , '' http://dx.doi.org/10.1215/21562261-1424875[_kyoto j. math . _ * 51 * no . 4 , ( 2011 ) 831854 ] , http://arxiv.org/abs/0904.1679[arxiv:0904.1679 [ math.rt ] ] . s. yanagida , `` whittaker vector of deformed virasoro algebra and macdonald symmetric functions , '' http://dx.doi.org/10.1007/s11005-016-0821-2[_lett phys . _ * 106 * no . 3 , ( 2016 ) 395431 ] , http://arxiv.org/abs/1402.2946[arxiv:1402.2946 [ math.qa ] ] . h. nakajima and k. yoshioka , `` instanton counting on blowup . i. 4-dimensional pure gauge theory , '' http://dx.doi.org/10.1007/s00222-005-0444-1 [ _ invent . _ * 162 * ( 2005 ) 313355 ] , http://arxiv.org/abs/math/0306198[arxiv:math/0306198 [ math.ag ] ] . h. nakajima and k. yoshioka , `` instanton counting on blowup . @xmath1150-theoretic partition function , '' http://dx.doi.org/10.1007/s00031-005-0406-0[_transform . groups _ * 10 * no . 3 - 4 , ( 2005 ) 489519 ] , http://arxiv.org/abs/math/0505553[arxiv:math/0505553 [ math.ag ] ] . s. yanagida , `` five - dimensional su(2 ) agt conjecture and recursive formula of deformed gaiotto state , '' http://dx.doi.org/10.1063/1.3505826[_j . * 51 * ( 2010 ) 123506 ] , http://arxiv.org/abs/1005.0216[arxiv:1005.0216 [ math.qa ] ] . s. yanagida , `` norm of the whittaker vector of the deformed virasoro algebra , '' http://arxiv.org/abs/1411.0462[arxiv:1411.0462 [ math.qa ] ] . j. ding and k. iohara , `` generalization and deformation of drinfeld quantum affine algebras , '' http://dx.doi.org/10.1023/a:1007341410987[_lett . phys . _ * 41 * ( 1997 ) 181193 ] . c . tan , `` an m - theoretic derivation of a 5d and 6d agt correspondence , and relativistic and elliptized integrable systems , '' http://dx.doi.org/10.1007/jhep12(2013)031[_jhep_ * 12 * ( 2013 ) 031 ] , http://arxiv.org/abs/1309.4775[arxiv:1309.4775 [ hep - th ] ] . bouwknegt and k. pilch , `` the deformed virasoro algebra at roots of unity , '' _ comm . _ * 196 * no . 2 , ( 1998 ) 249288 , http://arxiv.org/abs/9710026[arxiv:9710026 [ math.qa ] ] . p. bouwknegt and k. pilch , `` on deformed w algebras and quantum affine algebras , '' _ adv . * 2 * ( 1998 ) 357397 , http://arxiv.org/abs/math/9801112[arxiv:math/9801112 [ math.qa ] ] . m. fukuda , s. nakamura , y. matsuo , and r .- d . zhu , `` sh@xmath1151 realization of minimal model cft : triality , poset and burge condition , '' http://dx.doi.org/10.1007/jhep11(2015)168[_jhep_ * 11 * ( 2015 ) 168 ] , http://arxiv.org/abs/1509.01000[arxiv:1509.01000 [ hep - th ] ] . y. ohkubo , h. awata , and h. fujino , `` crystallization of deformed virasoro algebra , ding - iohara - miki algebra and 5d agt correspondence , '' http://arxiv.org/abs/1512.08016[arxiv:1512.08016 [ math - ph ] ] . h. awata , h. kubo , s. odake , and j. shiraishi , `` virasoro type symmetries in solvable models , '' in _ extended and quantum algebras and their applications to physics tianjin , china , august 19 - 24 , 1996_. 1996 . http://arxiv.org/abs/hep-th/9612233[arxiv:hep-th/9612233 [ hep - th ] ] . m. taki , `` on agt - w conjecture and q - deformed w - algebra , '' http://arxiv.org/abs/1403.7016[arxiv:1403.7016 [ hep - th ] ] . a. morozov and y. zenkevich , `` decomposing nekrasov decomposition , '' http://dx.doi.org/10.1007/jhep02(2016)098[_jhep_ * 02 * ( 2016 ) 098 ] , http://arxiv.org/abs/1510.01896[arxiv:1510.01896 [ hep - th ] ] . m. taki , `` refined topological vertex and instanton counting , '' http://dx.doi.org/10.1088/1126-6708/2008/03/048[_jhep_ * 03 * ( 2008 ) 048 ] , http://arxiv.org/abs/0710.1776[arxiv:0710.1776 [ hep - th ] ] . y. ohkubo , `` existence and orthogonality of generalized jack polynomials and its @xmath13-deformation , '' http://arxiv.org/abs/1404.5401[arxiv:1404.5401 [ math - ph ] ] . s. mironov , a. morozov , and y. zenkevich , `` generalized jack polynomials and the agt relations for the @xmath1152 group , '' http://dx.doi.org/10.1134/s0021364014020076[_jetp lett . _ * 99 * ( 2014 ) 109113 ] , http://arxiv.org/abs/1312.5732[arxiv:1312.5732 [ hep - th ] ] . n. h. jing , `` vertex operators and hall - littlewood symmetric functions , '' http://dx.doi.org/10.1016/0001-8708(91)90072-f[_adv . _ * 87 * no . 2 , ( 1991 ) 226248 ] . h. awata , h. kanno , t. matsumoto , a. mironov , a. morozov , a. morozov , y. ohkubo , and y. zenkevich , `` explicit examples of dim constraints for network matrix models , '' http://dx.doi.org/10.1007/jhep07(2016)103[_jhep_ * 07 * ( 2016 ) 103 ] , http://arxiv.org/abs/1604.08366[arxiv:1604.08366 [ hep - th ] ] .
|
in this thesis , we obtain the formula for the kac determinant of the algebra arising from the level @xmath0 representation of the ding - iohara - miki algebra .
this formula can be proved by decomposing the level @xmath0 representation into the deformed @xmath1-algebra part and the @xmath2 boson part , and using the screening currents of the deformed @xmath1-algebra .
it is also discovered that singular vectors obtained by its screening currents correspond to the generalized macdonald functions .
moreover , we investigate the @xmath3 limit of five - dimensional agt correspondence . in this limit ,
the simplest 5d agt conjecture is proved , that is , the inner product of the whittaker vector of the deformed virasoro algebra coincides with the partition function of the 5d pure gauge theory .
furthermore , the r - matrix of the ding - iohara - miki algebra is explicitly calculated , and its general expression in terms of the generalized macdonald functions is conjectured .
@xmath4 + phd thesis submitted to nagoya university + graduate school of mathematics + march 2017
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
observation of the kondo effect in a quantum dot ( qd ) @xcite has stimulated researches in the field of quantum transport , and recent experimental developments enable one to examine the kondo physics in a variety of systems , such as an aharonov - bohm ( ab ) ring with a qd and double quantum dots ( dqd ) . in these systems multiple paths for electron propagation also affect the tunneling currents , and the interference causes fano - type asymmetric line shapes . superconductivity also brings rich and interesting features into the quantum transport . competition between superconductivity and the kondo effect has been reported to be observed in carbon nanotube qd and in semiconductor nanowires . @xcite furthermore , interplay between the andreev scattering and the kondo effect has been studied intensively for a qd coupled to a normal ( n ) lead and superconductor ( s ) , theoretically @xcite and experimentally . @xcite so far , however , the andreev - kondo physics has been discussed mainly for a single dot . in this paper , we consider a dqd system with a t - shape geometry as shown in fig . [ modelndqdscom ] , and study how multiple paths affect the interplay at low temperatures , using the numerical renormalization group ( nrg ) method . @xcite golub and avishai calculated first , to our knowledge , the andreev transport through an ab ring with a qd,@xcite in which a similar interference effect is expected . however , the underlying kondo physics in such a combination with superconductivity and interference is still not fully understood , and is needed to be clarified precisely , as measurements are being not impossible . @xcite we find that the coulomb interaction in the side dot ( qd2 in fig . [ modelndqdscom ] ) suppresses destructive interference typical of the t - shape geometry , and it enhances substantially the tunneling current between the normal and superconducting ( sc ) leads in the kondo regime . this is quite different from the behavior seen in the normal transport in the same configuration fig . [ modelndqdscom ] ( b ) , for which the conductance is suppressed , and shows a wide minimum called a kondo valley as a result of strong interference by the kondo resonance in the side dot . @xcite the sc proximity penetrating into the interfacial dot ( qd1 in fig . [ modelndqdscom ] ) causes this stark contrast between the andreev and normal transports . it also changes the fano line shape in the gate - voltage dependence of the conductance . furthermore , we show that the proximity deforms the kondo cloud to make a singlet bond long , and it can be deduced from the fermi - liquid properties of the ground state . in sec . [ sec : model ] , we introduce the model and describe the effective hamiltonian in a large gap limit . in sec . [ sec : result ] , we show the numerical results and clarify the transport properties using the renormalized parameters . the fano structures in the gate - voltage dependence of the conductance are also discussed . a brief summary is given in the last section . we start with an anderson impurity connected to sc and normal leads , @xmath0 where @xmath1 @xmath2 describes the interfacial ( @xmath3 ) and side ( @xmath4 ) dots : @xmath5 the energy level , @xmath6 the coulomb interaction , @xmath7 , and @xmath8 the inter - dot hopping matrix element . @xmath9 describes the sc / normal lead , and @xmath10 is a @xmath11-wave bcs gap . @xmath12 is the tunneling matrix element between qd1 and the sc / normal lead . we assume that @xmath13 is a constant independent of the energy @xmath14 , where @xmath15 is the total number of @xmath16 s in the leads . throughout the work , we concentrate on a large gap limit @xmath17 . then the starting hamiltonian @xmath18 can be mapped exactly onto a single - channel model , which still captures the essential physics of the andreev reflection and makes nrg approach efficient , @xcite @xmath19 note that at @xmath17 the real and virtual excitations towards the continuum states outside the gap in the sc lead are prohibited . nevertheless , the proximity from the sc lead to the dot remains finite , and it induces a local static pair potential @xmath20 ( @xmath21 ) at qd1 . furthermore , the current can flow between the sc lead and the qd1 via @xmath22 . for ( a ) @xmath24 and ( b ) @xmath25 , for several side - dot repulsions @xmath26 . the parameters for qd1 are chosen to be @xmath27 , and @xmath28 which is the local sc gap defined by @xmath29 . ] we can calculate the conductance at zero temperature as a function of the level position @xmath23 of qd2 for different values of @xmath26 , using the kubo formula.@xcite in this paper we focus on the coulomb interaction in the side dot ( qd2 ) , assuming that @xmath30 in the following . the results of the conductance are shown in fig . [ con - ed2 ] for @xmath27 and @xmath31 . the coupling to the normal lead is chosen to be ( a ) @xmath24 and ( b ) @xmath25 . the conductance is enhanced for the kondo regime @xmath32 , where the wide kondo valley appears in the case of the normal transport . this is a novel phenomenon caused by the interplay between superconductivity and the kondo effect ; the local gap @xmath20 due to the proximity into qd1 leads to the andreev transport with destructive interference , but the introduction of @xmath26 suppresses the sc interference via qd2 , which in turn enhances the conductance . note that the couplings are symmetric @xmath33 for fig . [ con - ed2 ] ( a ) , and in this particular case the conductance increases with @xmath26 for any @xmath23 , except for the values @xmath34 and @xmath35 . outside of the kondo regime , the side dot is empty or doubly occupied , and the interference becomes no longer important . when the coupling to the normal lead is small @xmath36 as fig . [ con - ed2 ] ( b ) , the conductance in the kondo regime decreases after the peak reaches the unitary limit @xmath37 . this behavior can be related to a crossover from short - range to long - range kondo screening as illustrated in fig . [ modelndqdscom ] , and is discussed later again . we examine the behavior at the middle point @xmath38 of fig . [ con - ed2 ] in detail . the low - lying energy states show the fermi - liquid properties , @xcite and the conductance can be deduced from the renormalized parameters for the quasi - particles ( see eq . ( [ pcstg ] ) ) . at @xmath39 and @xmath27 for several values of @xmath26 . ( a ) @xmath24 and ( b ) @xmath25 . the dashed line is the conductance of a single dot ( @xmath40 ) , for which the horizontal axis should be interpreted as ( a ) @xmath41 and ( b ) @xmath42 . ] the conductance is plotted as a function of @xmath43 in fig . [ con - gs ] for ( a ) @xmath24 and ( b ) @xmath25 . we see that the peak shifts towards smaller @xmath22 as @xmath26 increases and will coincide in the limit of @xmath44 with the dashed line , which corresponds to the conductance without the side dot . it means that the interference caused by the side dot is suppressed completely for large @xmath26 , and in this limit the conductance reaches the unitary limit value for the symmetric couplings @xmath45 . the difference in the line shape of fig . [ con - ed2 ] ( a ) and that of fig . [ con - ed2 ] ( b ) at fixed @xmath22 reflects the position of the unitary - limit peak in fig . [ con - gs ] . in order to clarify the properties of the the ground state precisely , we consider a special case @xmath38 . then the hamiltonian @xmath46 can be written in terms of the interacting bogoliubov particles , which conserve the total charge , as shown in appendix.@xcite consequently , the low - energy states can be described by a local fermi liquid , the fixed - point hamiltonian@xcite of which can be written in the form @xmath47 here , @xmath48 is a local sc gap that emerges in qd2 via the self - energy correction due to @xmath26 , while @xmath29 as defined in eq . ( [ delta_d1 ] ) is caused by the direct proximity from the sc lead . @xmath49 is the renormalized value of the inter - dot hopping matrix element . we calculate these parameters from the fixed - point of nrg.@xcite then , the conductance @xmath50 and a staggered sum @xmath51 of the pair correlation are deduced from the phase shift , @xmath52 , of the bogoliubov particles , @xmath53 : ( a ) conductance , ( b ) @xmath49 , @xmath48 , ( c ) @xmath54 , @xmath55 , and @xmath56 . we choose @xmath27 , @xmath57 , and for filled ( open ) marks @xmath24 ( @xmath58 ) . inset of ( b ) : phase boundary , between singlet and doublet ground states , for an isolated dqd ( @xmath59 ) with a finite local sc gap @xmath29 . ] in fig . [ all ] , we show the @xmath26 dependence of the ground - state properties at @xmath38 for @xmath27 and @xmath57 . the coupling is chosen to be @xmath60 , and @xmath61 . the conductance for @xmath62 shows a peak as a function of @xmath26 , while for @xmath63 it increases simply towards the unitary limit . this corresponds to the difference that we see in fig . [ con - ed2](a ) and ( b ) at @xmath38 . figure [ all](b ) shows the renormalized parameters ( @xmath64 ) @xmath49 and ( @xmath65,@xmath66 ) @xmath67 . the ratio @xmath68 , which equals to the square root of the wavefunction renormalization factor @xmath69 ( see appendix ) , decreases monotonically from @xmath70 to @xmath35 with increasing @xmath26 , while the local sc gap @xmath48 becomes large for intermediate values of @xmath26 . the behavior of these fermi - liquid parameters implies that a crossover from weak to strong correlation regimes occurs around @xmath71 . the nature of the crossover can be related to a level crossing taking place in a _ molecule _ limit @xmath59 , where qd1 is decoupled from the normal lead . in this limit , the isolated dqd is described by a hamiltonian @xmath72 , which includes the local sc gap @xmath29 at qd1 . the ground state of the _ molecule _ is a singlet or doublet depending on @xmath73 and @xmath43 , as shown in the inset of fig . [ all](b ) . the ground state is a spin - singlet , if either @xmath73 or @xmath43 is small . in the opposite case , a spin - doublet becomes the ground state . the local moment in this doublet state emerges mainly at qd2 , because the correlation at qd1 is small in the present situation @xmath27 . we see in the phase diagram in fig . [ all](b ) that the transition takes place in this _ molecule _ limit at @xmath71 for @xmath57 , and it agrees well with the position where the conductance peak appears in fig . [ all](a ) . for finite @xmath74 , conduction electrons can tunnel from the normal lead to qd2 via qd1 . however , the sc correlation @xmath29 tends to make the local state at qd1 a singlet , which consists of a linear combination of the empty and doubly occupied states . thus , for large @xmath22 , the electrons at qd1 can not contribute to the screening of the moment at qd2 . in this situation , the kondo screening is achieved mainly by the conduction electrons tunneling to qd2 virtually via qd1 . this process is analogous to a superexchange mechanism , which can also be expected from the hamiltonian written in terms of the bogoliubov particles in appendix . from these observations we see that the conductance peak at @xmath71 in fig . [ all](a ) reflects the crossover from the short - range singlet to long - range one due to the superexchange screening process ( see also fig . [ modelndqdscom ] ) for the bogoliubov particles . note that the peak structure of the conductance vanishes for @xmath75 . the deformation of the kondo cloud can also be deduced from the results shown in fig . [ all](c ) . this is because the staggered pair correlation @xmath51 is related directly to the scattering phase shift @xmath52 of the bogoliubov particles , by the friedel sum rule eq . given in appendix . therefore , the value of @xmath51 reflects the changes occurring in the kondo clouds . particularly , a sudden change observed in @xmath51 around @xmath71 shows that the phase of the wavefunction shifts by @xmath76 during this change . this also explains the occurring of the crossover from the short - range to long - range screening . we can also calculate each correlation function @xmath77 directly with nrg based on the definition . the local sc correlations @xmath54 and @xmath55 have the same value in the noninteracting case @xmath78 , and thus in this particular limit there is no reduction in the amplitude of the proximity from qd1 to qd2 . the coulomb interaction @xmath26 causes the reduction , as both @xmath54 and @xmath55 decrease for small @xmath26 where the ground state is a singlet with a _ molecule _ character . for large @xmath26 , the sc correlation @xmath79 almost vanishes in qd2 , while @xmath54 shows an upturn and approaches the value expected for @xmath40 . therefore , the sc proximity into qd1 is enhanced when the kondo cloud expands to form a long - range singlet . then the tunneling current is not interfered much by the local moment at qd2 , and flows almost directly without using the path to the side dot . for several values of @xmath81 , where @xmath82 and @xmath57 . other parameters : ( a ) @xmath24 , @xmath83 . ( b ) @xmath25 , @xmath84 . ] so far , we have chosen the level of qd1 to be @xmath85 . the result obtained for different values of @xmath81 is plotted as a function of @xmath23 in fig . [ con - ed2_extra ] . we see that two asymmetric fano structures , each of which consists of a pair of peak and dip , emerge at @xmath86 and @xmath87 for @xmath80 , as the fermi level crosses the energy corresponding to the upper and lower levels of the atomic limit . the conductance peaks become sharper for @xmath62 as @xmath88 increases . et al _ studied the fano structure in the side - coupled dqd system with two normal leads,@xcite and showed that at low temperatures the conductance has only one pair of the peak and dip , which are separated widely by a fano - kondo plateau at @xmath89 . this type of the plateau was known earlier to appear in a qd embedded in an ab ring.@xcite in contrast , our result shows that the fano - kondo plateau vanishes , when one of the leads is a superconductor . this is because the kondo screening in this case is achieved by the long - range singlet bond due to the superexchange process , as a result of the competition between the sc proximity into qd1 and coulomb interaction at qd2 . we have studied andreev transport through the side - coupled dqd with nrg approach . we have found that the coulomb interaction in the side dot suppresses the destructive interference effect typical of the t - shape geometry , and enhances the tunneling current between the normal and sc leads . this novel phenomenon is caused by the interplay between the sc correlation and the kondo effect ; the sc proximity into qd1 pushes the kondo cloud towards the normal lead , and the conductance shows a peak of the unitary limit as the nature of the singlet changes from a short - range to long - range one . we have also clarified that two asymmetric fano structures appear in the gate - voltage dependence of the andreev transport , instead of a reduced single fano - kondo plateau which appears in the kondo regime of the normal transport . we thank k. inaba for valuable discussions . is also grateful to j. bauer and a. c. hewson for discussions . the work is partly supported by a grant - in - aid from mext japan ( grant no.19540338 ) . y.t . is supported by jsps research fellowships for young scientists . is supported by jsps grant - in - aid for scientific research ( c ) . the hamiltonian @xmath46 defined in eq . can be transformed , at @xmath38 , into the interacting bogoliubov particles , which conserve the total charge . for describing this property briefly , we rewrite @xmath46 using the logarithmic discretization of nrg,@xcite @xmath90 for @xmath91 , the operator @xmath92 describes the conduction electron in the normal lead , and @xmath93 is given by @xmath94 here , @xmath95 is the half - width of the conduction band . for the double - dot part , we use a notation @xmath96 for @xmath97 . correspondingly , @xmath98 and @xmath99 with @xmath100 at @xmath101 , the system has a uniaxial symmetry in the nambu pseudo - spin space,@xcite and the hamiltonian can be simplified by the transformation @xmath102 = \left [ \begin{array}{cc } u & -v \\ v & \ u \rule{0cm}{0.5 cm } \end{array } \right ] \left [ \begin{array}{c } f_{n\uparrow}^{\phantom{\dagger } } \\ ( -1)^{n-1 } f_{n\downarrow}^{\dagger } \end{array } \right]\ ; , \label{eq : bogo_b } \\ & u=\sqrt{\frac{1}{2}\left(1+\frac{\xi_{d1}}{e_{d1}}\right ) } , \quad v=\sqrt{\frac{1}{2}\left(1-\frac{\xi_{d1}}{e_{d1}}\right ) } \label{eq : bogo_factor_b } \;.\end{aligned}\ ] ] here , @xmath103 , @xmath104 , and @xmath29 as defined in eq . . then @xmath105 is transformed into a normal two - impurity anderson model for the bogoliubov particles @xmath106 . \label{eq : nrg_h_nt_bogo}\end{aligned}\ ] ] here , @xmath107 , and the total number of the bogoliubov particles , @xmath108 , is conserved . the equation implies that the low - energy excited states can be described by a local fermi - liquid theory . this is true also for the original hamiltonian @xmath46 , and it does not depend on the discretization procedure of nrg.@xcite to be specific , we assume that @xmath30 in the following . in this case , the bogoliubov particles feel a normal impurity potential @xmath109 at qd1 , and this potential causes the superexchange mechanism that makes the singlet - bond long range as discussed in sec . [ subsec : fl ] . the retarded green s function for the bogoliubov particle @xmath110 at qd2 takes the form @xmath111 where @xmath112 is the self - energy caused by the interaction @xmath113 . at zero temperature , the asymptotic form of the green s function for small @xmath114 is given by @xmath115 note that @xmath48 has a finite value even though @xmath116 , because @xmath117 . the value of the parameters @xmath48 and @xmath49 can be deduced from the fixed point of nrg.@xcite then , using the friedel sum rule for eq . , the local charge at the double dot can be calculated from the phase shift @xmath52 of the bogoliubov particles , @xmath118 the charge of the bogoliubov particles corresponds to the sc pair correlation for the original electrons @xmath92 . specifically for @xmath119 , it is transformed into the staggered sum @xmath51 given in eq . , by the inverse transformation of eq . . similarly , the conductance @xmath50 can be expressed in terms of the phase shift @xmath52 . furthermore , the free quasi - particles corresponding to the green s function given in eq . can be described by a hamiltonian , which is rewritten in terms of the original electron operators in eq . by the inverse transformation . d. goldhaber - gordon , h. shtrikman , d. mahalu , d. abusch - magder , u. meirav , and m. a. kanster , nature ( london ) * 391 * , 156 ( 1998 ) ; s. m. cronenwett , t. h. oosterkamp , and l. p. kouwenhoven , science * 281 * , 540 ( 1998 ) .
|
we study the transport through side - coupled double quantum dots , connected to normal and superconducting ( sc ) leads with a t - shape configuration .
we find , using the numerical renormalization group , that the coulomb interaction suppresses sc interference in the side dot , and enhances the conductance substantially in the kondo regime .
this behavior stands in total contrast to a wide kondo valley seen in the normal transport . the sc proximity penetrating into the interfacial dot pushes the kondo clouds , which screens the local moment in the side dot , towards the normal lead to make the singlet bond long .
the conductance shows a peak of unitary limit as the cloud expands .
furthermore , two separate fano structures appear in the gate - voltage dependence of the andreev transport , where a single reduced plateau appears in the normal transport .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the discovery of a resonance at the lhc @xcite that is compatible with the standard - model ( sm ) higgs boson @xcite marked a milestone in particle physics . the existence of the higgs boson is inherently related to the mechanism of spontaneous symmetry breaking @xcite while preserving the full gauge symmetry and the renormalizability of the sm @xcite . its mass , the last missing parameter of the sm , has been measured to be ( @xmath0 ) gev @xcite . the existence of the higgs boson allows the sm particles to be weakly interacting up to high - energy scales without violating the unitarity bounds of scattering amplitudes @xcite . this , however , is only possible for particular higgs - boson couplings to all other particles so that with the knowledge of the higgs - boson mass all its properties are uniquely fixed . the massive gauge bosons and fermions acquire mass through their interaction with the higgs field that develops a finite vacuum expectation value in its ground state thus hiding the still unbroken electroweak gauge symmetry . the minimal model requires the introduction of one weak isospin doublet of the higgs field and leads after spontaneous symmetry breaking to the existence of one scalar higgs boson , while non - minimal higgs sectors predict in general more than one higgs boson . within the sm with one higgs doublet the higgs mass is constrained by consistency conditions as the absence of a landau pole for the higgs self - coupling up to high energy scales and the stability of the electroweak ground state @xcite . if the sm is required to fulfill these conditions for energy scales up to the scale of grand unified theories ( guts ) of @xmath1 gev the higgs mass is constrained between 129 gev and about 190 gev @xcite . the last condition of vacuum stability can be relaxed by demanding the life - time of the ground state to be larger than the age of our universe @xcite . this reduces the lower bound on the higgs mass to about 110 gev @xcite . the measured value of the higgs mass indicates that our universe is unstable with a large lifetime far beyond the age of the universe . if the sm is extended to the gut scale , radiative corrections to the higgs - boson mass tend to push the higgs mass towards the gut scale , if it couples to particles of that mass order . in order to obtain the higgs mass at the electroweak scale the counter term has to be fine - tuned to cancel these large corrections thus establishing a very unnatural situation that requires a solution . this is known as the hierarchy problem @xcite . possible solutions are the introduction of supersymmetry ( susy ) @xcite , a collective symmetry between sm particles and heavier partners as in little higgs theories @xcite or an effective reduction of the planck and gut scales as in extra - dimension models at the tev scale @xcite . sm higgs bosons are dominantly produced via the loop - induced gluon - fusion process @xmath2 with top quarks providing the leading loop contribution @xcite . other production processes as vector - boson fusion @xmath3 @xcite , higgs - strahlung @xmath4 @xcite and higgs radiation off top quarks @xmath5 @xcite are suppressed by more than one order of magnitude . while the dominant gluon - fusion process allows for the detection of the higgs boson in the rare decay modes into @xmath6 , four charged leptons and @xmath7-boson pairs , other decay modes are only detectable in the subleading production modes as e.g. the main higgs - boson decay into bottom quarks in strongly boosted higgs - strahlung @xcite or the higgs - boson decay into @xmath8-leptons in the vector - boson - fusion process @xcite . supersymmetric extensions of the sm are motivated by providing a possible solution to the hierarchy problem if the supersymmetric particles appear at the few - tev scale @xcite . supersymmetry connects fermionic and bosonic degrees of freedom and thus links internal and external symmetries @xcite . it is the last possible symmetry - type of @xmath9-matrix theories , i.e. the last possible extension of the poincar algebra . the minimal supersymmetric extension of the sm ( mssm ) predicts the weinberg angle in striking agreement with experimental measurements if embedded in a gut @xcite . moreover , it allows to generate electroweak symmetry breaking radiatively @xcite and yields a candidate for dark matter if @xmath10-parity is conserved @xcite which renders the lightest supersymmetric particle stable . finally it increases the proton lifetime beyond experimental bounds in the context of supersymmetric guts @xcite . the mssm requires the introduction of two isospin doublets of higgs fields in order to maintain the analyticity of the superpotential and the anomaly - freedom with respect to the gauge symmetries @xcite . moreover , two higgs doublets are needed for the generation of the up- and down - type fermion masses . the higgs sector is a two - higgs - doublet model ( 2hdm ) sector of type ii . the mass eigenstates consist of a light ( @xmath11 ) and heavy ( @xmath12 ) scalar , a pseudoscalar ( @xmath13 ) and two charged ( @xmath14 ) states . the self - interactions of the higgs fields are entirely fixed in terms of the electroweak gauge couplings so that the self - couplings are constrained to small values . this leads to an upper bound on the light scalar higgs mass that has to be smaller than the @xmath15-boson mass @xmath16 at leading order ( lo ) . this is , however , broken by radiative corrections , which are dominated by top - quark - induced contributions @xcite . the parameter @xmath17 , defined as the ratio of the two vacuum expectation values of the scalar higgs fields , will in general be assumed to be in the range @xmath18 , where @xmath19 denotes the top ( bottom ) mass , consistent with the assumption that the mssm is the low - energy limit of a supergravity model @xcite and to avoid non - perturbative phenomena . if the soft susy - breaking parameters do not contain any complex phases the input parameters of the mssm higgs sector at lo are generally chosen to be the mass @xmath20 of the pseudoscalar higgs boson and @xmath17 . all other masses and the mixing angle @xmath21 between the scalar cp - even higgs states can be derived from these basic parameters ( and the top mass and susy parameters , which enter through radiative corrections ) . the radiative corrections can be approximated by the parameter @xmath22 , which grows with the fourth power of the top quark mass and logarithmically with the stop masses @xmath23 , supplemented by terms originating from soft susy - breaking parameters , i.e. the trilinear coupling @xmath24 , the higgsino mass @xmath25 and the third - generation squark mass @xmath26 with @xmath27 , @xmath28 where @xmath29 denotes the fermi constant and @xmath30 the @xmath31 top mass at the scale @xmath26 . these corrections are positive and reach a maximum ( related to @xmath32 ) for @xmath33 . they increase the squared mass of the light neutral higgs boson @xmath11 to @xmath34 in this approximation , the upper bound on @xmath35 is shifted from the tree level value @xmath16 up to @xmath36 145 gev . the mass of the lightest scalar state @xmath11 is given by @xmath37 \label{eq : hmass}\end{aligned}\ ] ] the masses of the heavy neutral and charged higgs bosons are determined by the sum rules ( valid for this approximation ) , @xmath38 the effective mixing parameter @xmath21 between the cp - even scalar higgs states can be derived as @xmath39 the radiative corrections to the mssm higgs sector have been calculated up to the two - loop level in the effective potential approximation and in the diagrammatic approach @xcite . the leading corrections are also known at the three - loop level within the effective potential approach @xcite . the corrections beyond next - to - leading order ( nlo ) are dominated by the qcd corrections to the top - quark - induced contributions . they decrease the upper bound on the light scalar higgs mass @xmath35 by about 10 gev to @xmath40 gev , while leaving a residual uncertainty of @xmath41 gev on the light scalar higgs mass . in the context of increasing lower mass bounds for the supersymmetric particles , a resummation of large logarithms related to the susy - particle masses has been performed for the calculation of the mssm higgs - boson masses @xcite . for large susy - particle masses this resummation may lead to effects of the order of 5 gev ( or larger for small values of @xmath42 ) on the light scalar higgs mass . the calculation of the radiative corrections has also been extended to the effective trilinear and quartic self - interactions of the mssm higgs bosons that are consistently defined at vanishing external momenta . the nlo corrections at @xmath43 and @xmath44 are available since a long time @xcite , where @xmath45 and @xmath46 . in the leading approximation in terms of the parameter @xmath22 the trilinear couplings of the neutral higgs bosons are given up to @xmath43 by @xcite @xmath47 which have been normalized to @xmath48 . similar but more involved expressions can be obtained for the quartic higgs self - couplings . some time ago the nlo corrections to the higgs self - couplings have been extended by the next - to - next - to - leading order ( nnlo ) corrections of @xmath49 @xcite , where @xmath50 denotes the strong coupling constant . the radiative corrections are large in general , while the nnlo part is of moderate size but reduces the theoretical uncertainties significantly to the few per - cent level . the couplings of the neutral higgs bosons to fermions and gauge bosons depend on the angles @xmath21 and @xmath51 . normalized to the sm higgs couplings , they are listed in table [ tb : hcoup ] . the pseudoscalar particle @xmath13 does not couple to gauge bosons at tree level , and its couplings to down ( up)-type fermions are ( inversely ) proportional to @xmath17 . .[tb : hcoup ] _ mssm higgs couplings to up- and down - type fermions and gauge bosons ( @xmath52 ) relative to the corresponding sm couplings . _ [ cols="<,^,^,^,^",options="header " , ] the final results for the mssm higgs branching ratios have been obtained with the public codes feynhiggs @xcite and hdecay @xcite using the sm input parameters of the previous section . [ fig : mssmnbr ] shows the neutral higgs branching ratios for two values of @xmath42 within the @xmath53 benchmark scenario @xcite that is defined as @xmath54 the kinks visible in these plots are due to the opening of new decay modes according to the sm and susy - particle masses of the final - state particles . [ fig : mssmcbr ] displays the corresponding charged higgs branching ratios within the @xmath53 benchmark scenario @xcite for two values of @xmath42 . the related uncertainties are not shown in the plots . ( 150,640)(0,0 ) ( -270,370.0 ) ( -20,370.0 ) ( -270,170.0 ) ( -20,170.0 ) ( -270,-30.0 ) ( -20,-30.0 ) ( 150,240)(0,0 ) ( -180,-20.0 ) ( 70,-20.0 ) the gluon - fusion mechanism @xcite @xmath55 dominates higgs - boson production at the lhc in the entire relevant higgs mass range . the gluon coupling to the higgs boson in the sm is mediated by triangular top- and bottom - quark loops , see fig . [ fg : gghlodia ] . since the yukawa coupling of the higgs particle grows with the quark mass , the form factor reaches a constant value for large loop quark masses . if the masses of heavier quarks beyond the third generation are fully generated by the higgs mechanism , these particles would add the same amount to the form factor as the top quark in the asymptotic heavy quark limit . thus gluon fusion can serve as a counter of the number of heavy quarks , the masses of which are generated by the conventional higgs mechanism . on the other hand within the three - generation sm gluon fusion will allow to measure the top quark yukawa coupling . this , however , requires a precise knowledge of the cross section within the sm with three generations of quarks . ( 180,100)(0,0 ) ( 0,20)(50,20)-35 ( 0,80)(50,80)35 ( 50,20)(50,80 ) ( 50,80)(100,50 ) ( 100,50)(50,20 ) ( 100,50)(150,50)5 ( 155,46)@xmath12 ( 25,46)@xmath56 ( -15,18)@xmath57 ( -15,78)@xmath57 + the partonic cross section can be derived from the gluonic width of the higgs boson at lowest order @xcite , @xmath58 with the scaling variables defined as @xmath59 , @xmath60 . the variable @xmath61 denotes the partonic c.m . energy squared and @xmath62 the renormalization scale . the amplitudes @xmath63 are given in eq . ( [ eq : ftau ] ) . in the narrow - width approximation the hadronic cross section can be cast into the form @xcite @xmath64 with the gluon luminosity @xmath65 at the factorization scale @xmath66 , and the scaling variable is defined , in analogy to the drell yan process , as @xmath67 , with @xmath68 specifying the total hadronic c.m . energy squared . the bottom - quark contributions interfere destructively with the top loop and decrease the cross section by about 10% at lo . ( 450,100)(-10,0 ) ( 0,20)(30,20)-33 ( 0,80)(30,80)33 ( 30,20)(60,20)-33 ( 30,80)(60,80)33 ( 30,20)(30,80)35 ( 60,20)(60,80 ) ( 60,80)(90,50 ) ( 90,50)(60,20 ) ( 90,50)(120,50)5 ( 125,46)@xmath12 ( 65,46)@xmath56 ( -10,18)@xmath57 ( -10,78)@xmath57 ( 15,48)@xmath57 ( 180,100)(210,100)33 ( 210,100)(270,100)36 ( 180,0)(210,0)-33 ( 210,100)(210,60)34 ( 210,0)(210,60 ) ( 210,60)(240,30 ) ( 240,30)(210,0 ) ( 240,30)(270,30)5 ( 275,26)@xmath12 ( 215,26)@xmath56 ( 165,-2)@xmath57 ( 165,98)@xmath57 ( 195,78)@xmath57 ( 275,98)@xmath57 ( 330,100)(360,100 ) ( 360,100)(420,100 ) ( 330,0)(360,0)-33 ( 360,100)(360,60)34 ( 360,0)(360,60 ) ( 360,60)(390,30 ) ( 390,30)(360,0 ) ( 390,30)(420,30)5 ( 425,26)@xmath12 ( 365,26)@xmath56 ( 315,-2)@xmath57 ( 315,98)@xmath69 ( 425,98)@xmath69 ( 345,78)@xmath57 + [ [ qcd - corrections . ] ] qcd corrections . + + + + + + + + + + + + + + + + in the past the ( two - loop ) nlo qcd corrections to the gluon - fusion cross section , fig . [ fg : gghqcddia ] , have been calculated including the full mass dependences @xcite . they consist of virtual corrections to the basic @xmath2 process and real corrections due to the associated production of the higgs boson with massless partons , @xmath70 these subprocesses contribute to the higgs production at @xmath71 . the virtual corrections rescale the lowest - order fusion cross section with a coefficient depending only on the ratios of the higgs and quark masses . gluon radiation leads to two - parton final states with invariant energy @xmath72 in the @xmath73 and @xmath74 channels at nlo . in general the hadronic cross section can be split into seven parts @xcite , @xmath75 \tau_{h } \frac{d{\cal l}^{gg}}{d\tau_{h } } + \delta \sigma_{gg } + \delta \sigma_{gq } + \delta \sigma_{q\bar{q } } + \delta \sigma_{qq } + \delta \sigma_{qq'}\ ] ] where the finite parts of virtual corrections @xmath76 and the real corrections @xmath77 , @xmath78 and @xmath79 ( same - flavour quark - antiquark initial states ) start to contribute at nlo , while @xmath80 ( same - flavour quark - quark and antiquark - antiquark initial states ) and @xmath81 ( different - flavour quark and antiquark initial states ) appear for the first time at nnlo . the renormalization scale @xmath62 of @xmath50 and the factorization scale @xmath66 of the parton densities are fixed properly , in general at @xmath82 . the quark - loop mass has been identified with the pole mass @xmath83 , while the qcd coupling @xmath50 and the parton density functions are defined in the @xmath31 scheme with five active flavours . we define the nlo @xmath84 factor as the ratio @xmath85 the nlo corrections are positive and large , increasing the gluon - fusion cross section at the lhc by about 6090% . the qcd corrections to the bottom - quark contributions are significantly smaller if the bottom mass is used in terms of the pole mass or the @xmath31 mass at the scale of the bottom mass . this choice is motivated by the numerical cancellation of squared and single logarithms of the relative qcd corrections at nlo @xcite . this feature modifies the destructive bottom - quark contribution to a reduction of the cross section by about @xmath86 at nlo . comparing the exact mass - dependent results with the expressions in the heavy - quark limit , it turns out that this asymptotic @xmath84 factor provides an excellent approximation even for higgs masses above the top - decay threshold . we explicitly define the approximation by @xmath87 where we neglect the @xmath88 quark contribution in @xmath89 , while the leading order cross section @xmath90 includes the full @xmath91 quark mass dependence . the comparison with the full massive nlo result is presented in fig . [ fg : gghapprox ] . the solid line corresponds to the exact cross section and the broken line to the approximate one . for higgs masses below @xmath36 1 tev , the deviations of the asymptotic approximation from the full nlo result are less than 15% , whereas for @xmath92 gev they amount to @xmath93 , if the full lo cross section is multiplied by the approximate k - factor . this property of the nlo corrections suggests this to be true also at higher orders , since it is a consequence of the dominating soft and collinear gluon effects in the qcd corrections . within the heavy top - quark limit the nnlo @xcite and n@xmath94lo @xcite qcd corrections have been calculated . the nnlo contributions increase the production cross section by about 20% beyond nlo , while the n@xmath94lo corrections range at the level of a few per - cent . these results indicate that the gluon - fusion cross section is under theoretical control despite the large size of the nlo corrections . this is further corroborated by the results obtained by a soft and collinear gluon resummation on top of the n@xmath94lo result . this resummation has been performed at the nnll level @xcite and the n@xmath94ll level @xcite in the heavy top - quark limit . quite recently also finite top - mass effects have been included in the resummation framework @xcite at the nll order , where they are known exactly . resummation effects beyond n@xmath94lo yield only a per - cent increase of the cross section for the central scale choice . however , they yield an approximation of effects beyond n@xmath94lo and contribute to a sophisticated estimate of the residual uncertainties . [ [ electroweak - corrections . ] ] electroweak corrections . + + + + + + + + + + + + + + + + + + + + + + + + the electroweak corrections to the gluon - fusion cross section have been computed approximately first . the leading top mass corrections of @xmath95 coincide with the corrections to the gluonic decay mode of eq . ( [ eq : hggelw ] ) and are thus small @xcite . this calculation has been refined by the determination of the nlo electroweak corrections due to light - fermion loops @xcite and finally by the full numerical integration of the exact nlo corrections to the top- and @xmath96-induced electroweak corrections @xcite . the electroweak corrections coincide with the ones to the @xmath97 decay . the nlo electroweak corrections have been extended by a calculation of the mixed qcd - electroweak corrections in the limit @xmath98 @xcite which can be attributed to the corrections of the effective @xmath99 coupling . however , it is unclear how reliable this approximation is in practice . due to the dominance of soft and collinear gluon effects the bulk of the electroweak corrections will factorize from the pure qcd corrections . in the following the electroweak corrections will be combined with the qcd corrections in factorized form . [ [ total - cross - section . ] ] total cross section . + + + + + + + + + + + + + + + + + + + + theoretical uncertainties in the prediction of the higgs cross section originate from three major sources , the dependence of the cross section on the parton densities , the unknown corrections beyond n@xmath94lo and the parametric uncertainties originating from the input value of the strong coupling @xmath50 and to a lesser extent of the top and bottom quark masses . the missing quark - mass effects beyond nlo have been estimated as less than 1% by an explicit large top - mass expansion of the nnlo corrections beyond the heavy top - quark limit @xcite . the total uncertainty of the prediction for the total gluon - fusion cross sections has been estimated as 4% and can dominantly be traced back to the renormalization- and factorization - scale dependence as well as the pdf+@xmath50 uncertainties @xcite . the renormalization- and factorization - scale dependence is depicted in fig . [ fg : gghscale ] for the known perturbative orders in the heavy top - quark limit . a significant reduction from @xmath100 to the few - per - cent level is visible from lo to n@xmath94lo . the results for the cross section are contained in the public codes higlu @xcite up to nnlo , sushi @xcite up to n@xmath94lo as well as ihix @xcite up to n@xmath94lo . [ [ transverse - momentum - distribution . ] ] transverse - momentum distribution . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + apart from the total cross section also distributions are relevant for analyzing the higgs properties . of particular interest in this context is the transverse - momentum distribution of the higgs boson that arises due to the additional radiation of gluons . the lo contributions are part of the real corrections to the gluon - fusion cross section at nlo . the first results were obtained a long time ago @xcite that include the full quark - mass dependence at lo . the nlo qcd corrections were calculated in the heavy top - quark limit @xcite that is expected to be reliable up to @xmath101-values below the @xmath102 threshold as can be inferred from a large top - mass expansion @xcite . recently a more rigorous inclusion of finite top - mass effects at nlo has been studied by using exact results partially @xcite . these calculations are extended by the recent derivation of the nnlo qcd corrections in the heavy - top limit @xcite . these predictions have been matched to a soft - gluon resummed expression that provides a regular prediction for small values of @xmath101 thus removing the spurious singularities in @xmath103 in fixed - order expressions @xcite . the consistent matching between the resummed and fixed order expressions has been performed at nnll+nnlo level @xcite . in addition finite quark - mass effects have been included in the resummed expressions at nll+nlo , too @xcite . the latter step requires to use different matching scales for the pure top - induced contributions and those that involve bottom loops . during the last years the full nlo expressions have been implemented in the powheg box @xcite providing in this way a reliable nlo event generator @xcite . a second matching to parton showers has been performed by combining sushi @xcite and the mc@nlo @xcite framework in the code amcsushi @xcite at nlo accuracy , too . the gluon - fusion mechanism @xcite @xmath104 dominates the neutral mssm higgs boson production at the lhc in the phenomenologically relevant higgs mass ranges for small and moderate values of @xmath17 . only for large @xmath17 can the associated @xmath105 production channel develop a larger cross section due to the enhanced higgs couplings to bottom quarks @xcite . analogous to the gluonic decay modes , the gluon coupling to the neutral higgs bosons in the mssm is built up by loops involving top and bottom quarks as well as squarks , see fig . [ fg : mssmgghlodia ] . ( 180,100)(0,0 ) ( 0,20)(50,20)-35 ( 0,80)(50,80)35 ( 50,20)(50,80 ) ( 50,80)(100,50 ) ( 100,50)(50,20 ) ( 100,50)(150,50)5 ( 155,46)@xmath106 ( 20,46)@xmath107 ( -15,18)@xmath57 ( -15,78)@xmath57 + the partonic cross sections can be obtained from the gluonic widths of the higgs bosons at lowest order @xcite : @xmath108 where the scaling variables are defined as @xmath109 , @xmath110 , and @xmath61 denotes the partonic c.m . energy squared . the amplitudes @xmath111 are defined in eqs . ( [ eq : mssmhgg ] , [ eq : mssmagg ] ) , and the mssm couplings @xmath112 can be found in tables [ tb : hcoup ] and [ tb : hsqcoup ] . in the narrow - width approximation the hadronic cross sections are given by @xmath113 with the gluon luminosity defined in eq . ( [ eq : gglum ] ) and the scaling variables @xmath114 where @xmath68 specifies the total hadronic c.m . energy squared . for small @xmath17 the top loop contribution is dominant , while for large values of @xmath17 the bottom quark contribution is strongly enhanced . if the squark masses are less than @xmath115 gev , their contribution is significant , and for squark masses beyond @xmath116 gev they can safely be neglected @xcite . [ [ qcd - corrections.-1 ] ] qcd corrections . + + + + + + + + + + + + + + + + in the past the full two - loop qcd corrections to the quark and squark loops of the gluon - fusion cross section were calculated @xcite . in complete analogy to the sm case they consist of virtual corrections to the basic @xmath117 process and real corrections due to the associated production of the higgs bosons with massless partons . thus the contributions to the final result for the hadronic cross section can in complete analogy to the sm case be classified as @xmath118 \tau_\phi \frac{d{\cal l}^{gg}}{d\tau_\phi } + \delta\sigma^\phi_{gg } + \delta\sigma^\phi_{gq } + \delta\sigma^\phi_{q\bar{q } } + \delta\sigma^\phi_{qq } + \delta\sigma^\phi_{qq ' } \label{eq : mssmgghqcd5}\ ] ] where @xmath119 and @xmath120 start to contribute at nnlo . the analytic nlo expressions for arbitrary higgs boson and quark masses are rather involved and can be found in @xcite . as in the sm case the ( s)quark - loop masses have been identified with the pole masses @xmath121 , while the qcd coupling and pdfs of the proton are defined in the @xmath122 scheme with five active flavours . the axial @xmath123 coupling can be regularized in the t hooft veltman scheme @xcite or its extension by larin @xcite , which preserves the chiral symmetry in the massless quark limit by the addition of supplementary counter terms and fulfills the non - renormalization theorem @xcite of the abj anomaly @xcite at vanishing momentum transfer . the same result can also be obtained with the scheme of ref . @xcite that gives up the cyclicity of the traces involving clifford matrices . similar to the sm the leading terms in the heavy - quark limit provide a reliable approximation for small @xmath17 up to higgs masses of @xmath124 tev with a maximal deviation of @xmath125 for @xmath126 $ \sim$}}~}5 $ ] in the intermediate mass range . the squark contributions in the heavy - squark limit coincide with the heavy - quark case apart from the mismatch related to the effective lagrangians of eqs . ( [ eq : leff ] , [ eq : leffsq ] ) . the genuine susy qcd corrections are the same as for the gluonic decay width as discussed in section [ sc : hgg ] . they are moderate for small values of @xmath42 @xcite , but can be large for large @xmath42 due to the contribution of the @xmath127 terms and a sizeable remainder beyond these approximate contributions @xcite . in the opposite limit , where the higgs mass is much larger than the quark mass , the analytic results coincide with the sm expressions for both the scalar and pseudoscalar higgs particles up to nlo at which the results for small quark masses are known @xcite . this coincidence reflects the restoration of the chiral symmetry in the massless quark limit . the leading logarithmic structure is the same as in the sm case , see eq . ( [ eq : hgglog ] ) . the non - abelian part of these logarithms has not been resummed so that the uncertainties for large values of @xmath42 are sizeable due to the dominance of the bottom - quark loops involving the large logarithms . the qcd corrections are positive and large , increasing the mssm higgs production cross sections at the lhc by up to about 100% . the effect of the pure squark loops ( without gluino exchange ) on the scalar higgs @xmath84 factors is of moderate size and lead to a maximal modification by about 10% . squark mass effects on top of the lo ones can amount to @xmath128 for large higgs masses @xcite . for the top - loop contributions alone the n@xmath94lo corrections can be used consistently . susy - electroweak corrections are unknown so far . [ [ transverse - momentum - distribution.-1 ] ] transverse - momentum distribution . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the results of the powheg implementation of the full lo matrix elements for the transverse - momentum distribution of ref . @xcite and amcsushi @xcite can also be used for the mssm , since they also include variable top and bottom yukawa couplings . the code of ref . @xcite includes squark - loop contributions in addition . a rigorous comparison between the powheg implementation and analytical resummation approaches @xcite as well as amcsushi has been performed @xcite . this comparison addressed in particular different approaches for the setting of the resummation scale of the bottom - loop contributions . in scenarios where the top loops provide the dominant contribution the differences between the codes are in the range of up to about 50% , while for scenarios with bottom - loop dominance they are larger and can reach 100% @xcite . these differences will only be reduced by calculating the nlo corrections to the transverse - momentum distribution including the full mass dependence . ( 120,110)(0,0 ) ( 0,0)(50,0 ) ( 50,0)(100,0 ) ( 0,100)(50,100 ) ( 50,100)(100,100 ) ( 50,0)(50,50)35 ( 50,50)(50,100)35 ( 50,50)(100,50)5 ( 105,46)@xmath12 ( -15,-2)@xmath69 ( -15,98)@xmath69 ( 55,21)@xmath96 ( 55,71)@xmath96 + at the lhc the second important higgs production channel is the vector - boson - fusion ( vbf ) mechanism ( see fig . [ fg : vvhlodia ] ) @xcite . for intermediate higgs masses the vector - boson - fusion cross section is about one order of magnitude smaller than the gluon - fusion one . the cross section can be approximated by the @xmath130-channel diagrams of the type shown in fig . [ fg : vvhlodia ] within @xmath131 accuracy , i.e. without any colour - cross talk between the quark lines , while interference effects for identical quark flavours and @xmath68-channel contributions are at the per - cent level after subtracting the corresponding higgs - strahlung component from the @xmath68-channel contributions @xcite . within the structure - function approach the leading order partonic vector - boson - fusion cross section @xcite can be cast into the form ( @xmath132 ) : @xmath133 ^ 2 [ q_2 ^ 2-m_v^2]^2 } \nonumber \\ & & \left\ { f_1(x_1,\mu_f^2 ) f_1(x_2,\mu_f^2 ) \left [ 2+\frac{(q_1 q_2)^2}{q_1 ^ 2q_2 ^ 2 } \right ] \right . \nonumber \\ & & + \frac{f_1(x_1,\mu_f^2)f_2(x_2,\mu_f^2)}{p_2q_2 } \left[\frac{(p_2q_2)^2}{q_2 ^ 2}- m_p^2+\frac{1}{q_1 ^ 2}\left(p_2q_1-\frac{p_2q_2}{q_2 ^ 2}q_1q_2\right)^2 \right ] \nonumber \\ & & + \frac{f_2(x_1,\mu_f^2)f_1(x_2,\mu_f^2)}{p_1q_1 } \left[\frac{(p_1q_1)^2}{q_1 ^ 2}- m_p^2+\frac{1}{q_2 ^ 2}\left(p_1q_2-\frac{p_1q_1}{q_1 ^ 2}q_1q_2\right)^2 \right ] \nonumber \\ & & + \frac{f_2(x_1,\mu_f^2)f_2(x_2,\mu_f^2)}{(p_1q_1)(p_2q_2 ) } \left[p_1p_2 - \frac{(p_1q_1)(p_2q_1)}{q_1 ^ 2 } - \frac{(p_2q_2)(p_1q_2)}{q_2 ^ 2 } \right . \nonumber \\ & & \left . \hspace*{4.5cm}+\frac{(p_1q_1)(p_2q_2)(q_1q_2)}{q_1 ^ 2q_2 ^ 2 } \right]^2 \nonumber \\ & & \left . + \frac{f_3(x_1,\mu_f^2)f_3(x_2,\mu_f^2)}{2(p_1q_1)(p_2q_2 ) } \left [ ( p_1p_2)(q_1q_2 ) - ( p_1q_2)(p_2q_1 ) \right ] \right\ } dx_1 dx_2 \frac{dp\!s_3}{\hat s } \label{eq : vvhlo}\end{aligned}\ ] ] where @xmath134 denotes the three - particle phase space of the final - state particles , @xmath135 the proton mass , @xmath136 the proton momenta and @xmath137 the momenta of the virtual vector bosons @xmath138 . the functions @xmath139 are the usual structure functions from deep - inelastic scattering processes at the factorization scale @xmath66 : @xmath140 \nonumber \\ f_2(x,\mu_f^2 ) & = & 2x \sum_q ( v_q^2+a_q^2 ) [ q(x,\mu_f^2 ) + \bar q(x,\mu_f^2 ) ] \nonumber \\ f_3(x,\mu_f^2 ) & = & 4 \sum_q v_qa_q [ -q(x,\mu_f^2 ) + \bar q(x,\mu_f^2 ) ] \label{eq : stfu}\end{aligned}\ ] ] where @xmath141 are the ( axial ) vector couplings of the quarks @xmath69 to the vector bosons @xmath142 : @xmath143 for @xmath144 and @xmath145 , @xmath146 for @xmath147 . @xmath148 is the third weak isospin component and @xmath149 the electric charge of the quark @xmath69 . in the past the nlo qcd corrections have been calculated first within the structure function approach @xcite . since , at lowest order , the proton remnants are color singlets , no color will be exchanged between the first and the second incoming ( outgoing ) quark line and hence the qcd corrections only consist of the well - known corrections to the structure functions @xmath139 . the final result for the nlo qcd - corrected cross section leads to the replacements @xmath150 \right . \nonumber \\ & & \left [ -\frac{3}{4 } p_{qq}(z ) \log \frac{\mu_f^2z}{q^2 } + ( 1+z^2 ) { \cal d}_1(z ) - \frac{3}{2 } { \cal d}_0(z ) \right . \nonumber \\ & & \left . \hspace*{6 cm } + 3 - \left ( \frac{9}{2 } + \frac{\pi^2}{3 } \right ) \delta(1-z ) \right ] \nonumber \\ & & \left . + \frac{1}{4 } g(y,\mu_f^2 ) \left [ -2 p_{qg}(z ) \log \frac{\mu_f^2z}{q^2(1-z ) } + 4z(1-z ) - 1 \right ] \right\ } \\ \delta f_2(x,\mu_f^2,q^2 ) & = & 2x\frac{\alpha_s(\mu_r)}{\pi}\sum_q ( v_q^2+a_q^2 ) \int_x^1 \frac{dy}{y } \left\ { \frac{2}{3 } [ q(y,\mu_f^2 ) + \bar q(y,\mu_f^2 ) ] \right . \nonumber \\ & & \left [ -\frac{3}{4 } p_{qq}(z ) \log \frac{\mu_f^2z}{q^2 } + ( 1+z^2 ) { \cal d}_1(z ) - \frac{3}{2 } { \cal d}_0(z ) \right . \nonumber \\ & & \left . \hspace*{3.0 cm } + 3 + 2z - \left ( \frac{9}{2 } + \frac{\pi^2}{3 } \right ) \delta(1-z ) \right ] \nonumber \\ & & \left . + \frac{1}{4 } g(y,\mu_f^2 ) \left [ -2p_{qg}(z ) \log \frac{\mu_f^2z}{q^2(1-z ) } + 8z(1-z ) - 1 \right ] \right\ } \\ \delta f_3(x,\mu_f^2,q^2 ) & = & \frac{\alpha_s(\mu_r)}{\pi } \sum_q 4 v_q a_q \int_x^1 \frac{dy}{y } \left\ { \frac{2}{3 } [ -q(y,\mu_f^2 ) + \bar q(y,\mu_f^2 ) ] \right . \nonumber \\ & & \left [ -\frac{3}{4 } p_{qq}(z ) \log \frac{\mu_f^2z}{q^2 } + ( 1+z^2 ) { \cal d}_1(z ) - \frac{3}{2 } { \cal d}_0(z ) \right . \nonumber \\ & & \left . \left . \hspace*{3 cm } + 2 + z - \left ( \frac{9}{2 } + \frac{\pi^2}{3 } \right ) \delta(1-z ) \right ] \right\}\end{aligned}\ ] ] where @xmath151 and the functions @xmath152 denote the altarelli parisi splitting functions , which are given by @xcite @xmath153 the plus distributions are defined as @xmath154 the physical scale @xmath155 is given by @xmath156 for @xmath157 . these expressions have to be inserted in eq . ( [ eq : vvhlo ] ) and the full result expanded up to nlo . the typical renormalization and factorization scales are fixed by the vector - boson momentum transfer @xmath158 . the size of the qcd corrections amounts to about 10% and is thus small @xcite as can be inferred from fig . [ fg : vbfcorr ] that displays the individual corrections to the cross section @xcite . these results have been extended by the calculation of the full nlo qcd corrections including interference contributions between the @xmath159-channels thus confirming the smallness of the additional contributions at nlo , too @xcite . the nlo electroweak corrections reduce the cross section by about 10% @xcite so that there is a significant interplay between the qcd and electroweak corrections , if the latter are defined in the @xmath29-scheme , see fig . [ fg : vbfcorr ] . for a reliable prediction of the vector - boson - fusion cross section both corrections have to be taken into account . the nlo qcd and electroweak corrections are also known exclusively and have been implemented in the public monte carlo programs hawk @xcite and vbfnlo @xcite . the full nlo results can also be generated with the mg5_amc@nlo framework @xcite and are available within the powheg box @xcite so that a full matching to parton showers can be used . the nnlo qcd corrections have been obtained in the structure - function approach giving rise to a per - cent effect on the total cross section @xcite , see fig . [ fg : vbfcorr ] . these are implemented in the public codes vbf@nnlo @xcite and provbf @xcite with the latter providing exclusive results . up to nnlo non - factorizing contributions have been shown to be small @xcite . the theoretical calculations have been extended to n@xmath94lo very recently @xcite . the n@xmath94lo corrections are tiny , but reduce the theoretical uncertainties significantly . ( 150,280)(0,0 ) ( -170,-10.0 ) ( 100,-10.0 ) ( 120,120)(0,0 ) ( 0,0)(50,0 ) ( 50,0)(100,0 ) ( 0,100)(50,100 ) ( 50,100)(100,100 ) ( 50,0)(50,50)35 ( 50,50)(50,100)35 ( 50,50)(100,50)5 ( 105,46)@xmath160 ( -15,-2)@xmath69 ( -15,98)@xmath69 ( 55,21)@xmath96 ( 55,71)@xmath96 + due to the absence of vector boson couplings to pseudoscalar higgs particles @xmath13 , only the scalar higgs bosons @xmath160 can be produced via the vector - boson - fusion mechanism at tree level ( see fig . [ fg : mssmvvhlodia ] ) . however , these processes are suppressed with respect to the sm cross section due to the mssm couplings ( @xmath161 ) , @xmath162 it turns out that the vector - boson - fusion mechanism is less relevant in the mssm , because for large heavy scalar higgs masses @xmath163 , the mssm couplings @xmath164 are very small apart from the decoupling regime where the light scalar higgs boson becomes sm - like . the relative pure qcd corrections are the same as for the sm higgs particle , i.e. @xmath165 @xcite . the genuine susy qcd corrections amount to less than a per cent and are thus small @xcite . some years ago the susy electroweak corrections have been determined @xcite . being moderate in size in general they can modify the vbf cross section by @xmath165 . ( 160,120)(0,-10 ) ( 0,100)(50,50 ) ( 50,50)(0,0 ) ( 50,50)(100,50)35 ( 100,50)(150,100)36 ( 100,50)(150,0)5 ( 155,-4)@xmath12 ( -15,-2)@xmath167 ( -15,98)@xmath69 ( 65,65)@xmath96 ( 155,96)@xmath96 + the higgs - strahlung mechanism @xmath168 ( see fig . [ fg : vhvlodia ] ) is important in the intermediate higgs mass range due to the possibility to tag the associated vector boson and to reconstruct the @xmath169 decay for strongly boosted higgs bosons by using jet - substructure techniques @xcite . its cross section is about one to two orders of magnitude smaller than the gluon - fusion cross section for the relevant higgs mass range . the partonic cross section can be expressed at lowest - order as @xcite @xmath170 where @xmath171 denotes the usual two - body phase - space factor and @xmath141 are the ( axial ) vector couplings of the quarks @xmath69 to the vector bosons @xmath142 , which have been defined after eq . ( [ eq : stfu ] ) . at leading order the partonic c.m . energy squared @xmath172 coincides with the invariant mass @xmath173 of the higgs vector - boson pair squared , @xmath174 . the hadronic cross section can be obtained from convolving eq . ( [ eq : vhvpart ] ) with the corresponding @xmath175-luminosity , @xmath176 with @xmath177 and @xmath68 the total hadronic c.m . energy squared and @xmath178\ ] ] the nlo qcd corrections are identical to the corresponding corrections to the drell yan process @xcite . the natural scale of the process is given by the invariant mass of the higgs vector - boson pair in the final state , @xmath158 . the nlo qcd corrections increase the total cross section by about 30% and are thus moderate @xcite , see fig . [ fg : vhcorr ] . some time ago the nnlo qcd corrections have been determined @xcite . for the @xmath179 final state there is a sizeable contribution from the loop - induced @xmath180 process , see fig . [ fg : ggzhdia ] . it contributes @xmath128 to the total cross section as can be inferred from fig . [ fg : vhcorr ] , while the rest of the nnlo corrections amounts to about 5% . recently the nlo qcd corrections to @xmath180 have been calculated in the heavy - top - quark limit @xcite . they increase this contribution significantly . the calculation of the qcd corrections has been supplemented by the full nlo electroweak corrections @xcite . within the @xmath29-scheme they decrease the cross section by about 10% and are thus of the same relevance as the nnlo qcd corrections , see fig . [ fg : vhcorr ] . the nlo qcd and electroweak corrections are implemented in the public program hawk @xcite including leptonic decays of the final - state vector boson . the nnlo qcd corrections are implemented in the public codes hvnnlo @xcite that includes final - state higgs decays into leptons and bottom quarks , mcfm @xcite , vhnnlo @xcite , vh@nnlo @xcite and nnlops @xcite . the nlo results have been matched to parton showers within the powheg box @xcite and the mg5_amc@nlo framework @xcite . ( 150,280)(0,0 ) ( -170,-10.0 ) ( 100,-10.0 ) ( 180,90)(40,0 ) ( 0,20)(50,20)-35 ( 0,80)(50,80)35 ( 50,20)(50,80 ) ( 50,80)(100,50 ) ( 100,50)(50,20 ) ( 100,50)(150,50)35 ( 150,50)(200,80)35 ( 150,50)(200,20)5 ( 205,76)@xmath15 ( 205,16)@xmath12 ( 25,46)@xmath56 ( -15,18)@xmath57 ( -15,78)@xmath57 ( 180,90)(-40,0 ) ( 0,20)(50,20)-35 ( 0,80)(50,80)35 ( 50,20)(50,80 ) ( 50,80)(100,80 ) ( 100,80)(100,20 ) ( 100,20)(50,20 ) ( 100,80)(150,80)35 ( 100,20)(150,20)5 ( 155,76)@xmath15 ( 155,16)@xmath12 ( 25,46)@xmath56 ( -15,18)@xmath57 ( -15,78)@xmath57 + ( 160,120)(0,-10 ) ( 0,100)(50,50 ) ( 50,50)(0,0 ) ( 50,50)(100,50)35 ( 100,50)(150,100)36 ( 100,50)(150,0)5 ( 155,-4)@xmath160 ( -15,-2)@xmath167 ( -15,98)@xmath69 ( 65,65)@xmath96 ( 155,96)@xmath96 + for the same reasons as in the vector - boson - fusion mechanism case , the higgs - strahlung off @xmath96 bosons , @xmath181 ( see fig . [ fg : mssmvhvlodia ] ) , is smaller for the scalar mssm higgs particles @xmath160 than for the sm higgs boson apart from the decoupling regime for the light scalar @xmath11 . the cross sections can be easily related to the sm cross sections , @xmath182 which is true up to nnlo qcd apart from the loop - induced @xmath183 process that involves the large top yukawa coupling in addition to the @xmath184 coupling , see fig . [ fg : ggzhdia ] . pseudoscalar couplings to intermediate vector bosons are absent so that pseudoscalar higgs particles can not be produced at tree level in this channel . the relative qcd corrections are the same as in the sm case and thus of moderate size @xcite in general up to nnlo . only the contamination from @xmath183 will be weighted differently due to the top yukawa coupling factor . its contribution ranges at about 20% . the genuine susy qcd corrections are small @xcite , while the genuine susy - electroweak corrections are unknown so far . ( 360,120)(0,-10 ) ( 0,100)(50,50 ) ( 50,50)(0,0 ) ( 50,50)(100,50)35 ( 100,50)(125,75 ) ( 125,75)(150,100 ) ( 150,0)(100,50 ) ( 125,75)(150,50)5 ( 155,46)@xmath12 ( -15,98)@xmath69 ( -15,-2)@xmath167 ( 65,65)@xmath57 ( 155,98)@xmath130 ( 155,-2)@xmath185 ( 250,0)(300,0)35 ( 250,100)(300,100)35 ( 350,0)(300,0 ) ( 300,0)(300,50 ) ( 300,50)(300,100 ) ( 300,100)(350,100 ) ( 300,50)(350,50)5 ( 355,46)@xmath12 ( 235,98)@xmath57 ( 235,-2)@xmath57 ( 355,98)@xmath130 ( 355,-2)@xmath185 + [ [ tbar - th - production . ] ] @xmath186 production . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + in the intermediate higgs mass range the cross section of the associated production of the higgs boson with a @xmath102 pair is smaller than those of the higgs - strahlung processes @xcite , but of significant size , since it allows to tag the additional @xmath102 pair and the higgs boson decays @xmath169 , @xmath187 as well as the rare photonic decay mode @xmath188 on the long time run . this process is generated by gluon gluon and @xmath175 initial states at leading order ( see fig . [ fg : httlodia ] ) . at the lhc the gluon gluon channel is dominant due to the enhanced gluon structure function analogous to the gluon - fusion mechanism . the nlo qcd corrections to @xmath186 production have been calculated and result in a moderate increase of the cross section by @xmath128 @xcite . the origin of this moderate size is the strong phase - space suppression of the massive three - particle threshold so that soft and collinear threshold effects are strongly diminished . the main parts of the qcd corrections originate from regions significantly above the production threshold and can be approximated by a fragmentation approach involving first producing a @xmath102 pair supplemented by the @xmath189 fragmentation in the high - energy limit @xcite . although this provides a bad approximation for the magnitude of the cross section itself it leads to a reasonable estimate of the relative qcd corrections @xcite . the full nlo results have recently been implemented in the powheg box @xcite , matched to sherpa @xcite and generated within the mg5_amc@nlo framework @xcite thus offering nlo event generators matched to parton showers . the nlo result has recently been improved by a soft and collinear gluon resummation based on the scet approach starting from the boosted final - state particle triplet @xcite leading to a further increase of the cross section by 5 - 10% . the residual scale dependence is reduced to the level of @xmath190 . recently the electroweak corrections have been calculated for @xmath186 production @xcite . they range a the per - cent level and are thus small . moreover , off - shell top - quark effects have been determined at nlo in qcd @xcite with leptonic top - quark decays and turn out to be small for the inclusive @xmath186 cross section . however , they play a role in certain regions of phase space and are thus of relevance for distributions . [ [ bbar - bh - production . ] ] @xmath191 production . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + higgs bremsstrahlung off bottom quarks does not play a significant role for the sm higgs boson , but yields an important constraint on the bottom yukawa coupling . its total cross section is of similar size as the @xmath186 production cross section . the results of @xmath186 production can be taken over for @xmath191 production . however , they have to be transformed to the four - flavour - scheme ( 4fs ) in order to avoid artificial large logarithms initiated by the bottom mass in the combination of the virtual and real corrections at nlo . in this way finite bottom - mass effects can be taken into account consistently . the nlo qcd corrections are positive and large . there is a decrease by about 10% due to top - yukawa induced contributions at nlo corresponding to the diagrams of fig . [ fg : bbhyt ] @xcite . ( 180,120)(40,-10 ) ( 0,0)(50,0)-35 ( 0,80)(50,80)35 ( 50,80)(80,80)33 ( 50,80)(50,60)32 ( 80,80)(100,100 ) ( 100,60)(80,80 ) ( 50,0)(50,60 ) ( 50,60)(100,30 ) ( 100,30)(50,0 ) ( 100,30)(150,30)5 ( 155,26)@xmath12 ( 105,96)@xmath88 ( 105,56)@xmath192 ( 40,36)@xmath130 ( -15,-2)@xmath57 ( -15,78)@xmath57 ( 180,120)(-40,-10 ) ( 0,20)(50,20)-35 ( 0,80)(50,80)35 ( 50,20)(50,80 ) ( 50,80)(100,80 ) ( 100,80)(100,20 ) ( 100,20)(50,20 ) ( 100,80)(130,80)33 ( 130,80)(150,100 ) ( 150,60)(130,80 ) ( 100,20)(150,20)5 ( 155,16)@xmath12 ( 155,96)@xmath88 ( 155,56)@xmath192 ( 40,46)@xmath130 ( -15,18)@xmath57 ( -15,78)@xmath57 + however , the @xmath101-integration of the final - state bottom quarks generates potentially large logarithmic contributions as part of the 4fs result , @xmath193 where the effective bottom densities are given by @xmath194 with a scale @xmath155 significantly smaller than the higgs boson mass as can be inferred from the @xmath101 distribution of the bottom quark in the final state @xcite . the logarithmic contributions contained in @xmath195 can be resummed by the dlglap evolution @xcite of the initial - state bottom pdfs . this requires to treat the bottom quark as a massless active flavour , i.e. to introduce the 5fs with bottom densities of the proton . ( 100,110)(-20,0 ) ( 0,100)(50,50 ) ( 50,50)(0,0 ) ( 50,50)(100,50)5 ( 105,46)@xmath12 ( -15,98)@xmath88 ( -15,-2)@xmath192 + the 5fs - calculation starts from the process @xmath196 at lo , see fig . [ fg : bb2hdia ] , yielding the lo cross section @xmath197 with the variable @xmath198 involving the hadronic c.m . energy squared @xmath68 and the parton luminosity @xmath199\ ] ] the 5fs relies on three approximations at lo , i.e. neglecting bottom - mass and off - shell effects and the transverse momenta of the initial - state bottom quarks . the last two conditions are resolved by adding the higher - order qcd corrections so that only the inclusion of bottom mass effects is not possible in the 5fs framework . in the small - mass limit the 4fs- and 5fs - calculations have to approach each other at higher orders so that the comparison of both schemes yields an estimate of missing higher - order corrections . using the running @xmath31 bottom mass @xmath200 the nlo qcd corrections to @xmath196 have been calculated @xcite , @xmath201 with @xmath202 , the coefficient functions @xmath203 + 1-z \right\ } \nonumber \\ \omega_{bg}(z ) & = & -\frac{1}{2 } p_{qg}(z ) \log \left ( \frac{\mu_f^2}{(1-z)^2 \tau s } \right ) - \frac{1}{8}(1-z)(3 - 7z)\end{aligned}\ ] ] and the parton luminosity @xmath204\ ] ] the calculation of the qcd corrections to @xmath196 has been extended to nnlo @xcite . the qcd corrections decrease the cross section by about 30% for the central scale choice equal to the higgs mass . at nnlo the @xmath205-initiated diagrams of the 4fs ( see fig . [ fg : httlodia ] ) contribute for the first time so that the kinematics of the spectator bottom quarks is restored . for a better comparison of both schemes a general analysis of the logarithms related to the @xmath101-integrated bottom quarks has been performed with the result that the proper factorization scale to be chosen for the bottom densities is significantly smaller than the higgs boson mass within the 5fs @xcite . this analysis has been extended to all orders for the logarithmic terms and confirms the smaller effective factorization scale in the 5fs @xcite . this decreases the 5fs result and makes it agree better with the nlo 4fs result . in order to combine both calculations within the 4fs and 5fs one has to avoid double counting of common contributions according to eq . ( [ eq : beff ] ) . this has been done empirically in the past by using the santander matching @xcite @xmath206 where the chosen weight @xmath207 pays attention to the fact that the common terms of both calculations are logarithmic . [ fg : bbhcomp ] shows the comparison of the nlo 4fs result with the nnlo 5fs and santander - matched cross section as a function of the higgs mass . this empirically matched result has been improved by two different procedures of a consistent matching of the 4fs and 5fs , one dubbed fonll @xcite and the other nlo+nnll@xmath208+ybyt @xcite . both approaches reexpress the quantities of one scheme in terms of the other one and use suitably modified sets of the pdf4lhc15 parton density functions that treat the particular region at the input scale of the size of the bottom mass consistent with the matching procedures @xcite . the perturbative counting within both approaches is different so that a comparison of both approaches is motivated to obtain an idea of their differences . both results are shown together with the santander - matched result in fig . [ fg : bbhmatch ] as a function of the higgs mass . both consistent matching procedures agree well within their respective uncertainties and come out slightly higher than the cross sections obtained by santander matching by about 20% . both consistently matched results indicate a tendency towards the original 5fs calculation for larger higgs boson masses . [ [ th - production . ] ] @xmath209 production . + + + + + + + + + + + + + + + + + + + + + + + + + + + + ( 100,100)(100,0 ) ( 0,80)(50,80 ) ( 0,20)(50,20)35 ( 50,80)(100,110 ) ( 50,80)(50,50)33 ( 50,20)(50,50 ) ( 100,20)(50,20 ) ( 50,50)(100,50 ) ( 100,50)(150,20 ) ( 100,50)(150,80)5 ( -60,48)@xmath210 ( 155,76)@xmath12 ( 155,18)@xmath130 ( -15,78)@xmath69 ( -15,18)@xmath57 ( 105,108)@xmath211 ( 55,65)@xmath7 ( 75,55)@xmath130 ( 105,18)@xmath192 ( 100,100)(-30,0 ) ( 0,80)(50,80 ) ( 0,20)(50,20)35 ( 50,80)(100,110 ) ( 50,80)(50,40)34 ( 50,20)(50,40 ) ( 100,20)(50,20 ) ( 50,40)(100,40 ) ( 50,60)(100,60)5 ( 105,56)@xmath12 ( 105,38)@xmath130 ( -15,78)@xmath69 ( -15,18)@xmath57 ( 105,108)@xmath211 ( 55,65)@xmath7 ( 105,18)@xmath192 + ( 100,100)(100,0 ) ( 0,80)(50,80 ) ( 0,20)(50,20 ) ( 50,80)(100,80 ) ( 50,80)(50,20)35 ( 50,20)(100,20 ) ( 100,20)(150,0 ) ( 100,20)(150,40)5 ( -60,48)@xmath212 ( 155,36)@xmath12 ( 155,-2)@xmath130 ( -15,78)@xmath69 ( -15,18)@xmath88 ( 105,78)@xmath211 ( 55,46)@xmath7 ( 75,25)@xmath130 ( 100,100)(-30,0 ) ( 0,80)(50,80 ) ( 0,20)(50,20 ) ( 50,80)(100,80 ) ( 50,80)(50,20)35 ( 50,20)(100,20 ) ( 50,50)(100,50)5 ( 105,46)@xmath12 ( 105,18)@xmath130 ( -15,78)@xmath69 ( -15,18)@xmath88 ( 105,78)@xmath211 ( 55,61)@xmath7 + ( 100,120)(100,0 ) ( 0,80)(50,50 ) ( 50,50)(0,20 ) ( 50,50)(100,50)35 ( 150,80)(100,50 ) ( 100,50)(125,35 ) ( 125,35)(150,20 ) ( 125,35)(150,50)5 ( -60,48)@xmath213 ( 155,46)@xmath12 ( 155,18)@xmath130 ( -15,78)@xmath69 ( -15,18)@xmath167 ( 155,78)@xmath192 ( 70,60)@xmath7 ( 100,100)(-30,0 ) ( 0,80)(50,50 ) ( 50,50)(0,20 ) ( 50,50)(100,50)35 ( 100,50)(150,80)5 ( 100,50)(125,35)33 ( 125,35)(150,20 ) ( 150,50)(125,35 ) ( 155,76)@xmath12 ( 155,18)@xmath130 ( -15,78)@xmath69 ( -15,18)@xmath167 ( 155,48)@xmath192 ( 70,60)@xmath7 + single - top quark production in association with a higgs boson proceeds in two different generic ways at lo , the dominant @xmath130-channel contribution and the subleading @xmath68-channel one , see fig . [ fg : thdia ] . it allows to test the sign of the top yukawa coupling , since there are interference effects between the higgs - boson couplings to top quarks and @xmath7 bosons . the @xmath130-channel suffers from the same problem as @xmath191 production , i.e. there is the alternative to treat it within the 4fs or 5fs the latter starting with bottom densities of the proton the dglap evolution of which resum potentially large logarithms arising in the 4fs . within the 5fs there is no interference between the @xmath130- and @xmath68-channel contributions up to nlo . the full nlo calculation has been performed by means of the mg5_amc@nlo framework @xcite . the qcd corrections to the @xmath130-channel contribution are small , i.e. @xmath165 , and of a similar size as the difference between the 4fs and 5fs at nlo . in order to reduce the difference between the 4fs and 5fs the central scale choice has been reduced to @xmath214 for the @xmath130-channel contributions , while the @xmath68-channel uses a central scale twice as large @xcite . the theoretical uncertainties , estimated from the scale dependence and the difference between both schemes , range at the level of 515% @xcite . including the pdf+@xmath50 uncertainties the total uncertainty can be estimated to be in the same ball park . [ [ tbar - tbbar - b - phi - production . ] ] @xmath215 production . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ( 360,120)(0,-10 ) ( 0,100)(50,50 ) ( 50,50)(0,0 ) ( 50,50)(100,50)35 ( 100,50)(125,75 ) ( 125,75)(150,100 ) ( 150,0)(100,50 ) ( 125,75)(150,50)5 ( 155,46)@xmath106 ( -15,98)@xmath69 ( -15,-2)@xmath167 ( 65,65)@xmath57 ( 155,98)@xmath216 ( 155,-2)@xmath217 ( 250,0)(300,0)35 ( 250,100)(300,100)35 ( 350,0)(300,0 ) ( 300,0)(300,50 ) ( 300,50)(300,100 ) ( 300,100)(350,100 ) ( 300,50)(350,50)5 ( 355,46)@xmath106 ( 235,98)@xmath57 ( 235,-2)@xmath57 ( 355,98)@xmath216 ( 355,-2)@xmath217 + the scalar higgs cross sections for higgs bremsstrahlung off heavy quarks @xmath155 can simply be related to the sm case at lo : @xmath218 this relation is also valid at nlo apart from diagrams involving closed top / bottom loops coupling to the higgs bosons , see fig . [ fg : bbhyt ] . the lo expressions for the pseudoscalar higgs boson @xcite are similarly involved as the scalar case and will not be presented here . the top quark coupling to mssm higgs bosons is suppressed with respect to the sm for @xmath219 . therefore higgs bremsstrahlung off top quarks @xmath220 is less important for mssm higgs particles . on the other hand higgs bremsstrahlung off bottom quarks @xmath221 will be the dominant higgs production channel for large @xmath17 due to the strongly enhanced bottom quark yukawa couplings @xcite . the qcd corrections for the scalar higgs bosons can be inferred from the sm cross section by eq . ( [ eq : tthmssm ] ) but with the properly rescaled diagrams of fig . [ fg : bbhyt ] . the nlo qcd corrections to pseudoscalar @xmath222 production have been calculated in ref . @xcite yielding a slightly larger but moderate increase of the cross section . the full susy - qcd corrections have been calculated quite recently for @xmath223 production @xcite and for @xmath224 production in the 4fs @xcite and the 5fs @xcite . being of moderate size for small values of @xmath42 , i.e. similar to the pure qcd corrections , they can be large for @xmath224 production due to the @xmath127 corrections to the bottom yukawa couplings . however , the remainder after absorbing the leading terms in the effective resummed yukawa couplings turns out to be small @xcite . the susy - electroweak corrections are known for @xmath225 production within the 5fs @xcite . the latter can be well approximated by the radiatively improved higgs masses and couplings and the electroweak @xmath127 terms for large values of @xmath42 . [ [ charged - higgs - production . ] ] charged higgs production . + + + + + + + + + + + + + + + + + + + + + + + + + ( 130,110)(300,0 ) ( 250,0)(300,0)35 ( 250,100)(300,100)35 ( 350,0)(300,0 ) ( 300,0)(300,50 ) ( 300,50)(300,100 ) ( 300,100)(350,100 ) ( 300,50)(350,50)5 ( 355,46)@xmath226 ( 235,98)@xmath57 ( 235,-2)@xmath57 ( 355,98)@xmath130 ( 355,-2)@xmath192 ( 295,-30)_(a ) _ ( 130,110)(-20,0 ) ( 0,100)(50,50 ) ( 50,50)(0,0 ) ( 50,50)(100,50)35 ( 100,50)(150,100)5 ( 100,50)(150,0)5 ( 155,96)@xmath227 ( 155,-4)@xmath226 ( -15,-2)@xmath167 ( -15,98)@xmath69 ( 60,60)@xmath228 ( 75,-30)_(b ) _ + ( 130,110)(70,0 ) ( 10,20)(50,20)-34 ( 10,80)(50,80)34 ( 50,20)(50,80 ) ( 50,80)(90,80 ) ( 90,80)(90,20 ) ( 90,20)(50,20 ) ( 90,80)(130,80)5 ( 90,20)(130,20)5 ( 0,18)@xmath57 ( 0,78)@xmath57 ( 55,85)@xmath229 ( 135,76)@xmath227 ( 135,16)@xmath226 ( 60,-10)_(c ) _ ( 130,110)(-30,0 ) ( 10,20)(50,20)-34 ( 10,80)(50,80)34 ( 50,20)(50,80 ) ( 50,80)(90,80 ) ( 90,80)(90,20 ) ( 90,20)(50,20 ) ( 90,80)(130,80)34 ( 90,20)(130,20)5 ( 0,18)@xmath57 ( 0,78)@xmath57 ( 55,85)@xmath229 ( 135,76)@xmath230 ( 135,16)@xmath226 ( 60,-10)_(d ) _ + ( 130,110)(50,0 ) ( 0,80)(50,80 ) ( 50,80)(50,20 ) ( 50,20)(0,20 ) ( 50,80)(100,80)5 ( 50,20)(100,20)5 ( 105,76)@xmath226 ( 105,16)@xmath227 ( -15,78)@xmath88 ( -15,18)@xmath192 ( 55,48)@xmath130 ( 45,-10)_(e ) _ ( 130,110)(-50,0 ) ( 0,80)(50,80 ) ( 50,80)(50,20 ) ( 50,20)(0,20 ) ( 50,20)(100,20)36 ( 50,80)(100,80)5 ( 105,76)@xmath226 ( 105,16)@xmath230 ( -15,78)@xmath88 ( -15,18)@xmath192 ( 55,48)@xmath130 ( 45,-10)_(f ) _ + ( 130,120)(10,0 ) ( 0,100)(50,100 ) ( 50,100)(100,100 ) ( 0,0)(50,0 ) ( 50,0)(100,0 ) ( 50,100)(50,70)33 ( 50,30)(50,0)33 ( 50,70)(100,70)5 ( 50,30)(100,30)5 ( 50,70)(50,30)5 ( 55,83)@xmath231 ( 55,13)@xmath231 ( 105,66)@xmath226 ( 105,26)@xmath227 ( 105,98)@xmath69 ( 105,-2)@xmath69 ( -15,98)@xmath69 ( -15,-2)@xmath69 ( 45,-20)_(g ) _ + charged higgs bosons are dominantly produced at the lhc in association with top quarks @xmath232 and the charge - conjugated process , see fig . [ fg : hcpro]a . for charged higgs - boson masses @xmath233 the dominant contribution to this process factorizes into @xmath102 pair production with the subsequent ( anti)top decay into a charged higgs and ( anti)bottom quark . this region is then suitably described by the qcd- and electroweak - corrected @xmath102 production cross section @xcite multiplied with the radiatively corrected branching ratio of the top decays @xcite . however , for larger charged higgs masses the top decays are kinematically closed and charged higgs bosons are produced in terms of the full process of fig . [ fg : hcpro]a . for the inclusive rate the final - state bottom quark gives rise to the options to calculate this process in the 4fs or the 5fs , where the latter starts from the process @xmath234 and the charge - conjugated process and thus uses bottom pdfs of the proton . both calculations have been performed up to nlo qcd resulting in moderate scale dependences @xcite . while the 5fs is available exclusively for all distributions involving the top quark and charged higgs boson , the 4fs allows to determine distributions involving the bottom quark in the final state in addition . both calculations have been combined by means of the santander matching of eq . ( [ eq : santander ] ) @xcite with the weight @xmath235 . for small values of @xmath42 the genuine susy - qcd corrections are of moderate size @xcite , while for large @xmath42 they are completely dominated by @xmath127-terms with a small remainder . thus for the compilation of the charged - higgs production cross section the effective bottom yukawa coupling @xmath236 ( combined with the top yukawa coupling factor @xmath237 ) has been used that can be translated into an effective @xmath42 value @xcite . the result for the continuum production cross section without susy - qcd corrections is shown in fig . [ fg : proch ] for two charged higgs masses as a function of @xmath42 . the qcd corrections enhance the 4fs cross section by about 60% @xcite , while they are moderate for the 5fs @xcite . ( 150,180)(0,0 ) ( -300,-90.0 ) ( -40,-90.0 ) an open problem for a long time has been the treatment of the intermediate region between resonant top quark decays and the continuum contribution for large charged higgs masses . this has been solved recently by a full nlo calculation in the 4fs within the complex - mass scheme for the intermediate top quarks @xcite . the nlo qcd corrections turn out to be large in this mass regime , too , while the results nicely interpolate between the low- and high - mass regimes as can be inferred from fig . [ fg : prochint ] . ( 150,350)(0,0 ) ( -90,-80.0 ) the second important charged higgs production process is charged higgs pair production in a drell yan type process ( see fig . [ fg : hcpro]b ) @xcite @xmath238 which is mediated by @xmath68-channel photon and @xmath15-boson exchange . the nlo qcd corrections can be taken from the drell yan process and are of moderate size as in the case of the neutral higgs - strahlung process discussed before @xcite . the genuine susy qcd corrections , mediated by virtual gluino and squark exchange in the initial state , are small @xcite . charged higgs pairs can also be produced from @xmath205 initial states by the loop - mediated process @xcite ( see fig . [ fg : hcpro]c ) @xmath239 where the dominant contributions emerge from top and bottom quark loops as well as stop and sbottom loops , if the squark masses are light enough . the nlo corrections to this process are unknown . this cross section is of similar size as the bottom - initiated process @xcite ( see fig . [ fg : hcpro]e ) @xmath240 which relies on the approximations required by the introduction of the bottom densities as discussed before and is known at nlo @xcite . qcd corrections are of significant size due to the @xmath127 terms related to the bottom yukawa coupling . the pure qcd corrections and the genuine susy qcd corrections can be of opposite sign . charged higgs bosons can be produced in association with a @xmath7 boson @xcite ( see fig . [ fg : hcpro]d ) @xmath241 which is generated by top - bottom quark loops and stop - sbottom loops , if the squark masses are small enough . this process is known at lo only . the same final state also arises from the process @xcite ( see fig . [ fg : hcpro]f ) @xmath242 which is based on the approximations of the 5fs . the qcd corrections have been calculated and turn out to be of moderate size @xcite . finally , charged higgs - boson pairs can be produced in vector - boson - fusion @xcite ( see fig . [ fg : hcpro]g ) @xmath243 the lo cross section is independent of @xmath42 and can be sizeable within the mssm . however , the calculation of ref . @xcite is not consistent with the parton picture , since small quark masses have been introduced for the accompanying quarks in order to regulate the collinear divergences of photon - exchange contributions . for a more reliable prediction qed - mass factorization has to be performed already at lo . -90 -90 all higgs boson production cross sections have been updated with the known higher - order corrections and the most recent parton density functions , i.e. the pdf4lhc15 sets @xcite , where nlo densities have been used consistently for nlo predictions and nnlo densities for nnlo predictions . using the same values of the input parameters as for the branching ratios discussed in section [ sc : br ] and their uncertainties a rigorous analysis has been performed to derive a sophisticated prediction of the central cross section values and their uncertainties . the results are shown in fig . [ fg : hprodcxn ] as a function of the higgs mass for 13 and 14 tev c.m . energy at the lhc . the size of the coloured bands represents the individual sums of the theoretical and parametric uncertainties . all production cross sections with results beyond nlo in qcd exhibit a small residual uncertainty in the few - per - cent range . only the cross sections for @xmath186 , @xmath244 and @xmath209 production develop larger uncertainties due to the problems discussed in the previous sections . the theoretical and parametric uncertainties of each production process have been added in quadrature . the gluon - fusion cross sections can be predicted with a total ( gaussian ) uncertainty of about 5% , the vector - boson - fusion and @xmath245 higgs - strahlung channels with less than 3% uncertainty and the @xmath179 higgs - strahlung channel with about 4% uncertainty due to the novel loop contributions from @xmath180 as discussed in section [ sc : higgsstrahlung ] . the uncertainties of @xmath186 production amount to about @xmath246 , for @xmath68- and @xmath130-channel @xmath209 production to about @xmath247 and for @xmath191 production to about @xmath248 . fig . [ fg : hcxnen ] shows the energy dependence of the higgs production cross sections for a higgs mass @xmath92 gev . it is visible that all cross sections develop a similar rising slope apart from @xmath186 and @xmath209 that grow stronger due to the larger phase - space suppression for smaller energies . -90 an analogous update of the production cross sections as for the sm case has been made also for the mssm higgs - boson production cross sections . the public code sushi @xcite has been used as the preferred choice since this includes the full nlo qcd corrections to the gluon - fusion cross section and the nnlo qcd corrections in the heavy top - quark limit for the top - loop contributionslo qcd corrections have been included in sushi @xcite . ] . moreover , electroweak corrections originating from light - fermion loops @xcite are taken into account in this program . for the @xmath249 production cross section sushi contains the nnlo qcd - corrected 5fs - cross section that agrees with the 4fs results within about 20% for the adopted scale choices susy - qcd corrections are included in the heavy susy - particle limit for the gluon - fusion process and in the @xmath127 approximation for @xmath250 production . the compiled cross sections within the @xmath53 scenario @xcite are shown in fig . [ fig : mssmcxn ] for two values of @xmath42 . the left plot exhibits the dominance of the gluon - fusion process for smaller values of @xmath42 and the right one the dominance of @xmath224 production for large values of @xmath42 . uncertainties have not been added to these results , but have been analyzed in detail in ref . @xcite . ( 150,240)(0,0 ) ( -280,-20.0 ) ( -30,-20.0 ) ( 100,100)(100,0 ) ( 0,80)(50,80)35 ( 0,20)(50,20)35 ( 50,20)(50,80 ) ( 50,80)(100,80 ) ( 100,80)(100,20 ) ( 100,20)(50,20 ) ( 100,80)(150,80)5 ( 100,20)(150,20)5 ( -60,48)@xmath210 ( 155,76)@xmath12 ( 155,16)@xmath12 ( -15,78)@xmath57 ( -15,18)@xmath57 ( 30,48)@xmath56 ( 100,100)(-20,0 ) ( 0,80)(50,80)35 ( 0,20)(50,20)35 ( 50,20)(50,80 ) ( 50,80)(100,50 ) ( 100,50)(50,20 ) ( 100,50)(150,50)5 ( 150,50)(200,80)5 ( 150,50)(200,20)5 ( 205,76)@xmath12 ( 205,16)@xmath12 ( -15,78)@xmath57 ( -15,18)@xmath57 ( 30,48)@xmath56 ( 146,46 ) ( 146,60 ) + ( 100,100)(100,0 ) ( 0,80)(50,80 ) ( 50,80)(100,80 ) ( 0,20)(50,20 ) ( 50,20)(100,20 ) ( 50,80)(50,20)35 ( 48,60)(100,60)5 ( 52,40)(100,40)5 ( -60,48)@xmath212 ( 105,78)@xmath69 ( 105,18)@xmath69 ( 105,56)@xmath12 ( 105,36)@xmath12 ( -15,78)@xmath69 ( -15,18)@xmath69 ( 20,48)@xmath96 ( 100,100)(-20,0 ) ( 0,80)(50,80 ) ( 50,80)(100,80 ) ( 0,20)(50,20 ) ( 50,20)(100,20 ) ( 50,80)(50,20)35 ( 50,50)(100,50)5 ( 100,50)(150,80)5 ( 100,50)(150,20)5 ( 105,78)@xmath69 ( 105,18)@xmath69 ( 155,76)@xmath12 ( 155,16)@xmath12 ( -15,78)@xmath69 ( -15,18)@xmath69 ( 20,63)@xmath96 ( 96,46 ) ( 96,60 ) + ( 100,100)(100,0 ) ( 0,80)(50,50 ) ( 50,50)(0,20 ) ( 50,50)(100,50)35 ( 100,50)(150,80)35 ( 100,50)(150,20)5 ( 120,60)(150,42)5 ( -60,48)@xmath213 ( 155,78)@xmath96 ( 155,38)@xmath12 ( 155,16)@xmath12 ( -15,78)@xmath69 ( -15,18)@xmath167 ( 60,60)@xmath96 ( 100,100)(-20,0 ) ( 0,80)(50,50 ) ( 50,50)(0,20 ) ( 50,50)(100,50)35 ( 100,50)(150,80)35 ( 100,50)(125,35)5 ( 125,35)(150,20)5 ( 125,35)(150,50)5 ( 155,78)@xmath96 ( 155,46)@xmath12 ( 155,16)@xmath12 ( -15,78)@xmath69 ( -15,18)@xmath167 ( 60,60)@xmath96 ( 121,31 ) ( 121,45 ) + ( 100,100)(100,0 ) ( 0,80)(50,80)35 ( 0,20)(50,20)35 ( 50,20)(50,80 ) ( 50,80)(100,80 ) ( 100,80)(100,50 ) ( 100,50)(100,20 ) ( 100,20)(50,20 ) ( 100,80)(150,80)35 ( 100,50)(150,50)5 ( 100,20)(150,20)5 ( 155,76)@xmath15 ( 155,46)@xmath12 ( 155,16)@xmath12 ( -15,78)@xmath57 ( -15,18)@xmath57 ( 30,48)@xmath56 ( 100,100)(-20,0 ) ( 0,80)(50,80)35 ( 0,20)(50,20)35 ( 50,20)(50,80 ) ( 50,80)(100,80 ) ( 100,80)(100,20 ) ( 100,20)(50,20 ) ( 100,80)(150,80)35 ( 100,20)(150,20)5 ( 125,20)(150,35)5 ( 155,76)@xmath15 ( 155,31)@xmath12 ( 155,16)@xmath12 ( -15,78)@xmath57 ( -15,18)@xmath57 ( 30,48)@xmath56 ( 121,16 ) ( 121,30 ) + ( 100,100)(100,0 ) ( 0,80)(50,80)35 ( 0,20)(50,20)35 ( 50,20)(50,40 ) ( 50,40)(50,60 ) ( 50,60)(50,80 ) ( 50,80)(100,80 ) ( 100,20)(50,20 ) ( 50,60)(100,60)5 ( 50,40)(100,40)5 ( -60,48)@xmath251 ( 105,78)@xmath130 ( 105,18)@xmath185 ( 105,56)@xmath12 ( 105,36)@xmath12 ( -15,78)@xmath57 ( -15,18)@xmath57 ( 100,100)(-20,0 ) ( 0,80)(50,80)35 ( 0,20)(50,20)35 ( 50,20)(50,50 ) ( 50,50)(50,80 ) ( 50,80)(100,80 ) ( 100,20)(50,20 ) ( 50,50)(100,50)5 ( 100,50)(150,80)5 ( 100,50)(150,20)5 ( 105,78)@xmath130 ( 105,18)@xmath185 ( 155,76)@xmath12 ( 155,16)@xmath12 ( -15,78)@xmath57 ( -15,18)@xmath57 ( 96,46 ) ( 96,60 ) + higgs - boson pair production is the first process that allows to obtain direct access to the trilinear self - coupling of the higgs boson and paves the way to the higgs potential at the origin of electroweak symmetry breaking . higgs boson pairs are dominantly produced in the gluon - fusion process and to a lesser extent in vector - boson fusion , double higgs - strahlung and double higgs bremsstrahlung off top quarks , see fig . [ fg : hhdia ] . the production cross sections are shown as a function of the collider energy for a higgs mass @xmath92 gev in fig . [ fg : hhcxns ] . gluon fusion dominates the production of higgs - boson pairs by more than an order of magnitude , while the production modes roughly follow the pattern of single - higgs boson production . since the diagrams involving the trilinear higgs coupling are only a subset for each process the sensitivity to the trilinear higgs coupling is reduced compared to the size of the individual cross sections . the sensitivities of the individual production cross sections to the trilinear higgs coupling are shown in fig . [ fg : hhcxns ] . the locations of the minima of the cross sections in terms of the value of @xmath252 are different for the production mechanisms @xcite . [ [ gluon - fusion . ] ] gluon fusion . + + + + + + + + + + + + + the gluon - fusion process is mediated by top- and to a lesser extent bottom - quark loops . there are box- and triangle contributions at lo , see fig . [ fg : hhdia]a . the box contributions are dominant with a destructive interference to the triangle diagrams @xcite . the sensitivity to the trilinear higgs coupling follows the rough behaviour @xmath253 so that the uncertainties of the production cross section is immediately translated to the uncertainties of the extraction of the trilinear self - coupling . in the past the nlo qcd corrections have been determined in the limit of heavy top quarks @xcite . similar to the single higgs case they increase the cross section by up to 100% . due to the fact that the invariant mass of the final - state higgs - boson pair is much larger than in the single - higgs case the heavy top - quark limit is expected to work less reliable in the higgs - pair case . the first attempt to estimate finite top - mass effects at nlo was by means of a systematic heavy - top expansion of the inclusive cross section @xcite that gave an estimate of about 10% for the finite top - mass effects beyond lo . the second attempt kept the virtual corrections in the heavy - top limit but included the real corrections exactly @xcite . this resulted in a 10%-decrease of the nlo cross section . very recently the full nlo calculation has been completed by numerical methods implying a decrease of the total cross section by about 14% at the lhc for a c.m . energy of 13 tev @xcite so that the heavy - top limit works still reasonably well for the total cross section . for large invariant higgs - pair masses the finite mass effects can reach a level of 20% . the nlo result has been extended by the nnlo qcd corrections in the heavy top - quark limit that lead to a rise of the total cross section by about 20% @xcite . within a heavy top - quark expansion nnlo top mass effects have been estimated to be at the 5%-level @xcite . finally a nnll soft and collinear gluon resummation has been added which adds a contribution of 5 - 10% beyond nnlo @xcite . including the most up - to - date theoretical status for the prediction of the total cross section the theoretical uncertainties due to the scale dependence is reduced to about 5% . together with the pdf+@xmath50 uncertainties one obtains a total uncertainty of about 10% for the total cross section at the lhc @xcite . recently a public code has been constructed for fully exclusive higgs - boson pair production via gluon fusion including nnlo qcd corrections at parton level @xcite . [ [ vector - boson - fusion . ] ] vector - boson fusion . + + + + + + + + + + + + + + + + + + + + higgs - boson pair production via vector - boson fusion is dominated by @xmath130-channel contributions as in the single - higgs case , see fig . [ fg : hhdia]b . the nlo qcd corrections can be taken from deep inelastic lepton - nucleon scattering analogous to single - higgs production @xcite . they increase the cross section by about 10% . within the same approach the nnlo qcd corrections have been obtained @xcite . they range at the per - cent level . the residual theoretical and parametric uncertainties amount to about 3 - 4% at the lhc @xcite . [ [ double - higgs - strahlung . ] ] double higgs - strahlung . + + + + + + + + + + + + + + + + + + + + + + + double higgs - strahlung proceeds along the same lines as single - higgs - strahlung , i.e. the higgs boson pair is produced in association with a @xmath7 or @xmath15 boson , see fig . [ fg : hhdia]c . the nlo and nnlo qcd corrections can be taken over from the corresponding calculation for the drell yan process , since the final state is only weakly interacting @xcite . the only difference to the drell yan process emerges from the additional loop - mediated @xmath254 process , see fig . [ fg : hhdia]c . this has been added to the nnlo qcd corrections . the qcd corrections increase the production cross sections by about 30% , while the @xmath255 adds another 2030% to @xmath256 production @xcite . the residual theoretical uncertainties range at the 3%-level for @xmath257 production and at the level of 4% for @xmath256 production @xcite . a fully differential calculation for @xmath257 final states has recently been completed @xcite . [ [ double - higgs - bremsstrahlung - off - top - quarks . ] ] double higgs bremsstrahlung off top quarks . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + higgs - boson pair production in association with top quarks is generated by analogous diagrams to the single - higgs case , see fig . [ fg : hhdia]d @xcite . the nlo qcd corrections have been calculated recently with the mg5_amc@nlo tool @xcite . they modify the total cross section by about 20% and reduce the scale dependence to a level of less than 5% . the total theoretical and parametric uncertainties for this production process amount to about @xmath258 at the lhc with c.m . energies of 13 and 14 tev @xcite . ( 100,100)(-30,-5 ) ( 0,100)(50,100)36 ( 0,0)(50,0)36 ( 50,100)(100,50 ) ( 100,50)(50,0 ) ( 50,0)(50,100 ) ( 100,50)(150,50)5 ( 150,50)(200,100)5 ( 150,50)(200,0)5 ( 165,76)@xmath259 ( 165,-5)@xmath260 ( 90,50)@xmath261 ( -15,-3)@xmath57 ( -15,78)@xmath57 ( 20,38)@xmath56 ( 100,100)(-150,-5 ) ( 0,100)(50,100)36 ( 0,0)(50,0)36 ( 50,100)(140,100 ) ( 140,100)(140,0 ) ( 140,0)(50,0 ) ( 50,0)(50,100 ) ( 140,100)(190,100)5 ( 140,0)(190,0)5 ( 165,76)@xmath259 ( 165,-5)@xmath260 ( -15,-3)@xmath57 ( -15,78)@xmath57 ( 20,38)@xmath56 in the mssm there are several possible neutral higgs - pair final states . the gluon - fusion mechanism yields @xmath262 final states that emerge from analogous diagrams as in the sm - higgs case . in the mixed scalar - pseudoscalar channel , i.e. @xmath263 , off - shell @xmath15-boson exchange contributes to the @xmath68-channel diagrams , too , see fig . [ fg : ggha ] . however , these final states are dominates by drell - yan production mechanisms , see fig . [ fg : hady ] . since the qcd corrections to mssm higgs pair production via gluon fusion are only known in the heavy - top - quark limit so far , reliable nlo predictions of the corresponding cross sections are only possible for small values of @xmath42 where the top loops provide the dominant contributions and for not too large external higgs masses . the qcd corrections are large and positive , increasing the cross sections by up to 100% @xcite . the mssm cross sections for the gluon - fusion processes range below 10 fb in all regions where none of the higgs bosons involved in the @xmath68-channel becomes resonant . however , for small values of @xmath42 and below the @xmath102-threshold there are sizeable regions where the heavy scalar higgs boson can become resonant and decays into @xmath264 final states , @xmath265 . in these regions the cross sections become large , since the dominant piece factorizes into the single heavy - scalar higgs production cross section and the branching ratio of the @xmath266 decay @xcite . mixed scalar - pseudoscalar higgs boson production is dominated by the drell yan process @xmath267 . the qcd corrections can be translated from the corresponding drell yan process and are of moderate size @xcite . the cross sections for these processes can reach the level of about 100 fb . ( 160,100)(0,0 ) ( 0,80)(50,50 ) ( 50,50)(0,20 ) ( 50,50)(100,50)35 ( 100,50)(150,80)5 ( 100,50)(150,20)5 ( 155,16)@xmath160 ( -15,18)@xmath167 ( -15,78)@xmath69 ( 70,60)@xmath15 ( 155,76)@xmath13 + in this review the decay widths and branching ratios of sm and mssm higgs bosons have been discussed . all relevant higher order corrections , which are dominated by qcd corrections , have been summarized according to the present state of the art . at the lhc the sm higgs particle is produced predominantly by gluon fusion @xmath2 , followed by vector - boson fusion @xmath268 ( @xmath52 ) and , to a lesser extent , higgs - strahlung off vector bosons , @xmath269 , and top quarks , @xmath270 . the cross sections of these production channels have been described including all known qcd and electroweak corrections , which are important in particular for the dominant gluon - fusion mechanism . in the mssm the neutral higgs bosons will mainly be produced via gluon fusion @xmath117 . however , through the enhanced @xmath88 quark couplings , higgs bremsstrahlung off @xmath88 quarks , @xmath271 , will dominate for large @xmath42 . all other higgs production mechanisms , i.e. vector - boson fusion and higgs - strahlung off vector bosons or @xmath102 pairs , will be less important than in the sm . g. aad _ et al . _ [ atlas collaboration ] , phys . * b716 * ( 2012 ) 1 ; s. chatrchyan _ et al . _ [ cms collaboration ] , phys . lett . * b716 * ( 2012 ) 30 . g. aad _ et al . _ [ atlas and cms collaborations ] , jhep * 1608 * ( 2016 ) 045 . p. w. higgs , phys . * 12 * ( 1964 ) 132 , phys . * 13 * ( 1964 ) 508 and phys . * 145 * ( 1966 ) 1156 ; f. englert and r. brout , phys . * 13 * ( 1964 ) 321 ; g. s. guralnik , c. r. hagen and t. w. kibble , phys . * 13 * ( 1964 ) 585 ; t. w. b. kibble , phys . * 155 * ( 1967 ) 1554 . g. t hooft , nucl . b * 35 * ( 1971 ) 167 ; g. t hooft and m. j. g. veltman , nucl . b * 44 * ( 1972 ) 189 . c. h. llewellyn smith , phys . lett . * 46b * ( 1973 ) 233 ; j. m. cornwall , d. n. levin and g. tiktopoulos , phys . d * 10 * ( 1974 ) 1145 erratum : [ phys . d * 11 * ( 1975 ) 972 ] ; b. w. lee , c. quigg and h. b. thacker , phys . * 38 * ( 1977 ) 883 and phys . d * 16 * ( 1977 ) 1519 . n. cabibbo , l. maiani , g. parisi and r. petronzio , nucl . b * 158 * ( 1979 ) 295 ; m. s. chanowitz , m. a. furman and i. hinchliffe , phys . lett . * 78b * ( 1978 ) 285 ; r. a. flores and m. sher , phys . rev . d * 27 * ( 1983 ) 1679 ; m. lindner , z. phys . c * 31 * ( 1986 ) 295 ; a. hasenfratz , k. jansen , c. b. lang , t. neuhaus and h. yoneyama , phys . b * 199 * ( 1987 ) 531 ; j. kuti , l. lin and y. shen , phys . * 61 * ( 1988 ) 678 ; m. lscher and p. weisz , nucl . b * 318 * ( 1989 ) 705 ; m. sher , phys . * 179 * ( 1989 ) 273 and phys . b * 317 * ( 1993 ) 159 addendum : [ phys . b * 331 * ( 1994 ) 448 ] ; g. altarelli and g. isidori , phys . b * 337 * ( 1994 ) 141 ; j. a. casas , j. r. espinosa and m. quiros , phys . b * 342 * ( 1995 ) 171 . g. degrassi , s. di vita , j. elias - miro , j. r. espinosa , g. f. giudice , g. isidori and a. strumia , jhep * 1208 * ( 2012 ) 098 ; d. buttazzo , g. degrassi , p. p. giardino , g. f. giudice , f. sala , a. salvio and a. strumia , jhep * 1312 * ( 2013 ) 089 . j. r. espinosa and m. quiros , phys . b * 353 * ( 1995 ) 257 . e. gildener and s. weinberg , phys . d * 13 * ( 1976 ) 3333 ; s. weinberg , phys . d * 13 * ( 1976 ) 974 and phys . d * 19 * ( 1979 ) 1277 ; l. susskind , phys . rev . d * 20 * ( 1979 ) 2619 . y. a. golfand and e. p. likhtman , jetp lett . * 13 * ( 1971 ) 323 [ pisma zh . eksp . * 13 * ( 1971 ) 452 ] ; d. v. volkov and v. p. akulov , phys . lett . * 46b * ( 1973 ) 109 ; j. wess and b. zumino , nucl . b * 70 * ( 1974 ) 39 . for reviews on supersymmetric theories , see p. fayet and s. ferrara , phys . * 32 * ( 1977 ) 249 ; h. p. nilles , phys . rept . * 110 * ( 1984 ) 1 ; r. barbieri , riv . nuovo cim . * 11n4 * ( 1988 ) 1 ; h. e. haber and g. l. kane , phys . * 117 * ( 1985 ) 75 . n. arkani - hamed , a. g. cohen and h. georgi , phys . b * 513 * ( 2001 ) 232 ; n. arkani - hamed , a. g. cohen , t. gregoire and j. g. wacker , jhep * 0208 * ( 2002 ) 020 ; n. arkani - hamed , a. g. cohen , e. katz , a. e. nelson , t. gregoire and j. g. wacker , jhep * 0208 * ( 2002 ) 021 . n. arkani - hamed , s. dimopoulos and g. r. dvali , phys . b * 429 * ( 1998 ) 263 ; i. antoniadis , n. arkani - hamed , s. dimopoulos and g. r. dvali , phys . b * 436 * ( 1998 ) 257 ; l. randall and r. sundrum , phys . * 83 * ( 1999 ) 3370 and phys . * 83 * ( 1999 ) 4690 . h. m. georgi , s. l. glashow , m. e. machacek and d. v. nanopoulos , phys . * 40 * ( 1978 ) 692 . r. n. cahn and s. dawson , phys . * 136b * ( 1984 ) 196 erratum : [ phys . * 138b * ( 1984 ) 464 ] ; k. i. hikasa , phys . lett . * 164b * ( 1985 ) 385 erratum : [ phys . b * 195 * ( 1987 ) 623 ] ; g. altarelli , b. mele and f. pitolli , nucl . b * 287 * ( 1987 ) 205 . s. l. glashow , d. v. nanopoulos and a. yildiz , phys . d * 18 * ( 1978 ) 1724 ; z. kunszt , z. trocsanyi and w. j. stirling , phys . b * 271 * ( 1991 ) 247 . r. raitio and w.w . wada , phys . * d19 * ( 1979 ) 941 ; j. n. ng and p. zakarauskas , phys . rev . * d29 * ( 1984 ) 876 ; z. kunszt , nucl . * b247 * ( 1984 ) 339 ; j. f. gunion , phys . lett . * b261 * ( 1991 ) 510517 ; w. j. marciano and f. e. paige , phys . * 66 * ( 1991 ) 24332435 . j. m. butterworth , a. r. davison , m. rubin and g. p. salam , phys . * 100 * ( 2008 ) 242001 . d. l. rainwater , d. zeppenfeld and k. hagiwara , phys . d * 59 * ( 1998 ) 014037 ; t. plehn , d. l. rainwater and d. zeppenfeld , phys . b * 454 * ( 1999 ) 297 . e. witten , phys . lett . * 105b * ( 1981 ) 267 . s. dimopoulos , s. raby and f. wilczek , phys . d * 24 * ( 1981 ) 1681 ; l. e. ibanez and g. g. ross , phys . * 105b * ( 1981 ) 439 . l. e. ibanez and g. g. ross , phys . * 110b * ( 1982 ) 215 . h. goldberg , phys . * 50 * ( 1983 ) 1419 erratum : [ phys . * 103 * ( 2009 ) 099905 ] ; j. r. ellis , j. s. hagelin , d. v. nanopoulos , k. a. olive and m. srednicki , nucl . b * 238 * ( 1984 ) 453 . s. dimopoulos and h. georgi , nucl . b * 193 * ( 1981 ) 150 . p. fayet , nucl . b * 90 * ( 1975 ) 104 , phys . lett . b * 64 * ( 1976 ) 159 and phys . b * 69 * ( 1977 ) 489 ; n. sakai , z. phys . c * 11 * ( 1981 ) 153 ; k. inoue , a. kakuto , h. komatsu and s. takeshita , prog . theor . phys . * 67 * ( 1982 ) 1889 , prog . theor . * 68 * ( 1982 ) 927 [ erratum - ibid . * 70 * ( 1983 ) 330 ] and prog . * 71 * ( 1984 ) 413 . see e.g. g. degrassi , s. heinemeyer , w. hollik , p. slavich and g. weiglein , eur . j. * c28 * ( 2003 ) 133 . f. zwirner , in * saariselkae 1991 , proceedings , physics and experiments with linear colliders , vol . 1 * 309 - 351 and cern geneva - th . 6357 [ hep - ph/9203204 ] . s. p. martin , phys . * d75 * ( 2007 ) 055005 ; r. v. harlander , p. kant , l. mihaila and m. steinhauser , phys . * 100 * ( 2008 ) 191602 , erratum _ ibid . _ * 101 * ( 2008 ) 039901 . p. draper , g. lee and c. e. m. wagner , phys . d * 89 * ( 2014 ) no.5 , 055023 ; e. bagnaschi , g. f. giudice , p. slavich and a. strumia , jhep * 1409 * ( 2014 ) 092 ; k. cheung , r. huo , j. s. lee and y. l. sming tsai , jhep * 1504 * ( 2015 ) 151 ; j. pardo vega and g. villadoro , jhep * 1507 * ( 2015 ) 159 ; g. lee and c. e. m. wagner , phys . rev . d * 92 * ( 2015 ) no.7 , 075032 ; h. bahl and w. hollik , eur . j. c * 76 * ( 2016 ) no.9 , 499 . barger , m.s . berger , a.l . stange and r.j.n . phillips , phys . rev . * d45 * ( 1992 ) 4128 ; w. hollik and s. pearanda , eur . j. * c23 * ( 2002 ) 163 ; a. dobado , m.j . herrero , w. hollik and s. pearanda , phys . * d66 * ( 2002 ) 095016 . m. brucherseifer , r. gavin and m. spira , phys . d * 90 * ( 2014 ) no.11 , 117701 ; m. brucherseifer , phd thesis , eth zrich , 2012 . l. j. hall , r. rattazzi and u. sarid , phys . d * 50 * , 7048 ( 1994 ) ; r. hempfling , phys . d * 49 * , 6168 ( 1994 ) ; m. carena , m. olechowski , s. pokorski and c. e. m. wagner , nucl . b * 426 * ( 1994 ) 269 ; d. m. pierce , j. a. bagger , k. t. matchev and r .- j . zhang , nucl . b * 491 * , 3 ( 1997 ) ; j. guasch , w. hollik and s. pearanda , phys . b * 515 * ( 2001 ) 367 ; g. dambrosio , g. f. giudice , g. isidori and a. strumia , nucl . b * 645 * , 155 ( 2002 ) ; a. j. buras , p. h. chankowski , j. rosiek and l. slawianowska , nucl . b * 659 * , 3 ( 2003 ) ; v. barger , h. e. logan and g. shaughnessy , phys . d * 79 * , 115018 ( 2009 ) ; n. d. christensen , t. han and s. su , phys . d * 85 * , 115018 ( 2012 ) . m. s. carena , d. garcia , u. nierste and c. e. m. wagner , nucl . b * 577 * , 88 ( 2000 ) . j. guasch , p. hfliger and m. spira , phys . d * 68 * , 115001 ( 2003 ) . d. noth and m. spira , phys . * 101 * , 181801 ( 2008 ) and jhep * 1106 * , 084 ( 2011 ) . l. mihaila and c. reisser , jhep * 1008 * , 021 ( 2010 ) ; a. crivellin and c. greub , phys . d * 87 * ( 2013 ) 015013 erratum : [ phys . d * 87 * ( 2013 ) 079901 ] . b. a. kniehl , phys . * 240 * ( 1994 ) 211 ; a. djouadi , phys . * 457 * ( 2008 ) 1 and phys . * 459 * ( 2008 ) 1 ; s. dittmaier and m. schumacher , prog . phys . * 70 * ( 2013 ) 1 ; j. baglio , a. djouadi and j. quevillon , rept . * 79 * ( 2016 ) no.11 , 116201 . m. spira , fortsch . * 46 * ( 1998 ) 203 . j. r. ellis , m. k. gaillard and d. v. nanopoulos , nucl . b * 106 * ( 1976 ) 292 . l. resnick , m. k. sundaresan and p. j. s. watson , phys . d * 8 * ( 1973 ) 172 . e. braaten and j. p. leveille , phys . d * 22 * ( 1980 ) 715 ; n. sakai , phys . d * 22 * ( 1980 ) 2220 ; t. inami and t. kubota , nucl . b * 179 * ( 1981 ) 171 ; m. drees and k. i. hikasa , phys . b * 240 * ( 1990 ) 455 erratum : [ phys . b * 262 * ( 1991 ) 497 ] and phys . d * 41 * ( 1990 ) 1547 . s. g. gorishnii , a. l. kataev and s. a. larin , sov . j. nucl . * 40 * ( 1984 ) 329 [ yad . * 40 * ( 1984 ) 517 ] ; s. g. gorishnii , a. l. kataev , s. a. larin and l. r. surguladze , mod . a * 5 * ( 1990 ) 2703 and phys . rev . d * 43 * ( 1991 ) 1633 ; a. l. kataev and v. t. kim , mod . lett . a * 9 * ( 1994 ) 1309 ; l. r. surguladze , phys . b * 341 * ( 1994 ) 60 ; k. g. chetyrkin , phys . b * 390 * ( 1997 ) 309 ; k. melnikov , phys . d * 53 * ( 1996 ) 5020 . k. g. chetyrkin and a. kwiatkowski , nucl . b * 461 * ( 1996 ) 3 ; s. a. larin , t. van ritbergen and j. a. m. vermaseren , phys . b * 362 * ( 1995 ) 134 . j. fleischer and f. jegerlehner , phys . d * 23 * ( 1981 ) 2001 . d. y. bardin , b. m. vilensky and p. k. khristova , sov . j. nucl . * 53 * ( 1991 ) 152 [ yad . fiz . * 53 * ( 1991 ) 240 ] ; a. dabelstein and w. hollik , z. phys . c * 53 * ( 1992 ) 507 ; b. a. kniehl , nucl . b * 376 * ( 1992 ) 3 . a. djouadi , d. haidt , b. a. kniehl , p. m. zerwas and b. mele , in * munich / annecy / hamburg 1991 , proceedings , @xmath272 collisions at 500 gev , pt . a * 11 - 30 . b. a. kniehl and m. spira , nucl . b * 432 * ( 1994 ) 39 ; a. kwiatkowski and m. steinhauser , phys . b * 338 * ( 1994 ) 66 erratum : [ phys . lett . b * 342 * ( 1995 ) 455 ] . k. g. chetyrkin , b. a. kniehl and m. steinhauser , phys . * 78 * ( 1997 ) 594 . l. mihaila , b. schmidt and m. steinhauser , phys . b * 751 * ( 2015 ) 442 . d. de florian _ et al . _ [ lhc higgs cross section working group collaboration ] , arxiv:1610.07922 [ hep - ph ] . c. s. li and r. j. oakes , phys . d * 43 * ( 1991 ) 855 ; a. mendez and a. pomarol , phys . b * 252 * ( 1990 ) 461 . a. djouadi and p. gambino , phys . d * 51 * ( 1995 ) 218 erratum : [ phys . d * 53 * ( 1996 ) 4111 ] . a. djouadi , j. kalinowski and m. spira , comput . commun . * 108 * ( 1998 ) 56 ; a. djouadi , m. m. mhlleitner and m. spira , acta phys . b * 38 * ( 2007 ) 635 . a. dabelstein , nucl . b * 456 * ( 1995 ) 25 ; r. a. jimenez and j. sola , phys . b * 389 * ( 1996 ) 53 ; j. a. coarasa perez , r. a. jimenez and j. sola , phys . b * 389 * ( 1996 ) 312 . a. djouadi , j. kalinowski and p. m. zerwas , z. phys . c * 70 * ( 1996 ) 435 ; s. moretti and w. j. stirling , phys . b * 347 * ( 1995 ) 291 erratum : [ phys . b * 366 * ( 1996 ) 451 ] . b. a. kniehl , nucl . b * 352 * ( 1991 ) 1 ; d. y. bardin , p. k. khristova and b. m. vilensky , sov . j. nucl . * 54 * ( 1991 ) 833 [ yad . fiz . * 54 * ( 1991 ) 1366 ] . b. a. kniehl and m. spira , z. phys . c * 69 * ( 1995 ) 77 . b. a. kniehl and m. spira , nucl . b * 443 * ( 1995 ) 37 . b. a. kniehl and m. steinhauser , phys . b * 365 * ( 1996 ) 297 and nucl . b * 454 * ( 1995 ) 485 . t. g. rizzo , phys . d * 22 * ( 1980 ) 722 ; w. y. keung and w. j. marciano , phys . rev . d * 30 * ( 1984 ) 248 . r. n. cahn , rept . * 52 * ( 1989 ) 389 . a. bredenstein , a. denner , s. dittmaier and m. m. weber , phys . d * 74 * ( 2006 ) 013004 and jhep * 0702 * ( 2007 ) 080 . s. boselli , c. m. carloni calame , g. montagna , o. nicrosini and f. piccinini , jhep * 1506 * ( 2015 ) 023 . a. denner , s. heinemeyer , i. puljak , d. rebuzzi and m. spira , eur . j. c * 71 * ( 2011 ) 1753 . s. heinemeyer _ et al . _ [ lhc higgs cross section working group collaboration ] , arxiv:1307.1347 [ hep - ph ] . w. hollik and j. h. zhang , arxiv:1011.6537 [ hep - ph ] and phys . d * 84 * ( 2011 ) 055022 . m. spira , a. djouadi , d. graudenz and p. m. zerwas , nucl . b * 453 * ( 1995 ) 17 . t. inami , t. kubota and y. okada , z. phys . c * 18 * ( 1983 ) 69 . a. djouadi , m. spira and p. m. zerwas , phys . b * 264 * ( 1991 ) 440 . k. g. chetyrkin , b. a. kniehl and m. steinhauser , phys . * 79 * ( 1997 ) 353 . p. a. baikov and k. g. chetyrkin , phys . * 97 * ( 2006 ) 061803 . m. a. shifman , a. i. vainshtein , m. b. voloshin and v. i. zakharov , sov . j. nucl . phys . * 30 * ( 1979 ) 711 [ yad . fiz . * 30 * ( 1979 ) 1368 ] . m. spira , jhep * 1610 * ( 2016 ) 026 . k. g. chetyrkin , b. a. kniehl and m. steinhauser , nucl . b * 510 * ( 1998 ) 61 . w. bernreuther and w. wetzel , nucl . b * 197 * ( 1982 ) 228 erratum : [ nucl . b * 513 * ( 1998 ) 758 ] ; s. a. larin , t. van ritbergen and j. a. m. vermaseren , nucl . b * 438 * ( 1995 ) 278 ; y. schrder and m. steinhauser , jhep * 0601 * ( 2006 ) 051 ; k. g. chetyrkin , j. h. khn and c. sturm , nucl . b * 744 * ( 2006 ) 121 . b. a. kniehl and a. v. kotikov , phys . b * 642 * ( 2006 ) 68 ; r. n. lee and i. s. terekhov , jhep * 1101 * ( 2011 ) 068 ; b. a. kniehl , a. v. kotikov , a. i. onishchenko and o. l. veretin , phys . * 97 * ( 2006 ) 042001 ; k. chetyrkin , p. baikov and j. khn , pos ll * 2016 * ( 2016 ) 010 . a. djouadi and p. gambino , phys . * 73 * ( 1994 ) 2528 ; k. g. chetyrkin , b. a. kniehl and m. steinhauser , nucl . b * 490 * ( 1997 ) 19 . u. aglietti , r. bonciani , g. degrassi and a. vicini , phys . b * 595 * ( 2004 ) 432 and hep - ph/0610033 ; g. degrassi and f. maltoni , phys . b * 600 * ( 2004 ) 255 . s. actis , g. passarino , c. sturm and s. uccirati , phys . b * 670 * ( 2008 ) 12 ; s. actis , g. passarino , c. sturm and s. uccirati , nucl . b * 811 * ( 2009 ) 182 . s. dawson , a. djouadi and m. spira , phys . * 77 * ( 1996 ) 16 . j. f. gunion , h. e. haber , g. l. kane and s. dawson , front . * 80 * ( 2000 ) 1 . m. mhlleitner and m. spira , nucl . b * 790 * ( 2008 ) 1 ; r. bonciani , g. degrassi and a. vicini , jhep * 0711 * ( 2007 ) 095 . s. gori and i. low , jhep * 1309 * ( 2013 ) 151 . m. mhlleitner , h. rzehak and m. spira , jhep * 0904 * ( 2009 ) 023 . h. e. haber and g. l. kane , phys . * 117 * ( 1985 ) 75 . g. t hooft and m. j. g. veltman , nucl . b * 44 * ( 1972 ) 189 ; p. breitenlohner and d. maison , commun . * 52 * ( 1977 ) 11 . s. a. larin , phys . b * 303 * ( 1993 ) 113 . m. spira , a. djouadi , d. graudenz and p. m. zerwas , phys . b * 318 * ( 1993 ) 347 . j. g. krner , d. kreimer and k. schilcher , z. phys . c * 54 * ( 1992 ) 503 ; d. kreimer , hep - ph/9401354 . k. melnikov , m. spira and o. i. yakovlev , z. phys . c * 64 * ( 1994 ) 401 . k. g. chetyrkin , b. a. kniehl , m. steinhauser and w. a. bardeen , nucl . b * 535 * ( 1998 ) 3 . s. l. adler , phys . * 177 * ( 1969 ) 2426 ; j. s. bell and r. jackiw , nuovo cim . a * 60 * ( 1969 ) 47 . d. g. sutherland , nucl . b * 2 * ( 1967 ) 433 ; m. j. g. veltman , proc . a * 301 * ( 1967 ) 107 . s. l. adler and w. a. bardeen , phys . * 182 * ( 1969 ) 1517 . j. brod , f. fugel and b. a. kniehl , phys . d * 78 * ( 2008 ) 011303 and nucl . b * 807 * ( 2009 ) 188 . v. v. sudakov , sov . jetp * 3 * ( 1956 ) 65 [ zh . fiz . * 30 * ( 1956 ) 87 ] . m. i. kotsky and o. i. yakovlev , phys . b * 418 * ( 1998 ) 335 ; r. akhoury , h. wang and o. i. yakovlev , phys . d * 64 * ( 2001 ) 113008 . r. v. harlander and m. steinhauser , phys . b * 574 * ( 2003 ) 258 and jhep * 0409 * ( 2004 ) 066 ; g. degrassi and p. slavich , nucl . b * 805 * ( 2008 ) 267 ; g. degrassi , s. di vita and p. slavich , eur . j. c * 72 * ( 2012 ) 2032 . r. v. harlander and f. hofmann , jhep * 0603 * ( 2006 ) 050 ; r. v. harlander , f. hofmann and h. mantler , jhep * 1102 * ( 2011 ) 055 ; g. degrassi , s. di vita and p. slavich , jhep * 1108 * ( 2011 ) 128 . g. degrassi and p. slavich , jhep * 1011 * ( 2010 ) 044 . c. anastasiou , s. beerli and a. daleo , phys . * 100 * ( 2008 ) 241806 . m. mhlleitner , h. rzehak and m. spira , pos radcor * 2009 * ( 2010 ) 043 and desy - proc-2010 - 01 . m. carena , s. heinemeyer , c. e. m. wagner and g. weiglein , eur . j. c * 26 * ( 2003 ) 601 . zheng and d.d . wu , phys . d * 42 * ( 1990 ) 3760 ; a. djouadi , m. spira , j. j. van der bij and p. m. zerwas , phys . b * 257 * ( 1991 ) 187 ; s. dawson and r. p. kauffman , phys . d * 47 * ( 1993 ) 1264 ; a. djouadi , m. spira and p. m. zerwas , phys . b * 311 * ( 1993 ) 255 ; k. melnikov and o. i. yakovlev , phys . b * 312 * ( 1993 ) 179 ; m. inoue , r. najima , t. oka and j. saito , mod . lett . a * 9 * ( 1994 ) 1189 ; j. fleischer , o. v. tarasov and v. o. tarasov , phys . b * 584 * ( 2004 ) 294 . r. harlander and p. kant , jhep * 0512 * ( 2005 ) 015 ; c. anastasiou , s. beerli , s. bucherer , a. daleo and z. kunszt , jhep * 0701 * ( 2007 ) 082 ; u. aglietti , r. bonciani , g. degrassi and a. vicini , jhep * 0701 * ( 2007 ) 021 . m. steinhauser , in * tegernsee 1996 , the higgs puzzle * 177 - 185 [ hep - ph/9612395 ] . c. sturm , eur . j. c * 74 * ( 2014 ) 8 , 2978 . p. maierhfer and p. marquard , phys . b * 721 * ( 2013 ) 131 . a. djouadi , p. gambino and b. a. kniehl , nucl . b * 523 * ( 1998 ) 17 ; g. degrassi and f. maltoni , nucl . b * 724 * ( 2005 ) 183 ; g. passarino , c. sturm and s. uccirati , phys . b * 655 * ( 2007 ) 298 ; s. actis , g. passarino , c. sturm and s. uccirati , nucl . b * 811 * ( 2009 ) 182 . a. djouadi , v. driesen , w. hollik and j. i. illana , eur . j. c * 1 * ( 1998 ) 149 . r. n. cahn , m. s. chanowitz and n. fleishon , phys . lett . * 82b * ( 1979 ) 113 ; l. bergstrm and g. hulth , nucl . b * 259 * ( 1985 ) 137 erratum : [ nucl . b * 276 * ( 1986 ) 744 ] . m. spira , a. djouadi and p. m. zerwas , phys . b * 276 * ( 1992 ) 350 . r. bonciani , v. del duca , h. frellesvig , j. m. henn , f. moriello and v. a. smirnov , jhep * 1508 * ( 2015 ) 108 ; t. gehrmann , s. guns and d. kara , jhep * 1509 * ( 2015 ) 038 . a. abbasabadi , d. bowser - chao , d. a. dicus and w. w. repko , phys . d * 55 * ( 1997 ) 5647 ; a. abbasabadi and w. w. repko , jhep * 0608 * ( 2006 ) 048 and phys . d * 71 * ( 2005 ) 017304 ; d. a. dicus and w. w. repko , phys . d * 87 * ( 2013 ) no.7 , 077301 ; l. b. chen , c. f. qiao and r. l. zhu , phys . b * 726 * ( 2013 ) 306 ; g. passarino , phys . b * 727 * ( 2013 ) 424 . y. sun , h. r. chang and d. n. gao , jhep * 1305 * ( 2013 ) 061 . g. gamberini , g. f. giudice and g. ridolfi , nucl . b * 292 * ( 1987 ) 237 . a. brignole and f. zwirner , phys . b * 299 * ( 1993 ) 72 ; s. heinemeyer and w. hollik , nucl . b * 474 * ( 1996 ) 32 ; k. e. williams and g. weiglein , phys . b * 660 * ( 2008 ) 217 ; k. e. williams , h. rzehak and g. weiglein , eur . j. c * 71 * ( 2011 ) 1669 ; m. krause , m. mhlleitner , r. santos and h. ziesche , arxiv:1609.04185 [ hep - ph ] . s. heinemeyer , w. hollik and g. weiglein , comput . phys . commun . * 124 * ( 2000 ) 76 ; t. hahn , s. heinemeyer , w. hollik , h. rzehak and g. weiglein , comput . commun . * 180 * ( 2009 ) 1426 . j. baglio , r. grber , m. mhlleitner , d. t. nhung , h. rzehak , m. spira , j. streicher and k. walz , comput . commun . * 185 * ( 2014 ) no.12 , 3372 . a. g. akeroyd , a. arhrib and e. m. naimi , eur . j. c * 12 * ( 2000 ) 451 erratum : [ eur . j. c * 14 * ( 2000 ) 371 ] and eur . j. c * 20 * ( 2001 ) 51 ; m. krause , r. lorenz , m. mhlleitner , r. santos and h. ziesche , jhep * 1609 * ( 2016 ) 143 . a. djouadi , j. kalinowski , p. ohmann and p. m. zerwas , z. phys . c * 74 * ( 1997 ) 93 . a. bartl , h. eberl , k. hidaka , t. kon , w. majerotto and y. yamada , phys . * b402 * ( 1997 ) 303 ; a. arhrib , a. djouadi , w. hollik and c. jnger , phys . * d57 * ( 1998 ) 5860 . e. accomando , g. chachamis , f. fugel , m. spira and m. walser , phys . d * 85 * ( 2012 ) 015004 . h. eberl , k. hidaka , s. kraml , w. majerotto and y. yamada , phys . d62 * ( 2000 ) 055006 . w. frisch , h. eberl and h. hlucha , comput . phys . commun . * 182 * ( 2011 ) 2219 . c. weber , k. kovarik , h. eberl and w. majerotto , nucl . phys . * b776 * ( 2007 ) 138 . a. djouadi , p. janot , j. kalinowski and p. m. zerwas , phys . b * 376 * ( 1996 ) 220 . r. y. zhang , w. g. ma , l. h. wan and y. jiang , phys . d * 65 * ( 2002 ) 075018 ; h. eberl , w. majerotto and y. yamada , phys . b * 597 * ( 2004 ) 275 ; t. ibrahim , phys . d * 76 * ( 2007 ) 075012 and phys . d * 77 * ( 2008 ) 065028 ; a. bharucha , a. fowler , g. moortgat - pick and g. weiglein , jhep * 1305 * ( 2013 ) 053 ; a. c. fowler and g. weiglein , jhep * 1001 * ( 2010 ) 108 ; s. heinemeyer and c. schappacher , eur . j. c * 75 * ( 2015 ) no.5 , 230 . e. gross , g. wolf and b. a. kniehl , z. phys . c * 63 * ( 1994 ) 417 erratum : [ z. phys . c * 66 * ( 1995 ) 321 ] ; a. djouadi , m. spira and p. m. zerwas , z. phys . c * 70 * ( 1996 ) 427 . l. g. almeida , s. j. lee , s. pokorski and j. d. wells , phys . d * 89 * ( 2014 ) no.3 , 033006 ; g. p. lepage , p. b. mackenzie and m. e. peskin , arxiv:1404.0319 [ hep - ph ] . n. gray , d. j. broadhurst , w. grafe and k. schilcher , z. phys . c * 48 * ( 1990 ) 673 ; k. g. chetyrkin and m. steinhauser , phys . * 83 * ( 1999 ) 4001 ; k. g. chetyrkin and m. steinhauser , nucl . b * 573 * ( 2000 ) 617 ; k. melnikov and t. v. ritbergen , phys . b * 482 * ( 2000 ) 99 ; p. marquard , a. v. smirnov , v. a. smirnov and m. steinhauser , phys . * 114 * ( 2015 ) no.14 , 142002 ; a. l. kataev and v. s. molokoedov , arxiv:1511.06898 [ hep - ph ] . c. w. bauer , z. ligeti , m. luke , a. v. manohar and m. trott , phys . d * 70 * ( 2004 ) 094017 . k. a. olive _ et al . _ [ particle data group collaboration ] , chin . c * 38 * ( 2014 ) 090001 . m. carena , s. heinemeyer , o. stal , c. e. m. wagner and g. weiglein , eur . j. c * 73 * ( 2013 ) no.9 , 2552 . s. dawson , nucl . b * 359 * ( 1991 ) 283 . d. graudenz , m. spira and p. m. zerwas , phys . * 70 * ( 1993 ) 1372 ; c. anastasiou , s. bucherer and z. kunszt , jhep * 0910 * ( 2009 ) 068 . m. spira , ph . d. thesis , rwth aachen , 1992 . j. butterworth _ et al . _ , j. phys . g * 43 * ( 2016 ) 023001 . s. catani , d. de florian and m. grazzini , jhep * 0105 * ( 2001 ) 025 ; r. v. harlander and w. b. kilgore , phys . d * 64 * ( 2001 ) 013015 and phys . * 88 * ( 2002 ) 201801 ; c. anastasiou and k. melnikov , nucl . phys . b * 646 * ( 2002 ) 220 ; v. ravindran , j. smith and w. l. van neerven , nucl . b * 665 * ( 2003 ) 325 ; s. marzani , r. d. ball , v. del duca , s. forte and a. vicini , nucl . b * 800 * ( 2008 ) 127 . t. gehrmann , m. jaquier , e. w. n. glover and a. koukoutsakis , jhep * 1202 * ( 2012 ) 056 ; c. anastasiou , c. duhr , f. dulat and b. mistlberger , jhep * 1307 * ( 2013 ) 003 ; c. anastasiou , c. duhr , f. dulat , f. herzog and b. mistlberger , jhep * 1312 * ( 2013 ) 088 ; w. b. kilgore , phys . d * 89 * ( 2014 ) 7 , 073008 ; y. li , a. von manteuffel , r. m. schabinger and h. x. zhu , phys . d * 90 * ( 2014 ) 5 , 053006 ; c. anastasiou , c. duhr , f. dulat , e. furlan , t. gehrmann , f. herzog and b. mistlberger , jhep * 1503 * ( 2015 ) 091 ; c. anastasiou , c. duhr , f. dulat , f. herzog and b. mistlberger , phys . rev . * 114 * ( 2015 ) 21 , 212001 . c. anastasiou , c. duhr , f. dulat , e. furlan , t. gehrmann , f. herzog , a. lazopoulos and b. mistlberger , jhep * 1605 * ( 2016 ) 058 . m. krmer , e. laenen and m. spira , nucl . b * 511 * ( 1998 ) 523 ; s. catani , d. de florian , m. grazzini and p. nason , jhep * 0307 * ( 2003 ) 028 ; s. moch and a. vogt , phys . b * 631 * ( 2005 ) 48 ; v. ravindran , nucl . b * 746 * ( 2006 ) 58 and nucl . b * 752 * ( 2006 ) 173 ; a. idilbi , x. d. ji , j. p. ma and f. yuan , phys . d * 73 * ( 2006 ) 077501 ; v. ahrens , t. becher , m. neubert and l. l. yang , eur . j. c * 62 * ( 2009 ) 333 ; d. de florian and m. grazzini , phys . b * 674 * ( 2009 ) 291 . d. de florian , j. mazzitelli , s. moch and a. vogt , jhep * 1410 * ( 2014 ) 176 ; m. bonvini and l. rottoli , phys . d * 91 * ( 2015 ) 5 , 051301 . s. catani , l. cieri , d. de florian , g. ferrera and m. grazzini , nucl . b * 888 * ( 2014 ) 75 . d. de florian and m. grazzini , phys . b * 718 * ( 2012 ) 117 ; m. bonvini and s. marzani , jhep * 1409 * ( 2014 ) 007 ; t. schmidt and m. spira , phys . rev . d * 93 * ( 2016 ) no.1 , 014022 ; m. bonvini , s. marzani , c. muselli and l. rottoli , jhep * 1608 * ( 2016 ) 105 . c. anastasiou , r. boughezal and f. petriello , jhep * 0904 * ( 2009 ) 003 . r. v. harlander and k. j. ozeren , phys . b * 679 * ( 2009 ) 467 and jhep * 0911 * ( 2009 ) 088 ; a. pak , m. rogal and m. steinhauser , phys . b * 679 * ( 2009 ) 473 and jhep * 1002 * ( 2010 ) 025 . m. spira , hep - ph/9510347 and nucl . instrum . a * 389 * ( 1997 ) 357 . r. v. harlander , s. liebler and h. mantler , comput . commun . * 184 * ( 2013 ) 1605 and arxiv:1605.03190 [ hep - ph ] . c. anastasiou , s. buehler , f. herzog and a. lazopoulos , jhep * 1112 * ( 2011 ) 058 . r. k. ellis , i. hinchliffe , m. soldate and j. j. van der bij , nucl . b * 297 * ( 1988 ) 221 ; u. baur and e. w. n. glover , nucl . b * 339 * ( 1990 ) 38 . c. r. schmidt , phys . b * 413 * ( 1997 ) 391 ; d. de florian , m. grazzini and z. kunszt , phys . * 82 * ( 1999 ) 5209 ; c. j. glosser and c. r. schmidt , jhep * 0212 * ( 2002 ) 016 ; v. ravindran , j. smith and w. l. van neerven , nucl . b * 634 * ( 2002 ) 247 . r. v. harlander , t. neumann , k. j. ozeren and m. wiesemann , jhep * 1208 * ( 2012 ) 139 ; t. neumann and m. wiesemann , jhep * 1411 * ( 2014 ) 150 . t. neumann and c. williams , arxiv:1609.00367 [ hep - ph ] . r. boughezal , f. caola , k. melnikov , f. petriello and m. schulze , jhep * 1306 * ( 2013 ) 072 and phys . * 115 * ( 2015 ) no.8 , 082003 ; x. chen , t. gehrmann , e. w. n. glover and m. jaquier , phys . b * 740 * ( 2015 ) 147 ; x. chen , j. cruz - martinez , t. gehrmann , e. w. n. glover and m. jaquier , jhep * 1610 * ( 2016 ) 066 . y. l. dokshitzer , d. diakonov and s. i. troian , phys . * 58 * ( 1980 ) 269 ; g. parisi and r. petronzio , nucl . b * 154 * ( 1979 ) 427 ; j. c. collins , d. e. soper and g. f. sterman , nucl . b * 250 * ( 1985 ) 199 ; s. catani and m. grazzini , nucl . b * 845 * ( 2011 ) 297 . g. bozzi , s. catani , d. de florian and m. grazzini , nucl . b * 737 * ( 2006 ) 73 ; h. mantler and m. wiesemann , eur . j. c * 73 * ( 2013 ) no.6 , 2467 ; m. grazzini and h. sargsyan , jhep * 1309 * ( 2013 ) 129 ; r. v. harlander , h. mantler and m. wiesemann , jhep * 1411 * ( 2014 ) 116 ; f. caola , s. forte , s. marzani , c. muselli and g. vita , jhep * 1608 * ( 2016 ) 150 . e. bagnaschi , r. v. harlander , h. mantler , a. vicini and m. wiesemann , jhep * 1601 * ( 2016 ) 090 . s. frixione , p. nason and c. oleari , jhep * 0711 * ( 2007 ) 070 ; s. alioli , p. nason , c. oleari and e. re , jhep * 1006 * ( 2010 ) 043 . e. bagnaschi , g. degrassi , p. slavich and a. vicini , jhep * 1202 * ( 2012 ) 088 . j. alwall , m. herquet , f. maltoni , o. mattelaer and t. stelzer , jhep * 1106 * ( 2011 ) 128 ; j. alwall _ et al . _ , jhep * 1407 * ( 2014 ) 079 . h. mantler and m. wiesemann , eur . phys . j. c * 75 * ( 2015 ) no.6 , 257 . d. a. dicus and s. willenbrock , phys . d * 39 * ( 1989 ) 751 . t. han , g. valencia and s. willenbrock , phys . * 69 * ( 1992 ) 3274 . t. figy , c. oleari and d. zeppenfeld , phys . d * 68 * ( 2003 ) 073005 ; t. figy and d. zeppenfeld , phys . b * 591 * ( 2004 ) 297 ; e. l. berger and j. m. campbell , phys . d * 70 * ( 2004 ) 073011 . v. n. gribov and l. n. lipatov , sov . j. nucl . phys . * 15 * ( 1972 ) 438 [ yad . fiz . * 15 * ( 1972 ) 781 ] ; g. altarelli and g. parisi , nucl . b * 126 * ( 1977 ) 298 ; y. l. dokshitzer , sov . jetp * 46 * ( 1977 ) 641 [ zh . * 73 * ( 1977 ) 1216 ] . m. ciccolini , a. denner and s. dittmaier , phys . * 99 * ( 2007 ) 161803 and phys . d * 77 * ( 2008 ) 013002 . a. denner , s. dittmaier , s. kallweit and a. mck , comput . commun . * 195 * ( 2015 ) 161 . k. arnold _ et al . _ , comput . commun . * 180 * ( 2009 ) 1661 , k. arnold _ et al . _ , arxiv:1107.4038 [ hep - ph ] and arxiv:1207.4975 [ hep - ph ] ; j. baglio _ et al . _ , arxiv:1404.3940 [ hep - ph ] . p. bolzoni , f. maltoni , s. o. moch and m. zaro , phys . * 105 * ( 2010 ) 011801 and phys . d * 85 * ( 2012 ) 035002 . m. cacciari , f. a. dreyer , a. karlberg , g. p. salam and g. zanderighi , phys . * 115 * ( 2015 ) no.8 , 082002 . r. v. harlander , j. vollinga and m. m. weber , phys . d * 77 * , 053010 ( 2008 ) ; j. r. andersen , t. binoth , g. heinrich and j. m. smillie , jhep * 0802 * ( 2008 ) 057 ; a. bredenstein , k. hagiwara and b. jger , phys . rev . d * 77 * ( 2008 ) 073004 . f. a. dreyer and a. karlberg , phys . * 117 * ( 2016 ) no.7 , 072001 . a. djouadi and m. spira , phys . d * 62 * ( 2000 ) 014004 . w. hollik , t. plehn , m. rauch and h. rzehak , phys . * 102 * ( 2009 ) 091802 ; t. figy , s. palmer and g. weiglein , jhep * 1202 * ( 2012 ) 105 . m. h. seymour , z. phys . c * 62 * ( 1994 ) 127 . t. han and s. willenbrock , phys . b * 273 * ( 1991 ) 167 . o. brein , a. djouadi and r. harlander , phys . b * 579 * ( 2004 ) 149 . l. altenkamp , s. dittmaier , r. v. harlander , h. rzehak and t. j. e. zirke , jhep * 1302 * ( 2013 ) 078 ; a. hasselhuhn , t. luthe and m. steinhauser , arxiv:1611.05881 [ hep - ph ] . m. l. ciccolini , s. dittmaier and m. krmer , phys . d * 68 * ( 2003 ) 073003 ; a. denner , s. dittmaier , s. kallweit and a. mck , jhep * 1203 * ( 2012 ) 075 . g. ferrera , m. grazzini and f. tramontano , jhep * 1404 * ( 2014 ) 039 and phys . b * 740 * ( 2015 ) 51 . j. m. campbell , r. k. ellis and c. williams , jhep * 1606 * ( 2016 ) 179 . o. brein , r. v. harlander and t. j. e. zirke , comput . commun . * 184 * ( 2013 ) 998 ; r. v. harlander , s. liebler and t. zirke , jhep * 1402 * ( 2014 ) 023 . w. astill , w. bizon , e. re and g. zanderighi , jhep * 1606 * ( 2016 ) 154 . w. beenakker et al . , phys . * 87 * ( 2001 ) 201805 . w. beenakker et al . , nucl . * b653 * ( 2003 ) 151203 ; l. reina and s. dawson , phys . rev . * 87 * ( 2001 ) 201804 ; s. dawson , l. h. orr , l. reina , and d. wackeroth , phys . * d67 * ( 2003 ) 071503 . r. frederix , s. frixione , v. hirschi , f. maltoni , r. pittau and p. torrielli , phys . b * 701 * ( 2011 ) 427 . s. dawson and l. reina , phys . d * 57 * , 5851 ( 1998 ) ; e. braaten and h. zhang , phys . d * 93 * ( 2016 ) no.5 , 053014 . m. v. garzelli , a. kardos , c. g. papadopoulos and z. trocsanyi , europhys . * 96 * ( 2011 ) 11001 ; h. b. hartanto , b. jger , l. reina and d. wackeroth , phys . rev . d * 91 * ( 2015 ) no.9 , 094003 . f. cascioli , s. hche , f. krauss , p. maierhfer , s. pozzorini and f. siegert , jhep * 1401 * ( 2014 ) 046 . a. kulesza , l. motyka , t. stebel and v. theeuwes , jhep * 1603 * ( 2016 ) 065 . a. broggio , a. ferroglia , b. d. pecjak , a. signer and l. l. yang , jhep * 1603 * ( 2016 ) 124 ; a. broggio , a. ferroglia , b. d. pecjak and l. l. yang , arxiv:1611.00049 [ hep - ph ] . y. zhang , w. g. ma , r. y. zhang , c. chen and l. guo , phys . b * 738 * ( 2014 ) 1 ; s. frixione , v. hirschi , d. pagani , h. s. shao and m. zaro , jhep * 1409 * ( 2014 ) 065 and jhep * 1506 * ( 2015 ) 184 . a. denner and r. feger , jhep * 1511 * ( 2015 ) 209 . s. dittmaier , m. krmer and m. spira , phys . d * 70 * ( 2004 ) 074010 ; s. dawson , c. b. jackson , l. reina and d. wackeroth , phys . d * 69 * ( 2004 ) 074027 ; m. wiesemann , r. frederix , s. frixione , v. hirschi , f. maltoni and p. torrielli , jhep * 1502 * ( 2015 ) 132 . d. l. rainwater , m. spira and d. zeppenfeld , hep - ph/0203187 ; m. spira , hep - ph/0211145 ; f. maltoni , z. sullivan and s. willenbrock , phys . d * 67 * ( 2003 ) 093005 ; e. boos and t. plehn , phys . rev . d * 69 * ( 2004 ) 094005 ; j. m. campbell _ et al . _ , hep - ph/0405302 . d. dicus , t. stelzer , z. sullivan and s. willenbrock , phys . d * 59 * ( 1999 ) 094016 ; c. balazs , h. j. he and c. p. yuan , phys . rev . d * 60 * ( 1999 ) 114001 . r. v. harlander and w. b. kilgore , phys . d * 68 * ( 2003 ) 013001 . f. maltoni , g. ridolfi and m. ubiali , jhep * 1207 * ( 2012 ) 022 erratum : [ jhep * 1304 * ( 2013 ) 095 ] ; m. lim , f. maltoni , g. ridolfi and m. ubiali , jhep * 1609 * ( 2016 ) 132 . r. harlander , m. krmer and m. schumacher , arxiv:1112.3478 [ hep - ph ] . s. forte , d. napoletano and m. ubiali , phys . lett . b * 751 * ( 2015 ) 331 and phys . b * 763 * ( 2016 ) 190 m. bonvini , a. s. papanastasiou and f. j. tackmann , jhep * 1511 * ( 2015 ) 196 and jhep * 1610 * ( 2016 ) 053 . f. demartin , f. maltoni , k. mawatari and m. zaro , eur . j. c * 75 * ( 2015 ) no.6 , 267 . s. dittmaier , p. hfliger , m. krmer , m. spira and m. walser , phys . d * 90 * ( 2014 ) no.3 , 035010 . wu , w. g. ma , h. s. hou , r. y. zhang , l. han and y. jiang , phys . b * 618 * ( 2005 ) 209 . s. dawson , c. b. jackson and p. jaiswal , phys . d * 83 * ( 2011 ) 115007 . s. dittmaier , m. krmer , a. mck and t. schlter , jhep * 0703 * ( 2007 ) 114 . p. nason , s. dawson and r. k. ellis , nucl . b * 303 * ( 1988 ) 607 and nucl . b * 327 * ( 1989 ) 49 erratum : [ nucl . b * 335 * ( 1990 ) 260 ] ; w. beenakker , h. kuijf , w. l. van neerven and j. smith , phys . d * 40 * ( 1989 ) 54 ; w. beenakker , w. l. van neerven , r. meng , g. a. schuler and j. smith , nucl . b * 351 * ( 1991 ) 507 ; m. czakon , a. mitov and s. moch , nucl . b * 798 * ( 2008 ) 210 ; p. brnreuther , m. czakon and a. mitov , phys . * 109 * ( 2012 ) 132001 ; m. czakon and a. mitov , jhep * 1212 * ( 2012 ) 054 and jhep * 1301 * ( 2013 ) 080 ; m. czakon , p. fiedler and a. mitov , phys . rev . lett . * 110 * ( 2013 ) 252004 ; m. czakon , p. fiedler , d. heymes and a. mitov , jhep * 1605 * ( 2016 ) 034 . w. beenakker , a. denner , w. hollik , r. mertig , t. sack and d. wackeroth , nucl . b * 411 * ( 1994 ) 343 . m. jezabek and j. h. khn , nucl . b * 314 * ( 1989 ) 1 ; c. s. li , r. j. oakes and t. c. yuan , phys . d * 43 * ( 1991 ) 3759 ; a. czarnecki and s. davidson , phys . d * 48 * ( 1993 ) 4183 ; j. m. campbell , r. k. ellis and f. tramontano , phys . rev . d * 70 * ( 2004 ) 094012 ; a. czarnecki and k. melnikov , nucl . b * 544 * ( 1999 ) 520 ; k. g. chetyrkin , r. harlander , t. seidensticker and m. steinhauser , phys . rev . d * 60 * ( 1999 ) 114015 ; i. r. blokland , a. czarnecki , m. slusarczyk and f. tkachov , phys . rev . * 93 * ( 2004 ) 062001 ; i. r. blokland , a. czarnecki , m. slusarczyk and f. tkachov , phys . rev . d * 71 * ( 2005 ) 054004 erratum : [ phys . d * 79 * ( 2009 ) 019901 ] ; a. czarnecki , j. g. krner and j. h. piclum , phys . d * 81 * ( 2010 ) 111503 ; j. gao , c. s. li and h. x. zhu , phys . * 110 * ( 2013 ) no.4 , 042001 ; m. brucherseifer , f. caola and k. melnikov , jhep * 1304 * ( 2013 ) 059 . a. denner and t. sack , nucl . b * 358 * ( 1991 ) 46 ; g. eilam , r. r. mendel , r. migneron and a. soni , phys . * 66 * ( 1991 ) 3105 ; t. kuruma , z. phys . c * 57 * ( 1993 ) 551 . w. peng , m. wen - gan , z. ren - you , j. yi , h. liang and g. lei , phys . d * 73 * ( 2006 ) 015012 erratum : [ phys . d * 80 * ( 2009 ) 059901 ] ; s. dittmaier , m. krmer , m. spira and m. walser , phys . d * 83 * ( 2011 ) 055005 . g. p. gao , g. r. lu , z. h. xiong and j. m. yang , phys . d * 66 * ( 2002 ) 015007 ; s. h. zhu , phys . d * 67 * ( 2003 ) 075006 ; t. plehn , phys . d * 67 * ( 2003 ) 014018 ; e. l. berger , t. han , j. jiang and t. plehn , phys . d * 71 * ( 2005 ) 115012 ; n. kidonakis , pos hep * 2005 * ( 2006 ) 336 . m. flechl , r. klees , m. krmer , m. spira and m. ubiali , phys . d * 91 * ( 2015 ) no.7 , 075015 . c. degrande , r. frederix , v. hirschi , m. ubiali , m. wiesemann and m. zaro , arxiv:1607.05291 [ hep - ph ] . k. d. lane , econf c * 8206282 * ( 1982 ) 222 ; e. eichten , i. hinchliffe , k. d. lane and c. quigg , rev . * 56 * ( 1984 ) 579 addendum : [ rev . * 58 * ( 1986 ) 1065 ] ; n. g. deshpande , x. tata and d. a. dicus , phys . d * 29 * ( 1984 ) 1527 . r. foot , h. lew and g. c. joshi , phys . d * 37 * ( 1988 ) 3161 ; s. willenbrock , phys . * d35 * ( 1987 ) 173 ; a. krause , t. plehn , m. spira and p.m. zerwas , nucl . * b519 * ( 1998 ) 85 ; y. jiang , l. han , w.g . ma , z.h . yu and m. han , j. phys . * g23 * ( 1997 ) 385 [ erratum - ibid . * g23 * ( 1997 ) 1151 ] ; y. jiang , w. g. ma , l. han , m. han and z. h. yu , j. phys . g * 24 * ( 1998 ) 83 ; o. brein and w. hollik , eur . j. * c13 * ( 2000 ) 175 . barrientos bendezu and b.a . kniehl , nucl . * b568 * ( 2000 ) 305 . h. hong - sheng , m. wen - gan , z. ren - you , j. yi , h. liang and x. li - rong , phys . * d71 * ( 2005 ) 075014 ; a. alves and t. plehn , phys . d * 71 * ( 2005 ) 115014 . barrientos bendezu and b.a . kniehl , phys . * d59 * ( 1999 ) 015009 ; o. brein , w. hollik and s. kanemura , phys . * d63 * ( 2001 ) 095001 . barrientos bendezu and b.a . kniehl , phys . d * 63 * ( 2001 ) 015009 . dicus , j.l . hewett , c. kao and t.g . rizzo , phys . d * 40 * ( 1989 ) 787 . w. hollik and s.h . zhu , phys . * d65 * ( 2002 ) 075015 ; j. zhao , c.s . li and q. li , phys . * d72 * ( 2005 ) 114008 . s. moretti , j. phys . g * 28 * ( 2002 ) 2567 . e. bagnaschi , r. v. harlander , s. liebler , h. mantler , p. slavich and a. vicini , jhep * 1406 * ( 2014 ) 167 . g. degrassi , p. p. giardino , f. maltoni and d. pagani , arxiv:1607.04251 [ hep - ph ] . j. baglio , a. djouadi , r. grber , m. m. mhlleitner , j. quevillon and m. spira , jhep * 1304 * ( 2013 ) 151 . j. baglio , a. djouadi and j. quevillon , rept . * 79 * ( 2016 ) no.11 , 116201 . e. w. n. glover and j. j. van der bij , nucl . b * 309 * ( 1988 ) 282 ; t. plehn , m. spira and p. m. zerwas , nucl . b * 479 * ( 1996 ) 46 erratum : [ nucl . b * 531 * ( 1998 ) 655 ] . s. dawson , s. dittmaier and m. spira , phys . d * 58 * ( 1998 ) 115012 . j. grigo , j. hoff , k. melnikov and m. steinhauser , nucl . b * 875 * ( 2013 ) 1 . r. frederix , s. frixione , v. hirschi , f. maltoni , o. mattelaer , p. torrielli , e. vryonidou and m. zaro , phys . b * 732 * ( 2014 ) 142 ; f. maltoni , e. vryonidou and m. zaro , jhep * 1411 * ( 2014 ) 079 . g. degrassi , p. p. giardino and r. grber , eur . j. c * 76 * ( 2016 ) no.7 , 411 ; s. borowka , n. greiner , g. heinrich , s. p. jones , m. kerner , j. schlenk , u. schubert and t. zirke , phys . rev . * 117 * ( 2016 ) no.1 , 012001 erratum : [ phys . * 117 * ( 2016 ) no.7 , 079901 ] ; s. borowka , n. greiner , g. heinrich , s. p. jones , m. kerner , j. schlenk and t. zirke , jhep * 1610 * ( 2016 ) 107 . d. de florian and j. mazzitelli , phys . b * 724 * ( 2013 ) 306 and phys . * 111 * ( 2013 ) 201801 ; j. grigo , k. melnikov and m. steinhauser , nucl . b * 888 * ( 2014 ) 17 . j. grigo , j. hoff and m. steinhauser , nucl . b * 900 * ( 2015 ) 412 . d. y. shao , c. s. li , h. t. li and j. wang , jhep * 1307 * ( 2013 ) 169 ; d. de florian and j. mazzitelli , jhep * 1509 * ( 2015 ) 053 . d. de florian , m. grazzini , c. hanga , s. kallweit , j. m. lindert , p. maierhfer , j. mazzitelli and d. rathlev , jhep * 1609 * ( 2016 ) 151 . l. s. ling , r. y. zhang , w. g. ma , l. guo , w. h. li and x. z. li , phys . d * 89 * ( 2014 ) no.7 , 073001 . h. t. li and j. wang , arxiv:1607.06382 [ hep - ph ] . m. moretti , s. moretti , f. piccinini , r. pittau and a. d. polosa , jhep * 0502 * ( 2005 ) 024 .
|
higgs physics at hadron colliders as the lhc is reviewed within the standard model ( sm ) and its minimal supersymmetric extension ( mssm ) by summarizing the present state - of - the - art of theoretical predictions for the production cross sections and decay rates .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
recently , economy has become an active research area for physicists . they have investigated stock markets using statistical tools , such as the correlation function , multifractal , spin - glass models , and complex networks @xcite . as a consequence , it is now found evident that the interaction therein is highly nonlinear , unstable , and long - ranged . all those companies in the stock market are interconnected and correlated , and their interactions are regarded as the important internal force of the market . the correlation function is widely used to study the internal inference of the market @xcite . however , the correlation function has at least two limitations : first , it measures only linear relations , although a linear model is not a faithful representation of the real interactions in general . second , all it says is only that two series move together , and not that which affects which : in other words , it lacks directional information . therefore participants located in hubs are always left open to ambiguity : they can be either the most influential ones or the weakest ones subject to the market trend all along . it should be noted that introducing time - delay can be a good remedy for these limitations . some authors use such concepts as time - delayed correlation and time - delayed mutual information , and these quantities construct asymmetric matrices by preserving directionality @xcite . in case that the length of delay can be appropriately determined , one can also measure the ` velocity ' whereby the influence spreads . in this paper , however , we rely on a newly - devised variant of information to check its applicability . information is an important keyword in analyzing the market or in estimating the stock price of a given company . it is quantified in rigorous mathematical terms @xcite , and the mutual information , for example , appears as meaningful choice replacing a simple linear correlation even though it still does not specify the direction . the directionality , however , is required to discriminate the more influential one between correlated participants , and can be detected by the transfer entropy ( te ) @xcite . this concept of te has been already applied to the analysis of financial time series by marschinski and kantz @xcite . they calculated the information flow between the dow jones and dax stock indexes and obtained conclusions consistent with empirical observations . while they examined interactions between two huge markets , we may construct its internal structure among _ all _ participants . let us consider two processes , @xmath0 and @xmath1 . transfer entropy @xcite from @xmath1 to @xmath0 is defined as follows : @xmath2 where @xmath3 and @xmath4 represent the states at time @xmath5 of @xmath0 and @xmath1 , respectively .. in terms of relative entropy , it can be rephrased as the _ distance _ from the assumption that @xmath1 has no influence on @xmath0 ( i.e. @xmath6 ) . one may rewrite eq . ( [ te1 ] ) as : @xmath7 from the property of conditional entropy . then the second equality shows that te measures the change of entropy rate with knowledge of the process @xmath1 . ( [ te2 ] ) is practically useful , since the te is decomposed into entropy terms and there has been already well developed technique in entropy estimation . there are two choices in estimating entropy of a given time series . first , the symbolic encoding method divides the range of the given dataset into disjoint intervals and assign one symbol to each interval . the dataset , originally continuous , becomes a discrete symbol sequence . marschinski and kantz @xcite took this procedure and introduced the concept called _ effective transfer entropy_. the other choice exploits the generalized correlation integral @xmath8 . prichard and theiler @xcite showed that the following holds for data @xmath9 : @xmath10,\ ] ] where @xmath11 determines the size of a box in the box - counting algorithm . we define the fraction of data points which lie within @xmath11 of @xmath12 by @xmath13 where @xmath14 is the heaviside function , and calculate its numerical value by the help of the box - assisted neighbor search algorithm @xcite after embedding the dataset into an appropriate phase space . the generalized correlation integral of order @xmath15 is then given by @xmath16 notice that @xmath17 is expressed as an averaged quantity along the trajectory @xmath12 and it implies a kind of ergodicity which converts an ensemble average @xmath18 into a time average , @xmath19 . temporal correlations are not taken into consideration since the daily data already lacks much of its continuity . it is rather straightforward to calculate entropy from a discrete dataset using symbolic encoding . but determining the partition remains as a serious problem , which is referred to as the _ generating partition problem_. even for a two - dimensional deterministic system , the partition lines may exhibit considerably complicated geometry @xcite and thus should be set up with all extreme caution @xcite . hence the correlation integral method is often recommended if one wants to handle continuous datasets without over - simplification , and we will take this route . in addition , one has to determine the parameter @xmath11 . in a sense , this parameter plays a role of defining the resolution or the scale of concerns , just as the number of symbols does in the symbolic encoding method . before discussing how to set @xmath11 , we remark on the finite sampling effect : though it is pointed out that the case of @xmath20 does not suffer much from finiteness of the number of data @xcite , then the positivity of entropy is not guaranteed instead @xcite . thus we choose the conventional shannon entropy , @xmath21 throughout this paper . there have been works done @xcite on correcting entropy estimation . these correction methods , however , can be problematic when calculating te , since the fluctuations in each term of eq . ( [ te2 ] ) are not independent and should not be treated separately @xcite . we actually found that a proper selection of @xmath11 is quite crucial , and decided to inactivate the correction terms here . a good value of @xmath11 will discriminate a real effect from zero . without _ a priori _ knowledge , we need to scan the range of @xmath11 in order to find a proper resolution which yields meaningful results from a time series . for reducing the computational time , however , we resort to the empirical observation that an airline company is quite dependent on the oil price while the dependency hardly appears in the opposite direction . fig . [ fig_te1 ] shows this unilateral effect : the major oil companies , chevron and exxon mobile , have influence over delta airline . from @xmath22 which maximizes the difference between two directions ( fig . [ fig_te2 ] ) , we choose the appropriate scale for analyzing the data . even in observing the temporal evolution , this value gives good discrimination through the whole period . in fig . [ fig_te1 ] , the influence seems reversed on very small length scales . the te , however , is known to increase monotonically under refinement of the partitions in many cases @xcite and the refined partition means the small length scale which is covered by the small @xmath11 in the correlation integral method . hence we regard this reversal as a finite sample effect in this paper , but it seems worth looking further into the characteristics of te analysis . and we set @xmath23 in eq . ( [ te1 ] ) since other values does not make significant differences . this study deals with the daily closure prices of 135 stocks listed on new york stock exchange ( nyse ) from 1983 to 2003 ( @xmath24 trading days , @xmath25 trading day ) , obtained through the website @xcite . we select stocks which is listed on nyse over the whole periods . the companies in a stock market are usually grouped into business sectors or industry categories , and our data contain 9 business sectors ( basic materials , utilities , healthcare , services , consumer goods , financial , industrial goods , conglomerates , technology ) and 69 industry categories . the following method shows how the information flows between the groups : suppose that we have a time series data @xmath26 , representing the daily closure price of a company at time @xmath5 . a stock market analysis usually prefers treating the log return value : @xmath27 to the original price itself , since it satisfies the additive property : @xmath28 . this log return transformation also make the result invariant under the arbitrary scaling of the input data . therefore , in order to measure the information transfer between two companies , say @xmath0 and @xmath1 , we create the log return time series @xmath29 and @xmath30 from the raw price data . then one can calculate the transfer entropies @xmath31 and @xmath32 between them from the equalities in the section 2 . for obtaining an overview of the market , we consider groups of similar companies . let @xmath0 be a company of the group @xmath33 , and @xmath1 be one of the group @xmath34 . the _ information flow index _ between these two groups is defined as a simple sum : @xmath35 in addition , we define the _ net information flow index _ to measure the disparity in influences of the two groups as : @xmath36 if @xmath37 is positive , we can say that the category @xmath33 influences to the category @xmath34 . we examine the market with two grouping methods . one is business sector , and the other is industry category . grouping into business sectors , however , does not exhibit clear directionality : the influence of the @xmath33 sector just alternates from that of the @xmath34 sector . in other words , the difference between @xmath37 and @xmath38 over the whole period is almost 0 ( zero ) . this unclarity comes from the fact that a business sector contains so many diverse companies that its directionality just cancels out . on the other hand , if we construct the asset tree through the minimum spanning tree , each business sector forms a subset of the asset tree and the subsets are connected mainly through the hub . then , it can be said that each of the business sectors forms a cluster @xcite and there are no significant direct links among them . hence we employ the industry category grouping , more detailed than the business sectors . we have to exclude the categories which contain only one element , and table [ category ] lists the remaining industry categories used in the analysis . as in our previous observation , it is verified again that oil companies and airline companies are related in a unilateral way : the category 20 , major oil & gas , has continuing influence over the category 19 , major airline , during the whole 14 periods under examination ( @xmath39 ) . one can easily find such relations in other categories : for example , the category 20 always influences on the categories 15 ( independent oil&gas ) , 22 ( oil&gas equipment&services ) , and 23 ( oil&gas refining&marketing ) . it also affects the category 27 ( regional airlines ) over 13 periods and maintains its power on the whole market during 11 periods ( fig . [ map ] ) . it is well - known that economy greatly depends on the energy supply and price such as oil and gas . transfer entropy analysis quantitatively proves this empirical fact . the top three influential categories ( in terms of periods ) are the categories 10 ( diversified utilities ) , 12 ( electric utilities ) and 20 . all of ten companies in the categories 10 and 12 are again related to the energy industry , such as those for holding , energy delivery , generation , transmission , distribution , and supply of electricity . on the contrary , an airline company is sensitive to the tone of the market . these companies receive information from other categories almost all the time ( category 19 : 11 periods , category 27 : 12 periods ) . the category 8 ( credit services ) and the category 9 ( diversified computer systems , including only hp and ibm in our data ) are also market - sensitive as easily expected . we calculated the transfer entropy with the daily data of the us market . the concept of transfer entropy provides a quantitative value of general correlation and the direction of information . thus it reveals how the information flows among companies or groups of companies , and discriminates the market - leading companies from the market - sensitive ones . as commonly known , the energy such as natural resources and electricity is shown to greatly affect economic activities and the business barometer . this analysis may be applied to predicting the stock price of a company influenced by other ones . in short , te proves its possibility as a promising measure to detect directional information . we suggest that the merits and demerits of te should be judged in details with respect to those of the classical methods like the correlation matrix theory . over the whole 14 periods . the degree of darkness represents the number of periods when @xmath33 is affected by @xmath34 , and @xmath40 s are left blank . for example , the category 10 affects almost all the other categories and is affected by the categories 12 and 25 in a few periods . the row of a market - leading category is bright on the average , while that of a market - sensitive one is dark.,scaledwidth=100.0% ]
|
in terms of transfer entropy , we investigate the strength and the direction of information transfer in the us stock market . through the directionality of the information transfer
, the more influential company between the correlated ones can be found and also the market leading companies are selected .
our entropy analysis shows that the companies related with energy industries such as oil , gas , and electricity influence the whole market .
, , , transfer entropy , information flow , econophysics , stock market + 05.20.gg , 89.65.gh , 89.70.+c
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
thurston ( * ? ? ? * chapter 5 ) showed that a holonomy representation @xmath0 of the group of a hyperbolic knot @xmath1 in @xmath2 can be deformed into a one - parameter family @xmath3 of representations to give a corresponding one - parameter family @xmath4 of singular complete hyperbolic manifolds , the hyperbolic _ cone - manifolds _ of a knot @xmath1 . let @xmath5 be a meridian of @xmath1 . kojima @xcite showed further that @xmath6 is totally determined by the action of @xmath7 which is a rotation of angle @xmath8 around the fixed axis of @xmath9 . a point on @xmath1 of the cone - manifold @xmath6 is in the core of a neighborhood isometric to a cylinder made of an angle @xmath8 wedge by identifying the two boundaries . the @xmath8 is called a _ cone - angle _ along @xmath1 . a point off @xmath1 has a neighborhood isometric to a neighborhood in @xmath10 . we consider the complete hyperbolic structure on the knot complement as the cone - manifold structure of cone - angle zero . as we mentioned , if we increase the cone - angle from zero and if we keep the angle small , we get a one - parameter family of hyperbolic cone - manifolds . similarly , for a link @xmath1 having @xmath11 components , we can get an @xmath11-parameter family of hyperbolic cone - manifolds of a link @xmath1 . in particular , for each two - bridge hyperbolic link , there exists an angle @xmath12 for each link @xmath1 such that @xmath6 is hyperbolic for @xmath13 , euclidean for @xmath14 , and spherical for @xmath15 $ ] @xcite . explicit volume formulae for hyperbolic cone - manifolds of knots and links are known only for a few cases . the volume formulae for hyperbolic cone - manifolds of the knot @xmath16 @xcite , the knot @xmath17 @xcite , the link @xmath18 @xcite , the link @xmath19 @xcite , and the link @xmath20 @xcite have been calculated . in @xcite a method of calculating the volumes of two - bridge knot cone - manifolds were introduced but without explicit formulae . the main purpose of the paper is to find explicit and efficient volume formula for hyperbolic twist knot cone - manifolds . the following theorem gives the formula for @xmath21 for even integers @xmath5 . for odd integers @xmath5 , we can replace @xmath21 by @xmath22 as explained in section [ sec : twist ] . so , the following theorem actually covers all possible hyperbolic twist knots . but for the volume formula , since the knot @xmath23 has to be hyperbolic , we exclude the case when @xmath24 . [ thm : main ] let @xmath23 be a hyperbolic twist knot . let @xmath25 , @xmath26 be the hyperbolic cone - manifold with underlying space @xmath27 and with singular set @xmath23 of cone - angle @xmath8 . then the volume of @xmath25 is given by the following formula @xmath28 where for @xmath29 , @xmath30 with @xmath31 is a zero of the complex distance polynomial @xmath32 which is given recursively by @xmath33 with initial conditions @xmath34 where @xmath35 . reflection ] a knot @xmath1 is a twist knot if @xmath1 has a regular two - dimensional projection of the form in figure [ fig : t2n ] . for example , figure [ fig : knot ] is knot @xmath36 . @xmath1 has 2 right - handed horizontal crossings and @xmath5 right - handed vertical crossings . we will denote it by @xmath21 . note that @xmath21 and its mirror image have the same fundamental group and hence have the same fundamental domain up to isometry in @xmath10 . it follows that @xmath37 and its mirror image have the same fundamental set up to isometry in @xmath10 and have the same volume . so , we will make no distinction between @xmath21 and its mirror image because we are calculating volumes . since the mirror image of @xmath21 is equivalent to @xmath22 , when @xmath5 is odd we will think @xmath22 as @xmath21 . hence a twist knot can be represented by @xmath23 for some integer @xmath11 . let us denote by @xmath38 the exterior of @xmath39 in @xmath27 . in ( * ? * proposition 1 ) , the fundamental group of two - bridge knots is presented . we will use the fundamental group of @xmath40 in @xcite . in @xcite , the fundamental group of @xmath40 is calculated with 2 left - handed horizontal crossings as positive crossings instead of two right - handed horizontal crossings . the following proposition is tailored to our purpose . [ thm : fundamentalgroup ] @xmath41 where @xmath42 . we remark here that @xmath43 of proposition [ thm : fundamentalgroup ] is the meridian which winds around the bottom arc of the twist knot in figure [ fig : t2n ] and @xmath44 is the one that does the top arc as in figure [ fig : t2n ] . let @xmath45 . given a set of generators , @xmath46 , of the fundamental group for @xmath47 , we define a set @xmath48 to be the set of all points @xmath49 , where @xmath50 is a representaion of @xmath47 into @xmath51 . since the defining relation of @xmath47 gives the defining equation of @xmath52 @xcite , @xmath52 is an affine algebraic set in @xmath53 . @xmath52 is well - defined up to isomorphisms which arise from changing the set of generators . we say elements in @xmath54 which differ by conjugations in @xmath51 are _ equivalent_. we use two coordinates to give the structure of the affine algebraic set to @xmath52 . equivalently , for some @xmath55 , we consider both @xmath50 and @xmath56 : for the complex distance polynomial , we use for the coordinates @xmath57,\ ] ] @xmath58,\ ] ] and for the a - polynomial , @xmath59 \text { , } \ \ \ \eta^{\prime}(t)=\left[\begin{array}{cc } m & 0 \\ t & m^{-1 } \end{array } \right ] . \end{array}\ ] ] since we are interested in the excellent component ( the geometric component ) of @xmath60 , in this subsection we set @xmath61 . given the fundamental group of a twist knot @xmath62 where @xmath42 , let @xmath63 and @xmath64 . then the trace of @xmath65 and the trace of @xmath66 are both @xmath67 . let @xmath68 be the complex distance between the axes of @xmath65 and @xmath66 in the hyperbolic space @xmath69 see ( @xcite , p. 68 ) for a formal definition of the complex distance ( width ) between oriented lines in @xmath69 the detailed formulae for calculation of @xmath70 can be found in the proof of theorem 4.3 . [ lem : swc ] for @xmath71 which satisfies @xmath72 and @xmath73 , @xmath74 @xmath75 from the structure of the algebraic set of @xmath60 with coordinates @xmath76 and @xmath77 we have the defining equation of @xmath60 . by plugging in @xmath78 into @xmath79 of that equation and changing the variables to @xmath80 and @xmath81 , we have the following theorem . [ thm : cpolynomial ] for @xmath80 , @xmath81 is a root of the following complex distance polynomial @xmath32 which is given recursively by @xmath82 with initial conditions @xmath83 note that @xmath84 , which gives the defining equations of @xmath60 , is equivalent to @xmath85 in @xmath86 by lemma [ lem : swc ] and @xmath85 in @xmath86 is equivalent to @xmath87 . we may assume @xmath88 , \end{array}\ ] ] @xmath89 , \ \ \ t=\left[\begin{array}{cc } \cos \frac { \alpha}{2 } & i e^{-\frac{\rho}{2 } } \sin \frac { \alpha}{2 } \\ i e^{\frac{\rho}{2 } } \sin \frac { \alpha}{2 } & \cos \frac { \alpha}{2 } \end{array } \right ] , \end{array}\ ] ] and let @xmath90 . then the equation , @xmath91 or @xmath92 gives the complex distance polynomial , where the third equality comes from the cayley - hamilton theorem . by direct computations , @xmath93 , @xmath94 , and @xmath95 have @xmath96 as a common factor . hence , all of @xmath97 s have @xmath96 as a common factor . actually , the common factor comes from the reducible representations . just as the a - polynomials , we left the common factor out of our complex distance polynomials . we divide @xmath97 by @xmath96 and denote @xmath98 by @xmath99 . mathematica _ for the calculations . let @xmath100 and @xmath101 , where @xmath102 is the word obtained by reversing @xmath103 . then @xmath104 and @xmath105 are longitudes which are null - homologus in @xmath40 . we use @xmath105 for this subsection and use both @xmath104 and @xmath105 in section [ sec : pytha ] to keep the original form in @xcite and to keep the familar @xmath105 . one can also deal with section [ sec : pytha ] with only @xmath105 or @xmath104 . define @xmath106 to be a subset of @xmath107 such that @xmath108 and @xmath109 are upper triangular . since every representation can be conjugated into this form , any element of @xmath54 is equivalent to an element of @xmath106 . by adding the equation stating that the bottom - left entry of the matrix corresponding to @xmath108 is zero ( the bottom - left entry of the matrix @xmath109 is already zero and the equation that the bottom - left entry of the matrix corresponding to @xmath108 is equal to zero is redundant in our setting ) , we have defining equations of @xmath106 and hence @xmath106 is an algebraic subset of @xmath54 . define an _ eigenvalue map _ @xmath110 given by taking the top - left entries of @xmath108 , @xmath111 , and of @xmath109 , @xmath79 . the closure of the image @xmath112 of an algebraic component @xmath113 of @xmath106 is an algebraic subset of @xmath114 . if the closure of the image @xmath112 is a curve , there is a unique defining polynomial of this curve up to constant multiples . the _ a - polynomial _ of the knot @xmath23 is defined by the product of all defining polynomials of image curves of @xmath106 . the a - polynomial of a knot can be defined up to sign @xcite . practically , if we let @xmath115 be the upper right entry of @xmath116 and @xmath117 be the upper left entry of @xmath108 , then the a - polynomial of the knot @xmath23 can be obtained by taking the resultant of @xmath118 and @xmath119 over @xmath44 , where the exponents @xmath120 and @xmath121 are chosen so that @xmath118 and @xmath119 become polynomials . in ( * theorem 1 ) , hoste and shanahan presented the a - polynomial of the twist knots . let @xmath122 and @xmath123 . if we let @xmath124 and @xmath125 , then @xmath126 , @xmath127 and we have the following lemma . [ lem : trace ] @xmath128 since @xmath129 we have @xmath130 [ def : longitude ] the _ complex length _ of the longitude @xmath104 or @xmath105 is the complex number @xmath131 modulo @xmath132 satisfying @xmath133 note that @xmath134 is the real length of the longitude of the cone - manifold @xmath25 . we prepare and prove theorem [ thm : pytha ] for @xmath25 with @xmath135 . for @xmath136 , the same pythagrean theorem is obtained by replacing @xmath66 and @xmath137 with @xmath65 and @xmath138 . we will use the oriented line matrix corresponding to a given matrix . one can refer to ( * ? ? ? * section v ) for oriented line matrices . denote by @xmath139 the line matrix corresponding to a matrix , @xmath140 , in @xmath86 . then @xmath141 . by sending common fixed points of @xmath66 and @xmath142 to @xmath143 and @xmath144 , we have @xmath145 , \ \ \ l_{\eta}=\left[\begin{array}{cc } e^{\frac{\gamma_{\alpha}}{2 } } & 0 \\ 0 & e^{-\frac{\gamma_{\alpha}}{2 } } \end{array } \right ] , \end{array}\ ] ] @xmath146,$ ] and the following line matrices @xmath147\\ & = \left[\begin{array}{cc } -i & 0 \\ 0 & i \end{array } \right ] , \end{aligned}\ ] ] @xmath148\\ & = \left[\begin{array}{cc } -i & 0 \\ 0 & i \end{array } \right ] , \end{aligned}\ ] ] which give the orientations of axes of @xmath66 and @xmath137 . ] figure [ fig : polygon ] is the fundamental polyhedron for @xmath149 . the double branched covering space of the polyhedron along @xmath150 and @xmath151 is the lens space @xmath152 . the fundamental polyhedron for the hyperbolic cone - manifold of @xmath153 can be obtained from the fundamental polyhedron for @xmath149 by deforming the cone - angle continuously . recall that a lambert quadrangle is a quadrangle with three right angles and one acute angle , not necessarily lying on a plane . you can consult ( * ? ? ? * of section vi ) or @xcite for the trigonometry of a lambert quadrangle or a right angled hexagon . let @xmath5 be the midpoint of @xmath154 . then the quadrangle @xmath155 is a lambert quadrangle with acute angle @xmath156 , which can be considered as a right angled hexagon which is a generalized right angled triangle . the six sides are @xmath157 . by applying the law of cosines to the hexagon , we get the formula in the following theorem geometrically and the same argument works for all twist knots . hence , we call the following theorem pythagorean theorem . now , we are ready to prove the following theorem which gives theorem [ thm : mpytha ] . recall that @xmath131 modulo @xmath132 is the _ complex length _ of the longitude @xmath104 or @xmath105 of @xmath25 . ( pythagorean theorem)[thm : pytha ] let @xmath25 be a hyperbolic cone - manifold and let @xmath68 be the complex distance between the oriented axes @xmath65 and @xmath66 . then we have @xmath158 suppose @xmath135 . @xmath159 where the first equality comes from @xcite , the sixth equality comes from the cayley - hamilton theorem , and the seventh equality comes from lemma [ lem : trace ] . let @xmath160 , @xmath161 , and @xmath162 . then @xmath163 , @xmath164 , and @xmath165 hence , @xmath166 which is equivalent to @xmath167 by solving for @xmath30 , we have @xmath168 by putting back @xmath169 , we have @xmath170 which is equivalent to @xmath158 pythagorean theorem [ thm : pytha ] gives the following theorem which relates the zeros of @xmath171 and the zeros of @xmath172 for @xmath173 , @xmath174 and @xmath175 . [ thm : mpytha ] let @xmath175 and @xmath173 . then the following formulae show that there is a one to one correspondence between the zeros of @xmath171 and the zeros of @xmath172 : @xmath176 with the same notation as in the proof of theorem [ thm : pytha ] , @xmath177 if we solve the above equation , @xmath178 for @xmath111 , we have @xmath179 we mention here that the proof can be done without referring to a - polynomial . we identified @xmath111 with a root of @xmath180 because a - polynomial is rather well - known . for @xmath135 and @xmath181 ( @xmath80 ) , @xmath180 and @xmath182 have @xmath183 component zeros , and for @xmath184 , @xmath185 component zeros . for each @xmath11 , there exists an angle @xmath12 such that @xmath25 is hyperbolic for @xmath13 , euclidean for @xmath14 , and spherical for @xmath15 $ ] @xcite . from the following equality [ equ : absl ] , when @xmath186 , which is equivalent to @xmath14 , @xmath187 . hence , when @xmath8 increases from @xmath143 to @xmath188 , two complex numbers @xmath30 and @xmath189 approach to a same real number . in other words , @xmath190 has a muliple root and hence @xmath191 has a multiple root by theorem [ thm : mpytha ] . denote by @xmath192 be the greatest common factor of the discriminant of @xmath193 over @xmath111 and the discriminant of @xmath194 over @xmath30 . then @xmath188 will be one of the zeros of @xmath192 . from theorem [ thm : mpytha ] , we have following equality , @xmath195 for the volume , we can either choose @xmath196 or @xmath197 . we choose @xmath111 with @xmath198 and hence we have @xmath31 by equality [ equ : absl ] . using the schlfli formula , we calculate the volume of @xmath199 for each component with @xmath198 and having one of the zeros @xmath188 of @xmath192 with @xmath12 on it . the component which gives the maximal volume is the excellent component @xcite . on the geometric component we have the volume of a hyperbolic cone - manifold @xmath25 for @xmath26 : @xmath200 where the first equality comes from the schlfli formula for cone - manifolds ( theorem 3.20 of @xcite ) , the second equality comes from the fact that @xmath134 is the real length of the longitude of @xmath25 , the third equality comes from the fact that @xmath201 for @xmath202 by equality [ equ : absl ] since all the characters are real ( the proof of proposition 6.4 of @xcite ) for @xmath202 , and @xmath203 is a zero of the discriminant @xmath192 . we note that the fundamental set of the two - bridge link orbifolds are constructed in @xcite . we also note that the explicit formulae for the chern - simons invariant of the twist knot orbifolds are presented in @xcite and the a - polynomials of twist knots are obtained from the complex distance polynomials in @xcite . table [ tab1 ] gives the approximate volume of @xmath204 for each n between @xmath205 and @xmath206 except the unknot and the torus knot and for each component with @xmath31 and having one of the zeros of @xmath192 with @xmath12 on it . we used simpson s rule for the approximation with @xmath207 intervals from @xmath143 to @xmath188 . in that way our approximate volume on the geometric component is the same as that of snappea up to four decimal points . the geometric volume is written one more time on the rightmost column . table [ table2 - 1 ] ( resp . table [ table2 - 2 ] ) gives the approximate volume of the hyperbolic twist knot cone - manifold , @xmath208 for @xmath11 between @xmath209 and @xmath206 ( resp . for @xmath11 between @xmath205 and @xmath210 ) and for @xmath211 between @xmath212 and @xmath213 , and of its cyclic covering , @xmath214 . we again used simpson s rule for the approximation with @xmath207 intervals from @xmath215 to @xmath188 . mathematica _ for the calculations . [ tab1 ] ll .volume of the hyperbolic twist knot cone - manifold , @xmath208 for @xmath11 between @xmath209 and @xmath206 and for @xmath211 between @xmath209 and @xmath213 , and of its cyclic covering , @xmath214 . [ cols="^,^,^",options="header " , ] the authors would like to thank prof . hyuk kim for his various helps and anonymous referees for their careful suggestions . hugh m. hilden , mara teresa lozano , and jos mara montesinos - amilibia . volumes and chern - simons invariants of cyclic coverings over rational knots . in _ topology and teichmller spaces ( katinkulta , 1995 ) _ , pages 3155 . world sci . publ . , river edge , nj , 1996 . alexander d. mednykh . trigonometric identities and geometrical inequalities for links and knots . in _ proceedings of the third asian mathematical conference , 2000 ( diliman ) _ , pages 352368 . world sci . publ . , river edge , nj , 2002 .
|
we calculate the volumes of the hyperbolic twist knot cone - manifolds using the schlfli formula .
even though general ideas for calculating the volumes of cone - manifolds are around , since there is no concrete calculation written , we present here the concrete calculations .
we express the length of the singular locus in terms of the distance between the two axes fixed by two generators . in this way
the calculation becomes easier than using the singular locus directly .
the volumes of the hyperbolic twist knot cone - manifolds simpler than stevedore s knot are known . as an application
, we give the volumes of the cyclic coverings over the hyperbolic twist knots .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
in this section the definitions of the basic concepts and the notation to be used throughout this contribution shall be presented . differentiable manifolds are denoted by italic capital letters @xmath0 and , to our purposes , all such manifolds will be connected causally orientable lorentzian manifolds of dimension @xmath1 . the signature convention is set to @xmath2 . @xmath3 and @xmath4 will stand respectively for the tangent and cotangent spaces at @xmath5 , and @xmath6 ( resp . @xmath7 ) is the tangent bundle ( cotangent bundle ) of @xmath8 . similarly the bundle of @xmath9-contravariant and @xmath10-covariant tensors of @xmath11 is denoted @xmath12 . if @xmath13 is a diffeomorphism between @xmath11 and @xmath14 , the push - forward and pull - back are written as @xmath15 and @xmath16 respectively . the hyperbolic structure of the lorentzian scalar product naturally splits the elements of @xmath3 into timelike , spacelike , and null , and as usual we use the term _ causal _ for the vectors ( or vector fields ) which are non - spacelike . to fix the notation we introduce the sets : @xmath17 the simplest example ( leaving aside $ \r^+$ ) of causal tensors are the causal 1-forms ( $ \equiv \dp_{1}(v)$ ) \cite{s - e}\footnote{see also bergqvist 's and senovilla 's contributions to this volume . } , while a general characterization of $ \dp^{+}_r\equiv \dp^{+}_r(v)$ is the following ( see \cite{sup } for a proof)$^1 $ : \begin{prop } $ { \bf t}\in \dp^{+}_r$ if and only if the components $ t_{i_1\dots i_r}$ of $ { \bf t}$ in all orthonormal bases fulfill $ t_{0\dots 0}\geq|t_{i_1\dots i_r}|$ , $ \forall i_1\dots i_r$ , where the $ 0$-index refers to the temporal component . \label{orthonormal } \end{prop } % \p see \cite{sup}.\n we are now ready to present our main concept , which tries to capture the notion of some kind of relation between the causal structure of $ v$ and $ w$. \begin{defi } let $ \f : v\rightarrow w$ be a global diffeomorphism between two lorentzian manifolds . we shall say that $ w$ is properly causally related with $ v$ by $ \f$ , denoted $ v\prec_{\f}w$ , if for every $ \x\in\z^{+}(v)$ we have that $ \f^{'}\x$ belongs to $ \z^{+}(w)$. $ w$ is said to be properly causally related with $ v$ , denoted simply as $ v\prec w$ , if $ \exists \f$ such that $ v\prec_{\f}w$. \label{prec } \end{defi } { \bf remarks } \begin{enumerate } \item this definition can also be given for any set $ \zeta\subseteq v$ by demanding that $ ( \f^{'}\x)_{\f(x)}\in \z^{+}(\f(x))\,\,\,\ , \forall \x\in\z^{+}(x)$ , $ \forall x\in\zeta$. \item two diffeomorphic lorentzian manifolds may fail to be properly causally related as we shall show later with explicit examples . \end{enumerate } \begin{defi } two lorentzian manifolds $ v$ and $ w$ are called causally isomorphic if $ v\prec w$ and $ w\prec v$. this shall be written as $ v\sim w$. \label{equiv } \end{defi } we claim that if $ v\sim w$ then their causal structure are somehow the same . let $ \g$ and $ \tilde{\g}$ be the lorentzian metrics of $ v$ and $ w$ respectively . by using \be \tilde{\g}(\f^{'}\x,\f^{'}\y)=\f^{*}\tilde{\g}(\x,\y ) , \label{pull - back } \ee we immediately realize that $ v\prec_{\f } w$ implies that $ \f^{*}\tilde{\g}\in\dp^{+}_2(v)$. conversely , if $ \f^{*}\t\in\dp^{+}_2(v)$ then for every $ \x\in\z^{+}(v)$ we have that $ ( \f^{*}\t)(\x,\x)=\t(\f^{'}\x,\f^{'}\x)\geq 0 $ and hence $ \f^{'}\x\in\z(w)$. however , it can happen that $ \z^{+}(v)$ is actually mapped to $ \z^-(w)$ , and $ \z^{-}(v)$ to $ \z^+(w)$. this only means that $ w$ { \em with the time - reversed orientation } is properly causally related with $ v$. keeping this in mind , the assertion $ \f^{*}\t\in\dp^{+}_2(v)$ will be henceforth taken as equivalent to $ v\prec_{\f}w$. \section{mathematical properties } let us present some mathematical properties of proper causal relations . \begin{prop } if $ v\prec_{\f}w$ then : \begin{enumerate } \item $ \x\in\z^{+}(v)$ is timelike $ \longrightarrow$ $ \f^{'}\x\in \z^{+}(w)$ is timelike . \item $ \x\in\z^{+}(v)$ and $ \f^{'}\x\in \z^{+}(w)$ is null $ \longrightarrow$ $ \x$ is null . \end{enumerate } \label{caus } \end{prop } \p for the first implication , if $ \x\in\z^{+}(v)$ is timelike we have , according to equation ( \ref{pull - back } ) , that $ \f^{*}\tilde{\g}(\x,\x)=\t(\f^{'}\x,\f^{'}\x)$ which must be a strictly positive quantity as $ \f^{*}\tilde{\g}\in\dp^{+}_2(w)$ \cite{s - e}. for the second implication , equation ( \ref{pull - back } ) implies $ 0=\f^{*}\t(\x,\x)$ which is only possible if $ \x$ is null since $ \f^{*}\t\in\dp^{+}_2(v)$ and $ \x\in\z^{+}(v)$ ( see again \cite{s - e}).\n \begin{prop } $ v\prec_{\f}w \hspace{2 mm } \longleftrightarrow \hspace{2 mm } \f^{'}\x\in\z^{+}(w)$ for all null $ \x\in\z^{+}(v)$. \label{null } \end{prop } \p for the non - trivial implication , making again use of ( \ref{pull - back } ) we can write : \ [ \f^{'}\x\in\z^{+}(w)\ \forall\x\ \mbox{null in}\ \z^{+}(v)\leftrightarrow\f^{*}\tilde{\g}(\x,\y)\geq 0\ \ \forall\ \x,\y\ \mbox{null in}\ \z^{+}(v ) \ ] which happens if and only if $ \f^{*}\tilde{\g}$ is in $ \dp^{+}_2(v)$ ( see \cite{s - e } property 2.4).\n \begin{prop}[transitivity of the proper causal relation]\hspace{0.1 cm } \\ if $ v\prec_{\f } w$ and $ w\prec_{\psi } u$ then $ v\prec_{\psi\circ\f}u$ \label{order } \end{prop } \p consider any $ \x\in \z^{+}(v)$. since $ v\prec_{\f } w$ , $ \f^{'}\x\in \z^{+}(w)$ and since $ w\prec_{\psi } u$ we get $ \psi^{'}[\f^{'}\x]\in\z^{+}(u)$ so that $ ( \psi\circ\f)^{'}\x\in\z^{+}(u)$ from what we conclude that $ v\prec_{\psi\circ\f } u$.\n therefore , we see that the relation $ \prec$ is a preorder . notice that if $ v\sim w$ ( that is $ v\prec w$ and $ w\prec v$ ) this does not imply that $ v = w$. nevertheless , one can always define a partial order for the corresponding classes of equivalence . next , we identify the part of the boundary of the null cone which is preserved under a proper causal relation . a lemma is needed first . recall that $ \x$ is called an `` eigenvector '' of a 2-covariant tensor $ { \bf t}$ if $ { \bf t}(\cdot , \x ) = \lambda \g ( \cdot , \x ) $ and $ \lambda$ is then the corresponding eigenvalue . \begin{lem } if $ { \bf t}\in \dp^{+}_2(x)$ and $ \x\in\z^{+}(x)$ then $ { \bf t}(\x,\x)=0\ \longleftrightarrow\x$ is a null eigenvector of $ { \bf t}$. \label{null - eigen } \end{lem } \p let $ \x\in\z^{+}(x)$ and assume $ 0={\bf t}(\x,\x)=t_{ab}x^{a}x^{b}$. then since $ t_{ab}x^{b}\in\dp^{+}_1(x)$ \cite{s - e } we can conclude that $ x_a$ and $ t_{ab}x^{b}$ must be proportional which results in $ x^{a}$ being a null eigenvector of $ t_{ab}$. the converse is straightforward.\n \begin{prop } assume that $ v\prec_{\f } w$ and $ \x\in\z^{+}(x),\ x\in v$. then $ \f^{'}\x$ is null at $ \f(x)\in w$ if and only if $ \x$ is a null eigenvector of $ \f^{*}\tilde{\g}(x)$. \label{cone } \end{prop } \p let $ \x$ be in $ \z^{+}(x)$ and suppose $ \f^{'}\x$ is null at $ \f(x)$. then , according to proposition \ref{caus } , $ \x$ is also null at $ x$. on the other hand we have $ 0=\tilde{\g}(\f^{'}\x,\f^{'}\x)=\f^{*}\tilde{\g}(\x,\x)$ and since $ \f^{*}\t|_{x}\in\dp^{+}_2(x)$ , lemma \ref{null - eigen } implies that $ \x$ is a null eigenvector of $ \f^{*}\t$ at $ x$.\n the vectors which remain null under the causal relation $ \f$ are called its { \em canonical null directions}. on the other hand , the null eigenvectors of $ { \bf t}\in\dp^{+}_2 $ can be used to classify this tensor , as proved in \cite{s - e}. as a result we have \begin{prop } if the relation $ v\prec_{\f}w$ has n linearly independent canonical null directions then $ \f^{*}\t=\lambda\g$. \label{conf } \end{prop } \p if there exist $ n$ independent canonical null directions , then $ \f^{*}\t$ has $ n$ independent null eigenvectors which is only possible if $ \f^{*}\t$ is proportional to the metric tensor $ \g$ ( \cite{s - e , sup}.)\n proposition \ref{conf } has an interesting application in the following theorem \begin{theo } suppose that $ v\prec_{\f}w$ and $ w\prec_{\f^{-1}}v$. then $ \f^{*}\t=\lambda\g$ and $ ( \f^{-1})^{*}\g=\fr{1}{(\f^{-1})^{*}\lambda}\t$ for some positive function $ \lambda$ defined on $ v$. \label{inv } \end{theo } \p under these hypotheses , using proposition \ref{caus } , we get the following intermediate results \bea \f^{'}\x\in \z^{+}(w)\ \mbox{null and $ \x\in\z^{+}(v)$}\longrightarrow\x\ \mbox{is null,}\nonumber\\ ( \f^{-1})^{'}\y\in\z^{+}(v)\ \mbox{null and $ \y\in\z^{+}(w)$}\longrightarrow\y\ \mbox{is null.}\nonumber \eea now , let $ \x\in \z^{+}(v)$ be null and consider the unique $ \y\in t(v)$ such that $ \x=(\f^{-1})^{'}\y$. then $ \y=\f^{'}\x$ and $ \y\in\z^{+}(w)$ as $ \f$ sets a proper causal relation and $ \x$ is in $ \z^{+}(v)$. hence , according to the second result above $ \y$ must be null and we conclude that every null $ \x\in \z^{+}(v)$ is push - forwarded to a null vector of $ \z^{+}(w)$ which in turn implies that $ \f^{*}\t=\lambda\g$. in a similar fashion , we can prove that $ ( \f^{-1})^{*}\g=\mu\t$ and hence $ ( \f^{-1})^{*}\lambda = 1/\mu$.\n \begin{coro } $ v\prec_{\f}w$ and $ w\prec_{\f^{-1}}v \,\ , \longleftrightarrow \,\ , \f$ is a conformal relation . \end{coro } \section{applications to causality theory } in this section we will perform a detailed study of how two lorentzian ma-\\ nifolds $ v$ and $ w$ such that $ v\prec_{\f}w$ share common causal features . to begin with , we must recall the basic sets used in causality theory , namely $ i^{\pm}(p)$ and $ j^{\pm}(p)$ for any point $ p\in v$ ( these definitions can also be given for sets ) . one has $ q\in j^+(p)$ ( respectively $ q\in i^+(p)$ ) if there exists a continuous future directed causal ( resp.\ timelike ) curve joining $ p$ and $ q$. recall also the cauchy developments $ d^{\pm}(\zeta)$ for any set $ \zeta\subseteq v$ \cite{ff , w , cond}. another important concept is that of future set : $ \a\subset v$ is said to be a future set if $ i^{+}(\a)\subseteq \a$. for example $ i^{+}(\zeta)$ is a future set for any $ \zeta$. all these concepts are standard in causality theory and are defined in many references , see for instance \cite{ff , w , cond}. \begin{prop } if $ v\prec_{\f } w$ then , for every set $ \zeta\subseteq v$ , we have $ \f(i^{\pm}(\zeta))\subseteq i^{\pm}(\f(\zeta))$ and $ \f(j^{\pm}(\zeta))\subseteq j^{\pm}(\f(\zeta))$. \label{set } \end{prop } \p it is enough to prove it for a single point $ p\in v$ and then getting the result for every $ \zeta$ by considering it as the union of its points . for the first relation , let $ y$ be in $ \f(i^{+}(p))$ arbitrary and take $ x\in i^{+}(p)$ such that $ \f(x)=y$. choose a future - directed timelike curve $ \g$ joining $ p$ and $ x$. from proposition \ref{caus } , $ \f(\g)$ is then a future - directed timelike curve joining $ \f(p)$ and $ y$ , so that $ y\in i^{+}(\f(p))$. the second assertion is proven in a similar way using again proposition \ref{caus}. the proof for the past sets is analogous.\n the converse of this proposition does not hold in general unless we impose some causality conditions on the spacetime . \begin{defi } a lorentzian manifold $ v$ is said to be distinguishing if for every neighbourhood $ u_p$ of $ p\in v$ there exist another neighbourhood $ b_p\subset u_p$ containing $ p$ which intersects every causal curve meeting $ p$ in a connected set . \label{distinguishing } \end{defi } we need some concepts of standard causality theory . for any $ p\in v$ one can introduce normal coordinates in a neighbourhood $ { \cal n}_p$ of $ p$ ( see , e.g. \cite{cond } ) . then the exponential map provides a diffeomorphism $ \exp:{\cal o}\subset t_p(v)\rightarrow { \cal n}_p$ where $ { \cal o}$ is an open neighbourhood of $ \vec{0}\in t_p(v)$. the interior of the future ( past ) light cone of $ p$ is defined by $ c^{\pm}_p=\exp(\mbox{int}(\z^{\pm}(p))\cap { \cal o})$ , and obviously $ c^{\pm}_p\subseteq i^{\pm}(p)$ \cite{cond}. other important issue deals with the chronology relation $ < \!\!<$ between two points . we have $ p<\!\!<q$ if there exist a future timelike curve joining $ p$ and $ q$. see \cite{kronheimer } for an axiomatic study of this relation . \begin{prop } let $ \g$ be a piecewise continuous curve of a distinguishing lorentzian manifold $ ( v,\g)$. then , $ \g$ is total with respect to $ < \!\!<$ if and only if $ \g$ is timelike . \end{prop } \p clearly if $ \g$ is timelike then $ \g$ must be a total set for the relation $ < \!\!<$ ( this is true for every spacetime ) . for the converse consider a curve $ \g$ which is total with respect to $ < \!\!<$ and let $ q\in\g$ be an arbitrary point of the curve . if we take a normal neighbourhood of $ q$ , $ { \cal n}_q$ then we can find a neighbourhood $ u_q$ of $ q$ which is intersected in a connected set by every causal curve meeting $ q$. now , if we pick up a point $ z\in\g\cap u_q$ we have that either $ q<\!\!<z$ or $ z<\!\!<q$. assuming the former we deduce that there exists a timelike curve $ \tilde{\g}$ joining $ q$ and $ z$ which implies that $ \tilde{\g}\cap u_q$ is a connected set . this property together with the distinguishability of $ v$ implies that $ \tilde{\g}$ must be a subset of $ u_q$ and hence $ \tilde{\g}\subset { \cal n}_q$ from what we conclude that $ \tilde{\g}\subset c_p$ ( \cite{cond } ) and hence $ z\in c_p$ $ \forall z\in\g\cap u_q$ which is only possible if $ \g\cap u_q$ is timelike . by covering $ \g$ with sets of the form $ \g\cap u_q$ , $ q\in \g$ we arrive at the desired result.\n \begin{prop } let $ \f : v\rightarrow w$ be a diffeomorphism with the property $ \f(i^{+}(p))\subseteq i^{+}(\f(p))\ \forall p\in v$. then if $ w$ is distinguishing , $ \f$ is a proper causal relation . a similar result holds replacing $ i^{+}$ by $ i^{-}$. \label{chronological } \end{prop } \p from the statement of this proposition is clear that $ \forall$ $ p , q$ of $ v$ such that $ p<\!\!<q$ then $ \f(p)<\!\!<\f(q)$. therefore every timelike curve $ \g$ of $ v$ is mapped onto a continuous curve in $ w$ total with respect to $ < \!\!<$ and hence timelike due to the distiguishability of $ w$. furthermore if the curve $ \g$ is future directed then $ \f(\g)$ must be also future directed since $ < \!\!<$ is preserved which is only possible if every timelike future - pointing vector is mapped onto a future - pointing timelike vector . as a consequence , if $ \k$ is a null vector , $ \f^{'}\k$ must be a causal vector ( to see it just construct a sequence of timelike future directed vectors converging to $ \k$ ) which proves that $ \f$ is a proper causal relation.\n the results for the cauchy developments are the following : \begin{prop } if $ v\prec_{\f } w$ then $ d^{\pm}(\f(\zeta))\subseteq \f(d^{\pm}(\zeta ) ) , \,\ , \forall \zeta\subseteq v$. \label{cauch } \end{prop } \p it is enough to prove the future case . let $ y\in d^{+}(\f(\zeta))$ arbitrary and consider any causal past directed curve $ \g^{-}_{\f^{-1}(y)}\subset v$ containing $ \f^{-1}(y)$. since the image curve by $ \f$ of $ \g^{-}_{\f^{-1}(y)}$ is a causal curve passing through $ y$ , ergo meeting $ \f(\zeta)$ , we have that $ \g^{-}_{\f^{-1}(y)}$ must meet $ \zeta$ from what we conclude that $ y\in\f(d^{+}(\zeta))$ due to the arbitrariness of $ \g^{-}_{\f^{-1}(y)}$.\n \begin{coro } if $ \s\subset w$ is a cauchy hypersurface then $ \f^{-1}(\s)$ is also a cauchy hypersurface of $ v$. \label{hyp } \end{coro } \p if $ \s$ is a cauchy hypersurface then $ d(\s)=w$ , and from proposition \ref{cauch } $ d(\s)\subseteq\f(d(\f^{-1}(\s)))$. since $ \f$ is a diffeomorphism the result follows.\n one can prove the impossibility of the existence of proper causal relations sometimes . for instance , from the previous corollary we deduce that $ v\prec w$ is impossible if $ w$ is globally hyperbolic but $ v$ is not . other impossibilities arise as follows . let us recall that , for any inextendible causal curve $ \g$ , the boundaries $ \partial i^{\pm}(\g)$ of its chronological future and past are usually called its future and past event horizons , sometimes also called particle horizons \cite{ff , w , cond}. of course these sets can be empty ( then one says that $ \g$ has no horizon ) . \begin{prop } suppose that every inextendible causal future directed curve in $ w$ has a non - empty $ \partial i^{-}(\g)$ ( $ \partial i^{+}(\g)$ ) . then any v such that $ v\prec w$ can not have inextendible causal curves without past ( future ) event horizons . \label{hor } \end{prop } \p if there were a future - directed curve $ \g$ in $ v$ with $ \partial i^{-}(\g)=\emptyset$ , $ i^{-}(\g)$ would be the whole of $ v$. but according to proposition \ref{set } $ \f(i^{-}(\g))\subseteq i^{-}(\f(\g))$ from what we would conclude that $ i^{-}(\f(\g))=w$ against the assumption . \n the class of future ( or past ) sets characterize the proper causal relations for distinguishing spacetimes as it is going to be shown next ( every statement for future objects has a counterpart for the past ) . \begin{lem } if $ \a$ is a future set then $ p\in\overline{\a}\longleftrightarrow i^{+}(p)\subseteq\a$. \label{closure } \end{lem } \p suppose $ i^{+}(p)\subseteq a$. then since $ c_p^{+}\subseteq i^{+}(p)$ and $ p\in \overline{c_p^{+}}$ we have that $ u_p\cap c_p^{+}\neq\emptyset$ for every neighbourhood $ u_p$ of $ p$ which in turn implies that $ u_p\cap\a\neq\emptyset$ and hence $ p\in\overline{\a}$. conversely , let $ p$ be any point of $ \overline{\a}$ then $ i^+(p)\subseteq i^+(\overline{a})=i^+(a)\subseteq a$.\n \begin{theo } suppose that $ ( w,\t)$ is a distinguishing spacetime . then a diffeomorphism $ \f:(v,\g)\rightarrow ( w,\t)$ is a proper causal relation if and only if $ \f^{-1}(\a)$ is a future set for every future set . $ \a\subseteq w$. \label{key } \end{theo } \p suppose $ \a\subseteq w$ is a future set , $ v\prec_{\f } w$ and take $ \f^{-1}(\a)\subseteq v$. proposition \ref{set } implies $ \f(i^{+}(\f^{-1}(\a)))\subseteq i^{+}(\f(\f^{-1}(\a)))=i^{+}(\a)\subseteq\a$ which shows that $ i^{+}(\f^{-1}(\a))\subseteq \f^{-1}(\a)$. conversely , for any $ p\in v$ take the future set $ i^{+}(\f(p))$ and consider the future set $ \f^{-1}(i^{+}(\f(p)))$. as $ \f(p)\in\overline{i^{+}(\f(p))}$ then $ p\in\overline{\f^{-1}(i^{+}(\f(p)))}$ and according to lemma \ref{closure } $ i^{+}(p)\subseteq\f^{-1}(i^{+}(\f(p)))$ so that $ \f(i^{+}(p))\subseteq i^{+}(\f(p))$. since this holds for every $ p\in v$ and $ w$ is distinguishing , proposition \ref{chronological } ensures that $ \f$ is a proper causal relation.\n this theorem has important consequences . \begin{prop } if $ v\sim w$ and both manifolds are distinguishing , then there is a one - to - one correspondence between the future ( and past ) sets of $ v$ and $ w$. \label{conserv } \end{prop } \vspace{-0.5 cm } \p if $ v\sim w$ then $ v\prec_{\f}w$ and $ w_{\prec\psi}v$ for some diffeomorphisms $ \f$ and $ \psi$. by denoting with $ { \cal f}_v$ and $ { \cal f}_w$ the set of future sets of $ v$ and $ w$ respectively , we have that $ \f^{-1}({\cal f}_w)\subseteq{\cal f}_v$ and $ \psi^{-1}({\cal f}_v)\subseteq{\cal f}_w$ , due to theorem \ref{key}. since both $ \f$ and $ \psi$ are bijective maps we conclude that $ { \cal f}_v$ is in one - to - one correspondence with a subset of $ { \cal f}_w$ and vice versa which , according to the equivalence theorem of bernstein \cite{bern } , implies that $ { \cal f}_v$ is in one - to - one correspondence with $ { \cal f}_w$.\n \section{causal transformations } in this section we will see how the concepts above generalize , in a natural way , the group of conformal transformations in a lorentzian manifold $ v$. \begin{defi } a transformation $ \f : v\longrightarrow v$ is called { \em causal } if $ v\prec_{\f}v$. \label{gg } \end{defi } the set of causal transformations of $ v$ will be denoted by $ \c(v)$. this is a subset of the group of transformations of $ v$ which is closed under the composition of diffeomorphisms , due to proposition \ref{order } , and contains the identity map . this algebraic structure is well - known , see e.g. \cite{semigroup } , and called subsemigroup with identity or submonoid . thus , $ \c(v)$ is a { \em submonoid } of the group of diffeomorphisms of $ v$. nonetheless , $ \c(v)$ usually fails to be a group . in fact we have , \begin{prop } every subgroup of causal transformations is a group of conformal transformations . \label{group } \end{prop } \p let $ g\subseteq \c(v)$ be a subgroup of causal transformations and consider any $ \f\in g$ , so that both $ \f$ and $ \f^{-1}$ are causal transformations . then $ \f$ is necessarily a conformal transformation as follows from theorem \ref{inv}.\n from standard results , see \cite{semigroup } , we know that $ \c(v)\cap\c(v)^{-1}$ is just the group of conformal transformations of $ v$ and there is no other subgroup of $ \c(v)$ which contains $ \c(v)\cap\c(v)^{-1}$. the causal transformations which are not conformal transformations are called proper causal transformations . it is now a natural question whether one can define infinitesimal generators of one - parameter families of causal transformations which generalize the `` conformal killing vectors '' , and in which sense . notice , however , that if $ \{\f_{s}\}_{s\in \r}$ is a one - parameter group of causal transformations , from the previous results the only possibility is that $ \{\f_{s}\}$ be in fact a group of conformal motions . on the other hand , things are more subtle if there are no conformal transformations in the family $ \{\f_s\}$ other than the identity , in which case it is easy to see that the ` best ' one can accomplish is that either $ g^{+}\equiv \{\f_{s}\}_{s\in \r^{+}}$ or $ g^{-}\equiv \{\f_{s}\}_{s\in \r^{-}}$ is in $ \c(v)$. if this happens one talks about maximal one - parameter submonoids of proper causal transformations . of course , it is also possible to define local one - parameter submonoids of causal transformations $ \{\f_s\}_{s\in i}$ for some interval $ i=(-\epsilon,\epsilon)$ of the real line assuming that $ \{\f_s\}_{s\in ( 0,\epsilon)}$ consists of proper causal transformations . in any of these cases , we can define the infinitesimal generator of $ \{\f_s\}$ as the vector field $ \xiv = d\f_{s}/ds|_{s=0}$. given that $ \f_{s}^{*}\g\in\dp_{2}$ for all $ s\geq 0 $ ( or for all non - positive $ s$ ) , one can somehow control the lie derivative of $ \g$ with respect to $ \xiv$. for instance , it is easy to prove that $ \lie\g ( \k,\k ) \geq 0 $ ( or $ \leq 0 $ ) for all null $ \k$ , clearly generalizing the case of conformal killing vectors . an explicit example of this will be shown in the next section . \section{examples } { \bf example 1 einstein static universe and de sitter spacetime . } let us take $ v$ as the einstein static universe \cite{ff } and $ w=\ss$ as de sitter spacetime . in both cases the manifold is $ \r\times s^{3}$ and hence they are diffeomorphic . by proposition \ref{hor } we know that $ v\not\prec w$ because every causal curve in de sitter spacetime possesses event horizons . however , the proper causal relation in the opposite way does hold as can be shown by constructing it explicitly . the line element of each spacetime is ( with the notation $ d\o^2=d\z^2+\sin^2\z d\phi^2 $ ) : \bea v : & & ds^{2}=dt^{2}-a^2(d\x^{2}+\sin^{2}\x d\o^{2})\ \nonumber\\ w : & & d\tilde{s}^{2}=d\bar{t}^{2}-\alpha^{2}\cosh^{2}(\bar{t}/\alpha ) ( d\bar{\x}^{2}+\sin^{2}\bar{\x } d\bar{\o}^{2})\ \nonumber \eea where $ \x,\z,\phi$ ( and their barred versions ) are standard coordinates in $ s^{3}$ and $ a,\alpha$ are constants . the diffeomorphism $ \p : w\rightarrow v$ is chosen as $ \{t = b\bar{t } , \x={\bar\x } , \z={\bar\z } , \phi={\bar\phi}\}$ for a constant $ b$. one can readily get $ \p^{*}\g$ \begin{eqnarray * } ( \p^{*}\g)_{ab}dx^{a}dx^{b}= b^{2}d\bar{t}^2-a^2(d\bar{\x}^{2}+\sin^{2}\bar{\x } d\bar{\o}^{2})\end{aligned}\ ] ] which on using proposition [ orthonormal ] shows that @xmath18 if @xmath19 and therefore @xmath20 is proper causal relation for those @xmath21 . + consider the following spacetimes : @xmath22 is the region of lorentz - minkowski spacetime with @xmath23 in spherical coordinates @xmath24 ; @xmath25 is the outer region of schwarzschild spacetime with @xmath26 in schwarzschild coordinates @xmath27 . define the diffeomorphism @xmath28 given by @xmath29 for an appropriate positive constant @xmath21 , so that we have @xmath30 by choosing @xmath21 and @xmath31 one can achieve @xmath32 _ whenever _ @xmath33 , while for @xmath34 @xmath13 fails to be a proper causal relation . actually @xmath35 due to corollary [ hyp ] as @xmath36 is globally hyperbolic but @xmath22 is not . take now the diffeomorphism @xmath37 defined by @xmath38 , so that @xmath39 reads @xmath40 from where we immediately deduce that @xmath41 for every @xmath42 as long as @xmath43 . we have thus proved that @xmath44 if @xmath33 , but not for @xmath34 . this is quite interesting and clearly related to the null character of the event horizon @xmath45 in schwarzschild s spacetime . + let us take as @xmath46 the flat friedman - robertson - walker ( frw ) spacetimes in standard frw coordinates @xmath47 with line element given by @xmath48 and assume that the source of einstein s equations is a perfect fluid with equation of state given by @xmath49 ( @xmath50 pressure , @xmath51 density , @xmath52 constant ) . then the scale factor is @xmath53 with constant @xmath54 . by straightforward calculations , it can be proven the following causal equivalences : w~_0 + w~v these causal equivalences are rather intuitive if we have a look at the penrose diagram of each spacetime ( figure [ steady ] ) . , b ) @xmath55 and c ) @xmath56 . notice the similar shape of diagram c ) with that of @xmath57 , and of the steady state part of @xmath58 with a ) and b ) @xcite . ] * example 4 ( vaidya s spacetime . ) * let us show finally an example of a submonoid of causal transformations . consider the vaidya spacetime whose line element is @xcite @xmath59 where @xmath60 is a null coordinate ( that is , @xmath61 is a null 1-form ) , and @xmath62 is a non - increasing function of @xmath60 interpreted as the mass . take the diffeomorphisms @xmath63 . then @xmath64 can be cast in the form @xmath65 hence , @xmath66 iff @xmath67 , which implies that @xmath68 are causal transformations , so that @xmath69 is a maximal submonoid of causal transformations . the differential equation for the infinitesimal generator @xmath70 of this submonoid is easily calculated and reads @xmath71 this is a particular case of a proper kerr - schild vector field , recently studied in @xcite . notice that schwarzschild spacetime is included for the case @xmath72const . , in which case @xmath73 is a killing vector . this may lead to a natural generalization of symmetries . in this work a new relation between lorentzian manifolds which keeps the causal character of causal vectors has been put forward . with the aid of this relation , we have introduced the concepts of causal relation and causal isomorphism of lorentzian manifolds which allow us to establish rigorously when two given lorentzian manifolds are causally indistinguishable regardless their metric properties . this tools could be also useful in order to find out the global causal structure of a given spacetime by just putting it in causal equivalence with another known spacetime . finally a new transformation for lorentzian manifolds , called causal transformation has been defined . these transformations are a natural generalization of the group of conformal transformations and their actual relevance is one of our main lines of future research . this research has been carried out under the research project upv 172.310-g02/99 of the university of the basque country . 99 g. bergqvist and j. m. m. senovilla _ null cone preserving maps , causal tensors and algebraic rainich theory . quantum grav.*18 * , 5299 - 5326 , ( 2001 ) j. m. m. senovilla _ super - energy tensors . _ quantum grav . * 17 * , 2799 - 2842 , ( 2000 ) . s. w. hawking and g. f. r. ellis _ the large scale structure of spacetime . _ cambridge university press , cambridge , ( 1973 ) . r. m. wald _ general relativity . _ the university of chicago press , ( 1984 ) . j. m. m. senovilla _ singularity theorems and their consequences . _ . grav . * 30 * , 701 - 848 , ( 1998 ) . e. h. kronheimer and r. penrose proc . . soc . * 63 * 481 - 501 ( 1967 ) see e. g. f. hausdorff _ set theory _ chelsea n.y . j. hilgert , k. h. hofmann and j. d. lawson _ lie groups , convex cones and semigroups . _ oxford sciencie publications , ( 1989 ) . vaidya _ proc . indian acad . _ * a*33 , 264 ( 1951 ) . b. coll , s. r. hildebrandt and j. m. m. senovilla . _ kerr schild symmetries . * 33 * , 649 - 670 , ( 2001 ) .
|
in this work we define and study the relations between lorentzian manifolds given by the diffeomorphisms which map causal future directed vectors onto causal future directed vectors . this class of diffeomorphisms , called _ proper causal relations _ , contains as a subset the well - known group of conformal relations and are deeply linked to the so - called causal tensors of ref.@xcite .
if two given lorentzian manifolds are in _ mutual _ proper causal relation then they are said to be causally isomorphic : they are indistinguishable from the causal point of view .
finally , the concept of causal transformation for lorentzian manifolds is introduced and its main mathematical properties briefly investigated .
alfonso garca - parrado and jos m. m. senovilla 0.5 cm _ departamento de fsica terica .
universidad del pas vasco .
+ apartado 644 , 48080 bilbao , spain _
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the aim of this paper is to elucidate a phenomenon that has been studied for the symmetric group @xmath2 by studying the underlying geometry . here we sketch the phenomenon , along with our geometric interpretation and generalization . for @xmath3 divisible by @xmath0 , recall that the @xmath0-divisible partition lattice @xmath4 is the poset of partitions of the set @xmath5 with parts divisible by @xmath0 , together with a unique minimal element @xmath6 when @xmath7 . in @xcite , calderbank , hanlon and robinson showed that for @xmath7 the top homology of the order complex @xmath8 , when restricted from @xmath9 to @xmath2 , carries the _ ribbon representation _ of @xmath2 corresponding to a ribbon with row sizes @xmath10 . wachs @xcite gave a more explicit proof of this fact . their results generalized stanley s @xcite result for the mbius function of @xmath11 , which generalized g. s. sylvester s @xcite result for 2-divisible partitions @xmath12 . ehrenborg and jung extend the above results by introducing posets of _ pointed partitions _ @xmath13 parametrized by a composition @xmath14 of @xmath15 with last part possibly @xmath16 , from which they obtain all ribbon representations . more importantly , they explain why specht modules are appearing by establishing a homotopy equivalence with another complex whose top homology is naturally a specht module . ehrenborg and jung construct their pointed partitions @xmath17 by distinguishing a particular block ( called the _ pointed block _ ) and restricting to those of type @xmath14 . they show that @xmath18 is homotopy equivalent to a wedge of spheres , and that the top reduced homology @xmath19 is the @xmath2-specht module corresponding to @xmath14 . their approach is to first relate @xmath13 to a selected subcomplex @xmath20 of the simplicial complex @xmath21 of ordered set partitions of @xmath22 with last block possibly empty . in particular , they use quillen s fiber lemma to show that @xmath18 is homotopy equivalent to @xmath20 . they then give an explicit basis for @xmath23 that identifies the top homology as a specht module . ehrenborg and jung recover the results of calderbank , hanlon and robinson @xcite and wachs @xcite by specializing to @xmath24 . taking a geometric viewpoint , one can consider @xmath21 as the barycentric subdivision of a distinguished facet of the standard @xmath15-simplex having vertices labeled with @xmath25 . as such , it carries an action of @xmath2 and is a balanced simplicial complex , with each @xmath20 corresponding to a particular type - selected subcomplex . under this identification , the poset @xmath26 corresponds to linear subspaces spanned by faces in @xmath20 . we propose an analogous program for all well - generated complex reflection groups by introducing _ well - framed _ and _ locally conical _ systems . we complete the program for all irreducible _ finite _ groups having a presentation of the form @xmath27 with @xmath28 for all @xmath29 . each such group has an irreducible faithful representation as a complex reflection group . the irreducible finite coxeter groups are precisely those with each @xmath30 , i.e. , those with a real form . the remaining groups are _ shephard groups _ , the symmetry groups of regular complex polytopes . the family of coxeter and shephard groups contains 21 of the 26 exceptional well - generated complex reflection groups . using shephard and todd s numbering , the remaining five groups are @xmath31 . let @xmath32 denote an @xmath33-dimensional @xmath34-vector space . a _ reflection _ in @xmath32 is any non - identity element @xmath35 of finite order that fixes some hyperplane @xmath36 , and a finite group @xmath37 is called a _ reflection group _ if it is generated by reflections . henceforth , we assume that @xmath38 acts irreducibly on @xmath32 . shephard and todd gave a complete classification of all such groups in @xcite . given a ( finite ) reflection group @xmath37 , we may choose a positive definite hermitian form @xmath39 on @xmath32 that is preserved by @xmath38 , i.e. , @xmath40 for all @xmath41 and @xmath42 . we always regard @xmath32 as being endowed with such a form , which is unique up to positive real scalar when @xmath38 acts irreducibly on @xmath32 . we let @xmath43 denote the associated norm , defined by @xmath44 for all @xmath45 . as a special case of a complex reflection group , consider a finite group @xmath46 that is generated by reflections through hyperplanes . by extending scalars , we consider @xmath38 as acting on @xmath47 , and regard @xmath38 as a reflection group . we will call a ( complex ) reflection group that arises in this way a _ ( finite ) real reflection group_. a subgroup @xmath37 naturally acts on the dual space @xmath48 via @xmath49 , and this action extends to the symmetric algebra @xmath50 . an important subalgebra of @xmath51 is _ the ring of invariants _ @xmath52 , whose structure actually characterizes reflection groups : a finite group @xmath37 is a reflection group if and only if @xmath52 is a polynomial algebra @xmath53 $ ] . when @xmath38 is a reflection group , such generators @xmath54 of algebraically independent homogeneous polynomials for @xmath52 are not unique , but we do have uniqueness for the corresponding _ degrees _ @xmath55 . it is well - known that the minimum number of reflections required to generate @xmath38 is either @xmath56 or @xmath57 . if there exists such a generating set @xmath58 with @xmath59 , we say that @xmath38 is _ well - generated _ and that @xmath60 is a _ well - generated system_. ( finite ) real reflection groups form an important class of well - generated reflection groups . another important family consists of symmetry groups of _ regular complex polytopes _ ; these are known as _ shephard groups _ , and were extensively studied by both shephard @xcite and coxeter @xcite . the ( complex ) reflecting hyperplanes of a real reflection group @xmath38 intersect the embedded real sphere @xmath61 to form the _ coxeter complex _ , a simplicial complex with many wonderful properties . we call the maximal simplices of a coxeter complex _ chambers_. the real cone @xmath62 over a chamber @xmath63 is called a _ ( closed ) weyl chamber _ , and one nice feature of the coxeter complex is its algebraic description when @xmath60 is a _ simple system _ , i.e. , when @xmath58 is the set of reflections through walls of a weyl chamber . in this case , the poset of faces for the complex has the alternate description @xcite as the poset of parabolic cosets @xmath64 ordered by reverse inclusion , i.e. , @xmath65 naturally , one can define such a poset @xmath66 for an arbitrary group @xmath38 with set of _ distinguished generators _ @xmath58 . in @xcite , babson and reiner show that the geometry of this general construction is still well - behaved when @xmath58 is finite and minimal with respect to inclusion . in particular , they show that such posets @xmath66 are _ simplicial posets _ , meaning that every lower interval is isomorphic to a boolean algebra . in fact , each @xmath66 is _ pure _ of dimension @xmath67 and _ balanced_. in other words , there is a coloring of the atoms using @xmath68 colors so that each maximal element lies above exactly one atom of each color . the natural coloring given by @xmath69 recall that when such a coloring is present , we can select subposets by restricting to particular colors . precisely , from each subset @xmath70 , we obtain the following subposet _ selected by @xmath71 _ : @xmath72 we will often write @xmath73 in place of @xmath66 , and @xmath74 in place of @xmath75 . when @xmath38 is a reflection group , the lattice of intersections of reflecting hyperplanes for @xmath38 under reverse inclusion is denoted @xmath76 , or simply @xmath77 . it is a subposet of the lattice @xmath78 of all @xmath34-linear subspaces of @xmath32 ordered by reverse inclusion . for a subset @xmath79 , we define @xmath80 these are the only notions of span and hull that will appear in this paper . the following definitions aim to extend the relation between @xmath77 and @xmath66 for simple systems of real reflection groups to well - generated systems for complex reflection groups . for @xmath60 a well - generated system , a _ frame _ @xmath81 is a collection of nonzero vectors with @xmath82 here , @xmath83 is the reflecting hyperplane for reflection @xmath84 . we say that @xmath85 is a _ framed system_. we will sometimes index a generating set @xmath58 and frame @xmath86 with @xmath87 instead of @xmath88 , writing @xmath89 and @xmath90 . this is usually done when @xmath38 is the symmetry group of a regular polytope and @xmath86 is chosen from the vertices of the barycentric subdivision in such a way that each @xmath91 corresponds to an @xmath29-dimensional face of the polytope ; see figures [ fig : our ] , [ fig : cubeexample ] , and section [ background ] . before stating the next definition , we make precise what is meant by ( piecewise ) @xmath92-linearly extending a map @xmath93 on vertices of a simplicial poset @xmath73 . identifying @xmath73 with a @xmath94-complex as in @xcite , each point @xmath95 is contained in a unique ( open ) cell @xmath96 . because @xmath73 is a simplicial poset , the cell admits a characteristic map @xmath97 that maps the standard @xmath98-simplex onto @xmath99 while restricting to a bijection between vertices ( @xmath16-cells ) . thus , there are unique scalars @xmath100 so that @xmath101 we extend @xmath102 to all of @xmath73 by defining @xmath103 we can now state the main definitions of this paper . [ def : well - framed ] let @xmath85 be a framed system . define @xmath104 by the following map on vertices @xmath105 where @xmath106 * we say that @xmath85 is _ well - framed _ if the equivariant map @xmath107 is an embedding . * if , in addition , each @xmath108 contains the image of at least one @xmath109-face under @xmath107 , then we say that @xmath85 is _ strongly stratified_. [ convention ] given a well - framed system @xmath85 , we will often identify the abstract simplicial poset @xmath66 and the geometric realization @xmath110 afforded by @xmath107 . observe that a system @xmath85 is well - framed if and only if for any positive real constants @xmath111 , the system @xmath112 is well - framed . [ example : coxeter ] let @xmath60 be a simple system , i.e. , an irreducible finite real reflection group @xmath38 with @xmath58 the set of reflections through walls of a weyl chamber . if @xmath38 is a weyl group , one can obtain a real frame by choosing @xmath86 to be the set of fundamental dominant weights . more generally , a real frame is obtained by choosing one nonzero point on each extreme ray of a weyl chamber corresponding to @xmath58 . the resulting system @xmath85 is strongly stratified and @xmath110 is homeomorphic to the coxeter complex via radial projection ; figure [ fig : embedding ] illustrates the construction for @xmath113 , the dihedral group of order 10 . . ] we say that @xmath60 is _ well - framed _ or _ strongly stratified _ if there exists a frame @xmath86 for which @xmath85 is well - framed or strongly stratified , respectively . for @xmath38 a real reflection group , the following example suggests that only simple systems @xmath60 can be well - framed by a real frame @xmath114 . in other words , the well - framed systems @xmath85 with @xmath86 real and @xmath38 a ( complexified ) finite real reflection group are completely characterized in example [ example : coxeter ] ; see corollary [ corollary : coxeter ] below . [ nonembedding ] let @xmath113 , the dihedral group of order @xmath115 , and let @xmath116 , where @xmath117 are the reflections indicated in figure [ fig : nonembedding ] . in this case , @xmath60 is not a simple system , as the corresponding hyperplanes @xmath118 do not form a weyl chamber ; see the shaded region . two real frames are shown in the figure , one with @xmath119 ( left ) and one with @xmath120 ( right ) . both fail to yield a well - framed system . for example , on the right in figure [ fig : nonembedding ] , when @xmath120 , the map @xmath121 is a double covering . that are not well - framed for @xmath113 , @xmath116 , and real @xmath122 . ] comparing figures [ fig : embedding ] and [ fig : nonembedding ] , we see that for a fixed @xmath38 , it is possible to have both good and bad choices for @xmath58 and @xmath86 . also observe that @xmath85 being well - framed is a global property , not a local one . for example , figure [ fig : nonembedding ] shows that for @xmath123 large , one has to check for intersections of the simplices @xmath124 and @xmath125 . though some systems @xmath60 do not give a well - framed triple @xmath85 for any _ real _ @xmath126 , one may still be able to choose a good frame @xmath127 , as in the following example . [ triangleexample ] let @xmath128 and let @xmath116 , where @xmath117 and @xmath129 are the reflections indicated in figure [ fig : triangleexample ] . as in example [ nonembedding ] , @xmath60 is not a simple system because the corresponding hyperplanes @xmath118 do not form a weyl chamber . if @xmath130 are real and as shown in figure [ fig : triangleexample ] , the triple @xmath85 is not well - framed . in fact , for every choice of a real frame @xmath131 , the resulting system @xmath85 is not well - framed . however , it is still possible to construct a well - framed system from @xmath60 using @xmath132 . let @xmath133 be the ( complex ) reflecting hyperplanes for @xmath134 , respectively , and choose coordinates so that @xmath135,\quad h_2=\mathbb c\left[\begin{matrix}0\\ 1\end{matrix}\right],\quad h_3=\mathbb c\left[\begin{matrix}\sqrt{3}\\ 1\end{matrix}\right].\ ] ] let @xmath136 \quad\text{and}\quad \lambda_1=i\left[\begin{matrix } 0 \\ 1\end{matrix}\right ] .\ ] ] then it is straightforward to verify that @xmath85 is a well - framed system . for example , the two segments @xmath137 + ( 1-t)\frac{1}{4}\left[\begin{matrix}\sqrt{3}\\ -1\end{matrix}\right]\ : \ 0\leq t\leq 1\right\}\\ \intertext{and } \sigma_2&=w\sigma_1=\left\{s\frac{i}{2}\left[\begin{matrix } \sqrt{3 } \\ -1\end{matrix}\right ] + ( 1-s)\frac{1}{2}\left[\begin{matrix } 0 \\ 1\end{matrix}\right ] \ : \ 0\leq s\leq 1\right\}\end{aligned}\ ] ] do not intersect , since they have distinct endpoints and there is no real solution to @xmath138 note also that the linear form @xmath139 is nonzero on @xmath140 and @xmath141 , implying that @xmath142 neither intersect @xmath140 nor @xmath141 . with @xmath128 and @xmath116 . the system @xmath60 is well - framed for some nonreal @xmath143 , but not for any real @xmath144 . ] though @xmath66 is generally only a boolean complex , existence of a well - framed system @xmath85 forces it to be a simplicial complex : [ proposition : simplicial ] if @xmath85 is well - framed , then @xmath145 is a balanced simplicial complex . the image of a face under @xmath107 is determined by its vertices . because @xmath107 is assumed to be an embedding , it follows that any two faces of @xmath73 with equal vertex sets must be the same face . as a corollary , we see that every well - framed system @xmath85 with @xmath38 a real reflection group and @xmath114 is of the type constructed in example [ example : coxeter ] . we make this precise in the following corollary , whose proof is straightforward and left to the reader . [ corollary : coxeter ] let @xmath60 be a well - generated system with @xmath38 a ( finite ) real reflection group . assume that @xmath114 is a real frame . then @xmath85 is well - framed if and only if @xmath146 are the extreme rays of a weyl chamber . moreover , if @xmath85 is well - framed , then @xmath60 is a simple system . we now come to the notion of _ support_. note that in the following definition , we could very well replace @xmath147 with @xmath148 , given convention [ convention ] . however , it will be helpful to distinguish between the two . let @xmath85 be well - framed . we define the _ support _ map @xmath149 and let @xmath150 viewed as a subposet of @xmath78 . as for @xmath76 , we will often write @xmath151 in place of @xmath152 . we start by observing that for a well - framed system , @xmath153 . [ pi : l ] for a well framed system @xmath85 , we have @xmath154 with equality if and only if @xmath85 is strongly stratified . consider a coset @xmath155 , and recall that @xmath156 . using the definition of @xmath86 , we also have that @xmath157 the inclusion follows . the second claim follows from the definition of a strongly stratified system by considering dimension . the main theorem of this section is that the equivariant support map for a well - framed system has a purely algebraic description . the mechanism enabling this characterization is the _ galois correspondence _ , an analogue of barcelo and ihrig s galois correspondence ; see @xcite , where they utilize the tits cone to establish a result generalizing the real case to all coxeter groups . in order to state the correspondence , we introduce the poset of _ standard parabolics _ @xmath158 ordered by inclusion , i.e. , @xmath159 note that @xmath38 acts on @xmath160 via conjugation . [ galois ] for @xmath85 a well - framed system , @xmath161 is an @xmath38-poset isomorphism , with inverse @xmath162 using the fact that @xmath107 is an equivariant embedding of a balanced complex , @xmath163 regarding the inverse , we have @xmath164 the promised algebraic interpretation of support is now encoded in an equivariant commutative diagram : [ commutative ] for a well - framed system @xmath85 , we have the following commutative diagram of equivariant maps : @xmath165 & \delta\ar@{.>>}[d]^-{}_-{}\ar@{->}[rrr]^-{\sim}_-{\rho } & & & \rho(\delta ) \ar@{->>}[d]^-{\linspan}_- { } & \\ gw_jg^{-1 } & { \par}\ar@<.8ex>[rrr]^-\fix_- { } & & & \pi_w \ar@<.8ex>[lll]^-\stab_-\sim \ar@<-.4ex>@{^{(}->}[r]^-{\iota}_- { } & \l_w } \ ] ] if @xmath60 is strongly stratified , then @xmath166 . this section introduces the main objects of this paper , the generalizations of ehrenborg and jung s pointed objects . recall that the _ ( closed ) star _ @xmath167 of a simplex @xmath96 in a simplicial complex @xmath168 is the subcomplex of all faces that are joinable to @xmath96 within @xmath168 . that is , @xmath169 we will write @xmath170 for the join @xmath171 . [ pointedobjects ] let @xmath85 be a well - framed system . let @xmath172 . the _ subcomplex pointed by @xmath173 _ is @xmath174 as a subposet of @xmath76 . for @xmath70 , let @xmath175 note that by theorem [ commutative ] , we have @xmath176 , showing that @xmath177 figure [ fig : our ] illustrates the construction of @xmath178 for @xmath179 and eight choices of @xmath71 and @xmath173 . we have written `` @xmath173 '' above each element of @xmath173 , and `` @xmath71 '' below each element of @xmath71 in the coxeter diagram @xmath180 of @xmath60 . labeling the generators @xmath181 , a vertex marked @xmath29 in @xmath180 represents the reflection @xmath182 whose hyperplane can be written as @xmath183 recall that @xmath38 is the symmetry group of the tetrahedron @xmath184 , so the barycentric subdivision @xmath185 of the boundary of @xmath184 is homeomorphic to the coxeter complex via radial projection . the vertex of @xmath73 marked with @xmath29 corresponds to @xmath91 . it happens that @xmath38 is also a shephard group , because @xmath184 is a regular polytope . in the later notation of shephard groups , vertex @xmath29 will correspond to a face @xmath186 in a distinguished _ base flag _ @xmath187 ( a chamber in the flag complex @xmath188 for the polytope @xmath184 ) . for @xmath189.,title="fig : " ] similarly , figure [ fig : cubeexample ] illustrates the construction for the hyperoctahedral group @xmath190 of @xmath191 signed permutation matrices . recall that this is the symmetry group of the @xmath15-cube @xmath184 , so its coxeter complex is a radial projection of the barycentric subdivision @xmath185 of the boundary of @xmath184 . again , we let @xmath29 denote our choice of @xmath91 , and in later notation , a face @xmath186 of a distinguished base flag @xmath192 in the flag complex @xmath188 of @xmath184 . for @xmath193 . ] recall that a map @xmath194 of posets is _ order - preserving _ if @xmath195 whenever @xmath196 , and _ order - reversing _ if @xmath197 whenever @xmath196 . a @xmath198-poset is a poset with a @xmath198-action that preserves order , and a map @xmath194 of such posets is _ @xmath198-equivariant _ if it is a mapping of @xmath198-sets , i.e. , if @xmath199 for all @xmath200 and @xmath201 . the order complex of a poset @xmath202 , i.e. , the simplicial complex of all totally ordered subsets of @xmath202 , is denoted @xmath203 . the _ face poset _ @xmath204 of a simplicial complex @xmath168 is the poset of all _ nonempty _ faces ordered by inclusion . finally , @xmath205 denotes homotopy equivalence , with added decoration to indicate equivariance . note that the barycentric subdivision @xmath206 is homeomorphic to @xmath168 . the aim of this section is to establish a sufficient condition for the order complex of @xmath207 to be equivariantly homotopy equivalent to @xmath178 . our main tool is a specialization if @xmath208 is an order - preserving @xmath198-equivariant map of @xmath198-posets such that each fiber @xmath209 is @xmath210-contractible . theorem [ webb ] specializes @xmath202 to @xmath204 and replaces @xmath211 with its _ opposite _ @xmath212 , using the fact that @xmath213 and @xmath214 . recall that @xmath212 is obtained from @xmath211 by reversing order . ] of thvenaz and webb s equivariant version of quillen s fiber lemma . [ webb ] let @xmath211 be a @xmath198-poset , and let @xmath168 be a simplicial complex with a @xmath198-action . if @xmath215 is an order - reversing @xmath198-equivariant map of @xmath198-posets such that the order complex @xmath216 is @xmath210-contractible for all @xmath217 , then @xmath218 . consider a simplicial complex @xmath168 and order - reversing map @xmath215 of posets . since @xmath219 is an order ideal in @xmath204 , it is the face poset @xmath220 of some subcomplex @xmath221 . call such a subcomplex @xmath222 a _ quillen fiber_. note that @xmath216 is homeomorphic to @xmath222 , so quillen s fiber lemma concerns contractibility of quillen fibers . the following definition of a _ locally conical system _ is central to our work . in particular , we will show that for such systems , theorem [ webb ] can be applied to establish the desired homotopy equivalence . a well - framed system @xmath85 is _ locally conical _ if for each nonempty @xmath172 , every quillen fiber @xmath223 of @xmath224 has a cone point . recall that a _ cone point _ @xmath225 of a simplicial complex @xmath168 is a vertex of @xmath168 that is joinable in @xmath168 to every simplex of @xmath168 , i.e. , every maximal simplex of @xmath168 contains @xmath225 . [ prop : contraction ] let @xmath168 be a simplicial complex with a @xmath198-action . if @xmath168 has a cone point , then @xmath168 is @xmath198-contractible . the union of all cone points must form a @xmath198-stable simplex of @xmath168 , whose barycenter @xmath225 is therefore a @xmath198-fixed point of the geometric realization @xmath226 . since @xmath225 lies in a common simplex with every simplex of @xmath168 , this space @xmath226 is star - shaped with respect to the @xmath198-fixed point @xmath225 , and a straight - line homotopy retracts @xmath226 onto @xmath225 in a @xmath198-equivariant fashion . before employing theorem [ webb ] , recall that @xmath227 and that the action of @xmath38 on @xmath73 preserves types . hence , the subcomplex @xmath228 is a @xmath229-poset . it follows that @xmath230 is an order - reversing @xmath229-equivariant map . [ equivalence ] let @xmath85 be a locally conical system . let @xmath172 be nonempty and @xmath70 . then @xmath231 is @xmath229-homotopy equivalent to @xmath178 . we apply theorem [ webb ] to the map @xmath232 let @xmath233 , and consider first the quillen fiber @xmath234 for the unrestricted map @xmath224 . by definition of locally conical system , @xmath222 has a cone point @xmath225 . since @xmath73 is balanced , the subcomplex @xmath222 is also balanced . it follows that @xmath225 is the unique vertex of @xmath222 of its type . choose @xmath235 with @xmath236 . by the construction of @xmath237 , the vertices of the join @xmath238 are contained in @xmath239 for some @xmath42 . because @xmath240 is a linearly independent set , @xmath241 implies that @xmath225 is a vertex of @xmath96 , and hence a vertex of @xmath178 . therefore , the restricted quillen fiber @xmath242 also has @xmath225 as a cone point . it follows from proposition [ prop : contraction ] that @xmath243 is @xmath244-contractible . by applying the homology functor to theorem [ equivalence ] , we have the following [ theorem : main ] let @xmath85 be a locally conical system . let @xmath172 be nonempty and @xmath70 . then @xmath245 and @xmath246 are isomorphic as graded @xmath247-modules . recall that a simplicial complex @xmath168 is _ cohen - macaulay _ ( over @xmath248 ) if for each @xmath249 we have @xmath250 whenever @xmath251 , where @xmath252 denotes the _ link _ : @xmath253 the complex @xmath168 is _ homotopy cohen - macaulay _ if for each @xmath249 we have @xmath254 whenever @xmath255 . homotopy cohen - macaulay implies cohen - macaulay ( over @xmath248 ) , and cohen - macaulay implies cohen - macaulay over any field ; see the appendix of @xcite . this section is devoted to establishing explicit descriptions for the modules in theorem [ theorem : main ] when @xmath73 is cohen - macaulay . we start with the following [ cm : typed ] let @xmath85 be a well - framed system . let @xmath256 . if @xmath73 is cohen - macaulay ( resp . homotopy cohen - macaulay ) , then @xmath178 is cohen - macaulay ( resp . homotopy cohen - macaulay ) . it is an easy exercise to show that stars inherit the cohen - macaulay ( resp . homotopy cohen - macaulay ) property . the nontrivial step is concluding that @xmath178 is cohen - macaulay ( resp . homotopy cohen - macaulay ) . this follows from a type - selection theorem for pure simplicial complexes ; see bjrner ( * ? ? ? * thm . 11.13 ) and bjrner , wachs , and welker @xcite . [ lemma : contractible ] let @xmath85 be a well - framed system . if @xmath257 , then @xmath178 is contractible . this is immediate from @xmath258 . [ homology ] let @xmath85 be a well - framed system . let @xmath256 , and assume that @xmath73 is cohen - macaulay . if @xmath257 , then the top homology @xmath259 is trivial ; otherwise , @xmath260 as virtual @xmath247-modules . specializing to type @xmath1 and to ehrenborg and jung s objects ( see section [ section : ehrenborg ] ) , lemma [ lemma : contractible ] translates to @xmath20 being contractible whenever @xmath14 ends with a zero ; this is precisely lemma 3.1 in @xcite . the condition that @xmath261 collapses to the condition that @xmath14 not end with zero . we also note that the virtual modules in are well - known to be the natural generalization of ribbon representations to all coxeter and shephard groups ; see @xcite and @xcite . though the proof of theorem [ homology ] is entirely standard , we first need a particular description of @xmath178 when @xmath262 . the following intermediate description is straightforward . [ parabolic : typed ] let @xmath60 be a well - generated system . then for @xmath256 we have @xmath263 when @xmath66 is a simplicial complex , one has that @xmath60 satisfies the _ intersection condition _ @xmath264 in fact , satisfying the intersection condition is equivalent to @xmath66 being a simplicial complex ; see ( * ? ? ? the following lemma is a straightforward application of the intersection property , and the desired description of @xmath178 is obtained by applying the lemma to . [ induced ] let @xmath60 be well - framed . if @xmath262 , then the map @xmath265 is a @xmath247-poset isomorphism . the proof of theorem [ homology ] now follows : the first claim follows from lemma [ lemma : contractible ] . when @xmath262 , the result follows from the description of @xmath178 obtained through lemmas [ parabolic : typed ] and [ induced ] by applying the standard argument using cohen - macaulayness and the hopf trace formula , as is detailed by solomon in @xcite . other good sources include @xcite and the notes of wachs @xcite . in the case of type @xmath1 , ehrenborg and jung constructed _ pointed objects _ @xmath266 and subcomplexes @xmath267 indexed by compositions @xmath268 of @xmath15 that have positive entries @xmath269 but _ nonnegative _ last entry @xmath270 . figure [ fig : ehrenborg : s3:poset ] shows ehrenborg and jung s @xmath271 and @xmath272 , each carrying an action of @xmath273 . in @xcite , they show that the top homology of @xmath20 is homotopy equivalent to the order complex of @xmath274 , and that the top homology is given by a ribbon specht module of @xmath2 with row sizes determined by @xmath14 . the aim of this section is to present the underlying geometry of ehrenborg and jung s objects by showing how our objects @xmath275 specialize to theirs . it will follow that applying our results to this specialization recovers their main results , even upgrading their homotopy equivalence to an equivariant one . the translations between objects of this section are well - known , and we will largely follow the discussion of aguiar and mahajan ; see @xcite . let @xmath276 and let @xmath277 be the usual generating set of adjacent transpositions , i.e. , @xmath278 the symmetric group @xmath9 is the symmetry group of the standard @xmath15-simplex @xmath279 with vertices labeled by @xmath5 . the barycentric subdivision @xmath280 of the boundary of @xmath279 is homeomorphic to the coxeter complex via radial projection . letting @xmath91 be the vertex of @xmath280 indexed by @xmath281 yields a well - framed system @xmath85 with @xmath282 . note that , in particular , @xmath86 lies on the _ distinguished face _ @xmath283 of @xmath279 that has vertex set @xmath22 ; see figure [ e : intro ] . we call @xmath85 the _ standard system for @xmath9_. with the subdivision of facet @xmath284 shaded.,title="fig : " ] a _ set composition _ of @xmath3 is an ordered partition @xmath285 of @xmath286 $ ] with nonempty blocks . the collection of all set compositions of @xmath3 form a simplicial complex @xmath287 under refinement , with chambers having @xmath3 singleton blocks ; see figure [ fig : ehrenborg : s3 ] . the _ type _ of a set composition is the composition of its block sizes , i.e. , @xmath288 given a composition @xmath14 of @xmath3 , the subcomplex of @xmath287 generated by the faces of type @xmath14 is denoted @xmath289 . the map @xmath290 obtained by letting @xmath291 and extending by the action of @xmath38 , is an equivariant isomorphism @xmath292 ; see figure [ fig : ehrenborg : map ] . under this isomorphism , a face @xmath293 of type @xmath14 corresponds to a face of type @xmath294 note also that those faces of @xmath295 in the star of @xmath283 either have @xmath3 contained in the last block or have last block equal to @xmath296 . the lattice of hyperplane intersections @xmath76 for @xmath38 also has a simple description obtained from set compositions . under the identification of @xmath237 and @xmath287 , the support map corresponds to forgetting the order on the blocks . that is , @xmath297 the induced partial order is given by refinement . a _ pointed set composition _ of @xmath15 is an ordered partition @xmath293 of @xmath298 $ ] with last block @xmath299 possibly empty . we denote the collection of all pointed set compositions of @xmath298 $ ] by @xmath21 . the previous discussion shows that by removing @xmath3 from blocks in elements of @xmath287 , the barycentric subdivision of the facet @xmath283 can be identified with @xmath21 ; see figures [ fig : ehrenborg : map ] and [ fig : ehrenborg ] . more accurately , @xmath21 is @xmath2-equivariantly isomorphic to @xmath300 . given a composition @xmath14 of @xmath15 with last part possibly @xmath16 , the corresponding selected complex is denoted @xmath301 , which is ehrenborg and jung s complex @xmath20 ; see figure [ fig : ehrenborg ] . by distinguishing terminal blocks before taking the image of @xmath20 under the support map , one obtains their pointed poset @xmath13 after removing any ( possibly distinguished ) empty blocks ; see figure [ fig : ehrenborg : s3 ] . and @xmath272 from figure [ fig : ehrenborg : s3:poset].,title="fig : " ] ehrenborg and jung distinguish a block by underlining it . thus , the above map is @xmath302 for all possible choices of @xmath303 ( with last part is allowed to be @xmath16).,title="fig : " ] the following proposition summarizes the correspondence outlined above . [ prop : conversion ] let @xmath85 be the standard system for @xmath9 , and let @xmath304 be a composition of @xmath15 with last part @xmath305 possibly @xmath16 . then the following diagram composed of @xmath306-equivariant maps is commutative : @xmath307 ^ -{}_-{\supp } \ar@{<->}[rrrrr]^-{\sim}_- { } & & & & & \delta_{\vec{c}}\ar@{->>}[d ] & { \scriptstyle{b_1\dash b_2\dash \cdots\dash b_k } } \ar@{|->}[d]^{\supp } \\ & \pi_{{\rm{des}}(\vec{c})}^{\{r_n\ } } \ar@{<.>}[rrrrr]^-\sim_- { } & & & & & { \pi_{\vec{c}}^\bullet } & { \scriptstyle{b_1|b_2|\cdots|\underline{b_k } } } } \ ] ] the main result of @xcite is the following [ thm : ej ] let @xmath268 be a composition of @xmath3 with last part @xmath305 possibly @xmath16 . then we have the following isomorphism of top ( reduced ) homology groups as @xmath2-modules : @xmath308 proposition [ prop : conversion ] shows that theorem [ thm : ej ] is implied by combining theorem [ c : thm ] or theorem [ s : thm ] below with theorem [ equivalence ] ( or theorem [ theorem : main ] ) . the main theorem of this section is that we can apply all previous results to finite irreducible coxeter groups . [ c : thm ] let @xmath60 be a simple system for a finite irreducible real reflection group @xmath38 , and let @xmath309 consist of one nonzero point from each extreme ray of a weyl chamber corresponding to @xmath58 . then the following hold : a. @xmath85 is strongly stratified.[c : stronglystratified ] b. @xmath237 is homeomorphic to the coxeter complex of @xmath60 via radial projection.[c : complex ] c. @xmath145 is homotopy cohen - macaulay . [ c : c - m ] d. @xmath85 is locally conical.[c : locallyconical ] properties - are well - known . the aim of this section is to establish . we will need the following [ lemma : proj ] let @xmath38 be a finite irreducible real reflection group , and let @xmath310 be nonzero vectors on extreme rays of a fixed weyl chamber . then the orthogonal projection of @xmath311 onto @xmath312 is nonzero . this follows from the claim that , for all @xmath313 , one has @xmath314 for any set @xmath315 of nonzero vectors on the extreme rays of a weyl chamber @xmath63 . to see this claim , let @xmath316 be the simple system of roots associated with @xmath63 . in particular , @xmath317 is orthogonal to @xmath318 and its direction is chosen so that @xmath319 . recall that the @xmath317 form an obtuse basis for @xmath32 , i.e. , a basis with @xmath320 it follows from ( * ? ? ? * ch.v , 5 , lemma 6 ) that @xmath321 . this implies the weak inequality , i.e. , @xmath322 for all @xmath313 . one can now obtain the desired strict inequality by using the connectivity of the coxeter diagram for @xmath38 and the fact that the @xmath317 are obtuse ; see ( * ? ? ? 8) for an outline . we will also need some basic facts regarding convexity of the coxeter complex ; see @xcite , particularly section 3.6 , for details and a more general treatment . let @xmath323 be a finite real reflection group , and let @xmath168 denote its coxeter complex . recall that @xmath168 is a _ chamber complex _ , meaning that all maximal simplices ( called _ chambers _ ) are of the same dimension , and any two chambers can be connected by a gallery . here , a _ gallery _ connecting two chambers @xmath63 , @xmath324 is a sequence of chambers @xmath325 with the additional property that consecutive chambers are adjacent , meaning that they share a codimension-1 face . a _ _ root _ _ in @xmath326 ( vectors perpendicular to reflecting hyperplanes ) are in canonical 1 - 1 correspondence with roots of @xmath168 . the notion of root extends to arbitrary thin chamber complexes by introducing the notion of a folding . the terminology is due to tits , who characterized abstract coxeter complexes as precisely those thin chamber complexes with `` enough '' foldings ; see ( * ? ? ? of @xmath168 is the intersection of @xmath168 with a closed half - space determined by a reflecting hyperplane , and a subcomplex of @xmath168 is called _ convex _ if it is an intersection of roots . each convex subcomplex @xmath327 is itself a chamber complex in which any two maximal simplices can be connected by a @xmath327-gallery . moreover , a chamber subcomplex @xmath327 is convex if and only if any shortest @xmath168-gallery connecting two chambers of @xmath327 is contained in @xmath327 . the main tool for proving theorem [ c : thm ] is an iterative method for detecting cone points . the following discussion and lemma make this precise . choose a nontrivial subset @xmath172 and consider a nontrivial simplex @xmath328 . choose @xmath329 to be a chamber of @xmath330 containing @xmath331 . from a sequence @xmath332 of distinct vertices of @xmath329 , we construct a descending sequence of convex subcomplexes @xmath333 where we set @xmath334 and define @xmath335 we call the sequence @xmath336 _ cone - approximating _ for the triple @xmath337 if the following two conditions hold : 1 . @xmath338 is a cone point of @xmath339 for @xmath340 . 2 . @xmath341 for @xmath340 . note that @xmath342 implies the existence of cone - approximating sequences , since @xmath173 is nontrivial . the main result is that any cone - approximating sequence can be extended to contain a vertex of @xmath96 : [ c : approx ] in the above setting , a maximal cone - approximating sequence @xmath343 for @xmath337 has @xmath344 . supposing @xmath345 , we have that @xmath346 . from this it follows that @xmath347 first note that @xmath348 is a convex subcomplex . set @xmath349 the _ distinguished chamber _ of @xmath348 containing @xmath96 , and let @xmath350 be another chamber of @xmath348 . since @xmath348 is convex , we have that @xmath351 is a convex chamber subcomplex of @xmath330 . thus , there is a gallery in @xmath352 that connects chambers @xmath353 and @xmath354 . therefore , there is a sequence of reflections @xmath355 that induces a gallery from @xmath356 to @xmath350 in @xmath348 . since @xmath355 ( sequentially ) stabilizes @xmath348 , meaning that @xmath357 for every reflection @xmath182 in @xmath358 , the sequence @xmath355 also stabilizes both @xmath359 and its orthogonal complement in @xmath360 . because @xmath361 is fixed by @xmath355 and lies strictly on one side of @xmath359 in @xmath360 , it follows that @xmath355 fixes the orthogonal complement pointwise . this implies that the orthogonal projection @xmath362 of the vector @xmath361 onto @xmath359 in @xmath360 is fixed by @xmath355 . from lemma [ lemma : proj ] we have @xmath363 we claim that this projection has nontrivial intersection with the cone over @xmath356 . that is , @xmath364 to see this , note first that @xmath365 forms a chamber for a coxeter complex . the claim now follows from the fact that pairs of walls in a chamber do not intersect obtusely ; indeed , writing @xmath366 for the reflections in two distinct walls of a chamber , we have @xmath367 and the dihedral angle formed by the walls is @xmath368 , where @xmath369 is the order of @xmath370 . by , the line @xmath371 intersects a face @xmath102 of @xmath356 that is minimal in the sense that no proper face of @xmath102 meets @xmath372 . therefore , if @xmath372 is fixed by a reflection , the face @xmath102 is fixed pointwise by the reflection . choose @xmath373 to be some vertex of @xmath102 . we conclude that any gallery in @xmath348 that contains @xmath356 has @xmath373 as a cone point . by connectivity , this implies that @xmath348 has @xmath373 as a cone point . as we also have @xmath374 and @xmath375 , we can append @xmath373 to obtain a longer approximating sequence . let @xmath172 be nonempty and @xmath376 . choose @xmath329 to be a chamber of @xmath330 containing @xmath331 . consider the quillen fiber @xmath377 for the map @xmath378 , as in of theorem [ equivalence ] , and let @xmath379 be a maximal cone - approximating sequence for @xmath337 . by lemma [ c : approx ] , @xmath380 has cone point @xmath361 that is also a vertex of @xmath96 . we want to show that @xmath361 is also a cone point for the quillen fiber . since @xmath381 and @xmath382 for some particular @xmath108 , it follows that @xmath383 , and hence @xmath384 let @xmath385 . we want to show that the join @xmath386 is also in the fiber . since @xmath387 , we need only show that @xmath388 . but this is clear , since @xmath389 and @xmath390 . _ shephard groups _ form an important class of complex reflection groups . they are the symmetry groups of regular complex polytopes , as defined by shephard @xcite . here we will follow coxeter s treatment @xcite . let @xmath184 be a finite arrangement of complex affine subspaces of @xmath32 , with partial order given by inclusion . we call its elements _ faces _ , denoting an @xmath29-dimensional face by @xmath391 . a @xmath16-dimensional face is called a _ vertex_. allowing _ trivial faces _ @xmath392 and @xmath393 , all other faces are called _ proper faces_. a totally ordered set of proper faces is called a _ flag_. the simplicial complex of all flags is called the _ flag complex _ and is denoted @xmath188 . this is the order complex of @xmath184 with its improper faces @xmath394 and @xmath32 omitted . such an arrangement @xmath184 is a _ polytope _ if the following hold : a. @xmath395.[poly:1 ] b. if @xmath396 and @xmath397 , then the open interval @xmath398 is connected , i.e. , its hasse diagram is a connected graph.[poly:2 ] c. [ poly:3 ] if @xmath396 and @xmath399 , then the open interval @xmath400 contains at least two distinct @xmath401-dimensional faces @xmath402 for each @xmath401 with @xmath403 . for @xmath184 a polytope , note that properties and enable one to extend any partial flag @xmath404 in @xmath188 to a maximal flag ( under inclusion ) of the form @xmath405 we call maximal flags in @xmath188 _ chambers_. if the group @xmath37 of automorphisms of @xmath184 acts transitively on the chambers of @xmath184 , then we say that @xmath184 is _ regular _ and that @xmath38 is a _ shephard group_. the complexifications of the two ( affine ) arrangements shown in figure [ starry ] are examples of regular ( complex ) polygons . both polygons have symmetry group @xmath406 , the dihedral group of order 10 . . ] if @xmath184 contains a pair of distinct vertices that are at the minimum distance apart ( among all pairs of distinct vertices ) with no edge of @xmath184 connecting them , then @xmath184 is _ starry _ ; see @xcite . for example , the first polytope of figure [ starry ] is _ starry _ , whereas the second is _ nonstarry_. from the work of coxeter , each shephard group @xmath38 is the symmetry group of two ( possibly isomorphic ) nonstarry regular complex polytopes ; see tables iv and v in @xcite . henceforth , we assume that all polytopes are nonstarry . though currently unmotivated , the importance of this assumption will be made clear by theorem [ s : thm ] . given a regular complex polytope @xmath184 and a choice of maximal flag @xmath407 called the _ base chamber _ , for each @xmath29 the group @xmath408 is generated by some reflection @xmath182 . choosing such an @xmath182 for each @xmath29 yields an associated set @xmath89 that generates @xmath38 . we call @xmath58 a set of _ distinguished generators_. in the case that @xmath184 has a real form , each @xmath182 is uniquely determined , and they give the usual coxeter presentation for @xmath38 . in general , coxeter shows that one can always choose the reflections @xmath182 so that for some integers @xmath409 the group has the following coxeter - like presentation with defining relations @xmath410 these relations are encoded by _ symbol _ @xmath411p_1[q_1]p_2\cdots p_{\ell-2}[q_{\ell-2}]p_{\ell-1},\ ] ] which is uniquely determined by @xmath38 up to reversal . the corresponding nonstarry polytopes are denoted @xmath412 the second called the _ dual _ of the first . if we denote one of the two polytopes by @xmath184 , then the other is denoted by @xmath413 . the two polytopes have dual face posets and isomorphic flag complexes . the complete classification of shephard groups is quite short : * the symmetry groups of real regular polytopes , i.e. , the coxeter groups with connected unbranched diagrams : types @xmath414 . * @xmath415p_1 $ ] with @xmath416 not both @xmath417 , and @xmath418 satisfying @xmath419 where @xmath420 if @xmath421 is odd . using shephard and todd s numbering , these groups are @xmath422 . * @xmath423 with @xmath424 . the group can be represented as @xmath191 permutation matrices with entries the @xmath425th roots of unity . * @xmath4263[3]3=g_{26}$ ] * @xmath4273[3]3=g_{25}$ ] * @xmath4273[3]3[3]3=g_{32}$ ] . the following list summarizes notation and assumptions that will remain fixed when dealing with shephard groups . * @xmath38 is a shephard group . * @xmath184 is a non - starry regular complex polytope with symmetry group @xmath38 . * @xmath188 is the flag complex of @xmath184 , consisting of all flags of ( proper ) faces . * @xmath428 is a chosen base flag in @xmath188 . * @xmath58 is a set of distinguished generators for @xmath38 corresponding to @xmath429 . one can choose @xmath58 to satisfy the presentation found in the classification above , but doing so is unnecessary . * let @xmath172 with @xmath430 and @xmath431 . then @xmath432 * @xmath76 is the lattice of hyperplane intersections for @xmath38 under reverse inclusion . the aim of this section is to present an analogue of theorem [ c : thm ] for shephard groups . if @xmath38 is the symmetry group of a _ real _ regular polytope @xmath184 , then the coxeter complex @xmath168 is obtained by intersecting the reflecting hyperplanes with the real sphere @xmath433 . a radial projection sends @xmath168 homeomorphically onto the barycentric subdivision of @xmath184 . moreover , it is a geometric realization of the ( a priori ) poset of _ standard cosets _ @xmath434 ordered by reverse inclusion . this section presents the analogous picture for shephard groups , as established by orlik @xcite and orlik , reiner , shepler @xcite . the _ vertices _ of a face @xmath102 of @xmath184 are the vertices of @xmath184 lying on @xmath102 , and the _ centroid _ @xmath435 of @xmath102 is the average of its vertices . centroids play an important role in what follows . a _ shephard system _ is a triple @xmath85 with the following properties : a. @xmath38 is the symmetry group of a nonstarry regular complex polytope @xmath436 . b. @xmath58 is a set of distinguished generators corresponding to a chosen base flag @xmath437 c. @xmath438 is defined by setting @xmath439 . we start by observing that such triples are framed systems . a shephard system is a framed system . recall from section [ background ] that for @xmath440 , the reflection @xmath441 stabilizes @xmath442 in particular , centroid @xmath443 of @xmath186 is fixed by all @xmath444 with @xmath445 . in other words , @xmath446 let @xmath185 denote the ( topological ) subspace of @xmath32 that consists of all real convex hulls of centroids of flags under inclusion , i.e. , @xmath447 we can now state the analogue of theorem [ c : thm ] for shephard groups : [ s : thm ] let @xmath85 be a shephard system . then the following hold : a. @xmath85 is strongly stratified.[s : stronglystratified ] b. @xmath448.[s : b ] c. @xmath145 is homotopy cohen - macaulay . [ s : c - m ] d. @xmath85 is locally conical.[s : locallyconical ] note that figure [ starry ] illustrates why the non - starry assumption is necessary ; indeed , @xmath107 fails to be an embedding in the starry case when @xmath449 and @xmath450 is a maximal flag . the remainder of this section explains - , while will be established in the next section . during our discussion , the reader should take note of our use of theorems [ barycentric ] and [ theorem : orliksolomon ] , two _ uniformly stated _ theorems that are _ proven case - by - case_. in particular , theorem [ barycentric ] relies on a theorem of orlik and solomon that says a shephard group and an associated coxeter group have the same discriminant matrices , a result relying on the classification of shephard groups ; see @xcite . however , up to the use of theorems [ barycentric ] and [ theorem : orliksolomon ] , our approach for theorem [ s : thm ] is case - free . for each shephard group , the invariant @xmath451 of smallest degree @xmath452 is unique , up to constant scaling . for example , if @xmath38 has a real form , then @xmath453 for some suitable set of coordinates . the _ milnor fiber of @xmath38 _ is defined to be @xmath454 , where @xmath451 is regarded as a map @xmath455 . in @xcite , orlik constructs a @xmath38-equivariant strong deformation retraction of the milnor fiber @xmath454 onto a simplicial complex @xmath456 homeomorphic to @xmath185 , which he shows is a geometric realization of the flag complex @xmath188 . [ barycentric ] let @xmath457 be a shephard group with invariant @xmath455 of smallest degree . then there exists a simplicial complex @xmath458 called the _ milnor fiber complex _ containing the vertices of @xmath184 such that 1 . 1 . there is an equivariant strong deformation retract @xmath459 . 2 . for each @xmath108 , the set @xmath460 is a subcomplex of @xmath456 , and @xmath461 restricts to a strong deformation retract of @xmath462 onto @xmath463 . [ skeleton ] let @xmath464 and @xmath465 denote the @xmath401-skeleton of each complex . then 1 . @xmath466 for all @xmath401 , and 2 . @xmath467 is a disjoint union . [ realization ] 1 . @xmath456 is @xmath38-equivariantly homeomorphic to @xmath185 , and 2 . @xmath185 is a geometric realization of the flag complex @xmath188 via @xmath468 parts ( 1 ) and ( 2 ) are ( * ? ? ? * thm 4.1(i)-(ii ) ) , while ( 3 ) uses the proof of ( * ? ? ? * thm 5.1 ) . a nice discussion of the related theory is found in @xcite . using the additional property that @xmath454 has an isolated critical point at the origin , orlik was able to describe the topology of the flag complex @xmath188 : [ cm ] @xmath188 is homotopy cohen - macaulay , and is homotopy equivalent to a wedge of @xmath469-spheres of dimension @xmath470 . we will establish and of theorem [ s : thm ] by combining theorem [ barycentric ] and theorem [ cm ] with the following [ parabolic ] let @xmath38 be a shephard group of @xmath184 , and let @xmath187 be a base flag with corresponding distinguished generating set @xmath58 . then the map @xmath471 is a @xmath38-equivariant isomorphism the crux of the proof is that @xmath472 the type function on @xmath188 is naturally given by @xmath473 notice that @xmath107 factors as @xmath474 where @xmath290 is as in theorem [ parabolic ] , and @xmath475 is provided by theorem [ barycentric](a ) . hence , the triple @xmath85 is well - framed and @xmath448 . employing theorem [ cm ] yields . all that remains for establishing - is to show that each @xmath108 contains the image under @xmath107 of a @xmath109-simplex of @xmath73 . this follows from the following beautiful theorem that merges work of orlik and solomon ( * ? ? ? * thm . 6 ) and orlik ( * ? ? ? 4.1(iii ) ) ; this amalgamation appears in the latter paper of orlik . [ theorem : orliksolomon ] let @xmath38 the symmetry group of a nonstarry regular complex polytope @xmath184 . let @xmath108 , and write @xmath476 . then there exists strictly positive integers @xmath477 such that @xmath478 and @xmath479 is the @xmath401-skeleton of the restricted milnor fiber complex @xmath480 . consider @xmath108 with @xmath481 . by theorem [ theorem : orliksolomon ] , @xmath482 is nonempty if @xmath483 , as this implies that the right side of is strictly positive . but this is clear , since @xmath484 for any set of basic invariants @xmath54 . ( in fact , @xmath485 , with equality if and only if @xmath38 is a real reflection group . ) by theorem [ barycentric ] , each @xmath486-simplex of @xmath487 corresponds to an @xmath486-simplex of @xmath488 . hence , @xmath489 is nonempty . since @xmath490 , it follows by considering dimension that @xmath236 for some @xmath491 . this section is dedicated to proving of theorem [ s : thm ] . throughout , @xmath85 will be a fixed shephard system , and @xmath391 will denote an @xmath29-dimensional face of @xmath184 . because we will need to work with faces of @xmath184 instead of centroids , we start with some straightforward results relating the two . the most important of these results says that centroids of a maximal flag ( together with the origin @xmath492 ) form an _ orthoscheme _ ; see @xcite . more precisely , we have the following [ prop : orthoscheme ] let @xmath184 be a regular complex polytope , and let @xmath493 be a maximal flag of faces . then the vectors @xmath494 form an orthogonal basis for @xmath32 . by taking partial sums in , we obtain the following [ orthoscheme_basis ] let @xmath495 be a maximal flag of a regular polytope @xmath184 . then the centroids @xmath496 form a basis for @xmath497 . two particularly important results follow from lemma [ orthoscheme_basis ] . [ orthospan ] @xmath498 for all @xmath499 . [ flagintersection ] if @xmath500 are two subflags of a fixed flag @xmath501 , then @xmath502 we are now ready to present the main tool that will be used in the proof of theorem [ s : thm ] : [ unique ] let @xmath108 of dimension @xmath503 , and suppose that @xmath504 and @xmath505 are two @xmath401-flags with @xmath506 set @xmath507 . for @xmath508 , if @xmath102 is a face such that @xmath509 , then one of the following holds : a. @xmath510 . b. @xmath511 for all @xmath512 . suppose that @xmath513 with @xmath514 and @xmath512 maximal . using the definition of a polytope , we can extend @xmath515 to a maximal flag @xmath501 . by doing so , corollaries [ orthospan ] and [ flagintersection ] imply that @xmath516 similarly , @xmath517 by comparing dimension , we have that @xmath518 . our first claim is that @xmath519 are minimal faces of @xmath102 containing @xmath520 . certainly the intersection is contained in @xmath521 ( and @xmath522 ) . moreover , it contains the centroid @xmath523 ( resp . @xmath524 ) , which can not be contained in any proper face of @xmath521 ( resp . @xmath522 ) by corollary [ orthospan ] . assume without loss of generality that @xmath525 . from the equalities of and , we have @xmath526 . the definition of a polytope implies the existence of a face @xmath527 , which must therefore necessarily contain both @xmath528 and @xmath524 . we claim that @xmath529 . this follows immediately if @xmath530 , since faces and centroids determine each other ; see theorem [ barycentric](3)(b ) , for example . the other case is slightly more work . suppose that @xmath531 . extend @xmath532 to a maximal flag @xmath533 and recall that the centroids for @xmath533 form an orthoscheme . combining corollary [ orthospan ] with proposition [ prop : orthoscheme ] , it follows that the vector of @xmath528 is perpendicular to @xmath532 . as @xmath524 is also in @xmath532 , this implies that @xmath534 is perpendicular to @xmath528 . hence , @xmath535 because @xmath531 , we therefore have @xmath536 this contradicts regularity , since there is a unitary @xmath42 with @xmath537 , i.e. , with @xmath538 . having established that @xmath539 , the minimality of @xmath522 forces equality . let @xmath172 be nonempty . identifying @xmath73 with @xmath188 , we have @xmath229 corresponds to @xmath540 , and @xmath541 with @xmath542 given by @xmath543 . let @xmath544 . we claim that for some face @xmath545 of @xmath184 , the quillen fiber @xmath546 has @xmath545 as a cone point . this implies the desired claim of @xmath85 being locally conical . from the definition of @xmath547 and theorem [ barycentric ] , we can write @xmath548 for some @xmath401-flag @xmath549 in @xmath330 . because @xmath501 can be extended to a flag in @xmath73 containing @xmath540 , we can fix a nontrivial face @xmath550 and choose @xmath551 such that one of the following holds : a. @xmath552 is the maximal face of @xmath501 that is weakly contained in @xmath102 , or b. @xmath552 is the minimal face of @xmath501 that weakly contains @xmath102 . we claim that @xmath552 is a cone point of @xmath222 . to establish this , suppose @xmath553 is another @xmath401-flag in @xmath330 with support @xmath554 . using lemma [ unique ] , we will show that @xmath555 suppose first that we are in case ( a ) , i.e. , @xmath556 . because @xmath557 can be extended to a maximal flag containing @xmath540 , either 1 . an element of @xmath557 is weakly contained in @xmath102 , or 2 . each element of @xmath557 strictly contains @xmath102 . however , ( 2 ) is not a valid possibility . supposing otherwise , we can extend @xmath557 by @xmath552 to obtain a strictly larger flag with support equal to @xmath554 . considering dimension yields a contradiction . in the case of ( 1 ) , lemma [ unique ] shows that @xmath558 . if we are instead in case ( b ) , i.e. , @xmath559 , we can use the same argument , applied to the dual polytope @xmath413 , to conclude that @xmath560 . the aim of this section is to relate ribbon representations of shephard groups to the group algebra , the exterior powers of the reflection representation , and to the coinvariant algebra . recall that the group algebra @xmath561 $ ] of a finite group @xmath198 over @xmath34 is semisimple , and thus the grothendieck group @xmath562 of the category of finite dimensional @xmath198-representations is the free @xmath248-module with basis the isomorphism classes of irreducible @xmath198-modules . the map sending each irreducible module to its character linearly extends to an isomorphism of @xmath563 and the space of complex class functions on @xmath198 . in fact , this is an isomorphism in the category of finite - dimensional hilbert spaces , where the form @xmath564 on @xmath563 is that for which the irreducible @xmath198-modules form an orthonormal basis , and the form @xmath39 on class functions is given by @xmath565 . in what follows , we make no notational distinction between elements of @xmath563 and class functions . let @xmath60 be a simple system , and @xmath566 be the sign representation , defined by @xmath567 for @xmath568 . in @xcite , solomon showed that @xmath569 for @xmath570 , where @xmath571 the @xmath38-modules @xmath572 are known as _ ribbon representations _ due to their alternate description when @xmath573 and @xmath574 for @xmath575 . considering this case , let @xmath70 , write @xmath576 with @xmath577 , and let @xmath578 be the ribbon skew shape corresponding to composition @xmath579 ; see figure [ tableau : fig ] . filling @xmath578 with @xmath580 in increasing order from southwest to northeast produces a tableau whose rows are stabilized by @xmath581 and whose columns are stabilized by @xmath582 . thus , @xmath583 is the _ young symmetrizer _ @xmath584 , and hence @xmath572 is the @xmath2-specht module of ribbon skew shape @xmath578 . . ] for a simple system or shephard system @xmath60 , define @xmath585 for @xmath70 . we call these _ ribbon representations _ , noting that in the case of coxeter groups @xmath586 regarding solomon s group algebra decomposition , applying mbius inversion to the equality @xmath587 yields [ cs : decomp ] for @xmath60 a simple system or shephard system , @xmath588 . another extension of a main theorem in @xcite is [ theorem : steinberg ] let @xmath60 be a simple system or shephard system . for @xmath70 , the ribbon representation @xmath589 contains a unique irreducible submodule isomorphic to @xmath590 and has no submodule isomorphic to @xmath591 for @xmath592 . one can follow the same proof as in @xcite , replacing with theorem [ cs : decomp ] . however , for the reader s convenience , we supply a simple and significantly shorter argument . since @xmath593 occurs in @xmath594 with multiplicity equal to its dimension @xmath595 , by theorem [ cs : decomp ] it will suffice to show @xmath596 for each @xmath70 . fixing @xmath70 , so that @xmath597 , it suffices to show for each subset @xmath598 that @xmath599 if @xmath600 , then @xmath601 is the regular representation and @xmath602 , so follows . assuming that @xmath603 , there are disjoint nonempty sets @xmath604 such that @xmath605 with each @xmath606 acting irreducibly on @xmath607 , thus yielding a decomposition @xmath608 on which each @xmath606 acts trivially on all factors except @xmath609 ; see ( * ? ? ? * ch . the resulting decomposition of the exterior power @xmath610 combined with frobenius reciprocity , implies that @xmath611 by a theorem of steinberg ( * ? ? ? * ch . 5 , 2 , exercise 3 ) , the @xmath606-modules @xmath612 are irreducible and distinct for @xmath613 . hence @xmath614 and follows . a central object in invariant theory is the coinvariant algebra @xmath615 of a finite subgroup @xmath37 . this is the graded quotient of @xmath51 by the ideal @xmath616 generated by homogeneous invariants of positive degree . recall that @xmath38 is a reflection group if and only if @xmath615 affords the regular representation as an ungraded module . this section concerns the decomposition of the coinvariant algebra of a shephard group @xmath38 into ribbon representations . more precisely , we give a determinantal expression for the multivariate generating function @xmath617 where @xmath618 is defined below , and @xmath619 denotes the graded inner product @xmath620 of an element @xmath621 and a graded @xmath38-module @xmath622 . for a shephard system @xmath60 and subset @xmath623 , define @xmath624}=w_{\{r_k\ : \ i\leq k\leq j\}},\ ] ] where @xmath625 is the trivial subgroup , we set @xmath626 , and the _ hilbert series _ of a graded module @xmath622 is the formal power series @xmath627 . recall that the generators of a shephard system @xmath60 inherit an indexing from the associated chosen flag of faces @xmath192 . however , in what follows it will be convenient to shift all indices by 1 , thus writing @xmath628 . [ lineardiagram ] let @xmath60 be a shephard system . then @xmath629}(q ) } & \frac{1}{w_{[1,2]}(q ) } & \cdots & \frac{1}{w_{[1,\ell]}(q ) } \\ t_1 - 1 & t_1 & \frac{t_1}{w_{[2,2]}(q ) } & \cdots & \frac{t_1}{w_{[2,\ell-1]}(q ) } \\ 0 & t_2 - 1 & t_2 & \cdots & \frac{t_2}{w_{[3 , \ell-2]}(q ) } \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & \cdots & 0 & t_{\ell}-1 & t_{\ell } \end{matrix}\right].\ ] ] for a ( finite ) coxeter system @xmath60 , the _ multivariate @xmath421-eulerian distribution _ is defined to be @xmath630 where @xmath631 is the usual length function on coxeter groups , and @xmath632 is the descent set of @xmath633 . in the case of real shephard groups , reiner @xcite , following stembridge @xcite , established theorem [ lineardiagram ] with @xmath634 in place of @xmath635 . thus , @xmath636 for real shephard groups . extending to other shephard groups is a problem of considerable interest . fix @xmath70 and consider the coefficient of @xmath618 in @xmath635 . by frobenius reciprocity , we have @xmath637 recall that for a reflection group @xmath638 , one has @xmath639 as graded @xmath198-modules . it follows that @xmath640 for any @xmath623 , and that the graded @xmath38-module @xmath615 affords graded character @xmath641 . therefore , @xmath642 consider now the right - hand side of . from the usual permutation expansion of the determinant , we have the following general equation : @xmath643 thus , the right - hand side of is equal to @xmath644}(q ) w_{[i_1 + 1,i_2 - 1]}(q)\cdots w_{[i_j+1,\ell]}(q ) } \prod_{\stackrel{1\leq i\leq \ell}{i\neq i_1,\ldots , i_j}}(t_i-1),\ ] ] which , using the fact that @xmath645 for @xmath399 , can be written as @xmath646 taking the coefficient of @xmath618 yields @xmath647 . in section [ section : well - framed ] we introduced well - framed and strongly stratified systems @xmath85 . the well - framed systems subsequently studied were additionally strongly stratified , locally conical , and produced cohen - macaulay complexes @xmath66 . however , there are well - framed systems @xmath60 lacking many of these properties , including that of @xmath648 being homotopy equivalent to @xmath228 . we discuss some of these here . consider @xmath279 the boundary of the standard @xmath15-simplex having vertices labeled by the set @xmath5 . its symmetry group is @xmath9 , and a minimal generating set of reflections corresponds to a spanning tree @xmath71 of the complete graph @xmath649 on @xmath5 by identifying an edge @xmath650 of @xmath71 with the transposition @xmath651 ; see @xcite . we will focus on the generating set @xmath652 of @xmath9 that corresponds to the star graph with center @xmath3 . we leave the proof of the following result to the reader . [ prop : star ] let @xmath279 be as above . let @xmath653 be the vertices of @xmath279 , indexed so that @xmath654 corresponds to the vertex labeled by @xmath29 . let @xmath655 be pairwise linearly independent over @xmath92 , and set @xmath656 . then @xmath657 is well - framed . for @xmath658 , a system @xmath657 as in proposition [ prop : star ] is neither locally conical nor strongly stratified . moreover , for @xmath659 , it is no longer true that @xmath228 is homotopy equivalent to @xmath648 for every @xmath256 with @xmath173 nonempty . these assertions are straightforward after first considering let @xmath660 be a system as in proposition [ prop : star ] with @xmath661 . the link of a @xmath662-simplex in @xmath73 has 6 vertices supporting 3 lines ; see figure [ fig : triangleexample ] . thus , the system is not locally conical . moreover , it fails to be strongly stratified , since the vertices of @xmath73 support only the 5 complex lines spanned by the vertices of @xmath663 . the link @xmath664 of a vertex is isomorphic to @xmath665 . the latter is isomorphic to the @xmath666 _ chessboard complex _ , which is known to be a 2-torus ; see ( * ? ? ? * example 3.1 ) and @xcite . on the other hand , @xmath667 is easily seen to be the face poset of the barycentric subdivision of @xmath668 , hence its order complex is a 2-sphere . this subsection provides evidence that our main results partially extend to the remaining well - generated reflection groups . [ theorem : well - generateds ] if @xmath60 is a well - generated system , then @xmath66 is a simplicial complex . our proof relies on each subgroup @xmath669 being _ parabolic _ , meaning the pointwise stabilizer of a subset @xmath670 . the following can be obtained from the classification by considering cases . [ theorem : parabolic ] let @xmath37 be well - generated by @xmath58 . then each standard parabolic @xmath671 is parabolic . more precisely , @xmath672 . to establish the intersection property ( see section [ section : homology ] ) , first note that @xmath673 for @xmath674 . thus , @xmath675 where the third equality follows from the obvious inclusion by comparing dimensions . consider the rank 1 reflection group @xmath676 generated by a primitive 6th root of unity @xmath677 . observe that @xmath678 is a minimal generating set with @xmath679 and @xmath680 simplicial . thus , a well - generated group @xmath38 and minimal generating set @xmath58 may yield a simplicial complex even when @xmath38 is not well - generated by @xmath58 . let @xmath38 be a well - generated group , and let @xmath58 be a minimal generating set of reflections under inclusion . is @xmath66 necessarily a simplicial complex ? for which non - well - generated groups @xmath38 is @xmath66 a simplicial complex for some @xmath58 ? let @xmath60 be a well - generated system . motivated by , define @xmath681 and let @xmath682 for @xmath256 . recall that we identify @xmath228 with the poset of faces of a simplicial complex , and thus @xmath683 is obtained from @xmath228 by removing its unique bottom element @xmath38 . call @xmath60 _ ( abstractly ) locally conical _ if for each @xmath256 with @xmath173 nonempty , every quillen fiber of @xmath684 has a cone point . note that if @xmath60 is ( abstractly ) locally conical , then @xmath231 is @xmath229-homotopy equivalent to @xmath228 for all @xmath256 with @xmath173 nonempty . [ conjecture : conical ] for each well - generated reflection group @xmath38 , there exists a well - generating @xmath58 for which @xmath60 is ( abstractly ) locally conical . further , we predict the following partial extension of theorems [ c : thm ] and [ s : thm ] . [ conjecture : main ] for each well - generated reflection group @xmath38 , there exists a generating set @xmath58 and a frame @xmath86 such that a. @xmath59 . b. @xmath85 is strongly stratified . c. @xmath85 is locally conical . it is well - known @xcite that the coxeter complex @xmath456 for a finite coxeter group is _ shellable _ , meaning that its facets can be ordered @xmath685 so that the subcomplex @xmath686 is pure of dimension @xmath687 for all @xmath688 . the question of whether the flag complex @xmath188 of a regular complex polytope @xmath184 is lexicographically shellable appears in ( * ? ? ? * question 16 ) and @xcite . by section [ section : shephard:2 ] , @xmath188 is isomorphic to @xmath66 for @xmath60 a shephard system for @xmath184 . it is straightforward to shell those of rank 2 , as they are connected graphs , and it is also straightforward for @xmath689 . those of coxeter type are shellable , as mentioned in question [ q : shelling ] . the author used a computer to produce shellings in the remaining cases : [ shellingthm ] let @xmath60 be a coxeter or shephard system . then @xmath66 is shellable . [ q : shelling ] is there a uniform way of shelling the flag complex @xmath188 of a regular complex polytope ? this would give a more direct proof that @xmath188 is homotopy cohen - macaulay . the following was inspired by a personal communication with taedong yun and ( * ? ? ? * section 8) . let @xmath60 be a coxeter or shephard system . is @xmath690 shellable for all @xmath256 ? this work forms part of the author s doctoral thesis at the university of minnesota , supervised by victor reiner , whom the author thanks for suggesting some of these questions . he is also grateful to marcelo aguiar and volkmar welker for their helpful suggestions and corrections , richard ehrenborg and jiyoon jung for making their work available , stephen griffeth , jia huang , gus lehrer , and vivien ripoll for helpful conversations , and michelle wachs for writing a set of truly wonderful notes @xcite . p. orlik and l. solomon , complexes for reflection groups , in `` algebraic geometry : proceedings '' ( l. libgober and p. wagreich , eds . ) , 193 - 207 , lecture notes in mathematics , vol . 862 , springer - verlag , berlin / new york , 1981
|
ehrenborg and jung @xcite recently related the order complex for the lattice of @xmath0-divisible partitions with the simplicial complex of _ pointed ordered set partitions _ via a homotopy equivalence .
the latter has top homology naturally identified as a specht module .
their work unifies that of calderbank , hanlon , robinson @xcite , and wachs @xcite . by focusing on the underlying geometry ,
we strengthen and extend these results from type @xmath1 to all real reflection groups and the complex reflection groups known as _ shephard groups_. [ section ] [ theorem]proposition [ theorem]lemma [ theorem]corollary [ theorem]conjecture [ theorem]definition [ theorem]example [ theorem]observation [ theorem]convention [ theorem]observations [ theorem]question [ theorem]remark l
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the topology of generic intersections of quadrics in @xmath0 of the form : @xmath1 where @xmath2 , @xmath3 has been studied for many years : for @xmath4 they are diffeomorphic to a triple product of spheres or to the connected sum of sphere products ( @xcite , @xcite ) ; for @xmath5 this is no longer the case ( @xcite , @xcite ) but there are large families of them which are again connected sums of spheres products ( @xcite ) . + let @xmath6 . the generic condition , known as _ weak hyperbolicity _ and equivalent to regularity , is the following : + _ if @xmath7 has @xmath8 or fewer elements then the origin is not in the convex hull of the @xmath9 with @xmath10 . _ + a crucial aspect of these varieties is that they admit natural group actions . all of them admit @xmath11 actions . their complex versions in @xmath12 @xmath13 ( now known as _ moment angle manifolds _ ) admit natural @xmath14 actions . the quotient is in both cases a polytope @xmath15 that determines completely the manifolds and the actions . we will use both notations @xmath16 and @xmath17 for these manifolds to the notations @xmath18 and @xmath19 to specify that some _ facets _ may be empty . in theorem 1 this would mean that the binding might be empty . in all our examples this will be explicit . ] . they both fall under the general concept of _ generalized moment angle complexes _ ( @xcite and @xcite ) . + an open book construction with these manifolds was used to provide a description of the topology of some cases not covered by the main theorem in @xcite ( see remark on page 281 ) . in @xcite it is a principal technique for proving the results about some families when @xmath5 . + in section 2 we recall this construction , underlining its consequences for moment - angle manifolds : + if @xmath15 is a simple convex polytope and @xmath20 is one of its facets , there is an open book decomposition of @xmath21 with binding @xmath22 and trivial monodromy . + when @xmath4 , we also give a topological description of the binding and leaves of the decomposition , in terms of the _ odd cyclic partitions _ of @xmath23 that classify the varieties . this description is almost complete in the case of moment - angle manifolds for @xmath4 : the leaf is the interior of a manifold that can be : + * a product @xmath24 , * a connected sum along the boundary of products of the form @xmath25 , * in some cases , in the connected sum there may appear summands of the form @xmath26 or of the exterior of an embedded @xmath27 in @xmath28 . the specific products that appear in the above formulae will be described precisely in [ pagec ] in section 3 . the proofs will be postponed to section 7 where they will follow from a more general theorem for the corresponding real manifolds that requires , as in @xcite , additional dimensional and connectivity hypotheses . the corresponding result in the real case amounts to the topological description of the _ half _ manifolds @xmath29 , complementing the results in @xcite . parts of these theorems follow directly from the results in @xcite , other parts require the use of many of the technical lemmas proved there . all these manifolds with boundary are also generalized moment - angle complexes . + in section 4 we recall some results in @xcite which state that the @xmath30-@xmath31 complex manifolds @xmath32 which correspond to configurations @xmath33 which satisfy an arithmetic condition ( condition ( * k * ) ) fibre ( possibly as seifert fibrations ) over toric varieties ( or orbifolds with simple singularities ) with fibre a complex torus . from this we conclude that the pages of the open books described in section 2 are complex manifolds of the form @xmath34 where @xmath35 is obtained from @xmath33 by suppressing one element of the configuration . + in section 5 we show that every moment - angle manifold @xmath36 admits an almost contact structure and , as a consequence , these manifolds also admit a quasi - contact structure . we recall that there is the conjecture @xcite that every almost contact manifold is in fact a contact manifold . + in section 6 we describe a new construction of compact contact manifolds @xmath37 in arbitrarily large odd dimensions . these manifolds are generalizations of the moment - angle manifolds @xmath36 that have been studied by v. gmez gutirrez and the second author @xcite . + even dimensional moment - angle manifolds can be given complex structures but admit symplectic structures only in a few well - known cases @xcite so it is surprising that many of the odd dimensional ones admit contact structures . in this respect we remark that using results of c. meckert @xcite and y. eliashberg @xcite it is possible to give a contact structure on these manifolds . see remark [ rmk4 ] in section 6 . + however our method is distinct and in some sense explicit without using open books . what we use is the heat flow method described in @xcite . in other words , we construct a positive confoliation @xmath38 which is conductive i.e. every point of the manifold can be connected by a suitable legendrian curve to a point where the form is positive ( see definition 4 in 6.3 in section 6 ) . + in section 7 we recall some old results about the topology of intersections of quadrics and use them to proof a new one from which theorem [ pagec ] of section 2 follows . let us recall the concept of an _ open book _ , introduced by h. e. winkelnkemper in 1973 ( see @xcite ) . an open book in a smooth manifold @xmath39 is a pair @xmath40 consisting of the following : * a proper submanifold @xmath41 of codimension two in @xmath39 with trivial normal bundle , so @xmath41 admits a neighborhood @xmath42 diffeomorphic to @xmath43 . * a locally trivial smooth fibration @xmath44 such that there exists a neighborhood @xmath42 of @xmath41 as above , in which @xmath45 is the normal angular coordinate . the submanifold @xmath41 is called the _ binding _ and the fibres of @xmath45 are the _ pages _ of the open book @xmath40 . + also we can define an open book as follows : let @xmath20 be a manifold with a boundary and @xmath46 a self diffeomorphism of @xmath20 that is the identity on @xmath47 . the mapping torus @xmath48 of @xmath46 , i.e. , the quotient of @xmath49 by the equivalence relation generated by @xmath50 , is a manifold with boundary @xmath51 . collapsing each circle @xmath52 , to a point , @xmath53 we obtain a manifold @xmath54 without boundary called the _ relative mapping torus _ of @xmath46 . + this manifold has an obvious open book structure whose binding is a copy of @xmath47 , the collapsed @xmath55 , and whose mapping to the circle is induced by the projection @xmath56 and so the pages are copies of @xmath20 . + we call _ monodromy _ of an open book @xmath40 in a manifold @xmath39 to any self diffeomorphism of a page @xmath20 , such that there is a diffeomorphism @xmath57 taking the obvious open book to @xmath40 . + passing to our varieties , @xmath58 admits a @xmath11 action obtained by changing the signs of the coordinates . the quotient is a simple polytope @xmath15 which can be identified with the intersection of @xmath58 and the first orthant of @xmath0 . it follows that @xmath58 can be reconstructed from this intersection by reflecting it on all the coordinate hyperplanes . + by a simple change of coordinates @xmath59 , this quotient can be identified with the @xmath60-dimensional convex polytope given by @xmath61 @xmath62 let @xmath63 be obtained from @xmath33 by adding an extra @xmath64 which we interpret as the coefficient of a new extra variable @xmath65 , so we get the manifold @xmath66 : @xmath67 @xmath68 let @xmath69 be the intersection of @xmath58 with the half space @xmath70 and @xmath71 the intersection of @xmath66 with the half space @xmath72 . + the boundary of @xmath71 is @xmath58 ( this shows that @xmath58 is always the boundary of a parallelizable manifold ) . + @xmath69 admits an action of @xmath73 by changing signs on all the variables except @xmath74 and the quotient is again @xmath15 . in other words , @xmath69 can be obtained from @xmath75 by reflecting @xmath15 on all the coordinate hyperplanes except @xmath76 . + consider also the manifold @xmath77 which is the intersection of @xmath58 with the subspace @xmath76 . @xmath77 is the boundary of @xmath69 . + @xmath78 acts on @xmath66 ( rotating the coordinates @xmath79 ) with fixed set @xmath77 . its quotient can be identified with @xmath69 . the map @xmath80 is a retraction from @xmath66 to @xmath69 which restricts to the retraction from @xmath58 to @xmath69 @xmath81 observe further that this retraction restricted to @xmath82 is homotopic to the identity : the homotopy preserves the coordinates @xmath83 , @xmath84 and folds gradually the half space @xmath85 of the @xmath65 , @xmath74 plane into the ray @xmath86 , @xmath70 preserving the distance to the origin . + so @xmath58 is the double of @xmath69 and @xmath66 is the double of @xmath82 , and @xmath66 is the open book with binding @xmath77 , page @xmath69 and trivial monodromy . [ bookr ] every manifold @xmath66 is an open book with trivial monodromy whose binding is @xmath77 and page @xmath69 . since the manifold @xmath19 can be considered for each @xmath87 as a manifold @xmath66 with repeated coefficient @xmath9 , then it is an open book with binding the manifold obtained by taking @xmath88 . this proves : [ bookc ] if @xmath15 is a simple convex polytope and @xmath20 is one of its facets , there is an open book decomposition of @xmath19 with binding @xmath22 and trivial monodromy . when @xmath4 the topology of @xmath89 can be described precisely ( see @xcite and section 8) and that includes also that of all the bindings . it can be expressed in terms of the _ cyclic partition _ associated to @xmath89 . we have a precise description of the topology of the leaves in most cases . for our purposes we will only need the case where the total manifold is a moment - angle manifold which is described in the following : [ pagec ] let @xmath4 , and consider the manifold @xmath89 corresponding to the cyclic partition @xmath90 . consider the open book decomposition of @xmath89 corresponding to the binding at @xmath91 , as given by theorem [ bookc ] . then the leaf of this decomposition is diffeomorphic to the interior of : * if @xmath92 , the product @xmath93 * if @xmath94 and @xmath95 , the connected sum along the boundary of @xmath96 manifolds : @xmath97 * if @xmath98 and @xmath99 , the connected sum along the boundary of @xmath100 manifolds : @xmath101 @xmath102 * if @xmath103 and @xmath104 , a connected sum along the boundary of two manifolds : @xmath105 where @xmath106 is the exterior of an embedded @xmath107 in @xmath28 . the proof of this theorem and of its real version will appear in section 7 . similar results can be given when @xmath5 for large families in the spirit of @xcite . to illustrate the variety of decompositions obtained we give now some examples , by direct application of the previous theorem ( which gives always the decomposition with binding at @xmath91 ) . the reader may get a better feeling if she or he looks at these examples in the light of the proof of the theorem in section 7 . an additional feature of these examples is that we obtain three different open book decompositions of the same moment - angle manifold : + let @xmath108 . the @xmath109-tuple in @xmath110 corresponding to the five roots of unity ( @xmath33 satisfies the weak hiperbolicity condition ) . 1 . consider @xmath111 where @xmath64 has multiplicity 3 . we obtain the moment - angle manifold @xmath112 + when @xmath91 we have a configuration @xmath63 , where now the multiplicity of @xmath64 is two . then the binding is @xmath113 + the page , by theorem [ pagec](b ) , is in this case the manifold @xmath114 2 . consider @xmath111 where now @xmath115 has multiplicity 3 . the moment - angle manifold is the same : @xmath116 + when @xmath91 the binding is , + @xmath117 and the page , by theorem [ pagec](d ) , is @xmath118 where @xmath106 is the complement of an embedding @xmath119 in @xmath120 . consider now @xmath111 where now @xmath121 has multiplicity 3 . the moment - angle manifold is again @xmath122 when @xmath91 the binding is @xmath123 and the page , by theorem [ pagec](d ) , is @xmath124 where @xmath106 is the complement of the ( unique , up to isotopy ) embedding of @xmath125 in @xmath120 . when we take @xmath126 or @xmath127 with multiplicity 3 we obtain , by symmetry the same open book decompositions as in examples @xmath128 and @xmath129 above , respectively . let : @xmath33 be an admissible configuration , + @xmath36 the moment - angle manifold corresponding to @xmath33 , + @xmath130 the page and @xmath131 the binding of the open book decomposition . [ cols="^,^,^,^ " , ] if we take the quotient of @xmath36 by the scalar ( or diagonal ) action of @xmath78 : @xmath132 we obtain a compact , smooth manifold @xmath133 . these manifolds are called by some authors @xmath30-@xmath31 manifolds . + it is known that various of these objects admit natural complex structures ( see , for example , @xcite , theorem 12.2 ) : when @xmath8 is even , both @xmath32 and @xmath134 have natural complex structures . when @xmath8 is odd , @xmath36 itself has a natural complex structure . + let @xmath135 denote the canonical projection . + consider now the open book decomposition of @xmath36 described above , corresponding to the variable @xmath136 . let @xmath35 be obtained from @xmath33 by removing @xmath64 . it is clear that the diagonal @xmath78-action on @xmath36 has the property that each orbit intersects the page @xmath130 in a unique point and every point of this page is intersected tranversally by the orbits . this implies that the restriction of the canonical projection @xmath137 to each page is a diffeomorphism onto its image @xmath34 . + for @xmath8 even we therefore obtain , by pulling - back the complex structure of @xmath34 : + [ rmk1 ] _ the page of the open book decomposition of @xmath36 with binding @xmath138 admits a natural complex structure which makes it biholomorphic to the complex manifold @xmath34 . _ + we can go a step further by using the results in @xcite : + it is shown there that for every @xmath33 the manifold @xmath32 has a holomorphic and locally - free action of @xmath139 whose orbits determine a tranversally khler foliation @xmath140 of complex dimension @xmath141 . the condition for the leaves of @xmath140 to be compact is the following : + an admissible configuration @xmath142 fulfills condition @xmath143 if and only if we may choose , for the ( real ) space of solutions of the system @xmath144 a basis with integer coordinates . + also it was shown in @xcite that every configuration @xmath33 can be made to satisfy condition @xmath143 by an arbitrarily small perturbation . the main result proved in @xcite is the following : [ cef ] let @xmath33 be an admissible configuration that satisfies condition @xmath143 . then + ( a ) the leaves of the foliation @xmath140 of @xmath32 are compact complex tori of complex dimension @xmath145 . + ( b ) the quotient space of @xmath32 by @xmath140 is a projective toric variety of complex dimension @xmath146 . we denote it by @xmath147 . + ( c ) the toric variety @xmath147 comes equipped with an equivariant orbifold structure . + ( d ) the natural projection @xmath148 is a holomorphic principal seifert bundle , with compact complex tori of complex dimension @xmath145 as fibers . + ( e ) the transversely khlerian form @xmath149 of @xmath32 projects onto a khlerian form @xmath150 of @xmath147 . the bundle @xmath151 is called a _ generalized calabi - eckmann fibration _ over @xmath147 . + if @xmath33 satisfies condition @xmath143 ( and therefore @xmath35 also ) the manifolds @xmath147 and @xmath152 are both toric orbifolds ( possibly singular ) and therefore both are algebraic varieties . as a consequence , @xmath153 is a quasi - projective variety , which ( when is nonsingular ) is a khler manifold and , in particular , symplectic . + therefore , we have shown : assume @xmath8 is even . then the leaf of the open book structure on @xmath36 is naturally a complex manifold . after a small perturbation of @xmath33 , this leaf admits a holomorphic ( seifert ) fibration over a quasitoric variety , with fibre a compact complex torus . the even dimensional moment - angle manifolds and the @xmath30-manifolds have the characteristic that , except for a few , well - determined cases , they do not admit symplectic structures . it was , therefore , surprising to us that the odd dimensional ones can have contact structures . in fact we can conjecture that they are all contact manifolds : assume @xmath154 is even . then , for every admissible configuration @xmath155 , the odd dimensional manifold @xmath36 is a contact manifold . here is a first example : the next theorem was proved by f. bourgeois in @xcite ( it is a corollary of theorem 10 in @xcite ) . [ bour ] if a closed manifold @xmath156 admits a contact structure , then so does @xmath157 . therefore , _ for @xmath158 , moment - angle manifolds such as @xmath159 admit a contact structure . _ + we will get closer to our conjecture by showing that odd - dimensional moment - angle manifolds admit structures that are weaker versions of the contact structure . a @xmath160-dimensional manifold @xmath156 is called _ almost contact _ if its tangent bundle admits a reduction to @xmath161 . it is known that every contact manifold is an almost contact manifold ( see @xcite ) . + on the other hand , d. martnez , v. muoz and f. presas in @xcite defined a quasi - contact manifold as follows . a @xmath160-dimensional manifold @xmath156 is called _ quasi - contact _ if it admits a closed @xmath129-form @xmath149 such that @xmath162 is a non - zero @xmath163-form all over the manifold . they proved that given an almost contact manifold @xmath156 and given @xmath164 , there exist a quasi - contact structure @xmath149 in @xmath156 such that @xmath165=\gamma$ ] . they also conjectured that every almost contact manifold is in fact a contact manifold . + now , we have : if @xmath8 is even , @xmath36 is an almost contact manifold and also a quasi - contact manifold . consider the fibration @xmath166 with fibre the circle , given by taking the quotient by the diagonal action . since @xmath167 is a complex manifold we have that the foliation defined by the diagonal circle action is transversally holomorphic . therefore , @xmath36 has an atlas modeled on @xmath168 with changes of coordinates of the charts of the form @xmath169 where @xmath170 and @xmath171 where @xmath172 is an open set in @xmath173 and , for each fixed @xmath174 the function @xmath175 is a biholomorphism onto an open set of @xmath176 . this means that the differential , in the given coordinates , is represented by a matrix of the form @xmath177,\ ] ] where @xmath178 denotes a column @xmath179-real vector and @xmath180 . the set of matrices of the above type form a subgroup of @xmath181 . by gram - schmidt this group retracts onto @xmath182 . + this shows that @xmath36 is an almost contact manifold and therefore also a quasi - contact manifold by the result in @xcite . @xmath183 1 . the conjecture in @xcite would imply our conjecture that every odd dimensional moment - angle manifold admits a contact structure . we are working on a direct proof of this . it would give support to the conjecture in @xcite . in section [ s6 ] we will construct contact structures in some manifolds related to moment - angle manifolds . the construction in @xcite is based on an open book decomposition of the manifold . we have still not related this open book structure with the one we have in the case of a moment - angle manifold . the definition of _ overtwistedness _ in higher dimensions was given by k. niederkrger based in the existence of a _ plastikstufe _ ( see @xcite ) . let @xmath184 be a cooriented @xmath160-dimensional contact manifold , and let @xmath185 be a closed @xmath179-dimensional manifold . a _ plastikstufe _ @xmath186 with singular set @xmath185 in @xmath156 is an embedding of the @xmath23-dimensional manifold @xmath187 that carries a ( singular ) legendrian foliation given by the @xmath188-form @xmath189 satisfying : * the boundary @xmath190 of the plastikstufe is the only closed leaf . * there is an elliptic singular set at @xmath191 . * the rest of the plastikstufe is foliated by an @xmath78-family of stripes , each one diffeomorphic to @xmath192 , which are spanned between the singular set on one end and approach @xmath193 on the other side asymptotically . a contact manifold is called _ @xmath194-overtwisted _ if it admits the embedding of a plastikstufe . in dimension three , the definition of @xmath194-overtwisted is identical to the standard definition of overtwistedness . following the ideas of f. presas in the article @xcite , k. niederkger and o. van koert proved the next interesting results ( see @xcite ) : + let @xmath195 be the unit sphere in @xmath12 with coordinates @xmath196 and let @xmath45 be the polinomial @xmath197 the @xmath188-form @xmath198 defines a contact structure on @xmath195 ( see ( * ? ? ? * proposition 7 ) ) . * corollary 4 ) every sphere @xmath199 with @xmath200 supports a ps - overtwisted contact structure . more precisely , on @xmath199 , with @xmath200 , exists a contact structure which admits the embedding of a plastikstufe ps@xmath201 ( with @xmath202 and @xmath203 ) . as a consequence of theorem [ bour ] , we have the next corollary : for @xmath158 , moment - angle manifolds such as @xmath204 are @xmath194-overtwisted contact manifolds . theorem 2 of @xcite was extended in @xcite to the case where the manifold is given by two quadratic forms which are not necessarily simultaneously diagonalizable . this includes the manifolds we construct now : + let @xmath205 and @xmath206 be two integers and @xmath142 be an admissible configuration with @xmath207 . now consider the manifold @xmath37 define by the following equations : @xmath208 where @xmath209 . @xmath37 has real dimension @xmath210 . + the topology of the manifolds @xmath37 is related to the topology of the associated moment - angle manifolds @xmath36 as shown by the following theorem . + the manifold @xmath37 is diffeomorphic to : @xmath211 where @xmath212 . we will show that for every set of @xmath213 positive numbers @xmath214 the @xmath188-form @xmath215+\sum_{j=1}^n\left[b_j\left(z_j\bar{z}_j-\bar{z}_jdz_j\right)\right]\right]\ ] ] can be deformed into a contact form on the manifold @xmath37 , by an arbitrarily small @xmath216 perturbation . for all @xmath218 the @xmath188-form @xmath38 restricted to the tangent space @xmath219 is a nontrivial form . we will only consider the case where @xmath220 and @xmath221 since the other cases are completely analogous . for @xmath218 , the linear function @xmath222 is trivial if and only if there exist @xmath223 and @xmath224 such that : @xmath225\\ \nonumber = t\left[2w_1dw_1 + \sum_{j=1}^n\left[\lambda_j\left(\bar{z}_jdz_j+z_jd\bar{z}_j\right)\right]\right]\\ \nonumber+\overline{t}\left[2\bar{w}_1d\bar{w}_1+\sum_{j=1}^n\left[\bar{\lambda}_j\left(z_jd\bar{z}_j+\bar{z}_jdz_j\right)\right]\right]\\ + \mu\left[\left(w_1d\bar{w}_1+\bar{w}_1dw_1\right)+\sum_{j=1}^n \left(z_jd\bar{z}_j+\bar{z}_jdz_j\right)\right ] . \end{aligned}\ ] ] comparing coefficients in equation we conclude that the unique point satisfying the last equations is the origin @xmath226 . however , the origin is not in @xmath227 . we conclude that @xmath38 is nontrivial . we will denote by @xmath228 the kernel of @xmath38 at the point @xmath229 . + the only vectors @xmath230 where @xmath231 are of the form : @xmath232 where @xmath223 and @xmath233 . + if @xmath234 is tangent to the sphere @xmath195 and @xmath235 then it follows that @xmath236 must be of the form @xmath237 with @xmath233 . + the condition @xmath238 , when @xmath239 and @xmath240 implies : @xmath241 which implies since @xmath239 that @xmath242 if @xmath243 we have : @xmath244 therefore , if @xmath245 implies @xmath246 since , by hypothesis , what is inside the parentheses is positive . + vectors of the form @xmath247 which are in @xmath248 , imply that @xmath245 . hence , if @xmath239 we must have that @xmath246 also . + for @xmath249 the vectors of the form @xmath234 with @xmath223 and @xmath224 are in @xmath250 . hence @xmath250 has real dimension three . + on the other hand , on the set of points with @xmath249 , the vectors of the form : @xmath251 are all in @xmath252 . + hence @xmath253 is a two dimensional vector space . we will denote this 2-dimensional vector space at the point @xmath254 by @xmath255 . + therefore , in the set of points of @xmath227 such that @xmath239 , the form @xmath38 is a contact form . + for a generic set of admissible configurations @xmath256 the intersection of @xmath227 with the complex hyperplane with equation @xmath249 is a real codimension - two submanifold @xmath257 of @xmath227 ( i.e. a submanifold of @xmath258 of dimension 2n-3 ) and @xmath259 . @xmath257 is essentially the moment - angle manifold @xmath260 where @xmath261 . for all @xmath263 the @xmath188-form @xmath38 restricted to the tangent space @xmath264 is a non trivial form . as in the case @xmath217 , we will consider @xmath265 for all @xmath266 and @xmath267 since the other cases are completely analogous . for @xmath263 , the linear function @xmath268 is trivial if and only if there exist @xmath269 and @xmath224 such that @xmath270\\ \nonumber = t\left[\sum_{r=1}^s 2w_rdw_r+\sum_{j=1}^n\left[\lambda_j\left(\bar{z}_jdz_j+z_j\bar{z}_j\right)\right]\right ] \\ \nonumber + \overline{t}\left[\sum_{r=1}^s 2\bar{w}_rd\bar{w}_r+\sum_{j=1}^n\left[\lambda_j\left(z_jd\bar{z}_j-\bar{z}_jdz_j\right)\right]\right ] \\ + \mu \left[\sum_{r=1}^s \left(w_rd\bar{w}_r+\bar{w}_rdw_r\right)+\sum_{j=1}^n\left(z_jd\bar{z}_j+\bar{z}_jdz_j\right)\right].\end{aligned}\ ] ] comparing coefficients we have that the points satisfying the last equations are of the form @xmath271 where at least two @xmath272 , @xmath273 not zero , @xmath274 , @xmath275 . in this cases , the equations @xmath276 and @xmath277 defines a brieskorn manifold and @xmath38 is a contact form on that manifolds @xcite , in particular is non trivial on it . + we conclude that @xmath38 is non trivial on @xmath37 . we will be denote by @xmath228 the kernel of @xmath38 at the point @xmath278 . + the unique vectors @xmath279 where @xmath280 are of the form : @xmath281 @xmath282 where @xmath223 and @xmath224 . + if @xmath234 is tangent to the sphere @xmath283 and @xmath284 for all @xmath266 then it follows that @xmath236 must be of the form @xmath285 the condition @xmath286 when @xmath284 for all @xmath287 and @xmath288 implies : @xmath289 which implies , since @xmath290 , that @xmath291 if @xmath292 we have : @xmath293 therefore , if @xmath245 implies @xmath246 since , by hypothesis , @xmath294 is positive . + vectors of the form @xmath234 which are in @xmath295 imply that @xmath245 . hence , if @xmath284 we must have that @xmath246 also . + when @xmath296 for all @xmath297 , the vectors of the form @xmath234 with @xmath223 and @xmath224 are in @xmath250 . hence @xmath250 has real dimension three . + on the other hand , on the set of points with @xmath296 for all @xmath297 , the vectors of the form @xmath298 are all in @xmath253 . + hence @xmath299 is a two dimemsional space . we will denote this @xmath129-dimensional vector space at the point @xmath300 by @xmath255 . + therefore , in the set of points of @xmath37 such that @xmath290 for all @xmath297 , the form @xmath38 is a contact form . + for a generic set of admissible configurations @xmath301 the intersection of @xmath37 with the complex hyperplanes @xmath302 is a submanifold @xmath257 of real codimension @xmath303 of @xmath37 . we introduce some definitions following the ideas of s. j. altschuler and l. f. wu in @xcite : the space of _ conductive confoliations _ , @xmath304 , is defined to be the subset of @xmath305 such that * @xmath38 is a positive confoliation : @xmath306 , where @xmath178 denotes de hodge operator ; * every point @xmath307 is accessible from a contact point @xmath308 of @xmath38 : there is a smooth path @xmath309\to { { \pmb{z}}^{^{\mathbb{c}}}_{({\pmb{\lambda}},n , s)}}$ ] from @xmath310 to @xmath311 with @xmath312 in the orthogonal complement of @xmath313 for all @xmath314 . [ aw ] if @xmath315 then @xmath38 is @xmath316 close to a contact form . from the above and the fact that @xmath257 does not separate @xmath37 , since it is of codimension two , it follows : let @xmath178 denote the hodge operator for a given riemannian metric on @xmath37 , then for the appropriate orientation of @xmath37 one has that @xmath317 for @xmath318 and if @xmath319 @xmath320 therefore @xmath38 is a positive confoliation on @xmath37 . we must show that every point is accessible from a contact point of @xmath38 . let @xmath321 . then , there exists a smooth parametrized curve @xmath309\to{{\pmb{z}}^{^{\mathbb{c}}}_{({\pmb{\lambda}},n , s)}}$ ] such that @xmath322 , @xmath323 if @xmath324 $ ] and @xmath312 is a nonzero vector such that @xmath325 . let us fix a riemannian metric @xmath326 . for @xmath327 let @xmath328 denote the @xmath129-dimensional subespace of @xmath329 which is orthogonal to @xmath330 at @xmath331 . + let us first show that there exists an open neighborhood @xmath332 of @xmath319 and a smooth and non - vanishing vector field @xmath333 defined on @xmath334 such that 1 . @xmath335 , 2 . @xmath336 for all @xmath337 , 3 . @xmath338 for all @xmath337 . indeed , let @xmath339 . then @xmath340 has dimension two if @xmath341 or @xmath340 has dimension one if @xmath342 is transverse to @xmath228 . + let @xmath343 be a non zero vector . extend this vector anchored at @xmath344 to a smooth vector field @xmath345 defined in a neighborhood @xmath346 of @xmath319 let @xmath347 be the orthogonal projection . consider the vector field defined on @xmath41 by @xmath348 . then @xmath349 is a smooth vector field and by continuity @xmath349 satisfies all the required properties in a possible smaller neighborhood @xmath334 . + to finish the proof of the lemma we have that by standard extension theorems ( partition of unity ) there exists an extension of @xmath349 to a nonsingular vector field @xmath350 defined on an open neighborhood @xmath351 of @xmath344 . the vector field defined by @xmath352 has the property that @xmath353 . + by multiplying the vector field @xmath354 by a positive constant @xmath355 , if necessary , we can assume that all the solutions of the differential equation defined by the vector field @xmath356 on @xmath357 are defined in the interval @xmath358 . + if @xmath309\to{\mathcal{u}}$ ] is the solution of the differential equation determined by @xmath356 and satisfying the initial condition @xmath322 then this parametrized curve satisfies the requirements of the theorem [ aw ] if @xmath359 is sufficiently small . the proposition implies that every point of @xmath257 can be joined , by a legendrian path of finite length , to a point where the form @xmath360 is positive . + since @xmath38 is a contact form on @xmath361 we have that any two points of @xmath361 can be joined by a legendrian curve . therefore every point of @xmath37 is _ accessible _ and @xmath38 defines a _ conductive confoliation _ in the sense of j. s. altschuler and l. f. wu ( @xcite ) . + these forms are also called _ transitive confoliations _ by y. eliashberg and w. p. thurston @xcite , since we can connect any point of the manifold to a point where the form @xmath38 is contact by a legendrian path of finite length . + therefore applying theorem [ aw ] we can deform @xmath38 to a contact form @xmath362 . furthermore @xmath362 can be chosen arbitrarily close to @xmath38 in the @xmath316 topology . we have the theorem : for a generic set of admissible configurations @xmath363 , the manifold @xmath37 is a contact manifold . in other words : let @xmath158 , @xmath364 and let @xmath33 be an admissible configuration . the manifolds @xmath365 where @xmath212 admit contact structures . [ rmk4 ] it was shown by c. meckert @xcite that the connected sum of contact manifolds of the same dimension is a contact manifold . it was pointed to us by dishant pancholi that this implies the manifolds @xmath37 have a contact structure . + indeed , the manifolds @xmath37 are connected sums of products of the form @xmath366 with @xmath23 even and @xmath145 odd , and @xmath367 . whitout loss of generality , we suppose that @xmath368 ( the other case is analogous ) then @xmath369 is an open book with binding @xmath370 and page @xmath371 . hence @xmath366 is an open book with binding @xmath372 and page @xmath373 . the page @xmath374 is paralellizable since it embeds as an open subset of @xmath375 , therefore , since @xmath376 is even it has an almost complex structure . furthermore , by hypothesis , @xmath377 hence by a theorem of y. eliashberg @xcite the page is stein and is the interior of an compact manifold with contact boundary @xmath372 . hence by a theorem of e. giroux @xcite @xmath366 is a contact manifold . however our construction is in some sense explicit since it is the instantaneous diffusion through the heat flow of an explicit 1-form which is a positive confoliation . + another interesting fact is that the manifolds @xmath37 also have an open book decomposition . however for these open book decompositions there does not exist a contact form which is supported in the open book decomposition like in giroux s theorem because the pages are not weinstein manifolds ( i.e manifolds of dimension @xmath163 with a morse function with indices of critical points lesser or equal to @xmath23 ) . in this last section we recall some old results about the topology of intersections of quadrics @xmath378 and the ideas behind their proofs . we use these to proof a new result about the topology of the manifolds with boundary @xmath379 from which theorem [ pagec ] of section 2 follows . we recall here the results of @xcite , whose proofs are equally valid for any intersection of quadrics and not only for @xmath4 : + let @xmath380 as before , @xmath15 its associated polytope and @xmath381 its facets obtained by intersecting @xmath15 with the coordinate hyperplanes @xmath382 ( some of which might be empty ) . + let @xmath383 be the reflection on the @xmath87-th coordinate hyperplane and for @xmath384 let @xmath385 be the composition of the @xmath383 with @xmath10 . + let also @xmath386 be the face obtained by intersecting the facets @xmath387 for @xmath388 . + the polytope @xmath15 , all its faces , and all their combined reflections on the coordinate hyperplanes form a cell decomposition of @xmath58 . then the elements @xmath389 are generators of the chain groups @xmath390 , where to avoid repetitions one has to ask @xmath391 ( since for any @xmath87 the intersection @xmath383 acts trivially on @xmath387 ) . + a more useful basis is given as follows : let @xmath392 and @xmath393 be the product of the @xmath394 with @xmath10 . the elements @xmath395 with @xmath391 are also a basis , with the advantage that @xmath396 is a chain subcomplex for every @xmath397 and , since @xmath394 annihilates @xmath387 and all its subfaces , it can be identified with the chain complex @xmath398 , where @xmath399 is the union of all the facets @xmath387 with @xmath10 . it follows that @xmath400 for the manifold @xmath69 we have the same elements , but we can not reflect them in the subspace @xmath76 . this means we miss the classes @xmath395 where @xmath401 and we get induces an epimorphism in homology and fundamental group . ] @xmath402 these splittings are consistent with the ones derived from the homotopy splitting of @xmath403 described in @xcite . even if it is not clear that they are the _ same _ splitting , having two such with different geometric interpretations is most valuable . let us recall the results of @xcite for @xmath4 . in this case , every intersection of quadrics is diffeomorphic to one of the following particular forms : + take @xmath90 a partition of @xmath23 , where we will not distinguish between a partition and any of its permutations preserving the cyclic order and we will think of the index @xmath87 in @xmath404 as an interger mod @xmath96 . corresponding to it we have the configuration @xmath33 consisting of the @xmath405-th roots of unity , the @xmath87-th one taken with multiplicity @xmath404 . in other words , we have @xmath406 in increasing order where the size of @xmath407 is @xmath404 and for every @xmath408 one has @xmath409 is the @xmath87-th of the @xmath405-th roots of unity . any configuration can be deformed into one of these by concentrating in one point all the coefficients that is possible without breaking the weak hyperboli - city condition . we think of the @xmath407 as classes of points that can be joined this way . + the pairs @xmath410 with non - trivial homology are those where @xmath397 consists of @xmath411 or @xmath412 consecutive classes , that is , those where @xmath397 is either one of the @xmath413 or one of their complements @xmath414 in @xmath415 . in those cases there is just one dimension where the homology is non - trivial and it is infinite cyclic . in the case of @xmath416 that homology group is in dimension @xmath417 where @xmath418 is the length of @xmath416 . to specify a generator consider the cell @xmath419 where @xmath420 and @xmath421 , @xmath422 are any two indices in the extreme classes of @xmath414 ( in other words , those contiguous to @xmath416 ) . + @xmath419 is non empty of dimension @xmath417 . it is not in @xmath423 , but its boundary is . therefore it represents a homology class in @xmath424 , which is actually a generator , and defines a generator @xmath425 of @xmath426 . since @xmath419 has exactly @xmath427 facets it is a @xmath417-simplex so when reflected in all the coordinate subspaces containing those facets we obtain a sphere , which clearly represents @xmath428 . + the class corresponding to @xmath414 is in dimension @xmath429 and is the poincar dual of the the one corresponding to @xmath416 . one gets again easily a representative which in this case is not a sphere , but which can be turned into one with a good choice and a surgery if @xmath94 . for our purpose we do not need to say more about this case . + the net result is that , if @xmath94 , all the homology below the top dimension can be represented by embedded spheres with trivial normal bundle which can be made disjoint inside @xmath430 and ( since the inclusion @xmath431 induces an epimorphism in homology due to the homotopy equivalence @xmath432 ) we represent all the classes in @xmath433 by spheres , the @xmath434-cobordism theorem shows that this manifold is a connected sum along the boundary of manifolds of the form @xmath435 . then its boundary @xmath58 is a connected sum of spheres products . knowing its homology we can tell the dimensions of those spheres : + _ if @xmath94 and @xmath58 is simply connected of dimension at least @xmath109 , then : _ @xmath436 when @xmath92 a simple computation shows that @xmath437 the topology of @xmath430 is implicit in the above proof , and since any @xmath58 with @xmath95 is such a @xmath66 so we have : + _ if @xmath77 is simply connected of dimension at least @xmath109 , and @xmath94 , @xmath95 then : _ @xmath438 the case @xmath98 has to be considered separately . the difference in the topology of @xmath69 between case @xmath439 and @xmath103 can be seen as follows : + as mentioned before , in the first case the map @xmath440 induces an epimorphism in homology . + this is not the case for @xmath98 . for example , take the case @xmath441 . here @xmath77 consists of four copies of @xmath78 while @xmath69 is a torus minus four disks and @xmath77 follow easily by looking at its faces . ] or , equivalently , a sphere minus four disks ( where all the homology comes from the boundary ) with a handle attached that carries the homology not coming from the boundary . + the main fact is that @xmath77 is given by @xmath442 classes , and has only @xmath443 homology generators below the top dimension , only half of which survive in @xmath69 . while the latter manifold has @xmath96 homology generators . + to be more precise , the removal of the element @xmath444 allows the opposite classes @xmath445 and @xmath446 to be joined into one without breaking the weak hyperbolicity condition . therefore @xmath77 has fewer such classes and @xmath447 , which gives a generator of @xmath448 , does not give anything in @xmath449 because there _ it is not a union of classes _ ( it lacks the elements of @xmath446 to be so ) . + the two classes in @xmath448 missing in @xmath449 are thus those corresponding to @xmath450 and @xmath451 , all the others contain both @xmath445 and @xmath446 and thus live in @xmath449 . + as shown above , these two classes are represented by embedded spheres in @xmath69 with trivial normal bundle built from the cells @xmath452 and @xmath453 by reflection , where @xmath454 the corresponding spheres are obtained by reflecting in the hyperplanes corresponding to elements in @xmath461 and @xmath462 , respectively . since these sets are disjoint , the spheres also intersect in a single point . + now , a neighborhood of the vertex @xmath457 in @xmath15 looks like the first orthant of @xmath463 where the faces @xmath452 and @xmath464 correspond to complementary subspaces . when reflected in all the coordinates hyperplanes of @xmath463 , one obtains a neighborhood of @xmath457 in @xmath69 where those subspaces produce neighborhoods of the two spheres . therefore the spheres intersect transversely in that point . ( this shows again that those two classes do not come from the boundary @xmath77 : any homology class coming from the boundary can be separated from any other homology class in @xmath69 and so has trivial homology intersection with it ) . + a regular neighborhood of the union of those spheres is diffeomorphic to their product minus a disk , specifically to @xmath465 if @xmath99 the rest of the classes coming from @xmath77 can be represented again by disjoint products @xmath466 so finally the @xmath434-cobordism theorem gives + _ if @xmath58 is simply connected of dimension at least @xmath467 , and @xmath98 , @xmath99 then : _ when @xmath98 and @xmath104 we have the additional complication that making @xmath76 we pass from the _ pentagonal _ @xmath69 to the _ triangular _ @xmath77 , which is not a connected sum but a product of three spheres and , furthermore , not all its homology below the middle dimension is spherical . all we can say from the above is that @xmath470 where the latter manifold carries all the homology coming from the boundary @xmath77 . + we need to be more precise : + the homology generators in @xmath69 are those corresponding to @xmath471 , @xmath472 , @xmath473 , @xmath474 and @xmath475 . out of these @xmath471 and @xmath476 give us the handle which is not in @xmath106 , so the classes in this part are @xmath472 , @xmath474 and @xmath475 all coming from its boundary , which is @xmath477 . they can all be killed by taking the union @xmath478 of @xmath106 with @xmath479 along their common boundary . this implies ( by the mayer - vietoris sequence ) that @xmath478 is a homology sphere . it is simply - connected if @xmath69 is so because this implies that @xmath106 is so too and that @xmath77 is connected . + so @xmath478 is a homotopy sphere and @xmath106 is the exterior of @xmath257 inside a homotopy sphere . adding the inverse of that homotopy sphere at the interior of @xmath106 does not change its diffeomorphism type but turns @xmath478 into a standard sphere therefore @xmath106 is diffeomorphic to the exterior of a @xmath482 in @xmath483 . it can be expected that this embedding can be assumed to be standard , but for the moment we have the following result : let @xmath4 , and consider the manifold @xmath58 corresponding to the cyclic decomposition @xmath90 and the half manifold @xmath29 . when @xmath94 assume @xmath58 and @xmath484 are simply connected and the dimension of @xmath58 is at least @xmath467 . then @xmath69 diffeomorphic to : * if @xmath92 , the product @xmath485 * if @xmath94 and @xmath95 , the connected sum along the boundary of @xmath96 manifolds : @xmath486 * if @xmath98 and @xmath99 , the connected sum along the boundary of @xmath100 manifolds : @xmath487 @xmath488 * if @xmath98 and @xmath104 , a connected sum along the boundary of two manifolds : @xmath489 where @xmath106 is the exterior of an embedded @xmath482 in @xmath483 . let @xmath4 , and consider the manifold @xmath58 corresponding to the cyclic partition @xmath90 . when @xmath94 assume @xmath58 and @xmath484 are simply connected and the dimension of @xmath58 is at least @xmath467 . consider the open book decomposition of @xmath66 given by theorem [ bookr ] . then the leaf of this decomposition is diffeomorphic to the interior of : * if @xmath92 , the product @xmath485 * if @xmath94 and @xmath95 , the connected sum along the boundary of @xmath412 manifolds : @xmath490 * if @xmath98 and @xmath99 , the connected sum along the boundary of @xmath100 manifolds : @xmath487 @xmath488 * if @xmath98 and @xmath104 , a connected sum along the boundary of two manifolds : @xmath489 where @xmath106 is the exterior of an embedded @xmath482 in @xmath483 . bahri , a. and bendersky , m. and cohen , f. r. and gitler , s. , the polyhedral product functor : a method of decomposition for moment - angle complexes , arrangements and related spaces " . , 1634 - 1668 , 2010 . gmez gutierrz , v. and lpez de medrano , s. , intersections of quadrics , 30 years later `` , _ in preparation_. see also lpez de medrano , s. , ' ' singularities of homogeneous quadratic mappings " , to appear in _ revista de la real academia de ciencias exactas , fisicas y naturales . _ serie a. ( doi : 10.1007/s13398 - 012 - 0102 - 6 ) . martnez , d. and muoz , v. and presas , f. , open book decompositions for almost contact manifolds " , _ proceedings of the xi fall workshop on geometry and physics , publicaciones de la rsme _ , vol . 6 , pp . 131 - 149 , 2004 .
|
we construct open book structures on moment - angle manifolds and give a new construction of examples of contact manifolds in arbitrarily large dimensions .
* key words : * open book decomposition , moment - angle manifolds , contact structures .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
er uma stars are a still enigmatic small subgroup of su uma - type dwarf novae [ for a review of dwarf novae , see @xcite ] , which have extremely short supercycle lengths ( @xmath0 the interval between successive superoutbursts ) of 1950 d [ for a review , see @xcite ] and regular occurrence of superoutbursts . only five definite members have been recognized up to now : er uma ( @xcite , @xcite , @xcite ) ; v1159 ori ( @xcite , @xcite ) ; rz lmi ( @xcite , @xcite ) ; di uma ( @xcite ) ; and ix dra ( @xcite ) . some helium - transferring cataclysmic variables have become recognized as helium counterparts " of er uma stars [ cr boo : @xcite ; v803 cen : @xcite , @xcite ] . from a theoretical side , er uma stars have been understood as a smooth extension of normal su uma - type dwarf nova toward higher mass - transfer rates ( @xmath1 ) @xcite . the exact origin of such a high mass - transfer rate is still a mystery . even considering a higher mass - transfer rate , the shortest period systems ( rz lmi and di uma ) are difficult to explain without a special mechanism of prematurely quenching a superoutburst @xcite . in recent years , there have been an alternative attempt to explain the er uma - type phenomenon . @xcite tried to explain the er uma - type phenomenon by considering a decoupling between the thermal and tidal instabilities [ see @xcite for details of the thermal - tidal instability model ] under extremely small binary mass - ratio ( @xmath2=@xmath3/@xmath4 ) conditions . @xcite speculated that repeated post - superoutburst rebrightenings in wz sge - type dwarf novae ( hereafter wz sge stars ) or large - amplitude su uma - type dwarf novae ( e.g. @xcite ; see @xcite for a recent observational review of wz sge - type stars ) . @xcite tried to explain er uma - type phenomenon by ( rather arbitrary ) introducing an inner truncation of the accretion disk and irradiation on the secondary star on a numerical model developed by @xcite . @xcite further tried to explain the unification idea by @xcite using the same scheme as in @xcite . although the results partly reproduced the characteristics of er uma stars and wz sge stars , they failed to quantitatively reproduce the light curves of these dwarf novae . from the observational side , the existence of a gap between distributions of er uma stars and usual " su uma - type dwarf novae has been a challenge . the shortest known @xmath0 in usual su uma - type dwarf novae had been 130 d ( yz cnc , see also table 1 in @xcite ) at the time of the initial proposition of er uma stars . although further works have slightly shortened this minimum @xmath0 [ ss umi : 84.7 d , @xcite ; bf ara : 83.4 d , @xcite ] , there still remains a undisputed gap . in addition to these usual su uma - type dwarf novae with the shortest @xmath0 s , there exists a seemingly different population of su uma - type dwarf novae with short @xmath0 s , but with infrequent normal outbursts . v503 cyg [ @xmath0 = 89 d , only a few normal outbursts in a supercycle @xcite ] and ci uma [ @xmath5 140 d , infrequent normal outbursts @xcite ; @xmath0 variable ? @xcite ] are the best - known examples . the relation , however , between these objects and er uma stars ( and short @xmath0 usual su uma - type dwarf novae ) are unknown . in most recent years , some instances of strong @xmath0 variations have been reported in er uma stars ( @xcite , @xcite ) . in this letter , we report on the dramatic changes in the outburst properties in v503 cyg . we examined the observations of v503 cyg posted to vsnet collaboration , http://www.kusastro.kyoto - u.ac.jp / vsnet/@xmath6 . ] and found an appreciable change of the outburst properties . the observations used @xmath7-band comparison stars , and typical errors of individual estimates are smaller than 0.3 mag , which will not affect the following discussion . the object has been well sampled by many observers around the world except for periods of solar conjunctions . table [ tab : burst ] lists the observed outbursts since 1997 . the outburst lengths listed in the table approximately correspond to the durations above @xmath8 15.5 . although occasional observational gaps introduced an uncertainty of a few days , most of these superoutbursts were well recorded . many of normal outbursts during the favorably observed seasons were recorded , although some outbursts must have been missed . in order to estimate the numbers of missed outbursts , we performed monte - carlo simulations . the fractions of missing simulated 1000 normal outbursts ( the maximum magnitude and the rate of decline have been adjusted to those of actual outbursts ) were 10% , 18% and 30% for the three representative epochs shown in figure [ fig : lc ] . these fractions of missing normal outbursts will not affect the discussion given in section [ sec : dis ] . cccccccc jd start & peak mag & length & type & jd start & peak mag & length & type + 2450545 & 14.2 & 3 & normal & 2451519 & 14.4 & 2 & normal + 2450574 & 13.7 & 3 & normal & 2451531 & 13.8 & 10 & super + 2450601 & 13.1 & 14 & super & 2451641 & 14.5 & 1@xmath9 & normal + 2450643 & 14.4 & 2 & normal & 2451666 & 14.5 & 2 & normal + 2450669 & 14.1 & 3 & normal & 2451673 & 14.4 & 3 & normal + 2450697 & 13.1 & 13 & super & 2451691 & 14.2 & 3 & normal + 2450719 & 14.5 & 2 & normal & 2451704 & 13.4 & 12 & super + 2450758 & 14.3 & 1@xmath9 & normal & 2451727 & 13.7 & 2 & normal + 2450776 & 13.8 & 12 & super & 2451736 & 14.0 & 2 & normal + 2450896 & 14.3 & 2 & normal & 2451744 & 14.6 & 3 & normal + 2450952 & 13.3 & 10 & super & 2451753 & 15.6 & 1 & normal + 2450989 & 14.4 & 3 & normal & 2451785 & 13.4 & 11 & super + 2451018 & 14.1 & 3 & normal & 2451813 & 14.8 & 2 & normal + 2451051 & 13.3 & @xmath108 & super & 2451819 & 14.8 & 2 & normal + 2451072 & 14.6 & 2 & normal & 2451826 & 14.5 & 2 & normal + 2451124 & 13.4 & 12 & super & 2451837 & 14.3 & 2 & normal + 2451220 & 14.0 & 1@xmath9 & normal & 2451848 & 13.6 & 1@xmath9 & normal + 2451237 & 14.0 & 1@xmath9 & normal & 2451865 & 13.8 & 2 & normal + 2451323 & 14.5 & 3 & normal & 2451873 & 13.6 & 11 & super + 2451337 & 14.4 & 3 & normal & 2451964 & 14.7 & 1@xmath9 & normal + 2451348 & 14.3 & 1 & normal & 2452014 & 14.1 & 1@xmath9 & normal + 2451359 & 14.4 & 1@xmath9 & normal & 2452046 & 13.5 & 13 & super + 2451372 & 13.6 & 12 & super & 2452114 & 13.3 & 3 & normal + 2451406 & 14.4 & 2 & normal & 2452135 & 13.0 & 14 & super + 2451428 & 14.1 & 3 & normal & 2452176 & 13.9 & 3 & normal + 2451435 & 14.2 & 2 & normal & 2452231 & 13.4 & 10 & super + 2451453 & 13.7 & 13 & super & 2452278 & 13.7 & 2 & normal + 2451479 & 14.4 & 1@xmath9 & normal & 2452321 & 13.8 & @xmath105 & super + 2451485 & 14.4 & 2 & normal & 2452397 & 13.8 & 19@xmath11 & super + 2451495 & 14.4 & 2 & normal & 2452466 & 13.7 & 3 & normal + + + ( 88mm,130mm)fig1.eps in the standard disk instability model , the recurrence time of normal outbursts ( @xmath12 ) is mainly governed by the diffusion process , while @xmath0 represents the increasing rate of net angular momentum in the accretion disk @xcite . if the quiescent viscosity parameter has a fixed value between various su uma - type dwarf novae , both @xmath12 and @xmath0 are unique functions of @xmath1 @xcite . this relation has been observationally confirmed in most of su uma - type stars @xcite . v503 cyg apparently violates this relation in its low frequency of normal outbursts ( figure [ fig : lc ] , upper panel ) , and several other stars ( v344 lyr , sx lmi ) have been proposed to be analogous to v503 cyg ( @xcite , @xcite ) . there must be an unknown suppression mechanism of normal outbursts in these systems . in 19992000 , v503 cyg showed a very frequent occurrence of normal outbursts ( minimum @xmath13 79 d , figure [ fig : lc ] , middle panel ) . this @xmath12 is just what is expected for a @xmath0 = 89 d usual su uma - type dwarf nova @xcite . this fact indicates that the usually outbursting su uma - type state and unusually outbursting ( in the sense of low frequency of normal outbursts ) v503 cyg - type state are interchangeable . since @xmath0 during this period was not appreciably different from the canonical @xmath0 = 89 d , there should have not been an appreciable change in the @xmath1 . the suppression mechanism of normal outbursts must have been somehow unlocked " during this period . in 20012002 , v503 cyg showed another different aspect ( figure [ fig : lc ] , lower panel ) . during this period , the number of normal outbursts in a supercycle dramatically decreased to @xmath141 . there is some hint of alternating occurrence of a superoutburst and a normal outburst with a period of 4080 d. such a sequence of outbursts is only known in rarely outbursting su uma - type dwarf novae [ cf . sw uma , v844 her cf . @xcite for a discussion ] , and is unprecedented in short @xmath0 systems . during this period , some normal outbursts have comparable peak magnitudes to those of superoutbursts . some of superoutbursts showed rather short durations , which seems to be incompatible with a high @xmath1 necessary to reproduce the short @xmath0 @xcite . these findings suggest that premature quenching of superoutbursts , as proposed by @xcite and @xcite , indeed occurred during this period , although v503 cyg ( orbital period = 0.0757 d ) is unlikely to have a small @xmath2 required in @xcite and @xcite . the overall light curve more or less resembles that of ci uma ( @xcite , @xcite ) . although exact mechanisms have not been yet identified , the present remarkable alternations between the outbursting states in v503 cyg support the presence of mechanisms of suppressing normal outbursts and premature quenching superoutbursts . the most important finding is that the effects of these mechanisms are temporarily variable even in the same object , and are not a fixed character of a certain system . this finding suggests that the shortest @xmath0 usual su uma stars and unusual v503 cyg - like stars can represent different aspects of the same system . among er uma stars , di uma can be a similar system with systematic state changes @xcite . the observed temporal variability of the suppressing / quenching mechanisms in the same object suggests that these mechanisms are not primarily governed by a fixed system parameter [ i.e. mass of the white dwarf @xcite ; @xmath2 @xcite etc . ] but more reflect state changes in the accretion disk . we are grateful to many amateur observers for supplying their vital visual and ccd estimates via vsnet . this work is partly supported by a grant - in - aid ( 13640239 , tk ) from the japanese ministry of education , culture , sports , science and technology . part of this work is supported by a research fellowship of the japan society for the promotion of science for young scientists ( mu ) .
|
we examined the vsnet light curve of the unusual su uma - type dwarf nova v503 cyg which is known to show a short ( 89 d ) supercycle length and exceptionally small ( a few ) normal outbursts within a supercycle . in 19992000 , v503 cyg displayed frequent normal outbursts with typical recurrence times of 79 d. the behavior during this period is characteristic to an usual su uma - type dwarf nova with a short supercycle length . on the other hand ,
v503 cyg showed very infrequent normal outbursts in 20012002 .
some of the superoutbursts during this period were observed shorter than usual .
the remarkable alternations of the outbursting states in v503 cyg support the presence of mechanisms of suppressing normal outbursts and premature quenching superoutbursts , which have been proposed to explain some unusual su uma - type outbursts .
the observed temporal variability of the suppressing / quenching mechanisms in the same object suggests that these mechanisms are not primarily governed by a fixed system parameter but more reflect state changes in the accretion disk .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
narrow , low - energy ( @xmath2 kev ) spectral lines in grbs have been reported using the data of several instruments ( e.g. , see review @xcite ) . based on these reports , the batse team expected lines to be easy to find and looked for them manually @xcite . the reality was different : lines were not obvious in burst spectra . in order to be sure that the manual search had not missed any lines , we implemented a comprehensive , automatic computer search @xcite . bursts with at least one spectrum with a normed signal - to - noise ratio ( snr ) @xcite near 40 kev above 5.0 are searched . for each burst , we form spectra from each individual spectral record , every pair , triple , etc . the spectra so formed overlap in many cases . once a burst is selected for the search , spectra are searched regardless of the presence of burst flux low snr spectra serve as controls . each spectrum is fit with a continuum model and then a series of fits are made adding narrow lines at a closely spaced grid of line centroids extending to 100 kev . line candidates are identified by a @xmath3 change @xmath4 of more than 20 , corresponding to a chance probability in a single spectrum of @xmath5 . the probability is calculated for two line parameters , intensity and centroid , since the intrinsic width is assumed narrow compared to the detector resolution . so far , 133,000 spectra formed from 12,000 spectral records from 117 bursts have been searched . most of these spectra have snr too low to support the detection of a line . only 16,000 have normed snr @xmath6 5 , which our simulations show is needed to have a reasonable sensitivity to lines similar to those found in the _ ginga _ data @xcite . the search identified several cases with @xmath4 values exceeding our significance threshold , but which we rejected either because they occur in background intervals or because they become insignificant when a better fit requiring a non - negative flux is made . these cases were located in spectra of low snr , and because of the large number of low - snr spectra , they are consistent with statistical fluctuations . several possible candidates with angles between the burst direction and detector normal exceeding @xmath7 are inconsistent with the data of other detectors . we believe these cases to be due to inadequacies in the detector model and we have set them aside . after these exclusions , our candidate list has 12 members with @xmath4 values ranging from the threshold of 20 to 31.8 ( @xmath8 ) . the normed snrs of the spectra in which these candidates are most significant range from 2.1 to 18.4 , with 10 normed snr values exceeding 5 . these snr values are reasonable for real features . the number of independent trials is a nebulous concept . while there are roughly 16,000 spectra sufficiently bright for there to be a high probability of detecting a real line , many of these spectra overlap , e.g. , starting a few records earlier or later , or differing in length by a few records . consequently the number of bright , independent spectra is one or more orders of magnitude below 16,000 . the number of independent energy resolution elements averages about 5 per spectrum . we estimate that the ensemble chance probability of the most - significant candidate is below @xmath9 , and probably much lower than this value . we therefore believe that few , if any , of these candidates are statistical fluctuations . of the 12 candidates , in all but one the best line fit is an emission feature with a centroid near 40 kev . the twelfth candidate is an absorption feature at 60 kev . typically the lowest energy of the data is about 20 kev , so we are unable to find lines much below 40 kev . we have previously presented results @xcite for a candidate in grb 940703 ( trigger 3057 ) which was usefully observed by only one batse spectroscopy detector ( sd ) . observations are useful only from sds in high - gain mode which point to within @xmath7 of the burst . since that time we have searched more bursts and begun more detailed analyses of the nine candidates that were usefully observed by more than one sd . these multiple - detector cases allow stringent tests of the reality of the candidates . ideally , the data from another detector will confirm the feature found by the automatic search . while some features may not be confirmed by another detector due to differing snrs between the detectors or from plausible statistical fluctuations , the data of all detectors must be consistent . grb 941017 ( trigger 3245 ) was usefully observed by sds 0 and 5 . the automatic search identified a candidate in the data of sd 0 during the rising portion of the burst ( fig . a spectrum is not available for sd 5 for the time interval with the best @xmath4 value for sd 0 . we therefore continued our analysis using spectra from a similar time interval ( 9.728 to 23.936 s ) which is available for both detectors . the results of fits to this interval are listed in table 1 . because of the non - optimum time interval , the significance of the feature in the data of sd 0 is reduced . the line was discovered in the data of sd 0 so we may regard the centroid as a priori determined when we analyze the data of sd 5 . a one parameter line fit yields a significance for the feature in sd 5 of @xmath10 . the joint fit ( fig . 2 ) yields a significance of @xmath11 , better than the value obtained with sd 0 alone despite the forced change to the common time interval . sd 5 provides evidence for the feature independent of the detector in which the feature was found by the automatic search . .line fits to the common time interval , grb 941017 . [ cols="^,^,^,^,^,^ " , ] to demonstrate the existence of a line , we must show that the data require a line independent of which reasonable continuum model is assumed @xcite . the above fits were made using the standard ` grb ' continuum function of band . however , if we use an alternative continuum model with a low - energy break _ in addition _ to the standard high - energy break , the significance of the line in sd 0 is reduced to @xmath12 ( @xmath13= 4% ) . while this model still involves a low - energy spectral feature ( fig . 3 ) , it means that for this burst we can not prove that the feature is a line . since our last report @xcite , we have completed the automatic search for bright bursts through may 1996 . we have identified 12 line candidates and have begun detailed analysis . because the 12 candidates appear as expected in the small fraction of spectra with high snr , their probability of appearing in the ensemble by chance is low . for all except one of the 12 candidates , the best line fit is an emission line with a centroid near 40 kev . our simulations show that we have useful sensitivity to _ ginga_-like absorption lines at 40 kev @xcite , so the lack of detections of absorption lines gives a constraint on their frequency . however , with the small number of bright bursts seen by batse and _ ginga _ , there is no significant discrepancy between the two instruments @xcite . observations made with multiple detectors have the strong advantage of redundantly verifying the existence of the features . we have now carefully investigated several candidate features , including some observed with multiple detectors . so far the evidence appears good for the existence of spectral features , however we do not want to make a blanket statement that batse has observed spectral features until we have carefully examined all 12 candidates . note that we are stating that the evidence appears good for the existence of _ spectral features . _ in the cases examined so far , we have not been able to demonstrate that the spectral features must be narrow lines an alternative explanation is possible : a low - energy break in addition to the normal break in the few 100 kev to 1 mev region .
|
we have developed an automatic search procedure to identify low - energy spectral features in grbs . we have searched 133,000 spectra from 117 bright bursts and have identified 12 candidate features with significances ranging from our threshold of @xmath0 5e@xmath15 to @xmath0 1e@xmath17 .
several of the candidates have been examined in detail , including some with data from more than one batse spectroscopy detector .
the evidence for spectral features appears good ; however , the features have not conclusively been shown to be narrow lines .
= = = 1=1=0pt = 2=2=0pt
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
europium hexaboride is part of the large and heterogeneous class of materials that exhibit colossal magnetoresistance ( cmr ) . the ferromagnetic transition in eub@xmath0 is accompanied by a dramatic change in resistivity . there is a large body of experimental data available on the magnetic and electric properties , but a thorough understanding is lacking . eub@xmath0 has a cubic unit cell with eu - ions at its vertices and a boron octahedron at its center . the material is ferromagnetic and shows two magnetic transitions : at @xmath1k and at @xmath2k @xcite . these have been associated with a spin reorientation and a ferromagnetic transition , respectively . neutron diffraction experiments @xcite have given a magnetic moment @xmath3 . this is exclusively due to the localized half - filled @xmath4-shell in the eu@xmath5 ions.@xcite electronic structure calculations @xcite , shubnikov - de haas and de haas - van alphen measurements @xcite show that eub@xmath0 is a semimetal . the fermi surface consist of two ellipsoidal pockets , one electron - like and one hole - like , centered on the @xmath6 point in the brillouin zone . the pockets contain very few carriers : hall effect measurements yield @xmath7 carriers per formula unit @xcite at low temperatures . small dilations of the boron octahedra cause overlap of the conduction and valence bands at the @xmath6 points rendering eub@xmath0 semimetallic . the carrier concentration decreases smoothly as temperature increases . the electrical resistivity is metallic in the ferromagnetic regime . it shows a sharp peak near @xmath8 . above this temperature , the resistivity decreases with an almost activated temperature dependence until it reaches a minimum at about 30k . at higher temperatures it increases and eventually starts to saturate at about room temperature . the application of a magnetic field produces sharp changes in the resistivity . close to the magnetic transition , _ negative _ magnetoresistance ( mr ) values of up to 100% have been observed @xcite . this decrease in resistivity is accompanied by a large decrease in the ( negative ) hall coefficient@xcite and an increase in the plasma frequency @xcite . the change in the plasma frequency is more gradual than the changes in resistivity and hall coefficient . in the ferromagnetic regime , on the other hand , the mr is large and _ positive _ : at 1.7 k resistivity changes of up to 700% have been observed in a transversal applied field of 7 t @xcite . the mr depends quadratically on the applied field strength at low temperatures@xcite . just above t@xmath9 and up to @xmath10 , the existence of magnetic polarons has been proposed as the cause of the stokes shift measured with spin flip raman scattering ( sfrs)@xcite . the resistivity is activated at these temperatures . however , this data contains a previously unremarked puzzle , in that the energy scale turns out to be considerably lower - by a factor of thirty - than expected based on reliable estimates of the exchange interaction . wigger et al.@xcite showed how the crossover between large positive and large negative mr from well below to well above the ferromagnetic transition can be explained by the dominance of orbital scattering at @xmath11 to spin scattering at @xmath12 . the model we shall use for the carrier transport in these regimes is similar to that of ref.@xcite and we shall thus suppress most of the details . the key feature of the model is its multiband nature - there are two types of carrier . in this paper we concentrate principally on the regime close to @xmath13 and analyze the evidence for the existence of magnetic polarons in europium hexaboride . we show how the sfrs results can be explained using a multiband model , resolving the conundrum of the anomalously small energy associated with the carrier spin flip . we model eub@xmath0 as a ferromagnetic semimetal with a low carrier density . both electrons and holes are itinerant and are coupled to the local moments @xmath14 . this can be described by the following general hamiltonian : @xmath15 here , the hopping parameter is roughly @xmath16 ev@xcite . @xmath17 is the itinerant carrier spin operator and the subindices @xmath18 and @xmath19 stand for electrons and holes respectively . @xmath20 ( @xmath21 ) is the on - site coupling between the spins of the electrons ( holes ) and the local moments . @xmath22 is the magnetic exchange between local moments . first of all we need to discuss what is the origin of the ferromagnetism and the order of magnitude of the magnetic couplings . ferro- and antiferro - magnetism of the insulating eu - chalcogenides ( eux , x@xmath23 o , s , se , te ) has been explained as due to superexchange interaction between neighbor eu ions @xcite through the anion between them . the density of carriers in the undoped chalcogenides is too low to expect any indirect rkky ( ruderman - kittel - kasuya - yosida ) interaction . the ferromagnetic interaction arises instead from the overlap between the 4@xmath4- and 5@xmath24-orbitals at different cations . this overlap leads to an effective exchange interaction in third order in perturbation theory@xcite . this does not apply directly to eub@xmath0 due to the different crystalline structure , but nevertheless one expects that the superexchange coupling @xmath22 is small . moreover , the increase of magnetic critical temperature and concomitant decrease of resistivity under high pressures @xcite has revealed that the magnetic exchange in eub@xmath0 is mainly due to the rkky interaction . therefore , @xmath22 in eq . [ eq : hamiltonian ] is negligible . the rkky magnetic exchange is mediated by the itinerant carriers via their coupling with the lattice magnetic moments . an effective heisenberg - like magnetic exchange can be written in terms of the local hund s like exchange coupling @xmath25 ( @xmath20 or @xmath21 ) @xcite @xmath26 where @xmath27 , @xmath28 is the density of carriers , and @xmath29 is the fermi energy . @xmath30 is an oscillating function of @xmath31 but is ferromagnetic for small @xmath31 . this is the relevant limit for europium hexaboride , as its low carrier density implies @xmath32 . to estimate the value of @xmath25 we use the mean field relation between t@xmath9 and j@xmath33 , @xmath34 , where @xmath35 is the coordination number for eu , and @xmath36 is the @xmath35-component of the local moments . using a critical temperature @xmath37 and a parabolic approximation to the bands , @xmath38 is found , consistent with reported data for isolated eu - ions@xcite . in this estimation we are considering that only one kind of carrier is responsible for the magnetism . when the local exchange coupling @xmath25 is large enough , carriers can be localized by ferromagnetic clusters and form composite objects called magnetic polarons . ferromagnetic polarons can exist in the low temperature phases of antiferromagnets but here we are interested in those formed in the paramagnetic phase . a necessary condition for the existence of magnetic polarons is that the density of carriers is very low compared to the inverse of the correlation volume , namely @xmath39 . when this condition is fulfilled , polarons are well - defined non - overlapping entities . there are two kinds of magnetic polarons : free and bound . a free magnetic polaron is a carrier localized in a ferromagnetic cluster embedded in a paramagnetic background . a carrier that is coupled strongly to local moments via a hund s like coupling tends to align the moments that are within a bohr radius . this causes a trapping potential that localizes the carrier . the potential can be enhanced by random fluctuations of the magnetization that produce an alignment of local moments in the carrier s vicinity@xcite . the carrier thus traps itself by the magnetization it causes . it could increase the alignment of the local moments and hence decrease its energy by localizing itself in a smaller volume . however , this would lead to an increase in kinetic energy . the quantity that determines the stability and size of these objects is therefore @xmath40 where @xmath25 is the coupling of the carrier spin to the local moments and @xmath41 is the hopping parameter . the ratio @xmath40 needs to be typically larger than one @xcite to guarantee stability of the free magnetic polaron . on the other hand , in bound magnetic polarons the main driving force trapping the carrier is not the local magnetic interaction but the electrostatic potential created by impurities . the formation of the ferromagnetic cluster described above does occur . however , it is a second order process , as the magnitude of hund s coupling is much smaller than the coulomb interaction . mean field @xcite and monte - carlo @xcite calculations have shown that magnetic polarons can exist within a temperature window above t@xmath9 whose width depends on the ratio @xmath40 . at higher temperatures , magnetic fluctuations are strong enough to destroy the magnetic polarons . below @xmath42 , the condition @xmath43 is not fulfilled and the polarons overlap . if a magnetic field is applied within the existence temperature window , the size of a polaron increases until eventually the polarons overlap and produce a ferromagnetic transition . free and bound magnetic polarons can be differentiated by their dynamics and the resistivity they cause . bound magnetic polarons are bound to an impurity in the system so the only way of transport is via an activated process : when the trapped carrier is `` ionized '' it is free to move until it is trapped by another impurity . therefore they produce a resistivity @xmath44 such that @xmath45 . in contrast , free magnetic polarons are able to move to adjacent areas when random fluctuations of the nearby spins produce an aligned region . there is not a barrier to overcome in this process . this transport mechanism has been called `` fluctuation - induced hopping '' @xcite and produces a metallic resistivity @xmath46 . magnetic polarons have been largely studied in connection with eu - chalcogenides ( euo , eus , euse , eute ) @xcite and diluted magnetic semiconductors such as cd@xmath47mn@xmath48te and pb@xmath47mn@xmath48te with @xmath31 , the concentration of magnetic ions , small . experimental evidence included photoluminescence spectra @xcite and magneto - optical experiments as spin flip raman scattering @xcite . the raman scattering spectrum shows an inelastic peak at low frequencies ( stokes shift ) which , for the diluted magnetic semiconductors , depends as follows on polaronic properties:@xcite @xmath49 where @xmath50 is the density of magnetic ions participating in the formation of polarons @xcite , and @xmath25 is the local exchange interaction between the @xmath51 itinerant electrons and the @xmath24 electrons localized in the mn ions . the low density of local moments makes for a small stokes shift , which - for cd@xmath47mn@xmath48se - is consistent with experiment@xcite . sfrs measurements done in eub@xmath0 have similarly revealed a zero - field peak in scattered intensity of the order of @xmath52 @xcite at @xmath53 , just above the magnetic critical temperature . the behavior of this peak with temperature and external magnetic field is consistent with the stability conditions theoretically established for magnetic polarons . free magnetic polarons are not expected to be stable in eub@xmath0 as @xmath25 and @xmath41 are comparable . moreover , we have argued above that the activated behavior of the resistivity is better explained by means of bound magnetic polarons . eu - site vacancies would produce the binding coulomb potential for electrons . we expect the energy of the stokes shift to be given by eq . [ eqn : stokesshift ] but now @xmath54 as eub@xmath0 has a local moment on every eu site in the cubic lattice . very few site vacancies are expected in this fairly clean material . mean - field theory@xcite predicts that bound magnetic polarons are fully spin polarised so @xmath55 . using this value and the energy of the stokes shift we obtain @xmath56mev this is far too low compared to the values reported in the literature for @xmath57 in isolated eu ions @xmath58mev @xcite . we are therefore left with a conundrum : the peak in the light scattering intensity follows all the trends calculated for an object with magnetic origin but the energy of that peak is almost two orders of magnitude smaller than expected . the solution to this problem lies in the fact that both polarised electrons and holes are found at the fermi energy . electrons and holes come from different b and eu orbitals and therefore their magnetic couplings to the localized spin in the eu @xmath59 orbitals , @xmath20 and @xmath21 respectively , can be very different . electronic structure calculations @xcite reveal that the hole pocket comes from the highest intraoctahedron b @xmath60 band . on the other hand , the electron pocket comes from bonding combinations of the cation @xmath24 orbitals pointing along the cartesian directions with some hybridization with the b atoms and some free - electron - like character on the ( 110 ) axes between the cations . in other words , the electron charge density distribution is mainly found around the eu ions while the holes are found around the b. therefore , the coupling of the electrons is expected to be much larger than that of the holes . consistently , fig . 10 in ref.@xcite shows a much larger majority - minority spin band splitting for electrons than for holes . in conclusion , we propose that the ferromagnetic ordering is produced by the itinerant electrons coupled to the localized spin in eu with @xmath61 , while the itinerant holes , much more weakly coupled ( @xmath62 ) , account for the sfrs stokes shift . a corollary of this identification is that there is likely a much higher energy feature in the sfrs , so far unobserved , that correspond to spin - flip of the electron state . in the following section we will see how the existence of both electrons and holes is necessary to explain other electronic properties of eub@xmath0 . at low temperatures the magnetoresistance is due to the presence of two types of carriers and we will call it `` orbital '' mr . there are two effects involved . the same physics that causes the hall effect is the most important cause of the mr . in addition to this , there is also a small shift of the bands with applied magnetic field that causes a small change in the carrier density . in a simple metal with one type of carrier and a simple fermi surface , there is no mr . a current that flows perpendicular to a magnetic field is initially deflected due to the transverse lorentz force . this causes an electric field , the hall field @xmath64 , to build up : @xmath65 the carriers that are deflected hit the edge of the sample and accumulate . the field that is thus built up counteracts the lorentz force . when it cancels the lorentz force , the current is undeflected . this is well known ; it means that there is no mr in a normal metal . the component of the current density that is parallel to the applied electric field is not affected by the magnetic field . therefore the resistivity remains unaffected as well . in a semimetal , on the other hand , there are by definition two kinds of carriers . these two kinds will almost invariably have different scattering times and different masses . [ eqn : hall_voltage ] shows that the hall voltage depends on the scattering time and the mass of the carrier . therefore , the hall voltages of the different kinds of carriers are also different . there is thus no voltage at which the two carriers will travel through the sample without deflection . the difference in the hall voltages is proportional to the applied magnetic field , so that the component of the current parallel to the applied electric field decreases with increasing field . the resistivity increases therefore when a magnetic field is applied , and the mr is therefore positive . we call this orbital mr , since it is due to the difference of the masses and scattering times of the two types of carriers . these properties derive from particularities of the atomic orbitals in the material . the mr can be calculated from a linearized boltzmann equation that includes a magnetic field under the assumption of a spherical fermi surface . this is a standard calculation , which can be found in @xcite and @xcite . we quote the result for the magnetoresistance : @xmath66 where @xmath67 and @xmath68 are the electron and hole conductivity respectively , and @xmath69 and @xmath70 are their mobilities , both in the absence of any magnetic field . the right hand side of the above formula is easily seen to vanish if both types of carriers have the same mass and scattering time : the difference of the mobilities in the numerator vanishes in that case . at low temperatures the orbital mr is largest : the scattering time is largest , so that the deflection is largest as well . as the scattering time decreases the effect becomes less important . for sufficiently small scattering times orbital mr becomes negligible . the second effect that causes mr in eub@xmath0 is the shifting of the bands when a magnetic field is applied . the shift is caused by the coupling @xmath25 of the carriers to the eu local moments . let us write the local moments as the sum of their average and the deviations therefrom : @xmath71 . we now use this expression in the hamiltonian in eq . [ eq : hamiltonian ] , and obtain a term that couples the carrier s energy to the magnetization . as the magnetization grows , the electron - like band is shifted to lower energies , and the hole - like band to higher energies . this causes an increase in the number of carriers as carriers spill over from one band into the other . the change in carrier density can be obtained from the following two requirements . firstly , the band - shift introduced by the change in magnetization @xmath72 is @xmath73 for electrons and @xmath74 for holes . secondly , the increase in the number of carriers of both kinds is equal : as the hole like band is shifted up , new holes are created as negatively charged particles spill into the electron - like band and vice versa . overall charge balance is maintained . we also assume a spherical fermi surface , an accurate assumption in the case of eub@xmath0 @xcite . we can use the requirement that charge neutrality be conserved to calculate the shift in the fermi level . the shift is then used to obtain the number of carriers by integrating the density of states up to the fermi level . the increase resulting from the shift of the bands is small , even at full saturation of the magnetization . its effect on the mr at low temperatures is then negligible . the magnetization as a function of applied field is obtained from a curie - weiss model . we included this change in carrier density due to the bands shifting in our model for the orbital mr . the contribution to the mr due to band shifting is opposite to that of the orbital effects . an external magnetic field increases the carrier density , so that it decreases the resistivity . since the magnetization is almost saturated at low temperatures , the carrier density does not change much with applied field , and the mr is affected only slightly . the orbital contribution to the mr dominates . we plot the mr obtained for the combined effect of the band shifts and the orbital effects in fig . [ fig : mrlowt ] . the temperature dependent scattering times at zero field are obtained from experimental data in @xcite . the electron and hole mobilities are obtained from the conductivity at zero field , the masses of the carriers@xcite and the carrier densities . the latter were obtained from the plasma frequency in @xcite . we introduced a small imbalance between the carrier densities of @xmath75 per unit cell , in accordance with @xcite . this imbalance is thought to arise from impurities . these numbers provide input to the model . our simple model , which depends only on parameters measured at zero field , can reproduce the large positive mr at low temperatures accurately . [ fig : mrlowt ] shows that it reproduces the magnitude of the mr well . it also predicts the ( nearly ) quadratic dependence in applied field . from fig . [ fig : mrlowt ] it is clear that the orbital contribution to the mr dwindles at higher temperatures . the model proposed will lose its validity near the magnetic transition and in the paramagnetic phase , when other effects dominate . the mechanisms that govern the low - temperature mr become insignificant near the ferromagnetic transition . close to @xmath13 the scattering time is so short that the positive orbital contribution to the mr is negligible . on the other hand , the shift of the bands caused by an applied magnetic field becomes substantial . we estimate that the carrier density changes by about @xmath77 as the applied field saturates the magnetization . this could explain only part of the negative mr in the critical regime . aditionally , near the critical point , spin fluctuations may provide a large contribution to the electrical resistance . the dominant modes near a ferromagnetic transition are those with small @xmath78 . they only produce substantial backscattering if @xmath79 is itself small @xcite . this is the case of eub@xmath0 as its carrier density is very small . the suppression of the spin fluctuations when a magnetic field is applied is largely responsible for the mr found in the critical regime . moreover , as shown in ref . @xcite ( see fig . 3 ) , the localization of the carriers in magnetic polarons further increases the magnetoresistance . in the temperature regime - just above @xmath13 - where the sfrs data gives evidence for bound magnetic polarons , the magnetoresistivity is large , and strongly negative , as expected since an applied magnetic field suppressed the magnetic polarons . at temperatures greater than about @xmath80 where the polarons are destabilized , we have a smaller ( but still negative ) mr dominated by local spin fluctuation scattering . as mentioned in the introduction , the negative mr is accompanied by a shift in the plasma frequency @xcite . the carrier density change produced by the band shifting alone is too small to account for that shift . this is in contrast to the results by wigger _ et al _ @xcite who claim consistency between the experiment and their calculation , though the functional dependence @xmath81 @xcite is not reproduced . there are two main differences between their model and ours : they consider eub@xmath0 to be a strongly compensated n - type magnetic semiconductor and use the same local coupling @xmath25 for electrons and holes . eub@xmath0 is a low carrier density ferromagnet with unusual properties : the resistance changes from metallic to activated and then to metallic as the temperature increases and the magnetoresistance changes sign close to @xmath42 . the activated region has been ascribed to the existence of bound magnetic polarons . we discuss their existence in the light of spin flip raman scattering measurements reported in ref.@xcite . we conclude that the signature seen by those experiments is due to the hund s like coupling of holes with the local spins while electrons are responsible for the magnetic ordering through the rkky interaction . this resolves the puzzle that the rkky transition temperature implies an exchange coupling of the carriers to the local moment of about 0.1 ev , about 30 times larger than the measured spin - flip energy of a carrier trapped in a bound polaron . the existence of bound magnetic polarons is also consistent with a large _ negative _ magnetoresistance above @xmath13 . the _ positive _ magnetoresistance in the ferromagnetic phase is also produced by the interplay of two kinds of carriers with different masses and scattering times . in the diluted magnetic semiconductors of the type cd@xmath47mn@xmath48se , the mn are antiferromagnetically coupled via superexchange . this is a shortrange interaction . at very low @xmath31 , the probability of having mn clusters is very low so @xmath82 . for larger @xmath31 , antiferromagnetically coupled clusters of mn form , so not all the mn ions will participate in the magnetization of the system . see , for instance @xcite .
|
eub@xmath0 is a low carrier density ferromagnet which exhibits large magnetoresistance , positive or negative depending on temperature .
the formation of magnetic polarons just above the magnetic critical temperature has been suggested by spin - flip raman scattering experiments .
we find that the fact that eub@xmath0 is a semimetal has to be taken into account to explain its electronic properties , including magnetic polarons and magnetoresistance .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
electromagnetic field enhancement in nanoplasmonic structures plays an important role in a number of applications such as nano antennas @xcite , single - photon emitters @xcite , and surface - enhanced spectroscopy techniques in general @xcite , including surface - enhanced raman spectroscopy ( sers ) @xcite . field localization is the key to understanding and further enhancing the resolution of sers , and the use of highly engineered surface plasmonic structures has been envisioned @xcite . strong field enhancement has so far been inherently connected to singularities @xcite . it is commonplace to study the plasmonic response using maxwell s equations with the bulk dielectric function of the metal , with an abrupt change at the air - metal interface @xcite . though clearly neglecting surface and confinement effects for the electrons in the metal , this is a celebrated approximation which predicts the existence of surface plasmons forming the heart of nanoplasmonics . however , when samples involve true nano - scale feature - sizes , the validity of this approximation is challenged and new phenomena , such as non - local response and spatial dispersion @xcite and inhomogeneous electron densities due to quantum confinement @xcite are expected to play an important role . in nanoscale metallic structures the electrons are subject to quantum confinement , which leads to a number of spectacular effects , such as the quantized conductance of atomic quantum - point contacts @xcite , friedel oscillations and confinement of electrons to quantum corrals on noble metal surfaces @xcite , and a size - dependent structure of nanocrystals @xcite . light emission from stm has been used to reveal plasmonic properties of electrons confined to triangular islands on surfaces of noble metals @xcite . all these studies clearly show that , at least at low temperatures , quantum confinement plays an important role leading to inhomogeneous electron densities at interfaces that differ from the piecewise constant bulk values . the qualitative optical consequences of an inhomogeneous electron density can be explored within lindhard s _ `` local density approximation '' _ for the dielectric function @xcite @xmath0 where @xmath1 accounts for the density variations relative to the bulk density of electrons @xmath2 , with @xmath3 being the bulk plasma frequency . to mimic the ohmic damping we have included a damping rate @xmath4 and in the optical regime @xmath5 @xcite . while several studies have considered the consequences of local density variations for the surface - plasmon dispersion relation , the overall effects have been reported to be minor and for planar surfaces the surface - plasmon resonance at @xmath6 appears unaffected by the detailed density profile at the metal - air surface @xcite . however , it remains an open question to which extent the detailed field distribution will be affected by a smoothly varying electron density in contrast to the discontinuous drop in the electron density assumed by the bulk approximation . zero - index phenomena in metamaterials are receiving increasing attention @xcite , and the recent results by litchinitser _ et al . _ @xcite show that for a metamaterial with the effective constitutive coefficients varying linearly , @xmath7 and @xmath8 , a pronounced field enhancement will occur at @xmath9 where @xmath10 . in this paper we show that plasmonic structures employing nature s own metals may support large field enhancement at the zero-@xmath11 surface appearing in the vicinity of the air - metal interface , where the real part of the polarization of the dilute electron gas will exactly balance the polarization of the vacuum background . as shown below , the particular position of the zero-@xmath11 surface depends on the frequency @xmath12 of the incident radiation , and may be either in air or in the metal , depending on the local geometry . the basic mechanism behind the field enhancement can be understood from the continuity condition for the electrical displacement field , which stipulates that @xmath13 be continuous ( here @xmath14 is the normal vector of a surface element ) . at the zero-@xmath11 surface the normal component @xmath15 thus has to diverge to keep the product @xmath16 finite . from eq . ( [ eq : lindhard ] ) it follows that in the regions where @xmath17 vanishes , there will still be a small , but finite imaginary part : @xmath18 . for an electrical field @xmath19 incident at an angle @xmath20 we thus expect that the normal component @xmath21 is enhanced by a factor proportional to @xmath22 on the zero-@xmath11 surface . the intensity enhancement can then be estimated as @xmath23 we emphasize that this mechanism of field - enhancement is generic and robust as it only depends on the assumed continuous negative - to - positive variation of the dielectric function at the metal - air interface , in combination with a maxwell boundary condition @xcite , rather than on the details of the excitation of strongly localized surface - plasmon resonances with pronounced sample - to - sample fluctuations . the low - temperature density profile for jellium is a function of @xmath24 only , with the radius of the free - electron sphere @xmath25 and @xmath26 being the bohr radius . the position - dependent electron density may be obtained self - consistently with the aid of density - functional theory @xcite , thus taking into account both exchange and correlation effects in the inhomogeneous electron gas . in ref . @xcite , results are tabulated for a range of values of @xmath24 . the resulting density @xmath27 is shown in fig . [ fig1](a ) for @xmath28 , a typical value for gold or silver . as seen , @xmath29 varies continuously on the length scale of the fermi wavelength rather than exhibiting a discontinuous drop at the surface of the metal ( @xmath9 ) . in the metal ( @xmath30 ) , there are friedel oscillations near the surface , while for @xmath31 the decay of the density is caused by the finite value of the work function @xmath32 , which allows electron wave functions to penetrate into the classically forbidden region . according to eq . ( [ eq : lindhard ] ) , the dielectric function will inherit the friedel oscillations in the metal , as well as approach the value in vacuum outside the metal surface . as a consequence , the dielectric function passes through zero just outside the metal surface [ green line in fig . [ fig1](a ) ] . similar spatial variations of the dielectric function have been reported in more elaborate studies of e.g. semiconductor quantum dots @xcite and recently also in doped semiconductors @xcite . in the case of @xmath33 the zero-@xmath11 surface appears at a distance of 0.25 nm from the surface ( @xmath9 ) . [ fig1](b ) illustrates the field enhancement in the case of a plane wave incident at an angle of @xmath34 . the numerical results were obtained by solving the wave equation @xmath35 with the aid of a finite - element method employing an adaptive mesh algorithm ( comsol multiphysics ) . a comparison with @xmath36 in fig . [ fig1 ] ( a ) shows that the field enhancement indeed occurs at the point where @xmath37 . we emphasize that this phenomenon has no counterpart within the classical bulk approximation of the dielectric function . for a realistic value of the damping ( @xmath38 ) the intensity may be enhanced by more than 3 orders of magnitude , which suggests that in a sers experiment the raman signal could be enhanced by 6 orders of magnitude @xcite without the need for any geometry - induced localized resonances . to further test the predictions of eq . ( [ scaling ] ) we have explored the dependence on the angle of incidence as well as the particular value of the plasmon damping . [ fig2 ] ( a ) illustrates how the peak height depends on damping . the enhancement is limited by the plasmonic damping with the slope of the dashed lines being in full accordance with the power ( -2 ) of @xmath4 in eq . ( [ scaling ] ) . panel ( b ) illustrates the angle dependence with the dashed line showing the @xmath39 dependence of eq . ( [ scaling ] ) , indicating a pronounced enhancement for oblique incidence . we next explore the field enhancement in the presence of strong confinement of electrons in metallic nanostructures . here we limit ourselves to a model of non - interacting electrons in wire geometries with nanoscale cross sections @xmath40 so that the problem effectively is two - dimensional . the energy states of the electron system are then given by @xmath41 with the transverse energy components being quantized and governed by a two - dimensional schrdinger equation @xmath42\psi_\nu(x , y)={\mathscr e}_\nu\psi_\nu(x , y).\ ] ] for the confinement potential @xmath43 we adjust the height in accordance with the work function of the metal @xmath44 , where @xmath45 is the chemical potential . in equilibrium the states are populated according to the fermi dirac distribution function and we find @xmath46 where @xmath47 is area and @xmath48 $ ] with @xmath49 being jonquire s polylogarithm function . equilateral triangular nano islands have been the subject of numerous investigations @xcite and in fig . [ fig3 ] we consider field enhancement in such a structure . [ fig3](a ) shows results within the bulk approximation , where an artificial rounding of the corners has been added to circumvent the effect of an otherwise divergent electrical field @xcite . clearly , field enhancement is modest since no pronounced surface plasmon resonances are excited . [ fig3](b ) on the other hand is based on the spatially continuously varying dielectric function of eqs . ( [ eq : lindhard ] ) and ( [ eq : jonquiere ] ) and shows a pronounced field enhancement along the zero-@xmath11 surface . we also note that while the geometry has arbitrarily sharp corners , the electron density and dielectric function themselves are still smooth , showing that quantum confinement provides a built - in smoothing even in the presence of sharp geometrical features . [ fig4](a ) and ( b ) show in more detail the field enhancement along the lines indicated in fig . [ fig3](b ) . for the results in fig . [ fig4](b ) , the peak height clearly matches the one - dimensional results in panel ( b ) of fig . [ fig2 ] for an incident angle of @xmath50 . similarly , the result within the bulk approximation resembles typical results , see e.g. ref . @xcite , i.e. the field enhancement is more modest . in fig . [ fig4](a ) , the incident wave propagates almost parallel to the zero-@xmath11 surface ( @xmath51 ) , thus causing a peak that is even higher at the two right - most corners of the triangle . on the other hand , at the left - most corner the wave has normal impact on the zero-@xmath11 surface ( @xmath52 ) so that no pronounced field enhancement is observed . it is interesting to note that the enhancement may occur either outside [ as in fig . [ fig1](b ) ] or inside [ as in fig . [ fig3](b ) ] the metal , depending on the local geometry and the frequency relative to the plasma frequency . we predict that a large non - resonant field enhancement may occur at zero - epsilon surfaces near the air - metal interface . theoretically , the enhancement occurs in a wide class of models where the discontinuous jump of the dielectric function of the usual bulk description of at an air - metal interface is replaced by a continuous function @xmath53 that has the bulk values as limiting values away from the interface . we discussed two microscopic models for such a continuous variation at the interface . the predicted field enhancement has therefore no counterpart in the classical bulk treatment of plasmonic field enhancement , though it is intimately related to the familiar boundary condition for the electrical displacement field . in particular , for light impinging on an interface we discussed the dependence of the field enhancement on the angle of incoming light and on the damping in the metal . furthermore , the enhancement constitutes another mechanism for the enhancement of raman signals at metal interfaces ( sers ) . interestingly , feigenbaum _ et al . _ have recently reported their experimental observation of zero - index driven field enhancement at the surface of conducting oxides ( i.e. with lower electron density compared to good metals ) @xcite . we emphasize that the decay of the electron density into vacuum ( or more generally into a dielectric ) could have important implications also for nano - gap structures ( such as the gap formed between the corners of two nearby triangles @xcite ) , where quantum tunneling of electrons will introduce a small but finite electron density inside the gap , thus also modifying the local dielectric function and the ability to support short - circuiting currents @xcite . we thank peter nordlander , natalia m. litchinitser , and vladimir m. shalaev for useful discussions . this work is financially supported by the danish council for strategic research through the strategic program for young researcher ( grant no : 2117 - 05 - 0037 ) , the danish research council for technology and production sciences ( grants no : 274 - 07 - 0080 & 274 - 07 - 0379 ) , as well as the fidipro program of the finnish academy . o. perez - gonzalez , n. zabala , a. g. borisov , n. j. halas , p. nordlander , and j. aizpurua , `` optical spectroscopy of conductive junctions in plasmonic cavities , '' _ nano lett . _ * 10*(8 ) , 3090 3095 ( 2010 ) . j. v. lauritsen , j. kibsgaard , s. helveg , h. topse , b. s. clausen , e. lgsgaard , and f. besenbacher , `` size - dependent structure of o@xmath54 nanocrystals , '' _ nat . nanotechnol . _ * 2*(1 ) , 53 58 ( 2007 ) . our qualitative conclusions do not change if a more sophisticated form for the dielectric function is adopted , such as the charge - conserving form due to n. d. mermin , _ phys . b _ * 1*(5 ) , 2362 ( 1970 ) . r. liu , q. cheng , t. hand , j. j. mock , t. j. cui , s. a. cummer , and d. r. smith , `` experimental demonstration of electromagnetic tunneling through an epsilon - near - zero metamaterial at microwave frequencies , '' _ phys . lett . _ * 100*(2 ) , 023903 ( 2008 ) . b. edwards , a. al , m. e. young , m. silveirinha , and n. engheta , `` experimental verification of epsilon - near - zero metamaterial coupling and energy squeezing using a microwave waveguide , '' _ phys . rev . lett . _ * 100*(3 ) , 033903 ( 2008 ) . r. j. pollard , a. murphy , w. r. hendren , p. r. evans , r. atkinson , g. a. wurtz , a. v. zayats , and v. a. podolskiy , `` optical nonlocalities and additional waves in epsilon - near - zero metamaterials , '' _ phys . rev . lett . _ * 102*(12 ) , 127405 ( 2009 ) . a. sundaramurthy , p. j. schuck , n. r. conley , d. p. fromm , g. s. kino , and w. e. moerner , `` toward nanometer - scale optical photolithography : utilizing the near - field of bowtie optical nanoantennas , '' _ nano lett . _ * 6*(3 ) , 355360 ( 2006 ) . j. nelayah , m. kociak , o. stphan , f. j. garca de abajo , m. tenc , l. henrard , d. taverna , i. pastoriza - santos , l. m. liz - marzn , and c. colliex , `` mapping surface plasmons on a single metallic nanoparticle , '' _ nat . phys . _ * 3*(5 ) , 348353 ( 2007 ) .
|
we point out an apparently overlooked consequence of the boundary conditions obeyed by the electric displacement vector at air - metal interfaces : the continuity of the normal component combined with the quantum mechanical penetration of the electron gas in the air implies the existence of a surface on which the dielectric function vanishes .
this , in turn , leads to an enhancement of the normal component of the total electric field .
we study this effect for a planar metal surface , with the inhomogenous electron density accounted for by a jellium model .
we also illustrate the effect for equilateral triangular nanoislands via numerical solutions of the appropriate maxwell equations , and show that the field enhancement is several orders of magnitude larger than what the conventional theory predicts .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the discovery of charged higgs bosons @xcite will provide a concrete evidence of the multi - doublet structure of the higgs sector . recent efforts have focused on their relevance to supersymmetry ( susy ) , in particular in the mssm , which incorporates exactly two higgs doublets , yielding after spontaneous ew symmetry breaking five physical higgs states : the neutral pseudoscalar ( @xmath11 ) , the lightest ( @xmath4 ) and heaviest ( @xmath12 ) neutral scalars and two charged ones ( @xmath13 ) . in much of the parameter space preferred by susy , namely @xmath14 and @xmath15 @xcite , the lhc will provide the greatest opportunity for the discovery of @xmath13 particles . in fact , over the above @xmath6 region , the tevatron ( run 2 ) discovery potential is limited to charged higgs masses smaller than @xmath16 @xcite . however , at the lhc , whereas the detection of light charged higgs bosons ( with @xmath17 ) is rather straightforward in the decay channel @xmath18 for most @xmath6 values , thanks to the huge top - antitop production rate , the search is notoriously difficult for heavy masses ( when @xmath19 ) , because of the large reducible and irreducible backgrounds associated with the main decay mode @xmath20 , following the dominant production channel @xmath21 @xcite . ( notice that the rate of the latter exceeds by far other possible production modes @xcite@xcite , this rendering it the only viable channel at the cern machine in the heavy mass region . ) the analysis of the @xmath20 signature has been the subject of many debates @xcite@xcite , whose conclusion is that the lhc discovery potential is satisfactory , but only provided that @xmath6 is small ( @xmath22 ) or large ( @xmath23 ) enough and the charged higgs boson mass is below 600 gev or so . a recent analysis @xcite has shown that the @xmath24 decay mode , indeed dominant for light charged higgs states and exploitable below the top threshold for any accessible @xmath6 @xcite , can be used at the lhc even in the large @xmath25 case , in order to discover @xmath13 scalars in the parameter range @xmath26 and 200 gev @xmath27 tev . besides , if the distinctive @xmath28 polarisation @xcite is used in this channel , the latter can provide at least as good a heavy @xmath13 signature as the @xmath20 decay mode ( for the large @xmath6 regime @xcite ) . at present then , it is the @xmath29 region of the mssm which ought to be explored through other decay modes , especially those where direct mass reconstruction is possible . the most obvious of these is the @xmath30 channel @xcite ( see also @xcite ) , proceeding via the production of a charged gauge boson and the lightest higgs scalar of the mssm , with the former on- or off - shell depending on the relative values of @xmath25 and @xmath31 . in fact , its branching ratio ( br ) can be rather large , competing with the bottom - top decay mode and overwhelming the tau - neutrino one for @xmath32 at low @xmath6 : see figs . [ fig : brs][fig : brh ] . besides , under the assumption that the @xmath4 scalar has previously been discovered ( which we embrace here ) , its kinematics is rather constrained , around two resonant decay modes , @xmath33 2 jets ( or lepton - neutrino ) and @xmath1 , an aspect which allows for a significant reduction of the qcd background . as demonstrated in ref . @xcite , signals of charged higgs bosons in the @xmath34 range can be seen in this channel , provided that 200 gev @xmath35 gev ( see also @xcite for an experimental simulation ) . the above lower limit on @xmath6 corresponds to the border of the exclusion region drawn from lep2 direct searches for the mssm @xmath4 scalar , whose mass bound is now set at @xmath36 gev or so @xcite . it is the purpose of this letter that of resuming the studies of ref . @xcite , by analysing the contribution to the background due to several irreducible processes , not considered there , whose presence could spoil the feasibility of charged higgs searches in the @xmath37 mode of the mssm . the plan of this paper is as follows . in the next section we discuss possible signals and backgrounds , their implementation and list the values adopted for the various parameters needed for their computation . section 3 is devoted to the presentation and discussion of the results . conclusions are in section 4 . we generate the signal cross sections by using the formulae of ref . that is , we implement the @xmath38 matrix element ( me ) for the process @xmath39 this nicely embeds both the @xmath40 subprocess of top - antitop production and decay , which is dominant for @xmath41 , as well as the @xmath42 + c.c . one of @xmath43-fusion and @xmath13-bremsstrahlung , which is responsible for charged higgs production in the case @xmath44 @xcite . the me of process ( [ signalme ] ) has been computed by means of the spinor techniques of refs . @xcite@xcite . in the @xmath45 channel , assuming high efficiency and purity in selecting / rejecting @xmath46-/non-@xmath46-jets , possible irreducible background processes are the following ( we consider only the @xmath47-initiated channels ) : 1 . the @xmath48 continuum ; 2 . @xmath49 production , especially when @xmath50 ; 3 . the qcd induced case @xmath51 ; 4 . and , finally , @xmath52 and @xmath53 intermediate states ; in which @xmath54 , plus their c.c . channels . once the top quark appearing in the above reactions decays , two @xmath3 bosons are present in each event . we will eventually assume the @xmath55 pair to decay semi - leptonically to light - quark jets , electrons / muons and corresponding neutrinos . furthermore , we will require to tag exactly three @xmath46-jets in the final state ( e.g. , by using @xmath56-vertex or high @xmath57 lepton techniques ) . the same ` signature ' was considered in ref . @xcite , where only the ` intrinsic ' @xmath58 background and the qcd noise due to ` @xmath59 + jet ' events were studied ( with jet signifying here either a @xmath46- , light - quark or gluon jet , the latter two mistagged for the former ) . both signal and background mes have been integrated numerically by means of vegas @xcite and , for test purposes , of rambo @xcite and metropolis @xcite as well . while proceeding to the phase space integration , one also has to fold in the @xmath60-dependent parton distribution functions ( pdfs ) for the two incoming gluons . these have been evaluated at leading - order , by means of the package mrs - lo(05a ) @xcite . the numerical values of the sm parameters are ( @xmath61 ) : @xmath62 @xmath63 @xmath64 @xmath65 as for the top width @xmath66 , we have used the lo value calculated within the mssm ( i.e. , @xmath67 gev if @xmath68 ) . concerning the mssm parameters , we proceed as follows . for a start , we assume that the mass of the lightest neutral higgs particle ( but not @xmath6 ) is already known , thanks to its discovery at either lep2 , tevatron ( run 2 ) or from early analyses at the lhc itself . thus , for us , @xmath69 is a fixed parameter , assuming for reference the following discrete values : e.g. , 90 , 100 , 110 , 120 and 130 gev . then we express all other higgs masses as a function of @xmath25 and @xmath6 . for the pseudoscalar higgs boson mass , the tree - level relation @xmath70 is assumed . radiative corrections then , of arbitrary perturbative order , are in practice embedded in the @xmath12 mass and the mixing angle @xmath71 . in general , notice that , at the ` renormalisation group improved ' one - loop level @xcite , it is only for very large values of the lightest stop mass and of the squark mixing parameters that @xmath2 can escape the lep2 bound in the low @xmath6 region , on which we will focus most of our attention . finally , notice that we develop our discussion at the parton level , without considering fragmentation and hadronisation effects . thus , jets are identified with the partons from which they originate and all cuts are applied directly to the latter . in particular , when selecting @xmath46-jets , a vertex tagging is implied , with a finite efficiency , @xmath72 , per each tag . moreover , we assume no correlations among multiple tags , nor do we include misidentification of light - quark ( including @xmath73-quark-)jets produced in @xmath3 decays as @xmath46-jets . as a preliminary exercise , we study the total production and decay cross sections before any cuts , as all our reactions are finite over their entire phase spaces ( recall that @xmath74 ) . this is done in figs . [ fig : cross90][fig : cross100 ] for the signal and the five background processes discussed in the previous section , for five values of @xmath6 , over the range 140 gev @xmath75 500 gev , for @xmath76 and 100 gev , in the channel @xmath77 , where @xmath78 or @xmath11 . ( of course , the @xmath79 and @xmath80 backgrounds have no dependence on any of the three parameters above[fig : cross100 ] account for the c.c . production modes as well . ] . ) as for the decay rates of the top ( anti)quark and the @xmath3 boson , for sake of simplicity , we take them equal to @xmath81 for the time being . the signal is always dominated by the qcd background and at large @xmath6 also by the ew ones . notice the local maxima of the signal rates at @xmath82 , as induced by the opening of the @xmath83 decay ( compare to figs . [ fig : brs][fig : brh ] ) , and the minima as well , due to the onset of the @xmath20 channel instead . in the reminder of our analysis , we assume semi - leptonic decays of @xmath55 pairs , as in ref . @xcite : i.e. , @xmath84 ( hereafter , jet refers to a non-@xmath46-jet and @xmath61 ) . however , as compared to that analysis , we make one simplification . namely , we assume that _ one _ top ( anti)quark and the @xmath3 boson generated in its decay have already been reconstructed , e.g. , by using the mass selection procedure advocated in ref . @xcite , either leptonically or hadronically . this allows us to greatly reduce the complexity of our numerical calculation while we believe substantially un - affecting the relative rates of signal and backgrounds ( in fact , all processes described produce the same final state and all involve at least one top quark ) . then we apply the following cuts on the remaining particles ( here , the label @xmath85 refers to the decay products of the second @xmath3 boson present in the event , which can be either light - quarks or leptons ) : @xmath86 on the transverse momentum ( including the missing one ) , @xmath87 on the pseudorapidity , and @xmath88 on the relative separation of @xmath46- and light - quark jets / leptons j , where @xmath89 is defined in terms of relative differences in pseudorapidity @xmath90 and azimuth @xmath91 , with @xmath92 , j@xmath93 . furthermore , we impose ( see also ref . @xcite ) latexmath:[\[\label{mhcut } @xmath46-jets , @xmath95 on the light - jet ( or lepton - neutrino ) pair ( recall that the @xmath3 can be off - shell ) , and , finally , @xmath96 around the top mass if three @xmath46 s are present in the event ( in addition to the one already used to reconstruct the top ( anti)quark ) . in such a case , one may assume that the charged higgs boson has predominantly been produced in the decay of a top ( anti)quark ( when @xmath97 ) . if instead only two appear , then one should conclude that the higgs has mainly been generated in a bremsstrahlung / fusion process ( because @xmath32 ) with a @xmath46-(anti)quark lost along the beam pipe . our @xmath38 production mechanism naturally allows one to emulate both dynamics in a gauge invariant fashion , including all interference effects . as already mentioned , however , we will assume a triple @xmath46-tagging , this implying an overall efficiency factor of @xmath98 multiplying our signal and background rates . ( thus , the third @xmath46-jet in eq . ( [ mtcut ] ) is actually non-@xmath46-tagged : it can be interpreted as the jet system satisfying neither eq . ( [ mhcut ] ) nor eq.([mwcut ] ) . ) we take @xmath99 , like in @xcite ( and assume 100% lepton identification efficiency ) . given the signal production rates before acceptance and selection cuts , it is clear that for such an @xmath72 even at high collider luminosity ( i.e. , @xmath100 fb@xmath101 per annum ) , hopes of disentangling the charged higgs boson of the mssm in the @xmath37 decay channel are only confined to the very low @xmath6 region . we will thus restrict ourselves to study in the reminder of the paper @xmath6 values which are , e.g. , below seven . the total signal rates after the cuts ( [ pt])([mtcut ] ) have been applied can be found in fig . [ fig : final ] , for the choices @xmath102 and @xmath103 , as a function of @xmath25 . for reference , we illustrate the ` borderline ' case @xmath104 gev . ( indeed , a lower @xmath2 value at @xmath105 is in contradiction with lep2 data , whereas higher masses induce a far too large suppression on br(@xmath106 ) : see fig . [ fig : brh ] . ) the trends in the figure are the consequence of two effects . on the one hand , the production cross section of @xmath107 + c.c . is roughly proportional to @xmath108 , so that its maxima occur at very low or very high @xmath6 . on the other hand , we have seen how the largest @xmath109 decay fraction is attained for @xmath110 . in the end , the largest values for @xmath111 + c.c . are obtained for @xmath112 : see fig . [ fig : final ] . unfortunately , such a @xmath6 value is already excluded in the mssm from lep2 data @xcite . for the optimal remaining choice , i.e. , @xmath105 , the annual rate never exceeds 140 events ( before any @xmath46-tagging efficiency but after acceptance and selection cuts ) . the maximum occurs at @xmath10 gev , significantly above the real threshold at @xmath113 gev . we now compare such a signal with the irreducible backgrounds 1.4 . , for the same choice of @xmath6 and @xmath2 ( where relevant ) . this is done in the upper half of fig . [ fig : last ] , at the level of total production rates . after the cuts ( [ pt])([mtcut ] ) are enforced , all background components in 2.4 . are overwhelmed by the signal in the vicinity of @xmath114 gev , whereas the @xmath0 continuum production is always larger than the @xmath5 resonant channel . thus , it is relevant to compare the last two processes in the ` reconstructed ' invariant mass @xmath115 , i.e. , that obtained from pairing the two @xmath46-jets fulfilling condition ( [ mhcut ] ) and the two light - quark jets ( or , alternatively , the lepton - neutrino pair ) satisfying eq . ( [ mwcut ] ) and not already reconstructing @xmath116 on their own and @xmath16 in association with any of the @xmath46 s ( see ref . the spectrum in this variable is presented in the lower half of fig . [ fig : last ] , for our ideal case @xmath114 gev ( and , again , @xmath105 and @xmath104 gev ) . for such mssm parameter combination , the charged higgs signal is well above the continuum for values of @xmath115 which are @xmath117 gev from @xmath25 . ( to vary @xmath25 and/or @xmath6 basically corresponds to rescale the solid line in the last plot by a constant factor , according to the rates in fig . [ fig : final ] . ) for reference , tab . [ tab : cuts ] presents the number of events of resonant and continuum @xmath0 production at the lhc , after 300 inverse femtobarns of collected luminosity , for @xmath118 , in the window @xmath119 40 gev , for the three values @xmath120 and 220 gev . although very small , a @xmath121 signal is generally observable above the @xmath122 continuum for @xmath25 around @xmath123 gev . our numbers are roughly consistent with those in ref . @xcite , if one considers that we neglect the finite efficiency of reconstructing one @xmath3 boson and the associated top ( anti)quark and since we have chosen somewhat different cuts . therefore , in the end , the dominant backgrounds remain ( in the @xmath7-tagged channel ) the @xmath124 + c.c . decay and the qcd noise involving misidentified gluons , i.e. , those already identified in ref . @xcite . in summary , in this paper , we have complemented a previous analysis @xcite of the production and decay of charged higgs bosons of the mssm at the lhc , in the channels @xmath107 and @xmath109 ( and charged conjugated modes ) , respectively , by considering several irreducible backgrounds in the @xmath7-tagged channel , i.e. , @xmath125 + c.c . @xmath126 + ` missing energy ' ( where the initial @xmath46-(anti)quark is usually lost along the beam pipe ) , which had not yet been considered . we have found that , after standard acceptance cuts and a kinematic selection along the lines of the one outlined in ref . @xcite , the dominant background among those considered here is the continuum production @xmath127 + c.c . however , the latter has been found to lie significantly below the signal in the only region where this is detectable : when @xmath128 and @xmath129 gev ( with @xmath2 around 100 gev , close to the latest lep2 constraints ) . thus , the chances of detecting the @xmath130 decay in such a ( narrow ) region of the mssm parameter space depend mainly on the interplay between this mode , the competing one @xmath131 and the qcd background with mistagged gluons , as are the latter two that clearly overwhelm the former ( recall the last figure in @xcite ) . we have carried out our analysis at parton level , without showering and hadronisation effects but emulating typical detector smearing . we are confident that its salient features should survive a more sophisticated simulation , such as the one presented in ref . besides , our results concerning the backgrounds can be transposed to the case of non - minimal susy models ( where the @xmath13 discovery potential can extend to a much larger portion of parameter space ) , such as those considered in ref . @xcite , so that also in these scenarios the irreducible backgrounds analysed here can be brought under control . the author is grateful to the uk - pparc for financial support . furthermore , he thanks d.p . roy for his remarks , which induced him to eventually considering the subject of this research . he also thanks k.a . assamagan for several useful discussions . finally , many conversations with k. odagiri are acknowledged , as well as many numerical comparisons against and the use of some of his programs . assamagan , atlas internal note atl - phys-99 - 013 ( 1999 ) ; k.a . assamagan , a. djouadi , m. drees , m. guchait , r. kinnunen , j.l . kneur , d.j . miller , s. moretti , k. odagiri and d.p . roy , contribution to the workshop ` physics at tev colliders ' , les houches , france , 8 - 18 june 1999 , hep - ph/0002258 ( to appear in the proceedings ) . e. barradas , j.l . diaz - cruz , a. gutierrez and a. rosado , _ phys . rev . _ * d53 * ( 1996 ) 1678 ; a. djouadi , j. kalinowski and p.m. zerwas , _ z. phys . _ * c70 * ( 1996 ) 435 ; e. ma , d.p . roy and j. wudka , _ phys . * 80 * ( 1998 ) 1162 . + @xmath25 ( gev ) & @xmath125 & @xmath132 & @xmath133 + @xmath134 & @xmath135 & @xmath136 & @xmath137 + @xmath123 & @xmath138 & @xmath139 & @xmath140 + @xmath141 & @xmath142 & @xmath143 & @xmath103 + + +
|
we analyse the chances of detecting charged higgs bosons of the minimal supersymmetric standard model ( mssm ) at the large hadron collider ( lhc ) in the @xmath0 mode , followed by the dominant decay of the lightest higgs scalar , @xmath1 .
if the actual value of @xmath2 is already known , this channel offers possibly the optimal final state kinematics for charged higgs discovery , thanks to the narrow resonances appearing around the @xmath3 and @xmath4 masses . besides , within the mssm
, the @xmath5 decay rate is significant for not too large @xmath6 values , thus offering the possibility of accessing a region of mssm parameter space left uncovered by other search channels .
we consider both strong ( qcd ) and electroweak ( ew ) ` irreducible ' backgrounds in the @xmath7-tagged channel to the @xmath8 production process that had not been taken into account in previous analyses . after a series of kinematic cuts ,
the largest of these processes is @xmath9 production in the continuum .
however , for optimum @xmath6 , i.e. , between 2 and 3 , the charged higgs boson signal overcomes this background and a narrow discovery region survives around @xmath10 gev . 0 pt .7truecm 33by .05 truein .05 truein 1 in 0.75 in 6.125 truein = 100000 # 1 # 2 # 3 _
phys .
lett . _ * # 1 * ( # 2 ) # 3 # 1 # 2 # 3 _ nucl .
phys . _ * # 1 * ( # 2 ) # 3 # 1 # 2 # 3 _ z. phys .
_ * # 1 * ( # 2 ) # 3 # 1 # 2 # 3 _ phys .
rev . _ * # 1 * ( # 2 ) # 3 # 1 # 2 # 3 _ phys .
rep . _
* # 1 * ( # 2 ) # 3 # 1 # 2 # 3 _ phys . rev .
lett . _ * # 1 * ( # 2 ) # 3 # 1 # 2 # 3 _ mod .
phys .
lett . _ * # 1 * ( # 2 ) # 3 # 1 # 2 # 3 _ rev .
mod .
phys . _ * # 1 * ( # 2 ) # 3 # 1 # 2 # 3 _ sov .
j. nucl .
phys . _ * # 1 * ( # 2 ) # 3 # 1 # 2 # 3 _ comp .
phys .
comm . _ * # 1 * ( # 2 ) # 3 # 1 # 2 # 3 * # 1 * , ( # 2 ) # 3 ral - tr-2000 - 005 + march 2000
+ * the @xmath0 decay channel + as a probe of charged higgs boson production + at the large hadron collider * + stefano moretti + _ rutherford appleton laboratory , _ + _ chilton , didcot , oxon ox11 0qx , uk . _
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
while the problem of electroweak symmetry breaking can be solved in the standard model ( sm ) by introducing one higgs boson , the minimal supersymmetric standard model ( mssm ) requires five physical higgses : a light cp - even ( @xmath6 ) , a heavy cp - even ( @xmath7 ) , a heavy cp - odd ( @xmath8 ) and two charged higgs bosons ( @xmath9 ) . therefore , the discovery of heavy neutral higgs bosons would be a major breakthrough in verifying the supersymmetric nature of the fundamental theory , which is one of the main physics goals of the large hadron collider project . + the most promising channel to discover the heavy susy higgses is the @xmath10 @xcite channel , where both the leptonic and hadronic decays of the tau can be exploited . this channel has been shown to cover large parts of the intermediate and high @xmath11 region of the mssm parameter space for an integrated luminosity of 30 @xmath12 . for low values of @xmath11 , the coupling of the higgs bosons to taus is not sufficiently enhanced and therefore this region is inaccessible for the @xmath13 channel . + in all studies of the sm channels ( meaning that the susy higgses decay into standard model particles ) , it is assumed that sparticles are too heavy to participate in the decay process . one should ask what would happen if some of the sparticles would be light and the decays of higgs bosons into these susy particles would be kinematically allowed . indeed , the existence of light neutralinos ( @xmath14 ) , charginos ( @xmath15 ) and sleptons ( @xmath16 ) seems favoured by a large number of supersymmetric models in order to explain electroweak symmetry breaking without large fine - tuning @xcite . also recent experimental results ( precision measurements at lep2 @xcite , muon @xmath17 @xcite ) may point towards the existence of light gauginos and sleptons . + light susy particles may jeopardize the higgs discovery potential of the sm channels , since their presence can drastically decrease the branching ratios of the higgses into sm particles . furthermore , pair and cascade production of light sparticles becomes an extra background to the higgs searches . on the other hand , higgs bosons decaying into sparticles might open new possibilities to explore regions of parameter space where sm decays would not be accessible @xcite . in this note we report on a study of this type of decay with the cms detector . we will focus on the decay of the heavy neutral higgses @xmath7 and @xmath8 into two next - to - lightest neutralinos , with each of the neutralinos in turn decaying as @xmath1 , i.e. into two ( isolated ) leptons + @xmath2 , so we get @xmath18 this results in a clear four lepton final state signature . we will show that , as is often the case for supersymmetric channels , susy backgrounds are more difficult to suppress than the sm backgrounds . of the latter , basically only @xmath19 survives after requiring four isolated leptons . of the susy backgrounds , sneutrino pair production and sparticle cascade decay production of neutralinos are the most dangerous processes . using a set of selection criteria as described in section 5 , we can clearly distinguish the signal from the background in the intermediate mass range 230 gev @xmath20 @xmath21 @xmath20 450 gev and for low and intermediate values of @xmath11 , depending on the values of the other mssm parameters . the remainder of this note is organised as follows : first we study the behaviour of the relevant branching ratios . then we describe the event generation , the signal versus background discrimination methods , and the discovery potential of the channel in the @xmath21 - @xmath11 plane . as a next step we investigate the effects of varying the other mssm parameter values . in the last section the results are summarized . the main difficulty in studying decay modes involving supersymmetric particles is the large amount of free parameters in the mssm . therefore most studies are carried out in the msugra or gmsb context in order to reduce the number of free parameters ; we will however stick to the more general mssm framework , to avoid too many model dependent assumptions . as free parameters , we take the mass of the cp - odd higgs @xmath21 , the higgs vev ratio @xmath5 , the higgsino mass parameter @xmath22 , the bino mass parameter @xmath23 , the wino mass parameter @xmath24 , the slepton mass @xmath25 and the squark / gluino mass @xmath26 . as a starting point for our studies , we will adopt the following framework : h & + & + * we consider light neutralinos and charginos , above the lep2 limits . initially , we fix @xmath23 at 60 gev , and using the renormalisation group relation @xmath24 @xmath27 2 @xmath23 , we can set @xmath24= 120 gev . we take @xmath28 . this large @xmath22 scenario is favoured in models where @xmath29 is the dark matter candidate , like msugra . in low @xmath22 scenarios , the decay of @xmath0 into leptons will be strongly suppressed . for large values of @xmath30 , @xmath0 is rather wino and @xmath29 is bino - like . therefore it approximately holds that @xmath31 and @xmath32 . the effects of varying these parameters will be discussed later on . * we also take sleptons to be light . in the most favourable case they would be lighter than @xmath0 , thereby allowing two - body decays into leptons . we will consider two scenarios : @xmath25 @xmath33 @xmath34 , where real decays of neutralinos into sleptons are allowed and @xmath25 @xmath35 @xmath34 , where only the virtual exchange is possible . * the masses of squarks and gluinos are kept at the 1 tev scale . in the mssm , it is natural that these sparticles are heavier than neutralinos and sleptons . in section 7 , we will investigate the effect of lowering the masses of squarks and gluinos . these parameter values and domains for @xmath22 , @xmath23 , @xmath24 , @xmath25 and @xmath26 will be used as default throughout this note . the exact values for @xmath22 and @xmath25 will be chosen after analysing and optimizing the @xmath36 cross sections through the mssm parameter space . after establishing the visibility in this optimal point , we will scan the area in @xmath37 around it to see how far the discovery region reaches . effects of varying the initial susy parameter values will be discussed . in order to determine the regions in mssm parameter space where sparticle decay modes may be accessible , we will first discuss the behaviour of the relevant branching ratios . the package hdecay @xcite is used to study the supersymmetric decay modes @xmath38 of the heavy higgses . the @xmath7 and @xmath8 couple preferably to mixtures of gauginos and higgsinos . the dominant mssm parameters controlling this process are @xmath21 , @xmath39 , @xmath24 and @xmath22 . + figs . [ fig : hd1 ] and [ fig : hd2 ] show the decays of @xmath7 and @xmath8 into neutralinos and charginos , in the case where @xmath23 = 60 gev@xmath40 = 120 gev , @xmath22 = -500 gev and @xmath11 = 5 . for @xmath41 @xmath20 500 gev , the probability for the heavy higgses to decay into susy particles can be as high as 20 % . in this mass region , the decay mode to @xmath42 has the highest branching ratio ( br ) , however it produces a final state with only two leptons . this mode would be in competition with numerous sm and susy backgrounds . the second best sparticle mode is @xmath43 . this channel can provide four leptons and is thus a priori more appropriate for obtaining a good signal to background ratio . the @xmath43 threshold is determined by our choice of @xmath24 ( @xmath44 ) = 120 gev ; the fall in br at @xmath41 @xmath35 350 gev is caused by the opening of the @xmath45 mode . the partial decay widths into supersymmetric particles remain the same as for lower values of @xmath41 , but due to the opening of the @xmath45 mode , the total decay width increases and the branching ratio into charginos / neutralinos decreases . for values of @xmath41 @xmath33 350 gev , the br of the cp - odd higgs ( @xmath8 ) into gauginos is substantially higher than in the cp - even ( @xmath7 ) case . this is due to the fact that for the cp - even higgs more couplings to sm particles are allowed , thus leading to a larger total decay width and smaller br s to sparticles . for high values of @xmath21 the br s are about the same for @xmath7 and @xmath8 since one reaches the decoupling regime . also for higher masses , other neutralino modes like @xmath46 , @xmath47 or @xmath48 may open up which will contribute to the four lepton signal . the next - to - lightest neutralino @xmath0 will decay into two fermions and the lightest neutralino : @xmath49 . these fermions will most often be quarks , leading to two jets and missing @xmath50 in the final state . to obtain a clean signature , we will only focus on the case where the neutralino decays into two leptons @xmath51 , where @xmath52 = @xmath53 or @xmath22 . this process is determined by the bino , wino and higgsino mass parameters @xmath23 , @xmath24 , @xmath22 , by @xmath11 and by the slepton masses @xmath25 . if sleptons are heavier than the @xmath0 , and as long as direct decays into a @xmath54 boson are not allowed ( or suppressed ) , only three - body decays @xmath51 will contribute . these decays are mediated by virtual slepton and @xmath54 exchange @xcite . therefore it is more favourable to have light sleptons in order to have larger br s . the decay branching ratios can be rather sensitive to the mssm parameters due to the fact that the @xmath54 and slepton exchange amplitudes may interfere constructively as well as destructively . ht & + & + in fig . [ fig : nd1 ] we show the br as a function of @xmath11 . sleptons ( including staus ) are taken at 250 gev and @xmath22 = -500 gev . @xmath0 is then rather wino and @xmath29 is bino - dominated . because of this , their coupling to the @xmath54 is dynamically suppressed , and the @xmath55 branching ratio will depend strongly on slepton masses . + if sleptons are lighter than the @xmath0 , direct two - body decays of the neutralino into a slepton - lepton pair are allowed , which may lead to large branching ratios . in fig . [ fig : nd2 ] the evolution of the br with @xmath11 is shown for @xmath24 = 200 gev , @xmath22 = 300 gev , @xmath56 = 200 gev , @xmath57 = 150 gev . this is however only valid in a rather limited region of the mssm parameter space , since often sneutrinos will be lighter than sleptons , causing the neutralinos to decay purely into invisible particles . + the fall of br(@xmath51 ) with @xmath5 in fig . [ fig : nd2 ] is compensated by a rise in br(@xmath58 ) . this means that allowing taus in the final state could possibly extend our discovery reach towards higher @xmath11 values . however , taus decay into leptons ( @xmath53 , @xmath22 ) in only @xmath435% of the cases , whilst the hadronic decay modes have detection efficiencies of @xmath430% @xcite . this , together with the fact there are up to four taus in the final state , makes that there is only a limited hope for a large improvement by including taus in the final state , but a dedicated study is needed . the signal events are generated with spythia @xcite . for low @xmath11 values ( @xmath59 ) , the gluon - gluon fusion mechanism @xmath60 dominates the production . due to the large coupling of the higgses to @xmath61 , the associated production @xmath62 dominates for @xmath11 @xmath35 5 @xcite . the cp - odd higgs is produced more than the cp - even one because the @xmath63 coupling is directly proportional to @xmath5 , whilst the @xmath64 coupling is proportional to @xmath65 . in the decoupling regime ( i.e. high values of @xmath21 ) , both couplings become equal . besides these two main processes , we also included the @xmath66 fusion and higgsstrahlung processes . we scanned the @xmath67 cross section in the ( @xmath21 , @xmath11 ) plane for different values of @xmath22 and @xmath25 . @xmath23 and @xmath24 were initially kept on 60 and 120 gev respectively , and no direct decays of neutralinos in sleptons were allowed . in figs . [ fig : sc1 ] and [ fig : sc2 ] , the plot of @xmath68 br for @xmath22 = -500 gev and @xmath25 = 250 gev is shown . values of @xmath11 @xmath20 30 - 40 and @xmath21 @xmath20 400 - 500 gev seem to be favoured . one also notices that the pseudoscalar higgs gives much higher cross sections than the scalar one . h & + & + for the background processes , pythia 6.136 @xcite was used with a few bugs fixed both in the susy and general code . the following sm backgrounds giving rise to four ( real or fake ) leptons in the final state have been generated : @xmath19 , @xmath69 , @xmath70 , @xmath71 , @xmath72 and @xmath45 . decays of @xmath73 into @xmath74 s have been included , since they might be dangerous due to their non - zero @xmath2 . for the susy backgrounds , we generated all pair production processes involving squarks , gluinos , sleptons , charginos and neutralinos . + + the cms detector response is simulated using the cmsjet fast monte carlo @xcite . the effects of pile - up at high luminosity running of lhc have not been included yet , but are expected to be minor in the four - lepton final state . in order to obtain a clear signal , we will have to discriminate between the signal events and background events that contain a similar four lepton final state . two categories of background have to be considered : standard model processes and susy backgrounds . + the main sm backgrounds are @xmath19 and @xmath45 production . they are dangerous because of their large cross sections at the lhc . in order to distinguish between events coming from the signal and from the sm background , we apply the following selection criteria : * we require two pairs of isolated leptons with opposite sign and same flavour , with a @xmath75 larger than 10 gev and within @xmath76 @xmath33 2.4 . the isolation criterion demands that there are no charged particles with @xmath75 @xmath35 1.5 gev in a cone of r = 0.3 rad around each lepton track , and that the sum of the transverse energy in the crystal towers between r = 0.05 and r = 0.3 rad is smaller than 3 gev . * all dilepton pairs of opposite sign and same flavour that have an invariant mass in the range @xmath77 gev are rejected ( z veto ) . demanding four tightly isolated leptons with a transverse momentum higher than 10 gev is a powerful requirement in fighting the @xmath45 and @xmath72 background . an explicit @xmath73 veto eliminates the @xmath19 , @xmath78 , @xmath70 production and all other backgrounds containing a @xmath73 boson . furthermore , to reduce @xmath19 we also require a minimal missing transverse energy of 20 gev . @xmath19 events where one of the @xmath73 s decays into taus , with the taus decaying leptonically , can however pass this criterion . + + the susy background is more complex . squark / gluino production is characterised by a large jet multiplicity ( 5 jets on average ) , a significant @xmath2 ( @xmath79 100 gev ) and jet transverse momenta that are large compared to the expectations for the signal . selecting events with few , rather soft jets ( e.g. @xmath80 2 jets , @xmath50 of the hardest jet below 100 gev ) and with @xmath2 @xmath33 130 gev allows us to eliminate most of these events . the @xmath81 threshold can be lowered to 50 gev if necessary . the squark / gluino - gaugino associated production can be eliminated this way too . if we assume @xmath82 = 1000 gev as in our default scenario , no squark / gluino events will survive the selection . in paragraph 7 , the effects of lighter masses will be discussed . + slepton - slepton production predominantly ends up in a 2-lepton final state . sneutrino - sneutrino production remains however as the dominant susy background . it could possibly be distinguished from the signal because of larger @xmath2 and larger @xmath83 of the leptons , as sneutrinos either decay into @xmath84 + @xmath85 ( leading to extra @xmath2 ) or into @xmath86 + @xmath87 ( leading to harder leptons ) . + pair production of heavier neutralinos and charginos will lead to more and harder jets and will often contain @xmath73 bosons in the final state . direct @xmath0-@xmath0 production gives the same signature as the signal , but the production cross section is much smaller due to the strongly suppressed coupling of gauginos to the @xmath73/@xmath88 intermediate state . + + in figures [ fig : sel7 ] - [ fig : sel2 ] , the distributions of the different kinematical variables for the signal and the total background ( sm + susy ) are plotted . the parameters of the considered case are : @xmath21 = 350 gev , @xmath11 = 5 , @xmath23 = 60 gev , @xmath24 = 120 gev , @xmath22 = -500 gev , @xmath25 = 250 gev , @xmath82 = 1000 gev . the dark shaded ( blue ) area is the part of the spectrum that is retained in the event selection . hp & + & + hp & + & + hp & + & + hp & + & + htp & + & + + in view of these distributions , we will apply the following search strategy : events are selected with @xmath2 smaller than 130 gev ( to suppress the susy background ) , but larger than 20 gev ( to suppress zz background ) . the @xmath75 of the hardest lepton should be less than 80 gev . the @xmath50 of the harderst jet in the event is taken smaller than 100 gev . in addition , we could also make a jet multiplicity requirement ( @xmath80 2 jets ) , but this seems to be needed only if squarks and/or gluinos would be light ( cfr . paragraph 7 ) . the four lepton invariant mass of the signal events should not exceed @xmath21 - @xmath89 . if the mass of the lightest neutralino is approximately known at the time of the analysis , one could set a limit at @xmath90 @xmath91 230 gev . + the number of signal and background events remaining after applying this selection is , for the considered case , given in table 1 . [ tab:1 ] .number of events after successive cuts ( at 100 @xmath12 ) . as parameters were used : @xmath21 = 350 gev , @xmath11 = 5 , @xmath23 = 60 gev , @xmath24 = 120 gev , @xmath22 = -500 gev , @xmath25 = 250 gev , @xmath82 = 1000 gev . [ cols="<,^,^,^,^,^,^",options="header " , ] the sparticle decay modes of the heavy neutral susy higgs bosons have been investigated . + the channel @xmath92 ( @xmath52 = @xmath53 , @xmath22 ) seems the most promising . in a rather large region of the mssm parameter space , a clean signal can be observed by selecting events with 4 isolated leptons in the final state . the main backgrounds are @xmath19 and sparticle pair production ( sneutrino , neutralino ) , but they can be sufficiently suppressed using appropriate selection criteria . extra backgrounds due to light squark / gluino production can also be kept under control by applying additional cuts . + in the most common case where direct decays of neutralinos to sleptons are not allowed , the @xmath93 channel seems to provide a detectable signal in the region between @xmath21 @xmath27 230 and 450 gev and for @xmath11 @xmath20 40 ( at 100 @xmath12 ) , in a scenario where @xmath24 @xmath27 120 gev , @xmath22 @xmath27 -500 gev and @xmath25 @xmath27 250 gev . + since the branching ratio of the @xmath94 into four leptons is determined by the interplay between a number of mssm parameters , the observability will also depend strongly on the values of @xmath22 , @xmath25 , @xmath23 and @xmath24 . large values of @xmath30 and low values of @xmath25 are favourable since they enhance the decay rate of the neutralinos into leptons . + + motivated by the low @xmath11 discovery potential of the @xmath95 channel , we also plan a similar study of the sparticle decay modes of the charged higgs bosons @xmath9 . the authors would like to thank abdel djouadi for helpful discussions . 9999999 r. kinnunen and d. denegri , cms note-1999/037 m. bastero - gil , g.kane and s. king , phys . rev . * b474 * ( 2000 ) 103 - 112 g. altarelli et al . , jhep 0106 ( 2001 ) 018 s. komine et al . , phys.lett . * b506 * ( 2001 ) 93 - 98 a. nikitenko and r. kinnunen , cms note-2001/031 h. baer et al . , phys . rev . * d50 * ( 1994 ) 316 - 324 a. djouadi , j. kalinowski and m. spira , hdecay , comput . phys . comm . * 108 * ( 1998 ) 5 h. baer and x. tata , phys * d47 * ( 1993 ) 2739 - 2745 s. mrenna , spythia , a supersymmetric extention of pythia , anl - hep - pr-96 - 63 d.dicus and s. willenbrock , phys . rev . * d39 * ( 1989 ) 751 t. sjostrand , comp . * 82 * ( 1994 ) 74 s. abdullin , a. khanov and n. stepanov , cmsjet , cms note/94 - 180 ; + _ http://cmsdoc.cern.ch/@xmath4abdullin/cmsjet.html _
|
we discuss the possibilities to observe the decays of heavy susy higgs bosons into supersymmetric particles at the lhc .
such an observation would be of interest either in a discovery search if sparticle modes are the dominant ones , or in a study of additional decay modes , bringing information on the susy scenario potentially at work .
we will focus on the most promising channel where the heavy neutral higgses decay into a pair of next - to - lightest neutralinos @xmath0 , followed by @xmath1 , thus leading to four isolated leptons + @xmath2 as the main final state signature .
a study with the cms detector shows that the background ( sm + susy ) can be sufficiently suppressed and that in the mass region between @xmath3 @xmath4 230 and 450 gev , for low and intermediate values of @xmath5 , the signal would be visible provided neutralinos and sleptons are light enough .
filip moortgat salavat abdullin daniel denegri
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
in the last several decades many new materials with intriguing properties were discovered and synthesized . these materials range from the high - temperature superconductors to magnetic materials , and the latter have already found real applications in electronic industry . typically , most of these materials contain elements from the 3@xmath0 or 4@xmath3 rows and their electronic structure is characterized by the presence of a partially filled narrow band across the fermi level . the fact that the narrow band is partially filled implies that there are many configurations with approximately equal weight rendering a one - particle description of the electronic structure problematic . indeed , it has been recognized for a long time that many of the intriguing properties of these materials originate from correlations among the electrons residing in the partially filled narrow band . the electrons are neither fully localized , like core electrons , nor itinerant , like @xmath4 or @xmath5 electrons in alkalis or conventional semiconductors such as silicon or diamond . this hybrid property poses a tremendous theoretical difficulty for an accurate description of the electronic structure , because due to the electrons partially itinerant character the problem can neither be treated as a purely atomic problem nor within a pure band picture . moreover , the interaction with other electrons can be very important . a large amount of work has been directed to solving the correlation problem of the above materials . the usual approach is to consider only the narrow bands near the fermi level and eliminate the degrees of freedom for the rest of the bands by the downfolding procedure , resulting in the well - known hubbard model which contains an effective on - site coulomb interaction , the hubbard @xmath6 . in general , the models represent multi - band systems containing interorbital as well as long - ranged part of coulomb interaction . the models can then be solved with various sophisticated low - energy solvers such as dynamical mean - field theory ( dmft ) @xcite or solvers for lattice models @xcite . an important issue in mapping the real system to a model hamiltonian is how to determine the one - particle kinetic energy term and the effective interaction or the hubbard _ u _ in the model . unlike the one - particle parameters that can be downfolded from the band structure , the hubbard _ u _ is much more elusive to determine and it is often treated as an adjustable parameter . a widely used scheme to calculate the hubbard _ u _ from first - principles is the constrained lda ( clda ) method @xcite . the clda method , however , is known from early on to yield values of @xmath6 , which are too large in some cases ( e.g. late transition metals ) . it has been argued that this arises from technical difficulty in including transitions of electrons between the @xmath0 and @xmath1 space contributing the screening processes . this oversight leads in some cases to a larger value of _ u _ @xcite . recent extensions of the clda method may be found in refs.@xcite and @xcite . another method for determining the effective interaction is a scheme based on the random phase approximation ( rpa ) . early attempts along this direction can be found in refs.@xcite . a combined clda and rpa method to circumvent the difficulty was also proposed @xcite . some years ago a scheme for calculating the hubbard _ u _ , called the constrained rpa ( crpa ) scheme @xcite , was proposed . the main merit of the crpa method over currently available methods is that it allows for a precise elimination of screening channels , which are instead to be included in a more sophisticated treatment of the model hamiltonian . this is a controlled approximation without any ambiguity , expected to become asymptotically exact if the @xmath1-space becomes well separated from the @xmath0 space . moreover , the effective screened interaction can be calculated as a function of @xmath7 and @xmath8 , i.e. , @xmath9 , independent of the basis functions . this allows easy access to obtaining not only on - site matrix elements but also off - site matrix elements as well as screened exchange matrix elements , which are usually taken to be the atomic value . another merit is the possibility of obtaining the frequency - dependent hubbard _ u _ , which may prove to be important . the crpa method has now been applied to a number of systems with success @xcite . although the crpa method is rather general , its applications to real systems have revealed a serious technical problem . the problem arises when the narrow band is entangled with other bands , i.e. , it is not completely isolated from the rest of the bands . in many materials , the narrow band of interest is entangled . even in simple materials such as the 3@xmath0 transition metals , the 3@xmath0 bands mix with the 4@xmath4 and 4@xmath5 bands . similarly , the 4@xmath3 bands of the 4@xmath3 metals hybridize with the more extended @xmath4 and @xmath5 bands . for such cases , it is not clear anymore which part of the polarization should be eliminated when calculating the hubbard @xmath6 using the crpa method . some procedures to overcome the problem of determining _ u _ for entangled bands have been attempted . one of these is to choose a set of band indices and define the bands of hubbard model as those bands corresponding to the chosen indices . another alternative is to introduce an energy window and define the hubbard bands to be those that have energy within the energy window . yet another alternative is to have a combination of energy window and band indices . these procedures , however , suffer from a number of difficulties . when choosing band indices it is inevitable that some of the states will have a character very different from that of the intended model . for example , in the case of 3@xmath0 transition metals , choosing five 3@xmath0 " bands will include at some @xmath10-points states which have little 3@xmath0 character , with a considerable 4@xmath4 component instead . moreover , the chosen bands will be awkward to model since they do not form smoothly connected bands . a similar problem is encountered when choosing an energy window . a combination of band indices and energy window proposed in ref . @xcite partially solves the problem but it suffers from arbitrariness . another procedure is , as we will discuss in detail later , to project the polarization to the orbitals of interest , e.g. , 3@xmath0 orbitals , but this procedure has been found to yield an unphysical result of negative static _ u_. in this work , we offer a solution to the problem of determining the hubbard _ u _ for entangled bands . the basic idea is to disentangle the narrow bands of interest from the rest and carry out the crpa calculation for the disentangled band structure , not using the original band structure . the disentangling procedure is described in sec.[sec : method ] . we apply the method to 3@xmath0 transition metals in sec.[sec : result ] and show that the method is numerically stable and yields reasonable values of @xmath6 . finally the paper is summarized in sec.[sec : summary ] . in the crpa method we first choose a one - particle subspace @xmath11 , which defines the model hamiltonian , and label the rest of the hilbert space by @xmath12 . we define @xmath13 to be the polarization within the @xmath0 subspace and the total polarization is written as @xmath14 . it is important to realize that the rest of the polarization @xmath15 is _ not _ * * the same as the polarization of the @xmath1 subspace because it contains polarization arising from transitions between the @xmath0 and @xmath1 subspaces . since @xmath13 is the polarization of the model hamiltonian , this polarization should be subtracted out from the total polarization when the effective parameter of the model is determined . the effective coulomb interaction @xmath16 should be calculated with the rest of the polarization @xmath17 : @xmath18^{-1}v\label{wr } \;,%\ ] ] where @xmath19 is the bare coulomb interaction . it can indeed be shown @xcite that the fully screened interaction is given by @xmath20^{-1}w_{r } \;.\ ] ] this mathematical identity ensures that @xmath16 can be interpreted as the effective interaction among the electrons residing in the @xmath0 subspace since the screening of @xmath16 by @xmath13 leads to the fully screened interaction . the matrix elements of @xmath16 in some localized functions can then be regarded as the frequency - dependent hubbard _ u_. it has been shown that the formula in eq.([wr ] ) is formally exact , provided @xmath17 is the difference between the exact polarization @xmath14 and the exact polarization of the @xmath0 subspace @xmath13 . in the crpa method , @xmath17 is calculated within the random - phase approximation . if the @xmath0 subspace forms an isolated set of bands , as for example in the case of the @xmath21 bands in srvo@xmath22 , the crpa method can be straightforwardly applied . however , in practical applications , the @xmath0 subspace may not always be readily identified . an example of these is provided by the 3@xmath0 transition metal series where the 3@xmath0 bands , which are usually taken to be the @xmath0 subspace , do not form an isolated set of bands but rather they are entangled with the 4@xmath4 and 4@xmath5 bands . to handle these cases we propose the following procedure . we first construct a set of localized wannier orbitals from a given set of bands defined within a certain energy window . these wannier orbitals may be generated by following the post - processing procedure of souza , marzari and vanderbilt @xcite or other methods , such as the preprocessing scheme proposed by andersen _ et al . _ within the nth - order muffin - tin orbital ( nmto ) method @xcite . we then choose this set of wannier orbitals as the generators of the @xmath0 subspace and use them as a basis for diagonalizing the one - particle hamiltonian , which is usually the kohn - sham hamiltonian in the local density approximation ( lda ) or generalized gradient approximation ( gga ) . the so obtained set of bands , which equivalently define the @xmath0 subspace , may be slightly different from the original bands defined within the chosen energy window , because hybridization effects between the @xmath0 and @xmath1 spaces are neglected . however , it is important to confirm that the dispersions near the fermi level well reproduces the original kohn - sham bands . from these bands we calculate the polarization @xmath23 as @xmath24 \ ; , \label{eq : pd}\ ] ] where @xmath25 , @xmath26 @xmath27 are the wavefunctions and eigenvalues obtained from diagonalizing the one - particle hamiltonian in the wannier basis . it would seem sensible to define the rest of the polarization as @xmath28 where @xmath14 is the full polarization calculated using the _ original _ ( kohn - sham ) wavefunctions and eigenvalues @xmath29 , @xmath30 @xmath31 , and calculate @xmath16 according to eq.([wr ] ) . we have found , however , that this procedure is numerically very unstable , resulting in some cases to unphysically negative static @xmath6 and a large oscillation as a function of frequency . this is understandable given that @xmath14 and @xmath32 are obtained from two different band structures , so that low energy screening channels associated with the @xmath0-@xmath0 transitions are not excluded from @xmath33 completely . due to the singular nature of the expression in eq.([wr ] ) these low - energy excitations can cause a large fluctuation in @xmath16 . another way of calculating @xmath33 is to project the wavefunctions to the @xmath0 space , @xmath34 where the projection operator @xmath35 is defined as @xmath36 the effective @xmath0 polarization may be expressed as @xmath37 \;,\ ] ] and @xmath38 can be used to calculate @xmath39 . we found that this procedure does not work either and is again unstable . this problem may be related to the fact that @xmath40 s are not orthogonal with each other and transitions between the states do not correspond to single particle - hole excitations . based on these observations we propose the following procedure . we define the @xmath1 subspace by @xmath41 which is orthogonal to the @xmath0 subspace constructed from the wannier orbitals . in practice it is convenient to orthonormalize @xmath42 and prepare @xmath43 basis functions . by diagonalizing the hamiltonian in this subspace a new set of wavefunctions @xmath44 and eigenvalues @xmath45 @xmath46 are obtained . as a consequence of orthogonalizing @xmath47 and @xmath48 , the set of @xmath1 bands @xmath45 are completely disentangled from those of the @xmath0 space @xmath49 , and they are slightly different from the original band structure @xmath50 . as we will see later , however , the numerical tests show that the disentangled band structure is close to the original one . the hubbard @xmath6 is calculated according to eq.([wr ] ) with @xmath51 , where @xmath52 is the full polarization calculated for the _ disentangled _ band structure . it is important to realize that the screening processes between the @xmath0 space and the @xmath1 space are included in @xmath6 , although the @xmath0-@xmath1 coupling is cut off in the construction of the wavefunctions and eigenvalues . as an illustration we apply the method to 3@xmath0 transition metals . the electronic structure calculations are done in the local density approximation @xcite of density functional theory @xcite with the full - potential lmto implementation @xcite . the wavefunctions are expanded by @xmath53 mtos and a @xmath54 * k*-mesh is used for the brillouin zone summation . spin polarization is neglected . more technical details are found elsewhere @xcite . figure [ fig : band](a ) shows the kohn - sham band structure of nickel . there are five orbitals having strong 3@xmath0 character at [ -5 ev:1 ev ] , crossed by a dispersive state which is of mainly 4@xmath4 character . using the maximally localized wannier function prescription with the energy window of [ -7 ev:3 ev ] , interpolated `` @xmath0 '' bands are obtained . the subsequent disentangling procedure gives the associated `` @xmath1 '' bands . comparing fig.[fig : band](b ) with ( a ) we can see that there is no anti - crossing between the @xmath0 bands and the @xmath1 bands in ( b ) . otherwise the two band structures are nearly identical . in order to see the impact of the disentanglement on the screening effects , we perform the full rpa calculation using the disentangled band structure . the fully screened coulomb interaction is compared with that for the original band structure in fig.[fig : w ] where the average of the five diagonal terms in the wannier basis @xmath55 is plotted , @xmath56 the two methods yield similar results the static values agree with each other within 0.2 ev , and the frequency dependence is weak at low frequencies . as frequency increases there is a sharp increase at @xmath5720 ev , where screening by plasmons becomes ineffective . these results assure that screening effects can be treated accurately with the disentangled band structure . the hubbard @xmath6 is calculated by the constrained rpa , namely , by replacing @xmath58 in eq.([eq : w ] ) with @xmath39 . the results are shown in fig.[fig : u ] . there is no large fluctuation against frequency , in contrast to the methods described in sec.[sec : method ] , and @xmath59 shows a stable behavior . as is expected , @xmath6 is significantly larger than @xmath58 at low frequencies . this implies proper elimination of @xmath60 screening processes is crucial . comparing with the previous results using a combined energy and band window @xcite , the agreement is reasonably good . a small difference between these two results at low frequency may be due to a small portion of @xmath0-@xmath0 screening presumably contained in the previous method @xcite , although it should be excluded from the crpa calculations . we carried out the calculations for a series of other 3@xmath0 metals as well and found in all the cases that ( i ) the present scheme is numerically stable and does not result in unphysical frequency dependence of @xmath6 , and ( ii ) the value of @xmath61 is close to that from the original band structure . the latter is confirmed in fig.[fig : static ] where the static values ( @xmath62 limit ) are summarized . concerning @xmath6 , the present method gives larger values compared to the previous results , particularly for early transition metals . since @xmath58 is nearly equal to each other in the two methods , the discrepancy is ascribed to the different treatment between @xmath2 and @xmath63 . we should note that @xmath2 in the previous method depends on the choice of the window . for a wider window , obviously we would obtain a better agreement . also , some states have a mixed character of 3@xmath0 and 4@xmath4 near the anti - crossing points . this makes elimination of the screening process difficult in the original band structure . the present scheme , on the other hand , enables us to determine @xmath6 without ambiguity . the @xmath0 bands are disentangled from the @xmath1 bands . consequently , the polarization in the @xmath0 space is well - defined and can be removed completely in @xmath64 . in the present formulation , small off - diagonal matrix elements of the kohn - sham hamiltonian between the @xmath0 space wavefunction @xmath65 constructed from the wannier orbitals and the @xmath1 space @xmath66 are ignored . this is the reason why the anti - crossing is avoided . if the energy of this hybridization point in the band dispersion is smaller than the screened coulomb energy and the energy scale of the interest , strictly speaking , one has to keep all of these hybridizing bands in the effective model , because the hybridization effects are non - perturbative . in the present case of transition metals , the energy crossing point of 4@xmath4 and 3@xmath0 bands are relatively larger than the screened coulomb energy scale and the low energy models constructed only from the 3@xmath0 wannier orbitals may give at least qualitatively reasonable description of the low energy physics . we have proposed a method to calculate the effective interaction parameters for the effective low - energy models of real materials when bands are entangled . the key point is to first properly orthogonalize the low - energy subspace contained in the models and the complementary high - energy subspace to each other . this orthogonalization by the projection technique enables the disentanglement of the bands . once the disentangled band structure is obtained , the constraint rpa method can be used to determine the partially screened coulomb interaction uniquely . numerical tests for 3@xmath0 metals show that the method is stable and yields reasonable results . the method is applicable to any system . applications to more complicated systems , such as interfaces of transition metal oxides are now under way . we thank s. biermann for fruitful discussions . this work was supported by grants - in - aid for scientific research from mext , japan under grant numbers 17064004 and 19019013 . tm acknowledges support from next generation supercomputing project , nanoscience program , mext , japan . for example , the path - integral renormalization group method has been applied by y. imai , i. solovyev and m. imada , phys . 95 * , 176405 ( 2005 ) ; y. imai and m. imada , j. phys . . jpn . * 75 * , 094713 ( 2006 ) ; y. otsuka and m. imada , j. phys . . jpn . * 75 * , 124707 ( 2006 ) . -@xmath1 hybridization switched off . the red lines show the @xmath0 states obtained by the maximally localized wannier scheme , while the blue lines are disentangled @xmath1 states . energy is measured from the fermi level . , title="fig:",width=340 ] -@xmath1 hybridization switched off . the red lines show the @xmath0 states obtained by the maximally localized wannier scheme , while the blue lines are disentangled @xmath1 states . energy is measured from the fermi level . , title="fig:",width=340 ] of nickel as a function of frequency . the diagonal term of the partially screened interaction in the wannier basis is calculated by the present method and compared with the published data of ref.@xcite . , width=340 ]
|
we present a first - principles method for deriving effective low - energy models of electrons in solids having entangled band structure .
the procedure starts with dividing the hilbert space into two subspaces , the low - energy part ( `` @xmath0 space '' ) and the rest of the space ( `` @xmath1 space '' ) . the low - energy model is constructed for the @xmath0 space by eliminating the degrees of freedom of the @xmath1 space .
the thus derived model contains the strength of electron correlation expressed by a partially screened coulomb interaction , calculated in the constrained random - phase - approximation ( crpa ) where screening channels within the @xmath0 space , @xmath2 , are subtracted .
one conceptual problem of this established downfolding method is that for entangled bands it is not clear how to cut out the @xmath0 space and how to distinguish @xmath2 from the total polarization .
here , we propose a simple procedure to overcome this difficulty . in our scheme , the @xmath0 subspace is cut out from the hilbert space of the kohn sham eigenfunctions with the help of a procedure to construct a localized wannier basis .
the @xmath1 subspace is constructed as the complementary space orthogonal to the @xmath0 subspace .
after this disentanglement , @xmath2 becomes well defined . using the disentangled bands , the effective parameters are uniquely determined in the crpa .
the method is successfully applied to 3@xmath0 transition metals .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
a gamma - ray burst ( grb ) event comprises two phases , prompt emission and afterglow . the prompt @xmath1-ray emission is usually highly variable , with many pulses overlapping within a short duration ( fishman & meegan 1995 ) . the power density spectra ( pdss ) of the light curves are typically a power law with a possible turnover at high frequencies ( beloborodov et al . 2000 ) . the light curves may be decomposed as the superposition of an underlying slow component and a more rapid fast component ( gao et al . the fast component tends to be more significant in high energies , and becomes less significant at lower frequencies ( vetere et al . 2006 ) . it has been shown that the external shock model has difficulty producing grb variability while maintaining a high radiative efficiency ( sari & piran 1997 ; cf . dermer & mitman 1999 ) . the detection of the steep decay phase following grb prompt emission ( tagliaferri et al . 2005 ) suggests that the prompt emission region is detached from the afterglow emission region ( zhang et al . this nails down the internal origin of grb prompt emission for the majority of grbs . for an internal origin of grb prompt emission , the variability is usually attributed to the erratic activity of the central engine ( e.g. , rees & mszros 1994 ; kobayashi et al . 1997 ) . it is envisaged that the ejecta launched from the central engine is composed of multiple shells with variable bulk lorentz factors . faster late shells catch up and collide with slower early shells . part of the kinetic energy of the ejecta is converted to energy of non - thermal particles in these internal shocks , a fraction of which is released as the observed non - thermal radiation . in this model , different variability timescales are related to the angular spreading time of colliding shells at different internal shock radii . in order to account for superposed slow and fast variability components , one has to assume that the central engine itself carries these two variability components in the time history of jet launching ( hascot et al . 2012 ) , whose physical origin is unclear . the internal shock model also suffers a list of criticisms ( e.g. , zhang & yan 2011 for a review ) , including low radiation efficiency ( e.g. , kumar 1999 ; panaitescu et al . 1999 ) , fast cooling ( ghisellini et al . 2000 ; kumar & mcmahon 2008 ) , spectrum . however , a requirement is that the emission region has to be large where the magnetic field is weak . this corresponds to an unconventional internal shock radius , but is consistent with the icmart model . ] , particle number excess ( daigne & mochkovitch 1998 ; shen & zhang 2009 ) , inconsistency with some empirical relations ( amati et al . 2002 ; zhang & mszros 2002 ; liang et al . 2010 ) , and overpredicting the brightness of the photosphere emission component ( daigne & mochkovitch 2002 ; zhang & peer 2009 ) . alternatively , the grb variability can be interpreted as locally doppler - boosted emission in a relativistic bulk flow , such as relativistic mini - jets ( lyutikov & blandford 2003 ; yamazaki et al . 2004 ) or relativistic turbulence ( narayan & kumar 2009 ; kumar & narayan 2009 ; lazar et al . 2009 ) in a bulk relativistic ejecta . some criticisms have been raised to these models . for example , relativistic turbulence damps quickly so that the emission from the turbulence can not be sustained ( zrake & macfadyen 2012 ) . the simulated light curves are composed of well - separated sharp pulses without an underlying slow component ( narayan & kumar 2009 ; lazar et al . 2009 ) . also the pulse was calculated to have a symmetric shape for the turbulence model ( lazar et al . 2009 ) , which is in contradiction with the data . recently , zhang & yan ( 2011 , hereafter zy11 ) proposed an internal - collision - induced magnetic reconnection and turbulence ( icmart ) model to explain prompt emission of grbs . like the traditional internal shock scheme , the icmart model envisages internal interactions of shells within the ejecta wind . the main difference is that the ejecta is poynting flux dominated , with the magnetization parameter @xmath2 in the collision region , where @xmath3 and @xmath4 are poynting flux and matter flux , respectively . this was motivated by the non - detection of a bright photosphere thermal component in grb 080916c ( zhang & peer 2009 ) and most other large area telescope grbs ( zhang et al . 2011 ) . for a helical magnetic field structure , the initial collisions only serve to distort the magnetic field configurations . as multiple collisions proceed , the field configurations would be distorted to a critical point when a cascade of reconnection and turbulence occurs . charged particles can be accelerated in these reconnection regions , leading to intense gamma - ray radiation . within this model , a grb light curve is supposed to have two variability components : a broad ( slow ) component that tracks central engine activity , and an erratic ( fast ) component with multiple sharp pulses superposed on the slow component , which is related to numerous reconnection sites during the icmart event . in this paper , we simulate grb light curves and their corresponding pdss within the framework of the icmart model . in section 2 we describe the basic model and the simulation method . the simulation results are presented in section 3 . section 4 summarizes the findings with some discussion . we first summarize the basic ideas of the icmart model ( zy11 ) . magnetized shells with initial @xmath5 are envisaged to collide , leading to distortion of magnetic field lines until a threshold is reached and a runaway magnetic dissipation is triggered . during such an `` avalanche''-like reconnection / turbulence cascade , it is envisaged that fast reconnection seeds in the moderately high @xmath0 regime would inject moderately relativistic outflows in the emission regions ( zy11 ; lyubarsky 2005 ) , which would excite relativistic turbulence . the turbulence would facilitate more reconnection events , which trigger further turbulence . the magnetic energy is converted to particle energy and efficient radiation . during the growth of the reconnection / turbulence cascade , the number of reconnection sites as observed at any instant increases rapidly with time , so that multiple mini - emitters contribute simultaneously to the observed gamma - ray emission . rapid evolution of individual reconnection sites leads to rapid variability of the observed grb light curves . the cascade stops as @xmath0 drops around or below unity when most magnetic energy is converted into radiation or kinetic energy . during the growth of an icmart event , turbulence is not quickly damped due to the continuous injection of particle energy from the reconnection events , which continuously drives turbulence . with these preparations , we can model the light curve of a grb within the framework of the icmart model . lacking full numerical simulations of magnetic turbulence and reconnection , in this paper we perform a monte carlo simulation based on some simplest assumptions . we define each reconnection event as a fundamental mini - emitter , which carries a local lorentz boost with respect to the bulk of the emission outflow . each reconnection event can be modeled as a pulse , which can be bright and spiky if the mini - emitter beams toward the observer , but dim and broad if the mini - emitter beams away from the observer s direction . the observed light curve is the superposition of the emission from all these mini - emitters . for simplicity , we assume that the characteristic brightness ( peak luminosity ) of each reconnection event in the rest frame of the reconnection outflow is the same . we also take the shape of each pulse as a gaussian form for simplicity ( e.g. , narayan & kumar 2009 ; lazar et al . our goal is to try to simulate the superposed slow and fast components , and the precise shape of each pulse does not matter too much . in any case , we note that the shape of a spike within the icmart model is mainly defined by the time history of each reconnecting mini - jet rather than the time history of an ideal eddy , so the pulse profile may not necessarily be symmetric with peak time . this is different from the previous models ( narayan & kumar 2009 ; lazar et al . 2009 ) that invoke relativistic turbulence . more importantly , the shape of a broad pulse in the model is asymmetric : the rising portion is defined by the timescale of the reconnection - turbulence cascade process , while the decay portion is controlled by high - latitude emission after the icmart cascade ceases . there are three rest frames in this model : the first is the rest frame of the mini - jet , i.e. the outflow of the individual reconnection event . these mini - jets are moving with a relative lorentz factor @xmath1 with respect to the jet bulk . we denote parameters in this frame as ( @xmath6 ) . the second frame is the rest frame of the jet bulk , which moves with a lorentz factor @xmath7 with respect to the central engine . we denote parameters in this frame as ( @xmath8 ) . the third one is the rest frame of the observer ( with the cosmological expansion effect ignored ) . the quantities within these three frames are connected through two doppler factors , i.e. , @xmath9^{-1}\ ] ] and @xmath10^{-1},\ ] ] where @xmath11 and @xmath12 are the corresponding dimensionless velocities with respect to @xmath7 and @xmath1 , respectively , @xmath13 is the latitude of the mini - jet with respect to the line of sight ( i.e. the angle between the line of sight and the radial direction of the bulk ejecta at the location of the mini - jet ) , and @xmath14 is the angle between the mini - jet direction and radial direction of the ejecta bulk within the comoving frame of the ejecta bulk . each reconnection event is supposed to give rise to a single pulse in the grb light curve . since several reconnection events may occur simultaneously , some pulses can superpose with each other . for a naive sweet - parker reconnection , one has ( e.g. , see zweibel & yamada 2009 and references therein ) @xmath15 where @xmath16 is the inflow velocity of the reconnection layers , @xmath17 is the outflow velocity , and @xmath18 and @xmath19 are the width and length of the reconnection layer , respectively . reconnection physics demands @xmath20 , so that @xmath21 . on the other hand , what defines the duration of the reconnection event is the thickness of the bunch of magnetic field lines that continuously approach each other , and we assume that it is also of the order of @xmath19 . as a result , in the bulk comoving frame ( the @xmath8 frame ) , the duration of each pulse can be approximated as @xmath22 . in the observer frame , this is translated to @xmath23 , which corresponds to the duration of a certain pulse in the observer frame . for simplicity , we assume that the radiation intensity arising from each reconnection event has the same spectral form , i.e. , the band function ( see band et al . 1993 ) , in the comoving frame of the mini - jet ( the @xmath6 frame ) , @xmath24 the observed flux can be calculated as @xmath25 where @xmath26 is the distance of the grb to the observer . in a high-@xmath0 flow , @xmath17 can eventually reach a relativistic speed ( with lorentz factor @xmath1 ) , and @xmath27 can reach a maximum value of @xmath28 ( e.g. lyubarsky 2005 and references herein ) . therefore , @xmath29 . the lorentz factor of the mini - jet is related to @xmath0 and would drop to unity when @xmath0 drops below unity . the detailed dependence is related to the complicated physics of relativistic reconnection . in this paper , we adopt @xmath30 ( i.e. , @xmath1 is proportional to the relativistic alfvn lorentz factor ) . we also investigated other dependences between @xmath1 and @xmath31 . the general conclusions regarding how the simulated light - curve properties depend on various parameters are essentially similar . in the rest of the paper , we only focus on the @xmath30 assumption . in the simulations , we fix the band function parameters as the following : @xmath32 , @xmath33 , and the peak frequency @xmath34 is chosen such that @xmath35 kev is satisfied , where 300 kev is the typical observed value of grb spectral peak , and @xmath36 is the average value of the product of the two doppler factors . based on these assumptions , we calculate the received flux in the detector band of _ swift _ burst alert telescope ( bat ; i.e. , 15 - 150 kev ) . in our monte carlo simulation , four random parameters have been introduced . they are : ( 1 ) comoving length of the reconnection region @xmath37 , which is assumed to either have a typical value or have a power - law distribution with index @xmath38 below a typical value ; ( 2 ) the mini - jet direction ( angle @xmath14 with respect to the bulk motion direction ) in the bulk comoving frame , which is taken as isotropic or a gaussian distribution with respect to @xmath39 ( see more discussion below ) ; ( 3 ) the latitude of a mini - jet @xmath13 with respect to the viewing direction , which is random within the cone of the jet opening angle ; and ( 4 ) the epoch when a mini - jet occurs , which is taken to satisfy a distribution of exponential growth with time , i.e. @xmath40 . the total number @xmath41 of the mini - jets is a free parameter , which is defined by the requirement that they dissipate most magnetic energy in the local emission regions , so that the local @xmath0 is brought to below unity after each icmart event . of the ejecta can be still above unity , if the filling factor @xmath42 , since the majority of magnetic energy is still not dissipated . a small @xmath43 seems to be required by the central engine study of lei et al . ( 2013 ) , who obtained @xmath0 values greater than the measured typical lorentz factors of grbs ( liang et al . assuming that the magnetic energy density is roughly uniform within the emission region , this number can be simply written as the ratio between the total dissipated volume ( i.e. , total volume multiplied by the filling factor @xmath43 ) and the volume of the region affected by each reconnection event that powers a mini - jet . within the @xmath44 cone , this number is @xmath45 where @xmath46 is the radius of the emission region from the central engine . other input parameters include the radius of the emission region @xmath46 , the jet opening angle @xmath47 , the initial values of @xmath7 , and @xmath0 ( which defines the initial @xmath1 ) . for each reconnection event , we assume that half of the dissipated magnetic energy is released in the form of photons , while the other half is deposited to the jet bulk and used to boost the kinetic energy of the bulk . therefore , @xmath7 , @xmath0 , and @xmath1 are all functions of time during each icmart event . the exponential growth of magnetic dissipation eventually ends when the local @xmath0 drops around or below unity . without numerical simulations , it is unclear how abrupt the ending process is . in this paper we just assume an abrupt cessation of the cascade process , so that the number of new mini - jets drops to 0 after a particular time . the observed `` tail '' emission after this epoch is therefore contributed by the high - latitude emission from other mini - jets not along the line of sight due to the `` curvature effect '' delay . this delay timescale is calculated as @xmath48 with respect to the last emission along the line of sight , where @xmath49 . we calculate the contribution of all the mini - jets within @xmath50 . although most of the received emission comes from the mini - jets within the @xmath44 cone , those mini - jets outside the @xmath44 cone make some contribution to the high - latitude emission . we calculate the delay timescale of each mini - jet , apply its doppler factor to calculate the amplitude and shape of the pulse , and superpose these mini - jets to get the curvature tail of each icmart event . we run a series of monte carlo simulations to generate sample light curves . we first focus on the light curves for only one icmart episode . the light curve of one grb could be then modeled by superposing multiple icmart events . we first take the following nominal parameters : @xmath51 cm , @xmath52 cm , @xmath53 , @xmath54 , and @xmath55 . considering an exponential growth , i.e. , that each reconnection seed would eject a bipolar outflow and would stir up the ambient medium to trigger two reconnection events , one may estimate the generation number of successive reconnection events , @xmath56 , through the requirement @xmath57 . the timescale for each generation in the bulk comoving frame may be estimated as @xmath58 s , which corresponds to an observer frame timescale @xmath59 s. this is the typical `` @xmath60-folding '' timescale . the total duration ( rising timescale ) of an icmart event is therefore @xmath61 times larger , i.e.m @xmath62 s , which we adopt in the simulations . we also assume that the observer s line of sight is along the jet axis , and we take a redshift @xmath63 for simplicity . for a power - law distribution of @xmath19 , in principle , @xmath19 can extend to much smaller values . in our simulations , reconnection regions with @xmath64 cm are not considered , since the observed durations of these events already meet the detector s variability limit . in the following we test various factors that may affect the shape of the light curves . we first test how the simulated light curve depends on the unknown distribution of @xmath14 in the bulk comoving frame . we first assume an isotropic distribution and calculate the light curve . the result is shown in figure [ fig : direction](a ) . one can immediately see that the light curve has a broad component , with some spiky small pulses superposed on top . the broad component is due to the contributions of all the mini - jets beaming toward random directions in the bulk motion rest frame . the rising of the broad pulse corresponds to the exponential growth of the number of mini - jets , while the decay is controlled by the high - latitude effect . + + since an icmart event corresponds to an event of destroying the initial ordered magnetic field , the magnetic configurations in the icmart region , even near the end of the cascade , should not be completely random . the initial magnetic field configuration should be parallel to the ejecta plane ( e.g. , spruit et al . 2001 ; zhang & kobayashi 2005 ) . this is because the toroidal component falls with radius much slower than the poloidal component . such a configuration should still leave an imprint on the @xmath14 distribution . we consider a distribution of @xmath14 that has a gaussian distribution with respect to the original field line direction , i.e. , @xmath39 . in figures [ fig : direction](b ) and [ fig : direction](c ) we show the gaussian angle to be @xmath65 and @xmath66 , respectively . one can see that the simulated light curves have progressively less flux as the distribution angle becomes smaller . this is because with a smaller distribution angle , only rare mini - jets could beam toward the observer , which have a relatively lower flux ( than the larger gaussian angle distribution ) with respect to the majority of mini - jets that beam away from the observer and only contribute to the background . the overall shape of the light curves does not differ significantly . we next compare the effect of lorentz factor contrast in the icmart region . we keep the initial value of the bulk lorentz factor @xmath7 constant , i.e. , @xmath53 , and vary @xmath67 . this corresponds to different values of the initial magnetization @xmath68 . in figure [ fig : gamma - ratio ] , we compare three sets of simulations , with ( a ) @xmath69 ; ( b ) @xmath70 and ( c ) @xmath71 . other parameters are the same as those adopted to calculate figure [ fig : direction ] , and the gaussian @xmath14-distribution model with typical angle @xmath65 has been adopted . we show that the light curves become progressively more erratic and spikier when the @xmath67 becomes larger . this is because a larger @xmath67 would give rise to larger @xmath72 , and thus a larger value of the total doppler factor @xmath73 . a larger @xmath67 also tends to give a more significant evolution of the parameters ( figure [ fig : evolution ] ) . initially , a constant @xmath74 corresponds to a constant @xmath75 cone , so that observed numbers of mini - jets are the same in all these cases . however a larger @xmath67 can give rise to a larger @xmath7 near the end of evolution , thus a smaller @xmath44 cone . the slow component is not as significant , so that the light curves become spikier . + + in order to show the evolution of the physical parameters during the icmart cascade event , in figure [ fig : evolution ] we display the evolution of the bulk lorentz factor @xmath7 , the mini - jet lorentz factor @xmath1 , and the emission region magnetization @xmath0 as a function of time . it can be seen that evolution is more significant for a larger @xmath67 ( and equivalently a larger @xmath68 ) . + + next , we test how the total number of mini - jets @xmath41 within the @xmath44 cone affects the light curves . according to equation [ eq : n ] , varying @xmath41 is effectively varying the filling factor @xmath43 . by varying @xmath41 , the total number of e - folding steps @xmath61 is slightly modified , as is the rising time @xmath76 . in figure 4 , we compare the simulated light curves for different @xmath41 values , i.e. , @xmath77 , and @xmath78 , respectively . it can be seen that in general the light curves appear smoother with increasing @xmath41 . this can be readily understood : the larger the @xmath41 , the more reconnection events happen simultaneously , so that more mini - jets beaming to different directions tend to enhance the slow component . the short - timescale structures are smeared out , and the light curves become smoother . + next , we explore the effect of the emission region radius @xmath46 . figures [ fig : r](a ) , [ fig : r](b ) , and [ fig : r](c ) show the results for @xmath79 cm , @xmath80 cm , and @xmath81 , respectively . one can see that the larger the @xmath46 , the longer and stronger the high - latitude emission tail . this is because the length of the high - latitude tail is defined by @xmath82 . we notice again that the rising time is the growth time of the cascade , which is the @xmath60-folding time of consuming most of the magnetic energy in the emission region , which is defined by the total number @xmath41 of the mini - jets and the characteristic scale @xmath19 of each mini - jet . since the rising and falling times are related to different parameters , the pulse is usually asymmetric ( e.g. , figure [ fig : r ] ) . the simulated light curve is more consistent with data if the emission radius @xmath46 is large . zy11 suggested that icmart events should happen at larger radii , say , @xmath83 cm , in order to reach the critical condition of triggering a reconnection / turbulence cascade . it is intriguing to see that such large - radius icmart events make light curves more resemble the observed ones . + + we also discuss the effect of different sizes of reconnection regions . we make two sets of simulations . in the first set , we vary @xmath19 while keeping @xmath46 constant . we also keep @xmath84 , so effectively , we are varying the filling factor @xmath43 . since @xmath85 , the rising time @xmath86 is modified correspondingly . the results are presented in figure [ fig : size1 ] , which shows the simulated light curves for @xmath87 , @xmath88 , and @xmath89 cm , respectively . it can be seen that the smaller the @xmath19 , the spikier the light curve . this is because a smaller @xmath19 corresponds to a shorter duration of each reconnection event . for the @xmath90 cm case , short - timescale structures are missing , and the light curve is very smooth . + + next , we keep both @xmath41 and @xmath43 constant . by varying @xmath19 , we are effectively varying @xmath46 as well , so that the ratio @xmath91 is a constant . the results are shown in figure [ fig : size2 ] , in which light curves for @xmath87 , @xmath88 , and @xmath89 cm are simulated . the general trend as discussed above is still there , but since @xmath46 is changed accordingly , the contrasts are less significant , namely , the smaller @xmath19 cases are less spiky and larger @xmath19 cases are less smooth with respect to the case where @xmath46 is fixed ( figure [ fig : size1 ] . since the decay phase is defined by @xmath46 ( section [ sec : r ] above ) , varying @xmath46 with @xmath19 also affects the length of the decaying phase . + + we next test the effect of size distribution of the reconnection regions . we try two possibilities : the power - law distribution with an index @xmath38 ( the kolmogorov type ) ( figure [ fig : size - distribution](a ) ) and a uniform distribution ( figure [ fig : size - distribution](b ) ) . one can see that the uniform distribution has a smoother shape . in this case , the observed small pulse width distribution is solely determined by the distribution of the doppler factors . for the power - law distribution case , an extra factor ( the intrinsic distribution ) plays a role to make small pulses , so that the light curves are spikier . + finally we calculate the light curves for different energy bands . we consider three cases here , below the peak of the band spectrum ( figure [ fig : bands](a ) ) , i.e. , 15 - 150 kev ( also the observation band for _ swift _ bat ) , above the peak ( 500 - 650 kev , figure [ fig : bands](b ) ) , and across the entire energy band ( 15 - 650 kev , figure [ fig : bands](c ) ) . the high - energy light curve is slightly narrower and spikier , as observed in real grbs . in general , the overall shape of the light curves does not differ significantly . + + as suggested by zy11 , a real grb light curve may consist of multiple icmart events . in figure [ fig : grb ] , we simulate three emission episodes and superpose them together to make a mock grb light curve . we have varied @xmath74 and @xmath67 around the values @xmath92 and @xmath93 , respectively , with small fluctuations in different episodes . other parameters are the same as those adopted in figure [ fig : direction ] with a @xmath65 gaussian @xmath14-distribution . the simulated light curve shows reasonable features as observed in some grbs . we note that in reality the parameters of different icmart events could be more different , so that a variety of light curves could be made , which may account for the diverse prompt emission light curves as observed . in order to test whether our simulated light curves mimic the observed ones , we also perform a pds analysis of our results . in order to get robust pds slopes , for each set of parameters , we perform 10 different monte carlo simulations to get 10 different light curves , derive the pds slope of each light curve , and calculate the average slope to stand for this particular set of parameters . some examples of pdss are presented in figure [ fig : pds ] . generally , the pdss can be fit with a power law , with indices generally steeper than @xmath94 . the averaged pds indices for all the cases corresponding to figures 1 , 2 , and 4 - 9 are collected in table 1 . observationally the pds slopes are steeper in softer bands ( e.g. _ swift _ ; guidorzi et al . 2012 ) than harder bands ( e.g. batse ; beloborodov et al . our simulations recover this trend . the presented pds values are taken from the _ swift _ band . it is encouraging to see that the simulated values are generally consistent with the _ swift _ data ( guidorzi et al . our simulations also show a turnover of pdss in the high - frequency regime with a steeper index . such a feature is seen in some grbs . + .pds slopes of simulated light curves [ cols="^,^",options="header " , ] from table 1 , one can see that various parameters can affect the slope of a pds . generally speaking , spikier light curves have more power in high frequencies and therefore have a shallower pds slope . most pds indices listed in table 1 can be understood this way . for figure [ fig : direction ] , it is seen that more isotropic distributions give steeper slopes . this is because the more isotropic cases give more mini - jets contributing to the broad component , and thus enhance the low - frequency power . similarly , as shown in figure [ fig : n ] , a smaller number @xmath41 gives richer spiky features , and therefore gives a shallower pds slope . the @xmath46-dependence ( figure [ fig : r ] ) can be understood as the following : a larger @xmath46 corresponds to a longer curvature decay tail , on top of which rapid variability can be observed , so that the pds slope is shallower . for the size effect ( figure [ fig : size1 ] ) , a smaller @xmath19 can give rise to pulses with shorter duration and hence , a more dominant high - frequency power and shallower pds ( figure [ fig : size1 ] ) . when both @xmath46 and @xmath19 co - vary , this effect is still relevant , but somewhat compensated by the @xmath46 effect ( figure [ fig : size2 ] ) . next , without a size distribution , the pds is steep ( figure [ fig : size - distribution](b ) ) . by introducing a size distribution , one has more contributions to short - time variability from smaller sizes , so the pds becomes shallower . finally , the light curves in a higher energy band are somewhat spikier ( figure [ fig : bands ] ) and hence have a shallower pds . this is consistent with the finding of guidorzi et al . ( 2012 ) and beloborodov et al . ( 2000 ) : using the _ swift _ bat data , guidorzi et al . ( 2012 ) obtained a steeper pds slope than beloborodov et al . ( 2000 ) , who used the batse data ( higher energy band ) to perform the analysis . it is interesting to investigate the change of pds slope due to the change of the initial lorentz factor contrast . as shown in figure [ fig : evolution ] , in principle one can have strong parameter evolution during one icmart event , which causes complicated evolution of the pds behavior . to avoid such strong evolution , we first fix @xmath53 , and vary @xmath67 so that the ratio @xmath95 evolves in the range of @xmath96 . in figure [ fig : pds1 ] , we present the pds slope as a function of @xmath95 . the triangles ( and dotted line ) are calculated by turning off parameter evolution ( i.e. , keeping @xmath1 and @xmath7 unchanged throughout ) , and the squares ( and solid line ) are calculated by turning on the parameter evolution ( figure [ fig : evolution ] ) . one can see that the pds slope becomes progressively shallower as @xmath97 increases . this is understandable , since a larger @xmath67 corresponds to a stronger fast emission component , and therefore the light curves are spikier ( see figure [ fig : gamma - ratio ] ) . one can tentatively draw the conclusion that a more magnetized outflow tends to make spikier light curves . since the final lorentz factor of the ejecta at the deceleration time is proportional to @xmath98 , and since observationally the lorentz factor at the onset of afterglow does not have a wide distribution ( e.g. , liang et al . 2010 ) , it is interesting to investigate how the pds slope depends on the lorentz factor contrast when @xmath98 is set to constant . in figure [ fig : pds2 ] , we present the case of @xmath99 for cases both without and with parameter evolution . the range of the contrast is set to @xmath100 ( i.e. @xmath101 , @xmath102 ) to @xmath103 ( i.e. , @xmath104 ) . the convention is the same as figure 10 . the dependence shows more complicated patterns . for the case without evolution ( triangles and dotted line ) , in general one can see decrease of pds slope when @xmath95 increases ( except the slight tilt at very large @xmath95 ) . this can be understood in the following way . as @xmath95 increases , one has two competing effects . the increase of @xmath67 tends to enhance the small timescale variability . on the other hand , the decrease of @xmath7 tends to enlarge the @xmath44 cone , so that many more mini - jets not beaming toward the observer could contribute to the slow component . the net result after competition is that the latter effect wins , so that the long - time variability is more enhanced , and hence , a steeper pds is obtained . this trend is overturned when @xmath67 exceeds @xmath74 near the end of the curve . when evolution is taken into account ( squares and solid line ) , the situation is even more complicated . when @xmath67 is small enough , the above - mentioned trend is retained . however , when @xmath67 becomes large enough , evolution of @xmath1 and @xmath7 becomes significant ( figure [ fig : evolution ] ) , so that quickly one can reach a regime with small @xmath1 and large @xmath7 . the average pds would be dominated by this late phase , so that the general trend is reversed from the no - evolution case . in reality , since a real grb light curve would usually be the superposition of multiple icmart events , the clean evolution expected in a single icmart event would be smeared out . in this paper we have simulated a sample of grb prompt emission light curves and pdss within the framework of the icmart model ( zy11 ) . this model was developed to model grbs whose jet composition is still somewhat poynting flux dominated in the emission region . this was motivated by the non - detection of the photosphere component in some grbs ( zhang & peer 2009 ; zhang et al . since the emission region has a moderately high @xmath0 , in order to generate a reconnection / turbulence cascade envisaged by zy11 , the energy dissipation region must have many locally lorentz - boosted emission regions , or mini - jets . the detected emission would be the superposition of emissions from all these mini - jets , which beam to random directions in the bulk comoving frame . other global magnetic dissipation models for grb prompt emission have been proposed in the literature ( e.g. , lyutikov & blandford 2003 ; giannios & spruit 2006 ) . if these models invoke runaway generation of mini - jets at a relatively large emission radius , then the simulations in this paper also apply to those scenarios . lacking detailed numerical simulations for a reconnection / turbulence cascade , we carried out a monte carlo simulation by inputting many mini - jets with certain directional and temporal distributions within the icmart scenario . we investigated the roles of the directional distribution , lorentz factor contrast , number of reconnection regions , emission radius , size of the mini - jet , mini - jet size distribution , energy dependence , etc . , in defining the light curves and their pdss . we adopt our simulation parameters according to observations ( e.g. , typical length of reconnection region @xmath52 cm corresponding to observed variability timescale @xmath59 s , 15 - 150 kev band for simulated light curves corresponding to _ swift _ bat band , and so on ) , as well as the requirements of the icmart model itseft ( e.g. , emission region radius @xmath51 cm in order to make sure that runaway reconnection can happen , and exponential growth of the number of reconnection events with time ) . within the icmart framework , most of our parameters are physically related to each other self - consistently . even though some simplified assumptions are introduced so that the light curves may not fully represent the complex physics in an icmart event , our simulated light curves nonetheless show some encouraging features that are consistent with the grb prompt emission data . the most noticeable feature is the superposition of an underlying slow component and more erratic fast component , which seems to be consistent with the data ( gao et al . 2012 ; vetere et al . the slow component is caused by the superposition of emission from all the mini - jets in the emission region , while the fast component is related to those mini - jets that happen to beam toward the observer . we follow the physics of an icmart event , including the exponential growth of the reconnection region , dissipation of the magnetic field energy ( so that @xmath0 drops with time ) , and acceleration of the bulk ejecta during the energy dissipation process and find that the erratic grb light curves as observed can be generally reproduced within the model . among all the model parameters , the lorentz factor contrast and the number of mini - jets play an important role in defining the `` spikiness '' of the light curve . we also derived the pds slopes of the simulated light curves , and found that they are generally consistent with the data . generally speaking , the larger the contrast @xmath95 ( keeping @xmath74 constant ) , the shallower the pds slope . besides grbs , the `` jet - in - the - jet '' scenario has been discussed in other astrophysical contexts . giannios et al . ( 2010 ) interpreted the fast tev variability of active galactic nuleus jets using the mini - jet scenario . yuan et al . ( 2011 ) applied the scenario to account for the gamma - ray flares of the crab nebula . compared with earlier work of narayan & kumar ( 2009 ) , kumar & narayan ( 2009 ) , and lazar et al . ( 2009 ) , the new ingredient introduced in our paper is the exponential growth of the number of mini - jets as a function of time , as envisaged in the icmart model ( zy11 ; see also stern & svensson 1996 ) . as a result , our model allows many mini - jets emitting simultaneously at any instant . this is the key ingredient to define the broad component of each icmart event . a grb light curve is composed of multiple icmart events ( figure 8) , which are controlled by the erratic central engine activity . in order to set up the monte carlo simulations , we had to introduce a number of assumptions . these include power - law distribution of the size of reconnection regions , gaussian shape of each pulse , same intrinsic radiation spectrum for all emitters , exponential growth of numbers of pulses with time , isotropic or gaussian distribution of the mini - jet directions , and so on . some factors are still missing . for example , in the comoving frame of the jet bulk but outside the mini - jets , there would also be particles that give rise to radiation . the effects of this inter - mini - jet emission should be investigated ( e.g. , lin et al . 2013 ) . the physical conditions of real grbs must be more complex than what is modeled here , so that one may not reproduce the full observational features of grbs with the simulations presented in this paper . nonetheless , our simulations show the encouraging results that the simulated light curves based on these simplified assumptions can indeed reproduce some key features of the observations , e.g. the slow and fast variability components and a variety of degree of spikiness of the light curves . by changing parameters ( e.g. , @xmath14-distribution , lorentz factor contrast , jet opening angle ) , diverse light curves can be generated , ranging from relatively smooth to relatively spiky ones . the pdss of the simulated light curves are also generally consistent with the data . all these suggest that the icmart model may be a good candidate to interpret grb prompt emission . within the icmart theoretical framework , the following constraints can be made to the model parameters . ( 1 ) to reproduce the general fast - rising slower decay shape of broad pulses , the emission radius should be relatively large ( @xmath105 cm and beyond ) . ( 2 ) since many grbs show high - amplitude rapid variability , the grb initial magnetization parameter @xmath68 in the emission region could be high ( e.g. , from several to hundreds ) . ( 3 ) the observed minimum variability timescale constrains that @xmath19 can not be too large and has to be @xmath106 cm . ( 4 ) in order not to smear these peaks by overgenerating mini - jets , one also requires a filling factor @xmath42 , suggesting that in these cases the global @xmath0 of the outflow after the icmart event may not drop to unity . ( 5 ) erratic light curves with multiple episodes suggest that the grb central engine acts multiple times to eject highly magnetized shells so that multiple icmart events can be generated within one burst . ( 6 ) the existence of smooth - pulse grbs suggests that in some cases the @xmath68 is not much larger than unity ( so that @xmath67 is not much larger than unity ) , or there are so many mini - jets operating simultaneously . other information ( e.g. polarization properties and prompt emission efficiency ) is needed to break the degeneracy . we thank kohta murase , pawan kumar , zi - gao dai , he gao , da - bin lin , and chun li for useful discussion , and an anonymous referee for very helpful suggestions . this work is partially supported by nsf through grant ast-0908362 . bo zhang acknowledges a scholarship from china scholarship council for support . amati , l. , frontera , f. , tavani , m. , et al . 2002 , , 390 , 81 band , d. , matteson , j. , ford , l. , et al . 1993 , , 413 , 281 beloborodov , a. m. , stern , b. e. , & svensson , r. 2000 , , 535 , 158 cho , j. , lazarian , a. , & vishniac , e. t. 2003 , turbulence and magnetic fields in astrophysics , 614 , 56 daigne , f. , & mochkovitch , r. 1998 , , 296 , 275 daigne , f. , & mochkovitch , r. 2002 , , 336 , 1271 dermer , c. d. , & mitman , k. e. 1999 , , 513 , l5 drenkhahn , g. & spruit , h. c. 2002 , , 391 , 1141 fishman , g. j. , & meegan , c. a. 1995 , , 33 , 415 gao , h. , zhang , b .- b . , & zhang , b. 2012 , , 748 , 134 giannios , d. , & spruit , h. c. 2006 , , 450 , 887 giannios , d. , uzdensky , d. a. , & begelman , m. c. 2010 , , 402 , 1649 ghisellini , g. , celotti , a. , & lazzati , d. 2000 , , 313 , l1 goldreich , p. , & sridhar , s. 1995 , , 438 , 763 guidorzi , c. , margutti , r. , amati , l. , et al . 2012 , , 422 , 1785 hascot , r. , daigne , f. , & mochkovitch , r. 2012 , , 542 , l29 kobayashi , s. , piran , t. , & sari , r. 1997 , , 490 , 92 kumar , p. 1999 , , 523 , l113 kumar , p. , & mcmahon , e. 2008 , , 384 , 33 kumar , p. , & narayan , r. 2009 , , 395 , 472 lazar , a. , nakar , e. , & piran , t. 2009 , , 695 , l10 lazarian , a. , & vishniac , e. t. 1999 , , 517 , 700 lei , w .- h . , zhang , b. , & liang , e .- w . 2013 , , 765 , 125 liang , e .- w . , yi , s .- x . , zhang , j. , et al . 2010 , , 725 , 2209 lin , d .- b . , gu , w .- m . , hou , s .- j . , liu , t. sun , m .- y . , lu , j .- f . 2013 , , 776 , 41 lyubarsky , y. e. 2005 , , 358 , 113 lyutikov , m. , & blandford , r. 2003 , arxiv : astro - ph/0312347 narayan , r. , & kumar , p. 2009 , , 394 , l117 panaitescu , a. , spada , m. , & mszros , p. 1999 , , 522 , l105 rees , m. j. , & meszaros , p. 1994 , , 430 , l93 sari , r. , & piran , t. 1997 , , 485 , 270 shen , r .- f . , & zhang , b. 2009 , , 398 , 1936 spruit , h. c. , daigne , f. , & drenkhahn , g. 2001 , , 369 , 694 stern , b. e. , & svensson , r. 1996 , , 469 , l109 tagliaferri , g. , goad , m. , chincarini , g. , et al . 2005 , , 436 , 985 uhm , z. l. , & zhang , b. 2013 , arxiv:1303.2704 vetere , l. , massaro , e. , costa , e. , soffitta , p. , & ventura , g. 2006 , , 447 , 499 yamazaki , r. , ioka , k. , & nakamura , t. 2004 , , 607 , l103 yuan , q. , yin , p .- f . , wu , x .- f . , et al . 2011 , , 730 , l15 zhang , b. , & mszros , p. 2002 , , 581 , 1236 zhang , b. , & kobayashi , s. 2005 , , 628 , 315 zhang , b. , fan , y. z. , dyks , j. , et al . 2006 , , 642 , 354 zhang , b. , & peer , a. 2009 , , 700 , l65 zhang , b. , & yan , h. 2011 , , 726 , 90 ( zy11 ) zhang , b .- b . , zhang , b. , liang , e .- w . , et al . 2011 , , 730 , 141 zrake , j. , & macfadyen , a. i. 2012 , , 744 , 32 zweibel , e. g. , & yamada , m. 2009 , , 47 , 291
|
in this paper , we simulate the prompt emission light curves of gamma - ray bursts ( grbs ) within the framework of the internal - collision - induced magnetic reconnection and turbulence ( icmart ) model .
this model applies to grbs with a moderately high magnetization parameter @xmath0 in the emission region .
we show that this model can produce highly variable light curves with both fast and slow components .
the rapid variability is caused by many locally doppler - boosted mini - emitters due to turbulent magnetic reconnection in a moderately high @xmath0 flow .
the runaway growth and subsequent depletion of these mini - emitters as a function of time define a broad slow component for each icmart event .
a grb light curve is usually composed of multiple icmart events that are fundamentally driven by the erratic grb central engine activity . allowing variations of the model parameters , one is able to reproduce a variety of light curves and the power density spectra as observed .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
gamma - ray burst ( grb ) 130427a may well be the burst with the most comprehensive afterglow follow - up , its multiwavelength monitoring covering radio , optical , x - ray , and @xmath2-ray frequencies , and extending from seconds to tens of days after trigger . the x - ray _ prompt _ emission ( up to 100 s ) was accompanied by the second brightest optical flash , monitored by raptor ( vestrand et al 2013 ) , with the optical afterglow light - curve displaying a steepening at 300 s and a flattening at 10 ks . the swift x - ray light - curve ( x - ray light - curve repository evans et al 2009 ) is consistent with a single power - law from 500 s to 5 ms . the fermi - lat @xmath2-ray light - curve ( tam et al 2013 ) displays a peak at 1020 s , simultaneous with the optical flash peak , and a steepening at 550800 s ( zhu et al 2013 ) . the vla radio light - curves ( laskar et al 2013 ) display a slow decay at 1 - 10 day . with such a rich dataset , grb afterglow 130427a demands a theoretical interpretation , done here in the framework of the external - shock model ( & rees 1997 ) where some relativistic ejecta , produced by the black - hole resulting from the core - collapse of a massive star , drive a _ forward - shock _ into the ambient medium while the ejecta are energized by the _ reverse - shock_. the synchrotron and inverse - compton emissions from both shocks are calculated assuming that electrons and magnetic field acquire a certain fraction of the post - shock energy . the shock - accelerated electrons are assumed to have a power - law distribution with energy ( hence the synchrotron and inverse - compton spectra are also power - laws ) , with a break at the cooling energy ( where the radiative - loss timescale equals the shock age ) . analytical treatments for the forward - shock emission have been provided by & rees ( 1997 ) , sari , piran & narayan ( 1998 ) , waxman , kulkarni & frail ( 1998 ) , granot , piran & sari ( 1999 ) , wijers & galama ( 1999 ) , chevalier & li ( 2000 ) , panaitescu & kumar ( 2000 ) , and for the reverse - shock by kobayashi ( 2000 ) . both shocks have been studied with 1-dimensional hydrodynamical codes by panaitescu & ( 1998 ) and kobayashi & sari ( 2000 ) , the former focusing on the two - shock synchrotron and inverse - compton emission , the latter on the dynamics of the shocks . to model the multiwavelength emission of grb afterglow 130427a , we employ a 1-dimensional code that follows the ejecta medium interaction , with the dynamics of each shock calculated from conservation of energy and using the shock jump - conditions ( blandford & mckee 1976 ) . after the onset of deceleration , the dynamics of the forward - shock is determined by the ejecta initial energy , injected energy ( rees & 1998 ) , and ambient medium density . the dynamics of the reverse - shock is determined by that of the shocked fluid and two properties of the incoming ejecta : their energy and lorentz factor . here , we consider that the ejecta add energy to the blast - wave as a power - law in observer time and that they have a single lorentz factor . the self - absorption and cooling frequencies of the synchrotron spectrum and the inverse - compton parameter are calculated self - consistently from the electron distribution and the magnetic field strength ( panaitescu & 2000 ) . radiative losses are also calculated from the electron distribution , but they are negligible for the following best - fit models . the emissions from both shocks are integrated over their motion and over the angle at which the fluid moves relative to the direction toward the observer . more details about this numerical model and its application to the multiwavelength emission of ten grb afterglows are given in panaitescu ( 2005 ) . the choice of the model features that may accommodate the temporal decay of the broadband emission of grb afterglow 130427a starts with the x - ray light - curve because its temporal decay index ( @xmath3 ) and spectral slope ( @xmath4 ) are the best determined : @xmath5 at 20 ks 5 ms ( fig [ homog ] ) and @xmath6 at mean time 24 ks ( kennea et al 2013 ) . these lead to @xmath7 , which is compatible with the value expected ( zero ) for the synchrotron emission from the forward - shock interacting with a homogeneous medium and for the x - ray being below the cooling frequency @xmath8 of the synchrotron spectrum . as the optical flux also decays at that time , the optical must be above the peak energy @xmath9 of the synchrotron spectrum , hence optical and x - ray are in the same spectral regime : @xmath10 . consequently , the intrinsic afterglow optical flux can be calculated from the x - ray flux : @xmath11 . for instance , the observed @xmath12jy implies that @xmath13 mjy , which is a factor 2.5 larger than the measured @xmath14 mjy , requiring @xmath15 mag of dust extinction in the host galaxy . the 10 kev 100 mev spectral slope @xmath16 indicates that @xmath8 is well above 10 kev . that the lat flux decays slower than in the x - ray ( @xmath17 at 500 s 50 ks ) indicates that @xmath8 is below the lat range ( otherwise , for @xmath18 mev , the model expectation is @xmath19 ) and that the electron radiative cooling is dominated by inverse - compton scatterings ( otherwise , for synchrotron - dominated electron cooling , @xmath20 and @xmath21 , incompatible with the observed @xmath22 ) . more exactly , for a compton - dominated electron cooling , the decay index of the synchrotron flux above @xmath8 is @xmath23 , which matches well the observed @xmath24 , with @xmath25 being the exponent of the power - law distribution of electrons with energy ( @xmath26 ) that is required by the forward - shock model , given the measured flux decay index @xmath27 below the cooling frequency . in summary , the optical , x - ray , and @xmath2-ray fluxes of grb afterglow 130427a , their decay indices , and the x - ray spectral slope , require that @xmath28 , if the afterglow emission is synchrotron from the forward - shock . under the assumption that the two microphysical parameters of the forward - shock ( @xmath29 and @xmath30 ) that quantify the post - shock fractional energy in the magnetic field and in electrons are constant , the forward - shock synchrotron light - curve at any frequency below the optical can be easily calculated from the optical light - curve , using the expected evolution of the synchrotron peak flux ( @xmath31 ) and peak energy ( @xmath32 ) for a homogeneous medium . if @xmath9 crosses the optical at some time @xmath33 , yielding an optical flux @xmath34 , then the radio flux at frequency @xmath35 is @xmath36 here , @xmath37 mjy is the intrinsic optical light - curve after 10 ks ( corrected for the above - inferred host extinction of @xmath15 mag ) and @xmath38 ( the forward - shock model requires that @xmath39 ) . the largest @xmath33 required by equation ( [ radio ] ) arises from the radio measurement with the highest @xmath40 ; taking the @xmath41 mjy measurement as an upper limit for the forward - shock radio flux , implies @xmath42 ks . this means that , for the forward - shock emission ( that accommodates the observed optical flux ) not to exceed the measured radio fluxes , the synchrotron peak should cross the optical at 23 ks . conversely , if the synchrotron peak crossed the optical before 23 ks , then the synchrotron flux from the forward - shock would violate vla measurements . that may be avoided if the magnetic field parameter @xmath29 decreases ( roughly as @xmath43 ) , and if energy injection in the forward - shock is allowed ( to match the optical and x - ray flux decays at 110 day , which are faster when @xmath29 decreases ) , but this scenario requires fine - tuning and we do not pursue it . fig [ homog ] illustrates the failure of the forward - shock synchrotron model with a homogeneous medium to accommodate simultaneously the radio , optical , x - ray , and @xmath2-ray fluxes of grb afterglow 130427a : while it can explain the optical afterglow emission after 10 ks and the x - ray flux at 50 s 5 ms ( excluding the second grb pulse , which is a feature that can not be accounted for by any type of external shock ) , this model over - predicts either the lowest or the highest frequency data . the closure relation @xmath7 is also compatible with the forward - shock model expectation for a wind - like medium ( with an @xmath44 particle density distribution with radius ) and for x - ray below the cooling frequency , provided that there is an energy injection in the forward - shock that slows its deceleration and the decay of the afterglow x - ray flux . if that energy injection is parametrized as @xmath45 in observer time ( a power - law flux decay requires that the dynamics of the forward - shock is a power - law in observer time ) , then @xmath46/2 $ ] , from where @xmath47 . numerically , we find that the best - fit to the x - ray emission after 500 s ( including all the gev data and the optical after 10 ks ) has @xmath48 and that forward - shock energy should increase by a factor @xmath49 until @xmath50 ms , to account for the observed x - ray flux decay . that means that the energy added to the forward - shock mitigates its deceleration after @xmath51 ks . it is important to note that the forward - shock interacting with a wind - like medium does not produce more radio emission than measured because the synchrotron peak flux decreases as @xmath52 ( instead of being constant , as for a homogeneous medium ) . the evolution of the synchrotron peak energy is the same as for a homogeneous medium ( @xmath32 ) , hence the radio flux expected from the optical emission is @xmath53 then , @xmath54 mjy and @xmath55 mjy require that the time when the synchrotron peak crosses the optical is @xmath56 ks . that brief flattening seen in the optical light - curve at 10 ks could be due to @xmath9 crossing the optical and is compatible with @xmath56 ks . the best - fit to the optical data after 10 ks , the x - ray after 500 s , and all gev measurements , with the forward - shock emission and for a wind - like medium is shown in fig [ wind ] , with a sequence of spectra shown in fig [ spek ] . the @xmath57 for 135 dof of that best - fit makes it statistically unacceptable ; the gev fit has the largest @xmath58 for 9 points , closely followed by the optical fit s @xmath59 for 39 points , with the largest contribution to the fit s @xmath60 arising from the x - ray data , @xmath61 for 79 points . the model light - curves follow well all flux trends and relative intensities except the brightness of the prompt emission until 50 s , but can not describe well the early gev light - curve and can not capture the fluctuations in the x - ray and optical measurements ( after 10 ks , optical data are from different instruments ) . compared to the parameters inferred for other afterglows by modelling their multiwavelength emission , the wind density of the best - fit shown in fig [ wind ] is very small , but not unprecedented ( chevalier , li & fransson 2004 ) . its parameter , @xmath62 , corresponds to a stellar mass - loss rate to terminal wind - velocity ratio ( @xmath63 ) that is 300 smaller than for a typical wolf - rayet ( wr ) star ( as the progenitor of long bursts with an associated type ic supernovae ) , for which @xmath64yr and @xmath65 cm / s . the reason for that low density is the requirement that the synchrotron peak crosses the optical after 3 ks and matches the optical flux detected at that time . for @xmath66 and for the fluid moving directly toward the observer , the forward - shock synchrotron peak energy and peak flux are @xmath67 @xmath68 imposing that @xmath69 ev and @xmath70 mjy , yields @xmath71 taking the ratio of these two equations leads to @xmath72 . the @xmath30 parameter that quantifies the typical electron energy corresponds to a total electron energy that is a fraction @xmath73 of the post - shock energy . equipartition with protons sets an upper limit , @xmath74 , thus @xmath75 for @xmath76 , from where @xmath77 . this wind density is about 20 times lower than the lowest value measured ( nugis & lamers 2000 ) for galactic wr stars and indicates a low mass loss - rate combined with a high wind velocity . provided that can happen at the end of a wr s life , it has a strong consequence on the medium in which that star resides , as following . owing to low wind density and high ejecta kinetic energy , the forward - shock that fits the late time broadband emission of grb afterglow 130427a is highly relativistic , having @xmath78 , hence the shock radius is @xmath79 pc . requiring that @xmath80 at the latest observation epoch ( 50 day ) is less than the size of the bubble blown by a wr star during its @xmath81 yr lifetime , @xmath82 pc ( cf . castor , mccray & weaver 1975 ) , with @xmath83 the medium density around the star , we find that @xmath84 for a wind with @xmath85 , where @xmath86 is the observer - frame epoch when the afterglow shock encounters the wind termination shock . such a low ambient density suggests that the progenitor of grb 130427a occurred in a supper - bubble ( scalo & wheeler 2001 ) blown by many preceding supernovae . there are two interesting facts related to the lat emission produced by the forward - shock synchrotron model shown in figs [ homog ] [ spek ] . first is that the scattering of the synchrotron emission ( at the peak of the spectrum ) by the forward - shock electrons ( of typical energy ) occurs near the klein - nishina ( kn ) regime . when the electron cooling is dominated by inverse - compton scatterings ( i.e. compton parameter @xmath87 ) , the cooling frequency satisfies @xmath88 . inclusion of the kn effect reduces the compton @xmath89 parameter , thus , taking into account the kn effect , increases @xmath8 and the synchrotron flux at @xmath90 : @xmath91 . in other words , the synchrotron emission from fast - cooling electrons increases when a competing radiative process ( inverse - compton ) is reduced ( by inclusion of the kn effect ) . for the forward - shock best - fit to grb afterglow 130427a , the lat range is above @xmath8 and @xmath87 ; inclusion of the kn effect reduces @xmath89 by about 10 and increases the 100 mev flux by an order of magnitude ( see fig [ wind ] ) . furthermore , as the electrons at the peak of their distribution with energy enter and exit the kn regime , the synchrotron light - curve at 100 mev displays more structure than when the kn effect is ignored . the second is that radiative cooling during one gyration time limits the energy that electrons acquire through first - order fermi acceleration to a corresponding synchrotron characteristic energy @xmath92 mev , independent of the magnetic field @xmath93 . for the best - fit parameters given in fig [ wind ] , the forward - shock has @xmath94 and @xmath95 , so the maximal synchrotron energy is @xmath96 mev ( see synchrotron spectrum cut - off in fig [ spek ] ) . at earlier times , that cut - off is higher , but the inverse - compton emission from the forward - shock takes over above 2 gev ( as shown by the @xmath97 s spectrum ) and can account for the higher - energy lat emission until about 10 ks , after which the inverse - compton flux is too low . interestingly , a hardening of the lat spectrum above several gev was identified by tam et al ( 2013 ) , from @xmath98 at 0.15 gev to @xmath99 at 5100 gev . tam et al ( 2013 ) have proposed that the harder high - energy component is inverse - compton , although we find that the observed spectrum above 5 gev is softer than the model expectation @xmath100 , corresponding to the gev range being below the peak of the upscattered spectrum . the estimation of the expected radio emission given in equation ( [ radios2 ] ) led to the conclusion that the forward - shock can not account for the optical afterglow emission prior to @xmath101 ks . also , the flat radio light - curve arising from the forward - shock interacting with a wind can not account for the radio emission at 110 day , which is slowly decaying . both these emissions are attributed to the reverse - shock ( see also laskar et al 2013 ) , as discussed below . we note that , after 10 ks , the existence of a reverse - shock is required by the energy injection into the forward - shock required by the measured decay index of the x - ray flux . the radio data are contemporaneous with the higher energy ( optical , x - ray , and @xmath2-ray ) afterglow emission accommodated by the forward - shock , thus , for the calculation of the reverse - shock emission , the dynamical parameters @xmath102 , @xmath103 , @xmath104 , and @xmath105 are fixed at the values determined from the forward - shock best - fit . the free parameters of the reverse - shock are the lorentz factor @xmath106 of the incoming ejecta ( which sets the post - shock energy density ) and the three microphysical parameters ( @xmath29 , @xmath30 , and @xmath107 ) that determine the synchrotron spectrum . the best - fit obtained with the reverse - shock emission to the 110 day radio data is shown in figs [ wind ] and [ spek ] . unfortunately , it has a large @xmath108 for 25 dof , because it underestimates the radio flux above 50 ghz . as shown in fig [ spek ] , those radio data can not be explained by the forward - shock either , if its microphysical parameters are constant . requiring the same microphysical parameters for both the reverse and forward shocks yields a much worse radio data fit , with @xmath109 . the best - fit to the early optical emission with a reverse - shock includes also the earlier x - ray data and all gev data , to ensure that the reverse - shock emission does not exceed what was observed . again , the dynamical parameters @xmath102 and @xmath105 are fixed to the values obtained for the forward - shock , but the energy @xmath110 carried by the incoming ejecta arriving at the blast - wave prior to 10 ks is only weakly constrained by the forward - shock fit to the optical and x - ray data after 10 ks , which sets an upper limit @xmath111 . with free micro - parameters , the best - fit with the reverse - shock to the early afterglow has @xmath112 for 136 dof , as it fails to account for the gev prompt emission prior to 100 s , although it explains well the early optical data and the x - ray data at 0.53 ks . we note that the reverse - shock magnetic parameter @xmath29 prior to 10 ks ( from fitting the early optical afterglow ) is 100 times larger than after 10 ks ( from modelling for the radio emission ) . if the reverse - shock microphysical parameters were held constant across 10 ks , then the fit to the radio emission would have a @xmath113 twice larger , thus a decrease in @xmath29 at 10 ks is required . that may mean that the ejecta arriving at the blast - wave later are less magnetized . as discussed in [ sypeak ] , to reconcile the radio and optical fluxes of afterglow 130427a requires that the peak of the forward - shock synchrotron crosses the optical after 3 ks . in turn , that requires ( [ tenouswind ] ) a very weak stellar wind , about 300 less tenuous ( @xmath85 ) than for the average galactic wr star . then , the forward - shock radius is @xmath114 pc , while the wind bubble radius should be @xmath115 pc . thus , if the circumstellar medium is sufficiently dense , it possible that @xmath116 . alternatively , if the stellar wind had the average density , the wind termination shock could be encountered by the forward - shock at 3 ks , provided that the burst is embedded in a hot , highly pressurized environment ( chevalier et al 2004 ) . at frequencies below the cooling break , the afterglow light - curve should display a flattening when the forward - shock crosses the wind termination shock , transiting from the @xmath117 free wind to the quasi - homogeneous shocked wind . to be self - consistent , the interpretation of the 3 ks optical light - curve flattening as the blast - wave encountering the wind termination shock should attribute the entire afterglow emission to the same shock . then , the peak of the synchrotron spectrum must be below optical at all times when a decaying optical flux is measured , a model which overproduces radio emission , if the optical afterglow originates in the forward - shock ( as shown in [ sypeak ] ) . the subsequent steepening of the optical light - curve at 20 ks can not originate in the ambient medium stratification because , outside the termination shock , the shocked wind and circumstellar medium are still homogeneous . instead , that light - curve steepening should be attributed to the cooling frequency falling below the optical , which yields a steepening of the power - law flux decay by @xmath118 ( consistent with that measured for the optical light - curve of 130427a at 20 ks ) , and a softening of the optical spectrum by @xmath119 ( consistent with the reddening reported by perley et al 2013 , after 10 ks ) . the fortuitous temporal coincidence of the cooling frequency falling below optical just after the blast - wave arrives at the free - wind termination shock is not required if the discontinuity in the ambient medium structure that yields the 3 ks optical light - curve flattening is caused by an internal interaction within an unsteady stellar wind or by the interaction between the winds of two stars . in the former scenario , considered analytically by chevalier & imamura ( 1983 ) and in the context of grb afterglows by ramirez - ruiz et al ( 2005 ) , a stronger wind produced by the wr star prior to its core - collapse interacts with a slower wind ejected previously . in the latter scenario , proposed by mimica & giannios ( 2011 ) to be the source for more diverse afterglow light - curves , the grb progenitor is in a dense stellar cluster , where the mean distance between stars is below 1 pc , and the wr wind interacts with the weaker wind of a nearby o star or a later type . in either scenario , after the interaction with the shocked wind(s ) , which yields a light - curve flattening , the blast - wave goes into an @xmath117 wind , which is the earlier wr wind or the wind of the nearby star , producing a light - curve steepening , with the flux decay index @xmath120 returning to the value it had during the interaction with the free wr wind . only the dense cluster scenario provides a natural explanation for the very weak wind inferred here from modelling the afterglow 130427a : the wind of a b star located within 1 pc of the grb progenitor . however , this scenario can not explain why that weak wind extends over tens of pcs ( as required by the duration of the afterglow , [ tenouswind ] ) despite the more powerful winds of nearby , earlier type stars . thus , the 3 ks flattening and 20 ks steepening seen in the optical light - curve of 130427a could originate from a forward - shock interacting with the more complex ambient medium resulting from an internal wind interaction provided that microphysical parameters evolve such that this model does not exceed the 110 day radio measurements . alternatively , radio emission is not overproduced if the entire afterglow emission arises from the reverse - shock , and the optical light - curve flattening and steepening could result from the changing dynamics of the reverse - shock when the shocked - wind shell is crossed . such light - curve features could also be due to variations in the density and lorentz factor of the incoming ejecta , without any need for a non - uniform ambient medium . however , a model where the entire afterglow emission arises from the same shock ( reverse or forward ) does not provide a natural explanation for the colour evolution displayed by 130427a , which becomes bluer after 3 ks ( vestrand et al 2013 ) , when the optical light - curve flattens , and redder after 10 ks ( perley et al 2013 ) , when the optical light - curve steepens . in contrast , the hybrid reverse - forward shock model explains naturally both the 3 ks spectral hardening , as due to the harder ( below the synchrotron peak energy ) forward - shock emission emerging from under the softer reverse - shock emission , and the following spectral softening , caused by the peak energy of the forward - shock synchrotron spectrum falling below optical . the closure relations expected between the forward - shock synchrotron flux decay index and spectral slope suggest a homogeneous ambient medium for grb afterglow 130427a . although long grbs arise from massive stars that drive powerful winds , a homogeneous medium is possible if the afterglow emission is produced in the shocked wind . however , this afterglow s ( 10 ks ) optical flux to ( 110 day ) radio flux ratio and its slowly decaying radio light - curves disfavour that type of mbient medium . instead , for an unevolving constant magnetic field parameter , the synchrotron spectrum peak flux is constant and the radio emission should have been brighter and slowly rising . with some fine - tuning of the evolutions of those micro - parameters , it may be possible to reduce the forward - shock model radio flux below measurements , while still accounting for the observed optical and x - ray light - curves . a wind - like medium ( @xmath44 ) is the more natural expectation for a massive star as the grb progenitor . the forward - shock emission still can not account for the radio data because the expected radio light - curve is flat , however , a wind - like medium yields a decreasing synchrotron spectrum peak flux , making it easier to keep the forward - shock radio emission below radio measurements . to explain the optical and x - ray flux decay after 10 ks with the forward - shock synchrotron emission , a moderate energy injection into the forward - shock is required , increasing the shock energy by a factor 4 from 10 ks to 1 ms . the agent of that energy injection should be some ejecta that arrive at the forward - shock at that time , which provides a natural explanation for the afterglow radio emission : the reverse - shock that crosses the incoming ejecta . the reverse - shock must have been operating at even earlier times because the high early - optical to late - radio flux ratio precludes a forward - shock origin of the optical afterglow emission prior to 10 ks . such a reverse - to - forward shock switch for the origin of the optical emission , occurring at few ks , is supported by the optical afterglow becoming bluerat that time ( vestrand et al 2013 ) , when the forward - shock emission , with a spectrum @xmath121 in the optical , begins to dominate the softer reverse - shock emission , with a spectrum @xmath122 . as the peak of the forward - shock synchrotron spectrum falls below optical at about 10 ks , the optical afterglow should become redder after 10 ks , as was observed by perley et al ( 2013 ) . however , for the reverse - shock to explain the 100 s few ks optical afterglow and the 110 day radio afterglow emission , the properties of the reverse - shock ( microphysical parameters , kinetic energy and lorentz factor of the incoming ejecta ) must change around 10 ks . furthermore , vestrand et al ( 2013 ) have shown that the reverse - shock can also account for the optical flash ( up to 100 s ) and the gev light - curve peak , but for microphysical different than after that peak . for this hybrid reverse - forward shock model , we find that the x - ray flux of grb afterglow 130427a is accounted mostly by the forward - shock emission , from the tail of the first grb pulse ( 50100 s ) up to 5 ms , excluding the second grb pulse at 100 - 500 s. the reverse - shock may have had a significant contribution to the early x - ray emission , at 500 s 2 ks . both shocks give comparable gev emissions . as shown in fig [ wind ] , the radio emission from the forward - shock is expected to overshine that from the reverse - shock at 30 day ( or somewhat later , if energy injection continues after 1 ms ) , yielding a flat flux @xmath123 mjy until @xmath124 day , when the peak of the synchrotron spectrum falls below 10 ghz . if that flat radio flux is not seen , then the magnetic field parameter @xmath29 of the forward - shock must be decreasing , so that the peak flux of the forward - shock synchrotron spectrum falls - off faster than the @xmath52 expected for @xmath125const . the relative dimness of the radio afterglow suggests that the peak of the synchrotron spectrum has crossed the optical range at 10 ks . an immediate consequence is that the wind - like ambient medium is a factor 20 less dense than the most tenuous wind measured for galactic wr stars . we can not provide a good argument for why grb 130427a s progenitor had such a low mass - loss rate to wind - speed ratio ( @xmath126 ) , but note that , owing to the weak wind , the afterglow remains highly relativistic and travels @xmath127 pc until the last observation epoch ( 50 day ) . for such a large afterglow radius to remain inside the free wr wind ( i.e. within the wind termination shock ) , the grb progenitor must have been embedded in a very tenuous medium , suggesting a supper - bubble blown by preceding supernovae and stellar winds . owing to tenuous ambient medium , the afterglow transverse size , @xmath128 pc , is unusually large , and implies a source apparent diameter of @xmath129 mas , which may be resolved with radio interferometry . if the gev emission of grb afterglow 130427a arises from the forward - shock , then the up - scattering of the synchrotron emission occurred at the onset of the kn regime , where the reduction of the electron scattering cross - section lowers the compton parameter , increases the synchrotron cooling - break frequency , and increases the synchrotron flux above that break ( i.e. in the lat range ) . furthermore , lat must have measured the forward - shock inverse - compton emission at photon energies above a few gev . this work was supported by an award from the laboratory directed research and development programme at the los alamos national laboratory and made use of data supplied by the uk swift science data centre at the university of leicester .
|
+ the complex multiwavelength emission of grb afterglow 130427a ( monitored in the radio up to 10 days , in the optical and x - ray until 50 days , and at gev energies until 1 day ) can be accounted for by a hybrid reverse - forward shock synchrotron model , with inverse - compton emerging only above a few gev .
the high ratio of the early optical to late radio flux requires that the ambient medium is a wind and that the forward - shock synchrotron spectrum peaks in the optical at about 10 ks .
the latter has two consequences : the wind must be very tenuous and the optical emission before 10 ks must arise from the reverse - shock , as suggested also by the bright optical flash that raptor has monitored during the prompt emission phase ( @xmath0 100 s ) .
the vla radio emission is from the reverse - shock , the swift x - ray emission is mostly from the forward - shock , but the both shocks give comparable contributions to the fermi gev emission .
the weak wind implies a large blast - wave radius ( @xmath1 pc ) , which requires a very tenuous circumstellar medium , suggesting that the massive stellar progenitor of grb 130427a resided in a super - bubble .
radiation mechanisms : non - thermal relativistic processes shock waves
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the canonical edwards - anderson ( ea ) model ising spin glasses ( isgs ) in dimension @xmath11 have been the subject of very many numerical studies . there is now consensus supported by analytic arguments that the two archetype models , the isgs on square lattices with near neighbor interactions having distributions which are either gaussian or bimodal ( @xmath0 ) , have zero temperature transitions @xcite . for the gaussian case , where the interaction distribution is continuous and the ground state is unique , there is now also general consensus concerning the low temperature thermodynamic limit ( thl ) behavior and exponents . in the bimodal case there is an `` effectively continuous energy level distribution '' regime coming down from high temperatures and ending with a crossover at a size dependent temperature @xmath2 to a ground state dominated regime @xcite . interpretations differ considerably concerning the critical exponents for the bimodal interaction case . we will give a new analysis of accurate numerical monte carlo data up to size @xmath12 on the bimodal system in the thl and up to @xmath13 on the gaussian system . we use the temperature @xmath14 or the inverse temperature @xmath15 when it is convenient . we first discuss discuss the specific heat using data from the simulations and independent data down to low temperatures and large sizes from refs . @xcite and @xcite . then we analyse the simulation data for other observables to obtain reliable and accurate estimates for the critical exponents of the thl regime , using both the conventional scaling variable @xmath14 and a novel scaling variable compatible with the generic scaling approach for isgs introduced in @xcite , adapted to a situation where @xmath16 . we find that the gaussian model and the bimodal model in the thl are not in the same universality class . the @xmath5d gaussian model is relatively clear - cut . because @xmath17 and the interaction distribution is continuous , there is a unique ground state ( for each sample ) and the low temperature excitation distribution has no gap . the fact that the ground state is unique necessarily implies that for all @xmath3 , as @xmath18 , @xmath19 and the reduced susceptibility @xmath20 . with @xmath14 chosen as the critical scaling variable , the standard thermodynamic limit low temperature critical expressions are @xmath21 and @xmath22 because the critical exponent @xmath4 is strictly zero . the critical behavior of both the observables at low temperature is governed by the single exponent @xmath23 , which is related to the stiffness exponent through @xmath24 . accurate zero temperature domain wall stiffness measurements to large sizes @xcite show that @xmath25 , i.e. @xmath26 . in the @xmath5d bimodal case the situation is complicated by two factors . first , the ground state is not unique but is massively degenerate ; the zero - temperature entropy per spin is @xmath27 @xcite . secondly , the distribution of excited state energy levels is not continuous but increases by steps of @xmath28 ; in particular there is an energy gap @xmath28 between the ground state and the first excited state . one can write @xcite the `` nave '' leading low temperature finite size specific heat expression @xmath29 where @xmath30 , @xmath31 are the sample - averaged entropies of the first excited state and the ground state . setting @xmath32 a crossover temperature can then be defined by @xcite @xmath33 which separates the critical behavior in the low temperature ground state dominated regime ( with @xmath34 ) and a @xmath35 regime where the whole ensemble of higher energy states dominate the thermodynamics @xcite . an explicit phenomenological expression for @xmath2 derived from eq . 5 of ref . @xcite , which is consistent with the raw data points @xcite for @xmath36 is @xmath37 a much simpler droplet - based expression from @xcite is @xmath38 . @xmath2 decreases with increasing @xmath3 because the degeneracy of the excited states increases faster with @xmath3 than that of the ground state . we will assume @xcite that in the @xmath35 thl regime the data can be analysed in the same way as if the energy level distribution were continuous . with this assumption the @xmath35 regime will have `` effectively continuous '' energy level distribution critical exponents with an effective ordering temperature @xmath39 still zero . the ground state dominated regime at @xmath40 is a finite size effect which disappears in the infinite @xmath3 limit . a non - zero @xmath4 is to be expected _ a priori _ for a system with a strong ground state degeneracy , unless each individual ground state is isolated in phase space which is not the case @xcite in the bimodal isg . a droplet analysis of ground state measurements on large sized samples @xcite show that @xmath41 , broadly consistent with a number of finite temperature simulation estimates @xcite . however it has been claimed that in the @xmath35 regime the bimodal isg can be considered to be effectively in the same universality class as the gaussian isg @xcite , meaning that the effective exponents are again @xmath42 and @xmath26 . in view of the basic definition of @xmath4 in terms of the short range limit of the spin - spin correlation function @xmath43 $ ] , this claim is rather surprising . a major difficulty in establishing the limiting @xmath44 $ ] behavior for isgs in dimension @xmath5 @xcite consists in finding an appropriate and reliable extrapolation procedure from simulation data necessarily restricted in size and in temperature because of the need to achieve good thermal equilibration at large sizes . this is a problem that we will address . the simulations were performed using the houdayer cluster method @xcite in combination with the exchange monte carlo @xcite method . in the cluster step we first pick a random site @xmath45 and compute its overlap @xmath46 , where @xmath47 and @xmath48 denote the spin for two different replicas . we then build an equal-@xmath49 cluster along the nearest neighbor interactions and flip all cluster spins in both replicas . we used four replicas which turned out to be remarkably efficacious . on each iteration the replicas are paired at random , then , for each pair , a cluster update is performed , and the usual heat - bath spin update and exchange . for all systems we used @xmath50 . the number of temperatures were more than 250 for the smallest systems starting at @xmath51 . with increasing system size the number of temperatures was decreased and @xmath52 increased . for the largest system ( bimodal @xmath53 with @xmath12 ) @xmath54 temperatures were used with @xmath55 . the exchange rate was always at least @xmath56 for all systems and temperatures . the systems were deemed equilibrated when the average @xmath57 for the systems at @xmath58 appeared stable between runs . the number of equilibration steps increased with system size , for the bimodal @xmath12 this took about @xmath59 steps . after equilibration , at least @xmath60 measurements were made for each sample for all sizes , taking place after every cluster - sweep - exchange step . the usual observables were registered , the energy @xmath61 , the correlation length @xmath62 , the spin overlap moments @xmath63 , @xmath64 . correlations @xmath65 between the energy and some observables @xmath66 were also registered . thermodynamic derivatives could then be evaluated through the usual @xmath67 . error estimates of observables and derivatives were done with the bootstrap method . sizes studied were @xmath68 , @xmath69 , @xmath70 , @xmath71 , @xmath72 , @xmath73 , @xmath74 , @xmath75 , @xmath76 , @xmath77 , @xmath78 for bimodal interactions , and up to @xmath13 for the gaussian interactions , with @xmath79 samples ( @xmath53-interactions ) for all sizes . the size dependence of the ground state energy per spin @xmath80 for the 2d gaussian isg has been shown @xcite to follow the simple critical finite size scaling rule @xmath81 with a @xmath82 consistent with the estimate from ground state domain wall stiffness measurements @xcite . standard scaling arguments @xcite would suggest that the low temperature specific heat should behave as @xmath83 but because of the continuous interaction distribution , in addition to critical excitations there are always single spin excitations . these lead to a term @xmath84 which dominates the gaussian low temperature specific heat as noted by ref . @xcite . specific heat data for the bimodal model were calculated through the present simulations ; data extending to a much lower temperature range and larger sizes have already been measured using the sophisticated pfaffian arithmetic technique by lukic _ @xcite and by thomas _ @xcite , and we are very grateful to be able to quote these results _ in extenso_. the data for the two models are shown , see fig . [ fig1 ] and fig . [ fig2 ] , in the form of plots of the derivative @xmath85 against @xmath86 . this non - conventional form of plot happens to be particularly instructive . a low temperature limit @xmath87 appears as a straight line through the origin with slope @xmath88 , while a low temperature limit of the `` nave '' bimodal ground state dominated form eq . appears as a straight line with intercept @xmath89 and slope @xmath90 . the gaussian data are almost independent of @xmath3 for the whole temperature range . physically this occurs because the specific heat in isgs is predominately a near neighbor effect . the curve tends to a slope @xmath91 corresponding to @xmath92 in the low @xmath14 limit , in agreement with the conclusion of ref . @xcite . for the bimodal model there is first a high temperature and/or high @xmath3 envelope curve corresponding to the effectively continuous @xmath93 regime . in this regime finite size effects are very weak : the specific heat is almost independent of @xmath3 as in the gaussian . the curves for the two models are of similar form but are not identical . in the large @xmath3 , low @xmath14 limit of this envelope curve , the bimodal data as shown in fig . [ fig2 ] indicate @xmath94 in agreement with the conclusions drawn in ref . @xcite based on droplet excitation arguments . for each @xmath3 the data curve peels off the large @xmath3 envelope curve below an @xmath3-dependent temperature which can be identified with the start of the effectively continuous to ground state dominated regime crossover centered at @xmath2 . finally for each @xmath3 in the low temperature range @xmath95 the specific heat links up to the `` nave '' limit of eq . . ( it should be noted that because of the logarithmic derivative , temperature independent @xmath3-dependent factors do not show up in this plot ) . the crossover can be seen to be gentle for small @xmath3 , becoming sharp for large @xmath3 . defining @xmath2 as the location of the maximum positive slope on this plot , the crossover temperatures can be clearly identified and are consistent with @xmath96 . an anomalous limit of the form @xmath97 which has been proposed by some authors @xcite following ref . @xcite is inconsistent with the data in fig . [ fig2 ] for all @xmath3 and @xmath14 ( see also @xcite ) . an intermediate @xmath3 regime where @xmath98 as proposed in ref . @xcite or @xmath99 as proposed in ref . @xcite appear to be valid only for a limited range of @xmath14 and @xmath3 . for isgs with non - zero critical temperatures finite size scaling analyses at and close to the critical temperature are used to estimate critical exponents . for the @xmath5d bimodal isg , because of the crossover to the ground state dominated regime this approach is ruled out and the critical exponents must be estimated using the entirely different strategy of thl measurements . the standard renormalization group theory ( rgt ) scaling variable for models with non - zero ordering temperatures is @xmath100 . this obviously can not be used when @xmath101 ; by convention the scaling variable used in the literature for @xmath5d isgs is the un - normalized temperature @xmath14 . this is only a convention ; it is perfectly legitimate to use other conventions . thus , when considering the canonical @xmath102d ising ferromagnet , baxter @xcite remarks `` when @xmath101 it is more sensible to replace @xmath103 by @xmath104 '' . ( in fact for the particular @xmath102d model scaling without corrections over the entire temperature range follows if a related scaling variable @xmath105 is chosen @xcite ) . below we will introduce another scaling variable appropriate for isgs with @xmath106 , but for the moment we follow this traditional @xmath107 @xmath5d isg convention . the critical exponents are defined through the leading thl expressions for the reduced susceptibility and the second moment correlation length within this convention : @xmath108 and @xmath109 in the limit @xmath110 . for all data which fulfil the condition ( either in the bimodal and gaussian models ) @xmath111 with @xmath112 , observables such as @xmath113 and @xmath114 depend on @xmath14 but not on @xmath3 , and so correspond to the thl infinite size values @xmath115 and @xmath116 . the thl condition defines implicitly a crossover temperature @xmath117 . it turns out that in the bimodal @xmath5d isg the thl limit temperature @xmath117 is always higher than the corresponding crossover temperature to the ground state dominated regime @xmath2 defined above , so the thl data are always well in the effectively continuous regime . the thl data extrapolation to @xmath118 corresponds to estimates for the critical exponents in the successive limits @xmath119 $ ] so in the effectively continuous energy level regime , to be distinguished from the exponents defined taking the successive limits @xmath120 $ ] which would correspond to the `` finite size '' ground state dominated regime . there have been many previous studies having the aim of estimating the critical exponents and in particular @xmath4 for the bimodal model . mcmillan already in 1983 estimated @xmath121 from @xmath122 correlation data on one @xmath123 sample well in the effectively continuous regime @xcite . katzgraber and lee @xcite estimated @xmath124 from @xmath113 data . et al _ @xcite show a plot of @xmath125 against @xmath126 after an extrapolation to infinite @xmath3 using the technique of ref . they state `` fits of this curve lead to values of @xmath4 that are very small , between @xmath127 and @xmath128 , strongly suggestive of @xmath42 '' . however , this type of extrapolation to infinite @xmath3 is delicate , particularly in the bimodal @xmath5d case . in addition , the data displayed by @xcite on a @xmath129 plot extending over five decades on the @xmath130 axis are hard to fit with precision . katzgraber _ et al _ @xcite show a plot of @xmath131 which in principle is equivalent to the ref . @xcite plot but which provides a display much more sensitive to the value of @xmath4 ; they state cautiously `` for all system sizes and temperatures studied @xmath132 is always greater than @xmath133 , although an extrapolation to @xmath42 can not be ruled out '' , so that the possibility of the bimodal and gaussian isgs being in the same universality class `` can not be reliably proven '' . in refs . @xcite it is claimed that the gaussian and bimodal models are in the same universality class , which is surprising as `` the data are not sufficiently precise to provide a precise determination of @xmath4 , being consistent with a small value @xmath134 , including @xmath42 '' . all the estimates quoted so far can be considered to concern the effectively continuous regime . at zero or low temperatures , so in the ground state dominated regime , different sophisticated algorithms lead to the estimates @xmath135 @xcite , and to @xmath136 @xcite . in fig . [ fig3 ] and fig . [ fig4 ] , we show plots of @xmath137 against @xmath138 for the gaussian and bimodal models . these are _ raw _ data points having the high statistical precision of the present measurements . with the conventional definition of the critical exponents through @xmath139 and @xmath140 in the thl regime low-@xmath14 limit , the limiting slope @xmath141 at criticality as @xmath142 is by definition equal to @xmath143 . for the gaussian model the observed tendency of the slope is consistent with the limit of @xmath144 which must be the case for this nondegenerate ground state model . for the bimodal model the observed @xmath145 in the thl regime is not tending to @xmath5 but to a constant limit of @xmath146 . slight overshoots for each @xmath3 in both systems can be ascribed to @xmath113 and @xmath114 not reaching the thl condition at quite the same temperature . as stated above , very similar observations were made in ref . @xcite for the bimodal model . the present results thus confirm unambiguously that for the bimodal isg in the effectively continuous regime @xmath4 is not zero but is @xmath147 . thus the bimodal isg in the effectively continuous thl regime and the gaussian isg are not in the same universality class . it has been shown that in dimension @xmath148 also , gaussian and bimodal isgs are not in the same universality class @xcite , so the breakdown of universality in isgs appears to be general . estimating the exponents @xmath23 or @xmath149 is more difficult than for the exponent @xmath4 . as we have noted above , the standard rgt convention for models with finite temperature ordering is to use the scaling variable @xmath150 , which obviously can not be applied to models with @xmath151 , and for @xmath5d isgs the preferred convention in the literature has been to use the un - normalized scaling variable @xmath107 . in practice this is inefficient as the extrapolations towards the @xmath152 limit in order to estimate the values for the critical exponents are very ambiguous . for instance , when presenting @xmath14-scaled susceptibility data for sizes up to @xmath12 katzgraber _ et al _ @xcite state `` the [ susceptibility ] data for the bimodal case can be extrapolated to any arbitrary value including @xmath153 '' . we will introduce a novel scaling variable suitable for the @xmath5d isgs , applying the same principles as for isgs at higher dimensions @xcite , adapted to @xmath154 ordering : for spin glasses the relevant interaction strength parameter is not @xmath155 but is @xmath156 , so the natural dimensionless parameter is @xmath157 ( or alternatively @xmath158 for bimodal isgs ) . with the standard normalisation @xmath159 the natural inverse `` temperature '' in isgs is @xmath160 , not @xmath15 . this was recognized immediately after the edwards - anderson model was introduced , in high temperature series expansion ( htse ) analyses for isgs including @xmath5d models @xcite , but has since been overlooked in most simulation analyses . it is convenient to choose a scaling variable @xmath161 defined in such a way that @xmath162 at criticality and @xmath163 at infinite temperature . with an isg ordering at a finite inverse temperature @xmath164 , @xmath165 is an appropriate choice @xcite . when @xmath166 as in the @xmath5d isg case , @xmath167 has been used @xcite , but here we will prefer @xmath168 as it turns out to be efficient and the limits are easy to relate to those of the @xmath14 scaling convention . with non - zero @xmath39 the effective exponents at criticality do not depend on the choice of scaling variable ; this is not the case when @xmath169 , but a simple dictionary is given below relating the limiting derivatives for @xmath170 scaling to the exponents for the conventional @xmath14 scaling . the thl htse darboux @xcite format for observables @xmath171 is @xmath172 with @xmath173 in isgs @xcite . the htse isg susceptibility @xmath174 is naturally in this format , so for isg models with @xmath175 the thl susceptibility can be scaled in the wegner @xcite form @xmath176\ ] ] because the correlation function second moment @xmath177 htse is of the form ( see ref . @xcite for the ising ferromagnet ) @xmath178 and the second moment correlation length is defined through @xmath179 with @xmath180 the number of nearest neighbors , for consistency the appropriate correlation length variable for isg scaling is @xmath181 rather than @xmath182 ( whether @xmath183 is zero or not ) . this point has been spelt out in ref . @xcite . examples of applications of the scaling rules outlined here to other specific models ( both ferromagnets and isgs ) have been given elsewhere . a general discussion of ferromagnets and spin glasses is given in ref . @xcite , analyses of @xmath184d ising , xy and heisenberg ferromagnets in ref . @xcite , the @xmath5d ising ferromagnet is analysed in ref . @xcite , @xmath184d ising ferromagnets in @xcite , high dimension ising ferromagnets in ref . @xcite , and the @xmath5d villain fully frustrated model in ref . @xcite . the scaling of the binder cumulant @xmath185 is discussed in the appendix . the @xmath5d simulation data analysis and the extrapolations below are based on the derivatives @xmath186 in the thl regime where these derivatives are independent of @xmath3 and so equal to the infinite size derivatives . an advantage of the @xmath5d models is that in contrast to @xmath187 for the models with non - zero @xmath183 , for the @xmath5d isgs with @xmath188 there is no uncertainty in the definition of @xmath189 related to an uncertainty in the value of the ordering temperature . once the @xmath190 limits for the various derivatives have been estimated by extrapolation of the thl data for finite @xmath3 , there is a simple dictionary for translating into terms of the conventional @xmath14 scaling critical exponents @xmath23 and @xmath4 defined above : [ derivs ] @xmath191 the four derivatives of eq . are shown in figs . [ fig5 ] to [ fig12 ] . in contrast to the derivatives in which @xmath14 is used as the scaling variable , each derivative can be extrapolated in a fairly unambiguous manner to criticality , and always has an exact finite value at infinite temperature @xmath192 . the exact infinite temperature limits from the general high temperature scaling expansion expressions @xcite applied to scaling with @xmath170 are ( when @xmath193 ) : [ exact ] @xmath194 the extrapolation method is outlined in appendix ii . with @xmath144 and assuming @xmath195 @xcite , the predicted gaussian critical limits ( when @xmath196 ) for the derivatives are [ gderivs ] @xmath197 from the fitted thl data extrapolations ( see appendix ii ) the estimated gaussian critical limit values are @xmath198 , @xmath199 , @xmath200 , @xmath201 respectively . these values are fully consistent with the list above , validating the approach and the extrapolation procedure that we have used . for the bimodal data , the extrapolated thl limits ( when @xmath196 ) from the figures ( see appendix ii ) give estimates [ jderivs ] @xmath202 these are all significantly different from the gaussian values , confirming non - universality . when translated into the @xmath14 scaling convention , the 2d bimodal critical exponents from these measurements are @xmath203 and @xmath204 ( so @xmath205 . not only is the thl bimodal exponent @xmath4 different from the gaussian value but the @xmath23 value is different also . simulation data are presented for the gaussian and bimodal interaction distribution ising spin glasses in dimension two , which are known to order only at zero temperature . in order to facilitate extrapolations to zero temperature , a temperature scaling variable @xmath206 is introduced in addition to the traditional @xmath5d isg scaling variable @xmath107 . the gaussian simulation data are completely consistent with the well established critical behavior for this model , with exponents @xmath1 and @xmath207 @xcite . the bimodal specific heat simulation data supplemented by data from lukic _ et al _ @xcite and from thomas _ et al _ @xcite show clear crossovers from an effectively continuous energy level thermodynamic limit regime to a finite size ground state dominated regime at size dependent temperatures @xmath208 ( see ref . @xcite ) . the extrapolated thl simulation results tend to critical limits which correspond consistently to @xmath209 and @xmath204 , clearly different from the gaussian values . this demonstrates that the standard universality rules do not hold for @xmath5d isg models . in dimension @xmath148 also bimodal and gaussian isgs have been shown not to be in the same universality class either @xcite , strongly suggesting a lack of universality for isgs in each dimension ( presumably up to the upper critical dimension ) . against @xmath14 . sizes @xmath210 , @xmath75 , @xmath74 , @xmath73 , @xmath72 , @xmath71 top to bottom in the dip . curve : extrapolation.,width=288 ] [ fig1 ] against @xmath14 . full points : simulation data @xmath211 , @xmath75 , @xmath73 , @xmath71 , @xmath70 ( black , green , red , pink , cyan ) top to bottom . open points : pfaffian data ; red polygons @xmath212 , blue right triangles @xmath213 , black left triangles @xmath12 , brown diamonds @xmath13 ( all data from ref . @xcite ) , green down triangles @xmath214 , red up triangles @xmath215 , pink circles @xmath216 , all data fom ref . dashed diagonal red line @xmath217 , green diagonal line @xmath218.,width=288 ] [ fig2 ] against @xmath219 for @xmath210 , @xmath75 , @xmath74 , @xmath73 left to right . in this and all following figures both gaussian and bimodal , the color coding is : black , pink , red , blue , green , brown , cyan , olive for @xmath220 , @xmath77 , @xmath76 , @xmath75 , @xmath74 , @xmath73 , @xmath72 , @xmath71.,width=288 ] [ fig3 ] against @xmath219 for @xmath220 , @xmath77 , @xmath76 , @xmath75 , @xmath74 , @xmath73 left to right . same color coding as in fig . [ fig3].,width=288 ] [ fig4 ] against @xmath170 . sizes @xmath210 , @xmath75 , @xmath74 , @xmath73 , @xmath72 left to right . same color coding as in fig . [ fig3 ] . dashed line : extrapolation . red arrow : exact infinite temperature value . blue arrow : gaussian critical value.,width=288 ] [ fig5 ] against @xmath170 . sizes @xmath220 , @xmath77 , @xmath76 , @xmath75 , @xmath74 , @xmath73 , @xmath72 left to right . same color coding as in fig . [ fig3 ] . dashed line : extrapolation . red arrow : exact infinite temperature value . , width=288 ] [ fig6 ] against @xmath170 . sizes @xmath210 , @xmath75 , @xmath74 , @xmath73 left to right . same color coding as in fig . [ fig3 ] . dashed line : extrapolation . red arrow : exact infinite temperature value . blue arrow : gaussian critical value.,width=288 ] [ fig7 ] against @xmath170 . sizes @xmath220 , @xmath77 , @xmath76 , @xmath75 , @xmath74 , @xmath73 , @xmath72 left to right . same color coding as in fig . [ fig3 ] . dashed line : extrapolation . red arrow : exact infinite temperature value . , width=288 ] [ fig8 ] against @xmath170 , where @xmath221 is the binder cumulant . @xmath222 , @xmath75 , @xmath74 , @xmath73 left to right . same color coding as in fig . line : extrapolation . blue arrow : gaussian critical value . , width=288 ] [ fig9 ] against @xmath170 , where @xmath221 is the binder cumulant . sizes @xmath223 , @xmath77 , @xmath76 , @xmath75 , @xmath74 , @xmath73 left to right . same color coding as in fig . [ fig3 ] . line : , width=288 ] [ fig10 ] against @xmath170 . sizes @xmath210 , @xmath75 , @xmath74 , @xmath73 left to right . same color coding as in fig . [ fig3 ] . dashed line : extrapolation . red arrow : exact infinite temperature value . blue arrow : gaussian critical value.,width=288 ] [ fig11 ] against @xmath170 . sizes @xmath220 , @xmath77 , @xmath76 , @xmath75 , @xmath74 , @xmath73 , @xmath72 left to right . same color coding as in fig . [ fig3 ] . dashed line : extrapolation . red arrow : exact infinite temperature value . , width=288 ] [ fig12 ] against @xmath170 . sizes @xmath222 , @xmath75 , @xmath74 , @xmath73 , @xmath72 , @xmath71 top to bottom . same color coding as in fig . @xmath221 is the binder cumulant . line slope @xmath224 . , width=288 ] [ fig13 ] against @xmath170 . sizes @xmath223 , @xmath77 , @xmath76 , @xmath75 , @xmath74 , @xmath73 , @xmath72 , @xmath71 top to bottom . same color coding as in fig . @xmath221 is the binder cumulant . line slope @xmath225 . , width=288 ] [ fig14 ] the ferromagnetic binder cumulant has been extensively exploited in the fss limit regime very close to criticality for its properties as a dimensionless observable . in addition its thl properties can also be studied . in ising ferromagnets , the critical exponent for the second field derivative of the susceptibility @xmath226 ( also called the non - linear susceptibility ) , is @xcite @xmath227 the non - linear susceptibility @xmath226 is directly related to the binder cumulant , @xcite eq . 10.2 , through @xmath228 as @xmath229 scales with the critical exponent @xmath230 , the normalized binder cumulant @xmath231 scales with the thl regime critical exponent @xmath232 . in any @xmath233 ising system the infinite temperature ( independent spin ) limit for the binder cumulant is @xmath234 where @xmath235 is the number of spins ; @xmath236 for a hypercubic lattice . so @xmath231 has an infinite temperature limit which is strictly @xmath102 , and a large @xmath3 critical limit ( with corrections as for the other observables ) : @xmath237 exactly the same argument can be transposed to isgs ( see ref . @xcite for @xmath226 in isgs ) . in the particular case of a @xmath5d isg model with @xmath170 scaling , the critical value for the derivative @xmath238 of the binder cumulant thl data extrapolated to @xmath239 is @xmath240 where @xmath23 is once again the correlation length critical exponent in the @xmath14 scaling convention . the binder cumulant data plotted in the eq . form are shown for the two models in figs . [ fig13 ] and [ fig14 ] . the thl envelope curves can be seen by inspection . the derivatives of these curves have already been shown in figs . [ fig9 ] and [ fig10 ] . it has been suggested that if two models have the same function when @xmath241 is plotted against @xmath242 , it is a proof of universality . however , because both @xmath243 and @xmath62 are controlled by just the same exponent @xmath23 this is questionable . as the data sets do not extend to infinite size , to estimate the critical @xmath239 limit values from the thl derivative data in figs . [ fig5 ] to [ fig12 ] , an extrapolation must be made . there is no definitive method to extrapolate so as to be sure to obtain exact values of the critical exponents , though data to still larger sizes would make the task easier . the most economical choice for extrapolation is to assume that the thl derivative data continue to evolve smoothly and regularly when an extrapolation is made towards @xmath239 through the smaller @xmath170 region where no thl data are for the moment available . to do this , for each derivative observable @xmath145 with @xmath244 we collect together the thl data points for all the sizes @xmath3 up to @xmath245 and make standard polynomial fits with @xmath184 or @xmath148 terms @xmath246 or @xmath247 ( in fits with larger numbers of terms the fit parameter values become unstable ) . assuming that each polynomial fit curve extended to @xmath248 is a good approximation to the true behavior , each @xmath249 is an estimate for the critical limit value . the @xmath249 values for @xmath184 or @xmath148 parameter fits turn out to be similar . in figs . [ fig15 ] , [ fig16 ] , [ fig17 ] and [ fig18 ] the data and fits are shown for @xmath250 , @xmath251 and @xmath252 and @xmath253 for both gaussian and bimodal models . the fits are automatic , so this procedure is objective and we assume that it is optimal for the available data . all the gaussian extrapolated critical values estimated in this way are close to those expected assuming the published exponents , @xmath42 and @xmath207 @xcite . this implies that the estimated bimodal critical values are also close to the true critical limits . we are very grateful to olivier martin and to alan middleton who generously allow us access to the specific heat data of ref . @xcite and of ref . @xcite respectively . we would like to thank alex hartmann for very helpful suggestions . the computations were performed on resources provided by the swedish national infrastructure for computing ( snic ) at the high performance computing center north ( hpc2n ) and chalmers centre for computational science and engineering ( c3se ) . 99 a. k. hartmann and a. p. young , phys . b * 64 * , 18404 ( 2001 ) . m. ohzeki and h. nishimori , j. phys . a : math . theor . * 42 * , 332001 ( 2009 ) . t. jrg , j. lukic , e. marinari , o. c. martin , phys . * 96 * , 237205 ( 2006 ) . j. lukic , a. galluccio , e. marinari , o. c. martin and g. rinaldi , phys . 92 * , 117202 ( 2004 ) . c. k. thomas , d. a. huse , and a. a. middleton , phys . lett . * 107 * , 047203 ( 2011 ) . i. a. campbell , k. hukushima , and h. takayama , phys . * 97 * , 117202 ( 2006 ) . h. rieger , l. santen , u. blasum , m. diehl , m. jnger , and g. rinaldi , j. phys . a * 29 * , 3939 ( 1996 ) ; * 30 * , 8795(e ) ( 1997 ) . a. c. carter , a. j. bray , and m. a. moore , phys . lett . * 88 * , 077201 ( 2002 ) . c. amoruso , e. marinari , o. c. martin , and a. pagnani , phys . lett . * 91 * , 087201 ( 2003 ) . j. houdayer and a. k. hartmann , phys . b * 70 * , 014418 ( 2004 ) . a. k. hartmann , a. j. bray , a. c. carter , m. a. moore , and a. p. young , phys . b * 66 * , 224401 ( 2002 ) j. poulter and j. a. blackman , phys . rev . b * 72 * , 104422 ( 2005 ) . k. t. creighton and a. a. middleton phys . rev . b * 77 * , 144418 ( 2008 ) . h. g. katzgraber , lik wee lee , and i. a. campbell , phys . rev . b * 75 * , 014412 ( 2007 ) . r. fisch , j. stat . phys . * 128 * , 1113 ( 2007 ) . a. k. hartmann , phys . b * 77 * , 144418 ( 2008 ) . w. l. mcmillan , phys . b * 28 * , 5216 ( 1983 ) . j. houdayer , eur . j. b * 22 * , 479 ( 2001 ) . h. g. katzgraber and lik wee lee , phys . b * 71 * , 134404 ( 2005 ) . k. hukushima and k. nemoto , j. phys . japan * 65 * , 1604 ( 1996 ) . i. a. campbell , a. k. hartmann and h. g. katzgraber , phys . b * 70 * 054429 ( 2004 ) . wang , phys . e * 72 * , 036706 ( 2005 ) . wang and r. h. swendsen , phys . b * 38 * , 4840 ( 1988 ) . l. saul and m. kardar , phys . e * 48 * , r3221 ( 1993 ) . w. atisattapong and j. poulter , new journal of physics * 10 * 09312 ( 2008 ) . r. j. baxter `` exactly solved models in statistical mechanics '' academic press ( london ) ( 1982 ) . h. g. katzgraber , i. a. campbell , and a. k. hartmann , phys . b * 78 * , 184409 ( 2008 ) . i. a. campbell and p. h. lundow , phys . rev . b * 83 * , 014411 ( 2011 ) . s. caracciolo _ et al . _ , lett . * 74 * , 2969 ( 1995 ) . m. palassini and s. caracciolo , phys . lett . * 82 * , 5128 ( 1999 ) . m. baity - jesi _ et al . b * 88 * , 224416 ( 2013 ) . f. parisen toldin , a. pelissetto , and e. vicari , phys . e * 82 * , 021106 ( 2010 ) . f. parisen toldin , a. pelissetto , and e. vicari , phys . e * 84 * , 051116 ( 2011 ) . p. h. lundow and i. a. campbell , phys . e * 91 * , 042121 ( 2015 ) , physica a * 434 * , 181 ( 2015 ) . r. r. p. singh and s. chakraverty , phys . b * 36 * , 559 ( 1987 ) . l. klein , j. adler , a. aharony , a. b. harris and y. meir , phys . b * 43 * , 11249 ( 1991 ) . d. daboul , i. chang , and a. aharony , eur . j. b * 41 * , 231 ( 2004 ) . j. g. darboux , j. math . pures appl . * 4 * , 377 ( 1878 ) . f. j. wegner , phys . rev . b * 5 * , 4529 ( 1972 ) . p. butera and m. comi , phys . b * 65 * , 144431 ( 2002 ) . i. a. campbell , k. hukushima , and h. takayama , phys . b * 76 * , 134421 ( 2007 ) . i. a. campbell and p. butera , phys . rev . b * 78 * , 024435 ( 2008 ) . p. h. lundow and i. a. campbell , phys . b * 83 * , 184408 ( 2011 ) . b. berche , c. chatelain , c. dhall , r. kenna , r. low , and j. c. walter , j. stat . p11010 ( 2008 ) . v. privman , p. c. hohenberg and a. aharony , `` universal critical - point amplitude relations '' , in `` phase transitions and critical phenomena '' ( academic , ny , 1991 ) , eds . c. domb and j.l . lebowitz , * 14 * , 1 .
|
a new analysis is given of numerical simulation data on the archetype square lattice ising spin glasses ( isg ) with a bimodal ( @xmath0 ) and gaussian interaction distributions .
it is well established that the ordering temperature of both models is zero .
the gaussian has a non - degenerate ground state so exponent @xmath1 and it has a continuous distribution of energy levels . for the bimodal model , above a size - dependent cross - over temperature @xmath2
there is a regime of effectively continuous energy levels ; below @xmath2 there is a distinct regime dominated by the highly degenerate ground state plus an energy gap to the excited states .
@xmath2 tends to zero at very large @xmath3 leaving only the effectively continuous regime in the thermodynamic limit .
we show that in this regime the critical exponent @xmath4 is not zero , so the effectively continuous regime @xmath5d bimodal isg is not in the same universality class as the @xmath5d gaussian isg .
the simulation data on both models are analyzed using a scaling variable @xmath6 suitable for zero temperature transition isgs , together with appropriate scaling expressions .
accurate simulation estimates can be obtained for the temperature dependence of the thermodynamic limit reduced susceptibility @xmath7 and second moment correlation length @xmath8 over the entire range of temperature from zero to infinity . the gaussian critical exponent from the simulations @xmath9 is in full agreement with
the well established value from the literature .
the bimodal exponent from the thermodynamic limit regime analysis is @xmath10 , once again different from the gaussian value .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
we propose a feature allocation model [ broderick , jordan and pitman ( @xcite ) ] to describe tumor heterogeneity using next - generation sequencing ( ngs ) data . we use a variation of the indian buffet process ( ibp ) [ @xcite ] . the feature allocation in our model is latent . that is , the features are not directly observed . we record point mutations as single nucleotide variants ( snvs ) , each of which is defined as a dna locus that possesses a variant sequence from that on the reference human genome . we use the feature allocation model to describe unobserved haplotypes , defined as a collection of single nucleotide variants ( snvs ) scaffolded on a homologous genome . in a tumor sample , having more than two haplotypes is evidence of heterogeneous cell subpopulations with a distinct genome . this is the case because humans are diploid and we would therefore only observe up to two haplotypes if all cells in a tumor sample were genetically homogeneous . in the proposed application of feature allocation models to inference for tumor heterogeneity , the haplotypes are the features and the snvs are the experimental units that select the features . the number of features is unknown . each tumor sample is composed of an unknown proportion of each of these haplotypes . the top level sampling model for the observed snv counts is then defined as binomial sampling with a proportion for each snv that is implied by this composition . in summary , we solve a deconvolution problem to explain the observed snv frequencies for each sample by compositions of latent haplotypes . heterogeneity in cancer tissue has been hypothesized over the past few decades [ @xcite ] and has been demonstrated elegantly using ngs technology over the past few years [ @xcite ] . genetic variation in a tumor occurs due to evolutionary processes that drive tumor progression . specifically , tumors include distinct clonal subpopulations of cells that arise stochastically by a sequence of randomly acquired mutations . substantial genetic heterogeneity between tumors ( inter - tumor ) or within a tumor ( intra - tumor ) can be explained by differences in clonal subpopulations and varying proportions of those subpopulations [ @xcite ] . for example , @xcite reported clonal genomic heterogeneity in breast cancers . data derived from ngs experiments include snvs , small indels and copy number variations [ @xcite ] . many researchers use snv data to investigate genes and genomic regions related to cancer phenotypes [ @xcite ] . in this paper , we utilize whole - genome sequencing data measuring variant allele fractions ( vafs ) at snvs to understand tumor heterogeneity by proposing inference on how haplotypes may be distributed within a tumor . in an ngs experiment , millions of short dna reads are generated and are then aligned to the reference genome . at certain positions of the genome , some or all of the mapped reads will show a sequence different from what is in the reference genome . at each genomic locus , the proportion of short reads bearing a variant sequence is called the vaf . if the vaf at a locus is nonzero , an snv may be `` called '' at that locus , based on statistical inference [ @xcite ] . the raw experimental data comprises the total number of reads ( @xmath0 ) that are mapped to the locus and the number of those reads ( @xmath1 ) indicating a variation from the reference sequence . then vaf @xmath2 . if a tumor sample is homogeneous , that is , having a single clone , the vaf values of all the snvs should be close to 0 , 0.5 or 1 , reflecting the three possible homozygous and heterozygous alleles ( i.e. , aa , ab , bb ) at any snv . different vaf values imply heterogeneity of the cellular genome in the tumor sample ( see figure [ fig : data_1 ] for an example ) . we propose to study inference to deconvolute the vafs from multiple snvs and back out the latent haplotypes . @c@ + + + we propose a bayesian feature allocation model to characterize such cellular heterogeneity in a way that explains the observed ngs data . we construct a matrix of binary features ( equivalently , haplotypes ) as shown in figure [ fig : demo_z ] . in the figure , columns correspond to haplotypes and rows correspond to snvs . we define haplotype @xmath3 by a binary vector @xmath4 of indicators of whether ( @xmath5 ) or not ( @xmath6 ) a variant sequence is observed at the snv @xmath7 . here we view snv as a genetic locus on which either a variant or reference dna sequence could be observed . figure [ fig : demo_z ] illustrates the definition of five haplotypes ( @xmath8 , columns ) with @xmath9 snvs ( rows ) . in the figure , black ( white ) indicates @xmath5 ( @xmath6 ) . for example , snv 1 in figure [ fig : demo_z ] possesses a variant sequence in the two haplotypes @xmath10 and @xmath11 . on the other hand , snv 9 possesses variant sequences in four haplotypes : @xmath12 and @xmath13 . a prior probability model on such a binary matrix @xmath14 $ ] is known as a feature allocation model . here , we assume that @xmath15 is unknown and place a prior on @xmath15 . assuming that samples are composed of different proportions of @xmath15 haplotypes , we aim to fit the observed vafs of the snvs to infer these proportions . for example , we may observe that one sample is dominated by haplotypes @xmath16 and @xmath13 , while another is dominated by haplotypes @xmath17 and @xmath18 . if the samples are from the same tumor , the differences in haplotypic compositions are evidence of _ intra - tumor _ heterogeneity ; on the other hand , differences in samples from different tumors imply _ inter - tumor _ heterogeneity . therefore , the proposed inference provides a unified framework to address inference for both biological concepts . importantly , the characterization of haplotypes is based on selected snvs only . otherwise inference for heterogeneity between tumors in different patients would not be biologically meaningful , as cellular genomes and haplotypes are not expected to be shared across patients . however , for tumors in the same class of disease , snvs in local disease - related genomic regions may be common to all or some of the tumors , thereby allowing for the proposed inference . there are currently few approaches that address the problem of tumor heterogeneity . @xcite and @xcite recognized that a tumor sample is a mixture of normal cells and tumor cells , and developed a method to estimate tumor purity levels for paired tumor - normal tissue samples using dna sequencing data . none of the two methods considers more than two samples or unpaired samples . purbayes [ @xcite ] accounts for intra - tumor heterogeneity , but it does not provide inference on the subpopulation configurations as inference on the latent matrix @xmath19 under the proposed model . pyclone [ @xcite ] , a recently published method , proposes inference to cluster snvs with different vafs . an underlying assumption of pyclone is that snvs can be arranged in clusters that inform about subclones . a key component of pyclone is the use of clustering models such as the dirichlet process for inference on these clusters . while such clusters are informative about heterogeneity , inference that is provided by pyclone is not meant to identify subclones or haplotypes . the primary aim of pyclone is inference on mutation clusters , defined as a group of snvs with similar variant allele fractions . in contrast , our proposed feature allocation model explicitly models the haplotypic genomes of subclones , allowing overlapping snvs shared between different subclones . we do not use nonoverlapping snv clusters as the building block for subclones . that is , instead of first estimating the snv clusters and then constructing subclones based on clusters , we directly infer the subclonal structure based on haplotypes . we show in later examples the distinction between pyclone and our proposed method . the remainder of the paper is organized as follows : section [ sec : model_sel ] describes the proposed bayesian feature allocation model and a model selection criterion to select the number of subclones . section [ sec : simulation ] describes simulation studies . sections [ sec : example ] and [ sec : lung_cancer ] report data analyses of in - house data sets to illustrate inter - tumor heterogeneity and intra - tumor heterogeneity , respectively . the last section concludes with a final discussion . and @xmath20 . let @xmath21 denote an @xmath22 matrix of count data from an ngs genome sequencing experiment , with @xmath23 denoting the number of reads that bear a variant sequence at the location of snv @xmath7 for tissue sample @xmath24 , @xmath25 and @xmath26 . we assume a binomial sampling model . let @xmath27 denote the total number of reads in sample @xmath24 that are mapped to the genomic location of snv @xmath7 . we assume @xmath28 in figure [ fig : demo_bin ] , @xmath20 and @xmath29 . we do not model @xmath27 , that is , we treat @xmath27 as fixed , and only consider a sampling model for @xmath23 conditional on @xmath27 ( modeling @xmath27 would not contribute any information on tumor heterogeneity based on snvs ) . conditional on @xmath27 , the observed counts @xmath23 are independent across @xmath7 and @xmath24 . the model in ( [ eq : like ] ) is illustrated in figure [ fig : demo_bin ] . we build a prior probability model for @xmath30 in two steps , using the notion that each sample is composed of a mixture of different haplotypes . and each haplotype , in turn , is characterized by the haplotypes consisting of the snvs . let @xmath31 denote the proportion of haplotype @xmath3 in sample @xmath24 and let @xmath32 denote a latent indicator of the event that snv @xmath7 bears a variant sequence for haplotype @xmath3 . note that @xmath33 corresponds to a black block in figure [ fig : demo_z ] . then @xmath30 is written as a sum over @xmath15 latent haplotypes @xmath34 the construction of the haplotypes , including the number of haplotypes , @xmath15 , and the indicators @xmath35 are latent . the key term , @xmath36 , indirectly infers haplotypes by explaining @xmath30 as arising from sample @xmath24 being composed of a mix of hypothetical haplotypes which do ( @xmath5 ) or do not ( @xmath6 ) include a mutation for snv @xmath7 . the indicators @xmath35 are collected in a @xmath37 binary matrix @xmath38 . the number of latent haplotypes , @xmath15 , is unknown . conditional on @xmath15 , the binary matrix @xmath38 describes @xmath15 latent tumor haplotypes present in the observed samples . joint inference on @xmath15 , @xmath39 and @xmath40 explains tumor heterogeneity . in addition , we introduce a background haplotype , labeled as haplotype @xmath10 , which includes all snvs . the background haplotype accounts for experimental noise and haplotypes that appear with negligible abundance . specifically , @xmath41 in ( [ eq : model_p ] ) relates to this background haplotype , with @xmath42 being the relative frequency of observing a mutation at an snv due to noise and artifact ( we assume equal frequency for all snvs ) and @xmath43 being the proportion in sample @xmath24 . the prior on @xmath43 is defined later . for @xmath42 , we assume @xmath44 with @xmath45 to inform a small @xmath42 value a priori . we start the prior construction with a prior for the number of haplotypes , @xmath15 . we consider a geometric distribution , @xmath46 where @xmath47 . conditional on @xmath15 , we use a feature allocation model for a binary matrix @xmath19 . we first define the model for any given @xmath15 and start with feature - specific selection probabilities , @xmath48 let @xmath49 . the selection probabilities are used to define @xmath50 as @xmath51 where @xmath52 is the number of snvs bearing variant sequences for haplotype @xmath3 . a limit of the model , as @xmath53 , becomes a constructive definition of the indian buffet process ( ibp ) [ @xcite ] . the model is symmetric with respect to arbitrary indexing of the snvs , simply because of the symmetry in ( [ eq : cond_z ] ) and ( [ eq : p_mu ] ) . note that @xmath54 is possible with positive prior probability . next , we consider a prior distribution for abundances associated with the haplotypes defined by @xmath19 . the haplotypes are common for all tumor samples , but the relative weights in the composition ( [ eq : model_p ] ) are different across tissue samples . we assume dirichlet priors for the relative weights @xmath31 , defined as follows . let @xmath55 denote an ( unscaled ) abundance level of haplotype @xmath3 in tissue sample @xmath24 . we assume @xmath56 for @xmath57 and @xmath58 . we then define @xmath59 as the relative weight of haplotype @xmath3 in sample @xmath24 . this is equivalent to @xmath60 for @xmath61 , where @xmath62 . recall the binomial sampling likelihood ( [ eq : like ] ) with success probability , @xmath30 . given @xmath15 , @xmath19 and @xmath63 , we define @xmath30 in ( [ eq : model_p ] ) . in words , @xmath30 is determined by @xmath15 , @xmath19 and @xmath40 with the earlier describing the latent haplotypes and the latter specifying the relative abundance of each haplotype in sample @xmath24 . let @xmath64 , where @xmath65 . markov chain monte carlo ( mcmc ) posterior simulation proceeds by sequentially drawing random numbers for the parameters in @xmath66 . given @xmath15 , such mcmc simulation is straightforward . specifically , gibbs sampling transition probabilities are used to update @xmath35 , and metropolis hastings transition probabilities are used to update @xmath67 and @xmath42 . it is possible to improve the mixing of the markov chain by updating all columns in row @xmath7 of the matrix @xmath19 jointly by means of a metropolis hastings transition probability that proposes changes in the entire row vector @xmath68 . the construction of transition probabilities that involve a change of @xmath15 is more challenging , since the dimension of @xmath19 and @xmath69 changes as @xmath15 varies . we use a reversible jump ( rj ) mcmc algorithm for posterior simulation [ @xcite ] . we first define a proposal distribution @xmath70 for @xmath15 , and then introduce a proposal distribution @xmath71 for @xmath66 conditional on the proposed @xmath72 . the latter potentially involves a change in dimension of the parameter vector . we found that high posterior correlation of @xmath19 and @xmath63 conditional on @xmath15 greatly complicated the construction of a practicable rj scheme . to overcome this , we use an approach similar to @xcite . we split the data into a minimal training set @xmath73 with @xmath74 , @xmath75 , and a test data set , @xmath76 with @xmath77 etc . in the implementation we use @xmath78 generated from @xmath79 . let @xmath80 denote the posterior distribution under @xmath15 using the training sample . we use @xmath81 in two instances . first , we replace the original prior @xmath82 by @xmath83 and , second , we also use it as proposal distribution @xmath84 . the test data is then used to evaluate the acceptance probability . the strategy can be characterized as model comparison by fractional bayes factors [ @xcite ] and is related to a similar approach proposed in @xcite for model comparison with intrinsic bayes factors . both are originally proposed for model comparison with noninformative priors , but can be modified to facilitate mcmc across models as we need it here . [ cols="^,^ " , ] tumors are heterogeneous tissues . the traditional way to identify this heterogeneity has been to sequence multiple samples from the tumor . using such data to study the coexistence of genetically different subpopulations across tumors and within a tumor can shed light on cancer development . identifying subpopulations within a tumor can lead to clinically important insights . for example , @xcite found that a chemotherapy affects subclonal heterogeneity in chronic lymphocytic leukemia . they also observed that the presence of a certain subpopulation may adversely affect clinical outcome . we have proposed a model - based approach based on a feature allocation model . the feature allocation model allows us to impute inference about different components of tumor tissues based on ngs data . the identified components are not necessarily unique because there might be other possible solutions which can lead to the same hypothetical mutation frequencies . instead of reporting a single solution , the proposed approach provides a full probabilistic description of all possible solutions as a coherent posterior probability model over @xmath15 , @xmath19 and @xmath63 . a number of extensions are possible for the present model . first , the number of snvs examined in this paper was relatively limited ( about 100 ) , although the total number of snvs that were found in the whole exome of a tissue is on the order of about 50,000 . other than computational complexity , there is no bar in principle on expanding the model to analyze the whole snv complement of the exome . it could also be instructive to quantify the cellular diversity of the tumor based on findings from various regions of the exome . another important extension of the model is in the basic representation of subclones and haplotypes . the current model uses a binary matrix to record whether a variant sequence for an snv is present or absent in a haplotype . a variation of the model could instead record for each subclone whether an snv is absent ( @xmath6 ) , heterozygous ( @xmath5 ) or homozygous ( @xmath85 ) . that is , @xmath19 would become a trinary matrix . other extensions of the model are to consider each snv position to have four possible bases , @xmath86 , to introduce dependence among mutations or to formally model the noise in variant calling . each of these extensions is currently in development . for example , incorporating explicit error probabilities in variant calls is possible . similar to our previous work [ @xcite ] , we could replace the binomial likelihood ( [ eq : like ] ) in the proposed model with a bernoulli likelihood , for each read , where the probability associated with a read depends on quality scores of base calling and read mapping . we will consider this extension as future work . tumor genome sequencing projects have typically looked for specific genes to be mutated or not . the inherent assumption here , so far unproven , is that the overall effect of carcinogenesis could be explained by a handful of changes in a small number of genes . our model takes the opposite approach and allows us to examine the whole genome ( or exome ) and , by considering vaf patterns , to construct reasonable models for the tissue . we believe this holistic approach to the analysis might provide more robust conclusions and biomarkers than the gene - by - gene approach . church , d. m. , schneider , v. a. , graves , t. , auger , k. , cunningham , f. , bouk , n. , chen , h .- c . , agarwala , r. , mclaren , w. m. , ritchie , g. r. , _ et al . _ ( 2011 ) . modernizing reference genome assemblies . _ plos biology _ * 9 * , 7 , e1001091 . gerlinger , m. , rowan , a. , horswell , s. , larkin , j. , endesfelder , d. , gronroos , e. , martinez , p. , matthews , n. , stewart , a. , tarpey , p. , varela , i. , phillimore , b. , begum , s. , mcdonald , n. , butler , a. , jones , d. , raine , k. , latimer , c. , santos , c. , nohadani , m. , eklund , a. , spencer - dene , b. , clark , g. , pickering , l. , stamp , g. , gore , m. , szallasi , z. , downward , j. , futreal , p. , and swanton , c. ( 2012 ) . intratumor heterogeneity and branched evolution revealed by multiregion sequencing . _ n engl j med . _ * 366 * , 10 , 883892 . ji , y. , xu , y. , zhang , q. , tsui , k .- w . , yuan , y. , norris jr , c. , liang , s. , and liang , h. ( 2011 ) . bm - map : bayesian mapping of multireads for next - generation sequencing data . _ biometrics _ * 67 * , 4 , 12151224 . kanehisa , m. , goto , s. , furumichi , m. , tanabe , m. , and hirakawa , m. ( 2010 ) . for representation and analysis of molecular networks involving diseases and drugs . _ nucleic acids research _ * 38 * , suppl 1 , d355d360 . landau , d. , carter , s. , stojanov , p. , mckenna , a. , stevenson , k. , lawrence , m. , sougnez , c. , stewart , c. , sivachenko , a. , wang , l. , wan , y. , zhang , w. , shukla , s. , vartanov , a. , fernandes , s. , saksena , g. , cibulskis , k. , tesar , b. , gabriel , s. , hacohen , n. , meyerson , m. , lander , e. , neuberg , d. , brown , j. , getz , g. , and wu , c. ( 2013 ) . evolution and impact of subclonal mutations in chronic lymphocytic leukemia . _ cell _ * 152 * , 4 , 714726 . li , h. , handsaker , b. , wysoker , a. , fennell , t. , ruan , j. , homer , n. , marth , g. , abecasis , g. , durbin , r. , _ et al . _ the sequence alignment / map format and samtools . _ bioinformatics _ * 25 * , 16 , 20782079 . mckenna , a. , hanna , m. , banks , e. , sivachenko , a. , cibulskis , k. , kernytsky , a. , garimella , k. , altshuler , d. , gabriel , s. , daly , m. , _ et al . _ the genome analysis toolkit : a mapreduce framework for analyzing next - generation dna sequencing data . _ genome research _ * 20 * , 9 , 12971303 . navin , n. , krasnitz , a. , rodgers , l. , cook , k. , meth , j. , kendall , j. , riggs , m. , eberling , y. , troge , j. , grubor , v. , _ et al . _ ( 2010 ) . inferring tumor progression from genomic heterogeneity . _ genome research _ * 20 * , 1 , 6880 . roth , a. , khattra , j. , yap , d. , wan , a. , laks , e. , biele , j. , ha , g. , aparicio , s. , bouchard - ct , a. , and shah , s. p. ( 2014 ) . pyclone : statistical inference of clonal population structure in cancer . _ nature methods _ . russnes , h. g. , navin , n. , hicks , j. , and borresen - dale , a .- insight into the heterogeneity of breast cancer through next - generation sequencing . _ the journal of clinical investigation _ * 121 * , 10 , 38103818 . su , x. , zhang , l. , zhang , j. , meric - bernstam , f. , and weinstein , j. n. ( 2012 ) . purityest : estimating purity of human tumor samples using next - generation sequencing data . _ bioinformatics _ * 28 * , 17 , 22652266 . teh , y. w. , grr , d. , and ghahramani , z. ( 2007 ) . stick - breaking construction for the indian buffet process . in _ proceedings of the international conference on artificial intelligence and statistics 11 . wheeler , d. a. , srinivasan , m. , egholm , m. , shen , y. , chen , l. , mcguire , a. , he , w. , chen , y .- j . , makhijani , v. , roth , g. t. , _ et al . _ . the complete genome of an individual by massively parallel dna sequencing . _ nature _ * 452 * , 7189 , 872876 .
|
we develop a feature allocation model for inference on genetic tumor variation using next - generation sequencing data . specifically , we record single nucleotide variants ( snvs ) based on short reads mapped to human reference genome and characterize tumor heterogeneity by latent haplotypes defined as a scaffold of snvs on the same homologous genome . for multiple samples from a single tumor , assuming that each sample is composed of some sample - specific proportions of these haplotypes , we then fit the observed variant allele fractions of snvs for each sample and estimate the proportions of haplotypes . varying proportions of haplotypes across samples is evidence of tumor heterogeneity since it implies varying composition of cell subpopulations .
taking a bayesian perspective , we proceed with a prior probability model for all relevant unknown quantities , including , in particular , a prior probability model on the binary indicators that characterize the latent haplotypes .
such prior models are known as feature allocation models .
specifically , we define a simplified version of the indian buffet process , one of the most traditional feature allocation models .
the proposed model allows overlapping clustering of snvs in defining latent haplotypes , which reflects the evolutionary process of subclonal expansion in tumor samples .
./style / arxiv - general.cfg , ,
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
near - infrared , long baseline interferometry is sensitive to the distribution of dust around the nearest young stars on scales of the order of 1 au , and provides a powerful probe of models of disks and envelopes of such stars . the herbig ae - be stars are pre - main sequence , emission line objects that are the intermediate mass ( @xmath3 ) counterparts of t tauri stars ( hillenbrand _ et al . _ 1992 ) . we also observed the fu orionis object v1057 cyg , expected to have a strong disk signature due to the high accretion rate of such objects . while the evolutionary status of the fu orionis objects remains unclear , they are believed to be t tauri stars undergoing an episode of greatly increased disk accretion , involving a brightening of @xmath4 magnitudes . v1057 cyg , whose outburst began in 1969 - 70 , is the only fu orionis object for which a pre - outburst spectrum is available , confirming its pre - main sequence nature ( grasdalen 1973 ) . until now , only one fu orionis object , fu orionis itself , has been resolved by long baseline optical interferometry ( malbet _ et al . _ 1998 ) , and v1057 cyg was chosen for study as the next - brightest such object accessible to pti . we selected a sample of 5 sources from the thesis of millan - gabet , chosen to satisfy the observing limitations of pti , and to avoid known binaries ( with the exception of mwc 147 , whose companion is too faint to affect the current measurements ) . details of the instrument are described in colavita _ et al . _ table i describes our final sample . 0.8 cm llllllll + * name * & * alternate * & * ra ( @xmath5 ) * & * dec ( @xmath5 ) * & * @xmath6 * & * @xmath7 * & * spec * & d , pc + & & & & & & + + hbc 330 & v594 cas & 00 43 @xmath8 & + 61 54 40.100 & 9.9 & 5.7 & b8e&650 + hd 259431 & mwc 147 & 06 33 @xmath9 & @xmath10 19 19.984 & 8.7 & 5.7 & b6pe&800 + mwc 297 & nz ser & 18 27 @xmath11 & @xmath12 49 52 & 9.5 & 3.1 & o9e&450 + hd 179218 & mwc 614 & 19 11 @xmath13 & + 15 47 15.630 & 7.4 & 5.9 & b9&240 + hd 190073 & v1295 aql & 20 03 @xmath14 & + 05 44 16.676 & 7.8 & 5.8 & a2pe&280 + hbc 300 & v1057 cyg & 20 58 @xmath15 & + 44 15 28.4 & 11.6 & 5.7 & &575 + observations of each source were interweaved with nearby calibrator stars , chosen to exclude known binaries and variable stars . system visibility was determined based upon observations of the calibrators and models of the calibrator ( e.g. size based upon multiwavelength photometry ) . the measured raw source visibilities were then divided by the system visibility . the resulting calibrated visibilities @xmath16 are presented in table ii . our reported visibilities are a wideband average produced synthetically from five narrowband channels . as a consistency check , sources were calibrated first relative to one calibrator , then relative to another , and the results compared to avoid problems with unknown binarity . the stellar contribution to @xmath16 is subsequently removed , assuming the observed spatial distribution of emission on the sky is the sum of an unresolved point source of known flux , and an extended circumstellar contribution . for the herbig stars , mst estimated the fractions of the infrared emission due to the star and due to circumstellar emission at k. in table ii we list the fraction @xmath17 of emission due to circumstellar matter , while that of the star is @xmath18 . for v1057 cyg , we will assume all the infrared emission is circumstellar . table ii also gives @xmath19 for the circumstellar contribution , where @xmath20 . because our program stars all have large infrared excesses , the corrections for stellar light are generally small . upper limits to the visibility squared were determined for sources lacking fringes , based upon the sensitivity of the detection algorithm and measuring the system visibility with a nearby calibrator . figures 1 - 2 show some of the measured individual visibilities @xmath16 for our resolved sources . 0.8 cm lcccccc + * source * & * baseline*&*@xmath16 * & @xmath17&@xmath21 + & & & & + + v594 cas & nw & @xmath22 & @xmath23&@xmath24 + mwc 147 & nw & @xmath25 & @xmath26&@xmath27 + mwc 147 & ns & @xmath28 & @xmath26&@xmath29 + v1057 cyg & nw & @xmath30 & @xmath31 & @xmath32 + mwc 297 & nw , ns & @xmath33 & @xmath34&@xmath35 + mwc 614 & nw , ns & @xmath33 & @xmath36&@xmath37 + v1295 aql & ns & @xmath33 & @xmath38&@xmath37 + fringes were obtained for a total of four sources , although for one of these , mwc 297 , there are insufficient data to produce a calibrated measurement . thus , we treat mwc 297 as an upper limit . based upon the observed circumstellar visibilities @xmath21 , table iii gives approximate source sizes based upon a circular gaussian and a uniform disk model : @xmath39 here @xmath40 , @xmath41 the projected baseline , @xmath42 is the fwhm in radians , @xmath43 is the uniform disk diameter in radians , and @xmath44 is a bessel function . the baseline lengths are 110 m in ns , and 85 m in nw . error bars include uncertainties in our measurements and in the stellar and circumstellar fluxes , but not in the distance . 0.8 cm lccccc + * source * & * baseline * & & + & & * ( mas ) * & * ( au ) * & * ( mas ) * & * ( au ) * + + v594 cas & nw & @xmath45&@xmath46&@xmath47&@xmath48 + mwc 147 & nw & @xmath49&@xmath50&@xmath51&@xmath52 + mwc 147 & ns & @xmath53&@xmath54&@xmath55&@xmath56 + v1057 cyg & nw & @xmath57&@xmath58&@xmath59&@xmath60 + mwc 297 & nw & @xmath61&@xmath62&@xmath63&@xmath64 + mwc 614 & nw & @xmath65&@xmath66&@xmath67&@xmath68 + v1295 aql & ns & @xmath69&@xmath70&@xmath71&@xmath72 + for our observations with the largest range of hour angles and projected baseline orientation , v1057 cygni is consistent with a circularly symmetric source . as an fu ori type object , there is little doubt that its infrared excess comes from a circumstellar disk and not a spherical distribution of dust . more modeling is necessary to put limits on the possible orientation of the disk . our measurements of mwc 147 in the ns baseline are consistent with those of akeson et al . 2000 . however , the new measurement in the nw baseline is inconsisitent with that of the ns baseline if the source is indeed a circularly - symmetric distribution on the sky . because the baselines have differring orientations , the difference can be accounted for by an asymmetric source distribution , such as a tilted disk . we wish to confirm the new nw measurement and perform further modeling of this source . we have resolved three young stellar objects at milli - arc second scales , two for the first time ( the herbig be star v594 cas , and the fu orionis star v1057 cyg ) . presumably we are sampling the distribution of warm dust close to the stars . however , these data alone are insufficient to fully constrain the sources , and other explanations besides circumstellar disks ( e.g. a binary companion ) are possible . no significant variation of the visibility is seen as a function of hour angle on the sky , suggesting a symmetric distribution on the sky . however , for mwc 147 , the measurements in two different baselines suggest an asymmetric distribution , such as a tilted disk . this is consistent with recent measurements by eisner _ et al . _ ( 2003 ) for a similar sample of herbig stars , three of which appear to have disks with significant inclinations . this work has made use of software produced by the michelson science center at the california institute of technology . f.p.w . is grateful to the observatoire de la cte dazur for a poincar fellowship , and to the nsf international researchers fellows program for financial support . 0.4truecm akeson , r.l . , ciardi , d.r . , van belle , g.t . , creech - eakman , m.j . , + & lada , e.a . 2000 , apj , 543 , 313 + colavita , m.m . , wallace , j.k . , hines , b.e . , _ et al . _ 1999 , apj , 510 , 505 + eisner , j.a . , lane , b.f . , akeson , r.l . , hillenbrand , l.a . , & sargent , a.i . 2003 , apj , ( in press ) + grasdalen , g.l . 1973 , apj , 182 , 781 + hillenbrand , l.a . , strom , s.e . , vrba , f.j . , & keene , j. 1992 , apj , 397 , 613 + malbet , f. , berger , j .- p . , colavita , m.m . , _ et al . _ 2000 , apj , 543 , 313 + millan - gabet , r. , schloerb , f.p . , & traub , w.a . 2001 , apj , 546 , 358 +
|
we present observations of a sample of herbig aebe stars , as well as the fu orionis object v1057 cygni .
our k - band ( @xmath0 ) observations from the palomar testbed interferometer ( pti ) used baselines of 110 m and 85 m , resulting in fringe spacings of @xmath1 and @xmath2 , respectively .
fringes were obtained for the first time on v1057 cygni as well as v594 cas .
additional measurements were made of mwc147 , while upper limits to visibility - squared are obtained for mwc297 , hd190073 , and mwc614 .
these measurements are sensitive to the distribution of warm , circumstellar dust in these sources .
if the circumstellar infrared emission comes from warm dust in a disk , the inclination of the disk to the line of sight implies that the observed interferometric visibilities should depend upon hour angle .
surprisingly , the observations of millan - gabet , schloerb , & traub 2001 ( hereafter mst ) did not show significant variation with hour angle .
however , limited sampling of angular frequencies on the sky was possible with the iota interferometer , motivating us to study a subset of their objects to further constrain these systems .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
cerium , the first of the lanthanide series ( rare earths ) with an inner 4f - electron exhibits a broad range of unusual electronic , magnetic and structural properties in the solid state @xcite . most intriguing is the transformation under pressure @xcite between two face - centered - cubic ( fcc ) phases called @xmath6-ce and @xmath7-ce . while @xmath6-ce is stable at room temperature ( @xmath8 ) and ambient pressure ( @xmath9 ) , it transforms to @xmath7-ce at @xmath10 gpa . this apparent isostructural phase transition @xmath11 is accompanied by a @xmath1215% volume reduction and a change in magnetism . while in the @xmath6-phase the magnetic susceptibility @xmath13 follows a curie weiss law which is attributed to the magnetic moment of the 4f - electron , it is a quasi @xmath8-independent in the @xmath7-phase and reminiscent of pauli paramagnetism @xcite . the isostructural nature of the @xmath11 transition is at variance with the landau theory of phase transitions @xcite which would require a change of space group symmetry . since the establishing of the fcc structure for both phases @xcite various microscopic theories have been proposed . nowadays two theoretical models are well represented in the literature . a mott - transition scenario for 4f - states is based on the concept of a localized non bonding state of the 4f - electron in @xmath6-ce and an extended metallic bonding state in the @xmath7-phase @xcite . the increased binding capability due to the metallic f state is considered then as the reason for the marked volume reduction in the @xmath7-phase @xcite . in the following this model has been supported by band structure calculations @xcite . on the other hand more recent inelastic neutron scattering experiments on @xmath7-ce found that the magnetic form factor is quite different @xcite from the one calculated @xcite under the assumption that the 4f states are itinerant . the absence of extended 4f states in the solid is also supported by many - electron calculations on the ce dimer @xcite where it is found that the 4f - electrons do not participate in the chemical bond . the second widely recognized theoretical model is based on spin fluctuations . these are the driving mechanism in the kondo - volume - collapse ( kvc ) model of the @xmath11 transition @xcite . in first approximation the 4f - electron stays localized in both phases . however in the @xmath7-phase the hybridization between the 4f - electrons and the conduction electrons ( anderson impurity hamiltonian , see e.g. ref . ) is much more intense than in the @xmath6-phase . this interaction leads to a screening of the 4f local moments in the energy ground state . the volume dependence of the hybridization coupling or equivalently of the kondo - temperature @xmath14 is used to obtain the equation of state . the interpretation of the @xmath11 transition in ce within the kvc scenario @xcite has found strong support by the analysis of electron spectroscopy data @xcite . notwithstanding much effort , both the mott transition and the kvc model remain under debate @xcite and there is no consensus on the driving mechanism of the phase transition @xcite . a common feature is the absence of any symmetry breaking at the transition as would be required by landau theory @xcite . on the basis of thermodynamical data it has been suggested @xcite that the @xmath11 transformation is in fact a first order phase transition which becomes of second order beyond a tricritical point . therefore , @xmath7-ce should have lower symmetry than @xmath6-ce . however a distorted lattice has been discarded by x - ray diffraction experiments @xcite . a mechanism of symmetry lowering without lattice distortion ( positions of the ce nuclei fcc in the @xmath6 and @xmath7 phase ) has been suggested by the present authors @xcite . the main concepts are the support of quadrupolar electronic charge - density fluctuations by the 4f - electrons in the @xmath6-phase on a compressible fcc lattice and the collective orientational ordering of the quadrupolar densities in the @xmath7 phase @xcite on four simple cubic ( sc ) sublattices . the space group symmetry lowering at the transition @xmath11 is @xmath0 , the local density symmetry is @xmath15 . it is accompanied by a uniform lattice contraction so that the fcc structure is conserved . although this latter aspect is isostructural " , the symmetry lowering of the electronic structure is fully consistent with landau theory . in reciprocal space the active electronic density mode condenses at the @xmath16 point of the brillouin zone . a phase transition @xmath0 occurs in solid c@xmath5 ( fullerite ) @xcite . here the symmetry lowering is due to the orientational ordering of the c@xmath5 molecules . at the transition the cubic lattice constant contracts discontinuously @xcite , the center of mass points of the molecules still occupy an fcc lattice . from the latter point of view , the phase transition would be isostructural " . however , the molecular order on four sc sublattices entails an order of the constituent c atoms and hence the @xmath2 structure leads to characteristic reflections in x - ray @xcite and neutron scattering @xcite experiments . the problem of measuring the @xmath2 structure directly in ce is complicated by the fact that the ce nuclei remain on an fcc lattice . although the quadrupolar charge density ordering could be detectable by synchrotron radiation , the intensity of the additional @xmath2 reflections is likely to be too weak and has not yet been uncovered @xcite . on the other hand there have been two recent experimental results which are very specific and point to the relevance of the quadrupolar ordering at the @xmath11 transition . time - differential perturbed angular correlation ( tdpac ) experiments in solid ce in a pressure range up to 8 gpa detect an appreciable electric field gradient ( efg ) in @xmath7-ce which is almost four times larger than in the cubic @xmath6 phase and close to values in the noncubic phases @xmath17 and @xmath18 @xcite . this finding rules out the @xmath19 symmetry in @xmath7-ce and evidences in support of the antiferroquadrupolar order suggested by theory @xcite . the other experimental support of the antiferroquadrupolar ordering is more indirect but very specific too . phonon dispersion measurements by inelastic x - ray scattering on elemental ce across the @xmath11 transition reveal strong changes in the dispersion shape @xcite . in particular a pronounced softening of certain phonon branches is found in the @xmath7-phase toward the @xmath16 point of the brillouin zone . given the above mentioned fact that the discussion on the validity of the mott transition or kvc scenario remains open , there have been in the last decade increased experimental efforts in investigating the lattice dynamics at the phase transition @xmath11 @xcite . from these studies it results that the lattice vibrations play an important role at the @xmath11 phase transition and that the inclusion of lattice dynamics in the theoretical description is of paramount importance . in this respect we recall that anomalous elastic behavior at the @xmath11 transition was originally discovered in ultrasound experiments @xcite . from inelastic neutron scattering experiments @xcite performed on single crystal of @xmath6-ce it was concluded that premonitory effect of the @xmath20 transition are present in the phonon dispersion curves of @xmath6-ce at room temperature . however , until recently the relevance of these results has not been sufficiently appreciated , while all research efforts were concentrated on the electronic properties . the anomalies of the elastic properties at the @xmath11 transition in ce [ 34,21,33 ] have their counterpart at the fcc@xmath21sc transition in c@xmath5 fullerite @xcite . in particular there is a striking resemblance of the pressure dependence of the bulk modulus in ce with corresponding experimental results in c@xmath5 @xcite . the content of the paper is as follows . in sec . [ sec : model ] we recall the main features of the model of interacting 4f - electron quadrupolar densities on a compressible fcc lattice . the hamiltonian and the resulting condensation scheme from the disordered @xmath6-phase to the quadrupolar ordered @xmath7-phase are described . the coupling to the lattice is included . next ( sec . [ sec : phase_trans ] ) we extend the model by adding an external hydrostatic pressure . various quantities that characterize the phase transition ( order parameter , susceptibility and correlation length ) are derived as functions of temperature and pressure . in sec . [ sec : el_res ] we calculate the dynamic displacement displacement correlation function , taking into account the coupling of the quadrupolar electron densities to the crystal lattice . in the static limit the inversion of the displacement displacement correlation function tensor leads to the elastic constants c@xmath22 and @xmath4 . while in c@xmath22 the coupling to quadrupolar electron density fluctuations leads to remarkable anomalies at the @xmath1 phase transition , the coupling is absent in c@xmath23 . the results of the theory are compared with experiments . conclusions of the paper are presented in sec . [ sec : con ] . here we will remind the model hamiltonian which describes electronic quadrupolar charge density fluctuations on a compressible fcc lattice . in previous work we have first treated quadrupolar charge fluctuations due to solely 4f - electrons @xcite . later on we have taken into account conduction electrons @xcite as well as f- and d - electron intra site correlations @xcite . since these extensions can be cast into an effective hamiltonian with essentially the same structure as the one originally studied @xcite , we will restrict ourselves here to 4f - electrons as a generic case . we consider @xmath24 ce atoms located on a non rigid fcc lattice with nuclear positions @xmath25 , @xmath26 , 2 , ... , @xmath24 . here @xmath27 denote the equilibrium positions and @xmath28 the displacements due to lattice vibrations . we assume that core and valence electrons follow adiabatically the nuclear displacements . the hamiltonian comprises three parts : @xmath29 where @xmath30 stands for the 4f - electrons , @xmath31 for the lattice and @xmath32 for the coupling of both . explicitly one has @xmath33 where @xmath34 is the single particle potential and @xmath35 the quadrupole - quadrupole interaction on the rigid lattice . we assume that the 4f - electrons have coordinates @xmath36 and are localized on spheres centered at @xmath27 and with radii @xmath37 , @xmath381.378 a.u . the single particle potential is due to the cubic crystal field in presence of spin - orbit coupling . one obtains @xmath39 where @xmath40 and @xmath41 are the eigenvalues and eigenstates , @xmath42 , 2 , ... , 14 . the angular part of the wave function of the 4f - electron ( angular momentum quantum number @xmath43 ) at site @xmath44 is described by the @xmath45 spin orbitals @xmath46 , @xmath47 , where @xmath48 . for details see refs . . the electronic quadrupolar interaction is given by @xmath49 where @xmath50 , @xmath511 , 2 , 3 are quadrupolar density operators and @xmath52 interaction matrix elements between nearest neighbors on the fcc lattice . the density operators are defined by @xmath53 with @xmath54 here we restrict ourselves to the three quadrupolar functions @xmath55 , @xmath56 , 2 , 3 , which transform as the irreducible representation @xmath57 of the cubic point group @xmath58 ( @xmath59 ) . these symmetry adapted functions ( saf s ) are linear combinations of spherical harmonics belonging to the manifold @xmath60 and are tabulated in ref . . they transform as the cartesian components @xmath61 , @xmath62 , @xmath63 for @xmath56 , 2 , 3 , respectively . defining fourier transforms @xmath64 @xmath65 we obtain @xmath66 with @xcite @xmath67 } \nonumber \\ \label{m10}\end{aligned}\ ] ] here @xmath6 , @xmath68 and @xmath7 are quadrupole - quadrupole interaction coefficients on the fcc lattice . the quantities @xmath69 , and @xmath70 , @xmath71 , @xmath72 is the cubic lattice constant , account for the fcc lattice structure . at the @xmath16 point of the brillouin zone for @xmath73 , the interaction matrix @xmath74 becomes diagonal . the star of @xmath75 has the arms @xmath76 , @xmath77 , @xmath78 . for @xmath79 one has @xmath80 , and similarly for @xmath81 and @xmath82 with permutation of indices @xmath83 . hence the quadrupolar interaction becomes attractive with the largest twofold degenerate eigenvalue @xmath84 at each arm of @xmath85 . as we have shown previously @xcite this interaction is compatible with a symmetry lowering @xmath0 , characterized by an orientational order of the quadrupolar densities on four different sc sublattices ( see fig . [ fig0 ] ) . the corresponding condensation scheme reads : @xmath86 here the superscript @xmath87 stands for a thermal expectation value and @xmath88 is the order parameter amplitude . since the three arms of @xmath85 are involved one speaks of a triple-@xmath89 antiferroquadrupolar order ( afq ) . notice that a condensation scheme similar to eqs . ( [ m11a ] ) , ( [ m11b ] ) also holds for the phase transitions with the symmetry @xmath0 in nao@xmath90 @xcite and in c@xmath5-fullerite @xcite . although icosahedral symmetry of the c@xmath5 molecule implies that not quadrupoles but higher order multipoles ( @xmath91 , 10 , ... ) determine the orientational interactions , the corresponding saf s transform as irreducible representations of symmetry @xmath57 of the cubic point group . a compelling mathematical reason to consider the symmetry lowering @xmath0 as a candidate for the isostructural phase transition in ce is the fact that it leads to a lattice contraction while the center of mass points still occupy an fcc lattice . in addition the transition is of first order . indeed the number of symmetry elements is thereby reduced by a factor 3 ( from 48 to 16 ) , which implies the existence of a third order cubic invariant in free energy @xcite . the lattice dynamics is described by the phonon hamiltonian @xmath92 where @xmath93 is the kinetic energy and @xmath94 the potential energy in harmonic approximation . in fourier space one has @xmath95 @xmath96 . the displacements @xmath97 and the conjugate moments @xmath98 are related to the variables in real space by @xmath99 @xmath100 where @xmath101 is the atomic mass . one has the usual commutation rules @xmath102=[p , p]=0 $ ] and @xmath103 = i \hbar \delta_{\vec{q } \vec{k } } \delta_{ij } . \label{m15}\end{aligned}\ ] ] in the long wavelength limit the dynamical matrix @xmath104 is given by @xmath105 } \nonumber \\ \label{m16}\end{aligned}\ ] ] where @xmath106 are the bare elastic constants in absence of coupling to the quadrupolar electronic fluctuations . the coupling between quadrupolar charge density fluctuations and lattice dynamics has been derived previously @xcite . in the long wavelength regime for the lattice displacements and for quadrupolar fluctuations near the @xmath16 point of bz we have @xmath107 here the prime on the sum over @xmath7 indicates the following restrictions : for @xmath108 , @xmath109 ; @xmath110 , @xmath111 ; @xmath112 , @xmath113 . the coupling matrix is given by @xmath114 here @xmath115 , @xmath116 refers to the 12 nearest neighbors of @xmath44 on the fcc lattice , @xmath117 is the first order derivative with respect to the lattice displacement component @xmath118 of the quadrupole - quadrupole interaction @xmath119 . the structure of the fcc lattice implies that @xmath120 , where for @xmath108 or @xmath121 , @xmath122 ; for @xmath123 or @xmath124 , @xmath125 ; for @xmath112 or @xmath121 , @xmath126 . within the quadrupolar model we obtain that @xmath127 . carrying out the summation over @xmath128 and exploiting the symmetry of the lattice , we rewrite eq . ( [ m17 ] ) as @xmath129 under the proviso of the summation restriction over @xmath7 we define for @xmath130 or @xmath131 @xmath132 and hence @xmath133 only longitudinal lattice displacements or equivalently longitudinal lattice strains @xmath134 occur on the right hand side of eq . ( [ m19 ] ) . since in addition @xmath135 for @xmath130 and @xmath131 occur on the same footing , the coupling @xmath32 leads to a striction of the lattice with conservation of cubic symmetry @xcite . notice that there is no coupling to transverse lattice displacement waves or equivalently to shear strains @xmath136 , @xmath137 , in @xmath32 . in the next section we will investigate the influence of the quadrupolar density fluctuations on the low frequency lattice dynamics . in order to study the tripple-@xmath89 antiferroquadrupolar phase transition on a compressible lattice as a function of temperature and pressure , we first recall some concepts of the underlying free energy . the phase transition is of first order . we calculate the order parameter as a function of temperature and pressure . we show that the transition temperature increases linearly with pressure . finally we study the order parameter susceptibility in the disordered and the ordered phase . taking into account the quadrupole - quadrupole interaction on a rigid fcc lattice we have written the helmholtz free energy as a landau expansion in terms of the order parameter amplitude @xmath88 @xcite : @xmath138 here @xmath139 is the free energy of the disordered phase which has to be calculated with the cubic crystal field @xmath34 , eq . ( [ m3 ] ) . the coefficients of the order parameter terms are @xmath140 , \label{fe23 } \\ b & = & -t\,x_{123}^{(3)}\,/[x^{(2)}]^3 , \label{fe24 } \\ c & = & { \frac{t}{8(x^{(2)})^4}}\ , \left [ \,9(x^{(2)})^2-x_{1111}^{(4)}-6x_{1122}^{(4)}+ { \frac{24(x_{123}^{(3)})^2}{x^{(2 ) } } } \right ] , \nonumber \\ \label{fe25}\end{aligned}\ ] ] where @xmath141 is the quadrupolar interaction and @xmath8 the temperature in unergy units ( @xmath142 ) . the quantities @xmath143 , @xmath144 and @xmath145 are single particle thermal expectation values @xcite , which are calculated by means of the single particle potential . numerical values of relevant parameters are given in table 1 . . parameters @xmath143 , @xmath144 , @xmath6 , @xmath146 , @xmath147 , @xmath148 , @xmath149 , @xmath150 calculated from ref . . @xmath151 and @xmath152 estimated from experiment @xcite . [ table1 ] [ cols="^,^ " , ] in case of a compressible lattice @xcite the hamiltonian @xmath32 , eq . ( [ m21 ] ) , gives the free energy term @xmath153 , \label{fe28}\end{aligned}\ ] ] where @xmath154 etc . are the lattice strains . since @xmath127 , this term favors @xmath155 , i.e. a volume contraction . the lattice hamiltonian @xmath31 leads to the elastic contribution @xmath156 , \label{fe29}\end{aligned}\ ] ] where @xmath157 is the volume per atom . in presence of an applied external pressure @xmath9 , we consider the gibbs free energy @xcite @xmath158 minimizing @xmath159 with respect to @xmath134 for @xmath130 and @xmath131 , we get an isostructural contraction of the cubic lattice @xmath160 \kappa_l . % ( 29 ) \label{fee29}\end{aligned}\ ] ] here @xmath161 is the compressibility . isostructural contraction of the lattice means that there is no symmetry breaking associated with the change of the center of mass positions of the ce atoms at the @xmath11 transition . we completely agree with conclusions from high - pressure and high - temperature x - ray diffraction experiments that the structure remains fcc across the @xmath11 transformation and retains crystallographic orientation during the transformation @xcite . within the present theory a symmetry change occurs solely in the quadrupolar electronic charge densities whereby the @xmath2 ordering of the latter is compatible with the fcc structure of the lattice . eliminating @xmath134 from eq . ( [ fe30 ] ) and retaining only linear terms in @xmath9 we rewrite @xmath159 as @xmath162/n = f_0/n + a ' \rho^2 + b \rho^3 + c'\rho^4 , \nonumber \\ % ( 30 ) \label{fe31}\end{aligned}\ ] ] where @xmath163 , % ( 31 ) \label{fe32a}\end{aligned}\ ] ] with @xmath164 and where @xmath165 with @xmath166 . the occurrence of the cubic invariant in the free energy implies that the phase transition is of first order . the coexistence condition @xmath167 establishes a relation between the transition temperature @xmath168 and the corresponding pressure @xmath151 : @xmath169 . % ( 34 ) \label{fe33}\end{aligned}\ ] ] in first approximation @xmath168 increases linearly with pressure , in agreement with the experimental @xmath170 phase diagram of ce @xcite . notice that the analogue of eq . ( [ fe33 ] ) also holds for c@xmath5 fullerite @xcite , where experiments @xcite show a linear pressure dependence of @xmath168 . with the parameters of table [ table1 ] we obtain @xmath171 k. in accordance with the experimental situation @xcite we consider a fixed temperature @xmath172 and a variable pressure . then @xmath173 , @xmath174 , corresponds to the disordered phase ( in casu @xmath6-ce ) , while @xmath175 , @xmath176 corresponds to the ordered phase ( @xmath7-ce ) . minimization of @xmath159 at @xmath168 and @xmath177 with respect to @xmath88 leads to the order parameter amplitude @xmath178 since @xmath179 , the order parameter @xmath88 has to be negative and hence the @xmath180 sign has to be chosen in front of the square root in eq . ( [ fe33b ] ) . the discontinuity of the order parameter on the transition line , eq . ( [ fe33 ] ) , is given by @xmath181 where we remind that @xmath168 is an implicit function of @xmath151 . with the parameters of table [ table1 ] we obtain @xmath182 . here and in the following we assume that the quantities @xmath183 , @xmath184 and @xmath185 , all which are single expectation values , are not affected by moderate variations of pressure near @xmath151 . in the ordered phase we obtain by means of eqs . ( [ fe33b ] ) and ( [ fe33 ] ) : @xmath186 where @xmath187 . since @xmath188 , the order parameter amplitude increases in absolute value with the increment of pressure above @xmath151 . [ fig1 ] we have calculated the order parameter amplitude by means of eq . ( [ fee37 ] ) . here and below the cubic lattice constant @xmath72 stands for @xmath189 in the ordered phase and @xmath190 in the disordered phase . in the following we will need the wave number dependent order parameter susceptibilities @xmath191 in the disordered phase and in the ordered phase . the susceptibility is defined in terms of the fluctuations @xmath192 of the local order parameter from its average value : @xmath193 where @xmath71 for @xmath194 , respectively . hence , @xmath195 we first consider the disordered phase . in case of a rigid lattice we obtain within molecular field theory @xmath196^{-1}_{\alpha \alpha } . \label{fe34}\end{aligned}\ ] ] we recall that @xmath197 is the quadrupolar interaction matrix eq . ( [ m10 ] ) while @xmath198 is the @xmath199 unit matrix . the single particle expectation value @xmath200 has the same value for @xmath113 , and 3 . since we consider order parameter fluctuations near the phase transition , only wave vectors @xmath201 near the star of @xmath202 are relevant . in accordance with the condensation scheme ( [ m11a ] ) , ( [ m11b ] ) the matrix @xmath197 becomes diagonal at @xmath203 , where @xmath204 . expansion about @xmath205 gives @xmath206 similar expressions with cyclic permutation of indices are obtained for @xmath207 and @xmath208 near @xmath209 and @xmath210 , respectively . in case of a non - rigid lattice we see from eq . ( [ fe32b ] ) that the applied pressure acts as addition to the quadrupolar interaction at @xmath202 . replacing in eq . ( [ fe35 ] ) @xmath141 by @xmath211 , we rewrite for the order parameter susceptibility in the disordered phase at temperature @xmath168 and pressure @xmath212 : @xmath213 } . % ( 42 ) \label{fee40}\end{aligned}\ ] ] here the correlation length @xmath214 is defined by @xmath215^{1/2 } . % ( 43 ) \label{fee41}\end{aligned}\ ] ] expressions similar to eq . ( [ fee40 ] ) are obtained for @xmath216 and @xmath217 near @xmath218 and @xmath210 , respectively , and with @xmath219 replaced by @xmath220 and @xmath221 , respectively . making use of eqs . ( [ fe33 ] ) and ( [ fe32b ] ) , we obtain the correlation length @xmath222 } \right]^{1/2 } , % ( 44 ) \nonumber \\ \label{fee42}\end{aligned}\ ] ] where @xmath223 . with increasing pressure @xmath224 in the disordered phase , the correlation length reaches its maximum value @xmath225^{1/2 } . ( 45 ) \label{fee43}\end{aligned}\ ] ] toward the onset of the phase transition . likewise the susceptibility becomes maximum at @xmath226 and @xmath227 : @xmath228 in the ordered phase the order parameter susceptibility is given by @xmath229 . \label{fe40}\end{aligned}\ ] ] applying mean - field theory @xcite we obtain for the quadrupolar model on a rigid lattice : @xmath230^{-1}_{\alpha \alpha } , % ( 48 ) \nonumber \\ \label{fe41}\end{aligned}\ ] ] where @xmath231 . \quad \label{fe42}\end{aligned}\ ] ] numerical calculation show that @xmath232 . in expression ( [ fe40 ] ) @xmath88 is the order parameter amplitude . proceeding now as before we consider a nonrigid lattice and an external pressure . using eqs . ( [ fe32b ] ) and ( [ fe35 ] ) we define the correlation length in the ordered phase at temperature @xmath168 and pressure @xmath233 : @xmath234^{1/2 } . % ( 50 ) \nonumber \\ \label{fe43}\end{aligned}\ ] ] here the subscript @xmath235 stands for ordered quadrupolar phase . the corresponding order parameter susceptibility for the component @xmath236 is then given by @xmath237 } . % ( 51 ) \label{fe44}\end{aligned}\ ] ] notice that the discontinuity of the order parameter at the first order phase transition leads to a drop of the correlation length and hence of the order parameter susceptibility . with @xmath238 given by eq . ( [ fe33a ] ) , we obtain @xmath239 } \right]^{1/2 } , % eq . ( 52 ) \nonumber \\ \label{fee49}\end{aligned}\ ] ] and since @xmath240 , @xmath241 . the correlation length in the ordered phase is calculated by means of eqs . ( [ fe43 ] ) , ( [ fee37 ] ) and ( [ fe33 ] ) . the result reads @xmath242^{-1/2 } \label{fee50}\end{aligned}\ ] ] where @xmath243 numerical evaluation of the quadrupolar model shows that @xmath244 . hence the correlation length decreases with increasing pressure in the ordered phase . using eq . ( [ fee42 ] ) for @xmath245 and eq . ( [ fe43 ] ) for @xmath246 we have calculated the squared correlation length @xmath247 as a function of pressure . the plot is shown in fig . [ fig2 ] . from eqs . ( [ fee40 ] ) and ( [ fe44 ] ) for @xmath248 we see that fig . [ fig2 ] also describes the pressure dependence of the order parameter susceptibility at @xmath226 . the expressions of the order parameter susceptibility will be used in the next section where we study the temperature and pressure dependence of the elastic response near the first order phase transition . we will derive the static displacement displacement response function matrix @xcite by using the well known green s functions techniques @xcite . inversion of the response function in the long wavelength limit leads to the elastic constants . the fourier transform of the retarded green s function of two operators @xmath249 and @xmath147 is defined by @xmath250 \rangle , \label{m22}\end{aligned}\ ] ] with frequency @xmath251 , @xmath252 . the skew brackets @xmath253 stand for a thermal average with the system s hamiltonian @xmath254 . in heisenberg representation the time dependence reads @xmath255 , @xmath256 . we quote the equation of motion @xcite : @xmath257 \rangle + \langle \langle [ a,{\cal h } ] ; b \rangle \rangle_z , \label{m23}\end{aligned}\ ] ] with the identity @xmath258 ; b \rangle \rangle_z = - \langle \langle a ; [ b,{\cal h } ] \rangle \rangle_z . \label{m24}\end{aligned}\ ] ] the dynamic displacement displacement green s function is defined by @xmath259 with lattice displacements and conjugate momenta specified in eqs . ( [ m14a ] ) , ( [ m14b ] ) . we recall that the hamiltonian is given by eq . ( [ m1 ] ) , where the parts have been specified subsequently in sect . [ sec : model ] . in the following we treat @xmath260 as a dynamic variable . thereby we retain only the fluctuation contribution of the order parameter variable , writing @xmath261 applying twice the equation of motion ( [ m23 ] ) to @xmath262 we use the commutation rules eq . ( [ m15 ] ) as well as the fact that the electronic variables @xmath263 and @xmath264 commute with the lattice variables @xmath265 and @xmath266 . the result reads @xmath267 here the summation is understood over the repeated index @xmath268 . we recall that @xmath260 has been defined by eq . ( [ m20 ] ) . in a similar way we obtain by using eq . ( [ m24 ] ) @xmath269 taking the static limit @xmath270 in eqs . ( [ m26 ] ) and ( [ m27 ] ) and combining the results we find @xmath271 . \nonumber \\ \label{m28}\end{aligned}\ ] ] here the subscript @xmath272 stands for @xmath270 . we introduce the static susceptibilities @xcite @xmath273 and rewrite eq . ( [ m28 ] ) as @xmath274 . \nonumber \\ \label{m31}\end{aligned}\ ] ] we remind that in the long wavelength limit the static displacement susceptibility is related to the elastic constants @xcite @xmath275 by @xmath276 where @xmath277 is the mass density . hence eq . ( [ m31 ] ) allows us to calculate the elastic constants in presence of the coupling @xmath32 between electronic and lattice degrees of freedom . we recall that in absence of the coupling the bare elastic constants are given by @xmath278 [ see eq . ( [ m16 ] ) ] . for cubic crystals we have in voigt s notation the elastic constants @xmath279 , @xmath280 , @xmath281 . with @xmath282 and @xmath283 , ( [ m31 ] ) reduces in the long wavelength limit to @xmath284^{-1 } . % ( 67 ) \label{m33}\end{aligned}\ ] ] here we have made use of eqs . ( [ m32 ] ) and ( [ m16 ] ) . on the basis of the hamiltonian , eq . ( [ m1 ] ) , the obtained expressions for @xmath285 and @xmath3 are rigorous results . since @xmath286 , the quadrupolar - elastic coupling , eq . ( [ m19 ] ) , leads to a reduction of the elastic constant @xmath3 in comparison with the bare quantity @xmath287 . this reduction as a consequence of the @xmath32 coupling is responsible for the relative softening of the corresponding @xmath288[001 ] phonon branch in @xmath6-ce @xcite . in contradistinction to @xmath3 , the shear elastic constant @xmath4 is not affected by the coupling of the lattice to quadrupolar electronic density fluctuations . as we have shown in sec . ii , there is no coupling to shear strains @xmath289 in @xmath32 , eq . ( [ m19 ] ) . hence taking @xmath290 in eq . ( [ m31 ] ) , we obtain for @xmath291 and @xmath292 : @xmath293 or equivalently by means of eqs . ( [ m16 ] ) and ( [ m32 ] ) @xmath294 . pressure experiments exhibit a discontinuity of the shear modulus @xcite at the phase transition . however , this effect is solely due to the lattice contraction and a concomitant change of interatomic forces . due to the absence of a direct coupling between lattice shears and orientational density fluctuations there are no precursor effects . we want to study the anomalous behavior of the elastic constant @xmath3 near the quadrupolar ordering transition . from eq . ( [ m33 ] ) it follows that the important quantity is the four - point ( four factors @xmath88 ) susceptibility @xmath295 . at high temperature and in the long wavelength regime such that the length scale of order parameter fluctuations is large in comparison with the interatomic spacing we use classical statistical mechanics . we then have @xmath296 where @xmath297 in appendix we show that in the limit @xmath298 @xmath299 with @xmath236 , 3 . here @xmath191 is the order parameter susceptibility for the component @xmath7 . cubic symmetry implies that @xmath300 we calculate the right hand side of eq . ( [ m36 ] ) by using @xmath301 where @xmath302 is the volume of the crystal and @xmath24 the number of atoms . so far the considerations of the present section are general and hold for the disordered as well as for the ordered phase . since the expressions of the order parameter susceptibility in the disordered and in the ordered phase , eqs . ( [ fee40 ] ) and ( [ fe44 ] ) , respectively , exhibit the same wave vector dependence , we can treat both cases simultaneously . since the integrand in eq . ( [ m45 ] ) does not depend on @xmath303 , the integration over @xmath304 in the interval @xmath305 yields @xmath306 . on the other hand the integrand vanishes for large values of @xmath307 . hence we integrate over a circle and extend the radius to @xmath308 . taking into account eq . ( [ m38 ] ) , we obtain @xmath309 where @xmath310 stands for @xmath311 in the disordered phase and for @xmath312 in the ordered quadrupolar phase . substitution of the result into eq . ( [ m33 ] ) gives the longitudinal elastic constant @xmath313^{-1 } , % ( 75 ) \label{m58}\end{aligned}\ ] ] where we have defined @xmath314 with @xmath315 , eq . ( [ fee42 ] ) , in the disordered phase and @xmath316 , eq . ( [ fee50 ] ) , in the ordered phase . near the phase transition in the disordered phase we have @xmath317^{-1 } . % ( 77 ) \label{m73}\end{aligned}\ ] ] the increase of @xmath245 or equivalently of @xmath318 with @xmath223 leads to a decrease of @xmath3 which reaches its minimum value @xmath319^{-1 } % ( 78 ) \label{m74}\end{aligned}\ ] ] at @xmath320 . on the other hand at the onset of the phase transition one has @xmath321^{-1 } . % ( 79 ) \label{m75}\end{aligned}\ ] ] the discontinuity of the order parameter and the concomitant drop of the correlation length @xmath241 results in a positive jump of @xmath3 at the first order phase transition : @xmath322}{[1 + \xi_{d}(t_1,p_1 ) ] [ 1 + \xi_{q}(t_1,p_1 ) ] } . % ( 80 ) \label{m80n}\end{aligned}\ ] ] here we have approximated @xmath72 in the prefactor by @xmath323 . in the ordred phase the decrease of the correlation length @xmath246 with increasing pressure leads to an increase of @xmath324^{-1 } . % ( 81 ) \label{m76}\end{aligned}\ ] ] the scenario described by eqs . ( [ m73])([m76 ] ) is in full agreement with experiment . indeed the pioneering experiments on elastic properties of ce under pressure by voronov et al . @xcite as well as recent high resolution ultrasonic measurements @xcite show that the propagation velocity of longitudinal ultrasonic waves decreases when the phase transition toward the @xmath7 phase is approached with increasing pressure in the @xmath6 phase . at the first order phase transition the longitudinal sound velocity exhibits a stepwise increase . with increasing pressure in the @xmath7 phase the longitudinal sound velocity increases continuously . a corresponding behavior of the bulk modulus as a function of pressure was deduced from high - resolution neutron and synchrotron x - ray powder diffraction @xcite experiments . it was argued that the softening of the bulk modulus in the @xmath6 phase with increasing pressure @xmath325 is a direct consequence of the softening of @xmath3 . we close this section by observing that in c@xmath5 fullerite there occurs a marked lowering of the bulk modulus @xmath147 if at fixed @xmath8 the fcc@xmath21sc phase transition is approached with increasing pressure @xcite . the phenomenon is attributed to orientational reordering of the c@xmath5 molecules . given the similarities of the lattice related phenomena at the @xmath11 phase transition in ce and the orientational phase transition in c@xmath5 fullerite , we conclude that these transitions are isomorphic . from the mathematical point of view @xcite all essential aspects ( @xmath0 ) are the same , while the constituents ( electronic quadrupolar densities versus icosahedral molecules ) are different . the elastic properties of a model of interacting 4f - electron quadrupolar densities on a compressible fcc lattice have been investigated as a function of temperature and pressure . on the basis of previous theoretical work @xcite this model , supported by recent nuclear spectroscopy experiments @xcite , is taken as representative for the @xmath1 isostructural " phase transition in ce . in particular , we have studied by analytical theory the pressure dependent anomalies of the elastic constant @xmath3 at the phase transition from the quadrupolar orientationally disoredered phase which we identify with the @xmath6-phase ( space group @xmath19 ) to the quadrupolar orientationally ordered phase which we identify with the @xmath7-phase ( space group @xmath2 ) . as a result we find that @xmath3 ( equivalently the longitudinal sound velocity ) decreases by approaching the phase transition from the disoredered phase with increasing pressure @xmath224 , at the first order phase transition @xmath3 exhibits a positive jump , in the ordered phase @xmath3 increases continuously with pressure @xmath326 . on the other hand we find that the elastic constant @xmath4 ( equivalently the shear sound velocity ) exhibits no precursor effects near the transition . these theoretical results are in full qualitative agreement with experiments [ 34,21,22,33 ] on elastic anomalies at the @xmath1 isostructural " phase transition in solid ce . we notice that the hamiltonian @xmath32 , eq . ( [ m17 ] ) , which is quadratic in the electronic quadrupolar order parameter variables and linear in the lattice displacements , accounts as well for the elastic anomalies as for the isostructural " lattice contraction . the hamiltonian is reminiscent from the compressible ising model @xcite which is quadratic in the spin variables and linear in the lattice displacements . however , in the present case symmetry properties are more subtle and account for the interplay between antiferroquadrupolar order in the electronic densities and isostructural " lattice contraction . here a remark on symmetry reduction at the phase transition is in order . the quadrupolar interaction has its origin in the repulsive coulomb interaction between 4f - electrons on neighboring atoms . the ordering of the electronic quadrupoles on four sc sublattices reduces the repulsion and acts as an effective attraction . in reciprocal space the quadrupolar interaction matrix @xmath74 , eq . ( [ m10 ] ) , has negative eigenvalues at the @xmath16-point of the brillouin zone of the fcc lattice . phonon dispersions of ce measured by synchrotron radiation show pressure dependent anomalies related to the @xmath1 transition at the @xmath16-point of the brillouin zone @xcite . the explanation of these experiments is an outstanding challenge for theoretical work . a further problem is the quantitative improvement of the theory which should lead to increase the magnitude of the coupling coefficient @xmath146 in eq . ( [ m19 ] ) for @xmath32 . at present we believe that the theory should be extended by including the intrasite coupling between 4f - electrons and conduction electrons ( say 5d ) and a possible influence of this coupling on the interatomic bonding . an indication of the relevance of such a mechanism is provided by recent work on the ce dimer @xcite . likely such an extension of the theory would also contribute to elucidate the concept of a mott transition @xcite versus a kondo scenario @xcite at the @xmath1 transition . the authors acknowledge useful discussions with k. parlinski , m. krisch , f. decremps , r.m . pick , g. roth , g. gntherodt , a.v . tsvyashchenko and e.v . . financial support has been provided by the research group theory of condensed matter , university of antwerp and by the institut fr kristallographie , rwth aachen . here we express @xmath295 as a function of the order parameter susceptibility @xmath191 . starting from eqs . ( [ m35 ] ) and ( [ m70n ] ) we first notice that @xmath327 where @xmath328 we rewrite eq . ( [ m35 ] ) as @xmath329 . \nonumber \\ \label{a3}\end{aligned}\ ] ] we then approximate the four - point function @xmath330 by the factorization scheme @xmath331 or equivalently @xmath332 substituting this result into eq . ( [ a3 ] ) and using eq . ( [ fe39n ] ) we obtain @xmath333 taking the limit @xmath334 leads to the result of eq . ( [ m36 ] ) . d.c . koskenmaki and k.a . gschneidner , jr . , _ handbook on the physics and chemistry of rare earths _ gschneidner , jr . , and l. eyring ( amsterdam : north - holland , 1978 ) , p. 337 . p. w. bridgman , proc . arts sci . * 76 * , 55 ( 1948 ) . macpherson , g.e . everett , d. wohlleben , m.b . maple , phys . * 26 * , 20 ( 1971 ) . l. d. landau and e. m. lifshitz . : _ statistical physics _ , ( pergamon , bristol , 1995 ) , vol . 5 . a. w. lawson , t .- y . tang , phys . rev . * 76 * , 301 ( 1949 ) . b. johansson , phil . mag . * 30 * , 469 ( 1974 ) . b. johansson , i.a . abrikosov , m. aldn , a.v . ruban , and h.l . skriver , phys . lett . * 74 * , 2335 ( 1995 ) . murani , s.j . levett , and j.w . taylor , phys . lett . * 95 * , 256403 ( 2005 ) . a. hjelm , j. trygg , o. eriksson , b. johansson , and j. wills , phys . b * 50 * , 4332 ( 1994 ) . x. cao and m. dolg , mol . * 101 * , 1967 ( 2003 ) . roos , r. lindh , p .- a . malmqvist , v. veryazov , and p .- o . widmark , j. phys . a * 112 * , 11431 ( 2008 ) . nikolaev , phys . rev . a * 84 * , 012512 ( 2011 ) . allen , r.m . martin , phys . * 49 * , 1106 ( 1982 ) ; j.w . allen and l.z . liu , phys . b * 46 * , 5047 ( 1992 ) . m. lavagna , c. lacroix , and m. cyrot , phys . a * 90 * , 210 ( 1982 ) . p. fulde , _ electron correlations in molecules and solids _ ( springer heidelberg , 1995 ) . liu , j.w . allen , o. gunnarsson , n.e . christensen , and o.k . andersen , phys . b * 45 * , 8934 ( 1991 ) . b. johansson , a. v. ruban , and i. a. abrikosov , phys . . lett . * 102 * , 189601 ( 2009 ) . b. amadon , s. biermann , a. georges , and f. aryasetiawan , phys . lett . * 96 * , 066402 ( 2006 ) . j .- rueff , j .- iti , m. taguchi , c.f . hague , j .- mariot , r. delaunay , j .- kappber , and n. jaonen , phys . lett . * 96 * , 237403 ( 2006 ) . g.eliashberg , h.capellmann , jetp lett . * 67 * , 125 ( 1998 ) . jeong , t.w . darling , m. j. graf , th . proffen , r. h. heffner , y. lee , t. vogt , j. d. jorgensen , phys . lett . * 92 * , 105702 ( 2004 ) . lipp , d. jackson , h. cynn , c. aracne , w. j. evans , and a. k. mcmahan , phys . * 101 * , 165703 ( 2008 ) . a.v . nikolaev and k.h . michel , eur . j. b * 9 * , 619 ( 1999 ) . a.v . nikolaev and k.h . michel , eur . j. b * 17 * , 15 ( 2000 ) . a.v . nikolaev and k.h . michel , phys . b * 66 * , 054103 ( 2002 ) . r. sachidanandam and a.b . harris , phys . 67 * , 1467 ( 1991 ) ; p.a . heiney , j.e . fischer , a.r . mcghie , w.j . romanow , a.m. denenstein , j.p . mccauly jr . , a.b . smith iii , and d.e . cox , ibid . * 67 * , 1468 ( 1991 ) . david , r.m . ibberson , j.c . matthewman , k. prassides , t.j.s . dennis , j.p . hare , h.w . kroto , d.r.m . walton , nature , * 353 * , 147 ( 1991 ) . david , r.m . ibberson , t.j.s . dennis , j.p . hare , k. prassides , europhys . lett . * 18 * , 219 ( 1992 ) . heiney , g.b.m . vanghan , j.e . fischer , n. coustel , d.e . cox , j.r.d . copley , d.a . neumann , w.a . kamitakahara , k.m . creegan , d.m . cox , j.p . mccauley , jr . , and smith iii , phys . b * 45 * , 4544 ( 1992 ) . f. decremps , l. belhadi , d.l . farber , k.t . moore , f. occelli , m. ganthier , a. polian , d. antonangeli , c. m. aracne - ruddle , and b. amadon , phys . lett . * 106 * , 065701 ( 2011 ) . tsvyashchenko , a.v . nikolaev , a.i . velichkov , a.v . salamatin , l.n . fomicheva , g.k . ryasny , a.a . sorokin , o.i . kochetov , m. budzynski , and k.h . michel , phys . b * 82 * , 092102 ( 2010 ) ; a.v . tsvyashchenko , a.v . nikolaev , a.i . velichkov , a.v . salamatin , l.n . fomicheva , g.k . ryasny , a.a . sorokin , o.i . kochetov and m. budzynski , jetp * 111 * , 627 ( 2010 ) . m. krisch , d.l . farber , r. xu , d. antonangeli , c. m. aracne , a. beraud , t .- c . chiang , j. zarestky , d. y. kim , e. i. isaev , r. ahuja , and b. johansson , pnas ( usa ) * 108 * , 9343 ( 2011 ) . f. decremps , d. antonangeli , b. amadon , and g. schmerber , phys . b * 80 * , 132103 ( 2009 ) . f.f.voronov , l.f.vereshchagin and v.a.goncharova , doklad . nauk sssr , * 135 * , 1104 ( 1960 ) , engl soviet phys . doklady , * 135 * , 1280 ( 1960 ) . c. stassis , t. gould , o.d . mcmasters , and k.a . gschneidner , jr . , and r.m . nicklow , phys . b * 19 * , 5746 ( 1979 ) . w. schranz , a. fuith , p. dolinar , h. warhanek , m. haluska , and h. kuzmany , phys . lett . * 71 * , 1561 ( 1993 ) . a. lundin and b. sundqvist , phys . b * 53 * , 8329 ( 1996 ) bradley and a.p . cracknell , _ the mathematical theory of symmetry in solids _ , ( clarendon , oxford , 1972 ) . p. zielinski and k. parlinski , j. phys . c : solid state phys . * 17 * , 3287 ( 1984 ) . michel , j.r.d . copley , d.a . neumann , phys . 68 * , 2929 ( 1992 ) . d. lamoen , k.h . michel , phys . b * 48 * , 807 ( 1993 ) . moore , l. belhadi , f. decremps , d.l . faber , j.a . bradley , f. occelli , m. gauthier , a. polian , and c.m . aracne - ruddle , acta materialia * 59 * , 6007 ( 2011 ) . samara , j.e . schirber , b. morosin , l.v . hansen , d. loy , and a.p . sylwester , phys . lett . * 67 * , 3136 ( 1991 ) . g. kriza , j .- c . ameline , d. jrome , a. dworkin , h. swarc , c. fabre , d. schtz , a. rassat , p. bbernier , and a. zahab , j. phys . i ( france ) * 1 * , 1361 ( 1991 ) . p.m. chaikin and t.c . lubensky , _ principles of condensed matter physics _ , cambridge university press , cambridge ( 1995 ) . w. gtze and k.h . michel , in _ dynamical properties of solids _ , horton and a.a . maradudin eds . , north - holland publ . comp . 1974 , ch . 9 , zubarev , soviet physics uspekhi * 3 * , 320 ( 1960 ) . , t. gowers , editor , princeton university press ( 2008 ) , p. 27 . h. wagner and j. swift , z. phys . * 239 * , 182 ( 1970 ) . d.j . bergman and b.i . halperin , phys . b * 13 * , 2145 ( 1976 ) . k. gofryk , e.k.h . salje , d.j . safarik , j.l . smith , x. ding , a.c . lawson , p.s . riseborough , b. mihaila , and j.c . lashley ( submitted ) .
|
starting from a model of 4f - electron generated quadrupolar densities on a compressible fcc lattice , the elastic anomalies at the @xmath0 phase transition are studied by means of analytical theory .
the model is taken as representative for the @xmath1 phase transition in ce .
the coupling of the ( linear ) lattice displacements to the square of the quadrupolar orientational density fluctuations renormalizes the elastic constants .
the condensation of the quadrupolar densities into the orientationally ordered @xmath2 structure is studied as function of temperature and pressure .
precursor effects of the transition lead to an anomalous softening of the elastic constant @xmath3 while @xmath4 exhibits no such softening .
the theoretical results are in excellent qualitative agreement with pressure experiments on the elastic constants ( equivalently on sound velocities ) at the @xmath1 transition in ce .
lattice dynamical analogies in theory and striking similarities in experimental results with the @xmath0 transition in c@xmath5 fullerite are discussed .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
ngc891 is one of the best known and studied nearby edge - on spiral galaxies . it is at the distance of 9.5 mpc , is classified as a sb / sbb , and it is often referred to as a galaxy very similar to the milky way @xcite . because of its very high inclination ( [email protected]@xmath3 , * ? ? ? * ) it is very suitable for the study of the distribution and kinematics of the gas above the plane . ngc891 has been the subject of numerous studies at different wavelengths that have led to the detection of various halo components : an extended radio halo @xcite , an extended layer of diffuse ionised gas ( dig ) ( e.g. * ? ? ? * ) and diffuse extra - planar hot gas @xcite . also `` cold '' ism components have been detected in the halo such as hi @xcite , dust @xcite and co @xcite . here we concentrate on the neutral gas and present results from recent third generation hi observations obtained with the westerbork synthesis radio telescope ( wsrt ) . ngc891 was first studied in hi in the late seventies with the wsrt and the presence of neutral gas seen in projection out of the plane was reported @xcite . subsequently , a new study with higher sensitivity showed that the extra - planar emission was very extended , up to 5 kpc from the plane . 3d modeling indicated that such emission was produced by a thick layer of neutral gas rotating more slowly than the gas in the disk @xcite . since then , several other studies have confirmed the presence of neutral gas in the halos of spiral galaxies . it has been detected in edge - on or nearly edge - on systems ( e.g. ugc7321 , @xcite and ngc253 , @xcite ) , as well as in galaxies viewed at different inclination angles ( e.g. ngc2403 , * ? ? ? indications of vertical gradients in rotation velocity have been found in several galaxies also in the ionised gas ( e.g. ngc891 , @xcite , @xcite ; ngc5055 , @xcite ) . in the first part of the paper we present the new hi observations of ngc891 together with a 3d modeling of the hi layer . in the second part ( section 3 ) we study the kinematics of the extra - planar gas and , in the third one ( section 4 ) , we present results from a dynamical model of the extra - planar gas . the new observations of ngc891 have been carried out during the second semester of 2002 with the westerbork synthesis radio telescope ( wsrt ) and a total integration time of about 200 hrs . with this long integration time we reach a sensitivity ( r.m.s . noise per channel = 0.22 mjy / beam at 28@xmath4 km s@xmath1resolution ) that is about a factor 4 better than the previous observations of @xcite . a complete presentation of the observations will be given in oosterloo et al . ( in preparation ) . figure 1 ( right panel ) shows the new total hi map of ngc891 at 28@xmath5 ( @xmath61.3 kpc ) . hi emission is detected at a projected distance of as far as 15 kpc from the plane ( see the spur in the n - w side of the disk ) . the size of the disk itself ( in the plane ) is very similar to that reported by @xcite and @xcite suggesting that we have possibly reached the edge of the hi disk , especially on the n - e side . the emission above the plane , instead , is significantly more extended than in previous observations and almost everywhere extends up to 10 kpc above and below the plane . figure 2 shows the position - velocity plot along the major axis of ngc891 at 28@xmath5 resolution with the rotation curve ( white squares ) overlaid . the rotation curve was derived with the method described in section 3 using only the n - e ( approaching ) side of the galaxy . the kinematics of the receding side within a radius of @xmath66 kpc is very similar to that of the approaching one . at larger radii the velocity is apparently declining . however , we can not be sure that the gas in this extension is at the line of nodes and , therefore , the derivation of a rotation curve in that region is not possible . in the inner regions of the galaxy ( about 1 - 2 kpc ) we confirm the presence of a fast rotation disk or ring ( see also * ? ? ? the total hi map of ngc891 ( figure 1 ) shows extended emission in the direction perpendicular to the plane . is this emission coming from gas located in the halo of the galaxy or is it the result of projection effects ? here we address this question using a 3d modeling technique similar to that of @xcite . in figure 3 we compare some of the observed channels maps of ngc891 ( right hand column ) with models of the hi layer . from the left hand column they are : 1 ) a warp along the line of sight , i.e. a change of the inclination angle of the disk , from 90 to 60 degrees , in the outer parts ; 2 ) a flaring ( increasing thickness ) of the outer disk from a fwhm of 0.5 kpc up to @xmath66 kpc ; 3 ) a two - component model with thin disk + thick ( fwhm@xmath66 kpc ) disk corotating ; 4 ) and 5 ) two - component models with the thick disk rotating more slowly ( 35 km s@xmath1 ) than the disk . the models in columns 4 and 5 differ only for the radial density distribution of the thick component : one ( 5 ) is the same ( scaled ) as that of the gas in the disk and the other ( 4 ) has a depression in the central regions . of the models reported here , only those in columns 4 and 5 give a reasonable reproduction of the data . in particular the warp model does not reproduce the shape of channel maps at 374 and 432 km s@xmath1 , while the flare and corotating model do not reproduce the thin channels of the two top rows ( at 275.5 and 308.5 km s@xmath1 ) . this clearly indicates that the extra - planar emission in ngc891 is produced by a thick ( fwhm@xmath66 kpc ) layer of hi rotating more slowly than the gas in the plane . the halo is possibly relatively denser than the disk in the outer parts ( with a radial density distribution somewhat in between the models in column 4 and 5 ) . in the previous section we have presented some simple galaxy models showing that the emission above the plane in ngc891 comes from gas which is located in the halo region and rotates more slowly ( is lagging ) with respect to the gas in the plane . here we quantify this lag by deriving the 2d rotation velocity field ( rotation surface ) or rotation curves at different heights from the plane . a rotation curve of a spiral galaxy seen edge - on is usually derived using an envelop - tracing method i.e. by taking the highest measured velocity along the line of sight as the rotation velocity ( e.g. * ? ? ? this method was , for instance , used to derive the rotation curve shown in figure 2 . in an ideal situation ( high s / n , galaxy perfectly edge - on , pure rotation , etc . ) the highest velocity measured along the line of sight is indeed produced by the gas at the line of nodes ( i.e. rotation velocity ) . however , the real situation can be very different . in particular it is possible that the emission from the gas at the line of nodes is below detection and , as a consequence , the rotation velocity is underestimated . this could easily happen when observing the gas above the plane where the density is lower and especially in cases of central hidepressions or holes . figure 4 ( left panel ) shows the rotation velocity surface ( contours with constant @xmath7 ) of ngc891 derived with the envelop - tracing method for the n - e side of the galaxy . can this rotation velocity field be fully trusted ? in order to test this , we have constructed model galaxies with known ( flat ) rotation velocity in r and z. figure 4 ( central and right panels ) shows two rotation velocity surfaces obtained from two of these models . these surfaces are obtained using exactly the same fit and fitting parameters as for the one of ngc891 ( left panel ) . the two models have different radial density distributions for the extraplanar gas . these were obtained from the data from above ( n - w side , model a ) and below ( s - e side , model b ) the disk . the results show that the fit works very well in a large part of the galaxy giving a constant value of the velocity in r and z ( uniform lowest gray level ) . the fit does not return the input constant rotation velocity with z only in the inner regions outlined by the dashed lines in figure 4 . there is very little difference between the results obtained with the two different radial density distributions . these results were used to exclude the inner regions in the 2d rotation velocity field of ngc891 ( left panel ) . figure 5 shows the results for ngc891 plotted as rotation curves at various distances from the plane after the uncertain inner points have been removed . the solid line shows the ( smoothed ) rotation curve in the plane while the dashed lines outline the value of the flat part of the rotation curves at the different heights . the halo of ngc891 appears to corotate up to 1.3 kpc , then it starts to lag with respect to the disk , the lagging increasing with height from the plane . given the limited angular resolution , the apparent corotation below 1.3 kpc may be the effect of beam - smearing . the gradient in rotation velocity is roughly 15 km s@xmath1 kpc@xmath1 . the west and east side of the galaxy do not show significant differences . a detailed description of this procedure and results will be given in a forthcoming paper ( fraternali , in preparation ) . the presence of neutral gas in the halo of spiral galaxies is still unexplained . it can be either the result of a galactic fountain @xcite and/or of accretion from the intergalactic medium ( e.g. * ? ? ? the processes involved are probably non - stationary and require a dynamical modeling of the medium above the plane ( see @xcite for a stationary model ) . as a first approach to the problem we have considered a ballistic ( particle - based ) model for the gas expelled from the plane as a result of star formation activity ( see also * ? ? ? we have considered a multicomponent potential with stellar and gaseous exponential disks , a r@xmath8 bulge and a double power - law dark matter halo with adjustable flatness ( in analogy to the galactic models of * ? ? ? a c++ code was written to integrate the orbits of the particles and project them along the line of sight . the particles are shot vertically with different kick velocities from 0 to a maximum v@xmath9 that can depend on r. for each @xmath10 ( time interval ) the position and velocity of the particle is projected along the line of sight and the integration is carried out until the particle falls back to the plane . the output of the code is a model cube that can be compared with the data cubes . these models can be applied to external galaxies viewed at any inclination angle as well as to the milky way ( internal projection ) . here we show a preliminary application to ngc891 . details and improvements of the model to take into account also non - ballistic effects and gas accretion from the intergalactic medium will be presented in a forthcoming paper ( fraternali & binney , in preparation ) . figure 6 shows the total hi map and some representative channel maps for ngc891 , compared with the outputs of two ballistic models . the two models are for a maximum and a minimum disk fit of the rotation curve with b band m / l ratios of 7 and 1 respectively . the parameters of the models have been `` tuned '' to reproduce as well as possible the vertical distribution of the gas and the shape of the channel maps . this figure shows that both ballistic models can qualitatively reproduce the data . however , there seem to be problems in reproducing the thinness of the channel maps around 300 km s@xmath1 i.e. there is not enough lagging in the halo . this is not the effect of the potential considering that little differences are visible when considering extreme choices of the potential shape . this preliminary analysis suggests that the pure ballistic galactic fountain models fail to explain the amount of lagging in galactic halos and that other phenomena such as gas accretion may be important . new hi observations of the edge - on spiral galaxy ngc891 show the presence of extraplanar neutral gas up to distances of 15 kpc from the plane . we have used these data to derive , for the first time , a 2d rotation velocity field ( in r and z ) of an edge - on spiral galaxy . we find that the extra - planar gas is corotating with the gas in the disk up to about 1.3 kpc ; beyond that it rotates more slowly by about 15 km s@xmath1 kpc@xmath1 . dynamical models of ngc891 show that a pure ballistic galactic fountain can qualitatively reproduce the data . however , problems with missing low angular momentum material suggest that other mechanisms such as accretion may also play a role . allen , r.j . , sancisi , r. , baldwin , j.e . 1978 , , 62 , 397 barnab m. , ciotti l. , fraternali f. , sancisi r. 2004 , this conference bregman , j.n . 1980 , , 236 , 577 bregman , j.n . , pildis , r.a . 1994 , , 420 , 570 boomsma , r. , oosterloo t. , fraternali f. , van der hulst j.m . , sancisi r. 2004 , , accepted collins , j.a . , benjamin , r.a . , rand , r.j . 2002 , , 578 , 98 dehnen , w. , binney , j. 2001 , mnras , 294 429 dettmar , r.j . 1990 , , 232 , l15 garcia - burillo , s. , guelin , m. , cernicharo , j. , dahlem , m. 1992 , , 266 , 21 fraternali , f. , oosterloo , t. , sancisi , r. , van moorsel , g. , 2001 , , 562 , l47 fraternali , f. , oosterloo , t. , boomsma r. , swaters r. , sancisi r. 2003 , `` recycling intergalactic and interstellar matter '' , iau , symposium no . 217 , 44 howk , j.c . , savage , b.d . 1999 , , 117 , 2077 van der hulst , t. , sancisi , r. 1988 , , 95 , 1354 hummel , e. , dahlem , m .. van der hulst , j.m . , sukumar , s. 1991 , , 246 , 10 van der kruit , p.c . l. 1981 , , 99 , 298 van der kruit , p.c . , searle , l. 1981 , , 95 , 105 matthews , l.d . , wood , k. 2003 , , 593 , 721 oort , j.h . 1970 , , 7 , 381 pildis a.r . , bregman j.n . , schombert j.m . 1994 , , 423 , 190 rand , r.j . 1997 , , 474 , 129 rand , r.j . 2000 , , 537 , l13 rupen , m.p . 1991 , , 102 , 48 1979 , , 74 , 73 shapiro , p.r . , & field g.b . 1976 , , 205 , 762 sofue , y. , rubin , v. 2001 , ara&a , 39 , 137 swaters , r.a . , sancisi , r. , & van der hulst , j.m . 1997 , , 491 , 140 wakker , b.p . , & van woerden h. 1997 , , 35 , 217
|
we present neutral hydrogen observations of the nearby edge - on spiral galaxy ngc891 which show extended extra - planar emission up to distances of 15 kpc from the plane .
3d modeling of the galaxy shows that this emission comes from halo gas rotating more slowly than the gas in the disk .
we derive the rotation curves of the gas above the plane and find a gradient in rotation velocity of @xmath015 km s@xmath1 kpc@xmath1 .
we also present preliminary results of a galactic fountain model applied to ngc891 .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the green s functions of qcd are the fundamental building blocks of hadron phenomenology @xcite . their infrared behaviour is also known to contain essential information about the realisation of confinement in the covariant formulation of qcd , in terms of local quark and gluon field systems . the landau gauge dyson - schwinger equation ( dse ) studies of refs . @xcite established that the gluon propagator alone does not provide long - range interactions of a strength sufficient to confine quarks . this dismissed a widespread conjecture from the 1970 s going back to the work of marciano , pagels , mandelstam and others . the idea was revisited that the infrared dominant correlations are instead mediated by the faddeev - popov ghosts of this formulation , whose propagator was found to be infrared enhanced . this infrared behaviour is now completely understood in terms of confinement in qcd @xcite , it is a consequence of the celebrated kugo - ojima ( ko ) confinement criterion . this criterion is based on the realization of the unfixed global gauge symmetries of the covariant continuum formulation . in short , two conditions are required by the ko criterion to distinguish confinement from coulomb and higgs phases : ( a ) the massless single particle singularity in the transverse gluon correlations of perturbation theory must be screened non - perturbatively to avoid long - range fields and charged superselection sectors as in qed . ( b ) the global gauge charges must remain well - defined and unbroken to avoid the higgs mechanism . in landau gauge , in which the ( euclidean ) gluon and ghost propagators , @xmath0 are parametrised by the two invariant functions @xmath1 and @xmath2 , respectively , this criterion requires @xmath3 the translation of ( b ) into the infrared enhancement of the ghost propagator ( 2b ) thereby rests on the ghost / anti - ghost symmetry of the landau gauge or the symmetric curci - ferrari gauges . in particular , this equivalence does not hold in linear covariant gauges with non - zero gauge parameter such as the feynman gauge . as pointed out in @xcite , the infrared enhancement of the ghost propagator ( 2b ) represents an additional boundary condition on dse solutions which then lead to the prediction of a conformal infrared behaviour for the gluonic correlations in landau gauge qcd consistent with the conditions for confinement in local quantum field theory . in fact , this behaviour is directly tied to the validity and applicability of the framework of local quantum field theory for non - abelian gauge theories beyond perturbation theory . the subsequent verification of this infrared behaviour with a variety of different functional methods in the continuum meant a remarkable success . these methods which all lead to the same prediction include studies of their dyson - schwinger equations ( dses ) @xcite , stochastic quantisation @xcite , and of the functional renormalisation group equations ( frges ) @xcite . this prediction amounts to infrared asymptotic forms @xmath4 for @xmath5 , which are both determined by a unique critical infrared exponent @xmath6 with @xmath7 . under a mild regularity assumption on the ghost - gluon vertex @xcite , the value of this exponent is furthermore obtained as @xcite @xmath8 the conformal nature of this infrared behaviour in the pure yang - mills sector of landau gauge qcd is evident in the generalisation to arbitrary gluonic correlations @xcite : a uniform infrared limit of one - particle irreducible vertex functions @xmath9 with @xmath10 external gluon legs and @xmath11 pairs of ghost / anti - ghost legs of the form @xmath12 when all @xmath13 , @xmath14 . in particular , the ghost - gluon vertex is then infrared finite ( with @xmath15 ) as it must @xcite , and the non - perturbative running coupling introduced in @xcite via the definition @xmath16 approaches an infrared fixed - point , @xmath17 for @xmath18 . if the ghost - gluon vertex is regular at @xmath19 , its value is maximised and given by @xcite @xmath20 comparing the infrared scaling behaviour of dse and frge solutions of the form of eqs . ( [ infrared - gh_gl ] ) , it has in fact been shown that in presence of a single scale , the qcd scale @xmath21 , the solution with the infrared behaviour ( [ kappaz = kappag ] ) and ( [ genir ] ) , with a positive exponent @xmath22 , is unique @xcite . because of its uniqueness , it is nowadays being called the _ scaling solution_. this uniqueness proof does not rule out , however , the possibility of a solution with an infrared finite gluon propagator , as arising from a transverse gluon mass @xmath23 , which then leads to an essentially free ghost propagator , with the free massless - particle singularity at @xmath24 , _ i.e. _ , @xmath25 for @xmath5 . the constant contribution to the zero - momentum gluon propagator , @xmath26 , thereby necessarily leads to an infrared constant ghost renormalisation function @xmath2 . this solution corresponds to @xmath27 and @xmath28 . it does not satisfy the scaling relations ( [ kappaz = kappag ] ) or ( [ genir ] ) . this is because in this case the transverse gluons decouple for momenta @xmath29 , below the independent second scale given by their mass @xmath23 . it is thus not within the class of scaling solutions considered above , and it is termed the _ decoupling solution _ in contradistinction @xcite . the interpretation of the renormalisation group invariant ( [ alpha_minimom ] ) as a running coupling does not make sense in the infrared in this case , in which there is no infrared fixed - point and no conformal infrared behaviour . without infrared enhancement of the ghosts in landau gauge , the global gauge charges of covariant gauge theory are spontaneously broken . within the language of local quantum field theory the decoupling solution can thus only be realised if and only if it comes along with a higgs mechanism and massive physical gauge bosons . the schwinger mechanism can in fact be described in this way , and it can furthermore be shown that a non - vanishing gauge - boson mass , by whatever mechanism it is generated , necessarily implies the spontaneous breakdown of global symmetries @xcite . early lattice studies of the gluon and ghost propagators supported their predicted infrared behaviour qualitatively well . because of the inevitable finite - volume effects , however , these results could have been consistent with both , the scaling solution as well as the decoupling solution . recently , the finite - volume effects have been analysed carefully in the dyson - schwinger equations to demonstrate how the scaling solution is approached in the infinite volume limit there @xcite . comparing these finite volume dse results with latest @xmath30 lattice data on impressively large lattices @xcite , corresponding to physical lengths of up to @xmath31 fm in each direction , finite - volume effects appear to be ruled out as the dominant cause of the observed discrepancies with the scaling solution . the lattice results are much more consistent with the decoupling solution which poses the obvious question whether there is something wrong with our general understanding of covariant gauge theory or whether we are perhaps comparing apples with oranges when applying inferences drawn from the infrared behaviour of the lattice landau gauge correlations on local quantum field theory ? the latter language is based on a cohomology construction of a physical hilbert space over the indefinite metric spaces of covariant gauge theory from the representations of the becchi - rouet - stora - tyutin ( brst ) symmetry . but do we have a non - perturbative definition of a brst charge ? the obstacle is the existence of the so - called gribov copies which satisfy the same gauge - fixing condition , _ i.e. _ , the lorenz condition in landau gauge , but are related by gauge transformations , and are thus physically equivalent . in fact , in the direct translation of brst symmetry on the lattice , there is a perfect cancellation among these gauge copies which gives rise to the famous neuberger @xmath32 problem . it asserts that the expectation value of any gauge invariant ( and thus physical ) observable in a lattice brst formulation will always be of the indefinite form @xmath32 @xcite and therefore prevented such formulations for more than 20 years now . in present lattice implementations of the landau gauge this problem is avoided because the numerical procedures are based on minimisations of a gauge fixing potential w.r.t . gauge transformations . to find absolute minima is not feasable on large lattices as this is a non - polynomially hard computational problem . one therefore settles for local minima which in one way or another , depending on the algorithm , samples gauge copies of the first gribov region among which there is no cancellation . for the same reason , however , this is not a brst formulation . the emergence of the decoupling solution can thus not be used to dismiss the ko criterion of covariant gauge theory in the continuum . from the finite - volume dse solutions of @xcite it follows that a wide separation of scales is necessary before one can even hope to observe the onset of an at least approximate conformal behaviour of the correlation functions in a finite volume of length @xmath33 . what is needed is a reasonably large number of modes with momenta @xmath34 sufficiently far below the qcd scale @xmath35 whose corresponding wavelengths are all at the same time much shorter than the finite size @xmath33 , @xmath36 it was estimated that this requires sizes @xmath33 of about @xmath37 fm , especially for a power law of the ghost propagator of the form in ( [ infrared - gh_gl ] ) to emerge in a momentum range with ( [ scales ] ) . a reliable quantitative determination of the exponents and a verification of their scaling relation ( [ kappaz = kappag ] ) on the other hand might even require up to @xmath38 fm @xcite . as an alternative to the brute - force method of using ever larger lattice sizes for the simulations might therefore be to ask what one observes when the formal limit @xmath39 is implemented by hand . this should then allow to assess whether the predicted conformal behaviour can be seen for the larger lattice momenta @xmath34 , after the upper bound in ( [ scales ] ) has been removed , in a range where the dynamics due to the gauge action would otherwise dominate and cover it up completely . therefore , the ghost and gluon propagators of pure @xmath30 lattice landau gauge were studied in the strong coupling limit @xmath40 in @xcite . in this limit , the gluon and ghost dressing functions tend towards the decoupling solution at small momenta and towards the scaling solution at large momenta ( in units of the lattice spacing @xmath41 ) as seen in figure [ beta0props ] . the transition from decoupling to scaling occurs at around @xmath42 , independent of the size of the lattice . the observed deviation from scaling at @xmath43 is thus not a finite - size effect . the high momentum branch can be used to attempt fits of @xmath44 and @xmath45 in ( [ infrared - gh_gl ] ) and the data is consistent with the scaling relation ( [ kappaz = kappag ] ) . with some dependence on the model used to fit the data , good global fits are generally obtained for @xmath46 , with very little dependence on the lattice size . for the scaling solution one would expect the running coupling defined by ( [ alpha_minimom ] ) to approach its constant fixed - point value in the strong - coupling limit , and this is indeed being observed for the scaling branch @xcite : the numerical data for the product ( [ alpha_minimom ] ) levels at @xmath47 for large @xmath48 . as expected for an exponent @xmath22 slightly smaller than the value in ( [ kappa_c ] ) , see @xcite , this is just below the upper bound given by ( [ alpha_c ] ) , @xmath49 for @xmath30 . when comparing various definitions of gauge fields on the lattice , all equivalent in the continuum limit , one furthermore observes that neither the estimate of the critical exponent @xmath50 nor the corresponding value of @xmath51 are sensitive to the definition used @xcite . this is in contrast to the decoupling branch for @xmath52 , which is very sensitive to that definition . different definitions , at order @xmath53 and beyond , lead to different jacobian factors . this is well known from lattice perturbation theory where , however , the lattice slavnov - taylor identities guarantee that the gluon remains massless at every order by cancellation of all quadratically divergent contributions to its self - energy . the strong - coupling limit , where the effective mass in ( [ decoupling ] ) behaves as @xmath54 , therefore shows that such a contribution survives non - perturbatively in minimal lattice landau gauge . this contribution furthermore depends on the measure for gauge fields whose definition from minimal lattice landau is therefore ambiguous . one might still hope that this ambiguity will go away at non - zero @xmath55 , in the scaling limit . while this is true at large momenta , it is not the case in the infrared , at least not for commonly used values of the lattice coupling such as @xmath56 or @xmath57 in @xmath30 , as demonstrated in @xcite . it would obviously be desirable to have a brst symmetry on the lattice which could then provide lattice slavnov - taylor identities beyond perturbation theory . in principle , this could be achieved by inserting the partition function of a topological model with brst exact action into the gauge invariant lattice measure . because of its topological nature , this gauge - fixing partition function @xmath58 will be independent of gauge orbit and gauge parameter . the problem is that in the standard formulation this partition function calculates the euler characteristic @xmath59 of the lattice gauge group which vanishes @xcite , @xmath60 neuberger s @xmath32 problem of lattice brst arises because we have then inserted zero instead of unity ( according to the faddeev - popov prescription ) into the measure of lattice gauge theory . on a finite lattice , such a topological model is equivalent to a problem of supersymmetric quantum mechanics with witten index @xmath61 . unlike the case of primary interest in supersymmetric quantum mechanics , here we need a model with non - vanishing witten index to avoid the neuberger @xmath32 problem . then however , just as the supersymmetry of the corresponding quantum mechanical model , such a lattice brst can not break . in landau gauge , with gauge parameter @xmath62 , the neuberger zero , @xmath63 , arises from the perfect cancellation of gribov copies via the poincar - hopf theorem . the gauge - fixing potential @xmath64 $ ] for a generic link configuration @xmath65 thereby plays the role of a morse potential for gauge transformations @xmath66 and the gribov copies are its critical points ( the global gauge transformations need to remain unfixed so that there are strictly speaking only @xmath67sites@xmath68 factors of @xmath69 in ( [ eulerc ] ) ) . the morse inequalities then immediately imply that there are at least @xmath70 such copies in @xmath71 on the lattice , or @xmath72 in compact @xmath73 , and equally many with either sign of the faddeev - popov determinant ( _ i.e. _ , that of the hessian of @xmath64 $ ] ) . the topological origin of the zero originally observed by neuberger in a certain parameter limit due to uncompensated grassmann ghost integrations in standard faddeev - popov theory @xcite becomes particularly evident in the ghost / anti - ghost symmetric curci - ferrari gauge with its quartic ghost self - interactions @xcite . due to its riemannian geometry with symmetric connection and curvature tensor @xmath74 for @xmath71 , in this gauge the same parameter limit leads to computing the zero in ( [ eulerc ] ) from a product of independent gauss - bonnet integral expressions , @xmath75 for each site of the lattice . this corresponds to the gauss - bonnet limit of the equivalent supersymmetric quantum mechanics model in which only constant paths contribute @xcite . the indeterminate form of physical observables as a consequence of ( [ gaussbonnet ] ) is regulated by a curci - ferrari mass term . while such a mass @xmath10 decontracts the double brst / anti - brst algebra , which is well - known to result in a loss of unitarity , observables can then be meaningfully defined in the limit @xmath76 via lhospital s rule @xcite . the 0/0 problem due to the vanishing euler characteristic of @xmath71 is avoided when fixing the gauge only up to the maximal abelian subgroup @xmath77 because the euler characteristic of the coset manifold is non - zero . the corresponding lattice brst has been explicitly constructed for @xmath30 @xcite , where the coset manifold is the 2-sphere and @xmath78 . this indicates that the neuberger problem might be solved when that of compact @xmath73 is , where the same cancellation of lattice gribov copies arises because @xmath79 . a surprisingly simple solution to this problem is possible , however , by stereographically projecting the circle @xmath80 which can be achieved by a simple modification of the minimising potential @xcite . the resulting potential is convex to the above and leads to a positive definite faddeev - popov operator for compact @xmath73 where there is thus no cancellation of gribov copies , but @xmath81 , for @xmath82 gribov copies . as compared to the standard lattice landau gauge for compact @xmath73 their number is furthermore exponentially reduced . this is easily verified explicitly in low dimensional models . while @xmath82 grows exponentially with the number of sites in the standard case as expected , the stereographically projected version has only @xmath83 copies on a periodic chain of length @xmath84 and @xmath85 on a @xmath86 lattice of size @xmath87 in coulomb gauge , for example , and in both cases their number is verified to be independent of the gauge orbit . the general proof of @xmath81 with stereographic projection which avoids the neuberger zero in compact @xmath73 @xcite follows from a simple example of a nicolai map @xcite . applying the same techniques to the maximal abelian subgroup @xmath77 , the generalisation to @xmath88 lattice gauge theories is possible when the odd - dimensional spheres @xmath89 , @xmath90 , of its parameter space are stereographically projected to @xmath91 . in absence of the cancellation of the lattice artifact gribov copies along the @xmath73 circles , the remaining cancellations between copies of either sign in @xmath71 , which will persist in the continuum limit , are then necessarily incomplete , however , because @xmath92 . for @xmath30 this program is straightforward . one replaces the standard gauge - fixing potential @xmath64 $ ] of lattice landau gauge by @xmath93 $ ] , via gauge - transformed links @xmath94 , where @xmath95 = 4 \sum_{x,\mu}\left(1-{\mbox{\small $ { \displaystyle \frac{1}{2}}$ } } { \operatorname{tr}}u^g_{x\mu}\right ) \quad \mbox{and } \quad \label{modlg } \widetilde v_u[g ] = -8 \sum_{x,\mu } \ln\left(\frac{1}{2}+\frac{1}{4}{\operatorname{tr}}u^{g}_{x,\hat{\mu } } \right)\;.\ ] ] the standard and stereographically projected gauge fields on the lattice are defined as @xmath96 the gauge - fixing conditions @xmath97 and @xmath98 are their respective lattice divergences , in the language of lattice cohomology , @xmath99 and @xmath100 . a particular advantage of the non - compact @xmath101 is that they allow to resolve the modified lattice landau gauge condition @xmath98 by hodge decomposition . this provides a framework for gauge - fixed monte - carlo simulations which is currently being developed for the particularly simple case of @xmath30 in 2 dimensions . in the low - dimensional models mentioned above it can furthermore be verified explicitly that the corresponding topological gauge - fixing partition function is indeed given by @xmath102 as expected from @xmath103 . the proof of this will be given elsewhere . comparisons of the infrared behaviour of qcd green s functions as obtained from lattice landau gauge implementations based on minimisations of a gauge - fixing potential and from continuum studies based on brst symmetry have to be taken with a grain of salt . evidence of the asymptotic conformal behaviour predicted by the latter is seen in the strong coupling limit of lattice landau gauge where such a behaviour can be observed at large lattice momenta @xmath104 . there the strong coupling data is consistent with the predicted critical exponent and coupling from the functional approaches . the deviations from scaling at @xmath52 are not finite - volume effects , but discretisation dependent and hint at a breakdown of brst symmetry arguments beyond perturbation theory in this approach . non - perturbative lattice brst has been plagued by the neuberger @xmath32 problem , but its improved topological understanding provides ways to overcome this problem . the most promising one at this point rests on stereographic projection to define gauge fields on the lattice together with a modified lattice landau gauge . this new definition has the appealing feature that it will allow gauge - fixed monte - carlo simulations in close analogy to the continuum brst methods which it will thereby elevate to a non - perturbative level . r. alkofer and l. von smekal , phys . * 353 * , 281 ( 2001 ) . l. von smekal , r. alkofer and a. hauck , phys . rev . lett . * 79 * , 3591 ( 1997 ) . l. von smekal , a. hauck and r. alkofer , annals phys . * 267 * , 1 ( 1998 ) . r. alkofer and l. von smekal , nucl . phys . a * 680 * , 133 ( 2000 ) . lerche and l. von smekal , phys . d * 65 * , 125006 ( 2002 ) . d. zwanziger , phys . d * 65 * , 094039 ( 2002 ) . j. m. pawlowski , d. f. litim , s. nedelko and l. von smekal , phys . lett . * 93 * , 152002 ( 2004 ) ; aip conf . proc . * 756 * , 278 ( 2005 ) . r. alkofer , c. s. fischer and f. j. llanes - estrada , phys . b * 611 * , 279 ( 2005 ) . j. c. taylor , nucl . b * 33 * , 436 ( 1971 ) . c. s. fischer and j. m. pawlowski , phys . d * 75 * , 025012 ( 2007 ) . c. s. fischer , a. maas and j. m. pawlowski , arxiv:0810.1987 [ hep - ph ] . n. nakanishi and i. ojima , _ covariant operator formalism of gauge theories and quantum gravity _ , 27 of lecture notes in physics , world scientific ( 1990 ) . c. s. fischer , a. maas , j. m. pawlowski and l. von smekal , annals phys . * 322 * , 2916 ( 2007 ) ; pos * lat2007 * , 300 ( 2007 ) . a. sternbeck , l. von smekal , d. b. leinweber and a. g. williams , pos * lat2007 * , 340 ( 2007 ) . a. cucchieri and t. mendes , pos * lat2007 * , 297 ( 2007 ) . h. neuberger , phys . b * 175 * , 69 ( 1986 ) ; _ ibid . _ * 183 * , 337 ( 1987 ) . a. sternbeck and l. von smekal , arxiv:0811.4300 [ hep - lat ] . a. sternbeck and l. von smekal , pos * lattice2008 * , 267 ( 2008 ) . m. schaden , phys . rev . * d59 * , 014508 ( 1998 ) . l. von smekal , m. ghiotti and a. g. williams , phys . d * 78 * , 085016 ( 2008 ) . d. birmingham , m. blau , m. rakowski , g. thompson , phys . rept . , 129 ( 1991 ) . l. von smekal , d. mehta , a. sternbeck , a. g. williams , pos * lat2007 * , 382 ( 2007 ) .
|
the infrared behaviour of qcd green s functions in landau gauge has been focus of intense study .
different non - perturbative approaches lead to a prediction in line with the conditions for confinement in local quantum field theory as spelled out in the kugo - ojima criterion .
detailed comparisons with lattice studies have revealed small but significant differences , however . but are nt we comparing apples with oranges when contrasting lattice landau gauge simulations with these continuum results ?
the answer is yes , and we need to change that .
we therefore propose a reformulation of landau gauge on the lattice which will allow us to perform gauge - fixed monte - carlo simulations matching the continuum methods of local field theory which will thereby be elevated to a truly non - perturbative level at the same time . *
landau gauge qcd : functional methods versus lattice simulations international conference on _ selected problems of modern theoretical physics _ ( spmtp08 ) , bogoliubov laboratory of theoretical physics , dubna , russia , 23 27 june , 2008 . ] * lorenz von smekal + _ centre for the subatomic structure of matter , school of chemistry and physics , + the university of adelaide , sa 5005 , australia _
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
massive stars play a key role in our universe . they drive the chemical evolution of galaxies by synthesising most of the heavy elements . their strong stellar winds , radiation feedback , powerful supernova explosions and long gamma ray bursts shape the interstellar medium . they are thought to have played an essential role in reionising the universe after the dark ages and are visible up to large distances . unfortunately , our understanding of the formation and evolution of the most massive stars in the local universe is incomplete . recently it was established that most of the massive stars in the milky way are actually part of a binary star system and that more than @xmath0 of them will exchange mass with a companion during their life @xcite . our understanding of these stars is further hampered by two major controversies . the first one , the cluster age problem , concerns the ages of the youngest star clusters . emerging star clusters are expected to form stars in a time span shorter than the lifetime of their most massive members @xcite . in contrast , the most luminous stars in two of the richest young clusters in our galaxy , the arches and quintuplet clusters , show an apparently large age range . their hydrogen- and nitrogen - rich wolf - rayet ( wnh ) stars appear significantly younger than most of their less luminous o stars . similar age discrepancies are observed in other young stellar systems such as the cygnus ob2 association and the star clusters pismis 24 and ngc 6611 . the second controversy , the maximum stellar mass problem , concerns the stellar upper mass limit . such a upper mass limit is theoretically motivated by the eddington - limit which may prevent stellar mass growth by accretion above a certain mass . observationally , an upper mass limit of about @xmath7 is derived from the individual stellar mass distributions of the arches and the r136 clusters @xcite and from a broader analysis of young stellar clusters @xcite . this result is questioned by a recent analysis of very massive stars in the core of r136 , in which stars with initial masses of up to about @xmath9 are found @xcite . furthermore , recently detected ultra - luminous supernovae in the local universe are interpreted as explosions of very massive stars e.g. sn 2007bi is well explained by a pair - instability supernova from an initially @xmath10 star @xcite . here , we show that both controversies can be resolved by considering a time dependent stellar mass function in young star clusters that accounts for stellar wind mass loss and binary mass exchange . we perform detailed population synthesis calculations of massive single and binary stars that include all relevant physical processes affecting the stellar masses and compare them to observed present day mass functions of the arches and quintuplet clusters . our methods and the observations of the mass functions of the arches and quintuplet clusters are described in sec . [ sec : methods ] . we analyse the arches and quintuplet clusters in sec . [ sec : analysis - arches - quintuplet ] to derive cluster ages and identify possible binary products by fitting our models to the observed mass functions . stochastic sampling effects are investigated in sec . [ sec : stochastic - sampling - effects ] and the implications of our findings for the upper mass limit are explored in sec . [ sec : upper - mass - limit ] . we discuss our results in sec . [ sec : discussion ] and give final conclusions in sec . [ sec : conclusions ] . we analyse the arches and quintuplet clusters in two steps . first , we model their observed stellar mass functions to e.g. determine the initial mass function ( imf ) slopes and the cluster ages . we set up a dense grid of single and binary stars , assign each stellar system in the grid a probability of existence given the initial distribution functions ( cf . [ sec : init - distr - fcts ] ) and evolve the stars in time using our rapid binary evolution code described ( cf . [ sec : binaryc ] ) . present - day mass functions are then constructed from the individual stellar masses at predefined ages . this ensures that all the relevant physics like stellar wind mass loss and binary mass exchange , which directly affects stellar masses , is factored in our mass functions . second , we investigate stochastic sampling effects to e.g. compute the probability that the most massive stars in the arches and quintuplet clusters are binary products . to that end , we randomly draw single and binary stars from initial distribution functions until the initial cluster masses are reached and , again , evolve the drawn stellar systems with our rapid binary evolution code . the set - up of these monte carlo experiments is described in detail in sec . [ sec : mc - experiments ] . the initial distribution functions used in the above mentioned steps are summarised in sec . [ sec : init - distr - fcts ] and an overview of the observations of the arches and quintuplet clusters to which we compare our models is given in sec . [ sec : observations ] . we bin mass functions in a non - standard way to compare them to observations the binning procedure is described in sec . [ sec : binning - procedure ] . the details of our population synthesis code are described in @xcite and @xcite . here , we briefly summarise the most important methods and assumptions that are used to derive our results . we use a binary population code to evolve single and binary stars and follow the evolution of the stellar masses and of other stellar properties as a function of time . our code is based on a rapid binary evolution code @xcite which uses analytic functions @xcite fitted to stellar evolutionary models with convective core overshooting @xcite to model the evolution of single stars across the whole hertzsprung - russell diagram . we use a metallicity of @xmath13 . stellar wind mass loss is applied to all stars with luminosities larger than @xmath14 @xcite . the mass accretion rate during mass transfer is limited to the thermal rate of the accreting star . binaries enter a contact phase and merge if the mass ratio of accretor to donor is smaller than a critical value at the onset of roche lobe overflow @xcite . when two main sequence stars merge , we assume that @xmath15 of the total mass is lost and that @xmath15 of the envelope mass is mixed with the convective core @xcite . photometric observations of star clusters can not resolve individual binary components . in order to compare our models to observations we assume that binaries are unresolved in our models and determine masses from the combined luminosity of both binary components utilising our mass - luminosity relation . hence unresolved , pre - interaction binaries contribute to our mass functions . we concentrate on main sequence ( ms ) stars because stars typically spend about @xmath16 of their lifetime in this evolutionary phase ; moreover , our sample stars used for comparison are observationally colour - selected to remove post - ms objects . if a binary is composed of a post - ms and a ms star , we take only the ms component into account . we assume that primary stars in binaries and single stars have masses @xmath17 distributed according to a power law initial mass function ( imf ) with slope @xmath18 , @xmath19 in the mass range @xmath20 to @xmath21 ( where @xmath22 is a normalisation constant ) . secondary star masses , @xmath23 , are taken from a flat mass ratio distribution , i.e. all mass ratios @xmath24 are equally probable @xcite . the initial orbital periods for binaries with at least one o - star , i.e. a primary star with @xmath25 , mass ratio @xmath26 and a period @xmath27 are taken from the distribution of stars in galactic open clusters @xcite . the initial periods of all other binaries follow a flat distribution in the logarithm of the orbital period @xcite . orbital periods are chosen such that all interacting binaries are taken into account , i.e. the maximum initial orbital separation is @xmath28 ( @xmath29 ) . binaries with wider orbits would be effectively single stars . to address stochastic sampling , we perform monte carlo simulations of star clusters and investigate the probability that the most massive star in a star cluster is a product of binary evolution as a function of cluster mass , age , binary fraction and imf slope . we assume that all stars are coeval and that every star cluster forms from a finite supply of mass with stellar masses stochastically sampled from initial distribution functions . while single stars are sampled from the initial mass function , binary stars are chosen from a larger parameter space of primary and secondary masses and orbital periods . this larger parameter space is better sampled in clusters of higher mass . we draw single and binary stars for a given binary fraction from the initial distribution functions of primary mass , secondary mass and orbital period until a given initial cluster mass , @xmath30 , is reached . here we consider only stars with masses in the range of @xmath20@xmath21 . the true cluster masses are therefore larger if stars below @xmath31 are added . including these stars according to a kroupa imf @xcite , increases the true cluster mass by @xmath32 and @xmath33 for high mass ( @xmath34 ) imf slopes of @xmath35 and @xmath36 , respectively . after the stellar content of a cluster is drawn , we evolve the stars in time to analyse whether the most massive stars at a given cluster age result from close binary interaction . repeating this experiment @xmath37 times provides the probability that the most massive cluster star formed from binary interactions , how long it takes on average until the most massive star is a product of binary evolution , @xmath38 , and how many stars have on average a mass larger than that of the most massive cluster star ( @xmath39 ) which did not accrete from a companion , @xmath40 . the most massive star that did not accrete from a companion can be a genuine single star or a star in a binary where binary mass transfer has not yet happened . from here on we refer to this star as ` the most massive genuine single star ' . we evolve and distribute stars as described in secs . [ sec : binaryc ] and [ sec : init - distr - fcts ] . to compare our monte carlo simulations with observations of the arches and quintuplet clusters , we need to know the corresponding cluster masses @xmath41 in our monte carlo experiments . we use imf slopes of @xmath42 as later determined in sec . [ sec : comparison - mass - functions ] for both clusters . the observations used for comparison are complete for masses @xmath43 , corresponding to @xmath44 and @xmath45 stellar systems in the arches and quintuplet and integrated masses of stars more massive than @xmath46 of @xmath47 and @xmath48 , respectively . our best fitting monte carlo models of the central regions of arches and quintuplet with primordial binary fractions of @xmath49 and @xmath50 have @xmath51 stellar systems with an integrated ( initial ) mass of @xmath52 and @xmath53 stellar systems with an integrated initial mass of @xmath54 , respectively . these models correspond to initial cluster masses of @xmath55 and @xmath56 , respectively , in stars with @xmath57 . we assume that binaries are resolved in our monte carlo calculations , contrary to when we model mass functions in order to compare to observed mass functions . this is because we make theoretical predictions and are thus interested in individual masses of all stars regardless of them being in a binary or not . the observed present - day mass functions of the arches and quintuplet clusters were obtained from naos / conica ( naco ) photometry at the vlt . the arches cluster was observed in 2002 over a field of view ( fov ) of @xmath58 . the centre of the quintuplet cluster was imaged with naco over a fov of @xmath59 in 2003 and 2008 , which allowed the construction of a membership source list from proper motions . both data sets were obtained in the @xmath60 ( @xmath61 ) and @xmath62 ( @xmath63 ) passbands . the colour information is used to remove likely blue foreground interlopers , red clump and giant stars towards the galactic center line of sight . details can be found in @xcite for the arches cluster and in for the quintuplet cluster . in the case of the arches cluster , the known radial variation of the extinction is removed prior to individual mass determination employing the extinction law of @xcite . masses are then derived from the @xmath62 magnitudes of each star by comparison with a @xmath64 geneva isochrone . in the case of the quintuplet , the better photometric performance allowed all sources to be individually dereddened to a @xmath65 padova ms isochrone using the recently updated near - infrared extinction law towards the galactic center line of sight @xcite . as detailed in , isochrone ages of @xmath66 and @xmath67 do not significantly alter the shape and slope of the constructed mass function . all mass determinations are based on solar metallicity evolution models . with the aim to minimise any residual field contamination , only the central @xmath68 or @xmath69 of the arches and @xmath70 or @xmath71 of the quintuplet ( at an assumed distance of @xmath72 to the galactic center ; @xcite ) were selected to construct the mass functions . for the arches cluster , this radial selection corresponds approximately to the half - mass radius , which implies that the mass projected into this annulus is of the order of @xmath73 . in the quintuplet cluster , the total mass is estimated to be @xmath74 within the considered @xmath71 radius . the mass functions in the central regions of arches and quintuplet have slopes that are flatter than the usual salpeter slopes , most likely because of mass segregation . the most massive stars in the arches and quintuplet clusters are hydrogen and nitrogen rich wnh stars . as reliable masses can not be derived for these wolf - rayet ( wr ) stars from photometry alone , and as several of the wrs in the quintuplet suffered from saturation effects , the most massive stars are excluded from the mass functions . this affects @xmath75 wnhs stars with uncertain masses in arches and @xmath66 ( plus @xmath76 post - ms , carbon rich wr stars ) in quintuplet . these wnh stars are expected to contribute to the high mass tail of the arches and quintuplet mass functions . following @xcite , we employ a binning procedure that renders the observed mass functions independent of the starting point of the bins . we shift the starting point by one tenth of the binsize and create mass functions for each of these starting points . we use a fixed binsize of @xmath77 to ensure that the number of stars in each bin is not too small and does not introduce a fitting bias @xcite . each of these ten mass functions with different starting points is shown when we compare our mass functions to observations . this procedure results in lowered number counts in the highest mass bins because only the most massive stars will fall into these bins as seen in the power - law mass function ( black dotted lines in fig . [ fig : observations ] ) where a kink is visible around @xmath78 ( left panel ) and @xmath79 ( right panel ) respectively ( cf . convolution of a truncated horizontal line with a box function with the width of the bins ) . this kink is caused by the binning procedure . importantly , the observations , our models and the power - law mass functions in figs . [ fig : observations ] and [ fig : sfh ] are binned identically to render the mass functions comparable . for a meaningful comparison of the modelled with the observed mass functions , the star cluster and the observations thereof need to fulfil certain criteria . they should be * between @xmath80 and @xmath81 in age such that the wind mass loss peak in the mass function is present ( see below and * ? ? ? * ) , * massive enough such that the mass function samples the largest masses * homogeneously analysed , with a complete _ present - day _ mass function above @xmath82 . both , the arches and quintuplet clusters fulfil all criteria and are therefore chosen for our analysis . other possible star clusters , which can be analysed in principle , are the galactic center cluster , ngc 3603 yc , westerlund 1 and r136 in the large magellanic cloud . trumpler 14 and trumpler 16 in the galactic carina nebula are not massive enough and rather an ob star association with stars of different ages , respectively . for westerlund 1 @xcite and ngc 3603 yc @xcite , present - day mass functions were recently derived . a brief inspection of these results shows that both clusters may be suitable for an analysis as performed here for the arches and quintuplet clusters . we will investigate this further in the near future . possibly , ngc 3603 yc is too young such that its mass function is not yet altered enough by stellar evolution to apply our analysis . the initially most massive stars in a cluster end their life first . this depopulates the high mass end of the stellar mass function . before that , however , massive stars lose a significant fraction of their initial mass because of strong stellar winds ; e.g. our @xmath21 star at solar metallicity loses about @xmath83 during core hydrogen burning . stellar wind mass loss shifts the top of the mass function towards lower masses and a peak accumulates near its high mass end ( figs . [ fig : models]a and [ fig : models]b ) . the location of the peak depends strongly on the cluster age and provides a clock to age - date a star cluster . stars in close binary systems exchange mass with their companion either by mass transfer or in a stellar merger . _ a fraction _ of stars gain mass , producing a tail at the high mass end of the mass function ( hatched regions in fig . [ fig : models]b ) which extends beyond the most massive single - stars ( fig . [ fig : models]a ) by up to a factor of about two . the mass gainers _ appear _ younger than genuine single stars because their convectively mixed stellar core grows upon mass accretion and mixes fresh fuel into their centre , thereby turning their clock backwards . furthermore , the most massive gainers reach masses which , when interpreted as single stars , have lifetimes that are shorter than the cluster age they are the massive counterpart of classical blue straggler stars @xcite . the mass functions of the cores of the arches ( @xmath84 ) and quintuplet ( @xmath85 ) clusters reveal both the stellar wind mass loss peak and the tail because of binary mass exchange . compared to a power - law , we find that the arches and quintuplet mass functions are overpopulated in the ranges @xmath86 to @xmath87 ( @xmath88@xmath89 ) and @xmath90 to @xmath91 ( @xmath92@xmath93 ) , respectively ( fig . [ fig : observations ] ) . these peaks are well reproduced by our models ( figs . [ fig : models ] and [ fig : observations ] ) . we can thus determine the cluster age because among the stars in the peak are the initially most massive stars that are ( a ) still on but about to leave the main sequence and ( b ) unaffected by binary interactions ( we refer to them as turn - off stars in analogy to their position close to the turn - off in a hertzsprung - russell diagram ) . the majority of stars in the peak are turn - off stars but there are small contributions from unresolved and post - interaction binaries ( see * ? ? ? a correction for wind mass loss then reveals the initial mass of the turn - off stars and hence the age of the cluster . we can correct for wind mass loss by redistributing the number of excess stars in the peak @xmath94 such that the mass function is homogeneously filled for masses larger than those of the peak stars up to a maximum mass , the initial mass of the turn - off stars @xmath95 . the number of excess stars is then @xmath96 where @xmath97 is the present - day mass of the turn - off stars , which can be directly read - off from the upper mass end of the peak , and @xmath98 the initial mass function as defined in eq . ( [ eq : gammadef ] ) . the initial mass of the turn - off stars @xmath95 and hence the cluster age follows from integrating eq . ( [ eq : number - excess - stars ] ) , @xmath99 the normalisations , @xmath22 , of the mass functions to be filled up with the excess stars , @xmath94 , ( dotted , power - law functions in fig . [ fig : observations ] ) are @xmath100 and @xmath101 for arches and quintuplet , respectively , with slopes of @xmath42 in both cases ( see discussion below for why the mass functions are so flat ) . it is difficult to read - off the exact value of @xmath97 from the observed mass functions because of the binning . but from binning our modelled mass functions in the same way as the observations , we know that @xmath97 corresponds to the mass shortly after the peak reached its local maximum ( cf . the vertical dashed lines in figs . [ fig : models ] and [ fig : observations ] ) . depending on the exact value of @xmath97 , @xmath102@xmath103 in arches and @xmath104@xmath105 in quintuplet , we find @xmath106@xmath107 and @xmath76@xmath108 excess stars in the peaks of the arches and quintuplet mass functions , respectively . these numbers of excess stars result in turn - off masses @xmath95 of @xmath109@xmath110 and @xmath111@xmath112 and hence ages of @xmath113@xmath114 and @xmath115@xmath116 for the arches and quintuplet clusters , respectively . these are only first , rough age estimates that will be refined below and their ranges stem from the uncertainty in reading - off @xmath97 from the observed mass functions . from the difference between the initial and present day masses of the turn - off stars in arches and quintuplet , we can directly measure the amount of mass lost by these stars through stellar winds . the turn - off stars in arches lost about @xmath106@xmath117 and the turn - off stars in quintuplet about @xmath118@xmath119 during their ms evolution . this is a new method to measure stellar wind mass loss which does not require measurements of stellar wind mass loss rates and can therefore be used to constrain these . more accurately , we determine the ages of the arches and quintuplet clusters by fitting our population synthesis models ( sec . [ sec : methods ] ) to the observed mass functions . first , we fit power - law functions to the observed mass functions in mass regimes in which they are observationally complete and not influenced by stellar wind mass loss ( @xmath120 and @xmath121 , respectively ) . binary effects are also negligible because stars with such masses are essentially unevolved at the present cluster ages . this fit gives the normalisation and a first estimate of the slope of the mass function . we then vary the mass function slope , the cluster age and the primordial binary fraction in our models simultaneously such that the least - square deviation from the observations is minimised . our best - fit models are shown in fig . [ fig : observations ] together with the observed mass functions . we find slopes of @xmath42 , ages of @xmath122 and @xmath123 and primordial binary fractions of @xmath49 and @xmath50 for the arches and quintuplet cluster , respectively . the binary fractions are less robust and may be the same within uncertainties because we do not take the uncertain masses of the wnh stars into account . while our mass function fits contribute to the age uncertainties by only @xmath124 , its major part , @xmath125 and @xmath126 for arches and quintuplet , respectively , is due to observational uncertainties in stellar masses of @xmath127 ( sec . [ sec : discussion ] ) . massive stars tend to sink towards the cluster cores because of dynamical friction ( mass segregation ) , thereby flattening the mass function of stars in the core . the derived mass function slopes of @xmath42 are flatter than the typical salpeter slope of @xmath36 @xcite because we investigate only the mass segregated central regions of both clusters , i.e. a subsample of stars biased towards larger masses . in our models ( fig . [ fig : models ] ) , the tail of the arches mass function contains about @xmath128 unresolved , pre - interaction binaries with @xmath129 ( @xmath130 ) and about @xmath32 with @xmath131 ( @xmath132 ) . for quintuplet , the fraction of unresolved , pre - interaction binaries is about @xmath32 with @xmath133 ( @xmath134 ) and about @xmath15 with @xmath135 ( @xmath136 ) . the binary fraction among the rejuvenated binary products in the tails is about @xmath137 in our arches and @xmath0 in our quintuplet model , where the remaining stars are single star binary products , i.e. merger stars . previously estimated ages for the arches and quintuplet clusters lie in the range @xmath138@xmath139 and @xmath138@xmath140 , respectively . within these ranges , the age discrepancy between the most luminous cluster members , the wn and the less luminous o stars , accounts for about @xmath141 and @xmath142 , respectively , which is eliminated by our method . our ages of @xmath122 and @xmath123 for the arches and quintuplet clusters , respectively , agree with the ages derived from the o stars and dismiss the proposed younger ages from the brightest stars as a result of neglecting binary interactions . the most famous member of the quintuplet , the pistol star , is such an example because it appears to be younger than @xmath143 assuming single - star evolution @xcite . the initial mass of the primary star , the mass ratio and the orbital period of a binary system determine when mass transfer starts , with more massive and/or closer binaries interacting earlier . stochastic effects caused by the limited stellar mass budget prevent the formation of all possible binaries in a stellar cluster , i.e. binaries with all possible combinations of primary mass , mass ratio and orbital period . the likelihood that a binary in a given cluster interacts , e.g. after @xmath64 , and that the binary product becomes then the most massive star depends thus on the number of binary stars in that cluster , hence on the total cluster mass . using monte carlo simulations , we investigate the influence of stochastic sampling and binary evolution on the most massive stars in young star clusters ( cf . [ sec : mc - experiments ] ) . the galactic star cluster ngc 3603yc contains ngc 3603-a1 , a binary star with component masses @xmath144 and @xmath145 in a @xmath146 orbit @xcite . an initially @xmath147 binary in a @xmath146 orbit starts mass transfer @xmath148 after its birth according to the non - rotating models of . this is the time needed for the @xmath149 primary star to fill its roche lobe as a result of stellar evolutionary expansion . this time provides an upper age estimate for ngc 3603 yc . after mass transfer , the secondary star will be the most massive star in the cluster . were ngc 3603-a1 in a closer orbit , it could already be a binary product today . to find the probability that the most massive star in a cluster of a given age is a binary product , we investigate how many close binaries are massive enough to become the most massive star by mass transfer . were the cluster a perfect representation of the initial stellar distribution functions , we could use these functions to derive the probability directly . however , the finite cluster mass and hence sampling density must be considered for comparison with real clusters . returning to the example of ngc 3603 yc , were the cluster of larger total mass , its binary parameter space would be better sampled and its most massive star might already be a binary interaction product . for perfect sampling , i.e. infinite cluster mass , the time until a binary product is the most massive star tends towards zero . the idea that the most massive star in a star cluster may be a binary product resulted from the first discovery of blue straggler stars @xcite . it was proposed that blue stragglers might stem from binary mass transfer and/or stellar collisions @xcite . stellar population synthesis computations including binary stars then showed that this is indeed possible . here , we show using the binary distribution functions of @xcite that the formation of blue stragglers by binary interactions prevails up to the youngest and most massive clusters and quantify it for the arches and quintuplet clusters . in fig . [ fig : average - times - until - interaction ] , we show the average time @xmath38 after which the most massive star in a star cluster is a product of binary evolution as a function of the cluster mass @xmath30 for two different primordial binary fractions @xmath150 . the error bars are @xmath151 standard deviations of @xmath37 monte carlo realisations . the slope of the mass function is @xmath42 , appropriate for the mass - segregated central regions of both arches and quintuplet . the more massive a star cluster , i.e. the more stars populate the multidimensional binary parameter space , the shorter is this average time because the probability for systems which interact early in their evolution is increased . for less massive clusters @xmath38 increases and the statistical uncertainty grows . for example if @xmath152 , there are only about @xmath153 binaries in which at least one star has a mass above @xmath46 ( for @xmath42 and @xmath154 ) . the same reasoning holds for different binary fractions : the higher the binary fraction , the more binaries and hence a shorter average time until the most massive star results from binary interactions . until the most massive star in a star cluster is a product of close binary evolution as a function of cluster mass for two primordial binary fractions , @xmath155 , and a mass function slope @xmath42 . for steeper , salpeter - like mass functions see fig . [ fig : average - times - until - interaction-2 ] . the error bars are the standard deviation of @xmath37 realisations of each cluster . the star symbols indicate the age and _ central _ cluster mass of arches and quintuplet as derived in this work.,scaledwidth=46.0% ] with a salpeter mass function ( @xmath36 , * ? ? ? * ) the average time until the most massive star is a binary product increases compared to @xmath42 ( fig . [ fig : average - times - until - interaction-2 ] ) because there are fewer massive binaries that interact to form the most massive star . assume a @xmath65 old star cluster has a mass function slope of @xmath36 , a total mass in stars above @xmath31 of @xmath156 ( i.e. a true cluster mass of @xmath157 if stars below @xmath31 follow a kroupa imf ; see sec . [ sec : mc - experiments ] ) and a primordial binary fraction of @xmath158 . from fig . [ fig : average - times - until - interaction-2 ] , we can then read - off after which time the most massive star is expected to be a binary product , namely after @xmath159 . but for a steeper mass function with a salpeter slope of @xmath36 . the binary parameter space spanned by the initial mass ratios and initial orbital separations for massive primary stars is now less populated , resulting in increased average times until the most massive star is a binary product . similarly , the standard deviations increase.,scaledwidth=46.0% ] the central regions of the arches and quintuplet clusters have masses of @xmath160 and @xmath161 in stars more massive than @xmath31 ( sec . [ sec : mc - experiments ] ) and ages of @xmath162 and @xmath123 ( sec . [ sec : comparison - mass - functions ] ) , respectively . from fig . [ fig : average - times - until - interaction ] , we expect that the most massive star in the arches cluster is a binary product after @xmath163 and after @xmath164 in the quintuplet cluster . in fig . [ fig : prob - numbers ] we show the probability that the most massive star is a binary product and the average number of stars that are more massive than the most massive genuine single star as a function of cluster age for different cluster masses and two different binary fractions . the imf slope is @xmath42 . the corresponding probabilities and average numbers for a salpeter ( @xmath36 ) mass function are shown in fig . [ fig : prob - numbers-2 ] . again , the error bars are @xmath151 standard deviations from @xmath37 monte carlo experiments . returning to the above mentioned example star cluster ( @xmath156 ) : from fig . [ fig : prob - numbers-2 ] , we find that the most massive star is a binary product with a probability of @xmath165 and that the most massive @xmath166 stars are expected to be binary products for the exemplary cluster age of @xmath65 . given the ages of the arches and quintuplet clusters , we find a probability of @xmath167 that the most massive star in each cluster is a binary product , with the most massive @xmath168 and @xmath169 stars being products of binary evolution in arches and quintuplet , respectively . this is compatible with the number of wnh stars in the cores of arches and quintuplet , which are the most luminous and hence most massive stars in these clusters , implying they are massive blue stragglers . data from two star clusters provide the current evidence for the existence of an upper stellar mass limit around @xmath7 : the arches cluster in the galactic center @xcite and the r136 cluster in the large magellanic cloud @xcite . however , according to our analysis an upper mass limit can not be derived from the arches cluster because ( a ) it is too old , hence the most massive stars already exploded , and ( b ) its present - day high mass star population is dominated by binary products . the situation might be different in the r136 cluster : current age estimates lie in the range @xmath20@xmath170 @xcite . in the following we assume that the cluster is young enough such that even the most massive stars have not yet evolved off the main sequence , to explore what we can learn from r136 about a possible stellar upper mass limit . four stars in r136 with initial masses of @xmath8@xmath9 appear to exceed the currently discussed upper mass limit of @xmath7 @xcite . either these stars were born with masses exceeding @xmath7 or gained mass from other stars e.g. by binary interactions ( this work ) or dynamically induced stellar mergers . from our monte carlo simulations ( sec . [ sec : stochastic - sampling - effects ] ) , we can not judge with high enough confidence whether the most massive star in r136 is expected to be a binary product or not because of the uncertain age of r136 . r136 has an imf with approximately a salpeter slope @xmath36 @xcite and its cluster mass is @xmath171@xmath172 . from our monte carlo simulations of star clusters with binary fractions of @xmath50 and cluster masses @xmath173 of @xmath174 and @xmath175 ( fig . [ fig : prob - numbers-2 ] ) , we find that the most massive star is expected to be a binary product after @xmath141 with a probability of @xmath176 and @xmath177 , respectively . the probabilities increase to @xmath178 and @xmath179 , respectively , for a cluster age of @xmath64 and are larger than @xmath180 for an age of @xmath181 . so if the cluster is older than about @xmath64 , the most massive star is likely a binary product ( note that our calculations are for a metallicity of @xmath13 while the r136 cluster in the large magellanic cloud has a lower metallicity so the above numbers will slightly change for the appropriate metallicity but are good enough for this estimate ) . because it is not clear whether the most massive star in r136 is a binary product or not , we explore both possibilities . with monte carlo simulations , we investigate the likelihood of finding the observed @xmath182@xmath9 stars @xcite in r136 . we randomly sample r136-like star clusters for different adopted stellar upper mass limits @xmath183 using the observed imf slope @xcite of @xmath36 , a binary fraction of @xmath0 and that r136 contains about @xmath184 stellar systems more massive than @xmath46 @xcite . we then compute the average number of stars that are initially more massive than a given mass @xmath185 , @xmath186 , and the probability that at least one star is more massive than @xmath185 , @xmath187 , by repeating each experiment @xmath37 times ( the quoted errors are @xmath151 standard deviations ) . the average numbers and probabilities for the case that binary interactions did not yet take place are summarised in table [ tab : upper - mass - limit - single - stars ] . for the case that binary interactions already took place , we assume that all massive binaries with initial periods @xmath188 interact by mass transfer ( which happens within @xmath138@xmath189 ) and that the post - interaction mass is @xmath16 of the total binary mass . the corresponding average numbers and probabilities for this case can be found in table [ tab : upper - mass - limit - binary - stars ] . in both tables . [ tab : upper - mass - limit - single - stars ] and [ tab : upper - mass - limit - binary - stars ] , we also give the results for less massive clusters with @xmath190 and @xmath191 stellar systems initially exceeding @xmath46 . through binary mergers , stars of up to @xmath192 can be produced if the star formation process stops at an upper mass of @xmath7 . however , this scenario requires equal mass o - type binaries which are rare . we find that with an upper mass of @xmath7 , the probability of forming stars in excess of @xmath193 in r136 is zero of the total system mass , i.e. @xmath194 for @xmath195 . with an upper mass limit of @xmath196 , the probability of forming at least one star of mass @xmath197 increases to @xmath198 , and for an upper mass limit of @xmath199 , the probability of forming at least one star exceeding @xmath193 and @xmath192 is @xmath200 and @xmath201 , respectively . so , @xmath202 provides a lower limit on the maximum stellar birth mass . it is also unlikely that the upper mass limit exceeds @xmath203 because then the probability of forming one star above @xmath203 by binary mass transfer increases to @xmath204 but such massive stars are not observed . we conclude that an upper mass limit in the range of about @xmath11@xmath203 is needed to explain the most massive stars in r136 by binary evolution . dynamically induced stellar coalescence was proposed as a mechanism to produce the very massive stars in r136 . however , n - body simulations of dynamically induced stellar coalescence typically only produce one to two stars exceeding @xmath202 for r136 adopting an upper mass limit of @xmath7 @xcite , i.e. fewer than observed in r136 . furthermore , the rate of dynamically induced mergers in these simulations should be viewed as an upper limit only . observational results indeed favour a larger half - mass radius , hence a lower density compared to the simulation assumptions . similarly , adopting the recent measurements of the orbital distributions of massive binaries further decreases the number of possible dynamical mergers that can overcome the @xmath7 limit by a factor of @xmath205 to @xmath206 . it appears thus unlikely that dynamically induced stellar coalescence is sufficiently efficient to explain the origin of the very massive stars in r136 if the upper mass limit is @xmath7 . as mentioned above , it is also possible that the four massive stars in r136 were born with their deduced initial masses and did not gain mass by other means . this provides then an upper limit on the maximum stellar birth mass . the most massive star found in r136 has an initial mass of @xmath207 @xcite hence , the upper mass limit has to be at least of this order , i.e. @xmath208 . this initially @xmath9 star allows us to exclude an upper mass limit of @xmath209 with @xmath210 confidence because we expect to find @xmath211 stars that initially exceed @xmath12 in this case while no such star is observed . however , it becomes more difficult to exclude an upper mass limit of @xmath12 or less because the probability of finding _ no _ star that initially exceeds @xmath203 ( @xmath212 ) is about @xmath213 ; in other words , no star would initially exceed @xmath203 in about every tenth r136-like star cluster for an upper mass limit of @xmath12 . the probability increases further to @xmath214 for an upper mass limit of @xmath215 . we conclude that stochastic sampling effects are important even in the richest massive star clusters in the local group . altogether , we find that current data does not exclude an upper mass limit as high as @xmath216@xmath12 if binary interactions are neglected . however , the most massive star in r136 is a binary product with a probability of @xmath217@xmath50 . including effects of close binary evolution , an initial stellar upper mass limit of at least @xmath202 is required to explain the observed stars with apparent initial masses of about @xmath192 . the upper mass limit is thus in the range @xmath11@xmath12 , thereby solving the maximum mass problem . there are several sources of uncertainty that affect theoretical and observed mass functions and hence e.g. our cluster ages derived from them . it is important to understand the uncertainties to estimate their influence on our conclusions and the derived quantities . the conclusion that binary effects shape the upper end of the stellar mass function remains unaffected . in sec . [ sec : uncertainties - models ] , we discuss modelling uncertainties because of the fitting procedure , stellar wind mass loss , binary star evolution and rotation . observational uncertainties like the influence of different reddening laws on derived stellar masses of stars in the galactic center are discussed in sec . [ sec : uncertainties - observations ] . we discuss the influence of dynamical interactions on stellar mass functions in sec . [ sec : dynamical - effects ] . star formation histories that are different from single starbursts are considered in sec . [ sec : sfh ] to investigate whether such scenarios are also consistent with the observed age spread among the most massive stars and the resulting stellar mass functions . stars in the wind - mass - loss peak of the mass function will very soon leave the main sequence . the mass of these turn - off stars is a sensitive function of cluster age especially for massive stars , which radiate close to the eddington limit . massive stars have lifetimes which depend only weakly on mass and hence a small change in age corresponds to a large change in mass . we can not reproduce the observed mass functions of arches and quintuplet if we change the age of our models in fig . [ fig : observations ] by more than @xmath218@xmath219 . we therefore adopt @xmath219 as the age uncertainty associated with our fitting . the initial binary fraction is best constrained by the number of stars in the mass function tail because it consists only of either post - interaction or pre - interaction , unresolved binaries . in contrast , the wind - mass - loss peak changes little with the binary fraction . our observational sample is limited by the exclusion of wnh stars in both arches and quintuplet because no relieable masses of the wnh stars could be determined ( see sec . [ sec : observations ] . we thus can not determine the primordial binary fractions accurately , especially because the tail of the quintuplet mass function is not very pronounced . increasing the age of our quintuplet model by @xmath220 allows for @xmath49 binaries while maintaining a satisfactory , albeit slightly inferior to the best , fit to the mass function . both clusters are thus consistent with having the same primordial binary fraction . our wind mass loss prescription slightly underestimates stellar winds compared to the latest predictions . compared to the most recent stellar evolution models of that use the prescriptions of , we find that our turn - off masses agree to within @xmath221 for initial masses @xmath222 , while in more massive stars our turn - off mass is up to @xmath223 larger , mainly because of the applied wolf - rayet wind mass loss rates in . the widths of the bins in our model mass functions are @xmath224 , i.e.masses differ by about @xmath15 from bin to bin . the observed mass functions have bin sizes of @xmath225 , i.e. masses are different by @xmath226 from bin to bin . wind mass loss prescriptions that lead to stellar masses at the end of the ms that differ by only a few percent result in indistinguishable mass functions our mass functions and conclusions are therefore essentially independent of whether the empirical wind mass loss prescription of or the theoretical prescription of are used . augmenting our wind loss rate by @xmath0 , we find that an initially @xmath227 star has a turn - off mass of about @xmath228 which matches the recent stellar models by ( compared to @xmath229 in our standard model ) . with the enhanced wind mass loss rate , the arches wind - mass - loss peak corresponds to an initially @xmath230 star with a main - sequence lifetime of @xmath231 , compared to @xmath232 and @xmath114 respectively in our standard model . the quintuplet wind - mass - loss peak comes from initially @xmath233 stars for which the uncertainty in wind mass loss is @xmath234 . our quintuplet age estimate is thus robust with respect to the wind mass loss rate uncertainty . our understanding of binary star evolution in general is subject to uncertainties . uncertainties that directly influence the shape of the mass function tails are discussed in @xcite . a further , more quantitative discussion of uncertainties in binary star evolution is found in @xcite and @xcite . here , we restrict ourselves to ms stars , i.e. to mergers of two ms stars and mass transfer onto ms stars . mergers that involve a post - ms star form a post - ms object and are thus not considered here . we assume that two ms stars merge if the mass ratio of the accretor to donor star is less than @xmath235 . this threshold is calibrated against the detailed binary models of and is of limited relevance to our results : if a binary does not merge but instead transfers mass ( or vice versa ) , the accretor becomes massive because the mass transfer efficiency of ms stars is high . in either case , the mass gainer will be a massive star @xcite . the expected binary fraction of stars in the tail of the mass functions however changes : a lower critical mass ratio leads to fewer ms mergers and hence to a higher binary fraction and vice versa . the amount of rejuvenation of ms mergers is determined by the amount of mixing of fresh fuel into the core of the merger product and determines by how much the lifetime of the merger product is prolonged . the more rejuvenation , the longer the remaining ms lifetime and the more mergers are expected to be found . we assume that a fraction of @xmath15 of the envelope is mixed into the core , resulting into fairly short remaining ms lifetimes of the merger products compared to the assumption of complete mixing used in the original @xcite code . recent simulations of massive mergers seem to support the mild mixing as used in our work ( * ? ? ? * and references therein ) . the mass transfer efficiency is important for our results . the more of the transferred mass is accreted during rlof , the larger the final mass of the accreting star . the maximum reachable mass of any accretor is given by the total mass of the binary ( i.e. at most twice the mass of the donor star ) and the larger the overall mass transfer efficiency , the more binary products exceed the most massive genuine single star . in our models , we limit the mass accretion rate to the thermal timescale of the accretor , which results in higher mass transfer efficiencies the larger the mass ratio and the closer the binary ( see * ? ? ? * ) . this idea is motivated by detailed binary evolution models . the initial distributions of primary masses , mass ratios and orbital separations determine the relative fraction of stars that will merge , transfer mass etc . it turns out that the distribution of orbital separations influences the incidence of binary products most @xcite because initially close binaries transfer mass on average more efficiently than wider binaries . a distribution of initial orbital separations that favours close binaries therefore leads to on average more massive binary products than distribution functions that favour initially wider binaries . a more quantitative assessment of the above quoted uncertainties in binary evolution and initial binary distribution functions reveals that a population of ms stars with luminosities @xmath236 ( i.e. o- and b - stars ) contains about @xmath237 binary products if continuous star formation is assumed @xcite . all in all , depending on the exact assumptions regarding binary evolution , there will be more or fewer stars in the tail of the mass function . however , the peak - tail structure never disappears unless it is assumed that neither ms mergers nor rlof are able to increase stellar masses which is unphysical . mixing induced by stellar rotation increases the fuel available to a star and increases its lifetime . the amount of mixing grows with increasing mass , increasing rotation rate and decreasing metallicity and may contribute to the observed age spreads and mass function tails in arches and quintuplet . the models of show that the ms lifetime of a @xmath238 star lengthens by @xmath239 and @xmath240 for initial rotational velocities of @xmath241 and @xmath242 , respectively . assuming that the present day distribution of rotational velocities of galactic o- and b - stars approximately represents the initial distribution , no more than @xmath15 and less than @xmath243 of stars would have initial rotation rates exceeding @xmath241 and @xmath244 , respectively , and are thus expected to be influenced significantly by rotational mixing ( see table 2 in * ? ? ? * and references therein ) . this is small compared to the @xmath245 of all o - stars that undergo strong binary interaction during their main - sequence evolution @xcite . the present day distribution of rotational velocities is probably altered e.g. by binary star evolution such that some of the fast rotators are expected to have gained their fast rotation by binary interactions ( @xcite , but see also @xcite ) . in this respect , the expected fraction of genuine galactic single stars that are significantly affected by rotational mixing is even smaller than the above quoted fractions . the effect of rotation is thus only of limited relevance to our results compared to binary star interactions . there are two steps involved in determining stellar masses from photometric observations that contribute to the uncertainties of the derived individual stellar masses . the first step involves the conversion of the observed apparent magnitudes ( fluxes ) to absolute magnitudes and luminosities , respectively , taking amongst others the distance and extinction into account . the second step involves the conversion of luminosities to stellar masses . this step relies upon mass - luminosity relations that depend on ( in general a priori unknown ) stellar ages and the applied stellar models . in this section , we estimate the uncertainty on individual stellar masses introduced by these two steps for stars in arches and quintuplet . once we know the uncertainties , we can apply them to the turn - off masses derived from the wind mass - loss peak in the mass function to find the corresponding uncertainty in cluster age . in the upper panel of fig . [ fig : mass - uncertainty ] we show main - sequence mass - luminosity relations of milky way stars of different ages as used in our code ( based on * ? ? ? not knowing the exact age of a star , but only a probable age range ( here @xmath246 ) , introduces an uncertainty , @xmath247 , on the derived individual stellar masses ( cf . lower panel in fig . [ fig : mass - uncertainty ] ) . the more evolved a star and the more massive , the larger is the uncertainty . if we additionally include the uncertainty in the luminosity ( here @xmath248 ; see below ) the uncertainty in the stellar mass grows to @xmath249 . the uncertainty in luminosity is the dominant contribution here . , and by the combined effect of unknown stellar ages and uncertain luminosities , @xmath250 . the upper panel illustrates how we estimate these uncertainties for a star with luminosity @xmath251 using the main - sequence mass - luminosity relations of different ages from our code @xcite . the lower panel shows the uncertainties as a function of mass.,scaledwidth=46.0% ] the uncertainties presented in fig . [ fig : mass - uncertainty ] are tailored to stars in the arches cluster where stellar masses have been derived by @xcite using a @xmath64 isochrone ( mass - luminosity relation ) whereas our analysis reveals an age of about @xmath114 hence the @xmath246 age range for the mass - luminosity relations . deriving individual stellar masses using two different extinction laws , find that stellar masses can deviate by up to @xmath128 when either a @xcite extinction law or the traditional @xcite law towards the galactic center are used . as the appropriate extinction law towards the galactic center line of sight is still a matter of debate , we adopt uncertainties of @xmath248 on luminosities to be in line with the work of . the situation is similar in quintuplet : we find slightly smaller uncertainties on derived stellar masses for uncertainties of @xmath248 on luminosities and an age range of @xmath252 than given in fig . [ fig : mass - uncertainty ] . to be conservative , we adopt that individual stellar masses are uncertain by @xmath127 ( i.e. @xmath253 ) in both clusters , which agrees with the diversity of mass estimates in the literature for stars in arches . find stellar masses that are up to @xmath128 less than those of our analysis for a @xcite extinction law . an independent study of the arches present - day mass function @xcite shows a similar peak - tail structure at the high mass end using a different extinction law and the derived masses are comparable to those of our analysis @xcite . masses in excess of @xmath7 i.e. larger than our masses have been suggested for the most luminous stars in the arches cluster by @xcite . hence , an uncertainty of @xmath127 applied to our adopted stellar masses covers the complete range of suggested masses for stars in arches . the overall structure of the mass functions of arches and quintuplet a stellar - wind peak and binary tail is robust to the mentioned uncertainties . we also find the peak - tail structure in the mass function of and in mass functions constructed from the photometric data of @xcite and , respectively , using isochrones of different ages . the whole mass functions shift in mass , but the relative structure stays the same . the reason is that different extinction laws or isochrones systematically influence all stars in a similar way and do not introduce differential effects . applying @xmath127 uncertainties on stellar masses leads to turn - off masses of the arches and quintuplet of about @xmath254 and @xmath255 , with associated age uncertainties of @xmath256 and @xmath126 respectively . the observed present - day mass functions are influenced by dynamical cluster evolution ( * ? ? ? * and references therein ) . flat mass function slopes of the order of @xmath257 , compared to a salpeter imf slope of @xmath258 , are likely a consequence of mass segregation in which massive stars sink towards the cluster center for the arches cluster , show that a dynamical model with a standard salpeter imf explains the steepening of the imf slope towards larger distances from the cluster center . while the conclusion is more elusive for the dispersed quintuplet population , the similarity of the mass function slopes in the inner regions of both clusters and the older age of quintuplet suggest that similar processes have shaped the present - day mass function of this cluster as well . we further investigate whether dynamical interactions in star clusters can also cause the peak in the observed mass functions . mass segregation flattens the high - mass end of the mass function without producing a peak @xcite while dynamical ejection of stars works on all stars with an ejection efficiency monotonically increasing with mass @xcite as confirmed by the observed galactic fraction of runaway o - stars which is larger than that of runaway b - stars @xcite . such a smoothly increasing ejection efficiency does not create a peak in the mass function but can give rise to a tail which is however not seen in the mass functions of n - body simulations of the arches cluster @xcite . star formation is not an instantaneous process but lasts a finite amount of time . observations show that there is an empirical relationship between the duration of star formation and the crossing time of star clusters @xcite . for example , in westerlund 1 and ngc 3603yc the age difference among stars less massive than about @xmath259 and @xmath260 shows that they were formed within at most @xmath261 and @xmath220 @xcite , respectively , which compare well to the cluster crossing times of @xmath262 and @xmath220 @xcite . the core of the arches cluster has a radius of about @xmath263 @xcite and a velocity dispersion of @xmath264 @xcite at a distance of @xmath265 corresponding to a crossing time , and hence a duration of star formation , of about @xmath266 . a similar estimate for the quintuplet cluster is more uncertain because the core radius and especially the velocity dispersion are not well known . if we assume that the observed central region of quintuplet with a radius of about @xmath267 corresponds to the core radius and that the velocity dispersion is about @xmath268 , the crossing time is @xmath269 . the mass functions produced by such short periods of star formation are indistinguishable from an instantaneous starburst . the apparent age spread among the most luminous stars in arches and quintuplet is about @xmath20@xmath270 and hence much larger than the estimated star formation periods . in appendix [ sec : sfh - cont ] we investigate whether a star formation history that is different from a single starburst can also explain the observed peak - tail structure in the mass functions of arches and quintuplet . we find that this is possible with e.g. a two - stage starburst but the age spread among the most massive stars would then be inconsistent with observations . massive stars rapidly change their mass , thereby altering the stellar mass function . stellar wind mass loss reduces stellar masses such that stars accumulate near the high mass end of present day mass functions , creating a bump whose position reveals the mass of the turn - off stars and hence the age of young star clusters @xcite . binary stars are frequent and important for massive star evolution because of binary mass exchange @xcite : mass transfer and stellar mergers increase stellar masses and create a tail of rejuvenated binary products at the high mass end of mass functions @xcite . we model the observed mass functions of the young arches and quintuplet star clusters using a rapid binary evolution code to identify these two features and to address two pressing controvercies : 1 . _ the cluster age problem : _ the most massive stars in arches and quintuplet , the wnh stars , appear to be younger than the less massive o - stars . this is not expected from star cluster formation @xcite but well known in older clusters under the blue straggler phenomenon . the maximum mass problem : _ a stellar upper mass limit of @xmath7 is observationally determined @xcite . in contrast , the supernova sn 2007bi is thought to be a pair - instability supernova from an initially @xmath10 star @xcite and four stars greatly exceeding this limit are found in the r136 cluster in the large magellanic cloud @xcite . we identify the peak and tail in the observed mass functions of the arches and quintuplet clusters . by fitting our models to the observations , we determine the ages of arches and quintuplet to be @xmath1 and @xmath2 , respectively . this solves the age problem because the most massive stars in arches and quintuplet are rejuvenated products of binary mass transfer and the ages derived from these stars are therefore significantly underestimated . while our age error bars are still large mostly because of uncertain absolute magnitudes our method removes the ambiguity in the age determination . for the age determination of older star clusters , blue stragglers are eminently disregarded when isochrones are fitted to the turn - off in hertzsprung - russell diagrams . our analysis shows that for young star clusters , where the higher fraction of interacting binaries produces even more blue stragglers , they obviously need to be disregarded as well in order to derive the correct cluster age . even without modelling the observed mass functions , the ages and also the mass lost by the turn - off stars during their main - sequence evolution can be determined from the mass function alone . refilling the mass function above the present day mass of the turn - off stars with the number of excess stars in the wind - mass - loss peak gives the initial mass of the turn - off stars and hence the cluster age . the derived ages agree with the more accurate ages from detailed modelling of the observed mass functions . according to this new method , the turn - off stars in arches lost @xmath106@xmath117 of their initial @xmath109@xmath110 and @xmath118@xmath119 of their initial @xmath111@xmath271 in quintuplet . this is the first direct measurement of stellar wind mass loss which does not rely on derivations of stellar wind mass loss rates . monte carlo experiments to investigate the effects of stochastic sampling show that the most massive star in the arches and quintuplet clusters is expected to be a rejuvenated product of binary mass transfer after @xmath3 and @xmath4 , respectively . at their present age , the probability that the most massive star in arches and quintuplet is a product of binary mass exchange is @xmath167 and the most massive @xmath168 and @xmath169 stars in arches and quintuplet , respectively , are expected to be such rejuvenated binary products . our findings have implications for the maximum mass problem . the arches cluster is older than previously thought and its most massive stars are most likely binary products . the mass function is thus truncated by finite stellar lifetimes and not by an upper mass limit . to constrain a potential stellar upper mass limit , we consider the massive cluster r136 in the large magellanic cloud which is thought to be so young that its initially most massive stars are still alive today . we find that the most massive star is a binary product with a probability of @xmath272 , depending on the exact , albeit yet uncertain cluster age ( sec . [ sec : upper - mass - limit ] ) . assuming binaries already interacted , a stellar upper mass limit of at least @xmath202 is needed to form the observed @xmath8@xmath9 stars in r136 . it can also not exceed @xmath203 because then the probability of forming stars above e.g. @xmath203 becomes larger than about @xmath273 but such stars are not observed . assuming that no binary interactions changed the masses of the very massive stars in r136 , a stellar upper mass limit of up to @xmath216@xmath12 can not be fully excluded because of stochastic sampling even in this rich star cluster . the upper mass limit is thus likely in the range @xmath11@xmath12 , thereby solving the maximum mass problem . we conclude that the most massive stars in the universe may be the rejuvenated products of binary mass transfer . because of their extreme mass and luminosity , radiation feedback from these stars is crucial to observable properties of young stellar populations , to the state of the interstellar medium around young stellar clusters and even to the reionization of the universe after the big bang . our results have strong implications for understanding star - forming regions nearby and at high redshift as observationally derived fundamental properties like initial mass functions are based on the assumption that the brightest stars are single and less massive than @xmath7 . these very massive stars are thought to die as pair - instability supernovae and produce huge , so far unaccounted contributions to the chemical enrichment of nearby and distant galaxies and their final explosions may be observable throughout the universe . understanding the most massive stars in young nearby star clusters is an essential step towards investigating these exciting phenomena which shape our cosmos . we thank the referee , dany vanbeveren , for carefully reading our manuscript and constructive suggestions . f.r.n.s . acknowledges the fellowships awarded by the german national academic foundation ( studienstiftung ) and the bonn - cologne graduate school of physics and astronomy . r.g.i . would like to thank the alexander von humboldt foundation . s.d.m acknowledges support by nasa through hubble fellowship grant hst - hf-51270.01-a awarded by the space telescope science institute , which is operated by the association of universities for research in astronomy , inc . , for nasa , under contract nas 5 - 26555 and the einstein fellowship program through grant pf3 - 140105 awarded by the chandra x - ray center , which is operated by the smithsonian astrophysical observatory for nasa under the contract nas8 - 03060 . b.h . and a.s . acknowledge funding from the german science foundation ( dfg ) emmy noether program under grant sto 496 - 3/1 . cccccccc & & & & & & & & @xmath274 & @xmath275 & @xmath276 & @xmath187 & @xmath276 & @xmath187 & @xmath276 & @xmath187@xmath277 & @xmath278 & @xmath279 & @xmath280 & @xmath281 & @xmath282 & @xmath283 & @xmath167 & @xmath11 & @xmath284 & @xmath285 & @xmath286 & @xmath287 & @xmath288 & @xmath167 & @xmath289 & @xmath290 & @xmath291 & @xmath292 & @xmath293 & @xmath294 & @xmath167 & @xmath241 & @xmath295 & @xmath296 & @xmath297 & @xmath298 & @xmath299 & @xmath282 & @xmath191 & @xmath300 & @xmath301 & @xmath302 & @xmath303 & @xmath304 & @xmath305 & @xmath216 & @xmath306 & @xmath307 & @xmath308 & @xmath309 & @xmath310 & @xmath311 & @xmath312 & @xmath313 & @xmath314 & @xmath315 & @xmath316 & @xmath317 & @xmath318 & @xmath319 & @xmath320 & @xmath321 & @xmath284 & @xmath322 & @xmath211 & @xmath323@xmath37 & @xmath278 & @xmath324 & @xmath325 & @xmath326 & @xmath167 & @xmath327 & @xmath167 & @xmath11 & @xmath328 & @xmath329 & @xmath330 & @xmath331 & @xmath332 & @xmath167 & @xmath289 & @xmath333 & @xmath334 & @xmath335 & @xmath336 & @xmath337 & @xmath282 & @xmath241 & @xmath300 & @xmath338 & @xmath339 & @xmath340 & @xmath304 & @xmath341 & @xmath191 & @xmath313 & @xmath342 & @xmath166 & @xmath343 & @xmath344 & @xmath345 & @xmath216 & @xmath320 & @xmath346 & @xmath328 & @xmath347 & @xmath348 & @xmath349 & @xmath312 & @xmath350 & @xmath351 & @xmath290 & @xmath352 & @xmath353 & @xmath354 & @xmath319 & @xmath355 & @xmath356 & @xmath295 & @xmath357 & @xmath315 & @xmath358@xmath319 & @xmath278 & @xmath359 & @xmath360 & @xmath361 & @xmath167 & @xmath362 & @xmath167 & @xmath11 & @xmath363 & @xmath364 & @xmath310 & @xmath365 & @xmath366 & @xmath167 & @xmath289 & @xmath367 & @xmath368 & @xmath339 & @xmath369 & @xmath370 & @xmath331 & @xmath241 & @xmath320 & @xmath371 & @xmath372 & @xmath373 & @xmath374 & @xmath323 & @xmath191 & @xmath375 & @xmath376 & @xmath333 & @xmath377 & @xmath315 & @xmath378 & @xmath216 & @xmath379 & @xmath380 & @xmath306 & @xmath381 & @xmath333 & @xmath382 & @xmath312 & @xmath383 & @xmath384 & @xmath355 & @xmath385 & @xmath320 & @xmath386 & @xmath319 & @xmath387 & @xmath388 & @xmath387 & @xmath388 & @xmath387 & @xmath388@xmath216 & @xmath278 & @xmath389 & @xmath390 & @xmath391 & @xmath282 & @xmath392 & @xmath167 & @xmath11 & @xmath295 & @xmath393 & @xmath317 & @xmath394 & @xmath395 & @xmath167 & @xmath289 & @xmath396 & @xmath397 & @xmath398 & @xmath399 & @xmath400 & @xmath336 & @xmath241 & @xmath375 & @xmath401 & @xmath333 & @xmath402 & @xmath359 & @xmath403 & @xmath191 & @xmath383 & @xmath404 & @xmath320 & @xmath405 & @xmath406 & @xmath407 & @xmath216 & @xmath387 & @xmath388 & @xmath387 & @xmath388 & @xmath387 & @xmath388@xmath241 & @xmath278 & @xmath284 & @xmath408 & @xmath409 & @xmath410 & @xmath411 & @xmath167 & @xmath11 & @xmath300 & @xmath412 & @xmath279 & @xmath413 & @xmath414 & @xmath415 & @xmath289 & @xmath375 & @xmath416 & @xmath295 & @xmath417 & @xmath418 & @xmath419 & @xmath241 & @xmath387 & @xmath388 & @xmath387 & @xmath388 & @xmath387 & @xmath388@xmath11 & @xmath278 & @xmath367 & @xmath420 & @xmath421 & @xmath422 & @xmath423 & @xmath287 & @xmath11 & @xmath387 & @xmath388 & @xmath387 & @xmath388 & @xmath387 & @xmath388 cccccccc & & & & & & & & @xmath274 & @xmath275 & @xmath276 & @xmath187 & @xmath276 & @xmath187 & @xmath276 & @xmath187@xmath216 & @xmath278 & @xmath359 & @xmath424 & @xmath425 & @xmath282 & @xmath426 & @xmath167 & @xmath11 & @xmath290 & @xmath427 & @xmath428 & @xmath429 & @xmath430 & @xmath167 & @xmath289 & @xmath300 & @xmath431 & @xmath279 & @xmath432 & @xmath433 & @xmath341 & @xmath241 & @xmath320 & @xmath434 & @xmath435 & @xmath436 & @xmath339 & @xmath437 & @xmath191 & @xmath438 & @xmath439 & @xmath300 & @xmath440 & @xmath435 & @xmath441 & @xmath216 & @xmath383 & @xmath442 & @xmath355 & @xmath385 & @xmath320 & @xmath443 & @xmath312 & @xmath444 & @xmath445 & @xmath438 & @xmath446 & @xmath375 & @xmath416 & @xmath319 & @xmath444 & @xmath447 & @xmath383 & @xmath448 & @xmath379 & @xmath449 & @xmath450 & @xmath451 & @xmath452 & @xmath383 & @xmath453 & @xmath383 & @xmath454 & @xmath455 & @xmath451 & @xmath456 & @xmath444 & @xmath457 & @xmath444 & @xmath458 & @xmath184 & @xmath451 & @xmath459 & @xmath451 & @xmath460 & @xmath451 & @xmath461 & @xmath462 & @xmath387 & @xmath463 & @xmath387 & @xmath464 & @xmath387 & @xmath463 & @xmath465 & @xmath387 & @xmath388 & @xmath387 & @xmath388 & @xmath387 & @xmath388 & @xmath466 & @xmath387 & @xmath388 & @xmath387 & @xmath388 & @xmath387 & @xmath388@xmath191 & @xmath278 & @xmath166 & @xmath467 & @xmath468 & @xmath167 & @xmath469 & @xmath167 & @xmath11 & @xmath470 & @xmath471 & @xmath472 & @xmath293 & @xmath473 & @xmath282 & @xmath289 & @xmath396 & @xmath474 & @xmath166 & @xmath475 & @xmath476 & @xmath477 & @xmath241 & @xmath355 & @xmath478 & @xmath333 & @xmath479 & @xmath480 & @xmath481 & @xmath191 & @xmath383 & @xmath482 & @xmath375 & @xmath483 & @xmath306 & @xmath204 & @xmath216 & @xmath444 & @xmath484 & @xmath379 & @xmath485 & @xmath350 & @xmath486 & @xmath312 & @xmath451 & @xmath487 & @xmath383 & @xmath482 & @xmath438 & @xmath488 & @xmath319 & @xmath451 & @xmath489 & @xmath444 & @xmath490 & @xmath383 & @xmath491 & @xmath450 & @xmath451 & @xmath459 & @xmath451 & @xmath452 & @xmath444 & @xmath457 & @xmath455 & @xmath387 & @xmath388 & @xmath387 & @xmath388 & @xmath387 & @xmath388 & @xmath184 & @xmath387 & @xmath388 & @xmath387 & @xmath388 & @xmath387 & @xmath388 & @xmath462 & @xmath387 & @xmath388 & @xmath387 & @xmath388 & @xmath387 & @xmath388@xmath241 & @xmath278 & @xmath389 & @xmath492 & @xmath493 & @xmath167 & @xmath494 & @xmath167 & @xmath11 & @xmath495 & @xmath496 & @xmath497 & @xmath336 & @xmath286 & @xmath287 & @xmath289 & @xmath320 & @xmath498 & @xmath435 & @xmath436 & @xmath339 & @xmath499 & @xmath241 & @xmath500 & @xmath501 & @xmath350 & @xmath502 & @xmath306 & @xmath503 & @xmath191 & @xmath383 & @xmath504 & @xmath438 & @xmath505 & @xmath350 & @xmath506 & @xmath216 & @xmath444 & @xmath507 & @xmath383 & @xmath508 & @xmath379 & @xmath509 & @xmath312 & @xmath451 & @xmath461 & @xmath444 & @xmath510 & @xmath511 & @xmath512 & @xmath319 & @xmath451 & @xmath513 & @xmath451 & @xmath456 & @xmath451 & @xmath460 & @xmath450 & @xmath387 & @xmath388 & @xmath387 & @xmath388 & @xmath387 & @xmath388 & @xmath455 & @xmath387 & @xmath388 & @xmath387 & @xmath388 & @xmath387 & @xmath388@xmath289 & @xmath278 & @xmath514 & @xmath515 & @xmath516 & @xmath287 & @xmath517 & @xmath167 & @xmath11 & @xmath306 & @xmath518 & @xmath519 & @xmath520 & @xmath310 & @xmath365 & @xmath289 & @xmath500 & @xmath501 & @xmath320 & @xmath521 & @xmath295 & @xmath522 & @xmath241 & @xmath511 & @xmath523 & @xmath438 & @xmath524 & @xmath525 & @xmath434 & @xmath191 & @xmath451 & @xmath452 & @xmath383 & @xmath384 & @xmath379 & @xmath509 & @xmath216 & @xmath451 & @xmath459 & @xmath451 & @xmath526 & @xmath444 & @xmath527 & @xmath312 & @xmath387 & @xmath388 & @xmath387 & @xmath388 & @xmath387 & @xmath388 & @xmath319 & @xmath387 & @xmath388 & @xmath387 & @xmath388 & @xmath387 & @xmath388@xmath11 & @xmath278 & @xmath528 & @xmath402 & @xmath529 & @xmath429 & @xmath169 & @xmath167 & @xmath11 & @xmath379 & @xmath530 & @xmath300 & @xmath531 & @xmath532 & @xmath533 & @xmath289 & @xmath383 & @xmath534 & @xmath355 & @xmath535 & @xmath320 & @xmath321 & @xmath241 & @xmath451 & @xmath536 & @xmath383 & @xmath537 & @xmath500 & @xmath201 & @xmath191 & @xmath387 & @xmath463 & @xmath387 & @xmath464 & @xmath387 & @xmath464 & @xmath216 & @xmath387 & @xmath388 & @xmath387 & @xmath388 & @xmath387 & @xmath388@xmath278 & @xmath278 & @xmath355 & @xmath538 & @xmath333 & @xmath539 & @xmath315 & @xmath343 & @xmath11 & @xmath383 & @xmath540 & @xmath355 & @xmath541 & @xmath320 & @xmath521 & @xmath289 & @xmath451 & @xmath542 & @xmath451 & @xmath526 & @xmath444 & @xmath543 & @xmath241 & @xmath387 & @xmath388 & @xmath387 & @xmath388 & @xmath387 & @xmath388 here , we investigate changes in the mass function due to a star formation scenario which deviates from a true starburst in order to understand whether the observed mass functions of arches and quintuplet can be reproduced without binaries . we analyse two scenarios : ( a ) a period of prolonged but constant star formation rate and ( b ) two instantaneous starbursts separated in time . the latter scenario ( b ) not only represents a two stage starburst within one cluster but also two merged star clusters where stars in each cluster are coeval . this situation most probably applies to the massive star cluster r136 in the large magellanic cloud which is thought to be a double cluster in the process of merging @xcite . we compute mass functions for the star formation scenarios ( a ) and ( b ) ( which include single , true starbursts ) and search for parameter values that minimise the least - square deviation , of the modelled ( @xmath544 ) from the observed ( @xmath545 ) mass functions of the arches and quintuplet clusters assuming poisson uncertainties , i.e. @xmath546 , @xmath547 where @xmath94 is the number of mass bins . exemplary star formation scenarios are described in table [ tab : sf - models ] and the resulting mass functions are compared to observations in fig . [ fig : sfh ] . among these examples are those star formation scenarios that lead to the best fits ( models a2 , a4 , q2 and q4 ) . ccccccp9 cm sf model & @xmath548 & @xmath549 & @xmath150 & @xmath550 & @xmath551 & description a1 & & & & @xmath552 & @xmath553 & power law mass function truncated at the most massive observed star ; power law index @xmath42a2 & @xmath205 & & @xmath49 & @xmath554 & @xmath555 & single starburst at @xmath556a3 & @xmath557 & @xmath558 & @xmath559 & @xmath560 & @xmath561 & constant sf between @xmath556 and @xmath562a4 & @xmath563 & @xmath558 & @xmath559 & @xmath564 & @xmath565 & two starbursts at @xmath556 and @xmath562q1 & & & & @xmath566 & @xmath567 & power law mass function truncated at the most massive observed star ; power law index @xmath42q2 & @xmath568 & & @xmath50 & @xmath235 & @xmath555 & single starburst at @xmath556q3 & @xmath568 & & @xmath559 & @xmath569 & @xmath570 & single starburst at @xmath556q4 & @xmath571 & @xmath572 & @xmath559 & @xmath573 & @xmath574 & constant sf between @xmath556 and @xmath562 from table [ tab : sf - models ] and the top panels of fig . [ fig : sfh ] it is evident that simple power - law mass functions ( models a1 and q1 ) fit the observed mass functions of arches and quintuplet much worse than the best single starburst models including binary stars ( a2 and q2 ) . especially the mass region around the wind - mass - loss peak is not fitted well by models a1 and q1 ( see @xmath575 in table [ tab : sf - models ] ) . the mass functions of the arches and quintuplet clusters do not follow simple power laws . in the bottom panels of fig . [ fig : sfh ] , we also present models of the observed mass functions of the arches and quintuplet clusters _ without _ binary stars . we do not find satisfactory models which fit peak and tail simultaneously with a single starburst without binaries . model a3 for example fits the peak due to mass - loss well ( @xmath576 ) but fails to explain the high mass end of the observed mass function ( @xmath577 ) . we can improve this situation by adding a younger stellar population that fits the tail . such a scenario is given by model a4 which fits the peak and the total mass function . however , this two - component model has an age spread of @xmath578 which is more than twice as large as the observed age discrepancy of about @xmath579 in the arches cluster and much larger than the expected star formation duration . a two component solution is not needed to model the observed mass function of the quintuplet cluster because the tail of the mass function is not very pronounced . consequently our models q3 and q4 predict no or a too small age spread contrary to the observations . the age spread of @xmath239 of model q4 might be compatible with the above estimated star formation duration given the quite uncertain core radius and velocity dispersion of quintuplet . the single starburst model q3 is shown to illustrate the difference between the mass functions with ( q2 ) and without ( q3 ) binaries . the tail of the mass function is however underestimated in the observed mass function in fig . [ fig : sfh ] because no self - consistent mass determination for the three wnh stars in the core of quintuplet is available . if the tail were visible , we could of course model it by an additional younger population as is in arches . in summary , we conclude that we can reproduce the mass functions of arches and quintuplet without binaries but with freedom in the star formation history . however , the best fit star formation parameters ( e.g. the age spread of @xmath578 ) are inconsistent with other observables . our single starburst models which include binaries are thus the only models which fulfil all observational constraints . these models are also consistent with a star formation duration of the order of the crossing time of the cluster .
|
massive stars rapidly change their masses through strong stellar winds and mass transfer in binary systems .
the latter aspect is important for populations of massive stars as more than @xmath0 of all o - stars are expected to interact with a binary companion during their lifetime .
we show that such mass changes leave characteristic signatures in stellar mass functions of young star clusters which can be used to infer their ages and to identify products of binary evolution .
we model the observed present day mass functions of the young galactic arches and quintuplet star clusters using our rapid binary evolution code .
we find that shaping of the mass function by stellar wind mass loss allows us to determine the cluster ages to @xmath1 and @xmath2 , respectively .
exploiting the effects of binary mass exchange on the cluster mass function , we find that the most massive stars in both clusters are rejuvenated products of binary mass transfer , i.e. the massive counterpart of classical blue straggler stars .
this resolves the problem of an apparent age spread among the most luminous stars exceeding the expected duration of star formation in these clusters .
we perform monte carlo simulations to probe stochastic sampling , which support the idea of the most massive stars being rejuvenated binary products .
we find that the most massive star is expected to be a binary product after @xmath3 in arches and after @xmath4 in quintuplet .
today , the most massive @xmath5 stars in arches and @xmath6 in quintuplet are expected to be such objects .
our findings have strong implications for the stellar upper mass limit and solve the discrepancy between the claimed @xmath7 limit and observations of fours stars with initial masses of @xmath8@xmath9 in r136 and of sn 2007bi , which is thought to be a pair - instability supernova from an initial @xmath10 star . using the stellar population of r136 ,
we revise the upper mass limit to values in the range @xmath11@xmath12 .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
a family of geometrically integral algebraic varieties defined over @xmath0 is said to satisfy the hasse principle if any variety in the family has a point in @xmath0 as soon as it has points in every completion of @xmath0 . quadrics are among the first examples of families satisfying this property . in dimension @xmath1 , chtelet surfaces constitute a family of varieties for which the hasse principle is known to fail in general . the chtelet surfaces that we consider are proper smooth models of affine surfaces @xmath2 where @xmath3 $ ] is a separable polynomial of degree @xmath4 or @xmath5 . for these surfaces it follows from work of colliot - thlne , sansuc and swinnerton - dyer @xcite that all failures of the hasse principle are accounted for by the brauer manin obstruction , a cohomological obstruction based on the brauer group of the surface . in particular it is known that the hasse principle holds whenever @xmath6 is irreducible over @xmath0 , which is the generic situation , or when @xmath6 has a linear factor over @xmath0 . in the remaining case , when @xmath6 is a product of two irreducible quadratic polynomials over @xmath0 , counter - examples to the hasse principle can arise . a great deal of recent work has been directed at the quantitative arithmetic of rational varieties , the aim being to count @xmath0-rational points of bounded height on the variety , assuming that it contains a zariski dense set of @xmath0-rational points . in this paper we seek instead to vary the varieties in a family and measure how often counter - examples to the hasse principle emerge . in the setting of chtelet surfaces we have already noted that a random surface of the form will satisfy the hasse principle . with this in mind , we place ourselves in the family where @xmath6 factorises as a product of two diagonal quadratic polynomials over @xmath0 , which will generically be irreducible over @xmath0 . for any @xmath7 , let @xmath8 denote the chtelet surface defined by the equation @xmath9 with @xmath10 the fact that the brauer manin obstruction is the only obstruction to the hasse principle for this family goes back to earlier work of colliot - thlne , coray and sansuc @xcite , wherein it is shown that @xmath11 . furthermore , in ( * ? ? ? c ) , a @xmath12-parameter family of counter - examples to the hasse principle is exhibited . this family , which we will meet again in [ s : ct ] , is given by @xmath13 for any positive integer @xmath14 . it generalises the particular case @xmath15 first discovered by iskovskikh @xcite . our chief object is to give a finer quantitative treatment of this circle of ideas . as we vary over rational coefficients , producing surfaces @xmath8 that are defined over @xmath0 , we will investigate the proportion of surfaces that * have points everywhere locally , * have @xmath0-rational points , * fail the hasse principle . one of the issues that we shall need to address is how best to parameterise the chtelet surfaces that are of interest to us . poonen and voloch @xcite have discussed similar questions in the setting of projective hypersurfaces @xmath16 of degree @xmath17 , assuming that @xmath18 and @xmath19 . let @xmath20 denote the set of valuations of @xmath0 and let @xmath21 be the completion of @xmath0 at @xmath22 , following the convention that @xmath23 . in ( * ? 3.6 ) it is shown that the proportion of @xmath24 that are everywhere locally soluble converges to an euler product @xmath25 , where each @xmath26 is the proportion of @xmath24 for which @xmath27 . this is the exact analogue of our theorem [ c : cor1 ] for hypersurfaces . they conjecture , furthermore , that when @xmath28 the proportion of @xmath24 that have @xmath0-rational points should converge to this constant @xmath29 . this is shown ( see ( * ? ? ? 3.4 ) ) to follow from the conjecture of colliot - thlne that the brauer manin obstruction to the hasse principle is the only one for smooth , proper , geometrically integral varieties over @xmath0 which are geometrically rationally connected . in fact , for non - singular @xmath30 , with @xmath31 , the natural map @xmath32 is an isomorphism . thus the brauer manin obstruction to the hasse principle is generically empty when @xmath31 . bhargava @xcite has undertaken an extensive investigation of the case @xmath33 , where it is well - known that the hasse principle can fail . in this setting one gets different behaviour to that predicted for fano hypersurfaces . the main result of @xcite shows that that the hasse principle fails for a positive proportion of plane cubics defined over @xmath0 . a hypersurface of degree @xmath17 in @xmath34 has @xmath35 possible coefficients . poonen and voloch use affine space @xmath36 to parameterise these hypersurfaces , associating to each vector in @xmath37 a hypersurface @xmath24 defined over @xmath0 . one disadvantage of this approach is that two different vectors may produce the same @xmath24 . in our work we will handle this issue by identifying a parameter space for the chtelet surfaces @xmath8 in . consider the open set of @xmath7 for which holds . it will be convenient to work with the affine equation @xmath38 rather than . we will consider two equations as leading to the same chtelet surface if there is @xmath39-transformation in the variables @xmath40 which takes one into the other . on replacing @xmath41 by @xmath42 , for example , we see that @xmath43 , for any @xmath44 . thus we are led to consider instead the open set @xmath45\in { \mathbb{p}}^3 : \mbox{\eqref{eq : stipulate } holds}\}.\ ] ] in a similar fashion , it is not hard to see that @xmath46 , for any @xmath47 . furthermore , for @xmath48 , we have @xmath49 , where @xmath50 is the norm of any @xmath51 . the underlying transformation in the latter case has matrix @xmath52 , with @xmath53 in conclusion , we have produced an action of @xmath54 on @xmath55 , where @xmath56 denotes the linear algebraic group @xmath57 $ ] associated to a field @xmath58 . building on this , we note that @xmath59 , where @xmath60 this leads to a further action on @xmath55 by the @xmath61-dimensional algebraic group @xmath62 . we therefore obtain a group action @xmath63 , where @xmath64 is the algebraic @xmath0-group @xmath65 given @xmath66\in u$ ] , the orbit of @xmath67 under @xmath68 produces the same surface @xmath8 . in our analysis we will restrict attention to elements of @xmath69 . we will be interested in three basic subsets , given by @xmath70\in { \mathcal{m } } : x_{a , b , c , d}({\mathbb{q}}_v ) \neq \emptyset ~\forall v\in \omega \right\},\\ { \mathcal{m}}_{\mathrm{glob } } & = \left\ { [ a , b , c , d]\in { \mathcal{m } } : x_{a , b , c , d}({\mathbb{q } } ) \neq \emptyset \right\},\\ { \mathcal{m}}_{\mathrm{br } } & = { \mathcal{m}}_{\mathrm{loc}}\setminus { \mathcal{m}}_{\mathrm{glob}}.\end{aligned}\ ] ] the elements of @xmath71 are precisely the surfaces for which there is a non - empty brauer manin obstruction to the hasse principle . using the naive exponential height function @xmath72 on @xmath73 , with norm @xmath74 on @xmath75 , we wish to study the cardinalities @xmath76 as @xmath77 . our first result is the following . [ t : m - local ] we have @xmath78 for any @xmath79 , where if @xmath80 is the density of points in @xmath81 for which the associated surface has @xmath21-points then @xmath82 here , as throughout our work , the implied constant is allowed to depend at most on the choice of parameter @xmath79 . theorem [ t : m - local ] will be established in [ s : asymptotic - nloc ] , where an explicit description of the factors @xmath80 will appear . once combined with an estimation of the total number @xmath83 of elements in @xmath84 with height at most @xmath85 , we will also establish the following result . [ c : cor1 ] we have @xmath86 where @xmath87 in particular 83.3% of the elements of @xmath84 are everywhere locally soluble . in [ s : s - global ] we will characterise when a surface @xmath88 in @xmath89 actually belongs to @xmath90 . the brauer group @xmath91 will play a fundamental rle in this analysis , for which purpose we recall some basic facts here . for any field @xmath92 , each element @xmath93 gives rise to an evaluation map @xmath94 . class field theory gives the exact sequence @xmath95 for any @xmath93 we then have the commutative diagram @xmath96 \ar[d]^{\mathrm{ev}_{\mathcal{a } } } & x({\mathbb{a}}_{\mathbb{q } } ) \ar[d]^{\mathrm{ev}_{\mathcal{a } } } \\ \operatorname{br}({\mathbb{q } } ) \ar[r ] & \bigoplus_v \operatorname{br}({\mathbb{q}}_v ) \ar[r ] & { \mathbb{q}}/{\mathbb{z}}}\ ] ] where @xmath97 denotes the adles and @xmath98 . let @xmath99 denote the composed map . then it follows that @xmath100 for all @xmath93 . we write @xmath101 in our setting , @xmath102 has order @xmath1 and so @xmath103 takes the values @xmath61 or @xmath104 in @xmath105 an obstruction to the hasse principle arises if and only if @xmath106 , that is to say , if and only if for any @xmath107 there exists @xmath93 such that @xmath108 moreover , a generator for @xmath109 is given by the quaternion algebra @xmath110 on . fundamental to our work is the observation that is impossible if there exists a valuation @xmath22 for which @xmath103 takes both values @xmath61 and @xmath104 , as a function of @xmath111 . thus , in order to obtain counter - examples to the hasse principle , it will be necessary for @xmath112 to be locally constant . in [ s : lower - upper ] we shall build on [ s : s - global ] to establish the following asymptotic formula for the density of surfaces which provide counter - examples to the hasse principle . [ thm2 ] let @xmath79 . there exists @xmath113 such that @xmath114 it follows from theorems [ t : m - local ] and [ thm2 ] that @xmath115 satisfies the asymptotic behaviour predicted by poonen and voloch @xcite for fano hypersurfaces . moreover , once coupled with theorem [ c : cor1 ] , theorem [ thm2 ] shows that 83.3% of the elements of @xmath84 are soluble over @xmath0 . it should be stressed , moreover , that our argument provides a completely explicit algorithm for determining whether or not a given chtelet surface @xmath8 of the form gives a counter - example to the hasse principle , without explicitly needing to work with elements of @xmath116 . the authors are grateful to professors colliot - thlne and fouvry for suggesting this problem to us , to pierre le boudec for comments on an earlier draft and to tim dokchitser for help with verifying numerically the proof of lemma [ lem : tau-2 ] . while working on this paper the first author was supported by an _ iuf junior _ and _ anr project ( pepr ) _ , while the second author was supported by _ erc grant _ 306457 . we will need to choose representative coordinates for the parameter space @xmath84 that we met in the introduction . two sets that will feature heavily in our work are @xmath117 where @xmath118 denotes the mbius function . let @xmath119 denote the set of relatively prime @xmath5-tuples of integers . we define @xmath120 it is clear , in view of the invariance under the action of the maps , that we have a @xmath121 bijection between @xmath84 and our set of representatives @xmath122 . when it comes to determining whether or not elements of @xmath122 produce surfaces @xmath8 with @xmath1-adic points , a tedious number of subcases arise , depending on the residue classes modulo @xmath5 of the coefficients . in order to reduce the number of cases that need to be considered we note that for any @xmath123 , we have @xmath124 . but @xmath125 is left invariant under the action of the map @xmath126 defined in . hence we may write @xmath127 the elements of @xmath122 have non - zero coordinates . since we identify @xmath128 with @xmath129 in @xmath122 , it will suffice to take representative coordinates in which @xmath130 . the sets in which we are interested therefore take the shape @xmath131 for @xmath132 . among the elements of these sets we will be interested primarily in those coefficients which give rise to chtelet surfaces @xmath8 which have points everywhere locally , or points globally . let us therefore define the sets @xmath133 for @xmath132 . having defined suitable representatives for the spaces parameterising our chtelet surfaces , we proceed to introduce some further notation . given @xmath134 , we define the binary quadratic forms @xmath135 let @xmath136 and @xmath137 . then @xmath138 , with @xmath139 . we henceforth write @xmath140 where @xmath141 and @xmath142 in particular we have @xmath143 and @xmath144 where @xmath145 . returning to the family of surfaces @xmath8 arising as proper smooth models of , in gauging solubility over the completed field @xmath21 , for any @xmath146 , it will suffice to work with a non - empty zariski open subset @xmath147 of @xmath8 . indeed , according to ( * ? ? ? * lemme 3.1.2 ) , the set @xmath148 has dense image in @xmath149 for the topology defined by the natural topology of @xmath21 . in this way we see that it suffices to examine the local solubility of the affine equation @xmath150 where @xmath151 are as in . our first order of business is a precise characterisation of when an element can locally be written as the sum of two squares . let @xmath152 and let @xmath153 denote the non - zero elements in @xmath21 which can be written as a sum of two squares in @xmath21 . firstly we note that @xmath154 . when @xmath22 is a prime @xmath155 then @xmath156 . next when @xmath22 is a prime @xmath157 then @xmath158 is the set of @xmath159 for which the @xmath160-adic valuation @xmath161 of @xmath162 is even . finally when @xmath163 we have @xmath164 , where @xmath165 frequent use will be made of this characterisation in [ s : s - local ] and [ s : s - global ] . we will reserve the letter @xmath160 for denoting a prime number . our work will involve various standard arithmetic functions , including @xmath166 , the generalised divisor function @xmath167 and the euler totient function @xmath168 . in addition to this we will also meet the function @xmath169 for any @xmath170 , so that @xmath171 . note that @xmath172 . finally , let @xmath173 we will typically abbreviate @xmath174 by @xmath175 and @xmath176 by @xmath177 . we will also require an estimate for the average order of the divisor function , as it ranges over the values of a binary linear form . a crucial aspect of the following result is the uniformity in the coefficients of the linear form that it enjoys . [ lem : divisor ] let @xmath178 with @xmath179 , let @xmath180 and let @xmath79 . we have @xmath181 where the implied constant depends at most on @xmath182 . suppose without loss of generality that @xmath183 . the bound follows from previous work of the authors ( * ? ? ? 1 ) when @xmath184 . alternatively it follows from the trivial bound @xmath185 . we close this section with a modified result of le boudec ( * ? ? ? * lemma 2 ) , concerning pairs of integers constrained to satisfy congruences modulo @xmath186 and modulo @xmath187 , with @xmath188 . we are interested in the cardinality @xmath189 for suitable @xmath190 . the following result allows us to approximate @xmath191 by the expected main term . [ lem : lb ] let @xmath178 with @xmath179 and @xmath192 , let @xmath193 and let @xmath194 . we have @xmath195 where @xmath196 as we have mentioned , this result is based on work of le boudec ( * ? ? ? * lemma 2 ) , which corresponds to @xmath197 and general closed sets @xmath198 in place of @xmath199 $ ] and @xmath200.$ ] we have decided to include a sketch of the proof for completeness , the key idea being to break the @xmath201 into congruence classes modulo @xmath202 and then use the orthogonality of additive characters @xmath203 to detect these congruences . since @xmath186 and @xmath187 are coprime , there exist @xmath204 such that @xmath205 . we therefore have @xmath206 where @xmath207 here , recalling that @xmath208 , we have @xmath209 where @xmath210 is the ramanujan sum , satisfying @xmath211 . in particular @xmath212 and @xmath213 are seen to be independent of @xmath214 . we denote by @xmath215 the contribution from @xmath216 or @xmath217 . then bounding the sums over @xmath218 using the standard bound for linear exponential sums , we see that @xmath219 in the notation of the lemma . summing over @xmath220 we deduce that @xmath221 moreover , it is clear that @xmath222 note that @xmath223 . combining this with we swiftly arrive at the statement of the lemma . as we have seen , in order to determine whether or not @xmath8 is soluble over @xmath21 , for any @xmath146 , it will suffice to work with the equation . furthermore , when considering solubility over @xmath224 , it suffices to establish the existence of solutions @xmath225 for which @xmath226 are coprime . in order to summarise the local solubility criterion for @xmath8 it will be convenient to introduce a new symbol . for any odd prime @xmath160 and any @xmath227 with @xmath228 , we set @xmath229}= \begin{cases } \left(\frac{n / p^{\nu}}{p}\right ) , & \mbox{if $ 2\mid \nu$},\\ 1 , & \mbox{if $ 2\nmid \nu$ } , \end{cases}\ ] ] where @xmath230 is the usual legendre symbol . this symbol takes values in @xmath231 . finally , let @xmath232 denote the sign of any real number @xmath162 , which we extend to vectors in the obvious way . we may now record the following result , in which @xmath136 and @xmath137 . [ lem : x - local ] let @xmath134 . then the following hold : * @xmath233 if and only if @xmath234 ; * @xmath235 if @xmath236 ; * if @xmath237 for @xmath157 then @xmath235 if and only if * * @xmath238 or @xmath239 only when these @xmath160-adic orders are zero , * * either @xmath240}=1 $ ] or @xmath241}=1 $ ] ; * @xmath242 if and only if @xmath243 coprime @xmath244 such that @xmath245 . one recalls here that @xmath138 , with @xmath139 . thus @xmath246 for any prime @xmath160 and @xmath235 whenever @xmath247 . that the three conditions are necessary is self - evident . turning to sufficiency , we begin by establishing ( i ) , noting that @xmath248 for any @xmath134 . the case in which @xmath249 or @xmath250 is positive is trivial , so we study only the case @xmath251 and @xmath252 , with ( i ) being satisfied . in this case @xmath253 and @xmath254 , whence @xmath255 condition ( iv ) is a direct consequence of the criterion for solubility of @xmath8 over @xmath256 . solubility over @xmath224 is automatic for @xmath155 . it therefore remains to consider solubility over @xmath224 when @xmath257 . for ease of notation we will henceforth write @xmath258 for @xmath259 . we claim that there exist coprime @xmath226 such that @xmath260 . this will clearly suffice to show that @xmath261 is non - empty when @xmath236 , as required for condition ( ii ) . recall that @xmath262 and @xmath263 . if @xmath264 then the claim is satisfied by taking @xmath265 . similarly , if @xmath266 then we may take @xmath267 . the remaining cases to consider are when @xmath268 with @xmath269 , and lastly when @xmath270 with @xmath271 . for both of these the choice @xmath272 is satisfactory , which thereby concludes the proof of the claim . for the remainder of the proof we suppose that @xmath273 , with @xmath257 . we wish to show that under the hypotheses of the lemma we can always find coprime @xmath226 such that precisely one of @xmath274 or @xmath275 is odd . this will then imply that @xmath276 is even as required for condition ( iii ) . suppose first that @xmath277 . then by hypothesis one of either @xmath278 or @xmath279 is a non - zero square modulo @xmath160 . there are two cases to consider . suppose first that @xmath280 but @xmath281 . we can find coprime @xmath226 such that @xmath282 . on the other hand @xmath275 is even since @xmath279 is not a square modulo @xmath160 . next suppose that @xmath283 , with @xmath284 . we may choose coprime @xmath226 so that @xmath285 . then if @xmath286 one sees that @xmath287 is a divisor of @xmath288 by . it therefore follows from that @xmath289 , as required . we now suppose that @xmath290 but @xmath291 . if @xmath292 then it is clear that we can find @xmath226 , with @xmath293 , such that @xmath275 is odd . moreover , we will have @xmath294 since @xmath160 divides precisely one of @xmath295 or @xmath296 . if @xmath281 then @xmath297 for any coprime @xmath298 suppose @xmath299 , with @xmath300 and @xmath301 . if @xmath302 is odd then we can ensure that @xmath274 odd by taking @xmath265 . alternatively , if @xmath302 is even then @xmath303 by hypothesis . hence we can force @xmath274 to be odd by considering solutions of the form @xmath304 . the case in which @xmath305 but @xmath306 follows by symmetry . it remains to consider the possibility @xmath290 and @xmath306 . we will consider the cases @xmath307 or @xmath308 , the remaining cases following by symmetry . suppose first that @xmath300 and @xmath309 with @xmath310 . if @xmath302 ( resp . @xmath311 ) is odd then we are done on taking @xmath265 ( resp . @xmath267 ) . suppose that @xmath302 and @xmath311 are both even . then our hypothesis ensures that one of the legendre symbols @xmath312 or @xmath313 is equal to @xmath12 . supposing without loss of generality that it is the former , we easily make @xmath274 odd with @xmath297 . turning to the case @xmath308 , we write @xmath300 and @xmath314 with @xmath315 and @xmath316 not both even . if @xmath317 are of opposite parities then the situation is easy and we can proceed by taking @xmath265 . finally , if @xmath316 are both odd then by hypothesis they must be unequal . suppose without loss of generality that @xmath318 . then in fact @xmath319 . taking @xmath320 we easily deduce that @xmath321 and @xmath322 , as required . this concludes the proof of the lemma . lemma [ lem : x - local ] gives us a means of characterising the elements of @xmath323 , in the notation of . before proceeding to the proof of theorem [ t : m - local ] , we will need a finer description of case ( iv ) in lemma [ lem : x - local ] . this will depend intimately on the possible residue classes of @xmath324 modulo @xmath5 , with @xmath325 odd . the various constraints are obtained by taking @xmath326 in the second part of lemmas [ lem:1 - 1][lem:3 - 2 ] in [ s:2adic ] , where the description of @xmath327 exactly provides necessary and sufficient conditions for the non - nullity of @xmath328 . throughout this section we will assume familiarity with the notation introduced in [ s : prelim ] . among the set of @xmath329 we require a means of sifting for those @xmath128 such that @xmath8 actually has points in @xmath0 . our principal tool comes from the theory of descent , as formulated by colliot - thlne , coray and sansuc ( * ? ? ? this ensures that given any @xmath329 , we have @xmath330 if and only if there exists @xmath331 , with @xmath332 , such that the smooth variety @xmath333 has points in @xmath21 for every @xmath152 . this in turn is equivalent to the existence of @xmath334 , with @xmath335 , such that the corresponding pair of equations with @xmath336 replaced by @xmath337 is everywhere locally soluble . note that , by ( * ? ? ? * lemme 3.1.2 ) , we see that in checking local solubility for the pair of equations it suffices to replace @xmath338 and @xmath339 by @xmath340 and @xmath341 , respectively , as given by . we proceed to simplify the set of allowable @xmath342 somewhat . on carrying out a suitable change of variables it suffices , without loss of generality , to consider the existence of @xmath334 for which @xmath343 for @xmath344 . recall that @xmath138 , with @xmath139 , in the notation of . suppose that @xmath335 for @xmath334 satisfying the above @xmath160-adic constraints . if @xmath345 then we must have @xmath346 . alternatively , if @xmath347 , then @xmath348 or @xmath349 . given such @xmath342 let us put @xmath350 to be the product of primes @xmath347 for which @xmath351 . then we may assume that @xmath352 and @xmath353 , where @xmath354 are square - free integers comprised of primes @xmath345 . on multiplying the first equation by @xmath355 and the second by @xmath356 , and making a further change of variables , we see that it suffices to check local solubility with @xmath357 replaced by @xmath358 , where @xmath359 . in particular , we have the relation @xmath360 . we summarise our remarks in the following result . [ lem:1 ] let @xmath329 . then @xmath361 if and only if there exists @xmath362 and @xmath363 , with @xmath364 and @xmath365 such that the smooth variety @xmath366 has solutions in @xmath21 for every @xmath146 . the thrust of lemma [ lem:1 ] is that our task of determining elements of @xmath367 in has become a purely local problem . in fact , the varieties @xmath368 are known to satisfy the hasse principle and any @xmath0-point on @xmath368 gives rise to a @xmath0-point on @xmath8 . we shall henceforth refer to @xmath368 as torsor equations . in reality , as explained in @xcite , any universal torsor over @xmath8 is birational to @xmath369 , for a smooth conic @xmath370 defined over @xmath0 . we proceed to study the sets @xmath371 for each @xmath152 . when determining solubility over @xmath224 it will suffice to consider the existence of solutions @xmath372 with @xmath226 coprime . recall from [ s : intro ] our discussion of the brauer group @xmath116 , which is generated by a single element , given on by the quaternion algebra @xmath373 . given any local point @xmath374 in @xmath375 , we have @xmath376 according to our discussion of , in order for counter - examples to the hasse principle to arise , we will need @xmath103 to be locally constant for each @xmath377 , in which case we will show that there is a unique torsor @xmath378 which has @xmath21-points . it will be convenient to retain our shorthand notation @xmath258 for @xmath259 in what follows . the case in which @xmath160 is an odd prime not appearing in the factorisation of @xmath379 is straightforward , as the following result shows . [ lem : p1 ] let @xmath362 and let @xmath380 , with @xmath381 . then @xmath382 . if @xmath383 then the conclusion is trivial . suppose now that @xmath257 . the second paragraph in the proof of lemma [ lem : x - local ] shows that there exists @xmath226 such that @xmath260 . but then @xmath384 for @xmath344 , which concludes the proof . since @xmath385 are square - free and entirely composed of primes congruent to @xmath4 modulo @xmath5 , lemma [ lem : p1 ] completely handles the solubility question over @xmath224 for @xmath386 . turning to @xmath387 , it remains to consider the possibility that @xmath388 , together with the possibility that @xmath347 and @xmath389 . we may assume that @xmath390 with @xmath391 and @xmath392 . then the condition demands that @xmath393 in particular we have @xmath394 when @xmath236 and @xmath395 when @xmath237 . recall that @xmath145 . we proceed to establish the following result . [ lem : p2 ] let @xmath329 and let @xmath257 with @xmath396 . let @xmath397 with @xmath392 . then one of the following must hold : * if @xmath398+[\frac{-c'd'}{p}]{\leqslant}0 $ ] then there exists a unique @xmath399 with @xmath400 such that @xmath401 ; * if @xmath398=[\frac{-c'd'}{p}]=1 $ ] and @xmath402 , then @xmath403 ; furthermore , @xmath404 if and only if @xmath405 ; * if @xmath398=[\frac{-c'd'}{p}]=1 $ ] and @xmath406 , then @xmath407 unless @xmath408 is even and @xmath409 , or @xmath410 is even and @xmath411 ; likewise , @xmath412 unless @xmath413 is even and @xmath414 , or @xmath415 is even and @xmath416 . in particular , the nullity of @xmath417 depends only on @xmath418 and @xmath419 . [ rem:1 ] let @xmath160 be an odd prime . lemmas [ lem : p1 ] and [ lem : p2 ] together show that whenever @xmath235 , there exists @xmath420 , with @xmath421 , such that @xmath422 . our arguments will have a similar flavour to the proof of lemma [ lem : x - local ] . beginning with case ( i ) we suppose without loss of generality that @xmath423=-1 $ ] . now any solution @xmath218 must have @xmath424 , whence the choice @xmath425 ( resp . @xmath426 ) is not admissible when @xmath402 ( resp . @xmath406 ) . finally we note that the case @xmath427 is admissible when @xmath402 by lemma [ lem : p1 ] , and the case @xmath428 is admissible when @xmath406 since lemma [ lem : x - local ] implies that @xmath398=1 $ ] . let us consider case ( ii ) , wherein we have @xmath429 and @xmath398=[\frac{-c'd'}{p}]=1 $ ] . let us set @xmath430 with @xmath431 and @xmath432 . it is clear that @xmath427 leads to a torsor with @xmath160-adic points , by lemma [ lem : p1 ] . the choice @xmath433 is permissible if and only if there exist coprime @xmath218 such that @xmath434 for @xmath344 . if @xmath435 then implies that there are no such solutions . alternatively , suppose that @xmath436 . suppose first that @xmath437 . in particular we have @xmath280 . if @xmath438 then we may choose @xmath218 such that @xmath282 , in which case implies that @xmath439 . if @xmath440 then we choose @xmath218 so that @xmath441 and it will follow that @xmath439 . when @xmath442 we argue according to parity . without loss of generality we may suppose that @xmath443 and @xmath444 . if @xmath445 then it suffices to take @xmath265 . if @xmath316 are of opposite parity , with say @xmath446 , we consider @xmath447 chosen so that @xmath448 . then it follows that @xmath449 since @xmath450 . finally , the case @xmath451 is impossible . in case ( iii ) we must have @xmath428 or @xmath452 . by symmetry it will suffice to consider the case @xmath428 . we wish to determine when there exist coprime @xmath218 such that @xmath274 is odd and @xmath275 is even . suppose first that @xmath305 , so that @xmath280 . let @xmath453 . if @xmath454 is even we may choose @xmath218 such that @xmath285 , in which case it follows from that @xmath455 is even . likewise , if @xmath454 is odd then @xmath291 and so we also have @xmath292 . thus we may choose @xmath218 such that @xmath456 is even , in which case @xmath457 is odd . we suppose now that @xmath458 , say , with @xmath300 and @xmath301 . if @xmath459 , write @xmath309 with @xmath460 . if @xmath302 is odd then we may take @xmath461 if @xmath462 is even then we choose @xmath463 such that @xmath464 , which is satisfactory . if , on the other hand , we write @xmath314 for @xmath465 and @xmath466 , then the situation is more complicated . suppose that @xmath302 is odd and @xmath467 is even . then it suffices to take @xmath265 . if @xmath302 and @xmath467 are both odd then @xmath468 by lemma [ lem : x - local ] . supposing that @xmath318 then the argument used in the last part of the proof of lemma [ lem : x - local ] shows that @xmath320 suffices . if @xmath302 and @xmath467 are both even then @xmath469 and @xmath470 , resulting in a case that is easy to handle . finally if @xmath302 is even and @xmath467 is odd , a case that requires @xmath471 and @xmath472 , then @xmath473 and it is impossible to find suitable @xmath218 . the case in which @xmath474 is symmetric . this completes the proof of the lemma . we now turn to the set @xmath476 , given @xmath329 . recall the notation for the sign function @xmath232 . we may assume that @xmath477 and ( i ) holds in lemma [ lem : x - local ] . on recalling that @xmath478 and @xmath479 are positive integers , we see that the constraint demands that @xmath480 and @xmath481 share the same sign . let @xmath482 . we wish to determine when there exist @xmath483 such that @xmath484 for @xmath344 , which will then ensure that @xmath476 is non - empty . we write @xmath485 for @xmath476 with @xmath486 . suppose first that @xmath487 . taking @xmath488 and @xmath489 shows that @xmath490 . if , furthermore , @xmath491 or @xmath492 then @xmath493 . finally , if @xmath494 then @xmath495 , as can be seen by taking @xmath496 and @xmath497 . next we consider the case @xmath498 . here , for @xmath499 , we seek the existence of @xmath218 with @xmath497 such that @xmath500 this occurs if and only if @xmath492 and @xmath501 . for @xmath502 we require instead the existence of @xmath218 with @xmath497 such that @xmath503 this occurs if and only if @xmath504 and @xmath505 . finally we note that the case in which @xmath506 and @xmath507 does not enter into consideration . we summarise the situation in table [ table1 ] , in which the final column lists the possible signs of @xmath508 which give rise to a non - empty set @xmath509 .@xmath476 for @xmath510 [ cols="^,^,^,^,^,^",options="header " , ] once combined with our formula for @xmath511 , this therefore concludes the proof of the lemma . the numerical value of @xmath511 is @xmath512 . let @xmath513 denote the cardinality on the right hand side of when @xmath514 . tim dokchitser has kindly implemented a computer algorithm for calculating the ratios @xmath515 for small values of @xmath58 . taking @xmath516 leads to the numerical value @xmath517 , which agrees quite closely with lemma [ lem : tau-2 ] . the conclusion of theorem [ c : cor1 ] is now available . combining theorem [ t : m - local ] and lemma [ lem : m - total ] , we conclude that @xmath518 with @xmath519 as in . it follows from and that @xmath520 applying and lemma [ lem : tau-2 ] , we deduce that @xmath521 with @xmath522 given by . the latter euler product converges very rapidly and has numerical value @xmath523 whence @xmath524 , up to 7 decimal places . this completes the proof of theorem [ c : cor1 ] . the goal of this section is to establish theorem [ thm2 ] . the argument will begin along similar lines to the treatments of @xmath83 and @xmath526 , but the analysis is ultimately more involved . whereas our earlier work relied upon the basic estimate in lemma [ lem : basic ] for the number of integers in an interval which are coprime to a given integer , the treatment of @xmath527 will require more sophisticated tools from analytic number theory . let @xmath528 . we define an arithmetic function @xmath529 multiplicatively on prime powers via @xmath530 let @xmath531 where @xmath532 is given by . one finds that @xmath533 we begin with an estimate for the average order of @xmath534 . applying the burgess bound for character sums , it follows that @xmath538 for any integer @xmath539 . if @xmath540 then implies that @xmath541 hence @xmath542 the remaining sum over @xmath543 is easily seen to be @xmath544 , by an application of the selberg delange method ( see @xcite , for example ) . this concludes the proof . at a certain point in our argument it will be useful to approximate the function @xmath545 in by the simpler arithmetic function @xmath546 we will need an asymptotic formula for the average order of @xmath547 , where @xmath548 is an arbitrary multiplicative arithmetic function satisfying @xmath549 for primes @xmath160 . this is achieved in the following result . [ lem : z - sum ] let @xmath79 and let @xmath548 be a multiplicative arithmetic function satisfying . for @xmath550 and @xmath551 , we have @xmath552 where @xmath553 and @xmath554 is the convergent euler product @xmath555 we consider the associated dirichlet series @xmath556 say , for @xmath557 . since @xmath558 , one sees that @xmath559 where @xmath560 is the dirichlet @xmath561-function associated to the real character @xmath562 modulo @xmath5 and @xmath64 may be continued as a holomorphic function to the half - plane @xmath563 and is bounded absolutely in this region . hence it follows that @xmath564 , where @xmath565 in particular @xmath566 in the notation of the lemma . our starting point in the proof of theorem [ thm2 ] is , followed by the changes of variables , and . in the present situation a further change of variables will be expedient . define @xmath569 we now write @xmath570 for @xmath571 satisfying @xmath572 . the union of this constraint with and is equivalent to @xmath573 and @xmath574 with @xmath575 constrained to satisfy @xmath576 for @xmath577 . note that since @xmath578 it automatically follows that @xmath579 is coprime to @xmath580 . in what follows it will be convenient to redefine under our various transformations we may now write @xmath582 with @xmath583 in this new notation we must proceed to consider the constraints recorded in [ s : s - global ] which are both necessary and sufficient to have @xmath584 for all @xmath152 , but @xmath585 . our key tool is lemma [ lem:1 ] and the associated calculations . for the infinite valuation we must have @xmath586 by part ( i ) of lemma [ lem : x - local ] and lemma [ lem : easy ] . for the primes @xmath247 there are no additional constraints arising . the situation is more complicated for the primes @xmath587 . according to lemma [ lem : easy - ish ] we seek constraints under which there exists a unique choice of @xmath399 with @xmath421 and @xmath422 , where @xmath378 is given in lemma [ lem:1 ] . suppose that @xmath345 , so that @xmath237 . then lemma [ lem : x - local](iii ) implies that does nt hold , that @xmath588 and that @xmath589 if @xmath590 and @xmath591 , for @xmath592 . the constraint in lemma [ lem : p2](iii ) gives either @xmath593 and @xmath594 for @xmath595 , or @xmath596 and @xmath597 for @xmath595 . the constraint in lemma [ lem : p2](i ) translates as @xmath598}+ { \left[}\frac{-{\varepsilon}_3{\varepsilon}_4 \ell_{3}\ell_{4}c''d''}{p}{\right]}{\leqslant}0.\ ] ] it follows from that @xmath599 and @xmath600 can not both be odd . on recalling from that @xmath601 are coprime , we see that @xmath602 if and only if @xmath603 . similarly , @xmath604 if and only if @xmath605 , since @xmath606 are coprime . combining all these conditions therefore leads to the description that @xmath588 , with one of the following * @xmath593 and @xmath594 for @xmath595 ; * @xmath596 and @xmath597 for @xmath595 ; * @xmath603 and @xmath605 , with @xmath607 * @xmath603 and @xmath608 , with @xmath609 * @xmath610 and @xmath605 , with @xmath611 for given parameters @xmath612 , the constraints arising from primes @xmath587 for which @xmath345 force the vector @xmath613 to lie in one of finitely many congruence classes modulo @xmath160 . let us denote the set of possible classes by @xmath614 here @xmath615 and @xmath616 for each prime @xmath160 . although @xmath617 also depends on @xmath618 and @xmath619 , its cardinality does not , as the following result shows . next we suppose that @xmath625 , still with @xmath626 recall that @xmath627 , with @xmath628 given by . then according to lemma [ lem : p2](ii ) we must avoid the the case @xmath629 if @xmath405 and @xmath630 note that @xmath405 and @xmath277 if and only if @xmath631 and @xmath632 . but for such primes @xmath633 and @xmath634 are equal . hence , since @xmath587 , we see that the constraint is equivalent to @xmath635 if @xmath631 . suppose now that @xmath347 and @xmath636 . in particular @xmath405 and lemma [ lem : p2 ] implies that we must have @xmath637+[\frac{-{\varepsilon}_3{\varepsilon}_4 c'd'}{p}]{\leqslant}0 $ ] . we can not have both @xmath638 and @xmath639 being odd . moreover , we must have @xmath640 . we deduce that precisely one of @xmath641 or @xmath642 is odd . hence we have the pair of conditions @xmath643=-1,\ ] ] or @xmath644=-1.\ ] ] in accordance with this we are now led to introduce the set @xmath645 for given parameters @xmath612 , with @xmath646 , the constraints arising from primes @xmath587 for which @xmath636 force the vector @xmath613 to lie in one of finitely many congruence classes modulo @xmath160 . let us denote the set of possible classes by @xmath647 the analogue of lemma [ lem : card - t ] is the following easy result . for any @xmath128 constrained as above , our work so far has shown that there exists a unique @xmath362 satisfying for some @xmath363 with @xmath364 , such that @xmath649 for every valuation @xmath377 . suppose that @xmath650 for @xmath651 . for the prime @xmath514 , it remains to distinguish precisely when @xmath328 is non - empty but @xmath652 is empty . this is equivalent to demanding that @xmath653 in the notation of [ s:2adic ] . an inspection of lemmas [ lem:1 - 1][lem:3 - 2 ] shows that for a given choice of @xmath654 , the @xmath1-adic constraints placed on @xmath655 take the shape of a union of particular congruence classes modulo @xmath656 . let us denote by @xmath657 the set of all possible classes modulo @xmath656 that can arise . then the pair of constraints @xmath242 and @xmath658 are equivalent to demanding that @xmath659 for given @xmath654 . in particular this constraint implies that @xmath660 is odd . although @xmath657 depends on numerous parameters , including the residue class of @xmath342 modulo @xmath5 , its cardinality is independent of all of these . we will set@xmath661 this constant is equal to the density of points @xmath128 for which there exist coprime integers @xmath244 for which @xmath662 , yet for every choice of coprime @xmath244 one never has both @xmath663 and @xmath664 . using our work in [ s:2adic ] , as an analogue of lemma [ lem : tau-2 ] , it is possible to calculate a numerical value for @xmath665 . we have chosen not to pursue this here , however , it being sufficient to note that @xmath666 since @xmath667 . we are now ready to return to the expression for @xmath527 . it will be convenient to put @xmath668 let us define the function @xmath669 where @xmath670 are given by . one notes that @xmath671 and @xmath672 in the notation of , with @xmath673 and @xmath674 . in particular @xmath545 is only supported on positive integers coprime to @xmath675 , which are built from primes congruent to @xmath4 modulo @xmath5 . summarising our investigation so far , our analogue of is @xmath676 where @xmath677 ( resp . @xmath678 , @xmath679 ) is given by ( resp . , ) and @xmath680 the conditions of summation here are restricted to @xmath681 such that , and hold , with @xmath682 and @xmath683 the definitions of @xmath617 and @xmath684 ensure that the product @xmath685 is coprime to @xmath686 . likewise implies that @xmath685 is odd . hence is redundant . in our analysis of it will frequently be useful to reduce the allowable ranges for the various parameters appearing in the outer summations . two quantities that will feature heavily in this process are @xmath687 recall the definitions of @xmath688 . we proceed to establish the following result . let us denote the expression that is to be estimate by @xmath694 suppose first that @xmath695 . then there are @xmath696 choices for @xmath697 . recall the definition of @xmath698 , for any @xmath170 . rankin s trick and allow us to deduce that @xmath699 as required . let us next consider the case @xmath700 . then our argument in [ s : asymptotic - nloc ] , which was used to restrict the size of the parameters appearing in , easily gives @xmath701 , again using rankin s trick and to handle the sum over @xmath702 and @xmath619 . finally we must consider the case @xmath703 in this case , since @xmath704 and the sum over @xmath705 is absolutely convergent , we obtain @xmath706 using rankin s trick again , we see that @xmath707 since @xmath708 for @xmath709 . hence @xmath710 as required . a trivial upper bound is given by @xmath711 applying lemma [ lem : r1r2 ] with @xmath700 and @xmath712 , we may henceforth restrict attention to parameters in for which @xmath713 with satisfactory overall error @xmath714 . we wish to break the sum into congruence classes modulo @xmath715 . we shall introduce the set @xmath716 , say , of @xmath717 for which @xmath718 these two conditions imply that @xmath719 we may therefore write @xmath720 note from that @xmath721 . invoking we obtain a sum over @xmath543 which a priori runs over all integers up to order @xmath722 . define the @xmath1-dimensional lattice @xmath723 we then have @xmath724 where @xmath725 it will be crucial to show that there is a negligible contribution from large values of @xmath543 in this sum . this will be achieved using lemma [ lem : g ] . according to heath - brown ( * lemma 2 ) we have @xmath726 recall the definition of @xmath727 . the overall contribution to @xmath728 from @xmath729 is therefore seen to be @xmath730 note here that @xmath731 . we break the summation over @xmath543 into @xmath732 dyadic intervals @xmath733 . lemma [ lem : g ] therefore yields the contribution @xmath734 it remains to sum this over the remaining parameters subject to . taking @xmath735 and @xmath736 in lemma [ lem : r1r2 ] , one arrives at the overall contribution @xmath737 , which is satisfactory for theorem [ thm2 ] . we may now focus our attention on the contribution from @xmath738 in , which we denote by @xmath739 . we will need to take care of the coprimality conditions @xmath740 in @xmath741 , retaining the condition @xmath742 . recall that modulo @xmath715 the first condition is already implied by the definition of @xmath716 , as is the fact that @xmath743 . in this way we deduce that @xmath744 where a change of variables yields @xmath745x , [ k_1,k_3]y)\in \mathsf{\lambda}_{z}\\ x { \leqslant}p/([k_1,k_2]c),~ y { \leqslant}p/([k_1,k_3]d)\\ ( x , y ) { \equiv}(x_0,y_0){\,(\operatorname{mod}{m})}\\ \gcd(xy , z)=1 \end{array } \right\}.\ ] ] here , if @xmath746}\in { \mathbb{z}}$ ] denotes the multiplicative inverse of @xmath747 $ ] modulo @xmath715 for @xmath748 , then @xmath749}c_0 $ ] and @xmath750}d_0 $ ] . finally , we note that @xmath751 is automatically coprime to @xmath752 , since @xmath753 satisfies this property . we will also need to reduce the ranges of summation for @xmath754 in this expression . recall from that @xmath755 for any @xmath543 . to achieve our goal we invert the summation over @xmath543 and @xmath756 , finding that the overall contribution to @xmath739 from @xmath757 is @xmath758 where @xmath759y - { \varepsilon}_2{\varepsilon}_3\ell_2\ell_2''\ell_3\ell_3''2^{\beta+\gamma } \tilde b [ k_1,k_2]x.\ ] ] recall . summing over @xmath760 , we may bound this contribution using lemma [ lem : divisor ] by @xmath761[k_1,k_3]cd } + \frac{p^{1+{\varepsilon}}}{k_1}\right)\\ & \ll \frac{t_1^{\varepsilon}p^4(\log p)^3}{abcd \sqrt{t_2 } } + p^{3+{\varepsilon}}.\end{aligned}\ ] ] applying lemma [ lem : r1r2 ] with @xmath735 and @xmath700 , this therefore shows that @xmath757 contribute @xmath762 to @xmath527 . this is satisfactory for theorem [ thm2 ] . likewise , the same argument shows that parameters with @xmath763 make a satisfactory overall contribution to @xmath527 . our work so far has therefore shown that we can approximate @xmath739 by @xmath764 with acceptable error . to handle @xmath765 we call upon lemma [ lem : lb ] , noting that @xmath766 in , with @xmath767 and @xmath768,\\ a_2 & = { \varepsilon}_4\ell_1\ell_1''\ell_4\ell_4''2^\delta \tilde a [ k_1,k_3],\end{aligned}\ ] ] and furthermore , @xmath769c } , \quad v=\frac{p}{[k_1,k_3]d}.\ ] ] in particular it is clear that @xmath770 and so all the conditions are met for an application of lemma [ lem : lb ] . this gives @xmath771[k_1,k_3]cd } \right|\ll \frac{\tau_3(z)p}{k_1z}+ \tau_3(z)(\log zm)^3 + e,\ ] ] where @xmath772 we need to sum these three error terms over the remaining parameters to check that they ultimately make a negligible contribution . the first term on the right of is easy to deal with . lemma [ lem : r1r2 ] with @xmath695 and @xmath736 easily shows that it contributes @xmath773 overall . recall that @xmath755 and @xmath774 has average order @xmath775 . the contribution to @xmath776 from the second term on the right of is therefore seen to be @xmath777 we note here that @xmath778 . hence once summed over the remaining parameters using lemma [ lem : r1r2 ] with @xmath695 and @xmath736 , we obtain the satisfactory overall contribution @xmath779 . it remains to handle the contribution to @xmath776 from @xmath780 . let us write @xmath781 and @xmath782 . then since @xmath783 it follows that @xmath784 . taking @xmath755 and carrying out the sums over @xmath754 and @xmath785 , this contribution is seen to be @xmath786 we will need to sort the sum according to the greatest common divisor @xmath787 , writing @xmath788 . the final condition on @xmath789 forces the vectors @xmath790 in which we are interested to lie on a rank @xmath1 integer sublattice of determinant @xmath791 . heath - brown ( * ? ? ? * lemma 2 ) therefore shows that the number of @xmath790 is @xmath792 substituting this into the above the second term here is seen to contribute @xmath793 to @xmath776 , which once combined with lemma [ lem : r1r2 ] therefore leads to a satisfactory overall contribution . finally , the first term contributes @xmath794 to @xmath776 , since @xmath778 . this too is found to be satisfactory once summed over all the remaining parameters using lemma [ lem : r1r2 ] . having handled the contribution from the error terms in , we are now free to approximate @xmath765 by the expected main term . having done so , furthermore , it is convenient to extend the summations over @xmath754 to infinity , which we may do with acceptable error using lemma [ lem : r1r2 ] . it henceforth suffices to consider the quantity @xmath795[k_1,k_3]},\end{aligned}\ ] ] in place of @xmath776 . we may carry out the summations over @xmath754 , finding that @xmath796[k_1,k_3]}=\frac{6}{\pi^2 { \varphi}_2^*(mz)\psi(\tilde a \tilde b)},\ ] ] in the notation of and . hence @xmath797 since @xmath798 . rather than carrying out the sum over @xmath543 directly , which intimately depends on @xmath799 , we shall first show that @xmath545 can be approximated by the function @xmath800 defined in . to this end we exame the sum @xmath801 for given parameters @xmath802 and @xmath803 . our aim is to show that @xmath804 which leads to a satisfactory overall contribution once summed over the remaining parameters , via lemma [ lem : r1r2 ] with @xmath695 and @xmath805 . to establish the desired bound for @xmath806 we deduce from and that @xmath807 with @xmath808 . this allows us to write @xmath809 for @xmath810 we let @xmath811 ( resp . @xmath812 ) denote the contribution to @xmath806 from @xmath813 ( resp . @xmath814 ) . we begin by estimating @xmath811 . for this it will be convenient to write @xmath815 , where @xmath6 is given multiplicatively at prime powers by @xmath816 for given @xmath817 with @xmath539 , this allows us to deduce that @xmath818 by the burgess bound for character sums . equipped with this it is straightforward to conclude that @xmath819 turning to the contribution from @xmath814 and writing @xmath820 , we have @xmath821 note that @xmath822 is necessarily odd in this summation . rearranging terms we are led to an inner sum of the form @xmath823 for suitable real numbers @xmath824 with modulus at most @xmath12 . breaking the @xmath822 sum into dyadic intervals and applying heath - brown s large sieve for real characters ( * ? ? ? * cor . 4 ) , we obtain @xmath825 taking @xmath826 and combining this with our estimate for @xmath811 , we therefore arrive at the desired bound for @xmath806 in . having shown that we may safely approximate @xmath545 by @xmath827 in @xmath828 , we proceed to swap the sums over @xmath829 and @xmath543 . in this way we are led to consider the sum @xmath830 the asymptotic evaluation of @xmath831 is the object of the following result . [ lem : calculate - s ] let @xmath832 where @xmath833 then we have latexmath:[\[\sum_{m , n}\sum_{{\ensuremath{\boldsymbol\ell } } } \sum_{m'',n '' } \sum_{\beta,\gamma,\delta}\sum_{{\ensuremath{\boldsymbol\ell } } '' } \frac{p^2}{cdm^2 } \sum_{t_0}\sum_{z } \frac{1}{z\psi(z ) } since @xmath835 and @xmath836 are coprime , we have @xmath837 . writing @xmath815 , where @xmath6 is given by , we obtain @xmath838 we recall that @xmath839 . making the obvious change of variables , and and using the mbius function to detect coprimality conditions , the inner sum becomes @xmath840 where @xmath841 denotes the number of positive integers @xmath842 and @xmath843 such that @xmath844\mid x , \quad [ k , k_1]\mid y,\ ] ] and @xmath845 . note here that @xmath846 and @xmath847 are automatically both coprime to @xmath752 since @xmath480 and @xmath481 are . making a further change of variables , @xmath841 is equal to the number of positive integers @xmath848e_1a)$ ] and @xmath849e_2b)$ ] such that @xmath850 and @xmath851 , for integers @xmath852e_1}a_0 $ ] and @xmath853e_2}b_0 $ ] . taking a trivial upper bound for @xmath841 , we find that @xmath854[k , k_2]e_1e_2ab}.\end{aligned}\ ] ] note that @xmath855 for any @xmath856 . in the usual way , combining these estimates with lemma [ lem : r1r2 ] shows that the overall contribution from @xmath857 is @xmath858 , for any @xmath79 . hence we may assume that @xmath859 in . we estimate @xmath841 using lemma [ lem : basic ] twice , with @xmath860 and @xmath861 . this is valid provided that @xmath862e_1a)\gg t_1mz^{\varepsilon}$ ] and @xmath863e_2b)\gg t_1mz^{\varepsilon}$ ] . recall from and that @xmath864 and @xmath865 . it therefore suffices to have @xmath866 , which obviously holds . hence we may conclude that @xmath867[k , k_2]e_1e_2abm^2 } \left(1+o \left(\frac{1}{t_1^{1-{\varepsilon}}}\right)\right).\ ] ] the overall contribution from the error term here , once substituted into and summed over the remaining parameters , is easily found to be @xmath868 . similarly , the shape of the main term allows us to extend the summations over @xmath58 and @xmath385 to infinity with acceptable error . we are therefore led to approximate @xmath831 by @xmath869[k , k_2]}.\end{aligned}\ ] ] the inner sum over @xmath870 here has already been calculated in . finally , by calculating euler factors , one sees that the sums over @xmath871 evaluate to @xmath872 it now follows that @xmath873 agrees with expression in the statement of the lemma . armed with lemma [ lem : calculate - s ] , we now return to our expression for @xmath828 . let @xmath874 furthermore , we observe that @xmath875 say . then , in place of @xmath828 , our work so far has shown that we may work with @xmath876 where @xmath877 is given in the statement of lemma [ lem : calculate - s ] . we have now arrived at the final stages of the argument , where it is necessary to evaluate the sum over @xmath543 asymptotically . this is achieved by taking @xmath878 of order @xmath879 in lemma [ lem : z - sum ] . in particular it is clear that @xmath880 . let us write @xmath881 and @xmath882 . since clearly holds , we deduce that @xmath883 one notes that @xmath884 and @xmath885 . hence the overall contribution from the error term here is satisfactory for theorem [ thm2 ] , once summed over the remaining parameters satisfying using lemma [ lem : r1r2 ] . finally , using this result a final time we can eliminate and so extend the summation over the outer parameters to infinity , with acceptable error . returning to , we have therefore shown that the statement of theorem [ thm2 ] holds , with @xmath886 we proceed to study this constant , with the goal of showing that it is positive . the set @xmath716 was defined to be the @xmath888 for which holds . since @xmath656 , @xmath889 and @xmath890 are pairwise coprime , we clearly have @xmath891 furthermore , lemma [ lem : card - t ] implies that @xmath892 whereas @xmath893 is given by lemma [ lem : card - t ] . recall the expression for the non - zero constant @xmath665 . returning to our formula for the constant , in view of the definition of @xmath688 , we sum over the six possible choices for @xmath894 to get @xmath895 next we undertake the summation over @xmath619 , where @xmath679 is given by . a simple calculation reveals that @xmath896 likewise , with reference to the definition of @xmath677 and lemma [ lem : card - t ] , it is straightforward to check that the sum over @xmath702 evaluates to @xmath897 where @xmath898 is given by and @xmath899 noting that @xmath900 and @xmath901 , we may therefore conclude that @xmath902 these sums over @xmath903 are absolutely convergent and positive , as one checks by considering the associated euler products . this therefore shows that @xmath904 , as required to complete the proof of theorem [ thm2 ] .
|
chtelet surfaces provide a rich source of geometrically rational surfaces which do not always satisfy the hasse principle .
we investigate the frequency that such counter - examples arise over the rationals .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
after many years of investigating hadron - hadron and heavy ion collisions , the study of hadron production remains an active and important field of research . the lack of detailed knowledge of the microscopic mechanisms has led to the use of many different models , often from completely opposite directions . thermal models , based on statistical weights for produced hadrons @xcite , are very successful in describing particle yields at different beam energies @xcite , especially in heavy ion collisions . these models assume the formation of a system which is in thermal and chemical equilibrium in the hadronic phase and is characterised by a set of thermodynamic variables for the hadronic phase . the deconfined period of the time evolution dominated by quarks and gluons remains hidden : full equilibration generally washes out and destroys large amounts of information about the early deconfined phase . the success of statistical models implies the loss of such information , at least for certain properties , during hadronization . it is a basic question as to which ones survive the hadronization and behave as messengers from the early ( quark dominated ) stages , especially if these are strongly interacting stages . in the case of full thermal and chemical equilibrium , relativistic statistical distributions can be used , leading to exponential spectra for the transverse momentum distribution of hadrons . on the other hand , experimental data at sps and rhic energies display non - exponential behaviours at high @xmath1 . one explanation of this deviation is connected to the power - like hadron spectra obtained from perturbative qcd descriptions : the hadron yield from quark and gluon fragmentation overwhelms the thermal ( exponential ) hadron production . however , this overlap is not trivial . one can assume the appearance of near - thermal hadron distributions , which is similar to the thermal distribution at lower @xmath1 , but it has a non - exponential tail at higher @xmath1 . a stationary distribution of strongly interacting hadron gas in a finite volume can be characterized by such a distribution ( or strongly interacting quark matter ) , which will hadronize into hadron matter . tsallis distributions satisfy such criteria @xcite . in the next section we will review the tsallis distribution and emphasize the properties most relevant to particle yields . neglecting quantum statistics , the entropy of a particle of species @xmath2 is given by @xcite @xmath3 where the mean occupation numbers , @xmath4 , are given by @xmath5 with @xmath6 being the degeneracy factor of particle @xmath2 . the total number of particles of species @xmath2 is given by an integral over phase space of eq . ( [ boltzmann ] ) : @xmath7 the transition to the tsallis distribution makes use of the following substitutions @xcite @xmath8^{1\over 1-q } , \label{subb}\end{aligned}\ ] ] which leads to the standard result @xcite @xmath9^{-{1\over q-1 } } , \label{tsallis}\ ] ] which is usually referred to as the tsallis distribution @xcite . as these number densities are not normalized , we do not use the normalized @xmath10-probabilities which have been proposed in ref . @xcite . in the limit where @xmath11 this becomes the boltzmann distribution : @xmath12 the particle number is now given by @xmath13 note that @xmath14 is the maximum value that still leads to a convergent integral in eq . ( [ tsallis_number ] ) . a derivation of the tsallis distribution , based on the boltzmann equation , has been given in ref . @xcite . a comparison between the two distributions is shown in fig . ( [ tsallis_boltzmann ] ) , where it can be seen that , at fixed values of @xmath15 and @xmath16 , the tsallis distribution is always larger than the boltzmann one if @xmath17 . taking into account the large @xmath1 results for particle production we will only consider this possibility in this paper . as a consequence , in order to keep the particle yields the same , the tsallis distribution always leads to smaller values of @xmath15 for the same set of particle yields . , keeping the temperature and chemical potential fixed , for various values of the tsallis parameter @xmath10 . the value @xmath14 is the maximum value that still leads to a convergent integral in eq . [ tsallis_number ] . chemical potential @xmath16 , keeping the temperature and the energy fixed . , title="fig:",scaledwidth=47.0%,height=377 ] , keeping the temperature and chemical potential fixed , for various values of the tsallis parameter @xmath10 . the value @xmath14 is the maximum value that still leads to a convergent integral in eq . [ tsallis_number ] . chemical potential @xmath16 , keeping the temperature and the energy fixed . , title="fig:",scaledwidth=47.0%,height=377 ] the dependence on the chemical potential is also illustrated on the right of fig . [ tsallis_boltzmann ] for a fixed temperature @xmath15 and a fixed energy @xmath18 . as one can see , the tsallis distribution in this case increases with increasing @xmath10 . the tsallis distribution for quantum statistics has been considered in ref . the parameter @xmath10 plays a central role in the tsallis distribution and a physical interpretation is needed to appreciate its significance . to this end we follow the analysis of ref . @xcite and write the tsallis distribution as a superposition of boltzmann distributions @xmath19 where the detailed form of the function @xmath20 is given in @xcite . the parameter @xmath21 is the standard temperature as it appears in the boltzmann distribution . it is straightforward to show @xcite that the average value of @xmath22 is given by the tsallis temperature : @xmath23 while the fluctuation in the temperature is given by the deviation of the tsallis parameter @xmath10 from unity : @xmath24 which becomes zero in the boltzmann limit . the above leads to the interpretation of the tsallis distribution as a superposition of boltzmann distributions with different temperatures . the average value of these ( boltzmann ) temperatures is the temperature @xmath15 appearing in the tsallis distribution . this is the interpretation of the tsallis temperature that we will follow . the other parameter in the tsallis distribution , @xmath10 , describes the spread around the average value of the ( boltzmann ) temperature @xmath15 . for @xmath25 we have an exact boltzmann distribution , for values of @xmath10 which deviate from 1 , we have a corresponding deviation . from this point of view the tsallis distribution describes a distribution of ( boltzmann ) temperatures . a deviation from @xmath25 means that a spread of temperatures is needed instead of a single value . /d.o.f . of the fits as a function of the tsallis parameter @xmath10.,scaledwidth=80.0%,height=377 ] in order to identify the energy dependence of the deviation from ideal gas behaviour , thermal fits were performed on yields measured at the cern sps in central pb - pb collisions at 40 agev , 80 agev and 158 agev ( using the same data as analyzed in @xcite ) and yields measured at rhic in central au - au collisions at @xmath26 = 130 agev ( using the same data as analyzed in @xcite ) and at @xmath26 = 200 agev . in the cern sps fits , the thermal parameters @xmath15 , @xmath27 , @xmath28 and @xmath29 were fit to the data , while @xmath30 and @xmath31 were fixed by the initial baryon - to - charge ratio and strangeness content in the colliding system , respectively . in the case of the rhic analysis we again fit @xmath15 , @xmath27 , @xmath31 and @xmath28 to the data . the use of mid - rapidity data here led to the relaxing of the constraints on @xmath31 and @xmath30 typical in analyzes of 4@xmath32 data . instead , @xmath30 was set to zero as justified by the observed @xmath33 ratio . the following expression was used to calculate primordial particle yields , @xmath34^{-1/(q-1)},\ ] ] where @xmath35 is the number of valence strange quarks and anti - quarks in species @xmath2 . the value @xmath36 obviously corresponds to complete strangeness equilibration . all calculations were done using the thermus package @xcite . the most surprising result of our analysis is shown in fig . ( [ chi ] ) : the quality of the fits , as measured by the @xmath37 , improves at first as the tsallis parameter @xmath10 increases . it reaches a minimum value around @xmath38 1.07 for sps beam energy of 158 agev . this behaviour is repeated at other sps energies with the minima at slightly different values of @xmath10 , i.e. 1.08 for 80 agev and about 1.05 at 40 agev beam energy . this behaviour is not seen at rhic energies . clearly , changes in the tsallis parameter @xmath10 have only a small negligible effect on the @xmath0 values at rhic energies , of course , this still leaves open the possibility for @xmath10 values larger than 1 @xcite . however on the sps data the effect is substantial and changes the interpretation substantially . one possible interpretation is that at sps energies fluctuations in the freeze - out temperature are substantial . recently @xcite a coalescence model with a tsallis distribution for quarks was used to fit the transverse momenta spectra measured at rhic . this fit does not include decays from resonances and therefore can not be compared directly to ours since decays can substantially modify the transverse momenta , also the emerging hadrons are not in a tsallis - type equilibrium gas , which is an assumption of the present analysis . the authors obtained values for the tsallis parameter @xmath10 which are remarkably similar for all particle species considered , i.e. @xmath39 which can not be excluded by our analysis . the freeze - out temperature @xmath15 decreases , as expected , with increasing values of @xmath10 . this can be understood from the fact that the tsallis distribution is always larger than the boltzmann one ( as long as @xmath17 ) . hence , in order to match the same particle yields one has to adjust @xmath15 to lower values . this is seen at all energies in fig . however , the drop in @xmath15 , turns out to be quite drastic numerically . in fact , the decrease in particle numbers has to be compensated by increases in all other thermodynamic variables . the ( modest ) increase in the baryon chemical potential is shown in fig . [ temp ] on the right hand side . the strangeness non - equilibrium factor @xmath40 as shown in fig . [ gammas ] . it is interesting to note that the tsallis distribution leads to a much better chemical equilibrium than the corresponding boltzmann distribution with @xmath41 . in all cases considered the @xmath40 is very close to the chemical equilibrium value of 1 . .,title="fig:",scaledwidth=47.0%,height=377 ] .,title="fig:",scaledwidth=47.0%,height=377 ] as a function of the tsallis parameter @xmath10.,scaledwidth=80.0%,height=377 ] clearly , the use of the tsallis distribution in relativistic heavy ion collisions calls for a reevaluation of the understanding gained from previous analyses @xcite . the authors acknowledge support from the hungary - south africa scientific cooperation programme and hungarian otka grant nk62044 . we acknowledge useful discussions with h.g . miller , t.s . bir , p. vn and g.g . barnafldi . 10 e. fermi , progress theor . phys . , * 5 * ( 1950 ) 570 . pomeranchuk , dokl . 78 ( 1951 ) 889 . w. heisenberg , naturwissenschaften , * 39 * ( 1952 ) 69 . r. hagedorn , nuovo cimento , * 35 * ( 1965 ) 395 . j. cleymans , h. oeschler , k. redlich , s. wheaton , phys . c * * 73 ( 2006 ) 034905 . j. cleymans , p. braun - munzinger , h. oeschler and k. redlich , nucl . phys . a * 697 * ( 2002 ) 902 . a. andronic , p. braun - munzinger , j. stachel , nucl . phys . a * 772 * ( 2006 ) 167 . f. becattini , j. manninen , m. gadzicki phys . c * 73 * ( 2006 ) 044905 . see e.g. s.r . de groot , w.a . van leeuwen , ch.g . van weert , _ relativistic kinetic theory _ , north holland 1980 . c. tsallis , j. stat . phys . * 52 * ( 1988 ) 479 . c. tsallis , r.s . mendes , a.r . plastina , physics a * 261 * ( 1998 ) 534 . bir , g. purcsel , phys . * 95 * ( 2005 ) 162302 . a.m. teweldeberhan , a.r . plastino , h.g . miller , phys . a * 343 * ( 2005 ) 71 . plastino , a. plastino , h.g . miller , phys . lett . a * 343 * ( 2005 ) 71 . f. buyukkilic , d. demirhan , phys . lett . a * 181 * ( 1993 ) 24 . f. buyukkilic , d. demirhan , a. gulec , phys . lett . a * 197 * ( 1995 ) 209 . g. wilk and z. wlodarczyk , phys . rev * 84 * ( 2000 ) 2770 . s. wheaton , j. cleymans , m. hauer , computer physics communications * 180 * ( 2009 ) 84 . j. cleymans , b. kmpfer , m. kaneta , s. wheaton and n. xu , phys . c * 71 * ( 2005 ) 054901 . bir , k. rmssy k and g.g . barnafldi , j. phys . g * 35 * 044012 t.s . bir , k. rmssy , arxiv:0812.2985 [ hep - ph ] .
|
hadron yields in high energy heavy ion collisions have been fitted with thermal models using standard ( extensive ) statistical distributions .
these models give insight into the freeze - out conditions at varying beam energies and lead to a systematic consistent picture of freeze - out conditions at all beam energies . in this paper
we investigate changes to this analysis when the statistical distributions are replaced by non - extensive tsallis distributions for hadrons .
we investigate the particle yields at sps and rhic energies and obtain better fits with smaller @xmath0 for the same hadron data , as applied earlier in the thermal fits for sps energies but not for rhic energies .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
to maintain activities of central starbursts or active galactic nuclei in galaxies , fueling of molecular gas toward the center is necessary . bar is one of the possible mechanisms , for the gas fueling . many numerical simulations of gas motion in the bar potential show that bars can accumulate gas toward the central region ( e.g. , athanassoula 1992 ) . observational evidence of gas accumulation toward the central kilo - parsec region has also been shown . for example , sakamato et al . ( 1999 ) and sheth et al . ( 2005 ) showed that barred spiral galaxies have higher central concentration of molecular gas than non - barred spiral galaxies . furthermore , kuno et al . ( 2007 ) showed that the higher central concentration in barred spirals is made by gas accumulation within a region on the order of the bar length and found a correlation between bar strength and the degree of central concentration . on the other hand , observations of molecular gas with high angular resolution sufficient to resolve the central structure less than 100 pc are still limited ( e.g. , schinnerer et al . 2003 ; garca - burillo et al . such observations are necessary to understand how the gas flows into the nucleus and relates with central activities . for example , the size of the twin peaks often seen in the central region of barred galaxies is less than several hundred pc ( kenney et al . therefore , we need such high angular resolution to investigate the distribution and dynamics of molecular gas in the central region . maffei 2 is a barred spiral galaxy that belongs to the ic 342/maffei group . since maffei 2 is located behind the galactic plane , its optical features are not clear . however , the bar and spiral structures can be seen clearly in a near infrared image ( jarrett et al . 2003 ) . many co observations have so far been done ( ishiguro et al . 1989 ; kawabe et al . 1991 ; israel , baas 2003 ; mason , wilson 2004 ; kuno et al . the co map obtained with the nobeyama 45-m telescope shows structures typical of the distribution of molecular gas in barred spiral galaxies ; central peak , offset ridges along the leading side of the large - scale bar , condensations of molecular gas at the end of the bar , and spiral arms from the end of the bar . they are symmetric against the center . the motion of molecular gas in the bar is also consistent with that seen in barred spiral galaxies . namely , the large velocity change is seen across the bar implying that the gas moves along the offset ridges toward the center . since maffei 2 is one of the nearest barred spiral galaxies , we can investigate the distribution and dynamics of molecular gas in the center and the bar in detail . furthermore , since maffei 2 has a moderate starburst at the center as suggested from ir and radio observations ( rickard , harvey 1984 ; turner , ho 1994 ) , observations of maffei 2 with high angular resolution are helpful to understand the mechanism of the gas fueling toward the galactic center by bars and relation with a starburst phenomenon in barred spiral galaxies . in this paper , we present the results of @xmath0co(10 ) , @xmath0co(21 ) , cs(21 ) lines and 103 ghz continuum observations of maffei 2 with the nobeyama millimeter array ( nma ) . we discuss the central structure of molecular gas and star formation in maffei 2 . we adopt 2.8 mpc as the distance of mafei 2 throughout this paper ( karachentsev et al . observations in @xmath0co(10 ) , @xmath0co(21 ) , and cs(21 ) were made from dec . 2002 to jan . 2004 using nma . 103 ghz continuum data was also obtained simultaneously with @xmath0co(10 ) . figure 1 shows the field of views of nma on the co map of maffei2 obtained with the nobeyama 45-m telescope ( kuno et al . the receiver frontends were sis mixers . system temperatures ( dsb ) were 100 200 k for cs(21 ) , 150 300 k for co(10 ) , and 200 350 k for co(21 ) . the uwbc spectro correlator ( okumura et al . 2000 ) which covers 512 mhz with 256 channels was used as backend ( effective frequency resolution : 4 mhz ) . the bandwidth and the frequency resolution correspond to 1563 km s@xmath8 and 12.2 km s@xmath8 at 98 ghz , 1322 km s@xmath8 and 10.3 km s@xmath8 at 115 ghz , 666 km s@xmath8 and 5.2 km s@xmath8 at 230 ghz . the data of @xmath0co(10 ) and cs(21 ) were obtained with antenna configurations ab , c , and d , while only c and d were used for @xmath0co(21 ) . the field center is ( @xmath9 , @xmath10 ) = ( , + ) for all observations . phase and amplitude calibrations were performed every 20 25 minutes using a nearby quasar nrao 150 ( 0359 + 509 ) . the quasar was also used for band - pass calibration . the absolute flux of nrao 150 was measured by comparing with uranus and neptune . the uncertainty of absolute flux is estimated to be about 20 % . we mapped the @xmath0co data with uniform weight , while natural weight was used for cs(21 ) and 103 ghz continuum . we achieved @xmath11 synthesized beam in @xmath0co(10 ) , @xmath12 beam in @xmath0co(21 ) . the beam sizes correspond to 23 pc @xmath13 20 pc and 31 pc @xmath13 19 pc at the distance of maffei 2 . for cs(21 ) and 103 ghz continuum , the beam sizes are @xmath14 and @xmath15 , which correspond to 68 pc @xmath13 49 pc and 67 pc @xmath13 50 pc , respectively . the observational data are listed in table 1 . the missing fluxes of the lines are estimated by comparing single dish observations . flux within the field of view of nma ( @xmath16 ) in the @xmath0co(10 ) map is 3200 jy km s@xmath8 after correction for primary beam attenuation . on the other hand , the total flux within the @xmath17 beam measured with a single - dish telescope is 3513 jy km s@xmath8 ( sargent et al . the missing flux is estimated to be about 10 % . therefore , most of the total flux was recovered in our @xmath0co(10 ) map . for @xmath0co(21 ) , the flux within the field of view of nma ( @xmath18 ) is 6850 jy km s@xmath8 . the total flux within the @xmath19 beam measured by sarget et al . ( 1985 ) is 14880 jy km s@xmath8 . the total flux of nma is only 46 % of the single dish measurement . however , the total flux within the @xmath20 and @xmath21 beams measured by israel and baas ( 2003 ) is 4700 jy km s@xmath8 and 8527 jy km s@xmath8 , respectively . these results seem to be consistent with ours . when we convolve our data to the @xmath20 beam , the peak flux is 3468 jy km s@xmath8 which corresponds to 74 % of the single dish measurement . total flux of cs(21 ) line is 68 jy km s@xmath8 . total flux density of 3 mm continuum is 38 mjy . unfortunately , there are no single dish data of cs(21 ) and 103 ghz continuum . figure 2 shows the integrated intensity maps of @xmath0co(10 ) and @xmath0co(21 ) . although the @xmath0co(10 ) map is consistent with the previous map ( ishiguro et al . 1989 ; kawabe et al . 1991 ) , our map shows more detailed structures . in the bar region , the offset ridges along the leading side of the bar can be seen . the width of the ridge ( fwhm ) deconvolved with the beam size is @xmath22 which corresponds to 50 pc at the distance of maffei2 . the offset ridges are truncated at about @xmath23 from the center on both sides . at the edges of the offset ridges , they have bifurcation toward the trailing side . it is interesting that the same feature is seen in the @xmath0co(10 ) map of ic 342 ( meier et al . 2000 ) . the offset ridges connect to strong peaks at about @xmath24 ( @xmath25 pc ) north and south of the center . they can be regarded as the same feature as the twin peaks found by kenney et al . ( 1992 ) in some barred spiral galaxies . the size of both peaks ( fwhm ) is about @xmath26 ( 54 pc @xmath13 136 pc ) . the center derived from 2 mass data ( , + ( j2000 ) : jarrett et al . 2003 ) is located between the peaks . at the peaks , the offset ridges , especially the northern ridge , show large curvature . since the inclination angle of maffei 2 is fairly large ( @xmath27 : hurt et al . 1996 ) , some structures overlap along the line of sight . such structures mixed in the peaks can be separated in the velocity domain , as shown in section 3.2 . the distribution of @xmath0co(21 ) emission is consistent with that of @xmath0co(10 ) emission . the curve of the ridges at the peaks is more apparent in @xmath0co(21 ) , especially in the southern ridge . although the size of the peaks is about the same as that in @xmath0co(10 ) , they are closer to the center . in both lines , the emissions are weak at the center . assuming a conversion factor from co(1 0 ) intensity to column density of h@xmath28 @xmath29h@xmath30 of @xmath31 @xmath32 [ k km s@xmath8]@xmath8 ( nakai , kuno 1995 ) , the molecular gas mass within the field of view ( @xmath34 pc ) is estimated to be about @xmath35 . this is smaller than the values reported in the previous observations ( e.g. , ishiguro et al . 1989 ) because of the difference in the adopted distance and the conversion factor . if we use the same distance and conversion factor , our result is consistent with previous studies . the peak flux of @xmath0co(10 ) is 47.3 jy beam@xmath8 km s@xmath8 which corresponds to the column density of @xmath36 @xmath32 or the surface density of about 2700 @xmath37 pc@xmath38 . since the width of the ridges ( fwhm ) in the map is about 50 pc , the thickness must be smaller than that . if we assume that the thickness and inclination angle of the gas disk are 50 pc and 67 deg , respectively , a lower limit of the average volume density within the beam is derived to be @xmath39 @xmath7 . the peak flux density of @xmath0co(10 ) at the twin peaks is about 600 mjy beam@xmath8 which corresponds to @xmath40 k. on the other hand , the peak flux density of @xmath0co(21 ) is about 2.8 jy beam@xmath8 which corresponds to @xmath41 k. although these values are the average within the beam and a lower limit of the kinetic temperature of the molecular gas , the temperature is higher than that of gmcs in our galaxy . we will discuss the physical properties of molecular gas including observations of other lines in a forthcoming paper . figure 3 shows the integrated intensity map of cs(21 ) . the cs map also show twin peaks . the peaks coincide with the co peaks . the northern peak is much stronger than the southern side . figure 4 shows the 3 mm continuum map . there is a peak about @xmath42 north of the center with an extended structure along the declination . there is a weak peak about @xmath43 south of the center . the overall size is @xmath44 ( @xmath45 ) . the 3 mm continuum is thought to trace star - forming regions , since it is dominated by free - free emission ( condon 1992 ) . actually , these structures coincide with those in other tracers of star - forming activity , such as the 2 cm and 6 cm continua ( turner , ho 1994 ) , 10.8 @xmath46 m ( telesco et al . 1993 ) . however , the correlation with h@xmath47 is not good . the h@xmath47 peak in ishiguro et al . ( 1989 ) is located about @xmath48 north of the 103 ghz peak . the reason for the bad correlation between free - free emission and h@xmath47 is thought to be dust extinction in h@xmath47 . a similar trend is seen in m82 ( matsushita et al . 2005 ) . the total flux density of the 3 mm continuum is 38 mjy . the peak flux density within the @xmath15 beam is about @xmath49 of the total flux density . therefore , most of the emission is concentrated in the peak . we estimate the production rates of lyman continuum photons , @xmath50 , using the flux according to condon ( 1992 ) . the result derived from the total flux density is @xmath51 where @xmath52 and @xmath53 are electron temperature and distance , respectively . this is about two times larger than that derived from the 6 cm continuum ( turner , ho 1994 ) after correcting the difference of the adopted distance . as compared with the starburst galaxy m82 , the derived @xmath50 of maffei 2 is comparable with that of an individual peak in m82 measured with the @xmath54 beam ( matsushita et al . 2005 ) . the massive star formation rate ( @xmath55 ) derived from the production rate of lyman continuum photons using the equation ( 24 ) in condon ( 1992 ) is @xmath56 . on the other hand , the star formation rate derived from the infrared luminosity of @xmath57 ( rickard , harvey 1984 ) using the equation ( 26 ) in condon ( 1992 ) is @xmath56 , if the distance we adopted is used . they show good agreement . figures 5 and 6 are the channel maps of @xmath0co(10 ) and @xmath0co(21 ) . they show quite similar features . near the systemic velocity of maffei 2 ( -24 km s@xmath8 : kuno et al . 2007 ) , two peaks are seen north and south of the center , shown by a cross in the figures . for a circular motion , the emission should be observed near the minor axis at systemic velocity . the peaks , however , are located near the major axis , indicating large non - circular motion . the peaks move toward the north with decreasing radial velocity keeping the separation of the peaks almost constant . on the other hand , they move toward the south with increasing radial velocity . position - velocity diagrams are helpful to understand the complex velocity structure in maffei 2 . figure 7 shows the position - velocity diagrams of @xmath0co(21 ) parallel to the major axis of maffei 2 ( p.a . = @xmath58 : hurt et al . it is apparent that the central structure is separated into two parallel components in the p v diagrams ( b - c - d , b-c-d ) . it implies that there are two components along the line of sight . the radial velocities at the center of both components are not systemic velocity of maffei 2 ( -24 km s@xmath8 ) . there is no emission at systemic velocity at the center . the parallel components are clearly separated in the p v diagram along the minor axis as shown in figure 8 . since the parallel structures in the p v diagrams are symmetric against the 2 mass center , the center is thought to coincide with the dynamical center . for the northern component ( upper side ) in figure 7(c ) , a b corresponds to the offset ridges of the large - scale bar ( a b in figure 7(a ) for the southern component ) . if we assume that the spiral arms of maffie 2 are trailing arm , the eastern side against the major axis of the galaxy is far side and the large - scale bar approaches us closing to the center for the northern side of the galaxy ( figure 1 ) . therefore , if the gas moves along the large - scale bar , the observed blue - shift of the radial velocities from the systemic velocity means inward motion . the offset ridge connects to the parallel structure at b ( b for the southern component ) . the radial velocity along the parallel structure changes from blue - shift to red - shift before the structure crosses the minor axis ( b c ) , although the point should be on the minor axis for a circular motion . the same feature but in the opposite sense can be seen for the southern component . furthermore , the parallel components are connected by a weak oval structure , as indicated in figure 7b . there are bifurcations in the oval structure at c and c. the radial velocity increases closing to the center from the points to d and d. as mentioned in the previous section , the p v diagrams along the major axis ( figure 7 ) shows two components almost overlapping along the line of sight . these structures can be regarded as a spiral structure embedded in a weak oval structure . we made integrated intensity maps of each component of the parallel structure in figure 7 separately using the channel map . figure 9 shows the integrated intensity map overlaid on the hst image ( f814w ) obtained from the hst archive . they look like the two - arm spiral structure continues from the offset ridges along the large - scale bar in the co map obtained with the nro 45-m telescope . the structure can be seen in both @xmath0co(10 ) and @xmath0co(21 ) . the spiral structure is symmetric against the center determined from the 2mass data . the northern arm coincides with the dust lane in the hst image very well near the center , while the dust lane along the southern arm is not clear . this is because the northern arm is on the near side ( western side against the major axis of the galaxy ) near the center . it is apparent that the oval structure in figure 7b is not an expanding ring . the part of the oval structure that overlaps with the northern arm is on the near side near the center , as mentioned above . the radial velocity of the part is red - shift from the systemic velocity of maffei 2 ( around c in figure 7c ) . on the other hand , if the ring is shrinking , the velocity toward the center must be higher than 50 km s@xmath8 . in that case , the oval structure collapses into the nucleus in less than @xmath59 yr . this seems to be unlikely . the oval structure can be naturally understood as an oval motion of molecular gas . the oval structure connects the spiral structure . the schematic figure of the spiral and oval structures is shown in figure 10 . here , we assume inclination and position angle derived for the galactic disk . similar structures with the offset ridges and oval structure are often seen in numerical simulations ( e.g. , athanassoula 1992 ) . these structures are made from @xmath11 and @xmath12 orbits in a bar potential . similar structure has been found in ngc 4303 ( schinnerer et al . 2002 ; koda , sofue 2006 ) . the complex p v diagrams can be understood with our interpretation . the offset ridges along the large - scale bar contact with the x2 orbit at b and b in figure 7 . the constant gradient of the radial velocity b c and b c in figure 7 is attributed to the flow of the molecular gas along the spiral arms around the center . the spiral arms diverge from the oval structure toward the center . this corresponds to c d and c d in the p v diagrams . the deviation from the systemic velocity becomes max there , since the gas stream line closes to the line of sight . the sizes of the structure shown in figure 10 were derived from the p v diagrams along the major and minor axes and the channel map as follows : ( 1 ) the separation along the major axis between the spiral arms derived from figure 7(b ) ( @xmath25 pc ) . ( 2 ) the separation along the minor axis between the spiral arms derived from figure 8 ( @xmath60 pc ) . ( 3 ) the separation along the major axis direction between the positions where radial velocity equals to the systemic velocity . if we assume that most of the gas moves along the spiral arms , the spiral arms are perpendicular to the line of sight at these positions . this can be measured in channel map at the systemic velocity of maffei 2 ( @xmath61 -24 km s@xmath8 ) ( figure 5 and 6 ) . ( @xmath62 pc ) . ( 4 ) the separation along the minor axis between the positions where radial velocity equals the systemic velocity derived from figures 5 and 6 . ( @xmath63 pc ) . ( 5 ) the length of the oval structure along the major axis derived from figure 7(b ) ( @xmath64 pc ) . martini et al . ( 2003a ; 2003b ) show from hst images that galaxies with strong bars often have a grand - design nuclear dust spiral structure , which appears to be the continuation of the dust lanes along the leading edges of a large - scale bar toward the center . on the other hand , observations of such spiral structures of molecular gas are still rare . the distribution of molecular gas in maffei 2 can be regarded as such a spiral structure . although maffei 2 is classified as sab(rs ) in rc3 , it seems to be due to the unclear optical image . according to buta and block ( 2001 ) , maffei 2 should be classified as sb(rs ) . therefore , our result is consistent with martini(2004 ) s finding that grand - design spiral structure is only found in sb(s ) or sb(rs ) . some numerical simulations show that when there is a massive concentration at the center , such as a super massive black - hole , a nuclear spiral structure is formed ( fukuda et al . 1998 ; ann , lee 2004 ; maciejewski 2004 ) . this is because the mass concentration removes the rigid rotation part from the rotation curve . as a result , iilr ( inner inner lindblad resonance ) , where a circum - nuclear ring is formed , disappears or another resonance ( nuclear lindblad resonance ) appears inside the iilr . the mass of the central concentration assumed in numerical simulations to make the spiral structure is 0.1 % 1 % of the total mass of the host galaxies . the observational data of molecular gas has the advantage that it can be used to investigate the kinematics of the central structure . from the p v diagram along the major axis ( figure 7 ) , the rotation velocity is estimated to be about 140 km s@xmath8 at @xmath65 pc . the central mass within @xmath3 35 pc is estimated to be @xmath4 from the rotation velocity assuming spherical distribution of mass . the mass is about 0.5 % of the total dynamical mass of maffei 2 ( mason , wilson 2004 ) . even if the rotation velocity is overestimated due to the non - circular motion , it seems to be more than 0.1 % of the dynamical mass ( koda , wada 2002 ) . the mass fraction of the central mass concentration in maffei 2 is comparable to the prediction of the simulations . the rotation velocity still rising toward the center at @xmath65 pc . it is desirable to obtain the inner rotation curve with higher spatial resolution and higher sensitivity observations to know the mass distribution in the inner region . the molecular gas changes the direction of motion with a large velocity change when it enters the offset ridge along the large - scale bar . then , the gas flows toward the center along the offset ridge . since the radial velocity is almost constant along the straight ridges , as seen in figure 7 , the streaming velocity along the ridge seems to be constant . ( the component of the pattern speed of the bar is relatively small as compared with the velocity along the bar . ) the velocity along the ridge is estimated to be 70 - 120 km s@xmath8 assuming the inclination angle and position angle of the galactic disk . here , we assumed a pattern speed of the bar of 53.5 km s@xmath8 kpc@xmath8 which was adopted from hurt et al . ( 1996 ) and made a correction for the difference of the adopted distance . the velocity is comparable with those in ngc 253 ( sorai et al . 2000 ) and ngc 1530 ( regan et al . 1997 ) . using the velocity and surface density of molecular gas at the radius of 200 pc in the offset ridge , the mass flow of molecular gas along the ridges is derived to be 712 @xmath66 . this is comparable with the estimation for ngc 1530 by regan et al . ( 1997 ) . if the molecular gas in the flow is accumulated in the center directly , it takes only @xmath67 yr to concentrate all of the molecular gas in the disk on the oval structure . however , only a portion of the flowing gas must be accumulated in the central region as estimated by regan et al . . they evaluated it to be 20 % of the flowing gas . there are concentrations of molecular gas where the offset ridges connect to the spiral structure ( figures 7 and 9 ) . the mass of the condensations within 65 pc @xmath13 94 pc is estimated to be @xmath68 from the @xmath0co(10 ) intensity using a conversion factor of @xmath31 @xmath32 [ k km s@xmath8]@xmath8 ( nakai , kuno 1995 ) . the peaks correspond to @xmath11/@xmath12 orbit - crowding region . we compare the position of the peaks with the peaks in the cs(21 ) and 3 mm continuum maps . since the northern peak is much stronger than the southern peak in the cs(21 ) and 103 ghz maps , we discuss the northern peak here . the peak of the 3 mm continuum which is a tracer of star - forming regions is located downstream of the co peak . the offset between the co and 3 mm peaks is about 20 pc , while the cs peak coincides with the co peak . we can estimate the time scale for the gas to move from the cs peak to the 3 mm peak along the ridge using the streaming velocity measured at the offset ridge , 70 120 km s@xmath8 , assuming that the peaks are near the point where the spiral structure is perpendicular to the line of sight ( cf . figure 10 ) . the time scale is about 1.6 2.8 @xmath69 yr . on the other hand , free fall time of the dense gas traced by cs ( @xmath6 @xmath7 ) is estimated to be @xmath70 yr . this is comparable to the time for the gas to move from the cs peak to the 3 mm peak . from these results , the following scenario can be described : the molecular gas in @xmath11 orbit moves along the offset ridge in the large bar . at the @xmath11/@xmath12 orbit - crowding region , the gas contacts with @xmath12 orbit and is compressed by shock and the density rises abruptly there . once the dense gas is formed , it evolves into stars with free - fall time . it is interesting to compare the structure with that seen in ngc 6951 ( kohno et al . kohno et al . ( 1999 ) show that the condensations of dense gas traced by hcn in ngc 6951 are located downstream of more diffuse gas traced by co. they showed that the time scale of gravitational instability and flowing time of the gas along the ridges are comparable ( @xmath71 yr ) . this result supports the idea that the dense gas is made by gravitational instability during the flow along the ridge . we speculate that the difference of our results from kohno et al . ( 1999 ) may be caused by the difference in the size of the structure . the structure found in ngc 6951 is several times larger than that in maffei 2 . the size of the co peak is more than hundreds pc , while the condensations in maffei 2 are comparable to the galactic gmcs . therefore , the molecular clouds should be treated as particles in ngc 6951 . in that case , since the viscosity of the gas is small , shock may not play an important role in dense gas formation at the @xmath11/@xmath12 orbit - crowding region , while the gravitational instability does . on the other hand , the clump of molecular gas whose size is comparable with a gmc is compressed at the crossing point in maffei 2 . as a result , strong shock may occur and dense gas is formed at a burst there . we made @xmath0co(10 ) , @xmath0co(21 ) , cs(21 ) lines and 103 ghz continuum observations of maffei 2 . using the data set , we investigated the distribution and kinematics of molecular gas in the central region . the results are summarized as follows : + ( 1 ) we revealed the spiral structure of molecular gas in the central region . the offset ridges of molecular gas along the leading side of the large - scale bar continue to the spiral structure embedded in the weak oval structure which is regarded as a x2 orbit . the size of these structures is less than @xmath3 100 pc . the spiral structure continues toward the center at least until @xmath3 35 pc diverging from the oval structure . + ( 2 ) the central mass within @xmath3 35 pc is estimated to be @xmath4 from the rotation curve , which corresponds to 0.5 % of the dynamical mass of maffei 2 . the percentage is consistent with the theoretical predictions to make such a spiral structure of molecular gas . + ( 3 ) the amount of the molecular gas which flows along the offset ridges of the large - scale bar is estimated to be 712 @xmath66 . this is an upper limit of the mass flow of molecular gas into the nucleus . + ( 4 ) our results imply that dense gas is formed at the crossing points of @xmath11 and @xmath12 orbits . massive stars are formed from the dense gas and star - forming regions appear after free - fall time of the dense gas ( @xmath72 yr ) . we thank the nma staff for their kind support and encouragement . this research used the facilities of the canadian astronomy data centre operated by the national research council of canada with the support of the canadian space agency . & cs(21 ) & 103 ghz cont . & @xmath0co(10 ) & @xmath0co(21 ) + array configuration & ab , c , d & ab , c , d & ab , c , d & c , d + velocity coverage ( km s@xmath8 ) & 1563 & & 1322 & 666 + velocity resolution ( km s@xmath8 ) & 30.5 & & 10.3 & 7.8 + synthesized beam ( @xmath73 , @xmath74 ) & @xmath75 & @xmath76 , -35 & @xmath77 , 27 & @xmath78 , -34 + synthesized beam ( pc ) & @xmath79 & @xmath80 & @xmath81 & @xmath82 + rms noise of channel map ( mjy beam@xmath8 ) & 8 & 2 & 40 & 130 + total flux within field of view ( jy km s@xmath8 ) & 68 & 38 & 3200 & 6850 +
|
distribution and kinematics of molecular gas in the central region of the barred spiral galaxy maffei 2 were investigated using a data set of @xmath0co(10 ) , @xmath0co(21 ) , cs(21 ) lines and 103 ghz continuum .
we found that the offset ridges along the kpc - scale bar continue to the central spiral structure embedded in the weak oval structure which is regarded as @xmath12 orbit in the bar potential .
the spiral structure continues toward the center diverging from the oval structure .
the size of these structures is less than @xmath2 pc . the mass concentration within @xmath3 35 pc is estimated to be @xmath4 .
the high mass concentration is consistent with theoretical predictions concerning the creation of such a nuclear spiral structure . a comparison with the tracers of dense gas and star - forming region suggests that the dense molecular gas traced by cs(21 ) line is formed at the crossing points of @xmath11 and @xmath12 orbits and the star - forming region appears after @xmath5 yr which is comparable with the free - fall time of the dense gas traced by the cs line ( @xmath6 @xmath7 ) .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
dihadron azimuthal correlation has been a successful tool in understanding the interactions between jet and medium , and in extracting the properties of the sqgp . over the years , the correlation analyses have been carried out in various regions of transverse momentum ( @xmath0 ) for the triggers and partners . many interesting features have been discovered . in the high @xmath0 region , the correlation distributions show narrow peaks around @xmath1 ( near - side ) and @xmath2 ( away - side ) @xcite , consistent with fragmentation of jets escaping the dense medium with small energy loss . in the low @xmath0 region , the correlation distributions are dominated by a double hump structure around @xmath3 at the away - side ( the cone ) @xcite and a structure elongated along the @xmath4 but centered around @xmath1 ( the ridge ) @xcite , characteristic of a complicated response of the medium to energy deposited by the quenched jets . in the meanwhile , many theoretical models @xcite have been proposed to interpret the data . but to date , a complete and consistent picture accommodating the vast amount data is still missing . our goal is to provide a brief overview of the dihadron correlation results , with an eye towards the reciprocal relation between jet quenching and medium response , and discuss several insights distilled from the data . in general the dihadron correlations depend on the @xmath0 of both hadrons in the pair , and the full characterization of their modification patterns have to be studied differentially as function of trigger @xmath0 ( @xmath5 ) and partner @xmath0 ( @xmath6 ) . such a survey study has been carried out recently by the phenix @xcite and star collaboration @xcite . [ fig:1a ] summarizes dihadron @xmath7 distribution in a broad transverse momentum space , which shows many distinctive features appearing at different @xmath0 regions ( indicated by the circles and lines ) . these features fits well into a simple two - component picture as illustrated in fig . [ fig:1b ] separately for both the near- and away - side : a jet fragmentation component that dominates for @xmath8 gev/@xmath9 , and a medium response component that dominates at @xmath10 gev/@xmath9 . the rich @xmath0 dependent correlation patterns simply reflect the competition between fragmentation of survived jets and medium response to quenched jets on both the near- and away - side . the observed patterns are rather complicated in fig . [ fig:1a ] since 1 ) the medium response and jet fragmentation have very different angular distribution and very different spectral slope , 2 ) the shapes of the medium response are also quite different between the near- and away - side . a new variable @xmath11 was introduced recently to describe the medium response at low @xmath0 @xcite . @xmath11 quantify the medium modification of hadron pair yield from the expected yield , in a way similar to @xmath12 for describing the modification of single hadron yield . the hadron pair yield is proportional to the dijet yield , and in the absence of nuclear effects , it should scale with @xmath13 , and @xmath14 . [ fig:2 ] shows @xmath11 as a function of pair proxy energy ( @xmath15 ) for the near- ( left panel ) and away - side ( right panel ) . the star autocorrelation result @xcite is shown as a single point at @xmath16 gev/@xmath9 . in contrast to a constant suppression at large @xmath17 , the pair yields are not suppressed or even enhanced at @xmath18 gev/@xmath9 . this enhancement directly reflects the energy transport that redistribute energy of the quenched jets to low @xmath0 hadrons ( medium response ) . we would like to point out that @xmath17 is a natural variable for the near - side correlation since it approximates the jet energy , and data show an approximate scaling in @xmath17 . however , even the data points for the away - side tend to group together , probably because the medium response is a function of jet energy , which increase monotonously with @xmath19 . the transition from jet fragmentation dominated to medium dominated region in dihadron correlation happens around @xmath20 gev/@xmath9 , a region similar to that for the single particle from soft physics ( hydrodynamics+ recombination ) dominated region to hard physics ( jet ) dominated region . naturally , we expect the physics important for single particle production should play an important role for the dihadron correlation . [ fig:3 ] shows schematically the @xmath0 dependence of the modifications of the single particle yield ( via @xmath12 ) and hadron pair yield ( via @xmath11 ) . their @xmath0 dependence trend are quite different , especially at low @xmath0 , which can be explained qualitatively as follows . even though jet production dominates single particle yield at @xmath21 gev/@xmath9 in @xmath22 collisions , the strong energy loss and collective flow modify the @xmath0 distribution by shifting hard hadrons to lower @xmath0 and pushing soft hadrons to higher @xmath0 . this reshuffling changes single - hadron and correlated hadron - pair yield , hence the @xmath12 and @xmath11 shape . indeed , several theoretical models suggest that collective flow and recombination play a significant role in modifying the angular shape , spectra slope and particle composition of the correlated pairs @xcite . @xmath11 provides a mean to quantify the contribution of jet fragmentation hadrons or jet induced hadrons in this @xmath0 region . previously , the modification of dihadron yield is characterized with @xmath23 ( ratio of per - trigger yield between au+au and @xmath22 ) @xcite . @xmath23 is a good variable at high @xmath0 , since most triggers come from jets and most jets fragment into at most one trigger , such that per - trigger yield ( pty ) is a good representation of per - jet yield . however at lower @xmath0 region , non - fragmentation triggers from soft production mechanisms or medium response mechanisms become important . these triggers tend to dilutes the @xmath23 , since they either has no correlation or non - jet like correlation ( such as ridge ) . [ fig:4 ] illustrate the dilution effects for near - side @xmath4 correlation . we estimate dilution factor ( @xmath24 ) for 3 - 4 gev/@xmath9 triggers based on their correlations with 5 - 10 gev/@xmath9 hadrons as shown by the inserted panel : requiring 5 - 10 gev/@xmath9 hadrons ensures the pairs are dominated by the jet fragmentation ( left panel of the insert ) , thus deviation of @xmath23 from one for soft triggers reflects the level of dilution ( the red arrow ) . once the dilution factor is corrected , we subtract out the jet fragmentation contribution and obtain the ridge distribution ( black circles ) . the estimated ridge contribution is approximately flat , consistent experimental data at large @xmath4 . however , this dilution effect was not observed in some star analysis @xcite , which shows that the pty@xmath25 subtracted by the estimated ridge before any dilution correction already equals pty@xmath26 . in most correlation analyses and model calculations , it was normally assumed that one hadron ( `` trigger '' ) comes from the jet , and the second hadron ( `` partner '' ) comes from either fragmentation or feedback , i.e. only jet - jet and jet - medium pairs are considered . in this picture , the trigger comes from a jet that is biased to the surface , which losses some energy and fragments outside the medium . the fragments contribute to the near - side jet peak , and the feedback of the lost energy gives rise to the near - side ridge . in parallel , the away - side jet is quenched as it traverses a longer medium , contributing to the away - side cone . this picture does not include the medium - medium pairs ( both hadrons come from medium feedback of quenched jets ) . these pairs could be important at intermediate and low @xmath0 , since each jet can induce more correlated hadrons via jet quenching than via fragmentation . for example , most medium response models induce correlation by local heating of medium by the jet , such as momentum kick , jet deflection , mach - cone , etc @xcite , which are very effective in generating large yield of correlated hadron pairs . in addition , the whole overlap volume contributes to the observed medium - medium pairs , while both jet - jet and jet - medium pairs suffer a strong suppression . this point is illustrated by fig . [ fig:5 ] , which shows the typical geometrical origin for the three types of correlated pairs . the jet fragmentation contribution is proportional to the number of survived jet ( @xmath27 , i.e. the constant suppression level at large @xmath0 , @xmath28 in most central bin . ) , while the medium response is proportional to the number of quenched jets ( @xmath29 ) . for jet - jet pairs , both hadron are emitted from the surface ( tangential emission ) ; for jet - medium pairs , the jet hadron is emitted from the surface ( surface emission ) and the other from the whole volume ; for medium - medium pairs , both hadrons are emitted from the whole volume . the production rate for jet - jet , jet - medium and medium - medium pairs scale approximately with @xmath30 , @xmath31 and @xmath32 . clearly , if @xmath33 , the medium - medium pairs becomes dominant . recently , several analyses have been carried out to quantify the properties of the near - side ridge and away - side cone structures @xcite . the data show very similar properties between the ridge and the cone , i.e. both have similar slope and bulk like particle compositions , and both are important up to 4 gev/@xmath9 . these similarities suggests that their production mechanisms are connected . the medium - medium pairs from quenched jets are natural candidates for creating these similarities . because medium - medium pairs come from quenched jets originated deep inside the medium , they contribute to both the near - side and away - side on a equal footing . the near - side pairs could contain correlations among mach cone particles , and away - side pairs could also contain correlation between the ridge and mach cone particles ( see ref . @xcite for a possible realization ) . a strong modification of the away - side correlation was also observed at the top sps energy ( @xmath34 gev ) @xcite . the strong away - side broadening has been used to argue for a similar interpretation ( such as mach cone ) as for the rhic results . however a quantitative analysis of the energy dependence of the modification patterns ( see fig.[fig:6 ] ) shows that the yield of medium response are quite different between rhic and sps energies . in fact the near - side yield drop by almost factor of 8 going from 200 gev to 17 gev while the away - side shoulder yield drops by factor of 2 in the same energy range . but there are little dependence of the yields on @xmath35 in the away - side head region , where the jet fragmentation is important . to quantify the energy dependence of away - side shape , we calculate the ratio of the yield density in the head region to that in shoulder region , @xmath36 @xcite , in fig.[fig:7]a . the @xmath36 increases with decreasing collision energy , with a ratio slightly above one in sps energy . this value is comparable with that obtained for rather peripheral ( @xmath37 ) in au+au collisions at 200 gev ( fig.[fig:7]b ) . these results suggest a much weaker medium response in sps energy ( the ridge almost disappeared and cone strongly suppressed ) than that at rhic , but a similar jet fragmentation contribution , probably related to smaller energy loss and stronger cronin effects at lower energy @xcite . cl & lastly , it was shown that the mach - cone angle found from the three particle ( 3-p ) correlation ( 1.4 rad ) is significantly larger than the two particle ( 2-p ) correlation analysis ( 1.1 rad ) . sps also seems to see a 3-p correlation signal @xcite . however one should realized that , it is possible that the kinematics of jets contributing to 3-p signal is different from those contributing to the 2-p signal . it is likely that most jet have multiplicity @xmath38 that the 3-p only samples a small fraction of all jets that contribute to the dihadron correlations . due to the surface bias and steeply falling parton spectra , the observed high @xmath0 single hadrons and dihadron pairs mainly come from those jets that suffer minimal interaction with the medium . this energy loss bias limits their usefulness as tomography tools . on the other hand , medium responses are directly sensitive to the energy loss and energy dissipation processes used to model the high @xmath0 production . for example the collisional energy loss would imply that momentum kick dominates the low @xmath0 pairs , the radiative energy loss would favor for the gluon feedback mechanism . finally , the jet quenching and medium responses are modeled separately in most theoretical calculations . a unified framework , including both jet quenching and medium response , which can describe the full @xmath39 evolution of the jet shape and yield at both near- and away - side is required to understand the details of the parton - medium interactions . 99 j. adams _ et al . _ [ star collaboration ] , phys . rev . lett . * 95 * , 152301 ( 2005 ) j. adams _ et al . _ [ star collaboration ] , phys . lett . * 97 * , 162301 ( 2006 ) a. adare _ et al . _ [ phenix collaboration ] , phys . c * 77 * , 011901(r ) ( 2008 ) a. adare _ et al . _ [ phenix collaboration ] , arxiv:0801.4545 [ nucl - ex ] . m. j. horner [ star collaboration ] , j. phys . g * 34 * , s995 ( 2007 ) s. s. adler _ et al . _ [ phenix collaboration ] , phys . lett . * 97 * , 052301 ( 2006 ) j. adams _ et al . _ [ star collaboration ] , phys . c * 75 * , 034901 ( 2007 ) n. armesto , c. a. salgado and u. a. wiedemann , phys . rev . lett . * 93 * , 242301 ( 2004 ) , phys . rev . c * 72 * , 064910 ( 2005 ) ; i. vitev , phys . b * 630 * , 78 ( 2005 ) ; i. m. dremin , jetp lett . * 30 * , 140 ( 1979 ) v. koch , a. majumder and x. n. wang , phys . * 96 * , 172302 ( 2006 ) j. casalderrey - solana , e. v. shuryak and d. teaney , hep - ph/0602183 . t. renk and j. ruppert , phys . rev . c * 73 * , 011901 ( 2006 ) ; c. b. chiu and r. c. hwa , phys . c * 74 * , 064909 ( 2006 ) ; a. d. polosa and c. a. salgado , phys . c * 75 * , 041901(r)(2007 ) ; p. romatschke , phys . c * 75 * , 014901 ( 2007 ) ; a. majumder , b. muller and s. a. bass , phys . rev . lett . * 99 * , 042301 ( 2007 ) ; c. y. wong , phys . c * 76 * , 054908 ( 2007 ) ; e. v. shuryak , phys . c * 76 * , 047901 ( 2007 ) ; v. s. pantuev , arxiv:0710.1882 [ hep - ph ] ; j. putschke , eur . j. c * 49 * , 57 ( 2007 ) . s. afanasiev _ et al . _ [ phenix collaboration ] , arxiv:0712.3033 [ nucl - ex ] . a. frantz , [ phenix collaboration ] these proceedings . j. jia and r. lacey , arxiv:0806.1225 [ nucl - th ] . m. ploskon [ ceres collaboration ] , nucl . a * 783 * , 527 ( 2007 ) a. adare _ et al . _ [ phenix collaboration ] , arxiv:0801.4555 [ nucl - ex ] . j. g. ulery , these proceedings .
|
a brief review of the @xmath0 dependence of the dihadron correlations from rhic is presented .
we attempt to construct a consistent picture that can describe the data as a whole , focusing on the following important aspects , 1 ) the relation between jet fragmentation of survived jet and medium response to quenched jets , 2 ) the possible origin of the medium response and its relation to intermediate @xmath0 physics for single hadron production , 3 ) the connection between the near - side ridge and away - side cone , 4 ) and their relations to low energy results .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
small trapped fermi gases with contact or short - range interactions have attracted a great deal of attention recently @xcite . using lithium or potassium , for example , equal - mass two - component systems can be realized experimentally by occupying two different hyperfine states . for typical experimental conditions , @xmath11-wave or higher partial wave interactions between two like atoms ( say , two spin - up atoms ) and between two unlike atoms ( a spin - up and a spin - down atom ) are negligibly small . furthermore , by tuning an external magnetic field in the vicinity of a fano - feshbach resonance , the @xmath0-wave scattering length @xmath12 can be adjusted to essentially any value @xcite . in this paper , we consider the regime where the @xmath0-wave scattering length is much larger than the range @xmath3 of the underlying two - body model potential . in the limit that @xmath3 goes to zero and @xmath12 goes to infinity , the unitary regime is realized . in this regime , the only meaningful length scale of the system is given by the oscillator length @xmath13 that characterizes the external confining potential @xcite . throughout , we assume a spherically symmetric harmonic potential with angular trapping frequency @xmath8 ( i.e. , @xmath14 with @xmath15 denoting the atom mass ) . from a theoretical point of view , small harmonically trapped fermi gases with central short - range interactions are particularly appealing since they can be treated with comparatively high accuracy by a variety of methods , including techniques that have been developed in the context of atomic physics , nuclear physics and quantum chemistry problems @xcite . for the harmonically trapped equal - mass system , the center of mass degrees of freedom separate . furthermore , the relative orbital angular momentum quantum number @xmath5 , the projection quantum number @xmath16 and the parity @xmath6 are good quantum numbers . this implies that the hilbert space can be divided into subspaces , which significantly reduces the complexity of the calculations compared to , for example , systems confined to move within a box with periodic boundary conditions @xcite . the harmonically trapped fermi gas consisting of two spin - up and two spin - down atoms with vanishing angular momentum has been treated by a variety of techniques in the literature ( see refs . @xcite for reviews ) . the ground state energy and ground state properties of the @xmath1 system in the zero - range limit , for example , are by now well characterized @xcite . much less , however , is known about the excitation spectrum @xcite , which contains both natural and unnatural parity states , i.e. , states with parity @xmath17 and @xmath18 , respectively . while a good portion of the excitation spectrum of the @xmath1 system with natural parity has been determined throughout the crossover and at unitarity @xcite , little is known about the properties of states with unnatural parity . moreover , the energy spectra of the @xmath2 , @xmath9 and @xmath10 systems have not yet been characterized in detail . this paper presents extensive benchmark results for the @xmath1 and @xmath2 energies of natural and unnatural parity states at unitary . in addition , we present portions of the energy spectra of the @xmath9 and @xmath10 systems . we then use the energy spectra of the @xmath1 and @xmath2 systems at unitarity to determine the fourth - order virial coefficient @xmath19 of the trapped system in the low - temperature regime . the fourth - order virial coefficient enters into the virial equation of state , which allows for the determination of the universal thermodynamics of two - component fermi gases in the temperature regime down to about half the fermi temperature @xmath20 @xcite . for the temperature regime in which we have convergence , i.e. , for @xmath21 , where @xmath22 denotes the temperature and @xmath23 boltzmann s constant , we find that the fourth - order virial coefficient @xmath19 of the trapped system is negative and decreases monotonically with increasing temperature . if we assume that @xmath19 continues to change monotonically with increasing temperature in the medium- and high - temperature regime , our results predict that the fourth - order virial coefficient of the trapped system and through application of the local density approximation ( lda)that of the homogeneous system approach a negative value in the high - temperature limit . this is in contrast to recent results @xcite based on the equation of state , determined both experimentally and calculated via a diagrammatic monte carlo technique . these studies predict that the fourth - order virial coefficient of the homogeneous system is positive . the discrepancy would be resolved if the fourth - order virial coefficient of the trapped system was changing non - monotonically with temperature , allowing for a sign change of @xmath19 in the medium- or high - temperature regime . analogous non - monotonic behavior was found for one of the third - order virial coefficients of the trapped unequal - mass two - component fermi gas at unitarity @xcite . while we do not have access to sufficiently large portions of the energy spectra of the @xmath1 and @xmath2 systems to determine the fourth - order virial coefficient of the trapped system in the medium- and high - temperature regimes ( thereby preventing us from drawing definite conclusions ) , our results illuminate a number of aspects related to the determination of the virial coefficients from few - body energy spectra . section [ sec_system ] introduces the system under study , reviews the stochastic variational approach , and presents details regarding our implementation . compact expressions for the relevant matrix elements for natural and unnatural parity states are presented in appendix [ appendix ] . section [ sec_energies ] ( see also supplementary material @xcite ) summarizes our extrapolated zero - range energies for the @xmath1 , @xmath2 , @xmath9 and @xmath10 systems in tabular form and discusses their characteristics . section [ sec_virial ] uses the @xmath1 and @xmath2 energies to determine the fourth - order virial coefficient of the trapped system at unitarity . lastly , sec . [ sec_summary ] concludes . we consider a two - component fermi gas with @xmath24 spin - up and @xmath25 spin - down atoms of mass @xmath15 with @xmath26 . we assume that the atoms are confined by a spherically symmetric trapping potential with angular frequency @xmath8 . furthermore , we assume that the spin - up and spin - down atoms interact through a short - range interaction potential @xmath27 , where @xmath28 ( @xmath29 ) denotes the position vector of the @xmath30 atom measured relative to the center of the trap and @xmath31 , and that atoms with like spins do not interact . the model hamiltonian @xmath32 then reads @xmath33 where @xmath34 throughout , we are interested in the regime where the @xmath0-wave scattering length @xmath12 of the interspecies interaction potential @xmath35 becomes infinitely large . for the @xmath36 and @xmath37 systems , semi - analytical solutions are known if @xmath35 coincides with the zero - range @xmath38-function potential @xcite . for @xmath39 systems with @xmath40 , however , no such semi - analytical solutions are known . to determine the eigenenergies of @xmath39 systems with @xmath41 and @xmath42 , we separate off the center of mass motion and resort to a numerical technique , the stochastic variational approach @xcite . in this approach , it is convenient to model the interactions between the unlike atoms through a gaussian potential @xmath43 with depth @xmath44 ( @xmath45 ) and range @xmath3 @xcite , @xmath46.\end{aligned}\ ] ] to treat the unitary system , we adjust the depth @xmath47 of @xmath48 for a given @xmath3 such that the two - body system in free space supports one zero - energy @xmath0-wave bound state but no deep - lying bound states . to determine the zero - range energies , we consider a number of @xmath3 , @xmath49 , and extrapolate the finite - range energies to the @xmath4 limit ( see sec . [ sec_energies ] for examples ) . we take advantage of the fact that the hamiltonian @xmath32 separates into the center of mass hamiltonian @xmath50 and the relative hamiltonian @xmath51 , @xmath52 . in the following , we consider the relative hamiltonian @xmath51 and use the stochastic variational approach to determine the eigenenergies and eigenstates of the schrdinger equation @xmath53 . here , we explicitly indicate the dependence of the eigenenergies on @xmath24 and @xmath25 but , for notational simplicity , not that of the hamiltonian and the wave function . to compact the notation , we write @xmath51 as @xmath54 , where @xmath55 denotes the kinetic energy operator associated with the relative motion , and @xmath56 the contribution of the confining potential associated with the relative degrees of freedom . the stochastic variational approach is a basis set expansion approach that writes the relative wave function @xmath57 of a given state in terms of a set of basis functions @xmath58 @xcite , @xmath59 here , the @xmath60 denote expansion coefficients and @xmath61 an anti - symmetrization operator that ensures that the wave function is anti - symmetric under the exchange of any pair of like fermions . in eq . ( [ eq_expand ] ) , @xmath62 denotes the number of basis functions . as with other basis set expansion approaches , the ritz variational principle ensures that the lowest energy as well as the higher - lying energies obtained by the stochastic variational approach are rigorous upper bounds to the exact eigenenergies of the system @xcite . in the following , we introduce the basis functions used in this work , which have good orbital angular momentum @xmath5 , projection quantum number @xmath16 and parity @xmath6 ; here , @xmath5 , @xmath16 and @xmath6 are associated with the relative motion . following refs . @xcite , we write the basis functions @xmath58 as a product of a correlated gaussian [ second line of eq . ( [ eq_basis1 ] ) ] and a `` prefactor '' [ first line of eq . ( [ eq_basis1 ] ) ] that carries the angular momentum @xmath5 of the system , @xmath63_{lm } \nonumber \\ \times \exp \left ( - \frac { \vec{x}^t \underline{a}_k \vec{x}}{2 } \right).\end{aligned}\ ] ] here , @xmath64 collectively denotes the @xmath65 jacobi vectors @xmath66 , where @xmath67 . the notation @xmath68_{lm}$ ] indicates that the spherical harmonics @xmath69 and @xmath70 are coupled to form a function with angular momentum @xmath5 and projection quantum number @xmath16 . for states with natural parity , i.e. , for states whose parity is given by @xmath71 , we choose @xmath72 and @xmath73 @xcite . for states with unnatural parity ( @xmath74 ) , i.e. , for states whose parity is given by @xmath18 , we choose @xmath72 and @xmath75 @xcite . the basis functions that describe unnatural parity states with @xmath76 have a slightly different form since the construction of states with @xmath76 and @xmath77 requires the coupling of three spherical harmonics with @xmath78 , @xmath79 and @xmath80 @xcite . the matrix @xmath81 is symmetric and positive - definite , and has dimensions @xmath82 . the @xmath83 independent elements of @xmath81 are treated as variational parameters and optimized semi - stochastically . the three - dimensional vectors @xmath84 and @xmath85 , referred to as global vectors since they depend on all @xmath65 jacobi vectors , are defined through @xmath86 and similarly for @xmath85 . the vectors @xmath87 and @xmath88 are optimized semi - stochastically , where @xmath89 and similarly for @xmath88 . a key benefit of the basis functions given in eq . ( [ eq_basis1 ] ) is that the overlap matrix element @xmath90 , the matrix element for the kinetic energy operator @xmath91 , and the matrix element for the confining potential @xmath92 reduce to compact expressions @xcite . here , it is understood that the integration is performed over all @xmath93 jacobi coordinates and that @xmath58 is characterized by @xmath81 , @xmath87 and @xmath88 while @xmath94 is characterized by @xmath95 , @xmath96 and @xmath97 . moreover , a compact expression can also be found for the matrix elements @xmath98 associated with the atom - atom interaction if @xmath35 is modeled by the gaussian potential @xmath48 . appendix [ appendix ] summarizes explicit expressions of the matrix elements with natural parity ( any @xmath5 ) and unnatural parity ( @xmath74 ) . the matrix elements for states with @xmath99 symmetry can be found in refs . @xcite . we note that the overlap matrix element @xmath100 between two different basis functions does not vanish , i.e. , the basis set employed is not orthogonal . this implies that the determination of the eigenenergies amounts to the diagonalization of a generalized eigenvalue problem defined by the hamiltonian and overlap matrices @xcite . while one might think , at first sight , that the non - orthogonality of the basis functions could introduce numerical instabilities , it has been shown in previous work @xcite that numerical instabilities due to linear dependence issues can be avoided completely for the systems of interest in this work if the basis sets are chosen carefully . our strategy to optimize the large number of non - linear variational parameters is quite simple @xcite . we start with a reference basis set , which could consist of just one basis function or as many as several 100 or 1000 basis functions . we then enlarge this reference basis set by one basis function , which is chosen from a large number of trial basis functions , typically between several 100 and several 1000 . each trial function is characterized by a different set of variational parameters . to decide which trial basis function to keep , we calculate the energy for each of the enlarged trial basis sets , which consist of the reference basis set plus one of the trial basis functions , and choose the one that results in the largest reduction of the energy of the state of interest . the state of interest could be the ground state or an excited state . the procedure is repeated till the basis set is sufficiently complete to describe the state of interest with the desired accuracy . when optimizing a state whose energy is nearly degenerate with that of another state or when optimizing highly excited states , some care needs to be exercised . in the former case , we find it advantageous to optimize two or more states simultaneously . in the latter case , we find it beneficial to start with a basis set that provides a reasonably accurate description of the lower lying part of the energy spectrum . the advantage of our optimization procedure is that the basis set is optimized for a particular state or a particular subset of states . correspondingly , we work with comparatively small basis sets . the energies of the @xmath1 and @xmath2 systems at unitarity ( see table [ tab1 ] and supplemental material ) are obtained using basis sets that consist of 700 - 3400 basis functions , while the energies of the @xmath9 and @xmath10 systems ( see supplemental material ) are obtained using basis sets that consist of 1500 - 3800 basis functions . one key purpose of this paper is to elucidate how we determine a large portion of the energy spectrum of trapped two - component fermi systems with @xmath101 and 5 , and to tabulate the extrapolated zero - range energies . we believe that the tabulation of the energies is useful as these energies provide much needed highly accurate benchmark results that can be used to assess the accuracy and validity regime of alternative approaches . we anticipate that the tabulated energies will also prove useful in other applications . figure [ fig1 ] shows an example of our basis set optimization system at unitarity with @xmath102 symmetry and @xmath103 . solid and dashed lines show the quantity @xmath104 , where @xmath105/e^{\rm{rel}}_{3,1}(n_b \rightarrow \infty)$ ] , for states 1 and 12 as a function of @xmath106 . dotted lines show the extrapolation to the @xmath107 limit . the inset shows a blow - up of the small @xmath106 region . , width=264 ] for the @xmath2 system with @xmath102 symmetry and @xmath108 at unitarity . solid and dashed lines show the fractional difference @xmath104 for the ground state ( state 1 ) and state 12 @xcite , respectively , between the relative energy @xmath109 for a basis set of size @xmath62 and the energy for an infinite basis set . the dotted lines in fig . [ fig1 ] show the extrapolation to the @xmath107 limit . it can be seen that the ground state energy converges notably faster than the excited state energy . the energies for @xmath110 and @xmath111 are @xmath112 for state 1 and @xmath113 for state 12 , respectively . the basis set errors for these basis sizes are 0.0002% and 0.003% , respectively , i.e. , the energies of states 1 and 12 lie respectively @xmath114 and @xmath115 above the extrapolated energies for the infinite basis set . the low - lying states of the @xmath2 system with @xmath102 symmetry at unitarity converge relatively quickly with increasing @xmath62 . the convergence is slower for most other states and , in general , we choose the size of our basis sets for the @xmath1 and @xmath2 systems such that the basis set extrapolation error is smaller than @xmath116 % . figure [ fig2 ] exemplarily illustrates the range dependence for the relative energy of the @xmath2 system with @xmath102 symmetry at unitarity . squares show the relative eigenenergies @xmath117 for various ranges @xmath3 of the underlying two - body interaction potential for ( a ) the ground state ( state 1 ) , ( b ) state 12 , and ( c ) state 5 ; @xmath62 is the largest basis set considered . the energies provide variational upper bounds and the estimated basis set extrapolation error is indicated by errorbars ; in panels ( a ) and ( b ) , the basis set extrapolation error is smaller than the symbol size and thus not visible . in panels ( a ) and ( b ) , solid lines show linear fits to the energies @xmath118 [ the fit shown in panel ( a ) includes the energies for the five smallest @xmath3 values ] . , width=264 ] system with @xmath102 symmetry . figure [ fig2](a ) shows the range dependence of the ground state energy , fig . [ fig2](b ) shows the range dependence of the energy associated with state 12 , and fig . [ fig2](c ) shows the range dependence of the energy for a state that depends comparatively weakly on @xmath3 ( state 5 ) . in figs . [ fig2](a ) and [ fig2](b ) , the energies vary to a very good approximation linearly with @xmath3 for sufficiently small @xmath119 . this finding is in agreement with earlier work @xcite . for the ground state [ see fig . [ fig2](a ) ] , the range dependence is quite weak and linear behavior is only observed for @xmath120 . in fig . [ fig2](c ) , the zero - range energy agrees to within @xmath121 with the energy of the non - interacting system . this , combined with the very weak dependence of the energy on @xmath3 and the fact that the energy approaches the zero - range limit from below , suggests that this state is not affected by @xmath0-wave scattering but only by higher - partial wave scattering . in the zero - range limit , energy shifts associated with higher - partial wave scattering processes vanish . our interpretation is corrobated by a perturbative calculation along the lines of that performed in refs . @xcite , which utilizes zero - range contact interactions . for the @xmath2 system with @xmath122 symmetry , we find , in agreement with our results based on the stochastic variational approach , that there exists one state with relative energy @xmath123 and six states with relative energy @xmath124 that are independent of @xmath12 . we refer to states that are unaffected by @xmath0-wave interactions as unshifted states . we find that a relatively large number of states fall into this category . their existence and likelihood of occurance has already been discussed for the @xmath37 and @xmath1 systems in the literature @xcite . for the @xmath37 system , e.g. , all unnatural parity states are unaffected by @xmath0-wave interactions in the zero - range limit . for the @xmath1 and @xmath2 systems , unnatural parity states can be affected by @xmath0-wave interactions in the zero - range limit . the only exception are states with @xmath99 symmetry , which are unshifted . this behavior can be intuitively understood within a picture that utilizes angular momentum coupling . to construct a state with @xmath99 symmetry , the coupling of three finite angular momenta is needed . these angular momenta can be envisioned as being each associated with one of the three jacobi vectors that characterize the @xmath125 system . as a consequence , the @xmath0-wave interactions are effectively turned off by the nodal structure of the wave function . for @xmath126 , this argument predicts that states with @xmath99 symmetry can be affected by @xmath0-wave interactions since the system is characterized by one more jacobi vector than angular momenta needed to ensure the @xmath99 symmetry . indeed , this prediction is in agreement with our results from the perturbative and stochastic variational calculations . table [ tab1 ] summarizes our extrapolated zero - range energies @xmath127 , @xmath128 , for states with @xmath102 symmetry at unitarity that are affected by @xmath0-wave interactions . the zero - range energies are obtained by calculating the energies of a given state for several ranges @xmath3 between @xmath129 and by then fitting these energies for the largest basis set considered by a linear function . .relative energies @xmath130 for the @xmath2 system with @xmath122 symmetry [ only states that are affected by @xmath0-wave interactions are included ; each energy is @xmath131-fold degenerate ] . the first column indicates the state number ( st . the second column shows the extrapolated zero - range energy @xmath127 at unitarity ; the uncertainty is estimated to be 0.1 % or smaller . the third column indicates the dependence of the energy at unitarity on the range @xmath3 of the gaussian potential @xmath48 . we assume a linear dependence and write @xmath132 . the fourth column shows the @xmath133 value determined from the energy ; the value of @xmath133 is only shown for the lowest rung of a ladder , i.e. , for states with @xmath134 . the last column shows @xmath135 of the non - interacting state that is `` paired '' with the interacting state when determining @xmath136 ( see sec . [ sec_virial ] ) . there exist 1 and 6 unshifted states with energy @xmath137 and @xmath138 , respectively . [ cols="^,^,^,^,^",options="header " , ] [ tab_virial ] where the @xmath139 are defined in terms of the canonical partition functions @xmath140 and @xmath141 of the interacting @xmath39 system and the single - component system with @xmath24 atoms , respectively , @xmath142 the temperature - dependent canonical partition functions @xmath140 and @xmath141 , @xmath143\end{aligned}\ ] ] and @xmath144,\end{aligned}\ ] ] are determined by the total energies @xmath145 and @xmath146 of the interacting two - component and non - interacting single - component systems , respectively . it is important to note that the energies @xmath145 and @xmath146 contain the center of mass energy . the summation over @xmath147 in eqs . ( [ eq_partint ] ) and ( [ eq_partni ] ) extends over all states allowed by symmetry . for @xmath148 , the sum in eq . ( [ eq_partni ] ) can be performed analytically , yielding eq . ( [ eq_q1 ] ) . in the high - temperature limit , one finds for systems with zero - range interactions at unitarity that @xcite @xmath149 the second - order virial coefficient of the trapped system at unitarity takes the simple form @xcite @xmath150\end{aligned}\ ] ] or , performing the infinite sum , @xmath151 the solid line in fig . [ fig_b2 ] shows the second - order virial coefficient @xmath152 , eq . ( [ eq_b2compact ] ) , as a function of @xmath153 . of the trapped two - component fermi gas at unitarity as a function of the inverse temperature @xmath153 . the solid line shows @xmath152 , eq . ( [ eq_b2compact ] ) , while the dash - dot - dotted , dash - dotted and dashed lines show @xmath152 obtained by setting @xmath154 in eq . ( [ eq_b2sum ] ) to 0 , 1 and 10 , respectively . the solid horizontal line shows the high - temperature limit @xmath155 . , width=245 ] in the high - temperature ( small @xmath153 ) limit , @xmath152 approaches the constant @xmath155 , @xmath156 ( solid horizontal line in fig . [ fig_b2 ] ) , which can be obtained by taylor - expanding eq . ( [ eq_b2compact ] ) . to illustrate the convergence of @xmath152 with increasing energy cutoff , dash - dot - dotted , dash - dotted and dashed lines show @xmath152 obtained by setting @xmath154 in eq . ( [ eq_b2sum ] ) to 0 , 1 and 10 , respectively . for a finite energy cutoff , it can be seen that @xmath152 goes to @xmath157 in the small @xmath153 region as opposed to @xmath156 . as expected , a larger energy cutoff provides an accurate description of @xmath152 over a larger temperature range , i.e. , down to a smaller inverse temperature @xmath153 . the relative three - body energies at unitarity and for vanishing @xmath0-wave scattering length @xmath12 can be written as @xmath158 ( see sec . [ sec_energies ] ) and @xmath159 , respectively . performing the sum over @xmath160 analytically , the interaction piece @xmath161 of the trapped three - body system at unitarity takes the form @xcite @xmath162.\end{aligned}\ ] ] using large @xmath163 and @xmath164 , a fully converged pointwise representation of @xmath165 is obtained @xcite ( see solid line in fig . [ fig_b3 ] ) . of the trapped two - component fermi gas at unitarity as a function of the inverse temperature @xmath153 . the solid line shows @xmath166 with @xmath163 and @xmath164 set to very large values ( see ref . @xcite for details ) while the dash - dot - dotted , dash - dotted and dashed lines show @xmath166 obtained by limiting @xmath163 and @xmath164 in eq . ( [ eq_b3sum ] ) such that @xmath167 , @xmath168 and @xmath169 , respectively . the solid horizontal line shows the high - temperature limit @xmath170 , eq . ( [ eq_b3hight ] ) . , width=245 ] using the analytical forms for @xmath171 and @xmath152 , eqs . ( [ eq_q1 ] ) and ( [ eq_b2compact ] ) , we find that @xmath172 diverges as @xmath173 in the high - temperature limit . this divergence is cancelled by a divergence of @xmath174 of opposite sign . as a result , @xmath165 is well behaved in the small @xmath153 ( high @xmath22 ) limit . a careful analysis of the high - temperature behavior gives @xcite @xmath175 ( see horizontal solid line in fig . [ fig_b3 ] ) . to illustrate the convergence of @xmath166 with increasing @xmath163 and @xmath164 [ see eq . ( [ eq_b3sum ] ) ] , dash - dot - dotted , dash - dotted and dashed lines in fig . [ fig_b3 ] show @xmath166 calculated using @xmath161 from eq . ( [ eq_b3sum ] ) with @xmath163 and @xmath164 chosen such that @xmath176 , @xmath177 and @xmath169 , respectively . no cutoff is imposed in calculating @xmath178 . in these calculations , we include the same number of @xmath133 and @xmath135 in evaluating @xmath161 , i.e. , each interacting @xmath133 value is paired with the corresponding non - interacting @xmath135 value . figure [ fig_b3 ] shows that the cutoff introduces a divergence in @xmath165 . this divergence arises because the cutoff alters the high - temperature behavior of @xmath174 , which implies that the divergencies of @xmath172 and @xmath174 no longer cancel . importantly , @xmath165 is converged in the low - temperature ( large @xmath153 ) regime even for a relatively small cutoff . this allows us to use the converged low - temperature tail to constrain @xmath165 in the high - temperature regime . extrapolating @xmath165 ( calculated using a cutoff of @xmath179 ) to the high - temperature limit , we find @xmath180 , which deviates by less than 2% from the exact value . the validity of the employed extrapolation scheme crucially hinges on the fact that the functional form of @xmath166 changes `` predictably '' as @xmath153 changes from the low- to the medium- to the high - temperature regime . for example , if @xmath166 changed sign in the medium- or high - temperature regime , as is the case for the coefficient @xmath181 that characterizes the behavior of two identical fermions and one lighter fermion ( with a mass ratio from @xmath182 to @xmath183 ) @xcite , the extrapolation employed above would predict the incorrect high - temperature limit of @xmath181 . the interaction piece @xmath184 of the fourth - order virial coefficient can be expressed analogously to @xmath161 . in particular , we write the energies at unitarity in terms of the @xmath133 ( see sec . [ sec_energies ] and the supplemental material for a listing of the @xmath133 values ) and perform , as in the three - body case above , the sum over the hyperradial quantum number @xmath160 analytically . since both natural and unnatural parity states of the four - body systems are affected by the @xmath0-wave interactions , the @xmath133 values corresponding to both natural and unnatural parity states need to be included when evaluating @xmath184 . the reference piece @xmath185 diverges as @xmath186 in the high - temperature limit . this divergence must be cancelled by a divergence of @xmath184 of opposite sign . dash - dot - dotted , dash - dotted and dashed lines in fig . [ fig_b4 ] show @xmath187 at unitarity obtained by using the full expression for @xmath185 and by limiting the sums over @xmath188 and @xmath5 in @xmath184 such that @xmath167 , @xmath189 and @xmath168 , respectively . of the trapped two - component fermi gas at unitarity as a function of the inverse temperature @xmath153 . the dash - dot - dotted , dash - dotted and dashed lines show @xmath19 obtained by limiting @xmath133 to be smaller than @xmath190 , @xmath191 and @xmath192 , respectively . the dotted line shows our attempt to extrapolate to the high - temperature limit ; this extrapolation assumes that @xmath19 changes `` predictably '' from the low- to the medium- to the high - temperature regime . the inset shows the same data as the main figure . in addition , the solid horizontal line shows the high - temperature limit @xmath193 determined by applying the lda to the fourth - order virial coefficient predicted for the homogeneous system @xcite . , width=245 ] for the largest cutoff , our calculation includes 169 and 89 @xmath133 values associated with shifted states [ not counting the @xmath131-multiplicity ] of the harmonically trapped @xmath1 and @xmath2 systems with zero - range interactions , respectively . figure [ fig_b4 ] shows that @xmath19 is negative in the low - temperature ( large @xmath153 ) regime and that neither @xmath19 nor its first or second derivatives with respect to @xmath153 change sign in the regime where @xmath19 is converged . this motivates us to extrapolate the converged part of @xmath19 to the medium- and high - temperature regime ( see dotted line in fig . [ fig_b4 ] ) , yielding @xmath194 . the lda predicts that the virial coefficient @xmath195 of the homogeneous system is related to the high - temperature limit of the @xmath196th order virial coefficient of the trapped system via @xcite @xmath197 application to our extrapolated @xmath193 yields @xmath198 . this value for the homogeneous system differs in both sign and magnitude from the values @xmath199 @xcite and @xmath200 @xcite determined from experimental data . these experimental values have been found to be consistent with the equation of state determined by a diagrammatic path integral monte carlo approach @xcite . given the disagreement between our value and that reported in the literature , we speculate that the fourth - order virial coefficient of the trapped system changes sign in the medium- or high - temperature limit , implying that the applied extrapolation scheme does not predict the correct medium- and/or high - temperature behavior of @xmath19 . if this conclusion is correct , it would follow that the determination of the medium- and high - temperature behavior of the fourth - order virial coefficient of the trapped systems requires , if determined via the microscopic energy spectra , knowledge of large portions of the energy spectra of the @xmath1 and @xmath2 systems . this suggests that other approaches , based on feynman diagrams or based on simulating the finite temperature behavior directly numerically , may be more suitable than the approach pursued here for determining the temperature - dependence of @xmath19 . we also analyzed the low - temperature tail of @xmath201 . using the @xmath9 and @xmath10 energies from the supplemental material , we find that the fifth - order virial coefficient of the trapped fermi gas at unitarity is positive in the low - temperature limit . high precision measurements of the equation of state in the high - temperature regime might reveal if @xmath201 changes sign as a function of temperature . more generally , we find that the low - temperature limit of @xmath202 at unitarity is fully determined by the low - temperature behavior of @xmath203 and @xmath171 . to arrive at this result , we derive explicit expressions for @xmath204 and @xmath205 for @xmath206 , and determine the low - temperature behavior of all terms that enter into @xmath204 and @xmath205 . using the ground state energies at unitarity for trapped two - component fermi gases with up to @xmath207 ( with @xmath208 or 1 ) @xcite , we find that @xmath204 falls off faster than @xmath205 with decreasing @xmath22 , thus allowing us to obtain analytic expressions for the leading order low - temperature behavior of @xmath202 : @xmath209\exp[-(2n-3/2 ) \tilde{\omega}]/(2n)$ ] and @xmath210 for @xmath211 . thus , the sign of @xmath212 in the low - temperature regime is @xmath213 for @xmath214 . for @xmath215 , we have checked that these analytical predictions agree with our numerically determined virial coefficients in the low - temperature regime . while the sign of @xmath212 in the low - temperature regime may not allow one to draw conclusions about @xmath216 , it is interesting , at least from a theoretical point of view , that the sign and functional form of @xmath212 in the low - temperaure regime are fully determined by @xmath171 and @xmath152 . this paper considered the energy spectra of small trapped two - component fermi gases with vanishing and finite angular momentum as well as natural and unnatural parity . large portions of the energy spectra of the @xmath1 and @xmath2 systems at unitarity were determined as a function of the range of the underlying two - body model potential and extrapolated to the zero - range limit . the extrapolated zero - range energies are expected to be universal , i.e. , independent of the underlying gaussian model potential . portions of the energy spectra of the @xmath9 and @xmath10 systems at unitarity were also determined . the energies were obtained by solving the relative schrdinger equation using the stochastic variational approach . compact expressions for the relevant matrix elements were presented in the appendix . the @xmath1 and @xmath2 energies at unitarity were then used to determine the low - temperature behavior of the fourth - order virial coefficient @xmath19 of the trapped fermi gas . the high - temperature limit of the fourth - order virial coefficient enters into the universal virial equation of state . the present study suggests that much larger portions of the microscopic energy spectra are needed to predict the high - temperature limit of @xmath19 . in our view this is unfortunate . despite this , we believe that the analysis presented illuminates important characteristics relevant to the determination of the virial coefficients . we gratefully acknowledge support by the aro . kmd and db acknowledge hospitality of the int where part of this work was conducted . we also gratefully acknowledge communication by the mit - amherst collaboration prior to publication of refs . this appendix summarizes the expressions for the overlap , kinetic energy , trap potential , and interaction potential matrix elements for states with natural parity ( any @xmath5 ) and unnatural parity ( @xmath74 ) . for notational simplicity , we omit the subscripts of the matrix @xmath81 and the vectors @xmath87 and @xmath88 , and consider the matrix elements between the unsymmetrized basis functions @xmath217 and @xmath218 characterized by @xmath219 and @xmath220 , respectively [ see eq . ( [ eq_basis1 ] ) of sec . [ sec_system ] ] . the matrix elements have been derived in the literature @xcite and are summarized here for completeness . before providing explicit expressions for the matrix elements , we introduce a number of auxiliary quantities that are utilized in subsecs . [ appendixa ] and [ appendixb ] . the product of @xmath218 and @xmath217 can be conveniently written in terms of the matrix @xmath221 , @xmath222 we further define the scalars @xmath223 and @xmath224 ( @xmath225 or @xmath226 ) , @xmath227 and @xmath228 note that the order of the primed and unprimed vectors @xmath229 and @xmath230 matters . we further define the scalars @xmath231 and @xmath232 ( @xmath225 or @xmath226 ) , @xmath233 and @xmath234 where the diagonal elements of the matrix @xmath235 are given by the inverse of the masses associated with the jacobi vectors and the off - diagonal elements of @xmath235 are zero . in eq . ( [ eq_r ] ) , @xmath236 denotes the trace operator . the scalars @xmath237 and @xmath238 ( @xmath239 and @xmath240 ) have a similar structure to @xmath231 and @xmath232 , @xmath241 and @xmath242 the matrix @xmath243 is defined as @xmath244 where @xmath245 is the @xmath246-dimensional vector that relates the distance vectors @xmath247 to the jacobi vectors @xmath248 , @xmath249 lastly , we define the total mass @xmath250 , @xmath251 for natural parity states , we use @xmath72 and @xmath73 in eq . ( [ eq_basis1 ] ) , which implies that @xmath218 and @xmath217 are independent of @xmath252 and @xmath253 , respectively . in the following , we assume that @xmath218 and @xmath217 are characterizd by the same @xmath5 and @xmath6 values . under these assumptions the overlap matrix element is given by @xmath254 where @xmath255 is a @xmath5-dependent constant that enters into all matrix elements and thus cancels when calculating expectation values . the kinetic energy matrix element reads @xmath256 the matrix element for the trapping potential reads @xmath257 lastly , the interaction matrix element for the gaussian potential can be written as @xmath258 \rangle.\end{aligned}\ ] ] the expression for the matrix element @xmath259 | \psi \rangle$ ] reduces to that for the overlap matrix element if the matrices @xmath260 and @xmath261 are replaced by @xmath262 and @xmath263 , respectively . for unnatural parity states with @xmath74 , we use @xmath72 and @xmath75 in eq . ( [ eq_basis1 ] ) . in the following , we assume that @xmath218 and @xmath217 are characterizd by the same @xmath5 and @xmath6 values . under these assumptions the overlap matrix element is given by @xmath264 where @xmath265 is a @xmath5-dependent constant that enters into all matrix elements and thus cancels when calculating expectation values . the kinetic energy matrix element reads @xmath266 \left ( \rho_{11}\rho_{22}-\rho_{12 } \rho_{21 } \right ) + \nonumber \\ 2\rho_{11 } \left(\rho_{11 } s_{22}+\rho_{22}s_{11}-\rho_{12}s_{21}-\rho_{21}s_{12 } \right ) \}.\end{aligned}\ ] ] the matrix element for the trapping potential reads @xmath267 \left ( \rho_{11}\rho_{22}-\rho_{12 } \rho_{21 } \right ) + \nonumber \\ 2\rho_{11 } \left(\rho_{11 } \tilde{s}_{22}^{(pq)}+\rho_{22}\tilde{s}_{11}^{(pq)}- \rho_{12}\tilde{s}_{21}^{(pq)}-\rho_{21}\tilde{s}_{12}^{(pq ) } \right ) \}.\end{aligned}\ ] ] as in the natural parity case , the expression for the interaction matrix element for the gaussian potential can be related to that of the overlap matrix element by making the appropriate substitutions . the states are counted separately for each @xmath268 symmetry . in counting the states , we do include the unshifted states but we do not account for the @xmath269 degeneracy of the energies ( see column 1 of table [ fig1 ] for the state number ) .
|
equal - mass two - component fermi gases under spherically symmetric external harmonic confinement with large @xmath0-wave scattering length are considered . using the stochastic variational approach , we determine the lowest 286 and 164 relative eigenenergies of the @xmath1 and @xmath2 systems at unitarity as a function of the range @xmath3 of the underlying two - body potential and extrapolate to the @xmath4 limit .
our calculations include all states with vanishing and finite angular momentum @xmath5 ( and natural and unnatural parity @xmath6 ) with relative energy up to @xmath7 , where @xmath8 denotes the angular trapping frequency of the external confinement .
our extrapolated zero - range energies are estimated to have uncertainties of 0.1% or smaller .
the @xmath1 and @xmath2 energies are used to determine the fourth - order virial coefficient of the trapped unitary two - component fermi gas in the low - temperature regime .
our results are compared with recent predictions for the fourth - order virial coefficient of the homogeneous system .
we also calculate small portions of the energy spectra of the @xmath9 and @xmath10 systems at unitarity .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
a phase transformation in a liquid phase can be affected when it is subject to high intensity acoustic waves . the rarefaction pressure swing of the wave can nucleate bubbles or induce a liquid - gas transition , @xcite called acoustic cavitation . several theoretical and experimental studies have shown that wave propagation in melts and supercooled liquids causes periodic phase transformation resulting in enhanced crystallisation . @xcite for instance , liquid helium exposed to a high intensity focused ultrasound field undergoes liquid - solid transformation . @xcite nucleation of solid helium was observed to happen over the compression cycle followed by a decay and finally melting during the rarefaction cycle . likewise , there is a body of works , mainly experimental , reporting the effect of an ultrasound field on crystallisation in a supersaturated solution . @xcite the latter is usually referred as the sonocrystallisation process . depending on the acoustic pressure magnitude and frequency , sonocrystallisation can yield a high nucleation rate and produce much finer crystals with a narrower crystal size distribution ( csd ) compared to the conventional ( silent ) cooling crystallisation . @xcite the mechanism by which ultrasound affects nucleation in a supersaturated solution is uncertain but it has been mainly attributed to the cavitation phenomenon . @xcite enhancement in nucleation rate was , however , experimentally observed @xcite in an ultrasound field that is weak enough to inhibit cavitation . two main types of ultrasound - induced cavitation are inertial and stable cavitation . inertial cavitation is the event when tiny cavities or dissolved gases in the liquid grow rapidly due to the rarefaction created by the ultrasound wave and collapse violently in the compression cycle of the ultrasound wave . this collapse generates enormous shock waves travelling with a speed of about @xmath0 and a magnitude of up to @xmath1 as well as a temperature rise at the centre of the bubble to about @xmath2 . @xcite this can also lead to a significant temperature variation at a rate of @xmath3 . @xcite all these effects happen locally and over a very short period of time , i.e. spatially and temporally on scales of the volume of a bubble and nano - seconds respectively . @xcite in the case of an asymmetric collapse , e.g. due to an oscillation and implosion of a bubble in the vicinity of a solid surface , a jet of fluid , at speeds greater than @xmath4 , is generated which can also influence the crystallisation process . @xcite both the direct acoustic field and the indirect effects associated with cavitation influence the thermodynamics and kinetics of nucleation . considering the effect of static pressure on nucleation , ford @xcite modelled the pressure dependent homogeneous nucleation in a gas mixture using a statistical mechanics approach . within the framework of classical nucleation theory ( cnt ) , kashchiev et al . @xcite proposed a model estimating the pressure dependent nucleation rate of condensed phase in a solution . this model does not consider the effect of pressure on the excess free energy as it was based on a cluster boundary defined by the equimolar dividing surface ( eds ) . with regard to modelling the effect of acoustic cavitation on crystallisation or solidification , the influence of radiated pressure from a collapsing bubble on the thermodynamics of ice formation was studied by saclier et al . @xcite louisnard et al . @xcite , however , proposed a segregation hypothesis where mass transportation due to the emitted shock wave from an inertial cavitation is the main factor leading to high nucleation rates observed experimentally . they suggested the mass diffusion mechanism and its effect on the kinetics of nucleation as the key factor rather than the effect of pressure oscillation on the thermodynamic state . here we show that this may only be the case if a cluster is defined by an eds . nevertheless , pressure fluctuation affects both the nucleation work and kinetics simultaneously . if we model the kinetics of nucleation with the cluster dynamics approach , i.e. the master equation , @xcite it is determined by means of aggregative and non - aggregative mechanisms . aggregative mechanisms include nucleation , growth and ageing , that give rise to the flux of cluster concentration along the size axis @xmath5 . the non - aggregative mechanism accounts for change in composition ( concentration of clusters ) driven by mass flux along the space parameter axis . these two fluxes together determine the cluster distribution over time . we show that an acoustic wave can affect both processes which creates a coupled problem . however , depending on the magnitude and wavelength of the pressure fluctuation , the non - aggregative process , i.e. mass flux due to pressure gradient across space between adjacent systems within the bath , might be negligible . @xcite this work aims to investigate the effect of pressure fluctuation on aggregative mechanism in particular . this allows us to study the effect of pressure variation on nucleation , the early stage of growth and also the ostwald - ripening phenomenon . furthermore , focusing on the aggregative mechanism we only need to know the local pressure fluctuation in the region of interest , i.e. dynamic pressure in the system , which can be emitted from any type of acoustic source , e.g. a planar or focused transducer or radiated pressure from either stable or inertial cavitation . nevertheless , the development accounts for the effects associated with the wave propagation including temperature change , e.g. due to absorption , too . this makes it possible to apply this formulation to investigate the effect of pressure fluctuation in the old phase , emitted from any acoustic source , on the thermodynamics and kinetics of a first order phase change . accounting for the non - aggregative effect of an acoustic wave and combining it with the present work to resolve the coupled problem is the subject of a forthcoming paper . to accomplish our objectives , we use the gibbs droplet model in a generic format to estimate the clustering work for both equilibrium and non - equilibrium clusters ( sec . [ sec : workcluster ] ) . we develop equations to determine the number of molecules in both the new phase core , and the surface of a cluster defined by a non - eds ( sec . [ sec : nsigma ] ) . we then demonstrate the significance of these improvements in reproducing the excess free energy of clusters obtained from statistical mechanics simulations and consequently estimating a nucleation rate by employing experimental data of water droplet formation from the vapour phase ( sec . [ sec : waterdropletresults ] ) . using this model , we then show that the effect of pressure on kinetics is size - dependent and sensitive to the placement of dividing surface , especially for small clusters ( sec . [ sec : trannucl ] ) . it was previously shown @xcite that the isothermal effect of pressure on nucleation work depends on the excess number of molecules in a nucleus and therefore is size - dependent too . we finally report in sec . [ sec : resultdisc ] the effect of magnitude and frequency of acoustic waves on nucleation work and kinetics in an aqueous solution if we use a non - eds cluster and compare it with results predicted by the cnt . mechanical work is required to convert the old phase into the metastable state and start the formation of a new phase . the new phase is characterised as a cluster of molecules with a density that differs from the mother phase . this work becomes maximum in the case of formation of a critical cluster which is the cluster in unstable thermodynamic equilibrium with the old phase . cluster formation work depends on the thermodynamic state and constraints applied to the old phase . we consider the system as a volume element coupled to a heat and particle bath . the phase change takes place within this system . the choice of heat and particle bath essentially means that temperature and volume of system remain constant and the old phase in the system has the same chemical potential as of the bath . this set of constraints is usually experimentally favoured and will be adopted in the following analysis . the system initially consists of the homogeneous old phase . after cluster formation , the system includes three phases , namely the core of cluster taking the new phase , the old phase surrounding the new phase and an interface phase . the new phase is considered as a homogeneous closed phase . the interface phase lies on an arbitrary dividing surface between the new and old phase considered as a gibbs geometrical surface , i.e. a zero volume layer . the properties of the old phase are displayed below with no suffix whereas the suffices @xmath6 label the new and interface phases , respectively . the reversible work of creating a cluster is equal to the change in the free energy of the system and is given by @xcite @xmath7 where @xmath8 is a difference in chemical potentials of old and new phases at temperature @xmath9 , pressure @xmath10 and composition @xmath11 of the old phase , @xmath12 and @xmath13 are the number of molecules in the new and interface phase , respectively , and @xmath14 is the cluster volume . @xmath15 is the thermodynamic grand potential . likewise , @xmath16 is the grand potential associated with the interface phase which is also represented by @xmath17 where @xmath18 is the interfacial surface area and @xmath19 is the surface tension . the size of cluster ( in molecules ) is equal to @xmath20 and its volume ( for a single component cluster ) is given by @xmath21 where @xmath22 is the specific volume of the new phase . the difference between the chemical potential of the old phase and the bulk new phase evaluated at the temperature and pressure of the old phase reads @xmath23 and similarly , @xmath24 . for the sake of briefness in the notation , independent variables , i.e. @xmath25 , will not be displayed unless it is required . nevertheless , we shall note that they could vary over time in the system and time and space in the bath . in the case of a condensed new phase , the cluster can be considered practically incompressible . thus , the work of formation of a condensed cluster becomes @xmath26 and substituting @xmath27 with @xmath28 and rearranging the above equation gives @xmath29 to be able to use these equations , @xmath30 and @xmath13 should also be determined for a generic dividing surface . this is discussed in secs . [ sec : nsigma]-[sec : noncriticalcluster ] . for the system containing a new cluster within the old phase , we can write @xcite @xmath31 where @xmath32 represents the entire system including all three phases , and @xmath33 and @xmath34 are the molecular number density of the new and old phase , respectively ( see figure [ fig : bath_system ] ) . this equation can be re - arranged to @xmath35 where the left hand side ( lhs ) is invariant with respect to the choice of the dividing surface . if we choose an eds , we will have by definition @xmath36 @xcite and therefore @xmath37 where @xmath38 is the volume of a cluster defined using the eds . given that lhs is invariant with the choice of surface , the equation for an arbitrary surface becomes @xmath39 which yields and @xmath40 are the number of molecules in the system before and after cluster formation , respectively . refer to the text for details . ] @xmath41 and substituting @xmath42 and @xmath43 where @xmath44 is the size of an eds - defined cluster , this equation simplifies to @xmath45 where @xmath46 . this is a generic equation and valid for any shape of cluster . we can write @xmath47 where @xmath48 is the excess number of molecules in the cluster of volume @xmath14 comparing to the same volume of old phase . this quantity is independent of the choice of dividing surface . in the case of a condensed new phase we have @xmath49 and consequently @xmath50 . however , when the new phase is less denser than the old phase , e.g. bubble formation , @xmath51 and @xmath48 become negative . as shown in appendix [ sec : appxnsigma ] , for cubic and spherical clusters we have @xmath52 with @xmath53 where @xmath54 is a dimensionless quantity that distinguishes an arbitrary dividing surface from the eds . here @xmath55 is the radial separation between the eds and the arbitrary surface and @xmath56 is the radius of a molecule in the new phase , considered to be a sphere . @xmath57 is the shape factor which equals to unity for a spherical cluster . henceforth , we assume the cluster is spherical . the total size of a cluster then reads @xmath58 . depending on the density of new and old phases and the location of the dividing surface , @xmath13 can become positive or negative . this model satisfies the following conditions @xmath59 they imply that for large clusters the number of molecules in the core becomes dominant and the eds becomes acceptable for defining the boundary of a cluster . however , for a small cluster for which a core with bulk properties does not exist , the contribution of interface phase takes on an important role which can be modelled through interface terms with non - zero @xmath13 ( @xmath60 in our model ) . if the arbitrary surface is selected such that it coincides with the surface of tension , then @xmath61 and in the limits of @xmath62 the separation length converges to the tolman length @xmath63 and subsequently @xmath64 . now we need to obtain @xmath65 which depends on the condition of a cluster . this is addressed in the secs . [ sec : criticalcluster ] and [ sec : noncriticalcluster ] . the critical cluster is in an unstable thermodynamic equilibrium with the old phase and satisfies the following conditions:@xcite @xmath66 , and the well - known laplace equation @xmath67 where the asterisk denotes the properties of the critical cluster . substituting these relations in eq . [ eq : workcond ] gives the work of formation of the critical cluster as follows @xmath68 given that @xmath69 , this equation may be reformulated as @xmath70 the last two terms in the above equation essentially represent the excess helmholtz free energy of the interface phase of the critical cluster size @xmath71 , i.e. @xmath72 . @xcite the grand potential of the interface phase can be written as a function of @xmath13 @xcite or the area of the cluster , basically a function of @xmath12 . in any case , we can plausibly consider @xmath73 . therefore , the taylor series expansion of @xmath16 about @xmath5 reads @xmath74 evaluating this equation at the critical cluster and inserting the results in the first formula of eq . [ eq : criticalwork ] gives @xmath75 if a cluster is defined by the eds and the capillarity approximation is imposed , the above equation simplifies to the nucleation work given by the cnt . the equality of chemical potentials of all phases may not hold for a non - critical cluster , i.e. a non - equilibrium cluster , which makes it a complicated situation to analyse . for a non - critical cluster we assume @xmath76 . this assumption is justified if diffusive exchange of molecules from interface phase to the new phase is faster than the diffusion of molecules towards the interface from the old phase . @xcite this gives @xmath77 and subsequently eq . [ eq : workcond ] transforms to @xmath78 now we need to determine the quantity @xmath79 for a non - critical cluster . this is not a trivial problem and needs knowledge from statistical or molecular models . nevertheless , the following two methods have previously been used to estimate this quantity using continuum thermodynamics.@xcite in the first method , we use the maxwell relationship of @xmath80 , under isothermal conditions , and obtain the exact equation @xmath81 : see appendix [ sec : appxchempot ] for the derivation . the last term is the difference between the inner pressure of a non - critical cluster and a critical cluster for the same pressure of the old phase @xmath10 . inserting this into eq . [ eq : workfinaldelp ] gives @xmath82 given that @xmath83 is in the order of few @xmath84 , we can approximate the second term in the above equation by @xmath85 as @xmath86 is relatively small comparing to @xmath87 . therefore this equation simplifies to @xmath88 which can also be written as @xmath89 where @xmath90 . making a comparison between eq . [ eq : workfinaldelpv1appr0 ] and eq . [ eq : workcond ] reveals that this approximation essentially sets @xmath91 to zero . this condition is a result of mathematically cancelling the pressure term against the supersaturation term in the former equations while not enforcing the physical equilibrium conditions . evaluating this equation for the critical cluster @xmath71 yields eq . [ eq : criticalworkform2 ] as anticipated . this tells that the work of formation of a non - critical cluster can be reasonably approximated by the equation that determines the work of formation of a critical cluster . in the second method , the pressure difference between the inside and outside of a cluster is approximated using the generalised laplace equation @xmath92 . @xcite this method basically assumes that laplace equation could be extended to sub - critical and supercritical clusters . employing this approximation transforms eq . [ eq : workfinaldelp ] to @xmath93 and given @xmath94 it follows that @xmath95 where @xmath96 . this equation simplifies to the following relationship by using eq . [ eq : omegataylor ] @xmath97 both eqs . [ eq : workfinaldelpv1appr ] and [ eq : workfinaldelpv2nn ] give an approximation of the cluster formation work over the entire range of cluster size . they become identical for an eds cluster . however , evaluating eq . [ eq : workfinaldelpv2nn ] for the critical non - eds cluster gives a nucleation work which is different from the nucleation work obtained from the exact eq . [ eq : criticalwork2 ] . since we are interested in a formulation that estimates cluster formation work for both critical and non - critical non - eds clusters , eq . [ eq : workfinaldelpv1appr ] suits our needs better and will be utilised in this work . it should be noted that nucleation work is a physical property of the system and is independent of the location of dividing surface . consequently , the desired formulation has to agree with the result of the exact eq . [ eq : criticalwork2 ] for the critical cluster with non - zero @xmath13 . in spite of this , the work of formation of a cluster varies depending on the way it is being identified . to elaborate on this , we keep the cluster size @xmath5 constant and compare the work of formation of a classical cluster ( identified with eds and capillarity approximation ) with a generic cluster of the same size . the identical cluster size implies equivalent `` bulk '' work ( i.e. @xmath98 , note that this is different from volume work as volume depends on @xmath12 not @xmath5 ) whereas the excess free energy is different . the ratio of excess free energies is determined by @xmath99 considering @xmath100 where @xmath101 and @xmath102 where @xmath103 is the planar surface tension between two phases in equilibrium , this equation simplifies to @xmath104 we can use an eds cluster to define the effective surface free energy as @xmath105 . considering that the majority of simulations conducted by means of molecular dynamics ( md ) or other statistical mechanical approaches report results for eds defined clusters , this choice also allows us to make a comparison between our @xmath106 and excess free energy with their counterparts in those works . setting @xmath107 yields @xmath108 . substituting @xmath109 from eq . [ eq : fsratiosimp ] gives @xmath110 the first term accounts for temperature and concentration dependence of the effective surface tension and the second term describes the curvature dependence of the effective surface tension . the fact that concentration influences the excess free energy and consequently the effective surface tension , according to our definition , was demonstrated and formulated in different works too . @xcite now we specify the dividing surface and associated surface tension which will be used in this work . given eq . [ eq : nsigma ] we can use any arbitrary surface which can more precisely approximate the excess free energy . utilising the eds ( @xmath111 ) to define a cluster along with the capillarity approximation yields the conventional form of the cnt which is unsuccessful in explaining many experimental observations . this can be improved if the size dependency of surface tension is accounted for . if the cluster s boundary is identified by the surface of tension , the curvature dependence of the surface tension could be accounted for by the tolman equation@xcite or other polynomial expansion models.@xcite it has been shown that by choosing the surface of tension ( @xmath112 ) and employing the tolman equation to correct the surface tension , we can achieve a better agreement with some experiments . @xcite md or dft simulations are usually required to determine an appropriate tolman length . nevertheless , the tolman equation is useful for larger clusters but is expected to break down for small clusters . therefore , we opt not to employ it in this work , instead we define a non - eds surface identified by the parameter @xmath60 , and the size - independent surface tension of @xmath113 ( denoted as _ the new surface _ in this work ) . this leads to @xmath114 and consequently @xmath115 . fitting @xmath106 of eq . [ eq : gammaeff ] to the effective surface tension obtained from statistical mechanical methods yields the value of @xmath60 . in the case of nucleation of water droplets discussed in sec . [ sec : waterdropletresults ] , @xmath60 is estimated with the aid of excess free energies of clusters obtained from statistical mechanics simulations using the tip4p/2005 molecular model . for the crystallisation exercise , however , we performed simulations at different values of @xmath60 to investigate its influence on the kinetics of crystallisation under pressure fluctuation . we should point out that kashchiev @xcite introduced a non - equimolar dividing surface identified by the condition @xmath116 , termed the conservative surface . for the conservative surface , the surface tension is also size - independent and equals the macroscopic planar value @xmath113 . the question of whether the new surface adopted in this work coincides with the conservative surface is out of the scope of the current paper and will be addressed elsewhere . having determined the specifications of the dividing surface , the excess number of molecules and clustering work , we can study the process of cluster formation and determine the size of the nucleus and nucleation work when the old phase is exposed to an acoustic wave . furthermore , the kinetics of nucleation under this circumstance should be calculated . this is discussed in secs . [ sec : perteffect ] and [ sec : trannucl ] . as the acoustic wave propagates in the bath , it causes pressure fluctuation , temperature variation and mass transportation due to a spatial pressure gradient . @xcite the effect of variation of pressure , temperature and composition on the work of formation of a cluster then should be evaluated . the total differential of @xmath117 is calculated by differentiating eq . [ eq : workcond2 ] which gives @xmath118 and subsequently , this re - arranges to @xmath119 to compare with the silent condition , we are interested in evaluating the effect of acoustic wave on the work of formation of the same size cluster . this is obtained by setting @xmath120 in the above equation which simplifies to @xmath121 the change in the chemical potential of the old phases with respect to pressure and temperature can be estimated using a gibbs - duhem relation @xmath122 where @xmath123 and @xmath124 are the partial molecular volume and entropy of the old phase . likewise for the interface phase @xmath125 and for the new phase @xmath126 : see appendix [ sec : appxchempot ] for the derivation of the latter equation . substituting @xmath127 and @xmath128 in the previous equation then gives @xmath129 dt \nonumber \\ & & -\left[\nu(n_n+n_\sigma)-n_n \nu_n \right ] dp,\end{aligned}\ ] ] which can be written as @xmath130 where @xmath131 is the excess entropy gained by the system through the formation of a cluster of size @xmath5 and @xmath132 is defined in eq . [ eq : nexcess ] . in the case of isothermal acoustic wave propagation , the effect of a pressure perturbation on the work of forming an @xmath5-sized cluster is determined by @xmath133 finally , integrating eq . [ eq : workdiffcondfinal2 ] gives the work required to form clusters at a temperature , pressure and composition which differ from the reference state . this is expressed as @xmath134 where @xmath135 is the work required to create an @xmath5-sized cluster while the system is at the reference thermodynamic state ( @xmath136 ) . if the reference condition is chosen such that @xmath137 , where @xmath138 is the equilibrium fractional concentration of monomers , then @xmath139 and @xmath140 . however , in sonocrystallisation experiments a supersaturated solution is usually made first , and then an acoustic wave is introduced . therefore , it is practically desirable to choose the supersaturated state in silent condition , i.e. prior to application of acoustic wave , as the reference state . the difference in chemical potentials at the reference state needs then to be obtained . for the crystallisation process , we can write the partial molecular chemical potential of solute species in an ideal solution as @xcite @xmath141 where @xmath142 is the boltzmann constant . it is presumed that the solution is sufficiently dilute , such that the activity can be estimated by the concentration of solute molecules . @xmath143 is the chemical potential of the pure liquid at the same temperature and pressure and does not depend on the composition @xmath11 . consequently , at constant pressure and temperature of the reference state we can write @xmath144 where @xmath138 is evaluated at @xmath145 and @xmath146 . substituting the relationships describing the critical cluster in sec . [ sec : criticalcluster ] together with @xmath147 @xmath148 ( see appendix [ sec : appxchempot ] ) and @xmath149 , in eq . [ eq : workconddifftot ] , we can determine the variation in nucleation work due to change in properties of the old phase as follows @xmath150 which applying equilibrium conditions simplifies to @xmath151 this equation is akin to eq . [ eq : workconddiff ] being evaluated at the critical cluster size which reads @xmath152 where @xmath153 and @xmath154 are the excess quantities evaluated for a critical cluster . evaluating eq . [ eq : workdiffcondfinalbetainteg ] at the size of a critical cluster at the reference condition gives the integral form of this equation . the fact that @xmath155 implies that the work required for cluster formation becomes a maximum for a critical cluster . the size of critical cluster is then the extremum of the equilibrium equation which can be found by solving @xmath156 for @xmath5 . differentiating eq . [ eq : workdiffcondfinalbetainteg ] with respect to @xmath5 gives @xmath157 where @xmath158 and @xmath159 where @xmath160 and @xmath161 are derivatives of @xmath13 and @xmath12 with respect to the cluster size @xmath5 and are given in appendix [ sec : appxnsigma ] . using eq . [ eq : workfinaldelpv1appr ] for the reference state and identifying clusters with _ the new surface _ yields @xmath162 this is a complete equation for calculation of the variation of work required for cluster formation with respect to the size of cluster at different thermodynamic states . the size of the critical cluster is the root of @xmath163 . the generic solution for the critical cluster size in an arbitrary state depends on material properties and change in density and entropy of all phases with pressure and temperature , respectively . the solution for special cases though can be derived . the case of an isothermal process with incompressible old and new phases is considered and discussed below . the absorption of propagating acoustic waves in a medium mainly depends on the viscosity of the medium and the wavelength . wave propagation in an aqueous medium can be considered as an isothermal process since the absorption is low , especially during a short exposure . additionally , if the partial molecular density of the old and new phases are pressure independent , eq . [ eq : work11 ] simplifies to @xmath164 where @xmath165 is the variation in pressure . replacing @xmath5 with @xmath166 and setting this relationship equal to zero gives a polynomial equation with @xmath12 unknown . we can numerically solve this equation and obtain @xmath167 and consequently @xmath71 . if we define the cluster by eds , this equation simplifies and gives the analytic solution for the size of nucleus ( @xmath168 ) as follows @xmath169 where @xmath170 . in the crystallisation process , @xmath171 is the ratio of solute molar concentration to the equilibrium molar concentration at the reference state at initial time instant . this equation demonstrates the effect of pressure fluctuation and variation in composition on the size of eds - defined nuclei . this equation for eds clusters was first derived by kashchiev and van rosmalen . @xcite having determined @xmath71 , the nucleation work is given by eq . [ eq : workfinaldelpv1appr ] for the cluster size of @xmath71 . we established the thermodynamics of equilibrium and non - equilibrium clusters based on the gibbs droplet model with an arbitrary dividing surface . the conservation of mass was used to determine the number of molecules in the interface phase ( @xmath13 ) as a function of the cluster size @xmath5 . we also calculated the effective surface tension of this arbitrary surface and demonstrated its size and chemical potential dependencies . the new development may resemble a classical model with a variable surface tension as a function of the cluster size and chemical potentials of the new and old phases . finally , the effect of pressure and temperature variation on the thermodynamics of non - eds clusters were studied . in sec . [ sec : trannucl ] we proceed to develop the kinetics of cluster growth and decay subject to such thermodynamics . cluster formation is a transient phenomenon with a certain lifetime which depends on size . the szilard model explains the cluster formation as a result of a series of consecutive attachments and detachments of single monomers . it describes the kinetics of nucleation , the early stage of growth and even the ostwald - ripening regime @xcite as they are mainly driven by gaining and losing monomers . the szilard model is expressed by @xmath172 where @xmath173 and @xmath174 are attachment and detachment frequencies at time @xmath175 , @xmath176 is the concentration of @xmath5-sized clusters , and @xmath177 reflects the non - aggregative change in the concentration of the cluster size @xmath5 in an open system . @xmath178 is the inwards flux of @xmath5-sized clusters to the system from the bath and @xmath179 is the outwards flux of @xmath5-sized clusters to the bath from the system . the szilard model is a discrete equation . the truncated second order taylor expansion of this discrete equation about point @xmath5 produces the continuous form of the szilard model which is known as the fokker - planck equation ( fpe ) and reads @xmath180}{\partial n}\right ) \nonumber \\ & & + k(n , t)-l(n , t),\end{aligned}\ ] ] where @xmath181 and @xmath182 are given by @xmath183 @xmath181 is the drift velocity along the size axis , known as the mean growth rate , specifying the rate of deterministic incrementation of the cluster size @xmath5 . @xmath182 is the rate of random change of cluster size along the size axis ( dispersion of cluster size along the size axis ) . the fpe is computationally favoured if the concentration of large clusters is desired . however , because of approximation in the derivation of fpe , it is inaccurate with respect to the szilard equation at small clusters . therefore a hybrid model is envisaged to take advantages of both discrete and continuous description of the cluster dynamics @xcite . subsequently , the cluster size axis @xmath5 is divided up to two sections , a discrete part @xmath184 and a continuous part @xmath185 $ ] where @xmath186 is the boundary between discrete and continuous sections and @xmath187 is the largest cluster size postulated . @xmath186 is chosen such that the simulation results are independent of this choice and the fpe numerically converges to the result of szilard model . the boundary condition of the continuity of cluster flux is applied at the transition point between discrete and continuous model . the cluster flux along the size axis is defined as @xmath188 in this study we assume that the system conserves mass . as such , for both discrete and continuous models we get @xmath189 . having determined the concentration of different clusters by using the hybrid model as well as determining the nucleus size @xmath71 as the root of eq . [ eq : work11 ] and [ eq : work11_isotherm ] , we can calculate the nucleation rate . by definition , the nucleation rate is the rate of appearance of supercritical clusters per unit volume in the system . this is given by @xmath190 where @xmath191 . this is the generic definition of the nucleation rate and can be used for the non - stationary state of the old phase produced due to acoustic wave propagation . the unknown quantities that should be determined now are the attachment and detachment frequencies . monomer attachment to a condensed - phase cluster depends on the state of the old phase . the governing mechanism of monomer attachment is mass transfer . this usually occurs through three main mechanisms @xcite : i ) direct impingement of molecules , ii ) volume or surface diffusion of molecules and iii ) transfer of molecules through the interface of cluster with old phase . the direct impingement is the governing mass transport mechanism when the old phase is gaseous . in the case of homogeneous nucleation in liquid or solid solutions , the main method of monomer attachment is volume diffusion . the interface - transfer method plays an important role for nucleation of clusters , solids or liquids , in a condensed old phase , e.g. melts or solutions . the diffusion mechanism may depend on the cluster size , e.g. for small clusters interface transfer may be dominant and once the cluster has grown enough the volume transfer becomes more important . if the homogeneous nucleation of solids in a dilute solution exposed to acoustic wave is the matter of concern , we postulate that volume diffusion is the main monomer attachment mechanism . volume diffusion can be modelled based on two different approaches : i ) continuum approach , i.e. modelling the conservation of condensable mass in a supersaturated solution , and ii ) atomic approach , i.e. using a random - walk model to determine the probability of collision of a monomer with a cluster and estimating the attachment frequency accordingly . using the first approach , the attachment frequency of monomers to a spherical @xmath5-sized cluster in the condensed phase is given by @xcite @xmath192 where @xmath193 where @xmath194 is the sticking coefficient which is nearly unity in a dilute solution and @xmath195 is the diffusivity of a monomer in the old phase . here we assumed that both cluster and monomers are mobile and diffusing through the medium . this is implemented by using the effective diffusivity and radius for collision between a monomer and an @xmath5-sized cluster , as shown by smoluchowski . the diffusivity of a cluster was estimated based on the stokes - einstein equation . @xmath196 resembles the collision kernel of a monomer with an @xmath5-mer in the smoluchowski coagulation equation . this notion may be employed to generalise this equation for the case of non - spherical clusters by making a modification of the collision kernel using the fractal dimension of the cluster.@xcite these equations are valid for both discrete and continuous cluster size variable @xmath5 . the rate at which monomers detach from an @xmath5-sized cluster depends on the characteristics of the clusters rather than properties of the bulk new phase . this rate can be estimated following the zeldovich method which integrates the thermodynamics under equilibrium condition into the cluster dynamics . at the thermodynamic equilibrium state , a balance between the number of monomers gained and lost by two adjacent clusters on the size axis , i.e. @xmath197 , should hold . the generalised form of the zeldovich method for the case of time - variable supersaturation and a quasi - equilibrium condition , reads@xcite @xmath198 and this equation for the case of continuous cluster size @xmath5 becomes @xmath199 for the sake of brevity , the time variable @xmath175 will not be noted in the following equations while all parameters are considered to be time - dependent . substituting eq . [ eq : work9 ] into above equation results in @xmath200 the minus sign before the integrals in eq . [ eq : work9 ] are removed here by reversing the integration limits . this equation manifests the effect of a change in temperature and pressure on the detachment frequency of monomers from a cluster of size @xmath5 . in our case where we are interested in investigating the effect of an acoustic wave and cavitation on nucleation and growth , this equation gives the full picture within the framework of cluster dynamics by accounting for the effect of pressure fluctuation and temperature variation due to absorption or cavitation of a bubble and mass transportation via pressure diffusion . if we use the same reference state and _ the new surface _ as before , after some manipulations we obtain @xmath201 } \nonumber \\ & & \times \exp \left ( \dfrac{2}{3}\dfrac { a_0 \gamma_{\scriptscriptstyle \infty } } { k_b t } n_n^{-\tfrac{1}{3}}h^{'}(n_n ) \right ) \nonumber \\ & & \times \exp \left(\dfrac{1}{k_bt } \int^{t_0}_t \delta s^\prime_{exc}dt\right ) \nonumber \\ & & \times \exp \left(\dfrac{1}{k_bt } \int^{p_0}_p\delta n^\prime_{exc } \nu dp \right).\end{aligned}\ ] ] so far , we considered the cluster size @xmath5 to be a continuous variable . it is shown in appendix [ sec : appxdisctform ] that equations derived for the detachment frequency for the case of continuous @xmath5 can also be used for the case of discrete representation of cluster formation work . consequently eqs . [ eq : detachfreq_cont1 ] and [ eq : detachfreq_cont2 ] can be employed in conjunction with the szilard model , too . the nucleation work and nucleus size in an incompressible solution which is exposed to an acoustic wave was studied in sec . [ sec : isothermnucl ] . here we are interested in calculating the attachment and detachment frequencies under this condition . given the volume diffusion mechanism , the diffusivity and concentration of monomers are the main factors affecting the attachment rate of monomers to an @xmath5-mer . the effect of pressure on diffusivity is almost negligible due to weak pressure dependence of viscosity and incompressiblity of solution . concentration of monomers can be spatially influenced because of mass transportation due to pressure diffusion . this effect is negligible in low and medium pressure magnitudes . nevertheless , in strong acoustic fields and specially in the vicinity of an oscillating surface , e.g. near the wall of an inertially collapsing bubble , mass transportation can be significant and should be accounted for . an acoustic wave propagating in a solution alters the thermodynamic state and consequently changes the detachment frequency , as demonstrated in eq . [ eq : detachfreq_cont2 ] . in the case of an isothermal condition and pressure independent partial molecular density , this equation simplifies to @xmath202 subsequently , approximating molar concentration with the concentration of monomers , i.e. @xmath203 where @xmath204 is the solubility at the reference state , we arrive at @xmath205 and the latest assumption is justified since the concentration of monomers in the system at the initial time is significantly greater than that of @xmath5-mers . we have established all the required equations to determine the kinetics of nucleation while accounting for the effect of fluctuations in the thermodynamic state of the old phase . in the first part of this section we examine the new development by applying it to the test case of water droplet nucleation from the gas phase . the model of water was chosen given the fact that homogeneous nucleation of vapour is very well studied both experimentally and theoretically . subsequently , having validated the model and numerical implementation , we will evaluate the effect of an acoustic wave on crystallisation in an aqueous solution in sec . [ sec : crystallisationresults ] . we will use these results to explain some experimental trends reported in the literature . since the majority of experimental works do not define all the necessary parameters of both the acoustic field and crystallisation , a direct comparison with the sonocrystallisation data seems impractical . in addition , acoustic cavitation usually happens prior to or concurrent with crystallisation which is often not characterised in experiments . for the numerical computations , the fpe ( eq . [ eq : fpeq ] ) is discretized and solved together with the discrete szilard equation ( eq . [ eq : szilardeq ] ) using a variable ode solver . the details of numerical implementation are expressed elsewhere.@xcite we define the dimensionless time variable @xmath206 here as follows : @xmath207 where @xmath208 in the case of attachments governed by the volume diffusion process . regarding the initial condition , following our previous discussion we assume only monomers are present in the system and bath initially . the presence of @xmath5-sized clusters , @xmath209 , in the initial condition may change the nucleation rate by less than one order of magnitude.@xcite consequently , the considered initial condition is adequately reasonable for our work . for all the simulations , we consider supersaturation is time - varying and the system is closed . the excess free energies of water droplets of different sizes have recently been calculated by means of a statistical mechanical approach at a temperature @xmath210.@xcite considering that calculations using the tip4p/2005 molecular model could successfully estimate the surface energy in agreement with experiments,@xcite we use the calculations of lau et al . @xcite to validate our model following the ensuing procedure : i ) we deduce the values of @xmath60 by comparing the effective surface tension determined by our model , eq . [ eq : gammaeff ] , with those obtained from statistical mechanics , and ii ) use these results to calculate nucleation rates and compare them against experimental results at @xmath211 obtained by brus et al .. @xcite figure [ fig : gammaeff ] shows the effective surface tension at different values of @xmath60 calculated at experimental supersaturation @xmath212 at @xmath211 . we can see that at the very small clusters of size @xmath213 , the best fit is achieved at @xmath214 whereas for larger clusters ( @xmath215 ) the curve with @xmath216 happens to give the best agreement with statistical mechanic results . this may suggest that @xmath60 is size - dependent analogous to the tolman length @xcite for the surface of tension as the dividing surface . nevertheless , since with @xmath216 we achieve acceptable approximation of size - dependent surface energy with respect to statistical mechanical simulations over a wide range of cluster size , we take this value for our calculations at this condition . the efficacy of this choice will be evaluated by comparing the calculated kinetics of nucleation with the experiments . the same procedure was repeated for all experimental supersaturations at @xmath211 reported by brus et al . @xcite and we observed the same trend showing that @xmath60 is larger for small clusters ( @xmath217 ) and decreases for larger clusters ( @xmath215 ) . likewise , in the case of the surface of tension as the dividing surface , the similar tendency of size - dependence of tolman length , i.e. inverse relationship with droplet size , and supersaturation dependence at a constant temperature were also reported . @xcite having determined the parameter @xmath60 for a range of supersaturations at temperature @xmath211 , we can now calculate the kinetics of water droplet nucleation by solving the hybrid model . considering that for the gaseous old phase , clusters are much smaller than the mean free path in the gas phase , the attachment rate should correspond to a particle flux modelled by the gas kinetic theory . consequently , the attachment frequency reads @xcite @xmath218 where @xmath219 . based on the definition of a gibbs surface , the mass of a cluster of size @xmath5 is given by @xmath220 , where @xmath221 is the mass of a monomer in the new phase , but the radius of cluster is defined by @xmath12 . subsequently , if we use @xmath222 instead of @xmath223 given in eq . [ eq : kfattachfreq ] , all the previous equations are applicable and can be used for this exercise . the time non - dimensionalisation coefficient also changes to @xmath224 where @xmath225 is the equilibrium vapour pressure . at @xmath211 , we have @xmath226 . the physicochemical properties of a vapour mixture are determined by equations provided in table 1 of brus et al .. @xcite the hybrid model is then numerically solved and the nucleation rate is calculated . figures [ fig : zsupnucovertime ] and [ fig : sovertime ] depict variation in concentration of supercritical clusters ( @xmath227 ) and supersaturation over time , respectively , for two cases where clusters are defined by the eds ( @xmath111 ) and the surface with @xmath216 . the equilibrium monomer concentration of @xmath228 is used to determine the concentrations @xmath229 in the supersaturated state . the nucleation time is identified by the appearance of the first ten supercritical clusters , i.e. @xmath230 , which happens around @xmath231 and @xmath232 for these eds and non - eds cases , respectively , and are indicated by vertical dashed - dotted lines , see figure [ fig : zsupnucovertime ] . the system is closed and therefore the total mass is a constant . as a result , the condensation depletes the monomer supersaturation and terminates the nucleation and monomer - driven growth stages around @xmath233 for the classical cluster case . at this moment the concentration of monomers drops drastically ( @xmath234 ) whereas it lasts longer for the non - eds clusters . this implies that this new model with @xmath216 predicts lower nucleation rate than the cnt . the sharp fall marked by the vertical line at the right hand side is due to the way we define @xmath227 , see above , and does not hold physically . since supersaturation drops to almost unity , the critical cluster size mathematically tends to infinity and therefore all the previously made clusters become subcritical which brings about this abrupt drop of @xmath227 . repeating these calculations for all experimental supersaturations , we determine the stationary nucleation rates for all these conditions . these results together with the experimental nucleation rate and values obtained by becker - dring ( bd ) model are plotted in figure [ fig : jsimexpbd ] . the main difference between our model and the bd is due to the non - eds definition of clusters in our model which allows a more accurate estimation of the excess free energy whereas the bd model calculates the stationary nucleation rate using a classical definition of a cluster . this also leads to a different critical cluster size and consequently a different nucleation rate . additionally , the hybrid szilard and fpe model determine the kinetics of nucleation for a stationary as well as a time variable non - stationary system . this is important because in practice the supersaturation imposed on the system is always time variable . we used the same physicochemical properties of the vapour mixture and experimental supersaturations in the bd . the agreement between predicted nucleation rate and experimental values are very good . though the dividing surface we used to define clusters has the property of size - independent surface tension , the effective surface tension of this surface is size , temperature and supersaturation dependent , see eq . [ eq : gammaeff ] . this is attributed to the fact that in our model , a cluster of size @xmath5 can take on different combinations of @xmath12 and @xmath13 due to the arbitrary placement of dividing surface contrary to a cluster defined by eds or surface of tension . therefore we are able to reproduce @xmath106 by choosing the location of the dividing surface appropriately , see figure [ fig : gammaeff ] . it seems that this important characteristic corrects some of the shortcomings of cnt , at least for water at @xmath211 . we should not , however , dismiss the chance of a coincidental close agreement between our numerical results and experimental values since this model does not account for non - idealities in the gas phase and compressibility of the liquid phase . the lack of molecular simulation results at other temperatures does not allow us at this stage to extend these calculations to lower temperatures . nevertheless , we deduced the values of @xmath60 from experimental data of wlk et al.,@xcite shown in figure [ fig : lambfitted ] . we can see the gradual descent of @xmath60 with temperature rise . a similar trend was observed and reported for the tolman length too . @xcite ) at different @xmath60 calculated by eq . [ eq : gammaeff ] at @xmath211 . @xmath235 : statistical mechanical simulations@xcite at cluster size of @xmath236 . solid black curve shows the best fit to statistical mechanical simulations . ] values over time . vertical lines labelled @xmath237 and @xmath238 indicate the beginning of the nucleation stage in models with @xmath111 and @xmath216 , respectively . the unlabelled vertical line indicates the end of nucleation and monomer - driven growth of supercritical clusters in the case of @xmath111 while nucleation is still ongoing in the case of @xmath216 . this is due to a faster nucleation rate for @xmath111 which leads to quicker depletion of the imposed supersaturation of monomers . ] values over time . see caption of figure [ fig : zsupnucovertime ] for details . ] . nucleation rate calculated by our new model using @xmath60 values determined from statistical mechanical calculation of gabriel et . al.@xcite at the data points of brus 2008 and 2009 ( solid line with @xmath239 and dashed line with @xmath240 , respectively ) . the experimental results of brus et . 2008 and 2009@xcite are also shown ( solid line with @xmath241 and dashed line with @xmath242 , respectively ) . nucleation rate determined by bd model at the data points of brus 2008 and 2009 ( @xmath235 and @xmath243 , respectively ) . ] calculated from the experimental nucleation rate@xcite for water nucleation at different temperatures . the error bars show the range in @xmath60 at a specific temperature as a function of supersaturation . the lower and upper limits correspond to the smallest and largest experimental supersaturations at a specific temperature , respectively . ] we demonstrated in this section that using a specific non - eds dividing surface together with a gibbs droplet model in a general format can better predict the kinetics of nucleation compared to the cnt . this supports our intention to study the effect of pressure fluctuations on the nucleation process by means of a more generic model than the cnt . these results are elaborated in sec . [ sec : crystallisationresults ] . the effect of pressure fluctuation on the clustering work and the kinetics of nucleation in the system are determined by eqs . [ eq : workdiffcondpress ] and [ eq : detachfreq_cont2 ] , respectively . employing these equations under isothermal condition in our hybrid cluster dynamics model , we can study nucleation , the early stage of growth and also the ostwald - ripening phenomenon in a system undergoing pressure variation . we consider a single closed system in the bath with a time varying pressure . the local pressure can then be written as @xmath244 where @xmath245 is the ambient pressure at the reference state and @xmath246 is the acoustic pressure in the system with magnitude @xmath247 and frequency @xmath248 . here we report simulations performed using different sets of parameters . acoustic pressure and frequency are varied from @xmath249 to @xmath250 and @xmath251 to @xmath252 , respectively . this range of acoustic parameters pertains to experimental amplitude and frequency of ultrasound waves generated by different ultrasonic transducers , e.g. planar and high intensity focused , in sonocrystallisation experiments . further , we studied the effect of the parameter @xmath60 on the kinetics of nucleation under isothermal pressure perturbation too . with regard to the solution properties , we use the generic physicochemical properties of a sparingly soluble salt in an aqueous solution at room temperature ( @xmath253 ) provided in table 6.1 of reference . following this reference , we consider the new phase to be denser than the old phase with a typical value of @xmath254 . this gives @xmath49 . furthermore , we have the time non - dimensionalisation constant of @xmath255 . unless otherwise stated , all the following simulations are conducted with @xmath256 which is an average value of @xmath60 formerly obtained for water droplet formation at @xmath257 . this choice is improvised assuming the surface energy of clusters in a dilute aqueous solution shows a similar size dependence at the same temperature as water droplets . nevertheless , we investigate the effect of different values of @xmath60 in sec . [ sec : simlambdaeff ] . initially , we investigate the effect of the magnitude of static pressure on crystallisation . this is obtained by setting @xmath258 and keeping @xmath259 constant . figure [ fig : zsupnucovertime_lam35 ] shows changes in concentration of supercritical clusters over time at different pressure magnitudes of @xmath260 and @xmath261 . the dashed vertical lines illustrate the nucleation time lag at different pressure magnitudes . we can see that in the case of positive @xmath51 , @xmath262 has an inverse relation with pressure magnitude . for example , the nucleation time lag reduces by more than six orders of magnitude as the pressure magnitude increases only by one order from @xmath263 to @xmath261 ( @xmath264 which gives @xmath265 ) . a similar trend between the pressure magnitude @xmath266 and the experimental lifetime of superheated xenon , oxygen and argon liquids was reported in the literature too . @xcite . vertical lines labelled @xmath267 to @xmath238 indicate the beginning of the nucleation stage for different static pressures of old phase . static pressure decreases from the black curve ( @xmath261 ) at the top to the red curve at the bottom ( @xmath249 ) . ] . ] and with @xmath256 . around @xmath268 the concentration of supercritical clusters becomes a maximum and starts to decline whereas the mean size of supercritical clusters increases and plateaus shortly after . ] and with @xmath256 . around @xmath269 the supersaturation approaches unity , the concentration of supercritical clusters becomes a maximum and starts to decline . the mean size of supercritical clusters drops too but a ripening process could not be identified . ] the change in the supersaturation over time is depicted in figure [ fig : sovertime_lam35 ] . an increase in pressure magnitude amplifies the depletion rate of imposed monomers in a closed system . this fast nucleation rate leads to smaller supercritical clusters on average . at the highest pressure , the ostwald ripening regime starts at roughly @xmath268 where the concentration of supercritical clusters reaches its maximum and declines afterwards . we can see that the average size of supercritical clusters , however , increases after this instance which is due to the absorption of depleted monomers from smaller clusters by larger clusters , see figure [ fig : zsn_zsnavg_pac100mpa ] . the ostwald ripening , however , could not be observed when @xmath270 , see figure [ fig : zsn_zsnavg_pac1mpa ] . in this case the concentration and the average size of supercritical clusters increase and sharply drop together . as a result , we expect to see a cluster size distribution ( csd ) with a smaller mean value and a broader distribution in a higher magnitude excitation than a lower one . ) , over time at the static pressure @xmath249 with @xmath256 . the black dashed line shows the time variable size of the critical cluster . ] with @xmath256 . the black dashed line shows the time variable size of the critical cluster . ] with @xmath256 . the black dashed line shows the time variable size of the critical cluster . ] ) at two static pressures of @xmath261 , the left vertical axis , and @xmath250 , the right vertical axis . refer to the text for details . ] the contour plots in figures [ fig : zsn_tau_n_pac1mpa]-[fig : zsn_tau_n_pac100mpa ] show the size - weighted cluster size distribution at three different static pressures . the time - variable size of critical cluster is overlaid on each plot . we can obviously see that the size of critical clusters follow the same trend as supersaturation over time . the initial critical cluster sizes are @xmath271 and @xmath272 at pressure magnitudes of @xmath273 and @xmath261 , respectively . furthermore , these plots illustrate that the size of critical and mean size of supercritical clusters inversely correlate with pressure magnitude ( when @xmath49 ) . reading the concentration of clusters at the end of the nucleation period , i.e. @xmath234 , from this contour , we obtain the csd under these conditions , depicted in figure [ fig : z_n_pac100_50mpa ] . this figure shows that the mean of the csd becomes smaller as pressure increases . this is attributed to a short nucleation period due to a fast nucleation rate which causes a significant reduction in the time difference between the birth time of different stable supercritical clusters . furthermore , the distribution becomes broader at the higher pressure magnitude which is due to enhancement of the ripening process with pressure rise ( when @xmath49 ) . as we have seen so far , a relatively high magnitude of pressure is required to influence the nucleation process . static pressure can be manipulated within this range and even higher experimentally using a high pressure chamber . in terms of the ultrasonic pressure oscillation at such magnitudes , a focused transducer is required as available flat transducers are unable to generate such a strong pressure field . a high intensity focused ultrasound transducer operating at high driving frequencies , e.g. @xmath274 , can generate high magnitude pressure oscillation at focus in water . @xcite such a strong high frequency acoustic wave , however , becomes distorted and turns into shock due to nonlinearities of the transducer and the wave medium . nevertheless , our main objective in this work is to develop and study a theoretical approach for such applications and therefore we will approximate pressure oscillation by a sinusoidal wave in the following simulations . furthermore , we only account for the direct acoustic field and exclude the emitted pressure from the potential acoustic cavitation which may occur at a setting of the acoustic field . nevertheless , incorporating the bubble dynamics into our model , we can estimate the thermodynamics and kinetics of nucleation stimulated by the bubble dynamics too . the simulation results of nucleation in the same aqueous solution exposed to an acoustic wave with @xmath275 and frequencies of @xmath276 and @xmath252 are presented below . comparing to the static pressure condition , pressure oscillation leads to a smaller effective pressure magnitude which lowers the effective nucleation rate . this point is observed in figure [ fig : sovertime_lam35_allfreq ] where the nucleation stage ends at @xmath277 at driving frequencies of @xmath278 and @xmath279 , respectively , whereas it is still ongoing in higher frequency oscillations . the main reason for this behaviour is the variation in the nucleation work due to pressure oscillations , see figure [ fig : delomegastarovertime_lam35_allfreq ] , and subsequently the detachment frequency . equation [ eq : workdiffcondpress ] shows that in an isothermal process , pressure can impede or facilitate nucleation depending on the sign of @xmath280 . when @xmath48 is positive ( i.e. the formation of a condensed phase ) , an isothermal increase in reference pressure reduces the nucleation work and consequently the depletion rate , eq . [ eq : detachfreq_isotherm2 ] , which gives a higher nucleation rate and vice versa . this also influences the concentration of supercritical clusters , shown in figure [ fig : zsn_tau_n_pac50mpa_allfreq_lam35 ] , such that @xmath281 reduces as frequency increases . the csd contour plots for two frequencies are shown in figures [ fig : zsn_tau_n_pac50mpa_100khz ] and [ fig : zsn_tau_n_pac50mpa_2mhz ] . comparing them with the csd at static pressure of @xmath275 , we observe that supercritical clusters become more numerous , i.e. we have nonzero concentrations at @xmath282 . as we discussed above , their concentration , however , is reduced due to pressure oscillations . overall , an acoustic wave causes reduction in the magnitude of the csd at the end of nucleation and moves the mean of the csd to a larger @xmath5 as frequency goes up . with @xmath256 . ] with @xmath256 . the legend is the same as that of figure [ fig : sovertime_lam35_allfreq].,width=302 ] and @xmath283 with @xmath256 . ] and @xmath283 with @xmath256 . ] and static pressure of @xmath284 . ] and static pressure of @xmath285.the legend is the same as that of figure [ fig : sovertime_alllamp1 ] . ] ) at a static pressure of @xmath250 and at different @xmath60 values . ] at different @xmath60 values for a range of cluster sizes . ] . [ fig : govertime_alllam ] we demonstrated that by choosing a suitable @xmath60 value , we could correctly predict the water droplet nucleation rate . to study the effect of an acoustic wave on crystal nucleation , we employed the size - independent @xmath256 . here we perform a sensitivity analysis of the parameter @xmath60 including the case of @xmath111 representing the eds cluster . figures [ fig : sovertime_alllamp1 ] and [ fig : sovertime_alllamp100 ] show variation in supersaturation over time at two different pressure magnitudes and different @xmath60 values . given @xmath49 , a negative @xmath60 basically implies that the dividing surface is placed beyond the eds . the variation of supersaturation over time at different @xmath60 and pressure magnitudes is depicted in figures [ fig : sovertime_alllamp1 ] and [ fig : sovertime_alllamp100 ] . at a small static pressure magnitude , the effect of pressure on the thermodynamics and kinetics of nucleation is negligible and @xmath60 influences the kinetics through @xmath286 and @xmath287 in the first two terms of eq . [ eq : detachfreq_isotherm2 ] . we observed at low pressure magnitudes , a nucleation rate increase as @xmath60 drops whereas at high magnitude static pressure , due to the role of the last term in eq . [ eq : detachfreq_isotherm2 ] , the inverse trend was identified . this change in the nucleation rate influences the csd at different @xmath60 values . for instance , at @xmath283 and at the end of nucleation stage , we see that the mean of the csd is shifted towards a smaller @xmath5 ( figure [ fig : z_n_pac50mpa_alllam ] ) . this difference becomes more noticeable at higher pressure magnitudes . in the case of pressure fluctuation with non - zero frequency , the effect of the location of dividing surface on nucleation is more clear . inspecting eq . [ eq : detachfreq_isotherm2 ] shows that a non - eds cluster can affect the kinetics of nucleation through the values of @xmath288 and @xmath287 . for the eds cluster , @xmath286 and @xmath287 are constant and size - independent ( equal to @xmath289 and @xmath290 , respectively ) . however , for a non - eds cluster these quantities are variable and size - dependent , see figure [ fig : hprime_n ] . this influences the pressure effect on the depletion rate and nucleation rate consequently , as shown in figure [ fig : govertime_alllam ] for two different supercritical clusters . the simulation results at @xmath283 and @xmath291 , shown in figure [ fig : sovertime_alllam ] , displays a variable supersaturation over time for both non - eds cases whereas it is roughly non - oscillatory for the eds cluster . this is particular to this combination of supersaturation and pressure magnitude as we observed a fluctuating supersaturation for the case of eds clusters either at lower initial supersaturation or higher pressure magnitude . this is explained by the inverse relationship between pressure and supersaturation such that the pressure effect becomes more significant at lower supersaturations @xcite and therefore imposes variation in detachment frequency and supersaturation eventually . the gibbs formalism is often employed to determine the thermodynamics of a phase transformation . this model assumes that the new phase forms a cluster of molecules separated from the old phase by a sharp , i.e. zero volume , interface phase . for large clusters , the deviations between the core new phase modelled with continuum properties and the real structure of the new phase are physically associated with the interface phase ( and its excess free energy ) . however , this does not hold so readily for small clusters of the size of few molecules ; for instance the density of the core new phase deviates from the bulk condensed phase density . nevertheless , we have shown that the gibbs model can overcome some of these difficulties associated with the thermodynamics of small clusters if a non - eds is utilised to define a cluster . for a given cluster size , moving a dividing surface essentially modifies the size of the core new phase and its thermodynamics . furthermore , the specification of the dividing surface influences the excess helmholtz free energy of the interface phase , given by @xmath90 , and consequently the effective surface tension : see eq . [ eq : gammaeff ] . the dividing surface is the unphysical element of the model and its corresponding surface tension is defined to make the free energy of the interface phase independent of the location of the dividing surface.@xcite derivations in the paper are valid for any dividing surface , including the eds and surface of tension , and their associated size - dependent surface tension @xmath292 . computation of the excess free energy of the surface @xmath293 requires the knowledge about the size of interface phase @xmath13 and the surface tension . equation [ eq : nsigma ] is developed to calculate the size of the core new phase and the interface phase for any location of a dividing surface relative to the conventional eds . selecting @xmath292 requires a suitable model of the size - dependent surface tension but many of those available models often break down in the limit of small clusters . this issue becomes more significant in the case of sonocrystallisation process : the critical cluster size ( for a condensed new phase ) decreases as the pressure magnitude increases . therefore , we defined _ the new surface _ which is identified as follows : i ) this surface is characterised by the size - independent surface tension @xmath113 , and ii ) the surface is positioned such that we obtain a reference excess free surface energy for the clusters . this was achieved by equating @xmath106 ( obtained from eq . [ eq : gammaeff ] when setting @xmath294 ) to the effective surface tension obtained from statistical mechanical simulations and solving for the parameter @xmath60 . we showed that even a size - independent @xmath60 and associated non - eds clusters can reasonably well reproduce the excess free energy of different cluster sizes obtained from statistical mechanical simulations and successfully predict the kinetics of water droplet formation . in addition , the effect of pressure variation on the cluster formation kinetics was studied . we demonstrated that this effect is cluster size - dependent . this is introduced by the term @xmath295 in eq . [ eq : detachfreq_cont2 ] which is illustrated in figure [ fig : hprime_n ] as well . in contrast to eds clusters used in the cnt , the effect of pressure on the work of cluster formation and consequently the detachment rate varies with the size of non - eds clusters ( the work and detachment rate can be decreased or increased depending on the sign of @xmath60 ) and tends towards the predictions of the cnt in the limit of large clusters . for an eds cluster we have @xmath296 and therefore @xmath297 which becomes negligible if the difference in molecular density of the old and new phases is small . this impairs the effect of pressure on the thermodynamics of phase transformation . in contrast , a non - eds cluster gives a non - unity @xmath287 , especially for a small cluster size , see figure [ fig : hprime_n ] , and the thermodynamic effect of a pressure variation can be more substantial . additionally , the size of a condensed critical cluster inversely correlates to the pressure magnitude , see dashed black curves in figures [ fig : zsn_tau_n_pac1mpa]-[fig : zsn_tau_n_pac100mpa ] . this together with the size - dependence of @xmath287 may explain some sonocrystallisation experimental observations revealing the improvement in a nucleation rate for a scenario with @xmath298 @xcite while the the conventional form of the cnt which uses eds clusters is incapable of doing so . with regard to the effect of pressure oscillation on a phase transformation , especially when the new phase is condensed and incompressible , pressure fluctuation can in general enhance or diminish the nucleation rate of the new phase by changing the nucleation barrier . equations [ eq : workdiffcondfinal2 ] and [ eq : nucworkdiff3 ] demonstrate the effect of pressure on the work of non - critical and critical cluster formation , respectively . we can see that for a denser new phase , pressure elevation reduces the nucleation barrier and consequently favours nucleation kinetics whereas pressure reduction increases the nucleation barrier and consequently lessens the probability of nucleation . the inverse trend happens for a new phase with @xmath299 . as a result , the nucleation rate goes up in a half cycle of acoustic waves but diminishes in the other half cycle . this is due to variation in detachment frequency with pressure oscillation as shown in figure [ fig : govertime_alllam ] for a crystallisation process . in a half cycle , the detachment frequency is lower than the attachment frequency which leads to cluster growth whereas in the other half cycle detachment frequency becomes larger than attachment frequency which promotes the decay of the cluster . this leads to a time variable nucleation rate which alters the size distribution of supercritical clusters . the effect of variation in the static pressure on nucleation kinetics is binary , either enhancement or attenuation , however the acoustic wave produces both effects over a cycle . a precise experiment on the nucleation of solid helium from liquid helium conducted by chavanne et . @xcite illustrated a similar observation over a cycle of acoustic irradiation by a hemi - spherical focused ultrasound transducer . the thermodynamics and kinetics of a phase transformation in a closed system exposed to an acoustic field and governed only by the aggregative mechanism has been investigated . adding the non - aggregative effect of an acoustic wave into the developments made in this work and solving the coupled problem will be the subject of a forthcoming paper . this work was supported by the epsrc [ grant number ep / i031480/1 ] . the number of molecules in the interface phase of a cluster with an arbitrary shape can be determined by eq . [ eq : interfacenofinal ] . for a spherical cluster this equation reads @xmath300 where @xmath301 is the radius of a cluster defined by the equimolar surface . considering @xmath302 , it follows @xmath303 substituting @xmath304 where @xmath305 is the average intermolecular distance in the bulk of new phase , in the above equation yields @xmath306 where @xmath307 and @xmath308 . we can write @xmath309 as follows @xmath310 where @xmath311 and @xmath312 are dimensionless quantities . for a cubic cluster with the length of @xmath313 , we have @xmath314 . plugging this in eq . [ eq : interfacenofinal ] and after some algebra we arrive in @xmath315 where @xmath316 and @xmath317 is the shape factor and @xmath308 . therefore we can use the formula in the form of eq . [ eq : interfacenocube ] for both spherical and cubic clusters with shape factors of @xmath318 and @xmath317 respectively . having determined the number of excess molecules and utilising eq . [ eq : nexcess ] , the number of molecules in the new phase ( core ) is obtained by @xmath319 therefore , in general we can write @xmath320 where @xmath321 . if @xmath322 , i.e. @xmath323 , the number of molecules in the interface and new phase can be approximated with second order error ( @xmath324 ) as follows @xmath325 these equations give @xmath12 and @xmath13 as a function of @xmath44 which is the size of the eds - defined cluster . we are , however , interested in determining these quantities and the size of cluster as the function of either the core size or the number of molecules in the interface . in this regard , we start with eq . [ eq : coreno ] and solve it for @xmath44 while employing @xmath308 as follows @xmath326 which gives @xmath327 by substituting this relationship in eq . [ eq : interfacenocube ] , we obtain @xmath13 and the cluster size in the following format @xmath328 where @xmath329 . for the case of a condensed new phase , if the dividing surface is placed beyond the eds ; this gives @xmath330 and subsequently @xmath331 . on the other hand , if the surface is enclosed in the eds , we have @xmath332 . finally , we need to determine the derivatives of @xmath12 and @xmath13 with respect to @xmath5 as they are required in eq . [ eq : work9 ] . using the last two equations , we have @xmath333 \dfrac{d n_n}{d n},\end{aligned}\ ] ] furthermore @xmath334 which gives @xmath335 eventually plugging eq . [ eq : diffnn1 ] into eq . [ eq : diffnsigma ] gives @xmath336 the gibbs - duhem equation when temperature and composition for the new phase are kept constant is as follows : @xcite @xmath337 . integrating both sides given that the new phase is condensed , we arrive at @xmath338 this equation can be rearranged as @xmath147 @xmath339 . differentiating both sides of this equation while the partial molar volume is kept constant yields : @xmath340 . the gibbs - duhem relationship for the new phase also reads : @xmath341 . combining the last two equations gives the differential form of the chemical potential of the new phase at pressure @xmath10 of old phase which was used in the text . in addition , evaluating eq . [ eq : chempotdiff ] at the pressure of a critical cluster @xmath342 gives : @xmath343 the equilibrium condition for a critical cluster yields @xmath344 . substituting this in eq . [ eq : chempotdiffcritical ] and using eq . [ eq : chempotdiff ] gives @xmath345 in the discrete representation of the cluster formation work , eq . [ eq : detachfreq_disc ] is used to determine detachment frequency . so , @xmath346 should be determined where @xmath347 denotes the dependency of work on all other parameters , e.g. pressure , temperature and composition , which is omitted here to avoid long relations . the two terms of the integrand are obtained with the aid of eq . [ eq : workdiffcondfinal ] . the integrand then becomes this is the same as @xmath360 in eq . [ eq : diffnn1 ] and [ eq : work9 ] . plugging @xmath361 into eq . [ eq : nclusterdisc3 ] we calculate @xmath362 which reads the same as @xmath160 in eq . [ eq : diffnsigma1 ] and [ eq : work9 ] . performing the integration gives exactly the same results already achieved for the case of variation in work as a function of continuous @xmath5 shown in eq . [ eq : work9 ] . considering the second order binomial truncation is applied , then all the equations derived previously to calculate detachment frequency are valid and can be used for the discrete representation of @xmath5 , too . 56ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop * * , ( ) @noop * * , ( ) in @noop _ _ ( , ) pp . @noop * * , ( ) @noop _ _ ( , ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop _ _ ( , ) @noop ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop _ _ ( , ) @noop _ _ , structure of matter series ( , ) @noop _ _ , wiley international edition ( , ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop ( ) @noop _ _ , advances in theoretical chemistry ( , ) @noop _ _ , lecture notes in physics ( , ) @noop _ _ ( , ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) in @noop _ _ , vol . ( , ) pp . @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( )
|
a phase transformation in a metastable phase can be affected when it is subjected to a high intensity ultrasound wave . in this study we determined the effect of oscillation in pressure and temperature on a phase transformation using the gibbs droplet model in a generic format .
the developed model is valid for both equilibrium and non - equilibrium clusters formed through a stationary or non - stationary process .
we validated the underlying model by comparing the predicted kinetics of water droplet formation from the gas phase against experimental data in the absence of ultrasound .
our results demonstrated better agreement with experimental data in comparison with classical nucleation theory .
then , we determined the thermodynamics and kinetics of nucleation and the early stage of growth of clusters in an isothermal sonocrystallisation process .
this new contribution shows that the effect of pressure on the kinetics of nucleation is cluster size - dependent in contrast to classical nucleation theory .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the experimental discovery of persistent currents in mesoscopic rings pierced by a magnetic flux,@xmath5 earlier proposed theoretically,@xmath6 has revealed interesting new effects . the currents measured in metallic and semiconducting rings , either in a single ring or an array of many rings , generally exhibit an unexpectedly large amplitude , i.e. , larger by at least one order of magnitude , than predicted by theoretical studies of electron models with either disorder or electron - electron interaction treated perturbatively.@xmath7 it has been suggested that the interactions and their interplay with disorder are possibly responsible for the large currents observed , expecting that the effect of the interactions could counteract the disorder effect . however , no consensus has yet been reached on the role of the interactions . in order to gain theoretical insight , it is desirable to perform numerical calculations which allow to consider both interactions and disorder directly in systems with sizes varying from small to large . analytical calculations usually involve approximations which mainly provide the leading behavior of the properties for large system sizes . persistent currents in mesoscopic rings strongly depend on the system size , since they emerge from the coherence of the electrons across the entire system . hence , it is most important to study the size dependence of the current beyond leading order in microscopic models , for a complete understanding of the experimental results . exact diagonalization was used to calculate persistent currents in systems with very few lattice sites.@xmath8 in this work , we use the density matrix renormalization group ( dmrg ) algorithm,@xmath9 to study a simplified model incorporating interactions and a single impurity , accounting for disorder , in larger system sizes . we consider a system of interacting spinless electrons on a one - dimensional ring , with a single impurity , and penetrated by a magnetic field . we study an intermediate range of system sizes , where analytical results obtained by bosonization techniques for large system sizes , do not yet fully apply . without impurity , and at half - filling , the system undergoes a metal - insulator transition from a luttinger liquid ( ll)@xmath10 to a charge density wave ( cdw)@xmath11 groundstate . the persistent current of the interacting system with an impurity was studied before with the dmrg , in the ll phase.@xmath12 here we study the persistent current , and also the drude weight characterizing the conducting properties of the system , in both the ll and the cdw phase , investigating the interplay between the impurity and the interactions in the two phases . in mesoscopic systems the separation between metallic and insulating behavior is not always obvious , since the localization length can be of the order or significantly larger than the system size . hence , a finite drude weight and a current can be observed in the cdw phase of a mesoscopic system . it is therefore of great interest to characterize the persistent current and the drude weight in both the ll and the cdw phases of mesoscopic systems . although the simple model that we consider is not the most appropriate to describe the experimental situation , we hope to obtain useful information for the understanding of the more realistic systems . under a jordan - wigner transformation,@xmath13 the system considered is equivalent to a spin-1/2 xxz chain with a weakened exchange coupling . hence , our results also provide insight into the spin transport in this type of systems . the hamiltonian describing a system of spinless fermions on a ring pierced by a magnetic flux , with repulsive interactions and a single hopping impurity , or defect , is given by , @xmath14 where @xmath15 is the hopping term , @xmath16 contains the magnetic flux @xmath17 in units of the flux quantum @xmath18 , @xmath2 measures the strength of the defect with values between @xmath19 and @xmath20 , ( @xmath21 corresponding to the defectless case ) , and @xmath22 is the interaction term , with @xmath23 representing the nearest neighbor coulomb repulsion , and @xmath24 , where @xmath25 and @xmath26 are the spinless fermion operators acting on the site @xmath27 of the ring . we consider a system of @xmath3 sites , with @xmath3 even , and at half - filling , when @xmath28 particles are present . the lattice constant is set to one and periodic boundary conditions , @xmath29 , are used . via the gauge transformation @xmath30 , the flux can be removed from the hamiltonian , but in the impurity term where the flux is trapped , and the quantum phase @xmath31 is encoded in a twisted boundary condition @xmath32 . it is then clear that the energy is periodic in @xmath31 with period @xmath33 , i.e. , it is periodic in the flux @xmath17 threading the ring with period @xmath34.@xmath35 after a jordan - wigner transformation , eqs . ( 2 ) and ( 3 ) can be rewritten , respectively , as @xmath36 and @xmath37with @xmath38 and @xmath39 , and the boundary conditions @xmath40 and @xmath41 . hence , the model ( 1 ) of spinless fermions is equivalent to a spin-1/2 xxz chain with a weakened exchange coupling , and twisted boundary conditions in the transverse direction . the half - filled case corresponds to total spin projection @xmath42 . the persistent current generated on a ring pierced by a magnetic flux , at temperature @xmath43 , can be obtained from the ground state energy @xmath44 , by taking the derivative with respect to @xmath45 , @xmath46 for the spinless fermion system , eqs . ( 2 ) and ( 3 ) , @xmath47 corresponds to the ground state value of the charge current operator @xmath48 , while for the xxz chain , eqs . ( 4 ) and ( 5 ) , it corresponds to the ground state value of the spin current operator @xmath49 . as a consequence of the periodicity of the energy , the current is also periodic in @xmath31 , with period @xmath33 . hence , it can be expressed as a fourier series , @xmath50 and the behavior of the current can be analyzed in terms of the coefficients @xmath51.@xmath12 in the noninteracting case ( @xmath52 ) , it has been found that for large system sizes , the current is invariant under the defect transformation @xmath1,@xmath53 i.e. , @xmath54 we shall investigate the existence of this kind of invariance in the interacting case ( @xmath23 ) , both in the ll and the cdw phases . the drude weight was proposed by kohn as a relevant quantity to distinguish between a metal and an insulator.@xmath55 it is defined as @xmath56 where @xmath57 is the location of the minimum of @xmath58 , which depends on the parity of the number of electrons , i.e. , @xmath59 or @xmath60 for , respectively , an odd or an even number of electrons . for the spinless fermion system @xmath61 represents the charge - stiffness and measures the inverse of the effective mass of the charge carriers.@xmath62 in a metallic conductor @xmath61 tends to a finite value whereas in an insulator @xmath61 vanishes with the system size @xmath3 , when @xmath4 . in the insulating state the drude weight decays as @xmath63 , where @xmath64 measures the localization length . for the xxz chain the drude weight represents the spin - stiffness . in a model of free fermions ( @xmath52 ) with no impurity ( @xmath21 ) , it is straightforward to see that the leading behavior of the persistent current @xmath47 in the system size @xmath3 , has a saw - tooth like shape with slope @xmath65 , where @xmath66 is the fermi velocity . thus , the amplitude of the current scales with @xmath67 , vanishing in the limit @xmath68 . the discontinuity in @xmath47 , that results from a degeneracy of energy levels associated to the translation symmetry,@xmath69 appears at @xmath70 or @xmath60 for , respectively , an even or an odd number of electrons . in the presence of an impurity ( @xmath71 ) , bosonization@xmath72 and conformal field theory@xmath73 calculations predict that the shape of the current @xmath47 is rounded off , and its amplitude decreases with increasing strength of the scatterer potential , still vanishing with the system size as @xmath67 . the impurity lifts the degeneracy of the energy levels and the current then varies continuously.@xmath69 the model with interactions ( @xmath74 ) and without defect ( @xmath21 ) , is solvable by the bethe ansatz for periodic boundary conditions ( @xmath70)@xmath75 and also for twisted boundary conditions ( @xmath76).@xmath77 at half - filling , the system exhibits a metal - insulator transition , which occurs at @xmath78 . for @xmath79 , the system is in a gapless ll phase , while for @xmath80 it is in a gapped cdw state.@xmath81 the ll phase is characterized by a power - law decay of the correlations . bosonization predicts that in an homogeneous ll , the leading behavior of the persistent current in the system size @xmath3 , has a saw tooth like shape with slope @xmath82 , where @xmath83 is the velocity of current excitations.@xmath84 since translation invariance is preserved in the presence of interactions , the discontinuity in the current still exists for finite @xmath0.@xmath69 a bethe ansatz calculation shows that the drude weight of an homogeneous ll in the thermodynamic limit , has a finite value , which decreases with increasing strength of the interaction @xmath85.@xmath62 the ll state is strongly affected by the presence of an impurity,@xmath86 and bosonization yields that the current then vanishes as @xmath87 , with @xmath88.@xmath72 the study of the ll phase with @xmath71 , performed with the dmrg,@xmath12 has in fact found this kind of behavior . the cdw phase is characterized by a localization length @xmath64 , which is associated to the energy gap . from the work of baxter,@xmath89 the drude weight in the gapped phase , is expected to behave as @xmath63 , vanishing for an infinite system size . this behavior implies that although the system is insulating in the infinite system size limit , for a finite system , provided @xmath90 is not too large , @xmath61 is still finite and a current can be observed . the localization length can then be extracted from the size dependence of the drude weight . we use the dmrg to numerically calculate the groundstate energy of the spinless fermion system as a function of the magnetic flux , @xmath58 , for fixed interaction @xmath0 and impurity strength @xmath2 , in rings up to @xmath91 sites , keeping up to @xmath92 density matrix eigenstates per block.@xmath93 the dmrg is applied to the hamiltonian ( 1 ) after performing the gauge transformation , which removes the flux into a twisted boundary condition.@xmath94 the states of the system are characterized by the quantum numbers associated to the eigenvalues of the local occupation number @xmath95 and the total number of particles @xmath96 operators , which commute with the hamiltonian ( 1).@xmath97 for each set of @xmath3 , @xmath2 and @xmath0 , we obtained the groundstate energy @xmath98 for @xmath99 values of @xmath31 in the periodicity interval @xmath100 , and using chebyshev interpolation@xmath101 we determined the corresponding current ( 6 ) and drude weight ( 9 ) , by numerical differentiation . we developed a dmrg algorithm for complex hamiltonian matrices , which allowed to calculate the detailed form of the persistent current @xmath102 as a function of @xmath31,@xmath12 and to obtain the drude weight . in a previous approach the dmrg was used to calculate the so called phase sensitivity @xmath103 , which is the difference of the groundstate energy at flux @xmath70 and @xmath60 , and can be considered a crude measure of the persistent current.@xmath104 although the calculation of @xmath103 requires considerably less computational effort than the calculation of @xmath105 , because then the hamiltonian matrix is real , the phase sensitivity does not provide information on the shape of the current and the value of the drude weight . we now present the results obtained for the persistent current and the drude weight , where we take @xmath106 and the interaction @xmath0 is in units of @xmath107 . 1 and 2 , exhibit @xmath108 plotted versus @xmath31 , for respectively , @xmath109 and @xmath110 , which correspond , respectively , to the ll and the cdw phase , considering different impurity strengths @xmath2 , on a fixed system size @xmath111 . we can see that the effect of the impurities , in both phases , is to reduce the intensity of the current , and to round off the shape of @xmath47 . the amplitude of the current decreases rapidly with increasing values of @xmath112 , and also with increasing strength of the interaction @xmath0 . 3 shows that the invariance of the current with respect to the defect , described in eq . ( 8) , found for large system sizes in the noninteracting case , is also observed for the interacting case , both in the ll and in the cdw phases , the system size @xmath3 required to reach the invariance being larger for larger interaction @xmath0 . 4 displays the drude weights associated to the systems with different interactions @xmath0 and impurities @xmath2 , fixed @xmath111 , of the currents presented in figs . 1 and 2 . as one would expect the drude weight decreases with increasing @xmath112 , and also increasing @xmath0 . figs . 5 and 6 , present @xmath108 plotted for several system sizes @xmath3 , respectively , for @xmath109 , in the ll phase , and @xmath110 , in the cdw phase , with @xmath2 fixed at @xmath113 . we observe that the current vanishes faster than @xmath67 in both phases , exhibiting a different behavior in each phase , @xmath47 vanishing much faster with @xmath3 in the cdw than in the ll phase . in order to analyze the behavior of the current in more detail we have numerically evaluated the coefficients @xmath51 ( for @xmath114 ) of the fourier expansion ( 7 ) . the first ( @xmath115 ) and the second ( @xmath116 ) fourier coefficients of the current , for @xmath109 and @xmath110 , are shown in fig . one can clearly see that the coefficients @xmath117 and @xmath118 behave similar to each other in both phases . however , their behavior in the ll and cdw phase is distinct . in the ll phase , the fourier coefficients show a power - law decay with @xmath3 , with the second order coefficient decaying faster , i.e. , with a larger exponent , than the first one . in the cdw phase , the fourier coefficients show a dominant exponential decay with @xmath3 , with the second order coefficient also decaying faster , i.e. , with a smaller localization length , than the first one . we observe that for longer rings , stronger interactions and also stronger impurities , the current is increasingly more precisely described by its first fourier component . in the ll phase , this in fact corresponds to the asymptotic behavior predicted by bosonization in the large @xmath3 limit , that is @xmath119 . however , for the system sizes considered here this asymptotic regime is not reached and the current displays a more complex behavior . the current is composed of a few fourier components with decreasing weight . 8 presents the first fourier coefficient of the current for different values of the interaction , @xmath109 , @xmath120 , @xmath121 , fixed @xmath122 , from which we extract the dependence of @xmath117 on @xmath3 , in the intermediate range of sizes considered . we observe that for @xmath109 , in the ll phase , the first coefficient of the current varies as @xmath123 @xmath124 , with @xmath125 , while for @xmath126 and @xmath110 , in the cdw phase , it varies as @xmath127 , respectively , with @xmath128 , @xmath129 , and @xmath130 , @xmath131 . the exponent @xmath132 is given by the slope of the straight line in fig . 8.a , and the length @xmath133 and the exponent @xmath134 were carefully adjusted in order to obtain the best collapse of the data in fig.8.b , on a plot of @xmath135 vs @xmath136 . the drude weights characterizing the systems with different interactions @xmath0 , fixed @xmath122 , are presented in fig 10 clearly shows that the results obtained for the drude weight confirm the conducting behavior shown by the first coefficient of the currents in fig.8 . we observe that , for @xmath109 the drude weight varies with the system size as @xmath137 , with @xmath138 , while for @xmath126 and @xmath110 it varies as @xmath139 , respectively , with @xmath140 , @xmath141 and @xmath142 , @xmath143 . the exponent @xmath144 is given by the slope of the straight line in fig . 10.a , and the localization length @xmath64 and the exponent @xmath145 were carefully adjusted in order to obtain the best collapse of the data in fig.10.b , on a plot of @xmath146 vs @xmath90 . the exponents and localization lengths characterizing the drude weight are a little different from those characterizing the first fourier component of the current , as one would expect , since the drude weight contains the contribution from the various fourier components . one sees that the localization length @xmath64 and the exponent @xmath145 decrease with increasing strength of the interaction @xmath0 . also , concerning the impurity influence , fig . 4 implies that the exponent @xmath144 in the ll phase increases , and the localization length @xmath64 in the cdw phase decreases , with increasing @xmath147 . as mentioned , in the large @xmath3 limit the current is expected to behave as its first fourier component , which in the ll phase implies that the exponent @xmath132 should be identified with @xmath148 as calculated from bosonization,@xmath72 where @xmath149 is the ll parameter , calculable from the bethe ansatz.@xmath150 for @xmath109 this leads to @xmath151 , which is much larger than our value of @xmath132 . we should note that the size dependence found for the first fourier component of the current , and the drude weight , characterizes the behavior of an intermediate and limited range of system sizes . if one would consider a larger range of systems , in the ll phase , one would most probably see the data for the larger @xmath3 bending down , crossing to an asymptotic power - law behaviour with the exponent approaching @xmath152 . this was observed in ref . @xmath153 , where the behavior of the first few fourier components of the current in the ll phase was discussed in detail , with data taken for larger values of @xmath3 and stronger interaction and impurity strengths . also , in the cdw phase we consider systems in an intermediate regime where the localization length is larger or near the system size.@xmath154 for larger systems , the power factors that occur in the first fourier component of the current and the drude weight may decline,@xmath155 possibly leaving a pure exponential behavior in the asymptotic regime . from the results obtained , we observe that the system with @xmath109 and @xmath122 , is characterized by an exponent @xmath156 , which is generated by the interplay of the electron interaction with the impurity , and @xmath61 exhibits then a power - law decay with @xmath3 , which implies vanishing in the limit @xmath4 . on the other hand , the systems with @xmath126 and @xmath110 , fixed @xmath122 , are characterized by a localization length @xmath64 , which decreases with increasing interaction and impurity strength , and @xmath61 exhibits now an exponential decay with @xmath3 , also vanishing as @xmath4 . hence , we find that both in the ll and the cdw phases , with an impurity in the system , the effect of the interaction is to decrease the current and the drude weight . as referred before , our results also provide insight into the spin transport in a spin-1/2 xxz chain with a weak link , and a similar behavior to the one above is implied for the spin current and stiffness . so , in the gapless xy phase , the spin stiffness decays with a power - law , vanishing in the limit @xmath4 , while in the gapped ising phase , it decays exponentially , also vanishing as @xmath4 . comparing our results for the persistent current in the ll phase , with those obtained in ref . @xmath153 , we have similarly found that the current vanishes faster than @xmath67 . one observes that the model parameters strongly influence when the last asymptotic regime described by bosonization is reached . a calculation of the finite - size corrections to the spin stiffness in a pure spin-1/2 xxz chain,@xmath157 has revealed a size dependence in the gapped phase that has a similar form to the one found here . the result that the drude weight in a ring in the gapless phase with an impurity drops to zero , is in agreement with a previous result obtained for a spin chain,@xmath158 and with renormalization group arguments , which state that the impurity term is relevant leading to a transmission cut.@xmath159 the renormalization group studies find that either a weak barrier or a weak link lead to an insulating state for repulsive interactions , while in the noninteracting case those are marginal perturbations . in turn , our work shows that there is an invariance of the current under the defect transformation @xmath160 in the interacting system , as for the noninteracting system , and that implies that a strong link will also reduce the current and the drude weight . the observation that with an impurity in the system , the interaction always leads to an additional decrease of the current and the drude weight is in agreement with previous results by other authors,@xmath8 and can be understood as it is more difficult to move correlated electrons in a scattering potential than independent electrons . we have studied the behavior of the persistent current and the drude weight on a mesoscopic ring pierced by a magnetic flux . we considered a model of spinless fermions with repulsive interactions and a hopping impurity , which is also equivalent to a spin-1/2 xxz chain with a weakened exchange coupling . using a powerful numerical method , the dmrg with complex fields , we have calculated the detailed form of the current as a function of the magnetic flux , which enabled us to investigate the corrections to the large system - size limit , and also allowed to obtain the drude weight . we show that the system at half - filling , changes from an algebraic to an exponential behavior as the interaction increases , corresponding to a change from a ll to a cdw phase . we find that the analytical predictions of bosonization for the ll phase , are not yet fully observed in the intermediate range of system sizes considered . in addition we observe that the invariance of the current under the defect transformation @xmath160 , seen in the noninteracting system , is also verified in the interacting system , in both phases . hence , an isolated strong link is not only useless for increasing the persistent current ( as might have been expected ) , but it rather destroys coherence and reduces the current . the behavior determined for the current is consistent with the behaviour determined for the drude weight , the ll phase being characterized by an exponent @xmath156 , which results from the interplay of the interactions with the impurity , while the cdw phase is characterized by a localization length @xmath64 , which decreases with increasing interaction and impurity strength . we find that , both in the ll and the cdw phase , with a defect in the system the interactions always suppress the current , and the drude weight drops to zero in the limit @xmath4 . away from half - filling there is no metal - insulator transition in the pure case , and the system is always metallic . hence , one does not expect to observe then a change in the current and the drude weight from an algebraic to an exponential decay . nevertheless , one still expects to observe that in the system with an impurity , the current and the drude weight decrease with increasing impurity and interaction strengthes . therefore , within the model considered , the interactions can not explain the results observed in the experiments . @xmath81 in the xxz chain , a transition occurrs at @xmath176 , from an xy model , for @xmath177 , to an ising model , for @xmath178 , @xmath176 corresponding to the heisenberg model . a gap opens up in the spin - excitation spectrum , that is equivalent to the gap in the charge - excitation spectrum of the spinless fermion system . @xmath183the number @xmath184 of eigenstates required to achieve convergence , of 1 part in @xmath185 , for the ground state energy @xmath186 , increases with the system size , e.g. , @xmath187 for @xmath188 and @xmath189 for @xmath190 . @xmath202 for different @xmath0 and @xmath122 . ( a ) @xmath203 vs @xmath201 , for @xmath109 , the line represents @xmath204 . ( b ) @xmath205 vs @xmath3 , for @xmath126 and @xmath121 , the lines represent @xmath127 , with @xmath133 and @xmath134 dependent on @xmath0 . @xmath61 for different @xmath0 and @xmath122 . ( a ) @xmath206 vs @xmath201 , for @xmath109 , the line represents @xmath137 ; ( b ) @xmath207 vs @xmath3 , for @xmath126 and @xmath121 , the lines represent @xmath139 , with @xmath64 and @xmath145 dependent on @xmath0 .
|
we study the persistent current and the drude weight of a system of spinless fermions , with repulsive interactions and a hopping impurity , on a mesoscopic ring pierced by a magnetic flux , using a density matrix renormalization group algorithm for complex fields .
both the luttinger liquid ( ll ) and the charge density wave ( cdw ) phases of the system are considered . under a jordan - wigner transformation ,
the system is equivalent to a spin-1/2 xxz chain with a weakened exchange coupling .
we find that the persistent current changes from an algebraic to an exponential decay with the system size , as the system crosses from the ll to the cdw phase with increasing interaction @xmath0 .
we also find that in the interacting system the persistent current is invariant under the impurity transformation @xmath1 , for large system sizes , where @xmath2 is the defect strength .
the persistent current exhibits a decay that is in agreement with the behavior obtained for the drude weight .
we find that in the ll phase the drude weight decreases algebraically with the number of lattice sites @xmath3 , due to the interplay of the electron interaction with the impurity , while in the cdw phase it decreases exponentially , defining a localization length which decreases with increasing interaction and impurity strength .
our results show that the impurity and the interactions always decrease the persistent current , and imply that the drude weight vanishes in the limit @xmath4 , in both phases .
pacs : 71.10.pm , 73.23.ra , 73.63.-b
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
light - induced 11-_cis _ @xmath1 all-_trans _ isomerization in retinal is a paradigmatic example of an important ultrafast photochemical reaction in biology @xcite . the photoreaction is the first step in dim - light vision , and its high quantum yield , and formation of all-_trans _ product within 200 fs , contribute to the high efficiency of the phototransduction cascade @xcite . the importance of retinal photoisomerization has made its mechanism the subject of numerous experimental and computational studies . a related biological process , _ cis - trans _ isomerization in retinoic acid , has become of increasing interest in its role in zebrafish hindbrain development@xcite . to study retinal dynamics , current pump - probe experiments use ultrashort femtosecond laser pulses to excite the retinal from the ground ( @xmath2 to the first excited ( @xmath3 ) electronic state . the subsequent isomerization dynamics is then followed using a series of probe pulses , providing important insights into the sub-200 fs timescale of photoproduct formation @xcite , the coherent dynamics of all-_trans _ photoproduct @xcite , and most recently , the role of conical intersections in determining the reaction rate @xcite . significantly , retinal photoisomerization occurs with high quantum yield ( @xmath4 ) , making the first step in visual phototransduction highly efficient @xcite . measurements of the quantum yield @xmath5 as a function of excitation wavelength @xmath6 show a maximum at 500 nm and a small decline of @xmath75% as the laser wavelength increases from 500 nm to 570 nm @xcite . a detailed time - dependent computational study of retinal photoisomerization using a multilevel redfield theory @xcite based on a minimal two - state , two - mode ( 2d ) model of retinal chromophore@xcite provided insights into the role of avoided crossings , conical intersections , and dissipative dynamics in ultrafast energy conversion in biomolecules @xcite . in addition , they qualitatively reproduce many salient features of isomerization dynamics of retinal in rhodopsin , including the high quantum yield . there and elsewhere the quantum yield is defined dynamically , as the probability of forming the all-_trans _ product starting from the initial _ cis _ wavepacket created by an ultrafast franck - condon excitation from the ground _ cis _ state . this time - dependent wavepacket view @xcite is well - suited to describe time - domain experiments on rhodopsin , in which ultrafast @xmath8 excitation creates a wavepacket on a highly excited state . however , in nature , the excitation involves natural light . such light has a very short coherence time ( 1.32 fs for sunlight ) compared to the fs laser pulses used experimentally @xcite , and is incident on the molecule for a far longer time . as a result , after some time , which is dependent on a number of conditions@xcite , the molecule is prepared in a mixture of stationary states that exhibits no coherent time evolution@xcite . furthermore , these stationary states are , from a zeroth - order viewpoint , a linear superposition of _ cis _ and _ trans _ configurations . the existence and some unusual properties ( such as microsecond lifetimes ) of stationary eigenstates of this kind in large polyatomic molecules were examined experimentally as early as 1977 @xcite . the questions then arise as to how to extract and understand the quantum yield in the case of incoherent light excitation , and how to describe the _ cis - trans _ isomerization of retinal from a stationary eigenstate viewpoint that is appropriate for vision under natural light conditions . a substantial step towards this major goal is provided in this paper . here , we address this issue by first generalizing the concept of the quantum yield to the case of incoherent light excitation of a polyatomic molecule . we note that in the limit of rapid decoherence of initial coherences in the reduced density matrix , the photoreaction quantum yield is stationary . this definition is then applied to calculate the quantum yield of _ cis - trans _ photoisomerization of retinal using a minimal two - state , two - mode model@xcite . we find excellent overall agreement between our calculated quantum yield and the previous time - dependent result of stock and hahn @xcite . in addition , our calculations , in both the presence and absence of relaxation , agree with the observed value of the quantum yield at 500 nm and qualitatively reproduce its observed decline with increasing excitation wavelength @xcite . these findings demonstrate the role of localization properties of the stationary states in determining the quantum yields , a crucial feature in the process induced with natural light . note that there are various definitions of the quantum yield that are adopted in time - dependent studies . below , we consistently utilize the parameters and approach of ref . , which has become something of a `` standard model '' for basic retinal dynamics . this imposes a number of consistency requirements that are discussed in sect . [ summary ] . before proceeding further , we emphasize that incoherent excitation of biomolecules embedded in condensed - phase environments involves two processes that occur simultaneously : ( a ) the creation of the stationary states by the incoherent light , and ( b ) the relaxation between the stationary states caused by the interaction with the environment . in this work , we treat these processes sequentially , assuming that the stationary states are formed first and then relax due to the system - environment coupling . this allows for a better assessment of their individual roles . the eigenstates formed on stage ( a ) are considered in secs . ii and iii ; the effects of relaxation are considered in sec . iv and , significantly , shown to have a negligible effect on the quantum yield of retinal photoisomerization treated within in the two - state , two - mode model @xcite . the results , as seen below , motivate a future study of simultaneous incoherent excitation and relaxation in this and related model retinal systems . we extract the quantum yield for the stationary case from the standard time dependent result , hence exposing their relationship . reference defines the photoreaction quantum yield from the time - dependent view as @xmath9 where the time - dependent populations of 11-_cis _ and all-_trans _ isomers are defined as @xcite @xmath10 here @xmath11 is the reduced density matrix of the retinal subsystem , @xmath12 11-_cis _ or @xmath12 all-_trans _ , and @xmath13 are the projection operators , which divide the full range of the isomerization coordinate @xmath14 $ ] into the _ cis _ ( @xmath15 $ ] ) and _ trans _ regions ( @xmath16 $ ] ) , as illustrated in fig . 1 . in eq . ( [ populations ] ) , @xmath17 is the heaviside step function , and the operators @xmath18 project onto the ground and excited diabatic electronic states @xmath19 and @xmath20 . qualitatively , @xmath21 can be thought of as projecting onto the adiabatic ground state reactant _ cis _ region and @xmath22 onto the adiabatic ground state product _ trans_. in this definition , the adiabatic excited state populations @xmath23 and @xmath24 are neglected assuming that these populations have decayed to zero in the long - time limit . alternative definitions [ e.g. , including @xmath23 in the numerator and @xmath24 in the denominator of eq . ( [ y ] ) ] are certainly possible and will be explored in future work @xcite , and would necessitate a refitting of the potential parameters to the new definition of the time - dependent quantum yield . as an example , consider impulsive fc excitation of retinal from its ground electronic and vibrational states to the first excited electronic state @xmath25 . here the density matrix of the system at time zero is given by @xcite @xmath26 where @xmath27 is the ground state with zero quanta in the torsional and coupling modes @xmath28 and @xmath29 . it is convenient to express the density matrix in the system eigenstate basis defined by @xmath30 where @xmath31 and @xmath32 are time - independent eigenvalues and eigenfunctions . in this basis , with frequencies @xmath33 , @xmath34 . hence the population dynamics is given by @xmath35 since the time - dependence arises from the oscillatory behavior of the coherences , ( i.e. , terms with @xmath36 ) , if @xmath37 then the state populations are time independent . an important example of such a case is molecular excitation with incoherent light [ eq . ( [ p1t_trans ] ) ] , which populates , after some time , the eigenstates of @xmath38 without any subsequent coherent dynamics @xcite . similarly , environmentally induced effects cause the loss of coherences . in this case , retaining only the diagonal terms in eq . ( [ p1t_trans ] ) , we have @xmath39 this expression may be made more physically transparent by noting that the diagonal elements of the reduced density matrix following impulsive fc excitation at @xmath40 correspond to the linear absorption spectrum of the molecule described by the spectral lineshape function @xcite @xmath41 , where @xmath42 is the excitation frequency measured from the ground vibrational state of the _ cis_-isomer with energy @xmath43 ( we assume that this is the only vibrational state populated prior to excitation ) . then it follows from eq . ( [ rho_t0 ] ) that @xmath44 where @xmath45 is the transition dipole moment operator , and we set @xmath46 a.u . without loss of generality [ since the @xmath47 prefactors multiplying state populations cancel out in the expression for @xmath48 in eq . ( [ y ] ) ] . therefore , eq . ( [ pdiag ] ) may be rewritten in the form @xmath49 hence , as expected , only the eigenvalues @xmath31 that have non - negligible probability of being excited by incoherent light make a contribution to the stationary populations in eq . ( [ pi ] ) . given eq . ( [ pi ] ) , the time - dependent definition of the quantum yield ( [ y ] ) extends to the frequency domain by defining @xmath50 with the `` pre - averaged '' populations @xmath51 given by eq . ( [ pi ] ) . alternatively , we can first define a frequency - dependent _ trans _ / _ cis _ probability ratio @xmath52 where @xmath53 are the expectation values of the projection operators in eq . ( [ pi ] ) . the resulting frequency - dependent quantum yield ( [ qy2omega ] ) can be averaged with the normalized spectral line shape function to give @xmath54 it is clear that the definitions ( [ qy1 ] ) and ( [ qy2 ] ) , termed the pre - averaged " and post - averaged " quantum yield , are not equivalent , but both provide a physically meaningful measure of ( frequency - dependent ) photoreaction efficiency . specifically , the pre - averaged definition ( [ qy1 ] ) is consistent with the original definition of hahn and stock [ eq . ( [ y ] ) ] @xcite , whereas the post - averaged definition admits a clear physical interpretation as the degree of _ cis _ vs. _ trans _ character of a collection of eigenstates independently excited by incoherent light . expressions related to eqs . ( [ qy1 ] ) and ( [ qy2 ] ) were previously obtained in references and , where the long - time limits of electronic state populations were associated with the phase volumes occupied by the wavepackets evolving on the upper and lower adiabatic potential energy surfaces . equations ( [ p1t_trans ] ) , ( [ pi ] ) , ( [ qy1 ] ) , and ( [ qy2 ] ) form the central tool for the computations below . they generalize the quantum yield in eq . ( [ y ] ) to the frequency domain relevant to stationary eigenstates , and provide a theoretical basis for the computational study of photoinduced isomerization in model retinal as described below . the essential physics comes from the recognition that the established time dependent definition [ eq . ( [ y ] ) ] relies entirely on the projections onto domains of @xmath55 to define _ cis _ vs. _ trans _ configurations . indeed , each of eqs . ( [ y ] ) , ( [ qy1 ] ) and ( [ qy2 ] ) have essentially the same meaning , the fraction of population in the ground adiabatic state that is , in the long time limit , in the _ trans_-configuration , disregarding any initial population in ground - state _ cis _ well . this is achieved in eq . ( [ y ] ) by placing all population in an initial wavepacket on the excited electronic surface . equations ( [ qy1 ] ) and ( [ qy2omega ] ) achieve this by putting all population initially into a stationary mixture on the excited potential surface . what the latter two equations emphasize is that in the presence of incoherent light and decoherence , the appropriate states to consider in the long time limit are the stationary eigenstates of the hamiltonian . they , as seen below , will allow stationary state insight into the efficiency of the isomerization process . below , we apply this formulation to evaluate the quantum yield for the primary photoreaction in rhodopsin using the two - state two - mode model@xcite , which was previously applied @xcite to linear absorption @xcite , raman @xcite , and femtosecond pump - probe @xcite spectra of retinal in rhodopsin . the two - state two - mode model and its multidimensional ( 25-mode ) extension @xcite have since been used to explore quantum dynamics and coherent control of _ cis - trans _ photoisomerization of retinal chromophore in rhodopsin @xcite . we note that the model has the following advantages over the one - dimensional scenario ( e.g. ref . ): ( 1 ) the present model is two - dimensional , including a relevant bend degree of freedom ; ( 2 ) it accounts for the conical intersection between the ground and the first excited electronic states of retinal ; ( 3 ) it reproduces many salient features of the experimentally measured isomerization dynamics , including the ultrafast 200 fs timescale , the transient pump - probe spectra , and the energy storage of the photoreaction . we emphasize that the two - state two - mode model and the associated definition of the quantum yield [ eq . ( [ y ] ) ] were previously parametrized so as to reproduce various properties of the photoreaction@xcite of retinal in rhodopsin . it neglects the effects of solvation , which are known to dramatically affect the photoisomerization dynamics in solution @xcite , but does include , due to the parameter fit , some features of the interaction of the two modes with the remaining molecular background . our study is thus restricted to _ cis - trans _ photoisomerisation of retinal in rhodopsin . the wavefunctions @xmath56 in eq . ( [ pi ] ) are the eigenfunctions of the 2d model hamiltonian given by @xcite @xmath57 where @xmath58 is the kinetic energy operator , @xmath55 is the tuning mode ( or generalized reaction coordinate ) corresponding to low - frequency torsional modes , and @xmath59 is the coupling mode that corresponds to high - frequency unreactive modes . the second term on the right - hand side of eq . ( [ model ] ) is the interaction potential in a basis spanned by the diabatic electronic functions @xmath60 with @xmath61 @xcite . a plot of the adiabatic pess obtained by diagonalizing the potential energy term in eq . ( [ model ] ) is shown in fig . 1 . the model parameters @xmath62 , @xmath63 , @xmath64 , @xmath65 , and @xmath6 , chosen to reproduce the femtosecond dynamics of retinal in rhodopsin @xcite , are ( in ev ) : @xmath66 , @xmath67 , @xmath68 , @xmath69 , @xmath70 , @xmath71 , @xmath72 , and @xmath73 . a total of 900 eigenvalues and eigenvectors of @xmath38 were calculated and used to assemble the matrix elements of the projector operators and the lineshape function in eqs . ( [ qy1 ] ) and ( [ qy2 ] ) . the converged @xmath74 was in agreement with results in ref . . interestingly , we found that the spectrum could be classified as integrable , insofar as the nearest neighbor distribution of energy levels @xmath75 shows a structure that can be fit with a poisson distribution @xmath76 with the local mean spacing @xmath77 @xmath78 ( see , e.g. , ref . ) . such distributions are becoming of increasing interest due , e.g. to a recent proposal @xcite regarding chaos and transport in biological processes . the upper panel of figure 1 shows torsional profiles of the adiabatic potential energy surfaces ( pes ) obtained by diagonalizing the hamiltonian matrix ( [ model ] ) at @xmath79 . the pes profiles exhibit a conical intersection at @xmath80 and @xmath79 , clearly visible in the two - dimensional plot in the lower panel of fig 1 . the _ cis _ isomer of retinal is localized in the potential well of the lower diabatic pes ( @xmath81 ) on the left - hand side . photoexcitation by incoherent light ( represented by a green arrow ) populates a number of stationary eigenstates ( grey lines ) with mixed _ cis_-_trans _ character . the quantum yield is determined by projecting these eigenstates onto their respective _ cis _ and _ trans _ regions of configuration space as discussed above . figure 2 shows the frequency - dependent quantum yield @xmath82 given by eq . ( [ qy2omega ] ) as a function of excitation energy ( measured from the ground vibrational state of _ cis_-isomer ) . here [ from eq . ( [ qy2omega ] ) ] , pure _ cis _ states correspond to @xmath83 and pure _ trans _ states correspond to @xmath84 ; thus , the magnitude of @xmath82 reflects the _ cis _ or _ trans _ character of a particular eigenstate with energy @xmath85 . as expected , all molecular eigenstates that occur below the minimum energy of the _ trans _ well ( 11,000 @xmath78 , see fig . 2 ) have negligible quantum yields due to the absence of _ trans _ eigenstates in this low energy range . above the 11,000 @xmath78 threshold , the quantum yield is a rapidly varying irregular function of @xmath86 , reflecting the strong mixing between the _ cis _ and _ trans _ components by the full hamiltonian . such mixed _ cis - trans _ eigenstates are conceptually similar to the long - lived eigenstates of mixed singlet - triplet character observed in pentacene @xcite . interestingly , a closer inspection of fig . 2 , as shown in the figure insert , reveals the presence of purely _ cis _ ( or _ trans _ ) eigenstates with @xmath87 ( or 1 ) , which are qualitatively similar to the electronically localized eigenstates in vibronically coupled systems such as pyrazine@xcite . the effect has recently been analyzed in the context of geometric phase - induced localization @xcite . however , such `` extreme states '' may well be a result of low dimensionality of the model , as is indeed the case in pyrazine@xcite . table i lists the values of the quantum yield computed using the stationary eigenstates of the 2d model ( the corresponding @xmath88 is shown by the green line in fig . 2 ) @xcite . both pre - averaged and post - averaged results ( @xmath89 and @xmath90 ) agree extremely well with the previous time - dependent wavepacket result@xcite of @xmath91 . the fact that the stationary quantum yields agree so well with time - dependent calculations @xcite makes clear how the quantum yield is directly manifest in the stationary eigenstates . moreover , we also note agreement with the measured value @xcite of @xmath92 . the agreement with experiment is a natural consequence of the two - state two - mode model @xcite being parametrized to reproduce the measured quantum yield in time - dependent calculations . in order to explore the effect of @xmath74 on the calculated quantum yields , we replaced the 2d model lineshape function in eqs . ( [ qy1 ] ) and ( [ qy2 ] ) by the experimentally measured absorption profile of retinal in rhodopsin @xcite shown by the red line in fig . 2 . the result gives @xmath93 , in worse agreement with experiment than the value obtained with @xmath88 . this confirms that the potential surfaces are optimized for behavior in the domain shown in green in fig . 2 , but would require further work to properly represent the _ cis / trans _ branching in other energy regions . mathies and co - workers @xcite also measured @xmath94 observing a decline in the photoreaction efficiency below 500 nm . for comparison , we calculate the wavelength - resolved stationary quantum yields by dividing the entire @xmath6 interval into 10 nm - wide bins , and average eqs . ( [ qy1 ] ) and ( [ qy2 ] ) over the eigenstates with energies falling into a particular bin . as with the overall quantum yield , two related , but not identical , definitions are possible . in the first , one calculates the averaged _ cis _ and _ trans _ populations @xmath95 where @xmath96 is the center frequency corresponding to @xmath97-th bin , and define : @xmath98 as an alternative , we directly average the frequency - dependent quantum yield ( [ qy2 ] ) over the entire bin @xmath99 where @xmath100 is a normalization factor that serves to correct for the change in absorption intensity due to the varying @xmath6 . figure 3 compares the wavelength dependent quantum yield with the experimental results in the 500 - 570 nm range ( ref . ) . while the observed quantum yield declines monotonically with @xmath6 and stays constant below @xmath101 nm , our theoretical values oscillate over the entire range of @xmath6 . above 500 nm , the calculated quantum yields tend to decline with @xmath6 , in qualitative agreement with experiment ( the only exception being the value of @xmath102 at 540 nm ) . as expected from the above analysis ( see table i ) , switching lineshape functions has a dramatic effect on the calculated quantum yields . in contrast with the results presented in table i , however , using the experimental @xmath74 improves the overall agreement with experiment , particularly at @xmath103 nm . also shown in the inset of fig . 3 is the `` bare '' frequency - dependent quantum yield , @xmath104 here , the absence of the spectral lineshape function [ present in eqs . ( [ qy1 ] ) and ( [ qy2 ] ) ] reveals the variation in _ cis _ / _ trans _ character of molecular eigenstates with zero oscillator strengths , which do not contribute to the physical quantum yield ( [ y2lambda ] ) . the @xmath105 [ inset of fig . 3 ] is seen to generally decline with increasing @xmath6 . this behavior reflects the appearance of _ trans _ states at energies above 11,000 @xmath78 , and that their density in this region is smaller than that of _ cis _ states ( see fig . thus , the _ trans _ character of molecular eigenstates can be expected to decrease with decreasing @xmath86 , as is implicit in the experimental results . while the downward trend in the wavelength dependence of @xmath106 ( inset of fig . 3 ) is somewhat more pronounced than that observed for @xmath107 and @xmath102 ( fig . 3 ) , we note that the former can not be directly compared with experiment . this is because the bare quantum yield [ eq . ( [ ybare ] ) ] does not include the variation of the spectral lineshape function @xmath74 with @xmath64 . our computed time - independent definitions of the quantum yield [ eq . ( [ qy1 ] ) and ( [ qy2 ] ) ] assume that the eigenstate populations @xmath108 do not depend on time . in reality , however , the populated eigenstates undergo relaxation due to the interaction with the environment . to elucidate the effect of the relaxation on the quantum yield , we calculated the transition rates @xmath109 between eigenstates @xmath32 and @xmath110 using fermi s golden rule @xcite @xmath111 and @xmath112 where @xmath113^{-1}$ ] is the bose distribution at temperature @xmath114 k , @xmath115 is the boltzmann s constant , and @xmath116 is the spectral density of the bath modes representing low - frequency , non - reactive vibrational modes @xcite coupled to degree of freedom @xmath117 . following stock and co - workers @xcite , we adopt an ohmic spectral density @xmath118 with @xmath119 , @xmath120 ev , @xmath121 , and @xmath122 ev . the bath operators @xmath123 in eqs . ( [ relax1])-([relax2 ] ) are given by @xcite @xmath124 and @xmath125 . figure 4(a ) shows the time dependence of the expectation values @xmath126 and @xmath127 in eq . ( [ populations ] ) obtained by propagating the rate equations parametrized by the transition rates given by eqs . ( [ relax1])-([relax2 ] ) with the initial condition @xmath128 , _ i.e. _ assuming fully incoherent excitation with natural light . a substantial fraction of population at @xmath40 resides in the excited diabatic electronic states [ @xmath129 . the interaction with low - frequency bath modes leads to dissipation of the electronic and vibrational energy , manifested in the decay of the excited - state populations @xmath130 and @xmath131 . the decay is accompanied by a growth of the ground - state populations @xmath132 and @xmath133 shown by the solid lines in fig . 4(a ) . figure 4(b ) plots the time dependence of the quantum yield given by eq . ( [ y ] ) . remarkably , the quantum yield shows only a weak time dependence , with deviations from the asymptotic value of 0.62 not exceeding 3% over the time interval studied ( 0 3 ps ) . thus , while the individual _ cis _ and _ trans_-populations evolve in time , the value of the quantum yield , defined as their ratio via eq . ( [ qy1 ] ) , remains constant . analysis of the system - bath coupling matrix elements [ eq . ( [ relax1 ] ) ] shows that the matrix elements involving the torsional degree of freedom @xmath55 are small compared to those of the coupling mode @xmath59 . we can therefore expect that relaxation of the initial `` bright '' eigenstates populated by incoherent fc excitation ( see below and fig . 5 ) is driven by the interaction of the coupling mode with the bath oscillators . indeed , as shown in fig . 4(b ) , neglecting the @xmath55-component of the system - bath coupling ( dashed curve ) has little effect on the time variation of the quantum yield . the dominant role played by the coupling mode in the relaxation process is at the heart of our arguments presented below that analyze the lack of time dependence of the quantum yield . in order to gain insight into relaxation dynamics , we plot in fig . 5 the transient linear absorption spectrum of the two - state , two - mode model [ e.g. the populations @xmath134 during the various stages of the relaxation process . at time zero , an incoherent mixture of `` bright '' eigenstates is assumed to follow fc excitation from the ground state . the three dominant `` bright '' eigenstates that account for over 40% of all @xmath40 excited - state population are @xmath135 , @xmath136 , and @xmath137 , and are focused upon below . as time proceeds , the eigenstates begin to relax due to the interaction with the phonon bath . at @xmath138 fs , the population is seen to be spread over three lower - lying manifolds of states , which are separated from the initial `` bright '' manifold by a constant energy gap . the dominant eigenstates populated in the decay of the bright state @xmath135 , are shown in fig . we observe that during the course of relaxation , the population `` branches out '' into different final eigenstates until it finally settles in a steady state characterized by a stationary eigenstate distribution shown in the lowermost panel of fig . 5 . the steady state defines the asymptotic ( @xmath139 ) limit of the quantum yield , and is thus of particular importance to the theoretical description . according to fermi s golden rule ( [ relax1 ] ) , the relaxation rates are determined by ( a ) the magnitude of the system - bath coupling matrix element , and ( b ) the value of the spectral density at the transition frequency @xmath140 . the existence of multiple intermediate relaxation stages in fig . 6 is a consequence of a fixed cutoff frequency of the bath @xmath122 ev . because the ohmic spectral density function peaks at @xmath141 and decays quickly away from the maximum @xcite , relaxation to the eigenstates separated from the initial state by the energy window @xmath142 is allowed ( provided that the corresponding coupling matrix elements are non - zero ) , whereas relaxation to the eigenstates outside the energy window can not occur via single phonon - emission . however , such lower - lying eigenstates are populated via multiple phonon emission ( or relaxation cascade ) , as illustrated in fig . 6 . as for the quantum yield , relaxation influences the quantum yield through the time dependence of the populations @xmath143 which changes the eigenstates that contribute to the evolving quantity : @xmath144 therefore , in order to understand the time dependence of the quantum yield , it is necessary to follow both the time evolution of the populations @xmath143 and their localization properties , as manifest in the matrix elements of @xmath145 and @xmath146 . to examine the localization , we define the probability density @xmath147 for eigenstate @xmath32 projected onto the diabatic electronic state @xmath61 @xmath148 where @xmath149 are the diabatic components of the eigenstate @xmath32 described by the wavefunction @xmath150 where the direct - product basis @xmath151 is chosen in such a way as to diagonalize the torsional vibronic coupling hamiltonian ( [ model ] ) at @xmath152 ( no vibronic coupling ) . in this latter case , the problem reduces to that of two non - interacting electronic states , and the eigenstates ( [ expansion1 ] ) factorize into products of basis functions that depend on the @xmath55 and @xmath59 coordinates @xmath153 this expression for the system eigenstates provides a reasonable zeroth - order approximation to the true eigenstates of the vibronic coupling hamiltonian ( [ model ] ) in the energy region below the conical intersection ( @xmath154 @xmath78 ) @xcite . here , the low - lying eigenstates of the full vibronic hamiltonian are localized in their own potential wells @xcite ( as clearly observed in fig . 2 ) . however , as the energy increases above the conical intersection , the eigenstates become strongly mixed by the vibronic coupling , and delocalize over the whole configuration space . an example of such behavior is shown in fig . 7 , which shows the probability density of the bright eigenstate @xmath135 together with that of the ground state @xmath155 and the eigenstates populated in the first few relaxation stages ( green arrows in fig . the bright state has a large probability amplitude in the _ cis_-region of the first excited diabatic state , which maximizes its overlap with the ground _ cis_-state , to enhance excitation . the lower - lying states @xmath156 and @xmath157 start to experience the repulsive wall of the @xmath158 diabatic potential in the _ cis_-region of configuration space ( see fig . 1 ) . as a result , the population is completely transferred away from the repulsive region to the ground _ cis _ and excited _ trans _ regions . there is no apparent preference for either _ cis _ or _ trans _ regions during these initial relaxation stages due to the strong mixing of all degree of freedom : as shown in fig . 4(a ) , both @xmath21 and @xmath22 increase monotonically with time . after being transferred one step below below the initial state , the population continues to relax to lower - lying eigenstates while preserving its profile along the reaction coordinate . this can be attributed to the particular form of the system - bath coupling , which primarily depends on the @xmath59 coordinate and hence does not strongly alter the @xmath55-profiles of the eigenstates coupled by the bath . in particular , we verified that the amount of _ cis _ and _ trans _ character of most of the states involved in the relaxation process is approximately constant . exceptions do occur because of the strong vibronic coupling , which may occasionally change the @xmath55 distribution of some eigenstates due to the coupling mediated by the @xmath59 coordinate . however , this situation is an exception rather than the rule : as shown in fig . 8 , only two out of @xmath720 eigenstates involved in the first stage of relaxation ( @xmath159 and @xmath160 ) are dramatically different in their _ trans_-character ( as quantified by the matrix element @xmath161 ) from the other states . as a result , the quantum yield varies only weakly with time . a dramatic change in the relaxation mechanism occurs when the eigenstate energy falls below that of the conical intersection . figure 9 shows that the eigenstates to which relaxation then occurs become localized in their corresponding _ cis- _ and _ trans_- wells . the population of the initial ( delocalized ) eigenstate @xmath162 relaxes to two eigenstates , one of which ( @xmath163 ) is strongly localized in the _ trans_-well and the other ( @xmath164 ) has more probability amplitude in the _ cis_-well ( see supplementary material @xcite ) . as the population gets partitioned between the two eigenstates , the quantum yield remains unaltered again due to the @xmath55-independent nature of the system - bath coupling , which tends to conserve the angular probability density of all eigenstates involved . finally , we consider the final stages of the relaxation process depicted in fig . 9 . these represent bath - induced transitions between localized eigenstates , and can be understood by noting that the eigenstates are given to zeroth order by eq . ( [ zeroth_order ] ) . for simplicity , let us assume that the value of the quantum yield before relaxation begins is determined by a single eigenstate @xmath165 . after relaxation is over , the population is transferred to the eigenstate @xmath166 . taking the matrix elements of @xmath146 and using eq . ( [ zeroth_order ] ) , we find @xmath167 because @xmath146 does not depend on @xmath59 , and similarly @xmath168 but the system - bath coupling operator does not depend on @xmath55 , and hence can not change the number of quanta in the torsional mode @xmath169 , so the right - hand sides of eqs . ( [ loc1 ] ) and ( [ loc2 ] ) are equal . this result suggests that in the low - energy regime of localized eigenstates , the quantum yield of _ any _ process that depends on a single reaction coordinate should be time - independent , provided that the system - bath coupling does not depend on that coordinate . this is clearly illustrated in fig . 9 : the localized state @xmath163 has 3 peaks along the @xmath59 coordinate , corresponding to 2 quanta in the @xmath59-mode ( @xmath170 ) . the system - bath coupling changes @xmath171 from 2 to 1 and then from 1 to 0 , while leaving the @xmath55 distribution unchanged ( see also figs . 1 and 2 of supplementary material @xcite ) . the final eigenstate @xmath172 is vibrationally `` cold '' with respect to the coupling mode , but remains highly excited along the torsional coordinate . a few closing remarks are in order concerning the asymptotic ( @xmath139 ) value of the quantum yield that corresponds to the population distribution shown in the lowermost panel in fig . the steady state is obtained by propagating the rate equations of motion for the populations , which is a numerically efficient procedure that scales quadratically with the number of eigenstates @xcite . the nature of the steady state is determined by the form of the system - bath coupling , and the initial conditions . we emphasize that this steady - state solution does _ not _ have the form of the boltzmann distribution that would be normally expected of pauli - type rate equations in the limit @xmath139 @xcite . rather , in accord with ref . , our steady - state distribution corresponds to a metastable state , which will eventually tunnel to the _ cis_-well ( at least if the well is one - dimensional ) . the large barrier height in the two - state two - mode model makes the tunneling timescale extremely long compared to any other timescale of interest in this system . as a possible direction of future research , it would be interesting to explore whether such metastable steady states can be obtained without propagating the dynamical equations of motion . if so , the determination of the quantum yield from the stationary eigenstates becomes a straightforward task via eq . ( [ qy1 ] ) with the matrix elements @xmath108 replaced by their steady - state values . we have considered a time - independent approach to the quantum yield of _ cis - trans _ photoisomerization , here applied to model retinal in rhodopsin . the need for this approach arises due to the recognition that natural processes take place in incoherent light ( e.g. , sunlight with a coherence time of 1.32 fs ) and environmental decoherence , which produce mixtures of stationary hamiltonian eigenstates . here we have recast one of the standard time - dependent definitions of the quantum yield in terms of time - independent quantities , the eigenstates of the hamiltonian and associated dipole transition matrix elements . the quantum yield is then shown to be a direct reflection of the _ cis _ vs. _ trans _ character of the individual stationary eigenstates of the system and the associated dipole transition matrix elements from the ground electronic state . further , applied to a model of retinal , this approach gives excellent results for the quantum yield , fully in agreement with experiment . interestingly , relaxation from the initially prepared stationary mixture does not alter the quantum yield , a consequence of both the _ cis / trans _ partitioning of the stationary states and the system - bath coupling in this well established minimalist model of retinal isomerization . ideally , we should consider a full treatment of all modes of retinal in a proper rhodopsin environment , define the quantum yield and other observable properties , fit the retinal potential parameters to experiment , and compare time - dependent computational results to the stationary state results associated with incoherent light excitation . however , such and extensive computational study is not required to support the main result of this paper , i.e. , that the stationary eigenstates of the system hamiltonian provide an alternative and important way to understand features affecting the quantum yield in incoherent light . rather , we adopt the basic two - dimensional model of hahn and stock @xcite . restricting attention to this model necessitates , by requirements of consistency , that if we adopt their two - dimensional potentials and associated system parameter fits to experimental data , that we also must maintain their definition of quantum yield , with the associated neglect of @xmath130 and @xmath131 . alternative definitions of the quantum yield would have resulted in different values of the system parameter fits and different stationary eigenstates of the hamiltonian . hence , the successful computational results demonstrated here do motivate a more extensive calculation of retinal hamiltonian eigenstates for a retinal model including all degrees of freedom@xcite . such work is in progress . * acknowledgments*. this work was supported by the natural sciences and engineering research council of canada , and by the us air force office of scientific research under contract numbers fa9550 - 10 - 1 - 0260 and fa9550 - 13 - 1 - 0005 . table i. calculated quantum yields for retinal photoisomerization . both pre - averaged ( @xmath107 ) and post - averaged ( @xmath102 ) results are shown . @xmath88 normalized lineshape function calculated within the 2d model ; @xmath173 normalized lineshape function based on the measured absorption spectrum of retinal in rhodopsin @xcite . the time - dependent wavepacket result from ref . is given in the last column . mathies , r. a. ; lugtenburg , j. in _ handbook of biological physics , volume 3 : molecular mechanisms in visual transduction , _ eds . d. g. stavenga , w. j. degrip and e. n. pugh jr . , elsevier science press , pp . 55 - 90 ( 2000 ) . polli , d. ; alto , p. ; weingart , o. ; spillane , k. m. ; manzoni , c. ; brida , d. ; tomasello , g. ; orlandi , g. ; kukura , p. ; mathies , r. a. ; garavelli , m. ; cerullo , g. conical intersection dynamics of the primary photoisomerization event in vision . _ nature ( london ) _ * 2010 * , _ 467 _ , 440 - 443 . kim , j. e. ; tauber , m. j. ; mathies , r. a. analysis of the mode - specific excited - state energy distribution and wavelenth - dependent photoreaction quantum yield in rhodopsin . _ biophys . j. _ * 2003 * , _ 84 _ , 2492 - 2501 . zewail , a. h. ; orlowski , t. e. ; jones , k. e. radiationless relaxation in `` large '' molecules : experimental evidence for preparation of true molecular eigenstates and born - oppenheimer states by a coherent laser source . usa _ * 1977 * , _ 74 _ , 1310 - 1314 . mller , u. ; stock , g. flow of zero - point energy and exploration of phase space in classical simulations of quantum relaxation dynamics . ii . application to non - adiabatic processes . _ j. chem . phys . _ * 1999 * , _ 111 _ , 77 - 88 . abe , m. ; ohtsuki , y. ; fujimura , y. ; domcke , w. optimal control of ultrafast _ cis - trans _ photoisomerization of retinal in rhodopsin via a conical intersection . _ j. chem . phys . _ * 2005 * , _ 123 _ , 144508 - 1 - 144508 - 10 . leitner , d. m. ; kppel , h. ; cederbaum , l. s. statistical properties of molecular spectra and molecular dynamics : analysis of their correspondence in no@xmath174 and c@xmath174h@xmath175 . _ j. chem . phys . _ * 1996 * , _ 104 _ , 434 - 443 . joubert - doriol , l ; ryabinkin , i. g. ; izmaylov , a. f. geometric phase effects in low - energy dynamics near conical intersections : a study of the multidimensional linear vibronic coupling model . _ j. chem . phys . _ * 2013 * , _ 139 _ , 234103 - 1234103 - 10 . christopher , p. s. ; shapiro , m. ; brumer , p. overlapping resonances in the coherent control of radiationless transitions : internal conversion in pyrazine . _ j. chem . phys . _ * 2005 * , _ 123 _ , 064313 - 1 - 064313 - 9 . doukas , a. g. ; junnarkar m. r. ; alfano r. r. ; callender r. h. ; kakitani t. ; honig b. fluorescence quantum yield of visual pigments : evidence of subpicosecond isomerization rates . _ proc . natl . _ * 1984 * , _ 81 _ , 4790 - 4794 . fig . 1 : ( upper panel ) ground and first excited diabatic potential profiles along the reaction coordinate @xmath55 for retinal photoisomerization . the green upward arrow illustrates laser excitation , the downward arrows illustrate the partitioning of the eigenstates into _ cis _ and _ trans _ ( by the projection operators , see text ) . the _ cis _ and _ trans _ regions of configuration space are separated by the red dashed line . ( lower panel ) adiabatic pess for retinal as functions of the torsional coordinate @xmath55 and the coupling mode @xmath59 orthogonal to it . the pess are obtained by diagonalizing the hamiltonian in eq . ( [ model ] ) ( without the kinetic energy term ) . 2 : frequency dependence of the stationary quantum yield . superimposed on the plot are the linear absorption spectrum of the 2d model ( green ) and the experimental absorption spectrum spectrum of retinal in rhodopsin adapted from fig . 2 of ref . 23 , both normalized to unity at their respective maxima . the inset shows an expanded view of the frequency - dependent quantum yield in the region of maximum absorption at @xmath176 @xmath78 ( @xmath177 nm ) fig . 3 : wavelength dependence of the stationary quantum yield . diamonds results for @xmath88 calculated from the two - state two - mode model ; dashed ( dash - dotted ) lines results obtained for @xmath107 ( @xmath102 ) and the experimentally measured @xmath74 @xcite , circles experiment . the error bars are smaller than the size of the circles . the inset shows the bare frequency - dependent quantum yield ( [ ybare ] ) calculated _ without _ the spectral lineshape function @xmath74 . the experimental quantum yield stays constant below @xmath177 nm . the results for @xmath88 are shown only in those spectral regions where @xmath88 does not vanish ( see fig . 2 ) . fig . 4 : ( a ) time dependence of _ cis _ and _ trans _ state populations ( see eq . [ populations ] ) . the excited - state populations @xmath23 and @xmath24 ( dashed lines ) decay in time due to the interaction with the environment ( see text ) . ( b ) time dependence of the quantum yield given by eq . ( [ y ] ) . these results are obtained by solving the equations of motion for the diagonal elements of the density matrix parametrized by the transition rates given by eqs . ( [ relax1])-([relax2 ] ) . the quantum yield remains constant ( within 3% of the asymptotic value of 0.62 ) over the time interval @xmath178 fs . fig . 5 : snapshots of eigenstate populations @xmath143 . at @xmath40 , the bright eigenstates ( mostly 512 , 507 , and 508 ) are populated by fully incoherent , impulsive fc excitation ( see eq . 6 ) . at later times , interaction with the bath causes the population of the bright states to decay through several cascades ( middle panels ) . the resulting steady - state eigenstate distribution is plotted in the lowermost panel . fig . 6 : the network of dominant relaxation pathways starting from the dominant bright state @xmath135 ( the highest peak in the upper panel of fig . 5 ) . green arrows show the most efficient pathway leading to the @xmath172 eigenstate ( the highest peak in the lower panel of fig . 6 ) . see text for details . fig . 7 : probability density ( eq . [ probden ] ) for the ground ( @xmath81 , upper panels ) and the first excited ( @xmath158 , lower panels ) diabatic electronic states as a function of the torsional coordinate @xmath55 and the coupling mode @xmath59 . only the eigenstates above the conical intersection are shown . note the similarity of the @xmath55-profiles of the eigenstates involved in the relaxation process , which implies no change in the quantum yield . the lower left figure illustrates that _ cis / trans _ partitioning of delocalized eigenstates above the conical intersection ( marked by the star ) does not change qualitatively during relaxation . note changes in color scale from panel to panel . fig . 8 : localization properties of the eigenstates in the first relaxation cascade . ( upper panel ) : the populations of the first - cascade eigenstates at @xmath179 fs after the initial excitation . ( lower panel ) : the diagonal matrix element of the projection operator @xmath146 quantifying the amount of population in the _ trans_-region of configuration space . note that all eigenstates significantly populated in the first stage of relaxation have similar localization properties with @xmath180 . notable exceptions include states @xmath159 and @xmath160 with @xmath181 and @xmath182 . same as in fig . 7 but for the eigenstates in the vicinity of and below the conical intersection . note the different localization properties of the eigenstates involved in the relaxation process . the lower left figure illustrates the transition from delocalized to localized eigenstates below the conical intersection ( marked by the star ) . note changes in color scale from panel to panel .
|
_ cis - trans _ isomerization in retinal , the first step in vision , is often computationally studied from a time dependent viewpoint .
motivation for such studies lies in coherent pulsed laser experiments that explore the isomerization dynamics .
however , such biological processes take place naturally in the presence of incoherent light , which excites a non - evolving mixture of stationary states . here
the isomerization problem is considered from the latter viewpoint and applied to a standard two - state , two - mode linear vibronic coupling model of retinal that explicitly includes a conical intersection between the ground and first excited electronic states .
the calculated quantum yield at 500 nm agrees well with both the previous time - dependent calculations of hahn and stock ( 0.63 ) and with experiment ( @xmath0 ) , as does its wavelength dependence .
significantly , the effects of environmental relaxation on the quantum yield in this well - established model are found to be negligible .
the results make clear the connection of the photoisomerization quantum yield to properties of stationary eigenstates , providing alternate insights into conditions for yield optimization .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
fetal distress is generally used to describe the lack of the oxygen of the fetus , which may result in damage or death if not reversed or the fetus is delivered immediately . thus the emergent action by discriminating the fetal distress is an important issue in the obstetrics . in particular , it is important to discriminate the normal fetal heart rate(fhr ) and two types of fetal distress groups(the presumed distress and the acidotic distress ) and reduce the wrong diagnosis rate in which about 65% of the presumed distress fetus(not serious but in need of careful monitoring ) are diagnosed as the acidotic distress fetus(serious , needing an emergent action ) , experiencing a useless surgical operation . in this paper , we try to discriminate the normal and pathologic fetal heart rate group with a robust and reliable method . + the cardiovascular system of the fetus is a complex system . the complex signal , the heart rate , from the complex system contains enormous information about the various functions . the incoming information , on which various heart activities are projected , can not be linearly separated into each function . but we may be able to estimate the signal from a pathologic fetal heart which contains information on the weakness or a complete loss of a particular function . based on this concept , multiple time scale characteristic of heart dynamics have received much attention . as a quantitative method , the multi - scale entropy , which quantifies multi - time scale complexity in the heart rate , was introduced and widely applied to the classification between normal and pathologic groups and also different age groups . it was found that in a specific time scale region , the normal and the pathologic adult heart rate groups and the different age effects on the heart rhythms are significantly distinguished @xcite . + the multiple - scale characteristic in the fetal heart rate is due to the autorhythmicity of its different oscillatory tissues , its interaction with other neural controller and the maternal circulatory system and other neural or hormonal activities , which have a wide range of time scales from secondly to yearly@xcite . + in our previous work on the multiple - scale analysis of heart dynamics , we extended the analysis with the time scale to both the event and time scales@xcite . we found that the event scale in heart dynamics plays more important role than the time scale in classifying the normal and pathologic adult groups . in this paper , therefore , we will investigate characteristic event or time scales of heart dynamics of the normal and the fetal distress group in order to determine the criteria for classifying the normal and the fetal distress groups . + previous works on the fetal heart rate were based on the various nonlinear measures such as the multi - scale entropy , approximate entropy , power spectral density and detrended fluctuation analysis @xcite . they were able to significantly differentiate the mean of normal and pathologic fhr groups but their classification performance was poor , which prevent practical applications of these methods . in this paper , we investigate the typical scale structure of each fhr group by scanning both the event and time scale regions in order to find an appropriate scale region for classifying the normal and the pathologic fetal groups in an optimal way . + we introduce a new analysis method , called the unit time block entropy(utbe ) , which scans both the event and time scale regions of the heart rate based on symbolic dynamics to find the characteristic scale region of the normal and the pathologic heart rate groups@xcite . this method matches the unevenly sampled rr interval data length and the measurement time of the heart rate data set simultaneously , where the rr interval means the time duration between consecutive r waves of the electrocardiogram(ecg ) . in most previous studies the number of rr interval sequences of all data set are fixed in spite of a large variability in the measurement time@xcite . using the utbe method , we can directly compare the entropy of data sets without any ambiguity caused by the nonstationarity and noise effect that inevitably appears in the data set with different data length or measurement time . we find that the normal and two pathologic fhr groups are reliably discriminated . + in section 2 , we will introduce the normal and pathologic fhr data set and their linear properties . in section 3 , the new analysis method , called unit time block entropy(utbe ) , is introduced and applied to the fetal heart rate data set . in section 4 , we show that the normal , the presumed distress and the acidotic distress fhr groups can be discriminated through the utbe method in some characteristic scale regions . finally , we end with the conclusion . the rr interval time of three fetuses . ( a ) a normal fetus . ( b ) a presumed fetus . ( c ) a distress fetus . ( d ) the mean and standard deviation of normal , presumed and distress rr interval sequences in units of seconds . ( e ) the log - log plot of rr interval acceleration . the log - log distribution of rr acceleration shows a power - law distribution.,title="fig : " ] + the rr interval time of three fetuses . ( a ) a normal fetus . ( b ) a presumed fetus . ( c ) a distress fetus . ( d ) the mean and standard deviation of normal , presumed and distress rr interval sequences in units of seconds . ( e ) the log - log plot of rr interval acceleration . the log - log distribution of rr acceleration shows a power - law distribution.,title="fig : " ] the rr interval time of three fetuses . ( a ) a normal fetus . ( b ) a presumed fetus . ( c ) a distress fetus . ( d ) the mean and standard deviation of normal , presumed and distress rr interval sequences in units of seconds . ( e ) the log - log plot of rr interval acceleration . the log - log distribution of rr acceleration shows a power - law distribution.,title="fig : " ] + the fetal heart rate is acquired from 77 pregnant women who were placed under computerized electronic fetal monitoring during the ante partum and intra partum periods . the fetal heart rate is digitized with the data received with the corometrics 150 model(corometrics , connecticut , usa ) through the catholic computer - assisted obstetric diagnosis system(ccaod ; dobe tech , seoul , korea ) . the computerized electronic fetal heart rate monitoring was done on the fetal heart rate data two hours before the delivery , from each pregnant woman without any missing data . first , 77 pregnant women are divided into two groups ; 36 women into the normal fetus group , who showed the normal heart rate tracing , and 41 women with the abnormal fetal heart rate tracing(severe variable , late deceleration , bradycardia , or tachycardia , and delivery by caesarean section ) . the heart rate tracing has been done using as a standard criteria to determine normal or pathologic states of a fetus . after an immediate delibery by a caesarean section , the umbilical artery @xmath0 of the fetus is examined . then , 41 women were further divided into the presumed distress group with 26 women whose umbilical artery @xmath0 was higher than 7.15 , and the acidotic distress group with 15 women whose umbilical artery @xmath0 was lower than 7.15 and the base excess was lower than @xmath1 . by this umbilical artery @xmath0 test we can retrospectively know whether a fetus was in dangerous situation or not . in this work , the wrong diagnosis rate is @xmath2 , 26 women undertaken a useless surgical operation . the improvement of this wrong diagnosis rate through the presurgical classification of different groups based on the fhr data is the main goal of this work . + in order to treat errors in the measuring equipment and ectopic beats we selected the interval that did not have any missing data , and manually treated the ectopic beats using a linear interpolation , which is less than @xmath3 of all heart beats selected . we replaced the ectopic beats with resampled data around the vicinity of @xmath4 using the average sampling frequency of each heart rate sequence . the resampling , which is usually used for even sampling from the unevenly measured heart beat , is restrictively applied for replacing ectopic beats . however , this method inevitably produces a number of unexpected artificial effects on the original rr sequences . for example , it can cause the distortion of the short term correlation of heart rate data@xcite . thus a new method is developed and applied to overcome these problems as in the following section . + before applying the nonlinear method , we investigate conventional linear properties such as the mean and the standard deviation and its student t - test between healthy and pathologic groups(normal , presumed distress and acidotic distress ) . in fig.1(a)-(c ) , the rr interval time of three fetuses are presented . the normal fetus shows a typical characteristic of the small , fast variations , while the pathologic fetus shows relatively slower , larger variations . in fig.1(d ) and ( e ) , linear properties such as the mean , the standard deviation and the log - log distribution of rr acceleration from the normal , the presumed distress and the acidotic distress groups are presented . the result of the t - test on the mean and the standard deviation distributions shows that three fhr groups are distinguishable by these two linear properties , resulting in significant p - values for the mean and the standard deviation[p - value(for the mean / the standard deviation ) : the normal and the presumed distress(@xmath5 ) , the normal and the acidotic distress(@xmath6 ) , the presumed distress and the acidotic distress(0.28/0.903 ) ] . although their significant difference in the mean and the standard deviation , their classification performance based on the sensitivity and specificity is poor . in the next section , we will compare the linear properties with the nonlinear ones obtained from the utbe and the multi - scale entropy method . in fig.1(e ) , the rr interval acceleration shows a power - law distribution . this suggests that the rr interval acceleration rather than rr interval has a nonlinear characteristics which can be best utilized in classifying the different fhr groups . the investigation of linear properties such as the mean and the standard deviation of the rr sequence shows that the heart dynamics of the normal fetus group is faster than that of the pathologic fetus group , leading to a significant difference in the mean . however , the total measurement time shows a large variability for the 77 subjects studied . in fig.[fig : fig1 ] , the mean and the standard deviation of the measurement time of all rr interval sequence is @xmath7 and the time difference between the shortest one and the longest one is @xmath8 . in treating the rr interval sequence , we typically face this contradictory situation . if we try to match the number of rr interval sequence in advance , the measurement time for all rr interval sequences becomes different . if we try to match the measurement time in advance , the number of rr interval sequence becomes different . this naturally occurs for unevenly sampled data such as the rr interval sequence . in order to solve this problem , we used the unit time block entropy(utbe ) , which simultaneously matches both the measurement time and the number of rr interval sequence for all subjects . the variation of measurement time after fixing the number of rr interval sequence in healthy and pathologic fhr groups . the mean and standard deviation of @xmath9 subjects is @xmath10(min ) , it shows a large variation . ] the utbe estimates the entropy of the symbolic sequence composed at the specific event and time scale of a heart beat sequence . since it matches both the measurement time and the number of words of the alphabet from the rr sequence , we can make a direct comparison of all involved fhr groups reliably . + first , we define a word sequence from the rr interval sequence . to construct a word , we use a unit time windowing method , in which each word is constructed in a given unit time window , so the number of symbols involved in each word can be different between windows . + the rr interval sequence is given by @xmath11 and the rr interval acceleration is by @xmath12 . a word is defined as follows @xmath13 a word @xmath14 composed by a unit time windowing method consists of n(i ) symbols . the number of symbols , n(i ) , is different at each unit time scale @xmath15 , satisfying @xmath16 . a symbol @xmath17 , which is used to construct a word , is 1 for @xmath18 larger than @xmath19 and 0 , otherwise . here , @xmath15 is a unit time for constructing a word and @xmath19 is a rr interval threshold or a rr acceleration threshold for binary symbolization . thus a word @xmath14 contains both information about the time scale and the event scale of the heart rate . therefore , it defines a state of specific scale events of the cardiac system during a unit time . the event scale varies with 20 steps from 5% to 95% of the cumulative rank for all rr acceleration values from normal and pathologic hr data set . the time scale varies with 20 steps from 1 second to 10 seconds , which 1 second is sufficiently short to avoid an empty word . + fig.[fig : fig2 ] briefly illustrates how to compose a word sequence from a rr interval sequence . in the block windowing case , a word is defined regularly with the block size n=2 and the block is shifted to the one step next . in the unit time windowing case , a word is defined with a unit time @xmath15 and the unit time windows shifted by 0.5 second to define the next word , allowing the window overlap . the window shifting time , 0.5 second , is appropriately chosen to accumulate an enough number of sampled words . in this way , the unit time widowing method can match the number of words and the total measurement time of different rr intervals sets , while the block windowing method can not . through this procedure we can investigate the complexity of the symbolic rr sequence composed at each time and event scale combination . + with the above word constructing method , we can compose a word sequence from a rr interval sequence . let @xmath20 be a finite , nonempty set which we refer to as an alphabet . a string or word over w is simply a finite sequence of elements of w. an alphabet w has the following relation , @xmath21 where @xmath22 is the set of all words over w , including the empty word , @xmath23 , and @xmath24 the set of all words over w of length n or less and @xmath25 the set of all words over w of length n. the number of possible words of @xmath24 and @xmath25 are @xmath26 and @xmath27 , respectively . and s is a s - ary number , which in this paper is binary(s=2 ) . and the number of possible symbols in a word , @xmath28 , varies with the unit time @xmath15 , which is used to construct a word . @xmath29 is the smallest rr interval time and @xmath30 is an integer number . since @xmath30 depends on @xmath29 , which is sensitive on the ectopic beat or short term noisy beats , noise reduction should be done carefully in preprocessing . the @xmath14 word sequence composed by the unit time windowing method is a natural generalization of one based on the block windowing method . + unit time windowing and block windowing method : two different ways to compose a word sequence from a rr interval sequence . the upper case presents how to compose a word sequence from a rr interval sequence with the unit time windowing method , on the other hand , the lower case presents how to compose a word sequence with block windowing method . the former focus on unit time to define a word but the latter focus on the number of symbols , in this example , the block size is n=2 , to define a word . ] + based on this unit time windowing method , we calculate the symbolic entropy , called the utbe , which quantifies complexity of a word sequence at specific event and time scale region . the entropy , @xmath31 , is a function of the time scale @xmath15 and the event scale @xmath19 . @xmath32 where @xmath33 is the estimation of the frequency of word @xmath14 and @xmath34 is the occurrence number of a word @xmath14 and @xmath35 is the total number of sampled words . to calculate the exact probability of a word @xmath14 , infinite words are considered with @xmath36 then for s=2 , utbe varies between a lower bound and a upper bound at each time scale . the lower bound occurs for the completely regular case and the upper bound occurs for the completely random case where all words are equally probable . + in this paper , to construct an appropriate word ensemble from a rr sequence , the unit time scale is chosen up to 5 steps(about 2.8 seconds ) for the utbe . at this time scale , the largest number of symbols contained in a word from our data set is 8 , which contribute to 2.9% of the population . the number of possible words is @xmath37 and the number of sampled words from our rr interval sequence is 3,313 at this unit time scale . in the followings , we show that among two scales , the event scale is more significant for classification of healthy and two pathologic fhr groups than the time scale . + + + in fig.4 , we applied the utbe method to the three fhr groups and show the result of the search for all event and time scale regions . in order to compare utbe distributions between the normal and two pathologic groups , all p - values of the student t - test in the @xmath38 parameter plane are presented in fig . 4(a - f ) . since the scale characteristics of three groups are not known a priori , we scanned all event and time scales for optimization of the classification . for the presumed distress and the acidotic distress groups in the fig.[fig : fig4](a ) and fig.[fig : fig4](d ) , the event sequences composed by only the 50 - 80% cumulative rank region in the rr interval acceleration are significantly distinguished over all time scales(@xmath39 ) . this suggests that the presumed distress and the acidotic distress groups have relatively different characteristics scale complexity on this specific rr interval acceleration region . these two groups could not be discerned by their linear properties[the p - value of ( mean / standard deviation):(0.28/0.903 ) ] . for the normal and the acidotic distress groups in fig.[fig : fig4](b ) and fig.[fig : fig4](e ) , the event sequences composed by most rr interval acceleration scales except the 50 - 75% cumulative rank region are distinguished in their complexity(@xmath40 ) , while in event sequences composed by only narrow rr interval scale regions , with 5% and 95% cumulative rank ones , show the difference . it suggests that the rr interval acceleration of the normal and the acidotic distress group contains significant information on different internal dynamics of the cardiac system , while the rr interval does not . for the normal and the presumed distress groups in fig.[fig : fig4](c ) and fig.[fig : fig4](f ) , most rr interval scales could not discriminate these two groups except narrow regions , with 5 - 35% ranks and 90 - 95% cumulative ranks , while a wide rr acceleration scale region below the 50% cumulative rank significantly discriminates these two groups(@xmath41 ) . + the above results suggest that the normal , the presumed distress and the acidotic distress groups exhibit characteristic scales with typical dynamics in those scale regions . the information on the characteristic event and time scale regions of three different groups can be used to determine the optimal parameters for the classification of different group . between two types of scales , the event scale is more effective than the time scale in classifying these groups . when two groups are discriminated at an event scale region , these groups are also well discriminated over most time scales . these results suggest that the rr interval acceleration contains typical information about cardiac dynamics of three groups , whereas the rr interval does not . since slow or fast acceleration of the cardiac system is associated with fetal vagal activity and the motherly - fetal respiratory exchange system , it may provide some clues to which functional difference of cardiac systems causes such difference between healthy and pathological groups . + in fig.[fig : fig4](g)-(i ) , the best sensitivity and specificity in the @xmath42 parameter plane is presented for the four measures of statistics and complexity for comparison , respectively . + the sensitivity determines that when a utbe value is given for classifying two groups(a , b ) as a threshold , what percentage of the subjects involved in the group a is correctly classified by the given threshold utbe value . the specificity determines that when a utbe value is given for classifying two groups as a threshold , what percentage of the subjects involved in the other group b is correctly excluded from the group a. so if the sensitivity and specificity are all @xmath43 , the two groups are completely classified by the given threshold . here , the best sensitivity and specificity are achieved after calculating them at all points of the ( @xmath44 ) parameter space , varying the threshold from the minimum utbe to the maximum utbe value of the calculated utbe set@xcite . here , the best sensitivity and specificity is determined as the highest values along the diagonal in the plane of the sensitivity and specificity . as a result , utbe using the rr interval acceleration as a threshold provides the best performance in classification of the presumed distress and the acidotic distress , the normal and the acidotic distress , and the normal and the presumed distress groups . surprisingly , for the normal group and two types of distress groups , utbe using the rr interval acceleration as a threshold completely discriminates these groups with sensitivity 100% , and the specificity 100% , which could not be archived with two linear properties and the utbe using the rr interval threshold . for the presumed distress and the acidotic distress groups , both utbes lead to the same result(sensitivity=71.4% , specificity=72% ) . ( a ) the apen with error bar for different fhr groups . the normal(bold solid line ) has the highest complexity and the presumed(dotted line ) and the acidotic fhr(bold dotted line ) have the lower apen values at each time scale . ( b ) the best sensitivity and specificity of all fhr groups . the best sensitivity and specificity is ( 94% , 95% ) for the normal group and the presumed distress group(solid line ) , ( 85%,86% ) for the normal group and the acidotic distress group(bold dotted line ) and ( 48%,48% ) for the presumed group and acidotic distress group(dotted line).,title="fig:",scaledwidth=45.0% ] ( a ) the apen with error bar for different fhr groups . the normal(bold solid line ) has the highest complexity and the presumed(dotted line ) and the acidotic fhr(bold dotted line ) have the lower apen values at each time scale . ( b ) the best sensitivity and specificity of all fhr groups . the best sensitivity and specificity is ( 94% , 95% ) for the normal group and the presumed distress group(solid line ) , ( 85%,86% ) for the normal group and the acidotic distress group(bold dotted line ) and ( 48%,48% ) for the presumed group and acidotic distress group(dotted line).,title="fig:",scaledwidth=45.0% ] in this section , we compare the performances of the utbe and the multiscale entropy(mse ) , which has been widely used in the hear rate analysis@xcite . the multi - scale entropy calculates the approximate entropy(apen ) or the sample entropy at different scales with heart rate data , which measures the regularity of a given data . since the same data length(n=4061 ) is used in order to remove the dependency on the data length , the approximate entropy is chosen instead of the sample entropy . first , in order to calculate the apen , the rr sequence of length n(=4061 ) is divided into segments of length n and the mean value is calculated for each segment . with the coarse - grained sequence at each scale n , the apen is computed with the following , the parameters ; the embedding dimension m=2 and the delay @xmath45@xcite . in the calculation of the multi - scale entropy , we use two types of r values ; one is determined from the original data at n=1[@xmath46 and the other is variable to be determined at all n scales[@xmath47 . by using the variable r(n ) we can remove the effect of variation due to the coarse - graining process , where sd(n ) denotes the standard deviation of the coarse - grained sequence at a scale n@xcite . since the multi - scale entropy for two cases lead to the similar results , we present here one for the first case . in fig.5(a ) and ( b ) , we present the performance of multi - scale entropy in the classification of three fetal heart rate groups . in fig 5.(a ) , the best p - values for each pair of groups in the student t - test are @xmath48 for the normal vs the acidotic distress , @xmath49 for the normal vs the presumed distress and p=0.143(n=1 ) for the presumed distress vs the acidotic distress . this result shows that the mean multi - scale entropy values of each group are significantly different between the normal and two pathologic groups , but the mean multi - scale entropy values of two pathologic groups are not distinguishable . in order to check the possibility of classification , we investigate the sensitivity and specificity as in the utbe . fig.5(b ) presents the best sensitivities and specificities selected from the calculation in all scales . the best sensitivity and specificity is @xmath50 at the scale(n=4 ) of the normal and the presumed distress case , @xmath51 for the normal and the acidotic distress case and @xmath52 for the presumed distress and the acidotic distress case . the classification performance of multi - scale entropy is not better than that of the utbe in all classification cases from the three groups . this is because utbe searches all the event and time scales to find the optimal classification of different characteristics of healthy and two pathologic data , while multi - scale entropy searches only the time scale . ( a ) the normal and the pathologic fhr groups are distinguished at a specific time and event scale(3sec,5% rank ) . here , the two groups are divided by the threshold(utbe=7.8bit ) . ( b ) the presumed fhr and the acidotic fhr groups are divided by the threshold(utbe=1.4 ) . the marker `` e '' indicates the three fetuses in each group , whose umbilical artery phs are closest to 7.15.,title="fig:",scaledwidth=45.0% ] ( a ) the normal and the pathologic fhr groups are distinguished at a specific time and event scale(3sec,5% rank ) . here , the two groups are divided by the threshold(utbe=7.8bit ) . ( b ) the presumed fhr and the acidotic fhr groups are divided by the threshold(utbe=1.4 ) . the marker `` e '' indicates the three fetuses in each group , whose umbilical artery phs are closest to 7.15.,title="fig:",scaledwidth=45.0% ] in this section , using the difference in the characteristic event scale between the normal and two pathologic groups , we distinguish the normal , the presumed and acidotic groups systematically . in fig.4(e ) and ( f ) , the normal group is significantly distinguished from the presumed distress group and the acidotic distress group in the lower event scale region . in fig.4(d ) , the presumed distress group and the acidotic distress group are distinguished in the relatively higher event scale region . thus , with these characteristics , we can make a strategy to systematically differentiate these groups based on the utbe . first , as in fig.6(a ) , we separate the normal and pathologic groups by a threshold of 7.8 in the utbe , which is determined as the smallest utbe value of the normal group , at the specific time and event scale(3 seconds,@xmath53 rank),respectively . from this procedure , we can differentiate the normal and the pathologic groups with a 100% accuracy . then , for the fetuses deviating from the normal group , usually having the lower utbe , we try to separate the presumed and the acidotic distress groups . in this case , we determine the threshold utbe as 1.4 bits in the time and event scale , which are the first step(1 sec ) and the tenth step(about 50% rank ) , respectively . these time and event scales are selected from the scan of the parameter space as in fig.4(d ) , in which these two groups are distinguished well on the scale region above the tenth event scale . with this threshold , 9 acidotic fetuses out of 14 are distinguished . however , some ambiguity still remains . the clinical determination of the presumed distress fetus and the acidotic distress fetus was carried out by the umbilical artery phs . in fig.6(b ) , we mark the three fetuses with e in each group , who have the umbilical artery phs closest to the threshold value of phs , 7.15 . the range of umbilical artery phs measured is from 6.863 to 7.38 . the phs values of the six fetuses are 7.229 , 7.24 and 7.226 in the presumed distress group and 7.15 , 7.16 and 7.144 in the acidotic distress group . since all the pathologic fetuses were delivered by caesarean surgery after several signs of distress(severe variable , later deceleration , bradycardia , or tachycardia ) , it is not certain if all the acidotic distress fetuses would go to the emergency state or all the presumed distress would be in the safety state . therefore , if the fetuses marked with e are excluded in this analysis , the characteristic of two groups can be clarified more clearly . as a result , most acidotic distress fetuses are less complex than the most presumed distress fetuses at the time and event scale regions . in the comparison of three groups , two pathologic groups are less complex than the normal group , while in the comparison of two pathologic groups the acidotic distress group is less complex than the presumed group . we find that in order to distinguish the normal and pathologic fetuses a small event scale(5% rank ) and a relatively large time scale(3 sec ) of fetal heart dynamics is useful . on the other hand , in order to distinguish the presumed distress and the acidotic distress fetuses a large event scale(about 50 % rank ) and a relatively small time scale(1 sec ) of fetal heart dynamics is more appropriate . with these scale regions , we were able to reduce the wrong diagnosis rate from @xmath54 to @xmath55 . in this paper , we investigated the event and time scale structure of the normal and pathologic groups . we also introduced and calculated the utbe method in the appropriate event and time scale region to distinguish the three groups . to extract meaningful information from the data set , the scale structure of the rr interval acceleration is found to be more helpful than that of the rr interval . in the comparison of the utbe over all event and time scale regions , we found that the normal , the presumed distress and the acidotic distress groups have relatively different event scale structures in the rr interval acceleration . in particular , for the normal and two pathologic groups the utbe from the rr acceleration threshold completely classifies these groups in a chosen scale regions , although both linear properties and utbe using the rr interval threshold performs worse . in the case of the presumed distress and the acidotic distress groups , it also provides better classification performance than other measures . the comparison with the multi - scale entropy also shows that the utbe method performs better . it is due to the fact that the utbe approach searches the event and time scale region , while the multi - scale entropy method searches only the time scale . + based on the difference in the scale structure , we are able to systematically distinguish three fhr groups . the normal and the pathologic groups are separated in a small event scale(5% rank ) and a large time scale(3 sec ) . then , the fetuses deviating from the normal group are separated into the presumed fetuses and the acidotic fetuses at a relatively large event scale(50% rank ) and a small time scale(1sec ) . from these scale regions , we reduce the wrong diagnosis rate significantly . + the results suggest that the utbe approach is useful for finding the characteristic scale difference between healthy and pathologic groups . in addition , we can make a more reliable comparison between all fetuses by simultaneous matching of the measurement time and the number of words . this approach can be applied to the other unevenly sampled data taken from complex systems such as biomedical , meteorological or financial tick data . in this study , we also reconfirm that the more pathological a fetus is the less complex its dynamics , following the pathological order , from the acidotic distress , the presumed distress and to the normal fetus . this indicates that in spite of the peculiarity of the fetal cardiac system , the generic notion of the complexity loss can be applied to the fetal cardiac system . + but these results come from a retrospective test under the well elaborated condition . in a practical view point , the prospective test is necessary to confirm the selected scale regions and its classification performance . we will further test these results with more subjects for the purpose of practical application of this analysis method . @xmath56 we thank dr . sim and dr.chung in dep . obstetrics and gynecology , medical college , catholic univ . for their assistance on clinical data and helpful comments . this work has been supported by the ministry of education and human resources through the brain korea 21 project and the national core research center program . 99 j. altimiras , comp . physiol . a * 124*,447 ( 1999 ) s. havlin , s. v. buldyrev , a. bunde _ et al . _ , physica a * 273 * , 46 ( 1999 ) m. costa , a. goldberger and c .- k . peng , phys . rev . lett . * 89 * , 068102 ( 2002 ) d. r. chialvo , nature * 419 * , 263 ( 2002 ) m. costa , a. goldberger and c - k . peng , comput . * 29 * , 137 ( 2002 ) m. costa and j. a. healey , comput . cardiol . * 30 * , 705 ( 2003 ) m. costa and a. goldberger and c .- k . peng , phys . e * 71 * , 021906 ( 2005 ) c. e. wood and h. tong , am . j. physiol . * 277 * , r1541 ( 1999 ) u. c. lee , s. h. kim and s. h. yi , phys . rev . e**71 * * , 061917 ( 2005 ) g. magenes , m. g. signorini and d. arduini , proc . the second joint embs / bmes conference , october 23 - 26 , 2002 g. magenes , m. g. signorini and m. ferrario _ et al . _ , the 25th annual international conference of the ieee / embs , sep . 17 - 21 , 2003 m. g. signorini , g. magenes and s. cerutti__et al . _ _ , ieee trans . biomed . 50 * , 365 ( 2003 ) m. e. d. gomes , h. n. guimaraes , a. l. p. ribeiro and l. a. aguirre , comput . 32 * , 481 ( 2002 ) the sensitivity is a true positive rate defined as @xmath57 , and the specificity is a true negative rate defined as @xmath58 . ( a : true positive , b : false positive , c : false negative , d : true negative ) . more details are available at http://bmj.bmjjournals.com/ or see t. greenhalgh , bmj * 315 * , 540 ( 1997 ) v. v. nikulin and t. brismar , phys . * 92 * , 089803 ( 2004 )
|
recently , multiple time scale characteristics of heart dynamics have received much attention for distinguishing healthy and pathologic cardiac systems . despite structural peculiarities of the fetal cardiovascular system , the fetal heart rate(fhr )
displays multiple time scale characteristics similar to the adult heart rate due to the autorhythmicity of its different oscillatory tissues and its interaction with other neural controllers . in this paper , we investigate the event and time scale characteristics of the normal and two pathologic fetal heart rate groups with the help of the new measure , called the unit time block entropy(utbe ) , which approximates the entropy at each event and time scale based on symbolic dynamics .
this method enables us to match the measurement time and the number of words between fetal heart rate data sets simultaneously .
we find that in the small event scale and the large time scale , the normal fetus and the two pathologic fetus are completely distinguished .
we also find that in the large event scale and the small time scale , the presumed distress fetus and the acidotic distress fetus are significantly distinguished .
event scale , time scale , symbolization , fetal heart rate , dynamics , complexity 87.19.hh , 87.10.+e , 89.75.-k
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
during the past decade study of nonlinear behavior of magnetic crystals has been attracted large attention , specially it accompany with the progress in some other fields such as development of theory of nonlinear differential equation , achieving new laboratory results and also potential applications in other branches of science and technology [ 1 , 2 ] . particles with spin @xmath0 are more interesting among the other nano particles [ 3 , 4 ] . this is because of existing of complexity in their behavior due to their multipole dynamic spin excitations . in such systems , the number of necessary parameters for complete description of macroscopic properties increases up to 4s , that s stands for magnitude of system spin . also it worthwhile , the process of achieving classical spin equations and dynamic multipoles is based on coherent states that are obtained in @xmath1 group[5 ] . we consider unitary anisotropic hamiltonian as form of : @xmath2 which , @xmath3 are spin operators in lattice @xmath4 , and @xmath5 is anisotropy coefficient . this is hamiltonian of one dimensional ferromagnetic spin chain observed in compositions like @xmath6 [ 6 ] . in this paper the goal is to obtain classical equation for stated hamiltonian and finding the answer of spin wave for small linear excitations upper than the ground state . coherent states issued nearest approximation to classical state i.e. pseudo classical , because they minimize uncertainty principles . for this reason , in section 2 , coherent states for spin @xmath7 developed that are the same as coherent states in su(3 ) group . to obtain classical hamiltonian , we need average values of spin operator ; so in section 3 , these values and classical hamiltonian equation are derived . in section 4 , hamiltonian equation computed in previous section is substituted in classical equations of motion resulted from using feynman path integral on coherent states , and then we acquire spin wave equations and dispersion equations of dipole and quadupole branches for small linear excitation above the ground state , and finally we calculate soliton answers of linearized equations . coherent states are special quantum states that their dynamic is very similar to behavior of their classical system . the kind of coherent state that is used in a problem depends on symmetry of existent operators . with considering existent symmetry in operators of hamiltonian ( 1 ) , coherent states in su(3 ) group is used for accurate description and considering all multipole excitations . in this group , ground state considered as @xmath8 and its single - site coherent state is written as : @xmath9 is wigner function for spin @xmath7 and two angles @xmath10 and @xmath11 determine alignment of classical spin vector . angle @xmath12 determines direction of quadruple torque around the spin vector . parameter g specifies change of length of average value of quadruple torque and also of magnitude of spin vector . lagrangian can be obtained by use of feynman path integral for declared coherent states as:[8 ] @xmath13 where @xmath14 and h is classical energy of system obtained by averaging hamiltonian ( 1 ) on coherent states ( 2 ) . two other terms appear when acquiring lagrangian of spin system . the first is kinetic term that has berry phase characteristics issued from quantum interference of instanton paths and has important role in quantum phenomenons such as spin tunneling and the second is boundary term that depends on boundary values of path . both of term have no role in classical dynamic of spin excitations and so are not considered here . average spin values in su(3 ) group written as:[9 ] @xmath15 by averaging hamiltonian ( 1 ) and using ( 4 ) , the continuous limit of classical hamiltonian obtained as:[4 ] @xmath16 to obtain classical equation of motion , the above classical hamiltonian is substituted in motion equations resulted from lagrangian equation : @xmath17 these equations completely describe nonlinear dynamics of hamiltonian of problem up to quadrupole excitation . solutions of these equations are magnetic solitons . these equations result landau - lifshitz equation if quadrupole excitations ignored @xmath18 . so these equations are more general in comparison with landau - lifshitz and have more degree of freedom . it s noteworthy that solution of these equations has different range of solitons . for small linear excitation from ground stste , classical equations of motion change to : @xmath19 to obtain dispersion equations , functions @xmath20 and g are considered as plane waves and their substitution in linearized equations result in dispersion equation for spin wave near the ground state : @xmath21\end{aligned}\ ] ] from the above equation , it is obvious that both dipole and quadruple branches of unitary hamiltonian are dispersive in presence of linear excitations . to compute soliton answers of equations ( 7 ) , we define variable @xmath22 such as @xmath23 . in this case above equations convert to below nonlinear equations . @xmath24 the first equation is third order differential equation . so change of dipole moment in hamiltonian ( 1 ) is not of the form of soliton . solution of this equation has the following forms : @xmath25\end{aligned}\ ] ] the second equation is nonlinear klein - gordon equation and shows change of average value of quadruple excitation that its solution is of the form of hylomorphic solitons . these solitons are like q - ball solitons . the reason of this name is because of they cause matter have appropriate form . also these solitons are of the kind of non topologic ones because their boundary values in ground and infinity are the same from the topological point of view . if rewrite nonlinear klein - gordon equation ( 9 ) as : @xmath26 where @xmath27 numerical solution of ( 11 ) is plotted in figure ( 1 ) . in this computation we consider @xmath28 and @xmath29 . analytical solution of above nonlinear klein - gordon equation is the following form : @xmath30\end{aligned}\ ] ] where c is constant . in this paper , we study semi - classic theory for spin systems with spin @xmath31 that contain anisotropic exchange terms . it is shown that for anisotropic ferromagnet , value of average quadruple torque is not constant ( @xmath32 ) and its dynamic contains rotational term around classical spin vector ( @xmath33 ) and another dynamics that relates to change of length of quadruple torque . there are no such excitations in regular magnets and their dynamics is achieved by use of average value of heisenberg spin hamiltonian . also it is shown that soliton solutions are of the kind of non topologic hilomorphic solitons for quadruple excitations . o. abdulloev , kh . muminov , accounting of quadrupole dynamics of magnets with spin , proceedings of tajikistan academy of sciences , n.1 , 1994 , p.p . 28 - 30 ( in russian ) . v. g. makhankov , m. a. granados , and a. v. makhankov , `` generalized coherent states and spin @xmath34 systems , '' journal of physics a , vol . 29 , no . 12 , 2005
|
we discuss system with non - isotropic non - heisenberg hamiltonian with nearest neighbor exchange within a mean field approximation process .
we drive equations describing non - heisenberg non - isotropic model using coherent states in real parameters and then obtain dispersion equations of spin wave of dipole and quadrupole branches for a small linear excitation from the ground state . in final , soliton solution for quadrupole branches for these linear equations obtained .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
exotic isotopes along the neutron and proton drip lines are important for our understanding of the formation of elements and they constitute tests of our understanding of nuclear structure . the proton- and neutron - rich regimes in the chart of nuclei are therefore the focus of existing and forthcoming experimental facilities around the world @xcite . the emergence of new degrees of freedom is one important feature of these systems ; exemplified , e.g. , by the discovery of several nuclear halo states along the drip lines @xcite . halo states in nuclei are characterized by a tightly bound core with weakly attached valence nucleon(s ) . universal structures of such states can be considered a consequence of quantum tunneling , where tightly - bound clusters of nucleons behave coherently at low energies and the dynamics is dominated by relative motion at distances beyond the region of the short - range interaction . in the absence of the coulomb interaction , it is known that halo nuclei bound due to a large positive s - wave scattering length will show universal features @xcite . in the case of proton halo nuclei , however , the coulomb interaction introduces an additional momentum scale @xmath1 , which is proportional to the charge of the core and the reduced mass of the halo system . the low - energy properties of proton halos strongly depend on @xmath1 . halo effective field theory ( eft ) is the ideal tool to analyze the features of halo states with a minimal set of assumptions . it describes these systems using their effective degrees of freedom , i.e. core and valence nucleons , and interactions that are dictated by low - energy constants @xcite . for s - wave proton halo systems there will be a single unknown coupling constant at leading order , and this parameter can be determined from the experimental scattering length , or the one - proton separation energy . obviously , halo eft is not intended to compete with _ ab initio _ calculations that , if applicable , would aim to predict low - energy observables from computations starting with a microscopic description of the many - body system . instead , halo eft is complementary to such approaches as it provides a low - energy description of these systems in terms of effective degrees of freedom . this reduces the complexity of the problem significantly . by construction , it can also aid to elucidate the relationship between different low - energy observables . furthermore , halo eft is built on fields for clusters , which makes it related to phenomenological few - body cluster models @xcite . the latter have often been used successfully for confrontation with data for specific processes involving halo nuclei . a relevant example in the current context is the study of proton radiative capture into low - lying states states of @xcite . a general discussion of electromagnetic reactions of proton halos in a cluster approach was given in @xcite . the emphasis of an eft , however , is the systematic expansion of the most general interactions and , as a consequence , the ability to estimate errors and to improve predictions order by order . the structure and reactions of one- and two - neutron halos have been studied in halo eft over the last years ( see , e.g. , refs . however , concerning charged systems only unbound states such as @xmath2 @xcite and @xmath3 @xcite have been treated in halo eft . in this letter , we apply halo eft for the first time to one - proton halo nuclei . we restrict ourselves to leading order calculations of systems that are bound due to a large s - wave scattering length between the core and the proton . the manuscript is organized as follows : in sec . [ sec : theory ] , we introduce the halo eft and discuss how coulomb interactions are treated within this framework . in the following section , we present our results and calculate , in particular , the charge form factor and charge radius at leading order . furthermore , we derive expressions for the radiative capture cross section . we apply our general formulae to the excited @xmath0 state of and compare our numerical results with existing data for this system . we conclude with an outlook and a discussion on the importance of higher - order corrections . in halo eft , the core and the valence nucleons are taken as the degrees of freedom . for a one - proton halo system , the lagrangian is given by @xmath4 here @xmath5 denotes the proton field with mass @xmath6 and @xmath7 the core field with mass @xmath8 , @xmath9 denotes the leading order ( lo ) coupling constant , and the dots denote derivative operators that facilitate the calculation of higher order corrections . the covariant derivative is defined as @xmath10 , where @xmath11 is the charge operator . the resulting one - particle propagator is given by @xmath12^{-1}~.\ ] ] for convenience , we will also define the proton - core two - particle propagator @xmath13^{-1}~,\ ] ] where @xmath14 denotes the reduced mass of the proton - core system . we include the coulomb interaction through the full coulomb green s function @xmath15 . the dashed line denotes a core propagator , the solid line a proton propagator and the wave line denotes the exchange of a coulomb photon.,width=302 ] where @xmath16 is the coulomb four - point function defined recursively in fig . [ fig : fourpointgamma ] . to distinguish coordinate space from momentum space states we will denote the former with round brackets , i.e. @xmath17 . in coordinate space , the coulomb green s function can be expressed via its spectral representation @xmath18 where we define the coulomb wave function through its partial wave expansion @xmath19 here we have defined @xmath20 and @xmath21 with the coulomb momentum @xmath22 and also the pure coulomb phase shift @xmath23 for the coulomb functions @xmath24 and @xmath25 , we use the conventions of ref . the regular coulomb function @xmath24 can be expressed in terms of the whittaker m - function according to @xmath26 with the @xmath27 defined as @xmath28}}{2(2l+1)!}~.\ ] ] we shall also need the irregular coulomb wave function , @xmath25 , which is given by @xmath29 where @xmath30 is the whittaker w - function and the coefficient @xmath31 is defined as @xmath32 to obtain the fully - dressed two - particle propagator , that includes strong and coulomb interactions , we calculate the irreducible self - energy shown in fig . [ fig : irredbubble ] . @xmath33 the expression above is known and is given by @xmath34 this integral was solved in ref . @xcite , using dimensional regularization in the power divergence subtraction ( pds ) scheme , as @xmath35 with a divergent part @xmath36 @xmath37-\frac{m_\mathrm{r}\mu}{2\pi}~,\ ] ] where @xmath38 is the space dimension , @xmath39 is the euler constant , and @xmath40 is the pds regulator . the function @xmath41 is defined as @xmath42 with @xmath43 being the polygamma function . note that the divergent part in eq . ( [ eq : jdivergent ] ) is energy independent . this will become important later when the derivative of @xmath44 , with respect to the energy , will be required . the coupling constant @xmath9 can be determined by matching to a two - body observable , such as the coulomb corrected proton - core scattering length @xcite : @xmath45 since we stay at leading order , however , an explicit expression for @xmath9 will not be required for the calculation of electromagnetic observables in the next section . .,height=113 ] in our calculation of the charge form factor , we follow the derivation of the deuteron form factor presented in ref . @xcite . the form factor is obtained by calculating the matrix element @xmath46 for momentum transfer @xmath47 in the breit frame , where no energy is transferred by the photon . it was shown in ref . @xcite that this matrix element can be expressed as in our eq . ( [ eq : ff1 ] ) compared to eq . ( a10 ) of @xcite is that we have an extra @xmath48 in our definition of the irreducible self - energy . ] @xmath49 where @xmath50 denotes the irreducible three - point function shown in fig . [ figlogamma ] , and @xmath51 is the derivative of the self - energy with respect to the total energy evaluated at the energy @xmath52 , where @xmath53 is the proton separation energy or core - proton binding energy . with the proton - core mass ratio @xmath54 , the three - point function @xmath50 is given by @xmath55~ , \label{eqlogamma}\end{aligned}\ ] ] and the derivative of the self - energy can be written as @xmath56 evaluating @xmath50 at zero momentum transfer , by using eq . ( [ eq : cgfspectral ] ) and orthonormality of the wavefunctions , and comparing with eq . ( [ eq : dsigmae ] ) shows that the charge form factor is properly normalized to one in this limit . we find that eq . ( [ eqlogamma ] ) can be simplified by writing the coulomb green s function for negative energy using the whittaker w - function . this is achieved by demanding proper asymptotics and using that only the s - wave part can contribute to propagation to zero separation , that is @xmath57}{\rho'\rho}\right ) \nonumber\\ & = & i\frac{m_{\mathrm{r}}\gamma(1+k_{\mathrm{c}}/\gamma_0)}{2\pi}\frac{w_{-k_{\mathrm{c}}/\gamma_0,1/2}(2\gamma_0r)}{r}~ , \label{eq : simplcoulgreen}\end{aligned}\ ] ] where we have introduced the binding momentum @xmath58 . the resulting integral is then @xmath59\nonumber\\ & & \times w_{-k_{\mathrm{c}}/\gamma_0,1/2}(2\gamma_0 r)^2~ , \label{eqbintegralnprime}\end{aligned}\ ] ] where @xmath60 are the spherical bessel functions . once the parameters of the proton halo system are fixed , the equation @xmath61 is used to calculate the charge form factor and the corresponding charge radius numerically . we have calculated these quantities for the excited @xmath0 state of , which has a proton separation energy of @xmath62@xcite . note that the proton separation energy is the only non - trivial experimental input at lo . the charge form factor is related to the charge radius via the expansion @xmath63 and we find for the charge radius squared @xmath64 since the proton and core are treated as structureless fields in halo eft , this quantity corresponds to the charge radius difference according to @xmath65 where @xmath66 is the charge radius squared corresponding to the particle @xmath67 , , @xmath68 . analog to the deuteron case , the charge radii of proton and enter at higher orders in the calculation via counter terms . the error of the eft can be estimated by comparing the momentum scale @xmath69 of the halo with the break - down scale @xmath70 of the eft . the latter is given by the closest interfering state . for the halo system the break - down scale is given by the bound state , @xmath71 below the @xmath0 state @xcite . thus , the expected lo error is @xmath72 for the halo state in @xmath73 , which is comparable to the lo error of a pionless eft calculation for the two - nucleon system . our approach can easily be applied to low - energy radiative capture . the differential cross section for this reaction is @xmath74 where @xmath75 is the vector amplitude for the sum of the diagrams shown in fig . [ fig : capture ] , where a proton is captured by a core while a real photon is emitted . the relative momentum of the proton - core system is @xmath76 and the four - momentum of the photon is @xmath77 , with associated polarization vectors @xmath78 and @xmath79 . the factor @xmath80 is the wavefunction renormalization , or lsz reduction factor . the vector amplitude @xmath75 can be expressed as the integral @xmath81~,\end{aligned}\ ] ] where the @xmath82 has emerged from the feynman rule of the vector photon coupling and acts on the coulomb wavefunction due to a partial integration . evaluating the angular integrals and multiplying with the polarization vector , the integral is simplified to @xmath83\nonumber\\ & & + \big[(f\to 1-f),~(z_{\mathrm{c}}\to 1)\big]\bigg|^2~ , \label{eq : captureintsimpl}\end{aligned}\ ] ] where the angles @xmath84 and @xmath85 will be integrated over to give the total cross section . for a given physical system , we can solve the integral in eq . ( [ eq : captureintsimpl ] ) numerically using eq . ( [ eq : simplcoulgreen ] ) for the coulomb green s function . ) @xmath86 is presented by the solid black line . the theoretical result is compared with the data by chow _ _ @xcite and morlock _ et al . _ @xcite shown by blue triangles and green dots , respectively . the calculation by bennaceur _ @xcite is shown by the dashed curve.,width=377 ] radiative capture into low - lying states of has been measured by rolfs _ @xcite , chow _ et al . _ @xcite and by morlock _ _ @xcite . in fig . [ fig : sfactor ] we show the astrophysical s - factor , defined as @xmath87 the figure shows the halo eft results of our lo calculation compared to experimental data for capture into the @xmath0 excited state and a phenomenological calculation using the shell model embedded in the continuum . at threshold , we find that @xmath88 . our lo results are slightly low , but consistent with the experimental data , within the expected 50% error . we anticipate that the next - to - leading order correction will increase the radiative capture cross section through the appearance of a finite effective range at this order . it can also be noted that the results agree qualitatively with the predictions obtained in the shell model embedded in the continuum @xcite . in this work , we have shown that coulomb effects can be included in halo eft , and that thereby static and dynamical observables of proton halo nuclei become accessible . we have calculated the charge radius and the radiative proton capture cross section of s - wave proton halo nuclei at leading order in halo eft . our results can be applied to any one - proton halo system whose interaction is dominated by s - waves . in particular , the excited @xmath0 state in is known to have a large s - wave component . we have calculated the charge radius for this system . while this observable is not yet experimentally accessible , this result provides a prediction for _ ab initio _ calculations using modern nucleon - nucleon interactions . in addition , we have compared our results for radiative capture into the excited @xmath0 state of with experimental data and found good agreement within the expected error . furthermore , we found that halo eft gives the same qualitative behavior for this observable as previous calculations that have employed phenomenological models . for a quantitative description of the experimental data , higher order corrections are required . in a future publication , we will address how these corrections are included within halo eft in the presence of coulomb interactions . the size of these contributions will strongly be affected by the relative size of the effective range and the coulomb momentum @xmath89 , which provides an additional scale in systems with coulomb interactions . our calculation is also a first step towards a calculation of properties of within halo eft . this system requires the inclusion of two low - energy constants at leading order since it interacts dominantly in the p - wave @xcite . finally , our approach might prove useful for heavier systems whose static observables can be calculated using _ ab initio _ approaches , but for which continuum properties are not accessible within the same framework due to the computational complexity . in this scenario , _ ab initio _ predictions of , e.g. , the one - proton separation energy could be used to fix the halo eft parameters , which in turn could be used to predict continuum observables such as the radiative capture cross section . in the case of neutron halos , such an approach was recently carried out to predict novel features in the calcium isotope chain using halo eft @xcite . we thank h. esbensen and s. knig for helpful discussions , p. mohr and k. bennaceur for supplying relevant data . this work was supported by the swedish research council ( dnr . 2010 - 4078 ) , the european research council under the european community s seventh framework programme ( fp7/2007 - 2013 ) / erc grant agreement no . 240603 , the office of nuclear physics , u.s . department of energy under contract nos . de - ac02 - 06ch11357 , by the dfg and the nsfc through the sino - german crc 110 , by the bmbf under contract 05p12pdfte , and by the helmholtz association under contract ha216/emmi .
|
we use halo effective field theory to analyze the universal features of proton halo nuclei bound due to a large s - wave scattering length . with a lagrangian built from effective core and valence - proton fields ,
we derive a leading - order expression for the charge form factor . within the same framework we also calculate the radiative proton capture cross section .
our general results at leading order are applied to study the excited @xmath0 state of fluorine-17 , and we give results for the charge radius and the astrophysical s - factor .
halo nuclei , charge radius , radiative capture , effective field theory
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
detecting the qcd phase transition is of primary interest in current nuclear and particle physics . experiments have shown that a new form of strongly interacting matter is created in high - energy nuclear collisions at the relativistic heavy ion collider ( rhic ) top energy and the large hadron collider ( lhc ) energies @xcite . theoretical arguments and experimental signals imply that this matter is a quark - gluon plasma ( qgp ) created well above a transition temperature at almost zero baryon chemical potential . the lattice qcd calculations have confirmed that the transition from hadronic matter to qgp is a crossover at zero baryon density @xcite . the next challenge is to discover the first and/or second - order phase transition of qcd matter . several effective model and approximate calculations of qcd suggest the existence of the first - order phase boundary and the critical point at finite chemical potentials @xcite . various observables , such as collective flows , particle ratios , moment of the distributions of conserved charges , have been measured at various incident energies to find evidences of a phase transition and a critical point @xcite . particularly , we shall focus on the observation of collapse of directed flow by the star collaboration @xcite , which might be a signal of a first - order phase transition at high baryon density region between hadronic matter and quark gluon plasma @xcite . the collective transverse flow @xcite has been utilized to explore the properties of hot and dense matter , since it reflects the properties of the equation of state ( eos ) in the early stages of nuclear collisions @xcite . the existence of bounce off , one of the collective flows , in heavy - ion collisions was first suggested in hydrodynamics @xcite and examined in experiments at bevalac @xcite at lawrence berkeley national laboratory . later , the presence of the softest point near the phase transition in the eos @xcite was discussed as a signal for the first - order phase transition , where the softest point is a local minimum of the ratio of pressure to energy density @xmath2 as a function of energy density , leading to a small sound velocity defined by @xmath3 . in ref . @xcite , baryon density dependence of the ratio @xmath2 was investigated within a quasi - particle model . the softest point of the crossover eos at vanishing chemical potential is not very pronounced , but it is predicted that the eos with a first order phase transition exhibits a very pronounced softest point at large chemical potentials . considerations of heavy ion collisions at rhic energy have raised the question whether the physics of the often speculated first - order phase transition or the `` softest point '' should better be studied at moderate energies @xcite in which the collapse of the directed flow at @xmath47 - 9 gev was speculated . presumably , the eos of baryon rich matter is the softest at moderate energy densities of a few gev/@xmath5 . particularly , the excitation function of the directed flow slope with respect to rapidity @xmath6 decreases at @xmath7 @xcite and is predicted to exhibit a minimum at a certain collision energy in hydrodynamical calculations using an eos with a first - order qcd phase transition @xcite , where the @xmath8 is defined as the first fourier component in the azimuthal angle distribution with respect to the reaction plane , @xmath9 . the negative slope of @xmath8 , called the third flow @xcite or the anti - flow of the nucleon @xcite emerges as a consequence of a tilted ellipsoid with respect to the beam axis from which negative flow builds up if matter passes through the softening point . thus the collapse of the directed flow slope to a negative value might signal the first - order phase transition from hadron phase to quark - gluon phase , and was recently observed in the beam energy scan ( bes ) program performed at rhic @xcite . a negative slope of proton @xmath8 at midrapidity has been found in the microscopic transport models rqmd @xcite , urqmd @xcite , and phsd / hsd @xcite in which sign change is purely geometrical and only happens at large impact parameters and sufficiently higher collision energies . note that such microscopic transport models do not show a negative slope of the proton @xmath8 at midrapidity at bombarding energies of @xmath10 gev @xcite , and thus the negative slope of the proton @xmath8 at @xmath11 and 19.6 gev observed by the star collaboration @xcite is incompatible with the predictions by the standard hadronic transport models . we also note that the hadronic transport model with momentum dependent mean field significantly improves the description of the directed flow data from e895 @xcite and na49 data @xcite , but inclusion of the hadronic mean field does not lead to the negative proton @xmath6 @xcite . this gives indirect evidence of the phase transition around such collision energies . the excitation function of the directed flow slope was investigated also in a transport + hydrodynamics hybrid approach @xcite , where it was found that there is no sensitivity of the directed flow on the eos , and there is no minimum in the excitation function of the directed flow slope . in contrast , strong sensitivities of the directed flow to the eos are found in a three - fluid model @xcite . the three - fluid calculations indicate that the crossover deconfinement transition is consistent with the directed flow data of energy range up to @xmath12 gev . however , the phsd transport model which incorporates crossover eos does not show the experimentally observed minimum @xcite . thus it is not yet clear whether the negative slope of @xmath8 signals the softening of the eos in hybrid approaches . in the present paper , we investigate the directed flow in the bes energy region within the microscopic transport model jam @xcite by imposing attractive orbits for each two - body scattering to simulate effects of a softening of the eos . the hadronic transport model jam @xcite has been developed based on resonance and string production and their decay , which is similar to other transport models @xcite . secondary products from decay can interact with each other by binary collisions . a detailed description of the jam model can be found in ref . @xcite . we take into account nuclear eos effects within a microscopic transport approach by changing the standard stochastic two - body scattering style , which is normally implemented so as not to contribute to the pressure . for example , we can simulate repulsive @xmath13 potential effects by allowing only repulsive orbits in the two - body collisions @xcite , instead of choosing the scattering angle randomly as in a standard cascade . it is reported that directed flow at bevalac and alternating gradient synchrotron ( ags ) energies are well described by this approach @xcite . later , attractive orbits were introduced to effectively incorporate the softening of the eos @xcite guided by the virial theorem @xcite . in this way , different treatment of scatterings can modify the eos . we impose attractive orbits for each two - body hadron - hadron scattering to reduce the pressure of the system . the pressure of a system , in which particles are interacting with each other only by two - body scattering , is given by the virial theorem as @xcite @xmath14 where @xmath15 corresponds to the free streaming contribution . the second term represents the pressure generation from all two - body scatterings between the pair of particles @xmath16 and @xmath17 , where @xmath18 is the momentum transfer and @xmath19 and @xmath20 are the coordinate of colliding particles . @xmath21 is the volume of the system , and @xmath22 is a time interval over which the system is measured . thus , pressure generation by the two - body collisions is related to the scattering style ; the repulsive orbit @xmath23 enhances the pressure , while the attractive orbit @xmath24 reduces the pressure . note that an attractive potential softens the eos @xcite , which leads to an attractive orbit . attractive orbits are realized in the simulation as follows . each orbit is selected randomly as in the standard simulation , but in the case where the orbit is repulsive , we change it to an attractive one by exchanging the momentum of two particles in the two - body cener - of - mass ( c.m . ) thus the scattering rate remains the same . while in reality modification of the scattering style should depend for example , on variables such as local energy density , we impose a modified scattering style for all hadron - hadron @xmath25 scatterings in order to see an effect of the softening , instead of trying to fit the data . thus there is no adjustable free parameter in our current approach unlike ref . @xcite in which repulsive trajectories are selected for colliding baryons with some probability in order to generate more pressure at ags energies . an energy density dependent implementation of attractive orbits will be examined shortly in sec . [ sec : eosdep ] we now discuss directed flows in the bes energy region . in the simulation , we choose the impact parameter range @xmath26 fm for mid - central and @xmath27 fm for central collisions for the star data @xcite . -27 gev from the jam cascade mode ( dashed lines ) and the jam cascade with attractive orbits ( solid lines ) in comparison with the star data @xcite.,width=340 ] in fig . [ fig : starv1mid ] we show the calculated directed flow @xmath8 of protons and pions in mid - central collisions from the standard jam cascade ( dotted lines ) and the jam cascade with attractive orbits ( solid line ) in au+au collisions at @xmath28 and 27 gev in comparison with the data from star collaboration @xcite . the standard jam cascade calculation agrees with the 7.7 gev data . however , it is seen that @xmath8 from the standard jam cascade calculations for beam energies of 11.5 and 19.6 gev yields much larger @xmath8 than the star data . the proton slope from jam turns out to be negative at @xmath29 gev . this is because of the geometrical reason pointed out in ref . @xcite , and is not related to the softening of the eos within our transport approach . we note that our results are consistent with other microscopic transport approaches @xcite . by comparison , attractive orbit scatterings drastically reduce the @xmath8 slope , and explain the star data at @xmath30 as shown in fig . [ fig : starv1mid ] ( solid lines ) ; at @xmath11 and @xmath31 gev , the @xmath8 slope becomes almost zero and negative , respectively . at lower energy @xmath32 , results with attractive orbits are far from the data , and there should not be large eos softening . from this analysis , we find that the softening of the eos affects the directed flow of protons at midrapidity and should emerge in the beam energy range of @xmath30 , but its effects should be small at @xmath32 . since na49 data at @xmath33 gev may also indicate evidence of softening of the eos @xcite , the onset beam energy of the softening might be lower than 10 gev . therefore , detailed experimental studies are needed around the beam energies of @xmath34 gev . unfortunately , the eos softening effects are not easy to see when the @xmath8 slope is already negative in the standard cascade . as seen in fig . [ fig : starv1mid ] , the proton @xmath8 slope at @xmath35 and pion @xmath8 slopes are negative in the standard cascade from geometrical non - qgp effects @xcite and from absorption by baryons @xcite , respectively . it should be noted , however , that jam with attractive orbits overestimates the negative slope of the proton @xmath8 indicating the need to reharden the eos , i.e. ; matter created at this collision energy reaches well above the transition region or weak softening of the eos due to less net baryonic density . , but for central collisions ( 0 - 10% ) . , width=340 ] we now discuss the directed flows in central collisions in order to distinguish the geometrical effects from phase transition . since sign change of the proton @xmath8 is purely geometrical and only happens at large impact parameters in standard hadronic transport models , it is possible to find the effects of the softening sharply in central collisions . we show directed flows in central collisions in fig . [ fig : starv1cent ] . star data on the proton @xmath8 do not show negative slope for central collisions . the standard jam cascade describes well the data at 7.7 gev , indicating that the hadronic description may be reasonable at 7.7 gev . it is seen that the proton @xmath8 at 27 gev from both jam with attractive orbits and standard jam simulation yield negative slope . thus 7.7 and 27 gev data do not show a hint of the softening of the eos within our analysis . on the other hand , one sees that jam with attractive orbits again quite reasonably describes the data at 11.5 and 19.6 gev , while the standard jam cascade overestimates the data . therefore , star data on the proton directed flow for both central and mid - central collisions indicate evidence of the softening of the eos . gev . in the jam calculations , momentum dependent hadronic mean - field potentials are included ( jam / ms ) . the dotted lines correspond to the result from the standard jam / ms model , while the solid lines are for jam / ms with attractive orbit results . symbols show star data @xcite . , width=340 ] it is also necessary to examine the influence of the nuclear mean field on the directed flow at midrapidity , since the mean field can also modify the flows . nuclear mean fields of hadrons are included based on the framework of simplified version of relativistic quantum molecular dynamics ( rqmd / s ) in ref . density dependent skyrme - type and momentum dependent yukawa potentials are employed as in ref . @xcite , but with slightly different parameter sets which yields the incompressibility of @xmath36 mev @xcite . in fig . [ fig : ms11 ] , we show the calculated results of the directed flow of protons and pions at @xmath11 gev from jam with momentum dependent potentials ( jam / ms ) together with the star data . the mean field slightly reduces the proton directed flow , but the basic trend is the same as the jam cascade result . it is interesting to see that attractive orbits supplemented by the mean field yields negative slope , and provides a better description of the data than the cascade calculation at midrapidity . and 27 gev . full ( open ) circles and full ( open ) diamonds represent the pressures @xmath37 at from standard jam ( jam with attractive orbits ) at 7.7 and 27 gev , respectively . triangles and boxes represent the transverse part of the pressure @xmath38 for the jam standard at 7.7 and 27 gev , respectively . the dashed and sold lines represent the eos from hadron gas and the eos with a first - order phase transition used in ref . , width=321 ] we would like to see how much pressure is suppressed by imposing attractive orbits . the free streaming part of the local isotropic pressure @xmath39 can be computed from the the energy - momentum tensor @xmath40 as @xmath41 , with the projector of @xmath42 , where @xmath43 is a hydrodynamics velocity defined by the landau and lifshitz definition that may be solved iteratively @xcite . the pressure difference @xmath44 from the free streaming @xmath39 caused by the two - body collision between particles @xmath16 and @xmath17 at the space - time coordinates of @xmath45 and @xmath46 is estimated based on the formula given by ref . @xcite : @xmath47 where @xmath48 is the lorentz invariant local particle density , @xmath49 is the proper time interval between successive collisions , and @xmath50 is the energy - momentum change of the particle @xmath16 . we extract the `` effective eos '' by accumulating statistics by computing local pressure @xmath51 and energy density @xmath52 at each collision point in the jam simulation in the central region of the reaction zone specified by the longitudinal region @xmath53 fm and the transverse radius of less than 3 fm in au+au collisions at @xmath54 , 11.5 , 19.6 , and 27 gev . we have checked that the volume dependence on the eos extracted from the simulation is very weak . in fig . [ fig : eosatt ] , pressure @xmath37 is plotted as a function of energy density @xmath55 from jam simulations as well as the ideal hadron gas eos and the eos with a first - order phase transition ( eos - q ) @xcite at vanishing baryon chemical potential . we first discuss the eos in the standard jam simulation . we see some beam energy dependence of the effective eos as reported in ref . @xcite in which the eos was extracted to be @xmath56 from ags to super proton synchrotoron ( sps ) energies that is quite similar to our results . there is a deviation of the effective eos from the hadron resonance gas eos at higher energy densities , which is mainly due to the nonequilibrium evolution of the system since the high energy density parts are extracted from early times where the system is far from the equilibrium state . in particular , pressure in the compression stage of the reaction tends to be much higher than the values expected from the equilibrium eos . we note that the transverse pressure @xmath57 , where @xmath58 is the energy - momentum tensor at a local rest frame , is lower than the equilibrium hadron resonance gas eos in high energy density regions as seen in our results in fig . [ fig : eosatt ] which are also reported in ref . it is expected that the deviation of the effective pressure in jam compared to the equilibrium hadron gas mainly comes from the bulk viscous pressure and/or chemical composition . note that the difference between transverse and longitudinal pressure ( shear - stress tensor ) should not contribute to the isotropic pressure defined through the isotropic projection of the energy - momentum tensor . in the compression phase , the bulk pressure should give a positive correction to the equilibrium pressure , but in the expansion stage , the sign should change . judging by fig . [ fig : eosatt ] , such sign change is , at least , not very pronounced . so it indicates that the main contribution to the hardening of the eos at high energy densities in jam is from the chemical composition ; namely , the system at early stages of the reaction is highly out of the chemical equilibrium state . at lower energy densities , transverse pressure @xmath38 is close to the isotropic pressure @xmath37 , showing the kinetic equilibration of the system , and transverse pressure departs from isotropic pressure above @xmath59 gev/@xmath5 depending on the beam energy . chemical equilibrium of the system in the time evolution of hadron transport models was investigated in ref . @xcite , and it was found that it reaches at late times , which is consistent with the finding here that the effective eos from standard jam is close to the ideal hadron resonance eos at lower energy densities . after equilibration , pressure from standard jam approaches values close to the ideal hadron resonance gas eos which is seen in fig . [ fig : eosatt ] when the energy density is less than 1 gev/@xmath5 . as energy density drops further , the standard jam yields slightly less pressure than that of the ideal gas eos , because of the chemical freeze - out @xcite . the effective eos from jam with attractive orbits is compared with the eos from the standard jam simulation in fig . [ fig : eosatt ] . when attractive orbits are selected for all two - body scatterings in jam , we see a significant reduction of the pressure , yielding a similar amount of softening in the transition region as eos - q , although our effective eos does not exhibit a sharp first - order phase transition , since we do not impose any specific condition to allow for an attractive orbit . we will examine which part of the eos is responsible for the collapse of proton directed flow in the next sections . let us now examine where the negative slope is generated . time evolution of the sign weighted directed transverse momentum integrated over the rapidity range of @xmath60 , @xmath61 for baryons is displayed in fig . [ fig : v1evol ] in semicentral au+au collision for both the standard jam cascade and jam with attractive orbits . in the standard jam cascade , directed flow of baryons rises in the early states of the reaction before two nuclei pass through each other , and decreases with time . at late times , it rises slowly again with time . integrated over midrapidity for baryons in semicentral au+au collisions for @xmath62 gev from ( a ) standard jam and ( b ) jam with attractive orbits simulations . time evolutions of normalized net - baryon density and energy density from the standard jam calculations are shown in the panels ( c ) and ( d ) , respectively . baryon density and energy density are averaged over a cylindrical volume of transverse radius 3 fm and longitudinal distance of 1 fm centered at the origin . those particles which have not interacted yet are not included in the calculations of @xmath63 , @xmath64 , and @xmath55 . , width=302 ] in the jam cascade with attractive orbits , on the other hand , it is observed that directed flow is strongly modified to be negative by the reduction of pressure at early times , especially at lower energies , and directed flow always increases as a function of reaction time . we note that the slope of directed flow stays negative also in the rescattering stages of the reaction ; the period long after two nuclei pass through each other . as expected , directed flow becomes smaller as collision energy becomes higher , due to less interaction time @xcite . the range of beam energies covered by bes is quite interesting . the crossing times of the two gold nuclei in the c. m. frame are approximately 3.27 , 2.15 , 1.25 , and 0.906 fm/@xmath65 at @xmath66 and 27 gev , respectively . thus , hadronic rescatterings start in au+au collisions before passing through two nuclei at @xmath54 gev , because crossing time is much longer than the hadronization time from string fragmentation which is typically 1 fm/@xmath65 . on the other hand , at @xmath29 gev , most of the hadronic rescattering among produced particles occur after two nuclei pass through each other . in ref . @xcite , a wiggle structure in the rapidity dependence of proton directed flow was predicted at @xmath67 gev in which initial nucleon - nucleon collisions are well isolated from the late hadronic rescatterings , and it is argued that a wiggle structure appears as a result of the correlation between the position of a nucleon and its stopping power due to initial glauber type nucleon - nucleon collisions . the negative directed flow seen in the hadronic transport models at @xmath29 gev is due to the same reason . at energies @xmath68 gev , this correlation is contaminated by meson - baryon collisions , since mesons and baryons start interacting with each other before the two nuclei pass though each other . we now look at the effects of rescatterings between mesons and baryons in the bes energy region . in fig . [ fig : v1bb ] , we show the rapidity dependence of @xmath8 for both protons and pions for the jam simulation without meson - baryon , and meson - meson collisions . we see the wiggle structure in the rapidity dependence of the proton @xmath8 for all of the beam energies , and pion directed flow is very small but slightly positive . the appearance of wiggle structure at lower beam energies when one switches off meson - baryon scatterings may be partly due to the similar mechanism as pointed out by ref . @xcite . thus we conclude that baryon - baryon collisions alone generates negative proton directed flow at midrapidity , and meson - baryon collisions bring the proton directed flow to the positive side and the pion directed flow to the negative side . this implies that strong negative directed flow must be generated at the initial stages of nuclear collisions to get the negative proton flow at freeze - out . , but jam cascade simulations with only baryon - baryon collisions . , width=302 ] in order to examine further in detail how @xmath8 is generated , we plot in fig . [ fig : v1evolbb ] the time evolution of @xmath69 from the jam simulations by switching off all baryon - meson and meson - meson collisions ( jam bb only ) . the upper panel of fig . [ fig : v1evolbb ] shows the time evolutions of @xmath63 from jam simulations with only baryon - baryon collisions . it shows that positive directed flow is first generated before two nuclei pass though each other , and then it becomes negative in the expansion stages of the reaction . the lower beam energy yields larger negative directed flow . this is because of the effect of spectators as well as the increasing number of baryon - baryon collisions at lower beam energies . the average number of baryon - baryon collisions @xmath70 in au+au semicentral collisions is larger at lower beam energy : @xmath71 , and 310 for @xmath28 and 27 gev , respectively . in the middle panel of fig . [ fig : v1evolbb ] , we display the time evolution of @xmath63 in jam bb only with attractive orbits . as in the case of full simulation , strong negative directed flow is generated in the early stages of the reaction and it rises in time , then it stays the same value at later times because of the absence of hadronic rescatterings . the question then arises , which part of the reaction stage is more relevant for the negative directed flow of protons ? for protons and pions in semicentral au+au collisions for @xmath11 gev from jam simulation in which attractive orbits are imposed at different time intervals ; the solid line corresponds to the simulation with attractive orbits for time @xmath72 fm/@xmath65 , the dashed line for @xmath73 fm/@xmath65 , and the dotted line for @xmath74 fm/@xmath65 . , width=321 ] to try to answer the question , we plot in fig . [ fig : v1tcut ] the rapidity dependence of @xmath8 in au+au semicentral collision for three different jam simulations at @xmath11 gev by noting that the crossing time of two nuclei is about 2.15 fm/@xmath65 : ( 1 ) jam with attractive orbits only for compression stages of the reaction until two nuclei reach full overlap , i.e. time less than 1 fm/@xmath65 ; ( 2 ) jam with attractive orbits at the reaction time between 1 and 2 fm/@xmath65 which corresponds to the time range with largest baryon densities . ( 3 ) jam with attractive orbits at times later than 2 fm/@xmath65 ; and as we expect , when attractive orbits are imposed at times later than 2 fm/@xmath65 , it is too late to generate negative flow , and the result is almost identical to the standard jam simulation . furthermore , it is very interesting to see that the effect of attractive orbits at times earlier than 1 fm/@xmath65 is small and its effect alone can not explain the strong suppression of the flow . to see this point more clearly , time evolution of @xmath63 is displayed in fig . [ fig : v1tcuteos ] . one see that strong negative directed flow generated in the earliest stages of the reaction quickly disappears with a much faster rate than the one shown in the lower panel of fig . [ fig : v1evol ] . thus initial scatterings at the compression stage of the reaction are not important in generating negative @xmath8 . finally , it is shown that what is responsible for the strong suppression of the proton flow is the effect of attractive orbits in the time interval @xmath75 fm/@xmath65 , which coincides with the highest baryon density in the course of the reaction at @xmath11 gev . this effect can be further confirmed in the time evolution of @xmath63 in fig . [ fig : v1tcuteos ] . attractive orbits at this stage are important to suppress the rise of @xmath8 due to the hadronic rescatterings . for baryons ( upper panel ) and effective eos ( lower panel ) in au+au collision at @xmath11 gev from jam simulations in which attractive orbits are imposed at different time intervals ; the circles correspond to the simulation with attractive orbits for time @xmath72 fm/@xmath65 , the squares for @xmath73 fm/@xmath65 , and the triangles for @xmath74 fm/@xmath65 . , title="fig:",width=321 ] for baryons ( upper panel ) and effective eos ( lower panel ) in au+au collision at @xmath11 gev from jam simulations in which attractive orbits are imposed at different time intervals ; the circles correspond to the simulation with attractive orbits for time @xmath72 fm/@xmath65 , the squares for @xmath73 fm/@xmath65 , and the triangles for @xmath74 fm/@xmath65 . , title="fig:",width=302 ] effective eos for each simulation are plotted together with the eos of the standard jam simulation in the lower panel of fig . [ fig : v1tcuteos ] . the effect of the attractive orbits for @xmath76 fm/@xmath65 is the slight reduction of pressure at high energy densities ; on the other hand , attractive orbits for @xmath77 fm/@xmath65 reduce the pressure at lower energy densities . attractive orbits at @xmath78 fm/@xmath65 strongly reduces the pressure at high energy densities which results in the collapse of proton directed flow . however , it does not necessarily imply that the equilibrium eos at high energy density at high baryon density needs to be very soft for the negative @xmath8 , since most of pressures at energy densities above 4 gev/@xmath5 are extracted from preequilibrium stages of the reaction in the jam simulation . nonequilibrium effects need to be examined . nevertheless , it suggests the need of a nonstandard dynamical effect which is related to the reduction of pressure at high baryon densities . this analysis strongly suggests the importance of reaction dynamics at high baryon density . the time period of @xmath78 fm / c at @xmath11 gev relevant to the negative flow corresponds to the preequilibrium stage in the jam hadronic transport approach , even though produced hadrons start to interact with each other . we do not have partonic interactions , unlike the phsd model @xcite . the effects of partonic interactions in the early stages of the reaction should be examined elsewhere in order to understand the dynamical effects on the directed flow . in order to see systematics on the use of the attractive orbits in the scattering style , it is important to check other flow harmonics such as elliptic flow @xmath79 . in fig . [ fig : v2ch ] , jam cascade results are compared with the pseudorapidity dependence of @xmath80 for charged hadrons in midcentral ( 10 - 40% ) au+au collisions at @xmath28 and @xmath81 gev @xcite . it is seen that @xmath80 at midrapidity is not modified by the attractive orbits scattering style , but it underestimates the data about 2030% . therefore , we do not see any softening effects on @xmath80 within our approach . underestimation of @xmath80 by our approach suggests the need of partonic interactions in the early stages of the reactions . we have also studied other inclusive hadronic observables such as transverse momentum distributions and rapidity distributions and found that the effect of attractive orbits on them is very small . it is also important to examine other observables such as the net - baryon number cumulants in the same energy range @xcite . if the softening of the eos comes from criticality around the critical point , divergence of cumulants appears as oscillating behavior as a result of smearing by the finite quark mass @xcite or finite volume @xcite . thus it is an interesting question whether dynamical model calculations with the eos softening can describe the observed nonmonotonic behavior of cumulant ratios @xcite . recently , it was shown that jam with attractive orbits as well as nuclear mean - field effects does not describe the observed large enhancement of cumulant ratios @xcite . so far , we implement attractive orbits in jam for all hadron - hadron @xmath25 scatterings without any restrictions . as a result , our equation of state is soft for all energy densities as shown in fig . [ fig : eosatt ] . we now explore the eos dependence of the directed flow . for this purpose , instead of imposing attractive scattering all the way , we select attractive orbits at each two - body scattering with the probability @xmath82 given by @xmath83 where @xmath84 is a pressure as a function of energy density @xmath55 from a given eos , and @xmath39 is the local pressure at the collision point computed from the energy - momentum tensor as in sec . [ sec : eos ] . the qcd equation of state at high baryon densities is not well understood . as a first step , we use the eos which does not depend on baryon density for simplicity , since our purpose here is to check the systematics of our approach that modifies the scattering style . we test the eos with crossover from lattice qcd ( @xmath85-v1.1 ) taken from ref . @xcite , and the eos with a first - order phase transition similar to that of eos - q with the modification of the slope in the qgp phase to @xmath86 ( instead of the massless ideal gas eos @xmath87 ) so that pressure at high energy densities is consistent with the lattice eos as shown in fig . [ fig : jameos ] . we also show in fig . [ fig : jameos ] the results which are extracted from jam simulations to ensure that our simple approach is consistent with a given eos . it is seen that our simple approach works very well to modify the eos of the system . for protons and pions in semicentral au+au collisions for @xmath11 gev from jam simulations with different eos . the solid line presents the jam result with the eos with fist - order phase transition , and the dashed line presents the jam result with the crossover eos . the dotted line is for the standard jam result.,width=321 ] in fig . [ fig : v1eos ] , we plot the rapidity dependence of the directed flow of protons ( upper panel ) and pions ( lower panel ) in au+au semicentral collision at @xmath11 gev obtained from the eos with crossover and first - order phase transition . while pion flow is not sensitive to the eos , one sees that both eoss yield the suppression of proton directed flow compared to the standard cascade simulation , since both eoss are softer than the effective eos from the standard jam cascade simulation as compared in fig . [ fig : v1eos ] . proton flow at midrapidity @xmath88 is not sensitive to the eos , but two eos give different behavior for larger rapidities indicating that the softening point of the current eos is responsible to the rapidity @xmath89 at @xmath11 gev , and explicit eos dependence of the directed flow may be observable in the experiments . it remains for further work to establish the eos dependence of the directed flow by utilizing a fully baryon density dependent eos . the interactions used here do not employ any baryon density dependent interactions , @xmath90 , as one may want to use in relativistic mean - field models of high baryon density matter . it is also possible that @xmath91 matter with slightly nonuniversal scalar attraction can easily cause a first - order phase transition without mentioning any high baryon density qcd . we will present a detailed systematic study of the eos dependence of the directed flow elsewhere . in summary , we have investigated the effect of the softening of the eos on the directed flow of protons and pions within a microscopic transport approach . the transport model jam with standard stochastic two - body scattering style predicts the large positive slope of proton @xmath8 at collision energy below @xmath92 gev , and the negative slope of proton @xmath8 only at higher collision energy @xmath93 gev , which disagree with the star data . however , softening effects of the eos simulated by attractive orbit scatterings lead to a dramatic change in the dynamics , and yield significant reduction of proton @xmath8 which well describe the star data around the minimum of @xmath6 at @xmath94 gev . the softening effects were not needed in the present approach at lower energies , @xmath32 . we found that attractive orbit scattering style does not modify elliptic flow at mid - rapidity . we show that this softening effect is needed only at early stages of the reaction where the system reaches the high baryon density state at midrapidity . we also proposed a simple recipe to simulate a given eos within a hadronic transport model , and compared two different eos . we saw an eos dependence of the proton directed flow in the forward rapidities . more detailed systematic studies are needed , using a fully baryon density dependent eos , in order to draw a conclusion that the minimum of @xmath6 is a result of the softening of the eos which may be caused by a first - order phase transition @xcite . a possible scenario to fully explain the beam energy dependence of the directed flow may be described as follows . we assume that there exists the softest point in the energy density range reachable at @xmath95 . hadrons will feel an attractive force when they go across the surface of the soft region . this additional force can be simulated by introducing the attractive orbit scatterings among hadrons , as we have demonstrated in the present work , and negative @xmath6 emerges . one may need to introduce new degrees of freedom other than hadrons to understand the rehardening at higher energies . it seems obvious to infer a softening of the eos from the experimentally observed collapse of net - proton flow when the c.m . energy is increased from 7 to 11 gev . however , the statement of a discovery of the `` softening '' of the eos from the net - proton @xmath8 data shows even more convincing evidence for the `` phase transition '' as we observe the rebound at higher energies ; namely the star - observed second change of sign of the @xmath8 values of the net protons at @xmath96 gev back to positive @xmath8 at higher energies @xcite . this shows that the soft region is overcome , and the directed flow picks up steam again , due to the rehardening of the eos at considerably larger energy densities . in the near future , a more detailed analysis of the softening effect should be addressed by employing a realistic eos which is consistent with the lattice qcd result . because of the nonequilibrium evolution , the pressure generation due to the two - body collision @xmath44 depends in the current study not only on the difference between the equilibrium eoss , but also on the dynamical evolution of the system . perhaps a more justified way of fixing the eos would be to look at a fully equilibrated system in a box , and then determine the probabilities for attractive orbits as a function of the energy of the colliding particles , so that a given ( equilibrium ) eos would be reproduced ; then the microscopic dynamics would tell how the system looks in out - of - equilibrium situations . it is expected that properties of the eos at high baryon density may be probed sensitively by using the flow . future experiments such as the bes ii at rhic @xcite , fair @xcite , nica @xcite , and j - parc @xcite should clarify this point at lower collision energies @xmath97 gev . we would like to thank adrian dumitru for valuable comments . h.s . thanks nu xu , zangbu xu , declan keane , and paul sorensen for numerous useful discussions . thanks the frankfurt institute of advanced studies where part of this work was done . this work was supported in part by the grants - in - aid for scientific research from jsps ( no . 15k05079 , no . 15h03663 and no . 15k05098 ) , the grants - in - aid for scientific research on innovative areas from mext ( no . 24105001 and no . 24105008 ) , and by the yukawa international program for quark - hadron sciences . has received funding from the european union s horizon 2020 research and innovation programme under marie sklodowska - curie grant agreement no . 655285 and from the helmholtz international center for fair within the framework of the loewe program launched by the state of hesse . l. adamczyk _ et al . _ [ star collaboration ] , phys . rev . lett . * 112 * , no . 16 , 162301 ( 2014 ) . i. arsene _ et al . _ [ brahms collaboration ] , nucl . phys . a * 757 * , 1 ( 2005 ) ; b. b. back _ et al . _ , nucl . phys . a * 757 * , 28 ( 2005 ) ; j. adams _ et al . _ [ star collaboration ] , nucl . phys . a * 757 * , 102 ( 2005 ) ; k. adcox _ et al . _ [ phenix collaboration ] , nucl . phys . a * 757 * , 184 ( 2005 ) . y. aoki , g. endrodi , z. fodor , s. d. katz and k. k. szabo , nature * 443 * , 675 ( 2006 ) . d. h. rischke , prog . * 52 * , 197 ( 2004 ) ; m. a. stephanov , prog . suppl . * 153 * , 139 ( 2004 ) [ int . j. mod . a * 20 * , 4387 ( 2005 ) ] ; k. fukushima and t. hatsuda , rept . phys . * 74 * , 014001 ( 2011 ) ; k. fukushima and c. sasaki , prog . * 72 * , 99 ( 2013 ) . h. stoecker , nucl . a * 750 * , 121 ( 2005 ) . h. g. baumgardt , j. u. schott , y. sakamoto , e. schopper , h. stoecker , j. hofmann , w. scheid and w. greiner , z. phys . a * 273 * , 359 ( 1975 ) . w. scheid , h. mller and w. greiner , phys . lett . * 32 * , 741 ( 1974 ) ; j. kapusta and d. strottman , phys . b * 106 * , 33 ( 1981 ) ; h. stoecker _ et al . _ , c * 25 * , 1873 ( 1982 ) . h. a. gustafsson _ et al . lett . * 52 * , 1590 ( 1984 ) . y. b. ivanov , e. g. nikonov , w. noerenberg , a. a. shanenko and v. d. toneev , heavy ion phys . * 15 * , 117 ( 2002 ) ; v. d. toneev , e. g. nikonov , b. friman , w. norenberg and k. redlich , eur . j. c * 32 * , 399 ( 2003 ) . l. p. csernai and d. rohrich , phys . b * 458 * , 454 ( 1999 ) . l. p. csernai , a. anderlik , c. anderlik , v. k. magas , e. molnar , a. nyiri , d. rohrich and k. tamosiunas , acta phys . hung . a * 22 * , 181 ( 2005 ) . j. brachmann , s. soff , a. dumitru , h. stoecker , j. a. maruhn , w. greiner , l. v. bravina and d. h. rischke , phys . c * 61 * , 024909 ( 2000 ) . j. brachmann , a. dumitru , h. stoecker and w. greiner , eur . phys . j. a * 8 * , 549 ( 2000 ) . h. appelshauser _ et al . _ [ na49 collaboration ] , phys . rev . lett . * 80 * , 4136 ( 1998 ) . c. alt _ et al . _ [ na49 collaboration ] , phys . c * 68 * , 034903 ( 2003 ) . m. isse , a. ohnishi , n. otuka , p. k. sahu and y. nara , phys . c * 72 * , 064908 ( 2005 ) . y. nara , n. otuka , a. ohnishi , k. niita and s. chiba , phys . c * 61 * , 024901 ( 2000 ) . h. sorge , phys . c * 52 * , 3291 ( 1995 ) . s. a. bass _ et al . . part . * 41 * , 255 ( 1998 ) . m. bleicher _ et al . _ , j. phys . g * 25 * , 1859 ( 1999 ) . m. gyulassy , k. a. frankel and h. stoecker , phys . b * 110 * , 185 ( 1982 ) . d. e. kahana , d. keane , y. pang , t. schlagel and s. wang , phys . lett . * 74 * , 4404 ( 1995 ) ; d. e. kahana , y. pang and e. v. shuryak , phys . rev . c * 56 * , 481 ( 1997 ) . h. sorge , phys . lett . * 82 * , 2048 ( 1999 ) . p. danielewicz and s. pratt , phys . c * 53 * , 249 ( 1996 ) . s. a. bass , c. hartnack , h. stoecker and w. greiner , phys . c * 51 * , 3343 ( 1995 ) doi:10.1103/physrevc.51.3343 [ nucl - th/9501002 ] . p. f. kolb , j. sollfrank and u. w. heinz , phys . b * 459 * , 667 ( 1999 ) ; p. f. kolb , j. sollfrank and u. w. heinz , phys . c * 62 * , 054909 ( 2000 ) ; h. song and u. w. heinz , phys . c * 77 * , 064901 ( 2008 ) . l. v. bravina _ et al . _ , phys . c * 60 * , 024904 ( 1999 ) ; phys . c * 78 * , 014907 ( 2008 ) .
|
we analyze the directed flow of protons and pions in high - energy heavy - ion collisions in the incident energy range from @xmath0 to 27 gev within a microscopic transport model .
standard hadronic transport approaches do not describe the collapse of directed flow below @xmath1 gev .
by contrast , a model that simulates effects of a softening of the equation of state describes well the behavior of directed flow data recently obtained by the star collaboration @xcite .
we give a detailed analysis of how directed flow is generated .
particularly , we found that softening of the effective equation of state at the overlapping region of two nuclei , i.e. the reaction stages where the system reaches high baryon density state , is needed to explain the observed collapse of proton directed flow within a hadronic transport approach .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the system of degenerate coupled multi - field kdv equations is given as @xcite-@xcite @xmath6 where @xmath7 and @xmath8 . in @xcite-@xcite , it was shown that this system is also a degenerate kdv system of rank one . in a previous work @xcite we focused on the equation ( [ eqn1 ] ) for @xmath9 . we reduced this equation into an ode @xmath10 where @xmath11 is a polynomial function of degree four . we analyzed all possible cases about the zeros of @xmath11 . due to this analysis we determined the cases when the solution is periodic or solitary . when the polynomial has one double @xmath12 and two simple zeros @xmath13 , @xmath14 with @xmath15 , or one triple and one simple zeros , the solution is solitary . other cases give periodic or non - real solutions . by using the jacobi elliptic functions @xcite , we obtained periodic solutions and all solitary wave solutions which rapidly decay to some constants , explicitly . we have also shown that there are no real asymptotically vanishing traveling wave solutions for @xmath9 . indeed we have the following theorem for the degenerate coupled @xmath0-kdv equation , @xmath16 . the degenerate coupled @xmath0-kdv equation can be reduced to the equation @xmath17 by taking @xmath0 functions as @xmath18 , @xmath19 , , @xmath20 , where @xmath21 in ( [ eqn1 ] ) . here @xmath22 is a polynomial of @xmath3 of degree @xmath23 . if we apply the asymptotically vanishing boundary conditions to ( [ genpoly ] ) , we have @xmath24 [ thmrealtravel ] when @xmath25 odd , we have @xmath26 , and then the degenerate coupled @xmath0-kdv equation has real traveling wave solution with asymptotically vanishing boundary conditions , but when @xmath25 even , the constant @xmath27 . hence the equation does not have a real traveling wave solution with asymptotically vanishing boundary conditions . * proof . * consider the degenerate coupled @xmath0-kdv equation ( [ eqn1 ] ) . let @xmath18 , @xmath19 , , @xmath20 where @xmath21 . by using the first @xmath28 equations we obtain all functions @xmath29 , @xmath30 as a polynomial of @xmath31 . we get @xmath32 where @xmath33 are constants , @xmath34 and @xmath35 are polynomials of @xmath3 of degree @xmath36 and @xmath37 , @xmath38 , @xmath39 , respectively . now use the @xmath0th equation . we obtain @xmath40 integrating above equation once we get @xmath41 by using @xmath42 as an integrating factor , we integrate once more . finally , we obtain @xmath43 where @xmath44 is a polynomial of @xmath3 of degree @xmath36 , @xmath45 . when @xmath0 is odd , the coefficient of @xmath46 , that is @xmath47 , is positive , and when @xmath0 is even , @xmath47 is negative . applying asymptotically vanishing boundary conditions , we get @xmath48 , where @xmath49 . hence for @xmath25 odd , the degenerate coupled @xmath0-kdv equation has real traveling wave solution with asymptotically vanishing boundary conditions , but when @xmath25 even , it does not . @xmath50 in this work , we study the equation ( [ eqn1 ] ) for @xmath51 which is @xmath52 in detail . it is clear from theorem [ thmrealtravel ] that unlike the case @xmath9 , we have real traveling wave solution with asymptotically vanishing boundary conditions in @xmath51 case . here the system ( [ l=3a])-([l=3c ] ) reduces to a polynomial of degree five , @xmath53 where @xmath54 , and @xmath55 are constants . when the degree of the polynomial in the reduced equation is equal to five or greater it is almost impossible to solve them . as far as we know there is no known method to solve these equations . we shall introduce two methods to solve such equations . the first one is based on the chebyshev s theorem @xcite which is used recently to solve the einstein field equations for a cosmological model @xcite , @xcite . by using the chebyshev s theorem we give several solutions of the reduced equations for @xmath51 and also for arbitrary @xmath0 . the second method is based on the factorizing the polynomial @xmath22 as product of lower degree polynomials . in this way we make use of the reduced equations of lower degrees . we have given all possible such solutions for @xmath51 . the layout of our paper is as follows : in sec . ii , we study the behavior of the solutions in the neighborhood of the zeros of @xmath56 and discuss the cases giving solitary wave solutions . in sec . iii , we find exact solutions of the system ( [ l=3a])-([l=3c ] ) by a new method proposed for any @xmath57 which uses the chebyshev s theorem and analyze the cases in which we may have solitary wave and kink - type solutions . in sec . iv , we present an alternative method . particularly , we find the solutions of the system ( [ l=3a])-([l=3c ] ) . here we obtain many solutions for @xmath51 including solitary wave , kink - type , periodic , and unbounded solutions . we present some of them in the text but the rest of the solutions are given in the appendices a and b. here we will analyze the zeros of @xmath56 in ( [ p_5(f ) ] ) . * ( i ) * if @xmath59 is a _ simple zero _ of @xmath56 we have @xmath60 . taylor expansion of @xmath56 about @xmath13 gives @xmath61 from here we get @xmath62 and @xmath63 . hence we can write the function @xmath31 as @xmath64 thus , in the neighborhood of @xmath65 , the function @xmath31 has local minimum or maximum as @xmath66 is positive or negative respectively since @xmath63 . + * ( ii ) * if @xmath59 is a _ double zero _ of @xmath56 we have @xmath67 . taylor expansion of @xmath56 about @xmath13 gives @xmath68 to have real solution @xmath3 , we should have @xmath69 . from the equality ( [ doublezero ] ) we get @xmath70 which gives @xmath71 where @xmath72 is a constant . hence @xmath73 as @xmath74 . the solution @xmath3 can have only one peak and the wave extends from @xmath75 to @xmath76 . + * ( iii ) * if @xmath59 is a _ triple zero _ of @xmath56 we have @xmath77 . taylor expansion of @xmath56 about @xmath13 gives @xmath78 this is valid only if both signs of @xmath79 and @xmath80 are same . hence , to obtain real solution @xmath3 we have the following two possibilities : @xmath81 @xmath82 and @xmath83 + @xmath84 @xmath85 and @xmath86 + if @xmath82 and @xmath87 then we have @xmath88 which gives @xmath89 where @xmath90 is a constant . thus @xmath73 as @xmath91 . let @xmath85 and @xmath92 . then @xmath93 which yields @xmath94 where @xmath95 is a constant . thus @xmath73 as @xmath91 . * ( iv ) * if @xmath59 is a _ quadruple zero _ of @xmath56 , then we have @xmath96 . in this case taylor expansion of @xmath56 about @xmath13 gives @xmath97 this is valid only if @xmath98 . then we have @xmath99 which gives @xmath100 where @xmath101 is a constant . thus @xmath73 as @xmath91 . * ( v ) * if @xmath59 is a _ zero of multiplicity 5 _ of @xmath56 , then we have @xmath102 . this is valid only if @xmath103 . so we obtain the solution @xmath3 as @xmath104 where @xmath105 is a constant . hence @xmath73 as @xmath106 . we analyze all possible cases about the zeros of @xmath56 that may give solitary wave solutions . here in each cases we will present the sketches of the graphs of @xmath56 . real solutions of @xmath107 occur in the shaded regions . * ( 1 ) one double and three simple zeros . * having one double and three simple zeros ] in figure 1.*(b ) * , @xmath13 , @xmath14 , and @xmath108 are simple zeros and @xmath12 is a double zero . the real solution occurs when @xmath3 stays between @xmath13 and @xmath12 or @xmath12 and @xmath14 . at @xmath13 , @xmath109 hence graph of the function @xmath3 is concave up at @xmath110 . at double zero @xmath12 , @xmath111 as @xmath106 . hence we have a solitary wave solution with amplitude @xmath112 . similarly at @xmath14 , @xmath113 , hence graph of the function @xmath3 is concave down at @xmath114 . therefore , we also have a solitary wave solution with amplitude @xmath115 . now consider the graph * ( d ) * in figure 1 . for @xmath14 we have @xmath116 thus graph of the function is concave up at @xmath114 . at double zero @xmath108 , @xmath117 as @xmath106 . hence we have a solitary wave solution with amplitude @xmath118 . in other cases we have periodic solutions . * ( 2 ) two double and one simple zeros . * having two double and one simple zeros ] in figure 2.*(f ) * , @xmath13 and @xmath14 are double zeros and @xmath12 is a simple zero . the real solution occurs when @xmath3 stays between @xmath12 and @xmath14 . for @xmath12 we have @xmath119 thus graph of the function is concave up at @xmath120 . at double zero @xmath14 , @xmath121 as @xmath106 . hence we have a solitary wave solution with amplitude @xmath122 . now consider the graph * ( g ) * in figure 2 . here @xmath12 and @xmath14 are double zeros and @xmath13 is a simple zero . the real solution occurs when @xmath3 stays between @xmath13 and @xmath12 or @xmath12 and @xmath14 . for @xmath13 we have @xmath123 thus graph of the function is concave up at @xmath110 . at double zero @xmath12 , @xmath111 as @xmath106 . hence we have a solitary wave solution with amplitude @xmath112 . the other cases give kink , anti - kink type or unbounded solutions . * ( 3 ) one triple and two simple zeros . * having one triple and two simple zeros ] consider the graph * ( h ) * in figure 3 . here @xmath12 and @xmath14 are simple zeros and @xmath13 is a triple zero . the real solution occurs when @xmath3 stays between @xmath13 and @xmath12 . for @xmath12 we have @xmath124 so graph of the function @xmath3 is concave down at @xmath120 . at triple zero @xmath13 , @xmath73 as @xmath106 . hence we may have a solitary wave solution with amplitude @xmath125 . in the graph in figure 3.*(i ) * , @xmath13 and @xmath14 are simple zeros and @xmath12 is a triple zero . the real solution occurs when @xmath3 stays between @xmath13 and @xmath12 . for @xmath13 we have @xmath109 so graph of the function @xmath3 is concave up at @xmath110 . at triple zero @xmath12 , @xmath111 as @xmath106 . hence we may have a solitary wave solution with amplitude @xmath126 . the other case gives periodic solution . * ( 4 ) one quadruple and one simple zeros . * having one quadruple and one simple zeros ] in the graph figure 4.*(l ) * , @xmath13 is a simple zero and @xmath12 is a quadruple zero . the real solution occurs between @xmath13 and @xmath12 . at @xmath13 we have @xmath109 , so graph of the function @xmath3 is concave up at @xmath110 . at quadruple zero @xmath12 , @xmath111 as @xmath106 . hence we may have a solitary wave solution with amplitude @xmath112 . the other case gives unbounded solution . * ( 5 ) one double and one simple zeros . * having one double and one simple zeros ] consider the graph figure 5.*(n)*. here @xmath13 is a simple zero and @xmath12 is a double zero . the real solution occurs between @xmath13 and @xmath12 . at @xmath13 we have @xmath109 , so graph of the function @xmath3 is concave up at @xmath110 . at double zero @xmath12 , @xmath111 as @xmath106 . hence we have a solitary wave solution with amplitude @xmath112 . the other case gives unbounded solution . to sum up , we can give the following proposition for @xmath51 case . [ prozeroanalysis ] equation ( [ p_5(f ) ] ) may admit solitary wave solutions when the polynomial function @xmath56 admits ( i ) one double and three simple zeros ( ii ) two double and one simple zeros ( iii ) one triple and two simple zeros ( iv ) one quadruple and one simple zeros ( v ) one double and one simple zeros . the chebyshev s theorem is given as following @xcite . [ chebthm ] let @xmath127 , @xmath128 , @xmath129 , @xmath72 , @xmath130 be given real numbers and @xmath131 . the antiderivative @xmath132 is expressible by means of the elementary functions only in the three cases : @xmath133 the term @xmath134 is called a differential binomial . note that the differential binomial may be expressed in terms of the incomplete beta function and the hypergeometric function . let us define @xmath135 . then we have @xmath136 where @xmath137 is the incomplete beta function and @xmath138 is the hypergeometric function . our aim is to transform the system ( [ eqn1 ] ) to @xmath139 by taking @xmath140 . we can apply chebyshev s theorem to this equation if we assume that @xmath141 reduces to the form @xmath142 , where @xmath143 . for @xmath51 , let @xmath144 in ( [ p_5(f ) ] ) , then the equation becomes @xmath145 where @xmath146 to apply the chebyshev s theorem [ chebthm ] we assume that @xmath147 reduces to the following form , @xmath148 where @xmath127 , @xmath128 , @xmath129 are the constants in theorem [ chebthm ] , @xmath149 , and @xmath150 is a constant . here we present the cases mentioned in the proposition [ prozeroanalysis ] . other cases are given in appendix a. * let @xmath151 . this form corresponds to the case of one simple or one simple and two double zeros . we have @xmath152 so @xmath153 here @xmath154 , @xmath155 , and @xmath156 . hence @xmath157 for @xmath158 , from ( [ casea1 ] ) , by letting @xmath159 we obtain @xmath160 here choose @xmath161 , @xmath162 , the integration constant @xmath163 , plus sign in ( [ casea1 ] ) , we have @xmath164 and the graph of this solution for @xmath165 is given in figure 6 + ] + this is a kink - type solution . * let @xmath166 . this form corresponds to the case of one simple and one quadruple zeros . we have @xmath167 so @xmath168 here @xmath154 , @xmath169 , and @xmath170 . hence + @xmath171 for @xmath158 , from ( [ casea2 ] ) , by letting @xmath159 we obtain @xmath172 here choose @xmath173 , @xmath162 , the integration constant @xmath163 , plus sign in ( [ casea2 ] ) we have @xmath174 and the graph of this solution for @xmath175 is given in figure 7 + ] * let @xmath176 . this form corresponds to the case of one simple and one double zeros . we have @xmath177 so @xmath178 here @xmath179 , @xmath180 , and @xmath181 . hence @xmath182 for @xmath183 , from ( [ casea3 ] ) , we obtain @xmath184 here choose @xmath185 , @xmath186 , the integration constant @xmath163 , and plus sign in ( [ casea3 ] ) we obtain @xmath187 + and the graph of this solution is given in figure 8 + ] + this is clearly a solitary wave solution . * let @xmath188 . this form corresponds to the case of one double and one triple zeros . we have @xmath189 so @xmath190 here @xmath179 , @xmath169 , and @xmath191 . hence @xmath192 for @xmath183 , from ( [ casea4 ] ) , we obtain @xmath193 here choose @xmath173 , @xmath162 , @xmath163 , and plus sign in ( [ casea4 ] ) we have @xmath194 and the graph of this solution is given in figure 9 + ] * let @xmath195 . this form corresponds to the case of one simple and one quadruple zeros . we have @xmath196 so @xmath197 here @xmath198 , @xmath169 , and @xmath181 . hence @xmath199 for @xmath183 , from ( [ casea5 ] ) , by letting @xmath159 we obtain @xmath200 + take @xmath173 , @xmath162 , @xmath163 , and plus sign in ( [ casea5 ] ) we get @xmath201 and the graph of this solution for @xmath202 is given in figure 10 + ] when we have @xmath1 for the reduced equation it becomes quite difficult to solve such equations for @xmath203 . for this purpose , we shall introduce a new method which is based on the factorization of the polynomial @xmath204 , @xmath203 . let the polynomial @xmath22 have @xmath23 real roots i.e. @xmath205 , @xmath47 is a constant . define a new function @xmath206 so that @xmath207 . hence we have @xmath208 . by taking @xmath209 where @xmath210 , @xmath211 , we get a system of ordinary differential equations . solving this system gives the solution of the degenerate coupled @xmath0-kdv equation . for illustration , we start with a differential equation where we know the solution . consider @xmath212 let @xmath213 so @xmath214 take @xmath215 where @xmath216 . we start with solving the equation ( [ exalternative1 ] ) . we have @xmath217 where @xmath218 and @xmath219 . let @xmath220 . this gives us @xmath221 . after taking the integral we obtain @xmath222 so that @xmath223 now we insert this solution into eq.([exalternative2 ] ) . let @xmath224 . hence we get @xmath225 where @xmath226 , @xmath227 , and @xmath228 is a constant . solving ( [ exampleintegral ] ) and using ( [ solnex ] ) gives @xmath229 which was obtained in @xcite . here @xmath230 is the jacobi elliptic function and @xmath231 is the elliptic modulus satisfying @xmath232 . now we apply our method to the case when the polynomial @xmath2 is of degree @xmath233 . we have several possible cases , but here we shall give the case when the polynomial ( [ p_5(f ) ] ) has five real roots . other cases are presented in appendix b. * case 1 . * if ( [ p_5(f ) ] ) has five real roots we can write it in the form @xmath234 where @xmath235 and @xmath236 are the zeros of the polynomial function @xmath56 . now define a new function @xmath206 so that @xmath207 . hence ( [ p_5(f)second ] ) becomes @xmath237 * i. * take @xmath238 in @xcite we have found the solutions of eq.([caseanewmethod1 ] ) . one of the solutions is @xmath239 where @xmath231 is the elliptic modulus satisfying @xmath240 . we use this solution in the equation ( [ caseanewmethod2 ] ) and get @xmath241 where @xmath242 and @xmath243 . for particular values ; @xmath244 , @xmath245 , @xmath246 , @xmath247 , @xmath173 , @xmath248 , and choosing plus sign in ( [ rhocasea ] ) we get the graph of the solution @xmath31 given in figure 11 ) with ( [ rhocasea ] ) for particular parameters ] this is a solitary wave solution . * * ii.**take @xmath249 where @xmath250 . eq.([casebnewmethod1 ] ) has a solution @xmath251 where @xmath231 satisfies @xmath252 and @xmath253 is a constant . we use this solution in eq.([casebnewmethod2 ] ) and get @xmath254 where @xmath255 is a constant and @xmath256 and @xmath257 . for particular values ; @xmath258 , @xmath259 , @xmath245 , @xmath260 , @xmath261 , @xmath262 , @xmath248 , and choosing plus sign in ( [ rhocaseb ] ) we get the graph of the solution @xmath31 given in figure 12 ) with ( [ rhocaseb ] ) for particular parameters ] this solution is periodic . + * iii . * take @xmath263 where @xmath250 . consider first the equation ( [ casecnewmethod1 ] ) . + we have @xmath264 this equality can be reduced to @xmath265 where @xmath266 , @xmath267 . by making the change of variables @xmath268 , we get @xmath269 where @xmath253 is a constant . thus we obtain @xmath270 which yields @xmath271 now we use this result in ( [ casecnewmethod2 ] ) , @xmath272 where @xmath255 is a constant and @xmath273 and @xmath274 . for particular values ; @xmath275 , @xmath276 , @xmath277 , @xmath278 , @xmath279 , @xmath280 , @xmath281 , and choosing plus sign in ( [ rhocasec ] ) , we get the graph of the solution @xmath31 given in figure 13 ) for particular parameters ] * iv . * take @xmath282 where @xmath250 . consider the eq.([casednewmethod1 ] ) . we have @xmath283 integrating both sides gives @xmath284 we use this result in the equation ( [ casednewmethod2 ] ) @xmath285 where @xmath255 is a constant and @xmath286 for particular values , @xmath258 , @xmath287 , @xmath288 , @xmath289 , @xmath279 , @xmath280 , and @xmath281 , and choosing plus sign in ( [ rhocased ] ) we get the graph of the solution @xmath31 given in figure 14 ) for particular parameters ] this is a solitary wave solution . we study the degenerate @xmath0-coupled kdv equations . we reduce these equations into an ode of the form @xmath290 , where @xmath291 is a polynomial function of @xmath3 of degree @xmath23 , @xmath57 . we give a general approach to solve the degenerate @xmath0-coupled equations by introducing two new methods that one of them uses chebyshev s theorem and the other one is an alternative method , based on factorization of @xmath22 , @xmath57 . particularly , for the degenerate three - coupled kdv equations we obtain solitary - wave , kink - type , periodic , or unbounded solutions by using these methods . this work is partially supported by the scientific and technological research council of turkey ( tbitak ) . + * appendix a. other cases using chebyshev s theorem for degenerate three - coupled kdv equation * * let @xmath292 . this form corresponds to the case of one simple zero or three simple zeros . we have @xmath293 so @xmath294 here @xmath154 , @xmath295 , and @xmath181 . so we have @xmath296 hence we can not obtain a solution through the chebyshev s theorem . * let @xmath297 . this form corresponds to the case of one triple or two simple and one triple zeros . we have @xmath298 so @xmath299 here @xmath300 , @xmath155 , and @xmath181 . so we have @xmath301 hence we can not obtain a solution through the chebyshev s theorem . * let @xmath302 . this form corresponds to the case of one double and one triple zeros . we have @xmath303 so @xmath304 here @xmath300 , @xmath169 , and @xmath156 . hence @xmath305 for @xmath158 , from ( [ caseap3 ] ) , by letting @xmath159 we obtain @xmath306 * let @xmath307 . this form corresponds to the case of a zero with multiplicity five . we have @xmath308 so @xmath309 here @xmath310 , @xmath169 , and @xmath311 . hence @xmath312 for @xmath183 , from ( [ caseap4 ] ) , by letting @xmath159 we obtain @xmath313 here we get @xmath314.\ ] ] * let @xmath315 . this form corresponds to the case of one simple zero . we have @xmath316 so @xmath317 here @xmath310 , @xmath318 , and @xmath181 . we have @xmath319 hence we can not obtain a solution through the chebyshev s theorem . * case 2 . * if ( [ p_5(f ) ] ) has three real roots we can write it in the form @xmath320 where @xmath13 , @xmath12 , and @xmath14 are the zeros of the polynomial function @xmath56 of degree five and @xmath321 . now define a new function @xmath206 so that @xmath207 . hence ( [ p_5(f)threerealroots ] ) becomes @xmath322 * i. * take @xmath323 where @xmath250 . consider the first equation above . we have @xmath324 we know from case 1.ii that eq.([case2inewmethod1 ] ) has a solution @xmath325 where @xmath253 is a constant . we use this solution in ( [ case2inewmethod2 ] ) and we get @xmath326 where @xmath255 is a constant and @xmath327 and @xmath257 . * ii . * take @xmath328 where @xmath250 . consider first the equation ( [ case2iinewmethod1 ] ) . it has a solution @xmath329 as it is found in case 1.iii . we use this solution in ( [ case2iinewmethod2 ] ) and get @xmath330 where @xmath255 is a constant and @xmath331 and @xmath332 . * iii . * take @xmath333 where @xmath250 . consider the equation ( [ case2iiinewmethod1 ] ) . it has a solution @xmath334 as it is obtained in case 1.iv . we use this result in eq.([case2iiinewmethod2 ] ) @xmath335 where @xmath255 is a constant and @xmath336 and @xmath337 * iv . * take @xmath338 where @xmath250 . consider the first equation above . we have @xmath339 where @xmath340 . let @xmath341 . hence the above equality becomes @xmath342 which gives @xmath343 hence after some simplifications and taking the integration constant zero we get @xmath344 we use this result in ( [ case2ivnewmethod2 ] ) @xmath345 where @xmath255 is a constant and @xmath346 and @xmath347 * case 3 . * if ( [ p_5(f ) ] ) has just one real root we can write it in the form @xmath348 where @xmath13 is the zero of the polynomial function @xmath56 of degree five and the constants @xmath349 , @xmath350 are so that @xmath351 for real @xmath3 . * i. * take @xmath353 where @xmath250 . consider the first equation above . it has a solution @xmath354 as it is obtained in case 1.iv . we use this result in ( [ case3inewmethod2 ] ) and get @xmath355 where @xmath255 is a constant . here we do not get a simpler expression than the original equation . hence it is meaningless to use the method of factorization of the polynomial for this case . take @xmath356 where @xmath250 . here for simplicity , we will use the form @xmath357 instead of the polynomial function of degree four above . since @xmath358 , we will take @xmath359 to get an irreducible polynomial . if @xmath360 , then take @xmath361 . from @xmath362 @xmath363 we have @xmath364 where @xmath365 . then we use this function @xmath3 in ( [ case3iinewmethod2 ] ) and get @xmath366 where @xmath367 . now take @xmath360 with @xmath361 . in this case we have @xmath368 which gives the solution @xmath369 now use the equation ( [ case3iinewmethod2 ] ) . we have @xmath370 where @xmath255 is a constant . solving the above equation gives @xmath371 -b\arctan\big(\frac{2a_1a_3}{a_3 ^ 2-a_1 ^ 2+a_2 ^ 2}\big ) \big\}=\pm \sqrt{\mu}\xi+c_2,\ ] ] where @xmath372 and @xmath373
|
traveling wave solutions of degenerate coupled @xmath0-kdv equations are studied . due to symmetry reduction
these equations reduce to one ode , @xmath1 where @xmath2 is a polynomial function of @xmath3 of degree @xmath4 , where @xmath5 in this work . here
@xmath0 is the number of coupled fields .
there is no known method to solve such ordinary differential equations when @xmath5 . for this purpose ,
we introduce two different type of methods to solve the reduced equation and apply these methods to degenerate three - coupled kdv equation .
one of the methods uses the chebyshev s theorem . in this case
we find several solutions some of which may correspond to solitary waves .
the second method is a kind of factorizing the polynomial @xmath2 as a product of lower degree polynomials .
each part of this product is assumed to satisfy different odes .
+ * keywords : * traveling wave solution , degenerate kdv system , chebyshev s theorem , alternative method .
[ section ] [ section ] [ section ] [ section ] [ section ]
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
polymers are repetitive chains of elementary blocks called monomers ( or monomer units ) . copolymers are inhomogeneous polymers , in the sense that each monomer unit carries a charge , and the charge is distributed along the chain in a _ disordered _ way . it is well known that when the medium surrounding the copolymer is made of two solvents , separated for example by an interface , and the solvents interact with the monomers according to the value of the charge , the typical behavior of the copolymer may differ substantially from the case in which the medium is homogeneous . on copolymers there is an extremely extended literature , given above all their practical relevance , see for example @xcite and @xcite and references therein . moreover , for a realistic model of the interface , one should consider the possibility of the presence of impurities or fluctuations in the interface layer , as in @xcite . as an extreme , but very important example , one could consider also the case in which the interactions at the interface are essentially the only relevant ones , see @xcite . in order to be more concrete , let us introduce a specific model , which is just a particular example of the general class we consider . it is based on the process @xmath4 , a simple random walk on @xmath5 , with @xmath6 and @xmath7 , @xmath8 , a sequence of iid random variables with @xmath9 . the process @xmath10 has to be interpreted as a directed polymer in @xmath11dimension , and @xmath12 as its law in absence of any interaction with the environment ( _ free polymer _ ) . the polymer environment interaction depends on the _ charges _ @xmath13 and on four real parameters @xmath14 : without loss of generality we will assume @xmath15 , @xmath16 and @xmath17 to be non negative . let us set @xmath18 and let us introduce the family of boltzmann measures indexed by @xmath19 @xmath20 with the convention that @xmath21 for any @xmath22 such that @xmath23 . the superscript @xmath24 will be often omitted . the model is completely defined once @xmath25 is given : being interested in the disordered case , we choose for instance @xmath25 an realization of an iid family , of law @xmath26 , of symmetric variables taking value @xmath27 . we invite the reader to look at figure [ fig : modello ] in order to get an intuitive idea on this model . = 7 cm [ c][l]@xmath22 [ c][l]@xmath28 [ c][l]@xmath29 [ c][l]@xmath30 [ c][l]@xmath31 [ c][l]@xmath32 [ c][l]@xmath33 [ c][l]@xmath34 [ c][l]@xmath35 [ c][l]@xmath36 [ c][l]*water * [ c][l]*oil * [ c][l]@xmath37 [ c][l]@xmath38 [ c][l]@xmath39 [ c][l]@xmath40 [ c][l]@xmath41 an important observation on the model we have introduced is that its hamiltonian may be easily rewritten in terms of the sequence @xmath42 , defined by setting @xmath43 and , for @xmath44 , @xmath45 ( @xmath46 with probability one for every @xmath47 since @xmath10 is recurrent ) and in terms of the sign of the excursions , that , conditionally on @xmath48 , are just an independent sequence of iid symmetric variables taking values @xmath27 . of course the ( strong ) markov property of @xmath10 immediately yields also that @xmath49 is an iid sequence . we are now going to introduce the general class of models that we consider . these models are based on a real valued free process @xmath50 , with law @xmath12 that satisfies the following properties : 1 . the sequence @xmath51 of successive returns to @xmath29 is an infinite sequence with @xmath52 , so @xmath36 is a process starting from @xmath29 and for which @xmath29 is a recurrent state . moreover @xmath48 is a renewal sequence , that is @xmath53 is a sequence of iid random variables , and we set @xmath54 . 2 . for some @xmath55 and for some function @xmath56 which is slowly varying at infinity ( see below for the definition and properties of slowly varying functions ) @xmath57 in particular @xmath58 for every @xmath59 . 3 . for @xmath44 such that @xmath60 we set @xmath61 . then conditionally on @xmath48 , @xmath62 is an iid sequence of symmetric random variables taking values @xmath27 . conventionally we complete the sequence @xmath63 ( i.e. , we choose the @xmath64 for @xmath47 such that @xmath65 ) by tossing independent fair coins and , always by convention , we stipulate that @xmath66 . of course , conditionally on @xmath48 , @xmath63 is just independent fair coin tossing . 4 . conditionally on @xmath48 , @xmath67 is an independent sequence of random vectors . moreover the law of @xmath68 , conditionally on @xmath48 , depends on @xmath48 only via the value of @xmath69 . note that this property implies that , conditionally on @xmath70 , the process @xmath71 has the same distribution as the original process @xmath72 . with some abuse of language , we will call this property the _ renewal property of @xmath36_. the free process @xmath36 is put in interaction with an environment via the _ charges _ @xmath13 . the definition of @xmath73 is still as in , provided one suppresses the superscript rw in the left hand side . the process @xmath36 constructed in section [ sec : premodel ] starting from @xmath10 corresponds to the case of @xmath74 and @xmath75 . we recall that a function @xmath76 is slowly varying ( at infinity ) if @xmath56 is a measurable function from @xmath77 to @xmath78 such that @xmath79 for every @xmath80 . one of the properties of slowly varying functions is that both @xmath81 and @xmath82 are @xmath83 for every @xmath84 . an example of slowly varying function is @xmath85 , for @xmath86 , but also @xmath87 , for @xmath88 , as well as any positive function for which @xmath89 . a complete treatment of slowly varying functions is found in @xcite , but these functions are needed for us to in order to work in a reasonably general and well defined set up : we will use no fine property of slowly varying function since in most of the cases rough bounds on @xmath90 will suffice . [ rem : s ] in short , one may think of building @xmath36 by first assigning the return times to zero according to a renewal process . the excursions are then _ glued _ to the renewal points , but essentially the only relevant aspect of the excursions for us is the sign , that is chosen by repeated tossing of a fair coin . note again that the energy of the model depends only on @xmath48 and on @xmath91 , and not on the details of the excursions in the upper or lower half plane . the last property in the list above , the renewal property of @xmath36 , makes a bit more precise what the excursions of the process really are : this property is very useful to have a nice pictorial vision of the process , but it is rather inessential for us . we will use it only in stating theorem [ th : correlazioni ] since this way it turns out to be somewhat nicer and more intuitive , but the essence of our analysis lies in @xmath48 . note also that if @xmath92 the energy does not depend on @xmath91 . in this case we are dealing with a pure pinning model and insisting on @xmath36 taking values in @xmath93 is a useless limitation . [ rem : s1 ] we have chosen @xmath29 to be a recurrent state of @xmath36 because it simplifies a little the notations , but everything carries over to the case in which @xmath29 is transient . the sequence @xmath94 is chosen as a typical realization of an iid sequence of random variables , still denoted by @xmath95 . we make the very same assumptions on @xmath96 and , in addition , @xmath97 and @xmath96 are independent . the law of @xmath25 will be denoted by @xmath26 , and the corresponding expectation by @xmath98 . a further assumption on @xmath25 is the following _ concentration inequality _ : there exists a positive constant @xmath99 such that for every @xmath28 , for every lipschitz and convex function @xmath100 with @xmath101 and for @xmath102 @xmath103\big\vert \ge t\right ) \ , \le \ , c \exp \left ( -\frac{t^2}{c \vert g \vert ^2 _ { \rm lip}}\right),\ ] ] where @xmath104 is the lipschitz constant of @xmath105 with respect to the euclidean distance on @xmath106 . of course such an inequality implies that @xmath107 and @xmath108 are exponentially integrable : without loss of generality we assume @xmath107 and @xmath108 to be centered and of unit variance . in practice , it is sufficient to check that the concentration inequality holds for @xmath94 and for @xmath96 separately . in fact , suppose that @xmath109\big\vert \ge t\right ) \ , \le \ , c \exp \left ( -\frac{t^2}{c \vert g \vert ^2 _ { \rm lip}}\right),\ ] ] for every @xmath110 and @xmath111 , and similarly for @xmath96 . then , @xmath112\big\vert \ge u\right)\le \\ { { \ensuremath{\mathbb p } } } \left(\big\vert { { \ensuremath{\mathbb e } } } [ g({{\underline{{\omega}}}})\vert { \omega}]-{{\ensuremath{\mathbb e } } } [ g({{\underline{{\omega}}}})]\big\vert \ge \frac u2\right ) + { { \ensuremath{\mathbb p } } } \left(\big\vert g({{\underline{{\omega}}}})-{{\ensuremath{\mathbb e } } } [ g({{\underline{{\omega}}}})\vert { \omega}]\big\vert \ge \frac u2\right).\end{gathered}\ ] ] using twice , once for @xmath113 $ ] and once for @xmath114 , noting that @xmath115 uniformly in @xmath94 , one obtains immediately . the concentration inequality is known to hold with a certain generality : its validity for the gaussian case and for the case of bounded random variables is by now a classical result ( @xcite ) . while of course such an inequality requires @xmath116 < \infty$ ] for some @xmath117 , a complete characterization of the distributions for which holds is , to our knowledge , still lacking , but among these distributions there are , for example , the cases in which the laws of @xmath107 and @xmath118 satisfy the log sobolev inequality , see @xcite , @xcite and @xcite , therefore , in particular , whenever @xmath107 and @xmath118 are continuous variables with positive densities of the form @xmath119 , with @xmath120 and @xmath121 when restricted to @xmath122 , for some @xmath123 . we have chosen to work assuming concentration because it provides a unified rather general framework in which proofs are at several instances much shorter ( and , possibly , more transparent ) . however , as it will be clear , several results hold under weaker assumptions . on the other hand we deal only with the polymer pinned at the endpoint , that is , constrained to @xmath124 : we have chosen this case for the sake of conciseness , but we could have decided for example to leave the endpoint free . under the above assumptions on the disorder the _ quenched free energy _ of the system exists , namely the limit @xmath125 exists @xmath126almost surely and in the @xmath127 sense . the existence of this limit can be proven via standard super additivity arguments ( we refer for example to @xcite for details ) . we stress that the concentration inequality implies immediately that @xmath128 is not random , but such a result may be proven under much weaker assumptions , see e.g. @xcite . a simple but fundamental observation is that @xmath129 the proof of this is elementary : if we set @xmath130 for @xmath131 and @xmath132 we have @xmath133 \\ = \ , \frac { \lambda}n { \sum_{n=1}^n \left({{\ensuremath{\mathbb e } } } \left[{\omega}_1 \right]+h\right ) } + \frac { { \widetilde}{\lambda}}n \left ( { { \ensuremath{\mathbb e } } } \left[{\widetilde}{\omega}_n \right ] + { \widetilde}h \right ) + \frac 1n \log { { \ensuremath{\mathbf p } } } \left ( { \omega}_n^+\right)\ , \stackrel{n \to \infty}{\longrightarrow}\ , { \lambda}h,\end{gathered}\ ] ] where we have applied the fact that , by , @xmath134 , for @xmath135 . the observation , above all if viewed in the light of its proof , suggests the definition @xmath136 and the following partition of the parameter space ( or _ phase diagram _ ) : * the localized region : @xmath137 ; * the delocalized region : @xmath138 . along with this definition we observe that @xmath139 where @xmath140 of course the normalization constant @xmath141,\ ] ] changes , but @xmath142 so that the @xmath126a.s . and @xmath143 asymptotic behavior of @xmath144 are given by @xmath145 . we will always work with @xmath146 , in order to conform with most of the previous mathematical literature . for the model we introduced there is a very vast literature , mostly in chemistry , physics and bio physics . it often goes under the name of _ copolymer with adsorption _ , see e.g. @xcite and references therein , and such a name clearly reflects the superposition of two distinct polymer environment interactions : * the monomer solvent interaction , associated to the charges @xmath94 . some monomers prefer one solvent and some prefer the other one . since the charges are placed in an inhomogeneous way along the chain , energetically favored trajectories need to stick close to the interface . whether pinning actually takes place or not depends on the interplay between energetic gain and entropic loss associated to localization ( trajectories that stay close to the interface have a much smaller entropy than those which wander away ) . + if @xmath147 and @xmath148 only this interaction is present and we will call the model simply _ copolymer_. * the monomer interface interaction , associated to @xmath96 . this interaction leads to a pinning ( or depinning ) phenomenon with a more direct mechanism : trajectories are energetically favored if they touch the interface _ as often as possible _ at points where @xmath149 , avoiding at the same time the points in which @xmath150 . also in this case , a non trivial energy / entropy competition is responsible for the localization / delocalization transition . + when @xmath92 and @xmath151 we will refer to the model as _ pinning _ model . the copolymer model has received a lot of attention : we mention in particular @xcite , in which it was first introduced and the replica method was applied in order to investigate the transition . rigorous work started with @xcite , followed by @xcite : these works deal with the case @xmath152 and @xmath107 symmetric and taking only the values @xmath27 ( _ binary charges _ ) . it turns out that in such a case there is no transition and the model is localized for every @xmath147 . in general , one may distinguish between results concerning the free energy and results on pathwise behavior of the polymer . about the first point , we mention that in @xcite the model with @xmath153 has been considered , still with the choice of binary charges , and the existence of a transition has been established , along with estimates on the critical curve and remarkable limiting properties of the free energy of the model in the limit of weak coupling ( @xmath15 small ) . improved estimates on free energy and critical curve may be found in @xcite . in the physical literature one can find a number of conjectures , mostly on the free energy behavior , that are far from clarifying the phase diagram . in this respect , it is interesting to mention that recent numerical simulations ( @xcite ) show that the critical line is different from that predicted in the theoretical physics literature , which means that the localization mechanism is still poorly understood . disordered pinning has been extensively studied in the physical literature , see e.g. @xcite and @xcite ( see also @xcite for more recent references ) , but much less in the mathematical one . however the model has started attracting attention lately , see @xcite , @xcite and @xcite . we should stress that there is no agreement in the physical literature on several important issues for disordered pinning . for instance , it is still unclear whether the critical curve coincides with the so called annealed curve . about the study of path behavior , there is a basic difference between the localized and the delocalized phase : in the first case , since @xmath154 , the interaction produces an exponential modification of the free polymer measure , and therefore _ large deviation _ techniques apply very naturally . the path behavior of the copolymer model in the localized phase has been considered in @xcite , for @xmath152 and binary charges , while in @xcite also the case @xmath155 is taken into account . in @xcite the focus is on the gibbsian characterization of the infinite volume polymer measure ( in the localized phase ) . the delocalized phase is more subtle , due to the fact that @xmath156 , and path delocalization estimates involve estimates on _ moderate deviations _ of the free energy . results on the path behavior in the delocalized phase have been obtained only recently in @xcite , both for the copolymer and for the disordered pinning model . in the present work , we consider the localized phase of the general model defined in section [ sec : model ] and formula . in addition to giving new results , our approach provides also a setting to reinterpret in a simpler way known results for copolymer and pinning models . the free energy is everywhere continuous and almost everywhere differentiable , by convexity , so in particular @xmath157 is an open set . however , one can go much beyond that , as our first theorem shows : [ th : cinf ] @xmath158 is infinitely differentiable in @xmath157 . an interesting problem is to study the regularity properties of @xmath159 at the boundary between @xmath157 and @xmath160 , where it is non analytic . this corresponds to investigating the order of the localization / delocalization transition . recently , an important step in this direction was performed in @xcite and @xcite where it was proved , in particular , that the first derivatives of @xmath159 are continuous on the boundary . in other words , the ( de)localization transition is at least of second order . as it will become clear in section [ sec : reg ] , the smoothness of the free energy in @xmath161 boils down to a property of exponential decay of ( average ) correlations . for this implication we essentially rely on @xcite , where a similar result has been proven in the context of disordered ising models . let us therefore state the decay of correlation property . we say that @xmath162 is a bounded local observable if @xmath162 is a real bounded measurable local function of the path configurations . in the sequel @xmath163 will denote the _ support of _ @xmath162 , that is the intersection of all the subsets @xmath164 of the form @xmath165 , with @xmath166 , such that @xmath162 is measurable with respect to the @xmath167algebra @xmath168 . we have [ th : correlazioni ] for every @xmath169 there exist finite constants @xmath170 such that the following holds for every @xmath171 : * ( exponential decay of correlations . ) for every couple of bounded local observables @xmath162 and @xmath172 we have @xmath173 \\ \le\ , c_1 \vert a\vert_\infty\vert b\vert_\infty \exp(-c_2 d(\mathcal s(a),\mathcal s(b ) ) ) \end{gathered}\ ] ] where @xmath174 if @xmath175 . * ( influence of the boundary . ) for every bounded local observable @xmath162 and @xmath176 , such that @xmath177 , we have @xmath178\le c_1 \vert a\vert_\infty \exp(-c_2d(\mathcal s(a),\{k\})).\ ] ] * for every bounded local observable @xmath162 the following limit exists @xmath126almost surely : @xmath179 [ rem : a.s . ] from one may easily extract an almost sure statement . choose two bounded local observables @xmath162 and @xmath172 and set @xmath180 , i.e. @xmath181 . take the limit @xmath182 in to obtain @xmath183\le \exp(-c_2 k/4),\ ] ] for @xmath184 sufficiently large . therefore the fubini tonelli theorem and yield @xmath185\ , < \ , \infty.\ ] ] the series appearing in the left hand side is therefore @xmath126a.s . this implies that there exists a random variable @xmath186 , @xmath187 @xmath126a.s . , such that @xmath188 for every @xmath184 . we consider now the question of whether or not knowing that @xmath169 does mean that the path of the polymer is really tight to the interface . even if this question has not been treated for the general model we are considering here , the techniques used in @xcite , @xcite and @xcite , see also @xcite , may be applied directly and one obtains for example that , in the case @xmath189 , for every @xmath169 there exist finite constants @xmath190 such that for every @xmath28 , @xmath191 , and @xmath192 @xmath193 or one can obtain an analogous @xmath126a.s . result , which is a bit more involved to state @xcite . the reason for revisiting this type of results , besides generalizing them to our case , is that they are only bounds and we would like to find estimates that are sharp to leading order . a notable exception is the case of some of the results in ( * ? ? ? 5.3 and th . 6.1 ) where the precise asymptotic size of the largest excursion ( and of the maximum displacement of the chain from the interface ) is obtained . this result is a bit surprising since it depends on a certain annealed decay exponent . this exponent turns out to be different from the decay exponent one finds for @xmath126a.s . estimates , see discussion after our theorem [ th : zu ] . the argument of the crucial point of the proof of ( * ? ? ? 5.3 ) looks obscure to us and we propose here a different one , based on decay of correlations . of course we present our results in terms of _ excursion lengths_. recall the definition of the return times @xmath48 in section [ sec : model ] . for every @xmath194 let us set @xmath195 , with @xmath196 equal to the value @xmath47 such that @xmath197 . so @xmath198 is the left margin of the excursion to which @xmath184 belongs and @xmath199 is the length of such an excursion . two distinct questions can be posed concerning polymer excursions : one may be interested about the typical length of a given excursion or , more globally , about the typical length of the _ longest _ excursion , @xmath200 . denote by @xmath201 the left shift on @xmath94 , like for @xmath36 : @xmath202 . about the first problem , we can prove : [ th : as ] take @xmath203 and let @xmath25 be the two sided sequence of iid random variables , @xmath204 , with law @xmath26 . for every @xmath117 there exist random variables @xmath205 , @xmath206 , such that @xmath207 and @xmath208 for every @xmath28 , every @xmath209 and every @xmath191 : for the first inequality , the lower bound , we require also @xmath210 . note that , in the definition of @xmath211 , the fact that @xmath25 is a doubly infinite sequence is completely irrelevant , since the polymer measure depends only on @xmath212 . the introduction of two sided disorder sequences , which might seem a bit unnatural , is needed here to have the almost sure result uniformly in @xmath213 and @xmath214 . about the maximal excursion , we have : [ th : zu ] the following holds : 1 . the limit @xmath215 \ , : = \ , \mu ( { { \underline{v } } } ) , \ ] ] exists and satisfies the bounds @xmath216 . moreover , for every @xmath217 there exists @xmath218 such that @xmath219 , for @xmath220 and @xmath221 . 2 . fix @xmath169 . for every @xmath222 we have that @xmath223 and @xmath224 in the localized region , under rather general conditions on the law @xmath26 , we can prove that @xmath225 , see appendix [ sec : mu ] . notice therefore the gap between and the result in proposition [ th : as ] , so that the largest excursion appears to be achieved in atypical regions . when the law of @xmath34 is symmetric , one can also prove ( see appendix [ sec : mumu ] ) that @xmath226 is equivalently given by @xmath227.\ ] ] this coincides with the expression given in @xcite , where only the case @xmath228 , @xmath152 and @xmath107 taking values in @xmath229 was considered . in this section , we investigate finite size corrections to the infinite volume limit of the quenched average of the free energy , and the behavior of its disorder fluctuations . about the first point , it is quite easy to prove ( see also ( * ? ? ? * proposition 2.6 ) ) [ th : nfin1 ] there exists @xmath230 such that , for every @xmath24 and @xmath171 , one has @xmath231 this bound is somehow optimal in general , in the sense that it is possible to prove a lower bound of order @xmath232 if @xmath24 is in the _ annealed region _ , i.e. , the sub region of @xmath160 where @xmath233 . however , in the localized region we can go much farther : [ th : nfin2 ] assume that @xmath169 . then , there exists @xmath234 such that , for every @xmath171 , one has @xmath235 about fluctuations , in @xcite it was proven that , for @xmath228 , @xmath152 and @xmath107 taking values in @xmath229 , the free energy satisfies in the large volume limit a central limit theorem on the scale @xmath236 . here , we generalize this result to the entire localized region . our proof employs basically the same idea as in @xcite ; however , the use of concentration of measure ideas , plus a more direct way to show that the limit variance is not degenerate , allow for remarkable simplifications . the precise result is the following : [ th : clt ] if @xmath169 the following limit in law holds : @xmath237 with @xmath238 . for compactness we introduce a notation for the hamiltonian @xmath239 and for @xmath240 we set @xmath241.\ ] ] moreover , we set @xmath242 the proof of theorem [ th : correlazioni ] is based on the following lemma , which is somehow similar in spirit to lemmas 4 and 5 in @xcite . [ lemma : ritorni ] for every @xmath169 , there exist constants @xmath243 such that , for every @xmath244 , @xmath245 , @xmath246 and @xmath247 , @xmath248 moreover , let @xmath249 be two independent copies of the copolymer , distributed according to the product measure @xmath250 . then , @xmath251 it will be clear from the proof that the constant @xmath252 in may be chosen smaller , but arbitrarily close to @xmath226 . the quantitative estimate on the constant @xmath252 in that one can extract from the proof given below is instead substantially worse and certainly not optimal . _ proof of theorem [ th : correlazioni]_. let @xmath253 and @xmath254 . we assume that @xmath255 @xmath256 , otherwise holds trivially with @xmath257 . without loss of generality , we take @xmath258 . then , letting @xmath259 be the event @xmath260 one can write @xmath261 since , by a simple symmetry argument based on the renewal property of @xmath36 , one can show that the above average vanishes , if conditioned to the complementary of the event @xmath259 . at this point , using one obtains @xmath262 \le 2 c_1 \vert a\vert_\infty\vert b\vert_\infty e^{-c_2(b_1-a_2)},\end{aligned}\ ] ] which is statement of the theorem . as for we observe that , since we are assuming that @xmath263 , one has the identity @xmath264 where we recall that @xmath265 . therefore , there exist positive constants @xmath266 and @xmath267 such that @xmath268 \ , = \ , { { \ensuremath{\mathbb e } } } \left [ \left \vert \frac { { { \ensuremath{\mathbf e } } } _ { n , { { \underline{{\omega}}}}}\left ( a\right){{\ensuremath{\mathbf e } } } _ { n , { { \underline{{\omega}}}}}\left ( \delta_k\right)- { { \ensuremath{\mathbf e } } } _ { n , { { \underline{{\omega}}}}}\left ( a\,\delta_k\right ) } { { { \ensuremath{\mathbf e } } } _ { n , { { \underline{{\omega}}}}}\left ( \delta_k\right ) } \right\vert \right ] \\ \le \ , c k^c { { \ensuremath{\mathbb e } } } \left[\zeta({{{\widetilde}\omega}}_k ) \left\vert { { \ensuremath{\mathbf e } } } _ { n , { { \underline{{\omega}}}}}\left ( a\right){{\ensuremath{\mathbf e } } } _ { n , { { \underline{{\omega}}}}}\left ( \delta_k\right)- { { \ensuremath{\mathbf e } } } _ { n , { { \underline{{\omega}}}}}\left ( a\,\delta_k\right)\right\vert \right ] \\ \le \ , c k^c \left({{\ensuremath{\mathbb e } } } \left [ \left\vert { { \ensuremath{\mathbf e } } } _ { n , { { \underline{{\omega}}}}}\left ( a\right){{\ensuremath{\mathbf e } } } _ { n , { { \underline{{\omega}}}}}\left ( \delta_k\right)- { { \ensuremath{\mathbf e } } } _ { n , { { \underline{{\omega}}}}}\left ( a\,\delta_k\right)\right\vert ^2 \right ] { { \ensuremath{\mathbb e } } } \left[\zeta({{{\widetilde}\omega}}_k)^2 \right]\right)^{1/2 } \\ \le c^\prime k^{c } \vert a \vert_\infty e^{-c_2 d(\mathcal s(a),\{k\})},\end{gathered}\ ] ] where in the first inequality we have applied lemma [ th : lbpk ] , in the second the cauchy schwarz inequality and in the third theorem [ th : correlazioni ] , formula . since @xmath162 is a local observable , @xmath269 as @xmath270 and therefore the proof of is complete . finally , is a consequence of the decay of the influence of boundary conditions expressed by . note in fact that states that @xmath271 is a cauchy sequence in @xmath272 . therefore it convergence in @xmath273 toward a limit random variable that we denote @xmath274 . therefore holds if we set @xmath275 and , by the fubini tonelli theorem , this clearly implies that @xmath276<\infty,\ ] ] so the series in the expectation is @xmath126a.s . convergent , which implies the almost sure convergence of @xmath271 , that is . @xmath277 _ proof of lemma [ lemma : ritorni ] , equation _ . it is immediate to realize that , letting @xmath278 and @xmath279 , @xmath280,\end{gathered}\ ] ] where @xmath90 was defined in section [ subs : fe ] and @xmath281 is the left shift . indeed , it suffices to apply lemma [ prop : spezz ] with @xmath282 and @xmath283 . thanks to part 1 of theorem [ th : zu ] and to the exponential integrability of @xmath118 , one has @xmath284 -_n1n@xmath285 =( ) = : > 0 , @xmath284 so that one obtains from @xmath286 @xmath287 _ proof of lemma [ lemma : ritorni ] , equation _ . in this proof the positive constants , typically dependent on @xmath24 , will be denoted by @xmath288 . the constants @xmath289 and @xmath252 are taken from , but since @xmath252 is repeated several times we set @xmath290 . let us first define , for @xmath291 , @xmath292 , @xmath293 for @xmath294 and @xmath295 . then , for @xmath296 we let @xmath297 . we will refer to the interval @xmath298 as to the @xmath299 excursion from zero of the walk @xmath300 , and to @xmath301 as to its length , see figure [ fig : figura ] . ( note that the @xmath302 and the @xmath303 excursions may have an endpoint outside @xmath304 . ) = 12 cm [ c][l]@xmath305 [ c][l]@xmath306 [ c][l]@xmath307 [ c][l]@xmath308 [ c][l]@xmath309 [ c][l]@xmath310 [ c][l]@xmath311 [ c][l]@xmath312 [ c][l]@xmath313 [ c][l]@xmath314 [ c][l]@xmath315 [ c][l]@xmath316 [ c][l]@xmath317 [ c][l]@xmath318 [ c][l]@xmath319 [ c][l]@xmath320 [ c][l]@xmath321 [ c][l]@xmath322 [ c][l]@xmath323 [ c][l]@xmath324 [ c][l]@xmath325 [ c][l]@xmath326 [ c][l]@xmath327 [ c][l]@xmath328 [ c][l]@xmath329 [ c][l]@xmath330 [ c][l]@xmath331 [ c][l]@xmath332 [ c][l]@xmath333 [ c][l]@xmath334 [ c][l ] @xmath22 [ c][c ] @xmath335 [ c][l ] @xmath336 [ c][l]@xmath337 [ c][l]@xmath22 the basic observation is that , as we will prove in a moment , there exists @xmath338 independent of @xmath24 such that@xmath339 uniformly in @xmath171 , @xmath340 , for some finite constant @xmath341 . in words , this means that with high probability at least @xmath342 of the interval @xmath304 is covered by excursions of @xmath36 whose length is smaller than @xmath343 ( we will call them _ short excursions _ ) . to prove , set @xmath344 , @xmath345 and @xmath346 so that the condition in the probability in the left hand side of reads @xmath347 . obviously , the number @xmath348 of excursions entirely contained in @xmath349 and of length at least @xmath343 ( _ long excursions _ ) is at most @xmath350 , and one can ( very roughly ) bound above the number of possible ways one can place them in the stretch @xmath304 by @xmath351 these facts , together with a simple application of lemma [ prop : spezz ] and eq . , allow to bound above the left hand side of by @xmath352 where the last inequality follows , if @xmath353 is sufficiently large , from the stirling formula . we stress that the constant @xmath289 is the one appearing in . the factor @xmath354 in just takes care of the possible location of @xmath315 and @xmath355 in @xmath356 . for ease of notation , let @xmath357 be the event @xmath358 that the two walks do not touch zero at the same time between @xmath305 and @xmath306 , and for @xmath291 let @xmath359 be the event @xmath360 then , from eq . , if @xmath361 , one has @xmath362 to estimate the last term in let us notice that if the event @xmath363 occurs then , denoting by @xmath364 the union of the short excursions of @xmath300 , the set @xmath365 contains at least @xmath366 sites . as a consequence , recalling that short excursions do not exceed @xmath343 in length , @xmath367 contains at least @xmath368 sites . in words , if @xmath369 then either @xmath370 and @xmath371 belongs to a short excursion of @xmath336 , or the same holds interchanging the roles of @xmath337 and @xmath336 . one can rewrite @xmath372 as the disjoint union @xmath373 , where @xmath374 and , of course , at least one among @xmath375 and @xmath376 contains @xmath377 points . therefore , using also the symmetry between @xmath337 and @xmath336 , one has @xmath378 where @xmath379 is the subset of @xmath304 which satisfies the following properties : 1 . @xmath370 for every @xmath380 ; 2 . for every @xmath380 there exist @xmath381 such that * @xmath382 * @xmath383 * @xmath384 if @xmath385 and @xmath386 . note that , since we are working on @xmath363 , @xmath384 when @xmath387 : this prescription is not contained in the definition of @xmath379 . one can now write ( see also fig . [ fig : trem ] ) @xmath388 in the second step we have decomposed the probability by summing over all _ a priori _ admissible configurations @xmath389 of the set @xmath379 , i.e. , all possible subsets of @xmath304 containing at least @xmath390 sites . we are now going to relax the constraint given by @xmath391 , but estimating the corresponding ratio of probabilities . we claim in fact that , given @xmath392 and @xmath393 , we have @xmath394 where @xmath395 where @xmath396 is a positive constant that depends on @xmath90 and on the value of @xmath397 . the bound , which will be applied with @xmath398 , is proven as follows . first observe that @xmath399 and in the ratio in the right hand side we may factor the expression containing the copolymer energy term , that is we can set @xmath92 and we can restrict ourselves to considering @xmath400 . we write : @xmath401 where @xmath402 may be chosen , with a very rough estimate , equal to @xmath403 . since @xmath404 , we obtain . therefore from , using , we can extract @xmath405 = 12 cm [ c][l]@xmath406 [ c][l]@xmath407 [ c][l]@xmath408 [ c][l]@xmath409 [ c][l]@xmath336 [ c][l]@xmath337 [ c][l]@xmath305 [ c][l]@xmath410 [ c][l]@xmath411 [ c][l]@xmath412 [ c][l]@xmath413 [ c][l]@xmath414 [ c][l]@xmath306 [ c][l]@xmath22 the remaining problem now is that @xmath415 is random and can get arbitrarily close to @xmath37 . this however does not happen too often : in fact there exists @xmath416 such that , for every @xmath169 , @xmath417 for every value of @xmath418 . the bound follows from a direct ( large deviation ) estimate on the binomial random variable with parameters @xmath419 , which can be made arbitrarily small , and @xmath420 . it suffices for example that @xmath421 . thanks to @xmath422 , implies that , on the complementary of @xmath423 , @xmath424 together with and , this completes the proof of . _ proof of theorem [ th : cinf]_. thanks to the ascoli - arzel theorem , it is sufficient to show that , for every integer @xmath184 , the @xmath426 derivative of @xmath427 with respect to any of the parameters @xmath428 is bounded above uniformly in @xmath28 . for definiteness , let us show this property for @xmath429 the above derivative is given by @xmath430 which is expressed through truncated correlation functions ( ursell functions ) defined as @xmath431 where the sum runs over all partitions @xmath432 of @xmath433 into subsets @xmath434 . starting from the property of decay of correlations for every pair of bounded local observables , one can prove by induction over @xmath435 that @xmath436 \\ \!\!\!\!\!\!\!\!\!\!\!\!\!\ ! & & \qquad \le c_1^{(k)}\vert a_1\vert_\infty\ldots \vert a_k\vert_\infty \exp(-c_2^{(k)}d(\mathcal s(a_1),\ldots,\mathcal s(a_k ) ) ) , \nonumber\end{aligned}\ ] ] for some finite positive constants @xmath437 where , if @xmath438 is the smallest interval including the supports of @xmath439 , then @xmath440 . a proof that the exponential decay of average two points correlations implies exponential decay of the average @xmath22point truncated correlations can be found for instance in @xcite , in the context of disordered @xmath441dimensional spin systems in the high temperature or large magnetic field regime ( see remark [ rem : g ] below for a sketch of the proof ) . in @xcite , explicit bounds for the constants @xmath437 are also given , which is not needed in our case . the proof in @xcite , which is basically an application of hlder s inequality , can be transposed to the present context almost without changes , to yield . then , after an application of the cauchy - schwarz inequality , it is immediate to realize that the sum in converges uniformly in @xmath28 , for every @xmath184 . @xmath442 [ rem : g]a sketch of how is deduced from goes as follows . consider the case @xmath443 and assume that @xmath444 ( which is just the case we need in view of ) and , without loss of generality , let @xmath445 . then consider the simple identities @xmath446 from the first identity and we obtain @xmath447 \le 3c_1 e^{-c_2 ( n_3-n_2)},\end{aligned}\ ] ] while from the second identity we obtain the bound @xmath448 on the same quantity . therefore @xmath449\le 3c_1e^{-c_2((n_3-n_2)+(n_2-n_1))/2},\end{aligned}\ ] ] which is just the statement of in this specific case . _ part 1 . _ the existence of the limit follows from the subadditivity of @xmath450\right\}_n$ ] , which is an immediate consequence of the renewal property ( of @xmath12 ) and of the iid property ( of @xmath25 ) : for every @xmath451 , @xmath452 , we have in fact @xmath453 \ , & \le \ , { { \ensuremath{\mathbb e } } } \left [ \frac{(1+e^{-2\lambda\sum_{n=1}^m({\omega}_n+h)})(1+e^{-2\lambda\sum_{n = m+1}^n({\omega}_n+h ) } ) } { z_{m , { { \underline{{\omega}}}}}^{{\underline{v}}}z_{n - m , \theta^m { { \underline{{\omega}}}}}^{{\underline{v } } } } \right]\\ & = { { \ensuremath{\mathbb e } } } \left [ \frac{1+e^{-2\lambda\sum_{n=1}^m({\omega}_n+h ) } } { z_{m , { { \underline{{\omega}}}}}^{{\underline{v } } } } \right ] { { \ensuremath{\mathbb e } } } \left [ \frac{1+e^{-2\lambda\sum_{n=1}^{n - m}({\omega}_n+h ) } } { z_{n - m , { { \underline{{\omega}}}}}^{{\underline{v } } } } \right ] . \end{split}\ ] ] the inequality @xmath454 is an immediate consequence of the elementary lower bound @xmath455 a more refined lower bound on @xmath456 , valid in the localized region , follows from the concentration inequality : call @xmath457 the event @xmath458 since for @xmath28 sufficiently large @xmath459 with @xmath460 and @xmath353 suitable positive constants , one has @xmath461 \ , \le \ , 2\exp\left(-n{\textsc{f}}({{\underline{v}}})/4\right)+ \frac { { { \ensuremath{\mathbb e } } } \left[\zeta({{{\widetilde}\omega}}_n)^{-1}{\mathbf{1}}_{\ { \ , e_n^\complement\ } } \right]}{{{\ensuremath{\mathbf p } } } \left ( { \omega}_n^+\right ) } \\ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\ ! & & \qquad\qquad\qquad\qquad\qquad \le 2\exp\left(-n{\textsc{f}}({{\underline{v}}})/4\right)+ \kappa'_1\exp \left(-\kappa'_2 n{\textsc{f}}({{\underline{v}}})^2 /\max({\lambda } , { \widetilde}{\lambda})^2\right),\end{aligned}\ ] ] which immediately implies @xmath462 and , for @xmath145 sufficiently small , @xmath463 . finally , @xmath464 is an immediate consequence of jensen s inequality . @xmath465 _ part 2 . _ throughout this proof @xmath169 , so that @xmath462 , and we set @xmath466 . we start with the proof of @xmath467 which clearly implies . for @xmath468 we set @xmath469 so that @xmath470 , @xmath471 ranging up to @xmath472 , and we have @xmath473 let us estimate the terms in the sum by writing first @xmath474 as the disjoint union of the sets @xmath475 with @xmath476 and @xmath477 . recalling the notations of section [ sec : conv ] , we have the bound @xmath478 this is an immediate consequence of the renewal property of @xmath36 . notice that once we take the expectation of both sides of we may set @xmath479 in the right hand side and therefore , by , for every @xmath22 and every @xmath480 , @xmath481 sufficiently large , we have that @xmath482 indeed , the factor @xmath483 is negligible for @xmath348 large since , with probability at least of order @xmath484 , @xmath485 does not exceed @xmath486 , see , so that it does not modify the exponential behavior . therefore , for @xmath28 sufficiently large and @xmath487 sufficiently small @xmath488 going back to we see that @xmath489 and is proven . [ rem : cutoff ] it is immediate to realize by looking at the proof that @xmath490 with @xmath491 as @xmath492 . let us then turn to proving @xmath493 let us set @xmath494 and @xmath495 for @xmath496 . we now consider the family of events @xmath497 , recall the definition of @xmath498 in , and we observe that @xmath499 . in words , we are simply going to compute the probability that the walk makes at least one excursion of length exactly equal to @xmath500 at @xmath501 prescribed locations : the excursions have to start at @xmath502 for some @xmath371 , see figure [ fig : lb ] . therefore @xmath503 where the last step defines @xmath504 ( we anticipate that , by theorem [ th : correlazioni ] , formula , @xmath504 is negligible , details are postponed to lemma [ th : rn ] below ) . we need therefore a lower bound on @xmath505 : we will find a lower bound on this quantity that depends on @xmath25 only via @xmath0 and @xmath506 with @xmath507 and this will allow the direct use of independence when taking the expectation with respect to @xmath26 . we use the explicit formula @xmath508 = 11.5 cm [ c][l]@xmath29 [ c][l]@xmath28 [ c][l]@xmath509 [ c][l]@xmath510 [ c][l]@xmath511 [ c][l]@xmath512 [ c][l]@xmath513 [ c][l]@xmath500 we proceed by finding an upper bound on the denominator and we first observe that , by , the event @xmath514 has @xmath126probability tending to @xmath37 for @xmath515 and @xmath516 : this is simply due to the fact that @xmath517 . we will actually choose @xmath99 larger , so that to guarantee that @xmath518 \ge 1-n^{-2}$ ] for @xmath519 large , cf . remark [ rem : cutoff ] . of the requirements defining the event @xmath520 we now take advantage only of the fact that there exists a return to zero at distance at most @xmath521 from both sites @xmath502 and @xmath522 , for every @xmath371 . by lemma [ th : lbpk ] we have that there exists @xmath523 such that for @xmath524 @xmath525 so that from there exist @xmath526 such that @xmath527 where the inequality holds for @xmath519 sufficiently large . notice that the random variables @xmath528 are independent , since @xmath529 depends on @xmath530 with @xmath22 only in @xmath531 . going back to , we have @xmath532 we take now the @xmath126expectation of both sides of : by independence , by the choice of @xmath99 in the definition of @xmath533 and by lemma [ th : rn ] @xmath534\right ) \ , + \ , { { \ensuremath{\mathbb p } } } \left[a_{n}^\complement\right ] \ , + \ , { { \ensuremath{\mathbb e } } } \left[\left\vert r_n({{\underline{{\omega}}}})\right\vert\right ] \\ & = \ , \prod_{j=1}^{j_n } \left ( 1- { { \ensuremath{\mathbb e } } } \left[q_{n , j}({{\underline{{\omega } } } } ) \right]\right)\ , + \ , o(1/n^2 ) . \end{split}\ ] ] we are reduced to estimating @xmath535 $ ] . by , for @xmath487 sufficiently small and @xmath28 sufficiently large , we have @xmath536\,\ge \ , \exp\left(- \mu ( { { \underline{v } } } ) ( 1+{\varepsilon}/2 ) a_n^-\right)\,\ge\ , n^{-1 + { \varepsilon}/4}.\ ] ] from this one gets @xmath537 by applying the markov inequality and borel cantelli lemma we conclude the proof of . @xmath538 [ th : rn ] for every @xmath539 we have @xmath540 \ , = \ , 0,\ ] ] with @xmath504 defined in . _ observe that a direct application of theorem , formula , yields that for @xmath541 we have @xmath542 \\ \le\ , c_1 \exp\left(-c_2 \frac{(\log n)^2}2\right ) .\end{gathered}\ ] ] one now applies iteratively this inequality starting from @xmath543 , down to @xmath544 , obtaining that @xmath545 $ ] is bounded above by @xmath546 . @xmath547 it makes use of the identity @xmath548 _ upper bound . _ we first observe that , reasoning as in section [ sec : maximal ] , we have @xmath549 where @xmath550 ( recall that we are working here with two sided disorder sequences , so that the @xmath25 variables may have negative indices ) . now we claim that for every @xmath117 there exists @xmath551 , @xmath126a.s . finite , such that @xmath552 for every @xmath553 and @xmath554 such that @xmath555 and every @xmath556 . of course implies that for every @xmath117 @xmath557 by combining and one directly obtains the upper bound in . we are therefore left with the proof of . this follows by observing first that @xmath558 where @xmath559 is the disorder sequence reflected around the origin : @xmath560 . therefore , with @xmath561 , we have @xmath562 the leading terms are the last two : by definition of @xmath145 , for every @xmath117 there exists @xmath563 , @xmath126almost surely finite , such that for @xmath553 and @xmath554 larger than @xmath563 both @xmath564 and @xmath565 are larger than @xmath566 and this easily yields the existence of @xmath567 such that @xmath568 for every @xmath569 . the remaining term in the last line of is treated by an analogous splitting of the sum and by applying the law of large numbers one sees that it gives a negligible contribution for @xmath214 sufficiently large . finally the first two terms in the right hand side of are both vanishing as @xmath570 diverges by the properties of @xmath90 , cf . , and by the fact that @xmath108 is integrable , so that @xmath571 , @xmath126a.s .. @xmath572 _ lower bound . _ by selecting in the sum in only the _ excursion _ from @xmath184 to @xmath573 we have that we can write for @xmath574 @xmath575 and we will use also @xmath576 where @xmath577 is the event @xmath578 ( with the conventions of section [ sec : conv ] for @xmath579 ) . thanks to lemma [ th : lbpk ] , one can write for some @xmath523 @xmath580 which , together with and , implies @xmath581^{-1 } -{{\ensuremath{\mathbf p } } } _ { n,{{\underline{{\omega}}}}}(o^{\complement}_{k , s}).\end{aligned}\ ] ] proceeding in analogy with the proof of the upper bound , and using in addition lemma [ th : lbpk ] to bound @xmath582 _ above _ , one can show that @xmath583 diverges @xmath126a.s . as @xmath584 , for every @xmath117 . therefore , keeping in mind that @xmath585 , the first term on the r.h.s . of is bounded _ below _ by @xmath586 if @xmath214 is larger than @xmath587 , where @xmath588 is a suitable @xmath126a.s . finite number , and the same quantity may then be bounded below by @xmath589 where @xmath590 is a @xmath126a.s . positive random variable . on the other hand , from the upper bound in proposition [ th : as ] one easily obtains @xmath591 taking @xmath487 small enough , this immediately implies the lower bound in proposition [ th : as ] for @xmath592 . the restrictions on @xmath184 is of course due to having chosen in the first step of the proof the @xmath36_excursion _ from @xmath184 to @xmath573 . but we may as well choose the @xmath36_excursion _ from @xmath593 to @xmath594 , and the argument may be repeated yielding the same bound , except for different @xmath25 dependent constants , for every @xmath184 ranging from @xmath595 to @xmath596 . therefore the proof is complete for @xmath184 in @xmath597 and this is the whole set of sites smaller than @xmath28 if @xmath210 . _ proof of proposition [ th : nfin1]_. just note that @xmath599 where the first inequality is a consequence of the renewal property of @xmath36 and the last one of lemma [ th : lbpk ] . inequality then immediately follows . @xmath600 _ proof of theorem [ th : nfin2]_. it is sufficient to prove that there exists @xmath601 such that , for every @xmath602 , @xmath603 it is convenient to define , for @xmath604 , @xmath605 which , in view of , just corresponds to @xmath606 if @xmath607 . note that , since @xmath608 , one has @xmath609 ( cf . lemma [ lemma : derivne0 ] below ) and that @xmath610 provided that @xmath611 . indeed , since @xmath612 is convex in @xmath613 one has @xmath614 from which immediately follows , since @xmath615 . essentially the same argument shows that there esists a smooth ( e.g. differentiable ) path @xmath616 , with @xmath617 , such that @xmath618 , @xmath619 , and such that @xmath620 for every @xmath621 . note that at @xmath622 the desorder dependence disappears and we have simply a homogenous pinning model which , thanks to , is in the localized phase . for this model it is easy to prove ( this can be extracted , for instance , from appendix a of @xcite ) that @xmath623 on the other hand , we will prove in a moment that @xmath624 uniformly for @xmath625 . of course , eqs . - immediately give . to prove , let us compute for instance the derivative with respect to @xmath613 : with obvious notations , @xmath626 which , thanks to eq . , is bounded above by @xmath627 @xmath628 being bounded above uniformly for @xmath629 belonging to the path . similar estimates hold for the derivatives with respect to @xmath630 and therefore follows . @xmath631 the first step in the proof of theorem [ th : clt ] is to show that the variance of @xmath632 is not trivial in the infinite volume limit : [ lemma : nontriv ] if @xmath169 , then @xmath633 _ proof of lemma [ lemma : nontriv ] . _ the upper bound is an immediate consequence of the deviation inequality applied to @xmath634 . indeed , it is immediate to verify that in this case @xmath635 , for some constant @xmath266 . to obtain the lower bound , we employ a martingale method analogous to that developed in @xcite . suppose first that @xmath607 . observe that , if @xmath636 $ ] and @xmath637-{{\ensuremath{\mathbb e } } } \left[x_{n,{{{\widetilde}\omega}}}|{{{\widetilde}\omega}}_1,\ldots,{{{\widetilde}\omega}}_{k-1}\right],\ ] ] then @xmath638\ge \sum_{k=1}^n { { \ensuremath{\mathbb e } } } \left[\left({{\ensuremath{\mathbb e } } } \left(y^{(n)}_k|{{{\widetilde}\omega}}_k\right)\right)^2\right],\end{aligned}\ ] ] with the convention that @xmath639={{\ensuremath{\mathbb e } } } \left [ x_{n,{{{\widetilde}\omega}}}\right]$ ] if @xmath640 . next , observe that @xmath641= { \widetilde}\lambda \,{{\ensuremath{\mathbb e } } } \left[{{\ensuremath{\mathbf e } } } _ { n,{{\underline{{\omega}}}}}(\delta_k)|{{{\widetilde}\omega}}_k\right]\ge0 , \end{aligned}\ ] ] and that @xmath642\right|\le { \widetilde}\lambda^2.\end{aligned}\ ] ] at this point , one can employ the following ( @xcite ) let @xmath643 , @xmath644 and let @xmath645 be the set of functions @xmath646 and , for every probability law @xmath647 on @xmath648 , @xmath649 then , @xmath650 is convex in @xmath651 and @xmath652 for @xmath653 , provided that @xmath647 is not concentrated on a single point . identifying @xmath654 , @xmath655 $ ] and @xmath647 with the law of @xmath118 , using the convexity of @xmath656 and recalling , one obtains immediately @xmath657 to conclude the proof of lemma [ lemma : nontriv ] , it suffices therefore to show that [ lemma : derivne0 ] if @xmath169 , @xmath658 _ proof of lemma [ lemma : derivne0 ] . _ it is enough to prove that there exists @xmath659 sufficiently small such that @xmath660 indeed , we have @xmath661 where in the first inequality we employed . using stirling s approximation , it is clear that , if @xmath659 is small enough , the right hand side of decays exponentially in @xmath28 . @xmath662 in the case @xmath148 , the condition @xmath663 implies @xmath664 and the proof of lemma [ lemma : nontriv ] proceeds analogously , with the role of @xmath96 being played by @xmath94 and replaced by @xmath665 @xmath666 the second step in the proof of theorem [ th : clt ] is the observation that ( this follows for instance from lemma [ th : lbpk ] ) @xmath667 for some @xmath668 independent of @xmath25 . therefore , keeping also into account the upper bound in , one obtains the following approximate subadditivity property : @xmath669 for some constant @xmath670 and @xmath28 large enough . this , together with , implies that the following limit exists : @xmath671 next , employing repeatedly , one obtains the decomposition @xmath672 for every @xmath25 ( we assumed for simplicity that @xmath28 is multiple of @xmath673 ) . therefore , if @xmath674 , @xmath675 where @xmath676 tends to zero in law and @xmath126a.s .. the summands in are iid random variables and lyapunov s condition , which in this case reads simply @xmath677 = 0\ ] ] for some @xmath678 , follows immediately since the deviation inequality implies that @xmath679 $ ] is bounded uniformly in @xmath28 . the central limit theorem is proven . recall the notations in section [ sec : conv ] . [ th : lbpk ] there exists @xmath523 such that for every @xmath681 , @xmath682 and every @xmath25 we have that @xmath683 _ proof . _ we write @xmath684 and @xmath685 using and the definition of slow variation for @xmath56 , one finds that @xmath686 uniformly in @xmath687 , so that the expression can be bounded above by @xmath688 for some constant @xmath670 independent of @xmath25 , which directly yields . @xmath689 [ prop : spezz ] let @xmath690 , @xmath691 with @xmath692 , and let @xmath693 @xmath694 be events depending only on @xmath695 , i.e. , @xmath696 . then , @xmath697 _ proof . _ this is elementary : just rewrite the probability in as @xmath698 and notice that one obtains an upper bound for it if one constrains the walk to touch zero at @xmath699 in the denominator @xmath700 . at that point , the probability factorizes thanks to the renewal property of the random walk and one obtains just the right hand side of . [ [ sec : mu ] ] in this section we sketch a proof of the fact that , under some reasonable conditions on the law @xmath26 , the strict inequality @xmath225 holds in the localized region . since @xmath702 in @xmath157 , let us assume for definiteness that @xmath151 ( in the alternative case , the role of @xmath96 in the following is played by @xmath94 ) . in analogy with @xcite , we assume that one of the following two conditions holds for the iid sequence @xmath96 : * c * : _ continuous random variables_. the law of @xmath118 has a density @xmath703 with respect to the lebesgue measure on @xmath93 , and the function @xmath704 exists and is at least twice differentiable in a neighborhood of @xmath705 . this is true in great generality whenever @xmath703 is positive , for instance in the case of @xmath706 , with @xmath707 a polynomial bounded below . _ bounded random variables_. the random variable @xmath118 is bounded , @xmath708 assume first that condition * c1 * holds . given @xmath709 , let @xmath710 be the law obtained from @xmath26 shifting the law of @xmath711 so that @xmath712=-\varepsilon$ ] . if @xmath659 is small enough , thanks to the assumed regularity of the function @xmath713 in one has @xmath714 for some finite constant @xmath715 . then , applying the jensen inequality , @xmath716\,= \\ \frac1n\log{\widetilde}{{\ensuremath{\mathbb e } } } _ n \left [ \frac { \left(1+e^{-2\lambda\sum_{n=1}^n({\omega}_n+h)}\right ) } { z_{n , { { \underline{{\omega}}}}}^{{\underline{v } } } } e^{\log\left({{\,\text{\rm d}}{{\ensuremath{\mathbb p } } } } /{{\,\text{\rm d}}{\widetilde}{{\ensuremath{\mathbb p } } } _ n}\right ) } \right ] \\ \ge \ , \frac 1n{\widetilde}{{\ensuremath{\mathbb e } } } _ n \left ( \log \frac{{\,\text{\rm d}}{{\ensuremath{\mathbb p } } } } { { \,\text{\rm d}}{\widetilde}{{\ensuremath{\mathbb p } } } _ n}\right)-\frac1n{{\ensuremath{\mathbb e } } } \,\log z^{{{\underline{v}}}'}_{n,{{\underline{{\omega}}}}},\end{gathered}\ ] ] so that @xmath717 where @xmath718 is obtained from @xmath24 replacing @xmath719 with @xmath720 . since @xmath159 is smooth in the localized region and the derivative @xmath721 is not zero ( cf . ) , for @xmath659 sufficiently small one has @xmath722 in the case of condition * c2 * , the proof of is slightly more complicated . instead of shifting the law of @xmath711 , @xmath710 is obtained by tilting it : @xmath723 the estimate on the relative entropy is still valid , with a different constant @xmath715 , while the proof that @xmath724 for some positive @xmath266 , although rather intuitive in view of the previous case ( note in fact that , for @xmath659 small , the tilting produces a shift of @xmath725 of the average of @xmath711 ) , still requires some care . we do not report here other details , which the interested reader can easily reconstruct from the proof of lemma 3.3 in @xcite . @xmath726 here we prove that , if the law of @xmath107 is symmetric , then the definition of @xmath226 is equivalent to . in fact , @xmath728}\\ \stackrel{s\stackrel{{{\ensuremath{\mathbf p } } } } { \sim}-s}= { { \ensuremath{\mathbb e } } } \frac 1{{{\ensuremath{\mathbf e } } } \left[\exp\left(2{\lambda}\sum_{n=1}^n({\omega}_n+h)\delta_n+{\widetilde}{\lambda}\sum_{n=1}^n({{{\widetilde}\omega}}_n+{\widetilde}h)\delta_n\right ) \delta_n\right]}\\ \le { { \ensuremath{\mathbb e } } } \frac 1{{{\ensuremath{\mathbf e } } } \left[\exp\left(2{\lambda}\sum_{n=1}^n({\omega}_n - h)\delta_n+{\widetilde}{\lambda}\sum_{n=1}^n({{{\widetilde}\omega}}_n+{\widetilde}h)\delta_n\right ) \delta_n\right]}\stackrel{\omega\stackrel{{{\ensuremath{\mathbb p } } } } { \sim}-{\omega}}={{\ensuremath{\mathbb e } } } \frac 1{z^{{\underline{v}}}_{n,{{\underline{{\omega}}}}}},\end{gathered}\ ] ] where in the inequality we simply used the fact that @xmath729 , see section [ sec : model ] . therefore , @xmath730\ , & \le\ , \frac 1n \log { { \ensuremath{\mathbb e } } } \left [ \frac { 1+e^{-2\lambda\sum_{n=1}^n({\omega}_n+h ) } } { z_{n , { { \underline{{\omega}}}}}^{{\underline{v } } } } \right ] \\ & \le \ , \frac 1n \log { { \ensuremath{\mathbb e } } } \left [ \frac 1 { z_{n , { { \underline{{\omega}}}}}^{{\underline{v } } } } \right]+\frac{\log 2}n , \end{split}\ ] ] and the claim follows in the limit @xmath731 we are very grateful to an anonymous referee for his observations and , in particular , for having pointed out that our decay of correlation estimates yield the almost sure result of th . [ th : correlazioni ] ( formula ) . we would like to thank also francesco caravenna for interesting discussions on the content of section 5 . this research has been conducted in the framework of the gip anr project jc05_42461 ( _ polintbio _ ) . m. ledoux , _ measure concentration , transportation cost , and functional inequalities _ , summer school on singular phenomena and scaling in mathematical models , bonn , 1013 june 2003 . available online : http://www.lsp.ups-tlse.fr/ledoux/
|
we analyze the localized phase of a general model of a directed polymer in the proximity of an interface that separates two solvents .
each monomer unit carries a charge , @xmath0 , that determines the type ( attractive or repulsive ) and the strength of its interaction with the solvents .
in addition , there is a polymer interface interaction and we want to model the case in which there are impurities @xmath1 , called again charges , at the interface .
the charges are distributed in an inhomogeneous fashion along the chain and at the interface : more precisely the model we consider is of quenched disorder type . + it is well known that such a model undergoes a localization / delocalization transition .
we focus on the localized phase , where the polymer sticks to the interface .
our new results include estimates on the exponential decay of correlations , the proof that the free energy is infinitely differentiable away from the transition and estimates on finite size corrections to the thermodynamic limit of the free energy per unit site .
other results we prove , instead , generalize earlier works that typically deal either with the case of copolymers near an homogeneous interface ( @xmath2 ) or with the case of disordered pinning , where the only polymer
environment interaction is at the interface ( @xmath3 ) .
moreover , with respect to most of the previous literature , we work with rather general distributions of charges ( we will assume only a suitable concentration inequality ) and we allow more freedom on the law of the underlying random walk .
+ + 2000 _ mathematics subject classification : 60k35 , 82b41 , 82b44 _
+ + _ keywords : directed polymers , copolymers , copolymers with adsorption , disordered pinning , localized phase _
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
investigations focusing on global changes in star - formation activity and stellar - mass build - up in field galaxies have provided significant insight into the physical evolution of galaxies and their constituent stellar populations . it has now been well - established that the global star - formation rate density has declined by roughly an order of magnitude since @xmath7 11.5 ( e.g. , lilly 1996 ; madau 1996 ; steidel 1999 ; hopkins 2004 ; prez - gonzlez 2005 ; schiminovich 2005 ; colbert 2006 ) . recent investigations into the details of this evolution have shown that the star - formation history of a given galaxy depends strongly on its stellar mass ( e.g. , cowie 1996 ; juneau 2005 ; bundy 2006 ; noeske 2007a , b ; zheng 2007 ) ; the peak star - formation epoch for the most massive galaxies occured earlier in cosmic history than it did for galaxies with lower masses . x - ray studies of normal late - type galaxies ( i.e. , those that are not dominated by luminous active galactic nuclei [ agns ] ) have shown that emission provides a useful , relatively - unobscured measure of star - formation activity ( e.g. , bauer 2002a ; cohen 2003 ; ranalli 2003 ; colbert 2004 ; grimm 2003 ; gilfanov 2004a ; persic 2004 ; persic & rephaeli 2007 ; however , see barger 2007 ) . in normal galaxies , emission originates from binaries , supernovae , supernova remnants , hot ( @xmath0 ) interstellar gas , and o - stars ( see , e.g. , fabbiano 1989 , 2006 for reviews ) . sensitive and studies of local late - type galaxies have shown that high - mass binaries ( hmxbs ) and low - mass binaries ( lmxbs ) typically dominate the total non - nuclear power output ( e.g. , zezas 2002 ; bauer 2003 ; soria & wu 2003 ; swartz 2003 ; jenkins 2005 ; kilgard 2005 ; however , see , e.g. , doane 2004 ) . observations indicate that the integrated emission from hmxb and lmxb populations trace galaxy star - formation rate ( sfr ) and stellar mass ( @xmath3 ) , respectively . for example , using observations of 32 local galaxies , colbert ( 2004 ) found that the summed non - nuclear point - source emission from a given galaxy ( @xmath8 ) can be approximated as , where @xmath9 and @xmath10 are constants . therefore , galaxies having relatively high star - formation rates per unit mass ( specific star - formation rates , ssfrs ) generally have dominant point - source contributions from hmxbs ( e.g. , late - type star - forming galaxies ) , while those with relatively low ssfrs have point - source emission primarily from lmxbs ( e.g. , massive early - type galaxies ) . if the binary populations are similarly dominating the normal - galaxy power output over a significant fraction of cosmic time , then there should be a rapid increase in the globally - averaged luminosity of normal star - forming galaxies with cosmic look - back time in response to the increasing global star - formation rate density ( e.g. , ghosh & white 2001 ) . in this scenario , hmxbs trace the immediate star - formation rate of a galaxy and lmxbs trace its star - formation history with a lag of a few gyr . with the advent of deep and surveys ( see , e.g. , brandt & hasinger 2005 for a review ) , it has become possible to study the properties of normal galaxies out to @xmath11 and @xmath12 , respectively ( see , e.g. , hornschemeier 2000 , 2003 ; alexander 2002 ; georgakakis 2007 ; georgantopoulos 2005 ; lehmer 2006 , 2007 ; kim 2006 ; tzanavaris 2006 ; rosa - gonzalez 2007 ) . evidence for a global increase in the activity with redshift for normal late - type galaxies has since been mounting . studies of the luminosity functions of detected normal galaxies have found that @xmath13 evolves as over the redshift range @xmath7 01.4 ( e.g. , norman 2004 ; georgakakis 2006 ; ptak 2007 ; tzanavaris & georgantopolous 2008 ) . additionally , stacking analyses have enabled investigations of more representative optically - selected galaxy populations over the majority of cosmic history ( @xmath7 0.14 ; e.g. , brandt 2001 ; hornschemeier 2002 ; nandra 2002 ; georgakakis 2003 ; reddy & steidel 2004 ; laird 2005 , 2006 ; lehmer 2005a ) . these studies have found that the average luminosities of normal late - type galaxies increases with redshift out to @xmath7 1.43 . for example , using a @xmath01 ms exposure of a subregion within the deep field - north , hornschemeier ( 2002 , hereafter h02 ) tentatively observed a factor of @xmath023 increase in @xmath14 from @xmath15 to 1.4 for @xmath16 galaxies . despite these promising initial constraints , the details of the evolution of normal late - type galaxy populations , including dependences on the physical properties of galaxies ( e.g. , optical luminosity , stellar mass , environment , and star - formation rate ) , have remained unexplored . in this paper , we aim to improve significantly upon constraints for the evolution of normal late - type galaxies ( e.g. , h02 ) . we study for the first time how the properties of late - type field galaxies evolve as a function of optical luminosity , stellar mass , and star - formation rate over the redshift range of @xmath17 . we construct late - type galaxy samples located in two of the most well - studied extragalactic survey fields , the @xmath02 ms _ chandra _ deep field - north (; alexander 2003 ) and the extended deep field - south ( ) , which is composed of the central @xmath01 ms deep field - south ( cdf - s ; giacconi 2002 ) and four flanking @xmath0250 ks observations ( lehmer 2005b ) . these deep fields ( hereafter cdfs ) reach detection limits of @xmath0@xmath18 in the most sensitive regions and @xmath19@xmath20 over the majority of the cdfs ; these levels are sufficient to detect moderately - powerful sources ( @xmath21 ) at @xmath1 and @xmath22 , respectively . therefore , the cdfs comprise an unprecedented data set for effectively studying the emission and evolution of cosmologically distant normal galaxies with minimal contamination from powerful agns . the galactic column densities are @xmath23 @xmath24 for the ( lockman 2004 ) and @xmath25 @xmath24 for the ( stark 1992 ) . all of the fluxes and luminosities quoted throughout this paper have been corrected for galactic absorption . unless stated otherwise , we quote optical magnitudes based upon the ab magnitude system for photometry and the vega magnitude system everywhere else . in the band , we make use of three standard bandpasses : ( soft band [ sb ] ) , ( hard band [ hb ] ) , ( full band [ fb ] ) . throughout this paper , we make estimates of stellar mass and star - formation rates using a kroupa ( 2001 ) initial mass function ( imf ) ; when making comparisons between these estimates and those quoted in other studies , we have adjusted all values to correspond to our adopted imf . @xmath26 = 70 , @xmath27 = 0.3 , and @xmath28 = 0.7 are adopted throughout this paper ( e.g. , spergel 2003 ) , which imply a look - back time of 7.7 gyr at @xmath29 . we constructed an optically selected sample of late - type galaxies within the cdfs to use for our subsequent analyses . we restricted our galaxy selection to regions of the cdfs where _ hubble space telescope _ ( ) observations were available to allow the best possible morphological classifications . the observations in the cdfs have been carried out via the great observatories origins deep survey ( goods ; giavalisco 2004a ) and galaxy evolution from morphology and seds ( gems ; rix 2004 ; caldwell 2005 ) programs ; these surveys cover @xmath090% of the -observed regions of the cdfs with the advanced camera for surveys ( acs ) . in the goods and gems regions , photometry was available for four ( @xmath30 , @xmath31 , @xmath32 , and @xmath33 ) and two ( @xmath31 and @xmath33 ) acs passbands , respectively . we began building our sample by selecting all galaxies having @xmath34 . this initial selection criterion was motivated by ( 1 ) the availability of deep @xmath33 band imaging over all of the cdfs , ( 2 ) the fact that the @xmath33 emission probes mass - tracing rest - frame optical light redward of the 4000 break for galaxies at @xmath35 , which constitutes the large majority of our sample , ( 3 ) the availability of reliable spectroscopic and photometric redshifts ( see details below ) , and ( 4 ) the high multiwavelength detection fractions for these sources , which allow us to determine informative rest - frame spectral energy distributions ( seds ) . in order to isolate most effectively distant detected agns , we further restricted our sample to include only galaxies that were located in the most sensitive areas of the cdfs where the point - spread function ( psf ) was small . we therefore chose to include sources having optical positions that were within 60 of at least one of the six aimpoints in the cdfs2 ms and @xmath01 ms , respectively , as well as table 1 of lehmer ( 2005b ) for the @xmath0250 ks . ] ; the corresponding total areal footprint is @xmath00.18 futhermore , we removed obvious galactic stars that were identified via optical spectral properties or the presence of obvious diffraction spikes in the @xmath33 band images . under these restrictions , we found 6905 galaxies . we cross - correlated our initial sample of 6905 galaxies with available spectroscopic and photometric redshift catalogs ( e.g. , barger 2003 ; le fevre 2004 ; szokoly 2004 ; wirth 2004 ; wolf 2004 ; mobasher 2004 ; mignoli 2005 ; vanzella 2005 , 2006 ; grazian 2006 ; ravikumar 2007 ; silverman 2008 ) . all galaxies that did not have spectroscopic redshifts were located in the where highly accurate ( median @xmath37 for galaxies with @xmath38 ) photometric redshifts were available via combo-17 ( classifying objects by medium - band observations in 17 filters ; wolf 2004 ) . in total , 6683 ( @xmath097% ) of our sources had either spectroscopic or photometric redshifts . visual inspection of the 222 galaxies in the without redshifts indicated that these sources were mainly faint galaxies near bright stars , as well as a handful of sources that were subgalactic features within relatively nearby galaxies . whenever possible , we adopted spectroscopic redshifts as the most accurate redshifts for our galaxies . using the redshift information , we filtered our sample to include only sources with @xmath39 in the @xmath02 ms , @xmath40 in the @xmath01 ms , and @xmath41 in the @xmath0250 ks regions of the ; these redshift limits represent the largest distances at which we would expect to identify moderately luminous ( @xmath42 ) agns effectively in each respective field . in total , 3259 galaxies remained after filtering our sample based on redshift properties . we used spectroscopic redshifts for 1351 ( @xmath041% ) galaxies and photometric redshifts for the remaining 1908 ( @xmath059% ) galaxies . the optical - color distribution for field galaxies has been shown to be bimodal , separating `` red '' and `` blue '' galaxy populations ( e.g. , strateva 2001 ; hogg 2002a ; blanton 2003 ; baldry 2004 ) . studies of the color - magnitude relation for distant galaxy populations have shown that this color bimodality is observed to persist out to at least ( see , e.g. , bell 2004a ; faber 2007 ; labbe 2007 ) , thus providing an excellent quantifiable means for separating late - type and early - type galaxy populations . we therefore filtered our galaxy sample to include only sources that had blue rest - frame optical colors , as expected for late - type galaxies that contain young stellar populations . to estimate rest - frame @xmath43 colors and absolute @xmath44 , @xmath45 , and @xmath46 band magnitudes ( @xmath47 , @xmath48 , and @xmath49 , respectively ) for each of the 3259 galaxies in our sample , we utilized the available optical / near - ir data . for the , we used the @xmath44 , @xmath45 , @xmath46 , @xmath50 , @xmath51 , @xmath52 , and @xmath53 photometric catalogs from capak ( 2004 ) , as well as irac imaging ( 3.6 , 4.5 , 5.8 , and 8.0@xmath54 m ; fazio 2004 ) from the goods ( dickinson , in preparation ) . for the , we used ( 1 ) the 17-bandpass photometry available through combo-17 , ( 2 ) @xmath55 and @xmath56 imaging from the eso imaging survey ( olsen 2006 ) , ( 3 ) @xmath55 , @xmath57 , and @xmath58 from the musyc collaboration ( taylor in preparation ; see also , moy 2003 ; gawiser 2006 ) , and ( 4 ) irac data from the goods and simple teams ( dickinson , in preparation ; van dokkum , in preparation ) . using these data , we constructed a rest - frame near - uv to near - ir sed for each galaxy . for these seds , the rest - frame @xmath44 , @xmath45 , and @xmath46 filters are well - sampled at all relevant redshifts , with the exception of sources at @xmath59 where the wavelength range of the available data is sparse at rest - frame @xmath46 band . for sources at @xmath60 , we linearly interpolated our sed to cover the @xmath46 filter . we convolved these seds with johnson @xmath44 , @xmath45 , and @xmath46 filter curves and computed rest - frame absolute magnitudes for each respective filter following equation 5 of hogg ( 2002b ) . for sources in the , these computed absolute magnitudes are consistent with those presented by wolf ( 2004 ) . in figure 1 ( _ small filled circles _ ) , we present rest - frame @xmath43 colors versus @xmath49 for our galaxies in six redshift ranges . for clarity , we also show inset histograms giving the distribution of rest - frame @xmath43 colors for each redshift interval . we utilized the rest - frame @xmath43 color to divide roughly populations of red and blue galaxies . use of the rest - frame @xmath43 color was motivated by bell ( 2004a ) , who note that the @xmath44 and @xmath46 bandpass pair straddle the 4000 break , which is particularly sensitive to age and metallicity variations of galactic stellar populations . the dashed lines in figure 1 show the empirically - determined redshift - dependent color divisions that separate blue and red galaxy populations ; we calculated these divisions following @xmath61 5 of bell ( 2004a ) : @xmath62 galaxies having rest - frame @xmath43 less than the values provided by equation 1 are often referred to as `` blue - cloud '' galaxies , while those with rest - frame @xmath43 greater than the division are called `` red - sequence '' galaxies . the redshift - dependence of the blue - cloud / red - sequence galaxy division is thought to be largely due to the evolution of the mean age and dust content of the blue - cloud population , with a smaller contribution from changes in metallicity . in total , we found 2502 blue - cloud galaxies and 757 red - sequence galaxies . generally , the blue - cloud and red - sequence populations are composed of late - type and early - type galaxies , respectively ( see , e.g. , bell 2004b , 2005 ; mcintosh 2005 ) . in support of this point , we have created figure 2 , which shows the fraction of galaxies in our sample having blue - cloud ( _ dashed histogram _ ) and red - sequence ( _ dotted histogram _ ) colors as a function of galaxy srsic index @xmath63 . we utilized the hu@xmath64ssler ( 2007 ) srsic indices , which were computed using the gems @xmath33 images and the galfit ( peng 2002 ) two - dimensional light - profile fitting program . light - profile studies of large galaxy samples have found empirically that a srsic cutoff of @xmath65 can roughly discriminate between late - type and early - type galaxies ( e.g. , blanton 2003 ; shen 2003 ; hogg 2004 ) . galaxies with @xmath66 are generally late - type galaxies , while the majority of galaxies with @xmath67 are early - types ( _ vertical line _ in fig . 2 ) . figure 2 shows that there is reasonable agreement between late - type and early - type galaxy populations selected using rest - frame optical colors and srsic indices . ) for our initial sample of @xmath17 01 galaxies in the having blue - cloud ( _ dashed histogram _ ) and red - sequence ( _ dotted histogram _ ) colors . the solid vertical line indicates @xmath65 , the empirical cutoff between late - type ( @xmath66 ) and early - type galaxies ( @xmath67 ) . , width=359 ] we refined further our division between late - type and early - type morphologies by visually inspecting the @xmath33-band images of our entire sample of 3259 galaxies to see if there were obvious cases where the rest - frame optical colors provided an inaccurate morphological classification . for example , late - type galaxies that are highly inclined to our line of sight may experience significant reddening of the young ( and blue ) disk population and will have red - sequence colors . also , due to variations of the stellar populations in galaxies of a given morphological class , there will be some scatter in rest - frame @xmath43 color near the division of the red - sequence and blue - cloud regimes . this will lead to a number of `` green '' galaxies that are misclassified morphologically when the classification is based solely on rest - frame color . we stress that such a visual inspection is only to correct source classifications that were _ obviously misclassified _ by simply using color - magnitude diagrams , and our resulting sample should not be considered a morphologically - selected sample . based on our visual inspection , we found 76 obvious early - type galaxies with blue - cloud colors , and 142 obvious late - type galaxies with red - sequence colors ; we reclassified these sources as late - type and early - type galaxies , respectively . we inspected histograms of the fraction of blue - cloud and red - sequence galaxies that were reclassified as a function of redshift and find no obvious redshift - related trend . in figure 3 , we show @xmath33-band postage - stamp images of ten obvious early - type galaxies with blue - cloud colors ( _ top panels _ ) , as well as ten obvious late - type galaxies with red - sequence colors ( _ bottom panels _ ) . after reclassifying these objects , we were left with 2568 late - type galaxies and 691 early - type galaxies with . since we were interested in studying the properties of the late - type galaxy populations , we hereafter refer to our sample of 2568 late - type galaxies as our _ main sample _ , which we use in subsequent analyses . the primary goal of this study is to investigate the evolution of normal ( i.e. , non - agn ) late - type galaxies and to determine how this evolution depends upon three intrinsic physical properties : optical luminosity , stellar mass , and star - formation rate . below , we describe how we estimated each of these physical properties for the galaxies in our main sample . we note that the populations of late - type galaxies that we are investigating here have been selected via their intrinsic physical properties , which may have changed significantly from the observed epoch to the present day . for example , it is expected that a significant fraction of the late - type galaxies in our main sample will evolve into early - type galaxies via mergers or passive stellar evolution ( e.g. , bell 2007 ) . ideally , we would like to study the evolution of the properties of late - type galaxies while controlling for such changes in the physical nature of each galaxy . however , since the details of this evolution are highly complex and not well understood for a given galaxy , such a task is beyond the scope of this paper . we therefore investigate the evolution of normal late - type galaxy populations in relation to their observed intrinsic physical properties . in order to study late - type galaxy samples selected using an observable quantity , we made use of the @xmath45-band luminosity ( @xmath2 ) . in @xmath61 2.3 , we computed absolute @xmath45-band magnitudes @xmath48 for galaxies in our main sample using photometrically - derived seds . the @xmath45-band emission from a given late - type galaxy will be significantly influenced by large populations of old ( @xmath68100 myr ) stars ( measured by the stellar mass ) as well as the younger and less - numerous massive stars that reside in low - obscuration star - forming regions ( measured by the star - formation rate ) . in figure 4@xmath69 , we show @xmath2 ( expressed in solar units ; where ) versus redshift for galaxies in our main sample . the redshift - dependent selection limit of @xmath2 for our sample is set by our @xmath34 criterion , and at @xmath70 ( @xmath1 ) this limit corresponds to roughly ( ) . for comparison , in figure 4@xmath69 , we have plotted the values of @xmath2 for the milky way ( mw ) and local galaxies m101 , m82 , and the luminous infrared galaxy ( lirg ) ngc 3256 . for the mw , we adopted @xmath71 as the approximate total disk - plus - bulge @xmath45-band luminosity ( see table 15.2 of gilmore 1990 ) . for the local galaxies , we utilized optical photometry from the third reference catalog of bright galaxies ( rc3 ; de vaucouleurs 1991 ) and adopted distances from the iras revised galaxy sample ( sanders 2003 ) , which gives distances of 6.7 , 3.6 , and 35.4 mpc and implies corresponding @xmath45-band luminosities of @xmath72 , @xmath73 , and @xmath74 for m101 , m82 , and ngc 3256 , respectively . as discussed in @xmath61 1 , the emission from lmxb populations scales with galaxy stellar mass . it is therefore useful to select late - type galaxies via their stellar masses as a means for estimating the lmxb contribution to their emission . to estimate the stellar mass ( @xmath75 ) of each of our galaxies , we exploited the tight correlation between rest - frame optical color and stellar mass - to - light ratio ( e.g. , bell & de jong 2001 ; bell 2003 ; kauffmann 2003a ; see also , borch 2006 ) . for this calculation , we used a combination of rest - frame @xmath76 colors and rest - frame @xmath58-band luminosities . @xmath58-band luminosities were computed by fitting all available optical / near - ir photometric data to a grid of 276 synthetic spectra generated by the pgase stellar population synthesis code ( fioc & rocca - volmerange 1997 ) . these templates assume a single formation epoch with an exponentially decaying star - formation history ( time constants @xmath77 1 , 2 , 4 , and 7 gyr ) and a kroupa ( 1993 ) imf ; the template grid spans ages of . for each galaxy in our main sample , we convolved the best - fit template spectrum with the @xmath58-band filter function to approximate the rest - frame @xmath58-band luminosity , @xmath78 . this method allowed us to account for significant curvature in the rest - frame near - ir seds of our sources where photometric data points did not always overlap with the rest - frame @xmath58-band . we note that 2546 ( @xmath099.1% ) of our late - type galaxies have at least one photometric data point lying at rest - frame wavelengths between 1 and 6 @xmath54 m , which significantly constrains the sed in the rest - frame @xmath58-band . we adopted the prescription outlined in appendix 2 of bell ( 2003 ) to estimate @xmath75 using rest - frame @xmath76 color and @xmath58-band luminosity : @xmath79 the numerical constants in equation 2 were supplied by table 7 of bell ( 2003 ) and are appropriate for our choice of @xmath76 color and @xmath78 ; the normalization has been adjusted by @xmath80 dex to account for our adopted kroupa ( 2001 ) imf ( see @xmath61 1 ) . we compared our stellar - mass estimates with those computed by borch ( 2006 ) for 1758 ( @xmath068% of our main sample ) galaxies in the combo-17 survey and find excellent agreement between methods , with an overall scatter of @xmath00.2 dex . -band luminosity @xmath2 ( _ a _ ) , stellar mass @xmath3 ( _ b _ ) , and star - formation rate sfr ( _ c _ ) versus redshift for sources in our main sample . for comparison , in each panel we have indicated the properties of the mw and local galaxies m101 , m82 , and ngc 3256 . in figure 4@xmath81 , we have included only sfrs for the 888 late - type galaxies in our main sample that have 24@xmath54 m counterparts ( see @xmath61 3.3 for details ) ; the dashed curve indicates roughly the sfr detection limit for sources without 24@xmath54 m counterparts . detected normal galaxies and agns have been indicated with open circles and diamonds , respectively . the thick gray rectangles indicate regions where galaxy populations were selected for stacking ( see @xmath824.2 and 5.1 ) . for each galaxy . @xmath3 and sfr were computed following equations 2 and 3 , respectively . ] in figure 4@xmath83 , we show @xmath3 versus redshift for our sample of late - type galaxies . we note that our stellar - mass estimates are broadly limited by the @xmath34 criterion used in our sample selection . at @xmath70 ( @xmath84 ) our samples are representative for late - type galaxies with @xmath85 ( @xmath86 ) . for comparison , in figure 4@xmath83 we have plotted stellar masses for the mw and local galaxies m101 , m82 , and ngc 3256 . we adopted a stellar mass of @xmath87 @xmath88 for the mw ( e.g. , hammer 2007 ) . for the local galaxies , we estimated @xmath3 using a similar relation to equation 2 , except using @xmath76 color and @xmath45-band luminosity ( see appendix 2 of bell 2003 for details ) ; for this computation , we utilized @xmath76 colors from rc3 and values of @xmath2 determined in @xmath61 3.1 . this computation gives stellar masses of @xmath89 , @xmath90 , and @xmath91 @xmath88 for m101 , m82 , and ngc 3256 , respectively . since the emission from normal late - type galaxies is known to be strongly correlated with sfr , it is of particular interest to understand how changes in sfr have contributed to the evolution of the normal late - type galaxy population . to calculate sfrs for the galaxies in our main sample , we utilized estimates of both the dust - uncorrected ultraviolet luminosities ( @xmath92 ) originating from young stars and the infrared luminosities (; @xmath93 ) from the dust that obscures uv light in star - forming regions ( see , e.g. , kennicutt 1998 for a review ) . the former quantity was computed following @xmath94(2800 ) , where @xmath95(2800 ) is the rest - frame 2800 monochromatic luminosity ( see @xmath61 3.2 of bell 2005 ) . @xmath96(2800 ) was approximated using the template seds that were constructed in @xmath61 3.2 . the latter quantity ( @xmath93 ) was computed using observed - frame 24@xmath54 m flux densities ( i.e. , rest - frame 24@xmath54m/@xmath97 ) from observations with the mips ( rieke 2004 ) camera onboard . we note that generally the dust - obscured star - formation activity , probed here using @xmath93 , can be measured using either the uv spectral slope or optical nebular recombination lines ( e.g. , balmer emission lines and ) ; however , the data available for our @xmath98 01.4 late - type galaxies lack the rest - frame uv and spectroscopic coverage needed to provide such useful measurements . over the cdfs , deep 24@xmath54 m observations were available for the goods fields ( @xmath99 30 @xmath54jy , 6@xmath100 ) . for the remaining area ( covering the outer regions of the ) , shallower 24@xmath54 m observations ( @xmath99 120@xmath101jy ) were available through swire ( lonsdale 2003 ) . we matched the positions of the 2568 late - type galaxies in our main sample with those from the available 24@xmath54 m source catalogs , requiring that the and centroids be offset by no more than 15 . we found successful matches for 888 ( @xmath035% ) of our late - type galaxies . for these sources , we converted 24@xmath54 m flux densities to @xmath93 following the methods outlined in papovich & bell ( 2002 ) . briefly , we utilized the entire grid of 64 infrared seds provided by dale ( 2001 ) to estimate the mean conversion factor @xmath102 , which transforms observed - frame 24@xmath54 m luminosity @xmath103(24@xmath104 ) to @xmath93 as a function of redshift [ i.e. , @xmath105 . the mean conversion factor spans the tight range of over the entire redshift range . we note that different choices of infrared seds yield similar results at @xmath35 , but become significantly discrepant at @xmath11 ( see , e.g. , fig . 2 of papovich & bell 2002 ) . using our estimates of @xmath92 and @xmath93 , we calculated star - formation rates for galaxies in our sample using the following equation : @xmath106 where @xmath93 and @xmath92 are expressed in units of the solar bolometric luminosity ( @xmath107 ) . equation 3 was adopted from @xmath61 3.2 of bell ( 2005 ) and was derived using pgase stellar - population models , which assumed a 100 myr old population with constant sfr and a kroupa ( 2001 ) imf ( see bell 2003 for further details ) . we note that for the majority of our galaxies @xmath108 , and for sources with @xmath109 and @xmath110 , the median ir - to - uv ratio @xmath111 = 2.8 and 7.1 , respectively . our @xmath112 galaxies have median @xmath113 , which is characteristic of objects that are found in the nearby @xmath114 mpc universe . by contrast , our @xmath110 galaxies have median @xmath115 , similar to the local lirg population found at . in figure 4@xmath81 , we show the distribution of sfrs for the 888 galaxies in our main sample that had 24@xmath54 m counterparts . we note that for sources in our main sample that were within the goods regions , where the 24@xmath54 m observations are most sensitive , the infrared detection fraction drops from @xmath0100% for galaxies with @xmath116 to @xmath020% for galaxies with @xmath117 . this demonstrates that our sfr completeness is limited primarily by the 24@xmath54 m sensitivity limit and that our sample of @xmath34 late - type galaxies is highly representative of galaxy populations above the apparent redshift - dependent sfr limit shown in figure 4@xmath81 ( _ dashed curve _ ) . at @xmath70 ( @xmath1 ) , this limit corresponds to sfr @xmath118 ( sfr @xmath119 ) . for comparison , in figure 4@xmath81 we have plotted the sfrs for the mw and local galaxies m101 , m82 , and ngc 3256 . the sfr of the mw was taken to be @xmath02 ( mckee & williams 1997 ) , and the sfrs of the local galaxies were computed using equation 3 . values of @xmath96(2800 ) were approximated from seds available through the nasa / ipac extragalactic database ( ned ) and @xmath93 was taken from the iras revised bright galaxy sample ( sanders 2003 ) . we find sfr = 2.5 , 6.2 , and 32.2 for m101 , m82 , and ngc 3256 , respectively . as a consistency check on our uv - plus - infrared sfr estimates sfr(uv+ir ) , we calculated radio - derived sfrs using 1.4 ghz observations , sfr(1.4 ghz ) , following equation 7 of schmitt ( 2006 ) , which we adjusted to be consistent with our adopted kroupa ( 2001 ) imf . we matched sources in our main sample to 1.4 ghz catalogs , which were derived from observations using the vla in the ( @xmath030 @xmath54jy ; richards 1998 ) and the atca in the ( @xmath060 @xmath54jy ; afonso 2006 ; rovilos 2007 ) . using a matching radius of 15 , we found a total of 54 radio sources coincident with our late - type galaxies . we found that 48 ( @xmath089% ) of the radio - detected sources had 24@xmath54 m counterparts , allowing for reasonable comparison between derived sfrs . for these sources we found reasonable agreement between sfrs derived from uv - plus - infrared and radio measurements , with a mean ratio of sfr(1.4 ghz)/sfr(uv+ir ) @xmath120 . a large number of late - type galaxies ( 840 sources ) in our main sample were detected in the 24@xmath54 m observations that were not detected at 1.4 ghz . all but three of these sources had sfr(1.4 ghz ) upper limits that were consistent with that expected from estimates of sfr(uv+ir ) ( see @xmath61 4.1 for further details ) . we utilized the multiwavelength observations in the cdfs to obtain a census of the active galaxies in our main sample . we began by matching the optical positions of our galaxies to the positions of point sources in the cdf catalogs of alexander ( 2003)2 ms and @xmath01 ms . ] for the @xmath02 ms and @xmath01 ms and lehmer ( 2005b)250 ks . ] for the @xmath0250 ks . for a successful match , we required that the optical and centroids be displaced by no more than 1.5 times the radius of the positional error circles ( confidence ) , which are provided in each respective catalog . we note that for a small number of the galaxies in our sample at @xmath121 , moderately luminous off - nuclear sources ( e.g. , ultraluminous sources [ ulxs ] ) that are intrinsically related to the galaxies may lie outside of our adopted matching radius ; we utilized the off - nuclear source catalog of lehmer ( 2006 ) to identify such galaxies and assign properties ( see details below ) . the source catalogs were generated using wavdetect ( freeman 2002 ) with false - positive probability thresholds of and @xmath122 for the alexander ( 2003 ) and lehmer ( 2005b ) point - source catalogs , respectively . however , as demonstrated in @xmath61 3.4.2 of alexander ( 2003 ) and @xmath61 3.3.2 of lehmer ( 2005b ) , legitimate lower significance sources , detected by running wavdetect at a false - positive probability threshold of , can be isolated by matching with relatively bright optical sources ; therefore , when matching our late - type galaxies to -detected sources , we utilized this technique . the sky surface density for late - type galaxies in our main sample ranges from @xmath023000 deg@xmath123 in the to @xmath08800 deg@xmath123 in the @xmath0250 ks . the large difference between these source densities is primarily due to differences in applied redshift cuts ( see @xmath61 2.2 ) . given the fact that the positional uncertainties are generally small ( @xmath1915 ) for sources within 60 of the aim points , as is the case for sources in our main sample , the corresponding estimated number of spurious matches is small . we estimate that when using wavdetect with a false - positive probability threshold of to search for sources in three bandpasses ( , , and ) , we expect @xmath01.8 spurious matches . when including the off - nuclear sources from lehmer ( 2006 ) , we expect an additional @xmath00.5 false sources ; this brings our total spurious matching estimate to @xmath02.3 sources for our main sample . using the matching criteria above , we find that 225 late - type galaxies are detected in at least one of the , , or bandpasses . out of these 225 galaxies , 12 are known off - nuclear sources from lehmer ( 2006 ) . since only one off - nuclear source is detected for each host galaxy , we assume that the off - nuclear point - source dominates the total emission from each host galaxy . we therefore adopted the properties presented in table 1 of lehmer ( 2006 ) for each off - nuclear host galaxy . ) versus the logarithm of the flux ( @xmath124 ) for detected sources in our main sample . off - nuclear sources catalogued by lehmer ( 2006 ) have been highlighted with open squares . the shaded region represents sources with effective photon indices @xmath125 ; we classified these sources as agn candidates ( see @xmath61 4 , criterion 1 ) . for reference , we have plotted lines corresponding to @xmath126 0.5 , 1 , and 2 ( _ dashed lines _ ) . , width=351 ] -band magnitude versus @xmath124 for detected sources in our main sample . open circles represent sources that were classified as agn candidates by criterion 1 ( see @xmath61 4.1 and fig . 5 ) , and filled circles represent all other sources ; open squares have the same meaning as in figure 5 . the shaded area represents the region where @xmath127 ; we classified sources in this region as agn candidates ( see @xmath61 4.1 , criterion 2 ) . for reference , the dashed lines represent @xmath128 , @xmath129 , and 1 . , width=351 ] the unprecedented depths of the cdfs allow for the individual detection of luminous normal galaxies over the entire redshift range of our main sample ( @xmath17 01.4 ) ; however , the majority of the detected sources in even the most - sensitive regions of the cdfs will be distant ( @xmath130 ) agns ( e.g. , bauer 2004 ) , which we want to separate from our normal late - type galaxy sample . we identified agn candidates using four primary criteria , which utilize ( 1 ) hardness to identify luminous obscured sources , ( 2 ) to optical flux ratios to identify additional relatively - unobscured agns , ( 3 ) the sfr correlation to identify additional lower - luminosity agns that are significantly influencing the total emission , and ( 4 ) the combination of radio and infrared properties to identify additional compton - thick agns and radio - loud agns . as a final check on our agn identifications we utilized optical spectroscopic information to identify sources with obvious agn signatures ( see criterion 5 below ) . in the sections below we provide details of each criteria . 1 . _ x - ray hardness_.one unique signature of moderately obscured ( @xmath131 @xmath24 ) agn activity is a hard spectrum . for normal galaxies , the collective emission from binaries dominates the total power output . on average , these sources have observed power - law seds with spectral slopes of for lmxbs ( e.g. , church & baluciska - church 2001 ; irwin 2003 ) and for hmxbs and ulxs ( e.g. , sasaki 2003 ; liu & mirabel 2005 ; liu 2006 ) ; the presence of a significant hot interstellar gas component will steepen the resulting spectral slope ( i.e. , produce larger @xmath132 ) . to identify obscured agns in our sample effectively , we flagged sources having effective photon indices of @xmath125 as agn candidates . we determined @xmath133 using the to hardness ratio @xmath134 , where @xmath135 is the count rate for each bandpass . a few sources have only detections . since these sources were not detected in the bandpass , our most sensitive bandpass , there must be a significant contribution from the bandpass such that @xmath136 ( @xmath137 ) . we therefore classified these sources as agn candidates . in figure 5 , we show the @xmath138 hardness ratio versus the logarithm of the flux @xmath139 for the detected sources in our main sample . the shaded area highlights the region corresponding to @xmath125 . sources that had only upper limits on @xmath138 that lie in the shaded region were not classified as agn candidates , while those with lower limits on @xmath134 that were in the shaded region were flagged as likely agns . we note that occasionally an luminous ulx that is too close to the nucleus of its host galaxy to be identified as off - nuclear may have @xmath140 and therefore be classified as an agn candidate via this criterion ; however , since only one identified off - nuclear source has @xmath140 ( _ open squares _ in fig . 5 ; n.b . , upper limits are not included ) , we do not expect such rare sources to significantly affect our results . using criterion 1 , we identified a total of 71 obscured agn candidates . x - ray to optical flux ratio_.detailed analyses of the spectra of luminous agns in the @xmath01 ms show that the intrinsic agn power - law photon index is relatively steep , ( e.g. , tozzi 2006 ) . therefore , luminous agns having column densities of @xmath141 @xmath24 will often have effective photon indices of @xmath142 and would not have been classified as potential agns by criterion 1 . in order to identify luminous agns with @xmath141 @xmath24 , we utilized the to optical flux ratio ( @xmath143 ) as a discriminator of agn activity ( e.g. , maccacaro 1988 ; hornschemeier 2000 ; bauer 2004 ) . we identified sources with @xmath144 ( see criterion 3 below for justification ) as unobscured agn candidates . in figure 6 , we show the @xmath50-band magnitude versus @xmath124 for sources in our main sample ; the shaded area shows the region where @xmath145 . sources that were classified as agn candidates via criterion 1 are denoted with open symbols . in total , 43 detected sources satisfied criterion 2 , and 16 of these sources were uniquely identified using this criterion ( i.e. , not identified by criterion 1 ) . versus sfr for detected sources in our sample that have 24@xmath54 m counterparts . open circles represent sources that were classified as agn candidates via criteria 1 and 2 ( see @xmath61 4.1 and figs . 5 and 6 ) , and filled circles represent all other sources ; open squares have the same meaning as in figure 5 . we have shown the -sfr relations calibrated by bauer ( 2002b ) , ranalli ( 2003 ) , gilfanov ( 2004a ) , and persic & rephaeli ( 2007 ) ; each respective curve has been annotated in the figure . the shaded area above the thick dashed line represents the region where @xmath146 is three times larger than its value predicted by persic & rephaeli ( 2007 ) ; we classified detections that lie in this region as agn candidates ( see @xmath61 4.1 , criterion 3 ) . , width=359 ] 3 . _ x - ray to sfr correlation_.taken together , criteria 1 and 2 provide an effective means for identifying agns that are affected by large absorption column densities ( criterion 1 ) and those that are notably overluminous for a given optical luminosity ( criterion 2 ) . however , these criteria will still miss moderately luminous unobscured agns that have @xmath147 ( see , e.g. , peterson 2006 ) . although an accurate classification for _ all _ such sources is currently not possible , the situation can be mitigated using the available multiwavelength data . we therefore exploited the correlation between luminosity @xmath4 and sfr ( see @xmath61 1 ) to identify additional agn candidates in our main sample that have significant excesses over what is expected based on the -sfr correlation . in order to calculate the rest - frame luminosity ( @xmath148 ; where @xmath149 and @xmath150 are the photon - energy lower and upper bounds , respectively ) of a source having a power - law sed , we used the following equation : @xmath151 where @xmath152 is the observed - frame emission in the bandpass and @xmath153 is the luminosity distance . using equation 4 and an adopted photon index of @xmath154 , we calculated luminosities for detected sources in our sample . in figure 7 , we show the logarithm of the luminosity @xmath155 versus sfr ( computed following equation 3 ) for galaxies in our main sample that had 24@xmath54 m counterparts ( see @xmath61 3.3 ) . several estimates of the correlation have been shown for reference ( bauer 2002a ; ranalli 2003 ; gilfanov 2004a ; persic & rephaeli 2007 , hereafter pr07 ) ; these correlations have been corrected for differences in bandpass and sed as well as adopted imfs . hereafter , we adopt the correlation from pr07 for comparisons ; however , the use of other correlations would yield similar results and conclusions . open squares show the locations of the galaxies hosting off - nuclear sources , which appear to be preferentially located near the correlation . open circles indicate sources that were identified as agn candidates via criteria 1 and 2 . generally , these agns have @xmath146/sfr @xmath156 times that predicted by the pr07 correlation ( _ thick dashed line _ ) , which corresponds to a factor of @xmath682.5 times the rms scatter of the pr07 correlation . we therefore classified all sources in this regime ( _ shaded region _ ) as agn candidates . we note that sources having only upper limits were not classified as agns . however , sources that were detected in the bandpass that had only upper limits on the sfr were classified as agn candidates if they were within the shaded region of figure 7 . using this criterion , we identified 101 potential agn candidates , of which 33 were unique to criterion 3 . for the 124 detected sources that were not classified as agns , we found that 19 sources ( @xmath015% ) had spectral properties indicative of agns ( criterion 1 ) and only one source ( @xmath00.8% ) had an to optical flux ratio elevated in comparison to normal galaxies ( criterion 2 ) . versus redshift for the 225 detected sources in our main sample . open circles indicate all of the agn candidates that were isolated using the four criteria outlined in @xmath61 4.1 , and filled circles represent potential normal galaxies ; open squares have the same meaning as in figure 5 . the estimated detection limits for the @xmath02 ms , the @xmath01 ms , and the @xmath0250 ks have been indicated with dotted , dashed , and dot - dashed curves , respectively . , width=359 ] 4 . _ infrared and radio properties_.in addition to the detected sources , we also expect there to be additional agns that lie below the detection threshold that are not identified here . in @xmath61 3.3 , we noted using 48 sources that the sfrs derived from the uv - plus - infrared emission are generally consistent with those derived from radio ( 1.4 ghz ) observations ; however , there are a few sources that are significantly scattered ( i.e. , by more than a factor of five times the intrinsic scatter ) outside of the correlation . these sources have either ( 1 ) excess infrared emission due to reprocessed dust emission from a highly - obscured ( @xmath157 @xmath24 ) moderately luminous ( intrinsic @xmath158@xmath159 ) compton - thick agn ( e.g. , daddi 2007a , b ) or ( 2 ) an excess of radio emission due to prominent radio - emitting jets ( e.g. , snellen & best 2001 ) . the reasonable agreement between sfr(uv+ir ) and sfr(1.4 ghz ) suggests that such agns are not prevalent in our late - type galaxy sample . ideally we would also like to explore how uv - corrected sfrs compare to the uv - plus - infrared sfr to search for potential compton - thick agns following the approach used by daddi ( 2007a , b ) for @xmath160 galaxies at @xmath161 ; however , since our galaxies generally lie at much lower redshifts that the daddi ( 2007a , b ) sources , we are unable to constrain well uv spectral slopes with our available photometry , which would be required to make dust - corrections to the observed uv sfrs ( see @xmath61 3.3 for discussion ) . for our sample of 24@xmath54m - detected galaxies , we identified three radio - excess galaxies with sfr(1.4 ghz)/sfr(uv+ir ) @xmath162 and three ir - luminous galaxies with sfr(1.4 ghz)/sfr(uv+ir ) @xmath163 . using these criteria we identified three new agn candidates , two radio luminous ( and ) and one ir luminous ( ) ; one of these sources , the radio luminous source , was detected in the and bandpasses . as an additional test , we utilized irac photometry ( see @xmath61 2.3 ) to search for infrared power - law sources having near - ir spectral properties characteristic of luminous agns ( e.g. , alonso - herrero 2006 ; donley 2007 ) . when searching for power - law sources , we adopted the criteria discussed in donley ( 2007 ) . we found that no sources in our sample satisfied these criteria , which is consistent with the finding that most irac power - law sources reside at @xmath11 . _ optical spectroscopy_.as a final check on our agn classifications , we searched the optical spectroscopic catalogs available for cdf sources ( e.g. , barger 2003 ; le fevre 2004 ; szokoly 2004 ; wirth 2004 ; vanzella 2005 , 2006 ) to isolate additional luminous agns in our sample . in total , we found 15 galaxies in our main sample that were classified as agns via optical spectroscopy and all of these sources had been identified as agn candidates by the previous criteria . we note that the majority of the detected agns have moderate luminosities ( intrinsic @xmath164@xmath159 ) and therefore often have high - excitation agn emission lines that are too faint with respect to stellar emission to be identified via optical spectroscopy ( e.g. , moran 2002 ) . therefore , it is not surprising that we do not find any additional agns using this criterion . to summarize , in the band we have detected a total of 225 ( @xmath09% ) late - type galaxies out of the 2568 sources in our main sample . using the criteria presented above , we classified 121 detected sources as agn candidates ( with an additional two undetected agn candidates via criterion 4 ) . the remaining 104 -detected sources that we do not classify as agn candidates are considered to be normal late - type galaxies , and we include these galaxies in subsequent stacking analyses ( see details in @xmath61 5 below ) . thus we use 2447 late - type galaxies in our stacking analyses . in figure 8 , we show @xmath155 versus redshift for detected sources in our main sample , and in table 1 ( _ available electronically _ ) we summarize their properties . agn candidates are denoted with open circles and normal galaxies are plotted with filled circles . we note that the above criteria are not completely sufficient to classify all detected sources that are truly agns as agn candidates ( see , e.g. , peterson 2006 ) . such a misclassification is possible for low - luminosity agns that are only detected in the more sensitive bandpass , which have emission too weak for an accurate classification . in @xmath61 5.2 , we use the agn fraction as a function of luminosity to argue quantitatively that we do not expect misclassified agns ( detected only in the bandpass ) and low - luminosity agns below the detection limit to have a serious impact on our results . ccccccccc [email protected] & 17.19 & 0.15@xmath166 & 10.39 & 10.49 & @xmath1675.89 & e - cdf - s 03 & 217 & @xmath16515.06 + [email protected] & 22.17 & 0.15@xmath166 & 8.85 & 8.44 & 0.23 & e - cdf - s 03 & 217 & @xmath16515.45 + [email protected] & 17.06 & 0.25@xmath166 & 10.94 & 11.07 & 1.40 & e - cdf - s 03 & 206 & @xmath167 @xmath16515.15 + [email protected] & 20.14 & 0.47@xmath168 & 10.14 & 10.43 & @xmath16719.93 & e - cdf - s 02 & 226 & @xmath16513.46 + [email protected] & 19.58 & 0.51@xmath166 & 10.84 & 10.54 & @xmath16713.17 & e - cdf - s 03 & 239 & @xmath16515.02 + cccccccccc @xmath169 9.510.0 & @xmath170 & 120 & 9 & 213.3 & 158.7 & 54.9 & 8.6 & 9.6 & 3.0 + @xmath169 9.510.0 & @xmath171 & 363 & 1 & 176.7 & 152.6 & 25.3 & 4.8 & 6.8 & 0.9 + + @xmath169 10.010.5 & @xmath172 & 28 & 2 & 66.9 & 46.5 & 20.2 & 5.2 & 5.3 & 2.1 + @xmath169 10.010.5 & @xmath173 & 33 & 3 & 82.9 & 68.5 & 14.9 & 5.9 & 6.7 & 1.5 + @xmath169 10.010.5 & @xmath174 & 67 & 7 & 204.2 & 144.7 & 59.8 & 9.3 & 9.7 & 3.8 + @xmath169 10.010.5 & @xmath175 & 158 & 6 & 347.5 & 240.0 & 109.1 & 10.7 & 11.3 & 4.4 + @xmath169 10.010.5 & @xmath176 & 100 & 3 & 128.0 & 109.0 & 18.7 & 6.0 & 7.5 & 1.2 + @xmath169 10.010.5 & @xmath177 & 71 & 2 & 154.1 & 93.5 & 60.6 & 6.3 & 6.1 & 3.2 + @xmath169 10.010.5 & @xmath178 & 62 & 2 & 156.3 & 107.4 & 48.8 & 6.4 & 6.9 & 2.6 + @xmath169 10.010.5 & @xmath179 & 57 & 3 & 105.9 & 64.8 & 41.2 & 4.4 & 4.5 & 2.2 + the majority of the normal late - type galaxies that make up our main sample were not detected individually in the bandpass . in order to study the mean properties of these sources , we implemented stacking analyses of galaxy populations selected by their physical properties ( see @xmath61 3 ) . we divided our main sample into subsamples ( to be used for stacking ) of normal late - type galaxies selected by both physical properties ( i.e. , @xmath45-band luminosity , stellar mass , and star - formation rate ) and redshifts . in figure 4 , we have highlighted the divisions of our sample with thick gray rectangles , and for normal late - type galaxies in each region , we used stacking analyses to constrain average properties . for each of the subsamples defined above , we performed stacking in each of the three standard bandpasses ( i.e. , sb , hb , and fb ; see @xmath61 1 ) . we expect these bandpasses to sample effectively power - law emission originating from binaries ( i.e. , hmxbs and lmxbs ) with a minor contribution from hot interstellar gas in the sb for late - type galaxies at @xmath180 . in our analyses , we used data products presented in alexander ( 2003 ) for the @xmath02 ms and @xmath01 ms and lehmer ( 2005b ) for the @xmath0250 ks ( see footnotes 15 and 16 ) . our stacking procedure itself was similar to that discussed in @xmath61 3 of steffen ( 2007 ) . this procedure differs from past stacking analyses ( e.g. , lehmer 2007 ) in how the local background of each stacked sample is determined , and produces results that are in good agreement with the method discussed in @xmath61 2.2 of lehmer ( 2007 ) . for completeness , we have outlined this procedure below . using a circular aperture with radius @xmath181 , we extracted source - plus - background counts @xmath182 and exposure times @xmath183 ( in units of cm@xmath36 s ) for each galaxy using images and exposure maps , respectively . for a given source , we used only pointings with aimpoints ( see footnote 12 ) that were offset from the source position by less than 60 ; hereafter , we refer to this maximum offset as the inclusion radius , @xmath185 . has the same meaning as it did in lehmer ( 2007 ) . ] since the psf increases in size with off - axis angle and degrades the sensitivity for sources that are far off - axis , our choices of @xmath186 and @xmath185 have been chosen to give the maximal stacked signal with the majority of the psfs being sampled by our stacking aperture ( see @xmath61 2.2 and fig . 3 of lehmer 2007 for justification ) . we estimate , based on stacked images of @xmath98 optically luminous ( @xmath187 ) late - type galaxies , that @xmath1910% of the normal galaxy emission originates outside of our @xmath188 radius aperture ( at ) for all stacked samples . for galaxies that were within 60 of more than one of the cdf aimpoints , we added source counts and exposure times from all appropriate images and exposure maps , respectively ; however , there were very few sources in our main sample that met this criterion . using background maps ( see @xmath61 4.2 of alexander 2003 for the @xmath02 ms and @xmath01 ms and @xmath61 4 of lehmer 2005b for the @xmath0250 ks ) and exposure maps , we measured local backgrounds @xmath189 and exposure times @xmath190 within a 30 pixel @xmath191 30 pixel ( @xmath0@xmath192 ) square , centered on each source with the 15 radius circle masked out . here again , if a source was within 60 of more than one of the cdf aimpoints , we summed the local backgrounds and exposure times . we estimated the expected number of background counts in each circular aperture @xmath193 by scaling the background counts within the square by the relative exposure times of the circular aperture and the square ( i.e. , @xmath194 ) . this approach is similar to scaling the background counts in the square by the relative areas of the circular aperture and the square ; however , by using exposure times , we are able to account more accurately for spatial variations in pixel sensitivity due to chip gaps , bad pixels , and vignetting . furthermore , comparisons between this method and the monte carlo method used in lehmer ( 2007 ) to compute @xmath193 give excellent agreement and are most convergent for large numbers of monte carlo trials . when stacking galaxy populations , we excluded sources that were ( 1 ) classified as agn candidates ( via the criteria outlined in @xmath61 4.1 ) , ( 2 ) within 10of an unrelated source detected in the catalogs , ( 3 ) within the extent of extended sources ( see bauer 2002b , @xmath61 3.4 of giacconi 2002 , and @xmath61 6 of lehmer 2005b ) , and ( 4 ) located within 3of another late - type galaxy in our main sample . we note that we include detected normal galaxies when stacking our samples , since we are interested in the average properties of the normal late - type galaxy population . we tested the effects of including such sources by stacking samples both with and without detected sources included , and find similar results for both cases . typically , the detected sources contribute of the counts in the total stacked signal . for each of the subsamples of normal late - type galaxies outlined in figure 4 ( _ gray rectangles _ ) , we determined stacked source - plus - background ( @xmath195 ) and background counts ( @xmath196 ) to determine net counts ( @xmath197 ) . for each stacked sample , we required that the signal - to - noise ratio [ s / n @xmath198 ; where @xmath199 and @xmath200 be greater than or equal to 3 ( i.e. , @xmath6899.9% confidence ) for a detection . for stacked samples without significant detections , 3@xmath100 upper limits were placed on the source counts . we converted the net counts obtained from each stacked sample to absorption - corrected fluxes and rest - frame luminosities using a power - law sed with @xmath201 . due to the fact that our 15 radius stacking aperture encircles only a fraction of the psf , our 15 radius circular aperture contains an encircled - energy fraction of @xmath0100% , 80% , and 100% for the sb , hb , and fb , respectively ; however at @xmath202 , this fraction decreases to @xmath030% , 25% , and 25% , respecively . ] for sources at relatively large off - axis angle , we calculated aperture corrections @xmath203 for each stacked source @xmath204 . since we are calculating average counts from the summed emission of many sources of differing backgrounds and exposure times , we used a single , representative exposure - weighted aperture correction , @xmath205 . this factor , which was determined for each stacked sample , was calculated as follows : @xmath206 where @xmath207 . the average aperture corrections ( @xmath205 ) for sources in our main sample were @xmath01.6 , 1.8 , and 1.7 for the sb , hb , and fb , respectively . using our adopted sed , we estimated observed mean fluxes using the following equation : @xmath208 where @xmath209 is a bandpass - dependent count - rate to flux conversion factor that incorporates both the sed information as well as galactic extinction using the column densities listed in @xmath61 1 . these mean fluxes were then converted to rest - frame luminosities following equation 4 , assuming a photon index of @xmath154 . ) versus the logarithm of the to optical flux ratio ( @xmath210 ) for 44 stacked samples selected via observed properties : @xmath2 ( _ filled circles _ ) , @xmath3 ( _ filled squares _ ) , and sfr ( _ filled triangles _ ) . the characteristic mean error bar for each quantity is given in the lower right - hand corner . the median logarithm of the to optical flux ratio is indicated with a vertical dotted line ( @xmath211 ) . the median effective photon index for the samples that were detected in both the sb and hb is indicated with a horizontal dotted line ( @xmath212 ) . we note that all of the stacked samples with both sb and hb detections have @xmath142 , consistent with normal galaxies . for the several stacked samples that have only sb detections , we have indicated lower - limits on @xmath132 . the shaded regions and corresponding boundaries ( _ dashed lines _ ) represent areas where detected sources were classified as agn candidates ( for details , see discussion of criteria 1 and 2 in @xmath61 4.1 ) ; note that additional criteria were used to identify potential agns when generating our samples of normal late - type galaxies ( see criteria 35 in @xmath61 4.1 ) . , width=359 ] using the stacking analysis methods discussed in @xmath61 4.2 , we stacked the late - type galaxy samples presented in figure 4 ( stacked samples are denoted with _ thick gray rectangles _ ) . these samples were selected using @xmath2 , @xmath75 , and sfr , which include 14 , 17 , and 13 stacked samples ( 44 total ) , respectively . in table 2 ( _ available electronically _ ) , we tabulate our stacking results . we found significant ( i.e. , s / n @xmath68 3 ) detections in the and bandpasses for all stacked samples . in the bandpass , 15 out of the 44 stacked samples were detected , and these samples generally constitute the most optically luminous and massive galaxies , as well as those galaxies that are most actively forming stars . in figure 9 , we show the effective photon index ( @xmath132 ) versus the logarithm of the to optical mean flux ratio ( @xmath213 ) for our 44 stacked samples ( _ filled symbols _ ) that were selected via their observed properties . effective photon indices were estimated using hb - to - sb count - rate ratios ( i.e. , @xmath214 ) . all 44 stacked samples have to optical flux ratios and spectral slopes consistent with normal galaxies ( _ unshaded region _ in fig . 9 ) , suggesting that these samples are not heavily contaminated by an underlying population of agns . ) . these samples have and and lie at @xmath215 , a redshift where large - scale sheets of galaxies and agns have been isolated previously in the ( e.g. , gilli 2003 ) .. these galaxies have spectral slopes that are either statistically scattered towards @xmath216 ( see characteristic error bar in fig . 9 ) or may have some non - negligible contribution from heavily - obscured agns . ] we find a median logarithm of the to optical flux ratio of @xmath217 ( _ vertical dotted line _ ) , and for samples that were detected in both the hb and sb , the median effective photon index is @xmath218 ( _ horizontal dotted line _ ) . these values are characteristic of galaxies dominated by binary populations . -band mean luminosity ratio ( @xmath219 ) , ( _ b _ ) the to stellar - mass mean ratio ( @xmath220 ) , and ( _ c _ ) the to - star - formation - rate mean ratio ( @xmath221sfr ) versus redshift ( _ filled symbols and curves _ ) for stacked normal late - type galaxy samples selected by @xmath2 ( fig . 4@xmath69 ) , @xmath3 ( fig . 4@xmath83 ) , and sfr ( fig . 4@xmath81 ) , respectively . for comparison , in each panel we have indicated the properties of the mw and local galaxies m101 , m82 , and ngc 3256 . quoted luminosities correspond to the bandpass and were calculated following the methods described in @xmath61 4.2 , assuming a power - law sed with photon index of @xmath201 . symbols and curves correspond to unique ranges of @xmath2 , @xmath3 , and sfr , which are annotated in each respective figure . for reference , in figures 10@xmath69 and 10@xmath83 we have plotted the corresponding values of @xmath222 and @xmath223 , respectively , for normal late - type galaxies in the local universe ( _ open symbols _ ) using the s01 sample . in figure 10@xmath83 we show the expected lmxb contribution based on gilfanov ( 2004b ; _ dashed line _ ) . finally , in figure 10@xmath81 we show the local -sfr relation and its dispersion ( _ dashed line with shading _ ) derived by pr07 and corrected for our choice of imf and @xmath201 . we note that roughly all of our data points in figure 10@xmath81 lie @xmath0 dex below the pr07 relation , which is likely due to systematic differences in how @xmath4 and sfr were determined between studies . , width=343 ] in figure 10 , we show the logarithm of the ratio of the luminosity ( hereafter , @xmath4 ) to each physical property ( i.e. , @xmath219 , @xmath223 , and @xmath221sfr ) versus redshift for our samples . each relevant quantity for the mw and local galaxies m101 , m82 , and ngc 3256 have been shown for comparison . we adopt @xmath224 as the approximate luminosity of the mw ( grimm 2002 ) . for the local galaxies , we utilized the luminosities from shapley ( 2001 ; hereafter s01 ) for m101 ( @xmath225 ) and m82 ( @xmath226 ) and lira ( 2002 ) for ngc 3256 ( @xmath227 ) . we have corrected the luminosities to be consistent with our use of the bandpass and our choice of @xmath201 . for the purpose of comparing our results to those for nearby late - type galaxy populations , we made use of the s01 sample of 183 normal local ( @xmath228 100 mpc ) spiral and irregular galaxies . these galaxies were observed in the band using the ipc and hri ( fabbiano 1992 ) and agns , with luminous x - ray emission or spectral signatures indicative of agn activity , have been excised from the sample . to avoid the inclusion of early - type s0 galaxies , we chose to utilize 139 normal late - type galaxies from the s01 sample with morphological types @xmath229 ( hubble types ) . the s01 sample covers ranges of @xmath2 and @xmath3 that are well - matched to our main sample and are representative of late - type galaxies in the local universe . in order to compute mean luminosities for samples that were directly comparable with our results , we divided the s01 sample into the same intervals of @xmath2 and @xmath3 that were used for our main sample ( see figs . 4@xmath69 and 4@xmath83 ) . since several of the s01 galaxies had only upper limits available , we computed mean luminosities and errors using the kaplan - meier estimator available through the astronomy survival analysis software package ( asurv rev . 1.2 ; isobe & feigelson 1990 ; lavalley 1992 ) ; the kaplan - meier estimator handles censored data sets appropriately . when computing these mean luminosities , we filtered the s01 samples appropriately into distance intervals to avoid the malmquist bias . in figures 10@xmath69 and 10@xmath83 , we show the corresponding values of @xmath230 and @xmath223 , respectively , for the s01 sample with open symbols . by contrast , the sfrs of the local sample are generally too low ( @xmath19110 ) to provide a meaningful comparison with our distant 24@xmath54m - detected sources . this is due to the strong positive evolution of the star - formation rate density with redshift ( see @xmath61 1 ) , which makes sfrs that are common for galaxies in our sample ( @xmath68110 ) comparatively rare at @xmath231 . from figures 10@xmath69 and 10@xmath83 , it is apparent that there is significant positive redshift evolution in @xmath222 and @xmath223 over the redshift range of . for each of the six total selection ranges of @xmath2 and @xmath3 , the redshift progression of luminosities is inconsistent with a constant at the @xmath23299.9% confidence level . for the most optically luminous ( @xmath233320 @xmath234 ) and massive ( @xmath235 120 @xmath234 ) late - type galaxies at @xmath236 , @xmath237 and @xmath238 are measured to be larger than the local values of s01 by factors of @xmath239 and @xmath240 , respectively . such values are consistent with the @xmath0@xmath241 evolution of @xmath13 found for the normal late - type galaxy population , which has been constrained using largely the most optically luminous and massive galaxies ( georgakakis 2006 ; ptak 2007 ; tzanavaris & georgantopoulos 2008 ) . the above results confirm the increase in @xmath237 with redshift found by h02 . in past studies , @xmath237 has been used as a proxy for star - formation activity ( e.g. , ptak 2001 ; h02 ; lehmer 2005a ) , despite the fact that @xmath2 is likely to be somewhat sensitive to sfr ( see discussion in @xmath61 3.1 ) . as discussed in @xmath61 1 , @xmath4 has been shown to be strongly correlated with galaxy sfr . correlation studies of spiral galaxies in the local universe have also found strong correlations between @xmath4 and @xmath242 such that @xmath243 ( s01 ; fabbiano & shapley 2002 ) . the nonlinear relationship between and @xmath45-band emission is thought to be due to the increase in dust obscuration with star - formation activity , which attenuates light from the @xmath45-band more effectively than it does in the band . using our sample of 47 normal late - type galaxies with both and 24@xmath54 m detections , we found that @xmath237 was positively correlated with the uv dust - extinction measure ( @xmath93+@xmath244)/@xmath92 ( kendall s @xmath245 ; @xmath6899.99% confidence level ) , thus providing support for this hypothesis . furthermore , it has been shown that @xmath237 is correlated with @xmath246 , suggesting that more intense emission is associated with hotter ir colors , which is indicative of intense obscured star - formation activity . therefore , the increase in @xmath237 with redshift ( see fig . 10@xmath69 ) observed for our galaxies is likely due to an increase in their star - formation activity . upon comparing @xmath2 with @xmath3 and sfr for our galaxies , we find that @xmath2 is well correlated with both @xmath3 ( kendall s @xmath247 ) and sfr ( @xmath248 ) ; however , the scatter in the @xmath2@xmath3 relation is significantly smaller than it is for the @xmath2sfr relation ( @xmath00.2 dex versus @xmath00.4 dex , respectively ) implying that @xmath2 traces more effectively @xmath3 rather than sfr . it is therefore not surprising that we see similar redshift evolution of @xmath238 and @xmath237 . in figure 10@xmath83 , we show the estimated lmxb contribution to @xmath249 ( _ dashed line _ ) based on table 5 of gilfanov ( 2004b ) . this value is @xmath0510 times lower than all mean values of @xmath250 , suggesting that on average lmxbs play a fairly small role in the emission from our stacked samples . furthermore , late - type galaxies in the local universe with similar stellar masses are often found to have hmxb emission that is @xmath0210 times more luminous than that expected from lmxbs ( see _ open symbols _ in fig . 10@xmath83 and fig . 3 of gilfanov 2004c ) . for galaxy samples selected via sfr , we find no evidence for significant evolution in @xmath221sfr ( fig . 10@xmath81 ) . we note that _ if _ the majority of the flux from our stacked samples originated from detected sources , then the agn selection criteria 3 and 4 from @xmath61 4.1 may potentially bias these results . however , we find that the more populous and unbiased undetected source populations dominate the stacked signals at the @xmath6880% level ( see column 14 of table 2 ) . for each of the three ranges of sfr , @xmath221sfr is consistent with a constant value , and has a best - fit ratio of @xmath221sfr @xmath251 ( @xmath252 for 12 degrees of freedom ) . these results suggest that the contribution from lmxbs is small and that the integrated emission from our late - type galaxies is dominated by hmxbs . this implies that the evolution of our late - type galaxy samples is likely due to changes in star - formation activity . cccccc @xmath2 & 11 & 25.37 & @xmath253 & @xmath254 & @xmath255 + @xmath3 & 14 & 25.73 & @xmath256 & @xmath257 & @xmath258 + sfr & 10 & 9.72 & @xmath259 & @xmath260 & @xmath261 sfr versus the logarithm of the luminosity @xmath262 for normal late - type galaxies . detected sources from our main sample are indicated as small gray dots . sources that were not detected in the bandpass but were detected in either the or bandpasses are shown as upper limits . results from our stacking analyses of late - type galaxies selected via observed sfr are shown as large filled circles , squares , and triangles , which have the same meaning as in figure 10@xmath81 ; @xmath263 lbgs that were both uncorrected and corrected for agn contamination have been shown plotted as a filled star and open star , respectively . for comparison , we show the local galaxy sample from pr07 , which includes normal late - type galaxies ( _ open circles _ ) and ultraluminous infrared galaxies ( ulirgs ; _ open squares _ ) ; the best - fit pr07 relation is shown as a solid curve . the sfrs for the mw and local galaxies m101 , m82 , and ngc 3256 are indicated ( _ horizontal dashed lines _ ) . , width=377 ] since the emission from our late - type galaxies is dominated by star - formation processes , we note that our @xmath3-selected stacking results provide a relatively unobscured measure of the star - formation activity per unit stellar mass ( i.e. , the ssfr ; see fig . 10@xmath83 ) . we find that at @xmath264 the emission per unit stellar mass is a factor of @xmath0 larger for galaxies with versus that observed for galaxies with . at @xmath7 1 , we find that @xmath265 is larger than its local value ( s01 ) by factors of @xmath266 and @xmath267 for late - type galaxies with and , respectively . these results are broadly consistent with observed differences in the mean ssfrs found by zheng ( 2007 ) for @xmath264 galaxies of comparable stellar masses , and imply that the lower - mass galaxies are undergoing more significant stellar mass growth over than more massive galaxies . in order to quantify the dependences of the luminosity on redshift and physical properties , we performed multivariate parametric fitting to our stacked data . for each galaxy sample selected via @xmath2 , @xmath3 , and sfr , we investigated the redshift evolution of the luminosity . for this analysis , we fit our data to a power - law parametric form : @xmath268 where @xmath269 is a place holder for each of the three physical properties ( @xmath2 , @xmath3 , and sfr ) used for our sample selections , and @xmath270 , @xmath45 , and @xmath271 are fitting constants . for each sample , we utilized our stacking results , equation 7 , and @xmath272 fitting to estimate the constants @xmath270 , @xmath45 , and @xmath271 . for our adopted three - component model , we constrained @xmath270 , @xmath45 , and @xmath271 using 90% confidence errors ( @xmath273 ) . the s01 local data points were not used for these fits due to differences in galaxy selection , instrument calibration , and agn identification . versus @xmath274 for late - type galaxies with @xmath275 at @xmath276 ( _ filled circles _ ) and @xmath98 0.61 ( _ open triangles _ ) . for reference , we have indicated the corresponding agn fraction for lyman break galaxies at @xmath263 ( _ filled stars _ ; see @xmath61 5 for further details ) . we have indicated the median detection limit for galaxies in each redshift range ( _ downward - pointing arrows along the x - axis _ ) . the solid curve represents the best - fit relation for @xmath277 , which was fit using all late - type galaxies with @xmath98 01 and @xmath278 in our main sample . for reference , we have shown the estimated agn fraction for galaxies with @xmath279 , @xmath280 , and @xmath281 ( _ dotted curves _ ; see @xmath61 5.2 for further details ) . , width=351 ] in table 3 , we tabulate our constraints on @xmath272 , @xmath270 , @xmath45 , and @xmath271 for these fits . we find that this particular choice ( i.e. , eqn . 7 ) of parameterization does not provide acceptable fits for galaxy samples selected via @xmath2 and @xmath3 . however , for galaxy samples selected via sfr , we find a good fit for this parameterization ( @xmath282 for ten degrees of freedom ) . we constrain the evolution of @xmath262/sfr to be independent of or at most weakly dependent on redshift [ @xmath283 . based on radio observations of distant star - forming galaxies with sfr @xmath0 in the and , barger ( 2007 ) reported that the upper limits for undetected sources were below the level expected from the -sfr correlaiton , thus suggesting that the correlation may not hold in the high - redshift universe . however , our stacking results suggest that the -sfr correlation _ does _ in fact hold for average galaxies with sfr = ( sfr = ) out to @xmath284 ( @xmath5 ) . for illustrative purposes we have created figure 11 , which shows @xmath285 sfr versus @xmath285 for normal star - forming galaxies selected from several different sources including detected galaxies from our main sample ( _ filled gray circles and limits _ ) , stacked galaxies from this study ( _ filled black symbols _ ) , local galaxies from pr07 ( _ open circles _ ) , local ultraluminous infrared galaxies from pr07 ( ulirgs ; _ open squares _ ) , and stacked @xmath263 lyman break galaxies ( lbgs ; _ stars _ ; see @xmath61 6 for details ) . for all data used in this plot , we have normalized sfrs appropriately to be consistent with our adopted kroupa ( 2001 ) imf and have adjusted luminosities to correspond to the band using a @xmath154 power - law sed . the best - fit -sfr correlations for local galaxies from pr07 ( _ solid curve _ ) and @xmath17 01.4 late - type galaxies from this study ( @xmath262/sfr = 39.87 ; _ dotted curve _ ) have been shown for reference . for comparison , we have shown the sfrs for the mw and local galaxies m101 , m82 , and ngc 3256 . in this section we estimate the contribution to our stacked signals from contaminating agns that have luminosities below our detection limit . this analysis is similar in nature to that in @xmath82 3.1 and 3.2.2 of lehmer ( 2007 ) , which was performed for early - type galaxies . we implement the observed cumulative agn fraction @xmath286 : the fraction of galaxies harboring an agn with luminosity of @xmath274 or greater . hereafter , we compute @xmath286 by taking the number of candidate agns in a particular galaxy sample with a luminosity of @xmath287 or greater and dividing it by the number of galaxies in which we could have detected an agn with luminosity @xmath274 . the latter number is computed by considering the redshift of each galaxy and its corresponding sensitivity limit , as obtained from spatially varying sensitivity maps ( see @xmath61 4.2 of alexander 2003 and @xmath61 4 of lehmer 2005b ) ; these sensitivity maps were calibrated empirically using sources detected by wavdetect at a false - positive probability threshold of @xmath288 . the quantity @xmath286 is not only a function of @xmath274 , but is also dependent on the selection of the galaxy sample considered : in our case redshift and the physical property @xmath269 ( i.e. , @xmath2 , @xmath3 , and sfr ) of the galaxy population may plausibly play a role in @xmath286 . in our analyses , we assume that each of the respective dependencies ( i.e. , @xmath287 , @xmath289 , and @xmath269 ) are independent of each other , such that @xmath290 , where @xmath291 , @xmath292 , and @xmath293 represents the functional dependence of the cumulative agn fraction for each indicated variable @xmath274 , @xmath289 , and @xmath269 , respectively . we made use of the bandpass because of its ability to probe relatively unattenuated emission in a regime of the spectrum where we expect there to be minimal emission from normal galaxies ( see also criterion 1 of @xmath61 4.1 for further details ) . in total 62 ( @xmath051% ) of our 121 detected agn candidates had detections ; we use these agns in our agn fraction analyses . versus redshift for late - type galaxy samples with @xmath294 and @xmath295 . we find no significant evolution of @xmath296 over the redshift range . , width=351 ] we began constructing @xmath286 by estimating the shape of @xmath277 using late - type galaxies with and @xmath275 , an optical luminosity regime where we have a relatively large number of sources and are sufficiently complete out to @xmath297 ( see fig 4@xmath69 ) . we split this sample into two subsets about @xmath22 , to test whether there is substantial evolution in the shape and normalization of @xmath277 over this redshift range . in figure 12 , we show @xmath277 for galaxies in the redshift ranges @xmath276 ( @xmath298 ; _ filled circles _ ) and ( @xmath299 ; _ open triangles _ ) . from figure 12 , we see that the overall shape and normalization of @xmath277 for late - type galaxies with @xmath300 is similar for galaxies at @xmath298 and @xmath301 . we fit the shape of @xmath277 using least - squares fitting of the luminosity dependent cumulative agn fraction using all galaxies from with @xmath300 . for these fits , we found that the data were well - fit by an exponential function , which we parameterized as @xmath302 , where @xmath69 and @xmath83 are fitting constants . by construction , this function is only valid for @xmath303 , which is @xmath012 orders of magnitude less luminous than a typical stacked luminosity of our late - type galaxy samples ( see table 2 and fig . we find best - fit values of @xmath304 and @xmath305 ; in figure 12 ( _ thick black curve _ ) , we show our best - fit relation for @xmath277 . we constrained further the redshift evolution of @xmath286 [ i.e. , the shape of @xmath296 ] by dividing our main late - type galaxy sample into five nearly independent redshift bins ( with ) and calculating @xmath286 for fixed ranges of @xmath274 and @xmath269 . in figure 13 , we show @xmath296 as a function of redshift for late - type galaxies with @xmath295 and @xmath294 , the approximate completeness limit at @xmath307 ( see fig . 4@xmath69 ) . using these data and @xmath272 fitting [ assuming a @xmath308 dependence ] , we constrained the redshift evolution of @xmath296 to be proportional to @xmath309 ; similar results were found for different ranges of @xmath274 and @xmath2 . this result differs from that observed for early - type galaxies , where the agn fraction and mean agn emission has been found to evolve as @xmath0@xmath310 ( e.g. , brand 2005 ; lehmer 2007 ) . ) late - type galaxy agn number density is constant over . for example , @xmath16 has been shown to fade by @xmath01 mag since @xmath264 ( e.g. , lilly 1995 ; wolf 2003 ; faber 2007 ) , suggesting that there are fewer optically luminous late - type galaxies in the local universe than at @xmath311 . ] we therefore conclude that there is little redshift evolution in the late - type galaxy agn fraction over the redshift range , and hereafter we assume that @xmath296 remains roughly constant out to @xmath84 . to constrain the overall dependence of @xmath286 on @xmath269 [ i.e. , @xmath312 , we calculated the cumulative agn fractions for late - type galaxy samples with @xmath269 by holding the ranges of @xmath274 and @xmath289 fixed and varying @xmath269 . in figure 14 , we show @xmath313 versus @xmath314 ( fig . 14@xmath69 ) , @xmath315 ( fig . 14@xmath83 ) , and @xmath316 ( fig . 14@xmath81 ) for @xmath317 at @xmath318 ( _ dashed curves _ ) and ( _ dotted curves _ ) . we calculated @xmath319 for intervals of @xmath269 where we are approximately complete at @xmath22 ( for the @xmath276 interval ) and @xmath297 ( for the interval ; see fig . 4 ) . for each sample , we again utilized least - squares fitting to approximate the @xmath320 dependence of @xmath319 . these fits were performed using all data over the redshift range assuming a functional dependence of @xmath321 . in each panel of figure 14 , we show the best - fit solutions for @xmath319 with the gray lines . we find that the agn fraction is strongly dependent on @xmath2 ( fig . 14@xmath69 ) and @xmath3 ( fig . 14@xmath83 ) , such that more optically - luminous and massive galaxies have larger agn fractions ; this result is consistent with other studies of the agn host galaxies ( e.g. , kauffmann 2003b ; nandra 2007 ; silverman 2007 ) . also , the agn fraction seems to be mildly dependent on the galaxy sfr ; however , this is likely due to the fact that the sfr is larger on average for more massive galaxies . based on the above estimates of the shapes of @xmath277 , @xmath296 , and @xmath319 , we approximated empirically @xmath286 for each choice of @xmath269 following : @xmath322 + c\log f_{\rm phys } , % \ ] ] where @xmath323 represents the normalization of each relation based upon the value of @xmath286 at @xmath324 and , , and sfr @xmath325 , for galaxy samples selected via @xmath2 , @xmath3 , and sfr , respectively . we note that only @xmath323 and @xmath81 are dependent upon our choice of @xmath269 ; we find for the set of physical parameters @xmath326 [ @xmath2 , @xmath3 , sfr ] that @xmath327 [ @xmath328 , @xmath329 , 0.20 ] and @xmath330 [ 1.2 , 0.67 , 0.66 ] . for reference , in figure 12 we have shown curves of @xmath286 for @xmath331 , @xmath280 , and @xmath281 ( _ dotted curves _ ) . to estimate the agn contamination expected for each of our stacked samples presented in @xmath61 5.1 , we followed closely the procedure in @xmath61 3.2.2 of lehmer ( 2007 ) . for completeness , we outline this procedure below . for each stacked sample , we used equation 8 to compute cumulative agn fractions @xmath286 . in figure 15@xmath69 , we show @xmath286 for two of our @xmath2-selected samples : @xmath332 galaxies with @xmath33310@xmath334 ( _ dashed curve _ ) and @xmath335 galaxies with @xmath33610@xmath337 ( _ solid curve _ ) . we then converted @xmath286 for each sample to a differential agn fraction @xmath338 ( i.e. , the luminosity dependent fraction of galaxies harboring agns within discrete luminosity bins of width @xmath339 ; see fig . 15@xmath83 ) . for each sample , we calculated the luminosity dependent fraction of galaxies that were below our @xmath274 detection limit @xmath340 ( fig . 15@xmath81 ) . we then multiplied @xmath338 and @xmath340 to estimate the luminosity dependent fraction of galaxies harboring an agn that was undetected in the exposures @xmath341 ( i.e. , @xmath342 ; see fig . 15@xmath343 ) ; these agns would _ not _ have been removed for our stacking analyses . finally , we approximated the total agn contamination @xmath344 of each stacked sample using the following summation : @xmath345 where the summation is over all bins of @xmath339 in the range @xmath346 3945 . we converted each value of @xmath347 to estimated values of @xmath348 and @xmath344 by assuming the contaminating agn emission roughly follows an sed described by a power - law . in order to constrain the average photon index of the power - law , we stacked all detected agns in our stacked samples with luminosities below @xmath349 . for these agns , we find a stacked effective photon index of @xmath350 . if the intrinsic value of the photon index is @xmath154 , at the median redshift of our main sample ( @xmath351 ) , @xmath352 corresponds to an intrinsic column density of @xmath353 @xmath354 @xmath24 . if we assume @xmath355 describes well the effective sed of the undetected agns in our stacked sample , we find that agn contamination can account for ( median of @xmath05% ) of the emission from our stacked samples , suggesting that agns are not providing a significant contribution to our stacked results . we note that the sed used in this calculation has an important effect on the overall estimate of the agn contamination . since our estimates for contamination in the and bandpasses decrease as @xmath132 decreases , the amount of contamination in our samples may be underestimated if our choice of @xmath132 is too flat ; however , we find that for conservative choices of @xmath132 ( i.e. , @xmath356 ) that are representative of even unobscured agns ( for reference , see fig . 5 ) , the agn contamination remains low ( @xmath1940% ) and has no material effect on our results . it is also worth noting that if there exists a large population of intrinsically luminous yet heavily obscured agns in our high - redshift galaxy samples that are undetected in the cdfs , then our conclusions above could be somewhat different . if such a population were present and had significant influence on our stacked results , then we would expect to find stacked spectra that were relatively flat ( @xmath125 ) ; however , as discussed in @xmath61 5.1 , we find that very few of our stacked samples could have such flat spectra in our stacked samples , suggesting that such a population , if present , does not have a strong effect on our results . or greater ) , @xmath286 , versus @xmath357 for two of our stacked samples : @xmath332 galaxies with @xmath33310@xmath334 ( _ dashed curve _ ) and @xmath335 galaxies with @xmath35810@xmath337 ( _ solid curve _ ) . each curve was computed following equation 8 . ( _ b _ ) differential agn fractions ( i.e. , the fraction of galaxies harboring an agn in discrete bins of width @xmath359 ) , @xmath338 , versus @xmath357 . ( _ c _ ) fraction of late - type galaxies for which we could _ not _ have detected an agn with a luminosity of @xmath274 , @xmath340 , versus @xmath357 . ( _ d _ ) fraction of galaxies harboring agns in our optically luminous faded samples that would remain undetected due to sensitivity limitations , @xmath342 , versus @xmath357 ; these galaxies would not have been removed from our stacking analyses . ] in table 2 ( cols . 2527 ) , we have provided the estimated fractional agn contribution to each stacked sample [ i.e. , @xmath360 for the fb , sb , and hb using the technique described above and an assumed @xmath361 . we find that the estimated agn contamination is most significant for galaxy samples with large values of @xmath2 , @xmath3 , and sfr . as noted in @xmath61 1 , the global star - formation rate density has been observed to increase with redshift out to . investigations of the most - distant lbgs at show that the star - formation density peaks around and gradually declines toward higher redshifts ( e.g. , steidel 1999 ; giavalisco 2002 , 2004b ; bouwens 2004 , 2005 ; dickinson 2004 ) . to investigate whether the mean activity from normal late - type galaxies follows a similar trend , we study the properties of a sample of @xmath362 lbgs , which were identified as @xmath44-band `` dropouts '' through the goods project ( see lehmer 2005a for details ) . we filtered the goods @xmath263 lbg sample ( lee 2006 ) to include only lbgs that ( 1 ) were within the central @xmath040 of the @xmath02 ms and @xmath01 ms , and ( 2 ) had rest - frame @xmath45-band luminosities that were similar to the most optically luminous ( @xmath233 320 @xmath363 ) late - type galaxies used in this study . @xmath45-band luminosities were calculated by applying @xmath364-corrections to the @xmath33-band flux ( from goods ) , where @xmath364-corrections were derived using an sed appropriate for lbgs ( see @xmath61 2.2 of lehmer 2005a for details ) . we found that 85 @xmath263 lbgs from the lehmer ( 2005a ) sample satisfied these two selection criteria . we identified three detected lbgs , which all had @xmath365 . due to their high luminosities , we classified these sources as obvious agns . after removing these three detected agns from our @xmath263 lbg sample , we performed stacking analyses as described in @xmath61 4.2 . we found a significant ( @xmath366 ) detection in the bandpass , which corresponds roughly to rest - frame emission . assuming an intrinsic power - law spectrum with a photon index of @xmath201 , we found a mean luminosity of for our @xmath263 lbgs , a value that agrees well with previous studies ( e.g. , lehmer 2005@xmath69 ; laird 2006 ) . -band mean luminosity ratio @xmath222 versus the age of the universe ( for our adopted cosmology , the current age of the universe is 13.47 gyr ) for star - forming galaxies with @xmath233 @xmath363 . for reference , redshift has been plotted along the top axis . we have included mean values of @xmath237 for the s01 local sample of late - type galaxies ( _ open circles _ ) , our stacked samples at @xmath98 ( _ large filled circles _ ) , and @xmath263 lbgs that were both uncorrected ( _ filled star _ ) and corrected ( _ open star _ ) for agn contamination . at @xmath7 , we have plotted @xmath237 for individual late - type galaxies from our main sample ( _ small filled circles _ and _ upper limits _ ) ; the mean values and errors for these galaxies , computed using asurv , has been indicated . ] since our lbgs reside in the high - redshift universe , agn contamination is expected to have a more significant effect on these results than it did for our late - type galaxies . the median luminosity detection limit is and @xmath0@xmath159 for the and bandpasses , respectively . to estimate the agn contamination , we followed the approach outlined in @xmath61 5.2 , which made use of the luminosity dependent agn fraction ( @xmath286 ) . in figure 12 ( _ filled stars _ ) , we show the luminosity dependent agn fraction @xmath367 for @xmath6 lbgs . we note that at @xmath368 , the cumulative agn fraction for @xmath6 lbgs is a factor of @xmath034 times larger than that computed for our late - type galaxy sample with similar optical luminosities . using the functional form for @xmath286 presented in equation 8 , but with a times larger normalization factor for @xmath263 lbgs ( i.e. , @xmath369 34 @xmath323 ) , we find that agn emission may plausibly account for of the stacked counts . in figure 16 , we show the to@xmath45-band mean luminosity ratio for @xmath370 320 @xmath234 star - forming galaxies ( i.e. , late - type galaxies and lbgs ) as a function of the age of the universe . together , these data span @xmath085% of cosmic history ( i.e. , out to @xmath371 ) . as presented in @xmath61 5.1 , /@xmath2 shows significant evolution over the redshift range @xmath17 01.4 , and after correcting for agn contamination , we find that /@xmath2 is similar for @xmath263 lbgs and @xmath1 late - type galaxies . this result suggests that the non - agn emission for the most luminous star - forming galaxies may flatten near , which has been predicted roughly from simulations of how the normal - galaxy emission is expected to respond due to global changes in the star - formation rate density ( e.g. , ghosh & white 2001 ) . to test whether the -sfr correlation is similar for lbgs as we found for late - type galaxies at , we approximated absorption - corrected sfrs for the @xmath6 lbgs using uv band emission . these sfrs were approximated following sfr = @xmath372 , where is the absorption - correction factor ( see giavalisco 2004b ) . as described in @xmath61 3.3 , we approximated the uv luminosity following @xmath94(2800 ) ; however , here @xmath96(2800 ) was derived using our adopted lbg sed . we find that the mean sfr for optically - luminous @xmath6 lbgs is . after correcting the mean stacked luminosity for agn contamination , we find an to sfr ratio of @xmath262/sfr @xmath0 , which is suggestively lower than its value for late - type galaxies ( i.e. , @xmath262/sfr ) . we note , however , that the estimation of the mean sfr for these galaxies is based solely on the 2800 emission and is therefore highly uncertain ; further constraints on the mean infrared luminosity of these sources would help considerably . for reference , in figure 11 we have plotted @xmath285 sfr versus @xmath262 for the lbgs both uncorrected ( _ filled star _ ) and corrected ( _ open star _ ) for agn contamination . it is interesting to note that the agn - corrected lbg sample appears to have @xmath4 and sfr values similar to those of local ulirgs , which are relatively underluminous for their derived sfrs . pr07 suggest that galaxies with sfrs that begin to exceed @xmath6850100 , similar to the ulirgs , may have properties that are completely dominated by hmxbs . in such systems , other emitting populations that may normally be significant in more quiescent galaxies ( e.g. , lmxbs , hot gas , supernovae and their remnants , etc . ) would collectively contribute only a negligible fraction of the total emission , thus causing the overall galaxies to appear underluminous compared with the sfr correlation . we note , however , that our results presented in @xmath61 5.1 suggest that hmxbs likely dominate the x - ray emission from galaxies with sfrs much lower than those of ulirgs . another possibility is that as the absorption within the most actively star - forming galaxies increases with sfr ( see discussion in @xmath61 5.1 ) , the emission from point - source populations may become significantly obscured . deep observations of local lirgs and ulirgs with and have found significant absorption in their point - source populations ( see , e.g. , lira 2002 ; zezas 2002 , 2003 ) thus providing some support for this possibility . future investigations into the nature of the populations of ulirgs could help resolve this issue . we have investigated the emission from 2568 normal late - type galaxies over the redshift range @xmath17 01.4 that lie within the deep fields ( cdfs ) . our late - type galaxy sample was constructed primarily using color - magnitude diagrams , which incorporated rest - frame @xmath43 color and absolute @xmath46-band magnitudes , to isolate blue late - type galaxies ( see @xmath61 2 for details ) . in total , 225 ( @xmath09% ) of our late - type galaxies were detected individually in the band . based on and optical spectral properties , to optical flux ratios , the correlation between luminosity and star - formation rate , and comparisons between infrared and radio properties , we infer that 121 ( @xmath053% ) of the detected late - type galaxies are dominated by agn emission . the remaining 104 detected galaxies had and multiwavelength properties consistent with those of normal late - type galaxies with emission dominated by binaries ( hmxbs and lmxbs ) . to study the emission and evolution from large representative populations of late - type galaxies ( i.e. , including the undetected sources ) , we utilized stacking analyses of galaxy populations with agn candidates removed . we stacked normal - galaxy samples that were selected via their rest - frame @xmath45-band luminosity ( @xmath2 ) , stellar mass ( @xmath3 ) , and star - formation rate ( sfr ) in redshift bins ( see @xmath61 5.1 ) . furthermore , we compared these results with those found for a sample of @xmath6 lbgs from lehmer ( 2005a ) . in the points below , we summarize our key findings : \1 . we obtained significant detections in the and bandpasses for all of our stacked samples . we estimated that lmxbs and low - level agns provide only low - level contributions to the stacked emission from our samples and that hmxbs constitute the dominant emitting component . therefore , for these galaxies the emission is tracing primarily star - formation activity . normal late - type galaxy samples selected via @xmath2 and @xmath3 show significant ( at the @xmath23299.9% confidence level ) evolution in their average properties from @xmath15 to 1.4 . for the most optically luminous ( @xmath373 ) and massive ( ) late - type galaxies at @xmath236 , @xmath237 and @xmath238 are measured to be larger than their local values by factors of and , respectively . we find that late - type galaxies of lower stellar mass generally have larger to stellar - mass mean ratios ( @xmath238 ) than their higher - mass analogs . over @xmath7 0.21 , galaxies with @xmath374 are a factor of times more luminous per unit stellar mass than galaxies with @xmath375 . \3 . we characterized the properties of 888 24@xmath54m - detected late - type galaxies in our sample . the 24@xmath54 m data allowed us to select galaxy samples selected via their sfrs . for these samples , we found that the luminosity is well - predicted by a constant to sfr ratio , similar to the -sfr correlation reported by previous authors ( e.g. , pr07 ) . this implies that the -sfr correlation holds out to at least @xmath70 , 1 , and 1.4 for galaxies with sfr @xmath0 2 , 10 , and 50 , respectively , and supports the idea that the strong evolution observed for normal late - type galaxies selected via @xmath2 and @xmath3 is likely due to strong changes in sfr . the properties of our most optically - luminous ( @xmath233 320 @xmath363 ) late - type galaxies at @xmath236 are comparable to those for @xmath376 lbgs with similar optical luminosities , once undetected agn contamination in the lbg population has been accounted for . this suggests that there may plausibly be a flattening in the @xmath237@xmath289 relation for optically - luminous star - forming galaxies between @xmath377 1.43 . we estimate a mean sfr of @xmath060 for these lbgs . we find that the observed mean luminosity is suggestively underluminous based on the -sfr correlation prediction ; this result is similar to that found for local ulirgs with comparable sfrs . + the above results can be improved greatly through ( 1 ) the study of late - type galaxy populations in other existing multiwavelength extragalactic survey fields that contain observations , ( 2 ) additional observations of the cdfs to even better sensitivity levels than are currently available , and ( 3 ) observations with future missions with imaging capabilities that are complementary to those of . the areal footprint of the cdf regions used in this study totals @xmath00.18 deg@xmath36 ( see @xmath61 2.1 for details ) , which severely limits the number of galaxies being studied at @xmath378 . the redshift range spans @xmath02.5 gyr of cosmic look - back time , compared with the @xmath06.6 gyr over , that we are most effectively studying above . several complementary deep and wide extragalactic survey fields that have recently been conducted or are in progress can improve the present situation . three such ideal survey fields are the @xmath0200 ks all - wavelength extended groth strip international survey ( aegis ; e.g. , nandra 2005 ; davis 2006 ) , the @xmath050 ks cosmic evolution survey ( c - cosmos ; pi : m. elvis ; see also , scoville 2007),50 ks acis - i exposures ; in these regions , the total exposure will reach @xmath0200 ks due to the overlaping exposures . however , the analyses presented in this paper utilize only high - quality imaging at off - axis angles of @xmath1960 ( see @xmath61 2.1 for additional justification ) , where there is very little overlap in the c - cosmos pointings . we therefore do not consider the overlapping regions of the c - cosmos exposures in this discussion . for additional information regarding c - cosmos , see http://cfa-www.harvard.edu/hea/cos/c-cosmos.html . ] and the @xmath05 ks noao deep wide - field survey ( ndwfs ; e.g. , murray 2005 ) . our study shows that identifying and removing contaminating agns is a crucial ingredient to studying the stacked properties of normal - galaxy samples ( see @xmath61 5.2 ) . in order to avoid significant levels of agn contamination in shallower wide - area surveys such as aegis , c - cosmos , and ndwfs , studies of normal galaxy populations must be limited to lower redshift intervals . if we require a detection limit of @xmath379 over regions of each survey where the sensitivity is optimal ( i.e. , @xmath1960 from the aim - points ) , then we estimate that aegis , c - cosmos , and ndwfs could effectively be used to study normal late - type galaxy populations at redshifts less than @xmath00.5 , @xmath00.25 , and @xmath00.1 , respectively . when factoring in the areal coverage of these fields , we estimate that aegis and c - cosmos will contain @xmath0250350 late - type galaxies with at ( i.e. , a factor of @xmath03 larger than the cdfs ) and that aegis , c - cosmos , and ndwfs taken together will have @xmath0100250 similar galaxies at @xmath380 ( i.e. , a factor of @xmath022 times larger than the cdfs ) . studying these galaxy populations would significantly improve constraints on the properties of @xmath381 late - type galaxies . deeper observations of the cdfs would provide additional insight into the properties of the normal late - type galaxy populations presented in this study and enable us to extend our analyses to higher redshifts . typical galaxies in our sample have mean luminosities of . in the @xmath02 ms , where our sensitivity is greatest , we expect that galaxies with @xmath382 should be detectable out to @xmath332 . out of the 11 late - type galaxies with @xmath383 at @xmath378 , we find detections for 8 of them ( @xmath072% ) . deeper observations over the cdfs will allow for the detection of such galaxies out to progressively higher redshifts . for exposures of @xmath05 ms and @xmath010 ms , a source with @xmath384 should be detectable out to @xmath12 and @xmath385 , respectively . if we pessimistically assume an detection fraction of @xmath070% , we estimate that @xmath030 and @xmath050 normal galaxies would be detected individually for each respective deep exposure . the individual detection of these galaxies would allow for improved constraints on both the luminosity functions of normal late - type galaxies at higher redshifts and the low - luminosity agn contributions to our stacked signals . at higher redshifts , an @xmath05 ms and @xmath010 ms exposure would enable us to effectively perform studies of normal late - type galaxies , similar to those presented in this paper , out to @xmath386 and @xmath387 , respectively . such a data set would provide , for the first time , a reliable constraint on the emission from normal star - forming galaxies near the peak of the global star - formation rate density at @xmath7 1.53 . in addition to the improvement that additional observations could provide , future missions such as and should enable new scientific investigations of distant normal galaxies . will be able to place significant spectral constraints for sources with fluxes down to @xmath0@xmath388 , a level fainter than many of the average fluxes derived from our samples . is planned to provide imaging with @xmath001 per resolution element and will easily probe to flux levels of @xmath0@xmath389 ( @xmath390 at @xmath297 ) . at these levels , the first detailed investigations of the evolution of the normal late - type galaxy luminosity function can be performed effectively out to @xmath11 without confusion problems due to the crowding of large numbers of sources . furthermore , these observations will allow for new constraints to be placed on the populations of all normal galaxies in the observable universe that are offset by more than @xmath00.8 kpc from their host - galaxy nuclei . we thank robin ciardullo , doug cowen , mike eracleous , caryl gronwall , bin luo , brendan miller , tim roberts , and ohad shemmer for useful suggestions , which have improved the quality of this paper . we also thank the referee for carefully reviewing the manuscript and providing detailed comments . we gratefully acknowledge financial support from x - ray center grant g04 - 5157a ( b.d.l . , a.t.s . ) , the science and technology facilities council fellowship program ( b.d.l ) , the royal society ( d.m.a . ) , the emmy noether program of the deutsche forschungsgemeinscaft ( e.f.b . ) , nasa ltsa grant nag5 - 13102 ( d.h.m . ) , the fellowship program ( f.e.b . ) , and nsf grant ast 06 - 07634 ( d.p.s . ) .
|
we report on the evolution over the last @xmath09 gyr of cosmic history ( i.e. , since @xmath1 ) of late - type galaxy populations in the deep field - north and extended deep field - south ( and , respectively ; jointly cdfs ) survey fields .
our late - type galaxy sample consists of 2568 galaxies , which were identified using rest - frame optical colors and morphologies .
we utilized stacking analyses to investigate the emission from these galaxies , emphasizing the contributions from normal galaxies that are not dominated by active galactic nuclei ( agns ) . over this redshift range
, we find significant increases ( factors of @xmath0510 ) in the mean luminosity ratio ( /@xmath2 ) and the mean ratio ( /@xmath3 ) for galaxy populations selected by @xmath2 and @xmath3 , respectively . when analyzing galaxy samples selected via sfr , we find that the mean ratio ( @xmath4/sfr ) is consistent with being constant over the entire redshift range for galaxies with , thus demonstrating that emission can be used as a robust indicator of star - formation activity out to @xmath5 .
we find that the star - formation activity ( as traced by luminosity ) per unit stellar mass in a given redshift bin increases with decreasing stellar mass over the redshift range , which is consistent with previous studies of how star - formation activity depends on stellar mass .
finally , we extend our analyses to lyman break galaxies at @xmath6 and estimate that /@xmath2 at @xmath6 is similar to its value at @xmath1 .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
in a 1959 paper m. e. fisher introduced the general bond - decorated ising model as one example of a set of exactly solvable transformations of spin-1/2 ising models.@xcite a bond - decorated ising model has an `` arbitrary statistical mechanical system '' inserted in every original ising bond . the partition function of this decorated model is related to the partition function of the original or bare ising model by the addition of a prefactor and a renormalization of the coupling constants and magnetic moments ( the ising model is supposed to be in a parallel magnetic field).@xcite knowledge of the partition function of a given ising model thus allows one to obtain the partition function of any bond - decorated version of that ising model . more recently , streka and jaur have used the method of bond - decoration to investigate the thermodynamics of mixed ising - heisenberg chains in parallel magnetic fields where the decorating unit is a spin dimer or trimer with anisotropic heisenberg coupling ( see fig . [ 4:fig : isingheisenberg]).@xcite the partition function of these chains is readily obtained from the known partition function of the ising chain and the energy levels of the decorating unit . therefore , they could calculate exact magnetic properties and theoretically show , for example , the existence of magnetization plateaus in certain bond - alternating chains . the ising - heisenberg chain discussed in ref . ( showing only two unit cells ) . the bonds of the ising chain ( 1-@xmath2-@xmath3- ) are decorated with heisenberg dimers @xmath4(2 - 3 ) , ( @xmath5-@xmath6 ) , the partition function of this chain is exactly solvable . ] the convenience of the decorated ising chain as a theoretical model for spin chains derives from the relative ease with which exact solutions are obtained , in contrast for example with the pure heisenberg chain , for which no exact partition function has been found . up to now , this property of solvability has been the prime motive for the study of these chains in the literature . indeed , in ref . the decorated ising chain was considered as a substitute for the intractable heisenberg model , and in ref . the principle reason for introducing ising bonds to replace the more reasonable heisenberg bonds in a chain of cu^2+^ ions was the desire to obtain a solvable model . this approach can be applied to any type of heisenberg chain with a repeating unit : replace enough preferably ferromagnetic heisenberg bonds by ising bonds to obtain a decorated ising chain that is solved easily and , in some cases , exhibits thermodynamic properties that are qualitatively comparable with those of the original chain.@xcite however , the role of the decorated ising chain in the field of one - dimensional magnetism is not confined to that of a simplified model of realistic quantum spin chains . in this paper we show that some new molecular rings and chains are real examples of decorated ising systems . concretely , we treat a [ dycumocu ] infinite chain@xcite and a ( dycr)@xmath8 tetrameric ring.@xcite these compounds were recently synthesized in the course of the ongoing synthetic efforts to make new and better single - chain magnets ( scms ) and single - molecule magnets ( smms ) , whose characteristic property is a blocking or slow relaxation of magnetization at low temperatures . we will not be concerned here with these dynamical aspects of their magnetism , but only with their static magnetic properties . a necessary property of smms and scms is a magnetic anisotropy . one line of approach is to introduce anisotropy by means of lanthanide ions , whether or not in combination with transition metal ions.@xcite the two compounds considered here are products of this approach , with dysprosium as lanthanide ion . the dy@xmath9 ion plays a crucial role in these systems ; the nature of the ground state of this ion in its ligand environment determines whether the system is a decorated ising system or not and consequently , whether its partition function is exactly solvable or not . the ground kramers doublet of dy@xmath9 must have complete uniaxial magnetic anisotropy - factor of the kramers doublet is not zero ; for example @xmath10 and @xmath11 . ] and must be separated from excited kramers doublets by an amount that is large compared with the exchange coupling ( typically , this separation must be 100 @xmath12 or more).@xcite the required information on the ground and excited doublets of the dy@xmath9 monomer can be derived from _ ab initio _ calculations on the monomer complex , isolated from the polynuclear compound.@xcite the [ dycumocu ] chain and the ( dycr)@xmath8 ring are shown to be decorated ising chains in an arbitrarily directed magnetic field . the magnetic properties , in particular powder magnetization and susceptibility , are calculated with the help of the transfer - matrix method , which is a bit more general and convenient for numerical computation than the renormalization of the ising parameters , which was used by fisher and by streka and jaur . the results compare well with experiment and allow to determine values for the exchange coupling constants . the excited crystal field kramers doublets ( or stark levels ) of dy@xmath9 are not included in the decorated ising model . because of their relatively low energetic position , these kramers doublets can have a non - negligible contribution to the magnetic properties of the chain . we account for this in a first approximation by adding this contribution , as calculated _ ab initio _ for the monomeric dy@xmath9 complex , to the results of the decorated ising model . part of the decorated ising chain . in each ising bond is inserted an arbitrary statistical mechanical system@xcite , called decorating unit , that interacts only with the two ising spins @xmath13 at the vertices of the bond . the ising spin variables commute with the hamiltonian and have definite values in the eigenstates of the chain.,width=325 ] the decorated ising chain may be divided into units delimited by the ising spins , as in fig . [ 4:fig : decisingchain ] . the hamiltonian of the decorated ising chain ( or ring ) of length @xmath14 may accordingly be written as follows : @xmath15 where the subhamiltonians @xmath16 correspond to units of the chain . if the chain is closed into a ring , periodic boundary conditions apply by identifying the last with the first spin : @xmath17 . the ising spins @xmath13 commute with the subhamiltonians and the subhamiltonians commute with each other : @xmath18=0 \quad \text{and } \quad [ \hat{h}_i,\hat{h}_j]=0.\ ] ] it is further assumed that there is no direct interaction between the decorating units themselves , i.e. , two different @xmath16 have no variables in common , except for an @xmath13 when they are neighbors . in writing eq . ( [ 4:eq : hamdecising ] ) we made use of the fact that the ising spins are conserved variables and may be considered as parameters rather than operators . in the case of spin-1/2 , they take on the values @xmath19 and @xmath20 , so that to each decorating unit there correspond four different subhamiltonians @xmath21 .. in fact , any parameter that takes on a finite number of values can serve as @xmath13 in eq . ( [ 4:eq : hamdecising ] ) . in practice however , @xmath13 represents most commonly a kramers doublet , which is most often described as an effective spin-1/2 . this is the case for the examples considered in this paper . ] since there is no direct interaction between the decorating units of the chain , the subhamiltonians @xmath16 in eq . ( [ 4:eq : hamdecising ] ) work for a given set of @xmath13 values on disjunct spaces . an eigenfunction of @xmath22 is then simply a direct product of @xmath14 independent eigenfunctions , one of each @xmath16 . the corresponding energy is : @xmath23 where @xmath24 is the @xmath25-th energy level of @xmath21 . following ref . we write the partition function of one decorating unit for fixed @xmath13 and @xmath26 on the bond vertices : @xmath27,\ ] ] where @xmath28 . the total partition function for the chain is then given by @xmath29 this is the well - known form of a transfer - matrix solution.@xcite each decorated bond ( or pair of neighboring ising spins ) is represented by a transfer matrix @xmath30 whose elements are the @xmath31 , the values of @xmath13 labeling the rows and the values of @xmath26 labeling the columns . explicitly for the spin-1/2 decorated ising chain : @xmath32 since most applications deal either with rings or very long chains , the boundaries can be identified ( @xmath17 ) and expression ( [ 4:eq : partitionfunction ] ) can be written as the trace of a matrix product : @xmath33 this is the most general expression for the partition function of a decorated ising chain . if the chain has translational symmetry the partition function is expressed in terms of eigenvalues.@xcite suppose that the chain is a repetition of identical decorated ising bonds , then the @xmath30 are all the same , so that @xmath34 where the @xmath35 are the eigenvalues of @xmath36 . according to perron s theorem on positive matrices , the eigenvalue @xmath37 with largest modulus is real , positive and nondegenerate.@xcite this yields the simple but exact result for the free energy per unit cell of the infinite chain : @xmath38 if a unit cell of the chain is spanned by @xmath39 decorated bonds instead of one then it is only necessary to combine @xmath39 transfer matrices into one new transfer matrix @xmath40 , with largest eigenvalue @xmath41 , to be used in eqs . ( [ 4:eq : partfunc_periodic ] ) and ( [ 4:eq : freenergy_infinite ] ) , where @xmath14 is to be replaced by @xmath42 . this situation arises , for example , when the local easy axes on the ising ions are not parallel but canted with respect to each other . the solution in terms of transfer matrices , eq . ( [ 4:eq : partfunc_transmatr ] ) , is not limited to decorated spin-1/2 ising chains . it is valid for chains having ising spins of any multiplicity or a combination of ising spins with different multiplicities , in which case the dimension of the transfer matrix is different from two by two . another advantage of the transfer matrix method is that it can be readily extended to include next - nearest - neighbor bonds between the ising spins . to this end , the transfer matrix has to be enlarged so that it does not jump from one ising spin to the next , but from one _ pair _ of ising spins to the next pair.@xcite we are interested in this paper in molecular chains or rings that can be described by a decorated ising model . this means that their low - energy spectrum can be modeled to satisfactory accuracy by an effective hamiltonian that has the properties described in section [ 4:sec : decising ] . it has been noted there that the composition of the decorating unit is basically arbitrary and we need therefore not consider the properties of that part . instead , our attention goes here to the molecular realization of the ising spin [ @xmath13 in eq . ] . we focus on because this ion is used in the two examples studied below , but the discussion applies equally well to several other trivalent lanthanides and also to some transition metal ions . it is known that lanthanide ions in a coordination environment are often well described by crystal field theory applied to the ground @xmath43 level . one assumes that @xmath44 , @xmath45 , and @xmath46 remain good quantum numbers . is a kramers ion that belongs to the second half of the lanthanide series , whose ground level is ( @xmath47 ) , with associated land factor @xmath48 . this multiplet splits into eight kramers doublets by the crystal field perturbation ( except for high - symmetric environments belonging to the @xmath49 , @xmath50 , or @xmath51 point groups , which split the multiplet in less than eight levels ) . it will be useful to view each kramers doublet as an effective spin-1/2 with its own @xmath1-factors ( 3 in number ) and corresponding magnetic axes . for example , take the kramers doublet @xmath52 ( of the level ) , quantized with respect to the @xmath53 axis . its @xmath1 factors are @xmath54 and @xmath10 . if the action of the crystal field on is such that the lowest kramers doublet is separated from the next one by an energy that is large compared to the energy of interaction with the magnetic field and the exchange interaction with neighboring ions , we can omit all excited kramers doublets from the hamiltonian and keep only the lowest doublet . in this way is described by a spin of 1/2 and every interaction in which it takes part enters the hamiltonian as a linear combination of the three spin operators @xmath55 , @xmath56 , and @xmath57 . if we now want to be an ising spin , defined by the first commutation relation in eq . , it is clear that only one of these spin components , say @xmath57 or simply @xmath58 , may actually appear in the hamiltonian . in other words , there must be no interaction that creates an off - diagonal matrix element between the two components of the kramers doublet . this can be shown to be true with high accuracy if the lowest kramers doublet is @xmath52 . the two interactions of importance here are the zeeman interaction with the magnetic field and the exchange interaction with other magnetic ions . the zeeman hamiltonian follows directly from the @xmath1 factors of the kramers doublet ( vide supra ) , and is @xmath59 where @xmath60 is the @xmath53 component of the applied magnetic field . note that the field may be applied in any direction , but it is only the @xmath53 component that interacts with the kramers doublet because @xmath10 . the vanishing of @xmath61 and @xmath62 in @xmath52 follows from the selection rule stating that a vector operator ( the magnetic moment in this case ) can not connect states for which @xmath63 differs by more than one unit . we will sometimes refer to the @xmath53 axis as the anisotropy axis , to stress that it is the only magnetic axis with nonvanishing @xmath1 factor . to evaluate the effect of exchange interaction , we must first take a closer look at the composition of the kramers doublet . in terms of the russel - saunders states @xmath64 we have @xmath65 in a basic ( super)exchange process between two magnetic centers , one electron of each center takes part . if we look at one center , the process removes an electron with certain spin projection ( up or down ) and puts it back on the center either with the same or with opposite spin projection.@xcite this gives rise to the selection rule @xmath66 . if the exchange interaction is to connect both components of the kramers doublet in eq . , at least five successive processes are needed , for @xmath67 . in other words , not one but five electron spins have to be flipped to connect @xmath68 with @xmath69 . if the basic exchange process ( i.e. the one for which @xmath70 ) occurs in , say , @xmath71th order of perturbation theory then an off - diagonal matrix element between the two components can only appear in @xmath72th order of perturbation theory . it is therefore reasonable to assume that the off - diagonal matrix element is negligibly small compared to the diagonal matrix elements ( @xmath73 ) so that the effect of exchange interaction on the kramers doublet is accurately described by the @xmath57 spin operator only . note that we derived the selection rule for exchange interaction on the basis of the spin quantum number @xmath74 only , without paying attention to the angular momentum quantum number @xmath75 , although @xmath75 changes even more than @xmath74 between the states of the kramers doublet . the existence and precise form of a selection rule for @xmath75 depends on the spatial symmetry of the exchange problem under consideration . @xmath76 is therefore not as useful as @xmath77 for predicting the vanishing of certain matrix elements of exchange interaction . note however that , even in the lowest symmetry , there is a maximum to the amount that @xmath75 can change in the basic exchange process described in the previous paragraph : within the @xmath78 orbitals , a one - electron process can bring about at most a change of @xmath79 . so we would have , in general , that a basic exchange process can induce the following changes in a lanthanide state : @xmath80 thus at least two steps of this kind are needed to bridge @xmath81 , but at least five are needed to bridge @xmath67 . so in this case , the selection rule of @xmath74 gives the stronger result and leads to the conclusions reached in the last paragraph on the matrix elements of exchange interaction in the kramers doublet . we can now derive the precise form of that part of the effective hamiltonian that refers to the exchange interaction between a ion [ with ground state ] and another magnetic center . we consider two cases , which we shall encounter in the examples in sections [ 4:sec : chain ] and [ 4:sec : ring ] : in the first case the other center is another ion ; in the second case the other center is an ion with an isotropic spin moment @xmath82 . consider first the exchange interaction with another ion . we assume that the second has the same property of uniaxiality as the first one and that it also shares all other relevant properties discussed in the previous paragraphs . then both ions are represented by a spin-1/2 doublet with a local anisotropy axis @xmath83 ( @xmath84 ) and we already know that the effect of exchange interaction in each doublet is proportional to @xmath85 . therefore the exchange hamiltonian is necessarily of the form @xmath86 where it is understood that the first spin belongs to ion 1 and the second to ion 2 . note that @xmath87 and @xmath88 need not be parallel with each other . as a second case , consider the interaction between ( anisotropy axis @xmath87 ) and an isotropic spin @xmath89 . the latter may typically be a transition metal ion with quenched orbital momentum . we found that the exchange processes that contribute do not change the spin projection on the ion ( @xmath90 , quantization axis @xmath87 ) , and that this result is independent of the exchange partner . it is also known that every exchange process commutes with the total spin ( the matrix elements involved are spin - independent matrix elements of kinetic , potential , and coulomb energy @xcite ) , so that @xmath91 . it follows then that @xmath92 , or , the exchange hamiltonian commutes with the @xmath93 component of @xmath89 . the simplest expression compatible with this requirement is @xmath94 where @xmath58 naturally represents and @xmath46 represents @xmath89 . the interaction is of ising form with the anisotropy axis of as ising axis . note that , when @xmath95 , higher powers of @xmath96 may enter the hamiltonian . considering exchange interaction as a perturbation however , one can usually assume that the lowest - order contribution , eq . , is the leading term . there are other cases conceivable , for example dipole - dipole interaction between the moment of and a neighboring moment . one will always find , as above , that the hamiltonian is a product of @xmath57 ( belonging to ) and a part that belongs to the other ion and whose form depends on the kind of the other ion and on the details of the interaction . we have now obtained that a ion , if its ground state is @xmath52 , fairly well separated from excited states , interacts with the magnetic field and with neighboring ions as an ising spin-1/2 , in the sense that the interaction is always proportional to @xmath57 , as expressed in the eqs . , , and . this means that a chain - like molecular structure having ions of this kind at regular positions in the chain would meet the requirements of a decorated ising chain , given in part by eqs . and . it remains of course to be shown that @xmath52 can indeed be the ground state of a coordinated ion in a polynuclear complex . at first sight , this seems rather unlikely . @xmath52 is an eigenstate of cylindrical symmetry . within lanthanide @xmath97 states , the crystal field is effectively cylindrical if there is , at least , an eightfold rotation axis ( @xmath98 ) or rotation - inversion axis ( @xmath99 ) . @xcite @xmath99 symmetry has been obtained , for example , in mononuclear bis(phtalocyaninato ) sandwich complexes of the lanthanides . @xcite even when such high symmetry is attained , the ground state is not necessarily the cylindrical doublet with highest @xmath100 value . @xcite apart from that , the symmetry of the coordination sphere of a lanthanide ion in a polynuclear , possibly heterometallic , complex or chain is usually much lower or even completely absent . such is the case for the two examples considered in this paper . on the basis of symmetry alone , there is thus no reason to expect @xmath52 to be an eigenstate , let alone the ground state . nevertheless , recent _ ab initio _ calculations have revealed the unexpected result that the ground state of several low - symmetry complexes of is very close to @xmath52 . @xcite they used the multiconfigurational , wavefunction - based casscf / rassi - so method , to obtain accurate wavefunctions for several of the lowest kramers doublets . calculation of the principal @xmath1 factors of these states gives an indication of their composition . it was found in several cases that the ground doublet has @xmath101 close to , but lower than 20 , and @xmath61 and @xmath62 close to 0 . this corresponds to a doublet mainly composed of @xmath52 . with evidence of _ ab initio _ calculations it is thus possible to identify in certain coordination environments as an ising spin-1/2 ( to a good approximation ) . we will use this information to identify the compounds in sections [ 4:sec : chain ] and [ 4:sec : ring ] as decorated ising chains ( or rings ) . to conclude this section we remark that one can not deduce from the vanishing of two @xmath1 factors alone that a kramers doublet will behave as an ising spin . it will , of course , in its interaction with the magnetic field [ eq . ] , but this will not , in general , be true for the exchange interaction . take again the example of , supposing the ground state is the kramers doublet @xmath102 . it is perfectly uniaxial because @xmath10 and @xmath103 . a closer look at the expansion of @xmath102 in terms of the russel - saunders states @xmath104 shows , however , that the selection rules in eq . permit a matrix element to exist between @xmath105 and @xmath106 , therefore introducing non - ising terms ( i.e. , @xmath55 and @xmath56 ) in the same order of perturbation theory as the ising term in the exchange hamiltonian . there exists a similarity between the spectra of the decorated and undecorated ising chains at which we want to take a closer look here . we suppose infinite , periodic chains , or periodic , even - membered rings ( in odd - membered rings spin - frustration complicates the picture ) . we also limit ourselves to chains with ising spins of 1/2 . the undecorated , or simple , ising chain in a magnetic field @xmath107 parallel with the @xmath53 axis is given by eq . and @xmath108 where @xmath13 is the @xmath53 component of @xmath109 ( @xmath110 ) . we assume , without loss of generality , @xmath111 . the eigenstates of the chain are spin _ configurations _ like ( @xmath112 ) , etc . of the @xmath113 eigenstates only two distinct ones can be the ground state : the ferromagnetic ( f ) and the antiferromagnetic ( af ) : @xmath114 when @xmath115 time - reversal symmetry makes every state degenerate with the state formed by flipping all the spins . this degeneracy is meant to be implied in , where only one of two states is shown in each case . only the two af states remain degenerate when @xmath116 . when @xmath117 a ground state level crossing occurs from af to f when @xmath107 is increased . at the point of crossing ( @xmath118 ) the two af states are degenerate together with all states derived from an af state by flipping one or more down - spins up . however , no other state than af or f can be the ground state at any other value of @xmath107 . thus the ground state of the ising chain is either f or af , and they are degenerate , together with an infinite number of other states , at the crossing point . we now decorate the ising bonds with identical but arbitrary units to obtain a periodic decorated ising chain . the spectrum is given by eq . . since the chain is periodic , the spectrum of the individual units @xmath16 is independent of @xmath119 and the energies may be written as @xmath120 , with @xmath71 ranging over the eigenstates of @xmath16 . there are four sets of @xmath120 : @xmath121 , @xmath122 , @xmath123 , and @xmath124 , in an obvious notation . notice that the eigenstates of this chain can still be classified according to the configuration of the _ ising _ spins : ( @xmath112 ) etc . , which follows from the fact that all the @xmath13 and @xmath22 form a commuting set of observables . an interesting question is whether the same rules hold for the ground state of the decorated ising chain as did for the simple ising chain . the answer is yes ; the ground state is either f or af ( referring to the ising spin configuration ) and a crossing between them is possible , with the same number and kind of degenerate states as in the simple ising chain . to show this , we have to consider only the lowest eigenstate belonging to each of the @xmath113 possible ising spin configurations . in these states every unit is in its lowest possible state for the given orientation of the neighboring ising spins : @xmath125 ( we assume this energy to be nondegenerate ) , so that the total energy of the chain state is @xmath126 where @xmath127 denotes the number of pairs of neighboring ising spins that are both spin up , etc . for example , in the f configuration in eq . , @xmath128 ( periodic boundary conditions are assumed ) , while in the af configuration , @xmath129 . the eigenstates we have just described , with energy , are in an obvious one - to - one correspondence with the eigenstates of the simple ising chain . the ground state is found by minimizing with respect to the @xmath130 , under the restrictions @xmath131 the first relation states that the total number of ising spins ( or , equivalently , unit cells ) is @xmath14 . the second relation follows from the fact that , in a cyclic spin configuration , every @xmath132 pair must eventually be followed by a @xmath133 pair , possibly after a number of @xmath134 pairs . another restriction is that whenever both @xmath127 and @xmath135 are not zero , @xmath136 [ and by eq . also @xmath137 must be at least one . using we can rewrite eq . as @xmath138\bigr)\\ & + n_{\downarrow\downarrow}\bigl(\varepsilon_1(\downarrow\downarrow)- \frac{1}{2}[\varepsilon_1(\uparrow\downarrow ) + \varepsilon_1(\downarrow\uparrow)]\bigr)\\ & + \frac{n}{2 } [ \varepsilon_1(\uparrow\downarrow)+\varepsilon_1(\downarrow\uparrow ) ] , \end{split}\ ] ] where we see that only the average @xmath139/2 $ ] of the `` antiparallel '' energies enters the equation . the last term is a constant and can be discarded for the purpose of relative energy considerations . we can now derive the values of @xmath127 and @xmath135 for the ground state of the chain , keeping in mind that @xmath125 is a function of the magnetic field @xmath140 . suppose then , first , that @xmath141 . time reversal symmetry asserts that @xmath142 and @xmath143 . it is simple to see that , depending on the relative ordering of @xmath144 and @xmath145 , @xmath146 is minimal in the f configuration ( @xmath128 or @xmath147 ; @xmath148 ) when @xmath149 or in the af configuration ( @xmath150 ; @xmath151 ) when @xmath152 ( we exclude the possibility of equality of both energies from the discussion ; @xmath153 would correspond , in the simple ising chain , with @xmath154 ) . when @xmath155 , time reversal symmetry is not operative , and we have , in general , four different energies @xmath125 . the equation shows that the configuration that minimizes @xmath146 is determined by the sign of the two terms in round brackets ; if both are positive , then @xmath150 ( af configuration ) ; if at least one of them is negative , then either @xmath128 or @xmath147 ( f configuration ) , depending on whether respectively @xmath144 or @xmath156 is lower . finally , the magnetic field can induce a transition from the af to an f ground state configuration , say with all ising spins up . this happens when @xmath157,\ ] ] and @xmath158 . at this point , the ground state configurations are all those for which @xmath159 , exactly the same as in the simple ising chain . we find thus a complete analogy between the simple and the decorated ising chain as far as the ground state ising spin configuration is concerned . the only possible configurations are the fully antiferromagnetically aligned and the fully ferromagnetically aligned configurations . no `` intermediate '' configuration can be the ground state . the only exception is the crossing point between af and f , where there is a high degeneracy of configurations . these conclusions are independent of the nature of the decorating unit . although the decorated ising model predicts that the af and f ground states are both doubly degenerate ( in zero field ) , this degeneracy is not a result of the spatial symmetry : in the cyclic group @xmath160 , the two af components combine into irreducible representations ( irreps ) @xmath161 and @xmath107 , while the two f components transform as two @xmath161 irreps . introduction of neglected terms in the hamiltonian , that destroy the ising property , could split these ground state components . the decorated ising chain affords two new kinds of ground state level crossings , not present in the simple ising chain . the first of these is the transition between one f configuration of the ising spins and the other : @xmath162 . this transition takes place when @xmath142 . this level crossing , induced by the magnetic field , can for example be encountered in frustrated ising - heisenberg chains.@xcite a second kind of new ground state transition arises from the crossing of levels _ within _ a decorating unit . a ground state crossing can result in which the ising spin configuration remains the same but the state corresponding to @xmath125 crosses with the state corresponding to @xmath163 . more precisely this happens in the f configuration when @xmath164 and in the af configuration when either @xmath165 or @xmath166 , or both . the previous paragraphs have shown that we do not need to consider configurations other than f and af for the ground state . level crossings are usually connected with the presence of good quantum numbers . for the ising - type crossings , the relevant conserved quantities are the @xmath14 ising spins @xmath167 . the crossing of energy levels within the decorating unit should be associated with a conserved variable that is _ internal _ to that unit , much the same as in isolated molecules . in section [ 4:sec : chain ] we will encounter an example where both transitions ising type and internal type occur in a magnetic field . in the following sections we will be comparing our theoretical results with measurements performed on powder samples of the crystalline compounds . in this section we consider the powder averaging of magnetization for the example of the simple ising chain . let @xmath168 and @xmath169 be the polar angles of the magnetic field vector with respect to a molecular reference frame , then the free energy is a function of @xmath168 , @xmath169 , and the strength of the field , @xmath107 : @xmath170 . the projection of the magnetization on the field direction @xmath171 is @xmath172 averaging over one hemisphere gives the powder magnetization @xmath173 let us see how the powder averaging affects the magnetization curve for a simple ising chain . take a spin-1/2 infinite antiferromagnetic ising chain with anisotropy axes parallel with each other and with the @xmath53 axis , and uniaxial @xmath1-factors ( @xmath10 and @xmath174 ) . this could for example be realized by a chain of identical units ( see section [ 4:sec : dy ] ) . the hamiltonian is given by eq . , substituting @xmath175 where @xmath176 is the @xmath53 component of the magnetic field , and @xmath117 . defining @xmath177 and @xmath178 , the magnetization , which has only a nonzero @xmath53 component , is@xcite @xmath179}{\sqrt{\sinh^2[b\cos\theta/2t]+e^{-j / t}}}.\ ] ] the projection on the field direction [ eq . ] is @xmath180 . plugging this in eq . and substituting @xmath181 yields the powder magnetization of the ising chain @xmath182}{\sqrt{\sinh^2[b u/2t]+e^{-j / t}}}u\,du.\ ] ] because the hamiltonian in eq . does not depend on @xmath169 , this variable has been integrated out in eq . . ] ] figs . [ 4:fig : magnising_zas ] and [ 4:fig : magnising_powder ] show plots of magnetization versus magnetic field , for coupling constant @xmath183 . the step - like appearance of @xmath184 is associated with the ground state crossing that occurs at @xmath185 . at that point , the antiferromagnetic ground state ( or rather ground state ising doublet ) is replaced by the ferromagnetic state ( all spins up ) . consequently , the magnetization jumps from zero to the saturation value of 0.5 , as seen in fig . [ 4:fig : magnising_zas ] . the magnetization of a powder sample of the same ising chain is shown in fig . [ 4:fig : magnising_powder ] . before the crossing point , @xmath186 behaves qualitatively the same as @xmath184 . after the crossing point however , @xmath186 is seen to reach only slowly its saturation value , which is half of the saturation value of @xmath184 , viz . the limiting curve of @xmath186 as @xmath187 can be calculated exactly from eq . : @xmath188 ( this is of course only valid for the antiferromagnetic case @xmath189 . ) in fig . [ 4:fig : magnising_powder ] , this limiting curve is very closely approximated by the curve at @xmath190 . clearly , the sharp step of @xmath184 transforms in the powder to the concave form displayed by @xmath186 . this is understood from the fact that , in a powder , for a given field @xmath191 , there is always a fraction of molecules that is not magnetized ( in the sense that they are in the antiferromagnetic ground state ) because they are oriented so with respect to the field , that @xmath192 ( see fig . [ 4:fig : magnising_zas ] ) . the powder saturates only when every molecule is fully magnetized , and this happens only for @xmath193 . therefore , @xmath186 ( at @xmath194 ) does not abruptly saturate at the crossover point , but increases slowly to saturation . in deriving the exchange hamiltonian in section [ 4:sec : dy ] we assumed that only the lowest kramers doublet on took part . this is certainly a good approximation when the gap between the lowest and the second - lowest kramers doublet is much larger then the strength of the exchange interaction . however , the excited kramers doublets often have to be taken into account to a certain degree of approximation if a comparison with experimental data on susceptibility and magnetization is desired . the crystal field splitting of the level is of the order of @xmath195 at room temperature . this gives rise to two effects : ( i ) a thermal population of excited kramers doublets , and ( ii ) a modification of the lowest kramers doublet as a function of the applied magnetic field by interaction with the excited doublets . effect ( i ) is mainly visible in the temperature dependence of @xmath196 , where @xmath197 is the powder magnetic susceptibility : for a single center , @xmath196 increases monotonically with increasing temperature , from the value of the ground doublet at 0 k to the saturation value of at higher temperatures . effect ( ii ) gives rise to temperature - independent paramagnetism ( tip ) . it is visible at temperatures sufficiently low so that only the ground doublet is occupied . it contributes a linear increase of @xmath196 with @xmath49 and a linear increase of the magnetization @xmath186 with the applied field @xmath107 . in the simplest approximation , the contribution of the excited kramers doublets to the magnetic properties of the chain is equal to the contribution they have to the properties of the single , isolated ion in the same ligand environment it has in the chain . let @xmath198 and @xmath199 denote susceptibility and magnetization derived from the decorated ising chain model , and let @xmath200 denote the susceptibility of the center and @xmath201 the magnetic moment induced by @xmath107 in the ground doublet of the center , then the corrected properties are ( supposing one ion per unit cell ) [ 4:eq : corrections ] @xmath202 the last equation assumes that only the ground doublet of is occupied . this is correct at the temperature at which magnetization curves are usually recorded ( e.g. , 2@xmath203 ) . if necessary , corrections due to other magnetic ions can be added in the same way . @xmath200 and @xmath204 can be obtained from multiconfigurational _ ab initio _ calculations @xcite , or , in oligonuclear complexes , experimentally by replacing certain magnetic ions with diamagnetic ions.@xcite the equations are evidently correct in the limit of vanishing exchange interactions . the assumption we make is that the exchange interactions are sufficiently small for these corrections to remain valid . a more accurate approach should take into consideration the fact that the excited kramers doublets also participate in the exchange interaction with neighbors . this is however out of the scope of the decorated ising model . we now turn our attention to two actual examples of decorated ising models based on : a chain , treated in this section , and a four - ring treated in the following section . the problem is approached as follows : the hamiltonian for the chain in a magnetic field is formulated , with the help of the considerations in section [ 4:sec : dy ] . values of the @xmath1 factors of the magnetic ions are taken directly from _ ab initio _ calculations , reported elsewhere.@xcite this leaves the exchange coupling constants as parameters of the model , to be fitted by comparison with experimental magnetization and susceptibility data . ( in section [ 4:sec : ring ] , the direction of the anisotropy axis contributes one extra parameter . ) @xmath0 chain , showing type of exchange interactions and labeling of exchange constants . see ref . for the complete molecular structure.[4:fig : chainscheme ] ] the [ dycumocu ] chain was recently synthesized and details of its chemical composition and structure are given in ref . . the crystal structure was found to consist of parallel linear chains each made of [ dycumocu ] unit cells . [ 4:fig : chainscheme ] shows how the metal ions are connected by ligand bridges . multiconfigurational casscf / rassi - so calculations have been performed on each of the four metal ions in their ligand environment , suitably disconnected from the rest of the chain ( details of the calculations can be found in ref . and the accompanying supplementary information ) . most important for us is that the center was found to have a ground kramers doublet , separated by 141 @xmath12 from the second doublet , and characterized by complete uniaxial anisotropy : @xmath205 ( actually @xmath61 and @xmath62 were calculated about 0.03 , which is small enough to be ignored . ) the value of @xmath206 shows that this doublet is only slightly perturbed from the @xmath52 doublet of the level , the latter having @xmath207 . together with the fact that the energy gap to the second kramers doublet is about ten times larger than the exchange interaction ( as we will find later ) , these results indicate that the ion will behave as an ising spin , as described in section [ 4:sec : dy ] . as a side - note we may add that the total splitting of the level was calculated to be 560 @xmath12 , which is indeed of the order of room - temperature @xmath195 ( see section [ 4:sec : corrections ] ) . both ( @xmath208 ) and ( @xmath209 ) have a spin = 1/2 , orbitally nondegenerate ground state , well separated ( @xmath210 ) from higher states . the two ions in the unit cell reside in almost identical environments@xcite and have therefore virtually the same properties . the calculated @xmath1 factors are tetragonal : @xmath211 for and @xmath212 for . to avoid unnecessary complications we will regard these ions as isotropic spins with root - mean - square @xmath1 factors @xmath213 this approximation will not have important consequences for the magnetic properties , which are largely dominated by the high moment anyway . we introduce exchange interaction between metal ions directly connected by ligand bridges . interacts with its three neighbors via the ising hamiltonian eq . . interaction between the isotropic spins is given by the heisenberg hamiltonian @xmath214 . [ 4:fig : chainscheme ] shows the exchange configuration , with single bonds representing ising interaction and double bonds representing heisenberg interaction . note that we have approximated the dy - cu@xmath215 and dy - cu@xmath216 coupling strengths to be equal ( @xmath217 ) , following the approximate local symmetry of the dy - cu pairs.@xcite it is now possible to see that the [ dycumocu]@xmath0 polymeric chain is indeed an experimental realization of an ising - heisenberg chain where the [ cumocu ] trimeric heisenberg units decorate the dy - dy bonds and the ising spins separate the [ cumocu ] heisenberg trimers from each other . the total hamiltonian in a magnetic field @xmath140 is then given by eq . and @xmath218 where @xmath219 is shorthand for @xmath220 and @xmath221 is the @xmath101 factor of [ eq . ] . the @xmath53 axis is the anisotropy axis of the center . we have not specified its direction with respect to the chain axis but this is not important here because there are no other axes in the problem ( the anisotropy axes are parallel by translational symmetry and we have assumed and isotropic ) . all the spins in eq . are spins of 1/2 . the hamiltonian exhibits some symmetry . it is rotationally invariant around @xmath53 , if @xmath140 is rotated simultaneously . we may therefore restrict @xmath140 to lie in a plane through @xmath53 , say the @xmath222 plane . this simplifies calculation of the powder magnetization eq . : one has to integrate only over @xmath168 . when @xmath140 is directed along the @xmath53 axis , the @xmath53 component of the total spin _ in _ each decorating unit is conserved : @xmath223=0.\ ] ] we also note that in this case the zeeman hamiltonian commutes _ almost _ with @xmath22 . it would commute exactly when @xmath224 , for then the last term in eq . reduces to @xmath225 . we let the length of the chain go to infinity : @xmath226 . to solve for the thermodynamic properties we are only required to find the eigenvalues @xmath227 of eq . ( see section [ 4:sec : decising ] ) , with @xmath228 , corresponding to the @xmath229 possible states of the [ cumocu ] spin unit . this is done by 4 numerical @xmath230 matrix diagonalizations , one for each @xmath231 pair . @xmath0 . ] we can now compare the theory with experiment . powder magnetization ( at 2@xmath203 ) and susceptibility data have been recorded.@xcite we recall that we have to correct the theoretical curves before comparing with experiment according to eq . . the corrections are provided by the _ ab initio _ calculations.@xcite @xmath204 turns out to be 0.03@xmath232 ; the accompanying correction in eq . is never more than 2.5% of @xmath186 . we ignore this correction . we do however correct @xmath197 as in eq . . the theoretical curve contains the correction for the contribution of excited kramers doublets . ] closest agreement with experiment was found for the following values of the exchange constants ( plots in figs . [ 4:fig : chainmp ] and [ 4:fig : chainchip ] ) : @xmath233 these were obtained by a least - squares fit of @xmath197 followed by a small manual adjustment to improve the fit of @xmath186 while not distorting that of @xmath197 appreciably . ( the least - squares fit of @xmath197 gave @xmath234 , @xmath235 , @xmath236 , @xmath237 . ) there are , as far as we know , no data in the literature with which to compare the values in eq . . however , an experimental study is available of a ( , ) dinuclear complex in which the bridging ligand is the same as in this chain.@xcite the authors found a ferromagnetic interaction . a superficial analysis of the susceptibility curve in that paper , using the ising hamiltonian we use in this paper , yields @xmath238 . the other values in are difficult to assess . for a discussion of these values and their relation with the molecular structure as well as some evidence from dft calculations , we refer to ref . . certainly , no confidence should be attached to the numbers in decimal places in . the correction refers to the contribution of excited kramers doublets , eq . . ] the effect of the excited kramers doublets of is most clearly seen in the @xmath196 curve ( fig . [ 4:fig : chainchitp ] ) . the curve shows a steady increase above 50@xmath203 which is not predicted by our decorated ising model , but is indeed due to the thermal population of the kramers doublets that originate from the level . we can obtain the expected high - temperature limit of @xmath196 by considering the metal ions as independent spins . the susceptibility components @xmath239 of an angular momentum multiplet @xmath44 with principal @xmath1-factors @xmath240 ( @xmath241 ) are given by @xcite @xmath242 summing over ( , @xmath243 ) , , , and ( all are isotropic ) gives @xmath244 a similar calculation , only including the lowest kramers doublet of , with @xmath1-factors as in eq . , gives 13.2@xmath245 . the correction supplied by the _ ab initio _ calculations to account for this difference , is seen to cover nicely the high - temperature part of the experimental curve . one notices that @xmath196 shows a slight depression around 40@xmath203 which is not entirely reproduced by the theory . this might indicate a failure of the simple approximation we used to include the excited kramers doublets . eq . is certainly correct at very high temperatures , when the exchange interactions are irrelevant , and at very low temperatures , when the excited kramers doublets are not occupied . if these two regions do not overlap , however , there is a temperature window between , in which excited doublets start to get occupied while exchange interaction is not quite negligible yet . in that case , the exchange interaction of the occupied excited doublet(s ) with other ions should be taken into account . such an interaction of antiferromagnetic type could possibly depress @xmath196 as observed . we shall now describe some features of the spectrum of the chain , paying attention to the properties described in section [ 4:sec : eigenstates ] . consider the chain without magnetic field . the exchange parameters in eq . predict a ground state that has an af ising spin configuration . this is in accordance with the susceptibility measurements , which show that @xmath246 as @xmath247 , requiring a nonmagnetic ground state ( fig . [ 4:fig : chainchitp ] ) . the ground state is indeed nonmagnetic because @xmath248 is the time - reversed state of @xmath249 . let @xmath74 denote an eigenvalue of @xmath250 [ eq . ] : @xmath251 . @xmath252 is conserved so @xmath74 may be used to label the eigenstates @xmath253 ( we may leave out the index @xmath119 because all units of the chain are identical ) . for the ground state , we find @xmath254 in @xmath248 and @xmath255 in @xmath249 . the powder magnetization ( see also fig . [ 4:fig : chainmp ] ) is compared with the projections of @xmath256 on the field direction [ eq . ] , for three different directions of the field ; @xmath168 is the angle between @xmath140 and the @xmath53 axis . ] same as fig . [ 4:fig : chainmcomponents ] but at lower temperature and to higher field . ] [ eq . ] in a magnetic field _ parallel _ with the @xmath53 axis ( @xmath257 ) . circles indicate ground state level crossings . the ground state of the chain is af in zero field ( left ) , switches to f at 0.64@xmath258 , and undergoes an internal level crossing at 6.3@xmath258 , marked by a change of the internal quantum number @xmath74 from 1/2 to 3/2 . both crossings can be seen in the @xmath257 magnetization curve in fig . [ 4:fig : chainmcomponentslowt ] . note that the energy curves appear as straight lines , although , with the exception of @xmath259 , they are not exactly straight , because the zeeman hamiltonian does not _ completely _ commute with the total hamiltonian . all @xmath121 decrease with increasing field strength because the large magnetic moment of dominates the smaller magnetic moments of the decorating unit . ] since the ground state is af , we might expect that in a magnetic field a crossover will occur to an f ground state . this is indeed what happens . the convex increase of @xmath186 in fig . [ 4:fig : chainmp ] points to a flip of the spins to a parallel configuration . this is inferred from the value of the magnetization , which approaches 6@xmath260 at 5@xmath258 . the [ cumocu ] unit alone can only contribute a maximum of @xmath261 . the strong increase must come from the contribution of the large moments . the behavior of magnetization along certain directions of applied field is shown in fig . [ 4:fig : chainmcomponents ] . the af @xmath262 f transition is most clearly seen when the field is applied along @xmath53 ( @xmath257 ) ; the transition occurs below 1@xmath258 . after 1@xmath258 , @xmath184 reaches an approximately constant plateau at @xmath263 . the saturation value of magnetization in direction @xmath264 is @xmath265 . this gives @xmath266 for @xmath257 , which shows that @xmath184 has not quite reached its maximum at 5@xmath258 yet . the positions of level crossings become more sharply defined on lowering the temperature ( fig . [ 4:fig : chainmcomponentslowt ] ) . here we also see that @xmath184 undergoes a second transition at 6.3@xmath258 , after which it reaches saturation . this transition is connected with a level crossing _ in _ the [ cumocu ] unit ( see section [ 4:sec : eigenstates ] ) from @xmath255 to @xmath267 , as opposed to the first transition , at 0.64@xmath258 , which is of the ising type , described by eq . . the latter is the analogue of the transition in the af simple ising chain ( fig . [ 4:fig : magnising_zas ] ) , while the `` internal '' transition has no such analogue but is unique to the decorated ising chain . the relevant energy level diagram is shown in fig . [ 4:fig : chainlevelcrossings ] . note that , for fields not parallel to @xmath53 ( for example , @xmath268 in fig . [ 4:fig : chainmcomponentslowt ] ) , @xmath74 is not a quantum number and the internal level crossing turns into an avoided crossing . this does not apply for the ising level crossing because the ising spins are always conserved . only when the field is applied perpendicular to @xmath53 ( @xmath269 in fig . [ 4:fig : chainmcomponentslowt ] ) does the af @xmath262 f transition not occur because the spins do not interact with perpendicular fields . @xmath0 , without correction for contribution of excited kramers doublets . the powder @xmath196 ( see also fig . [ 4:fig : chainchitp ] ) is compared with the cartesian components of @xmath270 . @xmath53 is the direction of the anisotropy axis of , @xmath271 is any direction perpendicular to @xmath53 . @xmath272 . ] the low - temperature limit of the powder magnetization in fig . [ 4:fig : chainmcomponentslowt ] may be compared with that of the simple ising chain in fig . [ 4:fig : magnising_powder ] . the resemblance is clear ; the decorated chain is different in the small linear increase of @xmath186 before the transition , and the more linear approach to saturation , which lies at ( @xmath273 . both are due to tip interaction in the [ cumocu ] unit , the effect of which is most clearly seen in the @xmath274 curve in fig . [ 4:fig : chainmcomponentslowt ] . to conclude this section we remark that the mentioned similarity with the magnetization of the simple ising chain is a consequence of the very high magnetic moment of the spins in comparison with the [ cumocu ] unit . the dominance of is most dramatically shown in the components of @xmath196 ( fig . [ 4:fig : chainchitcomponents ] ) . an application of eq . shows that the high - temperature limit of @xmath275 is @xmath276 , while that of @xmath277 is @xmath278 . cr@xmath8 molecule indicating numbering of atoms and exchange coupling constants . the boxes show the orientation of the local anisotropy axes on dy sites , when viewed from the poles of the @xmath279 and @xmath280 axes . the @xmath281 axis points out of the center of the scheme.,width=325 ] as a second example we describe in this section the application of the decorated ising model to a ring - shaped molecule.@xcite dy@xmath8cr@xmath8 consists of alternating and ions forming a closed ring . the four ions lie in a plane . the ions are positioned alternatingly above and below this plane , `` decorating '' the dy - dy bonds . the molecule has @xmath282 symmetry , the ions lying on @xmath283 axes and the ions lying on the mirror planes . we choose a molecular reference frame @xmath284 so that @xmath281 coincides with the @xmath285 axis and @xmath279 and @xmath280 coincide with the two @xmath283 axes of @xmath282 ( fig . [ 4:fig : ringscheme ] ) . _ ab initio _ calculations have been performed in the same way as for the [ dycumocu ] chain.@xcite from these , we take again the @xmath1-factors of the ground doublet of and of the isotropic ground state spin multiplet of ( @xmath286 , @xmath287 ) : @xmath288 the kramers doublet is again very close to the @xmath289 state , permitting the use of the ising model . however , the same calculation predicted the second kramers doublet at 30@xmath290 , not very high compared with exchange interaction , which we found in the previous section @xmath291 . this should be seen as a warning that our treatment of the excited kramers doublets as `` innocent '' may not be entirely correct here , and thus may lead to discrepancies with experiment . in this respect we must also note that , for such small excitation energies , the results of the _ ab initio _ calculations are not always conclusive on the nature of the ground state kramers doublet . in the present case , for instance , another set of calculations produced a ground state kramers doublet on that is not uniaxial as in eq . , having relatively large transversal @xmath1-factors.@xcite the decorated ising model would be unusable in this case . we find however that , assuming the axial g - factors in eq . , is an interesting example of a decorated ising ring , for which qualitative agreement with experimental magnetic properties can be obtained . direct ligand bridges connect each with two neighboring ions and two neighboring ions . exchange interaction between these pairs is introduced [ eqs and ] . the hamiltonian is then given by eq . : @xmath292 , and @xmath293 where @xmath294 denotes the ising spin-1/2 variable on dy@xmath295 and @xmath296 denotes the projection of the spin of cr@xmath295 on the magnetic anisotropy axis of dy@xmath295 ( for numbering , see fig [ 4:fig : ringscheme ] ) . similarly , @xmath297 is the projection of the magnetic field on the anisotropy axis of dy@xmath295 . @xmath221 is the @xmath101 factor of [ eq . ] . an interesting difference with the [ dycumocu ] chain is that here , in , the four anisotropy axes @xmath298 are not , in general , parallel , a result of point symmetry instead of translational symmetry . the orientation of the local anisotropy axis on , being one of the @xmath1-tensor main axes , is restricted by the local @xmath283 symmetry to be either parallel with , or orthogonal to the local @xmath283 axis . the first possibility can be excluded on the basis of the experiment ; with the @xmath298 pointing radially outwards at each dy@xmath295 , the ground state of the whole molecule is necessarily nonmagnetic , because the local moments add up to zero , independent of whether the ground state is f or af with respect to the ising spins . the experimental susceptibility measurement however indicates a magnetic ground state ( nonzero intercept on the vertical axis in fig . [ 4:fig : ringchitp ] ) . we must therefore choose the second case and let the anisotropy axis on each dy be orthogonal to the local @xmath283 axis and make an angle of @xmath299 with the molecular @xmath281-axis ( see fig . [ 4:fig : ringscheme ] ) . by applying the symmetry elements of @xmath282 to one of these anisotropy axes , one obtains the other three . when @xmath300 the four axes are parallel and point in the same direction as @xmath281 . we note that the _ ab initio _ calculations yielded @xmath301 . we will need some flexibility in our model however , so we leave @xmath299 as a parameter that will be determined from comparison with experiment . in terms of the molecular coordinate system , the projections on the local anisotropy axes are a function of @xmath299 : @xmath302 the same relations hold for the magnetic field , after replacing @xmath303 by @xmath107 . the fact that only exchange interactions of ising type appear in eq . makes it possible to find analytical solutions of the eigenvalues and the partition function . from eqs . and we see that the part of @xmath16 that involves @xmath303 is a projection of @xmath304 on the vector @xmath305 where @xmath83 is the unit vector along the anisotropy axis of dy@xmath295 ( the superscripts @xmath298 on @xmath13 are left out from now on ) . the vector defines the quantization axis of @xmath304 , which depends on the states on the neighboring sites ( @xmath13 , @xmath26 ) . the stronger the coupling ( @xmath217 ) with dy , the stronger will be the deviation of the quantization axis from the direction of @xmath140 . the eigenvalues of @xmath16 are then @xmath306 where @xmath307 , and @xmath308 is the length of the vector in eq . . some remarks should be made on the solutions . eqs . and ( replace @xmath303 by @xmath107 ) show that the spectrum in eq . is not the same for every unit @xmath119 , as it was in the [ dycumocu ] chain , unless @xmath140 is applied along the @xmath281 axis . this means that also the transfer matrices @xmath30 will be different and that we have to use eq . instead of eq . for the partition function . a second remark concerns the quantum number @xmath74 . the lowest energy in eq . is always given by @xmath309 , but note that the axis to which this quantization refers is not invariant ; in particular , it changes with strength and direction of applied field , so that @xmath74 does not represent a real conserved quantity that could be responsible for level crossings of the `` internal '' type . such crossings do not occur in . we conclude the solution by finding the partition function @xmath310 . substituting eq . in eq . we find @xmath311}{\sinh [ \beta b_i/2]}\\ & \quad \times \exp[\beta(j_2 s_is_{i+1 } + \mu_\mathrm{b } g_\mathrm{dy } s_i b^{z_i})]\end{aligned}\ ] ] with @xmath30 as defined in eq . , we obtain the partition function @xmath312 let us now compare the theoretical results with experiment . a great amount of information on the values of the parameters @xmath299 , @xmath217 and @xmath313 can be obtained by inspection of the powder @xmath196 curve ( fig . [ 4:fig : ringchitp ] ) . the nonzero intercept @xmath314 indicates a magnetic ground state.@xcite now from the general theory we know that the ground state is either f ( @xmath315 ) or af ( @xmath316 ) with respect to the spins ; for a periodic ring or chain , is valid here because we are considering the eigenstates of in the _ absence _ of magnetic field , in which case the ring is effectively cyclic symmetric . ] af is nonmagnetic so we decide that the ground state must be f. incidentally , we can precisely delineate the regions in parameter space where the ground state is f or af : [ 4:eq : ring_groundstates ] @xmath317 a second piece of information comes from the increase of @xmath196 with increasing temperature . this is partly but not completely due to the occupation of excited kramers doublets , as one can show by subtracting the contribution of the latter , obtained from the _ ab initio _ calculations ( not shown here ) . there must still be an antiferromagnetic interaction to explain the increase . since the are already known to be ferromagnetically aligned , the only possibility is that the spins couple antiferromagnetically with , or @xmath318 . with this information , we can determine the angle @xmath299 . at 0@xmath203 , @xmath196 is determined by the magnetic moment in the ground state only.@xcite in the f ( @xmath315 ) state , @xmath319 by symmetry and @xmath320 so @xmath321 . with the help of eqs . and the fact that , in the ground state , @xmath309 in eq . , we can evaluate eq . to find @xmath322 this is a strictly decreasing function of @xmath299 that can be used to derive @xmath299 from the experimental value @xmath323 , and the knowledge that @xmath324 . this gives @xmath325 . we derive values for @xmath217 , @xmath313 , and @xmath299 by a least - squares fitting of @xmath197 . as before , a correction for the contribution of excited kramers doublets is provided by the _ ab initio _ calculations and applied following eq . . the fitting yields @xmath326 the comparison with experiment is shown in figs . [ 4:fig : ringchitp ] and [ 4:fig : ringmp ] . note that the magnetic properties are reported per mole ( @xmath197 and @xmath196 ) or per molecule ( @xmath186 ) of and not per dycr unit . note also that @xmath324 , that @xmath313 satisfies eq . , and that @xmath299 agrees with the value derived above . has been added to the theoretical curve , according to eq . . ] the agreement of magnetization curves ( fig . [ 4:fig : ringmp ] ) is not as good as it was for the [ dycumocu]@xmath0 chain , although the qualitative properties seem to correspond . in particular , we mention the strong linear increase of @xmath186 at higher fields ( @xmath327 ) , which is due to the gradual orientation of the spins to the magnetic field [ see discussion connected with eq . ] , and , to a smaller extent , also to the correction of 0.5@xmath232 , a non - negligible linear contribution to magnetization , which is due to the low - lying excited kramers doublets . as was mentioned before , the discrepancies are not unexpected given a low - lying first excited kramers doublet of , which could undermine the assumptions underlying the decorated ising model . note also that we could not take the _ ab initio _ value of 37@xmath328 for @xmath299 . leaving @xmath299 as a parameter can be seen as a partial compensation for the inaccuracies of the model and the _ ab initio _ results . we have shown that the decorated ising model is a valid model for the magnetic properties of certain lanthanide - containing magnetic compounds , if the crystal field spectrum of the lanthanide ion satisfies certain properties . the most important of these is the requirement of a ground state kramers doublet with completely uniaxial magnetic anisotropy ( this statement is simplified , see section [ 4:sec : dy ] for the correct details ) . it is a remarkable fact that precisely this property has been established by multiconfigurational _ ab initio _ calculations on several centers that are part of polynuclear molecular magnets . perhaps the best known example is the triangle , where the ising properties of were used to explain the nature of the ground state.@xcite we have focused on as lanthanide ion because this is a much - used lanthanide in current synthetic research in molecular magnetism ( witness both compounds in this paper ) and because computational results showing that it meets the requirements for an ising spin are available . however , there is no reason to assume that the findings are unique to . we expect that other lanthanides with high momentum ( e.g. , ) will exhibit the same uniaxial anisotropy in certain ligand environments and that examples of decorated ising chains based on lanthanides other than will be found in the future . we thank liviu ungur for providing results of the _ ab initio _ calculations . we thank the referee for useful suggestions and comments . w. v.d.h . acknowledges financial support from the research foundation - flanders ( fwo ) .
|
it is shown that the bond - decorated ising model is a realistic model for certain real magnetic compounds containing lanthanide ions . the lanthanide ion plays the role of ising spin .
the required conditions on the crystal - field spectrum of the lanthanide ion for the model to be valid are discussed and found to be in agreement with several recent _ ab initio _
calculations on centers . similarities and differences between the spectra of the simple ising chain and the decorated ising chain
are discussed and illustrated , with attention to level crossings in a magnetic field .
the magnetic properties of two actual examples ( a [ dycumocu]@xmath0 chain and a ring ) are obtained by a transfer - matrix solution of the decorated ising model .
@xmath1-factors of the metal ions are directly imported from _
ab initio _ results , while exchange coupling constants are fitted to experiment .
agreement with experiment is found to be satisfactory , provided one includes a correction ( from _ ab initio _ results ) for susceptibility and magnetization to account for the presence of excited kramers doublets on .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
two - dimensional ( @xmath7 ) lattice spin models , besides being interesting on their own right as models of ferromagnets can be viewed as mathematically well defined scheme of the nonperturbative regularization of quantum continuum field theories like @xmath0 principal chiral model or @xmath8 nonlinear sigma model . being a discrete theory lattice models can be simulated on computers by monte - carlo method , and this is nowadays the most important tool of obtaining physical results near the continuum limit . among major analytical methods one could mention the strong coupling expansion and the perturbation theory ( pt ) . the pt is essentially the only analytical tool which provides systematic expansion of different physical quantities in the region of weak coupling , i.e. in the region relevant for the construction of the continuum limit . however , at the best pt is only applicable for studying short - distance quantities and the most important and interesting phenomena , like the mass gap generation can not be described in its frameworks . when attempting to go beyond pt in the region of weak bare coupling , one runs into various mathematical problems , the most important being an absence of any analytical control over exponentially small contributions and , hence the absence of any reliable method to study long - distance physics where nonperturbative effects dominate even if they are exponentially suppressed . these facts impelled people to look for different , though equivalent representations which would allow to study the long - distance physics . one of such popular and promising representations is known as a dual formulation and is based on a certain non - classical change of variables . this formulation deals with dual lattice and appears to be very fruitfull for abelian models @xcite . namely , within such formulation 1 ) have been obtained various important analytical results on the long - distance dynamics of abelian spin fields and 2 ) the topological structure of the vacuum is transparent and can be more easily studied . in the context relevant to this paper we would like to mention the dual of the abelian @xmath4 model @xcite which has been used to prove the existence of a soft phase at low temperatures with power - like decay of the correlation function in two - dimensional @xmath4 model @xcite . in this case the dual of the @xmath4 model is a local theory for certain discrete variables . the conventional dual transformations @xcite for non - abelian models are perfectly well defined in mathematical terms and also lead to a local dual theory for integers which label irreducible representations of the symmetry group . nevertheless , these transformations are not complete if one compares to the abelian case . first of all , the resulting dual variables are not independent but are subject to some constraints known as triangular conditions , and as such they can not really be associated with elements of a dual lattice ( of course , introduction of the dual lattice is not the crucial point but rather matter of convenience ; e.g. , in the abelian case one can work with dual variables on the original lattice ) . it is to say , however that it is not really clear what are true independent degrees of freedom of the duals of non - abelian models . secondly , such formulation is so mathematically involved that today it is even far from obvious if it can be useful for any kind of the study of the model , except for the strong coupling expansion . in particular , the positivity of the boltzmann weight remains unclear ( see discussion in @xcite ) . even the precise definition of what can be termed the dual boltzmann weight is not really obvious , so the possibility of numerical simulations is at present rather uncertain . most importantly , however is that it is not clear how one could proceed in an analytical study of the model at low - temperatures . ] . on the other hand , there exists representation of @xmath0 and @xmath1 spin models in terms of link variables @xcite , and this representation can be formulated directly on the dual lattice . it is a first goal of the present paper to use the link representation for the derivation of an exact dual formulation of non - abelian spin models . here we give two such formulations which appear to be quite different from the conventional formulation mentioned above . in our opinion the most essential advantages of our formulations are that 1 ) one of them can be definitely used for the monte - carlo simulations in @xmath7 and 2 ) they are much more suitable for an analytical investigation of the model in the low - temperature region . the second fact follows from the properties of the link formulation and we refer for the details of why it is so to our papers @xcite . in the last of those papers we have already presented a dual of @xmath7 @xmath5 spin model and proposed approximate representation for the dual partition function at low temperatures . another important feature of one of our formulations is that it carries a close analogy with the abelian @xmath9 model . for instance , for @xmath7 @xmath5 model there are 3 independent degrees of freedom per ( dual ) site . at low temperatures it possesses an analog of the vortex spin - wave representation , etc . the deficiency of our approach is that it can be straightforwardly applied only to models in which spins are elements of the lie group . we do not know any obvious generalization of link formulation to , say , @xmath8 nonlinear sigma models . low - temperature properties of @xmath7 non - abelian models are crucial for construction of their continuum limit . it is commonly recognized that models possess no phase transition , correlation function has exponential decay at any coupling and models are asymptotically free . despite being more than twenty five years old this expectation has not been proven rigorously . on the contrary , certain percolation theory arguments suggest that all non - abelian models have soft low - temperature phase with power - like decay of the correlation function @xcite . it is thus another important motivation of the present investigation , namely to get deeper insight into the nature of the mass gap in non - abelian spin models . for example , in many papers devoted to @xmath7 non - abelian models it is written that `` there is a nonperturbative mass gap generation at arbitrarily small couplings '' . it is not really clear , however what is precise meaning of this `` nonperturbative generation '' . it can not be a simple consequence of the link decorrelation which happens in @xmath10 models . then , one could ask if this `` nonperturbative generation '' follows from the existence of some non - trivial background of defects like vortices of the @xmath4 model or is due to the strong but smooth disorder of non - abelian spins , e.g. like center vortices @xcite . as is well known , the dual formulation of abelian models have been extremely useful in clarifying all these important physical problems @xcite . we think that the dual formulation given here is a good starting point for obtaining reliable analytical results in the low - temperature limit of non - abelian models . this is our next goal to develop a technique within dual formulation of non - abelian models which would allow to investigate them in the limit of the weak coupling . in this context we study here the following two approaches on the simplest example of @xmath5 spin model * we derive an asymptotic expansion of the dual boltzmann weight when @xmath11 uniformly valid in fluctuations of the dual variables . the same procedure is also done for the two - point correlation function . as will be seen , at low temperatures the boltzmann weight converges to a certain gaussian ensemble the fact which ensures a possibility to define an analog of the vortex spin - wave representation for the partition and correlation functions in the semiclassical limit . this step provides a major simplification of the model as @xmath11 but even in this case it can not be yet solved exactly . the formulation obtained permits very simple calculation of the leading perturbative contribution to the correlation function . this contribution results in the power - like decay of the correlation function . * the second approach consists in replacing @xmath5 matrix elements by @xmath6 ones in the vicinity of the identity element of @xmath5 . we shall give a proof that such replacement is valid at low temperatures , compute the asymptotics of the dual boltzmann weight and present simple numerical evidence that practically for all configurations this way of calculations produces very reasonable approximation for the original boltzmann weight . this paper is organized as follows . in the section 2 we review briefly the conventional dual transformations . after introduction of the link formulation we develop original approach to the dual transformations and present two forms of the duals of the principal chiral model . in the section 3 we investigate the low - temperature properties of the dual boltzmann weights of @xmath5 model in two dimensions along the lines described above . some technical details of this investigation are given in the appendix . in the section 4 we summarize our results and outline some perspectives for future investigations . also , we calculate here the leading perturbative contribution to the correlation function . we work on a @xmath2-dimensional hypercubic lattice @xmath12 with lattice spacing @xmath13 and a linear extension @xmath14 . we impose either free or periodic boundary conditions ( bc ) . let @xmath15 ; @xmath16 and @xmath17 denote the haar measure on @xmath18 . @xmath19 and @xmath20 will denote the character and dimension of irreducible representation @xmath21 of @xmath18 , correspondingly . we treat models with local interactions , i.e. interactions between nearest neighbours . let @xmath22 $ ] be a real invariant function on @xmath18 such that @xmath23 \mid \ \leq h(i ) \label{huleq}\ ] ] for all @xmath24 and coefficients of the character expansion of @xmath25 @xmath26 \ = \ \int du \ \exp \left ( \beta h[u ] \right ) \chi_r(u ) \label{charexpdef}\ ] ] exist . the partition function ( pf ) of the principal chiral model with the symmetry group @xmath27 is defined as @xmath28 \right ] \ . \label{pfdef}\ ] ] conventional dual transformations can be defined as a sequence of transformations consisting of the following steps : 1 . fourier expansion of the boltzmann weight @xmath29 . this is an essentially character expansion on the group . 2 . exact integration over original degrees of freedom . as a result one obtains a set of constraints on the summation variables ( which label representations of the group @xmath18 and matrix elements of @xmath30 ) in the character expansion . solution of the constraints in terms of new dual variables . performing the first step one finds for the pf @xmath31 \right ] \prod_{x\in\lambda } q(x ) \ , \label{pfst1}\ ] ] where @xmath32 denotes links of the lattice and @xmath33 is a group integral . let @xmath34 $ ] be a matrix element of @xmath24 in representation @xmath35 . then @xmath33 can be written as @xmath36 d_{r(x - e_n , e_n)}^{m(x - e_n , e_n)k(x - e_n , e_n)}[u^{\dagger } ] \right ] \ . \label{qxdef}\ ] ] the second step in the most general settings ( for arbitrary graph and for all @xmath18 ) has been accomplished in @xcite . through the rest of this subsection we consider only the simplest cases of @xmath37 and @xmath38 in two dimensions . these examples already show a marked difference between abelian and non - abelian models , on the one hand . on the other hand , we want to have relatively simple formulae which can be compared with our formulations . for the @xmath9 model we have @xmath39 \right ) \ , \label{qxu1}\ ] ] where @xmath40 is the kronecker delta . in @xmath0 case the result for @xmath33 in @xcite is expressed in terms of the haar intertwiners . we re - state this result in more familiar form expanding @xmath33 in the clebsch - gordan ( cg ) series @xmath41 where @xmath42 , @xmath43 , @xmath44 , @xmath45 and similar notations are used for @xmath46 and @xmath47 . the third step can be readily accomplished for the @xmath9 model . we skip all the details of the derivation which are well known and can be found , for example in @xcite . the result for the pf reads @xmath48 where the product runs over all links of the dual lattice @xmath49 and the dual boltzmann weight is given by @xmath50 \ , \label{dualbwxy}\ ] ] which in the case of the standard choice @xmath51 leads to @xmath52 with @xmath53 - the modified bessel function . unfortunately , situation is much more difficult for non - abelian case . @xmath33 for @xmath5 contains not just one but 4 conditions : 2 kronecker delta s which constrain magnetic numbers @xmath46 and @xmath47 @xmath54 and 2 triangular conditions on representations @xmath55 we are not aware of any attempts in the literature to resolve these constraints simultaneously for all lattice sites . of course , this does not cause much problems for the strong coupling expansion where only lowest values of @xmath56 contribute . but situation becomes hopeless in the low temperature region where probably all @xmath56 become relevant . eventually , this reduces to the problem of finding the appropriate summation technique when @xmath11 which we could not solve . moreover , @xmath33 as defined in ( [ qxsu2 ] ) is not strictly positive quantity : on a number of configurations it is negative . we do not know how to perform resummation over the magnetic numbers to get positive boltzmann weight . most probably , only complete summation over all magnetic numbers will produce positive quantity . one of the possible ways of how to proceed further is to go over to the triangular lattices identifying representations with the length of the sides of triangles and magnetic numbers with the orientation of the triangles , and we have undertaken some attempts to go along this line . detailed description of this procedure is however beyond the scope of the present paper . we turn now to a different approach to duality transformations based on so - called link representation for partition and correlation functions . that is , the three steps formulated in the beginning of the previous subsection are replaced by the following ones 1 . change of variables in the partition function ( [ pfdef ] ) @xmath57 integration over original degrees of freedom . this integration generates a set of constraints on the link matrices @xmath58 known as the bianchi identities . 2 . implementation of constraints into the partition function by making use of the invariant delta - function on the group . this step introduces new degrees of freedom which can be associated with plaquettes of the original lattice and which label the irreducible representations of the group @xmath18 . at this step one can go over to the dual lattice so that the new variables belong to the elements of the dual lattice . integration over link variables . these two steps give already some dual version of the principal chiral model . 3 . before integration over link variables one can decouple summations over matrix indices employing certain orthogonality relations for the cg coefficients ( or for some other objects ) of the group and multiplying the group matrices entering bianchi identity in some specially ordered way . though details depend on the dimension of the lattice the result is that only independent dual variables remain in the theory and their interaction is confined to a subset of all hypercubes of the dual lattice . the first step of this program has been accomplished twenty years ago in @xcite . we therefore restrict ourselves only to the description of the result . the pf ( [ pfdef ] ) can be exactly reformulated in terms of link variables as ( with periodic bc ) @xmath59 \right ] j[v ] \ , \label{lpf}\ ] ] with jacobian @xmath60 $ ] given by @xmath61 = \prod_{p\in\lambda } j[v(p ) ] \ \prod_{n=1}^d j[h(n ) ] \ , \label{jacobdef}\ ] ] where @xmath62 $ ] and @xmath63 $ ] are given by @xmath64 \ = \ \sum_{\ { r \ } } d(r ) \chi_r \left ( \prod_{l\in p}v(l ) \right ) \ , \label{jacob}\ ] ] @xmath65 \ = \ \sum_{\ { r \ } } d(r ) \chi_r \left ( \prod_{l_n=1}^lv(l_n ) \right ) \ , \ l=(x , e_n ) \ . \label{holonconstr}\ ] ] @xmath66 is a product over all plaquettes of lattice @xmath67 , the sum over @xmath21 is sum over all representations of @xmath68 , @xmath69 is the dimension of @xmath35-th representation . the group character @xmath70 depends on a product of the link matrices @xmath71 along a closed path ( plaquette in our case ) : @xmath72 the expression @xmath73 is the invariant group delta - function which reflects the fact that the product of @xmath74 around plaquette equals @xmath75 @xmath76 this constraint is called local bianchi identity while the constraint on holonomy operators @xmath77 is usually called global bianchi identity . the only solution of two constraints ( [ bloc ] ) and ( [ bglob ] ) is a pure gauge ( [ linkfunc ] ) which restores the equivalence of the standard and the link formulations . on the lattice with free bc the global constraint should be omitted . furthermore , in the thermodynamic limit ( tl ) the contribution from the constraints on the holonomy operators vanishes very fast . in the perturbation theory in @xmath7 it vanishes like inverse powers of @xmath14 @xcite , and similar behaviour is expected for @xmath78 . since we are eventually interested in the tl we neglect these global constraints ( in fact , the presence of global constraints does not cause any principal problems as will be seen from the procedure described below but their absence makes the technical details somewhat simpler ) . thus , for both free and periodic bc we work with the pf @xmath59 \right ] \prod_{p\in\lambda } j[v(p ) ] \ . \label{linkpffin}\ ] ] the two - point correlation function in representation @xmath79 takes the form @xmath80 where @xmath81 is some path connecting points @xmath82 and @xmath83 and @xmath84 if along the path the link @xmath85 points positive direction and @xmath86 , otherwise . for more details on the link formulation we refer the reader to our paper @xcite , where we have developed a weak coupling expansion for @xmath0 spin models using the link representation . here we are going to accomplish the step 2 described in the previous subsection . the idea that the true dual variables in the link formulation are the fourier conjugate to the local bianchi identity has been proposed also in @xcite but the formulae have been explicitely written only for abelian @xmath9 model in @xmath7 . we thus want to reformulate the model ( [ linkpffin ] ) on the dual lattice only in terms of discrete variables that are in our case representations @xmath87 and magnetic quantum numbers @xmath88 . we shall use the following conventions for the dual lattice . @xmath89 will denote an object dual to site @xmath82 , i.e. @xmath2-cell of the dual lattice , @xmath90 - @xmath91-cell of the dual lattice , @xmath92 - @xmath3-cell of the dual lattice and so on . by definition , object dual to k - th cell is @xmath93-cell . as follows from ( [ linkpffin ] ) and from the definition of the character which enters the group delta - function in ( [ jacob ] ) @xmath94 on the dual lattice the pf may be written as @xmath95 \sum_{m_i(\omega_p)}^{d[r(\omega_p ) ] } \right ] \prod_{\omega_l } \xi_0(\omega_l ) \ . \label{pfsundual}\ ] ] due to the trace there are 4 variables @xmath46 at each @xmath92 , thus @xmath96 . the dual weight @xmath97 is given by the following one - link integral @xmath98 } \ \prod_{\nu = 1}^{d-1 } \left [ v_{r(\omega_p)}^{m_im_{i+1 } } \ v_{r(\omega_p^{\prime})}^{\dagger \ n_in_{i+1 } } \right ] _ { \nu } \ , \label{dualweight}\ ] ] where @xmath99 is a matrix element of @xmath35-th representation . @xmath3-cells @xmath92 and @xmath100 share @xmath91-cell @xmath90 . @xmath101 runs over @xmath91 such pairs . similar form for the two - point correlation function in the representation @xmath79 reads @xmath102 \sum_{m_i(\omega_p)}^{d[r(\omega_p ) ] } \right ] \nonumber \\ \times \prod_{\omega_l\in \overset{\star}c_{xy } } \xi^{s_is_{i+1}}_j(\omega_l ) \ \prod_{\omega_l\notin \overset{\star}c_{xy } } \xi_0(\omega_l ) \ , \label{2func}\end{aligned}\ ] ] where @xmath103 and the link integral on @xmath104 is @xmath105 } \ \prod_{\nu = 1}^{d-1 } \left [ v_{r(\omega_p)}^{m_im_{i+1 } } \ v_{r(\omega_p^{\prime})}^{\dagger \ n_in_{i+1 } } \right ] _ { \nu } \ v_j^{s_is_{i+1 } } \ . \label{2funcdualw}\ ] ] here , @xmath106 is a path dual to the path @xmath81 between points @xmath82 , @xmath83 on the original lattice . to vizualize last formulae let us give expression for the pf in two dimensions with @xmath38 . in @xmath7 we have @xmath107 , @xmath108 and @xmath109 . placing as usually dual sites in the centers of original plaquettes we obtain @xmath110 \prod_{l\in\overset{\star}\lambda } \xi_0(l ) \ , \label{pflsu2}\ ] ] where the dual weight @xmath111 becomes @xmath112 } \ v_{r(x)}^{m_i(x ) \ m_{i+1}(x ) } \ v_{r(x+e_n)}^{\dagger \ t_i(x+e_n ) \ t_{i+1}(x+e_n ) } \ . \label{linkint}\ ] ] it is very easy to make an integration in ( [ linkint ] ) expanding the result into the cg series . one then finds the following representation for the dual weight of @xmath5 model @xmath113 \ c_{r_1 m_1 \ j k}^{r_2 t_2 } \ c_{r_1 m_2 \ j k}^{r_2 t_1 } \ , \label{xio2dexact}\ ] ] where we have denoted @xmath114 , @xmath115 , @xmath116 and so on . for the standard choice @xmath117=\chi_{1/2}(v)$ ] the coefficients @xmath118 $ ] can be computed explicitely @xmath119 \ \frac{2j+1}{\beta } i_{2j+1}(2\beta ) \ . \label{cj2d}\ ] ] for the correlation function in eq.([2func ] ) using the cg expansion one gets @xmath120 last expressions can be compared with the dual of the @xmath9 model given in ( [ dualxy ] ) and with the standard dual formulation of @xmath5 model described by formulae ( [ pfst1 ] ) and ( [ qxsu2 ] ) . certainly , the dual weight for @xmath5 model @xmath121 as given in ( [ linkint])-([cj2d ] ) is very close to what we have for the dual weight in @xmath9 model : 1 ) dual variables , i.e. representations @xmath122 and magnetic numbers @xmath123 , live on sites of the dual lattice ; 2 ) the full gibbs measure is factorized into product of dual weights @xmath121 over links of the dual lattice . unlike the standard dual formulation there is no complicated triangular conditions on representations : triangular constraint on @xmath124 in ( [ xio2dexact ] ) is only one - link problem which does not lead to any significant complications . nevertheless , there is also a difference from the abelian case . as follows from the properties of the cg coefficients we have one constraint on the magnetic numbers on every link , namely summation over @xmath125 in ( [ xio2dexact ] ) produces the following condition @xmath126 this shows that not all dual variables are in fact independent , and this is the main difference from the @xmath9 model . there is no simple way to make a change of variables such that one gets only independent variables and keeps the locality of interaction . in the next subsection we present a different way of the construction of the dual theory from the link formulation in such a way that the dual weight depends only on independent dual variables . we finish this subsection with the brief description of some important features of @xmath121 : * as follows from the properties of the coefficients of the expansion @xmath118 $ ] the series in @xmath124 in ( [ xio2dexact ] ) gives directly the strong coupling expansion of the model written in closed and compact form . much less trivial task is to get weak coupling expansion for @xmath121 since all @xmath124 in the series ( [ xio2dexact ] ) become relevant . * on all configurations @xmath127 @xmath128 is strictly positive @xmath129 . though we could not prove it rigorously this claim is supported by the following facts : 1 ) the first term in the strong coupling expansion is strictly positive , thus at sufficiently small @xmath130 where the series converges very fast @xmath128 is positive ; 2 ) the leading term of the asymptotic expansion of @xmath128 at large @xmath130 is strictly positive on all configurations ; 3 ) numerical computations of @xmath128 on a number of configurations and in a wide region of @xmath130 also support this conclusion ( we have checked this statement using mathematica on more than @xmath131 configurations ) . if @xmath129 on all configurations this gives a chance for a numerical monte - carlo simulations of the dual model . * the dominant contribution to @xmath128 at large @xmath130 comes from diagonal components of the rotation matrices , non - diagonal contribution is suppressed roughly as @xmath132^{-1}$ ] . this is , of course a consequence of the fact that when @xmath11 the link matrix performs only small fluctuations around unity . in turn , this property gives a possibility to compute low - temperature asymptotic expansion of @xmath128 . this will be subject of the next section . the main idea of the following approach is to introduce the independent dual variables from the beginning , namely to find a representation for the group invariant delta - function in terms of quantities which would be free of any conditions . we explain the realization of this idea for @xmath7 @xmath5 model and then show how it can be extended to other groups and to higher dimensions . we start with the representation for @xmath5 matrix elements in which the dependence on the magnetic numbers enters only through the clebsch - gordan coefficients @xmath133 @xmath134 where @xmath135 is expressed through the spherical harmonics @xmath136 and generalized characters @xmath137 of @xmath5 @xmath138 generalized character of rank @xmath139 in representation @xmath35 can be defined through the relation @xmath140 we remind that the invariant measure in this parameterization is @xmath141 and the fundamental trace becomes @xmath142 the basic formula which we need is the following representation for @xmath5 characters @xmath143 @xmath144 where @xmath145 is the legendre polynomial and @xmath146 this representation can be easily proved if one uses orthogonality relations for the spherical harmonics and then the addition theorem for the generalized characters . as the next step , we divide all sites of the dual lattice into sets of even and odd sites . site @xmath147 is even if both @xmath148 and @xmath149 are even or odd , and is odd if one of @xmath150 is odd . thus , all even sites are surrounded by odd sites and vice - versa . using cyclic properties of the trace we can couple the four matrices entering local bianchi identities in two different ways , namely @xmath151 \ , \label{chareven}\ ] ] if the site @xmath82 is even and @xmath152 \ , \label{charodd}\ ] ] if the site @xmath82 is odd . we introduced here obvious notations @xmath153 with help of ( [ su2char ] ) and ( [ chareven ] ) , ( [ charodd ] ) the jacobian @xmath60 $ ] defined in ( [ jacobdef ] ) and ( [ jacob ] ) can be written on the dual lattice as @xmath61 = \prod_{x , \mbox{even } } j[v(x ) ] \ \prod_{x , \mbox{odd } } j[v(x ) ] \ , \label{jacobdual}\ ] ] where in even sites @xmath154 while for odd sites @xmath155 a set of variables @xmath156 is chosen to be dual variables at site @xmath82 . finally , we decompose all plaquettes of the dual lattice into sets of even and odd plaquettes in such a way that all even plaquettes are surrounded by odd ones and vice - versa . due to the special coupling of matrices as described above , the integration over link matrices is now coupled only within every , for example even plaquette . this is shown in fig.[matrcoupl ] . substituting ( [ su2deltaeven ] ) and ( [ su2deltaodd ] ) in ( [ lpf ] ) , re - denoting notations for links @xmath157 uniformly for all even plaquettes as shown in fig.[matrcoupl ] we obtain dual expression for the partition function @xmath158 \ \nonumber \\ & \times & \prod_{p,\mbox{even } } b_0\left [ \xi ( x),\xi ( x+e_1 ) , \xi ( x+e_2 ) , \xi ( x+e_1+e_2 ) ; \ \beta \right ] \ , \label{dual2su2}\end{aligned}\ ] ] where the dual boltzmann weight @xmath159\ ] ] is given by @xmath160 \right \ } \ \phi ( \xi ( x);v_1 v_2 ) \\ & \times & \phi ( \xi ( x+e_2);v_2^{\dagger}v_3 ) \ \phi ( \xi ( x+e_1+e_2);v_3^{\dagger } v_4^{\dagger } ) \ \phi ( \xi ( x+e_1);v_4 v_1^{\dagger } ) \ . \nonumber \end{aligned}\ ] ] dual expression for the correlation function can be easily re - constructed from eq.([corfuncdef ] ) and last formulae . consider , for simplicity the shortest path between points @xmath161 and @xmath162 . the corresponding dual path is a set of links which point direction @xmath163 . let @xmath164 be even . since every link belongs to one and only one even plaquette we obtain @xmath165 \ \nonumber \\ & \times & \prod_{p^{\prime } } b_0(p^{\prime } ) \ \sum_{s_1=-j}^{j } \cdots \sum_{s_r =- j}^{j } \ \prod_{p(c ) } b_j(p(c);s_{i-1}s_is_{i+1 } ) \ , \label{corrfd1}\end{aligned}\ ] ] where @xmath166 denotes a set of even plaquettes which do not contain links @xmath167 and @xmath168 - a set of even plaquettes which contain such links . in notations of fig.[matrcoupl ] we have @xmath169 \right \ } \ v_j^{s_{i-1}s_i}(l_1 ) \ v_j^{s_{i}s_{i+1}}(l_3 ) \\ & & \times \phi ( \xi ( x);v_1 v_2 ) \ \phi ( \xi ( x+e_2);v_2^{\dagger}v_3 ) \ \phi ( \xi ( x+e_1+e_2);v_3^{\dagger } v_4^{\dagger } ) \ \phi ( \xi ( x+e_1);v_4 v_1^{\dagger } ) \ . \nonumber \end{aligned}\ ] ] finally , the disorder operator in the dual representation can be readily obtained from its expression in the link formulation . let @xmath170=-h[v(l)]$ ] . consider the following disorder operator @xmath171 \right ] \ , \label{disparamdef}\ ] ] where @xmath172 $ ] . expectation value of this operator can be written as a ratio of two partition functions . with help of the change of variables @xmath173 this ratio can be presented as the following expectation value in the dual formulation @xmath174 generalization of these representations to higher dimensions is straightforward . in this case we couple link matrices in such a way as to decouple integration over them into a set of integrations within each even @xmath2-cell of the dual lattice . then the partition function can be written as @xmath175 \nonumber \\ & \times & \prod_{h_d , \mbox{even } } b(h_d ) \ , \label{dualsu2d}\end{aligned}\ ] ] where the dual boltzmann weight reads @xmath176 \right \ } \right ] \prod_{\omega_p \in h_d } \phi ( \xi ( \omega_p ) ; \gamma ( \omega_p ) ) \ . \label{dualbwdef}\ ] ] here , @xmath177 are dual variables which belong to @xmath3-cells of the dual lattice and @xmath178 are combined angles arising from multiplication of @xmath91-cell matrices ( original link matrices ) which belong to the same plaquette trace . to our opinion , the dual representation presented above bears the closest analogy to the abelian case . for example , in two - dimensional model there are three independent dual variables per site as could be expected on general grounds . further , as follows from the eq.([dualbw ] ) the interaction between dual variables is local though it is not exactly the same as in abelian models : in addition to nearest - neighbour interaction , there is an interaction over diagonals of even plaquettes . the deficiency of this dual representation is that the dual weight is in general complex valued . this makes the numerical mc computations impossible . on the other hand , we believe it is well suited for an analytical investigation of the model at low temperatures . in the described above scheme the realization of ideas leading to the local dual formulation relies on the special representation for @xmath5 matrices ( [ vrmn ] ) , ( [ su2matr ] ) . this representation appears to be very efficient at large @xmath130 as we shall show in the next section . we do not know if similar parameterization exists for any group @xmath18 . therefore , for arbitrary group @xmath18 we proceed as follows . instead of ( [ su2char ] ) we use unitarity relations for the cg coefficients of @xmath1 or @xmath0 group @xcite @xmath179 which allows to present group characters as @xmath180 where @xmath181 we then proceed exactly as for @xmath5 . for the sake of simplicity let us consider the two - dimensional theory . identifying @xmath182 as new dual variables the pf can be written as @xmath183 \prod_{p,\mbox{even } } \ b(p ) \ , \label{dualsun}\ ] ] where the dual boltzmann weight takes the form @xmath184 \right \ } \ g ( \xi ( x);v_1 v_2 ) \\ & \times & g ( \xi ( x+e_2);v_2^{\dagger}v_3 ) \ g ( \xi ( x+e_1+e_2);v_3^{\dagger } v_4^{\dagger } ) \ g ( \xi ( x+e_1);v_4 v_1^{\dagger } ) \ . \nonumber\end{aligned}\ ] ] substituting ( [ sungdef ] ) into the last equation one can express @xmath185 in terms of one - link integrals defined in ( [ dualweight ] ) . since the cg coefficients of @xmath0 group can be chosen real @xcite , the dual weight in this representation is also real . we have not studied if it is positive . one of the most important application of the dual formulation is the investigation of the low - temperature region of two - dimensional non - abelian models . the first step in this investigation is to establish an asymptotic expansion for dual weights at large @xmath130 . clearly , it is necessary to get asymptotics uniformly valid in all fluctuations of dual variables . it turns out that such asymptotics can indeed be constructed , and this is , in our opinion one of the most important advantages of our dual formulation . calculation of the asymptotic expansion essentially relies on the fact that when @xmath11 the link matrix performs only small fluctuations around unity both in the finite volume and , most importantly in the thermodynamic limit . note , that this is not the case for the original ( @xmath30 ) degrees of freedom : in the large volume limit their fluctuations are not bounded . in this subsection we construct an effective low - temperature theory for the dual formulation ( [ dual2su2 ] ) . the corresponding dual weight is given by eq.([dualbw ] ) . we modify it by introducing sources @xmath186 which can be related to the correlation function @xmath187 + j_k(l)\omega_k(l ) \right ] \right \ } \ \phi ( \xi ( x);v_1 v_2 ) \\ & \times & \phi ( \xi ( x+e_2);v_2^{\dagger}v_3 ) \ \phi ( \xi ( x+e_1+e_2);v_3^{\dagger } v_4^{\dagger } ) \ \phi ( \xi ( x+e_1);v_4 v_1^{\dagger } ) \ , \nonumber \end{aligned}\ ] ] where @xmath188 it is well known ( see , e.g. @xcite ) that when @xmath11 the link matrix performs only small fluctuations around unit matrix . our next calculations relay on this fact . let @xmath117 $ ] has a unique maximum at @xmath189 . we also require that @xmath117 $ ] possesses character expansion . then , in the vicinity @xmath189 and in the parameterization ( [ vrmn ] ) , ( [ su2matr ] ) one can always construct the following expansion @xmath190 \ = \ h[i ] -\frac{\gamma}{2}\sin^2\frac{\omega ( l)}{2 } + { \cal o}(\omega^3 ) \ . \label{hexp}\ ] ] for the standard choice @xmath117={\rm tr } v_{1/2}=2\cos\frac{\omega}{2}$ ] one finds @xmath191=2 $ ] and @xmath192 . next , we need uniform asymptotic expansion of the function @xmath193 . we derive such an expansion in the appendix a. the leading term of this expansion takes the form @xmath194 where the classical angular momentum @xmath195 is introduced in ( [ clangm ] ) . as follows from the semiclassical expansion ( [ phisemiclexp ] ) , the remainder is actually bounded as @xmath196 for all @xmath195 . therefore , corrections @xmath197 and higher are in fact negligible in the limit @xmath11 so we treat them perturbatively . lastly , we substitute all these expressions into ( [ dualbwsrc ] ) , express combined angles @xmath198 , @xmath199 , @xmath200 in terms of link angles and perform integration . we omit all these simple but rather cumbersome calculations and give only the final result . introducing notations @xmath201 - 3l^2\ln ( \gamma\beta ) \right ] \ , \label{constant}\ ] ] @xmath202 and making the change of variables @xmath203 so that @xmath204 takes integer values , we present the result for the dual partition function at large @xmath130 in the form @xmath205 \exp [ s_{eff } ] \ . \label{dualeff}\ ] ] an expression for the effective action @xmath206 is too large to be given here in full , therefore we present only its expansion . rescaling sources as @xmath207 the effective action can be expanded as @xmath208 we have computed first three terms in the above expansion . here we give @xmath209 and @xmath210 @xmath211 @xmath212 @xmath213 @xmath214 we have introduced here the following notations @xmath215 @xmath216 due to the term @xmath217 in the measure , summation over @xmath204 can be extended to include point @xmath218 . we use poisson summation formula , perfrom change of variables @xmath219 and make the following expansion @xmath220 this expansion is valid when @xmath221 . we call this expansion semiclassical approximation . it allows to define an obvious analog of the vortex spin - wave representation for the pf @xmath222 \\ & \times & \exp \left [ s_{eff}(r_k(x ) ) + 2i\pi \sqrt{\gamma\beta}\sum_x m(x ) \left ( \sum_kr^2_k(x ) \right ) ^{1/2 } \right ] \ . \nonumber \end{aligned}\ ] ] here we investigate the dual model given by eq.([pflsu2 ] ) with the corresponding dual weight ( [ linkint ] ) . our approach is to replace the matrix elements in the integrand of ( [ linkint ] ) by the @xmath6-like matrix elements . justification of such replacement will be given in the course of calculations . for @xmath5 matrix elements we use the wigner @xmath2-function parametrized by euler angles @xmath223 for the sake of simplicity we consider here only the standard action @xmath117=\sum_nd_{1/2}^{nn}$ ] , i.e. the fundamental character which in this parameterization reads @xmath224 the invariant measure on the group takes the form @xmath225 substituting last expressions into ( [ linkint ] ) one can exactly integrate over @xmath226 and @xmath227 angles . this gives @xmath228 to get the asymptotics when @xmath11 we use the fact ( described in the previous subsection ) that the link matrix in this region performs small fluctuations around unit matrix , namely @xmath229 it seems to be unfeasible task to solve saddle - point equation for the integrand of ( [ xio1int ] ) . we therefore take a different approach to the problem , and replace this integrand by its asymptotics at @xmath230 such that @xmath231 here @xmath232 is classical angular momentum defined in eq.([clangm ] ) . the last condition states that the main contribution to the integral comes from diagonal components of @xmath5 matrix elements , non - diagonal contributions can be treated perturbatively . to be precise , we adjust the following three approximations : 1 . we use the following asymptotics for the modified bessel function @xmath233 when @xmath234 and such that @xmath235 . it leads to @xmath236 due to the property ( [ fluct ] ) the remainder is bounded like @xmath196 . 2 . the integration region over @xmath198 is extended to infinity after proper change of variables ( this introduces only exponentially small corrections which can be properly bounded ) . the main approximation concerns the behaviour of matrix elements @xmath237 in the vicinity @xmath230 . the standard results on the asymptotic expansion of @xmath237 assert that when @xmath238 , @xmath239 the @xmath5 matrix elements approach the @xmath6 matrix elements @xcite @xmath240 where @xmath241 is the bessel function . we have found , however that this approximation is oversimplified . in particular , it does not lead to the correct gaussian distribution of the dual boltzmann weight that is clearly seen from the effective theory obtained above . moreover , eq.([iso2asympt ] ) states that , e.g. the asymptotics of all diagonal elements is the same . to get more precise asymptotics one can use representation of @xmath242-function in terms of the jacobi function . the latter can be treated with help of the tricomi expansion into degenerate hypergeometric functions as described in @xcite , section 10.14 . finally , we use uniform tailor expansion for the degenerate hypergeometric functions @xcite , section 6.13 . this leads to the following asymptotics @xmath243 where all notations are given in ( [ finnote ] ) . it can be easily checked with help of mathematica that the last equation gives better approximation to @xmath242-function compared to eq.([iso2asympt ] ) , especially at large values of @xmath244 and is valid , in fact for all values of @xmath35 . moreover , it gives very reasonable approximation in rather wide region of @xmath198 , except region @xmath245 . in the appendix b we give independent and simpler proof of this asymptotic expansion calculating also the first correction @xmath246 to eq.([iso2likeasympt ] ) . the final result is given in eq.([dfuncasympt ] ) . we term this asymptotics as @xmath6-like approximation of @xmath5 matrix elements . the only ( but essential ) difference from the exact @xmath6 matrix element is the presence of @xmath247 in the argument of the bessel function . combining eqs.([bessas1 ] ) , ( [ iso2likeasympt ] ) we arrive finally at the following asymptotic representation for the one - link integral @xmath248 where we have used notations @xmath249 @xmath250 making change of variables @xmath251 in the last integral and extending the integration region over @xmath83 to infinity the last integral can be written as @xmath252 where @xmath195 and @xmath253 are defined in eq.([finnote ] ) . calculating the last integral we find @xmath254 \nonumber \\ \times \ i_k\left(\frac{r_1\sin\theta_1 r_2\sin \theta_2}{2\beta } \right ) \ , \label{xio2das}\end{aligned}\ ] ] where @xmath255 . again , we have compared the present asymptotics with the exact expression given by eq.([xio2dexact ] ) using matemathica . suprisingly , if @xmath130 is sufficiently large the asymptotic formula gives very reasonable approximation for all configurations we have studied . it is interesting to stress that the same asymptotics can be obtained directly from ( [ xio2dexact ] ) , where the cg coefficients are replaced by their semiclassical limit by ponzano - regge @xcite . the method presented above has however two advantages . first of all , we have computed corrections to the leading behaviour ( [ iso2likeasympt ] ) , therefore @xmath256-corrections to the asymptotic expansion of one - link integral can be computed in a straightforward manner . corrections to the ponzano - regge formula are in general unknown . secondly , we find that our method is much simpler . in this article we proposed dual formulation of non - abelian spin models . our approach to the dual transformations is summarized in the beginning of the section 2.2 . main formulae of the present paper , eqs.([pfsundual])-([2funcdualw ] ) , ( [ dual2su2])-([dualbw1 ] ) give two versions of the dual formulation . to demonstrate the usefulness of our formulations we have computed the low - temperature asymptotics of the dual boltzmann weights for two - dimensional model . we reckon that the results for the dual weights provide significant simplification of the partition function at large values of @xmath130 . nevertheless , even in this case the model is still too complicated to be solved exactly . in our next paper @xcite we consider some physical applications of the dual representation . in particular , we attempt to give a saddle - point solution of the dual model and derive effective @xmath10 model for the two - point correlation function . also , we calculate continuum limit of the dual representation and solve classical continuum equations of motion . some simple consequences which worth noting can be obtained from ( [ vortspwrepr ] ) . let us perform a shift of variables in the integrand of ( [ vortspwrepr ] ) @xmath257 this shift allows to extract the main contribution from the sources to the free energy which emerges from the gaussian term in the effective action , i.e. @xmath209 given by eq.([sefflead ] ) @xmath258 \tilde{z}(j,\beta ) \ , \label{zjl}\ ] ] where @xmath259 is a remaining part of the pf . here @xmath260 and @xmath261 are link green functions introduced in @xcite . in @xcite we show that the following sources @xmath262 and @xmath263 , otherwise , can be related to the correlation function in the representation @xmath79 . substituting these sources into ( [ zjl ] ) one finds after simple analysis @xmath264 \ , \label{gjlead}\ ] ] where @xmath265 , for @xmath266 . by no means we claim that this result establishes power - like decay of the correlation function : in fact , we do not know the @xmath195-dependence of the remainder . therefore , at the moment we can only claim that the shift ( [ shift ] ) allows to extract the main perturbative contribution . we would like to stress that the shift ( [ shift ] ) is precisely the same as the one used in @xcite . unfortunately , we do not know if the methods of @xcite can be applied to the partition function ( [ zjl ] ) to bound the remainder : the essential difference from the @xmath4 model is that even in the absence of vortices both full and effective dual actions are complex . nevertheless , we think that such possibility deserves further investigations . in this appendix we calculate the asymptotic expansion of the function @xmath267 , where @xmath268 is given by eq.([phidef ] ) @xmath269 here , @xmath145 is the legendre polynomial and @xmath270 is given in eq.([combangle ] ) . @xmath5 matrix is taken in the parameterization ( [ vrmn ] ) , ( [ su2matr ] ) . also , we define and compute the semiclassical limit of @xmath271 . to this end we introduce classical angular momentum as @xmath272 we are interested in the expansion of @xmath193 when @xmath230 uniformly valid in all variables @xmath273 . because @xmath274 and @xmath226 are compact variables the problem of uniformity concerns mainly the value of the variable @xmath35 . we face two situations : 1 ) @xmath35 is fixed and 2 ) @xmath239 . in the second case we look for the asymptotic expansion such that @xmath275 . as the first step we express the generalized characters entering @xmath193 through the associated legendre functions of the first kind taken on the cut @xmath276 $ ] @xmath277^{1/2 } { \rm p}^{-\lambda-1/2}_{2 r+1/2 } ( \cos\frac{\omega}{2 } ) \ . \label{chileg}\ ] ] when @xmath35 is fixed , we construct tailor expansion of @xmath193 . it is more convenient to expand in powers of @xmath278 rather than in powers of @xmath198 . from the representation ( [ chileg ] ) it follows that @xmath279^{1/2 } \left ( \frac{y}{2 } \right ) ^{\lambda } \left [ 1 - \frac{r^2-(\lambda + 1)^2}{2\lambda + 3}\frac{y^2}{2 } + { \cal{o}}(y^4 ) \right ] \ . \label{chiexp}\ ] ] substituting last expression into eq.([phidef1 ] ) , it is straightforward to find the following expansion @xmath280 where the coefficients of the expansion @xmath281 are @xmath282 @xmath283 \ , \label{f2}\ ] ] @xmath284 \nonumber \\ & + & \frac{3}{5}(r^2 - 4)-r^2 + 1 - \frac{3}{5}\sqrt{r^2 - 4}\sqrt{r^2 - 9 } + \sqrt{r^2 - 1}\sqrt{r^2 - 4 } \}\ . \label{f3}\end{aligned}\ ] ] when @xmath285 we expand coefficients @xmath286 in powers of @xmath195 . this defines the semiclassical expansion of @xmath193 when @xmath238 , @xmath266 such that @xmath275 @xmath287 it would seem that the bound @xmath288 in ( [ phiexp ] ) and , therefore in ( [ phisemiclexp ] ) does not hold uniformly in @xmath195 . it turns out however that the bound is correct both for fixed @xmath35 and for @xmath239 . to prove this we need an expansion for the associated legendre functions at large @xmath35 uniformly valid in the neighbourhood of the point @xmath289 . such an expansion is given by macdonald s formula @xcite , section 3.5 @xmath290^{-\mu } \\ & \times & \left \ { j_{\mu}(\alpha ) + \sin^2\frac{\omega}{2}\left [ \frac{\alpha}{6}j_{\mu + 3}(\alpha ) - j_{\mu + 2}(\alpha ) + \frac{1}{2\alpha}j_{\mu + 1}(\alpha ) \right ] + { \cal o}(\omega^4 ) \right \ } \ , \nonumber \end{aligned}\ ] ] where @xmath291 . the ratio of factorials in ( [ chileg ] ) is expanded as @xmath292^{1/2 } = r^{\lambda + 1/2}\left [ 1-\frac{\lambda ( \lambda^2+\frac{3}{2}\lambda + \frac{1}{2})}{6r^2 } + { \cal o}(r^{-4 } ) \right ] \ . \label{ratiofact}\ ] ] with these results , and using recursion relations for the bessel functions to reduce order to @xmath293 we obtain the representation for the generalized character of the form @xmath294^{1/2 } \ b(r,\lambda ; \omega,\partial_h)\mid_{h=1 } \ j_{\lambda + \frac{1}{2}}(\alpha h ) \ . \label{chiasymp}\ ] ] the function @xmath295 has the following asymptotic expansion @xmath296 + { \cal o } \left ( \sin^4\frac{\omega}{4 } \right ) \ , \label{basymp}\ ] ] where @xmath297 . substituting last expressions into eq.([phidef1 ] ) we can easily calculate all the sums and derivatives . this leads to @xmath298 re - expanding last formula in powers of @xmath299 one sees that eq.([phirwexp ] ) coincides with eq.([phisemiclexp ] ) . due to the bound @xmath300 in ( [ legasymp ] ) , the same bound holds in ( [ phirwexp ] ) and , consequently in ( [ phisemiclexp ] ) and in ( [ phiexp ] ) . we thus conclude that the tailor expansion ( [ phiexp ] ) does provide expansion at small @xmath198 which is valid both for fixed values of @xmath35 and for @xmath239 . here we compute the asymptotic expansion of the @xmath5 matrix elements @xmath237 in the classical region @xmath301 uniformly valid in the vicinity of the point @xmath289 for all allowed values of @xmath302 and @xmath303 , @xmath304 is held fixed . to get such an asymptotics we first present @xmath242-function in terms of hypergeometric function @xmath305 @xmath306^{\frac{1}{2 } } \left(\sin \frac{\omega}{2 } \right)^k \left(\cos \frac{\omega}{2 } \right)^{-p } \nonumber \\ \times f(s+k+1,-s - p;k+1;\sin^2 \frac{\omega}{2 } ) \ , \label{dasf}\end{aligned}\ ] ] where @xmath307 if @xmath308 , @xmath309 otherwise , and @xmath310 as is seen from the arguments of the hypergeometric function the infinite series in @xmath311 terminates so that right - hand side of ( [ dasf ] ) is polynomial in @xmath312 @xmath313^{\frac{1}{2 } } \left(\sin \frac{\omega}{2 } \right)^k \left(\cos \frac{\omega}{2 } \right)^{-p } \nonumber \\ \times \sum_{l=0}^{r+\frac{1}{2}(p - k)}(-1)^l \frac{(\sin^2\frac{\omega}{2})^l}{\gamma ( k+1+l)l!}{\cal f}_l(x , y ) \ , \label{dseries}\end{aligned}\ ] ] where we have introduced the following notations @xmath314 @xmath315 @xmath316 the second step consists in expanding the ratio of gamma functions . this can be done with help of the following formula @xmath317 where @xmath318 are the generalized bernoulli polynomials . important point concerns the large expansion parameter we use . we take not merely classical angular momentum @xmath319 but rather quantities @xmath82 and @xmath83 defined above . such a choice gives more accurate asymptotics valid in a wider region of parameters . then , in the case of quantity @xmath320 the series in ( [ gratio ] ) terminates because @xmath321 and representation ( [ gratio ] ) becomes exact . it leads to @xmath322 ^ 2}{(l+k - s_1)!(l+k - s_2 ) ! } \nonumber \\ \times b_{s_1}^{(k+l+1)}(\frac{1}{2}(k+1 ) ) b_{s_2}^{(k+l+1)}(\frac{1}{2}(k+1)+l ) \ \label{flxyexp}\end{aligned}\ ] ] what is essentially the desired expansion at large @xmath82 and @xmath83 . it follows from the last representation that @xmath323 \ . \label{flxyasym}\end{aligned}\ ] ] for @xmath324 following the same procedure one finds @xmath325 \ . \label{akxasym}\ ] ] substituting last expressions into ( [ dseries ] ) we get after some algebra @xmath326 here we have extended summation over @xmath85 to infinity since this introduces corrections of the order @xmath327 or less . recalling now the series representation for the bessel function @xmath328 we can easily sum up all series in the last formula . finally , we arrive at the following asymptotic expansion for @xmath242-function @xmath329 + o ( \sin^4\frac{\omega}{2 } ) \ } \ . \label{dfuncasympt}\end{aligned}\ ] ] we have introduced here the following notations : @xmath330 r. savit , phys.rev.lett . 39 ( 1977 ) 55 ; rev.mod.phys . 52 ( 1980 ) 453 . j. frhlich , t. spencer , commun.math.phys . 81 ( 1981 ) 527 . h. pfeiffer , j.math.phys . 44 ( 2003 ) 2891 . g. batrouni , m.b . halpern , phys.rev . d30 ( 1984 ) 1775 . o. borisenko , v. kushnir , a. velytsky , phys.rev . d62 ( 2000 ) 025013 . o. borisenko , v. kushnir , low - temperature behaviour of @xmath7 lattice @xmath5 spin model , proc . of nato workshop `` integrable structures of exactly solvable two - dimensional models of quantum field theory '' , ed . by s. pakuliak , and g. von gehlen , kluwer academic publishers , 2001 , 55 . a. patrascioiu , e. seiler , j.statist.phys . 69 ( 1992 ) 573 ; j.statist.phys . 106 ( 2002 ) 811 - 826 . o. borisenko , p. skala , phys.rev . d62 ( 2000 ) 014502 . t. banks , j. kogut , r. myerson , nucl.phys . b121 ( 1977 ) 493 . m. gpfert , g. mack , commun.math.phys . 81 ( 1981 ) 97 ; 82 ( 1982 ) 545 . varshalovich , a.n . moskalev , v.k . khersonskii , quantum theory of angular momentum , world scientific publishing co.pte.ltd . , singapore - new jersey - hong kong , 1988 . vilenkin , a.u . klimyk , representation of lie groups and special functions , kluwer academic publishers , dordrecht - boston - london , vol.316 , 1995 . j. bricmont and j .- r . fontaine , j.stat.phys . 26 ( 1981 ) 745 . vilenkin , special functions and theory of group representations , nauka , moscow , 1991 . h. bateman , a. erdelyi , higher transcendental functions , new york - toronto - london , 1953 ; russian edition , nauka , moscow , vol.1,2 , 1973 . g. ponzano , t. regge , semiclassical limit of racah coefficients , in spectroscopic and group theoretical methods in physics , north - holland publ . co. , amsterdam , 1968 , 1 ; k. schulten , r.g . gordon , journal of math.phys . , v16 ( 1975 ) , 1971 . o. borisenko , v. kushnir , saddle - point solution of the dual of @xmath7 @xmath5 chiral model , in preparation .
|
non - abelian lattice spin models with symmetry group @xmath0 or @xmath1 can be formulated in terms of link variables which are subject to the bianchi constraints .
using this representation we derive exact and local dual formulation for the partition function of such models on a cubic lattice in arbitrary dimension @xmath2 .
locality means that the dual action is given by a sum over some subset of hypercubes of the dual lattice and the interaction between dual variables ranges over one given hypercube .
dual variables are in general discrete - valued and live on @xmath3-cell of the dual lattice , in close analogy with the @xmath4 model .
we use our construction to study in details the dual of @xmath5 principal chiral model in two dimensions .
we give dual expressions also for two - point correlation function in arbitrary representation and for the free energy of defects .
leading terms of the asymptotic expansion of the dual boltzmann factor are computed and it is proven that at low temperatures it converges to a certain gaussian distribution uniformly in all fluctuations of dual variables . this result enables us to define the semiclassical limit of the dual formulation and to determine an analog of the vortex spin - wave representation for the partition function .
such representation is used to extract leading perturbative contribution to the correlation function which shows power - like decay at weak coupling .
we also present some analytical evidences that the low - temperature limit of the dual formulation is completely described by @xmath6-like approximation of @xmath5 matrix elements .
* dual formulations of non - abelian spin models : local representation and low - temperature asymptotics * * o. borisenko , v. kushnir * +
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
newtonian gravity theory has served physics and technology faithfully for well over three centuries . nevertheless , it has long been known that it is only an approximation to a relativistic gravitation theory , usually identified with einstein s 1915 general relativity ( gr ) . gr has correctly predicted subtle effects in the dynamics of the solar system , in the celebrated hulse - taylor double pulsar , and has anticipated the existence of those exotic denizens of the universe , black holes , now confirmed by myriad observations of galactic nuclei , compact galactic x - ray sources , etc . the nonlinear aspects of gravitation in gr are crucial to these new settings : gravitation is more difficult than newton supposed . but up until a few decades ago astronomers took comfort in the belief that if one sidesteps situations with extremely strong gravitational fields , or with extremely rapid motions , then newtonian theory is a good approximation to the truth . but is that so ? consider the situation in the realm of galaxies . speeds are low compared to light s , and gravitational fields are weak there . so newtonian physics _ should _ describe motions extremely well . yet the accelerations of stars and gas clouds in the outskirts of spiral galaxies , and of galaxies swimming in the large clusters of galaxies , well exceed newtonian predictions made on the basis of the matter actually visible in these systems . those accelerations are inferred by combining doppler measured velocities with geometric assumptions , e.g. that in a particular spiral galaxy stars move in circles on a plane . the geometric assumptions can be checked , at least statistically , and the acceleration discrepancy is found to be real and generic . sometimes the problem is characterized thus : rotation or random velocities are much bigger than expected . i stress acceleration because it is the quantity which directly measures the strength of the gravitational field . in any case , what is wrong ? it was realized by ostriker and peebles @xcite that massive halos would stabilize disk galaxies against the bar forming instabilities which , in newtonian theory , are endemic to systems with little velocity dispersion . since about third of the spiral galaxies have no significant bar , this justified the hypothesis that disk galaxies ( including spirals ) are often embedded in massive but invisible halos . thus started the trend to resolve the acceleration discrepancy by postulating the existence of much dark matter ( dm ) in systems ranging from the very tenuous dwarf spheroidal galaxies with visible masses @xmath1 to the great clusters of galaxies with observed masses in the @xmath2 ballpark @xcite , in brief in any system where an acceleration discrepancy exists . the dm s role is to provide the missing gravitational pull to account for the excessive accelerations ( an easy introduction to the dm paradigm is provided by khalil and mu~ noz @xcite ) . but thirty years of astronomical exploration and laboratory experiments @xcite have yet to provide independent evidence of dm s existence , e.g. @xmath3 rays from its decay . is there an alternative to dm ? this review focuses on the modified newtonian dynamics scheme put forward by milgrom @xcite as an alternative to the systematic appeal to dm . to pass judgment on dm and alternatives to it , it is well to take stock of the two overarching empirical facts in the phenomenology of disk galaxies . first , ever since the work of bosma as well as rubin and coworkers @xcite , it has been clear that gas clouds in the disks of spiral galaxies , which serve as tracers of the gravitational potential , circle around each galaxy s center with a ( linear ) velocity which first rises as one moves out of the center , but hardly drops as the radius grows to well beyond the visible disk s edge . yet a falloff of the velocity with radius was naively expected because , to judge from the light distribution in spiral galaxies , most of their visible mass is rather centrally concentrated . newtonian gravitation would thus predict that `` rotation curves '' ( rc ) should drop as @xmath4 outside the bright parts of these galaxies , but this is not seen in over a hundred extended rcs measured with sufficient precision @xcite . in fact in many spiral galaxies , particularly those possessing high surface brightness , the extended rc becomes flat away from the central parts ( fig . [ figure : fig1 ] ) . is plotted vs galactocentric radius @xmath5 in kiloparsecs ( kpc ) ; 1 kpc @xmath6 3000 light years . it is seen that the rcs are flat to well beyond the edges of the optical disks ( @xmath7 kpc ) . graph from ref . , reprinted with permission from the annual review of astronomy and astrophysics , volume 39 ( c)2001 by annual reviews www.annualreviews.org,width=264 ] second , as originally pointed out by tully and fisher ( tf ) @xcite , the rotation velocity @xmath8 and the blue band luminosity @xmath9 of a galaxy are simply correlated ( fig . [ figure : fig2 ] ) . in a more informative form @xcite , the tf law states that for disk galaxies the luminosity in the near infrared band , @xmath10 ( a good tracer of stellar mass ) , is proportional to the fourth power of the rotation velocity in the flat part of the rc , with a universal proportionality constant . in units of @xmath11 vs. the asymptotic rotational velocity @xmath8 in km / s . the straight line ( mine ) has slope exactly 4 and corresponds to mond s prediction , eq . ( [ tflaw ] ) , for a reasonable mass ( in units of @xmath12 ) to @xmath13 ratio @xmath14 . graph reproduced from ref . by permission of the american astronomical society.,width=226 ] the flat extended rc s have provided the logic underpinning the dm paradigm as follows . in the flat part of a rc , the rotation velocity is @xmath15 independent , so the centripetal acceleration goes as @xmath16 . thus in the plane of the galaxy the gravitational field must be decreasing as @xmath16 . according to poisson s equation of newtonian theory , such a gravitational field , if assumed spherically symmetric , must be sourced by a mass distribution with @xmath17 profile ( isothermal sphere model ) . since the visible mass density ( in stars and gas ) in the inner disk drops much faster than this , it is consistent to assume that the total mass distribution in the outer parts is quasispherical . the conclusion is that each spiral galaxy must be immersed in a roundish dm halo with mass density profile tending , at large @xmath15 , to @xmath17 . but questions plague the halo hypothesis . the halo dm , though much searched for , has never been detected directly @xcite . cosmogonic simulations of dm halo formation within newtonian physics predict that such a halo would have a density profile behaving like @xmath16 for small @xmath15 and gradually steepening to @xmath18 asymptotically . only by fine tuning the halo s parameters is it feasible to match this kind of profile to the observed flat rcs as well as would the isothermal sphere ( the need to fine tune parameters of a halo model in order to fit the shape of the galaxy s rc was already clear long ago @xcite ) . and the predicted @xmath16 `` cusp '' in the density profile is also observationally problematic @xcite . the dm halo hypothesis is evidently an attempt to resolve the acceleration discrepancy within orthodox gravitation theory . but in the coming to terms with this discrepancy , suspicion fell on newtonian gravity already early in the game . zwicky , who had exposed the acceleration discrepancy in clusters of galaxies @xcite , opined much later that the discrepancy may reflect a failure of conventional physics @xcite . concrete proposals for departures from the newtonian inverse square law in various settings were put forward by a number of workers @xcite . all of these still regarded gravitation as a linear interaction with the strength of the field proportional to its source s mass . as milgrom realized @xcite , such a modification of the gravity law , if relevant on the scales of galaxies , is incompatible with the tf law . for one it would imply that a mass @xmath19 at @xmath20 generates at @xmath21 the acceleration field @xmath22 , with @xmath23 depending only on fundamental constants and on the relative positions of source and field points in accordance with the principle of homogeneity of space . then , obviously , for matter orbiting with speed @xmath8 at radius @xmath5 in the outskirts of a galaxy ( with its source mass , @xmath24 , concentrated at a fairly definite distance @xmath5 ) , the acceleration , @xmath25 should be equal to @xmath26 . this would require not only @xmath27 , but also @xmath28 , the last incompatible with the tf law . then again @xmath29 with the correct dimensions , @xmath30 , can not be built by using only @xmath31 and the scale of length @xmath32 at which gravity departs from newtonian form . and inclusion of mass in the construction ( presumably the source s mass ) would be against the spirit of linearity . the bottom line is that linear gravity , even if non - newtonian , is incompatible with the tf law . milgrom @xcite proposed a novel paradigm which can be interpreted as reflecting non - newtonian as well as nonlinear character of gravity already at the nonrelativistic level . he called the scheme modified newtonian dynamics ( mond ) . the essential part of it is the relation @xmath33 between the acceleration @xmath34 of a particle and the ambient conventional newtonian gravitational field @xmath35 . if the function @xmath36 were unity , this would be usual newtonian dynamics . milgrom assumes that the positive smooth monotonic function @xmath36 approximately equals its argument when this is small compared to unity ( deep mond limit ) , but tends to unity when that argument is large compared to unity ( fig . [ figure : fig3 ] ) . the @xmath37 is a natural constant , approximately equal to @xmath38 . it is a fact that the centripetal accelerations of stars and gas clouds in the outskirts of spiral galaxies tend to be below @xmath37 . function : the `` simple '' function @xmath39 ( solid ) and the `` standard '' function @xmath40 ( dotted).,width=226 ] how does the mond formula ( [ mond ] ) help ? consider again stars or gas clouds orbiting in the disk of a spiral galaxy ( mass @xmath24 ) with speed @xmath41 at radius @xmath15 from its center . we must identify @xmath42 with the centripetal acceleration @xmath43 . sufficiently outside the main mass distribution we may estimate @xmath44 . and at sufficiently large @xmath15 , @xmath42 will drop below @xmath37 and we shall be able to approximate @xmath45 . putting all this together gives @xmath46 from which we may conclude two things . first , @xmath41 well outside the main mass distribution becomes independent of @xmath15 , that is , the rc flattens at some value @xmath47 . second , from the coefficients follows @xmath48 introducing the ratio of mass to luminosity @xmath49 , @xmath50 , we have the added prediction @xmath51 . although @xmath50 in the blue band varies somewhat with galaxy color , this last results explains the tf law , fig . [ figure : fig2 ] . theoretically @xmath50 is much less variable in the near infrared ( k ) band @xcite , which accounts for the extra sharpness of the tf relation in that band ( see fig . 2 in ref . ) . in units of the solar mass @xmath12 vs. the observed asymptotic rotational velocity @xmath47 in km / s . for each of the 60 galaxies from ref . ( deep blue circles ) the mass _ in stars _ comes from a fit of the _ shape _ of the rc with mond , whereas for the eight dwarf spirals ( light blue circles ) the mass in stars ( relatively small ) is inferred in ref . directly from the luminosity . the green line , with slope 4 , is mond s prediction , eq . ( [ tflaw ] ) . graph reproduced from ref . by permission of the american astronomical society.,width=226 ] the impressive agreement of eq . ( [ tflaw ] ) with observations is most clear from the baryonic tf law @xcite , fig . [ figure : fig4 ] . in full agreement with mond , the observed baryonic mass in spiral galaxies is accurately proportional to the fourth power of the asymptotic rotation velocity . mond s single formula thus unifies the two overarching facts of spiral galaxy phenomenology . mond is similarly successful in explaining the detailed _ shapes _ of rcs . [ figure : fig5 ] shows the measured rcs of fifteen galaxies together with the mond fits based on surface photometry in the optical band and radio measurements . the only parameter adjusted in these impressive fits is the disk s stellar mass , or equivalently , the stellar mass - to - luminosity ratio @xmath50 . the trend of the so determined values of @xmath50 with galaxy color jibes with that predicted by stellar evolutionary models @xcite ; in this mond does better than dm halo models @xcite . to make similarly successful fits without just `` putting dm where it is needed '' , dm theorists must adjust two extra parameters apart from @xmath50 in standardized halo models . as stressed in a recent critical reappraisal @xcite , those galaxies where one - parameter mond fits do worse than halo fits often display complicating factors that could mitigate its less than striking performance . in disk galaxies mond is unquestionably more economical , and thus more falsifiable , than the dm paradigm . when applied to many galaxies , the mond fits mentioned serve to determine @xmath37 with some accuracy ; the value @xmath52 is often used @xcite . it is significant that @xmath37 as deduced from rc fits agrees well with the value for @xmath37 obtained by comparing @xmath53 with the empirical coefficient in the tf law : the roles of @xmath37 are different in the two subjects , and they are tied together only in mond . milgrom @xcite notes a few additional conceptually distinct roles played by @xmath37 in extragalactic astrophysics . an early example is that @xmath37 sets a special scale of mass surface density @xmath54 , and wherever in a system the actual surface mass density drops below @xmath55 , newtonian gravitational behavior gives way to mond dynamics @xcite . ( in galaxies like ours this occurs some way out in the disk , and that is why the rc may exhibit a brief drop before becoming flat asymptotically ) . on this basis milgrom predicted that were disk galaxies with surface mass density everywhere below @xmath55 to exist , they should show especially large acceleration discrepancies . a population of such galaxies became known in the late 1980 s @xcite , and all facets of milgrom s prediction were subsequently confirmed @xcite . mond obviously predicts that in these so called low surface brightness galaxies , the shapes of the rcs , which tend to be _ rising _ throughout the visible disk , should be independent of the precise way @xmath56 switches from @xmath57 to unity for @xmath58 . constraints of space forbid me from delving into other mond successes , e.g. for elliptical and dwarf spheroidal galaxies . but excellent reviews of these and other phenomenological aspects of mond exist @xcite . it is well , however , to stress one prominent empirical difficulty faced by the mond formula ( [ mond ] ) . clusters of galaxies exist which comprise hundreds of galaxies of various kinds moving in their joint gravitational field with velocities of up to @xmath59 . the newtonian virial theorem ( gravitational potential energy of a quiescent system equals minus twice its kinetic energy ) can be used to estimate a cluster s mass since the kinetic term is proportional to the total mass while the potential one goes like the square of mass . in this way it is determined that the total gravitating mass in a typical cluster is 5 - 10 times the mass actually seen in galaxies as well as in hot x - ray emitting gas , an oft constituent of clusters . a similar story is told by analyses of the hydrostatics of the hot gas in light of its measured temperature . finally , the gravitational lensing by several clusters has led to estimates of their masses commensurate with the mentioned diagnosis . when the mond formula , or even better , a mond analog to the virial theorem @xcite , is used to estimate the masses of clusters , one finds an improvement but not a full resolution to the acceleration discrepancy . clusters still seem to contain a factor of two more matter than actually observed in all known forms @xcite . the optimist will stress that mond has alleviated the discrepancy without even once overcorrecting for it ; the pessimist @xcite will view this finding as damaging mond s credibility . but one should keep in mind that clusters may contain much invisible matter of rather prosaic nature , either baryons in a form which is hard to detect optically , or massive neutrinos @xcite . however , the option that clusters contain non - baryonic cold dm between the galaxies , while logically possible , seems hardly justifiable in view of mond s overall philosophy . what is the physical basis of mond s success in unifying a lot of extragalactic data ? there are certain things the mond formula can not be . first , it is unlikely to be just a recipe for the way dm is distributed spatially in astronomical objects , as sometimes proposed @xcite . such proposals strive to explain how the scale @xmath37 enters into spiral galaxy properties through some regularity in the cosmogony of dm halos . but strong arguments exists against an intrinsic scale of the halos @xcite . and it is hard to see how such an explanation could give an account of the multiple roles of @xmath37 in different systems @xcite . the mond formula , conceived as _ exact _ , can not be a generic modification of the inertia aspect of newton s second law @xmath60 ( here @xmath61 is the sum of _ all _ forces acting on the particle including the newtonian gravity force ) , an alternative proposed originally by milgrom @xcite ( see a latter elaboration in ref . ) . to see why consider a binary system with unequal masses @xmath19 and @xmath62 evolving under mutual gravity alone . the time derivative of @xmath63 , as calculated from eq . ( [ mond ] ) , does not vanish in general if at least one of the accelerations is comparable to or smaller than @xmath37 , since the @xmath36 s will generally not be equal . so in the proposed interpretation , the mond formula does not conserve momentum . by the same token it does not conserve angular momentum , nor energy . milgrom sidesteps the problem by stipulating that the mond formula is only valid for test particles moving on a given background , e.g. , stars moving in the collective gravitational field of a galaxy . we may inquire more generally , does the mond formula , or some closely related one , represent a modification of inertia in test particle motion ? to comply with the conservation laws we likely want the formula to arise from a lagrangian . the kinetic part of the lagrangian should give us the @xmath64 part with whatever `` corrections '' are required . however , it is a theorem that no mond - like dynamics exists that simultaneously has a newtonian limit for @xmath65 ( all accelerations are large ) , is galilei invariant , and is derivable from a _ local _ action @xcite . ( `` local '' means that the relevant lagrangian can be written as a single integral over volume . ) this prohibition is even more stringent from a relativistic standpoint @xcite . accordingly milgrom introduced a _ nonlocal _ action , i.e. one which is a functional of complete orbits , but can not be reduced to an integral over a lagrangian @xcite . this approach does not quite reproduce formula ( [ mond ] ) generically , but that formula is recovered for the special case of circular orbits . of course circular stellar orbits _ a la _ mond is really all that one needs to analyze data on spiral galaxies . however , it is known that elliptical galaxies comprise highly radial stellar orbits too , yet elliptical galaxies also seem to be well described by mond @xcite , and so the nonlocal modified inertia approach might be found wanting here . on the plus side one may mention the pioneer anomaly @xcite . the pioneer 10 and pioneer 11 spacecraft , as they travelled almost radially in the outer parts of the solar system , were found to be subject to an anomalous sunward acceleration of order @xmath66 ( not far from @xmath37 ) which does not fall off measurably with distance from the sun . were such an acceleration to reflect a generic gravitational field , it would have affected the outer planet ephemerides to an intolerable extent @xcite . ( an ephemerides is the calculated position of a celestial body as a function of time , past or future . ) by contrast , milgrom s nonlocal theory does predict different modifications of newtonian dynamics for radial ( pioneer ) and circular ( planetary ) orbits , though details have yet to be worked out @xcite . milgrom has also speculated @xcite that modified inertia may have its origin in an effective interaction of bodies with the vacuum . this is motivated by the well known fact that an object swimming through a fluid has its inertial mass increased by interacting with fluid degrees of freedom . of course , by local lorentz invariance inertial motion through the vacuum is possible as usual . however , accelerated motion is distinguished from inertial motion ( even before we come to inertia ) by the perception that the accelerated system is immersed in a thermal bath whose temperature is proportional to the acceleration ( unruh radiation @xcite ) . accordingly , milgrom suggests that accelerated motion with respect to the vacuum may shape the inertial characteristics of objects in a way compatible with mond . an alternative speculation @xcite is that mond style modified inertia and its particular scale @xmath37 may be generated by a symmetry in a higher dimensional spacetime , just as we think of ordinary ( newtonian or einstenian ) inertia as reflecting lorentz symmetry in minkowski spacetime . in one implementation of this program , with the extremely symmetric desitter spacetime playing the role of the embedding spacetime , a relation between the cosmological constant and @xmath37 surfaces , which is actually approximately satisfied if the observed acceleration of the hubble expansion is interpreted as reflecting a nonzero cosmological constant . what if , instead of the modified inertia interpretation , we follow milgrom s alternative proposal @xcite to regard mond as a modification of newtonian gravity ? this entails a change in the r.h.s . of @xmath60 , with the new gravitational force possibly depending nonlinearly on its sources can such implementation be protected from violation of the conservation laws ? an affirmative answer to this query is readily available by starting from a modification of the lagrangian of newtonian gravity with milgrom s @xmath37 playing the role of characteristic scale , to wit @xmath67d^3x . \label{lagrangian}\ ] ] here @xmath68 is some positive function , @xmath69 is the matter s mass density and @xmath70 , the field in the theory , is to be identified with the gravitational potential that drives motion , i.e. @xmath71 . the lagrangian s kinetic part is the most general one that depends only on first derivatives of @xmath70 , and is consistent with the isotropy of space ( only @xmath72 appears ) . this aquadratic lagrangian theory ( aqual ) @xcite reduces to newton s in the limit @xmath73 , when it readily leads to poisson s equation for @xmath70 . more generically it yields the equation @xmath74&=&4\pi g\rho , \label{aqual } \\ \mu(\surd x ) & \equiv & f'(x ) . \label{f}\end{aligned}\ ] ] one identifies the @xmath36 function here with mond s @xmath36 . comparison of the aqual equation ( [ aqual ] ) with poisson s for the same @xmath69 shows that @xmath75 where @xmath76 is some calculable vector field . now in highly symmetric situations ( spherical , cylindrical or plane symmetry ) , use of gauss theorem with the aqual equation shows that @xmath77 has to vanish . in light of the relation @xmath78 , aqual then reproduces the mond formula ( [ mond ] ) exactly . however , the curl term does not usually vanish in situations with lower symmetry ( milgrom and brada exhibit a curious exception @xcite ) . we conclude that the mond formula is generically an approximation to the solution of the aqual equation . how good an approximation ? work by milgrom and brada @xcite has shown that often the curl is just a 5 - 10% correction . this explains how the mond formula manages to be so successful while not constituting by itself a physically consistent theory . now almost all mond fits to observed rcs are calculated using that formula exclusively . it should thus be clear that modest departures of empirical rcs from mond s predictions do not constitute evidence against the paradigm . the era of rc modeling with the full aqual eq . ( [ aqual ] ) is still in its early stages @xcite . i mentioned in sec . 4 that the mond formula is inconsistent with the conservation laws . the aqual theory is free of this problem . this needs no special proof ( although explicit proofs exist @xcite ) : a theory based on a lagrangian which does _ not _ depend _ explicitly _ on coordinates , directions and time ( all true for aqual s lagrangian ) automatically conserves momentum , angular momentum and energy . at the level of the first integral ( [ integral ] ) of the aqual equation , it is the curl term which protects aqual gravity from violating the conservation laws . aqual consistently embodies the weak equivalence principle ( all bodies , regardless of inner structure , and starting from the same initial conditions , follow the same path in a gravitational field ) . as remarked by milgrom @xcite , the mond formula is ambiguous in this respect . in a star accelerations of individual ions greatly exceed @xmath37 in magnitude . thus for each ion @xmath36 should be very close to unity , and we would guess that @xmath79 for the star as a whole . accordingly , if regarded as a collection of ions , the star would follow a newtonian orbit in the galaxy s field . this conclusion is entirely at variance with the evidence from the flat rcs , or from the dictate of the mond formula when applied to the star regarded as structureless . aqual predicts that the center of mass ( cm ) of a collection of particles , no matter what the internal gravitational field , will follow the orbit determined by @xmath80 with @xmath70 the external potential coming from the aqual equation @xcite . as mentioned , this acceleration will approximate that predicted by the mond equation . aqual does indeed supply a physical basis for mond . a system with sub-@xmath37 internal gravitational field immersed in an external field @xmath81 is predicted by aqual to exhibit quasi - newtonian internal dynamics @xcite . originally milgrom conjectured this `` external field effect '' to explain why no acceleration discrepancy is evident in the loose open star clusters residing in the milky way @xcite : the galaxy s field , still stronger than @xmath37 in its inner parts , suppresses the mond behavior expected in light of the weak internal fields . by contrast , if the said system is immersed in an external field @xmath82 , which is , however , _ stronger _ than its internal fields , the internal dynamics is predicted to be quasi - newtonian , but with the effective gravitational constant @xmath83 @xcite . a useful product of aqual is a formula for the force between two masses in motion about each other , e.g. a binary galaxy . the mond formula can not be used directly here because , as mentioned in sec . 4 , it would suggest that the binary s center of mass should accelerate ( nonconservation of momentum ) . instead , @xmath70 has to be computed from the aqual equation , and then @xmath84 has to be integrated over one galaxy s volume to obtain the force on it , as effectively done by milgrom @xcite . for point masses @xmath19 and @xmath62 separated by distance @xmath15 , the force is @xmath85 with @xmath86^{1/2}$ ] ; milgrom gives the graph of @xmath87 . in the newtonian limit ( @xmath88 ) , @xmath89 . in the deep mond limit ( @xmath90 ) , milgrom @xcite shows analytically that the force is @xmath91 $ ] . this is most elegantly computed by exploiting the discovery that in that limit the aqual equation is conformally invariant , an approach that also yields the exact virial relation in the deep mond regime @xcite . aqual permits a study of galaxy disc stability in the mond paradigm . i mentioned in sec . i that newtonian disk instabilities motivated the idea that dm halos surround disk galaxies . in the mond paradigm there is not dm , so how do a fraction of the spiral galaxies in the sky avoid forming bars ? using a mixture of mond formula and aqual equation arguments , milgrom @xcite showed analytically that mond enhances local stability of disks against perturbations as compared to the newtonian situation . brada and milgrom @xcite combined an @xmath92-body code with a numerical solver for the aqual equation to verify this and to demonstrate that mond enhances global stability . both works conclude that the degree of stabilization saturates in the deep mond regime ; this means no disk in mond is absolutely stable . christodoulou @xcite and griv and zhytnikov @xcite reach similar conclusions with @xmath92-body simulations based on just the mond formula . recently tiret and combes @xcite studied _ evolution _ of bars in spiral disks using @xmath92-body simulations built on the aqual equation . they find that bars actually form more rapidly in mond than in dm halo newtonian models , but that mond bars then weaken as compared to those in newtonian models . the long term distribution of bar strengths predicted by mond is in better agreement with the observed distribution than that obtained from newtonian halo models . aqual is also important for studies of two - body relaxation in stellar systems . two - body relaxation is the process whereby distant gravitational encounters of pairs of stars in a galaxy or a cluster redistribute the system s energy and help it to approach some sort of equilibrium . in newtonian theory two - body relaxation is excruciatingly slow . but as binney and ciotti show @xcite , two - body relaxation in the framework of aqual proceeds faster . in fact , if the system is deep in the mond regime , the speedup factor is the square of the acceleration discrepancy . a related effect is dynamical friction , whereby a massive object moving through a collection of lighter bodies , e.g. a globular star cluster coursing among the stars of its mother galaxy , loses energy by virtue of its kicking the background objects gravitationally as it sweeps by them . as a result the heavy object settles in the gravitational well confining it . in aqual for systems deep in the mond regime , dynamical friction is speeded up over the newtonian value by one power of the acceleration discrepancy @xcite . both of the above phenomena open a new window onto the mond - newtonian gravity dichotomy . _ suppose _ aqual is the correct nonrelativistic description of gravity . in regard to rcs of spiral galaxies , newtonian theory with dm could be `` saved '' by imagining the dm in each galaxy to be distributed in just such a way that the resulting total @xmath93 is just the @xmath70 generated in aqual theory by the baryonic ( visible ) matter . admittedly , the required dm distribution might be peculiar ( negative density somewhere @xcite , or incompatible with cosmogonic simulations ) . but leaving these options aside , observational uncertainties might make it difficult to tell the two theories apart , as recent fits of rcs of nearby galaxies based on high quality data testify @xcite . yet consideration of the mentioned dissipative effects might remove the effective degeneracy between the two models of a galaxy offered by the two theories . in view of the generic prediction of newtonian cosmogonic simulations that the dm distribution in a galaxy rises to a cusp at the center ( see sec . i ) , newtonian dynamical friction should have caused the globular clusters of many a dwarf spheroidal galaxy to already have merged at its center and there lost their integrity @xcite . while globular clusters are indeed absent from several such dwarf companions of the milky way , e.g. draco , sextans and carina @xcite , the fornax dwarf spheroidal has five globular clusters near , but _ not at _ , its center . goerdt , moore , et al . @xcite regard this as showing that the dm near the core of fornax has a flat density profile , in contradiction to the dm simulations . in fact , there is other growing evidence negating dm density cusps @xcite . now , some of the dwarf spheroidal satellites of the galaxy including fornax are in the deep mond regime . following binney and ciotti @xcite , snchez - salcedo , reyes - iturbide and hernandez @xcite point out that the mentioned speedup of settling by dynamical friction in aqual should long ago have caused the globular clusters in fornax to amalgamate at its center . aqual thus has a problem explaining the observations of fornax ; however , as we see the dm alternative has its problems too . can mond give a picture of the late stages of galaxy formation ? a step in this direction was taken by nipoti , londrillo and ciotti @xcite on the basis their earlier work with @xmath92-body codes incorporating the full aqual equation @xcite . they simulate the dissipationless collapse of a cloud of particles which ends up in the mond or deep mond regimes . they note that phase mixing ( the disappearance of structure in the initial conditions due to commingling of particle orbits ) proceeds slower than in the counterpart newtonian simulations , and that the final velocity distribution is anisotropic and favors radial orbits . current thought in cosmogony is that elliptical and spheroidal galaxies are the outcome of anisotropic collapse _ after _ star formation is complete . nipoti et al . note that the end - products of some of their simulations resemble , in their surface density and velocity dispersion profiles , the dwarf - elliptical and dwarf - spheroidal galaxies in the sky . but other final states have trouble fitting in all the accepted empirical correlations between luminosity , radius and velocity dispersion of elliptical galaxies . why is a relativistic version of mond important ? although most astronomical systems with significant acceleration discrepancy are nonrelativistic in the extreme , there are two important exceptions . cosmology , which is important in itself and as a framework for the process of galaxy formation , is widely regarded as requiring the presence of much dm on the largest scales , and cosmology is universally believed to require relativistic treatment . gravitational lensing by galaxies and clusters of galaxies has become a salient astronomical tool in the last two decades , and , since it discloses acceleration discrepancies , it is commonly regarded as supporting the need for dm . since light propagates with speed @xmath94 , nonrelativistic mond can not even begin to formulate the problem of gravitational lensing . the above cases drove the relativistic formulation of aqual @xcite , which i shall refer to as raqual . raqual retains einstein s equations of gr as a tool for deriving the spacetime metric @xmath95 from the energy - momentum tensor of the matter ( one of whose components represents matter density ) . raqual further stipulates that matter and radiation play out their dynamics , not in the arena of @xmath95 , but in that of @xmath96 , where @xmath97 denotes a scalar field with dimensions of squared velocity . since all matter is treated alike in this respect , the exquisitely tested weak equivalence principle is safeguarded , but since the metric governing gravitational field dynamics is different from the one experienced by matter , the strong equivalence principle fails . this is essential , for otherwise we should end up back with gr as the relativistic gravity theory , and gr in the nonrelativistic limit leads to newtonian theory , not to mond . how is @xmath97 determined in raqual ? one takes the lagrangian for @xmath97 to be the covariant version of that for @xmath70 in aqual , eq . ( [ lagrangian ] ) . this means replacing @xmath98 as well as @xmath99 , where @xmath100 is the determinant of the matrix of components of @xmath95 . but in raqual one does not include a term in lieu of @xmath101 ; the coupling between @xmath97 and matter , which is to provide the source of the equation for @xmath97 in accordance with its lagrangian , is automatically generated by the @xmath102 factor in the metric @xmath103 with which we built the _ matter _ s lagrangian . for time independent situations with nonrelativistic matter sources , the @xmath97 equation reduces to eq . ( [ aqual ] ) with @xmath104 . however , it would be a mistake to conclude that in raqual @xmath97 is the gravitational potential . in gr for weak gravity and nonrelativistic motion , the potential in which matter moves , @xmath93 , is related to the metric by @xmath105 ( @xmath106 is the time coordinate ) . because , in this linearized theory , the most relevant einstein equation reduces to poisson s , @xmath93 coincides with the usual newtonian potential . in raqual @xmath95 still satisfies einstein - like equations , so the above relation applies as well . but the stipulation that matter s dynamics go forward in the metric @xmath103 determines the nonrelativistic potential in which matter moves , now called @xmath70 , to be @xmath107 . writing in linearized theory @xmath108 , we have @xmath109 . now to first approximation @xmath110 ; thus to leading order @xmath111 consequently , in raqual the gravitational potential traced by the matter s motions is the sum of the newtonian potential and the aqual scalar field , both generated by the same baryonic matter . the @xmath97 does the job of the dm s potential in conventional approaches , but its source is ordinary matter . let us compare raqual with the successful aqual . we take the @xmath36 in eq . ( [ aqual ] ) with @xmath104 to be approximately equal to its argument for @xmath112 , and to level off at a value @xmath113 when @xmath114 . then for strong @xmath115 poisson s equations gives @xmath116 so that the true potential is @xmath117 . this means gravity is newtonian with effective gravitational constant @xmath118 . for @xmath112 , the raqual equation tells us that @xmath119 is much stronger than @xmath120 . consequently , by eq . ( [ sum ] ) , @xmath121 so that in the weak field regime raqual s gravitational field is close to aqual s . thus in raqual we have a relativistic theory which in the nonrelativistic limit inherits the good traits of mond by way of aqual . however , on the relativistic front raqual hit two roadblocks . already the original paper @xcite remarks that perturbations of @xmath97 from a static background for which @xmath112 , e.g. @xmath97 waves travelling in the reaches of a galaxy , propagate superluminally , i.e. , outside the light cone of @xmath103 . nowadays one hears the view that superluminality by itself is not sufficient reason for causal problems @xcite . this is by no means the standard view , and so the specter of acausality motivated disillusionment with raqual @xcite , and formulation of a two - scalar field theory , phase - coupled gravity ( pcg ) @xcite intended to prevent superluminality . in the pcg lagrangian the new scalar @xmath122 contributes a quadratic kinetic part , just as does @xmath97 ; there is also a simple coupling between @xmath122 and the kinetic term for @xmath97 which gives the theory its mond character . although pcg generates an intolerable drift of the kepler constant @xmath123 in the solar system , and marginally contradicts the observed precession of mercury s perihelion @xcite , the theory has merits for both galaxy dynamics and cosmology @xcite . however it may be , raqual and pcg were both finally disqualified as consistent representations of gravity by their inability to cope with observations of gravitational lenses . discovered in 1979 , gravitational lenses have been intensively studied ever since . in strong gravitational lensing a cluster of galaxies ( but sometimes a single galaxy ) forms a few images of a single quasar in the background by bending the incoming light rays with its gravitational field . this provided a novel tool for determination of masses of large astronomical systems by an elementary use of gr . by the late 1980 s it was apparent that masses of clusters so determined were larger than the observed baryonic mass in them , and commensurate with the masses determined from motions of the galaxies comprising them . the novelty was that the acceleration discrepancy was now put in evidence by a relativistic phenomenon . many pounced upon the finding as added support for dm s presence in clusters . raqual can not deal with the new evidence ; the reason is instructive . maxwell s equations are known to be conformally invariant : replacement of the metric @xmath95 with which they are formulated by @xmath124 does not change the form of the equations . it follows that study of light propagation can not distinguish between the metrics @xmath95 and @xmath103 of raqual . accordingly , although maxwell s equations are supposed to be written with @xmath103 , the metric for studying gravitational lensing might just as well be @xmath95 . but this last obeys einstein s equations whose sources are the baryonic matter s energy - momentum tensor and that of @xmath97 . however , for systems as distended as clusters or galaxies , this last is negligible . the light bending , then , is predicted by raqual to be nearly the same as predicted by gr _ without _ dm . yet there is observational evidence that the inner parts of clusters gravitationally lense light more strongly than would be expected from gr _ with no _ dm . the same problem recurs in pcg , in which the two metrics are , again , conformally related in the same manner as in raqual . it is thus quite clear that if a scalar field is to be the vehicle for mond effects , then regardless of the form of its dynamics , the relation between the metrics @xmath103 and @xmath95 must be non - conformal . i attempted @xcite to construct such a coupling out of scalar field alone , to wit @xmath125 with @xmath126 and @xmath127 functions of the invariant @xmath128 . however , sanders and i found that avoidance of propagation of gravitational waves which are superluminal with respect to the matter metric @xmath103 requires that @xmath129 , while @xmath130 is required so that both metrics have lorentzian signatures @xcite . we then calculated that for equal sources of einstein s equations , the light bending in the proposed modification of raqual is _ weaker _ than that in gr . as mentioned , in systems of the scale of galaxies and clusters of galaxies , the scalar field provides very little energy or momentum density as compared to the matter . thus in regard to light ray bending , the new theory performs worse than raqual . actually the above conclusion could be avoided if @xmath131 is a _ timelike _ 4-vector . however , we would expect that deep inside or near a galaxy , or a massive cluster of galaxies which is certainly virialized , the scalar field , whose principal source is the matter therein , should be quasistatic . then @xmath131 is expected to be spacelike , i.e. to give lensing weaker than needed . the problem is thus still present . to overcome it sanders @xcite proposed to replace @xmath131 in its above role by a nondynamical vector @xmath132 which in an isotropic cosmological model points precisely in the time direction , and approximately so in the presence of mass concentrations . he further related the metrics by @xmath133 sanders further supposed the action for the scalar field @xmath134 , which supersedes @xmath97 , to be of raqual form , or more generally with the function @xmath68 depending also on the scalar @xmath135 . the timelike nature of the vector is conducive to enhancement of the lensing , and such a theory , named stratified theory , can cope with the observations of gravitational lensing . however , it was clear to sanders that this is just a toy theory since a globally timelike vector field , such as @xmath136 , determines a universal preferred frame of reference , an aether . additionally , since the vector field is supposed constant , the theory is not covariant . the mentioned problems are avoided by making the vector field a dynamical one , as first done in t@xmath0v@xmath137s , my tensor - vector - scalar theory of gravity @xcite . t@xmath0v@xmath137s retains the relation ( [ gtilde ] ) between a gravitational metric @xmath95 and a physical ( or observable ) metric @xmath103 . the matter lagrangian is built exclusively with @xmath103 ; this guarantees that the weak equivalence principle will be obeyed precisely . the conventional gr einstein - hilbert lagrangian is used to give dynamics to @xmath95 , which then trivially induce dynamics for @xmath103 . the scalar field @xmath134 is provided with a raqual type lagrangian . for convenience this is couched in terms of a quadratic form in the 4-gradient of @xmath134 in interaction with a second scalar field , @xmath122 , which also occurs in a potential - like term . ( however , no derivatives of @xmath122 enter , so @xmath122 is not dynamical , and if eliminated , the explicit aquadratic lagrangian for @xmath134 shows up . ) the mentioned quadratic form is not the usual one , but rather @xmath138 ; it is introduced to forestall the superluminal propagation that afflicts raqual . the kinetic part of the @xmath136 lagrangian is quadratic in its first derivatives , and exactly analogous to that for a gauge field . this is not the only quadratic form possible , but it is the one in which the einstein equations for @xmath95 will not have second derivatives of @xmath136 in their sources . as mentioned , it is imperative that @xmath136 be a timelike vector . in t@xmath0v@xmath137s this is accomplished by introducing , alongside the kinetic term , a lagrange multiplier term of the form @xmath139 which forces the norm of @xmath136 to be negative ( and @xmath136 to be of unit length as a bonus ) . whereas raqual is a one parameter ( @xmath37 ) theory with one free function ( @xmath36 ) , t@xmath0v@xmath137s has one free function , @xmath140 [ distinct from the @xmath68 in eq . ( [ lagrangian ] ) ] that determines the _ shape _ of the potential - like term , and three parameters in addition . one of these is a scale of length @xmath32 in the @xmath134 lagrangian that sets the strength of the potential , while a dimensionless one , @xmath141 , determines the scale of its argument . the third parameter , @xmath142 , also dimensionless , sets the strength of the @xmath136 lagrangian . one can form the acceleration scale @xmath143 which parallels the role of milgrom s constant in mond and aqual . it is found that the limit of t@xmath0v@xmath137s with @xmath144 , @xmath145 and @xmath146 amounts to gr . in fact @xmath65 in this limit , so that all accelerations are to be regarded as strong , and gravitation as conventional . how about nonrelativistic motion ? let @xmath147 be the coeval value of @xmath134 in the cosmological model in which the nonrelativistic system is embedded . this will be the asymptotic boundary value of @xmath134 for any local solution of the @xmath134 equation . from a linearized version of einstein s equations , and use of the relation ( [ gtilde ] ) , it follows @xcite that the effective nonrelativistic gravitational potential is @xmath148 ( this corrects a sign error in refs . and ) . here @xmath93 comes from the usual poisson equation while @xmath134 is found from the scalar equation ; in both the source is the baryonic mass density @xmath69 . thus far all studies have assumed that @xmath149 and @xmath150 , in which case eq . ( [ sum2 ] ) essentially coincides with eq . ( [ sum ] ) in raqual . for this same reason any time variation of @xmath151 @xcite should be negligible . this is of particular significance for the above derived value of @xmath37 which , strictly speaking , includes a factor @xmath151 [ for example eq . ( 6.7 ) of ref . ] . for a static situation the equation for @xmath134 takes a form like eq . ( [ aqual ] ) , with @xmath36 ( essentially @xmath152 ) expressed in terms of @xmath153 through t@xmath0v@xmath137s s free function @xmath140 . by a suitable choice of @xmath68 one can reproduce any @xmath36 function that would be relevant in aqual , with the scale thereon agreeing with @xmath37 in eq . ( [ a0 ] ) . by analogy with our discussion of raqual we see that t@xmath0v@xmath137s can have mond phenomenology and also a newtonian limit . the transition between newtonian and mond regimes takes place as @xmath153 sweeps through the value @xmath37 , near the point where @xmath154 . actually for systems strongly departing from sphericity , the situation is not as clear , but it seems that at the nonrelativistic level t@xmath0v@xmath137s implements the mond paradigm overall . when it comes to details , my original choice for @xmath68 @xcite can be criticized : while it leads to flat outer rcs , it does not give a satisfactory account of the transition part of the rcs of several well measured galaxies , including our own @xcite . by contrast , the mond formula with the `` simple '' variant of migrom s @xmath36 ( fig . [ figure : fig3 ] ) leads to very good fits for many rc s @xcite . forms of the t@xmath0v@xmath137s function @xmath68 which would do a correspondingly good job have been proposed by zhao and famaey @xcite , sanders @xcite and famaey , et al . @xcite . sanders has also proposed a variation on t@xmath0v@xmath137s involving a second scalar @xmath122 with pcg - like dynamics instead of raqual ones @xcite . this bi - scalar tensor vector theory ( bstv ) has three free functions and a free parameter . the theory is not especially needed to obviate superluminal propagation since t@xmath0v@xmath137s seems to do well on this @xcite . however , bstv , like pcg , is a more appropriate frame for generating cosmological evolution of @xmath37 . since numerically @xmath155 , it is often argued that @xmath37 must be determined by cosmology , and should thus vary on a hubble timescale @xcite . as mentioned , @xmath37 in t@xmath0v@xmath137s is essentially set by the parameters @xmath32 and @xmath141 ; it should be nearly constant in the expanding universe . it is otherwise in bstv . discrimination between constant and evolving @xmath37 may be possible with good rcs of disk galaxies at redshifts @xmath1565 . such curves are just now coming into range . in fact , the data in ref . put the @xmath157 galaxy bzk-15504 right on the mond derived tully - fisher law with the standard value for @xmath37 @xcite . thus the meagre data available today are consistent with no @xmath37 evolutions on the hubble timescale . recently zlosnik , ferreira and starkman @xcite have clarified the relation between t@xmath0v@xmath137s and the so called einstein - aether gravity theories in which a timelike unit vector field ( but no scalar ) plays a role alongside the metric @xcite . they do this by re - expressing t@xmath0v@xmath137s solely in terms of the observable metric @xmath103 and the vector field . the @xmath134 is eliminated with the help of the constraint imposed through the lagrange multiplier term . there emerges an einstein - aether like theory in which the metric @xmath103 also satisfies einstein - like equations . however , in contrast to orthodox einstein - aether theories , the vector kinetic action in the theory in question is a generic quadratic form in the vector s derivatives contracted into polynomials of the vector s components . this form of t@xmath0v@xmath137s is more complex than the original one , but there are some circumstances in which it may be more convenient in ferreting out consequences of the theory . in t@xmath0v@xmath137s in either form the vector is normalized with respect to @xmath95 , not @xmath103 . zlosnik , et al . have also proposed a variant tensor - vector theory with a timelike vector normalized with respect to @xmath103 , and which also has mond like behavior @xcite . this is constructed by taking the vector s action as a function @xmath158 of @xmath159 , the quadratic form in the derivatives of the vector field of orthodox einstein - aether theories . the zlosnik , et al . theory has four parameters : a length scale and three dimensionless parameters . the form of the free function @xmath160 can be deduced approximately from the requirement that mond arise in the nonrelativistic quasistatic limit , and from the stipulation that a cosmology built on this theory shall have an early inflationary period and an accelerated expansion at late times . in summary , the relativistic implementations of mond involve either one or three free functions ; those with one free function have either three or four free parameters . as stressed earlier , the particular structure of t@xmath0v@xmath137s reflects the desire to encompass in the mond paradigm the observation that what passes , from a dynamical point of view , for dm in galaxies and clusters of galaxies also lenses light to a commensurate degree . how does t@xmath0v@xmath137s measure up to the task ? now the measured gravitational lensing by galaxies and clusters of galaxies takes place over cosmological distances . the light rays in t@xmath0v@xmath137s are null geodesics of the observable metric @xmath103 ; to compute @xmath103 one needs @xmath161 and @xmath95 . isotropic cosmological models in t@xmath0v@xmath137s closely resemble the corresponding cosmological models of gr ; they sport a @xmath162 pointed precisely in the time direction and feature very slow change of @xmath134 @xcite . a consequence is that @xmath103 in a t@xmath0v@xmath137s isotropic cosmological model differs little from the metric in the corresponding gr cosmology . thus the cosmological facet of gravitational lensing is very much like in gr . the second facet is the local bending of light rays in a mass vicinity . as in isotropic cosmological models , so in static situations , the solution of the vector s equation has @xmath162 pointed precisely in the time direction . to compute the bending in linearized theory , the approximation commonly used in the business , one also needs the scalar field @xmath134 and the metric @xmath95 , both to first order in @xmath93 . the final result is that the line element takes the form @xmath163 with @xmath70 given by eq . ( [ sum2 ] ) . note that the same potential @xmath70 appears in both terms of this isotropic form of the line element . hence light ray bending , which leans on both to equal degree , measures the same gravitational potential as do dynamics which are sensitive only to the temporal part of the line element . this mirrors the situation in gr whose line element is obtained by sending @xmath164 in eq . ( [ le ] ) . thus half of the problem that plagues raqual and similar theories is overcome : the acceleration discrepancy makes itself felt equally through the gravitational lensing as through the dynamics . the second half of the problem revolves about the mass distribution that generates @xmath70 . in gr @xmath70 is all there is , and its laplacian , as determined from the lensing observations or from the dynamics , will give the _ total _ ( baryonic plus dark ) mass distribution directly . because dm is not seen directly , this prediction can only be judged by the plausibility of the derived distribution of dm . by contrast in t@xmath0v@xmath137s the measured @xmath70 is to be decomposed into two parts in the manner of eq . ( [ sum2 ] ) , with each generated by the same _ baryonic _ mass density @xmath69 , one part through poisson s equation , and the second through a highly nonlinear aqual type equation . evidently , in a t@xmath0v@xmath137s model of a gravitational lense , the baryonic matter will be distributed differently from the total matter in a gr model of the same lense . and when the lensing system is not spherically symmetric , the centers of the two distributions may be offset . chiu , ko and tian @xcite have explored light ray bending by a pointlike mass @xmath24 in t@xmath0v@xmath137s . they note that the deflection angle in the deep mond regime [ impact parameter @xmath165 approaches a constant , as might have been expected from naive arguments , but is less predictable in the intermediate regime @xmath166 . thus calculations of lensing based on a mixture of mond and gr motives @xcite can easily mislead . chiu et al . work out the lens equation in t@xmath0v@xmath137s , which controls the amplifications of the various images in the strong lensing of a distant source , and remark that for two images the difference in amplifications is no longer unity as in gr , and may depend on the masses . with a photometric survey like the sloan digital survey it may be possible to check for this effect . finally , these authors work out the gravitational time delay in t@xmath0v@xmath137s , which could be of use in interpreting differential time delays in doubly imaged variable quasars . more phenomenologically oriented , zhao , bacon , taylor and horne @xcite compare t@xmath0v@xmath137s predictions with a large sample of galaxy strong lenses which each produce two images of a quasar . they model the galaxy baryonic mass distribution either by point masses or by the popular hernquist profile . galaxy masses are estimated by comparing observations both with predicted image positions and with predicted amplifications ratios ; the two methods are found to give consistent results , themselves well correlated with the luminosities of the galaxies . the corresponding mass - to - light ratios are found to be in the normal range for stellar populations , with some exceptions . how frequently should strong lensing occur ? this question is taken up in the context of t@xmath0v@xmath137s by chen and zhao @xcite . again modeling the mostly elliptical galaxies with hernquist profiles , they compute the probability of two images occurring as a function of their separation . their prediction falls somewhat beneath the frequency observed in the lensing surveys , though they consider this still acceptable . the predictions are sensitive to the assumed mass profile , as well as to the assumed shape of the @xmath36 function , which chen and zhao assume to switch brusquely from linear in the argument to unity . gravitational lensing by the colliding galaxy clusters 1e0657 - 56 has been claimed to give theory independent proof of dm dominance at large scales @xcite . in this system ( fig . [ figure : fig6 ] ) a smaller cluster , the `` bullet '' , has crashed through a larger one and the intracluster gas of both has been stripped by the collision , the bullet s gas trailing behind its galaxy component . weak lensing ( distorted but unsplit images ) mapping shows the lensing mass to be concentrated in the two regions containing the galaxies , rather than in the two clouds of stripped gas which contains the lion s share of baryonic mass @xcite . collisionless dm would indeed move together with the galaxies . hence the inference that much dm continues to accompany the bullet . angus , famaey and zhao @xcite ( see also ref . ) point out that in the very asymmetric system 1e0657 - 56 , mond , aqual and t@xmath0v@xmath137s all predict substantially different gravitational field distributions ( compare eqs . ( [ mond ] ) and ( [ integral ] ) ) , a situation which confuses clowe et al.s `` theory independent '' inference . whereas angus , famaey and zhao consider it possible to explain the lensing with a reasonable purely baryonic matter distribution , a later paper by angus , shan , et al . @xcite concludes that dark matter is needed after all . this is hardly surprising ; as we saw in sec . 3 , pure mond does not fully account for the acceleration discrepancy in the dynamics of quiescent galaxy clusters @xcite . but dm models of the bullet clusters within gr are not without their problems . farrar and rosen @xcite note that the relative velocity of the clusters is too high as compared to those seen in dm simulations of structure formation . to remove the contradiction they propose that a non - gravitational attraction of a new sort acts only between clumps of dm . but is assuming existence of dm together with a new interaction specific to it more parsimonious than a modification of standard gravity such as mond ? with my original choice of @xmath68 , t@xmath0v@xmath137s cosmological models with baryonic matter content alone can be very similar to the corresponding gr models by virtue of the scalar @xmath134 s energy density remaining relatively small @xcite . in particular chiu et al . @xcite note that these t@xmath0v@xmath137s models give a reasonable relation between redshift and angular distances , and are thus as effective as gr models in providing the scaffolding for the analysis of cosmologically distant gravitational lenses . how would changing @xmath68 affect the evolution of a cosmological model ? this is studied exhaustively by bourliot et al . @xcite following an earlier exploration by skordis et al . bourliot et al . display a large set of variants of my @xmath68 for which , while the scalar energy density may not be negligible , it tracks or mimicks the behavior of another energy component of the model , e.g. the radiation s during radiation dominance . they also characterize shapes of @xmath68 which can lead to future singularities in cosmology , and which presumably should be avoided . mond critics have always held up the complicated power spectrum of cosmological perturbations ( background radiation or baryons ) as a proof that dm is needed on the cosmic level ; after all the spectrum is said to be well fit by the `` concordance '' dm model of the universe . this argument took for granted that mond could never measure up to the test . with t@xmath0v@xmath137s on the scene one can face the question technically . in a massive work skordis has provided the full covariant formalism for evolution of cosmological perturbations in t@xmath0v@xmath137s @xcite . and using this skordis et al . @xcite have shown that , without invoking dark matter , t@xmath0v@xmath137s can be made consistent with the observed spectrum of the spatial distribution of galaxies and of the cosmic microwave radiation if one allows for contributions by massive neutrinos ( in the still allowed mass range ) and the cosmological constant . in this approach the role of dark matter in standard cosmology is taken over by a feedback mechanism involving the scalar field perturbations . dodelson and liguori @xcite have independently calculated perturbation growth , and stressed that it is rather the vector field in t@xmath0v@xmath137s which is responsible for growth of large scale structure without needing dm for this @xcite . it thus seems there is potential in t@xmath0v@xmath137s to do away with cosmic dm , as well as with dm in galaxies . much attention is commanded today by the mystery of the `` dark energy '' , the agent responsible for the observed acceleration of the hubble expansion . such agent is obligatory in the context of gr cosmological models . quite naturally many have wondered whether t@xmath0v@xmath137s could obviate this need . diaz - rivera , samushia , and ratra @xcite have found exact desitter solutions of t@xmath0v@xmath137s cosmology which can represent either early time inflation epochs or the late time acceleration era . in another t@xmath0v@xmath137s study , hao and akhoury @xcite have concluded that with a suitable choice of the t@xmath0v@xmath137s function @xmath68 , the scalar field can play the role of dark energy . and zhao @xcite maintains that with zhao and famaey s choice of @xmath68 @xcite , cosmological models can be had that evolve at early times like those of standard cold dark matter cosmology , and display late time acceleration with the correct present hubble scale , all this without assuming dm or dark energy . for the related einstein - aether theory , zlosnik et al . @xcite remark that with suitable choice of their theory s @xmath158 , the vector field can both drive early inflation as well as double for dark energy at late times . it may be unnecessary to go far from earth to tell t@xmath0v@xmath137s , bstv and similar theories apart from gr ; the solar system can be turned into a sieve for the correct modified gravity . between any two bodies in the solar system there is an extremum point , strictly a saddle point , of some relevant field . in aqual this would be the @xmath70 potential and in t@xmath0v@xmath137s the @xmath134 field . near such a point the gradient of the said field is small , and departures from newtonian gravity are significant ( @xmath36 in aqual falls short of unity , and the derivative of @xmath68 in t@xmath0v@xmath137s is driven away from the pole which signals newtonian behavior ) . a detailed study by magueijo and me @xcite with t@xmath0v@xmath137s shows that the anomalous regions , e.g. between sun and jupiter and earth and moon , are small , but gives some hope that with the fine guidance of space probes like esa s lisa - pathfinder ( scheduled for liftoff in october 2009 ) , it may be possible to diagnose mond - like effects . well away from the saddle points , but still in the region occupied by the planets , t@xmath0v@xmath137s predicts small departures of @xmath134 from @xmath16 form , i.e. small departures from newtonian behavior as encoded in @xmath70 @xcite . similar predictions issue from bstv @xcite . these are constrained by the observed constancy of kepler s constant @xmath123 out to the orbits of the major planets , and by the absence of significant departures from gr s predictions for the precessions of the perihelia of mercury and the asteroid icarus . according to sanders , t@xmath0v@xmath137s can satisfy these constraints with choices of the function the @xmath68 proposed by refs . and , which , as mentioned in sec . 7 , also lead to correct galaxy rcs . bstv can do just as well when its extra parameter , @xmath167 , is chosen properly @xcite . both t@xmath0v@xmath137s and bstv with the above choices predict that beyond 100 astronomical units ( au ) from the sun , there is a fairly @xmath15-independent attractive component of the sun s gravitational field of strength @xmath168 . the pioneer anomaly , as measured at distances beyond 20 au , is also @xmath15-independent but stronger . is this evidence for mond ? unfortunately , the interpretation of the pioneer anomaly as an almost constant sunward acceleration seems to clash with limits on the variation of kepler s constant set with spacecraft at the distance of uranus and neptune @xcite and with the latest ephemerides of the outer planets @xcite . it thus seems prudent to suspend judgment until a prosaic origin for pioneer anomaly , e.g. drag by unknown matter , can be excluded with high probability . the above focuses on the nonrelativistic limit . how do relativistic mond theories fare with regard to the lowest order relativistic effects in the solar system ? as for any metric theory of gravity , these effects can be calculated from the post - newtonian ( ppn ) parameters of the theory of choice . for any metric theory it is possible to parametrize the first and second order departures of the metric from minkowski form in terms of a set of ten intuitive looking potentials @xcite . the newtonian potential @xmath93 figures in the list , and occurs in the corrections both linearly as well as squared ; two others are @xmath169 where @xmath170 denotes the fluid velocity in the system . the post - newtonian ( ppn ) parameters are the dimensionless coefficients multiplying the diverse potentials in the correction . some examples will clarify this . the correction to the space - space part of the metric , @xmath171 , is @xmath172 ; it defines the ppn parameter @xmath3 . the correction to the temporal metric component , @xmath173 , starts with the terms @xmath174 , which define the ppn parameter @xmath175 , and also includes a @xmath176 term whose coefficient brings in a further ppn parameter , @xmath177 . each time - space metric component @xmath178 gets corrections proportional to the @xmath179 and @xmath180 ( @xmath181 denotes the velocity of the chosen coordinate system with respect to the cosmological matter comoving frame ) . with these last terms come two additional ppn parameters , @xmath182 and @xmath183 , which are called preferred frame parameters . a third one , @xmath184 , is associated with a correction of the form @xmath185 to @xmath173 . there are four more ppn parameters for a total of ten . the parameters @xmath175 and @xmath3 are both unity for gr and for some of it competitors . they have also been computed to be unity in t@xmath0v@xmath137s @xcite , which thus fares as well as gr in reference to gravitational light bending , perihelia precessions of the planets , and the radar time delay . sanders @xcite has provided a simple argument that @xmath175 and @xmath3 are always unity in a class of theories ( including bstv ) which start from the disformal relation ( [ gtilde ] ) . next in order of relevance are the preferred frame parameters @xmath186 and @xmath184 . these vanish in gr ; after all , gr has no preferred frames . in t@xmath0v@xmath137s or bstv at least one of the @xmath187s should be nonvanishing because the vector @xmath132 establishes a locally preferred frame , with the vector pointing out the time direction in that frame . sanders argues heuristically that the @xmath187s should be strongly suppressed in both theories @xcite . but explicitly computing the @xmath187 s for relativistic mond theories should be a high priority because they are subject to tight experimental bounds . the odyssey in search of a relativistic embodiment of the mond paradigm has led , not to one relativistic mond , but to many . t@xmath0v@xmath137s by itself exemplifies a family of theories . first there is the freedom in the choice of the function @xmath68 , which is still only modestly constrained @xcite . next , the coefficient of the second term in the factor @xmath188 in the scalar s lagrangian can be changed to any other negative number . in t@xmath0v@xmath137s the lagrangians for @xmath134 and @xmath136 are formulated upon the metric @xmath189 ; a different but related theory emerges if one uses instead the background of @xmath190 . additionally one can replace the aquadratic lagrangian for @xmath134 by a pcg - style lagrangian for two scalars , as indeed done in sanders bstv . finally , one can altogether dispense with the scalar fields and go the way of the zlosnik - ferreira - starkman theory @xcite . while the lack of uniqueness of relativistic mond is a nuisance , it does add needed flexibility to the search for a `` final '' fundamental theory with which to underpin the mond paradigm . it is already clear that some of the above mentioned theories may not be in full accord with the facts . for example , the published t@xmath0v@xmath137s has an exact mond limit at low accelerations , yet as mentioned in sec . 3 , mond can not handle the dynamics of clusters of galaxies without invoking additional unseen matter . an interesting escape is suggested by sanders @xcite . in bstv cosmology , just as in the pcg one @xcite , the @xmath122 field can undergo oscillations which generate bosonic particles early on . if the bstv parameters are right , these bosons can be trapped by clusters ( but not by galaxies ) , and can thus comprise the additional unseen matter . the charm of this resolution is that this new cold dark matter emerges from the modified gravity theory itself , and is not a separate invention . relativistic mond as here described has developed from the ground up , rather than coming down from the sky : phenomenology , rather than pure theoretical ideas , has been the main driver . actually a large industry flourishes on the sidelines with imaginative ideas from first principles regarding the essence of mond . i have not touched here on these motley approaches because they have given so little that is observationally viable . neither have i dwelt here on modified gravity theories that are not mond motivated or oriented . but the time may be ripe for turning to a more deductive approach to mond . now that we have some idea of what constitutes a viable relativistic mond theory , it should be easier to single out theoretical frameworks which might yield a promising candidate for the fundamental mond theory either as a limiting case , as an effective theory , after dimensional reduction , etc . to give an example , let us recall that brans - dicke modified gravity plus maxwellian electromagnetism in the real world can both be recovered by dimensional reduction of 5-d einstein gravity theory . t@xmath0v@xmath137s has the same number of degrees of freedom as that pair of theories ; specifically , its vector field has three degrees of freedom on account of the normalization condition , and the electromagnetic vector potential also has three on account of gauge freedom . might some variant of t@xmath0v@xmath137s arise from dimensional reduction of a pure gravity theory in 5-d ? if so , this would both ameliorate the common feeling that t@xmath0v@xmath137s is unduly complicated , and point the way to a lode for theories which might do better justice to the observations in the spirit of the mond paradigm . i thank mordehai milgrom , bob sanders and stacy mcgaugh for many useful remarks on the original manuscript , and the last two as well as douglas clowe for providing figures . research on this subject has been supported by grant 694/04 from the israel science foundation established by the israel academy of sciences and humanities . finzi , a. , 1963 , mon . not . soc . , * 127 * , 21 ; tohline , j. e. , 1982 , in _ internal kinematics and dynamics of galaxies _ , edited by a. athanassoula ( dordrecht : reidel ) , p.205 ; sanders , r. h. , 1984 , astron . astroph . , * 136 * , l21 ; and 1986 , astron . astroph . , * 154 * , 135 ; goldman , i. , 1986 , astron . astroph . , * 170 * , l1 ; kuhn , j. r. and kruglyak , l. , 1987 , astroph . , * 313 * , 1 ; and others . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ jacob bekenstein _ received his ph . d. ( 1972 ) from princeton university . he worked at the ben gurion university in israel , where he became full professor in 1978 and the arnow professor of astrophysics in 1983 . in 1990 he moved to the hebrew university of jerusalem where he is the polak professor of theoretical physics . a member of the israel academy of sciences and humanities , of the world jewish academy of sciences , and of the international astronomical union , bekenstein is a laureate of the rothschild prize and of the israel national prize . his scientific interests include gravitational theory , black hole physics , relativistic magnetohydrodynamics , galactic dynamics , and the physical aspects of information theory . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
|
no more salient issue exists in contemporary astrophysics and cosmology than that of the elusive `` dark matter '' . for many years already milgrom s paradigm of modified newtonian dynamics ( mond ) has provided an alternative way to interpret observations without appeal to invisible dark matter .
mond had been successful in elucidating economically the dynamics of disk galaxies of all scales , while doing less well for clusters of galaxies ; in its original form it could not address gravitational lensing or cosmology . after reviewing some of the evidence in favor of mond , i recollect the development of relativistic formulations for it to cope with the last deficiency .
i comment on recent work by various groups in confronting t@xmath0v@xmath0s , a relativistic embodiment of mond , with observational data on gravitational lensing and cosmology . throughout
i ask what sort of physics can be responsible for the efficacy of mond , and conclude with an appraisal of what theoretical developments are still needed to reach a full description of the world involving no unobserved matter .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
ever since the discovery of the first high - redshift quasar ( schmidt 1965 ) , quasars have maintained their title as the objects with the highest known redshift ; the present record holder is a quasar at @xmath6 ( schmidt , schneider , & gunn 1991 ) . nevertheless , the highest known redshifts of galaxies have followed closely behind , with bright radio galaxies having been found up to @xmath7 ( lacy et al . 1994 ; rawlings et al . 1996 ) ; more recently , galaxies with high star formation rates have been identified from interstellar absorption lines at @xmath8 to 3 ( steidel et al . 1996 ) and from the ly@xmath0 emission line at @xmath9 ( hu & mcmahon 1996 ) . in fact , if quasars are related to supermassive black holes that formed in the centers of high - redshift galaxies , we should expect that many galaxies already existed before the first quasars appeared . in any ` bottom - up ' theory where the observed structure in the universe forms by hierarchical gravitational collapse , and the primordial density fluctuations extend to sufficiently small scales , the first galaxies to form must have had much smaller masses than the present galaxies . the first stars should have formed in systems with velocity dispersions of @xmath10 or lower , corresponding to the lowest temperatures ( @xmath11 k ) that allow cooling and dissipation of the gas by atomic processes ( systems with even lower virial temperatures can cool and dissipate through molecular hydrogen , but this cooling process should be suppressed by photodissociation of the molecules after emission of a number of uv photons that is much smaller than that needed to reionize the universe ; see haiman , rees , & loeb 1996 ) . these systems would be very unlikely to form quasars , because even a small fraction of their baryons turning into stars should provide sufficient energy ( via ionization , stellar winds or supernovae ) to expel the remaining gas from the shallow potential well ( e.g. , couchman & rees 1986 ; dekel & silk 1986 ) . deeper potential wells , forming at later epochs , are probably needed to form supermassive black holes in galactic centers . even if all the baryons were converted into stars very efficiently in these early dwarf galaxies , with a baryonic mass @xmath12 , their total stellar luminosity would be much smaller than in @xmath13 galaxies at present , simply due to their small mass . since only a small fraction of the baryons in these systems is likely to turn into stars before the gas is ejected , the total stellar mass in the first galaxies to form in the universe should be much smaller than @xmath14 . this implies that a supernova in one of these first galaxies to form will be far brighter than the galaxy itself . thus , the brightest probes of the era when the reionization of the intergalactic medium ( igm ) started should be supernovae in very small galaxies , caused by the death of the same stars responsible for the first ionizing photons . in this paper , we shall estimate the number of supernovae that should have taken place in these galaxies and should be observable at very high redshift , and their apparent magnitudes . the scenario where these first small galaxies caused the reionization of the universe is strongly supported by recent evidence that the metal abundance in the ly@xmath0 forest absorption lines with @xmath15 is typically @xmath16 , from observations of civ lines ( see tytler et al . 1995 , songaila & cowie 1996 and references therein ) . there is relatively little uncertainty in the number of uv photons that were emitted by the stars that produced a given mass of heavy elements , because the heavy elements originate from the supernovae resulting from the same stars that emit most of the ionizing photons ( although the c / o ratio is more uncertain because carbon is more abundantly produced in lower mass stars ) . according to the most recent calculations ( madau & shull 1996 and references therein ) , the energy of lyman continuum photons emitted is 0.2% of the rest - mass energy of the heavy elements produced . thus , the energy emitted in ionizing photons per baryon is @xmath17 mev , where @xmath18 is the average metallicity of all baryons in the universe , so only @xmath19 is needed to have emitted one ionizing photon for each baryon . if @xmath20 , then 20 ionizing photons must have been emitted per baryon when the heavy elements were made . furthermore , if these were the photons responsible for reionizing the universe , then each baryon must have recombined 20 times on average during the reionization epoch . this is a reasonable number , because a fraction of these photons were probably absorbed in the systems where the stars were born before the gas was expelled , and those that escaped could also have been absorbed in lyman limit systems before the universe became transparent . thus , there is no need to invoke ejection of gas by more massive galaxies that can accrete the ionized igm to explain a metallicity @xmath21 . we can now calculate the number of supernovae that were required for enriching the gas in the igm to the average metallicity of @xmath22 that is observed in the ly@xmath0 forest at @xmath23 . this number should depend only on the igm density and the supernova yields , and should be independent of any other details related to the theory for galaxy formation and the type of galaxies that ejected the enriched gas . recent simulations of cold dark matter models show that the absorption lines in the ly@xmath0 forest can be identified with the igm , with density fluctuations caused by gravitational collapse , and that most of the baryons should be in the igm in these models ( cen et al . 1994 , hernquist et al . 1996 , miralda - escud et al.1996 ) . thus , it is reasonable to assume that the high - z supernovae enriched all the baryons in the universe to a mean at metallicity at least as high as that of the ly@xmath0 forest . since each supernova produces an average of @xmath24 of heavy elements ( with uncertainties depending on the assumed initial mass function and supernova models ; see woosley & weaver 1995 ) , this implies that a supernova took place at high redshift for each @xmath25 of baryons in the universe . we shall assume a high baryon density @xmath26 , in agreement with the primordial deuterium abundance measured by burles & tytler ( 1996 ) and by tytler , fan , & burles ( 1996 ) . notice that this implies that most of the baryons at the present time are dark , so many more baryons than those we observe in galaxies had to be enriched at high redshift . assuming the @xmath27 cosmological model , the total mass of baryons in a redshift shell of width @xmath28 around us is @xmath29 ^ 2/(1+z)^{3/2}\ , \delta z$ ] , where @xmath30 is the present hubble constant . with the above rate of supernovae per baryon mass ( assumed to take place within the epoch corresponding to the redshift shell @xmath28 ) , and taking into account that the supernovae within the shell would be seen by us over a time interval @xmath31 , the total supernova rate observed over all the sky is @xmath32 ^ 2 { \ , { \rm yr}}^{-1 } ~.\ ] ] or , for @xmath33 , about 1 supernova per square arc minute per year . most of these supernovae should be type ii which , if the progenitor is a red supergiant , have a plateau of the luminosity in their lightucurves from 1 to 80 days after the explosion , with @xmath34 erg / s ( woosley & weaver 1986 ) . a note of caution should be made here , in that the low - metallicity progenitors of these early supernovae could be very different from the high metallicity counterparts . as illustrated by the case of sn1987a , it is probably not possible at this stage to predict the type of supernovae we should expect from this first generation of stars ; in particular , the possibility that some supernovae might reach higher intrinsic luminosities than regular type iis should be kept in mind . a duration of 80 days would be redshifted to more than a year at redshifts @xmath35 , where the supernovae from stars responsible for the reionization of the universe should occur . at any random time there should therefore be one or more supernovae per square arc minute visible in the sky . of course , many more supernovae should have occurred in more massive galaxies at later epochs ( producing the metals in stars and interstellar gas at present ) , and possibly in small systems that ejected their gas and continued to enrich the igm . the main difficulty in detecting these supernovae is obviously the extremely faint flux expected . supernovae should be the brightest objects in the universe at very high redshift , but they are of course much fainter than quasars and , as the redshift increases , the bolometric fluxes decrease at least as rapidly as @xmath36 . one should point out , however , that when observing at a fixed , long wavelenth such that the supernovae are observed on the rayleigh - jeans part of the spectrum , the flux actually becomes brighter as @xmath37 , in the limit of high redshift . to estimate the apparent magnitudes of the supernovae , we assume a blackbody spectrum , which is a sufficiently close approximation for our purpose . in figure 1 we show the apparent magnitudes in several bands for a luminosity @xmath38 erg / s , and temperatures @xmath39 k and @xmath40 k. the high temperature is reached @xmath5 1 day after the explosion , when the luminosity drops to the value in the plateau part of the lightcurve ; during the next few days the temperature cools , reaching a value near @xmath41 k after about a week , and then it stays constant until the luminosity starts decreasing . immediately after the explosion , when the shock reaches the surface of the star , the luminosity and temperature can be much higher and the apparent flux can be brighter by a factor @xmath42 relative to the values in fig . 1 , but this phase only lasts for @xmath43 hour . the apparent magnitudes have been calculated for the @xmath27 model with @xmath44 km / s / mpc , from the flux at the central wavelengths of the bands , using the central wavelengths and zero - magnitude fluxes given in allen ( 1973 ) . the high - redshift cutoff in the curves in figure 1 indicate the redshift where the supernova light would be absorbed by the ly@xmath0 forest . at high redshift , the supernovae must obviously be searched in the infrared . the faintest galaxy surveys from the ground have reached magnitudes @xmath45 ( cowie et al . 1994 ) . from fig . 1 , type ii supernovae would have similar magnitudes at @xmath46 , although searching for variable objects should require more telescope time than simple object identification . in order to find supernovae at redshifts higher than known quasars , fainter fluxes by a factor of @xmath47 need to be detected . this might be achieved with implementation of adaptive optics on large ground - based telescopes ; in the longer term , the new generation space telescope should certainly be capable to observe these supernovae ( see mather & stockman 1996 ) . we emphasize again the large uncertainty in predicting the types and absolute magnitude of supernovae from these early generation of stars . one possibility to detect these supernovae before more powerful telescopes and cameras in the infrared can be built is to use the magnifying power of gravitational lensing in galaxy clusters . the deflection angles of the most massive clusters of galaxies are as large as @xmath48 , with critical lines of total length of several arc minutes . the cross section for magnifying a source by more than a factor @xmath49 is @xmath50 , or 0.01 square arc minutes for @xmath51 . thus , any rich lensing cluster should have a 1% chance of having a high - redshift supernova magnified by a factor larger than 10 at any time . the highly magnified images would always appear in pairs around critical lines , and would be simple to identify only from their positions and colors given a lensing model of the cluster ( see miralda - escud & fort 1993 ) . later , variability would have to be detected to distinguish the supernova from a faint , compact galaxy . we have suggested in this paper that , under reasonable assumptions , supernovae should be the brightest objects in the universe beyond some redshift , in particular during the early phases of the reionization . the supernovae might therefore be the first observational evidence we shall have of this epoch , when the very faint apparent magnitudes expected are observable . the other observable signature of this epoch may be the 21 cm absorption or emission by the neutral intergalactic gas ( scott & rees 1994 ; madau , meiksin & rees 1996 ) . the uv and heavy elements abundance inferred from quasar absorption lines allow us , as we have seen , to draw quantitative conclusions about the minimum number of high - mass stars formed beyond @xmath3 : to produce a metallicity @xmath22 requires one supernova for each @xmath52 of baryons . this inference is quite robust , being insensitive to the details of structure and galaxy formation at high redshift , which of course depend on the cosmological assumptions . however , the total mass of stars formed depends on the imf , and is therefore much more uncertain . for a standard imf , @xmath53 of stars need to be formed to produce one supernova , so 2% of the baryons should have been turned into stars by the time the igm reached this level of enrichment . notice that this is equivalent to @xmath54 of the _ observed _ baryons in galaxies today , given our adopted value of @xmath55 which implies that only @xmath5 10% of the baryons are in known stars and gas in galaxies . it is of course quite possible that the imf was different for these early stars , given the different physical environment ( a higher ambient temperature , absence of heavy elements to act as coolants and provide opacity , and no significant magnetic fields ) . direct clues to the slope of the high - mass imf may come from ( for instance ) the relative ionization levels of h and he and heavy elements , which depend on the background radiation spectrum shortward of the lyman limit , or from relative abundances of heavy elements relative to carbon . conceivably , all the early stars might be of high - mass , so that no coeval low - mass stars survive ; at the other extreme , the early imf could have been much steeper than the standard one , in which case there could be many pregalactic brown dwarfs . would any of these `` population iii '' stars be observable today ? let us consider the observational consequences of the simplest assumption : that the early imf was the same as in the solar neighborhood . in that case , most of the present luminosity from the population iii would arise from red giants and stars at the tip of the main - sequence , with @xmath56 . where should these stars be today ? after the first galaxies ejected all their gas back to the igm , the stars that had been formed should have remained in orbit near the center of the dark matter halos . the stars then behave as collisionless matter as the halos merge with larger objects , until the present galaxies are formed . we would therefore expect that these stars would at present be distributed approximately like the dark matter in galactic and cluster halos , and in addition there should be some surviving galaxies from that epoch which have not merged into much larger objects ( or have survived in orbit after merging with a large halo , having escaped tidal disruption ) and still have the population iii stars in their centers . the halos of stars formed in this way around galaxies might be somewhat more centrally concentrated than the dark matter , if many mergers take place with only a moderate increase of the halo mass at each merger ( so that dynamical friction is effective after each merger and it can bring the stars near the center of the newly formed halo before tidal disruption occurs ) . in fact , particles that start near the centers of halos that merge tend to end up near the center of the merger product ( e.g. , spergel & hernquist 1992 ) . the known halo stars have a very steep density profile , @xmath57 , and their total mass is @xmath58 ( e.g. , morrison 1993 ) . this mass is comparable to the total mass we would expect in the halo in the population iii stars , if the total mass of the halo of our galaxy is @xmath59 , with a baryon fraction of 10% , and if 2% of the baryons formed population iii stars . therefore , if the stellar mass function in the first galaxies was normal , a sizable fraction of the halo stars should have originated there ( this is not surprising , because it is derived from the assumption that the halo stars created their own metal abundance ) . it seems difficult that the process of dynamical friction alluded to above can result in the steep slope of the halo stars . however , the halo density profile might become shallower at large radius ( see hawkins 1983 and norris & hawkins 1991 for current observational evidence on this possibility ) , and a second halo population in the outer part of the galaxy ( @xmath60 kpc ) might be the remnant of the population iii . these halo stars could be found in the hubble deep field ( hdf , williams et al . if the stellar mass of this outer halo is @xmath61 , there should be @xmath62 stars near the main - sequence turnoff , i.e. , we expect a few stars in the hdf ( with area 4.4 arcmin@xmath63 ) ; these would have colors @xmath64 , @xmath65 at distances of 100 kpc . from fig . 2 in flynn , gould , & bahcall , we see that there is at least one stellar object with these characteristics in the hdf . several other observations may help to test the existence of the population iii stars . an outer stellar halo would also imply a certain number of high - velocity stars near the solar neighborhood . a stellar population may be found in the halos of external galaxies , with density profiles similar to the dark matter . sackett et al . ( 1994 ) found a luminous halo in the galaxy ngc 5907 with @xmath66 ( i.e. , about ten times more light than what we expect for the population iii ) . planetary nebulae could also be found in nearby halos of galaxies or galaxy groups ; several of them were reported recently by by theuns & warren 1996 ) in the fornax cluster . there is also the possibility that the imf in the early galaxies produced a large number of brown dwarfs . in this case , a large fraction of the baryons could have been turned into brown dwarfs , and these could be detected in ongoing microlensing experiments towards the lmc ( see paczyski 1996 ) . if the baryon fraction in the universe is 10% , the optical depth of these brown dwarfs toward the lmc could be as high as a few times @xmath67 . finally , we notice that the metallicity distribution of the population iii stars is difficult to predict . if only a small fraction of the neutral igm collapsed to galaxies before the reionization , then the gas in these galaxies could reached high metallicities and formed stars , and the metallicity could have diluted in the igm when the gas was ejected . at the same time , the metal abundance of the igm after reionization could be highly inhomogeneous , so some galaxies formed later could have very low metallicities . therefore , it is difficult to predict even if the average metal abundance of the population iii stars should be higher , lower or similar to the more centrally concentrated halo stars , let alone the distribution of these metallicities . as the observational techniques improve our ability to detect extremely faint sources , and higher redshift objects can be searched for to continue unravelling the history of galaxy formation , supernovae should become the brightest observable sources . these supernovae created the heavy elements that were expelled to the igm , and their progenitor stars are the most likely sources of the photons that reionized the universe . the expected rates of these supernovae , calculating under the assumption of a high baryon density ( @xmath68 ) , and an average metal production of @xmath21 , is as high as 1 supernova per square arc minute per year . to detect the supernovae , the flux limits of the faintest sources detectable with our telescopes will probably need to be pushed by another @xmath69 magnitudes , although the first examples might be discovered at brighter fluxes behind clusters of galaxies , using the lensing magnification . any low - mass stars that were formed in the first small galaxies where these supernovae took place should be observable today . we have argued that , if the imf in these galaxies was similar to the present one in our galactic disk , the population iii stars are likely to account for a large fraction of the stars in our galactic halo , although most of them should be in an as yet undetected outer halo with a shallower density profile than the known , inner stellar halo . allen , c. w. 1973 , _ astrophysical quantities _ ( london : athlone press ) burles , s. , & tytler , s. 1996 , submitted to science ( astroph 9603069 ) cen , r. , miralda - escud , j. , ostriker , j. p. , & rauch , m. 1994 , apj , 437 , l9 couchman , h. m. p. , & rees , m. j. , 1986 , mnras , 221 , 53 cowie , l. l. , gardner , j. p. , hu , e. m. , songaila , a. , hodapp , k .- w . , & wainscoat , r. j. 1994 , apj , 434 , 114 dekel , a. , & silk , j. 1986 , apj , 303 , 39 flynn , c. , gould , a. , & bahcall , j. n. 1996 , apj , 466 , l55 haiman , z. , rees , m. j. , & loeb , a. 1996 , apj , submitted ( astroph-9608130 ) hawkins , m. s. 1983 , mnras , 206 , 433 hernquist , l. , katz , n. , weinberg , d. h. , & miralda - escud , j. 1996 , apj , 457 , l51 hu , e. m. , & mcmahon , r. g. 1996 , nature , 382 , 231 lacy , m. , et al . 1994 , mnras , 271 , 504 madau , p. , meiksin , a. , & rees , m. j. 1996 , apj , submitted ( astroph 9608010 ) madau , p. , & shull , j. m. 1996 , , 457 , 551 mather , j. , & stockman , h. 1996 , nasa report . miralda - escud , j. , & fort , b. 1993 , apj , 417 , 5 miralda - escud , j. , cen , r. , ostriker , j. p. , & rauch , m. 1996 , apj , 471 , 582 morrison , h. l. 1993 , aj , 106 , 578 norris , j. , & hawkins , m. s. 1991 , apj , 380 , 104 paczyski , b. 1996 , ara&a , 34 , xxx rawlings , s. , lacy , m. , blundell , k. m. , eales , s. a. , bunker , a. j. , & garrington , s. t. 1996 , nature , 383 , 502 sackett , p. d. , morrison , h. l. , harding , p. , & boroson , t. a. 1994 , nature , 370 , 441 schmidt , m. 1965 , apj , 141 , 1295 schneider , d. p. , schmidt , m. , & gunn , j. e. 1991 , aj , 101 , 2004 scott , d. , & rees , m. j. 1990 , mnras , 247 , 510 songaila , a. , & cowie , l. l. 1996 , aj , in press ( astro - ph 9605102 ) spergel , d. n. , & hernquist , l. 1992 , apj , 397 , l75 steidel , c. c. , giavalisco , m. , pettini , m. , dickinson , m. , & adelberger , k. l. 1996 , apj , 462 , l17 theuns , t. , & warren , s. j. 1996 , submitted to mnras ( astro - ph 9609076 ) tytler , d. , fan , x .- m . , burles , s. , cottrell , l. , davis , c. , kirkman , d. , & zuo , l. 1995 , in qso absorption lines , ed . g. meylan , p. 289 tytler , d. , fan , x .- m . , & burles , s. 1996 , nature , 381 , 207 williams , r. , et al . 1996 , science with the hubble space telescope ii , eds . p. benvenuti , f. d. macchetto , & e. j. schreier ( baltimore : stsci ) , in press woosley , s. e. , & weaver , t. a. 1995 , apjs , 101 , 181 woosley , s. e. , & weaver , t. a. 1986 , ara & a , 24 , 205
|
recent evidence on the metal content of the high - redshift ly@xmath0 forest seen in quasar spectra suggests that an early generation of galaxies enriched the intergalactic medium ( igm ) at @xmath1 .
we calculate the number of supernovae that need to have taken place to produce the observed metallicity .
the progenitor stars of the supernovae should have emitted @xmath2 ionizing photons for each baryon in the universe , i.e. , more than enough to ionize the igm .
we calculate that the rate of these supernovae is such that about one of them should be observable at any time per square arc minute .
their fluxes are , of course , extremely faint : at @xmath3 , the peak magnitude should be @xmath4 with a duration of @xmath5 1 year .
however , these supernovae should still be the brightest objects in the universe beyond some redshift , because the earliest galaxies should form before quasars and they should have very low mass , so their luminosities should be much lower than that of a supernova .
we also show that , under the assumption of a standard initial mass function , a significant fraction of the stars in the galactic halo should have formed in the early galaxies that reionized and enriched the igm , and which later must have merged with our galaxy .
these stars should have a more extended radial distribution than the observed halo stars .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
in recent years , doped manganese oxides remained at the forefront of theoretical and experimental research@xcite . the main source of interest in theses systems is the phenomenon of colossal magnetoresistance ( cmr ) , which they exhibit , and that is likely to have important technological applications . in the meantime , the underlying basic physics remains elusive , and probably involves the strongly - correlated nature of the doped magnetic oxides . the cmr in doped manganates is observed for intermediate hole - doping levels , typically @xmath0 , in the temperature region around the transition between low - temperature metallic ferromagnetic ( fm ) and high - temperature insulating paramagnetic phases . in addition to double - exchange ferromagnetism@xcite , the cmr compounds also possess pronounced antiferromagnetic ( afm ) tendencies , as evident from the afm spin ordering with nel temperatures of about 100 - 200k , observed@xcite at the doping end - points ( @xmath1 and @xmath2 ) . this antiferromagnetism is of a superexchange origin@xcite . the manganates physics involves several degrees of freedom of substantially different nature , including localized core spins @xmath3 of mn ions , fermionic degrees of freedom associated with conduction @xmath4-electrons , lattice distortions , _ etc_. in such systems , the presence of competing interactions ( such as fm and afm ) often gives rise to phase separation@xcite , whereby areas of different phases are stabilized in a structurally and stoichiometrically homogeneous sample . in the case of the manganates , it has even been suggested@xcite that phase separation into insulating paramagnetic and metallic fm phases may explain the resistivity peak observed near the curie temperature . in the present paper , we focus on the low - temperature regime where the presence of phase separation in the appropriate manganate systems has been directly verified , _ e.g. , _ by means of scanning tunneling microscopy ( stm ) on thin films@xcite . transport measurements reveal metastability and history dependence near the percolation threshold ( @xmath5 ) , confirming phase separation in both film@xcite and crystalline@xcite samples . using simple microscopic models@xcite it can readily be shown@xcite that the hole concentration @xmath6 indeed controls the balance between the fm and afm tendencies of the system . once @xmath6 is tuned away from the optimal cmr doping region , the homogeneous fm metallic state no longer corresponds to the energy minimum . instead , energy can be gained by changing the magnetic ordering , carrier density , bandstructure , and/or orbital state in _ part _ of the system , making the sample inhomogeneous@xcite . the surface tension between different phases@xcite then competes against the long - range interactions present in the system in the form of electrostatic forces@xcite or long - range crystal strain fields@xcite . these require that the system remains homogeneous at least _ on average _ on the appropriate length scale ( such as the debye hckel screening length ) , resulting in a periodic arrangement of nano- or mesoscopic regions of different phases@xcite . the geometry of the ensuing inhomogeneous ( phase separated ) state is at the focus of our present study . early studies of phase separation in double - exchange superexchange systems@xcite implicitly assumed that the effects of the discrete lattice are unimportant , and consequently treated the problem within the continuum based , long - wavelength , approach . within this framework , the optimal phase separation geometry at small values of the fm volume ( or area ) fraction @xmath7 ( also the average magnetization per site ) is obviously that of spherical ( in two dimensions , circular ) fm droplets located at the sites of a packed hexagonal ( triangular ) super - lattice . to the best of our knowledge , only the three - dimensional case was treated in detail , with the implication that in two dimensions the situation is similar . when the system parameters are varied in such a way that @xmath7 increases beyond @xmath8 , the geometry changes to that of spherical afm droplets in an otherwise fm matrix . the change generally occurs via a direct `` geometrical phase transition''@xcite without any intervening regime characterized by both phases forming infinite connected shapes ( such as filaments and planar slabs in three dimensions or stripes in two dimensions)@xcite . this latter conclusion is important , since such slab or stripe arrangements , if realized , would have been characterized by peculiar and potentially useful properties such as history - dependent anisotropy of the ground state resistivity . however , the continuum treatment , which is at the basis of this result , is not valid beyond the region of very small values of @xmath9 . indeed , recent studies suggest@xcite that the boundary between the two phases is abrupt on the lattice - spacing scale ( _ i.e. , _ of the type commonly associated with ising spin systems ) . such a boundary can not be adequately described in the continuum limit , and its surface tension depends on its orientation with respect to the crystalline axes@xcite . this directional dependence of boundary energies should in turn affect the droplet shape ( generally favoring diamond - shaped droplets in two dimensions)@xcite , the arrangement of droplets in space , and ultimately the way the geometry of phase separation evolves with varying @xmath7 . this is apparently a generic property of electronic phase separation , found also within the frameworks of falikov kimball@xcite and @xmath10 ( ref . ) models . in the present paper , we revisit the problem within the framework of a single - orbital double - exchange superexchange hamiltonian ( with infinite hund s coupling @xmath11 ) , augmented by a long - range coulomb interaction term . using a variational hartree - fock approach , we compute the energies of various two - dimensional droplet and stripe phases corresponding to a fm area fraction @xmath12 , and determine the optimal configuration . our most important finding is that while a droplet lattice exists at low doping levels , a striped arrangement has a lower energy and is therefore stabilized over a broad region of the phase diagram . as anticipated from our previous results concerning the orientational dependence of the boundary energy@xcite , we find that diamond droplets and diagonal stripes are the preferred geometries for the fm regions of the inhomogeneous states . these conclusions gain further support from unrestricted hartree - fock calculations which we have carried out using monte - carlo simulated annealing on moderate size clusters . the simulations also demonstrate the existence of inhomogeneous states comprised of afm droplets ( or stripes ) embedded in a fm background ( @xmath13 ) , at higher doping levels . while our results pertain to the two - dimensional case , it is likely that qualitatively our conclusions would also apply to three - dimensional systems . specifically , we suggest that a phase separated state with filament or slab geometry ( rather than a lattice of droplets ) is realized for a certain range of parameters in three dimensions . in addition , we find that the typical droplet size and stripe width do not exceed several lattice constants . this means that the motion of the charge carriers is strongly quantized , rendering droplets midway between metallic bulk and magnetic polarons@xcite and giving rise to singularities in the stripe energy associated with the quantisation of the transverse kinetic energy . this important property was not included in the earlier work@xcite , which assumed sufficiently large length scales for such quantum effects to be negligible . our approach , on the other hands , allows one to explore the crossover between the regime of singly - occupied magnetic polarons , which appear for strong coulomb and afm couplings , and the more conventional phase separation behavior where each metallic droplet is populated by many charge carriers . the paper is organized as follows : in sec . [ sec : model ] , we introduce the model and briefly review the physics underlying phase separation and magnetic polaron formation in the absence of a long - range force . a brief description of the calculational methods which were implemented in order to include the effects of the long - range coulomb repulsion appears in sec . [ sec : methods ] , while the mass of details is relegated to the appendices . [ sec : results ] contains a detailed description of our hartree fock and monte - carlo results . we conclude with a brief discussion of the results in the context of current experimental and theoretical work ( sec . [ sec : conclu ] ) . while an arrangement of conducting and insulating stripes in doped manganate films has not yet been observed , we suggest that present experimental knowledge should allow for a meaningful and successful research effort in this direction . the starting point for the following calculation is the two - dimensional double - exchange hamiltonian , generalized to include the superexchange coupling and the long - range coulomb interaction , @xmath14 here @xmath15 is the nearest - neighbor hopping amplitude and @xmath16 annihilates a conduction electron of spin @xmath17 at site @xmath18 of a square lattice . @xmath19 denotes the core spin made of three d - shell electrons ( @xmath20 ) localized at site @xmath21 , whose afm superexchange interaction with neighboring core spins is given by the second term in @xmath22 . the third term arises from hund s coupling between the core spins and the conduction electrons , where the spin operator for the conduction electrons on site @xmath21 involves the pauli matrices @xmath23 . it is this term , in conjunction with the fact that the hopping preserves the electronic spin , which gives rise to the double - exchange mechanism . this favors a fm spin configuration in order to reduce the conduction electrons kinetic energy@xcite . the last term includes the coulomb interaction among the conduction electrons , whose average density is @xmath6 , and a neutralizing uniform positive background , created by the donors . owing to the long - range nature of the coulomb interaction , the atomic - scale inhomogeneities of this background in real systems ( created by chemical substitution ) are not expected to be important from the point of view of our main purpose of comparing the energies of various inhomogeneous phases . this is because such energies always involve integration over volume . in using the simplified model , ( [ eq : ham_orig ] ) , we neglect some additional physics characteristic of the cmr manganates@xcite . this includes the presence of two ( rather than one ) conduction electron @xmath4-bands and the electron - lattice coupling . the logics behind this simplification is summarized , _ e.g. , _ in ref . : it is assumed that the mechanism for phase separation ( charge ordering ) is the competition between ferro- and antiferromagnetism in the presence of a long - range coulomb repulsion [ all contained in eq . ( [ eq : ham_orig ] ) ] . once the charge ordering is established , in a real system the orbital ordering ( and the lattice distortions ) would follow , leading to a quantitative renormalization of the parameter values . the model , eq . ( [ eq : ham_orig ] ) , is however expected to suffice for a qualitative study of the generic features of the phase diagram while its simplicity allows to maintain clarity of analysis . further arguments regarding the expected model - independence of our conclusions shall be given in sec . [ sec : conclu ] . the relatively large value @xmath24 of the mn core spins means that their fluctuations are small , particularly in the @xmath25 limit considered here . in the following we assume @xmath26 and treat the core spins classically . consequently , the effective hamiltonian governing the physics of the conduction electrons is determined by the configuration of the classical spins @xmath27 . as far as the hund s coupling is concerned the manganates are characterized by a moderate bare @xmath28 . however , they also include a strong hubbard on - site repulsion , @xmath29 , which significantly renormalizes @xmath11 towards the strong coupling limit@xcite . therefore , while we omit the hubbard interaction from our hamiltonian ( [ eq : ham_orig ] ) , we model its effects by taking @xmath30 . band theory calculations@xcite suggest that typical values of the hopping amplitude @xmath15 in the cmr manganates lie between 0.3 ev and 0.5 ev . the value of @xmath31 can be roughly estimated from the experimentally observed nel temperatures in the fully doped or undoped ( with no conduction @xmath4 electrons or with no holes ) case@xcite , @xmath32 , corresponding to @xmath33 . the long - range coulomb interaction strength , @xmath34 , for thin films is evaluated as @xmath35 . here , @xmath36 is the electron charge and @xmath37 is the lattice spacing . the _ effective _ dielectric constant @xmath38 is given by the average of dielectric constant @xmath39 of the substrate and that of the air , @xmath40 . among the substances which can be used as substrates for manganate films , lanthanum aluminate and neodymium gallate have@xcite @xmath41 and @xmath42 , yielding @xmath43 ev and @xmath44 ev respectively . dielectric properties of the third possible substrate , strontium titanate , are strongly dependent on temperature , with@xcite @xmath39 changing from 24000 ( corresponding to @xmath45 mev ) at 4.2k to @xmath46 ( @xmath47 ev ) at 300k . this suggests a possibility of experimentally varying the value of @xmath34 by using different substrates and/or changing temperature . theoretical investigations of the double - exchange superexchange competition have a history of more than 40 years . it is by now firmly established@xcite that this competition is resolved not via a second - order phase transition from the fm state to a uniform state with a helical or canted magnetic ordering , but rather via separation of the sample into regions characterized by different spin arrangements and conduction electron bandstructures . we will be interested in the case of phase separation into fm and afm regions with abrupt ising - type boundaries@xcite between them . this means that the resulting configuration of @xmath19 remains collinear , with all core spins either parallel or anti - parallel to a selected direction . thus , it is possible to denote a spin state simply by @xmath48 , and the hamiltonian of the conduction electrons becomes a function of @xmath49 . the large hund s exchange coupling then forces the conduction electrons spins to polarize in parallel with the core spins , resulting@xcite in the following distribution of hopping amplitudes for a given spin configuration @xmath50 : @xmath51 after these simplifications , the hamiltonian takes the form @xmath52 the model , eq . ( [ eq : ham_simp ] ) , on the ( bi - partite ) square lattice is invariant under the particle - hole transformation @xmath53 , and @xmath54 , where @xmath55 takes the values 0,1 on the two sublattices . as a result we note that in the following , @xmath6 acquire the more general meaning of a carrier density , i.e. , either the electronic density or the hole density relative to the half - filled state . we will now briefly review the ground state properties of the hamiltonian ( [ eq : ham_simp ] ) at @xmath56 . when the carrier density @xmath6 is finite , the ground state of the system at @xmath57 is uniform and fm . with increasing @xmath31 beyond a certain critical value @xmath58 , this uniform fm state eventually becomes destabilized , and a non - uniform ground state is obtained instead . in this _ phase separated _ state only part of the system is occupied by the fm phase . we will be interested in the case in which the other part is a simple nel antiferromagnet with zero charge - carrier density . a variational study@xcite shows that in two dimensions such a phase separated state may be realized only for @xmath59 . at higher values of @xmath31 the magnetic ordering in either the electron - rich or the electron - poor regions of the sample differs from that of a ferromagnet or a nel antiferromagnet . in the fm region of a macroscopically phase separated state . ] thermodynamic equilibrium between macroscopic fm and afm regions means that the thermodynamic potentials in the two phases are equal , @xmath60 where @xmath61 and @xmath62 @xmath63 is the density of conduction electron states in the fm region . since the latter is large , @xmath63 can be taken to be the two - dimensional tight - binding density of states , and boundary corrections may be neglected . by solving eqs . ( [ eq : tdstability][eq : omegaafm ] ) for the fermi energy , @xmath64 , one readily obtains the carrier density in the fm region , @xmath65 , via @xmath66 with the result depicted in fig . [ fig : xfmtd ] . the system remains in a uniform fm state as long as @xmath67 . the fraction of the system area ( or volume ) , occupied by the fm phase , is given by @xmath68 . the critical value , @xmath69 , for the onset of phase separation is then determined by the condition @xmath70 . in addition to the macroscopic phase separation as described above , the double exchange - superexchange competition can also be resolved via an altogether different scenario ( formation of magnetic polarons ) . when a single electron ( or hole ) is lodged into an antiferromagnetically ordered double - exchange magnet with zero charge - carriers , ( @xmath1 ) , a free ( self - trapped ) magnetic polaron , or ferron@xcite , is formed around it . it is essentially a microscopic fm region , containing one charge carrier , in an otherwise afm system . since the propagation of charge is unimpeded in the fm region it acts as a potential well for the sole carrier , which occupies the lowest bound state inside the well . the polaron binding energy , @xmath71 , ( with respect to the state where the afm order is unperturbed ) can be be easily estimated@xcite . we consider the case of a diamond - shaped polaron , with @xmath72 sites along each side ( see fig . [ fig : geometry ] , upper left ) . for @xmath73 we find @xmath74 where the first two terms are the ground - state energy of the charge carrier and the last one represents the superexchange contribution . expression ( [ polenergy ] ) should be minimized with respect to @xmath75 , resulting in @xmath76 here , the coefficient of the second term depends on the geometry of the fm micro - region ( _ e.g. _ , for a round polaron one would have obtained @xmath77 instead of @xmath78 ) . the above expressions are valid in the @xmath79 regime ( yielding @xmath80 ) , where it is easy to verify an important statement which is expected to hold for all @xmath31 . namely , if in the absence of a coulomb interaction @xmath34 , a second carrier is added to the system , it is energetically favorable for the two charge carriers to occupy the two lowest bound states in a _ shared _ fm micro - region rather than to form two independent polarons . this conclusion is verified by calculating the binding energy of the ( diamond - shaped ) doubly - occupied polaron @xmath81 which clearly satisfies @xmath82 . this trend continues when further charge is added , and at @xmath83 the binding energy ( per carrier ) of the @xmath84-carrier polaron decreases toward the limiting value @xmath85 , @xmath86 which is the energy gain per carrier associated with the macroscopic phase separated state . the latter can be evaluated as @xmath87 where @xmath88 and @xmath89 are the energies per site of the fm and afm phases . using eqs . ( [ eq : tdstability ] ) and ( [ eq : tdxfm ] ) to evaluate @xmath64 and @xmath65 we find , in the limit @xmath90 , @xmath91 the inequality , @xmath92 , implies that for any finite carrier density , at @xmath56 , the double - exchange superexchange competition is resolved via macroscopic phase separation . notwithstanding the preceding discussion , its conclusion may change if a realistically strong coulomb interaction @xmath34 is included , favoring a large spatial separation between the charge carriers . indeed , as we demonstrate in the following , a polaronic state arises in the regime of large @xmath34 and small carrier density . it is the extreme limit of a broad range of inhomogeneous states which originate from the frustration of macroscopic phase separation by long - range forces . the study of this intermediate region of parameters lies at the focus of the remaining part of the paper . since the typical size of the resulting fm regions is rather small one needs to take into account the effects of quantization of the charge carrier motion . at the same time , some of the results obtained for the macroscopic phase - separated system , such as the directional dependence of the boundary energy@xcite , still offer important guidance to the understanding of the inhomogeneous configurations . next , we outline the methods used to treat this intermediate regime which is characterized by a combination of both traditional phase separation and magnetic polaron ( quantized ) physics . given the hamiltonian ( [ eq : ham_simp ] ) , our task is to find the configuration of core spins in the ground state . however , there is a vast multitude of possible spin configurations amongst which the ground state is to be sought , making it impossible to explore all of them . nevertheless , previous studies of similar or related systems@xcite suggest several families of highly symmetrical configurations as reasonable ground state candidates . the two main types of spin configurations studied in this work are fm droplets in an afm background , and alternating fm and afm stripes , as illustrated in fig . [ fig : geometry ] . a uniform fm phase , in which the double - exchange mechanism completely overcomes the superexchange , is also considered . [ cols="^,^",options="header " , ] calculating the energy of the conduction electrons in a given configuration of core spins is a difficult problem . here we suffice with the hartree - fock ( hf ) approximation , which gives an upper - bound to their ground state energy . since we are dealing with periodic spin configurations , the hf equations for the whole system can be rewritten as an effective eigenvalue problem within a single unit cell . the superexchange contribution to a configuration s energy is simply calculated by counting the number of fm and afm bonds in a unit cell . based on previous analytical results@xcite and numerical investigations@xcite the considered droplets are either diamond or square shaped , and are chosen to form either a triangular or square super - lattice ( see fig . [ fig : geometry ] ) . several droplet phases are possible by combining different droplet shapes and super - lattice types . in addition , one has variational freedom to specify @xmath75 , the size of the fm droplets , and @xmath84 , the number of conduction electrons in each one of the droplets . the distance between the droplets , @xmath93 , is then uniquely determined by the type of super - lattice , by @xmath84 , and by the average density of conduction electrons , @xmath6 . the energy of each droplet phase is found by minimizing its energy density , @xmath94 with respect to the variational parameters , @xmath75 , and @xmath84 . here @xmath95 is the hf energy of the conduction electrons inside a unit cell containing a single droplet , and @xmath96 is the afm energy _ per site_. the details of the hf calculation appear in appendix [ app : hartree ] . the main source of complication is the necessity to take into account the hartree interaction between electrons belonging to different droplets in the infinite super - lattice . this is done by employing ewald s summation method ( see appendix [ app : ewald ] ) . the afm coupling energy per lattice site is @xmath97 where @xmath98 is the area of a unit cell . two types of stripe phases were considered : diagonal and bond - aligned . additional variational freedom comes from the need to specify @xmath99 , the stripe width , and @xmath65 , the ( average ) density of conduction electrons within the fm stripe . just as for the droplet phase , the stripe phase energy is found by minimizing its energy density @xmath100 with respect to @xmath99 and @xmath65 . here @xmath101 and @xmath102 are , respectively , the conduction electrons number and energy per unit cell , and @xmath103 is the afm energy per site . a unit cell in diagonal stripes is only one lattice spacing long in the direction along the stripe , and two spacings long in bond - aligned stripes ( see fig . [ fig : geometry ] ) ; its width equals the stripe periodicity . therefore , @xmath104 @xmath101 , together with @xmath6 , uniquely determine the distance between stripes @xmath105 . in a similar manner to the case of the droplet phase , when calculating the hf energy one needs to take into account the hartree interaction between the infinite number of unit cells in the systems . moreover , the extended nature of the states along the stripes means that it is necessary to consider also the fock exchange between different unit cells on the same stripe . a detailed account of the way this is done is presented in appendix [ app : hartree ] . the afm spin coupling energy per unit area for both diagonal and bond - aligned stripe phases is @xmath106,\ ] ] where @xmath68 is the fraction of fm regions in the system . by comparing the energies of all the above mentioned phases , a phase diagram is constructed , depicting the nature of the ground state as a function of the external parameters , @xmath6 , @xmath107 and @xmath108 . we have supplemented the calculation of the hf energy of various variational configurations by an _ unrestricted _ hf calculation using monte - carlo simulated annealing . in this method , the energy of a finite - sized system is minimized with respect to the full configuration space of core spins , rather than a special subset of spin textures . in each monte - carlo step the energy of a given configuration of classical core spins is evaluated using the hf approximation . a spin configuration is accepted as the system s new state if the change in energy from the current state satisfies the metropolis condition . the temperature is slowly decreased until the system reaches a stable , low energy configuration . if the temperature is decreased slowly enough , the final state is the hf approximation of the ground state . the underlying assumption of the present study is that the system indeed separates into fm and afm regions with an abrupt boundary between them . therefore , the mc simulation needs to explore only such configurations , improving the convergence time . this can be achieved by setting all the spins on one sub - lattice to the `` up '' state , and incrementally flipping the spins on the other sub - lattice . an additional improvement comes from a new algorithm used to decide which spin to flip . at first , a spin is chosen randomly . it is flipped if the resulting state satisfies the metropolis criterion . if the spin is located near a fm - afm boundary then its neighbors are added to a queue of spins to be tested for flipping . after all the spins in the queue have been tested for a flip , a new spin is chosen randomly . requiring that a spin be added to the queue no more than once , prevents the simulation from repeating itself , thus maintaining ergodicity . in our calculations , the system contained @xmath109 sites arranged periodically on a torus . a linear annealing schedule was employed over @xmath110 mc sweeps , and an identical number of sweeps at the lowest temperature allowed the system to thermalize into the ground state . the temperatures started from above @xmath111 at the beginning of the annealing schedule to below @xmath112 at its end . even though this method minimizes the system s energy with respect to an unrestricted configuration space , it has a number of disadvantages when compared to the variational hf method , applied to only a number of special configurations . first , its periodicity is fixed ; in our case it is @xmath113 sites along each axis . in addition , the presence of long range interactions causes the simulations to converge very slowly . nevertheless , it provides an important reference point with which the the variational hf results may be contrasted , especially in order to confirm that the variational manifold contains the most relevant configurations . in the present section , we present and analyze our numerical results . the coupling constants @xmath31 and @xmath34 are measured in units of @xmath15 , by setting the hopping amplitude @xmath114 . the hf energies of all the considered phases were calculated in the parameter range @xmath115 , @xmath116 , and for three values of @xmath34 , namely @xmath117 . we chose to concentrate on this region in the @xmath118 plane for two reasons . as already mentioned , a previous estimate@xcite sets @xmath119 as the upper limit for the realization of a fm nel afm ( as opposed to other types of magnetic ordering ) phase - separated state in two - dimensions . secondly , our calculations indicate that the line @xmath120 , crosses @xmath121 at @xmath122 , see fig . [ fig : phsdgm ] . the region below this line in the @xmath118 plane corresponds to configurations in which the fm phase occupies more than half the system area . while the stripes phases , which we consider , continue to be relevant in this region , we expect ( and confirm in our mc simulations ) that the phases of fm droplets ought to be replaced by configurations of afm droplets embedded in a fm background . the latter turn out to be more involved computationally and were left out of the present study . we also wish to note that the above values of @xmath123 and @xmath124 , are sensitive to the details of the considered model . therefore , while experimentally , percolation of the metallic phase at low temperatures is observed in manganates with @xmath125 , we expect our qualitative conclusions to apply to more complicated models of manganates , as long as phase separation into fm and afm phases is possible . we begin our review of the results by discussing the phase diagram and presenting general arguments for its structure . we then move on to consider the details of the most dominant phases . . ] the main result of our calculation is the phase diagram , fig . [ fig : phsdgm ] , derived from the variational hf approach and depicting the system s ground state configuration as a function of the parameters , @xmath6 , @xmath31 and @xmath34 . it demonstrates that diamond droplets in a triangular formation is the preferred phase at low densities , while diagonal stripes are prevalent at higher values of @xmath6 . this stability of a striped phase is the most important result of our calculation . the striped arrangement is expected to possess unusual and potentially useful properties ( see sec . [ sec : conclu ] below , where we also mention possible directions of experimental search for the stripe phase in cmr manganates ) . when the coulomb interaction strength @xmath34 is increased , the transition between droplets and stripes occurs at higher @xmath6 . as discussed above , our variational approach becomes insufficient below the line @xmath120 , as we do not allow for a phase of afm droplets embedded in a fm background . such a phase is expected to appear near the transition to the uniform fm state . this conclusion is supported by the unconstrained hf results presented below . we are unable , though , to map in detail the boundary between the stripe and droplet phases in this parameter regime . the general features of the phase diagram can be explained by simple energy considerations . the preference of diagonal stripes and diamond droplets is a direct result of the lower energy of diagonal boundaries , as previously established by two of the authors@xcite . the appearance of a triangular droplet lattice at low densities is akin to the physics giving rise to the wigner crystal in a dilute gas of electrons . next , we elaborate on the reasons and nature of the transition between the droplet and stripe phases . to this end , let us examine how the energy difference between the two phases evolves with @xmath6 . as @xmath6 increases , the distance between droplets or stripes diminishes , but our hf results indicate that the size of these fm regions and the electron density within them , @xmath65 , vary slowly in the vicinity of the phase transition . the combined difference between the kinetic and magnetic energies per electron of the two phases , @xmath126 , depends on @xmath65 and on the size and shape of the fm regions , but not on the distance between them . we therefore conclude that at the qualitative level , changes of @xmath126 with @xmath6 can not be the driving force behind the transition . instead , we concentrate on the doping dependence of the difference in the coulomb energy per electron between droplets and stripes , @xmath127 . the coulomb energy contains contributions coming from the interaction between charges within a single super - lattice unit cell and between different cells . the neutrality of each unit cell ( due to the positive background ) implies that the dominant contribution to the coulomb energy per electron originates from the intra - cell component . simple dimensional analysis allows us to obtain an estimate for its behavior . the amount of positive background charge within a droplet unit cell is @xmath128 , @xmath93 being the inter - droplet spacing . thus , the coulomb potential due to the positive background is @xmath129 . the interaction between electrons within a droplet generates @xmath130 , where @xmath75 is the droplet size . since @xmath131 , we have @xmath132 the coulomb energy in the stripe phase takes a different form . the amount of charge per unit length is @xmath133 where @xmath99 and @xmath105 are the stripe width and the distance between stripes , correspondingly . the background potential is then @xmath134 and the potential due to electrons in the same stripe is @xmath135 . together they give @xmath136 where in this case @xmath137 . consequently , the difference in coulomb energy per electron between the droplet and stripe phases has the form @xmath138 , \label{dphi}\end{aligned}\ ] ] as a function of @xmath7 , demonstrating the difference between the two possible branches . ] and @xmath139 as a function of @xmath6 as obtained from the hf calculation for @xmath140 and various values of @xmath31 . below each plot are colored bands showing the ground state configuration at the corresponding @xmath6 values : green ( light)- droplets , blue ( dark ) - stripes . the transitions occur when @xmath141 . ] where @xmath142 and @xmath143 are numerical constants characterizing the geometry of droplets and stripes , respectively . it is implicitly assumed in eq . ( [ dphi ] ) that in the transition region between the phases @xmath65 is the same for both configurations ( the hf calculation shows that this is correct up to @xmath144 ) . the transition itself takes place at @xmath145 , satisfying @xmath146 where we have used the constancy of @xmath126 near the transition . if @xmath147 then @xmath127 is a monotonously increasing function of @xmath7 ( in the physical range @xmath148 ) , see fig . [ fig : dphith ] . in this case at most a single solution , @xmath145 , exists to condition ( [ eq : transition ] ) , implying that the droplet phase is preferred when @xmath149 , while stripes occur for @xmath150 ; the area of the droplet phase increases with @xmath34 . on the other hand , if @xmath151 , @xmath127 acquires a maximum and two solutions , @xmath152 and @xmath153 , may appear . under such conditions a reentrant behavior follows , i.e. , droplets are preferred when @xmath154 _ or _ @xmath155 , and stripes are realized in the region @xmath156 , which grows with increasing @xmath34 . we note that in any case the existence of a solution to condition ( [ eq : transition ] ) , crucially depends on the value of @xmath126 . it is the latter which reflects the features taken into account for the first time in the present work ( _ viz . _ , the orientational dependence of the boundary energy and the quantization of the carrier motion . ) fig . [ fig : dphi ] shows hf results for @xmath127 and @xmath157 as a function of @xmath6 for various values of @xmath31 . transitions between droplet and stripe phases occur when @xmath141 . the division into discontinuous segments is due to changes in the properties of the stripes or droplets respectively ( the optimal values of @xmath99 , @xmath75 , and @xmath65 , see below ) . however , the transitions generally do not occur at these points of discontinuity , leading to our previous assertions concerning the constancy of @xmath126 and the dominant role of the coulomb interaction in the vicinity of the transition . the first two transitions at @xmath158 and @xmath159 are near a maximum in @xmath127 , demonstrating the @xmath151 branch behavior . whereas these transitions occur on one continuous segment , the third transition occurs on a different segment , where only one solution exists . other one solution transitions are shown for @xmath160 and @xmath161 . in general , a triangular lattice of diamond shaped droplets proved to be energetically more favorable than the other types of droplet phases . as noted before this is a consequence of the directional dependence of the fm - afm boundary energy and the minimization of the inter - droplet coulomb energy . [ fig : ditr ] shows the optimal droplet size , @xmath75 , and the number of conduction electrons per droplet , @xmath84 , as deduced from the variational hf calculation . increasing the strength of the coulomb repulsion has the obvious effect of decreasing the droplet size . specifically , for the case of @xmath162 the variational study yields very small ( @xmath163 ) singly occupied droplets in the regime of low @xmath6 and intermediate to large @xmath31 . comparing their energy to the other types of inhomogeneous states reveals that these _ magnetic polarons _ are in fact the lowest energy configuration in this region of parameters , see the hf phase diagram , fig . [ fig : phsdgm ] . ( top ) and number of electrons per droplet @xmath84 ( bottom ) . ] fig.[fig : diastr ] shows the optimal stripe width @xmath99 and conduction electron density @xmath65 for diagonal stripes . the latter are more favorable than their bond - aligned counterparts due to the orientation dependence of the boundary energy . one striking feature in these hf results is the existence of abrupt transitions in the stripe width . a small increase in @xmath6 may lead to a discontinuous change in @xmath99 . on the other hand , increasing @xmath31 typically leads to changes in @xmath99 which are less steep . the electron density within the fm stripes , @xmath65 , varies , in general , very slowly with @xmath6 , and increases with @xmath31 . ( top ) and @xmath65 ( bottom ) . the number of partially filled bands within a stripe changes across each black contour on the top panel . ] we use the condition of thermodynamic equilibrium , eq . ( [ eq : tdstability ] ) , between a diagonal fm stripe and its afm environment to explain these features . the kinetic energy contribution to the stripe s thermodynamic potential is determined by its non - interacting spectrum consisting of @xmath99 bands ( corresponding to the quantization of transverse electron motion within the stripe ) @xmath164 where @xmath165 is the bandwidth of band @xmath166 @xmath167 the resulting density of states in band @xmath168 is then @xmath169 which together with the chemical potential @xmath64 determines the number of electrons @xmath170 per unit length in the band . since we are interested in relatively low doping levels we consider the lower @xmath171 bands for which @xmath172 and the non - interacting electronic contribution to the total energy is @xmath173 using these terms , the thermodynamic potential in the fm stripes is @xmath174 where the second term is the magnetic energy , taking into account the structure of the boundaries . @xmath175 remains the same as for an infinite afm region , eq . ( [ eq : omegaafm ] ) . [ fig : xfmth ] shows @xmath176 evaluated at the fermi energy @xmath64 which solves @xmath177 as a function of @xmath31 _ and _ @xmath99 . in diagonal stripes of width @xmath99 as imposed by thermodynamic equilibrium . ] points of non - analyticity occur whenever the chemical potential increases beyond the bottom of a band , @xmath178 , so that the carriers begin to fill this additional band . these non - analyticities result in a corrugated landscape for @xmath179 , shown in fig . [ fig : xfmth ] , whereby several values of @xmath99 may correspond to the same @xmath65 . thus , a small increase in @xmath6 may drive an abrupt change in @xmath99 but leave @xmath65 constant . this transition is accompanied by a change in the number of partially filled bands within the stripe , as depicted by the black contours in fig . [ fig : diastr ] . although our monte - carlo results are not sufficient for constructing the phase diagram , they yield convincing evidence that the phases included in the variational hf calculation are indeed the appropriate variational phases to consider . some examples of ground states obtained by mc simulated annealing are given in fig . [ fig : montecarlo ] . note that the moderate cluster size used in the simulation induces finite - size effects apparent , for example , in the imperfections of the stripe configurations . these results , in addition to results from other simulations done at other parameter values , agree with the general structure of the phase diagram in fig . [ fig : phsdgm ] . moreover , the unrestricted nature of the mc method yields also configurations with afm droplets in a fm background ( bottom right of fig . [ fig : montecarlo ] ) . as mentioned before , such states were not considered in the variational hf approach because of the relative difficulty in calculating their hf energy . nevertheless , there are reasons to believe , as is confirmed by the simulations , that such a phase indeed exists around the transition line between the striped and uniform fm phases , where @xmath180 . , @xmath181 , @xmath182 . top right : @xmath183 , @xmath184 , @xmath185 . bottom left : @xmath186 , @xmath181 , @xmath185 . bottom right : @xmath187 , @xmath188 , @xmath185 . dark lines outline fm - afm boundaries.,title="fig : " ] , @xmath181 , @xmath182 . top right : @xmath183 , @xmath184 , @xmath185 . bottom left : @xmath186 , @xmath181 , @xmath185 . bottom right : @xmath187 , @xmath188 , @xmath185 . dark lines outline fm - afm boundaries.,title="fig : " ] + , @xmath181 , @xmath182 . top right : @xmath183 , @xmath184 , @xmath185 . bottom left : @xmath186 , @xmath181 , @xmath185 . bottom right : @xmath187 , @xmath188 , @xmath185 . dark lines outline fm - afm boundaries.,title="fig : " ] , @xmath181 , @xmath182 . top right : @xmath183 , @xmath184 , @xmath185 . bottom left : @xmath186 , @xmath181 , @xmath185 . bottom right : @xmath187 , @xmath188 , @xmath185 . dark lines outline fm - afm boundaries.,title="fig : " ] + the main finding of this paper concerns the geometry of the low temperature phase - separated state in a two - dimensional double - exchange magnet . we did not invoke any lattice or orbital degrees of freedom but instead concentrated on the effects of the ubiquitous long range coulomb interaction . we verified that when the relative area occupied by the fm phase , @xmath189 , is sufficiently large , a _ striped arrangement _ ( rather than a droplet super - lattice ) is stabilized . our results also confirm the expectation , based on a previous analysis of the directional dependence of the fm - afm boundary energy@xcite , that diamond - shaped droplets and diagonal stripes are preferred over their square and bond - aligned counterparts . indications to this effect are also present in other numerical studies of double - exchange models@xcite . the stability of the stripe phase should not come as a surprise . in fact , even in the earlier studies , which considered the continuum limit@xcite , it was noted that the energies of the stripe and droplet configurations can be very close , although no parameter window was found where stripes would correspond to the lowest energy configuration . as a result , it is conceivable , as indeed was shown in ref . , that a stripe phase may be stabilized , even in this limit , once physics due to some sort of additional degrees of freedom is taken into account . stripes also occur naturally in other models , such as @xmath10 or hubbard@xcite , which involve a competition between the afm nature of a parent undoped state and the kinetic energy of the doped charge carriers . long - range afm interaction was found to favor stripes in the fm ising model@xcite . regarding the case of a pure double - exchange system with coulomb repulsion considered here , it has already been argued@xcite that the correct treatment of the boundaries between the fm and afm regions is likely to tilt the balance in favor of a striped arrangement . in the present work , we considered the experimentaly relevant case of nanometer - size fm inclusions ( comprising only a few lattice periods ) . beside mapping the evolution of the geometry of the inhomogeneous system , we addressed the long - standing question regarding the stability of free magnetic polarons . as expected , we find polaronic behavior in the region of small carrier concentration @xmath190 and strong coulomb interaction . away from this regime , individual magnetic polarons coalesce into larger fm areas . we were able to span the entire intermediate regime between the coventional phase separation ( where the quantized character of the carrier motion becomes unimportant ) and an array of free magnetic polarons ( for which the notion of thermodynamic equilibrium between fm and afm phases becomes irrelevant ) . we emphasize that the two main physical ingredients underlying our findings , namely , the quantized electronic motion in small fm regions and the directional dependence of the boundary energy , can be viewed as largely model - independent . therefore , our present conclusions can be expected to stand for any double - exchange model with a long range interaction , including the case when the latter originates from crystal strain fields@xcite . the bulk of our study was carried out using a variational hf approximation for the energy of various droplet and stripe phases . it was supplemented by unconstrained hf calculations on moderate size clusters , implemented via monte - carlo simulated annealing . the hf approximation is expected to gain accuracy whenever the ratio of electrostatic energy to kinetic energy is small . throughout the range of parameters studied by us , this ratio never exceeds 0.15 . moreover , since we deal with the case of fully polarized electronic spins , the spatial part of the many - body wave - function is antisymmetric . this fact reduces correlation corrections to the hf result which stem from the tendency of any pair of electrons , owing to their mutual repulsion , to be more distant from each other than the hf wave - function would indicate . we close with a brief discussion of the experimental situation . to the best of our knowledge , a conclusive experimental observation of metallic stripes in phase - separated films of cmr materials is yet to be made . we note , however , that stripe - like charge ordering on the atomic scale ( charge density wave ) was observed in a variety of manganates . this includes films with different doping levels@xcite , as well as ceramic@xcite and single crystal@xcite samples in the insulating state above @xmath191 . in addition , it was suggested@xcite that the phase separated state in a three - dimensional system may acquire a filament structure . nevertheless , we argue that it would be desirable to synthesize manganate films whose phase - separated state clearly exhibits metallic stripes . in addition to illustrating our theoretical picture , such systems are expected to display unique and potentially useful properties , some of which were not previously observed . one of these is an anisotropic conductance , whereby the stripes direction determines a low - resistivity axis , which ought to be amenable to reorientation by , _ e.g. _ , applying a voltage . in general , one expects to find history - dependent resistance and memory effects akin to , and probably more pronounced than those observed earlier in phase separated films@xcite , for which not evidence for stripes was reported . when the sample composition gets close to the one corresponding to a stable striped arrangement , weak perturbations such as external electric or magnetic fields may be sufficient to change the geometry of the fm regions from droplets to stripes , with a drastic change in transport properties in the form of colossal electroresistance due to dielectrophoresis@xcite and large low - temperature magnetoresistance . which manganate system could potentially exhibit a metallic stripe order ? in general , in order to look for such a state one is interested to explore the parameter space by changing the average carrier concentration @xmath6 , the metallic area fraction @xmath7 , and the strength @xmath34 of the coulomb interaction@xcite . while @xmath6 is determined by the dopant concentration , the experimentally measurable quantity @xmath7 depends , in our model , on the ratio of the afm coupling @xmath31 to the hopping @xmath15 . the latter may be affected by , _ e.g. , _ the choice of the rare earth ion . an example of a system which apparently allows control over the value of @xmath7 is @xmath192 . three - dimensional crystals of this compound with @xmath6 between 0.25 and 0.5 , are metallic for @xmath193 , with no signatures of phase separation at low temperatures@xcite . at @xmath194 , the system is phase - separated@xcite , and exhibits robust insulating behavior , presumably corresponding to well - separated metallic droplets in an insulating matrix . the properties of the phase - separated state change as one decreases the value of @xmath195 , and at @xmath196 it is possible to observe conduction paths formation and switching as a result of an applied current@xcite . similar behavior is also found in thin films of the same compound , which at least for sufficiently large values of @xmath195 are phase - separated@xcite , as reflected in their peculiar dielectric and transport properties@xcite . these findings prompt us to suggest looking for signatures of stripes in @xmath192 films by systematically varying @xmath195 , and with it , as indicated above , the relative area @xmath7 of the metallic phase . we expect stripes to appear around the point where the areas of metallic an insulating phases are equal to each other . besides @xmath192 , there are other hole - doped manganate systems which may exhibit a stripe geometry of phase separation , see ref . . in addition , we expect our results to be relevant for some electron - doped manganates@xcite , as well as possibly for eu - based magnetic semiconductors@xcite . the hf equations for @xmath84 interacting , spin polarized , electrons , may be written in matrix form as an effective eigenvalue equation , which needs to be solved self - consistently@xcite @xmath197 where the effective hamiltonian matrix is given by @xmath198 the indices @xmath199 , @xmath200 and @xmath201 indicate positions on the lattice , @xmath202 is the single - particle part of the hamiltonian , and @xmath203 is the interaction energy of a particle at site @xmath204 and a particle at site @xmath205 . @xmath206 are the eigenvectors in lattice - site representation , each indexed by label @xmath207 and with @xmath208 as its eigenvalue . the self - consistent solution yields the hf ground state energy , given by @xmath209 where the summation is over the @xmath84 states with lowest eigenvalues @xmath208 . for the hamiltonian , eq . ( [ eq : ham_simp ] ) , considered in the present study the single - particle term is @xmath210 with an implicit dependence , given by eq . ( [ teff ] ) , of @xmath211 on the configuration of the core spins @xmath212 . the second term in eq . ( [ hfsinglepart ] ) reflects the interaction between the conduction electrons and a _ continuous _ neutralizing positive background of density @xmath213 , where @xmath214 is the system area , via the coulomb potential @xmath215 . noting that the eigenvectors are normalized to unity , @xmath216 , and that the @xmath217 and @xmath218 terms in eq . ( [ hhf ] ) are equal and opposite , we are led to analyze the following hf hamiltonian @xmath219 . \label{eq : fullmatrix}\ ] ] we are interested in cases where the core spins configuration is periodic , such that the system can be divided into @xmath220 unit cells , each containing an identical configuration of spins on @xmath221 sites . let the super - lattice vectors @xmath222 identify the location of the unit cells . a position @xmath199 on the lattice can then be written as @xmath223 , where @xmath224 is the position within the unit cell @xmath225 , containing @xmath199 . the spin periodicity implies that @xmath226 between sites @xmath227 and @xmath228 depends only on @xmath21 , @xmath18 , and the super - lattice vector @xmath229 connecting the two unit cells , _ i.e. _ , @xmath230 . as a consequence of bloch s theorem this means that the energy eigenfunctions , expressed in the @xmath231 representation , take the form @xmath232 , with eigenenergies @xmath233 , where @xmath234 is defined within the first brillouin zone of the reciprocal super - lattice . the `` band '' index @xmath168 runs from @xmath235 to @xmath221 , and @xmath236 is normalized to unity within a single unit cell . written in the @xmath237 basis , the hamiltonian becomes block diagonal , where the matrix elements of the block connecting states with the same @xmath238 are given by @xmath239 \left[\delta_{ij}\sum_{i'}v_{ii'}^{h } \left|\phi_{bi'}\left(\mathbf{k}'\right)\right|^{2 } -v_{ij}\left(\mathbf{k}-\mathbf{k'}\right ) \phi_{bj}^{*}\left(\mathbf{k}'\right ) \phi_{bi}\left(\mathbf{k}'\right)\right ] . \label{hfhk}\ ] ] here @xmath240 is the fourier transform of the hopping amplitudes @xmath241 between sites @xmath21 and @xmath18 in unit cells separated by a super - lattice vector @xmath242 . we also introduced @xmath243 where @xmath244 is the area of a unit cell . the chemical potential , @xmath64 , is defined by @xmath245 $ ] , with @xmath246 denoting the step function . the hf ground - state energy of the conduction electrons is @xmath247\\ & \times & \theta\left[\mu-\varepsilon_{b}(\mathbf{k})\right ] -\frac{1}{2}unx\int\frac{d\mathbf{r}}{\left|\mathbf{r}\right| } , \label{eq : fullenergy}\end{aligned}\ ] ] where in the last term we have taken the limit @xmath248 . this diverging contribution is canceled by the self - interaction of the positive background , evaluated in the same limit @xmath249 consequently , the total energy ( not including the contribution of the anti - ferromagnetic interaction between the core spins ) is given by the sum over @xmath168 and @xmath238 in eq . ( [ eq : fullenergy ] ) . in order to evaluate the matrix elements of the hf hamiltonian , we need a method to calculate the infinite super - lattice sums in eqs . ( [ eq : kinsum])-([eq : hartreesum ] ) . the first of these is trivial since hopping is allowed only between nearest - neighbor sites within a unit cell or between adjacent ones . in two dimensions this leaves at most five terms to the sum . on the other hand , the coulomb interaction is long - ranged , and an infinite number of terms needs to be included in eqs . ( [ eq : focksum ] ) and ( [ eq : hartreesum ] ) . the hartree interaction matrix ( [ eq : hartreesum ] ) includes two diverging contributions , one coming from the interaction with the average electronic density and the other from the interaction with the positive uniform background . the two contributions cancel each other . in order to demonstrate this and extract the remaining finite piece we employ ewald summation ( see appendix [ app : ewald ] ) . the main identity of this method , directly applicable to the evaluation of the first term in eq . ( [ eq : hartreesum ] ) , is @xmath250 as before , @xmath242 are the super - lattice vectors and @xmath251 is the unit cell area . here , @xmath252 are the reciprocal super - lattice vectors , and @xmath253 is an arbitrary constant , chosen to minimize the number of relevant terms in both sums controlled by the complementary error function @xmath254 . note that the divergence which stems from summing over large @xmath225 vectors in the left hand side of eq . ( [ eq : ewaldquote ] ) is encoded in the @xmath255 term on the right hand side . this divergence is canceled by the integral over the whole system in eq . ( [ eq : hartreesum ] ) . this can be readily seen by using eq . ( [ eq : ewaldquote ] ) with @xmath256 , to write it as @xmath257 consequently we find for the hartree matrix @xmath258 where @xmath259 . when the core spins are arranged in fm droplets separated by an afm ordered background , the conduction electrons can not hop from one unit cell to the other , _ i.e. _ , @xmath260 . consequently @xmath261 is independent of @xmath262 , see eq . ( [ eq : kinsum ] ) . under such a condition it is easy to verify that the hf eigenfunctions and eigenenergies are @xmath262-independent as well . to prove this assertion , let us assume that it is true and show that it leads to a @xmath238-independent hf hamiltonian , hence closing the argument self - consistently . since the hartree term in the hf hamiltonian , eq . ( [ hfhk ] ) , depends on @xmath234 only through the hf eigenfunctions it obviously fulfills the requirement . to complete the demonstration we note that the same is true for the fock term since it satisfies @xmath263 moreover , eq . ( [ fockdrop ] ) implies that in the case of fm droplets the calculation of the fock term involves only a finite sum ( over the @xmath221 states within each droplet ) . this is a direct consequence of the vanishing overlap between electronic states in different droplets . when the core spins are arranged in a striped configuration , hopping is allowed between unit cells along the direction of the stripes . in other words , if we decompose the super - lattice vectors as @xmath264 , where @xmath265 and @xmath266 are primitive vectors along and off the stripe direction , respectively , then @xmath267 . as a result @xmath261 , depends only on the @xmath262 component along the stripes , _ i.e. _ , @xmath268 . it follows then , using the same reasoning presented above for the droplet case , that the hf eigenfunctions and eigenenergies depend only on @xmath269 , and the fock term takes the form @xmath270\phi_{bi}^{*}(k_a')\phi_{bj}(k_a ' ) \\ & \times & \sum_{n_a } \frac{1-\delta_{ij}\delta_{n_a,0 } } { |{\bf r}_{ij}+n_a{\bf a}|}e^{in_a ( k_a - k_a ' ) a } , \label{fockstripe}\end{aligned}\ ] ] where @xmath271 is the number of unit cells along the stripe , @xmath272 , and @xmath273 is the vector connecting sites @xmath21 and @xmath18 within a unit cell . in contrast to the hartree term where the interaction decays slowly , the exponential factor in the fock exchange , eq . ( [ fockstripe ] ) , ensures that the series converges relatively fast . hence , the infinite sum is well approximated by assuming a long but finite stripe . in our calculation , we used @xmath274 , and verified that larger values change the ground state energy by an insignificant amount . note that the logarithmic divergence of the @xmath275 sum in the case @xmath276 , is integrable , and vanishes upon the summation over @xmath277 . the development ( based on ref . ) of ewald s summation method begins with defining the function @xmath278 where the vectors @xmath279 correspond to the @xmath220 points of a two - dimensional lattice of area @xmath214 . @xmath280 is a periodic function of @xmath281 , with the periodicity of the lattice . therefore , it can be expanded into the following fourier series @xmath282 where @xmath283 are the reciprocal lattice vectors , and @xmath284 here , @xmath244 is the area of a unit cell . using eqs . ( [ fdef])-([appeniden ] ) and the identity @xmath285 we obtain @xmath286 where the integral was split into two at an arbitrary positive value @xmath253 . finally , calculating the integrals leads to eq . ( [ eq : ewaldquote ] ) , where the divergent piece of the original sum is given by the @xmath255 term ( and when @xmath287 also the @xmath288 term ) in the new representation . the remaining part of the infinite sums over @xmath252 and @xmath225 is rapidly converging at a rate which is optimized by an appropriate choice of @xmath253 . c. martin , a. maignan , m. hervieu , and b. raveau , phys . rev b * 60 * , 12191 ( 1999 ) ; j. f. mitchell , c. d. ling , j. e. millburn , d. n. argyriou , a. berger , and m. medarde , j. appl . phys . * 89 * , 6618 ( 2001 ) . d. p. arovas , g. gmez - santos , and f. guinea , phys . b * 59 * , 13569 ( 1999 ) ; j. l. alonso , l. a. fernndez , f. guinea , v. laliena , and v. martn - mayor , phys . rev . b * 63 * , 064416 ( 2001 ) ; m. yu . kagan , k. i. kugel , a. l. rakhmanov , and d. i. khomskii , fiz . nizk . temp . * 27 * , 815 ( 2001 ) [ low temp . phys . * 27 * , 601 ( 2001 ) ] ; n. d. mathur and p. b. littlewood , solid state comm . * 119 * , 271 ( 2001 ) ; d. khomskii and l. khomskii phys . b * 67 * , 052406 ( 2003 ) ; r. s. fishman , f. popescu , g. alvarez , j. moreno , t. maier , and m. jarrell , new j. phys . * 8 * , 116 ( 2006 ) . s. k. mishra , r. pandit , and s. satpathy , j. phys . : condens . matter * 11 * , 8561 ( 1999 ) . m. uehara , s. mori , c. h. chen , and s .- w . cheong , nature * 399 * , 560 ( 1999 ) . m. fath , s. freisem , a. a. menovsky , y. tomioka , j. aarts , and j. a. mydosh , science * 285 * , 1540 ( 1999 ) ; a. biswas , m. rajeswari , r. c. srivastava , t. venkatesan , r. l. greene , q. lu , a. l. de lozanne , and a. j. millis , phys . rev . b * 63 * , 184424 ( 2001 ) . f. x. hu and j. gao , phys b * 69 * , 212413 ( 2004 ) ; j. gao and f. x. hu , appl . * 86 * , 092504 ( 2005 ) ; a. masuno , t. terashima , y. shimakawa , and m. takano , appl . phys . lett . * 85 * , 6194 ( 2004 ) ; i. pallecchi , l. pellegrino , e. bellingeri , a. s. siri , and d. marr , phys . rev . b * 71 * , 014406 ( 2005 ) ; a. bhattacharya , m. eblen - zayas , n. e. staley , a. l. kobrinskii , and a. m. goldman , phys . rev . b * 72 * , 132406 ( 2005 ) ; n. takubo and k. miyano , phys . rev . b * 76 * , 184445 ( 2007 ) . e. s. vlakhov , k. a. nenkov , t. g. donchev , e. s. mateev , r. a. chakalov , vacuum * 76 * , 249 ( 2004 ) . a. asamitsu , y. tomioka , h. kuwahara , and y. tokura , nature ( london ) , * 388 * , 50 ( 1997 ) ; y. yuzhelevski , v. markovich , v. dikovsky , e. rozenberg , g. gorodetsky , g. jung , d. a. shulyatev , and ya . m. mukovskii , phys . b * 64 * , 224428 ( 2001 ) ; v. markovich , g. jung , y. yuzhelevski , g. gorodetsky , a. szewczyk , m. gutowska , d. a. shulyatev , and ya . m. mukovskii , phys . b * 70 * , 064414 ( 2004 ) ; markovich , g. jung , y. yuzhelevsky , g. gorodetsky , y. m. mukovskii , eur . j. b * 48 * , 41 ( 2005 ) ; g. garbarino , m. monteverde , c. acha , p. levy , m. quintero , t. y. koo , s. w. cheong , physica b 354 , 16 ( 2004 ) ; l. ghivelder and f. parisi , phys . b * 71 * , 184425 ( 2005 ) ; h. jain , a. k. raychaudhuri , y. m. mukovskii , and d. shulyatev , appl . phys . lett . * 89 * , 152116 ( 2006 ) ; h. sakai and y. tokura , preprint arxiv:0803.3922 ( 2008 ) . m. tokunaga , y. tokunaga , and t. tamegai , phys . lett * 93 * , 037203 ( 2004 ) . e. l. nagaev , zh . pisma red . * 16 * , 558 ( 1972 ) [ jetp lett . * 16 * , 394 ( 1972 ) ] ; v. a. kashin and e. l. nagaev , zh . . fiz . * 66 * , 2105 ( 1974 ) [ sov . jetp , * 39 * , 1036 ( 1974 ) ] . it was argued@xcite , however , that the combined effects of a long - range coulomb interaction , polarizability , and the interaction of electrons with acoustical phonons may stabilize a slab structure in three dimensions . n. datta , a. messager , and b. nachtergaele , j. stat . phys . * 99 * , 461 ( 2000 ) ; r. lemaski , j. k. freericks , and g. banach , phys . lett . * 89 * , 196403 ( 2002 ) . e. eisenberg , r. berkovits , d. a. huse , and b. l. altshuler , phys . b * 65 * , 134437 ( 2002 ) . e. l. nagaev , zh . pisma red . * 6 * , 484 ( 1967 ) [ jetp lett . * 6 * , 18 ( 1967 ) ] ; zh . fiz . * 54 * , 228 ( 1968 ) [ sov . jetp * 27 * , 122 ( 1968 ) ] ; j. magn . magn . mater . * 110 * , 39 ( 1992 ) , and references therein . s. satpathy , z. s. popovi , and f. r. vukajlovi , phys . lett . * 76 * , 960 ( 1996 ) ; j. e. medvedeva , v. i. anisimov , o. n. mryasov , and a. j. freeman , j. phys . : condens . matter * 14 * , 4533 ( 2002 ) . e. a. giess , r. l. sandstrom , w. j. gallagher , a. gupta , s. l. shinde , r. f. cook , e. i. cooper , e. j. m. osullivan , j. m. roldan , a. p. segmller , and j. angilello , ibm j. res . develop . * 34 * , 916 ( 1991 ) ; j. krupka , r. g. geyer , m. kuhn , j. h. hinken , ieee trans . microwave theory tech . * 42 * , 1886 ( 1994 ) . t. konaka , m. sato , h. asano , and s. kubo , j. supercond . * 4 * , 283 ( 1991 ) ; j. konopka and i. wolff , ieee trans . microwave theory tech . * 40 * , 2418 ( 1992 ) . e. sawaguchi , a. kikuchi , and y. kodera , j. phys . jpn * 17 * , 1666 ( 1962 ) ; t. sakudo and h. unoki , phys . * 26 * , 851 ( 1971 ) . p. w. anderson and h. hasegawa , phys . 100 * , 675 ( 1955 ) . v. m. pereira , j. m. b. lopes dos santos , and a. h. castro neto , preprint arxiv:0804.3094 ( 2008 ) . h. meskine , t. saha - dasgupta , and s. satpathy , phys . * 92 * , 056401 ( 2004 ) ; h. meskine and s. satpathy , j. phys . : condens . matter * 17 * , 1889 ( 2005 ) . m. umehara , phys . b * 54 * , 5523 ( 1996 ) . s. kumar and p. majumdar , phys . * 91 * , 246602 ( 2003 ) ; _ ibid . _ * 92 * , 126602 ( 2004 ) ; s. dong , h. zhu , x. wu , and j .- liu , appl . . lett . * 86 * , 022501 ( 2005 ) ; v. b. shenoy , t. gupta , h. r. krishnamurthy , and t. v. ramakrishnan , phys . lett . * 98 * , 097201 ( 2007 ) . e. l. nagaev , phys . b * 64 * , 014401 ( 2001 ) . e.g. , _ s. r.white and d. j. scalapino , phys lett . * 80 * , 1272 ( 1998 ) ; _ ibid . _ * 81 * , 3227 ( 1998 ) ; g. hager , g. wellein , e. jeckelmann , and h. fehske , phys . b , * 71 * , 075108 ( 2005 ) ; r. citro and m. marinaro , eur . j. b * 20 * , 343 ( 2001 ) ; _ ibid . _ * 22 * , 343 ( 2001 ) ; m. yu and h. q. lin , int . j. mod b * 19 * , 299 ( 2005 ) . u. lw , v. j. emery , k. fabricius , and s. a. kivelson , phys . lett . * 72 * , 1918 ( 1994 ) . s. mori , c. h. chen , and s .- w . cheong , nature * 392 * , 473 ( 1998 ) ; s. cox , j. singleton , r. d. mcdonald , a. migliori , and p. b. littlewood , nature materials * 7 * , 25 ( 2008 ) ; l. sudheendra , v. moshnyaga , e. d. mishina , b. damaschke , t. rasing , and k. samwer , phys . b * 75 * , 172407 ( 2007 ) . j. c. loudon , s. cox , a. j. williams , j. p. attfield , p. b. littlewood , p. a. midgley , and n. d. mathur , phys . 94 * , 097202 ( 2005 ) , and references therein . renner , g. aeppli , h. m. ronnow , mater . sci . eng . c * 25 * , 775 ( 2005 ) . j. tao , d. niebieskikwiat , m. b. salamon , and j. m. zuo , phys . lett . * 94 * , 147206 ( 2005 ) . s. dong , h. zhu , and j .- liu , phys . rev . * b76 * , 132409 ( 2007 ) . regarding the value of @xmath34 , see discussion in sec . [ sec : model ] . e.g. _ , i. g. deac , j. f. mitchell , and p. schiffer , phys . rev . b * 63 * , 172408 ( 2001 ) ; d. n. argyriou , u. ruett , c. p. adams , j. w. lynn , and j. f. mitchell , new j. phys . * 6 * , 195 ( 2004 ) , and references therein r. p. rairigh , g. singh - bhalla , s. tongay , t. dhakal , a. biswas , and a. f. hebard , nature physics * 3 * , 551 ( 2007 ) ; g. singh - bhalla , s. selcuk , t. dhakal , a. biswas , and a. hebard , preprint arxiv:0707.4411 ( 2007 ) . one of the candidates is @xmath289 . in crystalline form , the @xmath195-dependence of its properties was studied extensively [ y. tomioka and y. tokura , phys . b * 66 * , 104416 ( 2002 ) ] . results obtained on patterned films at @xmath290 [ t. wu and j. f. mitchell , phys . b * 74 * , 214423 ( 2006 ) ] suggest current - driven formation of conducting stripes . these include both lightly - doped compounds such as ca@xmath291sm@xmath292mno@xmath293 with @xmath294 [ m. respaud , j. m. broto , h. rakoto , j. vanacken , p. wagner , c. martin , a. maignan , and b. raveau , phys . rev . b * 63 * , 144426 ( 2001 ) ] , and lamno@xmath293 doped with tetra - valent ions [ _ e.g. , _ la@xmath291te@xmath292mno@xmath293 or la@xmath291ce@xmath292mno@xmath293 , with @xmath295 @xmath4-electrons per mn site ; see p. raychaudhuri , c. mitra , p. d. a. mann , and s. wirth , j. appl . phys . * 93 * , 8328 ( 2003 ) ; g. tan , p. duan , g. yang , s. dan , b. cheng , y . zhou , h. lu , and z. chen , j. phys . : * 16 * , 1447 ( 2004 ) ] .
|
we consider a model with competing double - exchange ( ferromagnetic ) and super - exchange ( anti - ferromagnetic ) interactions in the regime where phase separation takes place .
the presence of a long range coulomb interaction frustrates a macroscopic phase separation , and favors microscopically inhomogeneous configurations .
we use the variational hartree - fock approach , in conjunction with monte - carlo simulations to study the geometry of such configurations in a two - dimensional system .
we find that an array of diamond shaped ferromagnetic droplets is the preferred configuration at low electronic densities , while alternating ferromagnetic and anti - ferromagnetic diagonal stripes emerge at higher densities .
these findings are expected to be relevant for thin films of colossal magneto - resistive manganates .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
shannon s entropy quantifies information @xcite . it measures how much uncertainty an observer has about an event being produced by a random system . another important concept in the theory of information is the mutual information @xcite . it measures how much uncertainty an observer has about an event in a random system * x * after observing an event in a random system * y * ( or vice - versa ) . mutual information is an important quantity because it quantifies not only linear and non - linear interdependencies between two systems or data sets , but also is a measure of how much information two systems exchange or two data sets share . due to these characteristics , it became a fundamental quantity to understand the development and function of the brain @xcite , to characterise @xcite and model complex systems @xcite or chaotic systems , and to quantify the information capacity of a communication system @xcite . when constructing a model of a complex system , the first step is to understand which are the most relevant variables to describe its behaviour . mutual information provides a way to identify those variables @xcite . however , the calculation of mutual information in dynamical networks or data sets faces three main difficulties@xcite . mutual information is rigorously defined for random memoryless processes , only . in addition , its calculation involves probabilities of significant events and a suitable space where probability is calculated . the events need to be significant in the sense that they contain as much information about the system as possible . but , defining significant events , for example the fact that a variable has a value within some particular interval , is a difficult task because the interval that provides significant events is not always known . finally , data sets have finite size . this prevents one from calculating probabilities correctly . as a consequence , mutual information can often be calculated with a bias , only @xcite . in this work , we show how to calculate the amount of information exchanged per unit of time [ eq . ( [ mir_introduction ] ) ] , the so called mutual information rate ( mir ) , between two arbitrary nodes ( or group of nodes ) in a dynamical network or between two data sets . each node representing a d - dimensional dynamical system with @xmath0 state variables . the trajectory of the network considering all the nodes in the full phase space is called `` attractor '' and represented by @xmath1 . then , we propose an alternative method , similar to the ones proposed in refs . @xcite , to calculate significant upper and lower bounds for the mir in dynamical networks or between two data sets , in terms of lyapunov exponents , expansion rates , and capacity dimension . these quantities can be calculated without the use of probabilistic measures . as possible applications of our bounds calculation , we describe the relationship between synchronisation and the exchange of information in small experimental networks of coupled double - scroll circuits . in previous works of refs . @xcite , we have proposed an upper bound for the mir in terms of the positive conditional lyapunov exponents of the synchronisation manifold . as a consequence , this upper bound could only be calculated in special complex networks that allow the existence of complete synchronisation . in the present work , the proposed upper bound can be calculated to any system ( complex networks and data sets ) that admits the calculation of lyapunov exponents . we assume that an observer can measure only one scalar time series for each one of two chosen nodes . these two time series are denoted by @xmath2 and @xmath3 and they form a bidimensional set @xmath4 , a projection of the `` attractor '' into a bidimensional space denoted by @xmath5 . to calculate the mir in higher - dimensional projections @xmath5 , see supplementary information . assume that the space @xmath5 is coarse - grained in a square grid of @xmath6 boxes with equal sides @xmath7 , so @xmath8 . mutual information is defined in the following way @xcite . given two random variables , * x * and * y * , each one produces events @xmath9 and @xmath10 with probabilities @xmath11 and @xmath12 , respectively , the joint probability between these events is represented by @xmath13 . then , mutual information is defined as @xmath14 @xmath15 = @xmath16}$ ] , @xmath17 = @xmath18}$ ] , and @xmath19}$ ] . for simplification in our notation for the probabilities , we drop the subindexes @xmath20 , @xmath21 , and @xmath22 , by making @xmath23 , @xmath24 , and @xmath25 . when using eq . ( [ is ] ) to calculate the mutual information between the dynamical variables @xmath2 and @xmath3 , the probabilities appearing in eq . ( [ is ] ) are defined such that @xmath26 is the probability of finding points in a column @xmath9 of the grid , @xmath27 of finding points in the row @xmath10 of the grid , and @xmath28 the probability of finding points where the column @xmath9 meets the line @xmath10 of the grid . the mir was firstly introduced by shannon @xcite as a `` rate of actual transmission '' @xcite and later more rigorously redefined in refs . it represents the mutual information exchanged between two dynamical variables ( correlated ) per unit of time . to simplify the calculation of the mir , the two continuous dynamical variables are transformed into two discrete symbolic sequences @xmath2 and @xmath3 . then , the mir is defined by @xmath29 where @xmath30 represents the usual mutual information between the two sequences @xmath2 and @xmath3 , calculated by considering words of length @xmath31 . the mir is a fundamental quantity in science . its maximal value gives the information capacity between any two sources of information ( no need for stationarity , statistical stability , memoryless ) @xcite . therefore , alternative approaches for its calculation or for the calculation of bounds of it are of vital relevance . due to the limit to infinity in eq . ( [ original_mir ] ) and because it is defined from probabilities , the mir is not easy to be calculated especially if one wants to calculate it from ( chaotic ) trajectories of a large complex network or data sets . the difficulties faced to estimate the mir from dynamical systems and networks are similar to the ones faced in the calculation of the kolmogorov - sinai entropy , @xmath32 @xcite , ( shannon s entropy per unit of time ) . because of these difficulties , the upper bound for @xmath32 proposed by ruelle @xcite in terms of the lyapunov exponents and valid for smooth dynamical systems ( @xmath33 , where @xmath34 represent all the @xmath9 positive lyapunov exponents ) or the pesin s equality @xcite ( @xmath35 ) proved in ref . @xcite to be valid for the large class of systems that possess a srb measure , became so important in the theory of dynamical systems . our upper bound [ eq . ( [ i_c ] ) ] is a result equivalent to the work of ruelle . one of the main results of this work ( whose derivation can be seen in sec . [ methods_mir ] ) is to show that , in dynamical networks or data sets with fast decay of correlation , @xmath36 in eq . ( [ is ] ) represents the amount of mutual information between @xmath2 and @xmath3 produced within a special time interval @xmath37 , where @xmath37 represents the time for the dynamical network ( or data sets ) to lose its memory from the initial state or the correlation to decay to zero . correlation in this work is not the usual linear correlation , but a non - linear correlation defined in terms of the evolution of spatial probabilities , the quantity @xmath38 in sec . [ mixing ] . therefore , the mutual information rate ( mir ) , between the dynamical variables @xmath2 and @xmath3 ( or two data sets ) can be estimated by @xmath39 in systems that present sensitivity to initial conditions , e.g. chaotic systems , predictions are only possible for times smaller than this time @xmath37 . this time has other meanings . it is the expected time necessary for a set of points belonging to an @xmath7-square box in @xmath5 to spread over @xmath40 and it is of the order of the shortest poincar return time for a point to leave a box and return to it @xcite . it can be estimated by @xmath41}. \label{t}\ ] ] where @xmath42 is the largest positive lyapunov exponent measured in @xmath40 . chaotic systems present the mixing property ( see sec . [ mixing ] ) , and as a consequence the correlation @xmath43 always decays to zero , surely after an infinitely long time . the correlation of chaotic systems can also decay to zero for sufficiently large but finite @xmath44 ( see supplementary information ) . @xmath37 can be interpreted to be the minimum time required for a system to satisfy the conditions to be considered mixing . some examples of physical systems that are proved to be mixing and have exponentially fast decay of correlation are nonequilibrium steady - state @xcite , lorenz gases ( models of diffusive transport of light particles in a network of heavier particles ) @xcite , and billiards @xcite . an example of a `` real world '' physical complex system that presents exponentially fast decay of correlation is plasma turbulence @xcite . we do not expect that data coming from a `` real world '' complex system is rigorously mixing and has an exponentially fast decay of correlation . but , we expect that the data has a sufficiently fast decay of correlation ( e.g. stretched exponential decay or polynomially fast decays ) , implying that the system has sufficiently high sensitivity to initial conditions and as a consequence @xmath45 , for a reasonably small and finite time @xmath44 . the other two main results of our work are presented in eqs . ( [ i_c_intro ] ) and ( [ icl_intro ] ) , whose derivations are presented in sec . [ methods_bounds ] . the upper bound for the mir is given by @xmath46 where @xmath42 and @xmath47 ( positive defined ) represent the largest and the second largest lyapunov exponent measured in @xmath40 , if both exponents are positive . if the @xmath9-largest exponent is negative , then we set @xmath48 . if the set @xmath40 represents a periodic orbit , @xmath49 , and therefore there is no information being exchanged . the quantity @xmath50 is defined as @xmath51 where @xmath52 is the number of boxes that would be covered by fictitious points at time @xmath37 . at time @xmath53 , these fictitious points are confined in an @xmath7-square box . they expand not only exponentially fast in both directions according to the two positive lyapunov exponents , but expand forming a compact set , a set with no `` holes '' . at @xmath44 , they spread over @xmath40 . the lower bound for the mir is given by @xmath54 where @xmath55 represents the capacity dimension of the set @xmath40 @xmath56 , \label{tildad}\ ] ] where @xmath57 represents the number of boxes in @xmath5 that are occupied by points of @xmath40 . @xmath50 is defined in a way similar to the capacity dimension , thought it is not the capacity dimension . in fact , @xmath58 , because @xmath55 measures the change in the number of occupied boxes in @xmath5 as the space resolution varies , whereas @xmath50 measures the relative number of boxes with a certain fixed resolution @xmath7 that would be occupied by the fictitious points ( in @xmath5 ) after being iterated for a time @xmath37 . as a consequence , the empty space in @xmath5 that is not occupied by @xmath40 does not contribute to the calculation of @xmath55 , whereas it contributes to the calculation of the quantity @xmath50 . in addition , @xmath59 ( for any @xmath7 ) , because while the fictitious points form a compact set expanding with the same ratio as the one for which the real points expand ( ratio provided by the lyapunov exponents ) , the real set of points @xmath40 might not occupy many boxes . denote by @xmath60 a mixing transformation that represents how a point @xmath61 is mapped after a time @xmath37 into @xmath40 , and let @xmath62 to represent the probability of finding a point of @xmath40 in @xmath63 ( natural invariant density ) . let @xmath64 represent a region in @xmath5 . then , @xmath65 , for @xmath66 represents the probability measure of the region @xmath64 . given two square boxes @xmath67 and @xmath68 , if @xmath69 is a mixing transformation , then for a sufficiently large @xmath37 , we have that the correlation @xmath70 - \mu[i^{\prime}_1]\mu[i^{\prime}_2]$ ] , decays to zero , the probability of having a point in @xmath64 that is mapped to @xmath71 is equal to the probability of being in @xmath64 times the probability of being in @xmath71 . that is typically what happens in random processes . if the measure @xmath72 is invariant , then @xmath73=\mu(\sigma_{\omega})$ ] . mixing and ergodic systems produce measures that are invariant . we consider that the dynamical networks or data sets to be analysed present either the mixing property or have fast decay of correlations , and their probability measure is time invariant . if a system that is mixing for a time interval @xmath37 is observed ( sampled ) once every time interval @xmath37 , then the probabilities generated by these snapshot observations behave as if they were independent , and the system behaves as if it were a random process . this is so because if a system is mixing for a time interval @xmath37 , then the correlation @xmath38 decays to zero for this time interval . for systems that have some decay of correlation , surely the correlation decays to zero after an infinite time interval . but , this time interval can also be finite , as shown in supplementary information . consider now that we have experimental points and they are sampled once every time interval @xmath37 . the probability @xmath74 of the sampled trajectory to follow a given itinerary , for example to fall in the box with coordinates @xmath75 and then be iterated to the box @xmath76 depends exclusively on the probabilities of being at the box @xmath75 , represented by @xmath77 , and being at the box @xmath76 , represented by @xmath78 . therefore , for the sampled trajectory , @xmath79 . analogously , the probability @xmath80 of the sampled trajectory to fall in the column ( or line ) @xmath9 of the grid and then be iterated to the column ( or line ) @xmath10 is given by @xmath81 . the mir of the experimental non - sampled trajectory points can be calculated from the mutual information @xmath82 of the sampled trajectory points that follow itineraries of length @xmath31 : @xmath83 due to the absence of correlations of the sampled trajectory points , the mutual information for these points following itineraries of length @xmath31 can be written as @xmath84 , \label{original_mir_sampled1}\ ] ] where @xmath85 = @xmath86}$ ] , @xmath87 = @xmath88}$ ] , and @xmath89}$ ] , and @xmath90 , @xmath91 , and @xmath77 represent the probability of the sampled trajectory points to fall in the line @xmath9 of the grid , in the column @xmath10 of the grid , and in the box @xmath75 of the grid , respectively . due to the time invariance of the set @xmath40 assumed to exist , the probability measure of the non - sampled trajectory is equal to the probability measure of the sampled trajectory . if a system that has a time invariant measure is observed ( sampled ) once every time interval @xmath37 , the observed set has the same natural invariant density and probability measure of the original set . as a consequence , if @xmath40 has a time invariant measure , the probabilities @xmath26 , @xmath27 , and @xmath28 ( used to calculate @xmath36 ) are equal to @xmath90 , @xmath91 , and @xmath77 . consequently , @xmath92 , @xmath93 , and @xmath94 , and therefore @xmath95 . substituting into eq . ( [ original_mir_sampled ] ) , we finally arrive to @xmath96 where @xmath36 between two nodes is calculated from eq . ( [ is ] ) . therefore , in order to calculate the mir , we need to estimate the time @xmath37 for which the correlation of the system approaches zero and the probabilities @xmath26 , @xmath27 , @xmath28 of the experimental non - sampled experimental points to fall in the line @xmath9 of the grid , in the column @xmath10 of the grid , and in the box @xmath75 of the grid , respectively . consider that our attractor @xmath1 is generated by a 2d expanding system that possess 2 positive lyapunov exponents @xmath42 and @xmath47 , with @xmath99 . imagine a box whose sides are oriented along the orthogonal basis used to calculate the lyapunov exponents . then , points inside the box spread out after a time interval @xmath101 to @xmath102 along the direction from which @xmath42 is calculated . at @xmath44 , @xmath103 , which provides @xmath37 in eq . ( [ t ] ) , since @xmath104 . these points spread after a time interval @xmath101 to @xmath105 along the direction from which @xmath47 is calculated . after an interval of time @xmath44 , these points spread out over the set @xmath40 . we require that for @xmath106 , the distance between these points only increases : the system is expanding . imagine that at @xmath44 , fictitious points initially in a square box occupy an area of @xmath107 . then , the number of boxes of sides @xmath7 that contain fictitious points can be calculated by @xmath108 . from eq . ( [ t ] ) , @xmath109 , since @xmath8 . we denote with a lower - case format , the probabilities @xmath110 , @xmath111 , and @xmath112 with which fictitious points occupy the grid in @xmath5 . if these fictitious points spread uniformly forming a compact set whose probabilities of finding points in each fictitious box is equal , then @xmath113 ( @xmath114 ) , @xmath115 , and @xmath116 . let us denote the shannon s entropy of the probabilities @xmath112 , @xmath110 and @xmath111 as @xmath117 , @xmath118 , and @xmath119 . the mutual information of the fictitious trajectories after evolving a time interval @xmath37 can be calculated by @xmath120 . since , @xmath121 and @xmath116 , then @xmath122 . at @xmath44 , we have that @xmath109 and @xmath123 , leading us to @xmath124 . therefore , defining , @xmath125 , we arrive at @xmath126 . we defining @xmath50 as @xmath127 where @xmath52 being the number of boxes that would be covered by fictitious points at time @xmath37 . at time @xmath53 , these fictitious points are confined in an @xmath7-square box . they expand not only exponentially fast in both directions according to the two positive lyapunov exponents , but expand forming a compact set , a set with no `` holes '' . at @xmath44 , they spread over @xmath40 . using @xmath128 and @xmath129 in eq . ( [ d ] ) , we arrive at @xmath130 , and therefore , we can write that @xmath131 to calculate the maximal possible mir , of a random independent process , we assume that the expansion of points is uniform only along the columns and lines of the grid defined in the space @xmath5 , i.e. , @xmath132 , ( which maximises @xmath133 and @xmath134 ) , and we allow @xmath28 to be not uniform ( minimising @xmath135 ) for all @xmath9 and @xmath10 , then @xmath136}. \label{is_lower}\ ] ] since @xmath137 , dividing @xmath138 by @xmath139 , taking the limit of @xmath140 , and reminding that the information dimension of the set @xmath40 in the space @xmath5 is defined as @xmath141=@xmath142}}{\log{(\epsilon)}}$ ] , we obtain that the mir is given by @xmath143 since @xmath144 ( for any value of @xmath7 ) , then @xmath145 , which means that a lower bound for the maximal mir [ provided by eq . ( [ almost_true1 ] ) ] is given by @xmath146 but @xmath58 ( for any value of @xmath7 ) , and therefore @xmath97 is an upper bound for @xmath98 . to show why @xmath97 is an upper bound for the maximal possible mir , assume that the real points @xmath40 occupy the space @xmath5 uniformly . if @xmath147 , there are many boxes being occupied . it is to be expected that the probability of finding a point in a line or column of the grid is @xmath148 , and @xmath149 . in such a case , @xmath150 , which implies that @xmath151 . if @xmath152 , there are only few boxes being sparsely occupied . the probability of finding a point in a line or column of the grid is @xmath153 , and @xmath149 . there are @xmath57 lines and columns being occupied by points in the grid . in such a case , @xmath154 . comparing with @xmath122 , and since @xmath155 and @xmath59 , then we conclude that @xmath156 , which implies that @xmath151 . notice that if @xmath157 and @xmath158 , then @xmath159 . in order to extend our approach for the treatment of data sets coming from networks whose equations of motion are unknown , or for higher - dimensional networks and complex systems which might be neither rigorously chaotic nor fully deterministic , or for experimental data that contains noise and few sampling points , we write our bounds in terms of expansion rates defined in this work by @xmath160 } , \label{define_exp_rates}\ ] ] where we consider @xmath161 . @xmath162 measures the largest growth rate of nearby points . in practice , it is calculated by @xmath163 , with @xmath164 representing the largest distance between pair of points in an @xmath7-square box @xmath9 and @xmath165 representing the largest distance between pair of the points that were initially in the @xmath7-square box but have spread out for an interval of time @xmath101 . @xmath166 measures how an area enclosing points grows . in practice , it is calculated by @xmath167 , with @xmath168 representing the area occupied by points in an @xmath7-square box , and @xmath169 the area occupied by these points after spreading out for a time interval @xmath101 . there are @xmath57 boxes occupied by points which are taken into consideration in the calculation of @xmath170 . an order-@xmath171 expansion rate , @xmath172 , measures on average how a hypercube of dimension @xmath171 exponentially grows after an interval of time @xmath101 . so , @xmath173 measures the largest growth rate of nearby points , a quantity closely related to the largest finite - time lyapunov exponent @xcite . and @xmath174 measures how an area enclosing points grows , a quantity closely related to the sum of the two largest positive lyapunov exponents . in terms of expansion rates , eqs . ( [ t ] ) and ( [ i_c ] ) read @xmath175}$ ] and @xmath176 , respectively , and eqs . ( [ d ] ) and ( [ icl ] ) read @xmath177 and @xmath178 , respectively . from the way we have defined expansion rates , we expect that @xmath179 . because of the finite time interval and the finite size of the regions of points considered , regions of points that present large derivatives , contributing largely to the lyapunov exponents , contribute less to the expansion rates . if a system has constant derivative ( hyperbolic ) and has constant natural measure , then @xmath180 . there are many reasons for using expansion rates in the way we have defined them in order to calculate bounds for the mir . firstly , because they can be easily experimentally estimated whereas lyapunov exponents demand huge computational efforts . secondly , because of the macroscopic nature of the expansion rates , they might be more appropriate to treat data coming from complex systems that contains large amounts of noise , data that have points that are not ( arbitrarily ) close as formally required for a proper calculation of the lyapunov exponents . thirdly , expansion rates can be well defined for data sets containing very few data points : the fewer points a data set contains , the larger the regions of size @xmath7 need to be and the shorter the time @xmath37 is . finally , expansion rates are defined in a similar way to finite - time lyapunov exponents and thus some algorithms used to calculate lyapunov exponents can be used to calculate our defined expansion rates . to illustrate the use of our bounds , we consider the following two bidirectionally coupled maps@xmath181 where @xmath182 $ ] . if @xmath183 , the map is piecewise - linear and quadratic , otherwise . we are interested in measuring the exchange of information between @xmath184 and @xmath185 . the space @xmath5 is a square of sides 1 . the lyapunov exponents measured in the space @xmath5 are the lyapunov exponents of the set @xmath40 that is the chaotic attractor generated by eqs . ( [ network_maps ] ) . the quantities @xmath186 , @xmath97 , and @xmath98 are shown in fig . [ figure4 ] as we vary @xmath187 for @xmath183 ( a ) and @xmath188 ( b ) . we calculate @xmath36 using in eq . ( [ is ] ) the probabilities @xmath28 in which points from a trajectory composed of @xmath189 samples fall in boxes of sides @xmath7=1/500 and the probabilities @xmath26 and @xmath27 that the points visit the intervals @xmath190 of the variable @xmath191 or @xmath192 of the variable @xmath193 , respectively , for @xmath194 . when computing @xmath186 , the quantity @xmath37 was estimated by eq . ( [ t ] ) . indeed for most values of @xmath187 , @xmath195 and @xmath196 . for @xmath197 there is no coupling , and therefore the two maps are independent from each other . there is no information being exchanged . in fact , @xmath49 and @xmath198 in both figures , since @xmath199 , meaning that the attractor @xmath40 fully occupies the space @xmath5 . this is a remarkable property of our bounds : to identify that there is no information being exchanged when the two maps are independent . complete synchronisation is achieved and @xmath97 is maximal , for @xmath200 ( a ) and for @xmath201 ( b ) . a consequence of the fact that @xmath202 , and therefore , @xmath203 . the reason is because for this situation this coupled system is simply the shift map , a map with constant natural measure ; therefore @xmath204 and @xmath28 are constant for all @xmath9 and @xmath10 . as usually happens when one estimates the mutual information by partitioning the phase space with a grid having a finite resolution and data sets possessing a finite number of points , @xmath36 is typically larger than zero , even when there is no information being exchanged ( @xmath197 ) . even when there is complete synchronisation , we find non - zero off - diagonal terms in the matrix for the joint probabilities causing @xmath36 to be smaller than it should be . due to numerical errors , @xmath205 , and points that should be occupying boxes with two corners exactly along a diagonal line in the subspace @xmath5 end up occupying boxes located off - diagonal and that have at least three corners off - diagonal . the estimation of the lower bound @xmath98 suffers from the same problems . our upper bound @xmath97 is calculated assuming that there is a fictitious dynamics expanding points ( and producing probabilities ) not only exponentially fast but also uniformly . the `` experimental '' numerical points from eqs . ( [ network_maps ] ) expand exponentially fast , but not uniformly . most of the time the trajectory remains in 4 points : ( 0,0 ) , ( 1,1 ) , ( 1,0 ) , ( 0,1 ) . that is the main reason of why @xmath97 is much larger than the estimated real value of the @xmath206 , for some coupling strengths . if a two nodes in a dynamical network , such as two neurons in a brain , behave in the same way the fictitious dynamics does , these nodes would be able to exchange the largest possible amount of information . we would like to point out that one of the main advantages of calculating upper bounds for the mir ( @xmath186 ) using eq . ( [ i_c ] ) instead of actually calculating @xmath186 is that we can reproduce the curves for @xmath97 using much less number of points ( 1000 points ) than the ones ( @xmath189 ) used to calculate the curve for @xmath186 . if @xmath183 , @xmath207 can be calculated since @xmath208 and @xmath209 . we illustrate our approach for the treatment of data sets using a network formed by an inductorless version of the double - scroll circuit @xcite . we consider four networks of bidirectionally diffusively coupled circuits . topology i represents two bidirectionally coupled circuits , topology ii , three circuits coupled in an open - ended array , topology iii , four circuits coupled in an open - ended array , and topology iv , coupled in an closed array . we choose two circuits in the different networks ( one connection apart ) and collect from each circuit a time - series of 79980 points , with a sampling rate of @xmath210 samples / s . the measured variable is the voltage across one of the circuit capacitors , which is normalised in order to make the space @xmath5 to be a square of sides 1 . such normalisation does not alter the quantities that we calculate . the following results provide the exchange of information between these two chosen circuits . the values of @xmath7 and @xmath101 used to course - grain the space @xmath5 and to calculate @xmath174 in eq . ( [ define_exp_rates ] ) are the ones that minimises @xmath211 and at the same time satisfy @xmath212 , where @xmath213 represents the number of fictitious boxes covering the set @xmath40 in a compact fashion , when @xmath44 . this optimisation excludes some non - significant points that make the expansion rate of fictitious points to be much larger than it should be . in other words , we require that @xmath174 describes well the way most of the points spread . we consider that @xmath101 used to calculate @xmath214 in eq . ( [ define_exp_rates ] ) is the time for points initially in an @xmath7-side box to spread to 0.8@xmath215 . that guarantee that nearby points in @xmath40 are expanding in both directions within the time interval @xmath216 $ ] . using @xmath217 produces already similar results . if @xmath218 , the set @xmath40 might not be only expanding . @xmath37 might be overestimated . results for experimental networks of double - scroll circuits . on the left - side upper corner pictograms represent how the circuits ( filled circles ) are bidirectionally coupled . @xmath219 as ( green online ) filled circles , @xmath97 as the ( red online ) thick line , and @xmath98 as the ( brown online ) squares , for a varying coupling resistance @xmath220 . the unit of these quantities shown in these figures is ( kbits / s ) . ( a ) topology i , ( b ) topology ii , ( c ) topology iii , and ( d ) topology iv . in all figures , @xmath55 increases smoothly from 1.25 to 1.95 as @xmath220 varies from 0.1k@xmath5 to 5k@xmath5 . the line on the top of the figure represents the interval of resistance values responsible to induce almost synchronisation ( as ) and phase synchronisation ( ps).,width=264,height=264 ] @xmath36 has been estimated by the method in ref . since we assume that the space @xmath5 where mutual information is being measured is 2d , we will compare our results by considering in the method of ref . @xcite a 2d space formed by the two collected scalar signals . in the method of ref . @xcite the phase space is partitioned in regions that contain 30 points of the continuous trajectory . since that these regions do not have equal areas ( as it is done to calculate @xmath97 and @xmath98 ) , in order to estimate @xmath37 we need to imagine a box of sides @xmath221 , such that its area @xmath222 contains in average 30 points . the area occupied by the set @xmath40 is approximately given by @xmath223 , where @xmath57 is the number of occupied boxes . assuming that the 79980 experimental data points occupy the space @xmath5 uniformly , then on average 30 points would occupy an area of @xmath224 . the square root of this area is the side of the imaginary box that would occupy 30 points . so , @xmath225 . then , in the following , the `` exact '' value of the mir will be considered to be given by @xmath219 , where @xmath226 is estimated by @xmath227 . the three main characteristics of the curves for the quantities @xmath219 , @xmath97 , and @xmath98 ( appearing in fig . [ figure4_letter00 ] ) with respect to the coupling strength are that ( i ) as the coupling resistance becomes smaller , the coupling strength connecting the circuits becomes larger , and the level of synchronisation increases followed by an increase in @xmath219 , @xmath97 , and @xmath98 , ( ii ) all curves are close , ( iii ) and as expected , for most of the resistance values , @xmath228 and @xmath229 . the two main synchronous phenomena appearing in these networks are almost synchronisation ( as ) @xcite , when the circuits are almost completely synchronous , and phase synchronisation ( ps ) @xcite . for the circuits considered in fig . [ figure4_letter00 ] , as appears for the interval @xmath230 $ ] and ps appears for the interval @xmath231 $ ] . within this region of resistance values the exchange of information between the circuits becomes large . ps was detected by using the technique from refs . @xcite . to analytically demonstrate that the quantities @xmath97 and @xmath186 can be well calculated in stochastic systems , we consider the following stochastic dynamical toy model illustrated in fig . [ toy_model ] . in it points within a small box of sides @xmath7 ( represented by the filled square in fig . [ toy_model](a ) ) located in the centre of the subspace @xmath5 are mapped after one iteration of the dynamics to 12 other neighbouring boxes . some points remain in the initial box . the points that leave the initial box go to 4 boxes along the diagonal line and 8 boxes off - diagonal along the transverse direction . boxes along the diagonal are represented by the filled squares in fig . [ toy_model](b ) and off - diagonal boxes by filled circles . at the second iteration , the points occupy other neighbouring boxes , as illustrated in fig . [ toy_model](c ) , and at the time @xmath232 the points do not spread any longer , but are somehow reinjected inside the region of the attractor . we consider that this system is completely stochastic , in the sense that no one can precisely determine the location of where an initial condition will be mapped . the only information is that points inside a smaller region are mapped to a larger region . at the iteration @xmath31 , there will be @xmath233 boxes occupied along the diagonal ( filled squares in fig . [ toy_model ] ) and @xmath234 ( filled circles in fig . [ toy_model ] ) boxes occupied off - diagonal ( along the transverse direction ) , where @xmath235 for @xmath236=0 , and @xmath237 for @xmath238 and @xmath239 . @xmath240 is a small number of iterations representing the time difference between the time @xmath37 for the points in the diagonal to reach the boundary of the space @xmath5 and the time for the points in the off - diagonal to reach this boundary . the border effect can be ignored when the expansion along the diagonal direction is much faster than along the transverse direction . at the iteration @xmath31 , there will be @xmath241 boxes occupied by points . in the following calculations we consider that @xmath242 . we assume that the subspace @xmath5 is a square whose sides have length 1 , and that @xmath100 , so @xmath104 . for @xmath243 , the attractor does not grow any longer along the off - diagonal direction . the time @xmath232 , for the points to spread over the attractor @xmath1 , can be calculated by the time it takes for points to visit all the boxes along the diagonal . thus , we need to satisfy @xmath244 . ignoring the 1 appearing in the expression for @xmath245 due to the initial box in the estimation for the value of @xmath37 , we arrive that @xmath246 . this stochastic system is discrete . in order to take into consideration the initial box in the calculation of @xmath37 , we pick the first integer that is larger than @xmath247 , leading @xmath37 to be the largest integer that satisfies @xmath248 the largest lyapunov exponent or the order-1 expansion rate of this stochastic toy model can be calculated by @xmath249 , which take us to @xmath250 therefore , eq . ( [ tm_t ] ) can be rewritten as @xmath251 . the quantity @xmath50 can be calculated by @xmath252 , with @xmath232 . neglecting @xmath253 and the 1 appearing in @xmath254 due to the initial box , we have that @xmath255 $ ] . substituting in the definition of @xmath50 , we obtain @xmath256 . using @xmath37 from eq . ( [ tm_t ] ) , we arrive at @xmath257 where @xmath258 placing @xmath50 and @xmath42 in @xmath259 , give us @xmath260 let us now calculate @xmath186 . ignoring the border effect , and assuming that the expansion of points is uniform , then @xmath261 and @xmath262 . at the iteration @xmath232 , we have that @xmath263 . since @xmath264 $ ] , we can write that @xmath265 . placing @xmath37 from eq . ( [ tm_t ] ) into @xmath36 takes us to @xmath266 . finally , dividing @xmath36 by @xmath37 , we arrive that @xmath267 \nonumber \\ & = & \log{(2)}(1-r ) . \label{tm_ist}\end{aligned}\ ] ] as expected from the way we have constructed this model , eq . ( [ tm_ist ] ) and ( [ tm_ic ] ) are equal and @xmath268 . had we included the border effect in the calculation of @xmath97 , denote the value by @xmath269 , we would have typically obtained that @xmath270 , since @xmath47 calculated considering a finite space @xmath5 would be either smaller or equal than the value obtained by neglecting the border effect . had we included the border effect in the calculation of @xmath186 , denote the value by @xmath271 , typically we would expect that the probabilities @xmath28 would not be constant . that is because the points that leave the subspace @xmath5 would be randomly reinjected back to @xmath5 . we would conclude that @xmath272 . therefore , had we included the border effect , we would have obtained that @xmath273 . the way we have constructed this stochastic toy model results in @xmath274 . this is because the spreading of points along the diagonal direction is much faster than the spreading of points along the off - diagonal transverse direction . in other words , the second largest lyapunov exponent , @xmath47 , is close to zero . stochastic toy models which produce larger @xmath47 , one could consider that the spreading along the transverse direction is given by @xmath275 , with @xmath276 $ ] . in terms of the order-1 expansion rate , @xmath173 , our quantities read @xmath176 , @xmath277}$ ] , and @xmath178 . in order to show that our expansion rate can be used to calculate these quantities , we consider that the experimental system is uni - dimensional and has a constant probability measure . additive noise is assumed to be bounded with maximal amplitude @xmath278 , and having constant density . our order-1 expansion rate is defined as @xmath279}. \label{define_exp_rates_sup}\ ] ] where @xmath280 measures the largest growth rate of nearby points . since all it matters is the largest distance between points , it can be estimated even when the experimental data set has very few data points . since , in this example , we consider that the experimental noisy points have constant uniform probability distribution , @xmath281 can be calculated by @xmath282}. \label{define_exp_rates_sup1}\ ] ] where @xmath283 represents the largest distance between pair of experimental noisy points in an @xmath7-square box and @xmath284 represents the largest distance between pair of the points that were initially in the @xmath7-square box but have spread out for an interval of time @xmath101 . the experimental system ( without noise ) is responsible to make points that are at most @xmath164 apart from each other to spread to at most to @xmath165 apart from each other . this points spread out exponentially fast according to the largest positive lyapunov exponent @xmath42 by @xmath285 substituting eq . ( [ delta ] ) in ( [ define_exp_rates_sup1 ] ) , and expanding @xmath286 to first order , we obtain that @xmath287 , and therefore , our expansion rate can be used to estimate lyapunov exponents . as rigorously shown in @xcite , the decay with time of the correlation , @xmath43 , is proportional to the decay with time of the density of the first poincar recurrences , @xmath288 , which measures the probability with which a trajectory returns to an @xmath7-interval after @xmath101 iterations . therefore , if @xmath288 decays with @xmath101 , for example exponentially fast , @xmath43 will decay with @xmath101 exponentially fast , as well . the relationship between @xmath43 and @xmath289 can be simply understood in chaotic systems with one expanding direction ( one positive lyapunov exponent ) . as shown in @xcite , the `` local '' decay of correlation ( measured in the @xmath7-interval ) is given by @xmath290 , where @xmath291 is the probability measure of a chaotic trajectory to visit the @xmath7-interval . consider the shift map @xmath292 . for this map , @xmath293 and there are an infinite number of possible intervals that makes @xmath294 , for a finite @xmath101 . these intervals are the cells of a markov partition . as recently demonstrated by [ p. pinto , i. labouriau , m. s. baptista ] , in piecewise - linear systems as the shift map , if @xmath7 is a cell in an order-@xmath101 markov partition and @xmath295 , then @xmath296 and by the way a markov partition is constructed we have that @xmath297 . since that @xmath298 , we arrive at that @xmath299 , for a special finite time @xmath101 . notice that @xmath297 can be rewritten as @xmath300 . since for this map , the largest lyapunov exponent is equal to @xmath208 , then @xmath301 , which is exactly equal to the quantity @xmath37 , the time interval responsible to make the system to lose its memory from the initial condition and that can be calculated by the time that makes points inside an initial @xmath7-interval to spread over the whole phase space , in this case @xmath302 $ ] . imagine a network formed by @xmath303 coupled oscillators . uncoupled , each oscillator possesses a certain amount of positive lyapunov exponents , one zero , and the others are negative . each oscillator has dimension @xmath0 . assume that the only information available from the network are two @xmath304 dimensional measurements , or a scalar signal that is reconstructed to a @xmath304-dimensional embedding space . so , the subspace @xmath40 has dimension @xmath305 , and each subspace of a node ( or group of nodes ) has dimension @xmath304 . to be consistent with our previous equations , we assume that we measure @xmath306 positive lyapunov exponents on the projection @xmath40 . if @xmath307 , then in the following equations @xmath305 should be replaced by @xmath308 , naturally assuming that @xmath309 . in analogy with the derivation of @xmath97 and @xmath98 in a bidimensional projection , we assume that if the spreading of initial conditions is uniform in the subspace @xmath5 . then , @xmath310 represents the probability of finding trajectory points in @xmath304-dimensional space of one node ( or a group of nodes ) and @xmath311 represents the probabilities of finding trajectory points in the @xmath305-dimensional composed subspace constructed by two nodes ( or two groups of nodes ) in the subspace @xmath5 . additionally , we consider that the hypothetical number of occupied boxes @xmath254 will be given by @xmath312 . then , we have that @xmath313 , which lead us to @xmath314 similarly to the way we have derived @xmath98 in a bidimensional projection , if @xmath40 has more than 2 positive lyapunov exponents , then @xmath315 to write eq . ( [ ic_hd ] ) in terms of the positive lyapunov exponents , we first extend the calculation of the quantity @xmath50 to higher - dimensional subspaces that have dimensionality 2q , @xmath316 where @xmath317 are the lyapunov exponents measured on the subspace @xmath5 . to derive this equation we only consider that the hypothetical number of occupied boxes @xmath254 is given by @xmath318 . we then substitute @xmath50 as a function of these exponents ( eq . ( [ dk ] ) ) in eq . ( [ ic_hd ] ) . we arrive at @xmath319 consider a network whose attractor @xmath1 possesses @xmath320 positive lyapunov exponents , denoted by @xmath321 , @xmath322 . for a typical subspace @xmath5 , @xmath42 measured on @xmath5 is equal to the largest lyapunov exponent of the network . just for the sake of simplicity , assume that the nodes in the network are sufficiently well connected so that in a typical measurement with a finite number of observations this property holds , i.e. , @xmath323 . but , if measurements provide that @xmath324 , the next arguments apply as well , if one replaces @xmath325 appearing in the further calculations by the smallest lyapunov exponent , say , @xmath326 , of the network that is still larger than @xmath42 , and then , substitute @xmath327 by @xmath328 , and so on . as before , consider that @xmath306 . then , for an arbitrary subspace @xmath5 , @xmath329 , since a projection can not make the lyapunov exponents larger , but only smaller or equal . defining @xmath330 since @xmath331 , it is easy to see that @xmath332 so , @xmath97 , measured on the subspace @xmath40 and a function of the @xmath305 largest positive lyapunov exponents measured in @xmath40 , is an upper bound for @xmath333 , a quantity defined by the @xmath305 largest positive lyapunov exponents of the attractor @xmath1 of the network . therefore , if the lyapunov exponents of a network are know , the quantity @xmath333 can be used as a way to estimate how much is the mir between two measurements of this network , measurements that form the subspace @xmath5 . notice that @xmath97 depends on the projection chosen ( the subspace @xmath5 ) and on its dimension , whereas @xmath333 depends on the dimension of the subspace @xmath40 ( the number 2q of positive lyapunov exponents ) . the same happens for the mutual information between random variables that depend on the projection considered . equation ( [ icu_new ] ) is important because it allows us to obtain an estimation for the value of @xmath97 analytically . as an example , imagine the following network of coupled maps with a constant jacobian @xmath334 where @xmath335 $ ] and @xmath336 represents the connecting adjacent matrix . if node @xmath10 connects to node @xmath9 , then @xmath337 , and 0 otherwise . assume that the nodes are connected all - to - all . then , the @xmath303 positive lyapunov exponents of this network are : @xmath338 and @xmath339}$ ] , with @xmath340 . assume also that the subspace @xmath5 has dimension @xmath305 and that @xmath305 positive lyapunov exponents are observed in this space and that @xmath341 . substituting these lyapunov exponents in eq . ( [ icu_new ] ) , we arrive at @xmath342 we conclude that there are two ways for @xmath333 to increase . either one considers larger measurable subspaces @xmath5 or one increases the coupling between the nodes . this suggests that the larger the coupling strength is the more information is exchanged between groups of nodes . for arbitrary topologies , one can also derive analytical formulas for @xmath333 in this network , since @xmath321 for @xmath343 can be calculated from @xmath327 @xcite . one arrives at @xmath344 where @xmath345 is the @xmath9th largest eigenvalue ( in absolute value ) of the laplacian matrix @xmath346 . concluding , we have shown a procedure to calculate mutual information rate ( mir ) between two nodes ( or groups of nodes ) in dynamical networks and data sets that are either mixing , or present fast decay of correlations , or have sensitivity to initial conditions , and have proposed significant upper ( @xmath97 ) and lower ( @xmath98 ) bounds for it , in terms of the lyapunov exponents , the expansion rates , and the capacity dimension . since our upper bound is calculated from lyapunov exponents or expansion rates , it can be used to estimate the mir between data sets that have different sampling rates or experimental resolution ( e.g. the rise of the ocean level and the average temperature of the earth ) , or between systems possessing a different number of events . additionally , lyapunov exponents can be accurately calculated even when data sets are corrupted by noise of large amplitude ( observational additive noise ) @xcite or when the system generating the data suffers from parameter alterations ( `` experimental drift '' ) @xcite . our bounds link information ( the mir ) and the dynamical behaviour of the system being observed with synchronisation , since the more synchronous two nodes are , the smaller @xmath47 and @xmath347 will be . this link can be of great help in establishing whether two nodes in a dynamical network or in a complex system not only exchange information but also have linear or non - linear interdependences , since the approaches to measure the level of synchronisation between two systems are reasonably well known and are been widely used . if variables are synchronous in a time - lag fashion @xcite , it was shown in ref . @xcite that the mir is independent of the delay between the two processes . the upper bound for the mir could be calculated by measuring the lyapunov exponents of the network ( see supplementary information ) , which are also invariant to time - delays between the variables . * acknowledgments * m. s. baptista was partially supported by the northern research partnership ( nrp ) and alexander von humboldt foundation . m. s. baptista would like to thank a. politi for discussions concerning lyapunov exponents . rubinger , e.r . v. junior and j.c . sartorelli thanks the brazilian agencies capes , cnpq , fapemig , and fapesp .
|
the amount of information exchanged per unit of time between two nodes in a dynamical network or between two data sets is a powerful concept for analysing complex systems .
this quantity , known as the mutual information rate ( mir ) , is calculated from the mutual information , which is rigorously defined only for random systems . moreover ,
the definition of mutual information is based on probabilities of significant events .
this work offers a simple alternative way to calculate the mir in dynamical ( deterministic ) networks or between two data sets ( not fully deterministic ) , and to calculate its upper and lower bounds without having to calculate probabilities , but rather in terms of well known and well defined quantities in dynamical systems . as possible applications of our bounds
, we study the relationship between synchronisation and the exchange of information in a system of two coupled maps and in experimental networks of coupled oscillators .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
it is conventionally thought that the thermal x - ray components of neutron stars are originated from the initial residual heat when the stars cool @xcite , even before the discovery of galactic x - ray sources and pulsars . however , we will focus on this old problem in the regime of quark stars since there is no clear observational evidence to rule out quark stars or neutron stars . we demonstrate in this paper that the observed thermal emission of isolated pulsars could be well understood in a solid quark star model . the study of quark matter phases , both hot and cold , has been an interesting topic of research in recent years . in an astrophysical context , quark stars composed by cold quark matter have yet not been ruled out by the measured properties of pulsar - like compact stars @xcite . it has recently been proposed that realistic quark matter in compact stars could exist in a solid state @xcite , either as a super - solid or as a normal solid @xcite . the basic conjecture of normal solid quark matter is that de - confined quarks tend to form quark - clusters when the temperature and density are relatively low . below a certain critical temperature , these clusters could be in periodic lattices immersed in a degenerate , extremely relativistic , electron gas . note that even though quark matter is usually described as weakly coupled , the interaction between quarks and gluons in a quark - gluon plasma is still very strong @xcite . it is this strong coupling that could cause the quarks to cluster and form a solid - like material . in fact , various kinds of observations could put constraints on the state of the matter in a pulsar , and the cooling behavior is suggested since 1960s , even before the discovery of pulsars . can contemporary observations of x - ray thermal emitting and cooling pulsars be understood in the proposed solid quark star ( hereafter sqs ) model ? this is a question we try to answer in the paper , and we address that sqss could not be ruled out by the thermal observations . moreover , sqss might provide an opportunity to evaluate pulsar moments of inertia , if the thermal emission is also powered by spin . in this paper , we argue that contemporary observations of thermal x - ray emitting pulsars are consistent with the assumption that these sources are in fact sqss . the present analysis and calculation will concentrate on the observational temperature range , i.e. a few hundreds of to a few tens of ev . nonetheless , a phenomenological scenario for the whole thermal history of a strange quark star is also outlined , as described in 2.1 . the stellar residual heat would be the first energy source supporting the x - ray bolometric luminosities of sqss . 2.2 is involved on this topic , including both of the partial contributions of the lattices and the electrons . other energy reservoirs could exist as the stellar heating processes . this will be discussed in 2.3 , where different heating mechanisms are introduced , according to the different x - ray pulsar manifestations . in 3 , we compare the observations and the predictions given by the sqs pulsar model . our conclusions as well as further discussions are presented in 4 . mev ) when the star is born , and the state of the star could be the fluid of individual quarks . _ stage 2 _ : individual quark cluster phase . as the temperature decreases , individual quarks tend to form quark clusters because of the strong coupling between them . the state of the star could then be the fluid of quark clusters . _ stage 3 _ : solid quark star phase . as the temperature drops to a melting temperature , the fluid of quark clusters tend to solidify to form periodic lattice structure , such as bcc structure.,scaledwidth=80.0% ] the cooling process of a quark star can be quite complicated to model when starting from the birth of the star . theoretical uncertainties make it difficult to predict the exact temperatures where phase transitions occur . for illustrative purposes , we present the following scenario where cooling of a quark star takes place in approximately three stages ( see fig . [ fig : stg ] ) . the first stage occurs just after the birth of the quark star if its initial temperature is much higher than @xmath4 k ( 10 mev ) . the emission of neutrinos and photons will lead to fast cooling . hence , the star quickly enters the second stage where de - confined quarks begin to form quark clusters . as the temperature drops further , the fluid solidifies and the star is said to enter the third cooling stage or the sqs phase . here , quark clusters form periodic lattice structures ( e.g. the bcc structure ) . it is uncertain exactly how long the star will spend in each phase , or even if it will ever enter all three states . if the initial temperature is just around @xmath5 k or lower , stage 1 may be short lived or even non - existent . the quark - gluon plasma could be strongly coupled at birth and quark - clusters could be present immediately after formation . it is even possible that the melting temperature of solid quark matter could be @xmath6 k or higher . hence , a quark star would enter stage 3 immediately . the mechanisms for the emission of thermal photons and neutrinos in each specific stage would be quite different . a hot quark star ( in stages 1 or 2 ) would be a good radiator for thermal equilibrium photons with energy more than @xmath7 mev @xcite . meanwhile , the intense release of thermal energy would stimulate the generation and radiation of electron - positron pairs from the hot bare quark surface . the annihilation of the electron - positron pairs would generate the emission of photons inversely , and this plasma could be optically thick enough to produce a black body spectrum @xcite . the emission of neutrinos would be given rise to by the pair production process as well as the plasma process induced by the ultra - relativistic degenerate electron gas inside the star @xcite . the basic urca process would be another neutrino radiation component for stage 1 . when the star cools down and becomes solidified ( in stage 3 ) , thermal equilibrium photons ( usually soft x - ray ) could also be emitted from the bare sqs surface as free electrons transit in the levels of the energy bands of solid quark matter @xcite . the electron - phonon interaction as well as the interaction between electrons themselves might result in a metal - like spectrum . @xcite analyzed the spectrum of rx j1856 phenomenologically , and they did not conclude that there are significant differences between the metal - like spectrum and black body spectrum . hence , a black body thermal spectrum would be a good approximation for sqss . the neutrino emissivity of clustered quark matter is theoretically so far unknown . the pair production and the plasma process would also lead the neutrino cooling for sqss , if the stars enter the stage 3 at high temperatures , such as @xmath6 k. nevertheless , in the observational low temperature range @xmath8 k , the neutrino luminosity would be low enough so that photon cooling would be the dominant process for sqss . in general , in the low temperature range ( @xmath8 k ) which is the case considered by the present paper , the sqss cool down by losing thermal photons , and the sources of the energy that enable the thermal emission are firstly the residual thermal energy of the stars and secondly the energy input caused by the processes of stellar heating . the energy relation would then be described as @xmath9 where @xmath10 is the stellar bolometric x - ray luminosity , @xmath11 is the release rate of the stellar residual thermal energy , and @xmath12 is the luminosity of stellar heating . provided that the volume of a sqs is a constant , then the stellar residual thermal energy @xmath13 would only be the function of the stellar temperature @xmath14 , @xmath15 where @xmath14 is the value in the star s local reference frame . the heat capacity of the star @xmath16 comprises of the partial contribution of the lattice structure @xmath17 and that of the degenerate electrons @xmath18 , or @xmath19 . following debye elastic medium theory , the characteristic of the lattice heat capacity of solid quark matter could be evaluated by debye temperature , @xmath20 where @xmath21 is the reduced planck constant , and @xmath22 is the boltzmann constant . @xmath23 is debye cut - off frequency ( i.e. the maximum frequency of the wave that could propagate in a medium ) , which equals debye wave number @xmath24(@xmath25 is the number density of classical particles , or quark clusters for solid quark matter ) times the average sound speed in the medium , i.e. @xmath26 . for a sqs , the average sound speed could be the light speed approximately . a linear equation of state , extended to be used for a quark star , indicates that the pressure @xmath27 , where @xmath28 is the mass density of a quark star and @xmath29 is the speed of light . so an estimate could be @xmath30 . @xmath31 , where @xmath32 denotes the baryon number density of solid quark matter in the unit of @xmath33 . @xmath33 is the baryon number density of normal nuclear matter and equals 0.17 @xmath34 . we consider @xmath35 could be the typical values for a sqs . note that in the following calculation in this paper , we will adopt @xmath36 , since the variation of @xmath32 between @xmath37 would not cause the results to vary in orders . @xmath38 is the number of valence quarks in a quark cluster . we may expect that @xmath39 , since @xmath40 if quark-@xmath41-like clusters are formed @xcite , and @xmath38 could even be conjectured to be in the order of @xmath42 . debye temperature @xmath43 of a sqs is then @xmath44 k , which could be even higher than the temperature when a quark star is born . hence , the heat capacity in the low - temperature limit or the temperature - cube law , is applicable for the lattice in the third stage , i.e. @xmath45 where @xmath46 is the heat capacity per classical particle ( or quark cluster when referring to the solid quark matter)@xcite . thus @xmath47 , where @xmath48 is the total number of clusters in a star . for the stellar electron gas , those electrons distributed in the vicinity of the fermi surface will contribute significant heat capacity . the electron heat capacity would be evaluated by @xmath49 where @xmath50 is the number of electrons in a star , and @xmath51 is the fermi energy of the degenerate electron gas . in the extremely relativistic case , @xmath52 , in which @xmath53 is the number density of electrons . according to the calculation using the bag model , @xmath53 is typically chosen as one part in @xmath54 of the baryon number density of strange quark matter @xcite . in the 3 times nuclear density case , @xmath55 @xmath56 . @xmath51 is hence in the order of @xmath57 mev . in fig . [ fig : cv ] , the heat capacities of these two components are plotted . as can be seen , the electron heat capacity is much larger than that of the lattice . noting that it is really ambiguous about the melting temperature of a sqs , though we extended the calculation to 10 mev . we can now evaluate the cooling time scale for a sqs , if the star is only powered by its residual thermal energy . we choose the stellar mass of 1.4@xmath58 as a typical case . we then have @xmath59 where @xmath60 is the local stellar radius , @xmath61 is the stefan - boltzmann constant . on the right hand side of the equation ( [ eq : timesl ] ) , the first energy loss component origins from the radiation of thermal photons , the second and the third terms are the neutrino luminosities induced by the pair production and the interior ultra - relativistic electrons , respectively . the calculation about these two neutrino luminosities refers to the analytic formulae given by @xcite . the concerning about the high temperature here makes the consideration on the neutrino cooling necessary . although the actual neutrino emissivity of clustered quarks is hitherto unknown , these two components of neutrino emission could really take place for a sqs . thus , one could obtain an upper bound on the residual - heat - powered cooling time scale for a sqs via equation ( [ eq : timesl ] ) . the calculation showed that a 1.4@xmath58-sqs could only be sustained in @xmath6218 d , when it cools down from @xmath5 k to @xmath54 k. this would be the intrinsic distinction for sqss from neutron stars . neutron star cooling is mainly residual thermal energy powered , while thermal x - ray emission and cooling processes of sqss could however be sustained by heating processes . heating processes may play a significant role in the thermal evolution or the visibility of pulsars in the soft x - ray band . pulsars with different magnetospheric properties may be undergoing different heating mechanisms . luminous nonthermal radiation and bright pulsar wind nebulae ( pwne ) , these observational manifestations may imply active magnetospheres . strong star wind or relativistic particle flow would be ejected from the poles of such a pulsar . a certain amount of backflow of such plasma induce the stellar heating , or an energy input could take place at the polar caps and disperse to the bulk of the star @xcite . the heat flow @xmath63 could thus be @xmath64 where @xmath65 and @xmath66 are the radius and the surface temperature of the polar caps in the local reference frame respectively . we adopted the one - dimensional approximation for the gradient of temperature , i.e. @xmath67 . the luminosity of the return - current stellar heating @xmath12 could be a function of the spin energy loss rate @xmath68 and could generally be in the form of a power law . moreover , a phenomenological study on the bolometric luminosity and @xmath68 reveals that the power index could be 1/2 or 1 , namely @xmath69 or @xmath70 ( see appendix a for the details ) . the metal - like interior of a sqs could imply that its thermal conductivity @xmath71 could be the sum of the partial components of phonons , electrons , and static impurities , or @xmath72 where the subscripts ` p ' , ` e ' , and ` imp ' denote the partial components mentioned above , respectively . the electron thermal conductivity @xmath73 could be the dominant component for the metal - like solid material @xcite . so phonon contribution @xmath74 would be neglected , and , as a preliminary model , we omit the partial component of static impurities . the electron thermal conductivity would be written as @xmath75 where @xmath76 is the partial component contributed by the collision between electrons , while @xmath77 is that contributed by the collision between phonons and electrons . for their analytic formulae , we refer to @xcite with assuming that the results remain hold for sqss . some pulsars , showing steady long - term soft x - ray fluxes , being inert in nonthermal emission , lacking the proves on the existences of the associated pwne , ( and sometimes ) owning excesses of x - ray luminosities relative to the spin energy loss rates , could alternatively be inactive pulsar candidates . due to their little magnetospheric manifestations , sqss will not suggest the thermal x - ray radiation of these sources is of spin origin . one possible way to understand the origin of the energy is the accretion in the _ propeller _ regime . in this regime , a shell of atmosphere of matter may form surrounding the star . matter closing to the inner boundary of the shell may interact directly with the rotating stellar magnetosphere , as a result of which most of the matter will be expelled outward @xcite . nevertheless , a certain fraction of the accreting material , described by the accretion efficiency @xmath78 , may diffuse starward and fall onto the surface of the star finally . in the propeller regime , the stellar magnetosphere radius or alfvn radius @xmath79 would be on one hand larger than its corotation radius @xmath80 , but is on the other hand somewhat smaller than the light cylinder radius @xmath81 , i.e. @xmath82 where @xmath83 is the gravitational constant , @xmath84 , @xmath60 , and @xmath85 are the stellar mass , radius , and the magnetic strength at the poles , respectively . the accretion rate @xmath86 could be scaled by eddington accretion rate @xmath87 ; so @xmath88 could denote the accretion rate in this unit , i.e. @xmath89 . during the accretion of the material which can reach the stellar surface eventually , the energy of gravitation would be released . furthermore , when the two - flavor baryonic matter impact upon the surface of a sqs , it will burn into the three - flavor strange quark matter phase , and the latent heat of @xmath90 mev @xmath91 mev per baryon could be released in the phase transition @xcite . the luminosity of stellar heating in this situation could then be @xmath92 where @xmath93 is the mass of a proton . llccccccc + no . & source & t ( kyr ) & @xmath94 ( mk ) & @xmath95 ( km ) & @xmath96 ( mk ) & @xmath97 ( km ) & @xmath98 ( @xmath99 ergs s@xmath100 ) & refs . + + 1 & psr b0531 + 21 & 1 & @xmath101 & @xmath102 & & & @xmath103 & ( 1 ) + & ( crab ) + 2 & psr j1811 - 1925 & 2 & @xmath104 & & & & & ( 2 ) + & ( in g11.2 - 0.3 ) + 3 & psr j0205 + 6449 & 0.82 - 5.4 & @xmath105 & @xmath106 & & & @xmath107 & ( 3 ) + & ( in 3c 58 ) + 4 & psr j1119 - 6127 & @xmath108 & @xmath109 & @xmath110 & & & @xmath111 & ( 4 ) + & ( in g292.2 - 0.5 ) + 5 & rx j0822 - 4300 & 2 - 5 & @xmath112 & @xmath113 & @xmath114 & @xmath115 & @xmath116 & ( 5 ) + & ( in pup a ) + 6 & psr j1357 - 6429 & @xmath117 & @xmath118 & @xmath119 & & & @xmath120 & ( 6 ) + 7 & rx j0007.0 + 7303 & 10 - 30 & @xmath121 & @xmath122 & & & @xmath123 & ( 7 ) + & ( in cta 1 ) + 8 & psr b0833 - 45 & 11 - 25 & @xmath124 & @xmath125 & @xmath126 & @xmath127 & @xmath128 & ( 8) + & ( vela ) + 9 & psr b1706 - 44 & @xmath129 & @xmath130 & @xmath131 & & & @xmath132 & ( 9 ) + & ( in g343.1 - 02.3 ) + 10 & psr b1823 - 13 & @xmath133 & @xmath134 & @xmath135 & & & @xmath1360.30 & ( 10 ) + 11 & psr j0538 + 2817 & @xmath137 & @xmath138 & @xmath139 & & & @xmath140 & ( 11 ) + & ( in s147 ) + 12 & psr b2334 + 61 & @xmath141 & @xmath142 & @xmath143 & & & @xmath144 & ( 12 ) + & ( in g114.3 + 0.3 ) + 13 & psr b1916 + 14 & @xmath145 & @xmath146 & @xmath147 & & & @xmath148 & ( 2 ) + 14 & psr b0656 + 14 & @xmath149 & @xmath150 & @xmath151 & @xmath152 & @xmath153 & @xmath154 & ( 13 ) + & ( in monogem ring ) + 15 & psr j0633 + 1746 & @xmath155 & @xmath156 & @xmath157 & @xmath158 & @xmath159 & @xmath160 & ( 14 ) + & ( geminga ) + 15@xmath161 & & & 0.482 & @xmath162 & & & & ( 15 ) + 16 & psr b1055 - 52 & @xmath163 & @xmath164 & @xmath165 & @xmath166 & @xmath167 & @xmath140 & ( 14 ) + 17 & psr j2043 + 2740 & @xmath168 & @xmath169 & @xmath170 & & & @xmath171 & ( 16 ) + 18 & 1e 1207.4 - 5209 & @xmath172 & @xmath173 & @xmath174 & @xmath175 & @xmath176 & @xmath177 & ( 17,18 ) + & ( in pks 1209 - 51/52 ) + 19 & cxou j232327.9 & 0.3 & @xmath178 & @xmath179 & & & @xmath180 & ( 19 ) + & + 584842 ( in cas a ) + 20 & cxou j085201.4 & @xmath181 & @xmath182 & @xmath183 & & & @xmath184 & ( 20 ) + & -461753 ( in g266.2 - 1.2 ) + 21 & psr j1852 + 0040 & & @xmath185 & @xmath186 & & & @xmath187 & ( 21 ) + & ( in kes 79 ) + 22 & psr j1713 - 3949 & & 4.4 & 2.4 & & & 15 & ( 22 ) + & ( in g347.3 - 0.5 ) + 23 & rx j1856.5 - 3754 & @xmath188 & @xmath189 & @xmath190 & & & @xmath191 & ( 23 ) + 24 & rx j0720.4 - 3125 & @xmath192 & @xmath193 & @xmath194 & & & @xmath195 & ( 24 ) + 25 & rbs 1223@xmath196 & @xmath197 & 1.04 & 0.8 & & & @xmath198 & ( 25 ) + 26 & rx j0420.0 - 5022 & @xmath149 & @xmath199 & 1.4 & & & 2.7@xmath200 & ( 26 ) + 27 & rx j0806.4 - 4123 & & @xmath201 & @xmath1360.6 & & & @xmath202 & ( 27 ) + 28 & rx j1605.3 + 3249 & & 1.07 & @xmath1361.1 & & & @xmath203 & ( 25 ) + 29 & rbs 1774@xmath204 & & 1.04 & @xmath1361.1 & & & @xmath203 & ( 25 ) + + @xmath196 1rxs j130848.6 + 212708 @xmath204 1rxs j214303.7 + 065419 diagram for the x - ray pulsar sample . the solid lines indicate the death lines both for a typical pulsar ( with mass of 1.4@xmath205 and a radius of 10 km ) and low - mass sqss ( with mass of @xmath206 ) , with surface magnetic field @xmath207 g and potential drop in open field line region @xmath208 v. the death lines move up if decreasing @xmath209 and/or increasing @xmath210 @xcite . the hexagonals indicate the focused sources associated with detectable supernova remnants ( snrs ) , while the pentagons indicate those which exhibit without evident snrs . the triangles indicate 4 xdins with measurable temporal parameters , and the diamonds mark the ccos . the timing parameters for rx j0822 - 4300 are from @xcite , while the rest are from atnf pulsar catalog@xmath211 .,scaledwidth=70.0% ] we concentrate on those x - ray sources , which demonstrate significant thermal emission , own ordinary magnetic fields @xmath212 g , own comparatively young ages @xmath213 yrs , and have spins of a few tens of milliseconds to a few seconds . we mainly refer to the collation made by @xcite on cooling neutron stars , by @xcite on x - ray pulsars , by @xcite on x - ray dim isolated neutron stars ( xdins ) , and by @xcite on central compact objects ( ccos ) . the sample exhibited in table [ tab : data ] thus comprises top 17 active pulsar candidates , 7 xdins ( no . 23 - 29 ) and 6 ccos ( no . 5 and 18 - 22 ) . we note that the values of the surface temperatures adopted are determined by black body fits , according to the way that a sqs emits thermal photons . the xdins and ccos are considered to be magnetosphere - inactive pulsar candidates , since their quiescent manifestations on nonthermal radiation . ccos , moreover , would only be seen in the soft x - ray band without the evidences on the existences of the associated pwne . additionally , we note that the emission of the energetic crab pulsar is much likely to be overwhelmed by the nonthermal component originating from its luminous plerion , which hampers the detection to the stellar surface thermal radiation . as an estimate , we , however , adopted a @xmath214 upper limit to crab s surface temperature and an inferred radius with assuming a 2 kpc distance between crab and the earth @xcite . rx j0822 - 4300 , the central stellar remnant in puppis a , used to be analyzed by @xcite basing on the _ rosat _ observations in 1990s . they could not confirm whether the x - ray structures surrounding the star belong to the supernova remnant ( snr ) or are induced by the probable active magnetosphere . recent observations on it by _ chandra _ and _ xmm - newton _ telescopes , however , did not reveal the presence of the associated plerion , indicating an inert magnetosphere @xcite . because of the ambiguity of this source , we temporarily make it as one of the active source candidates . furthermore , we emphasize these x - ray sources that own measurable spin parameters in the @xmath215 diagram ( fig . [ fig : ppdot ] ) , by which a distribution of them can be read . other 10 x - ray sources with constraints on the upper limits of their bolometric luminosities are also denoted in the diagram @xcite . besides the death line for a typical pulsar , those for low - mass sqss are also indicated in the diagram . these death lines set boundaries of the ` graveyards ' for pulsars with different mass @xcite . the timing parameters are from atnf pulsar catalog @xcite , except for rx j0822 - 4300 , which refers to @xcite . for magnetosphere - active pulsar candidates , sqss would suggest themselves to reproduce the cooling processes , while , for magnetosphere - inactive pulsar candidates , possible approaches to understand their current x - ray luminosities would also be proposed by sqss . these two will be presented in 3.1 and 3.2 . clllcc + no . & source & @xmath216 ( s@xmath100 ) & @xmath217 ( s@xmath218 ) & @xmath219 ( ergs s@xmath100 ) & @xmath220 + + 1 & psr b0531 + 21 ( crab ) & 30.225437 & -3.862@xmath221 & 7.7@xmath222 & 4.1@xmath223 + 2 & psr j1811 - 1925 & 15.463838 & -1.052@xmath224 & & + 3 & psr j0205 + 6449 & 15.223856 & -4.495@xmath224 & 5.8@xmath225 & 4.6@xmath223 + 4 & psr j1119 - 6127 & 2.452508 & -2.419@xmath224 & 1.7@xmath226 & 8.5@xmath227 + 5 & rx j0822 - 4300 & 13.2856716499(3 ) & -2.6317(3)@xmath224 & 2.7@xmath226 & 2.8@xmath227 + 6 & psr j1357 - 6429 & 6.020168 & -1.305@xmath224 & 4.4@xmath225 & 1.2@xmath227 + 7 & rx j0007.0 + 7303 & 3.165922 & -3.623@xmath228 & 3.5@xmath229 & 8.8@xmath223 + 8 & psr b0833 - 45 ( vela ) & 11.194650 & -1.567@xmath224 & 1.4@xmath225 & 4.5@xmath223 + 9 & psr b1706 - 44 & 9.759978 & -8.857@xmath228 & 4.2@xmath225 & 1.1@xmath227 + 10 & psr b1823 - 13 & 9.855532 & -7.291@xmath228 & 3.2@xmath225 & 1.1@xmath227 + 11 & psr j0538 + 2817 & 6.985276 & -1.790@xmath230 & 4.3@xmath226 & 9.3@xmath200 + 12 & psr b2334 + 61 & 2.018977 & -7.816@xmath230 & 1.8@xmath222 & 1.7@xmath200 + 13 & psr b1916 + 14 & 0.846723 & -1.523@xmath230 & 1.8@xmath222 & 5.9@xmath200 + 14 & psr b0656 + 14 & 2.598137 & -3.713@xmath230 & 1.1@xmath231 & 1.7@xmath232 + 15 & psr j0633 + 1746 ( geminga ) & 4.217640 & -1.952@xmath230 & 3.2@xmath225 & 9.9@xmath227 + 16 & psr b1055 - 52 & 5.073371 & -1.501@xmath230 & 6.9@xmath226 & 1.5@xmath232 + 17 & psr j2043 + 2740 & 10.402519 & -1.374@xmath230 & 7.1@xmath229 & 3.5@xmath227 + + @xmath196 @xmath233 g @xmath234 ) if two isotropic black body emission components the hot component for polar caps and the warm one for the bulk of the star are defined for sqss , then the equation describing cooling processes would be written as ( cf . equation ( [ eq : coolinggeneral ] ) ) @xmath235 and the relation between @xmath14 and @xmath66 is given by equation ( [ eq : h ] ) . the luminosity of stellar heating @xmath12 could either follow the 1/2-law or the linear - law ( see appendix a ) . cooling behaviors would thus be calculated by assuming sqss rotate as orthogonal rotators and slow down as a result of magnetic dipole radiation with magnetic field strength at the poles of @xmath236 g. we provided a parameter space to fit the observational temperature - age data , and the comparison between the observations and expectations are shown in figs . [ fig : ccos ] ( for 1/2-law case ) and [ fig : cclr ] ( for linear - law case ) . some pulsars demonstrate two black body components in their thermal spectra , e.g. vela pulsar and the three musketeers , implying that the temperature inhomogeneity on these pulsars could be significant . we thus carried out a temperature - difference fit at the same time , as have been shown in the _ right _ panels in both figs . [ fig : ccos ] and [ fig : cclr ] . it is worthy of noting that , for geminga , if the photon index of the power - law ( pl ) component is thawed when fitting its phase - resolved spectra , the entry of the hot black body component could not improve the fits remarkably , or it may even becomes an artifact . this may mean the fluctuation of the magnetospheric emission during a spin cycle might mislead the understanding on its x - ray spectra , as analyzed by @xcite and their results are denoted by no . 15@xmath161 in table [ tab : data ] . if this situation holds , the surface temperature fluctuation on geminga could be tiny so that undetectable . we , hence , set a rough upper limit to the temperature difference for geminga by an order lower than its surface temperature , as denoted by 15@xmath161 in the ` temperature difference - age ' figures . the proposed relation between the rotational kinetic energy loss rate and the bolometric luminosity could inversely provide direct measurement to the moments of inertia of the active x - ray sources . the results are exhibited in table [ tab : moi ] and could be understood in the sqs regime , especially low - mass sqss ( cf . the adoption of the coefficients @xmath237 and @xmath238 in figs . [ fig : ccos ] and [ fig : cclr ] ) . we note that for sqss with 0.01@xmath205 , 0.1@xmath205 , 1.0@xmath205 and 2.0@xmath205 , the moments of inertia @xmath239 ( i.e. values scaled by @xmath240 g @xmath241 ) are @xmath242 , @xmath620.01 , @xmath620.55 and @xmath621.74 respectively . ccc + mass & @xmath243 @xmath205 yr@xmath100 ) & @xmath244 ergs s@xmath100 ) + + @xmath245 & 2.7 & 1.7@xmath246 + @xmath247 & 5.7 & 4.7@xmath246 + @xmath248 & 12.4 & 2.0 + + as has been discussed in 2.3.2 , one probable energy source to be responsible for the soft x - ray emission of magnetosphere - inactive pulsar candidates lies in the accretion in the _ propeller _ regime . the accretion could be either to the interstellar medium or to the fallback disk . in this regime , however , the stellar magnetosphere radius could hardly be given , even though the sources spindown rates are detectable . because the spindown rate in this case would be dominated by the accretion behavior . equation ( [ eq : coolinggeneral ] ) for this case could be specified as @xmath249 with considering equation ( [ eq : polarpowerd ] ) and the stellar residual thermal energy is too small to be taken into account . table [ tab : lum ] lists the luminosities of sqss under the accretion scenario . it could be concluded that the sources current observational thermal x - ray luminosities would be interpreted by considering the parameters @xmath88 , the accretion rate in the unit of that of eddington , and @xmath78 , the accretion efficiency , such as @xmath250 , @xmath251 . we collate the thermal observations of 29 x - ray isolated pulsars and , in the sqs regime , for the magnetospherically active pulsar candidates , establish their cooling processes ( figs . [ fig : ccos ] and [ fig : cclr ] ) , while for the magnetospherically inactive or dead pulsar candidates , interpret the x - ray luminosities under the accretion scenario . ( table [ tab : lum ] ) . fitting to the thermal spectra for x - ray pulsars using the black body model often results in such small emission sizes that could naturally be interpreted by the sqs model . a recent study on the cassiopeia a cco shows that the fitted emission size is still significantly smaller than a typical neutron star radius , even if the hydrogen atmosphere model is employed in the fitting @xcite . the uncertain estimates on the distances of pulsars may introduce considerable errors in the fitting about the x - ray source emission sizes . however , if these obtained emission radii can be trusted , sqss , because of the possibility of being low - mass , could provide an approach to understand these observational manifestations . we note that for sqss with mass of @xmath252 , @xmath253 and @xmath254 , their radii are @xmath621.8 , @xmath623.8 and @xmath628.3 km respectively . on the other hand , a linkage between pulsar rotational kinetic energy loss rates and bolometric x - ray luminosities is explored by sqss ( see appendix a ) , and the resulting estimates on pulsar moments of inertia is exhibited in table [ tab : moi ] . we hence conclude that the phenomenological sqs pulsar model could not be ruled out by the thermal observations on x - ray isolated pulsars , though a full depiction on the thermal evolution for a quark star in all stages could hardly be given nowadays , as the lack of the physics in some extreme conditions ( as has been discussed in 2.1 ) . sqss have significant distinguishable interiors with neutron stars , but the structures of the magnetosphere between these two pulsar models could be similar . therefore , sqss and neutron stars would have similar heating mechanisms , such as the bombardment by backflowing particles and the accretion to surrounding medium . however , a full comparison between sqss and neutron stars including their cooling processes as well as the heating mechanisms has beyond the scope of this paper , and should be interesting and necessary in the future study . the various performances of x - ray pulsars may indicate their current states or properties and imply their possible evolutionary history . in addition , if these x - ray sources are actually sqss , the possible formation mechanism could be an interesting topic . we thus extend a discussion in 4.3 . spin has always been a significant energy source for active pulsars , by which multiwave bands nonthermal radiation are driven , including those of the pulsar themselves as well as those of the surrounding plerions . some phenomenological studies demonstrate certain regularities for such an active pulsar population . a brief summary is that the nonthermal x - ray luminosity and spin energy loss rate own a relation of @xmath255 @xcite , and the @xmath256-ray luminosity is proportional to the square root of @xmath68 or @xmath257 @xcite . we , additionally , note that @xmath258 , where @xmath259 is the potential drop along the open field region . therefore , it seems that for such a population the more younger the larger @xmath68 and @xmath259 that a pulsar owns ( cf . [ fig : edot ] , _ bottom _ panel ) and the more luminous of the radiation in the hard x - ray and @xmath256-ray bands . besides nonthermal emission , spin may also be an energy origin for pulsar thermal radiation as the relations of @xmath2 or @xmath260 could exist observationally ( see appendix a for details ) . sqss predominantly follow these relations to accomplish their cooling processes , since their residual thermal energy could be quite inadequate to sustain a long term x - ray thermal emission . xdins and ccos could be representative populations of magnetosphere - inactive pulsars . nevertheless , their energy origin that could power the x - ray emission is still an enigma . the bolometric luminosities of 1e 1207.4 - 5209 , rx j1856.5 - 3754 and rx j0720.4 - 3125 exceed their spin energy loss rates by a factor of @xmath261 , @xmath262 and @xmath263 , respectively ; they appear not to be powered by spins . ccos could be a group of weakly - magnetized and long - initial - period pulsars @xcite , because of which their potential drops along the open field lines could be much less than @xmath264 v , so that the primary plasma could not be fully accelerated , resulting in inactive magnetospheres . the black body fits to the thermal spectra of ccos always result in extremely small emission sizes @xcite , implying the possible existence of low - mass sqss . therefore , the @xmath68 of ccos might be very low , so that effective magnetospheric emission and plerions could not be driven . ccos could thus be the representatives of a group of natal inactive pulsars . accretion to the fallback ejecta of the associated snrs could power their soft x - ray luminosities , and this scenario have been mentioned by several authors ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? xdins , owning comparatively larger spindown ages and stronger magnetic fields , might be the descendants of magnetars . considering similar properties between them @xcite , one could not exclude that xdins are still being powered by the decaying magnetic fields . however , observations have revealed the probable existence of residual disks around such pulsars , and thus their radiation may be of accretion origin @xcite . in this case , xdins could be the evolved products of active pulsars , and thanks to the accretion so that they are still visible after their death . most xdins have absorption features in their spectra , being similar to those of ccos . hence , if they are really undergoing accretion in the propeller regime , their spindown rates @xmath265 could then be of accretion origin rather than causing by the magnetic dipole braking . thus their magnetic fields could be much lower than the values determined by canonical magnetic dipole radiation , since the absorption features could be electron cyclotron lines . therefore , the common properties of absorption features between xdins and ccos could imply that xdins might be older ccos . xdins and ccos have not manifested themselves as radio pulsars , their radio - quiet demonstrations could be an intrinsic property rather than being of beaming origin , since they might be dead or natal inactive pulsars . so they could be grouped as radio - quiescent pulsars . for rx j0007.0 + 7303 , an active pulsar candidate , its lack of radio signal could be the result of an unfavorable geometry , or its radio beam sweeps away from the earth as has been analyzed by @xcite . the similar thermal manifestations of accreting pulsars ( e.g. xdins and ccos ) and cooling pulsars may have caused a confusion about distinguishing between the two classes @xcite . however , the activeness of the magnetosphere could provide a way to achieve an identification . if pulsars are in fact sqss with rapid rotation , cooling pulsars are then likely to be undergoing the spindown - powered heating evolution , and multi - bands nonthermal emission originating from luminous magnetospheres and even pwne will accompany such coolers during the processes . in contrast , accreting pulsars may have not demonstrated such characteristics observationally . in this context , the cco in the snr cassiopeia a ( source no . 19 in table [ tab : data ] ) should be an accretor because of the lack of pwn and nonthermal power - law component @xcite . possible low - mass compact stars is a direct consequence of the suggestion that pulsars could be quark stars @xcite . however , the probable existence of quark stars , spanning large mass range , could naturally raise such a question : how did these stars be created ? indeed , this could be another severe problem given by the observations . being different from solar - mass quark stars ( which could originate from core - collapse supernovae ) , low - mass quark stars could be the central remnants left by the detonation of the accretion - induced collapse ( aic ) of white dwarfs ( wds ) @xcite . the combustion from the hadronic matter to the strange quark matter could be the engine of such an explosion . during this phase transition , as high as @xmath266 mev per baryon could be released as the latent heat @xcite . the combustion could hence be in such a high efficiency that % 10 of the rest mass is liberated . for a wd approaching the chandrasekhar limit , its mass and radius are @xmath267 , @xmath268 cm respectively . the gravitational energy is then @xmath269 ergs . for a successful explosion of such a wd , the necessary mass of a quark star @xmath270 would hence be obtained by @xmath271 as @xmath272 . so , in this case , low - mass quark stars with a few tenth of or a few hundredth of solar mass could form , although one should be aware that the exact value of the quark star mass depends on the region where the detonation wave inside the wd can sweep though as well as the latent heat of the phase transition , which would be strongly model - dependent . these nascent quark stars would keep bare if they has not suffered massive super - eddington accretion @xcite . an accreted ion ( e.g. a proton ) should gain enough kinetic energy to penetrate the coulomb barrier of a quark star with mass @xmath273 and radius @xmath274 , or @xmath275 . the coulomb barrier @xmath276 is model - dependent , which could varies from @xmath7 mev to even @xmath277 mev @xcite . considering @xmath278 for low - mass quark stars , a mass lower - limit @xmath279 would then be determined as @xmath280 where the density @xmath281 , with @xmath282 the nuclear density , and @xmath283 mev . apart from the formation of the low - mass quark stars discussed here , other astrophysical processes , such as the collision between two strange stars , could also be the potential sources to give birth to the quark stars with smaller mass than that of the sun @xcite . ( _ top _ panel ) and @xmath284 ( _ bottom _ panel ) of active pulsar candidates . in the _ top _ panel , the fits are carried out both for group a ( including top 17 pulsars listed in table [ tab : data ] ; they are marked by dark points and their numbers ) and group b ( including the members in group a and the other 10 sources whose upper limits on the bolometric luminosities could be defined observationally ; the 10 sources are marked by grey points and letters ) . red lines are the fitted results for group a , while blue lines are those for group b. in both groups , solid lines provide the best fits to the data , while the dashed lines give the fits by freezing @xmath285 at 1 . we note that the 10 sources with upper limits on their bolometric luminosities defined are taken from @xcite , and they are a. b1509 - 58 , b. b1951 + 32 , c. b1046 - 58 , d. b1259 - 63 , e. b1800 - 21 , f. b1929 + 10 , g. b0540 - 69 , h. b0950 + 08 , i. b0355 + 54 , j. b0823 + 26.,scaledwidth=70.0% ] ccccc + group & @xmath285@xmath196 & @xmath286 & corr . coef.@xmath204 & @xmath287 ( d.o.f)@xmath288 + + a. & @xmath289 & @xmath290 & & 0.3277(14 ) + & ( 1 ) & @xmath291 & 0.7487 & 0.8607(15 ) + b. & @xmath292 & @xmath293 & & 0.6081(24 ) + & ( 1 ) & @xmath294 & 0.7730 & 0.9964(25 ) + + @xmath196 values in the parenthesis are frozen during the fits . the errors are in 95% confidence level . @xmath204 correlation coefficient of the data between bolometric luminosity and @xmath68.@xmath288 reduced @xmath295 , or @xmath295 per degree of freedom ( d.o.f ) . in the solid quark star(sqs ) regime , cooling of magnetosphere - active pulsars depends predominantly on the stellar heating as a result of the lack of sufficient residual heat as has been analyzed in 2.2 . the pulsar activity induced stellar heating would have intrinsically set up a linkage between the x - ray bolometric luminosity and the spin energy loss rate . we , hence , present here a phenomenological study on such a relation . the sample of active pulsars firstly includes the top 17 sources listed in table [ tab : data ] . @xcite summarized x - ray pulsars , in which other 10 sources upper limits on the bolometric luminosities could be defined . these 10 pulsars are meanwhile considered . [ fig : edot ] ( _ top _ panel ) illustrates these sources bolometric luminosities @xmath296 as a function of their spin energy loss rates @xmath68 , and table [ tab : fit ] lists the fitted parameters led by the function of @xmath297 the fits are carried out both only for the top 17 pulsars taken from table [ tab : data ] ( as group a ) and for the whole sample with the other 10 sources included at the same time(as group b ) . the best fits , moderate clearly , suggest a 1/2-law on the relation between pulsar bolometric x - ray luminosities and the spin energy loss rates , or @xmath298 with a coefficient @xmath299 in the unit of ergs@xmath300 s@xmath301 . if @xmath285 is fixed at 1 , a linear - law is then implied , i.e. @xmath302 with a coefficient @xmath303 , or @xmath238 is the conversion efficiency . the fits obtain @xmath304 , which is similar to the nonthermal x - ray case @xcite . for the 1/2-law case , the conversion efficiency turns out to be the function of @xmath68 , which would be @xmath305 which could in turn , besides the fits , provide a reference to the value of the coefficient @xmath237 by a natural constraint of @xmath306 . taking b0823 + 26 , the minimum @xmath68 in our sample as an example , its @xmath237 would be less than @xmath307 ergs@xmath300 s@xmath301 . additionally , we note here that we also present a age-@xmath68 relation in the _ bottom _ panel of fig . [ fig : edot ] , which illustrates a natural trend for spin - powered pulsars . the authors are grateful to prof . fredrick jenet for his valuable comments and advices , especially for his efforts on improving the english expression of this paper . we wish to thank mr . weiwei zhu for his detailed introduction to the observational situation about the sources psr j1811 - 1925 and psr b1916 + 14 . the authors also thank the colleagues in the pulsar group of peking university for the helpful discussion . this work is supported by nsfc ( 10778611 , 10973002 ) , the national basic research program of china ( grant 2009cb824800 ) , and by lcwr ( lhxz200602 ) .
|
we present a theoretical model for the thermal x - ray emission properties and cooling behaviors of isolated pulsars , assuming that pulsars are solid quark stars .
we calculate the heat capacity for such a quark star , including the component of the crystalline lattice and that of the extremely relativistic electron gas .
the results show that the residual thermal energy can not sustain the observed thermal x - ray luminosities seen in typical isolated x - ray pulsars .
we conclude that other heating mechanisms must be in operation if the pulsars are in fact solid quark stars .
two possible heating mechanisms are explored .
firstly , for pulsars with little magnetospheric activities , accretion from the interstellar medium or from the material in the associated supernova remnants may power the observed thermal emission . in the propeller regime
, a disk - accretion rate @xmath01 % of the eddington rate with an accretion onto the stellar surface at a rate of @xmath1 could explain the observed emission luminosities of the dim isolated neutron stars and the central compact objects .
secondly , for pulsars with significant magnetospheric activities , the pulsar spindown luminosities may have been as the sources of the thermal energy via reversing plasma current flows . a phenomenological study between pulsar bolometric x - ray luminosities and the spin energy loss rates
presents the probable existence of a 1/2-law or a linear law , i.e. @xmath2 or @xmath3 .
this result together with the thermal properties of solid quark stars allow us to calculate the thermal evolution of such stars .
thermal evolution curves , or cooling curves , are calculated and compared with the ` temperature - age ' data obtained from 17 active x - ray pulsars .
it is shown that the bolometric x - ray observations of these sources are consistent with the solid quark star pulsar model . _
pacs : _
97.60.gb , 97.60.jd , 95.30.cq pulsars ; neutron stars ; elementary particles
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
in this paper , we consider the following stochastic differential equation ( sde ) driven by lvy noises : @xmath18 where @xmath4 is a @xmath19-dimensional lvy process , and @xmath20 is a continuous vector field such that for any @xmath9 , @xmath21 holds for some constant @xmath22 . it is a standard result that in this case the sde ( [ ito - sde ] ) enjoys the unique strong solution . denote by @xmath0 the semigroup associated to ( [ ito - sde ] ) . if the initial value @xmath23 is distributed as @xmath24 , then for any @xmath16 , the distribution of @xmath25 is @xmath26 . we are concerned with the exponential contractivity of the map @xmath27 with respect to the standard @xmath1-wasserstein distance @xmath28 for all @xmath29 . given two probability measures @xmath24 and @xmath30 on @xmath6 , the standard @xmath1-wasserstein distance @xmath28 for all @xmath2 ( with respect to the euclidean norm @xmath31 ) is given by @xmath32 equipped with @xmath28 , the totality @xmath33 of probability measures having finite moment of order @xmath34 becomes a complete metric space . the following result is well known . [ theorem1 ] suppose that there exists a constant @xmath35 such that @xmath36 then , for any @xmath37 and @xmath16 , @xmath38 the proof of this result is quite straightforward , by simply using the synchronous coupling , which is also called the basic coupling or the coupling of marching soldiers in the literature ( see , e.g. , @xcite , definition 2.4 and @xcite , example 2.16 ) . the reader can refer to @xcite , page 2432 , the proof of theorem [ theorem1 ] for the case of diffusion processes . ( [ diss-1 ] ) is the so - called uniformly dissipative condition , which seems to be a limit in applications . for diffusion processes , it follows from @xcite , theorem 1 , or @xcite , remark 3.6 ( also see @xcite , theorem 3.6 ) that ( [ w2 ] ) holds for any probability measures @xmath24 and @xmath30 if and only if ( [ diss-1 ] ) holds for all @xmath39 , @xmath40 . the first breakthrough to get rid of such restrictive condition in this direction for @xmath41-wasserstein distance @xmath42 was done recently by eberle in @xcite , at the price of multiplying a constant @xmath43 on the right - hand side of ( [ w2 ] ) . see @xcite , corollary 2.3 , for more details , and @xcite , theorem 1.3 , for related developments on @xmath1-wasserstein distance @xmath28 with all @xmath2 on this topic . however , the corresponding result for sdes driven by lvy noises is not available yet now . indeed , we will see later that in this case we need a completely different idea for the construction of the coupling processes , and a new approach by using the coupling argument , in particular the more delicate choice of auxiliary functions . throughout this paper , we assume that the driving lvy process has a symmetric @xmath5-stable process as a component . that is , let @xmath30 be the lvy measure of the process @xmath4 , then @xmath44 where latexmath:[$c_{d,\alpha}=2^\alpha\gamma((d+\alpha)/2)\uppi^{-d/2 } measure of a symmetric @xmath5-stable process or fractional laplacian , that is , @xmath46 denote by @xmath47 the surface measure of the unit sphere in @xmath6 . our main contribution of this paper is as follows . [ main - result ] assume that for any @xmath9 , @xmath48 holds with some positive constants @xmath11 , @xmath12 , @xmath13 and @xmath14 . then for all @xmath7 or for all @xmath49 $ ] with @xmath50 there exists a positive constant @xmath51 , such that for any @xmath52 the following two statements hold : if @xmath53 , then for all @xmath39 , @xmath40 and any @xmath16 , @xmath54 if @xmath55 , then for all @xmath39 , @xmath40 and any @xmath16 , @xmath56,\ ] ] where @xmath22 is a positive constant depending on @xmath57 , @xmath11 , @xmath12 , @xmath58 and @xmath34 . theorem [ main - result ] above does provide new conditions on the drift term @xmath8 such that the associated semigroup @xmath0 is exponentially contractive with respect to the @xmath1-wasserstein distance @xmath28 for all @xmath37 . in particular , when @xmath7 , the conclusion of theorem [ main - result ] is the same as that of @xcite , theorem 1.3 , for diffusion processes ; while for @xmath49 $ ] we need the restrictive condition ( [ main - result-1 - 1 ] ) ; see remark [ main-1 - 1 ] for a further comment . indeed , ( [ main - result-1 - 1 ] ) is natural in the sense that , when @xmath49 $ ] the drift term plays the dominant role or the same role ( just in case that @xmath59 ) for the behavior of sdes driven by symmetric @xmath5-stable processes , see , for example , @xcite for ( dirichlet ) heat kernel estimates and @xcite for dimensional free harnack inequalities on this topic . similarly , in considering the exponential contractivity of sde ( [ ito - sde ] ) , we need ( [ main - result-1 - 1 ] ) to control the locally non - dissipative part of the drift term . note that ( [ main - result-1 - 1 ] ) holds true when @xmath60 are small enough . to show the power of theorem [ main - result ] , we consider the following example about the sde driven by symmetric @xmath5-stable processes with @xmath61 , which yields the exponential contractivity of the semigroup @xmath0 with respect to the @xmath1-wasserstein distance @xmath28 @xmath62 for super - convex potentials . [ 1-example-2 ] let @xmath4 be a symmetric @xmath5-stable process in @xmath6 with @xmath61 , and @xmath63 with @xmath64 and @xmath65 . then there exists a constant @xmath66 such that for all @xmath37 , @xmath39 , @xmath40 and @xmath16 , @xmath67.\ ] ] note that the uniformly dissipative condition ( [ diss-1 ] ) fails for example [ 1-example-2 ] ; see , for example , ( [ cg-1 ] ) below . that is , one can not deduce directly from theorem [ theorem1 ] the exponential contractivity with respect to the @xmath68-wasserstein distance @xmath69 for all @xmath2 . the remainder of this paper is arranged as follows . in the next section , we will present the coupling by reflection mixed with the synchronous coupling for the sde ( [ ito - sde ] ) driven by lvy noise , and also prove the existence of coupling process associated with this coupling ( operator ) . section [ section3 ] is mainly devoted to the proof of theorem [ main - result ] . for this , we need more delicate choice of auxiliary functions and some key estimates for them , which are different between @xmath7 and @xmath49 $ ] . the sketch of the proof of example [ 1-example-2 ] is also given here . it is easy to see that the generator of the process @xmath70 acting on @xmath71 is @xmath72 in this part , we construct a coupling operator for the generator @xmath73 above . for any @xmath39 , @xmath74 and @xmath75 , we set @xmath76 it is clear that @xmath77 has the following three properties : @xmath78 and @xmath79 , that is , @xmath80 ; @xmath81 ; @xmath82 and @xmath83 . next , for any @xmath84 , let @xmath85 now , let @xmath58 be the constant appearing in ( [ main - result-1 ] ) . we will split the construction of the coupling operator into two parts , according to @xmath9 with @xmath86 or with @xmath87 . first , for any @xmath84 and @xmath9 with @xmath86 , we define @xmath88 \\ & & { } + \int _ { \{|z|\le a{|x - y| } \ } } \bigl(f(x+z , y+z)-f(x , y)-\bigl\langle \nabla_xf(x , y ) , z\bigr\rangle \mathbf{1}_{\ { \\ & & \hspace*{63pt}{}-\bigl\langle\nabla_yf(x , y ) , z\bigr\rangle \mathbf{1}_{\{|z|\le 1\ } } \bigr ) \biggl(\nu(\mathrm{d } z)-\frac{c_{d,\alpha}}{|z|^{d+\alpha}}\ , \mathrm{d}z \biggr ) \\ & & { } + \int _ { \{|z|>a{|x - y| } \ } } \bigl(f(x+z , y+z)-f(x , y)-\bigl\langle \nabla_xf(x , y ) , z\bigr\rangle \mathbf{1}_{\{|z|\le1\ } } \\ & & \hspace*{63pt}{}-\bigl\langle\nabla_yf(x , y ) , z\bigr\rangle \mathbf{1}_{\{|z|\le1\ } } \bigr ) \nu(\mathrm{d}z ) \\ & & { } + \bigl\langle\nabla_x f(x , y ) , b(x)\bigr\rangle+ \bigl \langle\nabla_y f(x , y ) , b(y)\bigr\rangle,\end{aligned}\ ] ] where @xmath89 is a constant determined by later . on the other hand , for any @xmath84 and @xmath90 with @xmath87 , we define @xmath91 we can conclude the following . the operator @xmath92 defined by above is the coupling operator of the operator @xmath73 given by ( [ ope-0 ] ) . since @xmath93 is a linear operator , it suffices to verify that @xmath94 where , on the left - hand side , @xmath95 is regarded as a bivariate function on @xmath96 , that is , @xmath97 for all @xmath98 . for any @xmath9 with @xmath87 , it is trivial to see that ( [ coup - proof ] ) holds true , and so we only need to verify that for @xmath90 with @xmath86 . first , we have @xmath99 \\ & & { } + \int _ { \{|z|\le a{|x - y| } \ } } \bigl(f(x+z)-f(x)-\bigl\langle \nabla f(x ) , z\bigr\rangle \mathbf{1}_{\{|z|\le1\ } } \bigr ) \biggl(\nu ( \mathrm{d}z)- \frac { c_{d,\alpha}}{|z|^{d+\alpha } } \,\mathrm{d}z \biggr ) \\ & & { } + \int _ { \{|z| > a{|x - y| } \ } } \bigl(f(x+z)-f(x)-\bigl\langle \nabla f(x ) , z\bigr\rangle \mathbf{1}_{\{|z|\le1\ } } \bigr ) \nu(\mathrm { d}z ) \\ & & { } + \bigl\langle b(x),\nabla f(x)\bigr\rangle.\end{aligned}\ ] ] by ( a1 ) and ( a2 ) , we know that the measure @xmath100 is invariant under the transformation @xmath101 . this , along with ( a2 ) and the equality above , leads to @xmath102 this completes the proof . \(1 ) here , we give an interpretation of the construction of the coupling operator @xmath103 above . if @xmath87 , we use the synchronous coupling . if @xmath86 , then the coupling operator @xmath92 constructed above consists of two parts . fix any @xmath104 . if @xmath105 , then we adopt the coupling by reflection by making full use of the rotationally invariant measure @xmath106 ; while for the remainder term , we use the synchronous coupling again , where the components maintain at each step the same length of jumps ( i.e. , from @xmath107 to @xmath108 ) with the biggest rate @xmath109 when @xmath110 , and with the rate @xmath111 when @xmath105 . for the coupling by reflection for brownian motion and diffusion processes , we refer to . \(2 ) recently , the coupling property of lvy processes has been developed in @xcite . the corresponding property for ornstein uhlenbeck processes with jumps also has been successfully studied in @xcite . unlike lvy processes and ornstein uhlenbeck processes with jumps , it is impossible to write out an explicit expression for transition functions of the solution to the sde ( [ ito - sde ] ) with general drift term @xmath112 . this observation indicates that all the approaches in @xcite are not efficient in the present setting . this difficulty will be overcome by constructing proper coupling operators for the markov generator corresponding to the solution of the sde ( [ ito - sde ] ) , as done in @xcite . however , different from @xcite which deals with the corresponding coupling property by making full use of large jumps part of lvy processes , here to consider the exponential contractivity of the associated semigroups @xmath0 with respect to wasserstein distances we need a new construction of the coupling operator . as seen from propositions [ key1 ] and [ key2 ] below , the coupling for small jumps part of lvy processes [ i.e. , the coupling by reflection as mentioned in ( 1 ) ] is key for our purpose . in this part , we will construct a coupling process associated with the coupling operator @xmath92 . for this , we will frequently talk about the martingale problem for the operator @xmath73 given by ( [ ope-0 ] ) and the coupling operator @xmath92 . let @xmath113 be the space of right continuous @xmath6-valued functions having left limits on @xmath114 , equipped with the skorokhod topology . for @xmath115 , denote by @xmath25 the projection coordinate map on @xmath113 . a probability measure @xmath116 on the skorokhod space @xmath113 is said to be a solution to the martingale problem for @xmath117 with initial value @xmath118 if @xmath119 and for every @xmath120 @xmath121 is a @xmath116-martingale . the martingale problem for @xmath117 is said to be well - posed if it has a unique solution for every initial value @xmath118 . similarly , we can define a solution to the martingale problem for the coupling operator @xmath92 on @xmath122 . note that , in @xcite an equivalence is proved between the existence of weak solutions to sdes with jumps and the existence of solutions to the corresponding martingale problem , by using a martingale representation theorem . recently , kurtz @xcite studied equivalence between the uniqueness ( in sense of distribution ) of weak solutions to a class of sdes driven by poisson random measures and the well - posed solution to martingale problems for a class of non - local operators using a non - constructive approach . note that in our setting the sde ( [ ito - sde ] ) has the pathwise unique strong solution . according to @xcite , theorem 1 , page 2 , the weak solution to the sde ( [ ito - sde ] ) enjoys the unique ( in sense of distribution ) weak solution . this , along with @xcite , corollary 2.5 , yields that the martingale problem for @xmath117 is well posed . let @xmath123 be the constants in the definition of the coupling operator @xmath92 . for any @xmath39 , @xmath40 and @xmath124 , set @xmath125 then , for any @xmath39 , @xmath40 and @xmath126 , we have @xmath127 { \mu } ( x , y,\mathrm{d}u_1,\mathrm{d}u_2 ) \\ & & \qquad { } + \bigl\langle\nabla_x f(x , y),b(x)\bigr\rangle+ \bigl\langle \nabla_y f(x , y),b(y)\bigr\rangle.\end{aligned}\ ] ] furthermore , for any @xmath128 , by ( a2 ) , @xmath129 which implies that @xmath130 is a continuous function on @xmath96 . note that @xmath112 is a continuous function on @xmath6 . according to @xcite , theorem 2.2 , there is a solution to the martingale problem for @xmath93 , that is , there exist a probability space @xmath131 and an @xmath132-valued process @xmath133 such that @xmath133 is @xmath134-progressively measurable , and for every @xmath84 , @xmath135 is an @xmath134-local martingale , where @xmath136 is the explosion time of @xmath133 , that is , @xmath137 let @xmath138 . then @xmath139 and @xmath140 are two stochastic processes on @xmath141 . since @xmath93 is the coupling operator of @xmath73 , the generator of each marginal process @xmath139 and @xmath140 is just the operator @xmath73 , and hence both distributions of the processes @xmath139 and @xmath140 are solutions to the martingale problem of @xmath73 . in particular , by our assumption and the remark in the beginning of this subsection , the processes @xmath139 and @xmath140 are non - explosive , hence one has @xmath142 a.s . therefore , the coupling operator @xmath93 generates a non - explosive process @xmath133 . let @xmath143 be the coupling time of @xmath139 and @xmath140 , that is , @xmath144 then @xmath143 is an @xmath145-stopping time . define a new process @xmath146 as follows : @xmath147 for any @xmath148 and @xmath16 , @xmath149 by the optimal stopping theorem and the facts that both @xmath150 and @xmath140 are solutions to the martingale problem of @xmath73 , @xmath151 and @xmath152 are martingales and so is @xmath153 ( see , e.g. , @xcite , section 3.1 , page 251 ) . since the martingale problem for the operator @xmath73 is well - posed , @xmath153 and @xmath140 are equal in the distribution . therefore , we conclude that @xmath154 is also a non - explosive coupling process of @xmath70 such that @xmath155 for any @xmath156 and the generator of @xmath154 before the coupling time @xmath143 is just the coupling operator @xmath93 . in particular , according to @xcite , lemma 2.1 , we know that for any @xmath39 , @xmath40 and @xmath157 , @xmath158 and @xmath159 where @xmath160 is the expectation of the process @xmath154 with starting point @xmath107 . we first assume that @xmath7 . for any @xmath161 , define @xmath162 $ ; \vspace*{3pt}\cr a\mathrm{e}^{c_2(r-2l_0)}+b(r-2l_0)^2 + \bigl(1-\mathrm{e}^{-2c_1l_0}-a \bigr ) , & \quad $ r\in [ 2l_0,\infty)$,}\ ] ] where @xmath163 @xmath164 is a positive constant such that @xmath165 , that is , @xmath166 and @xmath167 is a positive constant determined by later . with the choice of the constants @xmath168 and @xmath169 above , it is easy to see that @xmath170 . then we have : [ key1 ] assume that @xmath7 . then there exists a constant @xmath171 such that for any @xmath9 , @xmath172 \(1 ) in this part , we treat the case that @xmath104 with @xmath86 . first , for any @xmath173 , by ( a3 ) , @xmath174 and so @xmath175 therefore , @xmath176 \\ & & \qquad\quad { } + \psi'\bigl(|x - y|\bigr ) \frac{\langle b(x)-b(y),x - y\rangle}{|x - y|}.\end{aligned}\ ] ] it is easy to see that @xmath177 such that @xmath178 , @xmath179 and @xmath180 on @xmath181 . then , for any @xmath182 , @xmath183 where in the inequality we have used the fact that @xmath180 on @xmath181 . hence , according to the definition of @xmath184 and the inequality above , for all @xmath185 with @xmath86 and @xmath105 with @xmath89 , we have @xmath186 then we deduce that for any @xmath39 , @xmath40 with @xmath86 , @xmath187 c_1\mathrm{e}^{-c_1|x - y|}|x - y|,\end{aligned}\ ] ] where in the inequality @xmath188 denotes the first coordinate of @xmath189 , that is , @xmath190 , both equalities above follow from the rotationally invariant property of the measure @xmath100 , and in the last inequality we have used ( [ main - result-1 ] ) and the fact that @xmath191 . now , taking @xmath192 we find that for any @xmath39 , @xmath40 with @xmath86 , @xmath193 since @xmath194 and @xmath195 on @xmath181 , @xmath196,\ ] ] which along with the estimate above yields that for any @xmath39 , @xmath40 with @xmath86 , @xmath197 where @xmath198 . \(2 ) second , we consider the case that @xmath9 with @xmath199 . for any @xmath39 , @xmath40 with latexmath:[$l_0 < @xmath178 , @xmath201 on the other hand , also by ( [ main - result-1 ] ) and the fact that @xmath202 , for any @xmath9 with @xmath203 , @xmath204 |x - y|^{\theta-1}.\end{aligned}\ ] ] next , we consider the function @xmath205 on @xmath206 . it is easy to see that due to the definitions of the constants @xmath168 and @xmath169 , there is a unique @xmath207 such that @xmath208 and @xmath209= \frac { -2b}{c_2 } \biggl[1-\log \frac{2(c_1+c_2)}{c_2 } \biggr].\ ] ] since @xmath210 is large enough such that @xmath211 we have @xmath212 , which implies that @xmath213 for all @xmath214 . in particular , @xmath215 for any @xmath9 with @xmath203 . that is , for any @xmath9 with @xmath203 , @xmath216 combining both estimates above with the definition of @xmath217 , we finally conclude that there is a constant @xmath218 such that for any @xmath39 , @xmath40 with @xmath87 , @xmath219 this along with the conclusion of part ( 1 ) yields the desired assertion . next , we turn to the case of @xmath49 $ ] . for this , we first take the constant @xmath220 in the definition of the coupling operator @xmath93 , and then change the test function @xmath217 as follows , which is different from that in the case @xmath7 . for any @xmath161 , we define @xmath221 $ ; \vspace*{3pt}\cr a\mathrm{e}^{c_0(r-2l_0)}+b(r-2l_0)^2 + \bigl(2l_0-c(2l_0)^{1+\alpha}-a \bigr ) , & \quad $ r\in [ 2l_0,\infty)$,}\ ] ] where @xmath222,\qquad c_0= \frac{10\alpha}{l_0}.\ ] ] due to the choice of the constants above , @xmath223 and @xmath224 for all @xmath161 . [ key2 ] assume that @xmath49 $ ] . if @xmath225 then there exists a constant @xmath171 such that for any @xmath9 with @xmath226 , @xmath172 we mainly follow the proof of proposition [ key1 ] , and here we only present the main different steps . for @xmath9 with @xmath86 , we have @xmath227 & & \quad=\frac{1}{2 } \biggl[\int _ { \{|z|\le ( { 1}/{4}){|x - y| } \ } } \bigl(\psi \bigl ( \bigl|x - y+\bigl(z-\varphi_{x , y}(z)\bigr ) \bigr| \bigr)+\psi \bigl ( \bigl|x - y-\bigl(z- \varphi_{x , y}(z)\bigr ) \bigr| \bigr ) \\[-2pt ] & & \hspace*{12pt}\qquad{}-2\psi\bigl(|x - y|\bigr ) \bigr)\frac{c_{d,\alpha}}{|z|^{d+\alpha } } \,\mathrm{d } z \biggr ] \\[-2pt ] & & \qquad { } + \psi'\bigl(|x - y|\bigr ) \frac{\langle b(x)-b(y),x - y\rangle}{|x - y|}.\end{aligned}\ ] ] since @xmath228 such that @xmath178 , @xmath179 and @xmath180 on @xmath229 , one can follow the proof of ( [ proof - one ] ) , and get that for all @xmath185 with @xmath230 and @xmath231 , @xmath232 & & \quad\leq4\psi '' \biggl(\frac{3}{2}|x - y| \biggr ) \frac{\langle x - y , z\rangle ^2}{|x - y|^2}.\end{aligned}\ ] ] then we follow the argument of ( [ proof - two ] ) and deduce that for any @xmath39 , @xmath40 with @xmath233 , @xmath234|x - y|.\end{aligned}\ ] ] by assumption ( [ con - d1 ] ) , we know that for all @xmath104 with @xmath233 , @xmath235 next , we turn to the case that @xmath9 with @xmath199 . for any @xmath39 , @xmath40 with @xmath236 , by ( [ main - result-1 ] ) and @xmath178 , @xmath237 on the other hand , also by ( [ main - result-1 ] ) and the fact that @xmath202 , for any @xmath9 with @xmath203 , @xmath238 |x - y|^{\theta-1}.\end{aligned}\ ] ] now , we consider again the function @xmath239 on @xmath206 . it is easy to see that due to the definitions of the constants @xmath168 and @xmath169 , there is a unique @xmath207 such that @xmath208 and @xmath240= \frac { -2b}{c_0 } \biggl[1-\log \biggl(2+\frac{\alpha}{l_0c_0 } \biggr ) \biggr].\ ] ] noticing that @xmath241 , we get @xmath242 and so @xmath212 , which implies that @xmath213 for all @xmath214 . in particular , @xmath243 for any @xmath9 with @xmath203 . that is , for any @xmath9 with @xmath203 , @xmath244 according to both estimates above and the definition of @xmath217 , we finally conclude that there is a constant @xmath218 such that for any @xmath39 , @xmath40 with @xmath87 , @xmath219 this along with the conclusion above yields the desired assertion . [ main-1 - 1 ] according to the argument above , we can easily improve ( [ con - d1 ] ) , for example , by taking @xmath245 for @xmath246 $ ] and changing the integral domain @xmath247 in the definition of the coupling operator @xmath93 into @xmath248 with some proper choices of @xmath249 , @xmath250 $ ] and @xmath251 . for simplicity , here we just set @xmath252 , @xmath253 , @xmath254 and @xmath255 . we divide the proof of theorem [ main - result ] into two parts . proof of theorem [ main - result ] for @xmath86 or @xmath53 we will make full use of the coupling process @xmath256 constructed in section [ section22 ] . denote by @xmath257 and @xmath160 the distribution and the expectation of @xmath256 starting from @xmath107 , respectively . for any @xmath16 set @xmath258 , and for @xmath259 define the stopping time @xmath260\bigr\}.\ ] ] for any @xmath39 , @xmath40 with @xmath261 , we take @xmath262 large enough such that @xmath263 . let @xmath217 be the function given in proposition [ key1 ] if @xmath7 or the function given in proposition [ key2 ] if @xmath49 $ ] . then @xmath264 therefore , @xmath265\leq\psi(r_0 ) \mathrm{e}^{-\lambda t}.\ ] ] since the coupling process @xmath256 is non - explosive , we have @xmath266 a.s . as @xmath267 , where @xmath143 is the coupling time of the process @xmath268 . thus , by fatou s lemma , letting @xmath267 in the above inequality gives us @xmath269\leq\psi(r_0)\mathrm{e}^{-\lambda t}.\ ] ] thanks to our convention that @xmath270 for @xmath271 , we have @xmath272 for all @xmath271 , and so @xmath273 that is , @xmath274 as a result , if @xmath86 , then for any @xmath37 and @xmath16 , @xmath275 where the first inequality follows from the definitions of the test function @xmath217 in propositions [ key1 ] and [ key2 ] . now for any @xmath9 with @xmath276 , take @xmath277 + 1\geq2 $ ] . we have @xmath278 set @xmath279 for @xmath280 . then @xmath281 and @xmath282 ; moreover , ( [ proof.2 ] ) implies @xmath283 for all @xmath284 . therefore , by ( [ small ] ) and ( [ proof.2 ] ) , @xmath285 in particular , the proof of the first assertion for @xmath53 in theorem [ main - result ] is completed . on the other hand , from ( [ small ] ) and the conclusion above , we also get the second assertion for @xmath55 with @xmath286 and all @xmath16 , or with @xmath287 and @xmath288 . next , we turn to : proof of theorem [ main - result ] for @xmath199 and @xmath55 for @xmath87 , we use the synchronous coupling and the assertion of theorem @xmath289 for @xmath86 . in detail , with ( [ ito - sde ] ) , let @xmath290 be the coupling process on @xmath96 such that its distribution is the same as that of @xmath256 constructed in section [ section22 ] . we now consider @xmath291 where @xmath292 and @xmath293 is the coupling time . for @xmath271 , we still set @xmath294 . therefore , the difference process @xmath295 satisfies @xmath296 note that the equality above implies that @xmath297 is a continuous function on @xmath298 such that @xmath299 . as a result , @xmath300 still denoting by @xmath301 , we get from ( [ main - result-1 ] ) that @xmath302 which implies that @xmath303 since @xmath55 and the continuity of @xmath304 on @xmath298 . therefore , for any @xmath9 with @xmath87 , @xmath37 and @xmath305 , we have @xmath306 \\ & \le & c_1\mathbb{e } \bigl[|x_{t_{l_0}}-y_{t_{l_0}}|\mathrm{e}^{-\lambda ( t - t_{l_0 } ) } \bigr ] \\ & \le & c_1 l_0\exp(\lambda t_0 ) \mathrm{e}^{-\lambda t},\end{aligned}\ ] ] where in the first inequality we have used ( [ small ] ) , and the last inequality follows from ( [ rrr2 ] ) and the fact that @xmath307 . in particular , we have for all @xmath276 and @xmath305 , @xmath308 combining with all conclusions above , we complete the proof of the second assertion in theorem [ main - result ] . we finally present the following . proof of example [ 1-example-2 ] in this example , @xmath309 it follows from the proof of @xcite , example 5.3 , that for any @xmath90 , @xmath310 then , ( [ main - result-1 ] ) holds with @xmath311 , @xmath312 and any positive constants @xmath313 . in particular , ( [ main - result-1 - 1 ] ) holds for all @xmath49 $ ] and @xmath314 small enough . then the required assertion is a direct consequence of theorem [ main - result ] . the author would like to thank professor feng - yu wang and the referee for helpful comments and careful corrections . financial support through nsfc ( no . 11201073 ) , jsps ( no . 26@xmath31504021 ) , nsf - fujian ( no . 2015j01003 ) and the program for nonlinear analysis and its applications ( no . irtl1206 ) are gratefully acknowledged .
|
coupling by reflection mixed with synchronous coupling is constructed for a class of stochastic differential equations ( sdes ) driven by lvy noises . as an application
, we establish the exponential contractivity of the associated semigroups @xmath0 with respect to the standard @xmath1-wasserstein distance for all @xmath2 .
in particular , consider the following sde : @xmath3 where @xmath4 is a symmetric @xmath5-stable process on @xmath6 with @xmath7 .
we show that if the drift term @xmath8 satisfies that for any @xmath9 , @xmath10 holds with some positive constants @xmath11 , @xmath12 , @xmath13 and @xmath14 , then there is a constant @xmath15 such that for all @xmath2 , @xmath16 and @xmath9 , @xmath17.\ ] ] ./style / arxiv - general.cfg
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
a bright soft x - ray enhancement in the monoceros and gemini constellations was first detected in rocket experiments over thirty years ago @xcite , and resolved into a shell - like structure by _ heao-2 _ a decade later @xcite . spectral studies with _ rosat _ confirmed that this so - called ` monogem ring ' is a supernova remnant , probably in the adiabatic expansion phase @xcite . the distance is poorly constrained by evolutionary arguments self - consistent sedov - taylor models were found by @xcite at all distances between 100 and 1300 pc but distances around 300 pc are preferred on grounds of supernova energetics . at this distance , the model age is 86,000 years , explosion energy is @xmath1 erg , and current radius is 66 pc . the inferred interstellar medium density is @xmath2 , typical of the hot interstellar medium . the radio pulsar psr b0656 + 14 lies very close in projection to the center of the monogem ring , and an association between the objects seems natural @xcite . the pulsar is young , with a characteristic spin - down timescale of 110 kyr @xcite , pulsed non - thermal optical and x - ray emission , and unpulsed thermal x - ray emission consistent with a @xmath3 year - old cooling neutron star @xcite . but because the pulsar distance was estimated from interstellar dispersion measurements @xcite to be 760 pc more than twice the best estimate for the remnant and a proper motion measurement appeared to show the pulsar moving towards the center of the remnant @xcite , a physical association has been widely regarded as unlikely ( * ? ? ? * for example ) . new very long baseline interferometric measurements ( brisken et al . 2003 ) lead us to reconsider this negative conclusion . the distance , now accurately known from parallax to be @xmath4 pc , is much lower than previously believed , and the proper motion , 44 mas / yr , is about 40% smaller . as we will show , these measurements , together with a re - examination of the x - ray data , convincingly demonstrate that the pulsar and remnant were born in a single supernova explosion about @xmath3 yrs ago . the monogem ring ( fig . 1 ) shows significant deviations from circular symmetry , with what may be a blow - out to the west , away from the galactic plane , and a missing quadrant in the northwest , where foreground absorption is relatively high @xcite . because of the lack of symmetry , the morphological center of the ring is somewhat uncertain . for example , at low energy ( e.g. , the _ rosat _ r1 band image displayed as fig . 6 in plucinsky et al . 1996 ) , the portion of the ring in the galactic plane is not readily visible , making the remnant appear smaller and offset slightly away from the plane ; the arc is evident in higher energy observations ( fig . 1 , and also note fig . 8 of plucinsky et al . 1996 ) , suggesting the importance of greater absorption on lines of sight near the plane . in projection , the pulsar , at @xmath5 , lies well within the ring . to test the consistency of its position with the morphological center of the ring , we have fitted ` by eye ' a circle to the incomplete ring , centered on the pulsar . this fit , shown in fig . 1 , demonstrates that the pulsar s position is not inconsistent with the morphological center of the ring , within admittedly large uncertainties . with the measured proper motion , 44 mas / yr , the pulsar has moved about @xmath6 in @xmath3 yrs , from roughly @xmath7 . this inferred birth position appears slightly further from the current center of the ring , but not unacceptably so . it is also important to distinguish the morphological center of the remnant from its true dynamical center : we should expect an old remnant expanding into a medium with density decreasing away from the galactic plane to have its apparent center offset from its explosion center in the direction away from the plane , just as observed @xcite . in the end , we conclude that the pulsar position at birth was within a few degrees of the current morphological center of the ring . we note that the _ a priori _ likelihood of an unrelated background source projected within the ring falling this close to the center is about 5% . however , on its own the positional coincidence is a strong argument neither for nor against an association . the strongest argument against a physical association between the pulsar and remnant has been the pulsar distance , estimated at 760 pc from the measured column density of free electrons . sedov modeling formally allows a wide range of distances for the monogem ring , but a distance of about 300 pc has been preferred . although estimates of pulsar distances through interstellar dispersion are fairly crude ( the quoted uncertainty was 25% @xcite ) the distance discrepancy appeared significant . the small parallax distance , @xmath4 pc , thus came as a surprise . ( we note , though , that the greater distance implied an unacceptably large thermal x - ray luminosity : an attempt to directly estimate the distance from x - ray modeling yielded @xmath8 pc or less @xcite . ) with this precise pulsar distance available , we consider whether the distance estimate for the remnant can also be improved . a strong interstellar o vi absorption feature at @xmath9 km / s has been detected in the star 15 mon , in the southern region of the ring @xcite . the parallax distance of 15 mon is @xmath10 pc @xcite . because o vi at such high velocity is rarely found in the disk , and the velocity is similar to that expected from a remnant at this stage , we consider this to be a secure upper limit to the distance of the monogem ring . other evidence suggests the ring is no farther than the pulsar . the absorption is low , @xmath11 @xcite , compared to @xmath12 for the pulsar @xcite . the pulsar dispersion measure is much higher than expected at a distance of 290 pc , suggesting a line of sight through a highly ionized region . it is also unlikely that the remnant is significantly closer than the pulsar . at an assumed distance of 300 pc , the inferred explosion energy is already low : @xmath1 erg . a distance below 230 pc would imply an implausibly low energy , below @xmath13 erg . we conclude that the pulsar and remnant distances agree to within @xmath14 pc . indeed , since the shell radius is @xmath15 pc , the evidence is strong that the pulsar is currently within the expanding supernova shell . finally , we must consider the relative ages of the pulsar and remnant . the remnant age , from sedov modeling , has been estimated at 86,000 yrs for a distance of 300 pc . two lines of evidence suggest a comparable age for the neutron star . first , its current temperature is in good agreement with standard cooling models for a @xmath3 yr old neutron star @xcite . second , the characteristic spin - down timescale for the pulsar is @xmath16 yrs . the second argument deserves more comment . the age of a pulsar slowing with a torque proportional to a constant power @xmath17 of its frequency , @xmath18 , can be expressed as @xmath19,\ ] ] where @xmath20 is the initial spin period . for magnetic - dipole braking , the `` braking index '' @xmath21 , so if the pulsar is born spinning much faster than its current period , @xmath22 , while the age is overestimated if the pulsar were born near its current period . for a handful of very young pulsars , where the braking index can be measured directly , @xmath23 @xcite . for middle - aged pulsars , the good agreement between timing ages and so - called kinetic ages ( @xmath24 , where @xmath25 is the height above the galactic plane ) requires a mean braking index near 3 @xcite . we acknowledge that the timing age can be misleading . several putative pulsar - snr associations suggest large discrepancies between timing and true ages : for example , psr j1012@xmath265226 , in the @xmath27 kyr old remnant pks 1209@xmath2651/52 , has a spin - down age of @xmath28 kyr @xcite , and psr j1811@xmath261925 , with a spin - down age of 24 kyr , has been associated with the snr of ad 386 @xcite . nevertheless , a recent comparison of kinetic and timing ages for 21 pulsars under 10 myr in age found a typical discrepancy of only @xmath29 @xcite . we conclude that both the timing and cooling ages of the pulsar are consistent with the remnant age . the angular position , distance , and age of the pulsar and remnant are all in excellent agreement with the hypothesis that both were born at the same location , @xmath30 pc away , @xmath3 yrs ago . the likelihood that this is merely a positional and temporal coincidence is extremely low . crudely estimating the galactic type ii supernova rate at one per century and the region within 300 pc of earth as @xmath31 of the area of the galactic disk , the mean supernova interval in this region is about 100 kyr . the chance that two unrelated supernovae occur within @xmath32 kyr , within @xmath33 on the sky , and within 50 pc in radial distance is vanishingly small . perhaps the most persuasive argument for an association comes from assuming the opposite . if this @xmath3 yr old remnant is not that of the supernova that formed the @xmath3 yr old pulsar , then where is the remnant of that supernova ? many old remnants are of course invisible because of distance or environment , but if the monogem ring and pulsar were formed in different supernovae then they occurred in very close physical and temporal proximity . the monogem ring itself is evidence that a remnant of age @xmath3 years expanding into this particular environment is visible at 300 pc . we conclude that a single supernova , 300 pc away and a hundred thousand years ago , formed both psr b0656 + 14 and the monogem ring . supernova remnants are generally believed to be acceleration sites for cosmic rays @xcite , and in this case it is intriguing to note that the monogem ring may hold a clue to a long - standing puzzle : the origin of the steepening in the primary cosmic - ray spectrum at about @xmath34 ev ( 3 pev ) , called the ` knee . ' between about @xmath35 and a few times @xmath36 ev , the differential cosmic ray energy spectrum is well - described by a broken power law , proportional to below the knee and above the knee . ( a recent review can be found in @xcite . ) possible explanations for the knee are extremely varied , including loss of the most energetic particles from the galaxy , unknown physical processes in the development of the atmospheric shower through which the cosmic rays are detected , or , most likely , a termination in the acceleration process . a typical cut - off energy for acceleration in supernova remnants is @xmath37 ev , where @xmath38 is the nuclear charge , though the exact cut - off will vary by perhaps an order of magnitude with variations in the explosion energy , magnetic field , interstellar medium density , and age @xcite . recently , erlykin and wolfendale ( ew ) have drawn attention @xcite to the sharpness of the knee feature , which has been a challenge for models in which the break arises from propagation effects or from a stochastic superposition of multiple sources with varying high - energy cut - offs . ew propose that around the knee a single nearby source dominates the cosmic ray spectrum , which is otherwise a smooth superposition of the contributions from many supernova remnants throughout the galaxy and from whatever sources provide the higher energy cosmic rays , up to @xmath39 ev and beyond . at energies of a few pev , this single source alone produces 60% of the flux at earth . ( this possibility has also recently been considered by other authors ( * ? ? ? * for example ) . ) although the data are not yet conclusive , ew have identified the knee as most likely due to oxygen nuclei , with a smaller second break at about 10 pev due to iron nuclei . the shape and amplitude of the knee feature have led ew to predict @xcite that the single source is a 90100 kyr old supernova remnant between 300 and 350 pc from earth , expanding in an under - dense medium . the match with the properties of the monogem ring is striking , though almost certainly in part coincidental considering the large remaining uncertainties in cosmic ray acceleration and diffusion models . nevertheless , an important objection to the single source model for the knee feature has been the lack of a suitable source @xcite . that objection now appears to have been removed . is supported by the nsf under grant ast-0098343 , r.a.b . is supported by nasa atp grant nag5 - 12128 , and a.g . is supported by enterprise ireland under grant sc/2001/322 . this research has made use of the _ rosat _ all - sky survey data , which have been processed at mpe , and of the telescopes of the national radio astronomy observatory , a facility of the nsf operated under cooperative agreement by associated universities , inc . we thank an anonymous referee for suggestions that improved the manuscript . , m. a. c. , lindegren , l. , kovalevsky , j. , hoeg , e. , bastian , u. , bernacca , p. l. , crz , m. , donati , f. , grenon , m. , van leeuwen , f. , van der marel , h. , mignard , f. , murray , c. a. , le poole , r. s. , schrijver , h. , turon , c. , arenou , f. , froeschl , m. , & petersen , c. s. 1997 , a&a , 323 , l49
|
the monogem ring is a bright , diffuse , @xmath0-diameter supernova remnant easily visible in soft x - ray images of the sky .
projected within the ring is a young radio pulsar , psr b0656 + 14 .
an association between the remnant and pulsar has been considered , but was seemingly ruled out by the direction and magnitude of the pulsar proper motion and by a distance estimate that placed the pulsar twice as far from earth as the remnant . here we show that in fact the pulsar was born very close to the center of the expanding remnant , both in distance and projection .
the inferred pulsar and remnant ages are in good agreement .
the conclusion that the pulsar and remnant were born in the same supernova explosion is nearly inescapable .
the remnant distance and age are in remarkable concordance with the predictions of a model for the primary cosmic ray energy spectrum in which the ` knee ' feature is produced by a single dominant source .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
is considered the prototype for a class of evolved binaries with peculiar properties . the effective temperatures and luminosities of the members of this class suggest that they are in the post - asymptotic giant branch ( post - agb ) phase of their evolution , but their evolutionary path is thought to be severely affected by the presence of a close companion ( see * ? ? ? * for a review ) . like the other members of its class , hr 4049 shows a significant infrared ( ir ) excess and a time - variable optical and ultraviolet ( uv ) deficit @xcite , which suggests the presence of a massive circumbinary disk . in addition , its photosphere shows a severe depletion in refractory elements , but it has roughly solar abundances in volatiles . this unusual depletion pattern is generally attributed to the formation of dust ( incorporating refractory elements ) in a circumbinary disk , followed by the re - accretion of the depleted gas onto the star @xcite . since the circumbinary disk plays an important role in determining the properties of hr 4049 , it has been studied extensively ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? while the ir excess is a clear indication of the presence of dust in a stable disk around the system , it has been difficult to determine the nature of the dust in the disk . indeed , it is not even clear whether the dust in the disk is oxygen - rich or carbon - rich . while the ir spectrum shows the clear presence of gas that is typically associated with oxygen - rich environments ( e.g. co@xmath0 , h@xmath0o , oh ; @xcite ; see also ( * ? ? ? * paper i hereafter ) ) there is no trace of corresponding oxygen - rich dust features such as silicates or oxides . instead , the spectral energy distribution ( sed ) of the disk resembles a 1150 k black body down to sub - millimeter wavelengths @xcite , while also showing strong emission features due to polycyclic aromatic hydrocarbons ( pahs ) . this pah emission however does not originate from the disk , but from what appear to be bipolar lobes @xcite . the 1150 k black body sed can be reproduced by a wall model " , in which the circumbinary disk is vertically extended and the dust is very optically thick , effectively producing a radiating inner wall at a temperature of @xmath5 k , with a scale height of @xmath63 au at the inner wall distance of 10 au @xcite . because of the high opacity , the dust beyond the inner rim is cold and does not contribute to the ir excess . however , the wall model has been challenged by recent interferometric observations by @xcite that show a more extended distribution of material emitting strongly in the ir than the wall model can accommodate . instead , these authors suggest emission from optically thin dust with smooth opacity profiles . in paper i , we present a detailed analysis of the molecular emission in the mid - ir spectrum and conclude that neither of these models is fully consistent with the properties of the molecular gas ( in particular co@xmath0 , h@xmath0oand co ) . indeed , while we find that the gas originates from a radially extended disk , we also determine that it is very optically thick across most of the mid - ir . radiative trapping by this optically thick gas then results in a warmer and more homogeneous temperature structure than previously considered for the disk . additional clues to the geometry and properties of the disk can be inferred from observations in the near - ir at high spectral resolution . @xcite presented such observations , examining three regions of the near - ir spectrum of hr 4049 at 2.3 , 3.0 and 4.6 @xmath1 m . they detect many co , oh and h@xmath0olines and identify distinct components in the system . they propose that the absorption in the co overtone originates from gas in keplerian rotation along the inner rim of the disk in the wall model . furthermore , they suggest that the gas is slowly streaming out over the edge of the wall and over the disk , causing the more complex emission - absorption line profiles in the 4.6 @xmath1mregion . finally , they found evidence for a cold gas component . clearly , the near - ir observations contain a lot of information about the properties , geometry and kinematics of the gas disk . it is thus important to investigate whether we can reconcile these data with the disk model inferred from mid - ir observations . therefore , we re - examine the near - ir data presented by @xcite . this paper is organized as follows . in section [ sec : observations ] we briefly describe the observational data . we present our analysis of the spectrum in section [ sec : analysis ] , discuss our results in section [ sec : discussion ] and present our conclusions in section [ sec : conclusion ] . the data we discuss in this paper are high - resolution ( @xmath7 ) observations in the near - ir ( @xmath8 m ) carried out with the phoenix spectrograph @xcite from the national optical astronomy observatory ( noao ) mounted on gemini south . these observations primarily targeted co fundamental and overtone lines and were described and discussed previously by @xcite . a large number of telluric lines are present in the near - ir ; fig . [ fig : overtone_fits ] shows an illustrative atmospheric radiance spectrum from hitran @xcite on the web . additional observations of a hot star ( without stellar lines in this wavelength range ) at the same airmass as hr 4049 were used to divide out these lines . this telluric correction is generally adequate ; however , some residuals remain due to imperfect cancellations , especially in wavelength ranges where there are a lot of lines ( e.g. near 2.319 @xmath1 m ; see fig . [ fig : overtone_fits ] ) . in most cases though , the residuals are significantly smaller than the depth of the lines we study here . the telluric lines in the hot star spectrum were also used to achieve a very accurate wavelength calibration , with residuals of typically 0.25 km s@xmath9 . for more details about these observations or the data reduction aspects , we refer to @xcite . for each individual data segment , we then performed a heliocentric velocity correction and normalized the data to the continuum . next , we used a weighted mean ( adopting the signal - to - noise , s / n , ratios provided by * ? ? ? * ) to merge all the individual segments into a single , final spectrum . finally , we rebinned the resulting spectrum onto a constant resolution wavelength grid for later comparison to our model spectra . we begin our analysis of these data with the co overtone lines in the 2.3 @xmath1mregion . there are several good reasons for this . first , in this range , the spectrum reveals simple , pure absorption profiles with relatively little blending between the lines . since they are intriniscally much weaker , these lines will furthermore have a much lower optical depth than the fundamental lines for a given column density , thus offering the best prospects to reliably determine the temperatures and column densities . finally , since the gas in this region of the spectrum is absorbing continuum radiation , its location is constrained : it must be directly along the line of sight to the dusty disk and/or the central star ; note however that the stellar continuum will only contribute @xmath625% of the flux at this wavelength . we will then use our results from the 2.3 @xmath1mrange to better interpret the emission - absorption spectrum at 4.6 @xmath1 m . the combined spectrum of hr 4049 in the 2.3 @xmath1mrange is shown in fig . [ fig : overtone_fits ] . we remind the reader that @xcite used these data to produce an excitation diagram for co , and found that the absorption originates from two distinct layers of co gas in local thermodynamic equilibrium ( lte ) : a warm layer ( @xmath10 k , @xmath11 10@xmath12 @xmath13 ) and a cold one ( @xmath14 k , @xmath15 10@xmath16 @xmath13 ) . we first used these parameters to create a plane - parallel slab model in which 1200 k black body radiation ( representing the dust ) is absorbed by the warm layer of co gas which is in turn absorbed by the cold layer of co gas . we used the co line list from @xcite and adopted a gaussian line profile with a width of 10 km s@xmath9 . this corresponds to the measured full width at half maximum ( fwhm ) for the overtone lines ; note that @xcite suggest 16 km s@xmath9 ; however , this results in lines that are too broad ( see also sect . [ sec : discussion_kinematics ] ) . when we compared this model to the observations ( see fig . [ fig : overtone_fits ] ) , we found that the model reproduces neither the depths nor the relative strength ratios of the individual rovibrational lines . this discrepancy could be due to the use of a different line list ; at any rate , it warrants an independent analysis . [ [ excitation - diagram ] ] excitation diagram + + + + + + + + + + + + + + + + + + first , we constructed our own excitation diagram for the co first overtone @xmath17 lines . we fit a gaussian to each co line and integrated these to determine the equivalent width ( @xmath18 ) . then we determined the population of each level using @xmath19 where @xmath20 is the column density ( in @xmath13 ) , @xmath21 is the oscillator strength for a rovibrational transition ( obtained from the co line list by * ? ? ? * ) , @xmath22 is the central wavelength of the transition and both @xmath18 and @xmath22 are measured in @xcite . we present the resulting excitation diagram in fig . [ fig : co_excitation ] . qualitatively , our results are similar to those obtained by @xcite , in the sense that our excitation diagram is consistent with two layers of optically thin co in lte a warm and a cold one . however , we find very different values for the temperatures and column densities of the co gas : our `` hot '' layer has a temperature of 970 @xmath23 40 k and a column density of ( 1.19 @xmath23 0.04)@xmath24 @xmath13 . the temperature we determine is nearly twice that found by @xcite and the column density is two orders of magnitude larger . for the `` cold '' layer , we find a temperature of 40 @xmath23 10 k and a column density of ( 6.1 @xmath23 0.5)@xmath25 @xmath13 . here , the temperature and column density are relatively similar to those determined by @xcite . lines indicated by diamonds , the @xmath26 points indicated by crosses and the @xmath27co points indicated by squares . the gray triangles indicate the points on the excitation diagram presented by @xcite . the solid black lines indicate the best fits to the different components . the dashed line shows the extrapolation of the lines originating from the ground state to those from the first vibrationally excited level.,scaledwidth=50.0% ] if we extrapolate the hot component to higher energies , we find that it also fits the overtone lines originating from the first vibrationally excited state ( i.e. the @xmath26 lines ) relatively well . this suggests that the co may be in or close to vibrational lte . for the @xmath27co lines , we find a column density of ( 5.5 @xmath23 0.6)@xmath28 10@xmath16 @xmath13 . this yields a @xmath29c/@xmath27c ratio of 2.1 @xmath23 0.9 , which is a little lower the value of 6@xmath30 reported by @xcite . [ [ model - spectra ] ] model spectra + + + + + + + + + + + + + as an independent check , we modeled the co absorption at 2.3 @xmath1musing the same methods employed to build the spectrafactory database @xcite . for these models , we began once again with co line lists from ( * ? ? ? * including the @xmath27co , c@xmath16o and c@xmath12o isotopologues ) . from these , we calculated optical depth profiles assuming a population in lte and a gaussian intrinsic line profile with a width of 0.5 km s@xmath9(consistent with the @xmath31 value derived by * ? ? ? note though that since the lines are optically thin , the precise value of the intrinsic line width in our approach does not matter much as long as the line width is much smaller than a resolution element . we tested a few different model configurations . initially , we attempted to fit a model consisting of a single layer of lte gas absorbing a 1200 k black body , however , we found that this model did not reproduce all of the co absorption and found strong residuals , especially for the low @xmath32 lines . thus , it seems that at least two layers of co gas were required ; this is consistent with the results from the excitation diagrams . we thus modified our model to include two layers of gas in a slab geometry absorbing a background dust continuum . in our fit to the co absorption , we used gas temperatures between 50 and 1000 k in increments of 100 k and column densities between 10@xmath33 and 10@xmath34 @xmath13 in increments of log@xmath20 = 0.2 for each of the gas layers . we varied log[/ ] from 0 to 2 ; log[/ ] and log[/ ] from 0 to 3 in increments of 0.2 . we also varied the radial velocity ( @xmath35 ) between -30 and -40 km s@xmath9 in increments of 1 km s@xmath9 and smoothed our line profiles to yield lines with widths between 2 and 20 km s@xmath9 in increments of 2 km s@xmath9 to represent the observed line broadening ( due to e.g. the rotation of material in a disk ) . we compared each model to the entire 2.3 @xmath1 m spectral region and calculated @xmath36 , the reduced @xmath37 statistic for each model . in table [ table : nirfits ] , we present the parameters from our best fit model and in fig . [ fig : overtone_fits ] , we compare this model to the spectrum . we note that this model reproduces the co absorption very well . in addition , the resulting best - fit temperatures ( 50 k and 900 k ) agree well with those we determined from our excitation diagram ( 40 k and 970 k ) and also the column densities are comparable ( log@xmath38 versus log@xmath39 for the cold layer and log@xmath40 versus log@xmath41 in the hot ) ; this corresponds to a maximum optical depth of @xmath42 . we find a @xmath36 of 3.54 for our best model however , indicating significant residuals . l c c & model & excitation diagram + temperature ( k ) & 900 @xmath23 50 & 970 @xmath23 40 + [ 1.1ex ] log @xmath20 ( co ) & 19.2 @xmath23 0.1 & 19.08 @xmath23 0.01 + [ 1.1ex ] log @xmath20 ( h@xmath0o ) & 19.4 @xmath23 0.1 & @xmath43 + [ 1.1ex ] temperature ( k ) & 50 @xmath23 25 & 40 @xmath23 10 + [ 1.1ex ] log @xmath20 ( co ) & 18.2 @xmath23 0.1 & 18.7@xmath44 + [ 1.1ex ] @xmath29c@xmath45c & 1.6@xmath46 & 2.1 @xmath23 0.9 + [ 1.1ex ] @xmath33o@xmath47o & 16@xmath48 & @xmath43 + [ 1.1ex ] @xmath33o@xmath49o & 16@xmath50 & @xmath43 + [ 1.1ex ] & & velocity analysis + @xmath35 ( km s@xmath9 ) & -33.0 @xmath23 0.5 & -32.7 @xmath23 0.2 + @xmath51 ( km s@xmath9 ) & 10 @xmath23 1 & 11.0 @xmath23 0.5 + [ table : nirfits ] note that our model does not only reproduce the ground state @xmath17 transitions , but also the @xmath26 lines ( see e.g. the r18 , r24 and r28 lines in fig . [ fig : overtone_fits ] ) , again suggesting that the co gas is near vibrational lte . note however that the critical density for the vibrational levels of co is @xmath52 @xmath53 @xcite , which is much higher than any realistic estimate of the density in this environment . as described in paper i , the disk is extremely gas rich and significant radiative trapping is occurring . the gas may thus be radiatively thermalized . the residual spectrum still exhibits several additional absorption features ; some of these are telluric ( see fig . [ fig : overtone_fits ] ) , however , others are consistent with h@xmath0oat the radial velocity of hr 4049 . these features are much shallower than the co lines which makes them much less obvious . we thus recalculated our model for this region with the addition of h@xmath0ousing the line list from ( * ? ? ? * including the h@xmath54o and h@xmath55o isotopologues ) . we began with the assumption that h@xmath0owas in the same layers as the co ( keeping the same layers reduced the number of free parameters ) and varied the column density in each layer between 10@xmath33 and 10@xmath34 @xmath56 . the best fit for these models indicated an absence of h@xmath0oin the cold molecular layer . given the temperature of this cold layer , water may not exist in the gas phase here . thus , we removed h@xmath0ofrom our cold co layer and only included it in the hot layer . like the hot co , we find a high column density for h@xmath0oin this region of the spectrum ( log @xmath20 = 19.4 ) , but still corresponding to optically thin lines ( @xmath57 ) . armed with a good characterization of the absorbing gas in the 2.3 @xmath1mregion , we now turn our attention to the lines near 4.6 @xmath1 m . we show the spectrum of hr 4049 in this range in fig . [ fig : co_fundamental ] . in contrast to the pure absorption lines in the co overtone region , the spectrum here is dominated by emission bands ( of both co and h@xmath0o ) , many of which show a superposed absorption component as well . note that some of these absorption features are not just self - absorption of the emission lines , but actually absorb continuum radiation as well . such a spectrum , dominated by emission , is somewhat surprising at first sight given our results from above . the overtone lines at 2.3 @xmath1mrevealed the presence of a hot gas in front of the continuum emission from the star and dust disk . all other things being equal , the same gas should also produce absorption in the fundamental lines since they originate from the same ( ground ) state . in fact , since the fundamental lines are intrinsically @xmath6100 times stronger than the overtone lines and since the @xmath61200 k black body dust emission is almost twice as strong at 4.6 @xmath1mthan it is at 2.3 @xmath1 m , we would expect much stronger , saturated absorption lines , as we show in fig . [ fig : co_fundamental ] . using the same parameters as for the 2.3 @xmath1mregion , we find that in this range , both the co and h@xmath0olines would be very optically thick ( @xmath58 for co and @xmath59 for h@xmath0o ) and thus appear as broad , saturated lines . this is in stark contrast with the observations . thus , the emission must originate from gas that is not residing in the same line of sight as the gas detected in the overtone and must cover a much larger area to completely fill in the expected absorption . @xcite interpreted this in the framework of the wall model and suggested that the gas in absorption was along the inner wall of the disk , while the gas in emission originated from above the cold dust surface . however , this is not a viable explanation anymore given our current understanding of the disk from interferometric and mid - ir spectroscopic observations : the disk is radially extended , and is warm over a large radial distance ( * ? ? ? * paper i ) . furthermore , the gas emitting in the co fundamental must be quite warm as well , as evidenced by the high @xmath32 lines and the presence of @xmath60 and @xmath61 transitions . one way to explain these observations is by considering that the line of sight that we probe at 2.3 @xmath1mis such that the co is not absorbing the dust emission , but rather the stellar continuum . given our line depths at 2.3 @xmath1mand that about 25% of the flux at those wavelengths originate from the star , this could certainly not be ruled out . the total flux we observe is then the sum of this absorbing line of sight and the total emission ( gas + dust ) of the disk . at the gas temperatures we determined , the gas in the disk would not emit much at 2.3 @xmath1 m , thus the main effect would be veiling of the absorption lines by the dust . at 4.6 @xmath1mhowever , the starlight contributes only 5% to the continuum , so the depths of the absorption lines would be much more reduced due to veiling . at the same time , the @xmath61000 k gas would also emit very efficiently at 4.6 @xmath1mand any remaining absorption would be more easily filled in by emission from hot gas . however , this scenario does not work . if the co gas in the overtone lines would be absorbing only the stellar radiation , our line of sight through the disk would only probe material at a zero radial velocity since any gas motions would be perpendicular to the line of sight . a detailed analysis of the overtone line profiles ( see sect . [ sec : discussion_kinematics ] ) shows that the higher @xmath32 lines have much broader line profiles and thus at least a fraction of the absorption originates from gas with a significant radial velocity ; thus , this gas must be absorbing non - stellar radiation . therefore , we must conclude that the gas in the overtone region is absorbing dust emission as well . to explain the emission in the fundamental region , there must then be a significant amount of hot gas located in lines of sight that do not intersect the dust emission . at this point , there only seem two plausible locations for that gas . first , the gas could originate from above ( and/or below ) the disk mid - plane , while most of the dust providing the continuum emission would be located near the mid - plane . as noted by @xcite , the dust settling time for a gas - rich disk is short typically 150 years for the case of hr 4049 ; thus , the dust distribution is most likely concentrated near the mid - plane . at the same time , if the disk is in hydrostatic equilibrium , the high gas temperatures imply that the disk must still have a large scale height . it is thus possible that the dust emission is predominantly originating from the mid - plane while most of the gas emission is coming from above the mid - plane . however , there must still be enough ( small ) dust grains mixed in with the gas at appreciable vertical distances from the mid - plane to explain the phase - dependent extinction @xcite . furthermore , if the gas is being heated by the dust , the dust must be able to intercept a sufficient amount of stellar radiation , which would not be possible if all the dust is located at the mid - plane . a second possibility is that there is hot gas located inside the dust disk , in the dust - free central cavity . since the inner boundary of the disk is determined by the sublimation temperature of the dust and since the dissociation temperature for co ( and h@xmath0o ) is much larger than the dust sublimation temperature , some gas must indeed be located inside the disk . some support for an origin inside the disk can be found in the larger widths of the emission components compared to the absorption ( by hot gas ) in the overtone ( see sect . [ sec : discussion_kinematics ] ) , pointing to larger velocities for the emitting component than for the absorbing gas . finally , we should also briefly discuss the absorption components in the emission lines . while most emission lines show some evidence for an absorption component , it is most pronounced in those lines involving the ground vibrational state . furthermore , the low @xmath32 lines show the strongest absorption , including absorption of the continuum radiation . thus , this absorption component must originate from a much cooler component ( such as the cold component we observe in absorption at 2.3 @xmath1 m ) and can not be due to self - absorption by the gas ( which would not result in continuum absorption ) . since some of the absorption lines are slightly blue shifted compared to the emission component , this cold gas may be in an outflow , possibly related to the bipolar lobes . we also investigate the kinematic properties of the gas . as noted in sect . [ sec : analysis2 ] , the lines at 2.3 @xmath1mappear narrower than the velocity width reported by @xcite . when we fitted all the overtone lines from the ground state with gaussians , we determined that the average full - width at half maximum velocity was @xmath62 km s@xmath9rather than their @xmath63 km s@xmath9 . as an additional check , we also incorporated the line width as a parameter in our full model fit which confirmed this result . note that this also agrees with @xcite , who determined that the low @xmath32 lines were unresolved when observed with an instrumental resolution of 10 km s@xmath9 . km s@xmath9 * ? ? ? the lower plot compares the @xmath64 as a function of lower energy . the blue dot - dashed horizontal line indicates the weighted mean of all the @xmath51 measurements and the blue dotted lines indicate the uncertainties on this measurement . the red dashed line indicates a linear fit between these parameters , with the dotted red lines showing the uncertainties in the fit . error bars on the velocity measurements are determined with a monte carlo technique.,scaledwidth=50.0% ] to further examine the kinematics of the gas in the disk , we fit unconstrained gaussian profiles to only the isolated co overtone transitions from the ground vibrational state ( i.e. those that are not blended with lines from hot bands , other isotopologues or h@xmath0olines ) and examined the resulting radial velocities and line widths . we find that on average , the lines have radial velocities of -32.7 @xmath23 0.2 km s@xmath9and a range between -35.5 and -31.1 km s@xmath9 . we also find an average line width of 11.0 @xmath23 0.5 km s@xmath9and a range between 8.5 and 13.4 km s@xmath9 . interestingly , both the radial velocity and width measurements show some clear variations as a function of the lower state energy of the lines involved ( see fig . [ fig : overtone_v ] ) . [ [ radial - velocities ] ] radial velocities + + + + + + + + + + + + + + + + + in fig . [ fig : overtone_v ] , we recover a trend previously noted by ( * ? ? ? * their fig . 10 , but there for the absorption component in the _ fundamental _ lines ) : the low @xmath32 lines are blue shifted compared to the higher @xmath32 lines which are closer to the system velocity . this was attributed to a slow outflow of gas from the system . to compare both absorption characteristics , we show the profiles of the overtone and fundamental lines that exhibit absorption for lines originating from the same level in fig . [ fig : fundamental_v_overtone ] . again , we selected only the lines which had relatively clean absorption profiles with as little blending as possible in both the fundamental and the overtone . it is clear that the centers of the absorption lines do not correspond well to each other ; the @xmath65 lines are the only ones with a clear overlap . in addition , the variations in the fundamental lines are different from the overtone : for the overtone , the low @xmath32 lines are at the system velocity and the largest deviations from the system velocity are found for the r5 and r10 lines ; for the fundamental , it is only the lowest @xmath32 lines that are blue shifted . these observations are hard to explain consistently . for instance , the absorption in the fundamental could on first sight be explained by a cold outflow from the system . indeed , with the deepest absorption ( the r2 line ) originating from the @xmath66 level , this blue shifted gas must have @xmath67 . however , at such temperatures , there would still be a significant population in the @xmath65 state and we would expect to also see the same blue shifted absorption quite strongly in the r3 line but this is not the case . that points to a low - density gas that is not even in rotational lte and may be more characteristic of interstellar gas than circumstellar gas . in the overtone lines , the case for a systematic effect is even less clear ( see fig . [ fig : overtone_v ] ) , with some low @xmath32 lines at the system velocity and others blue shifted . it is thus not clear whether this represents a true effect ( e.g. we underestimate blending by other lines ) or rather points to instrumental or calibration issues . for completeness , we note that these shifts are only seen in the co fundamental of the main isotopologue ; there are no discernible trends in the radial velocities of the absorption components of the hot bands and other isotopologues of co. [ [ line - widths ] ] line widths + + + + + + + + + + + an interesting trend becomes obvious when looking at the widths of the lines as a function of the lower state energy ( bottom panel of fig . [ fig : overtone_v ] ) : overtone absorption lines originating from higher energies are broader than the lower energy lines . a linear fit yields a slope of @xmath68 km s@xmath9 per cm@xmath9 . since only hot gas can contribute to the lines at higher energies and since cold gas dominates the lower energy lines , we conclude that the hotter gas has a larger rotational velocity than the cooler gas , consistent with keplerian motion in a disk that is hotter on the inside and cooler on the outside . note that thermal broadening is much smaller than the effect we measure here ( at @xmath70 k , @xmath71 km s@xmath9 ) . we will discuss this effect further below . we also note that in fig . [ fig : fundamental_v_overtone ] , the emission component is clearly broader than the overtone absorption . if the broadening mechanism is rotation in a keplerian disk and the overtone absorption begins at the interior of the dust disk , this once more suggests that the emission component originates on the interior of the dust disk . it is clear from our analysis above that the co ( and h@xmath0o ) gas probed in the near - ir traces a very complex , gas - rich environment . as pointed out by @xcite , hydrostatic gas pressure in a gas - rich disk will not only puff up the disk in the vertical direction , but it will also act in radial direction and tend to spread out the disk . indeed , also the near - ir observations point to such a radially extended disk , in agreement with our conclusions from the mid - ir observations in paper i. as we suggest in sect . [ sec : discussion_kinematics ] , the gas appears to have some temperature variation in which the gas closest to the binary system is hotter than the gas further away . if the gas and dust are mixed in , then this should also be true of the dust . if the dust were gray and in equilibrium with the stellar radiation field , the temperature in the disk would vary from 520 k to 370 k at 15 and 30 au respectively ( using a stellar temperature of 7500 k and a stellar radius of 31 r@xmath4 as described by * ? ? ? * ) or 640 and 450 at 10 and 20 au . the dust must be hotter than this to allow the @xmath72 k gas component to appear in absorption ; indeed , it must be hotter than the gas for the gas to appear in absorption . in paper i , we suggest that the optically thick gas in the disk traps and re - radiates the stellar radiation , thus the temperatures of the gas and dust in the disk will increase significantly compared to equilibrium values and they will become more homogeneous . the question thus remains : what type of temperature distribution can the disk have while still reproducing the sed we observe in hr 4049 ? as described by @xcite , the sed can be reproduced by the sum of equally weighted black bodies with temperatures between 880 and 1325 k. as well , when they assigned weights for the black bodies based on a power law distribution , they found that the temperatures could vary between 730 and 1238 k. in general , the maximum dust temperature possible is @xmath61300 k. if the inner rim of the dust component of the disk is determined by the sublimation temperature of the dust , this puts constraints on the type of dust in the disk . for instance , this excludes dust species with high dust condensation temperatures such as e.g. metallic iron or alumina . instead , we should expect the dust to be composed primarily of dust with a sublimation temperature close to 1300 k. we can use our measurements to estimate the ( radial ) size of the gas disk . if we assume that the overtone absorption originates from gas in keplerian rotation in the disk , the gas velocity at a distance @xmath73 from the center of mass is @xmath74 where @xmath75 is the gravitational constant , @xmath76 is the total mass of the binary system ( the stars are much closer to each other than the disk is to the stars so they can be treated as a point mass ) . since we find @xmath77 of @xmath78 km s@xmath9and @xmath79 of @xmath80 km s@xmath9 , we conclude that @xmath81 , compatible with the results by @xcite , who determined that the outer radius was @xmath82 based on their model fits to the interferometric observations and mid - ir sed . the value for @xmath83 is hard to determine , but is most likely between 10 au @xcite and 15 au @xcite ; this would then correspond to outer radii of @xmath84 and @xmath85 au and disk surface areas of @xmath86 and @xmath87 au@xmath88 respectively . the lower values compare especially well to what we determined for the co@xmath0gas in the mid - ir where we found an emitting area of 1300 au@xmath88 ( assuming an inclination angle of 60@xmath89 , paper i ) . thus , the circumbinary disk of hr 4049 is a radially extended , gas - rich disk . however , it is not a flat disk . indeed , the high temperatures of the gas ensure that hydrostatic pressure will still hold up the disk to appreciable vertical scale heights as well , but less than the @xmath90 that was determined by @xcite . from @xmath77 of the disk we are able to estimate a mass of the binary system . using an inclination angle of 60@xmath91 , we determine a deprojected @xmath77 of 15 @xmath23 4 km s@xmath9at the inner radius ; since this is the fwhm velocity , we find a tangential velocity of @xmath92 km s@xmath9for the gas in keplerian rotation at the inner radius . adopting an inner radius of 15 @xmath93 au , we find a total mass of @xmath94 m@xmath4 for the binary . if we then apply the mass function for the primary determined by ( * ? ? ? * @xmath95 = 0.158 @xmath23 0.004 ) , we calculate a mass of @xmath96 m@xmath4 for the primary and @xmath97 m@xmath4 for the secondary . if we use an inner radius of 10 au , we determine smaller masses for the binary and individual stars ( @xmath98 ) . the mass we find for the primary at an inner radius of 15 au is close to the @xmath99 m@xmath4 estimated by @xcite . it is thus interesting to contemplate the implications of such a low mass for the evolution of the primary . when a star is on the red giant branch ( rgb ) , it burns hydrogen in a shell and adds helium to the core . for a low - mass star , with a degenerate core , the temperature for helium ignition is attained when the core mass reaches 0.45 m@xmath4 . unless part of the core was removed during the common envelope phase , this suggests that either hr 4049 never ignited helium in its core or that it had a non - degenerate core on the rgb , which can ignite helium at lower masses . however , this latter scenario would require a much higher initial mass for hr 4049 , and would suggest that it has lost a great deal of material which is currently missing from the system . when low - mass stars terminate their evolution without igniting helium in their cores , they will end their lives as helium white dwarfs . however , hr 4049 is still a giant star . this suggests that something else may be occurring in this system . theoretical predictions show that hydrogen shell flashes occur on helium white dwarfs which have sufficient hydrogen atmospheres remaining ; these flashes are also thought to return the white dwarf to giant sizes for very short periods of time ( @xmath610@xmath100 yr , * ? ? ? the short duration of these flashes is inconsistent with the observational history of hr 4049 . perhaps it has lost most of its envelope , but has kept enough for it to sustain a longer - term hydrogen burning shell . to our knowledge , no evolutionary models of such an object have been attempted . based on our new analysis and re - interpretation of near - ir high - resolution spectoscopic data of hr 4049 , we find that the circumbinary disk surrounding the system is hot and radially extended , consistent with the results from the mid - ir observations described in paper iand by the interferometry described by @xcite . we find evidence that the fundamental co emission originates from within the central , dust - free cavity of the disk . in the absence of dust ( and thus ignoring radiation pressure ) , viscous dissipation will cause this gas to end up accreting onto one of the stars in the binary system . the gas in the circumbinary disk must play a significant role in determining the physical properties and geometry of the disk . future radiative transfer modeling should elucidate these effects . the mass of the system and the individual components are both very low and consistent with mass loss . in particular , the total mass of the system would correspond to a main sequence lifetime on the order of 10 gyr ; such an age is incompatible with the abundances of the system . therefore , the system must have lost a considerable amount of mass during its evolution . we acknowledge the support from the natural sciences and engineering research council of canada ( nserc ) . we would like to thank ken hinkle and sean brittain for their help in providing us with the gemini - phoenix data . this research has also made use of nasa s astrophysics data system bibliographic services and the simbad database , operated at cds , strasbourg , france . , joyce r.r . , et al . , 2003 , in : guhathakurta p. ( ed . ) , _ society of photo - optical instrumentation engineers ( spie ) conference series _ , vol . 4834 of _ society of photo - optical instrumentation engineers ( spie ) conference series _ , pp .
|
hr 4049 is a peculiar evolved binary which is surrounded by a circumbinary disk .
mid - infrared observations show that the disk is rich in molecular gas and radially extended . to study the properties of this disk , we re - analyzed a set of near - infrared observations at high spectral resolution obtained with gemini - phoenix .
these data cover absorption lines originating from the first overtone of co and from h@xmath0oin the 2.3 @xmath1 m region as well as more complex emission - absorption profiles from h@xmath0oand the fundamental mode of co near 4.6 @xmath1 m . by using an excitation diagram and from modeling the spectrum
, we find that most of the co overtone and h@xmath0oabsorption originates from hot gas ( @xmath2 k ) with high column densities , consistent with the mid - infrared data .
the strong emission in the wavelength range of the co fundamental furthermore suggests that there is a significant quantity of gas in the inner cavity of the disk .
in addition , there is a much colder component in the line of sight to the disk .
a detailed analysis of the overtone line profiles reveals variations in the line widths which are consistent with a radially extended disk in keplerian rotation with hotter gas closer to the central star .
we estimate the mass of the primary to be @xmath3 m@xmath4 and discuss the implications for its evolutionary status .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the scaling behavior of directed percolation is recognized as the paradigm of the critical behavior of several non - equilibrium systems which exhibits a continuous phase transition from an active state to an absorbing non - active state ( see for instance @xcite ) . the widespread occurrence of such systems in physics , biology , as well as catalytic chemical reactions is reflected by the well known universality hypothesis of janssen and grassberger that models which exhibit a continuous phase transition to a single absorbing state generally belong to the universality class of directed percolation @xcite . introducing additional symmetries the critical behavior differs from directed percolation . in particular particle conservation leads to the different universality class of absorbing phase transitions with a conserved field as pointed out in @xcite . in that work the authors introduced two models , the conserved lattice gas ( clg ) and the conserved threshold transfer process ( cttp ) . the latter one is a conserved modification of the threshold transfer process introduced in @xcite . both models display a continuous phase transition from an active to an inactive phase and are believed to belong to the same universality class @xcite . the steady - state scaling behavior of the clg model was investigated recently . the order parameter and its fluctuations were numerically examined in @xcite . the scaling behavior in an external field conjugated to the order parameter was considered in @xcite . furthermore a modified clg model was introduced which allows to determine analytically the steady - state mean - field scaling behavior of the universality class @xcite . on the other hand the scaling behavior of the cttp was investigated in low dimensional ( @xmath0 ) systems only @xcite and no external field was applied . therefore we consider in this work the cttp with and without an external field in various dimensions ( @xmath1 ) and determine a set of critical exponents . all obtained results coincides with those of the clg model , strongly supporting the universality hypothesis of @xcite . in this work we consider the cttp on simple cubic lattices of linear size @xmath2 in various dimensions with @xmath3 particles . the lattice sites may be empty , occupied by one particle , or occupied by two particles . empty and single occupied sites are considered as non - active whereas double occupied lattice sites are considered as active . in the latter case one tries to transfer both particles of each active site to randomly chosen empty or single occupied nearest neighbor sites . if no active sites exist the system is trapped forever in a certain configuration , a so - called absorbing state . in the following we denote the densities of active sites with @xmath4 and the density of particles on the lattice as @xmath5 , which is considered as the control parameter of the absorbing phase transition . the density of active sites @xmath4 is the order parameter of the absorbing phase transition , i.e. , it vanishes at the critical density @xmath6 according to @xmath7 with the reduced control parameter @xmath8 and the order parameter exponent @xmath9 . similar to equilibrium phase transitions it is possible in the case of absorbing phase transitions to apply an external field @xmath10 which is conjugated to the order parameter ( see for instance @xcite ) . as usually for continuous phase transitions the conjugated field has to destroy the disordered phase and the associated linear response function @xmath11 has to diverge at the critical point ( @xmath12 , @xmath13 ) . in the case of an absorbing phase transition the external field acts as a spontaneous creation of active particles , i.e. , the external field destroys the absorbing state and thus the phase transition itself . but considering absorbing phase transitions with particle conversation one has to take care that the external field does not change the particle number . a possible realization of the external field was developed in @xcite where the external field triggers movements of inactive particles which may be activated in this way . the external field @xmath10 is another relevant scaling field and for sufficiently small values of @xmath10 the order parameter scales as @xmath14 with the critical field exponent @xmath15 and the scaling function @xmath16 . choosing @xmath17 one recovers eq.([eq : order_par ] ) whereas @xmath18 leads at the critical density to @xmath19 in our simulations we start with randomly distributed particles . all active sites are listed and this list is updated in a randomly chosen sequence . in the case that an external field is applied the active particle creation is performed after each update step in order to mimic the external field . after a certain relaxation time the system reaches a steady state where the density of active sites at update step @xmath20 fluctuates around the average value @xmath21 which is interpreted as the order parameter @xmath22 ( see for instance figs.1 of @xcite ) . additionally to the order parameter we consider its fluctuations @xmath23 . \label{eq : def_fluc}\ ] ] approaching the transition point the fluctuations diverge for zero - field according to @xmath24 the fluctuation exponent @xmath25 fulfills the scaling relation @xcite @xmath26 where the exponent @xmath27 describes how the spatial correlation length diverges at the transition point . in the critical regime we assume that the fluctuations obey the scaling ansatz @xmath28 setting @xmath29 one recovers eq.([eq : fluc_crit ] ) for @xmath13 . analogous to equilibrium phase transitions the susceptibility is defined as the derivative of the order parameter with respect to the conjugated field @xmath30 setting @xmath31 one gets that the susceptibility diverges for zero - field as required according to @xmath32 furthermore , one yields the scaling relation @xmath33 which corresponds to the well known widom equation of equilibrium phase transitions . using this scaling relation one can calculate the value of the susceptibility exponent @xmath34 from the obtained values of @xmath9 and @xmath15 . notice that in contrast to the scaling behavior of equilibrium phase transitions the non - equilibrium absorbing phase transition is characterized by @xmath35 . as a function of the particle density for zero - field ( symbols ) and for @xmath36 and @xmath37 ( lines ) . the inset displays the order parameter fluctuations @xmath38 for zero field ( symbols ) and for @xmath39 , @xmath36 and @xmath37 ( lines ) . , width=302 ] at the beginning of our analysis we consider the scaling behavior of the order parameter for @xmath40 . system sizes up to @xmath41 for @xmath42 , @xmath43 for @xmath44 , and @xmath45 for ( @xmath46 ) are considered . in each cases we start the simulation with randomly distributed particles . after a certain transient regime the system reaches a steady state where the density of active particles fluctuates around an average value which is interpreted as the order parameter . in the steady state up to @xmath47 update steps for @xmath42 and @xmath48 for @xmath49 are performed to measure the average density of active sites . for zero - field this procedure is repeated for at least 10 different initial configurations in order to get an accurate estimation of the order parameter close to the critical point ( @xmath50 , @xmath13 ) . in fig.[fig : rho_a_1d ] we present the data of the one - dimensional order parameter at zero - field . approaching the transition point the corresponding correlation length increases and the system tends to the absorbing state if the correlation length is of the order of the system size . instead of a finite - size scaling analysis ( see for instance @xcite ) we take care of these finite - size effects in the way that we increase the system size before these finite - size effects occur and use only data from simulations that have not reached the absorbing state . as a function of the reduced particle density @xmath51 at zero - field for various dimensions @xmath52 . the dashed line corresponds to a power - law behavior according to eq.([eq : order_par ] ) for @xmath53 . for @xmath54 the data are shifted horizontally by a factor 1.5 in order to avoid an overlap . in the case of the four - dimensional model the dashed line corresponds to eq.([eq : order_par_dc_zero_field ] ) with @xmath55 . , width=302 ] as a function of the reduced particle density @xmath51 at zero - field for various dimensions @xmath52 . the dashed line corresponds to a power - law divergence [ eq.([eq : fluc_crit ] ) ] . for @xmath56 the fluctuations are maximal at the transition point but finite . , width=302 ] decreasing the particle density the order parameter decreases and vanishes at the transition point . to determine the critical indices one varies the critical density @xmath6 until one obtains asymptotically a straight line in a log - log plot the exponent is then obtained by a regression analysis . the values of the order parameter as a function of the reduced particle density @xmath57 are plotted in fig.[fig : rho_a_all_d ] . in all cases the asymptotic behavior ( @xmath58 ) of the order parameter obeys eq.([eq : order_par ] ) . for @xmath42 we get @xmath59 and @xmath60 . the latter value is smaller than the value @xmath61 estimated from significantly smaller system sizes ( @xmath62 ) @xcite . furthermore our value differs from @xmath63 obtained from simulations of the one - dimensional fix - energy manna sandpile model @xcite that is expected to belong to the same universality class . in the two dimensional case we obtain @xmath64 and @xmath65 . again the order parameter exponent differs slightly from the previously reported result @xmath66 obtained from simulations of small lattice sizes ( @xmath67 ) @xcite . but our value agrees with the estimate of the corresponding two - dimensional manna sandpile model @xmath68 @xcite . the estimates of the three dimensional model are @xmath69 and @xmath70 . all obtained critical exponents are listed in table [ table : critical_indicees ] . in fig.[fig : rho_a_fluc_all_d ] we present the order parameter fluctuations as a function of the control parameter at zero field . we observe for @xmath71 a power - law behavior according to eq.([eq : fluc_crit ] ) . using a regression analysis we get the estimates @xmath72 for @xmath42 , @xmath73 for @xmath44 , and @xmath74 for @xmath46 . for various dimensions . for @xmath75 the curves are shifted vertically in order to avoid overlaps . in the case of the four - dimensional model @xmath76 is plotted vs. @xmath77 with @xmath78 and @xmath79 ( see text ) . , width=302 ] in the following we analyze the order parameter as a function of the control parameter @xmath51 for different fields from @xmath80 up to @xmath81 . the applied field results in a smoothing of the zero - field curve , i.e. , the order parameter increases smoothly with the control parameter for @xmath82 ( see fig.[fig : rho_a_1d ] ) . according to the scaling ansatz of the order parameter [ eq.([eq : scal_ansatz ] ) ] we choose @xmath83 and get the scaling form @xmath84 thus one varies the exponent @xmath15 until the curves for different values of the driving field have to collapse onto the scaling function @xmath16 if one plots @xmath85 as a function of @xmath86 . convincing results are obtained for @xmath87 ( @xmath42 ) , @xmath88 ( @xmath44 ) , as well as @xmath89 ( @xmath46 ) and the corresponding scaling plots are shown in fig.[fig : rho_a_field_scal_all_d ] . next we consider the scaling behavior of the order parameter fluctuations @xmath90 . the fluctuation data for @xmath42 are shown for different values of the external field in the inset of fig.[fig : rho_a_1d ] . for finite fields the fluctuations display a peak . approaching the transition point ( @xmath91 ) this peak becomes a divergence signalling the critical point . in order to analyze the scaling behavior of the fluctuations we use the scaling ansatz eq.([eq : fluc_scal_ansatz ] ) and set @xmath92 @xmath93 using the above determined values of @xmath6 , @xmath15 and @xmath25 we get good data collapses confirming the accuracy of our analysis . ( see fig.[fig : rho_a_fluc_scal_all_d ] ) furthermore we determine the susceptibility exponent @xmath34 . using the scaling relation eq.([eq : widom ] ) one gets the estimates of the susceptibility exponents @xmath94 ( @xmath42 ) , @xmath95 ( @xmath44 ) , and @xmath96 ( @xmath46 ) . for various dimensions . for @xmath75 the curves are shifted vertically in order to avoid overlaps . the fluctuations diverges at the critical point for @xmath97 whereas a jump of the fluctuations is observed in higher dimensions at zero - field . in the in case of the four - dimensional model @xmath90 is plotted vs. @xmath98 with @xmath99 . , width=302 ] in the case of the four dimensional model we considered system sizes from @xmath100 up to @xmath101 . at least @xmath102 update steps were used to reach the steady state close to the transition point and @xmath103 update steps were performed to determine the order parameter and its fluctuations . . the data are rescaled according to eq.([eq : order_par_dc_zero_field ] ) . the assumed asymptotic scaling behavior ( dashed line ) is obtained for @xmath104 . , width=302 ] at the upper critical dimension @xmath105 the scaling behavior of the cttp is affected by logarithmic corrections similar to the clg model @xcite . as argued in @xcite the order parameter obeys in leading order the scaling ansatz @xmath106 where the exponents @xmath9 and @xmath15 are given by the corresponding mean - field values @xmath107 and @xmath108 @xcite , respectively . thus , for zero field the asymptotic scaling behavior of the order parameter obeys @xmath109 with @xmath110 . in our analysis we plot @xmath111 as a function of @xmath112 and vary the exponent @xmath113 as well as the critical density @xmath6 until one gets asymptotically a straight line . the best result is obtained for @xmath104 , @xmath114 and the corresponding plot is shown in fig.[fig : rho_a_4d_log ] . this value of @xmath113 differs from the corresponding value of the clg model @xmath115 @xcite . similar to the lower dimensions we consider the scaling behavior of the order parameter as a function of the control parameter for different external fields . choosing @xmath116 the scaling ansatz [ eq.([eq : scal_ansatz_dc ] ) ] yields in leading order @xmath117 where the scaling argument @xmath118 is given in leading order by @xmath119 with @xmath120 . varying the logarithmic correction exponents one gets for @xmath78 and @xmath79 a convincing data - collapse , which is shown in fig.[fig : rho_a_field_scal_all_d ] . using the values @xmath121 and @xmath79 we get the estimation @xmath122 which agrees with @xmath104 obtained from numerical simulations in zero - field . on the other hand this values differs from the corresponding estimations of the clg model @xmath115 , @xmath123 , and @xmath124 @xcite . furthermore we consider how the logarithmic corrections affect the scaling behavior of the fluctuations at the upper critical dimension . as pointed out in @xcite the order parameter fluctuations are expected to obey the scaling ansatz @xmath125 a good data collapse is observed for @xmath126 ( see fig.[fig : rho_a_fluc_scal_all_d ] ) which differs again from the corresponding value of the clg model @xmath127 @xcite . a modified version of the cttp with random neighbor hopping was recently introduced in @xcite . there , unrestricted particle hopping breaks long range correlations and the scaling behavior is characterized by the mean - field values @xmath128 , @xmath107 , and @xmath108 which are calculated analytically . in our simulations of the five dimensional model we considered system sizes from @xmath100 up to @xmath129 whereas system sizes from @xmath130 up to @xmath131 are used for @xmath54 . at least @xmath103 update steps were used to reach the steady state and @xmath103 update steps were performed to determine the order parameter and its fluctuations . the values of the order parameter are plotted in fig.[fig : rho_a_all_d ] and the obtained critical densities are @xmath132 for @xmath133 and @xmath134 for @xmath54 , respectively . in both dimensions the asymptotic scaling behavior of the order parameter is in agreement with the mean - field behavior @xmath107 . the fluctuations of the order parameter @xmath90 are plotted in fig.[fig : rho_a_fluc_all_d ] . analogous to the clg model the fluctuations are characterized by a jump at the transition point corresponding to @xmath135 @xcite . above the critical dimension , i.e. @xmath136 , the scaling behavior of the cttp is expected to obey again the scaling ansatzes eqs.([eq : scal_ansatz],[eq : fluc_scal_ansatz ] ) where the exponents are given by the mean - field values independently of the particular dimension . the obtained data collapse of the order parameter curves are presented in fig.[fig : rho_a_field_scal_all_d ] and confirm the above scenario . furthermore we consider the fluctuations above the upper critical dimension . according to the mean - field value @xmath135 @xcite we plot @xmath38 as a function of @xmath137 and the obtained data collapses are shown in fig.[fig : rho_a_fluc_scal_all_d ] . as a function of the dimension @xmath52 . the critical density of the mean - field solution is denoted by @xmath138 . the dashed line corresponds to a power - law behavior [ eq.([eq : rho_c_d ] ) ] with an exponent @xmath139 . , width=302 ] , @xmath25 , and @xmath15 of the cttp and clg model ( obtained from @xcite ) for various dimensions . the dashed lines are just to guide the eyes . , width=302 ] finally we address the question how the critical densities depends on the dimension . as can be seen from table[table : critical_indicees ] the critical density tends with increasing dimension to the mean - field value @xmath128 @xcite that corresponds to an infinite dimension . our analysis reveals that the critical densities approaches that mean - field value according to @xmath140 with @xmath141 ( see fig.[fig : rho_c_d ] ) . this behavior is different from that of clg models on simple cubic lattices which is characterized by an exponent @xmath142 @xcite . we investigated the steady state scaling behavior of the cttp model in various dimensions . the order parameter exponent , the fluctuation exponent and the external field exponent are determined and the corresponding values are listed in table[table : critical_indicees ] . for @xmath42 and @xmath44 our results of the order parameter exponents differ from previous simulations obtained from significantly smaller system sizes . our values of the critical exponents @xmath9 , @xmath25 , and @xmath15 agree within the error - bars with the corresponding exponents of the clg model ( see fig.[fig : clg_cttp_exp ] ) , strongly supporting the conjecture @xcite that both models belong to the same universality class . the picture is not so clear at the upper critical dimension @xmath105 . although the exponents are identical the logarithmic correction exponents of the cttp and clg model are different . this result is rather surprising since the logarithmic corrections exponents are a characteristic feature of the whole universality class ( see for instance @xcite ) . we think that more than statistical uncertainties this result is caused by systematic uncertainties of our analysis . in all cases we focused our attention to the leading order of the scaling behavior . taking corrections to the leading order into account may result in comparable values of the logarithmic correction exponents . further investigations are needed to clarify this point . llllll @xmath143 & @xmath144 & @xmath145 & @xmath146 & @xmath147 & @xmath34 + + @xmath148 & @xmath149 & @xmath150 & @xmath151 & @xmath152 & @xmath153 + @xmath154 & @xmath155 & @xmath156 & @xmath157 & @xmath158 & @xmath159 + @xmath160 & @xmath161 & @xmath162 & @xmath163 & @xmath164 & @xmath165 + @xmath166 & @xmath167 & @xmath168 & @xmath169 & @xmath170 & @xmath168 + @xmath171 & @xmath172 & @xmath148 & @xmath154 & @xmath173 & @xmath148 + @xmath174 & @xmath175 & @xmath148 & @xmath154 & @xmath173 & @xmath148
|
we analyze numerically the critical behavior of an absorbing phase transition in the conserved transfer threshold process . we determined the steady state scaling behavior of the order parameter as a function of both , the control parameter and an external field , conjugated to the order parameter .
the external field is realized as a spontaneous creation of active particles which drives the system away from criticality .
the obtained results yields that the conserved transfers threshold process belongs to the universality class of absorbing phase transitions in a conserved field .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
any network found in the literature is inevitably just a sampled representative of its real - world analogue under study . for instance , many networks change quickly over time and in most cases merely incomplete data is available on the underlying system . additionally , network sampling techniques are lately often applied to large networks to allow for their faster and more efficient analysis . since the findings of the analyses and simulations on such sampled networks are implied for the original ones , it is of key importance to understand the structural differences between the original networks and their sampled variants . a large number of studies on network sampling focused on the changes in network properties introduced by sampling . lee et al . @xcite showed that random node and link selection overestimate the scale - free exponent @xcite of the degree and betweenness centrality @xcite distributions , while they preserve the degree mixing @xcite . on the other hand , random node selection preserves the degree distribution of different random graphs @xcite and performs better for larger sampled networks @xcite . furthermore , leskovec et al . @xcite showed that the exploration sampling using random walks or forest - fire strategy @xcite outperforms the random selection techniques in preserving the clustering coefficient @xcite , different spectral properties @xcite , and the in - degree and out - degree distributions . more recently , ahmed et al . @xcite proposed random link selection with additional induction step , which notably improves on the current state - of - the - art . their results confirm that the proposed technique well captures the degree distributions , shortest paths @xcite and also the clustering coefficient of the original networks . lately , different studies also focus on finding and correcting biases in sampling process , for example observing the changes of user attributes under the sampling of social networks @xcite , analyzing the bias of traceroute sampling @xcite and understanding the changes of degree distribution and hubs inclusion under various sampling techniques @xcite . however , despite all those efforts , the changes in network structure introduced by sampling and the effects of network structure on the performance of sampling are still far from understood . real - world networks commonly reveal communities ( also link - density community @xcite ) , described as densely connected clusters of nodes that are loosely connected between @xcite . communities possibly play important roles in different real - world systems , for example in social networks communities represent friendship circles or people with similar interest @xcite , while in citation networks communities can help us to reveal relationships between scientific disciplines @xcite . furthermore , community structure has a strong impact on dynamic processes taking place on networks @xcite and thus provides an important insight into structural organization and functional behavior of real - world systems . consequently , a number of community detection algorithms have been proposed over the last years @xcite ( for a review see @xcite ) . most of these studies focus on classical communities characterized by higher density of edges @xcite . however , some recent works demonstrate that real - world networks reveal also other characteristic groups of nodes @xcite like groups of structurally equivalent nodes denoted modules @xcite ( also link - pattern community @xcite and other @xcite ) , or different mixtures of communities and modules @xcite . despite community structure appears to be an intrinsic property of many real - world networks , only a few studies considered the effects between the community structure and network sampling . salehi et al . @xcite proposed page - rank sampling , which improves the performance of sampling of networks with strong community structure . furthermore , expansion sampling @xcite directly constructs a sample representative of the community structure , while it can also be used to infer communities of the unsampled nodes . other studies , for example analyzed the evolution of community structure in collaboration networks and showed that the number of communities and their size increase over time @xcite , while the network sampling has a potential application in testing for signs of preferential attachment in the growth of networks @xcite . however , to the best of our knowledge , the question whether sampling destroys the structure of communities and other groups of nodes or are sampled nodes organized in a similar way than nodes in original network remains unanswered . in this paper , we study the presence of characteristic groups of nodes in different social and information networks and analyze the changes in network group structure introduced by sampling . we consider six sampling techniques including random node and link selection , network exploration and expansion sampling . the results first reveal that nodes in social networks form densely linked community - like groups , while the structure of information networks is better described by modules . however , regardless of the type of the network and consistently across different sampling techniques , the structure of sampled networks exhibits much stronger characterization by community - like groups than the original networks . we therefore conclude that the rich community structure is not necessary a result of for example homophily in social networks . the rest of the paper is structured as follows . in section [ sec : sampl ] , we introduce different sampling techniques considered in the study , while the adopted node group extraction framework is presented in section [ sec : nodegroups ] . the results of the empirical analysis are reported and formally discussed in section [ sec : analys ] , while section [ sec : conclusion ] summarizes the paper and gives some prominent directions for future research . network sampling techniques can be roughly divided into two categories : random selection and network exploration techniques . in the first category , nodes or links are included in the sample uniformly at random or proportional to some particular characteristic like the degree of a node or its pagerank score @xcite . in the second category , the sample is constructed by retrieving a neighborhood of a randomly selected seed node using random walks , breadth - first search or another strategy . for the purpose of this study , we consider three techniques from each of the categories . from the random selection category , we first adopt random node selection by degree @xcite ( rnd ) . here , the nodes are selected randomly with probability proportional to their degrees , while all their mutual links are included in the sample ( fig . [ subfig : rnd ] ) . note that rnd improves the performance of the basic random node selection @xcite , where the nodes are selected to the sample uniformly at random . rnd fits better spectral network properties @xcite and produces the sample with larger weakly connected component @xcite . moreover , it shows good performance in preserving the clustering coefficient and betweenness centrality distribution of the original networks @xcite . nevertheless , it can still construct a disconnected sample network , despite a fully connected original network . next , we adopt random link selection @xcite ( rls ) , where the sample consists of links selected uniformly at random ( fig . [ subfig : rls ] ) . rls overestimates degree and betweenness centrality exponent , underestimate the clustering coefficient and accurately matches the assortativity of the original network @xcite . the samples created with rls are sparse and the connectivity of the original network is not preserved , still rls is likely to capture the path length of the original network @xcite . last , we adopt random link selection with induction @xcite ( rli ) , which improves the performance of rls . in rli , the sample consists of randomly selected links as before , while also all additional links between their endpoints ( fig . [ subfig : rli ] ) . rli outperforms several other methods in capturing the degree , path length and clustering coefficient distribution . it selects nodes with higher degree than rls , thus the connectivity of the sample is increased @xcite . techniques from random selection category imitate classical statistical sampling approaches , where each individual is selected from population independently from others until desired size of the sample is reached . from the network exploration category , we first adopt breadth - first sampling @xcite ( bfs ) . here , a seed node is selected uniformly at random , while its broad neighborhood retrieved from the basic breadth - first search is included in the sample ( fig . [ subfig : bfs ] ) . the sample network is thus a connected subgraph of the original network . bfs is biased towards selecting high - degree nodes in the sample @xcite . it captures well the degree distribution of the networks , while it performs worst in inclusion of hubs in the sample quickly in the sampling process @xcite . bfs imitates the snowball sampling approach for collecting social data used especially when the data is difficult to reach @xcite . selected seed participant is asked to report his friends , which are than invited to report their friends . the procedure is repeated until the desired number of people is sampled . next , we adopt a modification of bfs denoted forest - fire sampling @xcite ( ffs ) . in ffs , the broad neighborhood of a randomly selected seed node is retrieved from partial breadth - first search , where only some neighbors are included in the sample on each step ( fig . [ subfig : ffs ] ) . the number of neighbors is sampled from a geometric distribution with mean @xmath0 , where @xmath1 is set to @xmath2 @xcite . ffs matches well spectral properties @xcite , while it underestimates the degree distribution and fails to match the path length and clustering coefficient of the original networks @xcite . however , ffs corresponds to a model by which one author collects the papers to cite and include them in the bibliography @xcite . the author starts with one paper , explores its bibliography and selects the papers to cite . the procedure is recursively repeated in selected papers until desired collection of citations is reached . last , we adopt expansion sampling @xcite ( exs ) , where the seed node is again selected uniformly at random , while the neighbors of the sampled nodes are included in the sample with probability proportional to @xmath3 where @xmath4 is the concerned node , @xmath5 the current sample and @xmath6 the neighborhood of nodes in @xmath5 ( fig . [ subfig : exs ] ) . expression @xmath7 denotes the expansion factor of node @xmath4 for sample @xmath5 and means the number of new neighbors contributed by @xmath4 . the parameter @xmath8 is set to @xmath9 @xcite . note that exs ensures that the sample consists of nodes from most communities in the original network and that the nodes that are grouped together in the original network , are also grouped together in the sample @xcite . exs imitates the modification of snowball sampling approach mentioned above , where for example we want to gather the data about individuals from different countries . thus , on each step we include in the sample the individuals , which knows larger number of others from various countries . the node group structure of different networks is explored by a group extraction framework @xcite with a brief overview below . let the network be represented by an undirected graph @xmath10 , where @xmath11 is the set of nodes and @xmath12 the set of links . next , let @xmath5 be a group of nodes and @xmath13 a subset of nodes representing its corresponding linking pattern ( i.e. , the pattern of connections of nodes from @xmath5 to other nodes @xcite ) , @xmath14 . denote @xmath15 and @xmath16 . the linking pattern @xmath13 is selected to maximize the number of links between @xmath5 and @xmath13 , and minimize the number of links between @xmath5 and @xmath17 , while disregarding the links with both endpoints in @xmath18 . for details on the group objective function see @xcite . the above formalism comprises different types of groups commonly analyzed in the literature ( fig . [ fig : groups ] ) . it consider communities @xcite ( i.e. , link - density community @xcite ) , defined as a ( connected ) group of nodes with more links toward the nodes in the group than to the rest of the network @xcite . communities are characterized by @xmath19 . furthermore , the formalism consider possibly disconnected groups of structurally equivalent nodes denoted modules @xcite ( i.e. , link - pattern community @xcite ) , defined as a ( possibly ) disconnected group of nodes with more links towards common neighbors than to the rest of the network @xcite . modules have @xmath20 . communities and modules represent two extreme cases with all other groups being the mixtures of the two @xcite , @xmath21 and/or @xmath22 . the reader may also find it interesting that the core - periphery structure is a mixture with @xmath23 , while the hub & spokes structure is a module with @xmath24 . the type of group @xmath5 can in fact be determined by the jaccard index @xcite of @xmath5 and its corresponding linking pattern @xmath13 . the group parameter @xmath25 @xcite , @xmath26 $ ] , is defined as @xmath27 communities have @xmath28 , while modules are indicated by @xmath29 . mixtures correspond to groups with @xmath30 . for the rest of the paper , we refer to groups with @xmath31 as community - like and groups with @xmath32 as module - like . groups in networks are revealed by a sequential extraction procedure proposed in @xcite . one first finds the group @xmath5 and its linking pattern @xmath13 with random - restart hill climbing @xcite that maximizes the objective function . next , the revealed group @xmath5 is extracted from the network by removing the links between groups @xmath5 and @xmath13 , and any node that becomes isolated . the procedure is then repeated on the remaining network until the objective function is larger than the @xmath33th percentile of the values obtained under the same framework in a corresponding erds - rnyi random graph @xcite . all groups reported in the paper are thus statistically significant at @xmath34 level . note that the above procedure allows for overlapping @xcite , hierarchical @xcite , nested and other classes of groups . section [ subsec : nets ] introduces real - world networks considered in the study . section [ subsec : orig ] reports the node group structure of the original networks extracted with the framework described in section [ sec : nodegroups ] . the groups extracted from the sampled networks are analyzed in section [ subsec : sampled ] . for a complete analysis , we also observe the node group structure of a large network with more than a million links in section [ subsec : large ] . clrr & & & + _ collab _ & high energy physics collaborations @xcite & @xmath35 & @xmath36 + _ pgp _ & pretty good privacy web - of - trust @xcite & @xmath37 & @xmath38 + _ p2p _ & gnutella peer - to - peer file sharing @xcite & @xmath39 & @xmath40 + _ citation _ & high energy physics citations @xcite & @xmath41 & @xmath42 + the empirical analysis in the following sections was performed on four real - world social and information networks . their main characteristics are shown in table [ tbl : nets ] . the _ collab _ @xcite is a social network of scientific collaborations among researchers , who submitted their papers to high energy physics theory category on the arxiv in the period from january @xmath43 to april @xmath44 . the nodes represent the authors , while undirected links denote that two authors co - authored at least one paper together . the _ pgp _ @xcite is a social network , which corresponds to the interaction network of users of the pretty good privacy algorithm collected in july @xmath45 . the nodes represent users , while undirected links indicate relationships between those , who sign each other s public key . the _ p2p _ @xcite is an information network , which contains a sequence of snapshots of the gnutella peer - to - peer file sharing network collected in august @xmath46 . the nodes represent hosts in the gnutella network , which are linked by undirected links if there exist connections between them . the _ citation _ @xcite is an information network , again gathered from the high energy physics theory category from the arxiv in the period from january @xmath43 to april @xmath44 and includes the citations among all papers in the dataset . the network consists of nodes , which represent papers , while links denote that one paper cite another . we first analyze the properties of groups extracted from the original networks summarized in table [ tbl : orig ] . the number of groups differs among networks , still the mean group size @xmath47 ( denoted @xmath48 ) is comparable across network types . groups @xmath5 in social networks consist of around @xmath49 nodes , while @xmath48 in information networks exceeds @xmath50 nodes . the mean linking pattern size @xmath51 ( denoted @xmath52 ) of social networks is comparable to @xmath48 . the latter relation @xmath53 is expected due to the pronounced community structure commonly found in social networks @xcite . on the other hand , @xmath54 is expected for information networks , due to the abundance of module - like groups . the characteristic group structure of networks is reflected in the group parameter @xmath25 . for social networks , its values are around @xmath55 , which indicates the presence of communities , modules and mixtures of these . in contrast to social networks , the information networks have @xmath25 closer to @xmath56 and consist mostly of module - like groups . to summarize , social networks represent people and interactions between them , like a few authors writing a paper together , therefore we can expect a larger number of community - like groups in these networks . on the other hand , in information networks the homophily is less typical and thus the structure of these networks seem better described by module - like groups . crrrrrrrr & & & & + & & & & & & + _ collab _ & @xmath57 & @xmath58 & @xmath59 & @xmath60 & @xmath61 & @xmath62 & @xmath63 & @xmath64 + _ pgp _ & @xmath65 & @xmath66 & @xmath67 & @xmath60 & @xmath68 & @xmath69 & @xmath70 & @xmath71 + _ p2p _ & @xmath72 & @xmath73 & @xmath74 & @xmath75 & @xmath76 & @xmath77 & @xmath78 & @xmath79 + _ citation _ & @xmath80 & @xmath81 & @xmath82 & @xmath83 & @xmath84 & @xmath77 & @xmath85 & @xmath86 + sampling techniques outlined in section [ sec : sampl ] enable setting the size of the sampled networks in advance . we consider sample sizes of @xmath87 of nodes from the original networks , that provides for an accurate fit of several network properties @xcite . table [ tbl : samplsoc ] and [ tbl : samplinf ] present the properties of the node group structure of sampled social and information networks , respectively . notice that rls and ffs show different performance than other techniques . the samples obtained with rls and ffs contain less groups , which consist of no more than @xmath88 nodes . additionally , almost all groups in these samples are modules , which reflects in the mean group parameter @xmath25 ( denoted @xmath89 ) approaching @xmath56 for all networks . to verify the above findings , we compute externally studentized residuals of the sampled networks that measure the consistency of each sampling technique with the rest . the residuals are calculated for each technique as the difference between the observed value of considered property and its mean divided by the standard deviation . the mean value and standard deviation are computed for all sampling techniques , excluding the observed one ( for details see @xcite ) . statistically significant inconsistencies between techniques are revealed by two - tailed student @xmath90test @xcite at @xmath91value of @xmath92 , rejecting the null hypothesis that the values of the considered property are consistent across the sampling techniques . statistical comparison of sampling techniques for the number of groups and the mean group parameter @xmath25 is shown on fig . [ fig : stau ] . we confirm that the samples obtained with rls and ffs reveal significantly less groups with significantly smaller @xmath89 than other sampling techniques . moreover , if we compare the number of links in the sampled networks , rls and ffs create samples that contain on average @xmath93 of links from the original networks . in contrast , the samples obtained with rnd , rli , bfs and exs consist of around @xmath94 of links from the original networks . as mentioned before , the sizes of all samples are @xmath87 of the original networks , thus the sampled networks obtained with rls and ffs are much sparser than others . in addition , the performance of rls and ffs can also be explained by their definition . since in rls we include only randomly selected links in the sample , the variance is very high , while it commonly contains a large number of sparsely linked components , whose structure is best described as module - like . on the other hand , the samples obtained with ffs consist of one connected component with a low average degree of @xmath95 . thus , the sparsely connected nodes also form groups , which are more similar to modules . due to the above reasons , we exclude rls and ffs from further analysis . we focus on rnd , rli , bfs , and exs , whose performance is clearly more comparable . the selected sampling techniques perform similarly across all networks as shown in table [ tbl : samplsoc ] for social and table [ tbl : samplinf ] for information networks . the samples consist of various number of groups , still in most cases less than the original networks . the mean sizes @xmath47 and @xmath51 are around @xmath96 , in contrast to groups with @xmath97 nodes on average in the original networks . still , @xmath98 irrespective of network type and the sampling technique , which implies stronger characterization by community - like groups , as already argued in the case of social networks in section [ subsec : orig ] . ccrrrrrrrrrr & & & & & + & & & & & & & + & / & & @xmath58 & @xmath59 & & @xmath61 & @xmath62 & @xmath85 & @xmath62 + & rnd & @xmath99 & @xmath100 & @xmath101 & @xmath102 & @xmath103 & @xmath104 & @xmath105 & @xmath106 + & rls & @xmath107 & @xmath108 & @xmath109 & @xmath110 & @xmath111 & @xmath77 & @xmath112 & @xmath113 + & rli & @xmath114 & @xmath101 & @xmath115 & @xmath116 & @xmath117 & @xmath118 & @xmath119 & @xmath120 + & bfs & @xmath121 & @xmath122 & @xmath123 & @xmath124 & @xmath125 & @xmath126 & @xmath127 & @xmath86 + & ffs & @xmath128 & @xmath129 & @xmath130 & @xmath131 & @xmath76 & @xmath77 & @xmath77 & @xmath132 + & exs & @xmath133 & @xmath134 & @xmath135 & @xmath136 & @xmath137 & @xmath138 & @xmath139 & @xmath140 + & / & & @xmath66 & @xmath67 & & @xmath68 & @xmath69 & @xmath141 & @xmath142 + & rnd & @xmath143 & @xmath144 & @xmath145 & @xmath146 & @xmath147 & @xmath148 & @xmath149 & @xmath150 + & rls & @xmath151 & @xmath152 & @xmath153 & @xmath154 & @xmath155 & @xmath138 & @xmath156 & @xmath157 + & rli & @xmath158 & @xmath144 & @xmath159 & @xmath160 & @xmath147 & @xmath161 & @xmath162 & @xmath163 + & bfs & @xmath164 & @xmath165 & @xmath166 & @xmath167 & @xmath168 & @xmath169 & @xmath170 & @xmath171 + & ffs & @xmath172 & @xmath100 & @xmath173 & @xmath131 & @xmath76 & @xmath77 & @xmath77 & @xmath132 + & exs & @xmath174 & @xmath175 & @xmath144 & @xmath176 & @xmath177 & @xmath178 & @xmath179 & @xmath180 + ccrrrrrrrrr & & & & & + & & & & & & & + & / & & @xmath73 & @xmath74 & & @xmath76 & @xmath77 & @xmath78 & @xmath79 + & rnd & @xmath181 & @xmath182 & @xmath183 & @xmath184 & @xmath185 & @xmath186 & @xmath187 & @xmath157 + & rls & @xmath188 & @xmath189 & @xmath190 & @xmath191 & @xmath76 & @xmath77 & @xmath77 & @xmath132 + & rli & @xmath192 & @xmath193 & @xmath194 & @xmath195 & @xmath196 & @xmath163 & @xmath197 & @xmath198 + & bfs & @xmath199 & @xmath200 & @xmath201 & @xmath202 & @xmath185 & @xmath140 & @xmath203 & @xmath204 + & ffs & @xmath205 & @xmath206 & @xmath207 & @xmath131 & @xmath76 & @xmath77 & @xmath77 & @xmath132 + & exs & @xmath208 & @xmath209 & @xmath210 & @xmath211 & @xmath196 & @xmath212 & @xmath213 & @xmath214 + & / & & @xmath81 & @xmath82 & & @xmath84 & @xmath77 & @xmath85 & @xmath86 + & rnd & @xmath215 & @xmath216 & @xmath217 & @xmath218 & @xmath219 & @xmath220 & @xmath221 & @xmath222 + & rls & @xmath223 & @xmath224 & @xmath225 & @xmath226 & @xmath227 & @xmath77 & @xmath77 & @xmath132 + & rli & @xmath228 & @xmath229 & @xmath230 & @xmath231 & @xmath232 & @xmath220 & @xmath233 & @xmath234 + & bfs & @xmath235 & @xmath236 & @xmath237 & @xmath238 & @xmath239 & @xmath240 & @xmath241 & @xmath242 + & ffs & @xmath243 & @xmath244 & @xmath245 & @xmath131 & @xmath76 & @xmath77 & @xmath77 & @xmath132 + & exs & @xmath246 & @xmath247 & @xmath248 & @xmath249 & @xmath250 & @xmath220 & @xmath251 & @xmath252 + indeed , the majority of groups found in sampled social networks are community - like , which reflects in the parameter @xmath253 . in sampled information networks the number of mixtures decreases and communities appear , thus @xmath25 is larger than in the original networks . [ fig : tau - hists ] shows a clear difference in the distribution of @xmath25 between the original and sampled networks . furthermore , to confirm that differences exist between the structure of the original and sampled networks , we compute externally studentized residuals , where we include the value of considered property of the original network in computing the mean over different sampling techniques . we compare the number of groups and the parameter @xmath89 for the original networks and their samples ( fig . [ fig : stauo ] ) . the results prove that the original networks contain a significantly larger number of groups with significantly smaller @xmath89 than the sampled networks . yet , larger parameter @xmath25 and consequently more community - like groups in sampled social networks and less module - like groups in sampled information networks indicate clear changes in the network structure introduced by sampling . we conclude that these changes occur regardless of the network type or the adopted sampling technique . + + notice that the largest @xmath25 and thus the strongest characterization by community - like groups is revealed in the sampled networks obtained with both random selection techniques , rnd and rli . in rnd nodes with higher degrees are more likely to be selected to the sample by definition , while rli is biased in a similar way @xcite . thus , densely connected groups of nodes have a higher chance of being included in the sampled network , while sparse parts of the networks remain unsampled . on the other hand , bfs and exs sample the broad neighborhood of a randomly selected seed node and thus the sampled network represents a connected component . in the case of bfs , all nodes and links of some particular part of the original network are sampled . the latter is believed to be representative of the entire network @xcite , yet bfs is biased towards sampling nodes with higher degree @xcite and overestimates the clustering coefficient , especially in information networks @xcite . on the other hand , exs ensures the smallest partition distance among several other sampling techniques , which means that nodes grouped together in communities of sampled network are also in the same community in the original network @xcite . therefore , the stronger characterization by community - like groups in sampled networks can also be explained by the definition and behavior of the sampling techniques . + due to the relatively high time complexity of the node group extraction framework , we consider only networks with a few thousand nodes . however , our previous study @xcite proved that the size of the original network does not affect the accuracy of the sampling . still , for a complete analysis , we also inspect the changes in node group structure introduced by sampling of a large _ notredame _ network with more than a million links . due to the simplicity and execution time , we present the analysis for two sampling techniques , rnd from random selection and bfs from network exploration category . we also limit the number of groups extracted from the networks to @xmath254 ( i.e. , we consider top @xmath254 most significant groups with respect to the objective function ) . the _ notredame _ data are collected from the web pages of the university of notre dame _ nd.edu _ domain in @xmath255 . the network contains @xmath256,@xmath257 nodes representing individual web pages , while @xmath258,@xmath259,@xmath260 links denote hyperlinks among them . table [ tbl : wnd ] shows the properties of groups , found in the original and sampled networks . the samples consist of smaller groups , still the mean size @xmath47 remains larger than the mean size @xmath51 . the majority of groups extracted from the original network are module - like , which reflects in the parameter @xmath25 slightly larger than @xmath56 . on the other hand , the changes introduced by sampling are clear , since the samples contain less modules , which is revealed by a larger parameter @xmath25 . these findings are consistent with the results on smaller networks from previous sections . the _ notredame _ as an information network expectedly consists of densely linked groups similar to modules , while the structure of sampled networks exhibits stronger characterization by community - like groups . that is again irrespective of the adopted sampling technique . crrrrrrrr & & & & + & & & & & & + / & @xmath254 & @xmath261 & @xmath262 & & @xmath263 & @xmath77 & @xmath264 & @xmath265 + rnd & @xmath254 & @xmath266 & @xmath81 & @xmath267 & @xmath268 & @xmath77 & @xmath269 & @xmath270 + bfs & @xmath254 & @xmath271 & @xmath272 & @xmath273 & @xmath274 & @xmath77 & @xmath275 & @xmath276 + in this paper , we study the presence of characteristic groups of nodes like communities and modules in different social and information networks . we observe the groups of the original networks and analyze the changes in the group structure introduced by the network sampling . the results first reveal noticeable differences in the group structure of original social and information networks . nodes in social networks form smaller community - like groups , while information networks are better characterized by larger modules . after applying network sampling techniques , sampled networks expectedly contain fewer and smaller groups . however , the sampled networks exhibit stronger characterization by community - like groups than the original networks . we have shown that the changes in the node group structure introduced by sampling occur regardless of the network type and consistently across different sampling techniques . since networks commonly considered in the literature are inevitably just a sampled representative of its real - world analogue , some results , such as rich community structure found in these networks , may be influenced by or are merely an artifact of sampling . our future work will mainly focus on larger real - world networks , including other types of networks like biological and technological . moreover , we will further analyze the changes in the node group structure introduced by sampling and explore techniques that could overcome observed deficiencies . this work has been supported in part by the slovenian research agency _ arrs _ within the research program no . p2 - 0359 , by the slovenian ministry of education , science and sport grant no . 430 - 168/2013/91 , and by the european union , european social fund . j. leskovec , j. kleinberg , c. faloutsos , graphs over time : densification laws , shrinking diameters and possible explanations , in : proceedings of the 11th acm sigkdd international conference on knowledge discovery and data mining , acm , 2005 , pp . 177187 . h. park , s. moon , sampling bias in user attribute estimation of osns , in : proceedings of the 22nd international conference on world wide web companion , international world wide web conferences steering committee , 2013 , pp . 183184 . a. lakhina , j. w. byers , m. crovella , p. xie , sampling biases in ip topology measurements , in : proceedings of the 22nd annual joint conference of the ieee computer and communications , vol . 1 , ieee , 2003 , pp . 332341 . a. s. maiya , t. y. berger - wolf , benefits of bias : towards better characterization of network sampling , in : proceedings of the 17th acm sigkdd international conference on knowledge discovery and data mining , acm , 2011 , pp . 105113 . l. ubelj , n. blagus , m. bajec , group extraction for real - world networks : the case of communities , modules , and hubs and spokes , in : proceedings of the international conference on network science , 2013 , pp .
|
any network studied in the literature is inevitably just a sampled representative of its real - world analogue .
additionally , network sampling is lately often applied to large networks to allow for their faster and more efficient analysis .
nevertheless , the changes in network structure introduced by sampling are still far from understood . in this paper
, we study the presence of characteristic groups of nodes in sampled social and information networks .
we consider different network sampling techniques including random node and link selection , network exploration and expansion .
we first observe that the structure of social networks reveals densely linked groups like communities , while the structure of information networks is better described by modules of structurally equivalent nodes .
however , despite these notable differences , the structure of sampled networks exhibits stronger characterization by community - like groups than the original networks , irrespective of their type and consistently across various sampling techniques .
hence , rich community structure commonly observed in social and information networks is to some extent merely an artifact of sampling .
complex networks , network sampling , node group structure , communities , modules + _ pacs : _ 64.60.aq , 89.75.fb , 89.90.+n
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the electron properties of two - dimensional ( 2d ) graphene , a single - layer carbon sheet , has attracted much attention by both theoreticians and experimentalists ( see @xcite and references therein ) . along with this related structures , namely graphene nanoribbons , are also under intensive investigation @xcite . one of the reason for this is that the long electron mean free path in graphene up to 1@xmath0 m opens a field of carbon - based nanoelectronics , where gnrs are used as interconnects in nanodevices . the unique electron mobility in graphene structures is caused by the strong bonding between the carbon atoms in the honeycomb lattice of graphene . this in turn prevents the replacing of the carbon atoms by alien ones . nevertheless , graphene is not immune to extrinsic disorder and its transport properties @xcite are very sensitive to impurities and defects @xcite . the theoretical problem of an impurity in 2d graphene was considered originally in @xcite . in the vicinity of the dirac points in @xmath1 space , which are peaks of the double cones of the fermi surface , the low - energy electronic excitations in gapless graphene are described by the equation of the effective mass approximation , which is formally identical to the 2d dirac equation for a massless neutrino , having the fermi speed @xmath2 . in the presence of an attractive impurity centre of charge @xmath3 screened by a medium of the effective dielectric constant @xmath4 the electron states are drastically different for the subcritical @xmath5 and supercritical @xmath6 regions of the strength of the coulomb interaction , where @xmath7 and @xmath8 are the 2d momentum of the impurity electron and that , having the speed @xmath9 , respectively . clearly , the super- and subcritical regimes can be reached for the dimensionless coulomb potential strength @xmath10 for @xmath11 , respectively . the difference of the subcritical and supercritical electron states is caused by their different behaviour in the vicinity of the impurity centre @xmath12 . the subcritical regime admits regular solutions to the dirac equations , while the wave functions corresponding to the supercritical case oscillate and do not have any definite limit . the physical reason for this is that at the subcritical strengths @xmath13 the centrifugal potential barrier prevents the electron `` fall to the centre '' @xcite , while the supercritical strengths @xmath14 provide the collapse . clearly , the continuum approach based on the dirac formalism becomes inapplicable . the lattice - scale physics dominates that in turn requires a regularization procedure , namely the cutoff of the coulomb potential at short distances @xmath15 , where @xmath16 is the c - c distance in graphene . the physics of the supercritical impurity electron in graphene @xcite closely resembles that of the relativistic electron in an atom having the nuclear charge @xmath17 @xcite . since , as it follows from below , only the supercritical regime is relevant to the impurity state in gnr , we focus on this case . numerical and analytical approaches developed on the tight - binding model of the graphene lattice and of the dirac equation , subject to the regularization procedure , respectively , undertaken originally by pereira @xcite et.al . have revealed the infinite number of the quasi - bound states , having the finite width , arising , as it was shown quasi - classically @xcite , from the collapsed states . if the requirement of the regularity of the wave functions in the vicinity of the source of the electron attraction is to be replaced by the less rigorous condition of its square integrability an infinite number of the strictly discrete energy levels were found to occur . the coulomb potential cutoff in the gapped @xcite and no cutoff in the gapless graphene @xcite induce the energy series bounded and unbounded from below , respectively . in the gnr , which in principle can be treated as a quasi-1d structure , we can expect completely different results . the strictly discrete bound states regular at the impurity centre @xmath18 are realized without the regularization procedure , in particular without the cutoff of the coulomb potential , preventing the collapse . consequential concerns lie in the well known fact that the reduction of the dimension of the structure increases the stability of the impurity electron . in units of the impurity rydberg constant @xmath19 the binding energy @xmath20 of the impurity electron in 3d bulk material is @xmath21 , in the narrow 2d quantum well @xmath22 @xcite , and in the thin quantum wire of radius @xmath23 much less than the impurity bohr radius @xmath24 , @xmath25 @xcite . besides , an extremely weak 3d atomic potential not providing bound electron states , transforms in the presence of a magnetic field into a quasi-1d system binding the electron @xcite . it is relevant to note that these atomic states arise under as weak as one likes magnetic fields i.e. the as large as one likes magnetic lengths playing the same role as the width of the gnr . note that the confinement attributed to the semiconductor thin films @xcite replaces the 3d coulomb potential @xmath26 by the effective 2d potential of the weaker singularity of the logarithmic character . it seems that in the quasi-1d gnr the effect of the attenuation of the potential singularity preventing the fall to the centre exceeds that of the vanishing of the 2d centrifugal potential barrier promoting the collapse . clearly a study of the impurity electron state in graphene structures is important on account of two reasons . first , these structures provide a realization in solid state physics of remarkable effects of quantum electrodynamics caused by a large `` fine structure constant '' @xmath27 @xcite . second , we expect a strong impact of impurities on the electronic systems not only for 2d graphene layers possessing an outstanding high electron mobility @xcite but in particular for impurity gnrs whose properties are not widely addressed in the literature yet . brey and fertig @xcite have shown that the energy spectrum of the electron in an armchair gnr bounded in @xmath28-direction is the sequence of the subbands formed by the branches of the continuous energies of the longitudinal unbounded @xmath29-motion emanating from the size - quantized energy levels @xmath30 , ( @xmath31 is the discrete lable ) , reflecting the ribbons @xmath28-confinement . the equation for the components @xmath32 of the dirac spinor relevant to the a and b sublattices of the graphene for the electron positioned far away from the impurity centre has the form @xmath33 showing that the armchair gnr manifests itself as gapped structure entailing the bound and unbound impurity states for the energies @xmath34 , respectively . of special interest is the narrow gnr of width @xmath35 for which @xmath36 , where @xmath37 is the radius of the bound electron state , being induced by the ribbon confinement @xmath35 . such a gnr provides the expected electron binding energy @xmath38 , where @xmath39 is a some function vanishing at @xmath40 , which is of interest and attractive because of two aspects . on the one hand the ribbon provides a considerable impurity binding energy which could be measured experimentally and on the other hand the impurity potential can be treated perturbatively and an analytical approach to the problem becomes feasible . a schematic form of the potentials @xmath41 provided in eqs . ( [ e : coulomb ] ) , ( [ e : pot ] ) , ( [ e : limit ] ) at @xmath42 and quasi - discrete @xmath43 ( [ e : rydberg ] ) and continuous @xmath44 ( [ e : outercont ] ) spectra adjacent to the ground @xmath45 ( discrete states ) and first @xmath46 and second @xmath47 size - quantized levels @xmath48 ( [ e : phi ] ) in the gnr of width @xmath35 . ] a comment concerning the from of the energy spectrum is in order . in the zeroth approximation of isolated size - quantized @xmath31-subbands i.e. in the single - subband approximation the slow longitudinal motion parallel to the boundaries is governed by the 2d coulomb potential averaged with respect to the `` fast '' transverse @xmath31-states . the energy spectrum consists of sequence of series of quasi - coulomb discrete @xmath49-levels and continuous sub - bands positioned below and above , respectively relatively to the size - quantized energy levels @xmath48 ( see fig . 1 ) . only the series of the impurity energy levels @xmath50 adjacent to the ground size - quantized energy level @xmath51 is strictly discrete . the @xmath49-series adjacent to the excited levels @xmath52 come into resonance with the states of the continuous spectra of lower subbands and in fact in the next multi - subband approximation turn into quasi - discrete resonant states ( fano resonances)@xcite . the corresponding resonant widths @xmath53 determine the auto - ionization rate and life - time @xmath54 of the resonant impurity states being of relevance to an experimental study . also to our knowledge an analytical approach to the problem of impurities in gnr providing the explicit dependencies of both discrete and especially quasi - discrete electron states on the width of the gnr @xmath55 and the position of the impurity centre within the gnr are not comprehensively available in the literature . in order to fill the above mentioned gap we perform an analytical study of the strictly discrete and resonant impurity states in a narrow armchair gnr . the impurity centre is positioned anywhere within the ribbon bound by the impenetrable boundaries . the width of the gnr is assumed to be much less than the radius of the impurity state . the complete 2d envelope wave function satisfying the massless dirac equation is expanded with respect to the basis formed by the 1d size - quantized subband wave functions describing the fast transverse motion bound by the boundaries of the gnr . the generated set of equations for the 1d quasi - coulomb wave functions relevant to the longitudinal slow motion is solved in the single- , two- and three - subband approximations , in which the ground , first and second excited subbands are involved . the mathematical method is based on the matching of the coulomb wave functions with those obtained by an iteration procedure at any point within the intermediate region bound by the ribbon width and the radius of the created coulomb state . both the real and imaginary parts of the complex energy levels are calculated in a single procedure . the dependencies of the binding energy and resonant energy shift and width on the width of the gnr and the position of the impurity centre are obtained in an explicit form . numerical estimates show that for a narrow gnr the binding energy and the resonant width are quite reasonable , to render the impurity electron states in gnr experimentally observable . our analytical results are in line with those calculated numerically and revealed in an experiment . we remark that our aim is to elucidate the physics of the impurity states in gnr by deriving closed form analytical expressions for their properties . we do not intend to compete with the results of computational studies . the paper is organized as follows . in section 2 the analytical approach based on the multi - subband approximation is described . the real quasi - coulomb functions of the discrete and continuous spectrum and the real energy levels determining the binding energies are calculated in the single - subband approximation in section 3 . the complex energies including the resonant shift and width associated to the first and second excited subbands are found in the double and three - subband approximation , respectively , in section 4 . in section 5 we discuss the obtained results . section 6 contains the conclusions . according to the above an analytical description of the stable and metastable impurity electron states in the narrow armchair gnr is of significant interest . it elucidates the underlying basic physics of the carbon - based nanodevices , in which the highly mobile electrons remain unbound in the 2d graphene monolayers while in their interconnects , namely in quasi-1d armchair gnrs these electrons are trapped by impurity centres . the latter could modify the overall transport properties . we consider a ribbon of width @xmath35 located in the @xmath56 plane and bounded by the lines @xmath57 the impurity centre of charge @xmath3 is displaced from the mid - point of the ribbon @xmath58 by the distance @xmath59 the equation describing the impurity electron at a position @xmath60 possesses the form of a dirac equation @xmath61 where the hamiltonian @xmath62 is given by @xmath63 + v(\vec{\rho})\hat{{\rm i}};\,p=\hbar v;\,v=10 ^ 6~\mbox{m / c}\ ] ] with @xmath64 composed by the hamiltonians relevant to the inequivalent dirac points @xmath65 and @xmath66 ( @xmath67 are the pauli matrixes ) presented originally in ref . the matrix @xmath68 in ( [ e : basic ] ) is the unit matrix and @xmath69 is the 2d coulomb impurity potential , @xmath4 is the effective dielectric constant related to the static dielectric constant @xmath70 of the substrate by @xcite @xmath71 the envelope wave four - vector @xmath72 consists of two vectors @xmath73 describing the motion of the electron in sublattices @xmath74 and @xmath75 of graphene @xmath76 each determined by the wave functions @xmath77 @xmath78 the total @xmath79 state implies the multiplication of the @xmath77 functions with the factors @xmath80 , respectively . the boundary conditions for the armchair ribbon require the total wave function to vanish at both edges @xmath81 for both @xmath79 superlattices @xcite @xmath82 where @xmath83 is the graphene superlattice constant . the basis wave vectors @xmath84 and the energies @xmath48 describing the transverse size - quantized @xmath28-states are derived from equation @xmath85 to obtain @xmath86;~ \vec{\phi}_{na } ( x)= \begin{array}{c } \begin{bmatrix } \varphi_{na}^{(+ ) } \\ 0 \\ \varphi_{na}^{(-)}\\ 0 \end{bmatrix } \end{array } ; ~ \vec{\phi}_{nb } = \begin{array}{c } \begin{bmatrix } 0\\ \varphi_{nb}^{(+ ) } \\ 0 \\ \varphi_{nb}^{(- ) } \end{bmatrix } \end{array}\ ] ] , where @xmath87 with @xmath88 \right ) \right ] \right \ } ; \nonumber\\ \varepsilon_n=|n-\tilde{\sigma}|\frac{\pi p}{d};~n=0,\pm1,\pm2,\ldots~ ; \quad\tilde{\sigma}=\frac{kd}{\pi}-\left [ \frac{kd}{\pi}\right]\end{aligned}\ ] ] we consider transverse states with positive energies @xmath89 in the armchair ribbon of width @xmath35 providing the gaped ( insulator ) structure @xmath90 @xcite . it follows from eq.([e : phi ] ) that the energy levels @xmath48 as a function of width @xmath35 are the oscillations describing by parameter @xmath91 imposed on the decreasing curve @xmath92 . below we ignore these oscillations keeping @xmath93 the boundary conditions ( [ e : bound ] ) after substitution of @xmath77 by @xmath94 , respectively , are satisfied . the wave vectors @xmath95 form orthonormal subsets , for which @xmath96 the boundary conditions ( [ e : bound ] ) imposed on the wave vector @xmath97 ( [ e : wavevect ] ) force us to expand the wave vectors @xmath73 in series @xmath98 with respect to the basis functions @xmath99 taking for the coefficients @xmath100 substituting the wave vector @xmath72 ( [ e : wavevect ] ) with the wave vectors @xmath73 and the wave functions @xmath101 into eq . ( [ e : basic ] ) and subsequently using the properties ( [ e : ortho ] ) we obtain by the standard method the set of equations for the wave functions @xmath102 @xmath103 @xmath104,\ ] ] where the potential @xmath105 is given by eq . ( [ e : coulomb ] ) . at @xmath106 @xmath107~;\,s=0,1,2,\ldots ; \ ] ] as expected in the limiting case @xmath108 eqs . ( [ e : set ] ) decompose into the sets describing the 1d coulomb states , while in the absence of the impurity centre @xmath109 we arrive at the wave functions @xmath110 and the energies @xmath111 of free electrons in the armchair nanoribbon @xcite . below we solve the set ( [ e : set ] ) in the adiabatic approximation implying the longitudinal @xmath29-motion governed by the quasi - coulomb potentials @xmath112 to be much slower than the transverse @xmath28-motion affected by the boundaries of the narrow ribbon . the coulomb potential ( [ e : coulomb ] ) is assumed to be small compared to the ribbon confinement . in the case of @xmath113 the lowest three sub - bands are specified by indices @xmath114 . the set ( [ e : set ] ) corresponding to these subbands becomes @xmath115=0~;\\ v_0^{(2 ) ' } + \frac{1}{p}\left(e - \varepsilon_0 -v_{00}\right)v_0^{(1 ) } -\frac{1}{p } \left [ v_{10}v_{1}^{(1)}+ v_{-10}v_{-1}^{(1)}\right]=0~;\\ v_1^{(1 ) ' } - \frac{1}{p}\left(e + \varepsilon_1 -v_{11}\right)v_1^{(2 ) } + \frac{1}{p}\left [ v_{01}v_{0}^{(2)}+ v_{-11}v_{-1}^{(2)}\right]=0~;\\ v_1^{(2 ) ' } + \frac{1}{p}\left(e - \varepsilon_1 -v_{11}\right)v_1^{(1 ) } -\frac{1}{p } \left [ v_{01}v_{0}^{(1)}+ v_{-11}v_{-1}^{(1)}\right]=0~;\\ v_{-1}^{(1 ) ' } - \frac{1}{p}\left(e + \varepsilon_{-1 } -v_{-1 - 1}\right)v_{-1}^{(2 ) } + \frac{1}{p}\left [ v_{0 - 1}v_{0}^{(2)}+ v_{1 - 1}v_{1}^{(2)}\right]=0~;\\ v_{-1}^{(2 ) ' } + \frac{1}{p}\left(e - \varepsilon_{-1 } -v_{-1 - 1}\right)v_{-1}^{(1 ) } -\frac{1}{p } \left [ v_{0 - 1}v_{0}^{(1)}+ v_{1 - 1}v_{1}^{(1)}\right]=0~ ; \end{array } \right \}\end{aligned}\ ] ] at the first stage we neglect the coupling between the states corresponding to the subbands with different @xmath116 the reason for this is that in the narrow ribbon of small width @xmath35 the diagonal potentials @xmath117 dominate the off - diagonal terms @xmath118 almost everywhere but for a small region @xmath119 ( see eq . ( [ e : onoff ] ) ) . in this case @xmath120 and the set ( [ e : set ] ) decomposes into independent subsets each specified by an index @xmath31 . the 1d impurity states are then governed by the potential @xmath121 @xmath122 the set ( [ e : set ] ) for @xmath123 with @xmath124 is solved by matching in the intermediate region the corresponding solutions @xmath125 one of which is valid in the inner region close to the impurity centre and the other represents a solution of the outer region distant from the centre . this method was originally developed by hasegava and howard @xcite in studies of excitons subject to strong magnetic fields and then successfully employed for the investigation of the impurity and exciton states in quantum wells @xcite , super - lattices @xcite and quantum wires @xcite . in the inner region + @xmath126 an iteration procedure is performed . the first integration of the set ( [ e : set ] ) , in which we neglect the terms consisting of the energies @xmath127 , with the trial functions @xmath128 gives @xmath129~;\quad f_{1,2}=\frac{2y}{d_{1,2}}\ ] ] where @xmath130 and where @xmath131 is the dimensionless strength of the coulomb potential . the function @xmath132 can be obtained from eq . ( [ e : step1 ] ) by replacing @xmath133 by @xmath134 and @xmath134 by @xmath135 . subsequent integration leads to the two independent particular solutions @xmath136 and @xmath137 corresponding to the relationships @xmath138 the linear combination of these solutions provides the general iteration functions , which read in the region @xmath139 @xmath140 where @xmath141 and where @xmath142 and @xmath143 are the arbitrary magnitude and phase , respectively . since the potentials ( [ e : pot ] ) satisfy @xmath144 = @xmath145 the wave two - vectors + @xmath146 are classified with respect to parity . further we focus on the even wave vectors for which @xmath147 where @xmath148 with @xmath149 ( @xmath150 is the pauli matrix ) . the condition of even parity imposed on the wave vector @xmath151 formed by the components ( [ e : iter ] ) , implies that the phases @xmath143 are equal to the half integer of @xmath152 . obviously , an alternative way to derive eq . ( [ e : iter ] ) is to solve eqs . ( [ e : set ] ) for @xmath153 and then to compare the resulting solutions @xmath154 , expanded in series up to the terms of the first order of @xmath155 , with those given by eqs . ( [ e : step1 ] ) . the calculated constant @xmath74 leads to the functions ( [ e : iter ] ) . \a ) _ discrete states _ the exact solutions to eqs . ( [ e : set ] ) at @xmath124 for @xmath156 in the region @xmath157 with @xmath158 are calculated by the same method employed in studies of a relativistic electron in hydrogen @xcite and super - heavy atoms with the nuclear charge number number @xmath17 @xcite @xmath159\\ \sinh\frac{\psi_n}{2}\tau^{-\frac{1}{2}}\left [ w_{\kappa,\mu}(\tau ) -\frac{\tanh \psi}{q}w_{\kappa + 1,\mu}(\tau ) \right ] \end{array } \right.\end{aligned}\ ] ] where @xmath160 and where @xmath161 is the whittaker function associated with the kummer function @xmath162 @xcite @xmath163 with @xcite @xmath164 the functions ( [ e : outer ] ) are normalized to @xmath165 . the asymptotic behavior of the outer functions ( [ e : outer ] ) at large distances @xmath166 follows from eqs . ( [ e : outer]),([e : whitt ] ) and ( [ e : kum ] ) @xmath167 in the region @xmath168 eqs . ( [ e : outer]),([e : whitt ] ) and ( [ e : kum ] ) lead to the expressions @xmath169 where @xmath170 and @xmath171 are arbitrary constants . matching the eqs . ( [ e : outer ] ) and ( [ e : iter ] ) in the overlapping intermediate region @xmath172 we impose the condition @xmath173 which yields @xmath174 using the properties of the arguments of the @xmath175-functions in eq . ( [ e : theta ] ) for a small parameter @xmath176 and for the quantum numbers @xmath177 @xcite ( see also ref . @xcite for details ) and choosing @xmath178 we arrive at the equation for the corrections @xmath179 @xmath180 -\ln ( n+\delta_{nn } ) \qquad \qquad \nonumber \\ + ~~ \psi(1+n)+\ln \frac{|n-\tilde{\sigma}|\pi d}{2d}+2c-1=0,\end{aligned}\ ] ] where @xmath181 is the psi - function ( logarithmic derivative of the @xmath175-function ) , @xmath182 is the euler constant . the corrections @xmath179 calculated from eq . ( [ e : discr ] ) determine the rydberg series of the discrete energy levels @xmath183 adjacent to the size - quantized energy level @xmath48 @xmath184;\ , & n=1,2,\ldots \\ \frac{\varepsilon_n}{\sqrt{1+\frac{q^2}{\delta_{n0}^2}}};~&n=0 \end{array } \right.\end{aligned}\ ] ] which allow to estimate the size of the coulomb state in eq . ( [ e : asimpt ] ) @xmath185 for @xmath186 and @xmath187 clearly from ( [ e : rydberg ] ) , the existence of the intermediate matching region @xmath188 is provided for excited state @xmath186 by the employed above small parameter @xmath189 and for the ground state @xmath190 by the condition @xmath191 the correction @xmath192 satisfies the transcendental equation @xmath193 + \ln \frac{|n-\tilde{\sigma}|\pi d}{2d}+c-1=0,\ ] ] while the corrections @xmath179 for the excited states @xmath186 can be calculated in an explicit form @xmath194\right\},\ ] ] \b ) _ continuous states _ since our approach to determine the wave function of the continuous states closely resembles that applied above for the wave functions of the discrete states only the basic points will be given below . setting in eqs . ( [ e : outer ] ) @xmath195 we obtain @xmath196\\ -{\rm i}&\sin\frac{\varphi_n}{2}t^{-\frac{1}{2}}\left [ w_{\tilde{\kappa},\mu}(t ) + { \rm i}\frac{\tan \varphi}{q}w_{\tilde{\kappa}+1,\mu}(t ) \right ] . \end{array } \right.\end{aligned}\ ] ] where @xmath197 the wave vectors ( [ e : outercont ] ) are normalized to @xmath198 . at large distances @xmath199 the wave functions ( [ e : outercont ] ) , ( [ e : whitt ] ) , ( [ e : kum ] ) have the asymptotic form of the outgoing waves @xmath200 further we introduce the real wave functions associated with the standing waves @xmath201,\ ] ] where the functions @xmath202 have the asymptotic form of the ingoing waves @xmath203 and @xmath204 and @xmath205 are the arbitrary magnitude and phase , respectively . in the region @xmath206 eqs . ( [ e : outercont ] ) , ( [ e : whitt ] ) and ( [ e : kum ] ) lead to @xmath207\ ] ] where @xmath208 ; \xi_{+,-}=\arg \gamma ( -2{\rm i } q)\mp \arg \gamma_{+,- } ; m(\varphi)=\frac{\sin\frac{\varphi}{2}+\cos\frac{\varphi}{2}}{1+\sin\varphi}.\ ] ] the wave functions @xmath209 can be obtained from eq . ( [ e : outercont1 ] ) by replacing @xmath204 by @xmath210 and @xmath211 by @xmath212 for @xmath213 the wave functions ( [ e : outercont1 ] ) read @xmath214 ; v_{n \mbox{out}}^{(2)}(t)=d_n\left [ \cos\chi_n + c_n\cot\lambda_n\sin\chi_n \right]\ ] ] with @xmath215 similar to the case of the discrete states we obtain the equation for the phase @xmath216 on equating the ratios @xmath217 taken for the iteration ( [ e : iter ] ) and outer functions ( [ e : outercont2 ] ) @xmath218 since @xcite @xmath219 eq . ( [ e : lambda ] ) acquires for @xmath220 an explicit form @xmath221 - 2c+1}.\ ] ] since at @xmath189 the phase @xmath222 ( @xmath223 is the riemann zeta function with @xmath224 ) is the value of the higher order of smallness @xmath225 we set in eq . ( [ e : omega ] ) @xmath226 . as expected , setting in the functions @xmath227 ( [ e : outercont1 ] ) @xmath228 and then matching these functions with the iteration functions @xmath229 ( [ e : iter ] ) we obtain the equation for the phase @xmath205 . substituting this result into equation @xmath230 determining the poles of the @xmath231 matrix with @xmath232 @xcite we arrive at eqs . ( [ e : eqn ] ) , ( [ e : discr ] ) for the discrete energy levels . below we consider the coupling between the ground @xmath45 and first excited @xmath46 states described by the system of the four upper eqs . ( [ e : set1 ] ) at @xmath233 applying the iteration method with the trial functions @xmath234 we arrive at two particular linear independent four - vectors , having the components @xmath235 calculated for @xmath236 . the linear combination of these vectors taken for @xmath237 $ ] provides the general expression for the iteration four - vector with the components @xmath238 where @xmath239 are an arbitrary constant and phase , respectively . the parameter @xmath240 \nonumber\\ & + & \sin\alpha_0 \left [ { \rm { si}}\left ( \frac{\pi}{2 } + \alpha_0\right)+ { \rm { si}}\left ( \frac{\pi}{2 } -\alpha_0\right ) \right],~\alpha_0=\frac{\pi x_0}{d}.\end{aligned}\ ] ] consisting of the integral sine @xmath241 @xcite , describes the coupling induced by the potentials @xmath242 ( [ e : pot ] ) . the functions @xmath243 can be obtained from the functions @xmath244 ( [ e : iter2 ] ) by mutual replacing @xmath245 equating the ratios of the functions of the continuous spectrum + ( [ e : outercont2 ] ) @xmath246 and the iteration functions ( [ e : iter2 ] ) @xmath247 and then matching the ratios of the functions of the discrete states ( [ e : outer1 ] ) @xmath248 and the iteration functions @xmath249 we obtain the set of equations where the functions @xmath251 are defined by eqs . ( [ e : qfirst ] ) , ( [ e : iter ] ) , ( [ e : outer1 ] ) , respectively . in the limiting case of negligible coupling the set ( [ e : setcoupl ] ) decomposes into two independent equations relevant to the discrete ( [ e : eqn ] ) , ( [ e : discr ] ) and continuous ( [ e : lambda ] ) states . solving the set ( [ e : setcoupl ] ) by the determinantal method we obtain the equation for @xmath252 , which is then expanded in series with respect to the parameter @xmath253 keeping at @xmath178 the terms of the first order @xmath254 we arrive at the equation for the phase @xmath255 in an explicit form @xmath256 substituting eq . ( [ e : cot ] ) into equation @xcite @xmath257 the complex quantum numbers @xmath258 introduced in eq . ( [ e : outer ] ) can be calculated , which in turn determine the complex energy levels @xmath259 adjacent to the size - quantized first excited energy level @xmath260 @xmath261 where the second term in the right - hand part is the rydberg series of the energy levels associated with the quasi - coulomb diagonal potential @xmath262 ( [ e : limit ] ) ( no coupling ) . the following notation in eq . ( [ e : compl ] ) for the resonant shift @xmath263 and resonant width @xmath264 both induced by the inter - subband @xmath265 interaction is used @xmath266 and @xmath267 in eqs . ( [ e : width ] ) and ( [ e : shift ] ) @xmath268 @xmath269 where the corrections @xmath270 can be calculated from eqs . ( [ e : discr ] ) ( [ e : ground ] ) ( [ e : exc ] ) . in the logarithmic approximation @xmath271 , @xmath272 the quantum number @xmath273 can be found from equation @xmath274,\ ] ] with ( [ e : phi ] ) for @xmath275 . in conclusion of this section note that the equation absolutely identical to eqs . ( [ e : compl ] ) - ( [ e : shift ] ) can be derived by matching the real iteration functions @xmath276 ( [ e : iter ] ) and complex functions of the continuous states @xmath277 ( [ e : outercont ] ) having the asymptotic form of the outgoing wave . in this section we consider the coupling between the discrete states adjacent to the highest size - quantized level @xmath278 and and the continuous states attributed to the low - lying levels @xmath279 and @xmath280 . below we neglect in the set ( [ e : set1 ] ) the off - diagonal potentials @xmath281 and @xmath282 describing the interactions of the @xmath265 subbands . extending the iteration procedure employed above for the single- and double- subband approximation to the present stage with the trial functions @xmath283 we arrive at two particular linear independent sixfold vectors @xmath284 calculated for @xmath285 . taking @xmath286 $ ] , we obtain the components @xmath287 of the total iteration sixfold vector @xmath288 @xmath289 where @xmath290 is determined in eq . ( [ e : iter ] ) and @xmath142 and @xmath143 are arbitrary constants and phases , respectively . the parameter @xmath291 ( [ e : couple ] ) , while @xmath292 \nonumber\\ + \sin 2\alpha_0 \left[\mbox{ci}(\pi + 2\alpha_0)- \mbox{ci}(\pi - 2\alpha_0 ) \right],~\alpha_0=\frac{\pi x_0}{d}\end{aligned}\ ] ] describes the coupling induced by the potentials @xmath293 ( [ e : pot ] ) . the functions @xmath294 can be obtained from the functions @xmath295 ( [ e : iter3 ] ) , respectively , by replacing @xmath296 . as mentioned above further we match the wave functions of the continuous spectrum @xmath297 having the asymptotic form of the outgoing wave @xmath298 ( [ e : outercont ] ) to give in turn the sixfold vector @xmath299 with the components @xmath300 \nonumber\\ -(1-c_n)\exp\left[-\imath \left(q\ln2k_n y + \xi_{- } \right ) \right],~n=0,1,\end{aligned}\ ] ] where @xmath301 and @xmath211 are given in eqs . ( [ e : outercont2 ] ) ( [ e : outercont1 ] ) and ( [ e : outercont ] ) . the wave functions @xmath302 can be obtained from the functions @xmath303 ( [ e : outercont3 ] ) , respectively , by replacing @xmath304 the wave functions @xmath305 have the form ( [ e : outer1 ] ) , in which @xmath306 ( [ e : outer ] ) and @xmath307 ( [ e : theta ] ) are calculated for @xmath47 . matching the sixfold wave vectors @xmath308 within the intermediate region by imposing the conditions @xmath309 where @xmath310 are given by eqs . ( [ e : outercont3 ] ) and ( [ e : outer ] ) and @xmath311 by eqs . ( [ e : iter3 ] ) we obtain @xmath312 - r_{-1 } q \gamma_{0,-1 } [ a_0^{(-)}\sin\zeta_{-1}+{\rm i } a_0^{(+)}\cos\zeta_{-1}]=0~;\\ r_1 [ a_1^{(- ) } \cos(q + \zeta_1 ) - { \rm i } a_1^{(+)}\sin(q + \zeta_1 ) ] - r_{-1 } q \gamma_{1,-1 } [ a_1^{(-)}\sin\zeta_{-1}+{\rm i } a_1^{(+)}\cos\zeta_{-1}]=0~;\\ r_{-1}\sin(\omega_{-1}-q - \zeta_{-1})-r_{1}q\gamma_{1,-1}\cos(\omega_{-1}-q - \zeta_{1})- r_{0}q\gamma_{0,-1}\cos(\omega_{-1}-q - \zeta_{0})=0~ , \end{array } \right \}\end{aligned}\ ] ] in eqs . ( [ e : setcouple1 ] ) @xmath313 \pm(1-c_{n})\exp\left[-{\rm i } \left(q\ln2k_n y + \xi_{- } \right)\right]\ ] ] and @xmath314 are introduced by eqs . ( [ e : iter ] ) and ( [ e : outer1 ] ) for @xmath47 , respectively . solving the set ( [ e : setcouple1 ] ) by the determinantal method we obtain the equations for the complex quantum numbers @xmath315 ( [ e : outer ] ) , in which we take for the phases @xmath316 and keep the terms of the first order of @xmath189 @xmath317 with @xmath318 and the quantum numbers @xmath319 are obtained from @xmath320\ ] ] with ( [ e : phi ] ) for @xmath278 . the complex quantum numbers @xmath321 calculated from eq ( [ e : image1 ] ) lead to the complex impurity energy levels adjacent to the size - quantized second excited sub - band @xmath278 @xmath322 where the resonant width @xmath323 and shift @xmath324 have the form @xmath325 and @xmath326 the coefficients @xmath327 are defined in eqs . ( [ e : width ] ) and ( [ e : shift ] ) . we define the binding energy of the electron @xmath328 in the @xmath43-th quasi - coulomb state associated with the @xmath31 size - quantized subband as the real part of the difference between the size - quantized energy @xmath48 ( [ e : phi ] ) of the free electron and the energy of the impurity electron @xmath183 given by eqs . ( [ e : rydberg ] ) , ( [ e : compl ] ) and ( [ e : compl ] ) for the ground @xmath45 , first @xmath46 and second @xmath47 excited subbands , respectively . since the resonant shifts @xmath329 ( [ e : shift ] ) , ( [ e : shiftn ] ) are of the order of @xmath330 with respect to the rydberg energies determined by the second terms in the right - hand parts of eqs . ( [ e : rydberg ] ) , ( [ e : compl ] ) and ( [ e : compl ] ) the binding energies read @xmath331;~&n=0 \end{array } \right.\end{aligned}\ ] ] where the corrections @xmath179 can be calculated in the single - subband approximation from eqs . ( [ e : ground ] ) and ( [ e : exc ] ) for the ground @xmath190 and excited @xmath186 impurity states , respectively . it follows from eqs . ( [ e : bind ] ) and ( [ e : phi ] ) that the binding energy @xmath332 and the oscillatory part of @xmath48 ( see @xmath91 in eq . ( [ e : phi ] ) ) decrease with increasing the ribbon width @xmath35 . in an effort to render our calculations close to an experimental setup , we take below for the estimates of the expected values the parameters @xmath333 @xcite and @xmath334 corresponding to the @xmath335 and sapphire , respectively , employed as substrates for gnr @xcite . the latter parameter @xmath155 is close to the limit caused by the condition @xmath336 ( see below eq.([e : rydberg ] ) ) . further we focus on the monotonic dependence @xmath92 and keep the the parameter @xmath337 in eq . ( [ e : phi ] ) for the levels @xmath48 to be @xmath338 . the binding energy @xmath339 ( [ e : bind ] ) of the ground state @xmath190 calculated for(@xmath340 ) as a function of the reciprocal width @xmath341 of the gnr . impurity is placed symmetrically to the boundaries @xmath342 . the parameter @xmath343 . ] the dependencies of the binding energies on the width of the gnr @xmath35 for the ground state for different strengths of the impurity potential are given in fig . 2 . these graphs , while ignoring the oscillations , are qualitatively completely in line with the data of the numerical calculations and experimental observations recently performed with the related coulomb systems . the exciton effects in the armchair gnrs were studied in frame of the tight - binding model @xcite and density functional theory @xcite , while han et.al . @xcite investigated experimentally the influence of the localized states in gnrs on the electron transport . the relation @xmath344 including oscillations @xcite have been found to occur . the differences between the impurity states considered here and the exciton and localized states prevent us from a detailed quantitative comparison . the dimensionless binding energy @xmath345 calculated from ( [ e : bind ] ) , ( [ e : ground ] ) , ( [ e : phi ] ) with @xmath346 for the ground state @xmath347 plotted as a function of the effective index of the corresponding subband @xmath348 and the relative impurity position @xmath349 in the gnr of width @xmath35 . ] the coulomb pattern of the energy levels ( [ e : rydberg ] ) enables to introduce the effective rydberg constant @xmath350 , the bohr radius @xmath351 and the mass @xmath352 for the impurity electron in gnr @xmath353 which additionally illustrate the physical reason of the bonding of the impurity electron , namely the quasi-1d geometry of the gnr . note that the bound states arise at any finite width @xmath354 . this result is qualitatively analogous to the effect of anti - diamagnetism caused by the influence of the magnetic field on the weakly bound atomic state . demkov and drukarev @xcite considered the 3d potential well of small radius and depth to provide the capturing of the electron . it was shown that the arbitrarily weak magnetic field @xmath75 induces the bound electron state with the binding energy @xmath355 . the common reason for this is that the finite width @xmath354 and magnetic length @xmath356 transform the graphene monolayer and atomic structure , respectively , into the quasi-1d systems , which are more favorable to generate bound states . the dependencies @xmath357 and @xmath344 correspond to the different dispersion laws namely @xmath358 and @xmath359 @xmath360 for the atomic ( @xmath361 ) and gnr ( @xmath362 ) electron , respectively . the dependence of the binding energy @xmath328 ( [ e : bind ] ) on the displacement of the impurity centre @xmath363 from the mid - point of the ribbon @xmath58 is contained in the corrections @xmath364 namely in the term @xmath365 in eqs . ( [ e : ground ] ) and ( [ e : exc ] ) , while @xmath328 as a function of the effective number of the subband @xmath366 is given by the sub - band threshold @xmath48 ( [ e : phi ] ) mainly and the term @xmath367 in the correction @xmath179 . the dimensionless binding energy @xmath368 as a function of the effective quantum number @xmath366 and relative displacement @xmath369 for the ground @xmath190 state is depicted in fig . 3 . clearly , the higher the subband i.e. the greater the value @xmath366 is the less the binding energy @xmath370 . also the binding energy decreases when the impurity shifts from the mid - point of the ribbon towards the boundaries . the latter conclusion coincides with those obtained for the quantum well in refs . @xcite . the inter - band coupling shifts the strictly discrete excited rydberg series @xmath371 ( [ e : rydberg ] ) calculated in single - subband approximation by an amount @xmath329 ( [ e : shift ] ) @xmath46 , and ( [ e : shiftn ] ) @xmath47 and transforms them to the quasi - discrete levels of width @xmath53 ( [ e : width ] ) @xmath46 , and ( [ e : widthn ] ) @xmath47 . note that the conclusions made on the base of the first and second excited subbands can be qualitatively extended to others . since the resonant shifts @xmath372 first are much less than the resonant widths @xmath373 at @xmath374 and second the resonant shifts do not change the discrete character of the energy spectrum ( [ e : rydberg ] ) we focus on the widths @xmath53 . it is clear from eqs . ( [ e : width ] ) , and ( [ e : widthn ] ) that the widths @xmath375 increase with decreasing the ribbons width @xmath35 . note that this dependence is opposite to that in a semiconductor narrow quantum well : the narrower the well is the less are the resonant widths @xcite . the reason for this is that in the quantum well the resonant width @xmath376 where the impurity rydberg constant @xmath377 and the binding energy @xmath328 do not depend on the well width @xmath35 , while the inte - rband energy distance @xmath378 increases and consequently the resonant width decreases with the narrowing of the quantum well . for the ribbon @xmath379 ( [ e : bind ] ) and the inter - subband coupling do not depend on the ribbon width @xmath35 and @xmath380 . the resonant width @xmath381 ( [ e : width ] ) of the ground impurity state @xmath347 relatively to the corresponding threshold @xmath382 ( [ e : phi ] ) @xmath383 versus the relative impurity position @xmath349 in the gnr of width @xmath35 providing the parameter @xmath343 . the parameter @xmath155 is taken to be @xmath340 . ] the dependence of the resonant widths @xmath384 ( [ e : width ] ) , calculated in the double - subband approximation , on the position of the impurity centre @xmath363 is described by the coupling parameter @xmath385 ( [ e : couple ] ) and the corrections @xmath270 ( [ e : ground ] ) and ( [ e : exc ] ) . the dependencies of the relative resonant widths @xmath386 ( [ e : width ] ) , on the dimensionless shift @xmath387 for the ground @xmath190 state are presented in fig.4 , in which the limitation on the parameter @xmath388 are caused by the condition imposed on @xmath389 placed below eq . ( [ e : rydberg ] ) . for the impurity positioned at the mid - point of the ribbon @xmath390 the resonant width and shift both vanish @xmath391 because of the even @xmath28-parity of the coulomb potential @xmath105 ( [ e : coulomb ] ) in eq . ( [ e : pot ] ) and opposite parities of the neighboring @xmath265 transverse @xmath28-states to give @xmath392 . both in the quantum well and in the ribbon the shift of the impurities from their mid - points eliminates the even @xmath28-parity of the potential @xmath105 ( [ e : coulomb ] ) in eq . ( [ e : pot ] ) , that leads to the coupling between the @xmath393 subbands . if the impurity displaces from the mid - point towards the boundaries @xmath394 the resonant widths @xmath264 ( [ e : width ] ) monotonically increases . this correlates completely with the analogous dependence found for the impurity states in the semiconductor quantum well @xcite . for small shifts @xmath395 in eq . ( [ e : couple ] ) we obtain for the parameter @xmath396 in eqs . ( [ e : width ] ) @xmath397 while for the impurity positioned close to the ribbon edge @xmath398 we obtain @xmath399 . note that the zeroth width of the first excited @xmath400 @xmath401 series in case of the symmetrical @xmath42 impurity position is a consequence of the double - subband approximation . in the multi- subband approximation the levels of the above mentioned series would acquire finite widths . the resonant width @xmath402 ( [ e : widthn ] ) of the ground impurity state @xmath347 relatively to the corresponding threshold @xmath278 ( [ e : phi ] ) @xmath403 versus the relative impurity position @xmath404 in the gnr of width @xmath35 using the parameter value @xmath343 . the parameter @xmath155 is taken to be @xmath405 . ] the dependence of the resonant widths ( [ e : widthn ] ) of the impurity states corresponding to the second excited subband @xmath47 on the position of the impurity centre is completely different from that related to the first excited subband @xmath46 . equation ( [ e : widthn ] ) shows that contributions to the resonant widths @xmath323 are caused by the coupling with the subbands @xmath406 ( [ e : couple ] ) and @xmath407 ( [ e : couple1 ] ) . note that the estimated contribution to the resonant width @xmath323 caused by the neglected coupling between the @xmath393 subbands is of the order of @xmath408 in the vicinity of the mid - point @xmath409 the subband @xmath46 contributes mostly @xmath410 , while for @xmath411 both subbands contribute @xmath412 . the position of the impurity @xmath413 , at which the effects of the subbands @xmath45 and @xmath46 on the resonant width @xmath323 are in balance is determined by the root @xmath414 of the equation @xmath415 to give the result @xmath416 . the coupling between the subbands @xmath417 provides the nonzero widths @xmath323 and shifts @xmath324 for any positions @xmath363 of the impurity . the width @xmath402 as a function of the impurity shift @xmath363 is given in fig.5 demonstrating the monotonic drop within the same regions as those corresponding to fig.4 . as mentioned above the presented method is valid under the conditions @xmath189 for the excited impurity states @xmath186 and @xmath418 ( [ e : ground ] ) for the ground state @xmath190 . under these conditions the radius of the impurity state considerably exceeds the width of the gnr so that the ribbon is narrow compared to the impurity size . it follows from eq . ( [ e : discr ] ) that a `` big logarithm '' can only hardly achieved @xcite i.e. the logarithmic approximation @xmath419 ensures the real smallness of @xmath389 . however the previous calculations related to the ground state of the quasi-1d diamagnetic exciton @xcite and present estimates show that a reasonably small parameter @xmath155 leads to values @xmath420 , which provide a quite accurate and adequate description of the ground impurity state in gnr . taking into account possible experiments we estimate the expected electron binding energy for the impurity centre placed at the middle point of the gnr of width 1 nm on the sapphire substrate as @xmath421 and on the @xmath335 substrate as @xmath422 . this is less than the data attributed to the @xmath423 substrate @xmath424 because of the relatively small screening of the impurity potential . also an estimate of the lifetimes @xmath54 yields for the impurity positioned at the mid - point of the gnr @xmath425 for the @xmath426 and sapphire substrate , respectively . for the @xmath427 substrate the screening of the impurity attraction is less , the lifetime is reduced and therefore less favourable for a corresponding experimental observation . a shift of the impurity centre @xmath428 generates lifetimes @xmath429 of the same order as @xmath430 at @xmath42 . the electrons captured onto such short - lived trap states will most likely contribute to the dc transport . however , the high - frequency response of such electrons may reveal the signatures of localization . clearly , the above considered quasi - rydberg series ( [ e : rydberg ] ) do not cover the total set of discrete states . the oscillations of the wave functions ( [ e : iter ] ) and ( [ e : outer1 ] ) caused by the logarithmic term are an indicator of additional energy levels positioned below the series ( [ e : rydberg ] ) . since the possible strong shift of these levels away from the threshold @xmath48 is against the spirit of the employed adiabatic perturbation theory @xmath431 describing the shallow energy levels we are limited to qualitative estimates based on the quasi - classical relativistic approach @xcite and @xcite . the dimensionless binding energy @xmath432 of the quasi - classical ground state @xmath433 found from ( [ e : class ] ) for @xmath434 and from ( [ e : phi ] ) with @xmath343 for @xmath435 versus the parameter @xmath155 . ] in an effort to elucidate the origin of this additional series let us consider the so called `` logarithmic '' energy levels governed by the logarithmic potential ( [ e : limit ] ) taken for @xmath436 . these levels can be calculated from the bohr - sommerfeld quantization rule @xmath437 where @xmath438 and @xmath439 are the classical relativistic momentum and turning point , respectively , with @xmath440 ; \nonumber\\ \mathcal{p}(y_0)&=&0.\end{aligned}\ ] ] equation ( [ e : bohrsomm ] ) admits an exact solution which provides for the energies @xmath441,~s_n=\frac{\varepsilon_n d}{2pq}=\frac{|n-\tilde{\sigma}|\pi}{2q},\ ] ] where @xmath442 is the modified bessel function @xcite . the binding energy @xmath443 with @xmath183 calculated from ( [ e : class ] ) reads @xmath444 both for @xmath445 and for @xmath446 . it follows that the weakness of the logarithmic singularity and smallness of the strength of the impurity potential @xmath431 seems not to provide the bonding of the quasi - classical relativistic electron @xmath447 , while a sufficiently strong attraction @xmath448 could produce a localized impurity state @xmath449 . the dependence of the binding energy of the quasi - classical ground state @xmath433 found from ( [ e : class ] ) on the parameter @xmath155 is depicted in fig.6 . the ground `` logarithmic '' level arises at the critical value @xmath450 and shifts towards lower energies to provide for the binding energy @xmath451 . the above can be considered as no more than only a qualitative evidence of existence of such additional states in gnr that have transformed from the collapsed states in the graphene monolayer governed by the 2d impurity potential @xmath452 @xcite . though the `` logarithmic '' and quasi - rydberg levels in principle correspond to the same region of the parameter @xmath453 any numerical comparison between the results for the quasi - rydberg series based on the dirac equation and those for the `` logarithmic '' levels derived from the quasi - classical method applied moreover to the ground state seems to be incorrect . the total set of the impurity states in gnr requires a further study of the equations ( [ e : set ] ) with the potential ( [ e : pot ] ) , having the logarithmic singularity in the vicinity of the impurity centre . we have developed an analytical adiabatic approach to the problem of bound and meta - stable ( fano resonances ) quasi - coulomb impurity states in a narrow gaped armchair graphene nanoribbon ( gnr ) . the width of the gnr is taken to be much less than the radius of the impurity . this adiabatic criterion implies a variable width of the gnr and simultaneously the smallness of the coulomb interaction relative to the size - quantized energy induced by the gnr . the energy spectrum of the impurity electron is a sequence of the series of the quasi - rydberg discrete and resonant states adjacent to the ground and excited size - quantized subbands , respectively . the binding energies and the resonant widths and shifts attributed to the inter - subband coupling are calculated in an explicit form in the single- and multi - subband approximation , respectively . the binding energies and the resonant widths both increase with decreasing the gnr width . as the impurity centre displaces from the mid - point of the gnr the binding energies decrease , while the resonant widths of the quasi - rydberg series associated with the first / second excited sub - bands increase / decrease , respectively . our analytical results are in complete agreement with those found by other theoretical approaches and in particular numerical studies . estimates of the expected values show that the bound and meta - stable impurity states in gnr can be observed experimentally . the authors are grateful to c. morfonios for technical assistance . financial support by the deutsche forschungsgemeinschaft is gratefully acknowledge .
|
an analytical study of discrete and resonant impurity quasi - coulomb states in a narrow gaped armchair graphene nanoribbon ( gnr ) is performed .
we employ the adiabatic approximation assuming that the motions parallel ( `` slow '' ) and perpendicular ( `` fast '' ) to the boundaries of the ribbon are separated adiabatically .
the energy spectrum comprises a sequence of series of quasi - rydberg levels relevant to the `` slow '' motion adjacent from the low energies to the size - quantized levels associated with the `` fast '' motion .
only the series attributed to the ground size - quantized sub - band is really discrete , while others corresponding to the excited sub - bands consist of quasi - discrete ( fano resonant ) levels of non - zero energetic widths , caused by the coupling with the states of the continuous spectrum branching from the low lying sub - bands . in the two- and three - subband approximation the spectrum of the complex energies of the impurity electron is derived in an explicit form .
narrowing the gnr leads to an increase of the binding energy and the resonant width both induced by the finite width of the ribbon . displacing the impurity centre from the mid - point of the gnr causes the binding energy to decrease while the resonant width of the first excited rydberg series increases . as for
the second excited series their widths become narrower with the shift of the impurity .
a successful comparison of our analytical results with those obtained by other theoretical and experimental methods is presented .
estimates of the binding energies and the resonant widths taken for the parameters of typical gnrs show that not only the strictly discrete but also the some resonant states are quite stable and could be studied experimentally in doped gnrs .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the concept to realize and investigate quantum processes with single ions in paul traps was subject to an impressive development over the last years . as for quantum information with single to a few ions , there has hardly been an experimental challenge that could withstand this development , consider for instance high fidelity quantum gates @xcite , multi - qubit entanglement @xcite , and error correction @xcite . there has been a recent development in ion trap technology that was primarily initiated by the necessity and possibility to scale up the established techniques to process larger numbers of ions . this idea how to scale up an ion trap computer is to create multiple trap potentials , each keeping only a small , processable number of ions @xcite . time dependent potentials can then be used to shuttle the ions between different sections . to generate such spatially separated potentials , the respective trap electrodes are segmented along the transport direction and supplied individually with time dependent voltages . there have also been proposals to incorporate the shuttling process into the quantum gate time by letting the ion cross laser beams or magnetic field arrangements @xcite . another application of the shuttling ability is the exact positioning of the ion in the mode volume of an optical cavity to perform cavity quantum electrodynamic experiments . recently , multi - trap configurations with locally controlled , individual trap frequencies were proposed to generate cluster states in linear ion traps to perform one - way quantum computations @xcite . this paper presents the implementation of a method using a single trapped ion to investigate local electric fields . the method profits from the outstanding accuracy of spectroscopic frequency measurements and relies on the ability to shuttle the ion in order to reach different probing sites . first , we describe the experimental system and the way the electric trapping and shuttling potentials are generated . after that , the measurement scheme using the ion as a local field probe is presented . then , the quantitative experimental results are discussed and compared to numerical simulations of the trapping fields . we confine a single , laser cooled @xmath0-ion in a linear paul trap . the radial confinement is generated by a harmonic radio - frequency pseudo potential resulting from a @xmath1 drive with an amplitude of about @xmath2 . this results in radial trap frequencies around @xmath3 . for the axial confinement , 32 pairs of electrodes are available . each of these 64 segments can be individually biased to an voltage @xmath4 , @xmath5 . in the trap region the experiments were performed in , the segments are @xmath6 wide and separated by @xmath7 gaps ( details can be found in @xcite ) . the voltage data preprocessed by a computer and transmitted to a home - built electronics device . this device houses an array of 16 serial input digital - to - analog converters ( dac , @xmath8 maximum clock frequency ) , each of which supplying four individual voltage lines , so that a data bus conveying 16 bit in parallel suffices to obtain the desired 64 outputs . the amplitude resolution of the converters is 16 bit with a measured noise level well below the last significant bit . the serial input design of the converters keeps the amount of data being transmitted in parallel at a minimum . this makes the design extendable to a large number of channels . in this work , we restrict ourselves to the investigation of harmonic potentials . this is rectified by the fact that a resting , laser cooled ion only experiences the very minimum of the potential . such potentials are generated by applying control voltages to the trap segments . thereby , each specific voltage configuration results in an axial trap with a given trap frequency ; the position of the potential minimum this is where the ion resides can be changed to shuttle the ion between different locations on the trap axis . doing so , two requirements have to be met by the time dependent voltages : ( i ) the amplitudes must range within the limits given by the dac electronics , @xmath9 in our case . ( ii ) each segment s voltage is to change as continuous as possible to avoid unfeasibly high frequencies . all these requirements are fulfilled by the field simulation and voltage calculation techniques presented in sections [ sec : app_fieldcalc ] and [ sec : app_regul ] , respectively . for the experiments presented here , it is advantageous to perform slow , adiabatic potential changes , and thus avoiding excess oscillations of the ion . for that reason , the update rate of digital to voltage converters was chosen with @xmath10 , and a parallel port connection between computer and dac device was used . in an alternative operation modus of the trap supply device , the data transmission and updated rate is significantly sped up by using a field programmable gate array . then we reach @xmath11 even when updating all 64 channels simultaneously . for the application presented here , it is sufficient to use a doppler cooled probe ion . the probe s spatial extent can then be estimated from the ion temperature and the consequential extent of the motional wave function to be @xmath12 . this value could be reduced to @xmath13 by sub - doppler cooling and a tighter confinement of the ion . to test both the numerical methods and the developed electronic devices for accuracy , we employ a single ion as a probe for the electrostatic potential . we exploit the fact that it is possible to exactly measure the trap frequency of the confining axial potential by spectroscopic means . consider , for example , a voltage configuration @xmath14 to be tested for ; this configuration is meant to result in a trap at position @xmath15 with frequency @xmath16 . then we can use time dependent potentials to shuttle the ion from a loading position to @xmath15 @xcite , whereby the final voltage configuration must be @xmath14 . the spectroscopically obtained trap frequency at @xmath15 can then be compared to the theoretically expected one , @xmath17 . the trap frequency is determined by measuring the difference frequency of the carrier and the first motional red sideband ( rsb ) excitation of the ion in a resolved sideband regime . for that , we excite the quadrupole transition @xmath18 at @xmath19 . as the carrier resonance frequency does not depend on the trap frequency , it is sufficient to measure the rsb frequency at different locations @xmath15 . the measurement scheme is illustrated in figure [ fig : spectroscopy_scheme ] . it consists of the following steps : \(i ) an initial voltage configuration is chosen to trap and cool the ion at the starting position @xmath20 . all lasers necessary for cooling , repumping , state preparation and detection are aligned to interact with the ion here . additionally , @xmath20 is the position where fluorescence emitted by the ion can be detected by a photomultiplier tube and a camera . \(ii ) the ion is shuttled to the probing position @xmath15 . this is accomplished by applying voltage configurations resulting in a series of harmonic potentials whose minimum positions lead the ion from @xmath20 to @xmath15 . for simplicity , these potentials were chosen such that their harmonic frequencies were almost constant and close to @xmath16 . \(iii ) applying a spectroscopy pulse at the probing position @xmath15 . now , the voltages are exactly @xmath14 . resting at @xmath15 , the ion is exposed to a spectroscopy pulse of fixed duration ( @xmath21 ) . the pulse is detuned from the ( carrier ) resonance by @xmath22 . this excites the ion with a probability @xmath23 into the upper state @xmath24 . \(iv ) shuttle the ion back to @xmath20 , inverting step ( ii ) . \(v ) having arrived back at the starting position , the state of the ion is read out by illuminating it on the cooling transition . whenever we detect a fluorescence level above a certain threshold , the ion is found in the ground state @xmath25 , while a low fluorescence level indicates that the ion has been excited to the state @xmath24 . the excitation probability for a specific detuning , @xmath26 , at the remote position @xmath15 is obtained by averaging over many repetitions of steps ( i ) to ( v ) . by varying @xmath22 , a spectrum of the quadrupole excitation at the remote position is obtained without moving any lasers or imaging optics but the spectroscopy laser used in step ( iii ) . this is a significant advantage , because that way even such trap positions can be investigated that can not be directly observed by the imaging devices or reached by all lasers . the frequency difference between the red sideband and the carrier transition yields the trap frequency . figure [ fig : spectra ] shows two rsb resonance peaks obtained at different positions within the trap volume . it is noticeable that each iteration cycle ( i - v ) including two ion transports results due to the binary nature of the projective readout in exactly one bit information about the spectrum . therefore , thousands of transports , each relying on the calculated potentials , are performed for the determination of one frequency @xmath16 . the transport , however , can be performed so fast ( @xmath27 ) that its contribution to the overall experiment duration is secondary ; this duration is still dominated by cooling and detection times ( @xmath28 ) . to implement the measurement scheme , we first calculated voltage sets @xmath29 , where @xmath30 covers the whole extent of the trap in steps of @xmath31 ( for arbitrary positions , the voltages can be interpolated ) . each set results in a certain , wanted frequency @xmath32 . that means that for each arbitrary position @xmath30 in the trap , there can be found a set of voltages resulting in a potential with its minimum at @xmath30 and with trap frequency @xmath32 . then , in order to shuttle the ion , we simply subsequently apply the voltage configurations for @xmath33 . the calculated voltages are tested with high axial resolution , i.e. in small steps of @xmath15 , in two far distant regions of the trap . doing this , both small local deviations should be detectable and the stability over the whole trap structure can be tested for . to see a variation in @xmath16 when increasing @xmath15 , it is of advantage that small variations of the trap frequency around its means value occur . this is a reliable way to ensure that the ion in fact probes the remote position . figure [ fig : remote_results ] shows the expected , simulated trap frequencies and the measured ones . the data are in excellent agreement with the predicted frequencies . on both ends of the investigated trap structure , the predicted course of @xmath32 is confirmed within the spectroscopic accuracy of @xmath34 . the mean deviation of all measured data points is only @xmath35 . note that the solid line shown in figure [ fig : remote_results ] is based solely on geometric data from a technical drawing of the trap ; there is no free parameter that was used to match the simulations with the measurement . the high accordance between the simulated and the real trap frequencies , as can be observed in the presented experiments , indicates the reliability of at least four independent contributions : first , the numerical field simulation is accurate . this simulation is independent from a specific set of voltages , but relies on a model of the three dimensional trap design . second , the physical realization of this geometry is very good , i.e. the manufacturing and assembling process of the micro trap is so precise that it is not limiting the field accuracy . this aspect is coming more an more into the experimental focus , since the miniaturization of the trap layouts still proceeds . for example , a possible axial misalignment @xmath36 of the two dc electrode wings with respect to each other would mainly result in an axial shift of the oscillations depicted in fig . [ fig : remote_results]a by approximately @xmath37 . additionally , the amplitude of these oscillations decreases with increasing @xmath36 . indeed , the residuals ( fig . [ fig : remote_results]b ) show a small oscillatory behavior . this oscillation signal is shifted by at most @xmath38 with respect to the trap frequency oscillation . such an effect could stem from a small misalignment of the two trap layers ( see ref . @xcite ) or from a small offset of the theoretical and the real , axial zero point of position . we attribute the observed shift to an offset error , since in this case , the amplitude of the oscillation is not decreased , as it is the case in the measured data from microscope pictures of the trap . ] . this shift is systematic and could be corrected for . third , the calculation of the voltages leading to the wanted potentials is highly reliable . this is also qualitatively confirmed by the successful shuttling process itself . from our measurement we find that the generation of the voltages and their supply to the trap electrodes is indeed highly accurate . we also conclude that trapping fields from the applied voltages are not significantly perturbed by background fields from stray charges . as the generation of trapping fields in the multi - segment trap was found to be accurate over the whole considered volume , it is possible to tailor potentials for special purposes . for quantum computing tasks , a harmonic potential of constant frequency is mostly required . simulations show that it is possible to obtain fixed frequency potentials with a relative frequency deviation on the order of @xmath39 . the fast transport of quantum information in harmonic wells also requires a high degree of control over the trap voltages @xcite . the application of optimal control methods can help optimizing tailored time dependent potentials . alternatively , one can feed - back information , gained from the measurement presented here , into the generation process to refine the results iteratively @xcite . however , the scope of possible applications of the described techniques is much wider than quantum computing tasks : ions in time dependent potentials were proposed to be used as a testbed for quantum thermodynamic processes @xcite or for quantum simulation @xcite . the presented method is just a first proof of principle of using single atoms as ultra - sensitive probes . the measurement principle is not restricted to the investigation of the electric trapping fields . only slight modifications allow for the detection of any electric or magnetic field with a nano - meter scaled quantum probe , as an alternative to cold atom sensors @xcite . then , a single atom field probe might be used as well for probing magnetic micro - structures with a relative accuracy better than 10@xmath40 . also , a study of decoherence and heating effects and their dependence on the ion - electrode separation or ion location is in reach . financial support from the dfg within the sfb / trr-21 , the german - israel science foundation and by the european commission within microtrap , emali and scala is acknowledged . to create a specific electric potential @xmath41 on the trap axis , it is necessary to find the right set of voltages @xmath42 being applied to the trap electrodes labeled by @xmath43 . the potential generated by such a set of voltages is the linear superposition of the @xmath44 individual electrodes , whereby the contribution of each electrode @xmath45 is weighted by the applied voltage @xmath4 . after subdividing the axial position into a grid of @xmath46 points @xmath47 , we can write the overall potential at any @xmath48 by @xmath49 here we introduced the electrode potential matrix @xmath50 . it describes the influence of the @xmath45-th electrode to the overall potential at @xmath51 . each row @xmath45 of @xmath52 can be seen as a position - dependent function , describing the potential generated by electrode @xmath45 ( in units of @xmath4 ) , when all other electrodes are set to zero voltage . this quantity @xmath52 is independent from the specific voltage and is solely given by the trap geometry , i.e. the shape and size of the electrode and its distance from @xmath48 , for instance . therewith , the potential generation can be logically divided into two parts : the matrix @xmath52 can be calculated independently from voltage constraints and independent from the desired potential . second , for each desired potential @xmath53 , there has to be found a set of voltages @xmath54 by inverting the matrix equation above . tackling the first problem , one recognizes that modern segmented trap geometries can be realized in such a geometric complexity that conventional simulation techniques like the finite element method ( fem ) fail . instead , we obtained the potentials by solving the boundary element problem of the segmented trap design . details can be found in @xcite . in the following , we address the problem how to obtain a set of voltages @xmath42 that generates a given potential @xmath55 when applied to the respective electrodes . the problem is formally solved by matrix inversion as @xmath56 . several circumstances make this straight forward ansatz unfeasible : first , there is often no exact solution to the problem , because @xmath53 is not an exactly realizable potential ( note that in general , @xmath57 is possible ) . in this case , an approximate solution has to be found . additionally , as a specific electrode does not effectively contribute to the potential at a far distant point , this electrode s voltage is ill - determined . these cases have to be treated adequately by the algorithm . we solved the inversion problem with a singular - value decomposition of the matrix @xmath52 to identify its critical , singular values . the real @xmath58 matrix @xmath52 is decomposed into the product @xmath59 of the unitary matrices @xmath60 ( @xmath61 ) and @xmath62 ( @xmath63 ) and the diagonal @xmath58 matrix @xmath64 with non - negative entries @xmath65 . this decomposition is part of many standard numerical libraries and can be performed for any input matrix @xmath52 . the inverse can then be written as @xmath66 this step is numerically trivial , because the unitary matrices are simply transposed and the entries of @xmath67 are given by @xmath68 . at this point , the advantage of the decomposition becomes obvious , since small values of @xmath69 indicate an ( almost ) singular , critical value . one way to overcome these singular values would be to cut off their diverging inverse values . instead , the tikhonov regularization @xcite method implies a more steady behavior as it makes the displacement @xmath70 . the latter expression behaves like the original @xmath68 for large values @xmath71 , has its maximum at @xmath72 and tends to zero for small , critical values @xmath73 . from this can be seen that the choice of the optimization parameter @xmath74 is a compromise between exactness and boundedness of the results . for @xmath75 , the exact solution ( if existent ) is obtained , whereas large values of @xmath74 guarantee small inverse values and thus bounded voltage results . therefore , we label the regularized quantities with index @xmath74 . the approximate solution @xmath76 is @xmath77 with @xmath78 being the regularized matrix with entries @xmath79 . before the problem of finding an optimal @xmath74 is addressed , another constraint regarding time dependent voltages , i.e. series of voltage configurations , has to be accounted for . while moving the ion by one step , i.e. from a position @xmath15 to @xmath80 , the control voltage should vary as less as possible . this is achieved by extending eqn . ( [ eq : phi1 ] ) , @xmath81 where @xmath82 represents any previous voltage set , providing trapping at @xmath15 . the second term in eqn . ( [ eq : phi2 ] ) contains a diagonal matrix @xmath83 with entries @xmath84 . @xmath85 tends to zero for @xmath86 , so that uncritical voltages are affected only little by the second term . for all critical voltages indicated by a value @xmath87 , however , the first term in eqn . ( [ eq : phi2 ] ) vanishes due to the regularization replacement and what remains is the contribution from @xmath88 , since then , @xmath89 . here , the choice of @xmath74 determines , how strong the algorithm tries to generate similar voltages in a ( time ) series of voltage sets . the algorithm described above minimizes @xmath90 with respect to the euclidian norm . that is , the potential @xmath55 is reproduced as good as possible under the constraint that solutions similar to the previous one are preferred . what remains is to find the proper value of @xmath74 . hereby , one has to make a tradeoff between the boundedness of the voltages and their continuity . under practical circumstances requiring @xmath91 for some maximal voltage @xmath92 , @xmath74 can be iteratively increased to fulfill this constraint on the one hand , and to obtain as continuous voltage sets as possible , on the other hand . j. benhelm et al . , nature physics * 4 * , 463 ( 2008 ) . h. hffner et al . , nature * 438 * , 643 ( 2005 ) . d. leibfried et al . , nature * 438 * , 639 ( 2005 ) . j. chiaverini et al . , nature * 432 * , 602 ( 2004 ) . d. leibfried et al , phys . a * 76 * , 032324 ( 2007 ) . d. kielpinski , c. monroe and d. j. wineland , nature * 417 * , 709 ( 2002 ) . s. schulz et al . , fortschr . phys . * 54 * , no . 8 - 10 , 648 ( 2006 ) . l. greengard and v. rokhlin , in : vortex methods , 121 , ( springer , berlin 1988 ) . k. nabors et al . , sci . * 16 * , 713 ( 1994 ) . j. eble et al . , accepted for publication in j. opt . b ( 2010 ) , arxiv:0912.2527 . k. singer et al . , arxiv:0912.0196 . s. schulz et al . , new j. phys . * 10 * , 045007 ( 2008 ) . g. huber et al . , new j. phys . * 10 * , 013004 ( 2008 ) . g. huber et al . , phys . * 101 * , 070403 ( 2008 ) . r schtzhold et al . lett . * 99 * , 201301 ( 2007 ) . m. johanning et al . , j. phys . b : at . mol . opt . phys . * 42 * , 154009 ( 2009 ) . m. murphy et al . , phys . rev . a * 79 * , 020301(r ) ( 2009 ) . h. wunderlich et . al . , phys . a * 79 * , 052324 ( 2009 ) . c. pozrikidis , a practical guide to boundary element methods with the software library bemlib , chapman and hall / crc , boca raton , fl , usa , ( 2002 ) . a. n. tikhonov and v. a. arsenin , solution of ill- posed problems , winston and sons , washington , usa ( 1977 ) .
|
we introduce a measurement scheme that utilizes a single ion as a local field probe . the ion is confined in a segmented paul trap and shuttled around to reach different probing sites . by the use of a single atom probe , it becomes possible characterizing fields with spatial resolution of a few nm within an extensive region of millimeters .
we demonstrate the scheme by accurately investigating the electric fields providing the confinement for the ion . for this
we present all theoretical and practical methods necessary to generate these potentials .
we find sub - percent agreement between measured and calculated electric field values .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the classical and quantum propagation of strings in curved spacetimes has attracted a great deal of interest in recent years . the main complication , as compared to the case of flat minkowski space , is related to the non - linearity of the equations of motion . it makes it possible to obtain the complete analytic solution only in a very few special cases like conical spacetime @xcite and plane wave / shock wave backgrounds @xcite . there are however also very general results concerning integrability and solvability for maximally symmetric spacetimes @xcite and gauged wzw models @xcite . these are the exceptional cases , generally the string equations of motion in curved spacetimes are not integrable and even if they are , it is usually an extremely difficult task to actually separate the equations , integrate them and finally write down the complete solution in closed form . fortunately there are different ways to `` attack '' a system of coupled non - linear partial differential equations . besides numerical methods , which will not be considered here , there are essentially two ways to proceed : an approximative method was first developed by de vega and snchez @xcite ( see also @xcite ) . the idea was to expand the target space coordinates around a special solution , usually taken to be the string center of mass , and then to ( try to ) solve the string equations of motion and constraints order by order . this can be done , at least up to first or second order in the expansion , for most of the known gravitational and cosmological spacetimes . for a review of the method and its applications , see for instance @xcite . another approximative method , which is however not un - related to the first one , was considered in refs.@xcite . it consists of a large scale - factor expansion in spatially flat frw - universes and an important result of the analysis was the discovery of `` extremely unstable strings '' of negative pressure in inflationary universes . as already mentioned there are a few examples where the equations of motion and constraints can actually be solved completely . if this is not possible one can try to find special , but exact , solutions by making an ansatz . if the ansatz is properly chosen , i.e. such that it exploits the symmetries of the spacetime , it may reduce the original system of coupled non - linear partial differential equations to something much simpler , for instance decoupled non - linear ordinary differential equations , and it may then be possible to find the complete solution of this reduced system . this method has been used for stationary open strings @xcite and for circular strings in a variety of curved spacetimes @xcite . when using this method , one somehow has to argue that the special solutions under consideration are representative in the sense that their physical properties are more general than the solutions themselves . this is done by comparing with the results obtained from the approximative methods , so it is very important to have , and to use , both kinds of methods . in this talk i will present material bases on both exact and approximative methods and i will try to take a physical point of view and address the question : what are the strings actually doing in the curved spacetimes ? the talk is organized as follows . in section 2 we consider circular strings in stationary axially symmetric backgrounds . the main result is the effective potential determining the circular string radius . the main part of the talk concerns circular strings in de sitter space , as indicated by the title . the material is based on refs.@xcite and is presented in section 3 . we first consider the energy and pressure of the different types of strings . we then turn to the mathematical solutions and their physical interpretation , following closely ref.@xcite . finally we consider briefly the propagation of small perturbations around the circular strings . in section 4 we consider circular strings in the background of the @xmath0 dimensional black hole anti de sitter spacetime , recently found by baados et . @xcite and in section 5 we compare with circular strings in ordinary black hole and anti de sitter spacetimes . in section 6 we first refine the approximative string perturbation approach of de vega and snchez @xcite by eliminating , from the beginning , the perturbations in the direction of the geodesic of the string center of mass . we then discuss some of the results obtained in ref.@xcite , using this method , and we compare with the results obtained for circular strings . the main conclusions of the talk are compactly summarized in tables i , ii , iii . in this section we consider circular strings embedded in stationary axially symmetric backgrounds . the analysis is carried out in @xmath0 dimensions , but the results will hold for the equatorial plane of higher dimensional backgrounds as well . to be more specific we consider the following line element : @xmath2 that will be general enough for our purposes here . the circular string ansatz , consistent with the symmetries of the background , is taken to be : @xmath3 where the three functions @xmath4 and @xmath5 are to be determined by the equations of motion and constraints . the equations of motion lead to : @xmath6 @xmath7 @xmath8 while the constraints become : @xmath9 @xmath10 this system of second order ordinary differential equations and constraints is most easily described as a hamiltonian system : @xmath11 supplemented by the constraints : @xmath12 the function @xmath5 introduced in eq . ( 2.2 ) does not represent a physical degree of freedom . it describes the `` longitudinal '' rotation of the circular string and is therefore a pure gauge artifact . this interpretation is consistent with eq . ( 2.6 ) saying that there is no angular momemtum @xmath13 . the hamilton equations of the two cyclic coordinates @xmath14 and @xmath15 are : @xmath16 as well as : @xmath17 where @xmath18 is an integration constant and we used eq . ( 2.6 ) . the two functions @xmath19 and @xmath5 are then determined by : @xmath20 @xmath21 that can be integrated provided @xmath22 is known . ( 2.6 ) and eqs . ( 2.9)-(2.10 ) , the hamilton equation of @xmath23 becomes after one integration : @xmath24 so that @xmath22 can be obtained by inversion of : @xmath25}}.\ ] ] for the cases that we will consider in the following , eq . ( 2.12 ) will be solved in terms of either elementary or elliptic functions . by definition of the potential @xmath26 eq . ( 2.11 ) , the dynamics takes place at the @xmath27axis in a @xmath28diagram . the first thing to do for a stationary axially symmetric background under consideration , is therefore to find the zeros of the potential . then one can try to solve the equations of motion and finally extract the physics of the problem we close this section by the following interesting observation : insertion of the ansatz , eq . ( 2.2 ) , using the results of eqs . ( 2.9)-(2.11 ) in the line element , eq . ( 2.1 ) , leads to : @xmath29 we can then identify the invariant string size as : @xmath30 @xmath0 dimensional de sitter space is a 3 dimensional hyperboloid embedded in 4 dimensional minkowski space : @xmath31,\;\;\;\;\;\;\;\ ; \eta_{\mu\nu}q^\mu q^\nu=1.\ ] ] the hyperboloid coordinates cover the whole manifold contrary to the comoving coordinates : @xmath32,\ ] ] and the static coordinates : @xmath33 for our purposes it is however most convenient to start out with the static coordinates since then we can use directly the results of section 2 for the circular strings . afterwards , we then have to transform the solutions back to the hyperboloid using the appropriate coordinate transformations inside and outside the horizon . in static coordinates we can now immediately write down the potential , eq . ( 2.11 ) , determining the dynamics of circular strings in de sitter space : @xmath34 see fig.1 . for @xmath35 the top of the potential is above the @xmath27axis and there will be oscillating solutions to the left of the barrier and bouncing solutions to the right . the bouncing solutions will re - expand towards infinity . for @xmath36 on the other hand , the top of the potential is below the @xmath27axis and therefore the potential does not act as a barrier . strings can expand from @xmath37 towards infinity and very large strings can collapse . finally if @xmath38 there is a ( unstable ) stationary string at the top of the potential as well as expanding and collapsing strings . it is instructive to compare with the circular string dynamics in minkowski space . in that case the potential is given by @xmath39 and only oscillating strings exist . physically the difference is not so difficult to understand . in flat minkowski space the dynamics of a circular string is determined by the string tension only , that will always try to contract the string . this leads to strings oscillating between @xmath37 and some maximal size depending on the energy . in de sitter space there is an opposite effect namely the expansion of the universe , that will try to expand the string . this gives rise to a region where the tension is strongest ( to the left of the barrier ) , a region where the expansion of the universe is strongest ( to the right of the barrier ) and there may even be an exact balance of the two opposite forces , implying the existence of a stationary solution . before coming to the mathematical solution of the equations of motion , we consider the energy and pressure of the strings . the spacetime string energy - momentum tensor is : @xmath40 after integration over a spatial volume that completely encloses the string , the energy - momentum tensor takes the form of a fluid : @xmath41 where , in the comoving coordinates introduced in eq . ( 3.2 ) : @xmath42 @xmath43 represent the energy and pressure , respectively . from these expressions we can actually get a lot of information about the energy and pressure , without using the explicit time evolution of the strings . from the coordinate transformations between comoving and static coordinates we get the explicit expressions @xcite : @xmath44 @xmath45 both the energy and the pressure now depend on the string radius @xmath23 and the velocity @xmath46 the latter can however be eliminated using eq . ( 2.11 ) and eq . ( 3.4 ) . let us first consider a string expanding from @xmath37 towards infinity . this corresponds to a string with @xmath47 and @xmath48 , see fig.1 . for @xmath37 we find @xmath49 and @xmath50 thus the equation of state is @xmath51 this is like ultra - relativistic matter . as the string expands , the energy soon starts to increase while the pressure starts to decrease and becomes negative , see fig.2 . for @xmath52 we find @xmath53 and @xmath54 thus not surprisingly , we have recovered the equation of state of extremely unstable strings @xmath55 @xcite . now consider an oscillating string , i.e. a string with @xmath56 in the region to the left of the potential barrier , fig.1 . the equation of state near @xmath37 is the same as for the expanding string , but now the string has a maximal radius : @xmath57 for @xmath58 we find : @xmath59 corresponding to a perfect fluid type equation of state : @xmath60 notice that @xmath61,$ ] with @xmath62 decribing a string at the bottom of the potential while @xmath63 decribes a string oscillating between @xmath37 and the top of the potential barrier ( in this case the string actually only makes one oscillation @xcite ) . for @xmath64 the equation of state , eq.(3.13 ) , near the maximal radius reduces to @xmath55 in the other limit @xmath65 we find , however , @xmath66 corresponding to cold matter . for the oscillating strings we can also calculate the average values of energy and pressure by integrating over a full period , see fig.3 . since a @xmath67-integral can be converted into a @xmath23-integral , using eq . ( 2.11 ) and eq . ( 3.4 ) , the average values can be obtained without using the exact @xmath67-dependence of the string radius . the average energy becomes : @xmath68 where @xmath69 is the period and @xmath70 is the complete elliptic integral of the third kind ( using the notation of gradshteyn @xcite ) . the average pressure is zero , thus in average the oscillating strings describe cold matter . in minkowski space , for comparison , it can easily be shown @xcite that the energy is constant while the pressure depends on the string radius . the average pressure is however zero , just as in de sitter space . ( this and the following subsection follow closely ref.@xcite . ) + we now come to the explicit mathematical solutions . in the general case equations ( 2.11 ) and ( 3.4 ) are solved by : @xmath71 where @xmath72 is the weierstrass elliptic @xmath72-function @xcite : @xmath73 with invariants : @xmath74 and discriminant : @xmath75 it is convenient to consider separately the 3 cases @xmath76 , @xmath77 and @xmath78 , see fig.1 . @xmath79 : in this case the weierstrass function reduces to a hyperbolic function and eq . ( 3.15 ) becomes : @xmath80 two real independent solutions are obtained by the choices @xmath81 and @xmath82 , respectively : @xmath83 @xmath84 notice that : @xmath85 these are the 2 solutions originally found by de vega , snchez and mikhailov @xcite , corresponding to @xmath86 and @xmath87 in their notation , respectively . the interpretation of these solutions as a function of the world - sheet time @xmath67 is clear from fig.1 : the solution , eq . ( 3.20 ) , expands from @xmath88 towards infinity and then contracts until it reaches its original size . the solution , eq . ( 3.21 ) , contracts from @xmath89 until it collapses . it then expands again until it reaches its original size . the physical interpretation , that is somewhat more involved , was described in ref.@xcite and will be shortly reviewed in subsection 3.2 . there is actually also a stationary solution for @xmath90 , i.e. a string sitting on the top of the potential , see fig.1 . this solution with constant string size @xmath91 was discussed in ref.@xcite and will not be considered here . @xmath92 : here two real independent solutions are obtained by the choices @xmath81 and @xmath93 , respectively : @xmath94 @xmath95 where @xmath96 is the imaginary semi - period of the weierstrass function . it is explicitly given by @xcite : @xmath97 notice that : @xmath98 where @xmath99 is the real semi - period of the weierstrass function @xcite : @xmath100 and @xmath101 and @xmath102 are the complete elliptic integrals of first kind . the interpretation of these solutions as a function of @xmath67 is clear from fig.1 : the solution , eq . ( 3.23 ) , oscillates between infinity and its minimal size @xmath103 at the boundary of the potential , while the solution , eq . ( 3.24 ) , oscillates between @xmath104 and its maximal size @xmath105 . the physical interpretation will be considered in subsection 3.2 . @xmath106 : in this last case two real independent solutions are obtained by the choices @xmath81 and @xmath107 , respectively : @xmath94 @xmath108 where @xmath109 takes the explicit form : @xmath110 notice that : @xmath111 where : @xmath112 it should be stressed that in this case the primitive semi - periods are @xmath113 and @xmath114 , i.e. @xmath115 spans a fundamental period parallelogram in the complex plane . the interpretation of the solutions , eqs . ( 3.28)-(3.29 ) , as a function of @xmath67 follows from fig.1 : both of them oscillates between zero size ( collapse ) and infinite size ( instability ) . the physical interpretations will follow in subsection 3.2 . @xmath79 : we first consider the @xmath116-solution , eq . the hyperboloid time is obtained by integrating eq . ( 2.10 ) and transforming back to the hyperboloid coordinates : @xmath117 when we plot this function ( fig.4 . ) we see that the string solution actually describes 2 strings ( i and ii ) @xcite , since @xmath67 is a two - valued function of @xmath118 . for both strings the invariant string size is : @xmath119 but string i corresponds to @xmath120-\infty,0[\;$ ] and string ii to @xmath1200,\infty[\;$ ] . therefore , @xmath121 corresponds to @xmath122 for string i , but to @xmath123 for string ii . more generally , when @xmath121 the invariant size grows indefinitely for string i , while it approaches a constant value for string ii . we conclude that string i is an unstable string for @xmath121 , while string ii is a stable string . more details about the connection between hyperboloid time and world - sheet time for these solutions can be found in ref.@xcite . let us consider now the comoving time of the solution @xmath116 in a little more detail : @xmath124 when we plot this function ( fig.4 . ) we find that @xmath67 is a three - valued function of @xmath125 . what happens is that the time interval @xmath1200,\infty[\;$ ] for string ii splits into two parts . these features are easily understood when returning to the effective potential , fig.1 . : string i starts at @xmath88 for @xmath126 , it then expands through the horizon @xmath127 at : @xmath128 and continues towards infinity for @xmath129 . string ii starts at infinity for @xmath130 and contracts through the horizon at : @xmath131 this behaviour , approaching the horizon from the outside , corresponds to the going backwards in comoving time - part of fig.4 . string ii then continues contracting from @xmath127 at : @xmath132 until it reaches @xmath133 at @xmath134 . we now consider briefly the @xmath135-solution , eq . ( 3.21 ) . in this case the hyperboloid time is given by : @xmath136 which is a monotonically increasing function of @xmath67 . the @xmath135-solution therefore describes only one string . the proper size is given by : @xmath137 we can also express this solution in terms of the comoving time : @xmath138 but since everything now takes place well inside the horizon , this will not really give us more insight . the string starts with @xmath89 for @xmath139 it then contracts until it collapses for @xmath140 and expands again and eventually reaches @xmath89 for @xmath141 @xmath92 : using the notation introduced in eqs . ( 3.11)-(3.12 ) the solutions , eqs . ( 3.23)-(3.24 ) , can be written as : @xmath142},\ ] ] @xmath143,\ ] ] where @xmath144 consider first the @xmath116-solution , eq . it is clear from eq . ( 3.26 ) and the periodicity in general that we have infinitely many branches @xmath145,\ ; [ 2\omega,4\omega],\; ... $ ] . we will see in a moment that each of these branches actually corresponds to one string , that is , the @xmath116-solution describes infinitely many strings . for that purpose we will need the hyperboloid time and the comoving time as a function of @xmath67 . both of them are expressed in terms of the static coordinate time @xmath14 , that is obtained by integrating eq . ( 2.10 ) : @xmath146 where @xmath147 and @xmath148 are the weierstrass @xmath147 and @xmath148-functions @xcite and @xmath149 is a real constant obeying @xmath150=\mu$ ] , i.e. @xmath149 is expressed as an incomplete elliptic integral of the first kind . the expression , eq.(3.44 ) , can be further rewritten in terms of theta - functions @xcite : @xmath151 and finally as : @xmath152 where @xmath153 . in the latter expression we have isolated all the real singularities in the first term . to be more specific we see that the static coordinate time is singular for @xmath154 , where @xmath155 is an integer , with the asymptotic behaviour : @xmath156 on the other hand @xmath157 is completely regular at the boundaries of the branches , i.e. for @xmath158 . these results can be easily translated to the hyperboloid time @xcite : @xmath159 where : @xmath160 notice that the singularities , eq . ( 3.47 ) , that originated from the zeros of @xmath161 have canceled in @xmath162 so that @xmath163 is finite . @xmath162 blows up for @xmath164 , like : @xmath165 where @xmath155 is again an integer . this demonstrates that the world - sheet time @xmath67 is actually an infinite valued function of @xmath118 , and that the solution @xmath116 therefore describes _ infinitely many strings _ ( see fig.5 ) . this should be compared with the @xmath90 case where we found a solution describing two strings . in that case the two strings were of completely different type and had completely different physical interpretations . in the present case we find infinitely many strings but they are all of the same type . in the branch @xmath166 $ ] ( say ) the string starts with infinite string size at @xmath167 . it then contracts to its minimal size @xmath103 and reexpands towards infinity at @xmath168 . this solution , and the infinitely many others of the same type , are unstable strings . the comoving time of the @xmath116-solution is given by @xcite : @xmath169 @xmath170 @xmath171 it can be shown that @xmath172 and @xmath173 . it follows that : @xmath174 the comoving time , eq . ( 3.51 ) , is singular at @xmath175 but regular at @xmath176 and similarly in the other branches , fig.5 . therefore , the interpretation of the string solution in the branch @xmath166 $ ] ( say ) , as seen in comoving coordinates , is as follows : the string starts with infinite size at @xmath177 . it then contracts and passes the horizon from the outside at @xmath178 . the string now continues contracting from the inside of the horizon at @xmath179 until it reaches the minimal size at : @xmath180 from now on the string expands again . it passes the horizon from the inside after finite comoving time and continues towards infinity for @xmath181 . it is an interesting observation that the comoving time is not periodic in @xmath67 , i.e. @xmath182 , although the string size is . explicitly we find : @xmath183 this means that @xmath184 really describes infinitely many strings with different invariant size at a given comoving time . to be more specific let us consider a fixed comoving time @xmath125 and the corresponding world - sheet times : @xmath185 where @xmath186 . taking for simplicity a comoving time @xmath187 we have ( see fig.5 . ) : @xmath188 to the lowest orders we find from eq . ( 3.51 ) : @xmath189 so that : @xmath190\ ] ] the invariant string sizes are then : @xmath191\ ] ] i.e. they are separated by a multiplicative factor . this expression , of course , is only valid as long as @xmath192 , so @xmath193 should not be too large . we now consider the @xmath135-solution , eq . ( 3.43 ) . in this case the dynamics takes place well inside the horizon . the possible singularities of the hyperboloid time @xmath194 and the comoving time @xmath195 therefore coincide with the singularities of the static coordinate time @xmath196 . the static coordinate time is again obtained from eq . ( 2.10 ) , which we first rewrite as : @xmath197 integration leads to : @xmath198 where @xmath199 is a complex constant obeying @xmath200 i.e. @xmath201=1/\nu$ ] . it follows that @xmath202 where @xmath149 is real and @xmath150=\mu$ ] . again we can express the static coordinate time in terms of theta - functions @xcite : @xmath203 or in terms of the jacobi zeta - function @xmath204 @xcite : @xmath205 where @xmath153 . in this form we see explicitly that @xmath196 consists of a linear term plus oscillating terms . the comoving time takes the form : @xmath206)+ht_+(\tau)&\nonumber\end{aligned}\ ] ] @xmath207 + \log\mid\frac{\omega\vartheta_4(\omega(\tau - y))\vartheta'_1(0 ) } { \pi\vartheta_1(\omega y)\vartheta_4(\omega\tau)}\mid . & \end{aligned}\ ] ] notice that the argument of the @xmath208 has no real zeros : @xmath209 + \log\mid\frac{\omega\vartheta'_1(0)}{\pi\vartheta_1(\omega y)}\mid & \nonumber\end{aligned}\ ] ] @xmath210 the static coordinate time and the cosmic time are therefore completely regular functions of @xmath67 , and it follows that the string solution @xmath135 , which is _ oscillating regularly _ as a function of world - sheet time @xmath67 , is also oscillating regularly when expressed in terms of hyperboloid time or comoving time . this solution represents one _ stable _ string . @xmath106 : the analysis here is very similar to the analysis of the @xmath211solution in the @xmath212case so we shall not go into it here . the results are summarized in table i , together with the results from the other cases . * table i * circular string evolution in de sitter spacetime . for each @xmath213 there exists two independent solutions @xmath116 and @xmath135 : [ cols= " < , < , < " , ] using the covariant approach of frolov and larsen @xcite , we have considered small perturbations propagating along the circular strings in black hole and cosmological spacetime backgrounds @xcite . in the case of the @xmath1 dimensional de sitter space introduce two normal vectors @xmath214 and @xmath215 perpendicular to the string world - sheet : @xmath216 and then consider only physical perturbations : @xmath217 where @xmath218 and @xmath219 are the perturbations as seen by an observer travelling with the circular string . after fourier expanding @xmath220 : @xmath221 it can be shown that @xcite : @xmath222 @xmath223 determining the comoving perturbations . these equations have been discussed in detail in refs.@xcite , so we shall just give one simple result here . for @xmath52 the brackets in eqs . ( 3.69)-(3.70 ) become negative . this means that the perturbations develop imaginary frequencies and grow indefinitely . however , by considering the detailed solutions it turns out that the perturbations grow with the same rate as the radius of the underlying circular string ( which by the way grows with the same rate as the universe ) so although the perturbations grow , the circular shape of the string is actually stable . more details can be found in refs.@xcite . we now consider the circular string dynamics in the 2 + 1 black hole anti de sitter ( bh - ads ) spacetime recently found by baados et . this spacetime background has arised much interest recently . it describes a two - parameter family ( mass @xmath224 and angular momentum @xmath225 ) of black holes in 2 + 1 dimensional general relativity with metric : @xmath226 it has two horizons @xmath227 and a static limit @xmath228 defining an ergosphere , as for ordinary kerr black holes . using the general formalism of section 2 , we can immediately read off the potential ( see fig.6 . ) : @xmath229 the potential , eq . ( 4.2 ) , has a global minimum between the two horizons : @xmath230 which is always negative , since we only consider the case when @xmath231 ( otherwise there are no horizons ) . for large values of @xmath23 the potential goes as @xmath232 and at @xmath37 we have : @xmath233 that can be either positive , negative or zero . notice also that the potential vanishes provided : @xmath234 there are therefore three fundamentally different types of solutions . : for @xmath235 there are two positive-@xmath23 zeros of the potential ( fig.6a ) . the smallest zero is located between the inner horizon and @xmath37 , while the other zero is between the outer horizon and the static limit . therefore , this string solution never comes outside the static limit . on the other hand it never falls into @xmath37 . the mathematical solution oscillating between these two positive zeros of the potential may be interpreted as a string travelling between the different universes described by the maximal analytic extension of the spacetime ( the penrose diagram of the @xmath0 dimensional bh - ads spacetime is discussed in refs.@xcite ) . such type of circular string solutions also exist in other stringy black hole backgrounds @xcite . : for @xmath236 there is only one positive-@xmath23 zero of the potential , which is always located outside the static limit ( fig.6b ) . the potential is negative for @xmath37 , so there is no barrier preventing the string from collapsing into @xmath37 . by suitably fixing the initial conditions the string starts with its maximal size outside the static limit at @xmath237 . it then contracts through the ergosphere and the two horizons and eventually falls into @xmath37 . if @xmath238 it may however still be possible to continue this solution into another universe as in the case * ( i)*. : @xmath239 is the limiting case where the maximal string radius equals the static limit . the potential is exactly zero for @xmath37 so also in this case the string contracts through the two horizons and eventually falls into @xmath37 . the exact and complete mathematical solution can be obtained in terms of elementary or elliptic functions , the details can be found in ref.@xcite . in all cases we find only bounded string size solutions and no multi - string solutions . see also table ii . we will now compare the circular strings in the @xmath0 dimensional bh - ads spacetime and in the equatorial plane of ordinary @xmath1 dimensional black holes . in the most general case it is natural to compare the spacetime metric , eq . ( 4.1 ) , with the ordinary @xmath1 dimensional kerr anti de sitter spacetime with metric components : @xmath240 @xmath241 where we have introduced the notation : @xmath242 here the mass is represented by @xmath224 while @xmath199 is the specific angular momentum , and a positive @xmath243 corresponds to de sitter while a negative @xmath243 corresponds to anti de sitter spacetime . in the equatorial plane @xmath244 the metric , eq . ( 5.1 ) , is in the general form of eq . ( 2.1 ) so that we can use the analysis of section 2 . in the most general case the potential is given by : @xmath245 i.e. the potential covers seven powers in @xmath23 . the general solution will therefore involve higher genus elliptic functions . it is furthermore very complicated to deduce the physical properties of the circular strings from the shape of the potential ( the zeros etc . ) since the invariant string size defined in eq . ( 2.14 ) is non - trivially connected to @xmath246 @xmath247 we have exactly solved the string dynamics in a number of spacetimes of the form , eq . ( 5.1 ) @xcite . in the cases of minkowski space , anti de sitter space , schwarzschild and schwarzschild anti de sitter space , fig.7 . , there are only bounded string size solutions and no multi - string solutions . in schwarzschild de sitter space , on the other hand , the dynamics outside the schwarzschild horizon is similar to the dynamics in `` pure '' de sitter space , so we find the complicated spectrum of oscillating , expanding and contracting strings and the multi - string solutions , see table ii . to obtain more insight about the string propagation in all these curved spacetimes we solved the string equations of motion and constraints by considering perturbations around the exact string center of mass solution , refining the approach originally developed by de vega and snchez @xcite . in an arbitrary curved spacetime of dimension @xmath248 , the string equations of motion and constraints , in the conformal gauge , take the form : @xmath249 @xmath250 for @xmath251 and prime and dot represent derivative with respect to @xmath148 and @xmath252 respectively . consider first the equations of motion , eq . . a particular solution is provided by the string center of mass @xmath253 : @xmath254 then a perturbative series around this solution is developed : @xmath255 after insertion of eq . ( 6.4 ) in eq . ( 6.1 ) , the equations of motion are to be solved order by order in the expansion . to zeroth order we just get eq . ( 6.3 ) . to first order we find : @xmath256 the first three terms can be written in covariant form @xcite , c.f . the ordinary geodesic deviation equation : @xmath257 however , we can go one step further . for a massive string , corresponding to the string center of mass fulfilling ( in units where @xmath258 ) : @xmath259 there are @xmath260 physical polarizations of string perturbations around the geodesic @xmath253 . we therefore introduce @xmath260 normal vectors @xmath261 : @xmath262 and consider only first order perturbations in the form : @xmath263 where @xmath220 are the comoving perturbations , i.e. the perturbations as seen by an observer travelling with the center of mass of the string . the normal vectors are not uniquely defined by eqs . in fact , there is a gauge invariance originating from the freedom to make local rotations of the @xmath264-bein spanned by the normal vectors . for our purposes it is convenient to fix the gauge taking the normal vectors to be covariantly constant : @xmath265 this is achieved by choosing the basis @xmath266 obeying the conditions given by eqs . ( 6.8 ) at a given point , and defining it along the geodesic by means of parallel transport . another useful formula is the completeness relation that takes the form : @xmath267 using eqs . ( 6.7)-(6.10 ) in eq . ( 6.6 ) , we find after multiplication by @xmath268 the spacetime invariant formula @xcite : @xmath269 since the last term depends on @xmath148 only through @xmath270 it is convenient to make a fourier expansion : @xmath271 then eq . ( 6.12 ) finally reduces to : @xmath272 which constitutes a matrix schrdinger equation with @xmath67 playing the role of the spatial coordinate . notice that in the case of constant curvature spacetimes , @xmath273 the `` potential '' in eq . ( 6.14 ) is obtained directly from the normalization equations ( 6.7 ) and ( 6.8 ) without any calculations at all . for the second order perturbations the picture is a little more complicated . since they couple to the first order perturbations we consider the full set of perturbations @xmath274 @xcite : @xmath275 where the source @xmath276 is bilinear in the first order perturbations , and explicitly given by : @xmath277 after solving eqs . ( 6.14)-(6.15 ) for the first and second order perturbations , the constraints , eq . ( 6.2 ) , have to be imposed . in world - sheet light cone coordinates @xmath278 the constraints take the form : @xmath279 where @xmath280 . the world - sheet energy - momentum tensor @xmath281 is conserved , as can be easily verified using eq . ( 6.1 ) , and therefore can be written : @xmath282 at the classical level under consideration here , the constraints are then simply : @xmath283 up to second order in the expansion around the string center of mass we find : @xmath284 and the conditions @xmath285 eq . ( 6.19 ) , then give a formula for the string mass . in the case of ordinary @xmath286 de sitter space , the first order perturbations , eq . ( 6.14 ) , are determined by @xcite : @xmath287 for @xmath288 the frequencies become imaginary and classical instabilities develop . after solving the second order perturbation equations , eqs . ( 6.15)-(6.16 ) , the mass formula is found @xcite : @xmath289 after quantization one finds that real mass states can only be defined up to some maximal mass @xcite . this is reminiscent of the classical string instabilities in de sitter space . in ordinary anti de sitter space and in the @xmath0 black hole ads ( which is locally , but not globally , ads ) the first order perturbation equations and mass formula take the form of eqs . ( 6.21)-(6.22 ) , but with @xmath290 replaced by @xmath291 ( or @xmath292 depending on notation ) . it follows that in these cases there are no instabilities neither classically nor quantum mechanically . the perturbation series approach is perfectly well - defined in these cases . it is interesting to compare the @xmath0 black hole ads with ordinary @xmath286 black hole ads . in secs . 4,5 we found that the circular string motion is very similar in these backgrounds ( see table ii ) . this was actually somewhat surprising since the backgrounds are really very different : the ordinary black holes have a strong curvature singularity at @xmath293 the @xmath0 black hole ads has not . the circular strings are however very special configurations and that could be the reason why we did not really see any qualitative differences . for generic strings we would expect some differences , however , and that is indeed what we find when using the string perturbation series approach . in the ordinary black hole ads spacetime the first order perturbation equations become @xcite : @xmath294 @xmath295 for the transverse and longitudinal perturbations , respectively . for the transverse perturbations , eq . ( 6.23 ) , the bracket is always positive , thus the frequencies are real and no instabilities arise , not even for @xmath296 for the longitudinal perturbations , on the other hand , the bracket can be negative . in that case imaginary frequencies develop . the ( @xmath297)-instability sets in at : @xmath298 the higher modes develop instabilities for smaller @xmath299 i.e. closer to the singularity . similar results are obtained in the black string background @xcite , see table iii . we have studied the string propagation in de sitter and black hole backgrounds . the main part of the talk concerned the dynamics of circular strings in de sitter space , with special interest in the physical interpretation of the results of the mathematical analysis . we then compared with results obtained in various black hole backgrounds ( @xmath0 bh - ads , schwarzschild - anti de sitter , schwarzschild - de sitter ) . finally , more insight about the strings in curved spacetimes was obtained using the string perturbation series approach . the main results and conclusions are summarized in tables i , ii , iii . + i would like to thank h.j . de vega and n. snchez for a very fruitful collaboration on the material presented in this talk . 11 h.j . de vega and n. snchez , _ phys . rev . _ * d42 * ( 1990 ) 3969 . de vega , m.r . medrano and n. snchez , _ nucl . phys . _ * b374 * ( 1992 ) 405 . de vega and n. snchez , _ phys . _ * b244 * ( 1990 ) 215 , _ phys . lett . _ * 65c * ( 1990 ) 1517 , _ ijmp _ * a7 * ( 1992 ) 3043 , _ nucl . * b317 * ( 1989 ) 706 and ibid 731 . + d. amati and c. klimcik , _ phys . _ * b210 * ( 1988 ) 92 . + m. costa and h.j . de vega , _ ann . _ * 211 * ( 1991 ) 223 and ibid 235 . + c. loust and n. snchez , _ phys . * d46 * ( 1992 ) 4520 . v.e . zakharov and a.v . mikhailov , _ jetp _ * 47 * ( 1979 ) 1017 . + h. eichenherr , in _ integrable quantum field theories _ , ed . j. hietarinta and c. montonen ( springer , berlin , 1982 ) . i. bars and k. sfetsos , _ mod . lett . _ * a7 * ( 1992 ) 1091 . de vega , j.r . mittelbrunn , m.r . medrano and n. snchez , `` the two - dimensional stringy black hole : a new approach and a pathology '' , _ par - lpthe _ * 93/14 * and `` the general solution of the 2-d sigma model stringy black hole and the complex sine - gordon equation '' , _ par - lpthe _ * 93/53*. h.j . de vega and n. snchez , _ phys . _ * b197 * ( 1987 ) 320 . mende , in _ string quantum gravity and the physics at the planck scale _ , ed . n. snchez ( world scientific , singapore , 1993 ) . a.l . larsen and v.p . frolov , _ nucl . _ * b414 * ( 1994 ) 129 . a.l . larsen and n. snchez , `` strings propagating in the 2 + 1 dimensional black hole anti de sitter spacetime '' , _ obs . de paris , demirm _ * 94013*. the contributions by h.j . de vega and n. snchez in _ string quantum gravity and the physics at the planck scale _ , ed . n. snchez ( world scientific , singapore , 1993 ) . n. snchez and g. veneziano , _ nucl . phys . _ * b333 * ( 1990 ) 253 . + m. gasperini , n. snchez and g. veneziano , _ nucl . phys . _ * b364 * ( 1991 ) 365 frolov , v.d . skarzhinsky , a.i . zelnikov and o. heinrich , _ phys . lett . _ * b224 * ( 1989 ) 255 . de vega , a.v . mikhailov and n. snchez , _ teor . _ * 94 * ( 1993 ) 232 . . de vega , a.l . larsen and n. snchez , `` infinitely many strings in de sitter spacetime : expanding and oscillating elliptic function solutions '' , _ obs . de paris , demirm _ * 93055*. a.l . larsen , `` circular string - instabilities in curved spacetime '' , _ obs . de paris , demirm _ * 93052 * , to appear in phys . rev . d. a. davidson , n.k . nielsen and y. verbin , _ nucl . _ * b412 * ( 1994 ) 391 . . de vega , a.l . larsen and n. snchez , `` semi - classical quantization of circular strings in de sitter and anti de sitter spacetime '' , in preparation . m. baados , c. teitelboim and j. zanelli , _ phys . * 69 * ( 1992 ) 1849 . gradshteyn and i.m . ryznik , _ table of integrals , series and products _ ( academic press inc , london , 1980 ) . m. abramowitz and i.a . stegun , _ handbook of mathematical functions _ ( dover publications inc , new york , 1970 ) . larsen , `` stable and unstable circular strings in inflationary universes '' , _ nordita _ * 94/14 p*. m. baados , m. henneaux , c. teitelboim and j. zanelli , _ phys . rev . _ * d48 * ( 1993 ) 1506 . horowitz and d.l . welch , _ phys . * 71 * ( 1993 ) 328 . . de vega and i.l . egusquiza , _ phys . * d49 * ( 1994 ) 763 .
|
string propagation is investigated in de sitter and black hole backgrounds using both exact and approximative methods .
the circular string evolution in de sitter space is discussed in detail with respect to energy and pressure , mathematical solution and physical interpretation , multi - string solutions etc .
we compare with the circular string evolution in the @xmath0 dimensional black hole anti de sitter spacetime and in the equatorial plane of ordinary @xmath1 dimensional stationary axially symmetric spacetime solutions of einstein general relativity . using an approximative string perturbation approach we consider also generic string evolution and propagation in all these curved spacetimes . + arne l. larsen +
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the one - electron models of solids are based on the study of schrdinger operator with periodic potential . there are a lot of studies on the periodic potential , in particular , for periodic point interactions , we can show the spectral set explicitly ( albeverio et . @xcite is the best guide to this field for readers ) . most fundamental case is the one - dimensional schrdinger operator with periodic point interactions , called the _ kronig penney model _ ( see kronig penney @xcite ) , given by @xmath5 where @xmath6 is a non - zero real constant , and @xmath7 is the dirac delta measure at @xmath8 . the positive sign of @xmath6 corresponds the repulsive interaction , while the negative one corresponds the attractive one . more precisely , @xmath0 is the negative laplacian with boundary conditions on integer points : @xmath9 where @xmath10 here @xmath11 and @xmath12 is the usual sobolev space of order @xmath13 on the open set @xmath14 . from sobolev s embedding theorem @xmath15 , every elements of @xmath16 are continuous ( classical sense ) and uniformly bounded functions . it is well - known that @xmath0 is self - adjoint and is a model describing electrons on the quantum wire . the spectrum of this model is explicitly given by @xmath17 where @xmath18 @xmath19 is so - called _ discriminant _ and @xmath20 can be regarded as an entire function with respect to @xmath21 . the spectrum @xmath22 of @xmath0 consists of infinitely many closed intervals ( spectral bands ) and is purely absolutely continuous . on the other hand , for the schrdinger operator @xmath23 on @xmath24 with decaying potential @xmath6 , the _ dispersive estimate _ for the schrdinger time evolution operator @xmath2 is stated as follows : @xmath25 where @xmath26 denotes the spectral projection to the absolutely continuous subspace for @xmath0 . the dispersive estimate is a quantitative representation of the diffusion phenomena in quantum mechanics , and is extensively studied recently , because of its usefulness in the theory of the non - linear schrdinger operator ( see e.g. journ sogge @xcite , weder @xcite , yajima @xcite , mizutani @xcite , and references therein ) . the estimate ( [ dispersive_estimate ] ) is also obtained in the case of the one - dimensional point interaction . sacchetti @xcite obtain ( [ dispersive_estimate ] ) when @xmath6 is one point @xmath27 potential , and so do kovak sacchetti @xcite when @xmath6 is the sum of @xmath27 potentials at two points . the motivation of the present paper is to obtain a similar estimate for our periodic model ( [ kronig - penney0 ] ) . though this problem is quite fundamental , we could not find such kind of results in the literature , probably because the deduction of the result requires a detailed analysis of the band functions , as we shall see below . since the spectrum of the schrdinger operator with periodic potential is absolutely continuous , one may expect some dispersion type estimate holds also in this case . however , there seems to be few results about the dispersive type estimate for the time evolution operator of the differential equation with periodic coefficients . an example is the paper by cuccagna @xcite , in which the klein gordon equation @xmath28 is considered , where @xmath29 is a smooth real - valued periodic function with period @xmath30 . cuccagna proves the solution @xmath31 satisfies @xmath32 for @xmath33 , where @xmath34 is some bounded discrete set . the peculiar power @xmath35 comes from the following reason . the integral kernel for the time evolution operator is written as the sum of oscillatory integrals @xmath36 where @xmath37+x'$ ] , @xmath38+y'$ ] , @xmath39,[y]\in\mathbf{z}$ ] , @xmath40 , and @xmath41-[y])/t$ ] . the function @xmath42 is called the _ band function _ for the @xmath43-th band , which is a real - analytic function of @xmath44 with period @xmath45 . in the large time limit @xmath4 , it is well - known that the main contribution of the oscillatory integral ( [ intro0 ] ) comes from the part nearby the _ stationary phase point _ @xmath46 ( the solution of @xmath47 ) , and the _ stationary phase method _ tells us the principal term in the asymptotic bound is a constant multiple of @xmath48 ( see e.g. stein ( * ? ? ? * chapter viii ) or lemma [ lemma_stein ] below ) . however , since @xmath42 is a periodic function , there exists a point @xmath49 so that @xmath50 . if the stationary phase point @xmath46 coincides with @xmath49 , then the previous bound no longer makes sense . instead , the stationary phase method concludes the principal term is a constant multiple of @xmath51 . since the integral kernel of our operator has the same form as ( [ intro0 ] ) ( see ( [ st01 ] ) below ) , we expect a result similar to ( [ cuccagna ] ) also holds in our case . let us formulate our main result . let @xmath0 be the hamiltonian for the kronig - penney model given in ( [ kronig - penney1 ] ) and ( [ kronig - penney2 ] ) . as stated above , the spectrum of @xmath0 has the band structure , that is , @xmath52 where the @xmath43-th band @xmath53 is a closed interval of finite length ( for the precise definition , see ( [ bloch13 ] ) below ) . our main result is as follows . [ theorem_main ] let @xmath54 be the spectral projection onto the @xmath43-th energy band @xmath53 . then , for sufficiently large @xmath43 , there exist positive constants @xmath55 and @xmath56 such that @xmath57 for any @xmath58 and any @xmath59 , where @xmath60 . the coefficients obey the bound @xmath61 as @xmath62 . the power @xmath63 in the first term of the coefficient in ( [ main1 ] ) is the same as in ( [ dispersive_estimate ] ) with @xmath64 and @xmath65 , since it comes from the states corresponding to the energy near the band center , which behaves like a free particle . this fact can be understood from the graph of @xmath66 ( figure [ figure_band_function ] ) . the part of the graph corresponding to the band center is similar to the parabola @xmath67 or its translation , which is the band function for the free hamiltonian @xmath68 . on the other hand , the power @xmath35 comes from part of the integral ( [ intro0 ] ) given by @xmath69 where @xmath70 is some open set including two solutions @xmath49 to the equation @xmath71 . notice that @xmath72 is an inflection point in the graph of @xmath66 ; see figure [ figure_band_function ] . the estimates for the coefficients are obtained from the lower bounds for the derivatives of @xmath42 . actually , we can choose @xmath70 so that @xmath73\setminus j_n}|\lambda_n''(\theta)|\geq c,\quad \inf_{\theta\in j_n}|\lambda_n^{(3)}(\theta)|\geq c n^{1/3},\ ] ] where @xmath74 is a positive constant independent of @xmath43 . by ( [ intro2 ] ) and the estimates for the amplitude function ( proposition [ proposition_amplitude1 ] ) , we can prove theorem [ theorem_main ] by using a lemma for estimating oscillatory integrals , given in stein s book ( see stein @xcite or lemma [ lemma_stein ] below ) . we can also prove the power @xmath35 is optimal , by considering the case @xmath75 ( so , @xmath49 is a stationary phase point ) , and applying the asymptotic expansion formula in the stationary phase method ( see e.g. stein ( * ? ? ? * page 334 ) ) . ( @xmath76 ) for @xmath77 . the range of @xmath42 is the @xmath43-th band @xmath53 . we find two inflection points of @xmath66 near @xmath78 , for every @xmath43 . , width=302 ] the physical implication of the result is as follows . by definition , the parameter @xmath41-[y])/t$ ] represents the _ propagation velocity _ of a quantum particle . the wave packet with energy near @xmath79 has the maximal group speed in the @xmath43-th band , and the speed of the quantum diffusion is slowest in that band . thus such state has a bit longer life - span ( in the sense of @xmath80-norm ) than the ordinary state has ; the state is in some sense a _ resonant state _ , caused by the meeting of two stationary phase points @xmath46 as @xmath81 tends to @xmath82 . it is well - known that the existence of resonant states makes the decay of the solution with respect to @xmath83 slower ( see jensen kato @xcite or mizutani @xcite ) . since the estimate ( [ main1 ] ) is given bandwise , it is natural to ask we can obtain the dispersive type estimate for the whole schrdinger time evolution operator @xmath2 , like cuccagna s result ( [ cuccagna ] ) . however , it turns out to be difficult in the present case , from the following reason . a reasonable strategy to prove such estimate is as follows . first , we divide the integral @xmath84 into two parts @xmath85 and the rest , where @xmath85 is given in ( [ intro1 ] ) with @xmath70 some open set including @xmath86 . next , we show the sum of @xmath85 converges and gives @xmath3 , and the sum of the rests also converges and gives @xmath87 . however , for fixed @xmath88 and @xmath89 , we find that our upper bound for @xmath90 is not better than @xmath91 , and the sum of the upper bounds does not converge ( see the last part of section 4 ) . one reason for this divergence is very strong singularity of our potential , the sum of @xmath27-functions . because of this singularity , the width of the band gap , say @xmath92 ( @xmath43 is the band number ) , does not decay at all in the high energy limit @xmath93 . implies @xmath94 as @xmath93 . ] then we can not take the open set @xmath70 so small , in the proof of theorem [ theorem_main ] . ] and the sum of the lengths @xmath95 diverges ; if this sum converges , we can use a simple bound @xmath96 to control the sum . thus we do not succeed to obtain a bound for the sum of @xmath90 at present . on the other hand , for the schrdinger operator @xmath97 on @xmath98 with real - valued periodic potential @xmath6 , it is known that the decay rate of the width of the band gap @xmath92 reflects the smoothness of the potential @xmath6 . hochstadt @xcite says @xmath99 if @xmath6 is in @xmath100 , and trubowitz @xcite says @xmath101 ( @xmath102 is some positive constant ) if @xmath6 is real analytic . so , if @xmath6 is sufficiently smooth , it is expected that we can control the sum of @xmath85 , and obtain the dispersive type estimate for the whole operator @xmath2 ( i.e. ( [ main1 ] ) without the projection @xmath54 ) . we hope to argue this matter elsewhere in the near future . the paper is organized as follows . in section 2 , we review the floquet bloch theory for our operator @xmath0 and give the explicit form of the integral kernel of @xmath2 . in section 3 , we give more concrete analysis for the band functions , especially give some estimates for the derivatives . in section 4 , we prove theorem [ theorem_main ] , and give some comment for the summability with respect to @xmath43 of the estimates ( [ main1 ] ) . in this section , we shall calculate the integral kernel of the operator @xmath2 by using the floquet - bloch theory . most results in this section are already written in another literature ( e.g. reed - simon ( * ? ? ? * xiii.16 ) and albeverio et . * iii.2.3 ) ) , but we shall give it here again for the completeness . first we shall calculate the generalized eigenfunctions for our model , i.e. , the solutions to the equations @xmath103 the condition ( [ bloch02 ] ) comes from the requirement @xmath104 , and we use the abbreviation @xmath105 in ( [ bloch03 ] ) . [ proposition_generalized_eigenfunction ] let @xmath106 , @xmath107 , and take @xmath108 so that @xmath109 . then , the equations ( [ bloch01])-([bloch03 ] ) have a solution @xmath110 of the following form . @xmath111 where @xmath112 and @xmath113 are constants . when @xmath114 , we interpret @xmath115 . the coefficients @xmath112 and @xmath113 satisfy the following recurrence relation . @xmath116 @xmath117 the matrix @xmath118 satisfies @xmath119 and the discriminant @xmath120 is @xmath121 the proof is a simple calculation , so we shall omit it . notice that @xmath122 is an entire function with respect to @xmath123 , since @xmath19 is an even function . next we shall calculate the _ bloch waves _ , the solution to ( [ bloch01])-([bloch03 ] ) with the quasi - periodic condition @xmath124 for some @xmath125 . [ proposition_blochwave ] 1 . for @xmath126 , there exists a non - trivial solution @xmath127 to ( [ bloch01])-([bloch03 ] ) satisfying the bloch wave condition ( [ bloch04 ] ) if and only if @xmath128 2 . when ( [ bloch09 ] ) holds , a solution @xmath110 to ( [ bloch01])-([bloch03 ] ) and ( [ bloch04 ] ) is given by ( [ bloch05 ] ) with the coefficients @xmath129 \(i ) it is easy to see a non - trivial solution to ( [ bloch01])-([bloch03 ] ) and ( [ bloch04 ] ) exists if and only if @xmath118 has an eigenvalue @xmath130 , and the latter condition is equivalent to ( [ bloch09 ] ) , since @xmath119 and @xmath131 . ( ii ) when ( [ bloch09 ] ) holds , the vector @xmath132 given in ( [ bloch10 ] ) is an eigenvector of @xmath118 with the eigenvalue @xmath130 . thus the second equation in ( [ bloch10 ] ) follows from ( [ bloch06 ] ) . proposition [ proposition_blochwave ] and the bloch theorem imply @xmath133 if and only if ( [ bloch09 ] ) holds for some @xmath125 , that is , @xmath134 as already stated in ( [ kronig - penney3 ] ) . for @xmath135 , we have @xmath136 if @xmath77 , the right hand side of ( [ bloch12 ] ) is larger than @xmath137 and ( [ bloch11 ] ) does not hold for @xmath135 . thus , there is no negative part in @xmath22 . if @xmath138 , then some negative value @xmath123 belongs to @xmath22 , and the corresponding @xmath139 is pure imaginary . however , we concentrate on the high energy limit in the present paper , and the existence of the negative spectrum does not affect our argument . so we sometimes assume @xmath77 in the sequel , in order to simplify the notation . in this case , the results for @xmath138 will be stated in the remark . by an elementary inspection of the graph of @xmath140 , we find the following properties . [ proposition_discriminant ] assume @xmath77 . then , 1 . @xmath141 and @xmath142 for @xmath143 . 2 . the equation @xmath144 has a unique solution @xmath145 in the open interval @xmath146 for @xmath147 . 3 . the equation @xmath148 has a unique solution @xmath149 in the open interval @xmath146 for @xmath143 , and @xmath150 . 4 . for convenience , we put @xmath151 . then , @xmath19 is monotone decreasing on @xmath152 $ ] for even @xmath43 , and monotone increasing on @xmath152 $ ] for odd @xmath43 . * remark . * when @xmath138 , we denote the solution to @xmath153 in @xmath154 by @xmath155 , and the solution to @xmath148 in @xmath154 by @xmath156 , for @xmath157 . [ cols="^,^ " , ] though the formulas ( [ der1])-([der6 ] ) are explicit , it is still not easy to obtain the precise lower bound for the derivatives of @xmath42 , especially when @xmath42 is near the band edge . for this reason , we employ the _ puiseux expansion _ of the inverse function @xmath158 , which makes our analysis clear . this kind of expansion is studied in the classical work by kohn @xcite . first let us analyze the asymptotics of @xmath155 and @xmath156 given in proposition [ proposition_discriminant ] . a related result is written in albeverio et . * theorem 2.3.3 ) . [ proposition_knln ] let @xmath155 and @xmath156 as in proposition [ proposition_discriminant ] and its remark . then , @xmath159 @xmath160 we assume @xmath77 for simplicity ( in the case @xmath138 , we only need to change the sign of @xmath161 given below ) . first we prove ( [ asymptotic_kn ] ) . the number @xmath155 is the solution to @xmath162 put @xmath163 . then ( [ ba02 ] ) is equivalent to @xmath164 then , for sufficiently large @xmath43 , it is easy to see that the solution to ( [ ba03 ] ) is the limit of the sequence @xmath165 given by$ ] , @xmath166 , @xmath167 and assume @xmath168 and @xmath169 . then @xmath170 has the unique fixed point in @xmath171 which is the limit of the sequence ( [ ba04 ] ) ' . if we take @xmath170 as in ( [ ba03 ] ) and @xmath172 $ ] , we can apply the contraction mapping theorem for sufficiently large @xmath43 . ] @xmath173 by a simple calculation , we have @xmath174 then twice substitution of ( [ ba05 ] ) into ( [ ba04 ] ) gives the formula ( [ asymptotic_kn ] ) ( three times substitution gives the same formula ) . next we prove ( [ asymptotic_ln ] ) . put @xmath175 . then the defining equation of @xmath156 @xmath176 is equivalent to @xmath177 then we can obtain the formula ( [ asymptotic_ln ] ) by using the following expansion recursively . @xmath178 in order to calculate the puiseux expansion of @xmath179 , first we calculate the taylor expansion of @xmath140 . [ proopsition_taylor_dk ] the taylor expansion of @xmath140 near @xmath149 is given as follows . @xmath180 @xmath181,\\ \label{d2 } d_2 & = & ( -1)^n\left[-1 - \left(v+\frac{v^2}{8}\right)(n\pi)^{-2}+o(n^{-4})\right],\\ \label{d3 } d_3 & = & ( -1)^n\left [ \left(v+\frac{v^2}{6}\right)(n\pi)^{-3}+o(n^{-5})\right],\\ \label{d4 } d_4 & = & ( -1)^n\left[\frac{1}{12}+ \left(\frac{v}{6}+\frac{v^2}{96}\right)(n\pi)^{-2}+o(n^{-4})\right],\\ \label{d5 } d_5 & = & ( -1)^n\left [ -\left(\frac{v}{6}+\frac{v^2}{60}\right)(n\pi)^{-3}+o(n^{-5})\right ] , \\ \label{d6 } d_6 & = & ( -1)^n\left[-\frac{1}{360}- \left(\frac{v}{120}+\frac{v^2}{2880}\right)(n\pi)^{-2}+o(n^{-4})\right].\end{aligned}\ ] ] notice that @xmath182 by definition . the results ( [ yn])-([d5 ] ) can be obtained by substituting the following formulas into ( [ d0k])-([d6k ] ) . from ( [ asymptotic_ln ] ) , we have @xmath183 next we shall calculate the puiseux expansion of @xmath158 near @xmath184 . since @xmath145 is the zero of order 2 of @xmath19 , @xmath158 is a double - valued function with respect to the complex variable @xmath89 . [ proposition_puiseux_k ] the puiseux expansion of @xmath179 near @xmath184 is written as follows . @xmath185 put @xmath186 and @xmath187 , where @xmath188 are the coefficients in ( [ taylor_dk ] ) . then the taylor expansion ( [ taylor_dk ] ) is rewritten as @xmath189 substituting the puiseux expansion ( [ puiseux_k ] ) into ( [ pk01 ] ) , and comparing the coefficient of each power @xmath190 , we have @xmath191 since the expansions of @xmath187 are obtained from ( [ d2])-([d6 ] ) , we can calculate the coefficients ( [ e1])-([e5 ] ) by solving the above equations . * remark . * 1 . the branch points of @xmath179 are @xmath184 , and @xmath192 is nearby @xmath193 by ( [ yn ] ) . thus the radius of convergence of the puiseux expansion ( [ puiseux_k ] ) is about @xmath194 , for sufficiently large @xmath43 . this fact justifies the calculus in the sequel . \2 . in the case @xmath195 , the expansion ( [ puiseux_k ] ) coincides with the puiseux expansion @xmath196 where @xmath43 is any integer satisfying @xmath197 . substituting @xmath198 into ( [ puiseux_k ] ) , we obtain the function @xmath199 . since the puiseux expansion ( [ puiseux_k ] ) has two branches , we obtain two functions @xmath200 notice that @xmath161 is positive for real @xmath44 , so @xmath190 in ( [ ba07 ] ) is uniquely defined as a positive function . for notational simplicity , we put @xmath201 then we have the following formulas , useful for the calculation below . @xmath202 then we obtain expansions of the band functions and their derivatives near the band edge , as follows . [ proposition_puiseux_lambda ] let @xmath203 , @xmath161 and @xmath204 as in ( [ ba07])-([zndef ] ) , and put @xmath205 . then the following expansions hold near @xmath206 ( or @xmath207 ) . 1 . @xmath208 + @xmath209 2 . @xmath210 3 . @xmath211 + @xmath212 4 . @xmath213 the proof can be done simply by substituting the expansion ( [ ba07 ] ) , ( [ asymptotic_ln ] ) and ( [ e1])-([e5 ] ) into @xmath214 and taking the derivative with respect to @xmath44 ( or @xmath204 ) repeatedly . the formulas ( [ formula1])-([formula5 ] ) help the calculation . by construction , we have @xmath215 for small @xmath204 . using ( [ ba09 ] ) and the expansion ( [ pu19 ] ) , we can find the asymptotics of the solution to @xmath71 . [ proposition_inflection ] the equation @xmath216 has a unique solution @xmath217 for sufficiently large @xmath43 . the asymptotics of @xmath49 is given as @xmath218 before the proof , we give a numerical result by using the explicit formulas ( [ der1])-([der6 ] ) , in figure [ figure_inflection ] . the result shows the formula ( [ asymptotics_theta0 ] ) gives a good approximation . near @xmath219 , @xmath220 . here we take @xmath221 , @xmath222 , @xmath223 , and @xmath224 . , width=226 ] put @xmath220 . we divide the interval @xmath225 $ ] into three intervals @xmath226,\quad i_2=[(n-1)\pi+\delta_n^{1/4 } , n\pi -\delta_n^{1/4}],\\ & & i_3=[n\pi -\delta_n^{1/4 } , n\pi].\end{aligned}\ ] ] for @xmath227 , we apply the first equality of ( [ ba09 ] ) with @xmath43 replaced by @xmath228 , so @xmath229 and @xmath230 . then the expansion ( [ pu19 ] ) for @xmath231 implies @xmath232 for @xmath227 , for sufficiently large @xmath43 . moreover , the formula ( [ inner_lambda2 ] ) given later in proposition [ proposition_inner_taylor ] shows @xmath232 for @xmath233 , for sufficiently large @xmath43 . for @xmath234 , we apply the second equality of ( [ ba09 ] ) , so @xmath235 and @xmath236 . then @xmath237 , and ( [ pu25 ] ) for @xmath238 implies @xmath239 for @xmath240 , for sufficiently large @xmath43 . thus @xmath241 is monotone decreasing on this interval , and ( [ pu19 ] ) for @xmath242 implies @xmath243 and @xmath244 for sufficiently large @xmath43 . , the latter fact can also be proved by the explicit value @xmath245 ; see ( ( [ pm11 ] ) below ) . ] thus there exists a unique @xmath246 $ ] with @xmath50 for sufficiently large @xmath43 , and @xmath247 . let us find more detailed asymptotics of @xmath49 . by ( [ pu19 ] ) for @xmath242 , @xmath248 if @xmath234 , then @xmath249 and @xmath250 by ( [ formula2 ] ) . so , if @xmath161 is the solution to ( [ ba11 ] ) , the expansions of the coefficients ( [ pu21])-([pu24 ] ) imply @xmath251 , and again by ( [ ba11 ] ) @xmath252 thus ( [ ba12 ] ) and ( [ formula2 ] ) imply @xmath253 since @xmath254 , we have the conclusion . let us prove @xmath42 is similar to the parabola @xmath67 near the band center , that is , for @xmath44 in the interval @xmath255.\ ] ] notice that the asymptotics ( [ asymptotics_theta0 ] ) implies @xmath49 is not in the interval ( [ inner06 ] ) for large @xmath43 . [ lemma_taylor_inner ] let @xmath256 given in ( [ ba01 ] ) , and @xmath44 in the interval ( [ inner06 ] ) . then , we have the following expansion . @xmath257 * remark . * the explicit forms of the derivatives are given in ( [ d0k])-([d6k ] ) . the taylor expansion of @xmath179 around @xmath258 is @xmath259 substituting the equality @xmath260 into ( [ ba15 ] ) and ( [ ba16 ] ) , we obtain the conclusion since @xmath261 where we used ( [ dkformula2 ] ) . [ proposition_inner_taylor ] let @xmath43 be a sufficiently large integer . then , for @xmath44 in the interval ( [ inner06 ] ) , we have the expansion of @xmath262 given in ( [ ba01 ] ) as follows . @xmath263 moreover , for @xmath264 we have @xmath265 let us rewrite the formula ( [ taylor_k_inner ] ) as @xmath266 when @xmath44 is in the interval ( [ inner06 ] ) , we have @xmath267 thus @xmath268 thus the first three terms in ( [ inner15 ] ) coincide the formula ( [ inner_k0 ] ) . let us show that the remainder term @xmath269 is negligible . by the differentiation of the inverse function , we can prove that @xmath270 the polynomial part is bounded uniformly with respect to @xmath43 . by the expansion ( [ puiseux_k ] ) and the monotonicity of @xmath271 , we see that @xmath271 satisfies @xmath272 and @xmath273 put @xmath274 @xmath275 , then @xmath276 lies between @xmath262 and @xmath277 , and by ( [ inner20])-([inner22 ] ) @xmath278 thus , we once have a rough estimate @xmath279 so the equation ( [ inner_k0 ] ) holds with the worse remainder term @xmath280 . however , this conclusion implies @xmath281 which also implies @xmath282 thus we have @xmath283 and ( [ inner_k0 ] ) holds . other estimates can be obtained by differentiating the formula ( [ taylor_k_inner ] ) . then we find the estimate for the remainder term becomes worse by the power @xmath284 per one differentiation . for example , the leading term in the remainder in ( [ inner17 ] ) is up to constant multiple @xmath285 differentiating this term , we get @xmath286 and the result is worse than @xmath287 by @xmath284 . as for @xmath269 , @xmath288 for the first term of ( [ inner25 ] ) , one @xmath289 in the numerator changed into @xmath290 by differentiation , and the estimate becomes worse by @xmath284 , because of ( [ inner16 ] ) . for the second term , that ` @xmath291 turned into @xmath292 ' makes two @xmath293 in the denominator ( see ( [ inner20 ] ) ) , one of which cancels with @xmath294 appeared next . thus the estimate also becomes worse by @xmath284 , in total . we can treat the other remainder terms similarly . we shall give the bound for the amplitude function in ( [ bloch16 ] ) , that is , @xmath295 where @xmath296 and @xmath297 given in ( [ ba01 ] ) . [ proposition_amplitude1 ] let @xmath298 given in ( [ amp01 ] ) . 1 . the function @xmath299 is bounded uniformly with respect to @xmath126 , @xmath296 and @xmath143 . 2 . put @xmath300 , @xmath301 , and @xmath302,\quad i_2 = [ ( n-1)\pi + \delta_1 , ( n-1)\pi + \delta_2],\\ & & i_3 = [ ( n-1)\pi + \delta_2 , n\pi - \delta_2],\quad i_4 = [ n\pi - \delta_2 , n\pi - \delta_1],\quad i_5 = [ n\pi - \delta_1 , n\pi].\end{aligned}\ ] ] for sufficiently large @xmath43 , the derivative @xmath303 obeys the following bound @xmath304 where @xmath74 is a positive constant independent of @xmath44 , @xmath88 , @xmath89 , and @xmath43 . especially , @xmath305 ) } = \int_{(n-1)\pi}^{n\pi}|a_n'(\theta , x , y)|d\theta\ ] ] is bounded uniformly with respect to @xmath43 , @xmath88 , @xmath89 . \(i ) first we prove @xmath306 is bounded uniformly with respect to @xmath43 and @xmath307 $ ] . for @xmath308 , we have @xmath309 , and @xmath310 by ( [ formula2 ] ) . then we have by the expansions ( [ taylor_dk ] ) and ( [ ba07 ] ) @xmath311 let @xmath312 as in ( [ zndef ] ) . since @xmath313 , @xmath314 thus @xmath315 and the right hand side is uniformly bounded . we can prove @xmath316 is uniformly bounded for @xmath317 in a similar way . for @xmath234 , the bounds ( [ inner16 ] ) and ( [ inner24 ] ) hold , so ( [ inner16 ] ) holds even if @xmath44 is replaced by @xmath139 . then @xmath318 thus @xmath316 is uniformly bounded on all the intervals @xmath319 . similarly we can prove @xmath320 and @xmath321 are uniformly bounded . the remaining factors are clearly bounded , so we have the conclusion . \(ii ) it is sufficient to show the three functions @xmath322 obey the bound ( [ bound_amplitude ] ) , since the derivatives of the remaining factors are bounded . first , by ( [ d1k ] ) , ( [ d2k ] ) and ( [ bloch17.5 ] ) @xmath323 for @xmath324 , the expansion ( [ amp02 ] ) implies @xmath325 for @xmath326 , by ( [ formula1 ] ) @xmath327 thus ( [ amp04 ] ) , ( [ amp05 ] ) and ( [ amp06 ] ) imply @xmath328 for @xmath329 . for @xmath330 , we have @xmath331 and by ( [ amp021 ] ) @xmath332 for large @xmath43 . thus ( [ amp04 ] ) , ( [ amp05 ] ) and ( [ amp07 ] ) imply @xmath333 for @xmath334 , for some positive constant @xmath74 independent of @xmath43 . for @xmath234 , we have by ( [ amp03 ] ) @xmath335 so ( [ amp04 ] ) implies @xmath336 for @xmath234 . similarly we can prove @xmath337 for @xmath233 and @xmath338 for @xmath227 , thus @xmath339 obeys the bound ( [ bound_amplitude ] ) . next , we shall estimate @xmath340 . by ( [ dkformula1 ] ) , ( [ d0k])-([d2k ] ) and ( [ bloch17.5 ] ) , we have @xmath341 this equality and ( [ amp02 ] ) , ( [ amp06 ] ) , ( [ amp07 ] ) and ( [ inner16 ] ) ( with @xmath44 replaced by @xmath139 ) gives the same conclusion for @xmath340 ( actually , we obtain a bit faster decay for @xmath342 ) . finally , by ( [ dkformula1 ] ) , ( [ d0k])-([d2k ] ) and ( [ bloch17.5 ] ) , @xmath343 this estimate is the same as ( [ amp04 ] ) . so the same conclusion holds for @xmath344 . the @xmath1 norm of the operator @xmath345 is just the supremum with respect to @xmath346 of the absolute value of the integral kernel @xmath347 given in ( [ bloch16 ] ) . put @xmath37+x'$ ] , @xmath38+y'$ ] , @xmath39,[y]\in \mathbf{z}$ ] , @xmath348 , and @xmath41-[y])/t$ ] , then ( [ bloch16 ] ) is rewritten as @xmath349 where @xmath299 is the amplitude function given in ( [ amp01 ] ) . the following lemma , taken from stein s book @xcite , gives the decay order of the oscillatory integral with respect to @xmath83 . [ lemma_stein ] let @xmath350 be a finite open interval and @xmath351 . let @xmath352 , and assume @xmath353 if @xmath354 , we additionally assume @xmath355 is a monotone function on @xmath171 . let @xmath356 and assume @xmath357 . then we have @xmath358 for any @xmath359 , where @xmath360 . * here , we say @xmath361 is monotone if @xmath362 ( @xmath363 ) or @xmath364 ( @xmath363 ) . the assumption ` @xmath171 is finite ' can be removed if the integral in the left hand side converges . the proof is done by integration by parts and a mathematical induction ( see stein @xcite ) . let @xmath49 as in proposition [ proposition_inflection ] . let @xmath367 be the maximum of the function @xmath368 . put @xmath369 and @xmath370 . put @xmath220 . then we have by ( [ asymptotics_theta0 ] ) and ( [ formula2 ] ) , @xmath371 since @xmath372 near the upper edge , we have by ( [ pm01])-([pm03 ] ) and ( [ pu14 ] ) ) is consistent with figure [ figure_lambda1 ] . ] @xmath373 if @xmath374 , then we have @xmath375 by the periodicity of the integrand , we can divide the integral ( [ st01 ] ) as the sum of two integrals so that @xmath376 is a monotone function on each interval . then we can apply lemma [ lemma_stein ] with @xmath354 , and we have by ( [ pm05 ] ) and proposition [ proposition_amplitude1 ] @xmath377 where @xmath378 $ ] and @xmath74 is a positive constant independent of @xmath43 . if @xmath379 , then we slide the interval of integration by periodicity , and divide it into four intervals @xmath380,\quad i_2 = \left [ n\pi - \delta_n^{1/3}/2 , n\pi + \delta_n^{1/3}/2 \right],\quad\\ i_3 = \left[n\pi + \delta_n^{1/3}/2 , n\pi + 2 \delta_n^{1/3 } \right],\quad i_4 = \left[n\pi + 2 \delta_n^{1/3 } , ( n+2)\pi -2 \delta_n^{1/3}\right].\end{aligned}\ ] ] put @xmath381 if @xmath227 or @xmath382 , then @xmath383 and by ( [ formula2 ] ) @xmath384 by ( [ pu19 ] ) , @xmath385 ( [ pm06 ] ) and ( [ pm07 ] ) imply ) and ( [ pm09 ] ) in figure [ figure_inflection ] . ] @xmath386 next , for @xmath387 , we have from ( [ formula2 ] ) @xmath388 and by ( [ pu25 ] ) @xmath389 for sufficiently large @xmath43 , since @xmath250 and @xmath390 . thus @xmath391 is monotone on the left half of @xmath392 . this fact and ( [ pm09 ] ) imply @xmath393 for sufficiently large @xmath43 , for some positive constant @xmath74 independent of @xmath43 . moreover , ( [ pm08 ] ) , ( [ pm10 ] ) and ( [ inner_lambda2 ] ) imply @xmath394 is also uniformly bounded from below on @xmath395 . then lemma [ lemma_stein ] implies @xmath396 and @xmath397 is @xmath87 , uniformly with respect to @xmath43 . moreover , for @xmath398 , we have @xmath399 , and ( [ formula2 ] ) and ( [ pm10 ] ) imply @xmath400 for sufficiently large @xmath43 , where @xmath74 is a constant independent of @xmath43 . thus lemma [ lemma_stein ] implies @xmath401 and the conclusion holds . we conclude the paper by arguing the summability of the bandwise estimates . if we fix @xmath88 , @xmath89 and take the limit @xmath93 , then @xmath402 for sufficiently large @xmath43 , by ( [ pm04 ] ) ( see also figure [ figure_lambda1 ] ) . so there always exists the stationary phase point @xmath46 ( the solution to @xmath47 ) nearby @xmath403 , for sufficiently large @xmath43 . for simplicity , assume @xmath77 in the sequel . when @xmath206 , we have by the formulas ( [ der1])-([der6 ] ) , ( [ dkformula2 ] ) , ( [ d1k ] ) and ( [ d2k ] ) @xmath404 thus , even if we cut out a small interval @xmath70 around @xmath206 , the bound for the integral over @xmath70 is not better than @xmath91 , which is not summable with respect to @xmath43 . however , we already know the sum @xmath405 converges in the strong operator topology in @xmath406 . thus it seems that the sum of the integral kernels converges only conditionally . we have to analyze the cancellation between the integral kernels for two adjacent bands more carefully , in order to obtain a better estimate . we hope to argue this subject in the future .
|
in this paper we consider the 1d schrdinger operator @xmath0 with periodic point interactions .
we show an @xmath1 bound for the time evolution operator @xmath2 restricted to each energy band with decay order @xmath3 as @xmath4 , which comes from some kind of resonant state .
the order @xmath3 is optimal for our model .
we also give an asymptotic bound for the coefficient in the high energy limit .
for the proof , we give an asymptotic analysis for the band functions and the bloch waves in the high energy limit .
especially we give the asymptotics for the inflection points in the graphs of band functions , which is crucial for the asymptotics of the coefficient in our estimate .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the decays of heavy hadrons have recently received much attention in the literature @xcite . from the experimental standpoint , these decays allow access to some of the fundamental parameters of the standard model , such as the elements of the cabibbo - kobayashi - maskawa ( ckm ) matrix . questions of cp violation , heavy - flavor oscillations and many others have added to this interest . from the theoretical standpoint , these processes , and the heavy hadrons themselves , allow various quark models of qcd , as well effective theories , to be tested . in particular , the heavy quark effective theory ( hqet ) has both been tested by experimental observations , and has played a major role in the extraction of @xmath6 from experimental data @xcite . much of the success of hqet has been in the treatment of decays from one heavy flavor to another , namely @xmath7 transitions . the effective theory is more limited in scope when applied to heavy - to - light transitions , such as @xmath8 or @xmath9 . neverthless , as we will outline in a later section , the scaling behavior of the form factors that describe various weak decays can be deduced @xcite . this , in principle , allows the form factors for @xmath10 transitions to be inferred from those for @xmath8 . in this article , we assume the validity of the heavy quark symmetry and examine the decays @xmath0 , @xmath1 , and the recently measured @xmath11 in a number of scenarios . in particular , we seek a scenario in which all of these decays are adequately described by a single set of form factors . a number of authors have performed similar analyses @xcite , using the decays @xmath12 and @xmath1 , with varying results . once we find a scenario that is satisfactory for the decays mentioned above , we examine , briefly , the decays @xmath3 . in this way , we hope to make reliable estimates of the absolute decay rates for these processes . in a later article , we will consider more details of these decays , such as forward - backward asymmetries . these decays , as well as the decay @xmath2 , are particularly interesting as the short distance operators responsible arise first at the one - loop - level , and are therefore sensitive to new physics beyond the standard model @xcite . however , in order for any effects due to such new physics to be clearly identified , the long distance contributions that arise in the hadronic matrix elements must be well understood . as witness to this , we point out that the inclusive rate @xmath13 is reasonably well understood , while the exclusive rate @xmath2 has been predicted to be anywhere from 2% to 40% of this inclusive rate @xcite . in the case of the nonleptonic and the rare decays , there arises the crucial issue of the form factors describing the @xmath14 transition . such form factors may be estimated in various models , from qcd sum rules , or by applying the scaling relations predicted by hqet to the form factors for the corresponding @xmath15 transition . the method suggested by hqet , while clearly model - independent , is itself somewhat problematic , as one must take into account the fact that predicting @xmath16 , for instance , requires that these relations be carefully extrapolated beyond the kinematic range accessible in the @xmath15 transitions . this is because the maximum @xmath17 in the @xmath15 transitions is 1.95 ( 0.95 ) gev@xmath18 , while the @xmath17 appropriate to the @xmath19 transition is 9.6 gev@xmath18 . the hqet symmetry predictions relate form factors at the same value of the kinematic variable @xmath20 ( or @xmath21 ) , not @xmath17 , where @xmath22 and @xmath23 are the four - velocities of the parent and daughter hadron , respectively . in this case , the extrapolation is from @xmath20 = 2.1 in @xmath24 to @xmath20 = 3.6 in @xmath25 . the corresponding numbers for the @xmath26 decays are 1.3 and 2.0 . the question of extrapolation also applies to the rare decays , and is particularly important for the decay @xmath2 , for which @xmath17 = 0 , but @xmath20 = 3.0 . for the nonleptonic decays , a second issue is that of the factorization approximation , which is commonly used to calculate the hadronic matrix elements required . this approximation is not very well founded in general theoretically , yet it appears to work well phenomenologically in the @xmath27 decays where it has been tested . nevertheless , application of the form factors of some model or effective theory to the decay @xmath25 , in conjunction with the factorization approximation , serves to probe both issues , and may fail due either to inadequate choice of form factors , failure of the factorization approximation , or both . the rest of this article is organized as follows . in the next section we describe briefly the effective hamiltonians and the appropriate hadronic matrix elements for the processes of interest . in section iii we use hqet to obtain relations among the form factors of interest . in section iv we discuss our fitting procedure , and present the results for the decays that we are considering . in section v we discuss possible limitations of our results , and suggest questions that may be of interest to both theorists and experimentalists . of the three processes we discuss , the semileptonic decays are perhaps the simplest to treat theoretically . the effective hamiltonian for these decays is @xmath28 the hadronic matrix elements for the decays @xmath0 are @xmath29 these decays are thus described in terms of six independent , _ a priori _ unknown form factors . the terms in @xmath30 and @xmath31 are unimportant when the lepton mass is ignored , since @xmath32 in experimental analyses , the form factors @xmath33 , @xmath34 , @xmath35 and @xmath36 are usually assumed to have the form @xmath37 where @xmath38 , and @xmath39 is a mass , usually taken to be that of the nearest resonance with the appropriate quantum numbers . to date , only the mass appropriate to @xmath33 has been measured , and its value measured by cleo is @xmath40 gev @xcite . the masses appropriate to @xmath34 , @xmath35 and @xmath36 are assumed to be 2.5 gev , 2.5 gev and 2.1 gev respectively . the @xmath41 have the values @xmath42 , @xmath43 , @xmath44 and @xmath45 , for @xmath33 , @xmath34 , @xmath36 and @xmath35 , respectively @xcite . neglecting penguin contributions , the effective hamiltonian for the nonleptonic decays of interest here is @xmath46,\end{aligned}\ ] ] where @xmath47,\nonumber\\ c_2 ( m_b ) & = & \frac{1}{2 } \left [ \left ( \frac { \alpha_s ( m_b ) } { \alpha_s ( m_w ) } \right)^{-6/23 } - \left ( \frac { \alpha_s ( m_b ) } { \alpha_s ( m_w ) } \right)^{12/23 } \right].\end{aligned}\ ] ] here , @xmath48 is the mass of the @xmath49 boson , and @xmath50 is the running coupling of qcd . this effective hamiltonian mediates two classes of @xmath27 nonleptonic decays . the first class contains a @xmath51 in the final state : @xmath52 where @xmath53 may be @xmath54 . the second class contains a light meson in the final state : @xmath55 ` @xmath56 ' @xmath57 where @xmath57 is now a charmonium state . to evaluate the matrix elements of the effective hamiltonian we employ the factorization assumption . by fierz rearrangement we rewrite the effective hamiltonian in a form which is suitable for use with this assumption . both terms of the effective hamiltonian contribute , in general , but for the decays in which we are interested , only the second term is of interest , and it may be written @xmath58 where @xmath59 is the number of colors . at this point , it has become customary to replace the coupling coefficient by a phenomenological constant @xmath60 , whose absolute value is measured to be approximately 0.24 . the sign of @xmath60 is not important for our discussion at this point . it will become important if long distance contributions to the dileptonic rare decays are included . for the decay @xmath61 , we therefore write , after using factorization , @xmath62 the hadronic matrix elements @xmath63 are analogs of those of the previous subsection , so that the form factors required for these matrix elements may , in principle , be obtained from the ` semileptonic ' decays of @xmath27 mesons into kaons . such decays do not take place in the standard model , but we can invoke heavy quark symmetries to relate the form factors needed to those for the semileptonic decays of @xmath51 mesons to kaons . the remaining matrix element is @xmath64 the decay constant @xmath65 can be obtained from the leptonic width of the appropriate charmonium vector resonance as @xmath66 where we have ignored lepton masses . in this way , we find @xmath67=0.382 gev , and @xmath68=0.302 gev . in the standard model , the effective hamiltonian for the decay @xmath69 is @xcite @xmath70b f^{\mu\nu},\ ] ] where @xmath71 is the electromagnetic field strength tensor , and the term in @xmath72 may be safely ignored . for the decay @xmath73 , the corresponding effective hamiltonian is @xmath74.\end{aligned}\ ] ] the wilson coefficients @xmath75 are as in the article by buras _ _ @xcite . for the discussion at hand , we have ignored contributions that arise from closed @xmath76 loops , although these may be easily included . the only new hadronic matrix elements that arise in the rare decays of interest are @xmath77 , \nonumber\\ \left < k^*(p^\prime,\epsilon)\left|\bar s\sigma_{\mu\nu } b\right|b(p ) \right>&=&\epsilon_{\mu\nu\alpha\beta}\left[g_+\epsilon^{*\alpha } \left(p+p^\prime\right)^\beta + g_-\epsilon^{*\alpha}\left(p - p^\prime\right)^\beta\right.\nonumber\\ & & \left.+h\epsilon^*\cdot p\left(p+p^\prime\right)^\alpha\left(p - p^ \prime\right)^\beta\right],\end{aligned}\ ] ] introducing four new form factors . due to the relation @xmath78 we can easily relate the matrix elements above to those in which the current is @xmath79 . experimentally , nothing is known about the form factors @xmath80 , @xmath81 nor @xmath82 . using the dirac matrix representation of heavy mesons , one may treat heavy - to - light transitions using the same trace formalism that has been applied to heavy - to - heavy transitions @xcite . in the effective theory , a @xmath51 meson traveling with velocity @xmath22 is represented as @xcite @xmath83 with an identical representation for a @xmath27 meson . the meson states of the effective theory are normalized so that @xmath84 the states of qcd and hqet are therefore related by @xmath85 in all that follows , we will represent the states of full qcd as @xmath86 , and the states of hqet as @xmath87 . let us consider transitions between such a heavy meson ( @xmath51 meson ) and a light meson ( kaon ) of spin @xmath88 , through a generic flavor changing current . in the effective theory , the matrix element of interest is @xmath89 @xmath90 is an arbitrary combination of dirac @xmath91 matrices , and @xmath92 is the fully symmetric , traceless , transverse , @xmath88-index tensor that represents the polarization of the state with spin @xmath88 . the matrix @xmath93 must be the most general that can be constructed from the kinematic variables available , and dirac @xmath91 matrices . the most general form for this is @xcite @xmath94\left(\matrix{1\cr\gamma_5}\right).\end{aligned}\ ] ] the @xmath95 s are uncalculable , nonperturbative functions of the kinematic variable @xmath96 , and the 1 ( @xmath97 ) is present if the resonance @xmath98 has natural ( unnatural ) parity . from this point on , let us limit the discussion to only two of the kaon resonances , namely the ground - state pseudoscalar kaon itself , and its vector counterpart , @xmath26 . the above then leads to @xmath99\gamma m_d(v)\right\}.\end{aligned}\ ] ] at this point , let us emphasize that the form factors @xmath95 are independent of the form of @xmath90 , and so are valid for decays through the left handed current ( @xmath100 ) , as well as for rare decays ( @xmath101 ) . these form factors are also independent of the mass of the heavy quark , and are therefore universal functions . thus , they are valid for @xmath102 decays , as well as for @xmath103 decays . this independence of the quark mass allows us to deduce , in a relatively straightforward manner , the scaling behavior of the usual form factors that describe these transitions @xcite . we illustrate this by examining one matrix element in detail . consider @xmath104 from these equations , one finds that @xmath105 since the @xmath95 do not scale with the mass of the heavy quark ( or meson ) , it is trivial to deduce that @xmath106 for the other transitions of interest , the form factors are as defined in the previous section , and the relationships between these and the @xmath95 are @xmath107 these expressions yield the additional scaling behavior @xmath108 eqns . ( [ formsgeneral ] ) and ( [ ff1 ] ) , and consequently eqns . ( [ formsscaling]-[xitof ] ) , contain all of the leading order @xmath109 dependence in the form factors , and are valid irrespective of the mass of the strange quark . if the strange quark could be treated as heavy , then we could think of the @xmath95 as arising from an infinite sum of terms in the @xmath110 expansion . the leading order forms ( in @xmath110 ) are also contained in these expressions . these expressions are also valid in the limit of a very light strange quark . isgur and wise @xcite have used the scaling of @xmath111 ( @xmath112 ) to say that this combination of form factors vanishes ( at leading order in @xmath113 ) , and suggest that one can write @xmath114 . implicit in this argument is the assumption that the strange quark is very light . in our formalism , this amounts to setting @xmath115 to zero , and we would automatically lose the full scope of our predictions . this is because we could then never recover the limit of a heavy strange quark , for in this limit , @xmath116 , where @xmath117 is the usual isgur - wise function . the strange quark is such that it may be treated as either heavy or light . we believe that neither the full heavy @xmath80 limit ( @xmath116 ) , nor the limit of a very light @xmath80 quark ( @xmath118 ) is completely satisfactory . we assume neither limit in our analysis , and therefore make full use of the predictions of the heavy quark effective theory , which are valid independent of the mass of the strange quark . this means that we retain the form factor @xmath115 in our discussion and treat it as a completely independent form factor , tying it to neither of the two limits discussed . at the risk of boring the overly patient reader to tears , and perhaps even to death , we list one more set of relationships among form factors , this time writing the usual form factors in terms of the @xmath95 . the relations are @xmath119,\nonumber\\ a_-&=&-\frac{1}{m_d^{3/2}}\left[\xi_3-m_d\left(\xi_6-\xi_4\right ) \right],\nonumber\\ g&=&\frac{1}{\sqrt{m_d}}\xi_6.\end{aligned}\ ] ] for the corresponding transitions with a @xmath120 quark ( and a @xmath27 meson ) in the initial state , all factors of @xmath109 above must be replaced by @xmath121 . using this , rearrangement of eqns . ( [ ff2 ] ) yields @xmath122 , \nonumber\\ f_-^b(v\cdot p)&=&\frac{1}{2}\left(\frac{m_b}{m_d}\right)^{1/2 } \left[f_-^d(v\cdot p)\left(1+\frac{m_d}{m_b}\right ) + f_+^d(v\cdot p)\left(\frac{m_d}{m_b}-1\right)\right ] , \nonumber\\ f^b(v\cdot p)&=&\left(\frac{m_b}{m_d}\right)^{1/2}f^d(v\cdot p ) , \nonumber\\ g^b(v\cdot p)&=&\left(\frac{m_d}{m_b}\right)^{1/2}g^d(v\cdot p ) , \nonumber\\ a_+^b(v\cdot p)&=&\frac{1}{2}\left(\frac{m_d}{m_b}\right)^{1/2 } \left[a_+^d(v\cdot p)\left(1+\frac{m_d}{m_b}\right ) + a_-^d(v\cdot p)\left(\frac{m_d}{m_b}-1\right)\right ] , \nonumber\\ a_-^b(v\cdot p)&=&\frac{1}{2}\left(\frac{m_d}{m_b}\right)^{1/2 } \left[a_-^d(v\cdot p)\left(1+\frac{m_d}{m_b}\right ) + a_+^d(v\cdot p)\left(\frac{m_d}{m_b}-1\right)\right],\end{aligned}\ ] ] where @xmath123 is the form factor appropriate to the @xmath102 transition , while @xmath124 is the form factor appropriate to the @xmath125 transition , and quantities on the left - hand - sides of eqns . ( [ ffmix ] ) are evaluated at the same values of @xmath96 as those on the right - hand - sides . omitted from each of eqn . ( [ ffmix ] ) is a qcd scaling factor , discussed below . ( [ ffmix ] ) illustrate two effects , namely the scaling of form factors in going from the @xmath51 system to the @xmath27 system , as well as the mixing of @xmath34 with @xmath31 , and @xmath33 with @xmath30 . the effect of this mixing is very important in going from @xmath51 transitions to @xmath27 transitions , as it introduces form factors that have not yet been measured experimentally , or to which experiments are not yet sensitive , namely @xmath30 and @xmath31 . in the rates for the semileptonic decays @xmath12 , terms dependent on @xmath30 and @xmath31 are proportional to the mass of the lepton , and thus play a miniscule role , except near @xmath126 . such terms may also be significant if the polarization of the charged lepton is measured . we close this section with a brief discussion of radiative corrections to the currents and matrix elements that we have discussed . in the limit of a heavy @xmath120 quark , the full current of qcd is replaced by @xcite @xmath127^{-\frac{6}{25}}.\ ] ] this arises from integrating out the @xmath120 quark , and matching the resulting effective theory onto full qcd at the scale @xmath128 , at one loop level . at the scale @xmath129 , we must also integrate out the @xmath130 quark , but there is also the effect due to running between @xmath128 and @xmath129 . the net effect of this is that the form factors @xmath95 appropriate to the @xmath10 transitions are related to those for the @xmath8 transitions by @xmath131^{-\frac{6}{25}}.\ ] ] the forms of the matrix elements that we discuss above are valid in the limit of infinitely heavy @xmath120 and @xmath130 quarks . for quarks of finite mass , there are clearly going to be corrections to the relations we have obtained . in other words , new form factors that appear first at order @xmath132 and @xmath133 will begin to make contributions . it is expected that such contributions will become more significant away from the ` non - recoil ' point , or for @xmath134 . this is particularly important for the @xmath14 decays , as @xmath96 can become very large . nevertheless , we will apply the relations we have found through all of the available phase space . it is our hope that by doing this , we will at least be able to indicate the suitability of hqet for these transitions . however , since we fit the form factors rather than attempt to calculate them in some model , some of these higher order effects may , in fact , have been included . all of the results we describe are for decays with @xmath135 ( or @xmath136 ) in the final state . the treatment of charged kaons would be identical , and we believe that our results in these channels would be similar in quality to those we obtain for neutral kaons . before describing the results of our fits , we must comment on how we treat the available data , particularly in the case of the semileptonic decays . very few of the experimental collaborations have extracted acceptance - corrected distributions @xcite . instead , the form factors are all assumed to be of the monopole form @xcite , and the parameters are then extracted from the monte carlo simulations , with acceptances and efficiencies folded in . because of this , we proceed in the inverse sense to generate some ` simulated data ' . we use the published monopole parameters for the form factors to generate @xmath137 spectra for the semileptonic decays , using the published uncertainties in the monopole parameters to generate uncertainties in the simulated data . in general , these errors are correlated , but we ignore this correlation . the simulated data generated in this way are completely smooth . we introduce an ` anti - smoothing ' by smearing the simulated data with a pseudo - randomly generated gaussian distribution of mean zero and standard deviation determined by the errors in the unsmeared simulated data . it is this smeared simulated data that we use for fitting . for the decays @xmath138 , we also include the ratios @xmath139 and @xmath140 in the fit , where @xmath141 are as defined in pdg @xcite . in addition , we must point out that the measured decay rates for @xmath142 are somewhat smaller than those for @xmath143 , while the published form factor parametrizations are for the average over charge states . to account for this , we rescale the values of the @xmath41 to correspond to the smaller rates for neutral kaons . it is these rescaled values that are cited in section [ sldecays ] , and that we use to generate the simulated data . for the nonleptonic decays , we fit to the pdg averaged decay widths for @xmath144 and @xmath145 . in the case of the latter , we also include the ratio @xmath139 . for this ratio , we take the averaged value of @xmath146 as calculated by gourdin _ masses and lifetimes of mesons are all taken from pdg @xcite , and we use @xmath147 , @xmath148 , @xmath149 , @xmath150 , @xmath151 gev , @xmath129=1.5 gev , @xmath152=177 gev . it is worth mentioning that the experimental choice of monopole form factors may be inappropriate , particularly for @xmath35 . in the limit of a heavy strange quark , one finds that @xmath153 , where @xmath117 is the isgur - wise function . if @xmath154 is assumed to be monopolar in @xmath17 , then simple pole dependence for @xmath35 is inappropriate . even in the case of a light strange quark , we find that @xmath155 , again indicating that the dependence on the kinematic variable is not simply a monopole form . however , for the range of @xmath17 accessible in @xmath156 decays , the decay rate is not sensitive to the kinematic dependence . the effect on the decay rate for the nonleptonic and rare decays being considered here , however , are quite significant . in our fitting , we have separated the decays containing a @xmath56 meson in the final state from those containing a @xmath26 meson in the final state . thus , for instance , we do not include data on ratios of rates like @xmath157 . our reason is that such ratios introduce correlations between the parameters of the form factors for the two sets of decays . one approximation used recently in the literature has been to treat the strange quark as heavy @xcite , so that the decays of interest can be treated in the heavy - to - heavy limit . in this limit , the form factors of eqn . ( [ ff1 ] ) may be written @xmath158 where @xmath117 is the isgur - wise function for heavy - to heavy transitions , and in this limit , @xmath159 . in particular , this limit means that @xmath160 . we have used this form in fitting the data , and have obtained reasonable fits to the differential decay rates in the semileptonic decays , as well as to the total decay rates in the nonleptonic decays . polarization ratios , however , are poorly reproduced . in the case of the nonleptonic decay @xmath161 , the ratio @xmath139 depends only on kinematics , and has a value of 0.43 , independent of the form chosen for @xmath117 . the experimental value is @xmath162 . in addition , the ratio @xmath140 in @xmath163 always has a value of about 0.4 , significantly different from the experimental value of [email protected] , and largely independent of the form chosen for @xmath154 . this indicates that the value of 0.43 obtained for the ratio @xmath139 in @xmath161 is not necessarily due to the breakdown of factorization in the heavy @xmath80 limit , as this limit does not even provide an adequate description of all measurements in the semileptonic decays . relaxing the strict heavy-@xmath80 limit , but constraining the form factors to be near this limit , does not help much , as the polarization observables are still not well reproduced . the conclusion that we draw from this is that the heavy-@xmath80 limit may give an acceptable description of unpolarized data , but may be dangerous when applied to polarization observables . all of the results we present are obtained by performing four kinds of fits , namely ( 1 ) include the semileptonic decays @xmath165 only ; ( 1 ) include the semileptonic decays as well as the nonleptonic decays @xmath16 ; ( 3 ) include the semileptonic decays , the nonleptonic decays @xmath16 , and the nonleptonic decays @xmath166 ; ( 4 ) include the measured decay rate for @xmath11 in the fit , as well as the three other decays . clearly , in the case of the decays to @xmath56 mesons , we need perform only three kinds of fits . in cases where a measured quantity is not included in the fit , we calculate that quantity using the fit parameters . we have explored two sets of parametrizations of the form factors . in the first scenario , which we refer to as the exponential scenario , each @xmath95 has one of the forms @xmath167}=a_i\exp{\left[-\frac{b_i}{2m_d}\left(q^2_{{\rm max } } - -q^2\right)\right]}\label{exp1},\\ \xi_i&=&a_i\exp{\left[-b_i\left(v\cdot p - m_{k^{(*)}}\right)^2 \right]}=a_i\exp{\left[-\frac{b_i}{4m_d^2}\left(q^2_{{\rm max}}-q^2 \right)^2\right ] } \label{exp2},\\ \xi_i&=&a_i\exp{\left[-b_i\left(v\cdot p\right)^2\right]}\label{exp3},\end{aligned}\ ] ] while in the second scenario , which we call the multipolar scenario , the forms chosen are @xmath168 with @xmath169 . in the exponential scenario , eqn . ( [ exp1 ] ) most closely corresponds to the forms that arise in some quark models , most notably that of isgur and collaborators @xcite . however , in such models , the exponential of eqn . ( [ exp1 ] ) is usually multiplied by a polynomial in @xmath96 , or equivalently , in @xmath17 . in the multipolar scenario , @xmath170 corresponds to a dipole form factor , while @xmath171 represents a monopole . in any one fit , we do not choose all the form factors to have the same form . this means that , for instance , in the case of the decays to @xmath56 mesons , for which there are two form factors , the second scenario corresponds to sixteen different possible combinations of forms for @xmath115 and @xmath172 . in each case , the @xmath173 and @xmath174 are the free parameters to be varied in the fit . there are therefore twelve free parameters in each fit , to be compared with five extracted and three assumed parameters in the measured semileptonic decays . since we have more free parameters than are extracted from the experimental data , one might expect that it should be very easy to account for all of the data . in fact , the number of free parameters poses some problems , as it means that the problem is not very well constrained . one consequence of this is that there appear to be several local minima for any particular choice of form factors . nevertheless , there are some combinations of form factors that simply do not provide adequate descriptions of the data , despite the large number of free parameters . when only the semileptonic decays are included in the fit , we find that almost any combination of forms for the form factors leads to reasonable results . the few combinations that do not provide good descriptions fail only in their description of the polarization observables . when we use the exponential forms , we find that we are not able to obtain an adequate description of all of the data simultaneously . in particular , when we include the nonleptonic decays in the fit , the prediction for the rare decay @xmath175 is significantly different from the measured rate . in some cases , however , we find that if we fit the semileptonic decays alone , omitting all of the nonleptonic decays , the prediction for the rare decays is of the right order of magnitude . one possible conclusion here is that factorization is not applicable to these nonleptonic decays , or that there are significant non - factorizable contributions to the amplitude . in contrast with the exponentials of the first scenario , many combinations of the ` multipolar ' forms of the second scenario lead to good descriptions of all the data simultaneously . one outstanding feature of all of our results in this scenario ( which also exists in the exponential scenario , but to a lesser extent ) can be easily understood by examining eqn . ( [ ff2 ] ) , where it is seen that @xmath176 is present in all of the form factors that describe the semileptonic decays @xmath177 . it is therefore not at all surprising that our results are most sensitive to this form factor . invariably , we have found that the best fits occur for @xmath178 linear in @xmath96 ( _ i.e. _ , @xmath179 ) , independent of the forms chosen for the other form factors . furthermore , the slope parameter @xmath180 is almost always negative , with values lying between -0.4 and -0.65 gev@xmath181 . the only positive values for @xmath180 occur when only the semileptonic decays are included . the fact that @xmath176 is so well constrained in our fits means that @xmath35 and @xmath36 are also quite well constrained . since these are the only two form factors that are needed for the decay @xmath2 , it is no surprise that we find little variation in the predictions for this decay rate as we vary the parametrizations of @xmath182 , @xmath183 and @xmath184 , provided that the choice of parametrization of @xmath176 is unchanged . for other forms of @xmath176 , we find that at most two of the semileptonic , nonleptonic or rare decays are well accomodated . for instance , if we choose a monopole form for @xmath176 , then in addition to the semileptonic decays , we find that we can accomodate either the nonleptonic decays , the rare decay , but not both . in addition , if we choose all form factors to be monopolar , we fail to find adequate descriptions of the polarization observables , particularly for the ratio @xmath185 measured in the nonleptonic decay . our results for the form factors are comparable with those of other authors . @xcite have found that a number of scenarios , including monopolar form factors , are unable to describe the nonleptonic measurements . _ @xcite have found that softening the scaling relations allows an adequate description , while in a second article , gourdin _ et al . _ @xcite have found that allowing the form factor @xmath35 to decrease linearly with @xmath17 allows an adequate description of the data . we find that @xmath35 is quadratic in @xmath17 , but the absolute value at @xmath186 for the @xmath187 transitions is larger than the absolute value at @xmath188 , in keeping with the results of @xcite , and quite different from pole models . for each scenario , we have selected a set of fits that we consider to be representative . the parameters for these fits are displayed in table [ fitparame ] for the exponential forms , and in table [ fitparam ] for the multipolar forms . in the first scenario , the results we have selected correspond to @xmath172 and @xmath182 as in eqn . ( [ exp1 ] ) , @xmath115 and @xmath183 as in eqn . ( [ exp2 ] ) , and @xmath184 and @xmath176 as in eqn . ( [ exp3 ] ) . in the second scenario , the @xmath189 have the values @xmath190 . .values of the parameters that result from four different fits , for the exponential scenario . in this table , fit 1 means that only @xmath0 is included in the fit ; fit 2 means @xmath0 and @xmath191 are included ; fit 3 means @xmath0 , @xmath191 and @xmath192 are included ; fit 4 means @xmath163 , @xmath193 , @xmath194 and @xmath2 are all included , and applies only to decays with @xmath26 s in the final state . [ fitparame ] [ cols="^,^,^,^,^",options="header " , ] given the failure of the exponential scenario to explain the @xmath195 data , one might be tempted to discard its predictions for the dileptonic decays . however , we see that the integrated rates are , for the most part , quite similar to those predicted in the multipolar scenario . the spectra that result from the two scenarios are very different , however , and the predictions for the relative amount of longitudinally polarized and transversely polarized @xmath26 produced are also somewhat different . the exponential forms predict @xmath196 , while in the multipolar scenario , the ratio ranges between 0.15 and 0.24 . the moral here may be that the exponential forms may be adequate for predicting total rates , but not decay spectra , nor polarization observables . we limit our discussion to the multipolar scenario in what follows . 0.25 in 0.25 in the predictions for the process @xmath197 are two to three orders of magnitude smaller than present experimental upper limits , while those for @xmath198 are smaller than the experimental limits by factors of four to six . we point out , however , that the calculated rates for these last two processes do not include possible contributions from charmonium resonances , which will certainly alter the shape of the lepton spectrum , and should also increase the total decay rate . an investigation of this effect will be left for a future article . however , in at least one experimental analysis , kinematic cuts are imposed on the total mass of the lepton pair , so that events that may arise from either of the first two vector charmonium resonances are excluded @xcite . in any case , our results suggest that the exclusive dileptonic decay to the @xmath26 should be observed in the near future . the results that we have obtained here again illustrate that the fit to the present @xmath15 semileptonic spectra alone is inadequate for providing information on @xmath187 processes . even when we include the nonleptonic decays , the predictions for different decay modes ( particularly @xmath199 with longitudinally polarized @xmath26 s ) are sensitive to the nonleptonic modes we include in the fit . if any of these predictions are to be taken seriously , we would suggest that most attention be paid to the predictions of fit 4 in the multipolar scenario , as this is the only scenario that adequately describes all of the data available . one of the features of the predicted spectra are the minima in the differential decay rates . these minima are the result of zeroes in the respective helicity amplitudes , and the question of whether or not these zeroes do indeed exist , and of their exact locations , will have to await a @xmath27-factory . however , long - distance effects , such as those that arise from charmonium resonances , or even from the charmonium continuum , will at least alter the positions of the zeroes , and may wash out the effect altogether . we have used the scaling predictions of hqet , together with the most recent data on @xmath0 semileptonic decays , to extract the form factors that describe the @xmath15 and @xmath14 processes . the latter we have applied to other processes , namely the nonleptonic decays @xmath200 and @xmath192 , as well as the rare decays @xmath11 and @xmath3 . in the case of the nonleptonic decays , we have assumed factorization of the transition amplitude is valid . we have also performed simultaneous fits of the semileptonic , nonleptonic and rare processes , and have found that hqet , together with factorization , provide an adequate framework for describing the observations . our predictions for the modes @xmath3 suggest that these should be measurable in the next generation of experiments , and certainly at the proposed @xmath27 factory . perhaps the greatest shortcoming of our fit procedure lies in how we handle the data , or simulated data , for the semileptonic decays . ideally , we should have attempted to fit our choices of form factors to the experimentally measured differential decay rates . as a second choice , our choices of form factors should have been input into the experimental monte - carlo programs to obtain the fit parameters . in any case , it is clear that the present data , particularly in the @xmath163 mode , are inadequate to sufficiently constrain the form factors . in addition , the differences in form factors between fit 1 and fits 2 , 3 and 4 are quite striking . the scenario that best describes all of the experimental data is the multipolar one , and in this scenario , we find that the universal form factor @xmath176 is linear in @xmath96 . using this scenario , we predict @xmath201 and @xmath202 . these numbers are consistent with other model calculations @xcite . we also predict @xmath203 in @xmath204 to be @xmath205 . to fully constrain the predictions of hqet , information on the form factors @xmath31 and @xmath30 is needed from the semileptonic decays . such information can only be obtained from high - precision measurements of the decay spectra at low values of @xmath17 , particularly for semileptonic decays to muons , as well as by measuring the polarization of the charged lepton , again preferably the muon . in addition , the precision and statistics in the @xmath17 spectrum must be improved so that the form factor parameters can be extracted from the data , particularly for the decays @xmath163 . perhaps the ideal experiment would be the equivalent of present cleo experiments , in which the machine is tuned to be a source of @xmath206 pairs , produced from the strong decays of the @xmath207 . for @xmath51 decays , the equivalent would be to produce a copious number of @xmath208 s , which can be realized at the proposed tau - charm factory . we gratefully acknowledge helpful conversations with n. isgur , particularly for some very useful discussions on scaling relations . we also thank c. carlson , a. freyberger , j. goity and k. protasov for discussions . w. r. acknowledges the support of the national science foundation under grant phy 9457892 , and the u. s. department of energy under contracts de - ac05 - 84er40150 and de fg05 - 94er40832 . w. r. also acknowledges the hospitality and support of institut des sciences nuclaires , grenoble , france , where much of this work was done , and of centre international des etudiants et stagiaires .
|
we examine the measured rates for the decays @xmath0 , @xmath1 and @xmath2 in a number of scenarios , in the framework of the heavy quark effective theory .
we attempt to find a scenario in which all of these decays are described by a single set of form factors .
once such a scenario is found , we make predictions for the rare decays @xmath3 .
while we find that many scenarios can provide adequate descriptions of all the data , somewhat surprisingly , we observe that two popular choices of form factors , namely monopolar forms and exponential forms , exhibit some shortcomings , especially when confronted with polarization observables .
we predict @xmath4 and @xmath5 .
we also make predictions for polarization observables in these decays .
= 10000 /#1#1 -0.5em /
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the melting curves of transition metals at pressures up to the megabar region are highly controversial , particularly for b.c.c diamond anvil cell ( dac ) measurements find that the melting temperature @xmath3 increases by only a few hundred k over the range 1 - 100 gpa @xcite , while shock experiments indicate an increase of several thousand k over this range @xcite . there have been several _ ab initio _ calculations on transition - metal @xmath3(p ) curves , and the predictions agree more closely with the shock data than with the dac data @xcite . a challenging case is molybdenum , where there are very large differences between dac and shock data @xcite , and where the shock data reveal _ two _ transitions , the one at high pressure ( @xmath4 380 gpa ) being attributed to melting , and the one at low pressure ( @xmath4 210 gpa ) to a transition from b.c.c . to an unidentified structure @xcite . we report here on new _ ab initio _ calculations of @xmath3(@xmath5 ) for mo , and on the @xmath1 and @xmath2 relations on the hugoniot . we also report preliminary information that may help in searching for the unidentified high - p solid phase of mo . we use density functional theory ( dft ) , which gives very accurate predictions for many properties of transition metals , including hugoniot curves @xcite . dft molecular dynamics ( m.d . ) was first used to study solid - liquid equilibrium 12 years ago @xcite , and several different techniques are now available for using it to calculate melting curves . in such calculations , no empirical model is used to describe the interactions between the atoms , but instead the full electronic structure , and hence the total energy and the forces on the atoms , is recalculated at each time step . there have been earlier dft calculations on the melting of mo , but the techniques used were prone to superheating effects @xcite . in the present work , we use the so - called `` reference coexistence '' techniques @xcite , which does not suffer from this problem . our work has several aims . first , we want to improve on the reliability and accuracy of the predicted melting curve of mo obtained from dft ; second , we use dft to predict the @xmath1 and @xmath2 relations on the shock hugoniot ; third , we want to identify the unknown solid phase of mo observed in shock experiments . our tests on the accuracy of dft for mo , and our extensive calculations of the mo melting curve will be reported in detail elsewhere @xcite , so we present only a summary here . however , our very recent calculations on the shock hugoniot will be presented in more detail . these are important , because temperature is very difficult to measure in shock experiments @xcite and dft gives a way of supplying what is missing in the shock data . our search for the unidentified solid structure of mo is at the exploratory stage , but we present results for phonon frequencies as a function of pressure , which allow us to rule out some possibilities . in the following , we summarise our tests on the accuracy of the dft techniques ( sec . [ sec : techniques_and_tests ] ) , and outline the reference coexistence technique . in sec . [ sec : results ] we present our results for the dft melting curve and hugoniot of mo up to 400 gpa , and the study of the phonon frequencies . discussion and conclusions are in sec . [ sec : discussion ] . . solid and dashed curves show gga(pbe ) and lda(ca ) fp - lapw results , respectively ; short - dashed and dotted curves show gga(pbe ) and lda(ca ) paw calculations , respectively . solid dots show experimental data @xcite . ] our dft calculations are performed mainly with the projector augmented wave ( paw ) implementation @xcite , using the vasp code @xcite , since paw is known to be very accurate . the main uncontrollable approximation in dft is the form adopted for the exchange - correlation functional @xmath6 . to test the accuracy of paw , and the effect of @xmath6 , we have compared our predictions for the @xmath1 relation of the b.c.c . mo crystal against low-@xmath7 experimental results ( fig . [ fig : wien2kvasp ] ) . the pressure predictions from paw using the perdew - burke - ernzerhof ( pbe ) @xcite and local - density approximation ( lda ) @xcite forms of @xmath6 deviate by @xmath8 % in opposite directions from the experimental data , but we adopt the pbe form , because the deviations in this case are rather constant . the pbe results of fig . [ fig : wien2kvasp ] were obtained with 4s states and below in the core and all other states in the valence set . inclusion of 4s states in the valence set makes no appreciable difference to the pbe results . we also tested the paw implementation itself by repeating the @xmath1 calculations with the even more accurate full - potential linearized augmented plane - wave ( fp - lapw ) technique @xcite , using the wien2k code @xcite . as shown in fig . [ fig : wien2kvasp ] , paw and fp - lapw results are almost identical . further confirmation for the accuracy of the paw implementation and the pbe functional comes from our comparisons of the calculated phonon dispersion relations for the ambient - pressure b.c.c . crystal with experiment ( fig . [ fig : vaspphonon ] ) . the `` reference coexistence '' technique for calculating _ ab initio _ melting curves has been described elsewhere @xcite , but we recall the main steps . first , an empirical reference model is fitted to dft m.d . simulations of the solid and liquid at thermodynamic states close to the expected melting curve . then , the reference model is used to perform simulations on large systems in which solid and liquid coexist , to obtain points on the melting curve of the model . in crucial third stage , differences between the reference and dft total energy functions are used to correct the melting properties of the reference model , to obtain the _ ab initio _ melting curve . in the present work , the total energy function of the reference model is represented by the embedded - atom model ( eam ) @xcite , consisting of a repulsive inverse - power pair potential , and an embedding term describing the d - band bonding . the detailed procedure for fitting @xmath9 to dft m.d . data will be reported elsewhere @xcite , but we note that we needed to re - fit the model in different pressure ranges . the simulations on solid and liquid mo in stable thermodynamic coexistence using the fitted reference model employed systems of 6750 atoms , and we checked that the results do not change if even larger systems are used . the protocols for preparing these simulated systems , and for achieving and monitoring stable coexistence were similar to those used in earlier work on the melting curve of cu @xcite . the procedures for correcting the melting curve of the reference model , which depend on calculations of the the free energy differences between the reference and dft systems , have been described and validated in earlier work @xcite . [ fig : melting ] shows the melting curve of our reference eam model , and the melting curve obtained from this by correcting for the differences between the dft and reference total - energy functions ; earlier _ ab initio_-based calculations of the mo melting curve due to moriarty and to belonoshko _ _ are also indicated @xcite . we also show points on the melting curve from dac and shock measurements @xcite . the differences between reference and corrected melting curves are only a few hundred k , so that the corrected curve should be very close to the melting curve that would be obtained from the ( pbe ) exchange - correlation functional if no statistical - mechanical approximations were made . because we avoid approximations of earlier _ ab initio_-based work , our present results should be a more accurate representation of the dft melting curve . up to 100 gpa , the differences between our results and earlier dft work are rather small , and we confirm that dft gives a much higher melting slope than that given by dac experiments . we obtain an accurate value of @xmath10 at @xmath11 by fitting our melting curve to the simon equation @xmath12 , with @xmath13 k , @xmath14 gpa , @xmath15 . the resulting @xmath11 value of @xmath16 k is close to the accepted experimental value @xmath17 k. our @xmath10 value of @xmath18 k gpa@xmath19 at @xmath11 agrees with an older experimental value of @xmath20 k gpa@xmath19 @xcite . our dft melting curve is consistent with the point obtained at @xmath21 gpa from shock measurements . however , we stress that the temperature of the `` experimental '' point was not measured , but estimated by considering superheating corrections to the shock - wave data @xcite . because of this , it is important to try and corroborate the estimated experimental temperature , and this can be done by _ ab initio _ calculations , as we explain next . _ ab initio _ relation on the hugoniot of mo up to pressure of 400 gpa ; dots reproduce experimental data of refs . _ right _ : @xmath22 _ ab initio _ relation on the hugoniot of mo up to pressure of 400 gpa ; dots reproduce experimental data of refs . @xcite and triangles the same results but corrected for superheating effects as given by ref . error bars show the uncertainties in pressure and temperature.,title="fig : " ] the pressure @xmath23 , volume @xmath24 and internal energy @xmath25 in the shocked state are related to the initial volume @xmath26 and internal energy @xmath27 by the well - known rankine - hugoniot formula @xcite : @xmath28 since the internal energy and pressure are given in terms of the helmholtz free energy @xmath29 by @xmath30 and @xmath31 , we can calculate the hugoniot from our dft simulations , provided we can calculate @xmath29 as a function of @xmath32 and @xmath7 . so far , we have done this only for the b.c.c . crystal in the harmonic approximation , in which @xmath33 . here , @xmath34 is the free energy of the rigid perfect crystal , including thermal electronic excitations , and @xmath35 is calculated from the phonon frequencies @xmath36 ( @xmath37 the wavevector , @xmath38 the branch ) . we calculate @xmath35 in the classical limit , in which @xmath39 per atom , with @xmath40 and @xmath41 is the geometric average of phonon frequencies over the brillouin zone . the methods used for to calculate @xmath42 and the frequencies @xmath36 were similar to those used in our earlier work on fe ( ref . @xcite ) . for a set of temperatures , we calculated @xmath34 at a set of volumes , and fitted the volume dependence with a third - order birch - murnaghan equation @xcite . the temperature dependence of the coefficients in this equation were then fitted with a third - order polynomial . the phonon frequencies were calculated at 12 volumes in the range @xmath43 @xmath44/atom , as explained elsewhere @xcite . the volume dependence of the average @xmath41 was then fitted with a third - order polynomial . to obtain @xmath45 and @xmath46 from our fitted free energy , for each value of @xmath24 we seek the @xmath7 at which the rankine - hugoniot equation is satisfied , and from this we obtain @xmath23 . for @xmath26 and @xmath27 , we used values from our gga calculations ; we checked that use of the experimental @xmath26 made no significant difference . comparison of our calculated @xmath45 with measurements of hixson _ et al . _ @xcite ( left panel of fig . [ fig : hugoniot ] ) shows excellent agreement . in the right panel , we compare our @xmath46 with both uncorrected results of hixson _ et al . _ and also with results corrected for superheating , and our results confirm their temperature estimates . ) , 77 gpa ( @xmath47 ) , 136 gpa ( @xmath48 ) , 274 gpa ( @xmath49 ) and 400 gpa ( @xmath50 ) . ] efforts have been made to identify the high-@xmath5/high-@xmath7 structure of mo indicated by shock experiments @xcite . _ used their theoretical prediction of b.c.c . @xmath51 phase transition at @xmath52 gpa and @xmath53 k to suggest the h.c.p structure . however , later calculations locate this transition at higher pressures ( @xmath54 gpa ) , and temperature stabilisation of h.c.p . over b.c.c . below melting seems improbable . furthermore , there are recent claims that under pressure b.c.c mo transforms first to f.c.c . rather than h.c.p . very recently , belonoshko _ et al . _ @xcite reported _ ab initio _ simulations on the f.c.c and a15 structures at temperatures and pressures near those of interest ( namely , @xmath55 gpa and @xmath56 k ) , and concluded that both structures are unlikely high-@xmath7 phases of mo . we have performed _ ab initio _ calculations similar to those of belonoshko _ et al . _ on the h.c.p . and @xmath57 structures ( @xmath58 gpa and @xmath59 k ) we find that temperature neither favours any of them respect to b.c.c . , thus they must be excluded as well . recently , a 2nd - order phase transition from cubic to rhombohedral has been observed in vanadium at @xmath60 gpa @xcite . this appears to be related to an earlier _ ab initio _ prediction of a phonon softening in v @xcite . this suggests that a similar structural transition might occur in mo . we have used our calculated phonon dispersion relations of mo over the range @xmath0 gpa to test this . [ fig : freqcomp ] depicts our results at 0 , 77 , 136 , 274 and 400 gpa , but we see that no phonon anomaly like that reported for v is observed at any wavevector . this indicates that we should rule out structures based on small distortion of b.c.c . an important outcome of the present work is improved dft calculations of the melting curve of mo over the pressure range @xmath0 gpa . in particular , our techniques allow us to avoid the superheating errors which appear to affect an independent recent dft study of mo melting @xcite . the accuracy of our calculations is confirmed by the very close agreement with experiment for the melting temperature @xmath3 and the melting slope @xmath10 at ambient pressure . the results fully confirm that the increase of @xmath3 by @xmath61 k over the range @xmath62 gpa predicted by dft is about 10 greater than that deduced from dac measurements . a second important outcome is that our calculations of the temperature along the shock hugoniot support earlier temperature estimates based on experimental data but corrected for possible superheating effects @xcite . this allows us to compare more confidently the point on the melting curve at @xmath63 gpa derived from shock data with our predicted melting curve , and we confirm that @xmath3 at this pressure is _ ca . _ 8650 k. this is far above any reasonable extrapolation of the dac data . concerning the search for the unknown crystal structure of mo indicated by shock experiments to exist above _ ca . _ 220 gpa , we have been able so far only to rule out some possibilities . our calculations of the phonon dispersion realations in the b.c.c . structure over the range @xmath0 gpa reveal no softening of any phonons , and no indication of any elastic instability . this means that the new crystal structure does not arise from small distortion of b.c.c . this is interesting in the light of the recent discovery of the elastic instability of b.c.c . v above @xmath64 gpa , predicted initially by dft , and observed very recently in x - ray diffraction experiments @xcite . it seems that the structural transition in mo is of a different kind . the large conflict between the melting curve of mo derived from dac measurements on one side and from shock experiments and dft calculations on the other side must be due either to a misinterpretation of the dac data or to a combination of serious dft errors and misinterpretation of shock data . given the accuracy of dft that we have been able to demonstrate ( low - temperature @xmath1 curve , hugoniot @xmath1 curve , ambient-@xmath5 phonons ) , we think there is little evidence for significant errors in dft , which is also in good accord with shock data . a possible explanation might be that the large temperature gradients and non - hydrostatic stress in dac experiments might give rise to flow of material giving the appearance of melting , even well below the thermodynamic melting temperature . we also note recent evidence that temperature measurement in dac experiments may be subject to previously unsuspected errors @xcite , though probably not of the size needed to resolve the conflict by themselves . the work was supported by epsrc grant ep / c534360 , which is 50% funded by dstl(mod ) , and by nerc grant ne / c51889x/1 . the work was conducted as part of a euryi scheme award to da as provided by epsrc ( see www.esf.org/euryi ) .
|
we report _ ab initio _ calculations of the melting curve and hugoniot of molybdenum for the pressure range @xmath0 gpa , using density functional theory ( dft ) in the projector augmented wave ( paw ) implementation .
we use the `` reference coexistence '' technique to overcome uncertainties inherent in earlier dft calculations of the melting curve of mo .
our calculated melting curve agrees well with experiment at ambient pressure and is consistent with shock data at high pressure , but does not agree with the high pressure melting curve from static compression experiments .
our calculated @xmath1 and @xmath2 hugoniot relations agree well with shock measurements .
we use calculations of phonon dispersion relations as a function of pressure to eliminate some possible interpretations of the solid - solid phase transition observed in shock experiments on mo .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
one - dimensional ( 1d ) integer - spin heisenberg antiferromagnets ( haf ) are well - known for the haldane energy gap @xcite between the singlet spin - liquid ground state and the lowest excited state , which is an @xmath8 triplet . the application of a magnetic field leads to zeeman splitting of the triplet and eventual vanishing of the gap @xmath9 at @xmath10 , where the energy of the lowest branch of the split triplet reaches the ground - state level . at this critical field , the expected quantum transition is to the tomonaga - luttinger spin liquid , @xcite in which spin correlations decay with characteristic power laws . this scenario remains robust in an array of 1d integer - spin haf chains against the introduction of interchain exchange . such a coupling reduces @xmath7 but , provided it is small , does not destroy the singlet ground state below @xmath7 . above @xmath7 , it leads to finite - temperature long - range order ( lro ) , which can be described as the three - dimensional bose - einstein condensation of the lower - branch triplets . @xcite the nature of the ordered state , including the robustness of the tomonaga - luttinger spin liquid , is of strong current interest . the first experimental evidence @xcite of a haldane gap was found in csnicl@xmath11 . this material has a relatively large interchain coupling : @xmath12 , where @xmath13 and @xmath14 are interchain and in - chain exchanges , respectively . as a consequence , n@xmath15el ordering occurs at 4.85 k. ni(c@xmath3h@xmath16n@xmath3)@xmath3no@xmath3(clo@xmath17 ) ( nenp ) , with the small @xmath18 ratio of @xmath19 , was the first haldane - gap antiferromagnet with no n@xmath15el ordering . @xcite magnetic susceptibility shows no anomaly indicative of ordering , at least down to 4 mk . @xcite naturally , this material was a promising candidate for the field - induced lro at fields above @xmath7 . however , it has been found that such an order is preempted by the staggered effective field arising from the staggered _ g _ tensors of the ni@xmath20 ions within each spin chain . to date , another spin-1 chain material ni(c@xmath1h@xmath2n@xmath3)@xmath3n@xmath4(pf@xmath5 ) ( ndmap ) remains the only laboratory model of a 1d haf array in which the nature of the field - induced lro states has been revealed by experiment . @xcite this distinction owes to the unusually low @xmath7 , which is readily accessible to many experimental probes , as well as the absence of a staggered field . ndmap has an orthorhombic structure with the lattice parameters @xmath21 , @xmath22 , and @xmath23 , with the antiferromagnetic spin chains running along the @xmath24 axis . @xcite according to inelastic neutron scattering , @xcite the in - chain exchange is @xmath25 k , and the relative strengths of the interchain exchanges are @xmath26 and @xmath27 along the @xmath28 and @xmath29 axes , respectively . easy - plane crystal - field anisotropy @xmath30 , where @xmath31 is the crystallographic @xmath24 axis and @xmath32 , @xcite is responsible for the strong anisotropy of both the gap @xmath9 and the magnetic phase diagram of the field - induced lro states , as has been measured by magnetic susceptibility , @xcite specific heat , @xcite magnetization , @xcite esr , @xcite and neutron scattering . @xcite the gap energies of the triplet , whose degeneracy is lifted by the crystal - field anisotropy , have been measured by inelastic neutron scattering @xcite to be @xmath33 mev , @xmath34 mev , and @xmath35 mev for the @xmath36 excitations quantized along the @xmath29 , @xmath28 , and @xmath24 axes , respectively . here @xmath37 is the quantization axis . the small splitting of a doublet into @xmath38 and @xmath39 is due to a weak in - plane anisotropy @xmath40 . because of the gap anisotropy , @xmath7 depends on the crystal orientation and ranges from about 4 t to 6 t. @xcite above @xmath7 , the ordered states are commensurate with the crystal lattice for all four magnetic - field directions studied by neutron diffraction . @xcite for field @xmath41 applied along the @xmath29 axis and along @xmath42 $ ] , the order is short - ranged and confined within each @xmath43 plane , whereas it is long - ranged and three - dimensional for @xmath44 and @xmath45 $ ] . the neutron experiments have detected no incommensurate modulation of the magnetization component parallel to the field . such a modulation has been predicted for the tomonaga - luttinger spin liquid , @xcite and its absence in ndmap can be understood as a consequence of a lack of axial symmetry , which is necessary for the existence of the tomonaga - luttinger spin liquid . @xcite it is important to note that even @xmath44 is not an axisymmetric field configuration for ndmap because of its unique geometry . in this material , the crystal - field anisotropy of ni@xmath20 is determined by the local symmetry of the nin@xmath46 octahedron . first , there is a weak in - plane anisotropy as described above . second and probably more important , the principal axis of the octahedron is tilted from the @xmath24 axis by 15@xmath47 toward the @xmath29 axis , with the tilt direction alternating from chain to chain . @xcite therefore , there is no magnetic - field direction that strictly satisfies axial symmetry . in this paper , we explore the field - induced lro states to 32 t in field and 150 mk in temperature by means of specific heat measurements . the present study greatly extends the @xmath41-@xmath48 parameter space for the phase diagram , which has been previously limited @xcite to 12 t in @xmath41 and 0.52 k in temperature @xmath48 except for a few isolated points obtained by neutron diffraction @xcite at lower temperatures with less precision . preliminary results have been reported in ref . . the single crystals of ndmap used in this work were grown from aqueous solutions by the method described in ref . . fully deuterated crystals were used to eliminate the nuclear specific heat of protons in the set of measurements extending to the lowest temperature , whereas a hydrogenous sample was used when adequate . the specific heat measurements to 18 t were performed in a superconducting magnet at temperatures down to 150 mk with a dilution refrigerator . the relaxation calorimeter for this setup has been described in ref . . for measurements from 20 t to 32 t , another relaxation calorimeter with a built - in @xmath49he refrigerator was used in a resistive bitter magnet . each sample for the @xmath44 configuration was attached with silver paint @xcite to the vertical face of a small silver bracket , whose horizontal face was in turn glued with a wakefield compound @xcite to the calorimeter platform . the sample for the @xmath50 configuration was mounted with silver paint on a piece of 0.13 mm - thick sapphire substrate , which in turn was glued with a wakefield compound to the platform . a total of three samples ranging in mass from 4 mg to 8 mg were used to cover the different field ranges and orientations . axis . ( a ) temperature dependence at constant fields . the inset gives an expanded view of the region near the specific heat peak . ( b ) magnetic - field dependence at constant temperatures . the lines are guides for the eye . ] the magnetic - field and temperature dependence of the specific heat of a deuterated sample is shown in fig . 1 for fields up to 18 t , applied along the @xmath24 axis of the crystal . at each field , the peak in the specific heat seen in fig . 1(a ) clearly indicates a phase transition . as can be seen in the inset , the transition temperature denoted by the peak position first increases with increasing field up to about 10 t and then decreases for fields up to 14 t , where it starts to increase again , making a shallow local minimum at around 14 t. furthermore , the peak height has a less pronounced minimum at roughly the same field . these features reveal the existence of a new phase boundary , which separates two field - induced ordered states at around 14 t. for temperatures less than 0.85 k , specific heat is shown as a function of the magnetic field in fig . again , the peak clearly indicates a phase transition . we have confirmed that the transition temperatures determined by such magnetic - field scans agree with those determined by the temperature scans . axis . the lines are guides for the eye . ] the specific heat for the same field direction at higher fields produced by the resistive magnet are shown in fig . 2 . these data were obtained with the hydrogenous sample . at 26 t , the nuclear contribution of the protons is visible at temperatures below 0.7 k. however , the peak due to transition stands out , since it occurs at a higher temperature . in this field region extending from 20 t to 32 t , the transition temperature obtained from the peak position varies only monotonically , and so does the peak height . axis . the anomaly at the lowest field of each curve is due to the proximity of the transition to the disordered spin - liquid phase . the uninteresting proton contribution raises the curve at 0.260.29 k with respect to those at higher temperatures . the lines are guides for the eye . ] to further investigate the phase diagram near 14 t , we have measured the specific heat of the same hydrogenous sample as a function of the magnetic field , again applied along the @xmath24 axis , but to higher fields than in fig . 1(b ) . in these measurements , a constant electric current was fed to the heater of the thermal reservoir of the calorimeter , allowing the temperature to rise monotonically with an increasing field as dictated by the magnetoresistance of the heater . as seen in fig . 3 , a plateau - like anomaly in the specific heat occurs at around 14 t , clearly indicating a phase transition . this anomaly was overlooked in our preliminary report , @xcite where the field range investigated was too narrow . within the experimental resolution , no corresponding feature is found in magnetization , which has been measured at 80 mk and 1.3 k using a pulsed magnet and is featureless up to 60 t except for an anomaly at @xmath7 . @xcite axis . the lines are guides for the eye . ] for magnetic fields applied along the crystallographic @xmath29 axis , which is perpendicular to the spin chains , another deuterated sample was used to measure the specific heat at temperatures below 2.7 k as a function of field , as shown in fig . , a transition to the field - induced ordered phase is clearly indicated by a peak at each temperature . for fields ranging from 6.2 t to 6.5 t , these measurements were supplemented by temperature sweeps similar to those shown in fig . no new phase boundary was found for this field direction up to 18 t. axis and the closed symbols for the field along the @xmath29 axis . the circles are for the deuterated samples , and the squares and diamonds for the hydrogenous sample . the lines are guides for the eye . ] the magnetic phase diagram determined from the positions of the specific - heat peaks and the plateau - like anomalies is shown in fig . 5 for the two magnetic - field directions studied . when the field is along the @xmath24 axis , the transition temperature exhibits a shallow but distinct local minimum at about @xmath6 t , and a new phase boundary extends nearly horizontally from this minimum . the small break in the phase boundary between the new high - field phase and the thermally disordered phase at around 20 t indicates that the hydrogenous sample has a slightly higher transition temperature than the deuterated sample . as stated earlier , the calorimeters used for the two samples were different . however , we have confirmed in another experiment the consistency of the temperature scales of the two calorimeters . therefore , the discontinuity in the phase boundary is not an artifact and indicates that deuterated ndmap in fact has a somewhat weaker exchange . when the field is along the @xmath29 axis , the transition temperature rises rapidly with increasing field , with no indication of a new phase . this is the first time more than one field - induced ordered phase has been found in a quasi-1d antiferromagnet for a given magnetic - field orientation . we believe it is significant that the field direction required for the new phase is along the crystalline @xmath24 axis . this direction , parallel to the spin chains , is quite unlike the @xmath29 , @xmath28 , and @xmath51 $ ] directions in that the ordering field rises very rapidly with temperature in the overall @xmath41-@xmath48 phase diagram . @xcite this feature probably exemplifies the underlying tendency of the spin chains in this field configuration to form a tomonaga - luttinger spin liquid , as has been presumed earlier . @xcite hence it is likely that identifying the new phase above @xmath52 and understanding the mechanism of the transition will shed light on the physics of the tomonaga - luttinger spin liquid . at present , however , with the absence of experimental data in the field region near and above @xmath52 other than the specific heat and magnetization , which shows no anomaly at @xmath52 , we can at best speculate on the nature of the transition . one possible scenario can be a transition involving spin rotation around the direction of the applied field , which is parallel to the spin chains . such an exotic transition has been observed for instance in a spin-1/2 chain material bacu@xmath53si@xmath53o@xmath54 , where it is presumably driven by a competition between an off - diagonal exchange and the magnetic field . @xcite although the dzyaloshinskii - moriya interaction , @xcite the likely candidate for the off - diagonal exchange , is usually quite small for ni@xmath20 and has not been detected in ndmap , a spin - rotation transition is an interesting possibility . recently , wang @xcite has considered a spin-1 chain with broken symmetry using a fermionic field theory and has predicted that a magnetic field larger than @xmath7 will restore an approximate axial symmetry and lead to a well - defined second transition from a commensurate phase to an incommensurate phase . unlike in the tomonaga - luttinger spin liquid , the excitations in the incommensurate phase will be gapped . however , the gap will be small and roughly equal to @xmath55 , when the field is applied along the chain direction @xmath24 . although wang has predicted an absence of anomaly in specific heat as well as magnetization at the second transition , which is other than first - order , it is worth exploring the possibility of the phase above @xmath52 being an incommensurate phase . and @xmath29 axes . the closed symbols are for the deuterated samples of the present work , where the circles are from the temperature - scan data and the diamonds from the field - scan data . the open circles are for hydrogenous samples from ref . . the lines are low - temperature fits of the phase boundaries of the deuterated samples to the expression @xmath56 . ] the bose - einstein condensation of triplets is believed to be a valid description of field - induced lro in all gapped antiferromagnets , regardless of the type of disordered ground state below @xmath7 and the type of the ordered state above it , provided that the lowest excitations are bosons . according to the bose - einstein condensation theory , @xcite the phase boundary in the @xmath41-@xmath48 parameter space obeys a power law @xmath56 . the field - induced ordering in the @xmath57 spin - dimer material tlcucl@xmath11 was the first case that was analyzed in terms of the bose - einstein condensation . @xcite however , the exponent @xmath58 of 0.50 determined @xcite for this material is smaller than the theoretical value of 2/3 obtained by a hartree - fock approximation for a quadratic dispersion for the triplets . the exponent for ni(c@xmath1h@xmath2n@xmath3)@xmath3n@xmath4(clo@xmath17 ) ( ndmaz ) , a spin-1 chain material @xcite similar to ndmap , has been reported to be 0.45 for the magnetic field applied along the @xmath24 axis . @xcite this value is also significantly smaller than the theoretical one . the four - fold improvement in the minimum temperature over the previous work @xcite for the field directions along the @xmath24 and @xmath29 axes allows us to determine the field dependence of the ordering temperature near @xmath7 with high accuracy . the relevant portions of the phase boundaries from fig . 5 are reproduced in fig . 6 , separately for the two field directions . these boundaries , shown with closed symbols , are for deuterated samples . excellent fits of the data below 0.77 k for @xmath44 and below 1.9 k for @xmath50 are obtained with the power law , giving @xmath59 and @xmath60 , respectively . the critical fields are @xmath61 t and @xmath62 t , respectively , along the @xmath24 and @xmath29 axes . recently , specific heat measurements of the @xmath63 bond - alternating - chain antiferromagnet ni(c@xmath64h@xmath65n@xmath17)no@xmath3(clo@xmath17 ) ( ntenp ) have found @xmath66 and 0.52 , respectively , for the field parallel and perpendicular to the spin chains . @xcite the similarity of these exponents to those for ndmap suggests that ntenp has similar field - induced ordered phases , although its low - field ground state is a spin - dimer singlet@xcite rather than a haldane spin liquid . including ours , all the experimental values found to date for the exponent @xmath58 are smaller than 2/3 predicted by the theory . however , the values for ndmap and ntenp for the magnetic field applied along the chain direction agree , within the combined uncertainties of the experiments and the calculation , with the quantum monte - carlo results by wessel @xmath67 . , @xcite who have found @xmath68 for @xmath57 spin dimers with a weak three - dimensional inter - dimer coupling and @xmath69 for two - leg spin-1/2 ladders with a weak three - dimensional inter - ladder coupling . their results support the bose - einstein condensation theory , according to the quantum monte - carlo calculation for three - dimensionally coupled @xmath57 spin dimers by nohadani @xmath67 . , @xcite who have shown that @xmath58 deviates downward from 2/3 , as the temperature range of the power - law fit is widened . furthermore , misguich and oshikawa @xcite have used a realistic dispersion for the triplets in tlcucl@xmath11 in their calculation of the field dependence of the bose - einstein condensation temperature @xmath70 and found good agreement with the experiment . @xcite it remains to be seen , however , whether the bose - einstein condensation theory can explain the strongly anisotropic @xmath58 for ndmap and ntenp , particularly the extremely small @xmath58 when the field is applied along the spin chains . in fig . 6 , we have included the phase boundaries for hydrogenous samples from the previous work , as shown with open symbols . @xcite again , the ordering temperature is somewhat higher for the hydrogenous samples except at the fields of 10 t and 12 t applied parallel to the @xmath24 axis , whereas @xmath7 is identical for a given field direction within our accuracy . this indicates that the interchain exchange @xmath13 is slightly weaker in deuterated ndmap than in hydrogenous ndmap , since decreasing @xmath13 lowers the ordering temperature directly but has little effect on @xmath7 , which is primarily determined by the in - chain exchange @xmath14 . a previous study , @xcite which compared the phase diagrams of deuterated and hydrogenous samples for @xmath41 along the @xmath28 axis , saw a much smaller but qualitatively similar difference . we propose that the smaller @xmath13 for deuterated ndmap is caused by the smaller zero - point motion of deuterons leading to less overlap of the electronic wavefunctions in the exchange paths involving hydrogens . in some classical antiferromagnets , such as the linear - chain compound ( ch@xmath11)@xmath71nmncl@xmath11 ( tmmc ) , deuteration has been reported to reduce the n@xmath15el temperature . @xcite in the quasi - two - dimensional organic salt @xmath72-(bedt - ttf)@xmath53cu[n(cn)@xmath53]br , substituting deuterium for the hydrogen of the ethylene groups has been shown to drive the system from a superconductor to an antiferromagnetic mott insulator . @xcite to our knowledge , however , the present work represents the first observation of an effect of deuteration in a haldane - gap antiferromagnet . in summary , the magnetic phase diagram of ndmap has been extended by specific - heat measurements to 150 mk in temperature and 32 t in the magnetic field . a new transition has been found at @xmath6 t , when the field is parallel to the spin chains . further study using techniques other than specific heat is needed to investigate the nature of the new phase , including the possibility that it is an incommensurate phase . in addition , we have determined the power - law exponents for the transition temperatures of the field - induced ordering from the spin - liquid state and compared them with the exponents for tlcucl@xmath11 , ndmaz , and ntenp and with the predictions of the bose - einstein condensation theory . finally , we have observed an effect of deuterium substitution on the ordering temperature . the result indicates that the substitution slightly decreases the interchain exchange @xmath13 but hardly affects the in - chain exchange @xmath14 . we thank s. haas , m. hagiwara , i. harada , m. w. meisel , d. a. micha , y. narumi , j. ribas , t. sakai , d. r. talham , and h. tanaka for useful discussions , and f. c. mcdonald , jr . , t. p. murphy , e. c. palm , and c. r. rotundu for assistance . this work was supported by the nsf through dmr-9802050 and the doe under grant no . de - fg02 - 99er45748 . a portion of it was performed at the national high magnetic field laboratory , which is supported by nsf cooperative agreement no . dmr-0084173 and by the state of florida . kk acknowledges support by a grant - in - aid for scientific research from the japan society for the promotion of science . o. avenel , j. xu , j. s. xia , m .- f . xu , b. andraka , t. lang , p. l. moyland , w. ni , p. j. c. signore , c. m. c. m. van woerkens , e. d. adams , g. g. ihas , m. w. meisel , s. e. nagler , n. s. sullivan , y. takano , d. r. talham , t. goto , and n. fujiwara , phys . b * 46 * , 8655 ( 1992 ) .
|
we have determined the magnetic phase diagram of the quasi - one - dimensional @xmath0 1 heisenberg antiferromagnet ni(c@xmath1h@xmath2n@xmath3)@xmath3n@xmath4(pf@xmath5 ) by specific heat measurements to 150 mk in temperature and 32 t in the magnetic field . when the field is applied along the spin - chain direction
, a new phase appears at @xmath6 t. for the previously known phases of field - induced order , an accurate determination is made of the power - law exponents of the ordering temperature near the zero - temperature critical field @xmath7 , owing to the four - fold improvement of the minimum temperature over the previous work .
the results are compared with the predictions based on the bose - einstein condensation of triplet excitations . substituting deuterium for hydrogen
is found to slightly reduce the interchain exchange .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
to personalize an experience or make an application more secure and less accessible to undesired people , we need to be able to distinguish a person from everyone else . it is done using marks from the users to identify them and block unauthorized access , or personalize it based on the trusted identity . to do so , many alternatives are on the table , such as keys , passwords and cards . the most secure options so far , however , are biometric features which can not be imitated by any other than the desired person himself . they are divided into behavioral features that the person can uniquely create or express , such as signatures , walking rhythm , and physiological characteristics that the person possesses , such as fingerprints and iris pattern . many works revolve around identification and verification of such data including , but not limited to , fingerprints @xcite , palmprints @xcite-@xcite , faces @xcite and iris patterns @xcite . iris recognition systems are widely used for security applications , since they contain a rich set of features and do not change significantly over time . they are also virtually impossible to fake . one of the first modern algorithms for iris recognition was developed by john daugman and used 2d gabor wavelet transform @xcite . in a more recent work , kumar @xcite proposed to use a combination of log - gabor , haar wavelet , dct and fft based features to achieve high accuracy . in @xcite , farouk proposed an scheme which uses elastic graph matching and gabor wavelet . each iris is represented as a labeled graph and a similarity function is defined to compare the two graphs . in @xcite , belcher used region - based sift descriptor for iris recognition and achieved a relatively good performance . pillai @xcite proposed a unified framework based on random projections and sparse representations to achieve robust and accurate iris matching . the reader is referred to @xcite for a comprehensive survey of iris recognition . in most of iris recognition works , the iris region is first detected and the iris is mapped to a rectangular region in polar coordinate . various iris segmentation algorithms are developed during the past few years @xcite . foreground segmentation approaches can also be used for iris segmentation @xcite , @xcite . it is worth mentioning that no segmentation is performed to extract iris region from the eye image in our work , which makes it much easier to implement . in this work , two sets of features are extracted from iris images , one of them being the recently introduced set of scattering - transform features and the other one being that of textural features to capture the texture information of irises . we believe that if these features are combined , they will provide a high discriminating power to conduct the recognition task . after the features are extracted , their dimensionality is reduced by applying pca and then minimum distance classifier is used to recognize new iris images . skipping the segmentation step makes our algorithm very fast and it can be easily implemented in electronic devices for real time applications using energy - efficient implementation and power management @xcite . this algorithm is tested on the well - known iit delhi iris database , and a very high accuracy rate is achieved . three sample iris images of the dataset used in this work are shown in figure 1 . the rest of the paper is organized as follows . section [ sectionii ] describes the features which are used in this work . section [ sectioniii ] contains the explanation of the classification scheme . the results of our experiments and comparisons with other works are in section [ sectioniv ] and the paper is concluded in section [ sectionv ] . extracting good features and image descriptors is one of the most important steps in many computer vision and object recognition algorithms . as a result , many researchers have focused on designing useful features which can be used for a variety of object recognition and image classification tasks . a good feature should have some degree of invariance with respect to translation , slight rotation and deformation . there are many popular features and image descriptors which are being used today , including scale invariant feature transform ( sift ) , histogram of oriented gradient ( hog ) , bag of words ( bow ) @xcite-@xcite , etc . geometrical features and sparsity - based features are also used for some biometric and medical applications in several works @xcite-@xcite . a new algorithm for feature selection for small datasets is presented in @xcite . recently , unsupervised feature learning algorithms have been in the spotlight , where the image is fed directly as the input to the deep neural network and the algorithm itself finds the best set of features from the image . for iris recognition , various features have been used by several researchers , including wavelet - based features , pca and lda . in this paper , a combined set of two features is used : some derived from the scattering transform , and the rest from the textural information of iris patterns . these features are introduced in the following subsections in more detail . the scattering operator is a locally translation - invariant descriptor which is proposed by stephane mallat and has achieved state - of - the - art recognition accuracy in several computer vision @xcite and audio classification @xcite problems . a scattering transform computes local image descriptors with a cascade of three operations : wavelet decompositions , complex modulus and a local averaging . the scattering coefficients are similar to those of the sift descriptor , but they contain more high - frequency information than sift . as discussed in @xcite , other image descriptors such as sift and multiscale gabor textons can be obtained by averaging the amplitude of wavelet coefficients , calculated with directional wavelets . this averaging provide some sort of local translation invariance , but it also reduces the high - frequency information . scattering transform recovers part of the high - frequency information lost by this averaging with co - occurrence coefficients having the similar invariance as those of the scattering transform . in most object recognition tasks , locally invariant features are preferred , since they provide robust representation of images . they can be seen as the averaged value of gradient orientation . using this averaging , some local deformation and translation will be tolerated . however , such process will reduce too much high - frequency information and therefore could greatly decrease discriminating capability . the scattering features provide richer descriptors for complex structures such as corners and multiscale texture variations . the scattering operator is designed in a way that it preserves the locally invariance property of sift , but it also recovers the lost high - frequency content of the images . suppose we have a signal @xmath0 . the first scattering coefficient is the average of the signal and can be obtained by convolving the signal with an averaging filter @xmath1 as @xmath2 . the scattering coefficients of the first layer can be obtained by applying wavelet transforms at different scales and orientations , and taking the magnitude and convolving it with a low - pass filter @xmath1 as shown below : @xmath3 and @xmath4 denote different scales and orientations and @xmath5 . note that removing the complex phase of wavelet will make these coefficients insensitive to local translation . now to recover the high - frequency information , which is eliminated from the wavelet coefficients of first layer by averaging , we can convolve @xmath6 by another set of wavelet at scale @xmath7 , taking the absolute value of wavelet and taking the average : @xmath8 is negligible at scales where @xmath9 . therefore the coefficients are calculated only for @xmath10 . the convolution with @xmath1 at the second layer removes high frequencies and yields second - order coefficients which are locally invariant to translation . these high - frequency information can be restored again by finer scale wavelet coefficients in the next layers . to obtain the scattering coefficients at the @xmath11-th layer , we have to perform the following procedure iteratively @xmath11 times : @xmath12 the output of scattering transform of the @xmath11-th layer has a size of @xmath13 where p denotes the number of different orientations . in other words there are @xmath13 transformed images at the output of the @xmath11-th layer . the transformed images of the first and second layers of scattering transform for a sample iris image are shown in figures 2 and 3 . these images are derived by applying bank of filters of 5 different scales and 6 orientations . scattering vector can be thought of as the cascade of convolution with wavelets , non - linear modulus and averaging operators which makes it very similar to the deep convolutional neural network @xcite . to derive scattering features , the scattering - transformed images of all layers up to @xmath14 are taken and the mean and variance of these images are calculated as scattering features which results in a vector @xmath15 of size @xmath16 . for mainstream applications , using two or three levels of scattering transform will be enough . textural , spectral and contextual features are the three fundamental pattern elements in recognition . in human interpretation of color photographs , textural features contain the spatial information of intensity variation in a single band @xcite . to mimic the human visual system , there are several features introduced to capture textural information of an image . among them , haralick features and local binary pattern ( lbp ) are two major groups of textural features . haralick textural features are derived from the co - occurrence matrix of image . local binary pattern are derived based on the relative comparison between each pixel and its neighboring pixels . there are also various modified versions of lbp features such as transition local binary patterns , direction - coded local binary patterns , volume local binary pattern ( vlbp ) . in this work , haralick features are used to capture textural information of the image . to extract haralick features , we first need to derive the co - occurrence matrix . the co - occurrence matrix measures the distribution of co - occurring intensity values for a given offset . if we represent the image as a two - dimensional function which maps pairs of coordinates to the intensity values , i.e. , @xmath17 , where @xmath18 and @xmath19 , and @xmath20 denotes the set of all possible grayscale levels . then the co - occurrence matrix @xmath21 of image @xmath22 with the offset @xmath23 or @xmath24can be defined as : @xmath25 where @xmath26 denotes the discrete dirac function . it should be noted that the co - occurrence matrix has a size of @xmath27 , where @xmath28 denotes the number of gray levels in the image . the offset @xmath23 depends on the direction @xmath29 which can be defined as : @xmath30 here we have derived the co - occurrence matrix for the offset @xmath31 . in our work , the textural features are extracted on block level . each image is divided into non - overlapping blocks of size @xmath32 and their co - occurrence matrices are derived and 14 features are extracted from them . more details about the derivation of these 14 textural features from co - occurrence matrix is provided in the appendix . then the features from different blocks are concatenated and formed a longer feature vector . if an image has a size of @xmath33 , the total number of textural features will be : @xmath34 in our work the textural features are derived in a slightly different way from the original paper @xcite , but they are very similar . here the co - occurrence matrix is found only for a single pixel horizontal shift ( corresponding to @xmath35 ) . after derivation of the set of scattering and textural features , we can concatenate them to form the feature vector of each iris image as : @xmath36^\intercal$ ] , where @xmath15 and @xmath37 denote the scattering and textural features respectively . principal component analysis ( pca ) , also known as karhunen - loeve transformation , is a powerful algorithm used for dimensionality reduction @xcite . given a set of correlated variables , pca transforms them into another domain such the transformed variables are linearly uncorrelated . this set of linearly uncorrelated variables are called principal components . pca is usually defined in a way that the first principal component has the largest possible variance and the second one has the second largest variance and so on . therefore after applying pca , we could only keep a subset of principal components with the largest variance to reduce the dimensionality . pca can be thought of as fitting a @xmath11-dimensional ellipsoid to a set of data , where each axis of the ellipsoid represents a principal component . there are also others dimensionality reduction algorithms which are designed based on pca such as kernel - pca and sparse - pca . pca has many applications in computer vision . eigenface is one representative application of pca in computer vision , where pca is used for face recognition @xcite . without going into too much detail , let us assume we have a dataset of @xmath38 iris images and @xmath39 denote their features . also let us assume that each feature has dimensionality of @xmath40 . to apply pca , all features need to be centered first by removing their mean : @xmath41 where @xmath42 . then the covariance matrix of the centered images is calculated : @xmath43 next the eigenvalues @xmath44 and eigenvectors @xmath45 of the covariance matrix @xmath46 are computed . suppose @xmath44 s are ordered based on their values . then each @xmath47 can be written as @xmath48 . we can reduce the dimensionality of the data by projecting them on the first @xmath49 principal vectors as : @xmath50 by keeping @xmath11 principal components , the percentage of retained variance can be found as : @xmath51 . hence one simple way to choose @xmath11 would be to pick a value such that the above ratio is less than @xmath52 , where @xmath52 is usually chosen between 95% to 99% . there are various classifiers which can be used for this task , including majority voting algorithm , support vector machine , neural network and minimum distance classifier . in this work minimum distance classifier has been used which is quite popular for template matching problems . one benefit of minimum distance classifier is that it does not need any training , making it much faster than most of the other classifiers . as long as the features are discriminative enough to separate different classes , the minimum distance classifier will provide high accuracy , otherwise using other classifiers would be a better option . minimum distance classifier finds the distance between the features of the training samples and those of an unknown subject , and picks the training sample with the minimum distance to the unknown as the answer . to put it in equation , if we show the features of the test subject as @xmath53 and those of the test sample @xmath54 with @xmath55 , the test subject is matched to the sample that satisfies the following : @xmath56\end{gathered}\ ] ] we have used euclidean distance as our distance metric . this section presents a detailed description of experimental results . before showing the results , let us describe the parameter values of our algorithm . for each image , scattering transform is applied up to two levels with a set of filter banks with 5 scales and 6 orientations , resulting in 391 transformed images . the mean and variance of each scatter - transformed image are used as features , resulting in 782 scattering features . the scattering features are derived using the software implementation provided by mallat s group @xcite . to extract textural features , each image is divided into 12 smaller blocks and 14 features are derived from any of them which results in a total of 168 textural features . therefore the concatenated feature vector has a length of 950 . then pca is applied to all features and the first 80 pca features ( retain above 99% of the initial features energy ) are used for recognition . minimum distance classifier is used for template matching . we have tested our algorithm on a popular iris database collected by iit delhi . this database contains 2240 iris images captured from 224 different people . the images of 21 people ( around 10% ) are used as a validation set to find the optimum value of the parameters of the algorithm , and the rest are used for evaluation . for each person , around half of the images are used for training and the rest for testing . figure 4 shows the recognition accuracy using different numbers of pca features . interestingly , even by using few pca features , we are able to get a very high accuracy rate . as it can be seen , using 80 pca features results in a accuracy rate above 99% , which will not increase by using more pca features . table 1 provides a comparison of the performance of the proposed scheme and those of other recent algorithms . the accuracy of the proposed scheme is reported as the highest rate achieved by 80 pca features . as it can be seen , using the combination of scattering and textural features , we are able to outperform previous approaches . this is mainly due to the richness of both scattering and haralick features which are able to capture high - frequency patterns of irises , providing a very high discriminating power . one main advantage of this scheme is that , it does not require segmentation of iris from eye images ( although the segmentation could improve the results for some difficult cases ) . [ h ] haar wavelet @xcite & 96.6% + log gabor filter by kumar @xcite & 97.19% + fusion @xcite & 97.41% + elastic graph matching @xcite & 98% + proposed scheme using 80 pca features & 99.2% + [ tblcomp ] the experiments are performed using matlab 2012 on a laptop with core i5 cpu running at 2.6ghz . the execution time for the proposed scheme is about 11ms for each image which is fast enough to be used for real - time applications . this paper proposed a set of scattering and textural features for iris recognition . the scattering features are extracted globally , while the textural features are extracted locally . scattering features are locally invariant and carry a great deal of high - frequency information which are lost in other descriptors such as sift and hog . the high - frequency information provides great discriminating power for iris recognition . principal component analysis is applied on features to reduce dimensionality . then minimum distance classifier is used to match new iris images with training images . this algorithm is tested on a well - known dataset , and a high accuracy rate is achieved which outperforms the previous best results achieved on this dataset . in the future , we will investigate to apply the proposed set of features to more challenging iris datasets and also other biometric recognition problems . the authors would like to thank stephane mallat s research group at ecole normale superieure for providing the software implementation of scattering transform . we would also like to thank iit delhi for providing the iris database @xcite . to find textural features from the co - occurrence matrix , we first need to find the following terms which are used for derivation of haralick features . @xmath57 denotes the normalized co - occurrence matrix which can be thought as a probability distribution . @xmath58 and @xmath59 denote the marginal probabilities along @xmath60 and @xmath61 . @xmath62 and @xmath63 denote the probabilities of @xmath64 and @xmath65 . @xmath66 denotes the entropy of @xmath67 and : @xmath68 using the above terms , the following 14 textural can be derived for each ^2 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{angular second moment } \\ & f_2= \sum_{k=0}^{n_g-1}k^2 p_{x - y}(k ) , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{contrast } \\ & f_3= \frac{\sum_{i}\sum_{j}ijp(i , j)-\mu_x\mu_y}{\sigma_x\sigma_y } , \ \ \ \ \ \text{correlation } \\ & f_4= \sum_{i}\sum_{j } ( i-\mu)^2p(i , j ) , \ \ \ \ \ \ \ \ \ \ \text{variance } \\ & f_5= \sum_{i}\sum_{j } \frac{1}{1+(i - j)^2 } p(i , j ) , \ \ \ \text{inverse diference moment}\\ & f_6= \sum_{k=2}^{2n_g}k p_{x+y}(k ) , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{sum average } \\ & f_7= \sum_{k=2}^{2n_g}(k - f_6)^2 p_{x+y}(k ) , \ \ \ \ \ \ \ \ \ \ \ \ \text{sum variance } \\ & f_8= -\sum_{k=2}^{2n_g } p_{x+y}(k ) \log(p_{x+y}(k ) ) , \ \text{sum entropy } \\ & f_9= -\sum_{i}\sum_{j } p(i , j ) \log(p(i , j ) ) , \ \ \text{entropy } \ \\ & f_{10}= \sum_{k=0}^{n_g-1}(k-\mu_{x - y})^2 p_{x - y}(k ) , \ \ \ \ \text{diference variance } \\ & f_{11}= -\sum_{k=0}^{n_g-1 } p_{x - y}(k ) \log p_{x - y}(k ) , \ \text{difference entropy } \\ & f_{12}= \frac{hxy - hxy1}{max\{hx , hy\ } } \\~\\ & f_{13}= \sqrt{1-exp{[-2(hxy2-hxy ) ] } } \\~\\ & f_{14}= \sqrt{\text{second largest eigenvalue of } \ q } \\\end{aligned}\ ] ] 1 ak . hrechak and ja . mchugh , `` automated fingerprint recognition using structural matching '' , pattern recognition 23.8 : 893 - 904 , 1990 . s. minaee , aa . abdolrashidi , `` multispectral palmprint recognition using textural features , '' ieee signal processing in medicine and biology symposium ( spmb ) , 2014 . s. minaee , aa . abdolrashidi , `` highly accurate multispectral palmprint recognition using statistical and wavelet features , '' ieee signal processing workshop , 2015 . w. zhao , r. chellappa , pj . phillips and a. rosenfeld , `` face recognition : a literature survey '' , acm computing surveys ( csur ) 35 , no . 4 : 399 - 458 , 2003 . kw bowyer , kp hollingsworth and pj flynn , `` a survey of iris biometrics research : 20082010 , '' handbook of iris recognition . springer london , 15 - 54 , 2013 . daugman , `` high confidence visual recognition of persons by a test of statistical independence , '' ieee transactions on pattern analysis and machine intelligence , 15.11 : 1148 - 1161 , 1993 . a. kumar and a. passi , `` comparison and combination of iris matchers for reliable personal authentication , '' pattern recognition , vol . 1016 - 1026 , mar . farouk , `` iris recognition based on elastic graph matching and gabor wavelets , '' computer vision and image understanding , elsevier , 115.8 : 1239 - 1244 , 2011 . c. belcher and y. du , `` region - based sift approach to iris recognition , '' optics and lasers in engineering , elsevier 47.1 : 139 - 147 , 2009 . pillai , vm . patel , r. chellappa and nk . ratha , `` secure and robust iris recognition using random projections and sparse representations '' , ieee transactions on pami , 33 , no . 9 : 1877 - 1893 , 2011 . z. he , t. tan , z. sun and x. qiu , `` toward accurate and fast iris segmentation for iris biometrics '' , ieee transactions on pami , 31 , no . 9 : 1670 - 1684 , 2009 . s. minaee and y. wang , `` screen content image segmentation using least absolute deviation fitting '' , ieee international conference on image processing , 2015 . s. minaee , h. yu and y. wang , `` a robust regression approach for background / foreground segmentation '' , arxiv preprint arxiv:1412.5126 , 2014 . m. hosseini , a. fedorova , j. peters and s. shirmohammadi , `` energy - aware adaptations in mobile 3d graphics '' , acm multimedia : 1017 - 1020 , 2012 . d. lowe , `` distinctive image features from scale - invariant keypoints , '' international journal of computer vision 60.2 , : 91 - 110 , 2004 . n. dalal and b. triggs , `` histograms of oriented gradients for human detection '' , ieee conference on computer vision and pattern recognition , vol . 1 , 2005 . g. csurka , c. dance , l. fan , j. willamowski and c. bray , `` visual categorization with bags of keypoints '' , eccv workshop , vol . 1 , no . 1 - 22 , 2004 . s. minaee , m. fotouhi and b.h . khalaj , a geometric approach for fully automatic chromosome segmentation``,signal processing in medicine and biology symposium , ieee , 2014 . u. srinivas , h. mousavi , c. jeon , v. monga , a. hattel and b. jayarao , ' ' shirc : a simultaneous sparsity model for histopathological image representation and classification , `` isbi , pp . 1118 - 1121 . ieee , 2013 . u srinivas , hs mousavi , v monga , a. hattel and b. jayarao , ' ' simultaneous sparsity model for histopathological image representation and classification `` , ieee transactions on medical imaging , 33(5 ) , 1163 - 1179 , 2014 . s. minaee , y. wang , yw . lui , ' ' prediction of longterm outcome of neuropsychological tests of mtbi patients using imaging features " , signal processing in medicine and biology symposium ( spmb ) , ieee , 2013 . j. bruna , s. mallat , `` classification with scattering operators , '' ieee conference on computer vision and pattern recognition , pp.1561 - 1566 , 2011 . j. andn , s. mallat , `` multiscale scattering for audio classification , '' ismir . j. bruna , s. mallat , `` invariant scattering convolution networks , '' ieee transactions on pattern analysis and machine intelligence , 35.8 : 1872 - 1886 , 2013 . haralick , k. shanmugam , and i. dinstein , `` textural features for image classification , '' ieee transactions on systems , man and cybernetics , 6 : 610 - 621 , 1973 . h. abdi and lj . williams , `` principal component analysis , '' wiley interdisciplinary reviews : computational statistics 2.4 : 433 - 459 , 2010 . turk and ap . pentland , `` face recognition using eigenfaces , '' ieee conference on computer vision and pattern recognition , 1991 .
|
iris recognition has drawn a lot of attention since the mid - twentieth century . among all biometric features , iris is known to possess a rich set of features .
different features have been used to perform iris recognition in the past . in this paper , two powerful sets of features
are introduced to be used for iris recognition : scattering transform - based features and textural features .
pca is also applied on the extracted features to reduce the dimensionality of the feature vector while preserving most of the information of its initial value .
minimum distance classifier is used to perform template matching for each new test sample .
the proposed scheme is tested on a well - known iris database , and showed promising results with the best accuracy rate of 99.2% .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
one of the main goals of string theory is the inclusion of the standard model ( sm ) of particle physics in an ultraviolet complete and consistent theory of quantum gravity . the hope is a unified theory of all fundamental interactions : gravity as well as strong and electroweak interactions within the @xmath7 sm . recent support for the validity of the particle physics standard model is the 2012 discovery of the `` so - called '' higgs boson . how does this fit into known string theory ? ideally one would have hoped to derive the standard model from string theory itself , but up to now such a program has not ( yet ) been successful . it does not seem that the sm appears as a prediction of string theory . in view of that we have to ask the question whether the sm can be embedded in string theory . if this is possible we could then scan the successful models and check specific properties that might originate from the nature of the underlying string theory . known superstring theories are formulated in @xmath0 space time dimensions ( or @xmath8 for m theory ) while the sm describes physics in @xmath2 . the connection between @xmath0 and @xmath2 requires the compactification of six spatial dimensions . the rather few superstring theories in @xmath0 give rise to a plethora of theories in @xmath2 with a wide spectrum of properties . the search for the sm and thus the field of so - called `` string phenomenology '' boils down to a question of exploring this compactification process in detail . but how should we proceed ? as the top - down approach is not successful we should therefore analyse in detail the properties of the sm and then use a bottom - up approach to identify those regions of the `` string landscape '' where the sm is most likely to reside . this will provide us with a set of `` rules '' for @xmath2 model constructions of string theory towards the constructions of models that resemble the sm . the application of these rules will lead us to `` fertile patches '' of the string landscape with many explicit candidate models . given these models we can then try to identify those properties of the models that make them successful . they teach us some lessons towards selecting the string theory in @xmath0 as well as details of the process of compactification . in the present paper we shall describe this approach to `` string phenomenology '' . in section [ sec : fivegoldenrules ] we shall start with `` five golden rules '' as they have been formulated some time ago @xcite . these rules have been derived in a bottom - up approach exploiting the particular properties of quark- and lepton representations in the sm . they lead to some kind of ( grand ) unified picture favouring @xmath9 and @xmath10 symmetries in the ultraviolet . however , these rules are not to be understood as strict rules for string model building . you might violate them and still obtain some reasonable models . but , as we shall see , life is more easy if one follows these rules . in section [ sec : minilandscape ] we shall start explicit model building along these lines . we will select one of those string theories that allow for an easy incorporation of the rules within explicit solvable compactifications . this leads us to orbifold compactifications of the heterotic @xmath11 string theory @xcite as an example . we shall consider this example in detail and comment on generalizations and alternatives later . the search for realistic models starts with the analysis of the so - called @xmath12-ii orbifold @xcite . we define the search strategy in detail and present the results known as the `` minilandscape '' @xcite , a fertile patch of the string landscape for realistic model building . we analyse the several hundred models of the minilandscape towards specific properties , as e.g. the location of fields in extra - dimensional space . the emerging picture leads to a notion of `` local grand unification '' , where some of the more problematic aspects of grand unification ( gut ) can be avoided . we identify common properties of the successful models and formulate `` lessons '' from the minilandscape that should join the `` rules '' for realistic model building . section [ sec : orbifoldlandscape ] will be devoted to the construction of new , explicit mssm - like models using all @xmath13 and certain @xmath14 orbifold geometries resulting in approximately 12000 orbifold models . then , in section [ sec : rulesorbifoldlandscape ] we shall see how the lessons of the minilandscape will be tested in this more general `` orbifoldlandscape '' . in section [ sec : generallandscape ] we shall discuss alternatives to orbifold compactifications , as well as model building outside the heterotic @xmath11 string . the aim is a unified picture of rules and lessons for successful string model building . section [ sec : nvconclusions ] will be devoted to conclusions and outlook . let us start with a review of the `` five golden rules for superstring phenomenology '' , which can be seen as phenomenologically motivated guidelines to successful string model building @xcite . the rules can be summarized as follows : we need 1 . spinors of @xmath10 for sm matter 2 . incomplete gut multiplets for the higgs pair 3 . repetition of families from geometrical properties of the compactification space 4 . @xmath15 supersymmetry 5 . @xmath16-parity and other discrete symmetries let us explain the motivation for these rules in some detail in the following . it is a remarkable fact that the spinor @xmath18 of @xmath10 is the unique irreducible representation that can incorporate exactly one complete generation of quarks and leptons , including the right - handed neutrino . thereby , it can explain the absence of gauge - anomalies in the standard model for each generation separately . furthermore , it offers a simple explanation for the observed ratios of the electric charges of all elementary particles . in addition , there is a strong theoretical motivation for grand unified theories like @xmath10 from gauge coupling unification at the gut scale @xmath19 gev . hence , the first golden rule for superstring phenomenology suggests to construct string models in such a way that at least some generations of quarks and leptons reside at a location in compact space , where they are subject to a larger gauge group , like @xmath10 . hence , these generations come as complete representations of that larger group , e.g. as @xmath18 of @xmath10 . the heterotic string offers this possibility through the natural presence of the exceptional lie group @xmath20 , which includes an @xmath10 subgroup and its spinor representation . furthermore , using orbifold compactification the four - dimensional standard model gauge group can be enhanced to a local gut , i.e to a gut group like @xmath10 which is realized locally at an orbifold singularity in extra dimensions . in addition , there are matter fields ( originating from the so - called twisted sectors of the orbifold ) localised at these special points in extra dimensions and hence they appear as complete multiplets of the local gut group , for example as @xmath18-plets of @xmath10 . on the other hand , the spinor of @xmath10 is absent in ( perturbative ) type ii string theories , which can be seen as a drawback of these theories . often this drawback manifests itself in an unwanted suppression of the top quark yukawa coupling . on the other hand , f - theory ( and m - theory ) can cure this through the non - perturbative construction of exceptional lie groups like e.g. @xmath21 . when two seven - branes with @xmath10 gauge group intersect in the extra dimensions , a local gut can appear at the intersection . there , the gauge group can be enhanced to a local @xmath21 and a spinor of @xmath10 can appear as matter representation . beside complete spinor representations of @xmath10 for quarks and leptons , the ( supersymmetric extension of the ) standard model needs split , i.e. incomplete @xmath10 multiplets for the gauge bosons and the higgs(pair ) . their unwanted components inside a full gut multiplet would induce rapid proton decay and hence need to be ultra - heavy . in the case of the higgs doublet , this problem is called the doublet triplet splitting problem , because for the smallest gut @xmath9 a higgs field would reside in a five - dimensional representation of @xmath9 , which includes beside the higgs doublet an unwanted higgs triplet of @xmath22 . this problem might determine the localisations of the higgs pair and of the gauge bosons in the compactification space : they need to reside at a place in extra dimensions where they feel the breaking of the higher - dimensional gut to the 4d sm gauge group . hence , incomplete gut multiplets , e.g. for the higgs , can appear . this is the content of the second golden rule . in this way local guts exhibit grand unified gauge symmetries only at some special `` local '' surroundings in extra dimensions , while in 4d the gut group seems to be broken down to the standard model gauge group . this allows us to profit from some of the nice properties of guts ( like complete representations for matter as described in the first golden rule ) , while avoiding the problematic properties ( like doublet triplet splitting ) . in the case of the heterotic string on orbifolds the so - called untwisted sector ( i.e. the 10d bulk ) can naturally provide such split @xmath10 multiplets for the gauge bosons and the higgs . in particular , when the orbifold twist acts as a @xmath23 in one of the three complex extra dimensions , one can obtain an untwisted higgs pair that is vector - like with respect to the full ( i.e. observable and hidden ) gauge group . combined with an ( approximate ) @xmath16-symmetry this can yield a solution to the @xmath24-problem of the mssm . furthermore , as all charged bulk fields originate from the 10d @xmath11 vector multiplet this scenario naturally yields gauge higgs unification . finally , an untwisted higgs pair in the framework of heterotic orbifolds can relate the top quark yukawa coupling to the gauge coupling and hence give a nice explanation for the large difference between the masses of the third generation compared to the first and second one . in order to achieve this , the top quark needs to originate either from the bulk ( as it is often the case in the minilandscape @xcite of @xmath12-ii orbifolds ) or from an appropriate fixed torus , i.e. a complex codimension one singularity in the extra dimensions . the triple repetition of quarks and leptons as three generations with the same gauge interactions but different masses is a curiosity within the standard model and asks for a deeper understanding . one approach from a bottom - up perspective is to engineer a so - called flavour symmetry : one introduces a ( non - abelian ) symmetry group , discrete or gauge , and unifies the three generations of quarks and leptons in , for example , a single three - dimensional representation of that flavour group . however , as the yukawa interactions violate the flavour symmetry , it must be broken spontaneously by the vacuum expectation value of some standard model singlet , the so - called flavon . this might explain the mass ratios and mixing patterns of quarks and leptons . the third golden rule for superstring phenomenology asks for the origin of such a flavour symmetry . the rule suggests to choose the compactification space such that some of its geometrical properties lead to a repetition of families and hence yields a discrete flavour symmetry . in this case , the repetition of the family structure comes from topological properties of the compact manifold . within the framework of type ii string theories , the number of families can be related to intersection numbers of d - branes in extra dimensions , while for the heterotic string it can be due to a degeneracy between orbifold singularities . in the latter case , one can easily obtain non - abelian flavour groups which originate from the discrete symmetry transformations that interchange the degenerate orbifold singularities , combined with a stringy selection rule that is related to the orbifold space group @xcite . in any case the number of families will be given by geometrical and topological properties of the compact six - dimensional manifold . superstring theories are naturally equipped with @xmath26 or 2 supersymmetry in 10d . however , generically all supersymmetries are broken by the compactification to 4d . the fourth golden rule suggests to choose a `` non - generic '' compactification space such that @xmath26 survives in 4d . examples for such special spaces are calabi yaus , orbifolds and orientifolds . motivation for this is a solution of the so - called `` hierarchy problem '' between the weak scale ( a tev ) and the string ( planck ) scale . supersymmetry can stabilize this large hierarchy . since such a supersymmetry appears naturally in string theory , we assume that @xmath26 supersymmetry will survive down to the tev - scale . apart from the gauge symmetries of string theory , we need more symmetries to describe particle physics phenomena of the supersymmetric standard model . these could provide the desired textures of yukawa couplings , explain the absence of flavour changing neutral currents , help to avoid too fast proton decay , provide a stable particle for cold dark matter and solve the so - called @xmath24-problem . we know that ( continuous ) global symmetries might not be compatible with gravitational interactions . hence , local discrete symmetries might play this role in string theory . one of these symmetries is the well - known matter parity of the minimal supersymmetric extension of the standard model ( mssm ) : it forbids proton decay via dim . 4 operators and leads to a stable neutral wimp candidate . other discrete gauge symmetries are required to explain the flavour structure of quark / lepton masses and mixings . as we have seen in our review in section [ sec : fivegoldenrules ] , the five golden rules @xcite naturally ask for exceptional lie groups . @xmath10 , although it is not an exceptional group , fits very well in the chain of exceptional groups @xmath28 . therefore , the @xmath11 heterotic string is the prime candidate and we choose it as our starting point . alternatives to obtain @xmath20 in string theory are m- and f - theory , where such gauge groups can appear in non - perturbative constructions . the implementation of the rules in string theory started with the consideration of orbifold compactifications of the @xmath11 heterotic string . this lead to the so - called `` heterotic brane world '' @xcite where toy examples have been constructed in the framework of the @xmath29 orbifold . there , the explicit `` geographical '' properties of fields in extra dimensions have been presented and the local guts at the orbifold fixed points were analysed , see e.g. fig . [ fig : gaugegrouptopo ] . a first systematic attempt at realistic model constructions @xcite was based on the @xmath12-ii orbifold @xcite of the @xmath11 heterotic string . this orbifold considers a six - torus defined by the six - dimensional lattice of @xmath30 modded out by two twists , each acting in four of the six extra dimensions : @xmath31 of order 2 ( @xmath32 ) and @xmath33 of order 3 ( @xmath34 ) , see fig . [ fig : thetaomegasector ] . in ref . @xcite the embedding of the twists into the @xmath11 gauge group was chosen in such a way that at an intermediate step there are local @xmath10 guts with localised @xmath18-plets for quarks and leptons . this choice can be motivated by rule i , as discussed in the previous section . further breakdown of the gauge group to @xmath7 is induced by two orbifold wilson lines @xcite . in this set - up , a scan for realistic models was performed using the following strategy : * choose appropriate wilson lines ( and identify inequivalent models ) * sm gauge group @xmath35 times a hidden sector * hypercharge @xmath36 is non - anomalous and in @xmath9 gut normalisation * ( net ) number of three generations of quarks and leptons * at least one higgs pair * exotics are vector - like w.r.t . the sm gauge group and can be decoupled using the above criteria , the computer assisted search led to a total of some 200 and 300 mssm - like models in refs . @xcite and @xcite , respectively . the models typically have additional vector - like exotics as well as unbroken @xmath37 gauge symmetries , one of which is anomalous . this anomaly induces an fayet iliopoulos term ( fi - term ) , hence a breakdown of the additional @xmath37 s and thus allows for a decoupling of the vector - like exotics . explicit examples are given in ref . @xcite as benchmark models . all fields of the models can be attributed to certain sectors with specific geometrical properties . in the present case there is an untwisted sector with fields in 10d ( bulk ) , as well as twisted sectors where fields are localised at certain points ( or two - tori ) in the six - dimensional compactified space . the @xmath38 twisted sector ( fig . [ fig : thetaomegasector ] ) has fixed points and thus yields fields localised at these points in extra dimensions that can only propagate in our four - dimensional space time . the @xmath31 and @xmath33 twisted sectors , in contrast , have fixed two - tori in extra dimensions . fields in these sectors are confined to six space time dimensions . many properties of the models depend on these `` geographic '' properties of the fields in extra dimensions . for example , yukawa couplings between matter and higgs fields and in particular their coupling strengths are determined by the `` overlap '' of the fields in extra dimensions . given this large sample of realistic models , we can now analyse their properties and look for similarities and regularities . which geometrical and geographical properties in extra dimensions are important for realistic models ? by construction , all the models have observable sector gauge group @xmath7 and possibly some hidden sector gauge group relevant for supersymmetry breakdown . there is a net number of three generations of quarks and leptons and at least one pair of higgs doublets @xmath39 and @xmath40 . the higgs triplets are removed and the doublet triplet splitting problem is solved . a first question concerns a possible `` @xmath24-term '' : @xmath41 and we shall start our analysis with the higgs system , following the discussion of ref . @xcite . the higgs system is vector - like and a @xmath24-term @xmath41 is potentially allowed . as this is a term in the superpotential we would like to understand why @xmath24 is small compared to the gut - scale : this is the so - called @xmath24-problem . to avoid this problem one could invoke a symmetry that forbids the term . however , we know that @xmath24 has to be non - zero . hence , the symmetry has to be broken and this might reintroduce the @xmath24-problem again . in string theory the problem is often amplified since typically we find several ( say @xmath42 ) higgs doublet pairs . in the procedure to remove the vector - like exotics ( as described above ) we have to make @xmath43 pairs heavy while keeping one light . in fact , in many cases the small @xmath24-parameter is the result of a specific fine - tuning in such a way to remove all doublet pairs except for one . we do not consider this as a satisfactory solution . fortunately , the models of the minilandscape are generically not of this kind . many minilandscape models provide one higgs pair that resists all attempts to remove it . this is related to a discrete r - symmetry @xcite that can protect the @xmath24-parameter in the following way : in some cases the discrete r - symmetry is enlarged to an approximate @xmath44 @xcite . symmetries can explain vanishing vacuum energy in susy vacua . ] therefore , a @xmath24-parameter is generated at a higher order @xmath45 in the superpotential @xmath46 , where the approximate @xmath44 is broken to its exact discrete subgroup . this yields a suppression @xmath47 , where @xmath48 is set by the fi parameter . the crucial observation for this mechanism to work is the localisation of the higgs pair @xmath39 and @xmath40 in agreement with our second golden rule : both reside in the 10d bulk originating from gauge fields in extra dimensions . furthermore , the higgs pair is vector - like with respect to all symmetries , gauge and discrete . this is related to the @xmath23 orbifold action in one of the two - tori . hence , each term in the superpotential @xmath49 also couples to the higgs pair , i.e. @xmath50 . as susy breakdown requires a non - vanishing vev of the superpotential the @xmath24-term is related to the gravitino mass , i.e. @xmath51 . this is a reminiscent of a field theoretical mechanism first discussed in ref . @xcite . among all quarks and leptons the top - quark is very special : its large mass requires a large top - quark yukawa coupling . many minilandscape models address this naturally via the localisation of the top - quark in extra dimensions : both @xmath52 and @xmath53 reside in the 10d bulk , along with the higgs pair . hence , we have gauge - yukawa unification and the trilinear yukawa coupling of the top is given by the gauge coupling . typically the top - quark is the only matter field with trilinear yukawa coupling . the location of the other fields of the third family is strongly model - dependent , but in general they are distributed over various sectors : the third family could be called a `` patchwork family '' . the first two families are found to be located at fixed points in extra dimensions ( fig . [ fig : thetaomegasector ] ) . as such they live at points of enhanced symmetries , both gauge and discrete . the discrete symmetry is the reason for their suppressed yukawa couplings . in the @xmath12-ii example shown in the figure two families live at adjacent fixed points in the third extra - dimensional torus : one family is located at @xmath54 , the other at @xmath55 and @xmath56 ( see fig . [ fig : thetaomegasector ] ) . technically , this is a consequence of a vanishing wilson line in the @xmath57 direction . this leads to a @xmath58 flavour symmetry @xcite . the two localised families form a doublet , while the third family transforms in a one - dimensional representation of @xmath58 . this set - up forbids sizeable flavour changing neutral currents and thus relieves the so - called `` flavour problem '' . furthermore , the geometric reason for small yukawa couplings of the first and second family is their minimal overlap with the bulk higgs fields . this leads to yukawa couplings of higher order and a hierarchical generation of masses based on the froggatt nielsen mechanism @xcite , where the fi - term provides a small parameter @xmath59 that controls the pattern of masses . in addition , the first two families live at points of enhanced gauge symmetries and therefore build complete representations of the local grand unified gauge group , e.g. as @xmath18-plets of @xmath10 . hence , they enjoy the successful properties of `` local grand unification '' outlined in the first golden rule . the question of supersymmetry breakdown is a complicated process and we shall try to extract some general lessons that are rather model - independent . specifically we would consider gaugino condensation in the hidden sector @xcite realized explicitly in the minilandscape @xcite , see also section [ sec : orbifoldlandscaperule4 ] . a reasonable value for the gravitino mass can be obtained if the dilaton is fixed at a realistic value @xmath60 . thus , the discussion needs the study of moduli stabilization , which , fortunately , we do not have to analyse here . in fact we can rely on some specific pattern of supersymmetry breaking which seems to be common in various string theories , first observed in the framework of type iib theory @xcite and later confirmed in the heterotic case @xcite : so - called `` mirage mediation '' . its source is a suppression of the tree level contribution in modulus mediation ( in particular for gaugino masses and a - parameters ) . the suppression factor is given by the logarithm of the `` hierarchy '' @xmath61 , which numerically is of the order @xmath62 . non - leading terms suppressed by loop factors can now compete with the tree - level contribution . in its simplest form the loop corrections are given by the corresponding @xmath63-functions , leading to `` anomaly mediation '' if the tree level contribution is absent . without going into detail , let us just summarise the main properties of mirage mediation : * gaugino masses and a - parameters are suppressed compared to the gravitino mass by the factor @xmath61 * we obtain a compressed pattern of gaugino masses ( as the @xmath64 @xmath63-function is negative while those of @xmath65 and @xmath66 are positive ) * soft scalar masses @xmath67 are more model - dependent . in general we would expect them to be as large as @xmath68 @xcite . the models of the minilandscape inherit this generic picture . but they also teach us something new on the soft scalar masses , which results in lesson 4 . the scalars reside in various localisations in the extra dimensions that feel susy in different ways : first , the untwisted sector is obtained from simple torus compactification of the 10d theory leading to extended @xmath69 supersymmetry in @xmath2 . hence , soft terms of bulk fields are protected ( at least at tree level ) and broken by loop corrections when they communicate to sectors with less susy . next , scalars localised on fixed tori feel a remnant @xmath70 susy and might be protected as well . finally , fields localised at fixed points feel only @xmath26 susy and are not further protected @xcite . therefore , we expect soft terms @xmath71 for the localised first two families , while other ( bulk ) scalar fields , in particular the higgs bosons and the stop , feel a protection from extended susy . consequently , their soft masses are suppressed compared to @xmath68 ( by a loop factor of order @xmath72 ) . this constitutes lesson 4 of the minilandscape . the 10d heterotic string compactified on a six - dimensional toroidal orbifolds provides an easy and calculable framework for string phenomenology @xcite . a toroidal orbifold is constructed by a six - dimensional torus divided out by some of its discrete isometries , the so - called point group . for simplicity we assume this discrete symmetry to be abelian . combined with the condition on @xmath26 supersymmetry in 4d one is left with certain @xmath13 and @xmath14 groups , in total 17 different choices . for each choice , there are in general several inequivalent possibilities , e.g. related to the underlying six - torus . recently , these possibilities have been classified using methods from crystallography , resulting in 138 inequivalent orbifold geometries with abelian point group @xcite . the orbifolder @xcite is a powerful computer program to analyse these abelian orbifold compactifications of the heterotic string . the program includes a routine to automatically generate a huge set of consistent ( i.e. modular invariant and hence anomaly - free ) orbifold models and to identify those that are phenomenologically interesting , e.g. that are mssm - like . a crucial step in this routine is the identification of inequivalent orbifold models in order to avoid an overcounting : even though the string theory input parameters of two models ( i.e. so - called shifts and orbifold wilson lines ) might look different , the models can be equivalent and share , for example , the same massless spectrum and couplings . the current version ( 1.2 ) of the orbifolder uses simply the massless spectrum in terms of the representations under the full non - abelian gauge group in order to identify inequivalent models . however , there are typically five to ten 1 factors and the corresponding charges are neglected for this comparison of spectra , because they are highly dependent on the choice of 1 basis . as pointed out by groot nibbelink and loukas @xcite one can easily improve this by using in addition to the non - abelian representations also the @xmath36 hypercharge as it is uniquely defined for a given mssm model . we included this criterion into the orbifolder . however , it turns out that using this refined comparison method the number of inequivalent mssm - like orbifold models increases only by 3% . using the improved version of the orbifolder we performed a scan in the landscape of all @xmath13 and certain @xmath14 heterotic orbifold geometries for mssm - like models , where our basic requirements for a model to be mssm - like are : * sm gauge group @xmath73 times a hidden sector * hypercharge @xmath36 is non - anomalous and in @xmath9 gut normalisation * ( net ) number of three generations of quarks and leptons * at least one higgs pair * all exotics must be vector - like with respect to the sm gauge group we identified approximately 12000 mssm - like orbifold models that suit the above criteria . given the large number of promising models we call them the `` orbifoldlandscape '' . a summary of the results can be found in the appendix in tabs . [ tab : mssmsummary ] and [ tab : mssmsummary2 ] . furthermore , the orbifolder input files needed to load these models into the program can be found at @xcite . the scan did not reveal any mssm - like models from orbifold geometries with @xmath74 , @xmath75 and @xmath76-ii point group . this is most likely related to the condition of @xmath9 gut normalisation for hypercharge . note that this search for mssm - like orbifold models is by far not complete . for example , we only used the standard @xmath14 orbifold geometries ( i.e. those with label ( 1 - 1 ) following the nomenclature of ref . in addition , our search was performed in a huge , but still finite parameter set of shifts and wilson lines . finally , the routine to identify inequivalent orbifold models can surely be improved further . hence , presumably only a small fraction of the full heterotic orbifold landscape has been analysed here . let us compare our findings to the literature . the @xmath12-ii ( 1 - 1 ) orbifold has been studied intensively in the past , see e.g. @xcite . also the minilandscape @xcite was performed using this orbifold geometry , see section [ sec : reviewminilandscape ] . in the first paper @xcite local @xmath10 and @xmath21 guts were used as a search strategy and thus one was restricted to four out of 61 possible shifts , resulting in 223 mssm - like models . in the second paper @xcite this restriction was lifted , resulting in almost 300 mssm - like models . they are all included in our set of 348 mssm - like models from @xmath12-ii ( 1 - 1 ) , see tab . [ tab : mssmsummary ] in the appendix . similar to @xmath12-ii , the @xmath77 orbifold geometry has been conjectured to be very promising for mssm model building @xcite . here , we can confirm this conjecture : we found 3632 mssm - like models from @xmath77 ( 1 - 1 ) the largest set of models in our scan . also from a geometrical point of view , the @xmath77 orbifold is very rich : there are in total 41 different orbifold geometries with @xmath77 point group , i.e. based on different six - tori and roto translations @xcite . we considered only the standard choice here , labelled ( 1 - 1 ) . hence , one can expect a huge landscape of mssm - like models to be discovered from @xmath77 . recently , groot nibbelink and loukas performed a model scan in all @xmath78-i and @xmath78-ii geometries @xcite . they also used a local gut search strategy ( based on @xmath9 and @xmath10 local guts ) and hence started with 120 and 108 inequivalent shifts for @xmath78-i and @xmath78-ii , respectively . their scan resulted in 753 mssm - like models . without imposing the local gut strategy our search revealed in total 1713 mssm - like models from @xmath78 , see tab . [ tab : mssmsummary ] . further orbifold mssm - like models have been constructed using the @xmath79-i orbifold geometry @xcite . this orbifold seems also to be very promising as we identified 750 mssm - like models in this case , see tab . [ tab : mssmsummary ] . finally , we confirm the analysis of ref . @xcite for the orbifold geometries @xmath12-i and @xmath13 with @xmath80 and standard lattice ( 1 - 1 ) . in the next section we will apply the strategies described by the `` five golden rules of superstring phenomenology '' to our orbifoldlandscape and search for common properties of our 12000 mssm - like orbifold models . thereby , we will see how many mssm - like models would have been found following the `` five golden rules '' strictly and how many would have been lost . hence , we will estimate the prosperity of the `` five golden rules '' . in the following we focus on the golden rules i - iv . a detailed analysis of rule v is very model - dependent and will thus not be discussed here . as discussed in sec . [ sec : firstgoldenrules ] at least some generations of quarks and leptons might originate from spinors of @xmath10 sitting at points in extra dimensions with local @xmath10 gut . hence , we perform a statistic on the number of such localisations in our 12000 mssm - like orbifold models . the results are summarised in tab . [ tab : mssmsummary2 ] and displayed in fig . [ fig : so10spinors ] . it turns out that 25% of our models have at least one local @xmath10 gut . furthermore , we find that some orbifolds seem to forbid local @xmath10 guts with @xmath18-plets ( for example @xmath12-i @xcite ) . on the other hand , the mssm - like models from @xmath12-ii and @xmath78-i ( 1 - 1 ) and ( 2 - 1 ) prefer zero or two localised @xmath18-plets of @xmath10 . three local @xmath18-plets are very uncommon , they mostly appear in @xmath77 . note that the number of local guts can be greater than three even though the model has a ( net ) number of three generations of quarks and leptons . obviously , an anti - generation of quarks and leptons is needed in such a case . the maximal number we found in our scan is four local @xmath10 guts with @xmath18-plets for matter in the cases of @xmath29 and @xmath77 orbifold geometries . in addition , we analyse our 12000 models for local @xmath9 guts with local matter in @xmath81-plets . the results are summarised in table [ tab : mssmsummary2 ] and displayed in fig . [ fig : localsu5guts ] . we find this case to be very common : almost 40% of our mssm - like models have at least one local @xmath81-plet of a local @xmath9 gut . next , we also look for local @xmath21 guts with @xmath82-plets . we find only a few cases , most of them appear in @xmath14 orbifold geometries , see table [ tab : mssmsummary2 ] . finally , we scan our models for localised sm generations ( i.e. localised left handed quark doublets ) transforming in a complete multiplet of any local gut group that unifies the sm gauge group . again , our results are listed in table [ tab : mssmsummary2 ] and visualised in fig . [ fig : localanyguts ] . we find most of our models , i.e. 70% , have at least one local gut with a localised sm generation . in summary , the first golden rule , which demands for local guts in extra dimensions in order to obtain complete gut multiplets for matter , is very successful : most of our 12000 mssm - like models share this property automatically , it was not imposed by hand in our search . since the higgs doublets reside in incomplete gut multiplets , they might be localised at some region of the orbifold where the higher - dimensional gut is broken to the 4d standard model gauge group . this scenario yields a natural solution to the doublet triplet splitting problem . the untwisted sector ( i.e. bulk ) would be a prime candidate for such a localisation , but there can be further possibilities . the numbers of such gut breaking localisations are summarised in table [ tab : mssmsummary ] and displayed in fig . [ fig : higgs ] . we see that gut breaking localisations are very common among our mssm - like models . only a very few models do not contain any gut breaking localisations that yield incomplete gut multiplets for at least one higgs . on the other hand , there are 4223 cases with one gut breaking localisation in most cases ( 4097 out of 4223 ) this is the bulk . in addition , there are many models that have more than one possibility for naturally split higgs multiplets , but in almost all cases the bulk is among them . note that most of our mssm - like models have additional exotic higgs - like pairs , mostly two to six additional ones . in contrast to the mssm higgs pair they often originate from complete multiplets of some local gut . on the other hand , we identified 1011 mssm - like models with exactly one higgs pair . cases with exactly one higgs pair , originating from the bulk might be especially interesting . in summary , the second golden rule , which explains incomplete gut multiplets for the higgs using gut breaking localisations in extra dimensions , is very successful as in the case of the first golden rule , most of our 12000 mssm - like models follow this rule automatically . the standard model contains three generations of quarks and leptons with a peculiar pattern of masses and mixings . this might be related to a ( discrete ) flavour symmetry . or @xmath22 is also possible . some of the models in our orbifoldlandscape realise this possibility , but we do not analyse these cases in detail here . ] from the orbifold perspective discrete flavour symmetries naturally arise from the symmetries of the orbifold geometry @xcite . however , certain background fields ( i.e. orbifold wilson lines @xcite ) can break these symmetries . the maximal number of orbifold wilson lines is six corresponding to the six directions of the compactified space . the orbifold rotation , however , in general identifies some of those directions . hence , the corresponding wilson lines have to be equal . for example , the @xmath74 orbifold allows for maximally three independent wilson lines . in general , one can say that the more wilson lines vanish the larger is the discrete flavour symmetry . on the other hand , non - vanishing wilson lines are generically needed in order to obtain the standard model gauge group and to reduce the number of generations to three . hence , it is interesting to perform a statistic on the number of vanishing wilson lines for our 12000 mssm - like orbifold models , see tab . [ tab : mssmsummary ] in the appendix and fig . [ fig : generations ] . there are orbifold geometries , like @xmath83 , @xmath12-i and @xmath79-i , apparently demanding for all possible orbifold wilson lines to be non - trivial in order to yield the mssm , see tab . [ tab : mssmsummary ] . these mssm - like models are expected to have no discrete , non - abelian flavour symmetries . on the other hand , there are several orbifold geometries that seem to require at least one vanishing wilson line in order to reproduce the mssm with its three generations , for example @xmath12-ii , @xmath29 , @xmath77 , @xmath84 and @xmath85 . in general , the case of vanishing wilson lines is very common : we see that in 75% of our mssm - like orbifold models at least one allowed orbifold wilson line is zero . in these cases non - abelian flavour symmetries are expected . for example , most of the mssm - like models from @xmath12-ii ( 1 - 1 ) have a @xmath86 flavour symmetry with the first two generations transforming as a doublet and the third one as a singlet @xcite . in summary , the third golden rule , which explains the origin of three generations of quarks and leptons by geometrical properties of the compactification space , is generically satisfied for our 12000 mssm - like orbifold models . by construction , i.e. by choosing the appropriate orbifold geometries , our 12000 mssm - like orbifold models preserve @xmath26 supersymmetry in four dimensions . this is expected to be broken by non - perturbative effects , i.e. by hidden sector gaugino condensation @xcite . here , we follow the discussion of @xcite where low energy supersymmetry breaking in the minilandscape of @xmath12-ii orbifolds was analysed . see also @xcite for a related discussion . in detail , our mssm - like models typically possess a non - abelian hidden sector gauge group with little or no charged matter representations . the corresponding gauge coupling depends via the one - loop @xmath63-functions on the energy scale . if the coupling becomes strong at some ( intermediate ) energy scale @xmath87 the respective gauginos condensate and supersymmetry is broken spontaneously by a non - vanishing dilaton @xmath88-term . assuming that susy breaking is communicated to the observable sector via gravity the scale of soft susy breaking is given by the gravitino mass , i.e.@xmath89 where @xmath90 denotes the planck mass and the scale of gaugino condensation @xmath87 is given by @xmath91 for every mssm - like orbifold model we compute the @xmath63-function of the largest hidden sector gauge group under the assumption that any non - trivial hidden matter representation of this gauge group can be decoupled in a supersymmetric way . furthermore , we assume dilaton stabilization at a realistic value @xmath60 . hence , we obtain the scale @xmath87 of gaugino condensation . our results are displayed in fig . [ fig : gauginocondenstaion ] . for an intermediate scale @xmath92 one obtains a gravitino mass in the @xmath93 range , which is of phenomenological interest . the models in the orbifoldlandscape seem to prefer low energy susy breaking . this result is strongly related to the heterotic orbifold construction : the @xmath11 gauge group in 10d is broken by orbifold shift and wilson lines , which are highly constrained by string theory ( i.e. modular invariance ) . therefore , both @xmath20 factors get broken and not only the observable one . it turns out that the unbroken gauge group from the hidden @xmath20 has roughly the correct size to yield gaugino condensation at an intermediate scale and hence low energy susy breaking . note that our analysis is just a rough estimate as various effects have been neglected , for example the decoupling of hidden matter , the identification of the gaugino condensation and ( string ) threshold corrections . these effects can in principle affect the scale of susy breaking even by 2 - 3 orders of magnitude . with these considerations we have only scratched the surface of the parameter space of potentially realistic models . in addition , we have used `` five golden rules '' as a prejudice for model selection and it has to be seen whether this is really justified . for general model building in the framework of ( perturbative ) string theory we have the following theories at our disposal : * type i string with gauge group @xmath94 * heterotic @xmath94 * heterotic @xmath11 * type iia and iib orientifolds * intersecting branes with gauge group @xmath95 as we explained in detail , our rule i points towards exceptional groups and hence towards the @xmath11 heterotic string . on the other hand , type ii orientifolds typically provide gauge groups of type @xmath96 or @xmath97 and products thereof . although we have @xmath98 gauge groups in these schemes , matter fields do not come as spinors of @xmath98 , but originate from adjoint representations . in the intersecting brane models based on @xmath95 gauge groups matter transforms in bifundamental representations of @xmath99 ( originating from the adjoint of @xmath100 ) . while this works nicely for the standard model representations , it appears to be difficult to describe a grand unified picture with e.g. gauge group @xmath9 . trying to obtain a gut yields a gauge group at least as large as @xmath101 and one has problems with a perturbative top - quark yukawa coupling . one possible way out is the construction of string models without the prejudice for guts , see e.g. @xcite . a comprehensive review on these intersecting brane model constructions can be found in the book of ibez and uranga @xcite or other reviews @xcite . these models have a very appealing geometric interpretation , see e.g.@xcite : fields are located on branes of various dimensions . thus , physical properties of the models can be inferred from the localisation of the brane fields in the extra dimensions and by the overlap of their wave functions , similar to the heterotic minilandscape . this nice geometrical set - up leads to attempts to construct so - called `` local models '' . here , one assumes that all particle physics properties of the model are specified by some local properties at some specific point or sub - space of the compactified dimensions and that the `` bulk '' properties can be decoupled . however , the embedding of the local model into an ultraviolet complete and consistent string model is an assumption and its validity remains an open question . further schemes include `` non - perturbative '' string constructions : * m - theory in @xmath8 * heterotic m - theory @xmath11 * f - theory these non - perturbative constructions are conjectured theories that generalize string theories or known supergravity field theories in higher dimensions . the low energy limit of m - theory is 11-dimensional supergravity . heterotic m - theory is based on a @xmath8 theory bounded by two @xmath0 branes with gauge group @xmath20 on each boundary and f - theory is a generalization of type iib theory , where certain symmetries can be understood geometrically . this non - perturbative construction allows for singularities in extra dimensions that lead to non - trivial gauge groups according to the so - called a - d - e classification . groups of the a - type ( @xmath102 ) and d - type ( @xmath98 ) can also be obtained in the perturbative constructions with d - branes and orientifold branes , while exceptional gauge groups can only appear through the presence of e - type singularities . this allows for spinors of @xmath10 and can produce a non - trivial top - quark yukawa coupling within an @xmath9 grand unified theory . in that sense , f - theory can be understood as an attempt to incorporate rule i within type iib theory . unfortunately , it is difficult to control the full non - perturbative theory and the search for realistic models is often based on local model building . many questions are still open but there is enough room for optimism that promising models can be embedded in a consistent ultraviolet completion . a general problem of string phenomenology is the difficulty to perform the explicit calculations needed to check the validity of the model . this is certainly true for the non - perturbative models , where we have ( at best ) some effective supergravity description . but also in the perturbative constructions we have to face this problem . we have to use simplified compactification schemes to be able to do the necessary calculations we need a certain level of `` berechenbarkeit '' . in our discussion we used the flat orbifold compactification that allows the use of conformal field theory methods . in principle , this enables us to do all the necessary calculations to check the models in detail . in the @xmath12-ii minilandscape this has been elaborated to a large extend . for the more general orbifold landscape , this still has to be done . other constructions with full conformal field theory control are the free fermionic constructions @xcite and the `` tensoring '' of conformal field theory building blocks : so - called gepner models @xcite . they share `` berechenbarkeit '' with the flat orbifold models , but the geometric structure of compactified space is less transparent . we have to hope that these simplified compactifications ( or approximations ) lead us to realistic models . in the generic situation one needs smooth manifolds , e.g. calabi - yau spaces , and some specific models have been constructed @xcite . however , these more generic compactifications require more sophisticated methods for computations that are only partially available , for example in order to determine yukawa couplings . more recently a simplification based on the embedding of line bundles has allowed the constructions of many models @xcite . still the calculational options are limited . it would be interesting to get a better geometric understanding of the compact manifold . at the moment the `` determination '' of couplings is based on a supergravity approximation using @xmath37 symmetries . these symmetries are exact in this approximation at the `` stability wall '' but are expected to be broken to discrete symmetries in the full theory . this is in concord with rule v asking for the origin of discrete symmetries . furthermore , this question has recently been analysed intensively within the various string constructions @xcite . we have seen that there is still a long way to go in the search for realistic particle physics models from string theory . there are many possible roads but we are limited by our calculational techniques . thus , in the near future we are still forced to make choices . here , we have chosen to follow `` five golden rules '' outlined in section [ sec : fivegoldenrules ] , which are mainly motivated by the quest for a unified picture of particle physics interactions . this strategy seems to require an underlying structure provided by exceptional groups pointing towards the @xmath11 heterotic string and f - theory . even given these rules , there are stumbling blocks because of the complexity of the compact manifolds . we can not resolve these problems in full generality : we have to use simplified compactification schemes or approximations . we have to hope that nature has chosen a theory that is somewhat close to these simplified schemes . of course , any method to go beyond this simplified assumptions should be seriously considered . however , there is some hope that this assumption might be justified : the orbifold models studied in this work have enhanced ( discrete ) symmetries that could be the origin of symmetries of the standard model , especially with respect to the flavour structure and symmetries relevant for proton stability as well as the absence of other rare processes . generically , these symmetries are slightly broken as we go away from the orbifold point . this gives rise to some hierarchical structures , for example for the ratio of quark masses in the spirit of froggatt and nielsen @xcite . the analysis of the minilandscape can be seen as an attempt to study these questions in detail . based on the availability of conformal field theory techniques we can go pretty far in the analysis of explicit models . a detailed analysis of the `` orbifoldlandscape '' has not been performed yet , but should be possible along the same lines . in section [ sec : orbifoldlandscape ] we started this enterprise of model building by constructing 12000 mssm - like models . in a next step , the detailed properties of promising models have to be worked out . especially the framework of the @xmath77 @xcite should provide new insight into the properties of realistic models and might teach us further key properties shared by successful models . one key property that we have learned is the geography of fields in the extra dimensions . the localisation of matter fields and the gauge group profiles in extra dimensions are essential for the properties of the low energy model . this is the first message of the heterotic orbifold construction and shared by the `` braneworld '' constructions in type ii string theory and f - theory . further lessons are : * the higgs pair is a bulk field . this allows for a convincing solution of the @xmath24-problem using a ( discrete ) r - symmetry and yields doublet triplet splitting . * a sizeable value of the top - quark yukawa coupling requires a sufficient overlap with the higgs fields in extra dimensions . thus , the top - quark should extend to the bulk as well . * the matter fields of the first and second generation should be localised in a region of the extra - dimensional space where they are subject to an enhanced gauge symmetry , like @xmath10 . this local gut forces them to appear as complete representations , e.g. as spinors of @xmath10 . furthermore , the geometrical structure can manifest itself in a discrete flavour symmetry . * the quest for low energy supersymmetry is the guiding principle in string model building . still , it has to be seen whether this is realised in nature . at the moment no sign of supersymmetry has been found at the lhc , although the value of the higgs mass is consistent with susy . the analysis of the models of the minilandscape and the location of the fields suggests a certain structure where even some remnants of extended supersymmetry ( for fields in the bulk ) seem to be at work . this picture of `` heterotic supersymmetry '' @xcite can hopefully be tested experimentally in the not too far future . this work was partially supported by the sfb transregio tr33 `` the dark universe '' ( deutsche forschungsgemeinschaft ) and the dfg cluster of excellence `` origin and structure of the universe '' ( www.universe-cluster.de ) . .statistics on mssm - like models ( using the search criteria listed in sec . [ sec : searchstrategy ] ) obtained from a random scan in all @xmath13 and certain @xmath14 heterotic orbifold geometries . the first column labels the geometry following the nomenclature from @xcite . the second column gives the number of inequivalent mssm - like models found in our scan . next , we give the maximal number of independent wilson lines ( wls ) possible for the respective orbifold geometry and in the fourth column we count the number of mssm - like models with a certain number ( i.e. 0,1,2,3,4 ) of vanishing wilson lines , see sec . [ sec : orbifoldlandscaperule3 ] . in the fifth column we count the number of locations with broken local gut such that higgs - doublets without triplets appear , see sec . [ sec : orbifoldlandscaperule2 ] . finally , in the last column we give the number of models without @xmath103 , i.e. without fi term . [ cols="^,^ , > , > , > , > , > , > , > , > , > , > , > , > , > , > , > " , ] h. p. nilles , ` arxiv : hep - th/0410160 ` [ hep - th ] , hep - th/0410160 . l. j. dixon , j. a. harvey , c. vafa , and e. witten , nucl.phys . * b261 * ( 1985 ) , 678 . l. j. dixon , j. a. harvey , c. vafa , and e. witten , nucl.phys . * b274 * ( 1986 ) , 285 . t. kobayashi , s. raby , and r .- j . zhang , nucl.phys . * b704 * ( 2005 ) , 3 , ` arxiv : hep - ph/0409098 ` [ hep - ph ] . w. buchmller , k. hamaguchi , o. lebedev , and m. ratz , nucl.phys . * b712 * ( 2005 ) , 139 , ` arxiv : hep - ph/0412318 ` [ hep - ph ] . w. buchmller , k. hamaguchi , o. lebedev , and m. ratz , phys.rev.lett . * 96 * ( 2006 ) , 121602 , ` arxiv : hep - ph/0511035 ` [ hep - ph ] . w. buchmller , k. hamaguchi , o. lebedev , and m. ratz , nucl.phys . * b785 * ( 2007 ) , 149 , ` arxiv : hep - th/0606187 ` [ hep - th ] . o. lebedev , h. p. nilles , s. raby , s. ramos - snchez , m. ratz , et al . , phys.lett . * b645 * ( 2007 ) , 88 , ` arxiv : hep - th/0611095 ` [ hep - th ] . o. lebedev , h. p. nilles , s. raby , s. ramos - snchez , m. ratz , et al . , phys.rev . * d77 * ( 2008 ) , 046013 , ` arxiv:0708.2691 ` [ hep - th ] . m. ratz , ` arxiv:0711.1582 ` [ hep - ph ] , arxiv:0711.1582 [ hep - ph ] . o. lebedev , h. p. nilles , s. ramos - snchez , m. ratz , and p. k. vaudrevange , phys.lett . * b668 * ( 2008 ) , 331 , ` arxiv:0807.4384 ` [ hep - th ] . t. kobayashi , h. p. nilles , f. plger , s. raby , and m. ratz , nucl.phys . * b768 * ( 2007 ) , 135 , ` arxiv : hep - ph/0611020 ` [ hep - ph ] . h. p. nilles , s. ramos - snchez , m. ratz , and p. k. vaudrevange , eur.phys.j . * c59 * ( 2009 ) , 249 , ` arxiv:0806.3905 ` [ hep - th ] . s. frste , h. p. nilles , p. k. vaudrevange , and a. wingerter , phys.rev . * d70 * ( 2004 ) , 106008 , ` arxiv : hep - th/0406208 ` [ hep - th ] . l. e. ibez , h. p. nilles , and f. quevedo , phys.lett . * b187 * ( 1987 ) , 25 . h. p. nilles , eur.phys.j . * c74 * ( 2014 ) . r. kappl , h. p. nilles , s. ramos - snchez , m. ratz , k. schmidt - hoberg , et al . , phys.rev.lett . * 102 * ( 2009 ) , 121602 , ` arxiv:0812.2120 ` [ hep - th ] . f. brmmer , r. kappl , m. ratz , and k. schmidt - hoberg , jhep * 1004 * ( 2010 ) , 006 , ` arxiv:1003.0084 ` [ hep - th ] . j. casas and c. muoz , phys.lett . * b306 * ( 1993 ) , 288 , ` arxiv : hep - ph/9302227 ` [ hep - ph ] . h. p. nilles , m. ratz , and p. k. vaudrevange , fortsch.phys . * 61 * ( 2013 ) , 493 , ` arxiv:1204.2206 ` [ hep - ph ] . c. froggatt and h. b. nielsen , nucl.phys . * b147 * ( 1979 ) , 277 . h. p. nilles , phys.lett . * b115 * ( 1982 ) , 193 . s. ferrara , l. girardello , and h. p. nilles , phys.lett . * b125 * ( 1983 ) , 457 . j. derendinger , l. e. ibez , and h. p. nilles , phys.lett . * b155 * ( 1985 ) , 65 . m. dine , r. rohm , n. seiberg , and e. witten , phys.lett . * b156 * ( 1985 ) , 55 . o. lebedev , h .- nilles , s. raby , s. ramos - snchez , m. ratz , et al . , phys.rev.lett . * 98 * ( 2007 ) , 181602 , ` arxiv : hep - th/0611203 ` [ hep - th ] . k. choi , a. falkowski , h. p. nilles , m. olechowski , and s. pokorski , jhep * 0411 * ( 2004 ) , 076 , ` arxiv : hep - th/0411066 ` [ hep - th ] . k. choi , a. falkowski , h. p. nilles , and m. olechowski , nucl.phys . * b718 * ( 2005 ) , 113 , ` arxiv : hep - th/0503216 ` [ hep - th ] . o. lebedev , h. p. nilles , and m. ratz , phys.lett . * b636 * ( 2006 ) , 126 , ` arxiv : hep - th/0603047 ` [ hep - th ] . v. nilles and j. plank , ccr * 42 * , 736 . o. lebedev , h. p. nilles , and m. ratz , ` arxiv : hep - ph/0511320 ` [ hep - ph ] , 211 , hep - ph/0511320 . o. loaiza - brito , j. martin , h. p. nilles , and m. ratz , aip conf.proc . * 805 * ( 2006 ) , 198 , ` arxiv : hep - th/0509158 ` [ hep - th ] . o. lebedev , v. lwen , y. mambrini , h. p. nilles , and m. ratz , jhep * 0702 * ( 2007 ) , 063 , ` arxiv : hep - ph/0612035 ` [ hep - ph ] . v. lwen and h. p. nilles , phys.rev . * d77 * ( 2008 ) , 106007 , ` arxiv:0802.1137 ` [ hep - ph ] . v. lwen and h. p. nilles , nucl.phys . * b827 * ( 2010 ) , 337 , ` arxiv:0907.4983 ` [ hep - ph ] , arxiv:0907.4983 [ hep - ph ] . s. krippendorf , h. p. nilles , m. ratz , and m. w. winkler , phys.lett . * b712 * ( 2012 ) , 87 , ` arxiv:1201.4857 ` [ hep - ph ] . m. badziak , s. krippendorf , h. p. nilles , and m. w. winkler , jhep * 1303 * ( 2013 ) , 094 , ` arxiv:1212.0854 ` [ hep - ph ] . m. fischer , m. ratz , j. torrado , and p. k. vaudrevange , jhep * 1301 * ( 2013 ) , 084 , ` arxiv:1209.3906 ` [ hep - th ] . h. p. nilles , s. ramos - snchez , p. k. vaudrevange , and a. wingerter , comput.phys.commun . * 183 * ( 2012 ) , 1363 , ` arxiv:1110.5229 ` [ hep - th ] . s. groot nibbelink and o. loukas , jhep * 1312 * ( 2013 ) , 044 , ` arxiv:1308.5145 ` [ hep - th ] . h. p. nilles and p. k. vaudrevange , _ the orbifoldlandscape : input files for the orbifolder _ , 2014 , ` http://www.th.physik.uni-bonn.de/ nilles / orbifoldlandscape/ ` . d. k. m. pena , h. p. nilles , and p .- k . oehlmann , jhep * 1212 * ( 2012 ) , 024 , ` arxiv:1209.6041 ` [ hep - th ] . j. e. kim and b. kyae , nucl.phys . * b770 * ( 2007 ) , 47 , ` arxiv : hep - th/0608086 ` [ hep - th ] . j. e. kim , j .- h . kim , and b. kyae , jhep * 0706 * ( 2007 ) , 034 , ` arxiv : hep - ph/0702278 ` [ hep - ph ] . s. ramos - snchez , fortsch.phys . * 10 * ( 2009 ) , 907 , ` arxiv:0812.3560 ` [ hep - th ] , ph.d.thesis ( advisor : h.p . nilles ) . k. r. dienes , phys.rev . * d73 * ( 2006 ) , 106010 , ` arxiv : hep - th/0602286 ` [ hep - th ] . k. r. dienes , m. lennek , d. senechal , and v. wasnik , phys.rev . * d75 * ( 2007 ) , 126005 , ` arxiv:0704.1320 ` [ hep - th ] . b. gato - rivera and a. schellekens , ` arxiv:1401.1782 ` [ hep - ph ] , arxiv:1401.1782 [ hep - ph ] . l. e. ibez and a. m. uranga , cambridge , uk : univ . pr . ( 2012 ) 673 p. r. blumenhagen , b. krs , d. lst , and s. stieberger , phys.rept . * 445 * ( 2007 ) , 1 , ` arxiv : hep - th/0610327 ` [ hep - th ] . g. honecker , m. ripka , and w. staessens , nucl.phys . * b868 * ( 2013 ) , 156 , ` arxiv:1209.3010 ` [ hep - th ] . a. e. faraggi , phys.lett . * b278 * ( 1992 ) , 131 . t. dijkstra , l. huiszoon , and a. schellekens , nucl.phys . * b710 * ( 2005 ) , 3 , ` arxiv : hep - th/0411129 ` [ hep - th ] . r. donagi , b. a. ovrut , t. pantev , and d. waldram , adv.theor.math.phys . * 5 * ( 2002 ) , 93 , ` arxiv : hep - th/9912208 ` [ hep - th ] . v. braun , y .- h . he , b. a. ovrut , and t. pantev , phys.lett . * b618 * ( 2005 ) , 252 , ` arxiv : hep - th/0501070 ` [ hep - th ] . l. b. anderson , j. gray , a. lukas , and e. palti , phys.rev . * d84 * ( 2011 ) , 106005 , ` arxiv:1106.4804 ` [ hep - th ] . l. b. anderson , j. gray , a. lukas , and e. palti , jhep * 1206 * ( 2012 ) , 113 , ` arxiv:1202.1757 ` [ hep - th ] . m. berasaluce - gonzalez , l. e. ibez , p. soler , and a. m. uranga , jhep * 1112 * ( 2011 ) , 113 , ` arxiv:1106.4169 ` [ hep - th ] . l. ibez , a. schellekens , and a. uranga , nucl.phys . * b865 * ( 2012 ) , 509 , ` arxiv:1205.5364 ` [ hep - th ] . m. berasaluce - gonzalez , p. camara , f. marchesano , d. regalado , and a. uranga , jhep * 1209 * ( 2012 ) , 059 , ` arxiv:1206.2383 ` [ hep - th ] . anastasopoulos , m. cveti , r. richter , and p. k. vaudrevange , jhep * 1303 * ( 2013 ) , 011 , ` arxiv:1211.1017 ` [ hep - th ] . g. honecker and w. staessens , jhep * 1310 * ( 2013 ) , 146 , ` arxiv:1303.4415 ` [ hep - th ] .
|
string theoretical ideas might be relevant for particle physics model building .
ideally one would hope to find a unified theory of all fundamental interactions .
there are only few consistent string theories in @xmath0 or @xmath1 space - time dimensions , but a huge landscape in @xmath2 .
we have to explore this landscape to identify models that describe the known phenomena of particle physics .
properties of compactified six spatial dimensions are crucial in that respect .
we postulate some useful rules to investigate this landscape and construct realistic models .
we identify common properties of the successful models and formulate lessons for further model building .
+ * geography of fields in extra dimensions : + string theory lessons for particle physics * * hans peter nilles@xmath3 and patrick k.s .
vaudrevange@xmath4 * + @xmath5 bethe center for theoretical physics + and + physikalisches
institut der universitt bonn , + nussallee 12 , 53115 bonn , germany + @xmath6 excellence cluster universe , technische universitt mnchen , + boltzmannstr .
2 , d-85748 , garching , germany + and + arnold sommerfeld center for theoretical physics , lmu + theresienstrae 37 , 80333 mnchen , germany
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
one of the principal problems in solar physics is understanding how the sun s corona is heated to very high temperatures . recent work on coronal loops indicates that they have physical properties that are difficult to reconcile with theoretical models . coronal loops with temperatures near 1mk are observed to persist longer than a characteristic cooling time , suggesting steady or quasi - steady heating ( e.g. , * ? ? ? * ; * ? ? ? steady heating models , however , can not reproduce the high electron densities observed in these loops @xcite . multi - thread , impulsive heating models have been proposed as a possible heating scenario ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? such models are motivated by our understanding of the energy release during magnetic reconnection in flares ( e.g. , * ? ? ? * ) . in these models impulsive heating leads to high densities and multiple , sub - resolution `` threads '' lead to long lifetimes relative to the cooling time for an individual loop . these models are severely constrained by the relatively narrow distributions of temperatures that are often observed in loops with apex temperatures near 1mk ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? a narrow distribution of temperatures suggests that the loop can contain only a few independent threads . one difficulty with fully testing coronal heating scenarios such as these with hydrodynamic models has been the spareness of data . previous work on loop evolution has generally focused on measurements imaging instruments ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , which have limited diagnostic capabilities . current solar observatories , however , allow for coronal loops to be observed in unprecedented detail . the euv imaging spectrometer ( eis ) on the _ hinode _ mission provides high spatial and spectral resolution observations over a very wide range of coronal temperatures . eis plasma diagnostics yield important constraints on the physical properties of coronal loops . the x - ray telescope ( xrt ) on _ hinode _ complements these observations with high spatial and temporal resolution observations of the high temperature corona . the multiple viewpoints of the twin _ stereo _ spacecraft allow for loop geometry , a critical parameter in the modeling , to be measured using the euv imagers ( euvi ) . the _ transition region and coronal explorer _ ( _ trace _ ) currently provides the highest spatial resolution images of the solar corona . in this paper we use _ stereo _ , _ hinode _ , and _ trace _ observations of an evolving loop in a post - flare loop arcade to make quantitative comparisons between a multi - thread , impulsive heating model and measured densities , temperatures , intensities and loop lifetimes . an important component of this work is the development of methods for integrating the different observations into hydrodynamic simulations of the loop . we find that it is possible to reproduce the extended loop lifetime , the high electron density , and the narrow differential emission measure ( dem ) with a multi - thread model provided the time scale for the energy release is sufficiently short . the model , however , does not reproduce the evolution of the high temperature emission observed with xrt . one goal of investigating the heating on individual loops is to motivate the modeling of entire active regions or even the full sun ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? it is possible , however , that there is not a single coronal heating mechanism that can be applied to all coronal loops . for example , it may be that steady heating is the dominant heating scenario on some fraction of coronal loops ( e.g. , * ? ? ? * ; * ? ? ? * ) . even if impulsive heating of the kind discussed here is only a minor contributor to the heating of the solar corona , this study provides important insights into the energy release during magnetic reconnection , a fundamental process in astrophysical and laboratory plasmas . in this section we provide an overview of the instruments and observations used in this study . a summary of the observations is shown in figure [ fig : summary ] . the loop considered here is a post - flare loop from a very small event ( goes class b2.5 ) that peaked around 19:00 ut on may 2 , 2007 . the eis instrument on _ hinode _ produces stigmatic spectra in two wavelength ranges ( 171212 and 245291 ) with a spectral resolution of 0.0223 . there are 1 and 2 slits as well as 40 and 266 slots available . the slit - slot mechanism is 1024 long but a maximum of 512 pixels on the ccd can be read out at one time . solar images can be made using one of the slots or by stepping one of the slits over a region of the sun . telemetry constraints generally limit the spatial and spectral coverage of an observation . see @xcite and @xcite for more details on the eis instrument . for these observations the 1 slit was stepped over the active region and 15s exposures were taken at each position . an area of @xmath0 was imaged in about 71 minutes . a total of 20 spectral windows were read out of the ccd and included in the telemetry stream . the raw data were processed using ` eis_prep ` to remove dark current , warm pixels , and other instrumental effects using standard software . during the processing the observed count rates are converted to physical units . intensities from the processed data are computed by fitting the observed line profiles with gaussians . the eis rasters are co - aligned to account for any spatial offsets ( see @xcite for a discussion ) . spacecraft jitter during the raster has not been accounted for . hinode _ housekeeping logs suggest relatively small displacements ( less than one pixel ) for the narrow field of view of interest here . for larger structures spacecraft jitter can be important . eis rasters in a number of different emission lines are shown in figure [ fig : eis ] , and show post - flare loops at various temperatures in the lower part of the active region . these rasters also indicate a brief data gap due to orbital eclipse . one limitation of these eis data is the lack of temporal information . better information on the temporal evolution of these loops is provided by the imaging instruments , such as the euvi @xcite on the _ solar terrestrial relations observatory _ ( _ stereo _ ) mission . the euvi is a normal incidence , multilayer telescope which can observe the sun in 4 wavelength bands centered at 284 , 195 , 171 , and 304 . euvi observes the full sun and therefore has reduced spatial resolution ( 1.6 pixels ) relative to the other observations that we consider here . there are two _ stereo _ spacecraft with identical instrument packages . the twin _ stereo _ spacecraft drift away from the earth at about 23@xmath1 per year . on may 2 , 2007 the separation between the spacecraft was small , about 6@xmath1 . the euvi images taken around the time of the eis raster are indicated in figure [ fig : summary ] . because of telemetry constraints the image cadence is limited . for these observations 171 images were taken at a relatively high cadence ( @xmath2s ) while the images at the other wavelengths were taken at lower cadences ( @xmath31200s ) . the raw data are processed using ` euvi_prep ` to produce calibrated , co - aligned images . images of the active region and flare from _ stereo b _ euvi are shown in figure [ fig : euvi ] . the xrt on _ hinode _ is a high cadence , high spatial resolution ( approximately 1 pixels ) grazing incidence telescope that images the sun in the soft x - ray and extreme ultraviolet wavelength ranges . temperature discrimination is achieved through the use of focal plane filters . because xrt can observe the sun at short wavelengths , xrt images can observe high temperature solar plasma very efficiently . the thinner xrt filters allow longer wavelength euv emission to be images and extend the xrt response to lower temperatures . further details on xrt are given in @xcite . as indicated in figure [ fig : summary ] , the principal xrt images taken during this time period were in three filters , ti - poly , al - thick , and be - thick , at a variable cadence . unfortunately , the exposure times on the be - thick images are too short for the images to be used for analyzing active region loops . the standard processing routine ` xrt_prep ` is used to remove the ccd bias , dark current , and calibrate the images . images are also `` dejittered '' using _ hinode _ spacecraft housekeeping data so that the images are co - aligned with respect to each other . _ hinode _ tracks solar rotation so there is no need to account for it in the images . example ti - poly images are shown in figure [ fig : xrt ] . these images have a field of view of @xmath4 . the _ trace _ instrument is a high resolution normal incidence telescope . the primary and secondary mirrors are divided into quadrants and a rotating shutter is used to select which quadrant is illuminated . three of the quadrants are coated with multilayers for imaging at euv wavelengths . the multilayer coatings have peak sensitivities at approximately 171 , 195 , and 284 . the fourth quadrant is coated with aluminum and magnesium fluoride for imaging very broad wavelength ranges near 1216 , 1550 , 1600 , and 1700 . images in all of the wavelengths are projected onto a single detector , a @xmath5 ccd . each ccd pixel represents a solar area approximately 0.5 on a side . the instrument is described in detail by @xcite . the initial in - flight performance is reviewed by @xcite and @xcite . during this period _ trace _ observed mainly in the 171 channel at a cadence of about 60s with occasional 1600 and white light context images . all of the images have a @xmath4 field of view . as shown in figure [ fig : summary ] , there are periodic data gaps in the trace data due to orbital eclipses . all of the _ trace _ images are processed using a standard application of ` trace_prep ` . additionally , the images are despiked and co - aligned with respect to each other to account for solar rotation and drifts in the pointing . a simple cross - correlation method is used for this purpose . the _ trace _ 171 images are very similar in appearance to the euvi 171 images . the primary goal of this study is to compare multi - thread , hydrodynamic simulations with the emission observed in an evolving coronal loop . as we will discuss in more detail later in the paper , hydrodynamic simulations involve solving the equations for the conservation of mass , momentum , and energy in the loop given some input heating rate . relating the heating rate to physical observables is a critical element of the modeling . our strategy is to measure the electron density with eis and use a family of hydrodynamic simulations to infer the required heating rate for this density . previous numerical simulations suggest that for a fixed loop length there is a power law relationship between the peak electron density and the input energy @xcite . since we want the loop length to be fixed , the other critical element of this modeling is an accurate measurement of the loop geometry , including the inclination . observations from the twin euvi instruments allow the loop geometry to be measured , and we use the _ stereo _ software package developed for this purpose @xcite . once the density and loop geometry have been determined we can perform hydrodynamic simulations and synthesize the expected emission during the entire evolution of the loop . the simulation results can then be compared with light curves determined from _ trace _ and xrt and in this section we discuss how these light curves are calculated . the distribution of temperatures in the loops is an important constraint on the modelings and in this section we also discuss the calculation of the differential emission measure distribution with the eis spectra . our analysis requires the identification of loops observed simultaneously with both eis and _ trace_. to facilitate this we wrote routines to co - align eis rasters with _ trace _ images and to display 24 bit color images using an eis raster for one color channel and a trace image for another color channel . an animation of these images allowed us to quickly identify times when the eis slit was co - spatial with a loop observed with _ _ trace _ s small field of view and frequent data gaps due to orbital eclipse make finding good data sets more difficult than anticipated . to optimize the co - alignment between eis and _ trace _ we had to allow for a roll angle between the images . this is in addition to the usual spatial shifts between the pointing information contained in the data headers . since the _ trace _ data was taken in the /x 171 channel we used the eis 184.536 raster for co - alignment . the eis intensities for the loop of interest are summarized in figure [ fig : eis_ints ] . once the region of interest was identified we manually selected spatial positions along the loop in the eis 195.119 raster . these points are used as spline knots to define the loop coordinate system ( @xmath6 ) , with @xmath7 along the loop and @xmath8 perpendicular to it ( see @xcite figure 3 ) . since these loop coordinates are not necessarily aligned to the ccd we have interpolated to determine the intensities along the selected segment . the loop segments displayed in figure [ fig : eis_ints ] are interpolated to 0.2 per pixel . using the loop coordinate system it is a simple matter to compute the intensity averaged along the loop segment . the coordinate system derived from 195.119 is used for all of the eis rasters . to further isolate the intensity of the loop we identify two background points and fit a single gaussian with a linear background to the selected region . background subtraction is essential to separating the intensity in the loop from the contribution of the ambient corona , but there is no unique method for computing it . analysis of euvi data , which has the advantage of providing two different lines of sight for a single loop , suggests that the background subtracted intensities can be computed consistently , although there can be considerable uncertainties for individual measurements @xcite . since we are interested in emission that is co - spatial we extract the same region from all of the eis rasters . the resulting intensities are shown in figure [ fig : eis_ints ] . to test how co - spatial the emission is at various temperatures , we have calculated a simple correlation coefficient between the background subtracted intensity in each line and 195.119 , which represents a middle ground between the highest and lowest temperature emission that is observed . for this loop well correlated , co - spatial emission is observed for and below . for the emission at higher temperatures ( xvi ) the correlation is poor . it is clear from the images shown in figures [ fig : eis ] and [ fig : eis_ints ] that the combination of high spatial resolution and good temperature discrimination allows eis to probe the interrelationship of emission at different temperatures . the loop intensities suggest that the emission at different temperatures is generally not co - spatial and that the dem in the loop of interest should be relatively narrow . this is shown more clearly in the `` multicolor '' image of this region presented in figure [ fig : eis_color ] . this image , which is a 24 bit image formed from rasters in 3 different emission lines , would be white in regions where the dem is broad and the emission is strong in all three lines . there are some composite regions that are cyan ( green @xmath9 blue ) , but the post - flare loop arcade is generally dominated by emission in the primary colors suggesting relatively narrow distributions of temperature in each loop . to investigate the temperature structure of this loop more quantitatively we compute the differential emission measure using the background subtracted loop intensities for the loop segment . the intensities are related to the differential emission measure by the usual expression @xmath10 where @xmath11 is the plasma emissivity and @xmath12 is the differential emission measure distribution . we consider a gaussian dem model @xmath13,\ ] ] which allows for a dispersion in the temperature distribution . since the density is an important parameter in determining the emissivities of many of these lines we leave it is a free parameter . to determine the best - fit parameters ( @xmath14 , @xmath15 , @xmath16 , @xmath17 ) we use a levenberg - marquardt technique implemented in the ` mpfit ` package the chianti 5.2 atomic physics database ( e.g. , @xcite ) is used to calculate the emissivities . the abundances of @xcite and the low density ionization fractions of @xcite are assumed . there are several subtleties to computing the emission measure parameters . one is the statistical uncertainty associated with each intensity . because the intensities are averaged over a significant area the statistical uncertainties are generally small . the systematic errors introduced by the background subtraction and the atomic data , however , are large , but difficult to estimate . we simply assume that the relative error is 20% of the measured intensity . another question is how to deal with the emission from lines such as 262.984 , that do not show evidence for the loop and have no measured intensity . for these lines the background subtracted intensity is zero . observations of these lines provide important constraints on the high temperature component of the dem and must be included . to account for possible errors in computing the background subtracted intensities for these lines we assume that the uncertainty in the intensity is 20% of the background instead of 20% of the loop intensity . since the background can be large this represents a substantial enhancement of the uncertainty . the resulting dem for this loop segment is shown in figure [ fig : eis_ints ] . the observed and computed intensities are given in table [ table : ints ] . from this analysis we obtain an electron density of @xmath18 . for this work we use the 203.826/202.044 lines to provide the bulk of the density sensitivity . recently @xcite have noted systematic discrepancies between the various density sensitive line ratios from and . in light of this we compared the densities inferred from 203.826/202.044 and 186.880/195.119 with those from 258.375/261.058 in a series of other active region and quiet sun observations . the lines , which were not included in this study , are relatively weak and not sensitive over as large a range as the and lines . however , the atomic data for is potentially more reliable than the atomic data available for the complex fe ions . we find that the densities derived from the and ratios are in excellent agreement and emphasize the ratio here . the densities inferred from 186.880/195.119 are as much as a factor of 3 higher . details of this analysis will be presented in a future paper . the dispersion in the temperature is found to be @xmath19 , which is comparable to other active region loop observations with eis @xcite . these lines observed in the quiet corona above the limb indicate much narrower temperature distributions ( @xmath20 , @xcite ) . here we find a dispersion in temperature that is several times greater , indicating that this loop is not strictly isothermal . the relatively intense , co - spatial emission observed from both and provides the best direct evidence for a distribution of temperatures in this loop . these lines have peak temperatures of formation that are about 1mk apart . the application of a delta function emission measure to equation [ eq : ints ] confirms that a single temperature model can not adequately reproduce the intensities observed in these emission lines . in searching for coronal loops observed with both eis and _ trace _ we co - aligned the eis rasters with the _ trace _ images for this period . thus to compute the _ trace _ intensities we use the same coordinate system and apply it to all of the _ trace _ images taken during the time of interest . since the co - alignment between eis and _ trace _ is not perfect , the spline knots selected in the eis raster are modified slightly to better align with the loop . background subtracted intensities are computed for each of the available _ trace _ images taken during this time using the same procedure that was used on the eis data . for each image the region defined by the spline knots was extracted , straightened , and averaged along the loop coordinate to produce the intensity as a function of the perpendicular coordinate . the intensity at each time refers to the background subtracted intensity integrated over the loop . the _ trace _ light curve is shown in figure [ fig : trace_ints ] . for most of this period the background subtracted loop intensities are in the noise and the evolution of this loop can be seen clearly . we have manually selected the region around the peak in the light curve and fit it with a single gaussian . this fitting yields a gaussian width of about 294s . to extract the three dimensional geometry of this loop we use the software package developed by markus aschwanden and the _ stereo _ team for this purpose . an application of this software is discussed in detail in @xcite and @xcite . the initial processing co - registers images from _ stereo a _ and _ b _ to account for differences in spacecraft roll angle and spatial resolution . the next step is to outline the loop in the _ stereo a _ image . the selected coordinates are projected onto the corresponding _ b _ image . since geometry of the loop has not yet been determined this projection yields a range of possible coordinates in the _ b _ image and the user selects the position of the loop within this range . once the coordinates of the loop have been selected in both images the geometry of the loop is determined from simple trigonometric relationships . due to the gap in the _ a _ data , an image pair at 21:16 ut , after the peak emission observed in this loop , is used , so there is some ambiguity in the loop identification . the selection of nearby structures in the loop arcade generally yield very similar results . the projection of the extracted loop geometry onto the euvi _ b _ image available at the peak of the loop emission ( 20:59 ut ) also outlines the observed loop very well . thus it does not appear that the calculation of the loop geometry is significantly impacted by the data gap . the selected loop and the projection of this loop in various planes is shown in figure [ fig : euvi_stereo ] . images taken with the xrt provide information on the evolution of the loop at high temperatures . we have computed an xrt light curve similar to that computed for _ trace_. to use the loop coordinates derived from eis we first co - align the xrt images with the eis 262.984 raster , which shares some common features . we then compute the background subtracted intensities by selecting two background points and doing a linear fit . the emission seen earlier in the event is much broader than what is observed trace and so we select a wider area to compute the intensities . the resulting xrt light curve is shown in figure [ fig : xrt_ints ] . as is indicated by the light curve , the xrt images clearly show strong emission in the region that is eventually occupied by the loops observed with eis and _ trace_. it is also clear , however , that xrt does not show any individual loops that are as narrow as those that are seen at cooler temperatures . consistent with this , the loop cross section measured with xrt is systematically wider than what is measured with eis and _ trace_. to illustrate these differences we have constructed multicolor images from xrt and _ trace_. these images , which are presented in figure [ fig : xrt_trace ] , use different color channels to display the xrt and _ trace _ data in the same picture . to illustrate the differences in morphology as the plasma cools we have offset the times of the selected images by one hour . we show examples of the loop cross sections in figure [ fig : xrt_trace_ints ] . this difference between high temperature and low temperature emission in flares and active regions is well documented ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , and may be further evidence for fine scale structure in the solar corona . we will discuss this point in more detail later in the paper . one of the primary paradoxes of coronal loops with temperatures near 1mk is the disparity between the rapid cooling suggested by the high electron densities and the relatively long observed lifetimes . eis density diagnostics allow us to make more rigorous comparisons between these time scales . densities inferred from observed intensities requires an accurate determination of the differential emission measure , the absolute instrumental calibration , and the loop geometry . this measurement also represents a lower bound on the density . the density inferred from density sensitive line ratios circumvents many of these problems . if we assume that the loop is cooling only through radiation the energy balance is simply @xmath21 where @xmath22 is the radiative loss function for an optically thin plasma . in the limit of no flows the energy is @xmath23 . the plasma pressure given by @xmath24 , with @xmath25 is the boltzmann constant . the radiative cooling time is defined as @xmath26 and is given by @xmath27 using the temperature and density derived from the eis dem analysis ( @xmath28 and @xmath29 ) and a radiative loss rate of @xmath30erg @xmath31 s@xmath32 @xcite we obtain a radiative cooling time of 341s . the radiative cooling time is not directly comparable to the loop lifetime that we have measured with _ trace_. if we make the additional assumptions that @xmath33 and @xmath34 we can relate the radiative cooling time ( @xmath35 ) to the timescale for changes in the temperature ( @xmath36 ) @xmath37 numerical simulations suggest that @xmath38 @xcite . finally , to compare with the observed loop lifetime we must incorporate the temperature changes into the trace temperature response curve . this yields @xmath39\ ] ] where @xmath40 is the peak temperature of the trace response and @xmath41 is the gaussian width of the trace response ( see @xcite equation 9 ) . this yields @xmath42 with @xmath43mk and @xmath44mk @xcite we obtain @xmath45s , which is smaller than the observed gaussian width of 294s . this mismatch between the predicted and observed loop lifetime is one of the key motivations for the multi - thread modeling of coronal loops . by assuming that the observed emission comes from a series of loops that are heated at different times it is clear that we can create a composite loop with the required lifetime . the challenge is to also match the relatively narrow dem and the lifetime of the loop as it is observed with xrt . to simulate the evolution of this loop we consider numerical solutions to the full hydrodynamic loop equations using the nrl solar flux tube model ( solftm ) . we adopt many of the same parameters and assumptions that were used in previous simulations with this code and we refer the reader to the earlier papers for additional details on the numerical model ( e.g. , @xcite ) . for example , we assume that the loop is symmetric and only simulate the evolution over half of the loop length . we also assume that each loop has a constant cross section . for this work we consider a heating function for each thread that is a simple spatially uniform heating rate @xmath46 where @xmath47 is a triangular envelope that peaks at time @xmath48 and has width @xmath49 , which we set to be 100s . the parameter @xmath50 is a small background heating rate that provides the initial loop equilibrium . as we mentioned earlier , our strategy for inferring the heating rates from the observations is to use the density determined from eis and the results from systematic hydrodynamic simulations . this will determine the peak heating rate for the ensemble of threads . the heating rate for the other threads will be determined by assuming a gaussian envelope for the heating function . to determine the relationship between the input heating rate and the densities and temperatures observed during the cooling phase of the loop evolution , we have performed 21 simulations with @xmath51 varying between @xmath52 and @xmath53erg @xmath54 s@xmath32 . using the parameters derived from the euvi observations , the loop length is fixed at 135 mm and the loop inclination is fixed at 68.5@xmath1 . for each simulation we average over the loop apex to determine a representative density and temperature . the simulation results are summarized in figure [ fig : apex ] . we find that the apex density at 1.3mk , which is the peak temperature in the dem , essentially scales as @xmath55 . this implies that for a fixed loop length the observed intensity is linearly proportional to the input energy . the relationship shown in figure [ fig : apex ] indicates that a heating rate of 0.8erg @xmath54 s@xmath32 is required to reproduce an apex density of @xmath56 . the density - heating rate relationship is valid for a single loop . for a multi - thread simulation we assume the heating rate for each thread is related to this peak heating rate by @xmath57,\ ] ] where @xmath58 determines the duration of the heating envelope . the parameter @xmath59 is chosen so that all of the times are positive . the heating events are spaced so that as the heating in the previous loop ends the heating in the next loop begins . this heating scenario is illustrated in figure [ fig : eflare ] for @xmath60 , 200 , and 300s . once the individual hydrodynamic simulations are run , we average over the loop apex at each time step to compute a representative temperature and density . these densities and temperatures are then used as inputs to the _ trace _ temperature response to calculate the expected count rates in the _ trace _ 171 , 195 , and 284 channels as function of time . the simulation times are shifted so that the peak in the 171emission corresponds to the observed peak . since we are not interested in resolving the differences in absolute calibration among the various instruments , we also introduce a scaling factor so that the peak simulated emission matches what is observed . the simulation results are shown in figure [ fig : trace_sim ] and indicate that @xmath61s simulation , which yields a simulated loop lifetime of @xmath62s , best matches the observations . we also use the simulated densities and temperatures as a function of time to compute the expected intensities in many of the emission lines that can be observed with eis . light curves from selected emission lines are shown in figure [ fig : eis_sim ] . since the absolute time for the simulation has been established through the comparisons with _ trace _ , we select the simulated eis intensities that correspond to the time of the eis observations and use them as inputs to the same differential emission measure code that was used to produce figure [ fig : eis_ints ] . the resulting simulated dem is shown in figure [ fig : eis_sim ] . the simulated intensities are given in table [ table : ints ] . the agreement between the observed and simulated differential emission measure is relatively good . the simulation captures the salient features of the observations , a relatively high density and a narrow dem . for these simulated intensities we do obtain a somewhat lower electron density ( @xmath63 for the simulation and 9.7 for the observation ) and peak temperature ( @xmath64 for the simulation and 6.11 for the observation ) . the dispersion in the dem also does not match the observation exactly ( @xmath65 and 5.48 ) . given the approximate nature of the simulations we consider these discrepancies to be small . the difference in the density comes about because we have inferred the heating rate from a family of single - loop hydrodynamic simulations but the emission is actually a composite from several threads . it is likely that iterating on this solution would yield better agreement with the observations , but this is unlikely to yield addition physical insights . finally , we have simulated the expected xrt emission for the open / ti - poly filter combination using the standard xrt software routine ` xrt_t_resp ` . the simulated and observed light curves are shown in figure [ fig : xrt_sim ] . this comparison presents the greatest challenge to the modeling . the modeled composite intensities , which have been scaled to match the observations at 20:00 ut , clearly do not extend back in time enough to cover the entire evolution of the emission in this region . the peak observed emission in this loop occurs at approximately 19:30 ut , before the simulation has even begun . given the diffuse nature of the xrt emission and the difficulty of isolating individual loops at high temperatures perhaps the simplest explanation may be that the xrt light curve includes the contributions of many loops in addition to the loop we isolated using the eis and _ trace _ data . the identification of individual loops at very high temperatures with xrt is likely to be hampered by the slow evolution of plasma during the conductive cooling phase . this is evident by the slow evolution of the threads in the simulation . in figures [ fig : eis_sim ] and [ fig : xrt_sim ] , for example , we see that the threads last for approximately 1 hour at high temperatures . at the lower temperatures , when radiative losses are much higher , the cooling is dominated by radiation and the evolution is much faster . in these simulations the threads last only for about 10 minutes in the _ trace _ 171 bandpass . this difference suggests that the differentiated loops seen at lower temperatures , such as those illustrated in figure [ fig : eis_color ] , would appear as a single structure in xrt . these differences are also related to the broad temperature response of xrt . the relatively narrow line emission imaged with eis and _ trace _ emphasizes small differences in temperature . alternatively , the inability of the model to reproduce the observed xrt emission may reflect inadequacies with the hydrodynamic simulations during the conductive phase of the cooling . it may be that the heating is not as impulsive as we have assumed . many previous studies have suggested a gradual decay in the heating ( e.g. , * ? ? ? these differences in the assumed heating may be related to the changes in the topology of the magnetic field during the evolution of the event . observations of post - flare loop arcades have shown that newly reconnected field lines relax from cusp - shaped to approximately semi - circular during the early phases of the cooling @xcite . the comparisons between the hot and cool emission shown in figure [ fig : xrt_trace ] clearly suggests that field line shrinkage is occurring in this event . the cool post - flare loops observed with _ trace _ are generally observed at the lowest heights of the arcade and do not overlap with the high temperature xrt emission seen at the top of the arcade . the implications of field line shrinkage on hydrodynamic simulations has not been investigated . in general , the conversion of magnetic energy into thermal energy through the process of magnetic reconnection is not well understood ( see , for example , @xcite ) . more detailed analysis of mhd simulations is needed to better understand the evolution of coronal loops after reconnection ( e.g. , * ? ? ? in this paper we have made use of the unprecedented opportunity to observe evolving coronal loops in detail . we have used _ stereo _ , eis , _ trace _ , and xrt data to constrain a multithread model of coronal heating and compare with observations . these comparisons indicate that it is possible to reproduce the high densities , long lifetimes , and relatively narrow emission measure distributions inferred from the data so long as the heating envelope of the heating is sufficiently narrow . the most challenging comparisons are with xrt , where the model fails to reproduce the extended lifetime of the emission at high temperatures . it is not clear if this is due to our inability to isolate narrow loops at high temperatures or to problems with the assumed envelope on the heating . recent analysis of the eis spectral range has identified , xv , xvi , and xvii emission lines that can be used in the analysis of high temperature plasma @xcite . these lines will provide additional information on plasma evolution during the conductive phase and have been incorporated into the latest eis observing sequences . new active region observations should be available during the rise of the next solar cycle . ultimately our goal is to apply the multithread modeling described here to non - flaring active region loops . it is encouraging that the simulated eis differential emission measure curve derived here is similar to those derived for lower density active region loops @xcite . it remains to be seen , however , that this models can also reproduce the loop evolution observed in _ trace _ and xrt . the launch of the atmospheric imaging assembly ( aia ) on the _ solar dynamics observatory _ ( _ sdo _ ) , which will combine full disk imaging , _ trace_-like spatial resolution , 10s cadences , and multiple filters , will greatly expand the number of useful active region observations that combine eis plasma diagnostics and loop evolution . the authors would like to thank jim klimchuk for helpful discussions on the time scales for radiative cooling , and markus aschwanden for assistance with the stereo loop geometry code . amy winebarger contributed significant improvements to the interface to the hydrodynamic code . hinode is a japanese mission developed and launched by isas / jaxa , with naoj as domestic partner and nasa and stfc ( uk ) as international partners . it is operated by these agencies in co - operation with esa and nsc ( norway ) . , s. , antiochos , s. k. , & klimchuk , j. a. 2002 , in esa special publication , vol . 505 , solmag 2002 . proceedings of the magnetic coupling of the solar atmosphere euroconference , ed . h. sawaya - lacoste , 207 , 200 , and 300s . the simulation times have been adjusted so that the peak emission in 171 matches what is observed . the magnitude of the simulated emission has also been scaled to match the observations . the lifetime of the loop is calculated from a gaussian fit to the composite light curve . the @xmath66s simulation most closely matches to observed loop lifetime . ] lrrrr 275.352 & 37.2 & 37.4 & 66.7 & 61.2 + 184.536 & 190.3 & 194.4 & 339.5 & 342.6 + 188.216 & 436.9 & 357.9 & 439.7 & 496.7 + 195.119 & 718.6 & 853.2 & 718.6 & 773.7 + 202.044 & 595.5 & 545.1 & 301.2 & 277.1 + 203.826 & 207.9 & 200.3 & 120.7 & 120.9 + 274.203 & 0.0 & 106.1 & 33.9 & 28.7 + 284.160 & 0.0 & 296.1 & 36.9 & 31.2 + 262.984 & 0.0 & 2.71 & 0.1 & 0.1 + [ table : ints ]
|
the high densities , long lifetimes , and narrow emission measure distributions observed in coronal loops with apex temperatures near 1mk are difficult to reconcile with physical models of the solar atmosphere .
it has been proposed that the observed loops are actually composed of sub - resolution `` threads '' that have been heated impulsively and are cooling .
we apply this heating scenario to nearly simultaneous observations of an evolving post - flare loop arcade observed with the euvi/_stereo _ , xrt/_hinode _ , and _ trace _ imagers and the eis spectrometer on _
hinode_. we find that it is possible to reproduce the extended loop lifetime , high electron density , and the narrow differential emission measure with a multi - thread hydrodynamic model provided that the time scale for the energy release is sufficiently short .
the model , however , does not reproduce the evolution of the very high temperature emission observed with xrt . in xrt
the emission appears diffuse and it may be that this discrepancy is simply due to the difficulty of isolating individual loops at these temperatures .
this discrepancy may also reflect fundamental problems with our understanding of post - reconnection dynamics during the conductive cooling phase of loop evolution .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the performance of atomic clocks has improved more than seven orders of magnitude since the first cesium clock was demonstrated by essen and parry over 50 years ago @xcite . in the past two decades , improvement has been driven by several major scientific developments . most recently , optical frequency combs , narrow line width lasers , and methods of confining atoms and ions without perturbing the clock frequency have led to optical clocks , with reference frequencies and corresponding @xmath7s orders of magnitude higher than previous systems . prior to optical clocks , the ground breaking techniques of laser cooling and trapping of atomic gases brought about dramatic improvements in atomic clock performance by enabling the creation of ultra - cold samples of atoms , providing longer interaction times and reduced systematic uncertainties . the first clocks based on cold atoms were atomic fountains , which are in some sense the natural progression of atomic beam clocks . atomic fountains are the basis of contemporary primary frequency standards , in which the microwave frequency of the hyperfine transition in cesium-133 is used as the reference for a clock @xcite . as the most precise and accurate frequency standards , cold - atom clocks ( optical or microwave ) find application in frequency metrology @xcite , precise tests of fundamental symmetries @xcite , and as test beds for quantum scattering @xcite , entanglement @xcite and information processing @xcite . however , the application of atomic frequency standards as clocks for timing applications is still dominated by clocks based on older but more mature technology , with far greater levels of engineering and evaluation . the workhorses of the timing community , commercial cesium beams and hydrogen masers , which do not incorporate any laser technology , run continuously and require little maintenance and upkeep , often running for years without requiring service . international atomic time ( tai ) utilizes data from hundreds of these clocks , and many timing labs rely exclusively on commercial cesiums and hydrogen masers for their local ensembles . it is to be expected that the best optical clocks will require significant engineering before being compatible with user - free operation in a timing ensemble . ( in fact , many of the best accuracy and stability evaluations carried out with optical clocks use measurements made at optical frequencies , without dividing the signal down to rf , which would be a requirement for most timing applications @xcite . ) on the other hand , atomic fountains are being used much more regularly for timing applications . as primary frequency standards , cesium fountains have been contributing to tai for close to a decade , with the number and precision of reports increasing over time , and recently lne - syrte began reporting a rubidium fountain to the bipm as a secondary standard @xcite . and many timing labs are relying more on fountain clocks for their local timescales , with atomic fountains in use or under development in at least 10 nations worldwide . even with this increased use , the role of fountain clocks for timing has typically been different from the commercial clocks that dominate ( in number ) timing ensembles . serving as primary frequency standards , the fountains provide calibrations of the frequencies of other clocks . in this role , they do not necessarily need to run continuously for months or years at a time without user intervention , though some have achieved close to continuous operation @xcite . and for a local timescale , it can suffice to use only a single primary standard , which ensures a constant long - term frequency , at least on the time frame of the accuracy evaluations . for example , the ptb has produced an excellent timescale based on a single cesium fountain @xcite . still , there are advantages to having multiple clocks . more clocks translate to more robustness and reliability . and it is sometimes the case that a fountain clock is characterized by comparing to a second one . for these reasons , most institutions with fountains have or are developing at least two ; lne - syrte has three and npl is working on a third ( in each case one of the three fountains uses rubidium ) . additional advantages of multiple clocks , particularly for timekeeping , are related to reducing noise . the white - frequency - noise level of an ensemble with @xmath8 clocks decreases as @xmath9 . and since continuously running atomic clocks exhibit non - stationary behavior , timing ensembles based on such clocks require multiple devices to optimize performance . the u.s . naval observatory ( usno ) , the largest contributor of clock data to the bipm , one of the official sources of time for the united states and the official time and frequency reference for critical infrastructure such as gps , has designed and built an ensemble of atomic fountain clocks that has been in operation for more than 2.5 years . the ensemble consists of four rubidium fountains housed in washington , dc @xcite . these clocks run continuously , with little operator intervention required , and are used to enhance the local timing ensemble , contributing to the usno master clock in a manner similar to commercial cesium beams and hydrogen masers . the specific role of the fountains is to provide a superior long - term frequency reference , in the past solely the job of the commercial cesiums , while the short - term reference continues to be provided by an ensemble of hydrogen masers . all four rubidium fountains have been regularly reported to the bipm for over two years as continuously running clocks , the first cold - atom clocks so reported . here we present the performance of these clocks over the first 2 years of operation , the first characterization of continuously running atomic fountains over such time frames . the four fountains in operation at usno are designated nrf2 , nrf3 , nrf4 and nrf5 . the systems were built in two generations , according to slightly different designs @xcite . all four fountains began continuous operation in a dedicated clock facility in march 2011 . since then , some of the improvements in design used for the second generation , nrf4 and nrf5 , have been incorporated as retrofits to the first generation , nrf2 and nrf3 . additionally , both routine and unexpected maintenance were required on occasion . since december of 2011 , fountain data for all four devices have been reported without interruption to the bipm , as clock type 93 . in may of 2012 , after the traditional evaluation period , the fountains were weighted in tai for the first time , and all four clocks have received maximum weight in every evaluation since . each fountain has a local oscillator ( lo ) comprised of a quartz crystal phase - locked to a dedicated maser , which serves as the reference for the 6.8 ghz drive for rubidium spectroscopy as well as the reference for a precise frequency synthesizer , an auxiliary output generator ( aog ) . every 20 seconds , the average relative frequency of the fountain and maser is used to steer the aog output , providing a continuous nominal 5 mhz signal that reflects the fountain frequency and is measured against the usno master clock and other clocks in the ensemble . during each fountain cycle , auxiliary data such as the number of atoms participating in the frequency measurement and the fraction of atoms making the transition between clock states are used to determine whether the fountain is operating normally . if these data fall outside of a predetermined healthy range , the fountain automatically switches to holdover mode , in which the maser serves as a flywheel ; in this case , the control computer applies a steer that is the median frequency difference between the fountain and maser over the previous hour . the same holdover steer is applied until the auxiliary data reflect healthy values and normal operation resumes . the robustness of the fountains and the ability to rely on a maser as a flywheel have resulted in a high uptime over the 2 year period . the percentage of time that each fountain has generated a good , steered output is 99.2% , 99.7% , 98.5% and 100% for nrf2 through nrf5 . a `` good output '' can include brief intervals where the system is in holdover ; so long as the interval corresponds to an averaging time for which the relative frequency variations of the maser and fountain are still white ( _ i.e. _ the frequency fluctuations are still limited by fountain performance ) . periods in holdover long enough that the steered output reflects maser behavior is treated as `` down time '' and is not included in analysis of fountain behavior ; it is handled in the data analysis by interpolating the frequency record and integrating to generate the phase record . the quickest degradation of performance of the ( nominal ) 5 mhz output signal is the occurrence of a problem with the processor controlling fountain operation ; if the steering algorithm is interrupted , the aog will no longer be updated , and its output frequency will reflect a steer determined from 20 s of averaging ( rather than the holdover value ) . processor refresh is usually a simple process . problems that are more difficult to solve , but do not show up as quickly in the steered output , include shutter failure and diode laser aging . aside from these three problems that have arisen , routine maintenance is required on occasion . this consists almost exclusively of occasional tweaks to optical alignment , sometimes not needed for months at a time . very rarely an adjustment to the vco on the reference maser or aog is required to maintain phase lock . there is no maintenance required for the laser frequency locks , which perform without incident . perhaps in part due to the refinement in design , the highest performing fountains have been nrf4 and nrf5 . each device exhibits a white - frequency noise level of @xmath10 and excellent long - term stability . ( there is an obvious enhancement to the microwave chain which should improve the short - term stability to a level of about @xmath11 , which was demonstrated in an engineering prototype , nrf1 . ) in fig . [ f.45good](a ) , the relative phase of nrf4 and nrf5 is plotted over an uninterrupted period of more than 1 year , after removing a relative frequency of @xmath12 and a meaningless relative phase . residual peak - to - peak phase deviations stay within about 1 ns for the entire interval . a limit on the relative frequency drift can be obtained by fitting a line to the frequency , with the result that the relative drift is zero at the level of @xmath13/day ( @xmath14/year ) . a sigma - tau plot , shown in ( b ) , includes overlapping allan deviation , total deviation and theo statistics , along with a white - frequency noise reference line . there is no indication of deviation from white - frequency noise behavior out to the total averaging time of more than 8 months . the average ( in)stability for each fountain is consistent with an overlapping allan deviation of @xmath15 at @xmath16 s , a total deviation of @xmath17 at @xmath18 s and a theo of @xmath19 at @xmath20 s ( values correspond to the white - frequency reference level divided by @xmath21 , not to the data points which lie below this line ) . figure [ f.45good ] ( c ) shows the time deviation between the two fountains for the @xmath22 year interval ; the average tdev for each fountain is about 20 ps at a day . it is likely that this represents record performance for microwave clocks . for each fountain . ( c ) time deviation for data in ( a).,scaledwidth=60.0% ] the data analyzed in fig . [ f.45good ] correspond to an interval when both fountains exhibited excellent stability , which also coincided with a period of time during which there was little or no user intervention . there were times during the two years when some final refinements were made to nrf4 , upgrades were made to nrf2 and nrf3 , and maintenance consisting of laser and shutter replacement were carried out on nrf2 and nrf3 . when the entire two year data set is analyzed without any restrictions , and with no corrections or re - evaluations of the frequency across the service or upgrade , we see long - term behavior that deviates from white frequency noise . figure [ f.pairwise ] shows the overlapping allan deviations for all pair - wise comparisons using the entire two - year data set . the long - term behavior in all cases shows similar non - stationary behavior . when the nrf4 vs nrf5 comparison in ( a ) is extended to longer averaging times using the theo deviation ( darker symbols ) , the most obvious fit is to random - walk frequency noise , with a @xmath23 dependence . the reference line in the figure corresponds to a random - walk level of @xmath24 per fountain . but , as we now show , the behavior leading to these stability plots is entirely consistent with discrete changes in frequency , at a rate that makes it impossible at this point to distinguish between inherent random - walk frequency noise and frequency changes arising from user intervention . in fig . [ f.45white](a ) , we show the relative phase of nrf4 and nrf5 over the entire two year period . the record of the relative phase shows two different linear regions with different slopes , indicating that the relative frequency changed ( the data in fig . [ f.45good ] correspond to the second frequency shown ) . this bears out when we adjust the data starting from 0.9 years ( mjd 55975 ) to the end by removing a single frequency , of size @xmath25 , in order to make the last part of the frequency record match the first . the subsequent data look white , as illustrated by the sigma - tau plot in fig . [ f.45white](b ) , indicating that the non - stationary behavior is indeed consistent with a single frequency change . similar behavior was observed for nrf3 . in fig . [ f.35white](a ) , we show the relative phase of nrf3 and nrf5 over the entire two year period . here , the relative frequency appears to take on three different values . adjusting the data at the two times when the frequency changed ( 0.6 yrs ( mjd 55840 ) and 1.1 yrs ( mjd 56030 ) ) , thereby adding two frequency adjustments , the resulting behavior is consistent with integration as white - frequency noise , as shown by the sigma - tau plot in fig . [ f.35white](b ) . so for nrf3 , nrf4 and nrf5 , the long - term behavior under continuous operation is consistent with rare , discrete frequency changes , at an average rate of 1 every 2 years and an average size of @xmath26 . we can unambiguously assign the frequency changes in fig . [ f.35white ] to nrf3 , as they appear as well in comparisons with nrf4 . the frequency change in fig . [ f.45white ] is more difficult to assign to one fountain or the other , because the other two fountains do not support a particular position . we suspect that the frequency change can be assigned to nrf4 , however , because the time at which the change occurred coincides within the error bar to a time at which some finishing touches were completed on nrf4 . similarly , one of the changes in frequency exhibited in fig . [ f.35white](a ) occurred close in time to one of the upgrades made to bring the design of nrf3 closer to that of the second generation . it is impossible to conclude that these frequency changes were caused by user intervention , but it is a possibility , and it is something we intend to continue analyzing . almost all of the most likely sources of significant frequency shifts in a fountain have been ruled out as drivers of the observed behavior the magnetic field is regularly measured and shows no correlated changes ; light shifts do not seem to be a problem ; the ambient temperature is monitored and is constant . it is hard to conclusively rule out a microwave effect , particularly within the microwave chain . it would be interesting to compare to primary standards , in terms of whether biases measured in accuracy evaluations change occasionally over time . because these frequency changes are rare , it is straightforward to recharacterize a fountain using the others as a reference . this introduces the possibility of producing a paper fountain timescale that conceivably integrates as white - frequency noise to on order of @xmath27 . we plan to consider this further . the fourth fountain , nrf2 , has exhibited the least stable behavior thus far . it has shown discrete frequency changes , of larger size and at a higher rate than those discussed above , as well as intervals that more likely corresponded to drift . some of the problematic behavior in nrf2 does not correlate with episodes of significant intervention . we hope to be able to improve the performance of nrf2 and in the process identify the cause of at least some of this non - stationary behavior . finally , we point out that these changes in the output frequencies of the fountains are not observable using other clocks or timescales at usno . neither the cesium ensemble , nor the maser ensemble , nor individual masers independently show that a fountain has changed frequency ; these events are only observed with inter - fountain comparisons . for years the usno master clock , the physical representation of utc(usno ) , has been generated by incorporating the long - term frequency stability of commercial cesium beams with the superior short - term stability of hydrogen masers . masers are well known to display time - variable drift and other behavior different from white - frequency noise after integrating to a level between low @xmath28s and high @xmath1s . each maser is re - characterized using the cesiums ; the sooner that non - stationary behavior is detected , the better . one role of the rubidium fountains is to improve on the process of maser re - characterization . in fig . [ f.usno_timescale ] we show the allan deviation of an exceptionally stable maser versus a crude fountain timescale , created by averaging the frequencies of the four rubidium fountains . the short term relative stability of @xmath29 at 1 s is consistent with a white frequency noise level of @xmath30 for the maser and @xmath5 for the fountain timescale , as expected for four fountains with an average white - frequency noise of @xmath31 . the white frequency noise level achieved with the usno ensemble of about 70 commercial cesium beams is about @xmath32 . so the fountain ensemble shows a factor of 10 improvement in short - term stability , which , in terms of averaging time , translates to a factor of 100 reduction . even a single fountain is more than 50 times better than the entire ensemble of cesiums . , corresponding to @xmath5 for the fountain timescale and @xmath30 for the maser.,scaledwidth=60.0% ] as stated previously , it should be possible to improve the short - term stability of each fountain to about @xmath33 , making the fountain timescale white frequency noise level @xmath34 . we expect that this would be the limit for our four - fountain ensemble . we have investigated using the same lo for a pair of fountains to enable reduction of the lo noise contribution . removing the entire lo noise contribution by differencing the fountain frequencies is possible for a paper clock and is useful for measuring atomic physics effects more quickly , but it can not be incorporated into a physical output . to generate a physical output with reduced lo noise , we implemented what we call `` cooperative steering '' , where two fountains steer the same lo . by synchronizing the load - measurement cycles so that the lo is always being measured and compensated for , the additional noise from not continually sampling the lo , the dick effect , can be reduced @xcite . because of the high quality of the los used for the rubidium fountains , the reduction in the dick effect does not improve the stability greatly , while using a single lo for different fountains increases risk significantly . this increase in operational risk led us to abandon this technique . as confidence in the reliability of the fountains has been gained , the clocks are now being used operationally as the long - term reference in the generation of utc(usno ) . the specific algorithm for including the fountains as well as a quantitative assessment of their impact on utc(usno ) is an area of ongoing research @xcite . eight cesium fountains worldwide contributed on some basis to tai during the 2 years we are considering . perhaps the best metric of long - term frequency stability of the usno rubidium fountains is by comparing to these primary standards @xcite . we do this by using the usno master clock , the physical realization of utc(usno ) , as a transfer oscillator to allow us to determine the frequency difference between a fountain and tai , @xmath35 . along with the values of the average frequency of the primaries compared to eal , @xmath36 , and the steers of eal to tai , @xmath37 , which are published monthly by the bipm in their circular t report , the frequency of a fountain or fountain timescale compared to the primaries can be obtained . figure [ f.primaries](a ) shows the monthly average of the frequency of nrf5 versus the average of the primaries . the line in the plot is a linear fit , @xmath38/day , showing that we can place a limit on the drift of nrf5 with respect to the primaries at the level of @xmath39/per day . in fig . [ f.primaries](b ) , a sigma - tau plot shows the stability of each fountain versus the primaries . /per day , demonstrating that nrf5 exhibits zero drift with respect to the world s primaries at the level of @xmath39/per day . ( b ) total deviation for each individual fountain vs the cesium fountains . theo deviation and sample error bars are included for nrf5.,scaledwidth=60.0% ] because of the continuous operation of our rubidium fountains , in addition to comparing to the monthly average of the primaries , we can obtain a measurement against each individual primary standard reported or against standards at a particular institution . this allows us to analyze trends with respect to specific standards labs . this is illustrated in fig . [ f.ind_primaries ] , where the relative frequency of nrf5 and each primary standard reported to tai over the 2 years is plotted . the ( three ) fountains at lne - syrte and ( the two ) at ptb are differentiated from the other labs reporting ( nist , npl and one report from nict ) . to summarize , the usno has put into operation an ensemble of continuously running rubidium fountain clocks to incorporate into timing applications and timescale generation in a manner similar to hydrogen masers and commercial cesium beam clocks . we have analyzed performance of the four fountains in the ensemble over the first two years of operation , demonstrating intervals reflecting record performance for microwave clocks and observing rare , discrete frequency changes . comparison to the world s primary standards suggest that the rubidium fountains provide a stable , long - term frequency reference on par with any local reference . we benefited from assistance from many members of the usno time service department , particularly paul koppang , jim skinner and demetrios matsakis . scott crane played a significant role in the design and construction of the fountains . atomic fountain development at usno has been funded by onr and spawar . 10 g. j. dick , j. d. prestage , c. a. greenhall and l. maleki , in _ proceedings of the 19th precise time and time interval ( ptti ) applications and planning meeting _ , redondo , california ( national aeronautics and space administration , washington , dc 1987 ) , p. 133
|
an ensemble of rubidium atomic fountain clocks has been put into operation at the u.s . naval observatory ( usno ) .
these fountains are used as continuous clocks in the manner of commercial cesium beams and hydrogen masers for the purpose of improved timing applications .
four fountains have been in operation for more than two years and are included in the ensemble used to generate the usno master clock .
individual fountain performance is characterized by a white - frequency noise level below @xmath0 and fractional - frequency stability routinely reaching the low @xmath1s .
the highest performing pair of fountains exhibits stability consistent with each fountain integrating as white frequency noise , with allan deviation surpassing @xmath2 at @xmath3 s , and with no relative drift between the fountains at the level of @xmath4/day . as an ensemble ,
the fountains generate a timescale with white - frequency noise level of @xmath5 and long - term frequency stability consistent with zero drift relative to the world s primary standards at @xmath6/day .
the rubidium fountains are reported to the bipm as continuously running clocks , as opposed to secondary standards , the only cold - atom clocks so reported . here
we further characterize the performance of the individual fountains and the ensemble during the first two years in an operational environment , presenting the first look at long - term continuous behavior of fountain clocks .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
in the field of atom interferometry , the improving sensitivity of inertial sensors @xcite is paving the way for many new applications in geophysics , navigation and tests of fundamental physics . most of these experiments are based on raman transitions @xcite to realize beamsplitters and mirrors , which manipulate the atomic wave - packets . among others , this technique has the advantage of an internal state labelling of the exit ports of the interferometer @xcite , enable an efficient detection methods . moreover , the atoms spend most of the time in free fall , with very small and calculable interactions with the environment . the inertial forces are then determined by the relative displacement of the atomic sample with respect to the equiphases of the laser beams , which realise a very accurate and stable ruler . this makes this technique suitable for high precision measurements , as required for instance for inertial sensors and for the determination of fundamental constants @xcite . a limit to the accuracy and the long term stability of these sensors comes from wave - front distortions of the laser beams . this wave - front distortion shift appears directly on the signal of an interferometer when the atoms experience different wave - fronts at each raman pulse . this effect thus depends on the actual trajectories of the atoms , so that a precise control of the initial position , velocity and temperature of the atomic clouds is required @xcite . a convenient technique to reduce this bias is to minimize the number of optical components in the shaping of the two raman laser beams and by implementing them in a retro - reflected geometry @xcite . indeed , as long as the two beams travel together , wave - front aberrations are identical for the two beams and thus have no influence on their phase difference . this geometry also provides an efficient way to use the * k * reversal technique , which allows to diffract the atomic wavepackets in one or the opposite direction and thus to separate effects of many major systematic errors such as gradients of magnetic fields or light shifts @xcite . the main drawback of this geometry arises from the presence of off - resonant raman transitions , which induce a light shift on the resonant raman transition and thus a phase shift of the atom interferometer . in the following , we investigate this effect called two photon light shift ( tpls ) @xcite . we first show that the tpls arises from several off - resonant transitions and evaluate each contribution . we then derive the impact onto the phase of an atom interferometer and use our gravimeter and gyroscope - accelerometer for quantitative comparisons . in particular we measure the systematic shifts and we investigate the influence on the long term stability . the study demonstrates that the precise control of experimental parameters , in particular the raman laser intensities and polarisations , is needed to reduce the influence of this effect for such interferometers . ) , cross the experiment and are reflected by a mirror , crossing twice a second quater - wave plate . the wave - plates are set in such a way that counter - propagating raman transitions are allowed but co - propagating raman transitions are forbidden.,width=321 ] the two experiments are using different alkali - metal atoms : @xmath0rb in the case of the gravimeter and @xmath1cs in the case of the gyroscope . as hyperfine structures , transition selection rules and raman laser setups are similar ( see figure [ schema ] ) , their results can be compared easily . the raman transitions couple the two hyperfine ground states of the alkaline atom ( labelled @xmath2 and @xmath3 ) via an intermediate state ( labelled @xmath4 ) and two lasers with frequencies ( labelled @xmath5 and @xmath6 ) detuned by @xmath7 on the red of the @xmath8 line . during the interferometer sequence , a bias magnetic field is applied along the direction of propagation of the raman laser beam to lift the degeneracy of the magnetic sublevel manifold . the two raman lasers are overlapped with orthogonal linear polarisations and delivered within the same polarisation maintaining optical fiber to the vacuum chamber . after the fiber , the raman beams pass through a quarter - wave plate to convert the initial linear polarisations into circular polarisations , noted @xmath9 for the raman laser at the frequency @xmath5 and @xmath10 for the orthogonal polarisation at @xmath6 . these beams are then retro - reflected through a quarter - wave plate to rotate the polarisation of each beam into its orthogonal polarisation ( @xmath11 , @xmath12 ) . for @xmath13 to @xmath13 transitions , there are two pairs of beams ( @xmath14 and @xmath15 ) , which can drive counter - propagating raman transitions with effective wave - vectors @xmath16 . then , the ground state @xmath17 is coupled with the excited state @xmath18 by the pair of raman laser ( @xmath19 ) and to the excited state @xmath20 with the pair of raman laser ( @xmath21 ) . we use the doppler effect to lift the degeneracy between the two resonance conditions . indeed , if the atoms have a velocity in the direction of propagation of the raman lasers , the doppler shifts are of opposite sign for the two counter - propagating transitions . the resonance condition for each of these couplings is @xmath22 , where @xmath23 is the hyperfine transition frequency , @xmath24 is the recoil energy and @xmath25 the doppler shift due to the atomic velocity @xmath26 in the reference frame of the apparatus . consequently , the detuning between the two resonances is @xmath27 , therefore we can discriminate between the two transitions when the doppler shift is large enough compared to the linewidth of the raman transition . this linewidth is characterised by the effective rabi frequency @xmath28 , which depends on the product of the two raman lasers intensities and inversely to the raman detuning @xmath7 @xcite . in this first part , we use the gyroscope - accelerometer experiment described in detail in @xcite . the experiment has been performed with a cloud of cold caesium atoms ( 1.2 @xmath29 ) prepared initially in the @xmath30 state . the atoms are launched at 2.4 m.s@xmath31 with an angle of @xmath32 with respect to the vertical direction . the raman lasers are implemented in the horizontal plane with a @xmath33 angle with the normal of the atomic flight direction . thus , to select only one raman transition for the interferometer , the frequency difference between the two raman lasers can be tuned to be resonant with either the @xmath34 or the @xmath35 transition . ) . the data have been recorded with laser parameters corresponding to a @xmath36 pulse of 135 @xmath37s duration . lines ( 1,1 ) correspond to the two counter - propagating transitions , line 2 to the copropagating transition between the two @xmath13 states and lines 3 to the copropagating magnetic sensitive transitions.,width=302 ] figure [ spectre ] shows the transition probability as a function of the detuning of the raman transition with respect to the hyperfine transition frequency . one can identify the two velocity selective counter - propagation transitions ( labelled 1 and 1 ) , whose widths reflect the velocity distribution of the atomic cloud . in addition to the counter - propagating transitions , we observe transitions due to residual co - propagating raman coupling , also detuned from resonance by a doppler shift ( lines 2 , 3 and 3 ) . when the frequency difference of the raman lasers is tuned to be resonant with one of the counter - propagating transitions , the second counter - propagating transition and the co - propagating ones induce a light shift ( tpls ) on the selected raman transition used for the interferometer . the tpls is the differential shift between the two atomic levels corresponding to the atomic states @xmath17 and @xmath38 involved in the atomic interferometer . the energy of the state @xmath17 is shifted of @xmath39 by the off - resonant @xmath40 transition detuned by @xmath41 ( eq . [ epg ] ) , while the energy of the state @xmath38 is shifted of @xmath42 by the off - resonant @xmath43 transition detuned by @xmath44 ( eq . [ epe ] ) . the two levels are shifted in opposite directions as illustrated in figure [ couplages ] , here for the case of a raman transition resonant with @xmath45 . and @xmath46 involved in the atomic interferometer . the two states are coupled together through the selected raman transition ( + * * k * * in this particular case ) . but each state is also coupled to an other one , through an off - resonant raman transition of opposite wave - vector.,width=321 ] tpls corrections are calculated from fourth order perturbation theory . in the case of our system , the level shifts @xmath39 and @xmath42 for a @xmath47 interferometer are given by : @xmath48 where @xmath28 is the effective rabi frequency corresponding to counter - propagating raman transitions . thus , the shift of the resonance condition depends on the sign of direction of the selected raman laser pair ( i.e. @xmath47 ) , quadratically on the rabi frequency and inversely to the doppler detuning : @xmath49 & = \frac{\omega_\mathrm{eff}^2}{\pm 8 \omega_\mathrm{d}}+\frac{\omega_\mathrm{eff}^2}{4 ( \pm 2 \omega_\mathrm{d}+4 \omega_\mathrm{r } ) } \end{array}\end{aligned}\ ] ] the frequency shift is measured from fits of the spectrum lines , as displayed in figure [ spectre ] , with different raman intensities ( @xmath50 is proportional to the product of the intensity of the two lasers ) . note that we discriminate it from the shifts independent of @xmath51 , like quadratic zeeman effect or ac stark shift , by alternating measurements @xmath34 and @xmath35 , leading to a differential determination of the effect . the difference in the resonance condition @xmath52 depends only on the tpls and the doppler effect . the doppler effect does not depend on the rabi frequency and can be determined by extrapolating @xmath53 to @xmath54 . the results of these measurements are displayed in figure [ tpls ] as a function of @xmath50 . the curve clearly shows the quadratic dependence of the frequency shift with @xmath28 . for this experimental configuration the doppler shift was 85 khz , and the value of @xmath55 for the largest @xmath28 ( @xmath56 khz at the center of the beam ) is 2.1 khz in good agreement with the expected 2.4 khz . the laser beams have a gaussian shape , with a 15 mm waist ( radius at @xmath57 in intensity ) . in order to evaluate the phase shift , we first measure the tpls for different atomic positions by scanning the resonance along their trajectory . as the raman beam are horizontal , the doppler shift is constant for the three pulses and the tpls varies only with the raman laser intensity . figure [ tplsposition ] displays the measured tpls , proportional to the laser intensity , which follows exactly the gaussian profile of the laser beams . in an ideal experiment , with perfect circular polarization and a raman detuning @xmath7 large compared to the hyperfine structure of the intermediate state ( 201 mhz in the case of the caesium atom ) , co - propagating transitions are forbidden . in a real experiment , with imperfect polarization and/or finite raman detuning , co - propagating transitions are slightly allowed , and will lead to additional tpls . imperfect polarization leads to a residual combination of @xmath58 and @xmath59 in the co - propagating beams and allows coupling between @xmath60 states . as the momentum exchanged @xmath61 , the doppler and recoil effect are negligible and the resonance condition is @xmath62 ( line 2 of figure [ spectre ] ) . the rabi frequency corresponding to the transition @xmath63 is determined experimentally using the residual co - propagating transition probability , @xmath64 at full raman laser power . it gives @xmath65 , which can be explained by an error of linear polarization of one of the raman laser of 2% in power . the detuning of this transition , compared to the two counter - propagating transitions , depends on the doppler and recoil shift : @xmath66 for @xmath67 khz and @xmath68 khz we find an effect due to this coupling smaller than 100 hz . the second source of residual co - propagating transitions stems from the coupling of @xmath70 by the co - propagating raman laser pairs ( @xmath71,@xmath72 ) and ( @xmath73,@xmath74 ) . because of the hyperfine splitting in the intermediate state , there are two paths for the raman transition . both transitions interfere destructively when the detuning compared to the intermediate state @xmath7 is larger than the hyperfine splitting of the intermediate state , and so in this case the transition strength is zero . however , in our experimental set up , with @xmath75 mhz and the @xmath76 mhz the ratio between the rabi frequency of counter - propagating transitions @xmath28 and the rabi frequency of the co - propagating @xmath77 transition @xmath78 is 6.1 , leading to a transition probability of 6.4 @xmath79 in good agreement with the experimental value ( see figure [ spectre ] ) . these transition resonance conditions depend on the magnetic field amplitude as @xmath80 , where @xmath81 khz / g for caesium . with a calculation similar to the one used to obtain eq.[nuls2cp ] , we deduce the two photon light shifts @xmath82 induced by the magnetically sensitive transitions for the @xmath47 case to be : @xmath83 the first term in eq . [ nuls2b ] is due to the coupling with @xmath84 whereas the second term is induced by the coupling with @xmath85 . it is clear from eq . [ nuls2b ] that a residual magnetic contribution appears in the half difference and creates a magnetic sensitivity in addition to the standard quadratic zeeman effect . this contribution to the two photon light shift is measured by changing the bias field in the raman interaction zone . using the differential method previously described , we show in figure [ tpls_b ] the variation of the total tpls with the bias field . the resonance around @xmath86 mg corresponds to the case where a magnetic co - propagating transition and the counter - propagating transition are resonant simultaneously . previous measurements have been performed with a 31 mg magnetic field bias . in the following we will consider interferometers constituted of three raman pulses in a @xmath87 sequence . if the tpls is constant during the interferometer it is equivalent to a fixed frequency shift of the raman transition . in that case , it is well known that no phase shift is introduced in the interferometer . on the contrary , a fluctuation of the tpls during the interferometer sequence leads to a phase shift given by : @xmath88 where @xmath89 is the sensitivity function of the atom interferometer , defined in @xcite . in the case where the interaction pulses are short enough that one can neglect the variation of the tpls during the pulses , and that the area of first and last pulse fulfils the @xmath90 condition , the two - photon interferometer phase shift can be approximated by : @xmath91 where @xmath92 and @xmath93 are the tpls and the rabi frequencies of the i - th pulse , respectively . one might notice that the frequency shift during the @xmath36 pulse does not contribute to the interferometer phase shift . moreover , as all components of the tpls , counter - propagating and co - propagating terms , increase as the square of the rabi frequencies , the interferometer phase shift scales linearly with the raman laser power . in the limit where the co - propagating transitions are negligible ( perfect polarization and very large raman detuning ) and the dominant source of tpls is due to the counter - propagating transition , the phase shift of the interferometer can be simplified as : @xmath94 where @xmath95 is the doppler shift for the i - th pulse . more generally , the interferometer phase shift can be calculated when the @xmath97 pulse condition is no longer fulfilled . this appears when the rabi frequency drifts due to changes in power or polarization of the raman lasers . generalising the formalism of the sensitivity function to the case where @xmath96 allows deriving the interferometer phase shift : @xmath98 & & -\frac{\delta\omega^{(3)}_\mathrm{tpls}}{\omega^{(3)}_\mathrm{eff}}\tan\left(\frac{\omega^{(3)}_\mathrm{eff}\tau^{(3)}}{2}\right ) \end{array}\end{aligned}\ ] ] usually , the rabi frequencies and pulse durations can be taken equal for the first and the last pulses ; the expression of the interferometer shift is then : @xmath99 as before , for a dominant counter - propagating transition , the previous expression can be simplified to : @xmath100 an other aspect of the influence of the tpls on the atomic phase shift concerns the stability of the experiment versus the experimental parameters fluctuations , in particular the raman laser power . as rabi frequency fluctuations are small in relative values ( typically smaller than 10 % ) , we can develop eq . [ tppsgeneral3 ] to first order in @xmath101 close to the usual conditions @xmath90 and find : @xmath102 similar calculations may be derived to extract the dependance on the duration of the pulse or the doppler detuning . as the stability of the latter parameters is much better controlled in cold atom interferometers , no measurable influence on the short term stability of the interferometer is expected . we first consider the case of the gravimeter developed at syrte and described in detail in @xcite . in this compact experimental set - up , cold @xmath103 atoms are trapped in a 3d mot in 60 ms and further cooled during a brief optical molasses phase before being released by switching off the cooling lasers . during their free fall over a few centimetres , the interferometer is created by driving the raman laser in the vertical direction , with pulses separated by free evolution times of @xmath104 ms . the first raman pulse occurs 17 ms after releasing the atoms . the doppler shift at the first pulse is thus relatively small , about 400 khz , and gets large , about 3 mhz , for the last pulse . with a rabi frequency of 40 khz , this leads to a tpls of about 22 mrad for counter - propagating transition . following the method of paragraph [ copro ] , we find a 11 mrad for co - propagating ones . this corresponds to a large shift for the gravity measurement of about @xmath105 . to measure the bias on the atomic interferometer phase due to tpls , we exploit its dependence with the rabi frequency . the principle of this measurement is based on a differential method , where one performs an alternating sequence of measurements of the interferometer phase with two different rabi frequencies @xmath28 and @xmath106 , but keeping the areas of the pulses constant by changing the duration of the pulses @xmath107 . the rabi frequency is modulated with the power of the raman lasers . in practice , the differential measurement is performed by alternating sequences of measurements with four different configurations : ( @xmath108 ) , ( @xmath109 ) , ( @xmath110 ) and ( @xmath111 ) . after averaging for 5 minutes , we extract the difference of the tpls between the two rabi frequencies with an uncertainty below 1 mrad . this measurement was repeated for various @xmath106 , keeping @xmath28 fixed . the phase differences are displayed in figure [ shiftgravi ] in according to the ratio of the rabi frequency @xmath112 . the results clearly demonstrate the linear dependence of the phase shift with the rabi frequency . the fit of the data allows to extract a 32 mrad shift , in very good agreement with the expected value ( 33 mrad ) , deduced from eq . deviations from the linear behavior and discrepancies ( up to @xmath113 ) between different measurements that correspond to the same ratio can not be explained by uncontrolled fluctuations in the rabi frequencies , as the simultaneous monitoring of the laser intensities showed stability at the per cent level during the course of the measurements . we demonstrated that these fluctuations were correlated with changes of the polarization of the raman beams , which modulate the contribution to the phase shift of the undesired co - propagating transitions . the line corresponds to the calculated shift.,width=321 ] we perform a complementary measurement by changing the duration of the first and last pulse simultaneously , while keeping @xmath28 constant . the acquisition is performed with a similar differential method than previously , but now alternating between different pulse durations and a pulse duration of 6 @xmath114 , when the @xmath97 pulse condition is fulfilled . the data are then shifted by the bias deduced from previous measurements with @xmath115 @xmath114 ( result of figure [ shiftgravi ] ) and displayed in figure [ shiftvstau ] . using eq . [ tppsgeneral2 ] , the fit of the data gives a rabi frequency of 39 khz , in very good agreement with the expected value ( 42 khz ) . a small deviation appears for long pulse durations when the areas of each of the two pulses is close to @xmath36 . in the case of our gravimeter , where cold atoms are dropped from rest , the tpls is very large , almost two orders of magnitude above the pursued accuracy . in principle , this effect can be measured accurately by alternating measurements with different rabi frequencies . but , it seems desirable to decrease the effect by operating with lower rabi frequencies , the drawback being an increased velocity selectivity of the raman pulses . a more stringent velocity selection , or smaller temperatures , are then required in order to preserve a good fringe contrast . in the case of a fountain gravimeter , where the atoms are launched upwards at a few m / s , the doppler shift at the first and last pulse is much larger , considerably reducing this effect . for the parameters of the stanford gravimeter @xcite ( long pulse duration of 80 @xmath37s and time between pulses of 160 ms ) , we find a phase shift of 0.8 mrad , which corresponds to @xmath116 . in the case of the gyro - accelerometer , the mean velocity of the wave - packet is not collinear with the effective raman wave vector . consequently , the atomic phase shift measured with the interferometer is sensitive to the rotation rate * @xmath117 * in addition to the acceleration * a*. as in the case of the gravimeter , the atomic phase is shifted by the tpls and by other systematics ( labelled @xmath118 ) , for instance the phase shift induced by the laser wave - front distortions @xcite . the total phase shift is expressed by : @xmath119 our gyroscope - accelerometer uses a double interferometer with two atomic clouds following the same trajectory but with opposite directions to discriminate between acceleration and rotation phase shifts . moreover , our experiment is designed to measure different axis of rotation and acceleration according to direction of propagation of the raman laser beams . we will illustrate the impact of the tpls on the interferometer in the configuration where the raman lasers are directed along the vertical direction . acquisitions have been recorded for a total interaction time between the first and the last pulse of 40 ms . to remove drifts from other sources , the phase shift is measured in a differential way with respect to the maximum rabi frequency available at the location of the @xmath36/2 pulses ( 33 khz ) . the line shows the calculated shift for acceleration signal.,width=321 ] the measurement is realized in the same way than for the gravimeter experiment by comparing atomic phases with high and low rabi frequency , changing the raman laser power but keeping the pulse duration constant to 7.6 @xmath37s . in order to enhance the tpls signal we decrease the time between pulses to 20 ms . indeed , the doppler effect is reduced and the available laser power on the side of the gaussian laser profile is increased . the first pulse occurs 15 ms before the apogee and the third 25 ms after the apogee , corresponding respectively to a doppler shift of about @xmath120344 khz and @xmath121574 khz . the rabi frequency for each pulse is approximately 33 khz , then the tpls expected from eq.[tpps ] is about 38 mrad for each interferometer . as this shift is similar for the two interferometers when the two atomic clouds perfectly overlap and experiment the same tpls , it bias the acceleration signal only . figure [ shiftgyro40 ] displays the variation of the acceleration and rotation signals with the rabi frequency . acceleration shift ( squares ) varies in a good agreement with expected shift ( continuous curve ) calculated from eq . [ tppsgeneral ] . rotation shift ( circles ) shows no dependance on the rabi frequency and illustrates that the rejection from the acceleration signal is efficient . nevertheless , fluctuations from expected behaviour are clearly resolved and repeatable . we attribute these deviations to wave - front distortions of the raman laser beams . indeed , when the atomic trajectories do no perfectly overlap , a residual bias appears on the rotation due to unperfected cancellation . this bias depends on the details of the wave - front distortions weighted by the actual atomic cloud distributions , and is modified when the rabi frequency is changed . in the usual conditions , with interrogation time of 80 ms , the acceleration shift is reduced to about 12 mrad thanks to the increase of the doppler shift and reduction of the rabi frequency on the side of the gaussian raman beams . we finally estimate the impact of the raman laser power fluctuations on the stability of the rotation signal in usual conditions ( interrogation time of 80 ms ) . we performed a complementary measurement by recording the interferometer signal and the raman laser power in the same time when applying a modulation of the laser powers of 10@xmath79 . the modulation is applied by attenuating the radio - frequency signal sends to the acousto - optic modulator used to generate the raman pulses . the power of the lasers was recorded during the third pulse thanks to a photodiode measuring the intensity on the edge of the laser beams . we found a small dependance of the rotation signal on the power fluctuation of @xmath122 rad.s@xmath31/@xmath79 , which can limit the long term stability of the gyroscope . as no dependance was expected , we attribute it again to non perfect superimposition of the two atomic clouds trajectories , leading to a different rabi frequencies experienced by the two clouds . we have shown that the use of retro - reflected raman lasers in atom interferometers induces off - resonant raman transitions , which have to be taken into account in order to achieve best accuracy and stability of interferometers . we have first quantitatively evaluated the effect on the resonance condition for each off - resonant line : the other counter - propagating transition , which can not be avoided in the retro - reflected design , and co - propagating transitions arising mainly from imperfections in the polarisation . then , we have measured the impact of this two photon light shift on the phase of two atom interferometers : a gravimeter and a gyroscope . in particular , we show that this shift is an important source of systematic errors for acceleration measurements . nevertheless , it can be measured accurately by modulating the raman laser power and/or the pulse durations . our study has also shown that it can impact the stability of the two sensors if the polarization and/or the power of the raman lasers fluctuate . the tpls appears as a drawback of using retro - reflected raman laser beams . but , as it can be well controlled , it does not reduce the benefit from this geometry , whose key advantage is to drastically limit the bias due to wave - front aberrations , which is larger and more difficult to extrapolate to zero . this study can be extended to other two photon transitions ( like bragg transitions ) and to other possible polarization configurations ( when using linear instead of circular polarizations ) , when using the doppler effect to select the transition . if the signal is generated from the subtraction of the phase shifts of two independent atomic clouds , e.g. gradiometers or gyroscopes , perfect common mode rejection , is required to suppress this effect . in our case this means a perfect overlap of atomic trajectories . finally , the two photon light shift can be drastically reduced by increasing the doppler effect and/or using colder atoms , allowing to reduce the rabi frequency during the raman pulses . by contrast , for set - up with intrinsic small doppler effect , as for space applications @xcite , this effect becomes extremely large and has to be taken into account in the design of the experiment . we express our gratitude to f. biraben for pointing out this effect and to p. cheinet for earlier contributions to the gravimeter experiment . we would like to thank the institut francilien pour la recherche sur les atomes froids ( ifraf ) , the european union ( finaqs contract ) , the dlgation gnrale pour larmement ( dga ) and the centre national dtudes spatiales ( cnes ) for financial supports . b. c. , j. l.g . and t. l. thank dga for supporting their work . 30 a. peters , k. y. chung , s. chu , metrologia * 38 * 25 ( 2001 ) . t. l. gustavson , a. landragin , m. a. kasevich , class . quantum grav . * 17 * 1 - 14 ( 2000 ) . b. canuel , f. leduc , d. holleville , a. gauguet , j. fils , a. virdis , a. clairon , n. dimarcq , ch.j . bord , a. landragin , and p. bouyer , phys . lett . * 97 * , 010402 ( 2006 ) . j. m. mcguirk , g. t. foster , j. b. fixler , m. j. snadden , m. a. kasevich , phys.rev . a * 65 * 033608 ( 2002 ) . m. kasevich , d.s . weiss , e.riis , k. moler , s. kasapi and s. chu , phys . lett . * 66 * 2297 ( 1991 ) . bord , phys . a * 140 * , 10 ( 1989 ) . j. b.fixler , g. t. foster , j. m. mcguirk and m. a. kasevich , science magazine * vol . 315 . 5808 * , 74 ( 2007 ) . g. lamporesi , a. bertoldi , l. cacciapuoti , m. prevedelli , and g. m. tino , phys . lett . * 100 * , 050801 ( 2008 ) . a. wicht , j.m . hensley , e. sarajlic and s. chu , physica scripta * t102 * , 82 ( 2002 ) . p. clad , e. de mirandes , m. cadoret , s. guellati - khlifa , c. schwob , f. nez , l. julien and f. biraben , phys . lett . * 96 * , 033001 ( 2006 ) . g. genevs _ ieee trans . on instr . and meas . * 54 * , 850 ( 2005 ) . j. fils , f. leduc , p. bouyer , d. holleville , n. dimarcq , a. clairon and a. landragin , eur.phys . j. d * 36*,257 - 260 ( 2005 ) . a. landragin and f. pereira dos santos , in _ proc . of the enrico fermi international school of physics _ * 168 * ( 2007 ) , in press , arxiv : 0808.3837 p. cheinet , f. pereira dos santos , t. petelski , j. le gout , j. kim , k. t. therkildsen , a. clairon and a. landragin appl . b * 84 * , 643 ( 2006 ) . d. s. weiss , b. c. young and s. chu , appl . b * 59 * , 217 ( 1994 ) . p. clad , e. de mirandes , m. cadoret , s. guellati - khlifa , c. schwob , f. nez , l. julien and f. biraben , phys . a * 74 * , 052109 ( 2006 ) . p. cheinet , b. canuel , f. pereira dos santos , a. gauguet , f. leduc , a. landragin , ieee trans . on instr . and meas . * 57 * , 1141 ( 2008 ) . j. le gout , t. e. mehlstubler , j. kim , s. merlet , a. clairon , a. landragin and f. pereira dos santos , appl . b * 92 * , 133 - 144 ( 2008 ) . see for instance apll . b * 84 * , number 4 , special issue : `` quantum mechanics for space application : from quantum optics to atom optics and general relativity '' , p. wolf _ _ , accepted for publication in experimental astronomy , arxiv : 0711.0304v5 , w. ertmer _ submitted for publication in experimental astronomy , and references therein .
|
we study the influence of off - resonant two photon transitions on high precision measurements with atom interferometers based on stimulated raman transitions .
these resonances induce a two photon light shift on the resonant raman condition .
the impact of this effect is investigated in two highly sensitive experiments : a gravimeter and a gyroscope - accelerometer .
we show that it can lead to significant systematic phase shifts , which have to be taken into account in order to achieve best performances in term of accuracy and stability .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
let @xmath0 be a qbf in _ prenex cnf ( pcnf ) _ where @xmath1 with @xmath2 is the prefix containing quantified propositional variables @xmath3 and @xmath4 is a quantifier - free cnf . given a pcnf @xmath0 , an _ unsatisfiable core ( uc ) _ of @xmath5 is an unsatisfiable pcnf @xmath6 such that @xmath7 and @xmath8 . the prefix @xmath9 is obtained from @xmath10 by deleting the quantified variables which do not occur in @xmath11 . a _ _ minimal unsatisfiable core ( muc ) _ _ of @xmath5 is an unsatisfiable core @xmath12 of @xmath5 where , for every @xmath13 , the pcnf @xmath14 is satisfiable . _ incremental solving _ is crucial for the computation of mucs in the context of propositional logic ( sat ) , e.g. @xcite . modifications of a cnf by adding and deleting clauses in incremental solving are typically implemented by _ selector variables _ and _ assumptions _ @xcite . an added clause @xmath15 is augmented by a fresh selector variable @xmath16 so that actually @xmath17 is added . via the solver api , the user assigns these variables as assumptions under which the cnf is solved to control whether a clause is effectively present in the cnf . different from the assumption - based approach , the sat solver zchaff @xcite provides an api to modify the cnf by adding and removing _ groups _ ( sets ) of clauses . clauses are associated with an integer i d of the group they belong to . in assumption - based incremental solving , clause groups may be emulated by augmenting all clauses in a group by the same selector variable . the user must specify the necessary assumptions via the api in all forthcoming solver invocations to enable and disable the right groups . in contrast to that , zchaff allows to delete groups by a single api function call . in terms of usability , we argue that incremental solving by a clause group api is less error - prone , more accessible to inexperienced users , and facilitates the integration of the solver in other tools . we present a novel clause group api of our qbf solver depqbf ( version 4.0 or later ) in the style of zchaff . different from zchaff , we implemented clause groups based on selector variables and assumptions to combine the conceptual simplicity of zchaff s api with state of the art assumption - based incremental solving . as a novel feature of our api , the handling of selector variables and assumptions is entirely carried out by the solver and is hidden from the user . our approach is applicable to any sat or qbf solver supporting assumptions . based on the novel clause group api of depqbf , we implemented a tool to compute mucs of pcnfs , a problem which has not been considered so far . results on benchmarks used in the qbf gallery 2014 illustrate the applicability of the clause group api for muc computation of pcnfs . depqbf is a solver for pcnfs based on the qbf - specific variant of the dpll algorithm @xcite with learning @xcite . since version 3.0 @xcite , depqbf supports incremental qbf solving via an api to add and remove clauses in a stack - based way ( cf . fig . 3 in @xcite ) . this api is suitable for solving incremental encodings where clauses added most recently tend to be removed again in subsequent solver calls , like reachability problems such as conformant planning @xcite or bounded model checking @xcite . the new clause group api of depqbf , however , allows to add and delete clauses _ arbitrarily _ , which is necessary for the incremental computation of mucs of pcnfs . we first present our novel approach to keeping selector variables invisible to the user , which is a unique feature of depqbf . to this end , we distinguish between selector variables and variables in the encoding . let @xmath18 be a sequence of pcnfs . we consider variables over which the pcnfs @xmath19 are defined as _ user variables _ because they are part of the problem encoding represented by @xmath20 . when solving @xmath20 incrementally , _ selector variables _ used to augment clauses in @xmath19 are not part of the original encoding . variables @xmath21 are stored in an array @xmath22 indexed by an integer i d @xmath23 of @xmath21 such that @xmath24 = v$ ] . user and selector variables reside in separate sections of @xmath22 :
|
we consider the incremental computation of minimal unsatisfiable cores ( mucs ) of qbfs . to this end , we equipped our incremental qbf solver depqbf with a novel api to allow for incremental solving based on clause groups .
a clause group is a set of clauses which is incrementally added to or removed from a previously solved qbf .
our implementation of the novel api is related to incremental sat solving based on selector variables and assumptions .
however , the api entirely hides selector variables and assumptions from the user , which facilitates the integration of depqbf in other tools .
we present implementation details and , for the first time , report on experiments related to the computation of mucs of qbfs using depqbf s novel clause group api .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
large scale magnetic fields are widespread in the universe . from galaxies to clusters of galaxies coherent magnetic fields are detected , with intensities that range from @xmath3gauss to tenth of @xmath3gauss . our galaxy as well as nearby galaxies show magnetic fields coherent on the scale of the whole structure , while in galaxy clusters the coherent length is much less than the cluster s size @xcite . a remarkable fact recently discovered by observations , is that high redshift galaxies also posses coherent fields with the same intensitis as present day galaxies @xcite . this result challenges the generally accepted mechanism of magnetogenesis , namely the amplification of a primordial field of @xmath4 gauss by a mean field dynamo @xcite acting during a time of the order of the age of the structure : either the primordial fields are more intense so the galactic dynamo saturates in a shorter time , or the dynamo does not work as it is currently thought . it is hoped that future observations of high redshift environments will shed more light on the features of primordial magnetic fields @xcite . in view of the lack of success in finding a primordial mechanism for magnetogenesis that produces a sufficiently intense field , either to feed an amplifying mechanism , or to directly explain the observations ( see refs . @xcite as recent reviews ) , researchers began to delve on magnetohydrodynamical effects that could compensate the tremendous dilution of the field due to flux conservation during the expansion of the universe . among the possibilities there is primordial turbulence @xcite . possible scenarios for it are the reheating epoch , the phase transitions ( at least the electroweak one ) and possibly the epoch of reionization , all dominated by out of equilibrium processes . a key ingredient to produce stable , large scale magnetic fields in three - dimensional mhd turbulence , is the transfer of magnetic helicity from small scales to large scales , at constant flux @xcite ( see also ref . @xcite and references therein ) . magnetic helicity , @xmath5 , is defined as the volume integral of the scalar product of the magnetic field @xmath6 with the vector potential @xmath7 @xcite . in three dimensions , and in the absence of ohmic dissipation , it is a conserved quantity that accounts for the non - trivial topological properties of the magnetic field @xcite , such as the twists and links of the field lines . unlike the energy that performs a natural , direct cascade , i.e. , from large scales toward small ones where it is dissipated , magnetic helicity has the remarkable property of _ inverse cascading _ , that is , magnetic helicity stored in small scales evolves toward larger scales @xcite . the fact that magnetic energy and magnetic helicity spectra are dimensionally related as @xmath8 @xcite produces a dragging of the former toward large scales , thus enabling the field to re - organize coherently at large scales . it must be stressed that in a cosmological context , the inverse cascade mentioned above operates on scales of the order of the particle horizon or smaller . this is due to the fact that turbulence is a causal phenomenon . magnetic helicity on the other hand can be induced at any scale , the topology of the fields then remains frozen if the scales are super - horizon and if there is no resistive decay . for subhorizon scales it is a sufficient condition for its conservation that the conductivity of the plasma be infinite @xcite . the interpretation of @xmath5 as the number of twists and links must be considered with care because from its very definition it is clear that @xmath5 is gauge dependent . in their seminal work , berger and field @xcite proved that if the field lines do not cross the boundaries of the volume of integration , i.e. , the field lines close inside the considered volume , then @xmath5 as defined _ is _ a gauge invariant quantity . these authors also addressed the case of open field lines , and wrote down a definition of gauge invariant magnetic helicity based on the difference of two such quantities for field configurations that have the same extension outside the considered volume . in this case the quantity obtained can be interpreted as the numbers of links inside the volume . in general it is not difficult to find early universe mechanisms that produce magnetic fields endowed with magnetic helicity : generation of helical magnetic fields has been already addressed in the framework of electroweak baryogenesis @xcite and of leptogenesis @xcite . the main problem is still in the low intensities obtained in more or less realistic scenarios . the magnetic fields we consider in this work are induced by stochastic currents of scalar charges created gravitationally during the transition inflation - reheating @xcite ( see @xcite for more details ) , and such field configuration is of open lines . in the light of the analysis of berger and field , we shall discuss a criterion by which the result obtained can be considered as gauge invariant . the fields induced are random , the mean value of the magnetic helicity is zero , but not the corresponding rms deviation . we assume that those fields are weak enough to neglect their backreaction on the source currents , and show that the rms magnetic helicity can be written as the sum of four sqed feynman graphs , one of them representing the mean value of @xmath0 and consequently identically null . the remaining three add to a non null value . we compute the value of the helicity for large scales and find that the number density of links scales with the distance @xmath9 from a given point as @xmath10 , which means that their fractal dimension is @xmath11 this number density takes into account defects due to both regular and random fields . we also calculate the value of @xmath0 due to regular fields on a large scale . in this case the number density scales as @xmath12 , the corresponding fractal dimension being @xmath2 . using the relation @xmath13 , we compare the associated helical intensity to the one obtained by computing directly the correlation function of the magnetic field at the same scale @xmath14 . we find that both expressions coincide , which means that the fields generated by the considered mechanism are indeed helical . we estimate the intensity of those smooth fields on a galactic scale , finding an intensity too small to seed the dynamo . we finally address the evolution of fields generated at scales of the order of the particle horizon at the end of reheating , through the inverse cascade of magnetic helicity mechanism , until matter - radiation equilibrium . this evolution is based on the assumption that during radiation dominance the plasma is in a ( mild ) turbulent state . we find that the number density of magnetic links scales as @xmath15 , the corresponding fractal dimension then being @xmath16 . the field intensity as well as the scale of coherence are in a range that could have and impact on the process of structure formation @xcite . we work with signature @xmath17 and with natural units , i.e. , @xmath18 , @xmath19 . we use the hubble constant during inflation , @xmath20 , which we assume constant , to give dimensions to the different quantities , i.e. we consider spacetime coordinates @xmath21 = h^{-1}$ ] , lagrangian density @xmath22 = h^{4}$ ] , four vector potential @xmath23 = h$ ] , field tensor @xmath24 = h^{2}$ ] , scalar field @xmath25=h$ ] . the paper is organized as follows : section [ sed ] contains a brief description of scalar electrodynamics in curved spacetime . in section [ mh ] we define magnetic helicity and describe briefly its main properties . in section [ mhrf ] we develope the formalism to study magnetic helicity of random fields and estimate its rms value in different scenarios : in subsection [ demh ] we compute the sqed feynman graphs that describe the magnetic helicity two - point correlation function . in subsection [ dim ] we provide some physical quantities relevant for our study . in subsection [ hm - tir ] we describe the transition inflation - reheating and quote some useful formulae for our work . in subsection [ gd ] we apply the analysis of berger and field to our fields and show the gauge invariance of our results . in subsection [ mhls ] we calculte the magnetic helicity rms value on large scales , and compute the density and fractal dimension of the distribution of defects . in subsection [ hmgal ] we compute the rms value of magnetic helicity due to solely smooth fields , and find that the fields induced by the mechanism considered in this work are completely helical , but very weak . finally in subsection [ hm - ss ] we analyze the evolution of fields induced on scales of the order of the horizon at reheating along radiation dominance . by considering conservation of magnetic helicity and assuming full inverse cascade is operative , we find at decoupling a magnetic field of intensity and coherence that could impact on the process of structure formation . in section [ dc ] we sumarize and discuss our results . we leave details of the calculations to the appendices . in curved spacetime the action for a charged scalar field coupled to the electromagnetic field is given by @xmath26 \label{a-1}\ ] ] with the lagrangian density @xmath27 \label{a-2}\ ] ] with @xmath28 being the metric tensor that for a spatially flat friedmann - robertson - walker spacetime reads @xmath29 , @xmath30 the covariant derivative , @xmath31 the scalar curvature , @xmath32 the coupling constant of the scalar field to the curvature and @xmath33 the electromagnetic field tensor . due to the conformal invariance of the electromagnetic field in the spatially flat frw universe , it is convenient to work with conformal time , defined as @xmath34 , with @xmath35 being the cosmological ( or physical ) time . the metric tensor then reads @xmath36 , with @xmath37 . we shall be dealing with fields in inflation , reheating and radiation dominance . in those epochs the scale factors in physical time are respectively @xmath38 , @xmath39 and @xmath40 , while in dimensionless conformal time , @xmath41 they read @xmath42 , @xmath43 and @xmath44 . we note that @xmath45 corresponds to the end of inflation ( in that epoch @xmath46 ) and consequently the scale factor at that moment is @xmath47 . we rescale the fields according to @xmath48 which means that @xmath49 , @xmath50 and @xmath51 are dimensionless . working with the coulomb gauge , @xmath52 ( considering also @xmath53 ) , we obtain after taking variations of action ( [ a-1 ] ) the following evolution equations for the scaled scalar and electromagnetic fields @xmath54@xmath55 with @xmath56 the electric current due to the scalar field , given by @xmath57 \label{a-7}\ ] ] which , writing the complex field in term of real fields as @xmath58 , can be expressed as @xmath59 \label{a-7b}\ ] ] in a first approximation we consider that the induced fields @xmath60 are weak enough to discard their coupling to the scalar field given by the last two terms in eq . ( [ a-5 ] ) and the last term in eq . ( [ a-6 ] ) . also , we consider minimal coupling of scalar fields to gravity , for it will produce maximal particle creation @xcite . ( [ a-5 ] ) then turns into the klein - gordon equation for a free field in frw universe , @xmath61 \varphi = 0 \label{a-5b}\ ] ] whose solutions are given in appendix [ apa ] . for a realistic evolution of @xmath62 dissipative effects must be taken into account . this we do by assuming that ohm s law in its usual form , @xmath63 , is valid . in the lack of a clear knowledge about the early universe plasma , we assume a traditional form for the electric conductivity considered in the literature , namely , that it is proportional to the plasma temperature , i.e. , @xmath64 , with @xmath65 for a relativistic plasma . this assumption amounts to adding to the l.h.s . of eq . ( [ a-6 ] ) a term of the form @xmath66 , and so the evolution equation for @xmath67 reads @xmath68 \mathcal{a}_{i}=\mathcal{j}_{i}\label{a-8}\ ] ] by taking curl of this equation we obtain the one corresponding to the magnetic field , i.e. @xmath68 \mathcal{b}_{i}=\epsilon_{ijk}\partial _ { j}\mathcal{j}_{k } \label{a-9}\ ] ] where @xmath69 is the levi - civita tensor density . ( [ a-8 ] ) and ( [ a-9 ] ) can be readily integrated to give the fields in terms of their sources , i.e. @xmath70 @xmath71 with @xmath72 the solution of ( see appendix [ apb ] ) @xmath73g_{ret}\left ( \bar{x},\bar{s},\eta , \tau \right ) = \delta \left ( \bar{x}-\bar{s}\right ) \delta \left ( \eta -\tau \right ) \label{a-12}\ ] ] the electric currents we are considering consist of scalar charges created due to the change of the universe s geometry during the transition inflation - reheating . this change makes the scalar field vacuum state during inflation to correspond to a particle state in the subsequent phase @xcite . this process of `` particle creation '' is an out - of - equilibrium one , the resulting particle currents being stochastic . in the case of a charged field the mean value of the electric current is zero , but not its rms deviation , which sources a random magnetic field , whose magnetic helicity we are going to compute . classically , magnetic helicity is defined as the volume integral of the scalar product between magnetic field and magnetic vector potencial @xcite , i.e. @xmath74 which in view of the dimensions of @xmath75 and @xmath76 it is already a dimensionlessl quantity . as stated in the introduction , it is a measure of the non - trivial topology of the magnetic field inside the volume @xmath77 , or in other words , eq . ( [ b-1 ] ) represents the number of twists and links of the field lines inside @xmath77 . this interpretation , however , must be considered with care , because @xmath78 is not gauge invariant unless the boundaries of the volume of integration are not intersected by @xmath79-lines , i.e. the field lines close inside @xmath77 . on the other hand if the magnetic flux across the boundaries of @xmath77 is not zero , as in our case , it is still possible to define a gauge invariant measure of the links of the field inside @xmath77 @xcite by considering the difference between two field configurations that coincide outside @xmath77 . we shall discuss this issue in a following paragraph . considering the conformal transformation of the fields and coordinates given by eq . ( [ a-4 ] ) and writing the physical volume as @xmath80 , eq . ( [ b-1 ] ) can be written as @xmath81 where we see that this is not diluted by the expansion , i.e. , expansion does not erase or create the topology of the field , as it is to be expected . in classical magnetohydrodynamics @xmath78 is one of the ideal invariants , i.e. it is a conserved quantity in the absense of ohmic dissipation @xcite . this means that magnetic helicity can not be created or destroyed by turbulence or non - dissipative evolution , being then a property of the magnetic field created at its birth . unlike other ideally conserved quantities in 3-dimensional magnetohydrodynamics , magnetic helicity performs an _ inverse cascade _ @xcite . this means that instead of travelling towards small scales ( large wavenumber ) where it would be dissipated , it makes its way toward large scales ( small wavenumbers ) , carrying with it a bit of magnetic energy . mathematically , this is expressed by the spectral relation quoted above , i.e. , @xmath82 . in a framework of decaying turbulence ( that could exist at the earliest epochs of the universe ) , there would be a self - organization of the magnetic field at large scales , with the total energy contained in the considered volume decaying as @xmath83 and coherence length increasing as @xmath84 @xcite . for primordial magnetogenesis this fact can be of great help in obtaining stronger fields than the ones found up to now to seed subsequent amplifying mechanisms , or even to directly explain the observations . but on the other hand , the conservation of @xmath85 crucially constraints the operation of further amplifying mechanisms such as the mean field dynamo @xcite . as we are dealing with random fields , generated from stochastic quantum electric currents whose mean value is zero , we must evaluate a rms value of the helicity by calculating a two - point correlation function given by @xmath86 where angle brackets denote stochastic and quantum average . we shall consider the volume of integration @xmath87 as the commoving space occupied by the structure of interest ( i.e. , a galaxy , a cluster , particle horizon at a certain epoch , etc . ) . we begin by writing @xmath88\ ] ] where @xmath89 with @xmath90 to avoid cumbersome notation , repeated momenta other than @xmath15 are assumed to be integrated over . the volume integrals can be readily evaluated , giving @xmath91 the different fields in ( [ c-3 ] ) can be written as @xmath92 and @xmath93 ( integration in @xmath94 is understood ) . the electric currents @xmath95 can be expressed in terms of the scalar fields as @xmath96 gathering all expressions , @xmath97 is written as @xmath98 after integrating out the time delta functions the magnetic helicity correlation function spectrum reads @xmath99 each scalar field can be decomposed in its positive and negative frequency components , that respectively include the annihilation and the creation operator , namely @xmath100 when replacing this decomposition in expression ( [ c-9 ] ) it can be seen that in each bracket the only terms that contribute to the mean value are @xmath101 which in the end combine to form the scalar positive frequency operator , @xmath102 the first term on the r.h.s . of expression ( [ c-11 ] ) gives @xmath103 while the second term gives @xmath104 when performing the products of the two brackets we obtain nine terms that can be represented by the following graphs ( full lines indicate scalar fields , dotted lines vector potential and dashed lines magnetic fields ) . , width=240 ] , width=240 ] , width=240 ] observe that of the two vertices of scalar electrodynamics , only one contributes to these graphs : the one similar to the qed vertex . this is so because we disregarded the backreaction of @xmath105 on the scalar fields . observe also that the first graph , being the product of two mean values , vanishes identically . the multiplicity of the `` square '' graph is 4 , of the `` cross '' diagram is 2 and of the `` two - bubble '' figure is also 2 . after replacing the expressions for the scalar positive frequency operators and solving all the dirac delta functions we obtain a @xmath106 , which means that the momentum of the electromagnetic field is conserved . writing each graph as @xmath107 we have the following expressions for the prefactor of the non - null graphs : @xmath108 \left [ \bar{\kappa}\cdot \left ( \bar{q}_{1}\times \bar{q}_{2}^{\prime } \right ) \right ] \label{c-15 } \\ & & \phi \left ( \bar{q}_{1},\tau _ { 1}\right ) \phi ^{\ast } \left ( -\bar{q}_{1},\tau _ { 2}^{\prime } \right ) \phi \left ( \bar{\kappa}-\bar{q}_{1},\tau_{2}\right ) \phi ^{\ast } \left ( \bar{q}_{1}-\bar{\kappa},\tau _ { 1}^{\prime}\right ) \notag \\ & & \phi \left ( \bar{p}-\bar{q}_{1},\tau _ { 1}\right ) \phi ^{\ast } \left ( \bar{q}_{1}-\bar{p},\tau _ { 2}\right ) \phi \left ( \bar{q}_{2}^{\prime } , \tau_{1}^{\prime } \right ) \phi ^{\ast } \left ( -\bar{q}_{2},\tau _ { 2}^{\prime}\right ) , \notag\end{aligned}\ ] ] for the square graph , which vanishes when the momenta integrals are peformed . \left [ \left ( \bar{p}-\bar{q}_{3}\right ) \cdot \left ( \bar{\kappa}\times \bar{q}_{1}\right ) -2\bar{q}_{1}\cdot \left ( \bar{p}\times \bar{q}_{3}\right ) \right ] \nonumber \\ & & \phi \left ( \bar{q}_{1},\tau _ { 1}\right ) \phi ^{\ast } \left ( -\bar{q}_{1},\tau _ { 2}^{\prime } \right ) \phi \left ( \bar{q}_{3},\tau _ { 2}\right)\phi ^{\ast } \left ( -\bar{q}_{3},\tau _ { 1}^{\prime } \right ) \label{c-16 } \\ & & \phi \left ( \bar{p}-\bar{q}_{1},\tau _ { 1}\right ) \phi ^{\ast } \left ( \bar{q}_{1}-\bar{p},\tau _ { 1}^{\prime } \right ) \phi \left ( \bar{\kappa}-\bar{p}-\bar{q}_{3},\tau _ { 2}\right ) \phi ^{\ast } \left ( -\bar{\kappa}+\bar{p}+ \bar{q}_{3},\tau _ { 2}^{\prime } \right ) \notag\end{aligned}\ ] ] for the `` cross '' diagram and @xmath110 ^{2 } \label{c-17 } \\ & & \phi \left ( \bar{q}_{1},\tau _ { 1}\right ) \phi ^{\ast } \left ( -\bar{q } _ { 1},\tau _ { 1}^{\prime } \right ) \phi \left ( \bar{q}_{3},\tau _ { 2}\right ) \phi ^{\ast } \left ( -\bar{q}_{3},\tau _ { 2}^{\prime } \right ) \notag \\ & & \phi \left ( \bar{p}-\bar{q}_{1},\tau _ { 1}\right ) \phi ^{\ast } \left ( \bar{q } _ { 1}-\bar{p},\tau _ { 1}^{\prime } \right ) \phi \left ( \bar{\kappa}-\bar{p}- \bar{q}_{3},\tau _ { 2}\right ) \phi ^{\ast } \left ( p+\bar{q}_{3}-\bar{\kappa } , \tau _ { 2}^{\prime } \right ) \notag\end{aligned}\ ] ] for the `` two bubbles '' figure . it is clear that for a given volume , the flux of magnetic field across its boundary is not zero ; in other words , the boundary of @xmath87 is not a magnetic surface . in principle , this fact renders @xmath78 as defined in ( [ b-1 ] ) , a gauge dependent quantity . as stated above , berger and field @xcite gave a gauge - independent measure of magnetic helicity suitable for this situation , based on the difference between the magnetic helicities of two field configurations that have a common extension outside the considered volume . the only constraint that definition must satisfy is that the sources of the fields must be bounded in order to guarantee that the surface at infinity is a magnetic one , which is equivalent to say that the magnetic field at infinity must vanish . the fact that in our model @xmath111 at infinity @xcite allows us to consider the boundary at infinity as a magnetic surface , in spite of the fact that the stochastic currents exist in the whole space ( cosmological particle creation is not restricted to the particle horizon @xcite ) . below , in section [ gd ] we discuss how to apply berger s criterion to our fields . in this subsection we provide some ( dimensional ) quantities that will be used later to obtain concrete values of magnetic intensities . we are interested in two scales : one is large , e.g. , of the order of the galactic commoving scale ( or larger ) for which , due to the fact that it remains outside the particle horizon during most of radiation dominance , we can consider diffusive evolution of the magnetic field . the other is of the order of the horizon at reheating , where the magnetic field may be subjected to an inverse cascade of magnetic helicity if the medium is turbulent . in view of the fact that reheating happens in a very short period of time , we can consider that during it the particle horizon remains practically constant and equal to the one in inflation , i.e. , @xmath112 . when the universe enters into the radiation dominated regime , the horizon grows as @xmath113 , a fact that in terms of the temperatures can be expressed as @xmath114 . in the presence of turbulence , the viscous dissipation scale is usually estimated as @xmath115 with @xmath116 the reynolds number . according to ref . @xcite , during reheating can be taken as @xmath117 and smaller for later epochs @xcite . concerning the galactic size we have that today a galactic commoving scale ( i.e. , a not - collapsed scale that contains all the matter of a milky way - like galaxy ) is @xmath118 mpc . for the hubble constant during inflation we consider @xmath119 ; the mass of the scalar field can be taken as @xmath120 ; we also have @xmath121 cm ; the electric conductivity takes the usual form considered in the literature , @xmath122 ^{1/4}\simeq 1.37~\times 10^{19/2}gev$ ] , @xmath123 , where we considered @xmath124 and @xmath125 . finally , the temperature at matter - radiation equilibrium can be taken as @xmath126 ev . thus the different quantities that appear in the formulae lay in te intervals @xmath127 when the transition from inflation to reheating occurs , the vacuum state of the fields during inflation turns into a particle state @xcite . mathematically , this means that the positive frequency modes of the field in the inflationary epoch can be expressed as a linear combination of positive and negative frequency modes in reheating . as we are dealing with a charged scalar field , electric charges will appear and , as there are equal numbers of positive and negative carriers the mean value of the induced electric current is zero , but not its rms deviation , which will source the stochastic magnetic field . therefore in order to evaluate @xmath128 we express the @xmath129 fields of inflation in terms of the ones in reheating by the usual bogoliubov transformation @xcite : @xmath130@xmath131 and @xmath132 being the bogoliubov coefficients satisfying the normalization condition @xmath133 . the different field products then read @xmath134 \notag\end{aligned}\ ] ] the first term represents the vacuum - to - vacuum transition , the second and third account for a mixing of positive and negative frequency modes , while the fourth , proportional to @xmath135 , is due to solely the negative frequency modes , i.e. to the transition vacuum to particle state . as the main contribution comes from this term from now on we consider only it . in a non - instantaneous transition the creation of small scale modes depends on the details of the transition , while for superhorizon modes details of the transition do not matter . for subhorizon modes we have @xcite @xmath136 while for @xmath137 this coefficient reads ( see appendix [ apa ] ) @xmath138 @xmath139 we see that the contribution of subhorizon modes to the magnetic helicity correlation function will be suppressed relative to the one of superhorizon ones . however , according to the results of ref . @xcite subhorizon fluctuations are responsible for a mildly turbulent flow on scales of the order of the horizon size , with reynolds numbers @xmath140 . an important comment about the infrared limit in expr . ( [ d-4a ] ) is in order . it is clear that that expression blows out for @xmath141 . this is due to the approximations made to solve klein - gordon equation during reheating . to give a physical lower limit we must consider the largest homogenous patch created during inflation as the largest possible scale , which according to refs . @xcite can be considered as about 10 times the horizon during inflation . in principle , the dependence of the rms value of @xmath85 on the gauge could be analised by adding to @xmath142 a term of the form @xmath143 to the r.h.s . of expression [ c-5 ] , @xmath144 being an arbitrary scalar function . then in @xmath145 there will appear a term proportional to @xmath146 \left[\mathcal{b}_j\left(\bar\kappa^{\prime } - \bar p^{\prime},\eta^{\prime}\right ) p^{\prime j}\right]$ ] that cancels indentically when the angular integrals are performed . this means that only `` on average '' our result is gauge invariant . if we consider a gauge term like @xmath147 , with @xmath148 an arbitrary magnetic correlation function , generally it will satisfy that @xmath149 , and consequently the associated magnetic helicity will not be statistically gauge independent . we then reason according to berger and field as follows . for any volume @xmath87 we were interested in ( a galaxy , a galaxy cluster , horizon at a certain epoch , etc ) , its surrounding region corresponds to the rest of the embedding space whose characteristic scale is very much larger than @xmath150 . if the magnetic correlation tends to zero from a certain scale on , i.e. , if for @xmath151 it is proportional to a power of @xmath15 , then for an observer inside @xmath87 the field outside is statistically equivalent to a vanishing field . consequently we could interpret @xmath152 as the difference between two magnetic field configurations : one being the calculated through the diagrams above , and the other a null configuration . to quantify this assertion , we begin by noting that the trace of the magnetic field correlation function is given by @xmath153 where we again assume integration over @xmath154 and over repeated momenta other than @xmath15 . using again decomposition ( [ c-10 ] ) for the scalar fields we have that from each mean value the only non - null contribution is @xmath155 the integration of the delta functions in eq . ( [ g-1 ] ) produce again a @xmath156 , i.e. momentum conservation . writing the trace of the magnetic field correlation as @xmath157 we obtain the following expression for the prefactor for trace of the magnetic field correlation function @xmath158 ^ 2 \phi\left(\bar q_1,\tau_1\right)\phi^{\ast}\left(-\bar q_1,\tau_2\right ) \phi\left(\bar\kappa - \bar q_1,\tau_1\right)\phi^{\ast}\left(\bar q_1-\bar\kappa , \tau_2\right ) \nonumber\end{aligned}\ ] ] as above , for any considered volume @xmath87 we are interested in , its surrounding region , @xmath159 corresponds to the rest of the embedding space and satisfies @xmath160 . the magnetic field configuration in those regions corresponds to superhorizon modes whose evolution is in general diffusive . therefore to estimate the rms magnetic intensity at a certain time on a comoving scale @xmath161 we use expressions ( [ apb-4 ] ) for the retarded propagators , ( [ apa-7b ] ) for the modes and ( [ d-4a ] ) for the bogoliubov coefficients . we then write @xmath162}{\left(1+\tau/2\right)^2 } \frac{\left[1-e^{-\sigma_0\left(\eta^{\prime } - \tau^{\prime}\right)/h}\right]}{\left(1 + \tau^{\prime}/2\right)^2}\label{g-4}\end{aligned}\ ] ] where @xmath163 is the lifetime of the stochastic electric current . as @xmath164 , it can be neglected in the denominator of each of the time integrals , which can then be estimated as @xmath165 \simeq \tau_f \left [ 1 + e^{-\sigma_0\eta / h}\right]\ ] ] we can estimate the order of magnitude of the integral in the momenta by counting powers . in the numerator we have 5 powers of @xmath15 and 5 powers of @xmath166 ( three from the integration meassure and two from the square of the cross product ) . in the denominator there are 3 powers of @xmath166 and 3 powers of @xmath15 . consequently the overall contribution of the integrals in the momenta is a factor @xmath167 . to estimate their numerical value we must filter out the length scales smaller than those corresponding to @xmath168 because they oscillate inside @xmath159 , and so the main contribution from the momentum integrals is @xmath169 . finally , taking the coincidence limit @xmath170 we have @xmath171 ^ 2\kappa_s^4\label{g-5}\ ] ] thus we see that for @xmath172 , i.e. , for large volumes as those outside the considered structure , the magnetic field vanishes and so does their associated magnetic helicity . therefore we can interpret our expression for the rms value of the magnetic helicity in a given volume @xmath87 in the spirit of the work by berger and field , as the substraction of two magnetic configurations , one of them being effectively zero . in this section we evaluate the magnetic helicity on large scales due to both smooth and fluctuating fields on that scale . from the form of the bogoliubov coefficients , eqs . ( [ d-3a])-([d-5a ] ) we see that the main contribution is due to the modes with @xmath173 , i.e. , eq . ( [ d-4a ] ) . in this case the modes are given by eq . ( [ apa-7b ] ) , whereby @xmath174\right\ } } { \left ( 1+\tau /2\right)\left ( 1+\tau^{\prime } /2\right)}\label{h-1}\ ] ] we are interested in the value of the magnetic helicity at present time . in this limit , and for small momenta , the retarded propagators are given by eq . ( [ apb-5 ] ) , namely @xmath175 . the lifetime of the electric currents was calculated in ref . @xcite in physical time ( see eq . ( 4.9 ) of that reference ) . in conformal ( dimensionless ) time it reads @xmath176 and from the reference mentioned just above we have @xmath177^{1/6}}{\left[\left(3/2\right)\left(h / m\right)^3\tau_0 ^ 2 + \left ( 9/16\right)^4 \right]^{1/3}}\label{h-2}\ ] ] with @xmath178 the duration of the transition inflation - reheating and where we considered that @xmath179 around that transition . considering the values for @xmath180 quoted in section [ dim ] and the smallest possible value for @xmath178 , namely the planck time , it is seen that @xmath181 , and for @xmath173 expr . ( [ h-2 ] ) gives @xmath182 observe that @xmath183 . this fact allows us to neglect @xmath184 in front of 1 in expr . ( [ h-1 ] ) reducing the time integrals in each of the non - trivial graph to simply @xmath185 . this simplification permits to write the sum @xmath186 as @xmath187\label{h-4}\ ] ] with @xmath188 the factor @xmath189 $ ] was solved in appendix [ apc ] giving the non - null result @xmath189= \left\{\bar\kappa\cdot \left[\bar q\times\left(\bar q_1 + \bar q_3\right)\right]\right\}^2 $ ] , with @xmath190 . replacing expression ( [ d-4a ] ) for the bogoliubov coefficients and ( [ h-3 ] ) for @xmath163 , the magnetic helicity correlation function ( [ h-4 ] ) for large scales and long times becomes @xmath191 \right\}^2}{q_1 ^ 3q_3 ^ 3q^3\vert \kappa - \bar q - \bar q_1 - \bar q_3\vert^3}\label{h-7}\ ] ] there remain the integrations over the momenta other than @xmath15 . as before , their contribution can be roughly estimated by counting powers as before . since we are considering scales such that @xmath192 , we can disregard @xmath15 in the denominator of expr . ( [ h-7 ] ) . then we have that the power of the scalar field momenta in the numerator is 13 ( nine from the integration meassures plus 4 from the square of the cross product ) while in the denominator their power is 12 ( three from each of the four bogoliubov factor ) , giving as result a factor of the form @xmath193 . as the maximum value allowed by the approximations is @xmath194 the result is @xmath195 according to definition ( [ c-1 ] ) there remain the integration over @xmath15 s , wich gives an extra factor @xmath12 , and the two integrations over the volume , which according to expr . ( [ c-4 ] ) each one gives a factor of @xmath196 . then the contribution of all non - null graphs to the magnetic helicity is @xmath197 finally , the rms value of the magnetic helicity in a comoving volume @xmath196 can be considered as simply the squareroot of expr . ( [ h-9 ] ) giving @xmath198 note that this helicity corresponds to regular as well as irregular fields on the considered volume , i.e. to the total magnetic field due to fluctuations up to the scale @xmath180 . the density of defects on a scale @xmath15 is found by multiplying expr . ( [ h-10 ] ) by @xmath12 , finding that it varies as @xmath10 . using the fact that a fractal of dimension @xmath199 embedded in a spherical volume has a number density that scales with the radius @xmath14 from an occupied point as @xmath200 we estimate the fractal dimension of the distribution of topological defects as being @xmath201 . in this section we evaluate the magnetic helicity and its corresponding magnetic field on large scales due to smooth fields only . the estimates will be the values that they would have today if they were not affected by the galaxy formation process . the expression we are seeking for is obtained directly from eq . ( [ h-7 ] ) , estimating the integrals in the momenta other than @xmath15 by filtering the frequencies higher than the one associated to the considered scale , say @xmath202 . we obtain @xmath203 i.e. , a scale independent number . as before , we estimate the magnetic helicity on the considered scale by simply taking the squareroot of expression ( [ d-1b ] ) and , after replacing @xmath204 from ( [ h-3 ] ) we obtain @xmath205 using the numbers given by ( [ c-151])-([c-171 ] ) we have on a galactic scale @xmath206 which is a very small number . from eq . ( [ d-3 ] ) we obtain the number density of links of the smooth field by multiplying by @xmath207 , obtaining @xmath208 , which means that the fractal dimension of the distribution is @xmath2 . this would mean that for large scales the number of defects becomes independent of the scale . it is interesting to estimate the order of magnitude of the coherent magnetic field associated to this helicity on a galactic scale . we can crudely do that by taking @xmath209 with @xmath210 , whereby @xmath211 , i.e. @xmath212 recalling that the physical field is @xmath213 and that 1 gev@xmath214 gauss we obtain that on a galactic scale the helical fields have an intensity of the order @xmath215 which is a very small value , that however agrees with previous estimates in the literature @xcite . observe however that expr . ( [ d-4 ] ) coincides with the squareroot of expr . ( [ g-5 ] ) when @xmath163 is replaced by expr . ( [ h-3 ] ) and the limit @xmath216 is taken . this means that the magnetic fields smooth on large scales induced by the mechanism considered in this paper are indeed helical . the important feature of magnetic helicity for large scale magnetogenesis is the fact that it performs an inverse cascade if the medium where it evolves is turbulent . we shall then make the hypothesis that after reheating the plasma is in a state of decaying turbulence , where self - organization of magnetic structures can happen . the intensity of the turbulence is determined by the reynolds number , which during reheating was estimated in ref . @xcite to be @xmath217 , while for times around electron - positron annihilation was calculated to be @xmath218 @xcite . therefore a decaying turbulence scenario in the early universe is possible , the turbulence being mild . turbulence is a causal phenomenon , i.e. , it happens on scales equal or smaller than the particle horizon . we must evaluate the intensity of magnetic helicity in regions of size of at most the particle horizon at the time @xmath163 , when the sources of magnetic field vanish . after that moment the evolution of the field intensity and coherence scale will be considered to be due to the inverse cascade of magnetic helicity . we shall analize that evolution in the light of the simple model discussed in ref . @xcite ( see also ref . @xcite ) , whereby the comoving coherence length of the magnetic field grows as @xmath219 and the total comoving energy @xmath220 contained in a given volume decays as @xmath221 . moreover , when the volume is fixed that law can be applied to the comoving magnetic energy density , i.e. , we can consider @xmath222 . those laws arise from the conservation of magnetic helicity during the process of inverse cascade . observe that the comoving coherence length remains always smaller than the comoving horizon , which after inflation grows as @xmath223 ( the physical horizon growing as @xmath224 ) . we begin by estimating the dimensionless time interval elapsed between the end of reheating and matter - radiation equilibrium . knowing that the horizon grows as @xmath225 , that interval can be estimated as @xmath226 . for the horizon at matter - radiation equilibrium we take it as @xcite @xmath227 . for reheating , we assume that the transition inflation - reheating was fast enough , and that the electric currents vanished in a very short time , @xmath163 , to assume that the horizon at the end of reheating is not very different from @xmath228 , with @xmath20 the hubble constant during inflation . therefore @xmath229 . using the values for @xmath20 quoted in section [ dim ] that period of time would be in the interval @xmath230 . to evaluate the magnetic helicity correlation we firstly note that the modes that will contribute the most are those such that @xmath231 . this is due to the form of the bogoliubov coefficients ( [ d-3])-([d-5 ] ) . because of the different intervals used to find the scalar field modes and the corresponding bogoliubov coefficients , we must consider two intervals : @xmath232 , with bogoliubov coefficient given by expr . ( [ d-4 ] ) and @xmath233 with bogoliubov coefficient given by expr . ( [ d-3 ] ) . in the first interval @xmath234 is an infrared cut - off that corresponds to the scal of the largest inflationary patch discussed above . in the second interval , the upper limit @xmath235 is considered for consistency with the approximations made in appendix [ apb ] to find the retarded propagator for large scales , eq . ( [ apb-4 ] ) . with these considerations in the momenta and for the short time periods we consider , we can disregard the time dependence in the corresponding mode functions , eqs . ( [ apa-7b ] ) and ( [ apa-10 ] ) because @xmath236 . therefore the mode functions during reheating to be used to evaluate the integrals in the momenta read @xmath237 for @xmath173 and @xmath238 for @xmath239 . the contribution of the `` cross '' plus the `` two bubbles '' diagrams is then similar to the corresponding to large scales , namely @xmath240\right\}^2 \vert\beta_{q_1}\vert^2 \vert\beta_{q_3}\vert^2\vert\beta_{q - q_1}\vert^2\vert\beta_{\kappa - q -q_3}\vert^2\label{e-1}\\ & & g^{+}\left(q_1,\tau_1,\tau_1^{\prime}\right)g^{+}\left(q_3,\tau_2,\tau_2^{\prime}\right ) g^{+}\left(q - q_1,\tau_1,\tau_1^{\prime}\right)g^{+}\left(\kappa - q - q_3,\tau_2,\tau_2^{\prime}\right)\nonumber\end{aligned}\ ] ] with @xmath241 . as the time dependence of the momenta has dissappeared , the time integrals involve only the expressions for the retarded propagators . for @xmath242 and time intervals @xmath243 they are given by expr . ( [ apb-6 ] ) below , i.e. , @xmath244 . the integration between @xmath245 and @xmath246 is then straightforward , giving each propagator a contribution of @xmath247 , the overall time contribution thus being @xmath248 . replacing the corresponding expressions for the bogoliubov coefficients and the modes , eq . ( [ e-1 ] ) can be written as @xmath249 with @xmath250 corresponding to the momentum interval @xmath173 and @xmath251 to @xmath252 . explicitly we have @xmath253\right\ } ^2}{q_1 ^ 3q_3 ^ 3q^3\vert \bar\kappa - \bar q - \bar q_1 - \bar q_3\vert^3}\label{e-2}\ ] ] and @xmath254\right\}^2}{q_1 ^ 9q_3 ^ 9q^9 \vert \bar\kappa - \bar q - \bar q_1 - \bar q_3\vert^9}\label{e-3}\ ] ] to roughly estimate all momentum integrals we again count powers . as @xmath255 we begin by neglecting the @xmath256 s in front of @xmath15 in both expressions . for @xmath257 we obtain a contribution of the form @xmath258 and considering the contribution that gives the largest value , namely @xmath259 we obtain @xmath260 for @xmath261 the counting of powers give a factor of the form @xmath262 . and in the corresponding interval @xmath263 we consider the contribution of the lower limit , i.e. again @xmath180 , as it gives the largest value . te result is @xmath264 an important comment is in order : according to the approximations made to solve the klein - gordon equation during reheating in appendix [ apa ] , the intervals @xmath265 and @xmath266 determined the two different sets of solutions that were used to calculate the bogoliubov coefficients ( [ d-4 ] ) and ( [ d-5 ] ) . taking the limit @xmath267 in both of those expressions , we see that they are not continuous in that limit , which means that the mentioned approximations break down . consequently both @xmath257 and @xmath261 found using that value of momentum must be considered as upper bounds to the possible realistic values . we see that the main contribution to the rms value of the magnetic helicity is due to @xmath261 , which after replacing the expression ( [ h-3 ] ) reads @xmath268 we estimate the magnetic helicity on a volume @xmath196 again by taking the squareroot of ( [ e-6 ] ) , and thus see that the number of links scales as @xmath269 and their density as @xmath270 . the associated fractal dimension is @xmath271 . we now turn our attention to the evolution of the magnetic field from its value induced at the end of reheating in a scale of the order of the horizon up to matter - radiation equilibrium . that evolution will be due to a possible turbulent state during radiation dominance , which would allow an inverse cascade of magnetic helicity . we thus estimate again the comoving magnetic energy density as @xmath272 , and for @xmath273 it is simply @xmath274 . at matter - radiation equilibrium we would then have a comoving magnetic field intensity of the order @xmath275 with a corresponding ( dimensionless ) comoving coherence length @xmath276 . to obtain the value of the physical field and coherence scale we must take into account the expansion of the universe , which for the field means a dilution by a factor of @xmath277 and for the coherence scale an expansion by a factor @xmath278 . adding also the dimensions through the corresponding powers of @xmath20 we finally have @xmath279 and @xmath280 . using the figures given above and in subsection [ dim ] we obtain at the beginning of matter dominance , a field intensity in the range @xmath281 with a corresponding coherence length of @xmath282 these values could impact the process of early structure formation @xcite . in this work we have investigated the generation of magnetic helicity in primordial magnetogenesis . we considered a specific mechanism for magnetic field generation , developed in refs . @xcite ( see also @xcite ) , where stochastic magnetic fields were induced by electric currents that appeared due to particle creation at the transition inflation - reheating . the charges correspond to a scalar field minimally coupled to gravity , as in this case the number of particles created is maximal . there would be support for such a field in the supersymmetric theory of particles @xcite . as the induced magnetic fields are random , the mean values of the different quantities are null , and consequently we had to calculate a rms deviation of @xmath0 . we could write it as the sum of four different sqed feynman graphs of differente multiplicities . our main result is that the fields induced by stochastic currents generated by cosmological particle creation are helical . on large scales as e.g. the galactic ones , we considered that the evolution of the magnetic helicity is diffusive because that scales are larger than the horizon during magnetogenesis , entering the horizon by the end of the radiation era or during matter dominance . we also investigated the generation of magnetic helicity at a scales with a size of the order of the horizon during reheating , where inverse cascade can be operative . we made the naive hypothesis that throughout radiation dominance the plasma flow is endowed with decaying turbulence and applied the model for magnetic energy and coherence scale evolution developed in refs . @xcite . the estimation of the resulting magnetic field intensities at equilibrium between matter and radiation gives values and coherence scales that could be of importance for structure formation @xcite . the importance of primordial magnetic fields endowed with magnetic helicity is that fields coherent on scales equal or shorter than the particle horizon , would self - organize on larger scales . this is due to the fact that when the plasma where the field evolves possesses some degree of ( decaying ) turbulence , magnetic helicity performs an inverse cascade instead of a direct one , thus thus self - organizing at large scales . for large scales , as e.g. the galactic ones , there can also operate the inverse cascade , but even if there were not such a process , the operation of further amplifying mechanisms , as galactic dynamos , would be crucially affected by the topological properties of the seed fields @xcite . in conclusion , generation of magnetic helicity in the early universe seems to be quite easily achieved in different scenarios @xcite . the problem still remains in the intensities . in the mechanism considered here the fields obtained are indeed helical , but their intensities on large scales are too small , or have a marginal value to have an astrophysical impact , while on smaller scales they can be important provided that inverse cascade of magnetic helicity is operative in the early universe . e.c . aknowledges suport from conicet , uba and anpcyt . thanks cnpq / capes for financial help through the procad project 552236/2011 - 0 . in this appendix we find the solutions of eq . ( [ a-5b ] ) in the two epochs of the universe considered in the paper , i.e. , inflation and reheating , and calculate the bogoliubov coefficients . the fourier transformed eq . ( [ a-5b ] ) reads @xmath283 \varphi _ { p}=0 \label{apa-1}\ ] ] whose solutions in each epoch will be labeled as @xmath284 ( inflation ) and @xmath285 ( reheating ) . the fact that they are different in each epoch tells that a given quantum state in the first epoch will not coincide with a same state in the subsequent epoch . more concretely , a vacuum state during inflation will appear as a particle state in reheating . mathematically this is expressed as @xcite @xmath286 the fact that @xmath287 , shows that the two quantum states are not equivalent . @xmath288 and @xmath289 are the bogoliubov coefficients @xcite . for modes corresponding to scales larger than the horizon size at the epoch of transition , that transition can be considered as instantaneous and the coefficients can be calculated by demanding continuity of @xmath290 and @xmath291 at that instant . this gives @xmath292 for scales shorter than the horizon , i.e. , @xmath293 , the calculation is more involved , as details of the transition do matter ( see ref . @xcite and references therein ) . for completion we only quote here the result and refer the reader to the references for the calculations . @xmath294}{p^5},\qquad p > 1 \label{apa-0d}\ ] ] where @xmath178 is the lasting of the transition . considering that the scale factor at the end of inflation is equal to unity , we can write @xmath295 and eq . ( [ apa-1 ] ) reads @xmath296 \varphi _ { p}^{i}=0 \label{apa-2}\ ] ] writing @xmath297 eq . ( [ apa-2 ] ) transforms into a bessel equation for @xmath298 and therefore the normalized , positive frequency solutions of eq . ( [ apa-2 ] ) are @xmath299 \label{apa-3}\ ] ] with @xmath300 . as we shall be considering the case @xmath301 we can approximate @xmath302 and in this case the hankel function has a polinomic expression , namely @xmath303 \label{apa-4}\ ] ] in this case @xmath304 and eq . ( [ apa-1 ] ) is given by @xmath305 \varphi _ { p}^{r}=0 \label{apa-5}\ ] ] which can not be reduced to a known equation , unless some approximations are made . in this sense we consider two situations : @xmath173 and @xmath306 . in both cases the modes correspond to wavelengths larger than the size of the particle horizon and hence detais of the transition between the two considered epochs do not matter . the corresponding bogoliubov coefficients can be calculated considering an instantaneous transition at @xmath245 , and demanding continuity of the modes and their first derivatives at that moment . in this limit ( [ apa-5 ] ) reads @xmath308 \varphi _ { p}^{r}=0 \label{apa-6}\ ] ] proposing a solution of the form @xmath309 $ ] eq . ( [ apa-7 ] ) transforms into a bessel equation for @xmath298 , and so the normalized , positive frequency solutions of eq . ( [ apa-6 ] ) are @xmath310 \nonumber\\ & = & -\sqrt{\frac{6}{\pi}}\frac{\exp\left[i\left ( 2m/3h\right)\left(1 + \tau/2\right)^3\right]}{\left ( 1+\tau/2\right)}\label{apa-7b}\end{aligned}\ ] ] using expr . ( [ apa-0c ] ) we obtain @xmath311 this case still corresponds to modes outside the particle horizon , but their form is different from the one corresponding to the previous momentum interval . ( [ apa-4 ] ) reads @xmath312 \varphi _ { p}^{r}=0 \label{apa-9}\ ] ] writing @xmath313 $ ] we again obtain a bessel equation for @xmath314 . in this case the normalized , positive frequency modes are @xmath315\nonumber\\ & = & -\frac{1}{\sqrt{2p}}e^{i2p\left(1+\tau /2\right)}\left[1-\frac{i}{4p\left(1+\tau /2\right)}\right ] \label{apa-10}\end{aligned}\ ] ] replacing again in expr . ( apa-0c ) we have @xmath316 the fourier transform of the retarded propagator for the electromagnetic field satisfies the following equation @xmath317 g_{ret}\left ( \kappa , \eta , \tau \right ) = \delta \left ( \eta -\tau \right ) \label{apb-1}\ ] ] whose homogeneous solutions are of the form @xmath318 , with @xmath319 . we therefore propose @xmath320 \theta \left ( \eta -\tau \right ) \label{apb-2}\]]which is continuous in @xmath321 . demanding @xmath322 gives @xmath323 , whence @xmath324 \theta \left ( \eta -\tau \right ) \label{apb-3}\ ] ] in this case we can take @xmath326 and thus @xmath327\theta \left ( \eta -\tau \right ) \label{apb-4}\ ] ] in the limit @xmath328 eq . ( [ apb-4 ] ) simply reads @xmath329 while for @xmath330 we have @xmath331 in this case @xmath333 and then @xmath334}{\kappa } \theta \left ( \eta -\tau \right ) \label{apb-7}\ ] ] which for finite @xmath15 and @xmath335 gives @xmath336 we must evaluate @xmath337 \label{apd-1}\]]with@xmath338@xmath339@xmath340we have that@xmath341and @xmath342defining @xmath343 we can write@xmath344and@xmath345for the different terms in the integrand we thus have@xmath346 \left [ \bar{\kappa}\cdot \left ( \bar{q}_{1}\times \bar{q}\right ) \right ] + \left [ \bar{\kappa}\cdot \left ( \bar{q}_{3}\times \bar{q}_{1}\right ) \right ] \left [ \bar{\kappa}\cdot \left ( \bar{q}_{3}\times \bar{q}\right ) \right ] \label{apd-9}\]]that integrates to zero because the first term is odd in @xmath347 and the second in @xmath348 . for the same reasons , the same happens with the term@xmath349 \left [ \bar{\kappa}\cdot \left ( \bar{q}_{1}\times \bar{q}% \right ) \right ] + 2\left [ \bar{q}\cdot \left ( \bar{q}_{1}\times \bar{q}% _ { 3}\right ) \right ] \left [ \bar{\kappa}\cdot \left ( \bar{q}_{3}\times \bar{q}% \right ) \right ] \label{apd-10}\]]there remains the term@xmath350 \left [ \bar{\kappa}\cdot \left ( \bar{q}_{1}\times \bar{q}% \right ) \right ] -\left [ \bar{\kappa}\cdot \left ( \bar{q}\times \bar{q}% _ { 3}\right ) \right ] ^{2 } \label{apd-11}\ ] ] we define @xmath351 hence @xmath352 ^{2}+2\left [ \bar{\kappa}\cdot \left ( \bar{q}\times \bar{q}_{+}\right ) \right ] \left [ \bar{\kappa}\cdot \left ( \bar{q}\times \bar{q}_{-}\right ) \right ] \label{apd-13}\]]the term odd in @xmath353 integrate to zero , hence there only remains the first one , i.e.@xmath352 ^{2 } \label{apd-14}\]]which is clearly non - null .
|
we study the possibility that primordial magnetic fields generated in the transition between inflation and reheating posses magnetic helicity , @xmath0 .
the fields are induced by stochastic currents of scalar charged particles created during the mentioned transition .
we estimate the rms value of the induced magnetic helicity by computing different four - point sqed feynman diagrams . for any considered volume ,
the magnetic flux across its boundaries is in principle non null , which means that the magnetic helicity in those regions is gauge dependent .
we use the prescription given by berger and field and interpret our result as the difference between two magnetic configurations that coincide in the exterior volume . in this case
the magnetic helicity gives only the number of magnetic links inside the considered volume .
we calculate a concrete value of @xmath0 for large scales and analyze the distribution of magnetic defects as a function of the scale .
those defects correspond to regular as well as random fields in the considered volume .
we find that the fractal dimension of the distribution of topological defects is @xmath1 .
we also study if the regular fields induced on large scales are helical , finding that they are and that the associated number of magnetic defects is independent of the scale . in this case
the fractal dimension is @xmath2 .
we finally estimate the intensity of fields induced at the horizon scale of reheating , and evolve them until the decoupling of matter and radiation under the hypothesis of inverse cascade of magnetic helicity .
the resulting intensity is high enough and the coherence length long enough to have an impact on the subsequent process of structure formation .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
silica aerogel is a unique material with a refractive index ( @xmath6 ) in the range between gases and liquids or solids . its refractive index can be easily controlled from @xmath7 to @xmath8 . as a result , the refractive index of the aerogel can be chosen such that for a given momentum interval in the few gev/@xmath3 region charged pions radiate cherenkov photons , while kaons stay below the cherenkov radiation threshold @xcite . in the belle experiment at kek @xcite , a threshold type cherenkov detector ( belle - acc ) @xcite which uses aerogel as a radiator , is operated , providing at @xmath9 a kaon identification efficiency of @xmath10% with a pion missidentification probability of @xmath11% @xcite . a new production method of hydrophobic aerogel with a high transmission length and @xmath6 in the interval between @xmath12 and @xmath13 was developed during the construction period of belle - acc @xcite . the improvement in quality allows the use of an aerogel radiator in a ring imaging cherenkov counter ( rich ) @xcite . in the hermes experiment at desy , a rich counter is used with a dual - radiator ( aerogel and gas ) , and mirrors to focus the cherenkov photons @xcite . a similar detector is also designed for the lhcb experiment at cern @xcite . we are studying the feasibility of a rich counter with an aerogel radiator for the belle - acc in the forward end - cap region @xcite . since this part is now optimized for the pion / kaon separation needed for tagging of the @xmath14 flavor , and covering the momentum range below @xmath15 , separation at high - momentum region of around @xmath5 is not adequate . this kinematic region is , however , very important for the studies of two - body decays such as @xmath16 , @xmath17 . in order to achieve a @xmath18/@xmath19 separation for a wider momentum range , a ring imaging - type of detector is needed . due to spatial restrictions , such a counter has to be of the proximity focusing type . to cover the identification in the lower momentum region ( around @xmath20 ) as well as in the region up to @xmath5 , the aerogel has to have a refractive index around @xmath21 . the first beam test of such a detector was carried out in 2001 at the kek - ps @xmath2 beam line @xcite . these tests used an array of multi - anode pmts ( hamamatsu r5900-m16 ) for photo - detection . the detected number of photoelectrons was @xmath22 per ring for a @xmath23 thick aerogel tile with @xmath21 , and the cherenkov angle resolution per photon was @xmath24 . these results were consistent with expectations . the number of detected photons was , however , rather low , partly because only @xmath25% of the detector surface was covered by the photo - cathodes , and partly because the transmission length of the aerogel with @xmath21 could not be made large enough . for the second beam test , we improved the aerogel transmission by optimizing the materials used in the production process . the active area fraction of the photon detector was increased by employing recently developed flat - panel pmts , hamamatsu h8500 . although this type of pmt is not immune to magnetic field , and therefore can not be applied in the belle spectrometer , we consider this device as an intermediate step in our development . the paper is organized as follows . we first present the experimental set - up with flat - panel pmts , briefly review the improvement in aerogel production , describe the measurements , and finally discuss the results . the photon detector for the tested prototype rich counter employed @xmath26 channel multi - anode pmts ( hamamatsu h8500 , so called flat - panel pmt ) because of their large effective area . 16 pmts were used in a @xmath27 array and aligned with a @xmath28 pitch , as shown in figure [ fig : flat - panel - pmt ] . the surface of each pmt is divided into @xmath26 ( @xmath29 ) channels with a @xmath30 pixel size . therefore , the effective area of photon detection is increased to @xmath31 . at the back of each pmt , an analog memory board is attached to read out multi - channel pmt signals , as described below . among @xmath32 pmts , @xmath11 pmts were delivered in january , 2002 , and the remaining pmts were delivered in october , 2002 . since the manufacture method of the pmt was still under development , they exhibit a large variation in quantum efficiency and gain . the quantum efficiency at 400 nm varies between @xmath32% and @xmath33% ; the gain varies from @xmath34 to @xmath35 when the maximal allowed high voltage of @xmath36 v is applied to the photo - cathode @xcite . the pmts from the later batch show a slightly better performance . the hydrophobic form of the aerogel radiator from novosibirsk @xcite is known to have a long transmission length . however we prefer hydrophobic aerogel than hydrophilic one for the application to a collider experiment . with a low refractive index ( @xmath37 ) , such an aerogel was developed for the belle - acc , and is characterized by a high transmission length ( @xmath38 mm at a wave length of 400 nm ) which was not achieved before . however , the transmission length of aerogel with a higher refractive index of @xmath21 fell below one half the value compared to the aerogel with @xmath39 . therefore , we reexamined the aerogel production technique in a joint development with matsushita electric works ltd . as a result , we found that the important factors determining the transmission length are the solvent and selection of the precursor to be used for its production . originally , we used methyl - alcohol for the solvent , and methyl - silicate as a precursor @xcite . when we applied di - methyl - formamide ( dmf ) @xcite , and changed the supplier of the precursor , we could improve the transmission length of the aerogel . figure [ fig : aero ] shows the refractive indices of aerogel and the relation to transmission length for samples which were used in this beam test . the refractive index was determined by measuring the deflection angle of laser light ( laser diode : @xmath40 ) at a corner of each aerogel tile ; the transmission length was measured with a photo - spectrometer ( hitachi u-3210 ) . in addition to the samples produced with the new technique at matsushita electric works ltd . and chiba university , samples from binp ( novosibirsk ) were tested @xcite ; for comparison , we also tested the samples used in the previous beam test . the thicknesses of the prepared aerogel samples ranged from @xmath41 to @xmath42 . various thickness of up to about @xmath43 were tested by stacking these samples . note that in the production of the aerogel samples at binp propenol was used as the solvent , and the resulting aerogel was hydrophilic . also note that the matsushita aerogel samples produced with the new technique have a very similar transmission length as the binp samples . the transmission length for @xmath44 samples used in the first beam test was around @xmath45 mm , but was increased to @xmath46 mm for matsushita s sample with the new production method . for the beam test , pions with momenta between @xmath47 and @xmath5 were used . beside the rich detector under study , counters for triggering , tracking and particle identification were employed . the set up of the aerogel rich is shown in figure [ fig : rich ] . two rich counters were placed in a light - shield box and tested simultaneously . each rich was composed of a layer of aerogel radiator and a photo - detection plane , parallel to the radiator face at a distance of @xmath48 . the upstream cherenkov counter was the detector under study ; the downstream counter was the one employed in the previous beam test . since the latter uses a well - known photo - detector , multi - anode pmts hamamatsu r5900 , we regarded it as a reference . particle identification was done to remove particles other than pions . two co@xmath49 gas cherenkov counters in the beam line were used to exclude electrons . also , an aerogel counter was equipped and used to exclude protons for the high - momentum region . this detector was also used to exclude muons for the low - momentum region around @xmath47 . the particle trajectories were measured with multi - wire proportional chambers ( mwpc ) at the upstream and downstream ends of the light - shield box . these @xmath50 mwpcs , with @xmath51 m diameter , gold - plated tungsten anode wires at @xmath52 pitch and with @xmath53 ar + @xmath54 @xmath55 gas flow , were read out by delay lines on the x and y cathode strips . the trigger signal was generated as a coincidence of signals from several @xmath50 plastic scintillation counters and anode signals from the mwpcs to ensure valid tracking information . for the beam test , a new read - out system was designed by using analog memory chips . the analog memory chip is based on a chip developed by h. ikeda @xcite for a cosmic - ray experiment . we borrowed the chips from nasda ( national space development agency of japan ) , and developed the chip control system . in the analog memory chip , the signals of 32 channels are preamplified , sampled in @xmath56s intervals , and stored in an 8 steps deep analog pipeline . figure [ fig : rich - readout ] shows a schematic view of the readout system with these analog memories . two 32 channel analog memories are attached to each 64 channel pmt . the memories corresponding to four pmts are controlled by a 256 channel memory controller . when the gate pulse is formed from the trigger signal , a control signal is sent from the controller to the analog memories . the difference in the value of the analog memory between the latest and the first memory content is fed to the output . the obtained output values of 256 channels are clocked into one signal train with a period of 10 @xmath57s per channel . each analog memory controller outputs the serial signal together with synchronized control signals . these signals are then read by a 12-bit vme adc ( dsp8112 , mtt co. ) with a conversion time of 5 @xmath57s . a reference rich was instrumented with multi - anode pmts , hamamatsu r5900-m16 , the same photon detector as used in the previous test @xcite . the quantum efficiency of the pmts is around 26% ( at 400 nm ) , and the gain was around @xmath35 with @xmath58 v applied to the photo - cathode . the pmts were grouped in a @xmath59 array at a @xmath60 mm pitch . due to a limited number of available pmts and read - out channels , only a part of the cherenkov ring was covered with photon detectors . most of the test measurements were performed with a @xmath61 beam at @xmath62 . to systematically evaluate the detector performance , data were taken with different aerogel samples with various transmission lengths and thicknesses . data were also taken by varying the @xmath61 momentum in the range from @xmath47 to @xmath63 . a few typical events are displayed in figure [ fig : rich - eventdisplay ] . the hits on pmts can be associated with the expected position of the cherenkov ring . the hit near the center of the ring is due to cherenkov radiation generated by the beam particle in the pmt window . the distribution of accumulated hits is shown in figure [ fig : typical1 ] . cherenkov photons from the aerogel radiator are clearly seen with a low background level . the background hit distribution on the photon detector is consistent with the assumption that it originates from cherenkov photons which were rayleigh scattered in the radiator . the pulse - height distribution of the cherenkov photons detected in one of the flat - panel pmt is shown in figure [ fig : adc ] . the raw data were corrected as follows . a common - mode fluctuation of the base line was subtracted and signals due to cross - talk in the read - out electronics were removed . the signal mainly containing one photoelectron is clearly separated from the pedestal peak . note , however , that this distribution differs considerably from tube to tube because of the large variation in performance , as described before . for further analysis we also applied a threshold cut to suppress the pedestal noise contribution . figure [ fig : typical2](a ) shows a typical distribution of the cherenkov - angle for single photons . the angular resolution was obtained from a fit of this distribution with a gaussian signal and a linear function for the background . figure [ fig : s2n ] shows the resolution in the cherenkov angle for the @xmath61 beam at @xmath62 and @xmath64 mm thick aerogel samples . the resolution was around @xmath65 mrad , independent of the refractive index . the main contributions to the resolution of the cherenkov angle come from the uncertainty in the emission point and from the pixel size of the pmt . the first contribution is estimated to be @xmath66 , where @xmath67 is the aerogel thickness , @xmath68 is the cherenkov angle and @xmath69 is the distance from an average emission point in the aerogel to the surface of the pmt . the second contribution is @xmath70 , where @xmath71 is the pixel size . the measured variation of the resolution with the thickness of aerogel is shown in figure [ fig : s2d ] . by comparing the measured resolution and the expected values , we observed a rather good agreement . there was , however , a discrepancy between the two , which can be accounted for by a contribution of about @xmath72 mrad . the discrepancy could arise from the effect of aerogel ( non - flat aerogel surface and possible non - uniformities in the refractive index due to position variation and chromatic dispersion ) , which are subject to further investigation . the uncertainty in the track direction is expected to be negligible at @xmath62 , but increases considerably at low momenta ( @xmath47 ) due to the effect of multiple - scattering , as can be seen in figure [ fig : s2p ] . figure [ fig : typical2](b ) shows a typical distribution of the number of hits within @xmath73 from the average cherenkov angle . the number of hits for the signal region was estimated by subtracting the background from the fits to the cherenkov - angle distribution . the number of detected photons ( @xmath74 ) depends on the aerogel thickness and the effect of scattering . it is expressed as + @xmath75 , where @xmath76 is the transmission length of the aerogel at an average wave length of @xmath77 nm and @xmath78 is quantum efficiency of the pmt . figure [ fig : n2d ] shows the dependence of the number of detected photons on the aerogel thickness . as expected from the above expression , the number of photons does not linearly increase with the aerogel thickness , but saturates due to the scattering effect in aerogel . figure [ fig : n2tr ] shows the dependence of the number of photons with transmission length . from the figure the benefit of the improvement in the transmission length of the @xmath1 aerogel from around @xmath45 mm , as used in the previous beam test , to @xmath46 mm using the new production technique becomes evident . the dependence on the pion momentum , displayed in figure [ fig : n2p ] , is fitted with the form expected from the cherenkov relations , and shows a good agreement . for pions with momenta above @xmath79 , the number of detected cherenkov photons was typically around 6 for aerogel samples with @xmath1 . the performance of the rich counter under study was compared in the same set - up with the performance of the reference counter with a well - known photon detector , hamamatsu r5900-m16 multi - anode pmts . since the two counters have a different active area fraction ( @xmath80% for the flat - panel pmts , and @xmath25% for the r5900-m16 pmts ) and a different acceptance , the comparison of the photon yields was made by normalizing to the full active surface . while the flat - panel yield for a particular case was @xmath81 , which resulted in @xmath82 if extrapolated to the full active area , the corresponding number for the r5900-m16 was @xmath83 . it appears that this difference is mainly due to the rather low quantum efficiency and amplification of some of the flat - panel tubes employed . this , in turn , causes inefficiencies in single photon detection with a given threshold . if the best tube in the set is normalized to the full acceptance , the corresponding number increases to @xmath84 , and we would expect about 8 photons per ring . finally , we estimate the performance of pion / kaon separation in the momentum range of around @xmath5 , which is of considerable importance for the belle experiment . if we take into account a typical measured value for the single - photon angular resolution , @xmath85 mrad , and the number of detected photons @xmath86 , typical for 20 mm thick aerogel samples with @xmath21 , we can estimate the cherenkov angle resolution per track to be @xmath87 mrad . this naive estimate is also confirmed by the direct measurement shown in figure [ fig : sep_pik_4gev ] . here , the track - by - track cherenkov angle is calculated by taking the average of the angles measured for hits around the predicted position of the cherenkov ring . from this we can deduce that at @xmath5 , where the difference of cherenkov angles for pions and kaons is @xmath88 mrad , a @xmath89 separation between the two is possible . as an additional cross check , we have also collected data with the pion beam of @xmath90 , which can be used to represent a kaon beam of @xmath5 ( apart from a slightly larger sigma due to multiple scattering ) . as can be seen from figure [ fig : sep_pik_4gev ] , the two peaks are well separated . thus , the proximity focusing aerogel rich seems to be promising for the upgrade of the belle pid system at the forward region . we report on the test beam results of a proximity focusing rich using aerogel as the radiator . to obtain larger photoelectron yields , we used flat - panel multi - anode pmt with a large effective area , and aerogel samples produced with a recently developed method which have a higher transmission length than before . we also developed a multi - channel pmt read - out system using analog memory chips . a clear cherenkov ring from the aerogel radiator could be observed , and the number of photons was enhanced compared to that in previous tests . we performed a systematic study of the detector using various samples of the aerogel . the typical angular resolution was around @xmath65 mrad and the number of detected photoelectrons was around @xmath72 . the pion / kaon separation at @xmath5 is expected to be around @xmath89 . however , we still have some issues which have to be solved for implementation in the belle spectrometer . the most important item is the development of a pmt which can be operated under a strong magnetic field ( 1.5 t ) . an example of a candidate for such a device is a multi - anode hybrid photodiode ( hpd ) or hybrid avalanche photodiode ( hapd ) . of course , for a good candidate , its ability to efficiently detect single photons on a large active area has to be demonstrated . the other item is mass production of the aerogel tiles . while we have demonstrated that the new production method significantly increases the transmission length of the @xmath21 aerogel , the production method has to be adapted to stable good - quality manufacturing . we will study these items at the next stage towards construction of a real detector . we would like to thank the kek - ps members for operation of accelerator and for providing the beam to the @xmath2 beam line . we also thank h. ikeda ( kek ) and the meisei co. for their help in preparing the read - out electronics , the matsushita electric works ltd . for the good collaboration in developing the new aerogel type , and hamamatsu photonics k.k . for their support in equipping the photon detector . we also thank a.bondar ( binp , novosibirsk ) for providing us excellent aerogel samples , and dr . t.goka of nasda for providing us their read - out chips . one of the authors ( t.m . ) is grateful to fellowships of the japan society for the promotion of science ( jsps ) for young scientists . this work was supported in part by a grand - in - aid for scientific research from jsps and the ministry of education , culture , sports , science and technology under grant no . 13640306 , 14046223 , 14046224 , and in part by the ministry of education , science and sports of the republic of slovenia . m.cantin et al . instr . meth . * 118 * ( 1974 ) 177 - 182 a.abashian et al . instr . and meth . a * 479 * ( 2002 ) 117 t.sumiyoshi et al . instr . and meth . a * 433 * ( 1999 ) 385 - 391 t.iijima et al . instr . and meth . a * 453 * ( 2000 ) 321 i. adachi et al . , instr . and meth . a * 355 * ( 1995 ) 390 ; t. sumiyoshi et al . , j. non - cryst . solids * 225 * ( 1998 ) 369 d.e.fields et al . instr . meth . a * 349*(1994 ) 431 - 437 ; r. de leo et al . , instr . and meth . a * 401 * ( 1997 ) 187 n.akopov et al . instr . meth . a * 479 * ( 2002 ) 511 - 530 t.ypsilantis and j.seguinot , nucl . meth . a * 368 * ( 1995 ) 229 - 233 t.iijima , `` aerogel cherenkov counter in imaging mode '' , jps meeting , tokyo metropolitan university , september 1997 . i. adachi et al . , `` test of a proximity focusing rich with aerogel as radiator '' , proceedings for the ieee nuclear science symposium , norfolk , va , november 10 - 15 , 2002 , trans . 50 ( 2003 ) 1142 , hep - ex/0303038 , ; t.iijima et al . instr . meth . a * 502*(2003 ) 231 - 235 hamamatsu photonics k.k . matsushita electric works ltd . has a japanese patent ( no . 2659155 ) for usage of dmf as solvent to make aerogel . buzykaev et al . instr . meth . a * 433 * ( 1999 ) 396 - 400 h.ikeda et al . a * 372 * ( 1996 ) 125 - 134 of the mean cherenkov angle . data were corrected with the procedure described in the text . in this figure , the pulse - height distribution for the high sensitive pmt is shown . for further analysis , we used the hits above a threshold adc value , 120 . , width=302 ] ) for @xmath64 mm thick aerogel samples as a function of the transmission length . @xmath74 is corrected for the refractive index to @xmath21 and @xmath39 respectively . the symbols correspond to the data and the curves are fits described in the text.,width=302 ] and @xmath90 . pions at @xmath90 are used to represent the kaon beam of @xmath5 . the angular resolutions for @xmath63 and @xmath90 are @xmath92 mrad and @xmath93 mrad and two peaks are separated by @xmath94 . , width=302 ]
|
a proximity focusing ring imaging cherenkov detector using aerogel as the radiator has been studied for an upgrade of the belle detector at the kek - b - factory .
we constructed a prototype cherenkov counter using a @xmath0 array of 64-channel flat - panel multi - anode pmts ( hamamatsu h8500 ) with a large effective area .
the aerogel samples were made with a new technique to obtain a higher transmission length at a high refractive index ( @xmath1 ) .
multi - channel pmts are read - out with analog memory chips .
the detector was tested at the kek - ps @xmath2 beam line in november , 2002 . to evaluate systematically the performance of the detector ,
tests were carried out with various aerogel samples using pion beams with momenta between 0.5 gev/@xmath3 and 4 gev/@xmath3 .
the typical angular resolution was around 14 mrad , and the average number of detected photoelectrons was around 6 .
we expect that pions and kaons can be separated at a 4@xmath4 level at @xmath5 .
aerogel , flat - panel pmt , ring imaging cherenkov counter , proximity focusing , particle identification , belle 29.40.ka
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the challenges faced by companies working in nowadays complex it environments pose the need for comprehensive and dynamic systems to cope with the information flow requirements @xcite , @xcite , @xcite . planning can not answer all questions : we must take a step further and discuss a model for application management . one of the possible approaches to deal with this problem , is to use the decision support system that is capable of supporting decision - making activities . in @xcite , we proposed the foundations of our decision support system for complex it environments . developing our framework , we examined the time , energy usage , qop , finance and carbon dioxide emissions . regarding financial and economic analyzes , we considered only _ variable _ costs . however , calculating total operating cost of a data center , one needs to take into account both _ fixed _ and _ variable _ costs , which are affected by complex and interrelated factors . in this paper , performing a financial analysis of security - based data flow , we present an improved method for measuring the total cost of its maintenance , exploring trade - offs offered by different security configurations , performance variability and economic expenditures . with the proposed approach it is possible to reduce the data center costs without compromising data security or quality of service . the main contributions of this paper are summarized as follows : * we enhanced previous studies on security management presented in @xcite , extending it with the analysis of fixed costs , * we proposed a full cost model for data centers , being an economic method of their evaluation , in which all costs of operating , maintaining , and disposing are considered important , * we prepared a case study , in which we : * * applied the developed , financial model to an example data center , in order to show how the model actually works , * * we evaluated the proposed economic scheme and analyzed the distribution of data center maintenance costs over five years , taking into account different security levels , and comparing them with reference to data center total costs , * * based on the results gathered with the introduced method , we calculated possible profits and return on investment values for over five years , in order to choose the best option ( considering the security of the information , along with high company incomes ) . the total cost of the data center is somewhat elusive to project accurately . there are many subtleties which can be overlooked or are simply unaccounted for ( or perhaps underestimated ) , over the operational life of a data center . looking to involve all existing elements of a modern data center into cost calculations , several approaches have been proposed in the literature ( @xcite , @xcite , @xcite ) . in @xcite the authors presented a way of predicting the total budget required to build a new data center . they distinguish three primary construction cost drivers , namely : power and cooling capacity and density , tier of functionality and the size of the computer room floor . they utilize proposed cost model , providing calculations for the example data center . researchers state that their cost model is intended as a quick tool that can be applied very early in the planning cycle to accurately reflect the primary construction cost drivers . however , proposed model makes some rigid assumptions about the data center ( such as the minimum floor space or the type of the utilized rack ) , making it quite inflexible . moreover , as the authors themselves admit , some significant costs were not included in this cost model ( for instance , the operational costs ) . an interesting approach to assessing and optimizing the cost of computing at the data center level is presented in @xcite . here , scientists consider five main components , that should be taken into account , while evaluating the total cost of a data center : the construction of the data center building itself , the power and cooling infrastructure , the cost of electricity to power ( and cool ) the servers and the cost of managing those servers . besides creating a cost model for the data center , the authors examine the influence of server utilization on total cost of a data center , and state that an effective way to decrease the cost of computing is to increase server utilization . another method for assessing the total cost of a data center is proposed in @xcite . an approach examined in @xcite is a part of research which seeks to understand and design next generation servers for emerging warehouse computing environments . the authors developed cost models and evaluation metrics , including an overall metric of performance per unit total cost of ownership . they identify four key areas for improvement , and study initial solutions that provide significant benefits . unlike our approach , @xcite focuses mainly on performance and calculates data center costs only on its basis . the method proposed in our paper considers different aspects at once , which makes it flexible and suitable for heterogeneous environments . although existing cost models for the data center include many components , none of them mention security as the significant factor . however , security influences data center costs as well : proper security management translates into better utilization of central resources as well as reduced systems management and administration . in this section , we present and describe the formulas utilized in the proposed analysis process - in particular , in economic and financial analyzes , extending them with the calculation of _ fixed _ costs . introduced equations are used to evaluate the financial aspect of the data center maintenance . performing the financial analysis , we took under consideration cost of power delivery , cost of cooling , software and hardware expenditures as well as personnel salaries ( both _ fixed _ and _ variable _ costs ) . proposed method of data center cost evaluation is the sum of the present values of all costs over the lifetime of a data center ( including investment costs , capital costs , financing costs , installation costs , energy costs , operating costs , maintenance costs and security assurance costs ) . as shown in figure [ fig : extended_financial_analysis ] , in the introduced financial and economic analyzes , we distinguished @xmath0 main components : _ cost of power delivery _ , _ cost of cooling infrastructure utilization _ and _ operational costs_. each of them can be further specified . firstly , we introduce the general formula used for calculating total energy costs . the following equation is further detailed in terms of cpu and server utilization : @xmath1 where : * @xmath2 - is the total amount of the utilized kilowatt - hours * @xmath3 - is the total amount of hours when the server was busy * @xmath4 - is the total amount of days when the server was busy * @xmath5 - is the cost of a one kwh in us dollars the above formula is the crucial point in computing the total cash outlay for the considered system . original equation is elaborated in the following sections . the design of the electrical power system must ensure that adequate , high - quality power is provided to each machine at all times . except the back - up system operational expenditures , data center spends a lot of money on current power consumption - utilized both for the compute , network and storage resources . at this point one should pay special attention to the relationship between the cpu utilization and energy consumed by other components of a working server . since the amount of the power consumed by a single machine consists of the energy usage of all its elements , it is obvious that a decrease in the energy consumed by the cpu will result in a lower energy consumption in total . thus , total cost of both power delivery and utilization can be summarized as follows : @xmath6 where : * @xmath7 - is the total amount of the utilized kilowatt - hours by the server * @xmath3 - is the total amount of hours when the server was busy * @xmath4 - is the total amount of days when the server was busy * @xmath5 - is the cost of a one kwh being aware of the price of one kwh , and knowing that cpu worked @xmath3 hours through @xmath8days , utilizing @xmath2 kilowatt - hours , it is fairly straightforward to calculate the total financial cost ( @xmath9 ) of its work , using the equation . before we start further evaluation of the energy consumed by the cpu , we need to make some assumptions about its utilization . let us introduce the simplified cpu utilization formula : @xmath10 where : * @xmath11 - stands for the cpu utilization , expressed in percentage * @xmath12 - defines our requirements , the actual busy time of the cpu ( seconds ) * @xmath13 - is the cpu capacity , the total time spent on analysis ( seconds ) usually , the cpu utilization is measured in percentage . the requirements specified in the above formula refer to the time we require from the cpu to perform an action . this time is also known as the _ busy _ time . cpu capacity can be expressed as the sum of the _ busy _ and _ idle _ time ( that is , the _ total _ time available for the cpu ) . going simple , one can say that over a 1 minute interval , the cpu can provide a maximum of @xmath14 of its seconds ( power ) . the cpu _ capacity _ can then be understood as _ busy time _ + _ idle time _ ( the time which was used plus the one which was left over ) . using the above simplifications , when going multi - core , cpu capacity should be multiplied by the number of the cpu cores ( @xmath15 ) . the presented equation can be further detailed as follows : @xmath16 = \frac{time_{_{session } } \ , \cdot \ , users}{time_{_{total}}}\ ; , \ ] ] where @xmath17 is expressed as @xmath18 . + supposing a specified cpu load and assuming the server is able to handle a defined number of users within a given time , we can calculate the _ idle _ time using the above equation . regarding the _ busy _ time , one should use the results obtained for the prepared , example model . this simple formula can be used for calculating the total energy consumed by the cpu . knowing the amount of energy utilized by the cpu , it is quite straightforward to assess costs incurred for the consumed energy . as the cooling infrastructure absorbs energy to fulfill its function , the cost of the cooling needs to be included in the total cost of the server maintenance . to obtain an approximate amount of the power consumed by the cooling , one can use the equipment heat dissipation specifications , most often expressed in british thermal units ( btu ) . specifications generally state how many btu are generated in each hour by the individual machine . therefore , the formula for calculating the cooling cost to keep the equipment in normal operating conditions , is given as follows ( values per server ) : @xmath19 where : * @xmath20 - is the amount of the btus generated by the cooling system * @xmath21 - is the total amount of the utilized kilowatt - hours by the cooling system * @xmath3 - is the total amount of hours when the cooling system was busy * @xmath4 - is the total amount of days when the cooling system was busy * @xmath5 - is the cost of a one kwh operational costs of the data center management depend on miscellaneous factors - among them one can enumerate salaries of the employees responsible for managing servers ( along with the number of employees needed to adequately maintain and operate the data center ) , prices of equipment , the amount of a reduction in the value of servers with the passage of time and finally on software and licensing costs . in order to determine the complete cost of a data center , software and licensing costs should be analyzed as well . the server one purchases may , or may not include an operating system . when it comes to selecting a server os , high - end server operating systems can be quite expensive . except the operating system , one also needs to budget for the software applications the server will need in order to perform its tasks . the dollar amounts can add up quite quickly in this area , depending on the role of the data center and the server itself . it is very common in high - end server applications to offer per core licensing for some editions of the software . dealing with hardware costs , one should be aware of server depreciation as well . hence , the annual hardware cost is in fact an amortization cost , calculated as follows : @xmath22 where : * @xmath23 - is the purchase cost of a server ( in us dollars ) * @xmath24 - is the average life - time of a single server ( in years ) when it comes to the brand new equipment , besides the amortization expenditures ( the overall costs associated with installing , maintaining , upgrading and supporting a server ) , one should consider purchasing costs as well . saying so , the total , annual hardware and software cost of a single machine can be estimated using the formula below : @xmath25 the day may come when data centers are self - maintaining , but until then , one will need personnel to operate and maintain server rooms . internal personnel of the data center usually consist of it staff , data center security personnel , the data center managers , facilities maintenance personnel and housekeeping personnel . if the data center contains tens - if not hundreds - of thousands of working machines , it is common to have more than one employee dealing with a given equipment . discussing personnel costs , it can be calculated as the total number of employees multiplied by the salary of the particular staff member ( to simplify our evaluation , we can consider the average salary for every employee in the enterprise ) : @xmath26 where : * @xmath27 - is the total number of it personnel * @xmath28 - is the total number of ordinary workers * @xmath29 - is the total number of the housekeeping and facilities maintenance personnel * @xmath30 - is the average salary in the enterprise ( per month ) once the data center is built , it still requires financial investment to ensure a high - quality , competitive services with guaranteed levels of availability , protection and support continuously for 24 hours a week . key elements in data center budgets are the power delivery system , the networking equipment and the cooling infrastructure . besides the above most - crucial factors , there exist additional costs associated with data center operation , such as personnel and software expenses . therefore , the real operating cost of the data center can be expressed as : @xmath31 where each of the defined components consists of further operational expenditures . this concludes the discussion on calculating the total cost of a data center maintenance , resulting in the following formula ( being the combination of all the above equations ) : @xmath32 it is significant to remember that @xmath33 refers to the annual hardware and software costs . therefore , when calculating the total cost of a data center maintenance , one needs to adjust this value individually , depending on the considered analysis time interval . to demonstrate the use of the proposed analysis scheme , we used the role - based access control approach , prepared an example data center scenario and analyzed it with the help of the introduced method . we made use of qop - ml @xcite , @xcite and created by its means the role based access control model to examine the quality of chosen security mechanisms in terms of financial impact of data center maintenance . before we perform the actual estimation of the data center maintenance cost , let us give some assumptions about the examined environment . consider a call center company located in nevada , usa , managing a typical it environment of @xmath34u server racks ( @xmath35 physical servers in total , @xmath36 physical servers per rack ) . given a specified load capacity , servers handle enterprise traffic continuously for @xmath37 hours . in our analysis , we assume that all the utilized applications are tunnelled by the tls protocol . in the considered access control method , users are assigned to specific roles , and permissions are granted to each role based on the users job requirements . users can be assigned any number of roles in order to conduct day - to - day tasks ( figure [ fig : callcenter ] ) . in order to emphasize and prove role s influence on data flow management and system s performance , we prepared and analyzed a simple scenario . this scenario refers to the real business situation and possible role assignment in the actual enterprise environment . given the example enterprise network infrastructure , consider having three roles : _ role1 _ , _ role2 _ and _ role3 _ with corresponding security levels : _ low _ , _ medium _ and _ high_. each server in the example call center is equipped with the intel xeon @xmath38 processor , being able to handle the required number of employees connections , regardless of the assigned rbac role . prepared scenarios are listed in table [ table : rbac_scenarios ] . qop - ml s security models used in our case study can be downloaded from qop - ml s project webpage @xcite . after introducing the environment , we present an overview of predicting the total budget required to manage an example data center , focusing on an introduced method for measuring its total cost and indicating possible gains . by the analysis of an example scenario , we try to confirm our thesis about the influence of security management on the total cost of the data center maintenance . 1 . _ cost of power delivery _ to calculate the total cost of power delivery , we performed both experimental and theoretical investigations . regarding experiments , we utilized dell ups to measure the current power consumption of a single dell poweredge r@xmath39 server performing operations defined in our scenario . in order to obtain the most accurate results of the cpu power consumption , we made use of our model , and performed analysis using the aqopa tool @xcite . to ensure the accuracy of gathered results , in our simulation , we utilized real hardware metrics , provided by the cmt @xcite for dell poweredge r@xmath39 server . 1 . _ server power consumption _ according to dell ups in the laboratory , poweredge r@xmath39 performing defined operations consumes on average @xmath40w . since the server handles enterprise s traffic continuously for @xmath37 hours , its annual power consumption is equal to @xmath41 kwhs . as stated by http://www.electricitylocal.com/states/nevada/las-vegas/ , the average industrial electricity rate in las vegas , nevada is @xmath42 cents @xmath43 ( 0.0756 $ ) per kwh . if the server works @xmath44 days a year , it will cost the company about @xmath45 $ . 2 . _ cpu power consumption _ when it comes to the cpu power consumption , we assumed its load to be equal to @xmath46 . we performed a simulation using prepared model , considering @xmath0 levels of users permissions ( which differ in security level ) . + in table [ table : cost_cpu_server ] we collected power consumption costs , both for the cpu and the server in total ( rounded up to the nearest dollar ) . as evidenced by table [ table : cost_cpu_server ] , considering only cost savings related to the consumed energy , it is possible to handle about @xmath47 ( when switching between the first and the third role ) and @xmath48 ( by changing the role from third to second ) more users with the exact cpu load . those figures , when put in the context of a large data center environment , quickly become very significant . cost of cooling infrastructure utilization _ in addition to the power delivered to the compute hardware , power is also consumed by the cooling resources . the load on the cooling equipment is directly proportional to the power consumed by the compute hardware . in such case , the cost of the cooling infrastructure is equal to the cost of the energy consumed by the server and its cpu . 3 . _ operational costs _ apart from electricity and cooling , calculating the total maintenance cost of a data center , one should take under consideration also the cost of its physical infrastructure , such as hardware amortization and the actual price of physical machines . besides the equipment cost , operational expenditures must be covered as well . determining the approximate , total cost of a whole data center , we assumed to use dell power edge r@xmath39 servers ( @xmath49 each ) . however , we have include neither the network , nor the storage footprint ( nor its equipment ) . regarding utilized software , we assumed that working machines have windows server data center edition installed ( whose price is equal to about @xmath50 dollars ) . except the os , workstations use some proprietary software , which can cost about @xmath51 $ on average ( @xcite , @xcite ) . ( we assume the cost of the software for a single machine to be equal to @xmath52 $ . ) the number of employees of the whole data center consists of security managers , system operators , call center employees and housekeeping and facility maintenance personnel , resulting in @xmath53 employees in total . as stated by http://swz.salary.com , the median salary of security manager in us is equal to about @xmath54 dollars per month ( at the time of writing ) . in our estimation , we used this value as the average salary in our call center company . in order to prove that a proper data flow management has a significant impact on data center maintenance costs , we tried to estimate them over @xmath55 years . although it might not be easily noticed at first glance , it turned out that our approach can bring meaningful savings and influence rapid _ return on investment _ ( _ roi _ ) increase . ( table [ table : r1r2compareme ] explores this concept in more detail . ) in our approach , economic profits come from the number of handled users - the more customers ( served clients ) , the higher company profits . with the exact cpu load , the same number of working machines is capable of handling a greater number of users . by switching between the _ strongest _ and _ weakest _ security mechanisms , the example call center company can achieve actual roi growth . our analysis showed , that it is possible to provide effective services and keep the utilization of hardware resources at a certain level . since we can accomplish given goal using _ weaker _ security mechanisms , in many situations it is wasteful to assign too many hardware resources to perform the given task . applying the proposed solution to the existing it environment , one can observe a serious increase in company incomes , while preserving the efficiency , utilization and flexibility of the existing computer hardware . one of the main pricing models of a call center company concerns _ cost per contact _ , where all costs are combined into an unit price . the total income is then based on the number of served clients , such as calls , emails or chat sessions . for the example call center , we assumed that on average , single served customer brings the income of about @xmath0 us dollars . as it was proven by the time analysis presented in @xcite , server working with _ permissions is able to handle about @xmath56 users within an hour having @xmath57% of cpu load . since we assumed that the number of users grows linearly , within @xmath37 hours , it gives us @xmath58 employees a day , resulting in @xmath59 connections a year per server . if we assume that we have at our disposal the whole data center , it will turn out that we can serve roughly @xmath60 users assigned _ permissions a year . with the exact cpu load , using _ role1 _ permissions , server is capable of dealing with @xmath61 users within an hour , which results in @xmath62 served customers a year per single machine and @xmath63 clients per whole data center . when we translate the above calculations to the incomes and outcomes of the company , we see that the proper information management brings a variety of economic advantages . according to our previous assumptions , given the incomes equal to @xmath64 $ , @xmath65 $ and @xmath66 $ and outcomes of @xmath67 $ , @xmath68 $ and @xmath69 $ for roles @xmath70 , @xmath71 and @xmath0 respectively , in table [ table : r1r2compareme ] we calculated actual profits and roi values of the company for over @xmath55 years . as it is summarized in table [ table : r1r2compareme ] , considering the first working year of the example call center , the efficiency of an investment is much bigger when we handle users using role s @xmath70 security mechanisms ( comparing to the role @xmath0 ) , and about @xmath72 times greater , considering roles @xmath71 and @xmath0 . real profits can be observed after the @xmath55 years of call center business activity . the analysis shows , that if the company used the first role permissions instead of those from role @xmath71 , it could gain about @xmath55 times more money . what is more , if we consider the third and the first role , profits grows rapidly , resulting in about @xmath14 times greater gain . since high roi values means that the investment gains compare favorably to investment cost , and the primary goal of the call center is a fast return on the investment , company should re - think implemented information flow mechanisms . as proved by our analyzes , the main drivers of data center cost are power and cooling . in contrast to operational costs , they represent _ variable _ costs , which vary over time and depend on many factors . as power consumption and electricity prices rise , energy costs are receiving more scrutiny from senior - level executives seeking to manage dollars . however , focusing on the financial aspect of the data center , one can not forget about the proper data management . in this paper , we utilized qop - ml to increase company incomes , without compromising data security or quality of service . the proposed analysis scheme provides new opportunities and possibilities , not only for measuring data center costs , but also for increasing incomes . optimization of the available computational power can be accomplished in many different ways : by modifying system configurations , switching between utilized security mechanisms , by the suitable selection of used applications and services and adequate application management . k.mazur , b.ksiezopolski and a.wierzbicki `` on security management : improving energy efficiency , decreasing negative environmental impact and reducing financial costs for data centers , '' _ mathematical problems in engineering _ , 418535 , 119 . kevin lim , parthasarathy ranganathan , jichuam chang , chadrakant patel , trevor mudge and steven reinhardt , understanding and designing new server architectures for emerging warehouse - computing environments , _ proceeding of the acm international symposium on computer architecture _ , beijing , china , june 2008 . d.rusinek , b.ksiezopolski and a.wierzbicki , `` aqopa : automated quality of protection analysis framework for complex systems '' , _ 14th international conference on computer information systems and industrial management applications _ , warsaw , springer : lncs , v.9339 , 475486 , 2015 .
|
information management is one of the most significant issues in nowadays data centers .
selection of appropriate software , security mechanisms and effective energy consumption management together with caring for the environment enforces a profound analysis of the considered system . besides these factors ,
financial analysis of data center maintenance is another important aspect that needs to be considered .
data centers are mission - critical components of all large enterprises and frequently cost hundreds of millions of dollars to build , yet few high - level executives understand the true cost of operating such facilities .
costs are typically spread across the it , networking , and facilities , which makes management of these costs and assessment of alternatives difficult .
this paper deals with a research on multilevel analysis of data center management and presents an approach to estimate the true total costs of operating data center physical facilities , taking into account the proper management of the information flow .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the detection of feii and mgii absorption lines at a redshift of @xmath2 in the optical spectrum of gb970508 ( metzger et al . 1997 ) , provided the first confirmation that @xmath0-ray bursts ( grbs ) originate at cosmological distances . most of the qualitative properties of cosmological grbs are explained by the fireball model ( see e.g. , goodman 1986 ; paczyski 1986 ; mszros & rees 1993 [ mr ] ) . in this model , a compact ( @xmath3 cm ) source releases an energy of @xmath4 erg over a duration @xmath5 seconds with a negligible baryonic contamination ( @xmath6 ) . unsteady activity of the source results in a wind composed of many thin layers ( fireball shells ) of varying energy and baryonic mass . within each shell the high energy - density at the source results in an optically thick @xmath1-pair plasma that expands and accelerates to relativistic speeds . after an initial acceleration phase , the radiation and thermal energy of the fireball plasma is converted into the kinetic energy associated with the radial motion of the protons . collisions between the shells can convert part of that kinetic energy into radiation and yield the primary grb via synchrotron emission and inverse - compton scattering ( paczyski & xu 1994 ; rees & mszros 1994 [ rm ] ; sari & piran 1997 [ sp ] ) . as the wind continues to expand , it impinges on the surrounding medium and eventually drives a relativistic blastwave in it , which heats fresh gas and accelerates electrons to relativistic speeds , thus producing the delayed afterglow radiation observed on time scales of hours to months ( van paradijs et al . 1997 ; bond 1997 ; djorgovski et al . 1997 ; mignoli et al . 1997 ; frail et al . 1997 ) via synchrotron emission ( wijers , rees , & mszros 1997;waxman 1997a , b ; vietri 1997a , b ) . the primary grb emission is more likely caused by internal shocks than the external shocks ( mr ) , since they can occur closer to the source and thus account for the rapid variability observed in many bursts ( rm ; sp ) . unsteady activity of the central source naturally results in faster shells overtaking slower ones in front of them , and hence in energy dissipation by internal shocks . the complex temporal structure observed in grbs then reflects the activity - history of their sources ( sp ; kobayashi , piran , & sari 1997 [ kps ] ) . it is often assumed that behind internal shocks , electrons are fermi accelerated with a near equipartition energy density and magnetic fields acquire nearly equipartition strength . the electrons cool by synchrotron emission and inverse - compton ( ic ) scattering off the synchrotron photons . under typical conditions , the time scale for ic scattering is shorter than the synchrotron cooling time . multiple scattering of the photons boosts a significant fraction of the radiation energy to frequencies above the @xmath1-pair creation threshold . the pairs produced in this process are also relativistic and cool rapidly by ic scattering . since the annihilation time of these pairs is longer than the hydrodynamic time in the comoving frame , they survive in the wind for a long time . although the creation of pairs and their subsequent cooling is likely to leave noticeable imprints on the emergent radiation spectrum , it has not been analyzed before in the grb literature . since the photon and electron densities decline rapidly with radius , the strength of these signatures can serve as a probe of the radius at which the internal shocks occur . the grb spectrum should also depend on the level of baryonic contamination in the wind . the extreme limit of pure energy release with no baryons was ruled out in the past based on the prediction that a point explosion of this type would lead to a roughly thermal spectrum ( goodman 1986 ) . one can place a lower limit on the baryonic mass in the fireball shells by requiring that internal shocks should occur before an external shock does ( this limit depends on the ambient medium density ) . on the other hand , an upper limit can be placed based on the variability time scale of the source and the condition that the shells be optically thin at the radius where internal shocks occur . in this _ letter _ we study in detail the emergent spectra from the collision of two fireball shells . in particular , we quantify the significance of the radiation processes which were previously ignored in the literature , such as multiple compton scattering , @xmath7 creation , and their subsequent cooling . we use the collision kernels , reaction rates , and the computational techniques given in pilla & shaham ( 1997 [ ps ] ) . more details about this calculation and an elaborate study of the spectral characteristics of relativistic shocks will be included in a subsequent publication ( pilla & loeb 1997 ) . in 2 we describe our model and specify the physical conditions in the emission region . in 3 we outline the relevant radiation processes and compute the model spectra . finally , 4 summarizes the main implications of this work . the typical fireball dynamics ( piran , shemi , & narayan 1993 , [ psn ] ; mszros , laguna , & rees 1993 ; sari , narayan , & piran 1996 [ snp ] ; sp ; and kps ) can be illustrated by considering a single shell of total energy @xmath8 erg , rest mass @xmath9 g , and initial radius @xmath10 cm . after a brief acceleration phase , the lorentz factor of the shell reaches a constant value @xmath11 at an observer - frame radius @xmath12 ( the protons are taken to be non - relativistic in the comoving frame , before the collision of shells ) . the energy of the shell is predominantly kinetic beyond this stage . outside the radius @xmath13 ( all radii in the present analysis are measured in the observer s frame ) , the comoving width of the shell increases linearly with radius ( psn ) . the comoving proton density scales as @xmath14 for shell radii @xmath15 and as @xmath16 for @xmath17 . the thomson optical depth of the shell is @xmath18 for @xmath19 , where @xmath20 cm is the radius at which the shell becomes optically thin to thomson scattering by its own electrons . here @xmath21 is thomson cross section and @xmath22 is the proton rest mass . for @xmath17 , the comoving width of the shell is @xmath23 . two photons which are emitted with a proper - time difference @xmath24 , reach the observer with a time separation @xmath25 ( sp ) . assuming that the radiative cooling time is much shorter than the light transit time through the system , one finds that the observed width of the radiation pulse @xmath26 and the radius of the emission region @xmath27 are related through @xmath28 , which in turn can be used to constrain the values of @xmath29 , and @xmath30 from observations . the emission spectrum will be nonthermal only if @xmath31 . now consider a wind of total duration @xmath32 , composed of many thin fireball shells of thickness @xmath33 ( @xmath34 may change from one shell to another ) . we assume that for @xmath17 there are regions in the wind where slower shells precede the faster ones , i.e. @xmath35 , where @xmath36 is the local lorentz factor . the spatial extent of the wind is typically @xmath37 ( in fact , @xmath38 also holds in general ) . it was shown by waxman & piran ( 1994 ) that under these conditions the wind layers are susceptible to rayleigh - taylor instability because a rarefied fluid shell is pushing against a denser one . the resulting turbulent mixing will complicate the shock structure and deform it away from a simple planar geometry . merging of a rarefied shell with a denser one might therefore be accompanied by the formation of `` fingers '' perpendicular to the shell walls , similar to the non - relativistic shock structure in supernova remnants ( e.g. , jun & norman 1996 ) . the combined shells would then break into bubbles of different sizes , and the energy dissipation would take place near the bubble walls , due to collisions among them or instabilities on their surfaces ( kamionkowski & freese 1992 ) . by assuming that the shells as well as the shock fronts remain planar , kps had found that the dissipation efficiency of internal shocks might obtain high values @xmath39 for reasonable wind parameters . we assume that similar efficiencies are achieved in the case of unstable mixing . since the combined area of the bubble walls greatly exceeds that of a planar shock , the electron acceleration efficiency in the present case is likely to be higher . however the temporal and spectral characteristics of the bursts might be different in the two cases . for a planar shock the accelerated electrons populate a thin layer around the shock front since their cooling time is much shorter than the transit time of sound waves across the shells . however , in the unstable mixing case we assume , to a first approximation , that the energy dissipation takes place nearly uniformly and simultaneously throughout the entire volume of the emission region . for the purpose of estimating the physical conditions involved , we take @xmath40 erg , @xmath41 g , @xmath42 cm , and a bulk lorentz factor of the emission region of @xmath43 . these values yield @xmath44 @xmath45 , @xmath46 , a comoving width of the post - shock shell @xmath47 cm , and an average lorentz factor of the protons in the comoving frame ( i.e. , after shock heating ) of @xmath48 . acceleration of electrons to relativistic energies and the presence of strong magnetic fields are essential for converting the energy dissipated by the shock waves into radiation . since the physics of neither of these processes is well understood , we parameterize the corresponding energy densities in units of their equipartition values . a magnetic equipartition parameter @xmath49 , corresponds to a field strength @xmath50^{1/2}10^{4}$ ] g , where @xmath51 ( all quantities are comoving , unless stated otherwise ) . we assume that the electrons are accelerated throughout the emission region . the amount of energy transferred from protons to electrons is uncertain , and we define the acceleration efficiency @xmath52 in such a way that the average lorentz factor of the electrons immediately after they are heated ( via fermi - type acceleration ) is @xmath53 , where @xmath54 is the electron rest mass . because the coulomb collision time @xmath55 , we can safely ignore collisional relaxation in our analysis ( here @xmath56 is the coulomb logarithm ) . collisionless acceleration of electrons can be efficient if they are tightly coupled to the protons and the magnetic field by means of plasma waves ( kirk 1994 ) . the typical alfvn speed in the plasma is @xmath57 . the larmor radius of an electron with a lorentz factor @xmath0 is @xmath58 cm , and that of a proton of equal lorentz factor , @xmath59 , is larger by a factor @xmath60 . the corresponding acceleration time scales ( e.g. , hillas 1984 ) are , @xmath61 sec and @xmath62 , respectively . synchrotron losses limit the maximum value of the electron lorentz factor to @xmath63 . the minimum value is determined by @xmath64 and the shape of the electron distribution . we assume that the fraction of electrons per unit lorentz factor @xmath0 has the form @xmath65 for @xmath66 . thus @xmath67 , where @xmath68 is the average lorentz factor at any given time . the energy density in electrons immediately after their acceleration is @xmath69 . in our model the radiation time scale is much shorter than the hydrodynamic expansion time scale in the comoving frame ( which leads to a radiative efficiency of nearly @xmath70 ) and the radiation density at the end of electron cooling is therefore @xmath71 . we compute the spectra for @xmath72 , @xmath73 , and @xmath74 . we assume that the radiation energy density is initially small , and hence the electrons start losing their energy via synchrotron emission . as the energy density of the emitted radiation builds up , cooling via ic scattering becomes important as well . the typical time scale for synchrotron or ic losses is @xmath75 , where @xmath76 in the synchrotron case and @xmath77 in the ic case . here @xmath78 is the radiation energy density and @xmath79 is a dimensionless constant of order unity . for typical grb conditions @xmath80 , where @xmath81 is the light transit time through the system . therefore electron cooling takes place only locally . we assume that the acceleration proceeds throughout the cooling phase in such a way that it maintains a steady power - law distribution with a constant index @xmath82 , while @xmath68 declines due to radiative losses . if @xmath83 , the synchrotron cooling time is long and multiple ic scatterings become important . each scattering in the thomson regime increases the photon energy by a factor @xmath84 ( loeb , mckee , & lahav 1991 ) , so that some of the photons are eventually boosted into the klein - nishina ( kn ) regime . as @xmath68 decreases , the up - scattered part of the spectrum spreads over many decades in frequency . the electron energy density @xmath85 changes at a rate = ( ) _ sy+ ( ) _ ic , [ energy - rate ] where @xmath86 , @xmath87 , @xmath88 , and @xmath89 is the average value of @xmath90 over the initial electron distribution . the ic cooling rate , @xmath91 , is derived below . the characteristic cooling time @xmath92 is a few times @xmath93 under typical conditions . if @xmath94 is the instantaneous energy density of radiation , then energy conservation implies @xmath95 ( which is constant ) , and by assumption @xmath96 . the instantaneous radiation spectrum can be characterized by @xmath97 so that @xmath98 is the fraction of the radiation energy density within an interval @xmath99 around @xmath100 ; here @xmath101 , @xmath102 is the photon frequency , and @xmath103 is planck s constant . the spectral evolution rate ( see pilla & loeb 1997 ) is derived from the equation u_(t)(,t)- ( , t)=_sy+ _ ic , [ spec - rate ] where we have used the fact that @xmath104 . the first term on the right hand side is the synchrotron emissivity from relativistic electrons ( rybicki & lightman 1979 ) which depends on @xmath105 ; the second term is _ ic= _ -1 ^ 1d(1-)__l^_max df_e(,t ) , [ eq : dvdt ] where @xmath106 is the cosine of the scattering angle , @xmath107 , @xmath108 , @xmath109 , and @xmath110 is the klein - nishina cross - section . by integrating both sides of equation ( [ eq : dvdt ] ) over all values of @xmath100 , we obtain @xmath111 on the left hand side , whereas on the right hand side we use the relation @xmath112 to convert the integral to be over @xmath113 and integrate over all values of @xmath114 , and @xmath106 . finally , we divide the result by 2 to avoid double counting of each scattering event , and obtain ( ) _ ic=-()_ic = _ -1 ^ 1d__min^_max df_e(,t)__min^_maxd_1(_1,t ) ( 1-)^2 , [ ic - rate ] where @xmath115 are the limiting energies of photons in the plasma . in the thomson regime @xmath116 and @xmath117 , and one finds ( ) _ ic==cn_e_tu _ ^2 , in agreement with the well - known result for @xmath118 ( loeb et al . the coupled equations ( [ energy - rate])([ic - rate ] ) are solved numerically for the radiation spectrum at the end of the cooling process ( i.e. , when @xmath119 ) . cooling ends after a relatively short time , @xmath120 , for the parameters of interest here , and the photons decouple from the electrons subsequently . an example for the time evolution of the spectrum is shown in figure 1a . it is evident that the radiation density above the @xmath1-pair creation threshold is substantial . the spectral evolution rate due to pair creation is described by @xmath121_{\pm}&= & u_{\gamma}(t)\frac{\partial}{\partial t}\left[\phi(\ee , t)\right]_{\pm } + \phi(\ee , t)\left(\frac{du_{\gamma}}{dt}\right)_{\pm}\nonumber\\ & & \nonumber\\ & = & \frac{u_{\gamma}}{t^{\prime}}{\cal i}_{1}(\ee , t)\equiv -\frac{u_{\gamma}}{4t^{\prime } } \int d\mu \int d\ee^{\prime } \frac{\phi(\ee , t)}{\ee}\frac{\phi(\ee^{\prime},t ) } { \ee^{\prime}}\frac{\ee}{\overline{\gamma}_{0}}(1-\mu)\frac{\sigma_{\pm } } { \sigma_{t } } , \label{eq : pairc}\end{aligned}\ ] ] where the integration is over the range @xmath122 and @xmath123 subject to the condition @xmath124 . here @xmath125 is the pair creation cross section ( ps ) and @xmath126 . in all examples considered in this _ letter _ , we find that @xmath127 for all relevant photon energies . therefore , pair creation comes into play after the original electrons cool and decouple . the total energy loss per unit time and volume due to pair creation is obtained by integrating equation ( [ eq : pairc ] ) over all values of @xmath100 , ( ) _ = _ _ min^_maxd_1(,t)_1(t ) . the rate at which pairs are created per unit volume is ( ) _ = _ _ min^ _ max d_2(,t)_2(t ) , where @xmath128 is same as @xmath129 except that @xmath130 is absent in its integrand . at the beginning of pair creation , the energy density of radiation is @xmath71 . the average lorentz factor of the newly created pairs is @xmath131 , since @xmath132 . because the pairs are born relativistic , they transfer almost all their energy to the radiation via ic scattering ( and also synchrotron emission if @xmath133 ) on a time scale of order a few @xmath134 . similar cascades of pair creation and cooling occur in active galactic nuclei ( svensson 1987 ) . throughout the pair cascade and cooling process the radiation spectrum evolves continuously but its total energy density remains nearly constant at a value close to @xmath71 . figure 1b shows an example of this evolution up to a time @xmath135 . because the hydrodynamic time scale in the comoving frame @xmath136 , pair processes can not continue to operate for times much longer than @xmath137 because of the decline in the densities due to the expansion of the fireball . we solved the coupled equations given above using the methods described by ps and obtained the model spectra for @xmath138 erg , @xmath139 , @xmath72 , @xmath73 , and @xmath74 . we assume a source redshift @xmath140 and include its effect on photon energy and flux . the total observable fluence for the above parameters is @xmath141 . figure 1 shows the entire spectral evolution ( in the comoving frame ) due to the synchrotron and ic cooling and pair cascade processes . we take @xmath41 g which yields @xmath142 ( which are consistent with empirical constraints , e.g. woods & loeb 1995 ) ; and a dissipation ( or shell collision ) radius of @xmath143 cm . note that the assumed values for @xmath144 and @xmath145 correspond only to a single emitting shell ; they should obviously be higher for the entire burst . for the above parameters , we get @xmath146 cm , @xmath147 , @xmath148 , and @xmath149 g. the characteristic time scale for synchrotron and ic cooling is @xmath150 . we find that after a time @xmath151 , only @xmath152 of the initial energy remains with the electrons . figure 1a illustrates that as a result of electron cooling , the location of the synchrotron peak shifts to lower energies and also becomes broader . later on , another peak develops at a much higher energy due to ic losses , with kn suppression in the very high - energy tail . figure 1b shows the evolution much later due to the pair - cascade process , which depletes the population of high - energy photons . the ic loss by these pairs produces a power law tail at yet higher energies . the low - energy synchrotron part of the spectrum is almost unaltered throughout this phase . the computation was stopped at @xmath153 at which time most of the photons leave the system . figure 2 shows the dependence of the emergent spectrum on the dissipation radius for a fixed value of @xmath11 ( panel [ a ] ) and dependence on @xmath11 for a fixed value of the dissipation radius ( panel [ b ] ) . panel ( a ) shows that the location of synchrotron peak moves up in energy as the radius decreases ( with an opposite trend for the ic peak ) ; hence the overall extent of the emission spectrum gets narrower at smaller radii . since @xmath154 and @xmath155 , as @xmath156 decreases the synchrotron frequency increases and the pair - production depletion of high - energy photons is enhanced . one of the model spectra in figure ( 2a ) is compared with the batse data through the empirical formula for the time - integrated flux given by band et al . ( 1993 [ b93 ] ) . in that case our model predicts @xmath157 more time - integrated flux at 10 kev than observed , which could be due to an oversimplified form of the electron distribution function which we have adopted . in our notation , the number of photons per unit interval of @xmath100 is @xmath158 . therefore , band s formula ( b93 ) reads @xmath159 & = & c_{2}\ee^{-\beta},\qquad\mbox{\rm otherwise . } \label{eq : band}\end{aligned}\ ] ] the constants @xmath160 are fixed by the requirement that this function be continuous at @xmath161 and the normalization @xmath162 . note that we have altered the signs of both indices @xmath163 and @xmath164 relative to the convention of b93 . the observed values of the parameters ( cf . table 4 in b93 ) are in the range @xmath165 ( although in a few cases @xmath163 is zero or negative ) , @xmath166 , and @xmath167(kev)@xmath168 . the majority of the bursts seem to have @xmath169 , @xmath170 , and @xmath171 a few hundred kev . from the model spectra in figure 2 , it is clear that the emission extends over a wide range of photon energies , from the optical to the tev regime . we expect more optical emission when the dissipation takes place at larger radii . it is very likely that internal shocks occur over a wide range of radii , thereby extending the spectrum to longer wavelengths , with simultaneous optical and x - ray emission during a grb . recent reports of nearly simultaneous detection of x - ray emission from gb960720 ( piro et al . 1997 ) , gb970815 ( smith et al . 1997 ) , and gb970828 ( remillard et al . 1997 ) are in qualitative agreement with our expectation for internal shocks . we have shown that the emission spectra from internal shocks are affected by synchrotron emission , multiple compton scatterings , and pair creation ( fig . 1 ) . our model spectra could mimic the observed batse spectra ( b93 ) for reasonable choices of the shell lorentz factor @xmath172@xmath173 and shock radius @xmath174@xmath175 ( fig . the depletion of high energy photons due to pair creation is a sensitive probe of both @xmath11 and @xmath156 . detection of the high energy trough present in figures ( 1b ) and ( 2 ) can therefore be used to constrain these parameters . the potential degeneracy between the values of @xmath11 and @xmath156 can be removed on the basis of variability data , as the characteristic variability time scale depends on a different combination of these parameters , @xmath176 . despite their high - energy cut - off , our model spectra extend all the way up to photon energies as high as 10 gev tev . we acknowledge discussions with m. kamionkowski , t. piran , m. a. ruderman , and e. woods . one of us ( rp ) thanks r. sari for many critical comments and stimulating discussions during the viii marcel grossmann meeting . this research was supported in part by nasa grants nag5 - 618 and -2859 ( for rp ) and nasa atp grant nag 5 - 3085 and the harvard milton fund ( for al ) . band , d. l. et al . 1993 , apj 413 , 281 ( b93 ) bond , h. e. 1997 , iau circ . 6654 djorgovski , s. g. et al . 1997 , nature , 387 , 786 frail , d. a. et al . 1997 , apj , 483 , l91 goodman , j. 1986 , apj 308 , l47 hillas , a. m. 1984 , ann . 22 , 425 jun , b. and norman , m. l. 1996 , apj 472 , 245 kamionkowski , m. and freese , k. 1992 , phys . rev . lett . 69 , 2743 kirk , j. g. , in _ plasma astrophysics _ , edited by a. o. benz & t. j. l. courvoisier , saas fee advanced course 24 ( springer verlag , new york , 1994 ) kobayashi , s. , piran , t. & sari , r. 1997 , astro - ph/970513 ( kps ) loeb , a. , mckee , c. f. , & lahav , o. 1991 , apj 374 , 44 mszros , p. & rees , m. j. 1993 , apj 405 , 278 ( mr ) mszros , p. , laguna , p. , & rees , m. j. 1993 , apj 415 , 181 metzger , m. r. et al . 1997 , iau circ . 6655 mignoli , m. et al . 1997 , iau circ . 6661 paczyski , b. 1986 , apj 308 , l43 paczyski , b. , & xu , g. 1994 , apj , 427 , 708 pilla , r. p. & loeb , a. 1997 , in preparation pilla , r. p. & shaham , j. 1997 , apj 486 , 903 ( ps ) piran , t. , shemi , a. , & narayan , r. 1993 , mnras 263 , 861 ( psn ) piro , l. et al . 1997 , a & a , in press ; astro - ph/9707215 rees , m.j . & mszros , p. 1994 , apj 430 , l93 ( rm ) remillard , r. et al . 1997 , iau circ . 6726 rybicki , g. b. & lightman , a. p. 1979 , _ radiative processes in astrophysics _ ( new york : john wiley ) sari , r. & piran , t. 1997 , mnras 287 , 110 ( sp ) sari , r. , narayan , r. , & piran , t. 1996 , apj 473 , 204 ( snp ) smith , d. a. et al . 1997 , iau circ . 6718 svensson , r. 1987 , mnras 227 , 403 van paradijs , j. , et al . 1997 , nature , 386 , 686 vietri , m. 1997a , apj , 478 , l9 1997b , apj letters , in press ; astro - ph/9706060 waxman , e. 1997a , apj letters , in press ; astro - ph/9704116 1997b , astro - ph/9705229 waxman , e. & piran , t. 1994 , apj 433 , l85 wijers , a. m. , rees , m. j. , & meszaros , p. 1997 , mnras , 288 , l51 woods , e. & loeb , a. 1995 , apj 453 , 583
|
unsteady activity of @xmath0-ray burst sources leads to internal shocks in their emergent relativistic wind .
we study the emission spectra from such shocks , assuming that they produce a power - law distribution of relativistic electrons and posses strong magnetic fields .
the synchrotron radiation emitted by the accelerated electrons is compton up - scattered multiple times by the same electrons .
a substantial component of the scattered photons acquires high energies and produces @xmath1 pairs .
the pairs transfer back their kinetic energy to the radiation through compton scattering .
the generic spectral signature from pair creation and multiple compton scattering is highly sensitive to the radius at which the shock dissipation takes place and to the lorentz factor of the wind .
the entire emission spectrum extends over a wide range of photon energies , from the optical regime up to tev energies . for reasonable values of the wind parameters ,
the calculated spectrum is found to be in good agreement with the burst spectra observed by batse .
submitted to _ apj letters _ , october 1997
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
mathematics is the art of abstraction and generalization . historically , `` numbers '' were first natural numbers ; then rational , negative , real , and complex numbers were introduced ( in some order ) . similarly , the concept of taking derivatives has been generalized from first , second , and higher order derivatives to `` fractional calculus '' of noninteger orders ( see for instance @xcite ) , and there is also some work on fractional iteration . however , when we add some number of terms , this number ( of terms ) is still generally considered a natural number : we can add two , seven , or possibly zero numbers , but what is the sum of the first @xmath0 natural numbers , or the first @xmath1 terms of the harmonic series ? in this note , we show that there is a very natural way of extending summations to the case when the `` number of terms '' is real or even complex . one would think that this method should have been discovered at least two hundred years ago and that is what we initially suspected as well . to our surprise , this method does not seem to have been investigated in the literature , or to be known by the experts , apart from sporadic remarks even in euler s work @xcite ( see equation ( [ eq : euler ] ) below ) . of course , one of the standard methods to introduce the @xmath2 function is an example of a summation with a complex number of terms ; we discuss this in section [ secfromaxtodef ] , equation ( [ eqgamma ] ) . since this note is meant to be an introduction to an unusual way of adding , we skip some of the proofs and refer the reader instead to the more formal note @xcite . some of our results were initially announced in @xcite . we start by giving natural conditions for summations with an arbitrary complex number of terms ; here @xmath3 , @xmath4 , @xmath5 , and @xmath6 are complex numbers and @xmath7 and @xmath8 are complex - valued functions defined on @xmath9 or subsets thereof , subject to some conditions that we specify later : ( s1 ) continued summation : : @xmath10 ( s2 ) translation invariance : : @xmath11 ( s3 ) linearity : : for arbitrary constants @xmath12 , @xmath13 ( s4 ) consistency with classical definition : : @xmath14 ( s5 ) monomials : : for every @xmath15 , the mapping @xmath16 is holomorphic in @xmath9 . @xmath17 right shift continuity : : if @xmath18 pointwise for every @xmath19 , then @xmath20 more generally , if there is a sequence of polynomials @xmath21 of fixed degree such that , as @xmath22 , @xmath23 for all @xmath19 , we require that @xmath24 the first four axioms ( s1)(s4 ) are so obvious that it is hard to imagine any summation theory that violates these . they easily imply @xmath25 for every @xmath26 , so we are being consistent with the classical definition of summation . axiom ( s5 ) is motivated by the well - known formulas @xmath27 and similarly for higher powers ; we shall show below that our axioms imply that all those formulas remain valid for arbitrary @xmath28 . finally , axiom @xmath17 is a natural condition also . the first case , in ( [ eqs5 ] ) , expresses the view that if @xmath7 tends to zero , then the summation `` on the bounded domain '' @xmath29 $ ] should do the same . in ( [ eqs5b ] ) , the same holds , except an approximating polynomial is added ; compare the discussion after proposition [ prop1 ] . it will turn out that for a large class of functions @xmath7 , there is a unique way to define a sum @xmath30 with @xmath19 that respects all these axioms . in the next section , we will derive this definition and denote such sums by @xmath31 . we call them `` fractional sums . '' to see how these conditions determine a summation method uniquely , we start by summing up polynomials . the simplest such case is the sum @xmath32 with @xmath33 constant . if axiom ( s1 ) is respected , then @xmath34 applying axioms ( s2 ) on the left and ( s4 ) on the right - hand side , one gets @xmath35 it follows that @xmath36 . this simple calculation can be extended to cover every sum of polynomials with a rational number of terms . [ prop1 ] for any polynomial @xmath37 , let @xmath38 be the unique polynomial with @xmath39 and @xmath40 for all @xmath19 . then : * the possible definition @xmath41 satisfies all axioms ( s1 ) to @xmath17 for the case that @xmath7 is a polynomial . * conversely , every summation theory that satisfies axioms ( s1 ) , ( s2 ) , ( s3 ) , and ( s4 ) also satisfies ( [ eqsumpoly ] ) for every polynomial @xmath42 and all @xmath43 with rational difference @xmath44 . * every summation theory that satisfies ( s1 ) , ( s2 ) , ( s3 ) , ( s4 ) , and ( s5 ) also satisfies ( [ eqsumpoly ] ) for every polynomial @xmath42 and all @xmath43 . to prove the first statement , suppose we use ( [ eqsumpoly ] ) as a definition . it is trivial to check that this definition satisfies ( s1 ) , ( s3 ) , ( s4 ) , and ( s5 ) . to see that it also satisfies ( s2 ) , consider a polynomial @xmath42 and the unique corresponding polynomial @xmath45 with @xmath46 and @xmath39 . define @xmath47 and @xmath48 . then @xmath49 , and @xmath50 . hence @xmath51 to see that ( [ eqsumpoly ] ) also satisfies @xmath17 , let @xmath52 be the linear space of complex polynomials of degree less than or equal to @xmath53 . the definition @xmath54 for @xmath55 introduces a norm on @xmath52 . if we define a linear operator @xmath56 via @xmath57 , then this operator is bounded since @xmath58 . thus , if @xmath59 is a sequence of polynomials with @xmath60 , we have @xmath61 . axiom @xmath17 then follows from considering the sequence of polynomials @xmath62 with @xmath63 and noting that pointwise convergence to zero implies convergence to zero in the norm @xmath64 of @xmath65 , and thus of @xmath66 . to prove the second statement , we extend the idea that we used above to show that @xmath67 . using ( s1 ) , we write for an integer @xmath68 @xmath69 where the left - hand side has a classical interpretation , using ( s1 ) , ( s2 ) , and ( s4 ) . rewriting the right - hand side according to ( s2 ) and using ( s3 ) , we get @xmath70 where the @xmath71 are polynomials of degree @xmath72 ( and all @xmath73 ) . now we argue by induction . if @xmath74 , the previous equation clearly determines @xmath75 and by linearity also the corresponding sum over arbitrary constants @xmath33 . once the value of the sum of any polynomial of degree @xmath72 is determined , the equality also determines the value of @xmath76 , and by linearity , the sum of every polynomial @xmath42 of degree @xmath77 . using ( s2 ) again , we see that the axioms ( s1 ) to ( s4 ) uniquely determine the sum @xmath78 if @xmath44 . as we have seen , equation ( [ eqsumpoly ] ) is a possible definition satisfying those axioms ; hence it is the only possible definition for @xmath44 . finally , it is clear how the restriction @xmath44 can be lifted by additionally assuming ( s5 ) ( equation ( [ eqsumpoly ] ) is already satisfied for @xmath79 by requiring just continuity in ( s5 ) ; holomorphy is required for @xmath80 ) . consider now an arbitrary function @xmath81 . if we are interested in @xmath82 for complex @xmath43 , we can write @xmath83 where @xmath26 is an arbitrary natural number . hence , @xmath84 what have we achieved by this elementary rearrangement ? in the last line , the first sum on the right - hand side involves an integer number of terms , so this can be evaluated classically . all the problems sit in the last sum on the right - hand side . the payoff is that we have translated the domain of summation by @xmath85 to the right . since ( [ eqheuristics ] ) holds for every integer @xmath85 , we can use @xmath17 to evaluate the limit as @xmath86 : if @xmath87 as @xmath86 for all @xmath5 , then @xmath17 implies that the limit as @xmath86 of the last sum should vanish . we get @xmath88 this is of course a special condition to impose on @xmath7 , but the same idea can be generalized . for example , if @xmath89 , then for @xmath90\subset{\mathbb{r}}^+$ ] , the values @xmath91 are approximated well by the constant function @xmath92 , with an error that tends to @xmath93 as @xmath86 : we say that @xmath94 is `` approximately constant . '' using ( s3 ) , @xmath95 for every @xmath26 . but by @xmath17 , the last sum vanishes as @xmath86 , while the first sum on the right - hand side has a constant summand and is evaluated using proposition [ prop1 ] . taking the limit @xmath86 in ( [ eqheuristics ] ) , it follows by necessity that @xmath96 before generalizing our definition further , we take courage by observing that this interpolates the factorial function in the classical way : we define @xmath97 and thus get @xmath98 using a well - known product representation of the @xmath2 function @xcite . it is now straightforward to use the heuristic calculation in ( [ eqheuristics ] ) together with proposition [ prop1 ] and axiom @xmath17 to derive a general definition : all we need is that the value of @xmath99 can be approximated by some sequence of polynomials @xmath100 of fixed degree for @xmath86 . some care is needed with the domains of definition : the example of the logarithm shows that it is inconvenient to restrict to functions which are defined on all of @xmath9 . all we need is a domain of definition @xmath101 with the property that @xmath102 implies @xmath103 . this leads to the following ( using the convention that the zero polynomial is the unique polynomial of degree @xmath104 ) . [ deffracsummable ] let @xmath105 and @xmath106 . a function @xmath107 will be called _ fractional summable of degree @xmath108 _ if the following conditions are satisfied : * @xmath109 for all @xmath110 ; * there exists a sequence of polynomials @xmath21 of fixed degree @xmath108 such that for all @xmath110 @xmath111 * for every @xmath112 , the limit @xmath113 exists , where @xmath114 is defined as in ( [ eqsumpoly ] ) . in this case , we will use the notation @xmath115 for this limit . moreover , we can define fractional products by @xmath116 whenever @xmath117 is fractional summable . note that this definition does not depend on the choice of the approximating polynomials @xmath21 : if @xmath118 is another choice of approximating polynomials , then @xmath119 for all @xmath110 , and hence for all @xmath120 since the set of polynomials of degree at most @xmath108 is a finite - dimensional linear space . as shown in proposition [ prop1 ] , sums of polynomials satisfy axiom @xmath17 . substituting @xmath93 for @xmath7 and @xmath121 for @xmath122 in @xmath17 proves that @xmath123 . moreover , this definition is the unique definition that satisfies axioms ( s1 ) to @xmath17 : definition [ deffracsummable ] satisfies all the axioms ( s1 ) to @xmath17 ( for suitable domains of definition ) , and it is the unique definition with this property ( for the class of functions that we are considering ) . we have already proved uniqueness above , by deriving definition [ deffracsummable ] from the axioms ( s1 ) to @xmath17 . it remains to prove that this definition indeed satisfies all the axioms . clearly , ( s3 ) and ( s5 ) are automatically satisfied . substituting the definition into ( s1 ) , ( s2 ) , and ( s4 ) , these axioms can be confirmed by a few lines of direct calculation . to prove @xmath17 , we use the definition and the other axioms ( in particular continued summation ( s1 ) ) and calculate @xmath124 this proves that definition [ deffracsummable ] satisfies all the axioms . now that we have a definition of sums with noninteger numbers of terms , it is interesting to find out how many of the properties of classical finite sums remain valid in this more general setting , and what new properties arise that are not visible in the classical case . one of the most basic identities for finite sums is the geometric series . for simplicity , let @xmath125 . then the function @xmath126 is approximately zero ( we have @xmath127 for every @xmath19 ) , and the definition reads @xmath128 thus , the formula for the geometric series remains valid for every @xmath120 . a similar calculation shows that the binomial series remains valid in the fractional case : for every @xmath129 and @xmath120 with @xmath130 , we have @xmath131 there are generalizations of ( [ eqgeometric ] ) to the case @xmath132 and of ( [ eqbinomial ] ) to the case @xmath133 : these involve a `` left sum '' as introduced in section [ secleftsummation ] . an example of a summation identity with more complicated structure is given by the series multiplication formula @xmath134 for every @xmath120 , given that all the three fractional sums exist ( see ( * ? ? ? * lemma 7 ) ; it generalizes the formula @xmath135 , and similarly for all positive integers @xmath3 . as shown in section [ secfromaxtodef ] , our definition interpolates the factorial by the @xmath2 function , @xmath136 an amusing consequence is @xmath137 this is because @xmath138 using @xmath139 . many basic fractional sums are related to special functions . as a first example , consider the harmonic series . since @xmath140 is approximately zero , the definition reads @xmath141 and in particular @xmath142 which was noticed already by euler ( * ? ? ? * , 19 ) . for general @xmath3 , the harmonic series can be expressed in terms of the so - called digamma function @xcite @xmath143 and the euler - mascheroni constant @xmath144 : one obtains @xcite @xmath145 note that the reflection formula @xcite for the digamma function becomes @xmath146 as a further generalization , it is convenient to consider the hurwitz @xmath147 function , traditionally defined by the series @xmath148 by analytic continuation , @xmath149 can be defined for every @xmath150 , except for a pole at @xmath151 . for @xmath152 , the hurwitz @xmath147 function equals the well - known riemann @xmath147 function : @xmath153 it turns out that the hurwitz @xmath147 function can be understood as a fractional power sum . it can be shown ( * ? ? ? * corollary 14 ) that for every @xmath154 and for all @xmath155 , @xmath156 a useful special case is @xmath157 note that such equations give in many cases intuitive ways to compute properties and special values of special functions . everybody knows the formula @xmath158 , so @xmath159 , and thus by ( [ eqzeta12 ] ) @xmath160 it follows that @xmath161 . similarly , we have @xmath162 ( in the second equality , we interchanged differentiation and fractional summation ; it is not hard to check that this is indeed allowed ) . since @xmath163 , this easily implies that @xmath164 . similarly , differentiating ( [ eqhurwitz ] ) @xmath165 times with respect to @xmath166 and arguing as before ( compare also ( * ? ? ? * sec . 6 ) ) , we obtain @xmath167 there are some classically unexpected special values like @xmath168 there is an identity for classical sums which is almost never mentioned , because it seems so trivial . consider the sum @xmath169 obviously , there are two formally correct possibilities to write this sum , @xmath170 classically , it is clear that @xmath171 . does this carry over to the fractional case ? there is a fundamental problem : our definition of fractional sums involves @xmath172 , i.e. , @xmath7 is evaluated near @xmath173 , and when @xmath174 is replaced by @xmath175 then @xmath7 would be evaluated near @xmath104 where the values may be unrelated . this will be discussed in the next section . looking back at the axioms given in section [ sectheaxioms ] , there is one axiom that could possibly be modified : in @xmath17 , limits as @xmath22 are considered , but one could equally well look at limits as @xmath176 . this way , one obtains an axiom of `` left shift continuity '' : @xmath177 left shift continuity : : if @xmath178 pointwise for @xmath19 , then @xmath179 more generally , if there is a sequence of polynomials @xmath21 of fixed degree such that @xmath180 for @xmath19 as @xmath86 , we require that @xmath181 repeating the calculations of section [ secfromaxtodef ] , one gets an alternative definition ) can determine a definition uniquely : only adding or subtracting @xmath26 to the upper summation boundary consists of adding or subtracting @xmath85 terms to the series , which can be done classically . ] which we do not state here formally : it is exactly the same as definition [ deffracsummable ] , except that in every limit , @xmath86 is replaced by @xmath176 . it can be shown that this definition is the unique one that satisfies axioms ( s1 ) , ( s2 ) , ( s3 ) , ( s4 ) , ( s5 ) , and @xmath177 . note that in general , the existence of @xmath182 and @xmath183 are independent , and if both left and right fractional sums exist , they may have different values . for example , for every @xmath5 with @xmath184 , we have @xmath185 in contrast to equation ( [ eqzeta12 ] ) . what we do have is the obvious relation @xmath186 fractional sums are not simply a new world with results that have no meaning in the classical context ; they allow us to derive identities that can be stated entirely in classical terms . some of these formulas are known and some seem to be new . of course , all these identities can in principle be computed without fractional sums . but proving them with the help of fractional sums is rather intuitive and simple , since most of the steps use fractional generalizations of basic , very well - known classical summation properties . as a first example , we show how to compute a closed - form expression for the infinite product @xmath187 for @xmath188 . it was first considered by borwein and dykshoorn in 1993 ( see @xcite ) . by taking logarithms , one gets @xmath189 consider the function @xmath190 that tends to @xmath191 as @xmath192 . according to definition [ deffracsummable ] , we have @xmath193.\end{aligned}\ ] ] thus , we get @xmath194 where we have used equation ( [ eqvlnv ] ) , index shifting , equation ( [ eqhurwitzdiff ] ) , equation ( [ eqgamma ] ) , and continued summation . by exponentiating , we finally get @xmath195 ) . the right - hand side corresponds to the lowest curve , while the other three curves ( from top to bottom ) are plots of approximations to @xmath196 ( finite products as in ( [ eq : defpx ] ) ) for @xmath197 , @xmath198 , and @xmath199 respectively . , scaledwidth=60.0% ] by application of this method , a large class of infinite products can be explicitly computed , which seems to include the class of products considered in @xcite . here is an example of a new identity : using the same steps as in the calculation above , one easily proves that for @xmath200 , @xmath201 resolving the definition and exponentiating , we get the following classical limit identity : @xmath202\nonumber\\ & = & 2^{-\frac 1 4 \ln 2 + 4 \zeta'\left(-1,\frac{x+1 } 2\right)-4\zeta'\left(-1,\frac x 2 + 1\right ) } \left ( \frac{\gamma\left(\frac{x+1 } 2 \right ) } { \gamma\left(\frac x 2 + 1\right ) } \right)^{-2x\ln 2}\times\nonumber\\ & & \times\quad e^ { 2 \zeta''\left(-1,\frac x 2 + 1\right)-2\zeta''\left(-1,\frac{x+1 } 2\right ) + x\left ( \zeta''\left(0,\frac { x+1 } 2\right)-\zeta''\left(0,\frac x 2 + 1\right ) \right ) } .\label{eqcomplicated}\end{aligned}\ ] ] again , we have used mathematica for a quick numerical check that is shown in figure [ fig : secondzeta ] . ) . the right - hand side corresponds to the uppermost curve , while the other three curves ( from bottom to top ) are plots of the left - hand side for @xmath198 , @xmath203 , and @xmath204 respectively . , scaledwidth=60.0% ] in this section , we consider the multiple gamma function @xmath205 , a generalization of the classical gamma function @xmath2 , defined for @xmath26 and @xmath19 by the recurrence formula ( compare @xcite ) @xmath206 these equations do not determine the functions @xmath205 uniquely , so one needs the additional bohr - mollerup - like condition that @xmath207 is positive and @xmath85 times differentiable on @xmath208 , and that @xmath209 is increasing ( see @xcite ) . for @xmath197 , this definition reproduces the classical gamma function : @xmath210 . the ( reciprocal of the ) special case @xmath211 is known as the barnes @xmath212 function @xmath213 by ( [ eqdefgamman ] ) , it satisfies @xmath214 more generally , @xmath215 so @xmath216 is the reciprocal of the product of @xmath205 , which means that @xmath217 is something like an @xmath85-fold product of the first @xmath218 natural numbers : @xmath219 while this equation only makes sense for @xmath220 , one can easily show that the definition of @xmath217 for @xmath19 is compatible with our definition for fractional sums and products , i.e. , that for every @xmath19 ( except for poles ) and @xmath221 , we have @xmath222 we will now show some properties of the multiple gamma function @xmath205 , specifically for the example @xmath211 , simply by using basic fractional sum identities , without using any special function properties of @xmath205 . by the multiplication formula ( [ eqseriesmultiplication ] ) , we have @xmath223 the last equality follows from equation ( [ eqhurwitzdiff ] ) . thus , we have found an explicit formula for @xmath224 in terms of derivatives of the hurwitz @xmath147 function . equation ( [ eqvlnv ] ) gives the special value @xmath225 these are of course very well - known results , but the calculations are strikingly simple . moreover , this example shows that there is a wide variety of interesting `` special functions '' that do not have to be defined separately , but can be treated in a unified manner by our theory of fractional sums . new generalizations comparable to @xmath226 include @xmath227 @xmath228 here , @xmath229 , @xmath230 , and @xmath231 are the euler - mascheroni , stieltjes , and catalan constants , respectively . again , these formulas have classical limit representations looking like equation ( [ eqcomplicated ] ) which we do not write down here explicitly . the paper `` on some strange summation formulas '' @xcite contains some formulas like ( [ eqgosper ] ) below . there might possibly be very short proofs for all these identities using fractional sums . the only problem is that there is one single step ( indicated by the question mark ) which we are unable to justify : it is basically an interchange of a fractional sum and an infinite series . we will now use the basic identity @xmath235 for every @xmath120 and @xmath26 , which can be shown in two different ways : the first possibility is to see that for every @xmath236 , @xmath237 since even and odd terms cancel each other . by continued summation and proposition [ prop1 ] , @xmath238 is a polynomial in @xmath3 , so equation ( [ eqsumxminusx ] ) must be valid for every @xmath120 . going back to the fractional sum in ( [ eqsb ] ) , the odd function @xmath240 is holomorphic in the entire complex plane , except for a pole at @xmath241 , so we can develop it into a power series . we get @xmath242 the next step is critical : we apply the fractional sum term - by - term . unfortunately , it is not clear that this manipulation is justified . @xmath243 equations ( [ eqreflection ] ) and ( [ eqsumxminusx ] ) yield @xmath244 this method only works for a certain class of functions which obviously contains @xmath92 from ( [ eqfn ] ) and other functions like @xmath245 , but which does _ not _ contain other simple functions like @xmath246 . it is an open question to give sufficient conditions for the validity of this method , i.e. , for justification of termwise fractional summation as in equation ( [ eqtermwise ] ) . we would like to thank several colleagues from the community of `` special functions and exotic identities '' for their encouragement and support , especially richard askey and mourad ismail . moreover , we are grateful to otto forster , irwin kra , armin leutbecher , john milnor , as well as to the seminar audiences in stony brook and mnchen for encouragement , interest , and helpful discussions . we would also like to thank the referees for helping us improve the exposition . l. euler , dilucidationes in capita postrema calculi mei differentialis de functionibus inexplicabilibus [ ] , _ memoires de lacademie des sciences de st .- petersbourg * 4 * _ ( 1813 ) , 88119 ; reprinted in _ opera omnia , series 1 _ , * 16 * , 133 ; also available at http://www.math.dartmouth.edu/@xmath247euler/. * markus mller * received his dr . nat . from technical university of berlin in 2007 , where he is currently working as a postdoc in quantum information theory . after playing around with fractional sums as a high - school student , he was lured away from math to physics by popular scientific articles on schrdinger s cat and quantum weirdness . since then , his main motivation has been the idea that the notion of information opens up an unexpected , fresh perspective on foundational problems of physics . this line of thought led him to work on quantum turing machines and kolmogorov complexity , concentration of measure , and generalized probabilistic theories beyond quantum theory . at the moment , he is preparing for some tough postdoc years abroad by skiing with friends and enjoying the alternative music occasions in berlin . * dierk schleicher * studied physics and computer science in hamburg and obtained his ph.d . in mathematics at cornell university . he enjoyed longer educational and research visits in princeton , berkeley , stony brook , paris , and toronto . after many years in mnchen , he became the first professor at the newly - founded jacobs university bremen in 2001 and built up the mathematics program there . his main research area is dynamical systems , especially complex dynamics : `` real mathematics is difficult , complex mathematics is beautiful . '' he has always been active in math circles and special programs for talented high school students , which is what lured him away from physics to mathematics . he was one of the main organizers of the 50th international mathematical olympiad ( imo ) in bremen / germany , in 2009 . he enjoys outdoor activities : kayaking , paragliding , mountain hiking , and more .
|
starting from a small number of well - motivated axioms , we derive a unique definition of sums with a noninteger number of addends .
these `` fractional sums '' have properties that generalize well - known classical sum identities in a natural way .
we illustrate how fractional sums can be used to derive infinite sum and special functions identities ; the corresponding proofs turn out to be particularly simple and intuitive .
`` god made the integers ; all else is the work of man . ''
leopold kronecker
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the strangeness meson photoproduction off the nucleon target is one of the most well - studied experimental and theoretical subjects to reveal the hadron production mechanisms and its internal structures , in terms of the strange degrees of freedom , breaking the flavor su(3 ) symmetry explicitly . together with the recent high - energy photon beam developments in the experimental facilities , such as lpes2 at spring-8 and clas12 at jefferson laboratory @xcite , higher - mass strange meson - baryon photoproducitons must be an important subject to be addressed theoretically for future studies on those reaction processes . in the previous works @xcite , the @xmath0 photoproduction was investigated , the born approximation being used with the regge contributions . in comparison with the preliminary experimental data @xcite , the theory reproduced the data qualitatively well , but the theoretical cross - section strength was underestimated in the vicinity of the @xmath4 . in the present talk , we want to report our recent study to explain this discrepancy observed in the previous work . based on the theoretical framework as employed in ref . @xcite , we include the nucleon resonances in the @xmath7-channel baryon - pole contribution . as for the nucleon resonances @xmath8 , we take into account nine of them , i.e. @xmath9 , @xmath10 , @xmath11 , @xmath12 , @xmath13 , @xmath14 , @xmath15 , @xmath9 , @xmath16 , and @xmath17 , in a full relativistic manner . we start with explaining the theoretical framework briefly and represent the important numerical results in our study . we note that the nucleon resonances are carefully taken into account in a full - relativistic manner , in addition to the born diagrams , @xmath18 and @xmath19 meson - exchanges in the @xmath20 channel , and @xmath21 and @xmath22 hyperon exchanges in the @xmath23 channel , which were already employed in ref . all the effective interaction vertices for the nucleon resonances are taken from ref . for instance , the invariant amplitudes for the spin-@xmath24 and spin-@xmath25 resonance contributions in the @xmath7 channel can be written as follows : @xmath26 \cr & & \frac{ie_q\mu_{n^*}}{2m_n}\gamma^{\mp}(\not{k_1}+\not{p_1}+m_{n^ * } ) \gamma^{\mp}\sigma^{\mu{i}}k_{1i}u_n(p_1){\varepsilon}_{\mu}(k_1 ) , \cr \mathcal { m}_{n^*}\left(\frac{3}{2}^\pm \right)&= & \frac{g_{{k^*}n^*\lambda}}{s - m_{n^*}^2 } { \varepsilon}_{\nu}^*(k_2){\bar u}_{\lambda}(p_2)({k_2^\beta}g^{{\nu}i}-k_2^ig^{\nu\beta } ) \frac{e_q}{2m_{k^*}}\gamma_i^{\pm}\delta_{\beta\alpha } \cr & & \left[\frac{\mu_{n^*}}{2m_n}\gamma_j\,\mp\,\frac{\bar \mu_{n^ * } } { 4m_n^2}p_{1j } \right]\gamma^{\pm}({k_1^\alpha}g^{{\mu}j}-{k_1^j } g^{\mu\alpha})u_n(p_1){\varepsilon}_{\mu}(k_1),\end{aligned}\ ] ] where @xmath27 stand for the @xmath28 momenta and @xmath29 for the helicity amplitude . the @xmath30 and @xmath31 denote the strong vector and tensor coupling strengths , respectively . the polarization vectors for the photon and @xmath32 are assigned as @xmath33 and @xmath34 . the @xmath35 controls the parity of the relevant resonances in the following way : @xmath36 relevant parameters for the resonances are estimated using the experimental and theoretical information @xcite . in figure [ fig1 ] , we show the total cross section for the present reaction process , i.e. @xmath37 . the numerical results are drawn separately for the cases including the @xmath2 ( left ) and @xmath3 ( right ) , varying the coupling constants @xmath30 . as shown in the figure , the cross - section enhancement is observed in the vicinity of the threshold region , if we choose the strong coupling strengths as @xmath38 . we verified that other nucleon resonances are not so effective to interpret the discrepancy . [ cols="^,^ " , ] in the present work , we have studied the @xmath39 photoproduction theoretically , employing the tree - level born approximation and nucleon - resonance contributions below the @xmath4 . among the nucleon resonances , the @xmath2 and @xmath3 play a dominant role to reproduce the experimental data . it also turns out that other @xmath8 contributions are not so effective to improve the theoretical results . we note that the nucleon resonances beyond the @xmath4 may contribute to the threshold enhancement , especially due to the @xmath40 , since the @xmath4 for the present reaction process is about @xmath41 mev . related works are under progress and appear elsewhere . the authors are grateful to a. hosaka for fruitful discussions . the present work is supported by basic science research program through the national research foundation of korea ( nrf ) funded by the ministry of education , science and technology ( grant number : 2009 - 0089525 ) . the work of s.i.n . was supported by the grant nrf-2010 - 0013279 from national research foundation ( nrf ) of korea . 99 leps2 ( http://www.hadron.jp ) and clas12 ( http://www.jlab.org/hall-b/clas12 ) y. oh and h. kim , phys . c * 74 * , 015208 ( 2006 ) . y. oh and h. kim , phys . rev . c * 73 * , 065202 ( 2006 ) . l. guo and d. p. weygand [ clas collaboration ] , arxiv : hep - ex/0601010 . y. oh , c. m. ko and k. nakayama , phys . rev . c * 77 * , 045204 ( 2008 ) . k. nakamura [ particle data group ] , j. phys . g * 37 * , 075021 ( 2010 ) . s. capstick and w. roberts , phys . rev . d * 58 * , 074011 ( 1998 ) . s. capstick and w. roberts , prog . * 45 * , s241 ( 2000 ) .
|
in this presentation , we report our recent studies on the @xmath0 photoproduction off the proton target , using the tree - level born approximation , via the effective lagrangian approach . in addition , we include the nine ( three- or four - star confirmed ) nucleon resonances below the threshold @xmath1 mev , to interpret the discrepancy between the experiment and previous theoretical studies , in the vicinity of the threshold region . from the numerical studies , we observe that the @xmath2 and @xmath3 play an important role for the cross - section enhancement near the @xmath4
. it also turns out that , in order to reproduce the data , we have the vector coupling constants @xmath5 and @xmath6 .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the origin of cosmic radiation and a mechanism trough which it could acquire enormous energies has intrigued many physicists and mathematicians along the last decades . in particular , in the year of 1949 enrico fermi @xcite proposed a very simple model that qualitatively describes a process in which charged particles were _ bounced _ via interaction with moving magnetic fields . this heuristic idea was then later modified to encompass in a suitable model that could give quantitative results on the original fermi s model . after that , many different versions of the model were proposed @xcite and studied in different context and considering many approaches and modifications @xcite . one of them is the well known fermi - ulam model ( fum ) . such model consists of a classical point - like particle moving between two walls in the total absence of any external field . one of the walls is considered to be fixed while the other one moves periodically in time . despite this simplicity , the structure of the phase space is rather complex and it includes a large chaotic sea surrounding kam islands and a set of invariant spanning curves limiting the energy of a bouncing particle . however , the introduction of innelastic collisions on this model is enough to destroy such a mixed structure and the system exhibits attractors . depending on the initial conditions and control parameters , one can observe a chaotic attractor characterised by a positive lyapunov exponent . by a suitable control parameter variation , the chaotic attractor might be destroyed via a crisis event @xcite . after the destruction , the chaotic attractor is replaced by a chaotic transient @xcite . in this paper , we revisit a dissipative version of the fermi - ulam model seeking to understand and describe the behaviour of the dynamics for the regime of high dissipation . moreover , we investigate a phenomenon quite common observed for a large variety of dissipative and nonlinear systems known as _ period doubling route to chaos_. this route shows a sequence of doubling bifurcations connecting regular - periodic to chaotic behaviour . the main goal of this paper is then to examine and show that the fermi - ulam model obey the same convergence ratio as that obtained by feigenbaum @xcite for the dynamics of the logistic map . such a constant value has been further called as `` the feigenbaum s @xmath0 '' . the organisation of this paper is as follows : in section [ sec2 ] we present all the details needed for the construction of the nonlinear mapping . our numerical results are discussed in section [ sec3 ] and our conclusions and final remarks are drawn in section [ sec4 ] . let us describe the model and obtain the equations of the mapping . the model we are dealing with consists of a classical point - like particle confined in and bouncing betwen two rigid walls . one of them is assumed to be fixed at the position @xmath1 and the other one moves periodically in time according to @xmath2 . thus , the moving wall velocity is given by @xmath3 where @xmath4 denotes the amplitude of oscillation and @xmath5 is the angular frequency of the moving wall , respectively . aditionally , the motion of the particle does not suffer influence of any external field . we assume that all the collisions with both walls are inelastic . thus , we introduce a restitution coefficient for the fixed wall as @xmath6 $ ] while for the moving wall we consider @xmath7 $ ] . the completely inelastic collision happens when @xmath8 and that a single collision is enough to terminate all the particle s dynamics . on the other hand , when @xmath9 , corresponding to a complete elastic collision , all the results for the nondissipative case are recovered @xcite . we describe the dynamics of the particle using a map @xmath10 where the dynamical variables are @xmath11 and @xmath12 is the velocity of the particle at the instant @xmath13 . the index @xmath14 denotes the @xmath15 collision of the particle with the moving wall . starting then with an initial condition @xmath11 , with initial position of the particles given by @xmath16 with @xmath17 , the dynamics is evolved and the map @xmath10 gives a new pair of @xmath18 at the @xmath19 collision . it is important to say that we have three control parameters , namely : @xmath4 , @xmath20 and @xmath5 and that the dynamics does not depend on all of them . thus it is convenient to define dimensionless and more appropriated variables . therefore , we define @xmath21 , @xmath22 and measure the time in terms of the number of oscillations of the moving wall , consequently @xmath23 . incorporating this set of new variables into the model , the map @xmath10 is written as @xmath24 where the expressions for both @xmath25 and @xmath26 depend on what kind of collision occurs . there are two different possible situations , namely : ( i ) multiple collisions with the moving wall and , ( ii ) single collision with the moving wall . considering the first case , the expressions are @xmath27 and @xmath28 , where @xmath29 is obtained as the smallest solution of @xmath30 with @xmath31 $ ] . a solution for @xmath30 is equivalent to have that the position of the particle is the same as that of the moving wall at the instant of the impact . the function @xmath32 is given by @xmath33 if the function @xmath32 does not have a root in the interval @xmath31 $ ] , we can conclude that the particle leaves the collision zone ( the collision zone is defined as the interval @xmath34 $ ] ) without suffering a successive collision . considering now the case ( ii ) , i.e. the case where the particle leaves the collision zone , the corresponding expressions are @xmath35 and @xmath36 , with the auxiliary terms given by @xmath37 finally , the term @xmath29 is obtained as the smallest solution of @xmath38 for @xmath39 . the expression of @xmath40 is given by @xmath41 both , eq . ( [ g ] ) and eq . ( [ f ] ) must be solved numerically . after some algebra it is easy to shown that the determinant of the jacobian matrix for the case ( i ) is given by @xmath42~ , \label{det_s}\ ] ] while for the case ( ii ) we obtain @xmath43~. \label{det_ns}\ ] ] we therefore can conclude based on the above result that area preservation is obtained only for the case of @xmath9 . in this section we discuss some numerical results considering the case of high dissipation . we call as high dissipation the situation in which a particle loses more than @xmath44 of its energy upon colision with the moving wall . moreover , we have considered as fixed the values of the damping coefficients @xmath45 and @xmath46 . to illustrate that the dynamics of the system exhibits doubling bifurcation cascade , it is shown in fig . [ fig1](a ) and ( b ) @xmath47 both plotted against @xmath48 . in ( c ) it is shown the lyapunov exponent associated to ( a ) and ( b ) . the damping coefficients used for the construction of the figures ( a ) , ( b ) and ( c ) were @xmath45 and @xmath46 . ] the behaviour of the asymptotic velocity plotted against the control parameter @xmath48 , where the sequence of bifurcations is evident . a similar sequence is also observed for the asymptotic variable @xmath49 , as it is shown in fig . [ fig1](b ) . note that all the bifurcations of same period in ( a ) and ( b ) happen for the same value of the parameter @xmath48 . at the point of bifurcations , one can also observe that the positive lyapunov exponent shows null value , as it is shown in fig . [ fig1](c ) . it is well known that the lyapunov exponents are important tool that can be used to classify orbits as chaotic . as discussed in @xcite , the lyapunov exponents are defined as @xmath50 where @xmath51 are the eigenvalues of @xmath52 and @xmath53 is the jacobian matrix evaluated over the orbit @xmath54 . however , a direct implementation of a computational algorithm to evaluate eq . ( [ eq4 ] ) has a severe limitation to obtain @xmath55 . even in the limit of short @xmath14 , the components of @xmath55 can assume very different orders of magnitude for chaotic orbits and periodic attractors , yielding impracticable the implementation of the algorithm . in order to avoid such problem , we note that @xmath56 can be written as @xmath57 where @xmath58 is an ortoghonal matrix and @xmath10 is a right up triangular matrix . thus we rewrite @xmath55 as @xmath59 , where @xmath60 . a product of @xmath61 defines a new @xmath62 . in a next step , it is easy to show that @xmath63 . the same procedure can be used to obtain @xmath64 and so on . using this procedure , the problem is reduced to evaluate the diagonal elements of @xmath65 . finally , the lyapunov exponents are given by @xmath66 if at least one of the @xmath67 is positive then the orbit is classified as chaotic . we can see in fig . ( [ fig1 ] ) ( c ) the behavior of the lyapunov exponents corresponding to both figs . ( [ fig1])(a , b ) . it is also easy to see that when the bifurcations happen , the exponent @xmath68 assumes the zero value at same values of the control parameter @xmath48 where the bifurcation hold . another interesting observation is that the lyapunov exponent , in some regions , assumes a constant and negative values , such behavior occurs because the eigenvalues of the jacobian matrix become complex numbers ( imaginary numbers ) ) . let us now use the points where the lyapunov exponent assumes the null value to characterise a very interesting property of the model . as it was shown by feigenbaum @xcite , along the bifurcations the dynamics exhibits an universal feature . it implies that all the bifurcations happen at the same rate of convergence for the bifurcation diagram changing from periodic to chaotic behavior . the procedure used to obtain the feigenbaum value @xmath0 consists of : let @xmath69 represents the control parameter value at which period-1 gives birth to a period-2 orbit , @xmath70 is the value where period-2 changes to period-4 and so on . in general the parameter @xmath71 corresponds to the control parameter value at which a period-@xmath72 orbit is born . thus , we write the feigenbaum s @xmath0 as @xmath73 the numerical value for the constant number @xmath0 obtained by eq . ( [ fei ] ) and considering a sufficient large values for @xmath14 is @xmath74 . considering the numerical data obtained through the lyapunov exponents calculation , the feigenbaum s @xmath0 obtained for the fermi - ulam model is @xmath75 . we have considered in our simulations only the bifurcations of fourth to eighth order since the numerical results are very hard to be obtained for higher orders in the bifurcations . our result is in a good agreement with the feigenbaum s universal @xmath0 . as a summary , we have studied a dissipative version of the fermi - ulam model . we introduce dissipation into the model through inelastic collisions with both walls . we have shown that for regimes of high dissipation , the model exhibits a sequence of doubling bifurcation cascade . for this cascade , we obtained the so called feigenbaum s number as @xmath75 . dfmo gratefully acknowledges conselho nacional de desenvolvimento cientfico e tecnolgico cnpq . edl is grateful to fapesp , cnpq and fundunesp , brazilian agencies .
|
we have studied a dissipative version of a one - dimensional fermi accelerator model .
the dynamics of the model is described in terms of a two - dimensional , nonlinear area - contracting map .
the dissipation is introduced via innelastic collisions of the particle with the walls and we consider the dynamics in the regime of high dissipation .
for such a regime , the model exhibits a route to chaos known as period doubling and we obtain a constant along the bifurcations so called the feigenbaum s number @xmath0 .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the dynamics of confined cavity fields interacting with the cavity wall is of great interest for the understanding of a variety of problems such as hadron bag models @xcite , sonoluminescence @xcite , cavity qed @xcite and black hole radiations @xcite . most previous studies on dynamical cavity concentrated on scalar or photon fields @xcite , which despite the apparent simplicity , exhibit rich and complex dynamics . in contrast , the system of a schrdinger particle in an oscillating spherical cavity has not gained as much attention . in this article , we study the generalized quantum geometric phase of a particle in resonance with the vibrating cavity . we will show that the geometric phase acquires sudden @xmath0-jumps when the particle makes transitions . the geometric phase of a quantum system has drawn much attention since it was first proposed by berry in 1984 @xcite . it reveals the geometric structure of the hilbert space of the system @xcite , and its existence in physical systems has been verified in many experiments @xcite , including electron diffraction from a screw dislocation @xcite and neutron interferometry @xcite . the phase effects in molecular physics can lead to energy splittings and shift quantum numbers . the geometric phase has also been shown to be intimately connected to the physics of fractional statistics , the quantized hall effect , and anomalies in gauge theory @xcite . as far as we know , our study represents the first calculation of the geometric phase of a resonating system , which evolves non - adiabatically and non - cyclically . we consider an infinite cylindrical or spherical potential well with oscillating boundary @xcite : @xmath1 where @xmath2 , with @xmath3^{-1}$ ] . the coordinates can be transformed to a fixed domain via @xmath4 , and , to preserve unitarity , the wavefunction is renormalized through @xmath5 , where @xmath6 ( 3/2 ) , for a cylindrical ( spherical ) well . since the full hamiltonian @xmath7 commutes with @xmath8 and @xmath9 , the wavefunction can be factorized : @xmath10 , where @xmath11 depends only on the angular variables . inside the cavity , the radial wavefunction satisfies @xmath12y,\ ] ] where @xmath13 is the particle mass , and @xmath14 , 2 and @xmath15 for cylindrical and spherical wells respectively . in this paper , we only consider the @xmath16 sector . the wavefunction described by eq . [ d.r.e ] evolves in time and acquires a time - dependent phase , which in general consists of a dynamical phase and a geometric one @xcite . when the dynamics is adiabatic and/or the evolution is cyclic , the geometric phase , or berry s phase , has been studied for many systems . since we are interested in the geometric phase of a non - adiabatic , non - cyclic system , we have to resort to a generalized method . following ref . @xcite , we first remove the dynamical phase from the wavefunction of the system : @xmath17 , the dynamical phase is @xmath18 where @xmath19 . the general setting geometric phase , or the pancharatnam phase @xmath20 is defined as the relative phase between the state at time @xmath21 and that at time @xmath22 , both with the dynamical phase removed . it can be obtained from the inner product of these two states @xmath23 our main goal in this paper is to study this geometric phase for a simple but nontrivial dynamical system . we have solved eq . [ d.r.e ] numerically and checked that the solution is stable and converges very well . in fig . 1 , we show the maximum energy of the particle as a function of the driving frequency @xmath24 , both having been scaled to be dimensionless : @xmath25 , and @xmath26 . the particle is initially in the ground state , and it is in resonance at specific values of @xmath27 . in each period of cavity vibration @xmath28 , the geometric phase acquires some changes , @xmath29 , and it exhibits oscillations at these resonances . we show three examples in fig . 2 for @xmath30 and 22.21227 , where the geometric phase acquires periodic changes of @xmath31 respectively . all the resonances shown in fig . 1 are associated with oscillations of @xmath32 with amplitudes of @xmath33 , @xmath34 . to understand the resonances and the associated geometric phases , we limit ourselves to the parameter regime where the hamiltonian can be truncated to a two - level system . specifically , if @xmath35 , and the initial state is the @xmath36 unperturbed eigenstate @xmath37 , with eigenenergy @xmath38 , then when the driving frequency corresponds to the energy difference between the initial state and another unperturbed eigenstate @xmath39 , with eigenenergy @xmath40 , _ i.e. _ , @xmath41 , the particle is expected to behave as in a two - level system . then the problem simplifies considerably , and we have obtained its solution with two analytic approaches : the su(2 ) method and the rotating - wave approximation ( rwa ) . following cheng _ et al._@xcite we first expand the time - dependent hamiltonian @xmath7 of a two - level system in the identity operator @xmath42 , the raising and lowering operators @xmath43 , and pauli spin matrix @xmath44 @xmath45 , \label{eq.h(t)}\ ] ] where @xmath46 , are in general complex functions of time . the evolution operator can then be written as @xmath47 where @xmath48 , and @xmath49 satisfies @xmath50 with the initial conditions @xmath51 for @xmath521 , 2 , or 3 . suppose that the two levels in which the system oscillates are @xmath37 and @xmath39 , so that the wavefunction is @xmath53 . furthermore , if the initial conditions are @xmath54 and @xmath55 , then @xmath56 the su(2 ) method is exact and reduces the problem to solving the ode s for @xmath57 . the spirit of the rwa is to retain only those terms in the hamiltonian that correspond to the resonance frequency . we first separate out the fast phase factors from the wavefunction : @xmath58 with @xmath59 . substituting eq . [ 2.level.phi ] in the schrdinger equation , we have @xmath60 where @xmath61,\ ] ] and @xmath62 is a constant that depends only on the states involved . as @xmath63 is small , we can expand @xmath64 as a series of @xmath63 . retaining all terms up to third - order , we have : @xmath65 \right . \nonumber \\ & + & { \epsilon^{2 } \over 4\omega } \left [ \left(\omega ^2 - 6\omega_{nk}^2 \right ) \cos \omega t - \left(\omega^2 + 2\omega_{nk}^2\right ) \right . \cos 3\omega t \nonumber \\ & - & \left . i\frac{7}{2 } \omega_{nk}\omega(\sin \omega t + \sin 3\omega t ) \right ] \right\ } . \label{w}\end{aligned}\ ] ] note that @xmath64 consists of oscillatory terms with various frequencies depending on @xmath24 . in the spirit of rwa , we keep only terms with the _ lowest _ frequency for each @xmath24 , the rationale being that @xmath66 and @xmath67 vary slowly in time and the contributions to @xmath68 from high frequency terms cancel on average over such a long time scale . it is clear then from eq . [ w ] that @xmath68 is largest if @xmath69 , with @xmath70 an integer , because of the emergence of zero frequency terms . for these driving frequencies , @xmath68 is large and effective in inducing transitions , and we have resonances . at or close to a resonance , eq . [ w ] simplifies tremendously , and closed - form solutions for eq . [ matrix.eq.b ] can be obtained easily . for example , when @xmath71 , @xmath72 , and we have rabi oscillations @xcite : @xmath73 , \\ \label{c_n } c_{n}(t ) & = & \gamma e^{-i\delta \omega t/2}\sin \chi t /\chi , \end{aligned}\ ] ] where we took @xmath74 , @xmath75 , and @xmath76 . the maximum value of @xmath77 , @xmath78 $ ] , is a lorentzian with a fwhm of @xmath79 . the solutions for higher @xmath70 resonances differ only in the widths @xmath80 , where @xmath81 with @xmath82 . the fwhm of the resonances @xmath83 are narrower for larger @xmath70 , and the oscillation periods @xmath84 are longer . the energy of the system is given in rwa by @xmath85 , \end{aligned}\ ] ] where @xmath86/4\chi^2 $ ] . the energy of the particle therefore exhibits fast but small oscillations due to the @xmath87 factor , modulated by a large but slow oscillation of period @xmath88 . it follows that @xmath89 which is again a universal lorentzian for all resonances . the rwa results are virtually identical with those of the su(2 ) method . fig . 1 can be understood completely in terms of overlapping series of the @xmath90 resonances , and the positions of the peaks are in almost exact agreement with the rwa or su(2 ) predictions . the resonance line shapes are well fitted by the lorentzian eq . [ transform.emax ] , though the widths are underpredicted for the @xmath91 resonances , as shown in table 1 . for example , the @xmath92 resonance has a much larger width than predicted by rwa and su(2 ) , which is mainly due to the involvement of other states during transition , such as a @xmath93 process , which is second order in perturbation theory and so affects the @xmath94 transitions more severely . occasionally , resonances at similar frequencies may overlap and lead to broadened widths . at a resonance , the rabi oscillations of the particle as predicted in rwa ( eq . [ approx.et.1 ] ) can be seen explicitly in fig . 2 , where the energy of the particle vs. time is shown for @xmath95 and @xmath96 . the rwa also predicts that at exact resonances , @xmath97 , and therefore the oscillation periods should be inversely proportional to the widths of the resonances , @xmath98 . as shown in the last column of table 1 , this is clearly borne out in the numerical data , where @xmath99 is listed and is found to be closed to 1 for all resonances , deviating by less than 10% for even those with strong mixing . using the rwa and two - level approximation , we can calculate the dynamical phase easily . removing the dynamical phase with the help of eq . [ approx.et.1 ] , we have @xmath100 where @xmath101 . for @xmath69 resonances , if we choose @xmath102 to be an integral multiple , @xmath103 , of the cavity oscillation period @xmath104 , we get @xmath105 , \nonumber\ ] ] where @xmath106 . in particular , if @xmath107 , @xmath108 where @xmath109 , with @xmath110 . there are sudden approximate @xmath0 jumps in @xmath111 at @xmath112 , as shown for example in fig . 3 for @xmath113 . since @xmath114 , the phase change in each cycle ( @xmath115 ) is @xmath116 at an exact @xmath117 resonance , @xmath118 and @xmath119 , and so @xmath120.\ ] ] therefore , the geometric phase oscillates with an amplitude of @xmath33 and period of @xmath121 . both of these rwa predictions are in excellent agreement with the numerical data , as shown in fig . 2 . note that the @xmath0-jumps and the functional forms of the geometric phases are independent of @xmath63 and @xmath122 , as long as they are nonzero . to gain more insight into the geometric phase for a two - level system , we have studied a simple model of a magnetic field rotating around a spin-1/2 particle . suppose an electron of charge @xmath123 and mass @xmath124 is placed at the origin , in the presence of a magnetic field @xmath125\;,\ ] ] which has a constant magnitude @xmath126 but its direction sweeps out a cone with an opening angle @xmath127 , @xmath128 , at a constant angular speed @xmath24 . the hamiltonian of the system is given by @xmath129 where @xmath130 and * s * is the spin matrix . the system can be solved analytically @xcite . the instantaneous eigenspinors of @xmath7 with eigenenergies @xmath131 respectively are @xmath132 and @xmath133 suppose the electron spin is initially parallel to @xmath134 , and we consider the case when the particle makes a transition to spin down along the instantaneous direction of @xmath135 . this happens with unit probability if @xmath136 , provided that @xmath137 . the state vector at any time is then @xcite @xmath138 where @xmath139 the pancharatnam phases comparing the initial state with the state at time @xmath22 for different values of @xmath127 are shown in fig . ( [ ebb0k100w60]-[ebb0k100.2w90 ] ) . the sudden @xmath0-jump and a two - period oscillation can be seen in the case when @xmath140 , @xmath141 being an even integer , such as the case in fig . ( [ ebb0k100w60 ] ) . when @xmath141 is an odd integer , the pancharatnam phase performs a single - period oscillation with no @xmath0-jump , as shown in fig . ( [ ebb0k101w61 ] ) . it has neither single - period nor two - period oscillation when other values of @xmath127 are used , fig . ( [ ebb0k100.2w90 ] ) . if @xmath142 , with @xmath103 an integer and @xmath143 , @xmath144 which has the same form as eq . [ beta0.vib ] , where @xmath124 is an integer , @xmath145 , and @xmath146 for the case @xmath147 and @xmath148 , @xmath149 \ , & { \rm for}\ ; 0<\sin\alpha<1/2,\\ -{1\over 2 } \left [ \omega ( t_1+\tau ) - \omega ( t_1 ) \right ] \pm \pi\ , & { \rm for}\ ; 1/2<\sin\alpha<1\;. \label{beta1.k1 } \end{array } \right.\ ] ] in the limit @xmath150 , @xmath151 which has the same form as eq . [ beta1 ] . in the cyclic limit , _ i.e. _ , @xmath152 , @xmath153 and thus the geometric phases are @xmath154 and @xmath155 . since the geometric phases in the two models - an electron in a rotating magnetic field and a particle in a vibrating cavity - are remarkably similar , we conjecture that the main features of the generalized geometric phase we calculated , especially the @xmath0-jumps , are universal for a particle in transition from one state to another . similar features about the geometric phase have also been obtained in @xcite . here we present a geometrical model that helps to visualize the evolution of the geometric phase of a two - level system . it is clear from eq . [ simp.psi ] that the state vector traces out a path on the unit sphere defined by the angular coordinates @xmath156 and @xmath157 . note that we have used a time - dependent basis @xmath158 , @xmath159 , which mark the north and south poles respectively on the unit sphere . the excitation condition of our system therefore gives us the trajectory of the state on this unit sphere , @xmath160 which is a spiral curve . the solid angle subtended at the origin by the spiral curve and the geodesic , @xmath161 , up to any @xmath162 is @xmath163 which coincides with eq . [ omega.spin ] , in the limit @xmath164 . therefore , we see from eq . [ beta_0.spin ] and eq . [ beta1.k1 ] that the geometric phases @xmath111 and @xmath165 are simply the solid angles subtended by the spirial curve , @xmath166 , from @xmath167 and @xmath168 up to @xmath169 respectively , divided by @xmath170 . this picture is a generalization of berry s @xcite for adiabatic and cyclic evolution corresponding to trajectories with constant @xmath171 , and the solid angle is simply @xmath172 . even the @xmath0-jump can be represented in this model by a jump of the particle from one sphere to another if its trajectory happens to reach the south pole , which depends on the value of the angle @xmath127 . the double - sphere picture reminds one of the underlying su(2 ) structure . in summary , we have reported the first calculation of the quantum geometric phase of a physical system in resonance - that of a particle in a vibrating cylindrical or spherical cavity , and we have shown that it acquires sudden @xmath0-jumps when the particle makes transitions from one state to another . we have derived analytic expressions in the rwa and su(2 ) methods , which give excellent description of the energy , wavefunctions , and the geometric phases at these resonances . we found remarkably similar properties of the geometric phases for the simple system of an electron in a rotating magnetic field , which led us to conjecture that the main features of the generalized geometric phases and especially the @xmath0-jump we found are universal . we have also developed a geometrical model to help visualize these phases . moore , j. math . phys . * 11 * , 2679 ( 1970 ) ; p.w . milonni , _ the quantum vacuum _ ( academic press , new york , 1993 ) ; n.d . birrell and p.c.w . davies , _ quantum fields in curved space _ ( cambridge university press , cambridge , 1982 ) . for example , c.k . law , phys . lett . * 73 * , 1931 ( 1994 ) ; v.v . dodonov and a.b . klimov , phys . a * 53 * , 2664 ( 1996 ) , and references therein ; a. lambrecht , m.t . jaekel and s. reynaud , phys . rev . lett . * 77 * , 615 ( 1996 ) ; c.k . cole and w.c . schieve , phys . a * 52 * , 4405 ( 1995 ) ; p. meystre _ et al . _ , j. opt . b * 2 * , 1830 ( 1985 ) ; ying wu _ et al . _ , phys . a * 59 * , 3032 ( 1998 ) ; ying wu _ et al . _ , phys . a * 59 * , 1662 ( 1998 ) ; k.w . chan , master thesis , the chinese university of hong kong ( unpublished ) , 1999 ; k. colanero and m .- c . chu , phys . rev . e * 62 * , 8663 ( 2000 ) . .scaled widths ( @xmath173 ) of the lowest @xmath174 resonances for cylindrical cavity with @xmath175 . full numerical results are compared to the rwa values . the widths multiplied by the rabi oscillation period , both extracted from the numerical data , are also shown in the last column . corresponding numbers for spherical cavity with @xmath176 are shown in parentheses . [ cols="^,^,^,^,^",options="header " , ]
|
we study the general - setting quantum geometric phase acquired by a particle in a vibrating cavity . solving the two - level theory with the rotating - wave approximation and the su(2 ) method
, we obtain analytic formulae that give excellent descriptions of the geometric phase , energy , and wavefunction of the resonating system .
in particular , we observe a sudden @xmath0-jump in the geometric phase when the system is in resonance .
we found similar behaviors in the geometric phase of a spin-1/2 particle in a rotating magnetic field , for which we developed a geometrical model to help visualize its evolution .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
frustration in the antiferromagnetic ( afm ) ordering of fcc and spinel lattices was recognized long ago by anderson in his analysis of the highly degenerate magnetic ground state of these structures.@xcite this so - called @xmath26 frustration can prevent the system from undergoing spin - glass ( sg ) or afm ordering down to temperatures much lower than the curie - weiss temperature , @xmath27,@xmath28 . it has also been shown , theoretically @xcite and experimentally , @xcite that the ground state degeneracy can be removed by atomic disorder leading to a sg type of ordering . the afm @xmath5 spinel zncr@xmath6o@xmath7 structure , in which the octahedral cr sites form corner - sharing tetrahedra , @xcite as well as the pyrochlore @xcite@xmath29@xcite and kagom@xcite structures , are excellent systems to study the _ geometric _ frustration phenomenon . zncr@xmath6o@xmath7 has a very high curie - weiss temperature , @xmath30 k , and a first order afm transition at @xmath31 @xmath32 k @xcite accompanied by a slight tetragonal crystal distortion ( @xmath33 ) . @xcite besides , recent interesting low-@xmath0 neutron diffraction experiments showed , in the ordered phase ( @xmath24 ) , the existence of a near dispersionless excitation at @xmath23 mev , and for @xmath34 , a continuous magnetic gapless density of states.@xcite in this work we report on the @xmath0-dependence ( @xmath1 - @xmath2 k ) of the electron paramagnetic resonance ( epr ) , magnetic susceptibility , @xmath3 , and specific heat , @xmath4 , in a single crystal of the @xmath5 spinel zncr@xmath35o@xmath7 of cubic structure ( @xmath36 , o@xmath37 ) and in the cd doped polycrystalline zn@xmath8cd@xmath9cr@xmath6o@xmath7 ( @xmath10 ) compounds . a polycrystalline isomorphous compound of znga@xmath6o@xmath7 was also used as a reference compound for the specific heat measurements . single crystals of zncr@xmath6o@xmath7 of typical size of @xmath382x2x2 mm@xmath39 were obtained by the method of solid state reaction between stoichiometric amounts of cr@xmath6o@xmath40 and zno in air.@xcite the crystals show natural growing ( 001 ) , ( 111 ) , and ( 011 ) faces that were checked by the usual laue method . polycrystalline cd doped zn@xmath8cd@xmath9cr@xmath6o@xmath7 ( @xmath41 ) and znga@xmath6o@xmath7 samples were prepared by the same method . the epr experiments were carried out in a conventional elexsys bruker x - band epr spectrometers using a te@xmath42 room temperature cavity . the sample temperature was varied by a temperature controller using helium and nitrogen gas flux systems . this set up assures one that the spectrometer sensitivity remains about the same over a wide range of @xmath0 . magnetization measurements have been taken in a quantum design @xmath43 squid mpms-5 t magnetometer . the specific heat was measured using the heat pulse method in a quantum design calorimeter using the qd - ppms-9 t measurement system . for the zncr@xmath6o@xmath7 single crystal , as the temperature is lowered from room-@xmath0 , the epr line broadens and its intensity goes through a maximum at about @xmath16 k with no measurable resonance shift . figure 1 shows the @xmath0-evolution of the epr spectra between @xmath44 k and @xmath45 k. for @xmath46 k the resonance distorts and small resonances modes emerge at low-@xmath0 . these modes do not depend on whether the epr spectra are taken under field cooling ( fc ) or zero field cooling ( zfc ) conditions . but , they depend on the size and shape of the sample and show a slight orientation dependence ( see inset ) . for @xmath47 k the epr spectra show a single isotropic resonance . for the cd doped samples the resonance also broadens , but the intensity increases down to @xmath48 k ( see below ) where , again , the resonance distorts and small resonance modes are seen . these modes also show the same spectra under fc and zfc conditions . figure 2 shows the @xmath0-dependence of the linewidth , @xmath49 , and @xmath15-value between @xmath50 k and @xmath2 k for the crystal of figure 1 . the linewidth broadens at low-@xmath0 , the @xmath15-value is @xmath0-independent and its value , @xmath15 @xmath51 @xmath52 , corresponds to that of cr@xmath18 ( @xmath53 , s @xmath19 ) ions @xmath15-value in a cubic site . @xcite@xmath54@xcite both , the @xmath15-value and linewidth are isotropic for @xmath55 @xmath56 k. for the cd doped samples similar resonance line broadening and @xmath15-values are obtained ( not shown ) . figure 3 presents the @xmath43 magnetic susceptibility , @xmath3 , corrected by the host diamagnetism in the range between @xmath1 k and @xmath2 k for the same crystal of figure 1 . fc and zfc measurements at @xmath57 @xmath58 koe and @xmath59 koe gave no difference for the susceptibility data . at low field @xmath3 shows the typical 3d afm ordering with @xmath60 @xmath61 @xmath45 k ) . the inset shows the sharp drop of the susceptibility at @xmath0 @xmath62 k. this temperature defines the nel temperature , @xmath63 , for the 3d long range afm ordering in zncr@xmath64o@xmath65 . the inset shows that , for @xmath66 , the susceptibility is field dependent , @xmath67 . this has been attributed to domain wall movement in the afm ordered state . @xcite figure 4 compares the magnetic susceptibility of the zncr@xmath6o@xmath7 single crystal of figure 3 with those of the polycrystalline zn@xmath8cd@xmath68cr@xmath6o@xmath7 ( @xmath10 ) samples . for @xmath55 @xmath69 k the data for the three compounds can be fitted to the usual curie - weiss law . the linear fit yields to an effective number of bohr magnetons @xmath70 @xmath71 @xmath72 , as expected for cr@xmath18 ( @xmath15 @xmath73 , s @xmath19 ) , and a curie - weiss temperature , @xmath74 @xmath75 k for zncr@xmath6o@xmath76 . in a molecular field approximation @xmath74 @xmath51 s(s+1 ) @xmath77 . for s @xmath19 , @xmath78 @xmath79 ( nearest - neighbors ) , and @xmath74 @xmath80 k we obtain @xmath81 @xmath82 k. the curie - weiss parameters for the zn@xmath8cd@xmath9cr@xmath6o@xmath7 ( @xmath10 ) samples are given in table i. for @xmath83 @xmath69 k figure 4 shows , however , that there is a significant difference between the pure and cd doped compounds . low ( high ) field zfc - fc measurements show , in the cd doped samples , the typical sg irreversibility ( reversibility ) for @xmath84 k ( see inset of figure 4 ) . these results and the large values found for @xmath85 @xmath86 ( see table i ) , clearly indicate that the cd doped samples develop a highly frustrated sg - type behavior with @xmath87 k. figure 5 presents the @xmath0-dependence of the resonance intensity , @xmath88 , for the zncr@xmath6o@xmath7 single crystal and the polycrystalline zn@xmath8cd@xmath68cr@xmath6o@xmath7 ( @xmath10 ) samples . using an epr standard , we found that the intensity of the resonance at room-@xmath0 , @xmath89 k ) , corresponds to the total amount of cr@xmath18 ions present in the samples . similar to the susceptibility data shown in figure 4 , here also we observe two @xmath0-regimes , above and below @xmath90 @xmath69 k. for @xmath83 @xmath69 k @xmath88 shows significant difference between the pure and cd doped compounds . for the cd doped samples we found that @xmath88 and @xmath3 correlate well above @xmath87 k ( not shown ) ; however , for the pure sample this correlation is only observed for @xmath55 @xmath69 k ( see inset in figure 5 ) . figure 6 presents the @xmath0-dependence of the specific heat , @xmath4 , for the zncr@xmath6o@xmath7 single crystal , the polycrystalline zn@xmath8cd@xmath9cr@xmath6o@xmath7 ( @xmath10 ) samples , and the reference compound znga@xmath91o@xmath7 . the inset show the strong effect that the cd impurities have on the afm transition of the pure compound zncr@xmath6o@xmath7 . the large reduction in the peak of the @xmath4 confirms the assignment of sg character for the transition observed at @xmath92 k in the susceptibility data for the cd doped samples . the transition temperatures , @xmath93 and @xmath94 , are in fairly good agreement with those extracted from the susceptibility data ( see inset in figure 4 ) . fields up to @xmath95 t , within the data resolution , did not affect @xmath4 and the afm and sg transitions temperatures . the above epr and magnetic susceptibility results show that the cubic @xmath96 spinel zncr@xmath6o@xmath7 and zn@xmath8cd@xmath9cr@xmath6o@xmath7 ( @xmath41 ) compounds present interesting magnetic behavior between @xmath1 k and @xmath2 k. a high-@xmath0 paramagnetic phase ( htpp ) , @xmath55 @xmath69 k for zncr@xmath6o@xmath7 and @xmath55 @xmath32 k for the cd doped samples , and a low-@xmath0 ordered phase ( ltop ) , @xmath83 @xmath32 k , afm for the pure and sg for the doped compounds . for zncr@xmath6o@xmath7 , a transition between these two regimes is observed in the interval between @xmath32 k and @xmath69 k. our high-@xmath0 epr results are in general agreement with those already reported for polycrystalline samples . @xcite@xmath29@xcite however , the low-@xmath0 epr data for our zncr@xmath6o@xmath7 single crystal are quite different from those reported in ref . 12 . as the temperature decreases in the htpp , the cr@xmath97 magnetic moments experience short range afm correlations . the evidence for it is that , for @xmath55 @xmath98 , the epr resonance shows no @xmath15-shift and a @xmath0-dependence of the line broadening expected for a short range magnetic interaction in afm materials above the nel temperature @xmath94 : @xcite @xmath99^{x } } \label{1}\ ] ] the solid line in fig . 2 shows the fitting of the data to eq . the fitting parameters are : @xmath100 , @xmath94 @xmath62 k , @xmath101 @xmath102 oe , and @xmath103 @xmath104 oe k@xmath105 . we should mention that , for @xmath106 , recent neutron diffraction measurements found a continuous gapless spectrum that was attributed to quantum critical fluctuations of small short range afm correlated domains.@xcite in the htpp of zncr@xmath6o@xmath7 , and for @xmath55 @xmath69 k , @xmath3 and @xmath88 follow the same @xmath0-dependence ( see inset in fig . 5 ) , indicating that all the cr@xmath97 ions that contribute to @xmath3 also participate in @xmath88 . however , for @xmath83 @xmath69 k , @xmath3 deviates from @xmath88 , and they show maximums at @xmath90 @xmath107 k and @xmath90 @xmath69 k , respectively ( see inset in fig . the maximum in @xmath3 is caused by afm correlations and indicates the onset of long range afm ordering . instead , the maximum in @xmath88 can be interpreted as transitions within thermally populated exited states . the observation of epr resonances in exited state levels of nearest - neighbor cr@xmath18 spin - coupled pairs diluted in the spinel znga@xmath6o@xmath7 has been reported by henning et . @xcite these authors were able to determine , from the observed @xmath88 , the energy separation between the first exited triplet state ( s @xmath108 ) and the ground singlet state ( s @xmath109 ) to be @xmath110 @xmath111 k. also , from the optical spectra of the cr@xmath18 spin - coupled pairs in znga@xmath6o@xmath7 a value of @xmath112 @xmath111 k was measured for @xmath113 . @xcite within the same scenario and taking into account the thermal population of all the exited states for the cr@xmath18 spin - coupled pairs ( @xmath114 - @xmath115 ; 3 , 2 , 1 , and 0 ; @xmath116 @xmath19 ) , the expected @xmath0-dependence of the total epr intensity , @xmath88 , in the three exited states levels at energies @xmath117 @xmath118 , and @xmath119 above the singlet ground state is given by:@xcite @xmath120 /z\ ] ] where @xmath121 is the partition function , the coefficients @xmath122 , @xmath123 , and @xmath124 are adjustable parameters proportional to the transition probability within each excited multiplet . the solid line in the inset of the figure 4 shows the @xmath0-dependence given by eq . 2 for @xmath125 , @xmath126 , @xmath127 and @xmath128 @xmath129 k. the value found for @xmath113 is larger than the one ( @xmath112 @xmath111 k ) found for isolated cr@xmath18 spin - coupled pairs in znga@xmath6o@xmath7.@xcite this difference is probably due to the different lattice parameters of zncr@xmath35o@xmath7 ( @xmath130 @xmath131 ) in znga@xmath6o@xmath7 ( @xmath130 @xmath132 ) . @xcite nevertheless , the value is in good agreement with that extracted from the curie - weiss temperature , @xmath74 , ( see above ) . as we pointed out elsewhere @xcite , one can also attribute the difference between the temperature dependence of the susceptibility and the epr intensity to the presence of non resonant low frequency modes that contribute spectral weight to the kramers kronig integral for the static susceptibility , @xmath133 but do not participate in the epr absorption . it is likely that such modes are seen in inelastic neutron scattering above @xmath63@xcite . the fact that the susceptibility and the epr intensity have a common temperature variation in the cd doped samples suggests that the non resonant , low frequency modes , if present , are not making a significant contribution to the susceptibility integral . figure 7 shows the @xmath134 plots obtained from the data of figure 6 for each studied sample . the contribution from the magnetic component is obtained from the difference with the data for the non magnetic reference compound znga@xmath6o@xmath7 . the entropy , @xmath135 , is obtained integrating these differences and gives approximately the multiplicity of the involved levels , @xmath112 @xmath136 . within the same scenario of cr@xmath18 spin - coupled pairs , the schottky anomaly for the spin - coupled pairs is given by : @xmath137\ ] ] figure 8 shows the fitting of the data to eq . the obtained value for @xmath138 @xmath139 k suggests the scheme of levels shown in figure 8 . the value found for @xmath113 is consistent with those values obtained independently from the curie - weiss temperature , @xmath74 , and epr intensity measurements in the htpp ( see above ) . in the ltop our epr experiments show the appearance of small resonance modes ( see figure 1 for @xmath83 @xmath140 k ) . we believe that their sample size and shape dependence and angular variation are , probably , more related to demagnetizing effects rather than to the tetragonal crystal distortion observed at @xmath48 k.@xcite besides , for @xmath24 the magnetic susceptibility increases at higher fields ( see inset of figure 3 ) and also , a small increase is oberved for @xmath83 @xmath141 k ( see figure 3 ) . these behaviors are similar to those observed in magnetization measurements of polycrystalline samples of zncr@xmath6o@xmath7 and they have been attributed to the presence of afm domains in the ltop.@xcite thus , we associate our resonance modes with afm domains that might be present in these materials as a consequence of their highly frustrated 3d long range afm magnetic structure . in conclusion , our epr and @xmath3 results in the @xmath5 spinel zncr@xmath35o@xmath7 show , between @xmath32 k and @xmath69 k , a transition from a long to a short range regime of afm correlations ( ltop - htpp ) . from the @xmath0-dependence of the epr intensity in the htpp an exchange parameter of @xmath81 @xmath142 k between the cr@xmath18 ( s @xmath19 ) spin - coupled pairs was extracted . this value is close to the one obtained independently from the curie - weiss temperature , @xmath74 , and from the schottky anomaly observed in the specific heat . thus , the magnetic properties of these strongly frustrated systems in the htpp can be described within a scenario involving just spin - coupling pairs of cr@xmath18 ( s @xmath19 ) . the sharp drop in @xmath3 at @xmath90 @xmath32 k , the peak in @xmath4 also at @xmath90 @xmath32 k , and the ordering temperature extracted from the broadening of the epr linewidth ( @xmath31 @xmath32 k ) confirmed the afm ordering at @xmath90 @xmath32 k in zncr@xmath6o@xmath7 . the resonance modes observed in the ltop and the field dependent susceptibility , @xmath67 , indicates the presence of afm domains in this material . finally , we found that the disorder caused by the cd impurities in zncr@xmath6o@xmath7 drives the system from an afm to a sg type of highly frustrated magnetic ordering . although a model based on isolated pairs can account for many of the magnetic properties of zncr@xmath64o@xmath65 , it does not include the interaction of the pairs with the surrounding ions . a widely used pair model that does include the effects of the interaction with neighboring spins is the constant coupling approximation @xcite . however , this model does not exploit the unique tetrahedral character of the cr sublattice in zncr@xmath64o@xmath65 . in particular , it predicts the same susceptibility for zncr@xmath64o@xmath65 as found in an unfrustrated simple cubic antiferromagnet , which has the same number of nearest neighbor interactions ( 6 ) and exhibits a conventional afm transition . this contradiction has led two of us ( ajga and dlh ) to develop a quantum tetrahedral mean field model @xcite . in this model , the energy levels of a four spin tetrahedral cluster are calculated exactly and the interaction with the neighboring ions is treated in the mean field approximation . good agreement with the susceptibility and the magnetic specific heat are obtained with nearest neighbor interaction , @xmath143 k , and next neighbor interaction , @xmath144 k@xcite . this work was supported by fapesp grants no 95/4721 - 4 , 96/4625 - 8 , 97/03065 - 1 , and 97/11563 - 1 so paulo - sp - brazil , and nsf - dmr no 9705155 , and nsf - int no 9602928 . ajga wants to thank the spanish mec for financial support under the subprograma general de formacin de personal investigador en el extranjero . f. hartmann - boutron , a. gerard , p. imbert , r. kleibergerard , and f. varret , comptes rendus de l academie des sciences paris , serie ii fascicule b * 268 * , 906 ( 1969 ) ; s .- h . lee , c. broholm , t.h . kim , w. rotcliff , s .- w . cheong , and q. huang , unpublished ; h. martinho , n.o . moreno , j.a . sanjuro , c. rettori , a.j . adeva , d.l . huber , s.b . oseroff , w. ratcliff , ii , s .- w . cheong , p.g . pagliuso , j.l . sarrao , and g.b . martins , paper dh-08 , 8th joint mmm - intermag conference , san antonio , jan .
|
the @xmath0-dependence ( @xmath1- @xmath2 k ) of the electron paramagnetic resonance ( epr ) , magnetic susceptibility , @xmath3 , and specific heat , @xmath4 , of the @xmath5 antiferromagnetic ( afm ) spinel zncr@xmath6o@xmath7 and the spin - glass ( sg ) zn@xmath8cd@xmath9cr@xmath6o@xmath7 ( @xmath10 ) is reported .
these systems behave as a strongly frustrated afm and sg with @xmath11 @xmath12 k and @xmath13 k @xmath14 k. at high-@xmath0 the epr intensity follows the @xmath3 and the @xmath15-value is @xmath0-independent .
the linewidth broadens as the temperature is lowered , suggesting the existence of short range afm correlations in the paramagnetic phase . for zncr@xmath6o@xmath7 the epr intensity and @xmath3 decreases below @xmath16 k and @xmath17 k , respectively .
these results are discussed in terms of nearest - neighbor cr@xmath18 ( s @xmath19 ) spin - coupled pairs with an exchange coupling of @xmath20 @xmath17 k. the appearance of small resonance modes for @xmath21 k , the observation of a sharp drop in @xmath3 and a strong peak in @xmath4 at @xmath22 k confirms , as previously reported , the existence of long range afm correlations in the low-@xmath0 phase .
a comparison with recent neutron diffraction experiments that found a near dispersionless excitation at @xmath23 mev for @xmath24 and a continuous gapless spectrum for @xmath25 , is also given .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the baryon asymmetry of the universe ( bau ) is one of the intriguing puzzles in our understanding of the universe . in order to generate the bau , three conditions , which have been pointed out by sakharov @xcite , need to be satisfied in the early universe : baryon number ( @xmath3 ) violation , @xmath4/@xmath1 violation and departure from equilibrium . the @xmath3-violation is actually replaced by @xmath5-violation since the @xmath5-conserving and @xmath6-violating sphaleron process can easily transfer @xmath3 number into @xmath7 number and _ vice versa _ before the electroweak phase transition @xcite . while several ideas have been proposed to produce bau by using @xmath3-violating interactions , leptogenesis provides a different mechanism for generating bau by using @xmath7-violating interactions @xcite . in such a case , instead of a baryon asymmetry , a lepton asymmetry is generated in the early universe via @xmath7-violating interactions . once the net lepton number is generated , it is transferred into the baryon number via the sphaleron process . such scenarios are naturally realized in seesaw models for neutrino mass generation . the @xmath7-violating and @xmath4/@xmath1-violating interactions are introduced for majorana mass terms and their yukawa couplings of right - handed neutrinos ( rhns ) . one of the most beautiful scenarios is the thermal leptogenesis that produces lepton asymmetry from thermally produced rhns . the thermal plasma after reheating produces a number of rhns . when the produced rhns are out of equilibrium , they decay asymmetrically into leptons vs. anti - leptons and thus generate net lepton number . it has been known that in this case , a large reheating temperature @xmath8 gev is required to obtain enough baryon - number - to - entropy ratio @xmath9 @xcite . if supersymmetry ( susy ) exists , however , one must take care of gravitino production for such a high reheating temperature @xcite . the affleck - dine ( ad ) mechanism @xcite is an attractive baryogenesis scenario in susy models that does not necessarily require high reheating temperature and hence the tension with gravitino production is circumvented.gev . ] in susy models , it is quite generic to have many flat directions , so the baryon asymmetry can be efficiently generated along such flat directions . in particular , if one considers the @xmath0 flat direction , whose non - renormalizable potential is generated by integrating out rhns , sufficient lepton asymmetry can be produced at relatively small reheat temperature . in this case , however , one difficulty arises . for the desired baryon asymmetry , the @xmath0 direction must be very flat during lepton number generation , which results in a hierarchically light neutrino @xcite . while this is not excluded since currently there is no lower bound on the lightest neutrino mass , such a hierarchical structure of neutrino masses may not be natural from the viewpoint of model building . one reason for a hierarchically small neutrino mass in the ad mechanism stems from the fact that the rhn mass scale is responsible for both leptogenesis and neutrino mass generation . thus the flatness of the @xmath0 direction is directly related to the lightness of neutrino . one way to relax this constraint is to make the rhn mass dynamical . if the rhn mass scale is very large during lepton number generation but become small during the current universe , the lightest neutrino can have a rather large mass . as pointed out in @xcite , such a dynamical rhn mass can be realized in a peccei - quinn ( pq ) symmetric model to solve the strong cp problem @xcite . if the rhns are charged under pq symmetry , their masses are generated by the pq symmetry breaking @xcite . thus , the rhn mass is time - dependent and it is determined by the dynamics of the pq field . during inflation , the hubble - induced susy breaking potential holds the pq field at the planck scale , and thus pq symmetry is broken at the planck scale . after inflation and lepton number generation , the hubble - induced susy breaking effect becomes as small as ordinary susy breaking contributions , so the pq field starts to oscillate and settles down at the current value of the pq breaking scale . the rhns also have planck scale masses from the inflation era to the lepton number generation era , and become lighter afterwards . therefore , it is possible to simultaneously accommodate very efficient ad leptogenesis when the pq scale is around planck scale and rather larger neutrino mass in the current universe . in this work , we revisit the scenario of ad leptogenesis with a varying pq scale , and provide its concrete realization in the dine - fischler - srednicki - zhitnitsky ( dfsz ) type model @xcite . in this model , the pq breaking scalar fields provide a solution to the strong cp problem , generate rhn masses and the superpotential @xmath2-term for the higgsino sector . in addition , the superpotential term responsible for the @xmath2-term plays an important role to maintain the generated lepton asymmetry during pq field oscillation . the isocurvature contribution from axion dark matter is naturally suppressed by the large pq scale during inflation . in this case , one additional crucial feature resides in the pq field decay . as stated above , once the hubble - induced susy breaking effect becomes comparable to the ordinary susy breaking terms , the pq breaking field , which is called saxion , starts to oscillate with respect to the current pq scale . the saxion dominates the universe right after the reheating process , so its decay produces large amount of entropy . hence the final baryon asymmetry is sensitive to the saxion decay , which depends on how the pq sector couples to the standard model particles . in the dfsz case , the saxion decay is dependent on the @xmath2-term . therefore the resulting baryon asymmetry after saxion decay is linked to electroweak fine - tuning , which is determined by the @xmath2-term . in sec . [ sec : ad ] , we analyze ad leptogenesis along the @xmath0 flat direction with a dynamical pq breaking scale and then estimate the baryon asymmetry taking account of the dilution from saxion decay . in sec . [ sec : pq_dyn ] , we investigate the dynamics of pq breaking scalar fields and then examine conservation of lepton asymmetry during the saxion oscillation . in sec . [ sec : susy_scale ] we present the main results in the form of contours of required @xmath10 values in the @xmath2 vs. @xmath11 plane , which show a relation between the baryon asymmetry and the electroweak fine - tuning . in sec . [ sec : pq ] , we discuss some cosmological implications of the pq sector : the axion isocurvature perturbation and axino production . we conclude in sec . [ sec : conc ] . in this section , we show how dynamical pq symmetry breaking accommodates enough baryon asymmetry and a relatively large neutrino mass . we consider the neutrino sector for ad leptogenesis and the pq breaking sector , which are described by the following superpotential , @xmath12 where @xmath13 is the rhn , @xmath7 is the lepton doublet , @xmath14 ( @xmath15 ) is the up - type ( down - type ) higgs doublet , @xmath16 and @xmath17 are the pq fields , @xmath18 is a singlet scalar , @xmath19 , @xmath20 and @xmath21 represent numerical coefficients , @xmath22 denotes the ( present ) pq breaking scale and @xmath23 the reduced planck scale . the pq charges and lepton numbers of these fields are given in table [ table : pqcharge ] . .u(1 ) charges of the fields . [ cols="^,^,^,^,^,^,^,^",options="header " , ] when @xmath16 obtains a large field value , the rhn becomes massive and can be integrated out to obtain the effective superpotential : @xmath24 the neutrino mass is generated by the see - saw mechanism at low energy @xmath25 : @xmath26 where @xmath27 gev and @xmath28 here and in the following , we assume that the lepton asymmetry is generated along the flattest @xmath0 direction , which corresponds to the smallest neutrino mass @xmath29 . as will become clear , @xmath16 is different from its low - energy vacuum expectation value ( vev ) @xmath22 in the early universe . at the present universe . ] in particular , @xmath16 is shown to be fixed at @xmath30 until the saxion begins to oscillate ( see sec . [ sec : pq_dyn ] ) . thus one can also define an effective scale in the early universe as @xmath31 in a different way , @xmath32 is expressed by @xmath33 thus @xmath32 is much larger than @xmath34 , which modifies the ordinary relation between low - energy neutrino mass and the efficiency of ad leptogenesis @xcite . in what follows , we consider the dynamics of the ad field @xmath35 , which parameterizes the @xmath0 @xmath36-flat direction as @xmath37 the scalar potential of the ad field in the model of eq . [ eq : lagpq ] reads @xmath38 , \\ & v_h = -c_h h^2|\phi|^2 + \left [ a_h h \frac{y_{\nu}^2}{8\lambda}\frac{\phi^4}{x}+{\rm h.c.}\right],\end{aligned}\ ] ] where @xmath39 denotes the contribution from the soft susy breaking and @xmath40 the hubble - induced terms . here @xmath41 and @xmath42 are of the soft mass scale ( @xmath43 few tev ) , @xmath44 is the hubble parameter , @xmath45 , @xmath46 and @xmath47 are @xmath48 coefficients . term . in this paper we simply assume that there is a hubble - induced @xmath49 term . ] for the moment , we assume that the pq field @xmath16 is fixed at @xmath50 until the ad field begins to oscillate . we examine the dynamics including the pq field in sec . [ sec : pq_dyn ] . from the hubble induced mass and @xmath51 , the ad field is stabilized at @xmath52 for @xmath53 . writing @xmath54 one finds @xmath55 where @xmath56 . the mass along the phase direction is then given by @xmath57 where we have used the fact that @xmath58 . the mass along the phase direction is of order @xmath44 , so the phase condensate rapidly rolls down to its minimum . since the phase minimum of this hubble - induced @xmath49 term differs from the soft susy breaking - induced @xmath49 term , the ad field obtains angular momentum in the complex plane , generating the lepton number . note that we have one more phase direction orthogonal to the above massive direction . this does not appear in the potential , so is a massless mode corresponding to the axion of the spontaneously broken pq symmetry . in the limit of @xmath58 , the massive mode is mostly @xmath59-like and the massless mode is mostly @xmath60-like . now let us evaluate the lepton number generated through the ad mechanism . the massive phase mode automatically cancels the imaginary part of the hubble induced @xmath49-term potential , so it does not significantly contribute to lepton number generation . thus , as in ordinary ad leptogenesis , lepton number is determined by the ordinary @xmath49-term which depends on the hidden sector susy breaking ( _ i.e. _ , gravitino mass in gravity mediation ) . the lepton number obeys the equation , term in the superpotential after integrating out @xmath13 , the pq number is exactly conserved ( except for the small instanton effect ) . ] @xmath61 the baryon number to entropy ratio is obtained as @xcite @xmath62 where @xmath63 represents an effective cp violating phase , and @xmath64 is the hubble parameter when the @xmath35 field starts to oscillate . taking account of thermal effects on the ad potential @xcite , the latter is determined by @xmath65 , \label{eq : hosc}\ ] ] where @xmath66.\ ] ] here , the @xmath67 are coupling constants of @xmath35 with particles in thermal background , @xmath68 and @xmath69 are real positive constants of order unity , and @xmath70 is the reheating temperature after inflation . from eq . ( [ eq : hosc ] ) , @xmath64 is nearly @xmath71 when the temperature is small . in the case of a large reheat temperature , the @xmath35 oscillation can commence earlier due to thermal effects which result in the second and third terms inside the bracket of eq . ( [ eq : hosc ] ) . the detailed physical aspects are explained in ref . @xcite and references therein . if @xmath72 tev and @xmath73 gev , such early oscillation occurs for @xmath74 gev . for an illustration , we show a formula for a case where the early oscillation does not occur . in such a case , @xmath75 , so the baryon - number - to - entropy ratio becomes @xmath76 where we also assume @xmath77 . in this scenario , however , saxion will dominate the universe , and its decay produces entropy dilution . in order to obtain the final baryon asymmetry after saxion decay , the entropy dilution must be taken into account . we will consider the entropy production in the following subsection . before closing this subsection , let us comment on the possible lepton number violation during the saxion oscillation . since the field value of @xmath16 can become small during its oscillation , the effect of the lepton number violation , induced by the effective superpotential or the corresponding @xmath49-term , may become large . here , as shown in sec . [ sec : pq_dyn ] , the @xmath2-term interaction in eq . , @xmath78 plays an important role . assuming @xmath79 , the lepton number violation during the saxion oscillation is small enough to maintain the generated lepton asymmetry by the ad mechanism . as we shall see in the next subsection , the @xmath2-term interaction also plays a key role to determine the saxion decay , and hence the final baryon asymmetry . we have discussed how the dynamical pq breaking scale can enhance the baryon asymmetry . for the final result , one crucial point to consider is the entropy production from saxion decay . in the dfsz model , saxion interactions with the standard model particles and their superpartners are realized in the @xmath2-term interaction . once @xmath16 and @xmath17 settle down to the current value of the pq symmetry breaking scale , @xmath80 , this superpotential generates the @xmath2-term , @xmath81 and also interactions between the axion superfield and the higgs supermultiplets . through this interaction , the saxion dominantly decays into higgsino states if they are kinematically allowed . its decay rate is approximately given by @xcite , we use the saxion decay rates including phase space and mixings in @xcite . ] @xmath82 note that we have used here @xmath83 under assumption of @xmath84 in the present universe , so quantities related to axion dark matter is determined by @xmath85 ( @xmath86 : domain wall number ) as the usual normalization . the decay temperature is @xmath87 if saxion decays into higgsino states are disallowed , it dominantly decays into the light higgs and gauge bosons . the decay rate in such a case is given by @xmath88 and the decay temperature becomes @xmath89 from the above decay temperature for each case , one finds the entropy dilution factor @xmath90 . \label{eq : dilut}\ ] ] here we have included the case where @xmath70 is small so that the saxion decays before the reheating process is over . the final baryon asymmetry is determined by the amount of asymmetry when the ad mechanism completes , eq . ( [ eq : bary - to - ent_wo_dilut ] ) and by the dilution factor , eq . ( [ eq : dilut ] ) : @xmath91 for the case where saxion dominantly decays into higgsinos , @xmath92 or for the case where saxion dominantly decays into light higgs and gauge bosons , @xmath93 from this it is easily seen that the observed baryon - number - to - entropy - ratio can be obtained for relatively large neutrino mass @xmath94 ev if the pq breaking scale is near the planck scale in the beginning and settles to @xmath95 gev at the present universe . we will see numerical results for some example parameter regions in sec . [ sec : susy_scale ] . in this section , we discuss the dynamics of the pq breaking fields in order to investigate the realization of the planck scale pq breaking in the early stage and lepton number conservation at the late stage . let us first examine the scalar potential of @xmath16 and @xmath17 for large @xmath44 to check if the pq scale is @xmath96 . we have to consider the supergravity potential which is given by @xmath97 where @xmath98 . we assume that the effect of the ad field @xmath35 is negligible . the khler potential and superpotential are given by @xmath99 where @xmath100 is the inflaton field . note that only @xmath16 has non - minimal coupling with the inflaton in @xmath101 . if @xmath102 , one can obtain a negative hubble - induced mass term for @xmath16 and thus @xmath16 develops a large vev . let us see the scalar potential in detail . in this discussion , @xmath18 obtains a mass of @xmath103 and @xmath104 ( @xmath105 ) , so its vev is zero up to of the order of the gravitino mass : @xmath106 . thus we can safely neglect the dynamics of @xmath18 . if @xmath107 during inflation , the inflaton energy is dominantly determined by the @xmath108-term potential , _ i.e. _ @xmath109 , @xmath110 and @xmath111 . one can simplify the inflaton potential : @xmath112 . in these circumstances , the scalar potential of @xmath16 and @xmath17 is given by @xmath113 let us define @xmath114 and @xmath115 . the extremum condition is obtained as @xmath116\nonumber\\ & & \times e^{(x^2+y^2)/m_p^2},\label{eq : dvdx}\\ 0=\frac{\partial v}{\partial y}&= & \eta^2(xy - f^2)\left(\frac{y}{m_p^2}(xy - f^2)+2x\right)e^{(x^2+y^2)/m_p^2}.\end{aligned}\ ] ] from the second equation , we find that @xmath117 since the second solution leads to a trivial solution @xmath118 we select the first solution . from eq . ( [ eq : dvdx ] ) , we obtain the solution for @xmath119 , @xmath120 for @xmath102 , @xmath119 develops an @xmath96 vev as long as @xmath121 . it is also evident that @xmath16 obtains a ( negative ) mass squared of the order of @xmath122 during the inflaton domination . as argued in the previous subsection , @xmath16 stays at @xmath123 until the hubble parameter drops down to @xmath124 . after that , the saxion begins a coherent oscillation around the minimum @xmath125 with an initial amplitude of @xmath126 . since the scalar field orthogonal to the @xmath108-flat direction ( saxion ) has a mass of @xmath127 , which is much higher than the soft mass scale , we can safely set @xmath128 to integrate out either @xmath16 or @xmath17 . then the scalar potential along the @xmath108-flat direction @xmath129 and the ad field @xmath35 reads @xmath130 where @xmath131 and @xmath132 are the soft susy breaking mass of @xmath16 and @xmath17 , respectively . the last term comes from @xmath133 ( [ eq : mu ] ) . the third and fourth terms act as the effective potential for @xmath16 and they prevent @xmath16 from being very small during the oscillation . let us denote by @xmath134 the maximum value of @xmath16 during each @xmath16 oscillation , which adiabatically becomes smaller due to the hubble expansion @xmath135 . then we can define @xmath136 , the minimum value of @xmath16 during each oscillation . for a large ad field value @xmath35 , the last term is important to determine @xmath136 . thus we can evaluate @xmath136 as @xmath137 . \label{eq : xmin}\end{aligned}\ ] ] since @xmath138 and @xmath139 just after the saxion oscillation , @xmath136 is generically much larger than the soft mass scale , meaning that the rhn masses can not be as small as the soft mass during saxion oscillation and hence the procedure to integrate out the rhn to obtain the effective potential of the ad field is justified . now let us consider the lepton number violation after the @xmath16 begins to oscillate . the lepton number follows : @xmath140 as discussed above , @xmath16 oscillates between @xmath134 and @xmath136 in a time scale @xmath141 where @xmath136 is given by eq . . the most dangerous @xmath7 violation may happen around @xmath142 at which the @xmath7-violating operator becomes large . the time interval @xmath143 during which @xmath144 is estimated from the equation of motion @xmath145 during this time interval , the @xmath7 number changes as @xmath146 using @xmath147 , we obtain @xmath148 since @xmath149 in the numerator decreases faster than @xmath134 in the denominator , this takes a maximum value just after the @xmath16 begins to oscillate @xmath150 . @xmath151 this must be smaller than 1 to ensure the conservation of lepton number . if thermal effects are neglected and @xmath152 , we have @xmath153 and it becomes @xmath154 thus , the lepton number violation during the saxion oscillation can be neglected as far as @xmath152 is satisfied . we have discussed how pq symmetry breaking accommodates the baryon asymmetry with a sizable neutrino mass when the pq scale varies during and after inflation . in this scenario , the entropy dilution from saxion decay indeed plays a substantial role for determining the final value of the baryon asymmetry . the saxion decay rate depends on its mass and the @xmath2-term as shown in eqs . ( [ eq : saxdec1 ] ) and ( [ eq : saxdec2 ] ) . in many cases , the saxion mass and @xmath2-term are related to the soft susy breaking scale . in particular , @xmath2-term is a measure of fine - tuning of the electroweak symmetry breaking . therefore , it leads us to discuss the soft susy scale and fine - tuning from the measured baryon asymmetry . since the saxion is linked to the axion which is the nambu - goldstone boson of broken pq symmetry , it is massless in the supersymmetric limit . when susy is broken , however , the saxion ( and also the axino ) acquires a mass . the saxion mass is typically of the gravitino mass order although it can be either larger or smaller than the gravitino mass in some models @xcite . on the other hand , as shown in the sec . [ sec : pq_dyn]b , the saxion mass ( _ i.e. _ @xmath131 ) is required to be smaller than the ad field mass in order to not spoil lepton number generation . in this regard , we consider a rather small saxion mass compared to the ad field mass , _ i.e. _ @xmath155 . in contrast to the saxion mass , the @xmath2-term is a supersymmetric parameter , so its origin can be different from the susy breaking . in models with pq symmetry breaking , the @xmath2-term can be generated from pq symmetry breaking through non - renormalizable interactions @xcite .- term generated by pq symmetry breaking is related to the soft susy scale . in this work , however , we are agnostic as to the origin of the pq scale @xmath22 in eq . ( [ eq : lagpq ] ) . ] in sec . [ sec : sax_dec ] , we have discussed the @xmath2-term generation via an interaction suppressed by the planck scale as shown in eq . ( [ eq : mu ] ) . in such a case , the @xmath2-term is typically @xmath156 . however , the suppression scale for this interaction can be different from the planck scale ( _ e.g. _ the grand unification scale ) , while the coupling constant ( @xmath21 ) for this interaction can be smaller than unity . for this reason , we will consider @xmath2 as an independent parameter of the model in the following discussions . in order to achieve successful electroweak symmetry breaking , the soft susy breaking scale and @xmath2-term must coincide with each other since they need to satisfy the relation , @xmath157 where @xmath158 is the soft mass term for the up - type higgs at the weak scale . for a natural model , these three quantities above need to be comparable to one another so that no dramatic cancellation takes place . if @xmath158 and @xmath2 are much larger than @xmath159 , on the other hand , fine - tuning arises . although it is hard to quantify the level of fine - tuning without specifying the whole susy spectrum , we can roughly see how much fine - tuning is required from the size of @xmath2-term ( or equivalently @xmath158 ) @xcite : @xmath160 the baryon asymmetry depends on the soft susy breaking scale when the ad mechanism works as described in eq . ( [ eq : bary - to - ent_wo_dilut ] ) . it is also dependent on the saxion decay rate which is determined by the saxion mass ( soft susy scale ) , the @xmath2-term and the pq breaking scale as shown in eqs . ( [ eq : bau_large ] ) and ( [ eq : bau_small ] ) . therefore , by requiring @xmath161 , we can obtain _ the relation between the lightest neutrino mass and @xmath2-term_. [ ev ] on the @xmath162 plane to reproduce the observed baryon asymmetry . in the left ( right ) panel we have taken @xmath163 @xmath164 while @xmath165 ( @xmath166 ) . the light ( dark ) red shaded region corresponds to @xmath167gev ( 1gev ) and the grey shaded region is excluded by the kamland - zen experiment . the light - gray shaded region is constrained by planck+bao @xcite . the light - purple shaded region indicates bound from sn1987a @xcite . the blue shaded region corresponds to @xmath168 . , title="fig:",width=302][ev ] on the @xmath162 plane to reproduce the observed baryon asymmetry . in the left ( right ) panel we have taken @xmath163 @xmath164 while @xmath165 ( @xmath166 ) . the light ( dark ) red shaded region corresponds to @xmath167gev ( 1gev ) and the grey shaded region is excluded by the kamland - zen experiment . the light - gray shaded region is constrained by planck+bao @xcite . the light - purple shaded region indicates bound from sn1987a @xcite . the blue shaded region corresponds to @xmath168 . , title="fig:",width=302 ] fig . [ fig : fig1 ] shows illustrative contours of neutrino masses which produce the desired baryon asymmetry , @xmath169 for given values of @xmath11 and @xmath2 . in the left panel , we take @xmath170 for which saxion decays into higgsino states are allowed while in the right panel , we take @xmath171 for which saxion can decay only into sm particles . in order to maintain the generated lepton asymmetry , @xmath165 and @xmath166 are taken respectively for each case . the gray shaded region shows parameter space where the lightest neutrino mass is larger than the kamland - zen bound @xcite . we also show a bound from planck+bao constraint on the sum of neutrino masses , @xmath172 ev @xcite . the light - purple shaded region shows the bound from sn1987a @xcite . the ( light-)red shaded region shows parameter space where the saxion decay temperature is smaller than 1 gev ( 10 gev ) . the blue shade indicates the region for which @xmath173 . we consider fixed @xmath174 gev since larger @xmath70 does not change or does suppress @xmath175 ( see eq . ( [ eq : nbs_final ] ) ) . in the case where @xmath175 is suppressed , it requires a smaller neutrino mass that is less attractive . from the figure , it is clearly shown that neutrino mass is large for large @xmath2 and small @xmath11 while it becomes smaller for small @xmath2 and large @xmath11 . this feature stems from the saxion decay temperature . the saxion decay temperature is enhanced by the @xmath2-term while suppressed by @xmath11 . it is also of great importance that small @xmath11 is good for obtaining a flatter direction during lepton number generation as shown in eq . ( [ eq : m _ * ] ) . for @xmath176 tev and @xmath177 gev , our model predicts a rather large neutrino mass so that it is constrained by recent neutrinoless double beta decay ( @xmath178 ) experiment . for this constraint , we take a conservative bound from kamland - zen , @xmath179 ev @xcite . from the lower - right corner ( @xmath180 gev , @xmath181 gev ) to the upper - left corner ( @xmath182 gev , @xmath183 gev ) , the resulting neutrino mass scans over @xmath184 ev . for large @xmath185 gev , in order to obtain a natural value of the lightest neutrino mass , @xmath186 ev in cases of both @xmath170 and @xmath171 , the @xmath2-term is required to be tens of tev . in such cases , the fine - tuning in the electroweak symmetry breaking is of permyriad ( @xmath187 ) order . for a smaller pq scale , @xmath181 gev , @xmath2 can be a few hundred gev to achieve @xmath188 ev , so the model is much less fine - tuned . if we constrain the @xmath2-term to make fine - tuning better than a percent level , _ @xmath189 gev@xmath190 gev , @xmath191 ev@xmath192 ev can be achieved for @xmath193 gev@xmath194 gev . this region is well - matched with the parameter space where @xmath2 can be determined by the planck suppressed interaction , eq . ( [ eq : mu ] ) . moreover , it is good for the _ mixed axion - higgsino dark matter scenario _ @xcite . before closing this section , it is worth noting that the saxion decay temperature can be smaller than @xmath195 gev for small @xmath2 and large @xmath11 as indicated by red shaded regions in fig . [ fig : fig1 ] . the region with @xmath196 gev may cause the saxion to decay after freeze - out of weakly interacting massive particle ( wimp ) dark matter and thus such late decay affects the wimp dark matter density due to entropy production and/or non - thermal dark matter production , although it is not possible to make a concrete analysis without a specific susy spectrum . moreover , for the region with @xmath197 gev , the saxion decays after coherent oscillation of axion commences , so it affects the axion dark matter density , too . in this section we discuss some cosmological implications related to the pq sector : axion isocurvature perturbation and axino production . there is a ( nearly ) massless goldstone boson from the spontaneous breakdown of the u(1)@xmath198 symmetry , the axion , which can also be interpreted as a massless majoron in our model @xcite . such a massless boson potentially causes several cosmological problems @xcite . in our setup , the pq symmetry is already broken during inflation and is not restored thereafter . thus there is no axionic domain wall problem . the pq scalar @xmath16 obtains a large vev of @xmath126 in our setup , hence the pq scale during inflation is much higher than that in the present universe . it significantly suppresses the axion isocurvature perturbation @xcite . since the massless axion mode almost consists of the phase component of @xmath16 for @xmath199 , the effective pq scale during inflation is simply given by @xmath200 . or @xmath15 has a larger field value than @xmath16 , the effective pq scale is given by @xmath201 or @xmath202 @xcite . this is not the case in our model . ] the magnitude of cdm isocurvature perturbation is then given by @xmath203 where @xmath204 denotes the hubble scale during inflation , @xmath205 denotes the initial misalignment angle of the axion and @xmath206 denotes the fraction of present axion energy density in the matter energy density : @xmath207 . the final axion density is given by @xcite @xmath208 here we have assumed that there is no dilution of the axion density due to the saxion decay . the planck constraint on the uncorrelated isocurvature perturbation @xcite reads @xmath209 this constraint is easily satisfied for most inflation models since @xmath210 in our scenario . the axino is the fermionic superpartner of the axion , consisting of the fermionic components of @xmath16 and @xmath17 with a small mixture of higgsino . it obtains a mass of @xmath211 . it has a relatively long lifetime if it is not the lightest susy particle ( lsp ) @xcite . its decay width is approximately given by @xmath212 which is comparable to the saxion . thus the axino can have significant impacts on cosmology . the dominant axino production process is the thermal one .. ] the axino thermal production in the dfsz model comes from the combination of higgsino decay / inverse decay , scatterings of higgs and weak gauge bosons and also top / stop scatterings @xcite . there it was found that the production is dominated at @xmath213 in general and the abundance is independent of the reheating temperature @xmath70 as long as @xmath214 . in our case , the saxion decay temperature @xmath215 can be lower than @xmath42 and hence there is a dilution of the preexisting axino abundance . first , let us consider the case @xmath216 . one of the main contributions may be the heavy higgs decay into the higgsino plus axino with the partial decay rate @xmath217 where the heavy higgs mass is assumed to be @xmath218 . the axino abundance is then estimated as @xmath219 in the opposite case @xmath220 , we must take the dilution factor into account . noting that @xmath221 during saxion domination , the resultant axino abundance is given by @xmath222 due to the dilution , the axino abundance can be suppressed . when the axino decays into lsp ( _ e.g. _ neutralinos ) , the lsp density is determined by its re - annihilation rate @xcite , @xmath223 where @xmath224 is the axino decay temperature . thus the dark matter density from axino decay highly depends on details of the axino decay as well as upon its annihilation rate at @xmath224 . in this paper we have reconsidered the ad leptogenesis in a scenario where the rhn mass is dynamical . if the rhn mass is generated by the pq field , it naturally takes hierarchically different values between the early universe and the present epoch . in particular , the pq scalar can be stabilized at the planck scale in the early universe until the lepton asymmetry is generated , which makes leptogenesis much more efficient than in the ordinary scenario . the predicted lightest neutrino mass to reproduce the observed baryon asymmetry can be close to the neutrino mass differences known from the neutrino oscillation data . it significantly relaxes the problem of the ordinary ad leptogenesis scenario in which the lightest neutrino mass should be hierarchically smaller than the other two neutrinos . in order to realize this scenario , we have considered the dfsz model which provide a solution to the strong @xmath1 problem and generates the @xmath2-term and the rhn mass . since the final baryon asymmetry depends on the saxion decay , it is related to the @xmath2-term and to the electroweak fine - tuning . as a result , a comparatively large mass of the lightest neutrino , @xmath225 ev , is predicted for @xmath193 gev@xmath194 gev and @xmath189 gev@xmath190 gev . in this model , therefore , it is possible to accommodate successful ad leptogenesis with a natural neutrino mass and natural electroweak symmetry breaking . in addition , due to the very large pq scale during inflation , the axion isocurvature perturbation is suppressed . this work is supported by ibs under the project code , ibs - r018-d1 [ kjb ] and the grant - in - aid for scientific research on scientific research a ( no.26247038 [ kh ] , no.26247042 [ kn ] , no.16h02189 [ kh ] ) , young scientists b ( no.26800121 [ kn ] , no.26800123 [ kh ] ) and innovative areas ( no.26104001 [ kh ] , no.26104009 [ kh and kn ] , no.15h05888 [ kn ] ) , and by world premier international research center initiative ( wpi initiative ) , mext , japan . the work of hb is supported in part by the us department of energy , office of high energy physics . a. d. sakharov , pisma zh . * 5 * , 32 ( 1967 ) [ jetp lett . * 5 * , 24 ( 1967 ) ] [ sov . usp . * 34 * , 392 ( 1991 ) ] [ usp . nauk * 161 * , 61 ( 1991 ) ] . doi:10.1070/pu1991v034n05abeh002497 v. a. kuzmin , v. a. rubakov and m. e. shaposhnikov , phys . lett . b * 155 * , 36 ( 1985 ) . doi:10.1016/0370 - 2693(85)91028 - 7 m. fukugita and t. yanagida , phys . lett . b * 174 * , 45 ( 1986 ) . doi:10.1016/0370 - 2693(86)91126 - 3 w. buchmuller , r. d. peccei and t. yanagida , ann . rev . nucl . part . sci . * 55 * , 311 ( 2005 ) doi:10.1146/annurev.nucl.55.090704.151558 [ hep - ph/0502169 ] . m. kawasaki , k. kohri , t. moroi and a. yotsuyanagi , phys . d * 78 * ( 2008 ) 065011 doi:10.1103/physrevd.78.065011 [ arxiv:0804.3745 [ hep - ph ] ] . i. affleck and m. dine , nucl . b * 249 * , 361 ( 1985 ) . doi:10.1016/0550 - 3213(85)90021 - 5 m. dine , l. randall and s. d. thomas , nucl . b * 458 * , 291 ( 1996 ) doi:10.1016/0550 - 3213(95)00538 - 2 [ hep - ph/9507453 ] . g. lazarides and q. shafi , phys . lett . b * 258 * , 305 ( 1991 ) . doi:10.1016/0370 - 2693(91)91090-i t. asaka , k. hamaguchi , m. kawasaki and t. yanagida , phys . b * 464 * , 12 ( 1999 ) doi:10.1016/s0370 - 2693(99)01020 - 5 [ hep - ph/9906366 ] ; phys . d * 61 * , 083512 ( 2000 ) doi:10.1103/physrevd.61.083512 [ hep - ph/9907559 ] . k. hamaguchi , h. murayama and t. yanagida , phys . d * 65 * , 043512 ( 2002 ) doi:10.1103/physrevd.65.043512 [ hep - ph/0109030 ] . t. asaka , m. fujii , k. hamaguchi and t. yanagida , phys . d * 62 * , 123514 ( 2000 ) doi:10.1103/physrevd.62.123514 [ hep - ph/0008041 ] . m. fujii , k. hamaguchi and t. yanagida , phys . rev . d * 63 * , 123513 ( 2001 ) doi:10.1103/physrevd.63.123513 [ hep - ph/0102187 ] . r. d. peccei and h. r. quinn , phys . lett . * 38 * , 1440 ( 1977 ) . doi:10.1103/physrevlett.38.1440 p. langacker , r. d. peccei and t. yanagida , mod . lett . a * 1 * , 541 ( 1986 ) . doi:10.1142/s0217732386000683 m. dine , w. fischler and m. srednicki , phys . b * 104 * , 199 ( 1981 ) ; a. r. zhitnitsky , sov . . phys . * 31 * , 260 ( 1980 ) [ yad . fiz . * 31 * , 497 ( 1980 ) ] . s. kasuya , m. kawasaki and f. takahashi , jcap * 0810 * , 017 ( 2008 ) doi:10.1088/1475 - 7516/2008/10/017 [ arxiv:0805.4245 [ hep - ph ] ] . r. allahverdi , b. a. campbell and j. r. ellis , nucl . b * 579 * , 355 ( 2000 ) doi:10.1016/s0550 - 3213(00)00124 - 3 [ hep - ph/0001122 ] . a. anisimov and m. dine , nucl . b * 619 * , 729 ( 2001 ) doi:10.1016/s0550 - 3213(01)00550 - 8 [ hep - ph/0008058 ] . k. hamaguchi , hep - ph/0212305 . t. goto and m. yamaguchi , phys . b * 276 * , 103 ( 1992 ) . doi:10.1016/0370 - 2693(92)90547-h e. j. chun , j. e. kim and h. p. nilles , phys . b * 287 * , 123 ( 1992 ) doi:10.1016/0370 - 2693(92)91886-e [ hep - ph/9205229 ] . e. j. chun and a. lukas , phys . b * 357 * , 43 ( 1995 ) doi:10.1016/0370 - 2693(95)00881-k [ hep - ph/9503233 ] . k. j. bae , h. baer , e. j. chun and c. s. shin , phys . d * 91 * , no . 7 , 075011 ( 2015 ) doi:10.1103/physrevd.91.075011 [ arxiv:1410.3857 [ hep - ph ] ] . j. e. kim and h. p. nilles , phys . * 138b * , 150 ( 1984 ) . doi:10.1016/0370 - 2693(84)91890 - 2 h. murayama , h. suzuki and t. yanagida , phys . lett . b * 291 * , 418 ( 1992 ) . doi:10.1016/0370 - 2693(92)91397-r t. gherghetta and g. l. kane , phys . b * 354 * , 300 ( 1995 ) doi:10.1016/0370 - 2693(95)00620-z [ hep - ph/9504420 ] . k. choi , e. j. chun and j. e. kim , phys . b * 403 * , 209 ( 1997 ) doi:10.1016/s0370 - 2693(97)00465 - 6 [ hep - ph/9608222 ] . n. abe , t. moroi and m. yamaguchi , jhep * 0201 * , 010 ( 2002 ) doi:10.1088/1126 - 6708/2002/01/010 [ hep - ph/0111155 ] . k. nakayama and n. yokozaki , jhep * 1211 * , 158 ( 2012 ) doi:10.1007/jhep11(2012)158 [ arxiv:1204.5420 [ hep - ph ] ] . k. j. bae , h. baer and h. serce , phys . d * 91 * , no . 1 , 015003 ( 2015 ) doi:10.1103/physrevd.91.015003 [ arxiv:1410.7500 [ hep - ph ] ] . h. baer , v. barger , p. huang , a. mustafayev and x. tata , phys . lett . * 109 * , 161802 ( 2012 ) doi:10.1103/physrevlett.109.161802 [ arxiv:1207.3343 [ hep - ph ] ] . a. gando _ et al . _ [ kamland - zen collaboration ] , phys . lett . * 117 * , no . 8 , 082503 ( 2016 ) addendum : [ phys . rev . lett . * 117 * , no . 10 , 109903 ( 2016 ) ] doi:10.1103/physrevlett.117.109903 , 10.1103/physrevlett.117.082503 [ arxiv:1605.02889 [ hep - ex ] ] . p. a. r. ade _ et al . _ [ planck collaboration ] , astron . astrophys . * 594 * , a13 ( 2016 ) doi:10.1051/0004 - 6361/201525830 [ arxiv:1502.01589 [ astro-ph.co ] ] . we take the limit from a recent review , j. e. kim and g. carosi , rev . phys . * 82 * , 557 ( 2010 ) doi:10.1103/revmodphys.82.557 [ arxiv:0807.3125 [ hep - ph ] ] . k. j. bae , h. baer and e. j. chun , jcap * 1312 * , 028 ( 2013 ) doi:10.1088/1475 - 7516/2013/12/028 [ arxiv:1309.5365 [ hep - ph ] ] . k. j. bae , h. baer , a. lessa and h. serce , jcap * 1410 * , no . 10 , 082 ( 2014 ) doi:10.1088/1475 - 7516/2014/10/082 [ arxiv:1406.4138 [ hep - ph ] ] . m. kawasaki and k. nakayama , ann . nucl . part . sci . * 63 * , 69 ( 2013 ) doi:10.1146/annurev - nucl-102212 - 170536 [ arxiv:1301.1123 [ hep - ph ] ] . s. kasuya , m. kawasaki and t. yanagida , phys . b * 409 * , 94 ( 1997 ) doi:10.1016/s0370 - 2693(97)00809 - 5 [ hep - ph/9608405 ] . a. d. linde and d. h. lyth , phys . b * 246 * , 353 ( 1990 ) . doi:10.1016/0370 - 2693(90)90613-b a. d. linde , phys . lett . b * 259 * , 38 ( 1991 ) . doi:10.1016/0370 - 2693(91)90130-i m. kawasaki , n. sugiyama and t. yanagida , phys . d * 54 * , 2442 ( 1996 ) doi:10.1103/physrevd.54.2442 [ hep - ph/9512368 ] . m. kawasaki and k. nakayama , phys . d * 77 * , 123524 ( 2008 ) doi:10.1103/physrevd.77.123524 [ arxiv:0802.2487 [ hep - ph ] ] . k. choi , e. j. chun , s. h. i m and k. s. jeong , phys . b * 750 * , 26 ( 2015 ) doi:10.1016/j.physletb.2015.08.041 [ arxiv:1505.00306 [ hep - ph ] ] . k. nakayama and m. takimoto , phys . b * 748 * , 108 ( 2015 ) doi:10.1016/j.physletb.2015.07.001 [ arxiv:1505.02119 [ hep - ph ] ] . l. f. abbott and p. sikivie , phys . lett . * 120b * ( 1983 ) 133 ; j. preskill , m. b. wise and f. wilczek , phys . * 120b * ( 1983 ) 127 ; m. dine and w. fischler , phys . lett . * 120b * ( 1983 ) 137 ; m. s. turner , phys . rev . d * 33 * ( 1986 ) 889 ; k. j. bae , j. h. huh and j. e. kim , jcap * 0809 * ( 2008 ) 005 ; l. visinelli and p. gondolo , phys . d * 80 * ( 2009 ) 035024 . p. a. r. ade _ et al . _ [ planck collaboration ] , doi:10.1051/0004 - 6361/201525898 arxiv:1502.02114 [ astro-ph.co ] . e. j. chun , phys . d * 84 * , 043509 ( 2011 ) doi:10.1103/physrevd.84.043509 [ arxiv:1104.2219 [ hep - ph ] ] . k. j. bae , k. choi and s. h. i m , jhep * 1108 * , 065 ( 2011 ) doi:10.1007/jhep08(2011)065 [ arxiv:1106.2452 [ hep - ph ] ] . k. j. bae , e. j. chun and s. h. i m , jcap * 1203 * , 013 ( 2012 ) doi:10.1088/1475 - 7516/2012/03/013 [ arxiv:1111.5962 [ hep - ph ] ] . k. y. choi , j. e. kim , h. m. lee and o. seto , phys . d * 77 * , 123501 ( 2008 ) doi:10.1103/physrevd.77.123501 [ arxiv:0801.0491 [ hep - ph ] ] . f. t. avignone , iii , s. r. elliott and j. engel , rev . * 80 * , 481 ( 2008 ) doi:10.1103/revmodphys.80.481 [ arxiv:0708.1033 [ nucl - ex ] ] . t. yanagida , conf . c * 7902131 * , 95 ( 1979 ) ; m. gell - mann , p. ramond and r. slansky , conf . c * 790927 * , 315 ( 1979 ) [ arxiv:1306.4669 [ hep - th ] ] ; p. minkowski , phys . b * 67 * , 421 ( 1977 ) . z. maki , m. nakagawa and s. sakata , prog . theor . phys . * 28 * , 870 ( 1962 ) . f. capozzi , g. l. fogli , e. lisi , a. marrone , d. montanino and a. palazzo , phys . d * 89 * , 093018 ( 2014 ) [ arxiv:1312.2878 [ hep - ph ] ] .
|
the affleck - dine leptogenesis scenario along the @xmath0 flat direction is reconsidered .
it is known that successful affleck - dine leptogenesis requires that the lightest neutrino mass is extremely small .
this situation can be significantly relaxed if the neutrino mass in the early universe is different from the present one .
we consider a supersymmetric dine - fischler - srednicki - zhitnitsky ( dfsz ) type model , which provides a solution to the strong @xmath1 problem and generates a susy @xmath2-term and right - handed neutrino masses .
if the pq scale during lepton number generation is much larger than the present value , leptogenesis is very efficient so that enough baryon number can be generated without introducing a hierarchically small neutrino mass .
the final baryon asymmetry is related to the @xmath2-term , and hence linked to the level of electroweak fine tuning .
we also show the pq breaking scalar dynamics that keeps a large pq breaking scale during inflation and lepton number generation .
the @xmath2-term generating superpotential plays an important role for preserving the lepton asymmetry during saxion oscillation . in this scenario ,
the axion isocurvature perturbation is naturally suppressed .
ctpu-16 - 44 ut-16 - 37 ipmu-16 - 0190
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
among the structures shown by interacting galaxies ( barnes & hernquist 1992 ) , the so - called collisional ring galaxies ( theys & spiegel 1976 , 1977 ) are a rare and most distinctive sub - class ( appelton & struck - marcell 1996 ) . collisional ring galaxies occur from an almost head - on collision between a smaller galaxy travelling along the minor axis of a larger galaxy passing through , or close to , its centre ( appleton & struck - marcell 1996 ) . the shock wave sweeps up and expels gas from the system , usually leaving a gas - poor galaxy behind ( freeman & de vaucoleurs 1974 ) . the pre - eminent example is the cartwheel galaxy ( eso350 - 40 ; am0035 - 335 ) at a distance of 124mpc . the cartwheel was discovered by zwicky ( 1941 ) , and studied in - depth by fosbury & hawarden ( 1977 ) and amram et al . the cartwheel shows a spectacular ring of star formation , with interior spokes surrounding the host galaxy . other well - known collisional systems include arp148 ( mayall s object ) at 148mpc ( smith 1941 ; burbidge , burbidge & prendrgast 1964 ; arp 1966 ) , ic 298 ( arp147 ; gerber , lamb & balsara 1992 ) at 135mpc , and am0644 - 741 ( eso34 - 11 or the `` lindsay - shapley ring '' ) at 87mpc ( lindsay & shapley 1960 ; graham 1974 ; arp & madore 1987 ) . this latter object is often considered to be a typical exemplar of evolved ringed galaxies ( e.g. higdon , higdon & rand 2011 ) . while star - forming events resulting from galactic collisions may be more common than usually thought ( e.g. block et al . 2006 ) , spectacular cartwheel - type galaxies , showing complete rings ( but not necessarily with evident spokes ) , appear to be very rare in volume - limited surveys . the previous nearest example was the vela ring galaxy ( am1006 - 380 = eso316 - 43 ) , at 72mpc ( dennefeld , lausten & materne 1979 ) . partial ring galaxies are more numerous . probably the nearest in the madore , nelson & petrillo ( 2009 ) catalogue is am0322 - 374 at 20mpc . ugc9893 , at 11mpc ( kennicutt et al . 2008 ) , is also listed as a collisional system , but it lacks a clear ring structure ( gil de paz & madore 2003 ) , appearing indistinct in available imagery . collisional ring galaxy systems can be identified in high- resolution , deep h@xmath1 imagery where the hii regions of the ring are readily visible . however , h@xmath1 galaxy surveys such as gil de paz & madore ( 2003 ) , james et al . ( 2004 ) , and kennicutt et al . ( 2008 ) have not added to the modest number of cartwheel - type galaxies currently known and there are @xmath820 convincing systems , e.g. marston & appleton 1995 . bosch et al . ( 2015 ) present the most recent detailed work on such systems for an apparently empty ring galaxy system eso474g040 at @xmath9mpc , thought to have arisen from a recent merger of two disk galaxies . collisional rings systems offer an excellent environment to study the evolution of density - wave induced starbursts ( marston & appleton 1995 ; romano , mayya & vorobyov 2008 ) , as well as the dynamic interactions between the individual galaxies ( e.g. smith et al . a closer example of a collisional ring system would allow its star formation , stellar distribution and evolution to be studied in detail , spatially , kinematically and chromatically . here we present the discovery of a new collisional ring galaxy at a distance of only 10mpc . our paper is organised as follows . in section 2 we present some background and describe the discovery of the ring around eso179 - 13 . in section 3 we present our follow - up high - resolution imaging and other archival multi- wavelength images . optical spectroscopy is presented in section 4 . parameters for the system , such as masses and star formation rates , are provided in section 5 . section 6 presents our discussion and give our summary in section 7 . eso179 - 13 is noted as an interacting double galaxy system in simbad , appearing to consist of a near edge - on late - type spiral with another more chaotic system @xmath380arcsec to the north - east . it was first identified by sersic ( 1974 ) and independently re - discovered by lauberts ( 1982 ) , longmore et al . ( 1982 ) and corwin , de vaucouleurs & de vaucouleurs ( 1985 ) . a detailed @xmath10-band image was published by laustsen , madsen & west ( 1987 ) . woudt & kraan - korteweg ( 2001 ) identify the most prominent galaxy as wkk7460 ( componenta hereafter ) , and wkk7463 for the secondary ( component b ) . a small , third galaxy is also identified as wkk7457 to the west of componenta ( componentc ) . it agrees with our own positional determination to 1arcsec . the system , centred on componenta , is entry 391 in the h@xmath1 catalogue of kennicutt et al . ( 2008 ) , but they do not list integrated h@xmath1 ( + [ nii ] ) flux or luminosity estimates . lee et al . ( 2011 ) , in a related paper , included this system in their galaxy evolution explorer ( galex ) ultraviolet imaging survey of local volume galaxies with a quoted @xmath11 = 15 , but no near - uv or far - uv fluxes are quoted . owing to its proximity to the galactic plane , the galaxy was not observed by galex prior to mission completion ( bianchi 2014 ) . componenta has been classified as an sb(s)m spiral in the rc3 catalogue ( de vaucouleurs et al . 1991 ) and similarly classified as sb(s)dm system ( t = 7.5 ) by buta ( 1995 ) . these independent classifications treat the system as a distinct , late - type spiral galaxy , albeit in an interaction . it can be best described as a magellanic dwarf or a magellanic spiral , similar to the lmc . a range of heliocentric radial velocities for componenta are available in the literature from both optical spectroscopic and hi measurements . the most recent optical results are strauss et al . ( 1992 ) who quote 775@xmath1236 kms@xmath0 , di nella et al . ( 1997 ) , 750@xmath1270 kms@xmath0 , and woudt et al . ( 1999 ) , 757@xmath1250 km s@xmath0 . these are in good agreement so we take an unweighted average of 761kms@xmath0 ( @xmath13=11kms@xmath0 ) for componenta . the earliest hi heliocentric velocities towards the system are 836@xmath125kms@xmath0 by longmore et al . ( 1982 ) , 840@xmath1220kms@xmath0 by tully ( 1988 ) and 843@xmath129kms@xmath0 by de vaucouleurs et al . the parkes hi ` hipass ' multibeam receiver programme gives 843@xmath125kms @xmath0 ( hipass catalogue ; meyer et al . 2004 ) where it is noted as the 60@xmath14 brightest in terms of integrated flux density ( koribalski et al . a deeper , pointed parkes observation was obtained by schrder et al . they give parameters for componenta , but note that a contribution from componentb is likely . their heliocentric hi radial velocity is 842kms@xmath0 from the hi profile mid - point , with a velocity width of 222kms@xmath0 at 20% of peak . the integrated hi flux density is [email protected]@xmath0 , in good agreement with the hipass value of 100 . all the hi velocities are consistent within small errors . these values include the combined contribution from all the gas in the full system . we adopt 842@xmath125kms@xmath0 for the combined hi system velocity . kennicutt et al . ( 2008 ) give a distance of 9mpc based on a virgocentric flow model distance . we adopt a slightly further distance of 10.0@xmath15 mpc , based on kennicutt et al . ( 2008 ) , but scaling from a hubble constant h@xmath16 of 75 to 68 kms@xmath0mpc@xmath0 due to the latest wmap ( hinshaw et al . 2013 ) and planck results ( ade et al . 2014 ) . at this distance , 1arcsec equates to 48.5pc . wouldt & kraan - korteweg ( 2001 ) give a modest colour excess of @xmath17(b - v)@xmath18 mag in the surrounding field , giving @xmath19 mag while schlafly & finkbeiner ( 2011 ) give @xmath20 mag for galactic extinction in this general direction , which we adopt hereafter . we will also use @xmath21 mag based on the reddening law from howarth ( 1983 ) . apart from the basic information above , the system has been little studied because of its location in a crowded , low - latitude area near the galactic plane ( @xmath22 - 8 degrees ) and proximity to the bright , 7.7 mag a0iv star , hd150915 , located 72arcsec due south . there are actually two bright stars here , but only hd150915 is listed in simbad . the second star is @xmath310arcsec north of hd150915 and @xmath32magnitudes fainter . it was undoubtedly blended with hd 150915 in the original photographic imagery . the low heliocentric velocity meant the system was discovered as a strong h@xmath1 emitter in the anglo - australian observatory ( aao ) / u.k . schmidt telescope ( ukst ) supercosmos h@xmath1 survey of the southern galactic plane ( shs ; parker et al . 2005 ) during searches for galactic planetary nebulae ( parker et al . 2006 ) . the bandpass of the survey filter ( parker & bland - hawthorn 1998 ) includes all velocities of galactic gas and emission from extragalactic objects out to the virgo and fornax galaxy clusters at @xmath31200kms@xmath0 . [ shs ] shows the 4 @xmath23 4arcmin discovery images from the shs survey data centred on the system . from left to right there is the h@xmath1 image , the matching broadband short - red ` sr ' image and the image obtained by taking the sr image from the h@xmath1 image ( effectively a continuum subtracted image ) . componenta is at the centre . componentb is labelled to the north - east while the smaller compact componentc is indicated with an arrow . the shs continuum subtracted image reveals a spectacular ring of emission knots not obvious , without context , even in the h@xmath1 image while the main componenta galaxy has almost disappeared . the ring is approximately ( but not perfectly ) centred on componenta . surprisingly , given its previous spiral morphological classification , componenta does not show up well in the shs continuum subtracted image and appears to show little star formation . this indicates that along with its apparently highly elongated nature ( so not an elliptical ) it is has been largely stripped of gas . the smaller , more irregular galaxy ( componentb ) shows extensive , clumpy h@xmath1 emission . a third , faint compact system ( componentc ) is seen 38arcsec to the west of componenta , identified as wkk7457 . this object is clear in the broad - band @xmath10 , @xmath24 and @xmath25-band supercosmos images and is also undergoing star - formation . the strong hi detection noted earlier indicates that a significant , neutral hi component still remains within the overall system environment . the lack of much star formation in componenta suggests that most of this hi may be located outside of its inner disk . discovery of this collisional ring system led us first to investigate and compile existing multi - wavelength data from the archives . we then obtained deeper , higher resolution imaging and spectroscopy of the major emission components ( see below ) , to provide estimates of some key physical parameters for this system . high resolution imagery at the blanco 4-m telescope at ctio was obtained in june 2008 using the wide field mosaic camera . we used five filters : @xmath26-band , h@xmath27 , h@xmath1 off - band ( 80 redward of h@xmath1 ) , , and a ( wider ) off - band filter centred at 5300 . the field of view is 30arcmin on a side and the plate scale 0.26arcsecpx@xmath0 . the seeing measured from the data frames was typically 1.2arcsec . exposures were 5 minutes for the emission - line and off - band filters , and 60s for the @xmath26-band filter . the airmass during the exposures averaged 1.3 . to subtract the continuum , the h@xmath1 off - band image was scaled down by a factor of 0.88 , and the off - band by a factor of 0.14 , to compensate for the filter widths . these factors were established by minimizing residuals for a set of selected field stars . for that works very well and results are in good agreement with the filter curve . for h@xmath1 this can be an issue if the chosen stars have h@xmath1 in absorption . this does not appear to be the case for the field stars chosen which provide a consistent scale - factor to @xmath28 . it is best to minimize stellar residuals rather than filter curves as stars are a more important contribution to the continuum than bound - free emission from hi regions . a montage of these ctio images is shown in fig . they reveal the distribution and variety of the emission structures as well as data for quantitative flux estimates . from these deeper , higher resolution ctio h@xmath1 and continuum subtracted images the distribution of ionised gas is seen to be far more complex than is evident from the shs discovery data . the leftmost panel is a @xmath29arcminute ctio h@xmath1 image with the off - band frame subtracted . we avoid the bright star to the south where ccd blooming is serious . emission associated with componentc and at the southern tip of componenta s disk previously gave the impression of a circular ring in the shs as the bright star obscured emission further south . the higher resolution ctio data reveals an elliptical emission ring with a major - axis diameter of 127arcsec ( @xmath36.2kpc ) . the inscribed oval was positioned to fit the prominent emission now seen to extend further south . the ellipse is not centred on componenta , but at 16h47m19.5s , @xmath3057@xmath3126 44 ( j2000 ) , @xmath313arcsec ( or @xmath30.6kpc ) south - south - east . some low - level h@xmath1 emission is now seen across componenta , surrounding componentb and with faint , localised emission features and blobs around the entire system . an interesting feature is the elongated nature of some of these emission knots in a north - east direction from a to b. the main emission features follow the equivalent h@xmath1 structures though at typically a third to a half of their native integrated pixel intensities . the mid panel is the matching ctio continuum subtracted image ( narrow on and off - band images used ) . both these images are presented at 90% linear scaling to reveal the full extent of the emission ( this sets the upper and lower limits based on the 90% pixel intensity level where a histogram of the data is created and the limits are set to display the percentage about the mean value ) . with full pixel range , saturated pixels from the 7.7th magnitude star hd150915 would prevent detail being seen . low level diffuse flux is seen for componenta but it is not conspicuous . this shows it has almost been completely cancelled out indicating componenta is mostly composed of normal starlight . the right hand panel is the off - band filter image ( 5300 ) that effectively represents the @xmath26-band starlight without the + h@xmath32 emission line contribution . this shows dust lanes along and at the south - west extremity of the disk , concentrated starlight from componentsa , b and c and an envelope of diffuse starlight around the entire system . this common envelope " was already noted by laustsen et al . ( 1987 ) from old @xmath10-band photographic data taken with the eso 3.6-m telescope in the 1980s . this raises an interesting question and also a possible explanation for why the apparent axial ratios of the highly elongated componenta differs so much from the surrounding oval ring . currently known collisional ring systems with low impact parameter have axial ratios that are broadly similar to that of the target galaxy . such arrangements are a natural consequence of the copious gas and ism being swept up and compressed by the resultant density wave having a similar radial ( and perhaps also vertical ) distribution as the target galaxy stellar content . this does not appear to be the case here unless what we are actually seeing is the residual , less - inclined bar of a more face - on disk system that extends out to the entire ring and whose presence can still be seen in the low - level diffuse starlight refereed to earlier . if componenta is not a residual bar but a distinct late - type disk galaxy ( as commonly assumed ) stripped of gas , then the gas distribution could have become less flattened somehow . alternatively the original target galaxy could have been stretched out along the direction of the impact towards componentb before , during or even after the bulls - eye collision occurred . though componentb is only 13@xmath33 of the mass ( in stars ) of componenta these effects could have been strengthened due to the low ` magellanic'-type masses of the system as a whole . finally , fig . [ ctio - rgb ] is a composite red , green blue ( rgb ) ctio image displaying the system s spectacular nature . the red channel is h@xmath1 , the green and the blue broadband v. componenta is mostly star - light with dust lanes evident in this higher resolution data . the few small hii knots appear red , likely due to dust extinction within the disk . based on the shs data , literature data and our new ctio images , table [ fund_data ] presents some key parameters for galaxy components a , b and c , including the central position of the best fit ellipse to the main emission ring . the position for componenta is tied to the 2mass ( skrutskie et al . 2006 ) near - ir imagery and astrometry . the positions refer to the centre of each assumed galaxy . the centre of componenta is derived from the dominant starlight in the @xmath24-band . the separation between the centres of componentsa and b is @xmath380arcsec . using the distance of 10mpc gives a projected separation between them of 3.9kpc . componenta has a major - axis of @xmath386arcsec ( 4.2kpc ) measured to the outer edge in the stellar continuum from off - band and @xmath26-band ctio images where a decline in the background is seen before the star - forming ring is encountered . when checked against the broad band @xmath24 , @xmath25 and 2mass images when autocut at the 90% pixel intensity level the disk appears to extend only to 70 - 80arcsec . at low isophotal thresholds starlight extends out to the star - forming ring both along the major and minor axes . componentb has a major axis of @xmath333arcsec ( 1.6kpc ) determined in the same way . the strongest h@xmath1 emission in b is also in an oval structure of nucleated knots @xmath332arcsec across . this is in excellent agreement with the value estimated from the concentrated starlight . starlight internal to this emission oval is clear in fig . [ ctio - rgb ] and b may extend to 70arcsec ( @xmath33.4kpc ) in low level , isolated clumps of star formation and diffuse starlight . the overall system envelope , including all assumed associated broad and narrow band emission and diffuse starlight as seen across the region in the right panel of fig . [ ctio ] , has a major diameter of 318arcsec and a minor axis of 198arcsec , giving a total system size of @[email protected] . [ cols="<,^,^,^,^,^",options="header " , ] [ summary_data ] eso197 - 13 is an important system that can be used to study collisional rings as it is much closer than any equivalent , opening new possibilities for analysis . system dynamics can be studied at unprecedented spatial resolution as 0.1arcsec ( e.g. hst and/or ao resolution ) is only 25pc . this is sufficient to explore details of the system - wide star formation process and to allow detection of individual stars . it would be a prime candidate to measure the star formation history in the regions recently traversed by the ring . during visual scans of the shs ( parker et al . 2005 ) , a spectacular new collisional ring galaxy ( or cartwheel analogue ) has been found , eso179 - 13 , which we have named kathryn s wheel . with an estimated distance of only 10.0mpc , it is in the local supercluster but has been previously missed due its location near the galactic plane and its projected proximity to a bright 7.7-magnitude star which interferes with observation . the velocity range of all optical emission components from the main ring , componentb and the published velocity for componenta is about 200kms@xmath0 this is completely encapsulated within the fwhm of 222kms@xmath0 reported for the hi profile at the 20% level centred at a heliocentric velocity of 842kms@xmath0 which we consider representative of the overall interacting system . at a distance of 10mpc , the system has an overall physical size of @xmath315kpc for all major components and the diffuse surrounding envelope , a total mass of @xmath4m@xmath5 ( stars + hi ) and a metallicity of [ o / h]@xmath3 - 0.4 , that classifies it as a magellanic - type system . it has a large reservoir of neutral gas and is the 60@xmath14 brightest galaxy in the hipass hi survey . by some margin it is also the nearest ring system known . this system offers an unprecedented opportunity to undertake detailed studies of this rare phenomenon , where two reasonably well matched galaxies have a close encounter centred close to the potential well of the primary . this paper used the simbad and vizier services operated at the cds , strasbourg , the nasa / ipac extragalactic data base ( ned ) and data products of the aao / ukst h@xmath1 survey , produced with the support of the anglo - australian telescope board and uk particle physics and astronomy research council ( now the stfc ) . this research used data from the wide - field infrared survey explorer , a joint project of the university of california , los angeles , and jet propulsion laboratory / california institute of technology , funded by the national aeronautics and space administration , and data from the 2mass , a joint project of the university of massachusetts and the infrared processing and analysis center california institute of technology , funded by the national aeronautics and space administration and the national science foundation . the authors acknowledge help of dr anna kovacevic at the ctio telescope . we are grateful for stimulating discussion with kate zijlstra . we thank an anonymous referee who provided excellent feedback to help improve the quality of this paper . amram p. , mended de oliveira c. , boulesteix j. , balkowski c. , 1998 , a&a , 330 , 881 appleton p.n . , struck - marcell c. , 1996 , fund . cosmic phys , 16 , 111 arp h.c . , 1966 , apj , 140 , 1617 arp h.c . , madore b. , 1987 , a catalogue of southern peculiar galaxies and associations . cambridge , new york : cambridge university press baldwin j.a . , phillips m.m . , terlevich r. , 1981 , pasp , 93 , 5 barnes j.e . , hernquist l. , 1992 , ara&a , 30 , 705 beichman c.a . , neugebauer g. , habing h.j . , clegg p.e . , chester t.j . , 1988 , the infrared astronomical satellite ( iras ) catalogs and atlases . volume 1 : explanatory supplement bell e.f . , mcintosh d.h . , katz n. , weinberg m.d . , 2003 , apjs , 149 , 289 bendo g.j . et al . , 2006 , apj , 652 , 283 bendo g.j . et al . , 2012 , mnras , 419 , 1833 bendo g.j . et al . , 2015 , mnras , 448 , 135 bianchi l. , 2014 , ap&ss , 354 , 103b block d.l . et al . , 2006 , nature , 443 , 832 boselli a. et al . , 2012 , a&a , 540 , 54 boselli a. , fossati m. , gavazzi g. , ciesla l. , buat v. , boissier s. , hughes t. , 2015 , a&a , 579 , 102 brosch n. , vaisanen p. , kniazev a.y . , moisav a. , 2015 , mnras , in press , arxiv:1506.00409 burbidge e.m . , burbidge g.r . , prendergast k.h . , 1964 , apj , 140 , 1617 buta r. , 1995 , apjs , 96 , 39 childress m. j. , vogt f. p. a. , nielsen j. , sharp r. g. , 2014 , ap&ss , 349 , 617 cluver m. et al . , 2014 , apj , 782 , 90 condon j.j . , 1992 , ara&a , 30 , 575 conn b.c . , pasquali a. , pompei e. , lane r.r . , chene a - n . , smith r. , lewis g.f . , 2011 , apj , 741 , 80 corwin h.g . , de vaucouleurs a. , de vaucouleurs g. , 1985 , univ . texas monogr . astron . , 4 , 1 davies j.i . et al . , 2012 , mnras , 419 3505 dennefeld m. , lausten s. , materne j. , 1979 , a&a , 74 , 123 de vaucouleurs g. , de vaucouleurs a. , corwin h.g . , buta r.j . , paturel g. , fouque p. , 1991 , third reference catalogue of bright galaxies , volume iii . new york : springer di nella h. , couch w.j . , parker q.a . , paturel g. , 1997 , mnras , 287 , 472 donghia e. , mapelli m. , moore b. , 2008 , mnras , 389 , 1275 dopita m. , et al . , 2010 , ap&ss , 327 , 245 doyle m.t . , et al , 2005 , mnras , 361 , 34 draine b.t . et al . , 2007 , apj . , 663 , 866 drew j.e . 2014 , mnras , 440 , 2036 driver s.p . , 1999 , apj , 526 , l69 drory n. , bender r. , feulner g. , hopp , u. , maraston c. , snigula j. , hill , g.j . , 2004 , apj , 608 , 742 engelbracht c. , et al . , 2005 , apj , 628 , l29 engelbracht c. , et al . , 2008 , apj , 678 , 804 few j.m.a . , madore b.f . , 1986 , mnras , 222 , 673 fosbury r.a.e . , hawarden t.g , 1977 , mnras , 178 , 473 freeman k.c . , de vaucouleurs g. , 1974 , apj , 194 , 569 frew d.j . , parker q.a . , 2010 , pasa , 27 , 129 frew d.j . , bojii i.s . , parker q.a . , 2013 , mnras , 431 , 2 frew d.j . , bojii i.s . , parker q.a . , pierce m.j . , gunawardhana m.l.p . , reid w.a . , 2014a , mnras , 440 , 1080 frew d.j . et al . , 2014b , mnras , 440 , 1345 gaustad j.e . , mccullough p.r . , rosing w. , van buren d. , 2001 , pasp , 113 , 1326 gerber r.a . , lamb s.a . , balsara d.s . , 1992 , apjl , 399 , l51 gil de paz a. , madore b.f . , 2003 , apjs , 147 , 29 graham j. , 1974 , observatory , 94 , 290 gunawardhana m.l.p . . , 2013 , mnras , 433 , 2764 hambly n.c . et al . , 2001a , mnras , 326 , 1279 henriques b. , maraston c. , monaco p. , fontanot f. , menci n. , de lucia g. , tonini c. , 2011 , mnras , 415 , 357 higdon j.l . , higdon s.j.u . , rand r.j . , 2011 , apj , 739 , 97 hinshaw g. et al . , 2013 , apjs , 208 , 19 howarth i. , 1983 , mnras , 203 , 301 ishihara d. et al . , 2010 , a&a , 514 , 1 james p.a . , 2004 , a&a , 414 , 23 jarrett t.h . , chester t. , cutri r. , schneider s.e . , huchra j.p . , 2003 , aj , 125 , 525 jarrett t.h . et al . , 2012 , aj , 144 , 68 kankare e. et al . , 2014 , a&a , 572 , a75 kennicutt r.c . , 1984 , apj , 287 , 116 kennicutt r.c . , bresolin f. , bomans d.j . , bothun g.d . , thompson i.b . , 1995 , aj , 109 , 594 kennicutt r.c . , , funes s.j . , jos g. , sakai s. , akiyama s. , 2008 , apjs , 178 , 247 kennicutt r.c . , evans n.j . , 2012 , ara&a , 50 , 531 koribalski b.s . et al . , 2004 , aj , 1 , 16 lauberts a. , 1982 , eso / uppsala survey of the eso(b ) atlas . garching : european southern observatory laustsen s. , madsen c. , west r.m . , 1987 , exploring the southern sky . garching : european southern observatory lee , j.c . 2011 , apjs , 192 , 33 li a. , draine b.t . , 2001 , apj , 554 , 778 lindsay e.m . , shapley h. , 1960 , observatory , 80 , 223 longmore a.j . et al . , 1982 , mnras , 200 , 325 madore b.f . , nelson e. , petrillo k. , 2009 , apjs , 181 , 572 marston a.p . , appleton p.n . , 1995 , aj , 109 , 1002 masci f. , 2013 , astrophysics source code library , 2010 masci f.j . , fowler j.w . , 2009 , astronomical data analysis software and systems xviii , 411 , 67 mcdonald i. , zijlstra a.a . , boyer m.l . , 2012 , mnras , 427 , 343 melbourne j. , boyer m.l . , 2013 , apj , 764 , 30 meyer m.j . et al . , 2004 , mnras , 350 , 1195 murphy t. , mauch t. , green a. , hunstead r.w . , piestrzynska b. , kels a.p . , sztajer p. , 2007 , mnras , 382 , 382 parker q.a . , bland - hawthorn j. , 1998 , pasa , 15 , 33 parker q.a . et al . , 2005 , mnras , 362 , 689 parker q.a . et al . , 2006 , mnras , 372 , 79 pilyugin l.s . , vilchez j.m . , thuan , t.x . , 2010 , apj 720 , 1738 planck collaboration : ade p.a.r . et al . , 2014 , a&a , 571 , 28 relao m. , kennicutt r.c . , eldridge j.j . , lee j.c . , verley s. , 2012 , mnras , 423 , 2933 rodgers a.w . , harding p. , bloxham g. , bessell m.s . , 1998 , pasp , 100 , 626 romano r. , mayya y.d . , vorobyov e.i . , 2008 , aj , 136 , 1259 ryder s.d . , walsh w. , malin d. , 1999 , pasa , 16 , 84 sabin l. et al . , 2013 , mnras , 431 , 279 schiminovich d. , wyder t.k . , martin d. , 2007 , apjs , 173 , 315 schlafly e.f . , finkbeiner d.p . , 2011 , apj , 737 , 103 schroder a. , kraan - korteweg r.c . , henning p.a . , 2009 , a&a , 505 , 1049 skrutskie m.f . et al . , 2006 , aj , 131 , 1163 sersic j.l . , 1974 , , 28 , 365 smith r.t . , 1941 , pasp , 53 , 187 smith r. , lane r.r . , conn b.c . , fellhauer m. , 2012 , mnras , 423 , 543 stark d.v . , kannappan s.j . , wei l.h . , baker a.j . , leroy a.k . , eckert k.d . , vogel s.n . , 2013 , apj , 769 , 82 staveley - smith l. , kim s. , calabretta m.r . , haynes r.f . , kesteven m.j . , 2003 , mnras , 339 , 87 strauss m.a . , huchra j.p . , david m. , et al . , 1992 , apjs , 92 , 29 taylor e.n . et al . , 2011 , mnras , 418,1587 theys j.c . , spiegel e.a . , 1976 , apj , 208 , 650 theys j.c . , spiegel e.a . , 1977 , apj , 212 , 616 tully r.b . , 1988 , nearby galaxies catalog . cambridge and new york : cambridge university press van der marel r. , kallivayalil n. , besla g. , 2009 , in proc . 256 , the magellanic system : stars , gas , and galaxies , j.t . van loon & j.m . oliveira , eds . , p 81 viironen k. , delgado - inglada g. , mampaso a. , magrini l. , corradi r.l.m . , 2007 , mnras , 381 , 1719 vogt f.p.a . , dopita m.a . , borthakur s. , verdes - montenegro l. , heckman t.m . , yun m.s . , chambers k.c . , 2015 , mnras in press ( arxiv:1504.03337 ) wright a.e . , griffith m.r . , burke b.f . , ekers r.d . , 1994 , apj . suppl . 91 , 111 woudt p.a . , kraan - korteweg r.c . , fairall a.p . , 1999 , a&a , 352 , 39 woudt p.a , kraan - korteweg r.c . , 2001 , a&a , 380 , 441 wright e.l . et al . , 2010 , aj , 140 , 1868 yamamura i. et al . , 2009 , ser . , 418 , 3 zwicky f. , 1941 , in theodore von krmn anniversary volume : contributions to applied mechanics and related subjects . pasadena : california institute of technology .
|
we report the discovery of the closest collisional ring galaxy to the milky way .
such rare systems occur due to `` bulls - eye '' encounters between two reasonably matched galaxies . the recessional velocity of about 840kms@xmath0 is low enough that it was detected in the aao / ukst survey for galactic h@xmath1 emission .
the distance is only 10.0mpc and the main galaxy shows a full ring of star forming knots , 6.1kpc in diameter surrounding a quiescent disk .
the smaller assumed bullet " galaxy also shows vigorous star formation .
the spectacular nature of the object had been overlooked because of its location in the galactic plane and proximity to a bright star and even though it is the 60@xmath2 brightest galaxy in the hi parkes all sky survey ( hipass ) hi survey .
the overall system has a physical size of @xmath315kpc , a total mass of @xmath4m@xmath5 ( stars + hi ) , a metallicity of [ o / h]@xmath6 , and a star formation rate of 0.2 - 0.5m@xmath5yr@xmath0 , making it a magellanic - type system .
collisional ring galaxies therefore extend to much lower galaxy masses than commonly assumed .
we derive a space density for such systems of @xmath7 , an order of magnitude higher than previously estimated .
this suggests kathryn s wheel is the nearest such system .
we present discovery images , ctio 4-m telescope narrow - band follow - up images and spectroscopy for selected emission components . given its proximity and modest extinction along the line of sight , this spectacular system provides an ideal target for future high spatial resolution studies of such systems and for direct detection of its stellar populations .
galaxies : groups : general galaxies : interactions galaxies : kinematics and dynamics galaxies : star formation galaxies : individual ( eso179 - 13 )
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the dynamical evolution of dense star clusters is a problem of fundamental importance in theoretical astrophysics . star clusters like open and globular clusters are among the simplest stellar systems : they are spherical , they contain no dust to confuse the observations and they appear to have no dark matter . moreover , they are dynamically old : a typical star in a globular cluster has completed some @xmath0 orbits since the cluster was formed and processes like gravothermal collapse and two - body relaxation occur on timescales comparable with their ages . thus , they provide the best physical realization of the gravitational n - body problem i.e. to understand the evolution of a system of n point masses interacting only by gravitational forces . in spite of the many advances made in the recent past , many aspects of the problem have remained unresolved like the production of exotic objects ( ferraro et al . 2012 ) , the importance of tidal - shocks in the long term evolution and survival of star clusters in the galaxy ( gnedin , lee & ostriker 1999 ) and the ability to retain dark remnants ( morscher et al . 2013 ; sippel & hurley 2013 ) . the most direct approach to the simulation of star clusters is through n - body simulations . in these kind of studies the gravitational forces of stars are directly computed and any additional ingredient like e.g. binaries , tidal field , stellar evolution , etc . can be easily incorporated . for this reason , in many cases n - body simulations represent the unique tool to face with complex topics within the gravitational n - body problem . however , several processes involved in the dynamical evolution of a star cluster occur on different timescales , so that a direct scaling of the result of an n - body simulation to larger number of particles is not possible ( baumgardt 2001 ) . although the grape series of special - purpose computers is steadily increasing in performance and the development of graphic processing units ( gpus ) computing , direct n - body simulation of the evolution of clusters with more than a few percent binaries and a moderate number of stars ( @xmath1 ) is still computationally expensive , with computational timescales of the order of months . until now only open clusters such as m67 and the arches cluster ( hurley et al . 2005 ; harfst , portegies zwart & stolte 2010 ) and loosely bound globular cluster objects such as palomar 4 and palomar 14 ( zonoozi et al . 2011 , 2014 ) have been modelled at the necessary level of sophistication . alternative numerical methods to simulate the evolution of star clusters have been developed in past years including fluid models ( larson 1970 ; angeletti & giannone 1977a , b ) , orbit - averaged fokker - planck methods ( cohn 1980 ; takahashi 1995 ) and monte carlo simulations ( hnon 1971 , hereafter h71 ; giersz 1998 ; joshi , rasio & portegies - zwart 2000 ) . monte carlo methods can be regarded as a hybrid between direct n - body integrations and numerical solutions of the fokker - planck equation . in this approach the system is modelled as a sample of `` superstars '' i.e. a subsample of stars sharing the same mass and integrals of motions . in spherical systems the motion of each superstar depends only on its energy and angular momentum and on the cluster potential , this last quantity being a unique function of positions and masses of the superstars . through an iterative algorithm it is therefore possible to follow the evolution of the system once suited perturbations to the integrals of motion of the superstars are applied to account for the effect of two - body interactions . within the family of monte carlo methods two approaches can be distinguished : _ i ) _ the _ orbit - following _ method ( also known as the `` princeton method '' ; spitzer & hart 1971 ) where the orbits of the superstars are directly computed and _ ii ) _ the _ orbit - averaged _ method ( the `` cornell method '' ; h71 ) where only the energies and angular momenta of superstars are monitored . while orbit - following methods are more suited to follow all those processes occurring on the dynamical timescale ( such as the evaporation of stars from the cluster , the violent relaxation , the tidal shocks and the phase of post - core collapse ) , orbit - averaged methods are computationally less expensive since perturbations to the integrals of motions need to be computed at time - steps that are short compared to the relaxation time and the time consuming integration of the orbits is not required moreover , it is particularly easy to add more complexity and realism to the simulations one layer at a time and they are particularly easy to be parallelised . for these reasons , orbit - averaged monte carlo simulations have been employed by a number of groups to study the dynamical evolution of globular clusters and the dense centers of galaxies . in recent years a particular effort have been made by these groups to include the effect of a mass spectrum ( giersz 2001 ) , stellar evolution ( joshi , nave & rasio 2001 ) , three- and four - body interactions ( giersz & spurzem 2003 ; fregeau & rasio 2003 ; fregeau et al . 2007 ) and a simplified treatment of a tidal field ( giersz et al . 2013 ; takahashi & baumgardt 2012 ) . one of the most complex process to be modelled in monte carlo simulations is the escape from a cluster in an external tidal field . in fact , while stars can escape from an isolated cluster only when they have positive energies , when the cluster moves within a tidal field the effective potential felt by a cluster star is perturbed and has a maximum at a distance ( called `` tidal radius '' ) which depends on the shape of the external field and on the orbital parameters of the cluster . if the star moves beyond this radius the gravitational attraction of the cluster will not balance the combined effect of the external tidal field and the centrifugal force and the star will escape from the cluster . for this reason the presence of the external field accelerates the escape process and consequently the whole structural evolution of the system ( spitzer 1987 ) . since the first pioneering studies by h71 , the effect of a steady external field has been modelled by removing stars able to reach an apocenter larger that the tidal radius . this last quantity was estimated considering the distance of the lagrangian point in the simple case of an external potential produced by a point mass on a cluster moving on a circular orbit . even in this simple case , however , this criterion represents only a rough approximation . indeed , the presence of the external field breaks the spherical symmetry of the effective potential and the size of the tidal radius depends on the direction of the star motion . in practice , stars can escape from the cluster only close to the direction of the galactic center through the so - called `` 2nd and 3rd lagrangian points '' . moreover , in real clusters , once a star reach the energy required to escape , it needs several crossing times to reach the right direction ( the so - called `` potential escapers '' ) thus producing a delayed escape . fukushige & heggie ( 2000 ) derived a simple prescription to estimate the timescale of escape as a function of the excess of energy with respect to the lagrangian point energy level . on the basis of this last result , giersz et al . ( 2013 ) adopted a delayed escape criterion which successfully account for this effect . the situation is more complex when eccentric orbits are considered : in this case , the hamiltonian is time - dependent and the tidal radius can be only instantaneously determined . in this situation , the aperture in the phase - space for a star to escape changes with time and stars can have only a limited amount of time to reach such an aperture . moreover , stars escaping from the cluster can be re - captured when the cluster expand during its motion away from the perigalacticon . to further complicate the picture , it has been shown that stars with prograde and retrograde motion escape with different efficencies ( read et al . 2006 ) and there are stars which permanently remain bound to the cluster outside the tidal radius ( `` non - escapers '' ; ross , mennim & heggie 1997 ) . finally , the potential of the milky way can not be realistically approximated as a point mass but consists of many non - spherical components . in a non - spherical potential orbits are in general non - planar and rapid changes of potential can produce compressive shocks which increase the kinetic energy budget of cluster stars ( gnedin & ostriker 1997 ) . in this paper we present a new orbit - averaged monte carlo code able to simulate the evolution of a star cluster moving on an eccentric orbit within a realistic external potential . in sect . 2 we describe the code and the modification made to the original algorithm described by h71 . in sect . 3 the recipies to account for the effect of the external field are outlined . sect . 4 in devoted to the description of the set of simulations and their comparison with n - body simulations . we summarize our results in sect . a detailed derivation of the tidal radius and effective potential in complex potentials is provided in the appendix . the code presented here is an updated version of the orbit - averaged monte carlo method extensively described in h71 ( see also stodolkiewicz 1982 , giersz 1998 and joshi et al . the basic idea of this approach is to consider the cluster as a sample of superstars characterized by mass ( @xmath2 ) , energy ( @xmath3 ) and angular momentum ( @xmath4 ) per unit mass generating a spherical symmetric potential ( @xmath5 ) . the evolution of the cluster is divided in time - steps ( @xmath6 ) of variable duration . at each time - step the following steps are performed 1 . the optimal time - step is determined ( see eq . 10 by joshi et al . 2000 ; see also sect . [ esc_sec ] ) ; 2 . a statistical realization of the cluster is performed by placing the superstars at random positions along their orbits . each star is placed at a given distance from the cluster center according to the inverse of the star velocity at that distance ( see below ; see also sect . 7 of h71 ) ; 3 . the cluster potential profile is evaluated according to the masses and positions of the superstars ( see below ) ; 4 . the mechanical work made by the ( internal+external ) potential change on the suparstars is calculated and corrections to the stars energies are applied ( see sect . [ shocks_sec ] ; see also sect . 4 of stodolkiewicz 1982 ) ; 5 . each superstar is assumed to interact with its nearest neighbor producing a perturbation on its energy and angular momentum ( see sect . 5 of h71 ) . stars satisfying the escape criterion ( see sect . [ esc_sec ] ) are removed from the simulation . the above steps are repeated until the end of the simulation . a detailed description of the algorithms adopted to perform the above steps is provided in h71 , giersz 1998 and joshi et al . ( 2000 ) and will not be repeated here . below we describe only the modifications to their approaches regarding the way the cluster potential is determined while in sect . [ ext_sec ] we extensively describe the adopted escape criterion . the most time - consuming step of the procedure outlined above is the distribution of the superstars across the cluster . this is done by extracting a variable @xmath7 ( with @xmath8 ) from the distribution @xmath9 where @xmath10 and @xmath11 are the pericenter and apocenter of the star orbit within the cluster potential and @xmath12 is the radial component of the star velocity at @xmath13 the values of @xmath10 and @xmath11 are first determined from the star energy and angular momentum and the cluster potential at the two zeros of the function @xmath14 then @xmath7 is extracted from @xmath15 using the von neumann rejection technique . h71 demonstrated that the above algorithm ensures that the probability to extract a given position is proportional to the time spent by the star in that position . according to h71 , the cluster potential at a given distance from the cluster center @xmath16 can be determined in a straightforward way from the positions and masses of the superstars using the poisson equation in its discrete integral form @xmath17 where @xmath18 and @xmath19 are the position and mass of the i - th superstar , @xmath20 is the total number of superstars and @xmath21 is the index such that @xmath22 . this approach is efficient and adaptive by definition i.e. in the densest regions of the cluster it provides a better sampling of the potential . however , from a computational point of view , the above procedure is quite expensive since it requires a cycle over the n superstars to find the index @xmath21 which must be repeated at least two times for each superstar to find @xmath10 , @xmath11 and @xmath12 . for this purpose we decided to calculate the cluster potential at the beginning of each time - step on a grid of @xmath23 evenly spaced radial steps . the potential at the distance @xmath16 is then determined by linearly interpolating between the two contiguous knots @xmath21 and @xmath24 . in this case the index @xmath21 is immediately found as @xmath25 . this simple modification speed up the entire process by a factor of ten . we developed two independent methods to define the potential in the grid knots which are used in different conditions . the first method ( hereafter referred as the _ fast method _ ) is to use eq . [ pot1_eq ] defining at each time - step a step - size of the grid @xmath26 where @xmath27 is the cluster core radius ( casertano & hut 1985 ) and @xmath28 is the superstar number density at @xmath18 calculated using the 50 nearest neighbor superstars . this is the fastest method and provides a good accuracy for most of the cluster evolution when clusters with moderate concentrations and a large number of superstars are considered . unfortunately , something is lost in the above modification : in the advanced stages of core collapse a large fraction of stars is contained within the innermost radial bins . in this situation , the spatial resolution of the grid is not adequate to follow the fast evolution of the cluster core and produces an unrealistic delay of the core collapse . moreover , another drawback of this method ( which is in common with the canonical method adopted by h71 ) is that the potential depends on the position of the superstars . as positions are randomly extracted , it is possible that fluctuations in the potential are present when a small number of superstars are considered . this effect is particularly strong in the cluster core where the potential is determined by few stars and can produce a `` spurious relaxation '' . h71 have shown that such an effect is negligible when a large number ( @xmath29 ) of particles is used . however , a pernicious effect is produced by such fluctuations when the correction for the mechanical work made by the potential ( step ( iv ) of the above scheme ; see stodolkiewicz 1982 ) is considered . indeed , fluctuations are erroneously interpreted as real potential changes introducing spurious corrections in the stars energies . on the long term , this produces a drift in the total cluster energy accelerating the relaxation process even when a number of superstars as large as @xmath30 is considered . for this reason we developed another method ( hereafter referred as the _ integral method _ ) to determine the cluster potential in the grid knots . consider a star with energy @xmath31 and angular momentum @xmath32 moving in the potential @xmath5 . when the star is in the j - th radial bin @xmath33 its radial component of the velocity will be given by eq . [ vr_eq ] and its radial component of the acceleration will be @xmath34 these quantities have been estimated using the potential and its derivative calculated in the previous time - step . in the approximation that the star moves with a uniformly accelerated motion , the time spent to cross the interval ( @xmath35 , @xmath36 ) is @xmath37 the probability to find the star in that interval will be @xmath38 where @xmath39 is the orbital period of the superstar . the potential at each point of the grid can be then calculated through the relation @xmath40 where @xmath21 is the index such that @xmath41 . this method has the advantage to depend only on the energies and angular momenta of the stars and not on the randomly extracted positions of the superstars . in practice , it is equivalent to compute the potential from an infinite number of randomly extracted positions thus eliminating the problem of fluctuations . moreover , it allows to estimate the orbital period of each superstar ( eq . [ per_eq ] ) which will be used in the escape algorithm ( see sect . [ esc_sec ] ) . there are two main drawbacks of this methods : first , to determine @xmath42 and @xmath43 it is necessary to use the potential profile calculated in the previous time - step . for this reason , at odds with the _ fast method _ , the potential can be computed only on grid of positions which is fixed in time . therefore , a small step - size is required from the beginning of the simulation to adequately follow the cluster evolution also in the advanced stages of core collapse . we found that a good sampling of the potential profile during the entire cluster evolution is provided by the choice of @xmath44 where @xmath45 is the cluster core radius ( see eq . [ rc_eq ] ) at the beginning of the simulation . second , the above method is computationally expensive and almost all the improvement provided by the adoption of the evenly spaced grid is lost . to optimize the speed of the simulation without losses of accuracy , during the simulation we adopt the _ fast method _ when @xmath46 , and switch to the _ integral method _ when the above condition is not satisfied . in this section we describe the treatment of the external tidal field in our monte carlo code . here we considered two kinds of external field potential : the first generated by a point mass galaxy with mass @xmath47 and the second by the analytical bulge+disk+halo potential defined in johnston , spergel & hernquist ( 1995 ; hereafter j95 ) . the orbit of the cluster within these potentials has been computed starting from its orbital energy and z - component of the angular momentum using a fourth - order runge - kutta algorithm providing an accuracy in terms of energy conservation better than @xmath48 during the entire evolution . the effect of dynamical friction has been neglected since it is expected to be negligible on the star cluster scale ( gnedin et al . 1999 ) . the presence of an external field imply an increase of star losses because of two main processes : escape of stars through the cluster boundaries and tidal shocks ( this last process occurring only when the external potential has a bulge / disk component ) . we discuss the algorithms to include the above processes in the following sections . and @xmath49 moving on a circular orbit at @xmath50 around a point - mass galaxy of @xmath51 . the dashed , solid and dotted lines ( red , black and blue in the online version of the paper ) indicate the effective potential along the x- , y- and z - axis , respectively.,width=328 ] as already introduced in sect . [ intro_sec ] , the presence of an external tidal field imply the presence of a tidal cut in the cluster potential . stars with enough energy can cross the cluster tidal radius and evaporate from the cluster on a timescale comparable to the star orbital period . a star orbiting around a cluster immersed in an external field feels an effective potential given by the combination of the cluster potential ( @xmath52 ) , the external field potential ( @xmath53 ) and a term linked to the angular motion of the cluster . @xmath54 where * @xmath55 * is the angular speed of the cluster and * r * is the position vector of the star in a reference frame centered on the center of mass of the galaxy+cluster system rotating with angular speed @xmath56 . note that the jacobi integral associated to the above effective potential is conserved only in the particular case of a cluster moving on a circular orbit within a spherical potential . however , when other orbits / potentials are considered , the timescale on which such an integral changes is longer than the dynamical time of most cluster stars so that we can assume it instantaneously conserved . consider a cartesian reference system centered on the cluster with the x - axis pointed toward the galaxy center , the y - axis parallel to the galactic plane and directed toward the cluster rotation and the z - axis perpendicular to the previous axes . the acceleration felt by the a star approaching the tidal radius @xmath57 will be @xmath58 where @xmath59is the position vector of the star , @xmath60 is the cluster mass and @xmath61 is the acceleration due to the cluster angular motion . the tidal radius is defined as the distance where the projection of the above acceleration on * r * is zero which corresponds to the radius at which the effective potential has a local maximum . note that , according to the above definition , both the tidal radius and the effective potential depend on the direction of the escape . it can be shown ( see appendix ) that the shortest and less energetic tidal radius occurs in correspondence of the x - direction ( i.e. the 2nd and 3rd lagrangian points ) while in the z - direction the effective potential is a growing function of the radius and no maxima exist ( see fig . [ pot ] ) . for this reason stars escape preferentially from the lagrangian points . to determine wether a star escapes from the cluster we defined three distinct criteria . the first criterion we adopted is that the star energy and angular momentum allow the motion across the tidal radius . for this purpose we extracted two random numbers ( @xmath62 and @xmath63 ) uniformly distributed between 0 and 1 and defined the coefficients @xmath64 these coefficient univocally define a direction of escape such that @xmath65 . the tidal radius and effective potential in the defined direction are then calculated ( see appendix ) as well as the mechanical work made by the external potential on the superstar ( @xmath66 ; see below ) . the first criterion is satisfied if @xmath67 the second criterion is based on the fact that the escaping star needs a timescale comparable to its orbital period ( @xmath68 ) to reach the tidal radius . this produces a delay in the escape process . this can be crucial when eccentric orbits are considered : in this case , @xmath57 changes with time and the star satisfies eq . [ crit1_eq ] only in a limited time interval . if such time interval is short compared to its orbital period , there is only a small probability for the star to escape . we then extracted a random number @xmath69 uniformly distributed between 0 and 1 and calculated the orbital period of the star to reach the tidal radius using eq . [ per_eq ] . we assumed that the the star can escape during the time - step @xmath6 if @xmath70 once the first two cirteria are satisfied the star can escape from the cluster . after this phase the star moves in the galactic potential as an independent satellite following an epicyclic orbit along either the trailing or the leading arm of the cluster tidal tails . during this motion , the distance of the star from the cluster center oscillates according to the orbital phase of the cluster . to definitively escape from the cluster attraction there is a typical timescale which depends on the star s energy and on the cluster orbital parameters . moreover , if the cluster follows an eccentric orbit its tidal radius grows when the cluster leaves the perigalacticon and it can possibly exceed the distance of the previously escaped star . in this case the star is re - captured . to account for this effect , when a star satisfies the two above mentioned criteria its position and velocity are calculated using eq . [ pos_eq ] ( @xmath71 ) and @xmath72 where @xmath73 and @xmath74 is a random number uniformly distributed between 0 and 1 . the above phase - space coordinates are transformed in the galactic reference system , added to the cluster systemic coordinates and the orbit of both the star and the cluster within the galactic potential are followed using a fourth - order hermite integrator with an adaptive timestep for an entire cluster orbital period . during its motion outside the cluster the distance of the star from the cluster and the tidal radius are calculated . the star is removed from the simulation if the conditions @xmath75 are satisfied during a cluster orbital period . in the above equations @xmath76 is the maximum tidal radius reached by the cluster during an orbital period , @xmath6 is the time - step , @xmath68 is the star orbital period within the cluster and @xmath69 is the random number extracted for the criterion in eq . [ crit2_eq ] . the last modification regards the time - step adopted in the simulation when the external field is present . in joshi et a. ( 2001 ) the time - step is defined to ensure small deflection angles when introducing the perturbations to the supertars energies and angular momenta . however , another requirement is that the jacobi integral should not significantly vary within the time - step . so , we adoted as time - step the minimum between the time - step defined by joshi et al . ( 2001 ) and 0.01@xmath77 , where @xmath77 is the cluster orbital period . when a cluster passes through the galactic disk or close to the galactic bulge the gravitational field of these two components exerts a compressive force which is superposed on the cluster s own gravitational field . this process is known as `` disk / bulge shocking '' ( ostriker , spitzer & chevalier 1972 ; aguilar , hut & ostriker 1988 ) . the theory of tidal shocks has been studied in the past by many authors and applied to orbit - averaged fokker - planck codes by gnedin & ostriker ( 1997 , 1999 ) and allen , moreno & pichardo ( 2006 ) . this effect can be viewed as a consequence of the mechanical work made by the external potential during the orbit . indeed , any variation of the potential shape during the cluster evolution produces a work on the stars which is equal to the instantaneous potential variation . the variation of the internal potential can be due to many processes ( secular dynamical evolution , evaporation of stars , stellar evolution , etc . ) and is taken into account using the prescriptions by stodolkiewicz ( 1982 ) . in time - dependent external potentials ( e.g. when an eccentric orbit and/or non - spherical potentials are considered ) also the external potential makes a work on the stars . this is particularly important when fast changes in the external potential occurs , like in the case of the disk crossing and the perigalactic passages . in this case , the binding energy per unit mass of the superstars located in the outskirts of the cluster changes on a timescale comparable to the dynamical time . this effect facilitates the escape of stars and can accelerate the cluster dynamical evolution . in analogy with stodolkiewicz ( 1982 ) , at each time - step we estimated the work made by the external potential on each superstar as the average between the potential changes at the positions of the superstar in two subsequent time - steps @xmath78/2\\\end{aligned}\ ] ] such a correction is updated during the orbit and temporary added to the energy of the superstar only to verify the escape criteria defined above . this correction can not indeed be permanently added the superstars energies since these are distributed across the cluster adopting the internal potential only ( i.e. assuming the cluster as isolated ) while the external potential is taken into account only in a subsequent step when the escape criteria are verified . [ cols="<,^,^,^,^,^,^,^,^,^ , > " , ] = 0 , 0.14 , 0.33 and 0.6 are drawn with solid , dashed , dotted and dot - dashed lines , respectively.,width=328 ] but for the set of simulations with 42000 particles.,width=328 ] = 2 , 3 , and 4 kpc are drawn with solid , dashed , and dotted lines , respectively.,width=328 ] we tested the prediction of our code with a set of collisional n - body simulations performed with nbody6 ( aarseth 1999 ) . all simulations ( both monte carlo and n - body ) were run with n=21000 and n=42000 single - mass particles as a compromise to ensure a large number statistics and to limit the computational cost of n - body simulations . the standard @xmath79 parameter has been used to control the time step and set an energy error tolerance of @xmath80 . with these choices we got a relative error in energy smaller than @xmath81 at the end of every simulation run . for each simulation the cluster mass contained within the apocentric tidal radius and the lagrangian radii have been calculated and compared . lagrangian radii have been calculated from the cumulative distribution of distances from the cluster center of the particles . here we considered the radii containing 1% , 2% , 5% , 10 - 90% of the cluster mass within one apocentric tidal radius . we considered three different cases : _ i ) _ an isolated cluster with a plummer ( 1911 ) profile ; _ ii ) _ a cluster with a mass @xmath82 , a king ( 1966 ) profile with @xmath83 and @xmath49 orbiting in a galactic potential generated by a point - mass of @xmath84 with an apogalacticon at @xmath85 , and _ iii ) _ the same cluster of ( ii ) orbiting in the bulge+disk+halo galactic potential defined by j95 . in cases _ ( ii ) _ and _ ( iii ) _ a set of simulations with different eccentricities and orbits have been considered . the whole set of simulations is summarized in table 1 . in fig . [ plum ] the evolution of the lagrangian radii of the isolated plummer ( 1911 ) model as a function of the initial half - mass relaxation time is compared with that predicted by the n - body simulation with n=8192 particles by baumgardt , hut & heggie ( 2002 ) . the agreement is excellent with only a small discrepancy near the core collapse for the innermost radii . the core collapse occurs after @xmath86 when the cluster have lost @xmath873% of its stars . both quantities are in good agreement with the results of baumgardt et al . the excellent agreement with the n - body simulation indicates that the monte carlo code well reproduces the relaxation process until the core collapse . another set of simulations have been performed considering an external tidal field generated by a point - mass galaxy . in such a potential stars feel the tidal cut but are not subject to disk shocks . these simulations are therefore suited to test the escape from the cluster boundary both in case of circular and eccentric orbits . simulations have been run until core collapse . the subsequent evolution is largely influenced by the presence of binaries which form during the maximum density phase . as our monte carlo code does still not account for this process it can not reproduce properly such an evolutionary stage . the evolution of the cluster mass and of the lagrangian radii for this set of simulations are compared with the results of n - body simulations in fig . [ pm_mass ] and [ pm_lag ] , respectively . it is apparent that in all the simulations with eccentricity @xmath88 the monte carlo tends to systematically overpredict the mass - loss rate with respect to n - body simulations , although such a discrepancy is always within 5% . the evolution of the lagrangian radii is also well reproduced during the entire cluster evolution . a different situation is for the simulation of the most eccentric ( @xmath89 ) orbit . in this case , while the evolution of the bound mass is well reproduced by our monte carlo code , the lagrangian radii are strikingly different . in particular , while the n - body simulation predict an overall contraction of the cluster , the monte carlo code predicts a quick expansion followed by a quick collapse of the core . a possible reason for such a discrepancy is that in this last orbit the tidal radius penetrates into the cluster at pericenter leaving a significant fraction of cluster stars free to escape . when this occurs , the criteria defined in sect . [ esc_sec ] are not adequate anymore and an unrealistically large fraction of stars evaporate from the cluster in a short amount of time . the loss of potential energy is larger than that in kinetic energy and the cluster expands , furthermore increasing the escape efficiency . as the number of stars in the cluster becomes smaller the relaxation process speeds up and the core quickly collapse . summarizing , it appears that the treatment of the external tidal field described in sect . [ esc_sec ] is effective when the escape rate of stars during a cluster orbital period is smaller than a critical value . in fig . [ pm_mass42 ] and [ pm_lag42 ] the comparison between the bound mass and lagrangian radii for the same set of simulations with 42000 particles are shown , respectively . also in this case , the agreement is good in all the simulations with moderate eccentricity ( @xmath88 ) both in terms of the mass and of the lagrangian radii evolution . again , for the most eccentric simulation , while the bound mass evolution is fairly well reproduced , the lagrangian radii of the monte carlo simulation show the same unproper behaviour already noticed in the simulations with 21000 particles . to properly estimate the range of validity of our code , we performed another set of simulations where a cluster with 21000 particles and the same structural characteristics of the previous simulations has been launched on circular orbits at different distances from the point - mass galaxy ( @xmath902,3 and 4 kpc ) . the results of such an experiment are shown in fig.s [ pm_testmass ] and [ pm_testlag ] . as expected , monte carlo simulations related to clusters moving at large galactocentric distances show a good agreement with n - body ones . on the other hand , as the tidal field becomes stronger the mass - loss rate predicted by the monte carlo code exceeds that that of the n - body simulations . as a consequence , the cluster internal dynamical evolution accelerates and the core collapse is anticipated . according to the escape criteria defined in sect . [ esc_sec ] , the timescale at which a star escapes from the cluster is of the order of a few dynamical times . while this timescale depends on the energy and angular momentum of each star and on the cluster potential , it is expected to scale with the global quantity @xmath91 . it is therefore useful to introduce the mass - loss rate per half - mass dynamical time @xmath92 the above parameter has been calculated during the cluster evolution and averaged from the beginning of the simulation to the core collapse ( see table 1 ) . to define a criterion of validity of our simulations we correlated the values of @xmath93 with the discrepancy between bound mass fraction predicted by monte carlo and n - body simulations measured at core - collapse in this set of simulations . we obtain @xmath93=-0.00007 , -0.00013 and -0.00040 and @xmath94=1.2% , 3.0% and 15.9% for the simulations at 4 , 3 and 2 kpc , respectively . on the basis of the above comparison and defining a resonable agreement at @xmath955% , we adopt a value of @xmath96 - 0.0002 as a conservative limit of validity of our code . the last set of simulations considers a cluster immersed in the three - components bulge+disk+halo external potential defined by j95 . we considered a cluster lying at an initial distance of 10 kpc from the galactic center with two different eccentricities and two different heights above the galactic plane . in each case the intensities of the tidal shocks are different since the cluster cross the disk and approaches its pericenter at different distances from the galactic center with different velocities . the three considered orbits in the x - y , x - z and r - z planes are shown in fig . the mass evolution of the three simulations are compared with those predicted by the associated n - body simulations in fig.s [ compl_mass ] and [ compl_mass42 ] for the set with 21000 and 42000 particles , respectively . it can be noted that in all the simulations with 21000 particles the agreement is good ( within 5% ) during the entire evolution . on the other hand , in simulations with 42000 particles a tendency of the monte carlo code to underestimate the cluster mass - loss rate is noticeable . this is particularly apparent in the simulations with eccentric orbits where the difference with respect to the prediction of the n - body simulation reach @xmath8710% at the core - collapse . such a discrepancy is in the opposite sense of what observed in simulations run within a point - mass potential , where the mass - loss rates were slightly overpredicted . note that in the three simulations the average mass - loss rate is @xmath97 i.e. below the critical limit where significant differences have been noticed in the simulations within a point - mass external potential . it is also interesting to note that the mass evolution of the two simulations with different heights above the galactic plane ( k - j95-e033 and k - j95-e033z ) yield to a quite similar residual mass after 12 gyr . this is not surprising since at the moment of the disk crossing , although the disk shocks are more intense in simulation k - j95-e033z because of the largest velocity of the cluster , the disk density is in both cases relatively small . also , the pericentric distance of both orbits is 5 kpc , significantly larger with respect to the bulge half - mass radius ( @xmath871.69 kpc ) . so , in the above cases , both disk and bulge shocks have only a little impact on the cluster structural and dynamical evolution . in fig.s [ compl_lag ] and [ compl_lag42 ] the evolution of the lagrangian radii of the three simulations are compared with the predictions of the corresponding n - body simulations for the sets with 21000 and 42000 particles , respectively . again the agreement is good during the entire cluster evolution . in this paper we presented a new implementation of the monte carlo method to simulate the evolution of star clusters including for the first time the effect of a realistic tidal field and the possibility to consider eccentric orbits . the effect of the external field has been taken into account considering both the process of evaporation through the cluster boundaries and the effect of tidal shocks . the adopted algorithm is based on the theory of evaporation taking into account for the random direction of the escape , the time delay due to the star orbital period and the occurrence of the re - capture process . the comparison with direct n - body simulations indicates an excellent agreement in an isolated cluster and a good agreement ( within 5 - 10% ) in clusters within a tidal field of moderate intensity , in terms of the evolution of both the mass and the lagrangian radii . this indicates that both the process of two - body relaxation and the evaporation of stars are accurate enough . on the other hand , a significant discrepancy is apparent when an extremely intense tidal stress is present ( @xmath98 ) . the reason of the disagreement relies on the failure of some ( maybe all ) of the criteria adopted to take into account the escape of stars when these apply to a significant fraction of cluster stars . it is worth noting that such an intense tidal field should be felt by only a small fraction of globular clusters ( only 2 out the 53 globular clusters considered by allen et al . 2006 , 2008 ) . so this code can provide an efficient tool to study the evolution of present - day globular clusters . on the other hand , the code developed here could fail to reproduce the initial stages of cluster evolution : in this case , the fast potential changes due to stellar evolution driven mass loss likely lead to a situation of roche - lobe overfilling where the critical mass loss rate is easily reached . consider that one of the basic assumptions of the monte carlo method is spherical symmetry which conflicts with the presence of the external field . in this situation star orbits are not expected to be planar and it is not guaranteed that two superstars with contiguous ranking in distance from the cluster center are neighbors . so , any treatment of the external field is expected to fail when somehow strong tides are considered . the performance of the code are also very good : the simulation k - pm - e0 - 21k presented here with 21000 particles takes @xmath8745 minutes ( `` clock on the wall '' time ) on a single five - years old asus machine equipped with a single intel core t5800@2ghz processor , while the corresponding n - body simulation takes @xmath8730 hours with a cluster node equipped with an intel xeon e5645 cpu ( 12 cores ) and a nvidia tesla m2090 ( 512 cuda cores ) . in spite of the relatively small considered number of particles , the above performance is good considering that the code include several cycles which can be in principle easily parallelized thus reducing the computational cost of simulations . assuming a scaling of the computation time with the number of particles as @xmath99 , it is possible to run simulations with @xmath100 with computation time of few days . the code already include other features to properly simulate the cluster evolution like the inclusion of a mass spectrum , the mass - loss driven by stellar evolution and the direct integration of three- and four - body interactions following the prescriptions by joshi et al . ( 2001 ) and fregeau et al . however , the predictions of the code with these implemented features have not been tested and are not presented here . a forthcoming paper will introduce these features in the next future . as acknowledges the prin inaf 2011 `` multiple populations in globular clusters : their role in the galaxy assembly '' ( pi e. carretta ) . amb whish to thank the lady davis foundation . we thank the anonymous referee for his / her helpful comments and suggestions . we warmly thank holger baumgardt to have provided his n - body simulations and zeinab khorrami , carlo nipoti , luca ciotti and enrico vesperini for useful discussions . 99 aarseth s. j. , 1999 , pasp , 111 , 1333 aguilar l. , hut p. , ostriker j. p. , 1988 , apj , 335 , 720 allen c. , moreno e. , pichardo b. , 2006 , apj , 652 , 1150 allen c. , moreno e. , pichardo b. , 2008 , apj , 674 , 237 angeletti l. , giannone p. , 1977 , ap&ss , 50 , 311 angeletti l. , giannone p. , 1977 , a&a , 58 , 363 baumgardt h. , 2001 , mnras , 325 , 1323 baumgardt h. , hut p. , heggie d. c. , 2002 , mnras , 336 , 1069 casertano s. , hut p. , 1985 , apj , 298 , 80 cohn h. , 1980 , apj , 242 , 765 ferraro f. r. , et al . , 2012 , natur , 492 , 393 fregeau j. m. , rasio f. a. , 2007 , apj , 658 , 1047 fregeau j. m. , grkan m. a. , joshi k. j. , rasio f. a. , 2003 , apj , 593 , 772 fukushige t. , heggie d. c. , 2000 , mnras , 318 , 753 giersz m. , 1998 , mnras , 298 , 1239 giersz m. , 2001 , mnras , 324 , 218 giersz m. , spurzem r. , 2003 , mnras , 343 , 781 giersz m. , heggie d. c. , hurley j. r. , hypki a. , 2013 , mnras , 431 , 2184 gnedin o. y. , ostriker j. p. , 1997 , apj , 474 , 223 gnedin o. y. , ostriker j. p. , 1999 , apj , 513 , 626 gnedin o. y. , lee h. m. , ostriker j. p. , 1999 , apj , 522 , 935 harfst s. , portegies zwart s. , stolte a. , 2010 , mnras , 409 , 628 henon m. , 1969 , a&a , 1 , 223 hnon m. h. , 1971 , ap&ss , 14 , 151 hernquist l. , 1990 , apj , 356 , 359 hurley j. r. , pols o. r. , aarseth s. j. , tout c. a. , 2005 , mnras , 363 , 293 johnston k. v. , spergel d. n. , hernquist l. , 1995 , apj , 451 , 598 joshi k. j. , nave c. p. , rasio f. a. , 2001 , apj , 550 , 691 joshi k. j. , rasio f. a. , portegies zwart s. , 2000 , apj , 540 , 969 king i. , 1962 , aj , 67 , 471 king i. r. , 1966 , aj , 71 , 64 larson r. b. , 1970 , mnras , 147 , 323 miyamoto m. , nagai r. , 1975 , pasj , 27 , 533 morscher m. , umbreit s. , farr w. m. , rasio f. a. , 2013 , apj , 763 , l15 ostriker j. p. , spitzer l. , jr . , chevalier r. a. , 1972 , apj , 176 , l51 plummer h. c. , 1911 , mnras , 71 , 460 read j. i. , wilkinson m. i. , evans n. w. , gilmore g. , kleyna j. t. , 2006 , mnras , 366 , 429 ross d. j. , mennim a. , heggie d. c. , 1997 , mnras , 284 , 811 sippel a. c. , hurley j. r. , 2013 , mnras , 430 , l30 spitzer l. , 1987 , princeton , nj , princeton university press spitzer l. , jr . , hart m. h. , 1971 , apj , 164 , 399 takahashi k. , 1995 , pasj , 47 , 561 takahashi k. , baumgardt h. , 2012 , mnras , 420 , 1799 stodolkiewicz j. s. , 1982 , aca , 32 , 63 zonoozi a. h. , haghi h. , kpper a. h. w. , baumgardt h. , frank m. j. , kroupa p. , 2014 , mnras , 440 , 3172 zonoozi a. h. , kpper a. h. w. , baumgardt h. , haghi h. , kroupa p. , hilker m. , 2011 , mnras , 411 , 1989 as discussed in sect . [ esc_sec ] , when the cluster is immersed in an external tidal field its stars feel an effective potential due to the combination of the internal cluster potential ( @xmath52 ) , the external field potential ( @xmath53 ) and a term linked to the cluster angular motion . such an effective potential does not have a spherical symmetry so that the distances of its local maxima ( tidal radii ) are direction dependent . in the next sections the tidal radius and the effective potential as a function of the direction of escape are derived both for the case of an external field generated by a point - mass and for the complex bulge+disk+halo potential by j95 . the external potential generated by a point - mass galaxy is @xmath101 where @xmath102 is the mass of the galaxy and @xmath16 the distance of a test particle from the point - mass . consider a star at @xmath16 orbiting around a cluster located at @xmath103 . in the cartesian reference frame centered on the cluster with the x - axis pointed toward the point - mass galaxy , the y - axis in the direction of the rotation and the z - axis in the direction of the angular momentum , be @xmath104 the position vector of the star . the distance of the star from the galaxy will be @xmath105 the tidal radius is defined as the point where the projection of the acceleration felt by the star ( eq . [ acc_eq ] ) on @xmath106 is zero . the acceleration is composed by three terms : one associated to the cluster potential , one associated to the external field and another associated to the cluster angular motion . the term associated to the external field is @xmath108\nonumber\\ & = & \frac{g m_{g}}{|{\bf r'}| |{\bf r}|^{5}}[x^{2}\left(-2 r_{cl}^{2}+|{\bf r'}|^{2}+4 x r_{cl}-3 x^{2}\right)+y^{2}\left(r_{cl}^{2}+|{\bf r'}|^{2}-2 x r_{cl}-3 y^{2}\right)+z^{2}\left(r_{cl}^{2}+|{\bf r'}|^{2}-2 x r_{cl}-3 z^{2}\right)]\nonumber\end{aligned}\ ] ] in the limit @xmath109 the above term can be approximated to @xmath110 . the acceleration due to the angular motion of the cluster is @xmath111 where @xmath55 is the angular speed of the cluster which , in an elliptic orbit with eccentricity @xmath112 and apogalactic distance @xmath113 is @xmath114 the third term of equation [ pmat_eq ] has null projection on @xmath59 since it is orthogonal to @xmath59 by definition . in the cluster reference system the remaining terms ( corresponding to the coriolis and the centrifugal+tidal acceleration ) have projection on @xmath59 @xmath115 . the total acceleration in the direction of @xmath59 at @xmath57 will be given by the sum of eq.s [ pmc_eq ] , [ pmf_eq ] and [ pma_eq ] @xmath116 where @xmath117 @xmath4 is the angular momentum of the superstar and @xmath74 is a random number uniformly distributed between 0 and 1 ( see eq . [ vel_eq ] ) . it is interesting to note that the coriolis acceleration is directed toward the direction of escape when the star is on a prograde orbit , while it is directed toward the cluster when retrograde orbits are considered . this is the reason why stars on prograde orbits escape more easily from the cluster ( see also hnon 1969 ; read et al . 2006 ) . note that over a large number of random extractions of @xmath74 the term @xmath121 has null mean . in this case the tidal radius will simply be @xmath122 at pericenter and in correspondence to the lagrangian point @xmath123 will be @xmath124 , and the above equation reduces to that reported by king ( 1962 ) . we consider the three components galactic model by j95 which consists of the superposition of an hernquist ( 1990 ) bulge , a miyamoto & nagai ( 1975 ) disk and a logarithmic halo @xmath127 with @xmath128 , @xmath129 , @xmath130 , @xmath131 , @xmath132 , @xmath133 and @xmath134 . here we adopted a convenient cylindrical reference system @xmath135 , @xmath136 is the distance from the galactic center and @xmath137 is the angle between the z - axis and the position vector . in this reference system the cluster will be in @xmath138 . define an alternative cartesian rotating reference frame centered in the cluster center with the x - axis pointed toward the galactic center , the y - axis parallel to the galactic plane and pointed toward the direction of the cluster rotation and the z - axis perpendicular to the other axes . consider a star in @xmath139 moving around a cluster immersed in the above external potential . the coordinate transformation to the first reference system are @xmath140 and @xmath105 in analogy with what done in sect . [ pm_sec ] , we calculate the projection of the acceleration felt by the star ( eq . [ acc_eq ] ) on @xmath106 . the internal acceleration is always given by eq . [ pmc_eq ] . the term associated to the external field can be calculated separately for the three galactic components @xmath141\nonumber\\ \frac{{\bf r'}}{|{\bf r'}| } \cdot ( { \bf r ' } \cdot { \bf \nabla } ) \nabla \phi_{d}&=&g m_{d } |{\bf r'}|\left\{\frac{1+\frac{a b^{2}}{(b^{2}+z_{cl}^{2})^{\frac{3}{2}}}(\tilde{x}^{2}cos^{2}\theta+\tilde{z}^{2}sin^{2}\theta ) } { \left[r_{cl}^{2}+\left(a+\sqrt{b^{2}+z_{cl}^{2}}\right)^{2}\right]^{\frac{3}{2}}}- 3\frac{\left(r_{cl}^{2}+\frac{2 a z_{cl}^{2}}{\sqrt{b^{2}+z_{cl}^{2}}}\right ) \tilde{x}^{2}+\frac{a^{2 } z_{cl}^{2}}{b^{2}+z_{cl}^{2}}(\tilde{x}^{2}cos^{2}\theta+\tilde{z}^{2}sin^{2}\theta ) } { \left[r_{cl}^{2}+\left(a+\sqrt{b^{2}+z_{cl}^{2}}\right)^{2}\right]^{\frac{5}{2}}}\right\}\nonumber\\ \frac{{\bf r'}}{|{\bf r'}| } \cdot ( { \bf r ' } \cdot { \bf \nabla } ) \nabla \phi_{h}&=&\frac{2 v_{0}^{2 } |{\bf r'}|}{(d^{2}+r_{cl}^{2})^{2}}[(d^{2}+r_{cl}^{2})(\tilde{y}^{2}+\tilde{z}^{2})+(d^{2}-r_{cl}^{2})\tilde{x}^{2 } ] . \label{complf_eq}\end{aligned}\ ] ] where @xmath142 the acceleration due to the angular motion of the star is given by eq . [ pmat_eq ] , where @xmath143 and @xmath144 is the z - component of the cluster angular momentum . as in the case of the point - mass potential , only the coriolis ( * @xmath145 * ) and the centrifugal+tidal ( * @xmath146 * ) terms of the acceleration have a projection on @xmath59 different from zero . so @xmath147- \dot{\theta}_{cl}^{2 } |{\bf r'}|(\tilde{x}^{2}+\tilde{z}^{2})+ \frac{2 l_{z } \dot{\theta}_{cl } |{\bf r'}|}{r_{cl}^{2 } } \tilde{y } ( \tilde{x}~cos\theta_{cl}-\tilde{z}~sin\theta_{cl})\nonumber\\ & & -2\dot{\theta}_{cl}(\tilde{x}~v_{z}-\tilde{z}~v_{x})-\frac{2 l_{z}}{r_{cl}^{2}}[\tilde{x}~v_{y}sin\theta_{cl}-\tilde{y } ( v_{x}sin\theta_{cl}+v_{z}cos\theta_{cl})+\tilde{z}~v_{y}cos\theta_{cl } ] . \label{compla_eq}\end{aligned}\ ] ] it is convenient to introduce the characteristic densities @xmath148\right\}. \label{complb_eq}\end{aligned}\ ] ] where @xmath149 is either the acceleration due to the i - th galactic component ( @xmath150 ) or the centrifugal+tidal acceleration , @xmath4 is the angular momentum of the superstar and @xmath74 is a random number uniformly distributed between 0 and 1 ( see eq . [ vel_eq ] ) . it is therefore possible to combine eq.s [ pmc_eq ] , [ complf_eq ] , [ compla_eq ] and [ complb_eq ] to obtain the projection of the total acceleration felt by the star on @xmath59 latexmath:[\[{\bf \nabla \phi_{eff}}=\frac{g m_{cl}}{|{\bf r'}|^{2}}-g \gamma the tidal radius will therefore be @xmath119 with @xmath152 again , the coriolis term has null mean over a large number of random extractions of @xmath74 , so we can calculate @xmath57 assuming @xmath153 . @xmath154 note that at the lagrangian point @xmath123 the above formula is equivalent to the estimate provided by allen et al .
|
we present a new implementation of the monte carlo method to simulate the evolution of star clusters .
the major improvement with respect to the previously developed codes is the treatment of the external tidal field taking into account for both the loss of stars from the cluster boundary and the disk / bulge shocks .
we provide recipes to handle with eccentric orbits in complex galactic potentials .
the first calculations for stellar systems containing 21000 and 42000 equal - mass particles show good agreement with direct n - body simulations in terms of the evolution of both the enclosed mass and the lagrangian radii provided that the mass - loss rate does not exceed a critical value .
[ firstpage ] methods : numerical methods : statistical stars : kinematics and dynamics globular clusters : general
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
polynomial series are of great importance in control theory . both continuous and discrete polynomial series are useful in approximating state and/or control variables , in modal reduction , optimal control , and system identification , providing effective and efficient computational methods @xcite . from recent years , the theory of control for discrete and continuous time is being unified and extended by using the formalism of time scales : see @xcite and references therein . looking to the literature on time scales , one understands that such unification and extension is not unique . two main directions are being followed : one uses @xmath1-derivatives while the other chooses @xmath2-derivatives instead . in this paper we adopt the more general notion of diamond-@xmath0 derivative @xcite , and give the first steps on a correspondent theory of polynomial series . as particular cases , for @xmath3 we get @xmath1-polynomial series ; when @xmath4 we obtain @xmath2-series . by choosing the time scale to be the real ( integer ) numbers , we obtain the classical continuous ( discrete ) polynomial series . diamond - alpha derivatives have shown in computational experiments to provide efficient and balanced approximation formulas , leading to the design of more reliable numerical methods @xcite . we claim that the combined dynamic polynomial series here introduced are useful in control applications . here we give only very short introduction ( with basic definitions ) on three types of calculus on time scales . for more information we refer the reader to @xcite . by a time scale , here denoted by @xmath6 , we mean a nonempty closed subset of @xmath7 . as the theory of time scales give a way to unify continuous and discrete analysis , the standard cases of time scales are @xmath8 , @xmath9 , or @xmath10 , @xmath11 . for @xmath12 , the forward jump operator @xmath13 and the graininess function @xmath14 are defined by @xmath15 and @xmath16 if @xmath17 ; @xmath18 . moreover , we have the backward operator @xmath19 and the backward graininess function @xmath20 defined by @xmath21 and @xmath22 if @xmath23 ; @xmath24 . in the continuous - time case , when @xmath8 , we have @xmath25 and @xmath26 for all @xmath27 . in the discrete - time case , @xmath28 , @xmath29 , and @xmath30 for each @xmath31 . for the composition between a function @xmath32 and functions @xmath33 and @xmath34 , we use the abbreviations @xmath35 and @xmath36 . a point @xmath37 is called left - scattered ( right - scattered ) if @xmath38 ( @xmath39 ) . a point @xmath37 is called left - dense ( right - dense ) if @xmath40 ( @xmath41 ) . the set @xmath42 is defined by @xmath43 $ ] if @xmath44 , and @xmath45 if @xmath46 ; the set @xmath47 by @xmath48 if @xmath49 , and @xmath50 if @xmath51 . moreover , @xmath52 @xmath53 and @xmath54 . for a function @xmath32 , we define the @xmath1-derivative of @xmath55 at @xmath56 , denoted by @xmath57 , to be the number , if it exists , with the property that for all @xmath58 , exists a neighborhood @xmath59 of @xmath56 such that for all @xmath60 , @xmath61 . function @xmath55 is said to be @xmath1-differentiable on @xmath42 provided @xmath57 exists for all @xmath56 . the @xmath2-derivative of @xmath55 , denoted by @xmath62 , is defined in a similar way : it is the number , if it exists , such that for all @xmath58 there is a neighborhood @xmath63 of @xmath64 such that for all @xmath65 @xmath66 . function @xmath55 is said to be @xmath2-differentiable on @xmath47 provided @xmath62 exists for all @xmath64 . the classical settings are obtained choosing @xmath67 and @xmath68 : 1 . let @xmath8 . then , @xmath69 and @xmath55 is @xmath1 and @xmath2 differentiable if and only if it is differentiable in the ordinary sense . 2 . let @xmath10 , @xmath11 . then , @xmath70 and @xmath71 always exist . it is possible to establish some relationships between @xmath1 and @xmath2 derivatives . @xcite ( a ) assume that @xmath32 is delta differentiable on @xmath42 . then , @xmath55 is nabla differentiable at @xmath37 and @xmath72 for all @xmath64 such that @xmath73 . ( b ) assume that @xmath32 is nabla differentiable on @xmath47 . then , @xmath55 is delta differentiable at @xmath37 and @xmath74 for all @xmath56 such that @xmath75 . a function @xmath76 is called rd - continuous provided it is continuous at right - dense points in @xmath6 and its left - sided limits exist ( finite ) at left - dense points in @xmath6 . the class of real rd - continuous functions defined on a time scale @xmath6 is denoted by @xmath77 . if @xmath78 , then there exists a function @xmath79 such that @xmath80 . the delta - integral is defined by @xmath81 . similarly , a function @xmath76 is called ld - continuous provided it is continuous at left - dense points in @xmath6 and its right - sided limits exist ( finite ) at right - dense points in @xmath6 . the class of real ld - continuous functions defined on a time scale @xmath6 is denoted by @xmath82 . if @xmath83 , then there exists a function @xmath84 such that @xmath85 . in this case we define @xmath86 . let @xmath10 , @xmath11 , and @xmath87 . then , one has : @xmath88 , @xmath89 . @xcite let @xmath90 , @xmath91 , and @xmath32 . the diamond - alpha derivative of @xmath55 at @xmath37 is defined to be the value @xmath92 , if it exists , such that for all @xmath58 there is a neighborhood @xmath59 of @xmath37 such that for all @xmath93 @xmath94\eta_{ts } + ( 1-{\mathbb{\alpha}})\left[f^{\rho}(t)-f(s)\right]\mu_{ts } -f^{{\diamondsuit_{\alpha}}}(t)\mu_{ts}\eta_{ts}\right|\leq \varepsilon|\mu_{ts}\eta_{ts}| \ , .\ ] ] we say that function @xmath55 is @xmath5-differentiable on @xmath95 , provided @xmath92 exists for all @xmath96 . @xcite[thsheng ] let @xmath32 be simultaneously @xmath1 and @xmath2 differentiable at @xmath96 . then , @xmath55 is @xmath5-differentiable at @xmath37 and @xmath97 , @xmath98 $ ] . the @xmath5-derivative is a convex combination of delta and nabla derivatives . it reduces to the @xmath1-derivative for @xmath99 and to the @xmath2-derivative for @xmath100 . the case @xmath101 has proved to be very useful in applications . for more on the theory of @xmath5-derivatives than that we are able to provide here , we refer the interested reader to @xcite . the same idea used to define the combined derivative is taken to define the combined integral . [ diaminteg ] let @xmath102 and @xmath103 . then , the @xmath5-integral of @xmath55 is defined by @xmath104 , where @xmath98 $ ] . in general the @xmath5-derivative of @xmath105 with respect to @xmath37 is not equal to @xmath106 @xcite . next proposition gives direct formulas for the @xmath5-derivative of the exponential functions @xmath107 and @xmath108 . for the definition of exponential and trigonometric functions on time scales see , , @xcite . [ diamexpder ] let @xmath6 be a time scale with the following properties : @xmath73 , and @xmath75 . assume that @xmath109 and for all @xmath12 one has @xmath110 , and @xmath111 . then , @xmath112e_p(t , t_0 ) \ , , \label{eq : a } \\ \hat{e}_p^{{\diamondsuit_{\alpha}}}(t , t_0)=\left[(1-{\mathbb{\alpha}})p(t)+ \frac{{\mathbb{\alpha}}p^{\sigma}(t)}{1-\mu(t)p^{\sigma}(t)}\right]\hat{e}_p(t , t_0 ) \ , , \label{eq : b}\end{gathered}\ ] ] for @xmath96 . firstly , recall that @xmath113 and @xmath114 . hence , @xmath115 , and then @xmath116 , from where it follows . similarly , we have that @xmath117 , and then @xmath118 , from where holds . [ cor1 ] let @xmath119 , @xmath120 , and @xmath121 for all @xmath12 . then , for @xmath96 , \(a ) @xmath122 @xmath123 ; \(b ) @xmath124 @xmath125 ; \(c ) @xmath126 @xmath127 ; \(d ) @xmath128 @xmath129 . let @xmath109 , @xmath120 , and @xmath130 for all @xmath12 . then , for @xmath96 , \(a ) @xmath131 @xmath132 ; \(b ) @xmath133 @xmath134 ; \(c ) @xmath135 @xmath136 ; \(d ) @xmath137 @xmath138 . let @xmath6 be an arbitrary time scale . let us define recursively functions @xmath139 , @xmath140 , as follows : @xmath141 similarly , we consider the monomials @xmath142 : they are the functions @xmath143 , @xmath140 , defined recursively by @xmath144 all functions @xmath145 are rd - continuous , all @xmath146 are ld - continuous . the derivatives of such functions show nice properties : @xmath147 , @xmath56 ; and @xmath148 , @xmath64 , where @xmath109 and derivatives are taken with respect to @xmath37 . we have that @xmath149 for @xmath8 , @xmath150 . finding exact formulas of @xmath145 or @xmath146 for an arbitrary time scale is , however , not easy . from @xcite we have the following result : let @xmath151 and @xmath152 . then , @xmath153 for all @xmath140 . next proposition gives explicit formulas for homogenous time scales with @xmath154 , @xmath155 a strictly positive constant . for that we need two notations of factorial functions : for @xmath156 we define @xmath157 and @xmath158 with @xmath159 and @xmath160 . let @xmath11 and @xmath10 . for @xmath140 the following equalities hold : \(a ) @xmath161 ; \(b ) @xmath162 . firstly , @xmath163 . next we observe that @xmath164 . hence , by the principle of mathematical induction , ( a ) holds for all @xmath140 . since @xmath153 , ( b ) is also true . let @xmath10 , @xmath165 . from the properties of factorial functions it follows : 1 . if @xmath166 , then @xmath167 ; 2 . if @xmath168 , then @xmath169 . in particular , when @xmath170 and @xmath9 , we have : 1 . if @xmath171 , then @xmath167 ; 2 . if @xmath172 , then @xmath169 . in the next section we need the following results . let @xmath10 , @xmath11 , and @xmath12 . then , for @xmath140 , the following holds : \(a ) @xmath173 ; \(b ) @xmath174 . [ prop3012 ] let @xmath10 , @xmath11 , @xmath119 . then , \(a ) @xmath175 for @xmath176 ; \(b ) @xmath177 for @xmath178 . let @xmath10 , @xmath11 , and @xmath119 . for @xmath176 , it is enough to notice that @xmath179 to prove ( a ) . equality ( b ) is proved in a similar way : for @xmath178 , we have : @xmath180 . @xcite let @xmath152 and @xmath140 . then , @xmath181 for @xmath182 . as a consequence of ( [ rel3 ] ) and equalities @xmath183 and @xmath184 , the following laws of differentiation of generalized monomials follow . [ derdor1 ] \(a ) @xmath185^{\nabla } = \sum_{j=0}^{k}(-1)^j\nu^j(t)h_{k - j}(t , t_0)$ ] ; \(b ) @xmath186^{\delta } = \sum_{j=0}^{k}\mu^j(t)\hat{h}_{k - j}(t , t_0)$ ] . [ exdor1 ] let @xmath6 be an homogenous time scale with @xmath187 , @xmath188 . let us recall that for @xmath189 we have @xmath8 and for @xmath170 we have @xmath9 . then , @xmath185^{\nabla } = \sum_{j=0}^{k}(-1)^jc^jh_{k - j}(t , t_0)$ ] , @xmath186^{\delta}= \sum_{j=0}^{k}c^j\hat{h}_{k - j}(t , t_0)$ ] . for @xmath8 : @xmath185^{\nabla } = \left[\hat{h}_{k+1}(t , t_0)\right]^{\delta } = \frac{(t - t_0)^k}{k!},$ ] + for @xmath9 : @xmath185^{\nabla}= \sum_{j=0}^{k}(-1)^jh_{k - j}(t , t_0 ) = \binom{t - t_0}{0}-\binom{t - t_0}{1 } + \cdots+(-1)^k\binom{t - t_0}{k},$ ] and @xmath186^{\delta } = \sum_{j=0}^{k}\hat{h}_{k - j}(t , t_0)$ ] . let @xmath119 . then , \(a ) @xmath190 ; \(b ) @xmath191 ; \(c ) @xmath192 . from the definition of @xmath5-derivative and corollary [ derdor1 ] , we have : @xmath193 . next , @xmath194 . [ tayth1 ] @xcite assume that @xmath55 is @xmath195 delta - differentiable on @xmath196 . let @xmath197 , @xmath12 . then , @xmath198 where @xmath199 . [ tayth2]@xcite assume that @xmath55 is @xmath195 times nabla differentiable on @xmath200 . let @xmath201 , @xmath12 . then , @xmath202 where @xmath203 . by a polynomial real series we usually understand a series of the form @xmath204 , where @xmath205 is a given sequence of polynomials in the variable @xmath37 and @xmath206 is a given sequence of real numbers . in the continuous case one has @xmath207 . for the time scales we are considering in this paper , we have @xmath208 or @xmath209 , and we speak about _ generalized power series on time scales _ @xcite . [ szereg ] let @xmath6 be a time scale and let us fix @xmath109 . by a _ @xmath1-polynomial series ( on @xmath6 , originated at @xmath210 ) _ we shall mean the expression @xmath211 , @xmath212 ; by a _ @xmath2-polynomial series ( on @xmath6 , originated at @xmath210 ) _ we mean @xmath213 , @xmath212 , where for each @xmath214 , @xmath215 . the sequence @xmath216 is called the corresponding sequence of the series . for any fixed @xmath217 , both type of series become ordinary number series . if they are convergent for @xmath37 we say that the polynomial series is convergent at @xmath37 . if @xmath9 , then for each @xmath218 , @xmath219 , the number series @xmath220 is convergent because it is finite . the same situation we have when @xmath221 : @xmath222 is finite , so convergent . in @xcite and @xcite it is proved the following : let @xmath109 . if the power series @xmath223 with the corresponding sequence of coefficients @xmath224 is convergent at @xmath225 and @xmath219 , then the polynomial series is convergent for all values of @xmath12 such that @xmath226 . two polynomial series of the same type can be added and multiplied by scalars giving the same type of series . we can define the @xmath1-derivative of @xmath1-polynomial series : @xmath227 . similarly , we have the @xmath2-derivative of @xmath2-polynomial series in the form @xmath228 . additionally , if the @xmath1-polynomial series is convergent for @xmath229 and if the @xmath2-polynomial series is convergent for @xmath230 , then their derivatives are also convergent on the same sets . from corollary [ derdor1 ] we obtain the following result . [ derser ] let @xmath231 , @xmath232 , and @xmath233 be a sequence such that @xmath234 for each @xmath235 . we have : \(a ) let @xmath236 . then , the series @xmath237 is convergent for @xmath238 , and @xmath239 exists and it is convergent for @xmath238 . \(b ) let @xmath240 . then , the series @xmath241 is convergent for @xmath242 , and @xmath243 exists and it is convergent for @xmath242 . let @xmath10 , @xmath11 . there is no problem with convergence ( i ) in points @xmath244 for series of the first type , ( ii ) at points @xmath245 for series of the second ( `` hat '' ) type , because such series are finite . in @xcite and @xcite one can find generalized series for an exponential @xmath246 with constant function @xmath120 : for @xmath12 and @xmath244 one has @xmath247 it follows that @xmath248 , which gives the rule @xmath249 . as in proposition [ diamexpder ] , let us consider now a time scale with @xmath73 . then , @xmath250 where @xmath251 if @xmath252 . this gives that @xmath253 and then @xmath254 is as in proposition [ diamexpder ] . in @xcite it is proved that @xmath255 for @xmath244 . then , we have that @xmath256 . we obtain that the @xmath1-derivative of @xmath257 with respect to @xmath37 is given by the formula @xmath258 , where @xmath259 if @xmath260 . this gives that @xmath261 and then @xmath262 is also as in proposition [ diamexpder ] . the diamond-@xmath0 derivative reduces to the standard @xmath1 derivative for @xmath3 and to the standard @xmath2 derivative for @xmath4 . the same `` weighted '' type definition is proposed for the diamond-@xmath0 integral . based on this simple idea , we introduce diamond type monomials . let us begin with the trivial remark that for any @xmath263 we can write @xmath264 . assume that @xmath55 is @xmath195 delta- and nabla - differentiable on @xmath196 and @xmath200 , respectively . let @xmath265 , @xmath12 . then , @xmath266 where @xmath267 , and @xmath268 with remainders @xmath269 and @xmath270 given as in theorems [ tayth1 ] and [ tayth2 ] . [ diam_ser ] let @xmath6 be a time scale and @xmath109 . by a _ combined - polynomial series ( on @xmath6 , originated at @xmath210 ) _ we shall mean the expression @xmath271 where @xmath12 and @xmath98 $ ] . if in ( [ diam_ser1 ] ) we put @xmath99 , then we have a @xmath1-polynomial series . for @xmath100 we obtain @xmath2-polynomial series . a combined - series is convergent if both types of polynomial series are convergent . for fixed @xmath119 we get usual number series , so we can say that the series originated at @xmath210 is convergent at @xmath37 if it is convergent as a number series . let @xmath10 , @xmath11 , and @xmath272 , @xmath273 be two real sequences with nonzero elements such that @xmath274 , @xmath275 . then , the combined - polynomial series @xmath276 @xmath98 $ ] , is convergent for all @xmath12 . based on proposition [ prop3012 ] , we consider : @xmath277 , when combined - series @xmath278 ; @xmath178 , when the first part is finite the second is convergent ; @xmath176 , when we have opposite situation to the previous one . let @xmath9 and @xmath279 . then , @xmath280 and @xmath281 for @xmath140 . additionally , @xmath282 for any @xmath283 . but this series is not convergent for @xmath284 . we have @xmath285 and @xmath286 . the series @xmath287 is convergent for any @xmath288 . for that let @xmath289 . then , @xmath290 for each fixed @xmath291 . the combined - polynomial series has the form @xmath292 and is convergent for @xmath283 . polynomial series have been used in the literature for solving a variety of problems in control . in this paper we define taylor series via diamond - alpha derivatives on time scales and provide the first steps on the correspondent theory . such a theory provides a general framework that is valid for discrete , continuous or hybrid series . we trust that the polynomial series here introduced are important in the analysis of control systems on time scales . the first author was supported by biaystok technical university grant s / wi/1/07 ; the second author by the r&d unit ceoc , via fct and the ec fund feder / poci 2010 . r. j. higgins and a. peterson . cauchy functions and taylor s formula for time scales @xmath293 . in _ proceedings of the sixth international conference on difference equations _ , pages 299308 , boca raton , fl , 2004 .
|
_ the objective of this paper is twofold : ( i ) to survey existing results of generalized polynomials on time scales , covering definitions and properties for both delta and nabla derivatives ; ( ii ) to extend previous results by using the more general notion of diamond - alpha derivative on time scales .
we introduce a new notion of combined - polynomial series on a time scale , as a convex linear combination of delta and nabla generalized series .
main results are formulated for homogenous time scales . as an example , we compute diamond - alpha derivatives on time scales for delta and nabla exponential functions
. _ * keywords : * time scales , diamond-@xmath0 derivatives , generalized polynomials , generalized series . *
2000 mathematics subject classification : * 40c99 , 39a13 , 40a30 .
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the x - ray emission from galactic x - ray binaries is known to exhibit aperiodic variability on a vast variety of time scales , from months down to milliseconds ( for reviews see , e.g. @xcite ) . the pca on board the _ rossi x - ray timing explorer _ ( rxte ) , has accumulated a large amount of data on the rapid aperiodic variability ( rav ) and its photon - energy dependence for a large number of x - ray binaries . this information contains valuable hints about the source of high - energy emission in these objects . since it is generally believed that the x - ray emission above @xmath0 10 kev is due to comptonization of soft photons in a hot , tenuous coronal gas , it is natural to assume that the rav of this emission component also reveals detailed information about the size scales , dominant physical ( heating and cooling ) mechanisms , and geometry of the comptonizing region . theoretical calculations of the expected time - dependent signatures of comptonization in a hot , tenuous , static corona had been done for the case of a homogeneous corona ( @xcite ) and for inhomogeneous temperature and density distributions in the corona ( kazanas , hua & titarchuk 1997 , hua , kazanas & titarchuk 1997 ) for the case of central injection of soft photons into a spherical region . this had been generalized to other geometries , including the case of a cool , outer accretion disc adjacent to an inner , hot , quasi - spherical two - temperature inflow by bttcher & liang ( 1998 ) . while such models , in particular in the case of central injection , representative of a slab - coronal geometry , could successfully reproduce the dominant spectral and timing features of some objects , such as cyg x-1 , they generally required that the corona on top of the geometrically thin , optically thick accretion disk should extend out to radii of @xmath1 , where @xmath2 is the schwarzschild radius of the accreting compact object , and that out to those radii , the rate of energy dissipation per unit volume would have to decrease only @xmath3 , implying that most of the energy is dissipated at large distance from the central object . this seems to be very hard to reconcile with realistic calculations of the vertical structure of accretion - disk coronae ( @xcite ) and the radial structure of optically thin , adaf - type flows ( @xcite ) , which indicate that the energy dissipation is concentrated much more strongly towards the disk surface and the central object , respectively . the required large coronal size scales were implied by the fact that , in these comptonization models , the maximum time lag achievable between hard and soft x - rays corresponds to the difference in photon diffusion time in the process of compton upscattering and is thus of the order of the light travel time through the corona . in order to avoid this size - scale problem , two alternative models for the hard time lags in galactic x - ray binaries have been developed . in the first one ( @xcite ) , an inner , hot coronal flow is assumed to be surrounded by a cool , outer disc . in irregular intervals , blobs of cool material are breaking lose from the cool disc and spiral inward through the hot coronal flow . ( during the inspiraling process they may most certainly be tidally disrupted into ring - like density perturbations , but for simplicity , we will continue to call these inward - drifting perturbations `` blobs '' . it has been shown in bttcher & liang [ 1999 ] that they maintain their integrity as isolated , dense regions , and do not evaporate during the inspiraling process ) . in this model , hard lags are due to the spectral hardening resulting from the inward - movement of the soft photon source into hotter regions of the coronal flow , combined with the gradual heating of the blob . thus , the maximum hard time lag achievable corresponds to the drift time scale of the blobs , which may be several orders of magnitude longer than the light travel time through the corona , thus allowing for a correspondingly smaller corona . in the second alternative model ( @xcite ) , hard x - ray flares are assumed to be produced by magnetic flares in a hot corona on top of a cool accretion disc . thus , in this model , the flare is produced by additional heating of the comptonizing material rather than a flare of the soft photon input . the required spectral hardening occurs naturally during the period of efficient energy release in the corona , and hard lags could be produced in a model consistent with the observed weakness of the compton reflection component if the active regions of the corona are driven away from the disc due to radiation pressure ( @xcite ) . except for the latter model , in which the coronal heating and cooling are the essential ingredients for modeling the hard x - ray spectral evolution , all the comptonization - based models mentioned above had so far been calculated under the assumption of a static corona with fixed electron temperature , which does not change in response to the increased soft photon input during flares . since it is generally believed that the coronal temperature at least in spectral states other than the very - low or off state of transient sources is determined by the energy balance between various heating mechanisms and cooling dominated by compton cooling , this may be an unrealistic assumption , as has been pointed out by several authors ( e.g. , @xcite ) . the coronal response to accretion - disc flares in the case of a slab - coronal geometry has recently been investigated for two special test cases by malzac & jourdain ( 2000 ) . they found that coronal cooling due to the increased soft photon input leads to rapid cooling of the coronal electron population and a pivoting of the x - ray spectrum around @xmath4 kev for typical parameters , with rather small amplitude variability in the _ rxte _ pca energy range . they conclude that the resulting variability patterns are inconsistent with the ones observed in most x - ray binaries . in this paper , i am presenting a detailed re - evaluation of variability models based on flaring activity in the soft photon input , taking into account self - consistently the coronal heating and cooling in response to the time - varying soft photon input . the models considered here are the slab - coronal geometry , the hot inner two - temperature flow with outer , cool accretion disc , and the inward - drifting blob model . the model of dynamical active coronal regions has been investigated in detail in poutanen & fabian ( 1999 ) , and has also been considered in malzac & jourdain ( 2000 ) . in section [ code ] , a brief description of the coupled monte - carlo / fokker - planck code used for this study is given . in sections [ slab ] , [ adaf ] , and [ blob ] the results concerning the slab - coronal geometry , the hot inner two - temperature flow with outer , cool accretion disc , and the inward - drifting blob case , respectively , are presented . section [ summary ] contains a brief summary and conclusions . for our time - dependent simulations of the radiation transfer and electron heating / cooling , the coupled monte - carlo compton scattering and fokker - planck electron dynamics code described in detail by bttcher & liang ( 2000 ) is used . this code employs a monte - carlo method similar to the large - particle method ( @xcite ) for the photon transport , including all relevant radiation mechanisms , such as compton scattering , cyclotron / synchrotron emission and absorption , bremsstrahlung emission and absorption , and @xmath5 absorption and pair production and annihilation . the evolution of the electron population is treated simultaneously through an implicit fokker - planck scheme which allows for arbitrary thermal and/or nonthermal electron distributions . at each time step , the radiative energy - loss rates are normalized self - consistently to the energy transfer between electrons and photons evaluated during the photon - transport calculation in the respective time step . electron acceleration / heating is possible through coulomb interactions with a pool of background , thermal protons or through resonant wave - particle interaction ( 2@xmath6 order fermi acceleration ) with alfvn and whistler wave turbulence , as well as through self - absorption processes . this means that unlike in most other simulations of related problems ( e.g. , @xcite ; li , kusunose , & liang 1996 ; dove , wilms , & begelman 1996 ; @xcite ) the coronal heating compactness is the self - consistent result of specific acceleration processes , and is not a pre - specified parameter . for the different geometrical and physical scenarios treated below , the corona is generally split up into 10 or 15 vertical or radial zones within which the physical conditions ( electron and proton densities , magnetic fields , etc . ) are assumed to be homogeneous . unless otherwise specified , a globally unordered magnetic field in equipartition with the background thermal proton plasma is assumed . the properties of all photons escaping the corona toward the observer are written into an event file , from which photon - energy dependent light curves and snapshot photon spectra at arbitrary time intervals can be extracted . let us first investigate the coronal response to an accretion - disc flare in a slab - coronal geometry ( e.g. , @xcite ) . such flares could be produced , e.g. , by an unsteady accretion flow through the inner portion of the disk , or by small - scale magnetic flares in the ionized surface layer of the disk . for this case , the soft photon input from the disc is represented by a thermal blackbody spectrum of @xmath7 kev , typical of the soft excess in the x - ray spectra of black - hole x - ray binaries in the low / hard state . an accretion disc flare is simulated by increasing the blackbody temperature to 0.5 kev ( i.e. increasing the soft photon compactness by a factor of 39 ) over a limited time interval @xmath8 . compton reflection of coronal radiation impinging onto the disc is taken into account using the green s functions of white , lightman , & zdziarski ( 1988 ) and lightman & white ( 1988 ) , and at any given time the accretion disc flux is enhanced by the amount of coronal flux absorbed within the disc . the corona is split up into 10 vertical zones of equal height @xmath9 . the total height of the corona is fixed to @xmath10 cm , its thomson depth is @xmath11 , and the temperature of the background proton plasma is @xmath12 mev . coulomb heating of the electrons is assumed to be the dominant electron acceleration mechanism . this yields an equilibrium electron temperature of @xmath13 kev . [ adc1 ] shows the energy - dependent light curves , the evolution of the average coronal temperature , and some snapshot energy spectra of the observable x - ray emission for a typical simulation , where @xmath14 . the accretion - disc flare leads to strong cooling of the coronal electrons . consequently , in agreement with the results of malzac & jourdain ( 2000 ) , at hard x - ray energies ( @xmath15 kev ) a broad dip rather than a flare results . the simulated hard x - ray light curves can be reasonably well fitted with a function @xmath16 where @xmath17 + e^{t - t_0 \over \tau_{\rm r } } \ , \theta [ t - t_0 ] \right).\ ] ] here , @xmath18 is the heaviside function . in a series of simulations , different input parameters , such as the coronal thomson depth , proton temperature , etc . have been varied . the most critical parameter in these simulations is the duration of the accretion disc flare , @xmath8 , compared to the light crossing time through the corona , @xmath19 . adc_table ] lists the relevant light curve fitting parameters according to eq . ( [ adc_lightcurves ] ) for some of those cases with the standard coronal parameters listed above . the fourier transformations of the light curves defined in eq . ( [ adc_lightcurves ] ) are straightforward , but somewhat lengthy . if one neglects terms of order @xmath20 , the complex phase of the fourier transform is given by @xmath21 where @xmath22 the resulting phase and time lags between the 3 10 kev and the 10 50 kev bands from the three representative simulations with @xmath23 s , @xmath24 s , and @xmath25 s , respectively , are shown in fig . [ adc_phaselags ] . the figure illustrates that a rather rich variety of phase and time lag phenomena can result from this accretion - disc flare scenario with a slab - coronal geometry . it appears to be consistent with a fourier - frequency independent time lag at low frequencies , breaking into a power - law with @xmath26 , where generally @xmath27 ( e.g. , @xcite for cyg x-1 ) . in the limit @xmath28 , the resulting time lags are dominated by the difference in the fit parameter @xmath29 for the different energy bands , which exhibits a photon energy dependence consistent with the logarithmic time lag dependence measured , e.g. , for cyg x-1 , grs 1915 + 105 , ( e.g. , @xcite ) and xte j1550 - 564 ( wijnands , homan , & van der klis 1999 ; cui , zhang , & chen 2000 ) . the photon - energy dependence of the parameter @xmath29 is illustrated in fig . [ t0_energy ] . note that this logarithmic energy dependence of the maximum time lag is related to the time scales for compton cooling and relaxation back to thermal equilibrium due to coulomb heating rather than due to the difference in photon diffusion time between hard and soft x - rays as in the case of a static corona . this naturally avoids the problem of the size - scale constraint for static - corona models for the phase and time lags in the rav of x - ray binaries and , instead , allows to place constraints on the proton temperature and density within the corona , which determine the coulomb heating time scale . assuming that both the electron and proton temperatures are non - relativistic , the coulomb heating time scale ( which is expected to be comparable to the maximum time lag between soft and hard x - rays ) can be estimated as @xmath30 ( @xcite ) . here , @xmath31 is the coronal proton density in units of @xmath32 @xmath33 , and @xmath34 is the dimensionless electron and proton temperature , respectively . ( [ tau_coulomb ] ) indicates that maximum time lags of order @xmath35 @xmath36 s can naturally occur in such a scenario . interestingly , fig . [ adc_phaselags ] also indicates that over a limited frequency range also negative phase lags ( i.e. soft lags ) may occur as a result of the coronal response to accretion - disc flares . whether this is a potential explanation for the negative and alternating phase lags observed in grs 1915 + 105 ( @xcite ) and xte j1550 - 564 ( @xcite ) remains to be investigated in more detail in future work . an important point is that in this scenario , the onset of an accretion - disc flare marks the onset of the episode of enhanced coronal cooling and thus the onset of the decay of the high - energy light curves . consequently , considering that in a realistic scenario there will be a rapid succession of such flares occurring throughout the disc , the hard x - ray light curves will exhibit maxima around the onsets of the accretion disc flares , with no appreciable offset between the maxima at different photon energies . this is consistent with the recent result of maccarone et al . ( 2000 ) that the peaks of the cross - correlation function between light curves at different energy bands observed in cyg x-1 are consistent with 0 time lag . this would be inconsistent with accretion - disc flaring scenarios in slab - coronal geometry if the observed phase lags were due to the time - dependent comptonization response in a static corona ( @xcite ) . an alternative accretion - flow geometry consists of a cool shakura - sunyaev ( 1973 ) type , thin disc in the outer regions of the flow , which exhibits a transition to an inner , quasi - spherical two - temperature flow , which produces hard x - rays via compton upscattering of self - generated synchrotron and bremsstrahlung photons as well as external soft photons from the surrounding cool accretion disc . the inner coronal flow could plausibly be a shapiro - lightman - eardley ( 1976 ) type two - temperature flow , an advection - dominated accretion flow ( adaf ; @xcite ) or the recently discovered solution of a convection - dominated flow ( @xcite ) . the timing properties which are subject of this paper , are rather insensitive to the details of the proton density and temperature profile within the inner coronal region . thus , for simplicity , the density and temperature profiles of the inner corona are fixed corresponding to an adaf , choosing @xmath37 , and the proton temperature , @xmath38 , equal to the virial temperature . such a two - phase accretion flow model has been suggested to explain the various luminosity states of x - ray binaries ( narayan , mcclintock , & yi 1996 ; @xcite ) , and has been very successful , in particular , to represent the x - ray spectrum of grs 1915 + 105 over a wide range of source luminosities ( e.g. , muno , morgan , & remillard 1999 ) . thus , for a parameter study , i start out from parameters in the range typically appropriate for spectral fits to grs 1915 + 105 . the inner disc temperature found in grs 1915 + 105 is typically 0.75 kev @xmath39 2 kev , and the inner disc radius is 10 km @xmath40 100 km ( @xcite ) . as default settings , for simplicity , the soft radiation input from the cool , outer disc is represented by a thermal blackbody with @xmath41 kev , and the transition radius between cool , outer disc and inner , hot , quasi - spherical flow is chosen as @xmath42 km . the thomson depth of the inner corona is fixed to @xmath43 . the event horizon is modelled as an absorbing inner boundary of the corona at @xmath44 km . independent of the value of the transition radius , one finds that a fraction @xmath45 of the accretion disc radiation intercepts the inner corona , if any point on the disc surface is assumed to emit isotropically , the corona is assumed to be spherical , and the disc has a @xmath46 temperature profile . a short accretion - disc flare is represented by a temporary increase of the disc blackbody temperature to twice its value during non - flaring episodes ( i.e. increasing the soft photon compactness by a factor of 16 ) . figs . [ adaf1 ] and [ adaf2 ] shows the energy - dependent light curves , evolution of the average coronal temperature , and some snapshot energy spectra from two simulations with standard parameters as quoted above , and @xmath47 s ( @xmath48 ) and @xmath49 s ( @xmath50 ) , respectively . the figures illustrate that in this case , the coronal response to the accretion - disc flares occurs on time scales at least an order of magnitude shorter than in the case of the slab geometry , due to the substantially higher proton densities in the hottest , central regions of the adaf . this , combined with the short light crossing and photon diffusion times through the corona , @xmath51 s , makes it very unlikely that substantial phase and time lags ( of order @xmath52 s ) can be produced by the thermal and comptonization response in such a geometry . in particular , comparing the medium and hard x - ray light curves in figs . [ adaf1 ] and [ adaf2 ] one finds that if ( e.g. in the case of a larger corona ) substantial time lags were produced in this geometry due to radiation transfer effects in the corona , they would be accompanied by substantial peak misalignments of the light curves in different energy bands , which is generally not observed ( e.g. @xcite for grs 1915 + 105 ; @xcite for cyg x-1 ) . the third flaring scenario investigated here is the model of inward drifting density perturbations ( `` blobs '' ) in a hot , quasi - spherical inner accretion flow , proposed by bttcher & liang ( 1999 ) . this model is based on the accretion - flow geometry of an outer , cool , optically thick accretion disc and an inner , hot , quasi - spherical two - temperature flow , as in the previous section . however , in this case , flaring activity is not dominated by flares of the outer accretion disc , but by the inward - drifting of individual blobs ( or ring - like density perturbations ) from the inner boundary of the cool disc through the hot corona , which will eventually disappear within the event horizon of the black hole . the spectral hardening required to produce hard phase and time lags is a consequence of the combined effects of the soft photon source ( the blob ) moving further inward toward the hotter and denser portions of the corona , and of the heating of the blob as it drifts inward . by default , the system of the outer cool disc and the central corona will be described by the same set of standard parameters , appropriate to reproduce typical x - ray spectra of grs 1915 + 105 in its high state , as in the previous section : @xmath43 , @xmath37 , virial proton temperature , @xmath53 km , @xmath54 kev . it is assumed that the blob surface starts out with the same temperature as the disc material , @xmath55 , and heats up as @xmath56 , where i investigate the special cases @xmath57 , @xmath58 , @xmath59 , and @xmath60 . the blob drifts inward with a radial drift velocity @xmath61 . in this general study , i choose @xmath62 . the real value of @xmath63 may in fact be somewhat lower . however , i choose a rather high radial drift velocity in order to keep the required simulation time at @xmath64 hr ( on a 667 mhz alpha workstation ) , since the code automatically adjusts the intrinsic time step of the simulation to a step size smaller than the time scale of the net physical heating / cooling rate . for lower values of @xmath65 the resulting timing behaviour is simply stretched over a correspondingly longer time interval , and the respective features in the fourier power and phase lag spectra below @xmath66 khz ( see below ) are shifted to lower frequencies by the same factor , thus increasing the resulting time lags accordingly ( since @xmath67 $ ] ) . a typical simulation result is illustrated in fig . [ blob1 ] for a value of @xmath68 ( i. e. @xmath69 ) . except for the highest energy band ( which lies beyond the rxte pca energy range ) , the light curves generally show a gradual rise up to a peak , corresponding to the disappearence of the blob , and a steep decline . since this decline happens on sub - ms time scales , the rapid variability at frequencies @xmath70 khz will be dominated by the gradual - rise phase , which can be parametrized by a functional form @xmath71 the critical fit parameters @xmath72 and @xmath73 for various values of @xmath74 are listed in table [ blob_table ] . generally , the value of @xmath73 increases with increasing photon energy . this increase becomes more pronounced for decreasing values of @xmath74 . @xmath72 decreases with increasing photon energy . for the fourier transform of eq . ( [ blob_lc ] ) , i did not find an analytical solution . [ blob_fourier ] shows the numerically calculated values of the complex fourier phase and the power spectra from light curves parametrized by eq . ( [ blob_lc ] ) , for various values of @xmath73 . @xmath75 ms has been chosen . the effect of decreasing @xmath72 by amounts typical of the variations between different energy channels ( see table [ blob_table ] ) is similar to , but much less pronounced than the effect of increasing @xmath73 . thus , the photon - energy dependent power spectral and phase lag features are dominated by variations in @xmath73 between different energy channels . the figure demonstrates that the inward - drifting blob model predicts ( a ) a very significant hardening of the power spectra at high frequencies with increasing photon energy and ( b ) a photon - energy and fourier - frequency dependent phase lag . [ blob_phaselags ] shows the phase and time lags calculated from the analytical parametrizations to the simulated light curves shown in fig . [ blob1 ] . one can see that at least in the parameter range investigated here the fourier - frequency dependent phase lag does generally not show a simple broken - power - law or double - broken - power - law behaviour as seen , e.g. , in cyg x-1 ( e.g. , @xcite ) . rather , this scenario leads to approximately constant time lags at low fourier frequencies and pronounced positive - lag features around the inverse of the blob drift time scale , @xmath76 , beyond which the lags turn negative in a small frequency interval , and begin to oscillate between negative and positive values . beyond @xmath66 khz , the inadequacy of our analytical description of the light curves ( eq . [ blob_lc ] ) on very short time scales ( @xmath77 ms ) might begin to influence the results , so that no predictions can be made at this point . it might be worth to point out again that all these typical frequencies of the features described above scale @xmath78 and will thus be shifted towards lower frequencies for lower values of the drift velocity . it is conceivable that the oscillating phase lag features seen in fig . [ blob_phaselags ] could be related to the peculiar , sometimes alternating phase lags apparently associated with the 0.2 10 hz qpos in grs 1915 + 105 ( @xcite ) and in xte j1550 - 564 ( @xcite ) . assuming that the size scale of the corona deduced from spectral fitting to grs 1915 + 105 ( @xmath79 @xmath80 km ) is correct , this would require that the inward - drift velocity of density perturbations is @xmath81 . it will be the subject of future work to determine whether such a choice of @xmath63 is realistic . a detailed re - analysis of the time - dependent radiation transfer through hot , comptonizing coronae in various models for small - scale high - energy flares in x - ray binaries has been presented . in contrast to previous work , the effect of coronal cooling due to the increased soft photon input during the flare has been taken into account self - consistently . in particular , the case of a slab corona with a flare of the underlying , optically thick accretion disc , the case of an inner , hot , quasi - spherical accretion flow with a transition to an outer , cool , optically thick accretion disc , and the case of the inward - drifting blob model have been investigated . in the slab - corona case , the resulting light curves look drastically different from the ones calculated under the assumption of a static corona . this had been pointed out previously by malzac & jourdain ( 2000 ) , who had used a similar model setup as used in the slab - corona - case section of this paper for two specific test cases , but did not investigate the consequences of their results for the phase and time lag spectra . in this paper , i have presented a systematic study of the temporal features resulting from such a scenario . the resulting phase and time lag spectra seem still quite well suited to explain many of the features observed , e.g. , in cyg x-1 . in particular , the overall shape of the time lag spectra , the photon - energy dependence of the time lags , and the small peak misalignment , consistent with 0 , between the light curves at different photon energy bands , can be reproduced in such a geometry . the model of a flaring disc surrounding a central hot , quasi - spherical two - temperature flow , can be ruled out for most objects in which significant phase and time lags have been observed since it would predict very small time lags ( of the order of the light - crossing time through the corona , i.e. @xmath82 ms ) , and light - curve peak misalignments of the order of the maximum time lag , which is generally not observed . the light curves predicted by the inward - drifting blob model look quite similar to the ones calculated under the assumption of a static corona . however , there are marked differences with respect to the details of the predicted phase and time lag spectra . most remarkably , the model ( including coronal cooling ) predicts oscillating phase and time lag features with a typical frequency corresponding to the inverse of the inward - drifting time scale . if these features are to be associated with the alternating phase lags observed during some observations of grs 1915 + 105 and xte j1550 - 564 and apparently associated with the 0.5 10 hz qpos , this would require an inward drift velocity of the density perturbations of @xmath83 . the alternative model by poutanen & fabian ( 1999 ) where the spectral variability is produced by a flare of the coronal heat input has been very successful in explaining the time - averaged photon spectrum and power , phase lag and coherence spectra of cyg x-1 , and was in agreement with the peak of the cross - correlation functions between light curves at different energy bands being consistent with 0 . the fundamentally different flaring scenarios variation of the soft photon input vs. variation of the coronal energy input should be distinguishable by virtue of their markedly different spectral evolution and the photon - energy dependence of the rms variability . in the case of a varying soft photon input , there is considerable variability at soft x - ray energies , and the spectral evolution resembles a pivoting of the hard x - ray spectrum around a typical energy of @xmath79 20 kev . thus , up to an energy of @xmath84 kev , the fractional rms variability should decline as a function of photon energy . in contrast , if the variability is powered by a varying energy input into the corona , the soft x - ray band is expected to show little variability , and the rms variability should increase with increasing photon energy . most interestingly , cyg x-1 shows both types of behavior in different observations ( @xcite ) . a consistently declining rms variability over the _ rxte _ pca energy range has been observed in gx 339 - 4 , while 1e 1740.7 - 2942 almost always shows an increasing rms variability with increasing photon energy ( @xcite ) . this may indicate that fundamentally different variability processes are at work in different objects and even within the same object at different times . in its original version , the poutanen & fabian ( 1999 ) model exclusively predicts positive phase lags , i.e. hard lags . the negative and even alternating phase lags seen in grs 1915 + 105 and xte j1550 - 564 seem to indicate that those lags are not produced by the variability of active coronal regions . ccccc @xmath25 & 3 10 & 0.56 & 0.38 & 0.50 @xmath25 & 10 50 & 6.00 & 4.99 & 1.28 @xmath25 & 50 500 & 10.6 & 1.37 & 1.30 @xmath24 & 3 10 & 0.95 & 0.38 & 0.59 @xmath24 & 10 50 & 4.80 & 1.31 & 1.12 @xmath24 & 50 500 & 2.89 & 1.16 & 2.18 @xmath85 & 3 10 & 2.05 & 7.48 & 1.38 @xmath85 & 10 50 & 1.77 & 1.04 & 2.04 @xmath85 & 50 500 & 1.25 & 0.87 & 3.07 [ adc_table ] cccc 0 & 1 3 & & 0.00 0 & 3 10 & 37.7 & 0.81 0 & 10 50 & 5.69 & 1.66 0.25 & 1 3 & 9.90 & 0.20 0.25 & 3 10 & 4.33 & 1.28 0.25 & 10 50 & 3.16 & 2.09 0.5 & 3 10 & 4.41 & 1.11 0.5 & 10 50 & 2.62 & 2.05 0.5 & 50 500 & 2.48 & 2.80 0.75 & 1 3 & 3.26 & 1.59 0.75 & 3 10 & 2.10 & 2.02 0.75 & 10 50 & 1.91 & 2.59 [ blob_table ]
|
the most popular models for the complex phase and time lags in the rapid aperiodic variability of galactic x - ray binaries are based comptonization of soft seed photons in a hot corona , where small - scale flares are induced by flares of the soft seed photon input ( presumably from a cold accretion disc ) .
however , in their original version , these models have neglected the additional cooling of the coronal plasma due to the increased soft seed photon input , and assumed a static coronal temperature structure . in this paper ,
our monte - carlo / fokker - planck code for time - dependent radiation transfer and electron energetics is used to simulate the self - consistent coronal response to the various flaring scenarios that have been suggested to explain phase and time lags observed in some galactic x - ray binaries .
it is found that the predictions of models involving slab - coronal geometries are drastically different from those deduced under the assumption of a static corona . however , with the inclusion of coronal cooling they may even be more successful than in their original version in explaining some of the observed phase and time lag features .
the predictions of the model of inward - drifting density perturbations in an adaf - like , two - temperature flow also differ from the static - corona case previously investigated , but may be consistent with the alternating phase lags seen in grs 1915 + 105 and xte j1550 - 564 .
models based on flares of a cool disc around a hot , inner two - temperature flow may be ruled out for most objects where significant fourier - frequency - dependent phase and time lags have been observed .
accepted for publication in _ the astrophysical journal _ , vol .
553 ( june 1 , 2001 )
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
to introduce the main theme of this paper we recall the following theorem of casselman @xcite . let @xmath0 be a non - archimedean local field whose ring of integers is @xmath9 . let @xmath10 be the maximal ideal of @xmath9 . let @xmath11 be a non - trivial additive character of @xmath0 which is normalized so that the maximal fractional ideal on which it is trivial is @xmath9 . [ thm : casselman ] let @xmath12 be an irreducible admissible infinite - dimensional representation of @xmath3 . let @xmath13 denote the central character of @xmath5 . let @xmath14 let @xmath15 @xmath16 1 . there exists a non - negative integer @xmath17 such that @xmath18 if @xmath6 denotes the least non - negative integer @xmath17 with this property then the epsilon factor @xmath19 of @xmath5 is up to a constant multiple of the form @xmath20 . ( here @xmath21 is the cardinality of the residue field of @xmath0 . ) 2 . for all @xmath22 we have @xmath23 . the assertion @xmath24 is sometimes referred to as _ multiplicity one theorem for newforms _ and the unique vector ( up to scalars ) in @xmath25 is called the _ newform _ for @xmath26 this is closely related to the classical atkin lehner theory of newforms for holomorphic cusp forms on the upper half plane @xcite . when @xmath27 we have a spherical representation and the newform is nothing but the spherical vector . newforms play an important role in the theory of automorphic forms . we cite two examples to illustrate this . first , the zeta integral corresponding to the newform is exactly the local @xmath7-factor associated to @xmath5 ( see @xcite for instance ) . in addition , newforms frequently play the role of being ` test vectors ' for interesting linear forms associated to @xmath5 . for example , the newform is a test vector for an appropriate whittaker linear functional . in showing this , explicit formulae for newforms are quite often needed . for instance , if @xmath5 is a supercuspidal representation which is realized in its kirillov model then the newform is the characteristic function of the unit group @xmath28 . this observation is implicit in casselman @xcite and is explicitly stated and proved in shimizu @xcite . since the whittaker functional on the kirillov model is given by evaluating functions at @xmath29 , we get in particular that the functional is non - zero on the newform . in a related vein @xcite and @xcite show that test vectors for trilinear forms for @xmath3 are often built from newforms . ( see also a recent expository paper of schmidt @xcite where many of these results are documented . ) in addition to casselman s theory for @xmath3 , newforms have been studied for certain other classes of groups . jacquet _ et al _ @xcite have developed a theory of newforms for _ generic _ representations of @xmath4 . in this setting , there is no satisfactory statement analogous to ( ii ) of the above theorem . however , in his recent thesis , mann @xcite obtained several results on the growth of the dimensions of spaces of fixed vectors and has a conjecture about this in general . for the group @xmath30 , @xmath31 a @xmath32-adic division algebra , prasad and raghuram @xcite have proved an analogue of casselman s theorem for irreducible principal series representations and supercuspidal representations coming via compact induction . in an unpublished work , brooks roberts has proved part of ( i ) of the above for representations of @xmath33 whose langlands parameter is induced from a two - dimensional representation of the weil deligne group of @xmath34 in a previous paper @xcite , we develop a theory of conductors and newforms for @xmath2 . in this paper we use the results of @xcite to carry out a similar program for the unramified quasi split unitary group@xmath35 . let @xmath36 crucial to our study of newforms are certain filtrations of maximal compact subgroups of @xmath37 let @xmath38 be the standard hyperspecial maximal compact subgroup of @xmath39 . let @xmath40 , where @xmath41 then @xmath42 and @xmath43 are , up to conjugacy , the two maximal compact subgroups of @xmath39 . we define filtrations of these maximal compact subgroups as follows . for @xmath17 an integer @xmath44 let @xmath45 @xmath16 let @xmath46 be an irreducible admissible infinite - dimensional representation of @xmath37 let @xmath47 denote the center of @xmath39 and let @xmath48 be the central character of @xmath49 . let @xmath50 be any character of @xmath51 such that @xmath52 on the center . let @xmath53 denote the conductor of @xmath54 for any @xmath55 @xmath50 gives a character of @xmath56 and also @xmath57 given by @xmath58 we define for @xmath59 , @xmath60 the space @xmath61 is defined analogously . we define the _ @xmath50-conductor _ @xmath62 of @xmath49 as @xmath63 we define the _ conductor @xmath64 _ of @xmath49 by @xmath65 where @xmath50 runs over characters of @xmath51 which restrict to the central character @xmath48 on @xmath47 . we deal with the following basic issues in this paper . 1 . given an irreducible representation @xmath49 , we determine its conductor @xmath66 a very easy consequence ( almost built into the definition ) is that the conductor depends only on the @xmath7-packet containing @xmath49 . we identify the conductor with other invariants associated to the representation . for instance , for @xmath2 we have shown @xcite that the conductor of a representation is same as the conductor of a minimal representation of @xmath3 determining its @xmath7-packet . we prove a similar result for @xmath1 in this paper . [ sec : comparison ] and [ sec : comparison - u11 ] . 3 . we determine the growth of the space @xmath67 as a function of @xmath17 . this question is analogous to ( ii ) of casselman s theorem quoted above . computing such dimensions is of importance in ` local level raising ' issues . see @xcite . we address the question of whether there is a multiplicity one result for newforms . it turns out that quite often @xmath68 , but this fails in general ( for principal series representations of a certain kind ) . in these exceptional cases the dimension of the space of newforms is two , but a canonical quotient of this two - dimensional space has dimension one ( see [ sec : multiplicity - one ] ) . 5 . are the newforms test vectors for whittaker functionals ? this is important in global issues related to newforms . we are grateful to benedict gross for suggesting this question to us . it turns out that our newforms are always test vectors for whittaker functionals . in the proofs we often need explicit formulae for newforms in various models for the representations . these formulas are interesting for their own sake . for example , if @xmath12 is a ramified supercuspidal representation of @xmath1 , then the newform can be taken as the characteristic function of @xmath69 where @xmath70 is regarded as a subspace of the kirillov model of a canonically associated minimal representation of @xmath3 ( cf . @xcite ) . we set up notation in [ sec : notation ] following that used in @xcite . we then briefly review the structure of @xmath7-packets for @xmath71 and @xmath35 in [ sec : packets - sl2-u11 ] . as this paper depends crucially on our previous paper @xcite on @xmath71 , we summarize the results of @xcite in [ sec : newforms ] . the heart of this paper is [ sec : unitary ] . in [ sec : defns - u11 ] we define the notion of conductor and then make some easy but technically important remarks on spaces of fixed vectors . the next two subsections deal respectively with sub - quotients of principal series representations and supercuspidal representations . in @xcite , we use kutzko s construction of supercuspidal representations of @xmath3 to obtain results for supercuspidals of @xmath2 . in this paper , we use these results , in turn , to obtain information for @xmath1 . in general , we will often reduce the proofs of statements concerning @xmath1 to those of the corresponding @xmath2 statements . in particular , we exploit the fact that @xmath2 is the derived group of @xmath1 and that @xmath72 has index two in @xmath73 in this way we avoid directly dealing with @xmath74-types and other intrinsic details for @xmath75 as much of the work has been done for @xmath2 in @xcite . finally , in [ sec : multiplicity - one ] , we briefly discuss a multiplicity one result fornewforms . we mention some further directions that arise naturally from this work . to begin with , it would be interesting to see how our theory of newforms and conductors bears upon known results about local factors for @xmath1 . in particular , are our conductors the same as ( or closely related to ) the analytic conductors appearing in epsilon factors ? also , is a zeta - integral corresponding to a newform of a representation equal to a local @xmath7-factor for the representation ? if @xmath7 is any non - archimedean local field let @xmath76 be its ring of integers and let @xmath77 be the maximal ideal of @xmath78 let @xmath79 be a uniformizer for @xmath7 , i.e. , @xmath80 let @xmath81 be the residue field of @xmath82 let @xmath32 be the characteristic of @xmath83 and let the cardinality of @xmath83 be @xmath84 which is a power of @xmath85 let @xmath86 be an element of @xmath87 . if @xmath88 is a positive integer , let @xmath89 denote the @xmath88th filtration subgroup @xmath90 of @xmath91 , and define @xmath92 . let @xmath93 denote the additive valuation on @xmath94 which takes the value @xmath95 on @xmath96 we let @xmath97 denote the normalized multiplicative valuation given by @xmath98 if @xmath99 is a character of @xmath94 we define the conductor @xmath100 to be the smallest non - negative integer @xmath88 such that @xmath99 is trivial on @xmath89 . let @xmath101 be a non - trivial additive character of @xmath7 which is assumed to be trivial on @xmath76 and non - trivial on @xmath102 for any @xmath103 the character given by sending @xmath104 to @xmath105 will be denoted as @xmath106 or simply by @xmath107 ( in all the above notations we may omit the subscript @xmath7 if there is only one field in the context . ) in the following , @xmath0 will be a fixed non - archimedean local field whose residue characteristic is odd and @xmath108 will be used to denote a quadratic extension of @xmath0 . we denote by @xmath109 the quadratic character of @xmath110 associated to @xmath111 by local class field theory . recall that the kernel of @xmath109 is @xmath112 , the norms from @xmath113 we will require the units @xmath114 and @xmath115 to be compatible in the sense that @xmath116 we let @xmath117 denote the group @xmath3 . let @xmath118 be the standard borel subgroup of upper triangular matrices in @xmath117 with levi subgroup @xmath119 and unipotent radical @xmath120 . let @xmath121 be the center of @xmath117 . let @xmath122 let @xmath123 be the standard borel subgroup of upper triangular matrices in @xmath124 with levi subgroup @xmath125 and unipotent radical @xmath120 . we set @xmath126 and @xmath127 and denote by @xmath128 and @xmath129 respectively the standard iwahori subgroups of @xmath124 and @xmath117 . suppose that @xmath111 is unramified , and let @xmath130 denote the non - trivial element of @xmath131 . we denote by @xmath39 the group @xmath35 , i.e. , the group of all @xmath132 such that @xmath133 we let @xmath134 be the standard upper triangular borel subgroup of @xmath39 with diagonal levi subgroup @xmath135 and unipotent radical @xmath120 . we note that the elements of @xmath135 are of the form @xmath136 for @xmath137 , and those of @xmath134 are of the form @xmath138 with @xmath139 and @xmath140 we let @xmath47 be the center of @xmath39 . then @xmath141 , where @xmath142 is the subgroup of norm one elements of @xmath113 denote by @xmath143 the standard iwahori subgroup of @xmath39 and by @xmath144 the standard hyperspecial maximal compact subgroup of @xmath39 . the following filtrations of maximal compact subgroups of @xmath124 will be important in our study of newforms . let @xmath145 and @xmath146 . let @xmath147 , where @xmath148 then @xmath149 and @xmath150 are , up to conjugacy , the two maximal compact subgroups of @xmath124 . for @xmath17 an integer @xmath44 @xmath151 we note that for @xmath152 the following inclusions hold up to conjugacy within @xmath124 : @xmath153 analogous results hold for the following filtration groups of @xmath39 : @xmath154 we note that the filtration subgroups for @xmath124 and @xmath39 are related by @xmath155 where @xmath156 in addition to @xmath157 , we will also make frequent use of the matrices @xmath158 @xmath159 and @xmath160 for any subsets @xmath161 we let @xmath162 = \left\ { \left ( \begin{array}{cc } a & b\\ c & d \end{array}\right ) : a \in a , b \in b , c \in c , d \in d \right\}.\ ] ] we denote @xmath163 $ ] by @xmath164 or simply by @xmath165 we let @xmath166 denote the lower triangular unipotent subgroup of @xmath124 and a similar meaning is given to @xmath167 and @xmath168 if @xmath169 is a closed subgroup of a locally compact group @xmath170 and if @xmath171 is an admissible representation of @xmath169 then @xmath172 denotes _ normalized _ induction , and @xmath173 denotes the subrepresentation of @xmath172 consisting of those functions whose support is compact mod @xmath174 the symbol @xmath175 will denote the trivial representation of the group in context . for any real number @xmath176 we let @xmath177 denote the least integer greater than or equal to @xmath176 and we let @xmath178 in this section we collect statements about the structure of @xmath7-packets for @xmath179 and @xmath180 . all the assertions made here are well - known and can be read off from a combination of labesse and langlands @xcite , gelbart and knapp @xcite and rogawski @xcite . if @xmath181 is an irreducible admissible representation of @xmath117 then its restriction to @xmath124 is a multiplicity free finite direct sum of irreducible admissible representations of @xmath124 which we often write as @xmath182 on the other hand , if @xmath5 is any irreducible admissible representation of @xmath124 then there exists an irreducible admissible representation @xmath181 of @xmath117 whose restriction to @xmath124 contains @xmath26 note that @xmath117 acts on the space of all equivalence classes of irreducible admissible representations of @xmath124 and an _ @xmath7-packet _ for @xmath124 is simply an orbit under this action . it turns out that , with the notation as above , the @xmath7-packets are precisely the sets @xmath183 appearing in the restrictions of irreducible representations @xmath181 of @xmath184 we now give some general statements concerning the @xmath7-packets for @xmath180 . the adjoint group of @xmath35 is @xmath185 , and hence @xmath186 and @xmath117 act via automorphisms on @xmath39 , hence act on the set of equivalence classes of irreducible representations of @xmath39 . rogawski ( @xcite , 11.1 ) defines an @xmath7-packet for @xmath39 to be an orbit under this action . if @xmath49 is an element of a non - trivial @xmath7-packet , then the other element of the @xmath7-packet is @xmath187 . if @xmath188 is an @xmath7-packet for @xmath39 , then the set of irreducible components of the restrictions of elements of @xmath188 to @xmath124 is an @xmath7-packet @xmath8 for @xmath124 . the direct sum @xmath189 is therefore the restriction of an irreducible admissible representation @xmath181 of @xmath184 this @xmath181 is unique up to twisting by a character . in practice , we will choose a convenient @xmath190 since @xmath191 , we obtain an action of @xmath117 on @xmath192 via the represen - tation @xmath181 . this section collects our results @xcite on conductors and newforms for @xmath2 . all these results , along with their complete proofs , can be found in @xcite . we now give our definition of the conductor of a representation of @xmath124 . the basic filtration subgroups of @xmath124 considered in this paper are @xmath194 and for @xmath152 , @xmath195 for all @xmath196 we let @xmath197 let @xmath12 be an irreducible admissible infinite - dimensional representation of @xmath198 let @xmath13 be the character of @xmath199 such that @xmath200 we let @xmath201 be any character of @xmath202 such that @xmath203 let @xmath204 denote the conductor of @xmath205 for any @xmath206 @xmath201 gives a character of @xmath207 and also @xmath208 given by @xmath209 we define @xmath210 the spaces @xmath211 are defined analogously . we note that @xmath212 for @xmath213 . we define the _ @xmath201-conductor _ @xmath214 of @xmath5 as @xmath215 we define the _ conductor @xmath6 _ of @xmath5 by @xmath216 where @xmath201 runs over characters of @xmath202 such that @xmath203 if @xmath201 is such that @xmath217 and @xmath218 ( resp . @xmath219 ) , then we call @xmath220 ( resp . @xmath221 ) a _ space of newforms _ of @xmath5 . in this case , we refer to a non - zero element of @xmath222 or @xmath223 as a _ newform _ of @xmath5 . let @xmath99 be a character of @xmath224 then @xmath99 gives a character of @xmath225 via the formula @xmath226 let @xmath227 stand for the ( unitarily ) induced representation @xmath228 it is well - known that @xmath227 is reducible if and only if @xmath99 is either @xmath229 or if @xmath99 is a quadratic character . there is an essential difference between the two kinds of reducibilities . if @xmath230 , then @xmath227 is the restriction to @xmath124 of a reducible principal series representation of @xmath117 . hence @xmath227 will have two representations in its jordan hlder series , namely the trivial representation and the steinberg representation which we will denote by @xmath231 if @xmath99 is a quadratic character , then @xmath227 is the restriction to @xmath124 of an irreducible principal series representation of @xmath117 and breaks up as a direct sum of two irreducible representations , which constitute an @xmath7-packet of @xmath124 . if @xmath232 we denote @xmath227 by @xmath233 and let @xmath234 we denote the @xmath7-packet by @xmath235 as mentioned in the introduction , one of the applications of newforms we have in mind is that they are test vectors for whittaker functionals . for principal series representations and in fact all their sub - quotients we consider the following @xmath236-whittaker functional ( see @xcite ) . for any function @xmath237 in a principal series representation @xmath227 we define @xmath238 where the haar measure is normalized such that @xmath239 ( unramified principal series representations).[prop : ps - sl2 ] @xmath16 let @xmath99 be an unramified character of @xmath110 and let @xmath227 be the corresponding principal series representation of @xmath198 we have @xmath240 2 m , & \mbox { if $ m \geq 1.$ } \end{array } \right.\ ] ] ( test vectors for unramified principal series representations)[cor : unramified - sl2-test ] @xmath16 for an unramified character @xmath99 of @xmath110 such that @xmath241 let @xmath242 be any non - zero @xmath74-fixed vector . then we have @xmath243 where @xmath244 is the standard local abelian @xmath7-factor associated to @xmath245 ( steinberg representation)[prop : stienberg - sl2 ] @xmath16 if @xmath246 is the steinberg representation of @xmath247 then the dimension of the space of fixed vectors under @xmath207 is given by @xmath248 2m-1 , & \mbox { if $ m \geq 1.$ } \end{array } \right.\ ] ] ( test vectors for the steinberg representation)[cor : steinberg - test ] @xmath16 let the steinberg representation @xmath246 be realized as the unique irreducible subrepresentation of @xmath249 the @xmath236-whittaker functional @xmath250 is non - zero on the space of newforms @xmath251 . ( ramified principal series representations)[prop : sl2-ps - ramified ] @xmath16 let @xmath99 be a ramified character of @xmath224 let @xmath252 be the corresponding principal series representation of @xmath198 let @xmath100 denote the conductor of @xmath245 1 . we have @xmath253 and further @xmath254 only for those characters @xmath201 such that @xmath255 on the group of units @xmath256 2 . if @xmath257 and @xmath258 then @xmath259 1 , & \mbox { if $ m = c(\chi),$}\\[.2pc ] 2(m - c(\chi))+1 , & \mbox { if $ m > c(\chi).$ } \end{array } \right.\ ] ] 3 . if @xmath260 and @xmath261 2 m , & \mbox { if $ m \geq 1 = c(\chi)$. } \end{array } \right.\ ] ] ( test vectors for ramified principal series representations ) [ cor : test - ram - ps]@xmath16 let @xmath99 be a ramified character of @xmath224 let @xmath252 be the corresponding principal series representation of @xmath198 assume that @xmath5 is irreducible . let @xmath262 denote the conductor of @xmath245 the space of newforms @xmath263 is one - dimensional and the whittaker functional @xmath250 is non - zero on this space of newforms . ( ramified principal series @xmath7-packets)[prop : ramified - sl2 ] @xmath16 let @xmath111 be a quadratic ramified extension . let @xmath264 be the corresponding @xmath7-packet . then we have for @xmath265@xmath266 m , & \mbox { if $ m \geq 1.$ } \end{array } \right.\ ] ] ( test vectors for ramified principal series @xmath7-packets ) [ cor : test - ramified - l - packets - sl2]@xmath16 let @xmath111 be a ramified quadratic extension and let @xmath264 be the corresponding @xmath7-packet . then one and only one of the two representations in the packet is @xmath236-generic , say , @xmath267 . then @xmath268 is @xmath269-generic . the whittaker functional @xmath250 is non - zero on the one dimensional space of newforms @xmath270 any @xmath269-whittaker functional is non - zero on the one - dimensional space of newforms for @xmath271 ( unramified principal series @xmath7-packet ) [ prop : unramfified - sl2]@xmath16 let @xmath111 be the quadratic unramified extension . let @xmath272 be the corresponding @xmath7-packet . exactly one of the two representations , say @xmath267 , has a non - zero vector fixed by @xmath149 . then the dimensions of the space of fixed vectors under @xmath207 and @xmath208 for the two representations are as follows : 1 . @xmath273 2 . @xmath274 3 . for @xmath275 , @xmath276 4 . for @xmath277 @xmath278 ( test vectors for unramified principal series @xmath7-packet ) [ cor : test - unramified - lpacket - sl2]@xmath16 let @xmath111 be the unramified quadratic extension , and let @xmath279 be the corresponding @xmath7-packet . then one and only one of the two representations in the packet is @xmath236-generic , namely @xmath267 ( using the notation of proposition [ prop : unramfified - sl2 ] ) . moreover , a @xmath236-whittaker functional is non - zero on the @xmath149-fixed vector in @xmath267 . the representation @xmath268 is not @xmath280-generic for any @xmath280 of conductor @xmath9 . it is @xmath281-generic and any @xmath281-whittaker functional is non - zero on the unique ( up to scalars ) @xmath150-fixed vector in @xmath271 we now consider supercuspidal representations of @xmath122 for this we need some preliminaries on how they are constructed . we use kutzko s construction @xcite of supercuspidal representations for @xmath117 and then moy and sally @xcite or kutzko and sally @xcite to obtain information on the supercuspidal representations ( @xmath7-packets ) for @xmath198 we begin by briefly recalling kutzko s construction of supercuspidal representations of @xmath117 via compact induction from very cuspidal representations of maximal open compact - mod - center subgroups . for @xmath282 , let @xmath283 be the principal congruence subgroup of @xmath284 of level @xmath285 . let @xmath286 let @xmath129 be the standard iwahori subgroup consisting of all elements in @xmath284 that are upper triangular modulo @xmath287 for @xmath282 , let @xmath288 \mathcal{p}^{l+1 } & 1 + \mathcal{p}^l \end{array}\right],\ ] ] and let @xmath289 we will let @xmath290 ( resp . @xmath291 ) denote either @xmath292 ( resp . @xmath284 ) or @xmath293 ( resp . @xmath129 ) . here @xmath293 is the normalizer in @xmath117 of @xmath294 in either case we let @xmath295 denote the corresponding filtration subgroup . @xcite[defn : very - cuspidal ] @xmath16 an irreducible ( and necessarily finite - dimensional ) representation @xmath296 of @xmath290 is called a very cuspidal representation of level @xmath282 if 1 . @xmath295 is contained in the kernel of @xmath297 2 . @xmath298 does not contain the trivial character of @xmath299 we say that an irreducible admissible representation @xmath181 of @xmath117 is minimal if for every character @xmath99 of @xmath110 we have @xmath300 [ thm : sc - kutzko ] there exists a bijective correspondence given by compact induction @xmath301 from very cuspidal representations @xmath298 of either maximal open compact - mod - center subgroup @xmath290 and irreducible minimal supercuspidal representations of @xmath184 moreover , every irreducible minimal supercuspidal representation of conductor @xmath302 ( resp . @xmath303 ) comes from a very cuspidal representation of @xmath292 ( resp . @xmath293 ) oflevel @xmath285 . following kutzko we use the terminology that a supercuspidal representation of @xmath117 is said to be _ unramified _ if it comes via compact induction from a representation of @xmath292 and _ ramified _ if it comes via compact induction from a representation of @xmath293 . we now take up both types of supercuspidal representations and briefly review how they break up on restriction to @xmath124 . we refer the reader to @xcite and @xcite for this . we begin with the unramified case . let @xmath298 be an irreducible very cuspidal representation of @xmath292 of level @xmath285 ( @xmath304 ) . let @xmath181 be the corresponding supercuspidal representation of @xmath184 let @xmath305 then we have @xmath306 where @xmath307 if @xmath308 or if @xmath309 and @xmath171 is irreducible , then @xmath310 is irreducible , hence so is @xmath311 . we thus have an unramified supercuspidal @xmath7-packet @xmath312 if @xmath309 and @xmath171 is reducible , then @xmath298 comes from the unique ( up to twists ) cuspidal representation of @xmath313 whose restriction to @xmath314 is reducible and breaks up into the direct sum of the two cuspidal representations of @xmath314 of dimension @xmath315 correspondingly , we have @xmath316 , and if we let @xmath317 and @xmath318 , then we obtain the unique supercuspidal @xmath7-packet @xmath319 of @xmath124 containing four elements . for the ramified case , let @xmath298 be a very cuspidal representation of @xmath293 of level @xmath285 ( @xmath320 ) and let @xmath181 be the corresponding supercuspidal representation of @xmath117 . let @xmath321 then @xmath316 for two irreducible representations @xmath322 ( @xmath323 ) of @xmath128 and @xmath324 conjugates one to the other , i.e. , @xmath325 . let @xmath326 and so @xmath327 . then the restriction of @xmath181 to @xmath124 breaks up into the direct sum of two irreducible supercuspidal representations as @xmath328 we call @xmath329 a _ ramified supercuspidal @xmath7-packet _ of @xmath198 to summarize , we have three kinds of supercuspidal @xmath7-packets for @xmath124 namely , 1 . unramified supercuspidal @xmath7-packets @xmath330 ; 2 . the unique ( unramified ) supercuspidal @xmath7-packet @xmath319 of cardinality four ; 3 . ramified supercuspidal @xmath7-packets @xmath329 . ( unramified supercuspidal @xmath7-packets of cardinality two ) [ prop : sc - unramified-2-sl2]@xmath16 consider an unramified supercuspidal @xmath7-packet @xmath330 determined by a very cuspidal representation @xmath298 of level @xmath285 of @xmath292 as above . the conductors @xmath331 are both equal to @xmath302 . the dimensions of the spaces @xmath332 and @xmath333 are as follows : 1 . for any @xmath201 such that @xmath334 we have @xmath335 2 . let @xmath334 and @xmath336 if @xmath285 is odd then for all @xmath337 .2pc ( a ) @xmath338 + \(b ) @xmath339 . 3 . let @xmath334 and @xmath336 if @xmath285 is even then for all @xmath337 .2pc ( a ) @xmath340 + \(b ) @xmath341 ( test vectors for unramified supercuspidal @xmath7-packets of cardinality two)[prop : sc - unramified-2-sl2-test ] @xmath16 let @xmath298 be a very cuspidal representation of @xmath292 which determines an unramified supercuspidal @xmath7-packet @xmath330 as above . assume that @xmath342 is realized in its kirillov model with respect to @xmath343 define two elements @xmath344 and @xmath345 in the kirillov model as follows : @xmath346 0 , & \mbox{if $ x \notin ( \mathcal{o}^{\times})^2,$ } \end{array } \right.\\[.2pc ] \phi_{\epsilon}(x ) & = \widetilde{\pi}(\gamma)\phi_1.\end{aligned}\ ] ] let @xmath347 . we have 1 . @xmath348 2 . if @xmath285 is even , then @xmath349 . in addition , @xmath5 is @xmath236-generic and any @xmath236-whittakerfunctional is non - zero on @xmath344 and vanishes on @xmath345 . furthermore , @xmath350 is not @xmath280-generic for any character @xmath280 of conductor @xmath351 . it is however @xmath352-generic , and any @xmath352-whittaker functional is non - vanishing on @xmath353 , which is a newform for @xmath354 3 . if @xmath285 is odd , then ( ii ) holds with @xmath5 and @xmath350 interchanged . ( unramified supercuspidal @xmath7-packet of cardinality four ) [ prop : sc - unramified-4-sl2]@xmath16 let @xmath298 denote a very cuspidal representation of @xmath292 of level @xmath309 such that @xmath355 let @xmath319 be the corresponding @xmath7-packet of @xmath198 then @xmath356 . moreover , 1 . let @xmath201 be any character such that @xmath357 if @xmath5 denotes any representation in the @xmath7-packet , then @xmath358 2 . let @xmath201 be any character such that @xmath359 and @xmath360 then for all @xmath361 we have .2pc ( a ) @xmath362 + \(b ) @xmath363 ( test vectors for unramified supercuspidal @xmath7-packets of cardinality four)[prop : sc - unramified-4-sl2-test ] @xmath16 with notation as above let @xmath319 be the unramified supercuspidal @xmath7-packet of cardinality four . let @xmath364 be the character of @xmath365 obtained from @xmath236 by identifying @xmath365 with @xmath366 . without loss of generality assume that @xmath367 is @xmath364-generic . then 1 . @xmath368 is @xmath236-generic , @xmath369 is @xmath352-generic , @xmath370 is @xmath269-generic , and @xmath371 is @xmath372-generic . 2 . assume that @xmath181 is realized in its @xmath236-kirillov model . the function @xmath344 of proposition [ prop : sc - unramified-2-sl2-test ] is a newform for @xmath368 . this further implies that @xmath373 is a newform for @xmath374 @xmath375 is a newform for @xmath370 and @xmath376 is a newform for @xmath377 finally , each of these newforms is a test vector for an appropriate whittaker functional comingfrom ( i ) . ( ramified supercuspidal @xmath7-packets ) [ prop : sc - ramified - sl2]@xmath16 let @xmath329 be a ramified supercuspidal @xmath7-packet of level @xmath285 as above . then @xmath378 . moreover , 1 . for any character @xmath201 of @xmath110 such that @xmath359 we have @xmath379 2 . let @xmath380 and @xmath336 for all @xmath381 we have @xmath382 ( test vectors for ramified supercuspidal @xmath7-packets ) [ prop : sc - ramified - sl2-test]@xmath16 let @xmath329 be a ramified supercuspidal @xmath7-packet coming from a very cuspidal representation @xmath298 of @xmath383 of level @xmath384 one and only one of the @xmath385 is @xmath236-generic , say @xmath369 . then @xmath371 is @xmath269-generic . let @xmath386 if @xmath344 and @xmath345 have the same meaning as in proposition [ prop : sc - unramified-2-sl2-test ] ( assuming that @xmath181 is realized in its kirillov model ) , we have 1 . @xmath387 and @xmath388 2 . any @xmath236-whittaker functional is non - zero on @xmath344 and similarly any @xmath269-whittaker functional is non - zero on @xmath345 . [ thm : conductor ] let @xmath5 be an irreducible admissible representation of @xmath122 let @xmath181 be a representation of @xmath389 whose restriction to @xmath124 contains @xmath26 assume that @xmath181 is minimal , i.e. , @xmath390 for all characters @xmath99 of @xmath224 then @xmath391 the next theorem relates the conductor of a representation @xmath5 of @xmath124 with the depth ( see @xcite ) @xmath392 of @xmath5 ( cf . @xcite ) . [ thm : conductor - depth - sl2 ] let @xmath5 be an irreducible representation of @xmath198 let @xmath392 be the depth of @xmath26 1 . if @xmath5 is any subquotient of a principal series representation @xmath227 , then @xmath393 2 . if @xmath5 is an irreducible supercuspidal representation , then @xmath394 we now define the basic filtration subgroups of @xmath39 as we did for @xmath124 in [ sec : newforms ] . let @xmath396 , @xmath397 , the standard hyperspecial subgroup of @xmath39 , and for @xmath398 , @xmath399 we let @xmath400 . let @xmath46 be an admissible representation of @xmath39 such that @xmath47 acts by scalars on @xmath70 . let @xmath50 be a character of @xmath51 such that @xmath401 ( where we have identified @xmath47with @xmath402 ) . for any such character @xmath50 and any subgroup @xmath403 of @xmath39 we define @xmath404 we define the _ @xmath50-conductor _ @xmath62 of @xmath49 to be @xmath405 we define the _ conductor _ @xmath64 of @xmath49 as @xmath406 if @xmath407 is such that @xmath408 and @xmath409 ( resp . @xmath410 ) , then we call @xmath411 ( resp . @xmath412 ) a _ space of newforms _ of @xmath413 . in this case , we refer to a non - zero element of @xmath411 or @xmath412 as a _ newform _ of @xmath413 . in this section , we will compute the dimension of @xmath414 for every irreducible admissible infinite - dimensional representation @xmath49 of @xmath39 and every character @xmath50 such that @xmath415 . we will often make use of the following fact . let @xmath5 be the restriction of @xmath49 to @xmath124 , and let @xmath416 by definition , the group @xmath207 acts on @xmath417 via the character @xmath201 , hence via @xmath50 . also , @xmath47 acts on @xmath417 via the character @xmath48 , hence via @xmath50 since @xmath418 . thus any @xmath419 acts on @xmath417 by multiplication by @xmath420 . in the light of ( [ eq : filtration ] ) , @xmath421 & = \mathcal{o}_f^{\times}/(\mathcal{o}_f^{\times})^2.\end{aligned}\ ] ] we may therefore take @xmath95 and @xmath422 as coset representatives for @xmath423 . hence if @xmath424 , then @xmath425 if and only if @xmath426 , i.e. , @xmath427 let @xmath428 be a character of @xmath429 let @xmath430 denote the principal series @xmath431 . according to @xcite , @xmath430 is irreducible except in the cases 1 . @xmath432 , 2 . @xmath433 . in case ( i ) , let @xmath434 be the character of @xmath402 defined by @xmath435 . then @xmath430 has two jordan hlder constituents , namely the one - dimensional representation @xmath436 and a square integrable representation @xmath437 . in case ( ii ) , @xmath430 is the direct sum of two irreducible representations @xmath438 and @xmath439 , which together form an @xmath7-packet of @xmath39 . we distinguish @xmath438 from @xmath439 by defining @xmath438 to be the summand that has a @xmath74-spherical vector , hence @xmath440 let @xmath441 . then the restriction of @xmath430 to @xmath124 is isomorphic to @xmath442 . it is easily seen then that the restriction to @xmath124 of any irreducible constituent of @xmath430 is itself irreducible unless @xmath99 is the character corresponding to some ramified quadratic extension @xmath443 . in this case @xmath444 decomposes as the direct sum @xmath445 . we now compute the conductors of the representations in the principal series of @xmath39 . [ thm : ps - u11 ] let @xmath428 be a character of @xmath429 . suppose that @xmath49 is an irreducible constituent of the principal series @xmath430 . 1 . if @xmath50 is a character of @xmath446 with @xmath418 , then @xmath447 if and only if @xmath448 or @xmath449 . moreover , @xmath450 1 , & \mbox{if $ \bar{\pi } = { \rm st}(\xi)$}. \end{array}\right.\end{aligned}\ ] ] 2 . suppose @xmath50 is as above . .2pc ( a ) if @xmath451 , @xmath428 is ramified , and @xmath452 , then @xmath453 m+1 , & \mbox{if $ m>0$}. \end{array } \right.\ ] ] + \(b ) for @xmath454 , we have @xmath455 \hskip -2.5pc \dim\left(\bar{\pi}^2(\bar{\chi})_{\bar{\eta}}^{\bar{k}_m}\right ) = \dim\left(\bar{\pi}^1(\bar{\chi})_{\bar{\eta}}^{\bar{k}'_m}\right ) & = \left\lceil\frac{m}{2}\right\rceil.\end{aligned}\ ] ] + \(c ) in all other cases , @xmath456 we may assume without loss of generality that @xmath428 is chosen so that @xmath49 is a subrepresentation of @xmath430 . let @xmath5 be the restriction of @xmath49 to @xmath124 . let @xmath50 be any character of @xmath51 with @xmath401 . let @xmath457 . since @xmath458 , @xmath459 we claim that @xmath460 precisely for @xmath461 or @xmath462 . the first part of ( i ) follows immediately from this claim , and the second follows from this together with the conductor calculations in [ sec : principal - sl2 ] . let @xmath463 . the only @xmath201 such that @xmath464 are @xmath465 and @xmath466 . hence we can not have @xmath460 unless @xmath50 equals @xmath467 on @xmath28 . we first prove that @xmath468 if and only if @xmath469 or @xmath449 in the case where @xmath470 and @xmath471 . since @xmath472 is contained in the restriction of @xmath430 to @xmath124 , which is isomorphic to @xmath473 , it is an easy consequence of the proofs of the statements in [ sec : principal - sl2 ] ( see @xcite ) that @xmath474 \mathbb{c}\bar{f}_w + \mathbb{c}\bar{f}_1 , & \mbox{if $ \bar{\chi}^2|_{\mathcal{o}_f^{\times } } = { \bf 1}$ , } \end{array } \right.\end{aligned}\ ] ] where @xmath475 \bar{\chi}(t)|t|_e^{1/2}\eta ( d ) = \bar{\chi}(td)|t|_e^{1/2 } , & \mbox{if $ g= \left(\begin{array}{cc } t & * \\ 0 & ^{s}t^{-1 } \end{array}\right ) w \left(\begin{array}{cc } a & b\\ c & d\end{array}\right)$ , } \end{array } \right.\\[.2pc ] \bar{f}_1 ( g ) & = \left\ { \begin{array}{ll } 0 , & \mbox{if $ g\notin\bar{b}k_c$,}\\[.2pc ] \bar{\chi}(t)|t|_e^{1/2}\eta ( d ) = \bar{\chi}(td)|t|_e^{1/2 } , & \mbox{if $ g = \left(\begin{array}{cc } t & * \\ 0 & ^{s}t^{-1 } \end{array}\right ) \left(\begin{array}{cc } a & b\\ c & d\end{array}\right)$. } \end{array } \right.\end{aligned}\ ] ] we now determine when @xmath476 lie in @xmath477 . in the light of ( [ eq : theta ] ) , this reduces to verifying whether @xmath478 acts as the scalar @xmath479 on these vectors . it is easily checked that @xmath480 \bar{\pi}(\theta)\bar{f}_1 & = \bar{\chi}(\epsilon_e)\bar{f}_1.\end{aligned}\ ] ] hence @xmath481 if and only if @xmath482 . this is equivalent to @xmath483 since @xmath50 and @xmath428 already agree on @xmath28 and @xmath402 ( by assumption ) and since @xmath484 is a representative for the non - trivial coset in @xmath485 . similarly , if @xmath486 , then @xmath487 if and only if @xmath488 , which is equivalent to @xmath489 since @xmath50 and @xmath462 already agree on @xmath28 and @xmath402 and since the non - trivial coset in @xmath485 is represented by @xmath115 . summarizing , we have that when @xmath471 and @xmath490 if and only if @xmath491 or @xmath449 . on the other hand , if @xmath492 , note that we may exchange @xmath428 and @xmath462 in the above proof since @xmath430 and @xmath493 have the same constituents . ( of course , exchanging @xmath428 and @xmath462 may make our assumption that @xmath49 is a subrepresentation of @xmath430 false . the only case in which this matters , however , is when @xmath494 , and in this case we are already done since @xmath495 on @xmath28 . ) then carrying out the proof _ mutatis mutandis _ , we obtain again that @xmath496 if and only if @xmath461 or @xmath462 . this establishes our claim if @xmath49 is in a singleton @xmath7-packet since for all @xmath497 , @xmath498 finally , suppose that @xmath499 . by the above , @xmath500 if and only if @xmath461 or @xmath462 . also , if @xmath50 is any character of @xmath51 , then since @xmath501 and @xmath502 , we have that @xmath503 but @xmath504 by proposition [ prop : unramfified - sl2 ] since @xmath505 . thus @xmath506 so again @xmath507 precisely for @xmath461 or @xmath462 . finally , conjugating by @xmath157 as above , one easily obtains the claim in the case @xmath508 . we now compute the dimensions of @xmath414 to prove ( ii ) . since @xmath509 and @xmath510 have the same irreducible constituents , we may assume that @xmath491 . ( as above , the representations @xmath437 present no problem here since in this case @xmath511 on @xmath51 . ) if @xmath471 the proof of ( i ) shows that @xmath512 . thus @xmath513 is 1 if @xmath514 and 2 if @xmath515 . the proof also shows that @xmath516 \dim ( \bar{\pi}^2(\bar{\chi}))_{\bar{\eta}}^{\bar{k}_0 } = \dim ( \bar{\pi}^1(\bar{\chi}))_{\bar{\eta}}^{\bar{k}'_0 } & = 0.\end{aligned}\ ] ] this shows that the formulae for the dimensions are valid when @xmath517 . suppose that @xmath518 . as with theorem 5.3 of @xcite , it follows from lemma 3.2.1 and the following proofs in 3.2 of @xcite that @xmath332 is the direct sum of @xmath472 together with certain two - dimensional spaces @xmath519 of the form @xmath520 ( @xmath521 ) , where @xmath522 \bar{\chi}(t)|t|_e^{1/2}\eta ( d ) = \bar{\chi}(td)|t|_e^{1/2 } , & \mbox{if $ g = \left(\begin{array}{cc } t & * \\ 0 & ^st^{-1 } \end{array}\right ) \left(\begin{array}{cc } 1 & 0\\ \varpi^i & 1 \end{array}\right ) \left(\begin{array}{cc } a & b\\ c & d\end{array}\right)$ , } \end{array } \right.\\[.3pc ] \hskip -4pc \bar{f}_{i,\epsilon } ( g ) & = \left\ { \begin{array}{l@{\quad}l } 0 , & \mbox{if $ g\notin\bar{b } \left(\begin{array}{cc } 1 & 0\\ \varpi^m\epsilon_f & 1 \end{array}\right ) k_m$,}\\[.3pc ] \bar{\chi}(t)|t|_e^{1/2}\eta ( d ) = \bar{\chi}(td)|t|_e^{1/2 } , & \mbox{if $ g= \left(\begin{array}{cc } t & * \\ 0 & ^st^{-1 } \end{array}\right ) \left(\begin{array}{cc } 1 & 0\\ \varpi^i\epsilon_f & 1 \end{array}\right ) \left(\begin{array}{cc } a & b\\ c & d\end{array}\right)$. } \end{array } \right.\end{aligned}\ ] ] we will now verify that whenever @xmath523 , 1 . @xmath519 is @xmath56-stable , and 2 . the subspace of @xmath519 on which @xmath56 acts via the character @xmath50 is one - dimensional . if this holds , then @xmath524 and the formulae for the dimension of @xmath414 follow easily from this equation and the dimension results of [ sec : principal - sl2 ] . the dimension of @xmath525 is computed analogously . we now show ( 1 ) and ( 2 ) . by ( [ eq : theta ] ) , this reduces to showing that @xmath519 is @xmath526-stable , and that the subspace of @xmath519 on which @xmath526 acts as the scalar @xmath527 is one - dimensional . let @xmath528 be either 1 or @xmath529 . then @xmath530 if @xmath531 , this is non - zero if and only if @xmath532 , i.e. , if and only if @xmath533 . this together with the fact that @xmath534 implies that @xmath535 is a multiple of @xmath536 . the exact multiple is determined by evaluating @xmath537 thus @xmath538 similarly , @xmath539 as claimed , @xmath526 stabilizes @xmath540 . moreover , the characteristic polynomial of @xmath526 on this two - dimensional space is @xmath541 the eigenvalues of @xmath526 on @xmath542 are therefore @xmath543 . it follows that the subspace of @xmath519 on which @xmath526 acts as the scalar @xmath479 is one - dimensional.@xmath544 now suppose that @xmath49 is an irreducible representation of conductor @xmath545 in the principal series of @xmath39 and that @xmath50 is such that @xmath546 . we consider the effect of the whittaker functional @xmath547 given by ( [ eqn : whittaker - ps ] ) on @xmath548 . ( test vectors for principal series representations ) [ prop : ps - test - u11]@xmath16 suppose that @xmath49 is an irreducible representation in the principal series of @xmath39 . let @xmath50 be a character of @xmath51 with @xmath418 such that @xmath549 . let @xmath550 . 1 . if @xmath451 , @xmath428 is ramified , and @xmath551 , then @xmath49 is @xmath236-generic . moreover , the space of vectors @xmath552 on which @xmath547 vanishes is one - dimensional . 2 . if @xmath499 , then @xmath49 is @xmath236-generic and @xmath547 is non - zero on the one - dimensional space of newforms @xmath553 . 3 . if @xmath508 , then @xmath49 is not @xmath236-generic , but it is @xmath554-generic . moreover , @xmath555 is non - zero on the one - dimensional space of newforms @xmath556 . 4 . in all other cases , @xmath49 is @xmath236-generic . in addition , if @xmath557 , then @xmath558 for any newform @xmath559 in @xmath477 . let @xmath5 be the restriction of @xmath49 to @xmath124 . note that since @xmath124 and @xmath39 have borel subgroups with the same unipotent radical ( namely , @xmath120 ) , the restriction of @xmath547 to any @xmath236-generic component of @xmath5 is a non - zero @xmath236-whittaker functional on that component , while its restriction to any non-@xmath236-generic component is @xmath560 . let @xmath561 . assume we are in case ( ii ) , ( iii ) , or ( iv ) . let @xmath562 be either @xmath563 or @xmath564 , according to the case , and let @xmath565 , i.e. , @xmath7 is either @xmath566 or @xmath567 . assume that @xmath559 is a non - zero vector in @xmath568 . by theorem [ thm : ps - u11 ] , the restriction of @xmath49 to @xmath124 is irreducible of conductor @xmath545 , and @xmath569 is one - dimensional . the statements in each of these cases now follow easily from the analogous results about @xmath5 in [ sec : principal - sl2 ] . suppose now that @xmath451 with @xmath428 ramified and @xmath570 . then @xmath5 has conductor @xmath571 and @xmath572 has dimension 2 . if @xmath5 is irreducible , then @xmath5 is @xmath236-generic according to corollary [ cor : test - ram - ps ] . also , according to the proof of theorem [ thm : ps - u11 ] ( and using its notation ) , @xmath573 . it follows from corollary [ cor : test - ram - ps ] that @xmath574 . since the image of @xmath547 has dimension 1 , @xmath547 must vanish on a one - dimensional subspace of @xmath552 . if @xmath5 is reducible , then as discussed in [ sec : principal - sl2 ] , @xmath5 decomposes as the direct sum of two representations @xmath369 and @xmath371 . moreover , only one of these representations , say @xmath369 , is @xmath236-generic by corollary [ cor : test - ramified - l - packets - sl2 ] . then @xmath547 vanishes on moreover , by corollary [ cor : test - ramified - l - packets - sl2 ] , for all non - zero hence , as in the preceding paragraph , the subspace of @xmath552 on which @xmath547 vanishes is one - dimensional.@xmath544 we now consider the supercuspidal representations of @xmath39 . let @xmath49 be such a representation . it is easily deduced from analogous results on @xmath117 and @xmath124 that @xmath49 is compactly induced from an irreducible representation of @xmath144 , @xmath575 , or @xmath143 . we will call @xmath49 an _ unramified ( resp . ramified ) supercuspidal representation _ of @xmath39 if its restriction to @xmath124 contains an unramified ( resp . ramified ) supercuspidal representation of @xmath198 suppose first that @xmath49 is ramified . let @xmath5 be the restriction of @xmath49 to @xmath124 . let @xmath369 be an irreducible component of the restriction of @xmath49 to @xmath124 . then @xmath369 is a ramified supercuspidal representation of @xmath124 . we extend @xmath369 to a representation of @xmath576 via the central character @xmath48 , also denoted by @xmath369 . then @xmath49 is contained in @xmath577 , and the restriction of @xmath577 to @xmath576 is @xmath578 . but conjugation by @xmath526 and @xmath324 have the same effect on @xmath124 so , by the discussion in the beginning of [ sec : sc - sl2 ] , @xmath369 and @xmath579 comprise an @xmath7-packet for @xmath124 . since @xmath580 , @xmath577 is irreducible and hence equal to @xmath49 . thus @xmath581 , where@xmath582 . from theorem [ thm : conductor - depth - sl2 ] , we see that the conductor of both @xmath369 and @xmath371 is @xmath583 , where @xmath584 is the depth of both @xmath369 and @xmath371 . we note that the depth of a twist of @xmath49 is no less than @xmath584 . to see this , let @xmath104 be a point in the bruhat tits building of @xmath39 ( which is the same as that of @xmath124 ) and let @xmath585 be a non - negative real number . then any vector in the twist of @xmath49 that is fixed by @xmath586 is fixed by @xmath587 since @xmath588 ( see @xcite ) . it follows that the depth of the twist of @xmath49 is no less than the depth of its restriction to @xmath124 . but this restriction is @xmath5 , which has depth equal to @xmath584 . on the other hand , we may select a character @xmath99 of @xmath39 such that @xmath589 on @xmath590 ( viewed as a subgroup of @xmath47 ) . if @xmath591 , then @xmath592 is trivial on @xmath593 , and it is easily seen that @xmath594 . define @xmath595 as @xmath99 ranges over all characters of @xmath39 . then we have @xmath596 . [ thm : sc - ram - u11 ] let @xmath46 be a ramified supercuspidal representation of @xmath39 . let @xmath50 be any character of @xmath51 with @xmath418 and @xmath597 . then we have @xmath598 and @xmath599 let @xmath5 be the restriction of @xmath49 to @xmath124 . set @xmath600 and @xmath457 . as discussed above , the restriction of @xmath49 to @xmath124 is the direct sum of two ramified supercuspidal representations @xmath601 , each of conductor @xmath545 . by proposition [ prop : sc - ramified - sl2 ] , @xmath602 is non - zero if and only if @xmath603 . hence if @xmath604 , @xmath605 since @xmath606 suppose @xmath603 . as in [ sec : principal - u11 ] , we compute @xmath607 using the fact ( [ eq : theta ] ) that @xmath414 is the subspace of @xmath608 on which @xmath478 acts as the scalar @xmath609 . since @xmath610 and the conjugation action of @xmath526 and @xmath324 are the same on @xmath124 , @xmath369 and @xmath371 form an @xmath7-packet according to [ sec : sc - sl2 ] . thus @xmath611 is the restriction to @xmath124 of a minimal ramified supercuspidal representation @xmath181 of @xmath117 . in particular , we have an action of @xmath117 on @xmath70 . let @xmath612 be the one - dimensional space @xmath613 . then according to the proof @xcite of proposition [ prop : sc - ramified - sl2 ] @xmath614 now @xmath478 intertwines @xmath615 and @xmath371 and takes @xmath616 to @xmath617 . therefore , @xmath618 let @xmath619 for @xmath620 . note that @xmath621 & = \omega_{\bar{\pi}}(\epsilon_e/\,^s\epsilon_e)\bar{\pi}\left ( \left(\begin{array}{ll } \epsilon_f & 0\\ 0 & \epsilon_f^{-1 } \end{array}\right ) \right).\end{aligned}\ ] ] thus @xmath622 acts via the scalar @xmath623 on @xmath624 . it follows that @xmath478 exchanges the one - dimensional spaces @xmath625 since @xmath626 & \bar{\pi}(\theta ) ( \bar{\pi}(\theta)\widetilde{\pi}(\beta)^iw ) = \bar{\pi}(\theta)^2 ( \widetilde{\pi}(\beta)^iw ) = \bar{\eta}(^s\epsilon_e ^{-1})^2\widetilde{\pi}(\beta)^iw = \widetilde{\pi}(\beta)^iw.\end{aligned}\ ] ] in particular , each @xmath627 is stabilized by @xmath478 . moreover , since @xmath622 acts via the scalar @xmath628 on @xmath627 , the eigenspaces of @xmath478 on @xmath627 corresponding to the eigenvalues @xmath629 must each be one - dimensional . hence the subspace of @xmath630 on which @xmath478 acts via the scalar @xmath631 has dimension @xmath632 , as required.@xmath544 suppose that @xmath46 is an unramified supercuspidal representation induced from a representation @xmath633 of @xmath144 . it is easily seen that the restriction @xmath5 of @xmath49 to @xmath124 is either 1 . an irreducible unramified supercuspidal representation of @xmath124 induced from @xmath74 if the restriction of @xmath633 to @xmath74 is irreducible , or 2 . the direct sum of two irreducible unramified supercuspidal representations of @xmath124 induced from @xmath74 if the restriction of @xmath633 to @xmath74 is isomorphic to @xmath634 , where @xmath367 and @xmath635 come from the two cuspidal representations of @xmath314 of dimension @xmath636 ( as in [ sec : sc - sl2 ] ) . in case ( ii ) , we note that if @xmath5 decomposes into the direct sum of @xmath637 and @xmath638 , then @xmath639 . as discussed in the ramified case , if @xmath640 as @xmath99 ranges over all characters of @xmath39 , then the conductors of the components of @xmath5 are @xmath641 . [ thm : sc - un - u11 ] let @xmath46 be an unramified supercuspidal representation of @xmath39 that is induced from @xmath144 , and let @xmath642 . let @xmath50 be any character of @xmath51 with @xmath418 and @xmath643 . then @xmath644 . 1 . if @xmath645 is odd , then @xmath646 \hskip -1.25pc\dim ( \bar{\pi}_{\bar{\eta}}^{\bar{k}'_m } ) & = \max\left\ { \left\lceil\frac{m - c(\bar{\pi})-1}{2}\right\rceil , 0\right\ } = \dim ( ( \bar{\pi}')_{\bar{\eta}}^{\bar{k}_m}).\end{aligned}\ ] ] 2 . if @xmath645 is even , then @xmath647 \hskip -1.25pc\dim ( ( \bar{\pi}')_{\bar{\eta}}^{\bar{k}'_m } ) & = \max\left\ { \left\lceil\frac{m - c(\bar{\pi})-1}{2}\right\rceil , 0\right\ } = \dim ( \bar{\pi}_{\bar{\eta}}^{\bar{k}_m}).\end{aligned}\ ] ] we give a proof only for case ( ii ) ( @xmath648 even ) as the proof for case ( i ) is easily obtained therefrom by interchanging the representations @xmath49 and @xmath649 . moreover , we prove only the first equality of each line as the second follows by conjugating by @xmath157 . let @xmath650 be the restriction of @xmath651 to @xmath124 . set @xmath652 and @xmath653 . now @xmath350 is a direct summand of the restriction to @xmath124 of a minimal unramified supercuspidal representation @xmath654 of @xmath117 . since @xmath181 is unramified , it follows from [ sec : sc - sl2 ] that @xmath655 is isomorphic to @xmath350 and hence that @xmath656 maps @xmath70 onto @xmath70 . ( here we view @xmath70 as a subrepresentation of @xmath657 . ) as discussed above , @xmath350 is either an irreducible unramified supercuspidal representation of conductor @xmath545 or the direct sum of two such representations @xmath658 and @xmath659 , where @xmath660 . by propositions [ prop : sc - unramified-2-sl2 ] and [ prop : sc - unramified-4-sl2 ] , the level @xmath285 of the inducing data for these representations is @xmath661 . as in the ramified case , we have @xmath662 if @xmath663 . suppose @xmath603 . by ( [ eq : theta ] ) , to find @xmath664 , we compute the dimension of the subspace of @xmath665 on which @xmath666 acts as the scalar @xmath609 . let @xmath667 . since @xmath668 is odd , it follows from propositions [ prop : sc - unramified-2-sl2 ] and [ prop : sc - unramified-4-sl2 ] and their proofs @xcite that @xmath669 and @xmath670 in fact , from the proof of proposition 3.3.4 in @xcite , it follows that for a certain vector @xmath671 , @xmath672 . if @xmath350 is irreducible , then since conjugation by @xmath324 and @xmath526 have the same effect on @xmath124 , @xmath656 and @xmath666 are both elements of the one - dimensional space @xmath673 . they are therefore equal up to scalars so @xmath674 . if @xmath350 is reducible , then we may further assume that @xmath675 by proposition [ prop : sc - unramified-4-sl2-test ] . in this case , @xmath656 and @xmath666 are both elements of the one - dimensional space @xmath676 so @xmath677 as above . as in the ramified case , @xmath678 acts via the scalar @xmath628 on @xmath679 . it follows that @xmath666 exchanges the one - dimensional spaces @xmath680 since @xmath681 & \bar{\pi}'(\theta)\left(\mathbb{c}\bar{\pi}'(\theta ) \phi\right ) = \bar{\pi}'(\theta)^2\left(\mathbb{c}\phi\right ) = \bar{\eta}(^s\epsilon_e ^{-1})^2\mathbb{c}\phi = \mathbb{c}\phi.\end{aligned}\ ] ] in particular , @xmath666 stabilizes @xmath612 . again as in the ramified case , these facts imply that the eigenspaces of @xmath666 on @xmath612 corresponding to the eigenvalues @xmath629 must each be one - dimensional . the same is clearly true of @xmath682 for @xmath683 . it follows that the subspace of @xmath665 on which @xmath666 acts via the scalar @xmath684 has dimension @xmath685 as required . the computation of @xmath686 is entirely analogous.@xmath544 now suppose that @xmath49 is a supercuspidal representation of @xmath39 of conductor @xmath545 . we consider the effect of a whittaker functional @xmath547 on @xmath548 . for this we need to choose the character @xmath50 somewhat carefully . let @xmath188 be the @xmath7-packet of @xmath39 containing @xmath687 then the restriction to @xmath124 of the direct sum of representations in @xmath188 is also the restriction to @xmath124 of a minimal supercuspidal representation @xmath181 coming via kutzko s construction . we require @xmath688 ( test vectors for supercuspidal representations ) @xmath16 suppose that @xmath49 is an irreducible supercuspidal representation of @xmath39 of conductor @xmath545 . let @xmath50 be a character of @xmath51 with @xmath689 and @xmath690 ( see above ) . let @xmath691 . 1 . if @xmath49 is ramified , then @xmath49 is @xmath236-generic . moreover , @xmath558 for all non - zero @xmath559 in @xmath477 or @xmath692 . 2 . if @xmath49 is unramified and induced from @xmath144 , let @xmath693 . + .2pc ( a ) if @xmath694 is odd , then @xmath49 is @xmath236-generic and @xmath649 is @xmath554-generic . moreover , @xmath558 for all non - zero @xmath695 and @xmath696 for all non - zero@xmath697 . + \(b ) if @xmath694 is even , then @xmath649 is @xmath236-generic and @xmath49 is @xmath554-generic . moreover , @xmath558 for all non - zero @xmath698 and @xmath696 for all non - zero @xmath699 . let @xmath5 be the restriction of @xmath49 to @xmath124 . as in proposition [ prop : ps - test - u11 ] , we note that the restriction of @xmath547 to any @xmath236-generic component of @xmath5 is a @xmath236-whittaker functional on that component , while its restriction to any non-@xmath236-generic component is @xmath560 . suppose first that @xmath49 is ramified ( case ( i ) ) . then @xmath5 decomposes as the direct sum @xmath700 of irreducible ramified supercuspidal representations of conductor @xmath545 . by proposition [ prop : sc - ramified - sl2-test ] , only one summand , say @xmath369 , is @xmath236-generic and we have that @xmath250 is non - zero on @xmath701 now @xmath692 is the space of vectors in @xmath702 on which @xmath478 acts as the scalar @xmath609 . as observed in the proof of theorem [ thm : sc - ram - u11 ] , @xmath478 exchanges @xmath703 and @xmath704 . therefore , @xmath477 can not lie in either @xmath705 or @xmath704 . in particular , if @xmath706 is written as @xmath707 with @xmath708 , then @xmath709 . since @xmath710 is @xmath236-generic and @xmath371 is not , we get @xmath711 we now give a proof in case ( ii ) . we only prove ( a ) as the proof of ( b ) is obtained by interchanging @xmath49 and @xmath649 . suppose that @xmath49 is unramified and induced from @xmath144 and that @xmath712 is odd . then @xmath5 is also unramified , induced from @xmath74 , and has conductor @xmath545 . as noted in the proof of theorem [ thm : sc - un - u11 ] , @xmath713 since the level of the inducing data of @xmath5 is @xmath714 , which is even , @xmath5 is @xmath236-generic by proposition [ prop : sc - unramified-2-sl2-test ] . moreover , @xmath715 , while @xmath716 . by the proof of theorem [ thm : sc - ram - u11 ] , @xmath478 exchanges @xmath717 and @xmath718 therefore , just as in the ramified case , if @xmath719 , then @xmath720 . it follows that @xmath721 the proof of the non - vanishing of @xmath722 is entirely analogous.@xmath544 we have only considered the unitary group @xmath35 for an unramified extension @xmath723 the entire series of results go through with some minor modifications if instead we considered ramified extensions . [ thm : conductor - otherinvariants - u11 ] let @xmath49 be an irreducible admissible supercuspidal representation of @xmath37 the relation between its conductor @xmath64 and its minimal depth @xmath648 is given by @xmath724 if @xmath5 is an irreducible subrepresentation of the restriction of @xmath49 to @xmath124 then @xmath725 this follows from theorems [ thm : sc - ram - u11 ] and [ thm : sc - un - u11].@xmath544 given an irreducible representation @xmath49 of @xmath39 and a character @xmath50 of @xmath51 such that @xmath447 , one can ask if we have @xmath726 . the answer is that this is often the case but is not true in general . indeed , we have @xmath726 unless @xmath49 is the principal series representation @xmath430 , where @xmath428 is ramified and @xmath727 . for these exceptional representations , the dimension of the space of newforms is two . nevertheless , in all cases we have proved that an appropriate whittaker functional is non - vanishing on some newform . this can be used to formulate a kind of a multiplicity one result if we consider the quotient of the space of newforms by the kernel of this whittaker functional . more precisely , if @xmath728 is a non - trivial additive character of @xmath0 of conductor either @xmath9 or @xmath729 such that @xmath49 is @xmath728-generic , and @xmath730 is a @xmath728-whittaker functional for @xmath49 , then we have @xmath731 another possibility is to consider some canonical non - degenerate bilinear form on the space @xmath732 and consider the orthogonal complement of the subspace @xmath733 as a candidate for a one - dimensional space of newforms . then the multiplicity one result is formulated as @xmath734 we thank benedict gross , dipendra prasad and paul sally jr . for some helpful correspondence . we thank brooks roberts for his interest in this work and for his generous comments on new avenues of thought that might stem from it . the second author is grateful for the warm and pleasant working experience at bucknell university where a substantial part of this work was completed . we thank the referee for a very careful reading of the manuscript . kutzko p c and sally jr . p , all supercuspidal representations of @xmath737 over a @xmath32-adic field are induced , representation theory of reductive groups ( utah : park city ) ( 1982 ) pp . 185196 ; _ prog . math . _ ( boston , ma : birkhaser ) ( 1983 ) vol .
|
let @xmath0 be a non - archimedean local field whose residue characteristic is odd . in this paper
we develop a theory of newforms for @xmath1 , building on previous work on @xmath2 .
this theory is analogous to the results of casselman for @xmath3 and jacquet , piatetski - shapiro , and shalika for @xmath4 . to a representation @xmath5 of @xmath1
, we attach an integer @xmath6 called the conductor of @xmath5 , which depends only on the @xmath7-packet @xmath8 containing @xmath5 .
a newform is a vector in @xmath5 which is essentially fixed by a congruence subgroup of level @xmath6 .
we show that our newforms are always test vectors for some standard whittaker functionals , and , in doing so , we give various explicit formulae for newforms .
= msam10 at 10pt = tibi at 10.4pt [ theore]*theorem * [ theore]proposition [ theore]remark [ theore]corollary [ theore]definition
|
You are an expert at summarizing long articles. Proceed to summarize the following text:
there has been an intense interest to understand the superconductivity of the recently discovered lafeaso.@xcite experiments have found values of the curie temperature ( t@xmath5 ) as large as 26 k for electron doping of lafeaso@xmath0f@xmath1 , 0.04 @xmath6 0.12@xcite . similar values of t@xmath5 are found for hole doping of la with sr but not with ca@xcite . neutron scattering@xcite and optical measurements@xcite find an antiferromagnetic ( afm ) ground state which has been confirmed by previous electronic structure calculations.@xcite the nature of the superconductivity has not been understood , though evidence suggests its unconventional character.@xcite the understanding of the normal - state electronic structure is important and serves as the foundation for understanding the superconductivity . one important question is what happens to the electronic structure when the extra electrons are added to the system via the fluorine dopants . a number of band structure studies have been performed to date to address these questions ; however , most of them use either the simple rigid - band picture of shifting the fermi energy in the band structure of the undoped system or the virtual crystal approximation.@xcite while these methods are expected to describe the rough picture , the actual positions of the dopants could make significant differences to the band structure as compared to the rigid - band shift or to the vca band structure , which is well known from the work on other systems.@xcite in this work , we investigate the band structure using full supercell calculations and study the changes in the fermi surface and the energetics with electron doping , with the fluorine substitution of the oxygen sites . lafeaso forms in the @xmath7 structure@xcite with ( fe@xmath8as@xmath9)@xmath10 layers lying between ( la@xmath11o@xmath12)@xmath10 layers , each of the atoms forming a square sublattice . half of the as atoms belonging to the feas layer occur above the center of the fe squares and the other half below it in an alternating pattern . they belong to a class of materials@xcite formed by one layer of a rare - earth atom with oxygen and another layer with late transition metal with a pnictogen atom . each fe atom , lying at the middle of a layer as seen in fig . [ figcrystal ] , is coordinated with four as atoms in distorted tetrahedral bonds above and below ; o also lies in a distorted tetrahedron of la atoms . the doping of la ( with sr ) or o ( with f ) is not in the magnetic feas layer but changes the magnetic properties nonetheless . experimental lattice parameters of @xmath13 = 4.035 and @xmath14 = 8.739 were used . the internal parameters were relaxed by total energy minimization , the results of which agreed with the values reported in the literature@xcite , viz . , @xmath15 = 0.142 and @xmath16 = 0.633 . electronic structure calculations were performed using the linearized augmented plane wave ( lapw ) method as implemented in the wien2k@xcite program . the unit cell contains two formula units and for studying the effects of the dopants we used two supercells , a 16-atom supercell ( four formula units ) formed by doubling the cell in the @xmath17 or @xmath18 direction and a 32-atom supercell ( eight formula unit ) formed by doubling the cell in the @xmath19 plane in each direction . these two supercells correspond , respectively , to 25% and 12.5% f doping when one o atom is replaced by f. calculations were also performed with the virtual crystal approximation ( vca)@xcite with the standard unit cell . these two methods were used to understand the effects of f doping on the o sites . in the vca the nuclear and the electron charge of the o atoms are increased continuously to approximate the additional electrons introduced by the f dopants . for example , a 5% concentration of f would change the nuclear and electronic charge of the o atoms from 8.0 to 8.05 . since superconductivity is expected to arise in the nonmagnetic ( nm ) state , we have focused on the electronic structure in the nm state . . , title="fig:",width=317 ] . , title="fig:",width=317 ] in order to understand the effect of electron doping , we first discuss the results for the density of states obtained from the supercell calculation of f - doped lafeaso . the density of states ( dos ) for lafeaso given in fig . [ figdos1]a shows la @xmath20 and @xmath21 states lying above the fermi level , while the o @xmath22 and as @xmath22 states occur below it . the o @xmath23 and as @xmath23 states lie well below , outside the range of the figure . the fe @xmath21 states hybridize with the as @xmath22 states , though the size of the as sphere in the lapw method leaves much of the as @xmath22 character outside the spheres , reducing its weight in the plot . this leaves the primary character of the bands observed in the calculated dos near @xmath24 as fe @xmath21 . strong fe - fe interactions cause the fe @xmath21 states not to split apart into @xmath25 and @xmath26 states . the positions of these states agree very well with those reported for the undoped lafeaso@xcite and lafeasp.@xcite a full supercell calculation with 25% f replacing o , shown in fig . [ figdos1]b , finds that the f @xmath22 levels lie far below @xmath24 and act only to add electrons to the system , appearing to cause a rigid shift of the bands . as mentioned by previous authors@xcite , although the total number of carriers increases , the electron doping shifts @xmath24 to a lower dos , making it hard to understand how the superconducting state can arise . however , while the dos has a minimum at @xmath24 , there is no evidence that the system is close to a metal - insulator transition.@xcite fe@xmath27as@xmath27o@xmath28f ) in violet and for the undoped material ( la@xmath27fe@xmath27as@xmath27o@xmath27 ) with rigid shift in black and ( b ) the corresponding fermi surfaces given on the @xmath29 plane . the symmetry points are for the supercell brillouin zone , which has the same symmetry points as in the original unit cell but with half the magnitudes for the @xmath30 and @xmath31 components.,title="fig:",width=226 ] fe@xmath27as@xmath27o@xmath28f ) in violet and for the undoped material ( la@xmath27fe@xmath27as@xmath27o@xmath27 ) with rigid shift in black and ( b ) the corresponding fermi surfaces given on the @xmath29 plane . the symmetry points are for the supercell brillouin zone , which has the same symmetry points as in the original unit cell but with half the magnitudes for the @xmath30 and @xmath31 components.,title="fig:",width=226 ] from the calculated dos ( fig . [ figdos1 ] ) , it might appear that the band structure for lafeaso is relatively unaffected by f doping , so that a rigid band shift of @xmath24 to accommodate the added electrons might be good enough to describe the states at the fermi energy . we find that while the overall shapes of the bands are about the same , there are enough differences in the states near @xmath24 to produce significant differences in the fermi surface for the doped case . the band structure has been plotted in fig . [ figbands2]a for the 32-atom supercell with one f atom on an o site and a calculation without f doping but with the bands rigidly shifted . in comparing the two cases , we have aligned the bands so that the energies of the deep oxygen core levels ( o 1@xmath23 and 2@xmath23 ) remain the same , in view of the fact that the deep core levels are very narrow in energy and they are not affected by the f substitution . comparing the two sets of bands , the bands with f doping are sometimes above the shifted bands and sometimes below , so a better agreement is not possible simply by shifting the bands further . an important difference is the increased splitting of bands halfway between @xmath32 and @xmath33 at @xmath24 . previous calculations@xcite have predicted that a rigid shift would lead to no separation between these two bands at @xmath24 , but the supercell calculations show that these two bands remain apart . turning now to the fermi surface , in the original brillouin zone of the standard unit cell , the fermi surface consists of two hole sheets around @xmath32 and two electron sheets around @xmath33 . all sheets now occur around the @xmath32 point of the supercell brillouin zone , since the original @xmath33 point gets folded to @xmath32 . most of the fermi sheets in the full calculation have larger radii than that predicted from a rigid shift as the bands move further away from @xmath32 as seen from fig . [ figbands2]b . thus the rigid band shift does not describe very well the changes in the fermi surface due to the doping . fe@xmath27as@xmath27o@xmath28f ) ( violet lines ) compared with the equivalent vca calculation ( black lines ) with changed o nuclear charge . ( b ) same as ( a ) except that the vca calculation was done with changed la nuclear charge ( black lines ) . no difference is seen between the two sets of band structures near @xmath24.,title="fig:",width=226 ] fe@xmath27as@xmath27o@xmath28f ) ( violet lines ) compared with the equivalent vca calculation ( black lines ) with changed o nuclear charge . ( b ) same as ( a ) except that the vca calculation was done with changed la nuclear charge ( black lines ) . no difference is seen between the two sets of band structures near @xmath24.,title="fig:",width=226 ] our calculations of a rigid shift of the bands show significant changes in the fermi surface compared to the full supercell calculation with the dopants included . in view of the fact that the states at @xmath24 are predominantly fe @xmath21 and the f dopants are far from the feas layers , one might expect that the dopants could affect the band structure near @xmath24 in two ways : ( a ) by changing the coulomb potential on different fe sites by different amounts depending on their locations or ( b ) by introducing the extra electrons in the fe layers which can then modify the on - site energies of different fe orbitals differently because of their selective occupation of the various fe(@xmath21 ) orbitals . quite interestingly , we find that there is a remarkable agreement between the vca and the supercell results for states close to @xmath24 ( fig . [ figbands1 ] ) . in both cases , we have the same number of electrons in the feas layer and this agreement does not change even if we introduce the extra carriers in the vca by changing the la nuclear charge instead of the o nuclear charge . this shows that the band structure is sensitive only to the electron concentration in the feas layer , so the coulomb shift due to the relative position of the f dopants is lost by the dielectric screening due to the intermediate la and as layers . by the same token , the rigid band shift does not describe the band structure accurately because of the different concentration of the electrons implicit in the rigid band shift vs. the full calculation . with and without the spin - orbit interaction.,title="fig:",width=207 ] with and without the spin - orbit interaction.,title="fig:",width=207 ] while comparisons of vca and rigid shifts of the bands are important to the fermi surface , spin - orbit effects can also change the details of the fermi surface . since small changes to the fermi surface can play an important role in superconductivity , spin - orbit effects can not be ignored . spin - orbit has not been investigated in lafeaso in any detail . band structure calculations for the 8-atom unit cell given in fig . [ figbands2]a , b show that spin - orbit lifts degeneracies for bands lying near @xmath24 . one can see that the splitting is larger at @xmath24 along @xmath34 and @xmath35 . significant changes occur along @xmath2 where the more dispersive fe @xmath36 band hybridizes with much less dispersive fe @xmath37 and @xmath38 bands , separating the third hole pocket ( with strong @xmath36 character ) from the rest of the fermi surface . the other bands at @xmath24 are relatively unchanged . the calculated fermi surface for the standard 8-atom unit cell corresponding to the undoped ( lafeaso)@xmath10 agrees well with previous calculations@xcite . we here show the fermi surface calculated using the vca for 12.5% f concentration in fig . [ figfermi1]a . the fermi surface consists of two cylindrical hole sheets lying along @xmath2 and two cylindrical electron sheets lying along @xmath4 . by doubling the unit cell in the both directions of the @xmath19 plane to form the supercell , the fermi surface undergoes band folding , as seen in fig . [ figfermi1]b . this causes the elliptical electron pockets around @xmath33 to now surround the two cylindrical hole sheets at @xmath32 . we note that the crystal symmetry of the supercell is the same as that as the smaller unit cell . therefore , a point in the brillouin zone , such as @xmath33 ( @xmath39,@xmath39,0 ) is used in both figures , but corresponds to the fraction of the reciprocal lattice vectors in each case . therefore the @xmath33 point in fig . [ figfermi1]a is not the same @xmath33 point in fig . [ figfermi1]b . at lower concentrations , there exists a hole cap around the @xmath3 point , as has been mentioned in previous calculations@xcite . plane of the supercell brillouin zone for the 32-atom supercell of lafeaso with ( a ) no f doping and ( b ) with f replacing one of the 8 o sites.,title="fig:",width=226 ] plane of the supercell brillouin zone for the 32-atom supercell of lafeaso with ( a ) no f doping and ( b ) with f replacing one of the 8 o sites.,title="fig:",width=226 ] adding electrons to the feas plane via f doping increases the size of the elliptical electron pockets and reduces the size of the hole pockets . full calculations performed in the 32-atom supercell with no f doping ( fig . [ figfermi3]a ) shows the two elliptical electron pockets surrounding the two nearly circular hole pockets . the smaller electron pocket and the larger hole pocket nearly overlap . when we replace one f for an o atom ( la@xmath27fe@xmath27as@xmath27o@xmath28f ) in fig . [ figfermi3]b , the overall shape of the fermi surface remains unchanged , but the electron pockets become larger while the hole pockets shrink . as we can see in the plot of the fermi surface ( fig . [ figfermi1 ] ) , the hole pockets and electron pockets are narrower in the @xmath29 plane and become larger in the @xmath40 plane . addition of electrons reduces the differences between the sizes of the pockets in these two planes , so the fermi surface looks more column - like in the @xmath2 or @xmath4 direction , consistent with previous calculations.@xcite .radius @xmath41 of the hole pockets in in the @xmath42 and @xmath43 planes at different doping levels @xmath17 . the full supercell calculations with f dopants agree with the vca results for the cases where we have compared them ( @xmath17=0 , 0.125 , and 0.25 ) . the third hole pocket on the second plane disappears beyond @xmath44 . [ cols="^,^,^,^,^,^",options="header " , ] [ tabl1 ] c|ccc|ccc x&&@xmath45 plane&&&@xmath46 plane & + & a ( )&b ( )&@xmath47&a ( )&b ( )&@xmath47 + 0.00&0.102&0.128&0.601&0.102&0.152&0.744 + 0.125&0.104&0.127&0.571&0.104&0.147&0.707 + 0.25&0.122&0.148&0.571&0.122&0.172&0.707 + [ tabl2 ] the size of the hole and electron pockets were calculated using the vca and the supercell calculation with f substitution on o sites and have been shown in tables i and ii . since the fermi surfaces obtained from the vca and the full supercell calculations are substantially the same , only one number is given for each concentration . the size of the electron and hole pockets were calculated along the @xmath35 direction . the hole pockets as shown in fig . [ figfermi1 ] are circular ( or nearly so ) lying along @xmath2 and consist of hybridized fe @xmath37 and @xmath38 states . the @xmath37 and @xmath38 orbitals are degenerate due to the point group symmetry of fe . this is consistent with previous calculations@xcite . there exists a third band which forms the cap around the @xmath3 point , mostly of @xmath36 character . this third fermi surface sheet disappears below @xmath24 with 7 - 8% electron doping . the electron pockets are elliptical with significant nesting characteristics . several proposed superconducting theories require understanding of the eccentricity of electron pockets which affects the fermi surface nesting and in addition may be important for magnetic instabilities@xcite . we list in table ii the calculated eccentricity as a function of electron doping using the standard definition @xmath47 = @xmath48 , where @xmath13 and @xmath49 are , respectively , the major and the minor axes . the two electron fermi surface sheets surround the @xmath4 points in the standard unit cell ( fig . [ figfermi1]a ) . the electron cylinders arise out of two bands , one of which is primarily of fe @xmath50 character and the other , of mixed fe @xmath37 and @xmath38 character . these two bands can be identified as those lying along m@xmath32 in fig . [ figbands2]a , b crossing about 0.25 ev above @xmath24 . the eccentricity of the ellipse arises due to different dispersion along different directions in the @xmath19 plane.@xcite with electron doping , the separation between these two bands decreases along @xmath35 , reducing the eccentricity . however , unlike what was seen in a rigid shift of the bands@xcite , the eccentricity never disappears or begins to increase with electron doping . , width=317 ] conventional theories of the superconductivity describe the superconducting state to arise from the fermi surface instability of the paramagnetic normal state , while density functional calculations show the ground state of the undoped material to be an antiferromagnetic metal . therefore the question arises as to whether the electron doping destabilizes the afm state in favor of a paramagnetic state thereby facilitating the formation of the superconducting state . to address this question , we have performed calculations of the total energy with and without electron doping in the supercell geometry and have shown these results in fig . [ figene ] along with the vca results . the results of the full supercell calculation and the vca energies agree quite well , which is consistent with the excellent agreement between their two band structures ( fig . [ figbands1 ] ) . we find that even though the afm state is stable for all dopant concentrations , the energy of the nm state is significantly reduced as compared to that of the afm state . these results suggest that the the electron doping might serve to destabilize the afm state in favor of the nonmagnetic state thereby facilitating superconductivity . in summary , from density - functional supercell calculations we have studied the changes in the fermi surface of lafeaso as a function of electron doping . important differences in the fermi surface were found from results obtained with the simple rigid - band shift , while the virtual crystal approximation yielded reasonable results . finally , our total energy results suggest that electron doping might provide an extra degree of stability to the superconducting state by making the afm normal state less favorable . p. blaha , k. schwarz , g.k.h . madsen , d. kvasnicka , and j. luitz in _ wien2k , an augmented plane wave plus local orbitals program for calculating crystal properties _ , edited by k. schwarz ( technische universitt wien , austria , 2001 ) .
|
we study the changes in the fermi surface with electron doping in the lafeaso@xmath0f@xmath1 superconductors with density - functional supercell calculations using the linearized augmented planewave ( lapw ) method .
the supercell calculations with explicit f substitution are compared with those obtained from the virtual crystal approximation ( vca ) and from a simple rigid band shift .
we find significant differences between the supercell results and those obtained from the rigid - band shift with electron doping , although quite remarkably the supercell results are in good agreement with the virtual crystal approximation ( vca ) where the nuclear charges of the o atoms are slightly increased to mimic the addition of the extra electrons . with electron doping ,
the two cylindrical hole pockets along @xmath2 shrink in size , and the third hole pocket around @xmath3 disappears for an electron doping concentration in excess of about 7 - 8% , while the two elliptical electron cylinders along @xmath4 expand in size
. the spin - orbit coupling does not affect the fermi surface much except to somewhat reduce the size of the third hole pocket in the undoped case .
we find that with the addition of the electrons the antiferromagnetic state becomes energetically less stable as compared to the nonmagnetic state , indicating that the electron doping may provide an extra degree of stability to the formation of the superconducting ground state .
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.