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galactic halos formed from lcdm initial conditions in n - body simulations have approximately 10% of their mass in orbiting sub - halos @xcite . the gravitational stirring and heating of a galactic disk and halo star clusters by dark sub - halos is tempered by the large numbers , high random velocities and broad distribution of sub - halos throughout the overall dark halo . however , very low velocity dispersion star streams in galactic halos are sensitive to the degree to which the dark matter in the halo is sub - structured into thousands of orbiting sub - halos . the sub - halos fold and chop the star - streams and gradually increase the velocity dispersion around the mean motion to about @xmath115% of the halo circular velocity , typically 30 , over a hubble time @xcite . consequently , cool star streams are sensitive indicators of the presence of the predicted dark matter substructure . about half a dozen of the currently known milky way streams qualify as cool ( most confidently , pal 5 , gd-1 , orphan , archeron and styx ) that is , having local velocity dispersions below about 10 , or , width less than about 0.1 radian as seen from the center of the host galaxy . a dark matter sub - halo crossing such a cool stream will lead to visible disturbances . unfortunately , the star count data often do not yet have sufficient local numbers to allow statistically significant measurements of density variations relative to the galactic foreground and background @xcite . the pan andromeda archeological survey ( pandas ) @xcite in one fell swoop provides deep and uniform data to a distance of about 150 kpc from m31 s center . the spectacular star stream north - west of m31 , more than 100 kpc long , was first displayed in its entirety in @xcite . the great length of the stream , as well as being fairly distant from the disturbing effects of the main body of m31 , make the stream an exceptionally interesting case for analysis of density variations . and , the star counts have sufficient signal to noise to allow reliable local surface density measurements of the stream . multiple image gravitational lensing by a galaxy of a background quasars is another probe of sub - halos . @xcite pointed out that the anomalous flux ratios relative to a smooth potential model that accurately predicts the locations of the images are the natural consequence of substructure in galaxy halos . a statistical analysis of available strong lens systems @xcite assuming nfw density profile sub - halo profiles found that about 2% of the halo mass ( with a very large uncertainty ) is in substructure . lensing is very sensitive to the central density profile which @xcite found to be close to an einasto profile , which is shallower than the nfw profile . the theoretical situation has become somewhat unclear , since a projection of a sub - halo rich n - body galaxy simulation @xcite finds image distortions generally smaller than those seen in strong lens systems . a better overall understanding and tighter limits requires better data on more strong lens systems @xcite . in this paper we analyze m31 s nw star stream for the density variations that sub - halos are expected to induce . we first trace the mean centerline of the stream . the luminosity function and total luminosity of the main part of the stream are estimated as an indication of the progenitor system . the stream s width and deviations from the centerline are measured . the measurement of the density along the stream gives the data which we use to test whether the stream is smooth or lumpy . an illustrative simulation is used to show the expected scale and degree of stream structure as the amount of substructure increases from the visible dwarfs to the thousands of halos that lcdm predicts . pandas has acquired @xmath2 and @xmath3 band images covering nearly 400 square degrees of m31-m33 with the cfht megaprime system . the images have a minimum s / n=10 at @xmath4 mag . details of data reduction are in @xcite and @xcite . we adopt the @xcite distance of 785 kpc ( dm of 24.47 mag ) for m31 , although we note a statistically consistent , but somewhat smaller , cepheid distance having a dm 24.32@xmath5 mag or 731 kpc @xcite . the adopted distance indicates that the survey reaches to m@xmath6 mag at high signal - to - noise . we use data up to and including january , 2010 . image extraction yields approximately 31 million stellar objects in the images . matching the colors and magnitudes of the stars to isochrones of varying metallicity at a fixed age of 12 gyr @xcite identifies m31 red giant branch ( rgb ) stars using the procedures of @xcite . since we are seeing m31 through our own galaxy , it turns out that more than 80% of the stars do not match rgb isochrones at the adopted distance . these stars are not included in the map , although all stars above the completeness limit are useful for the measurement of geometric biases of the images . the stars are individually extinction corrected with galactic dust maps @xcite . we are left with about 4.3 million nominal red giant branch stars to 24.5 mag which have been assigned metallicities in the range of [ fe / h ] = [ -3 , 0 ] . of these stars , approximately 25% are in the nw quadrant which is our area of interest . the galactic stars that happen to fall into the range of colors and magnitudes that are appropriate for m31 red giants create a slowly varying foreground which will be removed in the analysis of the stream density . note that a shorter distance would make the inferred luminosities smaller and cause the isochrone match to systematically yield larger metallicities . to a first approximation this simply offsets the metallicity range we analyze and does not otherwise affect the analysis . star streams are identified as spatially coherent over - densities , over a limited range of metallicities . to create the map we project the rgb stars onto a tangent plane , for which we chose a center at ra=0h 24 m and dec = + 44 which is near the center of the nw stream . this center is offset from m31 s center , to make the co - ordinates reasonably rectangular in the region of the stream analysis . without the rgb star selection no streams are readily visible in the halo . figure [ fig_m31 ] plots the sky distribution ( in the tangent plane co - ordinates ) of the [ fe / h]=[-0.6 , -2.4 ] stars which we find below comprise the stream . the same feature is clear in any map made with the low metallicity star sample . the low metallicity map , figure [ fig_m31 ] , has a number of geometric sampling variations that must be controlled to make a reliable measurement of local stream density . the simplest problem is that the ccd array has gaps between the individual chips of @xmath780 in the horizontal direction and @xmath713 in the vertical direction . the images were dithered to cover the smaller gaps , but the resulting depth is shallower than the surrounding area and no simple count correction procedure that did not increase the noise was found . these gaps are identified in the camera ccd co - ordinate system and masked out of the map . a more complicated geometric bias results from the catalog being built up from stacked local images in which the stellar photometry was done to create a local catalog . these catalogs were then matched in the areas of overlap to eliminate duplicates and produce a global catalog . the images are vignetted towards the edge of the array so the image depth drops at the field edge , however the vignetting is partially compensated in the catalog matching at the image edges . that is , if the surface density is corrected with the inverse of the mean star density over the array then the map has an excess at the edges . it is straightforward to devise a correction which flattens the average surface density . the correction works well over a range of about 20% in surface density , but undesirably amplifies the noise if applied at lower completeness levels . we apply a cut to the image where the local surface density in ccd image co - ordinates drops below 20% of the average . the resulting masked out area is about 15% of the image , a fraction which is not very sensitive to the precise value of the cut . the masked out regions are not included in the analysis and are readily visible in figure [ fig_m31 ] . field to field photometric depth variations will introduce artificial variations in the density of stars in the map . the depth of an image for a fixed exposure time depends on the sky transparency , brightness and image quality at the time of observation . below the completeness limit there is substantial field to field variation that leads to a random checkerboard appearance , but the pattern essentially disappears above the completeness limit . our measurement of the stream density subtracts the local mean background to reduce the problems of local depth variations . a second , more difficult , form of depth variations is field to field variations in the zero points of the two filter bands . we can set a limit on these by using the double measurements of star brightness in the overlap regions . for bright stars where the photon noise is small the standard deviation is 0.037 mag in @xmath2 band and 0.052 mag in @xmath3 band , with no discernible pattern over the field . the star catalog in our selected range of color and magnitude has nearly constant numbers with increasing depth near the completeness limit , so a 5% change in zero point leads to about a 5% change in numbers . the 5% should be considered an upper limit since the edges of the images are vignetted and have significant local calibration variations . across each 1 megaprime field there are a myriad of camera co - ordinate dependent photometric offsets @xcite which the current calibration does not fully take into account . these offsets lead to a change in zero point within each one degree pointing that is an additional @xmath72 - 3% variation . the masking procedure removes the worst of these variations , and the non - linear correction to flat star counts substantially reduces the variations within a field . what remains is below the shot noise of the star counts . the north - west stream , readily visible in the star map of figure [ fig_m31 ] , extends more than 100 kpc from m31 in projection . an ellipse is used to trace the stream , which corresponds to finding the lowest order fourier mode , that is , @xmath8 , rotated and centered for a best fit . note that the angle @xmath9 is the angle once the ellipse is transformed into a sphere and is not the geometric angle except along the major and minor axes . higher order fourier terms are not yet justified given the signal to noise of the stream within these data . the ellipse parameters are varied , along with the stream width and the background region width to maximize the mean over - density within @xmath10 of the ellipse over the lower branch of the stream , taken as the angular range @xmath11 $ ] . the stream region is shown in the mexican top - hat filtered figure [ fig_cutout ] . this filtered image , with an inner positive region of radius 0.4 and outer negative region of radius of 0.8 , with an area integral of zero , usefully illustrates the stream and noise properties near the filter length . the analysis is done on an unfiltered high resolution version of figure [ fig_m31 ] with pixels of 18 , ten times smaller than in figures [ fig_m31 ] and [ fig_cutout ] . we find that the best fit ellipse has a location of [ -1.25,+1.90 ] in the tangent plane co - ordinates of figure [ fig_m31 ] relative to our field center at [ 0h 24m,+44 ] . there is no particular reason for this fitting procedure to be constrained to have an ellipse centered on m31 . the ellipse has axes of @xmath12 ( or @xmath13 kpc ) at an angle of 42.25 w of n. the best fit occurs for an ellipse that is within the image data rather than for one that extends off the imaged area , even if the fitting procedure is started with a longer ellipse that extends out of the field . rr & @xmath14 + [ fe / h ] - 0.2 & stars kpc@xmath15 + -0.60 & 0.05 + -0.80 & 0.16 + -1.00 & 0.41 + -1.20 & 0.53 + -1.40 & 0.27 + -1.60 & 0.28 + -1.80 & 0.06 + -2.00 & 0.17 + -2.20 & 0.08 + -2.40 & 0.01 + the straightforward consequences of projection are that the stream distance from m31 must be 117 kpc or larger , and , the ellipse must be at least as circular as we found above . we have assumed that all the stars in the stream are at 785 kpc , the adopted distance to m31 . however , the distances of stars around the stream could vary roughly 100 kpc closer or further away , depending on how the stream is projected . such distance variations will cause a distance modulus error of nearly @xmath16 mag , which means that individual stars will be identified with isochrones of the incorrect metallicity . this would be a significant problem if the stream were a narrow range in metallicity . however , our unweighted selection of stars over the very broad range of [ fe / h]=[-0.6 , -2.4 ] corresponds to selecting a large rhombus in color - magnitude space that is 3.1 mag high in absolute magnitude . the upper limit to the distance differences of the stream relative to the body of m31 would blur out our selection of stream stars about 10% or so , but it is only a small dilution of the stream density . we measure the density perpendicular to the ellipse ridge line and display the result in figure [ fig_stream_width ] . the stream has a half width at half of the peak density of about 2.5 kpc or a full width of 5 kpc . the stream width is dominated by the physical width of the stream , but also includes a significant component from the stream location deviating slightly from our assumed elliptical shape . the stream subtends an angle ranging from about 0.1 radian to 0.05 radian , as seen from the center of m31 , although the stream is physically wider than the cool , but relatively close , streams in our own galaxy . the location of the centerline of the stream , calculated as the density weighted mean distance from the best fit ellipse , is shown in figure [ fig_stream_center ] . the error flags are calculated from the statistics of the star counts and confirms the impression of figure [ fig_cutout ] that the stream center varies , sometimes discontinuously , from the best fit ellipse . the variations are largest on the upper branch of the stream , @xmath17 , where it has large gaps . on the more continuous lower branch there is a statistically significant offset from the mean centerline at @xmath18 . the density is measured within a constant width region around the ellipse , @xmath19 ( @xmath20 kpc ) . we subtract the foreground / background measured at the same angular bin within the two adjacent 0.2 strips on either side of the stream . the range of metallicity in the stream is shown in table [ tab_slices ] in bins of width 0.2 in [ fe / h ] . table [ tab_slices ] gives the top of the metallicity bin in column 1 , the mean number of stars per kpc@xmath21 in column 2 . this table shows that the stream has its maximum density in the [ fe / h]@xmath22 bin . for our measurement of the stream density we will use the entire [ fe / h]=[-0.6,-2.4 ] range , uniformly weighted . in principle somewhat better signal to noise would be obtained if we weighted the stars with the metallicity distribution function . however , the flat distribution gives the most straightforward stream density measurement . the stream is broadly distributed in brightness with the mean number of stars [ 21 - 22 , 22 - 23 , 23 - 24 , 2@xmath23(24 - 24.5 ) ] @xmath3-band mag in the [ fe / h]=[-0.6 , -2.4 ] range being [ 0.62 , 0.78 , 0.55 , 0.47 ] per magnitude per kpc@xmath21 over the -70 to -10 angle range . the luminosity function of the stream is essentially flat over the absolute magnitude range @xmath24 as is usually the case for low metallicity rgb luminosity functions . physical properties of the stream are derived below . whether the nw stream is smooth or lumpy is the observational question of primary interest in this paper . this measurement needs to be handled carefully since the stream over - density is only about 15% of the total nominal rgb star density . the stream density will vary simply due to the @xmath25 fluctuations in the few thousand stars in each angular bin . the density measurement is done within the region shown in figure [ fig_cutout ] . the stream is defined as being a fixed width of the centerline and the local background is measured within the cutout region beyond the stream . the analysis region of both the stream and local background is shown in figure [ fig_cutout ] . the newly identified dwarf galaxy andxxvii @xcite , just above the upper branch of the stream in figure [ fig_m31 ] , is masked out to exclude it from the local background correction . the density along the stream , within a half - width of 0.4 on the sky subtracting the local background in the adjacent 0.2 regions , is shown in figure [ fig_stream_density ] at intervals of 2 in the ellipse angle , @xmath9 . the errors are computed from the @xmath25 of the star - counts in the stream and in the background region , added in quadrature . the @xmath26 test estimates the probability that a set of data is consistent with a model , which in our case is a constant density . we need to be aware that sampling and calibration variations could artificially create density variations . masking takes care of the ccd gaps and the low exposure regions of the images . the field flattening largely corrects the counts over the rest of the image . any residual photometric fluctuations will be measured in the control fields to estimate their size . the @xmath26 statistic applied to the @xmath11 $ ] range indicates that the chance that the stream has a uniform density is less than about @xmath27 . a binning angle of 2 corresponds to roughly 2 - 4 kpc , depending on location around the ellipse . we have measured the stream density variations in bins ranging from 1 to 8 finding that the variations are present at all these scales . since there are easily visible clumps and gaps for @xmath28 including the upper branch of the stream in the @xmath26 analysis would significantly increase @xmath26 , which of course could well be a real description of the larger stream . we prefer to present the most conservative statistic on the visibly smoothest part of the stream . we emphasize that even after restricting our statistical analysis to the lower branch , the segment is the longest stream segment available for study so far . moreover its orbit appears to keep it well away from the potential disturbances of the structure in the disk of m31 . rrrc & & + @xmath29 & @xmath30 & @xmath31 & @xmath32 + + + 1 & 2.18 & 59 & @xmath33 + 2 & 3.01 & 28 & @xmath34 + 4 & 3.58 & 14 & @xmath35 + + + 1 & 1.17 & 59 & 0.17 + 2 & 1.16 & 28 & 0.25 + 4 & 0.65 & 14 & 0.85 + + + 1 & 1.96 & 29 & 0.002 + 2 & 1.57 & 13 & 0.090 + 4 & 2.38 & 6 & 0.027 + we have varied virtually all stream and image parameters to check the result that the density variations are truly those of the star stream . reasonable changes in the path of the ellipse , its width and the local background make little difference , other than slightly reducing the mean density of the stream . the photometric zero point variations that remain in this version of the data are readily estimated . the mean background is approximately 12 stars kpc@xmath15 and the photometric variations are expected to produce 5% or smaller fluctuations coherent over scales of 1 degree in the image , or a standard deviation of about 0.6 stars kpc@xmath15 . the measured density fluctuations have a standard deviation of @xmath12.0 stars kpc@xmath15 , at least three times the size of the variations that the photometric zero points are expected to cause . a straightforward control sample is to measure the fluctuations at the same location in higher metallicity stars , [ fe / h]@xmath36 - 0.5 , in which the stream is not detected . figure [ fig_zhi ] shows the mexican top - hat filtered image in the high metallicity stars , demonstrating that the background fluctuations have a significantly different pattern but have a similar amplitude . the density around the stream is displayed in figure [ fig_density_zhi ] . the @xmath26 test for the significance of the density variations is reported in table [ tab_chi2 ] . no significant variations are found . a second control sample is to simply offset the measurement region away from the stream . we add 2 to the minor axis and 4 to the major axis and repeat the measurements , adjusting the @xmath9 range to approximately the same distance . the statistics are reported in table [ tab_chi2 ] finding that the probability that the variations are due to chance to be about 5% . moving the region around find finds that the @xmath26 values fluctuate a fair bit depending on position although nothing as significant as the nw stream itself ever emerges . these fluctuations are likely the result of shorter and lower significance star streams that are real entities . whether the upper branch of the stream in fig . [ fig_m31 ] or [ fig_cutout ] is the extension of the lower branch of stream is unclear in these data . the upper branch certainly fits within the same ellipse and has a shared metallicity distribution at the precision we can measure it . however , the upper branch is less a stream than three dominant segments separated by gaps of comparable size . it is of great interest to note that the segments are tilted with respect to the general path of the stream . simulations show that tilted segments are some of the dominant structures frequently seen in chopped streams , as first noted in @xcite , and as we show below . the total luminosity of the lower branch of the stream is found by summing the over - densities in one magnitude bins within a range of [ -70,-10 ] ecliptic angle , to find a stream segment luminosity of @xmath37 to @xmath38 mag . the luminosity is the sum over the rgb stars alone . to estimate the total luminosity we use the cumulative stellar luminosity function of a stellar population of similar age and metallicity , for which we chose the globular cluster m12 . m12 is an intermediate metallicity cluster with a well studied stellar luminosity function @xcite . using their distance modulus of 14.0 mag and their i band luminosity function we find about 58% of the light above @xmath39 mag . therefore the corrected total luminosity is @xmath40 . the mean surface brightness is @xmath41 . if we assume the stream is a uniform density cylinder of 5 kpc radius and a stellar mass - to - light ratio of , say , 3 in the @xmath3-band , then the volume stellar mass density is @xmath42 . this can be compared to the mass density in the dark matter halo at say 90 kpc , which for our adopted halo parameters is @xmath43 , that is , about a factor of about 240 higher than the stellar stream mass density . at 30 kpc the ratio is 3600 times more background halo dark matter than stream stellar mass . if dark matter from the progenitor is mixed into the stream then the mass density of the stream would be proportionally increased , lowering the local mass ratios but the stream is unlikely to be significantly self - gravitating anywhere . @xcite have raised the possibility that jeans instabilities can be the source of of density variations in the stream . the jeans length in the stream , @xmath44 evaluated with the stellar mass density and assuming an internal velocity dispersion of 10 is about 400 kpc . on the other hand if the velocity dispersion were as low as 1 then the formal jeans length would be 40 kpc , comparable to the longest scale of density variations in the stream . very low velocity dispersions are expected for slowly tidally stripped globular clusters . the presence of several globular clusters along the course of the stream @xcite and the possibility that andxxvii @xcite is the source of the stream suggests that the internal velocity dispersion is at the level that a dwarf galaxy would create , @xmath45 or more . the resulting jeans length is then roughly 120 kpc , the size of the stream . it is important to note that the stability analysis is far from complete . first , the disrupted object may have its own dark matter which has identical kinematics to the visible stars and would increase the mass density . second , the stability analysis needs to account for the very strong tidal field of the m31 dark halo . strong tides generally act to suppress much instability , particularly in largely radially oriented streams , which is likely the case for the nw stream . another mechanism that generates clumps in tidal streams is simply the pileup of stars at low velocity points of their epicyclic orbits @xcite . the lumps appear at @xmath46 times the tidal radius . since no progenitor object for the stream has been confidently identified , the tidal radius is very uncertain . however if we take a low luminosity dwarf galaxy of stellar plus a relatively low dark mass for a total progenitor dwarf mass of @xmath47 , then at a galactic radius of @xmath48 kpc , the tidal radius , @xmath49 , is approximately 3.2 kpc . the epicycle pileup separation will then be at a separation of 120 kpc . that scale compares to the size of the stream . the stellar mass in the stream alone , @xmath50 for a fairly minimal assumed mass - to - light ratio of 3 , gives about 40 kpc epicyclic clumping scale , which is at the upper end of range of what we see , but still leaves the smaller scale variations to be explained . we conclude that epicyclic pile - up is not a significant clumping factor for the nw stream . if the progenitor system had large internal density variations in a highly order velocity field they could be fed out into the stream to create lumps . however , this idea runs into trouble with our astrophysical knowledge of low mass galaxies . dwarf spheroidal galaxies are comprised largely of stars on randomly oriented orbits and have essentially no substructure , which when tidally disrupted would lead to a smooth stream . dwarf irregular galaxies do have substantial substructure in their gas and young stellar populations . the visible luminosity ( corrected for light below our observational limit ) in the lower branch of the stream is only @xmath51 . dwarfs with such low luminosities usually have comparable rotational and random velocities which would quickly blur out stellar structures deposited into a stream . given that @xcite associate 3 or 4 globular clusters with the nw stream , which would lead to a unusually high ratio of globular cluster luminosity to progenitor luminosity , it it possible that the nw stream is only part of a larger stream that has yet to be clearly identified . the structures within star forming dwarf galaxies are often low mass stellar associations and clusters that fairly quickly dissolve as a result of stellar mass loss and their own internal dynamics . spiral patterns are at best very chaotic in low mass dwarf irregulars . our assessment is that although carefully sub - structured young progenitor disk system could produce some of the stream lumpiness , appropriate astrophysical systems do not exist . star streams normally originate from the tidal dissolution of either dwarf galaxies or globular clusters . in both cases tidal fields pull off outer stars or sometimes dissolve the entire system and distribute the stars in nearby orbits around the galaxy . the effect of adding randomly orbiting dark matter sub - halos on highly idealized star streams has been studied previously @xcite , finding that the streams were folded , chopped and heated to about 30 over a hubble time . for the examination of the expected degree of lumpiness of star streams we need a more realistic model stream . the association of globular clusters with the nw stream @xcite indicates that a dwarf galaxy , likely andxxvii , is the source of the stream . the study of the effects of dark matter sub - halos on the stream is inherently statistical since we are unsure to what degree the visible stars and gas give a complete census of the sub - halos , as highlighted in the celebrated missing satellite problem @xcite . a detailed match to the orbit of the nw stream is not required for this very basic assessment of the action of sub - halos on a stream . the details of the stream creation process have relatively little to do with the subsequent sub - halo interactions with the stream . our test particle stream originates from particles orbiting in a constant mass plummer sphere to which particles in the core region are gradually given velocities that boost them out into the region where tides can carry them away . future studies will report a full suite of self - gravitating simulations . the simulation uses a galactic potential with a spherical nfw halo potential scaled to the results of the aquarius simulation @xcite . the rotation curve of m31 is somewhat higher than the milky - way , but beyond 30 kpc mass modeling finds that the two galaxies are very similar @xcite . for simplicity and comparability we therefore adopt the same disk - bulge potential for the disk and bulge as @xcite which has a miyamoto - nagai disk potential @xcite and a nuclear bulge modeled with a plummer sphere . the sub - halos are modeled as a collection of plummer spheres following the distribution in masses , orbits and internal structure of the @xcite simulations . the star stream particles are evolved in the combined potentials of the background galaxy and orbiting sub - halos . we use time steps of 0.14 myr , or about @xmath52 in a hubble time , to ensure that the quickly moving particles are integrated properly over the small scale potentials of the sub - halos . the simulations start with the plummer sphere @xmath53 progenitor located at @xmath54 100 kpc from the center in the plane of the disk , with @xmath55 . the initial velocity is chosen to be entirely tangential perpendicular to the plane of the disk with an angular momentum of about 50% of the circular value at that radius . in the absence of sub - halos there are no density variations in the simulated star streams . m31 contains nearly 30 known dwarf galaxies . to what degree could these dwarfs , along with their dark - halos , cause the density variations we are measuring ? for instance , what is the effect of the heaviest 5 sub - halos alone ? figure [ fig_nsub_xy ] shows the @xmath56 plane projection of the stream in the presence of 0 , 5 , 100 , and 1000 sub - halos as drawn from the @xmath57 mass distribution . including the 5 most massive sub - halos produces only large scale distortions of the stream relative to the smooth halo . with 100 sub - halos we see significant small scale variations and going to 1000 sub - halos adds yet more small scale structure . figure [ fig_nsub_den ] shows the density variations in angular bins assuming that the stream has a constant width , which is close to being equivalent to the m31 measurements shown in fig . [ fig_stream_density ] . in this particular simulation the progenitor remains bound , so there is a density peak near @xmath58 . away from the progenitor there is a clear trend of increasing disturbance in the density variations as the number of sub - halos increases . neither a smooth halo nor one with only the five most massive sub - halos , which roughly mimics the most massive dwarf galaxies , produce anywhere near the amount of substructure we see . one hundred sub - halos do a reasonable job in the outer parts of the tail but do not do much damage to the inner part of the tail . once we get to one thousand sub - halos we star seeing gaps and clumps essentially everywhere . deviations from a mean centerline are also a signal of sub - halo interactions . again the hundred halo simulation produces a few angled clumps but they are much more clearly present with the thousand halo simulation . we will report on the statistics of a much larger set of simulations elsewhere . in this illustrative study we find that of order of one thousand sub - halos are required to produce all the small scale structure on the scales seen in the m31 nw stream from an initially smooth stream . @xcite report a more complete dynamical analysis focused on streams at smaller radii , but the dynamical outcomes they describe are quite generally applicable . they also firmly conclude that a large population of dark matter sub - halos will induce gaps and clumps in streams . a larger statistical study will be reported elsewhere . a spectacular star stream in our own galaxy is the pal 5 stream @xcite which is kinematically cold and narrow @xcite . at a distance of 23.2 kpc the visible part is only 4 kpc long . it is very narrow , about 0.4 kpc , roughly 1/5 of the width of the m31 nw stream segments . the source of the stream is the outer halo globular cluster pal 5 with the stream having the highest surface density at the location of its progenitor . the density analysis of @xcite finds that there are density variations of the pal 5 stream above the shot noise in the northern part of the stream , but not the southern part . the significance of the variation is not given , but there are 9 of 25 points more than @xmath59 from the mean trend over about 1 kpc of length . this certainly constitutes the detection of a significant lump in the currently available 4 kpc length of the pal 5 stream , but on a smaller scale . @xcite were inspired by the narrow velocity and geometric distribution of the 75 carbon stars of the sagittarius stream @xcite to consider the lumpiness of the milky way halo . they develop a statistic based on the fifth through tenth fourier terms to compare smooth halo models to models with up to 256 sub - halos , being careful to consider models that allow for the main dwarf galaxies . when applied to the available sagittarius data they find that `` the degree of scattering is entirely consistent with debris perturbed by the lmc alone . '' if applied to the 100 kpc nw m31 stream , the tenth fourier term would be a @xmath9=36 variation , which is roughly comparable to a 16 binning where there still is real structure relative to a constant , but only on yet larger scales . @xcite considered kinematic data for the sagittarius stream in our galaxy and concluded that the halo could not be very lumpy , however we note that our simulations show that sub - halos tend to chop " streams leaving locally cold remnants with pieces at the same radius offset in velocity , which may be compatible with the @xcite figure 2 . kinematic tests will become very powerful in testing for substructure once large velocity samples are available . the 100 kpc nw stream in m31 is a coherent geometric structure whose orbit is well clear of the body of m31 making it a near ideal testing ground for the presence or absence of thousands of dark sub - halos predicted in lcdm n - body simulations . although the stream appears to be nearly a half ellipse over a common range of metallicities the upper branch of the stream is less well defined and has a number of clearly visible gaps . the lower branch is nearly complete and provides a much more conservative test for the presence of sub - halos . the main result of this paper is that the stream has highly significant density variations on virtually all scales from 2 kpc , up to about 20 kpc . the variation of density around the mean has a very low probability of being a chance statistical fluctuation , less than @xmath0 . we have been careful to measure the lumpiness relative to an averaged local background and masked out gaps and regions where the photometric uncertainties add significantly to the variations . as a control sample we take a higher metallicity set of stars at exactly the same location as the stream , which finds no significant density variations . relative to other known cool star streams , the m31 stream stands apart for its length , distance from the disturbing effects of the m31 disk , a variety of scales of substructure and its high statistical significance . it is interesting to note that m31 appears to have only the one well defined long stellar stream at large radius . the nw stream has quite a low total luminosity so it should not particularly stand out relative to other stellar streams that were created at large radius over the buildup of the m31 dark halo . several other stream fragments are visible in the full field map @xcite but none as long or coherent as the nw stream . if one assumed that other long streams likely formed over the lifetime of the halo then the absence of others at the present time indirectly suggests that they have been broken up by sub - halos @xcite . the measured nw stream lumpiness essentially rules out the possibility that it is a low mass star stream orbiting in a smooth galactic potential of a disk plus bulge plus dark halo . conversely , the nw stream density variations are compatible with the level of density changes that a large population of dark matter sub - halos induce . the details of the density variation are sensitive to the statistical distribution of the sub - structure but are unlikely to be specifically modeled for a single stream . as more streams are observed statistical modeling of the degree of substructure present should become possible . the present study is statistically consistent with a highly sub - structured dark halo , although we caution that the density data alone for a single stream is not a conclusive proof that halos are as sub - structured as lcdm simulations predict . overall , the density variations of the nw stream are strong circumstantial evidence that the predicted thousands of dark matter sub - halos are present in m31 s dark halo .
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the pan andromeda archeological survey ( pandas ) cfht megaprime survey of the m31-m33 system has found a star stream which extends about 120 kpc nw from the center of m31 .
the great length of the stream , and the likelihood that it does not significantly intersect the disk of m31 , means that it is unusually well suited for a measurement of stream gaps and clumps along its length as a test for the predicted thousands of dark matter sub - halos .
the main result of this paper is that the density of the stream varies between zero and about three times the mean along its length on scales of 2 to 20 kpc .
the probability that the variations are random fluctuations in the star density is less than @xmath0 . as a control sample
we search for density variations at precisely the same location in stars with metallicity higher than the stream , [ fe / h]=[0 , -0.5 ] and find no variations above the expected shot noise .
the lumpiness of the stream is not compatible with a low mass star stream in a smooth galactic potential , nor is it readily compatible with the disturbance caused by the visible m31 satellite galaxies .
the stream s density variations appear to be consistent with the effects of a large population of steep mass function dark matter sub - halos , such as found in lcdm simulations , acting on an approximately 10 gyr old star stream .
the effects of a single set of halo substructure realizations are shown for illustration , reserving a statistical comparison for another study .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the study of the gaugino sector of supersymmetry is a complex and important endeavour , which appears well suited to a linear collider of sufficient energy and luminosity . the main observables of interest are the masses of the @xmath1 and @xmath2 states and their production cross sections , including those with polarised beams . @xmath3 collisions offer two independent techniques for determining the mass of supersymmetric particles . these are the analysis of the energy spectrum of the sm particle produced in association with a lighter supersymmetric state in the two - body decays and the study of the pair production cross section near threshold . these techniques have already been extensively studied for lower centre - of - mass energies , @xmath4 , between 0.35 to 0.5 tev @xcite . in this note , we analyse the gaugino pair production and derive the statistical accuracy on their masses using both techniques and including the effects of initial state radiation ( isr ) , beamstrahlung ( bs ) and parton energy resolution for multi - tev @xmath3 collisions . we follow the evolution of these accuracies for fully hadronic final states from pure signal samples to realistic inclusive susy samples and validate the results obtained at generator level with analyses performed on fully simulated and reconstructed events . the study provides us with requirements on parton energy resolution which are complementary to those obtained from other processes , such as heavy susy higgs decays , since the kinematics of decays of gaugino pairs with large missing energy into pairs of escaping neutralinos does not benefit from the kinematic fits , which are instead applicable to processes where the full beam energy is deposited in the detector . the estimated mass accuracies can be compared in a next step to those required for the reconstruction of the gut scale susy parameters @xcite and the determination of the lightest neutralino contribution to the dark matter relic density in the universe @xcite . this comparison will provide us with well - motivated quantitative requirements on parton energy resolution in susy events . this study considers two scenarios in the constrained mssm ( cmssm ) model , which offer different experimental challenges . their parameters are given in table [ tab : modelpar ] . the first ( model i ) , adopted as a benchmark point for the clic cdr studies @xcite , has the lightest neutralino at 340 gev and the chargino and heavier neutralinos with masses in the range 640 to 917 gev ( see table [ tab : mass ] and the left panel of figure[fig : spectra ] ) . at @xmath4 = 3 tev all the gauginos are observables . the relatively low masses and the 3 tev centre - of - mass energy make cross sections sizable but the beamstrahlung effects more significant ( see table [ tab : modelpar ] ) . in the second ( model ii ) the lightest neutralino has a mass of 554 gev , while the other neutralinos and the charginos have masses in the range from 1064 to 1414 gev ( see table [ tab : mass ] and the right panel of figure[fig : spectra ] ) @xcite . at 3 tev , most gauginos are close to threshold for pair production and cross sections are small . this minimises the beamstrahlung effects , since the production cross section drops significantly when the beams lose energy due to radiation . the cross sections are given in table [ tab : xsec ] and figure [ fig : xsec ] . .parameters of the two cmssm models adopted in this study [ cols="<,^,^",options="header " , ] [ tab : scan ] we compute the cross section @xmath5 at various @xmath4 values for a set of closely spaced masses and obtain the derivative @xmath6 of the change of the cross section at each energy per unit of mass change . results are shown in figure [ fig : sens ] , which indicate that the maximum of the sensitivity to the mass is achieved near threshold . the number of scan points and the share of the statistics among them is optimised by studying the mass uncertainty obtained from the fit for different assumptions . we find that it is preferable to concentrate the luminosity in a small number of scan points . for example , the statistical accuracy on the mass of the @xmath7 in the model i varies from @xmath80.85 gev , obtained for a four - point scan ( 1310@xmath91950 gev ) , to @xmath80.45 gev , when the luminosity is split between just two points , one of which at the peak of the sensitivity ( @xmath4=1350 gev ) and the second close to threshold ( @xmath4=1310 gev ) . this confirms the findings of @xcite for lower sparticle masses and different luminosity spectrum . finally , we consider the option of operating the collider with polarised beams . results are summarised in table [ tab : scan ] . in all cases , except the @xmath10 , the mass accuracies obtained with a dedicated threshold scan improve on those resulting from the kinematic edge analysis at 3 tev by factors of 2 or more . the use of polarised beam further improves these accuracies , effectively compensating for the loss of sensitivity due to isr and bs . the determination of chargino and neutralino masses in high - mass susy scenarios with two - body decays into @xmath11 , @xmath12 and @xmath13 bosons provides us with a mean to quantify the effect of radiation , by isr and beamstrahlung , and parton energy resolution on the accuracy achievable in supersymmetric particle mass measurements at a multi - tev @xmath3 linear collider . in our analysis both fits to boson energy spectra and threshold scans are considered for fully hadronic final states . results from generator - level quantities are validated using fully simulated and reconstructed events in the @xmath14 and @xmath15 final states . not accounting for reconstruction efficiencies , estimated to be @xmath1660% in four jet final states , the mass of charginos and neutralinos can be determined from the kinematic edges of the boson energy in inclusive susy event samples to a relative accuracy in the range 0.3% to 1.0% ( 0.6% - 1.0% ) in absence of radiation and energy resolution effects to 0.8% to 1.7% ( 1.1% - 2.0% ) accounting for isr , bs and realistic energy resolution for the benchmark with particle masses in the range 600 - 900 gev ( @xmath171000 gev ) , respectively , with 2 ab@xmath18 of integrated luminosity at @xmath4 = 3 tev . the relative increase of the statistical uncertainty of the mass measurement is larger for the model i which has the sparticles masses far way from pair the production thresholds . however , in absolute terms the larger production cross sections in this model yield better statistical accuracy in the mass determination . by adopting the criterion that the degradation to the mass measurement statistical accuracy from the parton energy resolution should not exceed that induced by isr and bs , we derive the requirement of a relative energy resolution for jets , @xmath190.05 . if the accelerator can operate at energies below the nominal @xmath4 ( down to @xmath4=1310 gev for model i and @xmath4=2200 gev for model ii ) with comparable performance to collect about one third of the statistics at centre - of - mass energies close to the kinematic thresholds for sparticle pair production , the mass accuracies from these threshold scans improves by factors of 2 or more compared to those obtained from study of the kinematic edges at the maximum @xmath4 energy . the availability of polarised beam in the scan further improves these accuracies , effectively compensating for the loss of sensitivity due to the effect of isr and beamstrahlung . we are grateful to the colleagues who contributed to this study . in particular to jean - jacques blaising , sabine kraml and abdelhak djouadi for extensive discussion and their careful reading of the text . we are also thankful to by dieter schlatter for valuable suggestions on this note . y. li and a. nomerotski , arxiv:1007.0698 [ physics.ins-det ] . g. a. blair , a. freitas , h. u. martyn , g. polesello , w. porod and p. m. zerwas , acta phys . polon . b * 36 * ( 2005 ) 3445 [ arxiv : hep - ph/0512084 ] . e. a. baltz , m. battaglia , m. e. peskin and t. wizansky , phys . d * 74 * ( 2006 ) 103521 [ arxiv : hep - ph/0602187 ] . s. martin , private communication . m. battaglia , a. de roeck , j. r. ellis , f. gianotti , k. a. olive and l. pape , eur . j. c * 33 * ( 2004 ) 273 [ arxiv : hep - ph/0306219 ] . j. r. ellis , t. falk , g. ganis , k. a. olive and m. srednicki , phys . b * 510 * ( 2001 ) 236 [ arxiv : hep - ph/0102098 ] . a. djouadi , j. l. kneur and g. moultaka , comput . commun . * 176 * ( 2007 ) 426 [ arxiv : hep - ph/0211331 ] . m. muhlleitner , a. djouadi and y. mambrini , comput . commun . * 168 * ( 2005 ) 46 [ arxiv : hep - ph/0311167 ] . g. belanger , f. boudjema , a. pukhov and a. semenov , comput . commun . * 176 * ( 2007 ) 367 [ arxiv : hep - ph/0607059 ] . d. larson _ et al . _ , arxiv:1001.4635 [ astro-ph.co ] . t. sjostrand , s. mrenna and p. z. skands , jhep * 0605 * ( 2006 ) 026 [ arxiv : hep - ph/0603175 ] . f. e. paige , s. d. protopopescu , h. baer and x. tata , arxiv : hep - ph/0312045 . s. katsanevas , p. morawitz , comput . phys . commun . * 112 * ( 1998 ) 227 - 269 . [ hep - ph/9711417 ] . j. l. feng and d. e. finnell , phys . d * 49 * ( 1994 ) 2369 [ arxiv : hep - ph/9310211 ] . h. u. martyn and g. a. blair , arxiv : hep - ph/9910416 . f. james and m. roos , comput . commun . * 10 * ( 1975 ) 343 . h. braun _ et al . _ [ clic study team ] , clic - note-764 ( 2008 ) . m. skrzypek and s. jadach , z. phys . c * 49 * ( 1991 ) 577 . e. boos _ et al . _ [ comphep collaboration ] , nucl . instr . and meth . a * 534 * ( 2004 ) , 250 . s. catani , y. l. dokshitzer , m. olsson , g. turnock and b. r. webber , phys . b * 269 * ( 1991 ) 432 . m. a. thomson , nucl . instrum . meth . a * 611 * ( 2009 ) 25 [ arxiv:0907.3577 [ physics.ins-det ] ] . g. a. blair , econf c010630 ( 2001 ) e3019 .
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this note reports the results of a study of the accuracy in the determination of chargino and neutralino masses in two high - mass supersymmetric scenarios through kinematic endpoints and threshold scans at a multi - tev @xmath0 collider .
the effects of initial state radiation , beamstrahlung and parton energy resolution are studied in fully hadronic final states of inclusive susy samples .
results obtained at generator level are compared to those from fully simulated and reconstructed events for selected channels .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
extensible markup language ( xml ) has reached a great success in the internet era . xml documents are similar to html documents , but do not restrict users to a single vocabulary , which offers a great deal of flexibility to represent information . to define the structure of documents within a certain vocabulary , schema languages such as _ document type definition _ ( dtd ) or _ xml schema _ are used . xml has been adopted as the most common form of encoding information exchanged by web services @xcite . @xcite attribute this success to two reasons . the first one is that the xml specification is accessible to everyone and it is reasonably simple to read and understand . the second one is that several tools for processing xml are readily available . we add to these reasons that as xml is _ vocabulary - agnostic _ , it can be used to represent data in basically any domain . for example , we can find the _ universal business language _ ( ubl ) in the business domain , or the standards defined by the _ open geospatial consortium _ ( ogc ) in the geospatial domain . ubl defines a standard way to represent business documents such as electronic invoices or electronic purchase orders . ogc standards define _ web service interfaces _ and _ data encodings _ to exchange geospatial information . all of these standards ( ubl and ogc s ) have two things in common . the first one is that they use xml schema to define the structure of xml documents . the second one is that the size and complexity of the standards is very high , making very difficult its manipulation or implementation in certain scenarios @xcite . the use of such large schemas can be a problem when xml processing code based on the schemas is produced for a resource - constrained device , such as a mobile phone . this code can be produced using a manual approach , which will require the low - level manipulation of xml data , often producing code that is hard to modify and maintain . another option is to use an xml data binding code generator that maps xml data into application - specific concepts . this way developers can focus on the semantics of the data they are manipulating @xcite . the problem with generators is that they usually make a straightforward mapping of schema components to programming languages constructs that may result in a binary code with a very large size that can not be easily accommodated in a mobile device @xcite . although schemas in a certain domain can be very large this does not imply that all of the information contained on them is necessary for all of the applications in the domain . for example , in @xcite a study of the use of xml in a group of 56 servers implementing the _ _ ogc s sensor observation service ( sos ) specification _ _ revealed that only 29.2% of the sos schemas were used in a large collection of xml documents gathered from those servers . based on this information we proposed in @xcite an algorithm to simplify large xml schema sets in an application - specific manner by using a set of xml documents conforming to these schemas . the algorithm allowed a 90% reduction of the size of the schemas for a real case study . this reduction was translated in a reduction of binary code ranging between 37 to 84% when using code generators such as jaxb , xmlbeans and xbinder . in this paper we extend the schema simplification algorithm presented in @xcite to a more complete _ instance - based xml data binding _ approach . this approach allows to produce very compact application - specific xml processing code for mobile devices . in order to make the code as small as possible the approach will use , similarly to @xcite , a set of xml documents conforming to the application schemas . from these documents , in addition to extract the subset of the schemas that is needed , we extract other relevant information about the use of schemas that can be utilised to reduce the size of the final code . a prototype implementation targeted to android and the java programming language has been developed . the remainder of this paper is structured as follows . section 2 presents an introduction to xml schema and xml data binding . in section 3 , related work is presented . the _ instance - based data binding approach _ is presented in section 4 . section 5 overviews some implementation details and limitations found during the development of the prototype . section 6 presents experiments to measure size an execution times of the code generated by the tool in a real scenario . last , conclusions and future work are presented . in this section we present a brief introduction to the topics of xml schema and xml data binding . xml schema files are used to assess the validity of well - formed element and attribute information items contained in xml instance files @xcite@xcite . the term xml data binding refers to the idea of taking the information in an xml document and convert it to instances of application objects @xcite . an xml schema document contains components in the form of complex and simple type definitions , element declarations , attribute declarations , group definitions , and attribute group definitions . this language allows users to define their own types , in addition to a set of predefined types defined by the language . elements are used to define the content of types and when global , to define which of them are valid as top - level element of an xml document . xml schema provides a derivation mechanism to express subtyping relationships . this mechanism allows types to be defined as subtypes of existing types , either by extending or restricting the content model of base types . apart from type derivation , a second subtyping mechanism is provided through substitution groups . this feature allows global elements to be substituted by other elements in instance files . a global element e , referred to as _ head element _ , can be substituted by any other global element that is defined to belong to the e s substitution group . with _ xml data binding _ , an abstraction layer is added over the raw xml processing code , where xml information is mapped to data structures in an application data model . xml data binding code is often produced by using code generators that use a description of the structure of xml documents using some schema language . the use of generators potentially gives benefits such as increased productivity , consistent quality throughout all the generated code , higher levels of abstraction as we usually work with an abstract model of the system ; and the potential to support different programming languages , frameworks and platforms @xcite . although most of the generators available nowadays are targeted to desktop or server applications , several tools have been develop for mobile devices such as xbinder and codesysnthesis xsd / e , or for building complete web services communication end - points for resource constrained environments , such as gsoap @xcite . all of the tools mentioned before map xml schema structures to programming languages construct in a straightforward way , which is not adequate when large schemas sets are used . problems related with having large and complex schemas have been presented in several articles @xcite . for example , @xcite deal with problems of large schemas in schema matching in the business domain . in the context of schema and ontology mapping , @xcite states that current match systems still struggle to deal with large - scale match tasks to achieve both good effectiveness and good efficiency . @xcite , the work extended here , expose the problems related to using xml data binding tools to generate xml processing code for mobile geospatial applications . last , @xcite present an algorithm to extract fragments of large conceptual schemas arguing that the largeness of these schemas makes difficult the process of getting the knowledge of interest to users . when considering xml processing in the context of mobile devices , literature is focused in two main competing requirements : _ compactness _ ( of information ) and _ processing efficiency _ @xcite . to achieve compactness compression techniques are used to reduce the size of xml - encoded information @xcite . about processing efficiency , not much work has been done in the mobile devices field . a prominent exception in this topic is the work presented in @xcite , @xcite and @xcite . these articles are all related to the implementation of a middleware platform for mobile devices : the _ fuego mobility middleware _ @xcite , where xml processing has a large impact . the proposed _ xml stack _ provides a general - purpose xml processing api called _ xas _ @xcite , an xml binary format called _ xebu _ @xcite , and others apis such as _ trees - with - references _ ( reftrees ) and _ random access xml store _ ( raxs)@xcite . regarding the use of instance files to drive the manipulation of schemas , @xcite presents a review of different methods that use instance files for ontology matching . in the field of schema inference , instance files are used as well to generated adequate schema files that can be used to assess their validity ( e.g. @xcite ) . _ instance - based xml data binding _ , is a two - step process . the first step , _ instance - based schema simplification _ , extracts the information about how schema components are used by a specific application , based on the assumption that a representative subset of xml documents that must be manipulated by the application is available . the second step , _ code generation _ , consists of using all of the information extracted in the previous step to generate xml processing code as optimised as possible for a target platform . the whole process is shown in figure [ fig : flow - xmldatabinding ] , the inputs to the first step are a set of schemas and a set of xml documents conforming to them . the outputs will be the subset of the schemas used by the xml documents and other information about the use of certain features of the schemas that can be used to optimise the code in the following step . the outputs of the first step are the inputs of the code generation step . the two steps of the process are detailed in the following subsections . + the _ instance - based schema simplification _ step extracts the subset of the schemas used on a set of xml documents . the algorithm used to perform this simplification was first presented in @xcite and has been extended here to extracts other information that can be used to produce more compact xml processing code . the idea behind this algorithm , is depicted graphically in figure [ fig : simplificationalg ] . the figure shows to the left the graph of relationships between schemas components . the different planes represent different namespaces . links between schema components represent dependencies between them . to the right we have the tree of information items ( xml nodes ) contained in xml documents . for the sake of simplicity we show in the figure only the tree of nodes corresponding to a single document . an edge between an xml node and a schema component represents that the component describes the structure of the node . to simplify the figure we have shown only a few edges , although an edge for every xml node must exist . starting from a set of xml documents and the schema files defining their structure , it is possible to calculate which schema components are used and which are not . in doing so , the following information is also recorded : * _ types that are instanced in xml documents _ : for each xml node exists a schema type describing its structure . while xml documents are processed the type of each xml node is recorded . this way we can know which types are instanced and which are not . * _ types and elements substitutions _ : the subtyping mechanisms mentioned in section 2.1 allow the _ real _ or _ dynamic type _ of an element to be different from its _ declared type_. elements declared as having type a , may have any type derived from a in an xml document . in this case the real type must be specified with the attribute _ xsi : type_. something similar happens with substitution groups , although in this case the attribute _ xsi : type _ is not necessary . the information about xml nodes whose dynamic type is different from its declared type is recorded . * _ wildcards substitutions _ : the elements used to substitute wildcards are recorded . * _ elements occurrence constraints information _ : for all of the elements it is checked that if they allow multiple occurrences there is at least one document where several occurrences of the element are present . * _ elements with a single child _ : all of the elements that contain a single child are also recorded . + a more detailed view of the code generation process is shown in figure [ fig : flow_gen ] . the outputs of the schema simplification step are used as inputs to the _ schema processor _ , the component of the generator in charge of creating the data model that will be used later by the _ template engine_. the _ template engine _ combines pre - existing _ class templates _ with the data model to generate the final source code . the use of a template engine allows the generation of code for other platforms and programming languages by just defining new class templates . + a summary of the features of the code generation process that contribute to the generation of optimised code is listed next : * _ use of information extracted from xml documents _ : the use of information about schema use allows to apply the following optimisations : * * _ remove unused schema components _ : the schema components that are not used are not considered for code generation . by removing the unused components we can substantially reduce the size of the generated code . the amount of the reduction will depend on how specific applications make use of the original schemas . * * _ efficient handling of subtyping and wildcards _ : the number of possible substitutions of a type by its subtypes , and a head element of a substitution group by the members of the group can be bounded with the information gathered from the instances files . in the general case , where no instance - based information is available , generic code to face any possible type or element substitution must be written . limiting the number of possible substitutions to only a few allows the production of simpler and faster code . the same reasoning is applied to wildcards . * * _ inheritance flattening _ : by flattening subtyping hierarchies for a given type , i.e. , including explicitly in its type definition all of the fields inherited from base types and eliminating the subtype relationship with its parents , we can reduce the number of classes in the generated code . the application of this technique will not necessarily result in smaller generated code , as the fields defined in base types must be replicated in all of their child types , but it will have a positive impact in the work of the class loader because a lower number of classes have to be loaded while the application is executed . let us consider the case of the geospatial schemas introduced in section 1 . these schemas typically present deep subtyping hierarchies with six or more levels , as a consequence when an xml node of a type in the lowest levels of the hierarchy must be processed , all of its parent types must be loaded first . the technique of inheritance flattening has been widely explored and used in different computer science and engineering fields as is proven by the abundant literature found in the topic @xcite . * * _ adjust occurrence constraints _ : if an element is declared to have multiple occurrences it must be mapped to a data structure in the target programming language that allows the storage of the multiple instances of the elements , e.g. an array or a linked list . in practice if the element has at most one occurrence in the xml documents that must be processed by the application it can be mapped to a single object instance . using this optimisation the final code will make a better use of memory because instead of creating a collection ( array , linked list , etc . ) that will only contain a single object , it creates a single object instance . * _ collapse elements containing single child elements _ : information items that will always contain single elements can be replaced directly by its content . by applying this optimisation we can reduce the number of classes in the generated code , which will have a positive impact in the size of the final code , the amount of work that has to be done by the class loader , and the use of memory during execution . this optimization is used by mainstream xml data binding tools such as jibx and the xml schema definition tool . * _ disabling parsing / serialization operations as needed _ : some code generators always includes code for parsing and serialization even when only one of these functions is needed . for example , in the context of geospatial web services , most of the time spent in xml processing by client - side applications is dedicated to parsing , as messages received from the servers are potentially large . on the other hand , most of these services allows request to be sent to the server encoded in an http get request , therefore xml serialisation is not needed at all . * _ ignoring sections of xml documents _ : frequently , we are not interested in all of the information contained in xml files , ignoring the unneeded portions of the file will improve the speed of the parsing process and it may have a significant impact in the amount of memory used by the application . in addition , the following features not related directly with code optimisation are also supported : * _ source code based on simple code patterns _ : the generated source code is straightforward to understand and modify in case it is necessary . * _ tolerate common validation errors : _ occasionally , xml documents that are not valid against their respective schemas must be processed by our applications . in many cases , the validation errors can be ignored following simple coding rules . a detailed explanation of each of the features presented in this section can be found in @xcite . as mentioned before , the approach presented in this paper is based on the assumption that a representative set of xml documents exists . by _ representative we mean that these documents contain instances of all of the possible xml schema elements and types that will be processed by the application in the future . nevertheless , this subset might not always be available . in this case , we can still take advantage of the approach by building _ synthetic _ xml documents containing relevant information . whether xml processing code is produced manually or automatically developers typically have some knowledge of the structure of the documents that must be processed by the applications . therefore , we can use this knowledge to build sample xml documents that can be used as input to the algorithm . in case it were necessary , the final code can be manually modified later , or the sample files changed and used to regenerate the code . if we were using synthetic documents instead of actual documents some of the optimisations related to the information extracted from them should not be applied . the reason for this is that we do not have enough information about how the related schema features are used . for example , we can not apply optimisations such as the efficient handling of subtyping and wildcards , as we might not know all of the possible type substitutions . something similar happens with the adjustment of occurrence constraints . nevertheless , other optimisations such as inheritance flattening or removing unused schema components can be still safely applied . _ dbmobilegen _ ( dbmg for short ) is the current implementation of the _ instance - based xml data binding _ approach @xcite . it includes components implementing both the simplification algorithm and code generation process . it is implemented in java and relies on existing libraries such as _ _ eclipse xsd _ _ for processing xml schemas , _ _ freemarker _ _ as template engine library , and as well as the generated code , _ _ kxml _ _ for low - level xml processing . this tool produces code targeted to android mobile devices and the java programming language . the current implementation has some limitations . because of the complexity of the xml schema language itself , support for certain features and operations have been only included if it is considered necessary for the case study or applications where the tool has been used @xcite . some of these limitations are listed next : * _ serialization is not supported yet _ : the role of parsing for our sample applications and case studies is far more important than serialization . this is mainly because we have preferred to use http get to issue server requests wherever possible . * _ dynamic typing using xsi : type not fully supported _ : the mechanism of dynamic type substitution by using the _ xsi : type _ attribute has not been fully implemented yet , as the xml documents processed in the applications developed so far do not use this feature . in this section we present two experiments . the first one tries to test how much the size of the generated code can be reduced by using dbmg . the second one measures the execution times of generated code in a mobile phone . in this experiment we borrow the test case presented in @xcite that implements the communication layer for an sos client . sos is a standard web service interface defined to enhance interoperability between sensor data producers and consumers @xcite . the sos schemas are among the most complex geospatial web service schemas as they are comprised of more than 80 files and they contain more than 700 complex types and global elements @xcite . the client must process data retrieved from a server that contains information about air quality for the valencian community . this information is gathered by 57 control stations located in that area . the stations measure the level of different contaminants in the atmosphere . a set of 2492 xml documents was gathered from the server to be used as input , along with the sos schemas , to the instance - based data binding process . the source code generated by dbmg is compiled to the _ compressed jar _ format and compared with the final code generated by other generators : xbinder , jaxb and xmlbeans . the last two are not targeted to mobile devices but are used here as reference to compare the size of similar code for other types of applications . table [ generatedcode2 ] shows the size of the code produced with the different generators from the full sos schemas ( full ) and from the subset of the schemas used in the input instance files ( reduced ) . the reduced schemas are calculated applying the schema simplification algorithm to the full sos schemas . the last row of the table ( libs ) includes the size of the supporting libraries needed to execute the generated code in each case . |p1.5cm| p1.1cm| p1cm| p1.6cm| p1cm| ' '' '' ' '' '' & xbinder & jaxb & xmlbeans & dbmg + full & 3,626 & 754 & 2,822 & 88 + reduced & 567 & 90 & 972 & 88 + libs & 100 & 1,056 & 2,684 & 30 + figure [ fig : codegen_full ] shows the total size of xml processing code when using the full and reduced schemas . in both cases , we can see the enormous difference that exists between the code generated by dbmg and the code generated by other tools . + the size for dbmg is the same in both cases because it implicitly performs the simplification of the schemas before generating source code . it must be noted that serialisation is not still implemented in dbmg . we roughly estimate that including serialisation code will increase the final size in about 30% . in any case , the code generated by this tool is about 6 times smaller than the code generated by xbinder from the reduced schemas and about 30 times smaller than the code generated from the full schemas . one of the reasons for this difference in size is the lack of serialisation support in dbmg . another reason is that xbinder generates code to ensure all of the restrictions related to user - defined simple types . this is an advantage if we parse data obtained for a non - trusted source and the application requires the data to be carefully validated , but it is a disadvantage in the opposite case , as unneeded verification increase processor usage and memory footprint . in the case of dbmg , as it aggressively tries to lower final code size , these simple type restrictions are ignored and these types do not even have a counterpart in the generated code . when compared to jaxb , using the reduced schemas , the main difference in size is in the supporting libraries , as the code generated by jaxb is very simple . still , the code generated by dbmg is slightly smaller because the step of removing elements with single child elements and inheritance flattening eliminates a large number of classes . in all of the cases , xmlbeans has the largest size . this tool is mostly optimised for speed at the expense of generating a more sophisticated and complex code and the use of bigger supporting libraries . to test the performance of the generated code we will parse a set of 38 capabilities files obtained from different sos servers . the code needed to parse these files is generated and deployed to a htc desire android smartphone with a 1 ghz qualcomm quaddragon cpu and 576 mb of ram . the 38 files have sizes ranging from less than 4 kb to 3.5 mb , with a mean size of 315 kb and a standard deviation of 26.7 kb . as the size range is large and with the purpose of simplifying presentation we divide the files in two groups , those with a size below 100 kb , caps - s ( 30 files ) , and those with size equal to or higher than 100 kb , caps - l ( 8 files ) . to obtain accurate measures of the execution time for the code we selected the methodology presented in @xcite . this methodology provides a statistically rigorous approach to deal with all of the non - deterministic factors that may affect the measurement process ( multi - threading , background processes , etc . ) . as our goal is only to measure the execution times of xml processing code , we stored the files to be parsed locally to avoid interferences related to network delays . besides , to minimise the interference of data transfer delays from the storage medium all of the files below 500 kb were read into memory before being parsed . it was impossible to do the same for files with sizes above 500 kb because of the device memory restrictions . figures [ mobile_execution_small ] and [ fig : mobile_execution ] shows the execution times of code generated by dbmg . the figures also include the execution times needed by _ kxml _ , the underlying parser used by dbmg , to process the same group of files . the execution times for _ kxml _ were calculated by creating a simple test case where files are processed using this parser , but no action is taken when receiving the events generated by it . + + when files below 100 kb are processed it can be observed that the overhead added by the generated code is not high ( figure [ mobile_execution_small ] ) . nevertheless , we can see in figure [ fig : mobile_execution ] that when file size is above 1 mb , the overhead starts to be important ( > 1s ) . this happens because the large amount of memory that is required to store the information that is being processed forces the execution of the garbage collector with a high frequency . we have to keep in mind that code produced manually can have similar problems if it were necessary to retain most of the information read from the xml files in memory . the experiment described above was extended in @xcite to compare the code generated by dbmg with other data binding tools and to measure also the performance of this code when executed in a windows pc . the experiments showed that the execution times for the mobile devices were around 30 to 90 times slower than those for the personal computer . the experiments also showed that the code generated by dbmg was as fast as code generated by other data binding tools for the android platform . in this paper we have presented an approach to generate compact xml processing code based on large schemas for mobile devices . it utilises information about how xml documents make use of its associated schemas to reduce the size of the generated code as much as possible . the solution proposed here is based on the observation that applications that makes use of xml data based on large schemas do not use all of the information included in these schemas . a code generator implementing the approach that produces code targeted to android mobile devices and the java programming language has been developed . this tool has been tested in a real case study showing a large reduction in the size of the final xml processing code when compared with other similar tools generating code for mobile , desktop and server environments . nevertheless , this result must be looked at with caution as the magnitude of the reduction will depend directly from the use that specific applications make of their schemas . this work has been partially supported by the `` espaa virtual '' project ( ref . cenit 2008 - 1030 ) through the instituto geogrfico nacional ( ign ) ; and project geocloud , spanish ministry of science and innovation ipt-430000 - 2010 - 11 . d. beyer , c. lewerentz , and f. simon . impact of inheritance on metrics for size , coupling , and cohesion in object - oriented systems . in _ proceedings of the 10th international workshop on new approaches in software measurement _ , iwsm 00 , pages 117 , london , uk , 2000 . springer - verlag . c. bogdan chirila , m. ruzsilla , p. crescenzo , d. pescaru , and e. tundrea . towards a reengineering tool for java based on reverse inheritance . in _ in proceedings of the 3rd romanian - hungarian joint symposium on applied computational intelligence ( saci 2006 ) _ , pages 9637154 , 2006 . bungartz , w. eckhardt , m. mehl , and t. weinzierl . . in _ proceedings of the 8th international conference on computational science , part iii _ , iccs 08 , pages 213222 , berlin , heidelberg , 2008 . springer - verlag . a. cicchetti , d. d. ruscio , r. eramo , and a. pierantonio . automating co - evolution in model - driven engineering . in _ proceedings of the 2008 12th international ieee enterprise distributed object computing conference _ , pages 222231 , washington , dc , usa , 2008 . ieee computer society . a. georges , d. buytaert , and l. eeckhout . statistically rigorous java performance evaluation . in _ proceedings of the 22nd annual acm sigplan conference on object - oriented programming systems and applications _ , oopsla 07 , pages 5776 , new york , ny , usa , 2007 . acm . s. kbisch , d. peintner , j. heuer , and h. kosch . . in _ proceedings of the 24th international conference on advanced information networking and applications workshops , waina 10 _ , volume 0 , pages 508513 , los alamitos , ca , usa , 2010 . ieee computer society . j. kangasharju , s. tarkoma , and t. lindholm . xebu : a binary format with schema - based optimizations for xml data . in a. ngu , m. kitsuregawa , e. neuhold , j .- y . chung , and q. sheng , editors , _ web information systems engineering - wise 2005 _ , volume 3806 of _ lecture notes in computer science _ , pages 528535 . springer berlin / heidelberg , 2005 . e. rahm . towards large - scale schema and ontology matching . in z. bellahsene , a. bonifati , and e. rahm , editors , _ schema matching and mapping _ , data - centric systems and applications , pages 327 . springer berlin heidelberg , 2011 . a. tamayo , c. granell , and j. huerta . . in _ proceedings of the 2nd international conference and exhibition on computing for geospatial research and application ( com.geo 2011 ) _ , pages 17:117:9 , new york , ny , usa , 2011 . a. tamayo , c. granell , and j. huerta . . in _ proceedings of the 2nd international conference and exhibition on computing for geospatial research and application ( com.geo 2011 ) _ , pages 16:116:9 , new york , ny , usa , 2011 . a. tamayo , p. viciano , c. granell , and j. huerta . . in s. geertman , w. reinhardt , and f. toppen , editors , _ advancing geoinformation science for a changing world _ , volume 1 of _ lecture notes in geoinformation and cartography _ , pages 185209 . springer berlin heidelberg , 2011 . r. a. van engelen and k. a. gallivan . . in _ proceedings of the 2nd ieee / acm international symposium on cluster computing and the grid , ccgrid 02 _ , pages 128 , washington , dc , usa , 2002 . ieee computer society . a. villegas and a. oliv . a method for filtering large conceptual schemas . in _ proceedings of the 29th international conference on conceptual modeling _ , er10 , pages 247260 , berlin , heidelberg , 2010 . springer - verlag .
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xml and xml schema are widely used in different domains for the definition of standards that enhance the interoperability between parts exchanging information through the internet . the size and complexity of some standards , and their associated schemas ,
have been growing with time as new use case scenarios and data models are added to them . the common approach to deal with the complexity of producing xml processing code based on these schemas
is the use of xml data binding generators .
unfortunately , these tools do not always produce code that fits the limitations of resource - constrained devices , such as mobile phones , in the presence of large schemas .
in this paper we present _ instance - based xml data binding _ , an approach to produce compact application - specific xml processing code for mobile devices .
the approach utilises information extracted from a set of xml documents about how the application make use of the schemas .
[ languages and system , standards ]
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You are an expert at summarizing long articles. Proceed to summarize the following text:
lattice qcd is the ideal approach not only for computing @xmath3 meson spectrum , but also for hybrids and glueballs . however , the lattice technique is not free of systematic errors . the wilson gauge and quark actions suffer from significant lattice spacing errors , which are smaller only at very large @xmath4 , and very large lattice volume is required to get rid of finite size effects . there have been several quenched lattice calculations of hybrid meson masses , part of them are listed in tab . [ tab6 ] . in ref . @xcite , the wilson gluon action and quark action were used . in refs . @xcite , the authors used wilson gauge action and sw improved quark action . for the hybrid mesons containing heavy quarks @xmath5 , the nrqcd action@xcite and the lbo action@xcite have also been applied . there is also a recent work using the improved ks quark action@xcite . in this work , we employ _ both improved gluon and quark actions on the anisotropic lattice _ , which should have smaller systematic errors , and should be more efficient in reducing the lattice spacing and finite volume effects . we will present data for the @xmath2 hybrid mass and the splitting between the @xmath2 hybrid mass and the spin averaged s - wave mass for charmonium . details can be found in ref . the total lattice action is @xmath6 . the improved gluonic action @xmath7 is @xcite : @xmath8 where @xmath9 stands for a @xmath10 plaquette and @xmath11 for a @xmath12 rectangle . the sw improved action for quarks@xcite is @xmath13 \nonumber \\ & - & \kappa_s \sum_{x , j } [ { \bar \psi}(x ) ( 1-\gamma_j)u_{j}(x ) \psi ( x+\hat{j } ) \nonumber \\ & + & { \bar \psi } ( x ) ( 1+\gamma_j)u^{\dag}_{j}(x-\hat{j})\psi(x-\hat{j } ) ] \nonumber \\ & + & i { \kappa}_s { c}_s^{ti } \sum_{x , j < k } { \bar \psi } ( x ) \sigma_{jk}{\hat f_{jk}}(x)\psi ( x ) \nonumber \\ & + & i \kappa_s c_t^{ti } \sum_{x , j}{\bar \psi } ( x ) \sigma_{j4 } { \hat f_{j4 } } ( x ) \psi ( x ) , \label{s_q}\end{aligned}\ ] ] where @xmath14 stands for the clover - leaf construction@xcite for the gauge field tensor . tadpole improvement is carried out so that the actions are more continuum - like . on our pc cluster@xcite , the su(3 ) pure gauge configurations were generated with the gluon action in eq . ( [ s_g ] ) using cabibbo - marinari pseudo - heatbath algorithm . the configurations are decorrelated by su(2 ) sub - group over - relaxations . we calculated the tadpole parameter @xmath15 self - consistently . 90 independent gauge configurations at @xmath0 and @xmath16 on the @xmath17 lattice were stored . although such an ensemble is not very big , it is bigger than earlier simulations by ukqcd and milc collaborations@xcite on isotropic lattices . the quark propagator was obtained by inverting the matrix @xmath18 in @xmath19 in eq . ( [ s_q ] ) by means of bicgstab algorithm . the residue is of @xmath20 . we computed the correlation functions with various sources and sinks@xcite , at four values of the wilson hopping parameter ( @xmath21 , 0.4199 , 0.4279 , 0.4359 ) . in fig . [ mprf1t.eps ] , we plot the effective masses for the @xmath22 , @xmath23 , @xmath24 ordinary mesons and @xmath2 exotic meson at @xmath25 . for the ordinary mesons , we used the their corresponding operator as both source and sink . for the exotic meson , we tried two different cases : ( 1 ) the @xmath2 operator as both source and sink ; ( 2 ) the @xmath26 source and @xmath2 sink , which give consistent results within error bars . the cp - pacs , milc and ukqcd collaborations used the unimproved wilson gauge action to generate configurations . they had to work on very large @xmath27 ) , corresponding to very small @xmath28 0.1 fm ) , to get rid of the finite spacing errors . they had also to use very large lattices @xmath29 , to avoid strong lattice size effects at such small @xmath30 . in comparison , our lattices are much coarser ( @xmath31 fm ) , and the number of lattice sites is much smaller . our finite size effects could be ignored , for the physical size of the spatial lattice is @xmath32 and should be big enough . our results for the effective mass indicate the existence of a much wider plateau than in the previous work on isotropic lattices . by extrapolating the effective mass of the @xmath2 hybrid meson to the chiral limit , and using @xmath33 determined from the @xmath23 mass , we get @xmath34 mev . in tab . [ tab6 ] , we compare the results from various lattice methods . our result is consistent with the milc data@xcite , obtained using the wilson gluon action and clover quark action on much larger isotropic lattices and much smaller @xmath30 . we also show our results in tab . [ tab6 ] for the @xmath2 hybrid meson mass in the charm quark sector , using the method discussed in refs . our corresponding @xmath35 is obtained by tuning @xmath36=3067.6 mev , where on the right hand side , the experimental inputs @xmath37 mev and @xmath38 mev are used . the @xmath2 hybrid meson mass at our @xmath39 is @xmath40 mev , is consistent with the milc data@xcite . the splitting between the hybrid meson mass and the spin averaged s - wave mass [ @xmath41 , at our @xmath42 is @xmath43 mev , consistent with the cp - pacs data , obtained using the wilson gluon action and nrqcd quark action on much larger anisotropic lattices and much smaller @xmath30 . as a byproduct , we give the @xmath24 p - wave @xmath44 meson in the chiral limit , as well as their experimental values@xcite . if we assume that the pion is massive and @xmath45 is made of @xmath46 , the @xmath24 p - wave @xmath44 meson mass would be @xmath47 mev . to summarize , we have used the tadpole - improved gluon action and clover action to compute the hybrid meson masses on much coarser anisotropic lattices . the main results are given in tab . [ tab6 ] and compared with other lattice approaches . in our opinion , our approach is more efficient in reducing systematic errors due to finite lattice spacing .
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we study hybrid mesons from the clover and improved gauge actions at @xmath0 on the anisotropic @xmath1 lattice using our pc cluster .
we estimate the mass of @xmath2 light quark hybrid as well as the mass of the charmonium hybrid .
the improvement of both quark and gluonic actions , first applied to the hybrid mesons , is shown to be more efficient in reducing the lattice spacing and finite volume errors .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
a key ingredient in the understanding and modelling of galaxy evolution is the relationship between the large - scale star formation rate ( sfr ) and the physical conditions in the interstellar medium ( ism ) . most current galaxy formation and evolution models treat star formation using simple ad hoc parametrizations , and our limited understanding of the actual form and nature of the sfr - ism interaction remains as one of the major limitations in these models ( e.g. , navarro & steinmetz 1997 ) . measurements of the star formation law in nearby galaxies can address this problem in two important respects , by providing empirical recipes " that can be incorporated into analytical models and numerical simulations , and by providing clues to the physical mechanisms that underlie the observed correlations . the most widely applied star formation law remains the simple gas density power law introduced by schmidt ( 1959 ) , which for external galaxies is usually expressed in terms of the observable surface densities of gas and star formation : @xmath2 the validity of the schmidt law has been tested in dozens of empirical studies , with most measured values of @xmath3 falling in the range 1 @xmath4 2 , depending on the tracers used and the linear scales considered ( kennicutt 1997 ) . on large scales the star formation law shows a more complex character , with a schmidt law at high gas densities , and a sharp decline in the sfr below a critical threshold density ( kennicutt 1989 , hereafter k89 ) . these thresholds appear to be associated with large - scale gravitational stability thresholds for massive cloud formation ( e.g. , quirk 1972 ; fall & efstathiou 1980 ; k89 ) . at high gas densities , well above the stability threshold , the form of the schmidt law appears to be remarkably consistent from galaxy to galaxy , both in terms of its slope ( @xmath5 ) and the absolute sfr efficiency ( the coefficient @xmath6 in eq . [ 1 ] ) . studies of this kind offer the beginnings of a quantitative , physical prescription for the sfr that can be incorporated into galaxy formation and evolution models . this is the first of two papers which reinvestigate the form and physical nature of the star formation law , over a much larger range of galaxy types and gas densities than was possible previously . paper ii ( martin & kennicutt 1998 ) uses new ccd imaging of an hi and co selected sample of spiral galaxies to quantify the behavior of the star formation law within individual galaxies , and to test several models for the star formation law . this paper is concerned with the behavior of the star formation law on global scales , averaged over the entire star forming disk . such global laws , which treat galaxies in a single - zone approximation , provide less physical insight into the star formation process itself , but they provide very useful parametrizations ( recipes ) for galaxy evolution modelling . earlier work has shown that the global , disk - averaged star formation law is reasonably well represented by a schmidt law ( k89 ; buat , deharveng , & donas 1989 ; buat 1992 ; boselli 1994 ; deharveng 1994 ) . however these analyses have been hampered by small samples and by the small range of gas densities represented in those samples . in this paper we use newly available hi , co , and data to more than double the sample over previous studies , and fully cover the range of mean gas densities found in disks . we combine these data with published co , , and far - infrared ( fir ) measurements of luminous starburst galaxies , to investigate the nature of the schmidt law in higher density environments , thereby extending the total density range probed to nearly five orders of magnitude . our main goal is to test whether the millionfold range in observed sfrs , extending from quiescent gas - poor disks to nuclear starbursts , can be understood within a common empirical and physical framework . to investigate the global star formation law in normal disks , we searched the literature for normal galaxies with well - sampled hi and co measurements , and for which imaging or photometry are available . our analysis of this sample closely follows that described in k89 . to investigate the star formation law at higher densities , we compiled published co maps , fir photometry , and emission - line measurements for a sample of infrared - selected starburst galaxies . each data set is discussed separately below . previous studies of the disk - averaged star formation law have shown that the global sfr correlates most strongly with the total ( atomic @xmath7 molecular ) gas density ( e.g. , kenney & young 1988 ; k89 ; buat 1992 ; boselli 1994 ) . consequently our primary data set consists of normal spirals for which spatially - resolved hi , co , and data are available . a master list of candidate galaxies was compiled from the fcrao co survey ( young 1989 ; 1995 ) , supplemented by the co survey of sage ( 1993 ) . within these samples , we identified 61 galaxies which also have published hi maps , photometry , and inclinations less than 75 ( to avoid severe extinction problems in edge - on systems ) . total hi masses based on single - dish measurements are available for another 150 galaxies , but those data are unsuitable for the current application , because much of the hi is located well outside of the star forming disks , and it is essential to correlate the sfr and gas densities over the same physical region . however we do use some of these additional galaxies in 3.1 to examine the form of the sfr _ vs _ hi schmidt law . table 1 lists the 61 galaxies in the sample , the relevant surface densities , and references , as described below . when considering the sample properties as a whole the main selection criterion was availability of co and hi maps , so the galaxies should comprise a virtually unbiased set in terms of star formation properties . approximately 40% of the galaxies are members of the virgo cluster , selected from the co survey of kenney & young ( 1988 ) and the hi survey of warmels ( 1988 ) , and this sample contains most of the luminous spirals in the cluster core . the field galaxy subsample is more heterogeneous , and is significantly biased toward galaxies of hubble type sb and later , but it is unlikely that this selection biases the form of the star formation law . hi surface densities were taken mainly from the compilations of warmels ( 1988 ) , broeils & van woerden ( 1994 ) , and rhee & van albada ( 1995 ) , supplemented by individual measurements of a few galaxies ( table 1 ) . the mean hi surface densities , averaged within the optical radius of the disk , were derived from the surface density profiles given in those papers or the references therein . the disk radii are the corrected isophotal radii as given in the rc2 catalog ( de vaucouleurs , de vaucouleurs , & corwin 1976 ) . the mean densities used here differ from those that are often tabulated in the original papers , the latter usually being averaged within the inner _ half _ of the optical disk . total molecular hydrogen masses were taken from the young ( 1989 ; 1995 ) and sage ( 1993 ) surveys , and converted when needed to a common co / h@xmath1 conversion factor : @xmath8 @xmath9 ( k km s@xmath10 ) . the mean 2 surface densities were then determined , by averaging within the radii listed in table 1 . these average densities are meaningful only if the co emission is confined to the optical disk , and the measurements extend to a substantial fraction of optical radius . galaxies which were sampled to less than half of the optical radius were not included in our sample . integrated sfrs were derived from measurements of the emission - line flux , following the method described in kennicutt ( 1983 ) . most of the fluxes were taken from the surveys of kennicutt & kent ( 1983 ) , romanishin ( 1990 ) , and young ( 1996 ) . those data were supplemented with new calibrated ccd images obtained with a focal reducer camera on the steward observatory 2.3 m bok telescope , and with the 0.9 m and burrell schmidt telescopes at kitt peak national observatory . details of these observations are given in paper ii . the fluxes were corrected as needed for foreground extinction and [ nii ] emission , following the prescriptions in kennicutt ( 1983 ) . the original fluxes of kennicutt & kent ( 1983 ) have been corrected upwards by a factor of 1.16 to place them on a consistent zeropoint with more recent measurements ( romanishin 1990 ; kennicutt 1992 ) . the luminosities were then converted to total sfrs , using the updated calibration of kennicutt , tamblyn , & congdon ( 1994 ) : @xmath11 the luminosities used in equation ( 2 ) were corrected for internal extinction by 1.1 mag ( factor 2.8 ) , based on a comparison of free - free radio fluxes and fluxes of galaxies by kennicutt ( 1983 ) . the actual extinction varies within the sample , of course , which introduces significant scatter in the observed star formation law , as discussed later . while it would be much better to apply individual extinction corrections to each galaxy , determining the reddening or extinction from integrated spectra is problematic ( kennicutt 1992 ) , and would introduce uncertainties that are larger than the single average correction . it may be possible in the future to derive improved estimates of the extinction and sfr using measurements of near - infrared brackett or paschen recombination lines , but such data are not currently available . the imf used in this conversion is a salpeter function ( @xmath12 ) over @xmath13 . the salpeter imf was adopted in order to be consistent with the infrared - derived sfrs in the next section . adopting the extended miller - scalo function used in kennicutt ( 1983 ) would produce nearly identical sfrs ( only 8% lower ) . galaxy distances from young ( 1989 ) were used in this intermediate calculation , but the distances are irrelevant for most of this paper , because the schmidt law is analyzed in terms of distance - independent surface densities . finally , the mean sfr surface density ( units yr@xmath10 kpc@xmath14 ) was derived for each galaxy , by dividing the total sfr from equation ( 2 ) by the deprojected area within the corrected rc2 radius . through the remainder of this paper , we shall refer to this sfr per unit area as the sfr density " . in most galaxies the rc2 radius coincides approximately with the edge of the main -emitting disk ( k89 ) , so the sfr density as measured here corresponds roughly to the mean sfr per unit area within the active star forming disk . the derived sfr surface densities are listed in table 1 . the observed surface densities ( uncorrected for extinction ) can be derived from table 1 by the simple relation : @xmath15 , where @xmath16 is expressed in units of ergs sec@xmath10 pc@xmath14 . this conversion may be useful for readers who may wish to apply a different sfr calibration to the data compiled here . the mean gas densities of the normal spiral disks in our sample lie within a relatively narrow range , from 2 to 50 pc@xmath14 , and this seriously limits the dynamic range over which the behavior of the schmidt law can be evaluated . the density range can be extended to @xmath17100 pc@xmath14 by analyzing spatially - resolved measurements of individual disks ( paper ii ) , but above these densities measurements become unreliable for determining the sfr . for a typical gas - to - dust ratio found in nearby galaxies , the visual extinction reaches 1 mag for column densities @xmath18 @xmath9 , or @xmath19 pc@xmath14 ( e.g. , bohlin , savage , & drake 1978 ; caplan & deharveng 1986 ) . hence one expects the extinction at to become problematic for regions with mean gas surface densities above 50 100 pc@xmath14 . if we wish to study the nature of the star formation law in these dense regions , a star formation diagnostic other than must be used . large - scale star formation at much higher densities is commonly found in the centers of normal galaxies , and particularly in luminous infrared starburst galaxies . in order to analyze the star formation law in this regime , we searched the literature for high - resolution co and infrared measurements of starburst galaxies . since the starbursts are often concentrated in compact circumnuclear disks ( e.g. , scoville 1994 ; sanders & mirabel 1996 ; smith & harvey 1996 ) , high - resolution data are required in order to accurately determine the linear sizes of the starburst regions and the corresponding surface densities . our sample comprises 36 galaxies with high - resolution co data , most based on aperture synthesis mapping , and for which infrared measurements of the same region are available . the sample ranges from low - level nuclear starbursts in normal and peculiar galaxies such as ngc 253 , ic 342 , maffei 2 , and m82 ( @xmath20 ) to ultraluminous starburst galaxies with @xmath21 ( e.g. , arp 220 ) . care was taken to select objects in which the dust heating is dominated by a starburst , as determined from optical spectra spectra ( e.g. , armus , heckman , & miley 1989 ; veilleux 1995 ) and/or mid - infrared spectroscopy ( e.g. , lutz 1996 ) . objects with evidence for a strong agn component were excluded ( e.g. , ngc 1068 , ngc 7469 , mrk 231 , mrk 273 ) . total molecular gas masses in the starburst disks were derived from the co flux and distance , using the same co/2 conversion factor as for the normal galaxies . the validity of a constant conversion factor is highly questionable ( e.g. , wild 1992 ; downes , solomon , & radford 1993 ; aalto 1994 ; solomon 1997 ) , and we have adopted a uniform conversion factor strictly for the sake of simplicity . the impact of adopting a different conversion factor will be discussed later . the mean molecular surface densities were then derived , averaged within the radius of the central molecular disk as determined from the co maps . high - resolution hi observations are only available for a few of these galaxies , and in those cases the atomic fraction in the circumnuclear region is small , of order a few percent or less ( e.g. , garcia - barreto 1991 ; downes 1996 ; sanders & mirabel 1996 ) . this is not surprising given the very high column densities found in these regions . consequently we have ignored the hi component and approximate the molecular mass as the total gas mass in the starburst region . table 2 lists the galaxies in the sample , the radii of the disks , and their mean molecular surface densities . the sfrs for the starbursts were derived from measurements of their fir luminosities . these were taken from a variety of sources , as listed in table 2 . for about half of the sample , high - resolution maps at mid - infrared wavelengths are available , and when combined with iras fluxes for the galaxies as a whole they provide an accurate estimate of the fir luminosity in the central starbursts themselves ( telesco , dressel , & wolstencroft 1993 ; smith & harvey 1996 ) . for the other galaxies the fir luminosity of the starburst was derived from a combination of iras photometry and groundbased aperture photometry at 1020 , or from the iras fluxes alone , in cases where most of the total fir emission appears to originate in the central starburst . sfrs for three of the galaxies were derived from a combination of and infrared photometry , as noted in table 2 . in normal disk galaxies the relationship between the fir luminosity and the sfr is complex , because stars with a variety of ages can contribute to the dust heating , and only a fraction of the bolometric luminosity of the young stellar population is absorbed by dust ( e.g. , lonsdale & helou 1987 ; walterbos & greenawalt 1996 ) . however in the starbursts studied here , the physical coupling between the sfr and the ir luminosity is much more direct . young stars dominate the radiation field that heats the dust , and the dust optical depths are so large that almost all of the bolometric luminosity of the starburst is reradiated in the infrared . this makes it possible to derive a reasonable quantitative measure of the sfr from the fir luminosity . our calibration of the sfr/@xmath22 conversion is based on the starburst synthesis models of leitherer & heckman ( 1995 ) . their models trace the temporal evolution of the bolometric luminosity for a fixed sfr , metal abundance , and imf . we computed the sfr calibration using their continuous star formation " models , in which the sfr is presumed to remain constant over the lifetime of the burst . the models show that the l@xmath23/sfr ratio evolves relatively slowly between ages of 10 and 100 myr , the relevant range for most of these starbursts ( e.g. , bernlhr 1993 ; engelbracht 1997 ) . alternatively one can derive the conversion using a instantaneous burst " approximation , where it is assumed that star formation has ceased , but the calibration is sensitive to the presumed burst age and the ( questionable ) assumption of an instantaneous burst . adopting the mean luminosity for 10100 myr continuous bursts , solar abundances , the salpeter imf described earlier , and assuming that the dust reradiates all of the bolometric luminosity yields : @xmath24 this lies within the range of previously published calibrations ( @xmath25 yr@xmath10 @xmath26 . equation ( 3 ) yields sfrs that are 14% lower than the recent calibration of lehnert & heckman ( 1996 ) , and 22% lower than meurer ( 1997 ) . the sfr surface density was then calculated within the radius of the starburst region as determined from the co maps , or from the infrared maps if high - resolution co data were not available . the sizes of the regions defined in co and the infrared show excellent correspondence in cases where comparable resolution data are available ( telesco 1993 ; smith & harvey 1996 ) . table 2 lists the radii , gas densities , and sfr surface densities derived in this way . in 4 we analyze the composite properties of the normal disk and starburst samples , so it is important to confirm that the fir and -based sfrs are on a consistent zeropoint . matching aperture photometry for 18 of the galaxies in our sample is available from the compilations of puxley , hawarden , & mountain ( 1990 ) , telesco ( 1993 ) and smith & harvey ( 1996 ) , and these allow us to compare the emission - line and fir sfr scales on a self consistent basis . the fir - based sfrs were derived using equation ( 3 ) , while the -based sfrs were derived using equation ( 2 ) and a / ratio of 0.0103 , corresponding to case b recombination at @xmath27 k and @xmath28 @xmath29 ( osterbrock 1989 ) . no extinction corrections were applied to the data . figure 1 shows a comparison of the fir and -derived sfrs . the solid line shows the correlation expected if the two sets of sfrs were equivalent . the data in figure 1 closely follow this correlation , but the fir - derived sfrs are systematically higher by an average of [email protected] dex , as shown by the dashed line . this displacement could indicate a general inconsistency between the zeropoints of the and fir calibrations of the sfr , which might arise , for example , from errors in the fir luminosities ( many of them extrapolated from the mid - ir ) , or in the synthesis model that is used to convert the fir luminosities to sfrs . however there is physical justification for expecting that the fluxes would systematically underestimate the sfrs in many of these objects . the extinction in most regions is so large that one expects part of the ionizing radiation from the starburst to be absorbed by grains , and in some objects extinction of itself is probably significant ( e.g. , lutz 1996 ; goldader 1997 ) . the -derived sfr will also tend to be systematically lower than the fir - derived value if the starbursts are observed after the peak of the burst , because the dust heating is dominated by longer lived stars than the emission lines . we provisionally adopt the sfrs from equation ( 3 ) in the following analysis , on the tentative assumption that the fir - based sfrs are more reliable in these objects . however we will also explore the consequences of adopting the lower -based scale , and include this uncertainty in the analysis of the global schmidt law . individual uncertainties are not listed for the surface densities listed in tables 1 and 2 , because the predominant errors are systematic in nature and difficult to quantify . however it is important to be aware of nature of these uncertainties and their possible influence on the observed star formation law . for the normal spiral disks , with sfrs derived from luminosities ( table 1 ) , the dominant systematic errors are extinction variations , which introduce a scatter in the sfr densities , and uncertainty in the extrapolated imf , which could introduce an overall shift in the sfrs ( kennicutt 1983 ) . the dominant errors in the gas densities are variation in the co / h@xmath1 conversion factor , combined with the limited sampling of the co measurements in some galaxies ( sage 1993 ; young 1995 ) . a realistic estimate for the observational scatter in the sfrs is @xmath303050% , or @xmath300.150.3 dex ( kennicutt 1983 ) , and the uncertainties in the gas densities are probably comparable . we adopt an average uncertainty of @xmath300.2 dex in the following analysis . the systematic uncertainties in the sfrs and gas densities derived for the starburst galaxies ( table 2 ) are larger . in many cases the fir luminosities have been derived from a combination of high - resolution mid - infrared measurements and iras fir fluxes , and there can be substantial uncertainty in the extrapolation to a total fir flux . in other cases only integrated iras fluxes for the galaxies are available , and the presence of significant fir emission from the region outside of the central starburst will cause the starburst sfr to be systematically overestimated . the sfr will also be overestimated if the dust is heated partly by other sources , such as an active nucleus . another significant source of uncertainty in the sfrs inferred for individual starbursts is the use of a fixed continuous burst model , though the effect on the overall sfr scale should be lower . the gas densities in the starburst regions are also subject to systematic error as well , mainly through uncertainties in the co / h@xmath1 conversion factor ( e.g. , downes 1993 ; solomon 1997 ) . other smaller sources of uncertainty include the neglect of atomic gas and errors in the radii of the starbursts . the latter errors affect the inferred sfr and gas densities equally , and have less of an effect on the form of the schmidt law . the largest of these systematic uncertainties , the l@xmath31 sfr conversion and the co/2 conversion , could introduce errors in the sfr or gas density scales at the factor of 2 @xmath4 3 level ( 0.3 @xmath4 0.5 dex ) . in our analysis we adopt uncertainties @xmath32 dex in both parameters , with the asymmetry reflecting the greater likelihood that the systematic errors tend to lead to overestimates of the sfrs and gas densities . despite these uncertainties , the data provide very strong constraints on the form of the star formation law , because of the very large range of absolute densities and sfrs represented in the sample , 2 @xmath4 6 orders of magnitude depending on the subsample of interest . figure 2 shows the relationship between the disk - averaged sfr and total gas density ( atomic and molecular hydrogen ) for the 61 normal spirals in our sample . a clear correlation is apparent in the expected sense of increasing sfr with increasing gas densities , with a mean slope that is considerably steeper than a linear relation ( indicated by the dotted and dashed lines ) . however the scatter in the relation is large , up to a factor of 30 in sfr at a fixed gas density , and comparable to the total range in observed gas density . consequently the slope of the schmidt law is poorly constrained . a conventional least squares fit which minimizes ( logarithmic ) residuals in the sfr density yields @xmath33 . this slope lies in the middle of the range @xmath34 derived in previous studies with smaller samples ( buat 1989 ; k89 ; buat 1992 ; deharveng 1994 ) . a bivariate least squares regression , which takes into account the uncertainties in the gas densities as well , yields a much steeper fit @xmath35 . both fits are shown with solid lines in figure 2 . the large difference between these solutions is a direct reflection of the large dispersion in the disk - averaged sfr vs gas density relation , and the result underscores the conclusion that any schmidt law in these galaxies should be regarded as a _ very _ approximate parametrization at best . what is the physical origin of the large dispersion in figure 2 ? as discussed earlier , variations in extinction and the co/2 conversion introduce a scatter at about the @xmath300.2 dex level in the sfr and gas densities , as signified by the error bars in figure 2 . this can account for roughly half of the observed scatter in the star formation law . the remaining scatter must be real , reflecting a real variation in the mean schmidt law . such a variation is not entirely surprising , when one recalls that the local sfrs and gas densities span orders of magnitude within typical disks , and averaging over the entire disk will not necessarily preserve the form of a nonlinear local schmidt law . the problem is illustrated in figure 3 , which shows the radial sfr vs gas density profiles for 21 of the galaxies in our sample ( paper ii ) . each profile was produced by measuring the azimuthally averaged gas density and sfr density as a function of galactocentric radius , then plotting the resulting sfr vs gas density relation on a common scale . at high densities the sfrs are well represented by a shallow schmidt law ( @xmath36 ) , but the slope of the star formation law steepens abruptly below the threshold density . the disk - averaged sfrs plotted in figure 2 represent gross averages over these highly nonlinear relations , and the resulting global schmidt law exhibits a slope that is intermediate between the @xmath36 power - law dependence at high density and the steeper law in the threshold regime . the dispersion in figure 2 is introduced because the star formation in some galaxies is highly concentrated to the high - density part of the local schmidt law , while in other systems much of the star formation takes place near the threshold density ( see k89 ) . this underscores the caveat that disk - averaged schmidt law analyzed here contains little physical information about the underlying star formation law . however it does provide a convenient means of parametrizing the gross star formation properties of disks in simple one - zone evolution models . we defer further discussion of the spatially - resolved star formation law for paper ii . the data in figure 2 also provide useful information on the average global efficiency of star formation in local disks , the coefficient @xmath6 in equation ( 1 ) . the dashed and dotted lines in figure 2 correspond to constant sfrs per unit gas mass , in units of 1% , 10% , and 100% per @xmath37 yr . the choice of @xmath37 yr as a fiducial timescale is arbitrary , but it does correspond roughly to a typical orbital time in the disks . the lines are offset by a factor of 1.37 to include helium and heavy elements in the total gas mass . the median efficiency for the disks in figure 2 is 4.8% , i.e. , a typical present - day spiral galaxy converts 4.8% of the gas ( within the optical radius ) to stars over this period . the efficiencies can be expressed alternatively as gas consumption timescales , with the lines in figure 2 corresponding to timescales @xmath38 of 10 , 1 , and 0.1 gyr ( bottom to top ) . the median gas consumption time for the disks in this sample is 2.1 gyr , again referring to the star forming disks alone , and not including corrections for recycling of interstellar gas . recycling typically extends the actual consumption timescale by factors of 23 above the simple calculation ( kennicutt 1994 ) . most of the galaxies in figure 2 possess disk - averaged star formation efficiencies in the range 2 @xmath4 10% per @xmath37 yr , corresponding to gas consumption times of 1 @xmath4 5 gyr . however several galaxies are more extreme , and the full range of efficiencies is 0.8 @xmath4 60% per @xmath37 yr ( @xmath38 = 0.2 @xmath4 12 gyr ) . the shortest timescales correspond to optically - selected starburst galaxies such as ngc 1569 and ngc 3310 , while the low extremes are represented by early - type spirals such as m31 , ngc 2841 , and ngc 4698 , where the current sfrs are so low that the future consumption times , even for their modest gas supplies , are comparable to the hubble time . until now our attention has focussed solely on the relationship between the disk - averaged sfr and the total gas density , but we can also examine how the sfrs correlate with the average atomic and molecular gas densities , as shown in figure 4 . these comparisons include galaxies mapped in hi or co ( but not both ) , so the samples are considerably larger than shown in figure 2 . the left panel of figure 4 shows the sfr vs hi density relation for 88 galaxies with and hi data in common . the correlation is very reminiscent of the sfr vs total density relation shown in figure 2 , and in fact the correlation coefficients are nearly identical , 0.66 for the sfr @xmath4 hi relation vs 0.68 for the sfr @xmath4 hi@xmath72 relation . this is not entirely surprising , as hi accounts for approximately half of the total gas density on average . these results are consistent with previous analyses based on smaller samples by k89 , buat ( 1992 ) , deharveng ( 1994 ) , boselli ( 1994 ) , and boselli ( 1995 ) . the physical interpretation of the sfr vs hi schmidt law is not obvious , however . it may trace the physical influence of the atomic gas density on the sfr , but it could be that the sfr regulates the density of hi , through the photodissociation of molecular gas by hot stars ( shaya & federman 1987 ; tilanus & allen 1989 ) . the correlation between the -based sfrs and h@xmath1 density is much weaker , as shown in the right panel of figure 4 . this has been reported previously , and appears to hold independently of whether sfrs based on , uv continuum fluxes , or fir fluxes are analyzed ( buat 1992 ; boselli 1994 ) . such a poor correlation between the sfr and molecular gas densities is unexpected , and it has led some to suggest that variations in the co / h@xmath1 conversion factor are responsible for the scatter ( k89 ; boselli 1994 ; boselli 1995 ) . our data provide indirect support for this interpretation . several lines of evidence suggest that the galactic co/2 conversion factor is valid in regions with near - solar metallicity , but that it tends to systematically underestimate the h@xmath1 mass in metal - poor regions , such as are found in the outer disks of spirals or in low - luminosity galaxies ( e.g. , maloney & black 1988 ; kenney & young 1988 ; rubio 1993 ; wilson 1995 ) . to test whether this effect might be contributing to the scatter in figure 4 , we subdivided our sample by blue luminosity , with solid points denoting galaxies with l@xmath39 ( @xmath40 for h@xmath41 = 75 km s@xmath10 mpc@xmath10 ) and open circles representing fainter galaxies . the mean metal abundance in disks is well correlated with luminosity , so this provides an approximate separation of the galaxies by abundance , around a value of @xmath42 ( zaritsky , kennicutt , & huchra 1994 ) . figure 4 shows that the luminous , metal - rich spirals do show a much better defined sfr vs h@xmath1 density correlation , comparable in slope and scatter to the correlations with total and hi density . by contrast , the low - luminosity galaxies show essentially no correlation between the sfr and co - inferred h@xmath1 densities , with many co - weak galaxies showing unusually _ high _ sfrs . although this is hardly a conclusive result , it offers circumstantial evidence that variations in the co / h@xmath1 conversion factor are responsible for most of the scatter in the sfr vs molecular gas density relation . our conclusions are consistent with those of boselli ( 1994 ) and boselli ( 1995 ) , and the reader is referred to those papers for more detailed discussions of this problem . we can perform a parallel analysis for the infrared - selected starbursts , and the results are summarized in figure 5 . the comparison is directly analagous to that shown for the normal disks in figure 2 , except that the sfrs are derived from fir luminosities , and the sfrs are correlated with the h@xmath1 gas density alone ( the disks are expected to be overwhelmingly molecular , as discussed earlier ) . the sfrs and densities are averaged within the radii of the central molecular disks and starbursts , which have typical dimensions of order 1 kpc . the error bars indicate the typical uncertainties , as discussed in 2.3 . the starburst galaxies also show a well - defined schmidt law , in this case with a best fitting least squares slope @xmath43 ( bivariate regression ) or @xmath44 ( errors in sfrs only ) . the schmidt law is better defined than for the normal disks , but partly because there is a much larger dynamic range in sfr and gas densities in the starburst sample ; the dispersion in absolute sfr per per unit area at fixed gas density is only slightly lower in the starburst sample . star formation threshold effects are probably unimportant in the starburst disks , and this might also account for the somewhat tighter schmidt law among these objects . although the starburst disks exhibit a sfr vs gas density relation that is qualitatively similar in form to that seen in the normal spiral disks , the physical regime we are probing is radically different . the average gas surface densities here range from @xmath45 to @xmath46 pc@xmath14 , compared to a typical range of order 1 @xmath4 100 pc@xmath14 in normal disks ( figures 2 , 3 ) . the mean densities of the starburst disks are comparable instead to those of individual molecular cloud complexes in normal galaxies . for example , the largest hii / gmc complexes in m31 , m33 , and m51 have molecular masses and sizes corresponding to mean surface densities of 40 @xmath4 500 pc@xmath14 ( wilson & rudolph 1993 ; wilson & scoville 1992 ; nakai & kuno 1995 ) . this is comparable to the _ low end _ of the density range for the starbursts in figure 5 . the mean densities of some of the starbursts approach those of galactic molecular cloud cores , but extending over kiloparsec diameter regions . the star formation densities are just as extraordinary . for example , the central 10 pc core of the 30 doradus giant hii region contains @xmath47 in young stars , which corresponds to @xmath48 yr@xmath10 kpc@xmath14 if the star formation timescale is as short as @xmath49 yr ; the average sfr density averaged over the entire hii region is @xmath171 @xmath4 10 yr@xmath10 kpc@xmath14 . thus the regions we are studying have projected sfrs per unit area that approach the maximum limit observed in nearby optically - selected star clusters and associations ( meurer 1997 ) , but extending over regions up to a kiloparsec in radius . not surprisingly , the global star formation efficiencies in the starburst sample are much higher than in the normal disk sample ( e.g. , young 1986 ; solomon & sage 1988 ; sanders , scoville , & soifer 1991 ) . in figure 5 we show the same lines of constant star formation efficiency and gas consumption times as in figure 2 ( 1% , 10% , and 100% per @xmath37 yr ) . the median rate of gas consumption is 30% per 10@xmath50 yr , 6 times larger than for the normal disk samples , and the efficiencies reach 100% per @xmath37 yr for the most extreme objects . it is interesting to note that the shortest gas consumption times are comparable to the dynamical timescales of the parent galaxies , implying that the most luminous starbursts are forming stars near the limit set by the gas accumulation timescale ( lehnert & heckman 1996 ) . taken together , the normal disk and starburst samples span a dynamic range of approximately @xmath46 in gas surface density and over @xmath49 in sfr per unit area . figure 6 shows the composite relation , with the normal spirals shown as solid circles and the starbursts as solid squares . quite remarkably , the data are consistent with a common schmidt law extending over the entire density range . figure 6 shows that the normal disk and starburst samples occupy completely separate regimes in gas density and sfr per unit area , not a surprising result given the very different selection criteria for the two samples . but before we interpret the composite relation it is important to establish whether there is a smooth physical continuity between the normal disk and starburst regimes , and to confirm the consistency of the and fir - derived sfr scales . to this end we derived -based sfrs and gas densities for the central regions of 25 of the normal spirals in table 1 ( @xmath51 ) , using our images and published hi and co maps ( paper ii ) . the resulting sfr and gas densities are shown as open circles in figure 6 . these regions span the physical parameter space between the normal disks as a whole and the infrared - selected circumnuclear starburst regions . figure 6 shows that the gas densities and -derived sfrs of these regions fall on the composite schmidt law defined by the normal disk and starburst samples , and fill the transition region between the two physical regimes . the same conclusion can be drawn by comparing the sfrs of the infrared - selected starburst galaxies in figure 5 with the spatially - resolved sfrs of the normal disks shown in figure 3 ; the starbursts lie on the extrapolation of the high - density star formation laws observed in the spiral disks . this result , combined with the -fir comparison discussed earlier , gives us confidence that we are measuring the form of the star formation law on a self - consistent basis across the sample . the solid line in figure 6 shows a bivariate least - square fit to the composite relation defined by the normal disks and the starbursts ( but not including the open circles ) . in this case we applied equal weights to all of the data points , in order to avoid having the fit driven by the normal spirals in the lower left region of figure 6 . this yields a best fitting index @xmath52 ( bivariate regression ) or @xmath53 ( errors in sfrs only ) . these are nearly identical to the schmidt law fits for the starburst sample alone , which further confirms the consistency of the large - scale star formation laws in the two samples . the formal uncertainties listed here assume random errors of @xmath300.3 dex in the sfrs and gas densities , but it underestimates the full uncertainty in the schmidt law , because we have not accounted for the possibility of a systematic shift in the overall sfr or density scales for the starburst sample as a whole . the effect of such a shift is easily calculated . for example , reducing the sfrs for all of the starbursts by a factor of two , to match the calibration in figure 1 , would lower the best fitting index @xmath3 from 1.40 to 1.28 . likewise , lowering the gas masses in the starbursts by a factor of two , to take into account the possibility that the co/2 conversion factor is systematically lower , would _ increase _ @xmath3 by approximately the same amount , from 1.4 to 1.5 . this range of values provides a fairer estimate of the actual uncertainty in the composite schmidt law . folding together all of these uncertainties , we adopt as our final result : @xmath54 figure 6 shows that equation ( 4 ) provides an excellent parametrization of the global sfr , over a density range extending from the most gas - poor spiral disks to the cores of the most luminous starburst galaxies . this may account for why conventional galaxy evolution models , which usually are based on a schmidt law parametrization of the sfr , often produce realistic predictions of the gross star formation properties of galaxies . there are limitations to the schmidt law in equation ( 4 ) that should be borne in mind , however , when applying this recipe to galaxy evolution models or numerical simulations . although the full range of sfrs and gas densities are very well represented by a single power law with @xmath55 , the scatter in sfrs about the mean relation is substantial , @xmath300.3 dex rms , and individual galaxies deviate by as much as a factor of 7 . consequently equation ( 4 ) provides at most a statistical description of the global sfr , averaged over large samples of galaxies . another potential limitation for its application to simulations and models is the need to accurately specify the linear sizes of the relevant star forming regions . this is relatively straightforward for normal disks , where the scaling radius is comparable to the photometric radius of the galaxy or the edge of the active star forming disk . it may be more difficult to model in starbursts , however , where the intense star formation is usually concentrated in a region that is a few percent of the radius of the parent galaxy . fortunately the slope of the schmidt law is relatively shallow , and a modest error in the scaling radius will displace the inferred sfr and gas densities nearly along a line of slope @xmath56 , nearly parallel to the schmidt law itself . this is illustrated in figure 6 , where a short diagonal line shows the effect of changing the scaling radius by a factor of two ( for a fixed gas mass and total sfr ) . the schmidt law in figure 6 is so well defined that it is tempting to identify a simple , unique physical origin for the relation . however we find that a schmidt law is not the only simple parametrization that can reproduce the range of sfrs observed in this sample , and this serves as a caution against overinterpreting the physical nature of the empirical star formation law . in this section we briefly discuss the form of the schmidt law expected from simple gravitational arguments , and demonstrate that a simple kinematical model provides an equally useful recipe for modelling the large - scale sfr . numerous theoretical scenarios which produce a schmidt law with @xmath3 = 1 @xmath4 2 can be found in the literature ( larson 1992 and references therein ) . simple self - gravitational models for disks can reproduce the large - scale star formation thresholds observed in galaxies ( quirk 1972 ; k89 ) , and the same basic model is consistent with a schmidt law at high densities with index @xmath57 ( larson 1988 , 1992 ) . for example in a simple self - gravitational picture in which the large - scale sfr is presumed to scale with the growth rate of perturbations in the gas disk , the sfr will scale as the gas density divided by the growth timescale : @xmath58 where @xmath59 and @xmath60 are the volume densities of gas and star formation . the corresponding scaling of the projected surface densities will depend on the scale height distribution of the gas , with @xmath61 expected for a constant mean scale height , a reasonable approximation for the galaxies and starbursts considered here . although this is hardly a robust derivation , it does show that a global schmidt law with @xmath57 is physically plausible . in a variant of this argument , silk ( 1997 ) has suggested a generic form of the star formation law , in which the sfr surface density scales with the ratio of the gas density to the local dynamical timescale : @xmath62 where @xmath63 refers in this case to the local orbital timescale of the disk , and @xmath64 is the angular rotation speed . models of this general class have been studied previously by wyse ( 1986 ) and wyse & silk ( 1989 ) , though with different scalings of the gas density and separate treatment of the atomic and molecular gas . equation ( 6 ) might be expected to hold if , for example , star formation triggering by spiral arms or bars were important , in which case the sfr would scale with orbital frequency . to test this idea , we compiled rotation velocities for the galaxies in tables 1 and 2 , and used them to derive a characteristic value of @xmath63 for each disk . the timescale @xmath63 was defined arbitrarily as @xmath65 , the orbit time at the outer radius @xmath66 of the star forming region . the mean orbit time in the star forming disk is smaller than @xmath63 defined in this way , by a factor of 1 @xmath4 2 , depending on the form of the rotation curve and the radial distribution of gas in the disk . we chose to define @xmath63 and @xmath64 at the outer edge of the disk to avoid these complications . tables 1 and 2 list the adopted values , in units of @xmath37 yr . face - on galaxies or those with poorly determined ( rotational ) velocity fields were excluded from the analysis . figure 7 shows the relationship between the observed sfr density and @xmath67 for our sample . the solid line is not a fit but simply a line of slope unity which bisects the relation for normal disks . this alternate prescription for the star formation law provides a surprisingly good fit to the data , both in terms of the slope and the relatively small scatter about the mean relation . when compared over the entire density range the observed law is slightly shallower than predicted by equation ( 7 ) ( slope @xmath170.9 instead of 1 ) ; on the other hand the fit to the normal disk sample is as tight as a schmidt law . the zeropoint of the line corresponds to a sfr of 21% of the gas mass per orbit at the outer edge of the disk . since the average orbit time within the star forming disk is about half that at the disk edge , this implies a simple parametrization of the local star formation law : @xmath68 from a strictly empirical point of view , the schmidt law in equation ( 4 ) and the kinematical law in equation ( 7 ) offer two equally valid parametrizations for the global sfrs in galaxies , and either can be employed as a recipe in models and numerical simulations . it is unclear whether the kinematic model can fit the radial distribution of star formation as well as a schmidt law , and we plan to explore this in paper ii . the two parametrizations also offer two distinct interpretations of the observation that the star formation efficiency in central starbursts is much higher than found in quiescent star forming disks ( e.g. , young 1986 ; solomon & sage 1988 ; sanders 1991 ) . in the schmidt law picture , the higher efficiencies in starbursts are simply a consequence of their much higher gas densities . for a given index @xmath3 , the sfr per unit gas mass will scale as @xmath69 , and hence for the law observed here roughly as @xmath70 . the central starbursts have characteristic gas densities that are 100 @xmath4 10000 times higher than the average for normal disks , hence we would expect the global star formation efficiencies to be 6 @xmath4 40 times higher , as observed . in the alternative picture in which the sfr is presumed to scale with @xmath67 , the high sfrs and star formation efficiencies in starburst galaxies simply reflect the smaller physical scales and shorter dynamical timescales in these compact central regions . it is difficult to to differentiate between these alternatives with disk - averaged measurements alone , and since the global star formation law is mainly useful as an empirical parametrization , the distinction may not be important . deeper insight into the physical nature of the star formation law requires spatially resolved data for individual disks , of the kind that will be analyzed in paper ii . several individuals contributed to the large set of data analyzed in this paper , and it is a pleasure to thank them . the kpno data used in this paper were obtained as part of other projects in collaboration with r. braun , r. walterbos , and p. hodge . c. martin worked on the reduction of the spatially resolved data shown in figure 3 . i am also grateful to j. black , j. ostriker , s. sakai , p. solomon , s. white , and especially j. silk for comments and suggestions about early versions of this work . i am also grateful to the anonymous referee for several comments that improved the paper . some of the data used in this paper were obtained on the 2.3 m bok telescope at steward observatory . this research was supported by the national science foundation through grant ast-9421145 . fig . 1. a comparison of integrated sfrs derived from emission - line fluxes and far - infrared continuum luminosities , for 18 infrared - selected starburst galaxies . the solid line shows the relation expected from eqs . ( 2 ) and ( 3 ) . the dashed line is the best fitting mean relation . 2. relation between the disk - averaged sfr per unit area and gas density for 61 normal disk galaxies . the solid lines are least square fits to the schmidt law , as described in the text . the dashed and dotted lines correspond to constant global star formation efficiencies and gas consumption timescales , as indicated . fig . 4. correlation of the disk - averaged sfr per unit area with the average surface densities of hi ( left ) and 2 ( right ) . the 2 densities were derived using a constant co/2 conversion factor . in the right panel , solid circles denote galaxies with l@xmath39 , while open circles denote galaxies with l@xmath71 . 5. relation between the disk - averaged sfr per unit area and molecular gas density for 36 infrared - selected circumnuclear starbursts . the solid line shows a bivariate least squares fit to the schmidt law , as described in the text . the dashed and dotted lines correspond to constant global star formation efficiences and gas consumption timescales , as indicated . fig . 6. composite star formation law for the normal disk ( solid circles ) and starburst ( squares ) samples . open circles show the sfrs and gas densities for the centers of the normal disk galaxies . the line is a least squares fit with index @xmath72 . the diagonal short line shows the effect of changing the scaling radius by a factor of two . 7. relation between the sfr for the normal disk and starburst samples and the ratio of the gas density to the disk orbital timescale , as described in the text . the symbols are the same as in figure 6 . the line is a median fit to the normal disk sample , with the slope fixed at unity as predicted by equation ( 7 ) .
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measurements of , hi , and co distributions in 61 normal spiral galaxies are combined with published far - infrared and co observations of 36 infrared - selected starburst galaxies , in order to study the form of the global star formation law , over the full range of gas densities and star formation rates ( sfrs ) observed in galaxies .
the disk - averaged sfrs and gas densities for the combined sample are well represented by a schmidt law with index @xmath0 .
the schmidt law provides a surprisingly tight parametrization of the global star formation law , extending over several orders of magnitude in sfr and gas density .
an alternative formulation of the star formation law , in which the sfr is presumed to scale with the ratio of the gas density to the average orbital timescale , also fits the data very well .
both descriptions provide potentially useful recipes " for modelling the sfr in numerical simulations of galaxy formation and evolution .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the rossiter mclaughlin effect ( hereafter the rm effect ) is a phenomenon originally reported as a `` rotational effect '' in eclipsing binary systems by @xcite ( for the beta lyrae system ) and @xcite ( for the algol system ) . in the context of extrasolar planetary science , the rm effect is seen as a radial velocity anomaly during a planetary transit caused by the partial occultation of the rotating stellar disk ( see @xcite , @xcite , or @xcite , for theoretical descriptions ) . the radial velocity anomaly depends on the trajectory of the planet across the disk of the host star , and in particular on the alignment between that trajectory and the rotation field of the star . by monitoring this anomaly throughout a transit one can determine whether or not the planetary orbital axis is well - aligned with the stellar spin axis . in the solar system , the orbits of all 8 planets are known to be well - aligned with the solar equator , but this is not necessarily the case for exoplanetary systems , or for hot jupiters in particular . the key parameter is the sky - projected angle between the stellar spin axis and the planetary orbital axis , @xmath1 , and measurements of this `` misalignment angle '' for various exoplanetary systems will help to place the solar system in a broader context . specifically , measurements of the rm effect for exoplanetary systems are important because of the implications for theories of migration and hot jupiter formation . so far , measurements of @xmath1 for two systems have been reported ; and @xcite for hd 209458 , @xcite for hd 189733 . in both of those systems , the host star is very bright ( @xmath8 ) , facilitating the measurement . the observed values of @xmath1 for the two systems are small or consistent with zero , which would imply that the standard migration mechanism ( planet - disk interaction ) does not alter the spin - orbit alignment grossly during the planetary formation epoch . however , just these few examples are not enough for statistical constraints on other hot jupiter formation theories , including planet - planet interaction @xcite , the `` jumping jupiter '' model @xcite , or the kozai mechanism @xcite , which may lead hot jupiters to have significantly misaligned orbits . thus further measurements of the rm effect for other transiting systems are valuable . given that most of ongoing transit surveys target relatively faint ( @xmath2 ) host stars , it is also important to extend the reach of this technique to fainter stars . further observations for new targets would be useful to constrain planet formation theories , and more importantly , have a potential to discover large spin - orbit misalignments , which would be a challenge to some theoretical models . in this paper , we report the measurement of the rm effect and the constraint on @xmath1 for tres-1 ( @xmath9 ) which has a significantly fainter host star than those in previous studies ( @xmath3 ) . in addition to the fainter host star , this work differs from previous studies of the rm effect in that we have conducted simultaneous spectroscopic and photometric observations . this new strategy offers several potentially important advantages . first , the simultaneous photometry eliminates any uncertainty in the results due to the orbital ephemeris and the transit depth . although this did not turn out to be crucial for the present work , it will be useful for newly discovered targets which still have uncertainty in the times of transits . second , the transit depth might be expected to vary due to star spots or transient events , and indeed evidence for star spots was reported in hst / acs photometry for this system @xcite . thus simultaneous monitoring is useful to assess anomalies in the transit depth . finally , obtaining all of the data on a single night is useful to avoid systematic errors in radial velocities from long - term instrumental instabilities . moreover , although it need not be simultaneous , the photometry also helps to determine the limb - darkening parameter for the visual band , which can be used in the interpretation of the rm - affected spectra . in this way , we can determine the limb - darkening parameter of the host star directly from the data , instead of assuming a value based on stellar atmosphere models . we describe our observations in section 2 and report the results in section 3 . section 4 provides a discussion and summary of this paper . we observed the planet - hosting k0v star tres-1 on ut 2006 june 21 , the night of a predicted planetary transit , using the subaru 8.2 m telescope at mauna kea and the magnum 2 m telescope at haleakala , both in hawaii . the event is predicted as the 238th transit from the first discovery , namely @xmath10 ( @xmath11 : integer ) in the ephemeris by @xcite ; @xmath12.\end{aligned}\ ] ] the transit occurred shortly after midnight . we observed tres-1 during 5 hours bracketing the predicted transit time , through air masses ranging from 1.0 to 1.3 . the magnum 2 m telescope is located near the haleakala summit on the hawaiian island of maui @xcite . the magnum photometric observation was conducted in parallel with the subaru spectroscopic observation . we employed the multi - color imaging photometer ( mip ) using a @xmath13 pixel ccd with a @xmath14 band filter , covering 4750 @xmath15 6180 , and we set 9 dithering positions ( 3@xmath16 positions ) on the ccd . the mip has a @xmath17 arcmin@xmath18 field of view ( fov ) with a pixel scale of 0.277 arcsec / pixel . we used 2mass j19041058 + 3638409 as our comparison star for differential photometry . this star is close enough to fit in the mip field of view , and is known to be photometrically stable at a level sufficient for our study ( e.g. , @xcite ) . the exposure time was either 40 or 60 seconds according to observing conditions so that the photon counts are close to the saturation level of the ccd , with a readout / setup time of 60 seconds . we then reduced the images with the standard mip pipeline described in @xcite . we determined the apparent magnitudes of tres-1 and the comparison star using an aperture radius of 20 pixels . the typical fwhm of each star ranged from 1.4 to 1.9 arcsec ( from 5 to 7 pixels ) . we estimated the sky background level with an annulus from 20 to 25 pixels in radius centered on each star , and subtracted the estimated sky contribution from the aperture flux . then we computed the differential magnitude between tres-1 and the comparison star . after these steps , we decorrelated the differential magnitude from the dithering positions and eliminated apparent outliers from the light curve , most of which were obtained at the 9th dithering position . we do not find any clear correlations with other observing parameters . for the analysis of transit photometry , @xcite studied the time - correlated noise ( the so - called `` red noise '' ) in detail , and introduced a simple and useful method to account at least approximately for the effect of the red noise on parameter estimation . based on these studies , we used the following procedure to determine the appropriate data weights for the magnum photometry ( which are similar to that employed by @xcite ) . we first fitted the magnum light curve with the analytic formula given in @xcite and found the residuals between the data and the best - fitting model . using only the poisson noise as an estimate of the error in each photometric sample , we found @xmath19 ( @xmath20 : degrees of freedom ) , implying that the true errors are significantly in excess of the poisson noise . we also found the residuals in the early part of the night ( before @xmath21 [ hjd ] ) to be significantly larger than those from later in the night . this larger scatter could have been caused by shaking of the telescope by the stronger winds that occurred during the early part of the night . we thus estimated error - bars separately for the early part of the night and the later part of the night , as described below . first , we rescaled the error bars to satisfy @xmath22 ( step 1 ) , namely 0.00259 for the early data and 0.00189 for the late data . the light curve with these rescaled error bars is shown in the upper panel of fig . [ ourdata ] . next , in order to assess the size of time - correlated noise for the magnum data , we solved the following equations , @xmath23 where @xmath24 is the standard deviation of each residual and @xmath25 is the standard deviation of the average of the successive @xmath26 points . @xmath27 is called the white noise , which is uncorrelated noise that averages down as @xmath28 , while @xmath29 is called the red noise , which represents correlated noise that remains constant for specified @xmath26 . we calculated @xmath27 and @xmath29 for the choice @xmath30 ( corresponding to one hour ) , finding @xmath31 and @xmath32 for the early data . on the other hand , we found @xmath33 ( suggesting a smaller level of the red noise ) for the late data . the choice of @xmath30 is fairly arbitrary ; other choices of @xmath26 between 5 and 50 gave similar results . we then adjusted the error bars for the early night by multiplying @xmath34^{1/2 } \sim 3.6 $ ] ( step 2 ) . we did not change the error bars for the late night data . we adopted these rescaled uncertainties for subsequent fitting procedures . ( 70.5mm,70.5mm)figure1.eps lcc time [ hjd ] & value [ m s@xmath35 & error [ m s@xmath35 + 2453907.87018 & 18.7 & 14.0 + 2453907.88139 & 30.5 & 12.5 + 2453907.89262 & 54.6@xmath36 & 12.0 + 2453907.90384 & 24.3 & 10.4 + 2453907.91506 & 26.4 & 11.4 + 2453907.92628 & 30.4 & 10.9 + 2453907.93750 & 22.4 & 14.3 + 2453907.94873 & 2.9 & 11.0 + 2453907.95996 & -7.1 & 12.1 + 2453907.97119 & -22.3 & 13.3 + 2453907.98241 & -40.5 & 13.3 + 2453907.99364 & -39.2 & 13.0 + 2453908.00488 & -9.8 & 12.2 + 2453908.01610 & -30.5 & 13.8 + 2453908.02732 & -17.7 & 13.6 + 2453908.03854 & -24.7 & 12.2 + 2453908.04978 & -27.5 & 11.1 + 2453908.06100 & -38.2 & 13.3 + 2453908.07223 & -23.7 & 11.2 + 2453908.08345 & -23.0 & 9.6 + [ rvsummary ] we used the high dispersion spectrograph ( hds ) on the subaru telescope @xcite . we employed the standard i2a set - up of the hds , covering 4940 @xmath15 6180 with the iodine absorption cell for measuring radial velocities . the slit width of @xmath37 yielded a spectral resolution of @xmath38 45000 , and the seeing was between @xmath39 and @xmath40 . the exposure time for tres-1 was 15 minutes yielding a typical signal - to - noise ratio ( snr ) @xmath38 60 per pixel . in order to estimate systematic errors from short term instrumental variations , we also obtained spectra of the much brighter ( @xmath41 ) k0v star hd 185144 before and after the series of tres-1 exposures . this star is known to be stable in velocity @xcite . we obtained five 30 s exposures of hd 185144 , each having a snr of approximately 200 pixel@xmath42 . we processed the frames with standard iraf procedures and extracted one - dimensional spectra . next , we calculated relative radial velocity variations by the algorithm following @xcite . we used this algorithm because it properly takes into account the fairly large changes of the instrumental profile during the observations . the hds is known to experience appreciable instrumental variations even within a single night , reported in @xcite and @xcite . we estimated internal errors of the radial velocities from the scatter of the radial velocity solutions for 2 segments of the spectra . the typical errors are @xmath43 [ m s@xmath35 , which are reasonable values to be expected from the photon noise limit . note that we do not find any evidence of star spots or transient events during our photometric observation ( see fig . [ ourdata ] ) . for this reason , we have not accounted for possible systematic errors in the velocities due to star spots . we also reduced the hd 185144 spectra with the same method in order to check for systematic errors due to short - term instrumental instabilities . the rms of the radial velocity of hd 185144 is less than 5 [ m s@xmath35 , attesting to good instrumental stability . the resultant radial velocities of tres-1 and their errors are shown in table [ rvsummary ] and the lower panel of fig . [ ourdata ] . ( 120mm,50mm)figure2a.eps ( 120mm,50mm)figure2b.eps ( 120mm,50mm)figure2c.eps ( 120mm,50mm)figure3.eps l|cc|cc|cc & & & + & & & + parameter & value & uncertainty & value & uncertainty & value & uncertainty + ( all rv samples ) & & & + @xmath44 [ m s@xmath35 & @xmath45 & @xmath46 & @xmath45 & @xmath46 & @xmath47 & fixed + @xmath48 [ km s@xmath35 & @xmath49 & @xmath50 & @xmath51 & @xmath52 & @xmath49 & @xmath50 + @xmath1 [ deg ] & @xmath53 & @xmath54 & @xmath55 & @xmath56 & @xmath57 & @xmath58 + @xmath59 & @xmath60 & @xmath61 & @xmath60 & @xmath61 & @xmath62 & @xmath61 + @xmath63 & @xmath64 & @xmath65 & @xmath64 & @xmath65 & & + @xmath66 & @xmath67 & @xmath68 & @xmath67 & @xmath68 & @xmath69 & fixed + @xmath70 [ @xmath71 & @xmath72 & @xmath73 & @xmath72 & @xmath73 & @xmath74 & fixed + @xmath75 [ deg ] & @xmath76 & @xmath50 & @xmath76 & @xmath77 & @xmath78 & fixed + @xmath79 [ m s@xmath35 & @xmath49 & @xmath80 & @xmath81 & @xmath82 & @xmath83 & @xmath84 + @xmath85 [ m s@xmath35 & @xmath86 & @xmath87 & @xmath86 & @xmath87 & & + @xmath88 [ m s@xmath35 & @xmath89 & @xmath90 & @xmath89 & @xmath90 & & + @xmath91 [ hjd ] & @xmath92 & @xmath93 & @xmath92 & @xmath93 & & + @xmath94 [ hjd ] & @xmath95 & @xmath96 & @xmath95 & @xmath96 & & + @xmath97 [ hjd ] & @xmath98 & @xmath99 & @xmath98 & @xmath99 & & + @xmath100 [ hjd ] & @xmath101 & @xmath102 & @xmath101 & @xmath102 & @xmath103 & @xmath102 + ( without the outlier ) & & & + @xmath44 [ m s@xmath35 & @xmath104 & @xmath46 & @xmath104 & @xmath46 & @xmath47 & fixed + @xmath48 [ km s@xmath35 & @xmath49 & @xmath50 & @xmath105 & @xmath52 & @xmath49 & @xmath50 + @xmath1 [ deg ] & @xmath106 & @xmath107 & @xmath108 & @xmath54 & @xmath109 & @xmath110 + @xmath79 [ m s@xmath35 & @xmath111 & @xmath112 & @xmath113 & @xmath114 & @xmath115 & @xmath80 + [ result ] as described above , we have obtained 20 radial velocity samples and 184 @xmath14 band photometric samples taken simultaneously covering the transit . in addition , in order to search for an optimal solution of orbital parameters for tres-1 , we incorporate our new data with 12 previously published radial velocity measurements using the keck i telescope ( 7 by @xcite and 5 by @xcite ) and 1149 @xmath116 band photometric measurements spanning 3 transits using the flwo 1.2 m telescope @xcite . the uncertainties of the flwo data had already been rescaled by the authors such that @xmath22 for each transit ( namely , the step 1 has been done ) . for the step 2 , we find @xmath33 for these data , thus we did not modify these error bars further . we employ the analytic formulas of radial velocity and photometry including the rm effect given in @xcite and @xcite ( hereafter the ots formulae ) in order to model the observed data . note that based on the previous studies by @xcite and @xcite , we have learned that the ots formulae systematically underestimate the amplitude of radial velocity anomaly by approximately 10% . this is possibly because the radial velocity anomaly defined by @xcite and that measured by the analysis pipeline are different . thus we correct @xmath48 in the ots formulas by modifying @xmath117 . this correction presumably gives more realistic values for @xmath48 and @xmath1 , and has little influence on any of the other parameters . here we assume circular orbits of the star and the planet about the center of mass ( namely , @xmath118 ) . we adopt the stellar mass @xmath119 [ @xmath120 @xcite and the orbital period @xmath121 [ days ] and @xmath122 [ hjd ] @xcite . as a result , our model has 15 free parameters in total . eight parameters for the tres-1 system include the radial velocity amplitude @xmath44 , the sky - projected stellar rotational velocity @xmath48 , the misalignment angle between the stellar spin and the planetary orbit axes @xmath1 , the linear limb - darkening parameter for @xmath14 band @xmath59 , the same for @xmath116 band @xmath63 , the ratio of star - planet radii @xmath66 , the stellar radius @xmath70 , and the orbital inclination @xmath75 . here we assume that the limb - darkening parameters for the spectroscopic and photometric models are the same . lines where the radial velocities are measured . however , in principle the correspondence is not exact because the limb - darkening function may not be identical in the lines as opposed to the continuum , and because the influence of limb - darkening on the rv - measuring algorithm has yet to be investigated in detail . ] we also add three parameters for velocity offsets to the respective radial velocity dataset @xmath123 ( for our template spectrum ) , @xmath124 ( for @xcite ) and @xmath125 ( for @xcite ) , and four parameters for the times of mid - transit @xmath126 . in previous studies of the exoplanetary rm effect , it was possible and desirable to determine both @xmath48 and @xmath1 from the radial velocity data . in this case , there are two reasons to prefer an external determination of @xmath48 . first , the signal - to - noise ratio of the anomaly is smaller , because of the faintness of the host star . this makes it valuable to reduce the number of degrees of freedom in the model . second , the transit geometry is nearly equatorial , which introduces a very strong degeneracy between @xmath48 and @xmath1 , as explained by gaudi & winn ( 2007 ) . the alternative we have chosen is to adopt a value for @xmath48 based on previous observations , and use the radial - velocity anomaly to determine @xmath1 . ( we have also investigated our ability to determine both parameters , as described below . ) @xcite reported @xmath127 [ km s@xmath35 for the tres-1 host star from their analysis of the observed spectral line profiles ; this is the most reliable estimate for @xmath48 to date . we incorporate this information into our model by adding a term @xmath128 ^ 2 $ ] to the @xmath129 fitting statistic . thus our @xmath129 statistic is @xmath130 ^ 2 + \sum_{j=1}^{n_{f}=1333 } \left [ \frac{f_{j , obs } -f_{j , calc}}{\sigma_{j } } \right]^2 \nonumber\\ & + & \left [ \frac{v \sin i_s - 1.08}{0.30 } \right]^2,\end{aligned}\ ] ] where @xmath131 and @xmath132 represent the values calculated by the ots formulae with the above parameters . we find optimal parameters by minimizing the @xmath129 statistic of eq . ( 4 ) using the amoeba algorithm @xcite , and estimate confidence levels of the parameters using @xmath133 from the optimal parameter set . to assess the dependence of our results on the _ a priori _ constraint on @xmath48 , we also compute and compare the best - fit values and uncertainties by using another function : @xmath134 ^ 2 + \sum_{j=1}^{n_{f}=1333 } \left [ \frac{f_{j , obs } -f_{j , calc}}{\sigma_{j } } \right]^2,\ ] ] for reference . in addition , we note that the third radial velocity sample of our data ( @xmath135 [ hjd ] ) may appear to be an outlier , but it lies just about 3@xmath7 from a theoretical radial velocity curve ( e.g. , fig . [ rvwithapr ] ) . for clarity , we calculate eq . ( 4 ) and eq . ( 5 ) with and without that sample . the results for both @xmath129 statistics are presented in table [ result ] . the minimum @xmath129 is 1308.57 ( 1296.39 ) for eq . ( 4 ) and 1305.18 ( 1298.57 ) for eq . ( 5 ) with 1350 ( 1349 ) degrees of freedom , where the numbers in parentheses refer to the case without the outlier . in fig . [ rvwithapr ] , we present the radial velocities and the best - fit curve with ( the top figure , marked with `` a '' ) and without ( the middle figure , marked with `` b '' ) the _ a priori _ constraint . in addition , we also compute the best - fit curve without the constraint , but assuming that @xmath136 [ deg ] ( the bottom figure , marked with `` @xmath136 [ deg ] '' ) . [ vlcontour ] plots ( @xmath137 ) contours calculated with ( the left panel , marked with `` a '' ) and without ( the right panel , marked with `` b '' ) the _ a priori _ constraint . note that we only show here the results with the possible outlier in fig . [ rvwithapr ] and fig . [ vlcontour ] , since the same figures but without the outlier have basically similar appearance and have less information to show . as a result , we find @xmath138 [ deg ] and @xmath139 [ km s@xmath35 for tres-1 with the _ a priori _ constraint , and our findings except for @xmath1 are in good agreement with previous studies @xcite . on the other hand , the result without the _ a priori _ constraint , @xmath140 [ deg ] and @xmath141 [ km s@xmath35 , agrees with the above result within about 1@xmath7 for @xmath1 and @xmath48 , and is also consistent with the previous results for other parameters . in case we calculate @xmath129 without the constraint but assuming that @xmath136 [ deg ] , we find that the minimum @xmath129 is 1309.85 ( 1298.91 ) with 1351 ( 1350 ) degrees of freedom , and @xmath142 [ km s@xmath35 . consequently , our results for @xmath1 have fairly large uncertainties . we find at least that the orbital motion of tres-1b is prograde . additional radial velocity measurements during transits , and a more precise measurement of @xmath48 , would be desirable to pin down @xmath1 and help to discriminate between different modes of migration . ( 75mm,75mm)figure4.eps ( 60mm,60mm)figure5.eps we have presented simultaneous spectroscopy and photometry of a transit of tres-1b , exhibiting a clear detection of the rossiter - mclaughlin effect and consequent constraints on the alignment angle between the stellar spin and the planetary orbital axes . our philosophy has been to use all of the best data available at present . however , it is also interesting to examine how well we are able to determine the system parameters using _ only _ the data gathered on a single night using subaru and magnum . this is because for future studies of newly discovered transiting exoplanetary systems , higher - precision data from other observatories may not be available . we repeat the fitting procedure without the keck and flwo data but still assuming the system parameters other than @xmath48 , @xmath1 , @xmath59 , @xmath79 , and @xmath143 to be the values presented in @xcite and @xcite . we find almost the same values and uncertainties for the parameters above ( the right side of table [ result ] ) as before , indicating that a single night s data would have done almost as well as the full data set . using transit timing of the tres-1 system , @xcite reported a constraint on the existence of additional planets in the system . subsequently , @xcite pointed out that @xmath144 are consistent with their ephemeris at about the 2@xmath7 level , but occurred progressively later than expected . here we present @xmath145 in fig . our result is also consistent with the published ephemeris within 1.5@xmath7 , occurring only slightly later than expected from @xcite . in addition to producing transit timing variations , any additional bodies in the tres-1 system could excite the orbital eccentricity of the tres-1b planet , which could be detectable in the radial velocity measurements of the host star . thus it is worthwhile to put empirical constraints on the orbital eccentricity . for this purpose , we compute @xmath129 using eq . ( 4 ) for fixed values of ( @xmath146 ) over numbers of grid points to map out the allowed region in ( @xmath147 ) space . in fact the most appropriate parameters are @xmath148 and @xmath149 since their uncertainties are uncorrelated ( see fig . [ ewcontour ] ) . the resulting constraints are @xmath150 and @xmath151 . our result for @xmath152 is similar to that of @xcite , which was based on the timing of the secondary eclipse . the value of @xmath152 is consistent with zero within 1@xmath7 , and the value of @xmath153 is consistent with zero within 2@xmath7 . thus we do not find any strong evidence for a nonzero orbital eccentricity in the tres-1 system . there is another interesting application of our data . recently , @xcite studied the detectability of `` hot trojan '' companions near the l4/l5 points of transiting hot jupiters , through any observed difference between the time of vanishing stellar radial velocity variation ( @xmath154 ) and the time of the midpoint of the photometric transit ( @xmath155 ) . our strategy of simultaneous spectroscopic and photometric transit observations is ideally suited for searching for the hot trojan companions . for the tres-1 system , @xcite set an upper limit on the mass of the trojan companions @xmath156 at the 3@xmath7 level ( assuming the circular orbit ) , using the radial velocity samples of @xcite . here we compute @xmath157 using all available out - of - transit radial velocity samples with ( without ) the possible outlier , and both for the circular and the eccentric orbit . we find @xmath158 [ min ] ( circular ) and @xmath159 [ min ] ( eccentric ) . accordingly , we set constraints on the mass of the trojan companions , @xmath160 , which is defined as the difference in the mass at l4 ( @xmath161 ) and the mass at l5 ( @xmath162 ) ( namely , @xmath163 ) , through the relation ; @xmath164 where @xmath165 and @xmath166 indicate the uncertainties of @xmath160 and @xmath167 , respectively ( eq . ( 1 ) and eq . ( 2 ) in @xcite ) . note that we adopt @xmath168 [ @xmath169 which is determined by this work . we find @xmath170 [ @xmath6 ] ( circular ) and @xmath172 [ @xmath6 ] ( eccentric ) . as a result , we exclude the trojan companions near the l4 point more massive than @xmath173 [ @xmath6 ] if the orbit is circular , and @xmath174 [ @xmath6 ] if the orbit is allowed to be eccentric , both at the 3@xmath7 level . our constraint under the reasonable assumption of a circular orbit is more stringent than that of @xcite by a factor of 4 , because we have increased the number of radial velocity samples and because our data cover the critical phase to determine @xmath154 . consequently , we conclude that we do not find any sign of the existence of additional bodies in the tres-1 system at present . in this paper , we have placed a constraint on the sky - projected angle between the stellar spin axis and the planetary orbital axis for the tres-1 system , namely @xmath0 [ deg ] using all available data and information from previous studies . although we can not discriminate whether the spin - orbit angle in this system is well - aligned or not at this point , our constraint on @xmath1 clearly indicates the prograde orbital motion of tres-1b . the uncertainty is larger than in previous studies ( @xmath175 [ deg ] for hd 209458 and hd 189733 ) because the host star is significantly fainter in this case . although further radial velocity measurements during transit would be necessary to pin down @xmath1 more stringently , we have demonstrated for the first time that such measurements are possible for such a faint target . this is important because most of the newly discovered transiting planets from ongoing transit surveys will have relatively faint host stars . for example , the new targets that were discovered in 2006 , namely xo-1 @xcite , tres-2 @xcite , hat - p-1 @xcite , wasp-1 and wasp-2 @xcite , are all in this category . combining future measurements of @xmath1 in other transiting systems , we would be able to determine the distribution of @xmath1 for exoplanetary systems with useful statistical accuracy . we acknowledge close support of our observations by akito tajitsu , who is a support scientist for subaru hds . we appreciate the detailed and helpful critique of the manuscript by the referee , scott gaudi . we also thank kazuhiro yahata , shunsaku horiuchi , takahiro nishimichi , and hiroshi ohmuro for useful discussions . is supported by a japan society for promotion of science ( jsps ) fellowships for research . this work is supported in part by a grant - in - aid for scientific research from the jsps ( no.14102004 , 16340053 , 17740106 ) , and mext japan , grant - in - aid for scientific research on priority areas , `` development of extra - solar planetary science , '' and nasa grant nag5 - 13148 . we wish to recognize and acknowledge the very significant cultural role and reverence that the summit of mauna kea has always had within the indigenous hawaiian community .
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we report a measurement of the rossiter
mclaughlin effect in the transiting extrasolar planetary system tres-1 , via simultaneous spectroscopic and photometric observations with the subaru and magnum telescopes . by modeling the radial velocity anomaly that was observed during a transit ,
we determine the sky - projected angle between the stellar spin axis and the planetary orbital axis to be @xmath0 [ deg ] .
this is the third case for which @xmath1 has been measured in a transiting exoplanetary system , and the first demonstration that such measurements are possible for relatively faint host stars ( @xmath2 , as compared to @xmath3 for the other systems ) .
we also derive a time of mid - transit , constraints on the eccentricity of the tres-1b orbit ( @xmath4 ) , and upper limits on the mass of the trojan companions ( @xmath514 @xmath6 ) at the 3@xmath7 level .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
pioneering experiments on ultracold gases of atoms trapped in optical lattices allowed for a direct observation of quantum many - body phenomena , such as the quantum phase transition from mott phase to superfluid phase.@xcite optical lattices are realized by counterpropagating laser beams , which form a periodic potential.@xcite the bosonic particles located on the optical lattice gain kinetic energy when tunneling through the potential wells of neighboring sites of the periodic potential and they exhibit a repulsive interaction when a lattice site is occupied by more than one atom . a condensate of ultracold atoms can be driven from superfluid phase to mott phase by gradually increasing the intensity of the laser beams . the potential wells of the optical lattice are shallow for low laser - beam intensity . thus the bosonic particles can overcome the barrier easily and are delocalized on the whole lattice . however , for large intensity of the laser beams the potential wells are deep and there is little probability for the atoms to tunnel from one lattice site to another . this physical behavior can be described by the bose - hubbard ( bh ) model @xcite provided the gas of ultracold atoms is cooled such that only the lowest bloch band of the periodic potential has to be taken into account.@xcite the ground state of the bh model is superfluid when the local on - site repulsion between the atoms is small in comparison to the nearest - neighbor hopping strength whereas it is a mott state for integer particle density and large on - site repulsion compared to the hopping strength . due to these characteristics of the bh model the depth of the potential wells in optical lattices can be associated directly with the ratio of the on - site repulsion and the hopping strength . ultracold atoms confined in optical lattices provide a very clean experimental realization of a strongly correlated many - body problem and the internal physical processes are well understood in comparison to conventional condensed - matter systems . there is large experimental control over the system parameters , such as the particle number , lattice size , and depth of the potential wells . furthermore the sites of the optical lattice can be addressed individually due to the mesoscopic scale of the lattice.@xcite the quantum phase transition from mott phase to superfluid phase has been first observed experimentally for ultracold rubidium atoms trapped in a three - dimensional optical lattice @xcite and subsequently as well in optical lattices of two dimensions.@xcite the corresponding theoretical model , the two - dimensional ( 2d ) bh model , has already been investigated to some detail in literature . the phase diagram , which describes the quantum phase transition from mott phase to superfluid phase , has been investigated thoroughly at the mean - field level ( possibly including gaussian - fluctuation corrections).@xcite more accurate results for the phase diagram from quantum monte carlo @xcite ( qmc ) simulations , variational approaches,@xcite and strong - coupling perturbation theory@xcite are also available . the phase diagram for arbitrary integer fillings has been obtained recently using the so - called diagrammatic process chain approach.@xcite spectral functions of the two - dimensional bh model have been evaluated within a strong - coupling approach @xcite and a variational mean field approach.@xcite in the present paper we evaluate the border of the quantum phase transition from mott phase to superfluid phase for the first two mott lobes by means of the variational cluster approach ( vca),@xcite and show that this method provides quite accurately the boundaries of the mott phase , as compared with more demanding qmc simulations and perturbative expansions . in addition , we study in detail the spectral functions of the two - dimensional bh model in both the first and the second mott lobe , which require computing the green s function in real frequency domain . we also present the densities of states and momentum distributions corresponding to the spectral functions . finally , as a technical point , we present an extension of the so - called @xmath0-matrix formalism , which has been originally proposed for fermionic ( anticommutator ) green s functions,@xcite to bosonic ( commutator ) green s functions.@xcite as we show below , this extension is nontrivial due to the nonunitary nature of the bogoliubov transformation for bosonic particles . this paper is organized as follows . in sec . [ sec : model ] the bh model is introduced . section [ sec : method ] contains a short review on the variational cluster approach and the extension of the @xmath0-matrix formalism . section [ sec : results ] is devoted to the spectral properties of the bh model in two dimensions . here the phase diagram , spectral functions , densities of states , and momentum distributions are presented . finally , we summarize and conclude our findings in sec . [ sec : conclusion ] . the ( grand - canonical ) hamiltonian of the bh model @xcite is given by @xmath1 where @xmath2 is the nearest - neighbor hopping strength , @xmath3 is the local on - site repulsion , and @xmath4 is the chemical potential . the angle brackets in the first part of the hamiltonian specify to sum over pairs of nearest neighbors ( each pair counted once ) . the operator @xmath5 creates a particle at lattice site @xmath6 whereas @xmath7 annihilates a particle at site @xmath6 . the total particle number @xmath8 is conserved , since @xmath9 = 0 $ ] . the particles of the bh model are of bosonic character and thus the commutation relation @xmath10 = \delta_{ij}$ ] is satisfied . the first term of the hamiltonian models the hopping of a particle from lattice site @xmath11 to lattice site @xmath6 . the second part describes the local on - site repulsion , which remains zero when a lattice site is unoccupied or occupied by only one particle . however , it increases proportional to @xmath3 for each additionally added particle . we consider the on - site repulsion @xmath3 as unit of energy . the third part of the hamiltonian is necessary to perform calculations in grand - canonical ensemble , where the chemical potential @xmath4 controls the total particle number of the system . we use vca@xcite to evaluate the phase diagram and the spectral functions of the 2d bh model . vca is a variational extension of the cluster perturbation theory @xcite and is based on the self - energy functional approach ( sfa ) which has been originally proposed for fermionic systems by m. potthoff.@xcite vca has been extended to bosonic systems as well.@xcite the sfa is based on the fact that dyson s equation for the exact green s function is recovered at the stationary point of the grand potential @xmath12 $ ] considered as a functional of the self - energy @xmath13 . thus @xmath13 corresponds , at the stationary point , to the real physical self - energy . the self - energy functional @xmath12 $ ] can not be evaluated directly as it contains the legendre transform @xmath14 $ ] of the luttinger - ward functional.@xcite however , the functional @xmath14 $ ] just depends on the interaction term of the hamiltonian , i.e. , on the second term of eq.([eq : bhm ] ) , and is thus equivalent for all hamiltonians which share a common interaction part . due to this property @xmath14 $ ] can be eliminated from the expression of the self - energy functional @xmath12 $ ] . for this purpose an exactly solvable , so - called `` reference , '' system @xmath15 is constructed , which must be defined on the same lattice and must have the same interaction part as the original system @xmath16 . thus both the self - energy functional of the original system @xmath12 $ ] and the one of the reference system @xmath17 $ ] contain the same @xmath14 $ ] , which can be eliminated by comparison from the expressions of the two self - energy functionals . this yields for bosonic systems @xcite @xmath18 & = \omega^\prime[{\mathsf{\sigma } } ] & & - { \mbox{tr}}\,\ln(-({\mathsf{g}}_0^{\prime\,-1}-{\mathsf{\sigma } } ) ) \nonumber \\ & & & + { \mbox{tr}}\,\ln(-({\mathsf{g}}_0^{-1}-{\mathsf{\sigma } } ) ) \ ; \mbox { , } \label{eq : num : om5}\end{aligned}\ ] ] where quantities with prime correspond to the reference system and @xmath19 is the free green s function . the free green s function is defined as @xmath20 , where @xmath21 contains the hopping matrix and all other one - particle parameters of the hamiltonian except for the chemical potential @xmath4 , which is already treated separately in the definition . the symbol @xmath22 denotes a summation over bosonic matsubara frequencies and a trace over site indices . the self - energy functional @xmath12 $ ] given by eq.([eq : num : om5 ] ) is exact . in order to be able to evaluate the functional , the search space of the self - energy @xmath13 has to be restricted,@xcite which consists in an approximation . more precisely , the functional @xmath12 $ ] is evaluated for the subset of self - energies available to the reference system @xmath15 , see fig.[fig : num : vca ] . practically , this is achieved by varying the single - particle parameters of the reference hamiltonian in order to find the stationary point of the grand potential . thus the functional @xmath12 $ ] becomes a function of the set @xmath23 of single - particle parameters of @xmath24 @xmath25 leading to the stationary condition @xmath26 is restricted to self - energies @xmath27 which are accessible via the reference system @xmath15 . ( b ) lattice decomposition of a square lattice into @xmath28 site clusters . , scaledwidth=48.0% ] in vca the reference system is given by the decomposition of the total system into identical clusters , see fig.[fig : num : vca ] . in order to implement the @xmath0-matrix approach , we solve each cluster by means of the band lanczos method.@xcite the green s function of the total system is obtained via the relation @xmath29 which can be deduced from the dyson equation of the total system @xmath30 and the reference system @xmath31 . the self - energy can be eliminated and it follows that @xmath32 the expression in parenthesis defines the matrix @xmath33 with eq.([eq : num : gvca ] ) the grand potential @xmath34 can be rewritten as @xmath35 the decomposition of the @xmath36-site lattice into clusters of @xmath37 sites can be described by a superlattice . the original lattice is obtained by attaching a cluster to each site of the superlattice.@xcite a partial fourier transform from superlattice indices to wave vectors @xmath38 , which belong to the first brillouin zone of the superlattice , yields the total green s function @xmath39 due to the diagonality of @xmath40 in the superlattice indices its partial fourier transform does not depend on @xmath38 . the matrices in eq.([eq : num : cpteqfourier ] ) are now defined in the space of cluster - site indices and are thus of size @xmath41 . the @xmath36 wave vectors @xmath42 from the brillouin zone of the total lattice can be expressed as @xmath43 where @xmath44 belongs to both the reciprocal superlattice and the first brillouin zone of the total lattice.@xcite the frequency integration implicit in the expression for the grand potential , given in eq.([om ] ) , can be carried out analytically , yielding at zero temperature@xcite @xmath45 where @xmath46 and @xmath47 are the poles of the cluster green s function and total green s function , respectively . the number of clusters @xmath48 is denoted as @xmath49 . the poles @xmath46 of the cluster green s function can be readily obtained from the lanczos method , whereas the poles of the total green s function @xmath47 can be evaluated with the so - called @xmath0-matrix formalism , which was originally proposed for fermionic green s functions.@xcite here , we extend this formalism to the generic case , i.e. , we include bosonic green s functions . as we will see , this extension is nontrivial , since it involves non - unitary transformations . for zero temperature , the cluster green s function reads @xcite @xmath50 where @xmath51 is the ground state of the @xmath52 particle system , @xmath53 is its ( grand - canonical ) energy , and @xmath54 ( @xmath55 ) for bosonic ( fermionic ) green s functions . the first term on the right - hand side of eq.([num : gf ] ) describes single - particle excitations from the @xmath52 particle ground state and can thus be referred to as particle term , whereas the second part corresponds to single - hole excitations and can be called hole term . inserting the identity @xmath56 into each part of eq.([num : gf ] ) , where @xmath57 are the eigenvectors of the reference hamiltonian with corresponding eigenvalues @xmath58 , yields the lehmann representation of the green s function @xmath59 which can be cast into the form @xmath60 in eq.([num : gfqmatrix ] ) , we have introduced the following notation : @xmath61 @xmath62 and @xmath63 where @xmath64 is the hilbert space of an @xmath65 particle system . with @xmath66 the cluster green s function can be written in matrix notation @xmath67 with the help of this expression the vca green s function eq.([eq : num : cpteqfourier ] ) can be rewritten as @xmath68^{-1 } { \mathsf{s}}\,{\mathsf{q}}^\dagger \nonumber \\ & = { \mathsf{q } } \frac{1 } { { \mathsf{g}}^{\prime\,-1 } - { \mathsf{s}}\,{\mathsf{q}}^\dagger\,{\mathsf{v}}\,{\mathsf{q } } } { \mathsf{s}}\,{\mathsf{q}}^\dagger \;\mbox { , } \label{eqn : num : gvcarewritten}\end{aligned}\ ] ] where in the third step we expanded the fraction in a taylor series . the matrix @xmath69 is diagonal and contains the poles of the cluster green s function @xmath40 , see eq.([eq : num : gsmall ] ) . it can be written as @xmath70 with @xmath71 . plugging this into eq.([eqn : num : gvcarewritten ] ) yields @xmath72 we introduce the matrix @xmath73 . this matrix can be diagonalized as @xmath74 , where @xmath75 is a diagonal matrix containing the eigenvalues of @xmath76 and @xmath77 is the matrix of the eigenvectors of @xmath76 . the eigenvalue equation of the matrix @xmath76 can be rewritten as @xmath78 , where @xmath79 is the inverse of @xmath77 and not its transpose as @xmath76 is a non - symmetric matrix . from that we obtain @xmath80 therefore , the poles of the total green s function @xmath81 in eq.([eqn : num : gqmatrixlambda ] ) are the eigenvalues of the matrix @xmath76 . the matrices @xmath81 and @xmath82 are defined on the space of cluster - site indices . thus @xmath81 and @xmath82 are of size @xmath41 and depend on the wave vector @xmath83 , see eq.([eq : num : cpteqfourier ] ) . the matrix @xmath0 is of size @xmath84 , where @xmath85 is the dimension of the krylov space generated in the band lanczos method . due to the dependence of @xmath82 on @xmath83 the diagonalization of the matrix @xmath76 yields @xmath85 eigenvalues @xmath86 , which are used in eq.([eqn : num : omintzero ] ) . the diagonalization has to be repeated for all wave vectors @xmath83 . with that the grand potential @xmath34 can be evaluated . the crucial point is that for bosonic green s functions , the entries of the diagonal matrix @xmath87 can be both @xmath88 as well as @xmath89 , see eq.([eq : num : sdef ] ) . therefore , the eigenvalue problem is not symmetric.@xcite the factorization of the total lattice into clusters breaks the translational symmetry of the lattice . hence the total green s function would depend on two wave vectors @xmath42 and @xmath90 , which is certainly not correct for a periodic lattice . this has to be circumvented by a periodization prescription that provides a total green s function @xmath91 depending only on one wave vector @xmath42 . the periodization prescription proposed in ref . ( greens - function periodization ) reads as follows : @xmath92 where @xmath42 is a wave vector of the total lattice and @xmath93 refers to lattice sites @xmath94 of the cluster . the wave vectors @xmath38 in eq.([eq : num : cpteqfourierfull ] ) can be replaced by the total wave vectors @xmath42 as they just differ by a reciprocal superlattice wave vector , see eq.([eqn : kdecomposition ] ) . with eqs.([eqn : num : gqmatrixlambda ] ) and ( [ eqn : m ] ) the periodized green s function can be rewritten in matrix notation @xmath95 where the vector @xmath96 and its adjoint @xmath97 contain @xmath37 plane waves @xmath98 there exists as well an alternative periodization prescription where the self - energy @xmath13 is periodized.@xcite this self - energy periodization should prevent spurious gaps , which arise in the spectral function . however , at least for fermion systems , this procedure yields spurious metallic bands in the mott phase for arbitrarily large @xmath3 . since we do not observe any spurious gaps in the spectral function of the 2d bh model we use the periodization on the green s function defined in eq.([eq : num : cpteqfourierfull ] ) . with the wave - vector resolved green s function of the total system @xmath99 we are able to calculate the single - particle spectral function @xmath100 the density of states @xmath101 and the momentum distribution @xmath102 the frequency integration can be evaluated directly by means of the @xmath0-matrix formalism , which yields a sum of the residues of the green s function , see eq.([eqn : greenfunctionmatrix ] ) , corresponding to negative poles @xmath103 , @xmath104 the bh model exhibits a quantum phase transition from a mott to a superfluid phase when the ratio between the hopping strength and the on - site repulsion @xmath105 is increased or when particles are added to or removed from the system . the mott phase is characterized by an integer particle density , a gap in the spectral function and zero compressibility.@xcite , scaledwidth=48.0% ] the first two mott lobes of the 2d bh model obtained by means of vca are shown in fig.[fig : pd ] . we used the chemical potential @xmath106 as variational parameter , which ensures a correct particle density of the total system.@xcite in contrast to the one - dimensional results @xcite the mott lobes of the 2d bh model are round shaped . the gray shaded area in fig.[fig : pd ] presents the phase boundaries calculated within the process chain approach by n. teichmann _ et al . _ in refs . and , which are basically identical to the qmc results by b. capogrosso - sansone _ et al . _ , see ref . . the agreement is quite good for small hopping . however , vca seems to overestimate the critical value of the hopping @xmath107 , which determines the tip of the mott lobe . for the critical hopping of the first mott lobe , we obtain approximately @xmath108 and for the second one @xmath109 . latest process chain approach , @xcite qmc ( ref . ) and strong - coupling perturbation theory @xcite results yield @xmath110 and @xmath111 for the critical parameter of the first and second mott lobe , respectively . the spectral functions @xmath112 and the densities of states @xmath113 for parameters of the first mott lobe are shown in fig.[fig : spectrallobe1 ] . , left column , and density of states @xmath113 , right column , in the first mott lobe for the parameters ( a ) @xmath114 , @xmath115 , ( b ) @xmath116 , @xmath117 and ( c ) @xmath118 , @xmath119 . the captions of the subfigures refer to the marks in fig.[fig : pd].[fig : spectrallobe1],scaledwidth=48.0% ] the spectral function is displayed on the conventional path around the brillouin zone @xmath120 over @xmath121 to @xmath122 and back to @xmath123 , and we use an artificial imaginary - frequency broadening @xmath124 . a peculiarity of bosonic systems is that the hole band of the spectral function has negative spectral weight whereas the particle band has positive spectral weight . this follows from the definition of the bosonic green s function which has a negative sign in front of the hole term , see eq.([num : gf ] ) . in the figures we always plot the absolute value of the spectral function . the local density of states is defined as a wave - vector summation of @xmath112 , see eq.([eq : spe : dos ] ) . therefore we observe a negative peak in the density of states , which corresponds to the hole band of the spectral function . for bosonic green s functions the density of states is not a probability distribution , as it contains negative values . taking the absolute value would yield an all positive density of states , however , it would not be normed and is thus no probability distribution either . for small hopping , the gap in the spectral function is large and the bands are rather flat , i.e. , the width of the bands is small , see fig.[fig : spectrallobe1 ] . the corresponding density of states contains two well - separated peaks . for increasing hopping , the gap of the spectral function is decreasing and the width of the bands is increasing . pursuant to the spectral function , the peaks in the density of states become broader for increasing hopping . the intensity of the two bands is almost constant for small hopping independent of the wave vector @xmath42 , whereas for large hopping a large intensity can be observed at @xmath125 . the boundaries of the mott lobes correspond to the chemical potential of the state with one additional particle ( hole ) , which is obtained directly from the single - particle ( single - hole ) minimum excitation energy . for this reason , we evaluate the phase diagram in fig . [ fig : pd ] by taking the minimal gap of the spectral function for each @xmath105 , which always occurs at @xmath125 . the spectral functions and densities of states in the second mott lobe corresponding to the marks iv and v in fig.[fig : pd ] are shown in fig.[fig : spectrallobe2 ] . , left column , and density of states @xmath113 , right column , in the second mott lobe for the parameters ( a ) @xmath126 , @xmath127 and ( b ) @xmath128 , @xmath129 . the captions of the subfigures refer to the marks in fig.[fig : pd].[fig : spectrallobe2],scaledwidth=48.0% ] qualitatively they are very similar to the spectral functions and densities of states in the first mott lobe . particularly , the intensity distribution of the bands seem to be strongly related . yet the peaks of the density of states are larger due to the twice as large particle density within the second mott lobe and thus the absolute value of the spectral weight in the second mott lobe is larger than the one in the first mott lobe . the momentum distribution @xmath130 corresponding to the spectral functions in the first and second mott lobe are shown in fig.[fig : nofk ] . in ( a ) the first mott lobe and ( b ) the second mott lobe . the roman numerals in the legends refer to the parameters marked in fig.[fig : pd].[fig : nofk],scaledwidth=48.0% ] the particle density in the first mott lobe is one and thus @xmath130 is centered around one in fig.[fig : nofk ] . for the second mott lobe @xmath130 is centered around two , see fig.[fig : nofk ] . the particle density @xmath130 is extremely flat for small hopping whereas it is peaked at @xmath125 for large hopping , which is already a precursor for the bose - einstein condensation where all particles condense in the @xmath125 state . this behavior directly reflects the intensity distribution of the bands in the spectral function . there is excellent quantitative agreement between our vca results for the momentum distribution and results obtained by means of qmc and a strong - coupling perturbation theory with scaling ansatz,@xcite see fig.[fig : nofkcomparison ] . obtained by means of vca and a strong - coupling perturbation theory with scaling ansatz.@xcite the data identified with the letters sc correspond to the strong - coupling results . the relative deviations between vca results and strong - coupling results with scaling ansatz are shown in ( b ) . the roman numerals in the legends refer to the parameters marked in fig.[fig : pd].[fig : nofkcomparison],scaledwidth=48.0% ] we compare the momentum distributions for the parameters i ( @xmath114 ) and ii ( @xmath116 ) , and observe that the relative deviations between our vca results and an approach obtained by combining strong - coupling perturbation theory with a scaling ansatz @xcite are almost zero for small hopping @xmath114 and less than one percent for medium hopping @xmath116 . this latter methods is certainly more accurate than vca in the evaluation of the momentum distribution . however , it should be mentioned that the information about the critical point ( critical exponents and critical hopping strength @xmath107 ) have to be inserted `` by hand , '' in order to optimize the results . this information , in turn , must be extracted , e. g. , from a qmc calculation . on the other hand , our vca results are obtained directly without the need to introduce external parameters . in the present paper , we presented and discussed results obtained within the variational cluster approach for the spectral properties of the two - dimensional bose - hubbard hamiltonian . this is a minimal model to describe bosonic ultracold atoms confined in optical lattices,@xcite and it undergoes a quantum phase transition from a mott to a superfluid phase depending on the chemical potential @xmath4 , and the ratio between the hopping strength and the on - site repulsion @xmath105 . in particular , we determined the first two mott lobes of the phase diagram and found reasonable agreement with essentially exact results from qmc simulations and from the process chain approach . in particular , the variational cluster approach yields very good results for the phase boundaries apart from the region close to the lobe tip . here , strong - coupling expansions and qmc calculations are , clearly , much more accurate . yet it should be emphasized that the computational effort is considerably lower for vca than for qmc . furthermore , we evaluated spectral functions in the first and second mott lobe . an important aspect of vca is that the green s function of the system is obtained directly in the real frequency domain , which allows for a direct calculation of the spectral function . on the other hand , qmc quite generally provides correlation functions in imaginary time . imaginary - time correlation functions have to be analytically continued to real frequencies , which is a very ill - conditioned problem , as the data contain statistical errors . in qmc this analytical continuation is best carried out by means of the maximum entropy method . a very accurate dispersion ( without spectral weight ) has been also obtained by a strong - coupling expansion.@xcite the intensity distribution of the spectral weight is similar for the spectral functions of both mott lobes , leading to an evenly distributed spectral weight for small hopping strengths and to a distribution sharply peaked at @xmath131 for large hopping strengths . the latter indicates a precursor to the bose - einstein condensation occurring above a certain critical hopping . we also evaluated the densities of states and momentum distributions corresponding to the calculated spectral functions . we compared our vca results for the momentum distribution with strong - coupling perturbation - theory results , where a scaling ansatz has been used , and found excellent quantitative agreement . finally , as a technical point , we extended the @xmath0-matrix formalism to deal with bosonic green s functions , which , in contrast to the fermionic case , produces a non - symmetric eigenvalue problem . we are grateful to n. teichmann for providing the process chain approach data of the phase diagram used in fig.[fig : pd ] . we thank j. k. freericks for sending us the self - consistently solved strong - coupling results with scaling ansatz shown in fig.[fig : nofkcomparison ] . we made use of parts of the alps library ( ref . ) for the implementation of lattice geometries and for parameter parsing . m.k . wants to thank p. pippan for fruitful discussions . we acknowledge partial financial support from the austrian science fund ( fwf ) under the doctoral program `` numerical simulations in technical sciences '' w1208-n18 ( m.k . and w.v.d.l . ) and under project no . p18551-n16 ( e.a . ) .
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spectral properties of the two - dimensional bose - hubbard model , which emulates ultracold gases of atoms confined in optical lattices , are investigated by means of the variational cluster approach .
the phase boundary of the quantum phase transition from mott phase to superfluid phase is calculated and compared to recent work . moreover the single - particle spectral functions in both the first and the second mott lobe are presented and the corresponding densities of states and momentum distributions are evaluated . a qualitatively similar intensity distribution of the spectral weight can be observed for spectral functions in the first and the second mott lobe .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
cluster formation in complex fluids is a topic that has attracted considerable attention recently.@xcite the general belief is that a short - range attraction in the pair interaction potential is necessary to initiate aggregation and a long - range repulsive tail in order to limit cluster growth and prevent phase separation . an alternative scenario for cluster formation pertains to systems whose constituent particles interact by means of purely repulsive potentials . cluster formation in this case is counterintuitive at first sight : why should particles form clusters if there is no attraction acting between them ? the answer lies in an additional property of the effective repulsion , namely that of being _ bounded _ , thus allowing full particle overlaps . though surprising and seemingly unphysical at first , bounded interactions are fully legitimate and natural as effective potentials@xcite between polymeric macromolecular aggregates of low internal monomer concentration , such as polymers,@xcite dendrimers,@xcite microgels,@xcite or coarse - grained block copolymers.@xcite the growing interest in this type of effective interactions is also underlined by the recent mathematical proof of the existence of crystalline ground states for such systems.@xcite cluster formation in the fluid _ and _ in the crystal phases was explicitly seen in the system of penetrable spheres , following early simulation results@xcite and subsequent cell - model calculations.@xcite cluster formation was attributed there to the tendency of particles to create free space by forming full overlaps . the conditions under which ultrasoft and purely repulsive particles form clusters have been conjectured a few years ago@xcite and explicitly confirmed by computer simulation very recently.@xcite the key lies in the behavior of the fourier transform of the effective interaction potential : for clusters to form , it must contain negative parts , forming thus the class of @xmath1-interactions . the complementary class of potentials with purely nonnegative fourier transforms , @xmath2 , does not lead to clustering but to remelting at high densities.@xcite an intriguing feature of the crystals formed by @xmath1-systems is the independence of the lattice constant on density,@xcite a feature that reflects the flexibility of soft matter systems in achieving forms of self - organization unknown to atomic ones.@xcite the same characteristic has recently been seen also in slightly modified models that contain a short - range hard core.@xcite in this work , we provide an analytical solution of the crystallization problem and of the properties of the ensuing solids within the framework of an accurate density functional approach . we explicitly demonstrate the persistence of a single length scale at all densities and for all members of the @xmath1-class and offer thus broad physical insight into the mechanisms driving the stability of the clustered crystals . we further establish some universal structural properties of all @xmath1-systems both in the fluid and in the solid state , justifying the use of the mean - field density functional on which this work rests . we make a connection between our results and the harmonic theory of solids in the einstein - approximation . finally , we establish a connection with suitably - defined infinite dimensional models of hard spheres . the rest of this paper is organized as follows : in sec . [ dft : sec ] we derive an accurate density functional by starting with the uniform phase and establishing the behavior of the direct correlation functions of the fluid with density and temperature . based on this density functional , we perform an analytical calculation of the freezing characteristics of the @xmath1-systems by employing a weak approximation in sec . [ analytical : sec ] . the accuracy of this approximation is successfully tested against full numerical minimization of the functional in sec . [ compare : sec ] . in sec . [ harmonic : sec ] the equivalence between the density functional and the theory of harmonic crystals is demonstrated , whereas in sec . [ invpower : sec ] a connection is made with inverse - power potentials . finally , in sec . [ summary : sec ] we summarize and draw our conclusions . some intermediate , technical results that would interrupt the flow of the text are relegated in the appendix . in this work , we focus our interest on systems of spherosymmetric particles interacting by means of _ bounded _ pair interactions @xmath3 of the form : @xmath4 with an energy scale @xmath5 and a length scale @xmath6 , and which fulfill the ruelle conditions for stability.@xcite in eq . ( [ gen_poten : eq ] ) above , @xmath7 is some dimensionless function of a dimensionless variable and @xmath8 denotes the distance between the spherosymmetric particles . in the context of soft matter physics , @xmath3 is an effective potential between , e.g. , the centers of mass of macromolecular entities , such as polymer chains or dendrimers.@xcite as the centers of mass of the aggregates can fully overlap without this incurring an infinitely prohibitive cost in ( free ) energy , the condition of boundedness is fulfilled : @xmath9 with some constant @xmath10 . the interaction range is set by @xmath6 , typically the physical size ( e.g. , the gyration radius ) of the macromolecular aggregates that feature @xmath3 as their effective interaction . in addition to being bounded , the second requirement to be fulfilled by the function @xmath7 is that it decay sufficiently fast to zero as @xmath11 , so that its fourier transform @xmath12 exists . in three spatial dimensions , we have @xmath13 and , correspondingly , @xmath14 for the fourier transform @xmath15 of the potential , evaluated at wavenumber @xmath16 . our focus in this work is on systems for which @xmath15 is oscillatory , i.e. , @xmath3 features positive and negative fourier components , classifying it as a @xmath1-potential.@xcite though this work is general , for the purposes of demonstration of our results , we consider a particular realization of @xmath1-potentials , namely the generalized exponential model of exponent @xmath17 , ( gem-@xmath17 ) : @xmath18 , \label{gemn : eq}\ ] ] with @xmath19 . it can be shown that all members of the gem-@xmath17 family with @xmath20 belong to the @xmath1-class , see appendix a. a short account of the freezing and clustering behavior of the gem-4 model has been recently published.@xcite another prominent member of the family is the @xmath21 model , which corresponds to penetrable spheres@xcite with a finite overlap energy penalty @xmath5 . indeed , the explicitly calculated phase behavior of these two show strong resemblances , with the phase diagram of both being dominated by the phenomenon of formation of clusters of overlapping particles and the subsequent ordering of the same in periodic crystalline arrangements.@xcite in this work , we provide a generic , accurate , and analytically tractable theory of inhomogeneous phases of @xmath1-systems . let us start from the simpler system of a homogeneous fluid , consisting of @xmath22 spherosymmetric particles enclosed in a macroscopic volume @xmath23 . the structure and thermodynamics of the system are determined by the density @xmath24 and the absolute temperature @xmath25 or , better , their dimensionless counterparts : @xmath26 and @xmath27 with boltzmann s constant @xmath28 . as usual , we also introduce for future convenience the inverse temperature @xmath29 . we seek for appropriate and accurate closures to the ornstein - zernike relation@xcite @xmath30 connecting the total correlation function @xmath31 to the direct correlation function @xmath32 of the uniform fluid . one possibility is offered by the hypernetted chain closure ( hnc ) that reads as@xcite @xmath33 - 1 . \label{hnc : eq}\ ] ] an additional closure , the mean - field approximation ( mfa ) , was also employed and will be discussed later . our solution by means of approximate closures was accompanied by extensive @xmath34-monte carlo ( mc ) simulations . we measured the radial distribution function @xmath35 as well as the structure factor @xmath36 , where @xmath37 is the fourier transform of @xmath31 , to provide an assessment of the accuracy of the approximate theories . we typically simulated ensembles of up to @xmath38 particles for a total of @xmath39 monte carlo steps . measurements were taken in every tenth step after equilibration . of the gem-4 model as obtained by monte carlo simulation ( points ) , the hnc - closure ( solid lines ) and the mfa ( dashed lines ) , at various temperatures and densities . for clarity , the curves on every panel have been shifted upwards by certain amounts , which are given below in square brackets , following the value indicating the density @xmath40 . ( a ) @xmath41 and densities , from bottom to top : @xmath42 [ 0 ] ; @xmath43 [ 0.2 ] ; @xmath44 [ 0.4 ] ; @xmath45 [ 0.6 ] ; @xmath46 [ 0.8 ] . ( b ) @xmath47 and densities , from bottom to top : @xmath43 [ 0 ] ; @xmath45 [ 0.2 ] ; @xmath48 [ 0.4 ] ; @xmath49 [ 0.6 ] ; @xmath50 [ 0.8 ] . ( c ) @xmath51 and densities , from bottom to top : @xmath45 [ 0 ] ; @xmath49 [ 0.2 ] ; @xmath52 [ 0.4 ] ; @xmath53 [ 0.6 ] ; @xmath54 [ 0.8].,width=309 ] of the gem-4 model as obtained by monte carlo simulation ( points ) , the hnc - closure ( solid lines ) and the mfa ( dashed lines ) , at various temperatures and densities . for clarity , the curves on every panel have been shifted upwards by certain amounts , which are given below in square brackets , following the value indicating the density @xmath40 . ( a ) @xmath41 and densities , from bottom to top : @xmath42 [ 0 ] ; @xmath43 [ 0.2 ] ; @xmath44 [ 0.4 ] ; @xmath45 [ 0.6 ] ; @xmath46 [ 0.8 ] . ( b ) @xmath47 and densities , from bottom to top : @xmath43 [ 0 ] ; @xmath45 [ 0.2 ] ; @xmath48 [ 0.4 ] ; @xmath49 [ 0.6 ] ; @xmath50 [ 0.8 ] . ( c ) @xmath51 and densities , from bottom to top : @xmath45 [ 0 ] ; @xmath49 [ 0.2 ] ; @xmath52 [ 0.4 ] ; @xmath53 [ 0.6 ] ; @xmath54 [ 0.8].,width=309 ] of the gem-4 model as obtained by monte carlo simulation ( points ) , the hnc - closure ( solid lines ) and the mfa ( dashed lines ) , at various temperatures and densities . for clarity , the curves on every panel have been shifted upwards by certain amounts , which are given below in square brackets , following the value indicating the density @xmath40 . ( a ) @xmath41 and densities , from bottom to top : @xmath42 [ 0 ] ; @xmath43 [ 0.2 ] ; @xmath44 [ 0.4 ] ; @xmath45 [ 0.6 ] ; @xmath46 [ 0.8 ] . ( b ) @xmath47 and densities , from bottom to top : @xmath43 [ 0 ] ; @xmath45 [ 0.2 ] ; @xmath48 [ 0.4 ] ; @xmath49 [ 0.6 ] ; @xmath50 [ 0.8 ] . ( c ) @xmath51 and densities , from bottom to top : @xmath45 [ 0 ] ; @xmath49 [ 0.2 ] ; @xmath52 [ 0.4 ] ; @xmath53 [ 0.6 ] ; @xmath54 [ 0.8].,width=309 ] in fig . [ gofr : fig ] we show comparisons for the function @xmath55 as obtained from the mc simulations and from the hnc closure for a variety of temperatures and densities . it can be seen that agreement between the two is obtained , to a degree of quality that is excellent . tiny deviations between the hnc and mc results appear only at the highest density and for a small region around @xmath56 for low temperatures , @xmath41 . otherwise , the system at hand is described by the hnc with an extremely high accuracy and for a very broad range of temperatures and densities . we note that , although in fig . [ gofr : fig ] we restrict ourselves to temperatures @xmath57 , the quality of the hnc remains unaffected also at higher temperatures.@xcite in attempting to gain some insight into the remarkable ability of the hnc to describe the fluid structure at such a high level of accuracy , it is useful to recast this closure in density - functional language . following the famous percus idea,@xcite the quantity @xmath58 can be identified with the nonuniform density @xmath59 of an inhomogeneous fluid that results when a single particle is kept fixed at the origin , exerting an ` external ' potential @xmath60 on the rest of the system . following standard procedures from density functional theory , we find that the sought - for density profile @xmath59 is given by @xmath61}{\delta \rho(r)}\right\ } , \label{profile_dft : eq}\ ] ] where @xmath62 is the thermal de broglie wavelength and @xmath63 the chemical potential associated with average density @xmath64 and temperature @xmath25 . moreover , @xmath65 $ ] is the intrinsic excess free energy , a _ unique _ functional of the density @xmath59 . as such , @xmath65 $ ] can be expanded in a functional taylor series around its value for a uniform liquid of some ( arbitrary ) reference density @xmath66 . for the problem at hand , @xmath67 is a natural choice and we obtain@xcite @xmath68 = \beta f_{\rm ex}(\rho ) & - & \sum_{n=1}^{\infty}\frac{1}{n!}\int\int\cdots\int { \rm d}^3r_1{\rm d}^3r_2\ldots{\rm d}^3r_n \\ \nonumber & \times & c_0^{(n)}({\bf r}_1,{\bf r}_2,\ldots,{\bf r}_n;\rho ) \\ & \times & \delta\rho({\bf r}_1)\delta\rho({\bf r}_2)\ldots\delta\rho({\bf r}_n ) , \label{taylor_expand : eq}\end{aligned}\ ] ] where @xmath69 denotes the excess free energy of the _ homogeneous _ fluid , as opposed to that of the inhomogeneous fluid , @xmath65 $ ] , and @xmath70 the basis of the expansion given by eq . ( [ taylor_expand : eq ] ) above is the fact that @xmath65 $ ] is the generating functional for the hierarchy of the direct correlation functions ( dcf s ) @xmath71 . in particular , @xmath72 is the @xmath73-th functional derivative of the excess free energy with respect to the density field , evaluated at the uniform density @xmath64:@xcite @xmath74 } { \delta\rho({\bf r}_1)\delta\rho({\bf r}_2)\ldots \delta\rho({\bf r}_n)}\bigg|_{\rho ( . ) = \rho}. \label{cofrn : eq}\ ] ] as the functional derivatives are evaluated for a uniform system , translational and rotational invariance reduce the number of variables on which the @xmath73-th order dcf s @xmath71 depend ; in fact , @xmath75 is a position - independent constant and equals @xmath76 , where @xmath77 is the excess chemical potential.@xcite similarly , @xmath78 is simply the ornstein - zernike direct correlation function @xmath79 entering in eqs . ( [ oz : eq ] ) and ( [ hnc : eq ] ) above . the hnc closure is equivalent to jointly solving eqs . ( [ oz : eq ] ) and ( [ profile_dft : eq ] ) by employing an approximate excess free energy functional @xmath65 $ ] , arising by a truncation of the expansion of eq . ( [ taylor_expand : eq ] ) at @xmath80 , i.e. , discarding all terms with @xmath81 ; this is the famous ramakrishnan - yussouff second - order approximation.@xcite indeed , the so - called bridge function @xmath82 can be written as an expansion over integrals involving as kernels all the @xmath71 with @xmath81 and the hnc amounts to setting the bridge function equal to zero.@xcite whereas in many cases , such as the one - component plasma,@xcite and other systems with long - range interactions , the hnc is simply an adequate or , at best , a very good approximation , for the case at hand the degree of agreement between the hnc and simulation is indeed extremely high . what is particularly important is that the accuracy of the hnc persists for a very wide range of densities , at all temperatures considered . this fact has far - reaching consequences , because it means that the corresponding profiles @xmath83 that enter the multiple integrals in eq . ( [ taylor_expand : eq ] ) vary enormously depending on the uniform density considered . thus , it is tempting to conjecture that for the systems under consideration ( soft , penetrable particles at @xmath84 ) , not simply the integrals with @xmath81 vanish but rather the kernels themselves . in other words , @xmath85 calculated in the hnc and its mfa - approximation , @xmath86 , for a gem-4 model , at various densities indicated in the legends . the insets show the mfa approximation for the same quantity , which is density - independent . each panel corresponds to a different temperature : ( a ) @xmath41 ; ( b ) @xmath47 ; ( c ) @xmath51 . these are exactly the same parameter combinations as the ones for which @xmath55 is shown in fig . [ gofr : fig].,width=309 ] calculated in the hnc and its mfa - approximation , @xmath86 , for a gem-4 model , at various densities indicated in the legends . the insets show the mfa approximation for the same quantity , which is density - independent . each panel corresponds to a different temperature : ( a ) @xmath41 ; ( b ) @xmath47 ; ( c ) @xmath51 . these are exactly the same parameter combinations as the ones for which @xmath55 is shown in fig . [ gofr : fig].,width=309 ] calculated in the hnc and its mfa - approximation , @xmath86 , for a gem-4 model , at various densities indicated in the legends . the insets show the mfa approximation for the same quantity , which is density - independent . each panel corresponds to a different temperature : ( a ) @xmath41 ; ( b ) @xmath47 ; ( c ) @xmath51 . these are exactly the same parameter combinations as the ones for which @xmath55 is shown in fig . [ gofr : fig].,width=309 ] the behavior of the higher - order dcf s is related to the density - derivative of lower - order ones through certain sum rules , to be discussed below . hence , it is pertinent to examine the density dependence of the dcf @xmath32 of the hnc . in fig . [ cofr : fig ] we show the difference between the dcf @xmath87 and the mean - field approximation ( mfa ) to the same quantity : @xmath88 eq . ( [ cmfa : eq ] ) above is _ meaningless _ if the pair potential @xmath3 diverges as @xmath89 , because @xmath32 has to remain finite at all @xmath8 , as follows from exact diagrammatic expansions of the same.@xcite in fact , the form @xmath90 denotes the large-@xmath8 asymptotic behavior of @xmath32 . in our case , however , where @xmath3 lacks a hard core , the mfa - form for @xmath32 can not be a priori rejected on fundamental grounds ; the quantity @xmath91 remains bounded as @xmath92 . in fact , as can be seen in fig . [ cofr : fig ] , the deviations between the mfa and the hnc - closure are very small for @xmath93 . in addition , the differences between @xmath87 and @xmath94 drop , both in absolute and in relative terms , as temperature grows , see the trend in figs . [ cofr : fig](a)-(c ) . at fixed temperature , the evolution of the difference with density is nonmonotonic : it first drops as density grows and then it starts growing again at the highest densities shown in the three panels of fig . [ cofr : fig ] . motivated by these findings , we employ now a second closure , namely the above - mentioned mfa , eq . ( [ cmfa : eq ] ) . introducing the latter into the ornstein - zernike relation , eq . ( [ oz : eq ] ) , we obtain the mfa - results for the radial distribution function @xmath55 that are also shown in fig . [ gofr : fig ] with dashed lines . before proceeding to a critical comparison between the @xmath55 obtained from the two closures , it is useful to make a clear connection between the mfa and the hnc . as mentioned above , the members of the sequence of the @xmath73-th order direct correlation functions are not independent from one another ; rather , they are constrained to satisfy a corresponding hierarchy of sum rules , namely:@xcite @xmath95 in particular , for @xmath96 , we have @xmath97 where we have used the translational and rotational invariance of the fluid phase to reduce the number of arguments of the dcf s and we show explicitly the generic dependence of @xmath32 on @xmath64 . in the mfa , one assumes @xmath86 , with the immediate consequence @xmath98 eqs . ( [ sumrule : eq ] ) and ( [ deriv : eq ] ) imply that the integral of @xmath99 with respect to any of its arguments must _ vanish _ for _ arbitrary _ density . as @xmath99 has a complex dependence on its arguments , this is a strong indication that @xmath99 itself vanishes . in fact , both for the barrat - hansen - pastore factorization approximation for this quantity@xcite and for the alternative , denton - ashcroft @xmath16-space factorization of the same,@xcite the vanishing of the right - hand side of eq . ( [ sumrule : eq ] ) implies that @xmath100 . now , if @xmath99 vanishes , so does also its density derivative and use of sum rule ( [ inductive : eq ] ) for @xmath101 implies @xmath102 . successive use of the same for higher @xmath73-values leads then to the conclusion that in the mfa : @xmath103 we can now see that the accuracy of the hnc stems from the fact that for these systems we can write @xmath104 where @xmath105 is a small function at all densities @xmath64 , offering concomitantly a very small , and _ the only _ , contribution to the quantity @xmath106 . this implies that @xmath99 itself is negligible by means of eq . ( [ sumrule : eq ] ) . repeated use of eq . ( [ inductive : eq ] ) leads then to eqs . ( [ approx : eq ] ) and shows that the contributions from the @xmath107 terms , that are ignored in the hnc , are indeed negligible . the hnc is , thus , very accurate , due to the strong mean - field character of the fluids at hand.@xcite the deviations between the hnc and the mfa come through the function @xmath105 above . let us now return to the discussion of the results for @xmath55 and @xmath32 and the relative quality of the two closures at various thermodynamic points . referring first to fig . [ gofr : fig](c ) , we see that at @xmath51 both the hnc and the mfa perform equally well . the agreement between the two ( and between the mfa and mc ) worsens somewhat at @xmath47 and even more at @xmath41 . the mfa is , thus , an approximation valid for @xmath84 , in agreement with previous results.@xcite the reason lies in the accuracy of the low - density limit of the mfa . in general , as @xmath108 , @xmath32 tends to the mayer function @xmath109 - 1 $ ] . if @xmath84 , one may expand the exponential to linear order and obtain @xmath110 , so that the mfa can be fulfilled . in the hnc , it is implicitly assumed that all dcf s with @xmath81 vanish . in the mfa , this is also the case . the two closures differ in one important point , though : in the hnc , the second - order direct correlation function is _ not _ prescribed but rather _ determined _ , so that both the ` test - particle equation ' , eq . ( [ profile_dft : eq ] ) , and the ornstein - zernike relation , eq . ( [ oz : eq ] ) , are fulfilled . in the mfa , it is a priori assumed that @xmath86 , which is introduced into the ornstein - zernike relation and thus @xmath31 is determined . this is one particular way of obtaining @xmath55 in the mfa , called the ornstein - zernike route . alternatively , one could follow the test - particle route in solving eq . ( [ profile_dft : eq ] ) in conjunction with eq . ( [ taylor_expand : eq ] ) and the mfa - approximation , eq . ( [ cmfa : eq ] ) . in this case , the resulting expression for the total correlation function @xmath31 in the mfa reads as @xmath111 - 1 , \label{testpart : eq}\ ] ] with @xmath112 denoting the convolution . previous studies with ultrasoft systems have shown that the test - particle @xmath31 from the mfa is closer to the hnc - result than the mfa result obtained from the ornstein - zernike route.@xcite the discrepancy between the two is a measure of the approximate character of the mfa ; were the theory to be exact , all routes would give the same result . as a way to quantify the approximations involved in the mfa , let us attempt to _ impose _ consistency between the test - particle and ornstein - zernike routes . since @xmath86 in this closure , the exponent in eq . ( [ testpart : eq ] ) above is just the right - hand side of the ornstein - zernike relation , eq . ( [ oz : eq ] ) . thus , if we insist that the latter is fulfilled , we obtain the constraint @xmath113 - 1 , \label{hofr : eq}\ ] ] which is strictly satisfied only for @xmath114 . however , as long as @xmath115 , one can linearize the exponential and an identity follows ; the internal inconsistency of the mfa is of quadratic order in @xmath31 and it follows that the mfa provides an accurate closure for the systems at hand , as long as @xmath116 remains small . this explains the deviations between mfa and mc seen at small @xmath8 for the highest density at @xmath51 , fig . [ gofr : fig](c ) . the same effect can also be seen in fig . [ cofr : fig](c ) as a growth of the discrepancy between @xmath87 and @xmath94 at small @xmath8-values for the highest density shown . in absolute terms , however , this discrepancy remains very small . note also that discrepancies at small @xmath8-values become strongly suppressed upon taking a fourier transform , due to the additional geometrical @xmath117-factor involved in the three - dimensional integration . it can therefore be seen that the mfa and the hnc are closely related to one another : the hnc is so successful due to the strong mean - field character of the systems under consideration . this fact has also been established and extensively discussed for the case of the gaussian model,@xcite i.e. , the @xmath118 member of the gem-@xmath17 class . once more , the hnc and the mfa there are very accurate for high densities and/or temperatures , where @xmath119 and the system s behavior develops similarities with an ` incompressible ideal gas',@xcite in full agreement with the remarks presented above . subsequently , the mfa and hnc closures have been also successfully applied to the study of structure and thermodynamics of binary soft mixtures.@xcite of the gem-4 model at temperature @xmath47 and various densities , indicated below . for clarity , the curves have been shifted vertically by amounts shown in the square brackets , following the numbers that indicate the values of the density @xmath40 . from bottom to top : @xmath43 [ 0 ] ; @xmath45 [ 0.5 ] ; @xmath48 [ 1.0 ] ; @xmath49 [ 1.5 ] ; @xmath50 [ 2.0 ] . the points are results from monte carlo simulations and the dashed lines from the hnc . as the hnc- and mfa - curves run very close to each other , we show the mfa - result by the solid curve only for the highest density , @xmath50.,width=321 ] a crucial difference between the gaussian model , which belongs to the @xmath2-class , and members of the @xmath1-class , which are the subject of the present work , lies in the consequences of the mfa - closure on the structure factor @xmath120 of the system . since @xmath86 , the ornstein - zernike relation leads to the expression @xmath121 whereas for @xmath2-potentials @xmath120 is devoid of pronounced peaks that exceed the asymptotic value @xmath122 , for @xmath1-systems a local maximum of @xmath120 appears at the value @xmath123 for which @xmath15 attains its negative minimum . in fig . [ sofk : fig ] we show representative results for the system at hand , where it can also be seen that the hnc and the mfa yield practically indistinguishable results . in full agreement with the mc simulations , the location of the main peak of @xmath120 is _ density - independent _ , a feature unknown for usual fluids , having its origin in the fact that @xmath32 itself is density independent . associated with this is the development of a @xmath124-line,@xcite also known as _ kirkwood instability_,@xcite on which the denominator in eq . ( [ sofk : eq ] ) vanishes at @xmath123 and thus @xmath125 . the locus of points @xmath126 on the density - temperature plane for which this divergence takes place is , evidently , given by @xmath127 in the region @xmath128 ( equivalently : @xmath129 ) on the @xmath130-plane , the mfa predicts that the fluid is absolutely unstable , since the structure factor there has multiple divergences and also develops negative parts . this holds _ only _ for @xmath1-systems ; for @xmath2-ones the very same line of argumentation leads to the opposite conclusion , namely that the fluid is the phase of stability at high densities and/or temperatures . the latter conclusion has already been reached by stillinger and coworkers@xcite in their pioneering work of the gaussian model in the mid-1970 s , and explicitly confirmed by extensive theory and computer simulations many years later.@xcite however , stillinger s original argument was based on duality relations that are strictly fulfilled only for the gaussian model , whereas the mfa - arguments are quite general . having established the validity of the mfa for vast domains in the phase diagram of the systems under consideration as far as the _ uniform _ fluids are concerned , we now turn our attention to nonuniform ones . apart from an obvious general interest in the properties of nonuniform fluids , the necessity to consider deviations from homogeneity in the density for @xmath1-models is dictated by the @xmath124-instability mentioned above : the theory of the uniform fluid contains its own breakdown , thus the system has to undergo a phase transformation to a phase with a spontaneously broken translational symmetry . whether this transformation takes place _ exactly _ on the instability line or already at densities @xmath131 ( or temperatures @xmath132 ) and which is the stable phase are some of the questions that have to be addressed . density - functional theory of inhomogeneous systems is the appropriate theoretical tool in this direction . let us consider a path @xmath133 in the space of density functions , which is characterized by a single parameter @xmath134 ; this path starts at some reference density @xmath135 and terminates at another density @xmath136 . the uniqueness of the excess free energy functional and its dependence on the inhomogeneous density field @xmath136 allow us to integrate @xmath137/\partial\chi$ ] along this path , obtaining @xmath65 $ ] , provided that @xmath138 $ ] is known . a convenient parametrization reads as @xmath139 $ ] , with @xmath140 corresponding to @xmath135 and @xmath141 to @xmath136 . the excess free energy of the final state can be expressed as@xcite @xmath68 & = & \beta f_{\rm ex}[\rho_{\rm r } ] \\ & - & \int_0^{1 } { \rm d}\chi \int{\rm d}^3r c^{(1)}({\bf r};[\rho_{\chi } ] ) \delta\rho({\bf r } ) , \label{chi_path : eq}\end{aligned}\ ] ] where @xmath142)$ ] denotes the first functional derivative of the quantity @xmath143 $ ] evaluated at the inhomogeneous density @xmath133 , and @xmath144 . since @xmath142)$ ] is in its own turn a unique functional of the density profile , repeated use of the same argument leads to a functional taylor expansion of the excess free energy around that of a reference system , an expansion that extends to infinite order . for the _ particular _ choice of a _ uniform _ reference system , @xmath145 , we obtain then the taylor series of eq . ( [ taylor_expand : eq ] ) . in general , however , the reference system does not have to have the same average density as the final one , hence the uniform density @xmath64 there must be replaced by a more general quantity @xmath66 . the usefulness of eq . ( [ taylor_expand : eq ] ) in calculating the free energies of extremely nonuniform phases , such as crystals , is limited both on principal and on practical grounds . fundamentally , there is _ no small parameter _ guiding such an expansion , since the differences between the nonuniform density of a crystal and that of a fluid are enormous ; the former has extreme variations between lattice- and interstitial regions . hence , the very convergence of the series is in doubt.@xcite in practice , the direct correlation functions for @xmath146 are very cumbersome to calculate@xcite and those for @xmath147 are practically unknown.@xcite the solution is either to arbitrarily terminate the series at second order@xcite or to seek for nonperturbative functionals.@xcite in our case , however , things are different because , for the systems we consider , we have given evidence that the dcf s of order @xmath107 are extremely small and we take them at this point as vanishing . then , the functional taylor expansion of the free energy @xmath65 $ ] terminates ( to the extent that the approximation holds ) at second order . the taylor series becomes a finite sum and convergence is not an issue any more . let us , accordingly , expand @xmath65 $ ] around an _ arbitrary _ , homogeneous reference fluid of density @xmath148 , taking into account that the volume @xmath23 is fixed but the system with density @xmath136 contains @xmath22 particles , whereas the reference fluid contains @xmath149 particles and , in general , @xmath150 : @xmath68 & = & \beta f_{\rm ex}(\rho_0 ) - c_0^{(1)}(\rho_0)\int{\rm d}^3r[\rho({\bf r } ) - \rho_0 ] \\ \nonumber & - & \frac{1}{2}\int\int{\rm d}^3r{\rm d}^3r ' c_0^{(2)}(|{\bf r}-{\bf r'}|;\rho_0 ) \\ & \times & [ \rho({\bf r } ) - \rho_0][\rho({\bf r ' } ) - \rho_0 ] . \label{taylor : eq}\end{aligned}\ ] ] using @xmath151 and the sum rule ( [ inductive : eq ] ) for @xmath152 together with the vanishing of the excess chemical potential at zero density , we readily obtain @xmath153 formally substituting in eq . ( [ taylor : eq ] ) , @xmath154 and @xmath155 and making use of the fact that the excess free energy of a system vanishes with the density , we also obtain the dependence of the fluid excess free energy on the density : @xmath156 with the particle number @xmath149 _ of the reference fluid _ and the fourier transform @xmath157 of the interaction potential . introducing eqs . ( [ c1:eq ] ) and ( [ frex_liq : eq ] ) into the taylor expansion , eq . ( [ taylor : eq ] ) above , we obtain : @xmath68 & = & \frac{n_0}{2}\beta{\tilde v}(0)\rho_0 \\ \nonumber & + & \beta{\tilde v}(0)\rho_0 ( n - n_0 ) \\ \nonumber & + & \frac{\beta}{2}\int\int{\rm d}^3r{\rm d}^3r'v(|{\bf r } - { \bf r'}| ) \rho({\bf r})\rho({\bf r ' } ) \\ \nonumber & - & \beta\rho_0\int\int{\rm d}^3r{\rm d}^3r'v(|{\bf r } - { \bf r'}| ) \rho({\bf r } ) \\ & + & \frac{n_0}{2}\beta{\tilde v}(0)\rho_0 . \label{expand : eq}\end{aligned}\ ] ] introducing @xmath158 the fourth term above becomes : @xmath159 now the sum of the 1st , 2nd , 4th and 5th term in eq . ( [ expand : eq ] ) yields : @xmath160 \\ = \beta{\tilde v}(0)\left[n_0\rho_0 + \rho_0(n - n_0 ) - n\rho_0\right ] = 0.\end{aligned}\ ] ] this is a remarkable cancelation because then only the 3rd term in eq . ( [ expand : eq ] ) survives and we obtain : @xmath161 = \frac{1}{2}\int\int{\rm d}^3r{\rm d}^3r ' v(|{\bf r } - { \bf r'}|)\rho({\bf r})\rho({\bf r ' } ) , \label{dftmfa : eq}\ ] ] which is our desired result.@xcite the derivation above demonstrates that the excess free energy of _ any _ inhomogeneous phase for our ultrasoft fluids is given by eq . ( [ dftmfa : eq ] ) , irrespectively of the density of the reference fluid @xmath66 . this is particularly important because , usually , functional taylor expansions are carried out around a reference fluid whose density lies close to the average density of the inhomogeneous system ( crystal , in our case ) . however , in our systems this is impossible . the crystals occur predominantly in domains of the phase diagram in which the reference fluid is meaningless , because they are on the high density side of the @xmath124-line . it is therefore important to be able to justify the use of the functional and to avoid the inherent contradiction of expanding around an unstable fluid . in practice , of course , the higher - order dcf s do not exactly vanish , hence deviations from result ( [ dftmfa : eq ] ) are expected to occur , in particular at low temperatures and densities . nevertheless , the comparisons with simulations , e.g. , in refs . [ ] , [ ] and [ ] fully justify our approximation . a mathematical proof of the mean - field character for fluids with infinitely long - range and infinitesimally strong repulsions has existed since the late 1970 s , see refs . [ ] and [ ] . however , even far away from fulfillment of this limit , and for conditions that are quite realistic for soft matter systems , the mean - field behavior continues to be valid.@xcite the mean - field result of eq . ( [ dftmfa : eq ] ) has been put forward for the gaussian model at high densities,@xcite on the basis of physical argumentation : in the absence of diverging excluded - volume interactions , at sufficiently high densities any given particle sees an ocean of others the classical mean - field picture . the mean - field character of the gaussian model for moderate to high temperatures was demonstrated independently in ref . , we have provided a more rigorous justification of its validity , based on the vanishing of high - order direct correlation functions in the fluid . it must also be noted that the mean - field approximation has recently been applied to a system with a broad shoulder and a much shorter hard - core interaction , providing good agreement with simulation results@xcite and allowing for the formulation of a generalized clustering criterion for the inhomogeneous phases . an astonishing similarity exists between the mean - field functional of eq . ( [ dftmfa : eq ] ) and an exact result derived for infinite - dimensional hard spheres . indeed , for this case frisch _ et al._@xcite as well as bagchi and rice@xcite have shown that @xmath162 = -\frac{1}{2}\int\int{\rm d}^d r{\rm d}^d r ' f(|{\bf r } - { \bf r'}|)\rho({\bf r})\rho({\bf r ' } ) , \label{mayer : eq}\ ] ] where @xmath163 and @xmath164 is the mayer function of the infinite dimensional hard spheres . again , one has a bilinear excess functional whose integration kernel does not depend on the density ; in this case , this is minus the ( bounded ) mayer function whereas for mean - field fluids , it is the interaction potential itself , divided by the thermal energy @xmath165 . in fact , the mayer function and the direct correlation function coincide for infinite - dimensional systems and higher - order contributions vanish there as well,@xcite making the analogy with our three - dimensional , ultrasoft systems complete . accordingly , infinite dimensional hard spheres have an instability at some finite @xmath16 at the density @xmath166 , given by @xmath167 . this so - called _ kirkwood instability_@xcite is of the same nature as our @xmath124-line but hard hyperspheres are athermal , so it occurs at a single point on the density axis and not a line on the density - temperature plane . following kirkwood s work,@xcite it was therefore argued@xcite that hard hyperspheres might have a second - order freezing transition at the density @xmath166 expressed as @xmath168d^{1/6 } , \label{rhostar : eq}\ ] ] where the limit @xmath163 must be taken and @xmath169 is the value of the minimum of the bessel function @xmath170 as @xmath163 . note that @xmath171 as @xmath163 . later on , frisch and percus argued that most likely the kirkwood instability is never encountered because it is preempted by a first - order freezing transition.@xcite in what follows , we will show analytically that this is also the case for our systems , which might provide a finite - dimensional realization of the above - mentioned mathematical limit . as mentioned above , an obvious candidate for a spatially modulated phase is a periodic crystal . the purpose of this section is to employ density functional theory in order to calculate the freezing properties . under some weak , simplifying assumptions , the problem can be solved analytically . adding the ideal contribution to the excess functional of eq . ( [ dftmfa : eq ] ) , the free energy of any spatially modulated phase is obtained as @xmath172 & = & f_{\rm id}[\rho ] + f_{\rm ex}[\rho ] \\ \nonumber & = & k_{\rm b}t\int{\rm d}^3r\left[\ln\left[\rho({\bf r})\lambda^3\right ] -1\right ] \\ & + & \frac{1}{2}\int\int{\rm d}^3r{\rm d}^3r ' v(|{\bf r}-{\bf r'}|)\rho({\bf r})\rho({\bf r ' } ) . \label{totalfren : eq}\end{aligned}\ ] ] as we are interested in crystalline phases , we parametrize the density profile as a sum of gaussians centered around the lattice sites @xmath173 , forming a bravais lattice . in sharp contrast with systems interacting by hard , diverging potentials , however , the assumption of one particle per lattice site has to be dropped . indeed , it will be seen that @xmath1 systems employ the strategy of optimizing their lattice constant by adjusting the number of particles per lattice site , @xmath174 , at any given density @xmath64 and temperature @xmath25 . accordingly , we normalize the profiles to @xmath174 and write @xmath175 where the occupation variable @xmath174 and the localization parameter @xmath176 have to be determined variationally , and the lattice site density @xmath177 is expressed as @xmath178 contrary to crystals of single occupancy , thus , the number of particles @xmath22 and the number of sites @xmath179 of the bravais lattice do not coincide . in particular , it holds @xmath180 and we are interested in multiple site occupancies , i.e. , @xmath181 or even @xmath182 ; it will be shown that this clustering scenario indeed minimizes the crystal s free energy . it is advantageous , at this point , to express the periodic density profile of eq . ( [ gaussian_real : eq ] ) as a fourier series , introducing the fourier components @xmath183 of the same : @xmath184 where the set @xmath185 contains all reciprocal lattice vectors ( rlvs ) of the bravais lattice formed by the set @xmath173 . accordingly , the inverse of ( [ denfour : eq ] ) reads as:@xcite @xmath186 where the first integral extends over the elementary unit cell @xmath187 of the crystal and @xmath188 is the volume of @xmath187 , containing a single lattice site . the second integral extends over all space , where use of the periodicity of @xmath136 and its expression as a sum over lattice site densities , eq . ( [ gaussian_real : eq ] ) , has been made . using eq . ( [ orbital : eq ] ) we obtain @xmath189 note that the site occupancy @xmath174 does not appear explicitly in the functional form of the fourier components of @xmath136 , a feature that may seem paradoxical at first sight . however , for fixed density @xmath64 and any crystal type , the lattice constant and thus also the reciprocal lattice vectors @xmath190 are affected by the possibility of clustering , thus the dependence on @xmath174 remains , albeit in an implicit fashion . with the density being expressed in reciprocal space , the excess free energy takes a simple form that reads as @xmath191 the ideal term , @xmath192 $ ] , can also be approximated analytically , provided that the gaussians centered at different lattice sites do not overlap . let @xmath193 denote the lattice constant of any particular crystal . then , for @xmath194 , the ideal free energy of the crystal takes the form @xmath195 where the trivial last term will be dropped in what follows , since is also appears in the expression of the free energy of the fluid and does not affect any phase boundaries . putting together eqs . ( [ fexk : eq ] ) and ( [ fida : eq ] ) , we obtain a variational free energy per particle , @xmath196 , for the crystal , that reads as @xmath197 \\ & + & \frac{\rho^*}{2}\sum_{\bf y}\tilde\phi(y ) e^{-y^2/(2\alpha^ * ) } , \label{fren_var : eq}\end{aligned}\ ] ] where @xmath198 and @xmath199 . in the list of arguments of @xmath196 the first two are variational parameters whereas the last two denote simply its dependence on temperature and density . the free energy per particle , @xmath200 of the crystal is obtained by minimization of @xmath196 , i.e. , @xmath201 in carrying out the minimization , it proves useful to measure the localization length of the gaussian profile , @xmath202 , in units of the lattice constant @xmath193 instead of units of @xmath6 . to perform this change , we first express the average density @xmath64 of the crystal in terms of @xmath174 and @xmath193 as @xmath203 where @xmath204 is a lattice - dependent numerical coefficient of order unity . introduce now the quantity @xmath205 . using eq . ( [ den_a : eq ] ) above , we obtain @xmath206 this change of variables is just a mathematical transformation that simplifies the mathematics to follow ; all results to be derived maintain their validity also in the original representation . for a further discussion of this point , see also appendix b. next we make the simplifying approximation to ignore in the sum over reciprocal lattice vectors on the right - hand - side of eq . ( [ fren_var : eq ] ) above all the rlvs beyond the first shell , whose length is @xmath207 . this is justified already because of the exponentially damping factors @xmath208 $ ] in the sum . in addition , the coefficients @xmath209 themselves decay to zero as @xmath210 , with an asymptotic behavior that depends on the form of @xmath7 in real space . the length of the first shell of rlvs of any bravais lattice of lattice constant @xmath193 scales as @xmath211 , with some positive , lattice - dependent numerical constant @xmath212 of order unity . together with eqs . ( [ den_a : eq ] ) this implies that the length of the first rlv depends on the aggregation number @xmath174 as @xmath213 and using eq . ( [ amin_nc : eq ] ) , we see that the ratio @xmath214 takes a form that depends _ solely _ on the parameter @xmath215 , namely @xmath216 introducing eqs . ( [ amin_nc : eq ] ) and ( [ d : eq ] ) into ( [ fren_var : eq ] ) , we obtain another functional form for the variational free energy , @xmath217 , expressed in the new variables . it can be seen that upon making the transformation ( [ amin_nc : eq ] ) , the term @xmath218 $ ] delivers a contribution _ minus _ @xmath219 that exactly cancels the same term with a positive sign on the right - hand side of eq . ( [ fren_var : eq ] ) . accordingly , the _ only _ remaining quantity of the variational free energy that still depends on @xmath174 is the length of the first nonvanishing rlv , @xmath220 , whose @xmath174 dependence is expressed by eq . ( [ y1nc : eq ] ) above . putting everything together , we obtain @xmath221 - \ln z \right ] \\ \nonumber & + & \frac{1}{2}\rho^*\tilde\phi(0 ) \\ & + & \frac{\xi_1\rho^*}{2}\tilde\phi\left(y_1(n_c)\right)e^{-\gamma\zeta^2/2 } , \label{fbar : eq}\end{aligned}\ ] ] where @xmath222 is the coordination number of the _ reciprocal _ lattice . minimizing @xmath223 with respect to @xmath174 is trivial and using eq . ( [ d : eq ] ) we obtain @xmath224 where the prime denotes the derivative with respect to the argument . evidently , @xmath220 coincides with @xmath225 , the dimensionless wavenumber for which the dimensionless fourier transform of the interaction potential attains its negative minimum . the other mathematical solution of ( [ partialnc : eq ] ) , @xmath226 , can be rejected because it yields nonpositive second derivatives or , on physical grounds , because it corresponds to a crystal with @xmath227 , whose occurrence would violate the thermodynamic stability of the system . regarding second derivatives , it can be easily shown that @xmath228 and @xmath229 irrespective of @xmath215 . having shown the coincidence of @xmath220 with @xmath230 , we set @xmath231 in eq . ( [ fbar : eq ] ) above . further , we notice that the term @xmath232 $ ] on the right - hand side of eq . ( [ fbar : eq ] ) gives the ideal free energy of a uniform fluid of density @xmath233 and the term @xmath234 the excess part of the same , see eq . ( [ frex_liq : eq ] ) . subtracting , thus , the total fluid free energy per particle , @xmath235 , we introduce the difference @xmath236 , which reads as @xmath237 \\ & + & \frac{\xi_1\rho^*}{2}\tilde\phi(y_*)e^{-\gamma\zeta^2/2}. \label{deltaf : eq}\end{aligned}\ ] ] the requirement of no overlap between gaussians centered on different lattice sites restricts @xmath215 to be small ; a very generous upper limit is @xmath238 . for such small values of @xmath215 , the first term on the right - hand side of eq . ( [ deltaf : eq ] ) above is positive . this positivity expresses the entropic cost of localization that a crystal pays , compared to the fluid in which the delocalized particles possess translational entropy . this cost must be compensated by a gain in the excess term , which is only possible if @xmath239 . an additional degree of freedom is offered by the candidate crystal structures . the excess free energy is minimized by the direct bravais lattice whose reciprocal lattice has the maximum possible coordination number @xmath222 . the most highly coordinated periodic arrangement of sites is fcc , for which @xmath240 . therefore , in the framework of this approximation , the stable lattice is bcc . it must be emphasized , though , that these results hold as long as only the first shell of rlvs is kept in the excess free energy . inclusion of higher - order shells can , under suitable thermodynamic conditions , stabilize fcc in favor of bcc . we will return to this point later . choosing now @xmath193 as the edge - length of the _ conventional _ lattice cell of the bcc - lattice , we have @xmath241 and @xmath242 . evidently , the lattice constant of the crystal is density - independent , @xmath243 , contrary to the case of usual crystals , for which @xmath244 . the density - independence of @xmath193 is achieved by the creation of clusters that consist of @xmath174 particles , each of them occupying a lattice site . the proportionality relation connecting @xmath174 and @xmath233 follows from eq . ( [ y1nc : eq ] ) and reads as @xmath245 it remains to minimize @xmath223 ( equivalently , @xmath246 ) with respect to @xmath215 to determine the free energy of the crystal . we are interested , in particular , in estimating the ` freezing line ' , determined by the equality of free energies of the fluid and the solid.@xcite accordingly , we search for the simultaneous solution of the equations @xmath247 resulting into @xmath248 and @xmath249 . \label{dfzero : eq}\ ] ] substituting ( [ dfzero : eq ] ) into ( [ dfdg : eq ] ) and using @xmath250 and @xmath242 , we obtain an implicit equation for @xmath215 that reads as @xmath251 , \label{gamma : eq}\ ] ] and has two solutions , @xmath252 and @xmath253 . due to ( [ second_nc : eq ] ) and ( [ second_ncgamma : eq ] ) , the sign of the determinant of the hessian matrix at the extremum is set by the sign of @xmath254 ; using eqs . ( [ dfdg : eq ] ) , ( [ dfzero : eq ] ) , and ( [ gamma : eq ] ) we obtain @xmath255 which is positive for @xmath256 but negative for @xmath257 . only the first solution corresponds to a minimum and thus to freezing , whereas the second is a saddle point . within the limits of the first - rlv - shell approximation , the crystals formed by @xmath1-potentials feature thus a _ universal localization parameter _ at freezing : irrespective of the location on the freezing line and even _ of the interaction potential itself _ , the localization length @xmath258 at freezing is a fixed fraction of the lattice constant and the parameter @xmath259 attains along the entire crystallization line the value @xmath260 we can understand the physics behind the constancy of the ratio @xmath261 by examining anew the variational form of the free energy , eq . ( [ fren_var : eq ] ) . suppose we have a fixed density @xmath233 and we vary @xmath174 , seeking to achieve a minimum of @xmath196 . an increase in @xmath174 implies an increase in the lattice constant @xmath193 by virtue of eq . ( [ den_a : eq ] ) . the density profile takes advantage of the additional space created between neighboring sites and becomes more delocalized . this increase of the spreading of the profile brings with it an entropic gain which exactly compensates the corresponding loss from the accumulation of particles on a single site , expressed by the term @xmath219 in eq . ( [ fren_var : eq ] ) . expressing @xmath258 in units of @xmath193 , i.e. , working with the variable @xmath215 instead with the original one , @xmath262 , brings the additional advantage that @xmath263 becomes independent of the pair potential . the corresponding value of @xmath262 at freezing , @xmath264 , can be obtained from eqs . ( [ den_a : eq ] ) and ( [ proport : eq ] ) , and reads for the bcc - lattice as @xmath265 here , a dependence on the pair interaction appears through the value of @xmath230 . complementary to the localization parameter , we can consider the lindemann ratio @xmath266 at freezing,@xcite taking into account that for the bcc lattice the nearest neighbor distance is @xmath267 . employing the gaussian density parametrization , we find @xmath268 and thus @xmath269 . using ( [ gammaf : eq ] ) , the lindemann ratio at freezing , @xmath270 , is determined as @xmath271 this value is considerably larger than the typical value of @xmath272 usually quoted for systems with harshly repulsive particles , such as , e.g. , the bcc alcali metals and the fcc metals al , cu , ag , and au [ ] , but close to the value 0.160 found by stillinger and weber@xcite for the gaussian core model . the particles in the cluster crystal are quite more delocalized than the ones for singly - occupied solids . the clustering strategy enhances the stability of the crystal with respect to oscillations about the equilibrium lattice positions . the locus of freezing points @xmath273 is easily obtained by eqs . ( [ dfzero : eq ] ) and ( [ gamma : eq ] ) and takes the form of of a straight line : @xmath274 contrary to the lindemann ratio , which is independent of the pair potential , the freezing line does depend on the interaction potential between the particles . yet , this dependence is a particular one , as it rests exclusively on the absolute value of the fourier transform at the minimum , @xmath275 and is simply proportional to it . comparing with the location of the @xmath124-line from eq . ( [ lambda : eq ] ) , @xmath276 , we find that crystallization _ preempts _ the occurrence of the instability : indeed , at fixed @xmath277 , @xmath278 or , equivalently , at fixed @xmath233 , @xmath279 ; see also fig . [ phdg : fig ] . the transition is first - order , as witnessed by the jumps of the values of @xmath176 and @xmath183 at the transition , which are nonzero for the crystal but vanish in the fluid . this is analogous to the conjectured preemption of the kirkwood instability for infinite - dimensional hard spheres by a first - order freezing transition.@xcite the freezing properties of @xmath1-potentials are , thus , quite unusual and at the same time quite simple : the lattice constant is fixed due to a clustering mechanism that drives the aggregation number @xmath174 proportional to the density . the constant of proportionality depends solely on the wavenumber @xmath230 for which the fourier transform of the pair interaction has a negative minimum , eq . ( [ proport : eq ] ) . the freezing line is a straight line whose slope depends only on the value of the fourier transform of the potential at the minimum , eq . ( [ freeze : eq ] ) . the lindemann ratio at freezing is a universal number , independent of interaction potential and thermodynamic state . whereas the lindemann ratio is employed as a measure of the propensity of a crystal to melt , the height of the peak of the structure factor of the fluid is looked upon as a measure of the tendency of the fluid to crystallize . the hansen - verlet criterion@xcite states that crystallization takes place when this quantity exceeds the value 2.85 . for the systems at hand , the maximum of @xmath280 lies at @xmath230 , as is clear from eq . ( [ sofk : eq ] ) . using eq . ( [ freeze : eq ] ) for the location of the freezing line , we obtain the value @xmath281 on the freezing line as @xmath282^{-1 } \cong 3.542.\ ] ] this value is considerably larger than the hansen - verlet threshold.@xcite in the fluid phase , @xmath1-systems can therefore sustain a higher degree of spatial correlation before they crystallize than particles with diverging interactions do . this property lies in the fact that some contribution to the peak height comes from correlations from _ within _ the clusters that form in the fluid ; the formation of clusters already in the uniform phase is witnessed by the maxima of @xmath55 at @xmath283 seen in fig . [ gofr : fig ] and also explicitly visualized in our previous simulations of the model.@xcite these , however , do not contribute to intercluster ordering that leads to crystallization . at any rate , the hansen - verlet peak height is also a universal quantity for all @xmath1 systems , in the framework of the current approximation . moreover , both for the lindemann and for the hansen - verlet criteria , the @xmath1 systems are more robust than usual ones , since they allow for stable fluids with peak heights that exceed @xmath284 by 25% and for stable crystals with lindemann ratios that exceed @xmath272 by almost 90% . the density functional of eq . ( [ totalfren : eq ] ) is very accurate for the bounded ultrasoft potentials at hand.@xcite the modeling of the inhomogeneous density as a sum of gaussians is an approximation but , again , an accurate one , as has been shown by comparing with simulation results,@xcite see also section [ harmonic : sec ] of this work . the analytical results derived in the preceding section rest on one additional approximation , namely on ignoring the rlvs beyond the first shell . here , we want to compare with a full minimization of the functional ( [ totalfren : eq ] ) under the modeling of the density via ( [ gaussian_real : eq ] ) , so as to test the accuracy of the hitherto drawn conclusions on clustering and crystallization . ) , under the gaussian parametrization of the density , eq . ( [ gaussian_real : eq ] ) , redrawn from ref . [ ] . on the same plot , we show by the dotted line the approximate analytical result for the freezing line , eq . ( [ freeze : eq ] ) , as well as the @xmath124-line of the system , eq . ( [ lambda : eq]).,width=321 ] we work with the concrete gem-4 system , for which the minimization of the density functional has been carried out and the phase diagram has been calculated in ref . [ ] . in fig . [ phdg : fig ] we show the phase diagram obtained by the full minimization , compared with the freezing line from the analytical approximation , eq . ( [ freeze : eq ] ) , for this system . it can be seen that the latter is a very good approximation to the full result , its quality improving slowly as the temperature grows ; the analytical approximation consistently overestimates the region of stability of the crystal . moreover , whereas the approximation only predicts a stable bcc crystal , the high - density phase of the system is fcc . although bcc indeed is , above the triple temperature , the stable crystal immediately post - freezing , it is succeeded at higher densities by a fcc lattice , which our analytical theory fails to predict . of the gem-4 potential . main plot : a zoom at the region of @xmath12 in which the first few nonvanishing rlv shells of the cluster - crystals of fig . [ phdg : fig ] lie . the arrows denote the positions of the shells and the numbers in square brackets the numbers of distinct rlvs within each shell . these positions are the result of the full minimization of the density functional . downwards pointing arrows pertain to the direct bcc lattice and upwards pointing arrows to the direct fcc one . in agreement with clustering predictions , the positions of the rlvs are density - independent.,width=321 ] all these discrepancies can be easily understood by looking at the effects of ignoring the higher rlv shells from the summation in the excess free energy , eq . ( [ fren_var : eq ] ) . consider first exclusively the bcc lattice . in fig . [ phiofq : fig ] we show the locations of the bcc - rlvs , as obtained from the full minimization , by the downwards pointing arrows . it can be seen that the first shell is indeed located very closely to @xmath230 , as the analytical solution predicts . however , the next two rlv shells do have contributions and , due to their location on the hump of @xmath12 , the latter is positive . by ignoring them in performing the analytical solution , we are artificially lowering the free energy of the crystal , increasing thereby its domain of stability . the occurrence of a fcc - lattice that beats the bcc at high densities is only slightly more complicated to understand . a first remark is that the parameter @xmath262 increases proportionally to @xmath285 , see the following section . hence , the gaussian factors from rlvs beyond the first shell , @xmath286 $ ] , @xmath287 , gain weight in the sum as density grows . the cutoff for the rlv - sum is now provided rather by the short - range nature of @xmath12 than by the exponential factors . due to the increased importance of the contributions from the @xmath287-terms in the excess free energy sum , the relative location of higher rlvs becomes crucial and can tip the balance in favor of fcc , although the bcc - lattice has a _ higher _ number of rlvs in its first shell than the fcc . in fig . [ phiofq : fig ] we see that this is precisely what happens : the second rlv shell of the fcc is located fairly close to the first . in the full minimization , both of them arrange their positions so as to lie close enough to @xmath230 . now , a total of fourteen 1st- and 2nd - shell rlvs of the fcc can beat the twelve 1st - shell rlvs of the bcc and bring about a structural phase transformation from the latter to the former . the relative importance of the first and second neighbors is quantified by the ratio @xmath288 , \label{ratio : eq}\ ] ] where @xmath289 is the number of rlvs in the second shell . if @xmath290 does not decay sufficiently fast to zero as @xmath291 grows , then the fcc lattice might even win over the bcc everywhere , since then @xmath292 could be considerable even for values of @xmath262 close to freezing , which are not terribly high . in fact , the penetrable sphere model ( gem-@xmath17 with @xmath293 ) does not possess , on these grounds , a stable bcc phase at all.@xcite the prediction of the analytical theory on bcc stability has to be taken with care and is conditional to @xmath292 being sufficiently small . a quantitative criterion on the smallness of @xmath292 is model specific and can not be given in general . the determination of the stable phases of the gem-@xmath17 family and their dependence on @xmath17 can be achieved by employing genetic algorithms@xcite and will be presented elsewhere.@xcite notwithstanding the quantitative discrepancies between the simplified , analytically tractable version of dft and the full one , which are small in the first place , the central conclusion of the former remains intact : the rlvs of the crystals are density - independent . whereas the analytical approximation predicts that the length of the first rlv shell coincides with @xmath230 , the numerical minimization brings about small deviations from this prediction . however , by reading off the relevant values from fig . [ phiofq : fig ] , we obtain @xmath294 for the bcc and @xmath295 for the fcc - lattice of the gem-4 model . comparing with the ideal value @xmath296 , we find that the deviation between them is only a few percent . clustering takes place , so that the lattice constants of both lattices remain fixed , a characteristic that was also explicitly confirmed by computer simulations of the model.@xcite the use of a gaussian parametrization for the one - particle density profiles , eq . ( [ gaussian_real : eq ] ) , is a standard modeling of the latter for periodic crystals . this functional form is closely related to the harmonic theory of crystals.@xcite each particle performs oscillations around its lattice site , experiencing thereby an effective , one - particle site potential , @xmath297 that is quadratic in the displacement @xmath298 , for small values @xmath299 [ ] . here , we will explicitly demonstrate that the gaussian form with the localization parameter predicted from density functional theory coincides with the results obtained by performing a harmonic expansion of the said site potential . the formation of clustered crystals is a generic property of all @xmath1-systems , since the @xmath124-instability is common to all of them ; the form of the clusters that occupy the lattice sites , however , can be quite complex , depending on the details of the interaction . the gaussian parametrization ( [ gaussian_real : eq ] ) implies that for each of the @xmath174 particles of the cluster , the lattice site @xmath300 is an equilibrium position . in other words , the particular clusters we consider here are internally structureless . clusters with a well - defined internal order have been found when an additional hard core of small extent is introduced.@xcite a necessary requirement for the lack of internal order is that the laplacian of the interaction potential @xmath3 be finite at @xmath283 , as will be shown shortly . on these grounds , we impose from the outset on the interaction potential the additional requirement : @xmath301 where the primes denote the derivative with respect to @xmath8 . ( [ condition1:eq ] ) implies that @xmath302 must be _ at least _ linear in @xmath8 as @xmath89 . concomitantly , @xmath303 must be at least @xmath304 as @xmath89 . as a consequence , we have @xmath305 it can be easily checked that ( [ condition : eq ] ) is satisfied by all members of the gem-@xmath17 class for @xmath20 . it is also satisfied by the @xmath118 member , i.e. , the gaussian model , which does _ not _ display clustering because it belongs to the @xmath2-class . this is , however , no contradiction . as mentioned above , the condition ( [ condition : eq ] ) is _ necessary _ for the formation of structureless clusters and not a sufficient one . for @xmath1-potentials for which ( [ condition : eq ] ) is not fulfilled ( such as the fermi distribution models of ref . [ ] ) , this does not mean that clusters do not form ; it rather points to the fact that they possess some degree of internal order . the clustered crystals can be considered as bravais lattices with a @xmath174-point basis . accordingly , their phonon spectrum will feature 3 acoustic modes and @xmath306 optical modes , for which the oscillation frequency @xmath307 remains finite as @xmath308 . we are interested in the case @xmath182 , i.e. , deep in the region of stability of the crystal , where the clusters have a very high occupation number . consequently , the phonon spectra and the particle displacements will be dominated by the optical branches . further , we simplify the problem by choosing , in the spirit of the einstein model of the crystal,@xcite one specific optical phonon with @xmath309 as a representative for the whole spectrum . this mode corresponds to the relative partial displacement of two sublattices : one with @xmath310 particles on each site and one with just the remaining one particle per site . the two sublattices coincide at the equilibrium position and maintain their shape throughout the oscillation mode , consistent with the fact of an infinite - wavelength mode , @xmath311 . accordingly , the site potential felt by any one of the particles of the single - occupied sublattice , @xmath297 , can be expressed as @xmath312 , \label{vsite : eq}\ ] ] where @xmath313 is the relative displacement of the two sublattices . for brevity , we also define @xmath314 the taylor expansion of a scalar function @xmath315 around a reference point @xmath316 reads as @xmath317 setting @xmath318 and @xmath319 , we obtain the quadratic expansion of the site potential ; the constant @xmath320 is unimportant . for the linear term , we have @xmath321 where @xmath322 and @xmath323 are unit vectors . the sum in ( [ linear : eq ] ) vanishes due to lattice inversion symmetry ; the first term also , since @xmath324 . thus , the term linear in @xmath313 in the expansion of @xmath297 vanishes , consistently with the fact that @xmath325 is an equilibrium position . we now introduce cartesian coordinates and write @xmath326 , @xmath327 , and @xmath328 . the quadratic term in ( [ taylor3:eq ] ) takes the explicit form @xmath329 let us consider first the mixed derivative acting on @xmath330 , evaluated at @xmath331 . using eqs . ( [ wsite : eq ] ) and ( [ vsite : eq ] ) , we obtain @xmath332_{{\bf r } = 0 } \\ & + & \sum_{{\bf r}\ne 0}\frac{r_xr_y}{r^2 } \left[v''(r)-\frac{v'(r)}{r}\right ] . \label{mixed : eq}\end{aligned}\ ] ] the first term on the right - hand side of ( [ mixed : eq ] ) vanishes by virtue of ( [ condition : eq ] ) . the second one also vanishes due to the cubic symmetry of the lattice . clearly , the other two terms in ( [ explicit : eq ] ) with mixed derivatives vanish as well . for the remaining terms , the cubic symmetry of the lattice implies that all three second partial derivatives of @xmath333 at @xmath331 are equal : @xmath334 gathering the results , we obtain the expansion of @xmath297 to quadratic order in @xmath298 as @xmath335s^2 , \label{isotropic : eq}\ ] ] which is isotropic in @xmath298 , as should for a crystal of cubic symmetry . the one - particle motion is therefore harmonic ; as we consider @xmath182 , we set @xmath336 in ( [ isotropic : eq ] ) and we obtain the effective , one - particle hamiltonian @xmath337 in the form : @xmath338 with @xmath339 and the momentum @xmath340 and mass @xmath17 of the particle . the density profile @xmath341 of this single - particle problem is easily calculated as @xmath342 , yielding @xmath343 this is indeed a gaussian of a single particle , with a localization parameter @xmath344 ; the total density on a given site will be then just @xmath345 , in agreement with the functional form put forward in eq . ( [ gaussian_real : eq ] ) . it is useful to consider in detail the form of the localization parameter @xmath346 predicted by the harmonic theory . the parameter @xmath347 is expressed as a sum of the values of @xmath348 over the periodic set @xmath173 . for every function @xmath349 that possesses a fourier transform @xmath350 , it holds@xcite @xmath351 where @xmath185 is the set of rlvs of @xmath173 and @xmath352 is the density of _ lattice sites _ of @xmath173 . from ( [ nc : eq ] ) , @xmath353 . taking into account that the fourier transform of @xmath354 is @xmath355 , we obtain for the localization parameter of the harmonic theory the result @xmath356 the localization parameter must be , evidently , positive . ( [ alphah : eq ] ) manifests the impossibility for cluster formation if the fourier transform of the pair potential is nonnegative , i.e. , for @xmath2 interactions . in the preceding section , we showed within the dft formalism that if the potential is @xmath1 , this implies the formation of cluster crystals . harmonic theory allows us to make the opposite statement as well : if the potential is _ not _ @xmath1 , then there can be _ no clustered crystals_. therefore , an _ equivalence _ between the @xmath1 character of the interaction and the formation of clustered crystals can be established . moreover , eq . ( [ alphah : eq ] ) offers an additional indication as to why the rlv where @xmath357 is most negative is selected as the shortest nonvanishing one by the clustered crystals : this is the best strategy in order to keep the localization parameter positive . harmonic theory provides , therefore , an insight into the necessity of locating the first shell of the rlvs at @xmath123 from a different point of view than density functional theory does . the choice @xmath358 guarantees that the particles inhabiting neighboring clusters provide the restoring forces that push any given particle back towards its equilibrium position . the density functional treatment of the preceding sections establishes that the lattice constant is chosen by @xmath1-systems in such a way that the sum of _ intracluster _ and _ intercluster _ interactions , together with the entropic penalty for the aggregation of @xmath174 particles is optimized.@xcite the unlimited growth of @xmath174 is avoided by the requirement of mechanical stability of the crystal . indeed , for too high @xmath174-values , the lattice constant would concomitantly grow , so that the resulting restoring forces working against the thermal fluctuations , would become too weak to sustain the particles at their equilibrium positions . let us , finally , compare the result ( [ alphah : eq ] ) for the localization parameter with the prediction from dft . we consider the high - density crystal phase , for which @xmath262 is very large , so that the simplification that only the first shell of rlvs can be kept in ( [ fren_var : eq ] ) must be dropped . setting @xmath359 there , we obtain @xmath360 the function @xmath12 is short - ranged in reciprocal space , thus the sum in ( [ dftall : eq ] ) can be effectively truncated at some finite upper cutoff @xmath361 . then , there exists a sufficiently large density @xmath233 beyond which the parameter @xmath262 is so large that @xmath362 for all @xmath363 included in the summation . accordingly , we can approximate all exponential factors with uniti in ( [ dftall : eq ] ) , obtaining an algebraic equation for @xmath262 . reverting back to dimensional quantities , its solution reads as @xmath364 and is _ identical _ with the result from the harmonic theory , eq . ( [ alphah : eq ] ) . thus , density functional theory and harmonic theory become identical to each other at the limit of high localization . this finding completes and generalizes the result of archer,@xcite who established a close relationship between the mean - field dft and the einstein model for the form of the variational free energy functional of the system . consistently with our assumptions , @xmath176 indeed grows with density . in fact , since the set of rlvs in the sum of ( [ alphasdft : eq ] ) is fixed , it can be seen that @xmath176 is simply proportional to @xmath365 . this peculiar feature of the class of systems we consider is not limited to the localization parameter , and its significance is discussed in the following section . as can be easily confirmed by the form of the density functional of eq . ( [ totalfren : eq ] ) , the mean - field nature of the class of ultrasoft systems considered here ( both @xmath2- and @xmath1-potentials ) implies that the structure and thermodynamics of the systems is fully determined by the ratio @xmath366 between density and temperature and not separately by @xmath40 and @xmath367 . this is a particular type of scaling between the two relevant thermodynamic variables , reminiscent of the situation for systems interacting by means of inverse - power - law potentials @xmath3 having the form : @xmath368 for such systems , it can be shown that their statistical mechanics is governed by a a single coupling constant @xmath369 expressed as@xcite @xmath370 where @xmath371 is the space dimension . it would appear that inverse - powers @xmath372 satisfy precisely the same scaling as mean - field systems do but there is a condition to be fulfilled : inverse - power systems are stable against explosion , _ provided _ that @xmath373 ; this can be most easily seen by considering the expression for the excess internal energy per particle , @xmath374 , given by@xcite @xmath375 ! } \int_0^{\infty}r^{d-1-n}g(r;\rho , t){\rm d}r . \label{inten : eq}\ ] ] as @xmath376 for @xmath377 , we see that the integral in ( [ inten : eq ] ) converges only if @xmath373 ; a logarithmic divergence results for @xmath372 . a ` uniform neutralizing background ' has to be formally introduced for @xmath378 , to obtain stable pseudo one - component systems , such as the one - component plasma.@xcite since we aim at staying with genuine one - component systems throughout , we must strictly maintain @xmath373 . instead of taking the limit @xmath379 , we consider therefore a different procedure by setting @xmath380 with some arbitrary , finite @xmath381 . then the coupling constant @xmath369 becomes @xmath382 now take @xmath163 in this prescribed fashion , obtaining : @xmath383 which has precisely the same form as the coupling constant of our systems . in taking the limit @xmath163 , the exponent @xmath384 of the inverse - power potential diverges as well . it can be easily seen that in this case , the interaction @xmath3 of eq . ( [ powerlaw : eq ] ) becomes a hard - sphere potential of diameter @xmath6 . in other words , the procedure prescribed above brings us once more to infinite - dimensional hard spheres , because also the inverse - power interaction becomes arbitrarily steep . this is a very different way of taking the limit than in refs . [ ] : there , the interaction is hard in the first place and subsequently the limit @xmath163 is taken , whereas here , interaction and dimension of space change together , in a well - prescribed fashion . the fact that the statistical mechanics of ultrasoft fluids in three dimensions is determined by the same dimensionless parameter as that of a particular realization of hard spheres in infinite dimensions is intriguing . in a sense , ultrasoft systems are effectively high - dimensional , since they allow for extremely high densities , for which every particle interacts with an exceedingly high number of neighbors . they might , in this sense , provide for three - dimensional approximate realizations of infinite - dimensional models . this is yet another relation to infinite - dimensional systems , in addition to the one discussed at the end of sec . [ dft : sec ] . whether there exists a deeper mathematical connection between the two classes , remains a problem for the future . we have provided a detailed analysis of the properties of bounded , ultrasoft systems , with emphasis on the @xmath1-class of interaction potentials . after having demonstrated the suppression of the contributions from the high - order direct correlation functions of the fluid phases ( of order 3 and higher ) , we established as a consequence the accurate mean - field density functional for arbitrary inhomogeneous phases . though this functional has been introduced and successfully used in the recent past both in statics@xcite and in dynamics,@xcite a sound justification of its basis on the properties of the uniform phase was still lacking . the persistence of a _ single _ , finite length scale for the lattice constants of the ensuing solids of @xmath1-systems has been understood by a detailed analysis of the structure of the free energy functional . in the fluid , the same length scale appears since the position of maximum of the liquid structure factor is independent of density . the negative minimum of the interaction potential in fourier space sets this unique scale and forces in the crystal the formation of clusters , whose population scales proportionally with density . the analytical solution of an approximation of the density functional is checked to be accurate when confronted with the full numerical minimization of the latter . universal lindemann ratios and hansen - verlet values at crystallization are predicted to hold for all these systems , which differ substantially from those for hard matter systems . the analytical derivation of these results provides useful insight into the robustness of these structural values for an enormous variety of interactions . though the assumption of bounded interactions has been made throughout , recent results@xcite indicate that both cluster formation and the persistence of the length scale survive when a short - range diverging core is superimposed on the ultrasoft potential , provided the range of the hard core does not exceed , roughly , 20% of the overall interaction range.@xcite the morphology of the resulting clusters is more complex , as full overlaps are explicitly forbidden ; even the macroscopic phases are affected , with crystals , lamellae , inverted lamellae and ` inverted crystals ' showing up at increasing densities . the generalization of our density functional theory to such situations and the modeling of the nontrivial , internal cluster morphology is a challenge for the future . here , a mixed density functional , employing a hard - sphere and a mean - field part of the direct correlation function seems to be a promising way to proceed.@xcite finally , the study of the vitrification , dynamical arrest and hopping processes in concentrated @xmath1-systems is another problem of current interest . the recent ` computer synthesis ' of model , amphiphilic dendrimers that do display precisely the form of @xmath1-interactions discussed in this work,@xcite offers concrete suggestions for the experimental realization of the hitherto theoretically predicted phenomena . we are grateful to andrew archer for helpful discussions and a critical reading of the manuscript and to andras st for helpful comments . this work has been supported by the sterreichische forschungsfond under project no . p17823-n08 , as well as by the deutsche forschungsgemeinschaft within the collaborative research center sfb - tr6 , `` physics of colloidal dispersions in external fields '' , project section c3 . consider the inverse fourier transform of the spherically symmetric , bounded pair potential @xmath3 , reading as @xmath385 from ( [ invft : eq ] ) , it is straightforward to show that the second derivative of @xmath3 at @xmath283 takes the form @xmath386 evidently , if @xmath387 , then @xmath15 _ must _ have negative parts and hence @xmath3 is @xmath1 . for the gem-@xmath17 family , it is easy to show that @xmath388 for @xmath20 , thus these members are indeed @xmath1 , as stated in the main text . double - gaussian potentials of the form @xmath389 with @xmath390 , @xmath391 , which feature a local _ minimum _ at @xmath56 are also @xmath1 , for the same reason . notice , however , that @xmath387 is a sufficient , not a necessary condition for membership in the @xmath1-class . thus , there exist @xmath1-potentials for which @xmath392 . the introduction of the new variable @xmath215 instead of @xmath262 , eq . ( [ amin_nc : eq ] ) , and the subsequent new form @xmath223 of the variational free energy , eq . ( [ fbar : eq ] ) , are just a matter of convenience , which makes the minimization procedure more transparent . a free gift of the variable transformation is also the ensuing diagonal form of the hessian matrix at the extremum . fully equivalent results are obtained , of course , by working with the original variational free energy , @xmath196 , eq . ( [ fren_var : eq ] ) . here we explicitly demonstrate this equivalence . keeping , consistently , only the first shell of rlvs with length @xmath220 , @xmath196 takes the form : @xmath393 \\ \nonumber & + & \frac{\rho^*}{2}\tilde\phi(0 ) \\ & + & \frac{\xi_1\rho^*}{2}\tilde\phi(y_1 ) e^{-y_1 ^ 2/(2\alpha^ * ) } , \label{fren_new : eq}\end{aligned}\ ] ] where @xmath220 and @xmath174 are related via eq . ( [ d : eq ] ) . minimizations of @xmath196 with respect to @xmath174 and @xmath262 yield , respectively : @xmath394e^{-y_1 ^ 2/(2\alpha^ * ) } = 0 , \label{delnc : eq}\ ] ] and @xmath395 subtracting the last two equations from one another we obtain @xmath396 implying @xmath397 , as in the main text ( once more , @xmath226 is a formal solution that must be rejected on the same grounds mentioned in the text . ) from this property , eq . ( [ proport : eq ] ) immediately follows . introducing @xmath398 , we can determine @xmath264 on the freezing line by requiring the simultaneous satisfaction of the minimization conditions above and of @xmath399 . the latter equation yields @xmath400\right ] \\ = - \frac{\xi_1\rho^*}{2}\tilde\phi(y_*)e^{-y_*^2/(2\alpha^*_{\rm f } ) } , \label{delftildezero : eq}\end{aligned}\ ] ] which , together with eq . ( [ dela : eq ] ) , yields , after some algebra @xmath401 - 1 = \frac{2\alpha^*_{\rm f}}{y_*^2}. \label{onemore : eq}\ ] ] using eqs . ( [ amin_nc : eq ] ) and ( [ y1nc : eq ] ) , we obtain the equation for the @xmath263-parameter at freezing as @xmath402 = \frac{2}{\gamma_{\rm f}\zeta^2 } , \label{gamma_new : eq}\ ] ] which , upon setting @xmath241 and @xmath242 , yields eq . ( [ gamma : eq ] ) of the main text . alternatively , we can introduce the variable @xmath403 and rewrite eq . ( [ onemore : eq ] ) as @xmath404 - 1 = 2 t , \label{omega : eq}\ ] ] which delivers @xmath405 as a solution or , equivalently , @xmath406 , in agreement with eqs . ( [ gammaf : eq ] ) and ( [ alphaf : eq ] ) of the main text . strictly speaking , there is no freezing _ line _ on the density - temperature plane but rather a domain of coexistence , since it is equality of the grand potentials that determines phase coexistence . nevertheless , the condition of equality of the helmholtz free energies determines a line that runs between the melting- and crystallization lines and therefore serves as a reliable locator of the freezing transition . a very important exception is the hard - sphere potential , for which no harmonic expansion can be made . the fact that hard - sphere crystals do still have density profiles that are very well modeled by gaussians , calls for a different explanation there . it can be argued that the huge number of uncorrelated collisions with the neighbors , together with the central limit theorem , are responsible for the gaussianity of density profiles in hard - sphere solids .
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we demonstrate the accuracy of the hypernetted chain closure and of the mean - field approximation for the calculation of the fluid - state properties of systems interacting by means of bounded and positive - definite pair potentials with oscillating fourier transforms .
subsequently , we prove the validity of a bilinear , random - phase density functional for arbitrary inhomogeneous phases of the same systems . on the basis of this functional , we calculate analytically the freezing parameters of the latter .
we demonstrate explicitly that the stable crystals feature a lattice constant that is independent of density and whose value is dictated by the position of the negative minimum of the fourier transform of the pair potential .
this property is equivalent with the existence of clusters , whose population scales proportionally to the density .
we establish that regardless of the form of the interaction potential and of the location on the freezing line , all cluster crystals have a universal lindemann ratio @xmath0 at freezing .
we further make an explicit link between the aforementioned density functional and the harmonic theory of crystals .
this allows us to establish an equivalence between the emergence of clusters and the existence of negative fourier components of the interaction potential .
finally , we make a connection between the class of models at hand and the system of infinite - dimensional hard spheres , when the limits of interaction steepness and space dimension are both taken to infinity in a particularly described fashion .
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a big achievement in the 70s-80 s was to show that , inside the axiomatic formulation of quantum mechanics , based on _ positive operator valued measures _ and _ instruments_,@xcite a consistent formulation of the theory of measurements continuous in time ( _ quantum continual measurements _ ) was possible.@xcite the main applications of quantum continual measurements are in the photon detection theory in quantum optics ( _ direct , heterodyne , homodyne detection_).@xcite a very flexible and powerful formulation of continual measurement theory was based on stochastic differential equations , of classical type ( commuting noises , it calculus ) and of quantum type ( non commuting noises , hudson - parthasarathy equation).@xcite in this paper we start by giving a short presentation of continual measurement theory based on quantum sde s . we consider only the type of observables relevant for the description of homodyne detection and we make the mathematical simplification of introducing only bounded operators on the hilbert space of the quantum system and a finite number of noises . then , we introduce the spectrum of the classical stochastic process which represents the output and we study the general properties of the spectra of such classical processes by proving characteristic bounds due to the heisenberg uncertainty principle . finally , we present the case of a two - level atom , where the spectral analysis of the output can reveal the phenomenon of squeezing of the fluorescence light , a phenomenon related to the heisenberg uncertainty relations . let @xmath0 be the _ system space _ , the complex separable hilbert space associated to the observed quantum system , which we call system @xmath1 . quantum stochastic calculus and the hudson - parthasarathy equation@xcite allow to represent the continual measurement process as an interaction of system @xmath1 with some quantum fields combined with an observation in continuous time of these fields . let us start by introducing such fields . we denote by @xmath2 the hilbert space associated with @xmath3 boson fields , that is the symmetric _ fock space _ over the `` one particle space '' @xmath4 , and we denote by @xmath5 , @xmath6 , the _ coherent vectors _ , whose components in the @xmath7 particle spaces are @xmath8 . let @xmath9 be the canonical basis in @xmath10 and for any @xmath11 let us set @xmath12 . we denote by @xmath13 , @xmath14 , @xmath15 the _ annihilation , creation and conservation processes _ : @xmath16 the annihilation and creation processes satisfy the _ canonical commutation rules _ ( ccr ) ; formally , @xmath17=t\wedge s$ ] , @xmath18=0 $ ] , @xmath19=0 $ ] . let @xmath20 , @xmath21 , @xmath22 , @xmath23 , be bounded operators on @xmath0 such that @xmath24 and @xmath25 . we set also @xmath26 . then , the quantum stochastic differential equation @xcite @xmath27 with the initial condition @xmath28 , has a unique solution , which is a strongly continuous family of unitary operators on @xmath29 , representing the system - field dynamics in the interaction picture with respect to the free field evolution . the states of a quantum system are represented by statistical operators , positive trace - class operators with trace one ; let us denote by @xmath30 the set of statistical operators on @xmath0 . as initial state of the composed system `` system @xmath1 plus fields '' we take @xmath31 , where @xmath32 is generic and @xmath33 is a coherent state , @xmath34 . one of the main properties of the hudson - parthasarathy equation is that , with such an initial state , the reduced dynamics of system @xmath1 obeys a quantum master equation.@xcite indeed , we get @xmath35 , \qquad \eta_t:={\operatorname{tr}}_\gamma \left\ { u(t)\bigl ( \rho \otimes \varrho_\gamma(f)\bigr)u(t)^*\right\},\ ] ] where the liouville operator @xmath36 turns out to be given by @xmath37= \left(k-\sum_{kl}r_k^ * s_{kl}f_l(t)\right)\rho + \rho \left(k^*-\sum_{kj}\overline{f_j(t)}s_{kj}^{\;*}r_k\right ) \\ { } + \sum_k \left(r_k-\sum_l s_{kl}f_l(t)\right)\rho\left(r_k^*- s_{kl}^{\;*}\overline{f_l(t)}\right ) - { \left\vertf(t)\right\vert}^2\rho.\end{gathered}\ ] ] a particularly important case is @xmath38 , when @xmath36 reduces to @xmath37= -{\mathrm{i}}\left [ h-{\mathrm{i}}\sum_k f_k(t)r_k^*+ { \mathrm{i}}\sum_k\overline{f_k(t)}r_k,\,\rho \right ] \\ { } + \sum_k \left(r_k\rho r_k^*-\frac 1 2 r_k^*r_k\rho -\frac 1 2\rho r_k^*r_k\right).\end{gathered}\ ] ] it is useful to introduce also the evolution operator from @xmath39 to @xmath40 by @xmath41 with this notation we have @xmath42 $ ] . the key point of the theory of continual measurements is to consider field observables represented by time dependent , commuting families of selfadjoint operators in the heisenberg picture.@xcite being commuting at different times , these observables represent outputs produced at different times which can be obtained in the same experiment . here we present a very special case of family of observables , a field quadrature . let us start by introducing the operators @xmath43 @xmath44 $ ] and @xmath45 are fixed . the operators @xmath46 are selfadjoint ( they are essentially selfadjoint on the linear span of the exponential vectors ) . by using ccr s , one can check that they commute : @xmath47=0 $ ] ( better : the unitary groups generated by @xmath46 and @xmath48 commute ) . the operators have to be interpreted as linear combinations of the formal increments @xmath49 , @xmath50 which represent field operators evolving with the free - field dynamics ; therefore , they have to be intended as operators in the interaction picture . the important point is that these operators commute for different times also in the heisenberg picture , because @xmath51 this is due to the factorization properties of the fock space and to the properties of the solution of the hudson - parthasarathy equation . these `` output '' quadratures are our observables . they regard those bosons in `` field 1 '' which eventually have interacted with @xmath1 between time @xmath52 and time @xmath40 . commuting selfadjoint operators can be jointly diagonalized and the usual postulates of quantum mechanics give the probabilities for the joint measurement of the observables represented by the selfadjoint operators @xmath53 , @xmath54 . let us stress that operators of type with different angles and frequencies represent incompatible observables , because they do not commute but satisfy @xmath55=\frac{4{\mathrm{i}}\sin\left(\frac{t\wedge s}2 \left(\nu-\mu\right)\right)\sin \left(\theta-\phi+\frac{t\wedge s}2 \left(\nu-\mu\right)\right)}{\nu-\mu}\ , .\ ] ] when `` field 1 '' represents the electromagnetic field , a physical realization of a measurement of the observables is implemented by what is called an heterodyne / homodyne scheme . the light emitted by the system in the `` channel 1 '' interferes with an intense coherent monochromatic laser beam of frequency @xmath56 . the mathematical description of the apparatus is given in section 3.5 of ref . the commuting selfadjoint operators have a joint pvm ( projection valued measure ) which gives the distribution of probability for the measurement . anyway , at least the finite - dimensional distributions of the output can be obtained via an explicit and easier object , the _ characteristic operator _ @xmath57 , a kind of fourier transform of this pvm . for any real test function @xmath58 and any time @xmath59 we define the unitary weyl operator @xmath60 then , there exists a measurable space @xmath61 , a pvm @xmath62 ( acting on @xmath2 ) with value space @xmath61 , a family of real valued measurable functions @xmath63 on @xmath64 , such that @xmath65 , and , for any choice of @xmath66 , @xmath67 , @xmath68 , @xmath69\biggr\ } \\ { } = \int_\omega \exp\biggl\ { { \mathrm{i}}\sum_{j=1}^n \kappa_j \big [ x(t_j;\omega ) - x(t_{j-1};\omega)\big]\biggr\ } \xi_\vartheta^\nu({\mathrm{d}}\omega)\,,\end{gathered}\ ] ] where @xmath70 . let us stress that the pvm depends on the observables and , so , on the parameters @xmath71 and @xmath56 , while the choices of the trajectory space ( the measurable space @xmath61 ) and of the process @xmath72 are independent of @xmath71 and @xmath56 . then , we introduce the characteristic functional @xmath73 all the finite - dimensional probabilities of the increments of the process @xmath72 are determined by @xmath74\,,\end{gathered}\ ] ] where we have introduced the test function @xmath75 , with @xmath67 , @xmath68 . the fact that the theory gives in a simple direct way the distribution for the increments of the process @xmath72 , rather than its finite - dimensional distributions , is related also to the interpretation : the output @xmath72 actually is obtained by a continuous observation of the generalized process @xmath76 followed by post - measurement processing . starting from the characteristic functional it is possible to obtain the moments of the output process @xmath77 and to express them by means of quantities concerning only system @xmath1.@xcite let us denote by @xmath78 the expectation with respect to @xmath79 ; for the first two moments we obtain the expressions [ moments ] @xmath80={\operatorname{tr}}\left\{\left(z(t)+z(t)^*\right)\eta_t\right\},\ ] ] @xmath81 = \delta(t - s ) \\ { } + { \operatorname{tr}}\left\{\left(z(t_2)+z(t_2)^*\right)\upsilon(t_2,t_1)\left[z(t_1)\eta_{t_1}+\eta_{t_1 } z(t_1)^*\right ] \right\},\end{gathered}\ ] ] where @xmath82 , @xmath83 and @xmath84 in the classical theory of stochastic processes , the spectrum is related to the fourier transform of the autocorrelation function . let @xmath85 be a stationary real stochastic process with finite moments ; then , the mean is independent of time @xmath86={\operatorname{\mathbb{e}}}[y(0)]=:m_y$ ] , @xmath87 , and the second moment is invariant under time translations @xmath88={\operatorname{\mathbb{e}}}[y(t - s)y(0)]=:r_y(t - s ) , \qquad \forall t , s\in { \mathbb{r}}\,.\ ] ] the function @xmath89 , @xmath90 , is called the _ autocorrelation function _ of the process . obviously , we have @xmath91=r_y(t - s)-m_y^{\,2}$ ] . the _ spectrum _ of the stationary stochastic process @xmath85 is the fourier transform of its autocorrelation function : @xmath92 this formula has to be intended in the sense of distributions . for instance , if @xmath93\in l^1({\mathbb{r}})$ ] , we can write @xmath94 { \mathrm{d}}\tau\,.\ ] ] by the properties of the covariance , the function @xmath93 $ ] is positive definite and , by the properties of positive definite functions , this implies @xmath95 { \mathrm{d}}\tau\geq 0;\ ] ] then , also @xmath96 . by using the stationarity and some tricks on multiple integrals , one can check that an alternative expression of the spectrum is @xmath97.\ ] ] the advantage now is that positivity appears explicitly and only positive times are involved . expression can be generalized also to processes which are stationary only in some asymptotic sense and to singular processes as our @xmath77 . let us consider our output @xmath76 under the physical probability @xmath79 . we call `` spectrum up to time @xmath98 '' of @xmath77 the quantity @xmath99.\ ] ] when the limit @xmath100 exists , we can speak of _ spectrum of the output _ , but this existence depends on the specific properties of the concrete model . by writing the second moment defining the spectrum as the square of the mean plus the variance , the spectrum splits in an elastic or coherent part and in an inelastic or incoherent one : @xmath101 @xmath102 \right\vert}^2,\ ] ] @xmath103 \\ { } + \frac 1 t{\operatorname{var}}_\rho^{\vartheta,\nu}\left[\int_0^t \sin\mu t\,{\mathrm{d}}x(t ) \right ] , \end{gathered}\ ] ] let us note that @xmath104 by using the expressions for the first two moments we get the spectrum in a form which involves only system operators : [ spectrums ] @xmath105 @xmath106 \right\},\end{gathered}\ ] ] @xmath107 equations give the spectrum in terms of the reduced description of system @xmath1 ( the fields are traced out ) ; this is useful for concrete computations . but the general properties of the spectrum are more easily obtained by working with the fields ; so , here we trace out first system @xmath1 . let us define the reduced field state @xmath108 and the field operators @xmath109 let us stress that @xmath110 commutes with its adjoint and that @xmath111 . by using eqs . and and taking first the trace over @xmath0 , we get [ eq : sp2i(t ) ] @xmath112 to elaborate the previous expressions it is useful to introduce annihilation and creation operators for bosonic modes , which are only approximately orthogonal for finite @xmath98 : @xmath113 @xmath114= [ a_t^\dagger(\omega ) , a_t^\dagger(\omega^\prime)]=0 , \\ [ a_t(\omega ) , a_t^\dagger(\omega^\prime)]= \begin{cases } 1 & \text{for } \omega^\prime=\omega , \\ \frac{{\mathrm{e}}^{{\mathrm{i}}(\omega -\omega^\prime)t}-1 } { { \mathrm{i}}(\omega -\omega^\prime)t } & \text{for } \omega^\prime\neq \omega . \end{cases}\end{gathered}\ ] ] then , we have easily @xmath115 [ thetadependence ] @xmath116 @xmath117 independently of the system state @xmath118 , of the field state @xmath33 and of the hudson - parthasarathy evolution @xmath119 , for every @xmath71 and @xmath56 we have the two bounds @xmath120 the first bound comes easily from @xmath121 and eqs . . to prove the second bound , let us introduce the operator @xmath122 which satisfy the ccr @xmath123= 1 , \qquad [ b_t(\omega ) , b_t(\omega)]= [ b_t^\dagger(\omega ) , b_t^\dagger(\omega)]=0.\ ] ] then , we can write @xmath124 the usual tricks to derive the heisenberg - scrdinger - robertson uncertainty relations can be generalized also to non - selfadjoint operators.@xcite for any choice of the state @xmath125 and of the operators @xmath126 , @xmath127 ( with finite second moments with respect to @xmath125 ) the @xmath128 matrix with elements @xmath129 is positive definite and , in particular , its determinant is not negative . then , we have @xmath130 by taking @xmath131 , @xmath132 , + @xmath133 , we get @xmath134 but we can change @xmath135 in @xmath136 and we have also @xmath137 the two inequalities together give @xmath138 which is what we wanted . ref . introduces a class of operators for the electromagnetic field , called _ two - mode quadrature - phase amplitudes _ , which have the structure of our operators @xmath110 . anyway only two modes are involved , as if we fixed @xmath139 and @xmath56 . let us denote here those operators by @xmath140 . the paper explicitly constructs a class of quasi - free ( or gaussian ) field states @xmath141 for which @xmath142 . such states are called _ two - mode squeezed states_. more generally , one speaks of _ squeezed light _ if , at least in a region of the @xmath135 line , for some @xmath71 one has @xmath143 . if this happens , the heisenberg - type relation says that necessarily @xmath144 in such a way that the product is bigger than one . let us take as system @xmath1 a two - level atom , which means @xmath145 , @xmath146 ; @xmath147 is the _ resonance frequency _ of the atom . we denote by @xmath148 and @xmath149 the lowering and rising operators and by @xmath150 , @xmath151 , @xmath152 the pauli matrices ; we set also @xmath153 . we stimulate the atom with a coherent monochromatic laser and consider homodyne detection of the fluorescence light . the quantum fields @xmath2 model the whole environment . the electromagnetic field is split in two fields , according to the direction of propagation : one field for the photons in the forward direction ( @xmath154 ) , that of the stimulating laser and of the lost light , one field for the photons collected to the detector ( @xmath155 ) . assume that the interaction with the atom is dominated by absorption / emission and that the direct scattering is negligible : @xmath156 the coefficient @xmath157 is the natural _ line - width _ of the atom , @xmath158 is the fraction of fluorescence light which reaches the detector and @xmath159 is the fraction of lost light ( @xmath160).@xcite we introduce also the interaction with a thermal bath , @xmath161 and a term responsible of _ dephasing _ ( or decoherence ) , @xmath162 to represent a coherent monochromatic laser of frequency @xmath163 , we take @xmath164}(t)$ ] ; @xmath98 is a time larger than any other time in the theory and the limit @xmath100 is taken in all the physical quantities . the quantity @xmath165 is called _ rabi frequency _ and @xmath166 is called _ detuning_. the squeezing in the fluorescence light is revealed by homodyne detection , which needs to maintain phase coherence between the laser stimulating the atom and the laser in the detection apparatus which determines the observables @xmath46 ; this in particular means that necessarily we must take @xmath167 . the limit @xmath100 can be taken in eqs . and it is independent of the atomic initial state.@xcite the result is @xmath168 @xmath169 where @xmath170\,\rho_\textrm{eq}\big)\,\vec\sigma\big],\qquad \vec{s}=\begin{pmatrix}\cos\vartheta\\ \sin\vartheta\\0\end{pmatrix } , \\ \rho_\textrm{eq}=\frac{1}{2}\left(1+\vec{x}_\textrm{eq}\cdot\vec\sigma\right ) , \qquad \vec{x}_\textrm{eq } = -\gamma a^{-1}\ , \begin{pmatrix}0\\0\\1\end{pmatrix } , \\ a=\begin{pmatrix}\gamma\left(\frac{1}{2}+\overline{n}+2k_{\mathrm{d}}\right ) & \delta \omega&0\\ -\delta \omega & \gamma\left(\frac{1}{2}+\overline{n}+2k_{\mathrm{d}}\right)&\omega\\ 0&-\omega&\gamma\left(1 + 2\overline{n } \right)\end{pmatrix}.\end{gathered}\ ] ] examples of inelastic spectra are plotted for @xmath171 , @xmath172 , @xmath173 , and two different values of @xmath174 . the rabi frequency @xmath64 and @xmath175 are chosen in both cases to get good visible minima of @xmath176 below 1 . thus in this case the analysis of the homodyne spectrum reveals the squeezing of the detected light . also complementary spectra are shown to verify theorem 2.1 . one could also compare the homodyne spectrum with and without @xmath177 and @xmath178 , thus verifying that the squeezing is very sensitive to any small perturbation . a. barchielli , g. lupieri , _ quantum stochastic calculus , operation valued stochastic processes and continual measurements in quantum mechanics _ , j. math . * 26 * ( 1985 ) 22222230 . a. barchielli , _ measurement theory and stochastic differential equations in quantum mechanics _ , phys . a * 34 * ( 1986 ) 16421649 . v. p. belavkin , _ nondemolition measurements , nonlinear filtering and dynamic programming of quantum stochastic processes_. in a. blaquire ( ed . ) , _ modelling and control of systems _ , lecture notes in control and information sciences * 121 * ( springer , berlin , 1988 ) pp . 245265 . a. barchielli , _ direct and heterodyne detection and other applications of quantum stochastic calculus to quantum optics _ , quantum opt . * 2 * ( 1990 ) 423441 . h. j. carmichael , _ an open system approach to quantum optics _ notes phys . * m 18 * ( springer , berlin , 1993 ) . h. m. wiseman , g. j. milburn , _ interpretation of quantum jump and diffusion processes illustrated on the bloch sphere _ a * 47 * ( 1993 ) 16521666 . h. m. wiseman , g. j. milburn , _ quantum theory of optical feedback via homodyne detection _ 70 * ( 1993 ) 548551 . a. barchielli , _ continual measurements in quantum mechanics and quantum stochastic calculus_. in s. attal , a. joye , c .- a . pillet ( eds . ) , _ open quantum systems iii _ , lecture notes in mathematics * 1882 * ( springer , berlin , 2006 ) , pp . 207291 .
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when a quantum system is monitored in continuous time , the result of the measurement is a stochastic process .
when the output process is stationary , at least in the long run , the spectrum of the process can be introduced and its properties studied .
a typical continual measurement for quantum optical systems is the so called homodyne detection . in this paper
we show how the heisenberg uncertainty relations give rise to characteristic bounds on the possible homodyne spectra and we discuss how this is related to the typical quantum phenomenon of squeezing .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
quantum system restricted on a bounded domain has become more relevant for theoretical physics . there , the role of boundary conditions are very important not only for the long distance ( infrared ) regime but also for the short distance ( ultraviolet ) regime . mathematically , the correct framework to treat the boundary conditions in quantum theory is by means of the analysis of von neumann s self - adjoint extension of the hamiltonian operator @xcite . physically speaking , the variety of boundary conditions provided by the self - adjoint extension of the hamiltonian implies that the very rich structure of point interactions available in quantum theory . the analysis of self - adjoint extension of the hamiltonian , as the name suggests , is essentially based on the hamiltonian operator approach . however , in the feynman s path - integral approach , we do not know _ a priori _ how to incorporate the boundary conditions into the integration measure nor into the path - integral weight . as discussed in many textbooks ( see for example @xcite ) , the naive path - integral representation for a system on a bounded domain leads to a wrong boundary behavior and hence requires modification . the most rigorous way to incorporate the boundary conditions into the feynman kernel is to evaluate it by the operator formalism . however , the kernel evaluated by the operator formalism becomes the summation over the energy spectrum . in order to switch to the path - integral description , we have to perform resummation of the energy spectrum to the paths of the space . in general , this resummation is accomplished by _ trace formulae_. trace formulae provide a direct connection between quantum energy spectrum and classical length spectrum ( periodic orbits ) . however , this connection is in general an asymptotic relation valid for large wave numbers , just as in the case of the gutzwiller trace formula @xcite . there are only few cases where the trace formulae become identities . noteworthy among these are the poisson summation formula and the selberg trace formula @xcite , the former is the trace formula for the laplace operator on flat tori and the latter on riemannian manifolds with constant negative curvature . although in the operator formalism point interactions have been extensively discussed in the literature , the path - integral description of point interactions has not been fully understood yet . in mathematically speaking , this is mainly due to the lack of trace formulae suitable for the point interactions . in physically speaking , on the other hand , this is mainly due to the lack of our knowledge about the classical trajectories for a particle in the presence of point interactions . the aim of this paper is to fulfill a gap of the description for boundary conditions between the operator formalism and the path - integral formalism : we would like to propose a physically transparent prescription how to incorporate the boundary conditions obtained in the operator formalism into the path - integral description . to illustrate our idea in a simple setting , in this paper we will consider a one - particle quantum mechanics on a circle in the presence of a single point interaction . to begin with , let us first consider a quantum particle on a circle of circumference @xmath1 in the presence of a @xmath2-interaction described by the hamiltonian @xmath3 , where @xmath4 is the dimensionless coupling constant and prime ( @xmath5 ) indicates the derivative with respect to @xmath6 . ( here as in the following we are using units where @xmath7 . ) it is known that the @xmath2-interaction belongs to the so - called scale - independent subfamily of point interactions @xcite and is verified by the boundary conditions @xmath8\psi(0)$ ] and @xmath9\psi^{\prime}(0)$ ] @xcite , where @xmath10 is the square integrable wave function on an interval @xmath11 . although the feynman kernel of this system has been analyzed in the literature @xcite , the physical interpretation for the weight factors ( see below ) remains open . we would like to first address this issue . as ubiquitous in the scale - independent point interactions , in this @xmath2-interaction case the wave numbers are quantized in an integer step so that it is easy to rewrite the feynman kernel @xmath12 evaluated in the operator formalism into the path - integral representation with the help of the poisson summation formula . the resultant kernel takes the form @xmath13 where @xmath14<\pi$ ] and @xmath15 ( @xmath16 ) sign for @xmath17 ( @xmath18 ) . @xmath19 is the principal value of the inverse cosine . notice that the presence of a point interaction breaks the global translational invariance . as a consequence the kernel is the sum of partial amplitudes for the translational invariant and variant classes of trajectories , which are weighted by the factors @xmath20 and @xmath21 respectively . before going to discuss the physical meaning of the weight factors , we have to reveal the particle propagations described by . to this end , it should first be noted that @xmath22 leads to @xmath23 so that eq . becomes the well - known form of the one - particle feynman kernel on a circle with periodic boundary conditions . as discussed in many textbooks ( see for example ref.@xcite ) , in this @xmath22 case the kernel is the sum of partial amplitudes for transitions via classical paths distinguished by the homotopy class of @xmath24 , i.e. , the winding number . for nonzero @xmath25 , however , the classical trajectories of a particle are not so trivial due to the presence of @xmath2-potential , which acts as a point scatterer . when a particle reaches to the position of the point scatterer , there must be in general two possibilities : reflection or transmission . thus the paths for a particle interacting @xmath0-times to the point interaction must consist of @xmath26 distinct paths . as an example the classical world lines for @xmath27 and @xmath28 in are depicted in figure [ fig : world_lines ] . [ cols="^,^,^,^ " , ] we emphasize that the eigenvalues of the s - matrix depend only on the three parameters @xmath29 , @xmath30 and @xmath31 , whereas the eigenvectors depend on the full parameters of @xmath32 . the reason will be explained as follows : since the eigenvalues of the s - matrix on @xmath33 correspond to the energy spectrum of @xmath34 , we here explain why the energy spectrum of @xmath34 does not depend on @xmath35 and @xmath36 . we first point out that the parity operator @xmath37 is well - defined on @xmath34 . let us then consider the following ( singular ) unitary transformation : @xmath38 where @xmath39 which acts on the two - component vectors as follows : @xmath40 since the unitary transformation leaves the hamiltonian invariant , the energy spectrum remains the same but the boundary condition is changed as @xmath41 where @xmath42 thus , we found that the unitary transformation has an effect on @xmath43 by a rotation of the angle @xmath44 , i.e. @xmath45 this implies that the energy spectrum should depend on not @xmath43 but their invariant @xmath46 , which is identical to @xmath47 . thus the spectrum depends only on the three parameters @xmath48 . the main success of this work is the systematic description for a one - particle feynman kernel on a circle with a point interaction . the point is that we do not need any knowledge of the spectrum nor the complete set of energy eigenfunctions of the system ( except for the bound states ) . what we have to know is the classical trajectories of a particle and the one - particle scattering matrix . we are left with a number of questions , however . let us close with a few comments on these issues : 1 . _ more rigorous foundation of partial amplitude unitarity_. + we showed that the particle propagation scattered @xmath0-times from the point interaction should be weighted by the elements of @xmath49 . as a direct consequence of the unitarity of the s - matrix , these weight factors satisfies the relation @xmath50 , which we proposed to call _ partial amplitude unitarity_. as briefly discussed in section [ sec : examples ] , in the case of reflectionless subfamily of point interactions our partial amplitude unitarity and the laidlaw - dewitt theorem seem to be the same thing . however , the theorem is essentially based on the homotopy theory so that it could not be applied in general to the other subfamily of point interactions . we thought that the partial amplitude unitarity would provide a wider notion than the laidraw - dewitt theorem and should be derived from some fundamental properties of the feynman kernel such as the unitarity @xmath51 . + another related issue is an algebraic structure for the construction of classical trajectories for a particle on @xmath34 . as mentioned before , the classical trajectories for a particle interacting @xmath0-times to the point interaction consist of @xmath26 distinct paths . these trajectories are constructed from the more fundamental trajectories depicted in figure [ fig : s - matrix ] by gluing them under the multiplication rule of the matrix @xmath52 . this fact implies that , in spite of the presence of a point singularity , it might be possible to introduce some kind of notion analogous to the fundamental group to the space @xmath34 . however , we have no idea to treat these problems path - integral representation for the kernels_. + the feynman kernels derived in this paper have the forms which will be obtained after performing the path - integration . it is very interesting to investigate its path - integral representation . as mentioned in the introduction , boundary conditions are treated unambiguously in operator formalism by von neumann s theory of self - adjoint extension . however , in path - integral formalism we do not know _ a priori _ what kind of trajectories we should integrate over and what kind of `` classical action '' we should adopt as a weight . indeed , even in the system of a free particle in an infinitely deep well potential , the `` classical action '' we should adopt as a path - integral weight includes the so - called _ topological term _ that localizes at the boundaries @xcite . furthermore , this topological term is proportional to the planck constant so that it is no longer classical . + to that aim we have to consider the limit @xmath53 of the equation : @xmath54 \langle x|\mathrm{e}^{-ih\delta t}|x_{n-1}\rangle \cdots \langle x_{1}|\mathrm{e}^{-ih\delta t}|x_{0}\rangle,\end{aligned}\ ] ] where @xmath55 . we already know the exact form of each partial amplitude @xmath56 . in order to evaluate the right hand side we have to tackle with the @xmath57 products of both translational invariant and variant classes of infinite sums . we would like to report this issue elsewhere . s.o . would like to thank y. adachi and k. sakamoto for valuable conversations and the latter also for the collaborations on several issues relevant in this paper . m.s . would like to thank k. takenaga and t. nagasawa for useful discussions . this work is supported in part by the grant - in - aid for scientific research ( no.18540275 ( m.s . ) ) by the japanese ministry of education , science , sports and culture . 15 natexlab#1#1bibnamefont # 1#1bibfnamefont # 1#1citenamefont # 1#1url # 1`#1`urlprefix[2]#2 [ 2][]#2 , _ _ ( ) . , _ _ ( ) . , _ _ ( ) . , _ _ , * * ( ) . , _ _ , * * ( ) . , , _ _ , * * ( ) . , _ _ , * * ( ) . , , _ _ , * * ( ) . , _ _ , * * ( ) , http://arxiv.org/abs/quant-ph/9910062[arxiv:quant-ph/9910062 ] . , , _ _ , * * ( ) , http://arxiv.org/abs/quant-ph/0307002[arxiv:quant-ph/0307002 ] . , , _ _ , * * ( ) , http://arxiv.org/abs/quant-ph/0105066[arxiv:quant-ph/0105066 ] . , _ _ , * * ( ) , http://arxiv.org/abs/cond-mat/0501110[arxiv:cond-mat/0501110 ] . , _ _ , * * ( ) . , , _ _ , * * ( ) , http://arxiv.org/abs/0712.4353[arxiv:0712.4353[quant-ph ] ] . , _ _ , * * ( ) . , , _ _ , * * ( ) , http://arxiv.org/abs/quant-ph/0111057[arxiv:quant-ph/0111057 ] .
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we study a particle propagation on a circle in the presence of a point interaction .
we show that the one - particle feynman kernel can be written into the sum of reflected and transmitted trajectories which are weighted by the elements of the @xmath0-th power of the scattering matrix evaluated on a line with a point interaction . as a by - product we find three - parameter family of trace formulae as a generalization of the poisson summation formula .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
since their first detection by cobe , the cmb temperature fluctuations have become an essential tool for constraining cosmological parameters . from the beginning of 2000 , new experiments have released data set of good quality up to the third acoustic peak . better constraints have been obtained on several cosmological parameters . nevertheless , it has been shown that even with precise measurements of the power spectrum , it is nearly impossible to distinguish models with the same physical parameters on the last scattering surface . basically , some degeneracies are inherent to the cmb . we consider in this work x ray clusters as an independent way for constraining cosmological parameters . in the present analysis we consider the data from cobe , boomerang , maxima and dasi . we analysed the likelihood of inflationary models with @xmath0 , @xmath1 as free cosmological parameters . this corresponds to @xmath2 models tested.to derive the likelihood values for the models we considered , we used the method developed in bartlett et al . ( 2000 ) , and already used in le dour et al . ( 2000 ) . the results are presented as two - dimensional contours plots of the likelihood projected onto various parameters planes . this 2-d presentation has the advantage of showing clearly all the degeneracies between parameters . the left contour plots of figure 1 show some degeneracies between our parameters . basically , @xmath3 , @xmath4 and @xmath5 are strongly degenerated if only cmb data are used in parameter estimation . this means that one should consider another source of constraints , independent from cmb observations , for a better cosmological parameter estimation . we choose x - ray clusters as additional constraints . clusters are interesting object for which both the luminous baryonic mass ( x - ray emitting intracluster gas ) and the total gravitating mass can be determined . therefore , an upper limit on the fraction baryon , @xmath6 , can be estimated . results are not necessary in agreement allowing values between @xmath7 and @xmath8 . in this work , we consider the constraints given in sadat and blanchard 2001 @xcite for the baryon fraction : @xmath9 . the aim of this contribution being to show that clusters may help for breaking degeneracies shown by cmb parameters estimations , we did not consider constraints given by other groups . a better consideration of these different results on x - ray clusters will be found in douspis et al . ( 2001b ) . given the constraints of cmb and x - ray clusters , one could combined the two by multiplying the corresponding likelihood . the right figure of fig . 1 shows the constraints from the combined analysis in the same plane as the left figure . we can see that the degeneracies are broken . confidence intervals are now determined for @xmath3 and @xmath4 . for both parameters , low values are preferred : @xmath10 and @xmath11 at 99% cl . due to inherent degeneracies in the cmb it is nearly impossible to specify some of the cosmological parameter ; `` cross constraints '' are then necessary . this work is thus a preliminary view of the kind of constraints one would obtain in a near future , using last x ray satellite data , and in a less near future the cmb satellite data ( map and planck ) .
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we present the results of a combined analysis of cosmic microwave background ( cmb ) and x - ray galaxy clusters baryon fraction to deduce constraints over 6 inflationnary cosmological parameters .
such a combination is necessary for breaking degeneracies inherent to the cmb .
# 1_#1 _ # 1_#1 _ = # 1 1.25 in .125 in .25 in
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You are an expert at summarizing long articles. Proceed to summarize the following text:
a maximally extended and regular covering of the schwarzschild vacuum is now a fundamental part of any introduction to general relativity . almost always the covering is given by way of the implicit kruskal @xcite - szekeres @xcite procedure @xcite . however , as emphasized long ago by ehlers @xcite , an explicit maximally extended and regular covering of the schwarzschild vacuum is known and was first given by israel @xcite . unfortunately , despite the fact that these coordinates offer many advantages over the kruskal - szekeres coordinates , they are almost never used . in this paper i extend israel s procedure to the schwarzschild - de sitter class of vacua in four dimensions and then to arbitrary dimensions @xmath0 assuming a @xmath1 sphere as in the tangherlini generalization of the schwarzschild vacuum . moreover , it is made clear that these coordinates offer important advantages over the kruskal - szekeres procedure . these advantages include : an explicit representation of the line element that can be extended to arbitrary dimension , a simultaneous covering of both the black hole and cosmological horizons and derivation by direct integration of einstein s equations @xcite without recourse to coordinate transformations . we start with a spherically symmetric spacetime in coordinates @xmath2 where @xmath3 is a null four - vector so that the line element takes the form @xcite @xmath4 where @xmath5 is the metric of a unit two - sphere ( @xmath6 ) . further , setting @xmath7 ( so that trajectories of constant @xmath8 and @xmath9 are radial null geodesics affinely parameterized by @xmath10 ) it follows that @xmath11 . defining @xmath12 and rewriting @xmath13 as @xmath14 we set @xmath15 in ( [ generalmetric ] ) which remains a completely general spherically symmetric spacetime . the expansion of @xmath16 reduces to @xmath17 associated with the vector field @xmath18 we have @xmath19 , @xmath20 and @xmath21 the trajectories of constant @xmath22 and @xmath9 are radial null geodesics ( affinely parameterized by @xmath14 ) only for @xmath23 . note that if @xmath24 then @xmath25 is tangent to a radial null geodesic and clearly @xmath10 is again affine . the associated geodesics immediately follow as @xmath26 where @xmath27 is a constant . important examples of this special case are given below . for the spacetime ( [ generalmetric ] ) with @xmath15 , @xmath29 and @xmath30 where @xmath31 ( @xmath32 ) and @xmath33 ( @xmath34 ) are constants , it follows that @xmath35 @xmath36 and @xmath37 where @xmath38 @xmath39 is the ricci tensor , @xmath40 the ricciscalar and @xmath41 the weyl tensor . viewing @xmath28 as a constant of nature and @xmath33 a property of the vacuum , it is clear from ( [ lambda ] ) that the specification of @xmath31 does not determine the physical situation uniquely . since , by ( [ r(u , w ) ] ) , the axes are centered on @xmath42 , it is important to distinguish ranges in @xmath31 . writing @xmath43 with @xmath44 and @xmath45 it follows immediately from ( [ lambda ] ) that @xmath46 it follows from ( [ crange ] ) that all values of @xmath47 are determined uniquely by @xmath48 using the following ranges : @xmath49 for @xmath50 and @xmath51 for @xmath52 . the latter range is of no interest and is not discussed here . the one independent invariant derivable from the riemmann tensor without differentiation can be taken to be ( [ weylscalar ] ) and so with @xmath14 and @xmath10 extending over the reals , the vacua are maximally extended and regular for @xmath53 ( and for all finite @xmath54 if @xmath55 ) . in particular , note that @xmath56 with ( [ r(u , w ) ] ) it follows that @xmath57 from ( [ metric ] ) and ( [ r(u , w ) ] ) it follows that trajectories of constant @xmath22 and @xmath9 are radial null geodesics only for @xmath58 . more generally , the radial null geodesics that satisfy @xmath59 with @xmath60 given by ( [ metric ] ) and ( [ r(u , w ) ] ) , can be written down in terms of elementary functions ( as explained below ) . note that for these trajectories @xmath61 as @xmath62 ( @xmath63 ) and @xmath64 as @xmath65 . we distinguish the cases : @xmath55 ( de sitter ) , @xmath66 ( bertotti - kasner ) , @xmath67 ( schwarzschild ) , @xmath68 ( schwarzschild - de sitter ) , @xmath69 ( schwarzschild - anti de sitter ) and discuss them below . for the case @xmath67 the coordinates were first obtained by israel @xcite . not covered as special cases are anti de sitter space ( see appendix a ) and the degenerate schwarzschild - de sitter spacetime ( see appendix b ) . with @xmath55 , @xmath70 , @xmath71 and we have de sitter space . the metric simplifies to @xmath72 trajectories with four tangents @xmath73 are radial null geodesics ( so @xmath10 is affine for both radial null directions ) . we can write these geodesics in the form @xmath74 where @xmath27 is a constant . the negative cosmological horizons ( @xmath75 ) are then given by @xmath76 where the expansion @xmath77 ( @xmath78 ) vanishes . some details are shown in fig . [ desitterfig ] . with @xmath79 we have bertotti - kasner space . the metric simplifies to @xmath80 trajectories with four tangents @xmath81 are radial null geodesics ( so again @xmath10 is affine for both radial null directions ) but now @xmath82 . we can write these geodesics in the form @xmath83 where @xmath27 is a constant . the @xmath84 plane is like that of de sitter space now with @xmath85 ( again @xmath76 ) distinguishing the branches of the @xmath86 geodesics . this space shows that the birkhoff theorem does not extend directly to @xmath87 ( see the case @xmath88 below ) @xcite . with @xmath67 , @xmath89 , @xmath90 and we have the schwarzschild vacuum in israel coordinates @xcite . the metric simplifies to @xmath91 trajectories with four tangents @xmath92 are radial null geodesics so @xmath10 is not affine . now @xmath93 . we write these geodesics in the form @xmath94 where @xmath27 is a constant . some details are shown in fig . [ schwarzschild ] . again writing @xmath43 it follows that for @xmath95 there is another constant @xmath96 for which @xmath97 . the constant is given by @xmath98 with @xmath99 the trajectories @xmath100 are radial null geodesics ( with non - zero expansion ) tangent to the cosmological " horizons . note that @xmath101 as @xmath102 and @xmath103 as @xmath104 . now define @xmath105 and @xmath106 trajectories with four tangents @xmath107 where @xmath108 and @xmath109 where @xmath27 is a constant , are radial null geodesics . now we find that the expansion is given by @xmath110 where @xmath111 the choice @xmath112 reproduces the bertotti - kasner result and the choice @xmath113 reproduces the schwarzschild vacuum result both for @xmath114 as given above . for @xmath95 these geodesics can be given explicitly in terms of elementary functions and are reproduced in appendix c. some details are shown in fig . [ schwarzdesitter ] . for @xmath116 there are no cosmological horizons but equations ( [ adef ] ) through ( [ fdef ] ) hold as given above . again the associated radial null geodesics can be given explicitly in terms of elementary functions and are reproduced in appendix d. some details are shown in fig . [ antidesitterimage ] . this section parallels ii but extends the construction to arbitrary dimensions ( @xmath118 ) . write @xmath119 where @xmath120 is the metric of a unit @xmath1 sphere . again @xmath3 is tangent to null geodesics affinely parameterized by @xmath10 but now the associated expansion is given by @xmath121 further , associated with the vector field @xmath18 we again have @xmath19 and @xmath122 but now the associated expansion is @xmath123 the trajectories for which all coordinates are constant except @xmath14 are radial null geodesics ( affinely parameterized by @xmath14 ) only for @xmath23 . finally , as before , if @xmath24 then @xmath124 is tangent to a radial null geodesic with @xmath10 again affine and the associated geodesics immediately follow as @xmath125 where @xmath27 is a constant . this section generalizes section iii above . for dimensions @xmath126 consider the spaces described by @xmath127 where the function @xmath128 is given by @xmath129 and @xmath54 signifies the function @xmath130 with @xmath31 ( @xmath32 ) and @xmath33 ( @xmath34 ) constants . it follows that @xcite @xmath131 @xmath132 and @xmath133 where @xmath134 again , the one independent invariant derivable from the riemmann tensor without differentiation can be taken to be ( [ weylscalargen ] ) and so with @xmath14 and @xmath10 extending over the reals , the spaces described by ( [ metricsgen ] ) are maximally extended and regular for @xmath53 ( and for all @xmath54 if @xmath135 , see below ) . in particular , note that @xmath136 again viewing @xmath28 as a constant of nature and @xmath33 a property of the vacuum , the specification of @xmath31 does not determine the physical situation uniquely . this is discussed below . in view of the generality of the cases considered , the formulae given are remarkably simple . consider the hyper - spherically symmetric spacetime @xmath137 note that @xmath54 is now a coordinate . it is not difficult to show that the unique form of @xmath60 which satisfies ( [ riccigen ] ) is given by @xmath138 where @xmath139 is a constant ( which we take @xmath140 ) . this solution was apparently first discussed by tangherlini @xcite . the coordinates are , of course , defective at @xmath141 , but the associated roots depend on @xmath0 and are not so easy to find from ( [ curvaturef ] ) . the previous construction circumvents this unnecessary algebra as follows : for ( [ curvature ] ) with ( [ curvaturef ] ) we find @xmath142 and so from ( [ weylscalargen ] ) and ( [ weylsqcurv ] ) ( assuming , without loss in generality , the same coordinates for @xmath120)we have the relation @xmath143 which gives @xmath144 only for @xmath145 . now substituting for @xmath139 from ( [ mrelation ] ) into ( [ curvaturef ] ) and writing an associated root to @xmath141 as @xmath42 we obtain ( [ lambdagen ] ) . in this way we never need to deal with ( [ curvaturef ] ) and so all dimensions @xmath118 are equally difficult to deal with as regards the location of horizons . the previous section raises the issue of mass " . in spacetime with @xmath89 the concept of mass in the spherically symmetric case is well established @xcite . the geometrical mass in @xmath146 dimensions for spaces admitting a @xmath147sphere has been defined previously by way of the sectional curvature @xcite . whereas this is merely a formal definition , it is of interest to compare this definition with @xmath33 and @xmath139 . in dimension @xmath0 define @xmath148 in the special case @xmath149 it follows that @xmath150 more generally , for @xmath151 , it follows that @xmath152 where @xmath28 is given by ( [ lambdagen ] ) . we can , of course , replace @xmath139 by @xmath33 in ( [ geomasslambda ] ) by solving for @xmath33 from ( [ mrelation ] ) . we now consider some specific cases . we define hyper - spherical de sitter space by the requirement @xmath135 so that from ( [ weylscalargen ] ) and ( [ weylsqcurv ] ) we have @xmath153 and so @xmath154 the metric follows as @xmath155 trajectories with four tangents @xmath156 are radial null geodesics ( so @xmath10 is affine for both radial null directions ) . we can write these geodesics in the form @xmath157 where @xmath27 is a constant . the negative cosmological horizons ( @xmath158 ) are then given by @xmath76 where the expansion @xmath77 ( @xmath159 ) vanishes . the associated diagram is qualitatively the same as fig . [ desitterfig ] . with @xmath160 we have hyper - spherical bertotti - kasner space . the metric simplifies to @xmath161 trajectories with four tangents @xmath162 are radial null geodesics ( so again @xmath10 is affine for both radial null directions ) and again @xmath82 . we can also write these geodesics in the form ( [ bertottigeo ] ) . asymptotically flat static vacuum black holes ( admitting the @xmath1 sphere ) are unique @xcite and given by the tangherlini generalization of the schwarzschild vacuum @xcite . in our notation these correspond to the case @xmath67 . global , regular and explicit coordinates for these spaces have been given previously @xcite . some further discussion is given in appendix e. more generally , the radial null geodesics that satisfy @xmath163 with @xmath128 given by ( [ fgen ] ) and @xmath54 by ( [ rfngen ] ) , are of interest . for these trajectories , excluding the cases discussed above , it is clear that @xmath61 as @xmath164 and @xmath64 as @xmath165 ( @xmath166 ) . from ( [ lambdagen ] ) , writing @xmath43 we have @xmath167 . in the range @xmath168 there is another constant @xmath169 for which @xmath170 . the trajectories @xmath171 are radial null geodesics and are tangent to the cosmological " horizons . it is also clear that for @xmath172 the solutions to ( [ othernullhigh ] ) evolve in a fundamentally different way as in the case @xmath145 . in fact , the qualitative behavior of all solutions to ( [ othernullhigh ] ) can be obtained without explicit integration . the essential conclusion is that the figures given for @xmath145 hold , qualitatively , for @xmath173 . this is discussed in detail elsewhere @xcite . a maximally extended , explicit and regular covering of the schwarzschild - de sitter vacua in arbitrary dimension ( @xmath126 ) has been given . it has bee stressed that these coordinates offer important advantages over the kruskal - szekeres procedure that include : an explicit representation of the line element that can be extended to arbitrary dimension , a simultaneous covering of both the black hole and cosmological horizons and derivation by direct integration of einstein s equations without recourse to coordinate transformations . in view of the generality of the problem solved , the resultant formulae obtained are of a remarkably simple form . part of this work was reported at the the dark side of extra dimensions " workshop held in may 2005 at the banff international research station ( birs ) , banff , alberta . it is a pleasure to thank werner israel and don page for comments and taylor binnington for drawing the figures . this work was supported by a grant from the natural sciences and engineering research council of canada and was made possible by use of _ grtensoriii _ @xcite . anti - de sitter space is not contained in ( [ metric ] ) with ( [ r(u , w ) ] ) but can be given in the form ( [ generalmetric ] ) with @xmath15 and is included here for completeness . it is given by @xmath174 where the constant @xmath175 . equations ( [ ricci ] ) and ( [ ricciscalar ] ) of course hold and the space is conformally flat . trajectories with four tangents @xmath176 are radial null geodesics so @xmath10 is again affine . we can write these geodesics in the form @xmath177 where @xmath27 is a constant . with @xmath14 and @xmath10 extending over the reals , ( [ antidesitter ] ) is maximally extended and regular . the degenerate schwarzschild - de sitter black hole has @xmath178 like the bertotti - kasner space but it is not contained in ( [ metric ] ) with ( [ r(u , w ) ] ) but can be given in the form ( [ generalmetric ] ) with @xmath15 and is included for completeness . coordinates for degenerate black holes are seldom discussed @xcite . now @xmath179 with @xmath180 in addition to the radial null geodesics given by constant @xmath14 and @xmath58 , the other radial null geodesics can be given by @xmath181 where @xmath182 is the lambert w function @xcite with @xmath183 and @xmath184 where @xmath27 is a constant . some details are shown in fig . [ degenschwarzdesitterimage ] . for @xmath95 the radial null geodesics with tangents @xmath114 can be given in a simple form : @xmath185 is a constant of motion where @xmath186 and @xmath187 writing @xmath188 with @xmath189 the radial null geodesics with tangents @xmath114 can be given in the form @xmath190 where @xmath27 is a constant . in the present notation we set @xmath67 and so have ( [ tangherlini ] ) with @xmath191 where @xmath192 with @xmath145 we recover ( [ schw ] ) . even implicit forms of @xmath54 in the kruskal - szekeres procedure are not known for for all @xmath0 ( _ e.g _ @xmath193 @xcite ) . however here we need only substitute for @xmath0 and insert ( [ rfn ] ) into ( [ tangherlinif ] ) to give a maximally extended , explicit and regular coverings of the space . for example , with @xmath194 we have @xmath195 solutions to ( [ othernull ] ) with ( [ tangherlinif ] ) and ( [ rfn ] ) can be given explicitly in terms of elementary functions in some cases . for example , with @xmath194 we have m. d. kruskal , phys . 119 * , 1743 ( 1960 ) . g. szekeres , publ . debrecen * 7 * , 285 ( 1960 ) ( now available at general relativity and gravitation * 34 * , 2001 , ( 2002 ) ) . although usually obtained by way of coordinate transformations , the kruskal - szekeres coordinates can be obtained in an algorithmic way without guessing transformations . see , for example , k. lake in _ vth brazilian school of cosmology and gravitation _ edited by m. novello ( world scientific , singapore , 1987 ) . j. ehlers in _ relativity , astrophysics and cosmology . proceedings of the summer school held , 14 - 26 august , 1972 at the banff centre , banff , alberta _ edited by w. israel ( reidel , boston , 1973 ) . w. israel , phys . rev . * 143 * , 1016 ( 1966 ) . israel coordinates have been obtained independetly by d. w. pajerski and e. t. newman , j. math . * 12 * , 1929 ( 1971 ) and by t. klsch and t. strobl , class . * 13 * , 1191 ( 1996 ) arxiv:(gr - qc/9507011 ) . israel @xcite ( in the published form of his work ) introduced the coordinates by way of a transformation which , in his notation reads as @xmath197 . in this paper einstein s equations for vacua have been solved directly without prior coordinates . this is not possible for kruskal - szekeres coordinates wherein even at @xmath145 with @xmath198 the vacuum field equations are difficult to solve . see , for example , lake @xcite . we use einstein s equations ( with cosmological constant ) , geometrical units and a signature of @xmath1 . the sign conventions for @xmath10 and @xmath14 is that @xmath10 can decrease and @xmath14 increase only in spacelike directions . for convenience , explicit functional dependence is usually shown only on the first appearance of a function . but see k. lake , phys . d * 20 * , 370 ( 1979 ) , j. podolsky , gen . . grav . * 31 * , 1703 ( 1999 ) arxiv : gr - qc/9910029 , s. gao , phys . d * 68 * , 044028 ( 2003 ) arxiv:(gr - qc/0207029 ) and r. balbinot , s. fagnocchi , a. fabbri , s. farese and j. navarro - salas , phys . d * 70 * , 064031 ( 2004 ) arxiv:(hep - th/0405263 ) for a detailed discussion see w. rindler , physics letters a * 245 * , 363 ( 1998 ) . this is defined by @xmath199 . see , for example , r. m. corless , g. h. gonnet , d. e. g. hare , d. j. jeffrey , and d. e. knuth , advances in computational mathematics * 5 * , 329 ( 1996 ) . if the cosmological constant @xmath200 is defined ( in dimension @xmath0 ) by the vacuum field equations @xmath201 where @xmath202 is the @xmath0 dimensional einstein tensor , then @xmath203 . f. r. tangherlini , nuovo cimento * 27 * , 636 ( 1963 ) . see , for example , w. c. hernandez and c. w. misner , astrophys . j. * 143 * , 452 ( 1966 ) , m. e. cahill and g. c. mcvittie , j. math . phys , 11 , * 1360 * ( 1970 ) , e. poisson and w. israel , phys . rev d * 41 * , 1796 ( 1990 ) , t. zannias , phys . d * 41 * , 3252 ( 1990 ) and s. hayward , phys . d * 53 * , 1938 ( 1996 ) arxiv:(gr - qc/9408002 ) k. lake , arxiv:(gr - qc/0606005 ) s. hwang , geometriae dedicata * 71 * , 5 ( 1998 ) , g. w. gibbons , d. ida and t. shiromizu , prog . phys . suppl . * 148 * , 284 ( 2003 ) arxiv:(gr - qc/0203004 ) . k. lake , j. cosmol . jcap * 10 * 007 ( 2003 ) arxiv:(gr - qc/0306073 ) k. lake ( in preparation ) . this is a package which runs within maple . it is entirely distinct from packages distributed with maple and must be obtained independently . the grtensorii software and documentation is distributed freely on the world - wide - web from the address _ http://grtensor.org _ grtensoriii software is in development .
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maximally extended , explicit and regular coverings of the schwarzschild - de sitter family of vacua are given , first in spacetime ( generalizing a result due to israel ) and then for all dimensions @xmath0 ( assuming a @xmath1 sphere ) .
it is shown that these coordinates offer important advantages over the well known kruskal - szekeres procedure .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
numerous precision tests of the standard model ( sm ) and searches for its possible violation have been performed in the last few decades , serving as an invaluable tool to test the theory at the quantum level . they have also provided stringent constraints on many `` new physics '' ( np ) scenarios . a typical example is given by the measurements of the anomalous magnetic moment of the electron and the muon , where recent experiments reached the fabulous relative precision of 0.7 ppb @xcite and 0.5 ppm,@xcite respectively . these experiments measure the so - called gyromagnetic factor @xmath1 , defined by the relation between the particle s spin @xmath5 and its magnetic moment @xmath6 , = g , where @xmath7 and @xmath8 are the charge and mass of the particle . in the dirac theory of a charged point - like spin-@xmath9 particle , @xmath10 . quantum electrodynamics ( qed ) predicts deviations from dirac s value , as the charged particle can emit and reabsorb virtual photons . these qed effects slightly increase the @xmath1 value . it is conventional to express the difference of @xmath1 from 2 in terms of the value of the so - called anomalous magnetic moment , a dimensionless quantity defined as @xmath11 . the anomalous magnetic moment of the electron , @xmath12 , is rather insensitive to strong and weak interactions , hence providing a stringent test of qed and leading to the most precise determination of the fine - structure constant @xmath13 to date.@xcite on the other hand , the @xmath1@xmath2@xmath3 of the muon , @xmath14 , allows to test the entire sm , as each of its sectors contributes in a significant way to the total prediction . compared with @xmath12 , @xmath14 is also much better suited to unveil or constrain np effects . indeed , for a lepton @xmath15 , their contribution to @xmath16 is generally expected to be proportional to @xmath17 , where @xmath18 is the mass of the lepton and @xmath19 is the scale of np , thus leading to an @xmath20 relative enhancement of the sensitivity of the muon versus the electron anomalous magnetic moment . this more than compensates the much higher accuracy with which the @xmath1 factor of the latter is known . the anomalous magnetic moment of the @xmath0 lepton , @xmath21 , would suit even better ; however , its direct experimental measurement is prevented by the relatively short lifetime of this lepton , at least at present . the existing limits are based on the precise measurements of the total and differential cross sections of the reactions @xmath22 and @xmath23 at lep energies . the most stringent limit , @xmath24 at 95% confidence level , was set by the delphi collaboration,@xcite and is still more than an order of magnitude worse than that required to determine @xmath21 . in the 1990s it became clear that the accuracy of the theoretical prediction of the muon @xmath1@xmath2@xmath3 , challenged by the e821 experiment underway at brookhaven,@xcite was going to be restricted by our knowledge of its hadronic contribution . this problem has been solved by the impressive experiments at low - energy @xmath25 colliders , where the total hadronic cross section ( as well as exclusive ones ) were measured with high precision , allowing a significant improvement of the uncertainty of the leading - order hadronic contribution.@xcite as a result , the accuracy of the sm prediction for @xmath14 now matches that of its measurement . in parallel to these efforts , very many improvements of all other sectors of the sm prediction were carried on by a large number of theorists ( see refs . for reviews ) . all these experimental and theoretical developments allow to significantly improve the theoretical prediction for the anomalous magnetic moment of @xmath0 lepton as well . in this article we review and update the sm prediction of @xmath21 , analyzing in detail the three contributions into which it is usually split : qed , electroweak ( ew ) and hadronic . updated qed and ew contributions are presented in secs . [ sec : qed ] and [ sec : ew ] ; new values of the leading - order hadronic term , based on the recent low energy @xmath4 data from babar , cmd-2 , kloe and snd , and of the hadronic light - by - light contribution are presented in sec . [ sec : had ] . the total sm prediction is confronted to the available experimental bounds on the @xmath0 lepton @xmath1@xmath2@xmath3 in sec . [ sec : sm ] , and prospects for its future measurements are briefly discussed in sec . [ sec : conc ] , where conclusions are drawn . the qed part of the anomalous magnetic moment of the @xmath0 lepton arises from the subset of sm diagrams containing only leptons and photons . this dimensionless quantity can be cast in the general form:@xcite a_^qed = a_1 + a_2 ( ) + a_2 ( ) + a_3 ( , ) , [ eq : atauqedgeneral ] where @xmath27 , @xmath28 and @xmath29 are the electron , muon and @xmath0 lepton masses , respectively . the term @xmath30 , arising from diagrams containing only photons and @xmath0 leptons , is mass and flavor independent . in contrast , the terms @xmath31 and @xmath32 are functions of the indicated mass ratios , and are generated by graphs containing also electrons and/or muons . the functions @xmath33 ( @xmath34 ) can be expanded as power series in @xmath35 and computed order - by - order a_i = a_i^(2 ) ( ) + a_i^(4 ) ( ) ^2 + a_i^(6 ) ( ) ^3 + a_i^(8 ) ( ) ^4 + . only one diagram is involved in the evaluation of the lowest - order ( first - order in @xmath13 , second - order in the electric charge ) contribution it provides the famous result by schwinger @xmath36.@xcite the mass - dependent coefficients @xmath31 and @xmath32 , discussed below , are of higher order . they were derived using the latest codata@xcite recommended mass ratios : m_/m_e & = & 3477.48 ( 57 ) [ eq : rte ] + m_/m _ & = & 16.8183 ( 27 ) . [ eq : rtm ] the value for @xmath29 adopted by codata in ref . , @xmath37 mev , is based on the pdg 2002 result.@xcite it remained unchanged until very recently ( see refs . ) , when preliminary results of two new measurements ( from the belle@xcite and kedr@xcite detectors ) were reported . the central values of the new mass values are slightly lower than the current world average value , but agree with it within the uncertainties , which are approaching that of the world average value ( used in this work ) . seven diagrams contribute to the fourth - order coefficient @xmath38 , one to @xmath39 and one to @xmath40 . they are depicted in fig . [ fig : qed2 ] . as there are no two - loop diagrams contributing to @xmath41 that contain both virtual electrons and muons , @xmath42 . the mass - independent coefficient has been known for almost fifty years:@xcite a_1^(4 ) & = & + + ( 3 ) - 2 + & = & -0.328 478 965 579 193 78 , [ eq : a14 ] where @xmath43 is the riemann zeta function of argument @xmath44 . for @xmath45 , @xmath46 or @xmath0 , the coefficient of the two - loop mass - dependent contribution to @xmath47 , @xmath48 , with @xmath49 , is generated by the diagram with a vacuum polarization subgraph containing the virtual lepton @xmath50 . this coefficient was first computed in the late 1950s for the muon @xmath1@xmath2@xmath3 with @xmath51 , neglecting terms of @xmath52.@xcite the exact expression for @xmath53 was reported by elend in 1966.@xcite however , its numerical evaluation was considered tricky because of large cancellations and difficulties in the estimate of the accuracy of the results , so that common practice was to use series expansions instead.@xcite taking advantage of the properties of the dilogarithm @xmath54,@xcite the exact result was cast in ref . in a very simple and compact analytic form , valid , contrary to the one in ref . , also for @xmath55 ( the case relevant to @xmath56 and part of @xmath57 ) : a_2^(4)(1/x ) & = & - - + x^2 ( 4 + 3x ) + ( 1 - 5 x^2 ) + & & + + & & + x^4 . [ eq : ea24 ] for @xmath58 , gives @xmath59 ; of course , this contribution is already part of @xmath38 in . numerical evaluation of with the mass ratios given in eqs . ( [ eq : rte])([eq : rtm ] ) yields the two - loop mass - dependent qed contributions to the anomalous magnetic moment of the @xmath0 lepton,@xcite a_2^(4)(m_/m_e ) & = & 2.024 284 ( 55 ) , [ eq : ta24e ] + a_2^(4)(m_/m _ ) & = & 0.361 652 ( 38 ) . [ eq : ta24 m ] these two values are very similar to those computed via a dispersive integral in ref . ( which , however , contain no estimates of the uncertainties ) . equations ( [ eq : ta24e ] ) and ( [ eq : ta24 m ] ) are also in agreement ( but more accurate ) with those of ref . . adding up eqs . ( [ eq : a14 ] ) , ( [ eq : ta24e ] ) and ( [ eq : ta24 m ] ) one gets:@xcite c_^(4 ) = 2.057 457 ( 93 ) [ eq : tc2 ] ( note that the uncertainties in @xmath60 and @xmath61 are correlated ) . the resulting error @xmath62 leads to a @xmath63 uncertainty in @xmath41 . more than one hundred diagrams are involved in the evaluation of the three - loop ( sixth - order ) qed contribution . their analytic computation required approximately three decades , ending in the late 1990s . the coefficient @xmath64 arises from 72 diagrams . its exact expression , mainly due to remiddi and his collaborators , reads:@xcite a_1^(6 ) & = & ^2 ( 3 ) - ( 5 ) - ^4 + + + & & + ( 3 ) - ^2 2 + ^2 + + & & + + & = & 1.181 241 456 587 . [ eq : a16 ] this value is in very good agreement with previous results obtained with numerical methods.@xcite the calculation of the exact expression for the coefficient @xmath65 for arbitrary values of the mass ratio @xmath66 was completed in 1993 by laporta and remiddi @xcite ( earlier works include refs . ) . let us focus on @xmath67 ( @xmath68 , @xmath69 ) . this coefficient can be further split into two parts : the first one , @xmath70 , receives contributions from 36 diagrams containing either electron or muon vacuum polarization loops,@xcite whereas the second one , @xmath71 , is due to 12 light - by - light scattering diagrams with either electron or muon loops.@xcite the exact expressions for these coefficients are rather complicated , containing hundreds of polylogarithmic functions up to fifth degree ( for the light - by - light diagrams ) and complex arguments ( for the vacuum polarization ones ) they also involve harmonic polylogarithms.@xcite series expansions were provided in ref . for the cases of physical relevance . using the recommended mass ratios given in eqs . ( [ eq : rte ] ) and ( [ eq : rtm ] ) , the following values were recently computed from the full analytic expressions:@xcite a_2^(6)(m_/m_e , ) & = & 7.256 99 ( 41 ) [ eq : ta26evac ] + a_2^(6)(m_/m_e , ) & = & 39.1351 ( 11 ) [ eq : ta26elbl ] + a_2^(6)(m_/m_,)&= & -0.023 554 ( 51 ) [ eq : ta26mvac ] + a_2^(6)(m_/m_,)&= & 7.033 76 ( 71 ) . [ eq : ta26mlbl ] almost identical values were obtained employing the approximate series expansions of ref . : 7.25699(41 ) , 39.1351(11 ) , @xmath20.023564(51 ) , 7.03375(71).@xcite the previous estimates of ref . were different : 10.0002 , 39.5217 , 2.9340 , and 4.4412 ( no error estimates were provided ) , respectively ; they are superseded by the results in eqs . ( [ eq : ta26evac])([eq : ta26mlbl ] ) , derived from the exact analytic expressions . the estimates of ref . compare slightly better : 7.2670 , 39.6 , @xmath72 , 4.47 ( no errors provided ) . in the specific case of @xmath73 , the values of refs . and differ from because their derivations did not include terms of @xmath74 , which turn out to be unexpectedly large . the sums of eqs . ( [ eq : ta26evac])([eq : ta26elbl ] ) and ( [ eq : ta26mvac])([eq : ta26mlbl ] ) are a_2^(6)(m_/m_e ) & = & 46.3921 ( 15 ) , [ eq : ta26e ] + a_2^(6)(m_/m_)&= & 7.010 21 ( 76 ) . [ eq : ta26 m ] the contribution of the three - loop diagrams with both electron- and muon - loop insertions in the photon propagator was calculated numerically from the integral expressions of ref . , obtaining:@xcite a_3^(6)(m_/m_e , m_/m _ ) = 3.347 97 ( 41 ) . [ eq : ta36 ] this value disagrees with the results of refs . ( 1.679 ) and ( 2.75316 ) . combining the three - loop results of eqs . ( [ eq : a16 ] ) , ( [ eq : ta26e ] ) , ( [ eq : ta26 m ] ) and ( [ eq : ta36 ] ) one finds the sixth - order qed coefficient,@xcite c_^(6 ) = 57.9315 ( 27 ) . [ eq : tc3 ] the error @xmath75 induces a @xmath76 uncertainty in @xmath41 . the order of magnitude of the three - loop contribution to @xmath41 , dominated by the mass - dependent terms , is comparable to that of ew and hadronic effects ( see later ) . contrary to the case of the electron and muon @xmath1@xmath2@xmath3 , qed contributions of order higher than three are not known.@xcite ( an exception is the mass- and flavor - independent term @xmath77,@xcite which is however expected to be a very small part of the complete four - loop contribution . ) adding up all the above contributions and using the new value of @xmath13 derived in refs . and , @xmath78 , one obtains the total qed contribution to the @xmath1@xmath2@xmath3 of the @xmath0 lepton,@xcite a_^qed = 117 324 ( 2 ) 10 ^ -8 . [ eq : tqed ] the error @xmath79 is the uncertainty @xmath80 assigned to @xmath41 for uncalculated four - loop contributions . as we mentioned earlier , the errors due to the uncertainties of the @xmath81 and @xmath82 terms are negligible . the error induced by the uncertainty of @xmath13 is only @xmath83 ( and thus totally negligible ) . with respect to schwinger s contribution , the ew correction to the anomalous magnetic moment of the @xmath0 lepton is suppressed by the ratio @xmath84 , where @xmath85 is the mass of the @xmath86 boson . numerically , this contribution is of the same order of magnitude as the three - loop qed one . the analytic expression for the one - loop ew contribution to @xmath26 , due to the diagrams in fig . [ fig : ew1 ] , is:@xcite ^ew ( ) = , [ eq : ewoneloop ] where @xmath87 is the fermi coupling constant,@xcite @xmath88 , @xmath85 and @xmath89 are the masses of the @xmath90 , @xmath86 and higgs bosons , and @xmath91 is the weak mixing angle . closed analytic expressions for @xmath92 taking exactly into account the @xmath93 dependence ( @xmath94 higgs , or other hypothetical bosons ) can be found in refs . . following ref . , we employ for @xmath95 the on - shell definition,@xcite @xmath96 , where @xmath97,@xcite and @xmath85 is the theoretical sm prediction of the @xmath86 mass . the latter can be easily derived from the simple analytic formulae of ref . ( see also refs . ) , = , [ eq : fops ] ( on - shell scheme ii with @xmath98,@xcite @xmath99,@xcite and @xmath100 gev,@xcite ) leading to @xmath101 for @xmath102 . this result should be compared with the direct experimental value @xmath103,@xcite which corresponds to a very small @xmath89 . in any case , these shifts in the @xmath85 prediction induced by the variation of @xmath89 from 114.4 gev , the current lower bound at 95% confidence level,@xcite up to a few hundred gev , only change @xmath92 by amounts of @xmath104 . from , including the tiny @xmath105 corrections of refs . , for @xmath102 we get ^ew ( ) = 55.1(1 ) 10 ^ -8 . [ eq : ewoneloopn ] the uncertainty encompasses the shifts induced by variations of @xmath89 from 114 gev up to a few hundred gev , and the tiny uncertainty due to the error in @xmath29 . the estimate of the ew contribution in ref . , @xmath106 , obtained from the one - loop formula ( without the small corrections of order @xmath107 ) , is similar to our value in . however , its uncertainty ( @xmath108 ) is too small , and it does nt contain the two - loop contribution which , as we ll discuss in the next section , is not negligible . the two - loop ew contributions to @xmath16 ( @xmath45 , @xmath46 or @xmath0 ) were computed in 1995 by czarnecki , krause and marciano.@xcite this remarkable calculation leads to a significant reduction of the one - loop prediction . navely one would expect the two - loop ew contribution @xmath109 to be of order @xmath110 , but this turns out not to be so . as first noticed in the early 1990s,@xcite @xmath111 is actually quite substantial because of the appearance of terms enhanced by a factor of @xmath112 , where @xmath113 is a fermion mass scale much smaller than @xmath85 . the two - loop contribution to @xmath114 involves 1678 diagrams in the linear t hooft - feynman gauge@xcite ( as a check , the authors of refs . and employed both this gauge and a nonlinear one in which the vertex of the photon , the @xmath86 and the unphysical charged scalar vanishes ) . it can be divided into fermionic and bosonic parts ; the former , @xmath115 , includes all two - loop ew corrections containing closed fermion loops , whereas all other contributions are grouped into the latter , @xmath116 . the expressions of ref . for the bosonic part were obtained in the approximation @xmath117 , computing the first two terms in the expansion in @xmath118 , and expanding in @xmath95 , keeping the first four terms in this expansion ( this number of powers is sufficient to obtain an exact coefficient of the large logarithms @xmath119 ) . recent analyses of the ew bosonic corrections of the @xmath1@xmath2@xmath3 of the muon@xcite relaxed these approximations , providing analytic results valid also for a light higgs . considering the present @xmath120 gev lower bound,@xcite we can safely employ the results of ref . , obtaining , for @xmath102 @xmath121 . the neglected terms are of @xmath122 . the fermionic part of @xmath123 contains the contribution of diagrams with light quarks ; they involve long - distance qcd for which perturbation theory can not be employed . in particular , these hadronic uncertainties arise from two types of two - loop diagrams : the hadronic photon@xmath90 mixing , and quark triangle loops with the external photon , a virtual photon and a @xmath90 attached to them ( see fig . [ fig : ew2 ] ) . the hadronic uncertainties mainly arise from the latter ones . two approaches were suggested for their study : in ref . the nonperturbative effects where modeled introducing effective quark masses as a simple way to account for strong interactions . in view of the high experimental precision of the @xmath1@xmath2@xmath3 of the muon , a more realistic treatment of the relevant hadronic dynamics was introduced in ref . within a low - energy effective field theory approach , later on developed in the detailed analyses of refs . . however , from a numerical point of view , the discrepancy between the results provided by these two different approaches turns out to be irrelevant for the present interpretation of the experimental result of the muon @xmath1@xmath2@xmath3 , in spite of its precision . the use of effective quark masses for the study of the @xmath1@xmath2@xmath3 of the @xmath0 whose experimental precision is a far cry from that of the muon ! thus appears to be sufficient at present . the tiny hadronic @xmath124@xmath90 mixing terms can be evaluated either in the free quark approximation or via a dispersion relation using data from @xmath25 annihilation into hadrons ; the difference was shown to be numerically insignificant.@xcite references and contain simple approximate expressions for the contributions of the diagrams with fermion triangle loops shown in fig . [ fig : ew2 ] ( right ) . in general , for a lepton @xmath45 , @xmath46 or @xmath0 , neglecting small mass - ratios , they are a_l^ew ( ) _ f , [ eq : ew2lfd ] where @xmath125 is the third component of the weak isospin of the fermion @xmath126 in the loop , @xmath127 is its charge , @xmath128 its number of colors ( 3 for quarks , 1 for leptons ) , and c(f ) \ { ll ( ^2/m_l^2 ) + 5/6 & + ( ^2/m_l^2 ) - 8 ^2/27 + 11/18 & + ( ^2/m_f^2 ) -2 & + ( m_top^2/^2 ) + + + ( 5/18)(^2/m_top^2 ) -4/3 & . [ eq : ew2lfdlogs ] the contribution of the top - quark triangle loop diagram of fig . [ fig : ew2 ] ( right ) with the @xmath90 boson replaced by the neutral goldstone boson ( @xmath129 ) has also been included in this expression.@xcite it is clear from eqs . ( [ eq : ew2lfd])([eq : ew2lfdlogs ] ) that the logarithms @xmath130 cancel in sums over all fermions of a given generation , as long as @xmath131 , due to the no - anomaly condition @xmath132 valid within every generation.@xcite this does not occur for the third generation due to the large mass of the top quark . note that this short - distance cancellation does not get modified by strong interaction effects on the quark triangle diagrams.@xcite contrary to the case of the muon @xmath1@xmath2@xmath3 , where all fermion masses , with the exception of @xmath27 , enter in @xmath133 , the approximate expressions in show that this is not the case for the @xmath1@xmath2@xmath3 of the @xmath0 lepton . indeed , due to the high infrared cut - off set by @xmath29 , for @xmath134 does not depend on any fermion mass lighter than @xmath29 ; apart from @xmath29 , it only depends on @xmath135 and @xmath136 , the masses of the top and bottom quarks ( assuming @xmath137 ) . the charm contribution requires some care , as the crude approximation provided by for a charm lighter than the @xmath0 lepton is valid only if @xmath138 . clearly , this is not a good approximation , and the spurious shift induced by when @xmath139 is varied across the @xmath29 threshold is of @xmath140 . one possibility is to use with @xmath139 equal to @xmath29.@xcite better still , we numerically integrated the exact expressions for @xmath141 provided in ref . for arbitrary values of @xmath113 , obtaining a smooth dependence on the value of @xmath139 . for completeness we repeated this detailed analysis for all light fermions . as expected , the result depends very mildly on the values chosen for their masses . employing the values @xmath142 gev , @xmath143 gev , @xmath144 gev and @xmath145 gev , and adding to the contribution of the remaining fermionic two - loop diagrams studied in ref . , for @xmath146 we obtain @xmath147 . in this evaluation we also included the tiny @xmath122 contribution of the @xmath124@xmath90 mixing diagrams , suppressed by ( @xmath148 for quarks and ( @xmath149 for leptons , via the explicit formulae of ref . . the sum of the fermionic and bosonic two - loop ew contributions described above gives @xmath150 , a 14% reduction of the one - loop result . the leading - logarithm three - loop ew contributions to the muon @xmath1@xmath2@xmath3 were determined to be extremely small via renormalization - group analyses.@xcite we assigned to our @xmath0 lepton @xmath1@xmath2@xmath3 ew result an additional uncertainty of @xmath151 \!\sim\ ! o(10^{-9})$ ] to account for these neglected three - loop effects . adding @xmath152 to the one - loop value of we get our total ew correction ( for @xmath146 ) ^ew = 47.4 ( 5 ) 10 ^ -8 . [ eq : tew ] the uncertainty allows @xmath89 to range from 114 gev up to @xmath153 gev , and reflects the estimated errors induced by hadronic loop effects ( @xmath154 and @xmath155 can vary between 70 mev and 400 mev ) , neglected two - loop bosonic terms , and the missing three - loop contribution . it also includes the small errors due to the uncertainties in @xmath135 and @xmath29 . the value in is in agreement with the prediction @xmath156,@xcite with a reduced uncertainty . as we mentioned in sec . [ subsec : ew1 ] , the ew estimate of ref . , @xmath106 , mainly differs from in that it does nt include the two - loop corrections . in this section we will analyze @xmath157 , the contribution to the @xmath0 anomalous magnetic moment arising from qed diagrams involving hadrons . hadronic effects in ( two - loop ) ew contributions are already included in @xmath114 ( see the previous section ) . similarly to the case of the muon @xmath1@xmath2@xmath3 , the leading - order hadronic contribution to the @xmath0 lepton anomalous magnetic moment is given by the dispersion integral:@xcite @xmath158 where the kernel @xmath159 is a bounded function of energy monotonously increasing to unity at @xmath160 , and @xmath161 is the total hadronic cross section of the @xmath25 annihilation in the born approximation . in fig . [ fig : rat ] we plot the ratio of the kernels in the @xmath0 lepton and muon case . clearly , although the role of the low energies is still very important , the different structure of @xmath162 compared to @xmath163 , induced by the higher mass of the @xmath0 , results in a relatively higher role of the larger energies . the history of these calculations is not as rich as that of the muon . the first calculation performed in 1978 in ref . was based on experimental data available at that time below 7.4 gev , whereas at higher energies the asymptotic qcd prediction was used . ten years later , a rough estimate was made in ref . based on low energy @xmath25 data . in ref . the contribution of the @xmath164 meson was estimated by integrating the approximation obtained using the breit - wigner curve , while other contributions used the data . the accuracy of the calculation was considerably improved in refs . where , below 40 gev , only data were used . in ref . , data were only used below 3 gev ( together with the experimental parameters of the @xmath165 and @xmath166 family states ) . in our opinion this can significantly underestimate the resulting uncertainty . in addition , in the same reference , data from @xmath0 lepton decays were extensively used ; as it is known today , this leads to higher spectral functions than in @xmath25 case,@xcite and can therefore overestimate the result . the results of these calculations are summarized in table [ tab : atau ] . [ tab : atau ] for completeness , in the second part of table [ tab : atau ] we also show purely theoretical estimates . the analysis based on qcd sum rules performed in ref . gives results which strongly depend on the choice of quark and gluon condensates . qcd sum rules are also used in ref . . in ref . the authors use a nonlocal constituent quark model for the description of the photon vacuum polarization function @xmath167 at space - like momenta and obtain @xmath168 , close to the estimates based on the experimental data . they also show that a simpler model with constituent quark masses independent of momentum is strongly dependent on the values chosen for the quark masses . for example , with @xmath169 mev and @xmath170 550 mev their result is @xmath171 , i.e. , significantly smaller than the previous estimate . they could reproduce the value @xmath172 using @xmath173 mev . in a recent analysis using the instanton liquid model the author obtains @xmath174.@xcite all these estimates somewhat undervalue the hadronic contribution and have rather large uncertainties . we updated the calculation of the leading - order contribution using the whole bulk of experimental data below 12 gev , which include old data compiled in refs . , as well as the recent datasets from the cmd-2@xcite and snd@xcite experiments in novosibirsk , and from the radiative return studies at kloe in frascati@xcite and babar at slac.@xcite the improvement is particularly visible in the channel @xmath175 , where four new independent measurements exist in the most important @xmath164 meson region : cmd-2,@xcite snd,@xcite and kloe@xcite ( see fig . [ fig : pi ] ) . our result is a_^hlo = 337.5 ( 3.7 ) 10 ^ -8 [ eq : thlo ] ( we recently presented a preliminary estimate of this value in ref . ) . the breakdown of the contributions of different energy regions as well as their relative fractions in the total leading - order contribution are given in table [ tab : at ] . the contribution of the @xmath164 meson energy range is still important , but its relative weight is smaller than in the case of the muon anomaly , 51.3% compared to about 72%.@xcite the contributions of the narrow resonances ( @xmath165 and @xmath166 families ) are included in the corresponding energy regions . it is worth noting that uncertainties of the contributions from the hadronic continuum are larger than that of the very precise @xmath176 one . the overall uncertainty is 2.5 times smaller than that of the previous data - based prediction.@xcite [ tab : at ] the hadronic higher - order @xmath177 contribution @xmath178 can be divided into two parts : @xmath179 the first one is the @xmath82 contribution of diagrams containing hadronic self - energy insertions in the photon propagators . it was determined by krause in 1996:@xcite a_^hho()= 7.6 ( 2 ) 10 ^ -8 . [ eq : thhovac ] note that navely rescaling the muon result by the factor @xmath180 ( as it was done in ref . ) leads to the totally incorrect estimate @xmath181 ( the @xmath182 value is from ref . ) ; even the sign is wrong ! the second term , also of @xmath82 , is the hadronic light - by - light contribution . similarly to the case of the muon @xmath1@xmath2@xmath3 , this term can not be directly determined via a dispersion relation approach using data ( unlike the leading - order hadronic contribution ) , and its evaluation therefore relies on specific models of low - energy hadronic interactions with electromagnetic currents . actually , very few estimates of @xmath183 exist in the literature,@xcite and all of them were obtained simply rescaling the muon results @xmath184 by a factor @xmath180 . following this very nave procedure , the @xmath185 estimate varies between @xmath186\times ( m_{\tau}^2/m_{\mu}^2)= 23(11)\times 10^{-8 } $ ] , and @xmath187\times ( m_{\tau}^2/m_{\mu}^2)= 38(7)\times 10^{-8 } $ ] , according to the values chosen for @xmath188 from refs . and , respectively . these very nave estimates fall short of what is needed . consider the function @xmath71 , the three - loop qed contribution to the @xmath1@xmath2@xmath3 of a lepton of mass @xmath18 due to light - by - light diagrams involving loops of a fermion of mass @xmath189 ( see sec . [ subsec : qed3 ] ) . the exact expression of this function , computed in ref . for arbitrary values of the mass ratio @xmath66 , is rather complicated , but series expansions were provided in the same article for the cases of physical relevance . in particular , if @xmath190 , then @xmath191 . this implies that , for example , the ( negligible ) part of @xmath192 due to diagrams with a top - quark loop can be reasonably estimated simply rescaling the corresponding part of @xmath188 by a factor @xmath180 . on the other hand , to compute the dominant contributions to @xmath192 , i.e.those induced by the light quarks , we need the opposite case : @xmath193 . in this limit , @xmath71 does not scale as @xmath194 , and a nave rescaling of @xmath184 by @xmath180 to derive @xmath195 leads to an incorrect estimate . we therefore decided to perform a parton - level estimate of @xmath192 based on the exact expression for @xmath71 using the quark masses recently proposed in ref . for the determination of @xmath184 : @xmath196 mev , @xmath197 mev , @xmath198 gev and @xmath199 gev ( note that with these values the authors of ref . obtain @xmath200 , in perfect agreement with the value in ref . see also ref . for a similar earlier determination ) . we obtain a_^hho()= 5 ( 3 ) 10 ^ -8 . [ eq : thholbl ] this value is much lower than those obtained by simple rescaling of @xmath188 by @xmath180 . the up - quark provides the dominating contribution ; the uncertainty @xmath201 allows @xmath154 to range from 70 mev up to 400 mev . further independent studies ( following the approach of ref . , for example ) would provide an important check of this result . the total hadronic contribution to the anomalous magnetic moment of the @xmath0 lepton can be immediately derived adding the values in eqs . ( [ eq : thlo ] ) , ( [ eq : thhovac ] ) and ( [ eq : thholbl ] ) , a_^had = a_^hlo+ a_^hho()+ a_^hho ( ) = 350.1 ( 4.8 ) 10 ^ -8 . [ eq : thad ] errors were added in quadrature . we can now add up all the contributions discussed in the previous sections to derive the sm prediction for @xmath21 : a_^sm = a_^qed + a_^ew + a_^hlo + a_^hho ( ) + a_^hho ( ) , [ eq : sm ] where @xmath202 ( the sum of the hadronic contributions is given in ) . adding errors in quadrature , our final result is a_^sm = 117 721 ( 5 ) 10 ^ -8 . [ eq : nsm ] the present pdg limit on the anomalous magnetic moment of the @xmath0 lepton was derived in 2004 by the delphi collaboration from @xmath22 total cross section measurements at @xmath203 between 183 and 208 gev at lep2:@xcite -0.052 < a _ < 0.013 [ eq : exp_delphi1 ] at 95% confidence level . the authors of ref . also quote their result in the form of central value and error : a _ = -0.018 ( 17 ) . [ eq : exp_delphi2 ] comparing this result with ( their difference is roughly one standard deviation ) , it is clear that the sensitivity of the best existing measurements is still more than an order of magnitude worse than needed . a reanalysis of various measurements of the cross section of the process @xmath204 , the transverse @xmath0 polarization and asymmetry at lep and sld , as well as of the decay width @xmath205 at lep and tevatron , allowed to set a stronger model - independent limit:@xcite -0.007 < a _ 0.005 . other limits on @xmath21 can be found in refs . . in this article we reviewed and updated the sm prediction of the @xmath0 lepton @xmath1@xmath2@xmath3 . updated qed and electroweak contributions were presented , together with new values of the leading - order hadronic term , based on the recent low energy @xmath4 data from babar , cmd-2 , kloe and snd , and of the hadronic light - by - light contribution . these results were confronted in sec . [ sec : sm ] to the available experimental bounds on the @xmath0 lepton anomaly . as we already mentioned in the introduction , quite generally , np associated with a scale @xmath19 is expected to modify the sm prediction of the anomalous magnetic moment of a lepton @xmath15 of mass @xmath18 by a contribution @xmath206 . therefore , given the large factor @xmath207 , the @xmath1@xmath2@xmath3 of the @xmath0 lepton is much more sensitive than the muon one to ew and np loop effects that give contributions @xmath208 , making its measurement an excellent opportunity to unveil ( or just constrain ) np effects . another interesting feature can be observed comparing the magnitude of the ew and hadronic contributions to the muon and @xmath0 lepton @xmath1@xmath2@xmath3 . the ew contribution to the @xmath1@xmath2@xmath3 of the @xmath0 is only a factor of seven smaller than the hadronic one , compared to a factor of 45 for the @xmath1@xmath2@xmath3 of the muon . also , while the ew contribution to @xmath209 is only a factor of three larger than the present uncertainty of the hadronic contribution , this factor raises to 10 for the @xmath0 lepton . if a np contribution were of the same order of magnitude as the ew one , from a purely theoretical point of view , the @xmath1@xmath2@xmath3 of the @xmath0 would provide a much cleaner test of the presence ( or absence ) of such np effects than the muon one . indeed , if this were the case , such a np contribution to the @xmath0 lepton @xmath1@xmath2@xmath3 would be much larger than the hadronic uncertainty , which is currently the limiting factor of the sm prediction . unfortunately , the very short lifetime of the @xmath0 lepton makes it very difficult to determine its anomalous magnetic moment by measuring its spin precession in the magnetic field , like in the muon @xmath1@xmath2@xmath3 experiment.@xcite instead , experiments focused on high - precision measurements of the @xmath0 lepton pair production in various high - energy processes , comparing the measured cross sections with the qed predictions.@xcite as we can see from , the sensitivity of the best existing measurements is still more than an order of magnitude worse than that required to determine @xmath21 . nonetheless , the possibility to improve such a measurement is certainly not excluded . for example , it was suggested to determine the @xmath0 lepton @xmath1 factor taking advantage of the radiation amplitude zero which occurs at the high - energy end of the lepton distribution in radiative @xmath0 decays.@xcite this method requires a very good energy resolution and could perhaps be employed at a @xmath0-charm or @xmath210 factory also benefiting from the possibility to collect very high statistics . it is not clear whether the huge data samples at @xmath210 factories will result in a corresponding gain for the limits on @xmath21 . indeed , lep measurements were rather limited by systematic uncertainties , which were of the order of 2 - 3% for the discussed processes and , until now , experiments at @xmath210 factories have not yet reached such a level of accuracy in the absolute measurements of the total cross sections . however , a search for the @xmath0 lepton electric dipole moment at belle@xcite showed that with the appropriate choice of observables , using full information about events , the improvement in sensitivity can be proportional to the square root of luminosity , i.e. , determined mainly by statistics . one can hope that this is also the case with the determination of @xmath21 . a similar method to study @xmath21 using radiative @xmath86 decays and potentially very high data samples at lhc was suggested in ref . . yet another method would use the channeling in a bent crystal similarly to the suggestion for the measurement of magnetic moments of short - living baryons.@xcite this method has been successfully tested by the e761 collaboration at fermilab , which measured the magnetic moment of the @xmath211 hyperon.@xcite in the case of the @xmath0 lepton , it was suggested to use the decay @xmath212 , which would produce polarized @xmath0 leptons.@xcite in 1991 , when this suggestion was published , the idea seemed completely unlikely . however , in the era of @xmath210 factories , when the decay @xmath212 is already observed by the belle collaboration,@xcite and the possibility of a super-@xmath210 factory is actively discussed , this is no longer a dream . even more promising could be the realization of this idea in a dedicated experiment at a hadron collider with its huge number of @xmath210 mesons produced and a more suitable geometry . we believe that a detailed feasibility study of such an experiment , as well as further attempts to improve the accuracy of the theoretical prediction for @xmath21 , are quite timely . we would like to thank m. giacomini and f.v . ignatov for many valuable comments and collaborations on topics presented in this manuscript . we are greatly indebted to a. vainshtein for communications concerning the hadronic light - by - light contribution to @xmath21 and to f. jegerlehner for many fruitful discussions . we are also grateful to k. inami for an interesting discussion on the feasibility of @xmath21 measurements at @xmath210 factories . thanks the dipartimento di fisica , universit di padova and infn , sezione di padova , where part of this work was done , for its hospitality . the work of s.e . was supported in part by the grants of rfbr 06 - 02 - 04018 and 06 - 02 - 16156 as well as by the grant of dfg gz : 436 rus 113/769/0 - 2 . is grateful to the instituto de fsica da universidade federal da bahia , brasil , for the hospitality during a visit when this manuscript was finalized . the work of m.p . was supported in part by the european community s marie curie research training networks under contracts mrtn - ct-2004 - 503369 and mrtn - ct-2006 - 035505 . all diagrams were drawn with jaxodraw.@xcite g.w . bennett _ et al . _ , phys . d * 73 * ( 2006 ) 072003 ; phys . * 92 * ( 2004 ) 161802 ; phys . * 89 * ( 2002 ) 101804 ; _ ibid . _ * 89 * ( 2002 ) 129903 ; h.n . brown _ et al . _ , phys . rev . lett . * 86 * ( 2001 ) 2227 . g. gabrielse , d. hanneke , t. kinoshita , m. nio , and b. odom , phys . rev . lett . * 97 * ( 2006 ) 030802 . k. melnikov and a. vainshtein , _ theory of the muon anomalous magnetic moment _ , springer , 2006 ; m. passera , nucl . * 155 * ( 2006 ) 365 ; m. davier and w.j . marciano , annu . nucl . part . * 54 * ( 2004 ) 115 ; m. knecht , arxiv : hep - 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this article reviews and updates the standard model prediction of the @xmath0 lepton @xmath1@xmath2@xmath3 . updated qed and electroweak contributions
are presented , together with new values of the leading - order hadronic term , based on the recent low energy @xmath4 data from babar , cmd-2 , kloe and snd , and of the hadronic light - by - light contribution .
the total prediction is confronted to the available experimental bounds on the @xmath0 lepton anomaly , and prospects for its future measurements are briefly discussed .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the deuteron , the only @xmath5 nucleus , provides the simplist microscopic test of the _ conventional nuclear model _ , a framework in which nuclei and nuclear interactions are explained as baryons interacting through the exchange of mesons . with improved nucleon - nucleon force models from the 1990s @xcite , and advances in our understanding of relativistic bound state techniques , more accurate calculations of deuteron structure are possible . during the 1990s there have also been revolutionary improvements in our experimental knowledge of deuteron electromagnetic structure . the start of experiments at the thomas jefferson national accelerator facility ( jlab ) has now made available continuous high energy beams , with high currents and large polarization , along with new detector systems . several experiments have now significantly extended the energy and momentum transfer range of deuteron electromagnetic studies , including @xmath0 and @xmath6 for elastic @xmath3 scattering , and photodisintegration cross sections and polarizations . existing experimental proposals promise to continue this trend . other laboratories have also made several important measurements , generally at lower momentum transfer . in this context , a review of the deuteron electromagnetic studies , examining the current status of the agreement between experiments and theory , is appropriate . we attempt to cover our current knowledge of the deuteron electromagnetic structure , focussing on the recent jlab results , and prospects for the future . we do not consider experiments that use the deuteron as a neutron target , for example , or for studies of the ( extended ) gerasimov - drell - hearn sum rule , deep inelastic scattering , or baryon resonance production in nuclei . our interest is on experiments that probe the conventional picture of a nucleus as composed of baryons and mesons , and that probe how far models with these effective degrees of freedom can be extended . table [ dexptab ] is a summary of some of the jlab exeriments that fit this description , and that we will review in the sections below . the high precision , large momentum transfer measurements may be sensivitive to effects not incorporated in the conventional nuclear model . it seems self - evident that probes of short distances , well below the size of the nucleon , should require explicit consideration of the quark substructure of the nucleons . our review suggests that evidence for the appearance of these effects seems to depend on the nature of the reaction . in elastic scattering , where only the @xmath7 chanel is expicitly excited , a successful description is obtained using a relativistic description of the @xmath7 channel together with a minor modification of the short - range structure of the nucleon current ( see sec . [ sec : elasticscat ] ) . in photodisintegration by 4 gev photons , where hundreds of @xmath8 channels are explicitly excited , an efficient explanation seems to require the explicit use of quark degrees of freedom ( see sec . [ photodis ] ) . @lll experiment & reaction / observables & status + & _ elastic scattering _ & + 91 - 026 & @xmath0 & paper published @xcite + & @xmath1 & analysis in progress + 94 - 018 & @xmath0 & paper published @xcite + & @xmath6 & paper published @xcite + & _ electrodisintegration _ & + 89 - 028 & recoil proton polarization & analysis in progress + 94 - 004 & in - plane response functions & analysis in progress + 94 - 102 & high momentum structure & awaiting beam time + 00 - 103 & threshold @xmath9 & proposal + & _ photodisintegration _ & + 89 - 012 & cross sections & paper published @xcite + 89 - 019 & @xmath10 , @xmath11 , @xmath12 & paper published @xcite + 96 - 003 & cross sections & paper published @xcite + 93 - 017 & cross sections & analysis in progress + 99 - 008 & cross sections & analysis in progress + 00 - 007 & @xmath10 , @xmath11 , @xmath12 & awaiting beam time + 00 - 107 & @xmath10 , @xmath11 , @xmath12 & awaiting beam time + this review begins with a survey of deuteron wave functions , and then discusses the deuteron form factors , threshold electrodisintegration , and high energy deuteron photodisintegration . we also call attention to recent reviews by garon and van orden @xcite , and by sick @xcite . these reviews contain a discussion of the static properties of the deuteron and a survey of recent models of the nucleon form factors , two topics we have decided to omit from this work . both also have an extensive discussion of the deuteron form factors . calculations of deuteron form factors and photo and electrodisintegration to the @xmath7 final state require a deuteron wave function , the final state @xmath7 scattering amplitude ( if the transition is inelastic ) , and the current operator , all of which should be consistently determined from the underlying dynamics . deuteron wave functions used in the _ conventional nuclear model _ will be reviewed in this section . the nonrelativistic @xmath7 wave function of the deuteron can be written in terms of two scalar wave functions . in coordinate space the full wave function is @xmath13 where @xmath14 are the spherical harmonics normalized to unity on the unit sphere , @xmath15 and @xmath16 are the reduced @xmath17 and @xmath18-state wave functions , and the @xmath19 distinguishes this from other ( relativistic ) components of the wave function to be described below . the spin part of the wave function is @xmath20 introducing the familar compact notation for matrix operations on each of the two nucleon subspaces 1 and 2 @xmath21 where @xmath0 is any @xmath22 operator , we can show that @xmath23 these identities permit us to write the wave function ( [ rspacewf ] ) in a convenient operator form @xcite : @xmath24\;\chi^{1m}_{_{12 } } \label{rspacewf2}\end{aligned}\ ] ] in momentum space the deuteron wave function becomes @xmath25\;\chi^{1m}_{_{12 } } \ , . \label{ft1}\end{aligned}\ ] ] we use the same notation for both coordinate and momentum space wave functions . if @xmath26 and @xmath27 , then @xmath28 note the appearance of the factors @xmath29 , a feature of the symmetric definition ( [ ft1 ] ) . the normalization condition @xmath30 implies @xmath31=\int_0^\infty p^2\,dp \left[u^2(p ) + w^2(p)\right]\ ] ] the @xmath18-state probability , @xmath32 is an interesting measure of the strength of the tensor component of the @xmath7 force , even though it is a model dependent quantity with no unique measurable value @xcite . the best nonrelativistic wave functions are calculated from the schrdinger equation using a potential adjusted to fit the @xmath7 scattering data for lab energies from 0 to 350 mev . the quality of realistic potentials have improved steadily , and now the best potentials give fits to the @xmath7 data with a @xmath33/d.o.f @xmath34 . the paris potential @xcite was among the first potentials to be determined from such realistic fits , and it has since been replaced by the argonne v18 potential ( denoted by av18 ) @xcite , the nijmegen potentials @xcite , and most recently by the cd bonn potentials @xcite . the @xmath17 and @xmath18-state wave functions determined from three of these models are shown in figs . [ uwr ] and [ uwp ] . these figures also show @xmath17 and @xmath18 wave functions from two relativistic models to be discussed shortly . in the right panel of the second figure we plot the dimensionless ratios @xmath35 and @xmath36 , where the scaling functions , in units of gev@xmath37 , are @xmath38 here @xmath39 is the deutreron binding energy and @xmath40 the nucleon mass ( we used @xmath41 mev@xmath42 ) , @xmath43 gev@xmath42 , and @xmath44 gev@xmath42 . we emphasize that these scaling functions _ have absolutely no theoretical significance _ and were introduced merely to remove the most rapid momentum dependence so that the percentage difference between models can be more easily read from the ratio graph . we conclude that the five models shown are almost identical ( i.e.variations of less than 10% ) for momenta below about 400 mev , and that they vary by less than a factor of 2 as the momenta reaches 1 gev ( except near the zeros ) . the definition of the relativistic deuteron wave function depends in large part on the formalism used to treat relativity . in formalisms based on hamiltonian dynamics ( discussed in sec . [ hexamples ] ) the wave function in the deuteron rest frame can be taken to be identical to the nonrelativistic wave function , and no further discussion is necessary until the wave function in a moving frame is needed . in formalisms based on the bethe - salpeter equation @xcite , the covariant spectator equation @xcite , or on some other quasi - potential equation @xcite , the wave functions usually have additional components which do not vanish in the rest frame . in the relativistic spectator formalism @xcite , where one of the two bound nucleons is off - mass shell , the wave function is a sum of a positive energy component and a negative energy component @xcite @xmath45 where @xmath46 and @xmath47 are nucleon spinors for the off - shell particle ( particle 2 in this case ) with dirac index @xmath48 . the positive energy part has the same structure as the nonrelativistic wave function , with an @xmath17 and @xmath18-state component . the new negative energy wave function has the form @xmath49\,\chi^{1m}_{ab}\ , , \label{pwf}\end{aligned}\ ] ] where @xmath50 and @xmath51 are two additional @xmath52-state components of the wave function , and @xmath53 is the nuclear spin 0 wave function . the names of the @xmath52-state wave functions follow from the fact that @xmath54 couples to the spin triplet ( @xmath55 ) and @xmath56 to the spin singlet ( @xmath57 ) wave function . the equivalence of the two forms given in eq . ( [ pwf ] ) follows from identities like those given in eq . ( [ identity ] ) . two model relativistic @xmath17 and @xmath18-state wave functions were shown in figs . [ uwr ] and [ uwp ] . both models are based on relativistic one boson exchange model developed in ref . model iib is a revised version of the model of the same name originally described in ref . @xcite and model w16 is one of a family of models with varying amounts of off - shell sigma coupling that were introduced in connection the relativistic calculations of the triton binding energy described in ref . these models are described further in a number of conference talks @xcite . the relativistic @xmath52-state components are small , but can make important contributions to the deuteron magnetic form factor . as figs . [ uwr ] and [ uwp ] show , the large @xmath17 and @xmath18-state components of these relativistic wave functions are very close to their nonrelativistic counterparts . because of the very small value of the electromagnetic fine structure constant ( @xmath58 ) , elastic electron deuteron scattering is described to high precision by assuming that the electron exchanges a single virtual photon when scattering from the deuteron . in this one - photon exchange approximation @xcite elastic scattering is fully described by three deuteron form factors @xcite . in its most general form , the relativistic deuteron current can be written @xcite @xmath59-g_3(q^2)\frac{(\xi'^*\cdot q)(\xi\cdot q)}{2m_d^2}\right\ } \,(d^\mu+d'^\mu ) \nonumber\\ & & + g_m(q^2)\;[\xi^\mu(\xi'^*\cdot q ) -\xi'^{*\mu}(\xi\cdot q)]\biggr)\ , , \label{deutcurrent}\end{aligned}\ ] ] where the form factors @xmath60 , @xmath61 , are all functions of @xmath62 , the square of the four - momentum transferred by the electron , with @xmath63 and @xmath64 . [ in most of the following discussion we will suppress the explicit @xmath62 dependence of the form factors . ] in practice , @xmath65 and @xmath66 are replaced by a more physical choice of form factors @xmath67 with @xmath68 . at @xmath69 , the form factors @xmath70 , @xmath71 , and @xmath72 give the charge , magnetic and quadrupole moments of the deuteron @xmath73 the form factors can also be related to the helicity amplitudes of the deuteron current ( where helicity is the projection of the spin in the direction of the particle three - momentum ) . in the breit frame ( where the energy transfer @xmath74 is zero ) the polarizations of the incoming ( @xmath75 ) and outgoing ( @xmath76 ) deuteron are @xmath77 ( where the phases for the incoming deuteron follow the conventions of jacob and wick @xcite for particle 2 ) and the virtual photon polarization is @xmath78 hence , denoting the helicity amplitudes by @xmath79 , the three independent amplitudes are @xmath80 where @xmath81 . the scattering amplitude in the one - photon approximation is @xmath82\,\left < d'|j_\mu|d\right > \ , , \label{matrix}\end{aligned}\ ] ] where @xmath83 and @xmath84 are electron spinors with @xmath85 ( @xmath86 ) the momentum and helicity of the incoming ( outgoing ) electron , respectively . squaring ( [ matrix ] ) , summing over the final spins , and averaging over initial spins give the following result for the unpolarized differential cross section @xmath87 = { { d\sigma}\over{d\omega } } \bigg|_{ns}\,s(q^2,\theta ) \label{edxsect}\ ] ] where @xmath88 is defined by this relation , and @xmath89 is the cross section for scattering from a particle without internal structure ( @xmath90 is the mott cross section ) , and @xmath91 , @xmath92 , and @xmath93 are the electron scattering angle , the incident and final electron energies , and the solid angle of the scattered electron , all in the lab system . the structure functions @xmath0 and @xmath1 depend on the three electromagnetic form factors @xmath94 where the definitions of @xmath95 , @xmath96 , and @xmath97 should be clear from the context . while cross section measurements can determine @xmath0 , @xmath1 , and @xmath71 , separating the charge @xmath70 and quadrupole @xmath72 form factors requires polarization measurements . the polarization of the outgoing deuteron can be measured in a second , analyzing scattering . the cross section for the double scattering process can be written @xcite @xmath98\ , , \label{eq : ds}\end{aligned}\ ] ] where @xmath99 is the polarization of the incoming electron beam , @xmath100 the angle between the two scattering planes ( defined in the same way as the @xmath101 shown in fig . [ fig_cross ] ) , and @xmath102 and the @xmath103 are the vector and tensor analyzing powers of the second scattering . although there is a @xmath104 component to the vector polarization , the term is omitted from eq . ( [ eq : ds ] ) as there is no longitudinal vector analyzing power ; without spin precession , this term can not be determined . the polarization quantities @xmath105 and @xmath106 ( sometimes denoted @xmath107 , but we will reserve capital letters for target asymmetries ) are functions of the form factors and the electron scattering angle @xmath108^{1/2}g_m(g_c+ { \textstyle{1\over3}}\eta g_q)\,\tan{\textstyle{1\over2}}\theta\nonumber\\ s\,p_z=\;\;\,{\textstyle{2\over3}}\eta\bigl[(1+\eta)(1+\eta\sin^2 { \textstyle{1\over2}}\theta)\bigr]^{1/2}g^2_m \tan { \textstyle{1\over2}}\theta\,\sec{\textstyle{1\over2}}\theta\nonumber\\ - \sqrt{2 } s\ , t_{20 } = { \textstyle{8\over3 } } \,\eta\ , g_c g_q + { \textstyle{8\over9}}\,\eta^2 g_q^2 + { \textstyle{1\over3 } } \eta \bigl[1 + 2(1+\eta)\tan^2 { \textstyle{1\over2}}\theta\bigr ] g_m^2 \nonumber\\ \sqrt{3}\ , s \,t_{21 } = 2 \,\eta \bigl[\eta + \eta^2\sin^2 { \textstyle{1\over2}}\theta\bigr]^{1/2 } g_m g_q\ , \sec { \textstyle{1\over2}}\theta \nonumber\\ -\sqrt{3 } s \,t_{22 } = { \textstyle{1\over2 } } \,\eta \,g_m^2\end{aligned}\ ] ] the same combinations of form factors occur in the tensor polarized target asymmetry as in the recoil deuteron tensor polarization . of these quantities , @xmath109 has been most extensively measured ; it does not require a polarized beam or a measurement of the out of plane angle @xmath100 . for measurements of @xmath0 and @xmath2 at forward electron scattering angles , the @xmath71 terms are very small , and one may approximate @xmath0 and @xmath2 by @xmath110 introducing @xmath111 gives @xmath112 the minimum of @xmath113 is reached for @xmath114 . the node in the charge form factor , @xmath115 , occurs when @xmath116 , and @xmath117 , giving @xmath118 . this approximation also makes it clear that @xmath119 largely depends on the deuteron structure , rather than the nucleon electromagnetic form factors . in the nonrelativistic limit ( to be discussed shortly ) , both @xmath70 and @xmath72 are a product of the nucleon isoscalar electric form factor multiplied by the _ body form factor _ , which is an integral over products of the deuteron wave functions weighted by spherical bessel functions . hence , in this approximation , the nucleon electric form factor cancels in the ratio @xmath120 , and @xmath119 depends only on the deuteron wave function . we note that the relations above between the form factors and the observables are model independent , so it is possible to extract form factors from the data and compare directly to theoretical calculations . the most complete form factor determination appeared recently in ref . @xcite ( see also the analysis in ref . we will discuss the data below in sec . [ thyandexp ] , after we have reviewed the experiments and the theory . the initial measurements of elastic @xmath3 scattering were by mcintyre and hofstadter in the mid 1950s @xcite . since then many experiments have run at several laboratories ; the fits of ref . @xcite include 269 cross sections from 19 references , dating from 1960 to the presentan important feature of the recent fits of the world data is that the measured cross sections were refit rather than using extracted structure functions or form factors . this is necessary since most extractions of @xmath0 ( @xmath1 ) used corrections for contributions of @xmath1 ( @xmath0 ) to their cross sections from earlier data . in some cases alternative definitions ( or incorrect formulas ) have been given . a minor point is the definition of @xmath121 ; in some cases the recoil factor @xmath122 is included , while in our definition eq . ( [ mott ] ) it is not . the magnetic form factor @xmath71 can be in units of @xmath123 ( our convention ) , @xmath124 , or dimensionless , with magnitude of 1.714 , 0.857 , or 1.0 , respectively , at @xmath62 = 0 , and leading to modified coefficients in eq . ( [ aandb ] ) . buchanan and yearian @xcite have an alternate definition of @xmath70 , and @xmath72 , with @xmath125 . benaksas @xcite and galster @xcite include an extra factor @xmath126 in the magnetic terms in @xmath0 and @xmath1 , which changes the @xmath62 dependence of the magnetic form factor , though not its value at @xmath62 @xmath127 0 . ganichot @xcite and grossette @xcite both use a factor of @xmath128 , rather than @xmath129 , in their definitions of the mott cross section . cramer @xcite give a dimensionally incorrect formula for their @xmath130 ( @xmath127 @xmath131 ) , with explicitly stated energy factors of @xmath132 , as opposed to our factor of @xmath133 . . polarization experiments are much more difficult . the first results were published in 1984 and there are now only 20 published @xmath6 data points , and 19 points for other polarization observables . the fits of ref . @xcite include a slightly larger data base , with 340 points for momentum transfers up to @xmath134 of about 1.6 gev ; this misses only a handful of the largest momentum transfer slac and jlab data points . forward - angle cross section measurements suffice to determine @xmath0 , both because @xmath1 is small and because of the @xmath135 dependence . the magnetic form factor @xmath71 is determined from large angle measurements of @xmath1 , since the @xmath0 contribution vanishes as @xmath91 @xmath136 180@xmath137 . with @xmath138 and @xmath139 , we obtain the following relations at @xmath91 @xmath127 180@xmath140 : @xmath141 one can see that the beam energies needed for high @xmath62 measurements of @xmath1 are quite low , with @xmath142 @xmath127 0.65 , 1.02 , 1.35 , 1.67 , and 1.97 gev corresponding to @xmath62 @xmath127 1 , 2 , 3 , 4 , and 5 gev@xmath42 , respectively . note that throughout this review we use @xmath134 @xmath127 @xmath143 to avoid confusion with the magnitude of the three momentum transfer @xmath144 , and we use units of gev and gev@xmath42 , not @xmath145 or @xmath146 . accurate measurements require that @xmath62 be known accurately since @xmath0 and @xmath1 vary rapidly with @xmath62 . energy or angle offsets of a few times 10@xmath147 could lead to @xmath62 being off by up to 0.5% . for both @xmath0 and @xmath1 , this leads to offsets that increase with @xmath62 , reaching about 2% at @xmath62 @xmath127 1 gev@xmath42 and 4% at @xmath62 @xmath127 6 gev@xmath42 . while cross section measurements can determine @xmath0 , @xmath1 , and @xmath71 , separating the charge @xmath70 and quadrupole @xmath72 form factors requires polarization measurements , most often @xmath6 . coincidence detection of the scattered electron and deuteron , which suppresses the background and allows experiments to be performed with moderate resolution , is a common technique . several experiments have measured the structure function @xmath0 at small @xmath134 . of particular note are the high precision , 1 - 2% measurements from monterey @xcite , mainz @xcite , and saclay als @xcite . the only measurements at moderately large @xmath134 are from slac e101 @xcite , bonn @xcite and cea @xcite , plus the two recent jlab experiments in halls a @xcite and c @xcite . data for several experiments are shown in fig . [ adata ] and summarized in table [ tab : edelast ] ; see refs . @xcite , @xcite or @xcite for more extensive listings of data . .[tab : edelast ] some measurements of @xmath0 . symbols are given for data shown in the figures . [ cols="<,<,^,^ , < " , ] @xmath148have larger errors and are consistent with the other data sets . [ adata ] reveals an unfortunate history of certain measurements not agreeing to within the stated uncertainties . for example , at low @xmath134 the monterey and mainz data overlap well , but the overlap of mainz and saclay als data indicates problems . the four largest @xmath134 mainz points used rosenbluth separations , with @xmath0 largely determined from forward angles of 50@xmath140 , 60@xmath140 , 80@xmath140 , and 90@xmath140 at 298.9 mev . saclay @xmath0 data were extracted from measured cross sections using previous @xmath1 data . the closest corresponding saclay points , for the same scattering angles at a beam energy of 300 mev , have cross section about 7% smaller ; the difference is beyond the quoted experimental uncertainties . significant differences such as this are often obscured by semilog plots or not plotting all data sets . the body of data , aside from the lowest @xmath134 orsay point , suggests the correctness of the saclay measurments . theoretical predictions span the range between the two data sets , and do not help to determine which is correct . thus , a new high precision experiment in this @xmath62 range appears desirable . the agreement between data from cea , slac e101 , and bonn near 1 gev was also unsatisfactory . in discussing these measurements , we will compare to the trend of the data as determined by the saclay and jlab measurements . the cea data have large uncertainties , and are systematically low by about 1@xmath149 . this experiment measured scattered electrons in a shower counter and deuterons in a spectrometer that used a quadrupole magnet with a stopper blocking out the central weak field region . in such a case it is difficult to determine the solid angle precisely , and this uncertaintly might introduce systematic errors into this data . alternatively , since the spectra were not significantly wider than the elastic peak , it has been suggested that over - subtraction of background was a problem . however , the background rates were determined to be consistent with expected rates from random coincidences and target cell walls . bonn measured coincidence cross sections at large electron scattering angles , @xmath150 @xmath151 80@xmath137 - 140@xmath137 . using forward angle data from slac e101 , cea , and orsay , bonn determined @xmath0 and @xmath1 . slightly inconsistent results from the other experiments led to a small uncertainty on the bonn determination of @xmath0 . thus , it is only the largest @xmath62 point , for which there was only the large angle bonn data , that has very significant disagreement with other determinations of @xmath0 . finally , the lowest @xmath62 slac point is high . the disagreements between the cea , slac e101 , and bonn data were part of the motivation for two jlab experiments that determined a. hall a experiment e91 - 026 @xcite measured @xmath0 for @xmath62 from 0.7 to 6.0 gev@xmath42 . hall c experiment e94 - 018 @xcite measured @xmath0 in the same kinematics as its @xmath6 points , from 0.7 to 1.8 gev@xmath42 . the main advantages of these experiments over previous work include the continuous beam , large luminosities , and modern spectrometers . the hall a measurements @xcite used @xmath152 100 @xmath153a beams on a 15 cm cryogenic ld@xmath154 target , to achieve a luminosity of approximately 5 @xmath155 10@xmath156/cm@xmath42/s , and two approximately 6 msr spectrometers . the hall c measurements used the hms spectrometer along with the deuteron channel built to measure @xmath6 with the recoil polarimeter polder . a feature of this system is that the solid angles of the two spectrometers were well matched , to within a few percent . in the overlap region , the two jlab experiments show better precision than the earlier data and generally good agreement ; comparisons of theory to data should focus on these results , rather than the older data . however , these measurements also show a significant disagreement with each other . uncertainties in each experiment are dominated by systematics of approximately 5 - 6% , with statistical precisions near 1% . the hall c data are systematically larger than the hall a data by just over 2@xmath149 , slightly over 10% , and there appears to be a tendency of the data sets to diverge with increasing momentum transfer . this discrepancy will be decreased by a few percent , but not eliminated , by a correction @xcite to a lower , more accurate , beam energy in hall c during the experiment . it is unclear if the discrepancy can be further resolved . an important experimental point is the use in these experiments , and in many earlier ones , of @xmath158 elastic scattering to calibrate the solid angle acceptance ; a fit to the world @xmath158 cross section data is often used @xcite . however , recent high precision polarimetry results @xcite imply that @xmath159 is significantly smaller than previously believed , with @xmath159 dropping nearly linearly for @xmath62 from about 0.5 to 5.6 gev@xmath42 . refitting the world cross section data , with the jlab data for the form factor ratio , decreases @xmath160 but enhances @xmath161 by about 2% @xcite . the new fits imply that the @xmath158 cross section is generally a few percent larger than would have been calculated previously , less than the systematic uncertainties of most experiments , and too small to affect comparisons of measurements of the @xmath3 cross sections and @xmath0 . the effects on the theoretical deuteron form factor predictions will be addressed below . in summary , the structure function @xmath0 is reasonably well determined up to @xmath62 @xmath127 6 gev@xmath42 , if one neglects several poorer data points . there remain regions in which there are up to about 10% systematic discrepancies between data of different experiments ; the resolution of these problems is at present unclear . @llccl experiment & @xmath134 ( gev ) & symbol & # of & year and + & & & points & reference + stanford mark ii & 0.10 - 0.13@xmath162 & @xmath148 & 2 & 1964 @xcite + stanford mark iii & 0.48 - 0.68 & @xmath163 & 4 & 1965 @xcite + orsay & 0.20 - 0.28 & @xmath148 & 3 & 1966 @xcite + orsay & 0.34 - 0.44 & @xmath148 & 3 & 1966 @xcite + stanford mark iii & 0.44 - 0.63 & @xmath148 & 5 & 1967 @xcite + orsay & 0.14 - 0.48 & @xmath148 & 4 & 1972 @xcite + naval research lab & 0.11 & @xmath148 & 1 & 1980 @xcite + mainz & 0.25 - 0.39 & @xmath164 & 4 & 1981 @xcite + bonn & 0.71 - 1.14 & @xmath165 & 5 & 1985 @xcite + saclay als & 0.51 - 1.04 & @xmath166 & 13 & 1985 @xcite + slac npas ne4 & 1.10 - 1.66 & @xmath167 & 9 & 1987 @xcite + jlab hall a & 0.7 - 1.4 & @xmath168 & 6 & unpublished + @xmath148these data sets are not shown ( @xmath1 must be inferred from the publication ) . the highest @xmath62 measurements of the @xmath1 structure function come from slac npas experiment ne4 @xcite , which covered the @xmath62 range of 1.20 to 2.77 gev@xmath42 . these measurements extended the range of previous data from saclay @xcite ( which went to 1.1 gev@xmath42 ) , and from bonn @xcite ( which went to 1.3 gev@xmath42 , and gave the results for @xmath0 discussed above ) . there is good overlap in all but a few of the earliest @xmath1 measurements . measurements of @xmath1 were taken as part of e91 - 026 for @xmath62 @xmath127 0.7 to 1.4 gev@xmath42 , but analysis is not yet final . a summary of the world data is shown in table [ t20tab ] . the first polarization measurements were from an argonne / bates recoil polarimeter experiment @xcite and a novosibirsk vepp-2 experiment @xcite using a polarized gas jet target . in the gas jet experiment , a polarimeter measured the gas polarization after it passed through the interaction region . there were three second generation experiments . an argonne / novosibirsk vepp-3 measurement @xcite pioneered the use of storage cells , increased the internal target density about a factor of 15 over the gas jet alone , and pushed out to 0.58 gev , near the minimum in @xmath6 . because the polarization of the gas varies in the cell , due to wall and beam interactions , it was decided to normalize the gas polarization by setting the lowest @xmath134 datum , at 0.39 gev , to theory where the uncertainties are small . such internal targets in storage rings are now common . a bonn polarized target experiment @xcite had large uncertainties . at bates , the ahead deuteron polarimeter was used @xcite to determine @xmath6 in the range just past the minimum of @xmath6 to just past the node in @xmath70 . a continuation of the novosibirsk experiment had large uncertainties @xcite , and was never published . note that , to facilitate comparison between different experiments , the data are often `` corrected '' to an electron scattering angle of 70@xmath140 , but this adjustment and the uncertainty it introduces are small . over the past several years , internal target experiments at nikhef @xcite and novosibirsk @xcite have improved the precision of the lower @xmath169 data , over a range of @xmath170 0.3 - 0.8 gev . the improvements in novosibirsk include higher luminosity resulting from an improved atomic beam source and a modified beam tune that allows use of a higher impedance storage cell . jlab hall c e94 - 018 @xcite used the recoil polarimeter polder to measure to the highest @xmath62 , 1.72 gev@xmath42 . the overlap of the data is good , but apparent systematic shifts can be seen , as the nikhef and bates measurements are more negative than the jlab and novosibirsk measurements ; note that this is not a difference between polarized targets and recoil polarimeters . the issue of determing at what @xmath62 @xmath70 @xmath127 0 is affected by this difference . the bates data @xcite suggest a larger @xmath62 than do the novosibirsk @xcite and jlab @xcite data . the fits of ref . @xcite do not include the unpublished novosibirsk data @xcite , and average between the bates and jlab points . we do not discuss the data for @xmath171 and @xmath172 . because of their dependence on @xmath71 , they have not been as useful in providing new information as has @xmath6 . to test time reversal invariance , one measurement of the _ induced _ vector polarization was made @xcite . the observed result was consistent with zero . it is instructive to see how the deuteron form factors are related to the free nucleon form factors and the deuteron wave function in the nonrelativistic limit . because the deuteron is an isoscalar target , only the isoscalar nucleon form factors @xmath173 will contribute to the form factors . in the nonrelativistic theory , _ without exchange currents or @xmath174 corrections _ , the deuteron form factors are @xmath175\ , , \label{body}\end{aligned}\ ] ] where the _ body form factors _ @xmath176 , @xmath177 , @xmath178 , and @xmath179 are all functions of @xmath62 . if we choose to evaluate these in the breit frame , defined by @xmath180 then the relativistic and nonrelativistic momentum transfers are identical , @xmath181 , and the relativistic nucleon form factors can be used without corrections . note that , in this nonrelativistic limit , only the nucleon electric form factors contribute to the deuteron charge and quadrupole structure , while both nucleon form factors contribute to the deuteron magnetic structure . the nonrelativistic formulae for the body form factors @xmath18 involve overlaps of the wave functions , weighted by spherical bessel functions @xmath182 j_0\left(\tau \right)\nonumber\\ d_q(q^2 ) & = & { { 3}\over{\sqrt{2}\eta } } \int_0^{\infty } dr\ ; w(r ) \left[u(r ) - { { w(r)}\over{\sqrt{8 } } } \right ] j_2\left(\tau \right)\nonumber\\ d_m(q^2 ) & = & \int_0^{\infty } dr \,\bigl [ 2\,u^2(r ) - w^2(r ) \bigr ] j_0(\tau ) + \bigl[\sqrt{2}\,u(r)w(r ) + w^2(r)\bigr ] j_2(\tau)\nonumber\\ d_e(q^2 ) & = & { { 3}\over{2 } } \int_0^{\infty } dr\ , w^2(r)\ , \bigl [ j_0(\tau ) + j_2(\tau ) \bigr ] \label{bodyff}\end{aligned}\ ] ] where @xmath183 . at @xmath69 , the body form factors become @xmath184=1\nonumber\\ d_q(0 ) & = & { { m_d^2}\over{\sqrt{50 } } } \int_0^{\infty } r^2\,dr\ ; w(r ) \left[u(r ) - { { w(r)}\over{\sqrt{8 } } } \right ] \nonumber\\ d_m(0 ) & = & \int_0^{\infty } dr \,\bigl [ 2\,u^2(r ) - w^2(r ) \bigr]=2 - 3p_d \nonumber\\ d_e(0 ) & = & { { 3}\over{2 } } \int_0^{\infty } dr\ , w^2(r)= { { 3}\over{2 } } p_d\end{aligned}\ ] ] giving the nonrelativistic predictions @xmath185 with @xmath186 = 0.8798 the isoscalar nucleon magnetic moment . the experimental value of the deuteron magnetic moment ( in these units ) is 1.7139 , leading to a predicted @xmath18-state probability of @xmath187 . however this estimate can not be taken too seriously because the magnetic moment is _ very sensitive to relativistic corrections and interaction currents _ which can easily alter this result significantly . these contributions will be reviewed qualitatively later in this review . the study of deuteron form factors is complicated by the fact that they are a _ product _ of the _ nucleon isoscalar _ form factors , @xmath188 , and the _ body _ form factors , @xmath18 . the dependence of the deuteron form factors on older models of the nucleon form factors is well discussed in ref . a year ago the model of mergell , meissner and drechsel @xcite ( referred to as mmd ) gave a good fit , and could have been adopted as a standard . figure [ ges - study ] shows the mmd isoscalar electric and magnetic form factors divided by the familiar dipole form factor @xmath189 ( with @xmath62 in gev@xmath42 ) . note that the mmd model does not differ by more than 20% from the dipole over the entire @xmath62 range , suggesting that the dipole approximation works very well ( on the scale of the experimental errors see below ) . however , recent measurements of the proton charge form factor are producing a surprising result , and at the time this review was being completed the picture was begining to change . the recent jlab measurements of both the neutron and proton charge form factors now suggest that the isoscalar charge form factor may be well approximated by @xmath190 where @xmath191 is the dipole form factor and @xmath192 . this jlab model is a sum of the old galster @xcite fit for the nucleon charge form factor ( supported by the recent measurements @xcite ) and a linear approximation to the new jlab @xmath193 data @xcite ( from which the charge form factor is obtained by assuming that @xmath194 ) . figure [ ges - study ] shows that this form factor differs significantly from the dipole ( and also the previously favored mmd model ) , and may have a significant affect on the theoretical interpretation of the data . this will be discussed in sec . [ thyandexp ] below . [ the @xmath195 ratios shown in the figure will be discussed in sec . [ hexamples ] below . ] dividing the individual factors @xmath95 , @xmath96 , and @xmath97 [ introduced in eq . ( [ aandb ] ) ] by @xmath196 gives reduced quantities that are ( except for the weak dependence on the ratio of @xmath197 ) independent of the choice of nucleon form factor . the contribution of these reduced quantities , which we denote by @xmath198 , @xmath199 , and @xmath200 , to the total @xmath201 is shown in fig . [ a - study ] . the figure shows that the contribution of the magnetic term , @xmath200 , to the total @xmath202 is small for @xmath203 gev@xmath42 ( for most of the @xmath62 range it is less than a few percent , reaching 10% at @xmath204 and also near 4 gev@xmath42 ) . above @xmath62 of 4 gev@xmath42 it is larger , and very model dependent . this justifies the observation that the @xmath0 structure function can be well approximated by @xmath205 , as stated eariler in eq . ( [ tildeaapprox ] ) . [ note that the new jlab data for @xmath206 , discussed briefly above , may enhance the magnetic contributions to @xmath0 above 4 gev@xmath42 , but will not change these conclusions qualitatively ] . how well does this simple nonrelativistic theory explain the data ? the high @xmath62 data for @xmath0 provide the most stringent test . in fig . [ ahighq2 ] we compare the data for @xmath0 with calculations using the five nonrelativistic wave functions shown in figs . [ uwr ] and [ uwp ] . the calculations use eq . ( [ body ] ) with mmd isoscalar nucleon form factors and nonrelativistic body form factors given in eq . ( [ bodyff ] ) . in the right panel the data and models have been divided by the `` fit '' described in eq . ( [ scale ] ) below . it is easy to see that the nonrelativistic models _ are a factor 4 to 8 smaller than the data for @xmath207 gev@xmath42_. furthermore , since the difference between different deuteron models is substantially smaller than this discrepancy , it is unlikely that any _ realistic _ nonrelativistic model can be found that will agree with the data . if the nucleon isoscalar charge form factor were larger than the mmd model by a factor of 2 to 3 it might explain the data , but this is also unlikely since the variation between nucleon form factor models is substantially smaller than this . [ if we use the fit eq . ( [ jlabges ] ) to the jlab @xmath208 measurements the discrepancy will be even larger . ] we are forced to conclude that these high @xmath62 measurements _ can not be explained by nonrelativistic physics and present very strong evidence for the presence of interaction currents , relativistic effects , or possibly new physics . _ a detailed comparison of the nonrelativistic models with the three deuteron form factors , @xmath70 , @xmath71 , and @xmath72 is given in fig . [ gfactors ] . the functions used to scale the data and theory in the right - hand panels of the figure are @xmath209 where @xmath62 is in gev@xmath42 and @xmath210 . while some of the factors in these expressions are theoretically motivated ( note the presence of the dipole form of the nucleon form factor ) we do not attach _ any _ theoretical significance to these functional forms ; they merely provide a reasonably simple way to scale out the rapid exponential decreases from the form factors . figure [ gfactors ] shows that the nonrelativistic models do a good job of predicting the form factors to a momentum transfer @xmath211 gev , beyond which departures from the data and variations of the models make the agreement increasingly unsatisfactory . point seriously ; kinematic factors make it difficult to extract this point accurately and it is only one standard deviation from the theory . the large @xmath72 and small @xmath70 values for the points at 0.55 gev and 0.58 gev result from the @xmath6 data points , from @xcite and @xcite respectively , being about 1 standard deviation more negative than calculations and overlapping the negative limit for @xmath6 of @xmath212 . note that the tabulated uncertainty of @xmath72 at @xmath213 0.55 gev in ref . @xcite should be asymmetric , + 0.075/@xmath2140.713 , ( as shown in the fig . [ gfactors ] ) . ] however , careful comparison reveals that there are still ( small ) discrepancies between the data and the nonrelativistic theory , even at low @xmath62 . the data and curves from the lower panel of fig . [ ahighq2 ] are shown on an expanded logarithmic scale in fig . [ alowq2 ] . in the lowest @xmath62 range from about 0.15 to 0.4 gev@xmath42 the data lie _ below _ the nonrelativistic theory , and are larger than the nonrelativistic theory only for @xmath62 above 1 gev@xmath42 . the very low @xmath62 discrepancy seems to be due in part ( but not entirely ) to the columb distortion corrections that have been used recently to explain the deuteron radius @xcite . we will discuss these corrections in sec . [ nucsec ] below . before we turn to a detailed discussion of the possible explanations for the failure of nonrelativistic models to explain the form factors at high @xmath62 , we discuss the low momentum transfer results from the perspective of effective theories . the recent development of effective field theory provides a powerful method for theoretical study of low @xmath62 physics . we will briefly review these results here , and return to the discussion of the high @xmath62 results in the next section . effective field theory techniques exploit the fact that the physics at low energies @xmath215 ( or large distances @xmath216 ) can not be sensitive to the _ details _ of the interactions at very high energies @xmath217 ( or short distances ) . for example , a low energy long wavelength probe may detect the presence of a small scattering center , but can not resolve its structure ( much as the far - field of a collection of electric charges depends on only one parameter , the total charge ) . the parameters that depend on the short - range physics may be very important , but they can not be calculated and must be determined by a fit to the data . effective field theory works best if the distance scales of the ( unknown ) short - range physics and the ( known ) long - range physics are clearly separated . then for energies well below the scale of the short - range physics ( which we take to be @xmath218 ) , the short - range physics is treated systematically by expanding in powers of @xmath219 . in applications to the @xmath7 system , two scales have been discussed . the so - called `` pionless theory '' chooses @xmath220 , and therefore requires no theory of the @xmath221 interaction . this approach can work only at _ very _ low energies . the chiral theory chooses @xmath222 and attempts to describe @xmath7 scattering up to the @xmath223 mass scale using the known pion - nucleon interaction as given by chiral symmetry . ( more precisely , if the magnitude of the center of mass relative momentum @xmath224 , the nucleon lab kinetic energy will @xmath225 , which is @xmath226 mev for the pionless theory , and @xmath227 mev for chiral perturbation theory . ) the effective range theory introduced by bethe @xcite is an early version of what we now call the pionless effective theory . weinberg @xcite first applied modern chiral perturbation theory to @xmath7 scattering . he proposed making a chiral expansion of the @xmath7 potential , and then inserting this potential into the schrdinger equation . later kaplan , savage , and wise ( ksw ) @xcite criticized the consistency of this approach , and introduced an alternative organizational scheme , sometimes referred to as @xmath134 counting , in which the pion interaction is to be included as a perturbative correction ( as opposed to including it as part of the potential , and counting it to all orders , as proposed by weinberg ) . ksw applied this method to calculation of the deuteron form factors @xcite . it is now known that the tensor part of the one pion exchange interaction is too strong to be treated perturbatively , and recent work has focused on how to include the singular parts of one pion exchange in the most effective manner @xcite . in the following discussion we review the recent results from phillips , rupak , and savage ( prs ) , who give a nice account of the calculations of the deuteron form factors in a pionless theory @xcite . the effective lagrangian density for a pionless effective theory of the @xmath7 interaction in any channel ( the coupled @xmath228 for example ) is @xmath229 \nonumber\\ & + \cdots \ , , \label{eft1}\end{aligned}\ ] ] where @xmath230 is a ( nonrelativistic ) nucleon field operator and @xmath231 , @xmath232 , and the general coefficient @xmath233 ( which fixes the strength of the terms with @xmath234 derivatives ) are determined from data . the coefficients @xmath233 parameterize the strength and shape of the short range interaction . the scattering amplitude predicted by ( [ eft1 ] ) is a sum of bubble diagrams which can be regularized using the ksw dimensional regularization scheme with power law divergence subtraction @xcite . in lowest order ( lo ) this bubble sum is @xmath235 where @xmath236 is the magnitude of the nucleon three - momentum in the c.m . system , @xmath237 is regular at the pole @xmath238 , @xmath239 with @xmath39 the deuteron binding energy , @xmath240 is related to the asympotic normalization of the deuteron wave function , and the dependence of @xmath231 on the ( arbitrary ) renormalization point @xmath153 is dictated by the requirement that the overall result be _ independent _ of @xmath153 . the lo result is @xmath241 in terms of the effective range expansion @xmath242 with @xmath202 the scattering length and @xmath243 the effective range , the lo calculation gives @xmath244 contributions from the next to leading order ( nlo ) term @xmath232 changes the relations in ( [ zeq ] ) and ( [ req ] ) ; in particular , the wave function renormalization constant @xmath240 begins to differ from unity and the effective range @xmath243 to differ from zero . prs point out that the most stable results are obtained by constraining @xmath231 and @xmath232 to give the experimental values of the deuteron parameters @xmath245 fm and @xmath246 instead of @xmath247 and @xmath248 fm . this is because the asympotic deuteron wave function is fixed by @xmath247 and @xmath240 @xmath249 and _ it is the wave function and not the scattering _ that largely determines the deuteron form factors and other electromagnetic observables . @xmath250 using this approach , the lo charge form factor is given entirely by the asymptotic wave function ( [ asywave ] ) @xmath251 where @xmath252 is the magnitude of the three momentum transferred by the electron , and working in the breit frame ( where the differences between relativistic and nonrelativistic theory is a minimum ) is also equal to @xmath143 . prs show that expansion of the charge form factor up to nnlo terms is @xmath253 \nonumber\\ & - { { 1}\over{6}}\,r_n^2\,q^2\,g_c^{(0)}(q^2 ) + \cdots\end{aligned}\ ] ] and because the wave function is correctly normalized , there are no wave function effects beyond nlo ( the second term ) . at nnlo effects from the finite nucleon size , @xmath254 , similarly , the lo quadrupole form factor obtained by prs is @xmath255\ ] ] with @xmath256 the asymptotic @xmath257-state ratio , and the lo quadrupole moment , @xmath258 , equal to @xmath259 @xmath127 0.335 fm@xmath42 . the expansion of @xmath72 to nnlo is then @xmath261 \nonumber\\ & - { { 1}\over{6 } } \,r_n^2\ , q^2 g_q^{(0)}(q^2 ) + \cdots\end{aligned}\ ] ] note that the quadrupole moment at nlo includes a contribution @xmath262 from a four - nucleon - one - photon contact term , not determined by @xmath7 scattering , and is used to fit the experimental value of @xmath263 . prs suggest that the absence of this piece of short distance physics in conventional calculations may explain their underprediction of the quadrupole moment . the finite size of the nucleon again comes in at nnlo . with parameters largely set by other data , the deuteron charge , quadrupole , and magnetic form factors are well predicted up to about @xmath213 0.2 gev , as shown in fig . [ fig : eft ] . the approach seems to converge well , but beyond nnlo more parameters enter , and there is less predictive power . the great strength of the pionless effective theory is that strips away complexity , revealing the essential physics required to understand the low @xmath134 results , and showing ( for example ) the central importance of the asympotic s - state normalization @xmath240 . however , as expected , it clearly does not work for @xmath134 much beyond 0.4 to 0.5 gev . the theory with pions ( sometimes referred to as a `` pionful '' theory ) will work to higher @xmath62 @xcite . removal of divergences from these theories is under active study . we now return to discussion of the reasons for the failure of nonrelativistic theory at high @xmath62 . in sec . [ comparenr ] we showed that the naive nonrelativistic theory can not explain the deuteron form factor data for @xmath264 gev . in this section we classify the possible explanations for this failure , preparing the way for detailed discussions to follow in secs.[nucsec ] and [ quarksec ] . the differences between the data and the nonrelativistic theory can only be explained by a combination of the following effects * interaction ( or meson exchange ) currents ; * relativistic effects ; or * new ( quark ) physics . the only possibilities excluded from this list are variations in models of the nucleon form factors , or model dependence of the deuteron wave functions . in the previous section we argued that _ neither _ the current uncertainty in our knowledge of the nucleon form factors , _ nor _ the model dependence of the nonrelativistic deuteron wave functions are sufficient to provide an explanation for the discrepencies . possible interaction currents that might account for the discrepency are shown in fig . because the deuteron is an isoscalar system , the familiar large @xmath265 exchange currents are `` filtered '' out and only @xmath266 exchange currents can contribute to the form factors . the @xmath266 currents tend to be smaller and of a more subtle origin . the nucleon @xmath267-graphs , fig . [ mec]c , and the recoil corrections , fig . [ mec]d , are both of relativistic origin . ( the recoil graphs will give a large , incorrect answer unless they are renormalized @xcite . ) the two - meson exchange currents should be omitted unless the force also contains these forces . the famous @xmath268 exchange current is very sensitive to the choice of @xmath268 form factor , which is hard to estimate and could easily be a placeholder for new physics arising from quark degrees of freedom . in most calculations based on meson theory , the two pion exchange ( tpe ) forces and currents arising from crossed boxes are excluded , and , except for the @xmath268 current ( which we will regard as new physics ) , the exchange currents are of relativistic origin . additional relativistic effects arise from boosts of the wave functions , the currents , and the potentials , which can be calculated in closed form or expanded in powers of @xmath174 , depended on the method used . at low @xmath62 calculations may be done using effective field theories ( discussed in sec . [ sec : eft ] ) in which a small parameter is identified , and the most general ( i.e. exact ) theory is expanded in a power series in this small parameter . in these calculations relativistic effects are automatically included ( at least in principle ) through the power series in @xmath174 . _ hence any improvement on nonrelativistic theory using nucleon degrees of freedom leads us to relativistic theory . _ alternatively , one may seek to explain the discrepancy using quark degrees of freedom ( new physics ) . when two nucleons overlap , their quarks can intermingle , leading to the creation of new @xmath7 channels with different quantum numbers ( states with nucleon isobars , or even , perhaps , so - called `` hidden color '' states ) . these models require that assumptions be made about the behavior of qcd in the nonperturbative domain , and are difficult to construct , motivate , and constrain . at very high momentum transfers it may be possible to estimate the interactions using perturbative qcd ( pqcd ) . very little has been done using other approaches firmly based in qcd , such as lattice gauge theory or skrymions ( but see ref . @xcite ) . we are thus led to two different alternatives for explanation of the failure of nonrelativistic models . in one approach the nucleon ( hadron ) degrees of freedom are retained , and relativistic methods are developed that treat boost and interaction current corrections consistently . in another approach , quark degrees of freedom are used to describe the short range physics , and techniques for handling a multiquark system in a nonperturbative ( or perturbative ) limit are developed . these two approaches will be reviewed in the next two sections . while the discussion appears to be focused on the deuteron form factors , it is actually more general , and will be applied later to the treatment of deuteron photodisintegration . are these two approaches really different ? superficially , of course , the answer must be : yes ! however , qcd tells us that all physical states must be color singlets , and a basis of states that describes any color singlet state can be constucted from _ either _ quarks ( and gluons ) _ or _ hadrons ( this would not be true if colored states were physical ) . so at a deeper level it appears that either approach ( hadrons or quarks ) should work , and the best choice is the system that can describe the relevant physics more compactly . further discussion of this issue is clearly beyond the scope of this review . this long section is divided into six parts as follows : ( i ) introduction , ( ii ) overview of propagator dynamics , ( iii ) choice of propagator and kernel , ( iv ) examples of propagator dynamics , ( v ) overview of hamiltonian dynamics , and ( vi ) examples of hamiltonian dynamics . the inhomogeneous lorentz group , or the poincar group , is described by 10 generators : three pure rotations , three pure boosts , and four pure translations . if we require the interactions to be local and manifestly covariant under the poincar group , we are led to a local relativistic quantum field theory with particle production and annihilation @xcite . in this case the poincar transformations of all matrix elements can be shown to depend only on the kinematics ( i.e. they depend only on the masses and spins of the external particles ) . the disadvantage is that the number of particles is not conserved . if perturbation theory can be used , this approach is very successful , but in the nonperturbative regime of strong coupling meson theory it leads to an infinite set of coupled equations that can not be solved in closed form . numerical , nonperturbative solutions of field theory can be obtained in euclidean space for a few special cases @xcite . methods that limit the intermediate states to a _ fixed number of particles _ ( two nucleons in this case ) are more tractable , and all modern calculations are based on the choices depicted in the decision tree shown in fig . [ decision ] . in deciding which method to use , if is first necessary to decide whether or not to allow _ antiparticle , or negative energy _ nucleons to propagate as part of the virtual intermediate state . since nucleons are heavy and composite , so that their antiparticle states are very far from the region of interest , some physicists believe that intermediate states should be built only from positive energy nucleons , and that all negative energy effects ( if any ) should be included in the interaction . these methods are referred to collectively as _ hamiltonian dynamics _ and are represented by the left hand branch in the figure . unfortunately , it turns out that this choice precludes the possibility of retaining the properties of locality and manifest covariance enjoyed by field theory . alternatively , in order to keep the locality and manifest covariance of the original field theory , other physicists are willing to allow negative energy states into the propagators . these methods , represented by the right - hand branch of the figure , are referred to collectively as _ propagator dynamics_. including negative energy states tends to make calculations technically more difficult and harder to interpret physically , and those who advocate the use of hamiltonian dynamics do not believe the advantages of exact covariance justify the work it requires . unfortunately , these two methods are so fundamentally different that many physicists do not realize that the limitations of one may not apply to the other . for example , for some choices of propagator dynamics all 10 of the generators of the poincar group will depend only on the kinematics , and the poincar transformations of _ all amplitudes can be done exactly_. with hamiltonian dynamics this is not the case ; some of the 10 generators must depend on the interaction , and transformation of matrix elements under these `` dynamical '' transformations must be calculated . comparison of the two methods is therefore very difficult ; the language and issues of each are very different and one can be easily misled by the different appearance of the results . we can not discuss these issues in detail in this review , and refer the reader to two recent references that survey the subject @xcite . here we will give a short review of some recent calculations , and explain these differences as we go along . propagator calculations all start from the field theory description of two ( in this case ) interacting particles . while some may prefer to express the field theory as a path integral , it is also possible to adopt a more intuitive approach and imagine expanding the path intergral as a sum of feynman diagrams ( ignoring issues of convergence for the moment ) . in order to generate the deuteron bound state , which produces as a pole in the scattering matrix , it is necessary to sum an infinite class of diagrams , written as @xmath269 where @xmath270 is the _ kernel _ being iterated , @xmath271 the two body propagator , @xmath272 the scattering amplitude , and the other quantities are defined below . this sum is obtained in closed form by solving the integral equation @xmath273 if the series ( [ sumdiag ] ) is compared to a geometric series @xmath274 , then the solution to the integral equation ( [ rel1 ] ) can be compared to the sum of the geometric series @xmath275 . the geometric series converges only when @xmath276 , but its unique analytic continuation , @xmath275 , is valid for all @xmath277 . similiarly , it is assumed that the solution to ( [ rel1 ] ) is valid even when the series ( [ sumdiag ] ) diverges . and just as the geometric series has a pole at @xmath278 , the solution to ( [ rel1 ] ) will have a pole at @xmath279 , the square of the deuteron mass . the amplitudes @xmath280 , @xmath281 , and @xmath282 are all matrices in the @xmath7 spin - isospin space , and are functions of the four - momenta @xmath283 and @xmath284 , with @xmath285 and @xmath286 the momenta of the two particles ( labeled in fig . [ feynmanladder ] ) . the dimension of the volume integration is @xmath287 , normally either 3 + 1=4 ( 3 space + one time dimensions ) for the bethe - salpeter method , or 3 + 0=3 for the quasipotential methods described below . if eq . ( [ rel1 ] ) has a homogenous solution at some external four momentum @xmath288 , the scattering matrix will have an @xmath289 channel pole ( represented in fig . [ feynmanbound ] ) , signifying the existence of a deuteron bound state . the _ vertex _ function for the deuteron bound state satisfies the equation @xmath290 with covariant normalization condition @xmath291 the covariant bound state wave function is defined by @xmath292 one of the advantages of the propagator approach is that the construction of the current operator is comparatively straightforward . it follows ( at least in principle ) from summing all electromagnetic interactions with all the consituents everywhere in the ladder sum . for bound states described by the bethe - salpeter or spectator formalisms ( see the discussion below ) there are two diagrams , illustrated in fig . [ feynmancurrent ] , that can be written @xmath293 in the first term , @xmath294 is the sum of the neutron and proton currents [ recall eq . ( [ nnff ] ) ] and we have chosen particle 1 to interact with the photon ( always possible because of the antisymmetry of the wave function ) . the interaction current is @xmath295 , and assumes a comparatively simple form if the kernel is a sum of single particle exchanges . this case is illustrated in fig . [ feynmancurrent ] . current conservation , @xmath296 follows automatically @xcite from the bound state equation ( [ bound1 ] ) if the nucleon and interaction currents satisfy the following two - body ward - takahashi ( wt ) identities @xmath297\,\bigl\{g^{-1}(k;p_0)-g^{-1}(k';p'_0)\bigr\ } \nonumber\\ q_\mu \;i^\mu(k'_1,k'_2;k_1,k_2)&&= { \cal v}(k',k;p'_0)\,{\textstyle{1\over2}}[1+\tau_3 ] - { \textstyle{1\over2}}[1+\tau_3]\,{\cal v}(k',k;p_0)\ , . \label{current3 } \end{aligned}\ ] ] note the appearance of @xmath298 $ ] , the isoscalar charge operator in isospin space . the @xmath299 identity is the two - body version of the familiar one - body wt identity @xmath300 with @xmath301 ( 0 ) for the proton ( neutron ) and the undressed nucleon propagator normalized to @xmath302 . note that the constraint on the interaction current is _ not _ zero ( and hence the interaction current is _ not _ zero ) if the kernel depends on the isospin or the total four - momentum @xmath303 . to fully specify a propagator dynamics , one must choose a propagator , @xmath281 , a kernel , @xmath280 , and current operators @xmath294 and @xmath304 . four different progagators have been used in the study of the deuteron form factors . the bethe - salpeter ( bs ) equation @xcite uses a fully off shell propagator for two nucleons @xmath305 \left [ e_p^2-\left({w\over2}-p_0\right)^2\right]}}\ , , \end{aligned}\ ] ] where @xmath306 is the ( off - shell ) positive energy projection operator and the right hand expression is the propagator for identical particles in the rest frame ( with @xmath307 , and @xmath308 used as a shorthand for the four - vector @xmath52 in the rest frame ) . this choice of relativistic equation was the first to be introduced and is perhaps the best known . it retains the full integration over all components of the relative four - momentum @xmath236 , and all of the off - shell degrees of freedom ( 2 for spin @xmath155 2 for `` @xmath223-spin '' , where @xmath309 are positive energy @xmath83 spinor states and @xmath310 are negative energy @xmath311 spinor states ) of both of the propagating nucleons , for a total of @xmath312 spin degrees of freedom . the equation has inelastic cuts arising from the production of the exchanged mesons ( when energetically possible ) and additional singularities when the nucleons are off - shell . these can be removed by transforming the equation to euclidean space . however , the bs equation , when used in _ ladder _ approximation , does not have the correct one - body limit . numerical comparisons of solutions obtained from the sum of _ all _ ladder and crossed ladder exchanges with ladder solutions of the bs equation , carried out for scalar theories , have shown that the ladder sum is inaccurate and that the one - body limit requires inclusion of crossed exchanges @xcite . the bs equation has been solved in ladder approximation by tjon and his collaborators @xcite , and used to calculate the deuteron form factors @xcite . the fits to the @xmath7 phase shifts originally obtained from these works are unsatisfactory by today s standards . the spectator ( or gross ) equation ( denoted by @xmath17 ) @xcite restricts one of the two nucleons to its positive energy mass - shell . if particle one is on - shell , the spectator propagator is @xmath313 where @xmath314 . this has the effect of fixing the relative energy in terms of the relative three - momentum so as to maintain covariance and reduce the four dimensional integration to three dimensions [ @xmath315 in eq . ( [ rel1 ] ) ] . identical particles are treated by properly ( anti)symmetrizing the kernel . the restriction of one of the particles to its positive energy mass - shell also removes the @xmath310 states of one of the nucleons , reducing the number of spin degrees of freedom to 2@xmath1554=8 . a primary motivation and justification for this approach is that it has the correct one - body limit , and the three body generalization satisfies the cluster property @xcite . the equation also has a nice nonrelativistic limit that can be easily interpreted . numerical studies of scalar field theories @xcite show that the exact ladder and crossed ladder sum is better approximated by the ladder approximation to this equation than it is by the ladder approximation to the bs equation . the method can be extended to include gauge invariant electromagnetic interactions @xcite . its principle drawback is that the kernel has unphysical singularities which can only be removed by an _ ad - hoc _ prescription . results from this method will be reviewed in the next section . the internal momentum integration can also be restricted to three dimensions in such a way that , for equal mass particles in the rest frame , the relative energy is zero and the particles are equally off shell . the blankenbecler - sugar - logunov - tavkhelidze ( bslt ) equation @xcite can be defined so that the two propagating particles are on their positive energy mass - shell , reducing the number of spin-@xmath223-spin variables to 2@xmath1552=4 . however , this equation does not satisfy the cluster property . the approach of phillips , wallace , and mandelsweig @xcite , which we denote by pwm , also puts the particles equally off - shell , but includes all negative energy contributions . setting @xmath315 in eq . ( [ rel1 ] ) , the pwm propagator for equal mass particles in the c.m . system is @xmath316 with @xmath312 spin degrees of freedom . this propagator differs from bslt primarily by the presence of the additional @xmath70 term [ which contributes the last two terms in the curly brackets involving the `` mixed '' @xmath317 projection operators ] that includes contributions from crossed graphs approximately , and correctly builds in the one body limit . the retarded kernel to be used with this propagator , in ladder approximation , is @xmath318 perhaps the principle obstacle to implementing this method is that construction of current operators is problematic , and manifest poincar invariance is lost ( but wallace @xcite has recently shown how to compute boosts for scalar particles exactly ) . calculations using this method will be described in next section . we now turn to a description of two examples of propagator dynamics . _ van orden , devine , and gross _ [ vog ] . the spectator equation has been used to successfully describe @xmath7 scattering and the deuteron bound state @xcite , and this work uses these results to describe the deuteron form factors @xcite . the relativistic kernel used to describe the @xmath7 system consists of the exchange of 6 mesons [ @xmath319 , @xmath320 , @xmath149 , @xmath321 ( or @xmath322 ) , @xmath223 and @xmath323 . the model includes a form factor for the off - shell nucleon @xcite , giving a `` dressed '' single nucleon propagator of the form @xmath324 where @xmath325 is one of the parameters of the model . coupling constants and form factor masses ( 13 parameters in all ) are determined by a fit to the data and the deuteron wave functions are extracted @xcite . to insure current conservation , the one - nucleon current must satisfy the ward - takahashi identity @xmath326 and this requires an off - shell modification of the single nucleon current . the solution used by vog is @xmath327 where @xmath328 but is otherwise undefined [ in the applications described below , @xmath329 where @xmath191 is the dipole form factor of eq . ( [ dipole ] ) ] , and @xmath330 + { h(p'^2)\over h(p^2 ) } \left[{m^2-p^2\over p'^2-p^2 } \right]\nonumber\\ g_0(p',p)=\left({h(p^2)\over h(p'^2)}-{h(p'^2)\over h(p^2)}\right ) { 4m^2\over p'^2-p^2}\ , . \end{aligned}\ ] ] while the on - shell form of the current ( [ onej ] ) is fixed by the nucleon form factors , and the functions @xmath331 and @xmath332 are fixed by the wt identity ( [ wt ] ) , other aspects of the off - shell extrapolation of the current ( [ onej ] ) are _ not unique_. using this one - nucleon current , and recalling that there are no currents of the type shown in figs . [ mec](a ) or ( b ) [ we postpone discussion of the @xmath268 current ] , it was shown @xcite that the full two body current to use with the spectator equation is given by the diagrams shown in fig . [ scurrent ] . these diagrams are manifestly covariant , and _ automatically include effects from z - graphs or retardation _ illustrated in figs . [ mec](c ) or ( d ) . they are referred to as the complete impulse approximation ( cia ) to distinguish them from the relativistic impulse approximation ( ria ) , an approximate current used in earlier calculations @xcite . the ria is obtained by multiplying diagram [ scurrent](a ) by two , and is very close to the cia . _ phillips , wallace , divine , and mandelsweig _ [ pwm ] . this work is based on the mandelsweig and wallace equation @xcite , supplemented by contributions from the crossed graphs @xcite , as described above . it is sometimes referred to as the equal - time approach . a feature of this equation is that it includes the full strength of the @xmath267-graphs ; the @xmath317 contributions shown in eq . ( [ pweq ] ) are roughly twice as strong as the @xmath267-graph contributions included in the spectator equation . the pwm propagator is also explicitly symmetric , a convenience when applied to identical particles . in the published work reviewed here @xcite , the deuteron is described by a one boson exchange force using the parameters of the bonn - b potential with the exception of the @xmath149 meson coupling , which is adjusted to give the correct deuteron binding energy . lorentz invariance is broken by the approximation ; the boosts of the deuteron wave functions from their rest frames are treated approximately . the pwm current is a modification of eq . ( [ current1 ] ) . in the present work retardation effects [ like those illustrated in fig . [ mec](d ) ] are omitted from the current operator ; only one body terms and @xmath267-graph contributions are included . we now turn to a discussion of the other major approach detailed in fig . [ decision ] : hamiltonian dynamics . approaches based on hamiltonian dynamics start from a very different point than propagator dynamics , and this is one reason it is difficult to compare the two . while propagator dynamics starts from field theory ( which can be described as a quantum mechanics with an arbitrary number of particles ) , hamiltonian dynamics starts from quantum mechanics with a fixed number of particles . for a detailed review , see ref . @xcite . quantum mechanics begins with a hilbert space of states defined on a fixed space - like surface in four - dimensional space - time . the various options for choosing this space - like surface were classified by dirac in 1949 @xcite . the _ instant - form _ corresponds to choosing to construct states at a fixed time @xmath333 , and is the choice usually made in elementary treatments . alternatively , _ front - form _ quantum mechanics constructs states on a fixed - light front , customarily defined to be @xmath334 . ( we use units in which the speed of light , @xmath335 , is unity . ) more generally , the light - front may be chosen in any direction defined by @xmath336 , with @xmath337 and @xmath338 . finally , _ point - form _ quantum mechanics constructs states on a forward hyperboloid , with @xmath339 ; @xmath340 [ the limiting cases of @xmath341 gives the instant - form ( with @xmath342 ) , and @xmath343 the front - form ] . these three surfaces are shown pictorially in fig . [ qmsurfaces ] . [ while the point @xmath344 , @xmath345 is not on the hyperbolid , all distances between points on the hyperbolid are space - like . ] the poincar transformations are symmetries that leave all probabilities unchanged ; they must be unitary transformations ( with hermitian generators ) on the space of quantum states . the 10 generators of the full poincar group are the hamiltonian , @xmath346 , generator of time translations , three - momenta , @xmath347 , generators of spatial translations , angular momenta , @xmath348 , generators of rotations , and @xmath349 , generators of boosts . they satisfy the following commutation relations : @xmath350=[h , j^i]=[p^i , p^j]=0\ , , \qquad[j^i , x^j]=i\epsilon_{ijk } x^k\ , , { \rm for}\ ; x^i = j^i , p^i , k^i\nonumber\\ \fl&&[k^i , k^j]=-i\epsilon_{ijk}j^k\ , , \qquad[k^i , p^j]=-i\delta_{ij}h\ , , \qquad[k^i , h]=-ip^i \label{ccr1}\end{aligned}\ ] ] for each of the forms of quantum mechanics there is a subgroup of the poincar transformations that leave the states invariant on the fixed surface associated with that form . this is the kinematic subgroup , and the transformations in this subgroup will not depend on the dynamics ( since the dynamics describe how the states change away for the fixed surface ) . in the _ instant - form _ , space translations and rotations clearly leave the surface @xmath344 invariant . generators of these transformations form a subgroup of the poincar group , with commutation relations @xmath351=0\ , , \quad[j^i , j^j]=i\epsilon_{ijk } j^k\ , , \quad[j^i , p^j]=i\epsilon_{ijk } p^k \ , .\label{ccr2}\end{aligned}\ ] ] transformations of states under these transformations will not depend on the dynamics . the hamiltonian @xmath346 carries the states away from the initial fixed @xmath344 surface , and contains the dynamics . the other three generators ( the boosts ) will also , in general , depend on the dynamics because their commutators involve @xmath346 . the _ front - form _ surface @xmath352 is left invariant by translations in the @xmath353 , @xmath354 , and @xmath355 directions [ the generator of translations in the @xmath355 direction is @xmath356 because @xmath357 . it is also left invariant by rotations and boosts in the @xmath277 direction , and by the generalized boosts @xmath358 and @xmath359 . these 7 generators form a subgroup of the poincar group , with commutation relations @xmath360=[p^i , h^+]=[j^z , h^+]=[e^i , e^j]=[e^i , h^+]=[j^z , k^z]= [ k^z , p^i]=0 \nonumber\\ \fl&&[j^z , p^i]=i\epsilon_{ij}p^{j}\ , , \quad[j^z , e^i]=i\epsilon_{ij}e^{j}\ , , \quad [ e^i , p^j]=-i\delta_{ij}h^+\nonumber\\ \fl&&[k^z , h^+]=-ih^+ \ , , \quad[k^z , e^i]=-ie^i\ , , \label{ccr3}\end{aligned}\ ] ] where @xmath361 and @xmath362 , and @xmath363 , @xmath364 , and @xmath365 . the fact that the front - form kinematic subgroup includes _ seven _ generators , including the boost @xmath366 and generalized boosts @xmath367 , makes the front - form popular . but a principle motivation for using the front - form is that it is a natural choice at very high momentum , where the interactions single out a preferred direction ( the beam direction ) and the dynamics evolves along the light - front in that direction . the disadvantage is that the generators that contain dynamical quantities are @xmath368 and @xmath348 , and this means that angular momentum conservation must be treated as a dynamical constraint . finally , the _ point - form _ hyperbolid is left invariant by the homogeneous lorentz group itself , with commutation relations @xmath369=i\epsilon_{ijk } j^k\ , , \quad[j^i , k^j]=i\epsilon_{ijk}k^k\ , , \quad [ k^i , k^j]=-i\epsilon_{ijk}j^k\ , . \label{ccr4}\end{aligned}\ ] ] the hamiltonian and the momentum operators @xmath347 all carry the dynamical information . we see that each of the forms of quantum mechanics has a different set of kinematic generators , and in no case are they all kinematic . practitioners of hamiltonian dynamics sometimes speak as if it were impossible to treat the full poincar group kinematically . this is true only in the context of hamiltonian dynamics ; _ all _ of the generators are kinematic in the bs or spectator forms of progagator dynamics . dynamics is introduced whenever the states are propagated away from the surface on which they are initially defined . as in normal quantum mechanics , the deuteron will be an eigenstate that propagates in `` time '' without loss of probability ; it will be an eigenstate of the generalized hamiltonian . in the instant - form , the rest state @xmath370 is an eigenstate of the momentum operators @xmath371 and the bound state equation in the rest frame is @xmath372 in the front - form the rest state @xmath373 is an eigenstate of the operators @xmath374 and @xmath356 @xmath375 and the dynamical bound state equation is @xmath376 finally , in point - form the rest frame eigenfunction must satisfy the four dynamical equations @xmath377 where @xmath378 and @xmath379 . in applications , the dynamical equations ( [ instant ] ) , ( [ front ] ) and the @xmath380 component of ( [ point ] ) can all be taken to be the nonrelativistic schrdinger equation in the rest frame , so the same nonrelativistic phenomenology can be used for any of these forms of mechanics . to complete the calculation of the deuteron form factors using hamiltonian dynamics one must choose a current operator that conserves current , and construct the proper matrix elements of this operator between deuteron wave functions . this will be discussed next . the steps taken to construct the current and calculate the form factors depend on the form of quantum mechanics used , and the taste of the investigator involved . here we briefly describe recent work by five groups . _ forest , schiavilla , and riska _ [ fsr ] : based on the work of schiavilla and riska @xcite , forest and schiavilla @xcite have done an instant - form calculation of the deuteron form factors . the original work of ref . @xcite used on - shell matrix elements of the one body charge and current operators @xmath381u(p , s)\label{1body1}\end{aligned}\ ] ] expanded in powers of @xmath382 . here @xmath383 and @xmath384 are the dirac and pauli form factors , usually replaced by the familiar charge and magnetic form factors @xmath385 in the recent unpublished work of ref . @xcite the calculations have been done in momentum space , where the one body current operators have been evaluated without making any @xmath386 expansions , the relativistic kinetic energy @xmath387 has been used in place of the usual nonrelativistic expansion @xmath388 [ with the parameters of the av18 potential refitted ] , and the boost corrections to the deuteron wave functions have been included . this work also includes two - body charge operators from @xmath319 and @xmath223 exchange using methods developed by riska and collaborators @xcite . _ arenhvel , ritz , and wilbois _ [ arw ] : this recent calculation @xcite does a systematic @xmath382 expansion of relativistic effects that arise from the one body current operator and from contributions from meson exchanges . the current operator ( [ 1body1 ] ) is approximated by @xmath389 & $ \mu=0$\cr f_1\,{({\bf p}'+{\bf p})^i\over2 m } + g_m\ , { i\,[\,\sigma\times { \bf q}\,]^i\over2 m } + { \cal o}\left[\left({v / c}\right)^3\right ] & $ \mu = i$ } \label{1body}\end{aligned}\ ] ] where @xmath390 or @xmath391 ( with appropriate @xmath383 and @xmath384 ) , @xmath149 is the operator in the nuclear spin space , and @xmath392 is the _ three_-momentum transferred by the electron . this charge operator is correct to order @xmath174 , and the @xmath393 contribution to the current operator is given in ref . the @xmath394 correction term to the charge operator is referred to as the darwin - foldy term . the @xmath395 is the spin - orbit term . there are ambiguities in all calculations based on expansions in powers of @xmath174 . one ambiguity arises from the fact that the square of the three - momentum , @xmath396 , depends on the frame in which it is evaluated . in the breit frame , @xmath397 , while in the center of mass of the final deuteron ( the frame preferred by arw ) , @xmath398 ( recall that @xmath320 was defined in sec . [ dffdef ] ) . a second ambiguity surrounds the choice of @xmath399 versus @xmath383 . some experts @xcite advocate using @xmath399 ( because it is the correct charge operator ) in place of @xmath383 . the difference between @xmath383 and @xmath399 is of higher order . these ambiguities introduce theoretical uncertainty into any calculation . the size of this uncertainty depends on both the value of @xmath62 and the choice of nucleon form factors ; for example the difference between using @xmath383 or @xmath399 can be inferred from the ratios shown in fig . [ ges - study ] and is large for the recently measured jlab form factors and small for the mmd form factors . uncertainties of this kind do not arise if the calculation is done covariantly , or to all orders in @xmath174 . arw also include boost corrections originally derived to lowest order in @xmath382 by krajcik and foldy @xcite . the boost corrections can be written as an operation on the wave function of the form @xmath400 where @xmath401 with @xmath402\cdot { \bf p}\over8m^2 } \label{boost1 } \end{aligned}\ ] ] the boost associated with the kinetic energy and the spin and @xmath403 the boost associated with the potential . in ( [ boost1 ] ) , @xmath404 and @xmath405 are the relative coordinate and relative momentum of the nucleon pair , and @xmath406 is the three - momentum of the moving deuteron . arw use the values of @xmath403 worked out by friar @xcite , and also include relativistic effects from retardation , isobar currents , and meson exchange . to evaluate the latter a meson exchange model is needed , and arw use the interactions and parameters of the bonn obepq potentials [ only results from the obepq b potential are presented in the next section , although ref . @xcite includes results from all three obepq potentials ] . friar has emphasized that relativistic effects can be moved in and out of the wave functions and currents by unitary transformations @xcite , so that all of these effects are ambiguous unless fully defined by the theory . effects due to pair currents or recoil corrections , shown in fig . [ mec](c ) and ( d ) , do not appear to be included . arw state that their calculations should be good only up to @xmath407 1.2 gev@xmath42 . corrections to the charge operator to order @xmath174 obtained from instant - form dynamics and from the spectator form of propagator dynamics have been compared @xcite . in the cases studied , the same total result was obtained from the sum of _ all _ of the corrections , but the individual terms in the sum were found to have a very different form even when they appeared to come from the same physical effects . _ carbonell and karmanov _ [ ck ] : in this front - form calculation @xcite the direction of the light - front [ denoted by @xmath408 where @xmath409 is treated as an unphysical degree of freedom . wave functions and amplitudes may depend on @xmath410 but only those components of scattering matrix elements independent of @xmath410 will be physical . it is argued that this approach will give an explicitly covariant front - form mechanics @xcite . when applied to the deuteron form factors there are 11 spin invariants , three that are physical and 8 that depend on @xmath410 and are unphysical . in an exact calculation the 8 unphysical invariants would be zero , but in approximate calculations , such as that carried out in ref . @xcite , they will not be zero . the deuteron form factors can be extracted from the three physical invariants by projecting them from the general result , as derived in ref . @xcite . for the choice @xmath411 ( corresponding to choosing the front - form surface @xmath352 ) this method shows that the charge and quadrupole form factors can be extracted from the @xmath412 component of the current ( in common with other treatments ) , but also shows that the magnetic form factor _ can not be obtained only from this component _ and requires a different projection ( and includes contributions from contact terms ) . the rules for a general graph technique for calculating amplitudes in this formalism are given in ref . @xcite . using this method the deuteron wave function will in general have 6 components , only three of which have been found to be numerically large . in addition to the familiar @xmath17 and @xmath18-state components , the third large component is proportional to a new scalar function @xmath413 , and adds the term @xmath414 to the deuteron wave function displayed in eq . ( [ ft1 ] ) ( to obtain this form we renormalized the expression in ref . @xcite so that @xmath415 and used the transformations in ref . @xcite ) . in ref . @xcite @xmath413 is calculated perturbatively using the bonn potential from ref . @xcite without change of parameters . they find that @xmath413 is the largest of the three components for all momenta greater than 500 mev , and believe that the perturbative estimate is accurate to about 20% . the physical meaning of the @xmath413 contribution has been studied in threshold deuteron electrodisintegration , where it contributes about 50% of important pair term contributions . _ lev , pace , and salm _ [ lps ] : the lps @xcite calculation is a recent version of a series of light - front calculations that have assumed the light - front is fixed ( at @xmath334 ) . in the past , calculations with fixed light fronts have run into a problem with the loss of angular momentum conservation , and before we review the lps results we will discuss this issue . in calculating form factors with fixed light fronts it has been conventional to choose a coordinate system where @xmath416 and @xmath417 . current conservation is then satisfied if only one component of the current ( @xmath412 ) is non zero . consider the matrix elements of the deuteron current , @xmath418 , where @xmath419 ( @xmath420 ) are the helicities of the outgoing ( incoming ) deuterons . one consequence of the loss of _ manifest _ rotational invariance is that there are _ four _ independent matrix elements of the @xmath412 current related by the constraint @xmath421 this is a dynamical constraint often referred to as the `` angular condition '' @xcite . the deuteron form factors can be extracted from _ any _ choice of three of the matrix elements @xmath418 , and if condition ( [ angular ] ) is not satisfied each choice will yield different results . the form factors will not be uniquely determined unless the angular condition is satisfied . to avoid ( or solve ) this problem , lps work in the breit frame , where @xmath422 and @xmath423 . current conservation then requires that @xmath424 . a current operator that satisfies these conditions was constructed in ref . @xcite . for elastic scattering this operator has the form @xmath425 where @xmath426 is the free ( one body ) current operator , and @xmath427 and @xmath428 are rotations by @xmath429 about the @xmath353 axis , @xmath427 in the vector space and @xmath428 in the spinor space . note that the definition insures that @xmath424 as required by current conservation . using this current , lps have calculated the deuteron quadrupole moment , @xmath263 , to 2% accuracy @xcite . the calculation shown below in sec . [ thyandexp ] uses mmd nucleon form factors and the nijmegen ii deuteron wave functions . _ allen , klink , and polyzou _ [ akp ] : the deuteron form factors have also been recently calculated using the point - form of quantum mechanics @xcite . here there is no difficulity in writing down manifestly covariant matrix elements , but there is some ambiguity in deciding how to impose current conservation . akp work in the breit frame , choose a one body impulse current to describe the @xmath430 , and 2 components of the current , and introduce a two body current @xmath431 ( which need not be calculated ) to insure current conservation . new effects come from the way the wave functions are constructed in point form ( `` velocity '' states are constructed ) , and from the fact that momentum is now a dynamical generator , so that the momentum transferred to the nucleon _ inside _ the deuteron is not equal to the momentum transferred to the deuteron as a whole . they argue that the momentum transferred to each nucleon inside the deuteron is @xmath432 at momentum transfers @xmath433 gev@xmath42 this is a 25% increase , and leads to a large suppression of the form factors . this explains part of the decrease in the size of the form factors predicted by this model . the results reported below use the mmd nucleon form factors and the av18 @xmath7 potentials . we now turn to a brief review of methods using quark degrees of freedom . calculations based on quark degrees of freedom must confront the fact that the deuteron is at least a six quark system . since the six quarks are identical ( because of internal symmetries ) the system must be antisymmetrized , and it is not clear that the nucleon should retain its identity when in the presence of another nucleon . how does the clustering of the six quarks into the two three - quark nucleons appear at large distance scales ? how do we treat the confining forces in the presence of so many quarks ? the approach to these issues depends on whether of not @xmath62 is large enough to justify the use of perturbative qcd ( pqcd ) . at modest @xmath62 the momentum transferred by the gluons is small and the qcd coupling is too large for perturbative methods to be useful . in the nonperturbative regime calculations must be based on models . many papers have been written addressing these issues , and a complete review is beyond the scope of our discussion . here we mention only two contributions that give the flavor of the discussion . maltman and isgur @xcite studied the ground state of six quarks interacting through a @xmath434 potential previously used to explain the spectrum of excited nucleons , and found that there was a natural tendency for the quarks to cluster into two groups of three ( i.e. nucleons ) . they obtained a reasonable description of the deuteron , and confirmed that the short range @xmath7 repulsion could be largely understood in terms of quark exchange . later , de forest and mulders @xcite , using a very simple model , considered the effect of antisymmetrization on the structure of the form factor . their calculations suggest that the zeros seen in form factors could be a consequence of antisymmetration alone . they also show that the factorized form of the impulse approximation obtained from nonrelativistic ( and some relativistic ) theories , which gives the deuteron form factors as a product of a nucleon form factor and a nuclear ( or body ) form factor @xmath435 may not be a good description in the presence of antisymmetration . when quarks are exchanged between nucleons it is no longer possible to separate the _ nucleon _ structure from the _ nuclear _ structure . consideration of the quark exchange diagram shown in fig . [ ffexchange ] suggests a factorization formula of the form @xmath436 ^ 2)\right]^2\times d_2(q^2)\ , . \label{exff}\ ] ] because nucleons are composite and identical , either of the forms ( [ fac1 ] ) or ( [ exff ] ) ( or yet other relation ) might hold , and there is no clearly correct way to isolate the structure of the nucleon from the structure of the bound state . in model calculations these issues can be handled by separating the problem into two regions : at large separations ( @xmath437 ) it is assumed that the system separates into two nucleons interacting through one pion exchange , and at small distances ( @xmath438 ) the system is assumed to coalesce into a six - quark bag with all the quarks treated on an equal footing . in this review we report the results of a calculation by dijk and bakker @xcite , where references to other calculations of this type can also be found ( see also the work of buchmann , yamauchi , and faessler @xcite ) . _ dijk and bakker _ [ db ] : this calculation is based on the quark compound bag model introduced by simonov @xcite . here the six - quark wave function is assumed to be the sum of a hadronic part and a quark part . the hadronic part is a fully antisymmetrized product of two three - quark wave functions , each with the quantum numbers of a nucleon , and a relative @xmath7 wave function @xmath439 @xmath440\ , , \ ] ] where @xmath441 is the effective internucleon separation . the quark part is a sum of eigenstates @xmath442 of a confined 6-quark system @xmath443 where the confined 6-quark states are zero outside of a confining radius @xmath444 , which is a parameter of the calculation . in the applications , only one term @xmath445 needs to be included in the sum ( [ cbstates ] ) . the dynamical quantities determined by the calculation are the @xmath7 wave function @xmath446 and the spectroscopic coefficient @xmath447 which is a function of the energy @xmath142 . the @xmath7 scattering phase shifts and mixing parameters are determined by replacing the spectroscopic coefficients by boundary conditions on the surface @xmath444 and integrating the schrdinger equation for @xmath448 . the paris potential @xcite is used to describe the @xmath7 interaction in the peripheral region and is set to zero in the inner region . two models were developed ; in this review we report results from the fits to the arndt single energy 1986 @xcite solutions , which db denote qbc86 . this fit finds @xmath449 fm . calculation of the form factors requires an assumption about the form factor of the internal compound bag part of the wave function . they use @xmath450 with @xmath451 gev obtained from a fit to the @xmath0 and @xmath1 structure functions . results from this model are reported in sec . [ thyandexp ] below . if one believes the momentum transfer is high enough , perturbative qcd ( pqcd ) may be used to study the deuteron form factor and reactions . here it is assumed that the problem naturally factors into a hard scattering process in which the momentum transfer is distributed more or less equally to all of the six quarks , _ preceeded and followed _ by soft , nonperturbative scattering that sets the scale of the interaction but does not strongly influence its @xmath62 behavior . the @xmath62 behavior is therefore determined by the hard scattering , which can be calculated perturbatively . the formalism and method are reviewed in the seminal papers by brodsky and farrar @xcite and lepage and brodsky @xcite . these calculations of the form factor are all based on the diagrams shown in fig . [ pqcddiagrams ] . in the hard scattering , the momentum transfer @xmath134 is distributed to the six quarks through the five hard gluon exchanges , the last of which carries a momentum of approximately @xmath452 . if the spin factors are included with the quark propagators , the only large @xmath62 dependence comes from the @xmath453 of each gluon propagator , giving the _ counting rule _ for the hard scattering part @xmath454 @xmath455^{-(n_c-1)}\ , , \label{counting}\end{aligned}\ ] ] where @xmath456 is the number of constituent quarks ( 6 for the deuteron ) and @xmath457 the number of gluon propagators . this leads immediately to the prediction that the leading contribution to the deuteron form factor should go like @xmath458 , or that @xmath459 . the argument also shows that the perturbative result can not be expected to set in until @xmath460 to 1 gev , somewhere in the region of @xmath62 from 9 to 36 gev@xmath42 . ( all agree that pqcd must give correct predictions at sufficiently high @xmath62 , but how large this @xmath62 must be is a topic of considerable controversy @xcite . ) note that this simple argument does not set the scale of the form factor ; estimates can be obtained from detailed evaluation of more than 300,000 diagrams that contribute to the hard scattering @xcite . it turns out that this leading twist pqcd estimate is 10@xmath461 10@xmath462 times smaller than the measured deuteron form factor , implying large soft contributions to the form factor , in agreement with @xcite , and suggesting that pqcd should not be used as an explanation for the form factor . the calculation is extremely complicated and a confirmation , or refutation , is desirable . perturbative qcd also predicts the _ spin dependence _ of the hard scattering , and these predictions provide a more stringent test of the onset of pqcd . these spin dependent predictions have implications for the individual deuteron form factors , and these were first presented in ref . @xcite , and further developed in refs . application of these rules to hadronic form factors in general shows that * hadrons with an _ even _ number of quarks will be dominated by the _ longitudinal _ ( charge ) currents , while these with an _ odd _ number of constituents by _ transverse _ ( magnet ) currents . hence , the dominant form factors at large @xmath62 should be the nucleon magnetic form factors and the deuteron charge ( or quadruple ) form factors . * the dominant form factors at large @xmath62 are those that conserve helicity . when applied to the deuteron , these rules lead to the conclusion that the helicity amplitude @xmath463 [ c.f . ( [ dhelicity ] ) ] dominates at large @xmath62 ; the others are smaller by at least a power of @xmath134 . in particular , this implies that @xmath464 , where @xmath465 is the mass scale above which the nonleading terms can be neglected . hence @xmath466 unfortunately , this argument does not allow one to estimate the ratio @xmath467 , since @xmath1 is controlled by a different , independent helicity amplitude . in ref . @xcite an attempt was made to improve on the constraint ( [ t20prediction ] ) . these authors used the front - form , and evaluated the current in the light - front breit frame where the plus component of the momentum transfer @xmath468 . in this frame all three deuteron form factors may be written in terms of matrix elements of the @xmath469 component of the current @xmath470 where @xmath471^{-1}$ ] . perturbative qcd predicts that @xmath472 will dominate at large @xmath62 , and if this happens at a scale @xmath473 it follows from ( [ frontff ] ) that the form factors go in the ratio of @xmath474 . this leads to a prediction for @xmath467 and to a prediction for @xmath475 that differs from ( [ t20prediction ] ) at moderate @xmath62 . however , rotational invariance is not manifest in the light front , and there are _ four _ nonzero components of the @xmath412 current corresponding to deuteron helicity combinations of @xmath476 , @xmath477 , @xmath478 and @xmath479 that are related by the angular condition discussed in sec . [ hexamples ] above . carlson has recently shown @xcite that the angular condition places strong constraints on the possible subleading behavior of the helicity amplitudes . perturbative qcd predicts that the subleading amplitudes will go like @xmath480 where @xmath202 , @xmath481 , and @xmath335 are dimensionless constants of the order of unity , and @xmath465 is the scale at which pqcd begins working for the deuteron [ the @xmath281s in eq . ( [ brodg ] ) are identical to the @xmath482s in eq . ( [ angular ] ) ] . assuming that @xmath483 ( but making no assumption about the size of @xmath465 ) the angular condition in leading order becomes : @xmath484 solution of this equation therefore requires that @xmath485 and @xmath486 . under these conditions the relations ( [ frontff ] ) again produce _ only _ the result ( [ t20prediction ] ) . in this section we compare theory with experiment and draw conclusions from this comparison . our major conclusions will be restated and summarized again in sec . [ conclusions ] below . figure [ gc - relcorr ] shows how coulomb distortion of the incoming and outgoing @xmath3 plane waves effects the very low @xmath62 data ( extracted from ref . @xcite ) for the charge form factor , @xmath70 . the figure also compares this data with theory . corrections for coulomb distortion change the deuteron radius from an apparant 2.113 fm ( as measured in @xmath3 scattering ) to 2.130 fm ( after the correction ) @xcite . to remove the distortions from the data of ref . @xcite , we adjust @xmath70 by @xcite @xmath487 note that this decreases @xmath70 at very small @xmath134 giving a larger deuteron radius , but increases @xmath70 where the data have been extracted . the figure shows both the uncorrected and the coulomb corrected data normalized to the nonrelativistic av18 calculation with the mmd nucleon form factor . note that the difference between the coulomb corrected and uncorrected data is about half of the experimental error at @xmath488 0.5 gev . the figure also shows the size of relativistic and interaction current corrections that arise from the instant - form calculation arw of ref . @xcite , the front - form calculation lps of ref . @xcite , the point form calculation akp of ref . @xcite , and the cia and ria approximations from ref . these calculations were discussed in sec . [ pexamples ] and [ hexamples ] above . at the scale of the current experimental accuracy ( a few percent ) , the relativistic treatments _ differ noticeably_. they also differ from the eft calculation ( shown previously in fig . [ fig : eft ] ) which drops sharply below the data for @xmath489 0.2 gev . it is important that these calculations be systematically compared and the different physical content of these approaches be isolated and understood . in particular , it would be very interesting to know why the covariant cia and ria have more positive corrections than those obtained from the hamiltonian forms of dynamics . @llll model & dynamics & description & consistent current + vog @xcite & propagator & spectator & yes + pwm @xcite & propagator & modified mandelsweig - wallace & no + fsr @xcite & hamiltonian & instant - form ; no @xmath382 expansion & yes + arw @xcite & hamiltonian & instant - form ; with @xmath382 expansion & yes + ck @xcite & hamiltonian & front - form ; dynamical light - front & no + lps @xcite & hamiltonian & front - form ; fixed light - front & no + akp @xcite & hamiltonian & point - form & no + db @xcite & nonrelativistic & quark - cluster & yes + the high @xmath62 predictions for the eight models reviewed in secs . [ pexamples ] , [ hexamples ] , and [ npqcd ] are shown in figs . [ alowq27][rabt20 ] . the models are summarized in table [ tab : theories ] . these calculations give very different results . figure [ alowq27 ] shows the predictions for @xmath490 , with the model dependent @xmath268 exchange current intentionally omitted from all of the calculations . all of the models except the akp point - form calculation give a reasonable description of @xmath0 out to @xmath491 gev@xmath42 , beyond which they begin to depart strongly from each other and the data . taking into account that the @xmath268 exchange current _ could be added to any of these models , and that this contribution tends to increase a above @xmath491 gev@xmath42 _ ( for the sign of the @xmath268 coupling constant used in the discussion in the following paragraph ) , four models seem to have the right general behavior : the vog , fsr , arw and the quark model of db ( but there are no results for this model beyond @xmath492 gev@xmath42 ) . ironically , none of the models favored by the high @xmath62 data does as well at low @xmath62 as the three `` unfavored '' models shown in the right panels ( unless the platchkov @xcite data are systematically too low ) . another possibility suggested by effective field theory @xcite is that the asymptotic normalization of the relativistic deuteron wave functions is incorrect , and that a small adjustment in @xmath7 parameters to insure a good value for this constant would correct the problem . figure [ jlaba ] shows the effect of the new jlab measurements of the nucleon form factors on predictions for @xmath490 . these new form factors will _ decrease _ predictions for @xmath0 for momentum transfers in a region around @xmath492 gev@xmath42 ( by a factor of 2 for the model shown ) , increasing the descrepancy between predictions and the data . [ however , it may improve the prediction for those models ( pwm , ck , and lps ) that are currently too large in this region . ] the figure also shows how the @xmath268 exchange current could increase predictions at large @xmath62 . the difference between the solid and long dashed lines is due largely to the effect of the @xmath268 exchange current ( but is also do in small part to the fact that the cia result is slightly smaller that the ria ) . unfortunately , the size of the @xmath268 exchange current is very sensitive to the @xmath268 form factor , as discussed in ref . @xcite , and could even be too small to see at these momentum transfers ( if the current estimates of the @xmath268 form factor are too large ) . this is our reason for insisting that this contribution should be viewed as new physics not readily predictable within a meson model . finally , fig . [ rabt20 ] shows the predictions for the structure functions @xmath0 , @xmath1 , and @xmath2 for the eight models discussed . the lps calculation shows a large descrepency with the @xmath2 data , but the most striking feature of these plots is the _ large model dependence _ of the predictions for @xmath493 . the magnetic structure function provides the most stringent test , and the predictions are comparatively free of the @xmath268 exchange current ( which gives only a small contribution to @xmath1 ) . examination of the figure shows that the @xmath1 predictions of the pwm , arw , akp , ck models fare the worst . in all , taking the predictions for the three structure functions together , the best results are obtained with the fsr , vog , and db models . we conclude our comparison with theory by extending our discussion somewhat beyond the limits of previously published papers . to make the point clearly , focus on the vog calculation using the spectator equation , and recall that current conservation _ required _ that the single nucleon current in this approach , eq . ( [ onej ] ) , include a new form factor , @xmath494 . this form factor must satisfy the constraint @xmath328 , but is otherwise _ completely unspecified_. in the published vog calculation @xcite and in all of the plots shown so far , this form factor was taken to be the standard dipole , @xmath191 . figure [ fig : f3 ] shows how agreement between theory and experiment can be significantly improved by choosing a different form for the unknown form factor @xmath495 . this figure shows three theoretical predictions : ( i ) the `` standard '' vog ria prediction with @xmath329 and _ no _ @xmath268 exchange current , ( ii ) another model with _ no _ @xmath268 exchange current , but with a tripole @xmath495 of the form @xmath496 and ( iii ) a model with @xmath329 and a dipole @xmath268 form factor @xmath497 in both form factors @xmath62 is in gev@xmath42 . these form factors are shown in the bottom left panel of the figure . figure [ fig : f3 ] shows that a good agreement between theory and experiment can be obtained with the tripole @xmath495 without any @xmath268 exchange current , and that to some extent the @xmath268 exchange current can be substituted for a hard @xmath495 ( although @xmath495 is better than @xmath498 at improving all three observables simultaneously ) . we see that we have , in some sense , achived the goals of a theory of elastic @xmath3 scattering based on nucleon degrees of freedom . _ with small adjustments of unknown form factors associated with short range physics , the nn theory can describe all three form factors quite well_. it seems likely that any nucleon model with a _ consistent and complete _ description of the current ( c.f.table [ tab : theories ] ) can do as well . the reasonable results obtained from the fsr and db models are probably due to the fact that they have consistent , complete currents not based on an expansion in powers of @xmath174 ( which must fail at high @xmath62 ) . from the discussions in sec . [ thyandexp ] above , it is clearly of interest to extend measurements of @xmath0 to higher @xmath62 . an @xmath3 coincidence experiment is straightforward , but prohibitive timewise with present accelerators . the proposed 12 gev jlab upgrade allows one to take advantage of the approximate @xmath499 scaling of @xmath90 at constant @xmath62 and high energy @xcite . a large acceptance spectrometer such as mad would be very helpful . depending on the details of the upgrade , a one month experiment could provide data to @xmath62 of 8 gev@xmath42 . it might also be desirable to do a new , high precision experiment at _ low _ @xmath62 . the goal of this experiment would be to resolve the discrepancy between the data sets of refs . @xcite and @xcite , and to check the low @xmath62 limit of the relativistic calculations . these measurements require little time , but do require excellent control of systematic uncertainties , at the level of 1 2% , if they are to be meaningful . a hasty examination of fig . [ rabt20 ] might lead one to believe that the problems with @xmath1 are mainly theoretical , and that there is no need for new data . we believe this attitude would be inappropriate for two reasons . first , some of the calculations shown in the figures are in early stages of development , and will improve before any new data are available . the number of calculations is a reflection of the challenge that the high @xmath62 data present , and a reflection of interest in , and knowledge of relativistic methods that is emerging from the study of these measurements . second , and most important , measurements of @xmath1 vary by 5 orders of magnitude . the existence and position of the first minimum has not yet been firmly established , and the location and existence of a possible second minimum is unknown . these minima in @xmath1 ( if they exist ) result from cancellations of various physical effects and provide a very precise test of any theory . as indicated above , measurements of @xmath1 do not require high energy , but do require large scattering angles , as close as possible to 180@xmath137 . at 180@xmath137 , beam energies of 1 to 2 gev cover a @xmath62 range from 1.4 gev@xmath42 to 5 gev@xmath42 . the slac ne4 experiment was a heroic effort , run with @xmath3 coincidences . energy resolution was limited by thick targets , 20 - 40 cm long . there was a large but manageable background of @xmath500 in the @xmath501 spectrometer , about 3 or 4 to 1 , and an extremely large proton background from @xmath502 in the deuteron channel - up to two events per beam pulse were seen in the worst kinematics . since ne4 ran at beam currents of up to 50 ma instantaneous , with a duty factor of about 0.3 @xmath155 10@xmath147 , corresponding to an average current of about 15 @xmath153a , the jlab continuous beam structure essentially eliminates random coincidence backgrounds such as these . background coincidence reactions included @xmath503 and @xmath504 at 180@xmath140 , with the photon producing an electron detected in the electron channel . the spectrum of these photoreactions ends near the elastic peak , allowing the background contributions to be fit and determined . a jlab experiment would run with both larger spectrometer acceptance and higher luminosity to increase rates , but with a shorter target to reduce these backgrounds . the proposed configuration @xcite would use the hall a septum magnets to detect the forward - going deuterons at angles of 3 - 6@xmath140 , along with special electron channels to detect scattered electrons at about 160 - 170@xmath140 . based on the slac ne4 cross sections , a one month experiment can map out @xmath1 to about 6 gev@xmath42 . extending @xmath6 or other polarization observables to higher @xmath62 is quite difficult @xcite . there are three obvious possibilites . recoil tensor polarimetry requires a well calibrated polarimeter with a large figure of merit , but no such device exists . for example , polder , used in the jlab hall c @xmath6 experiment , relied on the @xmath505 reaction , for which the figure of merit decreases at larger energies , leading to a practical upper limit in @xmath62 , that was reached in e94 - 018 . hypomme @xcite is promising , but not well enough calibrated . the combination of polarized electron beams with recoil vector polarimeters is an untested possibility @xcite . with @xmath0 , @xmath1 , and @xmath71 known , @xmath104 is calculable and can calibrate the polarimeter analyzing power , while @xmath506 determines the form factor combination @xmath507 . ( the ratio of the two polarization components depends on this combination of form factors , times kinematic factors and divided by @xmath71 . ) since the jlab polarized source can provide 50 100 @xmath153a beams , there is no luminosity problem . the difficulty with this measurement is that the polarization components are expected to be small in the @xmath62 range of interest , @xmath151 0.01 , and @xmath508 @xmath509 5 10 . a one month measurement for one @xmath62 of @xmath151 2 - 2.5 gev@xmath42 can determine the _ polarization components _ well , but if these are small as expected , the _ form factors _ can only be extracted with factor of two uncertainties . an alternative is to use asymmetries from a polarized target . however , the reduced currents that can be used with polarized targets require a large acceptance detector such as clas , to make up for the lack of luminosity . also , although current polarized targets have moderately large deuteron vector polarizations , tensor polarizations are small . furthermore , the asymmetry varies as @xmath510 , with @xmath511 the polarization direction , so it is desirable to have complete azimuthal coverage , with @xmath511 in the direction of @xmath512 at one azimuthal angle , rather than being purely transverse . extensive beam time would be needed , either as an external polarized target experiments at jlab , or as an internal polarized target experiment at hermes . we note that a series of moderate @xmath62 measurements are planned with the mit bates blast detector @xcite , for @xmath62 from 0.1 to 0.9 gev@xmath42 . proposed next generation colliders , such as epic / erhic , are promising due to large planned luminosities ; for this experiment the lower proposed c.m . collision energies are desirable for ensuring exclusivity . the spin direction of the polarized deuteron beam must be controllable . in collider kinematics , the scattered electron and deuteron energies are close to their respective beam energies and are slow functions of @xmath62 , while the scattering angles for fixed @xmath62 vary slowly with the beam energies . thus , if an experiment is possible , it would attempt large azimuthal coverage of coincidence @xmath513 elastic scattering , with the outgoing particles at angles from a few to about 20@xmath137 from the beam line . comparison of theory and experiment leads to the following conclusions : * nonrelativistic quantum mechanics ( without exchange currents or relativistic effects ) is ruled out by the @xmath490 data at high @xmath62 . reasonable variations in nucleon form factors or uncertainties in the nonrelativistic wave functions can not remove the discrepancies . * in some relativistic approaches using @xmath7 degrees of freedom only , short range physics not calculable within the model ( @xmath495 or @xmath498 , for example ) can be adjusted to give good agreement with all the data . * different ways of calculating relativistic effects ( or meson exchange currents ) can give results that differ substantially from each other . even at low @xmath62 , where all calculations are constrained , these differences are larger than errors in the data . this is not understood , but may be due to the failure of some models to use realistic currents . * the deuteron form factors provide no evidence for the onset of perturbative qcd , but quark cluster models could explain the data . study of the experimental situation leads to the following conclusions : * a good database of @xmath0 , @xmath1 , and @xmath6 measurements has been obtained ; while discrepancies exist they are generally not large enough to affect the theoretical interpretation . * the minimum of @xmath1 is very sensitive to details of the models , and improved measurements of @xmath1 for @xmath62 in the region 1.5 - 4 gev@xmath42 are particularly compelling . it is important to accurately map out the zero in the @xmath1 structure function . * detailed disagreements between theories and different data sets suggests the need for precision studies at low @xmath62 . as the deuteron has no excited bound states , inelastic scattering experiments have largely consisted either of ( i ) measurements in which the final state mass , w , is very close to @xmath514 ( referred to as _ threshold electrodisintegration _ even when @xmath62 is very large because the final state is close to the @xmath7 scattering threshold ) , ( ii ) measurements near the quasifree peak ( defined by the condition that the `` spectator '' nucleon remain at rest ) , or ( iii ) deep inelastic scattering in which both @xmath62 and @xmath308 become very large . we will not discuss deep inelastic scattering in this review . for processes at modest energies near the quasielastic peak , a rough estimate of the cross section can be made using the _ unrealistic _ plane wave impulse approximation ( pwia ) in which all final state interactions are ignored . denoting the momentum of the outgoing struck nucleon by @xmath515 , the cross section in pwia is proportional to @xmath516 where @xmath517 is some combination of the squares of the electric and magnetic form factors of the nucleon , and @xmath518 is an average of the square of the momentum space wave function of the deuteron with internal relative momentum @xmath405 . were the pwia realistic , eq . ( [ pwia ] ) shows that inelastic scattering in quasielastic kinematics would provide a direct measure of the ( square ) of the deuteron wave function . while the pwia is overly simplistic , it does illustrate ( correctly ) one of the central justifications for quasielastic measurements . within the context of a more realistic dynamical theory , one can use response function separations and polarization observables to enhance the sensitivity to various model dependent _ nonobservables _ , such as momentum distributions , meson - exchange currents , and medium modifications . one strong recent interest has been to choose kinematics in which the unobserved nucleon has a large momentum ; the plane wave approximation shows that this configuration enhances sensitivity to initial - state short range correlations ( i.e. the wave function ) and possible quark effects . a number of these experiments have been carried out at various accelerators , but no experiments at jlab have yet reported results . thus , an experimental review of this topic is unwarrented at this time . however , because photodisintegration , electrodisintegration , and threshold electrodisintegration are closely related theoretically , we present the theoretical background for these processes below . in this subsection the cross section and polarization observables for electrodisintegration and photodisintegration to an @xmath519 final - state are reviewed briefly . ( we do not discuss pion and meson production . ) the electroproduction cross sections will be obtained first , and photoproduction will then be treated as a special case . the most general decomposition of the @xmath520 coincidence cross section was first discussed by donnelly and raskin @xcite . here we will follow the later work of ref . the cross section can be shown to have the form ( eq . ( 95 ) of ref . @xcite ) @xmath521 \nonumber\\ & & + s_{lt}\left[\cos\phi \tilde{r}^{({\rm i})}_{lt } + \sin\phi\tilde{r}^{({\rm ii})}_{lt}\right ] + 2 h \ , s_{t'}\tilde{r}^{({\rm ii})}_{t^{\prime } } \nonumber\\ & & + 2hs_{lt'}\left[\sin\phi\tilde{r}^{({\rm i})}_{lt^{\prime } } + \cos\phi\tilde{r}^{({\rm ii})}_{lt^{\prime}}\right]\biggl\}\ , , \label{crosssec}\end{aligned}\ ] ] where , in the lab frame , @xmath522 , and the nine response functions are functions of @xmath64 , @xmath523 , and the scattering angle @xmath524 of the final - state proton ( measured in a coincidence experiment and integrated over in an inclusive measurement ) . the ejectile plane is tilted at an angle @xmath101 with respect to the electron scattering plane , as illustrated in fig . [ fig_cross ] . the mott cross section and other variables are as defined in eq . ( [ mott ] ) . the electron kinematic factors are @xmath525 with @xmath526 . one of the virtues of eq . ( [ crosssec ] ) is that the response functions @xmath527 are _ covariant _ , and hence ( [ crosssec ] ) it can be used to describe the cross section in either the c.m . of the outgoing @xmath519 pair or the laboratory frame , provided we use the appropriate form of @xmath528 : @xmath529 where @xmath530 is the lab recoil factor @xmath531 with @xmath308 the invariant mass of the outgoing pair . in any frame the nine response functions of ( [ crosssec ] ) are related to the components of the deuteron response tensor @xmath532 @xmath533 with x = i or ii , and @xmath534 all of these quantities are written in the helicity basis , with @xmath535 the helicity of the virtual photon , @xmath536 and @xmath537 the helicities of particles 1 and 2 in the final - state , and @xmath538 the helicity of the initial deuteron . the matrix element of the helicity - basis current operator between helicity states is represented by @xmath539 . in cases where the deuteron target might be polarized _ only _ in the @xmath540 direction , and where _ only _ the polarization of the outgoing particle 1 might be measured , the spin density matrices for particle 1 in the final state is given by @xmath541 and that of the deuteron in the initial state by @xmath542 , where @xmath543 where @xmath544 is the @xmath540 component of the spin - one matrix , @xmath545 is the vector polarization of the deuteron target , @xmath105 is the direction of the polarization of outgoing particle 1 , measured with respect to the @xmath546 coordinate system shown in fig . [ fig_cross ] , and @xmath547 are the pauli matrices . note that only those response functions of type i ( denoted by the superscript ) are nonzero if all of the hadrons are unpolarized ; type ii response functions require ( for the cases considered here ) measurement of the polarization of the outgoing nucleon . further details and additional cases can be found in ref . @xcite . the familiar unpolarized inclusive cross section is easily obtained by integrating ( [ crosssec ] ) in the c.m . and summing over electron polarizations . the result is @xmath548 with @xmath549 for real photons the longitudinal components are absent , and the cross section simplifies ( there is no electron scattering plane and no electron kinematics ) . the most general polarization of the incoming photon , @xmath39 , is therefore a superposition of the circular polarization states @xmath550 , which we write as @xmath551 with @xmath552 . the expansion coefficients @xmath553 can therefore be written in terms of only three independent parameters @xmath554 with @xmath555 . the coincidence cross section for a polarized photon beam is then @xmath556 where @xmath557 and @xmath558 are the fractions of right circular and linear photon polarizations , respectively , and @xmath559 depends on the frame @xmath560 with @xmath530 defined in eq . ( [ defrecoil ] ) . we will return to the cross section ( [ photonxsec ] ) in sec . [ photodis ] . it is important to appreciate that these formulae for the cross section are _ exact _ relativistic results ( subject only to the one photon exchange approximation ) . all of our ignorance is confined to the hadronic matrix elements of the current @xmath561 and the structure functions ( [ rrela ] ) that are products of these currents . in much of the older literature , particularly for studies of the @xmath562 reaction from nuclei with mass number @xmath563 @xcite , the cross section is written @xmath564 where @xmath565 is a kinematic factor , @xmath566 is cross section for scattering of an electron from an `` off - shell '' proton , and @xmath567 is the proton spectal function ( which gives the probability of finding a proton with momentum @xmath236 and separation energy @xmath568 in the target nucleus ) . the proton momentum distribution is obtained by integrating the spectral function over the separation energy @xmath569 some early experiements focused on `` measuring '' the momentum distribution and the spectral function . while this picture has a nice physical interpretation [ it is motivated by the pwia , eq . ( [ pwia ] ) ] , and presenting data this way is sometimes useful , particularly in the early phases of the program , it is important to realize that the individual structure functions that enter the exact cross section ( [ crosssec ] ) are , in general , _ independent _ functions which are _ not _ proportional to each other , and that therefore eq . ( [ spectral ] ) is _ only an approximation _ to the cross section @xcite . attempts to refine the definitions of @xmath566 and @xmath567 can have limited value at best , and at worst can lead to many unproductive debates about the precise definition of the spectral function . calculation of the hadronic current matrix elements ( [ currents ] ) is complicated by requirement that the current be conserved , @xmath570 . for elastic scattering , where the initial and final states are identical , invariance under time inversion usually guarantees that even simple approximations to the dynamics will satisfy this constraint . but building in current conservation for inelastic processes usually requires consistent treatment of both final - state interactions and interaction currents . the failure of approximate calculations ( and the pwia in particular ) to satisfy current conservation is often seen as a serious obstacle . some _ ad hoc _ prescription of the kind introduced by de forest @xcite , is needed . we propose the simple prescription introduced recently in the study of deep inelastic scattering @xcite . suppose the _ current is composed of two parts @xmath571 . in general , _ neither _ of these two parts will satisfy current conservation _ alone _ ; that is @xmath572 for each @xmath361 . however , since the exact current satisfies current conservation , @xmath573 . we propose replacing each of the individual terms in the current by @xmath574 this procedure is covariant , guarantees that each component conserves current ( so that one can be calculated without knowing the other ) , and that their sum is unchanged : @xmath575 . perhaps the best argument can be found in ref . @xcite where it was shown ( for a very simple case ) that the born term defined in this way dominates the final - state interaction term in the deep inelastic limit , resolving a long standing puzzle . finally , note that @xmath576 ( where @xmath577 are the virtual photon polarization vectors satisfying @xmath578 ) so that the response tensor ( [ resptens ] ) is _ unaffected _ by the redefinition ( [ redef ] ) ! we now turn to a brief discussion of threshold electrodisintegration . threshold deuteron electrodisintegration measures the @xmath579 reaction in kinematics in which the proton and neutron , rather than remaining bound , are unbound with a few mev of relative kinetic energy in their center of mass system . if the final - state energy is low enough , the final state will be dominated by transitions to the @xmath580 final state , and will be a pure @xmath581 , @xmath582 , @xmath583 transition , similar to the @xmath584 transition . this transition is a companion to the @xmath1 structure function ; both are magnetic transitions and both are filters for exchange currents with only one isospin ( @xmath585 is @xmath586 and @xmath587 is @xmath588 ) . to see the similarity , compare the top right panel of fig . [ rabt20 ] with the threshold measurements shown in fig . [ thresh ] . both have a similar shape , and in both cases the uncertainties in the theoretical predicitions are large . the similarity of these two processes ( elastic and threshold inelastic ) also holds for the theory . these two processes can be used to separately determine the precise details of the @xmath266 and @xmath265 exchange currents . once the exchange currents are fixed , they can be used to predict the results of @xmath520 over a wide kinematic region . any theoretical approach that works for the form factors should also work equally well for threshold electrodisintegration , yet very few of the groups who have calculated form factors have also calculated the threshold process . this may be due in part to the fact that the @xmath265 interaction currents are larger than the @xmath266 interaction currents , making the threshold electrodisintegration calculation more difficult than the elastic calculation . a more definitive test of the various relativistic approaches discussed in the previous sections will be possible once the elastic calculations are extended to the threshold inelastic process . previous threshold electrodisintegration experiments have reported an _ average _ cross section that can be obtained theoretically by integrating the relative @xmath589 energy in the final state , @xmath590 , from 0 to 3 , or ( in some cases ) 0 to 10 mev . the unbound @xmath580 final state dominates at threshold ( because the @xmath591-@xmath592 scattering state is orthogonal to the deuteron state at threshold ) , but above threshold there are contributions from the @xmath591-@xmath592 scattering state , and eventually from the @xmath7 @xmath593 scattering states as well . to emphasize this magnetic transition , data have been taken at large electron scattering angles , mostly 155@xmath137 or 180@xmath137 . threshold electrodisintegration provides strong evidence ( perhaps the best we have ) for the existence of isovector exchange currents @xcite . the impulse approximation calculation of the transition to the @xmath580 final state has a zero arising from the negative interference between the @xmath594 and @xmath595 pieces of the transition that lead to a minimum at @xmath62 near 0.5 gev@xmath596 . this minimum is not seen in the data , and theoretical calculations of the @xmath265 exchange current contribution fill in the minimum and explain the data . experiments have measured @xmath9 threshold electrodisintegration to @xmath62 about 1.6 gev@xmath42 with better than 1 mev ( @xmath149 ) resolution , integrating up to @xmath590 @xmath127 3 mev @xcite , and to nearly 3 gev@xmath42 with 12 - 20 mev ( fwhm ) resolution , and integrating up to @xmath590 @xmath127 10 mev @xcite . where @xmath410 ( @xmath597 ) is the energy transfer ( scattered electron energy ) . the slac articles @xcite and first bates article @xcite use instead @xmath598 , where @xmath599 is the total @xmath519 kinetic energy in their c.m . frame , @xmath600 . ( a typographical error at one point in @xcite misidentifies @xmath599 as the energy of a nucleon , rather than the two nucleons . ) these cross sections are related by the jacobian @xmath601 which numerically ranges from about 1.3 to 2.3 for the data that we present . the articles showing the `` @xmath602 '' cross sections appear to plot the saclay cross sections as published , as `` @xmath603 '' cross sections , rather than converted . ] figure [ thresh ] shows the smooth rapid fall off of threshold electrodisintegration cross sections for @xmath62 up to about 1.2 gev@xmath42 , and the change in slope for the higher @xmath62 slac and bates data . there is good agreement between the various measurements , including those not shown , considering the change in scattering angle ( 155@xmath140 for saclay , 160@xmath140 for bates , 180@xmath140 for slac ) and integration region for @xmath590 ( 0 - 3 mev for saclay and bates data shown , 0 - 10 mev for slac ) . the figure also shows results from seven theoretical calculations . the oldest calculation shown in the figure is from mathiot @xcite . this model includes one and two body currents based on @xmath319 and @xmath223 exchange , and also contributions to the two - body current operator from the exitation of the @xmath604 . mathiot confirms that the one body current ( the impulse approximation ) produces a sharp minimum in the cross section at about @xmath605 0.5 gev@xmath42 ( in agreement with the recent schiavilla-1 body curve shown in the figure and in complete disagreement with the data ) and that the @xmath319 exchange current fills this in , shifting the minimum to about 1 gev@xmath42 . the @xmath223 exchange is also important , shifting the mimimum to @xmath407 1.4 gev@xmath42 . contributions from the electroexication of a @xmath604 are smaller , at least below @xmath606 gev@xmath42 . figure [ thresh ] shows that this calculation breaks down above @xmath607 gev@xmath42 , probably because it does not include many of the contributions included in modern calculations . more recent calculations , also based on hadronic degrees of freedom , are from leidemann , schmidtt , and arenhvel [ lsa ] @xcite and schiavilla . ( the calculations by schiavilla shown on the figure are based on the work of ref . @xcite , but use the more recent argonne av18 potential . these curves were also published in the review @xcite . ) in both of these cases the details are largely unpublished and the inelastic calculations are not as up - to - date as the corresponding elastic calculations recently done by the same groups , so it is premature to draw definite conclusions . the work of smejkal , truhlk , and gller [ stg ] @xcite obtains exchange current contributions from the @xmath608 system using a chiral lagrangian , and does a good job describing the data out to 2 gev@xmath42 . predictions from two quark cluster models , the early model of yamauchi , yamamoto , and wakamatsu [ yyw ] @xcite , and the more recent model of lu and cheng [ lc ] @xcite , are also shown in fig . [ thresh ] . these calculations both tend to have too much structure in the region of the shoulder , but do show that quark cluster models have the ability to describe the exchange currents needed to account for the data . the recent improvements in relativistic theory discussed in sec . [ nucsec ] will lead to a new generation of calculations that will rely on threshold electrodisintegration to provide details about the nature of the @xmath265 exchange currents . the most precise constraint on these currents comes from the @xmath609 transition , and this part of the transition is partly obscured by the poor energy resolution of the existing high @xmath62 measurements . a new and improved experiment at jlab with higher resolution would allow the threshold @xmath609 process to be better extracted , with a better resulting determination of the isovector exchange currents . it is also important to determine whether or not there is a minimum near 1.2 gev@xmath42 . this present indication of a minimum might be an artefact of the end of the saclay data @xcite vs. the start of the slac data @xcite , along with systematic uncertainties of these and the bates measurements @xcite . a high statistics high resolution measurement at large @xmath62 is feasible . measurements have been proposed @xcite with 1.5 mev resolution at a scattering angle of 160@xmath137 . the experiment would use hall a with the hrs spectrometer vacuum coupled to the scattering chamber , and a special cryotarget and collimators to enhance resolution and reduce backgrounds . measurements were proposed for 6 points from 1 to 3.7 gev@xmath42 . we note that slac ne4 simultaneously measured the threshold electrodisintegration along with the elastic structure function @xmath1 . because the large @xmath62 threshold inelastic cross section is typically an order of magnitude larger , the @xmath610 measurements see essentially only the inelastic processes , and @xmath611 is needed to determine the elastic scattering . if it is possible to maintain large solid angle for the elastic scattering , and good resolution for the threshold electrodisintegration , both data sets can be obtained simultaneously . deuteron photodisintegration was first investigated in the early 1930s , in order to understand the structure of the neutron . after the discovery of the neutron by james chadwick , attention turned to its mass and structure . was the neutron a fundamental particle , like the proton and electron , or was it a bound state of the electron and proton , different from the hydrogen atom ? if it was a bound state of the proton and electron , how were the electrons confined into the small nuclear volume ? conflicting experimental evidence on the neutron mass prevented resolution of the issue until 1934 , when chadwick and maurice goldhaber used deuteron photodisintegration @xcite to determine that the neutron mass was slightly heavier than that of the hydrogen atom . thus , the neutron , being heavier than the proton plus electron , was a fundamental particle , and there was no longer any basis for thinking electrons could be present in nuclei @xcite . subsequently , deuteron photodisintegration cross sections have served as a standard test case for nuclear theory . the effects , for example , of the @xmath604 resonance in cross sections for beam energies near 300 mev are pronounced , but until recently large discrepancies between different experimental data sets made precise tests of theories difficult @xcite . the 1960s-1970s saw the start of polarization measurements . the earliest data were intermediate energy measurements of the induced proton polarization @xcite and low energy measurements of the induced neutron polarization @xcite . large induced polarizations were observed @xcite soon afterward , particularly for energies above the @xmath604 resonance and for center of mass angles near 90@xmath140 . the combination of more extensive confirming measurements @xcite for @xmath612 about 350 - 700 mev , which could not be reproduced theoretically , and interest in dibaryons led to much excitement about deuteron photodisintegration in the late 1970s and early 1980s . there were many serious theoretical efforts , numerological studies involving inclusion of dibaryon resonances , and extensive experimental studies of cross sections and polarization observables . since _ both _ the recent deuteron form factor measurements _ and _ the recent high energy deuteron photodisintegration measurements have been made with 4 gev electron beams , it is sometimes assumed that the same theory should work for both . in this review we emphasize that this need not be the case . the kinematics of elastic electron - deuteron scattering and deuteron photodisintegration are very different , and the physics being explored by these two measurements is also very different . the implications of this remarkable feature of electronuclear physics , often not fully appreciated , will be discussed briefly in this section . @llll & @xmath613 & & @xmath614 + @xmath615 & @xmath616 ( 939 ) & @xmath617 & @xmath618 + @xmath619 & @xmath620 , @xmath621 , @xmath622 , @xmath162 & @xmath623 & @xmath624 , @xmath625 , @xmath626 + & @xmath627 , @xmath628 , @xmath629 , & & + & @xmath630 , @xmath631 , @xmath632 & & + @xmath633 & @xmath634 , @xmath635 , @xmath636 & @xmath637 & @xmath638 , @xmath639 , @xmath640 , + & & & @xmath641 , @xmath642 + @xmath643 & @xmath644 channels & @xmath645 & @xmath646 channels + @lclr band & mass range & members & number of + & & & channels + @xmath647 & 1878 & @xmath648 & 1 + @xmath649 & 2171 & @xmath650 & 1 + @xmath651 & 2464 2579 & @xmath652 , @xmath653 , @xmath654 & 13 + @xmath655 & @xmath162 2858 2872 @xmath162 & @xmath656 , @xmath657 , @xmath658 & 17 + @xmath659 & @xmath162 3155 3280 @xmath162 & @xmath660 , @xmath653 , @xmath661 , @xmath662 & 86 + @xmath663 & @xmath162 3452 3652 @xmath162 & @xmath664 , @xmath665 , @xmath666 , @xmath667 , & + & & @xmath668 , @xmath669 & 66 + @xmath670 & @xmath162 3832 3860 @xmath162 & @xmath671 , @xmath672 , @xmath673 , @xmath674 , & 52 + @xmath675 & @xmath162 4046 4440 @xmath162 & @xmath676 , @xmath677 , @xmath678 , @xmath679 , & + & & @xmath680 , @xmath681 , @xmath682 & 50 + the kinematics of elastic scattering and photodisintegration are compared in fig.[fig : dgpkin ] , which shows @xmath683 as a function of the photon ( real or virtual ) energy @xmath684 where @xmath685 and @xmath686 for real photons . the mass of the final excited state increases rapidly as @xmath353 decreases below its maximum allowed value of @xmath687 . for any energy @xmath74 or any @xmath62 , elastic @xmath3 scattering leaves the @xmath589 system bound , with no internal excitation energy added to the two nucleons . for quasifree scattering ( @xmath688 ) the mass of the final @xmath589 system grows with @xmath74 , and as @xmath353 decreases below 1 the mass grows more rapidly with @xmath74 . as @xmath353 @xmath136 0 , we approach the real photon limit . real photons produce the maximum value of @xmath308 of any given beam energy . with each 1 gev of beam energy @xmath689 increases by approximately 500 mev , driving the final state deep into the resonance region . the well established nucleon resonances , all of which can be excited by 4 gev photons , are listed in table [ tab : thresh ] , and the bands of thresholds at which these resonances are excited , either singly or in pairs , are listed in table [ tab : channels ] and shown in fig . [ fig : dgpkin ] . [ the fermi momentum of the struck nucleon , and the widths of the resonances , will average these thresholds over a wider kinematic region than shown . ] by @xmath612 = 1.2 gev , the final - state mass is already reaching @xmath308 = 2 gev , the nominal onset of deep inelastic scattering ( dis ) from a single nucleon if we assume one of the nucleons in the deuteron remains at rest . at @xmath612 = 4 gev , the final - state mass is approximately 4.5 gev , and at least 286 thresholds for the production of pairs of baryon resonances have been crossed ( and there are probably more from unseen or weakly established resonances ) . a photon energy of 4 gev corresponds to @xmath519 scattering with a laboratory kinetic energy of about 8 gev ! ( see also the kinematic argument given by holt @xcite . ) it is clearly very difficult ( if not impossible ) to construct a theory of high energy photoproduction in which all of the 24 established baryon resonances and their corresponding 286 production thresholds are treated microscopically . by contrast , elastic electron deuteron scattering requires a microscopic treatment of only _ one channel_. all of the 286 channels also contribute to elastic scattering , of course , but in this case they are not explicitly excited , and can probably be well described by slowly varying short - range terms included in a meson exchange ( or potential ) model . in photodisintegration , _ each of these channels is excited explicitly_. as we shall see below , it is sufficiently difficult to construct an adequate theory in the region of the @xmath604 resonance , so it is difficult to imagine this program being extended to a realistic treatment of many resonances , including intermediate states in which multiple mesons are present . it would instead appear that an alternate framework that _ averages over the effects of many hadronic states _ is needed . the alternatives are to use a glauber - like approach , or to borrow from our knowledge of dis and build models that rely on the underlying quark degrees of freedom . we will return to these issues in our review of the theory in sec . [ phototheory ] below . the photodisintegration cross section was previously given in sec . [ photodis1 ] . including the polarization observables defined by the density matrices of eq . ( [ density ] ) , and using the notation of ref . @xcite , the structure functions are @xmath690 where the coordinate system , shown in fig . [ dgpcoord ] ( a simplified and relabeled version of that given in fig . [ fig_cross ] ) is constructed from the incident photon direction @xmath691 and the outgoing proton direction @xmath692 . substituting the expansions ( [ polex ] ) into the cross section formula ( [ photonxsec ] ) , gives , in a notation suggested by ref . @xcite , @xmath693 \label{gdcross}\end{aligned}\ ] ] where @xmath694 is the differential cross section for unpolarized photons , and explicit expressions for the asymmetry parameters are given in table [ tab : polpara ] . note that the observables @xmath695 and @xmath10 are the real and imaginary parts of the same combination of amplitudes , so that an experimental measurement of both of these observables will fully determine this linear combination of amplitudes . for the induced polarization instead of @xmath10 . ] this is not true for any other pair of observables shown in the table . in the c.m . system @xmath696 where @xmath697 \ , .\end{aligned}\ ] ] the @xmath698s are defined in table [ tab : fs ] @lll amplitude & ref . @xcite@xmath699 & ref . @xcite + @xmath700 & @xmath701 & @xmath702 + @xmath703 & @xmath704 & @xmath705 + @xmath706 & @xmath707 & @xmath708 + @xmath709 & @xmath710 & @xmath711 + @xmath712 & @xmath713 & @xmath714 + @xmath715 & @xmath716 & @xmath717 + the _ single _ polarization observables are the induced proton polarization @xmath10 , the linearly polarized photon asymmetry @xmath718 , and the vector polarized ( along @xmath719 ) target asymmetry @xmath720 . the quantities @xmath10 and @xmath718 are defined by @xmath721 where @xmath722 refers to @xmath723 , and in the expression for @xmath718 the photon is polarized either in plane along the @xmath724 direction ( @xmath101=0 , denoted @xmath725 ) or out of plane along the @xmath719 direction ( @xmath726 , denoted @xmath727 ) . note that the opposite phase convention for @xmath718 ( @xmath728 - @xmath729 ) is often used , and is used in the figures below . double _ polarization transfer observables are @xmath730 and @xmath731 ( for linearly polarized photons ) , and @xmath732 ( for circularly polarized photons ) , with the subscripts on @xmath733 and @xmath734 giving the polarization direction in the final state . there have been many attempts to understand low energy deuteron photodisintegration using a conventional meson - baryon framework . since the first band of nucleon resonances is not excited until about 400 mev photon energy ( recall fig . [ fig : dgpkin ] ) it makes sense to describe the process below 400 mev using a model of coupled @xmath7 , @xmath735 and @xmath736 channels . laget @xcite showed that the prominant shoulder in the total cross section at @xmath737 = 300 mev can be largely explained by the mechanism in which a @xmath604 is photoproduced at a nucleon followed by the reabsorption of its decaying pion by the other nucleon . he also examined many other mechanisms , including rescattering up to second order , but did not do a full calculation of the final - state interaction . later leideman and arenhvel @xcite treated the @xmath7 , @xmath735 and @xmath738 as coupled channels and included final - state effects to all orders . tanabe and ohta @xcite followed with a more complete treatment of the final state which is consistent with three - body unitarity . in a number of conference talks , lee @xcite reported on coupled channel calculations using @xmath616 , @xmath604 and the @xmath620 ( roper ) resonances , which he extended to @xmath739 gev . his work suggests that final - state interactions significantly enhance the cross section for photon energies above 1 gev . the recent calculations by schwamb and arenhvel and collaborators @xcite include the @xmath7 , @xmath735 and @xmath740 channels , and also contributions from meson retardation , meson exchange currents , and the meson dressing of the nucleon lines required by unitarity . all parameters are fixed from nucleon - nucleon scattering and photoreactions such as @xmath604 excitation from the nucleon , so no new parameters are introduced into the calculations of the deuteron photodisintegration process itself . they obtain a reasonable description of @xmath7 scattering up to lab energies of 800 mev , particularly for the important @xmath741 partial wave , and emphasize that the consistent inclusion of retardation effects improves their results for photodisintegration . in another work it was shown that inclusion of the @xmath742 and @xmath743 resonances @xcite seems to have only a small effect below about 400 mev . these resonances enhance the total cross section by only 14% at 680 mev , although effects on double polarization observables can be more significant . hence there is some justification to limiting the theory of low energy photodisintegration to the channels considered in ref . @xcite . by comparison to the detailed and careful treatments developed for lower energies , calculations specifically designed to describe higher energy photodisintegration are less complete . an unpublished bonn calculation @xcite includes pole diagrams generated from @xmath319 , @xmath223 , @xmath320 , and @xmath410 exchange , plus the 17 well - established nucleon and @xmath604 resonances with mass less than 2 gev and @xmath482 @xmath744 5/2 ( listed in table [ tab : thresh ] ) . the calculation uses resonance parameters taken from the particle data group . nagornyi and collaborators @xcite have introduced a covariant model based on the sum of pole diagrams . the latest version @xcite gives the photodisintegration amplitude as a sum of contributions from only 4 feynman diagrams : three pole diagrams coming from the coupling of the photon to the three external legs of covariant @xmath745 vertex plus a contact interaction designed to maintain gauge invariance . this model divides the @xmath745 vertex into `` soft '' and hard parts , with the hard part designed to reproduce the pqcd counting rules and its strength determined by a fit to the data at 1 gev . the model has no final - state interactions or explicit nucleon resonance contributions . there is also a relativistic calculation of photodisintgration in born approximation using the bethe - salpeter formalism @xcite . it is found that the cross section is a factor of 2 to 10 times too small , and that relativistic effects are large . all of these models are rather crude , and taken together it is not clear what one should conclude from them . the calculations each appear to emphasize some aspects of what a comprehensive meson - baryon theory of photodisintegration should entail . perhaps we can say that conventional calculations that neglect final - state interactions seriously underestimate the cross section , but may be corrected in an approximate manner by introducing diagrams with poles in the @xmath289 channel . s - channel pole diagrams can be regarded as a crude approximation to the missing final - state interactions . more generally , using pole diagrams with form factors that have the correct behavior at high momentum transfer may also insure that meson - baryon theories of deuteron photodisintegration will also have the correct high energy behavior @xcite . perturbative qcd , discussed briefly in sec . [ pertqcd ] , provides explicit , testable predictions for the cross section and polarization observables . for the case of elastic scattering , the high energy ( virtual ) photon had to share its mommentum equally with all of the constitutents , leading to the typical diagram shown in fig . [ pqcddiagrams ] . photo ( or electro ) disintegration differs in that the momentum will not be distributed to _ all _ six quarks unless momentum is transferred to _ each _ nucleon , and this requires non - forward scattering , i.e. the angle @xmath746 between the three - momentum of the outgoing proton and the photon ( in the c.m . system ) must not be 0 or @xmath319 . more precisely , if @xmath737 is the energy of the photon in the lab system , the square of the momentum transferred to each nucleon @xmath747 ( where @xmath748 1,2 ) can be written @xmath749 where @xmath289 , the square of the c.m . energy , is @xmath750 and the limits in ( [ t1t2 ] ) are the result for @xmath751 . if @xmath752 the momentum transferred to each nucleon is balanced and the pqcd result is reached most rapidly . a `` typical '' pqcd diagram leading to large angle scattering is illustrated in fig . [ fig : dg - pqcd ] . in general we can not expect the cross section to follow pqcd unless the minimum of the two momentum transfers @xmath753 and @xmath754 is larger than some value @xmath755 at which pqcd holds . if the above conditions are satisfied , then the cross section and polarization observables should satisfy results predicted by pqcd : the constituent counting rules ( ccr ) and hadron helicity conservation ( hhc ) . the ccr @xcite predicts the energy dependence of scattering cross sections at fixed center of mass angle @xmath756 where @xmath287 is the total number of pointlike particles in the initial and final states of the reaction . for deuteron photodisintegration , @xmath757 ( there is one photon and 6 + 6=12 quarks ) , and because of eq . ( [ t1t2 ] ) @xmath758 may be replaced by @xmath289 , as is commonly done . for elastic @xmath3 scattering @xmath759 because there are electrons in both the initial and final state , and @xmath760 , the momentum transfered by the electron . in this case @xmath761 , and recalling eq . ( [ mott ] ) we have @xmath762 recovering the pqcd prediction @xmath459 . the second rule from pqcd is that the total helicity of the incoming and outgoing hadrons should be conserved @xcite @xmath763 where @xmath764 ( @xmath765 ) are the helicities of the initial ( final ) hadrons . in sec . [ pertqcd ] we discussed the implication of this rule for the deuteron form factors . hhc makes predictions for many spin observables , particularly vector polarizations @xcite . here it predicts that the amplitudes of table [ tab : fs ] will have the following behavior at large @xmath758 @xmath766 where we have used the fact that each helicity flip is suppressed by a power of @xmath758 . the implication of hhc for the observables given in table [ tab : polpara ] is summarized in table [ tab : dgppol ] . the limits at @xmath746 @xmath127 90@xmath140 on some of the observables require assumptions about relations between the helicity conserving variables @xcite . @lll observable & hhc limit & approach + & & to hhc + @xmath767 & @xmath768 & @xmath769 + @xmath10 & 0 & @xmath770 + @xmath720 & 0 & @xmath770 + @xmath718 & @xmath771 at 90@xmath140 & @xmath769 + @xmath730 & 0 & @xmath770 + @xmath11 & 0 & @xmath770 + @xmath12 & @xmath772 / ( d\sigma / d\omega ) \;\rightarrow 0 $ ] at 90@xmath140 & @xmath769 + @xmath773 & 0 & @xmath770 + @xmath774 & @xmath775 at 90@xmath140 & @xmath769 + in an attempt to include some of the soft physics omitted from pqcd , and to extend the region of applicability of pqcd down to lower momentum transfers , brodsky and hiller introduced the idea of reduced nuclear amplitudes ( rna ) @xcite . in this model the gluon exchanges that contribute to identifiable subprocesses ( such as nucleon form factors ) are collected together and their contributions replaced by experimentally determined nucleon form factors . it is hoped that the resulting expressions will correctly include much of the missing soft physics , and will therefore be valid to lower momentum transfers than the original pqcd expressions from which they were obtained . when applied to deuteron photodisintegration , the cross section is written @xmath776 where the phase space factor of @xmath777 comes for a careful reduction of the phase space factors in eqs . ( [ resptens ] ) , ( [ sigma0 ] ) , and ( [ t1t2 ] ) , instead of @xmath778 . ] @xmath779 is the reduced nuclear amplitude , @xmath780 and @xmath781 are the proton and neutron form factors with @xmath782 and @xmath783 the average momentum transferred to the proton and neutron @xmath784 and the square of the transverse momentum is @xmath785 the power of the `` extra '' factor of @xmath786 is fixed once the phase space and nucleon form factors have been taken into account . note that this model does not attempt to normalize @xmath787 . the rna form given in eq . ( [ rna ] ) is somewhat arbitrary , particularly in the specification of the form of @xmath788 .. this choice of @xmath789 leads to an increased energy dependence and worse agreement with the data . ] radyushkin @xcite has argued that the elementary process not accounted for by the nucleon form factors ( i.e. the absorption of a hard photon followed by exchange of a hard gluon with another quark ) should include nonperturbative contributions . the effect is to replace the @xmath790 factor in eq . ( [ rna ] ) by a smooth function @xmath791 , which is assumed to vary slowly in energy and angle . in the fits described below , @xmath792 will be taken to be a constant adjusted to fit the data , implying that @xmath793 instead of @xmath794 . alternatively , if the quark exchange mechanism shown in fig . [ fig : dg - pqcd ] is to be taken seriously , a more detailed calculation is possible . this is the motivation for the work of frankfurt , miller , sargsian , and strikman @xcite where the quark exchange diagram ( which the authors refer to as a quark rescattering diagram ) is calculated in front - form dynamics using model wave functions for the nucleons and the deuteron . the matrix element is written as a convolution of an elementary quark exchange interaction with the initial and final nucleon wave functions . the final nucleons are free and the distribution of the initial nucleons is given by the deuteron wave function . since the photon momentum is shared by the proton and neutron , there is little sensitivity to the high momentum part of the deuteron wave function . the elementary interaction is a quark exchange between the two nucleons , with the photon absorbed by one quark which then gives up its momentum through a hard gluon exchange with another quark . the authors show that this can be replaced approximately by the wide angle @xmath519 scattering cross section ( also dominated by quark exchange ) . the final formula obtained from these arguments is @xmath795 where @xmath796 was used in the fits to the data discussed below . the authors propose using experimental data for the @xmath519 cross section , but since data does not exist for the actual kinematic conditions needed , it must be extrapolated , and predictions for photodisintegration are given as a band corresponding to the uncertainties introduced by the extrapolations . the authors believe that their predictions should be valid for @xmath612 @xmath152 2.5 gev , and nucleon momentum transfers @xmath797 and @xmath798 @xmath152 2 gev@xmath42 . the high energy approaches described above all focus on the region where both @xmath782 and @xmath782 are large ( where perturbative arguments can serve as the foundation for the treatment ) . alternatively , we may ask what to expect when one of these momentum transfers is small ( but @xmath289 is still large ) . the authors in refs . @xcite develop a model ( which they refer to as the `` quark - gluon string model '' ) based on a reggie generalization of the nucleon exchange born term . here the exchanged nucleon is replaced by a nucleon reggie trajectory that represents the sum of a tower of exchanged nucleon resonances ( or the exchange of three quarks dressed by an arbitrary number of gluons ) . the energy dependence of the predicted cross section is @xmath799 where @xmath800 is the nucleon reggie trajectory , with @xmath801 , where @xmath758 is in units of gev@xmath42 . recent work @xcite emphasizes the importance of the nonlinear term in the regge trajectory . the model is intended to work for @xmath802 gev , with @xmath803 less than about 1 gev@xmath42 . we now turn to a review of the experimental data and to a comparison between theory and experiment . the world data set for cross sections ( except for the most recent experiments ) has been presented several times @xcite , and we will not review the normalization problems of older cross section data sets . table [ tab : dgppoldata ] presents an extensive list of the published polarization data . about 70 publications , starting in 1960 , present about 1200 data points for photodisintegration and the time reversed radiative capture reaction . table [ tab : dgppoldata ] generally lists photon lab energy and proton c.m . angle ( neutron c.m . angle for the @xmath804 data ) . all of the radiative capture experiments have measured @xmath805 , and for these experiments we usually give the neutron beam kinetic energy and outging photon c.m . angle ; for the iucf experiment we give their reported proton c.m . angle . matching c.m . energies leads to @xmath806 or @xmath807 at high energies . comparison of these data indicates serious problems with backgrounds and/or estimates of systematic uncertainties in a number of cases , as will become clear in figures in the sections below . we review some of the lower energy data in the next subsection . high energy experiments , with photon energy above @xmath1511 gev , are covered in the following subsection . in this section we review selected experiments with beam energies from about pion production threshold to several hundred mev . tagged photon facilities , with their improved knowledge of incident beam flux , have allowed significantly improved cross section measurements in this region since the 1980s . of particular note are extensive recent data sets from legs and mainz . figures [ fig : dgpobs300 ] and [ fig : dgpobs410 ] show angular distributions for the cross section , and @xmath718 , @xmath720 , and @xmath10 at photon energies near 300 and 450 mev , respectively . the legs tagged , backscattered , and linearly - polarized photon beam was used to determined cross sections and @xmath718 @xcite . five independent measurements used three detector systems , two targets , and two different laser frequencies . data were taken for @xmath612 = 100 - 315 mev with eight laboratory angles from 15 - 155@xmath140 . cross section statistical uncertainties range from a few percent to about 15% , with systematic uncertainties of 5% . one observation in the legs data is of pion contamination that may have been missed in earlier experiments , leading to increased cross sections . the mami mainz experiment @xcite used the glasgow photon tagger along with the large solid angle detector daphne to determine cross sections in the ranges @xmath612 = 100 - 800 mev and @xmath746 = 30 - 160@xmath140 . data were binned by 20 mev in energy and 10@xmath140 in angle . statistical uncertainties ranged from a few percent at lower energies to about 25% at the highest energies . systematic errors were also energy dependent , ranging from a few percent to several percent . the @xmath718 asymmetry @xcite was obtained by using a coherent bremsstrahlung radiator . agreement between the mainz and legs cross section results is generally better than 10% . the @xmath718 asymmetries also agree well with each other and with earlier results from yerevan @xcite . measurements from kharkov @xcite generally agree , except for a tendency to be slightly smaller at many beam energies . the vector polarized target asymmetry @xmath720 was measured at tokyo @xcite and at bonn @xcite . the two measurements generally agree , with the data appearing to follow , very roughly , a @xmath808 dependence at each energy , as do the calculations of @xcite . the induced polarization @xmath10 has been measured at a number of laboraties , with significant amounts of lower energy data from stanford @xcite , tokyo @xcite , and kharkov . there were several experiments at kharkov , including initial measurements @xcite , high - energy measurements @xcite , simultaneous measurements of @xmath718 , @xmath730 , and @xmath10 @xcite , and lower - energy measurements @xcite . the kharkov data do not have a desirable level of consistency . the simultaneous measurements of @xmath718 , @xmath730 , and @xmath10 were taken as single - arm data , away from the photon endpoint , and may suffer from backgrounds . polarizations below about 300 mev , including the intermediate energy neutron measurements @xcite , appear to be well reproduced by theories . there are numerous data , particularly at @xmath746 @xmath127 45@xmath140 , 78@xmath140 , 90@xmath140 , and 120@xmath140 , but there are few energies at which there are good angular distributions . the conclusion that polarizations are large , close to @xmath809 , and peak at about 500 mev near 90@xmath140 is beyond dispute . figures [ fig : dgpobs300 ] and [ fig : dgpobs410 ] show the good agreement of the recent mainz @xcite and older bonn @xcite calculations with the cross section and @xmath718 asymmetry . theory seems to agree with @xmath810 better than @xmath811 @xcite . the agreement is better at the lower energy , and the newer mainz calculation is generally in better agreement . however , there is difficulty , particularly at the higher energy , with @xmath720 and @xmath10 , both imaginary parts of the interference of amplitudes . the large induced polarizations above the @xmath604 resonance have remained a puzzle for almost 30 years , and are still not fully explained by the newest theories . the bonn calculation was in sufficiently good qualitative agreement with all observables ( @xmath812 , @xmath718 , @xmath720 , _ and _ @xmath10 ) for energies up several hundred mev for the authors to consider this @xmath10 puzzle solved . however , detailed examination of fig . [ fig : dgpobs410 ] shows that neither the shape nor strength of the angular distribution is accurately reproduced ; this will become clearer when we examine the energy dependence of @xmath10 at @xmath746 = 90@xmath140 below . figure [ fig : dgpsigmas ] shows the published high energy photodisintegration data , from experiments ne8 @xcite and ne17 @xcite at slac , and e89 - 012 @xcite and e96 - 003 @xcite at jlab . these experiments determine cross sections for @xmath746 @xmath151 36@xmath140 , 52@xmath140 , 69@xmath140,and 89@xmath140 at energies from about 0.7 to 5.5 gev ; there are also some backward angle data up to 1.8 gev from ne8 . these data overlap well ; the experiments , while all run by essentially the same collaboration , used three spectrometers in two experimental halls at two laboratories . there is also good overlap , variations of less than about 20% , with the highest energy mainz tagged photon data @xcite , and with older untagged data @xcite . tagged photon measurements at low energies provide an accurate measure of beam flux , and along with the measured proton angle and energy , can determine a missing mass that allows background rejection . at high energies , smaller cross sections can not be determined with the reduced flux of tagged photons . the bremstrahlung endpoint technique was used for all of the slac and jlab measurements shown in fig . [ fig : dgpsigmas ] . in the endpoint technique , the measured proton momentum vector determines the incident photon energy and neutron kinematics , _ assuming _ the reaction is two body photodisintegration . low momentum protons are cut from the analysis to prevent contamination from final states such as @xmath813 , while high momentum protons are cut to eliminate the larger uncertainty in the photon flux close to the photon endpoint . backgrounds are determined by radiator out and empty target measurements , and subtracted . events in the region beyond the endpoint can be used to check the subtraction . the increase in time required for this subtraction makes it prohibitive for the highest energy measurements ; for these the electrodisintegration background is calculated @xcite and subtracted . to determine the cross section , the incident photon flux is calculated using the method of ref . thick radiator corrections are typically about 15% for a radiator thickness of 6% of a radiation length . the main feature of the cross section data above about 1 gev is the @xmath794 ( @xmath814 ) fall off of the cross sections @xmath815 ( @xmath812 ) at @xmath746 = 90@xmath140 and 69@xmath140 , in agreement with the ccr and thus with perturbative qcd expectations . in contrast , the cross sections at the forward angles 36@xmath140 and 52@xmath140 fall off more slowly , with approximate @xmath816 scaling at lower energies until the onset of the @xmath794 behavior at about 4 and 3 gev beam energy , respectively . at each angle , the onset of the @xmath794 behavior corresponds to @xmath789 @xmath151 1 gev . the rna and the radyushkin estimates in fig . [ fig : dgpsigmas ] were normalized to the datum at 89@xmath137 and @xmath817 gev , fixing their one free parameter . the rna is then almost a factor of 2 too large at 36@xmath137 , and also much too large at lower energies , requiring that the soft physics missing from the rna interfere _ destructively _ with the leading terms . suggested angular dependences of @xmath818 ) @xcite would increase the rna curve further at the forward angles , worsening agreeement with the data . in contrast , the radyushkin estimate gives a somewhat better account of both the angular and energy dependence ( even though it only goes asympotically as @xmath814 ) , confirming that phase space and nucleon form factors are all that is needed to account for much of the kinematic variation of the cross section . while the apparent onset of scaling at the forward angles suggests this agreement is starting to break down , we conclude the present data are insufficient to uniquely fix the asymptotic energy dependence of the cross section . the cross section data are also reasonably well reproduced by the model of frankfurt , miller , strikman and sargsian [ fmss ] @xcite and the quark gluon string ( qgs ) model @xcite . the predictions of fmss are uncertain because ( a ) the high energy @xmath7 scattering data has an uncertain energy dependence reflecting the experimental errors , and ( b ) the extrapolation of the @xmath7 data required for the predictions introduces further errors . these two uncertainties combine to give the jagged region shown in fig . [ fig : dgpsigmas ] . the qgs model describes the forward angle data up to 4 gev reasonably well , even for values of @xmath803 exceeding the nominal limits of the model . the newer work @xcite predicts that the angular distributions will become increasingly symmetric at higher energies ; older estimates @xcite had predicted an increasing _ two experiments currently have unpublished data for cross sections at photon energies up to 2.5 gev . jlab hall b e93 - 017 @xcite used the clas with tagged photons to determine nearly complete angular distributions . the preliminary data agree well with earlier measurements , but are much more comprehensive than previous measurements in this energy range . hall a e99 - 008 @xcite has taken angular distributions at eight angles with @xmath746 @xmath127 30@xmath140 143@xmath140 , at energies of 1.67 , 1.95 , and 2.50 gev . [ fig : ad ] shows a sample angular distribution at approximately 1.6 gev . there are only three sets of polarization data for deuteron photodisintegration at energies near and above 1 gev . the induced polarization @xmath10 was measured at kharkov @xcite , the @xmath718 asymmetry was measured at yerevan @xcite , and @xmath10 and the polarization transfers @xmath11 and @xmath12 were measured at jlab @xcite . data for the energy dependence of these four observables at a fixed @xmath752 are compared with theory in figs . [ fig:89019py ] , [ fig:89019cxcz ] , and [ fig : sigfig ] . the kharkov measurements of @xmath10 at @xmath746 @xmath127 90@xmath140 and 120@xmath140 extend up to about 1.1 gev ( see fig . [ fig:89019py ] ) . these experiments were very difficult . the small duty factor at kharkov increases instantaneous background rates a factor of about 20,000 over those at jlab . these large backgrounds made it necessary to use multiple spark chambers to track particle trajectories . it was also difficult to calibrate the polarimeter . calibrations of a polarimeter are best done by measuring its analyzing power using the known @xmath819 elastic scattering reaction @xcite , but kharkov had no polarized beam . the kharkov measurements relied on a single elastic @xmath158 point to check false asymmetries in their polarimeter , and used analyzing powers from the literature . finally , the polarimeter had a rear trigger scintillator ; any inefficiencies in the scintillator would lead to false asymmetries . in contrast , the recent jlab experiment @xcite had little background , used @xmath820 calibrations to determine false asymmetries and analyzing powers at each kinematic setting , and had no rear trigger scintillator . given the clear disagreement of the kharkov data with the recent jlab data , and noting that one of us ( rg ) is a spokesperson of the jlab experiment , we conclude that the highest energy set of karkov data should not be trusted . the induced polarizations shown in fig . [ fig:89019py ] confirm our comments above concerning the bonn calculation . the large negative polarization near 500 mev is not reproduced , and the calculation is qualitatively incorrect at higher energies . the imaginary part of the amplitude appears to be a problem in these meson - baryon calculations ; presumably this arises from an inadequate treatment of resonances . taken together , these recoil polarization data only weakly confirm the predictions of hhc , which predicts that @xmath10 and @xmath11 should approach zero as @xmath821 , and that ( with additional assumptions @xcite about relations between the helicity conserving amplitudes at @xmath746 = 90@xmath140 ) @xmath822 and @xmath823 as @xmath821 . the highest energy polarization measurements of @xmath10 show that it is consistent with vanishing at energies above about 1 gev , the same energy at which the @xmath794 cross section scaling begins . [ in the radyushkin model @xmath10 should be zero because the amplitudes are all real if there is no gluon exchange @xcite . ] similiarly , the polarization transfer observables @xmath11 and @xmath12 both appear to peak near 1 gev , and decrease at higher energies . however , @xmath11 does not appear to vanish sufficiently rapidly ; the data might be inconsistent with hhc . so @xmath10 and @xmath12 ( and perhaps @xmath11 ) seem to have close to the correct behavior , but @xmath718 does not . the highest energy @xmath718 asymmetry measurements from yerevan , with data up to about 1.6 gev , give the immediate impression that there is a minimum near 1.2 gev and that above this energy the asymmetry is tending to increase towards 1 , although the data are also statistically consistent with a constant value of about 0.3 . in either case the trend is clearly not consistent with @xmath824 . in models based on meson - baryon degrees of freedom , the data indicate that the combined effects of resonances plus final - state interactions are small . calculations of @xmath10 in meson baryon theories @xcite indicate that @xmath10 at higher energies arises largely from resonance - background interference , with a small contribution from final - state interactions . calculations generally indicate that the @xmath604 resonance generates a large polarization , though only perhaps about 50% of the magnitude seen in the experimental data at @xmath746 @xmath127 90@xmath140 . the roper and @xmath742 have small effects , while the @xmath743 has a large effect @xcite . the @xmath825 , included only by the bonn group @xcite , also generates a large polarization . ( of the 17 resonances included in the bonn calculation @xcite , only those mentioned above had large effects on @xmath10 . ) as discussed above , it is hard to imagine that a theoretically acceptable high energy model based on hadronic degrees of freedom will be constructed in the near future . still , a modern relativistic calculation based on hadronic degrees of freedom would nonetheless be desirable . a calculation of @xmath10 has been done in the model of fmss @xcite . since the helicity amplitudes in the nucleon - nucleon scattering for this center of mass energy range are not uniquely determined , some modelling was needed . the calculation showns that @xmath10 is generally very small , going from a negative value at low energies to positive values at several gev beam energy , and is consistent with the trend of the experimental results . the polarization observable @xmath695 is also expected to be small , and opposite in sign to @xmath10 , while @xmath826 is expected to vanish . the rapid falloff of photodisintegration cross sections with energy makes extension of the measurements difficult . as beam energy increases from 4 to 5 , 6 , and 7 gev , the @xmath794 dependence reduces cross sections by factors of 6 , 30 , and 115 , respectively . only a few experiments are possible without the proposed 12 gev upgrade to jlab . as indicated in table [ dexptab ] , there are two approved experiments to continue the hall a recoil polarization measurements to additional angles , and to beam energies near 3 gev . measurements of the @xmath718 asymmetry are possible in jlab hall b , over a range of angles and energies well over 1 gev . in the longer term , the jlab 12 gev upgrade offers additional possibilities . the luminosity increase planned for hall b would allow precise polarization measurements to continue above 2 gev . the proposed mad spectrometer for hall a @xcite would give about a factor of 5 improvement in solid angle , and if it has the low backgrounds characteristic of the current hrs spectrometers , cross section measurements are possible up to 7 gev and polarization measurements are possible up to 4 gev . review of deuteron photodisintegration suggests the following : * a microscopic meson - baryon theory of deuteron photodisintegration must describe the @xmath7 interaction at high energies , including pion production and the contributions of hundreds of @xmath827 channels . it is unlikely that such a theory will be constructed in the foreseeable future . the data might indicate that the effect of many resonances is to increase the cross section and decrease the polarization observables ( by averaging over may phases ) . this suggests that it might be possible to construct an effective theory based on hadronic degrees of freedom . * for @xmath788 greater than about 1 gev@xmath42 , cross sections appear to follow the constituent counting rules , but it is expected that an absolute pqcd calculation of the size of the cross section would give a result much too small . similar observations may be made for other photoreactions , and it remains to be seen how this behavior arises , and if there is a general explanation for it . * the energy dependence of the photodisintegration cross sections has been shown to be potentially misleading indicator of the success of pqcd . models with asymptotic behavior which differ from pqcd fit the data as well or better than pqcd . further theoretical development and experimental tests of nonperturbative quark models would be desirable . the new high energy measurements of the deuteron form factors and the deuteron photodisintegration observables have motivated much theoretical work . in particular : * conventional meson theory works well in cases where all of the _ active _ hadronic channels that can contribute to a process are included in the calculations . this has been done for the deuteron form factors ( where only the @xmath7 channel is active ) , but not for high energy deuteron photodisintegration where 100 s of @xmath8 channels are active . at high energy any successful meson theory must include relativistic effects . * new approaches , probably using quark degrees of freedom , are needed for high energy deuteron photodisintegration . while photodisintegration ( as well as other reactions ) seem to follow the scaling laws , theoretical estimates using pqcd give cross sections orders of magnitude too small . the data do not support hadronic helicity conservation . thus scaling is no longer seen as sufficient evidence for the applicability of pqcd . there is good evidence that , in this energy region , the deuteron is undergoing a transition from a region in which conventional hadronic degrees of freedom describe the physics to a region in which quark degrees of freedom are more appropriate . we thank h. arenhvel , b. l. g. bakker , j. carbonell , w. h. klink , d. phillips , r. schiavilla , and g. salm and for supplying numerical values of their deuteron form factor calculations , i. sick for values of the coulomb distortion corrections , and m. sargsian and m. schwamb for supplying numerical values for their photodisintegration calculations . it is also a pleasure to thank a. afanasev , c. carlson , m. garon , t .- s . . lee , k. mccormick , g g petratos , j. w. van orden , a. radyushkin , and g. warren for several helpful discussions , and s. strauch for assisting with photodisintegration figures . this work was supported in part by the us department of energy . the southeastern universities research association ( sura ) operates the thomas jefferson national accelerator facility under doe contract de - ac05 - 84er40150 . rg acknowledges the support of the national science foundation from grant phy-00 - 98642 to rutgers university . 999 buck w w and gross f friar j l lancombe m , loiseau b , richard j m , vinh mau r , ct j , pirs p and de tourreil r wiringa r b , stoks v g j and schiavilla r stoks v g j , klomp r a m , terheggen c p f and de swart j j machleidt r , sammarruca f and song y machleidt r salpeter e e and bethe h a gross f ; ; logunov a a and tavkhelidze a n ; blankenbecler r and sugar r phillips d r , wallace s j , and devine n k gross f , van orden j w and holinde k stadler a and gross f gross f 1998 _ proceedings of the workshop on electronuclear physics with internal targets and the blast detector _ , may 2830 , mit experimental limits on possible two photon exchange are discussed in rekalo m p , tomasi - gustafsson e and prout d gross f arnold r e , carlson c e and gross f donnelly t w and raskin a s jones h jacob m and wick g c abbott d mcintyre j a and hofstadter r buchanan c d and yearian r benaksas d galster s ganichot d grosset@xmath828te b , drickey d , and lehmann p cramer r schulze m e dmitriev v f wojtsekhowski b b [ ] gilman r boden b the i garon m popov s g 1995 _ proceedings of the 8th international symposium on polarization phenomena in nuclear physics , bloomington , indiana , 1994 _ , aip conf . 339 ( aip , new york ) ferro - luzzi m bouwhuis m nikolenko d m prepost r , simonds r m and wiik b h gross f 1991 _ modern topics in electron scattering _ ( frois b and sick i ed ) p 219 jackson a j , lande a and riska d o thompson r h and heller h nyman e m and riska d o 1986 _ proceedings of weak and electromagnetic interactions in nuclei , heidelberg 1986 _ p 63 weinberg s 1995 _ the quantum theory of fields i _ ( cambridge university press ) simonov yu a and tjon j a ; nieuwenhuis t and tjon j a ; nieuwenhuis t 1995 ph.d . thesis university of utrecht ; savkli c , tjon j a and gross f keister b d and polyzou w n gross f and riska d o tjon j a 1995 _ xivth international conference on few body problem in physics , williamsburg , virginia , 1994 _ aip conf . proc . 334 ( aip , new york ) p 177 fleischer j and tjon j a ; ; hummel e and tjon j a ; ; mandelzweig v b and wallace s j ; wallace s j and mandelzweig v b wallace s 2001 talk at the _ int workshop on theories of nuclear forces and few - nucleon systems _ van orden j w , devine n and gross f dirac p a m schiavilla r and riska d forest j and schiavilla r 2001 ( private communication to be published ) schiavilla r , pandharipande v r and riska d o ; + carlson j , riska r o , schiavilla r and wiringa r b arenhvel h , ritz f and wilbois t ritz f , gller h , wilbois t and arenhvel h krajcik r a and foldy l l friar j l friar j l gross f 1978 proceedings of the _ 8th international conference on few body problems _ ( springer - verlag lec . notes in physics # 82 , zingl h ed ) , p. 46 . [ the factor @xmath829 in eq . ( 12 ) should read @xmath830 . ] carbonell j and karmanov v a karmanov v a and smirnov a v ; carbonell j , desplanques b , karmanov v a and mathiot j -f machleidt r , holinde k and elster ch lev f m , pace e and salm g grach i l and kondratyuk l a ; chung p l , coester f , keister b d and polyzou w n lev f m , pace e and salm g lev f m , pace e and salm g allen t w , klink w h and polyzou w n maltman k and isgur n maltman k and isgur n de forest t and mulders p j dijk h and bakker b l g buchmann a , yamauchi y and faessler a simonov yu a arndt r a brodsky s j and farrar g r ; lepage g p and brodsky s j ; brodsky s j and lepage g p isgur n and llewellyn - smith c radyushkin a v farrar g r , huleihel k and zhang h carlson c e and gross f brodsky s j and hiller j r kobushkin a and syamtomov a cao jun and wu hui - fang carlson c e ( private communication ) petratos g g , _ deuteron and helium form factor measurments at jefferson lab _ , contribution to the jefferson lab 12 gev upgrade ; ( private communication ) garon m gilman r 1988 _ jefferson lab workshop on physics & instrumentation with 6 - 12 gev beams _ p 275 kox s ( unpublished ) tomasi - gustafsson e ladygin v p tomasi - gustafsson e arnold r e , carlson c e and gross f zhou z mit bates blast experiment 00 - 03 hoekert j , riska d o , gari m and huffman a simon g g bernheim m auffret s lee k s schmitt w m arnold r g frodyma m mathiot j f leidemann w , schmitt k - m and arenhvel h carlson j and schiavilla r smejkal j , truhlk e and gller h yamauchi y , yamamoto r and wakamatsu m lu l c and cheng t s jourdan j , warren g , jefferson lab proposal 00 - 103 laget j m leidemann w and arenhvel h tanabe h and ohta k lee t - s h ; lee t - s h 1991 _ argonne national laboratory preprint _ phy-6886-th-91 ; lee t - s h 1991 _ argonne national laboratory preprint _ phy-6843-th-91 schwamb m and arenhvel h 2001 _ preprint _ nucl - th/0105033 ; schwamb m and arenhvel h ; schwamb m and arenhvel h ; schwamb m , arenhvel h , wilhelm p , and wilbois th schwamb m , arenhvel h , wilhelm p kang y , erbs p , pfeil w , and rollnik h 1990 _ abstracts of the particle and nuclear intersections conference _ , ( mit , cambridge , ma ) ; kang y 1993 ph.d . thesis bonn nagornyi s i , kasatkin yu a , and kirichenko i k [ ] dieperink a e l and nagornyi s i kazakov k yu and shulga d v _ preprint _ nucl - th/0101059 . kazakov k yu and shimovsky s eh carlson carl e matveev v carlson carl e and chachkhunashvili m afanasev a ( private communication ) ; sivers d ( private communication ) brodsky s j and hiller j r ; radyushkin a ( private communication ) frankfurt l l , miller g a , sargsian , m m and strikman m i frankfurt l l , miller g a , sargsian m m , and strikman m i kondratyuk l a grishina v yu white c tiator l and wright l e ; wright l e and tiator l matthews j l and owens r o napolitano j freedman s j belz j e myers h ching r and schaerf c baba k rossi p , de sanctis e , jlab proposal e93 - 017 gilman r , holt r , meziani z e jefferson lab proposal e99 - 008 ; schulte e to be published
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recent measurements of the deuteron electromagnetic structure functions @xmath0 , @xmath1 , and
@xmath2 extracted from high energy elastic @xmath3 scattering , and the cross sections and asymmetries extracted from high energy photodisintegration @xmath4 , are reviewed and compared to theory .
the theoretical calculations range from nonrelativistic and relativistic models using the traditional meson and baryon degrees of freedom , to effective field theories , to models based on the underlying quark and gluon degrees of freedom of qcd , including nonperturbative quark cluster models and perturbative qcd .
we review what has been learned from these experiments , and discuss why elastic @xmath3 scattering and photodisintegration seem to require very different theoretical approaches , even though they are closely related experimentally .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
cassels @xcite was challenged to determine when the sum of three consecutive cubes equals a square . he @xcite reduced the problem to finding integral points on the elliptic curve @xmath0 . using the arithmetic of certain quartic number fields , he obtained that the integral points on the above elliptic curve were @xmath1 , @xmath2 , @xmath3 , and @xmath4 . using the classical work of ljunggren @xcite and its generalizations ( see @xcite , @xcite , @xcite , and @xcite ) , luca and walsh @xcite considered the problem of finding the number of positive integer solutions to the diophantine equation @xmath5 , where @xmath6 is a positive integer . they proved that the number of positive integer solutions to @xmath5 is at most @xmath7 , where @xmath8 is the number of distinct prime factors of @xmath9 . in @xcite , chen considered the case where @xmath9 is a prime number greater than @xmath10 . he proved , in particular , that the diophantine equation @xmath5 has at most two positive integer solutions . recently , togb @xcite considered the more general diophantine equation @xmath11 where @xmath12 is a prime number and @xmath13 is an odd integer greater than @xmath14 . he proved the following theorem . [ to1 ] for any prime @xmath12 and any odd positive integer @xmath15 , the diophantine equation has at most seven positive integer solutions @xmath16 . using results obtained through magma , he then made the following conjecture on sharp bounds for the number of solutions to equation . [ to2 ] let @xmath12 be a prime and @xmath15 any odd positive integer . 1 . if @xmath17 , @xmath18 , @xmath19 , @xmath20 , @xmath21 , @xmath22 , @xmath23 , @xmath24 , @xmath25 , @xmath26 , or @xmath27 , then diophantine equation has at most one positive integer solution @xmath16 . 2 . if @xmath28 or @xmath29 , then diophantine equation has at most two positive integer solutions @xmath16 . 3 . if @xmath30 or @xmath31 , then diophantine equation has at most three positive integer solutions @xmath16 . the aim of this paper is to improve the bound on the number of solutions to the diophantine equation provided in theorem [ to1 ] , and to prove conjecture [ to2 ] in some cases . the main result of this paper is the following theorem . [ th1 ] let @xmath12 be a prime and let @xmath15 be an odd integer . 1 . if @xmath32 , then diophantine equation has at most one positive integer solution @xmath16 . 2 . suppose that @xmath33 or @xmath34 , where @xmath12 is odd . 1 . if @xmath35 or @xmath36 , then diophantine equation has at most three positive integer solutions @xmath16 . diophantine equation has at most one positive integer solution @xmath16 otherwise . 3 . suppose that @xmath37 , where @xmath12 is odd . 1 . if @xmath38 , @xmath19 , @xmath21 , @xmath24 , @xmath26 , or @xmath27 , then diophantine equation has at most one positive integer solution @xmath16 . 2 . if @xmath17 , @xmath20 , @xmath22 , @xmath23 , @xmath39 , or @xmath40 , then diophantine equation has at most two positive integer solutions @xmath16 . 3 . if @xmath28 or @xmath41 , then diophantine equation has at most three positive integer solutions @xmath16 . 4 . if @xmath42 , then diophantine equation has at most four positive integer solutions @xmath16 . 5 . if @xmath43 , then diophantine equation has at most six positive integer solutions @xmath16 . we will also prove the following result . [ th5 ] let @xmath12 be a prime and let @xmath15 be an even integer . 1 . if @xmath32 , then diophantine equation has at most two positive integer solutions @xmath16 . moreover , if @xmath44 and @xmath45 , then diophantine equation has at most one positive integer solution @xmath16 . suppose that @xmath46 or @xmath47 , where @xmath12 is odd . 1 . if @xmath44 , then diophantine equation has at most one positive integer solution @xmath16 . 2 . if @xmath48 , then diophantine equation has at most two positive integer solutions @xmath16 . 3 . suppose that @xmath49 , where @xmath12 is odd . 1 . if @xmath50 , then diophantine equation has at most one positive integer solution @xmath16 . 2 . if @xmath51 , then diophantine equation has at most two positive integer solutions @xmath16 . 3 . if @xmath52 , then diophantine equation has at most three positive integer solutions @xmath16 . if @xmath53 , then diophantine equation has at most four positive integer solutions @xmath16 . we present the results required to prove theorem [ th1 ] and theorem [ th5 ] . recall that if @xmath54 is a prime number , @xmath55 denotes the @xmath54-adic valuation of @xmath56 . let @xmath57 and @xmath58 be odd positive integers for which the equation @xmath59 has a solution in positive integers @xmath60 . let @xmath61 be the minimal positive solution to this equation and define @xmath62 for an odd integer @xmath63 , define @xmath64 and @xmath65 by @xmath66 luca and walsh proved the following result in @xcite regarding the solutions to the equation @xmath67 [ th2 ] 1 . if @xmath68 is not a square , then equation has no solution . if @xmath68 is a square and @xmath69 is not a square , then @xmath70 is the only solution to equation . if @xmath68 and @xmath69 are both squares , then @xmath70 and @xmath71 are the only solutions to equation . ljunggren proved the following result in @xcite . [ th3 ] let @xmath72 and @xmath58 be two positive integers . the equation @xmath73 has at most one solution in positive integers @xmath60 . let @xmath74 be a positive non - square integer , and let @xmath75 denote the minimal unit greater than 1 , of norm 1 , in @xmath76 $ ] . define @xmath77 for @xmath78 . togb , voutier , and walsh proved the following result in @xcite . [ th4 ] let @xmath74 be a positive non - square integer . there are at most two positive integer solutions @xmath60 to the equation @xmath79 . 1 . if two solutions such that @xmath80 exist , then @xmath81 and @xmath82 , except only if @xmath83 or @xmath84 , in which case @xmath81 and @xmath85 . 2 . if only one positive integer solution @xmath60 to the equation @xmath79 exists , then @xmath86 where @xmath87 for some square - free integer @xmath88 , and either @xmath89 , @xmath90 , or @xmath91 for some prime @xmath92 . we will make the theorem [ th1 ] more precise when @xmath74 is even . [ lem7 ] let @xmath74 be a positive non - square integer . suppose that @xmath93 where @xmath94 is a positive integer different from @xmath95 . then the equation @xmath79 has at most one positive solution @xmath60 . suppose that there exist two solutions to the equation @xmath79 . then there exist positive integer solutions @xmath96 and @xmath97 such that @xmath80 . it follows from theorem [ th4 ] that @xmath81 , @xmath82 , and @xmath98 , so @xmath99 . then @xmath100 since @xmath75 is a unit of norm @xmath14 in @xmath76 $ ] and @xmath93 , we obtain @xmath101 , so that @xmath102 is odd . then @xmath103 , which is a contradiction with . _ proof of theorem [ th1 ] . _ let @xmath32 , and let @xmath13 be an odd positive integer . let @xmath104 be positive integers such that @xmath105 . it is not difficult to see that @xmath106 divides @xmath107 and @xmath108 . let @xmath109 and @xmath110 . then we obtain @xmath111 since @xmath112 , there exist positive integers @xmath113 and @xmath114 such that @xmath115 , @xmath116 , and @xmath117 by lemma [ lem7 ] , this equation has at most one positive integer solution @xmath118 . let @xmath12 be an odd prime , and let @xmath13 be an odd positive integer . let @xmath104 be positive integers such that @xmath119 . we remark that @xmath120 or @xmath121 , so we consider two cases depending on the parity of @xmath107 , with each case yielding two equations . suppose first that @xmath107 is even , so we let @xmath122 . since @xmath12 is prime , we let @xmath123 . then we obtain @xmath124 since @xmath125 , there exist positive integers @xmath113 and @xmath114 such that either @xmath126 , @xmath127 , and @xmath128 or @xmath115 , @xmath129 , and @xmath130 suppose next that @xmath107 is odd . since @xmath12 is prime , we let @xmath131 . then we obtain @xmath132 since @xmath133 , there exist odd integers @xmath113 and @xmath114 such that either @xmath134 , @xmath135 , and @xmath136 or @xmath137 , @xmath138 , and @xmath139 we consider each of the above four equations separately to determine the number of positive integer solutions to equation . we next consider equation , which has at most one positive integer solution by theorem [ th3 ] . it follows from this equation that @xmath114 is odd and that @xmath113 is even if and only if @xmath141 . if @xmath142 or @xmath143 , then @xmath113 is odd , and we obtain @xmath144 . then equation has a solution only if @xmath17 , @xmath20 , @xmath23 , @xmath29 , @xmath2 , @xmath39 , @xmath22 , or @xmath36 . furthermore , equation has a solution only if @xmath49 . equation has at most two positive integer solutions by theorem [ th2 ] . since @xmath113 and @xmath114 are both odd , we have @xmath145 so @xmath146 and @xmath147 so @xmath148 . then @xmath149 or @xmath143 , and equation has at least one solution only if @xmath35 or @xmath36 . equation has at most two positive integer solutions by theorem [ th2 ] . since @xmath113 and @xmath114 are odd , we have @xmath150 so that equation has a solution only if @xmath28 , @xmath151 , @xmath25 , or @xmath152 . in particular , suppose that equation has two solutions , and let @xmath61 be the minimal positive solution of @xmath153 , so @xmath154 let @xmath155 and compute @xmath156 to obtain @xmath157 since we assume that two solutions exist to equation , @xmath68 and @xmath69 must both be squares by theorem [ th2 ] . it follows that there exist two positive integers @xmath158 and @xmath159 such that @xmath160 , @xmath161 , and @xmath162 this yields @xmath163 . since @xmath164 , we obtain @xmath165 so @xmath166 . it follows that @xmath167 , so @xmath168 or @xmath169 . therefore equation has at most two positive integer solutions only if @xmath30 or @xmath152 , and it has at most one positive integer solution only if @xmath28 or @xmath40 . furthermore , equation has a solution only if @xmath49 . + since the number of solutions to equations and depends on the value of @xmath170 , we first suppose that @xmath171 or @xmath34 , then equations and have no integer solution , equation has at most one solution , and has at most two positive integer solutions only if @xmath35 , or @xmath36 . therefore when @xmath171 or @xmath172 , equation has at most three positive integer solutions if @xmath35 , or @xmath36 , and it has at most one positive integer solution in all other cases . + we next suppose that @xmath37 . then equation has at most one positive integer solution . + if @xmath173 , then equation has at most one solution , has at most one solution and only if @xmath168 or @xmath174 , has no solution , and has at most one solution and only if @xmath175 . + if @xmath176 , then equation has at most one solution , has at most one solution and only if @xmath168 or @xmath143 , has no solution , and has at most two solutions and only if @xmath177 . + if @xmath178 , then equation has at most one solution , has at most one solution and only if @xmath168 or @xmath174 , has no solution , and has at most one solution and only if @xmath179 . + if @xmath146 , then equation has at most one solution , has at most one solution and only if @xmath168 or @xmath143 , has at most two solutions and only if @xmath168 or @xmath143 , and has at most one solution and only if @xmath180 . + _ proof of theorem [ th5 ] . _ if @xmath13 is even and @xmath12 is odd , we let @xmath181 . then @xmath182we let @xmath123 , and we obtain @xmath183since @xmath184 there exist positive integers @xmath113 and @xmath114 such that either @xmath185 , @xmath186 , and @xmath187or @xmath188 @xmath189 and @xmath190or @xmath191 @xmath192 and @xmath193or @xmath194 @xmath195 and @xmath196 if @xmath197 is a perfect square , then equation has no positive integer solution , otherwise it has at most one positive integer solution by lemma [ lem7].by theorem [ th3 ] , each of equations , , and has at most one solution . equation has a solution only if @xmath198 and @xmath199 , equation has a solution only if @xmath197 is odd and @xmath200 and equation has a solution only if @xmath197 is odd . since the number of solutions to equations and depends on the value of @xmath201 , we first suppose that @xmath171 or @xmath172 . then equations and have no integer solution . + if @xmath44 , then equation has at most one solution , has no solution , has no solution , and has no solution . if @xmath48 , then equation has at most one solution , has no solution , has no solution , and has at most one solution . + we now suppose that @xmath49 . then equations and have at most one positive integer solution . if @xmath44 , then equation has at most one solution , has at most one solution only if @xmath198 , has no solution , and has no solution . + if @xmath48 , then equation has at most one solution , has at most one solution only if @xmath198 , has at most one solution , and has at most one solution . + if @xmath13 is even and @xmath32 , we let @xmath181 . then @xmath202we let @xmath203 , and we obtain @xmath204since @xmath205 , there exist positive integers @xmath113 and @xmath114 such that @xmath206 , @xmath186 , and @xmath207which has no solution if @xmath197 is a perfect square and at most two solutions by theorem [ th4 ] . moreover , if @xmath197 is even and @xmath208 then by lemma [ lem7 ] equation has at most one solution . when we had finished writing the paper , we noticed that a proof of the result stated in lemma [ lem7 ] already existed within the proof of theorem 1 by luca and walsh in @xcite . our proof of lemma [ lem7 ] seems to be different than the proof of the result in @xcite .
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in this paper , we find a bound for the number of the positive solutions to the titled equation , improving a result of togb . as a consequence , we prove a conjecture of togb in a few cases .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the first of the above issues has been addressed very often in the past and we will restrict ourselves here to a brief summary . in the lf framework , non - trivial vacuum structure can reside only in zero - modes ( @xmath0 modes ) . since these are high - energy modes ( actually infinite energy in the continuum ) one often does not include them as explicit degrees of freedom but assumes they have been integrated out , leaving behind an effective lf - hamiltonian . the important points here are the following . if the zero - mode sector involves spontaneous symmetry breaking , this manifests itself as explicit symmetry breaking for the effective hamiltonian . in general , these effective lf hamiltonians thus have a much richer operator structure than the canonical hamiltonian . therefore , compared to a conventional hamiltonian framework , the question of the vacuum has been shifted from the states to the operators and it should thus be clear that the issues of renormalization and the vacuum are deeply entangled in the lf framework . despite widespread confusion on this subject , vector current conservation ( vcc ) is actually not a problem in the lf framework . many researchers avoid the subject of current conservation because the divergence of the vector current q_j^(q ) = q^-j^+ + q^+j^- - q__involves @xmath1 , which is quadratic in the constrained fermion spinor component j^-= ^_(-)_(- ) , and thus @xmath1 contains quartic interactions making it at least as difficult to renormalize as the hamiltonian . we know already from renormalizing @xmath2 that the canonical relation between @xmath3 and @xmath4 is in general not preserved in composite operators ( such as @xmath5 ) . it is therefore clear that a canonical definition for @xmath1 will in general violate current conservation since it does not take into account integrating out zero - modes and other high - energy degrees of freedom . however , in lf gauge @xmath6 , @xmath1 does not enter the hamiltonian and therefore one can address its definition separately from the construction of the hamiltonian . in fact , it is very easy to find a pragmatic definition which obviously guarantees manifest current conservation , namely j^-(q^+ ) & = & - ( 1 + 1 ) [ eq : j - def ] + j^-(q^+,q _ ) & = & - \ { - q__(q^+,q _ ) } ( 3 + 1)in 1 + 1 and 3 + 1 dimensions respectively . the corresponding expressions in coordinate space @xmath7 can be obtained by fourier transform . in summary , most importantly , vcc is no problem in lf quantization and is manifest at the operator level , provided @xmath1 is _ defined _ using eq . ( [ eq : j - def ] ) . since vcc can easily be made manifest ( by using the above construction ! ) , there is no point in _ testing _ its validity and it can not be used as a non - perturbative renormalization condition either . for a non - interacting theory , eq . ( [ eq : j - def ] ) reduces to the canonical definition of @xmath1 , but in general this is not the case when @xmath2 contains interactions or even non - canonical terms . note that ( as so often on the lf ) @xmath8 appears in the denominator of eq . ( [ eq : j - def ] ) . therefore , as usual , one should be very careful while taking the @xmath9 limit and while drawing any conclusions about this limit . an example of this kind are the pair creation terms in @xmath1 . naively they do not contribute for @xmath9 , since the @xmath10 pair emanating from @xmath1 necessarily carries positive @xmath8 . however , since @xmath1 often has very singular matrix elements for @xmath8 , such seemingly vanishing terms nevertheless survive the @xmath9 limit , which often leads to confusion . for an early example of this kind see ref . @xcite . a more recent discussion can be found in ref . a parity transformation , @xmath11 , @xmath12 leaves the quantization hyperplane ( @xmath13 ) in equal time ( et ) quantization invariant and therefore the parity operator is a kinematic operator in such a framework . it is thus very easy to ensure that parity is a manifest symmetry in et quantization by tracking parity at each step in a calculation . the situation is completely different in the lf framework , where the same parity transformation exchanges lf-`time ' ( @xmath14 ) and space ( @xmath15 ) directions , i.e. x^+ x^- and therefore the quantization hyperplane @xmath16 is _ not _ invariant . hence , the parity operator is a dynamical operator on the lf and , except for a free field theory , it is probably impossible to write down a simple expression for it in terms of quark and gluon field operators . thus , parity invariance is not a manifest symmetry in this framework . note that the situation is the other way round for the boost operator ( kinematic and manifest on the lf , dynamical and non - manifest in et ) . it thus depends on the physics application one is interested in and the symmetries that one considers the most important ones for that particular physics problem , which framework is preferable . for most applications of lf quantization , the lack of manifest parity actually does not constitute a problem in fact , one can view it as an opportunity rather than a problem . the important point here is the following : due to the lack of manifest covariance in a hamiltonian formulation , lf hamiltonians in general contain more parameters than the corresponding lagrangian . parity invariance may be very sensitive to some of these parameters . in order to illustrate this important point , let us consider the example of a 1 + 1 dimensional yukawa model @xcite @xmath17 this model actually has a lot in common with the kind of interactions that appear when one formulates qcd ( with fermions ) on a transverse lattice @xcite . the main difference between scalar and dirac fields in the lf formulation is that not all components of the dirac field are dynamical : multiplying the dirac equation @xmath18 by @xmath19 yields a constraint equation ( i.e. an `` equation of motion '' without a time derivative ) @xmath20 where @xmath21 for the quantization procedure , it is convenient to eliminate @xmath3 , using _ ( - ) = ( m_f+g_5 ) _ ( + ) from the classical lagrangian before imposing quantization conditions , yielding @xmath22 the rest of the quantization procedure very much resembles the procedure for self - interacting scalar fields . the above canonical hamiltonian contains a kinetic term for the fermions , a fermion boson vertex and a fermion 2-boson vertex . while the couplings of these three terms in the canonical hamiltonian depend only on two independent parameters ( @xmath23 and @xmath24 ) , it turns out that these terms are renormalized independently from each other once zero - mode and other high - energy degrees of freedom are integrated out . more explicitly this means that one should make an ansatz for the renormalized lf hamiltonian density of the form @xmath25 where the @xmath26 do not necessarily satisfy the canonical relation @xmath27 . however , this does not mean that the @xmath26 are completely independent from each other . in fact , eq.([eq : pren ] ) will describe the yukawa model only for specific combinations of @xmath26 . it is only that we do not know the relation between the @xmath26 . thus the bad new is that the number of parameters in the lf hamiltonian has increased by one ( compared to the lagrangian ) . the good news is that a wrong combination of @xmath26 will in general give rise to a parity violating theory : formally this can be seen in the weak coupling limit , where the correct relation ( @xmath27 ) follows from a covariant lagrangian . any deviation from this relation can be described on the level of the lagrangian ( for free massive fields , equivalence between lf and covariant formulation is not an issue ) by addition of a term of the form @xmath28 , which is obviously parity violating , since parity transformations result in @xmath29 for lorentz vectors @xmath30 ; i.e. @xmath31 . this also affects physical observables , as can be seen by considering boson fermion scattering in the weak coupling limit of the yukawa model . at the tree level , there is an instantaneous contact interaction , which is proportional to @xmath32 . the ( unphysical ) singularity at @xmath33 is canceled by a term with fermion intermediate states , which contributes ( near the pole ) with an amplitude @xmath34 . obviously , the singularity cancels iff @xmath35 . since the singularity involves the lf component @xmath8 , this singular piece obviously changes under parity . this result is consistent with the fact that there is no zero - mode induced renormalization of @xmath36 and thus @xmath35 at the tree level . this simple example clearly demonstrates that a ` false ' combination of @xmath37 and @xmath36 leads to violations of parity for a physical observable , which is why imposing parity invariance as a renormalization condition may help reduce the dimensionality of coupling constant space . of course there are an infinite number of parity sensitive observables , but not all of them are easy to evaluate non - perturbatively in the hamiltonian lf framework . furthermore , we will see below that some relations among observables , which seem to be sensitive to parity violations , are actually ` protected ' by manifest symmetries , such as charge conjugation , or by vcc . from the brief discussion of the ( to - be - canceled ) singularity above it seems that the most sensitive observable to look for violations of parity in qcd would be compton scattering cross sections between quarks and gluons ( or the corresponding fermions and bosons in other field theories ) because there one could tune the external momenta such that the potential singularity enters with maximum strength . however , this is not a very good choice : first of all non - perturbative scattering amplitudes are somewhat complicated to construct on the lf . secondly , quarks and gluons are confined particles , which makes @xmath38 compton scattering an unphysical process . a much better choice are any matrix elements between bound states . bound states are non - perturbative and all possible momentum transfers occur in their time evolution . therefore , any parity violating sub - amplitude would contribute at some point and would therefore affect physical observables . secondly , since one of the primary goal of lfqcd is to explore the non - perturbative spectra and structure of hadrons , matrix elements in bound states are the kind of observables for which the whole framework has been tailored . one conceivable set of matrix elements are those of the vector current operator @xmath39 . for example , consider the vacuum to meson matrix elements 0 | j^+ |n , p&= & p^+ f_n^(+ ) + 0 | j^- |n , p&= & p^- f_n^(- ) . [ eq : fpm ] by boost invariance , the couplings defined in eq . ( [ eq : fpm ] ) must be independent of the momenta . obviously , parity invariance requires @xmath40 , which makes it a useless relation for the purpose of parity tests . similar statements can be made about elastic formfactors but we will omit the details here . the basic upshot is that the same relations between the matrix elements of @xmath39 that arise from parity invariance can often also be derived using only vcc , i.e. such relations are in general ` protected ' by vcc one may also consider non - conserved currents , such as the axial vector current . however , there one would have to face the issue of defining the ` minus ' component before one can test any parity relations . parity constraints are probably very helpful in this case when constructing the minus components , but then those relations can no longer be used to help constrain the coefficients in the lf - hamiltonian . another class of potentially useful operators consists of the scalar and pseudoscalar densities @xmath5 and @xmath41 . obviously , if parity is conserved then at most one of the two couplings f_s&= & 0 | || n + f_p&= & 0 | |_5| ncan be nonzero at the same time , since a state @xmath42 can not be both scalar and pseudoscalar . what restricts the usefulness of this criterion is the fact that the same ` selection rule ' also follows from charge conjugation invariance ( @xmath43 and @xmath41 have opposite charge parity ! ) which is a manifest symmetry on the lf . therefore , only for theories with two or more flavors , where one can consider operators such as @xmath44 and @xmath45 , does one obtain parity constraints that are not protected by charge conjugation invariance . but even then one may have to face the issue of how to define these operators . parity may of course be used in this process , but then it has again been ` used up ' and one can no longer use these selection rules to test the hamiltonian . in summary , many parity relations are probably very useful to determine the lf representation of the operators involved , but may not be very useful to determine the hamiltonian . we have seen above that relations between @xmath46 and @xmath1 are often protected by vector current conservation and are therefore not very useful as parity tests . a much more useful parity test can actually be obtained by considering the ` plus'-component only : let us now consider a vector form factor , @xmath47 where @xmath48 , between states of opposite parity . when writing the r.h.s . in terms of one invariant form factor , use was made of both vector current conservation and parity invariance . a term proportional to @xmath49 would also satisfy current conservation , but has the wrong parity . a term proportional to @xmath50 has the right parity , but is not conserved and a term proportional to @xmath51 is both not conserved and violates parity . other vectors do not exist for this example . the lorentz structure in eq . ( [ eq : form ] ) has nontrivial implications even if we consider only the `` plus '' component , yielding @xmath52 1.cm ( 10,12)(0,1 . ) that this equation implies nontrivial constraints can be seen as follows : as a function of the longitudinal momentum transfer fraction @xmath53 , the invariant momentum transfer reads ( @xmath54 and @xmath55 are the invariant masses of the in and outgoing meson ) @xmath56 this quadratic equation has in general , for a given value of @xmath57 , two solutions for @xmath58 which physically correspond to hitting the meson from the left and right respectively . the important point is that it is not manifestly true that these two values of @xmath58 in eq . ( [ eq : formplus ] ) will give the same value for the form factor , which makes this an excellent parity test . in ref . @xcite , the coupling as well as the physical masses of both the fermion and the lightest boson where kept fixed , while the `` vertex mass '' was tuned ( note that this required re - adjusting the bare kinetic masses ) . figure [ fig : parity ] shows a typical example , where the calculation of the form factor was repeated for three values of the dlcq parameter k ( 24 , 32 and 40 ) in order to make sure that numerical approximations did not introduce parity violating artifacts . for a given physical mass and boson - fermion coupling , there exists a `` magic value '' of the vertex mass and only for this value one finds that the parity condition ( [ eq : formplus ] ) is satisfied over the whole range of @xmath57 considered . this provides a strong self - consistency check , since there is only one free parameter , but the parity condition is not just one condition but a condition for every single value of @xmath57 ( i.e. an infinite number of conditions ) . in other words , keeping the vertex mass independent from the kinetic mass is not only necessary , but also seems sufficient in order to properly renormalize yukawa@xmath59 . in the above calculation , it was sufficient to work with a vertex mass that was just a constant . however , depending on the interactions and the cutoffs employed , it ma be necessary to introduce counter - term functions@xcite . but even in cases where counter - term functions need to be introduced , parity constraints may be very helpful in determining the coefficient - functions for those more complex effective lf hamiltonians non - perturbatively . 99 m. burkardt , _ advances nucl . phys . _ * 23 * , 1 ( 1996 ) . w. a.bardeen et al . , , 1037 ( 1980 ) . s. dalley and b. van de sande , , 1088 ( 1999 ) ; , 065008 ( 1999 ) . m.burkardt , , 762 ( 1989 ) ; , 613 ( 1992 ) . s.j . brodsky and d.s . hwang , , 239 ( 1998 ) ; h .- choi and c.r . ji , , 071901 ( 1998 ) . m. burkardt , , 2913 ( 1996 ) . m. burkardt and h. el - khozondar , , 054504 ( 1999 ) . perry , lectures given at nato advanced study institute on confinement , duality and nonperturbative aspects of qcd , cambridge , u.k . , 1997 , hep - th/9710175 .
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issues that are specific for formulating fermions in light - cone quantization are discussed .
special emphasis is put on the use of parity invariance in the non - perturbative renormalization of light - cone hamiltonians .
# 1#2#3#4#1 * # 2 * , # 3 ( # 4 ) light - front ( lf ) quantization is the most physical approach to calculating parton distributions on the basis of qcd @xcite . before one can formulate qcd with quarks ,
it is necessary that one understands how to describe fermions in this framework .
this in turn requires that one addresses the following issues how is spontaneous symmetry breaking ( chiral symmetry ! ) manifested in the lf framework , where the vacuum appears to be trivial ?
is it possible to preserve current conservation and parity invariance in this framework ?
how does one formulate fermions on the transverse lattice , which seems to be a very promising approaches to pure glue lfqcd @xcite ?
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the hard - sphere fluid at a hard wall is a useful reference model for the solid - liquid interface between chemically dissimilar materials . its simple , but non - trivial , nature has made it a standard reference model to test theories of inhomogeneous fluids , such as integral equation theories and classical density - functional theories . as such , there have been a large number of simulation efforts to study the detailed thermodynamics and structure of this system , both to provide data for the testing of theoretical methods and to provide insight into the generic phenomenology of solid - liquid interfaces . of particular interest in such studies is the surface free energy , @xmath0 , which measures the work required to create a unit area of interface , and the excess adsorption , @xmath1 , which measures the number of particles in the interfacial region relative to that in a region of equal volume in the bulk . in terms of the single - particle density profile , @xmath2 , the excess adsorption is given by @xmath3\ ; dz\ ] ] where @xmath4 is the bulk fluid density and @xmath5 is the cartesian coordinate normal to the surface . the excess adsorption and the surface free energy are related through the gibbs adsorption equation @xmath6 where @xmath7 is the chemical potential . a more convenient relationship for use in molecular simulation studies is one derived from the gibbs - cahn procedure@xcite for the excess volume , @xmath8 : @xmath9 the excess adsorption is directly related to the excess volume by the relation@xcite @xmath10 for the hard - sphere / hard - wall system , we can express both @xmath0 and @xmath1 in dimensionless form : @xmath11 and @xmath12 , where @xmath13 and @xmath14 is the hard - sphere diameter . in what follows , we will drop the @xmath15 and assume that all quantities are in dimensionless form . for the hard - sphere fluid / hard - wall system , the values of @xmath0 and @xmath1 are dependent upon the choice of the reference point for measuring distance between the wall and the fluid spheres . in this work , we will adopt the `` edge - centered convention '' , where the coordinate of the center of a fluid sphere in contact with the wall is @xmath16 . in contract , a number of other studies - especially many of the early works - use the `` sphere - centered '' convention with @xmath17 begin the coordinate of the center of such a sphere . these two conventions give different values of the system volume and other characteristics , but the relationship between them is easy to establish . if we denote the interfacial free energy for a system using the edge - centered and sphere - centered conventions as @xmath0 and @xmath18 , respectively , then we have @xmath19 where @xmath20 is the bulk fluid pressure . correspondingly , if we denote the edge - centered and sphere - centered values of the excess adsorption as @xmath1 and @xmath21 , respectively , we have @xmath22 over the past four decades , there have been a number of simulation studies focused on the calculation of @xmath0 and @xmath1 for the hard - sphere / hard - wall system . one of the earliest is that of henderson and van swol@xcite , who calculate @xmath0 using a mechanical definition of the surface tension via the kirkwood - buff equation@xcite @xmath23\ ; dz \label{eq : kirkwood - buff}\ ] ] where @xmath24 and @xmath25 are the normal and transverse components of the pressure tensor and @xmath5 is the direction normal to the wall . in that early work , @xmath0 ( and also @xmath1 ) are calculated at relatively few values of the packing fraction @xmath26 . the statistical uncertainties in these calculation are also quite high , due both to convergence issues inherent in eq . [ eq : kirkwood - buff ] and computational power at the time . more recently , de miguel and jackson using an improved kirkwood - buff algorithm calculated @xmath0 with higher statistical precision than the results of ref . , but the values at the highest packing fractions are in disagreement with most other later studies . heni and lwen@xcite used a more accurate thermodynamic integration technique with square - barrier and triangular cleaving potentials to determine @xmath0 . while considerably improved over the henderson and van swol results , the relative statistical error in @xmath0 at the highest packing fraction studied ( near fluid - solid coexistence ) was still high ( nearly 10% ) . this thermodynamic integration method was improved upon by fortini and dijkstra@xcite with significant improvement in the overall statistical error . the most precise published measurements to date are those of laird and davidchack@xcite and recent calculations by yang , et al.@xcite . the former were calculated using the gibbs - duhem integration technique@xcite from data for the excess volume ( from which @xmath1 can be easily calculated ) , whereas the latter utilizes a grand - canonical transition matrix monte carlo simulation method . recently , we have refined the calculations in ref . to be more precise . these new results are given in the supplemental information.@xcite these simulation data are identical to those in ref . within the original statistical estimates , but are considerably more precise . for many theoretical and practical applications using the hard - sphere / hard - wall as a reference model , it is useful to have an accurate parametrized form of the wall surface tension @xmath27 and of excess adsorption @xmath28 that has a simple form and accurately accounts for the simulation data . in this work , we critically review previous theoretical and empirical expressions for @xmath0 and @xmath1 and propose our own highly accurate parameterization that fits the most accurate simulation data within the statistical error . the earliest functional form for @xmath0 is that derived from scaled particle theory ( spt):@xcite @xmath29 where @xmath26 is the packing fraction - the fraction of the total volume occupied by the spheres . . [ eq : spt ] can be obtained from the expression for @xmath18 given in refs . using the spt expression for the pressure and eq . [ eq : gammabar ] @xmath30 given an expression for @xmath27 , the corresponding expression for the excess adsorption can be obtained using a modified version of eq . [ eq : excess_v ] @xmath31 for the spt , the expression for @xmath1 is quite simple @xmath32 these spt expressions work well for very low packing fractions , but exhibit significant deviation from the simulation values at intermediate and high packing fractions - see fig . [ fig : comparison ] . comparison of the spt and wbii expressions for the hard - sphere / hard - wall @xmath0 with the simulation results of laird and davidchack.@xcite the error bars on the simulation data in this an all subsequent figures represent 95% confidence estimates.,scaledwidth=70.0% ] a more - accurate theoretical expression for @xmath0 can be derived from density - functional theory ( dft ) . it has been shown@xcite that an approximation for the surface tension of a hard - sphere fluid at a planar hard wall can be obtained within a class of dfts known as fundamental measure theories ( fmt).@xcite in these theories , the surface free energy can be found from the excess free energy density of a binary bulk ( homogeneous ) mixture . for the white bear mark ii ( wbii ) version of fmt one obtains @xmath33 the wbii performs significantly better than the spt expression and shows significant deviations only at the highest packing fractions - see fig [ fig : comparison ] . this wbii expression will be the starting point of our later parameterization . the excess adsorption within wbii can be obtained from eq . [ eq : gamma_partial ] using the carnahan - starling equation of state for hard spheres @xmath34 using @xmath35 and eq . [ eq : wbii ] in eq . [ eq : gamma_partial ] gives @xmath36 the spt and wbii theoretical expressions for @xmath0 and @xmath1 make convenient starting points for designing empirical expressions to represent the simulation data . using the spt form as a reference , henderson and plischke@xcite proposed the following empirical functional form for @xmath18 to fit the molecular - dynamics simulation results of ref . @xmath37 using eq . [ eq : gammabar ] together with the carnahan - starling equation of state ( eq . [ eq : cs ] ) gives @xmath38 with the corresponding equation for @xmath1 : @xmath39 more recently , urrutia@xcite , also starting with the spt expression , but using the more precise simulation results,@xcite suggested a more accurate expression @xmath40 the corresponding expression for @xmath0 is @xmath41 this expression has the property that the first three virial coefficients for @xmath27 are exact , as discussed in the next section . urrutia does not provide a corresponding expression for @xmath1 , but one can easily derive one using eq . [ eq : excess_v ] and the carnahan - starling equation of state @xmath42 another popular parameterization for any thermodynamic quantity is the so - called virial series , where the quantity of interest is expanded in a taylor series with respect to the density , pressure or packing fraction . with respect to the packing fraction , the virial expansion for @xmath0 can be written as @xmath43 the virial expansion coefficients , @xmath44 , for @xmath18 can be calculated from those for @xmath0 using the usual virial expansion coefficients , @xmath45 , for the pressure ( usually given as an expansion in @xmath4 ) : @xmath46 the coefficients , @xmath45 , are known analytically up to @xmath47 and have been calculated numerically up to @xmath48.@xcite using eq . [ eq : gammabar ] gives @xmath49 like the virial coefficients for the pressure,@xcite the virial coefficients @xmath50 can be written as a sum of cluster integrals@xcite and the first three coefficients of eq . [ eq : virial ] are known analytically.@xcite recently , yang , et al.@xcite used monte carlo sampling techniques to evaluate the cluster integrals for @xmath51 to 7 , obtaining approximate estimates for the exact cluster expansion expression for @xmath50 . in this work , the virial coefficients , @xmath44 for @xmath18 were determined . the virial equation for @xmath18 and the corresponding one for @xmath1 , truncated at @xmath52 , were shown to give very good agreement to their simulation results at low to intermediate packing fractions ( up to about @xmath53 = 0.4 ) . each of the theoretical and empirical parameterizations in the previous section can be expanded in a virial series . table i shows the virial expansion coefficients for @xmath0 for @xmath54 to 6 . ( @xmath55 is not shown because of the large statistical error in the monte carlo estimates of the exact coefficient . ) all parameterizations get the exact first two virial coefficients ( @xmath56 and @xmath57 ) correctly , but of the parameterizations presented so far , only that of henderson and plischke and that of urrutia also get the correct third virial coefficient ( @xmath58 ) . [ cols="^,^,^,^,^,^,^",options="header " , ] the henderson - plischke and urrutia parameterizations use as their starting point the spt expression for @xmath1 . in this work , we begin with the more accurate wbii expression to build an empirical parameterization . in the wbii , the first two virial coefficients of @xmath0 agree with the exact values . to preserve this , one can write @xmath59 where @xmath60 and @xmath61 are fitting parameters . for @xmath62 , the fitting function also gives the exact third virial coefficient . for this value of @xmath60 , a weighted least - squares fit to the simulation data gives @xmath63 . while this form is a good fit at low to moderate @xmath53 , there are still deviations beyond the statistical simulation uncertainties at the very highest packing fractions . the high packing fraction data can be well fit by introducing a high power term in the numerator of the first term on the right hand side of eq . [ eq : fitting ] - a similar approach was used by kolafa , labk and malijevsk in their development of a highly accurate equation of state for hard spheres.@xcite the actual high - power exponent ( @xmath64 ) used is not too important as long as the term does not significantly affect the value of @xmath0 at low to intermediate packing fractions - here we use @xmath65 : @xmath66 to develop our final parameterization , we note that fixing @xmath60 to give the correct third virial coefficient does not lead to an expression that fits the data at intermediate @xmath53 within the error bars . relaxing this condition gives us our final parameterization and the main result of this work . [ fig : gamma_diff ] shows the percentage relative deviation of eq . [ eq : dlr ] from the simulation values , along with the deviations for the other parameterizations considered here . only the new parameterization fits the data within the statistical error over the full range of @xmath53 : @xmath67 percentage deviation of the parameterizations for @xmath0 considered here from the simulation results.@xcite , scaledwidth=45.0% ] comparison of the spt , wbii and dlr parameterizations for the hard - sphere / hard - wall excess adsorption , ( @xmath1 ) with the simulation results.@xcite the inset shows the high packing fraction data at higher resolution . , scaledwidth=45.0% ] using the carnahan - starling equation of state and eq . [ eq : excess_v ] , the corresponding dlr expression for @xmath1 is @xmath68 however , this expression does not quite fit the data for @xmath1 within the error bars - presumably due to slight inaccuracies in the carnahan - starling equation of state ; however , slight modification of the high - order terms gives an expression that is accurate over the whole range of the simulation data within statistical error . @xmath69 fig . [ fig : adsorption_comparison ] shows the excess adsorption as a function of @xmath53 for the dlr , spt and wbii expressions ( eq . [ eq : dlr_gamma ] ) together with the simulation results@xcite . to compare the adsorption for all of the methods presented here , we plot in fig . [ fig : adsorption_diff ] the percentage deviation from the simulation values as functions of @xmath53 of all of the adsorption expressions presented here . note that the dlr expression is the only one to capture the simulation data within the statistical error over the entire @xmath53 range studied . percentage deviation from the simulation data@xcite for each of the parameterizations of the adsorption ( @xmath1 ) presented here.,scaledwidth=45.0% ] the thermodynamics of an inhomogeneous fluid at a wall can be characterized by the surface free energy @xmath0 and the excess adsorption @xmath1 , two thermodynamic quantities that are related by eq . [ eq : gibbs ] , the gibbs adsorption theorem . for a hard - sphere fluid at a planar hard wall both @xmath0 and @xmath1 have been measured very precisely in recent computer simulations . these simulation data can be employed as benchmark data for theories , such as scaled particle theory ( spt ) , or expressions derived from the white bear mark ii ( wbii ) density functional theory in certain limits , or from empirical parameterizations . each of the expressions for @xmath0 and @xmath1 that we have tested here perform well at low to intermediate packing fractions ( @xmath53 ) of the hard - sphere fluid . however , at higher values of @xmath53 close to freezing there are , however , significant deviations visible in all existing expressions see figs . [ fig : comparison][fig : adsorption_diff ] . these deviations are most prominent for the excess adsorption @xmath1 . because explicit expressions for @xmath0 and @xmath1 for the hard - sphere hard - wall model system can be useful in theoretical studies we suggest a new empirical parameterization for these quantities . to this end we start with the surface free energy @xmath0 . it is interesting to note that the known virial coefficients for @xmath0 are of limited use only in constructing an accurate expression . we employ the functional form of the surface free energy from wbii with some added fitting parameters . the key observation , however , is that an additional high power ( in @xmath53 ) term is required to reproduce the correct behavior of @xmath0 at high fluid densities . our parameterization for @xmath0 is given in eq . [ eq : dlr ] . we obtain the parameterization for the excess adsorption in two steps . first we employ a modified gibbs adsorption theorem , eq . [ eq : gibbs - cahn_v ] , and the carnahan - starling equation of state to find the functional form of @xmath1 from that of @xmath0 . because the carnahan - starling equation of state shows small deviation from simulation results at very high @xmath53 , we can improve the agreement of our parameterization with the simulation data by slightly adjusting the higher - order terms , which leads to our final parameterization for @xmath1 ( eq . [ eq : dlr_gamma ] ) . both new parameterizations for @xmath0 and @xmath1 agree within the very precise statistical estimates with simulation data over the entire fluid density range . an interesting question that arises is whether or not is it possible to obtain also improved expressions for the free energy of a homogeneous hard - sphere fluid and to derive a corresponding density functional for the inhomogeneous hard - sphere fluid based on the requirement of a high - power term ( in @xmath53 ) in @xmath0 and @xmath1 . bbl acknowledges support from the national science foundation ( nsf ) under grant che-0957102 . the improved simulation data@xcite were obtained using the alice high performance computing facility at the university of leicester .
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the inhomogeneous structure of a fluid at a wall can be characterized in several ways . within a thermodynamic description
the surface free energy @xmath0 and the excess adsorption @xmath1 are of central importance . for theoretical studies
closed expression of @xmath0 and @xmath1 can be very valuable ; however , even for a well - studied model system such as a hard - sphere fluid at a planar hard wall , the accuracy of existing expressions for @xmath0 and @xmath1 , compared to precise computer simulation data , can still be improved . here , we compare several known expressions for @xmath0 and @xmath1 to the most precise computer simulation data .
while good agreement is generally found at low to intermediate fluid densities , the existing parameterizations show significant deviation at high density . in this work ,
we propose new parameterizations for @xmath0 and @xmath1 that agree with the simulation data within statistical error over the entire fluid density range .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
unambiguous proof for disks in massive star formation is still missing . millimeter continuum observations suggest flattened structures without providing velocity information ( e.g. , @xcite ) , and molecular line studies suggest rotational motions but are often confused outflows and ambient gas ( e.g. , @xcite and beuther et al . , this volume ) . maser studies show disk signatures in some cases but are mostly not unambiguous as well ( e.g. , @xcite ) . the best evidence yet for genuine disk emission comes from ch@xmath1cn observations in iras20126 + 4104 @xcite . in this case , the velocity gradient defining the presence of the disk is aligned perpendicular to the bipolar outflow , consistent with the common disk / jet paradigm . to further investigate possible disk emission and its association with molecular jets , we used the submillimeter array ( sma ) to observe the jet tracer sio(54 ) and the hot - core tracer hcooch@xmath1(2019 ) in a massive star - forming region . the source iras18089 - 1732 is a young high - mass protostellar object ( hmpo ) which has been studied in detail over recent years . the source is part of a sample of 69 hmpos selected mainly via infrared color - color criteria and the absence of strong cm emission @xcite . iras18089 - 1732 is approximately at a distance of 3.6kpc and its bolometric luminosity is about @xmath5l@xmath2 @xcite . millimeter continuum observations reveal a massive core @xmath6m@xmath2 with h@xmath7o and ch@xmath1oh maser emission , and a weak 1mjy source is detected at 3.6 cm @xcite . as part of a single - dish co outflow study , wing emission indicative of molecular outflows was detected but the co map was too confused to define a bipolar outflow @xcite . during these observations , @xcite also observed sio(21 ) at 3 mm , and bipolar structure was detected in the north - south direction . furthermore , @xcite reported the detection of the hot - core - tracing molecules ch@xmath1cn and ch@xmath1oh . this letter focuses on the jet / disk observations and the ( sub-)mm continuum data . a description of the line forest observed simultaneously is presented in an accompanying paper ( beuther et al . , this volume ) . iras18089 - 1732 was observed with the sma between may and july 2003 in two different configurations with 3 to 5 antennas in the array . the phase reference center of the observations was r.a.[j2000 ] 18:11:51.4 and dec.[j2000 ] @xmath8:31:28.5 . the frequency was centered on the sio(54 ) line at 217.105ghz , the hcooch@xmath1(2019 ) line at 216.967ghz could be observed simultaneously in the same band . the hcooch@xmath1 line consists of 8 distinct components but is dominated by 4 of them which are separated by 2.5mhz ( corresponding to 3.5 km s@xmath9 ) . the correlator bandwidth at that time was 1ghz with a frequency resolution of 0.825mhz . we smoothed the sio(54 ) data to a spectral resolution of 3kms@xmath9 and the hcooch@xmath1(2019 ) data to 2kms@xmath9 to increase the signal - to - noise ratio . the continuum was constructed via averaging the line - free channels in the upper side - band . the beam size at 217ghz was @xmath10 and at 354 ghz @xmath11 . system temperatures in the 850@xmath0 m band were between 300 - 900k and in the 1 mm band around 200k . the continuum rms at 217ghz was @xmath12mjy and at 354ghz 40mjy . the flux calibration was estimated to be accurate to @xmath13 . for more details on the sma , the observations and data reduction , see the accompanying papers by ho , moran & lo and beuther et al . ( this volume ) . figure [ continuum ] compares the ( sub-)mm continuum observations and shows additional cm continuum and h@xmath7o and ch@xmath1oh maser data @xcite . even in the highest - spatial - resolution data at 850@xmath0 m , the dust emission remains singly peaked , i.e. , it does not split up into multiple sources as observed in other massive star - forming regions , e.g. , iras19410 + 2336 @xcite . nevertheless , in our 1 mm data we resolve elongated emission in the south and north - west , which demonstrates that iras18089 - 1732 has a compact mm core with extended halo emission ( fig . [ continuum ] ) . the halo emission is not seen in the 850@xmath0 m observations because of the reduced sensitivity and uv - coverage . while the weak 3.6 cm peak and the h@xmath7o maser position coincide exactly with the ( sub-)mm continuum peak , the ch@xmath1oh maser position is about @xmath14 to the south . the latter could indicate that there might be a second source at the position of the ch@xmath1oh maser which we can not distinguish . table [ para ] shows the derived peak and integrated fluxes ( @xmath15 and @xmath16 ) at 1 mm and 850@xmath0 m . comparing the sma 1 mm data with single - dish observations of the region @xcite , we find that about @xmath17 of the flux is filtered out in the interferometric data . it is difficult to derive a spectral index from the continuum images because the different uv - coverages filter out different amounts of flux . however , we can measure fluxes @xmath18 in the uv - plane . ideally , one would select the same regions in the uv - plane , but as this would reduce the amount of available data even more , it is reasonable to compare the values for matching baseline ranges ( in units of @xmath19 ) . we selected one range of short baselines ( @xmath20 , corresponding to spatial scales between @xmath21 and @xmath22 ) and one range of longer baselines ( @xmath23 , corresponding to spatial scales between @xmath24 and @xmath25 ) where there were sufficient data in both frequency bands : the flux values are shown in table [ para ] . the 3.6 cm flux is only 0.9mjy @xcite , and assuming free - free emission its contribution to the sub-(mm ) observations is negligible . assuming a power - law relation @xmath26 in the rayleigh - jeans limit with the dust opacity index @xmath27 , we find @xmath28 for short baselines corresponding to large spatial scales and @xmath29 for large baselines corresponding to small spatial scales . these values are lower than the canonical value of 2 @xcite and decrease to small spatial scales . the exact values should be taken with caution due to the large calibration uncertainty ( estimated to be within @xmath13 ) , but the trend of a low @xmath27 decreasing with decreasing spatial scales appears real . without mapping the selected sets of uv - data in the image - plane , we can not determine whether the smaller spatial scales correspond to the central core or to another unresolved structure . however , the 850@xmath0 m continuum image , which is based on larger baselines , is more compact than the 1.3 mm emission ( fig . [ continuum ] ) indicating that the largest baselines do trace the emission from the core center . a low value of @xmath27 has also recently been found in another high - mass star - forming region @xcite , and @xcite report that @xmath27 decreases in l1489 from 1.5 - 2 in the envelope down to @xmath30 at the inner peak . this trend may be due to grain growth within the central core / disk or high optical depth . assuming optically thin dust emission at mm wavelength , we calculate the mass and peak column density using the 1.3 mm data following the procedure outlined for the single - dish dust continuum data by @xcite . we use a grain emissivity index @xmath31 and a temperature typical for hot - core - like sources of 100k . the central core mass then is @xmath32m@xmath2 and the peak column density @xmath33 @xmath34 , corresponding to a visual extinction @xmath35 } = n_{\rm{h_2}}/(10^{21}\rm{cm}^{-2 } ) \sim 800 $ ] . as discussed by @xcite , the errors are dominated by systematics , e.g. , exact knowledge of @xmath27 or the temperature . we estimate the masses and column densities to be correct within a factor of 5 . figure [ channel ] presents a channel map of sio(54 ) , and we find blue and red emission ( systemic velocity 34.9kms@xmath9 , [ hcooch3 ] ) north of the continuum core . the sio emission by itself allows an interpretation of a bipolar outflow centered @xmath36 north of the core . however , we do not detect any sub - mm continuum source there ( @xmath37 m @xmath38 mass sensitivity @xmath39m@xmath2 ) , and a massive outflow without a sub - mm continuum source at the driving center is very unlikely . therefore , we favor the interpretation of an outflow emanating from the core toward the north with a position - angle ( p.a . ) of @xmath40 . the fact that we see blue and red sio emission toward the northern lobe indicates that the outflow axis is near the plane of the sky . for the red lobe , we find an increase in velocity with distance from the center of the core resembling the hubble - law within molecular outflows ( e.g. , @xcite ) . previous unpublished single - dish sio(21 ) observations with @xmath41 resolution also show a bipolar outflow in the north - south direction . the main difference is that the southern part of the sio(21 ) outflow is not observed at the higher frequency and higher spatial resolution with the sma . we can attribute this difference to several possible reasons . first , interferometers filter out the large scale emission , and without additional short spacing information , only the more compact jet - like emission is detected with the sma . second , the excitation of the 54 transition is higher with respect to the 21 transition . while the sio(21 ) line traces the lower temperature gas , the sio(54 ) line is sensitive to the more excited gas being associated with the more collimated component of the outflow . furthermore , an asymmetry in the small - scale distribution of the dense gas can also contribute to the differences of the large - scale sio(21 ) and small - scale sio(54 ) observations . the high - density - gas tracing molecular hcooch@xmath1(2019 ) emission is confined to the central core traced by the ( sub-)mm continuum ( fig . [ channel ] ) . a gaussian fit to the central spectrum results in a systemic velocity of @xmath42kms@xmath9 , and we observe a velocity gradient across the core . it is possible to fit the peak position of each spectral channel shown in figure [ channel ] to a higher accuracy than the nominal spatial resolution , down to 0.5hpbw/(s / n ) , with s / n the signal - to - noise ratio @xcite . we performed these fits in the uv - plane to avoid artifacts due to inverting and cleaning the data , the resulting positions are shown in figure [ pos_velo ] . the derived velocity gradient follows a p.a . of @xmath43 from blue to red velocities . the separation of the most blue and most red positions is @xmath44 , corresponding to @xmath45au . figure [ pos_velo ] also presents a position - velocity diagram for the fitted channels at the p.a . of @xmath46 confirming the space - velocity shift . this velocity gradient is neither in the direction of the sio outflow nor directly perpendicular to it . a velocity gradient solely due to a rotating disk would be perpendicular to the outflow . the sio outflow is near the plane of the sky complicating an accurate determination of its p.a . , and the outflow might also be affected by precession ( e.g. , iras20126 + 4104 , @xcite ) . regarding the hcooch@xmath1 emission , it is unlikely that this high - density tracer depicts a different outflow which would remain undetected by the sio . therefore , we regard neither the partially uncertain outflow p.a . nor a second outflow sufficient to explain the difference in p.a . between the sio and the hcooch@xmath1 emission . however , as shown in the accompanying paper ( beuther et al . , this volume ) , most molecular line data are influenced by the molecular outflow , and it is likely that hcooch@xmath1 is also affected by the outflow . furthermore , @xcite have shown that infall can influence the dense - gas velocity signature as well . assuming that the inner core contains a rotating disk , which has an orientation perpendicular to the outflow , and that the hcooch@xmath1 velocity signature is influenced by the disk , the outflow and infall , it is plausible that the observed axis of the hcooch@xmath1 velocity gradient is offset from the expected disk orientation . we do not observe the inner region of the disk where the outflow / jet is accelerated , but we may have detected the outer parts of the accretion - disk . assuming equilibrium between centrifugal and gravitational forces at the outer radius of the disk , we calculate the dynamical mass enclosed within the central @xmath47 to be @xmath48m@xmath2 ( with @xmath49 the unknown inclination angle between the disk plane and the line of sight , and an hcooch@xmath1 velocity shift . taking into account the 4 dominant hcooch@xmath1 line components spanning 3.5kms@xmath9 ( [ obs ] ) , the hcooch@xmath1 velocity shift is @xmath50 . ] of 4.25kms@xmath9 ) , of the order of the core mass derived from the dust emission ( table [ para ] ) . thus , in contrast to t tauri systems where the disk mass is negligible compared with the protostellar mass , in iras18089 - 1732 a considerable fraction of the enclosed mass seems to be part of a large accretion disk and/or rotating envelope . this result fits the picture that iras18089 - 1732 is in a very young evolutionary state , with the central ( proto)star still accreting material from the surrounding core / disk . the combined sio(54 ) and hcooch@xmath1(2019 ) observations toward iras18089 - 1732 support a massive star formation scenario where high - mass stars form in a similar fashion as their low - mass counterparts , i.e. , via disk accretion accompanied by collimated jets / outflows . the hcooch@xmath1 observations still barely resolve the disk / envelope structure and the interpretation is not entirely unambiguous , but the data indicate rotation which might stem at least partly from an accretion disk . higher resolution observations in different disk tracers are needed to investigate the disk / envelope conditions in more detail . furthermore , the continuum data indicate a lower @xmath27 than the standard value of 2 , and we observe a decreasing @xmath27 with decreasing spatial scales . as nearly all the gas observed in dust emission takes part in the assumed dynamical rotation observed in hcooch@xmath1 this low @xmath27 might be due to grain growth or high opacity within a disk - like structure ( e.g. , @xcite ) .
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sma observations of the massive star - forming region iras18089 - 1732 in the 1 mm and 850@xmath0 m band reveal outflow and disk signatures in different molecular lines .
the sio(54 ) data show a collimated outflow in the northern direction .
in contrast , the hcooch@xmath1(2019 ) line , which traces high - density gas , is confined to the very center of the region and shows a velocity gradient across the core .
the hcooch@xmath1 velocity gradient is not exactly perpendicular to the outflow axis but between an assumed disk plane and the outflow axis .
we interpret these hcooch@xmath1 features as originating from a rotating disk that is influenced by the outflow and infall .
based on the ( sub-)mm continuum emission , the mass of the central core is estimated to be around 38m@xmath2 . the dynamical mass derived from the hcooch@xmath1 data is 22m@xmath2 , of about the same order as the core mass .
thus , the mass of the protostar / disk / envelope system is dominated by its disk and envelope .
the two frequency continuum data of the core indicate a low dust opacity index @xmath3 in the outer part , decreasing to @xmath4 on shorter spatial scales .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
majority of stars reside in binary systems @xcite . the mechanism of planet formation around binary systems would be different from that around single stars not only because the environment of the protoplanetary disk would be affected by the binary companion but also because the binary companion would affect the long - term stability of the planet orbit . therefore , one of the most generic environments to be considered in the study of planet formation should be that of a binary . however , the major planet - formation scenarios that have been developed over the past decades , e.g. , core - accretion theory @xcite and disk instability theory @xcite , were mostly focused on the case of single stars . although a considerable work has been done about the effect of binary companions on the planet formation ( e.g. , * ? ? ? * ) and the long - term orbital stability ( e.g. , * ? ? ? * ) , this work was restricted to mostly theoretical studies and thus many details about the planet formation scenario remain uncertain . these details can be refined by the constraints provided by the sample of actually detected planetary systems . unfortunately , there exists only a total of 19 known planets in 17 binary systems and most of these planets exist under a similar environment , i.e. circumbinary planets orbiting very close binaries . to give details about the planet formation in binary systems , therefore , it is important to detect more of such planets residing under various environments . when a gravitational microlensing event is caused by a very wide binary object , the lensing behavior in the region around each lens component is approximated by chang - refsdal ( c - r ) lensing , which refers to the gravitational lensing of a point mass perturbed by a constant external shear @xmath0 @xcite . in the low shear regime ( @xmath1 ) , c - r lensing induces a small astroid - shape caustic around the lens . the lensing behavior of a very close binary , on the other hand , can be approximated by a point - mass plus quadrupole lensing . in this case , an astroidal caustic similar to the c - r lensing caustic is produced around the center of mass of the binary . for this reason , the event caused by a very wide or a very close binary lens is often referred to as the c - r lensing event . due to the existence of the caustic in the central region , c - r lensing events are identified by a short - term anomaly that appears near the peak of a very high - magnification event . although c - r lensing events are routinely detected from microlensing surveys , not much attention is paid to them because they are thought to be simply one type of numerous binary - lens events and thus of little scientific importance . in this work , we point out that c - r lensing events provide an important channel to detect planets in binaries , both in close and wide binary systems . in order to demonstrate that dense and high - precision coverage of a c - r lensing - induced perturbation can provide a strong constraint on the existence of a planet in the wide range of the planet parameters , we present analysis of the actually observed c - r lensing event ogle-2015-blg-1319 . the paper is organized as follows . in section 2 , we briefly describe the lensing properties in the cases where the lens is composed of a single , binary , and triple masses . we also describe the c - r lensing behavior . in section 3 , we estimate the detection efficiency of planets in binaries by conducting analysis of the lensing event ogle-2015-blg-1319 . we summarize the results and conclude in section 4 . when a point - mass lensing event occurs the lensing behavior is described by the lens equation @xmath2 where @xmath3 and @xmath4 denote the complex notations of the source and image positions , respectively , and @xmath5 represents the complex conjugate of @xmath6 . here all lengths are normalized to the angular einstein ring radius @xmath7 and the lens is positioned at the origin . solving the lens equation yields two solutions of image positions : one outside and the other inside the einstein ring . the magnification @xmath8 of each image @xmath9 is given by @xmath10 where @xmath11 is the jacobian of the lens equation evaluated at the image positions @xmath12 and @xmath13 is the determinant of the jacobian . since the individual microlensing images can not be resolved , the observed lensing magnification is the sum of the magnifications of the individual images , i.e. @xmath14 . for a point mass , the magnification is represented analytically by @xmath15 where @xmath16 is the lens - source separation with the length normalized to @xmath7 , @xmath17 is the einstein time scale , @xmath18 is the time of the closest lens - source approach , and @xmath19 is the lens - source separation at @xmath18 . for a rectilinear relative lens - source motion , the lensing light curve is characterized by a smooth and symmetric shape @xcite . when an event is produced by a lens composed multiple components , the lens equation is expressed as @xmath20 where @xmath21 and @xmath22 , and @xmath23 represent the location , mass fraction , and mass of each lens component , respectively . the notation @xmath24 denotes the number of the lens components , and thus @xmath25 for a binary lens . here all lengths ( @xmath26 and @xmath6 s ) are in units of the angular einstein radius corresponding to the total mass of the lens , @xmath27 . one of the most important properties of a binary lens that differentiate from those of a single lens is the formation of caustics , which represent the sets of source positions at which @xmath28 and thus the magnification of a point source becomes infinite . as a result , a binary - lensing light curve can exhibit strong deviations when a source approaches close to or passes over the caustic . caustics of a binary lens form a single or multiple sets of closed curves and each curve composed of concave curves that meet at cusps . the topology of the binary - lens caustic is broadly classified into 3 categories @xcite . in the case of a binary where the binary separation is greater than @xmath29{\epsilon_1 } + \sqrt[3]{\epsilon_2})^{3/2}$ ] ( wide binary ) , there exist two sets of 4-cusp caustics that are located close to the individual lens components . when the binary separation is smaller than @xmath29{\epsilon_1 } + \sqrt[3]{\epsilon_2})^{-3/4}$ ] ( close binary ) , the caustic is composed of 3 pieces , where the central caustic with 4 cusps is formed around the center of mass of the binary lens , and the other 2 triangular caustics are located away from the center of mass . in the intermediate separation region , there exists a single big caustic with 6 cusps . for the visual presentation of the binary caustic topology , see figure 3 of @xcite . in the extreme case where the binary separation is much greater or less than @xmath7 , the 4-cusp central caustic has an astroid shape which is symmetric with respect to the binary axis and the line vertical to the binary axis . for the exact description of the lensing behavior produced by a lens system of a planet orbiting a binary object , one needs the triple lens equation , i.e. eq . ( [ eq4 ] ) with @xmath30 . with the addition of a third lens component , the complexity of the lensing behavior greatly increases and caustics can exhibit self - intersection and nesting @xcite . ( upper panel ) and the mass ratio between the planet and the host @xmath31 ( lower panel ) . both the caustic size and the binary separation are normalized to the angular einstein radius corresponding to the total lens mass . ] a planetary lens corresponds to the extreme case of a binary lens where the mass of one of the lens components is much smaller than the other , i.e. @xmath32 and @xmath33 . in this case , the lens equation is approximated as @xmath34 where @xmath35 is the mass ratio between the planet and the host and @xmath36 represents the position of the planet with respect to the host . we note that the lengths of @xmath26 and @xmath6 s in eq . ( 5 ) are normalized to the angular einstein radius corresponding to the mass of the primary lens , i.e. @xmath37 . for a planetary lens , however , the mass of the primary dominates , i.e. @xmath38 , and thus @xmath39 . the planet induces two types of caustics , where one is located close to the host ( central caustic ) and the other is away from the host . the central caustic has an arrowhead shape and its size is related to the star - planet separation @xmath40 and the mass ratio by @xcite @xmath41 due to the location of the central caustic close to the host , perturbations induced by the central caustic of a planet always appear near the peak of the lensing light curve produced by the host of the planet @xcite . the lens equation of c - r lensing is expressed as @xmath42 which describes the lensing behavior around the primary lens with an external shear @xmath0 . here lengths are given in units of the einstein radius corresponding to the primary mass . the shear induces a caustic around the lens . the shape of the caustic is similar to the central caustic of a very wide binary lens and thus the c - r lensing provides a good approximation in describing the binary lensing behavior in the region around the caustic . for a very close binary lens , the lensing behavior around the center of mass is described by quadrupole lensing @xcite , which also induces an astroidal caustic similar to the c - r lensing caustic . to the first order approximation , the caustic size in units of the angular einstein radius of the binary - lens mass , @xmath7 , is related to the separation and the mass ratio between the binary - lens components by @xmath43 for a wide binary and @xmath44 for a close binary . a c - r lensing event can provide a channel to detect planets in binary systems . this is because both of the c - r lensing caustic and the central caustic induced by the planet occur in the same region around the primary lens and the size of the planet - induced central caustic can be comparable to the size of the c - r lensing caustic @xcite . in figure [ fig : one ] , we present the sizes of the c - r lensing caustic and the planet - induced central caustic as a function of the primary - companion separation . we note that a pair of caustics with separations @xmath45 and @xmath46 have the same size to a linear order and thus we present distributions for only wide binaries . the plot shows that the central caustic induced by planets located in the `` lensing zone '' , although the range varies depending on how the lensing zone is defined . ] is of considerable size compared to the size of the c - r lensing caustic . this suggests that the c - r lensing anomaly can be additionally affected by the planetary perturbation , enabling one to identify the presence of a planet in the binary system . in figure [ fig : two ] , we present the variation of the c - r lensing caustic and the lensing light curve affected by the presence of planets with various separations and mass ratios between the planet and the primary of the binary . see @xcite for more details about the caustic variation . we note that the c - r lensing channel enables detections of planets not only in close binaries but also in wide binaries . to be dynamically stable , a planet should be either in a circumbinary ( or p - type ) orbit , where the planet orbits the barycenter of the two stars of a close binary , or in a circumprimary ( s - type ) orbit , where the planet orbits just one star of a wide binary system . this condition of the planet existence in the binary system matches the lens system configuration of the proposed c - r lensing channel of planet detections . planets in binary systems can be detected by various methods such as transit ( e.g. , * ? ? ? * ) , eclipsing binary timing ( e.g. , * ? ? ? * ) , and radial - velocity methods ( e.g. , * ? ? ? due to the intrinsic nature of the methods , however , it is difficult to detect planets in circumprimary orbits and thus all planets in binaries detected by these methods , 19 in total , reside in circumbinary orbits . on the other hand , c - r lensing events can be produced by both close and wide binary systems and thus the proposed c - r lensing channel provides a unique channel to detect planets both in circumbinary and circumprimary orbits . in this section , we demonstrate the high efficiency of the proposed c - r lensing channel in detecting planets of binary systems . estimating the efficiency requires to consider various details of observational conditions such as photometric precision and cadence . in order to reflect realistic observational conditions , we estimate the detection efficiency for an example c - r lensing event that was actually observed by lensing experiments . the event used for our efficiency estimation is ogle-2015-blg-1319 . the event was analyzed in detail by @xcite and turned out to be an exemplary c - r lensing event caused either by a close or a wide binary lens . figure [ fig : three ] shows the light curve of the event reproduced based on the same data sets as those used in the previous analysis . the model light curve superposed on the data points is obtained from binary - lensing modeling based on one of the 8 degenerate solutions ( `` + + wide '' solution ) presented in @xcite . the inset in the lower panel shows the source trajectory with respect to the caustic , which has a very characteristic shape of a c - r lensing caustic . the anomaly caused by the c - r lensing caustic in the peak of the light curve was densely and precisely observed . this was possible because the event was predicted to be a high - magnification event before it reached the peak based on the light curve obtained from the survey observation conducted by the optical gravitational lens experiment ( ogle : * ? ? ? * ) and the microlensing observations in astrophysics ( moa : * ? ? ? * ; * ? ? ? * ) , enabling intensive observations of the peak by follow - up observation groups including the microlensing follow - up network ( @xmath47fun : * ? ? ? * ) and the robonet @xcite groups . we note that the event was also observed in space using two space telescopes _ spitzer _ and _ swift_. this enabled the determinations of the lens mass and the distance , but we do not use the space - based data because these data are irrelevant to our scientific purpose . the values of the binary separation and the mass ratio presented in @xcite are @xmath48 for the close binary solution and and @xmath49 for the wide solution . for other lensing parameters , see table 1 of @xcite . to show the constraint on the existence of a planet , we construct an `` exclusion diagram '' , which shows the probability of excluding the existence of a planet as a function of the planet separation and the mass ratio @xcite . we construct the exclusion diagram following the procedure of @xcite . in this procedure , we first introduce a planet with the parameters @xmath50 to the binary lens with the parameters @xmath51 . here @xmath51 represent the separation and the mass ratio between the binary components , while @xmath52 are the separation and the mass ratio between the primary of the binary and the planet . the angle @xmath53 denotes the orientation angle of the planet with respect to the binary axis connecting the binary lens components . we use @xmath51 that are determined from the binary - lensing modeling . for a given set of @xmath50 , we then search for other parameters that yield the best fit to the observed light curve and compute @xmath54 of the fit . we repeat this process for many different orientation angles . then , the probability of excluding a planet for a given @xmath52 is estimated as the fraction of the angles @xmath53 that result in fits with @xmath55 , where @xmath56 is the difference between the triple - lens ( i.e. binary + planet ) and the binary - lens models . as a criteria for the planet detection , we adopt a threshold value of @xmath57 , which is a generally agreed value for planets detected through the high - magnification channel @xcite . the probability of excluding a planet corresponds to the probability of detecting the planet , i.e. planet detection efficiency . and the mass ratio @xmath31 between the planet and the primary of the binary lens . the values marked in the upper @xmath58 axis and the right of @xmath59 axis represent the physical primary - planet separation in au and the mass of the planet in jupiter masses , respectively . the color coding represents the regions of different efficiencies that are marked in the legend . the planetary separation is expressed in units of the angular einstein radius corresponding to the binary lens mass . ] figure [ fig : four ] shows the constructed exclusion diagram which shows the planet detection efficiency as a function of the normalized separation @xmath60 and the mass ratio @xmath31 of the planet . since the microlens parallax of the event ogle-2015-blg-1319 was determined using the combined data taken from the ground and space , the physical sizes of the primary - planet separation and the mass of the planet were determined . by adopting the lens mass of @xmath61 , the angular einstein radius of @xmath62 mas , and the distance to the lens of @xmath63 kpc , that were determined by @xcite , i.e. ( + , + ) wide model , we convert @xmath60 and @xmath31 into the physical sizes of the primary - planet separation in au , i.e. @xmath64 , and the mass of the planet in jupiter masses ( @xmath65 ) , i.e. @xmath66 , and they are presented in the upper @xmath58 axis and the right of the @xmath59 axis , respectively . lccc jupiter & 1.0 10 au & 0.5 11 au & 0.4 18 au + saturn & 2.0 5 au & 1.0 7 au & 0.9 9 au + neptune & & 2.0 3.5 au & 1.8 4.0 au in table [ table : one ] , we present the ranges of planet detection for 3 different planets with masses corresponding to those of the jupiter , saturn ( @xmath67 ) , and neptune ( @xmath68 ) of the solar system . it is found that the detection efficiency is greater than @xmath69 for the jupiter- and saturn - mass planets located in the ranges of 1.0 10 au and 2.0 5 au from the host , respectively . the ranges with @xmath70 probability are 0.4 18 au , 0.9 9 au , and 1.8 4 au for the jupiter- , saturn- , and neptune - mass planets , respecively . these ranges encompass wide regions around snow lines where giants are believed to form . despite the high efficiency of c - r lensing events in detecting planets of binary systems , there exists no report of planet detection yet . the reasons for this can be ( 1 ) the rarity of c - r lensing events , ( 2 ) the unoptimized observational strategy for planet detections , and ( 3 ) the rarity of planets in the region of sensitivity . considering the existence of the already known microlensing planets detected through the repeating - event channel and those detected by other methods combined with the wide region of planet sensitivity of the c - r lensing channel , the reason ( 3 ) is unlikely to be the main reason of nondetection . to check the possibilities ( 1 ) and ( 2 ) , we investigate the lensing events reported by the ogle and moa surveys in 2015 season . from the systematic analyses of all high - magnification events with anomalies near the peaks based on the online data of the surveys , we find 12 c - r lensing events including ogle-2015-blg-1319 , moa-2015-blg-040/ogle-2015-blg-0318 , ogle-2015-blg-0697/moa-2015-blg-148 , ogle-2015-blg-0797 , ogle-2015-blg-0812 , ogle-2015-blg-0189 moa-2015-blg-085/ogle-2015-blg-0472 , ogle-2015-blg-0313/moa-2015-blg-047 , ogle-2015-blg-0033/moa-2015-blg-017 , moa-2015-blg-047 , ogle-2015-blg-0919 , and ogle-2015-blg-0863 . this indicates that the rarity of c - r lensing events is not the reason for the nondetection , either . however , we find that the coverage of the peak regions for all of the c - r lensing events except ogle-2015-blg-1319 was not dense enough to constrain the existence of a planet . therefore , it is likely that the main reason for the nondetection of planet is due to the unoptimized observational strategy . in other words , this suggests that planets can be detected through the proposed channel in abundance with an aggressive strategy to densely cover the peak regions of high - magnification events , e.g. , vigilant monitoring of high - magnification events and timely alerts of anomalies followed by prompt and intensive coverage of the anomalies by follow - up observations we pointed out that c - r lensing events could provide one with an important channel to detect planets in binary systems . we also pointed out that while other methods could detect planets only in circumbinary orbits the proposed c - r lensing channel could provide a unique channel to detect planets both in circumbinary and circumprimary orbits . we demonstrated the high sensitivity of the c - r lensing channel to planets in a wide range of planet parameter space by presenting the exclusion diagram for an actually observed c - r lensing event . we mentioned that an increased number of microlensing planets in binary systems could be detected with an aggressive strategy to densely cover the peak regions of high - magnification events . the sample of an increased number of microlensing planets in binaries will make it possible to provide important observational constraints that can give shape to the details of the formation scenario which has been restricted to the case of single stars . work by c. han was supported by the creative research initiative program ( 2009 - 0081561 ) of national research foundation of korea . we acknowledge the high - speed internet service ( kreonet ) provided by korea institute of science and technology information ( kisti ) . abt , h. a. 1983 , , 21 , 343 albrow , m. d. , beaulieu , j .- p . , caldwell , j. a. r. , et al . 2000 , , 535 , 176 bond , i. a. , abe , f. , dodd , r. j. , et al . 2001 , , 327 , 86 boss , a. p. 2012 , , 419 , 1930 cameron , a. g. w. 1978 , the moon and the planets , 18 , 5 cassan , a. , kubas , d. , beaulieu , j .- , et al . 2012 , nature , 481 , 167 chang , k. , & refsdal , s. 1979 , nature , 282 , 561 chang , k. , & refsdal , s. 1984 , nature , 132 , 168 chung , s .- han , c. , park , b .- , et al . 2005 , , 630 , 535 correia , a. c. m. , udry , s. , mayor , m. , laskar , j. , naef , d. , pepe , f. , queloz , d. , & santos , n. 2005 , , 440 , 751 dank , k. , & heyrovsk , d. 2015 , , 806 , 99 dominik , m. 1999 , , 349 , 108 doyle , l. r. , carter , j. a. , fabrycky , d. c. , et al . 2011 , science , 333 , 1602 erdl h. , & schneider p. 1993 , , 268 , 453 gaudi , b. s. , naber , r. m. , & sackett , p. d. 1998 , , 502 , l33 gaudi , b. s. , & sackett , p. d. 2000 , , 528 , 5 goldreich , p. , & ward , w. 1973 , , 183 , 1051 gould , a. , & loeb , a. 1992 , , 396 , 104 gould , a. , dong , s. , gaudi , b. s. , et al . 2010 , , 720 , 1073 gould , a. , udalski , a. , an , d. , et al . 2006 , , 644 , 37 gould , a. , udalski , a. , shin , i .- g . , et al . 2014 , science , 345 , 46 griest , k. , & safizadeh , n. 1998 , , 500 , 37 hayashi , c. , nakazawa , k. , & nakagawa y. 1985 , in protostars and planets ii , eds . d. c. black & m. s. matthew ( tucson : univ . arizona press ) , 1100 kim , s .- l . , lee , c .- u . , park , b .- , et al . 2015 , journal of the korean astronomical society , 49 , 37 kubas , d. , cassan , a. , dominik , m. , et al . 2008 , , 483 , 317 kuiper , g. p. 1951 proc . , 37 , 1 lee , d. w. , lee , c .- u . , park , b .- chung , s .- kim , y .- s . , , & han , c. 2008 , , 672 , 623 luhn , j. k. , penny , m. t. , & gaudi , b. s. 2016 , , 827 , 61 paczyski , b. 1986 , , 304 , 1 poleski , r. , skowron , j. , udalski , a. , et al . 2014 , , 795 , 42 pollack , j. b. , hubickyj , o. , bodenheimer , p. , lissauer , j. j. , podolak , m. , & greenzweig , y. 1996 , icarus , 124 , 62 qian , s .- b . , liu , l. , zhu , l .- y . , dai , z .- b . , lajs , e. f. , & baume , g. l. 2010 , , 401 , l34 raghavan , d. , mcalister , h. a. , henry , t. j. , et al . 2010 , , 190 , 1 rhie , s. h. 1997 , , 484 , 63 safronov , v. 1969 , evolution of the protoplanetary cloud and formation of the earth and planets ( moscow : nauka ) shin , i .- g . , han , c. , choi , j .- y . , hwang , k .- h . , jung , y .- k . , & park , h. 2015 , , 802,108 shvartzvald , y. , li , z. , udalski , a. , et al . 2016 , , submitted sumi , t. , abe , f. , bond , i. a. , et al . 2003 , , 591 , 204 szebehely , v. 1980 , celest . , 22 , 7 thebault , p. , & haghighipour , n. 2015 , planetary exploration and science : recent results and advances , eds . s. jin , n. haghighipour , w - 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chang - refsdal ( c - r ) lensing , which refers to the gravitational lensing of a point mass perturbed by a constant external shear , provides a good approximation in describing lensing behaviors of either a very wide or a very close binary lens .
c - r lensing events , which are identified by short - term anomalies near the peak of a high - magnification lensing light curves , are routinely detected from lensing surveys , but not much attention is paid to them . in this paper , we point out that c - r lensing events provide an important channel to detect planets in binaries , both in close and wide binary systems . detecting planets through the c - r lensing event channel is possible because the planet - induced perturbation occurs in the same region of the c - r lensing - induced anomaly and thus the existence of the planet can be identified by the additional deviation in the central perturbation . by presenting the analysis of the actually observed c - r lensing event ogle-2015-blg-1319
, we demonstrate that dense and high - precision coverage of a c - r lensing - induced perturbation can provide a strong constraint on the existence of a planet in the wide range of the planet parameters
. the sample of an increased number of microlensing planets in binary systems will provide important observational constraints in giving shape to the details of the planet formation scenario which has been restricted to the case of single stars .
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a good understanding of quantum chromodynamics ( qcd ) at finite temperature and baryon density is crucial for us to understand a wide range of physical phenomena . for instance , to understand the evolution of the universe in the first few seconds , one needs the knowledge of qcd phase transition at temperature @xmath3mev and very small baryon density . on the other hand , understanding the physics of neutron stars requires the knowledge of qcd at high baryon density and very low temperature @xcite . lattice simulation of qcd at finite temperature has been successfully performed in the past few decades ; however , no successful lattice simulation at high baryon density has been done due to the sign problem @xcite : the fermion determinant is not positively definite in presence of a nonzero baryon chemical potential @xmath0 . we thus look for some special cases which have a positively definite fermion determinant . one case is qcd at finite isospin chemical potential @xmath4 @xcite , where the ground state changes from a pion condensate to a bcs superfluid with quark - antiquark condensation with increasing isospin density . another case is the qcd - like theories @xcite where quarks are in a real or pseudoreal representation of the gauge group , including two - color qcd with quarks in the fundamental representation and qcd with quarks in the adjoint representation . while these cases do not correspond to the real world , they can be simulated on the lattice and may give us some information of real qcd at finite baryon density . for all these special cases , chiral perturbation theories predict a continuous quantum phase transition from the vacuum to the matter phase at baryon or isospin chemical potential equal to the pion mass , in contrast to real qcd where the phase transition takes place at @xmath0 approximately equal to the nucleon mass . the resulting matter near the quantum phase transition is a dilute bose condensate of diquarks or pions with weakly repulsive interactions @xcite . the equations of state and elementary excitations in such matter have been investigated many years ago by bogoliubov @xcite and lee , huang , and yang @xcite . bose - einstein condensation ( bec ) phenomenon is believed to widely exist in dense matter , such as pions and kaons can condense in neutron star matter if the electron chemical potential exceeds the effective mass for pions and kaons @xcite . however , the condensation of pions and kaons in neutron star matter is rather complicated due to the meson - nucleon interactions in dense nuclear medium . on the other hand , at asymptotically high density , perturbative qcd calculations show that the ground state is a weakly coupled bcs superfluid with the condensation of overlapping cooper pairs @xcite . it is interesting that the dense bcs superfluid and the dilute bose condensate have the same symmetry breaking pattern and thus are continued with one another . in condensed matter physics , this phenomenon was first discussed by eagles @xcite and leggett @xcite and is now called bec - bcs crossover . it has been successfully realized using ultracold fermionic atoms in the past few years @xcite . while the lattice simulations of two - color qcd at finite baryon chemical potential @xcite and qcd at finite isospin chemical potential @xcite have been successfully performed , we still ask for some effective models to link the physics of bose condensate and the bcs superfluidity . the chiral perturbation theories @xcite as well as the linear sigma models @xcite , which describe only the physics of bose condensate , do not meet our purpose . the nambu jona - lasinio ( njl ) model @xcite with quarks as elementary blocks , which describes well the mechanism of chiral symmetry breaking and low energy phenomenology of the qcd vacuum , is generally believed to work at low and moderate temperatures and densities @xcite . recently , this model has been used to describe the superfluid transition at finite chemical potentials @xcite for the special cases we are interested in this paper . one finds that the critical chemical potential for the superfluid transition predicted by the njl model is indeed equal to the pion mass @xcite , and the chiral and diquark condensates obtained from the mean - field calculation agree with the results from lattice simulations and chiral perturbation theories near the quantum phase transition @xcite . the njl model also predicts a bec - bcs crossover when the chemical potential increases @xcite . a natural problem arises : how can the fermionic njl model describe the weakly interacting bose condensate near the quantum phase transition ? in fact , we do not know how the repulsive interactions among diquarks or mesons enter in the pure mean - field calculations @xcite . in this paper , we will focus on this problem and show that the repulsive interaction is indeed properly included even in the mean - field calculations . this phenomenon is in fact analogous to the bcs description of the molecular condensation in strongly interacting fermi gases studied by leggett many years ago @xcite . fermionic models have been used to describe the bec - bcs crossover in cold fermi gases by the cold atom community . recently , it has been shown that we can recover the equation of state of the dilute bose condensate with correct boson - boson scattering length in the strong coupling limit , including the lee - huang - yang correction by considering the beyond - mean - field corrections @xcite . in appendix [ app1 ] , we give a summary of the many - body theoretical approach in cold atoms , which is useful for us to understand the theoretical approach and the results of this paper . in this paper , using two - color two - flavor qcd as an example and following the theoretical approach of the bec - bcs crossover in cold fermi gases @xcite , we examine how the njl model describes the weakly interacting bose condensate and the bec - bcs crossover . near the quantum phase transition point @xmath1 , we perform a ginzburg - landau expansion of the effective potential at the mean - field level , and show that the ginzburg - landau free energy is essentially the gross - pitaevskii free energy describing weakly interacting bose condensates via a proper redefinition of the condensate wave function . as a by - product , we obtain a diquark - diquark scattering length @xmath5 ( @xmath6 is the pion decay constant ) characterizing the repulsive interaction between the diquarks , which recovers the tree - level result predicted by chiral lagrangian @xcite . we also show analytically that the goldstone mode takes the same dispersion as the bogoliubov excitation in weakly interacting bose condensates , which gives a diquark - diquark scattering length identical to that in the gross - pitaevskii free energy . the mixing between the sigma meson and diquarks plays an important role in recovering the bogoliubov excitation . the results of in - medium chiral and diquark condensates predicted by chiral perturbation theory are analytically recovered . at high density , we find the superfluid matter undergoes a bec - bcs crossover at @xmath7 with @xmath8 being the mass of the sigma meson . at @xmath9 , we find that the chiral symmetry is approximated restored and the spectra of pions and sigma meson become nearly degenerated . well above the chemical potential of chiral symmetry restoration , the degenerate pions and sigma meson undergo a mott transition , where they become unstable resonances . because of the spontaneous breaking of baryon number symmetry , mesons can decay into quark pairs in the superfluid medium at nonzero momentum . the beyond - mean - field corrections are studied . the thermodynamic potential including the gaussian fluctuations is derived . it is shown that the vacuum state @xmath10 is thermodynamically consistent in the gaussian approximation , i.e. , all thermodynamic quantities keep vanishing in the regime @xmath10 even though the beyond - mean - field corrections are included . near the quantum phase transition point , we expand the fluctuation contribution to the thermodynamic potential in powers of the superfluid order parameter . to leading order , the beyond - mean - field correction is quartic and its effect is to renormalize the diquark - diquark scattering length . the correction to the mean - field result is shown to be proportional to @xmath11 . thus , our theoretical approach provides a new way to calculate the diquark - diquark or meson - meson scattering lengths in the njl model beyond the mean - field approximation . we also find that we can obtain a correct transition temperature of bose condensation in the dilute limit , including the beyond - mean - field corrections . the paper is organized as follows : in sec . [ s2 ] , we derive the general effective action of the two - color njl model at finite temperature and density , and determine the model parameters via the vacuum phenomenology . in sec . [ s3 ] , we investigate the properties of dilute bose condensate near the quantum phase transition at the mean - field level . in sec . [ s5 ] , the properties of matter at high density are discussed . beyond - mean - field corrections are studied in sec . we summarize in sec . natural units are used throughout . without loss of generality , we study in this paper two - color qcd ( the number of colors @xmath12 ) at finite baryon chemical potential @xmath0 . for vanishing current quark mass @xmath13 , two - color qcd possesses an enlarged flavor symmetry su@xmath14 [ @xmath15 is the number of flavors ] , the so - called pauli - gursey symmetry which connects quarks and antiquarks @xcite . for @xmath16 , the flavor symmetry su@xmath17 is spontaneously broken down to sp@xmath17 driven by a nonzero quark condensate @xmath18 and there arise five goldstone bosons : three pions and two scalar diquarks . for nonvanishing current quark mass , the flavor symmetry is explicitly broken , resulting in five pseudo - goldstone bosons with a small degenerate mass @xmath2 . at finite baryon chemical potential @xmath0 , the flavor symmetry su@xmath14 is explicitly broken down to su@xmath19su@xmath20u@xmath21 . further , a nonzero diquark condensate @xmath22 can form at large enough chemical potentials and breaks spontaneously the u@xmath21 symmetry . in two - color qcd , the scalar diquarks are in fact the lightest baryons , " and we expect a baryon superfluid phase with @xmath23 for @xmath24 . to construct a njl model for two - color two - flavor qcd with the above flavor symmetry , we consider a contact current - current interaction @xmath25 where @xmath26 ( @xmath27 ) are the generators of color su@xmath28 and @xmath29 is a phenomenological coupling constant . after the fierz transformation we can obtain an effective njl lagrangian density with scalar mesons and color singlet scalar diquarks @xcite , @xmath30,\end{aligned}\ ] ] where @xmath31 and @xmath32 are the charge conjugate spinors with @xmath33 and @xmath34 ( @xmath35 ) are the pauli matrices in the flavor space . the four - fermion coupling constants for the scalar mesons and diquarks are the same , @xmath36 @xcite , which ensures the enlarged flavor symmetry su@xmath14 of two - color qcd in the chiral limit @xmath37 . one can show explicitly that there are five goldstone bosons ( three pions and two diquarks ) driven by a nonzero quark condensate @xmath18 . with explicit chiral symmetry broken @xmath38 , pions and diquarks are also degenerate , and their mass @xmath2 can be determined via the standard method for the njl model @xcite . the partition function of the two - color njl model ( [ njl ] ) at finite temperature @xmath39 and baryon chemical potential @xmath0 is @xmath40[dq]\exp\left[\int dx\left({\cal l}_{\text{njl}}+\frac{\mu_{\text b}}{2}\bar{q}\gamma_{0}q\right)\right],\end{aligned}\ ] ] where we adopt the finite temperature formalism with @xmath41 , @xmath42 , and @xmath43 . the partition function can be bosonized after introducing the auxiliary boson fields @xmath44 for mesons and @xmath45 for diquarks . with the help of the nambu - gorkov representation @xmath46 , the partition function can be written as @xmath47[d\psi][d\sigma][d\mbox{\boldmath{$\pi$}}][d\phi^\dagger][d\phi]\exp\left(-{\cal a_{\text{eff}}}\right),\end{aligned}\ ] ] where the action @xmath48 is given by @xmath49 with the inverse quark propagator defined as @xmath50 here @xmath51 . after integrating out the quarks , we can reduce the partition function to @xmath52[d\mbox{\boldmath{$\pi$}}][d\phi^\dagger][d\phi]\exp\big\{-{\cal s}_{\text{eff}}[\sigma,\mbox{\boldmath{$\pi$}},\phi^\dagger,\phi]\big\}$ ] , where the bosonized effective action @xmath53 is given by @xmath54=\int dx\frac{\sigma^2(x)+\mbox{\boldmath{$\pi$}}^2(x)+|\phi(x)|^2}{4g}-\frac{1}{2}\text{tr}\ln{\bf g}^{-1}(x , x^\prime).\end{aligned}\ ] ] here the trace @xmath55 is taken over color , flavor , spin , nambu - gorkov and coordinate ( @xmath56 and @xmath57 ) spaces . the thermodynamic potential density of the system is given by @xmath58 . the effective action @xmath53 as well as the thermodynamic potential @xmath59 can not be evaluated exactly in our @xmath60 dimensional case . in this work , we firstly consider the saddle point approximation , i.e. , the mean - field approximation . then we investigate the fluctuations around the mean field . + _ ( i)mean - field approximation . _ in this approximation , all bosonic auxiliary fields are replaced by their expectation values . to this end , we write @xmath61 , @xmath62 and set @xmath63 . while @xmath64 can be set to be real , we do not do this first in our derivations . we will show in the following that all physical results depend only on @xmath65 . the zeroth order or mean - field effective action reads @xmath66.\ ] ] here and in the following @xmath67 with @xmath68 being the fermion matsubara frequency , and @xmath69 with @xmath70 . the inverse of the nambu - gorkov quark propagator @xmath71 is given by @xmath72 with the effective quark dirac mass @xmath73 . the mean - field thermodynamic potential @xmath74 can be evaluated as @xmath75\end{aligned}\ ] ] with the definitions of the function @xmath76 and the bcs - like quasiparticle dispersions @xmath77 where @xmath78 . the signs @xmath79 correspond to quasiquark and quasi - antiquark excitations , respectively . the integral over the quark momentum @xmath80 is divergent at large @xmath81 , and some regularization scheme should be adopted . in this paper , we employ a hard three - momentum cutoff @xmath82 . the physical values of the variational parameters @xmath83 ( or @xmath84 ) and @xmath64 should be determined by the saddle point condition @xmath85}{\delta \upsilon}=0,\ \ \ \ \ \frac{\delta{\cal s}_{\text{eff}}^{(0)}[\upsilon,\delta]}{\delta \delta}=0,\end{aligned}\ ] ] which minimizes the mean - field effective action @xmath86 . one can show that the saddle point condition is equivalent to the following green function relations @xmath87\ , \end{aligned}\ ] ] where the matrix elements of @xmath88 are explicitly given by @xmath89 with the help of the massive energy projectors @xcite @xmath90\ .\ ] ] here we have defined the notation @xmath91 for convenience . ( ii)derivative expansion . _ next , we consider the fluctuations around the mean field , corresponding to the bosonic collective excitations . making the field shifts for the auxiliary fields , @xmath92 we can express the total effective action as @xmath93}.\end{aligned}\ ] ] here @xmath94 is the fourier transformation of @xmath95 , and @xmath96 is defined as @xmath97 with the help of the derivative expansion @xmath98}=\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\text{tr}[{\cal g}\sigma]^n,\end{aligned}\ ] ] we can calculate the effective action in powers of the fluctuations @xmath99 . the first order effective action @xmath100 which includes linear terms of the fluctuations should vanish exactly , since the expectation value of the fluctuations should be exactly zero . in fact , @xmath100 can be evaluated as @xmath101\sigma(x)\nonumber\\ & + & \frac{1}{2}\text{tr}\left[i\gamma_5\left({\cal g}_{11}\mbox{\boldmath{$\tau$}}+{\cal g}_{22}\mbox{\boldmath{$\tau$}}^{\text t}\right)\right]\cdot\mbox{\boldmath{$\pi$}}(x)\nonumber\\ & + & \left[\frac{\delta}{2g}+\frac{1}{2}\text{tr}\left(i\gamma_5\tau_2 t_2{\cal g}_{12}\right)\right]\phi^\dagger(x)\nonumber\\ & + & \left[\frac{\delta^\dagger}{2g}+\frac{1}{2}\text{tr}\left(i\gamma_5\tau_2 t_2{\cal g}_{21}\right)\right]\phi(x)\bigg\}.\end{aligned}\ ] ] we observe that the coefficients of @xmath102 is automatically zero after taking the trace in dirac spin space . the coefficients of @xmath103 and @xmath104 vanish once the quark propagator takes the mean - field form and @xmath105 take the physical values satisfying the saddle point condition . thus , in the present approach , the saddle point condition plays a crucial role in having a vanishing linear term in the expansion . the quadratic term @xmath106 corresponds to the gaussian fluctuations . it reads @xmath107.\end{aligned}\ ] ] for the convenience of our investigation in the following , we will use the form of @xmath106 in the momentum space . after the fourier transformation , it can be written as @xmath108\bigg\}.\end{aligned}\ ] ] where @xmath109 with @xmath110 being the boson matsubara frequency and @xmath111 . here @xmath112 is the fourier transformation of the field @xmath113 , and @xmath114 is defined as @xcite @xmath115 + _ ( iii)gaussian fluctuations . _ after taking the trace in nambu - gorkov space , we find that @xmath106 can be written in the following bilinear form @xmath116 the matrix @xmath117 takes the following nondiagonal form @xmath118 the polarization functions @xmath119 ( @xmath120 ) are one - loop susceptibilities composed of the matrix elements the nambu - gorkov quark propagator , and can be expressed as @xmath121,\ \ \ \ \pi_{22}(q)=\frac{1}{2}\sum_k\text{tr}\left[{\cal g}_{11}(k)\gamma{\cal g}_{22}(p)\gamma\right],\nonumber\\ & & \pi_{12}(q)=\frac{1}{2}\sum_k\text{tr}\left[{\cal g}_{12}(k)\gamma{\cal g}_{12}(p)\gamma\right],\ \ \ \ \pi_{21}(q)=\frac{1}{2}\sum_k\text{tr}\left[{\cal g}_{21}(k)\gamma{\cal g}_{21}(p)\gamma\right],\nonumber\\ & & \pi_{33}(q)=\frac{1}{2}\sum_k\text{tr}\left[{\cal g}_{11}(k){\cal g}_{11}(p)+{\cal g}_{22}(k){\cal g}_{22}(p)+{\cal g}_{12}(k){\cal g}_{21}(p)+{\cal g}_{21}(k){\cal g}_{12}(p)\right],\nonumber\\ & & \pi_{13}(q)=\frac{1}{2}\sum_k\text{tr}\left[{\cal g}_{12}(k)\gamma{\cal g}_{11}(p)+{\cal g}_{22}(k)\gamma{\cal g}_{12}(p)\right],\ \ \ \pi_{31}(q)=\frac{1}{2}\sum_k\text{tr}\left[{\cal g}_{21}(k){\cal g}_{11}(p)\gamma+{\cal g}_{22}(k){\cal g}_{21}(p)\gamma\right],\nonumber\\ & & \pi_{23}(q)=\frac{1}{2}\sum_k\text{tr}\left[{\cal g}_{11}(k)\gamma{\cal g}_{21}(p)+{\cal g}_{21}(k)\gamma{\cal g}_{22}(p)\right],\ \ \ \pi_{32}(q)=\frac{1}{2}\sum_k\text{tr}\left[{\cal g}_{11}(k){\cal g}_{12}(p)\gamma+{\cal g}_{12}(k){\cal g}_{22}(p)\gamma\right],\end{aligned}\ ] ] where @xmath122 , @xmath123 and the trace is taken over color , flavor and spin spaces . using the fact that @xmath124 and @xmath125 , we can easily show that @xmath126 therefore , only five of the polarization functions are independent . at @xmath127 , their explicit form is shown in appendix [ app2 ] . for general case , we can show that @xmath128 and @xmath129 . thus , in the normal phase where @xmath130 , the matrix @xmath117 recovers the diagonal form . the off - diagonal elements @xmath131 and @xmath132 represents the mixing between the sigma meson and diquarks . at large chemical potentials where the chiral symmetry is approximately restored , @xmath133 , this mixing can be safely neglected . on the other hand , the matrix @xmath134 of the pion sector is diagonal and proportional to the identity matrix , i.e. , @xmath135 , \ \ \ \text{i , j}=1,2,3.\ ] ] this means pions are eigen mesonic excitations even in the superfluid phase . the polarization function @xmath136 is given by @xmath137.\end{aligned}\ ] ] its explicit form at @xmath127 is shown in appendix [ app2 ] . we find that @xmath138 and @xmath139 is different only to a term proportional to @xmath140 . thus , at high density where @xmath141 , the spectra of pions and sigma meson become nearly degenerate which represents the approximate restoration of chiral symmetry . + _ ( iv)goldstone s theorem . _ the u@xmath21 baryon number symmetry is spontaneously broken by the nonzero diquark condensate @xmath22 in the superfluid phase , resulting in one goldstone boson . in our model , this is ensured by the condition @xmath142 . from the explicit form of the polarization functions shown in appendix [ app2 ] , we find that this condition holds if and only if the saddle point condition ( [ saddle ] ) for @xmath84 and @xmath64 is satisfied . we thus emphasize that in our theoretical framework , the condensates @xmath84 and @xmath64 should be determined by the saddle point condition , and the beyond - mean - field corrections are possible only through the thermodynamics , i.e. , the equations of state . for a better understanding our derivation in the following , it is useful to review the vacuum state at @xmath143 . in the vacuum , it is evident that @xmath130 and the mean - field effective potential @xmath144 can be evaluated as @xmath145 the physical value of @xmath83 , denoted by @xmath146 , satisfies the saddle point condition @xmath147 and minimizes @xmath144 . the meson and diquark excitations can be obtained from @xmath148 , which in the vacuum can be expressed as @xmath149,\end{aligned}\ ] ] where @xmath150 are the real and imaginary parts of @xmath151 , respectively . the inverse propagators in vacuum can be expressed in a symmetrical form @xcite @xmath152 where @xmath153 , @xmath154 , and the function @xmath155 is defined as @xmath156 with @xmath157 . keeping in mind that @xmath146 satisfies the saddle point condition , we find that the pions and diquarks are nambu - goldstone bosons in the chiral limit , corresponding to the symmetry breaking pattern su@xmath158sp@xmath17 . using the gap equation of @xmath146 , we find that the masses of mesons and diquarks can be determined by the equation @xmath159 since the @xmath160 dependence of the function @xmath155 is very weak , we find @xmath161 and @xmath162 . since pions and diquarks are deep bound states , their propagators can be well approximated by @xmath163 with @xmath164 . the pion decay constant @xmath6 can be determined by the matrix element of the axial current , @xmath165\nonumber\\ & = & 2n_cn_fg_{\pi qq}m_*q_\mu i(q^2)\delta_{\text{ij}}.\end{aligned}\ ] ] here @xmath166 . thus , the pion decay constant can be expressed as @xmath167 finally , together with ( [ mesonmass ] ) and ( [ piondecay ] ) , we recover the well - known gell - mann oakes renner relation @xmath168 . .model parameters ( 3-momentum cutoff @xmath82 , coupling constant @xmath169 , and current quark mass @xmath13 ) and related quantities ( quark condensate @xmath170 , constituent quark mass @xmath146 and pion mass @xmath2 ) for the two - flavor two - color njl model ( [ njl ] ) . the pion decay constant is fixed to be @xmath171 mev . [ cols="^,^,^,^,^,^,^ " , ] on the other hand , we find from the explicit forms of the meson propagators in appendix [ app2 ] that the decay process @xmath172 is also possible at @xmath173 ( even though @xmath174 is small ) due to the presence of superfluidity . thus , we have another unusual mott transition in the superfluid phase . notice that this process is not in contradiction to the baryon number conservation law , since the u@xmath21 baryon number symmetry is spontaneously broken in the superfluid phase . quantitatively , this transition occurs when the meson mass becomes larger than the two - particle continuum @xmath175 for the decay process @xmath172 at @xmath176 . in this case , we have @xmath177 the two - particle continuum @xmath175 is also shown in fig.[fig9 ] . we find that the unusual mott transition does occur at another chemical potential @xmath178 which is also sensitive to the value of @xmath146 . the values of @xmath179 for the four model parameter sets are also shown in table.[mottmu ] . for reasonable model parameter sets , this value is in the range @xmath180 . this process can also occur in the 2sc phase of quark matter in the @xmath181 case @xcite . in the 2sc phase , the symmetry breaking pattern is su@xmath182u@xmath183su@xmath184 where the generator of the residue baryon number symmetry @xmath185 is @xmath186 corresponding to the unpaired blue quarks . thus the baryon number symmetry for the paired red and green quarks are broken and our results can be applied . to show this explicitly , we write down the explicit form of the polarization function for pions in the 2sc phase @xcite @xmath187,\ ] ] where @xmath188 is the propagator for the unpaired blue quarks . here @xmath189 is given by ( [ pionpo ] ) ( the effective quarks mass @xmath83 and the pairing gap @xmath64 should be given by the @xmath181 case of course ) and corresponds to the contribution from the paired red and green sectors . the second term is the contribution from the unpaired blue quarks . therefore , the unusual decay process is only available for the paired quarks . the investigations in sec . [ s3 ] and [ s4 ] are restricted in the mean - field approximation , even though the bosonic collective excitations are studied . in this section , we will include the gaussian fluctuations in the thermodynamic potential , and thus really go beyond the mean field . the scheme of going beyond the mean field is somewhat like those done in the study of finite temperature thermodynamics of the njl model @xcite ; however , in this paper we will focus on the beyond - mean - field corrections at zero temperature , i.e. , the pure quantum fluctuations . we will first derive the thermodynamic potential beyond the mean field which is valid at arbitrary chemical potential and temperature , and then briefly discuss the beyond - mean - field corrections near the quantum phase transition . the numerical calculations are deferred for future studies . in the gaussian approximation , the partition function can be expressed as @xmath190[d\mbox{\boldmath{$\pi$}}][d\phi^\dagger][d\phi]\exp\big(-{\cal s}_{\text{eff}}^{(2)}\big).\end{aligned}\ ] ] integrating out the gaussian fluctuations , we can express the total thermodynamic potential as @xmath191 where the contribution from the gaussian fluctuations can be written as @xmath192.\end{aligned}\ ] ] however , there is a problem with the above expression , since it is actually ill - defined : the sum over the boson matsubara frequency is divergent and we need appropriate convergent factors to make it meaningful . in the simpler case without superfluidity , the convergent factor is simply given by @xmath193 @xcite . in our case , the situation is somewhat different due to the introduction of the nambu - gorkov spinors . keep in mind that in the equal time limit , there are additional factors @xmath194 for @xmath195 and @xmath196 for @xmath197 . therefore , to get the proper convergent factors for @xmath198 , we should keep these factors when we make the sum over the fermion matsubara frequency @xmath199 in evaluating the polarization functions @xmath119 and @xmath138 . the problem in the expression of @xmath198 is thus from the opposite convergent factors for @xmath200 and @xmath201 . from the above arguments , we find that there is a factor @xmath193 for @xmath200 and @xmath202 for @xmath201 . keep in mind that the matsubara sum @xmath203 is converted to a standard contour integral ( @xmath204 ) . the convergence for @xmath205 is automatically guaranteed by the bose distribution function @xmath206 , we thus should treat only the problem for @xmath207 . to this end , we write the first term of @xmath198 as @xmath208.\end{aligned}\ ] ] using the fact that @xmath209 , we obtain @xmath210e^{i\nu_m 0^+}.\end{aligned}\ ] ] therefore , the well - defined form of @xmath198 is given by the above formula together with the other term @xmath211 associated with a factor @xmath193 . the matsubara sum can be written as the contour integral via the theorem @xmath212 , where @xmath213 runs on either side of the imaginary @xmath214 axis , enclosing it counterclockwise . distorting the contour to run above and below the real axis , we obtain @xmath215,\end{aligned}\ ] ] where the scattering phases are defined as @xmath216.\end{aligned}\ ] ] keep in mind the pressure of the vacuum should be zero , the physical thermodynamic potential at finite temperature and chemical potential should be defined as @xmath217 as we have shown in the mean - field theory , at @xmath127 , the vacuum state is restricted in the region @xmath10 . in this region , all thermodynamic quantities should keep zero , no matter how large the value of @xmath0 is . while this should be an obvious physical conclusion , it is important to check whether our beyond - mean - field theory satisfies this condition . notice that the physical thermodynamic potential is defined as @xmath218 , we therefore should prove that the thermodynamic potential @xmath219 keeps a constant in the region @xmath10 . for the mean - field part @xmath220 , the proof is quite easy . because of the fact that @xmath221 , the solution for @xmath83 is always given by @xmath222 . thus @xmath220 keeps its value at @xmath223 in the region @xmath224 . now we turn to the complicated part @xmath198 . since @xmath130 , all the off - diagonal elements of @xmath117 vanishes , and @xmath198 is reduced to @xmath225e^{i\nu_m 0^+}\nonumber\\ & + & \frac{3}{2}\sum_q\ln\left[\frac{1}{2g}+\pi_{\pi}(q)\right]e^{i\nu_m 0^+}\nonumber\\ & + & \sum_q\ln\left[\frac{1}{4g}+\pi_{\text d}(q)\right]e^{i\nu_m 0^+},\end{aligned}\ ] ] where @xmath226 and we should set @xmath130 and @xmath222 in evaluating the polarization functions . first , we can easily show that the contributions from the sigma meson and pions do not have explicit @xmath0 dependence and thus keep the same values as those at @xmath223 . in fact , since the effective quark mass @xmath83 keeps its vacuum value @xmath146 guaranteed by the mean - field part , all the @xmath0 dependence in @xmath227 is included in the fermi distribution functions @xmath228 . since @xmath229 , they vanish automatically at @xmath127 . in fact , from the explicit expressions for @xmath227 in appendix [ app2 ] , we can check that there is no @xmath0 independence in @xmath227 . the diquark contribution , however , has an explicit @xmath230 dependence through the combination @xmath231 in the polarization function @xmath232 . the diquark contribution ( at @xmath127 ) can be written as @xmath233.\end{aligned}\ ] ] making a shift @xmath234 , and noticing that fact @xmath235 , we obtain @xmath236 to show the above quantity is in fact @xmath0 independent , we separate it into a pole part and a continuum part . there is a well - defined two - particle continuum @xmath237 for pions at arbitrary momentum @xmath238 , @xmath239 the pion propagator has two symmetric poles @xmath240 when @xmath238 satisfies @xmath241 . thus in the region @xmath242 , the scattering phase @xmath243 can be analytically evaluated as @xmath244.\end{aligned}\ ] ] since @xmath245 , the thermodynamic potential @xmath246 can be separated as @xmath247-\sum_{\bf q}\int_{-\infty}^{-e_c({\bf q})}\frac{d\omega}{\pi}\delta_{\pi}(\omega,{\bf q}),\end{aligned}\ ] ] which is indeed @xmath0 independent . notice that in the first term the integral over @xmath238 is restricted in the region @xmath248 where @xmath249 is defined as @xmath250 . in conclusion , we have shown that the thermodynamic potential @xmath59 in the gaussian approximation keeps a constant in the vacuum state , i.e. , at @xmath10 and at @xmath127 . all other thermodynamic quantities such as the baryon number density keep zero in the vacuum . the subtraction term @xmath251 in the gaussian approximation can be expressed as @xmath252\nonumber\\ & & \ \ \ -\sum_{\bf q}\int_{-\infty}^{-e_c({\bf q})}\frac{d\omega}{2\pi}\left[\delta_\sigma(\omega,{\bf q})+5\delta_{\pi}(\omega,{\bf q})\right].\end{aligned}\ ] ] now we consider the beyond - mean - field corrections near the quantum phase transition point @xmath1 . notice that the effective quark mass @xmath83 and the diquark condensate @xmath64 are determined at the mean - field level , and the beyond - mean - field corrections are possible only through the equations of state . formally , the gaussian contribution to the thermodynamic potential @xmath198 is a function of @xmath253 and @xmath254 , i.e. , @xmath255 . in the superfluid phase , the total baryon density including the gaussian contribution can be evaluated as @xmath256 where the mean - field part is simply given by @xmath257 and the gaussian contribution can be expressed as @xmath258 the physical values of @xmath83 and @xmath65 should be determined by their mean - field gap equations . in fact , from the gap equations @xmath259 and @xmath260 , we obtain @xmath261 thus , we can obtain the derivatives @xmath262 and @xmath263 analytically . finally , @xmath264 is a continuous function of @xmath0 guaranteed by the properties of second order phase transition , and we have @xmath265 . next we focus on the beyond - mean - field corrections near the quantum phase transition . since the diquark condensate @xmath64 is vanishingly small , we can expand the gaussian part @xmath198 in powers of @xmath65 . notice that @xmath0 and @xmath83 can be evaluated as functions of @xmath65 from the ginzburg - landau potential and mean - field gap equations . thus to order @xmath266 , the expansion takes the form @xmath267 where the expansion coefficient @xmath268 is defined as @xmath269 using the definition of @xmath270 , we find that @xmath268 can be related to @xmath270 by @xmath271 thus , the coefficient @xmath268 vanishes , and the leading order of the expansion should be @xmath272 . as shown above , to leading order , the expansion of @xmath198 can be formally expressed as @xmath273 the method to derive the exact expression of the numerical factor @xmath274 is shown in appendix [ app3 ] . notice that the factor @xmath274 is in fact @xmath0 independent , thus the total baryon density to leading order is @xmath275 near the quantum phase transition point , the mean - field contribution is @xmath276 from the gross - pitaevskii free energy . the last term can be evaluated using the analytical result @xmath277 on the other hand , the total pressure @xmath278 can be expressed as @xmath279 thus we find that the leading order quantum corrections are totally included in the numerical factor @xmath274 . setting @xmath280 , we recover the mean - field results obtained in sec . [ s3 ] . including the quantum fluctuations , the equations of state shown in ( [ eos ] ) are modified to be @xmath281 this means , to leading order , the effect of quantum fluctuations is giving a correction to the diquark - diquark scattering length . the renormalized scattering length is @xmath282 generally , we have @xmath283 and the renormalized scattering length is smaller than the mean - field result . an exact calculation of the numerical factor @xmath274 can be performed using the method shown in appendix [ app3 ] . however , this needs huge numerical power and we defer it to future work @xcite . in this paper we will give an analytical estimation of @xmath274 based on the fact that the quantum fluctuations are dominated by the gapless goldstone mode . to this end , we approximate the gaussian contribution @xmath198 as @xmath284,\end{aligned}\ ] ] where @xmath285 is given by ( [ dipro ] ) and can be approximated by ( [ diproa ] ) . subtracting the value of @xmath198 at @xmath1 with @xmath130 , and using the result @xmath286 from the gross - pitaevskii equation , we find that @xmath274 can be evaluated as @xmath287 where the numerical factors @xmath288 and @xmath289 are given by @xmath290 ^ 2}.\end{aligned}\ ] ] here the dimensionless notations @xmath291 and @xmath292 are defined as @xmath293 and @xmath294 respectively . notice that the integral over @xmath292 is divergent and hence such an estimation has no prediction power due to the fact that the njl model is nonrenormalizable . however , regardless of the numerical factor @xmath295 , we find that @xmath296 . thus , the correction should be small in the nonlinear sigma model limit @xmath297 . while the effect of the gaussian fluctuations at zero temperature is to give a small correction to the diquark - diquark scattering length and the equations of state , it can be significant at finite temperature . in fact , as the temperature approaches the critical value of superfluidity , the gaussian fluctuations should dominate . in this part , we will show that to get a correct critical temperature in terms of the baryon density @xmath298 , we must go beyond the mean field . the situation is analogous to the nozieres schmitt - rink treatment of molecular condensation in strongly interacting fermi gases @xcite . the transition temperature @xmath299 is determined by the thouless criterion @xmath300 which can be shown to be consistent with the saddle point condition @xmath301 . its explicit form is a bcs - type gap equation @xmath302 meanwhile , the dynamic quark mass @xmath83 satisfies the mean - field gap equation @xmath303 to obtain the transition temperature as a function of @xmath298 , we need the so - called number equation given by @xmath304 , which includes both the mean - field contribution @xmath305 $ ] and the gaussian contribution @xmath306 . at the transition temperature where @xmath130 , @xmath198 can be expressed as @xmath307,\end{aligned}\ ] ] where the scattering phases are defined as @xmath308 $ ] for the diquarks , @xmath309 $ ] for the sigma meson and @xmath310 $ ] for the pions . obviously , the polarization functions should take their forms at finite temperature in the normal phase . the transition temperature @xmath299 at arbitrary baryon number density @xmath298 can be determined numerically via solving simultaneously the gap and number equations . however , in the dilute limit @xmath311 which we are interested in this section , analytical result can be achieved . keep in mind that @xmath312 when @xmath313 , we find that the fermi distribution functions @xmath314 vanish exponentially ( since @xmath315 ) and we obtain @xmath1 and @xmath222 from the gap eqs . ( [ tc1 ] ) and ( [ tc2 ] ) , respectively . meanwhile the mean - field contribution of the density @xmath316 can be neglected and the total density @xmath298 is thus dominated by the gaussian part @xmath270 . when @xmath317 we can show that @xmath318 and @xmath319 are independent of @xmath0 , and the number equation is reduced to @xmath320 since @xmath321 , the inverse diquark propagator can be reduced to @xmath322 in ( [ dipro ] ) . thus the scattering phase @xmath323 can be well approximated by @xmath324 $ ] with @xmath325 . therefore , the number equation can be further reduced to the well - known equation for ideal bose - einstein condensation , @xmath326\bigg|_{\mu_{\text b}=m_\pi}.\end{aligned}\ ] ] since the above equation is valid only in the low density limit @xmath313 , the critical temperature is thus given by the nonrelativistic result @xmath327^{2/3}.\end{aligned}\ ] ] at finite density but @xmath328 , there exists a correction to @xmath299 which is proportional to @xmath329@xcite . such a correction is hard to handle analytically in our model since we should consider simultaneously the corrections to @xmath83 and @xmath0 , as well as the contribution from the sigma meson and pions . in summary , we have examined the njl model description of weakly interacting bose condensate and bec - bcs crossover in qcd - like theories at finite baryon density . our main conclusions are as follows : + ( 1)near the quantum phase transition point @xmath1 , we have performed a ginzburg - landau expansion of the effective potential . at the mean - field level , the ginzburg - landau free energy is essentially the gross - pitaevskii free energy describing weakly repulsive bose condensates after a proper redefinition of the condensate wave function . the obtained diquark - diquark scattering length reads @xmath330 , which recovers the tree - level result predicted by chiral lagrangian . + ( 2)we have analytically shown that the goldstone mode near the quantum phase transition point takes the same dispersion as the bogoliubov excitation in weakly interacting bose condensates , which gives a diquark - diquark scattering length identical to that in the gross - pitaevskii free energy . the mixing between the sigma meson and the diquarks plays an important role in recovering the bogoliubov dispersion . + ( 3)the results of baryon number density and in - medium chiral and diquark condensates predicted by chiral perturbation theory are analytically recovered near the quantum phase transition point in the njl model . + ( 4)at high density , the superfluid matter undergoes a bec - bcs crossover at @xmath331 . at @xmath9 , the chiral symmetry is approximated restored and the spectra of pions and sigma meson become nearly degenerate . well above the chemical potential of chiral symmetry restoration , the degenerate pions and sigma meson undergo a mott transition , where they become unstable resonances . because of the spontaneous breaking of baryon number symmetry , mesons can decay into quark pairs in the superfluid medium at nonzero momentum . + ( 5)the general theoretical framework of the thermodynamics beyond the mean field is established . it is shown that the vacuum state in the region @xmath10 is thermodynamically consistent in the gaussian approximation , i.e. , all thermodynamic quantities keep vanishing for @xmath10 even though the gaussian fluctuations are included . + ( 6)near the quantum phase transition point , we find that the effect of the leading order beyond - mean - field correction is to renormalize the diquark - diquark scattering length . the correction to the mean - field result is estimated to be proportional to @xmath11 . our theoretical approach provides a new way to calculate the diquark - diquark or meson - meson scattering lengths in the njl model beyond the mean - field approximation . we also find that we can obtain a correct transition temperature of bose condensation in the dilute limit once the beyond - mean - field corrections are included . our studies can be generalized to describe pion condensation at finite isospin chemical potential @xmath4 @xcite and kaon condensation at finite strangeness chemical potential @xmath332 @xcite . in the njl model , pion condensation is shown to occur at @xmath333 when @xmath334 , and kaon is shown to condense at @xmath335 when @xmath336 @xcite . the generalization to pion condensation is straightforward . the obtained ginzburg - landau and gross - pitaevskii free energies are the same as those derived in this paper , if we replace @xmath337 . the results are valid both for @xmath12 and @xmath181 cases . at the mean - field level , the results for diquark condensation at @xmath12 and pion condensation at @xmath181 are formally identical in the njl model . significant difference may appear if we consider the beyond - mean - field corrections , since for the @xmath181 case the scalar diquarks are not pseudo - goldstone bosons . a calculation of the pion - pion scattering length in the @xmath338 channel can be performed within our theoretical framework . the calculations of kaon condensation and kaon - kaon scattering length are also possible , but somewhat complicated due to the large mass difference between the light and strange quarks . we are also interested in how the beyond - mean - field corrections modify the superfluid equations of state . as we learn from the knowledge of bec - bcs crossover in cold fermi gases , the superfluid equations of state can be strongly modified in the crossover regime @xcite , corresponding to the moderate baryon density in our case . this issue is also important to the color - superconducting quark matter @xcite at moderate density , i.e. , for quark chemical potential around @xmath339mev where the pairing gap can be of order @xmath340mev@xmath341 . the numerical works are in progress . * acknowledgement : * the work is supported by the alexander von humboldt foundation . l. he would like to thank dr . tomas brauner for valuable discussions and critical reading of the manuscript , and prof . dirk rischke for encouragement during the work . in this appendix , we briefly review the theory of molecular bose condensation in two - component fermi gases in the strong coupling limit . while there exist many theoretical approaches @xcite to deal with this problem , we employ the field theoretical approach @xcite parallel to that used in this paper . the lagrangian density of the system can be written as @xmath342 where @xmath343 denote the two - component ( nonrelativistic ) fermion fields with equal masses @xmath344 and chemical potentials @xmath345 . the gas is assumed to be dilute , and the coupling constant @xmath346 can be related to the s - wave fermion - fermion scattering length @xmath347 as @xmath348 where @xmath349 . in the dilute limit , we can take the limit @xmath350 in the final result . performing the hubbard - stratonovich transformation with the auxiliary boson field @xmath351 , and defining the nambu - gorkov representation @xmath352 , we can evaluate the partition function of the system as @xmath353[d\psi][d\phi^\dagger][d\phi]\exp{\left(-{\cal a}_{\text{eff}}\right)}$ ] , where @xmath354 and the inverse fermion propagator @xmath355 is given by @xmath356 then integrating out the fermionic degree of freedom , we get @xmath357[d\phi ] \exp{\left(-{\cal s}_{\text{eff}}\right)}$ ] where the bosonized effective action reads @xmath358=\int dx\frac{|\phi(x)|^2}{g}-\text{tr}\ln { \bf g}^{-1}(x , x^\prime).\end{aligned}\ ] ] first , we consider the mean - field theory where the auxiliary boson field @xmath359 is replaced by its expectation value @xmath360 . in the strong coupling limit @xmath361 , the fermion chemical potential @xmath345 approaches @xmath362 with @xmath363 being the molecular binding energy . since the pairing gap @xmath364 , we can expand the effective action in powers of @xmath365 , which resulting in a ginzburg - landau free energy functional @xmath366=\int dx\bigg[\delta^\dagger(x)\left ( \kappa\frac{\partial}{\partial\tau}-\gamma\mbox{\boldmath{$\nabla$}}^2\right)\delta(x)\nonumber\\ & & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + \ \ \alpha|\delta(x)|^2+\frac{1}{2}\beta|\delta(x)|^4\bigg].\end{aligned}\ ] ] the coefficients @xmath367 of the potential terms can be obtained from the mean - field thermodynamic potential @xmath368 $ ] which can be evaluated as @xmath369 where @xmath370 and @xmath371 . after a simple algebra , the coefficients @xmath372 and @xmath373 can be evaluated as @xmath374 from the expression of @xmath372 , we find that a quantum phase transition from vacuum to bose condensation takes place at @xmath375 . thus , near the phase transition , @xmath372 can be simplified as @xmath376 where @xmath377 is the boson chemical potential . further , setting @xmath378 , @xmath373 can be simplified as @xmath379 the coefficients @xmath380 of the kinetic terms can be obtained from the inverse boson propagator @xmath381 with @xmath130 . it can be evaluated as @xmath382 in the strong coupling limit , it can be well approximated as@xcite @xmath383 in summary , if we define the new condensate wave function @xmath384 by @xmath385 the ginzburg - landau free energy can be reduced to the gross - pitaevskii free energy of dilute bose gases , @xmath386=\int dx\bigg[\psiup^\dagger(x)\left ( \frac{\partial}{\partial\tau}-\frac{\mbox{\boldmath{$\nabla$}}^2}{2m_{\text b}}\right)\psiup(x)\nonumber\\ & & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\ \ \mu_{\text b}|\psiup(x)|^2+\frac{1}{2}\frac{4\pi a_{\text{bb}}}{m_{\text b}}|\psiup(x)|^4\bigg],\end{aligned}\ ] ] where @xmath387 is the boson mass and @xmath388 is the boson - boson scattering length . since @xmath361 , the interactions among the composite bosons are repulsive and weak . to study the beyond - mean - field corrections , we consider the fluctuations around the mean field . making the field shift @xmath389 , we can expand the effective action @xmath53 in powers of the fluctuations . the zeroth order term @xmath390 is just the mean - field result , and the linear terms vanish automatically guaranteed by the saddle point condition for @xmath64 . the quadratic terms , corresponding to gaussian fluctuations , can be evaluated as @xmath391 where the inverse boson propagator @xmath117 is given by @xmath392 and @xmath393 here the fermion green function @xmath88 is defined as @xmath394 $ ] and the bcs distribution functions @xmath395 and @xmath396 are used . in the strong coupling limit where @xmath397 , the matrix elements of @xmath117 can be analytically evaluated . we have @xcite @xmath398 where @xmath399 and @xmath400 is the minimum of the gross - pitaevskii free energy . together with the mean - field result for the boson density @xmath401 , we can show that the goldstone mode takes a dispersion relation given by @xmath402 which is just the bogoliubov excitation in a dilute bose condensate . to evaluate the thermodynamic potential beyond the mean field , we express the partition function in the gaussian approximation as @xmath403[d\phi]\exp\big(-{\cal s}_{\text{eff}}^{(2)}\big).\end{aligned}\ ] ] integrating out the gaussian fluctuations , the total thermodynamic potential can be expressed as @xmath404 where the contribution from the gaussian fluctuations can be evaluated as @xcite @xmath405e^{i\nu_m0^+}.\end{aligned}\ ] ] near the quantum phase transition point @xmath406 , we can expand @xmath198 in powers of @xmath65 . because of the properties of second order phase transition , the terms of order @xmath266 vanish . to leading order , the result is @xcite @xmath407 where the numerical factor @xmath408 . from the gross - pitaevskii free energy , we find that the pressure @xmath278 in the mean - field approximation can be expressed as @xmath409 thus , to leading order , the beyond - mean - field corrections renormalize the boson - boson scattering length @xmath410 . the new renormalized scattering length reads @xmath411 notice that this result is quite close to the exact result for the four body problem of @xmath412 @xcite . this means the quantum fluctuations are almost correctly included in the present theoretical approach . further , going beyond the leading order we find that we can fit @xmath198 to the functional form @xcite @xmath413 where @xmath414 and the dimensionless factors @xmath415 can be numerically determined . solving for the molecular chemical potential @xmath416 one obtains @xmath417,\end{aligned}\ ] ] with the coefficient @xmath418@xcite which is @xmath419 smaller than the lee - huang - yang result @xmath420 @xcite . in this appendix , we evaluate the explicit forms of the one - loop susceptibilities @xmath119 ( @xmath421 ) and @xmath136 . at arbitrary temperature , their expressions are rather huge . however , at @xmath127 , they can be written in rather compact forms . for convenience , we define @xmath422 in this appendix . _ ( i ) diquark sector . _ first , the polarization functions @xmath423 and @xmath424 can be evaluated as @xmath425,\nonumber\\ \pi_{12}(q)&=&n_cn_f\sum_{\bf k}\bigg[\left(\frac{u_{\bf k}^-v_{\bf k}^-u_{\bf p}^-v_{\bf p}^-}{i\nu_m+e_{\bf k}^-+e_{\bf p}^-}-\frac{u_{\bf k}^-v_{\bf k}^-u_{\bf p}^-v_{\bf p}^-}{i\nu_m - e_{\bf k}^--e_{\bf p}^-}+\frac{u_{\bf k}^+v_{\bf k}^+u_{\bf p}^+v_{\bf p}^+}{i\nu_m+e_{\bf k}^++e_{\bf p}^+}-\frac{u_{\bf k}^+v_{\bf k}^+u_{\bf p}^+v_{\bf p}^+}{i\nu_m - e_{\bf k}^+-e_{\bf p}^+}\right){\cal t}_+ \nonumber\\ & & \ \ \ \ \ \ \ \ \ \ \ \ \ \ + \left(\frac{u_{\bf k}^-v_{\bf k}^-u_{\bf p}^+v_{\bf p}^+}{i\nu_m+e_{\bf k}^-+e_{\bf p}^+}-\frac{u_{\bf k}^-v_{\bf k}^-u_{\bf p}^+v_{\bf p}^+}{i\nu_m - e_{\bf k}^--e_{\bf p}^+}+\frac{u_{\bf k}^+v_{\bf k}^+u_{\bf p}^-v_{\bf p}^-}{i\nu_m+e_{\bf k}^++e_{\bf p}^-}-\frac{u_{\bf k}^+v_{\bf k}^+u_{\bf p}^-v_{\bf p}^-}{i\nu_m - e_{\bf k}^+-e_{\bf p}^-}\right){\cal t}_-\bigg]e^{2i\theta},\end{aligned}\ ] ] where @xmath426 . here @xmath427 are factors arising from the trace in spin space , @xmath428 and @xmath429 are the bcs distribution functions defined as @xmath430 at @xmath431 , we find that @xmath432.\ ] ] thus , near the quantum phase transition point , we have @xmath433 . on the other hand , a simple algebra shows that @xmath434 therefore , the mean - field gap equation for @xmath64 ensures the goldstone s theorem in the superfluid phase . _ ( ii ) diquark - sigma mixing terms . _ the term @xmath131 standing for the mixing between the sigma meson and the diquarks reads @xmath435e^{i\theta},\end{aligned}\ ] ] where the factors @xmath436 are defined as @xmath437 one can easily find that @xmath438 , thus it vanishes when @xmath64 or @xmath83 approaches zero . at @xmath431 , we have @xmath439.\ ] ] thus the quantity @xmath440 defined in ( [ bexp ] ) can be evaluated as @xmath441 _ ( iii ) sigma meson and pions . _ the polarization function @xmath442 which stands for the sigma meson can be evaluated as @xmath443,\end{aligned}\ ] ] where the factors @xmath444 are defined as @xmath445 at @xmath431 and for @xmath130 , we find that @xmath446 finally , the polarization function @xmath136 for pions can be obtained by replacing @xmath447 . thus , when @xmath448 , the sigma meson and pions become degenerate and chiral symmetry is restored . in this appendix , we derive the expression of the taylor expansion of @xmath198 in terms of @xmath449 . as we have shown in sec . [ s5 ] , the leading - order term should be @xmath272 . thus , we need to evaluate the numerical factor @xmath274 . a key problem here is that the effective quark mass @xmath83 and the chemical potential @xmath0 are both functions of @xmath65 determined at the mean - field level . first , we expand the matrix elements of @xmath117 and @xmath134 in terms of @xmath450 . any of these elements denoted by @xmath451 is a function of @xmath0 , @xmath83 and @xmath450 . our method of expansion is as follows . we firstly expand @xmath452 in terms of @xmath450 formally with @xmath0 and @xmath83 being fixed parameters , i.e. , @xmath453 where @xmath454 and the expansion coefficients are defined as @xmath455 we then expand the coefficients @xmath456 ( @xmath457 ) at @xmath458 , using the fact that @xmath459 doing this we formally obtain @xmath460 finally , up to order @xmath461 , we have meanwhile , the thermodynamic potential @xmath198 can be expressed as @xmath466+\ln\det{\bf n}(q)\right\}e^{i\nu_m 0^+}-\frac{1}{2}\sum_q\left[2\ln{\cal d}_{\text d}^{*-1}(q)+\ln{\cal d}_\sigma^{*-1}(q)+3{\cal d}_\pi^{*-1}(q)\right]e^{i\nu_m 0^+}.\end{aligned}\ ] ] using the expansion ( [ c6 ] ) , we find that the factor @xmath274 is given by @xmath467 ^ 2-\frac{y_3(q)}{{\cal d}_{\pi}^{*-1}(q)}\right\}.\end{aligned}\ ] ] it is obvious that @xmath468 and @xmath469 where @xmath470 and @xmath471 are defined in ( [ bexp ] ) . on the other hand , since @xmath472 to leading order of @xmath473 , we can set @xmath474 in all equations and identify @xmath475 defined in ( [ bexp ] ) .
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qcd - like theories possess a positively definite fermion determinant at finite baryon chemical potential @xmath0 and the lattice simulation can be successfully performed .
while the chiral perturbation theories are sufficient to describe the bose condensate at low density , to describe the crossover from bose - einstein condensation ( bec ) to bcs superfluidity at moderate density we should use some fermionic effective model of qcd , such as the nambu jona - lasinio model . in this paper , using two - color two - flavor qcd as an example , we examine how the nambu jona - lasinio model describes the weakly interacting bose condensate at low density and the bec - bcs crossover at moderate density .
near the quantum phase transition point @xmath1 ( @xmath2 is the mass of pion / diquark multiplet ) , the ginzburg - landau free energy at the mean - field level can be reduced to the gross - pitaevskii free energy describing a weakly repulsive bose condensate with a diquark - diquark scattering length identical to that predicted by the chiral perturbation theories .
the goldstone mode recovers the bogoliubov excitation in weakly interacting bose condensates .
the results of in - medium chiral and diquark condensates predicted by chiral perturbation theories are analytically recovered .
the bec - bcs crossover and meson mott transition at moderate baryon chemical potential as well as the beyond - mean - field corrections are studied .
part of our results can also be applied to real qcd at finite baryon or isospin chemical potential .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
large scale alignments of quasar polarization vectors were uncovered by hutsemkers ( 1998 ) , looking at a sample of 170 qsos selected from the litterature and confirmed later on a larger sample ( hutsemkers & lamy 2001 ) . the departure to random orientations was found at significance levels small enough to merit deeper investigations . moreover , these alignments seemed to come from high redshift regions , implying that the underlying mechanism might cover physical distances of giga parsecs . a large survey of linear polarization was then started , with the long - term goal to characterize better the polarization properties of quasars , and a short - term goal to investigate the reality of the alignments . this work gives a preliminary analysis of the alignment effect for a total sample of 355 quasars , comprising new polarization measurements from observing runs between 2001 - 2003 and a new comprehensive compilation from the litterature . the sample was chosen from vron - cetty & vron ( 2000 ) and the sloan digital sky survey . among possible targets a preference was given to bright extragalactic sources having a higher probability to show stronger polarization : bal quasars , red quasars . the selection criteria of the sample are detailed elsewhere ( hutsemkers 1998 , hutsemkers et al . 2001 , 2004 ; sluse et al . 2004 ) . in order to avoid possible contaminations from the interstellar medium of the galaxy , only the objects at galactic latitude @xmath1 were selected . above this galactic latitude the interstellar polarization is smaller ( heiles 2000 ) . the observations were carried out at eso la silla 3.6 m efosc2 ( 2001 , 2002 , 2003 ) and paranal vlt / fors1 ( 2003 ) , in broad - band @xmath2 filter , using a wollaston prism and 4 positions of the half - wave plate to derive stokes parameters @xmath3 and @xmath4 ( hutsemkers et al . 2004 ; sluse et al . the photometry was done using the procedure defined by lamy & hutsemkers ( 1999 ) . photon errors on @xmath3 and @xmath4 are @xmath5 . the foreground polarization ( coming from both the instrument and the interstellar medium ) was computed from the observed field stars and subtracted from the quasar polarization . because this was not possible on every frame , we used stokes parameters averaged from the field stars observed during the same run . this average polarization is always small : @xmath6 . final errors were conservatively derived by quadratically adding photon and foreground errors yielding a total error on the qso polarization degree ca . 0.2 - 0.3 % . finally , because the polarization is a positive quantity only , it is debiased according to the wardle & kronberg method ( 1974 ) . to insure a robust measurement of the polarization direction we include only the objects having a polarization degree of @xmath7 and a maximum error in the polarization position angle of @xmath8 . the final sample of polarized quasars comprises 195 quasars in the north galactic pole region ( ngp ) and 160 quasars in the south galactic pole region ( sgp ) . figure 1 shows an aitoff projection of the full sample superimposed on the galactic star polarization map ( heiles 2000 ) . our sample spans the redshift range @xmath9 homogeneously , with a marginally better sampling in the range @xmath10 , mainly due to our bias towards brighter quasars . figure 2 shows a plot of the quasar sample polarization degrees against redshift ( left panel ) . the slight enhancement of highly polarized quasars at lower redshift is attributed to the fact that more radio - loud quasars were observed at low redshifts . overall the quasar polarization degrees do not correlate with redshift which is consistent with previous studies ( berriman et al . 1990 ) . in fig.2 right panel , the histograms of the linear polarization degree of the quasar sample ( black line ) and of the star sample ( heiles 2000 ) selected over the same region ( grey line ) are superimposed . the star polarization histogram shows much smaller polarization degrees ( 94 out of 1996 stars have a polarization degree of 0.5% or higher ) arguing in favor of minor contaminations from the interstellar medium on the quasar sample . in order to detect and assess the significance of the alignments over the complete 3d sample , we used two statistical tests designed for circular data . the detailed procedure is described in hutsemkers ( 1998 ) . the basic idea is to compute for each quasar a statistics @xmath11 taking into account the compactness of a group of @xmath12 neighbors ( in redshift space ) and the dispersion of their polarization direction . the more compact and aligned a group of @xmath12 neighbors , the smaller the statistics @xmath11 . the second statistical test is the andrew - wasserman test designed by bietenholz ( 1986 ) . both statistics can be calculated for the entire sample and the significance of a departure from homogeneity is assessed with monte - carlo reshuffling of the polarization direction over 3-d positions . the significance level is then computed as the probability that a random configuration has a smaller ( @xmath11 test ) or higher statistics ( andrew - wasserman test ) than the observed one . figure 3 right frame shows the logarithm of the significance level of the andrew - wasserman test versus the number of neighbors @xmath12 for three samples of quasars . it clearly shows that , as the size of the sample increases , the alignments are harder and harder to produce from random distributions for any value of @xmath12 . 3 left frame show the polarization map of the high redshift ngp sample ( top ) and the low redshift ngp sample ( bottom ) . the length of the bars is proportional to @xmath13 . objects of both high and low polarization degrees participate to the same alignments . an unambiguous alignment effect of the polarization direction is visible at high redshift , whereas another alignment emerges at low redshift in a different direction . our statistical tests are not invariant under polar coordinates . a way to avoid coordinate dependence is to parallel transport the angles along great circle on the celestial sphere prior to computing the statistics . this was done by jain et al . they confirm the significance of the alignments of the 213-quasar sample over very large regions of the ngp and sgp . the instrumental polarization can be discarded as a source of contamination because , it is very small on efosc2 ( @xmath140.1% ) and because the polarization degree and angle of quasars observed on different instruments are consistent within the errors . the contamination from the galaxy is not a dominant component . indeed , the interstellar polarization is usually much smaller than the polarization of quasars , and the interstellar polarization angles do not follow the quasar alignment directions . moreover it is difficult to explain the redshift dependence of the alignment directions assuming a foreground interstellar screen . if we accept the fact that the alignments are intrinsic to the sources , we have to face correlated polarizations over extreme scales of giga parsecs . exotic sources of polarization on cosmological scales could be invoked , such as pervading exotic particles ( harari & sikivie 1992 ; jain et al . 2004 ) or an intrinsic alignment of the axes of quasar central engines . future theoretical developments need to take these observed correlations into account . the ongoing survey is paramount to characterize the alignments of quasar polarization directions accross the sky and correlate them with the other large - scale dataset ( cmb , galaxy surveys , ... ) . it will also allow us to deepen our knowledge of the linear polarization properties of quasars , and to understand the different source of linear polarization between the different species . we definitely need to increase the sample of polarized quasars from a few hundreds to a few thousands over the next decade to progress significantly in the field . berriman , g. , schmidt , g. d. , west , s. c. , & stockman , h.s . , , 1990 , * 74 * , 869 . bietenholz m. f. , 1986 , , * 91 * , 1249 . harari , d. , & sikivie , p. 1992 b , * 289 * , 67 heiles , c. , 2000 , , * 119 * , 923 . hutsemkers , d. , sluse , d. , cabanac , r. , lamy , h. , quintana , h. , in prep . sluse , d. , hutsemkers , d. , lamy , h. , cabanac , r. , in prep . hutsemkers , d. , 1998 , a&a , * 332 * , 410 . hutsemkers , d. , & lamy , h. , 2001 , a&a , * 367 * , 381 . jain , p. , narain , g. , & sarala , s. , , * 347 * , 394 . lamy , h. & hutsemkers , d. , 1999 , msngr . , * 96 * , 25 . ( erratum : the messenger 97 , 23 ) wardle , j. f. c. & kronberg , p. p. , 1974 , , * 194 * , 249 .
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a survey measuring quasar polarization vectors has been started in two regions towards the north and south galactic poles . here , we review the discovery of significant correlations of orientations of polarization vectors over huge angular distances .
we report new results including a larger sample of the quasars confirming the existence of coherent orientations at redshifts @xmath0 .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the number of supernova remnants ( snrs ) with convincing evidence for interaction with ambient molecular clouds has increased considerably in recent years . the evidence ranges from a simple morphological relation to the detection of broad and/or shock - excited emission lines from various molecules . although circumstantial evidence could be very suggestive , it is the molecular lines from the shocked gas that are essential for understanding the physical and chemical processes associated with the molecular shock . in this regard , there are still only a few snrs adequate for the study of molecular cloud - shock interaction ; perhaps w28 ( arikawa et al . 1999 ) , w44 ( seta et al . 1998 ) , w51c ( koo & moon 1997 ) , 3c391 ( reach & rho 1996 , 1999 ) , and the classical source ic 443 ( denoyer 1979 ; tauber et al . 1994 and references therein ) . in this paper , we report the discovery of broad emission lines from the shocked co gas in hb 21 . hb 21 ( g89.0 + 4.7 ) is one of those snrs with mixed morphology , e.g. , shell - like in radio and center - filled in the x - ray ( rho & petre 1998 ) , where the center - filled , thermal x - ray emission is suggested to be due to interaction with molecular clouds . it has a nearly complete radio - continuum shell with an angular extent of @xmath9 ( hill 1974 ; tatematsu et al . 1990 , hereafter t90 ) . the shell is elongated along the northwest - southeast direction . the brightness distribution of the shell is not uniform , but enhanced in scattered areas . particularly noticeable features are the v - shaped northern boundary , a @xmath4-sized loop structure in the south central area , and the one in the central region of the eastern boundary ( see fig . 1 ) . optical nebulosity associated with hb 21 has not been detected in h@xmath10 or [ sii ] plates ( van den bergh 1978 ) . x - ray emission from hb 21 was detected by leahy ( 1987 ) and studied in detail by leahy & aschenbach ( 1996 ) . the distance to the snr is uncertain . we adopt 0.8 kpc following t90 , which is the distance to the cyg ob 7 complex ( humphreys 1978 ) . hb 21 has been a suspect for interaction with molecular clouds based on its radio appearance and the ambient molecular clouds ( erkes & dickel 1969 ; huang & thaddeus 1986 ; t90 ) . erkes & dickel ( 1969 ) suggested that the distorted boundaries with enhanced radio brightness might be the places where the snr is interacting with dense ambient gas . huang & thaddeus ( 1986 ) found that the giant molecular cloud associated with cyg ob 7 appears to be partially surrounding hb 21 ( see also dobashi et al . t90 obtained a higher - resolution ( @xmath11 ) co map of the eastern part of the snr and found that the eastern boundary of the snr appeared to be in contact with molecular and atomic clouds . they also made a coarsely - sampled map of the regions with enhanced radio emission and detected molecular clumps . but no direct evidence for the shocked _ molecular _ gas has been detected for hb 21 . the search for oh masers , which are known to be an indicator for the interaction between a snr and molecular cloud , gave negative results too ( frail et al . on the other hand , koo & heiles ( 1991 ) detected shocked h i gas moving at 40 to 120 in the southern part of the snr , although the limited angular resolution ( @xmath12 ) hindered any detailed study of the shocked gas . we detected broad co emission lines in the northern and southern parts of hb 21 , but not in the eastern part . we have found no evidence suggesting that the molecular clouds in the eastern part are interacting with the snr , although they appear to be located along the boundary of the snr on the sky . we summarize the observations and the results of molecular line observations in 2 and 3 , respectively . in 4 . we discuss the implications of our results on the interaction between the snr and molecular clouds . 5 summarizes the main results of our paper . @xmath0co j=21 line observations were carried out using the 12 m telescope of the national radio astronomy observatory at kitt peak in 1999 june and 2000 january . the fwhm of the telescope at 230 ghz was 27@xmath13 . we mapped the eastern half ( @xmath1 ) of the snr almost completely using the otf ( on - the - fly ) observing technique . we used two 256 channel filter banks ; one with 500 khz , and the other with 1 mhz resolution . we split each filter bank into two sections and observed two linear polarizations simultaneously . the velocity resolution and coverage of the 500 khz filter bank were 0.65 and 83 , while those of the 1 mhz filter bank were two times greater . typical system temperatures were 350450 k. during the observing run in january 2000 , we had some trouble because of telluric co j=21 emission , which appeared at @xmath145 . the emission was not cancelled out completely by usual position - switching observation and produced a hill - and - valley feature in spectra , the strength of which depends on elevation . we were able to avoid the contamination from telluric co emission by averaging out the contaminated velocity channels because the telluric co emission line is narrow ( 0.8 ) and the absolute strength of the ` hill ' and ` valley ' features are equal . we also obtained sensitive spectra of @xmath6co j=21 , @xmath0co j=10 and j=21 lines toward several peak positions . for co j=10 observations , we used 1 and 2 mhz filter banks , so that they have the same velocity resolution and coverage with those of co j=21 line observations . we also used the millimeter autocorrelator ( mac ) for the @xmath0co observations , which provided high resolution ( 0.25 after smoothing ) spectra . we have converted the observed temperatures ( @xmath15 ) to the main - beam brightness temperatures ( @xmath16 ) using the corrected main - beam efficiency provided by the nrao . additional observations of @xmath0co and @xmath6co j=10 lines were performed using the taeduk radio astronomy observatory ( trao ) 13.7 m telescope ( hpbw=49@xmath13 at 115 ghz ) in 2000 january and march . an sis receiver equipped with a quasi - optical sideband filter was used along with a 250 khz , 256-channel filter bank . the main beam efficiency was 0.41 at 115 ghz ( roh & jung 1999 ) , and the pointing accuracy was better than 10@xmath13 . typical system temperatures were about 750 k at 115 ghz and 450 k at 110 ghz . figure 1 shows the distribution of the integrated intensity of co j=21 emission . the velocity range is between @xmath17 and @xmath18 , which covers most of the emission . the overlaied contour map shows the 1420 mhz brightness distribution of hb 21 ( t90 ) . the overall distribution of co gas in figure 1 is not very different from the low - resolution map of t90 . but figure 1 shows much detailed structure because of its high resolution , high sensitivity , and complete sampling . figure 1 immediately shows that molecular gas is distributed mainly along the boundary of the snr . ( we observed the central area in figure 1 , which had not been covered in our co j=21 observations , in co j=10 line emission , and have detected only several small [ @xmath19 clumps other than some faint extension associated with the clouds in southern and eastern parts of the remnant . ) but the overall distribution has little correlation either with the distortion of snr boundaries or with the distribution of radio brightness ( cf . 3.2 ) . for the purpose of discussion , we divide the remnant into three areas ( fig . 2 ) : ( 1 ) eastern area ( ra@xmath20 20@xmath21 47@xmath22 ) where three relatively large ( @xmath23 ) clouds and several filamentary clouds are present , ( 2 ) northern area centered at ( 20@xmath21 46@xmath22 , 51@xmath24 00@xmath25 ) , where a small ( @xmath2 ) , very bright u - shaped cloud is noticeable , and ( 3 ) southern area centered at ( 20@xmath21 44@xmath22 , 49@xmath24 50@xmath25 ) , where clumpy and filamentary clouds with complicated structures are present . we detected broad ( 2040 ) emission lines from the clouds in the northern and southern areas , which will be discussed in detail in the next sections . in the following , we summarize the results on the eastern area . in the eastern area , there are three clouds centered at declinations @xmath26 49@xmath25 , @xmath27 15@xmath25 , and @xmath28 @xmath29 . we call these three clouds by clouds a , b , and c following t90 ( see fig . 2 for the location of these clouds ) . figure 3 shows the channel maps of the eastern area . the velocity ranges of the channel maps were chosen to show the essential features clearly . cloud a appears at @xmath30 to @xmath31 and is composed of two velocity components centered at + 6 and @xmath32 , respectively . the former component ( 6 ) , which is seen in figure 3a , is extended and the emission peaks at the southern part ( @xmath33 45@xmath25 ) of the cloud , while the latter component ( @xmath32 ) , which is seen in figures 3b and 3c , is spatially confined and comprises the northern part of the cloud . their maximum brightnesses are @xmath34 and 7 k , respectively . cloud b appears at @xmath35 to @xmath36 and is seen in figures 3c and 3d . the south central part of the cloud , e.g. , the region between @xmath3717@xmath25 , is bright and appears to be connected to cloud c. cloud c has two components at very different velocities , e.g. , one at + 2 to @xmath38 , which is seen in figures 3c and 3d , and the other at + 17 to + 10 , which is not shown in figure 3 but has a distribution similar to the other velocity component . according to the result of t90 , cloud c extends to @xmath39 . an interesting feature in figure 3 is the semi - circular loop that appears above cloud b in figure 3b . the ratio of the minor , which is along the ns direction , to the major axis is 0.8 . if it is at 0.8 kpc , the linear size of the semimajor axis would be @xmath40 pc . the velocity increases systematically from both ends to the northern top of the loop , which is consistent with an expanding loop . the top portion is redshifted with respect to the both ends by @xmath41 . if we assume that the ellipticity is due to projection , then the expansion velocity would be @xmath42 , so that the dynamical age of the ring is probably shorter than @xmath43 yrs . this is much greater than the age ( @xmath44 yrs , koo & heiles 1991 , scaled to 0.8 kpc adopted in this paper ) of hb 21 and , therefore , the loop might not be associated with hb 21 . we suspect that the loop is originated from some energetic phenomena in cloud b. a faint v - shaped structure that connects cloud b and the ends of the loop in figure 3b seems to indicate the association of the two . the cloud in the northern area is composed of a small ( @xmath2 ) , very bright , u - shaped cloud and several clumps scattered around it ( fig . 1 ; see also fig . 9 for enlarged view ) . figure 4 shows its velocity structure . there are several points to be made from figure 4 : first the u - shaped cloud is composed of several clumps , whose central velocities shift systematically from + 3 to @xmath31 as we move from ne to nw along the structure . the integrated intensity attains a maximum at ( 20@xmath21 46@xmath22 [email protected] , 51@xmath24 00@xmath25 00@xmath13 ) , which we call hb21:bml - n1 ( broad molecular line northern position 1 ) , or simply n1 . second , there are several other clumps in the field . these clumps , except the one near the southeastern corner at @xmath46 , appear over a wide ( @xmath47 ) velocity range . among them , the one at ( 20@xmath21 45@xmath22 55.@xmath450 , 51@xmath24 03@xmath25 30@xmath13 ) , which we call hb21:bml - n2 ( or n2 ) , appears over the widest ( @xmath48 ) velocity interval . third , there is a diffuse emission at @xmath49 to the ne of the cloud . its line is narrow ( 23 ) and it is part of a large ( @xmath50 ) cloud that appears to be connected to cloud a. we consider that the clumps aligned along the ne - sw direction in figure 4 are associated and call them cloud n , i.e. , cloud n does not include the diffuse emission in the northeastern area and the clump in the southeastern corner . t90 detected in this area only the diffuse molecular gas at @xmath51 to 6 and called it cloud d. ) as can be expected from figure 4 , most clumps in cloud n have broad emission lines . as an example , we show the spectra of n1 and n2 in the top frames in figure 5 , where we see that the spectrum of n1 is box - shaped and its full width ( at zero intensity ) is 30 km / s , while that of n2 is asymmetric and extends from @xmath52 to + 11 km / s . for comparison , the spectrum of the diffuse , extended structure in the northeastern part of this area has narrow ( 23 ) emission lines centered at @xmath49 , a sample of which is shown in the right bottom frame in figure 5 . we have obtained sensitive j=10 and j=21 spectra of @xmath0co and @xmath6co molecules at the two peak positions , n1 and n2 , and figure 5 shows the spectra . the molecule and transition are marked in each spectrum . the second spectrum from the top is @xmath0co j=21 emission convolved to the j=10 beam size ( @xmath53 ) to be compared with the j=10 spectra . the difference between the top and convolved spectra indicates that some velocity components , e.g. , the narrow component centered at + 4 of n1 and the broad component at @xmath54 of n2 , are confined to small areas . by comparing the j=10 and the convolved j=21 spectra , we notice that the ratio of j=21 to j=10 intensities is high and that it varies over the profile : for n1 , the ratio is between 1.2 and 2.0 in the central parts of the spectrum , while it increases at the wings , e.g. , @xmath55 at + 5 and @xmath5613 . for n2 , the ratio varies between 0.8 and 2.8 , and it is higher between @xmath38 and 0 . the ratios of the @xmath0co j=21 and j=10 integrated intensities @xmath57 are 1.6 and 1.7 for n1 and n2 , respectively . the @xmath6co j=21 line is clearly detected toward n1 , while it is marginally detected toward n2 . for n1 , the line has double peaks centered at + 1 and @xmath36 , while the @xmath0co j=21 line profile toward n1 is composed of several narrow peaks . the narrow peaks might indicate that the emission is from several , unresolved subclumps . presumably , the @xmath6co emission might be from these subclumps too , which is not apparent in the profile in figure 5 because of low signal - to - noise ratio and low velocity resolution . ( note that the velocity resolutions of the @xmath0co j=2 - 1 and @xmath6co j=2 - 1 lines are 0.25 and 0.68 , respectively . ) the ratios of @xmath0co j=21 to @xmath6co j=21 integrated line intensities @xmath58 are @xmath59 and @xmath60 for n1 and n2 , respectively . ( the errors are statistical errors . ) table 1 summarizes the line parameters of the peak positions , i.e. , their coordinate , velocity range ( @xmath61 , @xmath62 ) , co j=21 peak brightness temperature @xmath63 , @xmath57 , @xmath58 , and the ratio of @xmath0co j=10 to @xmath6co j=10 integrated line intensities @xmath64 . the detailed line diagnostics based on the observed line parameters in table 1 is discussed in 3.4 . in the southern part of the snr , the emission is detected at @xmath65 to @xmath66 . the velocity structure is shown in figure 6 . at positive velocities , we see several clouds with narrow lines come and go , e.g. , a diffuse cloud that extends @xmath67 along the ns direction centered at ( 20@xmath21 [email protected] , 50@xmath24 00@xmath25 ) between @xmath68 and + 5 . the co distribution at negative velocities is fairly complicated : the distribution is filamentary , and small ( @xmath5 or 0.3 pc ) , bright clumps are seen along the filamentary structure . the filamentary structure , which we call cloud s , appears to form a loop of @xmath69 in extent , elongated along the ns direction . the eastern part of the loop is particularly clumpy and has a semicircular shape ( see the channel map centered at @xmath70 ) . the clumps generally have broad ( @xmath71 ) lines . among them , three clumps marked by crosses in figure 6 have broadest ( 3040 ) lines and we show the @xmath0co j=21 and j=10 spectra at their peak positions , which we call s1 , s2 , and s3 from east to west ( see fig . 2 ) , in figure 7 . again we show the convolved j=21 spectra together , although the line shapes do not change significantly by convolution toward these peak positions . we have obtained some sensitive j=10 and j=21 spectra of @xmath0co and @xmath6co molecules at these peak positions , and the line parameters are listed in table 1 . note that @xmath57(=1.72.3 ) and @xmath72 toward s1 ) are similar to those of northern positions , while @xmath64 s , although they have large uncertainties , appear to be much greater than that of n1 . the broad co lines are presumably emitted from the shocked gas , where physical parameters vary greatly over a short distance scale . but still it would be worthwhile to estimate their excitation parameters based on elementary considerations . first , the observed @xmath7340 are significantly less than either the average ratio ( @xmath74 ; langer 1997 ) of @xmath0c/@xmath6c in the solar neighborhood or the terrestrial value ( 89 ) , implying that the @xmath0co j=21 lines are _ not _ optically thin . if we adopt @xmath0c/@xmath6c=67 and assume that the emission is thermalized , then the optical depth for the @xmath0co j=21 line @xmath753.4 . on the other hand , the @xmath64(@xmath76 ) close to or greater than the terrestrial value imply that @xmath0co j=10 lines are optically thin , except at n1 where j=10 and j=21 lines appear to have comparable optical depths . this , however , is not conclusive because of large uncertainties associated with @xmath64 . second , the large values of @xmath772.3 imply that the broad - line emitting region is warm and dense . for typical molecular clouds , where the j=2 level is subthermally excited , the ratio is usually less than unity . for example , molecular clouds in the local arm exhibit ratios ranging from 0.53 ( taurus ) to 0.75 ( orion a ) ( sakamoto et al . 1994 , 1997 ) . our spectra of ambient gas also show this , e.g. , see the spectra toward s2 in fig . 7 where the narrow component at + 2 has @xmath78 . but the large ratio is common for the shocked molecular gas in snrs ( see 4.1 ) . third , the low ( @xmath79 k ) brightness temperature of j=21 lines , regardless of their moderate optical depths , imply that the emitting region must be clumpy , i.e. , composed of subclumps , and the emission is beam - diluted . from the above considerations , we may conclude that the broad emission lines are from warm , dense clumps with significant column densities so that the 21 lines are optically thick . we have applied the large - velocity - gradient ( lvg ) model ( scoville & solomon 1974 ; goldreich & kwan 1974 ) to our broad co lines in order to derive their excitation parameters . the model assumes an uniform , spherical cloud with a constant velocity gradient ( @xmath80 ) . if the lines are emitted from the shocked region where temperature and density vary greatly , the resulting parameters may be considered as ` average ' values . since we have found that the emission is beam - diluted , we have used the line ratios , @xmath57 and @xmath58 , instead of brightness temperatures to determine the excitation parameters . according to our lvg analysis , the observed ratios are possible for @xmath81 k. figure 8a shows the result of our model computations when @xmath82 k , where curves of constant @xmath57 and constant @xmath58 are drawn in ( @xmath83 , @xmath84 ) plane . @xmath83 is the fractional abundance of @xmath0co relative to h@xmath85 ( @xmath86 ) per unit velocity gradient interval . the asterisks ( * ) mark the observed ratios toward the peak positions where both ratios are obtained , i.e. , n1 , n2 , and s1 . according to figure 8a , @xmath87@xmath88 @xmath89 and @xmath90@xmath91 pc ( ) @xmath92 at the three peak positions . there are multiple choices for n1 , e.g. , the same ratios are obtainable when @xmath93 @xmath89 and @xmath94 pc ( ) @xmath92 . we adopt the lower density because it is comparable to the densities in the other peak positions and because the density of @xmath95 @xmath89 appears to be too high for the co emission to explore . if the temperature becomes higher , both @xmath84 and @xmath83 need to be greater . in figure 8b , we plot the _ expected _ co j=21 radiation temperature @xmath96 $ ] which is just the brightness temperature when @xmath97 and the expected @xmath64 from the same lvg model . note that the expected radiation temperatures are much greater than the observed main - beam brightness temperature . we have estimated beam filling factors of ( 7.78.8)@xmath98 from the ratio of these two brightnesses . we have estimated the co column densities @xmath99@xmath100 @xmath101 at these peak positions by @xmath102 \delta v$ ] , where @xmath10318 is the velocity width . the excitation parameters derived from the lvg analysis are listed in table 2 . note that the @xmath64 expected from the lvg model differ from the observed ones : at n1 , the observed value is small by a factor of 2 , while , at s1 , it is large by a factor of @xmath104 . considering the weakness of @xmath6co j=10 lines and various uncertainties associated with different telescopes , however , it is not obvious if this difference is critical . we have made a crude estimate of the mass of the broad - line clouds as follows . if the co j=10 line emission is optically thin , then @xmath105 can be obtained from the co j=10 luminosity @xmath106 by @xmath107 $ ] where @xmath108 is the fraction of co molecules at @xmath109 level and the other coefficients have their usual meanings . in our case , co j=10 emission has less optical depth than the j=21 emission , but is not very optically thin , so that the above formula might yield an underestimate . what we have is the luminosity of co j=21 emission @xmath110 , which has moderate optical depth . but , since @xmath772.3 at the peak positions , we may obtain @xmath106 by assuming that @xmath111 where @xmath112 and @xmath113 are co @xmath11410 and @xmath11421 line frequencies respectively . finally we assume @xmath115 , which is a mean value of those ( 0.150.26 ) at the three peak positions obtained from the lvg analysis . we have found that the h@xmath85 masses of clouds n and s are @xmath116 and @xmath117 , respectively . the mass of the central u - shaped part of cloud n is @xmath118 while the masses of the small clumps in cloud s are @xmath1191 . broad co emission lines with large @xmath57 in clouds n and s strongly suggest that they are being shocked . the observed velocity width is as large as @xmath120 . note that , toward this direction ( @xmath121 ) , the lsr velocity permitted by the galactic rotation is @xmath122 , so that it is not impossible for broad lines to be produced by molecular clouds accidentaly aligned along the line of sight . but it is highly improbable that such alignment ( over a few kpcs ) occurs in very small ( 12@xmath25 ) areas on the sky . we also searched for protostellar candidates around the broad line emitting regions using the _ infrared astronomical satellite _ ( _ iras _ ) point source catalog , because broad lines can be emitted from the high - velocity gas associated with protostellar object too . we did not find any suspicious sources except one , iras 20444 + 4954 , which is located close to the s2 clump , i.e. , at ( @xmath123 , @xmath124 ) from the peak position in table 1 . the source has been detected in two iras wavebands , i.e. , 60 and 100 @xmath125 , with flux densities of @xmath126 jy and @xmath127 jy . we have found that the source is located within a small ( @xmath128 ) , bright ( @xmath129 k ) co j=21 core at @xmath130 , so that it might be a young stellar object associated with the @xmath67-sized , diffuse cloud in the northern part of the + 2.9 map in figure 6 , not with the s2 clump . also the velocity of the s2 clump is similar to those of the other fast - moving clumps in this area , which suggests that they have the common origin . therefore , the broad lines that we detected are almost certainly from the fast - moving molecular gas swept - up by the snr shock in hb 21 . another indication that the broad co lines are from the shocked gas is their high @xmath772.3 . as we have shown in 3.4 , the high ratio implies that the emitting gas is warm and dense , which might be manifestation of shock . indeed high @xmath57 is a common property of the broad lines from the shocked molecular gas in snrs : all six snrs that are known to have broad molecular emission lines , i.e. , w28 ( arikawa et al . 1999 ) , 3c391 ( reach & rho 1999 ) , w44 ( seta et al . 1998 ) , w51c ( koo & moon 1997 ) , hb 21 ( this paper ) , and ic 443 ( e.g. , van dishoeck et al . 1993 ) , have ratios greater than 1 , which implies that the broad co lines in these snrs are all emitted from warm and dense , shocked gas . meanwhile , the maximum brightness temperature ( @xmath131 k ) of co j=21 line in hb 21 is significantly less than those of other snrs even if it was obtained with a higher spatial resolution ( 0.1 pc ) , e.g. , it is 33 k for w28 ( reach & rho 2000b ) and ic 443 ( van dishoeck et al . 1993 ) when observed with a resolution of 0.2 pc . the much smaller co j=21 brightness temperature with comparable @xmath57 imply that either the shocked gas in hb 21 is composed of much smaller clumps or is less dense . on a large scale , hb 21 appears to be in contact with a giant molecular cloud ( gmc ) along its eastern boundary ( huang & thaddeus 1986 ; tatematsu et al . the gmc is 130 pc@xmath13270 pc in extent ( dobashi et al . 1994 ) , and the structure that we call clouds a , b , and c defines the western boundary of the gmc . t90 inspected the correlation between these clouds and the snr in detail , and concluded that cloud a might be interacting with the snr because it is located where the radio continuum boundary of the snr is distorted . on the other hand , they concluded that clouds b and c might not , because there is no indication of the interaction in the distribution of radio brightness . according to our high - resolution observations , however , there is little relationship between the boundaries of cloud a and the snr . instead , since the velocity of the ambient molecular gas around hb 21 might be negative ( see 4.3 ) , we consider that clouds b and c have a better chance of interaction . but we have detected broad co lines toward none of these eastern clouds . even if a snr is interacting with a molecular cloud , the broad lines may be absent , however . 3c 391 ( g31.9 + 0.0 ) , which appears to be located at the edge of a large molecular cloud , for example , has no broad co lines along the interface ( wilner , reynolds , & moffett 1998 ; reach & rho 1999 ) . but strong [ oi ] 63 @xmath133 m emission has been detected near the interface , which indicates that the snr is interacting with the molecular cloud ( reach & rho 1996 ) . this would happen if the shock is dissociative and molecules have not reformed . figure 9 shows an enlarged view of the northern area overlaied with a 325 mhz radio continuum map of the snr . note that there is an enhanced radio continuum emission elongated along the ne - sw direction in the central area . its peak position falls exactly inside the u - shaped part of cloud n. the positional coincidence suggests that the enhancement is possibly associated with cloud n. we have derived a spectral index ( flux@xmath134 ) of @xmath135 for the radio emission associated with the u - shaped central part using the 325 mhz and 1420 mhz maps . the derived index has a large uncertainty because of the confusing background " level , but it appears to be flatter than the mean spectral index @xmath136 of hb 21 ( willis 1973 ) . for cloud s , there is no obvious correlation between the co emission and radio continuum brightness , although the radio continuum appears to be bright around the cloud in general . it is not obvious , observationally or theoretically , what determines whether or not radio synchrotron emission becomes enhanced when sn shock hits a dense cloud . observationally , we see very limited correlation between radio continuum brightness and shocked molecular gas in some snrs , i.e. , the shocked molecular gas is not usually associated with radio continuum enhancement and vice versa ( see also chevalier 1999 ) . in ic 443 , for example , shocked molecular gas is distributed in a fragmentary , flattened , ring , which partly overlaps with the radio continuum shell ( e.g. , see dickman et al . 1992 for the shocked molecular gas and green 1986 for the radio continuum ) . but the radio continuum is brightest in the northeastern part of the shell where there is no shocked molecular gas , and the radio continuum is not particularly bright toward the shocked molecular gas , perhaps except around the southern part of the molecular ring ( see next ) . in 3c 391 , there is a shocked molecular clump in the southern part of the snr , but there is only a faint , local radio continuum peak at @xmath137 pc apart from the shocked clump , whose association can not be confirmed ( reach & rho 1999 ) . in w28 , on the other hand , there is a ridge of radio continuum emission associated with the shocked molecular gas ( arikawa et al . theoretically , it could be either the dense cloud or the surrounding intercloud medium where synchrotron emission becomes enhanced . if the shock propagating through the dense cloud is radiative , there will be a large compression of cosmic rays and magnetic field , which would increase the synchrotron emissivity ( van der lann 1962 ; blandford & cowie 1982 ) . but this mechanism may not work because the molecular shocks in old snrs are not ionizing shocks and , therefore , high energy particles may escape from the shocked region ( draine & mckee 1993 ; chevalier 1999 ) . on the other hand , the shocked intercloud medium surrounding the cloud , particularly the medium behind the cloud , could have enhanced synchrotron emissivity because of the increased magnetic field strength there ( e.g. , jones & kang 1993 ; mac low et al . 1994 ) . in hb 21 , the peak of the enhanced radio emission is located behind the shocked cloud ( fig . it is also noteworthy that the cloud has a u - shape , which is similar to what we would expect when a small cloud is swept up by a strong shock ( e.g. , klein , mckee , & colella 1994 ; mac low et al . these morphological characteristics seem to suggest that the enhanced emission is not physically associated with the shocked cloud but with the shocked surrounding intercloud medium . in some respects , the region around cloud n in hb 21 is similar to the flat ( @xmath138 ) spectral region in ic 443 , which is also located behind a shocked molecular cloud ( green 1986 ; keohane et al . 1997 ) . in ic 443 , keohane ( 1997 ) found that the flat spectral region is particularly bright in hard x - rays , which are most likely due to synchrotron radiation . they concluded that the enhanced hard x - ray emission and the flat spectral index is due to shock acceleration of cosmic rays behind dense clouds . more observational study is certainly needed for hb 21 in order to reveal the relationship between the northern u - shaped cloud and the enhanced radio emission . cold molecular gas around hb 21 in general has central velocities between @xmath139 and + 6 , which must be the velocity range of the preshock gas . in the areas where broad lines have been detected , there is diffuse molecular gas at positive ( 0 to + 5 ) velocities ( see figures 4 and 6 ) . if this gas represents the preshock gas , then , since the broad lines are centered and spread out mostly at negative velocities , the gas should have been shocked and accelerated toward us _ systematically_. if we take + 3 as the velocity of the preshock gas and if we take either the central or peak positions of the broad lines as the systematic velocities of the shocked gas , then _ the line - of - sight _ velocities of the shock would be @xmath140 and @xmath8 for clouds n and s , respectively . and , if we take @xmath141 and @xmath142 ( @xmath143=the radius of the snr ) as their projected distances from the center of the snr , then their deprojected velocities are @xmath144 and @xmath145 , respectively . such coasting clumps are indeed theoretically expected for the old snrs such as hb 21 : molecular clumps swept up by a snr blast wave are accelerated by the shock propagating into the clumps and also by the ram pressure of interclump gas ( e.g. mckee 1988 ) . the characteristic timescale for acceleration is @xmath146 where @xmath147 is the radius of the clump , which becomes @xmath148 yrs ( @xmath149 yrs ) for the clumps in hb 21 using @xmath150 pc and @xmath15125 . the wings of the broad lines may be attributed to the gas accelerated by shocks propagating from sides . but one difficulty in this scenario would be that there is no clear correlation in the distributions of the broad lines between the preshock and postshock gases . in the southern area , for example , the preshock gas ( 23 component ) is distributed in a filamentary cloud extended along the ns direction , while the broad - line emitting clumps are distributed over a much wider area to the sw of this cloud . also it appears rather awkward that there are no broad lines centered near the velocity of the preshock gas . alternatively , it could be the molecular gas at negative velocities that is associated with the snr , and , in the northern and southern areas , most , if not all , of the ambient gas may have been shocked . in this picture , the shocked gas may be coasting too , but not necessarily at high speeds because the velocity of the preshock gas is somewhere within the velocity range of the broad lines , e.g. , near the velocity of the peak . the shock velocity determined from the line width is @xmath8 . between the two possible interpretations , we prefer the latter because of the difficulty with the former mentioned above . but , since the difference in the shock velocities between the two interpretations is @xmath140 , the following discussion remains valid basically even if the velocity of the preshock gas is 0 to + 5 . ( the derived properties of the shocked gas in 3.4 should remain valid too in either case because the emission from the preshock gas might be in narrow lines and its contribution to the integrated intensity of broad lines is expected to be small . ) the _ observed _ shock velocity is @xmath8 . this is less that the critical velocity for the dissociation of molecules , which is 2550 depending on preshock density and magnetic field strength ( hollenbach & mckee 1980 ; draine , roberge , & dalgarno 1983 ) . hence , the shock might be a _ non - dissociating c - shock_. the observed integrated intensity of the co j=21 emission is ( 39)@xmath152 ergs @xmath101 s@xmath92 sr@xmath92 at the peak positions . if we consider the beam dilution ( 3.4 ) , the actual surface brightness may be greater by an order of magnitude , e.g. , ( 410)@xmath153 ergs @xmath101 s@xmath92 sr@xmath92 . this is much larger than the _ angle - averaged _ surface brightness predicted from shock model computations . draine & roberge ( 1984 ) , for example , computed surface brightnesses expected for steady - state c - shocks propagating through molecular gas with different preshock conditions . according to their result , the angle - averaged surface brightness of co j=21 emission varies from @xmath154 to @xmath155 ergs @xmath101 s@xmath92 sr@xmath92 for a 1020 shock propagating through a molecular cloud with @xmath156 to @xmath157 @xmath89 . larger shock velocity does not raise the surface brightness while higher preshock density may yield @xmath158 ergs @xmath101 s@xmath92 sr@xmath92 . the much higher surface brightness toward the peak positions would be possible if these are directions where we are observing the shock tangentially . we want to briefly discuss the non - detection of oh 1720 mhz masers in hb 21 , because such masers are known to indicate interaction of snrs with molecular clouds ( frail et al . firstly , the oh masers may require a very specific set of physical conditions that might not be realized in hb 21 . according to lockett , gauthier , & elitzer ( 1999 ) , the 1720 mhz masers arise only in c - shocks when @xmath159 - 125 k , @xmath160 , and oh column density of @xmath161@xmath162 @xmath101 . according to our result in 3.4 , the density of the shocked molecular gas in hb 21 appears to be much lower than required . secondly , it is not impossible that the oh maser emission , even if present , had been missed in the survey by the frail et al . ( 1996 ) who mapped the snrs in rectangular grids with full - beam ( or @xmath163 full - beam ) grid spacing . it would be worthwhile to search for oh masers toward the shocked co gas in hb 21 . koo & heiles ( 1991 ) detected shocked h i gas associated with hb 21 . the shocked h i gas moves at @xmath164123 and is confined to the southern part ( @xmath165 0@xmath2550@xmath24 30@xmath25 ) of the snr . the highest velocity component coincides with cloud s , although the angular resolution ( @xmath12 ) of the hi observation is too large for a detailed comparison . koo & heiles ( 1991 ) assumed that the shocked h i gas represents a cap portion of a large expanding h i shell and derived a mean _ ambient _ h i density of 3.7 @xmath89 ( when scaled to 0.8 kpc adopted in this paper ) . if the molecular shock has been driven by this h i shell , then we can roughly estimate the density of the shell as follows : we assume that the shocked molecular gas is confined to a thin slab and that the radiative h i shell has an uniform density of @xmath166 . then it is straightforward to show that @xmath166 is related to the density of the molecular cloud @xmath167 by @xmath168 ( for @xmath169 ) where @xmath170 and @xmath171 are the velocities of radiative shell and shocked molecular slab , respectively ( e.g. , see chevalier 1999 ) . for hb 21 , @xmath172 and @xmath173 . and if we take the h@xmath85 density of the cloud @xmath174 @xmath89 , the density of the h i shell would be @xmath175 @xmath89 . by comparing with the mean ambient density ( 3.7 @xmath89 ) , this implies a compression factor @xmath176 for the h i shell . such moderate compression would be obtained if the ambient magnetic field strength tangential to the shell is @xmath177 @xmath133 g where @xmath178 is the density of the ambient medium . ( the equation is obtained by assuming that ambient magnetic field is uniform and magnetic pressure dominates the pressure in the shell . see chevalier 1974 for a discussion . ) alternatively , the fast - moving h i gas could be the gas originally associated with the molecular clouds , i.e. , the atomic and molecular shocks may be produced when the snr shock hits a large molecular cloud . in this case the fast - moving h i gas represents the swept - up interclump medium or h i envelope of molecular cloud . high - resolution h i observation is needed to reveal the relation between the shocked atomic and molecular gases . we used archival data from _ iras _ to search for infrared emission associated with the remnant . in his catalog of infrared emission from supernova remnants , arendt ( 1989 ) called hb 21 a `` probable '' infrared source at 12 and 60 @xmath133 m , with total fluxes of @xmath179 and @xmath180 jy , respectively . the main source of uncertainty is confusion with unrelated emission in the galactic plane , which can not be easily separated in infrared images . using the _ iras _ _ sky survey atlas _ ( _ issa _ ; wheelock et al . 1993 ) , we created an image covering the region around hb 21 at 60 and 100 @xmath133 m . there is extensive emission to the south , east , and west of the remnant , but with no clear correlation with the radio or co image . the region toward the center of the remnant is relatively fainter than these edges , but the northern part of the remnant is also faint . it is not possible to tell whether the emission is related to a partial shell around the remnant or just fluctuations in the background emission . to search for infrared emission associated with the remnant in more detail , we obtained a dedicated _ iras _ hires ( aumann , fowler , & melnyk 1990 ) image for a @xmath181 field centered on hb 21 . hb 21 is visible in the _ iras _ hires images at all four wavelengths ( 12 , 25 , 60 , and 100 @xmath133 m ) . the 60 @xmath133 m image , where hb 21 is most prominent , is shown in figure 10 . comparing the _ iras _ and co images , it is evident that the southern filamentary cloud , cloud s , is detected as a long arc , with very similar location , shape and width . the 60 @xmath133 m surface brightness of the filament is typically 7 mjy sr@xmath92 , and its structure is clumpy , like that of the co emission . but the peaks of co and infrared emission do not match in detail , suggesting that the 60 @xmath133 m emission does not arise from the exact same regions as the co. in the northern area , there is also a good correspondence between the infrared and co emission . the general correspondence in the south and north , and partial overlap in the east , show that many of the infrared features around the edge of hb 21 are related to the remnant , although the infrared emitting regions differ in detail from the co emitting regions . the nature of the infrared emission from hb 21 could be either dust grains surviving the shock , or from spectral lines from shock - excited gas , or both . we have estimated the mean surface brightness and color of the co clouds by using several faint regions in the field as background . the results are summarized in table 3 . the far - infrared color ratio , @xmath182 , for clouds a , b , and c is almost identical to that of diffuse cirrus clouds in the solar neighborhood ( boulanger & prault 1988 ) . this suggests that the infrared emission from clouds a , b , and c is most likely due to dust heated at the surface of the molecular clouds by the interstellar radiation field ; specifically , it suggests that the infrared emission is not related to shock fronts into the clouds . on the other hand , clouds n and s have a significantly higher color ratio @xmath183 . this enhanced color , and the morphological correspondence with the broad molecular line emitting regions , suggests that the infrared emission from clouds n and s is due to shocks in propagating into the clouds . conversely , the normal infrared color of clouds a , b , and c is consistent with their being due to ambient molecular clouds . if the infrared emission from clouds n and s is due to dust , then the relatively higher @xmath184 could be due to smaller or warmer dust grains . for dust heated by the average interstellar radiation field in the solar neighborhood , about half of the emission at 60 @xmath133 m is thought to be due to small , transiently heated grains ( draine & anderson 1985 ; dsert , boulanger , & puget 1990 ) . thus , if the enhanced 60 @xmath133 m emission is due to dust grains , then clouds n and s may contain a larger fraction of small grains . an enhanced abundance of small grains would be expected if a significant fraction of larger grains were shattered behind the shock front . observations of local cirrus clouds with significant velocities revealed that @xmath185 is typical for clouds with @xmath186 km s@xmath92 , suggesting that grain shattering was significant in the shocks that accelerated local clouds to intermediate velocities ( heiles , reach , & koo 1988 ) . theoretically , significant shattering is not predicted for slow shocks such as inferred from the widths of the co lines , but faster shocks through somewhat lower - density interclump gas , with @xmath187 km s@xmath92 , could produce the enhanced 60 @xmath133 m emission ( jones , tielens , & hollenbach 1996 ) . spectral lines could contribute significantly to the infrared emission from clouds n and s. the most important lines in the _ iras _ passbands , based on infrared spectra of similar supernova remnants ( oliva et al . 1999 , cesarsky et al . 1999 , reach & rho 2000a ) , are [ ] 88 @xmath133 m in _ iras _ band 4 , [ ] 63 @xmath133 m in band 3 , [ ] 26 @xmath133 m in band 2 , and [ ] 12.8 @xmath133 m and h@xmath85 lines in band 1 . if we were to interpret all of the _ iras _ emission from cloud s as due to the ionic lines listed above , then , using the system response and bandwidths ( beichman et al . 1988 ) , we find that the brightest line would be [ ] 88 @xmath133 m , with intensity @xmath188 erg s@xmath92 @xmath101 sr@xmath92 . relative to this line , the other bright lines would have ratios @xmath18963/88=0.27 , @xmath18926/88=0.13 , and @xmath18912.8/88=0.26 . the implied brightness of the [ ] 63 @xmath133 m line can be easily produced by shocks with velocities @xmath190 km s@xmath92 into moderate - density ( @xmath191 @xmath89 ) gas ( hollenbach & mckee 1989 ) ; however , such shocks do not produce as much [ ] emission as observed , because the column density of highly - ionized gas is insufficient . slower shocks into denser gas can also produce [ ] 63 @xmath133 m lines this bright ( draine , roberge , & dalgarno 1983 ) . however , slower ( c - type ) shocks produce essentially no ionic line emission especially not an ion such as . nor would the slow shocks destroy grains adequately to produce significant gas - phase . therefore , if the infrared emission is from slow shocks , the _ iras _ 100 @xmath133 m band emission is from dust , and the _ iras _ 60 @xmath133 m band emission is from a mix of dust and the [ ] 63 @xmath133 m line . the nature of the _ iras _ 12 and 25 @xmath133 m band emission is more difficult to constrain . if the infrared emission is from slower shocks , such as inferred from the co observations , then there is likely a contribution from h@xmath85 lines . rotational lines are the dominant coolant for a range of molecular shocks ( rho et al . 2000 , reach & rho 2000a ) . for now , it is not possible to clearly tell what fraction of the infrared emission is from gas or dust . the nature of the infrared emission from hb 21 ( and other supernova remnants ) can be determined in the future using spectroscopy or narrow - band imaging . we have mapped the eastern half ( @xmath192 ) of the snr hb 21 in @xmath0co j=21 line emission almost completely . our map , which has been completely sampled with @xmath193 resolution , shows the detailed structure of molecular clouds in this area . we have detected broad co lines with large @xmath57 in the northern and southern parts of hb 21 , which is direct evidence for the interaction between molecular clouds and the snr . in the following , we summarize the main results of this paper : \(1 ) we detected shocked molecular clouds , clouds n and s , with broad ( 2040 ) co lines in the northern and southern parts of the snr . cloud n is composed of a small ( @xmath2 or 0.5 pc ) , very bright , u - shaped , clumpy part and several clumps scattered around it . cloud s is filamentary and appears to form an elongated loop of @xmath4 in extent . small ( @xmath5 or 0.3 pc ) , bright clumps are seen along the filamentary structure . the h@xmath85 masses of clouds n and s are @xmath116 and @xmath117 , respectively.(2 ) we have obtained sensitive j=10 and j=21 spectra of @xmath0co and @xmath6co molecules toward several peak positions of clouds n and s. they have @xmath772.3 and @xmath7340 with co j=21 main - beam brightness temperature less than 7 k. according to our lvg analysis , @xmath81 k , and , for @xmath82 k , @xmath194(37)@xmath195 @xmath89 and @xmath196(2.411 ) @xmath197 @xmath101 . the emitting region appears to fill a small ( 0.0770.088 ) fraction of the beam . ( 3 ) there is an enhanced radio emission which attains a maximum exactly inside the central u - shaped part of cloud n. the emission has a spectral index ( @xmath198 ) flatter than that of the whole remnant . the association of this emission with cloud n needs to be explored.(4 ) clouds n and s are visible in the _ iras _ hires images at all four wavelengths ( 12 , 25 , 60 , and 100 @xmath133 m ) . they have the far - infrared color ratio @xmath183 , which is significantly greater than that ( 0.20 ) of the other clouds in this area . this enhanced color , and the morphological correspondence with the broad molecular line emitting regions , suggests that the infrared emission from clouds n and s is due to shocks in propagating into the clouds.(5 ) along the eastern boundary of the snr , three relatively large ( @xmath23 or 3.5 pc ) clouds and several filamentary clouds are present . no broad co emission or enhanced 60/100 @xmath133 m color were detected in any of these clouds , and there is little relationship between the boundaries of the clouds and the snr . therefore , there is no strong evidence for the interaction of the snr with molecular clouds along the eastern boundary . arendt , r. g. 1989 , apjs , 70 , 181 arikawa , y. , tatematsu , k. , sekimoto , y. , & takahashi , t. 1999 , pasj , 51 , l7 aumann , h. h. , fowler , j. w. , & melnyk , m. 1990 , aj , 99 , 1674 beichman , c. a. et al . 1988 , _ infrared astronomical satellite ( iras ) catalogs and atlases : volume 1 . explanatory supplement _ , nasa rp-1190 ( nasa : washington , dc ) blandford , r. d. , & cowie , l. l. 1982 , apj , 260 , 625 boulanger , f. , & prault , m. 1988 , apj , 330 , 964 cesarsky , d. , cox , p. , pineau des forts , g. , van dishoeck , e. f. , boulanger , f. , & wright , c. m. 1999 , a & a , 348 , 945 chevalier , r. a. 1974 , apj , 188 , 501 1999 , apj , 511 , 798 denoyer , l. k. 1979 , apjl , 232 , l165 dsert , f. x. , boulanger , f. , & puget , j .- l . 1990 , a&a , 237 , 215 dickman , r. l. , snell , r. l. , ziurys , l. m. , & huang , y .- l . 1992 , apj , 400 , 203 dobashi , k. , bernard , j .- yonekura , y. , & fukui , y. 1994 , apjs , 95 , 419 draine , b. t. , & anderson , n. 1985 , apj , 292 , 494 draine , b. t. , & mckee , c. f. 1993 , araa , 31 , 373 draine , b. t. , roberge , w. g. , & dalgarno , a. 1983 , apj , 264 , 485 draine , b. t. , & roberge , w. g. 1984 , apj , 282 , 491 erkes , j. w. , & dickel , j. r. 1969 , aj , 74 , 840 frail , d. a. , goss , w. m. , reynoso , e. m. , giacani , e. b. , green , a. j. , & otrupcek , r. 1996 , aj , 111 , 1651 goldreich , p. , & kwan , j. 1974 , apj , 189 , 441 green , d. a. 1986 , mnras , 221 , 473 heiles , c. , reach , w. t. , & koo , b .- c . 1988 , apj , 332 , 313 hill , i. e. 1974 , mnras , 169 , 59 hollenbach , d. , & mckee , c. f. 1980 , apjl , 241 , l47 1989 , apj , 342 , 306 huang , y .- l . , & thaddeus , p. 1986 , apj , 309 , 804 humphreys , r. m. 1978 , apjs , 38 , 309 jones , t. w. , & kang , h. 1993 , apj , 402 , 560 jones , a. p. , tielens , a. g. g. m. , & hollenbach , d. j. 1996 , apj , 469 , 740 keohane , j. w. , petre , r. , gotthelf , e. v. , ozaki , m. , & koyama , k. 1997 , apj , 484 , 350 klein , r. l. , mckee , c. f. , colella , p. 1994 , apj , 420 , 213 koo , b .- c . , & heiles , c. 1991 , apj , 382 , 204 koo , b .- c . & moon , d .- s . 1997 , apj , 485 , 263 langer , w. d. 1997 , in co : twenty - five years of millimeter - wave spectroscopy , iau symp . latter , w. b. et al . ( dordrecht : kluwer ) , 98 leahy , d. a. 1987 , mnras , 228 , 907 leahy , d. a. , & aschenbach , b. 1996 , a&a , 315 , 260 lockett , p. , gautheir , e. , elitzur , m. 1999 , apj , 511 , 235 mac low , m .- mckee , c. f. , klein , r. i. , stone , j. m. , & norman , m. l. 1994 , apj , 433 , 757 mckee , c. f. 1988 , in supernova remnants and the interstellar medium , proceedings of iau colloquium no . roger and t.l . landecker ( cambridge : cambridge ) , 473 oliva , e. , moorwood , a. f. m. , drapatz , s. , lutz , d. , & sturm , e. 1999 , a & a , 343 , 943 reach , w. t. , & rho , j. 1996 , a&a , 315 , l277 . 1999 , apj , 511 , 836 . 2000a , aph0007148 . 2000b , in preparation rengelink , r. b. , tang , y. , de bruyn , a. g. , miley , g. k. , bremer , m. n. , rttgering , h. j. a. , & bremer , m. a. r. 1997 , a&as , 124 , 259 rho , j. , & petre , r. 1998 , apjl , 503 , l167 rho , j. , jarrett , t. , cutri , r. m. , & reach , w. t. 2000 , apj , submitted roh , d .- jung , j. h. 1999 , publications of the korean astronomical society , 14 , 123 sakamoto , s. , hasegawa , t. , handa , t. , hayashi , m. , & oka , t. 1997 , apj , 486 , 276 sakamoto , s. , hayashi , m. , hasegawa , t. , handa , t. , & oka , t. 1994 , apj , 425 , 641 scoville , n. z. , & solomon , p. m. 1974 , apjl , 187 , l67 seta , m. , hasegawa , t. , dame , t. m. , sakamoto , s. , oka , t. , handa , t. , hayashi , m. , morino , j. , sorai , k. , & usuda , k. s. 1998 , apj , 505 , 286 tatematsu , k. , fukui , y. , landecker , t. l. , & roger , r. s. 1990 , a&a , 237 , 189 ( t90 ) tauber , j. a. , snell , r. a. , dickman , r. l. , & ziurys , l. m. 1994 , apj , 421 , 570 van den bergh , s. 1978 , apjs , 38 , 119 van der laan , h. 1962 , mnras , 124 , 179 van dishoeck , e. f. , jansen , d. j. , & phillips , t. g. 1993 , a&a , 279 , 541 wheelock s. l. , gautier , t. n. , chillemi , j. , kester , d. , mccallon , h. , oken , c. , white , j. , gregorich , d. , boulanger , f. , and j. good 1994 , _ iras sky survey atlas : explanatory supplement_. ( jpl / caltech : pasadena ) willis , a. g. , 1973 , a&a , 26 , 237 wilner , d. j. , reynolds , s. p. , & moffett , d. a. 1998 , aj , 115 , 247 hb21:bml - n1 & ( 20 46 03.2 , 51 00 00 ) & @xmath199 & 3.6 & 1.6&@xmath59&@xmath200hb21:bml - n2 & ( 20 45 55.0 , 51 03 30 ) & @xmath201 & 3.1 & 1.7&@xmath202& ... hb21:bml - s1 & ( 20 44 37.2 , 49 47 10 ) & @xmath203 & 2.7 & 1.9&@xmath204&@xmath205hb21:bml - s2 & ( 20 44 31.0 , 49 55 20 ) & @xmath206 & 2.9 & 1.7& ... &@xmath207hb21:bml - s3 & ( 20 42 45.2 , 49 56 50 ) & @xmath208 & 6.9 & 2.3& ... &@xmath209 a & 1.1(0.2 ) & 1.1(0.1 ) & 4.2(0.2 ) & 20.8(2.3 ) & 0.20(0.03)b & 1.7(0.1 ) & 1.8(0.1 ) & 5.9(1.1 ) & 29.5(2.4 ) & 0.20(0.04)c & 1.7(0.1 ) & 1.7(0.1 ) & 8.0(1.1 ) & 38.0(2.4 ) & 0.21(0.03)n & 1.0(0.2 ) & 0.8(0.1 ) & 4.8(0.2 ) & 17.0(2.3 ) & 0.28(0.04)s & 1.2(0.1 ) & 1.2(0.1 ) & 6.8(1.1 ) & 26.3(2.4 ) & 0.26(0.05 )
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we have carried out @xmath0co j=21 line observations of the supernova remnant ( snr ) hb 21 in order to search for evidence of interaction with molecular clouds .
we mapped the eastern half ( @xmath1 ) of the snr almost completely .
molecular gas appears to be distributed mainly along the boundary of the snr , but the overall distribution has little correlation either with the distortion of the snr boundary or with the distribution of radio brightness . along the eastern boundary , where the snr was considered to be interacting with molecular clouds in previous studies ,
we have not found any strong evidence for the interaction .
instead we detected broad ( 2040 ) co emission lines in the northern and southern parts of the snr .
in the northern area , the broad - line emitting cloud is composed of a small ( @xmath2 or 0.5 pc ) , very bright , u - shaped part and several clumps scattered around it . there is a significant enhancement of radio emission with flat ( @xmath3 ) spectral index possibly associated with this cloud . in the southern area ,
the broad - line emitting cloud is filamentary and appears to form an elongated loop of @xmath4 in extent .
small ( @xmath5 or 0.3 pc ) , bright clumps are seen along the filamentary structure . we have obtained sensitive j=10 and j=21 spectra of @xmath0co and @xmath6co molecules toward several peak positions .
the intensity of @xmath0co j=21 emission is low ( @xmath7 k ) and the ratio of @xmath0co j=21 to j=10 integrated intensities is high ( 1.62.3 ) , which suggests that the emission is from warm , dense , and clumpy gas .
we have applied an lvg analysis to derive their physical parameters .
the detected broad co lines are believed to be emitted from the fast - moving molecular gas swept - up by the snr shock . the small ( @xmath8 ) shock velocity suggests that the shock is a non - dissociating c - shock .
we discuss the correlation of the shocked molecular gas with the previously detected , shocked atomic gas and the associated infrared emission .
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the atlas experiment@xcite is one of two general purpose collider detectors for the large hadron collider ( lhc ) at cern . the atlas detector consists of an inner detector employing silicon pixel , strip , and transition radiation tracking detectors , all in a solenoidal magnetic field of 2 tesla ; electromagnetic and hadronic calorimeters using liquid argon and scintillator tile detectors ; and a muon spectrometer . the muon spectrometer consists of a large air - core barrel and endcap toroid magnets with a @xmath1 between 2 - 6 @xmath2 , and four types of trigger and precision tracking detectors , described below . the muon spectrometer is designed to measure the transverse momentum ( @xmath3 ) of muons with @xmath4 gev with a resolution of 4% up to @xmath3 of 100 gev and increasing to 10% @ 1 tev . the atlas muon spectrometer consists of monitored drift tubes ( mdts ) for precision tracking in the spectrometer bending plane , resistive plate chambers ( rpcs ) and thin gap chambers ( tgcs ) for triggering in barrel and endcap , respectively , and cathode strip chambers ( cscs ) for precision measurements in the high - rate endcap inner layer where mdts would have occupancy problems . the magnet system consists of 3 sets of air - core toroids , each with 8 coils , 1 for the barrel , and 1 for each endcap . the barrel toroids coils are each 25 m @xmath5 7 m and the endcap coils are 9 m @xmath5 4 m . the magnetic field provides an approximately 1 t field at the center of each coils , but is non - uniform , especially in the barrel - endcap transition region . for track reconstruction , the field is mapped using a computer model of the field which is normalized to measurements from 1850 hall sensors mounted on spectrometer chambers . alignment measurements of the spectrometer are also critical for momentum determination and are accomplished with an optical alignment system of 12k sensors . measurements from these sensors allow a 3-dimensional reconstruction of chamber positions accurate to better than 50 @xmath0 m . in addition , the optical alignment system is complemented by alignment done with tracks . table [ table : muon_spectrometer ] gives a summary of the muon spectrometer detector components and fig . [ fig : layout ] shows the layout . .atlas muon spectrometer . [ cols="<,^,^,^,^",options="header " , ] [ table : muon_spectrometer ] the spectrometer is designed so that muons cross three layers of mdt chambers for the sagitta measurement . the track coordinate in the bending plane of the spectrometer is measured by the precision chambers with a resolution of 40 @xmath0 m . in comparison , the sagitta of a 1 tev muon will be about 500 @xmath0 m . the trigger chambers are placed on opposite sides of the middle mdt layer . the trigger chambers provide a trigger based on muon momentum in addition to identifying the bunch crossing time of the muon . the also provide the second coordinate measurement ( non - bending plane ) accurate to 5 - 10 cm . figure [ fig : layout ] shows the expected resolution of the muon spectrometer . for @xmath6 gev / c multiple scattering is the dominant contributor to the resolution . above 100 gev / c calibration and alignment of the spectrometer become the most significant factors in momentum resolution . atlas muon reconstruction is done using momentum measurements from both the inner detector spectrometer and the muon spectrometer . the two spectrometers nicely complement each other as inner detector measurements are better below 100 gev / c above which the muon spectrometer resolution is superior . high @xmath3 measurements with the muon spectrometer require very accurate mdt and alignment calibrations which will become particularly important in a few years when lhc reaches 7 tev energy per beam and higher luminosities . this paper shows a mix of results from 2010 and 2011 using the most up - to - date plots whenever possible . not all plots are available for 2011 so in some cases 2010 plots are used . there are three calibrations required for the mdts : timing offsets ( @xmath7 ) ; time - space ( @xmath8 ) functions ; and drift tube resolution functions@xcite . in order to obtain high quality calibrations for the mdts a special high statistics calibration data stream is extracted second - level trigger processors and sent for processing at three calibration centers at michigan , rome , and munich . this calibration stream provides 10 - 100x the rate of single muon tracks compared to regular atlas data . with this stream it is possible to do daily calibrations of the monitored drift tubes as well as detailed data quality monitoring . a timing offset represents the minimum measured drift time . i.e. the time of a muon passing at the wire of the drift tube . this time is not zero due to cables and other delays in the data acquisition system . figure [ fig : drifttime ] shows a typical time spectrum from an mdt . the @xmath7 fit is shown in blue , and the @xmath7 is defined as the half - way point of the rising edge . the falling edge represents hits at the tube wall . figure [ fig : drifttime ] shows statistical error on @xmath7 fits as a function of number hits in the time spectrum . we require at least 10000 hits for the fits yielding a typical error of 0.5 ns . the average drift speed is 20 @xmath0m / ns so this error corresponds to a 10 @xmath0 m error due to the @xmath7 measurement . fit shown in blue ; right : the statistical error on @xmath7 fits as a function of number hits in the time spectrum . , title="fig:",width=302 ] fit shown in blue ; right : the statistical error on @xmath7 fits as a function of number hits in the time spectrum . , title="fig:",width=302 ] s for all mdt chambers over a 2-month period in 2010 . right : tracking residuals from the middle layer of mdt chambers . , title="fig:",width=336 ] s for all mdt chambers over a 2-month period in 2010 . right : tracking residuals from the middle layer of mdt chambers . , title="fig:",width=268 ] figure [ fig : t0diff ] shows the distribution of the change in @xmath7 for all mdt chambers over a 2 month period from 2010 . there is small global drift of a fraction of a nanosecond , but the overall width is close to the typical statistical error of 0.5 ns . hence , the t0s are quite stable . the other main calibration is the time - to - space or @xmath8 function . this function gives the drift radius ( impact parameter ) of the hit based on the drift time . an example of an @xmath8 function is shown in fig . [ fig : rtfun ] . the function is non - linear since atlas uses a non - linear drift gas , @xmath9 due to its better aging characteristics in high - radiation environments . the @xmath8 function is determined by an iterative procedure looping over tubes hits and minimizing the tracking residuals of track segments reconstructed with hits within a single mdt chamber which have either 6 or 8 layers of mdt tubes . tracking residuals are the differences between the drift radii from the drift time and the radius from the track fit . typically 10000 tracks segments are used in the calibration of a single chamber . figure [ fig : rtfun ] shows the difference between @xmath8 functions for several chambers in the barrel . the differences in @xmath8 functions are due primarily to the temperature gradient within the atlas cavern ( about @xmath10 from top to bottom ) , as well as due differences in magnetic field within chambers . function ; right : differences between @xmath8 functions for several barrel chambers.,title="fig:",width=245 ] function ; right : differences between @xmath8 functions for several barrel chambers.,title="fig:",width=374 ] the precision of the @xmath8 function is shown by fig . [ fig : resvr ] which shows the mean of the tracking residuals as a function of the mdt tube radius . except for the region close to the wire , the mean residuals are within 20 @xmath0 m . tracking residuals from the middle layer of endcap chambers are shown in fig . [ fig : t0diff ] . the residual width of 96 @xmath0 m is typical for all chambers in atlas . figure [ fig : resvr ] shows the single tube resolution as a function of drift tube radius . the resolution is determined from the tracking residuals width with the fit errors subtracted . we see that the resolution is close to the 80 @xmath0 m for large radii . near the wire the resolution degrades to faster drift speed and fewer drift electrons . this plot was made with 2010 data . we expect improvements in the future by applying some addition timing corrections such as a hit - level magnetic field correction and by using tube - level @xmath7s . the alignment system system is designed to track chamber positions with a 40 @xmath0 m precision . the monitoring is done with optical sensors which is cross - checked by doing alignment with straight tracks from magnet - off runs . the barrel and endcap have separate alignment systems . figure [ fig : alignment ] shows measurements from the mean value of the `` false '' sagitta measured with straight tracks from magnet - off runs . straight tracks should have a sagitta of zero and hence this sagitta measurement gives the precision of the alignment system . the plot shows the sagitta as a function of @xmath11 with the black points for the barrel and the red and blue corresponding with to the endcap . the barrel achieves a resolution of 50 @xmath0 m , close to the design goal , whereas the endcap gives a resolution of around 110 @xmath0 m indicting that further improvements are necessary . sector 7 of the muon spectrometer.,width=302 ] figure [ fig : triggerocc ] shows the an occupancy plot for the barrel from collision data . as can be seen the coverage is quite uniform except for dead regions due to the support feet of the atlas detector . the geometric acceptance is about 80% . figure [ fig : triggereff ] shows the trigger efficiency as a function of @xmath3 and @xmath11 , respectively . the triggers are very efficient within geometric coverage of the trigger . the data show a slightly higher efficiency than monte carlo due to some analysis improvements in the data analysis which have not yet been introduced to the monte carlo code . and @xmath11.,width=302 ] ; right : trigger efficiency as a function @xmath11.,title="fig:",width=302 ] ; right : trigger efficiency as a function @xmath11.,title="fig:",width=302 ] figure [ fig : momresb ] shows the momentum resolution as a function of muon @xmath3 in the barrel region of atlas . the momentum resolution is derived from the width of the reconstructed z mass as well as by comparing single muons reconstructed by both the inner detector and muon spectrometer . monte carlo and inner detector measurements are used to derive the contributions of momentum resolution from energy loss in calorimeters , multiple scattering , and the intrinsic resolution of the spectrometer . the measured resolution is somewhat worse than the simulation . this result is from the preliminary calibrations for 2010 data , so we expect improvement from more refined calibrations in the future . the atlas muon spectrometer is working well with a high trigger efficiency and tracking resolution near design specifications . calibrations of the drift tubes are done daily using a high statistics data stream from the level-2 trigger processors . alignment is working well in barrel , but needs some improvement in the endcap . momentum resolution is near design specifications . we expect improvements from better calibrations and statistics from 2011 data . 9 `` the atlas experiment at the cern large hadron collider '' , the atlas collaboration , g aad _ et al _ 2008 jinst * 3 * s08003 , http://www.iop.org/ej/abstract/1748-0221/3/08/s08003 `` calibration model for the mdt chambers of the atlas muon spectrometer '' , p. bagnaia _ et al _ , atlas note atl - muon - pub-2008 - 004 , http://cdsweb.cern.ch/record/1089868
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the atlas muon spectrometer is designed to measure muon momenta with a resolution of 4% @ 100 gev / c rising to 10% @ 1 tev / c track momentum .
the spectrometer consists of precision tracking and trigger chambers embedded in a 2 t magnetic field generated by three large air - core superconducting toroids .
the precision detectors provide 50 @xmath0 m tracking resolution to a pseudo - rapidity of 2.7 .
the system also includes an optical monitoring system which measures detector positions with 40 @xmath0 m precision .
this paper reports on the calibration and performance of the atlas muon spectrometer .
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when complex systems join to form even more complex systems , the interaction of the constituent subsystems is highly random @xcite . the complex stochastic interactions among these subsystems are commonly quantified by calculating the cross - correlations . this method has been applied in systems ranging from nanodevices @xcite , atmospheric geophysics @xcite , and seismology @xcite , to finance @xcite . here we propose a method of estimating the most significant component in explaining long - range cross - correlations . studying cross - correlations in these diverse physical systems provides insight into the dynamics of natural systems and enables us to base our prediction of future outcomes on current information . in finance , we base our risk estimate on cross - correlation matrices derived from asset and investment portfolios @xcite . in seismology , cross - correlation levels are used to predict earthquake probability and intensity @xcite . in nanodevices used in quantum information processing , electronic entanglement necessitates the computation of noise cross - correlations in order to determine whether the sign of the signal will be reversed when compared to standard devices @xcite . reference @xcite reports that cross - correlations for @xmath0 calculated between pairs of eeg time series are inversely related to dissociative symptoms ( psychometric measures ) in 58 patients with paranoid schizophrenia . in genomics data , @xcite reports spatial cross - correlations corresponding to a chromosomal distance of @xmath1 million base pairs . in physiology , @xcite reports a statistically significant difference between alcoholic and control subjects . many methods have been used to investigate cross - correlations ( i ) between pairs of simultaneously recorded time series @xcite or ( ii ) among a large number of simultaneously - recorded time series @xcite . reference @xcite uses a power mapping of the elements in the correlation matrix that suppresses noise . reference @xcite proposes detrended cross - correlation analysis ( dcca ) , which is an extension of detrended fluctuation analysis ( dfa ) @xcite and is based on detrended covariance . reference @xcite proposes a method for estimating the cross - correlation function @xmath2 of long - range correlated series @xmath3 and @xmath4 . for fractional brownian motions with hurst exponents @xmath5 and @xmath6 , the asymptotic expression for @xmath2 scales as a power of @xmath7 with exponents @xmath5 and @xmath6 . univariate ( single ) financial time series modeling has long been a popular technique in science . to model the auto - correlation of univariate time series , traditional time series models such as autoregressive moving average ( arma ) models have been proposed @xcite . the arma model assumes variances are constant with time . however , empirical studies accomplished on financial time series commonly show that variances change with time . to model time - varying variance , the autoregressive conditional heteroskedasticity ( arch ) model was proposed @xcite . since then , many extensions of arch has been proposed , including the generalized autoregressive conditional heteroskedasticity ( garch ) model @xcite and the fractionally - integrated autoregressive conditional heteroskedasticity ( fiarch ) model @xcite . in these models , long - range auto - correlations in magnitudes exist , so a large price change at one observation is expected to be followed by a large price change at the next observation . long - range auto - correlations in magnitude of signals have been reported in finance @xcite , physiology @xcite , river flow data @xcite , and weather data @xcite . besides univariate time series models , modeling correlations in multiple time series has been an important objective because of its practical importance in finance , especially in portfolio selection and risk management @xcite . in order to capture potential cross - correlations among different time series , models for coupled heteroskedastic time series have been introduced @xcite . however , in practice , when those models are employed , the number of parameters to be estimated can be quite large . a number of researchers have applied multiple time series analysis to world indices , mainly in order to analyze zero time - lag cross - correlations . reference @xcite reported that for international stock return of nine highly - developed economies , the cross - correlations between each pair of stock returns fluctuate strongly with time , and increase in periods of high market volatility . by volatility we mean time - dependent standard deviation of return . the finding that there is a link between zero time lag cross - correlations and market volatility is `` bad news '' for global money managers who typically reduce their risk by diversifying stocks throughout the world . in order to determine whether financial crises are short - lived or long - lived , ref . @xcite recently reported that , for six latin american markets , the effects of a financial crisis are short - range . between two and four months after each crisis , each latin american market returns to a low - volatility regime . in order to determine whether financial crisis are short - term or long - term at the world level , we study 48 world indices , one for each of 48 different countries . we analyze cross - correlations among returns and magnitudes , for zero and non - zero time lags . we find that cross - correlations between magnitudes last substantially longer than between the returns , similar to the properties of auto - correlations in stock market returns @xcite . we propose a general method in order to extract the most important factors controlling cross - correlations in time series . based on random matrix theory @xcite and principal component analysis @xcite we propose how to estimate the global factor and the most significant principal components in explaining the cross - correlations . this new method has a potential to be broadly applied in diverse phenomena where time series are measured , ranging from seismology to atmospheric geophysics . this paper is organized as follows . in section ii we introduce the data analyzed , and the definition of return and magnitude of return . in section iii we introduce a new modified time lag random matrix theory ( tlrmt ) to show the time - lag cross - correlations between the returns and magnitudes of world indices . empirical results show that the cross - correlations between magnitudes decays slower than that between returns . in section iv we introduce a single global factor model to explain the short- or long - range correlations among returns or magnitudes . the model relates the time - lag cross - correlations among individual indices with the auto - correlation function of the global factor . in section v we estimate the global factor by minimizing the variance of residuals using principal component analysis ( pca ) , and we show that the global factor does in fact account for a large percentage of the total variance using rmt . in section vi we show the applications of the global factor model , including risk forecasting of world economy , and finding countries who have most the independent economies . in order to estimate the level of relationship between individual stock markets either long - range or short - range cross - correlations exist at the world level we analyze @xmath8 world - wide financial indices , @xmath9 , where @xmath10 denotes the financial index and @xmath11 denotes the time . we analyze one index for each of 48 different countries : 25 european indices @xcite , 15 asian indices ( including australia and new zealand ) @xcite , 2 american indices @xcite , and 4 african indices @xcite . in studying 48 economies that include both developed and developing markets we significantly extend previous studies in which only developed economies were included e.g . , the seven economies analyzed in refs . @xcite , and the 17 countries studied in ref . we use daily stock - index data taken from _ bloomberg _ , as opposed to weekly @xcite or monthly data @xcite . the data cover the period 4 jan 1999 through 10 july 2009 , 2745 trading days . for each index @xmath9 , we define the relative index change ( return ) as @xmath12 where @xmath11 denotes the time , in the unit of one day . by magnitude of return we denote the absolute value of return after removing the mean @xmath13 in order to quantify the cross - correlations , random matrix theory ( rmt ) ( see refs . @xcite @xcite and references therein ) was proposed in order to analyze collective phenomena in nuclear physics . @xcite extended rmt to cross - correlation matrices in order to find cross - correlations in collective behavior of financial time series . the largest eigenvalue @xmath14 and smallest eigenvalue @xmath15 of the wishart matrix ( a correlation matrix of uncorrelated time series with finite length ) are @xmath16 where @xmath17 , and @xmath18 is the matrix dimension and @xmath19 the length of each time series . the larger the discrepancy between ( a ) the correlation matrix between empirical time series and ( b ) the wishart matrix obtained between uncorrelated time series , the stronger are the cross - correlations in empirical data @xcite . many rmt studies reported equal - time ( zero @xmath20 ) cross - correlations between different empirical time series @xcite . recently time - lag generalizations of rmt have been proposed @xcite . in one of the generalizations of rmt , based on the eigenvalue spectrum called time - lag rmt ( tlrmt ) , ref . @xcite found long - range cross - correlations in time series of price fluctuations in absolute values of 1340 members of the new york stock exchange composite , in both healthy and pathological physiological time series , and in the mouse genome . we compute for varying time lags @xmath20 the largest singular values @xmath21 of the cross - correlation matrix of n - variable time series @xmath22 @xmath23 we also compute @xmath24 of a similar matrix @xmath25 , where @xmath22 are replaced by the magnitudes @xmath26 . the squares of the non - zero singular values of are equal to the non - zero eigenvalues of @xmath27 or @xmath27 , where by @xmath27 we denote the transpose of . in a singular value decomposition ( svd ) @xcite @xmath28 the diagonal elements of are equal to singular values of , where the and correspond to the left and right singular vectors of the corresponding singular values . we apply svd to the correlation matrix for each time lag and calculate the singular values , and the dependence of the largest singular value @xmath21 on @xmath20 serves to estimate the functional dependence of the collective behavior of @xmath29 on @xmath20 @xcite . we make two modifications of correlation matrices in order to better describe correlations for both zero and non - zero time lags . * the first modification is a correction for correlation between indices that are not frequently traded . since different countries have different holidays , all indices contain a large number of zeros in their returns . these zeros lead us to underestimate the magnitude of the correlations . to correct for this problem , we define a modified cross - correlation between those time series with extraneous zeros , @xmath30 here @xmath31 is the time period during which both @xmath22 and @xmath32 are non - zero . with this definition , the time periods during which @xmath22 or @xmath32 exhibit zero values have been removed from the calculation of cross - correlations . the relationship between @xmath33 and @xmath34 is @xmath35 * the second modification corrects for auto - correlations . the main diagonal elements in the correlation matrix are ones for zero - lag correlation matrices and auto - correlations for non - zero lag correlation matrices . thus , time - lag correlation matrices allow us to study both auto - correlations and time - lag cross - correlations . if we study the decay of the largest singular value , we see a long - range decay pattern if there are long - range auto - correlations for some indices but no cross - correlation between indices . to remove the influence of auto - correlations and isolate time - lag cross - correlations , we replace the main diagonals by unity , @xmath36 with this definition the influence of auto - correlations is removed , and the trace is kept the same as the zero time - lag correlation matrix . in fig . 1(a ) we show the distribution of cross - correlations between zero and non - zero lags . for @xmath37 the empirical pdf @xmath38 of the cross - correlation coefficients @xmath29 substantially deviates from the corresponding pdf @xmath39 of a wishart matrix , implying the existence of equal - time cross - correlations . in order to determine whether short - range or long - range cross - correlations accurately characterize world financial markets , we next analyze cross - correlations for @xmath40 . we find that with increasing @xmath20 the form of @xmath38 quickly approaches the pdf @xmath39 , which is normally distributed with zero mean and standard deviation @xmath41 @xcite . in fig . 1(b ) we also show the distribution of cross - correlations between _ magnitudes_. in financial data , returns @xmath42 are generally uncorrelated or short - range auto - correlated , whereas the magnitudes are generally long - range auto - correlated @xcite . we thus examine the cross - correlations @xmath43 between @xmath44 for different @xmath20 . in fig . 1(b ) we find that with increasing @xmath20 , @xmath45 approaches the pdf of random matrix @xmath39 more slowly than @xmath46 , implying that cross - correlations between index magnitudes persist longer than cross - correlations between index returns . in order to demonstrate the decay of cross - correlations with time lags , we apply modified tlrmt . fig . 2 shows that with increasing @xmath20 the largest singular value calculated for @xmath47 decays more slowly than the largest singular value calculated for . this result implies that among world indices , the cross - correlations between magnitudes last longer than cross - correlations between returns . in fig . 2 we find that @xmath48 vs. @xmath20 decays as a power law function with the scaling exponent equal to 0.25 . the faster decay of @xmath48 vs. @xmath20 for implies very weak ( or zero ) cross - correlations among world - index returns for larger @xmath20 , which agrees with the empirical finding that world indices are often uncorrelated in returns . our findings of long - range cross - correlations in magnitudes among the world indices is , besides a finding in ref . @xcite , another piece of `` bad news '' for international investment managers . world market risk decays very slowly . once the volatility ( risk ) is transmitted across the world , the risk lasts a long time . the arbitrage pricing theory states that asset returns follow a linear combination of various factors @xcite . we find that the factor structure can also model time lag pairwise cross - correlations between the returns and between magnitudes . to simplify the structure , we model the time lag cross - correlations with the assumption that each individual index fluctuates in response to one common process , the `` global factor '' @xmath49 , @xmath50 here in the global factor model ( gfm ) , @xmath51 is the average return for index @xmath52 , @xmath49 is the global factor , and @xmath53 is the linear regression residual , which is independent of @xmath49 , with mean zero and standard deviation @xmath54 . here @xmath55 indicates the covariance between @xmath42 and @xmath49 , @xmath56 . this single factor model is similar to the sharpe market model @xcite , but instead of using a known financial index as the global factor @xmath49 , we use factor analysis to find @xmath49 , which we introduce in the next section . we also choose @xmath49 as a zero - mean process , so the expected return @xmath57 , and the global factor @xmath49 is only related with market risk . we define a zero - mean process @xmath58 as @xmath59 a second assumption is that the global factor can account for most of the correlations . therefore we can assume that there are no correlations between the residuals of each index , @xmath60 . then the covariance between @xmath42 and @xmath61 is @xmath62 the covariance between magnitudes of returns depends on the return distribution of @xmath49 and @xmath42 , but the covariance between squared magnitudes @xmath63 indicates the properties of the magnitude cross - correlations . the covariance between @xmath64 and @xmath65 is @xmath66 the above results in eqs . ( [ cov1])-([cov2 ] ) show that the variance of the global factor and square of the global factor account for all the zero time lag covariance between returns and squared magnitudes . for time lag covariance between @xmath58 , we find @xmath67 here @xmath68 is the autocovariance of @xmath49 . similarly , we find @xmath69 here @xmath70 is the autocovariance of @xmath71 . in gfm , the time lag covariance between each pair of indices is proportional to the autocovariance of the global factor . for example , if there is short - range autocovariance for @xmath49 and long - range autocovariance for @xmath71 , then for individual indices the cross - covariance between returns will be short - range and the cross - covariance between magnitudes will be long - range . therefore , the properties of time - lag cross - correlation in multiple time series can be modeled with a single time series the global factor @xmath49 . the relationship between time lag covariance among two index returns and autocovariance of the global factor also holds for the relationship between time lag cross - correlations among two index returns and auto - correlation function of the global factor , because it only need to normalize the original time series to mean zero and standard deviation one . in contrast to domestic markets , where for a given country we can choose the stock index as an estimator of the `` global '' factor , when we study world markets the global factor is unobservable . at the world level when we study cross - correlations among world markets , we estimate the global factor using principal component analysis ( pca ) @xcite . in this section we use bold font for n dimensional vectors or @xmath72 matrix , and underscore @xmath11 for time series . suppose @xmath73 is the multiple time series , each row of which is an individual time series @xmath74 . we standardize each time series to zero mean and standard deviation 1 as @xmath75 the correlation matrix can be calculated as @xmath76 where @xmath77 is the transpose of @xmath78 , and the @xmath19 in the denominator is the length of each time series . then we diagonalize the @xmath79 correlation matrix @xmath80 @xmath81 here @xmath82 and @xmath83 are the eigenvalues in non - increasing order , @xmath84 is an orthonormal matrix , whose @xmath52-th column is the basis eigenvector @xmath85 of @xmath80 , and @xmath86 is the transpose of @xmath84 , which is equal to @xmath87 because of orthonormality . for each eigenvalue and the corresponding eigenvector , it holds @xmath88 according to pca , @xmath89 is defined as the @xmath52-th principal component ( @xmath90 ) , and the eigenvalue @xmath91 indicates the portion of total variance of @xmath78 contributed to @xmath92 , as shown in eq . ( [ pca_value ] ) . since the total variance of @xmath78 is @xmath93 the expression @xmath94 indicates the percentage of the total variance of @xmath78 that can be explained by the @xmath92 . according to pca ( a ) the principal components @xmath92 are uncorrelated with each other and ( b ) @xmath92 maximizes the variance of the linear combination of @xmath95 with the orthonormal restriction @xmath96 given the previous principal components @xcite . from the orthonormal property of @xmath84 we obtain @xmath97 where * i * is the identity matrix . then the multiple time series @xmath78 can be represented as a linear combination of all the @xmath98 @xmath99 the total variance of all time series can be proved to be equal to the total variance of all principal components @xmath100 next we assume that @xmath101 is much larger than each of the rest of eigenvalues which means that the first @xmath98 , @xmath102 , accounts for most of the total variances of all the time series . we express @xmath78 as the sum of the first part of eq . ( [ r2pc ] ) corresponding to @xmath103 and the error term combined from all other terms in eq . ( [ r2pc ] ) . thus @xmath104 then @xmath103 is a good approximation of the global factor @xmath49 , because it is a linear combination of @xmath42 that accounts for the largest amount of the variance . @xmath105 is a zero - mean process because it is a linear combination of @xmath106 which are also zero - mean processes ( see eq . ( [ z ] ) ) . comparing eqs . ( [ z ] ) and ( [ r2pc1 ] ) with @xmath107 we find the following estimates : @xmath108 using eq . ( [ pca_value ] ) we find that @xmath109 in the rest of this work , we apply the method of eq . ( [ duan ] ) to empirical data . next we apply the method of eq . ( [ duan ] ) to estimate the global factor of 48 world index returns . we calculate the auto - correlations of @xmath49 and @xmath110 , which are shown in figs . 3 and 4 . precisely , for the world indices , fig . 3(a ) shows the time series of the global factor @xmath49 , and fig . 3(b ) shows the auto - correlations in @xmath49 . we find only short - range auto - correlations because , after an interval @xmath111 , most auto - correlations in @xmath49 fall in the range of @xmath112 @xcite , which is the 95% confidence interval for zero auto - correlations , here @xmath113 . for the 48 world index returns , fig . 4(a ) shows the time series of magnitudes @xmath110 , with few clusters related to market shocks during which the market becomes fluctuates more . 4(b ) shows that , in contrast to @xmath49 , the magnitudes @xmath110 exhibit long - range auto - correlations since the values @xmath110 are significant even after @xmath114 . the auto - correlation properties of the global factor are the same as the auto - correlation properties of the individual indices , i.e. , there are short - range auto - correlations in @xmath49 and long - range power - law auto - correlations in @xmath110 @xcite . these results are also in agreement with fig . 1(b ) where the largest singular value @xmath48 vs. @xmath20 calculated for @xmath47 decays more slowly than the largest singular value calculated for . as found in ref . @xcite for @xmath115 , @xmath21 approximately follows the same decay pattern as cross - correlation functions . although a ljung - box test shows that the return auto - correlation is significant for a 95% confidence level @xcite , the return auto - correlation is only 0.132 for @xmath116 and becomes insignificant after @xmath111 . therefore , for simplicity , we only consider magnitude cross - correlations in modeling the global factor . we model the long - range market - factor returns * m * with a particular version of the garch process , the gjr garch process @xcite , because this garch version explains well the asymmetry in volatilities found in many world indices @xcite . the gjr garch model can be written as @xmath117 where @xmath118 is the volatility and @xmath119 is a random process with a gaussian distribution with standard deviation 1 and mean 0 . the coefficients @xmath120 and @xmath121 are determined by a maximum likelihood estimation ( mle ) and @xmath122 if @xmath123 , @xmath124 if @xmath125 . we expect the parameter @xmath126 to be positive , implying that `` bad news '' ( negative increments ) increases volatility more than `` good news '' . for the sake of simplicity , we follow the usual procedure of setting @xmath127 in all numerical simulations . in this case , the gjr - garch(1,1 ) model for the market factor can be written as @xmath128 we estimate the coefficients in the above equations using mle , where the estimated coefficients are shown in table . 1 . next we test the hypothesis that a significant percentage of the world cross - correlations can be explained by the global factor . by using pca we find that the global factor can account for 30.75% of the total variance . note that , according to rmt , only the eigenvalues larger than the largest eigenvalue of a wishart matrix calculated by eq . ( [ rmt ] ) ( and the corresponding @xmath129s ) are significant . to calculate the percentage of variance the significant @xmath129s account for , we employ the rmt approach proposed in ref . the largest eigenvalue for a wishart matrix is @xmath130 for @xmath8 and @xmath113 we found in the empirical data . from all the 48 eigenvalues , only the first three are significant : @xmath131 , @xmath132 , and @xmath133 . this result implies that among the significant factors , the global factor accounts for @xmath134 of the variance , confirming our hypothesis that the global factor accounts for most variance of all individual index returns . pca is defined to estimate the percentage of variance the global factor can account for zero time lag correlations . next we study the time lag cross - correlations after removing the global trend , and apply the svd to the correlation matrix of regression residuals @xmath135 of each index [ see eq . ( [ duan1 ] ) ] . our results show that for both returns and magnitudes , the remaining cross - correlations are very small for all time lags compared to cross - correlations obtained for the original time series . this result additionally confirms that a large fraction of the world cross - correlations for both returns and magnitudes can be explained by the global factor . the asymptotic ( unconditional ) variance for the gjr - garch model is @xmath136 @xcite . for the market factor the conditional volatility @xmath118 can be estimated by recursion using the historical conditional volatilities and fitted coefficients in eq . ( [ pr2d ] ) . for example , the largest cluster at the end of the graph shows the 2008 financial crisis . in fig . 5(a ) we show the time series of the conditional volatility of eq . ( [ pr2d ] ) of the global factor . the clusters in the conditional volatilities may serve to predict market crashes . in each cluster , the height is a measure of the size of the market crash , and the width indicates its duration . in fig . 5(b ) we show the forecasting of the conditional volatility of the global factor , which asymptotically converges to the unconditional volatility . next , in fig . [ cormarind ] we show the cross - correlations between the global factor and each individual index using eq . ( [ pc12cor ] ) . there are indices for which cross - correlations with the global factor are very small compared to the other indices ; 10 of 48 indices have cross - correlations coefficients with the global factor smaller than 0.1 . these indices correspond to iceland , malta , nigeria , kenya , israel , oman , qatar , pakistan , sri lanka , and mongolia . the financial market of each of these countries is weakly bond with financial markets of other countries . this is useful information for investment managers because one can reduce the risk by investing in these countries during world market crashes which , seems , do not severely influence these countries . we have developed a modified time lag random matrix theory ( tlrmt ) in order to quantify the time - lag cross - correlations among multiple time series . applying the modified tlrmt to the daily data for 48 world - wide financial indices , we find short - range cross - correlations between the returns , and long - range cross - correlations between their magnitudes . the magnitude cross - correlations show a power law decay with time lag , and the scaling exponent is 0.25 . the result we obtain , that at the world level the cross - correlations between the magnitudes are long - range , is potentially significant because it implies that strong market crashes introduced at one place have an extended duration elsewhere which is `` bad news '' for international investment managers who imagine that diversification across countries reduces risk . we model long - range world - index cross - correlations by introducing a global factor model in which the time lag cross - correlations between returns ( magnitudes ) can be explained by the auto - correlations of the returns ( magnitudes ) of the global factor . we estimate the global factor as the first component by using principal component analysis . using random matrix theory , we find that only three principal components are significant in explaining the cross - correlations . the global factor accounts for 30.75% of the total variance of all index returns , and 75.34% of the variance of the three significant principle components . therefore , in most cases , a single global factor is sufficient . we also show the applications of the gfm , including locating and forecasting world risk , and finding individual indices that are weakly correlated to the world economy . locating and forecasting world risk can be realized by fitting the global factor using a gjr - garch(1,1 ) model , which explains both the volatility correlations and the asymmetry in the volatility response to both `` good news '' and `` bad news . '' the conditional volatilities calculated after fitting the gjr - garch(1,1 ) model indicates the global risk , and the risk can be forecasted by recursion using the historical conditional volatilities and the fitted coefficients . to find the indices that are weakly correlated to the world economy , we calculate the correlation between the global factor and each individual index . we find 10 indices which have a correlation smaller than 0.1 , while most indices are strongly correlated to the global factor with the correlations larger than 0.3 . to reduce risk , investment managers can increase the proportion of investment in these countries during world market crashes , which do not severely influence these countries . based on principal component analysis , we propose a general method which helps extract the most significant components in explaining long - range cross - correlations . this makes the method suitable for broad range of phenomena where time series are measured , ranging from seismology and physiology to atmospheric geophysics . we expect that the cross - correlations in eeg signals are dominated by the small number of most significant components controlling the cross - correlations . we speculate that cross - correlations in earthquake data are also controlled by some major components . thus the method may have significant predictive and diagnostic power that could prove useful in a wide range of scientific fields . c. k. peng _ et al _ phys . e * 49 * , 1685 ( 1994 ) ; a. carbone , g. castelli , and h. e. stanley , physica a * 344 * , 267 ( 2004 ) ; l. xu _ et al _ , phys . e * 71 * , 051101 ( 2005 ) ; a. carbone , g. castelli , and h. e. stanley , phys . e * 69 * , 026105 ( 2004 ) . ftse 100 , dax , cac 40 , ibex 35 , swiss market , ftse mib , psi 20 , irish overall , omx iceland 15 , aex , bel 20 , luxembourg luxx , omx copenhagen 20 , omx helsinki , obx stock , omx stockholm 30 , austrian traded atx , athex composite share price , wse wig , prague stock exch , budapest stock exch indx , bucharest bet index , sbi20 slovenian total mt , omx tallin omxt , malta stock exchange ind , ftse / jse africa top40 ix ise national 100 , tel aviv 25 , msm30 , dsm 20 , mauritius stock exchange , nikkei 225 , hang seng , shanghai se b share , all ordinaries , nzx all , karachi 100 , sri lanka colombo all sh , stock exch of thai , jakarta composite , ftse bursa malaysia klci , psei - philippine se , mse top 20 .gjr - garch(1,1 ) coefficients of the global factor . the p - values and t - values comfirms that all these parameters are significant at 95% confidence level . the positive value of @xmath126 means bad news " has larger impact on the global market than good news " . we find @xmath137 , which is very close to 1 , and so indicate long - range volatility auto - correlations . [ cols="^,^,^,^,^ " , ] world financial index returns each of size @xmath113 ( a ) the empirical pdf of the coefficients of the cross - correlation matrix calculated between index returns with increasing @xmath20 quickly converges to the gaussian form . the normal distribution is the distribution of the pairwise cross - correlations for finite length uncorrelated time series , which is a normal distribution with mean zero and standard deviation @xmath138 . between ( b ) the empirical pdf of the coefficients of the matrix @xmath139 calculated between index volatilities approaches the pdf of the random matrix more slowly than in ( a ) . , title="fig:",scaledwidth=50.0% ] + world financial index returns each of size @xmath113 ( a ) the empirical pdf of the coefficients of the cross - correlation matrix calculated between index returns with increasing @xmath20 quickly converges to the gaussian form . the normal distribution is the distribution of the pairwise cross - correlations for finite length uncorrelated time series , which is a normal distribution with mean zero and standard deviation @xmath138 . between ( b ) the empirical pdf of the coefficients of the matrix @xmath139 calculated between index volatilities approaches the pdf of the random matrix more slowly than in ( a ) . , title="fig:",scaledwidth=50.0% ] + obtained from the spectrum of the matrices and @xmath140 versus time lag @xmath20 . with increasing @xmath20 , the largest singular values obtained for of returns decays more quickly than @xmath140 calculated for absolute values of returns . the magnitude cross - correlations decay as a power law function with the scaling exponent of @xmath141.,title="fig:",scaledwidth=60.0% ] + , and become insignificant after time lag @xmath111 , with no more than one significant auto - correlation for every 20 time lags . therefore , only short - range auto - correlations can be found in the global factor.,title="fig:",scaledwidth=60.0% ] + , and become insignificant after time lag @xmath111 , with no more than one significant auto - correlation for every 20 time lags . therefore , only short - range auto - correlations can be found in the global factor.,title="fig:",scaledwidth=60.0% ] , and is still larger than 0.2 until @xmath142 . for every time lag , the autocorrelation is significant even after @xmath114 . therefore long - range auto - correlations exist in the magnitudes of the global factor.,title="fig:",scaledwidth=60.0% ] + , and is still larger than 0.2 until @xmath142 . for every time lag , the autocorrelation is significant even after @xmath114 . therefore long - range auto - correlations exist in the magnitudes of the global factor.,title="fig:",scaledwidth=60.0% ] and each individual index @xmath42 , @xmath143 . the global factor has large correlation with most of the indices . however , there are indices that are not much correlated with the global factor . 10 of the 48 indices have a correlation smaller than 0.1 between the global factor , corresponding to the indices for iceland , malta , nigeria , kenya , israel , oman , qatar , pakistan , sri lanka , and mongolia . hence , unlike most countries , the economies of these 10 countries are more independent of the world economy.,scaledwidth=60.0% ]
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we propose a modified time lag random matrix theory in order to study time lag cross - correlations in multiple time series .
we apply the method to 48 world indices , one for each of 48 different countries .
we find long - range power - law cross - correlations in the absolute values of returns that quantify risk , and find that they decay much more slowly than cross - correlations between the returns .
the magnitude of the cross - correlations constitute `` bad news '' for international investment managers who may believe that risk is reduced by diversifying across countries .
we find that when a market shock is transmitted around the world , the risk decays very slowly .
we explain these time lag cross - correlations by introducing a global factor model ( gfm ) in which all index returns fluctuate in response to a single global factor . for each pair of individual time series of returns , the cross - correlations between returns ( or magnitudes )
can be modeled with the auto - correlations of the global factor returns ( or magnitudes ) .
we estimate the global factor using principal component analysis , which minimizes the variance of the residuals after removing the global trend . using random matrix theory ,
a significant fraction of the world index cross - correlations can be explained by the global factor , which supports the utility of the gfm .
we demonstrate applications of the gfm in forecasting risks at the world level , and in finding uncorrelated individual indices .
we find 10 indices are practically uncorrelated with the global factor and with the remainder of the world indices , which is relevant information for world managers in reducing their portfolio risk . finally , we argue that this general method can be applied to a wide range of phenomena in which time series are measured , ranging from seismology and physiology to atmospheric geophysics .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
nowadays , it is very clear how special relativity effects influence on measured data . the first celebrated example of this fact was the atmospheric muons decay explanation as a time dilation effect . this is the rossi - hall experiment @xcite . considering the mrzke - wheeler synchronization @xcite as the natural generalization to accelerated observers of einstein synchronization in special relativity , we wonder whether mrzke - wheeler effects influence on measured data in nature . this question is also motivated by the fact that recently the twin paradox was completely solved in ( 1 + 1)-spacetime by means of these effects @xcite and it is natural to ask for empirical confirmation . of course these effects comprehend the well known special relativistic ones for inertial observers as well as the new ones . these new effects can be seen as corrections of the special relativistic ones due to the acceleration of the involved observer . + a small deviation towards the sun from the predicted pioneer acceleration : @xmath0 for pioneer 10 and @xmath1 for pioneer 11 , was reported for the first time in @xcite . the analysis of the pioneer data from 1987 to 1998 for pioneer 10 and 1987 to 1990 for pioneer 11 made in @xcite improves the anomaly value and it was reported to be @xmath2 . this is known as the pioneer anomaly . + considering that mrzke - wheeler tiny effects are difficult to measure , we careful looked for some observational object for which the searched effect could be appreciable . this search led us to the pioneer 10 . in fact , through a simple analytic formula for the mrzke - wheeler map exact calculation developed in this letter , computing the acceleration difference between the mrzke - wheeler and frenet - serret coordinates for the earth s translation around the sun , we see that this mrzke - wheeler long range effect is between @xmath3 and @xmath4 of the pioneer anomaly value . unfortunately , due to statistical errors in the measured anomaly , it is not possible to confirm the influence of the mrzke - wheeler acceleration effect on the measured pioneer data . moreover , a recently numerical thermal model based on a finite element method @xcite has shown a discrepancy of @xmath5 of the actual measured anomaly and due to the mentioned statistical errors , it was concluded there that the pioneer anomaly has been finally explained within experimental error of @xmath6 of the anomaly value : + _ ... to determine if the remaining @xmath5 represents a statistically significant acceleration anomaly not accounted for by conventional forces , we analyzed the various error sources that contribute to the uncertainties in the acceleration estimates using radio - metric doppler and thermal models ... we therefore conclude that at the present level of our knowledge of the pioneer 10 spacecraft and its trajectory , no statistically significant acceleration anomaly exists . _ + although it is tempting to think that the @xmath5 discrepancy found in @xcite is due to a long range mrzke - wheeler acceleration effect , it can not be confirmed . we hope that the ideas presented here could encourage other research teams in the search for other observational objects that could finally answer the question posed in this letter . + consider the @xmath7-spacetime @xmath8 spanned by the vectors @xmath9 with the lorentz metric : @xmath10 respect to the basis @xmath11 . an observer is a smooth curve @xmath12 naturally parameterized with timelike derivative at every instant ; i.e. @xmath13 . we will say a vector is spatial if it is a linear combination of @xmath14 . a spatial vector @xmath15 is unitary if @xmath16 . + consider a timelike vector @xmath17 in @xmath8 ; i.e. @xmath18 . we define the scaled lorentz transformation @xmath19 : @xmath20 where @xmath21 is the orthocronous lorentz boost transformation sending @xmath22 to the unitary vector @xmath23 ; i.e. the original and transformed coordinates are in standard configuration ( @xmath24 , @xmath25 and @xmath26 are colinear with @xmath27 , @xmath28 and @xmath29 respectively where the prime denote the spatial transformed coordinates and the others denote the original spatial coordinates ) . the scaled lorentz transformation has the following properties : @xmath30 @xmath31 + a smooth map @xmath32 is a mrzke - wheeler map of the observer @xmath33 if it verifies : @xmath34 for every real @xmath35 , positive real @xmath36 and unitary spatial vector @xmath15 ( see figure [ mw_coord ] ) . this map @xcite , @xcite , @xcite is clearly an extension of the einstein synchronization convention for non accelerated observers ; i.e. it is the natural generalization of a lorentz transformation in the case of accelerated observers . + [ mwformula ] consider an observer @xmath12 . then , @xmath37 is a mrzke - wheeler map of the observer @xmath33 such that @xmath15 is a unitary spatial vector . _ proof : _ recall that for every @xmath17 such that @xmath18 we have that @xmath38 . this way , @xmath39 because @xmath40 . from the formula it is clear that @xmath41 is smooth . @xmath42 + the last mrzke - wheeler map formula was written for the first time in @xcite for @xmath43-spacetime where it was shown , in this particular case , that it is actually a conformal map . moreover , the twin paradox is solved in @xmath43-spacetime . in the general case treated here , the mrzke - wheeler map is no longer conformal . + as an example , consider the uniformly accelerated observer in @xmath44-spacetime along the @xmath45 axis : @xmath46 where @xmath35 is its natural parameter and @xmath47 such that @xmath17 is the observer acceleration . its mrzke - wheeler map is : @xmath48\sigma_{0 } \\ & & + r\cosh \left(\frac{s}{r}\right)\left[\cosh \left(\frac{s}{r}\right ) + \sinh \left(\frac{s}{r}\right)\frac{x}{r}\right]\sigma_{1 } \\ & & + \frac{r}{r}\sinh \left(\frac{r}{r}\right)\left [ y\ \sigma_{2 } + z\ \sigma_{3 } \right ] \\\end{aligned}\ ] ] where @xmath49 . in this example , it is interesting that besides @xmath41 restricted to the @xmath50 plane is a conformal map ( as it was expected from @xcite ) , it is also also conformal restricted to the @xmath51 plane . the pioneer 10/11 data is measured from earth s dsn antennas ( deep space network ) and we wonder whether this data is affected by earth s translation around the sun . we comment about earth s rotation at the end of the section . we model the earth s translation as the uniformly rotating observer @xmath52\ ] ] where @xmath35 is its natural parameter , @xmath53 is its lorentz contraction factor and @xmath54 . its mrzke - wheeler map is : @xmath55\sigma_{0 } \\ & & + \left [ r\cos\left(\frac{\omega}{ck}r\right ) + x\ \sqrt{\frac{1}{k^{2 } } - \left(\frac{r}{r}\sin\left(\frac{\omega}{ck}r\right)\right)^{2 } } \right]\vec{a}(s ) \\ & & + \frac{1}{k}\ y\ \vec{b}(s ) \\ & & + z\ \sqrt{\frac{1}{k^{2 } } - \left(\frac{r}{r}\sin\left(\frac{\omega}{ck}r\right)\right)^{2}}\ \sigma_{3 } \\\end{aligned}\ ] ] where @xmath49 . we have chosen the framing @xmath56 corresponding to the @xmath57 coordinates such that @xmath58 is the frenet - serret framing of the observer ( see figure [ frenetserret ] ) : @xmath59 @xmath60 this expression was also obtained in @xcite in the particular case @xmath61 . it is interesting to notice the oscillatory term of the above map . in order to compare the spatial mrzke - wheeler coordinates with the frenet - serret coordinates we consider the difference @xmath62 . because @xmath63 and @xmath64 , we have that @xmath65 and restricted to the region @xmath66 we have the approximation : + @xmath67\ \vec{a}(s ) \\ & & + \ y\ \vec{b}(s)+ z\ \sigma_{3 } \\\end{aligned}\ ] ] because the @xmath22 component is zero , we have the following transformation between the spatial mrzke - wheeler coordinates and the frenet - serret coordinates : @xmath68 where @xmath69 . because the pioneer s velocity and acceleration are very small respect to the natural scale of the problem @xmath70 , differentiating the above expression we have : @xmath71 where @xmath36 is the distance from the sun and @xmath17 is the acceleration . because the recorded pioneer data ( at least for pioneer 10 ) corresponds to the region between @xmath72 and @xmath73 , we can consider that @xmath74 where @xmath75 is the pioneer s distance from the sun and we have the following approximation : @xmath76 where @xmath77 is the pioneer s speed and @xmath78 is the angle between its radius vector from the sun and its velocity vector . computing the acceleration difference @xmath79 between the mrzke - wheeler and frenet - serret coordinates at the pioneer s maximal speed @xmath80 , we have the result : @xmath81 and we see that it is between @xmath3 and @xmath82 of the measured pioneer anomaly @xmath83 . + the calculated difference @xmath79 points towards the @xmath29 edge when @xmath84 and in the opposite direction when @xmath85 . this would contradict the claim that the anomaly always points towards the sun made in the data analysis @xcite and @xcite . however , in the data analysis made in @xcite , it is claimed that it can not be confirmed whether the anomaly is sunwards , contrary to the earlier claim . + finally , we would like to comment about a possible numerical analysis on the influence of earth s rotation on the measured data . in order to do so , we define the following framing dependent non abelian product of observers : @xmath86 this product is the generalization of the special relativistic velocities addition and has the following property : @xmath87 this way , the observer @xmath88 is the one we should consider and its mrzke - wheeler map is just the composition of the previously exactly calculated map of the uniformly rotating observer . unfortunately , the map gets really involved and the analysis must be done numerically . an analysis of the parameters involved in the rotation analysis , shows that the magnitude order of the long range mrzke - wheeler acceleration effect coincides with the one of the pioneer anomaly and should also be considered . although after strongly numerical evidence it is tempting to think that the @xmath5 discrepancy of the anomaly value found in @xcite is due to a long range mrzke - wheeler acceleration effect described in this letter , due to statistical errors in the measured anomaly it can not be neither confirmed nor neglected . we hope that the ideas presented here could encourage other research teams in the search for other observational objects that could finally answer whether mrzke - wheeler effects influence on measured data in nature . mrzke m , wheeler j , _ gravitation as geometry - i : the geometry of spacetime and the geometrodynamical standard meter _ gravitation and relativity , h .- y . chiu and w. f. hoffmann , eds . , w. a. benjamin , new york - amsterdam ( 1964 ) 40 .
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we wonder whether mrzke - wheeler effects influence on measured data in nature . through a formula developed in this letter for the calculation of the mrzke - wheeler map of a general accelerated observer , we study the influence of the mrzke - wheeler acceleration effect on the nasa s pioneer anomaly and found that it is about a fifth of the anomaly value . due to statistical errors in the measured anomaly ,
it is not possible to neither confirm nor neglect the influence of the mrzke - wheeler acceleration effect on the measured pioneer data .
we hope that the ideas presented here could encourage other research teams in the search for other observational objects that could finally answer the question posed in this letter .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
x - ray free - electron lasers ( fels ) offer a brilliant tool for science at atomic length and ultrafast time scales @xcite , and they have been realized with the operation of the free - electron laser in hamburg ( flash ) @xcite , the linac coherent light source ( lcls ) @xcite , and the spring-8 angstrom compact free electron laser ( sacla ) @xcite . the x - ray fel driving electron bunches are subject to several collective effects , e.g. , microbunching instabilities or coherent synchrotron radiation ( csr ) , which degrade the required high transverse and longitudinal beam brightness @xcite . these instabilities may not only result in significant deteriorations of the fel performance @xcite but also in coherent radiation effects @xcite such as coherent optical transition radiation ( cotr ) or csr in the optical wavelength range @xcite ( abbreviated as cosr ) . beam profile imaging dominated by coherent optical radiation leads to an incorrect representation of the transverse charge distribution @xcite and renders electron beam diagnostics with standard imaging screens , e.g. , otr screens , and all the related diagnostics such as emittance or bunch length diagnostics impossible . however , beam diagnostics with imaging screens are essential for single - shot measurements or in cases where two transverse dimensions are required , e.g. , in slice - emittance or longitudinal phase space measurements @xcite . microbunching instabilities associated with longitudinal electron bunch compression can be mitigated by introducing additional uncorrelated energy spread @xcite as successfully demonstrated by the operation of the laser heater system at the lcls @xcite . however , the microbunching gain suppression is not necessarily perfect , and the corresponding remaining small but existing level of cotr still hampers electron beam profile diagnostics using standard imaging screens ( e.g. , ref . the origin of coherent optical radiation effects is not only restricted to microbunching instabilities but can also be related to ultrashort spikes inside electron bunches or generated by intrinsically ultrashort electron bunches like at laser - plasma accelerators ( e.g. , ref . @xcite ) or at x - ray fels with ultra - low charge operation @xcite . transition radiation is emitted when a charged particle beam crosses the boundary between two media with different dielectric properties @xcite , hence transition radiation is emitted using any kind of imaging screen and thus precludes the stand - alone use of scintillation screens in the presence of coherent optical radiation effects ( e.g. , cotr ) . however , by using ( scintillation ) imaging screens in dedicated measurement configurations , cotr can be mitigated ( see , e.g. , ref . @xcite ) . in this paper , we discuss methods to suppress coherent optical radiation effects both by electron beam profile imaging in dispersive beamlines and by utilizing scintillation imaging screens in combination with several separation techniques . the experimental setup and observations of coherent optical radiation effects at flash are described in sec . [ sec : setup ] . in sec . [ sec : es ] we discuss the suppression of coherent optical emission in dispersive beamlines and present experimental results for cotr generated by a local ultrashort charge concentration . section [ sec : sep ] covers the suppression of coherent optical radiation effects by using scintillation screens in combination with separation techniques . the experimental results obtained with the temporal separation technique are presented in sec . [ sec : res ] , and a summary and conclusions are given in sec . [ sec : summary ] . the measurements presented in this paper have been carried out at flash , which is a self - amplified spontaneous emission ( sase ) fel @xcite for extreme - ultraviolet ( euv ) and soft x - ray radiation , driven by a superconducting radio - frequency ( rf ) linear accelerator @xcite . the schematic layout of flash is depicted in fig . [ fig : flash_1 ] , showing the injector , which is based on a laser - driven normal conducting rf gun , the superconducting accelerating structures , two magnetic bunch compressor chicanes , and the undulator magnet system . the positions of the experimental setups used for the measurements presented in this paper are indicated by green dots and arrows . the third - harmonic rf system ( denoted by l3 in fig . [ fig : flash_1 ] ) is dedicated to the linearization of the longitudinal phase space upstream of the first bunch compressor @xcite . in order to properly set up fel operation with applied third - harmonic rf linearizer , a lola - type @xcite transverse deflecting rf structure ( tds ) has been integrated in a dedicated setup for diagnosis of the longitudinal phase space @xcite close to the fel undulators . as depicted in fig . [ fig : flash_1 ] , the tds can either be operated in combination with imaging screens in the dispersive magnetic energy spectrometer or by using off - axis imaging screens operated with a fast kicker magnet in the non - dispersive main beamline during fel operation . technical details and performance measurements on the setup for longitudinal beam diagnostics can be found in refs . @xcite . transverse deflecting rf structures are widely used for electron bunch length and longitudinal profile measurements at present fels and provide high - resolution single - shot diagnostics @xcite . detailed descriptions of time - domain electron bunch diagnostics using a tds can be found in refs . @xcite . here we describe only the basic principles of longitudinal electron beam diagnostics that are required throughout this paper . the vertical betatron motion of an electron passing a vertical deflecting tds around the zero - crossing rf phase , neglecting intrinsic longitudinal - to - vertical correlations @xcite which are not relevant for the experiments presented throughout this paper , can be given by @xcite@xmath0 with the vertical shear ( streak ) function @xmath1 where @xmath2 is the angular - to - spatial element of the vertical beam transfer matrix from the tds at @xmath3 to any position @xmath4 , @xmath5 is the vertical beta function , @xmath6 is the vertical phase advance between @xmath3 and @xmath4 , and @xmath7 describes an intrinsic offset . the expression @xmath8 is the vertical kick strength with the peak deflection voltage @xmath9 in the tds , @xmath10 is the speed of light in vacuum , @xmath11 is the elementary charge , @xmath12 is the electron momentum , @xmath13 is the longitudinal position of the electron relative to the zero - crossing rf phase , and @xmath14 is the operating rf frequency . the expression in eq . ( [ eq : motion ] ) shows a linear mapping from the longitudinal to the vertical coordinate and allows longitudinal electron beam profile measurements by means of transverse beam diagnostics using imaging screens . the shear function @xmath15 determines the slope of this mapping and can be calibrated by measuring the vertical centroid offset of the bunch as a function of the tds rf phase . the electron bunch current is given by the normalized longitudinal bunch profile multiplied by the electron bunch charge . the bunch length ( duration ) is given by the root mean square ( r.m.s . ) value @xmath16 , where @xmath17 is the vertical r.m.s . beam size during tds operation , and @xmath18 is the intrinsic vertical r.m.s . beam size when the tds is switched off . both @xmath17 and @xmath18 can be determined by measurements , and the latter limits the achievable r.m.s . time resolution to @xmath19 @xcite . the screen stations in both the magnetic energy spectrometer and non - dispersive main beamline ( see fig . [ fig : flash_1 ] ) are each equipped with different imaging screens and a charge - coupled device ( ccd ) camera @xcite ( 1360@xmath201024 pixels with 12bit dynamic range and @xmath21 pixel size ) with motorized optics ( motorized macro lens with teleconverter mounted on a linear translation stage ) . the translation stage allows variable demagnification @xmath22 in the range between @xmath231.5 - 3 with spatial resolutions of better than @xmath24 . the imaging screen station in the energy spectrometer ( es - ccd in fig . [ fig : flash_1 ] ) is equipped with an otr screen ( aluminum coated silicon ) and two scintillation screens made of cerium - doped yttrium aluminum garnet ( yag : ce ) and bismuth germanate ( bgo ) , respectively . in the non - dispersive beamline , the screen station is operated with a fast kicker magnet ( k - ccd in fig . [ fig : flash_1 ] ) , which is able to deflect one bunch out of the bunch train at the bunch train repetition rate of flash @xcite of 10hz , and provides an otr screen and a cerium - doped lutetium aluminum garnet ( luag : ce ) scintillation screen . all screens are mounted at a 45@xmath25 angle ( the cameras at a 90@xmath25 angle ) with respect to the incoming electron beam . the scintillation screens have a thickness of @xmath26 . the experimental setup in the non - dispersive beamline is additionally equipped with a fast gated intensified ccd camera @xcite ( k - iccd in fig . [ fig : flash_1 ] , 1280@xmath201024 pixels with 12bit and @xmath27 pixel size ) , which has been used for the temporal separation technique ( see sec . [ sec : res ] ) . further technical details on the screen stations and camera systems can be found in refs . @xcite . microbunching instabilities at x - ray fels can lead to significant generation and amplification of density modulations in the optical wavelength range @xcite which may result in coherent optical radiation effects such as cotr . this has been observed by spectral measurements and characteristic ring - shaped light patterns at the lcls @xcite and flash @xcite , and renders accurate electron beam profile diagnostics using standard imaging screens impossible . first observations of cotr @xcite and microbunching in the frequency - domain ( coherent transition radiation around 10 @xmath28 @xcite ) at flash were made directly upstream of the collimator ( see fig . [ fig : flash_1 ] ) . electron beam profile imaging performed downstream of the collimator section @xcite , an achromatic bending system , resulted in considerably more prominent observation of coherent optical radiation effects and microbunching . the measurements presented in fig . [ fig : indi_2 ] show single - shot light patterns , generated by moderately compressed electron bunches , at the imaging screens in the non - dispersive main beamline at k - ccd directly upstream of the undulators . ring - shaped structures in the profiles , characteristic for cotr @xcite , are clearly visible in the images of figs . [ fig : indi_2_a ] and [ fig : indi_2_b ] , which have been recorded by using an otr and luag imaging screen , respectively . for both images a long - pass filter , blocking wavelengths below 780 nm , was used . the luminescence emission of the luag scintillation screen occurs below 700 nm @xcite and is thus well blocked by the 780-nm long - pass filter used during the measurements . hence , the light pattern in fig . [ fig : indi_2_b ] is due to cotr without contribution from scintillation light . complementary to the observation of cotr , the images in fig . [ fig : indi_3 ] show single - shot longitudinal phase space measurements in the magnetic energy spectrometer ( es - ccd ) . the measurements were done for accelerator settings typical for fel operation with applied third - harmonic rf linearizer system upstream of the bunch compressor chicanes , and they clearly indicate microbunching in the time - domain with modulation periods of about 25fs and 30fs , respectively . we note that a maximum modulation wavelength of @xmath29 ( @xmath30 ) was predicted theoretically in ref . @xcite and measured by spectroscopy of coherent transition radiation in ref . the energy - dependent beam trajectories in dispersive beamlines can be utilized as a magnetic energy spectrometer for charged particle beams . by combining such an energy spectrometer with the operation of a tds and using imaging screens to get two - dimensional transverse beam profiles , longitudinal phase space measurements ( see , e.g. , fig . [ fig : indi_3 ] ) with single - shot capability can be accomplished . the corresponding horizontal betatron motion , which should be perpendicular to the vertical shearing plane of the tds @xcite , can be written as @xmath31 with the intrinsic offset @xmath32 , the horizontal momentum dispersion @xmath33 and the relative momentum deviation @xmath34 . for relativistic electron beams with lorentz factors of @xmath35 , the electron beam energy is given by @xmath36 , and @xmath37 represents the relative energy deviation . the dispersion @xmath38 can be determined by measuring the horizontal centroid offset of the bunch as a function of the relative energy deviation . the dispersion in the magnetic energy spectrometer at es - ccd ( see fig . [ fig : flash_1 ] ) , which is generated by two subsequent dipole magnets with 5@xmath25 deflection each ( equivalent to a single dipole magnet with 10@xmath25 deflection ) , amounts to 750 mm ( nominal ) @xcite , whereas @xmath38 at k-(i)ccd due to the kicker magnet operation is negligible . in addition to the momentum dispersion introduced in the horizontal betatron motion , the longitudinal particle motion can be described by @xmath39 with the initial bunch length coordinate @xmath40 and the initial horizontal offset @xmath41 and slope @xmath42 . the transfer matrix elements @xmath43 describe the mapping from position @xmath3 to @xmath4 , i.e. , @xmath44 throughout the rest of this paper . the expression in eq . ( [ eq : longdisp ] ) does not affect the principle of longitudinal phase space diagnostics described by eqs . ( [ eq : motion ] ) and ( [ eq : disp ] ) , but results in the suppression of coherent optical emission as is shown in the following . the spectral and angular intensity distribution , denoted as @xmath45 with the three - dimensional wave vector @xmath46 , of transition ( synchrotron ) radiation emitted by an electron bunch with @xmath47 electrons and charge @xmath48 is given by ( e.g. , refs . @xcite ) @xmath49 where @xmath50 describes the intensity distribution of a single electron as a function of the transverse and longitudinal wavenumber @xmath51 and @xmath52 , respectively , and @xmath53 is the three - dimensional form factor of the electron bunch . the latter can be expressed by the fourier transform of the normalized charge density @xmath54 as @xmath55 where @xmath56 . normalized charge distributions without longitudinal - transverse correlations can be factorized as @xmath57 , and by taking into account @xmath58 , which is assumed in the following , we get @xmath59 with the transverse and longitudinal form factor @xmath60 and @xmath61 , respectively . for small observation angles @xmath62 ( small covered solid angles @xmath63 ) with respect to the central axis ( @xmath40-axis ) of the emitted radiation we have @xmath64 with the wavenumber @xmath65 , and the expression in eq . ( [ eq : spe ] ) reads @xmath66 the first term on the right - hand side is linear in @xmath67 and describes the contribution of incoherent radiation , whereas the second term scales with @xmath68 , which describes the coherent radiation part . in order to perform electron beam diagnostics with incoherent radiation , we demand that the total spectral radiation intensity in eq . ( [ eq : spec ] ) is dominated by the incoherent term , i.e. , @xmath69 . in following , we derive an analytical expression describing a general strong suppression of the longitudinal form factor at optical wavelengths in a magnetic energy spectrometer . a transverse form factor of @xmath70 , i.e. , full transverse coherence , at the imaging screens is assumed , which is the worst case scenario . the actual transverse form factor in the experiment will be reduced due to the finite beam size and observation angle @xcite . however , the suppression of the longitudinal form factor @xmath61 presented below is much stronger in the general case . a cutoff wavelength @xmath71 can be defined via @xmath72 , and beam diagnostics at wavelengths below @xmath73 becomes dominated by incoherent radiation . the cutoff wavelength initially depends on the charge distribution [ via eq . ( [ eq : form ] ) ] , and significant values of @xmath74 in the optical wavelength range can occur due to the existence of density modulations or charge concentrations at ultrashort length scales . however , following the analytical treatment of microbunching degradation in ref . @xcite , we show that the cutoff wavelength in magnetic energy spectrometers is entirely determined by the terms in eq . ( [ eq : longdisp ] ) with a corresponding strong suppression of coherent emission at optical wavelengths for common magnetic energy spectrometers used at present fels . the amount of density modulations in a normalized electron beam distribution @xmath75 with the phase space vector @xmath76 and @xmath77 can be quantified by a complex bunching factor @xmath78 as @xcite @xmath79 where @xmath65 is the wavenumber of the modulation . according to refs . @xcite , the evolution of the bunching factor @xmath80 $ ] along dispersive beamlines can be expressed by @xmath81 = b_0[k(s),s ] + \int_{s_0}^{s}{ds'\,k(s',s)b[k(s'),s ' ] } \ , , \label{eq : bu2}\ ] ] where @xmath82 $ ] is the bunching factor in the absence of collective beam interactions due to csr . the second term on the right - hand side of the integral equation with the kernel @xmath83 @xcite ( a complicated expression that is not relevant here ) describes the induced bunching due to csr interactions . as discussed in refs . @xcite and verified by numerical particle tracking simulations below , the bunching induced in a dipole magnet from the energy modulation generated in the same dipole magnet can be neglected with the kernel @xmath84 , and the bunching factor in eq . ( [ eq : bu2 ] ) becomes @xmath80 \approx b_0[k(s),s]$ ] . this is also the case in a magnetic energy spectrometer consisting of a single dipole magnet , and the resulting evolution of the total bunching factor for a given initial bunching @xmath85 $ ] can be expressed by @xcite @xmath86\approx&\ , b_0[k(s_0),s_0]\,\mathrm{exp}\left [ -\frac{k^2(s)\sigma_{\delta0}^2}{2 } r_{56}^2\right]\nonumber \\ & \times \mathrm{exp}\left [ -\frac{k^2(s)\varepsilon_0\beta_0}{2 } \left ( r_{51 } -\frac{\alpha_0}{\beta_0 } r_{52}\right)^2\right]\nonumber \\ & \times \mathrm{exp}\left[-\frac{k^2(s)\varepsilon_0}{2\beta_0}r_{52}^2 \right]\ , , \label{eq : bu3 } \end{aligned}\ ] ] where the motion in eq . ( [ eq : longdisp ] ) is taken into account , and an initial beam distribution @xmath87 that is uniform in @xmath40 and gaussian in @xmath41 , @xmath88 , and @xmath37 is assumed . the initial uncorrelated energy spread and geometrical horizontal emittance are denoted by @xmath89 and @xmath90 , respectively , and @xmath91 and @xmath92 are the initial horizontal lattice functions ( twiss parameters ) . the compression of the wavenumber by @xmath93^{-1}$ ] with the initial energy chirp @xmath94 can be neglected , i.e. , @xmath95 , since the @xmath96 generated by a single dipole magnet is rather small . in addition to the evolution of an initial bunching , energy modulations generated upstream of a magnetic energy spectrometer can initiate bunching and , according to ref . @xcite and by using eq . ( [ eq : bu3 ] ) , the induced bunching @xmath97 due to an initial energy modulation is given by @xmath98 where @xmath99 is the fourier amplitude of the initial energy modulation @xmath100 . fortunately , the bunching @xmath101 can be neglected due to the small @xmath96 ( see above ) and the additional suppression discussed in the following . equation ( [ eq : bu3 ] ) implies a suppression of initial bunching due to the coupling with the transverse phase space given in eq . ( [ eq : longdisp ] ) , and a suppression factor @xmath102 can be defined as @xmath103 where @xmath104 by comparing eqs . ( [ eq : form ] ) and ( [ eq : bun ] ) , and taking into account @xmath105=\rho(z)\rho[(x , x',\delta)]$ ] , the suppression factor can be expressed as @xmath106 ( cf . the analytical treatment in refs . @xcite ) , which describes the general suppression of coherent emission in a common magnetic energy spectrometer . assuming a maximum initial density modulation or an ultrashort electron bunch , both with @xmath107 , the cutoff wavelength ( defined via @xmath72 ) is given by [ cf . eq . ( [ eq : bu4 ] ) ] @xmath108 we note that the suppression for ultrashort electron bunches is simply given by the lengthening due to the transverse phase space parameters and longitudinal motion given in eq . ( [ eq : longdisp ] ) , which act like a low - pass filter . lcccc parameter & symbol & value & unit + beam energy & @xmath109 & 1000 & mev + lorentz factor & @xmath110 & 1957 & + electron bunch charge & @xmath111 & 150 & pc & + horizontal emittance ( normalized ) & @xmath112 & 1.0 & @xmath113 m + relative slice energy spread & @xmath89 & @xmath114 & + horizontal beta function & @xmath92 & 13.55 & m + horizontal alpha function & @xmath91 & 5.33 & + spatial - to - longitudinal coupling & @xmath115 & -0.174 & + angular - to - longitudinal coupling & @xmath116 & -0.089 & + momentum compaction factor & @xmath96 & 0.006 & m + [ tab : spec ] for initial density modulations . the theory curve ( solid red line ) is calculated for the full term in eq . ( [ eq : bu5 ] ) , and the approximation ( dashed green line ) is calculated for @xmath117 . the inset shows the wavelength range below @xmath118 on a logarithmic scale including the cutoff wavelength @xmath119 calculated for @xmath120 electrons . ] the analytical treatment has been verified by numerical simulations using the tracking code _ elegant _ @xcite with gaussian and uniform beam distributions ( @xmath121 particles ) including csr effects , and by using the parameters of the magnetic energy spectrometer at flash , summarized in table [ tab : spec ] . figure [ fig : supp_4 ] shows the suppression factor for both numerical simulations with initial density modulations ( @xmath122 peak amplitude ) and analytical calculations using eqs . ( [ eq : bu4 ] ) and ( [ eq : bu5 ] ) for the parameters of flash . the analytical calculations are in perfect agreement with the numerical simulations . the shown approximation is calculated by using @xmath117 , which is a good practical estimate ( @xmath123 for a single dipole magnet with bending angle @xmath124 ) . according to the full term in eq . ( [ eq : bu5 ] ) , the cutoff wavelength in the magnetic energy spectrometer at flash amounts to @xmath125 , which manifests a strong suppression of coherent optical emission . coherent emission does not only lead to intense radiation , which is described by means of the form factor @xmath74 in the intensity distribution given in eq . ( [ eq : spec ] ) , but also to an incorrect representation of the transverse charge distribution in beam profile imaging @xcite . the imaging of transverse beam distributions with optical systems , e.g. , by using an imaging screen , a lens , and a camera , is generally described by means of the intensity distribution of a point source in the image plane ( e.g. , ref . @xcite ) , which is the so - called point spread function . according to ref . @xcite , the image formation with optical transition radiation of a normalized three - dimensional charge distribution @xmath126 with @xmath67 electrons can be expressed by @xmath127 where @xmath128 describes the measured intensity distribution proportional to the absolute square of the total electric field @xmath129 evolved from the charge distribution , and @xmath130 corresponds to the imaged electric field of a single electron , which can be expressed by means of the fresnel - kirchhoff diffraction integral ( e.g. , ref . the second integral in eq . ( [ eq : field ] ) describes the coherent radiation part ( @xmath131 ) , and by taking into account @xmath132 with @xmath58 , the expression for image formation in eq . ( [ eq : field ] ) can be rewritten as [ cf . ( [ eq : spec ] ) ] @xmath133 the first integral in eq . ( [ eq : field2 ] ) simply describes the incoherent imaging as a convolution of the transverse charge distribution @xmath134 with the point spread function related term @xmath135 . in the case of a nonvanishing longitudinal form factor @xmath136 , the second integral in eq . ( [ eq : field2 ] ) contributes to the image formation and describes no longer a simple convolution with a point spread function , but rather takes into account the actual field distribution . thus , significant deviations in the measured transverse charge distribution can occur even with a small longitudinal form factor due to the second term @xmath137 in eq . ( [ eq : field2 ] ) , where @xmath138 . an example with initially inconspicuous cotr , impeding the electron beam diagnostics finally , is demonstrated in the following . figures [ fig : spike_5_a ] and [ fig : spike_5_b ] show single - shot images of longitudinal bunch profile measurements using the tds that were recorded in the non - dispersive main beamline at k - ccd and in the energy spectrometer at es - ccd , respectively . the images were measured under the same electron beam conditions with a bunch charge of @xmath139 and do not display any conspicuous features of cotr . however , as can be seen in fig . [ fig : spike_5_e ] , the corresponding longitudinal bunch profile taken at k - ccd comprises a much narrower spike with higher peak current . when increasing the bunch charge to @xmath140 , cotr emission became apparent at k - ccd [ fig . [ fig : spike_5_c ] ] , whereas the image in the energy spectrometer at es - ccd [ see fig . [ fig : spike_5_d ] ] did not show any coherent radiation effects . the cotr emission in fig . [ fig : spike_5_c ] ( we chose a single - shot image with low saturation of the ccd ) is clearly localized in the longitudinal electron bunch profile at a time coordinate of about 0.5ps . at the same time coordinate , the longitudinal phase space in fig . [ fig : spike_5_d ] exhibits a huge but narrow increase in energy spread ( the width in the time is limited by the tds resolution ) . from this we conclude that the single - shot image in fig . [ fig : spike_5_a ] already partially contains cotr as a consequence of a small but nonvanishing form factor @xmath74 [ cf . eqs . ( [ eq : spec ] ) and ( [ eq : field2 ] ) ] and that the cotr emission in fig . [ fig : spike_5_c ] seems most probably to be generated by a local ultrashort charge concentration such as a sharp spike inside the electron bunch . we note that the measurements presented in fig . [ fig : spike_5_e ] should give the same longitudinal electron bunch profiles , and the existing deviations can not be explained due to a worse resolution as is the case in sec . [ sec : lp ] . in order to demonstrate the local energy spread increase in figs . [ fig : spike_5_b ] and [ fig : spike_5_d ] with a reasonable signal - to - noise ratio ( snr ) , the longitudinal phase space measurements are presented with the yag imaging screen . the measurement performed with the otr imaging screen , presented in fig . [ fig : spike_5_f ] , shows the same strong cotr suppression ( but worse snr ) . as demonstrated in sec . [ sec : es ] , electron beam profile measurements can be accomplished in dispersive beamlines , such as magnetic energy spectrometers , with standard optical imaging systems as the emission of coherent optical radiation is strongly suppressed . however , linear accelerators consist mainly of beamlines which are in general designed to be dispersion - free , and imaging in energy spectrometers precludes measuring pure transverse beam profiles due to the dispersion . in this section , we discuss methods that suppress the impact of coherent radiation by separation from an incoherent radiation part . in the bunch compressors ( see ref . @xcite for experimental details ) . the spectral intensity of the incoherent part of transition radiation is indicated as dashed black line . ] the spectral intensity of transition ( synchrotron ) radiation emitted by an electron bunch consists of two terms that describe the incoherent ( @xmath141 ) and coherent ( @xmath142 ) radiation part [ cf . ( [ eq : spec ] ) or eq . ( [ eq : field2 ] ) ] . a spectral separation of these terms in electron beam profile imaging can be accomplished by restricting the imaging with wavelengths below the cutoff wavelength @xmath73 , i.e. , where the emission is dominated by incoherent radiation . spectral separation has been considered in ref . @xcite by using a scintillation screen in combination with a bandpass filter . however , this method requires a good knowledge and control of the expected spectra , and a vanishing form factor ( @xmath143 ) in the detectable wavelength range , which is not the general case as the spectra can vary strongly with the operation modes of a linear accelerator . this is demonstrated in fig . [ fig : spec_6 ] , in which spectral measurements of transition radiation in the visible and near - infrared wavelength range are presented for different compression settings at flash . the dashed black line represents the incoherent radiation part convoluted with the transmission of the optical setup . in contrast to the measurements presented in sec . [ sec : subseccotr ] , the measurements shown in fig . [ fig : spec_6 ] were performed upstream of the collimator section . we note that similar , reproducible measurements for uncompressed electron bunches , showing coherent radiation prominently at the micrometer scale , have been presented in ref . @xcite , and cotr for uncompressed bunches has been reported in ref . @xcite . in general , the probability of coherent emission decreases at shorter wavelengths , which is often not sufficiently reduced for optical wavelengths , and imaging with transition radiation in the euv region might be an option @xcite . in addition to the knowledge and control of the spectra , the imaging with euv radiation also requires dedicated detectors and optics , and a complete set - up in vacuum to prevent strong absorption in air . the luminescence of scintillation screens @xcite , which is a stochastic process , is inherently linear in the number of interacting electrons ( neglecting quenching and saturation effects ) , hence coherent radiation effects are not expected in pure scintillation light . however , transition radiation is also emitted at the boundary of vacuum and scintillator , and coherent optical radiation can still appear [ see , e.g. , fig . [ fig : indi_3_b ] ] . then , the total spectral and angular intensity distribution can be written as ( omitting the arguments @xmath144 in the intensity distributions @xmath145 ) @xmath146\mathcal{i}_o\ , , \label{eq : specfull}\ ] ] where @xmath147 and @xmath148 are related to scintillation light and transition radiation , respectively . as discussed in sec . [ sec : sepspec ] for otr imaging screens and with the same requirements and restrictions , spectral separation can also be applied when using scintillation screens ( @xmath149 ) . another method , particularly suited for scintillation screens , which have nearly isotropic emission , is to make use of the strong angular dependence of optical transition radiation ( e.g. , refs . @xcite ) and to perform electron beam profile imaging with radiation that is dominated by scintillation light , i.e. , @xmath150 $ ] in eq . ( [ eq : specfull ] ) . spatial separation can be achieved with imaging geometries having large angular or spatial offsets , e.g. , by using tilted imaging screens @xcite or central masks @xcite , where @xmath151 is suppressed sufficiently . however , just as for spectral separation , this method also requires good knowledge and control of the form factor , and dedicated imaging geometries . in addition , the resolution depends on the observation angle of the scintillation screen ( e.g. , ref . @xcite ) , which has to be taken into account in the layout of the imaging system . we note that an experiment on the spatial separation technique is currently being commissioned at flash . the fundamentally different light generation processes of scintillators and optical transition radiators result in clearly distinct temporal responses . the emission of transition radiation from relativistic electrons is instantaneous ( @xmath152 ) and prompt @xcite compared to the decay times ( @xmath153 ) of common scintillators ( e.g. , ref . accordingly , the temporal profiles of the otr pulses resemble the longitudinal electron beam profiles , whereas the temporal scintillation light pulses are fully dominated by the decay of the excited states in the scintillator . temporal separation makes use of the distinct temporal responses and allows to entirely eliminate otr , i.e. , the term @xmath148 in eq . ( [ eq : specfull ] ) which is time - dependent with @xmath154 , and , therewith , coherent optical radiation effects in electron beam profile imaging with scintillation screens when reading out a gated camera with a certain time delay after the prompt emission of otr . image recording with delayed readout ( e.g. , ref . @xcite ) can be accomplished with intensified ccd ( iccd ) cameras , where a control voltage in the intensifier between photocathode and micro - channel plate allows fast gating and exposure times of a few nanoseconds ( e.g. , refs . the experiments on the temporal separation technique at flash have been performed by using the iccd camera `` pco : dicam pro ( s20 ) '' @xcite in combination with the off - axis luag scintillation imaging screen in the non - dispersive main beamline at k - iccd , which has a decay time of @xmath155 @xcite . the cameras used for the presented measurements are able to readout images at the bunch train repetition rate of flash of 10hz , hence one bunch per bunch train can be measured with single - shot capability . further technical details on the equipment used for the measurements presented in the following can be found in sec . [ sec : screens ] and in refs . @xcite . the series of single - shot images in fig . [ fig : proof_7 ] present first proof - of - principle measurements on the temporal separation technique . the image shown in fig . [ fig : proof_7_a ] was recorded at k - iccd with an otr screen , whereas for figs . [ fig : proof_7_b ] and [ fig : proof_7_c ] a luag scintillation screen was used . the image shown in fig . [ fig : proof_7_c ] has been recorded with a time delay of @xmath156 , which is rather long compared to the emission time of otr but takes into account the large camera trigger - jitter that existed during the measurements . the image recorded with the otr screen and time delay simply showed background noise and is not presented here . the intensity distributions in fig . [ fig : proof_7 ] have been generated by moderately compressed electron bunches with a charge of 0.5nc and a beam energy of 700mev . figures [ fig : proof_7_a ] and [ fig : proof_7_b ] show a composite of cotr and cosr with a contribution of scintillation light in fig . [ fig : proof_7_b ] . the round - shaped light pattern on the right - hand side of figs . [ fig : proof_7_a ] and [ fig : proof_7_b ] is most probably due to synchrotron radiation generated upstream of the off - axis screens ( a polarizer was not available during the measurements ) , where the appearance in fig . [ fig : proof_7_b ] is reduced by the transparency of the luag screen . the image in fig . [ fig : proof_7_c ] , recorded with a time delay of @xmath156 , can be attributed purely to scintillation light allowing for a quantitative analysis of the transverse beam profiles . in contrast to spectral and spatial separation , the temporal separation technique provides a definite method to suppress coherent optical transition radiation without further relying on the wavelength - dependent longitudinal form factor . in addition , this technique inherently includes the suppression of secondary incoherent radiation sources such as synchrotron radiation generated from magnets directly upstream of the imaging screen or backward otr emitted from the second imaging screen boundary , whereas spectral components in the uv region or at shorter wavelengths may excite the scintillator , affecting the temporal separation . as is shown in ref . @xcite , however , potential synchrotron radiation sources can be identified and thus separated by adjusting the upstream magnets . furthermore , the coherent emission of otr at the second scintillator screen boundary is mitigated due to multiple scattering in the scintillator material as is described and demonstrated in refs . we note that the current implementation of the temporal separation technique presented throughout this paper utilizes fast iccd cameras , which are currently an order of magnitude more expensive than conventional ccd cameras . the proof - of - principle measurements on the temporal separation technique presented in fig . [ fig : proof_7 ] were carried out at k - iccd . however , a reference measurement to quantitatively prove this technique in terms of transverse beam profiles , as would be provided by a wire - scanner , which is insensitive to coherent effects , is not available at this position . in this section , we verify the method of temporal separation by investigations on the charge - dependent image intensities and comparisons with longitudinal bunch profiles recorded in the energy spectrometer at es - ccd . incoherent radiation is linear in the number of electrons contributing to the emission process ( cf . [ sec : supp ] ) , i.e. , linear in the electron bunch charge ( @xmath157 ) , and deviations caused by the nonlinear charge dependence of coherent radiation ( @xmath158 ) are ideally suited to verify the temporal separation technique . the integrated image intensities presented in fig . [ fig : charge_8 ] were measured for bunch charges between 0.13nc and 0.87nc at k - iccd for different imaging screen and readout configurations . each data point represents the average intensity of 20 background - corrected single - shot images and the error bars indicate the statistical r.m.s . image intensity fluctuations . up to an electron bunch charge of @xmath159 , the integrated intensity is linear ( solid black line ) in @xmath111 for all presented configurations . for higher bunch charges , deviations from the linear dependence appear in the configurations without delayed readout , i.e. , the form factor @xmath74 becomes significant in the visible wavelength range , which are caused by contributions from coherent optical radiation . the inset in fig . [ fig : charge_8 ] shows the bunch charge range from 0.55nc to 0.9nc more detailed . we note that the integrated intensity of the otr ( blue dots ) has actually been higher than presented for @xmath160 , because of camera saturation due to the strong optical emission and the corresponding underestimated integrated intensity . the large error bars , representing the r.m.s . jitter , indicate strong fluctuations due to the cotr . in the case of the luag imaging screen recorded with a time delay ( green diamonds ) , the dependence of the integrated intensity is entirely linear in the bunch charge , which verifies the power of the temporal separation technique . , the measurements in the magnetic energy spectrometer ( `` es - ccd ( yag ) '' ) are intended to provide an absolute reference measurement . ] as the emission of cotr is strongly suppressed in the magnetic energy spectrometer at flash ( see sec . [ sec : es ] ) , electron bunch profiles measured at the screen station es - ccd can serve as a reference for comparison with the temporal separation technique applied in the non - dispersive beamline at k - iccd . while the transverse bunch profiles can differ at both locations due to different twiss parameters and dispersion at es - ccd , longitudinal bunch compression does not take place in between , and longitudinal bunch profile measurements using the tds can be used for a direct comparison . the measurements presented in fig . [ fig : compression_9 ] show the mean r.m.s . electron bunch length of 20 single - shot images , including the statistical r.m.s . jitter indicated via error bars , for various acc1 rf phases measured at es - ccd and k - iccd by using the tds . the electron bunches were set up with an energy of 700mev and a bunch charge of 0.5nc . the rf phase of acc1 affects the energy chirp of the electron bunches upstream of the first bunch compressor and , accordingly , the final electron bunch lengths . the r.m.s . electron bunch lengths measured in the magnetic energy spectrometer at es - ccd ( black dots ) decrease almost linearly and do not possess large fluctuations . in contrast to the magnetic energy spectrometer at es - ccd , coherent optical emission is not suppressed in the non - dispersive beamline at k - iccd , leading to a sudden increase of the r.m.s . electron bunch lengths in combination with large fluctuations , represented by the large error bars ( statistical r.m.s . jitter ) , for acc1 rf phases @xmath161 measured with a luag screen without a certain time delay ( red squares ) , i.e. , without applied temporal separation . the electron bunch length measurements using an otr screen are omitted in fig . [ fig : compression_9 ] due to even larger deviations and fluctuations compared to the reference at es - ccd for acc1 rf phases @xmath161 . instead , the otr images ( single - shots ) for acc1 rf phases of 3.25deg and 3.75deg are presented in figs . [ fig : comp_10_a ] and [ fig : comp_10_b ] , respectively , with obvious coherent optical radiation effects in fig . [ fig : comp_10_b ] . due to the fact that the electron beam images shown in fig . [ fig : comp_10 ] are sheared vertically by means of the tds , the vertical coordinate implies time information ( see eq . [ eq : motion ] ) and the faint bunching visible in fig . [ fig : comp_10_a ] may be assigned to microbunching . figure [ fig : comp_10_c ] shows a single - shot image taken at k - iccd using a luag screen without time delay for an acc1 rf phase of 3.75deg . the image clearly shows contributions of coherent optical radiation similar to the image in fig . [ fig : comp_10_b ] . by imaging the luag screen with a time delay of 100ns , the obtained distribution shown in fig . [ fig : comp_10_d ] is acceptable without obvious contributions from coherent optical radiation . in addition , the corresponding electron bunch length measurements with applied temporal separation ( green diamonds ) in fig . [ fig : compression_9 ] are in perfect agreement with the reference measurements in the energy spectrometer at es - ccd ( black dots ) . the electron bunch durations for fel operation at flash are typically shorter than 150fs ( e.g. , ref . @xcite ) , and typical electron beam parameters are given in table [ tab : spec ] . the temporal separation technique , which has demonstrated accurate r.m.s . electron bunch length measurements in the presence of coherent optical radiation effects , gives confidence that single - shot measurements of longitudinal bunch profiles and , accordingly , electron bunch currents using temporal separation result in reliable results . the single - shot measurements presented in fig . [ fig : prof_11 ] ( cf . measurements shown in figs . [ fig : compression_9 ] and [ fig : comp_10 ] for the same acc1 rf phase settings ) have been recorded for an acc1 rf phases of 3.75deg in fig . [ fig : prof_11_a ] and for 4.05deg in fig . [ fig : prof_11_b ] , i.e. , in the presence of coherent optical radiation effects . the longitudinal electron bunch profiles taken in the non - dispersive beamline at k - iccd ( blue line ) with temporal separation show good agreement with the reference measurements at es - ccd ( red line ) , and the observed deviations are most probably due to slightly nonlinear amplification in the intensifier ( photocathode and micro - channel plate ) of the iccd camera . the reduced peak current with broadening in time in the case of k - iccd : time delay , which is apparent on the right - hand side ( @xmath162 ) of fig . [ fig : prof_11_b ] , can be explained by the different time resolutions of @xmath163 and @xmath164 achieved with the tds during the measurements for es - ccd and k - iccd , respectively . in order to compare the longitudinal bunch profiles with comparable resolution , a convolution has been applied for the measurement at es - ccd in fig . [ fig : prof_11_b ] by taking into account the actual time resolution . the longitudinal bunch profile after carrying out the convolution ( green dashed line ) is in good agreement with the bunch profile taken at k - iccd with applied temporal separation ( blue line ) . electron beam profile imaging is crucial for many applications in electron beam diagnostics at fels , and particularly required to perform single - shot diagnostics . however , the frequent appearance of coherent optical radiation effects , e.g. , cotr , in high - brightness electron beams impedes incoherent beam profile imaging with standard techniques . the theoretical considerations , numerical simulations , and experimental data presented in this paper show that coherent optical emission can be strongly suppressed by performing beam profile imaging in a magnetic energy spectrometer due to sufficient spatial - to - longitudinal coupling . however , energy spectrometers preclude measuring pure transverse beam profiles due to dispersion in the bending plane . for incoherent beam profile imaging in non - dispersive beamlines , we discussed methods to separate the incoherent radiation from scintillation screens and to simultaneously exclude coherent optical radiation from detection . in contrast to spectral and spatial separation , the temporal separation technique , utilizing an iccd camera , provides a definite method to suppress coherent optical transition radiation without knowledge of the longitudinal form factor . in terms of readout times and rates , iccd cameras have the same applicability as standard ccd cameras . by applying the temporal separation technique in the presence of coherent optical radiation , we demonstrated reliable measurements of longitudinal electron beam profiles , and measurements of r.m.s . electron bunch lengths in excellent agreement with reference measurements in a magnetic energy spectrometer . limitations may appear due to scintillator excitation by secondary coherent radiation sources . however , the presented experimental results prove the temporal separation technique as a promising method for future applications in beam profile diagnostics for high - brightness electron beams . we would like to thank the whole flash - team , and the engineers and technicians of the desy groups fla , mcs , and mvs for their great support . we also thank b. faatz , k. honkavaara , and s. schreiber for providing beam time , and y. ding and h. loos for fruitful discussions . in particular , we are deeply grateful to e.a . schneidmiller for careful reading of the manuscript and to z. huang for providing many helpful explanations . m. borland , y.c . chae , p. emma , j.w . lewellen , v. bharadwaj , w.m . fawley , p. krejcik , c. limborg , s.v . milton , h .- nuhn , r. soliday , and m. woodley , nucl . instrum . methods phys . res . , sect . a * 483 * , 268 ( 2002 ) . z. huang , a. brachmann , f .- j . decker , y. ding , d. dowell , p. emma , j. frisch , s. gilevich , g. hays , ph . hering , r. iverson , h. loos , a. miahnahri , h .- nuhn , d. ratner , g. stupakov , j. turner , j. welch , w. white , j. wu , and d. xiang , phys . rev . beams * 13 * , 020703 ( 2010 ) . h. loos , r. akre , a. brachmann , f .- j . decker , y. ding , d. dowell , p. emma , j. frisch , s. gilevich , g. hays , ph . hering , z. huang , r. iverson , c. limborg - deprey , a. miahnahri , s. molloy , h .- nuhn , d. ratner , j. turner , j. welch , w. white , and j. wu , proceedings of the 30th international free electron laser conference , gyeongju , korea , 2008 , thbau01 . r. akre , d. dowell , p. emma , j. frisch , s. gilevich , g. hays , ph . hering , r. iverson , c. limborg - deprey , h. loos , a. miahnahri , j. schmerge , j. turner , j. welch , w. white , and j. wu , phys . rev . beams * 11 * , 030703 ( 2008 ) . bane , f .- j . decker , y. ding , d. dowell , p. emma , j. frisch , z. huang , r. iverson , c. limborg - deprey , h. loos , h .- nuhn , d. ratner , g. stupakov , j. turner , j. welch , and j. wu , phys . rev . beams * 12 * , 030704 ( 2009 ) . y. ding , a. brachmann , f .- j . decker , d. dowell , p. emma , j. frisch , s. gilevich , g. hays , ph . hering , z. huang , r. iverson , h. loos , a. miahnahri , h .- nuhn , d. ratner , j. turner , j. welch , w. white , and j. wu , phys . 102 * , 254801 ( 2009 ) . z. huang , a. baker , c. behrens , m. boyes , j. craft , f .- j . decker , y. ding , p. emma , j. frisch , r. iverson , j. lipari , h. loos , and d. walz , , proceedings of the 24th particle accelerator conference , new york , usa , 2011 , thp183 . m. yan , c. behrens , ch . gerth , g. kube , b. schmidt , and s. wesch , proceedings of the 10th european workshop on beam diagnostics and instrumentation for particle accelerators , hamburg , germany , 2011 , tupd59 . sukhikh , g. kube , yu.a . popov , a.p . potylitsyn , d. krambrich , and w. lauth , proceedings of the 10th european workshop on beam diagnostics and instrumentation for particle accelerators , hamburg , germany , 2011 , weoa02 .
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high - brightness electron beams with low energy spread at existing and future x - ray free - electron lasers are affected by various collective beam self - interactions and microbunching instabilities .
the corresponding coherent optical radiation effects , e.g. , coherent optical transition radiation , impede electron beam profile imaging and become a serious issue for all kinds of electron beam diagnostics using imaging screens .
furthermore , coherent optical radiation effects can also be related to intrinsically ultrashort electron bunches or the existence of ultrashort spikes inside the electron bunches . in this paper
, we discuss methods to suppress coherent optical radiation effects both by electron beam profile imaging in dispersive beamlines and by using scintillation imaging screens in combination with separation techniques .
the suppression of coherent optical emission in dispersive beamlines is shown by analytical calculations , numerical simulations , and measurements .
transverse and longitudinal electron beam profile measurements in the presence of coherent optical radiation effects in non - dispersive beamlines are demonstrated by applying a temporal separation technique .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the observational evidence for the acceleration of the universe demonstrates that canonical theories of gravitation and particle physics are incomplete , if not incorrect . the next generation of astronomical facilities must therefore be able to carry out precision consistency tests of the standard cosmological model and search for definitive evidence of new physics beyond it . codex @xcite is a spectrograph planned for the european extremely large telescope ( e - elt ) . it should provide the first measurement of the cosmological redshift drift ( known as the sandage - loeb test @xcite ) ; a detailed feasibility study has been carried out by liske _ @xcite , and other aspects relevant for our work have been explored in @xcite . another of its goals is an improved test of the stability of nature s fundamental couplings such as the fine - structure constant @xmath1 and the proton - to - electron mass ratio @xmath2 . apart from the intrinsic importance of these measurements , they can be used ( under certain assumptions ) for detailed characterization of dark energy properties all the way up to redshift 4 . this was suggested in @xcite ( see also @xcite for a related approach ) , and an assessment in the context of codex ( and its predecessor espresso ) can be found in @xcite . we illustrate how codex can probe dark energy beyond the regime where it is dominating the universe s dynamics deep in the matter era . we introduce these two observational tools in sect . ii , and discuss them in the context of two representative classes of models in sects . iii - iv , highlighting their potential synergies . our conclusions are in sect . v. in realistic dynamical dark energy scenarios the ( presumed ) scalar field should be coupled to the rest of the model , unless one postulates a ( yet unknown ) symmetry to suppress these couplings . the relevant coupling here is the one between the scalar field and electromagnetism , which we assume to be @xmath3 where the gauge kinetic function @xmath4 is linear , @xmath5 @xmath6 , and the coupling @xmath7 is related to equivalence principle violations . local constraints are ( conservatively ) @xmath8 @xcite . independent constraints can be obtained from the cosmic microwave background @xcite , and are currently about one order of magnitude weaker . this form of @xmath4 can be seen as the first term of a taylor expansion , and given the tight low - redshift constraints on varying couplings and on equivalence principle violations it is a good approximation for the redshift range being considered . the assumption here is that the dark energy and the varying @xmath1 are due to the same dynamical field , as in the case of nonminimally coupled quintessence models . we will also assume a flat frw universe with @xmath9 , neglecting the radiation contribution since we are concerned with the low - redshift behavior . the evolution of @xmath1 is given by @xmath10 and since the evolution of the scalar field can be expressed in terms of the dark energy properties @xmath11 and @xmath12 as @xcite @xmath13 ( where the prime denotes the derivative with respect to @xmath14 , @xmath15 being the scale factor ) we finally obtain the evolution of @xmath1 in this class of models @xmath16 as expected the magnitude of the variation is controlled by the strength of the coupling @xmath7 . the sandage - loeb test @xcite is a measurement of the evolution of the redshift drift of extragalactic objects , obtained by comparing quasar absorption spectra taken at different epochs . in any metric theory of gravity the redshift drift @xmath17 in a time interval @xmath18 , or equivalently the spectroscopic velocity shift @xmath19 ( which is the directly measured quantity ) is @xmath20\,.\ ] ] this provides a direct measurement of the expansion history of the universe , with no model - dependent assumptions beyond those of homogeneity and isotropy . a positive drift is a smoking gun for a dark energy component accelerating the universe ; a deccelerating universe produces a negative drift . the lyman-@xmath1 forest ( and possibly other absorption lines , including metal ones ) is ideal for this measurement , but it can only be done at redshifts @xmath21 ( in what follows , we will assume measurements between @xmath22 and @xmath23 ) . this applies to ground - based facilities ; measurements at lower redshift would be highly desirable ( since they would probe the dark energy dominated epoch ) , but they would need to be done from space , and there is currently no envisaged space - based spectrograph with the required resolution and stability . _ @xcite have studied in detail the performance of the envisaged codex spectrograph , finding that the uncertainty in the spectroscopic velocity shift is expected to behave as @xmath24 where @xmath25 is the signal - to - noise of the spectra , and @xmath26 and @xmath27 and the number of the absorption systems and their respective redshifts . this assumes photon - noise - limited observations and holds for @xmath28 ; beyond that the last exponent becomes @xmath29 . in our analysis we will assume @xmath30 , 40 systems uniformly divided into 4 bins at @xmath31 and a time between observations of @xmath32 years . suppose that the above assumption regarding varying @xmath1 does not hold : the dark energy is due to a cosmological constant ( with @xmath33 ) , and the variation of @xmath1 is due to some other field with a negligible contribution to the universe s energy density . the bekenstein - sandvik - barrow - magueijo ( bsbm ) model @xcite is precisely of this type ( it has a varying @xmath1 field with an energy density that is no larger than that of radiation ) . if one neglects the recent dark energy domination one can find an analytic solution for the behavior of @xmath1 @xmath34 where @xmath35 gives the magnitude of the variation . this is sufficient for our purposes since we are mainly be interested in the matter - era behavior , but regardless of the bsbm motivation we can take this as a phenomenological parametrization . a logarithmic redshift dependence is typical for a dilaton - type scalar field in the matter - dominated era . this must satisfy the atomic clock bounds at @xmath36 . now @xmath37 ( @xmath38 being the hubble parameter in units of 100 km / s / mpc ) which according to @xcite is constrained to be @xmath39 one therefore finds @xmath40 . this is smaller than the value inferred from the published evidence for a time variation of @xmath1 @xcite ( but see also @xcite ) ; conservatively we will assume a variation of 2 parts per million at @xmath41 , in which case @xmath42 . although the difference is not big given the approximations being made , it does indicate that for the two measurements to be compatible the scalar field must freeze abruptly close to the present time @xcite ; the fact that we have neglected the effect of the onset of dark energy domination will not remove this difference . given the @xmath1 variation of eq . ( [ alphabsbm ] ) , one can show @xcite that ( erroneously ) assuming eq . ( [ coupling ] ) to hold would lead to the following reconstructed equation of state @xmath43}\right]^{-1}\,,\ ] ] where @xmath44 the above assumptions and conservative assumptions on @xmath45 and @xmath46 imply @xmath47 -@xmath46 parameter space leading to allowed values of @xmath48 ( marked in the contour lines ) compatible with a webb - like variation of @xmath1 and the local bounds on @xmath7 . ] ( bottom band ) and @xmath49 ( middle ) , compared to the standard @xmath0cdm case ( top band ) . the bands correspond the range @xmath50 , and the vertical error bars show the uncertainty in the sandage - loeb signal for a codex dataset with an observation time of 20 years . ] in fig . [ fig1 ] we show the region of the @xmath51 parameter space compatible with this range of @xmath48 , and in fig . [ fig2 ] we plot the sandage - loeb signal for some representative models . notice that for a given amount of @xmath1 variation ( that is , a value of @xmath35 ) , a larger coupling @xmath7 implies a smaller @xmath48 ( a slower - moving scalar field ) and vice - versa . large values of @xmath48 produce a sandage - loeb signal that codex can easily distinguish from @xmath0cdm , which would highlight the presence of an inconsistency in the assumptions . conversely small values of @xmath48 yield a sandage - loeb signal indistinguishable from @xmath0cdm , but such a @xmath48 implies a large @xmath7 which could be checked with forthcoming ( improved ) local constraints . in either case , on the assumption that the current evidence for variations is correct , codex in combination with local experiments can support or rule out this class of models . if the dark energy is a cosmological constant its contribution to the universe s energy budget is subdominant by redshift @xmath52 and negligible for @xmath53 . we now consider the opposite case , where the dark energy remains a significant fraction of the universe s energy density . this is realized by the early dark energy models of doran and robbers @xcite , in which the dark energy density parameter and equation of state are @xmath54 @xmath55 } \frac{d\ln\omega_{\phi}(a)}{d\ln a } + \frac{a_{eq}}{3(a + a_{eq } ) } \label{eq : edew}\ ] ] @xmath56 being the scale factor at matter - radiation equality ; we still assume a flat universe , so @xmath57 . ( solid ) and @xmath58 ( dashed ) ; the dotted line shows the standard @xmath0cdm . in the sandage - loeb plot the vertical error bars correspond to the uncertainty in the spectroscopic velocity shift for a codex dataset with an observation time of 20 years.,title="fig : " ] ( solid ) and @xmath58 ( dashed ) ; the dotted line shows the standard @xmath0cdm . in the sandage - loeb plot the vertical error bars correspond to the uncertainty in the spectroscopic velocity shift for a codex dataset with an observation time of 20 years.,title="fig : " ] , at redshift @xmath59 , as a function of @xmath7 and @xmath46 and assuming an early dark energy model with @xmath60 . the shaded region is ruled out by the bound of eq . ( [ clocks ] ) . ] the dark energy has a scaling behavior , approaching a finite constant @xmath61 in the past , while its equation of state @xmath62 tracks the dominant energy component . we assume that the early dark energy field is also coupling to electromagnetism and thus yielding a varying @xmath1 @xcite , and our discussion in sect . ii , and in particular eqs . ( [ darkside1]-[darkside2 ] ) also apply in this case . here our analysis is the opposite of that in the previous section : there we assumed a given amount of @xmath1 variation ; here we will assume a given amount of early dark energy , namely @xmath60 , consistent with current bounds allowing for possible @xmath1 variations @xcite . the local atomic clock bound is now @xmath63 regardless of @xmath61 . for @xmath46 significantly different from @xmath64 this places a model - dependent constraint on @xmath7 that is stronger than the model - independent @xmath65 , but large values of @xmath7 are possible by having @xmath46 sufficiently close to @xmath64 : with @xmath66 and @xmath67 the largest allowed value is @xmath68 ; this highlights the importance of atomic clock constraints , and shows that this model is almost indistinguishable from @xmath0cdm if one is limited to low - redshift observations . despite the change in the dark energy equation of state at redshift @xmath69few , the sandage - loeb test is unable to distinguish this model from @xmath0cdm , as can be seen in fig . nevertheless , this model can yield significant variations of @xmath1 , as shown in [ fig4 ] . codex s baseline sensitivity for @xmath1 measurements is around the @xmath70 level , and as good as @xmath71 for a few ideal systems ; this is enough to detect such variations for @xmath72 and use the measurements to reconstruct @xmath73 , as discussed in @xcite . we discussed two examples of codex s ability to probe the nature of dark energy beyond the regime where it is dynamically important and highlighted the importance of carrying out both the sandage - loeb test and accurate measurements of nature s fundamental couplings . all three theoretical pillars of the @xmath0cdm paradigm ( inflation , dark matter and dark energy ) rely on the presence of new , presently unknown physics . in the absence of strong indications for what this new physics is and where it can be found , it is important to search for it in multiple places , and codex will have a unique role to play in the @xmath74 redshift range . our analysis is simplified , but the goal is to illustrate the point at proof - of - concept level . a detailed study , with precise codex specifications , can be done when these are finalized . finally , we emphasize that these tests do not exist in isolation : synergies can be found with other cosmological experiments , including esa s euclid mission , which will probe lower redshifts . an analysis of these possibilities , in the context of a broader observational strategy , is left for future work . this work was done in the context of the joint master in astronomy of the universities of porto and toulouse , supported by project ai / f-11 under the crup / portugal cup / france cooperation agreement ( f - fp02/11 ) . we acknowledge the support of fundao para a cincia e a tecnologia ( fct ) , portugal , through grant ptdc / fis/111725/2009 . the work of cm is funded by a cincia2007 research contract , supported by fct / mctes ( portugal ) and poph / fse ( ec ) . 99 s. cristiani _ et al . _ , nuovo cim . * b122 * ( 2007 ) , 1159 . a. sandage , astrophys . j. * 136 * ( 1962 ) 319 . a. loeb , astrophys . j. * 499 * ( 1998 ) , 111 . _ , mon . not . * 386 * ( 2008 ) 1192 . corasaniti , d. huterer and a. melchiorri , phys . * d75 * ( 2007 ) 062001 . a. balbi and c. quercellini , mon . not . soc . * 382 * ( 2007 ) 1623 . n. j. nunes and j. e. lidsey , phys . * d69 * ( 2004 ) 123511 . d. parkinson , b. a. bassett , and j. d. barrow , phys . lett . * b578 * ( 2004 ) 235 . l. amendola , a. c. o. leite , c. j. a. p. martins _ et al . _ , arxiv:1109.6793 . k. a. olive and m. pospelov , phys . rev . * d65 * ( 2002 ) 085044 . g. r. dvali and m. zaldarriaga , phys . * 88 * ( 2002 ) 091303 . e. calabrese , e. menegoni , c. j. a. p. martins _ et al . _ , phys . * d84 * ( 2011 ) 023518 . h. b. sandvik , j. d. barrow , and j. magueijo , phys . * 88 * ( 2002 ) 031302 . t. rosenband _ et al . _ , science * 319 * ( 2008 ) , 1808 . m. t. murphy _ et al . _ , notes phys . * 648 * ( 2004 ) 131 . m. t. murphy , j. k. webb and v. v. flambaum , mon . not . * 384 * ( 2008 ) 1053 - 1062 . j. k. webb _ _ , phys . * 107 * ( 2011 ) 191101 . n. j. nunes , t. dent , c. j. a. p. martins , and g. robbers , mem . s. a. it . * 80 * ( 2010 ) 785 . m. doran and g. robbers , j. c. a. p. * 0606 * ( 2006 ) 026 .
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precision measurements of nature s fundamental couplings and a first measurement of the cosmological redshift drift are two of the key targets for future high - resolution ultra - stable spectrographs such as codex .
being able to do both gives codex a unique advantage , allowing it to probe dynamical dark energy models ( by measuring the behavior of their equation of state ) deep in the matter era and thereby testing classes of models that would otherwise be difficult to distinguish from the standard @xmath0cdm paradigm .
we illustrate this point with two simple case studies .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
chiral lagrangian for low lying pseudoscalar mesons@xcite@xcite as the most successful effective field theory is now widely used in various strong , weak and electromagnetic processes . to match the increasing demand for higher precision in low energy description of qcd , the applications of the low energy expansion of the chiral lagrangian is extended from early time discussions on the leading @xmath3 and next to leading @xmath1 orders to present @xmath0 order . for the latest review , see ref.@xcite . in the chiral lagrangian , there are many unknown phenomenological low energy constants ( lecs ) which appear in front of each goldstone field dependent operators and the number of the lecs increases rapidly when we go to the higher orders of the low energy expansion . for example for the three flavor case , the @xmath3 and @xmath1 order chiral lagrangian have 2 and 10 lecs respectively , while the normal part of @xmath0 order chiral lagrangian have 90 lecs . such a large number of lecs is very difficult to fix from the experiment data . this badly reduces the predictive power of the chiral lagrangian and blur the check of its convergence . the area of estimating @xmath0 order lecs is where most improvement is needed in the future of higher order chiral lagrangian calculations . a way to increase the precision of the low energy expansion and improve the present embarrassed situation is studying the relation between the chiral lagrangian and the fundamental principles of qcd . we expect that this relation will be helpful for understanding the origin of these lecs and further offer us their values . in previous paper @xcite , based on a more earlier study of deriving the chiral lagrangian from the first principles of qcd @xcite in which lecs are defined in terms of certain green s functions in qcd , we have developed techniques and calculated the @xmath3 and @xmath1 order lecs approximately from qcd . our simple approach involves the approximations of taking the large-@xmath4 limit , the leading order in dynamical perturbation theory , and the improved ladder approximation , thereby the relevant green s functions relate to lecs are expressed in terms of the quark self energy @xmath2 . the result chiral lagrangian in terms of the quark self energy is proved equivalent to a gauge invariant , nonlocal , dynamical ( gnd ) quark model@xcite . by solving the schwinger - dyson equation ( sde ) for @xmath2 , we obtain the approximate qcd predicted lecs which are consistent with the experimental values . with these results , generalization of the calculations to @xmath0 order lecs becomes the next natural step . considering that the algebraic derivations for those formulae to express lecs in terms of the quark self energy at @xmath1 order are lengthy ( they need at least several months of handwork ) , it is almost impossible to achieve the similar works for the @xmath0 order calculations just by hand . therefore , to realize the calculations for the @xmath0 order lecs , we need to computerize the original calculations and this is a very hard task . the key difficulty comes from that the formulation developed in ref.@xcite and exploited in ref.@xcite not automatically keeps the local chiral covariance of the theory and one has to adjust the calculation procedure by hand to realize the covariance of the results . to match with the computer program , we need to change the original formulation to a chiral covariant one . in ref.@xcite , we have built and developed such a formulation , followed by next several year s efforts , we now successfully encode the formulation into computer programs . with the help of these computer codes we can reproduce analytical results on the computer originally derived by hand in ref.@xcite within 15 minutes now . this not only confirms the reliability of the program itself , but also checks the correctness of our original formulae . based on these progresses , in this paper , we generalize our previous works on calculating the @xmath1 order lecs to computing the @xmath0 order lecs of chiral lagrangian both for two and three flavor pseudo - scalar mesons . this generalization not only produces new numerical predictions for the @xmath0 order lecs , but also forces us to reexamine our original formulation from a new angle in dealing with @xmath3 and @xmath1 order lecs . this paper is organized as follows : in sec.ii , we review our previous calculations on the @xmath3 and @xmath1 order lecs . then , in sec.iii , based on the technique developed in ref.@xcite , we reformulate the original low energy expansion used in ref.@xcite into a chiral covariant one suitable for computer derivation . in sec.iv , from present @xmath0 order viewpoint , we reexamine the formulation we taken before and show that if we sum all higher order anomaly part contributions terms together , their total contributions to the normal part of the chiral lagrangian vanish . this leads a change the role of finite @xmath1 order anomaly part contributions which originally are subtracted in the chiral lagrangian in ref.@xcite and now must be used to cancel divergent higher order anomaly part contributions . we reexhibit the numerical result of the @xmath1 order lecs without subtraction of @xmath1 order anomaly part contributions . in sec.v , we present general @xmath0 order chiral lagrangian in terms of rotated sources and express the @xmath0 order lecs in terms of the quark self energy . sec.vi is a part where we give numerical results for @xmath0 order lecs in the normal part of chiral lagrangian both for two and three flavor pseudo scalar mesons . in sec . vii , we apply and compare with our results to some individuals and combinations of lecs proposed and estimated in the literature , checking the correctness of our numerical predictions . sec.viii is a summary . in appendices , we list some necessary formulae and relations . with the analytical formulae for lecs of @xmath53 and @xmath54 for @xmath55 as functions of @xmath25 , we can suitably choose running coupling constant @xmath30 , solve sde ( [ eq0 ] ) numerically obtaining quark self energy @xmath25 , then calculate the numerical values of all @xmath3 and @xmath1 order lecs . to obtain the final numerical result in ref.@xcite , we have assumed @xmath69mev as inputmev for two - flavour case . for detail , see the discussion of eq.(58 ) ] to fix the dimensional parameter @xmath70 appear in running coupling constant @xmath30 and taken cutoff parameter @xmath51 appear in ( [ trln ] ) equal to infinity and 1gev respectively . the final obtained values are consistent with those fixed phenomenologically . eq.([trln ] ) is the starting point of our reformulation in this section . in ref.@xcite , we expand ( [ trln ] ) up to the @xmath1 order and obtain analytical result . this expansion is not explicitly chiral covariant , since the operator @xmath71 appears in the formula is not always covariant under the local chiral symmetry transformations . for example , when @xmath71 acts on a constant number 1 , it gives @xmath72 which is not covariant since @xmath73 itself behaves as the gauge field in the local chiral symmetry transformations . only when they combined into commutators , such as @xmath74 $ ] or @xmath75 $ ] , the covariance recovers back . therefore in the detail calculation , we need to confirm that all @xmath71s appear in the result do can be arranged into some commutators . this is a conjecture . in the original work of ref.@xcite , we have found that this conjecture is valid up to some terms with coefficients being expressed as integration over some total derivatives , i.e. form of @xmath76 . if we ignore these total derivative terms , up to order of @xmath1 , we can explicitly prove the conjecture . at the stage of our earlier works , we do not question the reason that why we can drop out those total derivative terms ( in fact , in eq.(74 ) of ref.@xcite , we have shown that in order to obtain the well - known pagels - stokar formula , a total derivative term must be dropped out ) . this leads the further discussions on the role of total derivative terms in the quantum field theory @xcite . later in this section , we will give the correct reason of dropping out those total derivative terms . arranging various @xmath71 into commutators is a very tricky and complex task which is very hard to be achieved by computer . in order to computerize the calculation , we need to find a way which can automatically arrange all @xmath71s into some commutators . this leads the developments given in ref.@xcite , where we have introduced @xmath77 in which @xmath78 ( i\bar{\nabla}_x^\mu)\bigg)\nonumber\\ & = & \frac{1}{2}(\nu\mu)\frac{\partial}{\partial k^{\nu } } -\frac{i}{3}(\lambda\nu\mu)\frac{\partial^2}{\partial k^{\lambda}\partial k^{\nu } } -\frac{1}{8 } ( \rho\lambda\nu\mu ) \frac{\partial^3}{\partial k^{\rho}\partial k^{\lambda}\partial k^{\nu } } + \frac{i}{30}(\sigma\rho\lambda\nu\mu ) \frac{\partial^4}{\partial k^\sigma\partial k^{\rho}\partial k^{\lambda}\partial k^{\nu}}\notag\\ & & + \frac{1}{144}(\delta\sigma\rho\lambda\nu\mu ) \frac{\partial^5}{\partial k^\delta\partial k^\sigma\partial k^{\rho}\partial k^{\lambda}\partial k^{\nu}}+o(p^7)\;,\\ & & \hspace{-2.3cm}f(z)=\sum^\infty_{n=2}\frac{z^{n-1}}{n!}\hspace{3 cm } [ ad(b)]^n(c)\equiv[\underbrace{b,[b,\cdots,[b}_{\mbox{n times}},c]\cdots]]\;,\notag\\ & & \hspace{-2.3cm}(\mu_n\mu_{n-1}\cdots\mu_2\mu_1 ) \equiv[\nx^{\mu_n},[\nx^{\mu_{n-1}},\cdots,[\nx^{\mu_2},\nx^{\mu_1}]\cdots]]\;,\notag \end{aligned}\ ] ] where the default set of lorentz indices for @xmath79 is the supperscripts , in some cases , we need subscript , we will use @xmath80 to denote the corresponding subscript for @xmath81 . note that in present notation for @xmath79 , we do nt explicitly write @xmath82s , but only their greek superscripts for short . if we use other symbols , such as @xmath83 appeared in @xmath84 and @xmath85 in @xmath86 , then we take definition that @xmath87 $ ] and @xmath88 $ ] . with the help of ( [ debt ] ) , ( [ d2ebt ] ) and ( [ d3ebt])-([d6ebt ] ) , as long as the @xmath118 are known , ( [ ebexp0 ] ) is known and we can substitute it back into ( [ db1 ] ) to calculate the real part of @xmath63 $ ] order by orders up to the @xmath0 order in the low energy expansion . to obtain @xmath119 , ( [ bdef ] ) tells us that the difficulty is the low energy expansion for @xmath120 . to achieve it , we expand the argument of @xmath120 as @xmath121 in which @xmath122 since we are only interested in the terms not higher than @xmath0 , we find that those traceless terms of @xmath123 and @xmath0 orders will not make contributions to the final result . so to save space and simplify the computations , we do not explicitly write down the detail structure of them , just represent these terms with symbol @xmath124 and remove traceless term in @xmath125 and @xmath126 . further introduce @xmath127 as , @xmath128 then @xmath129\bigg]_{t=0 } + \frac{1}{2!}\bigg[\frac{d^2}{d t^2}\sigma[a(t)]\bigg]_{t=0 } + \frac{1}{3!}\bigg[\frac{d^3}{d t^3}\sigma[a(t)]\bigg]_{t=0 } + \frac{1}{4!}\bigg[\frac{d^4}{d t^4}\sigma[a(t)]\bigg]_{t=0}+\ldots\;.\label{sigmaexp}\end{aligned}\ ] ] now , we need to know @xmath130\bigg]_{t=0}$ ] , using the following formula @xmath131&=&\sigma[s+a(t)]\bigg|_{s=0 } = e^{a(t)\frac{\partial}{\partial s}}\sigma(s ) e^{-a(t)\frac{\partial}{\partial s}}\bigg|_{s=0 } = e^{a(t)\frac{\partial}{\partial s}}\sigma(s)\bigg|_{s=0}\ ; , \end{aligned}\ ] ] then @xmath132\bigg]_{t=0 } = \bigg[\frac{d^m}{d t^m}e^{a(t)\frac{\partial}{\partial s } } \bigg]_{t=0}\sigma(s)\bigg|_{s=0}\;. \end{aligned}\ ] ] therefore to compute @xmath130\bigg]_{t=0}$ ] , we only need to calculate @xmath133_{t=0}\sigma(s)\bigg|_{s=0}$ ] which is just equivalent to replace @xmath134 in ( [ debt]),([d2ebt ] ) and ( [ d3ebt])-([d6ebt ] ) , followed by multiplying an extra factor @xmath135 at the r.h.s . and vanishing parameter @xmath136 after finishing all differential operations . follow this calculation road map , the detail calculation gives @xmath137\bigg]_{t=0 } & = & e^{ad(a_0\frac{\partial}{\partial s})}(f[ad(-a_0\frac{\partial}{\partial s } ) ] a_1)\sigma'(s+a_0)\bigg|_{s=0}=0\;,\\ \frac{1}{2}\bigg[\frac{d^2}{d t^2}\sigma[a(t)]\bigg]_{t=0 } & = & \frac{1}{2}e^{ad(a_0\frac{\partial}{\partial s})}\bigg[\bigg(e^{-a_0\frac{\partial}{\partial s } } \frac{d}{d t}e^{a(t)\frac{\partial}{\partial s}}\bigg|_{t=0 } f[ad(-a_0\frac{\partial}{\partial s})](a_1\frac{\partial}{\partial s } ) + \frac{d f}{d t}[ad(-a(t)\frac{\partial}{\partial s})]\bigg|_{t=0}(a_1\frac{\partial}{\partial s})\nonumber\\ & & + f[ad(-a_0\frac{\partial}{\partial s } ) ] ( a_2\frac{\partial}{\partial s})\bigg)\bigg]\sigma(s+a(t))\bigg|_{s=0}=- ( \mu,\nu)k_{\mu}\sigma'_k \frac{\partial}{\partial k^{\nu}}\ ; , \end{aligned}\ ] ] where @xmath138 . for more higher orders , we list the results of @xmath139\bigg]_{t=0}$ ] , @xmath140\bigg]_{t=0}$ ] , @xmath141\bigg]_{t=0}$ ] and @xmath142\bigg]_{t=0}$ ] in appendix [ ebexp ] . with these results , we finally obtain the low energy expansion of @xmath100 , @xmath143\tau -i ( d^{\mu}a_\omega^\nu - d^{\nu}a_\omega^{\mu})\gamma_{\mu}\gamma_{\nu}\gamf \tau + 4 s_\omega \tau \sk+2 ( \mu \nu ) \tau k_{\mu}\frac{\partial}{\partial k^{\nu } } + 4 ( \mu a_\omega^{\nu } ) \gamma_{\nu}\gamf \tau k_{\mu}\skp\notag\\ & & + 4 ( \mu a_\omega^{\nu } ) \gamma_{\nu}\gamf \tau \sk \frac{\partial}{\partial k^{\mu } } + 2i ( \mu a_{\omega\mu } ) \gamf \tau + 4i ( \mu a_\omega^{\nu } ) \gamf \tau k_{\nu}\frac{\partial}{\partial k^{\mu}}+4i ( \mu \nu ) \gamma_{\mu}\tau k_{\nu}\skp + 4 ( \mu \nu ) \tau k_{\mu}\sk \skp \frac{\partial}{\partial k^{\nu}}\;.\end{aligned}\ ] ] we list @xmath144 in appendix.[ebexp ] . with these explicit expressions for @xmath118 , using ( [ debt ] ) , ( [ d2ebt ] ) and ( [ d3ebt])-([d6ebt ] ) , we get ( [ ebexp0 ] ) and further substitute ( [ ebexp0 ] ) back into ( [ db1 ] ) , we can obtain the real part of @xmath63 $ ] order by orders up to the @xmath0 order in the low energy expansion . the analytical results of @xmath3 and @xmath1 orders are the same as those given by ( 34),(35 ) and ( 36 ) in ref.@xcite , except some total derivative terms which , as we mentioned before , can be ignored as long as we take finite cutoff @xmath51 . in the last section , we have introduced a chiral covariant method to calculate @xmath34 $ ] which is already computerized now . with the help of computer , for the @xmath3 and @xmath1 order analytical formulae in the low energy expansion , we can get results within 15 minutes , while for the @xmath0 order terms , we need roughly 13 hours to output all expansion results . from our general result ( [ seff1 ] ) , the term @xmath63 $ ] is the normal part . to get the full result of the chiral lagrangian , we need to calculate the remaining anomaly part contributions @xmath145-in_c\mathrm{tr}\ln[i\slashed{\partial}+j]$ ] . as the discussion of ref.@xcite , in 1980s there is a class of works ( see references given in @xcite ) identifying this part as the full chiral lagrangian , and in ref.@xcite we refer them as the anomaly approach of calculating lecs . in our previous work @xcite , we pointed out that this anomaly part contributions are completely canceled by the normal part contribution , left nontrivial pure @xmath25 dependent terms contribute to the chiral lagrangian . for the anomaly part contributions , the key is to calculate the @xmath18 field dependent term @xmath35 $ ] which , as we mentioned before , can be obtained by vanishing @xmath25 in @xmath34 $ ] . in practice , the limit was taken by first assuming @xmath25 being a constant mass @xmath146 and then letting @xmath147 . for @xmath3 order , this operation gives null result , while for @xmath1 order , it gives the result originally presented in anomaly approach . now in this work , naively what we need to do is to generalize the calculation to @xmath0 order . but to our surprise , we get many terms with divergent coefficients . checking the calculation carefully , we find that the reason of appearance of infinities is due to the fact that most of the coefficients in front of the @xmath0 order operators have dimension of @xmath148 which goes to infinity when we take limit @xmath149 . note that the @xmath0 terms may also have coefficients of @xmath150 which are finite in the limit of @xmath147 , although they vanish when we take @xmath151 . these terms are irrelevant to our discussion on the divergence of @xmath0 order terms and therefore we do not need to care about them . applying the argument on @xmath148 dependence of the @xmath0 order coefficients back to the @xmath3 and @xmath1 order results we discussed before , coefficients in front of @xmath3 order operators have dimension of @xmath152 which goes to zero , this explains the phenomena that anomaly approach can not produce @xmath3 order terms . for @xmath1 order , the coefficients in front of operators are dimensionless and therefore the @xmath146 dependence is at most logarithmic of form @xmath153 which implies existence of a logarithmic ultraviolet divergence . since we know that in the large @xmath4 limit , the @xmath1 order lecs ( non - contact coefficients ) are not divergent , the @xmath153 term then can not appear in the final expression of these lecs , therefore in @xmath1 order , anomaly approach leads finite result lecs . in general for a @xmath154 order operator , the corresponding coefficient should has dimension @xmath155 . this implies that the infinity in the anomaly part contributions will be a general phenomena , when we go to the higher orders of the low energy expansion , since the more higher the order is , the more negative powers of @xmath146 dependence the coefficient will have and these negative powers of @xmath146 will result in infinities as we take limit @xmath149 . the appearance of these high order infinities provides another evidence that the anomaly approach is not a correct formulation in calculating lecs , at least not for the @xmath0 and more higher order lecs . since high order divergence term is as an addition part of our general result ( [ seff1 ] ) , we can not avoid them in our computations . how to deal with these high order infinities from negative powers of @xmath146 ? there exists an alternative way , not relying on the low energy expansion , to examine this anomaly part contributions in which we must exploit the first equation of ( [ jomega ] ) and we find @xmath156-\mathrm{tr}\ln[i\slashed{\partial}+j ] & = & \ln\mathrm{det}[i\slashed{\partial}+j_{\omega}]-\ln\mathrm{det}[i\slashed{\partial}+j]\nonumber\\ & = & \ln\mathrm{det}\bigg[[\omega p_r+\omega^{\dag}p_l][j+i\slashed{\partial}][\omega p_r+\omega^\dag p_l]\bigg]-\ln\mathrm{det}[i\slashed{\partial}+j]\nonumber\\ & = & \ln\mathrm{det}\bigg[[\omega p_r+\omega^{\dag}p_l][\omega p_r+\omega^\dag p_l]\bigg]=\mathrm{tr}\ln\bigg[[\omega p_r+\omega^{\dag}p_l][\omega p_r+\omega^\dag p_l]\bigg]\;.\end{aligned}\ ] ] for our interests , we are only interested in the real part of it , then @xmath157-\mathrm{retr}\ln[i\slashed{\partial}+j ] & = & \frac{1}{2}\mathrm{tr}\ln\bigg[[\omega p_r+\omega^{\dag}p_l][\omega p_r+\omega^\dag p_l]\bigg]+\frac{1}{2}\mathrm{tr}\ln\bigg[[\omega p_r+\omega^{\dag}p_l]^\dag[\omega p_r+\omega^\dag p_l]^\dag\bigg]\nonumber\\ & = & \frac{1}{2}\mathrm{tr}\ln\bigg[[\omega p_r+\omega^{\dag}p_l][\omega p_r+\omega^\dag p_l]\bigg]+\frac{1}{2}\mathrm{tr}\ln\bigg[[\omega^\dag p_r+\omega p_l][\omega^\dag p_r+\omega p_l]\bigg]\nonumber\\ & = & \frac{1}{2}\mathrm{tr}\ln\bigg[[p_r+p_l][p_r+p_l]\bigg]=\frac{1}{2}\mathrm{tr}\ln 1=0\;.\end{aligned}\ ] ] which shows that the compact form of anomaly part contributions to normal part of the chiral lagrangian is zero ! the third method is first taking taylor expansion in terms the power of @xmath164 and @xmath165 which corresponds performing the low energy expansion and then finishing integration , finally vanishing @xmath146 , @xmath170 we see that there are negative power of @xmath146 terms which will cause divergence when we take limit @xmath171 . this is just what has happened for the high order terms in the anomaly part contributions . so if we calculate term by terms in above expansion , we will meet infinities which seems contradict with results obtained in first two methods . the only way left to escape this contradiction is to sum all these divergences together , to see that what will happen after the summation , we introduce a series @xmath172 in which @xmath173 and @xmath174 which will go to negative infinity when @xmath171 . with the help of relation @xmath175 and boundary condition @xmath176 , we find @xmath177\;,\notag\\ & & g'(x , c)=(x-1)e^c[-\mathbf{ei}(cx - c)+\mathbf{ei}(-c)]+\frac{1}{c}(e^{cx}-1)\;,\notag\\ & & g(x , c)=\frac{1}{2}(x-1)^2e^c[-\mathbf{ei}(cx - c)+\mathbf{ei}(-c)]+\frac{x-1}{2c}e^{cx}+\frac{1}{2c } + \frac{1}{2c^2}(e^{cx}-1)-\frac{x}{c}\;.\end{aligned}\ ] ] then ( [ i1 ] ) becomes @xmath178\nonumber\\ & = & \lim_{m\rightarrow0}\bigg[\frac{\lambda^4}{2}-c\lambda^2-\frac{1}{2}c^2\mathbf{ei}(-\frac{m^2}{\lambda^2 } ) + \frac{1}{2}(bm+c+m^2)^2[-\mathbf{ei}(\frac{-bm - c - m^2}{\lambda^2})+\mathbf{ei}(-\frac{m^2}{\lambda^2 } ) ] \nonumber\\ & & + \frac{1}{2}\lambda^2(-bm - c - m^2)e^{\frac{-bm - c - m^2}{\lambda^2 } } + \frac{1}{2}\lambda^2m^2e^{-\frac{m^2}{\lambda^2}}+\frac{1}{2}\lambda^4 ( e^{\frac{-bm - c - m^2}{\lambda^2}}-e^{-\frac{m^2}{\lambda^2}})+\lambda^2(bm+c)\bigg]\nonumber\\ & = & -\frac{1}{2}c^2\mathbf{ei}(-\frac{c}{\lambda^2 } ) -\frac{1}{2}\lambda^2ce^{-\frac{c}{\lambda^2}}+\frac{1}{2}\lambda^4e^{-\frac{c}{\lambda^2}}\;. \end{aligned}\ ] ] it is the same as the results obtained from first two methods , i.e. summing all those infinities together , we obtain correct finite result . with this new viewpoint on all anomaly part contributions , we need to modify our original numerical results on @xmath1 order lecs , since it takes into account of the finite values of anomaly part contributions and now we know that these nontrivial values must be used to cancel the infinities come from all higher order terms . in table i , we list our modified @xmath1 order lecs for cutoff @xmath1811000@xmath182mev . the @xmath183 variation of the cutoff is considered in our calculation to examine the effects of cutoff dependence and the result change can be treated as the error of our calculations . the result lecs are taken the values at @xmath184gev with superscript the difference caused at @xmath185gev and subscript the difference caused at @xmath186gev , i.e. , @xmath187 . the obtained values of the @xmath1 coefficients @xmath188 for three flavor quarks and @xmath189 for two flavor quarks + where @xmath190 for @xmath191 , @xmath192770mev and @xmath193 are given in ref.@xcite . since @xmath194 , we calculate @xmath195 instead of @xmath196 . + together with the experimental values given in ref.@xcite and our old result given in ref.@xcite for comparisons . + @xmath70 , @xmath51 and @xmath197 are in units of mev , and @xmath198 are in units of @xmath199 . [ cols="^,^,^,^,^,^,^,^,^,^,^,^,^",options="header " , ] in obtaining the result , we have taken the running coupling constant as model a given by ( 40 ) of ref.@xcite and the low energy value of this @xmath200 is already chosen well above the critical value to trigger the s@xmath201sb of the theory . it should be noted that @xmath200 depends on the number of quark flavors , so does for @xmath25 from sde . in fixing the @xmath70 we have taken input @xmath202mev . the reason of taking this value is that if the final @xmath203 is around value of 93mev , then our formula shows @xmath204 must be located around 87mev . in sec.vi , we will exhibit this phenomena explicitly . the general form of @xmath0 order chiral lagrangian was first introduced in ref.@xcite and then discussed in ref.@xcite . now we can express the normal part of it in terms of our rotated sources as what we have done in ( [ p4 ] ) for the @xmath1 order terms . considering that our computation is done under large @xmath4 limit , within this approximation , terms in the chiral lagrangian with two and more traces vanish when we not apply the equation of motion . to avoid unnecessary complicities , in this paper we only write down those terms with one trace @xmath205~+~o(\frac{1}{n_c})\bigg ] \label{sp6ours}\end{aligned}\ ] ] with @xmath206 being @xmath0 order operator we could get in our calculation and @xmath207 being corresponding coefficient , @xmath208 are consist of the most multi - traces terms . our computations then give the explicit expressions of @xmath207 in terms of quark self energy . the detail expressions are given in ( [ zsigma ] ) . and the definitions of operators @xmath206 for @xmath209 are given in table.ii , @xmath210 where some operators have @xmath211 in front of them to insure their coefficients being real . in ref.@xcite , @xmath0 order operator was denoted by @xmath212 for the case of n flavor with coefficient @xmath213 @xcite , @xmath214 for the case of three flavor with coefficient @xmath215 and @xmath216 for the case of two flavors with coefficient @xmath217 , @xmath218 consider that our parametrization of the @xmath0 order chiral lagrangian ( [ sp6ours ] ) is general to the case of n flavor quarks , there exist some relations among our coefficients and n flavor coefficients given in ( [ sp6ref ] ) . with the help of computer derivations , we have worked out these relations and list them in apendix[kz ] . as a check , we vanish the quark self energy @xmath25 in the codes which corresponds to take @xmath219 before other further calculations and find null @xmath0 result . this verify the analytical result discussed in sec.iv that the anomaly part contributions do not the contribute to normal part of chiral lagrangian . another consistency check is done for those operators which have two terms combined together by @xmath165 and @xmath220 symmetry requirements . for @xmath221 flavor case , such operators are @xmath222 , @xmath223 , @xmath224 , @xmath225 , @xmath226 , @xmath227 , @xmath228 , @xmath229 , @xmath230 , @xmath231 , @xmath232 , @xmath233 , @xmath234 , @xmath235 , @xmath236 , @xmath237 , @xmath238 , @xmath239 , @xmath240 , @xmath241 , @xmath242 , @xmath243 , @xmath244 , @xmath245 , @xmath246 , @xmath247 , @xmath248 , @xmath249 , @xmath250 , @xmath251 , @xmath252 . since in each of these operators , there are two terms , we can compute the coefficients in front of each terms and check if they are same . we have done the checks for all these operators and all obtain the same analytical expressions for the two terms in the same operator . this partly verifies the correctness of our result given in ( [ zsigma ] ) . from n flavors to three flavors , there are some extra constraints ( see ( b1 ) in ref.@xcite)which make some operators depending on others . further from three flavors to two flavors , there are also some extra constraints ( see ( b3 ) in ref.@xcite)which make some more operators depending on others . the sequence number for n flavors , three flavors and two flavors are different , their comparisons are list in table 2 in ref.@xcite . with all above preparations in previous sections , we now come to the stage of giving numerical values to the @xmath0 order lecs in the normal part of the chiral lagrangian . note the necessary input and process of the present computations for the @xmath0 order lecs are the same as those for the @xmath3 and @xmath1 order lecs given in the end of sec.iv , we list the numerical result in table.iii , as done in table.i , the result lecs are taken for the values at @xmath184gev with superscript the difference caused at @xmath185gev and subscript the difference caused at @xmath186gev , @xmath253 we further list result of the @xmath0 order lecs at @xmath254 in table.iv . consider that in the limit of @xmath254 , dropping out momentum total derivative terms in eq.(14 ) is problematic , we only take result lecs at @xmath254 as a reference . since the terms of three flavors and two flavors may have different sequence numbers , as done in ref.@xcite , we put them in the same line in our table . since the number of independent operators in the two flavors is smaller than that in the three flavors , there are some operators in three flavors being independent operators , but being dependent in two flavors , then these operators will not have their two flavor counter parts in our table , these leave the r.h.s . some empty blanks in the corresponding two flavor columns . for two flavor case , ref.@xcite further propose a new relation among operators , @xmath255 which implies that one of the operators appears in above formula should be further dependent operator . due to ignorance of the values of the coefficients in front of these operators , ref.@xcite arbitrarily chooses this operator being @xmath256 . now our computations show that @xmath257 , so original choice is not suitable . considering that @xmath258 in our computation , we instead now take @xmath259 as that dependent operator . @xmath259 now is a dependent operator , its name then is deleted in our table.iii . @xmath273 as we have mentioned in the introduction of this paper , present experiment data is far enough to fix the @xmath0 order lecs . but there do exist some combinations of the lecs which already have their experiment or model calculation values . usually , these lecs are labeled by dimensionless parameters with convention and @xmath274 are used to denote the renormalized lecs in some literatures.]of @xmath275 or @xmath276 . in this section , we collect those combinations of lecs in the literature which have their experiment or model calculation values and compare them with our numerical results obtained in the last section with finite cutoff , we have checked that qualitative feature of the comparisons results of this section will not change . ] as the check of our computations . from the investigation of @xmath277 scattering amplitudes , one can work out the values of some combinations of @xmath0 order lecs . ref.@xcite introduces following combinations , @xmath279 and gives the values of them by two theoretical methods of the resonance - saturation ( rs)@xcite and pure dimensional analysis ( nd ) which only accounts for the order of magnitude and in table.v . @xmath280 we see that all coefficients obtained from our calculations are consistent with those more precise rs results given in ref.@xcite . with our predictions for @xmath1 and @xmath0 order lecs , we can directly calculate the scattering lengths @xmath281 and slope parameters @xmath282 which relate to @xmath1 and @xmath0 order lecs through formulae given in appendix c and d. of ref.@xcite . we list experimental and our results in table.vi . in our results , as done in table.iii , we take @xmath283mev , but to match the result given in ref.@xcite where @xmath81 is taken at @xmath284gev , we also take @xmath285mev for comparison . we take two options , one only includes @xmath1 order contributions and the other combines in @xmath0 order contributions . for @xmath0 order contributions , for comparisons , we consider the cases of without and with @xmath286 coefficients . @xmath287 we see that the contributions from @xmath0 order lecs are rather small and only change the third digit of the result . further , ref.@xcite introduce coefficients in @xmath278 scattering @xmath288 in the table 1 . of ref.@xcite , in terms of @xmath289 , three constraints of @xmath0 order lecs are fixed from @xmath278 subthreshold parameters , @xmath277 amplitude and a resonance model . and in the table 2 . of ref.@xcite , in terms of @xmath290 , another three constraints of @xmath0 order lecs are fixed from the dispersive calculations and a resonance model , @xmath291 in which for l.h.s . of the table vii . , except @xmath292 , all other lecs or combinations of lecs obtained by us have the same signs and orders of magnitudes as those from ref.@xcite . while for r.h.s . of the table , our results are not consistent with those obtained from the dispersive calculations . in ref.@xcite , in dealing with the vector form factor of the pion , @xmath293 and @xmath294 are introduced into theory which relate to @xmath0 order lecs through @xmath295 while for the scalar form factor , people introduce @xmath296 and @xmath297 relate to @xmath0 order lecs by @xmath298 in ref.@xcite , discussion of the decay of @xmath299 further introduces @xmath300 and @xmath301 relate to @xmath0 order lecs by @xmath302 in ref.@xcite , a naive estimation of @xmath303 is made from scalar meson dominance ( smd ) of the pion scalar form - factor and @xmath304 is estimated through @xmath305 in @xmath306 measurements ( see eq.(8.11 ) in ref.@xcite ) . while in ref.@xcite , @xmath307 and @xmath308 are also estimated from the @xmath278 form factors . in table.viii , we list the numerical results for above combinations of lecs given by our calculations based on table.iii in last section and by ref.@xcite,@xcite,@xcite . @xmath309 from which we see that among ten parameters between our predictions and values given in the literature , four of them have the same orders of magnitudes and signs ( @xmath310 , @xmath311 , @xmath312 and @xmath313 ) , another one of them has different orders of magnitudes but the same signs ( @xmath303 in ref.@xcite ) , the left five of them have opposite signs ( @xmath314 , @xmath315 , @xmath316 , @xmath317 and @xmath303 in ref.@xcite ) . + further in fig.[vectorformfactor ] , we compare the experimental data for vector form factors collected in figure 4 . and figure 5 . of ref.@xcite with our results . in obtaining our numerical predictions , we have exploited the formula given by eq.(3.16 ) in ref.@xcite which especially depends on @xmath0 order lecs through @xmath318 defined in ( [ rvdef ] ) and we input the formula @xmath1 and @xmath0 lecs obtained in table.iii of the last section . order with lecs obtained in table.iii of this paper . the red dashed line is the result by vanishing @xmath0 order lecs in corresponding red solid curve . the blue dot - dashed curve corresponds to predictions from chiral perturbation up to @xmath1 order with lecs obtained in table.i of this paper . the blue dotted line is the result by vanishing @xmath1 order lecs in corresponding blue dot - dashed curve.the black x - axis of with @xmath319 corresponds to predictions from @xmath3 order chiral perturbation.,title="fig : " ] order with lecs obtained in table.iii of this paper . the red dashed line is the result by vanishing @xmath0 order lecs in corresponding red solid curve . the blue dot - dashed curve corresponds to predictions from chiral perturbation up to @xmath1 order with lecs obtained in table.i of this paper . the blue dotted line is the result by vanishing @xmath1 order lecs in corresponding blue dot - dashed curve.the black x - axis of with @xmath319 corresponds to predictions from @xmath3 order chiral perturbation.,title="fig : " ] from fig.[vectorformfactor ] , we see that @xmath0 order lecs explicitly improve the @xmath1 and @xmath3 order chiral perturbation predictions and making them being more consistent with experimental data . in ref.@xcite , discussion of the photon - photon collision @xmath320 introduces @xmath321 and @xmath322 relate to @xmath0 order lecs by @xmath323 while in ref.@xcite , calculation of the photon - photon collision @xmath324 introduces another type of @xmath321 and @xmath322 relate to @xmath0 order lecs by @xmath325 in table.ix , we list the numerical results for above combinations of lecs given by our calculations based on table.iii in last section and by ref.@xcite and @xcite . @xmath326 for which we see that among six parameters between our predictions and values given in the literature , except two have opposite signs , other four all have the same orders of magnitudes and signs . in ref.@xcite , through reanalysis of the radiative pion decay , a group of @xmath0 order lecs are fixed . @xmath327 we find that all lecs and combination of lecs from our predictions have the same signs and orders of magnitudes as those from experiment values . except above phenomenological estimations on the values of some lecs , there are model calculations for some others of them and most of these analysis use a ( single ) resonance approximation . in contrast , our calculations do not rely on the assumption of existence of resonances . in this subsection , we list down these calculation values we can collect from the literature and compare with our results . ref.@xcite estimates values of some lecs . @xmath328 for @xmath329 and @xmath330 , ref.@xcite gives the value for their combination @xmath331 which , compares to our result of @xmath332 , is at the same order of magnitude and has the same sign . for @xmath333 , there are several works to estimate its values , we list them in table.xii . @xmath334 where @xmath335 given in ref.@xcite and @xcite are in form of @xmath336 in unit of gev@xmath337 , we have transformed them into our expression of @xmath333 with @xmath338 . further , ref.@xcite exploits resonance lagrangian estimates values of lecs @xmath339 , @xmath340 , @xmath336 , @xmath341 , @xmath342 , @xmath343 . @xmath344 we find that our results are consistent with those obtained from resonance lagrangian . ref.@xcite estimates the value of @xmath345 and gives @xmath346 which is also consistent with our result of @xmath347 . in terms of resonance exchange , ref.@xcite proposes some relations among different @xmath0 order lecs , @xmath348 to check the validity of these relations for our results , in table . xiv , we write corresponding values obtained in our calculations @xmath349 we see that except @xmath350 , all the other lecs satisfy the relations . in this paper , we revise our original formulation of calculating lecs from the first principle of qcd to a chiral covariant one suitable to computerize . with the help of computer , we successfully obtain the analytical expressions for all the @xmath0 order lecs in the normal part of chiral lagrangian for pseudo scalar mesons on the quark self energy @xmath351 . the ambiguities for anomaly part contributions to the normal part of the chiral lagrangian are clarified and we prove that this part totally should vanish and therefore need not to be considered in our computations . since our calculation is done under large @xmath4 limit , only operators of @xmath0 order with one trace and some multi - traces from the equation of motion survive in our formulation . we set up relations among the coefficients in front of these operators and lecs defined in ref.@xcite . then with input of @xmath35287mev to fix the @xmath70 in the running coupling constant of @xmath353 appear in the kernel of sde and choose cutoff of the theory being @xmath265mev and @xmath254 , we calculate all @xmath0 order lecs numerically both for two flavor and three flavor cases . compare our result lecs with those combinations which we can find experimental or model calculation values in the literature , we find that except few of them have wrong signs , most of our predicted combinations of @xmath0 order lecs have the same signs and orders of magnitudes with experiment or model calculation values . this sets the solid basis for our @xmath0 order computations . for those combinations with wrong signs or wrong order of magnitudes with experiment values , we need further investigations . based on these obtained @xmath0 order lecs , we expect a very large number of predictions for various pseudo scalar meson physics in the near future . this work was supported by national science foundation of china ( nsfc ) under grant no.10875065 . we thank prof . y.p.kuang for the helpful discussions . s.weinberg , physica , * 96a * , 327 ( 1979 ) . j.gasser and h.leutwyler , ann . phys . * 158 * , 142 ( 1984 ) ; nucl . * b 250 * , 465 ( 1985 ) . j.bijnens , prog . part . . phys . * 58 * , 521(2007 ) . h. yang , 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l . ho and t. h. wu , nucl b684 * , 281(2004 ) . i. rosell , a. pich and j. j. sanz - cillero , jhep*01 * , 039(2007 ) . v. cirigliano , g. ecker , m. eidem@xmath356ller , r. kaiser , a. pich and j. portol@xmath357s , nucl . b753 * , 139(2006 ) . in this appendix , we first list down the @xmath359 , @xmath1 , @xmath123 and @xmath0 order low energy expansion result for @xmath358 used in ( [ ebexp0 ] ) . @xmath360(b_1 ) + \frac{d}{d t}e^{b(t)}\bigg|_{t=0}\bigg\ { 2\frac{d f}{d t}[ad(-b(t))]\bigg|_{t=0}(b_1 ) + 2f[ad(-b_0)](b_2)\bigg\}\notag\\ & & + e^{b_0}\bigg\{\frac{d^2 f}{d t^2}[ad(-b(t))]\bigg|_{t=0}(b_1 ) + 2\frac{d f}{d t}[ad(-b(t))]\bigg|_{t=0}(b_2)+f[ad(-b_0)](b_3)\bigg\}\;,\label{d3ebt } \end{aligned}\ ] ] @xmath361(b_1 ) + \frac{d^2}{d t^2}e^{b(t)}\bigg|_{t=0 } \bigg\{3\frac{d f}{dt}[ad(-b(t))]\bigg|_{t=0}(b_1 ) + 3f[ad(-a_0)](b_2)\bigg\}\nonumber\\ & & + \frac{d}{d t}e^{b(t)}\bigg|_{t=0}\bigg\ { 3\frac{d^2 f}{d t^2}[ad(-b(t))]\bigg|_{t=0}(b_1 ) + 6\frac{d f}{d t}[ad(-b(t))]\bigg|_{t=0}(b_2 ) + 3f[ad(-b_0)](b_3)\bigg\}\nonumber\\ & & + e^{b_0}\bigg\{\frac{d^3 f}{d t^3}[ad(-b(t))]\bigg|_{t=0}(b_1 ) + 3\frac{d^2 f}{d t^2}[ad(-b(t))]\bigg|_{t=0}(b_2)\nonumber\\ & & + 3\frac{d f}{d t}[ad(-b(t))]\bigg|_{t=0}(b_3 ) + f[ad(-b_0)](b_4)\bigg\}\;,\label{d4ebt } \end{aligned}\ ] ] @xmath362(b_1 ) + \frac{d^3}{d t^3}e^{b(t)}\bigg|_{t=0 } \bigg\{4\frac{d f}{dt}[ad(-b(t))]\bigg|_{t=0}(b_1 ) + 4f[ad(-a_0)](b_2)\bigg\}\notag\\ & & + \frac{d^2}{d t^2}e^{b(t)}\bigg|_{t=0 } \bigg\{6\frac{d^2 f}{d t^2}[ad(-b(t))]\bigg|_{t=0}(b_1 ) + 12\frac{d f}{d t}[ad(-b(t))]\bigg|_{t=0}(b_2 ) + 6f[ad(-b_0)](b_3)\bigg\}\nonumber\\ & & + \frac{d}{d t}e^{b(t)}\bigg|_{t=0 } \bigg\{4\frac{d^3 f}{d t^3}[ad(-b(t))]\bigg|_{t=0}(b_1 ) + 12\frac{d^2 f}{d t^2}[ad(-b(t))]\bigg|_{t=0}(b_2)+12\frac{d f}{d t}[ad(-b(t))]\bigg|_{t=0}(b_3)\nonumber\\ & & + 4f[ad(-b_0)](b_4)\bigg\}+e^{b_0}\bigg\{\frac{d^4 f}{d t^4}[ad(-b(t))]\bigg|_{t=0}(b_1 ) + 4\frac{d^3 f}{d t^3}[ad(-b(t))]\bigg|_{t=0}(b_2)\notag\\ & & + 6\frac{d^2 f}{d t^2}[ad(-b(t))]\bigg|_{t=0}(b_3 ) + 4\frac{d f}{d t}[ad(-b(t))]\bigg|_{t=0}(b_4 ) + f[ad(-b_0)](b_5)\bigg\}\;,\label{d5ebt}\end{aligned}\ ] ] @xmath363(b_1 ) + \frac{d^4}{d t^4}e^{b(t)}\bigg|_{t=0 } \bigg\{5\frac{d f}{dt}[ad(-b(t))]\bigg|_{t=0}(b_1 ) + 5f[ad(-b_0)](b_2)\bigg\}\nonumber\\ & & + \frac{d^3}{d t^3}e^{b(t)}\bigg|_{t=0 } \bigg\{10\frac{d^2 f}{d t^2}[ad(-b(t))]\bigg|_{t=0}(b_1 ) + 20\frac{d f}{d t}[ad(-b(t))]\bigg|_{t=0}(b_2 ) + 10f[ad(-b_0)](b_3)\bigg\}\nonumber\\ & & + \frac{d^2}{d t^2}e^{b(t)}\bigg|_{t=0 } \bigg\{10\frac{d^3 f}{d t^3}[ad(-b(t))]\bigg|_{t=0}(b_1 ) + 30\frac{d^2 f}{d t^2}[ad(-b(t))]\bigg|_{t=0}(b_2)\nonumber\\ & & + 30\frac{d f}{d t}[ad(-b(t))]\bigg|_{t=0}(b_3 ) + 10f[ad(-b_0)](b_4)\bigg\}+\frac{d}{d t}e^{b(t)}\bigg|_{t=0 } \bigg\{5\frac{d^4 f}{d t^4}[ad(-b(t))]\bigg|_{t=0}(b_1)\notag\\ & & + 20\frac{d^3 f}{d t^3}[ad(-b(t))]\bigg|_{t=0}(b_2 ) + 30\frac{d^2 f}{d t^2}[ad(-b(t))]\bigg|_{t=0}(b_3 ) + 20\frac{d f}{d t}[ad(-b(t))]\bigg|_{t=0}(b_4 ) \notag\\ & & + 5f[ad(-b_0)](b_5)\bigg\}+e^{b_0}\bigg\{\frac{d^5 f}{d t^5}[ad(-b(t))]\bigg|_{t=0}(b_1 ) + 5\frac{d^4 f}{d t^4}[ad(-b(t))]\bigg|_{t=0}(b_2)\notag\\ & & + 10\frac{d^3 f}{d t^3}[ad(-b(t))]\bigg|_{t=0}(b_3 ) + 10\frac{d^2 f}{d t^2}[ad(-b(t))]\bigg|_{t=0}(b_4)+5\frac{d f}{d t}[ad(-b(t))]\bigg|_{t=0}(b_5)\notag\\ & & + f[ad(-b_0)](b_6)\bigg\}\;.\label{d6ebt}\end{aligned}\ ] ] then , we list down the @xmath359 , @xmath1 , @xmath123 and @xmath0 order low energy expansion result for @xmath120 used in ( [ sigmaexp ] ) . note traceless terms in @xmath123 and @xmath0 orders are dropped out . @xmath364\bigg]_{t=0 } & = & \frac{1}{6}e^{ad(a_0\pps)}\bigg[e^{-a_0\pps}\frac{d^2}{d t^2}e^{a(t)\pps}\bigg|_{t=0}f[ad(-a_0\pps)](a_1\pps)\nonumber\\ & & + e^{-a_0\pps}\frac{d}{d t}e^{a(t)\pps}\bigg|_{t=0}\bigg\ { 2\frac{d f}{d t}[ad(-a(t)\pps)]\bigg|_{t=0}(a_1\pps ) + 2f[ad(-a_0\pps)](a_2\pps)\bigg\}\notag\\ & & + \bigg\{\frac{d^2 f}{d t^2}[ad(-a(t)\pps)]\bigg|_{t=0}(a_1\pps ) + 2\frac{d f}{d t}[ad(-a(t)\pps)]\bigg|_{t=0}(a_2\pps ) \nonumber\\ & & + f[ad(-a_0\pps)](a_3\pps)\bigg\}\bigg ] \sigma(s+a(t))\bigg|_{s=0}\nonumber\\ & = & -\frac{i}{3 } ( \mu \underline{\mu } \nu ) k_{\nu}\skpp + \frac{2i}{3 } ( \mu \nu \lambda ) k_{\mu}k_{\nu}\skpp \frac{\partial}{\partial k^{\lambda } } + \frac{2i}{3 } ( \mu \nu \lambda ) k_{\nu}\skp \frac{\partial^2}{\partial k^{\mu}\partial k^{\lambda } } + \frac{i}{3 } ( \mu \underline{\mu } \nu ) \skp \frac{\partial}{\partial k^{\nu}}\;,\end{aligned}\ ] ] @xmath365\bigg]_{t=0 } & = & e^{ad(a_0\pps)}\bigg[e^{-a_0\pps}\frac{d^3}{d t^3}e^{a(t)\pps}\bigg|_{t=0}~f[ad(-a_0\pps)](a_1\pps ) \nonumber\\ & & + e^{-a_0\pps}\frac{d^2}{d t^2}e^{a(t)\pps}\bigg|_{t=0 } \bigg\{3\frac{d f}{dt}[ad(-a(t)\pps)]\bigg|_{t=0}(a_1\pps ) + 3f[ad(-a_0\pps)](a_2\pps)\bigg\}\nonumber\\ & & + e^{-a_0\pps}\frac{d}{d t}e^{a(t)\pps}\bigg|_{t=0}\bigg\ { 3\frac{d^2 f}{d t^2}[ad(-a(t)\pps)]\bigg|_{t=0}(a_1\pps ) + 6\frac{d f}{d t}[ad(-a(t)\pps)]\bigg|_{t=0}(a_2\pps ) \nonumber\\ & & + 3f[ad(-a_0\pps)](a_3\pps)\bigg\ } + \bigg\{\frac{d^3 f}{d t^3}[ad(-a(t)\pps)]\bigg|_{t=0}(a_1\pps ) + 3\frac{d^2 f}{d t^2}[ad(-a(t)\pps)]\bigg|_{t=0}(a_2\pps)\nonumber\\ & & + 3\frac{d f}{d t}[ad(-a(t)\pps)]\bigg|_{t=0}(a_3\pps ) + f[ad(-a_0\pps)](a_4\pps)\bigg\}\bigg]_{t=0}\sigma(s+a(t))\bigg|_{s=0}\nonumber\\ & = & -\frac{1}{4 } ( \mu \nu ) ( \underline{\mu } \lambda ) k_{\nu}\skpp \frac{\partial}{\partial k^{\lambda } } + \frac{1}{2 } ( \mu \nu ) ( \lambda \rho ) k_{\mu}k_{\lambda}\skpp \frac{\partial^2}{\partial k^{\nu}\partial k^{\rho } } -\frac{1}{8 } ( \mu \nu \underline{\nu } \lambda ) k_{\lambda}\skpp \frac{\partial}{\partial k^{\mu } } \notag\\ & & -\frac{1}{8 } ( \mu \nu \underline{\mu } \lambda ) k_{\lambda}\skpp \frac{\partial}{\partial k^{\nu } } + \frac{1}{4 } ( \mu \underline{\mu } \nu \lambda ) k_{\nu}\skpp \frac{\partial}{\partial k^{\lambda } } + \frac{1}{4 } ( \mu \nu \lambda \rho ) k_{\nu}k_{\lambda}\skpp \frac{\partial^2}{\partial k^{\mu}\partial k^{\rho } } \notag\\ & & + \frac{1}{4 } ( \mu \nu \lambda \rho ) k_{\mu}k_{\lambda}\skpp \frac{\partial^2}{\partial k^{\nu}\partial k^{\rho } } + \frac{1}{4 } ( \mu \nu \lambda \rho ) k_{\lambda}\skp \frac{\partial^3}{\partial k^{\mu}\partial k^{\nu}\partial k^{\rho } } + \frac{1}{4 } ( \mu \nu ) ( \underline{\mu } \underline{\nu } ) \skpp \notag\\ & & + \frac{1}{4 } ( \mu \nu ) ( \underline{\mu } \lambda ) k_{\lambda}\skpp \frac{\partial}{\partial k^{\nu } } + \frac{1}{4 } ( \mu \nu ) ( \underline{\mu } \lambda ) \skp \frac{\partial^2}{\partial k^{\nu}\partial k^{\lambda } } + \frac{1}{8 } ( \mu \nu \underline{\nu } \lambda ) k_{\mu}\skpp \frac{\partial}{\partial k^{\lambda } } \notag\\ & & + \frac{1}{8 } ( \mu \nu \underline{\nu } \lambda ) \skp \frac{\partial^2}{\partial k^{\mu}\partial k^{\lambda } } + \frac{1}{8 } ( \mu \nu \underline{\mu } \lambda ) k_{\nu}\skpp \frac{\partial}{\partial k^{\lambda } } + \frac{1}{8 } ( \mu \nu \underline{\mu } \lambda ) \skp \frac{\partial^2}{\partial k^{\nu}\partial k^{\lambda } } \notag\\ & & -\frac{1}{6 } ( \mu \nu \underline{\nu } \lambda ) k_{\mu}k_{\lambda}\skppp -\frac{1}{6 } ( \mu \nu \underline{\mu } \lambda ) k_{\nu}k_{\lambda}\skppp + \frac{1}{3 } ( \mu \nu \lambda \rho ) k_{\mu}k_{\nu}k_{\lambda}\skppp \frac{\partial}{\partial k^{\rho } } + \frac{1}{3 } ( \mu \nu ) ( \underline{\mu } \lambda ) k_{\nu}k_{\lambda}\skppp\ ; , \end{aligned}\ ] ] @xmath366\bigg]_{t=0}\nonumber\\ & & \hspace{-0.4cm}=\frac{1}{5!}e^{ad(a_0\pps)}\bigg [ e^{-a_0\pps}\frac{d^4}{d t^4}e^{a(t)\pps}\bigg|_{t=0}~f[ad(-a_0\pps)](a_1\pps ) + e^{-a_0\pps}\frac{d^3}{d t^3}e^{a(t)\pps}\bigg|_{t=0 } \bigg\{4\frac{d f}{dt}[ad(-a(t)\pps)]\bigg|_{t=0}(a_1\pps)\nonumber\\ & & + 4f[ad(-a_0\pps)](a_2\pps)\bigg\}+e^{-a_0\pps}\frac{d^2}{d t^2}e^{a(t)\pps}\bigg|_{t=0 } \bigg\{6\frac{d^2 f}{d t^2}[ad(-a(t)\pps)]\bigg|_{t=0}(a_1\pps ) \nonumber\\ & & + 12\frac{d f}{d t}[ad(-a(t)\pps)]\bigg|_{t=0}(a_2\pps ) + 6f[ad(-a_0\pps)](a_3\pps)\bigg\}+e^{-a_0\pps}\frac{d}{d t}e^{a(t)\pps}\bigg|_{t=0 } \bigg\{4\frac{d^3 f}{d t^3}[ad(-a(t)\pps)]\bigg|_{t=0}(a_1\pps)\nonumber\\ & & + 12\frac{d^2 f}{d t^2}[ad(-a(t)\pps)]\bigg|_{t=0}(a_2\pps ) + 12\frac{d f}{d t}[ad(-a(t)\pps)]\bigg|_{t=0}(a_3\pps ) + 4f[ad(-a_0\pps)](a_4\pps)\bigg\}\notag\\ & & + \bigg\{\frac{d^4 f}{d t^4}[ad(-a(t)\pps)]\bigg|_{t=0}(a_1\pps ) + 4\frac{d^3 f}{d t^3}[ad(-a(t)\pps)]\bigg|_{t=0}(a_2\pps ) + 6\frac{d^2 f}{d t^2}[ad(-a(t)\pps)]\bigg|_{t=0}(a_3\pps)\notag\\ & & + 4\frac{d f}{d t}[ad(-a(t)\pps)]\bigg|_{t=0}(a_4\pps ) + f[ad(-a_0\pps)](a_5\pps)\bigg\}\bigg]\sigma(s+a(t))\bigg|_{s=0}\nonumber\\ & & \hspace{-0.4cm}=\mbox{traceless terms}\ ; , \end{aligned}\ ] ] @xmath367\bigg]_{t=0}\nonumber\\ & & \hspace{-0.4cm}=e^{ad(a_0\pps)}\bigg [ e^{-a_0\pps}\frac{d^5}{d t^5}e^{a(t)\pps}\bigg|_{t=0}[ad(-a_0\pps)](a_1\pps ) + e^{-a_0\pps}\frac{d^4}{d t^4}e^{a(t)\pps}\bigg|_{t=0 } \bigg\{5\frac{d f}{dt}[ad(-a(t)\pps)]\bigg|_{t=0}(a_1\pps)\notag\\ & & + 5f[ad(-a_0\pps)]\bigg|_{t=0}(a_2\pps)\bigg\ } + e^{-a_0\pps}\frac{d^3}{d t^3}e^{a(t)\pps}\bigg|_{t=0 } \bigg\{10\frac{d^2 f}{d t^2}[ad(-a(t)\pps)]\bigg|_{t=0}(a_1\pps ) \nonumber\\ & & + 20\frac{d f}{d t}[ad(-a(t)\pps)]\bigg|_{t=0}(a_2\pps ) + 10f[ad(-a_0\pps)](a_3\pps)\bigg\}\nonumber\\ & & + e^{-a_0\pps}\frac{d^2}{d t^2}e^{a(t)\pps}\bigg|_{t=0 } \bigg\{10\frac{d^3 f}{d t^3}[ad(-a(t)\pps)]\bigg|_{t=0}(a_1\pps ) + 30\frac{d^2 f}{d t^2}[ad(-a(t)\pps)]\bigg|_{t=0}(a_2\pps)\nonumber\\ & & + 30\frac{d f}{d t}[ad(-a(t)\pps)]\bigg|_{t=0}(a_3\pps ) + 10f[ad(-a_0\pps)](a_4\pps)\bigg\}\notag\\ & & + e^{-a_0\pps}\frac{d}{d t}e^{a(t)\pps}\bigg|_{t=0 } \bigg\{5\frac{d^4 f}{d t^4}[ad(-a(t)\pps)]\bigg|_{t=0}(a_1\pps ) + 20\frac{d^3 f}{d t^3}[ad(-a(t)\pps)]\bigg|_{t=0}(a_2\pps)\notag\\ & & + 30\frac{d^2 f}{d t^2}[ad(-a(t)\pps)]\bigg|_{t=0}(a_3\pps ) + 20\frac{d f}{d t}[ad(-a(t)\pps)]\bigg|_{t=0}(a_4\pps)+5f[ad(-a_0\pps)](a_5\pps)\bigg\}\notag\\ & & + \bigg\{\frac{d^5 f}{d t^5}[ad(-a(t)\pps)]\bigg|_{t=0}(a_1\pps ) + 5\frac{d^4 f}{d t^4}[ad(-a(t)\pps)]\bigg|_{t=0}(a_2\pps ) + 10\frac{d^3 f}{d t^3}[ad(-a(t)\pps)]\bigg|_{t=0}(a_3\pps)\notag\\ & & + 10\frac{d^2 f}{d t^2}[ad(-a(t)\pps)]\bigg|_{t=0}(a_4\pps ) + 5\frac{d f}{d t}[ad(-a(t)\pps)]\bigg|_{t=0}(a_5\pps ) + f[ad(-a_0\pps)](a_6\pps)\bigg\}\bigg]\sigma(s+a_0)\bigg|_{s=0}\nonumber\\ & & \hspace{-0.4cm}=\frac{1}{9 } ( \mu \underline{\mu } \nu ) ( \lambda \underline{\nu } \underline{\lambda } ) \skppp -\frac{1}{6 } ( \mu \underline{\mu } \nu \lambda ) ( \underline{\nu } \underline{\lambda } ) \skppp -\frac{1}{6 } ( \mu \nu \underline{\mu } \lambda ) ( \underline{\nu } \underline{\lambda } ) \skppp -\frac{1}{6 } ( \mu \nu \underline{\nu } \lambda ) ( \underline{\mu } \underline{\lambda } ) \skppp -\frac{4}{27 } ( \mu \nu \lambda ) ( \underline{\mu } \underline{\nu } \underline{\lambda } ) \skppp -\frac{4}{27 } ( \mu \nu \lambda ) ( \underline{\nu } \underline{\mu } \underline{\lambda } ) \skppp \notag\\ & & -\frac{1}{24 } ( \mu \nu \underline{\nu } \lambda ) ( \underline{\mu } \rho ) k_{\lambda}k_{\rho}\skpppp -\frac{1}{24 } ( \mu \nu \underline{\mu } \lambda ) ( \underline{\nu } \rho ) k_{\lambda}k_{\rho}\skpppp -\frac{1}{18 } ( \mu \underline{\mu } \nu ) ( \lambda \underline{\lambda } \rho ) k_{\nu}k_{\rho}\skpppp -\frac{1}{4 } ( \mu \nu \lambda \rho ) ( \underline{\lambda } \underline{\rho } ) k_{\mu}k_{\nu}\skpppp \notag\\ & & + \frac{1}{4 } ( \mu \nu \lambda \rho ) ( \underline{\nu } \underline{\lambda } ) k_{\mu}k_{\rho}\skpppp + \frac{1}{4 } ( \mu \nu \lambda \rho ) ( \underline{\mu } \underline{\lambda } ) k_{\nu}k_{\rho}\skpppp -\frac{1}{4 } ( \mu \underline{\mu } \nu \lambda ) ( \underline{\nu } \rho ) k_{\lambda}k_{\rho}\skpppp -\frac{7}{54 } ( \mu \nu \lambda ) ( \rho \underline{\nu } \underline{\rho } ) k_{\mu}k_{\lambda}\skpppp \notag\\ & & -\frac{2}{9 } ( \mu \nu \lambda ) ( \rho \underline{\nu } \underline{\lambda } ) k_{\mu}k_{\rho}\skpppp + \frac{2}{9 } ( \mu \nu \lambda ) ( \rho \underline{\mu } \underline{\nu } ) k_{\lambda}k_{\rho}\skpppp + \frac{2}{9 } ( \mu \nu \lambda ) ( \underline{\nu } \underline{\lambda } \rho ) k_{\mu}k_{\rho}\skpppp -\frac{2}{9 } ( \mu \nu \lambda ) ( \underline{\mu } \underline{\nu } \rho ) k_{\lambda}k_{\rho}\skpppp \notag\\ & & + \frac{11}{54 } ( \mu \underline{\mu } \nu ) ( \lambda \underline{\nu } \rho ) k_{\lambda}k_{\rho}\skpppp + \frac{5}{24 } ( \mu \nu \underline{\nu } \lambda ) ( \underline{\lambda } \rho ) k_{\mu}k_{\rho}\skpppp + \frac{5}{24 } ( \mu \nu \underline{\mu } \lambda ) ( \underline{\lambda } \rho ) k_{\nu}k_{\rho}\skpppp -\frac{1}{12 } ( \mu \nu ) ( \underline{\mu } \lambda ) ( \underline{\lambda } \rho ) k_{\nu}k_{\rho}\skpppp \notag\\ & & -\frac{1}{12 } ( \mu \nu ) ( \lambda \rho ) ( \underline{\mu } \underline{\lambda } ) k_{\nu}k_{\rho}\skpppp + \frac{1}{8 } ( \mu \nu \underline{\nu } \lambda ) ( \underline{\lambda } \rho ) k_{\rho}\skppp \frac{\partial}{\partial k^{\mu } } + \frac{1}{12 } ( \mu \nu \underline{\nu } \lambda ) ( \underline{\lambda } \rho ) k_{\mu}\skppp \frac{\partial}{\partial k^{\rho } } + \frac{1}{24 } ( \mu \nu \underline{\nu } \lambda ) ( \underline{\mu } \rho ) k_{\rho}\skppp \frac{\partial}{\partial k^{\lambda } } \notag\\ & & + \frac{1}{12 } ( \mu \nu \underline{\nu } \lambda ) ( \underline{\mu } \rho ) k_{\lambda}\skppp \frac{\partial}{\partial k^{\rho } } + \frac{1}{8 } ( \mu \nu \underline{\mu } \lambda ) ( \underline{\lambda } \rho ) k_{\rho}\skppp \frac{\partial}{\partial k^{\nu } } + \frac{1}{12 } ( \mu \nu \underline{\mu } \lambda ) ( \underline{\lambda } \rho ) k_{\nu}\skppp \frac{\partial}{\partial k^{\rho } } + \frac{1}{24 } ( \mu \nu \underline{\mu } \lambda ) ( \underline{\nu } \rho ) k_{\rho}\skppp \frac{\partial}{\partial k^{\lambda } } \notag\\ & & + \frac{1}{12 } ( \mu \nu \underline{\mu } \lambda ) ( \underline{\nu } \rho ) k_{\lambda}\skppp \frac{\partial}{\partial k^{\rho } } + \frac{1}{18 } ( \mu \underline{\mu } \nu ) ( \lambda \underline{\lambda } \rho ) k_{\rho}\skppp \frac{\partial}{\partial k^{\nu } } + \frac{1}{18 } ( \mu \underline{\mu } \nu ) ( \lambda \underline{\lambda } \rho ) k_{\nu}\skppp \frac{\partial}{\partial k^{\rho } } -\frac{1}{6 } ( \mu \nu \lambda \rho ) ( \underline{\lambda } \underline{\rho } ) k_{\nu}\skppp \frac{\partial}{\partial k^{\mu } } \notag\\ & & -\frac{1}{6 } ( \mu \nu \lambda \rho ) ( \underline{\lambda } \underline{\rho } ) k_{\mu}\skppp \frac{\partial}{\partial k^{\nu } } + \frac{1}{6 } ( \mu \nu \lambda \rho ) ( \underline{\nu } \underline{\lambda } ) k_{\rho}\skppp \frac{\partial}{\partial k^{\mu } } + \frac{1}{6 } ( \mu \nu \lambda \rho ) ( \underline{\nu } \underline{\lambda } ) k_{\mu}\skppp \frac{\partial}{\partial k^{\rho } } + \frac{1}{6 } ( \mu \nu \lambda \rho ) ( \underline{\mu } \underline{\lambda } ) k_{\rho}\skppp \frac{\partial}{\partial k^{\nu } } \notag\\ & & + \frac{1}{6 } ( \mu \nu \lambda \rho ) ( \underline{\mu } \underline{\lambda } ) k_{\nu}\skppp \frac{\partial}{\partial k^{\rho } } -\frac{1}{12 } ( \mu \underline{\mu } \nu \lambda ) ( \underline{\nu } \rho ) k_{\rho}\skppp \frac{\partial}{\partial k^{\lambda } } -\frac{2}{27 } ( \mu \nu \lambda ) ( \rho \underline{\nu } \underline{\rho } ) k_{\lambda}\skppp \frac{\partial}{\partial k^{\mu } } -\frac{4}{27 } ( \mu \nu \lambda ) ( \rho \underline{\nu } \underline{\rho } ) k_{\mu}\skppp \frac{\partial}{\partial k^{\lambda } } \notag\\ & & -\frac{4}{27 } ( \mu \nu \lambda ) ( \rho \underline{\nu } \underline{\lambda } ) k_{\rho}\skppp \frac{\partial}{\partial k^{\mu } } -\frac{4}{27 } ( \mu \nu \lambda ) ( \rho \underline{\nu } \underline{\lambda } ) k_{\mu}\skppp \frac{\partial}{\partial k^{\rho } } + \frac{4}{27 } ( \mu \nu \lambda ) ( \rho \underline{\mu } \underline{\nu } ) k_{\rho}\skppp \frac{\partial}{\partial k^{\lambda } } + \frac{4}{27 } ( \mu \nu \lambda ) ( \rho \underline{\mu } \underline{\nu } ) k_{\lambda}\skppp \frac{\partial}{\partial k^{\rho } } \notag\\ & & + \frac{4}{27 } ( \mu \nu \lambda ) ( \underline{\nu } \underline{\lambda } \rho ) k_{\rho}\skppp \frac{\partial}{\partial k^{\mu } } + \frac{4}{27 } ( \mu \nu \lambda ) ( \underline{\nu } \underline{\lambda } \rho ) k_{\mu}\skppp \frac{\partial}{\partial k^{\rho } } -\frac{4}{27 } ( \mu \nu \lambda ) ( \underline{\mu } \underline{\nu } \rho ) k_{\rho}\skppp \frac{\partial}{\partial k^{\lambda } } + \frac{4}{27 } ( \mu \underline{\mu } \nu ) ( \lambda \underline{\nu } \rho ) k_{\rho}\skppp \frac{\partial}{\partial k^{\lambda } } \notag\\ & & + \frac{2}{27 } ( \mu \underline{\mu } \nu ) ( \lambda \underline{\nu } \rho ) k_{\lambda}\skppp \frac{\partial}{\partial k^{\rho } } -\frac{2}{27 } ( \mu \nu \lambda ) ( \rho \underline{\mu } \underline{\rho } ) k_{\nu}\skppp \frac{\partial}{\partial k^{\lambda } } + \frac{4}{27 } ( \mu \nu \lambda ) ( \underline{\nu } \underline{\mu } \rho ) k_{\lambda}\skppp \frac{\partial}{\partial k^{\rho } } -\frac{2}{27 } ( \mu \underline{\mu } \nu ) ( \underline{\nu } \lambda \rho ) k_{\lambda}\skppp \frac{\partial}{\partial k^{\rho } } \notag\\ & & -\frac{1}{6 } ( \mu \nu ) ( \underline{\mu } \underline{\nu } ) ( \lambda \rho ) k_{\lambda}\skppp \frac{\partial}{\partial k^{\rho } } -\frac{1}{12 } ( \mu \nu ) ( \lambda \rho ) ( \underline{\mu } \underline{\lambda } ) k_{\nu}\skppp \frac{\partial}{\partial k^{\rho } } -\frac{1}{12 } ( \mu \nu ) ( \lambda \rho ) ( \underline{\lambda } \underline{\rho } ) k_{\mu}\skppp \frac{\partial}{\partial k^{\nu } } -\frac{1}{12 } ( \mu \nu ) ( \underline{\mu } \lambda ) ( \underline{\lambda } \rho ) k_{\nu}\skppp \frac{\partial}{\partial k^{\rho } } \notag\\ & & + \frac{1}{16 } ( \mu \nu \underline{\nu } \lambda ) ( \underline{\lambda } \rho ) \skpp \frac{\partial^2}{\partial k^{\mu}\partial k^{\rho } } -\frac{1}{16 } ( \mu \nu \underline{\nu } \lambda ) ( \underline{\mu } \rho ) \skpp \frac{\partial^2}{\partial k^{\lambda}\partial k^{\rho } } + \frac{1}{16 } ( \mu \nu \underline{\mu } \lambda ) ( \underline{\lambda } \rho ) \skpp \frac{\partial^2}{\partial k^{\nu}\partial k^{\rho } } -\frac{1}{16 } ( \mu \nu \underline{\mu } \lambda ) ( \underline{\nu } \rho ) \skpp \frac{\partial^2}{\partial k^{\lambda}\partial k^{\rho } } \notag\\ & & -\frac{1}{18 } ( \mu \underline{\mu } \nu ) ( \lambda \underline{\lambda } \rho ) \skpp \frac{\partial^2}{\partial k^{\nu}\partial k^{\rho } } -\frac{1}{8 } ( \mu \nu \lambda \rho ) ( \underline{\lambda } \underline{\rho } ) \skpp \frac{\partial^2}{\partial k^{\mu}\partial k^{\nu } } + \frac{1}{8 } ( \mu \nu \lambda \rho ) ( \underline{\nu } \underline{\lambda } ) \skpp \frac{\partial^2}{\partial k^{\mu}\partial k^{\rho } } + \frac{1}{8 } ( \mu \nu \lambda \rho ) ( \underline{\mu } \underline{\lambda } ) \skpp \frac{\partial^2}{\partial k^{\nu}\partial k^{\rho } } \notag\\ & & -\frac{1}{8 } ( \mu \underline{\mu } \nu \lambda ) ( \underline{\nu } \rho ) \skpp \frac{\partial^2}{\partial k^{\lambda}\partial k^{\rho } } -\frac{1}{9 } ( \mu \nu \lambda ) ( \rho \underline{\nu } \underline{\rho } ) \skpp \frac{\partial^2}{\partial k^{\mu}\partial k^{\lambda } } -\frac{1}{9 } ( \mu \nu \lambda ) ( \rho \underline{\nu } \underline{\lambda } ) \skpp \frac{\partial^2}{\partial k^{\mu}\partial k^{\rho } } + \frac{1}{9 } ( \mu \nu \lambda ) ( \rho \underline{\mu } \underline{\nu } ) \skpp \frac{\partial^2}{\partial k^{\lambda}\partial k^{\rho } } \notag\\ & & + \frac{1}{9 } ( \mu \nu \lambda ) ( \underline{\nu } \underline{\lambda } \rho ) \skpp \frac{\partial^2}{\partial k^{\mu}\partial k^{\rho } } -\frac{1}{9 } ( \mu \nu \lambda ) ( \underline{\mu } \underline{\nu } \rho ) \skpp \frac{\partial^2}{\partial k^{\lambda}\partial k^{\rho } } -\frac{1}{6 } ( \mu \nu ) ( \underline{\mu } \lambda ) ( \underline{\nu } \rho ) k_{\lambda}\skppp \frac{\partial}{\partial k^{\rho } } -\frac{1}{8 } ( \mu \nu ) ( \underline{\mu } \lambda ) ( \underline{\lambda } \rho ) \skpp \frac{\partial^2}{\partial k^{\nu}\partial k^{\rho } } \notag\\ & & -\frac{1}{8 } ( \mu \nu ) ( \underline{\mu } \lambda ) ( \underline{\nu } \rho ) \skpp \frac{\partial^2}{\partial k^{\lambda}\partial k^{\rho } } -\frac{16}{45 } ( \mu \nu \lambda ) ( \rho \underline{\nu } \sigma ) k_{\mu}k_{\lambda}k_{\rho}k_{\sigma}\skppppp -\frac{2}{5 } ( \mu \nu \lambda \rho ) ( \underline{\lambda } \sigma ) k_{\mu}k_{\nu}k_{\rho}k_{\sigma}\skppppp\notag\\ & & -\frac{1}{4 } ( \mu \nu \lambda \rho ) ( \underline{\lambda } \sigma ) k_{\nu}k_{\rho}k_{\sigma}\skpppp \frac{\partial}{\partial k^{\mu } } -\frac{1}{4 } ( \mu \nu \lambda \rho ) ( \underline{\lambda } \sigma ) k_{\mu}k_{\rho}k_{\sigma}\skpppp \frac{\partial}{\partial k^{\nu } } -\frac{2}{9 } ( \mu \nu \lambda ) ( \rho \underline{\nu } \sigma ) k_{\lambda}k_{\rho}k_{\sigma}\skpppp \frac{\partial}{\partial k^{\mu } } \notag\\ & & -\frac{2}{9 } ( \mu \nu \lambda ) ( \rho \underline{\nu } \sigma ) k_{\mu}k_{\lambda}k_{\sigma}\skpppp \frac{\partial}{\partial k^{\rho } } -\frac{1}{27 } ( \mu \nu \lambda ) ( \rho \underline{\nu } \sigma ) k_{\mu}k_{\lambda}k_{\rho}\skpppp \frac{\partial}{\partial k^{\sigma } } + \frac{1}{12 } ( \mu \nu \lambda \rho ) ( \underline{\nu } \sigma ) k_{\mu}k_{\lambda}k_{\sigma}\skpppp \frac{\partial}{\partial k^{\rho } } \notag\\ & & + \frac{1}{12 } ( \mu \nu \lambda \rho ) ( \underline{\mu } \sigma ) k_{\nu}k_{\lambda}k_{\sigma}\skpppp \frac{\partial}{\partial k^{\rho } } + \frac{1}{6 } ( \mu \nu \underline{\nu } \lambda ) ( \rho \sigma ) k_{\mu}k_{\lambda}k_{\rho}\skpppp \frac{\partial}{\partial k^{\sigma } } + \frac{1}{6 } ( \mu \nu \underline{\mu } \lambda ) ( \rho \sigma ) k_{\nu}k_{\lambda}k_{\rho}\skpppp \frac{\partial}{\partial k^{\sigma } } \notag\\ & & + \frac{1}{27 } ( \mu \nu \lambda ) ( \rho \underline{\mu } \sigma ) k_{\nu}k_{\rho}k_{\sigma}\skpppp \frac{\partial}{\partial k^{\lambda } } + \frac{1}{9 } ( \mu \nu \lambda ) ( \rho \underline{\rho } \sigma ) k_{\mu}k_{\nu}k_{\sigma}\skpppp \frac{\partial}{\partial k^{\lambda } } + \frac{1}{9 } ( \mu \underline{\mu } \nu ) ( \lambda \rho \sigma ) k_{\nu}k_{\lambda}k_{\rho}\skpppp \frac{\partial}{\partial k^{\sigma } } \notag\\ & & -\frac{1}{4 } ( \mu \nu ) ( \underline{\mu } \lambda ) ( \rho \sigma ) k_{\nu}k_{\lambda}k_{\rho}\skpppp \frac{\partial}{\partial k^{\sigma } } -\frac{1}{12 } ( \mu \nu ) ( \lambda \rho ) ( \underline{\lambda } \sigma ) k_{\mu}k_{\rho}k_{\sigma}\skpppp \frac{\partial}{\partial k^{\nu } } -\frac{5}{27 } ( \mu \nu \lambda ) ( \rho \underline{\nu } \sigma ) k_{\mu}k_{\rho}k_{\sigma}\skpppp \frac{\partial}{\partial k^{\lambda } } \notag\\ & & + \frac{5}{27 } ( \mu \nu \lambda ) ( \underline{\nu } \rho \sigma ) k_{\mu}k_{\lambda}k_{\rho}\skpppp \frac{\partial}{\partial k^{\sigma } } -\frac{1}{6 } ( \mu \nu \lambda \rho ) ( \underline{\lambda } \sigma ) k_{\mu}k_{\nu}k_{\sigma}\skpppp \frac{\partial}{\partial k^{\rho } } -\frac{1}{12 } ( \mu \nu \lambda \rho ) ( \underline{\lambda } \sigma ) k_{\nu}k_{\sigma}\skppp \frac{\partial^2}{\partial k^{\mu}\partial k^{\rho } } \notag\\ & & -\frac{1}{12 } ( \mu \nu \lambda \rho ) ( \underline{\lambda } \sigma ) k_{\mu}k_{\sigma}\skppp \frac{\partial^2}{\partial k^{\nu}\partial k^{\rho } } -\frac{4}{27 } ( \mu \nu \lambda ) ( \rho \underline{\nu } \sigma ) k_{\lambda}k_{\sigma}\skppp \frac{\partial^2}{\partial k^{\mu}\partial k^{\rho } } + \frac{2}{27 } ( \mu \nu \lambda ) ( \rho \underline{\nu } \sigma ) k_{\mu}k_{\lambda}\skppp \frac{\partial^2}{\partial k^{\rho}\partial k^{\sigma } } \notag\\ & & + \frac{1}{12 } ( \mu \nu \lambda \rho ) ( \underline{\nu } \sigma ) k_{\lambda}k_{\sigma}\skppp \frac{\partial^2}{\partial k^{\mu}\partial k^{\rho } } -\frac{1}{6 } ( \mu \nu \lambda \rho ) ( \underline{\nu } \sigma ) k_{\mu}k_{\lambda}\skppp \frac{\partial^2}{\partial k^{\rho}\partial k^{\sigma } } + \frac{1}{12 } ( \mu \nu \lambda \rho ) ( \underline{\mu } \sigma ) k_{\lambda}k_{\sigma}\skppp \frac{\partial^2}{\partial k^{\nu}\partial k^{\rho } } \notag\\ & & -\frac{1}{6 } ( \mu \nu \lambda \rho ) ( \underline{\mu } \sigma ) k_{\nu}k_{\lambda}\skppp \frac{\partial^2}{\partial k^{\rho}\partial k^{\sigma } } + \frac{1}{8 } ( \mu \nu \underline{\nu } \lambda ) ( \rho \sigma ) k_{\lambda}k_{\rho}\skppp \frac{\partial^2}{\partial k^{\mu}\partial k^{\sigma } } -\frac{1}{8 } ( \mu \nu \underline{\nu } \lambda ) ( \rho \sigma ) k_{\mu}k_{\rho}\skppp \frac{\partial^2}{\partial k^{\lambda}\partial k^{\sigma } } \notag\\ & & + \frac{1}{8 } ( \mu \nu \underline{\mu } \lambda ) ( \rho \sigma ) k_{\lambda}k_{\rho}\skppp \frac{\partial^2}{\partial k^{\nu}\partial k^{\sigma } } -\frac{1}{8 } ( \mu \nu \underline{\mu } \lambda ) ( \rho \sigma ) k_{\nu}k_{\rho}\skppp \frac{\partial^2}{\partial k^{\lambda}\partial k^{\sigma } } -\frac{4}{27 } ( \mu \nu \lambda ) ( \rho \underline{\mu } \sigma ) k_{\nu}k_{\rho}\skppp \frac{\partial^2}{\partial k^{\lambda}\partial k^{\sigma } } \notag\\ & & + \frac{1}{9 } ( \mu \nu \lambda ) ( \rho \underline{\rho } \sigma ) k_{\nu}k_{\sigma}\skppp \frac{\partial^2}{\partial k^{\mu}\partial k^{\lambda } } -\frac{1}{9 } ( \mu \nu \lambda ) ( \rho \underline{\rho } \sigma ) k_{\mu}k_{\nu}\skppp \frac{\partial^2}{\partial k^{\lambda}\partial k^{\sigma } } -\frac{1}{9 } ( \mu \underline{\mu } \nu ) ( \lambda \rho \sigma ) k_{\lambda}k_{\rho}\skppp \frac{\partial^2}{\partial k^{\nu}\partial k^{\sigma } } \notag\\ & & + \frac{1}{9 } ( \mu \underline{\mu } \nu ) ( \lambda \rho \sigma ) k_{\nu}k_{\rho}\skppp \frac{\partial^2}{\partial k^{\lambda}\partial k^{\sigma } } -\frac{1}{6 } ( \mu \nu ) ( \underline{\mu } \lambda ) ( \rho \sigma ) k_{\lambda}k_{\rho}\skppp \frac{\partial^2}{\partial k^{\nu}\partial k^{\sigma } } -\frac{1}{12 } ( \mu \nu ) ( \lambda \rho ) ( \underline{\lambda } \sigma ) k_{\mu}k_{\sigma}\skppp \frac{\partial^2}{\partial k^{\nu}\partial k^{\rho } } \notag\\ & & + \frac{1}{12 } ( \mu \nu ) ( \lambda \rho ) ( \underline{\lambda } \sigma ) k_{\mu}k_{\rho}\skppp \frac{\partial^2}{\partial k^{\nu}\partial k^{\sigma } } -\frac{1}{4 } ( \mu \underline{\mu } \nu \lambda ) ( \rho \sigma ) k_{\nu}k_{\rho}\skppp \frac{\partial^2}{\partial k^{\lambda}\partial k^{\sigma } } + \frac{4}{27 } ( \mu \nu \lambda ) ( \underline{\nu } \rho \sigma ) k_{\lambda}k_{\rho}\skppp \frac{\partial^2}{\partial k^{\mu}\partial k^{\sigma } } \notag\\ & & -\frac{4}{27 } ( \mu \nu \lambda ) ( \underline{\mu } \rho \sigma ) k_{\nu}k_{\rho}\skppp \frac{\partial^2}{\partial k^{\lambda}\partial k^{\sigma } } -\frac{2}{27 } ( \mu \nu \lambda ) ( \rho \underline{\nu } \sigma ) k_{\rho}k_{\sigma}\skppp \frac{\partial^2}{\partial k^{\mu}\partial k^{\lambda } } -\frac{1}{6 } ( \mu \nu \lambda \rho ) ( \underline{\lambda } \sigma ) k_{\mu}k_{\nu}\skppp \frac{\partial^2}{\partial k^{\rho}\partial k^{\sigma } } \notag\\ & & -\frac{4}{27 } ( \mu \nu \lambda ) ( \rho \underline{\nu } \sigma ) k_{\mu}k_{\sigma}\skppp \frac{\partial^2}{\partial k^{\lambda}\partial k^{\rho } } -\frac{4}{27 } ( \mu \nu \lambda ) ( \rho \underline{\nu } \sigma ) k_{\mu}k_{\rho}\skppp \frac{\partial^2}{\partial k^{\lambda}\partial k^{\sigma } } -\frac{1}{6 } ( \mu \nu \lambda \rho ) ( \underline{\lambda } \sigma ) k_{\rho}k_{\sigma}\skppp \frac{\partial^2}{\partial k^{\mu}\partial k^{\nu } } \notag\\ & & -\frac{1}{8 } ( \mu \nu \lambda \rho ) ( \underline{\lambda } \sigma ) k_{\nu}\skpp \frac{\partial^3}{\partial k^{\mu}\partial k^{\rho}\partial k^{\sigma } } -\frac{1}{8 } ( \mu \nu \lambda \rho ) ( \underline{\lambda } \sigma ) k_{\mu}\skpp \frac{\partial^3}{\partial k^{\nu}\partial k^{\rho}\partial k^{\sigma } } -\frac{1}{9 } ( \mu \nu \lambda ) ( \rho \underline{\nu } \sigma ) k_{\sigma}\skpp \frac{\partial^3}{\partial k^{\mu}\partial k^{\lambda}\partial k^{\rho } } \notag\\ & & -\frac{1}{9 } ( \mu \nu \lambda ) ( \rho \underline{\nu } \sigma ) k_{\rho}\skpp \frac{\partial^3}{\partial k^{\mu}\partial k^{\lambda}\partial k^{\sigma } } + \frac{1}{9 } ( \mu \nu \lambda ) ( \rho \underline{\nu } \sigma ) k_{\lambda}\skpp \frac{\partial^3}{\partial k^{\mu}\partial k^{\rho}\partial k^{\sigma } } -\frac{1}{9 } ( \mu \nu \lambda ) ( \rho \underline{\nu } \sigma ) k_{\mu}\skpp \frac{\partial^3}{\partial k^{\lambda}\partial k^{\rho}\partial k^{\sigma } } \notag\\ & & -\frac{1}{8 } ( \mu \nu \lambda \rho ) ( \underline{\nu } \sigma ) k_{\lambda}\skpp \frac{\partial^3}{\partial k^{\mu}\partial k^{\rho}\partial k^{\sigma } } -\frac{1}{8 } ( \mu \nu \lambda \rho ) ( \underline{\mu } \sigma ) k_{\lambda}\skpp \frac{\partial^3}{\partial k^{\nu}\partial k^{\rho}\partial k^{\sigma } } -\frac{1}{8 } ( \mu \nu \underline{\nu } \lambda ) ( \rho \sigma ) k_{\rho}\skpp \frac{\partial^3}{\partial k^{\mu}\partial k^{\lambda}\partial k^{\sigma } } \notag\\ & & -\frac{1}{8 } ( \mu \nu \underline{\mu } \lambda ) ( \rho \sigma ) k_{\rho}\skpp \frac{\partial^3}{\partial k^{\nu}\partial k^{\lambda}\partial k^{\sigma } } -\frac{2}{9 } ( \mu \nu \lambda ) ( \rho \underline{\mu } \sigma ) k_{\nu}\skpp \frac{\partial^3}{\partial k^{\lambda}\partial k^{\rho}\partial k^{\sigma } } -\frac{1}{9 } ( \mu \nu \lambda ) ( \rho \underline{\rho } \sigma ) k_{\nu}\skpp \frac{\partial^3}{\partial k^{\mu}\partial k^{\lambda}\partial k^{\sigma } } \notag\\ & & -\frac{1}{9 } ( \mu \underline{\mu } \nu ) ( \lambda \rho \sigma ) k_{\rho}\skpp \frac{\partial^3}{\partial k^{\nu}\partial k^{\lambda}\partial k^{\sigma } } -\frac{1}{8 } ( \mu \nu ) ( \underline{\mu } \lambda ) ( \rho \sigma ) k_{\rho}\skpp \frac{\partial^3}{\partial k^{\nu}\partial k^{\lambda}\partial k^{\sigma } } -\frac{1}{8 } ( \mu \nu ) ( \lambda \rho ) ( \underline{\lambda } \sigma ) k_{\mu}\skpp \frac{\partial^3}{\partial k^{\nu}\partial k^{\rho}\partial k^{\sigma } } \notag\\ & & -\frac{1}{8 } ( \mu \nu \lambda \rho ) ( \underline{\lambda } \sigma ) \skp \frac{\partial^4}{\partial k^{\mu}\partial k^{\nu}\partial k^{\rho}\partial k^{\sigma } } -\frac{1}{9 } ( \mu \nu \lambda ) ( \rho \underline{\nu } \sigma ) \skp \frac{\partial^4}{\partial k^{\mu}\partial k^{\lambda}\partial k^{\rho}\partial k^{\sigma } } + \frac{1}{6 } ( \mu \nu ) ( \lambda \rho ) ( \underline{\mu } \sigma ) k_{\nu}k_{\lambda}\skppp \frac{\partial^2}{\partial k^{\rho}\partial k^{\sigma } } \notag\\ & & -\frac{1}{3 } ( \mu \nu \lambda \rho ) ( \sigma \delta ) k_{\mu}k_{\nu}k_{\lambda}k_{\sigma}\skpppp \frac{\partial^2}{\partial k^{\rho}\partial k^{\delta } } -\frac{2}{9 } ( \mu \nu \lambda ) ( \rho \sigma \delta ) k_{\mu}k_{\nu}k_{\rho}k_{\sigma}\skpppp \frac{\partial^2}{\partial k^{\lambda}\partial k^{\delta } } -\frac{1}{4 } ( \mu \nu \lambda \rho ) ( \sigma \delta ) k_{\nu}k_{\lambda}k_{\sigma}\skppp \frac{\partial^3}{\partial k^{\mu}\partial k^{\rho}\partial k^{\delta } } \notag\\ & & -\frac{1}{4 } ( \mu \nu \lambda \rho ) ( \sigma \delta ) k_{\mu}k_{\lambda}k_{\sigma}\skppp \frac{\partial^3}{\partial k^{\nu}\partial k^{\rho}\partial k^{\delta } } -\frac{2}{9 } ( \mu \nu \lambda ) ( \rho \sigma \delta ) k_{\nu}k_{\rho}k_{\sigma}\skppp \frac{\partial^3}{\partial k^{\mu}\partial k^{\lambda}\partial k^{\delta } } -\frac{2}{9 } ( \mu \nu \lambda ) ( \rho \sigma \delta ) k_{\mu}k_{\nu}k_{\sigma}\skppp \frac{\partial^3}{\partial k^{\lambda}\partial k^{\rho}\partial k^{\delta } } \notag\\ & & -\frac{1}{4 } ( \mu \nu \lambda \rho ) ( \sigma \delta ) k_{\lambda}k_{\sigma}\skpp \frac{\partial^4}{\partial k^{\mu}\partial k^{\nu}\partial k^{\rho}\partial k^{\delta } } -\frac{2}{9 } ( \mu \nu \lambda ) ( \rho \sigma \delta ) k_{\nu}k_{\sigma}\skpp \frac{\partial^4}{\partial k^{\mu}\partial k^{\lambda}\partial k^{\rho}\partial k^{\delta } } -\frac{1}{6 } ( \mu \nu ) ( \lambda \rho ) ( \sigma \delta ) k_{\mu}k_{\lambda}k_{\sigma}\skppp \frac{\partial^3}{\partial k^{\nu}\partial k^{\rho}\partial k^{\delta}}\notag\\ & & + \mbox{traceless terms}\;. \end{aligned}\ ] ] finally , we list down the @xmath359 , @xmath1 , @xmath123 , @xmath0 order low energy expansion result for @xmath100 used in ( [ bexp ] ) , @xmath368 @xmath369 \gamf \tau + 24i ( \mu d^{\nu}s_\omega)\gamma_{\nu}\tau \frac{\partial}{\partial k^{\mu } } + 24 ( \mu d^{\nu}p_\omega ) \gamma_{\nu}\gamf \tau \frac{\partial}{\partial k^{\mu } } -24 a_\omega^{\mu}(\nu s_\omega)\gamma_{\mu}\gamf \tau \frac{\partial}{\partial k^{\nu}}\notag\\ & & -24 ( \mu a_\omega^{\nu } ) s_\omega \gamma_{\nu}\gamf \tau \frac{\partial}{\partial k^{\mu } } + 24i a_\omega^{\mu}(\nu p_\omega)\gamma_{\mu}\tau \frac{\partial}{\partial k^{\nu } } + 24i ( \mu a_\omega^{\nu } ) p_\omega \gamma_{\nu}\tau \frac{\partial}{\partial k^{\mu } } -24 s_\omega ( \mu a_\omega^{\nu } ) \gamma_{\nu}\gamf \tau \frac{\partial}{\partial k^{\mu } } \notag\\ & & -24 ( \mu s_\omega)a_\omega^{\nu}\gamma_{\nu}\gamf \tau \frac{\partial}{\partial k^{\mu } } + 24i p_\omega ( \mu a_\omega^{\nu } ) \gamma_{\nu}\tau \frac{\partial}{\partial k^{\mu } } + 24i ( \mu p_\omega)a_\omega^{\nu}\gamma_{\nu}\tau \frac{\partial}{\partial k^{\mu } } -6 ( \mu \nu \lambda \rho ) \gamma_{\lambda}\gamma_{\rho}\tau \frac{\partial^2}{\partial k^{\mu}\partial k^{\nu } } \notag\\ & & + 6 a_\omega^{\mu}(\nu \lambda a_\omega^{\rho } ) \gamma_{\mu}\gamma_{\rho}\tau \frac{\partial^2}{\partial k^{\nu}\partial k^{\lambda } } + 12 ( \mu a_\omega^{\nu } ) ( \lambda a_\omega^{\rho } ) \gamma_{\nu}\gamma_{\rho}\tau \frac{\partial^2}{\partial k^{\mu}\partial k^{\lambda } } + 6 ( \mu \nu a_\omega^{\lambda } ) a_\omega^{\rho}\gamma_{\lambda}\gamma_{\rho}\tau \frac{\partial^2}{\partial k^{\mu}\partial k^{\nu } } \notag\\ & & -6 a_\omega^{\mu}(\nu \lambda a_\omega^{\rho } ) \gamma_{\rho}\gamma_{\mu}\tau \frac{\partial^2}{\partial k^{\nu}\partial k^{\lambda } } -12 ( \mu a_\omega^{\nu } ) ( \lambda a_\omega^{\rho } ) \gamma_{\rho}\gamma_{\nu}\tau \frac{\partial^2}{\partial k^{\mu}\partial k^{\lambda } } -6 ( \mu \nu a_\omega^{\lambda } ) a_\omega^{\rho}\gamma_{\rho}\gamma_{\lambda}\tau \frac{\partial^2}{\partial k^{\mu}\partial k^{\nu } } \notag\\ & & + 6i ( \mu\nu(d^{\lambda}a_\omega^{\rho}-d^{\rho}a_\omega^{\lambda}))\gamma_{\lambda}\gamma_{\rho}\gamf \tau \frac{\partial^2}{\partial k^{\mu}\partial k^{\nu } } -3 ( \mu \nu \underline{\nu } \lambda ) \tau \frac{\partial^2}{\partial k^{\mu}\partial k^{\lambda } } -3 ( \mu \nu \underline{\mu } \lambda ) \tau \frac{\partial^2}{\partial k^{\nu}\partial k^{\lambda } } + 12 a_\omega^{\mu}(\nu \lambda a_{\omega\mu } ) \tau \frac{\partial^2}{\partial k^{\nu}\partial k^{\lambda}}\notag\\ & & + 24 ( \mu a_\omega^{\nu } ) ( \lambda a_{\omega\nu } ) \tau \frac{\partial^2}{\partial k^{\mu}\partial k^{\lambda } } + 12 ( \mu \nu a_\omega^{\lambda } ) a_{\omega\lambda}\tau \frac{\partial^2}{\partial k^{\mu}\partial k^{\nu } } -12 ( \mu \nu ) ( \underline{\mu } \underline{\nu } ) \tau \sk \skpp -12 ( \mu \nu ) ( \underline{\mu } \underline{\nu } ) \tau \skp^2 \notag\\ & & -6 ( \mu \nu ) ( \underline{\mu } \lambda ) \tau \frac{\partial^2}{\partial k^{\nu}\partial k^{\lambda } } + 8 ( \mu \underline{\mu } \nu ) a_\omega^{\lambda}\gamma_{\lambda}\gamf \tau k_{\nu}\skpp -8 ( \mu \underline{\mu } \nu ) a_\omega^{\lambda}\gamma_{\lambda}\gamf \tau \skp \frac{\partial}{\partial k^{\nu } } -24 ( \mu \underline{\mu } s_\omega)\tau \skp-24i ( \mu \underline{\mu } p)\gamf \tau \skp \notag\\ & & -48 ( \mu \nu s_\omega)\tau k_{\mu}k_{\nu}\skpp -48i ( \mu \nu p_\omega)\gamf \tau k_{\mu}k_{\nu}\skpp -24 ( \mu \nu s_\omega)\tau k_{\mu}\skp \frac{\partial}{\partial k^{\nu } } -24i ( \mu \nu p_\omega)\gamf \tau k_{\mu}\skp \frac{\partial}{\partial k^{\nu } } \notag\\ & & -24 ( \mu \nu s_\omega)\tau k_{\nu}\skp \frac{\partial}{\partial k^{\mu } } -24i ( \mu \nu p_\omega)\gamf \tau k_{\nu}\skp \frac{\partial}{\partial k^{\mu } } -24 ( \mu \nu s_\omega)\tau \sk \frac{\partial^2}{\partial k^{\mu}\partial k^{\nu } } -8i ( \mu \nu \underline{\nu } \lambda ) \gamma_{\mu}\tau k_{\lambda}\skpp \notag\\ & & + 8i ( \mu \nu \underline{\nu } \lambda ) \gamma_{\mu}\tau \skp \frac{\partial}{\partial k^{\lambda } } + 8 a_\omega^{\mu}(\nu \underline{\nu } \lambda ) \gamma_{\mu}\gamf \tau k_{\lambda}\skpp -8 a_\omega^{\mu}(\nu \underline{\nu } \lambda ) \gamma_{\mu}\gamf \tau \skp \frac{\partial}{\partial k^{\lambda } } + 8 ( \mu \nu \underline{\nu } \lambda ) \tau k_{\mu}k_{\lambda}\sk \skppp\notag\\ & & + 8 ( \mu \nu \underline{\mu } \lambda ) \tau k_{\nu}k_{\lambda}\sk \skppp + 24 ( \mu \nu \underline{\nu } \lambda ) \tau k_{\mu}k_{\lambda}\skp \skpp + 24 ( \mu \nu \underline{\mu } \lambda ) \tau k_{\nu}k_{\lambda}\skp \skpp + 6 ( \mu \nu \underline{\nu } \lambda ) \tau k_{\lambda}\sk \skpp \frac{\partial}{\partial k^{\mu } } \notag\\ & & + 6 ( \mu \nu \underline{\mu } \lambda ) \tau k_{\lambda}\sk \skpp \frac{\partial}{\partial k^{\nu } } -12 ( \mu \underline{\mu } \nu \lambda ) \tau k_{\nu}\sk \skpp \frac{\partial}{\partial k^{\lambda } } -6 ( \mu \nu \underline{\nu } \lambda ) \tau k_{\mu}\sk \skpp \frac{\partial}{\partial k^{\lambda } } -6 ( \mu \nu \underline{\mu } \lambda ) \tau k_{\nu}\sk \skpp \frac{\partial}{\partial k^{\lambda } } \notag\\ & & + 6 ( \mu \nu \underline{\nu } \lambda ) \tau k_{\lambda}\skp^2 \frac{\partial}{\partial k^{\mu } } + 6 ( \mu \nu \underline{\mu } \lambda ) \tau k_{\lambda}\skp^2 \frac{\partial}{\partial k^{\nu } } -12 ( \mu \underline{\mu } \nu \lambda ) \tau k_{\nu}\skp^2 \frac{\partial}{\partial k^{\lambda } } -6 ( \mu \nu \underline{\nu } \lambda ) \tau k_{\mu}\skp^2 \frac{\partial}{\partial k^{\lambda}}\notag\\ & & -6 ( \mu \nu \underline{\mu } \lambda ) \tau k_{\nu}\skp^2 \frac{\partial}{\partial k^{\lambda } } -24 ( \mu \nu ) s_\omega \tau k_{\mu}\skp \frac{\partial}{\partial k^{\nu } } + 24i ( \mu \nu ) p_\omega \gamf \tau k_{\mu}\skp \frac{\partial}{\partial k^{\nu } } -6 ( \mu \nu \lambda \rho ) \tau k_{\lambda}\frac{\partial^3}{\partial k^{\mu}\partial k^{\nu}\partial k^{\rho } } \notag\\ & & -6 ( \mu \nu \underline{\nu } \lambda ) \tau \sk \skp \frac{\partial^2}{\partial k^{\mu}\partial k^{\lambda } } -6 ( \mu \nu \underline{\mu } \lambda ) \tau \sk \skp \frac{\partial^2}{\partial k^{\nu}\partial k^{\lambda } } -12i ( \mu \nu ) ( \lambda a_{\omega\mu } ) \gamf \tau \frac{\partial^2}{\partial k^{\nu}\partial k^{\lambda } } -16 ( \mu \nu \underline{\nu } a_\omega^{\lambda } ) \gamma_{\lambda}\gamf \tau k_{\mu}\skpp \notag\\ & & -8 ( \mu \nu \underline{\nu } a_\omega^{\lambda } ) \gamma_{\lambda}\gamf \tau \skp \frac{\partial}{\partial k^{\mu } } -16 ( \mu \nu \underline{\mu } a_\omega^{\lambda } ) \gamma_{\lambda}\gamf \tau k_{\nu}\skpp -16 ( \mu \underline{\mu } \nu a_\omega^{\lambda } ) \gamma_{\lambda}\gamf \tau k_{\nu}\skpp -32 ( \mu \nu \lambda a^{\rho } ) \gamma_{\rho}\gamf \tau k_{\mu}k_{\nu}k_{\lambda}\skppp\notag\\ & & -16 ( \mu \nu \lambda a_\omega^{\rho } ) \gamma_{\rho}\gamf \tau k_{\nu}k_{\lambda}\skpp \frac{\partial}{\partial k^{\mu } } -8 ( \mu \nu \underline{\mu } a_\omega^{\lambda } ) \gamma_{\lambda}\gamf \tau \skp \frac{\partial}{\partial k^{\nu } } -16 ( \mu \nu \lambda a_\omega^{\rho } ) \gamma_{\rho}\gamf \tau k_{\mu}k_{\lambda}\skpp \frac{\partial}{\partial k^{\nu } } \notag\\ & & -8 ( \mu \nu \lambda a_\omega^{\rho } ) \gamma_{\rho}\gamf \tau k_{\lambda}\skp \frac{\partial^2}{\partial k^{\mu}\partial k^{\nu } } -8 ( \mu \underline{\mu } \nu a_\omega^{\lambda } ) \gamma_{\lambda}\gamf \tau \skp \frac{\partial}{\partial k^{\nu } } -16 ( \mu \nu \lambda a_\omega^{\rho } ) \gamma_{\rho}\gamf \tau k_{\mu}k_{\nu}\skpp \frac{\partial}{\partial k^{\lambda}}\notag\\ & & -8 ( \mu \nu \lambda a^{\rho } ) \gamma_{\rho}\gamf \tau k_{\nu}\skp \frac{\partial^2}{\partial k^{\mu}\partial k^{\lambda } } -8 ( \mu \nu \lambda a_\omega^{\rho } ) \gamma_{\rho}\gamf \tau k_{\mu}\skp \frac{\partial^2}{\partial k^{\nu}\partial k^{\lambda } } -8 ( \mu \nu \lambda a_\omega^{\rho } ) \gamma_{\rho}\gamf \tau \sk \frac{\partial^3}{\partial k^{\mu}\partial k^{\nu}\partial k^{\lambda } } \notag\\ & & -8i ( \mu \nu \lambda ) a_{\omega\nu}\gamf \tau \frac{\partial^2}{\partial k^{\mu}\partial k^{\lambda } } -12i ( \mu a_\omega^{\nu } ) ( \underline{\nu } \lambda ) \gamf \tau \frac{\partial^2}{\partial k^{\mu}\partial k^{\lambda } } -8i a_\omega^{\mu}(\nu \underline{\mu } \lambda ) \gamf \tau \frac{\partial^2}{\partial k^{\nu}\partial k^{\lambda } } -24 s_\omega ( \mu \nu ) \tau k_{\mu}\skp \frac{\partial}{\partial k^{\nu } } \notag\\ & & -24i p_\omega ( \mu \nu ) \gamf \tau k_{\mu}\skp \frac{\partial}{\partial k^{\nu } } -16 ( \mu \nu ) ( \underline{\mu } \lambda ) \tau k_{\nu}k_{\lambda}\sk \skppp -48 ( \mu \nu ) ( \underline{\mu } \lambda ) \tau k_{\nu}k_{\lambda}\skp \skpp + 12 ( \mu \nu ) ( \underline{\mu } \lambda ) \tau k_{\nu}\sk \skpp \frac{\partial}{\partial k^{\lambda } } \notag\\ & & -12 ( \mu \nu ) ( \underline{\mu } \lambda ) \tau k_{\lambda}\sk \skpp \frac{\partial}{\partial k^{\nu } } + 12 ( \mu \nu ) ( \underline{\mu } \lambda ) \tau k_{\nu}\skp^2 \frac{\partial}{\partial k^{\lambda } } -12 ( \mu \nu ) ( \underline{\mu } \lambda ) \tau k_{\lambda}\skp^2 \frac{\partial}{\partial k^{\nu } } \notag\\ & & -12 ( \mu \nu ) ( \underline{\mu } \lambda ) \tau \sk \skp \frac{\partial^2}{\partial k^{\nu}\partial k^{\lambda } } -16 ( \mu \nu \lambda ) a_\omega^{\rho}\gamma_{\rho}\gamf \tau k_{\mu}k_{\nu}\skpp \frac{\partial}{\partial k^{\lambda } } -24 ( \mu \nu ) ( \lambda a_\omega^{\rho } ) \gamma_{\rho}\gamf \tau k_{\mu}\skp \frac{\partial^2}{\partial k^{\nu}\partial k^{\lambda } } \notag\\ & & -16 ( \mu \nu \lambda ) a_\omega^{\rho}\gamma_{\rho}\gamf \tau k_{\nu}\skp \frac{\partial^2}{\partial k^{\mu}\partial k^{\lambda } } -24 ( \mu a_\omega^{\nu } ) ( \underline{\mu } \lambda ) \gamma_{\nu}\gamf \tau \skp \frac{\partial}{\partial k^{\lambda } } -48 ( \mu a_\omega^{\nu } ) ( \lambda \rho ) \gamma_{\nu}\gamf \tau k_{\mu}k_{\lambda}\skpp \frac{\partial}{\partial k^{\rho } } \notag\\ & & -24 ( \mu a_\omega^{\nu } ) ( \lambda \rho ) \gamma_{\nu}\gamf \tau k_{\lambda}\skp \frac{\partial^2}{\partial k^{\mu}\partial k^{\rho } } -4i ( \mu \nu \lambda a_{\omega\mu } ) \gamf \tau \frac{\partial^2}{\partial k^{\nu}\partial k^{\lambda } } -4i ( \mu \nu \lambda a_{\omega\nu } ) \gamf \tau \frac{\partial^2}{\partial k^{\mu}\partial k^{\lambda } } \notag\\ & & -4i ( \mu \nu \lambda a_{\omega\lambda } ) \gamf \tau \frac{\partial^2}{\partial k^{\mu}\partial k^{\nu } } -8i ( \mu \nu \lambda a_\omega^{\rho } ) \gamf \tau k_{\rho}\frac{\partial^3}{\partial k^{\mu}\partial k^{\nu}\partial k^{\lambda } } + 16i ( \mu \nu \lambda \rho ) \gamma_{\mu}\tau k_{\nu}k_{\lambda}\skpp \frac{\partial}{\partial k^{\rho } } \notag\\ & & + 16i ( \mu \nu \lambda \rho ) \gamma_{\mu}\tau k_{\lambda}\skp \frac{\partial^2}{\partial k^{\nu}\partial k^{\rho } } -24i ( \mu \nu ) ( \lambda \rho ) \gamma_{\mu}\tau k_{\lambda}\skp \frac{\partial^2}{\partial k^{\nu}\partial k^{\rho } } + 24i ( \mu \nu ) ( \lambda \rho ) \gamma_{\lambda}\tau k_{\mu}\skp \frac{\partial^2}{\partial k^{\nu}\partial k^{\rho } } \notag\\ & & -16i ( \mu \underline{\mu } \nu \lambda ) \gamma_{\nu}\tau k_{\lambda}\skpp -8i ( \mu \underline{\mu } \nu \lambda ) \gamma_{\nu}\tau \skp \frac{\partial}{\partial k^{\lambda } } + 16i ( \mu \nu \underline{\mu } \lambda ) \gamma_{\lambda}\tau k_{\nu}\skpp + 16i ( \mu \nu \underline{\nu } \lambda ) \gamma_{\lambda}\tau k_{\mu}\skpp\notag\\ & & -32i ( \mu \nu \lambda \rho ) \gamma_{\lambda}\tau k_{\mu}k_{\nu}k_{\rho}\skppp -16i ( \mu \nu \lambda \rho ) \gamma_{\lambda}\tau k_{\mu}k_{\nu}\skpp \frac{\partial}{\partial k^{\rho } } + 8i ( \mu \nu \underline{\mu } \lambda ) \gamma_{\lambda}\tau \skp \frac{\partial}{\partial k^{\nu } } -16i ( \mu \nu \lambda \rho ) \gamma_{\lambda}\tau k_{\mu}k_{\rho}\skpp \frac{\partial}{\partial k^{\nu } } \notag\\ & & -8i ( \mu \nu \lambda \rho ) \gamma_{\lambda}\tau k_{\mu}\skp \frac{\partial^2}{\partial k^{\nu}\partial k^{\rho } } + 8i ( \mu \nu \underline{\nu } \lambda ) \gamma_{\lambda}\tau \skp \frac{\partial}{\partial k^{\mu } } -16i ( \mu \nu \lambda \rho ) \gamma_{\lambda}\tau k_{\nu}k_{\rho}\skpp \frac{\partial}{\partial k^{\mu } } -8i ( \mu \nu \lambda \rho ) \gamma_{\lambda}\tau k_{\nu}\skp \frac{\partial^2}{\partial k^{\mu}\partial k^{\rho } } \notag\\ & & -8i ( \mu \nu \lambda \rho ) \gamma_{\lambda}\tau k_{\rho}\skp \frac{\partial^2}{\partial k^{\mu}\partial k^{\nu } } -16 a^{\mu}(\nu \lambda \rho ) \gamma_{\mu}\gamf \tau k_{\nu}k_{\lambda}\skpp \frac{\partial}{\partial k^{\rho } } -16 a^{\mu}(\nu \lambda \rho ) \gamma_{\mu}\gamf \tau k_{\lambda}\skp \frac{\partial^2}{\partial k^{\nu}\partial k^{\rho } } \notag\\ & & -16 ( \mu \nu \lambda \rho ) \tau k_{\mu}k_{\nu}k_{\lambda}\sk \skppp \frac{\partial}{\partial k^{\rho } } -48 ( \mu \nu \lambda \rho ) \tau k_{\mu}k_{\nu}k_{\lambda}\skp \skpp \frac{\partial}{\partial k^{\rho } } -12 ( \mu \nu \lambda \rho ) \tau k_{\nu}k_{\lambda}\sk \skpp \frac{\partial^2}{\partial k^{\mu}\partial k^{\rho } } \notag\\ & & -12 ( \mu \nu \lambda \rho ) \tau \frac{\partial^2}{\partial k^{\nu}\partial k^{\rho } } -12 ( \mu \nu \lambda \rho ) \tau k_{\nu}k_{\lambda}\skp^2 \frac{\partial^2}{\partial k^{\mu}\partial k^{\rho } } -12 ( \mu \nu \lambda \rho ) \tau k_{\mu}k_{\lambda}\skp^2 \frac{\partial^2}{\partial k^{\nu}\partial k^{\rho } } \notag\\ & & -12 ( \mu \nu \lambda \rho ) \tau \frac{\partial^3}{\partial k^{\mu}\partial k^{\nu}\partial k^{\rho } } + 24i ( \mu \nu ) ( \underline{\mu } \lambda ) \gamma_{\nu}\tau \skp \frac{\partial}{\partial k^{\lambda } } -48i ( \mu \nu ) ( \lambda \rho ) \gamma_{\mu}\tau k_{\nu}k_{\lambda}\skpp \frac{\partial}{\partial k^{\rho } } \notag\\ & & -24 ( \mu \nu ) ( \lambda \rho ) \tau \frac{\partial^2}{\partial k^{\nu}\partial k^{\rho } } -24 ( \mu \nu ) ( \lambda \rho ) \tau k_{\mu}k_{\lambda}\skp^2 \frac{\partial^2}{\partial k^{\nu}\partial k^{\rho}}\ ; , \end{aligned}\ ] ] @xmath370 @xmath371 @xmath373\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{2}=\int dk\bigg[2 \tau^3 -\frac{1}{3 } \tau^4 k^2 -12 \tau^4 \sk^2 -\frac{1}{18 } \tau^5 k^4 + \frac{7}{3 } \tau^5 k^2 \sk^2 + 8 \tau^5 \sk^4 + \frac{1}{180 } \tau^6 k^6 -\frac{2}{45 } \tau^6 k^4 \sk^2 -\frac{4}{5 } \tau^6 k^2 \sk^4 -\frac{16}{15 } \tau^6 \sk^6\bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{3}=\int dk\bigg[- \tau^3 + \frac{1}{2 } \tau^4 k^2 + 6 \tau^4 \sk^2 -\frac{1}{18 } \tau^5 k^4 -\frac{4}{3 } \tau^5 k^2 \sk^2 -4 \tau^5 \sk^4 + \frac{1}{360 } \tau^6 k^6 + \frac{1}{15 } \tau^6 k^4 \sk^2 + \frac{2}{5 } \tau^6 k^2 \sk^4 + \frac{8}{15 } \tau^6 \sk^6\bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{4}= \int dk\bigg[- \tau^3 + \frac{1}{6 } \tau^4 k^2 + 6 \tau^4 \sk^2 - \tau^5 k^2 \sk^2 -4 \tau^5 \sk^4+\frac{1}{360 } \tau^6 k^6 -\frac{1}{45 } \tau^6 k^4 \sk^2 + \frac{2}{5 } \tau^6 k^2 \sk^4 + \frac{8}{15 } \tau^6 \sk^6 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{5}= \int dk\bigg[\frac{1}{3 } \tau^3 -\frac{1}{6 } \tau^4 k^2 -2 \tau^4 \sk^2 + \frac{1}{3 } \tau^5 k^2 \sk^2 + \frac{4}{3 } \tau^5 \sk^4 + \frac{1}{1080 } \tau^6 k^6 + \frac{1}{45 } \tau^6 k^4 \sk^2 -\frac{2}{15 } \tau^6 k^2 \sk^4 -\frac{8}{45 } \tau^6 \sk^6 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{6}= \int dk\bigg[-\frac{4}{15 } \tau^3 + \frac{1}{3 } \tau^4 k^2 + \frac{2}{5 } \tau^4 \sk^2 -\frac{1}{10 } \tau^5 k^4 + \frac{2}{3 } \tau^5 k^4 \sk^2 \skp^2 -\frac{2}{5 } \tau^5 k^2 \sk^2 + \frac{1}{120 } \tau^6 k^6 + \frac{1}{15 } \tau^6 k^4 \sk^2 -\frac{1}{30 } \tau^6 k^6 \sk^2 \skp^2\notag\\ & & \hspace{0.4 cm } -\frac{4}{15 } \tau^6 k^4 \sk^4 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{7}= \int dk\bigg[-\frac{11}{5 } \tau^3 + 3 \tau^3 k^2 \skp^2 + \frac{13}{10 } \tau^4 k^2 + \frac{92}{15 } \tau^4 \sk^2 -18 \tau^4 k^2 \sk^2 \skp^2 -\frac{2}{9 } \tau^4 k^4 \skp^2 -\frac{17}{90 } \tau^5 k^4 -\frac{11}{5 } \tau^5 k^2 \sk^2 -\frac{8}{5 } \tau^5 \sk^4 \notag\\ & & + \frac{14}{9 } \tau^5 k^4 \sk^2 \skp^2 + 12 \tau^5 k^2 \sk^4 \skp^2 + \frac{1}{120 } \tau^6 k^6 + \frac{2}{15 } \tau^6 k^4 \sk^2 + \frac{2}{5 } \tau^6 k^2 \sk^4 -\frac{1}{30 } \tau^6 k^6 \sk^2 \skp^2 -\frac{8}{15 } \tau^6 k^4 \sk^4 \skp^2 -\frac{8}{5 } \tau^6 k^2 \sk^6 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{8}= \int dk\bigg[\frac{4}{15 } \tau^3 + \frac{1}{5 } \tau^4 k^2 -\frac{2}{5 } \tau^4 \sk^2 -\frac{1}{10 } \tau^5 k^4 + \frac{2}{3 } \tau^5 k^4 \sk^2 \skp^2 -\frac{2}{5 } \tau^5 k^2 \sk^2 + \frac{1}{120 } \tau^6 k^6 + \frac{1}{15 } \tau^6 k^4 \sk^2 -\frac{1}{30 } \tau^6 k^6 \sk^2 \skp^2\notag\\ & & \hspace{0.4 cm } -\frac{4}{15 } \tau^6 k^4 \sk^4 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{9}= \int dk\bigg[-\frac{23}{15 } \tau^3 + \frac{59}{60 } \tau^4 k^2 + \frac{17}{15 } \tau^4 \sk^2 -\frac{2}{9 } \tau^4 k^4 \skp^2 -\frac{7}{45 } \tau^5 k^4 + \frac{8}{9 } \tau^5 k^4 \sk^2 \skp^2 -\frac{2}{5 } \tau^5 k^2 \sk^2 + \frac{1}{120 } \tau^6 k^6 + \frac{1}{15 } \tau^6 k^4 \sk^2 \notag\\ & & \hspace{0.4cm}-\frac{1}{30 } \tau^6 k^6 \sk^2 \skp^2-\frac{4}{15 } \tau^6 k^4 \sk^4 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{10}= \int dk\bigg[-\frac{4}{5 } \tau^3 + \frac{19}{30 } \tau^4 k^2 + \frac{8}{15 } \tau^4 \sk^2 -\frac{2}{9 } \tau^4 k^4 \skp^2 -\frac{7}{45 } \tau^5 k^4 + \frac{8}{9 } \tau^5 k^4 \sk^2 \skp^2 -\frac{2}{5 } \tau^5 k^2 \sk^2 + \frac{1}{120 } \tau^6 k^6 + \frac{1}{15 } \tau^6 k^4 \sk^2 \notag\\ & & \hspace{0.4cm}-\frac{1}{30 } \tau^6 k^6 \sk^2 \skp^2-\frac{4}{15 } \tau^6 k^4 \sk^4 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{11}= \int dk\bigg[-\frac{2}{5 } \tau^3 + \frac{1}{30 } \tau^4 k^2 + \frac{4}{15 } \tau^4 \sk^2 + \frac{2}{9 } \tau^4 k^4 \skp^2 -\frac{1}{30 } \tau^5 k^4 -\frac{2}{3 } \tau^5 k^4 \sk^2 \skp^2 + \frac{2}{5 } \tau^5 k^2 \sk^2 + \frac{1}{120 } \tau^6 k^6 -\frac{1}{15 } \tau^6 k^4 \sk^2\notag\\ & & \hspace{0.4cm}-\frac{1}{30 } \tau^6 k^6 \sk^2 \skp^2 + \frac{4}{15 } \tau^6 k^4 \sk^4 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{12}=\int dk\bigg[\frac{16}{15 } \tau^3 -\frac{2}{5 } \tau^4 k^2 -\frac{14}{15 } \tau^4 \sk^2 + \frac{2}{9 } \tau^4 k^4 \skp^2 -\frac{1}{30 } \tau^5 k^4 -\frac{2}{3 } \tau^5 k^4 \sk^2 \skp^2 + \frac{2}{5 } \tau^5 k^2 \sk^2 + \frac{1}{120 } \tau^6 k^6 -\frac{1}{15 } \tau^6 k^4 \sk^2\notag\\ & & \hspace{0.4cm}-\frac{1}{30 } \tau^6 k^6 \sk^2 \skp^2 + \frac{4}{15 } \tau^6 k^4 \sk^4 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{13}= \int dk\bigg[-\frac{28}{15 } \tau^3 + \frac{9}{10 } \tau^4 k^2 -\frac{2}{9 } \tau^4 k^4 \skp^2 + \frac{4}{5 } \tau^4 \sk^2 -\frac{7}{45 } \tau^5 k^4 + \frac{8}{9 } \tau^5 k^4 \sk^2 \skp^2 -\frac{2}{5 } \tau^5 k^2 \sk^2 + \frac{1}{120 } \tau^6 k^6 + \frac{1}{15 } \tau^6 k^4 \sk^2\notag\\ & & \hspace{0.4cm}-\frac{1}{30 } \tau^6 k^6 \sk^2 \skp^2 -\frac{4}{15 } \tau^6 k^4 \sk^4 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{14}= \int dk\bigg[\frac{13}{15 } \tau^3 + \frac{23}{60 } \tau^4 k^2 -\frac{7}{15 } \tau^4 \sk^2 -\frac{2}{9 } \tau^4 k^4 \skp^2 -\frac{7}{45 } \tau^5 k^4 + \frac{8}{9 } \tau^5 k^4 \sk^2 \skp^2 -\frac{2}{5 } \tau^5 k^2 \sk^2 + \frac{1}{120 } \tau^6 k^6 + \frac{1}{15 } \tau^6 k^4 \sk^2\notag\\ & & \hspace{0.4cm}-\frac{1}{30 } \tau^6 k^6 \sk^2 \skp^2 -\frac{4}{15 } \tau^6 k^4 \sk^4 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{15}= \int dk\bigg[\frac{11}{5 } \tau^3 -3 \tau^3 k^2 \skp^2 -\frac{19}{30 } \tau^4 k^2 -\frac{92}{15 } \tau^4 \sk^2 + 18 \tau^4 k^2 \sk^2 \skp^2 + \frac{4}{9 } \tau^4 k^4 \skp^2 + \frac{11}{5 } \tau^5 k^2 \sk^2 + \frac{8}{5 } \tau^5 \sk^4-\frac{4}{3 } \tau^5 k^4 \sk^2 \skp^2 \notag\\ & & \hspace{0.4 cm } -12 \tau^5 k^2 \sk^4 \skp^2+\frac{1}{120 } \tau^6 k^6 -\frac{2}{15 } \tau^6 k^4 \sk^2 -\frac{2}{5 } \tau^6 k^2 \sk^4 -\frac{1}{30 } \tau^6 k^6 \sk^2 \skp^2 + \frac{8}{15 } \tau^6 k^4 \sk^4 \skp^2 + \frac{8}{5 } \tau^6 k^2 \sk^6 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{16}= \int dk\bigg[-\frac{11}{5 } \tau^3 + 3 \tau^3 k^2 \skp^2 + \frac{13}{10 } \tau^4 k^2 + \frac{92}{15 } \tau^4 \sk^2 -18 \tau^4 k^2 \sk^2 \skp^2 -\frac{4}{9 } \tau^4 k^4 \skp^2 -\frac{11}{45 } \tau^5 k^4 -\frac{11}{5 } \tau^5 k^2 \sk^2 -\frac{8}{5 } \tau^5 \sk^4 \notag\\ & & \hspace{0.4cm}+\frac{16}{9 } \tau^5 k^4 \sk^2 \skp^2 + 12 \tau^5 k^2 \sk^4 \skp^2 + \frac{1}{120 } \tau^6 k^6 + \frac{2}{15 } \tau^6 k^4 \sk^2 + \frac{2}{5 } \tau^6 k^2 \sk^4 -\frac{1}{30 } \tau^6 k^6 \sk^2 \skp^2 -\frac{8}{15 } \tau^6 k^4 \sk^4 \skp^2 -\frac{8}{5 } \tau^6 k^2 \sk^6 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{17}= \int dk\bigg[-\frac{11}{15 } \tau^3 + \frac{49}{60 } \tau^4 k^2 + \frac{3}{5 } \tau^4 \sk^2 -\frac{2}{9 } \tau^4 k^4 \skp^2 -\frac{17}{90 } \tau^5 k^4 + \frac{10}{9 } \tau^5 k^4 \sk^2 \skp^2 -\frac{8}{15 } \tau^5 k^2 \sk^2 + \frac{1}{90 } \tau^6 k^6 + \frac{4}{45 } \tau^6 k^4 \sk^2\notag\\ & & \hspace{0.4cm}-\frac{2}{45 } \tau^6 k^6 \sk^2 \skp^2 -\frac{16}{45 } \tau^6 k^4 \sk^4 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{18}= \int dk\bigg[-\frac{29}{15 } \tau^3 + \frac{67}{60 } \tau^4 k^2 + \frac{11}{15 } \tau^4 \sk^2 -\frac{2}{9 } \tau^4 k^4 \skp^2 -\frac{17}{90 } \tau^5 k^4 + \frac{10}{9 } \tau^5 k^4 \sk^2 \skp^2 -\frac{8}{15 } \tau^5 k^2 \sk^2 + \frac{1}{90 } \tau^6 k^6 + \frac{4}{45 } \tau^6 k^4 \sk^2\notag\\ & & \hspace{0.4cm}-\frac{2}{45 } \tau^6 k^6 \sk^2 \skp^2 -\frac{16}{45 } \tau^6 k^4 \sk^4 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{19}= \int dk\bigg[\frac{4}{3 } \tau^3 -\frac{2}{5 } \tau^4 k^2 -\frac{2}{3 } \tau^4 \sk^2 + \frac{1}{9 } \tau^4 k^4 \skp^2 -\frac{1}{45 } \tau^5 k^4 -\frac{4}{9 } \tau^5 k^4 \sk^2 \skp^2 + \frac{4}{15 } \tau^5 k^2 \sk^2 + \frac{1}{180 } \tau^6 k^6 -\frac{2}{45 } \tau^6 k^4 \sk^2\notag\\ & & \hspace{0.4cm}-\frac{1}{45 } \tau^6 k^6 \sk^2 \skp^2 + \frac{8}{45 } \tau^6 k^4 \sk^4 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{20}= \int dk\bigg[-\frac{4}{5 } \tau^3 + 2 \tau^3 k^2 \skp^2 + \frac{7}{10 } \tau^4 k^2 + \frac{58}{15 } \tau^4 \sk^2 -12 \tau^4 k^2 \sk^2 \skp^2 -\frac{2}{9 } \tau^4 k^4 \skp^2 -\frac{13}{90 } \tau^5 k^4 -\frac{22}{15 } \tau^5 k^2 \sk^2 -\frac{16}{15 } \tau^5 \sk^4 \notag\\ & & \hspace{0.4cm}+\frac{10}{9 } \tau^5 k^4 \sk^2 \skp^2 + 8 \tau^5 k^2 \sk^4 \skp^2 + \frac{1}{180 } \tau^6 k^6 + \frac{4}{45 } \tau^6 k^4 \sk^2 + \frac{4}{15 } \tau^6 k^2 \sk^4 -\frac{1}{45 } \tau^6 k^6 \sk^2 \skp^2 -\frac{16}{45 } \tau^6 k^4 \sk^4 \skp^2 -\frac{16}{15 } \tau^6 k^2 \sk^6 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{21}= \int dk\bigg[\frac{1}{15 } \tau^3 -\frac{1}{30 } \tau^4 k^2 + \frac{2}{9 } \tau^4 k^4 \skp^2 -\frac{4}{15 } \tau^4 \sk^2 -\frac{1}{45 } \tau^5 k^4 -\frac{4}{9 } \tau^5 k^4 \sk^2 \skp^2 + \frac{4}{15 } \tau^5 k^2 \sk^2 + \frac{1}{180 } \tau^6 k^6 -\frac{2}{45 } \tau^6 k^4 \sk^2 \notag\\ & & \hspace{0.4cm}-\frac{1}{45 } \tau^6 k^6 \sk^2 \skp^2+\frac{8}{45 } \tau^6 k^4 \sk^4 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{22}= \int dk\bigg[-2 \tau^3 k^2 \skp^2 + \frac{4}{5 } \tau^3 -\frac{58}{15 } \tau^4 \sk^2 -\frac{1}{5 } \tau^4 k^2 + 12 \tau^4 k^2 \sk^2 \skp^2 + \frac{1}{3 } \tau^4 k^4 \skp^2 + \frac{22}{15 } \tau^5 k^2 \sk^2 + \frac{16}{15 } \tau^5 \sk^4 -\frac{8}{9 } \tau^5 k^4 \sk^2 \skp^2 \notag\\ & & \hspace{0.4cm}-8 \tau^5 k^2 \sk^4 \skp^2+\frac{1}{180 } \tau^6 k^6 -\frac{4}{45 } \tau^6 k^4 \sk^2 -\frac{4}{15 } \tau^6 k^2 \sk^4 -\frac{1}{45 } \tau^6 k^6 \sk^2 \skp^2 + \frac{16}{45 } \tau^6 k^4 \sk^4 \skp^2 + \frac{16}{15 } \tau^6 k^2 \sk^6 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{23}= \int dk\bigg[-\frac{1}{15 } \tau^3 -\frac{1}{10 } \tau^4 k^2 + \frac{2}{9 } \tau^4 k^4 \skp^2 + \frac{4}{15 } \tau^4 \sk^2 -\frac{1}{45 } \tau^5 k^4 -\frac{4}{9 } \tau^5 k^4 \sk^2 \skp^2 + \frac{4}{15 } \tau^5 k^2 \sk^2 + \frac{1}{180 } \tau^6 k^6 -\frac{2}{45 } \tau^6 k^4 \sk^2 \notag\\ & & \hspace{0.4cm}-\frac{1}{45 } \tau^6 k^6 \sk^2 \skp^2+\frac{8}{45 } \tau^6 k^4 \sk^4 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{24}= \int dk\bigg[4 \tau^2 - \tau^3 k^2 -16 \tau^3 \sk^2 + \frac{4}{3 } \tau^4 k^2 \sk^2 + \frac{16}{3 } \tau^4 \sk^4\bigg]\;,\hspace{3 cm } \mathcal{z}_{25}= \int dk\bigg[4 \tau^2 - \tau^3 k^2 -4 \tau^3 \sk^2\bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{26}= \int dk\bigg[2 \tau^2 -8 \tau^3 \sk^2 + \frac{2}{3 } \tau^4 k^2 \sk^2 + \frac{8}{3 } \tau^4 \sk^4 \bigg]\;,\hspace{1cm}\mathcal{z}_{27}= \int dk\bigg[2 \tau^2 \bigg]\;,\hspace{1cm}\mathcal{z}_{28}= \int dk\bigg[-4 \tau^2 + \tau^3 k^2 + 4 \tau^3 \sk^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{29}= \int dk\bigg[2 \tau^2 \bigg]\;,\hspace{2.5cm}\mathcal{z}_{30}= \int dk\bigg[-\frac{1}{3 } \tau^3 k^2 \skp \bigg]\;,\hspace{2.5cm}\mathcal{z}_{31}= \int dk\bigg[-\frac{1}{3 } \tau^3 k^2 \skp \bigg];,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{32}= \int dk\bigg[-6 \tau^3 \sk + \frac{2}{3 } \tau^3 k^2 \skp + \frac{4}{3 } \tau^4 k^2 \sk + \frac{8}{3 } \tau^4 \sk^3 -\frac{2}{9 } \tau^4 k^4 \skp\bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{33}= \int dk\bigg[4 \tau^3 \sk + \frac{4}{3 } \tau^3 k^2 \skp -\frac{4}{3 } \tau^4 k^2 \sk -\frac{8}{3 } \tau^4 \sk^3 -\frac{2}{9 } \tau^4 k^4 \skp \bigg]\;,\hspace{2cm}\mathcal{z}_{34}= \int dk\bigg[\frac{4}{3 } \tau^3 \sk + \frac{1}{3 } \tau^3 k^2 \skp \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{35}= \int dk\bigg[-2 \tau^3 \sk + \frac{1}{3 } \tau^3 k^2 \skp \bigg]\;,\hspace{2cm}\mathcal{z}_{36}= \int dk\bigg[-\frac{2}{3 } \tau^3 k^2 \skp \bigg]\;,\hspace{2cm}\mathcal{z}_{37}= \int dk\bigg[\frac{2}{3 } \tau^3 \sk \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{38}=\int dk\bigg[-\frac{1}{3 } \tau^3 k^2 \skp + \frac{1}{9 } \tau^4 k^4 \skp + \frac{2}{3 } \tau^4 k^4 \sk \skp^2 -\frac{2}{3 } \tau^4 k^2 \sk + \frac{1}{9 } \tau^5 k^4 \sk -\frac{4}{9 } \tau^5 k^4 \sk^3 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{39}= \int dk\bigg[6 \tau^3 \sk -10 \tau^3 k^2 \sk \skp^2 -2 \tau^4 k^2 \sk -\frac{8}{3 } \tau^4 \sk^3 + \frac{1}{9 } \tau^4 k^4 \skp + \frac{2}{3 } \tau^4 k^4 \sk \skp^2 + \frac{40}{3 } \tau^4 k^2 \sk^3 \skp^2 + \frac{1}{9 } \tau^5 k^4 \sk \notag\\ & & \hspace{0.4cm}+\frac{2}{3 } \tau^5 k^2 \sk^3 -\frac{4}{9 } \tau^5 k^4 \sk^3 \skp^2 -\frac{8}{3 } \tau^5 k^2 \sk^5 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{40}= \int dk\bigg[- \tau^3 k^2 \skp + \frac{4}{3 } \tau^3 \sk + \frac{1}{9 } \tau^4 k^4 \skp + \frac{2}{3 } \tau^4 k^4 \sk \skp^2 -\frac{2}{3 } \tau^4 k^2 \sk + \frac{1}{9 } \tau^5 k^4 \sk -\frac{4}{9 } \tau^5 k^4 \sk^3 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{41}= \int dk\bigg[10 \tau^3 \sk -\frac{8}{3 } \tau^4 k^2 \sk -\frac{40}{3 } \tau^4 \sk^3 + \frac{1}{9 } \tau^5 k^4 \sk + \frac{4}{3 } \tau^5 k^2 \sk^3 + \frac{8}{3 } \tau^5 \sk^5 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{42}= \int dk\bigg[-10 \tau^3 \sk + \frac{4}{3 } \tau^4 k^2 \sk + \frac{40}{3 } \tau^4 \sk^3 + \frac{1}{9 } \tau^5 k^4 \sk -\frac{4}{3 } \tau^5 k^2 \sk^3 -\frac{8}{3 } \tau^5 \sk^5 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{43}= \int dk\bigg[10 \tau^3 \sk -2 \tau^4 k^2 \sk -\frac{40}{3 } \tau^4 \sk^3 + \frac{1}{9 } \tau^5 k^4 \sk + \frac{4}{3 } \tau^5 k^2 \sk^3 + \frac{8}{3 } \tau^5 \sk^5 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{44}= \int dk\bigg[-\frac{1}{12 } \tau^3 -\frac{1}{6 } \tau^3 \sk \skp -\frac{4}{3 } \tau^3 k^2 \skp^2 + \frac{29}{120 } \tau^4 k^2 + \frac{1}{3 } \tau^4 k^2 \sk^2 \skp^2 -\frac{29}{540 } \tau^5 k^4 + \frac{19}{90 } \tau^5 k^4 \sk^2 \skp^2 -\frac{1}{135 } \tau^5 k^4 \sk^3 \skp^3 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{45}= \int dk\bigg[-\frac{1}{3 } \tau^2 \skp^2 + \frac{2}{3 } \tau^3 \sk \skp + \frac{5}{6 } \tau^3 k^2 \skp^2 -\frac{1}{3 } \tau^3 k^2 \sk \skp^3 -\frac{2}{5 } \tau^3 + \frac{4}{3 } \tau^3 \sk^2 \skp^2 + \frac{7}{45 } \tau^4 k^2 -\frac{8}{45 } \tau^4 \sk^2+\frac{4}{9 } \tau^4 k^2 \sk^3 \skp^3 \notag\\ & & \hspace{0.4 cm } -\frac{4}{9 } \tau^4 \sk^3 \skp-\frac{1}{3 } \tau^4 k^2 \sk^2 \skp^2 -\frac{4}{9 } \tau^4 \sk^4 \skp^2 + \frac{1}{270 } \tau^5 k^4 + \frac{1}{45 } \tau^5 k^2 \sk^2 -\frac{1}{45 } \tau^5 k^4 \sk^2 \skp^2-\frac{2}{135 } \tau^5 k^4 \sk^3 \skp^3 -\frac{2}{15 } \tau^5 k^2 \sk^4 \skp^2\notag\\ & & \hspace{0.4 cm } -\frac{4}{45 } \tau^5 k^2 \sk^5 \skp^3 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{46}= \int dk\bigg[\frac{7}{36 } \tau^3 + \frac{5}{6 } \tau^3 \sk \skp -\frac{5}{18 } \tau^4 k^2 -\frac{2}{9 } \tau^4 k^2 \sk^2 \skp^2 + \frac{1}{9 } \tau^4 k^4 \skp^2 + \frac{2}{9 } \tau^4 k^4 \sk \skp^3 + \frac{1}{18 } \tau^5 k^4 -\frac{2}{9 } \tau^5 k^4 \sk^2 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{47}= \int dk\bigg[\frac{11}{12 } \tau^3 -\frac{1}{18 } \tau^3 \sk \skp + \frac{2}{9 } \tau^3 k^2 \skp^2 -\frac{113}{360 } \tau^4 k^2 -\frac{1}{9 } \tau^4 k^2 \sk^2 \skp^2 + \frac{31}{540 } \tau^5 k^4 -\frac{7}{30 } \tau^5 k^4 \sk^2 \skp^2 -\frac{1}{135 } \tau^5 k^4 \sk^3 \skp^3 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{48}= \int dk\bigg[\frac{35}{18 } \tau^3 -\frac{1}{9 } \tau^3 \sk \skp -\frac{14}{9 } \tau^3 k^2 \skp^2 -\frac{211}{180 } \tau^4 k^2 -\frac{4}{3 } \tau^4 \sk^2 + \frac{85}{18 } \tau^4 k^2 \sk^2 \skp^2 + \frac{1}{9 } \tau^4 k^4 \skp^2 + \frac{2}{9 } \tau^4 k^4 \sk \skp^3 \notag\\ & & \hspace{0.4cm}+\frac{59}{540 } \tau^5 k^4 + \frac{1}{3 } \tau^5 k^2 \sk^2-\frac{13}{30 } \tau^5 k^4 \sk^2 \skp^2 + \frac{1}{135 } \tau^5 k^4 \sk^3 \skp^3 -\frac{4}{3 } \tau^5 k^2 \sk^4 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{49}= \int dk\bigg[\frac{1}{36 } \tau^3 -\frac{1}{18 } \tau^3 \sk \skp + \frac{5}{9 } \tau^3 k^2 \skp^2 -\frac{3}{20 } \tau^4 k^2 -\frac{2}{3 } \tau^4 k^2 \sk^2 \skp^2 + \frac{1}{9 } \tau^4 k^4 \skp^2 + \frac{2}{9 } \tau^4 k^4 \sk \skp^3 + \frac{7}{135 } \tau^5 k^4\notag\\ & & \hspace{0.4cm}-\frac{1}{5 } \tau^5 k^4 \sk^2 \skp^2 + \frac{2}{135 } \tau^5 k^4 \sk^3 \skp^3 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{50}= \int dk\bigg[-\frac{2}{45 } \tau \sk \skppp -\frac{2}{15 } \tau \skp \skpp + \frac{2}{15 } \tau^2 \sk \skpp + \frac{7}{15 } \tau^2 \skp^2 + \frac{4}{15 } \tau^2 \sk^2 \skp \skpp + \frac{4}{15 } \tau^2 \sk \skp^3 + \frac{1}{18 } \tau^2 k^4 \skpp^2 -\frac{1}{45 } \tau^3 \notag\\ & & \hspace{0.4 cm } -\frac{2}{15 } \tau^3 \sk \skp + \frac{1}{3 } \tau^3 k^2 \skp^2-\frac{4}{15 } \tau^3 \sk^2 \skp^2 + \frac{4}{9 } \tau^3 k^2 \sk \skp^3 -\frac{8}{45 } \tau^3 \sk^3 \skp^3 + \frac{4}{27 } \tau^3 k^4 \skp^4 -\frac{1}{18 } \tau^4 k^4 \skp^2 -\frac{4}{27 } \tau^4 k^4 \sk \skp^3\notag\\ & & \hspace{0.4cm}-\frac{2}{27 } \tau^4 k^4 \sk^2 \skp^4 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{51}= \int dk\bigg[\frac{1}{135 } \tau \sk \skppp + \frac{1}{45 } \tau \skp \skpp -\frac{1}{45 } \tau^2 \sk \skpp -\frac{1}{45 } \tau^2 \skp^2 -\frac{2}{45 } \tau^2 \sk^2 \skp \skpp -\frac{2}{45 } \tau^2 \sk \skp^3 -\frac{1}{18 } \tau^2 k^4 \skpp^2 -\frac{22}{135 } \tau^3 \notag\\ & & \hspace{0.4 cm } + \frac{1}{45 } \tau^3 \sk \skp -\frac{1}{9 } \tau^3 k^2 \skp^2+\frac{2}{45 } \tau^3 \sk^2 \skp^2 + \frac{4}{135 } \tau^3 \sk^3 \skp^3 -\frac{4}{27 } \tau^3 k^4 \skp^4 + \frac{1}{24 } \tau^4 k^2 + \frac{1}{18 } \tau^4 k^4 \skp^2 -\frac{1}{6 } \tau^4 k^2 \sk^2 \skp^2\notag\\ & & \hspace{0.4cm}+\frac{4}{27 } \tau^4 k^4 \sk \skp^3 + \frac{2}{27 } \tau^4 k^4 \sk^2 \skp^4 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{52}= \int dk\bigg[\frac{1}{135 } \tau \sk \skppp + \frac{1}{45 } \tau \skp \skpp -\frac{1}{45 } \tau^2 \sk \skpp -\frac{1}{45 } \tau^2 \skp^2 -\frac{2}{45 } \tau^2 \sk^2 \skp \skpp -\frac{2}{45 } \tau^2 \sk \skp^3 -\frac{1}{18 } \tau^2 k^4 \skpp^2 + \frac{1}{270 } \tau^3 \notag\\ & & \hspace{0.4cm}+\frac{1}{45 } \tau^3 \sk \skp -\frac{1}{9 } \tau^3 k^2 \skp^2+\frac{2}{45 } \tau^3 \sk^2 \skp^2 + \frac{4}{135 } \tau^3 \sk^3 \skp^3 -\frac{4}{27 } \tau^3 k^4 \skp^4 + \frac{1}{18 } \tau^4 k^4 \skp^2 + \frac{4}{27 } \tau^4 k^4 \sk \skp^3 + \frac{2}{27 } \tau^4 k^4 \sk^2 \skp^4 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{53}= \int dk\bigg[-\frac{1}{18 } \tau^3 + \frac{17}{180 } \tau^4 k^2 -\frac{1}{18 } \tau^4 k^2 \sk^2 \skp^2 -\frac{4}{81 } \tau^4 k^4 \sk \skp^3 + \frac{1}{162 } \tau^4 k^6 \skp^4 + \frac{1}{54 } \tau^4 k^6 \sk^2 \skpp^2 -\frac{1}{36 } \tau^5 k^4 \notag\\ & & \hspace{0.4cm}+\frac{29}{270 } \tau^5 k^4 \sk^2 \skp^2 + \frac{29}{405 } \tau^5 k^4 \sk^3 \skp^3 + \frac{1}{81 } \tau^5 k^6 \sk \skp^3+\frac{2}{81 } \tau^5 k^6 \sk^2 \skp^4 + \frac{1}{540 } \tau^6 k^6 -\frac{2}{135 } \tau^6 k^6 \sk^2 \skp^2-\frac{8}{405 } \tau^6 k^6 \sk^3 \skp^3\notag\\ & & \hspace{0.4 cm } -\frac{4}{405 } \tau^6 k^6 \sk^4 \skp^4 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{54}= \int dk\bigg[-\frac{1}{6 } \tau^3 + \frac{5}{36 } \tau^4 k^2 -\frac{1}{6 } \tau^4 k^2 \sk^2 \skp^2 -\frac{4}{81 } \tau^4 k^4 \sk \skp^3 + \frac{1}{162 } \tau^4 k^6 \skp^4 + \frac{1}{54 } \tau^4 k^6 \sk^2 \skpp^2 -\frac{17}{540 } \tau^5 k^4 \notag\\ & & \hspace{0.4 cm } + \frac{7}{54 } \tau^5 k^4 \sk^2 \skp^2 + \frac{7}{81 } \tau^5 k^4 \sk^3 \skp^3 + \frac{1}{81 } \tau^5 k^6 \sk \skp^3+\frac{2}{81 } \tau^5 k^6 \sk^2 \skp^4 + \frac{1}{540 } \tau^6 k^6 -\frac{2}{135 } \tau^6 k^6 \sk^2 \skp^2 -\frac{8}{405 } \tau^6 k^6 \sk^3 \skp^3\notag\\ & & \hspace{0.4cm}-\frac{4}{405 } \tau^6 k^6 \sk^4 \skp^4 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{55}= \int dk\bigg[\frac{1}{18 } \tau^2 k^4 \skpp^2 -\frac{1}{5 } \tau^3 + \frac{1}{2 } \tau^3 k^2 \skp^2 + \frac{11}{9 } \tau^3 k^2 \sk \skp^3 -\frac{2}{9 } \tau^3 k^4 \sk^2 \skpp^2 + \frac{2}{27 } \tau^3 k^4 \skp^4 + \frac{37}{180 } \tau^4 k^2 + \frac{11}{45 } \tau^4 \sk^2 \notag\\ & & \hspace{0.4 cm } -\frac{1}{18 } \tau^4 k^4 \skp^2 -\frac{14}{9 } \tau^4 k^2 \sk^2 \skp^2-\frac{26}{81 } \tau^4 k^4 \sk \skp^3 -\frac{44}{27 } \tau^4 k^2 \sk^3 \skp^3 + \frac{1}{108 } \tau^4 k^6 \sk^2 \skpp^2 + \frac{1}{324 } \tau^4 k^6 \skp^4 + \frac{2}{27 } \tau^4 k^4 \sk^4 \skpp^2 \notag\\ & & \hspace{0.4cm}-\frac{4}{9 } \tau^4 k^4 \sk^2 \skp^4 -\frac{1}{30 } \tau^5 k^4 -\frac{1}{9 } \tau^5 k^2 \sk^2 + \frac{41}{135 } \tau^5 k^4 \sk^2 \skp^2 + \frac{22}{45 } \tau^5 k^2 \sk^4 \skp^2+\frac{1}{162 } \tau^5 k^6 \sk \skp^3 + \frac{182}{405 } \tau^5 k^4 \sk^3 \skp^3 \notag\\ & & \hspace{0.4 cm } + \frac{44}{135 } \tau^5 k^2 \sk^5 \skp^3 + \frac{1}{81 } \tau^5 k^6 \sk^2 \skp^4 + \frac{8}{27 } \tau^5 k^4 \sk^4 \skp^4 + \frac{1}{1080 } \tau^6 k^6 + \frac{1}{135 } \tau^6 k^4 \sk^2 -\frac{1}{135 } \tau^6 k^6 \sk^2 \skp^2 -\frac{8}{135 } \tau^6 k^4 \sk^4 \skp^2 \skp^3\notag\\ & & \hspace{0.4cm}-\frac{4}{405 } \tau^6 k^6 \sk^3 \skp^3-\frac{32}{405 } \tau^6 k^4 \sk^5-\frac{2}{405 } \tau^6 k^6 \sk^4 \skp^4 -\frac{16}{405 } \tau^6 k^4 \sk^6 \skp^4 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{56}= \int dk\bigg[-\frac{1}{18 } \tau^3 + \frac{5}{72 } \tau^4 k^2 -\frac{1}{18 } \tau^4 k^2 \sk^2 \skp^2 -\frac{4}{81 } \tau^4 k^4 \sk \skp^3 + \frac{1}{162 } \tau^4 k^6 \skp^4 + \frac{1}{54 } \tau^4 k^6 \sk^2 \skpp^2 -\frac{1}{45 } \tau^5 k^4 \notag\\ & & \hspace{0.4cm}+\frac{2}{27 } \tau^5 k^4 \sk^2 \skp^2 + \frac{4}{81 } \tau^5 k^4 \sk^3 \skp^3 + \frac{1}{81 } \tau^5 k^6 \sk \skp^3+\frac{2}{81 } \tau^5 k^6 \sk^2 \skp^4 + \frac{1}{540 } \tau^6 k^6 -\frac{2}{135 } \tau^6 k^6 \sk^2 \skp^2 -\frac{8}{405 } \tau^6 k^6 \sk^3 \skp^3\notag\\ & & \hspace{0.4cm}-\frac{4}{405 } \tau^6 k^6 \sk^4 \skp^4 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{57}= \int dk\bigg[-\frac{1}{36 } \tau^3 + \frac{5}{144 } \tau^4 k^2 -\frac{1}{36 } \tau^4 k^2 \sk^2 \skp^2 -\frac{2}{81 } \tau^4 k^4 \sk \skp^3 + \frac{1}{324 } \tau^4 k^6 \skp^4 + \frac{1}{108 } \tau^4 k^6 \sk^2 \skpp^2 -\frac{1}{90 } \tau^5 k^4 \notag\\ & & \hspace{0.4cm}+\frac{1}{27 } \tau^5 k^4 \sk^2 \skp^2 + \frac{2}{81 } \tau^5 k^4 \sk^3 \skp^3 + \frac{1}{162 } \tau^5 k^6 \sk \skp^3+\frac{1}{81 } \tau^5 k^6 \sk^2 \skp^4 + \frac{1}{1080 } \tau^6 k^6 -\frac{1}{135 } \tau^6 k^6 \sk^2 \skp^2 -\frac{4}{405 } \tau^6 k^6 \sk^3 \skp^3\notag\\ & & \hspace{0.4cm}-\frac{2}{405 } \tau^6 k^6 \sk^4 \skp^4 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{58}= \int dk\bigg[-\frac{1}{36 } \tau^3 + \frac{5}{144 } \tau^4 k^2 -\frac{1}{36 } \tau^4 k^2 \sk^2 \skp^2 -\frac{2}{81 } \tau^4 k^4 \sk \skp^3 + \frac{1}{324 } \tau^4 k^6 \skp^4 + \frac{1}{108 } \tau^4 k^6 \sk^2 \skpp^2 -\frac{1}{90 } \tau^5 k^4 \notag\\ & & \hspace{0.4 cm } + \frac{1}{27 } \tau^5 k^4 \sk^2 \skp^2 + \frac{2}{81 } \tau^5 k^4 \sk^3 \skp^3 + \frac{1}{162 } \tau^5 k^6 \sk \skp^3+\frac{1}{81 } \tau^5 k^6 \sk^2 \skp^4 + \frac{1}{1080 } \tau^6 k^6 -\frac{1}{135 } \tau^6 k^6 \sk^2 \skp^2 -\frac{4}{405 } \tau^6 k^6 \sk^3 \skp^3\notag\\ & & \hspace{0.4cm}-\frac{2}{405 } \tau^6 k^6 \sk^4 \skp^4 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{59}= \int dk\bigg[\frac{1}{18 } \tau^2 k^4 \skpp^2 -\frac{2}{5 } \tau^3 + \frac{2}{9 } \tau^3 k^2 \sk \skp^3 + \frac{2}{27 } \tau^3 k^4 \skp^4 -\frac{2}{9 } \tau^3 k^4 \sk^2 \skpp^2 + \frac{31}{180 } \tau^4 k^2 -\frac{5}{9 } \tau^4 k^2 \sk^2 \skp^2 -\frac{8}{27 } \tau^4 k^2 \sk^3 \skp^3\notag\\ & & \hspace{0.4 cm } -\frac{1}{18 } \tau^4 k^4 \skp^2 -\frac{26}{81 } \tau^4 k^4 \sk \skp^3-\frac{4}{9 } \tau^4 k^4 \sk^2 \skp^4 + \frac{2}{45 } \tau^4 \sk^2 + \frac{1}{324 } \tau^4 k^6 \skp^4 + \frac{1}{108 } \tau^4 k^6 \sk^2 \skpp^2 + \frac{2}{27 } \tau^4 k^4 \sk^4 \skpp^2\notag\\ & & \hspace{0.4 cm } -\frac{1}{45 } \tau^5 k^4 + \frac{32}{135 } \tau^5 k^4 \sk^2 \skp^2 + \frac{164}{405 } \tau^5 k^4 \sk^3 \skp^3 + \frac{8}{27 } \tau^5 k^4 \sk^4 \skp^4 -\frac{2}{45 } \tau^5 k^2 \sk^2+\frac{4}{45 } \tau^5 k^2 \sk^4 \skp^2 + \frac{8}{135 } \tau^5 k^2 \sk^5 \skp^3\notag\\ & & \hspace{0.4 cm } + \frac{1}{162 } \tau^5 k^6 \sk \skp^3 + \frac{1}{81 } \tau^5 k^6 \sk^2 \skp^4 + \frac{1}{1080 } \tau^6 k^6 -\frac{1}{135 } \tau^6 k^6 \sk^2 \skp^2 -\frac{4}{405 } \tau^6 k^6 \sk^3 \skp^3-\frac{2}{405 } \tau^6 k^6 \sk^4 \skp^4 + \frac{1}{135 } \tau^6 k^4 \sk^2\notag\\ & & \hspace{0.4 cm } -\frac{8}{135 } \tau^6 k^4 \sk^4 \skp^2-\frac{32}{405 } \tau^6 k^4 \sk^5 \skp^3 -\frac{16}{405 } \tau^6 k^4 \sk^6 \skp^4 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{60}= \int dk\bigg[\frac{5}{18 } \tau^3 -\frac{1}{20 } \tau^4 k^2 + \frac{5}{18 } \tau^4 k^2 \sk^2 \skp^2 -\frac{2}{81 } \tau^4 k^4 \sk \skp^3 + \frac{1}{324 } \tau^4 k^6 \skp^4 + \frac{1}{108 } \tau^4 k^6 \sk^2 \skpp^2 -\frac{1}{108 } \tau^5 k^4 \notag\\ & & \hspace{0.4 cm } + \frac{7}{270 } \tau^5 k^4 \sk^2 \skp^2 + \frac{7}{405 } \tau^5 k^4 \sk^3 \skp^3 + \frac{1}{162 } \tau^5 k^6 \sk \skp^3+\frac{1}{81 } \tau^5 k^6 \sk^2 \skp^4 + \frac{1}{1080 } \tau^6 k^6 -\frac{1}{135 } \tau^6 k^6 \sk^2 \skp^2\notag\\ & & \hspace{0.4cm}-\frac{4}{405 } \tau^6 k^6 \sk^3 \skp^3 -\frac{2}{405 } \tau^6 k^6 \sk^4 \skp^4 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{61}= \int dk\bigg[\frac{1}{18 } \tau^2 k^4 \skpp^2 + \frac{2}{9 } \tau^3 k^2 \sk \skp^3 -\frac{1}{15 } \tau^3 + \frac{2}{27 } \tau^3 k^4 \skp^4 -\frac{2}{9 } \tau^3 k^4 \sk^2 \skpp^2 + \frac{4}{45 } \tau^4 k^2 -\frac{2}{9 } \tau^4 k^2 \sk^2 \skp^2 \notag\\ & & \hspace{0.4 cm } -\frac{8}{27 } \tau^4 k^2 \sk^3 \skp^3 -\frac{1}{18 } \tau^4 k^4 \skp^2 -\frac{26}{81 } \tau^4 k^4 \sk \skp^3-\frac{4}{9 } \tau^4 k^4 \sk^2 \skp^4 + \frac{2}{45 } \tau^4 \sk^2 + \frac{1}{324 } \tau^4 k^6 \skp^4 + \frac{1}{108 } \tau^4 k^6 \sk^2 \skpp^2 \notag\\ & & \hspace{0.4 cm } + \frac{2}{27 } \tau^4 k^4 \sk^4 \skpp^2 -\frac{1}{45 } \tau^5 k^4 + \frac{32}{135 } \tau^5 k^4 \sk^2 \skp^2 + \frac{164}{405 } \tau^5 k^4 \sk^3 \skp^3 + \frac{8}{27 } \tau^5 k^4 \sk^4 \skp^4 -\frac{2}{45 } \tau^5 k^2 \sk^2+\frac{4}{45 } \tau^5 k^2 \sk^4 \skp^2 \notag\\ & & \hspace{0.4 cm } + \frac{8}{135 } \tau^5 k^2 \sk^5 \skp^3 + \frac{1}{162 } \tau^5 k^6 \sk \skp^3 + \frac{1}{81 } \tau^5 k^6 \sk^2 \skp^4 + \frac{1}{1080 } \tau^6 k^6 -\frac{1}{135 } \tau^6 k^6 \sk^2 \skp^2 -\frac{4}{405 } \tau^6 k^6 \sk^3 \skp^3\notag\\ & & \hspace{0.4cm}-\frac{2}{405 } \tau^6 k^6 \sk^4 \skp^4 + \frac{1}{135 } \tau^6 k^4 \sk^2 -\frac{8}{135 } \tau^6 k^4 \sk^4 \skp^2-\frac{32}{405 } \tau^6 k^4 \sk^5 \skp^3 -\frac{16}{405 } \tau^6 k^4 \sk^6 \skp^4 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{62}= \int dk\bigg[\frac{1}{45 } \tau^4 k^2 -\frac{4}{81 } \tau^4 k^4 \sk \skp^3 + \frac{1}{162 } \tau^4 k^6 \skp^4 + \frac{1}{54 } \tau^4 k^6 \sk^2 \skpp^2 -\frac{2}{135 } \tau^5 k^4 + \frac{4}{135 } \tau^5 k^4 \sk^2 \skp^2 \notag\\ & & \hspace{0.4cm}+\frac{8}{405 } \tau^5 k^4 \sk^3 \skp^3 + \frac{1}{81 } \tau^5 k^6 \sk \skp^3 + \frac{2}{81 } \tau^5 k^6 \sk^2 \skp^4 + \frac{1}{540 } \tau^6 k^6-\frac{2}{135 } \tau^6 k^6 \sk^2 \skp^2 -\frac{8}{405 } \tau^6 k^6 \sk^3 \skp^3 -\frac{4}{405 } \tau^6 k^6 \sk^4 \skp^4 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{63}= \int dk\bigg[\frac{1}{18 } \tau^3 -\frac{1}{90 } \tau^4 k^2 + \frac{1}{18 } \tau^4 k^2 \sk^2 \skp^2 -\frac{2}{81 } \tau^4 k^4 \sk \skp^3 + \frac{1}{324 } \tau^4 k^6 \skp^4 + \frac{1}{108 } \tau^4 k^6 \sk^2 \skpp^2 -\frac{1}{180 } \tau^5 k^4 \notag\\ & & \hspace{0.4 cm } + \frac{1}{270 } \tau^5 k^4 \sk^2 \skp^2 + \frac{1}{405 } \tau^5 k^4 \sk^3 \skp^3 + \frac{1}{162 } \tau^5 k^6 \sk \skp^3+\frac{1}{81 } \tau^5 k^6 \sk^2 \skp^4 + \frac{1}{1080 } \tau^6 k^6 -\frac{1}{135 } \tau^6 k^6 \sk^2 \skp^2\notag\\ & & \hspace{0.4cm}-\frac{4}{405 } \tau^6 k^6 \sk^3 \skp^3 -\frac{2}{405 } \tau^6 k^6 \sk^4 \skp^4 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{64}= \int dk\bigg[-\frac{1}{9 } \tau^3 k^2 \skp + \frac{4}{9 } \tau^3 k^2 \sk \skp^2 -\frac{2}{9 } \tau^3 k^4 \skp^3 + \frac{1}{9 } \tau^4 k^4 \skp + \frac{2}{9 } \tau^4 k^4 \sk \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{65}= \int dk\bigg[\frac{4}{3 } \tau^3 \sk -\frac{7}{9 } \tau^3 k^2 \skp -\frac{20}{9 } \tau^3 k^2 \sk \skp^2 -\frac{2}{9 } \tau^3 k^4 \skp^3 -\frac{1}{3 } \tau^4 k^2 \sk + \frac{1}{9 } \tau^4 k^4 \skp + \frac{2}{9 } \tau^4 k^4 \sk \skp^2 + \frac{4}{3 } \tau^4 k^2 \sk^3 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{66}= \int dk\bigg[-\frac{4}{9 } \tau^3 k^2 \skp -\frac{2}{9 } \tau^3 k^2 \sk \skp^2 -\frac{2}{9 } \tau^3 k^4 \skp^3 + \frac{1}{9 } \tau^4 k^4 \skp + \frac{2}{9 } \tau^4 k^4 \sk \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{67}= \int dk\bigg[\frac{5}{3 } \tau^2 - \tau^2 k^2 \skp^2 -\frac{4}{3 } \tau^3 \sk^2 -\frac{1}{3 } \tau^3 k^2 + 4 \tau^3 k^2 \sk^2 \skp^2 + \frac{1}{3 } \tau^4 k^2 \sk^2 -\frac{4}{3 } \tau^4 k^2 \sk^4 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{68}= \int dk\bigg [ \tau^2 + \tau^2 k^2 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{69}= \int dk\bigg[\frac{1}{3 } \tau^3 + \frac{2}{3 } \tau^3 k^2 \skp^2 -\frac{2}{3 } \tau^3 k^2 \sk \skp^3 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{70}= \int dk\bigg[- \tau^2 \sk \skpp - \tau^2 \skp^2 + \frac{5}{3 } \tau^3 \sk \skp -\frac{1}{2 } \tau^3 + 4 \tau^3 \sk^2 \skp^2 - \tau^3 k^2 \skp^2 + \frac{2}{3 } \tau^3 \sk^3 \skpp + \frac{1}{6 } \tau^4 k^2 + \frac{1}{3 } \tau^4 \sk^2 \notag\\ & & \hspace{0.4 cm } + \frac{4}{3 } \tau^4 k^2 \sk^2 \skp^2-\frac{4}{3 } \tau^4 \sk^3 \skp -\frac{4}{3 } \tau^4 \sk^4 \skp^2 + \frac{2}{9 } \tau^4 k^4 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{71}= \int dk\bigg[\frac{13}{6 } \tau^3 + \frac{1}{3 } \tau^3 \sk \skp + \frac{1}{3 } \tau^3 k^2 \skp^2 -\frac{7}{12 } \tau^4 k^2 -\frac{4}{3 } \tau^4 \sk^2 -\frac{5}{3 } \tau^4 k^2 \sk^2 \skp^2 + \frac{2}{9 } \tau^4 k^4 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{72}= \int dk\bigg[-2 \tau^3 + \tau^3 \sk \skp -3 \tau^3 k^2 \skp^2 + \frac{13}{24 } \tau^4 k^2 + \frac{4}{3 } \tau^4 \sk^2 + \frac{11}{6 } \tau^4 k^2 \sk^2 \skp^2 + \frac{2}{9 } \tau^4 k^4 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{73}= \int dk\bigg[-\frac{11}{12 } \tau^3 + \frac{2}{3 } \tau^3 k^2 \skp^2 + \frac{23}{48 } \tau^4 k^2 + \frac{2}{3 } \tau^4 \sk^2 + \frac{1}{12 } \tau^4 k^2 \sk^2 \skp^2 + \frac{1}{9 } \tau^4 k^4 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{74}= \int dk\bigg[\frac{1}{12 } \tau^4 k^2 + \frac{1}{3 } \tau^4 \sk^2 -\frac{1}{9 } \tau^4 k^4 \skp^2 + \frac{2}{3 } \tau^4 k^2 \sk^2 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{75}= \int dk\bigg[-\frac{1}{3 } \tau \skpp + \frac{2}{3 } \tau^2 \skp + 2 \tau^2 \sk \skp^2 + \frac{2}{3 } \tau^2 \sk^2 \skpp + 2 \tau^3 k^2 \sk \skp^2 -\frac{4}{3 } \tau^3 \sk^2 \skp + \frac{2}{3 } \tau^3 \sk -\frac{4}{3 } \tau^3 \sk^3 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{76}= \int dk\bigg[-\frac{2}{3 } \tau^2 \skp + 2 \tau^3 \sk -\frac{8}{9 } \tau^3 k^2 \skp + \frac{20}{9 } \tau^3 k^2 \sk \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{77}= \int dk\bigg[-\frac{2}{3 } \tau^2 \skp + \frac{1}{9 } \tau^3 k^2 \skp + \frac{2}{9 } \tau^3 k^2 \sk \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{78}= \int dk\bigg[-\frac{2}{3 } \tau^3 -\frac{1}{6 } \tau^3 k^2 \skp^2 + \frac{43}{120 } \tau^4 k^2 -\frac{1}{18 } \tau^4 k^4 \skp^2 -\frac{1}{9 } \tau^4 k^4 \sk \skp^3 -\frac{11}{180 } \tau^5 k^4 + \frac{23}{90 } \tau^5 k^4 \sk^2 \skp^2 + \frac{1}{45 } \tau^5 k^4 \sk^3 \skp^3 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{79}= \int dk\bigg[\frac{31}{15 } \tau^3 - \tau^3 k^2 \skp^2 + 2 \tau^3 k^2 \sk \skp^3 -\frac{13}{30 } \tau^4 k^2 + \frac{2}{5 } \tau^4 \sk^2 -\frac{8}{3 } \tau^4 k^2 \sk^3 \skp^3 -\frac{1}{45 } \tau^5 k^4 -\frac{2}{15 } \tau^5 k^2 \sk^2 \notag\\ & & \hspace{0.4 cm } + \frac{2}{15 } \tau^5 k^4 \sk^2 \skp^2 + \frac{4}{5 } \tau^5 k^2 \sk^4 \skp^2+\frac{4}{45 } \tau^5 k^4 \sk^3 \skp^3 + \frac{8}{15 } \tau^5 k^2 \sk^5 \skp^3 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{80}= \int dk\bigg[-\frac{4}{3 } \tau^3 + \frac{5}{6 } \tau^3 k^2 \skp^2 + \frac{3}{40 } \tau^4 k^2 -\frac{4}{3 } \tau^4 k^2 \sk^2 \skp^2 + \frac{1}{18 } \tau^4 k^4 \skp^2 + \frac{1}{9 } \tau^4 k^4 \sk \skp^3 + \frac{7}{180 } \tau^5 k^4 -\frac{11}{90 } \tau^5 k^4 \sk^2 \skp^2\notag\\ & & \hspace{0.4cm}+\frac{1}{15 } \tau^5 k^4 \sk^3 \skp^3 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{81}= \int dk\bigg[\frac{8}{3 } \tau^3 -2 \tau^3 k^2 \skp^2 -\frac{6}{5 } \tau^4 k^2 -\frac{4}{3 } \tau^4 \sk^2 + 5 \tau^4 k^2 \sk^2 \skp^2 + \frac{1}{9 } \tau^4 k^4 \skp^2 + \frac{2}{9 } \tau^4 k^4 \sk \skp^3 + \frac{1}{10 } \tau^5 k^4+\frac{1}{3 } \tau^5 k^2 \sk^2 \notag\\ & & \hspace{0.4 cm } -\frac{17}{45 } \tau^5 k^4 \sk^2 \skp^2+\frac{2i}{45 } \tau^5 k^4 \sk^3 \skp^3 -\frac{4}{3 } \tau^5 k^2 \sk^4 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{82}= \int dk\bigg[-6 \tau^3 \sk -\frac{2}{3 } \tau^3 k^2 \skp + \frac{4}{3 } \tau^4 k^2 \sk + \frac{8}{3 } \tau^4 \sk^3 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{83}= \int dk\bigg[-\frac{20}{3 } \tau^3 \sk + \frac{4}{3 } \tau^3 k^2 \skp + \frac{4}{3 } \tau^4 k^2 \sk + \frac{8}{3 } \tau^4 \sk^3 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{84}= \int dk\bigg[\frac{17}{15 } \tau^3 -\frac{47}{60 } \tau^4 k^2 -\frac{73}{15 } \tau^4 \sk^2 + \frac{1}{9 } \tau^5 k^4 + \tau^5 k^2 \sk^2 + \frac{4}{3 } \tau^5 \sk^4 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{85}= \int dk\bigg[-\frac{8}{5 } \tau^3 + \frac{11}{30 } \tau^4 k^2 + \frac{76}{15 } \tau^4 \sk^2 - \tau^5 k^2 \sk^2 -\frac{4}{3 } \tau^5 \sk^4 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{86}= \int dk\bigg[\frac{46}{15 } \tau^3 -\frac{43}{30 } \tau^4 k^2 -\frac{28}{5 } \tau^4 \sk^2 + \frac{1}{9 } \tau^5 k^4 + \tau^5 k^2 \sk^2 + \frac{4}{3 } \tau^5 \sk^4 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{87}= \int dk\bigg[\frac{4}{5 } \tau^3 + \frac{2}{15 } \tau^4 k^2 -\frac{26}{5 } \tau^4 \sk^2 -\frac{1}{9 } \tau^5 k^4 + \tau^5 k^2 \sk^2 + \frac{4}{3 } \tau^5 \sk^4 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{88}= \int dk\bigg[-4 \tau^2 \sk + \frac{8}{3 } \tau^3 \sk^3 \bigg]\;,\hspace{1.9cm}\mathcal{z}_{89}= \int dk\bigg[-4 \tau^2 \sk \bigg]\;,\hspace{1.9cm}\mathcal{z}_{90}=0\hspace{1.9cm}\mathcal{z}_{91}=0\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{92}= \int dk\bigg[6 \tau^3 \sk -10 \tau^3 k^2 \sk \skp^2 -2 \tau^4 k^2 \sk -\frac{8}{3 } \tau^4 \sk^3 + \frac{4}{9 } \tau^4 k^4 \sk \skp^2 + \frac{40}{3 } \tau^4 k^2 \sk^3 \skp^2 + \frac{1}{9 } \tau^5 k^4 \sk+\frac{2}{3 } \tau^5 k^2 \sk^3\notag\\ & & \hspace{0.4 cm } -\frac{4}{9 } \tau^5 k^4 \sk^3 \skp^2 -\frac{8}{3 } \tau^5 k^2 \sk^5 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{93}= \int dk\bigg[-\frac{2}{3 } \tau^3 \sk + \frac{4}{9 } \tau^4 k^4 \sk \skp^2 -\frac{2}{3 } \tau^4 k^2 \sk + \frac{1}{9 } \tau^5 k^4 \sk -\frac{4}{9 } \tau^5 k^4 \sk^3 \skp^2 \bigg]\;,\notag\\ & & \hspace{-0.5cm}\mathcal{z}_{94}= \int dk\bigg[\frac{2}{3 } \tau^3 \sk + \frac{4}{9 } \tau^4 k^4 \sk \skp^2 -\frac{2}{3 } \tau^4 k^2 \sk + \frac{1}{9 } \tau^5 k^4 \sk -\frac{4}{9 } \tau^5 k^4 \sk^3 \skp^2 \bigg]\;,\label{zsigma } \end{aligned}\ ] ] where @xmath374 as we mentioned before , since the momentum integrations in ( [ zsigma ] ) are convergent due to suppression factor @xmath375 , the integrand is only accurate up to some total derivatives , but if we insist on taking the limit of @xmath151 , the dropping out momentum space total derivatives becomes problematic . we choose these momentum space total derivatives to reduce the high order self energy derivatives as much as possible . for example , following relation with @xmath376 being zero or positive integers can be used to reduce the term with a @xmath377 to the terms with lower order self energy derivatives , @xmath378 similarly , we have a series relations to reduce the term with a @xmath379 , or a @xmath380 , a @xmath381 , a @xmath382 to the terms with lower order self energy derivatives , @xmath383 with the help of above relations , the highest self energy derivatives appear in the final result ( [ zsigma ] ) is @xmath380 . @xmath384 to help understanding mutual relation between the definition of symbols in our formulation and those in ref.@xcite , in table xiii , we give a comparison . @xmath385 in obtaining ( [ kzrelation ] ) , some dependent terms must be reduced into independent terms , in table xv , the first column is the dependent operators and the second column is the sum of its corresponding independent operators . @xmath386
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we present results of computing the @xmath0 order low energy constants in the normal part of chiral lagrangian both for two and three flavor pseudo - scalar mesons .
this is a generalization of our previous work on calculating the @xmath1 order coefficients of the chiral lagrangian in terms of the quark self energy @xmath2 approximately from qcd .
we show that most of our results are consistent with those we can find in the literature .
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this paper proposes a new perspective to maximum likelihood(ml ) decoding in error - correcting codes as rational maps and shows some relationships between coding theory and dynamical systems . in section [ sec : cs ] , [ sec : lc ] , and [ sec : ml ] , we explain notations and minimum prerequisites of coding theory ( e.g. , see @xcite ) . the main results are presented in section [ sec : result ] . a mathematical model of communication systems in information theory was developed by shannon @xcite . a general block diagram for visualizing the behavior of such systems is given by figure [ fig : channel ] . the source transmits a @xmath1-bit message @xmath2 to the destination via the channel , which is usually affected by noise @xmath3 . in order to recover the transmitted message at the destination under the influence of noise , we transform the message into a codeword @xmath4 by some injective mapping @xmath5 at the encoder and input it to the channel . then the decoder transforms an @xmath6-bit received sequence of letters @xmath7 by some decoding mapping @xmath8 in order to obtain the transmitted codeword at the destination . here we consider all arithmetic calculations in some finite field and in this paper we fix it as @xmath9 except for section [ sec : ag ] . as a model of channels , we deal with a binary symmetric channel ( bsc ) in this paper which is characterized by the transition probability @xmath10 ( @xmath11 ) . namely , with probability @xmath12 , the output letter is a faithful replica of the input , and with probability @xmath10 , it is the opposite of the input letter for each bit ( see figure [ fig : bsc ] ) . in particular , this is an example of memoryless channels . then , one of the main purposes of coding theory is to develop a good encoder - decoder pair @xmath13 which is robust to noise perturbations . hence , the problem is how we efficiently use the redundancy @xmath14 in this setting . a code with a linear encoding map @xmath5 is called a linear code . a codeword in a linear code can be characterized by its generator matrix @xmath15 where each element @xmath16 . therefore the set of codewords is given by @xmath17 here without loss of generality , we assume @xmath18 for all @xmath19 and @xmath20 . we call @xmath1 and @xmath6 the dimension and the length of the code , respectively . because of the linearity , it is also possible to describe @xmath21 as a kernel of a matrix @xmath22 whose @xmath23 row vectors are linearly independent and orthogonal to those of @xmath24 , i.e. , @xmath25 where @xmath26 means the transpose matrix of @xmath22 . this matrix @xmath22 is called a parity check matrix of @xmath21 . the dual code @xmath27 of @xmath21 is defined in such a way that a parity check matrix of @xmath27 is given by a generator matrix @xmath24 of @xmath21 . the hamming distance @xmath28 between two @xmath6-bit sequences @xmath29 is given by the number of positions at which the two sequences differ . the weight of an element @xmath30 is the hamming distance to @xmath31 , i.e. , @xmath32 . then the minimum distance @xmath33 of a code @xmath21 is defined by two different ways as @xmath34 here the second equality results from the linearity . it is easy to observe that the minimum distance is @xmath35 if and only if there exists a set of @xmath36 linearly dependent column vectors of @xmath22 but no set of @xmath37 linearly dependent column vectors . for a code @xmath21 with the minimum distance @xmath38 , let us set @xmath39 , where @xmath40 is the integer part of @xmath41 . then , it follows from the following observation that @xmath21 can correct @xmath42 errors : if @xmath43 and @xmath44 for some @xmath45 then @xmath46 is the only codeword with @xmath44 . in this sense , the minimum distance is one of the important parameters to measure performance of a code and is desirable to design it as large as possible for the robustness to noise . let us recall that , given a transmitted codeword @xmath46 , the conditional probability @xmath47 of a received sequence @xmath43 at the decoder is given by @xmath48 for a memoryless channel . maximum likelihood(ml ) decoding @xmath49 is given by taking the marginalization of @xmath47 for each bit . precisely speaking , for a received sequence @xmath50 , the @xmath51-th bit element @xmath52 of the decoded word @xmath53 is determined by the following rule : @xmath54 in general , for a given decoder @xmath8 , the bit error probability @xmath55 , where @xmath56 is one of the important measures of decoding performance . obviously , it is desirable to design an encoding - decoding pair whose bit error probability is as small as possible . it is known that ml decoding attains the minimum bit error probability for any encodings under the uniform distribution on @xmath57 . in this sense , ml decoding is the best for all decoding rules . however its computational cost requires at least @xmath58 operations , and it is too much to use for practical applications . from the above property of ml decoding , one of the key motivation of this work comes from the following simple question . is it possible to accurately approximate the ml decoding rules with low computational complexity ? the main results in this paper give answers to this question . let us first define for each codeword @xmath59 its codeword polynomial @xmath60 as @xmath61 then we define a rational map @xmath62 , @xmath63,$ ] by using codeword polynomials as @xmath64 where @xmath65 . this rational map plays the most important role in the paper . it is sometimes denoted by @xmath66 , when we need to emphasize the generator matrix @xmath24 of the code @xmath21 . for a sequence @xmath43 , let us take a point @xmath67 as @xmath68 where @xmath10 is the transition probability of the channel . then it is straightforward to check that @xmath69 . namely , the conditional probability of @xmath50 under a codeword @xmath59 is given by the value of the corresponding codeword polynomial @xmath70 at @xmath71 . therefore , from the construction of the rational map , ml decoding ( [ eq : bitml ] ) is equivalently given by the following rule @xmath72 in this sense , the study of ml decoding can be treated by analyzing the image of the initial point ( [ eq : ip ] ) by the rational map ( [ eq : map ] ) . some of the properties of this map in the sense of dynamical systems will be studied in detail in section [ sec : ds ] . we will also discuss in section [ sec : discussions ] that performance of a code can be explained by these properties . for the statement of the main results , we only here mention that this rational map has a fixed point @xmath73 for any generator matrix ( proposition [ pp : fp ] ) . let us denote the taylor expansion at @xmath0 by @xmath74 where @xmath75 is a vector notation of @xmath76 , @xmath77 is the jacobi matrix at @xmath0 , @xmath78 corresponds to the @xmath51-th order term , and @xmath79 means the usual order notation . the reason why we choose @xmath0 as the approximating point is related to the local dynamical property at @xmath0 and will be explained in section [ sec : p ] . by truncating higher oder terms @xmath79 in ( [ eq : taylor ] ) and denoting it as @xmath80 we can define the @xmath81-th approximation of ml decoding by replacing the map @xmath82 in ( [ eq : mlrule ] ) with @xmath83 , and denote this approximate ml decoding by @xmath84 , i.e. , @xmath85 let us remark that the notations @xmath86 and @xmath87 do not explicitly express the dependence on @xmath81 for removing unnecessary confusions of subscripts . we note that there are two different viewpoints on this approximate ml decoding . one way is that , in the sense of its precision , it is preferable to have an expansion with large @xmath81 . on the other hand , from the viewpoint of low computational complexity , it is desirable to include many zero elements in higher order terms . the next theorem states a sufficient condition to satisfy these two requirements . [ thm : thm ] let @xmath88 . if any distinct @xmath81 column vectors of a generator matrix @xmath24 are linearly independent , then the taylor expansion ( [ eq : taylor ] ) at @xmath0 of the rational map ( [ eq : map ] ) takes the following form @xmath89 where the @xmath51-th coordinate @xmath90 of @xmath91 is given by @xmath92 first of all , it follows that the larger the minimum distance of the dual code @xmath27 is , the more precise approximation of ml decoding with low computational complexity we have for the code @xmath21 with the generator matrix @xmath24 . especially , we can take @xmath93 . secondly , let us consider the meaning of the approximate map @xmath83 and its approximate ml decoding @xmath87 . we note that each value @xmath94 in ( [ eq : ip ] ) for a received word @xmath43 expresses the likelihood @xmath95 . let us suppose @xmath96 . then , from the definition of @xmath97 , each term in the sum of ( [ eq : hosum ] ) satisfies @xmath98 when @xmath99 , this term decreases(increases , resp . ) the value of initial likelihood @xmath100 . in view of the decoding rule ( [ eq : mlrule ] ) , this induces @xmath52 to be decoded into @xmath101 , and this actually corresponds to the structure of the code @xmath102 appearing in @xmath103 . in this sense , the approximate map @xmath104 can be regarded as renewing the likelihood ( under suitable normalizations ) based on the code structure , and the approximate ml decoding @xmath87 judges these renewed data . from this argument , it is easy to see that a received word @xmath105 is decoded into @xmath106 , i.e. , the codeword is decoded into itself and , of course , this property should be equipped with any decoders . we also remark that theorem [ thm : thm ] can be regarded as a duality theorem in the following sense . let @xmath21 be a code whose generator(resp . parity check ) matrix is @xmath24 ( resp . @xmath22 ) . as we explained in section [ sec : lc ] , the linear independence of the column vectors of @xmath22 controls the minimum distance @xmath33 and this is an encoding property . on the other hand , theorem [ thm : thm ] shows that the linear independence of the column vectors of @xmath24 , which determines the dual minimum distance @xmath107 , controls a decoding property of ml decoding in the sense of accuracy and computational complexity . hence , we have the correspondence between @xmath108 duality and encoding / decoding duality . in corollary [ corollary : ag ] , we will consider this duality viewpoint in a setting of geometric reed - solomon / goppa codes . we show the second result of this paper about the decoding performances of the approximate ml ( [ eq : apmlrule ] ) . for this purpose , let us first examine numerical simulations of the bit error probability on the bsc with the transition probability @xmath109 . we also show numerical results on bch codes with berlekamp - massey decoding for comparison . the results are summarized in figure [ fig : berrate ] . and @xmath110 . horizontal axis : transition probability @xmath10 . vertical axis : bit error probability . @xmath111 : 3rd order approximate ml decoding . @xmath112 : berlekamp - massaey decoding . @xmath113 : ml decoding . , width=302 ] and @xmath110 . horizontal axis : transition probability @xmath10 . vertical axis : bit error probability . @xmath111 : 3rd order approximate ml decoding . @xmath112 : berlekamp - massaey decoding . @xmath113 : ml decoding . , width=302 ] here , the horizontal axis is the code rate @xmath114 , and the vertical axis is the bit error probability . the plots @xmath115 ( @xmath111 resp . ) correspond to the 2nd ( 3rd resp . ) order approximate ml ( [ eq : apmlrule ] ) , and @xmath112 are the results on several bch codes ( @xmath116 ) with berlekamp - messey decodings . for the proposed method , we randomly construct a systematic generator matrix in such a way that each column except for the systematic part has the same weight ( i.e. number of non - zero elements ) @xmath117 . to be more specific , the submatrix composed by the first @xmath1 columns of the generator matrix is set to be an identity matrix in order to make the code systematic , while the rest of the generator matrix is made up of @xmath118 random matrices generated by random permutations of columns of a circulant matrix , whose first column is given by @xmath119 the reason for using random codings is that we want to investigate average behaviors of the decoding performance , and , for this purpose , we do not put unnecessary additional structure at encodings . the number of matrices added after the systematic part depends on the code rate , and the plot for each code rate corresponds to the best result obtained out of about 100 realizations of the generator matrix . also , we have employed @xmath120 and 3 for the generator matrices of the 3rd and the 2nd order approximate ml , respectively . moreover , the length of the codewords @xmath6 are assumed to be up to 512 . from figure [ fig : berrate ] , we can see that the proposed method with the 3rd order approximate ml ( @xmath111 ) achieves better performance than that of bch codes with berlekamp - massey(@xmath112 ) . it should be also noticed that the decoding performance is improved a lot from the 2nd order to the 3rd order approximation . this improvement is reasonable because of the meaning of the taylor expansion . next , let us directly compare the decoding performances among ml , approximate ml ( 3rd order ) and berlekamp - massey by applying them on the same bch code ( @xmath121 , @xmath110 ) . the result on the bit error probability with respect to transition probability is shown in figure [ fig : berprob ] . this figure clearly shows that the performance of the 3rd order approximate ml decoding is far better than that of berlekamp - massey decoding ( e.g. , improvement of double - digit at @xmath122 ) . furthermore , it should be noted that the 3rd order approximate ml decoding achieves a very close bit error performance to that of ml decoding . although we have not mathematically confirmed the computational complexity of the proposed approach , the computational time of the approximate ml ( 3rd order ) is much faster than ml decoding . this fact about low computational complexity of the approximate ml is explained as follows : non - zero higher order terms in ( [ eq : hosum ] ) appear as a result of linear dependent relations of column vectors of @xmath24 , however , linear dependences require high codimentional scenario . hence , most of the higher order terms become zeros . as a result , the computational complexity for the approximate ml , which is determined by the number of nonzero terms in the expansion , becomes small . in conclusion , these numerical simulations suggest that the 3rd order approximate ml decoding approximates ml decoding very well with low computational complexity . we notice that the encodings examined here are random codings . hence , we can expect to obtain better bit error performance by introducing certain structure on encodings suitable to this proposed decoding rule , or much more exhaustive search of random codes . one of the possibility will be the combination with theorem [ thm : thm ] . on the other hand , it is also possible to consider suitable encoding rules from the viewpoint of dynamical systems via rational maps ( [ eq : map ] ) . this issue is discussed in detail in section [ sec : discussions ] . in any case , finding suitable encoding structure for the proposed decodings is one of the important future problem . the paper is organized as follows . in section [ sec : ds ] , we study properties of the rational map ( [ eq : map ] ) in view of a discrete dynamical system and show some relationships to coding theory . we also show that this discrete dynamical system is related to a continuous gradient dynamical system @xmath123 . section [ sec : taylor ] deals with relationship between a generator matrix of a code and its taylor expansion ( [ eq : taylor ] ) . the proof of theorem [ thm : thm ] , which is a direct consequence of proposition [ pp : jacobi ] and proposition [ pp : fl0 ] , is shown in this section . in section [ sec : ag ] , we apply theorem [ thm : thm ] to geometric reed - solomon / goppa codes in corollary [ corollary : ag ] with a simple example by using the hermitian curve . finally , we discuss the future problems on this subject as an intersection of dynamical systems and coding theory . in this section , we discuss ml decoding from dynamical systems viewpoints . let @xmath24 be a @xmath124 generator matrix for a code @xmath21 . we begin with showing the following easy consequence of linear codes , which will be used frequently throughout the paper . [ lemma : gi ] for @xmath125 , let us denote subcodes of @xmath21 with @xmath126 and @xmath127 respectively by @xmath128 then @xmath129 . from the assumption on the generator matrix , we have @xmath18 and let @xmath130 be the number of 1 in @xmath131 . then we have @xmath132 where the symbol @xmath133 means the number of combinations for taking @xmath134 elements from @xmath81 elements . however these summations of combinations are obviously equal because @xmath135 therefore @xmath129 . next , let us characterize the codewords in @xmath21 and the non - codewords in @xmath136 by means of the rational map ( [ eq : map ] ) . it should be remarked that a codeword polynomial @xmath70 for @xmath65 with @xmath137 , takes its value @xmath138 here we identify a point @xmath139 @xmath140 , with a point @xmath141 by a natural inclusion @xmath142 , and this convention will be used frequently in the paper . we also define a point @xmath143 as a pole of the rational map ( [ eq : map ] ) if @xmath144 . the boundary and the interior of a set @xmath145 are denoted by @xmath146 and @xmath147 , respectively . [ pp : fp ] the followings hold for the rational map ( [ eq : map ] ) : 1 . @xmath148 is a fixed point . 2 . let @xmath141 . then @xmath149 is a fixed point if and only if @xmath150 . the set of poles is given by @xmath151 especially , @xmath152 . for the statement 1 , let us note that @xmath153 for each codeword @xmath45 . then lemma [ lemma : gi ] leads to @xmath154 for the statement 2 , let us suppose @xmath150 . then , from the remark before this proposition , @xmath155 if @xmath156 for each @xmath51 . hence @xmath157 . on the other hand , if @xmath158 , then @xmath144 . it means that @xmath149 is a pole and can not be a fixed point . for the statement 3 , let us first note that @xmath159 if and only if @xmath160 for any codeword @xmath45 , because @xmath161 for @xmath143 . therefore @xmath162 , since @xmath163 for @xmath164 . let @xmath165 such that @xmath166 and @xmath167 for @xmath168 . then , if there exists a codeword @xmath45 such that @xmath169 , then the value of its corresponding codeword polynomial is @xmath163 . so , @xmath170 . on the other hand , if there is no such codeword , then @xmath144 , it means @xmath159 . from this proposition , the rational map ( [ eq : map ] ) has information of not only all codewords @xmath21 as fixed points but also non - codewords @xmath171 as poles . we call these fixed points codeword fixed points . the following proposition shows that all of the codeword fixed points are stable . [ pp : stable ] let a parity check matrix do not have zero column vectors , i.e. , there exists no codeword with weight 1 . let @xmath149 be a codeword fixed point . then the jacobi matrix of the rational map ( [ eq : map ] ) at @xmath149 is the zero matrix . hence , @xmath149 is a stable fixed point . let us denote the @xmath51-th element of the rational map ( [ eq : map ] ) by @xmath172 and denote the derivatives of @xmath173 and @xmath22 with respect to @xmath174 by @xmath175 and @xmath176 , respectively , for the simplicity of notations . in what follows , we will also use these notations for higher order derivatives in a similar way ( like @xmath177 ) . then the derivative @xmath178 is given by @xmath179 since @xmath150 , we have @xmath180 . let us consider the two cases @xmath181 and @xmath182 , separately . for @xmath181 , we have @xmath183 . similarly we have @xmath184 if @xmath185 . hence , @xmath186 in this case . on the other hand , if @xmath187 , from the assumption on the parity check matrix , we have no codeword @xmath188 such that the only difference from @xmath149 occurs at the @xmath51-th bit element , i.e. , @xmath189 for @xmath190 but @xmath191 . therefore , @xmath184 , and it again leads to @xmath186 . the case @xmath182 can be proven by the similar way . let us next discuss properties of the fixed point @xmath192 . [ pp : jacobi ] let @xmath193 be a generator matrix . then , the jacobi matrix @xmath77 of the rational map ( [ eq : map ] ) at @xmath0 is determined by @xmath194 for all @xmath195 . for the proof of this proposition , we need the following lemma . [ lemma : gij ] for @xmath196 with @xmath185 , let us consider the following subcodes of @xmath21 @xmath197 then the followings hold 1 . if @xmath198 , then @xmath199 2 . if @xmath200 , then @xmath201 the statement is trivial when @xmath198 , so we suppose @xmath200 . in the following we adopt the @xmath202 arithmetic for elements in @xmath203 . we can express @xmath131 , by using some permutations of rows if necessary , as follows @xmath204 in the following we only deal with the case @xmath205 , but the modification to the case @xmath206 is trivial . now we have the following two cases + case i : there exists @xmath207 such that @xmath208 , + case ii : not case i , i.e. , @xmath209 for all @xmath207 . + in the case i , let us fix @xmath210 in a message @xmath2 with @xmath211 , which corresponds to @xmath127 , and consider the numbers of codewords with @xmath212 and @xmath213 from the remaining message bits @xmath214 . then , from the assumption , there exists at least one non zero element in @xmath215 . hence , the same argument in lemma [ lemma : gi ] shows that the numbers of codewords with @xmath212 and @xmath213 under a fixed @xmath216 are the same . by considering all the possibilities of @xmath210 with @xmath217 , it gives @xmath201 next , let us consider the case ii . again by using a permutation if necessary , we have the following expressions @xmath218 where @xmath219 and @xmath220 because of @xmath200 . then we have @xmath221 however , the same argument in lemma [ lemma : gi ] implies @xmath201 the @xmath222 element in the jacobi matrix @xmath77 is given by @xmath223 from lemma [ lemma : gi ] , @xmath224 because @xmath225 where the first term comes from the codewords with @xmath213 and the second term comes from the codewords with @xmath212 . it is also easy to observe that @xmath226 , and @xmath227 . therefore the diagonal elements are @xmath228 . next , let us consider the case @xmath185 . in this case , if we have @xmath198 , then , from lemma [ lemma : gij ] , @xmath229 . on the other hand , if we have @xmath200 , then @xmath230 . this concludes the proof of proposition [ pp : jacobi ] . two corollaries follow from proposition [ pp : jacobi ] which characterize the eigenvalues and the eigenvectors of the jacobi matrix @xmath77 , and it clearly determines the stability and the stable / unstable subspaces of @xmath0 . to this end , let us denote by @xmath231 a graph whose adjacent matrix is @xmath77 . namely , the nodes of @xmath231 are @xmath232 , and an undirected edge @xmath222 appears in @xmath231 if and only if @xmath233 . [ corollary : eigenvalue ] suppose the graph @xmath231 is decomposed into @xmath81 connected components @xmath234 let @xmath235 be the number of nodes in the component @xmath236 , @xmath237 . then all the eigenvalues of @xmath77 are given by @xmath238 where the eigenvalues @xmath239 are simple and the @xmath31 eigenvalue has @xmath240 multiplicity . from proposition [ pp : jacobi ] , any two nodes in a same connected component have an edge between them . hence , it is possible to transform @xmath77 into the following block diagonal matrix @xmath241 where @xmath242 is determined by compositions of column switching elementary matrices , and @xmath243 is an @xmath244 matrix all of whose elements are 1 . the statement of the corollary follows immediately . from now on , we treat a jacobi matrix of the block diagonal form ( [ eq : elementarymatrix ] ) . obviously , it gives no restriction since , if necessary , we can appropriately permute columns of the original generator matrix in advance . let us denote the set of eigenvalues of @xmath77 derived in corollary [ corollary : eigenvalue ] by @xmath245 where the successive @xmath246 after each @xmath235 has @xmath247 elements . in case of @xmath248 , we ignore the successive @xmath249 for @xmath235 ( i.e. , @xmath250 ) . [ corollary : eigenvector ] the jacobi matrix @xmath77 is diagonalizable . furthermore , the corresponding eigenvectors @xmath251 for ( [ eq : eigenvalue ] ) under this ordering are given by the following @xmath252 where @xmath253 here the vectors @xmath254 have @xmath235 elements and the first element starts at the @xmath255-th row . the bold type @xmath256 expresses that the remaining elements in @xmath257 are @xmath31 . in case of @xmath248 , we only have @xmath258 ( with @xmath259 ) . it is obvious from corollary [ corollary : eigenvalue ] . we finally mention a relationship to a continuous gradient dynamical system . let us denote by @xmath260 a polynomial obtained by removing @xmath261 from a codeword polynomial @xmath70 of @xmath45 . by using this notation , the map ( [ eq : map ] ) can be also described as @xmath262 then it follows that @xmath263 this proves the following proposition . [ pp : gradient ] the rational map ( [ eq : map ] ) maps a point @xmath264 to the direction of @xmath265 with a contraction rate @xmath266 for each element @xmath267 . especially , @xmath264 is a fixed point of ( [ eq : map ] ) if and only if it is a fixed point of the continuous gradient dynamical system @xmath123 . next , we study a relationship between higher order terms in ( [ eq : taylor ] ) and a generator matrix @xmath24 , and prove theorem [ thm : thm ] . for this purpose , the key proposition is given as follows . [ pp : fl0 ] let @xmath88 . if any distinct @xmath268 column vectors of @xmath24 are linearly independent , then @xmath269 . let us denote a higher order derivative of an @xmath51-th element @xmath270 with respect to variables @xmath271 by @xmath272 for the proof of proposition [ pp : fl0 ] , we need to study higher order derivatives @xmath273 and @xmath274 . let us at first focus on higher order derivatives @xmath273 . we begin with the following observation , which characterizes the numbers of subcodes by means of column vectors of @xmath24 . it is noted that we adopt the @xmath202 arithmetic for elements in @xmath203 . [ lemma : c01 ] suppose @xmath275 are distinct indices , and let @xmath276 be subcodes in @xmath21 . then the followings hold 1 . if @xmath277 , then @xmath278 . if @xmath279 , then @xmath280 and @xmath281 . by a suitable bit permutation , if necessary , the sum of @xmath282 can be expressed as @xmath283 then an original message @xmath284 and its codeword @xmath285 satisfy the following @xmath286 in the case 1 ( @xmath287 ) , it leads to @xmath288 , so the conclusion follows from the same argument in lemma [ lemma : gi ] . the case 2 is trivial from the above expression of @xmath289 . the following lemma classifies the value @xmath273 based on the column vectors of @xmath24 . [ lemma : h ] let @xmath88 . then @xmath290 if either of 1 there exist same indices in @xmath291 2 . @xmath277 is satisfied . otherwise , that is @xmath291 are all distinct and @xmath292 , @xmath293 the condition 1 immediately implies @xmath290 since the degree of each variable @xmath294 in @xmath295 is 1 . hence we assume all the indices are distinct . let us define the following subcodes @xmath296 then @xmath273 can be expressed by @xmath297 suppose @xmath277 . then , by lemma [ lemma : c01 ] , we have @xmath298 . on the other hand , when @xmath81 is odd ( or even , resp . ) , @xmath299 therefore it concludes @xmath290 . the statement for @xmath292 is similarly derived from lemma [ lemma : c01 ] . next , we try to classify the value @xmath274 . [ lemma : ic01 ] suppose @xmath300 are distinct indices and let us define two subcodes in @xmath21 by @xmath301 then the following classification holds 1 . if @xmath292 , then @xmath302 , @xmath303 . 2 . if @xmath304 , then @xmath305 . 3 . if @xmath306 , then @xmath307 , @xmath308 . since we have @xmath309 the case 1 and 3 are trivial . so , let us assume @xmath304 . the proof is similar to that of lemma [ lemma : gij ] . by using a suitable permutation , let us express @xmath310 as follows @xmath311 here we only deal with the case @xmath205 again , since the modification for @xmath312 follows immediately from the following case ii . we have two situations + case i : there exists @xmath313 such that @xmath314 . + case ii : not case i , i.e. , @xmath315 for all @xmath313 . + in case i , let us fix @xmath316 with @xmath211 , which corresponds to @xmath317 , and consider the numbers of codewords with @xmath318 or @xmath319 for the remaining message bits @xmath320 . from the assumption , the @xmath321-th element of the vector @xmath322 is 1 , and , by applying lemma [ lemma : c01 ] to the subvector from @xmath323-th to @xmath1-th elements , the numbers of codewords with @xmath318 or @xmath319 are the same for each @xmath316 . hence we have @xmath305 . the proof for case ii is almost parallel to that of lemma [ lemma : gij ] , so we omit it . let us introduce the following notations , which are similar to those in the proof of lemma [ lemma : h ] , @xmath324 then , for odd @xmath81 ( or even @xmath81 , resp . ) , we have @xmath325 [ lemma : ip ] let @xmath88 . then @xmath274 is classified as @xmath326 where each condition is given by + c0 : there exist same indices in @xmath291 + c1 : @xmath327 and @xmath304 ( here @xmath327 means not c0 " ) + c2 : @xmath327 and @xmath328 + c3 : @xmath327 and @xmath292 the condition c0 immediately implies @xmath329 . hence we assume all the indices are distinct . the remaining proof follows directly from lemma [ lemma : ic01 ] for each case . first of all , let us study the case @xmath330 . by using the notations introduced before the lemma , we have @xmath331 therefore the condition c1 implies @xmath329 by lemma [ lemma : ic01 ] . on the other hand , if we assume the condition c2 , then @xmath307 and @xmath308 from lemma [ lemma : ic01 ] . if @xmath81 is even , @xmath332 . similarly , if @xmath81 is odd , @xmath333 . hence , we have @xmath334 for the condition c2 . for the condition c3 , the role of @xmath335 and @xmath336 changes each other from lemma [ lemma : ic01 ] , so it just leads to the opposite sign in @xmath274 to that for the condition c2 . next , let us study the case @xmath337 . without loss of generality , let us suppose @xmath338 . then we have @xmath339 therefore the condition c1 implies @xmath329 by lemma [ lemma : ic01 ] ( or lemma [ lemma : gij ] for @xmath340 ) . for the condition c2 ( @xmath81 must be @xmath341 ) , we have @xmath342 and @xmath343 from lemma [ lemma : ic01 ] . hence , by the same calculation as that for @xmath330 , we have @xmath334 . finally , let us consider the condition c3 . for @xmath344 , we have @xmath345 and @xmath346 by lemma [ lemma : ic01 ] again so , @xmath347 . for @xmath340 , we can not use lemma [ lemma : ic01 ] because of @xmath348 . however , @xmath349 holds from the assumption @xmath350 . hence a direct calculation shows @xmath351 , which is the formula for @xmath340 . before proving proposition [ pp : fl0 ] , let us show the following two lemmas . the proofs of them are easy application of induction . [ lemma:1/a ] the derivative @xmath352 is given by @xmath353 where the summation for the @xmath1-th term ( @xmath354 ) is taken on all combinations @xmath355 for dividing @xmath291 into @xmath1 groups . here @xmath356 represent a decomposition of @xmath357 ; @xmath358 [ lemma : fl ] let @xmath88 . then the derivative @xmath359 is given by @xmath360 where @xmath361 and @xmath362 are a decomposition of @xmath357 ; @xmath363 and the summations are taken on all the combinations of the decompositions . now we prove proposition [ pp : fl0 ] . + let us assume that any distinct @xmath268 column vectors of @xmath24 are linearly independent . then , from lemma [ lemma : h ] , [ lemma:1/a ] , and [ lemma : fl ] , we have @xmath364 on the other hand , from lemma [ lemma : ip ] , @xmath274 is 0 . the proof is completed . + finally , we are in the position to prove theorem [ thm : thm ] . + the formula for the case @xmath340 is given by proposition [ pp : jacobi ] . let us assume @xmath365 . from proposition [ pp : fl0 ] , all nonlinear terms with orders less than @xmath81 are zero , so we only study the @xmath81-th nonlinear terms . from the assumption and similar argument in the proof of proposition [ pp : fl0 ] , the derivative @xmath366 is given by @xmath364 then , the classification in lemma [ lemma : ip ] shows that @xmath367 in c0 and c1 . moreover , the condition c3 does not occur from the assumption . hence , only nonzero terms are derived from the condition c2 . it should be noted that the set of indices @xmath368 satisfying the condition c2 with @xmath51 is exactly the same as @xmath369 . hence , lemma [ lemma : ip ] results in @xmath370 for @xmath371 . finally , we have the following @xmath372 the identity in ( [ eq : hosum ] ) is derived by just substituting @xmath373 . it completes the proof . the purpose of this section is to derive corollary [ corollary : ag ] . this corollary shows that usual techniques in algebraic geometry codes can be applied to control , not only the minimum distance of the code , but also the approximate ml decoding . for the definitions of basic tools in algebraic geometry such as genus , divisor , riemann - roch space , differential , and residue , we refer to @xcite . we also request basic knowledge of algebraic geometry codes in this section ( e.g. , see @xcite , @xcite and @xcite ) . let @xmath374 be a finite field with @xmath375 elements . let us first recall two classes of algebraic geometry codes called geometric reed - solomon codes and geometric goppa codes . let @xmath376 be an absolutely irreducible nonsingular projective curve over @xmath374 . for rational points @xmath377 on @xmath376 , we define a divisor on @xmath376 by @xmath378 . moreover , let @xmath379 be another divisor whose support is disjoint to @xmath380 . we assume that @xmath379 satisfies the following condition @xmath381 for the sake of simplicity . here @xmath382 is the genus of @xmath376 . geometric reed - solomon codes are characterized by the riemann - roch space associated to @xmath379 @xmath383 where @xmath384 is the set of nonzero elements of the function field @xmath385 , and @xmath386 is the principal divisor of the rational function @xmath387 . the fundamental fact that the riemann - roch space @xmath388 is a finite dimensional vector space leads to the following definition . _ the geometric reed - solomon code _ @xmath389 of length @xmath390 over @xmath374 is defined by the image of the linear map @xmath391 given by @xmath392 . on the other hand , geometric goppa codes are defined via differentials and their residues . let us denote the set of differentials on @xmath376 by @xmath393 , and define for each divisor @xmath380 @xmath394 where @xmath395 is the divisor of the differential @xmath396 . _ the geometric goppa code _ @xmath397 of length @xmath398 over @xmath374 is defined by the image of the linear map @xmath399 given by @xmath400 , where @xmath401 expresses the residue of @xmath396 at @xmath402 . the following propositions are an easy consequence of the riemann - roch theorem . ( e.g. , @xcite , @xcite , @xcite)[pp : grs ] 1 . the dimension of the geometric reed - solomon code @xmath389 is @xmath403 and the minimum distance satisfies @xmath404 . 2 . the dimension of the geometric goppa code @xmath397 is @xmath405 and the minimum distance satisfies @xmath406 . the codes @xmath389 and @xmath397 are dual codes . it should be noted that the minimum distances of these two codes are controlled by the genus of @xmath376 and the choice of the divisor @xmath379 , and , as a result , they induce appropriate linear independence on their parity check matrices . for an application of theorem [ thm : thm ] , we need to derive expanded codes over @xmath203 from geometric reed - solomon and geometric goppa codes over @xmath407 . let @xmath408 be a code over @xmath374 with length @xmath390 , and @xmath409 be a basis of @xmath203-vector space @xmath374 . this basis naturally induces the map @xmath410 by expressing each element @xmath411 in @xmath412 as coefficients of @xmath203-vector space . then the expanded code of @xmath408 over @xmath203 is defined by @xmath413 a relationship between @xmath414 and @xmath415 is given by the following proposition . if a code @xmath414 has parameters @xmath416 $ ] , where @xmath390 is the code length , @xmath1 is the dimension , and @xmath36 is the minimum distance , then its expanded code @xmath415 has the parameters @xmath417 $ ] . now we apply theorem [ thm : thm ] to geometric reed - solomon / goppa codes . let @xmath418 , and let @xmath419 and @xmath420 be the expanded codes over @xmath203 of a geometric reed - solomon code @xmath389 and a geometric goppa code @xmath397 over @xmath374 . then , we have the following corollary . [ corollary : ag ] the expanded geometric reed - solomon code @xmath419 has the minimum distance @xmath421 . furthermore , there exists an @xmath81-th order approximate ml decoding with @xmath422 such that @xmath423 . the first statement is the property of a geometric reed - solomon code and its expansion . the second statement follows from theorem [ thm : thm ] and the duality of @xmath389 and @xmath397 . ( e.g. , @xcite , @xcite , @xcite ) + let @xmath424 be a power of 2 . the hermitian curve @xmath425 is given by the homogeneous equation @xmath426 and its genus is @xmath427 , because there are no singular points . it is known that the number of rational points over @xmath374 is @xmath428 . let us fix @xmath429 as an example . then , the following is the list of the rational points on @xmath425 : @xmath430 where @xmath396 is a primitive element of @xmath431 and @xmath432 ( i.e. , @xmath433 ) . let us suppose @xmath434 ( hence the code length is @xmath435 ) , and @xmath436 , @xmath437 . a basis of the riemann - roch space @xmath388 with @xmath438 is given by @xmath439 then , we can explicitly show a generator matrix of the expanded geometric reed - solomon code @xmath389 over @xmath203 as follows @xmath440 it should be noted that this case leads to a self - dual code @xmath441 . hence , its parity check matrix @xmath22 is the same as @xmath24 , and a direct calculation proves that the minimum distance of this code is 4 . it means that we have the @xmath442-rd order ml decoding in corollary [ corollary : ag ] . the explicit forms of @xmath443 are given as @xmath444 where @xmath445 . to conclude this paper , we address the following comments and discussions , some of which will be important for designs of good practical error - correcting codes . codeword fixed points in @xmath21 are stable from proposition [ pp : stable ] , and non - codeword poles in @xmath171 are unstable from proposition [ pp : gradient ] in the sense that nearby points to a pole in @xmath171 leave away from the pole . let us recall that each @xmath6-bit received sequence @xmath43 and its initial point @xmath67 are related by ( [ eq : ip ] ) and it is characterized that @xmath50 is the closest point to @xmath97 in @xmath446 . hence , if the received sequence @xmath50 is a codeword , then @xmath97 may approach to the codeword fixed point @xmath50 . this obviously depends on whether or not @xmath97 is located in the attractor region of the codeword fixed point , although it is actually the case when @xmath10 is small enough because of the stability . so far , a general structure of the attractor region for each codeword fixed point is not yet known . however this is an important subject since it is indispensable to give an estimate of error probabilities of ml decoding and its approximation . similar arguments also hold for a non - codeword received sequence and its repelling property . let us recall that the ml decoding rule is given by ( [ eq : mlrule ] ) which checks the location of the image for an initial point @xmath97 to the point @xmath0 . hence the local dynamics around the point @xmath0 will be important for decoding process . in the following , we explain the local dynamics around @xmath0 in two different cases : the jacobi matrix @xmath77 at @xmath0 is ( i ) identity or ( ii ) not identity . in the case ( i ) , the local dynamics around @xmath0 is precisely determined by theorem [ thm : thm ] . as explained after theorem [ thm : thm ] , the nonlinear dynamics around @xmath0 is closely related to the encoding structure of the code and the decoding process . in the case ( ii ) , let us suppose that @xmath243 in corollary [ corollary : eigenvalue ] induces unstable eigenvalues @xmath447 and let us focus on its stable / unstable eigenspaces @xmath448 , respectively . from its eigenvector , @xmath449 is given by the 1 dimensional subspace spanned by @xmath450 this plays a role to make @xmath451 to be equal , and it reflects the fact @xmath452 . it means that the unstable subspace @xmath449 points to codewords directions , i.e. , @xmath21 . on the other hand , @xmath453 is spanned by the stable eigenvectors @xmath454 in corollary [ corollary : eigenvector ] . contrary to the unstable eigenvector , these eigenvectors play a role to generate different elements in @xmath451 and point to non - codewords directions under time reversal , i.e. , @xmath171 . therefore , the fixed point @xmath0 can be regarded as an indicator to codewords in the sense that non - codeword elements shrink and codeword elements expand around @xmath0 . in both cases , further nonlinear analysis of center / stable / unstable manifolds of @xmath0 will be useful for finding suitable encoding rules and estimating the decoding performance for the approximate ml . from the above argument on the fixed point @xmath0 , it seems to be appropriate to design a generator matrix to be hyperbolic at @xmath0 , because @xmath0 separates expanding and shrinking directions properly and these separations have an affect on the decoding performance . however , if @xmath0 is hyperbolic , then the coding rate must satisfy @xmath455 by corollary [ corollary : eigenvalue ] . namely , the hyperbolicity prevents a code to have a high coding rate greater than half , although this is not a strict restriction in particular applications like wireless communication channels . therefore it is necessary to have a center eigenspace at @xmath0 for a code with the rate @xmath456 . the normal form theory in dynamical systems ( e.g. , see @xcite ) enables us to transform a map into a simpler form by using a near identity transformation around a fixed point . one of the essential points is that nonresonant higher order terms can be removed from the original map by this transformation . theorem [ thm : thm ] can be interpreted from the viewpoints of normal forms in such a way that an algebraic geometry code gives only zero nonresonant terms in the expansion of @xmath82 at @xmath0 . then , it leads to the following natural question whether a code whose rational map does not have resonant terms , but has nonresonant terms which are not necessarily zeros in its expansion is a good error - correcting code or not through a near identity transformation . at least , this class of codes contains algebraic geometry codes as a subclass , and a similar statement to corollary [ corollary : ag ] holds through near identity transformations . it seems to be valuable to mention a relationship to ldpc codes @xcite , @xcite , which are a relatively new class of error - correcting codes based on iterative decoding schemes ( for a reference to this research region , see @xcite ) . the iterative decoding schemes mainly use so - called sum - product algorithm for ml decoding and deal with a marginalized conditional probability in ( [ eq : bitml ] ) as a convergent point . although this coding scheme gives a good performance in some numerical simulations , mathematical further understanding of the sum - product algorithm and ml decoding is desired to design better coding schemes . from the viewpoint of dynamical systems , it seems to be natural to formulate the sum - product algorithm or ml decoding itself as a certain map , and then analyze its mechanism . the strategy in this paper is based on this consideration . the authors express their sincere gratitude to the members of tin working group for valuable comments and discussions on this paper . this work is supported by jst presto program .
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this paper studies maximum likelihood(ml ) decoding in error - correcting codes as rational maps and proposes an approximate ml decoding rule by using a taylor expansion .
the point for the taylor expansion , which will be denoted by @xmath0 in the paper , is properly chosen by considering some dynamical system properties .
we have two results about this approximate ml decoding .
the first result proves that the order of the first nonlinear terms in the taylor expansion is determined by the minimum distance of its dual code . as the second result
, we give numerical results on bit error probabilities for the approximate ml decoding .
these numerical results show better performance than that of bch codes , and indicate that this proposed method approximates the original ml decoding very well .
maximum likelihood decoding , rational map , dynamical system + * ams subject classification .
* 37n99 , 94b35
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You are an expert at summarizing long articles. Proceed to summarize the following text:
inclusive semileptonic decays of @xmath1 mesons to charmed final states provide an avenue for measuring the cabibbo - kobayashi - maskawa ( ckm ) matrix element @xmath5 @xcite and for determining non - perturbative hadronic properties of the @xmath1 meson . in particular , the moments of the hadronic mass in @xmath6 decays calculated in the framework of the operator product expansion ( ope ) and the heavy quark effective theory ( hqet ) @xcite depend on the @xmath7-quark mass ( @xmath8 ) and a few non - perturbative matrix elements that also appear in the expression of the total semileptonic width . thus , measurements of the hadronic invariant mass moments @xcite allow the determination of these non - perturbative parameters from the data and reduce the theoretical uncertainty in the extraction of @xmath5 from measurements of the semileptonic branching fraction . an improved knowledge of @xmath8 also results in a more precise determination of @xmath9 from inclusive charmless semileptonic @xmath1 decays . this analysis uses @xmath10 events in which the hadronic decay of one @xmath1 meson is fully reconstructed . the semileptonic decay of the other @xmath1 is inferred from the presence of an identified lepton ( electron or muon ) amongst the remaining particles in the event . we calculate the first two moments of the hadronic invariant mass squared ( @xmath0 ) distribution @xcite directly from the measured spectrum after the effects of finite detector resolution have been removed using the singular value decomposition algorithm @xcite . the measurement described in this paper improves the results previously reported by the babar and cleo collaborations @xcite . the sensitivity to @xmath8 and other non - perturbative parameters is increased by lowering the minimum lepton energy threshold to 0.7 gev . finally , this analysis minimizes the dependence on particular @xmath6 model assumptions by calculating the moments directly from the unfolded @xmath0 spectrum . the data used in this analysis were taken with the belle detector @xcite at the kekb asymmetric energy @xmath11 collider @xcite . belle is a large - solid - angle magnetic spectrometer that consists of a three - layer silicon vertex detector , a 50-layer central drift chamber ( cdc ) , an array of aerogel threshold erenkov counters ( acc ) , a barrel - like arrangement of time - of - flight scintillation counters ( tof ) , and an electromagnetic calorimeter comprised of csi(tl ) crystals ( ecl ) located inside a super - conducting solenoid coil that provides a 1.5 t magnetic field . an iron flux - return located outside of the coil is instrumented to detect @xmath12 mesons and to identify muons ( klm ) . the data sample consists of 140 fb@xmath3 taken near the @xmath4 resonance , or @xmath13 @xmath14 events . another 15 fb@xmath3 taken at 60 mev below the resonance are used to estimate the non-@xmath14 ( continuum ) background . the off - resonance data is scaled by the integrated on- to off - resonance luminosity ratio corrected for the @xmath15 dependence of the @xmath16 cross - section . a generic @xmath14 monte carlo ( mc ) sample equivalent to about three times the integrated luminosity is used in this analysis . mc - simulated events are generated with evtgen @xcite and full detector simulation based on geant3 @xcite is applied . the decays @xmath17 and @xmath18 are generated using an hqet inspired form factor parameterization @xcite . the decays @xmath19 @xcite are simulated according to the leibovich - ligeti - stewart - wise ( llsw ) model @xcite ( both relative abundance and form factor shape ) . the @xmath20 model also includes non - resonant @xmath21 decays which are generated using the goity - roberts model @xcite . the model for the @xmath22 background is a hybrid mixture of exclusive modes and an inclusive component described by the de fazio - neubert model @xcite . light - cone sum rule form factors @xcite are used for @xmath23 , @xmath24 and @xmath25 . other exclusive modes are simulated according to the isgw2 model @xcite . qed bremsstrahlung in @xmath26 decays is included using the photos package @xcite . hadronic events are selected based on the charged track multiplicity and the visible energy in the calorimeter . the selection is described in detail elsewhere @xcite . we fully reconstruct the hadronic decay of one @xmath1 meson ( @xmath27 ) using the decay modes @xmath28 and @xmath29 @xcite . pairs of photons satisfying @xmath30 mev in the laboratory - frame and 118 mev/@xmath31 mev/@xmath32 ( @xmath33 around the @xmath34 mass ) are combined to form @xmath34 candidates . @xmath35 mesons are reconstructed from pairs of oppositely charged tracks with invariant mass within @xmath36 mev/@xmath32 ( @xmath37 ) of the nominal @xmath35 mass and a decay vertex displaced from the interaction point . candidate @xmath38 and @xmath39 mesons are reconstructed in the @xmath40 and @xmath41 decay modes , requiring their invariant masses to be within @xmath42 mev/@xmath32 of the nominal @xmath43 mass . candidate @xmath44 mesons are obtained by combining a @xmath39 candidate with a charged pion and requiring an invariant mass between 1.0 and 1.6 gev/@xmath32 . @xmath45 candidates are searched for in the @xmath46 , @xmath47 , @xmath48 , @xmath49 and @xmath50 decay modes . the @xmath51 and @xmath52 modes are used to reconstruct @xmath53 mesons . charmed mesons are selected in a window corresponding to @xmath54 times the mass resolution in the respective decay mode . mesons are reconstructed by pairing a charmed meson with a low momentum pion , @xmath56 . the decay modes @xmath57 and @xmath58 are used to search for neutral charmed vector mesons . for each @xmath27 candidate , the beam - energy constrained mass @xmath59 and the energy difference @xmath60 are calculated , @xmath61 where @xmath62 , @xmath63 and @xmath64 are the beam energy , the 3-momentum and the energy of the @xmath1 candidate in the @xmath4 frame . in @xmath59 and @xmath60 , the signal peaks at the nominal @xmath1 mass and zero , respectively . we define the signal region by the selections @xmath65 gev/@xmath32 and @xmath66 gev . if multiple candidates are found in a single event , the best candidate is chosen based on the proximity of @xmath67 , @xmath68 and @xmath69 to their nominal values , where @xmath68 is the reconstructed @xmath70 meson mass and @xmath69 is the difference between the reconstructed @xmath71 and @xmath70 meson masses . without making any requirement on the decay of the other @xmath1 meson , the number of @xmath72 ( @xmath73 ) tags in this region , after subtraction of continuum and combinatorial backgrounds , is @xmath74 ( @xmath75 ) , fig . [ fig:1 ] . distributions for charged and neutral @xmath27 candidates after requiring @xmath76 gev . no constraints are made on the signal side . the points with error bars are on - resonance data after subtraction of the scaled off - resonance data . the combinatorial background ( cross - hatched histogram ) is estimated using mc simulation . ] semileptonic decays of the other @xmath1 meson ( @xmath77 ) are selected by searching for an identified charged lepton ( electron or muon ) within the remaining particles in the event . electron candidates are identified using the ratio of the energy detected in the ecl to the track momentum , the ecl shower shape , position matching between track and ecl cluster , the energy loss in the cdc and the response of the acc counters . muons are identified based on their penetration range and transverse scattering in the klm detector . in the momentum region relevant to this analysis , charged leptons are identified with an efficiency of about 90% and the probability to misidentify a pion as an electron ( muon ) is 0.25% ( 1.4% ) @xcite . we further require electron ( muon ) candidates to originate from near the interaction vertex , have a laboratory - frame momentum greater than 0.3 gev/@xmath78 ( 0.6 gev/@xmath78 ) and satisfy @xmath79 ( @xmath80 ) , where @xmath81 is the polar angle in the laboratory - frame relative to the beam direction . if more than one charged lepton candidate is found in the event , we only keep the one with the highest momentum in the @xmath1 rest frame . electrons from photon conversion are vetoed by rejecting the event if the invariant mass of the electron candidate and another oppositely charged particle in the event is below 0.04 gev/@xmath32 and secondary vertex criteria are satisfied . if the charged lepton candidate is consistent with the decay @xmath82 ( _ i.e. _ , the invariant mass of the lepton candidate and another oppositely charged lepton in the event is between 3 gev/@xmath32 and 3.15 gev/@xmath32 ) , the event is also rejected . in @xmath72 tagged events , we require the lepton charge to be consistent with a prompt semileptonic decay of @xmath77 . in @xmath73 events , we make no requirement on the lepton charge . in electron events , we partially recover the effect of bremsstrahlung by searching for a photon with laboratory - frame energy @xmath83 gev within a 5@xmath84 cone around the electron direction at the interaction point . if such a photon is found , it is merged with the electron and removed from the event . the 4-momentum @xmath85 of the hadronic system @xmath86 recoiling against @xmath87 is determined by summing the 4-momenta of the remaining charged tracks and unmatched clusters in the event . we exclude tracks passing very far away from the interaction point or compatible with a multiply reconstructed track generated by a low - momentum particle spiraling in the central drift chamber . unmatched clusters in the barrel region must have an energy greater than 50 mev . higher thresholds are applied in the endcap regions . to improve the resolution in @xmath0 , we reject events with a missing mass larger than 3 gev@xmath88/@xmath89 . further improvement is obtained by recalculating the 4-momentum of the @xmath86 system , @xmath90 taking the neutrino 4-momentum @xmath91 to be @xmath92 , where @xmath93 is the missing 3-momentum . defined as the half width at half maximum , the resolution in @xmath0 obtained from @xmath94 is about 0.8 gev@xmath95 , compared to 1.4 gev@xmath95 in @xmath0 from @xmath85 . we consider the following contributions to the background in the @xmath0 spectrum : non-@xmath14 ( continuum ) background , combinatorial background , background from secondary or fake leptons and @xmath22 background . combinatorial background are true @xmath96 events for which reconstruction or flavor assignment of the tagged @xmath1 meson is not correct . the shapes of these background components in @xmath0 are determined from the mc simulation , except for the continuum where off - resonance data is used . the shape of the fake muon background is corrected by the ratio of the pion fake rate in the experimental data over the same quantity in the mc simulation , as measured using kinematically identified pions in @xmath97 decays . we derive the shape of the combinatorial background from the generic @xmath14 simulation by selecting events in which the reconstruction of @xmath27 does not correspond precisely to what was generated in the simulation . the continuum background is scaled by the integrated on- to off - resonance luminosity ratio , taking into account the cross - section difference . the mc - prediction of the combinatorial background is normalized to the data using the side - band region ( @xmath65 gev/@xmath32 and @xmath98 gev ) . the normalization of the secondary or fake lepton background is found from the data by fitting the electron ( muon ) momentum distribution @xmath99 @xcite in the @xmath1 meson rest frame in the range from 0.3 to 2.4 gev/@xmath78 ( 0.6 to 2.4 gev/@xmath78 ) . the @xmath100 component is normalized to the number of @xmath72 ( @xmath73 ) tags , assuming a branching fraction of @xmath101 ( @xmath102 ) for @xmath103 ( @xmath104 ) @xcite . the background in the @xmath0 spectrum is estimated separately in the four sub - samples , defined by the charge of @xmath27 ( @xmath72 , @xmath73 ) and the lepton type ( electron , muon ) . the purity of the @xmath6 signal depends on the sub - sample and the lepton energy threshold , typical values being around 75% . table [ tab:1 ] shows the numbers of signal events and purities for each combination of @xmath27 charge , lepton type and lepton energy threshold . c@cccc ' '' '' @xmath105 & @xmath72 electron & @xmath72 muon & @xmath73 electron & @xmath73 muon + ' '' '' 0.7 & @xmath106 ( 70.5% ) & @xmath107 ( 61.5% ) & @xmath108 ( 65.9% ) & @xmath109 ( 60.3% ) + 0.9 & @xmath110 ( 73.2% ) & @xmath111 ( 64.8% ) & @xmath112 ( 73.4% ) & @xmath113 ( 67.3% ) + 1.1 & @xmath114 ( 74.9% ) & @xmath115 ( 68.3% ) & @xmath116 ( 77.1% ) & @xmath117 ( 74.2% ) + 1.3 & @xmath118 ( 75.8% ) & @xmath119 ( 70.6% ) & @xmath120 ( 80.4% ) & @xmath121 ( 78.0% ) + 1.5 & @xmath122 ( 74.6% ) & @xmath123 ( 72.3% ) & @xmath124 ( 84.2% ) & @xmath125 ( 79.7% ) + 1.7 & @xmath126 ( 77.2% ) & @xmath127 ( 72.4% ) & @xmath128 ( 83.7% ) & @xmath129 ( 80.7% ) + ' '' '' 1.9 & @xmath130 ( 73.8% ) & @xmath131 ( 74.0% ) & @xmath132 ( 84.3% ) & @xmath133 ( 76.7% ) + we measure the @xmath0 spectrum in 45 bins in the range from 0 to 15 gev@xmath88/@xmath89 ( bin width 0.333 gev@xmath88/@xmath89 ) , which is shown in fig . [ fig:2 ] , and unfold the finite detector resolution in this distribution using the singular value decomposition ( svd ) algorithm @xcite . the unfolded @xmath0 spectrum has 15 bins in the range from @xmath134 to about 15 gev@xmath95 . the bin width is 1 gev@xmath95 , except around the narrow states @xmath70 , @xmath71 , @xmath135 and @xmath136 where smaller bin sizes are chosen . signal and the different background components , explained in more detail in the text . ] the unfolding is done separately in each sub - sample ( @xmath72 electron , @xmath72 muon , @xmath73 electron and @xmath73 muon ) . from the unfolded spectrum , we calculate the first moment and its statistical uncertainty squared , @xmath137 here , @xmath138 is the unfolded spectrum corrected for slightly different bin - to - bin efficiencies and @xmath86 is its covariance matrix , also determined by the svd algorithm . @xmath139 is the central value of the @xmath140-th bin of the unfolded spectrum . the second central and non - central moments , @xmath141 and @xmath142 are calculated from the same spectrum , substituting @xmath0 by @xmath143 and @xmath144 in eq . [ eq:1 ] , respectively . as the hadron mass moments are not expected to depend on the @xmath1 meson charge or the lepton type @xcite , we take the average over the four sub - sample results . we have tested the entire measurement procedure including event reconstruction , unfolding and moment calculation on mc simulated events and no significant bias has been observed over the full range of lepton energy thresholds . our measurements of @xmath145 , @xmath146 and @xmath142 for different lepton energy thresholds are shown in table [ tab:2 ] and fig . [ fig:3 ] . the sub - sample results for a given charge of @xmath27 ( @xmath72 , @xmath73 ) or lepton type ( electron , muon ) are compatible within their statistical uncertainty only . c@ccc ' '' '' @xmath105 ( gev ) & @xmath147 ( gev@xmath88/@xmath89 ) & @xmath148 ( gev@xmath149/@xmath150 ) & @xmath142 ( gev@xmath149/@xmath150 ) + ' '' '' 0.7 & @xmath151 & @xmath152 & @xmath153 + 0.9 & @xmath154 & @xmath155 & @xmath156 + 1.1 & @xmath157 & @xmath158 & @xmath159 + 1.3 & @xmath160 & @xmath161 & @xmath162 + 1.5 & @xmath163 & @xmath164 & @xmath165 + 1.7 & @xmath166 & @xmath167 & @xmath168 + ' '' '' 1.9 & @xmath169 & @xmath170 & @xmath171 + . the error bars indicate the statistical and total experimental errors . ] the different contributions to the systematic error are shown in tables [ tab:3][tab:5 ] . the total systematic error in table [ tab:2 ] corresponds to the quadratic sum of these components . l@ccccccc ' '' '' & + ' '' '' @xmath105 ( gev ) & 0.7 & 0.9 & 1.1 & 1.3 & 1.5 & 1.7 & 1.9 + ' '' '' secondary / fake leptons & 0.033 & 0.023 & 0.013 & 0.007 & 0.004 & 0.002 & 0.000 + combinatorial background & 0.006 & 0.004 & 0.003 & 0.002 & 0.002 & 0.002 & 0.000 + continuum & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 + ' '' '' @xmath172 background & 0.004 & 0.004 & 0.004 & 0.004 & 0.006 & 0.007 & 0.009 + ' '' '' @xmath173 & 0.008 & 0.007 & 0.007 & 0.007 & 0.006 & 0.005 & 0.003 + @xmath174 & 0.022 & 0.014 & 0.006 & 0.000 & 0.000 & 0.008 & 0.006 + @xmath175 & 0.024 & 0.017 & 0.007 & 0.004 & 0.004 & 0.004 & 0.004 + @xmath176 form factors & 0.013 & 0.013 & 0.012 & 0.011 & 0.010 & 0.008 & 0.006 + ' '' '' @xmath177 form factors & 0.003 & 0.002 & 0.002 & 0.001 & 0.001 & 0.001 & 0.004 + ' '' '' unfolding & 0.015 & 0.015 & 0.015 & 0.015 & 0.015 & 0.015 & 0.015 + binning & 0.001 & 0.001 & 0.001 & 0.001 & 0.001 & 0.000 & 0.001 + ' '' '' efficiency & 0.008 & 0.011 & 0.012 & 0.009 & 0.008 & 0.005 & 0.004 + ' '' '' total & 0.052 & 0.041 & 0.029 & 0.024 & 0.022 & 0.022 & 0.021 + l@ccccccc ' '' '' & + ' '' '' @xmath105 ( gev ) & 0.7 & 0.9 & 1.1 & 1.3 & 1.5 & 1.7 & 1.9 + ' '' '' secondary / fake leptons & 0.167 & 0.109 & 0.050 & 0.023 & 0.009 & 0.005 & 0.002 + combinatorial background & 0.028 & 0.018 & 0.009 & 0.005 & 0.003 & 0.002 & 0.001 + continuum & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.001 & 0.000 + ' '' '' @xmath172 background & 0.004 & 0.004 & 0.004 & 0.003 & 0.003 & 0.005 & 0.005 + ' '' '' @xmath173 & 0.013 & 0.010 & 0.007 & 0.004 & 0.002 & 0.002 & 0.003 + @xmath174 & 0.216 & 0.169 & 0.102 & 0.049 & 0.042 & 0.011 & 0.009 + @xmath175 & 0.168 & 0.125 & 0.058 & 0.041 & 0.024 & 0.004 & 0.004 + @xmath176 form factors & 0.029 & 0.028 & 0.024 & 0.019 & 0.017 & 0.016 & 0.007 + ' '' '' @xmath177 form factors & 0.013 & 0.009 & 0.006 & 0.003 & 0.004 & 0.001 & 0.004 + ' '' '' unfolding & 0.035 & 0.035 & 0.035 & 0.035 & 0.035 & 0.035 & 0.035 + binning & 0.001 & 0.001 & 0.000 & 0.001 & 0.001 & 0.001 & 0.002 + ' '' '' efficiency & 0.025 & 0.032 & 0.027 & 0.014 & 0.013 & 0.005 & 0.002 + ' '' '' total & 0.327 & 0.244 & 0.137 & 0.080 & 0.064 & 0.040 & 0.038 + l@ccccccc ' '' '' & + ' '' '' @xmath105 ( gev ) & 0.7 & 0.9 & 1.1 & 1.3 & 1.5 & 1.7 & 1.9 + ' '' '' secondary / fake leptons & 0.46 & 0.31 & 0.16 & 0.09 & 0.04 & 0.02 & 0.00 + combinatorial background & 0.08 & 0.05 & 0.04 & 0.03 & 0.01 & 0.01 & 0.00 + continuum & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 + ' '' '' @xmath172 background & 0.04 & 0.04 & 0.04 & 0.04 & 0.05 & 0.06 & 0.07 + ' '' '' @xmath173 & 0.07 & 0.07 & 0.06 & 0.05 & 0.04 & 0.03 & 0.02 + @xmath174 & 0.41 & 0.30 & 0.14 & 0.04 & 0.04 & 0.05 & 0.06 + @xmath175 & 0.38 & 0.28 & 0.12 & 0.08 & 0.06 & 0.04 & 0.04 + @xmath176 form factors & 0.15 & 0.14 & 0.13 & 0.12 & 0.10 & 0.09 & 0.04 + ' '' '' @xmath177 form factors & 0.04 & 0.03 & 0.02 & 0.01 & 0.01 & 0.00 & 0.03 + ' '' '' unfolding & 0.16 & 0.16 & 0.16 & 0.16 & 0.16 & 0.16 & 0.16 + binning & 0.01 & 0.01 & 0.01 & 0.00 & 0.00 & 0.01 & 0.00 + ' '' '' efficiency & 0.09 & 0.13 & 0.13 & 0.09 & 0.08 & 0.04 & 0.03 + ' '' '' total & 0.77 & 0.58 & 0.36 & 0.26 & 0.23 & 0.21 & 0.20 + the uncertainties related to the different background components in @xmath0 are estimated by varying the respective background normalization factors within @xmath178 standard deviation . we consider both variations of the @xmath179 branching fractions and form factor shapes . for the former , the ranges of variation are taken from ref . for the latter , the curvature @xmath180 in the form factor parametrization @xcite is varied within @xmath181 ( @xmath182 ) for @xmath17 ( @xmath183 ) @xcite . for @xmath17 , we also vary the form factor ratios @xmath184 and @xmath185 @xcite . the llsw model @xcite predicts the relative abundance and the form factor shape of the different components in @xmath19 only . to obtain the absolute branching fractions of the @xmath186 components and of @xmath187 , we use @xmath188 @xcite , the recent belle measurement of @xmath189 @xcite and the total semileptonic branching fraction @xcite . the uncertainty assigned to the @xmath186 branching fractions in tables [ tab:3][tab:5 ] reflects the uncertainty in these measurements and the change in the @xmath186 composition when varying the llsw parameters within their allowed range . the svd algorithm used to unfold the measured @xmath0 distribution requires the detector response matrix , _ i.e. _ , the distribution of measured versus true values of @xmath0 . we determine this matrix from the mc simulation . to study the systematics related to unfolding and a possible mismodeling of the detector response , we change the amount of bin - to - bin migration by varying the effective rank of the detector response matrix , the main tunable parameter of the svd algorithm . we have further studied a change of the binning of the unfolded distribution and the effect of disabling the bin - to - bin efficiency correction . due to overlapping events , the moment measurements corresponding to different lepton energy thresholds are highly correlated . systematic uncertainties are another source of correlation . we have estimated the correlations due to both sources using a toy mc approach based on 50,000 simulated measurements . the results for the self- and cross - correlation coefficients are given in tables [ tab:6][tab:10 ] . c@c|ccccccc & + & 0.7 & 0.9 & 1.1 & 1.3 & 1.5 & 1.7 & 1.9 + ' '' '' & 0.7 & 1.000 & 0.932 & 0.786 & 0.615 & 0.481 & 0.168 & 0.071 + & 0.9 & & 1.000 & 0.888 & 0.715 & 0.573 & 0.241 & 0.116 + & 1.1 & & & 1.000 & 0.849 & 0.693 & 0.363 & 0.194 + @xmath145 & 1.3 & & & & 1.000 & 0.804 & 0.470 & 0.254 + & 1.5 & & & & & 1.000 & 0.591 & 0.308 + & 1.7 & & & & & & 1.000 & 0.363 + ' '' '' & 1.9 & & & & & & & 1.000 + c@c|ccccccc & + & 0.7 & 0.9 & 1.1 & 1.3 & 1.5 & 1.7 & 1.9 + ' '' '' & 0.7 & 0.897 & 0.847 & 0.788 & 0.713 & 0.576 & 0.306 & 0.102 + & 0.9 & 0.777 & 0.843 & 0.804 & 0.726 & 0.608 & 0.356 & 0.144 + & 1.1 & 0.548 & 0.615 & 0.757 & 0.690 & 0.606 & 0.426 & 0.211 + @xmath145 & 1.3 & 0.328 & 0.371 & 0.483 & 0.718 & 0.599 & 0.476 & 0.260 + & 1.5 & 0.223 & 0.263 & 0.346 & 0.481 & 0.702 & 0.559 & 0.280 + & 1.7 & @xmath190 & @xmath191 & 0.035 & 0.126 & 0.237 & 0.846 & 0.296 + ' '' '' & 1.9 & @xmath192 & @xmath193 & @xmath194 & 0.040 & 0.075 & 0.228 & 0.865 + c@c|ccccccc & + & 0.7 & 0.9 & 1.1 & 1.3 & 1.5 & 1.7 & 1.9 + ' '' '' & 0.7 & 0.983 & 0.933 & 0.830 & 0.683 & 0.523 & 0.194 & 0.073 + & 0.9 & 0.890 & 0.974 & 0.910 & 0.765 & 0.606 & 0.264 & 0.117 + & 1.1 & 0.704 & 0.810 & 0.976 & 0.857 & 0.707 & 0.380 & 0.196 + @xmath145 & 1.3 & 0.508 & 0.601 & 0.774 & 0.980 & 0.800 & 0.479 & 0.258 + & 1.5 & 0.383 & 0.469 & 0.614 & 0.764 & 0.985 & 0.597 & 0.305 + & 1.7 & 0.079 & 0.137 & 0.273 & 0.403 & 0.539 & 0.994 & 0.357 + ' '' '' & 1.9 & 0.017 & 0.052 & 0.136 & 0.208 & 0.271 & 0.348 & 0.995 + c@c|ccccccc & + & 0.7 & 0.9 & 1.1 & 1.3 & 1.5 & 1.7 & 1.9 + ' '' '' & 0.7 & 1.000 & 0.939 & 0.838 & 0.698 & 0.534 & 0.167 & @xmath195 + & 0.9 & & 1.000 & 0.901 & 0.732 & 0.586 & 0.195 & @xmath196 + @xmath197 & 1.1 & & & 1.000 & 0.793 & 0.638 & 0.262 & 0.034 + @xmath198 & 1.3 & & & & 1.000 & 0.731 & 0.340 & 0.102 + & 1.5 & & & & & 1.000 & 0.484 & 0.146 + & 1.7 & & & & & & 1.000 & 0.296 + ' '' '' & 1.9 & & & & & & & 1.000 + c@c|ccccccc & + & 0.7 & 0.9 & 1.1 & 1.3 & 1.5 & 1.7 & 1.9 + ' '' '' & 0.7 & 1.000 & 0.932 & 0.784 & 0.601 & 0.442 & 0.111 & 0.017 + & 0.9 & & 1.000 & 0.877 & 0.684 & 0.524 & 0.168 & 0.051 + & 1.1 & & & 1.000 & 0.817 & 0.651 & 0.297 & 0.137 + @xmath142 & 1.3 & & & & 1.000 & 0.780 & 0.421 & 0.212 + & 1.5 & & & & & 1.000 & 0.557 & 0.270 + & 1.7 & & & & & & 1.000 & 0.346 + ' '' '' & 1.9 & & & & & & & 1.000 + we have measured the first , @xmath145 , and the second central and non - central moments , @xmath199 and @xmath142 , of the hadronic mass squared spectrum in @xmath6 decays for lepton energy thresholds ranging from 0.7 to 1.9 gev . using a toy mc approach , we have also evaluated the full covariance matrix for this set of measurements . it is expected that this measurement , combined with measurements of the semileptonic branching fraction , moments of the lepton energy spectrum in @xmath6 decays and possibly other moments , will lead to an improved determination of @xmath7-quark mass @xmath8 and the ckm matrix element @xmath5 @xcite . we thank the kekb group for the excellent operation of the accelerator , the kek cryogenics group for the efficient operation of the solenoid , and the kek computer group and the national institute of informatics for valuable computing and super - sinet network support . we acknowledge support from the ministry of education , culture , sports , science , and technology of japan and the japan society for the promotion of science ; the australian research council and the australian department of education , science and training ; the national science foundation of china and the knowledge innovation program of the chinese academy of sciences under contract no . 10575109 and ihep - u-503 ; the department of science and technology of india ; the bk21 program of the ministry of education of korea , the chep src program and basic research program ( grant no . r01 - 2005 - 000 - 10089 - 0 ) of the korea science and engineering foundation , the pure basic research group program of the korea research foundation , and the sbs foundation ; the polish state committee for scientific research ; the ministry of science and technology of the russian federation ; the slovenian research agency ; the swiss national science foundation ; the national science council and the ministry of education of taiwan ; and the u.s.department of energy . n. cabibbo , phys . * 10 * , 531 ( 1963 ) ; m. kobayashi and t. maskawa , prog . phys . * 49 * , 652 ( 1973 ) . d. benson , i. i. bigi , t. mannel and n. uraltsev , nucl . b * 665 * , 367 ( 2003 ) [ hep - ph/0302262 ] . gambino and n. uraltsev , eur . j. c * 34 * , 181 ( 2004 ) [ hep - ph/0401063 ] . d. benson , i. i. bigi and n. uraltsev , nucl . b * 710 * , 371 ( 2005 ) [ hep - ph/0410080 ] . c. w. bauer , z. ligeti , m. luke and a. v. manohar , phys . d * 67 * , 054012 ( 2003 ) [ hep - ph/0210027 ] . b. aubert _ et al . _ [ babar collaboration ] , phys . d * 69 * , 111103 ( 2004 ) [ hep - ex/0403031 ] . s. e. csorna _ et al . _ [ cleo collaboration ] , phys . rev . d * 70 * , 032002 ( 2004 ) [ hep - ex/0403052 ] . d. acosta _ et al . _ [ cdf collaboration ] , phys . rev . d * 71 * , 051103 ( 2005 ) [ hep - ex/0502003 ] . j. abdallah _ et al . _ [ delphi collaboration ] , eur . j. c * 45 * , 35 ( 2006 ) [ hep - ex/0510024 ] . we provide numbers for both definitions of the second moment , @xmath141 and @xmath142 , used in literature @xcite for the convenience of further applications though one can be derived from the other . a. hcker and v. kartvelishvili , nucl . instrum . a * 372 * , 469 ( 1996 ) [ hep - ph/9509307 ] . a. abashian _ et al . _ [ belle collaboration ] , nucl . instrum . meth . a * 479 * , 117 ( 2002 ) . s. kurokawa , nucl . instrum . a * 499 * , 1 ( 2003 ) , and other papers included in this volume . d. j. lange , nucl . instrum . a * 462 * , 152 ( 2001 ) . r. brun , f. bruyant , m. maire , a. c. mcpherson and p. zanarini , cern - dd / ee/84 - 1 . i. caprini , l. lellouch and m. neubert , nucl . b * 530 * , 153 ( 1998 ) [ hep - ph/9712417 ] . in this paper , the symbol @xmath200 refers collectively to the @xmath135 , @xmath136 , @xmath201 and @xmath202 states . a. k. leibovich , z. ligeti , i. w. stewart and m. b. wise , phys . d * 57 * , 308 ( 1998 ) [ hep - ph/9705467 ] . j. l. goity and w. roberts , phys . d * 51 * , 3459 ( 1995 ) [ hep - ph/9406236 ] . f. de fazio and m. neubert , jhep * 9906 * , 017 ( 1999 ) [ hep - ph/9905351 ] . p. ball and v. m. braun , phys . d * 58 * , 094016 ( 1998 ) [ hep - ph/9805422 ] . p. ball and r. zwicky , jhep * 0110 * , 019 ( 2001 ) [ hep - ph/0110115 ] . d. scora and n. isgur , phys . d * 52 * , 2783 ( 1995 ) [ hep - ph/9503486 ] . e. barberio and z. was , comput . commun . * 79 * , 291 ( 1994 ) . k. abe _ et al . _ [ belle collaboration ] , phys . d * 64 * , 072001 ( 2001 ) [ hep - ex/0103041 ] . in this paper , the inclusion of the charge conjugate decay is implied . k. hanagaki , h. kakuno , h. ikeda , t. iijima and t. tsukamoto , nucl . instrum . a * 485 * , 490 ( 2002 ) [ hep - ex/0108044 ] . a. abashian _ et al . _ , nucl . instrum . meth . a * 491 * , 69 ( 2002 ) . in this paper , quantities calculated in the @xmath1 meson rest frame are denoted by an asterisk . w. m. yao _ et al . _ [ particle data group ] , j. phys . g * 33 * , 1 ( 2006 ) . e. barberio _ et al . _ [ heavy flavor averaging group ( hfag ) ] , hep - ex/0603003 . b. aubert _ et al . _ [ babar collaboration ] , hep - ex/0602023 . d. liventsev _ et al . _ , phys . d * 72 * , 051109 ( 2005 ) .
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we present a measurement of the hadronic invariant mass squared ( @xmath0 ) spectrum in charmed semileptonic @xmath1 meson decays @xmath2 based on 140 fb@xmath3 of belle data collected near the @xmath4 resonance .
we determine the first , the second central and the second non - central moments of this spectrum for lepton energy thresholds ranging between 0.7 and 1.9 gev .
full correlations between these measurements are evaluated .
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in a number of recent works [ 1 - 3 ] we have calculated various hadronic correlation functions and compared our results to results obtained in lattice simulations of qcd [ 4 - 6 ] . the lattice results for the correlators , @xmath2 , may be used to obtain the corresponding spectral functions , @xmath3 , by making use of the relation g ( , t)=_0^d ( , t ) k ( , , t),where k ( , , t)=.the procedure to obtain @xmath3 from the knowledge of @xmath2 makes use of the maximum entropy method ( mem ) [ 7 - 9 ] , since @xmath2 is only known at a limited number of points . in our studies of meson spectra at @xmath4 and at @xmath5 we have made use of the nambu jona - lasinio ( njl ) model . the lagrangian of the generalized njl model we have used in our studies is l&=&(i - m^0)q+_i=0 ^ 8 [ ( ^i q)^2 + ( i _ 5 ^i q)^2 ] + & -&_i=0 ^ 8 [ ( ^i_q)^2 + ( ^i_5_q)^2 ] + & + & + + l_conf . here , @xmath6 is a current quark mass matrix , @xmath6=diag@xmath7 . the @xmath8 are the gell - mann ( flavor ) matrices and @xmath9 , with @xmath10 being the unit matrix . the fourth term is the t hooft interaction and @xmath11 represents the model of confinement used in our studies of meson properties . in the study of hadronic current correlators it is important to use a model which respects chiral symmetry , when @xmath12 . therefore , we make use of the lagrangian of eq . ( 1.3 ) , while neglecting the t hooft interaction and @xmath11 . thus , there are essentially three parameters to consider , @xmath13 , @xmath14 and a cutoff parameter @xmath15 , which restricts the momentum integrals so that @xmath16 . when we use the njl model to study matter at finite temperature , we introduce the temperature - dependent parameters @xmath17 and @xmath18 . ( we have also used a gaussian cutoff for the momentum integrals in our earlier work . ) these parameters have been adjusted to obtain fits to the spectral functions @xmath3 for @xmath19 and @xmath20 , which are the values of @xmath21 studied in the lattice simulations of qcd [ 10 ] . the temperature - dependent coupling constants and cutoff parameters of our work are analogous to the corresponding density - dependent parameters introduced in ref . [ 11 ] and [ 12 ] . further study of models with temperature - dependent and density - dependent parameters are of interest and a general theoretical formalism for the introduction of such dependencies should be considered . in this work we limit our study to the data for the vector correlator at @xmath0 [ 10 , 13 ] and therefore only need to specify @xmath18 and the momentum cutoff . ( we remark that the results for the scalar , vector , pseudoscalar and axialvector correlators are quite similar in the deconfined region . ) in figs . 1 and 2 we show the data obtained by the mem method at @xmath19 and @xmath20 for both pseudoscalar and vector correlators [ 10 , 13 ] . the second peaks in these correlators are known to be a lattice artifacts [ 13 ] . the organization of our work is as follows . in appendix a we present the formalism for calculation of pseudoscalar and vector correlation functions . in appendix a we discuss the calculation of the correlator in the case the quark and antiquark carry zero total momentum . in appendix b we show how the formalism is modified for the correlator calculated at finite momentum @xmath22 . in section ii we present the results of our calculation of the imaginary part of the correlator @xmath23 . ( since we place @xmath22 along the _ z_-axis this quantity may be written as @xmath24 in accord with the notation of ref . [ our results for @xmath23 will be presented for a series of values of @xmath25 in section ii . in section iii we present our result for the coordinate - dependent correlator @xmath26 which is proportional to the correlator defined in eq . ( 1 ) of ref . [ 10 ] , c(z)=12_-^dp_ze^ip_zz_0^d . we may also use the form c(z)=14_-^dp_ze^ip_zz_0^dp^2 . finally , in section iv we present our conclusion and further discussion . we make use of the formalism presented in appendices a and b to obtain values of the vector correlator at @xmath0 . here we take @xmath27 mev , since we have usually made comparison to lattice calculations made in the quenched approximation . in fig . 3 we present @xmath28 , for various values of @xmath25 , as function of @xmath29 . comparison may be made to the lattice data shown in fig . 2 [ 10 , 13 ] . ( we again note that our calculation does not reproduce the second peak in the lattice data which is known to be a lattice artifact [ 13 ] . ) the curves shown in fig . 3 are given for values of @xmath25 ranging from 0.10 gev to 2.10 gev in steps of 0.20 gev . for these calculations we have used @xmath30gev . our results for the various values of @xmath25 given in fig.3 may be compared to fig.20 of ref.[14 ] . we see that the results calculated by completely different methods are similar . for pseudoscalar states obtained by mem are shown [ 10,13 ] . the solid line is for @xmath19 and the dashed line is for @xmath20 . the second peak is a lattice artifact [ 13 ] . ] for vector states obtained by mem are shown [ 10 , 13 ] . see the caption of fig . 1 . the second peak is a lattice artifact [ 13 ] . ] is shown for various values of @xmath31 as a function of @xmath29 . starting with the topmost curve the values of @xmath32 in gev units are 0.10 , 0.30 , 0.50 , 0.70 , 0.90 , 1.10 , 1.30 , 1.50 , 1.70 , 1.90 and 2.10 . here we have used @xmath33 gev@xmath34 and @xmath30 gev . ] defined in eq.(1.5 ) is shown . the dotted line represents a fit using an exponential function . ] we have used the results of our calculations which were presented in fig . 3 to calculate @xmath26 of eq . ( 1.5 ) . the result of that calculation is shown in fig . we note that the simple assumption for the behavior of this correlator that is usually made , @xmath35 $ ] , is born out in our calculation for 1gev@xmath36gev@xmath37 . for the study of charmonium on the lattice , values found for the screening mass are given in ref . [ 14 ] . for the result shown in fig . 4 we obtain a screening mass of 1.02 gev which may be compared to the value of @xmath38@xmath39=1.27 gev . the dotted line represents an exponential fit to our result . recent theoretical work concerning the quark - gluon plasma has been discussed by shuryak @xcite . he notes that the physics of excited matter produced in heavy - ion collisions in the region @xmath40 is different from that of a weakly coupled quark - gluon plasma . that is due to the strong coupling generated by the bound states of the quasiparticles . these bound states appear as resonant structures when the imaginary parts of the hadronic correlators are extracted from lattice data using the maximum entropy method ( mem ) [ 4 - 9 ] . these resonances lead to very strong interactions between the quasiparticles and are , in part , responsible for very small mean free paths and collective flow in heavy - ion collisions . that flow may be described by hydrodynamics . indeed , shuryak suggests that ... if the system is macroscopically large , then its description via _ thermodynamics _ of its bulk properties ( like matter composition ) and _ hydrodynamics _ for space - time evolution should work " @xcite . in our present work we have presented a simple chiral model that is able to reproduce the resonances extracted from the lattice data via the mem procedure . we have calculated the imaginary parts of the vector correlator for various values of the total external momentum @xmath22 . we have also calculated the correlation function , @xmath26 , and find the simple exponential behavior @xmath35 $ ] for a limited range of _ z_. we have shown in an earlier work that exponential behavior with the appropriate screening mass may be obtained for the full range of _ z _ values if quite small values of @xmath15 ( of the order of 0.4 gev ) are used @xcite . finally , we note that an extensive discussion of screening masses appears in ref . @xcite . for ease of reference , we present a discussion of our calculation of hadronic current correlators taken from ref.[3 ] . the procedure we adopt is based upon the real - time finite - temperature formalism , in which the imaginary part of the polarization function may be calculated . then , the real part of the function is obtained using a dispersion relation . the result we need for this work has been already given in the work of kobes and semenoff @xcite . @xcite the quark momentum is @xmath41 and the antiquark momentum is @xmath42 . we will adopt that notation in this section for ease of reference to the results presented in ref . @xcite . ) with reference to eq.(5.4 ) of ref . @xcite , we write the imaginary part of the scalar polarization function as j_s(__p__^2 , t)=12n_c_s(__p__^0)^k_max ( ) + \{[1-n_1(k)-n_2(k ) ] ( _ _ p__^0-e_1(k)-e_2(k ) ) + -[n_1(k)-n_2(k ) ] ( _ _ p__^0+e_1(k)-e_2(k ) ) + -[n_2(k)-n_1(k ) ] ( _ _ p__^0-e_1(k)+e_2(k ) ) + -[1-n_1(k)-n_2(k ) ] ( _ _ p__^0+e_1(k)+e_2(k))}.here , @xmath43^{1/2}$ ] . relative to eq.(5.4 ) of ref . @xcite , we have changed the sign , removed a factor of @xmath44 and have included a statistical factor of @xmath45 . in addition , we have used a sharp cutoff , @xmath15 , for the momentum integral . we also note that n_1(k)=1e^e_1(k)+1,and n_2(k)=1e^e_2(k)+1.for the calculation of the imaginary part of the polarization function , we may put @xmath46 and @xmath47 , since in that calculation the quark and antiquark are on - mass - shell . in eq.(a1 ) the factor @xmath48 arises from a trace involving dirac matrices , such that _ s&=&-[(k+m_1)(k - p+m_2 ) ] + & = & 2p^2 - 2(m_1+m_2)^2,where @xmath49 and @xmath50 depend upon temperature . in the frame where @xmath51 , and in the case @xmath52 , we have @xmath53 . for the scalar case , with @xmath52 , we find j_s(p^2 , t)=(1-)^3/2 [ 1 - 2n_1(k)],where k^2=4-m^2(t),with @xmath16 . for pseudoscalar mesons , we replace @xmath48 by _ p&=&-[i_5(k+m_1)i_5(k - p+m_2 ) ] + & = & 2p^2 - 2(m_1-m_2)^2,which for @xmath52 is @xmath54 in the frame where @xmath51 . we find , for the @xmath38 mesons , j_p(p^2,t)=(1-)^1/2 [ 1 - 2n_1(k)],where @xmath55 , as above , with @xmath16 . thus , we see that the phase space factor has an exponent of 1/2 corresponding to a _ s_-wave amplitude . for the scalars , the exponent of the phase - space factor is 3/2 , as seen in eq.(a6 ) . for a study of vector mesons we consider _ ^v=[_(k+m_1)_(k - p+m_2)],and calculate g^_^v=4[p^2-m_1 ^ 2-m_2 ^ 2 + 4m_1m_2],which , in the equal - mass case , is equal to @xmath56 , when @xmath52 and @xmath51 . this result will be needed when we calculate the correlator of vector currents . note that , for the elevated temperatures considered in this work , @xmath57 is quite small , so that @xmath58 can be approximated by @xmath59 , when we consider the vector current correlation functions . in that case , we have j_v(p^2,t ) j_p(p^2,t).at this point it is useful to define functions that are not for @xmath60 : _ p(p^2,t)=(1-)^1/2[1 - 2n_1(k)],and _ v(p^2,t)=(1-)^1/2[1 - 2n_1(k)],for the functions defined in eq.(a14 ) and ( a15 ) we need to use a twice - subtracted dispersion relation to obtain @xmath61 , or @xmath62 . for example , _ p(p^2,t)=_p(0,t)+ [ _ p(p_0 ^ 2,t)-_p(0,t ) ] + + _ 4m^2(t)^^2 ds , where @xmath63 can be quite large , since the integral over the imaginary part of the polarization function is now convergent . we may introduce @xmath64 and @xmath65 as complex functions , since we now have both the real and imaginary parts of these functions . we note that the construction of either @xmath66 , or @xmath67 , by means of a dispersion relation does not require a subtraction . we use these functions to define the complex functions @xmath68 and @xmath69 . in order to make use of eq.(a16 ) , we need to specify @xmath70 and @xmath71 . we found it useful to take @xmath72 2 and to put @xmath73 and @xmath74 . the quantities @xmath75 and @xmath76 are determined in an analogous function . this procedure in which we fix the behavior of a function such as @xmath77 or @xmath77 is quite analogous to the procedure used in ref . @xcite . in that work we made use of dispersion relations to construct a continuous vector - isovector current correlation function which had the correct perturbative behavior for large @xmath78 and also described the low - energy resonance present in the correlator due to the excitation of the @xmath79 meson . in ref . @xcite the njl model was shown to provide a quite satisfactory description of the low - energy resonant behavior of the vector - isovector correlation function . we now consider the calculation of temperature - dependent hadronic current correlation functions . the general form of the correlator is a transform of a time - ordered product of currents , ic(p^2 , t)=d^4xe^ipx<>,where the double bracket is a reminder that we are considering the finite temperature case . for the study of pseudoscalar states , we may consider currents of the form @xmath80 , where , in the case of the @xmath38 mesons , @xmath81 and @xmath82 . for the study of scalar - isoscalar mesons , we introduce @xmath83 , where @xmath84 for the flavor - singlet current and @xmath85 for the flavor - octet current @xcite . in the case of the pseudoscalar - isovector mesons , the correlator may be expressed in terms of the basic vacuum polarization function of the njl model , @xmath86 . thus , c_p(p^2 , t)=j_p(p^2 , t),where @xmath87 is the coupling constant appropriate for our study of @xmath38 mesons . we have found @xmath88 by fitting the pion mass in a calculation made at @xmath4 , with @xmath89 gev . the result given in eq.(a18 ) is only expected to be useful for small @xmath90 , since the gaussian regulator strongly modifies the large @xmath90 behavior . therefore , we suggest that the following form is useful , if we are to consider the larger values of @xmath90 . = .(as usual , we put @xmath91 . ) this form has two important features . at large @xmath92 , @xmath93 is a constant , since @xmath94 is proportional to @xmath92 . further , the denominator of eq.(a19 ) goes to 1 for large @xmath92 . on the other hand , at small @xmath92 , the denominator is capable of describing resonant enhancement of the correlation function . as we have seen , the results obtained when eq.(a19 ) is used appear quite satisfactory . ( we may again refer to ref . @xcite , in which a similar approximation is described . ) for a study of the vector - isovector correlators , we introduce conserved vector currents @xmath95 with i=1 , 2 and 3 . in this case we define j_v^(p^2 , t)=(g^-)j_v(p^2 , t)and c_v^(p^2 , t)=(g^-)c_v(p^2 , t),taking into account the fact that the current @xmath96 is conserved . we may then use the fact that j_v(p^2,t ) = 13g_j_v^(p^2,t)and j_v(p^2,t)&= & 23(1-)^1/2[1 - 2n_1(k ) ] + & & j_p(p^2,t).(see eq.(a7 ) for the specification of @xmath97 . ) we then have c_v(p^2,t)=_v(p^2,t)11-g_v(t)j_v(p^2,t),where we have introduced _ v(p^2,t)&= & 23(1-)^1/2[1 - 2n_1(k ) ] + & & _ p(p^2,t ) . in the literature , @xmath98 is used instead of @xmath99 [ 4 - 6 ] . we may define the spectral functions _ v ( , t)=c_v ( , t),and _ p ( , t)=c_p ( , t ) , since different conventions are used in the literature [ 4 - 6 ] , we may use the notation @xmath100 and @xmath101 for the spectral functions given there . we have the following relations : _ p ( , t)=_p ( , t),and = _ v ( , t),where the factor 3/4 arises because , in refs . [ 4 - 6 ] , there is a division by 4 , while we have divided by 3 , as in eq.(a22 ) . here we extend the work of appendix a to consider case of finite three - momentum , @xmath102 . we consider the calculation of @xmath103 . the momenta @xmath104 and @xmath102 are the values external to the loop diagram . internal to the diagram , we have a quark of momentum @xmath105 leaving the left - hand vertex and an antiquark of momentum @xmath106 entering the left - hand vertex . it is useful to define e_1(k)&=&|+/2| + & = & ( k^2 + 4+kp)^1/2and e_2(k)&=&|-/2| + & = & ( k^2 + 4-kp)^1/2.here @xmath107 and @xmath108 . we have j_v(p^0 , , t)=12n_c_v(p^0)^k_max ( ) + \{[1-n_1(k)-n_2(k ) ] ( p^0-e_1(k)-e_2(k ) ) + -[n_1(k)-n_2(k ) ] ( p^0+e_1(k)-e_2(k ) ) + -[n_2(k)-n_1(k ) ] ( p^0-e_1(k)+e_2(k ) ) + -[1-n_1(k)-n_2(k ) ] ( p^0+e_1(k)+e_2(k))}.here , n_1(k)=1e^e_1(k)+1,and n_2(k)=1e^e_2(k)+1.in eq . ( b5 ) , the second and third terms cancel and the fourth term does not contribute . it is useful to rewrite @xmath109 using = 2||_x(-x),where x^2&=&^2 + & = & .we find ||=12kp||,and obtain j_p(p^0 , , t)=12n_c_p(p^0)(2)^2^k_max + 12e_1(k)e_2(k)[1-n_1(k)-n_2(k ) ] || + ( -x)d().we note there is a singularity when @xmath110 . that occurs when @xmath111 or @xmath112 . for our calculations we eliminate the point with @xmath112 when evaluating the angular integral over @xmath113 in the last expression . we obtain j_p(p^0 , , t)=n_c_p(p^0)^k_max k^2dk + .|_x , where _ x _ is obtained from eq . ( b9 ) , x=^1/2 99 bing he , hu li , c. m. shakin , and qing sun , phys . d * 67 * , 014022 ( 2003 ) . bing he , hu li , c. m. shakin , and qing sun , phys . d * 67 * , 114012 ( 2003 ) . bing he , hu li , c. m. shakin , and qing sun , phys . c * 67 * , 065203 ( 2003 ) . i. wetzorke , f. karsch , e. laermann , p. petreczky , and s. stickan , nucl . suppl . ) * 106 * , 510 ( 2002 ) f. karsch , s. datta , e. laermann , p. petreczky , and s. stickan , and i. wetzorke , nucl . a * 715 * , 701c ( 2003 ) f. karsch , e.laermann , p. petreczky , s. stickan , and i. wetzorke , phys . b * 530 * , 147 ( 2002 ) . m. asakawa , t. hatsuda and y. nakahara , nucl . a * 715 * , 863 ( 2003 ) t. umeda , k. nomura and h. matsufuru , hep - ph/0211003 . i. wetzorke , hep - ph/0305012 . ( invited talk at the seventh workshop on quantum chromodunamics , villefranche - sur - mer , france , jan . 6 - 10 , 2003 ) p. petreczky , j. phys.g * 30 * , s431-s440 ( 2004 ) . m. ruggieri , hep - ph/0310145 . r. casalbuoni , r. gatto , g. nardulli , and m. ruggieri , phys . d * 68 * , 034024 ( 2003 ) . p. petreczky , private communication . s. datta , f. karsch , p. petreczky and i. wetzorke , hep - lat/0312037 . a. das , _ finite temperature field theory _ ( world scientific , singapore , 1997 ) . e. shuryak , hep - ph/0312227 . r. l. kobes and g. w. semenoff , nucl . b * 260 * , 714 ( 1985 ) . c. m. shakin , wei - dong sun , and j. szweda , ann . of phys . ( ny ) * 241 * , 37 ( 1995 ) . hu li and c.m . shakin , hep - ph/0209136 . xiangdong li , hu li , c.m . shakin , and qing sun , nucl - th/0405035 . w. florkowski , acta phys . b * 28 * , 2079 ( 1997 ) .
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we have calculated spectral functions associated with hadronic current correlation functions for vector currents at finite temperature .
we made use of a model with chiral symmetry , temperature - dependent coupling constants and temperature - dependent momentum cutoff parameters .
our model has two parameters which are used to fix the magnitude and position of the large peak seen in the spectral functions . in our earlier work ,
good fits were obtained for the spectral functions that were extracted from lattice data by means of the maximum entropy method ( mem ) . in the present work
we extend our calculations and provide values for the three - momentum dependence of the vector correlation function at @xmath0 .
these results are used to obtain the correlation function in coordinate space , which is usually parametrized in terms of a screening mass .
our results for the three - momentum dependence of the spectral functions are similar to those found in a recent lattice qcd calculation for charmonium [ s. datta , f. karsch , p. petreczky and i. wetzorke , hep - lat/0312037 ] . for a limited range
we find the exponential behavior in coordinate space that is usually obtained for the spectral function for @xmath1 and which allows for the definition of a screening mass .
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even if we do not notice it , our life on the internet is influenced by recommendations . popular web sites such as amazon , netflix and youtube attempt to facilitate our navigation by suggesting us new possibly relevant items and thus increase our satisfaction and their profits @xcite . employed recommendation algorithms range from simple variants of `` buyers who choose item a also choose item b '' @xcite in amazon to more sophisticated techniques such as the singular value decomposition @xcite . even though many users still act independently of any automated assistance , the use of recommendation is on a rise . for example , the dvd rental company netflix estimated that 75@xmath0 of the rental choices of their users come from some form of recommendation @xcite . the rationale behind recommendation is to match the right customers with the right products . this task is particularly important and difficult for less popular items for which user patterns can not be easily identified . correct matching of little popular items is crucial for e - commerce studies have shown that from 20@xmath0 to 40@xmath0 of amazon s sales do not come from the best selling items @xcite . it has been suggested that if one ranks items according to their sales and thus constructs a so - called popularity - rank curve ( see an example in fig . [ fig : introduction]a ) , there is a long tail which comprises a large number of niche items @xcite . these niche items enjoy a higher profit margin compared to the small profit margin determined by a more competitive market of popular items , and can even boost the sales of other items by providing a convenient one - stop outlet to users @xcite . in this respect , recommendation algorithms seem to be the best candidate to explore the profit that hides in the long tail . recommendation working as intended thus contributes to : ( 1 ) increasing the diversity of recommended items , ( 2 ) distributing user attention more evenly among the items . in this case , recommendation would cause the long tail in fig . [ fig : introduction]a to gain more weight with time . however , figs . [ fig : introduction]b - d demonstrate that the opposite is found in reality . despite recommendation algorithms implemented at netflix , amazon , and movielens ( a web site for movie recommendation ) , the tail of the popularity distribution becomes shorter with time ( see also the evolution of the distribution in fig . simultaneously , the most popular items account for an increasing share of total sales . it is an adverse effect if some items become too dominant , similar to the emergence of over - dominant species in an ecosystem , which leads to the reduction of biodiversity and ultimately may lead the loss of the equilibrium in the system . evaluation of recommendation exclusively on the basis of accuracy - oriented metrics can not , by its nature , capture and explain this long - term behavior . when recommendation is iterated for a small number of rounds as in @xcite , the possible feedback between user choice and the recommender system is detected . these interactions between users and the system impact on its macroscopic properties and trend , similar to other physical systems . a more physical approach is thus needed to gain the first insights into the puzzle posed by fig . [ fig : introduction ] . to understand the long - term impact of recommender systems , one has to study the co - evolution of the recommendation and the online user - item network . the recommendation results affect the growth of the network and the change of the network meanwhile influence the future recommendation outcome . the effect is amplified with successive recommendations . in this paper , we investigate this issue by repeatedly applying recommendation on the user - item network . our focus is different from currently known recommendation studies which aim at short - term metrics such as accuracy @xcite , diversity @xcite and others @xcite . we demonstrate that the repeated use of usual recommendation algorithms makes the system reach a stationary state where user attention is concentrated on a few items instead of distributed over a broad range of items which is in agreement with the data presented in fig . [ fig : introduction ] . in other words , usual recommendation algorithms ultimately narrow user choice and reduce information horizons instead of widening them . we also observe a hysteresis phenomenon which implies that while recommendation naturally gives rise to hugely popular items , to revert this change is not possible because the uneven distribution of user attention is robust over a broad range of the recommendation algorithm s parameters . our observations directly challenge the role of recommendation for online retailers , as well as other applications of recommendation in search engines @xcite , online social networks @xcite , news media @xcite , and even suggestions of research papers @xcite . we finally show that before some items become too dominant , it is possible to make a compromise between recommendation accuracy and long - term effects of recommendation . these results can provide insights and motivations for the design of a next generation of recommender systems . * data . * the data used for the empirical study in fig . 1 and fig . s1 is described in the supplementary information . in the following simulation , we use two benchmark data sets : movielens ( an online movie rating and recommendation service ) and netflix ( a dvd rental service ) . the movielens data ( available at _ www.grouplens.org_ ) contains @xmath1 movies and @xmath2 users who have rated movies using the integer scale ranging from 1 ( worst ) to 5 ( best ) . to obtain an unweighted bipartite network , we represent any rating of 3 or more as a link between the respective user and item . the resulting network contains 82520 links . the average degree of users and items is @xmath3 and @xmath4 , respectively . the maximum degree of users and items is 509 and 558 , respectively . the netflix data is a subset of the original data set released for the purpose of the netflixprize ( available at _ www.netflixprize.com _ the subset contains 1014 users and 2049 movies chosen from the original data at random and all links among them ( since the input data uses the same rating scale as movielens , the threshold rating of @xmath5 is again used to determine whether a link is present or not ) . the resulting network contains 54093 links . the average degree of users and items is @xmath6 and @xmath7 , respectively . the maximum degree of users and items is 409 and 604 , respectively . * models and metrics . * to model the co - evolution of user choices and the recommendations generated from the recommender systems , we construct a model of recommendation ecosystem as follows . the real data described above are used as the initial configuration . the network evolves through a so - called rewiring process where each link is assigned a time stamp ( the initial time stamps are chosen at random ) . in each rewiring step , the oldest link of every user is redirected to a new item and assigned with the current time ( i.e. , it becomes the user s newest link ) . deleting the oldest links simulates the case where the recommendation results are generated based on recent historical record @xcite . we assume that when a new target item for a currently rewired link is being chosen , the user follows recommendation with probability @xmath8 and decides independently of recommendation with the complementary probability @xmath9 . we use the widespread item - based collaborative filtering ( icf ) as the recommendation algorithm ( see details below ) ; variants of this algorithm are employed by amazon and other major web sites @xcite . when a user follows recommendation , they select an item from their current recommendation list with probability inversely proportional to item rank in the list ( the motivation to use rank - reciprocal rather than equal probability for all listed items comes from @xcite ) . when a user acts independently of recommendation , they either choose an item at random ( which we refer to as random attachment , ra ) or they choose proportionally to item degree increased by one ( which we refer to as preferential attachment , pa ; item degree is incremented by one to make it possible for items which lose all their links to be chosen again ) . as the network evolves , degree of each user is preserved . on the other hand , network structure and item degree values change significantly by rewiring . network structure can be at any moment represented by a so - called network adjacency matrix @xmath10 whose element @xmath11 is one when user @xmath12 is currently connected with item @xmath13 and zero otherwise . we use the gini coefficient @xmath14 to measure inequality of the item popularity distribution during the network evolution . while this quantity has been originally proposed to quantify inequality of income or wealth distribution @xcite , it has also been used in other fields @xcite . the gini coefficient can be computed as @xmath15 where @xmath16 , the popularity of objects , has been sorted in the ascending order . the two extreme values of the gini coefficient are @xmath17 and @xmath18 which correspond to equal popularity of all items and zero popularity of all items but one , respectively . an increase of the gini coefficient thus corresponds to the item popularity distribution becoming more unequal . we also used other measures , such as the link share of 1@xmath0 most popular items and the herfindahl index , to study the rewiring model . results obtained with different inequality metrics are consistent with those obtained with the gini coefficient . * recommendation . * we use the widely spread item - based collaborative filtering ( icf ) as the main recommendation algorithm @xcite . in icf , the recommendation score of an item is computed based on the item s similarity with other items collected by a target user . the score of item @xmath13 for user @xmath12 reads @xmath19 where @xmath20 is the similarity of items @xmath13 and @xmath21 , and @xmath22 are elements of the network s adjacency matrix . we choose the following formula for item similarity @xmath23 where @xmath24 denotes the set of users who have collected item @xmath13 and @xmath16 denotes the degree of item @xmath13 . parameter @xmath25 can be continuously adjusted in the range @xmath26 $ ] which includes three classical cases : the common neighbor similarity which simply counts the number of users who have collected both items ( when @xmath27 ) , the cosine similarity which is sometimes referred to as the salton index ( when @xmath28 ) and the leicht - holme - newman similarity ( when @xmath29 ) @xcite . due to a lack of normalization , high - degree items are favored when @xmath27 . by contrast , low - degree items are favored when @xmath30 . equation ( [ simil ] ) thus gives us the opportunity to gradually move from recommendations biased towards high - degree items ( when @xmath27 ) to diversity - favoring recommendations biased towards low - degree items ( when @xmath29 ) . to obtain a recommendation list for a given user , all items that are currently not connected with this user are sorted according to their recommendation score in a descending order and finally the top @xmath31 items are kept on the recommendation list . we use @xmath32 here which is a common value in previous studies of recommender systems @xcite . considering only the top @xmath31 items is motivated by the fact that users in real online systems do not have time to inspect the list of all items ranked by their recommendation score ( as further reflected in the rank - reciprocal probability in our link rewiring process ) . * distribution of item popularity . * after a sufficient number of iteration steps in our model , the system reaches a stable state and the item degree distribution becomes stationary . we begin our analysis by comparing the distribution of item degree in the original data with the stationary outcome of the rewiring procedure . we assume here for simplicity that the users choose new items solely by recommendation ( i.e. @xmath33 ) . our implementation of item - based collaborative filtering employs a user similarity metric with one parameter given by eq . ( [ simil ] ) . three particular cases of our user similarity lead to well - known similarity measures : @xmath34 correspond to common neighbor similarity ( cn ) , cosine similarity ( cos ) , and leicht - holme - newman similarity ( lhn ) , respectively . the present parameterization allows us to continuously tune between recommendation that favors high - degree ( when @xmath25 is small ) and low - degree ( when @xmath25 is big ) items . this is well demonstrated by fig . [ fig : deg_distr ] which shows the item degrees in the original data compared to those after the rewiring procedures using cn and lhn . icf with cn ( cn - icf ) improves the popularity of the most popular items and essentially eliminates the long tail . this inferior outcome produced by an otherwise well accepted and popular recommendation method demonstrates the potential danger of recommendation for information diversity , similar to the loss of biodiversity in an ecosystem . on the other hand , lhn - icf strengthens the long tail but , as we shall see later , its recommendation accuracy is low . to obtain a quantitative comparison , we measure the gini coefficient over the item degrees . the higher the values , the more uneven the distributions , and the greater the loss of information diversity in the system . the gini coefficient corresponding to the distributions shown in fig . [ fig : deg_distr ] are @xmath35 , @xmath36 , @xmath37 ( movielens ) and @xmath38 , @xmath39 , @xmath40 ( netflix ) for the lhn - icf , original data , and cn - icf , respectively . the lhn - icf method leads to a remarkable equalization of the item popularity , while the cn - icf method further advances the degree heterogeneity . here we use @xmath32 . we show in fig . s2 for the results when other @xmath31 values are used . moreover , we study in fig . s3 the case where users are influenced by the similarity constraint when selecting items from the recommendation list . we want in this way to prevent all information from being washed away . * user reliance on recommendation . * in practice , users do not always follow recommendations . we thus consider the case with @xmath41 , i.e. users follow recommendation with a probability @xmath8 and otherwise ( with a probability @xmath9 ) choose an item according to preferential attachment ( as shown in fig . s4 and s5 , the resulting behavior is similar if preferential attachment is replaced by random or real choice of items ) . the value of @xmath8 thus characterizes users reliance on recommendation , and parameter @xmath25 controls the popularity bias of recommendations . while users recommendation lists are populated mostly with popular items when @xmath25 is large , the number of niche ( little popular ) items increases when @xmath25 is small . we repeatedly rewire the input data and again quantify the diversity of the system by the stationary value of the gini coefficient , @xmath42 . as shown in fig . [ fig : stationaryg]a - b for various @xmath8 values , @xmath42 is not sensitive to @xmath25 over a broad range @xmath43 $ ] where @xmath44 . once @xmath45 , @xmath42 decreases quickly and eventually reaches values lower than those produced by preferential attachment only . another important finding out of our expectations is that , recommendation can hurt information diversity even more than preferential attachment ( i.e. , icf can lead to higher @xmath42 than pa ) . this is because pa is not personalized and biased on popularity , but any items may be chosen ; recommendation algorithms are personalized , biased on popularity and only the top @xmath31 items identified by the algorithms are recommended . if the top @xmath31 items for different users significantly overlap , they will attract a large amount of links . this strong tendency of recommendation to decrease information diversity is a joint outcome of a popularity - favoring recommendation method and the fact that the users are recommended with a short list of items which further contributes to the winner takes it all " situation . we also consider a different recommendation algorithm @xcite which show remarkable similarity with the results for icf presented in this paper ( see results in fig . * the impact of data density . * the high density of these two data sets enables us to adjust the density of the network by removing some links . in fig . [ fig : stationaryg]c - d , we study the effect of data density on the resulting popularity inequality when @xmath33 ( see fig . s7 for the results with @xmath41 ) . we randomly remove links from the original network until the desired density is reached . note that the last link of the users will never be removed in this process . the effect of data density is particularly strong when recommendations are made with the diversity - favoring lhn - icf method which leads to low @xmath42 when the data density is high but does the opposite when the data density is low . this can be explained by the presence of objects with only a few links in low - density data : once those links are rewired to other objects , the respective object can not be any more recommended and it effectively disappears from the system , thus contributing to an increased popularity inequality and a high @xmath42 . nevertheless , a decrease in the data density does not always increase gini value . these observations may come from an interesting phenomenon , which can be shown by two different simulations . in the first case , users are randomly removed from the system and @xmath42 is found to increase with decreasing data density ( see fig . s8(c ) ) . in the second case , items are removed at random , and by contrast @xmath42 decreases with decreasing data density ( see fig . while these two scenarios look similar , the average item degree decreases in first case ( see fig . s8(a ) ) and is preserved in the second case ( see fig . s8(b ) ) . in other words , even though data density decreases , recommendation algorithms work equally effectively to distribute the popularity given the average item degree is preserved . this may come from the fact that the nature of an item can be reflected by the characteristics of individual users who have collected it , the degree of an item represents the amount of available information on it . as a result , when the average item degree is preserved , recommendation algorithm works effectively since there are sufficient information on the items . these results show that the degree of an item is not just merely related to its role in the network , but may also represents the amount of information we possess on it . the effectiveness of recommendation algorithms are thus strongly dependent on the average item degree . on the stationary gini coefficient ; links which are not drawn based on recommendation are drawn based on preferential attachment . ( c , d ) the effect of data density on the stationary gini coefficient for different recommendation methods ( here all links are drawn based on recommendation ) . the curve labeled original is the gini coefficient of the network after density modification and before rewiring . ] * hysteresis . * the task of choosing the recommendation parameter @xmath25 is made more important by a pronounced hysteresis phenomenon . [ fig : hysteresis ] shows that while a state with low inequality achieved with @xmath29 can be fully reverted to a high inequality state by changing @xmath25 to @xmath17 , the opposite is not true . in other words , once high concentration of item popularity has set in , it can only be partially corrected by the use of a diversity - favoring recommendation method . this is also related to the amount of information available on the objects after we lose information on some objects in a recommendation algorithm , we may not retrieve it again . effective recommendation can not be made for these items and the popularity remains unevenly distributed , even with diversity - favoring methods . in fig . s7 , we show that the hysteresis phenomenon exists also when @xmath41 . we thus conclude that @xmath42 depends on the system s initial condition , especially when it corresponds to highly heterogeneous item popularity . achieved with @xmath29 and high-@xmath42 achieved with @xmath27 , in ( a ) movielens and ( b ) netflix data . ( c ) and ( d ) show the effect of @xmath25 on the stationary gini ( icf method ) under different initial configurations . ] * trade - off between diversity and accuracy . * the previous figures show that the use of high @xmath25 can limit or even reverse the potential popularity - concentrating effect of recommendation . however , high @xmath25 generally leads to low recommendation accuracy because it tends to recommend objects of low degree @xcite . this motivates us to investigate the trade - off between the possibly decreased stationary gini coefficient @xmath42 and recommendation precision @xmath46 measured on the original input data before rewiring . to measure recommendation accuracy , we apply the standard evaluation procedure which is based on randomly dividing the network data into two parts : a training set @xmath47 comprising 90% of all links and a probe set @xmath48 comprising the rest . the training data @xmath47 is then used to compute recommendation lists for all users . the standard accuracy metric called _ precision _ is based on comparing these recommendation lists with the probe data @xmath48 : a good recommendation algorithm is expected to be able to reproduce a large part of @xmath48 based on @xmath47 @xcite . if for user @xmath12 , there are @xmath49 probe entries related to this user in @xmath12 s recommendation list , we say that the recommendation precision for this user is @xmath50 . by averaging this quantity over all users with at least one entry in the probe set @xmath48 , we obtain the overall recommendation precision @xmath51 . to further remove its possible dependence on the data division , we average precision over ten independent training set - probe set divisions . based on the training - probe sets division , one can also measure the short - term recommendation diversity which is simply the average degree of items that appear in the recommendation lists . s9 shows the relation between the long - term gini coefficient and short - term recommendation diversity . [ fig : sp_hy]a simultaneously plots @xmath42 and @xmath46 as a function of @xmath25 in the movielens data . one can see that the highest precision is achieved at @xmath52 which is a point where the stationary gini decreases quickly with @xmath25 . this gives us the possibility to lose some precision by increasing @xmath25 from its optimal value and in turn achieve a substantial decrease of @xmath42 . this is demonstrated by fig . [ fig : sp_hy]c where the desired value of @xmath42 is plotted on the horizontal axis : the resulting precision first decreases rather slowly from its highest value as the desired stationary gini coefficient is lowered . only when the desired @xmath42 is lower than the gini value in the original data , precision decreases sharply the situation is less favorable for the netflix data where precision is maximized at @xmath53 which lies in the region where @xmath42 changes rather slowly with @xmath25 as shown in fig . 5b . as a result , one has to substantially increase @xmath25 in order to achieve a significant decrease of @xmath42 . this manifests itself in fig . [ fig : sp_hy]d which lacks the gentle slope region seen in fig . [ fig : sp_hy]c . nevertheless , one can limit @xmath42 to the values seen in the original movielens and netflix data by sacrificing 17% and 12% of the optimal recommendation precision , respectively . these results show the possibility to compromise recommendation accuracy and long - term impacts on diversity for recommendation systems . in ( a ) movielens and ( b ) netflix data . panel ( c ) and ( d ) show the relation between the desired stationary gini coefficient and the short - term recommendation precision . ] by studying the co - evolution of the recommendations generated from the recommender systems and users choices , we demonstrate the long - term effect of recommendation on the distribution of item popularity . this novel approach to the evaluation of recommendation performance gives us the possibility to observe new phenomena . contrary to the common belief that recommendation helps to match niche items with users who may appreciate them and thus contribute to improving their recognition , we show that typical recommendation methods reinforce the position of already popular items at the cost of niche items . this is particularly true when recommendation algorithms optimized for their accuracy are used because these tend to favor popular items . our observations suggest that recommendation may divert the system to a state where a few items enjoy extraordinarily high levels of user attention . furthermore , we found a strong hysteresis effect which implies that this state is very robust and can be hardly changed back to a state with a more even popularity distribution , even with the help of a diversity - favoring recommendation algorithm of lesser accuracy . all these are adverse effects if one considers the interaction between users and the recommender system as an information ecosystem . we remark that although it is not ideal for buyers to receive recommendations of common popular items , such systems may still benefit the sellers by lowering the effective number of distinct items that they need to keep in their inventory and thus reducing the costs for logistics . to the best of our knowledge , our model is the first one to analyze the long - term influence of recommendation on the evolution of online systems . we tested many different choices of our model including examining different original data sets and another recommendation method , combining recommendation with random attachment instead of preferential attachment , preventing the real information from being washed out by repeated rewiring , and setting different lengths of the recommendation list . our finding of the adverse effect of a sub - optimal recommendation system on information ecosystems in the long run is in general still valid in these cases and thus needs to be seriously considered in practice . our work raises a number of questions which aim to further strengthen our understanding of the long term influence . for example , the model now assumes that all users accept the recommendation with the same probability . one can actually instead consider a scenario where experienced users search for items on their own and thus depend less on recommendation . we measure recommendation accuracy before the rewiring process . it would be interesting to monitor the recommendation accuracy during the network s evolution , and couple it with the rate at which users accept it ( the higher the accuracy , the higher the probability that users follow recommendation ) . these more complicated scenarios may make the results of the model quantitatively different from the current ones . a systematic study on these variances would be an interesting and important extension of the current model . our findings suggest a need for a next generation of recommender systems which would take into account both short - term and long - term goals . one might argue that the prime goal of a commercial system is to increase the profit by maximizing the recommendation accuracy and the long - term goal of enhancing or at least preserving the item diversity is secondary . our results suggest that this is a short - sighted approach as focusing on short - term performance of recommendation may ultimately lead to a system where the long tail has been decimated together with its economic potential . to overcome this problem , item diversity can be enhanced by sacrificing a small fraction of recommendation s short - term accuracy in exchange for higher long - term diversity . a detailed investigation of various approaches to study the long - term effects of recommendation as well as possible trade - offs between short- and long - term performance of recommendation are of great interest to both researchers and practitioners in the future . this work was partially supported by the youth scholars program of beijing normal university ( grant no . 2014nt38 ) , eu fp7 grant 611272 ( project growthcom ) and by the swiss national science foundation ( grant no . 200020 - 143272 ) . the work of chy is partially supported by the internal research grant rg 71/2013 - 2014r of hkied . 99 j. b. schafer , j. a. konstan and j. riedl , _ data min . discov . _ * 5 * , 115 ( 2001 ) . n . ziegler , s. m. mcnee , j. a. konstan and g. lausen , _ in proc . 14th international www conference , chiba , japan_. new york : acm . 22 - 32 ( 2005 ) . liang , h .- j . lai and y .- c . ku , _ j. manage . inform . syst . _ * 23 * , 45 ( 2007 ) j. a. konstan , b. n. miller , d. maltz , j. l. herlocker , l. r. gordon and j. riedl , _ commun . acm _ * 40 * , 77 ( 1997 ) . g. linden , b. smith and j. york , _ internet computing , ieee _ 7 , 76 - 80 ( 2003 ) . g. adomavicius and a. tuzhilin , _ ieee trans . _ , * 17 * , 734 ( 2005 ) . g. taks , i. pilzy , b. nmeth and d. tikk , _ acm sigkdd explorations newsletter _ * 9 * , 80 - 83 ( 2007 ) . netflix prize , http://techblog.netflix.com/2012/04/netflix-recommendations-beyond-5-stars.html . e. brynjolfsson , y. hu and m. d. smith , _ manag . sci . _ * 49 * , 1580 - 1596 ( 2003 ) . c. anderson , the long tail : why the future of business is selling less of more , hyperion ( 2006 ) . j. leskovec , l. a. adamic and b. a. huberman , _ acm trans . web _ * 1 * , 5 ( 2007 ) . h. yin , b. cui , j. li , j. yao and c. chen , _ proceedings vldb endowment _ * 5(9 ) * , 896 - 907 ( 2012 ) . a. zeng , c. h. yeung , m .- s . shang and y .- c . zhang , _ europhys . lett . _ * 97 * , 18005 ( 2012 ) . l. l , m. matus , c. h. yeung , y .- c . zhang , z .- k . zhang and t. zhou , _ phys . rep . _ * 519 * , 1 - 49 ( 2012 ) . t. zhou , z. kuscsik , j .- liu , m. medo , j. r. wakeling and y .- c . zhang , _ proc . natl acad . sci . _ * 107 * , 4511 - 4515 ( 2010 ) . j. l. herlocker , j. a. konstan , k. terveen and j. t. riedl , _ acm trans . * 22 * , 5 ( 2004 ) . l. page , s. brin , r. motwani and t. winograd , _ technical report . stanford infolab . _ * 1999 * , 66 ( 1999 ) . l. l , y .- c . zhang , c. h. yeung and t. zhou , _ plos one _ * 6 * , e21202 ( 2011 ) . h. kwak , c. lee , h. park and s. moon , _ in proc . 19th international www conference , raleigh , usa_. new york : acm . 591 - 600 ( 2010 ) . k. sugiyama and m .- y . kan , _ in proc . 10th annual joint conference on digital libraries , gold coast , australia_. new york : acm . 29 j. bennett and s. lanning , _ in proc . kdd cup and workshop ( acm ) , san jose , usa_. new york : acm . 35 ( 2007 ) . we first denote by @xmath54 the rank of item @xmath13 according to their increase in degree within a year @xmath55 . we further denote the number of active items in year @xmath55 as @xmath56 , one can find the normalized rank of item @xmath13 by @xmath57 . to ensure a fair comparison , popularity is rescaled to keep the area under the popularity - rank curves the same for the results from different years . this can be achieved by computing the normalized item popularity ( @xmath58 ) , given by @xmath59 , where @xmath60 is the degree of item @xmath13 in year @xmath55 . y. koren , _ commun . of acm _ * 53 * , 89 ( 2010 ) . r. lempel and s. moran , predictive caching and prefetching of query results in search engines . _ in proc . 12th international www conference , budapest , hungary_. new york : acm . 29 t. joachims , l. granka , b. pan , h. hembrooke , f. radlinski and g. gay , _ acm trans . syst . _ * 25 * , 7 ( 2007 ) . c. gini , variabilit e mutabilit ( in english : variability and mutability ) ( c. cuppini , bologna ) ( 1912 ) . l. wittebolle , et al . _ nature _ * 458 * , 623 - 626 ( 2009 ) . y. asada , _ popul . health metr . _ * 3 * , 7 ( 2005 ) . w. halffman and l. leydesdorff , _ minerva _ * 48 * , 55 - 72 ( 2010 ) . d. goldberg , d. nichols , b. oki and d. terry , _ commun . acm _ * 35 * , 61 - 70 ( 1992 ) . l. l and t. zhou , _ physica a _ * 390 * , 1150 ( 2011 ) .
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recommender systems daily influence our decisions on the internet .
while considerable attention has been given to issues such as recommendation accuracy and user privacy , the long - term mutual feedback between a recommender system and the decisions of its users has been neglected so far .
we propose here a model of network evolution which allows us to study the complex dynamics induced by this feedback , including the hysteresis effect which is typical for systems with non - linear dynamics . despite the popular belief that recommendation helps users to discover new things , we find that the long - term use of recommendation can contribute to the rise of extremely popular items and thus ultimately narrow the user choice .
these results are supported by measurements of the time evolution of item popularity inequality in real systems .
we show that this adverse effect of recommendation can be tamed by sacrificing part of short - term recommendation accuracy .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
magnetic reconnection is a process whereby the magnetic field line connectivity @xcite is modified due to the presence of a localized diffusion region . this gives rise to a change in magnetic field line topology and a release of magnetic energy into kinetic and thermal energy . reconnection of magnetic field lines is ubiquitous in laboratory , space and astrophysical plasmas , where it is believed to play a key role in many of the most striking and energetic phenomena . the most notable examples of such phenomena include sawtooth crashes @xcite and major disruptions in tokamak experiments @xcite , solar and stellar flares @xcite , coronal mass ejections @xcite , magnetospheric substorms @xcite , coronal heating @xcite , and high - energy emissions in pulsar wind nebulae , gamma - ray bursts and jets from active galactic nuclei @xcite . an exhaustive understanding of how magnetic reconnection proceeds in various regimes is therefore essential to shed light on these phenomena . in recent years , for the purpose of organizing the current knowledge of the reconnection dynamics that is expected in a system with given plasma parameters , a particular form of phase diagrams have been developed @xcite . these diagrams classify what `` phase '' of magnetic reconnection should occur in a particular system , which is identified by two dimensionless plasma parameters , the lundquist number @xmath4 and the macroscopic system size @xmath5 here , @xmath6 indicates the system size in the direction of the reconnecting current sheet , @xmath7 is the alfvn speed based on the reconnecting component of the magnetic field upstream of the diffusion region , @xmath8 is the magnetic diffusivity , and @xmath9 is the relevant kinetic length scale . this length scale corresponds to ( see , e.g. , * ? ? ? * ; * ? ? ? * ) @xmath10 \rho_\tau = c_s/\omega_{ci } & \mbox{for guide - field reconnection}. \end{array } \right.\ ] ] of course , @xmath11 is the ion plasma frequency , @xmath12 is the ion cyclotron frequency , and @xmath13 is the sound speed based on both the electron and ion temperatures . all the proposed phase diagrams @xcite exhibit a strong similarity and only a few minor differences . they are useful to summarize some of the current knowledge of the magnetic reconnection dynamics , but they lack fundamental aspects that can greatly affect the reconnection process ( some caveats in the use of these diagrams have been discussed by @xcite ) . for example , they do not take into account the dependence of the reconnection process on the external drive or on the magnetic free energy available in the system . an attempt to include these effects has been discussed by @xcite , who proposed to incorporate them by adjusting the definition of the lundquist number , eq . ( [ def_s ] ) , but this solution should be viewed only as a rough way to circumnavigate the problem . a further issue is that these diagrams do not consider the evolution of the reconnection process and predict reconnection rates wich are always fast ( the estimated reconnection inflow is always a significant fraction of @xmath7 ) . this , however , in not what is commonly observed in laboratory , space , and astrophysical plasmas , where magnetic reconnection exhibits disparate time scales and is often characterized by an impulsive behaviour , i.e. , a sudden increase in the time derivative of the reconnection rate ( see , e.g. , * ? ? ? * ; * ? ? ? * ) . here we propose a different point of view in which we include explicitly the effects of the external drive and the plasma viscosity ( neglected in all previous diagrams ) on the magnetic reconnection process by considering a four - dimensional parameter space . then , in this four - dimensional diagram we identify specific domains of parameters where the reconnection process exhibits distinct dynamical evolutions . in other words , in each of these domains the reconnection process goes through diverse phases characterized by different reconnection rates . this analysis leads us to evaluate in greater detail the dynamical evolution of a forced magnetic reconnection process , while collisionless effects have not been taken into account in the present work . we introduce the considered model of forced magnetic reconnection in sec . [ sec : taylor_model ] , whereas sec . [ sec : conditions ] is devoted to the presentation of the possible evolutions of the system and the conditions under which these different evolutions occur . in sec . [ sec : phase_diagrams ] we construct the parameter space diagrams that show which reconnection evolution is expected in a system with given characteristic parameters . finally , open issues are discussed in sec . [ sec : discussion ] . magnetic reconnection in a given system is conventionally categorized as spontaneous or forced . spontaneous magnetic reconnection refers to the case in which the reconnection arises by some internal instability of the system or loss of equilibrium , with the most typical example being the tearing mode . forced magnetic reconnection instead refers to the cases in which the reconnection is driven by some externally imposed flow or magnetic perturbation . in this case , one of the most important paradigms is the so - called `` taylor problem '' , which consists in the study of the evolution of the magnetic reconnection process in a tearing - stable slab plasma equilibrium which is subject to a small amplitude boundary perturbation . this situation is depicted in fig . [ fig1 ] , where the shared equilibrium magnetic field has the form @xmath14 with @xmath15 , @xmath16 and @xmath17 as constants , and the perfectly conducting walls which bound the plasma are located at @xmath18 . magnetic reconnection is driven at the resonant surface @xmath19 by a deformation of the conducting walls such that @xmath20 where @xmath21 is the perturbation wave number and @xmath22 is a small ( @xmath23 ) displacement amplitude . the boundary perturbation is assumed to be set up in a time scale that is long compared to the alfvn time @xmath24 , with @xmath25 , but short compared to any characteristic reconnection time scale . hence , the plasma can be considered in magnetostatic equilibrium everywhere except near the resonant surface at @xmath19 . the first and probably most important contribution to unveiling the behaviour of forced magnetic reconnection in taylor s model is due to @xcite , who showed that very small amplitude boundary perturbations cause an initial linear phase in which a current sheet builds up at the resonant surface , and successive phases in which the reconnection process evolves according to a linear resistive regime and a nonlinear rutherford regime @xcite . the scenario discussed by @xcite , which is characterized by a very slow evolution of the reconnection process , was complemented some years later by @xcite , who showed that larger perturbations may foster reconnection to proceed through the nonlinear regime according to a sweet - parker - like evolution @xcite , which only on the long time scale of resistive diffusion gives way to a rutherford evolution . the scenario outlined by @xcite is characterized by a reconnection evolution faster than that presented by @xcite , but it could still be slow for very small values of plasma resistivity , since in both the sweet - parker - like @xcite and rutherford @xcite regimes , the reconnection rate is strongly dependent on the resistivity , which is known to be extremely small in many laboratory fusion plasmas and space / astrophysical plasmas . however , recent works @xcite have shown that relatively large boundary perturbations lead to a different reconnection evolution in plasmas with small resistivity and viscosity . in these cases , after a linear inertial phase and an initial nonlinear regime characterized by a gradually evolving current sheet , the reconnection suddenly enters into a fast reconnection regime distinguished by the disruption of the current sheet due to the development of secondary magnetic islands ( usually called plasmoids @xcite ) . in addition to the works discussed above , which adopt a magnetohydrodynamic ( mhd ) description of the plasma , we emphasize that many other efforts have been devoted to investigate the tayor problem assuming mhd , two - fluid and kinetic descriptions ( see * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . indeed , the taylor problem has important applications besides being interesting from the point of view of basic physics . for instance , in laboratory fusion plasmas the taylor model represents a convenient way to study magnetic reconnection processes driven by resonant magnetic perturbations , while in astrophysical plasmas this model can be adopted to study magnetic reconnection forced by the motions of photospheric flux tubes . in this section we review the present understanding of the forced magnetic reconnection dynamics in taylor s model focusing on a visco - resistive plasma with @xmath1 greater than @xmath26 . as shown by @xcite , this dynamics always starts with a linear inertial phase in which a current sheet builds up at the resonant surface and shrinks inversely in time . concurrently , the current density at the @xmath27-point increases linearly in time . the reconnection rate during this phase can be evaluated by recalling that the current density is proportional to the out - of - plane electric field at the @xmath27-point , which is equal to @xmath28 for @xmath29 @xcite . here , @xmath30 stands for the magnetic flux function of the perturbed magnetic field in the reconnection plane ( @xmath31 ) , @xmath32 and @xmath33 indicate the characteristic time for viscous and resistive diffusion , respectively , while @xmath34 parametrizes the contribution of the external source perturbation to the gradient discontinuity of the magnetic flux function at the resonant surface . it is important to point out that this phase is characterized by a non - constant-@xmath30 behaviour of the magnetic flux function across the island . depending on whether or not this property persists until the beginning of the nonlinear regime , different scenarios may occur . if the boundary perturbation is such that @xcite @xmath35 after the inertial phase the reconnection process evolves trough a visco - resistive phase , which is a linear regime characterized by a constant-@xmath30 behaviour , i.e. , the perturbed magnetic flux function can be treated as a constant in @xmath36 over the width of the reconnection layer . during this phase the reconnection rate is given by @xmath37 where @xmath38 is the standard tearing stability parameter and @xmath39 is a characteristic time defined as @xcite @xmath40 with @xmath41 indicating the gamma function . ( [ reconnrate_vr ] ) is valid for @xmath42 @xcite and a magnetic island width much smaller than the linear layer width , i.e. , @xmath43 @xcite . if the perturbation is sufficient to drive the magnetic island into the nonlinear regime ( @xmath44 ) , the visco - resistive phase ends up into a rutherford evolution , whose island width growth is governed by the rutherford equation @xmath45 where @xmath46 @xcite . this is a very slow reconnection evolution in which the reconnection rate can be evaluated analytically in the two limits @xcite @xmath47 @xmath48 where @xmath49 if the boundary perturbation is such that @xcite @xmath50 the non - constant-@xmath30 behaviour characteristic of the inertial phase lingers until the nonlinear regime is entered . therefore , since in this case the magnetic island grows faster than the current can diffuse out of the reconnecting layer , the evolution of the reconnection process is distinguished by a strong current sheet at the resonant surface @xcite . the reconnecting current sheet turns out to be stable if the boundary perturbation is such that @xcite @xmath51 where the multiplicative constant @xmath52 depends on the critical inverse aspect ratio of the reconnecting current sheet ( specified later ) . in this case , the reconnection process follows a sweet - parker evolution ( modified by plasma viscosity @xcite ) , whose reconnection rate in taylor s model is @xcite @xmath53 finally , the sweet - parker type of evolution gives way to a rutherford evolution on the time scale of resistive diffusion . if the boundary perturbation satisfies @xcite @xmath54 and also the condition @xcite @xmath55 the reconnection process does not reach a stable sweet - parker regime , but a different situation occurs . a gradually thinning current sheet evolves until its aspect ratio reaches the limit that allows the plasmoid instability to develop . the growth of the plasmoids leads to the disruption of the current sheet , and therefore to a dramatic increase of the reconnection rate . the reconnection rate during this plasmoid - dominated phase has been evaluated in a statistical steady state as @xcite @xmath56 where @xmath57 is the critical inverse aspect ratio of the reconnecting current sheet . this quantity , whose value has been found to lie in the range @xmath58 - @xmath59 by means of numerical simulations @xcite , represents the threshold below which the reconnecting current sheet becomes unstable to the plasmoid instability @xcite . in this section we illustrate the domain of existence of the three different scenarios described before with the help of appropriated parameter space maps . for the sake of clarity we state again the three type of reconnection evolutions we are referring to : + ( 1 ) hahm - kulsrud scenario @xcite , + ( 2 ) wang - bhattacharjee scenario @xcite , + ( 3 ) our scenario @xcite . + since each of these scenarios includes different phases / regimes of reconnection , the concept of `` phase diagrams '' is intended here in a broader sense . due to this fact , they could also be defined in a more general way as `` scenario diagrams '' . this kind of diagrams can be constructed from the conditions summarized in the previous section . therefore , the possible evolutions of the reconnection process may be organized in a four - dimensional parameter space map with @xmath60 , @xmath61 , @xmath62 , and @xmath63 on the four axes . however , due to the difficulty in the visualization of such a four - dimensional diagram , it is convenient to consider two - dimensional slices for fixed values of two of the four parameters . let us first consider four two - dimensional slices with fixed values of the magnetic prandtl number and perturbation wave number . assuming that the hahm - kulsrud scenario ( which occurs if @xmath64 ) may hold until @xmath65 , the corresponding diagrams for ( a ) @xmath66 , @xmath67 , ( b ) @xmath66 , @xmath68 , ( c ) @xmath69 , @xmath67 , ( d ) @xmath69 , @xmath68 , are shown in figs . [ fig2](a ) - [ fig2](d ) . from this plots it is clear that the wang - bhattacharjee scenario is limited to a small range of values of the lundquist number and the source perturbation amplitude . increasing values of the the magnetic prandtl number and perturbation wave number extend the domain of existence of this possible type of evolution of the system . however , after a threshold value of the lundquist number ( identified by the intersection of the two black lines representing @xmath70 and @xmath71 ) , the wang - bhattacharjee scenario can not occur because it is not possible to obtain a stable sweet - parker - type evolution . in these cases , the hahm - kulsrud scenario is facilitated by very small perturbation amplitudes , whereas larger perturbations lead the system to a fast reconnection regime as described in sec . [ our_scenario ] . note that while previously proposed phase diagrams always predict fast reconnection @xcite , in clear contrast to what happens in nature , our diagrams show that reconnection proceeds very slowly ( region ( 1 ) ) if the source perturbation is not sufficiently large . let us now examine the effect of the plasma viscosity by considering the domain of existence of the different scenarios as a function of the parameters @xmath2 and @xmath1 . [ fig3](a ) shows the functions @xmath70 and @xmath71 for @xmath72 and @xmath73 . for @xmath74 the threshold for the plasmoids formation coincides with that for the nonlinear evolution characterized by a strong reconnecting current sheet . therefore , for @xmath74 an increase of the amplitude perturbation @xmath2 drives the system directly from scenario ( 1 ) to scenario ( 3 ) . this situation is depicted in fig . [ fig3](b ) , where it is clearly shown that the increase of the magnetic prandtl number has the effect of making possible or extending the domain of existence of scenario ( 2 ) . to clarify the effect of the plasma viscosity we also delineate the boundaries of the diverse evolutions ( 1)-(3 ) in a parameter space map @xmath75 ( as in fig . [ fig2 ] ) for fixed @xmath73 but different values of @xmath1 . this is shown in fig . [ fig4 ] for @xmath76 . the increase of the magnetic prandtl number extends the domain of existence of the slow reconnection scenario ( 1 ) at the expense of the fast reconnection scenario ( 3 ) . the area of existence of scenario ( 2 ) remains almost unchanged , but shifted towards higher values of the lundquist number . we now examine in more detail how the possible evolutions of the forced magnetic reconnection process depend on the wave number of the boundary perturbation . fig . [ fig5](a ) shows the thresholds @xmath70 and @xmath71 as a function of @xmath3 for fixed values of @xmath72 and @xmath77 . below a critical perturbation wave number @xmath78 ( corresponding to @xmath79 for the fixed parameters used in fig . [ fig5](a ) ) , every time the non - constant-@xmath30 magnetic island pass into the nonlinear regime , the evolution of the system leads to the plasmoid - dominated phase predicted in scenario ( 3 ) . the domains of existence of the possible evolutions ( 1)-(3 ) are illustrated in fig . [ fig5](b ) . the scenario discussed by wang and bhattacharjee happens only for a small range of ( @xmath80 ) parameters . not also that scenario ( 3 ) is facilitated for @xmath81 , while scenario ( 1 ) may occur for large amplitude boundary perturbations if @xmath82 . to better evaluate the effects of @xmath3 on the possible evolutions of the reconnection process , we plot in fig . [ fig6 ] the boundaries between the scenarios ( 1)-(3 ) in a parameter space map @xmath75 ( as in figs . [ fig2 ] and [ fig4 ] ) for fixed @xmath77 but different values of @xmath3 . the maximum area of existence of scenario ( 2 ) occurs for @xmath83 , while for @xmath84 and @xmath85 the scenario ( 2 ) appears for a very limited range of ( @xmath80 ) parameters . note also that the scenario ( 3 ) is greatly facilitated in the case of very large perturbation wave numbers ( @xmath84 ) while the scenario ( 3 ) is facilitated by relatively large amplitude perturbations with ( @xmath86 ) . the introduction of a new type of phase / scenario diagrams that include explicitly the effects of the external drive has allowed us to graphically organize in a detailed way the possible evolutions of forced magnetic reconnection processes in collisional plasmas . in contrast to previous versions of the phase diagrams @xcite , this new representation highlights regions of the parameter space ( @xmath80 , @xmath0 , @xmath1 ) in which reconnection is a slow diffusive process ( sec . [ sechk ] ) in addition to regions where reconnection can be fast ( secs . [ secwb ] and [ our_scenario ] ) . we recall that by fast we mean that the out - of - plane inductive electric field at the @xmath27-point is a significant fraction of the one evaluated upstream of the reconnection layer . we also emphasize that this kind of diagrams respond to the criticism moved by @xcite concerning the fact that these diagrams are not able of taking into account the dynamical evolution of the reconnection process from a slow to a fast regime inside of a given region of the parameter space . indeed , scenarios ( 1)-(3 ) describe the forced magnetic reconnection process from the current sheet formation all the way to their specific nonlinear evolution . we would like to remark that while the proposed parameter space diagrams represent a valid way to summarize the current knowledge of the forced magnetic reconnection dynamics in a collisional plasma , there are a number of conditions that may significantly affect the reconnection process which have not been addressed in this paper . for instance , two - fluid / kinetic effects should be considered if the length scale associated with the width of the reconnecting current sheet becomes of the order or smaller than the characteristic length scales of these effects . in fact , in antiparallel reconnection ( i.e. , in the absence of a guide magnetic field ) , hall effects @xcite are known to enhance the reconnection rate , as well as effects associated to finite electron inertia @xcite , electron pressure @xcite and ion gyration @xcite are known to increase the reconnection rate in the presence of a strong guide field . we would like to remark also that a common condition in many physical systems is the presence of velocity flows , which are known to suppress the reconnection @xcite or to alter the reconnection rate @xcite . in this case our analysis should be extended by considering also the effects of a plasma flow on the reconnection dynamics . similarly , also the effects of turbulence should be considered @xcite in order to obtain a more complete description of the magnetic reconnection dynamics . finally , it is important to recall that all the presented diagrams of magnetic reconnection are based on two - dimensional models and simulations . at present , the knowledge of how magnetic reconnection evolve in large three - dimensional systems is still far behind our understanding of what happens in two - dimensional systems . therefore , despite the great progress achieved in recent years @xcite , other work is needed in this direction before we can implement a phase diagram description of three - dimensional magnetic reconnection . the authors would like to acknowledge fruitful conversations with richard fitzpatrick and enzo lazzaro . this work was carried out under the contract of association euratom - enea and was also supported by the u.s . department of energy under contract no . de - fg02 - 04er-54742 .
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recent progress in the understanding of how externally driven magnetic reconnection evolves is organized in terms of parameter space diagrams .
these diagrams are constructed using four pivotal dimensionless parameters : the lundquist number @xmath0 , the magnetic prandtl number @xmath1 , the amplitude of the boundary perturbation @xmath2 , and the perturbation wave number @xmath3 .
this new representation highlights the parameters regions of a given system in which the magnetic reconnection process is expected to be distinguished by a specific evolution .
contrary to previously proposed phase diagrams , the diagrams introduced here take into account the dynamical evolution of the reconnection process and are able to predict slow or fast reconnection regimes for the same values of @xmath0 and @xmath1 , depending on the parameters that characterize the external drive , never considered so far .
these features are important to understand the onset and evolution of magnetic reconnection in diverse physical systems .
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hexagonal boron nitride ( @xmath0-bn ) has the same honeycomb lattice as graphite with two atoms per unit cell and similar lattice parameters . due to this similarity , boron nitride materials have attracted a growing interest in line with the development of low - dimensional carbon - related materials . similarly to carbon , bn materials can be synthesized either as nanotubes ( one - dimensional ( 1d ) form ) @xcite or as monolayers and/or multilayers ( two - dimensional ( 2d ) form).@xcite in the following we focus on this latter form . 2d layers of carbon , namely graphene sheets , display extraordinary electronic properties which open unanticipated routes for a new generation of electronic devices . however , the electron mobility of supported graphene typically falls short of that of suspended graphene , due to detrimental effects of substrate disorder and adsorbents . @xcite facing this problem , @xmath0-bn layers are of particular interest as support or capping layers of graphene . they combine several properties : they are insulating ( @xmath0-bn is a large gap semiconductor due to the polar bn bond ) , @xcite they display an especially compatible layered @xmath2 structure with that of graphene , they have a low concentration of charges impurities and they can be very flat due to an easy cleavage . owing to these properties , graphene transferred on bn layers displays an electron mobility at room temperature of , which is the highest reported value for a supported graphene @xcite and very close to that of suspended graphene . @xcite beyond the high mobility of graphene supported on bn , their excellent lattice matching is promising for the realization of heterostructures of these materials for vertical transport stacking , in which graphene layers act as tunable metallic electrodes for the bn quasi - ideal tunnel barrier . @xcite these promising perspectives have been demonstrated by pioneering experiments done using sheets mechanically exfoliated from both graphite and @xmath0-bn single crystals . in the future , @xmath0-bn and graphene based devices and heterostructures would most probably use chemical vapor deposited ( cvd ) polycrystalline films and sheets . their performances would only be achieved via an accurate control of the defects in both graphene and bn layers and of the layers engineering . while the electronic properties of graphene have been well described theoretically and investigated experimentally , this is not the case of bn layers and even of @xmath0-bn . this is due to both the scarcity of high quality materials and to the nature of their electronic properties dictated by the large gap . it is thus a basic issue to understand the spectroscopic properties of atomically thin @xmath0-bn layers and their intrinsic defects , which is the focus of this paper . in contrast to graphene , usual spectroscopic characterization techniques such as raman are not easy to manipulate or they provide poor information when used for @xmath0-bn . absorption and luminescence spectroscopies have been shown to be the most direct approach to investigate the electronic properties of bn materials , due to their large gap . to this aim , dedicated cathodolumnescence and photoluminescence experiments have been recently developed and applied to bn powders and single crystals . @xcite both theoretical calculations @xcite and the most recent excitation photoluminescence experiments on single crystals @xcite converge to establish the band gap of @xmath0-bn near . furthermore , it is now commonly accepted that @xmath0-bn optical properties are dominated by huge excitonic effects . the near - band - edge luminescence spectrum is composed of two series of lines . referring to measurements done on single crystals in ref . [ ] , they are defined as the @xmath3 and @xmath1 series . the four higher energy lines , labeled @xmath4 to @xmath5 , located between 5.7 and , are attributed to the excitons , whereas the lower energy ones , labeled @xmath6 to @xmath7 , between 5.4 and , are assigned to excitons trapped to structural defects . @xcite the excitons in @xmath0-bn are more of frenkel - type than of wannier - type ( as in others usual semiconductors , such as aln with a gap ) . _ ab initio _ calculations indeed predict that the spatial extension of the exciton wavefunction is of the order of one @xmath0-bn atomic layer.@xcite moreover the experimental stokes shift of observed for the @xmath5-line suggests its self - trapping , @xcite consistent with the very localized view of the frenkel exciton . to complete this view , the effect of a reduction in the @xmath0-bn thickness down to the atomic level has to be analyzed . up to now , only scarce studies deal with the optical properties of nanometer - thick bn layers . an optical absorption edge between 5.6 and at room temperature is reported , @xcite _ i.e. _ in the same range than in bulk @xmath0-bn . only two studies report near - band edge recombination luminescence , with no correlation to the bn layer thickness under investigation . @xcite in this paper we present the first study of the luminescence properties of single bn nanosheets , with well - known thickness , by combining atomic force microscopy ( afm ) and cathodoluminescence ( cl ) measurements . bn nanosheets were prepared by mechanical exfoliation of small @xmath0-bn crystallites of a polycrystalline powder . this material offers the advantage to give access at the same time to the intrinsic optical response of the crystallite as well as to the effect of grain boundaries and the crystallite thickness on this response . an advanced characterization of the starting bulk material is first presented and its near - band - edge recombinations observed by cl are discussed with respect to those of the single crystal . then the luminescence of the exfoliated bn sheets is presented and discussed as a function of their thickness . the bulk material exfoliated in this paper is the high purity trs bn@xmath8 st - gobain commercial power puhp1108 , used in cosmetic applications . the @xmath0-bn crystallites are already shaped as flakes with large ( 00.1 ) surfaces of typically diameter , which is particularly convenient for the exfoliation process . their thickness is about . one of these crystallites is shown in the scanning electron microscope ( sem ) image of . this powder was synthesized at high temperature from boric acid and a nitrogen source . our measurements on this powder are compared with the ones obtained from a single crystal of the best available quality , synthesized by a high - pressure high - temperature ( hpht ) crystal process @xcite and provided by taniguchi _ _ this reference sample is hereafter referred to as the hpht sample . exfoliation of few - layer @xmath0-bn was carried out by mechanical peeling following the same method used for graphene . @xcite the powder is applied to an adhesive tape , whose repeated folding and peeling apart separates the layers . these thin layers are then transferred on a si wafer covered with of , which is the optimal thickness for imaging bn flakes with maximum contrast . @xcite prior to the @xmath0-bn deposition , cr / au localization marks were formed on the wafer by means of uv - lithography using an az5214e photo - resist and joule evaporation . @xcite the marks facilitate the localization of the flakes for the different measurements to be achieved on a given flake . still prior to transferring the layers , the wafer is chemically cleaned with acetone and isopropanol , followed by several minutes of exposure to a strong plasma ( , @xmath9 ) . this last step eliminates most contaminants but also renders the surface hydrophilic,@xcite which has the drawback of facilitating the inclusion of a thin water layer between the flakes and the substrate.@xcite as we shall see later , this complicates the layer thickness measurement done by atomic force microscopy ( afm ) . the surface morphology of the @xmath0-bn exfoliated layers was investigated by atomic force microscopy with a dimension 3100 scanning probe microscope ( brukers ) operating in tapping mode with commercial mpp-11100 probes . the cathodoluminescence of the @xmath0-bn samples was analyzed at low temperature in the ( i ) spectroscopic , ( ii ) imaging and ( iii ) spectral mapping modes , using an optical system ( horiba jobin yvon sa ) installed on a high resolution jeol7001f field - emission scanning electron microscope . the samples are mounted on a gatan cryostat sem stage and cooled down to @xmath10 with a continuous flow of liquid helium . they are excited by electrons accelerated at with a beam current as low as . the cl emission is collected by a parabolic mirror and focused with mirror optics on the entrance slit of a -focal length monochromator . the all - mirror optics combined with a suitable choice of uv detectors and gratings ensures a high spectral sensitivity down to . a silicon charge - coupled - display ( ccd ) camera is used to record spectra in mode ( i ) and as well for the spectral mapping mode ( iii ) . in mode ( iii ) , also referred as the hyperspectral imaging acquisition mode , the focused electron beam is scanned step by step with the _ hjy cl link _ drive unit ( co - developed and tested at gemac ) and synchronized with the ccd camera to record one spectrum for each point . the spectrometer is also equipped with a uv photomultiplier on the lateral side exit for fast monochromatic cl imaging ( _ i.e. _ image of the luminescence at a given wavelength ) in mode ( ii ) . for all spectra reported in this paper , it was checked that the linewidths are not limited by the spectral resolution of the apparatus ( in the best conditions ) . in the case of @xmath0-bn exfoliated layers , cl was only performed in mode ( i ) and using a fast _ e_-beam scanning of a 1.5@xmath11 area on the sample instead of using a fixed focused beam , in order to minimize the irradiation dose and the _ e_-beam induced modifications . the @xmath0-bn crystallites are often made of a few single crystals separated by grain boundaries , as for another powder studied in a previous work.@xcite owing to the electron diffraction patterns recorded by transmission electron microscopy ( tem ) on the specimen presented in , each grain could be precisely orientated along the ( 00.1 ) zone axis ( not shown here , for experimental details , see ref . the tem analysis shows that the two main grains of the crystallite in are slightly tilted one with respect to the other . they are separated by the grain boundary labeled # 1 . -bn materials in the near - band edge region : at the top is a reference spectrum of a hpht high quality crystal , @xcite compared to # 1 , registered in the grain boundary area delimited by the red rectangle ( averaged over 15 spectra ) and to # 2 on the main grain of the crystallite in the area delimited by the green square ( averaged over 240 spectra ) , as labeled in . the specimen temperature is about . ] cl spectra recorded on different areas ( labelled # 1 and # 2 ) of the crystallite in are shown in . they are compared to the one recorded on the hpht reference sample and hereafter referred to as the reference spectrum . it is worth mentioning that this reference sample precisely displays the same exciton recombination energies than the ones observed in photoluminescence experiments done on a single crystal of the same origin as reported in ref . spectrum # 1 in is recorded at the grain boundary of the @xmath0-bn crystallite . it presents the same features as the reference sample , with clear @xmath3and @xmath1 series . the @xmath1 lines are slightly shifted toward lower energies by about when compared to the reference spectrum . since the s lines are not shifted , such a decrease of the emission energy is attributed to a different binding energy of the exciton to the grain boundary , rather than to a change of bandgap energy . moreover the @xmath3 and @xmath1 series appear broader than in the reference sample , possibly due to an inhomogeneous distribution of residual strain in the sample . this interpretation is supported by the slight fluctuations of the recombination energies of a few mev , observed from point to point in a cl mapping ( not shown here ) . moreover we remark that the relative intensities of @xmath3 and @xmath1 series are very different when compared to the reference spectrum . spectrum # 1 is dominated by the @xmath1 series . it is known that this series is linked to the presence of structural defects such as grain boundaries , dislocations , or stacking faults . @xcite to confirm this interpretation , monochromatic images have been registered , at energy centered on the @xmath5 line ( at , ) and on the d4 series ( at , ) . both @xmath1 and @xmath3 emission series are slightly enhanced ( by a factor of two in average ) at the crystallite edges . this enhancement confirms previous observations on single @xmath0-bn crystals by kubota _ _ , @xcite and is consistent with their interpretation based on a more efficient light scattering at the grain edges . more interestingly , a clear one - to - one correlation is observed between the @xmath1 series emission and the grain boundary locations ( ) . to better quantify this view and to dispose of a comparison criterion , we define the @xmath12 ratio as the ratio of amplitudes between the @xmath7 and the @xmath5 peaks ( at and , resp . 227 and ) . it can be viewed as an indicator of the defect density , as introduced by watanabe _ _ @xcite and in the same way than in conventional semiconductors for the impurities content . @xcite owing to cl spectral mapping facilities , the @xmath12 ratio is evaluated in each point of the studied @xmath0-bn crystallite of , and shown in . it exhibits a variation of two orders of magnitude ( 0.2 20 ) . this reveals the extremely high impact of structural defects in bulk @xmath0-bn . it is also remarkable that in the central part of the upper grain , the @xmath12 ratio is four times lower than in our reference hpht sample ( @xmath13 ) , suggesting a better structural quality . as the @xmath1 series almost vanishes , new intrinsic features are revealed , which were hindered by the @xmath1 series in previous studies . indeed in the cl spectrum # 2 of recorded from the central part of the grain , three new lines are detected at 5.619 , 5.457 and . they are shifted by @xmath14 ( with @xmath15 ) times ( ) from the @xmath16 band , a period close to the ( ) optical phonon mode at the center of the brillouin zone . @xcite they are also characterized by a monotonous decay in intensity upon increasing @xmath14 . we thus attribute these new lines to three phonon replica of the dominant @xmath3 exciton recombinations . we notice that the energy of the second phonon replica almost coincides with the @xmath7 line . as a consequence , the @xmath12 ratio as defined here , has an intrinsic lower value , which corresponds to the ratio between the @xmath17 replica and the fundamental @xmath3 line . the observation of phonon replica attests to the existence of a strong electron - phonon coupling occurring in this material , as evidenced previously on a deep impurity band . @xcite turning now to the sheets exfoliated from the bulk material studied in section iiia , several sheets have been studied from which we show three representative samples in . the surface of each @xmath0-bn flake was scanned by afm in the tapping mode in order to measure its thickness , consecutively to the cl studies . the sheet surface is not completely flat , with some prominent dots being visible . most probably they are adhesive tape glue residues from the exfoliation process . some small 10- size holes are also observable . it is not clear whether they were already in the bulk material or whether they are due to electron exposure as reported in the literature . indeed when etching under an electron beam in a transmission electron microscope , even at a low acceleration voltage , they are often observed . @xcite aside from this , the flakes appear to be well spread on top of the substrate . [ cols="<,^,^,^",options="header " , ] particular attention has been paid to the thickness determination because of the water adsorbed at the surface and trapped under the bn flakes . this problem is well known in the case of graphene on @xcite and already mentioned for @xmath0-bn on . @xcite due to the presence of the water layer , the height profiles recorded by afm on steps at the flake edges overestimate the @xmath0-bn thickness . the special geometry of sample a , which displays a folded part on itself , gives a key to solving this problem . the step height of the folded part provides a thickness measurement independent of the amount of water trapped between @xmath0-bn and , and this measurement has been used as the reference for the thickness measurement as follows . the @xmath0-bn flake thickness is found to be equal to a value of averaged over five measurements of the step height along the edge ( see ) . the standard deviation is found to be below , the interplanar distance along the @xmath18 axis in @xmath0-bn , indicating that the thickness of flake a is homogeneous . knowing the @xmath0-bn interplanar distance and assuming an integer value of the layer number , we consider that the exfoliated sample a is 6 atomic layers thick . then the water layer thickness is deduced by difference with the step edges and is found equal to @xmath19 over seven measurements . the thickness of samples b ( ) and c ( ) is measured assuming a homogenous water layer thickness over the wafer . the results are summarized in . one can notice that the areas scanned during the cl acquisition appear as dark rectangles in afm images , which indicates a lower height . this could be interpreted by an electron beam induced etching . as we can also observe a lower height on the , it is more probably a reduction of the volume of the amorphous silica under irradiation . @xcite the cl spectra of the three exfoliated @xmath0-bn layers presented above are shown in . first , we observe that the cl intensity decreases when decreasing the @xmath0-bn layer thickness . since a nanometer - thick @xmath0-bn layer is transparent to the electron beam , the electron - hole generation in such a thin layer is directly proportional to its thickness . in spite of this effect , exfoliated layers show a significant luminescence in comparison to their nanometric thickness , which is a result in itself . indeed in semiconductor nanostructures , the surface to volume ratio , being extremely large , often results in non radiative surface recombinations , responsible for the quenching of luminescence . @xcite our results indicate that such a surface effect is weak in @xmath0-bn , probably due to the @xmath2 character of the bonding combined with the strong localization of the exciton at the atomic layer scale . this is an advantage of this material compared to other semiconductors , for which passivating the surface is necessary for enhancing the luminescence . @xcite the cl spectra of exfoliated layers are dominated by the @xmath7 and @xmath20 lines related to the @xmath1 series . weaker and unresolved @xmath21 and @xmath22 lines correspond to the @xmath3 series , clearly attributed by comparison with the reference hpht spectrum . looking in more detail at the excitonic recombination emission , we observe a shift from for the reference spectrum to 5.479 , 5.477 and for exfoliated samples c ( ) , b ( ) and a ( ) respectively . for the 8 and -thick samples , the blueshift is observed for the defect - related @xmath1 lines but also for the @xmath3 lines , indicating a modification in the intrinsic exciton emission energy ( change in bandgap and/or in exciton binding energy ) . for the thinnest sample , the intrinsic @xmath3 line emission is not observed possibly because of a too low signal - to - noise ratio . -bn layer a ( ) , b ( ) and c ( ) compared to the normalized reference spectrum ( from the hpht crystal ) . the zero of the cl spectra was shifted for clarity . ] to interpret our results , it has to be emphasized first that a calculation including excitonic effects still needs to be performed to describe the influence of the number of atomic layers on @xmath0-bn optical properties . however , inside bulk @xmath0-bn crystals , the strong localization of the exciton wavefunction around a single atomic basal plane has been evidenced by arnaud _ et al . _ @xcite it indicates that the regime of quantum confinement is still far to be reached with bn sheets in the range of 6- , and that exciton recombination properties are probably close to the ones of bulk @xmath0-bn . it is also interesting to consider the theoretical work of wirtz _ et al . _ @xcite it deals with the optical response of a single sheet of @xmath0-bn as a function of the intersheet distance . when the interdistance between @xmath0-bn planes becomes large , the situation tends to uncoupled bn sheets of atomic thickness . the calculation results as follows : the exciton binding energy increases when the intersheet distance increases , but simultaneously the quasiparticle band gap increases , so that the net result is a weak blueshift , about , in the error bar of such calculations . similar effects have been predicted in the case of bn nanotubes when their diameter decreases , @xcite and more recently in nanoribbons . @xcite these arguments indicate that the change of the exciton recombination energy in monoatomic bn sheets should be weak compared to the bulk and should not exceed a few tenths of ev anyway . our experimental results show a slight increase ( @xmath23 ) in the @xmath1-series exciton recombination energy from the bulk to the thinnest sample ( ) , consistent with these theoretical predictions . further calculations as well as experiments for less than six layers would be of great interest in the future to see if larger effects are observed . when comparing the exfoliated materials with the bulk from which they were exfoliated , it is remarkable that excitonic recombinations of exfoliated layers are systematically dominated by the @xmath1 series . in the bulk source materials , various crystallites have been investigated . we found that the @xmath12 ratio varies from one crystallite to another but stays between 0.2 and 0.5 . by contrast , the @xmath12 ratio is found to be equal to about 4 in the 8 and sheets ( samples b and c ) and to about 10 in the sheet ( sample a ) , that is hundred times larger than in the bulk crystallites . these results are summarized in . as discussed previously , this increase of the @xmath12 ratio indicates a dramatic increase in the density of defects . they are probably induced by the exfoliation process , even if we can not exclude layer deformation induced by the water layer freezing arising during cl experiments at , or etching under the electron beam . these results raise the question of the suitability of the mechanical cleavage exfoliation process . @xmath0-bn has indeed been proven to be extremely fragile and subject , for instance , to easy stacking fault generation.@xcite the exfoliation process may be a violent mechanical experiment for such a material and can be obviously incriminated as the source of new defects responsible for the @xmath12 increase . the nature of the substrate is also questionable . the roughness and surface charges of the used substrate can affect the @xmath0-bn band structure and degrade some properties as for graphene . @xcite ratio in the three exfoliated layers , compared with the bulk material , consisting of the crystallites from which the layers were exfoliated . ] in conclusion , we have studied high quality @xmath0-bn crystallites by cathodoluminescence and compared them with exfoliated sheets in the near - band edge region around . first owing to the high quality of the crystallites , we could clearly discriminate the luminescence of high structural quality areas ( dominated by the @xmath3 series ) and that of defective zones ( dominated by the @xmath1 series ) . from this , for evaluating the structural defect density in @xmath0-bn samples , we define a simple quantitative parameter as the ratio of the intensity of the @xmath1 series over that of the @xmath3 series . furthermore the observation of high quality areas has revealed new intrinsic features , attributed to phonon replica of the excitonic recombinations . second , investigation of the emission of exfoliated sheets has shown that the reduction in thickness induces a slight increase of the exciton emission energy . these observations are consistent with theoretical results obtained on single layers and on bn nanotubes . this explanation still deserves to be confirmed as one can not exclude for the moment surface and strain effects . the dominant defect - related emission of exfoliated samples also questions the exfoliation process and the role played by the substrate . this indicates that special care should be taken in the sample preparation . presently , our measurements evidence that in @xmath0-bn , exfoliation can degrade some of the properties and should be used as a guideline to further optimize this process . further experiments on free - standing sheets are thus in progress , with the advantage to make possible correlations between the emission and structural features owing to tem observations . more generally this work reports luminescence spectra of a semiconductor only a few atomic layers thick , which is not common and can be of great interest for future optical applications . t. taniguchi and k. watanabe from nims , japan , are warmly acknowledged for providing one of their hpht crystals . the authors would like to thank s. pouget from cea grenoble for x - ray experiments , c.vilar from gemac for her technical help on the cathodoluminescence - sem set - up , and h. mariette from the npsc for fruitful discussions . the research leading to these results has received funding from the european union seventh framework programme under grant agreement n^o^604391 graphene flagship . it has been also supported by grant from the mission interdisciplinaire of the cnrs ( challenge `` graphene '' of the program g3n ) and by the federative research program `` graphene '' of onera . thanks cnano rhne - alpes and ile - de - france for financial support . 59ifxundefined [ 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 * * , ( ) link:\doibase 10.1021/nl1023707 [ * * , ( ) ] link:\doibase 10.1021/nl1022139 [ * * , ( ) ] link:\doibase 10.1021/nl4012529 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1021/nl2005115 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1002/pssr.201105190 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1063/1.3599708 [ * * , ( ) ] link:\doibase 10.1021/nl200758b [ * * , ( ) ] link:\doibase 10.1063/1.3665405 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.101.096802 [ * * , ( ) ] http://dx.doi.org/10.1038/nnano.2008.199 [ * * , ( ) ] link:\doibase 10.1126/science.1218461 [ * * , ( ) ] link:\doibase 10.1021/nl1039499 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1021/nl203249a [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1021/nn301940k [ * * , ( ) ] link:\doibase 10.1039/c2jm15109j [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1021/nl301061b [ * * , ( ) ] link:\doibase 10.1063/1.4731203 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1002/smll.201001628 [ * * , ( ) ] _ _ , @noop ph.d . thesis , ( ) link:\doibase 10.1063/1.3611394 [ * * , ( ) ] link:\doibase 10.1021/nl070613a [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevb.83.073201 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevlett.109.205502 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrevlett.108.075501 [ * * , ( ) ] link:\doibase 10.1103/physrevb.85.045440 [ * * , ( ) ] link:\doibase 10.1063/1.363379 [ * * , ( ) ] link:\doibase 10.1063/1.3519980 [ * * , ( ) ] link:\doibase 10.1557/jmr.2011.211 [ * * , ( ) ] link:\doibase 10.1021/jp9530562 [ * * , ( ) ] link:\doibase 10.1021/ja970754 m [ * * , ( ) ] link:\doibase 10.1021/nl302490y [ * * , ( ) ] link:\doibase 10.1063/1.2364885 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1063/1.3625922 [ * * , ( ) ]
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hexagonal boron nitride ( @xmath0-bn ) and graphite are structurally similar but with very different properties .
their combination in graphene - based devices is now of intense research focus , and it becomes particularly important to evaluate the role played by crystalline defects on their properties . in this paper , the cathodoluminescence ( cl ) properties of hexagonal boron nitride crystallites are reported and compared to those of nanosheets mechanically exfoliated from them .
first the link between the presence of structural defects and the recombination intensity of trapped excitons , the so - called @xmath1 series , is confirmed .
low defective @xmath0-bn regions are further evidenced by cl spectral mapping ( hyperspectral imaging ) , allowing us to observe new features in the near - band - edge region , tentatively attributed to phonon replica of exciton recombinations .
second the @xmath0-bn thickness was reduced down to six atomic layers , using mechanical exfoliation , as evidenced by atomic force microscopy . even at these low thicknesses ,
the luminescence remains intense and exciton recombination energies are not strongly modified with respect to the bulk , as expected from theoretical calculations indicating extremely compact excitons in @xmath0-bn .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the proposal of materials with simultaneous negative electric permittivity and magnetic permeability by veselago in 1967 @xcite has opened the door toward the design of novel and remarkable optical devices based on the use of metamaterials or photonic crystals , such as the perfect flat lens @xcite or the invisibility cloak @xcite . recently , we have shown how these negative electromagnetic properties can be revisited through the admittance formalism @xcite , which is widely used in the thin - film community @xcite and defined the computational rules for the effective indices and phase delays associated with wave propagation through negative - index layers @xcite . we have demonstrated that we can simulate the optical properties of negative index material ( nim ) layer by replacing it with a positive index material ( pim ) with the same effective index ( @xmath0 ) , provided that we use for this pim layer a * virtual * thickness * opposite * to that of the nim layer ( @xmath1 ) , which is reminiscent of optical space folding in complementary media @xcite . this computational rule is easily implementable in standard thin - film software and has allowed us to analyze the spectral properties of some standard multilayer stacks , such as the antireflection coating , the quarter - wavelength bragg mirror and the fabry - perot bandpass filter , in which one or more layers of these stacks involve negative index materials @xcite . among the presented results , the most spectacular concerns the large increase in the spectral bandwidth of a quarter - wavelength bragg mirror induced by the use of a negatively refracting material ( either the high - index layers or the low - index layers ) and the ability to tailor the phase properties of such multilayer structures by adjusting the number and the features of nim layers within the stack . the objective of this work is to define an optimization method for such improvements and to identify the design of a _ white _ fabry - perot , i.e. a multilayer cavity that spontaneously exhibits * resonant behavior over a very large spectral range*. let us consider a bragg mirror that contains @xmath2 alternated quarter - wavelength layers as described by the following formula incident medium / @xmath3 / substrate the refractive index of the semi - infinite glass substrate is denoted by @xmath4 , whereas that of the semi - infinite incident medium is denoted by @xmath5 . the incident medium and the substrate are non - absorbing positive - index materials . each layer of the stack can be composed of a positive ( high - index @xmath6 , low - index @xmath7 ) or a negative index material ( @xmath8 , @xmath9 ) , each of which should be non - absorbing . moreover , in this first approach we neglect the dispersion law of refractive indices for all materials under study . though this assumption may appear too simplistic in the case of negative index materials , recently published results @xcite showed that it was possible to efficiently control this dispersion in a wide spectral range . to determine the reflection properties of such a stack , we use the following basic formula @xmath10 where @xmath11 is the effective index of the incident medium , @xmath12 is the complex admittance of the stack , @xmath13 is the amplitude reflection coefficient , @xmath14 is the corresponding reflectance and @xmath15 is the phase change at the reflection . the effective index of a medium is given by the general relation @xcite @xmath16 where @xmath17 is defined by @xmath18 for a plane wave passing through a multilayer stack that contains @xmath2 layers , @xmath19 is an invariant quantity that is defined by the angle of incidence ( aoi ) @xmath20 in the incident medium @xmath21 where @xmath22 is the layer number , while @xmath17 depends on the layer and is defined , in the propagating mode , by ( @xmath23 ) @xmath24 consequently , we can rewrite relation ( [ eq : effectiveindexalpha ] ) in the form @xmath25 where @xmath26 is the vacuum impedance . relation ( [ eq : effectiveindextheta ] ) is independent of the type of material ( pim or nim ) within the layer because @xmath27 and @xmath28 are simultaneously negative in the case of negative index materials . the computation of the @xmath12 factor is based on the application of a recursive formula that links the admittances at two consecutive boundaries @xcite @xmath29 where @xmath30 is the phase delay introduced by the crossing of the layer @xmath22 . the initialization of this recursive formula occurs in the substrate where only the outgoing plane wave is present @xmath31 the phase delay @xmath30 is given by @xmath32 where @xmath33 is the effectivel thickness of the layer @xmath22 and @xmath34 a binary coefficient equal to + 1 ( -1 ) for a layer that consists of a positive index ( negative index ) material . this last relation justifies the statement in section [ sec : introduction ] and indicating that we can replace each nim with an equivalent pim that is characterized by a virtual negative thickness @xcite . all of the layers in the bragg mirror stack are quarter - wavelength ; hence , at zero aoi , we can write @xmath35 where @xmath36 is the central wavelength of the mirror . if we use a linear approximation of this last relation near @xmath36 , we have @xmath37 consequently , the recursive formula ( [ eq : recursiveformula ] ) becomes , at the same level of approximation @xmath38\ ] ] by initializing this new recursive formula using relation ( [ eq : initialization ] ) , we find @xmath39 with @xmath40^q\enskip\text{for } p=2q\\ \frac{\tilde{n}_h^2}{\tilde{n}_s}&\left[\frac{\tilde{n}_h^{2}}{\tilde{n}_l^{2}}\right]^q\enskip\text{for } p=2q+1 \end{aligned } \right . \label{eq : y0lambda0}\ ] ] and @xmath41^l-\frac{\tilde{n}_h}{\tilde{n}_s}\sum\limits_{l=0}^{p-1}\gamma_{p - l}\left[\frac{\tilde{n}_h}{\tilde{n}_l}\right]^l \label{eq : alambda0}\ ] ] by combining ( [ eq : reflectioncoefficient ] ) and ( [ eq : linearadmittance ] ) , we finally obtain analytical expressions for the reflectance @xmath14 and the spectral derivative of the phase change at the reflection @xmath42 , both at the design wavelength @xmath36 @xmath43 ^ 2\ ] ] @xmath44 a multilayer fabry - perot ( fp ) cavity is composed of a thin spacer ( with refractive index @xmath45 and thickness @xmath46 ) surrounded by two quarter - wavelength bragg mirrors deposited at the surface of a semi - infinite substrate . the overall round trip phase @xmath47 of this planar cavity is defined by @xmath48 where @xmath49 is the optical path difference corresponding to the round trip of the light in the spacer layer ( @xmath50 ) and @xmath51 ( @xmath52 ) the phase change at the reflection on the upper mirror ( lower mirror ) . a fabry - perot resonance is defined by a central wavelength @xmath36 for which the overall round - trip phase @xmath53 is a multiple of @xmath54 and by a spectral bandwidth @xmath55 which is given , at leading order approximation , by @xmath56_{\lambda_0}|}\cdot\frac{1-\sqrt{r_{\text{ls}}^+r_{\text{us}}^-}}{(r_{\text{ls}}^+r_{\text{us}}^-)^{\frac{1}{4}}}\ ] ] in other words , the spectral bandwidth of such a fp resonance can become extremely large if the linear dependence of the overall round - trip phase in the cavity @xmath57_{\lambda_0}=-\frac{4\pi n_{\text{sp}}e_{\text{sp}}}{\lambda_0 ^ 2}+\left.\frac{\partial\rho_{\text{us}}^-}{\partial\lambda}\right|_{\lambda_0}+\left.\frac{\partial\rho_{\text{ls}}^+}{\partial\lambda}\right|_{\lambda_0 } \label{eq : philineardependence}\ ] ] is equal or very close to zero . to fulfill this condition and thus obtain a resonant behavior over a very large spectral range , it is absolutely required that at least * one of the cavity mirrors includes nim layers * , which is the only way to obtain a * positive * linear dependence of the phase change at reflection @xcite . to determine the optimal design of such a * _ white _ multilayer fabry - perot cavity * , we systematically investigate the variation of @xmath58_{\lambda_0}$ ] of all the symmetric multilayer stacks described by the general formula air / h@xmath59 2l @xmath60h / glass when the refractive index @xmath6 of the high - index material varies in the range between 2.00 and 3.00 , while the other refractive indices remain constant ( @xmath5 = 1.00 , @xmath4 = 1.52 and @xmath7 = 1.48 ) . in the previous stack formula , @xmath61 is an integer equal to 1 , 2 or 3 , while h and l represent either pim ( h , l ) or nim ( @xmath62 , @xmath63 ) quarter - wavelength layers . the calculus is performed for zero aoi . for each stack formula that exhibits a cancellation of the spectral dependence of the overall round - trip phase at the design wavelength @xmath36 , we thus calculate its spectral transmittance @xmath64 , and the variation of the ratio between the square modulus @xmath65 of the electric field within the stack and the square modulus of the incident field @xmath66 , where @xmath67 is the coordinate along a vertical axis perpendicular to the substrate plane , whose origin is taken to be at the top of the stack . these two quantities are computed using a recursive relation between the electric fields at the boundaries , completed by the related initialization condition @xmath68\thinspace\vec{\mathcal{e}}_{j}\\ & \vec{\mathcal{e}}_{0}=(1+r)\vec{\mathcal{e}}_{0}^+ \end{aligned } \right.\ ] ] and extended in the thickness of each layer by @xmath69\thinspace\vec{\mathcal{e}}_{j}\\ \text{with}\enskip\delta_j(z)=\alpha_j(e_j - z)\end{gathered}\ ] ] to quantify and compare the strength and the spectral width of the resonance behavior of these various fabry - perot configurations , we introduce the merit factor @xmath70 @xmath71 where @xmath72 and @xmath73 are the limits of the selected spectral range ( here , @xmath72 = 600 nm and @xmath73 = 800 nm , cf . fig . [ fig : whitefp ] ) , and @xmath74 is the spectral dependence of the amplitude of the electric field in the middle of the spacer . the table [ tab : fabryperotconfigurations ] summarizes the results provided by this systematic screening and gives , for each optimized configuration , the stack formula , the refractive index of the h - layer , the value of the merit factor @xmath70 and the ratio @xmath75 between this merit factor and that of the corresponding all - pim cavity ( for all these cavities , the design wavelength @xmath36 is equal to 700 nm ) . .[tab : fabryperotconfigurations]main features of three optimized symmetric _ white _ fabry - perot configurations [ cols="<,^,^,^ " , ] 2l @xmath76 , @xmath6 = 2.61 ; red line : @xmath76l@xmath62 2l @xmath62l@xmath76 , @xmath6 = 2.19 ; blue line : @xmath77l@xmath62 2l @xmath62l@xmath77 , @xmath6 = 2.37 . ] figure [ fig : whitefp ] shows the spectral transmittance of these three optimized fabry - perot configurations ; the most attractive wide - band behavior seems provided by the first two listed ( m3 2l m3 and m5 2l m5 ) . the variations in the normalized square modulus of the electric field within these stacks are presented in fig . [ fig : pimm32lm3 ] for a standard all - pim m3 2l m3 configuration ( @xmath6 = 2.61 , @xmath7 = 1.480 ) and in fig . [ fig : nimpimm32lm3 ] for the optimized symmetric nim / pim fabry - perot configuration ( @xmath76 2l @xmath76 ) . the use of nim layers provides , as expected , a spectacular increase of the spectral range in which resonant behavior is achieved . = 2.61 , @xmath7 = 1.48 ) . ] 2l @xmath76 fabry - perot configuration ( the refractive indices of the nim layers are opposite to those of the standard configuration ) . ] we have shown how negative index materials can be used to design a novel type of planar multilayer cavity , which we propose to call a _ white _ fabry - perot cavity , and which is resonant over a very large range of wavelengths in comparison to standard devices . more detailed analysis demonstrates that the choice of the refractive index @xmath6 of each optimized configuration is quite tolerant , which is advantageous for possible practical implementations . moreover , by slightly modifying the features of the optimized m5 2l m5 fabry - perot cavity described in the previous section ( centering wavelength 600 nm , modified formula @xmath62l@xmath62l@xmath62 2l @xmath62l@xmath62l@xmath62 , @xmath6 = -2.24 for all high - index nim layers ) and by stacking 3 identical cavities linked by a pim low - index quarter - wavelength layer , we are able to obtain a filter with a nice rectangular profile , a bandwidth of approximately 90 nm , and a high level of rejection throughout the entire remaining spectral range ( approximately -30 db ) . the addition of negative index materials to the data - base of standard thin - film software allows us to define optimized designs for many filtering applications . v. g. veselago , `` the electrodynamics of substances with simultaneously negative values of @xmath78 and @xmath79 , '' sov . * 10 * , 509514 ( 1968 , russian text 1967 ) . j. b. pendry , negative refraction makes a perfect lens , phys . lett . * 85 * , 3966 - 3969 ( 2000 ) . d. schurig , j. j. mock , b. j. justice , s. a. cummer , j. b. pendry , a. f. starr , d. r. smith , `` metamaterial electromagnetic cloak at microwave frequencies '' . science * 314 * , 977980 ( 2006 ) . m. lequime , b. gralak , s. guenneau , m. zerrad , and c. amra , `` negative indices and the admittance formalism in multilayer optics , '' in optical interference coatings conference ( optical society of america , 2013 ) , paper tb9 . h. a. macleod , _ thin - film optical filters _ , 4th ed . ( crc press , 2010 ) . m. lequime , b. gralak , s. guenneau , m. zerrad , and c. amra , `` optical properties of multilayer optics including negative index materials , '' submitted for publication ( 2013 ) j. b. pendry and s. a. ramakrishna , `` focusing light using negative refraction , j. phys . [ condensed matter ] , * 15 * , 6345 - 6364 ( 2003 ) . z. h. jiang , s. yun , l. lin , j. a. bossard , d. h. werner , and t. s. mayer , `` tailoring dispersion for broadband low - loss optical metamaterials using deep - subwavelength inclusions , '' scientific reports * 3 * , 01571 ( 2013 ) . c. amra , ' ' first - order vector theory of bulk scattering in optical multilayers , `` j. opt . am . a * 10 * , 365 - 374 ( 1993 ) m. lequime and c. amra , ' ' broadband emitters within multilayer micro - cavities : optimization of the light extraction efficiency , " in _ optical interference coatings conference _ ( optical society of america , 2010 ) , paper td3 .
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the use of negative index materials is highly efficient for tayloring the spectral dispersion properties of a quarter - wavelength bragg mirror and for obtaining a resonant behavior of a multilayer fabry - perot cavity over a very large spectral range .
an optimization method is proposed and validated on some first promising devices .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
for the past decade or so , the study of first - order phase transitions in cosmology has been the focus of much interest due to their possible relevance to the physics of the early universe . some well - known examples are inflationary models @xcite , the quark - hadron transition @xcite and , more recently , the generation of the cosmological baryon asymmetry in the electroweak phase transition @xcite . in a first - order phase transition , the initial metastable phase decays to the stable phase by the nucleation of bubbles larger than a critical size . this decay may be triggered by either quantum or thermal fluctuations , depending on how the ambient temperature compares to the nucleation barrier @xcite . within a cosmological context , the cooling is provided by the expansion of the universe ; the long - wavelength modes of the order parameter responsible for the symmetry breaking transition are coupled to the `` environment '' , which is assumed to be in local thermal equilibrium at some temperature @xmath0 . ( here , we are mainly concerned with `` late '' transitions , for which the typical fluctuation time - scales are much shorter than the expansion rate . ) of great relevance to the understanding of the evolution of the phase transition is the determination of the bubble nucleation rate per unit volume . this is a well - known problem in classical statistical mechanics , with an long history @xcite . phenomenological field - theoretic treatments were developed by cahn and hilliard @xcite , and by langer @xcite in the context of a time - dependent coarse - grained ginzburg - landau model . classical homogeneous nucleation theory within a field theoretic context has been recently shown by numerical experiments to successfully predict the nucleation barrier @xcite . in the case of zero - temperature quantum field theory , the study of metastable vacuum decay was initiated with the work of voloshin , kobzarev , and okun @xcite , and was put onto firm theoretical ground by coleman and callan in the late seventies @xcite . finite temperature corrections to the vacuum decay rate were first considered by linde @xcite , who argued that temperature corrections to the nucleation rate are obtained recalling that finite temperature field theory ( at sufficiently high temperatures ) in @xmath1-dimensions is equivalent to @xmath2-dimensional euclidean quantum field theory with @xmath3 substituted by @xmath0 . thus , in @xmath1 dimensions , the nucleation rate is proportional to @xmath4 $ ] , where @xmath5 is the @xmath2-dimensional euclidean action evaluated at its extremum ( specifically , a saddle point ) , the critical bubble , or bounce , @xmath6 . the usual expression for the nucleation rate per unit volume used in the literature is @xcite @xmath7 } { \det ' [ - \nabla^{2 } + v_{\rm { eff}}''(\varphi_b(r , t),t ) ] } \right\}^ { \frac{1}{2 } } { \rm exp}\left [ - { { s_e^3 ( \varphi_b(r , t),t)}\over t}\right ] \ : , \label{badrate}\ ] ] where @xmath8 is the value of the field @xmath9 at the metastable minimum , and the prime in the determinant in the denominator is a reminder that one should omit the zero and negative eigenvalues , associated with the translation symmetry of the bubble and with its instability ( being a saddle point configuration ) , respectively . @xmath10 is the one - loop approximation to the finite temperature effective potential , and @xmath11 . there are three important points here . the first is that in order to estimate the determinantal prefactor ( the `` equilibrium '' part of the prefactor ; there is a dynamic factor which can not be obtained by using equilibrium arguments ) one usually proceeds by invoking dimensional arguments to approximate it by a term of order @xmath12 ( @xmath13 is the critical temperature ) . how good is this approximation ? clearly , in most cases it is impossible to evaluate the determinants exactly . but can one obtain a better approximation than the simple use of dimensional arguments ? the second , and most important , point is that the critical bubble configuration @xmath14 used to evaluate the nucleation barrier , denoted above by @xmath15 , was obtained from an effective potential which includes corrections coming from scalar loops . hence the temperature dependence in @xmath16 . we will show here that this procedure is not in general justified and is only a good approximation if the corrections from scalar loops are negligible . finally , in the expression for the temperature corrected barrier in eq . ( [ badrate ] ) , @xmath17 , one uses the _ temperature corrected _ effective potential , @xmath18 as opposed to the tree level potential . thus , it is claimed that @xmath17 is equivalent to the free energy of the temperature dependent bounce , given by @xmath19 \ : . \label{badact}\ ] ] as far as we know , apart from the work of affleck in the context of finite temperature quantum mechanics @xcite , this point has never been properly addressed in the literature . how did the temperature corrected potential appear in the exponent ? is the exponentiation of the massless modes sufficient ? in fact , most of the work done on cosmological phase transitions in which temperature effects are important ( including some by the present authors ) simply invokes linde s results . given the many applications of finite temperature vacuum decay in cosmology , we feel that this important question should not be left unscrutinized . this concern has also been expressed in recent works by csernai and kapusta @xcite , and by buchm@xmath20ller , helbig , and walliser @xcite . both works attempted to improve on linde s results , by generalizations of langer s work . csernai and kapusta obtained an expression for the dynamical prefactor by using a relativistic hydrodynamic approach , while buchm@xmath20ller _ et al . _ obtained an approximate expression for the decay rate in scalar electrodynamics ( and more recently , with z. fodor , in the standard electroweak model ) by integrating out the electromagnetic degrees of freedom from the partition function . however , a more detailed analysis of the nucleation barrier and how it compares to the usual result is still lacking . in this paper we address the three points raised above . we will be mostly interested in the regime in which thermal fluctuations are much larger than quantum fluctuations . this way we avoid the question of how to match continuously the two regimes , although we believe this to be a very important question @xcite . ( see also refs . @xcite and @xcite . ) the hope is that in most situations of interest the transition will be dominated by one or the other regime . by a saddle - point evaluation of the partition function in the case of a self - interacting scalar field , it is possible to show that the temperature corrections to the nucleation barrier can be interpreted as entropic contributions due to stable vibrational modes on the _ tree - level _ bounce configuration , @xmath6 . in other words , the first corrections to the energy of the critical bubble configuration come from temperature induced stable fluctuations on the bubble , which will modify its volume and surface energies . we will show that these corrections are given by the temperature corrected effective action evaluated at the tree - level bounce @xmath6 . in expression ( [ badrate ] ) , the bounce is obtained from the effective potential which includes scalar loops . the difference between the two nucleation barriers will be important whenever scalar loops are not negligible . we will show that they become particularly important within the so - called thin - wall limit , that is , in the vicinity of the critical temperature for the transition . this is perfectly consistent with the fact that large entropic corrections are expected near the critical temperature . we will obtain this result by a perturbative evaluation of the determinantal prefactor . in principle , the determinantal prefactor can be evaluated in two ways . clearly , the computation can be done directly if we know the eigenvalues related to a given bubble configuration . this method is not very useful in practice , since we in general do not know the eigenvalues . ( unless , of course , we obtain them numerically . ) writing down explicitly the eingenvalue equations , and using a thin - wall approximation to the bubble configuration , we show how the temperature corrections to the nucleation barrier originate from fluctuations about the critical bubble configuration . even though the thin - wall approximation is not very useful in realistic situations , there is no reason to believe that thicker wall bubbles will behave any differently . ( unless the transition becomes too weak , in which case nucleation of critical bubbles may not be the relevant mechanism for the transition @xcite . ) the second approach we use to evaluate the prefactor relies on a perturbative expansion of the determinants . within first - order , it is again possible to show how the prefactor accounts for the temperature corrections to the nucleation barrier . the paper is organized as follows . in section 2 we briefly review langer s formalism for obtaining nucleation rates , adapted to field theory at finite temperatures . that is , we obtain the partition function for the metastable phase plus a nucleating fluctuation by a saddle - point evaluation of the functional integral . in section 3 we show how the determinantal prefactor can account for the finite temperature corrections to the nucleation barrier . for simplicity , the calculation is performed in the context of the thin - wall approximation for the bubble profile , although in principle one could obtain results for any configuration . in sections 2 and 3 , for the sake of clarity , the discussion is somewhat oversimplified . we _ assume _ that the system is initially in a metastable state and study only the scalar degrees of freedom in the problem . this situation is not unrealistic , as it can be reproduced in numerical simulations of vacuum decay @xcite . in section 4 we study a model of a scalar field coupled to fermions . this example is particularly interesting as it illustrates how a thermal state evolves into a metastable state due to radiative corrections , very much like in the standard electroweak model . ( recall that in a cosmological context the cooling is provided by the expansion of the universe . ) a related problem has been recently studied by e. weinberg , in the context of massless scalar models for which symmetry breaking occurs due to radiative corrections . in the regime dominated by quantum fluctuations , weinberg showed how radiative corrections induce a metastable state and how it is possible to evaluate its decay rate@xcite . we show how to obtain an effective partition function for the scalar field by integrating out the fermionic modes , and proceed to obtain the decay rate . we then compare our results for the nucleation barrier to the results obtained using eq . ( [ badrate ] ) . conclusions are presented in section 5 and two appendices are included to clarify a few technical points . consider a scalar field model with four - dimensional euclidean action @xmath21 where @xmath22 is the euclidean lagrangian density given by @xmath23 and the potential in ( [ lphih ] ) has a metastable minimum at @xmath24 and a stable minimum at @xmath25 , as shown in fig . 1 . [ we will only consider potentials with two minima here . ] assume _ that the system is prepared initially in the metastable phase , without worrying for the moment about how this is done . ( see section 4 . ) the metastable phase will decay into the stable phase by the nucleation of bubbles larger than a critical size . ( for a review see , e.g. , @xcite . ) as is well - known , in order to study the decay of the false vacuum at finite temperature we impose the periodic boundary condition ( anti - periodic for fermions ) @xmath26 , so that the euclidean action becomes @xcite @xmath27=\int_0^{\beta}d\tau\int d^3x\left [ { 1\over 2 } \left ( { { \partial \phi}\over { \partial \tau}}\right ) ^2+{1\over 2 } \left ( \nabla\phi \right ) ^2 + v(\phi ) \right ] \ : . \label{seuclt}\ ] ] the partition function of the system is given by a functional integral over all possible field configurations weighted by their euclidean action , @xmath28 following langer @xcite , we describe the nucleation of bubbles of the stable phase inside the metastable phase under the assumption that a dilute gas approximation for these droplets is valid . unless the transition is weakly first - order , this should be a very good approximation to describe the early stages of the transition , when bubble collisions and other complicated kinetic effects can be neglected . the critical configuration is an extremum of the euclidean action , @xmath29 being thus a solution of the equation of motion , @xmath30 with boundary conditions , @xmath31 , and @xmath32 . coleman , glaser , and martin @xcite , have shown that the configuration with minimum energy , _ i.e. _ , the minimum of all the maxima , will have @xmath33-symmetry . as argued by linde @xcite , for sufficiently high temperatures the problem becomes effectively three - dimensional , and the saddle point will be given by the @xmath34-symmetric , or static , solution of @xmath35 with boundary conditions , @xmath36 and @xmath37 . note that the potential in ( [ eulersph ] ) is the _ tree - level _ potential . for future reference we note that when the false vacuum energy density [ @xmath38 is much smaller than the barrier height [ @xmath39 , see fig . 1 ] , the bubble radius @xmath40 is much larger than the wall thickness @xmath41 , where @xmath42 is a typical mass scale in the problem . in this case , the solution to eq . ( [ eulersph ] ) can be estimated by the so - called thin - wall approximation and is given by @xmath43 which describes a bubble of radius @xmath40 of the stable phase @xmath44 embedded in the metastable phase @xmath45 . @xmath46 describes the bubble wall separating the two phases . the main advantage of langer s approach is that in a dilute gas of bubbles one can infer the thermodynamics of the system from the knowledge of the partition function of a single bubble . we can write the partition function for the system with a bubble of the stable vacuum inside the metastable vacuum as @xmath47 where @xmath48 and @xmath49 are the partition functions of the system for the vacuum field configuration @xmath45 and for the bubble field configuration @xmath50 , respectively @xcite . the generalization of ( [ z1bubb ] ) for several bubbles is given by @xmath51 + z ( \varphi_{f } ) \frac{1}{2 ! } \left [ \frac{z(\varphi_{b})}{z(\varphi_{f } ) } \right]^{2 } + \ldots \nonumber \\ & \simeq & z(\varphi_{f } ) \exp \left [ \frac{z(\varphi_{b})}{z(\varphi_{f } ) } \right ] \ : . \label{zmany}\end{aligned}\ ] ] the proof of ( [ zmany ] ) can be found in the work of arnold and mclerran @xcite , who studied the properties of a dilute gas of sphalerons . they expressed the multiple - sphaleron configurations as the superposition of many single sphalerons , with partition function approximated as above . the partition functions in ( [ zmany ] ) can be evaluated by the saddle - point method , expanding the lagrangian field in ( [ zphih ] ) as @xmath52 for @xmath49 and @xmath53 for @xmath48 . @xmath54 and @xmath55 are small perturbations around the classical field configurations @xmath56 and @xmath57 , respectively . these perturbations around each configuration bring the temperature corrections to the nucleation barrier into the problem , as we shall see . up to 1loop order one keeps the quadratic terms in the fluctuations @xmath54 and @xmath55 in the expansion of the scalar field in the partition function ( [ zphih ] ) . this way one can write the following expressions for @xmath49 and @xmath48 , respectively , @xmath58 \eta \right\ } \label{zphib}\ ] ] and @xmath59 \zeta \right\ } \ : , \label{zphif}\ ] ] where @xmath60 and @xmath61 . performing the functional gaussian integrals in ( [ zphib ] ) and ( [ zphif ] ) one gets the following expression for the ratio between the partition functions , @xmath62 , appearing in ( [ zmany ] ) : @xmath63^{-\frac{1}{2 } } e^{-\delta s } \ : , \label{ratiodet}\ ] ] where @xmath64^{- \frac{1}{2 } } \equiv \int d \eta \exp \left\ { - \int_{0}^{\beta } d \tau \int d^{3 } x \frac{1}{2 } \eta [ m ] \eta \right\}$ ] and @xmath65 is the difference between the euclidean actions for the field configurations @xmath50 and @xmath45 . note that @xmath66 , and hence @xmath67 , _ does not include any temperature corrections . _ from ( [ zmany ] ) and ( [ ratiodet ] ) , the free energy of the system , @xmath68 can be written as @xmath69^{-\frac{1}{2 } } e^{-\delta s } \ : . \label{energy}\ ] ] as is well - known , the determinantal prefactor evaluated for the bounce configuration has one negative eigenvalue , signalling the presence of a metastable state , and also three zero eigenvalues related with the translational invariance of the bubble in three - dimensional space . because of the negative eigenvalue , the free energy @xmath70 is imaginary . as shown by langer @xcite ( see also ref . @xcite ) , the decay rate is proportional to the imaginary part of @xmath70 @xmath71 where @xmath72 is the single negative eigenvalue . in general it depends on non - equilibrium aspects of the dynamics , such as the coupling strength to the thermal bath . in this section we compute the ratio of the determinants appearing in the decay rate , and show how it provides a finite temperature correction to the nucleation barrier . first recall that for static field configurations @xmath67 is given by @xmath73 = \frac{\delta e}{t } \ : , \label{deltas}\ ] ] where @xmath74 is simply the nucleation _ energy _ barrier , that is , the energy of a critical nucleation within the metastable phase . for example , in the thin - wall approximation of eq . ( [ bubble ] ) , @xmath74 is @xmath75 where @xmath76 is the tree - level surface tension of the bubble wall ( _ i.e. _ with no corrections due to fluctuations around the bubble wall field configuration @xmath77 ) @xmath78 \ : . \label{sigma0}\ ] ] using ( [ deltas ] ) in eqs . ( [ energy ] ) and ( [ gamma ] ) we can write the decay rate as @xmath79^{-\frac{1}{2 } } { \rm exp } \left ( -{{\delta e } \over t}\right ) \ : . \label{gamma2}\ ] ] the determinantal prefactor in ( [ gamma2 ] ) will provide the temperature corrections to the nucleation barrier . this should come as no surprise , given that the determinant is obtained from integrating over thermally induced fluctuations about the tree - level bubble configuration . ( recall that we are only considering the regime in which thermal fluctuations are much larger than quantum fluctuations . ) once the negative and zero eigenvalues are taken care of , the positive eigenvalues are easily associated with entropic contributions to the activation energy due to stable deformations of the bubble s shape , as in classical nucleation theory . we now proceed to show how to incorporate temperature corrections to the nucleation barrier . this can be done without an explicit evaluation of the eigenvalues of the operators in the determinants . as we show next , all that we need is to separate consistently the positive eigenvalues from the negative and zero eigenvalues , and then show how the former can be exponentiated . in principle , the computation of the determinantal prefactor in ( [ energy ] ) can be performed by two different methods . the first involves obtaining directly ( analytically , or more realistically , numerically ) the eigenvalues for the determinants in eq . ( [ energy ] ) . the second method consists in developing a consistent perturbative expansion for the ratio of the determinants . we now examine both these possibilities . consider the eigenvalue equations for the differential operators that appear in the determinantal prefactor , @xmath80 \phi_{f}(i ) = \varepsilon_{f}^{2}(i ) \phi_{f}(i ) \label{valuephif}\ ] ] and @xmath81 \phi_{b}(i ) = \varepsilon_{b}^{2}(i ) \phi_{b}(i ) \ : . \label{valuephib}\ ] ] in momentum space one writes , @xmath82 , where @xmath83 , for bosons ( for fermion fields @xmath84 ) . from ( [ valuephif ] ) and ( [ valuephib ] ) one can write the determinant ratio in ( [ energy ] ) as @xmath85^{\frac{1}{2 } } & = & \exp \left\ { \frac{1}{2 } \ln \left [ \frac { \det ( -\box_{e } + v''(\varphi_{f}))_{\beta } } { \det ( -\box_{e } + v''(\varphi_{b}))_{\beta } } \right ] \right\ } = \nonumber \\ & = & \exp \left\ { \frac{1}{2 } \ln \left [ \frac { \prod_{n=- \infty}^{+ \infty } \prod_{i } \left ( \omega_{n}^{2 } + e_{f}^{2}(i ) \right ) } { \prod_{n= - \infty}^{+ \infty } \prod_{j } \left ( \omega_{n}^{2 } + e_{b}^{2}(j ) \right ) } \right ] \right\ } \ : . \label{ratio}\end{aligned}\ ] ] using the identity , @xmath86 and taking into account that we have in the denominator of ( [ ratio ] ) one negative and three zero eigenvalues , one obtains ( for details see appendix a ) @xmath85^{\frac{1}{2 } } & = & { \cal v}\frac{t^{4}}{i |e_{-}| } \frac{\beta \frac{|e_{-}|}{2 } } { \sin \left ( \beta \frac{|e_{-}|}{2 } \right ) } \left [ \frac{\delta e}{2 \pi t } \right]^{\frac{3}{2 } } \nonumber \\ & \times & \exp \left\ { \sum_{i } \left [ \frac{\beta}{2 } e_{f}(i ) + \ln \left ( 1 - e^{- \beta e_{f}(i ) } \right ) \right ] \right . \nonumber \\ & - & \left . \sum_{j } \ ; ' \left [ \frac{\beta}{2 } e_{b}(j ) + \ln \left ( 1 - e^{- \beta e_{b}(j ) } \right ) \right ] \right\ } \ : . \label{eigenratio}\end{aligned}\ ] ] the factor @xmath87^{\frac{3}{2 } } $ ] in the right hand side of eq . ( [ eigenratio ] ) comes from the contribution of the zero eigenvalues , which can be handled as in ref . @xcite , through the use of collective coordinates corresponding to the position of the bubble . @xmath88 is the volume of three space , while the prime in @xmath89 is a reminder that we have excluded the negative and the three zero eigenvalues from the sum . thus , the argument of the exponential incorporates only the contributions from the stable vibrational modes on the bubble , very much like in langer s result @xcite . substituting eq . ( [ eigenratio ] ) in ( [ gamma2 ] ) one obtains the following expression for the nucleation rate per unit volume , @xmath90 , as defined in eq . ( [ gamma ] ) : @xmath91 \ : , \label{endgamma}\ ] ] where we have denoted by @xmath92 the dimensionless factor @xmath93 note that the zero - mode contribution in the prefactor depends on the energy barrier of the critical nucleation , and not on its free energy as in eq . ( [ badrate ] ) . in eq . ( [ endgamma ] ) we have incorporated the exponential contribution from the prefactor into the definition of the temperature corrected nucleation barrier , which we call @xmath94 . within the thin - wall approximation we can write @xmath95 where @xmath96 - \nonumber \\ & - & t \int \frac{d^{3 } k}{(2 \pi)^{3 } } \ln \left [ 1 - e^{- \beta \sqrt { \vec{k}^ { 2 } + m^{2}(\varphi_{t } ) } } \right ] \label{deltau}\end{aligned}\ ] ] and @xmath97 - \right . \nonumber \\ & - & \left . \sum_{i } \left [ \frac { \beta}{2 } e_{f}(i ) + \ln \left ( 1 - e^{- \beta e_{f}(i ) } \right ) \right ] \right\ } \ : . \label{sigmat}\end{aligned}\ ] ] in eq . ( [ deltau ] ) we have substituted the discrete sums by integrals over momenta . [ for field configurations @xmath45 and @xmath44 , we have the continuum eigenvalues , @xmath98 and @xmath99 , respectively , with @xmath100 and @xmath101 , from eq . ( [ deltau ] ) since they can always be handled by the introduction of the usual counterterms @xcite . in eq . ( [ sigmat ] ) @xmath102 are the eigenvalues related with the bubble wall field configuration @xmath77 . thus , within the thin - wall approximation , the problem is reduced to the computation of these eigenvalues for a field configuration describing the bubble wall , a non - trivial task . it is possible , however , that the temperature corrections to the surface density are negligible . for example , in the context of the qcd transition , the surface density is obtained from lattice calculations , and is shown not to be very sensitive to temperature @xcite . it is easy to see from eq . ( [ deltau ] ) that @xmath103 is the usual 1-loop approximation to the finite temperature false vacuum energy density @xcite . the second term in the right hand side of ( [ sigmat ] ) comes from the finite temperature 1-loop contribution to the surface tension @xmath76 , due to thermal fluctuations on the bubble wall . thus , by exponentiating the contribution from the stable modes , the activation energy for the critical bubble becomes indeed an activation free energy . note however , that contrary to eq . ( [ badrate ] ) , the free - energy functional is evaluated for the tree - level bounce . in section 4 we will compare the results obtained with the two approaches . a second approach to the computation of the determinantal prefactor in ( [ gamma2 ] ) consists in developing a perturbative expansion for it . first we write the determinantal ratio as @xmath104^{\frac{1}{2 } } & = & \exp \left\ { \frac{1}{2 } { \rm tr } \ ; \ln \left[-\box_{e } + v''(\varphi_{f } ) \right]_{\beta } - \right . \nonumber \\ & - & \left . \frac{1}{2 } { \rm tr ' } \ ; \ln \left[-\box_{e } + v''(\varphi_{b } ) \right]_{\beta } \right\ } \ : , \label{fieldratio}\end{aligned}\ ] ] where we have used in ( [ fieldratio ] ) the identity @xmath105 and the prime in both sides denote that the negative and the zero modes have been omitted . ( they are treated as in previous section . ) we rewrite ( [ fieldratio ] ) as @xmath106^{\frac{1}{2 } } = \exp \left\ { - \frac{1}{2 } { \rm tr } \ ; \ln \bigl [ 1 + g_{\beta}(\varphi_{f } ) \left [ v''(\varphi_{b } ) - v''(\varphi_{f } ) \right ] \bigr ] \right\ } \ : , \label{expratio}\ ] ] where @xmath107 is just the propagator for the scalar field @xmath9 , with @xmath108 . expanding the logarithm in ( [ expratio ] ) in powers of @xmath109 $ ] , we obtain the graphic representation , @xmath110 \right \ } = \ols \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! + \olr + \olk + \ : \ : \ldots \ : , \label{graphic}\ ] ] where the dashed lines correspond to the background `` field '' @xmath111 $ ] and the internal lines denote the propagator @xmath112 . the expression ( [ graphic ] ) can be written as @xmath113 \right\ } & = & \sum_{m=1}^{+ \infty } \frac { ( -1)^{m+1 } } { m } \int d^{3 } x \left[v''(\varphi_{b } ) - v''(\varphi_{f } ) \right]^{m } \times \nonumber \\ & \times & \sum_{n= -\infty}^{+ \infty } \int \frac { d^{3 } k } { ( 2 \pi)^{3 } } \frac{1 } { \left [ \omega_{n}^{2 } + \vec{k}^{2 } + m^{2}(\varphi_{f } ) \right]^{m } } \ : . \label{traceln}\end{aligned}\ ] ] the sum in @xmath42 can be performed and one obtains @xmath114 \right\ } = \int d^{3 } x \sum_{n= -\infty}^{+ \infty } \int \frac { d^{3 } k } { ( 2 \pi)^{3 } } \ln \left [ 1 + \frac { v''(\varphi_{b } ) - v''(\varphi_{f } ) } { \omega_{n}^{2 } + \vec{k}^{2 } + m^{2}(\varphi_{f } ) } \right ] \ : . \label{endtrln}\ ] ] this expression can be further simplified by means of the identity , ( @xmath115 , and @xmath116 ) @xmath117 & = & \beta e_{b}(\vec{k } ) + 2 \ln \left ( 1 - e^{- \beta e_{b}(\vec{k } ) } \right ) - \nonumber \\ & - & \beta e_{f}(\vec{k } ) - 2 \ln \left ( 1 - e^{- \beta e_{f}(\vec{k } ) } \right ) \ : . \label{sumln}\end{aligned}\ ] ] the terms proportional to @xmath118 can be renormalized . we are left with the familiar temperature corrected effective potential ( we neglect terms coming from zero temperature quantum corrections ) @xcite , @xmath119 within the thin - wall limit , using ( [ sumln ] ) into ( [ endtrln ] ) , and substituting into eq . ( [ expratio ] ) , we obtain , from eq . ( [ gamma2 ] ) the temperature corrected barrier , @xmath120 where @xmath121 \label{deltaveff}\ ] ] and @xmath122 \ : . \label{fieldsigmat}\ ] ] expression ( [ deltaveff ] ) for the temperature corrected false vacuum energy density can be easily identified with @xmath103 , as defined in eq . ( [ deltau ] ) . also , expression ( [ fieldsigmat ] ) for @xmath123 can be identified with the temperature corrected surface tension of the bubble wall , by writing it as @xmath124 \ : , \label{defsigmat}\ ] ] with @xmath125 where @xmath126 is the free energy for the field configuration @xmath127 . @xmath123 , as given by ( [ defsigmat ] ) , is then the free energy difference ( per unit area of the bubble ) between the two configurations ( @xmath77 and @xmath45 ) , which defines the surface tension . as shown in ref . @xcite , the expressions for the free energies in ( [ defsigmat ] ) , at 1-loop order give ( [ fieldsigmat ] ) . in the previous two sections we have obtained the temperature effects on the nucleation barrier by _ assuming _ that the system becomes trapped in a metastable phase as the temperature drops below the critical temperature @xmath13 . this assumption is not very realistic , and was addopted so that we could stress the main points of the calculation without having to worry about the effects of interactions of the order parameter with other fields . however , in models of interest within a cosmological context , such as the standard electroweak model and some of its extensions , the universe becomes trapped in a metastable phase due to the interactions of the higgs field ( or effective scalar order parameter ) with other massless fields , such as gauge bosons or fermions . in fact , it is possible to formulate this statement in terms of conditions that a general effective potential must satisfy , if it is to develop a metastable phase below @xmath13 @xcite . basically , the condition states that the mass gap between the symmetric and broken - symmetric phases must be large enough so that massive fluctuations away from the symmetric minimum are suppressed . since the mass gap is given in terms of the ( temperature dependent ) vacuum expectation value of the higgs field and of its coupling to other fields , the condition states that the transition can not be too weakly first order . the question then is what effective potential should be considered when calculating the decay rate . as recently shown by e. weinberg for the zero - temperature coleman - weinberg models ( which exhibit symmetry breaking only due to radiative corrections ) , the effective potential relevant in the calculation of the bounce solution is obtained by integrating over all fields _ but _ the scalar field ; the radiative corrections coming form these fields modify the vacuum structure of the model , making metastability possible . in this section , we argue that the same procedure must be followed when calculating the nucleation rate at finite temperature . we first review , in general terms , how to take into account the interactions of the scalar field with other fields in the calculation of the decay rate . ( for details see , ref . @xcite . ) then we apply our method to an example involving fermions , explicitly comparing our results to those obtained from eq . ( [ badrate ] ) . let us consider a system described by a scalar field @xmath9 and a set of fields @xmath128 ( bosonic or fermionic fields ) which are coupled to @xmath9 . the partition function of the system is given by @xmath129 where @xmath130 is the euclidean action of the system and the functional integration is carried over field configurations subject to periodic boundary conditions , @xmath131 , for bosons or antiperiodic , @xmath132 , for fermion fields . if one integrates out the @xmath128 fields in ( [ zphixi ] ) , the partition function can be written as @xmath133 where @xmath134 @xmath135 can be viewed as an effective action for the scalar field @xmath9 , where only @xmath128-loop terms are included . note that these @xmath128-loop terms introduce finite temperature corrections in ( [ wfunc ] ) . @xmath136 in ( [ zphixi ] ) and ( [ wfunc ] ) includes renormalization counter - terms and , if one or more of the @xmath128-fields is a gauge field , @xmath137 also includes the gauge fixings and the corresponding ghost terms . the procedure is now , in principle , straightforward . we evaluate the partition function in eq . ( [ zwphi ] ) semiclassically by expanding the effective action @xmath135 around its extremum configuration , which will be the bounce . note that the bounce will include the radiative corrections coming from the fields that couple to @xmath9 , _ but not from @xmath9 itself . _ the determinantal prefactor can be evaluated as before , by considering the negative and zero eigenvalues separately from the positive eigenvalues . however , as pointed out by weinberg @xcite , @xmath135 can not always be obtained in closed form , being in general a nonlocal functional . he proposes to resolve this difficulty by considering a local action @xmath138 which is close enough to the original one . from a derivative expansion of @xmath135 , @xmath139 is found to be @xmath140 \ : , \label{wphi0}\ ] ] where @xmath141 includes the tree level potential @xmath142 and the 1-loop contributions coming _ only _ from the @xmath128-field integration . the bounce solution can then be obtained from ( [ wphi0 ] ) . @xmath139 should be a good approximation to @xmath135 as long as the typical interaction length scale set by the field(s ) @xmath128 is shorter than the scale for the nonlocality of @xmath135 . in what follows we will apply the above formalism to a specific example of a scalar field coupled to fermions . for large enough yukawa couplings , this model has been shown to satisfy the conditions for metastability specified in ref . [ gleiser ] . consider a model of a real scalar field @xmath9 coupled to fermion fields with a lagrangian density @xmath143 with @xmath142 given by @xmath144 where @xmath145 and @xmath146 are positive , dimensionless constants and @xmath147 is a ( mass)@xmath148 parameter . a large enough coupling to fermions guarantees that a metastable phase is possible as the system is cooled below @xmath13 ; the high temperature minimum of the 1-loop effective potential , @xmath149 , lies to the left of the maximum of the potential at @xmath150 , and thermal fluctuations away from the symmetric minimum are suppressed . numerical values for the couplings satisfying these conditions can be found in ref . [ gleiser ] . the partition function for this model is given by @xmath151 as the fermion fields appear quadratically in ( [ zphipsi ] ) , they can be integrated out , giving @xmath152 where @xmath153 - { \rm tr } \ln ( - \not{\ ! \partial } - i g \phi)_{\beta } \ : , \label{wbetaphi}\ ] ] and @xmath154 thus , @xmath139 can be written as @xmath155 \ : , \label{wbeta}\ ] ] where , for sufficiently smooth fields ( see appendix b ) , the effective potential obtained after integrating over the fermions and renormalizing is ( we will drop all zero temperature quantum corrections ) @xmath156 the temperature dependent term accounts for finite temperature corrections coming from fermion loops . @xmath157 is the potential we should use to compute the bounce . note that , neglecting the @xmath158 quantum corrections , the high - temperature limit of @xmath157 is approximately @xmath159 so that for @xmath160 , a condition which is easily satisfied for reasonable values of the couplings , the high - temperature minimum is @xmath161 . once we have the action @xmath139 , the bounce is obtained as a solution of @xmath162 thus , for a static , spherically symmetric configuration , the bounce configuration @xmath6 will be a solution of @xmath163 with boundary conditions , @xmath164 and @xmath165 . ( from now on @xmath45 and @xmath44 should be understood as being the minima of ( [ vbetaphi ] ) and not the minima of the tree - level potential @xmath166 . ) the procedure is now identical to that of sections 2 and 3 . having a bounce solution we can evaluate the partition function written in eq . ( [ zwscalar ] ) semiclassically , exactly as was done in eqs . ( [ zphib ] ) and ( [ zphif ] ) , by expanding around @xmath167 and @xmath168 . we then obtain , from eqs . ( [ energy ] ) and ( [ gamma ] ) , the nucleation rate , @xmath169^{-\frac{1}{2 } } e^{- \delta w_0 } \ : , \label{gammaw}\ ] ] where @xmath170 , with @xmath171 obtained above . @xmath172 is given by @xmath173 where @xmath174 was defined in eq . ( [ wbeta ] ) . in order to proceed , we must rewrite the determinantal prefactor explicitly isolating the negative and zero modes from the positive modes . this is done following the same steps of section 3 , although now we must handle the fermionic contribution to the determinants . the details of the perturbative expansion for the fermionic determinantal prefactor are given in appendix b. we can then write the nucleation rate per unit volume as @xmath175^{\frac{3}{2 } } \exp \left [ - \frac { \delta f(t ) } { t } \right ] \ : , \label{endgammaw}\ ] ] where @xmath94 , the bubble activation free energy in the 1-loop approximation , is given by @xmath176 \right\ } \ : . \label{dft}\ ] ] as usual , the sum over @xmath177 can be performed and we get , @xmath178 \ : , \label{veff}\ ] ] where the effective potential @xmath179 , is given by ( neglecting zero temperature quantum corrections ) @xmath180 where @xmath166 is the tree level potential ( [ vphi ] ) and the mass term appearing in the scalar loop contribution is @xmath181 in leading order in the fermion loops . it is instructive to contrast this result with that obtained for self - coupled scalars , eq . ( [ veffren ] ) . the coupling to fermions modifies the scalar mass propagating in the loops . this effect naturally improves the infrared behavior of the theory , and can be of importance in weak first - order transitions . we will say more about this later . note also a crucial difference between this expression for the free energy barrier and the expression for the free energy barrier in eq . ( [ badrate ] ) : here , the bounce is obtained with the effective potential that does not include the corrections coming from scalar loops . the corrections from scalar loops which appear in the last term of eq . ( [ hatveff ] ) are thermally induced fluctuations about the bounce solution computed with @xmath182 . in the usual expression for the nucleation barrier the bounce is obtained from the full effective potential including the scalar loops . the two expressions are definetely not equivalent , even though for small scalar self - couplings the differences are negligible . in order to illustrate the differences let us look at a specific example . in fig . 2 we contrast the two approaches by comparing the nucleation barriers as a function of the temperature for a fixed set of coupling constants . the barriers in the figure were obtained by a numerical integration of the bounce equation including the relevant loops according to each approach . for clarity let us call the nucleation barrier obtained in the usual approach , _ i.e. _ , by including the scalar loops in the bounce calculation , the scalar barrier . the nucleation barrier obtained without including the scalar loops we call the fermionic barrier . in fig . 2 we take @xmath183 , and @xmath184 . since @xmath145 controls the strength of the scalar corrections , we expect the differences between the two barriers to be noticeable . we find that this is indeed the case , noting that as we approach the critical temperature ( that is , as we move closer to the thin - wall limit ) the differences between the two barriers increase , with the scalar barrier _ always larger _ than the fermionic barrier . this is precisely what one expects if the scalar corrections are entropic corrections to the nucleation barrier . thus , the nucleation barrier used in expression eq . ( [ badrate ] ) is overestimated for large enough scalar corrections . finally , we point out two additional differences between the results . first note that the contribution from the zero modes to the prefactor depends on @xmath185 , as opposed to @xmath94 . this could be important for weak transitions in which the prefactor may play a relevant rle . most importantly , the expression for @xmath186 , eq . ( [ hatveff ] ) , differs from the usual 1-loop finite temperature effective potential by the mass term for the scalar field loops , @xmath187 . since we have used the stationary points of @xmath139 , eq . ( [ wbeta ] ) , as opposed to the stationary points of @xmath188 , as the effective `` background '' fields in the saddle - point evaluation of the partition function , the scalar field propagator carries the finite temperature mass @xmath189 . the propagator is dressed by the quantum corrections due to fermion loops . in the usual 1-loop finite temperature effective potential , the stationary points are obtained from the tree level action , with mass term for scalar loops , @xmath190 . this results in the usual negative mass terms related to the change in convexity of the effective potential between the inflection points , and , in the case of very shallow potentials , in bad infrared behavior near @xmath8 . the incorporation of the fermionic corrections to the scalar propagator , which is demanded by our method of calculation atenuates these problems . in the example above , the scalar mass gets dressed by fermionic loops , being given by @xmath191 , where @xmath166 is the tree - level potential ( [ vphi ] ) . the temperature term in @xmath192 works as the infrared regulator for small values of @xmath190 . this result is independent of the particular model studied . similar conclusions have been obtained in ref . @xcite for scalar electrodynamics . in this paper we examined in some detail the computation of false vacuum decay rates at finite temperatures in the regime in which quantum fluctuations are negligibly small compared to thermal fluctuations . we have shown that temperature corrections to the nucleation barrier can be obtained from a saddle - point evaluation of the partition function in a dilute gas approximation . in fact , the temperature corrections are simply due to the positive eigenvalues from stable fluctuations around the critical bubble . that is , they are the entropic contributions due to thermally induced deformations on the bubble . even though this result has been known in classical statistical mechanics for more than two decades @xcite , we believe that a consistent treatment within field theory is still lacking . although we left many questions unanswered , we hope to have clarified some of the issues involved in the calculation of finite - temperature decay rates . of particular importance is the fact that the bounce is _ not _ obtained from the full 1-loop corrected effective potential , but from the potential excluding the scalar loops . thus , for a self - interacting scalar , the bounce is obtained from the tree - level potential . the full finite temperature potential appears in the exponent only after properly accounting for the positive eigenvalues of the determinantal prefactor . that is , the scalar contributions account for entropic corrections to the nucleation barrier . we obtained a temperature corrected nucleation barrier which can differ from the usual result . we showed this to be particularly true for sufficiently large scalar self - couplings in the vicinity of the critical temperature for the transition . also , we found that the interaction with other fields gives rise to a potential which is better behaved in the infrared . ( see also ref . this result is the finite - temperature equivalent to what e. weinberg found for the zero - temperature case , once the integration over the other fields is performed @xcite . the reader may be wondering if our results will have any consequences to current work on the electroweak phase transition . the answer depends on the higgs mass . for a sufficiently light higgs it is consistent to neglect the contribution from scalar loops to the effective potential . in this case , the usual estimate for the nucleation barrier is a valid approximation . however , the situation may change for a heavier higgs . given that the experimental lower bound on the higgs mass is now above 60 gev , we believe it worthwhile to study this question in more detail , keeping in mind that the transition becomes weaker as the higgs mass increases . we would like to thank a. linde for many important discussions on these and related issues . we would like to thank the institute for theoretical physics in santa barbara where , during the program on cosmological phase transitions , this work begun . at itp this work was supported in part by a national science foundation grant no . phy89 - 04035 at itp . ( mg ) was supported in part by a national science foundation grant no . ( gcm ) aknowledges financial support from fapesp ( so paulo , brazil ) and ( ror ) from conselho nacional de desenvolvimento cientfico e tecnolgico - cnpq ( brazil ) . * figure 1 : * a typical asymmetric double - well potential . + * figure 2 : * a comparison of the nucleation barrier as a function of temperature , in units of mass parameter @xmath193 , obtained by including ( stars ) and excluding ( dots ) scalar loops in the computation of the bounce . the parameters in the tree - level potential are @xmath194 . + * figure 3 : * a comparison of the terms @xmath195 and @xmath196 appearing in appendix b. the parameters in the tree - level potential are @xmath197 . + where the prime in @xmath199 means that the negative eigenvalue , @xmath200 , and the three zero eigenvalues , @xmath201 , are now excluded from the product . the term for @xmath202 in @xmath203 , can be handled as in ref . @xcite , resulting in the factor @xmath204^{\frac{3}{2}}$ ] in eq . ( [ eigenratio ] ) . separating the @xmath202 modes both in the numerator and the denominator of ( [ aratio ] ) , and using the identity ( [ idenpi ] ) , we get , @xmath205^{\frac{3}{2 } } \exp\left\{\left(-4 + \sum_{i } - \sum_{j}\ ; ' \right ) { \rm ln } \prod_{n=1}^{+\infty } \omega_{n}^{2 } - { \rm ln } \left ( e_{-}^{2 } \right)^{1/2 } - { \rm ln } \left [ \frac{\sin(\frac{\beta}{2 } |e_{-}|)}{\frac{\beta}{2 } |e_{-}| } \right ] + \right . & + & \left . \left(\sum_{j}\ ; ' - \sum_{i}\right ) { \rm ln } \beta + \sum_{i } \left [ \frac{\beta}{2 } e_{f}(i ) + { \rm ln } \left ( 1 - e^{- \beta e_{f}(i ) } \right ) \right ] + \right . \nonumber \\ & - & \left . \sum_{j } \ ; ' \left [ \frac{\beta}{2 } e_{b}(j ) + { \rm ln } \left ( 1 - e^{- \beta e_{b}(j ) } \right ) \right ] \right\ } \ : . \label{aexplicity}\end{aligned}\ ] ] & + & \left . \sum_{i } \left [ \frac{\beta}{2 } e_{f}(i ) + { \rm ln } \left ( 1 - e^{- \beta e_{f}(i ) } \right ) \right ] - \sum_{j } \ ; ' \left [ \frac{\beta}{2 } e_{b}(j ) + { \rm ln } \left ( 1 - e^{- \beta e_{b}(j ) } \right ) \right ] \right\ } \label{aendexplicitly}\end{aligned}\ ] ] @xmath213^{2 } & = & \det ( - \not{\ ! \partial } -i g \varphi).\det ( - \not{\ ! \partial } + i g \varphi ) = \nonumber \\ & = & \det \left [ ( -\box_{e } + g^{2 } \varphi^{2 } ) 1_{4 \times 4 } - i g \gamma_{e}^{\mu } \partial_{\mu } \varphi \right ] \ : , \label{bident}\end{aligned}\ ] ] @xmath227_{\beta } + { \rm tr } \ln \left [ - \box_{e } + g^{2 } \varphi_{b}^{2 } - g \frac { \partial \varphi_{b}}{\partial r } \right]_{\beta } - \right . \nonumber \\ & - & \left . 2 { \rm tr } \ln \left [ - \box_{e } + g^{2 } \varphi_{f}^{2 } \right]_{\beta } \right\ } \ : . \label{bdetbf}\end{aligned}\ ] ] @xmath228 \right ] + \right . \nonumber \\ & + & \left . { \rm tr } \ln \left [ 1 + s_{\beta}(\varphi_{f } ) \left [ g^{2 } ( \varphi_{b}^{2 } - \varphi_{f}^{2 } ) - g \frac { \partial \varphi_{b}}{\partial r } \right ] \right ] \right\ } \ : , \label{bexpratio}\end{aligned}\ ] ] @xmath230 \right ] & = & \sum_{m= 1}^{+ \infty } \frac{(-1)^{m+1}}{m } \int d^{3}x \left [ g^{2 } ( \varphi_{b}^{2 } - \varphi_{f}^{2 } ) \pm g \frac { \partial \varphi_{b}}{\partial r } \right]^{m } \times \nonumber \\ & \times & \sum_{n= -\infty}^{+ \infty } \int \frac{d^{3 } k } { ( 2 \pi)^{3 } } \frac{1 } { \left [ \bar{\omega}_{n}^{2 } + \vec{k}^{2 } + g^{2 } \varphi_{f}^{2 } \right]^{m } } \ : , \label{btraceln}\end{aligned}\ ] ] where @xmath231 . as before , ( [ btraceln ] ) can be expressed as a graphic expansion similar to ( [ graphic ] ) , with the propagators @xmath112 replaced now by @xmath232 and the external lines given by @xmath233 or @xmath234 . the determinant factor in ( [ gammaw ] ) , coming from the functional integration of the scalar field , can be evaluated by the same methods of sec . 3 . in ( [ gammaw ] ) , the determinant term @xmath235_{\beta}$ ] , with @xmath236 , has a negative eigenvalue , @xmath200 , associated with the instability of the critical bubble , and the three zero eigenvalues , associated with the translational invariance of the bubble . these eigenvalues can be handled as usual , giving the preexponential term in ( [ endgammaw ] ) . the part of the determinant involving the positive eigenvalues can be written as an expansion exactly as in ( [ traceln ] ) , @xmath237^{- \frac{1}{2 } } = \exp \left\ { - \frac{1}{2 } { \rm tr } \ ; \ln \bigl [ 1 + \hat{g}_{\beta}(\varphi_{f } ) \left [ \hat{v}_{\psi}''(\varphi_{b } ) - \hat{v}_{\psi}''(\varphi_{f } ) \right ] \bigr ] \right\ } \ : , \label{bdetbos}\ ] ] @xmath239 \right\ } & = & \sum_{m=1}^{+ \infty } \frac { ( -1)^{m+1 } } { m } \int d^{3 } x \left[\hat{v}_{\psi}''(\varphi_{b } ) - \hat{v}_{\psi}''(\varphi_{f } ) \right]^{m } \times \nonumber \\ & \times & \sum_{n= -\infty}^{+ \infty } \int \frac { d^{3 } k } { ( 2 \pi)^{3 } } \frac{1 } { \left [ \omega_{n}^{2 } + \vec{k}^{2 } + m_{\beta}^{2 } ( \varphi_{f } ) \right]^{m } } \ : . \label{btrlnw}\end{aligned}\ ] ] the sum in @xmath42 in both ( [ btraceln ] ) and ( [ btrlnw ] ) can be performed as in eq . ( [ traceln ] ) ) . therefore , from eqs . ( [ bexpw ] ) , ( [ btraceln ] ) and ( [ btrlnw ] ) , we can write the relevant part of eq . ( [ gammaw ] ) as @xmath240^{- \frac{1}{2 } } e^{- \delta w_0 } = \left [ \frac { \det ( -\box_{e } + \hat{v}_{\psi}''(\varphi_{b}))_{\beta } } { \det ( -\box_{e } + \hat{v}_{\psi}''(\varphi_{f}))_{\beta } } \right]^{- \frac{1}{2 } } \ : \ : \frac { \det ( - \not{\ ! \partial } - i g \varphi_{b } ) _ { \beta } } { \det ( - \not{\ ! \partial } - i g \varphi_{f } ) _ { \beta } } \ : \ : e^{-\delta s } = } \nonumber \\ & & = { \cal v}\frac{t^{4}}{i |e_{-}| } \frac{\beta \frac{|e_{-}|}{2 } } { \sin \left ( \beta \frac{|e_{-}|}{2 } \right ) } \left [ \frac{\delta w_0 } { 2 \pi } \right]^{\frac{3}{2 } } \exp \left\ { - \delta s + \int d^{3 } x \sum_{n= -\infty}^{+ \infty } \int \frac{d^3 k}{(2 \pi)^3 } \left [ \ln \left ( 1 + \frac{g^2 ( \vp_{b}^{2 } - \vp_{f}^{2 } ) + g \frac{\partial \vp_{b}}{\partial r}}{\bar{\omega}_{n}^{2 } + \vec{k}^2 + g^2 \vp_{f}^{2 } } \right ) + \right . \nonumber \\ & & + \left . \left . \ln \left ( 1 + \frac{g^2 ( \vp_{b}^{2 } - \vp_{f}^{2 } ) - g \frac{\partial \vp_{b}}{\partial r}}{\bar{\omega}_{n}^{2 } + \vec{k}^2 + g^2 \vp_{f}^{2 } } \right ) - \frac{1}{2 } \ln \left ( 1 + \frac{m_{\beta}^2 ( \vp_{b } ) - m_{\beta}^{2 } ( \vp_{f}^{2 } ) } { \omega_{n}^{2 } + \vec{k}^2 + m_{\beta}^{2 } ( \vp_{f}^{2 } ) } \right ) \right ] \right\ } \ : , \label{bdeterminantal}\end{aligned}\ ] ] apart from the derivative terms @xmath243 , the momentum integral reproduces the finite temperature corrections to the the tree - level potential appearing in @xmath67 . when we wrote the expression for @xmath244 in eq . ( [ dft ] ) , these terms were not included in the effective potential @xmath245 . there are two reasons for negleting this term . first , due to the graphic expansion we used for the determinants , it is easy to see that at least at the tadpole level , their contribution cancels . since the tadpole gives the dominant temperature contribution to the potential , terms that depend on @xmath243 will be sub - dominant . second , it is possible to explicitly compare the terms @xmath246 and @xmath247 , by obtaining @xmath6 numerically . we have performed this comparison for the same set of parameters used in figs . 2 and 3 , and convinced ourselves that the derivative term will indeed be sub - dominant . a typical example is shown in fig . neglecting the term @xmath243 , we can use eqs . ( [ bdeterminantal ] ) and ( [ finaldeltaw ] ) to obtain the expression for @xmath94 in ( [ dft ] ) . m. gleiser and e. w. kolb , _ phys . lett . _ * 69 * , 1304 ( 1992 ) ; fermilab preprint no . pub-92/222-a ; _ int . phys . _ * c3 * , 773 ( 1992 ) ; m. gleiser , e. w. kolb , and r. watkins , _ nucl . phys . _ * b364 * , 411 ( 1991 ) . c. a. carvalho , g. c. marques , a. j. silva and i. ventura , _ nucl . * b265 * , 45 ( 1986 ) ; c. a. carvalho , d. bazeia , o. j. p. eboli , g. c. marques , a. j. silva and i. ventura _ phys . rev . _ * d31 * , 1411 ( 1985 ) ;
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we examine the computation of the nucleation barrier used in the expression for false vacuum decay rates in finite temperature field theory . by a detailed analysis of the determinantal prefactor
, we show that the correct bounce solution used in the computation of the nucleation barrier should not include loop corrections coming from the scalar field undergoing decay .
temperature corrections to the bounce appear from loop contributions from other fields coupled to the scalar field .
we compute the nucleation barrier for a model of scalar fields coupled to fermions , and compare our results to the expression commonly used in the literature .
we find that , for large enough self - couplings , the inclusion of scalar loops in the expression of the nucleation barrier leads to an underestimate of the decay rate in the neighborhood of the critical temperature .
pacs number(s ) : 98.80.cq , 64.60.qb .
+ e - mail : [email protected] ; [email protected] ; [email protected] .
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cosmological general relativity ( cgr ) is a 5-d time - space - velocity theory@xcite of the cosmos , for one dimension of cosmic time , three of space and one of the universe expansion velocity . cosmic time is taken to increase from the present epoch @xmath32 toward the big bang time @xmath33 , where @xmath3 is the hubble - carmeli time constant . the expansion velocity @xmath34 at the present epoch and increases toward the big bang . in this paper the cosmic time @xmath35 is held fixed ( @xmath36 ) and measurements are referred to the present epoch of cosmic time . this is a reasonable approach since the time duration over which observations are made is negligible compared to the travel time of the emitted light from the distant galaxy . hence , in this paper we examine the four - dimensional ( 4-d ) space - velocity of cgr . a general solution to the einstein field equations in the space - velocity domain is obtained , analogous to the friedmann - lematre - robertson - walker ( flrw ) solution of space - time cosmology . the main emphasis herein is to develop a cosmology having a scale factor @xmath37 dependent on the expansion velocity which in turn can be expressed in terms of the cosmological redshift @xmath38 . this will enable a set of well defined tools for the analysis of observational data where the cosmological redshift plays a central role . the resulting cosmology is used to model a small set of sne ia data . we will derive carmeli s cosmology as a special case where the scale factor is held fixed . two principal results of carmeli s cosmology is the prediction of the accelerated expansion of the universe@xcite and the description of spiral galaxy rotation curves without additional dark matter@xcite . we continue to support those results within this paper with a theoretical framework that accommodates a more varied parameter space . for our purposes , a vacuum mass density @xmath11 is defined in terms of a cosmological constant @xmath12 by @xmath39 where the carmeli gravitational coupling constant @xmath14 , where @xmath15 is the speed of light in vacuum and @xmath3 is the hubble - carmeli time constant . in a previous article@xcite we hypothesized that the observable universe is one of two black holes joined at their event horizons . from this perspective we show that the vacuum density of the observable universe and the universe black hole entropy have the relation @xmath40 , where @xmath41 is the bekenstein - hawking entropy@xcite of the black hole . we also will use an evolving two parameter equation of state @xmath42 , dependent on the scale factor @xmath37 , which allows for the evolution of the effect of dark energy on the pressure@xcite . assuming the matter in the universe to be isotropically distributed we will adopt a metric that is spatially spherical symmetric . furthermore , the spatial coordinates will be co - moving such that galaxies expanding along the same geodesic curve are motionless with respect to one another . in this manner we can compare observations between galaxies moving along different geodesic paths . a general derivation of the metric we will use was given by tolman@xcite and is taken in the simplified form defined by @xmath43 where @xmath1 is the metric tensor . the comoving coordinates are @xmath2 , where @xmath3 is the hubble - carmeli time constant , @xmath4 is the universe expansion velocity and @xmath5 , @xmath6 and @xmath7 are the spatial coordinates . assume that the functions @xmath8 and @xmath9 are functions of coordinates @xmath10 and @xmath5 . the constant @xmath3 is related to the hubble constant @xmath44 at zero distance and zero gravity by the relation @xmath45 where measurements of @xmath44 at very close ( local ) distances are used to determine the value of @xmath46 . at this writing the accepted value@xcite is @xmath47 therefore @xmath48 from ( [ eq : tolman - line - element ] ) the non - zero components of the metric tensor @xmath1 are given by @xmath49 the choice of the particular metric ( [ eq : tolman - line - element ] ) determines that the 4-velocity of a point moving along the geodesic curve is given by @xmath50 the universe expands by the null condition @xmath51 . for a spherically symmetric expansion one has @xmath52 . the metric ( [ eq : tolman - line - element ] ) then gives @xmath53 which yields @xmath54 to determine the functions @xmath8 and @xmath9 we need to solve the einstein field equations . the einstein field equations with a cosmological constant @xmath12 term are taken in the form @xmath55 where @xmath56 is the ricci tensor , @xmath57 is the energy - momentum tensor , @xmath58 is its trace and @xmath59 is carmeli s gravitational coupling constant given by @xmath60 where @xmath25 is newton s gravitational constant and @xmath15 is the speed of light in vacuum . if we add the tensor @xmath61 to the ricci tensor @xmath56 , the covariant derivative of the new tensor is still zero . that is @xmath62 since the covariant derivatives of the ricci tensor and the metric tensor are both zero . we will move the cosmological constant term from the left hand side ( l.h.s . ) to the right hand side ( r.h.s . ) of ( [ eq : field - eqs-0 ] ) to make it a component of the energy - momentum tensor giving @xmath63 since the covariant derivative of the energy - momentum tensor @xmath64 , the covariant derivative of the r.h.s . of ( [ eq : field - eqs-1 ] ) equals zero . the @xmath12 term is absorbed into a new energy - momentum tensor by the form @xmath65 , \label{eq : tuv_def}\ ] ] where @xmath18 is the mass density , @xmath19 is the pressure and @xmath66 is the 4-velocity . the @xmath67 factor on the r.h.s . of ( [ eq : tuv_def ] ) is an artifact of convenience in solving the field equations . taking the trace @xmath68 of the new energy - momentum tensor yields @xmath69 \label{eq : t - trace-1 } \\ \nonumber \\ & = & \tau^2 \rho - 3 \frac { \tau } { c } p - 4 \tau^2 \left ( \frac{\lambda } { \tau^2 \kappa } \right ) . \nonumber\end{aligned}\ ] ] define the vacuum mass density @xmath11 in terms of the cosmological constant @xmath12 , @xmath70 by this definition we are defining @xmath11 to be a negative density , unless @xmath12 is negative . this is contrary to the design of the standard model but is in keeping with the behar - carmeli cosmological model@xcite . then in terms of the vacuum mass density @xmath11 , define the effective mass density and effective pressure , @xmath71 and @xmath72 with the definitions for the effective mass density and the effective pressure the energy - momentum tensor ( [ eq : tuv_def ] ) becomes @xmath20 . \label{eq : t_uv_eff}\ ] ] in terms of the effective mass density and pressure the only non - zero components of @xmath73 are given by @xmath74 the trace of @xmath73 in terms of the effective mass density and pressure is @xmath75 substituting for the defined values of @xmath76 and @xmath77 , the trace ( [ eq : t - eff ] ) expands out to @xmath78 which is equal to ( [ eq : t - trace-1 ] ) . with our definition for the new energy - momentum tensor the field equations now take the form @xmath79 here we write out the nonvanishing components of the ricci tensor . a dot @xmath80 denotes partial differentiation with respect to @xmath81 and a prime @xmath82 denotes partial differentiation with respect to @xmath83 . this follows@xcite , ( sect . 4.3 . ) @xmath84 expanding the r.h.s . of the field equations ( [ eq : field - eqs - new ] ) yields @xmath85 we obtain our first independent field equation by multiplying ( [ eq : r11=t11-g11t/2 ] ) by @xmath86 and adding the result to ( [ eq : r00=t00-g00t/2 ] ) . this operation will eliminate the @xmath87 and @xmath88 terms leaving @xmath89 by multiplying ( [ eq : r22=t22-g22t/2 ] ) by @xmath90 and adding the result to ( [ eq : basic-1a ] ) , the @xmath91 and @xmath77 terms will be eliminated leaving @xmath92 = \kappa \tau^2 \rho_{eff}. \label{eq : basic-1b}\ ] ] the next basic independent field equation is obtained by substituting for the value of the expression @xmath93 in ( [ eq : basic-1b ] ) . to do that , multiply ( [ eq : r22=t22-g22t/2 ] ) by @xmath94 and move all other terms to the r.h.s . leaving @xmath95 substitute the expression in ( [ eq : basic-2a ] ) for its corresponding expression in ( [ eq : basic-1b ] ) . after eliminating some terms , combining other terms , multiplying both sides by @xmath96 and then simplifying , we obtain the next basic field equation @xmath97 we restate the last basic field equation we need , which is ( [ eq : r01=t01-g01t/2 ] ) , @xmath98 equations ( [ eq : basic-1b ] ) , ( [ eq : basic-2b ] ) , ( [ eq : basic-3 ] ) correspond to carmelis@xcite eqns . ( 4.3.31 ) , ( 4.3.29 ) and ( 4.3.30 ) respectively . equation ( [ eq : basic-3 ] ) can be partially integrated with respect to @xmath81 , keeping @xmath5 constant . integrating it we have @xmath99 with the result @xmath100 which can be put in the form @xmath101 where @xmath102 is an arbitrary function . the integration constants @xmath103 and @xmath104 are both evaluated at some particular value of coordinate @xmath105 so they can only be functions of coordinate @xmath5 . thus @xmath102 is a function of @xmath5 only . we can then write ( [ eq : int - mu-1b ] ) in the useful form @xmath106 since the l.h.s . of ( [ eq : int - mu-1c ] ) is positive definite , the r.h.s . is likewise , implying that the condition on @xmath107 is @xmath108 the solution ( [ eq : int - mu-1c ] ) is the same as@xcite , eqn . ( 4.3.16 ) . for the function @xmath9 we assume the general form @xmath109 where @xmath110 is a function of @xmath5 only and @xmath111 is a function of coordinate @xmath10 only . before proceeding with the solution of ( [ eq : basic-2b ] ) , which has the pressure @xmath77 , we first determine the relevant partial derivatives of ( [ eq : r = rrv ] ) , @xmath112 substituting for @xmath113 and the above derivatives into ( [ eq : basic-2b ] ) yields @xmath114 - \left ( r^{'}_r \right)^2 r^2_v \label{eq : basic-2b - sol-1a } \\ \nonumber \\ & = & -\kappa \frac{\tau}{c } \left ( \frac{\left ( r^{'}_r \right)^2 r^2_v } { 1 + f \left ( r \right ) } \right ) r^2_r r^2_v p_{eff}. \nonumber \end{aligned}\ ] ] multiply ( [ eq : basic-2b - sol-1a ] ) by @xmath115 , gather terms and simplify to obtain @xmath116 assuming the pressure @xmath77 is not a function of @xmath5 , then since the l.h.s . of ( [ eq : basic-2b - sol-1b ] ) is a function of @xmath81 only and the r.h.s . is a function of @xmath5 only , they both must equal a constant @xmath117 . from the r.h.s . we conclude that @xmath118 we next solve ( [ eq : basic-1b ] ) which contains the mass density @xmath76 . however , we need to first determine the derivatives of the function @xmath8 . using ( [ eq : int - mu-1c ] ) we obtain @xmath119 substituting for the functions and derivatives in ( [ eq : basic-1b ] ) we obtain in unsimplified form @xmath120 \nonumber \\ \nonumber \\ & & + \left ( \frac { 2 \dot{r}_v } { r_v } \right ) \left ( \frac { r_r \dot{r}_v } { r_r r_v } \right ) + \left ( \frac { r_r \dot{r}_v } { r_r r_v } \right)^2 + \frac { 1 } { r^2_r r^2_v } = \kappa \tau^2 \rho_{eff}. \label{eq : basic-1b - sol-1a } \end{aligned}\ ] ] this simplifies to @xmath121 - \frac { 1 } { r^2_r } = { \cal f } \left ( v , r \right ) , \nonumber\end{aligned}\ ] ] where in general @xmath122 is a function of @xmath4 and @xmath5 . for the function @xmath123 we now assume the simple form @xmath124 then we have for its derivatives , @xmath125 as we can see , @xmath126 implies that its terms along with the @xmath127 factor drops out of ( [ eq : cal - f - equation ] ) . substituting the solution values ( [ eq : r_r - solution ] ) , ( [ eq : r_r - prime ] ) and ( [ eq : r_r - doubleprime ] ) into the r.h.s . of ( [ eq : cal - f - equation ] ) and simplifying we obtain a first order differential equation of the function @xmath102 given by @xmath128 where @xmath129 because the r.h.s . of ( [ eq : cal - f - equation ] ) has become a function of @xmath5 only , while the l.h.s . is a funcion of @xmath4 only , so they both must equal the constant @xmath130 . by ( [ eq : f - function - form ] ) and ( [ eq : r_r - solution ] ) we obtain @xmath131 which implies that , for this solution of @xmath132 , @xmath133 if @xmath134 then ( [ eq : f(r)-inhomo - sol ] ) is the solution to the inhomogeneous differential equation ( [ eq : f - differential ] ) . if @xmath135 then the homogeneous solution is @xmath136 where the coordinate system is centered on the central mass @xmath137 . the general solution to ( [ eq : f - differential ] ) is the sum of ( [ eq : f(r)-inhomo - sol ] ) and ( [ eq : fr)-homo - sol ] ) , @xmath138 from ( [ eq : cal - f - equation ] ) we have @xmath139 to obtain a value for @xmath117 from ( [ eq : basic-1b - sol-1b ] ) , assuming @xmath140 and @xmath141 , multiply the l.h.s . by @xmath142 and rearrange the result to obtain @xmath143 where @xmath144 is the hubble parameterdefined by @xmath145 and @xmath146 is the critical density defined by @xmath147 notice that ( [ eq : fo - val-1a0 ] ) is true at any epoch of coordinate @xmath148 , so we will evaluate it at @xmath149 . we define the scale factor @xmath37 , the hubble parameter @xmath144 , the effective mass density @xmath76 and the critical density @xmath146 to have the values at the present epoch , @xmath150 where @xmath151 is the mass density and @xmath152 is the critical density at the present epoch @xmath34 . with the values from ( [ eq : rv0])-([eq : rho_c0 ] ) put into ( [ eq : fo - val-1a0 ] ) we have @xmath153 \label{eq : fo - val-1a } \\ \nonumber \\ & = & \frac{-1 } { c^2 \tau^2 } \left [ \omega_m + \omega_{\lambda } - 1 \right ] \nonumber \\ \nonumber \\ & = & -k , \nonumber\end{aligned}\ ] ] where @xmath154 where @xmath155 is the matter mass density parameter at the present epoch ( @xmath34 ) , @xmath156 is the constant vacuum mass density parameter , and @xmath157 is the curvature . note that the curvature @xmath157 has the dimension of @xmath158^{-2}$ ] . as will be shown , the type of spatial geometry , hyperbolic ( open ) , euclidean ( flat ) or spherical ( closed ) , is determined by the curvature @xmath157 which in turn depends on the mass and vacuum densities . this compares with the standard model where the type of geometry is determined by the dimensionless curvature parameter @xmath159 or @xmath160 , for open , flat or closed , respectively . the scale radius @xmath161 is given by @xmath162 with the value for @xmath117 from ( [ eq : fo - val-1a ] ) , with no central mass @xmath137 , @xmath113 takes the form @xmath163 for @xmath51 , ( [ eq : dr / dv-0 ] ) describes the isotropic expansion of the universe . using ( [ eq : emu ] ) for @xmath113 , the expansion can be described by @xmath164 @xmath165 to show the relation of the scale factor @xmath37 to the cosmological redshift @xmath38 , suppose an observer in a galaxy a measures the expansion velocities of two other galaxies `` g '' and `` g+@xmath166 '' which is nearby galaxy `` g '' . the velocities for the galaxies are @xmath167 and @xmath168 respectively . suppose that another observer in another galaxy o measures expansion velocities for galaxies `` g '' and `` g+@xmath166 '' and obtains the values @xmath169 and @xmath170 respectively . we assume that @xmath171 and @xmath172 are small compared to @xmath167 and @xmath169 , respectively , since galaxies `` g '' and `` g+@xmath166 '' are near each other . what can we say about the relationship between these measured velocities from galaxies a and o ? assume the distance to galaxy a is @xmath173 and the distance to galaxy o is @xmath174 . the galaxies are comoving which implies that the distance @xmath175 between them is constant . for the galaxy `` g '' , we integrate ( [ eq : drar = dvbv-00 ] ) between the points @xmath176 and @xmath177 @xmath178 for the galaxy `` g+@xmath166 '' we integrate ( [ eq : drar = dvbv-00 ] ) between the points @xmath179 and @xmath180 , @xmath181 subtracting ( [ eq : int - galax - g ] ) from ( [ eq : int - galax - g+dg ] ) we obtain @xmath182 after some manipulations of the integrals ( [ eq : int - galax - diff - vel ] ) reduces to @xmath183 where @xmath184 is the approximate value of the assumed slowly varying scale factor over the small velocity interval @xmath185 , and likewise for @xmath186 over the small interval @xmath187 . ( [ eq : int - delta - vo , delta - va ] ) can be put into the form @xmath188 if in galaxy a the distance @xmath189 is determined by the measurement of @xmath190 of the wavelength of photons from galaxy `` g + @xmath191 '' , then in galaxy o the corresponding measurement of the photons from the same galaxy will have a wavelength @xmath192 . the ratio of the two wavelengths is assumed to be given by ( [ eq : distance - ratio-1 ] ) , @xmath193 where @xmath38 is the cosmological redshift of the photon . if we take galaxy o to be the local galaxy then @xmath194 and we set the scale factor of the local galaxy to unity , @xmath195 . then setting @xmath196 for the velocity of a general galaxy a , ( [ eq : distance - ratio-3 ] ) can be put into the familiar form of the scale factor redshift relation , @xmath197 where @xmath198 where @xmath190 is the photon wavelength detected by distant galaxy a and @xmath192 is the wavelength observed in the local galaxy . by substituting the r.h.s . of ( [ eq : basic-1b - sol-1b ] ) into its relevant term in ( [ eq : basic-2b - sol-1b ] ) and simplifying we obtain @xmath199 we can obtain another expression for @xmath200 by differentiating ( [ eq : basic-1b - sol-1b ] ) with respect to ( w.r.t . ) @xmath148 which , after simplifying gives @xmath201 combining ( [ eq : r - doubledot-1 ] ) and ( [ eq : r - doubledot-2 ] ) and simplifying gives us the effective mass density continuity equation @xmath202 simplifying ( [ eq : eff - mass - continuity ] ) yields @xmath203 this mass density continuity equation ( [ eq : mass - continuity ] ) is identical in form to the energy density continuity equation of the standard flrw model except that here the rate of mass density change is w.r.t . expansion velocity @xmath4 while the standard model rate is w.r.t . cosmic time @xmath35 . we choose an evolving equation of state parameter@xcite @xmath204 such that the pressure @xmath19 is related to the mass density @xmath18 by @xmath205 where @xmath206 where @xmath30 and @xmath31 are constants . the second term on the r.h.s . of ( [ eq : we - def ] ) represents the evolution of the equation of state as a function of expansion velocity . in particular , this functional form allows for a state equation to vary from low dark energy influence when the scale factor @xmath37 was small into becoming dominated by dark energy at the current unity scale factor . substituting for @xmath19 from ( [ eq : p - w - rho ] ) into ( [ eq : mass - continuity ] ) we obtain @xmath207 ( [ eq : mass - w - cont ] ) can be put into the form @xmath208 which upon integration yields @xmath209 divide ( [ eq : rho = rho0*rv ] ) by @xmath152 to get the mass density parameter @xmath210 converting @xmath37 in terms of the redshift @xmath38 from ( [ eq : cosmo - redshift - relation ] ) , ( [ eq : omega = omega_m*r_v ] ) for @xmath211 becomes @xmath212 with the equation of state ( [ eq : p - w - rho ] ) the effective pressure becomes @xmath213 by substituting the r.h.s . of ( [ eq : basic-1b - sol-1b ] ) into its relevant term in ( [ eq : basic-2b - sol-1b ] ) and simplifying we obtain @xmath214 substitute for effective pressure from ( [ eq : eqn - of - state-1 ] ) into ( [ eq : r - doubledot-3 ] ) and simplify to obtain the scale factor acceleration equation @xmath215 r_v . \label{eq : r - doubledot-3 - 0}\ ] ] this can be put into the form @xmath216 where @xmath217 is given by ( [ eq : omega = omega_m*r_v ] ) and @xmath156 is the vacuum mass density parameter . ( [ eq : r - doubledot-3 - 2 ] ) exhibits a range of possible senarios for accelerating and decelerating expansions depending on @xmath204 and on the values for @xmath155 and @xmath156 . it was discovered experimentally that the expanding universe makes a transition from accelerating to decelerating@xcite . assume that the transition took place at velocity @xmath218 corresponding to a redshift @xmath219 . then , taking ( [ eq : r - doubledot-3 - 2 ] ) and setting @xmath220 at @xmath221 we have an expression for @xmath204 in terms of the transition redshift @xmath219 , @xmath222 \omega_{z_t } - 2 \omega_{\lambda } = 0 , \label{eq : w_e - formula}\ ] ] where @xmath223 . this expression can be used to obtain a value for @xmath219 in terms of the fitted parameters @xmath155 , @xmath156 , @xmath30 and @xmath31 . we will encounter the transition redshift in the section on modeling . the scale radius of the universe given by ( [ eq : scale - radius - r0-def ] ) can be put into the form @xmath224 the value of @xmath155 is the mass density at the present epoch of cosmic time @xmath32 , recalling that in cgr the cosmic time is measured from the present time @xmath32 increasing toward the big bang time @xmath225 . assuming the universe is a black hole@xcite of radius @xmath161 , then the event horizon surface area @xmath226 is given by @xmath227 then the bekenstein - hawking@xcite entropy @xmath41 of the black hole universe is given by @xmath228 where @xmath229 is boltzmann s constant and @xmath230 is planck s constant over @xmath231 . multiplying the r.h.s . of ( [ eq : s - entropy - critical - mass ] ) by @xmath232 and simplifying we obtain @xmath233 where @xmath234 , @xmath235 and @xmath236 is the cosmological planck mass density defined by @xmath237 where @xmath238 is the cosmological planck mass and @xmath239 is the cosmological planck length . by eqs . ( [ eq : s - entropy - critical - mass]-[eq : planck_length ] ) , since @xmath240 observationally , and since the entropy is always non - negative , then a positive vacuum mass density implies a positive cosmological planck mass density and visa - versa for a negative vacuum density . a deeper analysis into the relation between black hole entropy and the vacuum mass density is beyond the scope of this paper . ( [ eq : basic-1b - sol-1b ] ) can be put into the form @xmath241 where we used ( [ eq : rho_eff_fin ] ) for @xmath242 , @xmath217 is given by ( [ eq : omega = omega_m*r_v ] ) and @xmath243 is the curvature density parameter . we select the minus sign in the r.h.s . of ( [ eq : drv / dv ] ) because @xmath37 is assumed to be a decreasing function of @xmath4 . ( [ eq : dr / dv-0 ] ) describes the isotropic expansion of the universe . using ( [ eq : emu ] ) for @xmath113 , the expansion can be described by @xmath244 from ( [ eq : drv / dv ] ) we obtain @xmath245 substituting for @xmath246 from ( [ eq : dv = drv / c ] ) into the l.h.s . of ( [ eq : drar = dvbv ] ) and simplifying we derive @xmath247 from ( [ eq : cosmo - redshift - relation ] ) the scale factor @xmath37 is related to the cosmological redshift @xmath38 by @xmath248 differentiating ( [ eq : rv - z ] ) w.r.t . @xmath38 gives @xmath249 substituting for @xmath37 and @xmath250 from ( [ eq : rv - z ] ) and ( [ eq : drv / dz ] ) into ( [ eq : dvbv = drar ] ) and simplifying yields the differential for the comoving distance relation @xmath251 where @xmath211 is given by ( [ eq : omega = omega_m*z ] ) . the spatial geometry defined by the r.h.s . of ( [ eq : dzbz = drar ] ) is either hyperbolic ( open ) , euclidean ( flat ) or spherical ( closed ) depending on curvature @xmath252 , @xmath253 or @xmath254 , respectively . since @xmath157 is dependent on the mass and vacuum densities then the geometry of the universe is determined by @xmath155 and @xmath156 . the expansion defined by ( [ eq : drar = dvbv ] ) when substituted for the l.h.s . of ( [ eq : dzbz = drar ] ) and combined with ( [ eq : rv - z ] ) yields the differential equation for the expansion velocity @xmath255 integrating ( [ eq : dv / c ] ) we get for the expansion velocity as a function of redshift @xmath256 the hubble parameter defined by ( [ eq : h_v - def ] ) , using ( [ eq : drv / dv ] ) , is expressed in terms of velocity @xmath4 , @xmath257 where @xmath45 , @xmath217 . in terms of redshift , using ( [ eq : rv - z ] ) for @xmath37 , @xmath258 where @xmath211 is defined by ( [ eq : omega = omega_m*z ] ) . the comoving distance @xmath259 is the integral of ( [ eq : dzbz = drar ] ) and can be written in terms of the hubble parameter using ( [ eq : h - func - of - z ] ) , @xmath260 the transverse comoving distance @xmath261 is the coordinate distance @xmath5 which is obtained from the inversion of ( [ eq : comoving_distance ] ) , i.e. , @xmath262 . it takes the form @xmath263 a physical source of size @xmath264 subtends an observed angle of @xmath265 on the sky given by the relation @xmath266 where @xmath267 is the proper distance from the coordinate system origin to the source . since @xmath268 is the scale factor and @xmath269 is the coordinate distance , we define the angular diameter distance , @xmath270 in can be shownthat the source luminosity @xmath271 transforms due to the universe expansion as @xmath272 . then the flux @xmath41 from a source of luminosity @xmath271 at proper distance @xmath273 is given by @xmath274 where we used @xmath275 . the bolometric flux @xmath41 can also be defined for a source of luminosity @xmath271 at distance @xmath276 by @xmath277 eliminating @xmath41 between ( [ eq : s_bolo_flux_def1 ] ) and ( [ eq : s_bolo_flux_def2 ] ) we have the luminosity distance in terms of the redshift @xmath278 where we also used the l.h.s . of ( [ eq : comoving_distance ] ) , ( [ eq : d_m_def ] ) and the hubble parameter defined by ( [ eq : h - func - of - z ] ) , @xmath279 where @xmath211 from ( [ eq : omega = omega_m*z ] ) , @xmath280 we give a view of the cosmology by applying it to a small combined set@xcite total of 157 high redshift sne ia data , distance moduli and errors ( @xmath281 ) but not systematic errors . since we are expecting a scale factor transition from accelerated to decelerated expansion at low redshift we require that at the origin the scale factor acceleration @xmath282 . the standard distance modulus relation @xmath283 is given by @xmath284 where @xmath285 is the absolute magnitude of a standard supernova at the peak of its light - curve , @xmath286 is the luminosity distance ( [ eq : d_l_z - def ] ) and @xmath287 is an arbitrary zero point offset@xcite . for all our examples , the set of parameters are pre - selected by trial and error and then a final fit is made of the distance modulus relation to the data varying only the offset parameter @xmath287 . for the first example we take for the mass density parameter @xmath288 where @xmath289 is the baryon density parameter , and @xmath290 . from ( [ eq : h - value ] ) this gives @xmath291 . we use a value of @xmath292 from@xcite . therefore , our value for the mass density parameter is @xmath293 by a few trials we found a good fit for a value of the vacuum density parameter @xmath294 this defines an open universe since the curvature @xmath295 . we consider an evolving dark energy state with @xmath296 and @xmath297 . [ fig : scalfac - accel ] shows that the acceleration @xmath298 starts out positive at @xmath299 and transitions to negative around @xmath300 . the detail at the acceleration transition is displayed by fig . [ fig : scalfac - accel - trans - z ] . in fig . [ fig : mub - vs - z ] is the plot of the distance moduli for the sne ia data . the theoretical distance modulus ( [ eq : m(z)-theoretical ] ) is shown by the solid line . we obtained a fitted value @xmath301 with @xmath302 . figs . [ fig : trnsv - dist]-[fig : angdiam - dist ] display the various distance relations all with the same parameters from this first example . distances are in units of @xmath303 . for the second example we select no dark matter @xmath304 with positive vacuum density @xmath305 this again defines an open universe since @xmath306 . the equation of state parameters are @xmath307 and @xmath308 . for the third examle we select baryonic and dark matter @xmath309 with positive vacuum density @xmath310 where @xmath311 which gives a flat space curvature @xmath312 as in the standard flrw lambda - cold - dark - matter ( lcdm ) model . we use the same equation of state @xmath313 and @xmath308 as the second example . the results of these two examples are combined in the plots . [ fig : scalfac - accel - trans - z - darkmatterdarkenergy ] shows details of the acceleration @xmath298 . the thick curve is for @xmath314 , @xmath315 , @xmath316 and @xmath317 with transition @xmath318 . the thin curve is for @xmath319 , @xmath320 , @xmath316 and @xmath317 with transition @xmath321 as was reported@xcite . ( we actually varied @xmath31 to get a smaller @xmath322 and in the process honed in on @xmath323 . ) fig . [ fig : mub - vs - z - darkmatterdarkenergy ] shows the plots of the distance moduli for the sne ia data fitted to these two sets of parameters . the thick ( upper ) curve is for @xmath314 , @xmath315 , @xmath307 , @xmath317 and fitted offset @xmath324 with @xmath325 . the thin ( lower ) curve is for @xmath319 , @xmath320 , @xmath307 , @xmath317 and fitted offset @xmath326 with @xmath327 . the examples and parameters are summarized in table [ tb : examples-1 - 2 - 3 ] . for a brief summary , example ( ex . ) 2 has the smallest @xmath328 where @xmath329 , while ex . 1 has the largest @xmath330 , with ex . 3 in between with a @xmath331 . however , both ex.2 and ex . 3 have exceptionally high evolution ( @xmath296 , @xmath308 ) toward higher redshift , of what may be termed a `` anti - phantom '' energy , which might be characterized as physically unlikely because it has not been observed so far . on the other hand , ex . 1 has an evolution ( @xmath296 , @xmath297 ) of `` anti - dark '' energy toward higher redshift which cancels the dark energy development , which is physically more plausible since it involves dark energy for which there is observable evidence . thus , qualitatively , ex . 1 is given a best fit " rating overall . ref . ( riess , et al.@xcite , sect . 4.3 ) for a nice exposition on dark energy . these three examples provide a glimpse at the cosmology . consider that there are @xmath332 possible categories of the parameter space : @xmath155 with no dark matter or dark matter , @xmath156 , @xmath30 and @xmath31 each either negative , zero or positive . obviously , systematic analyses and larger data sets are required to narrow down the parameter space . .curve fit parameters . [ cols="^,^,^,^,^,^,^,^,^",options="header " , ] [ tb : examples-1 - 2 - 3 ] the 4-d space - velocity cosmology of carmeli@xcite is based on the premise of a constant scale factor @xmath37 . assume @xmath333 with @xmath334 the key equations are ( [ eq : basic-2b - sol-1b ] ) , ( [ eq : cal - f - equation ] ) and ( [ eq : f - differential ] ) . with the values eqs . ( [ eq : r - carmeli])-([eq : r - dub - prime - carmeli ] ) substituted into ( [ eq : cal - f - equation ] ) we get @xmath335 where @xmath130 is a constant since @xmath76 is assumed to be a constant with respect to @xmath4 and @xmath5 . however , it is assumed that @xmath76 is a function of the cosmic time . for the inhomogeneous case where @xmath336 the solution of ( [ eq : carmeli - f_o - equation ] ) is @xmath337 where @xmath117 is a constant . and , for the homogeneous case where @xmath338 the solution is @xmath339 where @xmath137 is a constant point mass centered at the origin of coordinates . the general solution is the sum of ( [ eq : f - carmeli - inhomo - sol ] ) and ( [ eq : f - homo - carmeli - sol ] ) , @xmath340 where the system of coordinates is centered on the central mass @xmath137 . using ( [ eq : f - general - carmeli - sol ] ) in ( [ eq : carmeli - f_o - equation ] ) gives the value @xmath341 from ( [ eq : basic-2b - sol-1b ] ) , with @xmath342 and @xmath343 we get @xmath344 substituting for @xmath107 from ( [ eq : f - general - carmeli - sol ] ) into ( [ eq : carmeli - basic-1 ] ) gives the effective pressure @xmath345 the value of @xmath117 is obtained from ( [ eq : basic-1b - sol-1b ] ) with @xmath346 and @xmath343 , so that at @xmath34 it gives us @xmath347 with this value for @xmath117 we obtain from ( [ eq : carmeli - pressure - val ] ) the effective pressure @xmath348 where @xmath349 is the central mass density , which at this point is a convenient mathematical construct . the function ( [ eq : int - mu-1c ] ) for @xmath113 is now given by @xmath350 where @xmath351 we make note that carmeli did not define an effective pressure , but simply used an ordinary pressure . in our analysis the effective pressure is used . we could setup an equation of state for carmeli cosmology that is defined , taking the central mass @xmath352 in ( [ eq : basic-2b - sol - carmeli ] ) , @xmath353 where @xmath354 . looking at ( [ eq : p - w - rho ] ) this implies that @xmath355 and @xmath356 . in carmeli cosmology the effective mass density @xmath357 , so we equate it with our definition ( [ eq : rho_eff_fin ] ) and obtain @xmath358 from which , with @xmath359 , we get @xmath360 where @xmath361 is the critical density defined by ( [ eq : rho_c_def ] ) . using ( [ eq : rho_v - def ] ) this yields @xmath362 allowing for the @xmath363 this is the relation for @xmath12 which was reported@xcite . however , this does not fix the value of @xmath12 but it appears to experimentally have the same order of magnitude@xcite . using ( [ eq : carmeli - rho_eff ] ) and ( [ eq : carmeli - rho_lambda ] ) , the effective mass density parameter @xmath242 is given by @xmath364 where @xmath365 and @xmath366 with no central mass , @xmath352 , and using eqs . ( [ eq : omega_eff_carmeli])-([eq : omega_lambda - carmeli ] ) then ( [ eq : int - mu - carmeli ] ) takes the form @xmath367 equation ( [ eq : int - mu - carmeli - form-1 ] ) is written in that form because @xmath368 at the present epoch . the curvature @xmath369 which implies that the carmeli universe has a hyperbolic ( open ) spatial geometry . setting @xmath51 in ( [ eq : dr / dv-0 ] ) for the expansion of the universe gives the differential equation @xmath370 upon integration , assuming @xmath371 , ( [ eq : expansion - carmeli ] ) yields for the expansion velocity @xmath372 inverting ( [ eq : expan - carmeli - solve ] ) we obtain the velocity - distance relation @xmath373 when the central mass @xmath374 , the expression ( [ eq : int - mu - carmeli ] ) takes the form @xmath375 then the differential equation for the universe expansion ( [ eq : dr / dv-0 ] ) takes the form @xmath376 the velocity - distance relation is then given by @xmath377 where the expansion is centered on the point mass @xmath137 . this model was recently described by hartnett@xcite . this is the basics of the cosmology . for @xmath368 the curvature @xmath378 defines an open universe . since the scale factor @xmath37 is assumed to be constant , a velocity - redshift relation must be obtained by other methods . the special relativistic doppler velocity - redshift relation @xmath379 is often used . however , since here the scale factor @xmath37 is constant , the redshift @xmath38 is assumed to be a function of the cosmic time . also , the evolution of the mass density parameter @xmath155 is assumed to be in the time domain and its variation also must be obtained by other methods . a further limitation is that in ( [ eq : expansion - carmeli ] ) , the restriction that @xmath380 requires @xmath381 approximately . this makes it difficult for high redshift data analysis@xcite . on the other hand , when cosmic time is added as a fifth dimension the cgr 5-d time - space - velocity model is applicable to galaxy dynamics because the cosmological redshift across a galaxy region is nearly constant@xcite . a general solution to the einstein field equations has been obtained for the four dimensional space - velocity cosmological general relativity theory of carmeli . this development provides the tools necessary for the analysis of astrophysical data . in particular , a redshift - distance relation is given , analogous to the standard flrw cosmology , and an evolving equation of state is provided . an analysis of high redshift sne ia data was made to show the efficacy of the cosmology . model examples were given with only baryonic matter and dark energy in hyperbolic space and with baryonic plus dark matter and dark energy in flat space . finally , carmeli s cosmology in 4-d was obtained . oliveira , f. j. , and hartnett , j. g. : carmeli s cosmology fits data for an accelerating and decelerating universe without dark matter or dark energy . found . . lett . * 19*(6 ) , 519 - 535 ( 2006 ) . arxiv : astro - ph/0603500 riess , a. g. , et al . : type ia supernova discoveries at @xmath382 from the hubble space telescope : evidence for past deceleration and constraints on dark energy evolution . j. * 607 * , 665 - 687 ( 2004 ) riess , a. g. , et al . : new hubble space telescope discoveries of type ia supernovae at @xmath383 : narrowing constraints on the early behavior of dark energy . astrophs . j. * 659*(1 ) ( 2007 ) . arxiv : astro - ph/0611572(2006 ) astier , p. , et al . : the supernova legacy survey : measurement of @xmath384 , @xmath385 and @xmath386 from the first year data set . & astrophs . * 447*(1 ) , 31 - 48 ( 2006 ) . doi : 10.1051/0004 - 6361:20054185 . arxiv : astro - ph/0510447
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in this paper the four - dimensional ( 4-d ) space - velocity cosmological general relativity of carmeli is developed by a general solution of the einstein field equations .
the tolman metric is applied in the form @xmath0 where @xmath1 is the metric tensor .
we use comoving coordinates @xmath2 , where @xmath3 is the hubble - carmeli time constant , @xmath4 is the universe expansion velocity and @xmath5 , @xmath6 and @xmath7 are the spatial coordinates .
we assume that @xmath8 and @xmath9 are each functions of the coordinates @xmath10 and @xmath5 .
the vacuum mass density @xmath11 is defined in terms of a cosmological constant @xmath12 , @xmath13 where the carmeli gravitational coupling constant @xmath14 , where @xmath15 is the speed of light in vacuum .
this allows the definitions of the effective mass density @xmath16 and effective pressure @xmath17 where @xmath18 is the mass density and @xmath19 is the pressure . then the energy - momentum tensor @xmath20 , \label{abs : t_uv_eff}\ ] ] where @xmath21 is the 4-velocity .
the einstein field equations are taken in the form @xmath22 where @xmath23 is the ricci tensor , @xmath24 is carmeli s gravitation constant , where @xmath25 is newton s constant and the trace @xmath26 . by solving the field equations ( [ abs : field - eqs - new ] ) a space - velocity cosmology
is obtained analogous to the friedmann - lematre - robertson - walker space - time cosmology .
we choose an equation of state such that @xmath27 with an evolving state parameter @xmath28 where @xmath29 is the scale factor and @xmath30 and @xmath31 are constants .
carmeli s 4-d space - velocity cosmology is derived as a special case .
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ensuring accountability for security violations is essential in a wide range of settings . for example , protocols for authentication and key exchange @xcite , electronic voting @xcite , auctions @xcite , and secure multiparty computation ( in the semi - honest model ) @xcite ensure desirable security properties if protocol parties follow their prescribed programs . however , if they deviate from their prescribed programs and a security property is violated , determining which agents should be held accountable and appropriately punished is important to deter agents from committing future violations . indeed the importance of accountability in information systems has been recognized in prior work @xcite . our thesis is that _ actual causation _ ( i.e. , identifying which agents actions caused a specific violation ) is a useful building block for accountability in decentralized multi - agent systems , including but not limited to security protocols and ceremonies @xcite . causation has been of interest to philosophers and ideas from philosophical literature have been introduced into computer science by the seminal work of halpern and pearl @xcite . in particular , counterfactual reasoning is appealing as a basis for study of causation . much of the definitional activity has centered around the question of what it means for event @xmath0 to be an actual cause of event @xmath1 . an answer to this question is useful to arrive at causal judgments for specific scenarios such as `` john s smoking causes john s cancer '' rather than general inferences such as `` smoking causes cancer '' ( the latter form of judgments are studied in the related topic of type causation @xcite ) . notably , hume @xcite identified actual causation with counterfactual dependence the idea that @xmath0 is an actual cause of @xmath1 if had @xmath0 not occurred then @xmath1 would not have occurred . while this simple idea does not work if there are independent causes , the counterfactual interpretation of actual causation has been developed further and formalized in a number of influential works ( see , for example , @xcite ) . even though applications of counterfactual causal analysis are starting to emerge in the fields of ai , model - checking , and programming languages , causation has not yet been studied in connection with security protocols and violations thereof . on the other hand , causal analysis seems to be an intuitive building block for answering some very natural questions that have direct relevance to accountability such as ( i ) _ why _ a particular violation occurred , ( ii ) _ what _ component in the protocol is blameworthy for the violation and ( iii ) _ how _ the protocol could have been designed differently to preempt violations of this sort . answering these questions requires an in - depth study of , respectively , explanations , blame - assignment , and protocol design , which are interesting problems in their own right , but are not the explicit focus of this paper . instead , we focus on a formal definition of causation that we believe formal studies of these problems will need . roughly speaking , explanations can be used to provide an _ account _ of the violation , _ blame assignment _ can be used to hold agents _ accountable _ for the violation , and protocol design informed by these would lead to protocols with better accountability guarantees . we further elaborate on explanations and blame - assignment in section [ sec : domains ] . formalizing actual causes as a building block for accountability in decentralized multi - agent systems raises new conceptual and technical challenges beyond those addressed in the literature on events as actual causes . in particular , prior work does not account for the program dynamics that arise in such settings . let us consider a simple protocol example . in the movie _ flight _ @xcite , a pilot drinks and snorts cocaine before flying a commercial plane , and the plane goes into a locked dive in mid - flight . while the pilot s behavior is found to be deviant in this case he does not follow the prescribed protocol ( program ) for pilots it is found to not be an actual cause of the plane s dive . the actual cause was a deviant behavior by the maintenance staff they did not replace a mechanical component that should have been replaced . ideally , the maintenance staff should have inspected the plane prior to take - off according to their prescribed protocol . this example is useful to illustrate several key ideas that influence the formal development in this paper . first , it illustrates the importance of capturing the _ actual interactions _ among agents in a decentralized multi - agent system with non - deterministic execution semantics . the events in the movie could have unfolded in a different order but it is clear that the actual cause determination needs to be done based on the sequence of events that happened in reality . for example , had the maintenance staff replaced the faulty component _ before _ the take - off the plane would not have gone into a dive . second , the example motivates us to hold accountable agents who exercise their choice to execute a deviant _ program _ that actually caused a violation . the maintenance staff had the choice to replace the faulty component or not where the task of replacing the component could consist of multiple steps . it is important to identify which of those steps were crucial for the occurrence of the dive . thus , we focus on formalizing _ program actions _ executed in sequence ( by agents ) as actual causes of violations rather than individual , independent events as formalized in prior work . finally , the example highlights the difference between deviance and actual causes a difference also noted in prior work on actual causation . this difference is important from the standpoint of accountability . in particular , the punishment for deviating from the prescribed protocol could be suspension or license revocation whereas the punishment for actually causing a plane crash in which people died could be significantly higher ( e.g. , imprisonment for manslaughter ) . the first and second ideas , reflecting our program - based treatment , are the most significant points of difference from prior work on actual causation @xcite while the third idea is a significant point of difference from prior work in accountability @xcite . the central contribution of this paper is a formal definition of _ program actions as actual causes_. specifically , we define what it means for a set of program actions to be an actual cause of a violation . the definition considers a set of interacting programs whose concurrent execution , as recorded in a log , violates a trace property . it identifies a subset of actions ( program steps ) of these programs as an actual cause of the violation . the definition applies in two phases . the first phase identifies what we call _ lamport causes_. a lamport cause is a minimal prefix of the log of a violating trace that can account for the violation . in the second phase , we refine the actions on this log by removing the actions which are merely _ progress enablers _ and obtain _ actual action causes_. the former contribute only indirectly to the cause by enabling the actual action causes to make progress ; the exact values returned by progress enabling actions are irrelevant . we demonstrate the value of this formalism in two ways . first , we prove that violations of a precisely defined class of safety properties always have an actual cause . thus , our definition applies to relevant security properties . second , we provide a cause analysis of a representative protocol designed to address weaknesses in the current public key certification infrastructure . moreover , our example illustrates that our definition cleanly handles the separation between joint and independent causes a recognized challenge for actual cause definitions @xcite . in addition , we discuss how this formalism can serve as a building block for causal explanations and exoneration ( i.e. , soundly identifying agents who should not be blamed for a violation ) . we leave the technical development of these concepts for future work . the rest of the paper is organized as follows . section [ sec : example ] describes a representative example which we use throughout the paper to explain important concepts . section [ sec : definitions ] gives formal definitions for program actions as actual causes of security violations . we apply the causal analysis to the running example in section [ sec : application ] . we discuss the use of our causal analysis techniques for providing explanations and assigning blame in section [ sec : domains ] . we survey additional related work in section [ sec : related ] and conclude in section [ sec : conclusion ] . in this section we describe an example protocol designed to increase accountability in the current public key infrastructure . we use the protocol later to illustrate key concepts in defining causality . [ [ security - protocol ] ] security protocol + + + + + + + + + + + + + + + + + consider an authentication protocol in which a user ( @xmath21 ) authenticates to a server ( @xmath31 ) using a pre - shared password over an adversarial network . @xmath21 sends its user - id to @xmath31 and obtains a public key signed by @xmath31 . however , @xmath21 would need inputs from additional sources when @xmath31 sends its public key for the first time in a protocol session to verify that the key is indeed bound to @xmath31 s identity . in particular , @xmath21 can verify the key by contacting multiple notaries in the spirit of _ perspectives _ @xcite . for simplicity , we assume @xmath21 verifies @xmath31 s public key with three authorized notaries@xmath41 , @xmath42 , @xmath43and accepts the key if and only if the majority of the notaries say that the key is legitimate . to illustrate some of our ideas , we also consider a parallel protocol where two parties ( @xmath22 and @xmath23 ) communicate with each other . we assume that the prescribed programs for @xmath31 , @xmath21 , @xmath41 , @xmath42 , @xmath43 , @xmath22 and @xmath23 impose the following requirements on their behavior : ( i ) @xmath31stores @xmath21 s password in a hashed form in a secure private memory location . ( ii ) @xmath21 requests access to the account by sending an encryption of the password ( along with its identity and a timestamp ) to @xmath31 after verifying @xmath31 s public key with a majority of the notaries . ( iii ) the notaries retrieve the key from their databases and attest the key correctly . ( iv ) @xmath31 decrypts and computes the hashed value of the password . ( v ) @xmath31 matches the computed hash value with the previously stored value in the memory location when the account was first created ; if the two hash values match , then @xmath31 grants access to the account to @xmath21 . ( vi ) in parallel , @xmath22 generates and sends a nonce to @xmath23 . ( vii ) @xmath23 generates a nonce and responds to @xmath22 . [ [ security - property ] ] security property + + + + + + + + + + + + + + + + + the prescribed programs in our example aim to achieve the property that only the user who created the account and password ( in this case , @xmath21 ) gains access to the account . [ [ compromised - notaries - attack ] ] compromised notaries attack + + + + + + + + + + + + + + + + + + + + + + + + + + + we describe an attack scenario and use it to illustrate nuances in formalizing program actions as actual causes . @xmath21executes its prescribed program . @xmath21 sends an access request to @xmath31 . an @xmath5 intercepts the message and sends a public key to @xmath21 pretending to be @xmath31 . @xmath21 checks with @xmath41 , @xmath42 and @xmath43 who falsely verify @xmath5 s public key to be @xmath31 s key . consequently , @xmath21 sends the password to @xmath5 . @xmath5 then initiates a protocol with @xmath31 and gains access to @xmath21 s account . in parallel , @xmath22sends a request to @xmath31 and receives a response from @xmath31 . following this interaction , @xmath22 forwards the message to @xmath23 . we assume that the actions of the parties are recorded on a _ log _ , say @xmath6 . note that this log contains a violation of the security property described above since @xmath5 gains access to an account owned by @xmath21 . first , our definition finds _ program actions as causes _ of violations . at a high - level , as mentioned in the introduction , our definition applies in two phases . the first phase ( section [ sec : definitions ] , definition [ definition : cause1 ] ) identifies a minimal prefix ( phase 1 , _ minimality _ ) of the log that can account for the violation i.e. we consider all scenarios where the sequence of actions execute in the same order as on the log , and test whether it suffices to recreate the violation in the absence of all other actions ( phase 1 , _ sufficiency _ ) . in our example , this first phase will output a minimal prefix of log @xmath6 above . in this case , the minimal prefix will not contain interactions between @xmath22 and @xmath23 after @xmath31 has granted access to the @xmath5 ( the remaining prefix will still contain a violation ) . second , a nuance in defining the notion of _ sufficiency _ ( phase 1 , definition [ definition : cause1 ] ) is to constrain the interactions which are a part of the actual cause set in a manner that is consistent with the interaction recorded on the log . this constraint on interactions is quite subtle to define and depends on how strong a coupling we find appropriate between the log and possible counterfactual traces in sufficiency : if the constraint is too weak then the violation does not reappear in all sequences , thus missing certain causes ; if it is too strong it leads to counter - intuitive cause determinations . for example , a weak notion of consistency is to require that each program locally execute the same prefix in sufficiency as it does on the log i.e. consistency w.r.t . program actions for individual programs . this notion does not work because for some violations to occur the _ order of interactions _ on the log among programs is important . a notion that is too strong is to require matching of the total order of execution of all actions across all programs . we present a formal notion of _ consistency _ by comparing log projections ( section [ sec : support - defs ] ) that balance these competing concerns . third , note that while phase 1 captures a minimal prefix of the log sufficient for the violation , it might be possible to remove actions from this prefix which are merely required for a program execution to progress . for instance note that while all three notaries actions are required for @xmath21 to progress ( otherwise it would be stuck waiting to receive a message ) and the violation to occur , the actual message sent by one of the notaries is irrelevant since it does not affect the majority decision in this example . thus , separating out actions which are _ progress enablers _ from those which provide information that causes the violation is useful for fine - grained causal determination . this observation motivates the final piece ( phase 2 ) of our formal definition ( definition [ definition : cause2a ] ) . finally , notice that in this example @xmath5 , @xmath41 , @xmath42 , @xmath43 , @xmath31 and @xmath22 deviate from the protocol described above . however , the deviant programs are not sufficient for the violation to occur without the involvement of @xmath21 , which is also a part of the causal set . we thus seek a notion of sufficiency in defining a set of programs as a joint actual cause for the violation . joint causation is also significant in legal contexts @xcite . for instance , it is useful for holding liable a group of agents working together when none of them satisfy the cause criteria individually but together their actions are found to be a cause . the ability to distinguish between joint and independent ( i.e. , different sets of programs that independently caused the violation ) causes is an important criterion that we want our definition to satisfy . in particular , phase 2 of our definition helps identify independent causes . for instance , in our example , we get three different independent causes depending on which notary s action is treated as a progress enabler . our ultimate goal is to use the notion of actual cause as a building block for accountability the independent vs. joint cause distinction is significant when making deliberations about accountability and punishment for liable parties . we can use the result of our causal determinations to further remove deviants whose actions are required for the violation to occur but might not be blameworthy ( section [ sec : domains ] ) . we present our language model in section [ sec : model ] , auxiliary notions in section [ sec : support - defs ] , properties of interest to our analysis in section [ sec : properties ] , and the formal definition of program actions as actual causes in section [ sec : cause1 ] . we model programs in a simple concurrent language , which we call @xmath7 . the language contains sequential expressions , @xmath1 , that execute concurrently in threads and communicate with each other through ` send ` and ` recv ` commands . terms , @xmath8 , denote messages that may be passed through expressions or across threads . variables @xmath9 range over terms . an expression is a sequence of actions , @xmath10 . an action may do one of the following : execute a primitive function @xmath11 on a term @xmath8 ( written @xmath12 ) , or send or receive a message to another thread ( written @xmath13 and @xmath14 , respectively ) . we also include very primitive condition checking in the form of @xmath15 . @xmath16 each action @xmath10 is labeled with a unique line number , written @xmath17 . line numbers help define traces later . we omit line numbers when they are irrelevant . every action and expression in the language evaluates to a term and potentially has side - effects . the term returned by action @xmath10 is bound to @xmath9 in evaluating @xmath18 in the expression @xmath19 . following standard models of protocols , @xmath20 and @xmath21 are untargeted in the operational semantics : a message sent by a thread may be received by any thread . targeted communication may be layered on this basic semantics using cryptography . for readability in examples , we provide an additional first argument to @xmath20 and @xmath21 that specifies the _ intended _ target ( the operational semantics ignore this intended target ) . action @xmath13 always returns @xmath22 to its continuation . primitive functions @xmath11 model thread - local computation like arithmetic and cryptographic operations . primitive functions can also read and update a _ thread - local state _ , which may model local databases , permission matrices , session information , etc . if the term @xmath8 in @xmath15 evaluates to a non - true value , then its containing thread gets stuck forever , else @xmath15 has no effect . we abbreviate @xmath23 to @xmath24 and @xmath25 to @xmath26 when @xmath9 is not free in @xmath1 . as an example , the following expression receives a message , generates a nonce ( through a primitive function @xmath27 ) and sends the concatenation of the received message and the nonce on the network to the intended recipient @xmath28 ( line numbers are omitted here ) . @xmath29 for the purpose of this paper , we limit attention to this simple expression language , without recursion or branching . our definition of actual cause is general and applies to any formalism of ( non - deterministic ) interacting agents , but the auxiliary definitions of log projection and the function @xmath30 introduced later must be modified . [ [ app : operational : semantics ] ] operational semantics + + + + + + + + + + + + + + + + + + + + + the language @xmath7 s operational semantics define how a collection of _ threads _ execute concurrently . each thread @xmath31 contains a unique thread identifier @xmath32 ( drawn from a universal set of such identifiers ) , the executing expression @xmath1 , and a local store . a _ configuration _ @xmath33 models the threads @xmath34 executing concurrently . our reduction relation is written @xmath35 and defined in the standard way by interleaving small steps of individual threads ( the reduction relation is parametrized by a semantics of primitive functions @xmath11 ) . importantly , each reduction can either be internal to a single thread or a _ synchronization _ of a ` send ` in one thread with a ` recv ` in another thread . we make the locus of a reduction explicit by annotating the reduction arrow with a _ label _ this is written @xmath37 . a label is either the identifier of a thread @xmath32 paired with a line number @xmath17 , written @xmath38 and representing an internal reduction of some @xmath12 in thread @xmath32 at line number @xmath17 , or a tuple @xmath39 , representing a synchronization between a ` send ` at line number @xmath40 in thread @xmath41 with a ` recv ` at line number @xmath42 in thread @xmath43 , or @xmath44 indicating an unobservable reduction ( of @xmath8 or @xmath15 ) in some thread . labels @xmath45 are called _ local labels _ , labels @xmath46 are called _ synchronization labels _ and labels @xmath44 are called _ silent labels_. an _ initial configuration _ can be described by a triple @xmath47 , where @xmath48 is a finite set of thread identifiers , @xmath49 and @xmath50 . this defines an initial configuration of @xmath51 threads with identifiers in @xmath48 , where thread @xmath32 contains the expression @xmath52 and the store @xmath53 . in the sequel , we identify the triple @xmath54 with the configuration defined by it . we also use a configuration s identifiers to refer to its threads . given an initial configuration @xmath55 , a run is a finite sequence of labeled reductions @xmath56 . a pre - trace is obtained by projecting only the stores from each configuration in a run . let @xmath56 be a run and let @xmath57 be the store in configuration @xmath58 . then , the pre - trace of the run is the sequence @xmath59 . if @xmath60 , then the @xmath32th step is an unobservable reduction in some thread and , additionally , @xmath61 . a trace is a pre - trace from which such @xmath44 steps have been dropped . the trace of the pre - trace @xmath59 is the subsequence obtained by dropping all tuples of the form @xmath62 . traces are denoted with the letter @xmath8 . to define actual causation , we find it convenient to introduce the notion of a log and the log of a trace , which is just the sequence of non - silent labels on the trace . a log is a sequence of labels other than @xmath44 . the letter @xmath6 denotes logs . given a trace @xmath63 , the log of the trace , @xmath64 , is the sequence of @xmath65 . ( the trace @xmath8 does not contain a label @xmath66 that equals @xmath44 , so neither does @xmath64 . ) we need a few more straightforward definitions on logs in order to define actual causation . given a log @xmath6 and a thread identifier @xmath32 , the projection of @xmath6 to @xmath32 , written @xmath67 is the subsequence of all labels in @xmath6 that mention @xmath32 . formally , @xmath68 we call a log @xmath69 a _ projected prefix _ of the log @xmath6 , written @xmath70 , if for every thread identifier @xmath32 , the sequence @xmath71 is a prefix of the sequence @xmath67 . the definition of projected prefix allows the relative order of events in two different non - communicating threads to differ in @xmath6 and @xmath69 but lamport s happens - before order of actions @xcite in @xmath69 must be preserved in @xmath6 . similar to projected prefix , we define projected sublog . we call a log @xmath69 a _ projected sublog _ of the log @xmath6 , written @xmath72 , if for every thread identifier @xmath32 , the sequence @xmath71 is a subsequence of the sequence @xmath67 ( i.e. , dropping some labels from @xmath67 results in @xmath71 ) . a _ property _ is a set of ( good ) traces and violations are traces in the complement of the set . our goal is to define the cause of a violation of a property . we are specifically interested in ascribing causes to violations of safety properties @xcite because safety properties encompass many relevant security requirements . we recapitulate the definition of a safety property below . briefly , a property is safety if it is fully characterized by a set of finite violating prefixes of traces . let @xmath73 denote the universe of all possible traces . a property @xmath74 ( a set of traces ) is a safety property , written @xmath75 , if @xmath76 . as we explain soon , our causal analysis ascribes thread actions ( or threads ) as causes . one important requirement for such analysis is that the property be closed under reordering of actions in different threads if those actions are not related by lamport s happens - before relation @xcite . for properties that are not closed in this sense , the _ global order _ between actions in a race condition may be a cause of a violation . whereas causal analysis of race conditions may be practically relevant in some situation , we limit attention only to properties that are closed in the sense described here . we call such properties reordering - closed or @xmath77 . two traces @xmath78 starting from the same initial configuration are called reordering - equivalent , written @xmath79 if for each thread identifier @xmath32 , @xmath80 . note that @xmath81 is an equivalence relation on traces from a given initial configuration . let @xmath82_\sim$ ] denote the equivalence class of @xmath8 . a property @xmath74 is called reordering - closed , written @xmath83 , if @xmath84 implies @xmath82_\sim \subseteq p$ ] . note that @xmath83 iff @xmath85 . in the sequel , let @xmath86 denote the _ complement _ of a reordering - closed safety property of interest . ( the subscript @xmath87 stands for `` violations '' . ) consider a trace @xmath8 starting from the initial configuration @xmath55 . if @xmath88 , then @xmath8 violates the property @xmath89 . a violation of the property @xmath89 is a trace @xmath88 . our definition of actual causation identifies a subset of actions in @xmath90 as the cause of a violation @xmath88 . the definition applies in two phases . the first phase identifies what we call _ lamport causes_. a lamport cause is a minimal projected prefix of the log of a violating trace that can account for the violation . in the second phase , we refine the log by removing actions that are merely _ progress enablers _ ; the remaining actions on the log are the _ actual action causes_. the former contribute only indirectly to the cause by enabling the actual action causes to make progress ; the exact values returned by progress enabling actions are irrelevant . the following definition , called phase 1 , determines lamport causes . it works as follows . we first identify a projected prefix @xmath6 of the log of a violating trace @xmath8 as a potential candidate for a lamport cause . we then check two conditions on @xmath6 . the _ sufficiency _ condition tests that the threads of the configuration , when executed at least up to the identified prefix , preserving all synchronizations in the prefix , suffice to recreate the violation . the _ minimality _ condition tests that the identified lamport cause contains no redundant actions . [ definition : cause1 ] let @xmath88 be a trace starting from @xmath91 and @xmath6 be a projected prefix of @xmath64 , i.e. , @xmath92 . we say that @xmath6 is the lamport cause of the violation @xmath8 of @xmath86 if the following hold : 1 . [ sufficiency1 ] * ( sufficiency ) * let @xmath31 be the set of traces starting from @xmath93 whose logs contain @xmath6 as a projected prefix , i.e. , @xmath94 . then , every trace in @xmath31 has the violation @xmath86 , i.e. , @xmath95 . ( because @xmath96 , @xmath31 is non - empty . ) 2 . [ minimality1 ] * ( minimality ) * no proper prefix of @xmath6 satisfies condition [ sufficiency1 ] . at the end of phase 1 , we obtain one or more minimal prefixes @xmath6 which contain program actions that are sufficient for the violation . these prefixes represent independent lamport causes of the violation . in the phase 2 definition below , we further identify a sublog @xmath97 of each @xmath6 , such that the program actions in @xmath97 are actual causes and the actions in @xmath98 are progress enabling actions which only contribute towards the _ progress _ of actions in @xmath97 that cause the violation . in other words , the actions not considered in @xmath97 contain all labels whose actual returned values are irrelevant . briefly , here s how our phase 2 definition works . we first pick a candidate projected sublog @xmath97 of @xmath6 , where log @xmath6 is a lamport cause identified in phase 1 . we consider counterfactual traces obtained from initial configurations in which program actions omitted from @xmath97 are replaced by actions that do not have any effect other than enabling the program to progress ( referred to as no - op ) . if a violation appears in all such counterfactual traces , then this sublog @xmath97 is a good candidate . of all such good candidates , we choose those that are minimal . the key technical difficulty in writing this definition is replacing program actions omitted from @xmath97 with no - ops . we can not simply erase any such action because the action is expected to return a term which is bound to a variable used in the action s continuation . hence , our approach is to substitute the variables binding the returns of no - oped actions with arbitrary ( side - effect free ) terms @xmath8 . formally , we assume a function @xmath99 that for line number @xmath17 in thread @xmath32 suggests a suitable term @xmath100 that must be returned if the action from line @xmath17 in thread @xmath32 is replaced with a no - op . in our cause definition we universally quantify over @xmath101 , thus obtaining the effect of a no - op . for technical convenience , we define a syntactic transform called @xmath102 that takes an initial configuration , the chosen sublog @xmath97 and the function @xmath101 , and produces a new initial configuration obtained by erasing actions not in @xmath97 by terms obtained through @xmath101 . [ def : dummify ] let @xmath54 be a configuration and let @xmath97 be a log . let @xmath99 . the dummifying transform @xmath103 is the initial configuration @xmath104 , where for all @xmath105 , @xmath106 is @xmath52 modified as follows : * if @xmath107 appears in @xmath52 but @xmath38 does not appear in @xmath97 , then replace @xmath108 with @xmath109 $ ] in @xmath52 . * if @xmath110 appears in @xmath52 but @xmath111 does not appear in @xmath97 and @xmath112 , then replace @xmath113 with @xmath114 $ ] in @xmath52 . we now present our main definition of actual causes . [ definition : cause2a ] let @xmath88 be a trace from the initial configuration @xmath54 and let the log @xmath92 be a lamport cause of the violation determined by definition [ definition : cause1 ] . let @xmath97 be a projected sublog of @xmath6 , i.e. , let @xmath115 . we say that @xmath97 is the actual cause of violation @xmath8 of @xmath86 if the following hold : 1 . [ sufficiency2 ] ( * sufficiency * ) pick any @xmath101 . let @xmath116 and let @xmath31 be the set of traces starting from @xmath117 whose logs contain @xmath97 as a projected sublog , i.e. , @xmath118 . then , @xmath31 is non - empty and every trace in @xmath31 has the violation @xmath86 , i.e , @xmath95 . [ minimality2 ] ( * minimality * ) no proper sublog of @xmath97 satisfies condition [ sufficiency2 ] . at the end of phase 2 , we obtain one or more sets of actions @xmath97 . these sets are deemed the independent actual causes of the violation @xmath8 . the following theorem states that for all safety properties that are re - ordering closed , the phase 1 and phase 2 definitions always identify at least one lamport and at least one actual cause . suppose @xmath86 is reordering - closed and the complement of a safety property , i.e. , @xmath119 and @xmath120 . then , for every @xmath88 : ( 1 ) our phase 1 definition ( definition [ definition : cause1 ] ) finds a lamport cause @xmath6 , and ( 2 ) for every such lamport cause @xmath6 , the phase 2 definition ( definition [ definition : cause2a ] ) finds an actual cause @xmath97 . ( 1 ) pick any @xmath88 . we follow the phase 1 definition . it suffices to prove that there is a log @xmath92 that satisfies the sufficiency condition . since @xmath121 , there is a prefix @xmath122 of @xmath8 s.t . for all @xmath123 , @xmath124 . choose @xmath125 . since @xmath122 is a prefix of @xmath8 , @xmath126 . to prove sufficiency , pick any trace @xmath127 s.t . @xmath128 . it suffices to prove @xmath129 . since @xmath128 , for each @xmath32 , @xmath130 for some @xmath131 . let @xmath132 be the ( unique ) subsequence of @xmath127 containing all labels from the logs @xmath133 . consider the trace @xmath134 . first , @xmath135 extends @xmath122 , so @xmath136 . second , @xmath137 because @xmath138 . since @xmath139 , @xmath129 . ( 2 ) pick any @xmath88 and let @xmath6 be a lamport cause of @xmath8 as determined by the phase 1 definition . following the phase 2 definition , we only need to prove that there is at least one @xmath115 that satisfies the sufficiency condition . we choose @xmath140 . to show sufficiency , pick any @xmath101 . because @xmath141 , @xmath97 specifies an initial prefix of every @xmath52 and the transform @xmath142 has no effect on this prefix first , we need to show that at least one trace @xmath127 starting from @xmath143 satisfies @xmath144 . for this , we can pick @xmath145 . second , we need to prove that any trace @xmath127 starting from @xmath146 s.t . @xmath147 satisfies @xmath129 . pick such a @xmath127 . let @xmath122 be the prefix of @xmath8 corresponding to @xmath6 . then , @xmath148 for each @xmath32 . it follows immediately that for each @xmath32 , @xmath149 for some @xmath150 . let @xmath132 be the unique subsequence of @xmath127 containing all labels from traces @xmath151 . let @xmath134 . first , because for each @xmath32 , @xmath152 , @xmath153 trivially . because @xmath6 is a lamport cause , it satisfies the sufficiency condition of phase 1 , so @xmath154 . since @xmath120 , and @xmath135 extends @xmath122 , @xmath136 . second , @xmath155 because @xmath156 and both @xmath135 and @xmath127 are traces starting from the initial configuration @xmath146 . hence , by @xmath119 , @xmath129 . our phase 2 definition identifies a set of program actions as causes of a violation . however , in some applications it may be necessary to ascribe thread identifiers ( or programs ) as causes . this can be straightforwardly handled by lifting the phase 2 definition : a thread @xmath32 ( or @xmath52 ) is a cause if one of its actions appears in @xmath97 . [ definition : cause2b ] let @xmath97 be an actual cause of violation @xmath86 on trace @xmath8 starting from @xmath157 . we say that the set @xmath158 of thread identifiers is a cause of the violation if @xmath159 . [ [ remarks ] ] remarks + + + + + + + we make a few technical observations about our definitions of cause . first , because lamport causes ( definition [ definition : cause1 ] ) are projected _ prefixes _ , they contain all actions that occur before any action that actually contributes to the violation . many of actions in the lamport cause may not contribute to the violation intuitively . our actual cause definition filters out such `` spurious '' actions . as an example , suppose that a safety property requires that the value @xmath160 never be sent on the network . the ( only ) trace of the program @xmath161 violates this property . the lamport cause of this violation contains all four actions of the program , but it is intuitively clear that the two actions @xmath162 and @xmath163 do not contribute to the violation . indeed , the actual cause of the violation determined by definition [ definition : cause2a ] does not contain these two actions ; it contains only @xmath164 and @xmath165 , both of which obviously contribute to the violation . second , our definition of dummification is based on a program transformation that needs line numbers . one possibly unwanted consequence is that our traces have line numbers and , hence , we could , in principle , specify safety properties that are sensitive to line numbers . however , our definitions of cause are closed under bijective renaming of line numbers , so if a safety property is insensitive to line numbers , the actual causes can be quotiented under bijective renamings of line numbers . third , our definition of actual cause ( definition [ definition : cause2a ] ) separates actions whose return values are relevant to the violation from those whose return values are irrelevant for the violation . this is closely related to noninterference - like security definitions for information flow control , in particular , those that separate input presence from input content @xcite . lamport causes ( definition [ definition : cause1 ] ) have a trivial connection to information flow : if an action does not occur in any lamport cause of a violation , then there can not be an information flow from that action to the occurrence of the violation . in this section , we model an instance of our running example based on passwords ( section [ sec : example ] ) in order to demonstrate our actual cause definition . as explained in section [ sec : example ] , we consider a protocol session where @xmath31 , @xmath21 , @xmath22 , @xmath23 and multiple notaries interact over an adversarial network to establish access over a password - protected account . we describe a formal model of the protocol in our language , examine the attack scenario from section [ sec : example ] and provide a cause analysis using the definitions from section [ sec : definitions ] . we consider our example protocol with eight threads named \{@xmath31 , @xmath21 , @xmath5 , @xmath41 , @xmath42 , @xmath43 , @xmath22 , @xmath23}. in this section , we briefly describe the protocol and the programs specified by the protocol for each of these threads . for this purpose , we assume that we are provided a function @xmath166 such that @xmath167 is the program that _ ideally should have been _ executing in the thread @xmath32 . for each @xmath32 , we call @xmath167 the _ norm _ for thread @xmath32 . the violation is caused because some of the executing programs are different from the norms . these actual programs , called @xmath168 as in section [ sec : definitions ] , are shown later . the norms are shown here to help the reader understand what the ideal protocol is and also to facilitate some of the development in section [ sec : domains ] . the appendix describes an expansion of this example with more than the eight threads considered here to illustrate our definitions better . the proof included in the appendix deals with timestamps and signatures . the norms in figure [ fig : norms1 ] and the actuals in figure [ fig : actuals1 ] assume that @xmath21 s account ( called @xmath169 in @xmath31 s program ) has already been created and that @xmath21 s password , @xmath170 is associated with @xmath21 s user i d , @xmath171 . this association ( in hashed form ) is stored in @xmath31 s local state at pointer @xmath172 . the norm for @xmath31 is to wait for a request from an entity , respond with its ( @xmath31 s ) public key , wait for a username - password pair encrypted with that public key and grant access to the requester if the password matches the previously stored value in @xmath31 s memory at @xmath172 . to grant access , @xmath31 adds an entry into a private access matrix , called @xmath74 . ( a separate server thread , not shown here , allows @xmath21 to access its account if this entry exists in @xmath74 . ) the norm for @xmath21 is to send an access request to @xmath31 , wait for the server s public key , verify that key with three notaries and then send its password @xmath170 to @xmath31 , encrypted under @xmath31 s public key . on receiving @xmath31 s public key , @xmath21 initiates a protocol with the three notaries and accepts or rejects the key based on the response of a majority of the notaries . for simplicity , we omit a detailed description of this protocol between @xmath21 and the notaries that authenticates the notaries and ensures freshness of their responses . these details are included in our appendix . in parallel , the norm for @xmath22 is to generate and send a nonce to @xmath23 . the norm for @xmath23 is to receive a message from @xmath22 , generate a nonce and send it to @xmath22 . each notary has a private database of _ ( public_key , principal ) _ tuples . the notaries norms assume that this database has already been created correctly . when @xmath21 sends a request with a public key , the notary responds with the principal s identifier after retrieving the tuple corresponding to the key from its database . the programs in this example use several primitive functions @xmath11 . @xmath173 and @xmath174 denote encryption and decryption of message @xmath175 with key @xmath176 and @xmath177 respectively . @xmath178 generates the hash of term @xmath175 . @xmath179 denotes message @xmath175 signed with the key @xmath176 , paired with @xmath175 in the clear . @xmath180 and @xmath181 denote the public and private keys of thread @xmath32 , respectively . for readability , we include the intended recipient @xmath32 and expected sender @xmath28 of a message as the first argument of @xmath182 and @xmath183 expressions . as explained earlier , @xmath32 and @xmath28 are ignored during execution and a network adversary , if present , may capture or inject any messages . the security property of interest to us is that if at time @xmath184 , a thread @xmath176 is given access to account @xmath185 , then @xmath176 owns @xmath185 . specifically , in this example , we are interested in case @xmath186 and @xmath187 . this can be formalized by the following logical formula , @xmath89 : @xmath188 here , @xmath189 is the state of the access control matrix @xmath74 for @xmath31 at time @xmath184 . note that , if any two of the three notaries had attested the @xmath5 s key to belong to @xmath31 , the violation would have still happened . consequently , we may expect three independent program causes in this example : \{@xmath5 , @xmath21 , @xmath31 , @xmath41 , @xmath42 } with the action causes @xmath97 as shown in figure [ fig : log1](c ) , \{@xmath5 , @xmath21 , @xmath31 , @xmath41 , @xmath43 } with the actions @xmath190 , and \{@xmath5 , @xmath21 , @xmath31 , @xmath42 , @xmath43 } with the actions @xmath191 where @xmath190 and @xmath191 can be obtained from @xmath97 ( figure [ fig : log1](c ) ) by considering actions for \{@xmath41 , @xmath43 } and \{@xmath42 , @xmath43 } respectively , instead of actions for \{@xmath41 , @xmath42}. the following theorem states that our definitions determine exactly these three independent causes . [ theorem - passwords2 ] let @xmath192 , and @xmath193 and @xmath168 be as described above . let @xmath8 be a trace from @xmath157 such that @xmath194 for each @xmath105 matches the corresponding log projection from figures [ fig : log1](a ) and [ fig : log2 ] . then , definition [ definition : cause2b ] determines three possible values for the program cause @xmath195 of violation @xmath88 : \{@xmath5 , @xmath21 , @xmath31 , @xmath41 , @xmath42 } , \{@xmath5 , @xmath21 , @xmath31 , @xmath41 , @xmath43 } , and \{@xmath5 , @xmath21 , @xmath31 , @xmath42 , @xmath43 } where the corresponding actual causes are @xmath196 and @xmath191 respectively . it is instructive to understand the proof of this theorem , as it illustrates our definitions of causation . we verify that our phase 1 and phase 2 definitions ( definitions [ definition : cause1 ] , [ definition : cause2a ] , [ definition : cause2b ] ) yield exactly the three values for @xmath195 mentioned in the theorem . we show that any @xmath6 whose projections match those shown in figure [ fig : log1](b ) satisfies sufficiency and minimality . from figure [ fig : log1](b ) , such an @xmath6 has no actions for @xmath23 , @xmath24 , @xmath44 , @xmath32 and only those actions of @xmath22 that are involved in synchronization with @xmath31 . for all other threads , the log contains every action from @xmath8 . the intuitive explanation for this @xmath6 is straightforward : since @xmath6 must be a ( projected ) _ prefix _ of the trace , and the violation only happens because of @xmath197 in the last statement of @xmath31 s program , every action of every program before that statement in lamport s happens - before relation must be in @xmath6 . this is exactly the @xmath6 described in figure [ fig : log1](b ) . formally , following the statement of sufficiency , let @xmath31 be the set of traces starting from @xmath198 ( figure [ fig : actuals1 ] ) whose logs contain @xmath6 as a projected prefix . pick any @xmath199 . we need to show @xmath129 . however , note that any @xmath127 containing all actions in @xmath6 must also add @xmath200 to @xmath201 , but @xmath202 . hence , @xmath129 . further , @xmath6 is minimal as described in the previous paragraph . phase 2 ( definitions [ definition : cause2a ] , [ definition : cause2b ] ) determines three independent program causes for @xmath195 : \{@xmath5 , @xmath21 , @xmath31 , @xmath41 , @xmath42 } , \{@xmath5 , @xmath21 , @xmath31 , @xmath41 , @xmath43 } , and \{@xmath5 , @xmath21 , @xmath31 , @xmath42 , @xmath43 } with the actual action causes given by @xmath196 and @xmath191 , respectively in figure [ fig : log1](c ) . these are symmetric , so we only explain why @xmath97 satisfies definition [ definition : cause2a ] . ( for this @xmath97 , definition [ definition : cause2b ] immediately forces @xmath203 . ) we show that ( a ) @xmath97 satisfies sufficiency , and ( b ) no proper _ sublog _ of @xmath97 satisfies sufficiency ( minimality ) . note that @xmath97 is obtained from @xmath6 by dropping @xmath43 , @xmath22 and @xmath23 , and all their interactions with other threads . we start with ( a ) . let @xmath97 be such that @xmath204 matches figure [ fig : log1](c ) for every @xmath32 . fix any dummifying function @xmath101 . we must show that any trace originating from @xmath205 , whose log contains @xmath97 as a projected sublog , is in @xmath86 . additionally we must show that there is such a trace . there are two potential issues in mimicking the execution in @xmath97 starting from @xmath206 first , with the interaction between @xmath21 and @xmath43 and , second , with the interaction between @xmath31 and @xmath22 . for the first interaction , on line 7 , @xmath207 ( figure [ fig : actuals1 ] ) synchronizes with @xmath43 according to @xmath6 , but the synchronization label does not exist in @xmath97 . however , in @xmath208 , the @xmath14 on line 10 in @xmath207 is replaced with a dummy value , so the execution from @xmath208 progresses . subsequently , the majority check ( assertion [ b ] ) succeeds as in @xmath6 , because two of the three notaries ( @xmath41 and @xmath42 ) still attest the @xmath5 s key . a similar observation can be made about the interaction between @xmath31 and @xmath22 . line 4 , @xmath209 ( from figure [ fig : log1](b ) ) synchronizes with @xmath22 according to @xmath6 , but this synchronization label does not exist in @xmath97 . however , in @xmath208 , the @xmath14 on line 4 in @xmath209 is replaced with a dummy value , so the execution from @xmath208 progresses . subsequently , @xmath31 still adds permission for the @xmath5 . next we prove that every trace starting from @xmath208 , whose log contains @xmath97 ( figure [ fig : log1](c ) ) as a projected _ sublog _ , is in @xmath86 . fix a trace @xmath127 with log @xmath69 . assume @xmath69 coincides with @xmath97 . we show @xmath129 as follows : 1 . since the synchronization labels in @xmath69 are a superset of those in @xmath97 , @xmath31 must execute line 10 of its program @xmath210 in @xmath127 . after this line , the access control matrix @xmath201 contains @xmath211 for some @xmath212 . when @xmath209 writes @xmath213 to @xmath201 at line 10 , then @xmath212 is the third component of a tuple obtained by decrypting a message received on line 7 . since the synchronization projections on @xmath69 are a superset of @xmath97 , and on @xmath97 @xmath214 synchronizes with @xmath215 , @xmath212 must be the third component of an encrypted message sent on line 10 of @xmath216 . the third component of the message sent on line 10 by @xmath5 is exactly the term `` @xmath5 '' . ( this is easy to see , as the term `` @xmath5 '' is hardcoded on line 9 . ) hence , @xmath217 . this immediately implies that @xmath129 since @xmath218 , but @xmath219 . last , we prove ( b ) that no proper subsequence of @xmath97 satisfies sufficiency. note that @xmath97 ( figure [ fig : log1](c ) ) contains exactly those actions from @xmath6 ( figure [ fig : log1 ] ) on whose returned values the last statement of @xmath31 s program ( figure [ fig : actuals1 ] ) is data or control dependent . consequently , all of @xmath97 as shown is necessary to obtain the violation . in particular , observe that if labels for @xmath31 ( @xmath220 ) are not a part of @xmath190 , then @xmath31 s labels are not in @xmath208 and , hence , on any counterfactual trace @xmath31 can not write to @xmath201 , thus precluding a violation . therefore , the sequence of labels in @xmath220 are required in the actual cause . by sufficiency , for any @xmath101 , the log of trace @xmath127 of @xmath208 must contain @xmath97 as a projected _ sublog_. this means that in @xmath127 , the assertion [ a ] of @xmath209 must succeed and , hence , on line 7 , the correct password @xmath221 must be received by @xmath31 , independent of @xmath101 . this immediately implies that @xmath5 s action of sending that password must be in @xmath97 , else some dummified executions will have the wrong password sent to @xmath31 and the assertion [ a ] will fail . extending this logic further , we now observe that because @xmath5 forwards a password received from @xmath21 ( line 4 of @xmath222 ) to @xmath31 , the send action of @xmath223 will be in @xmath97 ( otherwise , some dummifications of line 4 of @xmath216 will result in the wrong password being sent to @xmath31 , a contradiction ) . since @xmath223 s action is in @xmath97 and @xmath69 must contain @xmath97 as a _ sublog _ , the majority check of @xmath207 must also succeed . this means that at least two of @xmath224 must send the confirmation to @xmath21 , else the dummification of lines 8 10 of @xmath225 will cause the assertion [ b ] to fail for some @xmath101 . since we are looking for a minimal _ sublog _ therefore we only consider the send actions from two threads i.e. @xmath226 . at this point we have established that each of the labels as shown in figure [ fig : log1](c ) are required in @xmath97 . hence , @xmath227 .
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protocols for tasks such as authentication , electronic voting , and secure multiparty computation ensure desirable security properties if agents follow their prescribed programs .
however , if some agents deviate from their prescribed programs and a security property is violated , it is important to hold agents _ accountable _ by determining which deviations actually caused the violation .
motivated by these applications , we initiate a formal study of _ program actions as actual causes_. specifically , we define in an interacting program model what it means for a set of program actions to be an actual cause of a violation .
we present a sound technique for establishing program actions as actual causes .
we demonstrate the value of this formalism in two ways .
first , we prove that violations of a specific class of safety properties always have an actual cause .
thus , our definition applies to relevant security properties .
second , we provide a cause analysis of a representative protocol designed to address weaknesses in the current public key certification infrastructure .
security protocols , accountability , audit , causation
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You are an expert at summarizing long articles. Proceed to summarize the following text:
in this paper we consider the simplest of electrical networks , namely those that consist of only resistors . the electrical properties of such a network @xmath1 are completely described by the _ response matrix _ @xmath2 , which computes the current that flows through the network when certain voltages are fixed at the boundary vertices of @xmath1 . de verdire - gitler - vertigan @xcite and curtis - ingerman - morrow @xcite studied the _ inverse ( dirichlet - to - neumann ) problem _ for _ circular planar _ electrical networks . specifically , they considered networks embedded in a disk without crossings , with boundary vertices located on the boundary of the disk . the following theorem summarizes their results . 1 . any circular planar electrical network is electrically equivalent to some _ critical _ network , which is characterized by its _ medial graph _ being _ lenseless _ ( see ( * ? ? ? * thorme 2 ) ) . any two circular planar electrical networks having the same response matrix can be connected by simple local transformations : series - parallel , loop removal , pendant removal , and star - triangle transformations discussed in section [ ss : trans ] . furthermore , if both networks are critical , then only star - triangle transformations are required ( see ( * ? ? ? * thorme 4 ) or ( * ? ? ? * theorem 1 ) ) . the edge conductances of a critical circular planar electrical network can be recovered uniquely from the response matrix ( see ( * ? ? ? * theorem 2 ) or ( * ? ? ? * thorme 3 ) ) . the response matrices realizable by circular planar networks are the ones having all _ circular minors _ nonnegative ( see ( * ? ? ? * theorem 3 ) ) . the space @xmath3 of response matrices of circular planar networks has a stratification by cells @xmath4 where each @xmath5 can be obtained as the set of response matrices for a fixed critical network with varying edge weights ( see ( * ? ? ? * thorme 3 and 5 ) or ( * ? ? ? * theorem 4 ) ) . it is an open problem to extend these results to electrical networks embedded in surfaces with more complicated topology . in this paper we make progress towards understanding the inverse problem for networks embedded in a cylinder . our first main result is to construct a birational transformation we call the _ electrical @xmath0-matrix_. this transformation acts on the edge weights of a local portion of an electrical network embedded into the cylinder , preserving all the electrical properties of the network ( corollary [ cor : resp ] ) . furthermore , this electrical @xmath0-matrix satisfies the yang - baxter relation ( theorem [ thm : ryb ] ) , and is a close analogue of the `` geometric @xmath0-matrices '' of affine crystals , to be explained below . using the electrical @xmath0-matrix , we formulate the following general conjecture . a more precise version is given as conjecture [ conj : gen ] . * any cylindrical electrical network is electrically equivalent to a critical cylindrical electrical network . * any two cylindrical electrical networks @xmath6 and @xmath7 with the same _ universal response matrices _ are connected by local electrical equivalences . furthermore , if @xmath6 and @xmath7 are both critical , then only star - triangle transformations , and electrical @xmath0-matrix transformations are needed . * if a cylindrical electrical network is critical , then the edge conductances can be recovered up to the electrical @xmath0-matrix action . * the space @xmath8 of universal response matrices of cylindrical electrical networks has an infinite stratification by @xmath9 where each @xmath10 is a semi - closed cell that can be obtained as the set of universal response matrices for a fixed critical network with varying edge weights . the naive analogue of ( 4 ) does not hold see section [ ssec : tnn ] . the universal response matrix in the conjecture is the response matrix of the universal cover of the cylindrical network @xmath6 . roughly speaking , it allows us to not only measure the current flowing through the boundary vertices , but also how many times the current has winded around the cylinder . it may be possible to formulate this in a more electrically natural way by measuring magnetic fields . thus the key difference between the planar and the cylindrical cases is that even for a critical network on a cylinder the edge conductances may not be uniquely determined from the response matrix . this non - uniqueness comes from the existence of the electrical @xmath0-matrix , the action of which preserves both the underlying graph of the network and the response matrix , while changing the edge conductances . the action of the electrical @xmath0-matrix can on the one hand be thought of as a galois group , and on the other hand as a monodromy group . our second main result is to establish the above conjecture for a certain class of cylindrical networks we denote @xmath11 . these critical networks can be thought of as the `` purely cylindrical '' networks . there is no local configuration for which the star - triangle transformation can be applied in @xmath11 , but the electrical @xmath0-matrix generates an action of the symmetric group @xmath12 . we show in theorem [ thm : sol ] that for the networks @xmath11 the edge conductances are recovered uniquely from the universal response matrix modulo this @xmath12 action , and in particular the inverse problem has generically @xmath13 solutions . one way to formulate our main conjecture is that the networks @xmath11 exactly encapsulate the difference between the planar and cylindrical cases . the proof of theorem [ thm : sol ] occupies the technical heart of the paper : we express the edge conductances as limits of certain rational functions of the universal response matrix ( theorem [ thm : main ] ) . here we use crucially the work of kenyon and wilson @xcite . kenyon and wilson study _ groves _ in circular planar electrical networks . these are forests whose connected components contain specified boundary vertices . kenyon and wilson connect ratios of grove generating functions with the response matrix of the corresponding network . by a careful choice of grove generating functions , we can recover the desired edge conductances . recall that a matrix is _ totally nonnegative _ if every minor of it is nonnegative . there is a mysterious similarity between electrical networks and a different kind of networks arising in the theory of totally nonnegative matrices . in @xcite , we presented an approach to understanding this similarity via lie theory . whereas the theory of total nonnegative is intimately related to the class of semisimple lie groups ( @xcite ) , we suggested in @xcite that a different class of `` electrical lie groups '' is related to electrical networks . these electrical lie groups are certain deformed versions of the maximal unipotent subgroup of a semisimple lie group . the main ideas of the present work are also motivated by this analogy , though our philosophy here is more combinatorial in nature . the construction of the electrical @xmath0-matrix follows the techniques developed in @xcite . there we constructed , using purely network - theoretic methods , the geometric ( or birational ) @xmath0-matrix of a tensor product of affine geometric crystals for the symmetric power representations of @xmath14 . in this paper , we use electrical networks instead of the `` totally nonnegative '' networks of @xcite , but nevertheless the underlying combinatorics is developed in parallel . we plan to expand on this analogy in @xcite . our formulation of conjecture [ conj : gen ] is also motivated by the analogy with total nonnegativity . indeed , analogues of ( 1)-(5 ) ( and even the missing ( 4 ) ! ) for the _ totally nonnegative part of the rational loop group _ are established in @xcite . in particular , in @xcite we studied in detail from the totally nonnegative perspective , the networks @xmath11 , or more precisely , their medial graphs . our solution here to the inverse problem for the networks @xmath11 follows the strategy in @xcite , where edge weights are recovered by taking limits of ratios of matrix entries ; this approach was originally applied by aissen - schoenberg - whitney @xcite to classify _ totally positive functions_. the situation we consider here is technically much more demanding , involving rather intricate kenyon - wilson grove combinatorics . * acknowledgements . * we cordially thank michael shapiro for stimulating our interest in the problem . for more background on electrical networks , we refer the reader to @xcite . for our purposes , an electrical network is a finite weighted undirected graph @xmath1 , where the vertex set is divided into the _ boundary _ vertices and the _ interior _ vertices . the weight @xmath15 of an edge is to be thought of as the conductance of the corresponding resistor , and is generally taken to be a positive real number . a @xmath16-weighted edge would be the same as having no edge , and an edge with infinite weight would be the same as identifying the endpoint vertices . we define the _ kirchhoff matrix _ @xmath17 to be the square matrix with rows and columns labeled by the vertices of @xmath1 as follows : @xmath18 let @xmath19 be a square @xmath20 matrix , and @xmath21 . recall that the _ schur complement _ @xmath22 is the square @xmath23 matrix defined to be @xmath24-i,[n]-i } - m_{[n]-i , i}m_{i , i}^{-1 } m_{i,[n]-i}$ ] , where @xmath25 denotes the submatrix of @xmath19 consisting of the rows labeled by @xmath26 and the columns labeled by @xmath27 . we define the _ response matrix _ @xmath2 to be the square matrix with rows and columns labeled by the boundary vertices of @xmath1 , given by the schur complement @xmath28 where @xmath29 denotes the submatrix of @xmath27 indexed by the interior vertices . the response matrix encodes all the electrical properties of @xmath1 that can be measured from the boundary vertices . note that our kirchhoff and response matrices are the negative of those commonly used in the literature . we now discuss the local transformations of electrical networks which leave the response matrix invariant . the following proposition is well - known and can be found for example in @xcite . [ p : sp ] series - parallel transformations , removing loops , and removing interior degree 1 vertices , do not change the response matrix of a network . see figure [ fig : elie1 ] . the most interesting local transformation is attributed to kennelly @xcite . assume parameters @xmath30,@xmath31,@xmath32,@xmath33,@xmath34 and @xmath35 are related by @xmath36 or equivalently by @xmath37 then switching a local part of an electrical network between the two options shown in figure [ fig : elec11 ] does not affect the response of the whole network . [ prop : fm ] the sequence of @xmath38 transformations shown in figure [ fig : elec7 ] returns the conductances of edges to their original values . direct computation . in the terminology of kashaev , korepanov , and sergeev @xcite , the proposition states that the @xmath38 transformation is a solution of type ( @xmath39 ) to the _ functional tetrahedron equation_. a _ planar partition _ of @xmath41 = \{1,2,\ldots , n\}$ ] is a partition @xmath42 such that there do not exist @xmath43 such that @xmath44 belong to the same part of @xmath45 , and @xmath46 belong to the same part . a _ circular planar electrical network _ @xmath1 is an electrical network embedded into a disk , so that the intersection of @xmath1 with the boundary of the disk is exactly the boundary vertices of @xmath1 . we suppose that the boundary vertices of @xmath1 are exactly @xmath41 $ ] , and that these vertices are arranged in order around the boundary of the disk . a _ grove _ in @xmath1 is a spanning forest where each connected component intersects the boundary . a grove @xmath47 has boundary planar partition @xmath42 if the connected components @xmath48 of @xmath47 are such that @xmath49 contains the boundary vertices labeled by @xmath50 . the _ weight _ of the grove is the product of the weights of its edges . kenyon and wilson @xcite study the probability @xmath51 that a random grove of @xmath1 is of type @xmath45 , where the probability of a grove is proportional to its weight . let `` @xmath52 '' denote the partition of @xmath41 $ ] into singletons . [ t : upr ] ( * ? ? ? * theorem 1.1 , lemma 4.1 ) let @xmath6 be a finite circular planar electrical network . 1 . the ratio @xmath53 is an integer - coefficient polynomial in the @xmath54 , homogeneous of degree @xmath55 . 2 . suppose @xmath45 be a planar partition with parts of size at most two . then the polynomial of ( 1 ) depends only on @xmath54 , for @xmath56 s which are not isolated parts of @xmath45 . recall from @xcite that for a partition @xmath45 , we define @xmath57 where the sum is over spanning forests @xmath58 of the complete graph , for which the trees of @xmath58 span the parts of @xmath45 , and the product is over the edges of @xmath58 . let s recall * rule 1 * from @xcite . if a partition @xmath45 is non - planar then one can pick @xmath59 so that @xmath60 belong to one part of @xmath45 , and @xmath61 belong to another part of @xmath45 . arbitrarily subdivide the part containing @xmath30 and @xmath32 into two sets @xmath33 and @xmath35 so that @xmath62 and @xmath63 , and similarly obtain @xmath34 and @xmath64 . denote the remaining parts of the partition @xmath45 by `` rest '' . then the rule is @xmath65 iterating this rule transforms each non - planar partition @xmath45 into a linear combination of planar ones , and kenyon and wilson show that the coefficients do not depend on how rule 1 is applied . denote by @xmath66 the coefficient of a planar partition @xmath67 in the application of rule 1 to @xmath45 . we have @xmath68 . [ l : singles ] let @xmath69 be a cyclic interval . suppose that a planar partition @xmath67 is such that no part @xmath70 of @xmath67 contains two elements of @xmath69 . let @xmath45 be a possibly nonplanar partition such that @xmath67 occurs in the expansion of @xmath45 under repeated application of rule 1 ( until no more applications are possible ) . then @xmath45 does not contain any part @xmath70 such that @xmath71 . by ( * theorem 1.2 ) , the result of repeatedly applying rule 1 does not depend on the choices made when applying rule 1 . let us suppose that @xmath45 contains a part @xmath70 such that @xmath71 . in applying rule 1 , if we ever encounter that @xmath72 , we will first try to choose @xmath33 and @xmath35 so that @xmath73 or @xmath74 . this would guarantee that all the partitions occurring in rule 1 contain some part @xmath75 such that @xmath76 . this choice is impossible only if @xmath77 , which would in turn imply that @xmath31 or @xmath78 lies in @xmath69 . but in this case again all the partitions occurring in rule 1 contain some part @xmath75 such that @xmath76 . thus @xmath67 can not occur in the expansion of @xmath45 . [ l : combtype ] suppose @xmath67 is a partition such that the non - singleton parts of @xmath67 contain at most @xmath27 elements . then there is a constant @xmath79 such that when @xmath80 is expanded as a polynomial in the @xmath81 s the coefficient of any monomial in the @xmath81 s is less than or equal to @xmath79 . denote the set of boundary vertices by @xmath69 , and by @xmath82 the elements in the non - singleton parts of @xmath67 . applying lemma [ l : singles ] , we see that every @xmath45 , such that @xmath67 occurs in the rule 1 expansion of @xmath45 , have parts that contain at most one element from each cyclic component of @xmath83 . since the number of cyclic components is bounded by @xmath27 , we see that there is a bound , depending only on @xmath27 , on the number of combinatorial types of possible @xmath45 s . here combinatorial type means the partition one obtains when singletons in @xmath84 are removed , and only the relative orders of the remaining elements are remembered . it follows that one can find a constant @xmath79 so that the coefficient of @xmath67 in the rule 1 expansion of @xmath45 is bounded by @xmath79 , for any @xmath45 . next we note that for the sum of , the edges of @xmath58 determine @xmath58 , which in turn determines @xmath45 . so each monomial in the @xmath81 s occurs in at most one @xmath85 . thus the coefficient of any monomial in the @xmath81 s is bounded by @xmath79 . the ideas of this section follow closely the calculation of the `` whurl transformation '' in @xcite . in a special case the whurl transformation reduces to the _ birational _ , or _ geometric @xmath0-matrix _ of certain affine geometric crystals . this motivates the terminology of an _ electrical @xmath0-matrix_. fix @xmath86 and @xmath87 . we define an electrical network denoted @xmath11 which is embedded in a cylinder . it has @xmath88 boundary vertices all lying on the boundary of the cylinder , with @xmath89 on the left boundary component and @xmath90 on the right boundary component . there are @xmath91 internal vertices , denoted @xmath92 where @xmath93 and @xmath94 . for convenience , the boundary vertices @xmath89 are denoted @xmath95 and the boundary vertices @xmath90 are denoted @xmath96 . for each @xmath97 and @xmath98 , we have edges from @xmath99 to @xmath100 , and from @xmath99 to @xmath101 . here all lower indices are taken modulo @xmath102 . now we focus on the network @xmath103 . we label the edge weights of @xmath103 as follows . for each @xmath97 , we have edges with weights @xmath104 from @xmath105 to @xmath106 , weights @xmath107 from @xmath105 to @xmath108 , weights @xmath109 from @xmath110 to @xmath111 , and weights @xmath112 from @xmath108 to @xmath113 . here all indices are taken modulo @xmath102 . now define polynomials @xmath114 and @xmath115 as follows : @xmath116 and @xmath117 @xmath118 also define @xmath119 suppose @xmath120 . then @xmath121 @xmath122 now introduce additional parallel wires with parameters @xmath70 and @xmath123 from @xmath124 to @xmath125 . this is a special case of the local electrical equivalence for parallel resistors . ( here the parameters @xmath70 and @xmath123 should be considered formally , instead of as nonnegative real numbers . ) we may perform @xmath38 operations to move the parameter @xmath70 through the resistor network , as shown in figure [ fig : elec5 ] . [ l : unique ] there is a unique non - zero parameter @xmath70 , which is unchanged after moving through one revolution . let us denote by @xmath126 the value of the `` extra edge '' after @xmath127 star - triangle transformations . ( the fourth network in figure [ fig : elec5 ] shows the location of @xmath128 . ) we claim that the parameter @xmath126 is a ratio of two linear functions in the original @xmath129 . this is easily verified by induction using the star - triangle transformation . thus after @xmath88 star - triangle transformations the equation @xmath130 we obtain is either a linear or a quadratic equation , and zero is clearly one of the roots . therefore there is at most one non - zero solution , and as we shall see soon in the proof of theorem [ t : elecwhirl ] , a solution indeed exists . suppose we perform the sequence of @xmath38 operations of figure [ fig : elec5 ] , using the parameter @xmath70 of lemma [ l : unique ] ( which we still have to prove exists ) . then as illustrated in the final diagram of figure [ fig : elec5 ] , the `` extra edges '' with parameters @xmath70 and @xmath123 can be removed via the local electrical equivalence for parallel resistors . we define the _ electrical @xmath0-matrix _ to be the transformation @xmath131 induced on the edge weights by this transformation . [ t : elecwhirl ] the electrical @xmath0-matrix is given by @xmath132 we claim that the parameter @xmath133 is the parameter of lemma [ l : unique ] . define @xmath126 to be the parameter after @xmath105 pairs of @xmath134 and @xmath38 operations , and @xmath135 to be the parameter obtained from @xmath126 by one @xmath134 operation . then @xmath136 indeed , calculating by induction @xmath137 and similarly for @xmath138 . we may then calculate that the transformation is given by @xmath139 and similarly @xmath140 . we also have @xmath141 and similarly @xmath142 . [ prop : un ] the electrical @xmath0-matrix does not depend on where you attach the extra pair of edges , and is an involution . if one applies the @xmath38 transformations to successively push through two edges , with weights @xmath143 and @xmath144 negative of each other , then there is no net effect on the weights of the other edges involved . also , the weights on the two edges pushed through remain negatives of each other . thus if one pushes through @xmath70 to perform the electrical @xmath0-matrix , and then a @xmath123 , the latter will perform a transformation that will undo the first one . furthermore , a working choice of parameter @xmath70 at one location this way yields a working choice of the parameter at any other location . the uniqueness in lemma [ l : unique ] implies that all resulting electrical @xmath0-matrices are the same . let us now consider the network @xmath145 . the procedure of section [ ssec : r ] gives two different electrical @xmath0-matrices acting on @xmath145 : by acting on the part of the network involving vertices @xmath146 , or by acting on the part of the network involving the vertices @xmath147 . we denote these @xmath0-matrices by @xmath148 and @xmath149 respectively . [ thm : ryb ] the electrical @xmath0-matrix satisfies the yang - baxter equation @xmath150 first we note that to perform the electrical @xmath0-matrix , we can either add extra horizontal edges with conductances @xmath70 and @xmath123 between @xmath124 and @xmath125 , or we could split the vertex @xmath151 into three vertices and add vertical edges with conductances @xmath70 and @xmath123 between them ( see figure [ fig : elec16 ] ) . to perform the sequence of @xmath0-matrices @xmath152 , we will add horizontal edges for the first and third factor , but add vertical edges for the second factor . let the corresponding weights be @xmath70 , @xmath143 , @xmath153 as shown in figure [ fig : elec6 ] . according to lemma [ l : unique ] and theorem [ t : elecwhirl ] and proposition [ prop : un ] , these weights exist and are unique . now , apply the @xmath134 transformation as shown in figure [ fig : elec6 ] to obtain weights @xmath75 , @xmath154 and @xmath155 , and their negatives on the other side . we claim that if these new weights are pushed around the cylinder , they come out the same at the other end . this follows from proposition [ prop : fm ] . now if we push the weights @xmath70 , @xmath143 and @xmath153 through and apply the @xmath134 transformations , we obviously get again the weights @xmath75 , @xmath154 and @xmath155 . thus by lemma [ l : unique ] and theorem [ t : elecwhirl ] and proposition [ prop : un ] , while pushing @xmath75 , @xmath154 and @xmath155 through we are applying @xmath156 . since , by the electrical tetrahedron relation ( proposition [ prop : fm ] ) the two results are the same , the claim of the theorem follows . [ cor : braid ] the electrical @xmath0-matrix gives an action of the symmetric group @xmath12 on @xmath11 . follows from theorem [ thm : ryb ] and proposition [ prop : un ] . let @xmath6 be a cylindrical electrical network . thus @xmath6 is an electrical network embedded into a cylinder so that the intersection of @xmath6 with the boundary of the cylinder is exactly the boundary vertices of @xmath6 . let @xmath157 denote the universal cover of @xmath6 . it is an infinite periodic network embedded into an infinite strip . given finite sets @xmath158 of vertices of @xmath157 which eventually cover all vertices of @xmath157 , we obtain a sequence of _ truncations _ @xmath159 , @xmath160 of @xmath157 as follows . we let @xmath161 be the subgraph of @xmath157 consisting of all edges incident to a vertex in @xmath162 . furthermore , we declare a vertex of @xmath161 internal if it lies in @xmath162 and is internal in @xmath157 . each @xmath161 is a finite planar electrical network . we suppose that the boundary vertices of @xmath157 are numbered @xmath163 on one side of the boundary , and by @xmath164 on the other side of the boundary . ( we will always picture the infinite strip as vertical , with vertex labels increasing as we go downwards . ) we define the _ universal response matrix _ of @xmath6 to be given by @xmath165 where @xmath166 where @xmath56 denote vertices of @xmath157 . for sufficiently large @xmath1 , any two fixed vertices of @xmath157 will lie in @xmath161 . these limits exist and are finite , due to following lemma . if @xmath167 is an electrical network , and @xmath168 a subset of its vertices , the _ response matrix of @xmath169 _ is the response matrix obtained by declaring all the vertices in @xmath168 to be interior . [ lem : smon ] assume @xmath170 are two subsets of vertices of an electrical network @xmath167 , and assume @xmath105 and @xmath171 are two vertices not contained in @xmath172 . then @xmath173 in the response matrix of @xmath174 is at least as large as @xmath81 in the response matrix of @xmath169 . the schur complement with respect to some set @xmath168 can be taken as a sequence of schur complements with respect to single vertices in @xmath168 in some order ( see for example ( * ? ? ? * ( 3.7 ) ) ) . thus , it would suffice to prove the statement for @xmath175 consisting of a single vertex . in this case the claim is obvious however , since off - diagonal entries of a response matrix are nonnegative , and diagonal entries are non - positive . [ t : universalresponse ] the matrix @xmath165 is a well - defined infinite periodic matrix , which does not depend on which truncations @xmath161 are taken . there are several parts to this statement . the limits @xmath165 exist . note that for sufficiently large @xmath1 , @xmath176 can be calculated by taking @xmath81 of the network @xmath177 , which is obtained from @xmath157 by declaring that only the internal vertices of @xmath161 are internal in @xmath177 . the network @xmath177 is obtained from @xmath161 by adding extra boundary vertices attached only to boundary vertices of @xmath161 ( and by assuming @xmath1 is large enough , we may assume that these extra vertices are not incident to @xmath105 or @xmath171 ) . but @xmath176 is by definition calculated by measuring the current flowing through @xmath171 when vertex @xmath105 is set to one volt and all other boundary vertices are set to zero volts . since current does not flow between zero volt vertices it follows that @xmath178 . by lemma [ lem : smon ] the sequence @xmath179 is non - decreasing as @xmath180 , since each network is obtained from the previous one by declaring some extra vertices internal and taking the corresponding schur complement . the limits @xmath165 do not depend on the sequence of truncations . assume we have two different sequences @xmath158 and @xmath181 . since we know that each eventually covers all vertices , we know that for each @xmath105 there is a @xmath171 such that @xmath182 and @xmath183 . then applying lemma [ lem : smon ] we conclude that the two limits bound each other from above , and thus are equal . the limits @xmath165 are periodic : @xmath184 . indeed , take two sequences @xmath158 and @xmath181 , one obtained from the other by a shift on the universal cover by the period @xmath102 . we know that they give the same value of @xmath165 by the previous part . on the other hand , it is clear that the value one of them gives for @xmath165 is the value the other gives for @xmath185 . the limits @xmath165 are finite . any truncation of @xmath157 can also be viewed as a truncation of a finite cover @xmath186 $ ] ( obtained by lifting @xmath6 to a @xmath187-fold cover of the cylinder ) for a large enough @xmath187 . by lemma [ lem : smon ] the conductance @xmath176 is bounded from above by the same conductance in @xmath186 $ ] , which in turn is bounded from above by the same conductance in the original network @xmath6 . indeed , if @xmath188 are vertices in @xmath186 $ ] that cover @xmath171 , then @xmath189)$ ] . this can be seen as follows . using the linearity of the response and periodicity , the sum @xmath190)$ ] measures the current through @xmath191 ( for any @xmath192 ) when all vertices @xmath193 that cover @xmath105 have potential @xmath124 , and all other vertices have potential @xmath16 . but projecting onto @xmath6 by identifying all covers of a vertex we see that this current flow is exactly @xmath54 . in particular , @xmath194 ) \leq l_{ij}(g)$ ] for any @xmath195 . this shows that @xmath165 is bounded from above by @xmath54 , and thus if the latter is finite , so is the former . thus for fixed @xmath56 , we can approximate @xmath165 arbitrarily well by calculating @xmath176 for some large @xmath1 . the universal response matrix @xmath165 is invariant under the local electrical equivalences of @xmath1 . for any local electrical transformation in @xmath6 one can choose a sequence of truncations of @xmath157 containing completely several occurrences of this transformation . since the conductances in these truncations do not change , the limit is also invariant . [ cor : resp ] the electrical @xmath0-matrix preserves the universal response matrix . the only comment one needs to make is that the entries of the universal response matrix are limits of rational functions in the edge weights , and that these rational functions make sense formally even when negative conductances are used ( as in the derivation of the electrical @xmath0-matrix ) . let @xmath196 $ ] be subsets of the same cardinality . then @xmath197 is a _ circular pair _ if a cyclic permutation of @xmath198 is in order . a @xmath20 matrix @xmath19 is _ circular totally - nonnegative _ if the minor @xmath199 is nonnegative for every circular pair @xmath197 . curtis , ingerman , and morrow @xcite show that the response matrices of circular planar electrical networks are exactly the set of circular totally - nonnegative symmetric matrices for which every row sums to 0 . ( note that the response matrices in @xcite are the negative of ours . ) let us extend this to cylindrical electrical networks . put the total order @xmath200 on @xmath201 . let @xmath202 be two ordered subsets of the same finite cardinality . then @xmath197 is a _ cylindrical pair _ if a cyclic permutation of @xmath198 is in order . a matrix @xmath19 with rows and columns labeled ( and ordered ) with @xmath201 is _ cylindrically totally - nonnegative _ if the minor @xmath199 is nonnegative for every cylindrical pair @xmath197 . [ p : tnn ] suppose @xmath203 is the universal response matrix of a finite cylindrical electrical network . then @xmath204 is cylindrically totally - nonnegative . for each fixed cylindrical pair @xmath197 , and sufficiently large @xmath1 , the truncation @xmath161 is a finite circular electrical network including all the boundary vertices in @xmath205 and @xmath26 . but then @xmath206 , using curtis - ingerman - morrow s result . the converse to proposition [ p : tnn ] , namely , which cylindrically totally nonnegative matrices are realizable as universal response matrices , is more subtle . let @xmath207 be the lifts to the universal cover of a particular vertex in a finite cylindrical electrical network @xmath6 . then for a fixed @xmath105 , the ( doubly - infinite ) sequence @xmath208 must satisfy certain recursions or convergence properties . in the different but closely related setting of total nonnegative points of loop groups , the correct property is to ask for the generating function of @xmath209 to be a rational function ( see ( * ? ? ? * theorem 8.10 ) ) . we now assume we are given a network @xmath210 . the vertices have been labeled so that if we take the `` low '' edge ( from @xmath211 to @xmath212 ) at every step starting from @xmath105 we will end up at @xmath111 . note that a shortest path from one side of the cylinder to the other consists of exactly @xmath187 edges . from now on we consider groves in the universal cover @xmath157 of @xmath6 ( or in the truncations @xmath161 ) . the boundary partition of such a grove would be a planar partition of @xmath201 arranged on the two edges of an infinite strip ( or in the truncations of this ) . [ l : short ] suppose @xmath105 and @xmath213 are can be connected by a path with @xmath187 edges . there exists an integer @xmath19 such that there are no groves @xmath47 with the properties 1 . there is a grove component @xmath214 with boundary vertices @xmath215 and which uses an edge below ( resp . above ) any of the shortest paths from @xmath105 to @xmath213 . 2 . there are grove components with boundary vertices @xmath216 ( resp . @xmath217 ) . for a grove component @xmath214 let us call _ bad _ the edges it contains that are below the lowest path from @xmath105 to @xmath213 . assume the grove component @xmath214 has bad edges , and furthermore without loss of generality assume that it has a bad high edge . the case of a bad low edge is similar with the left and right sides of the network swapped . assume @xmath195 is the first index such that bad high edge has one of the @xmath211 as its right endpoint . we claim that the @xmath218 component has a bad high edge with right end having index @xmath219 or smaller . indeed , consider the unique path from @xmath220 to @xmath221 inside @xmath218 . in order to avoid touching the bad edge of @xmath214 it has to turn , diverting from the lowest shortest path from @xmath220 to @xmath221 . the first time it thus diverts gives a desired high edge . now , the index of the first bad high edge can not decrease indefinitely , and in fact one sees that the statement of the lemma holds for @xmath222 . the case of edges above the highest shortest path is similar . define the radii @xmath223 for @xmath224 by @xmath225 where we have denoted the conductance of the edge joining two vertices @xmath226 by @xmath227 . [ l : optimalpath ] let @xmath228 be an @xmath102-tuple of consecutive shortest paths , where @xmath229 connects @xmath230 to @xmath231 for each @xmath105 , and some fixed @xmath195 . suppose that 1 . all the @xmath229 have the same shape ; thus they high or low edges respectively at the same points along the path . @xmath232 . then the total weight @xmath233 is maximized exactly when the highest path is taken . the weight of a subgraph is simply the product of its edge weights . all the shortest paths are connected by switching from one side of a rhombus to the other , see figure [ fig : elec3 ] . if this parallelogram involves vertices with upper index @xmath234 , @xmath195 and @xmath235 , then the higher path has greater weight exactly when @xmath236 . let @xmath237 be the partition of @xmath201 with parts of size two @xmath238 for all @xmath239 , parts of size two @xmath240 for @xmath241 , and all other parts are singletons . thus in particular , @xmath124 is a singleton . we shall denote by @xmath242 the partition of @xmath201 obtained from @xmath237 by placing @xmath124 in the same part as @xmath243 to get a single part @xmath244 of size three . we shall suppose that the truncation @xmath161 includes the boundary vertices @xmath245 and also the boundary vertices @xmath246 . for @xmath247 the partitions @xmath237 and @xmath242 naturally gives rise to partitions of the boundary vertices of @xmath161 , where boundary vertices of @xmath161 which are not boundary vertices of @xmath157 are all considered singletons . we will still denote these partitions of boundary vertices of @xmath161 by @xmath237 and @xmath242 . [ thm : asw ] suppose that @xmath232 . then @xmath248 where @xmath30 is the weight of the high edge connected to the vertex @xmath124 of @xmath6 . let @xmath249 denote the vertex connected to both @xmath16 and @xmath124 . let @xmath250 be the edge joining @xmath124 to @xmath249 , so that @xmath250 has weight @xmath30 . let @xmath251 denote the other edge incident to @xmath124 . to approximate the lhs , we shall assume we have chosen @xmath252 . let @xmath47 be a grove with boundary partition either @xmath242 or @xmath237 . by lemma [ l : short ] , the grove component @xmath253 can not extend either above or below the set of edges contained in shortest paths from @xmath254 to @xmath255 . in particular , there is a bound on the number of choices of @xmath253 , not depending on @xmath27 or @xmath1 . in the following , we shall assume that we have fixed such a choice for @xmath253 . let @xmath47 be a grove with boundary partition @xmath237 . by lemma [ l : short ] , the grove component @xmath256 can not extend above the ( unique ) shortest path from @xmath16 to @xmath257 . in particular , the vertex @xmath249 must lie in @xmath256 . thus the edge @xmath250 is never used in @xmath47 , and @xmath258 is a grove with boundary partition @xmath242 . also by lemma [ l : short ] , the grove component @xmath259 can not extend either above or below the ( unique ) shortest path from @xmath260 to @xmath261 , and therefore must be exactly this shortest path . similar observations hold for a grove with boundary partition @xmath242 . in particular , the part of the grove above @xmath259 and the part below are essentially independent . let us denote by @xmath262 ( resp . @xmath263 ) the set of groves with boundary partition @xmath237 ( resp . @xmath242 ) and by @xmath264 ( resp . @xmath265 ) the total weight of that set of groves . let @xmath47 be a grove with boundary partition @xmath242 . first we observe that all but a constant number of grove components @xmath266 are shortest paths . furthermore , as @xmath105 goes from @xmath267 to @xmath27 , the shapes of the shortest paths are locally constant , and can only change when we encounter a grove component which is not a shortest path . furthermore , as we go down , the shape of the shortest path can only become lower . we shall call this part of @xmath47 the lower half of the grove . note that there are @xmath187 different shapes @xmath268 of shortest paths , listed from highest to lowest , see figure [ fig : elec4 ] . let @xmath269 denote the subset of groves where @xmath270 uses the edge @xmath250 , and let @xmath271 denote the subset of groves where @xmath270 does not use the edge @xmath251 . given @xmath272 , we shall now show that for sufficiently large @xmath27 , one has @xmath273 . let @xmath274 . then the shortest paths which occur in the lower half of @xmath47 can only use the shapes @xmath275 . pick @xmath27 sufficiently large that we are guaranteed to have @xmath276 grove components in the lower half which are shortest paths , where @xmath35 is a constant ( not depending on @xmath27 or @xmath1 ) we shall describe below . then there is some @xmath277 ( say pick the least such @xmath105 ) which occurs at least @xmath278 times . define a set of new groves @xmath279 by : 1 . removing @xmath280 of the @xmath277 shaped paths , and shifting the lower half of @xmath47 downwards to fill in the removed area ; 2 . replacing the grove components @xmath281 with shortest paths ; 3 . replacing @xmath282 with the shortest path from @xmath16 to @xmath257 union the edge @xmath250 ; 4 . adding @xmath283 new shortest paths @xmath284 which are @xmath285-shaped ; 5 . adding one extra `` transition '' grove component @xmath286 which is @xmath285-shaped above , but which correctly fits with @xmath287 below . figure [ fig : elec1 ] illustrates the procedure . in this case @xmath288 , @xmath120 , @xmath289 , @xmath290 . the @xmath270 component of the groves is shown in red , the part shifted down is shown in green . the two removed groves of shape @xmath277 and the two new groves of shape @xmath285 that replaced them are shown in brown . note that @xmath291 belongs to @xmath292 . since apart from the shortest paths the rest of the grove has only been modified at a bounded number of edges , by lemma [ l : optimalpath ] there is a constant @xmath293 not depending on @xmath294 , or @xmath39 such that @xmath295 . furthermore , each grove in @xmath292 can occur in this way in at most @xmath296 different ways . the @xmath297 counts the possible grove components @xmath281 and @xmath298 of which there is a universal bound for . the @xmath187 ways count the possible @xmath105 such that the shortest paths @xmath277 are being replaced by @xmath285 . setting @xmath299 , we conclude that @xmath300 . for every @xmath301 , the grove @xmath302 lies in @xmath263 , and this map is an injection from @xmath262 to @xmath263 which changes the weight of each grove by exactly @xmath30 . on the other hand every @xmath303 is in the image of this map . it follows that @xmath304 . letting @xmath305 , we obtain the statement of the theorem . the partition @xmath237 is a planar partition of @xmath201 of the type described in theorem [ t : upr](2 ) , where all but finitely many vertices are isolated . define @xmath306 by taking the polynomial of theorem [ t : upr](2 ) and replacing @xmath54 by @xmath165 . then we have @xmath307 and in particular , @xmath306 can be approximated arbitrarily well on some @xmath161 for very large @xmath1 . unfortunately , this is not the case for the partition @xmath242 , which contains a part of size three , for which theorem [ t : upr](2 ) can not be applied . instead one has @xmath308 for the sequence of polynomials @xmath309 of theorem [ t : upr](1 ) . these polynomials depend on @xmath242 , which is suppressed from the notation . [ p : polyest ] suppose @xmath310 . then @xmath311 is a polynomial in the @xmath81 s such that every monomial involves some @xmath81 where @xmath312 and @xmath313 $ ] . furthermore , there is some constant @xmath79 such that the coefficient of each monomial in @xmath309 is less than @xmath79 . the first statement follows from lemma [ l : singles ] and our choice of @xmath242 . the second statement is lemma [ l : combtype ] . the assymmetry of the roles of @xmath105 and @xmath171 in proposition [ p : polyest ] is accounted for by the fact that @xmath81 is symmetric . that is , we treat @xmath314 as the same variable . note that we already know @xmath315 approaches a limit , and it follows from theorem [ thm : asw ] that @xmath316 approaches a limit as well . [ l : ak ] for each @xmath27 and @xmath272 , one can find some @xmath33 such that @xmath317 fix @xmath27 . we first show that there is @xmath33 such that for all @xmath318 we have @xmath319 it is known that the polynomials @xmath309 have degree @xmath320 , see section [ sec : kwpoly ] . by proposition [ p : polyest ] , every monomial in @xmath321 has a factor @xmath81 where @xmath322 and @xmath323 , and has coefficient @xmath324 . thus @xmath325 by the proof of theorem [ t : universalresponse ] , we know that for each @xmath105 we have @xmath326 . thus it is possible to find @xmath33 large enough that @xmath327 giving us @xmath328 since only finitely many @xmath81 s appear in @xmath329 , for sufficiently large @xmath1 we have @xmath330 furthermore , for sufficiently large @xmath1 we have @xmath331 finally , we combine the three estimates , , and use @xmath332 . [ thm : main ] fix @xmath6 , satisfying @xmath232 and fix one of the edges connected to one of the vertices @xmath333 . there is a sequence of polynomials ( depending on the universal response matrix of @xmath6 ) , @xmath334 , and @xmath335 such that @xmath336 where @xmath30 is the weight of the chosen edge . by symmetry , it is enough to establish the formula for the high edge connected to @xmath124 . the polynomial @xmath337 is the one associated to @xmath237 from theorem [ t : upr](2 ) . the polynomial @xmath338 is chosen via lemma [ l : ak ] so that @xmath339 , where @xmath340 is chosen so that @xmath341 as @xmath342 . finally , we apply theorem [ thm : asw ] . [ lem : wh ] assume a cylindrical network @xmath343 is obtained by concatenating two networks @xmath344 and @xmath345 . then knowing @xmath8 and the universal response matrix of @xmath3 , one can recover the universal response matrix of @xmath346 . assume the conductances in @xmath8 are @xmath104 for the high edges and @xmath107 for the low edges . concatenate @xmath3 with a network @xmath347 with ( virtual ) conductances @xmath348 for low edges and @xmath349 for high edges . we claim that the response of the resulting network is equal to that of @xmath346 . indeed , connect the opposite vertices of @xmath350 and @xmath8 by edges with conductances @xmath40 and @xmath351 , without changing the response . changing resulting triangles into stars using the @xmath38 transformation , and letting @xmath352 , we see that there is an infinite conductance between opposite vertices and zero conductance between other pairs . thus the two @xmath353 networks effectively cancel each other out , and we are left with a network with the same response as @xmath346 . see figure [ fig : elec9 ] . [ thm : sol ] there are generically @xmath13 sets of edge conductances which produce a given universal response matrix for the network @xmath11 . all the solutions are connected by the @xmath12-action via the electrical @xmath0-matrix . it is easy to see from theorem [ t : elecwhirl ] that the electrical @xmath0-matrix swaps the radii @xmath223 . assume we have a solution for conductances in @xmath11 with given universal response matrix . apply the electrical @xmath0-matrix to reorder the radii @xmath223 in non - increasing order . then theorem [ thm : main ] allows us to recover the conductances in the leftmost @xmath353 part of the network . in particular , these conductances are the same for any solution . once we know that , we can use lemma [ lem : wh ] to recover the universal response matrix of the remaining @xmath354 part of the network . then we repeat the procedure . we see that once we require the radii to form a non - increasing sequence , the conductances are recovered uniquely by theorem [ thm : main ] . therefore all other solutions can be obtained from that one by action of @xmath12 , which is what we want . in the generic case when all radii are distinct , the orbit has size @xmath13 . [ rem : monodromy ] in the language of @xcite , theorem [ thm : sol ] says that the group @xmath12 generated by electrical @xmath0-matrices is exactly the _ monodromy group _ of the network @xmath11 . it is the group acting on the edge weights of @xmath11 obtained by transforming the network via local electrical equivalences back to itself . it is convenient to describe the general answer we expect using the language of _ medial graphs _ , see for example @xcite . draw _ wires _ through the electrical network so that they pass through each edge and connect inside each face as shown in the first two pictures in figure [ fig : elec17 ] . the third picture shows an example of an electrical network and its medial graph . note that the medial graph always has four - valent vertices , and that the _ wires _ of the medial graph `` go straight through '' each vertex . in the terminology of @xcite a circular planar electrical network is _ critical _ if its medial graph avoids _ lenses _ , which is equivalent to saying that every pair of wires crosses as few times as possible , given their respective homotopy types . if @xmath6 is a cylindrical electrical network , we say that @xmath6 is _ critical _ if the universal cover of @xmath6 satisfies this condition ; namely , the medial graph has wires which cross as few times as possible . let us call a cylindrical network _ canonical _ if the medial graph of its universal cover has the following form . first , there are three kinds of wires : ( i ) some wires connect points on opposite boundaries , ( ii ) some wires connect points on the same boundary , and ( iii ) some wires do not intersect the boundary at all and correspond to ( simple ) cycles around the cylinder . secondly , we require that the third kind of wires do not intersect the second kind , and furthermore , all points of intersection of wires of the first kind with themselves happen strictly before they intersect wires of the third kind . an illustration is given in figure [ fig : elec19 ] . * any cylindrical electrical network can be transformed using local electrical transformations ( those in section [ ss : trans ] and the electrical @xmath0-matrix ) into a critical cylindrical electrical network . * any two cylindrical electrical networks @xmath6 and @xmath7 with the same universal response matrices are connected by local electrical equivalences . furthermore , if @xmath6 and @xmath7 are both critical , then only star - triangle transformations , and electrical @xmath0-matrix transformations are needed . * if a cylindrical electrical network is critical canonical , then the conductances corresponding to all crossings involving wires of types ( i ) and ( ii ) can be recovered uniquely . the conductances corresponding to crossings of wires of type ( i ) and type ( iii ) can be recovered up to the electrical @xmath0-matrix action . * the space @xmath8 of universal response matrices of cylindrical electrical networks has an infinite stratification by @xmath9 where each @xmath10 is a semi - closed cell that can be obtained as the set of universal response matrices for a fixed critical network with varying edge weights . another way to phrase conjecture [ conj : gen](3 ) is that the _ monodromy group _ of a critical canonical cylindrical network is a symmetric group , generated by electrical @xmath0-matrices . see remark [ rem : monodromy ] . let us explain the semi - closed cells in conjecture [ conj : gen](5 ) . let @xmath6 be a critical canonical cylindrical electrical network . some edge weights can be recovered uniquely and these each give a @xmath355 in the parametrization . the remaining part of the network is essentially one of the networks @xmath11 , whose edge weights can be recovered uniquely up to the electrical @xmath0-matrix action ( theorem [ thm : sol ] ) . so the response matrices would be parametrized by the orbit space @xmath356 . however , we can pick a distinguished element in each orbit : namely the one where the radii @xmath223 are non - increasing . the corresponding response matrices would then be parametrized by @xmath357 , @xmath358 together with some collection of edge weights which can be freely chosen in @xmath355 . thus the universal response matrices of critical canonical cylindrical electrical network is parametrized by a semi - closed cell @xmath359 .
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in this paper we study the inverse dirichlet - to - neumann problem for certain cylindrical electrical networks . we define and study a birational transformation acting on cylindrical electrical networks called the electrical @xmath0-matrix .
we use this transformation to formulate a general conjectural solution to this inverse problem on the cylinder .
this conjecture extends work of curtis , ingerman , and morrow @xcite , and of de verdire , gitler , and vertigan @xcite for circular planar electrical networks .
we show that our conjectural solution holds for certain `` purely cylindrical '' networks . here
we apply the grove combinatorics introduced by kenyon and wilson @xcite .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the origin of high - energy cosmic rays is among the most interesting questions in astrophysics . the origin of a structure in the all - particle energy spectrum around 4 pev , the so - called knee , is generally believed to be a corner stone in the understanding of the astrophysics of high - energy cosmic rays . the knee is proposed to be caused by the maximum energy reached in cosmic - ray accelerators or due to leakage of particles from the galaxy . hence , an understanding of the origin of the knee reveals hints on the acceleration and propagation of cosmic rays . experimental access to the understanding of the sources , acceleration and propagation mechanisms is provided by detailed investigation of the arrival directions , energy spectra , and mass composition of the ultrarelativistic particles . while at energies below 1 pev cosmic rays can be measured directly at the top of the atmosphere , the strongly decreasing flux as function of energy requires large acceptances and exposure times for higher energies . presently they can be realized in ground based facilities only . there , the secondary products , generated by interactions of high - energy cosmic - ray particles in the atmosphere , the extensive air showers , are registered . it turns out that a correct description of the high - energy interactions in the atmosphere is crucial for a precise astrophysical interpretation of air shower measurements . to investigate cosmic rays from several @xmath0 ev up to beyond @xmath1 ev the air shower experiment kascade ( `` karlsruhe shower core and array detector '' ) @xcite is operated since 1996 . the experiment detects the three main components of air showers simultaneously . a @xmath2 m@xmath3 scintillator array measures the electromagnetic and muonic components . the 320 m@xmath3 central detector system combines a large hadron calorimeter with several muon detection systems . in addition , high - energy muons are measured by an 128 m@xmath3 underground muon tracking detector . a correct understanding of high - energy interactions in the atmosphere is indispensable for a good astrophysical interpretation of air shower data . the electromagnetic part of the showers is well understood and described by qed . for the air shower development the understanding of multi - particle production in hadronic interactions with a small momentum transfer is essential . due to the energy dependence of the coupling constant @xmath4 and the resulting large values for soft interactions , the latter can not be calculated within qcd using perturbation theory . instead , phenomenological approaches have been introduced in different models . for the numerical simulation of the development of air showers the program corsika @xcite is widely used . it offers the possibility to use different models to describe low and high - energy hadronic interactions . a principal objective of the kascade experiment is to investigate the air shower development in detail and test the validity of the models included in simulation codes such as corsika , using as much information as possible from the simultaneous measurement of the electromagnetic , muonic and hadronic components . with these investigations already several problems in existing codes could be pointed out and some interaction models ( or particular versions of them ) could be shown to be incompatible with the measured data @xcite . as an example of present activities , fig . [ ehne ] shows the hadronic energy sum as function of the number of electrons . presented are measured values compared to predictions of three different interaction models for showers induced by primary protons or iron nuclei . for a presentation of the data as function of the number of electrons one expects an enrichment of light primaries within the particular intervals , hence , the data should approach the values for the proton component . one recognizes that the data are compatible with the predictions of qgsjet and sibyll , while on the other hand nexus 2 predicts too less hadronic energy in most of the electron number range . from such distributions one can conclude that the present version of nexus is not compatible with the data . more detailed investigations of the models qgsjet 01 and sibyll 2.1 are presently in progress . the investigation of the arrival directions of cosmic rays improves the understanding of the propagation of the particles through the galaxy and their sources . model calculations of the diffusion process in the galactic magnetic field indicate that there could be an anisotropy on a scale of @xmath5 to @xmath6 depending on particle rigidity as well as strength and structure of the galactic magnetic field @xcite . diffusion models relate a rigidity dependent leakage of particles from the galaxy to the steepening ( or knee ) in the all - particle energy spectrum around 4 pev . thus , anisotropy measurements can provide substantial information on the origin of the knee . in kascade investigations @xcite attention has been drawn to a search for point sources and large - scale anisotropy . of special interest is the search for potential gamma - ray induced showers . since photons are not deflected in the galactic magnetic field , they are suitable for a direct search for the sources of high - energy particles . experimentally such an investigation is realized by the selection of muon poor showers . as function of primary energy . the kascade upper limit ( bold line ) is compared to results from the literature @xcite . model predictions @xcite for the total anisotropy as well as for the light and heavy component are also shown ( thin lines ) . ] a search for point sources was performed for primary photon candidates as well as for all ( charged ) particles . the search covers the whole sky , visible by kascade ( declination @xmath7 to @xmath8 ) . special attention was drawn to the galactic plane and known gamma - ray sources in the tev region . the search was complemented by an analysis of the most energetic showers registered and an investigation of the most photon - like primary particles . none of the searches reveals an indication for a significant excess of the flux . the distributions of the significance values over the whole sky follow the expectations for an isotropic cosmic - ray flux . the results for the analysis of the large - scale anisotropy are illustrated in fig.[aniso ] . there , the rayleigh amplitude is plotted versus the primary energy . the upper limit derived from the kascade data is compatible with other measurements from the literature and also with theoretical calculations @xcite for different elemental groups . the most direct experimental access to the astrophysics of cosmic rays is provided by measurements of their energy spectrum and mass composition . investigations of the kascade experiment reveal that the knee in the all - particle spectrum is caused by a turn - off of the light component ( protons and helium nuclei ) @xcite . this is implies an increase of the mean logarithmic mass ( @xmath9 ) as function of energy . while it is beyond doubt that the data reveal such an increase , the absolute values of @xmath9 depend on the observables investigated and on the interaction models used in the simulations to interpret the data @xcite . the differences amount to about @xmath10 . this indicates that further and more detailed investigations of high - energy interactions in the atmosphere are necessary for an unambiguous astrophysical interpretation of the observed data . presently , the most promising approach is the unfolding of energy spectra for individual elemental groups from the data of the electromagnetic and muonic component @xcite . systematic studies are performed with different hadronic interaction models available in corsika , for both , low and high - energy ( @xmath11 gev ) interactions . different unfolding methods are applied , in order to investigate the systematic effects introduced by the individual methods . the different methods result in similar flux values if the same code is used for the air shower simulation . but significant differences occur for different interaction models . this is illustrated in fig . [ espec ] , where recent results of an analysis applying the gold algorithm are depicted . the figure also shows the flux of primary protons as obtained in an analysis of unaccompanied hadrons @xcite . the flux is compatible with the proton flux as obtained from the unfolding procedure . for comparison , also the results of direct measurements at lower energies at the top of the atmosphere @xcite are given in the figure . the analyses indicate that , at present , the understanding of primary cosmic rays is limited by the insufficient knowledge of high - energy hadronic interactions in the atmosphere and not by too low statistics of the measurements or the systematic uncertainties introduced by different reconstruction methods . the present status of the kascade experiment to investigate high - energy cosmic rays in the knee region has been discussed . continuing efforts are taken in order to improve the understanding of high - energy interactions in the atmosphere . a search for point sources of charged particles as well as gamma rays did not reveal any significant excess . upper limits for the large scale anisotropy have been derived . the observations reveal an increase of the mean logarithmic mass as function of energy in the knee region . energy spectra for groups of elements have been derived . to extend the investigations to higher energies up to @xmath12 ev data taking with an enlarged array @xcite has started in 2003 . d. heck et al . , report fzka 6019 , + forschungszentrum karlsruhe 1998 ; + and http://www-ik.fzk.de/@xmath13heck/corsika . t. antoni et al . , j. phys . g 25 ( 1999 ) 2161 . j. milke et al . , proc . cosmic ray conf . , hamburg 1 ( 2001 ) 241 . j. milke et al . , nucl . phys . suppl . ) 122 ( 2003 ) 388 . hrandel , nucl . . suppl . ) 122 ( 2003 ) 455 . j. candia et al . , j. cosmol . and astropart . phys . 5 ( 2003 ) 3 . g. maier et al . , proc . cosmic ray conf . , tsukuba 1 ( 2003 ) 179 . g. schatz et al . , proc . cosmic ray conf . , tsukuba 4 ( 2003 ) 2293 . k. nagashima et al . , proc . cosmic ray conf . , adelaide 3 ( 1990 ) 180 . m. aglietta et al . , ap . j. 470 ( 1996 ) 501 ; proc . cosmic ray conf . , tsukuba 4 ( 2003 ) 182 . t. kifune et al . , j. phys . g 12 ( 1986 ) 129 . p. gerhard & r. clay , j. phys . g 9 ( 1998 ) 1279 . t. antoni et al . , astropart . ( 2002 ) 373 . hrandel et al . , proc . 16european cosmic - ray symp . , alcala de henares , ( 1998 ) 579 ; j. engler et al . 26int . cosmic ray . , salt lake city 1 ( 1999 ) 349 . t. antoni et al . , astropart . ( 2002 ) 245 . c. bttner et al . cosmic ray conf . , tsukuba 1 ( 2003 ) 33 . h. ulrich et al . , proc . cosmic ray conf . , hamburg 1 ( 2001 ) 97 . h. ulrich et al . b ( proc . suppl . ) 122 ( 2003 ) 218 . m. roth et al . , nucl . b ( proc . suppl . ) 122 ( 2003 ) 317 . m. roth et al . , proc . cosmic ray conf . , tsukuba 1 ( 2003 ) 139 . m. mller et al . , proc . cosmic ray conf . , tsukuba 1 ( 2003 ) 101 . hrandel , astropart . ( 2003 ) 193 .
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recent results of the kascade air shower experiment are presented in order to shed some light on the astrophysics of cosmic rays in the region of the knee in the energy spectrum .
the results include investigations of high - energy interactions in the atmosphere , the analysis of the arrival directions of cosmic rays , the determination of the mean logarithmic mass , and the unfolding of energy spectra for elemental groups .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
modern power networks are increasingly dependent on information technology in order to achieve higher efficiency , flexibility and adaptability @xcite . the development of more advanced sensing , communications and control capabilities for power grids enables better situational awareness and smarter control . however , security issues also arise as more complex information systems become prominent targets of cyber - physical attacks : not only can there be data attacks on measurements that disrupt situation awareness @xcite , but also control signals of power grid components including generation and loads can be hijacked , leading to immediate physical misbehavior of power systems @xcite . furthermore , in addition to hacking control messages , a powerful attacker can also implement physical attacks by directly intruding upon power grid components . therefore , to achieve reliable and secure operation of a smart power grid , it is essential for the system operator to minimize ( if not eliminate ) the feasibility and impact of physical attacks . there are many closely related techniques that can help achieve secure power systems . firstly , coding and encryption can better secure control messages and communication links @xcite , and hence raise the level of difficulty of cyber attacks . to prevent physical attacks , grid hardening is another design choice @xcite . however , grid hardening can be very costly , and hence may only apply to a small fraction of the components in large power systems . secondly , power systems are subject to many kinds of faults and outages @xcite , which are in a sense _ unintentional _ physical attacks . as such outages are not inflicted by attackers , they are typically modeled as random events , and detecting outages is often modeled as a hypothesis testing problem @xcite . however , this event and detection model is not necessarily accurate for _ intentional _ physical attacks , which are the focus of this paper . indeed , an intelligent attacker would often like to strategically _ optimize _ its attack , such that it is not only hard to detect , but also the most viable to implement ( e.g. , with low execution complexity as well as high impact ) . recently , there has been considerable research concerning data injection attacks on sensor measurements from supervisory control and data acquisition ( scada ) systems . a common and important goal among these works is to pursue the integrity of network _ state estimation _ , that is , to successfully detect the injected data attack and recover the correct system states . the feasibility of constructing data injection attacks to pass bad data detection schemes and alter estimated system states was first shown in @xcite . there , a natural question arises as to how to find the _ sparsest unobservable _ data injection attack , as sparsity is used to model the complexity of an attack , as well as the resources needed for an attacker to implement it . however , finding such an _ optimal attack _ requires solving an np - hard @xmath2 minimization problem . while efficiently finding the sparsest unobservable attacks in general remains an open problem , interesting and exact solutions under some special problem settings have been developed in @xcite @xcite @xcite . another important aspect of a data injection attack is its impact on the power system . as state estimates are used to guide system and market operation of the grid , several interesting studies have investigated the impact of data attacks on optimal power flow recommendation @xcite and location marginal prices in a deregulated power market @xcite @xcite . furthermore , as phasor measurement units ( pmus ) become increasingly deployed in power systems , network situational awareness for grid operators is significantly improved compared to using legacy scada systems only . however , while pmus provide accurate and secure sampling of the system states , their high installation costs prohibit ubiquitous deployment . thus , the problem of how to economically deploy pmus such that the state estimator can best detect data injection attacks is an interesting problem that many studies have addressed ( see , e.g. @xcite among others . ) compared to data attacks that target state estimators , physical attacks that directly disrupt power network physical processes can have a much faster impact on power grids . in addition to physical attacks by hacking control signals or directly intruding upon grid components , several types of load altering attacks have been shown to be practically implementable via internet - based message attacks @xcite . topological attacks are another type of physical attack which have been considered in @xcite . dynamic power injection attacks have also been analyzed in several studies . for example , in @xcite , conditions for the existence of undetectable and unidentifiable attacks were provided , and the sizes of the sets of such attacks were shown to be bounded by graph - theoretic quantities . alternatively , in @xcite and @xcite , state estimation is considered in the presence of both power injection attacks and data attacks . specifically , in these works , the maximum number of attacked nodes that still results in correct estimation was characterized , and effective heuristics for state recovery under sparse attacks were provided . in this paper , we investigate a specific type of physical attack in power systems called _ power injection attacks _ , that alter generation and loads in the network . a linearized power network model - the dc power flow model - is employed for simplifying the analysis of the problem and obtaining a simple solution that yields considerable insight . we consider a grid operator that employs pmus to ( partially ) monitor the network for detecting power injection attacks . since power injection attacks disrupt the power system states immediately , the timeliness of pmu measurement feedback is essential . furthermore , our model allows for the power injections at some buses to be `` unalterable '' . this captures the cases of `` zero injection buses '' with no generation and load , and buses that are protected by the system operator . under this model we study the open @xmath2 minimization problem of finding the sparsest unobservable attacks given any set of pmu locations . we start with a feasibility problem for unobservable attacks . we prove that the existence of an unobservable power injection attack restricted to any given set of buses can be determined with probability one by computing a quantity called the structural rank . next , we prove that the np - hard problem of finding the sparsest unobservable attacks has a simple solution with probability one . specifically , the sparsity of the optimal solution is @xmath0 , where @xmath1 is the `` vulnerable vertex connectivity '' that we define for an augmented graph of the original power network . meanwhile , the entire set of globally optimal solutions ( there can be many of them ) is found in polynomial time . we further introduce a notion of potential impacts of unobservable attacks . accordingly , among all the sparsest unobservable attacks , an attacker can easily find the one with the greatest potential impact . finally , given optimized pmu placement , we evaluate the sparsest unobservable attacks in terms of their sparsity and potential impact in the ieee 30 , 57 , 118 and 300-bus , and the polish 2383 , 2737 and 3012-bus systems . the remainder of the paper is organized as follows . in section [ secform ] , models of the power network , power injection attacks , pmus and unalterable buses are established . in addition , the minimum sparsity problem of unobservable attacks is formulated . in section [ secfeas ] we provide the feasibility condition for unobservable attacks restricted to any subset of the buses . in section [ secmin ] we prove that the minimum sparsity of unobservable attacks can be found in polynomial time with probability one . in section [ secnum ] , a pmu placement algorithm for countering power injection attacks is developed , and numerical evaluation of the sparsest unobservable attacks in ieee benchmark test cases and large - scale polish power systems are provided . conclusions are drawn in section [ secconc ] . we consider a power network with @xmath3 buses , and denote the set of buses and the set of transmission lines by @xmath4 and @xmath5 respectively . for a line @xmath6 that connects buses @xmath7 and @xmath8 , denote its reactance by @xmath9 as well as @xmath10 , and define its _ incidence vector _ @xmath11 as follows : @xmath12 based on the power network topology and line reactances , we construct a weighted graph @xmath13 where the edge weight @xmath14 . the power system is generally modeled by nonlinear ac power flow equations @xcite . in this paper , a linearized model - the dc power flow model - is employed as an approximation of the ac model , which allows us to find a simple closed - form solution to the problem from which we glean significant insights . under the dc model , the real power injections @xmath15 and the voltage phase angles @xmath16 satisfy @xmath17 , where @xmath18 is the _ laplacian _ of the weighted graph @xmath19 . we assume that @xmath9 is positive which is typically true for transmission lines ( cf . chapter 4 of @xcite ) . furthermore , the power flow on line @xmath20 from bus @xmath7 to bus @xmath8 equals @xmath21 . we consider attackers inflicting power injection attacks that alter the generation and loads in the power network . we denote the power injections in normal conditions by @xmath22 , and denote a power injection attack by @xmath23 . thus the post - attack power injections are @xmath24 . we consider the use of pmus by the system operator for monitoring the power network in order to detect power injection attacks . with pmus installed at the buses , we consider the following two different sensor models : 1 . a pmu securely measures the voltage phasor of the bus at which it is installed . a pmu securely measures the voltage phasor of the bus at which it is installed , as well as the current phasors on all the lines connected to this bus . we denote the set of buses with pmus by @xmath25 , and let @xmath26 be the total number of pmus , where @xmath27 denotes the cardinality of a set . without loss of generality ( wlog ) , we choose one of the buses in @xmath28 to be the angle reference bus . we say that a power injection attack @xmath29 is _ unobservable _ if it leads to _ zero _ changes in all the quantities measured by the pmus . with the first pmu model described above , we have the following definition : an attack @xmath30 is unobservable if and only if @xmath31 where @xmath32 denotes the @xmath33 sub - vector of @xmath34 obtained by keeping its @xmath35 entries whose indices are in @xmath28 . is a weighted laplacian matrix , the elements of @xmath36 sum to @xmath37 . ] with the second pmu model described above , for any bus @xmath38 , it is immediate to verify that the following three conditions are equivalent : 1 . there are no changes of the voltage phasor at @xmath7 and of the current phasors on all the lines connected to @xmath7 . there are no changes of the voltage phasor at @xmath7 and of the power flows on all the lines connected to @xmath7 . 3 . @xmath39 $ ] , there is no change of the voltage phasor at @xmath40 , where @xmath41 $ ] is the closed neighborhood of @xmath7 which includes @xmath7 and its neighboring buses @xmath42 . thus , for forming unobservable attacks , the following two situations are equivalent to the attacker : * the system operator monitors the set of buses @xmath28 with the second pmu model ; * the system operator monitors the set of buses @xmath43 $ ] with the first pmu model , where @xmath43 $ ] is the closed neighborhood of @xmath28 which includes all the buses in @xmath28 and their neighboring buses @xmath44 . thus , the unobservability condition with the second pmu model is obtained by replacing @xmath28 with @xmath43 $ ] in . wlog , we employ the first pmu model in the following analysis , and based on the discussion above all the results can be directly translated to the second pmu model . in forming an unobservable attack , an attacker generally has two objectives : minimize execution complexity and maximize its impact on the grid . note that these two objectives can be competing interests that are not simultaneously achievable . we will first focus on finding the minimum execution complexity for an attack to be unobservable , which constitutes the main part of this work . among attacks with the minimum complexity , we then find the one with the maximum impact . for an attack vector @xmath29 , we use its zero norm @xmath45 to model its execution complexity . this is because attackers are typically resource - constrained , and can choose only a limited number of buses to implement attacks . for minimizing attack complexity , an attacker is interested in finding the sparsest attacks that satisfy the unobservability condition : @xmath46 since @xmath47 , a more compact form of is as follows : @xmath48 where @xmath49 denotes the complement of @xmath35 , and @xmath50 is the submatrix of @xmath51 formed by choosing all its rows and a set of columns @xmath52 . we now note that problem is np - hard : specifically , as a special case of the cospark problem of a matrix @xcite problem resembles a security index problem discussed in @xcite , which has been proven to be np - hard . under some special problem settings for data injection attacks , problems of this type have been shown to be solvable exactly in polynomial time @xcite @xcite @xcite . in general , low complexity heuristics have been developed for solving @xmath2 minimization problems ( e.g. , @xmath53 relaxation ) . we now generalize our model to allow a subset of buses to be `` unalterable buses '' , meaning that their nodal power injection can not be changed by attackers . this allows us to model the following scenarios : * a `` zero injection '' bus that simply connects multiple lines without nodal generation or load , and hence its power injection is always zero and can not be changed . * a `` protected '' bus by the system operator , and its power injection is not accessible by the attacker . we denote the set of unalterable buses by @xmath54 . the other buses @xmath55 are termed `` alterable '' buses . generalizing , the sparsest unobservable attack problem is established as follows : @xmath56 when @xmath57 , reduces to . generalizing , eq . has the following equivalent form : @xmath58 _ given the locations of the sensors _ @xmath28 , we now introduce a variation of the graph @xmath19 that will prove key to developing the main results later . [ gmdef ] given a set of buses @xmath59 , @xmath60 is defined to be the following augmented graph based on @xmath19 : 1 . @xmath60 includes all the buses in @xmath19 , and has one additional unalterable dummy bus . 2 . define an augmented set @xmath61 that contains @xmath28 and the unalterable dummy bus . @xmath60 includes all the edges of @xmath19 , and an edge is added between every pair of buses in @xmath61 , and its weight can be chosen arbitrarily as any positive number . we note that the dummy bus is only connected to the set of sensors @xmath28 . we observe the following key facts . first , an unobservable attack in the original graph @xmath19 leads to zero changes in all the voltage phase angles in @xmath28 . thus , any line between a pair of buses in @xmath28 would see a zero change of the power flow on it . it is then clear that the added dummy bus and lines in @xmath60 do not lead to any power flow changes in the network under any unobservable attack . we thus have the following lemma : [ lemaug ] an attack is unobservable by @xmath28 in @xmath19 if and only if it is unobservable by @xmath28 in @xmath60 . this allows us to work with the augmented graph @xmath60 instead of @xmath19 . it is clear that the weights of the added edges in @xmath60 do not matter for lemma [ lemaug ] to hold . in this section , we address the following question whose solutions will be useful in solving the minimum sparsity problem : assuming that the attacker can only alter the power injections at a subset of the buses , denoted by @xmath62 , does there _ exist _ an attack that is unobservable by a set of pmus @xmath28 ? for any given @xmath63 , a feasible non - zero attack @xmath64 must satisfy @xmath65 . in other words , it must not alter the power injections at the buses in @xmath66 . from , there exists an unobservable non - zero attack if and only if @xmath67 since @xmath68 , we have that is equivalent to @xmath69 where @xmath70 is the submatrix of @xmath51 formed by its rows @xmath66 and columns @xmath52 . an illustration of is depicted in figure [ feasfig ] , where the submatrix formed by the shaded blocks represents @xmath70 . from , we have the following lemma on the feasibility condition of unobservable attacks . [ lemfeas1 ] given @xmath63 and @xmath28 , there exists an unobservable non - zero attack if and only if @xmath70 is column rank deficient . to analyze when this column rank deficiency condition , @xmath71 , is satisfied , we start with the following observations based on the fact that @xmath51 is the laplacian of the weighted graph @xmath19 . 1 . the _ signs _ ( @xmath72 , @xmath73 , or @xmath37 ) of the entries of @xmath51 are fully determined by the _ network topology _ : @xmath74 2 . the _ values _ of the non - zero entries of @xmath51 are determined by the _ line reactances _ when all the line reactances in the power network are known , so are the entries of the submatrix @xmath70 , and it is immediate to compute whether @xmath77 . _ without _ knowing the exact values of any line reactances , we will show that whether @xmath77 can be determined almost surely by computing the _ structural rank _ of @xmath70 , defined as follows @xcite . [ indent ] a set of independent entries of a matrix @xmath78 is a set of nonzero entries , no two of which lie on the same line ( row or column ) . the structural rank of a matrix @xmath78 , denoted by @xmath79 , is the maximum number of elements contained in at least one set of independent entries . a basic relation between the structural rank and the rank of a matrix is the following @xcite , @xmath80 in the literature , structural rank is also termed `` generic rank '' @xcite . specifically , we consider _ generic _ power grid parameters , i.e. , we assume that the line reactances @xmath81 are independent , but not necessarily identical random variables drawn from continuous probability distributions . we assume that the reactances are bounded away from zero from below ( as lines do not have zero reactances in practice ) . as such , the analysis in this work is along the line of _ structural properties _ as in @xcite and @xcite , and we will develop results that hold _ with probability one_. we believe the independence ( but not identically distributed ) assumption is sufficiently general in practice . in particular , there are uncertainties in factors that influence the reactance of a line ( e.g. the distance that a line travels , the degradation of a line over time ) . these uncertainties can be modeled as independent ( but not identically distributed ) random variables , leading to the model employed in this paper . clearly , @xmath70 is always column rank deficient when @xmath82 . next , we discuss the case of @xmath83 . we begin with the special case @xmath84 , for which we have the following lemma whose proof is relegated to appendix [ secprfleml ] : [ leml ] let @xmath85 be the laplacian of a connected graph @xmath19 with strictly positive edge weights . for any set of node indices @xmath86 , denote by @xmath87 the submatrix of @xmath51 formed by its rows @xmath88 and columns @xmath88 . then @xmath89 , @xmath87 is of full rank . note that lemma [ leml ] holds deterministically without assuming generic edge weights of the graph . for the case of @xmath84 , we let @xmath90 , and lemma [ leml ] proves that @xmath91 . this implies the intuitive fact that there exists no attack restricted to @xmath63 that is unobservable by a set of pmus @xmath92 . now , we address the general case of arbitrary @xmath63 and @xmath28 . we have the following theorem demonstrating that having @xmath93 _ almost surely guarantees _ @xmath94 . the proof is relegated to appendix [ secprffeasthm ] . [ feasthm ] for a connected weighted graph @xmath95 , assume that the edge weights are independent continuous random variables strictly bounded away from zero from below , and denote the laplacian of @xmath19 by @xmath85 . then , any @xmath96 submatrix of @xmath51 , with @xmath97 , has a rank of @xmath98 with probability one if it has a structural rank of @xmath98 . from theorem [ feasthm ] , with @xmath83 , if @xmath99 , we have with probability one that @xmath100 , and there exists no attack restricted to @xmath63 that is unobservable by a set of pmus @xmath28 . it has been known in the literature that ( see e.g. , @xcite ) , a full structural rank of a matrix leads to a full rank matrix with probability one , as long as the nonzero entries in the matrix are drawn independently from continuous probability distributions . however , it is worth noting that this is not sufficient for proving theorem [ feasthm ] . this is because , as in theorem [ feasthm ] , we are interested in matrices that are _ submatrices of a graph laplacian _ : even with the edge weights of the graph drawn independently , the entries in these submatrices are _ correlated _ due to the special structure of a graph laplacian . such correlation leads to technical difficulties for the proof , which can be overcome as shown in appendix [ secprffeasthm ] . we note that the structural rank of a matrix can be computed in polynomial time by finding the maximum bipartite matching in a graph @xcite . since whether an entry of @xmath51 is non - zero is solely determined by the topology of the network , we have the following corollary . given @xmath63 and @xmath28 , whether a non - zero unobservable attack exists can be determined with probability one based solely on the knowledge of the grid topology . in this section , we study the problem of finding the sparsest unobservable attacks given any set of pmus @xmath28 ( cf . ) . as remarked in section [ secmm ] , @xmath2 minimization such as is np - hard . we will show that the sparsest unobservable attack can in fact be found in _ polynomial time with probability one_. we first introduce a key concept a _ vulnerable vertex cut_. we then state our main theorem that yields an explicit solution for the sparsest unobservable attack problem . we prove that this solution both upper and lower bounds the optimum of , hence proving the theorem . we start with the following basic definitions : a vertex cut of a connected graph @xmath19 is a set of vertices whose removal renders @xmath19 disconnected . the vertex connectivity of a graph @xmath19 , denoted by @xmath101 , is the size of the minimum vertex cut of @xmath19 , i.e. , it is the minimum number of vertices that need to be removed to make the remaining graph disconnected . from the definition of the augmented graph @xmath60 in section [ sec : aug ] , since all the buses in @xmath61 ( containing @xmath28 and the dummy bus ) are pair - wise connected , we have the following lemma : [ auglem ] for any vertex cut of the augmented graph @xmath60 , there is no pair of the buses in @xmath61 that are disconnected by this cut . accordingly , we introduce the following notations which will be used later on : [ ntn1 ] given a vertex cut of @xmath60 , we denote the set of buses disconnected from @xmath61 after removing the cut set by @xmath102 . the cut set itself is denoted by @xmath103 . with the vertex cut @xmath103 , @xmath60 is partitioned into three subgraphs : 1 . @xmath102 , which does not contain any bus in @xmath61 , i.e. , @xmath104 . 2 . @xmath103 , which is the vertex cut set itself , and may contain buses in @xmath61 . 3 . @xmath105 $ ] , which contains ( not necessarily exclusively ) all the remaining buses in @xmath61 after removing the cut set . an illustrative example with a cut @xmath103 of size 2 is depicted in figure [ 3spgen ] in section [ upsec ] . we note that there is a slight abuse of notation in @xmath106 : in general , a cut set does not necessarily consist of exactly all the neighboring nodes of @xmath107 . nonetheless , as will be shown in the remainder of the paper , we need only care about the _ minimum _ cut set , which indeed consists of exactly all the neighboring nodes of @xmath107 , namely , @xmath106 . leveraging the above notation , we now introduce a key type of vertex cut on @xmath60 . a vulnerable vertex cut of a connected augmented graph @xmath60 is a vertex cut @xmath103 for which @xmath108\vert \ge \vert n({\mathcal}{s})\vert+1 $ ] . in other words , the number of _ alterable _ buses in @xmath109 $ ] is no less than the cut size plus one . the reason for calling such a vertex cut `` vulnerable '' will be made exact later in section [ upsec ] . the basic intuition is the following . in order to have @xmath110 ( unobservability ) , the key is to have the phase angle changes on the cut @xmath103 be zero , with power injection changes ( which can only happen on the alterable buses ) restricted in @xmath109 $ ] . as will be shown later , this can be achieved if a cut @xmath103 is `` vulnerable '' as defined above . we note that it is possible that no vulnerable vertex cut exists ( e.g. , in the extreme case that all buses are unalterable ) . accordingly , we define the following variation on the vertex connectivity . the vulnerable vertex connectivity of an augmented graph @xmath60 , denoted by @xmath111 , is the size of the minimum vulnerable vertex cut of @xmath60 . if no vulnerable vertex cut exists , @xmath111 is defined to be @xmath112 . we note that the concepts of vulnerable vertex cut and vulnerable vertex connectivity do not apply to the original graph @xmath19 . we immediately have the following lemma : [ kplem ] if a vulnerable vertex cut exists , then @xmath113 . suppose a vulnerable vertex cut exists , and @xmath114 . denote the minimum vulnerable vertex cut by @xmath103 ( cf . notation [ ntn1 ] ) . now consider the set @xmath28 : it is a vertex cut of @xmath60 that separates the dummy bus and @xmath115 . because there are at least @xmath116 alterable buses in @xmath109\subseteq n[\bar{m}^c]$ ] , @xmath28 is also a _ vulnerable vertex cut_. this contradicts the minimum vulnerable vertex cut having size at least @xmath117 . we now state the following theorem that gives an explicit solution of the sparsest unobservable attack problem in terms of the vulnerable vertex connectivity @xmath111 . [ mainthm ] for a connected grid @xmath118 , assume that the line reactances @xmath119 are independent continuous random variables strictly bounded away from zero from below . given any @xmath28 and @xmath54 , the minimum sparsity of unobservable attacks , i.e. , the global optimum of , equals @xmath120 with probability one . we note that finding the minimum vulnerable vertex connectivity of a graph is computationally efficient . for polynomial time algorithms we refer the readers to @xcite and @xcite . in particular , vertex cuts are enumerated @xcite starting from the minimum and with increasing sizes , until a minimum vulnerable vertex cut is identified . we now prove theorem [ mainthm ] by upper and lower bounding the minimum sparsity of unobservable attacks in the following two subsections . we show that _ any _ vulnerable vertex cut @xmath103 provides an upper bound on the optimum of as follows . [ upthm ] for a connected grid @xmath19 and a set of pmus @xmath28 , for any vulnerable vertex cut of @xmath60 denoted by @xmath103 ( cf . notation [ ntn1 ] ) , there exists an unobservable attack of sparsity no higher than @xmath121 . a vulnerable vertex cut @xmath103 partitions @xmath60 into @xmath102 , @xmath103 and @xmath122}$ ] , with @xmath123 . similarly to the range space interpretation of the sparsest unobservable attack , it is sufficient to show that there exists a non - zero vector in the range space of @xmath124 such that i ) it has a sparsity no higher than @xmath121 , and ii ) non - zero power injections occur only at the alterable buses . by re - indexing the buses , wlog , i ) let @xmath125 , and ii ) let @xmath124 have the following partition as depicted in figure [ multicol ] : 1 . the top submatrix @xmath126 is an @xmath127 matrix . the middle submatrix ( which will be shown to be @xmath128 ) consists of all the remaining rows , each of which has at least one _ non - zero _ entry . 3 . the bottom submatrix is an _ all - zero _ matrix . in particular , from the definition of the laplacian , the middle submatrix of @xmath124 , as described above , is exactly @xmath128 because its row indices correspond to those buses not in @xmath102 but connected to at least one bus in @xmath102 . from the definition of the vulnerable vertex cut , @xmath108\vert \ge \vert n({\mathcal}{s})\vert+1 $ ] now , pick any set of @xmath129 alterable buses in @xmath130 $ ] , denote this set by @xmath63 , and denote the other buses in @xmath109 $ ] by @xmath131\backslash { \mathcal}{a}$ ] . clearly , @xmath132 . therefore , @xmath133 ( which is a submatrix of @xmath134{\mathcal}{s}}$ ] ) has @xmath135 columns but only @xmath136 rows , and is hence column rank deficient . now , we let @xmath137 be a non - zero vector in the null space of @xmath133 : @xmath138 then , we construct an attack vector @xmath139 : it has some possibly non - zero values at the indices that correspond to @xmath63 , and has _ zero values at all other indices . _ thus , @xmath140 theorem [ upthm ] explains our terminology of a `` vulnerable vertex cut '' , since if a vertex cut is vulnerable , it leads to an unobservable attack . if a vulnerable vertex cut of @xmath60 exists , applying theorem [ upthm ] to the _ minimum _ one , we have that the optimum of is upper bounded by @xmath120 . if no vulnerable vertex cut exists , @xmath141 is a trivial upper bound . we now provide a graph - theoretic interpretation of theorem [ upthm ] . as shown in figure [ multicol ] and [ 3spgen ] , all the buses can be partitioned into three subsets @xmath142 and @xmath105 $ ] , corresponding to the row indices of the top , middle and bottom submatrices of @xmath124 , respectively . @xmath103 is a vulnerable vertex cut of @xmath60 that separates @xmath102 from @xmath105 $ ] . the sparse attack @xmath29 ( cf . ) is formed by injecting / extracting power at @xmath121 alterable buses in @xmath109 $ ] , such that the phase angle changes at @xmath143 are all zero . note that @xmath144 . the example with @xmath145 in figure [ 3spgen ] illustrates a @xmath146-sparse attack with power injection / extractions at ( assumed alterable ) buses @xmath147 and @xmath148 , such that the phase angle changes at @xmath143 are all zero . we end this subsection by introducing a notion of `` potential impact '' of unobservable attacks . we make the following observation : as long as an attacker takes control of all the power injections in a vulnerable vertex cut @xmath103 ( assuming they are alterable ) , it can always _ cancel out the effects of anything that happens within @xmath109 $ ] _ on the measurements taken in @xmath149 . thus , by taking control of all the buses in @xmath103 , an attacker can successfully _ hide _ from the system operator a power injection attack with a zero norm as large as @xmath150 the potential impact of unobservable attacks associated with a vulnerable vertex cut @xmath103 is defined as @xmath151|$ ] . definition [ impactdef ] is one characterization of attack impact based solely on graph theoretic properties . in practice , there are many different notions of attack impact depending on , e.g. , the interpretation of the attacks and the operating objective of the system . employing definition [ impactdef ] , we can _ differentiate _ the potential impacts of multiple sparsest unobservable attacks _ with the same sparsity_. an illustration is depicted in figure [ impfig ] . in this example , two vulnerable vertex cuts both of size two , @xmath152 and @xmath153 , are enclosed by solid ovals . accordingly , both cuts enable 3-sparse unobservable attacks . however , their potential impacts are significantly different . cut @xmath154 only disconnects one other bus , namely @xmath155 from the set of pmus @xmath28 , and hence its potential impact equals @xmath156| = 3 $ ] . in comparison , cut @xmath157 disconnects all the vertices above @xmath157 from @xmath28 , and hence its potential impact equals @xmath158| \gg 3 $ ] . with this definition of potential impact , it is then natural for an attacker to _ seek the sparsest unobservable attack with the greatest potential impact_. as an immediate byproduct of the analysis of potential impact , by letting @xmath159 , we obtain the _ maximum _ potential impact of all unobservable attacks in a power network : [ maxcor ] for a connected power grid @xmath118 , given any @xmath28 denoting the pmu locations , the maximum potential impact among all the unobservable attacks equals @xmath160|$ ] . we first define the following property of a matrix @xmath161 , which will be shown to be equivalent to having @xmath162 . [ cond1def ] @xmath163 we have the following lemma whose proof is relegated to appendix [ secprfcond1 ] : [ cond1equ ] property [ cond1def ] is equivalent to having @xmath162 . we now prove the lower bounding part of theorem [ mainthm ] , namely , with probability one , all unobservable power injection attacks @xmath36 must have @xmath164 . the key idea is in showing that the equivalence between property [ cond1def ] and a full structural rank ( cf . lemma [ cond1equ ] ) implies a connection between the vulnerable vertex connectivity and the feasibility condition of unobservable attacks ( cf . lemma [ lemfeas1 ] ) . we focus on @xmath60 and consider its corresponding laplacian @xmath51 . suppose there exists a power injection attack @xmath165 such that @xmath166 denote the buses with non - zero power injection changes by @xmath62 , and hence @xmath65 . from , @xmath167 , and @xmath168 , implying that @xmath70 is column rank deficient . we first consider the case that a vulnerable vertex cut exists , i.e. , @xmath169 . the proof for the case of @xmath170 follows similarly . for notational simplicity , we will use @xmath171 instead of @xmath111 in the remainder of the proof . [ [ if - a - vulnerable - vertex - cut - exists - i.e .- barkappa - infty ] ] if a vulnerable vertex cut exists , i.e. , @xmath172 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + we will prove that , for all @xmath173 with @xmath174 , _ @xmath70 is of full column rank with probability one _ , i.e. , can only happen with probability zero . from lemma [ kplem ] , @xmath175 . it is then sufficient to prove for the `` worst cases '' with @xmath176 , i.e. , @xmath177 and @xmath70 is a square matrix . from theorem [ feasthm ] and lemma [ cond1equ ] , it is sufficient to show that _ @xmath70 satisfies property [ cond1def ] _ , and hence is of full rank with probability one . recall from the definition of the laplacian @xmath51 that , for any column ( or row ) of @xmath51 , @xmath178 , its non - zero entries correspond to bus @xmath179 and those buses that are connected to bus @xmath179 . with this , we now prove that @xmath70 satisfies property [ cond1def ] . consider any set of @xmath7 ( @xmath180 ) buses in @xmath66 , denoted by @xmath181 . \i ) if @xmath182 : based on the definition of the laplacian @xmath51 , the @xmath7 columns of @xmath183 _ that correspond to the buses @xmath181 themselves _ each has at least one non - zero entry . ii ) if @xmath184 : we prove that @xmath185 must contain at least @xmath171 buses . this is because , otherwise , @xmath186 , contradicting that @xmath171 is the minimum size of vulnerable vertex cuts for the following reasons : 1 . @xmath187 , and thus @xmath188 has at least @xmath189 alterable buses . 2 . @xmath186 implies that @xmath190 , and thus @xmath185 is a vertex cut that separates @xmath181 and @xmath191 . 3 . because @xmath184 and @xmath28 are pairwise connected in @xmath60 , @xmath192 $ ] . thus , @xmath191 and @xmath28 are disjoint . from 1 ) , 3 ) , and the fact that @xmath186 , we observe that @xmath185 is a _ vulnerable vertex cut _ of size @xmath193 , contradicting @xmath171 being the vulnerable vertex connectivity . now , based on the definition of the laplacian @xmath51 , the @xmath194 submatrix _ @xmath195 must have at least @xmath196 columns each of which has at least one non - zero entry for the following reasons : _ * the @xmath7 columns of @xmath195 that correspond to the buses @xmath181 themselves each has at least one non - zero entry . * as @xmath181 are connected to at least @xmath171 other buses , each one of these @xmath171 neighbors of @xmath181 corresponds to one column of @xmath195 that has at least one non - zero entry . accordingly , the @xmath197 submatrix @xmath183 has at least @xmath7 columns each of which has at least one non - zero entry . summarizing i ) and ii ) , @xmath70 satisfies property [ cond1def ] , and is thus of full column rank with probability one . therefore , can only happen with probability zero . [ [ if - no - vulnerable - vertex - cut - exists - i.e .- barkappa - infty ] ] if no vulnerable vertex cut exists , i.e. , @xmath198 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + if @xmath199 , i.e. , all buses have pmus , then clearly no unobservable attack exists . we now focus on @xmath200 . suppose @xmath201 . consider the set @xmath202 containing @xmath28 and the dummy bus . @xmath203 ( cf . ) implies that @xmath204 $ ] , and thus @xmath205 $ ] has at least @xmath206 alterable buses . since @xmath207 separates the dummy node and @xmath208 , @xmath28 is a _ vulnerable vertex cut_. this contradicts the nonexistence of a vulnerable vertex cut . therefore , @xmath209 . in this case , the same proof as in the above case i ) when a vulnerable vertex cut exists applies , and can only happen with probability zero . with the proofs of upper and lower bounds , we have now proved theorem [ mainthm ] . in addition , from the proof of theorem [ upthm ] , we have a _ constructive solution _ of the sparsest unobservable attack in polynomial time . we conclude this section by noting the following fact similar to that in section [ secfeas ] : the minimum sparsity of unobservable attacks is fully determined with probability one by the _ network topology , the locations of the alterable buses , and the locations of the pmus_. @ l @ l @ p8.0 cm @ + + + & if no unobservable attack exists given the current set of pmus @xmath28 , stop . + & step 1 : find all the minimum vulnerable vertex cuts of @xmath210}$ ] ; + & among them , find the cut with the greatest potential impact , + & denoted by @xmath211})$ ] . + & step 2 : among all the buses disconnected from @xmath43 $ ] by @xmath211})$ ] + & as well as those in the cut set @xmath211})$ ] , place the next pmu + & at the one such that the resulting maximum potential impact + & among all the remaining unobservable attacks is minimized . + in this section , we evaluate the sparsest unobservable attacks and their potential impacts when the system operator deploys pmus at optimized locations . we first provide an efficient algorithm for optimizing pmu placement by the system operator . next , we provide comprehensive evaluation of our analysis and algorithms in multiple ieee power system test cases as well as large - scale polish power systems . our matlab codes are openly available for download . we have seen in section [ secmin ] that the minimum sparsity and potential impacts of unobservable attacks are determined fully by the network topology , the locations of the alterable buses , and the pmu placement . note that , unlike network states and parameters which can vary over short and medium time scales , the transmission network topology and the alterable buses typically stay the same over relatively long time scales . this motivates the system operator to optimize the pmu placement according to this information . for the best performance in countering power injection attacks , the system operator wants to _ raise the minimum sparsity of unobservable attacks , as well as mitigate the maximum potential impact of unobservable attacks_. algorithm 1 ( cf . table [ alg1table ] ) is developed for the system operator to greedily place pmus to pursue both objectives . in this algorithm , we have assumed that the _ second pmu model _ in section [ secsensemod ] is employed , and the algorithm can be adapted to the first pmu model by replacing @xmath43 $ ] with @xmath28 . algorithm 1 is essentially a successive cut / attack elimination procedure . the purpose of step 1 is to identify the sparsest unobservable attack with the greatest potential impact . specifically , step 1 can be performed as follows : 1 . assign arbitrarily one of the buses in @xmath28 as the _ source _ node ; 2 . for each of the buses in @xmath212 $ ] , assign it as the _ destination _ node , and compute all the minimum vulnerable vertex cuts that separate such a source - destination pair . 3 . among all the computed source - destination vertex cuts that have the same minimum size , compute their corresponding potential impacts , and select the minimum vertex cut with the greatest potential impact , denoted by @xmath211})$ ] . we note that all the minimum vulnerable vertex cuts can be enumerated in polynomial time ( c.f . @xcite ) . in our numerical evaluation using matlab on a laptop with intel core i7 3.1-ghz cpu and 8 gb of ram , it takes less than @xmath213 seconds on average for every pmu placed for the ieee 300 bus systems . this per - pmu time increases to about 50 seconds for the polish 3012 bus system . in step 2 , our primary goal is to ensure that the cut set @xmath211})$ ] found in step 1 _ does not remain a legitimate vertex cut _ after placing the next pmu . this can be achieved by placing the next pmu among the buses disconnected from @xmath43 $ ] by @xmath211})$ ] as well as those in @xmath211})$ ] . among such candidate buses , we choose the one that renders the _ minimum _ maximum potential impact among all the remaining unobservable attacks ( cf . corollary [ maxcor ] ) had the next pmu been placed at it . we evaluate our results in the ieee 30-bus , ieee 57-bus , ieee 118-bus , ieee 300-bus , polish 2383-bus , polish 2737-bus , and polish 3012-bus systems . the evaluation is performed based on the software toolbox matpower @xcite . in each of these systems , we apply algorithm 1 to generate a set of pmu locations greedily , with the number of pmus @xmath35 increasing from one until all attacks become observable . moreover , from algorithm 1 , for all @xmath35 , the minimum sparsity of unobservable attacks as well as the maximum potential impact among the sparsest unobservable attacks are found ( cf . step 1 in algorithm 1 ) . we assume that all buses are alterable in the test cases . in general , for a given set of pmus , one can also search for the maximum potential impact among all @xmath214-sparse unobservable attacks for any given sparsity @xmath214 , ( as opposed to evaluate that among the sparsest attacks only as in algorithm 1 ) . however , this problem is np - hard in @xmath214 . in light of this , we selectively focused on some level of sparsity of unobservable attacks that is _ not _ minimally sparse , and evaluated their maximum potential impacts . specifically , the minimum sparsity of unobservable attacks and the maximum potential impact among these sparsest attacks both as functions of the number of pmus @xmath35 are plotted for the ieee 30 and 118-bus power systems and the polish 3012-bus system , in figures [ eval30 ] , [ eval118 ] and [ eval3012 ] respectively . in addition , * for the ieee 30-bus system , the maximum potential impact among all 2-sparse , 3-sparse , 4-sparse and 5-sparse unobservable attacks for the entire range of @xmath35 are plotted . ( note that the minimum sparsity of unobservable attacks does not exceed 3 for all @xmath35 ) . * for the ieee 118-bus system , the maximum potential impact among all 3-sparse attacks when @xmath215 is plotted . ( note that for @xmath216 the minimum sparsity of unobservable attacks is 2 ) . we make the following observations which appear in all seven of the evaluated systems : * in all seven systems , all the attacks become observable with _ less than a third _ of the buses installed with pmus ( assuming the second pmu model ) . the average percentage of the number of pmus needed to have full network observability equals @xmath217 . while this number resembles a well - known estimate of such percentage to be one third @xcite , it also demonstrates the efficacy of algorithm 1 in pmu placement . * the topologies of the tested power systems tend to allow sparse power injection attacks . in other words , the vertex connectivity of these power networks is often small . furthermore , there are often many unobservable attacks with the same minimum sparsity : this is why even after adding a lot more pmus into the network , with each addition eliminating the previous sparsest attack , the minimum sparsity of an unobservable attack can still remain the same . * while there are many unobservable attacks with the same sparsity , the potential impacts among them can vary significantly . moreover , as more pmus are added , the maximum potential impact among all the sparsest unobservable attacks drops quickly until it reaches the minimum sparsity . similar behavior is demonstrated for all the @xmath214-sparse unobservable attacks ( @xmath218 ) for the ieee 30-bus system as shown in figure [ eval30 ] . we have studied physical attacks that alter power generation and loads in power networks while remaining unobservable under the surveillance of system operators using pmus . given a set of pmus , we have first shown that the existence of an unobservable attack that is restricted to any given subset of the buses can be determined with probability one by computing the structural rank of a submatrix of the network laplacian @xmath51 . next , we have provided an explicit solution to the open problem of finding the sparsest unobservable attacks : the minimum sparsity among all unobservable attacks equals @xmath0 with probability one . the constructive solution allows us to find all the sparsest unobservable attacks in polynomial time . as a result , @xmath0 is a fundamental limit of this minimum sparsity that is not only explicitly attainable , but also unbeatable by all possible unobservable attacks . we have then introduced a notion of potential impacts of unobservable attacks . for the system operator to raise the minimum sparsity while simultaneously mitigating the maximum potential impact of all unobservable attacks , we have devised an efficient algorithm of greedily placing the pmus . with optimized pmu deployment , we have evaluated the sparsest unobservable attacks and their potential impacts in the ieee 30 , 57 , 118 , 300-bus systems and the polish 2383 , 2737 , 3012-bus systems . finally , while this work has studied a static system model and power injection attacks , extension to dynamic systems , measurements and power injection attacks remains an interesting future direction , for which we expect that similar insights will apply . first , we denote the laplacian of the _ induced subgraph _ @xmath219 $ ] by @xmath220 . denote the number of connected components of the induced subgraph @xmath219 $ ] by @xmath221 . by properly re - indexing the nodes , we have @xmath222 where @xmath220 is a block - diagonal matrix whose each block @xmath223 is _ positive semidefinite _ and corresponds to one connected component of @xmath219 $ ] , @xmath224 and @xmath225 is diagonal , which we write in a block diagonal form whose each block @xmath226 is itself a diagonal matrix with _ non - negative _ entries , @xmath227 since the original graph @xmath19 is connected , each connected component of the induced subgraph @xmath219 $ ] must be connected to at least one node in @xmath228 . this implies the following fact : now , for any non - zero vector @xmath229 , we write it as a concatenation of @xmath221 sub - vectors : @xmath230^t~ [ \bm{x}_{\mathcal}{i}^2]^t \ldots [ \bm{x}_{\mathcal}{i}^c]^t]^t,\end{aligned}\ ] ] where the length of each sub - vector @xmath231 follows the size of the sub - matrix @xmath232 . 1 . if @xmath234 , then immediately @xmath235 . if @xmath236 , then @xmath237 , which implies @xmath238 namely , @xmath239 is in the null space of @xmath232 . note that as @xmath232 corresponds to a single connected component of @xmath219 $ ] , the dimension of the null space of @xmath232 is one , and is spanned by the all one vector @xmath240^t$ ] with the appropriate length . thus , @xmath239 must be in the form of @xmath241 , for some @xmath242 . from fact 1 , @xmath243 has non - negative diagonal entries with at least one of them strictly positive , and we have @xmath244^t \bm{d}_{\mathcal}{i}^j\bm{x}_{\mathcal}{i}^j > 0 $ ] , and hence @xmath245 . [ lemcond1 ] for a matrix @xmath161 , if the following conditions are satisfied , @xmath315,i } \text { has at least one } { \nonumber}\\ & ~~~~~~~~~~~~~~~~~~\text{non - zero entry , } \label{cond13}\end{aligned}\ ] ] \ii ) assume that the lemma is true for all @xmath317 . for @xmath318 : first , because the upper left @xmath319 submatrix of @xmath78 satisfies the induction assumption , each of @xmath78 s first left @xmath320 columns must contain at least one non - zero entry . from , the last column of @xmath78 has at least one non - zero entry . thus , the case of @xmath321 in property [ cond1def ] holds for @xmath78 . * if the last @xmath7 rows of @xmath78 are selected to form @xmath323 , ( i.e. @xmath325 ) , from , the columns @xmath326 each has one non - zero entry , namely , @xmath327 . * otherwise , there exists a row @xmath328 , @xmath329 , which is not selected in @xmath323 ( cf . figure [ cond1fig ] ) . in this case , the row indices of @xmath323 can be partitioned into two subsets : @xmath330 and @xmath331 . on the one hand , note that the upper left @xmath332 submatrix of @xmath78 satisfies the induction assumption . thus , _ among the first @xmath333 columns _ of the rows @xmath334 , there exists @xmath179 columns each of which has one non - zero entry . on the other hand , from , @xmath335 are all non - zero , and none of these non - zero entries appears in the first @xmath333 columns . therefore , there exist @xmath7 columns of @xmath323 such that each of them has at least one non - zero entry . for @xmath343 , from the induction assumption b ) , there exists a non - zero permutated diagonal for the @xmath344 submatrix @xmath345,[1:n]}$ ] : wlog , assume that it corresponds to @xmath346 ( cf . figure [ cond1fig2 ] . ) now , we use proof by contradiction , and assume that @xmath323 does _ not _ have a non - zero permuted diagonal . then , the sub - row @xmath347}$ ] must be all zero , because otherwise any non - zero entry within @xmath347}$ ] will form a non - zero permuted diagonal of @xmath323 with @xmath348 . from property [ cond1def ] , the @xmath349 row of @xmath323 must have at least one non - zero entry . wlog , assume that @xmath350 is non - zero . then , the sub - row @xmath351}$ ] must be all - zero , because otherwise any non - zero entry within @xmath351}$ ] will form a non - zero permuted diagonal of @xmath323 with @xmath350 and @xmath352 . from property [ cond1def ] , the first two rows of @xmath323 must have at least two columns each of which has at least one non - zero entry . since @xmath353 , there is at least one more column of @xmath354,[1,n]}$ ] that has at least one non - zero entry . wlog , assume that the sub - column @xmath354,2}$ ] has at least one non - zero entry , ( cf . the shaded area in figure [ cond1fig2 ] ) . similarly , consider the @xmath355 row of @xmath323 . note that the submatrix @xmath356 , [ 1:r-1]}$ ] satisfies , and , and hence satisfies property [ cond1def ] by lemma [ lemcond1 ] . from the induction assumption ii ) , @xmath356 , [ 1:r-1]}$ ] has a non - zero permuted diagonal . then , the sub - row @xmath357}$ ] must be all - zero , because otherwise any non - zero entry within @xmath357}$ ] will form a non - zero permuted diagonal of @xmath323 with the non - zero permuted diagonal of @xmath356 , [ 1:r-1]}$ ] and @xmath358 . therefore , the submatrix @xmath359,[n , n]}$ ] must be all - zero . this implies that there are only @xmath360 ( instead of @xmath7 ) columns of @xmath323 each of which has at least one non - zero entry , and hence contradicts with property [ cond1def ] . yue zhao ( s06 , m11 ) is an assistant professor of electrical and computer engineering at stony brook university . he received the b.e . degree in electronic engineering from tsinghua university , beijing , china in 2006 , and the m.s . and ph.d . degrees in electrical engineering from the university of california , los angeles ( ucla ) , los angeles in 2007 and 2011 , respectively . his current research interests include smart grid , renewable energy integration , optimization theory , stochastic control , and statistical signal processing . andrea goldsmith ( s90 , m93 , sm99 , f05 ) is the stephen harris professor in the school of engineering and a professor of electrical engineering at stanford university . she was previously on the faculty of electrical engineering at caltech . goldsmith co - founded and served as cto for two wireless companies : wildfire.exchange , which develops software - defined wireless network technology for cloud - based management of wifi systems , and quantenna communications , inc . , which develops high - performance wifi chipsets . she has previously held industry positions at maxim technologies , memorylink corporation , and at&t bell laboratories . she is a fellow of the ieee and of stanford , and has received several awards for her work , including the ieee comsoc edwin h. armstrong achievement award as well as technical achievement awards in communications theory and in wireless communications , the national academy of engineering gilbreth lecture award , the ieee comsoc and information theory society joint paper award , the ieee comsoc best tutorial paper award , the alfred p. sloan fellowship , the wice outstanding achievement award , and the silicon valley / san jose business journal s women of influence award . she is author of the book `` wireless communications '' and co - author of the books `` mimo wireless communications '' and `` principles of cognitive radio , '' all published by cambridge university press , as well as an inventor on 28 patents . she received the b.s . , m.s . and ph.d . degrees in electrical engineering from u.c . berkeley . goldsmith has served as editor for the ieee transactions on information theory , the journal on foundations and trends in communications and information theory and in networks , the ieee transactions on communications , and the ieee wireless communications magazine as well as on the steering committee for the ieee transactions on wireless communications . she participates actively in committees and conference organization for the ieee information theory and communications societies and has served on the board of governors for both societies . she has also been a distinguished lecturer for both societies , served as president of the ieee information theory society in 2009 , founded and chaired the student committee of the ieee information theory society , and chaired the emerging technology committee of the ieee communications society . at stanford she received the inaugural university postdoc mentoring award , served as chair of stanford s faculty senate in 2009 , and currently serves on its faculty senate , budget group , and task force on women and leadership . h. vincent poor ( s72 , m77 , sm82 , f87 ) received the ph.d . degree in electrical engineering and computer science from princeton university in 1977 . from 1977 until 1990 , he was on the faculty of the university of illinois at urbana - champaign . since 1990 he has been on the faculty at princeton , where he is the dean of engineering and applied science , and the michael henry strater university professor of electrical engineering . he has also held visiting appointments at several other institutions , most recently at imperial college and stanford . his research interests are in the areas of information theory , stochastic analysis and statistical signal processing , and their applications in wireless networks and related fields such as smart grid . among his publications in these areas is the recent book mechanisms and games for dynamic spectrum allocation ( cambridge university press , 2014 ) . poor is a member of the national academy of engineering and the national academy of sciences , and is a foreign member of the royal society . he is also a fellow of the american academy of arts and sciences and of other national and international academies . he received a guggenheim fellowship in 2002 and the ieee education medal in 2005 . recent recognition of his work includes the 2014 ursi booker gold medal , the 2015 eurasip athanasios papoulis award , the 2016 john fritz medal , and honorary doctorates from aalborg university , aalto university , hkust , and the university of edinburgh . first , for a matrix @xmath246 with a full structural rank , we define an equivalent term , `` a non - zero permuted diagonal '' , for a set of @xmath247 independent entries ( cf . definition [ indent ] ) . this term is based on the following intuition : for example , for @xmath161 , a non - zero permuted diagonal ( i.e. , a set of @xmath3 independent entries ) corresponds to a permutation function @xmath248 , such that @xmath249 . for a @xmath253 submatrix of @xmath51 with a non - zero permuted diagonal , we denote it by @xmath254 . we denote the set of row indices of @xmath51 that are selected in forming @xmath254 by @xmath255 , and similarly the set of selected column indices by @xmath256 : @xmath257 clearly , if @xmath258 , @xmath254 is of full rank from lemma [ leml ] . now , consider the case that @xmath259 , where @xmath260 . in other words , @xmath88 denotes the common indices that appear in both the row indices @xmath261 and the column indices @xmath262 , @xmath263 denotes the indices that appear in @xmath261 but not in @xmath262 , and @xmath264 denotes the indices that appear in @xmath262 but not in @xmath261 . wlog , @xmath254 has the form as in figure [ ijk11 ] , in which the common row and column indices @xmath88 are located in the upper left part of @xmath254 , and @xmath254 consists of four blocks @xmath265 . since @xmath254 has a non - zero permuted diagonal , there exists a permutation function @xmath266 , such that @xmath267 . in other words , the mapping @xmath268 forms a bijection between @xmath269 and @xmath270 , such that @xmath271 we now consider the following two cases : wlog , assume that the entry @xmath274 in @xmath273 is on a non - zero permuted diagonal of @xmath254 , i.e. , @xmath275 . as in figure [ ijk12 ] , we partition @xmath254 into @xmath276 , [ 1:t ] } , \bm{b}'_{t+1 , [ 1:t ] } , \bm{b}'_{[1:t ] , t+1}$ ] and @xmath274 . because @xmath274 is on a non - zero permuted diagonal of @xmath254 , the @xmath277 submatrix @xmath276 , [ 1:t]}$ ] has a non - zero permuted diagonal . from the induction assumption , @xmath276 , [ 1:t]}$ ] is of full rank with probability one . when @xmath276 , [ 1:t]}$ ] is of full rank , let @xmath278 , [ 1:t]}\bm{b}'_{[1:t ] , t+1}.\end{aligned}\ ] ] thus , @xmath276 , t+1 } = \bm{b}'_{[1:t ] , [ 1:t]}\bm{\alpha}$ ] . then , @xmath254 is rank - deficient if and only if @xmath279 } \bm{\alpha } \label{rankdef1}\end{aligned}\ ] ] note that , except for @xmath280 itself , there are only three other entries in the laplacian @xmath51 that are correlated with @xmath274 : @xmath281 however , as @xmath282 , _ none _ of the above three entries is selected into the submatrix @xmath254 . therefore , _ @xmath274 is independent to all other entries in @xmath254 _ , and is hence independent to @xmath283 } \bm{\alpha}$ ] . because @xmath274 is drawn from a continuous distribution , the probability that is satisfied is zero . as a result , @xmath254 is of full rank with probability one . wlog , assume that the entry @xmath287 in @xmath286 is on a non - zero permuted diagonal of @xmath254 , i.e. , @xmath288 . as in figure [ ijk21 ] , we partition @xmath254 into @xmath289 } , \bm{b}'_{[2:t+1 ] , [ 1:t ] } , \bm{b}'_{1 , t+1}$ ] and @xmath290,t+1}$ ] . because @xmath287 is on a non - zero permuted diagonal of @xmath254 , the @xmath277 submatrix @xmath290 , [ 1:t]}$ ] has a non - zero permuted diagonal . from the induction assumption , @xmath290 , [ 1:t]}$ ] is of full rank with probability one . when @xmath290 , [ 1:t]}$ ] is of full rank , let @xmath291 , [ 1:t]}\bm{b}'_{[2:t+1 ] , t+1}. \label{alphadef}\end{aligned}\ ] ] thus , @xmath290 , t+1 } = \bm{b}'_{[2:t+1 ] , [ 1:t]}\bm{\alpha}$ ] . then , @xmath254 is rank - deficient if and only if @xmath292 } \bm{\alpha } \label{rankdef2}\end{aligned}\ ] ] note that @xmath293 . we have @xmath294 where @xmath295 , and @xmath296 is _ independent _ to @xmath287 . substitute for @xmath297 in , we have @xmath298 note that @xmath287 is _ independent to @xmath299 , and independent to the right hand side of _ . because @xmath287 is drawn from a continuous distribution , if @xmath300 , the probability ( conditioned on @xmath300 ) that is satisfied is zero . next , we prove that the probability of @xmath301 is zero . from , if @xmath301 , @xmath302 , t+1 } = \sum_{j=1}^t\alpha_j\bm{b}'_{[2:t+1 ] , j } { \nonumber}\\ \leftrightarrow~ & \bm{b}'_{[2:t+1 ] , 1 } + \bm{b}'_{[2:t+1 ] , t+1 } = \sum_{j=2}^t\alpha_j\bm{b}'_{[2:t+1 ] , j}.\end{aligned}\ ] ] thus , @xmath301 implies that @xmath290 , 1 } + \bm{b}'_{[2:t+1 ] , t+1}$ ] is in the range space of @xmath290 , [ 2:t]}$ ] . note that , all the entries in the two vectors @xmath290 , 1}$ ] and @xmath290 , t+1}$ ] are mutually independent non - diagonal entries of @xmath51 . now , consider that we make the following _ change of distributions _ of certain entries in @xmath254 ( and also @xmath51 correspondingly ) : * as in figure [ ijk22 ] , @xmath307 , the _ only _ entry in @xmath290 , [ 2:t]}$ ] that is correlated with @xmath308 and @xmath309 is @xmath310 where @xmath311 . note that @xmath312 depends on @xmath313 only via the _ sum _ @xmath305 . * @xmath314 , @xmath308 and @xmath309 are independent to all the entries in @xmath290 , [ 2:t]}$ ] . the joint distribution of @xmath290 , 1}$ ] and @xmath290 , [ 2:t]}$ ] after the change of distributions is equal to the joint distribution of @xmath290 , 1 } + \bm{b}'_{[2:t+1 ] , t+1}$ ] and @xmath290 , [ 2:t]}$ ] before the change . we now note that , after the change of distributions , the @xmath277 matrix @xmath290 , [ 1:t ] } = [ \bm{b}'_{[2:t+1 ] , 1 } ~\bm{b}'_{[2:t+1 ] , [ 2:t]}]$ ] still satisfies the induction assumption , and is hence of _ full rank with probability one_. thus , before the change of distributions , @xmath290 , 1 } + \bm{b}'_{[2:t+1 ] , t+1}$ ] falls in the range space of @xmath290 , [ 2:t]}$ ] _ with probability zero_. therefore , @xmath301 with probability zero , hence the probability that is satisfied is zero . as a result , @xmath254 is of full rank with probability one .
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physical security of power networks under power injection attacks that alter generation and loads is studied .
the system operator employs phasor measurement units ( pmus ) for detecting such attacks , while attackers devise attacks that are _ unobservable _ by such pmu networks .
it is shown that , given the pmu locations , the solution to finding the sparsest unobservable attacks has a simple form with probability one , namely , @xmath0 , where @xmath1 is defined as the vulnerable vertex connectivity of an augmented graph .
the constructive proof allows one to find the entire set of the sparsest unobservable attacks in polynomial time .
furthermore , a notion of the potential impact of unobservable attacks is introduced . with optimized pmu deployment
, the sparsest unobservable attacks and their potential impact as functions of the number of pmus are evaluated numerically for the ieee 30 , 57 , 118 and 300-bus systems and the polish 2383 , 2737 and 3012-bus systems .
it is observed that , as more pmus are added , the maximum potential impact among all the sparsest unobservable attacks drops quickly until it reaches the minimum sparsity .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
as a complement to detailed simulations of wolf - rayet ( wr ) winds ( e.g. , * ? ? ? * ; * ? ? ? * ) , we have developed a set of simple analytic models using diffusive cak - type line driving with frequency redistribution @xcite . in this paper , we apply these models to a preliminary study of the @xmath0 relation for wr stars , which is important for the study of wr stars as progenitors of long - duration gamma - ray bursts @xcite . the details of the basic model are discussed in @xcite . we compare our results with those from @xcite ( hereafter vdk ) using their wn star parameters ( see figure [ vdfig ] ) . the metal abundances were provided by a. heger ( private communication ) from a 25 @xmath5 evolution model based on solar abundances from @xcite . while the @xmath2 from these abundances is smaller than the traditional @xmath6 , uncertainties in other model parameters are expected to dominate over uncertainties in abundance . figure [ vdfig ] shows the metallicity dependence of our wnl model , as compared to the model in vdk . a least - squares fit to our result over @xmath3 gives @xmath7 , where @xmath8 , very close to the vdk result of @xmath9 . it appears that our @xmath1 flattens more quickly than those of vdk , perhaps because fe saturates more quickly with our more complete line list . calculations to higher @xmath2 need to be done to confirm that there is indeed a flattening , however . log @xmath1 vs. log @xmath10 for a typical wnl star ( @xmath11 ) . plusses and lines are from @xcite , figure 2 . crosses are from this paper . ] we have applied analytic wr wind models to the study of @xmath1 as a function of @xmath2 . our preliminary results for a wnl - type star show a power - law dependence of @xmath1 with @xmath2 over @xmath12 , with index @xmath8 , similar to the the findings vdk . however , our @xmath1 seems to flatten more quickly . we plan to cover a wider range in @xmath2 to confirm this and to perform the same analysis for wc stars . a. onifer would like to thank alexander heger for helpful discussions and valuable data . portions of this work were performed under the auspices of the u.s . department of energy by los alamos national laboratory under contract no .
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to better understand wolf - rayet stars as progenitors of gamma - ray bursts , an understanding of the effect metallicity has on wolf - rayet mass loss is needed . using simple analytic models , we study the @xmath0 relation of a wn star and compare the results to similar models .
we find that @xmath1 roughly follows a power law in @xmath2 with index 0.88 from @xmath3 and appears to flatten by @xmath4 .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the current tally of gravitationally lensed quasars now stands at nearly 70 ( kochanek et al . the various searches which have contributed to this total have naturally also discovered examples of the apparently rarer phenomenon of binary quasars , of which only @xmath1 systems have been documented ( mortlock , webster , & francis 1999 ; kochanek et al . 2002 ) . while the gravitationally lensed quasars have been intensively studied , binary quasars have not received nearly as much attention , even though binary quasars may provide invaluable insight into various aspects of the quasar phenomenon . of particular importance are the cases where the pair members exhibit strikingly different characteristics , perhaps making it possible to deduce what aspects of the agn environment are responsible for presently little - understood quasar behavior such as radio - loudness , broad or narrow absorption lines , or very red colors . in addition , physically close binaries also supply limits , statistically at least , on the timescales involved because once within @xmath2 galaxy radius ( @xmath3kpc ) , dynamical considerations limit the binary lifetimes ( mortlock et al . 1999 ) . in an ongoing infrared imaging survey of quasars at keck observatory , we have discovered that sdss j233646.2 - 010732.6 ( hereafter sdss 2336 - 0107 ) is a double with a separation of 167 . resolved spectra show that component a is a standard quasar with a blue continuum and broad emission lines . component b is a broad absorption line ( bal ) quasar , specifically , a bal qso with prominent absorption from and metastable , making it a member of the `` felobal '' class ( becker et al . 1997 ; 2000 ) . the number of known felobals has increased dramatically in the last five years ( becker et al . 2000 ; menou et al . 2001 ; hall et al . 2002 ) , including a gravitationally lensed example ( lacy et al . 2002 ) and the binary discussed here , suggesting that this type of object , once thought to be rare , may in fact be fairly common and simply overlooked in most quasar surveys . the presence of this bal quasar in a relatively small separation binary adds to the mounting evidence that the bal phenomenon is not simply due to viewing a normal quasar at a specific orientation , but rather that bals are an evolutionary phase in the life of many , if not all , quasars , and is associated with conditions found in interacting systems . we adopt @xmath4 , @xmath5 , and @xmath6 . sdss 2336 - 0107 was first identified by the sloan digital sky survey ( sdss ; york et al . 2000 ) using an early version of the quasar target selection algorithm ( richards et al . 2002 ) , as a quasar with z=1.285 and having modest broad absorption features ( schneider et al . it was also earmarked as an `` extended '' quasar in the sdss images , meaning that it was slightly resolved . quasars at @xmath7 are unresolved in sdss images , so the extended nature made it a target in our imaging survey for lensed quasars at keck observatory . on 2001 october 30 , we obtained deep imaging of sdss 2336 - 0107 , totaling 350s in the @xmath8 band using the near infrared camera ( nirc , matthews & soifer 1994 ) on the keck i 10 m telescope in 06 seeing . the image scale is 015 per pixel . the reduced image reveals the double nature of sdss 2336 - 0107 ( figure 1 ) and three galaxies ( g1 , g2 , and g3 ) in the field . photometry of the two components was measured with daophot / iraf , using a bright standard star as a psf model . the short exposure standard star psf clearly differed from the psf of the longer exposure , dithered images of the binary quasar , so we also used component b as a psf model , even though the wings of the psf can not be determined easily in this way . in both cases , the residuals after subtraction suggest that the shapes of two components are not quite identical and that there is extended diffuse light at @xmath8 around a , perhaps indicating that the data are just beginning to detect the host galaxies of one or both . the separation of the two components is determined to be 1673 in @xmath8 and 1681 in @xmath9 . the sdss images were obtained in relatively poor seeing , ranging from 25 in @xmath10 to 16 in @xmath11 . despite the seeing , the images prove adequate for photometry of the two components using daophot / iraf , a nearby bright star serving as the psf model and photometric zeropoint . in @xmath12 , the psf fits yield relatively small photometric errors ; the separations range from 15 to 17 , averaging @xmath13 , consistent to a fraction of an sdss pixel ( 0396 ) with the separation measured in the nirc images , lending credibility to the extracted photometry . the redder component b is not reliably detected in @xmath10 , but an upper limited is derived from the residuals after subtracting a single psf at the location of component a. table 1 lists the optical and ir photometry , corrected for galactic reddening ( schlegel , finkbeiner , & davis 1998 ) . we obtained resolved spectra of the two components of sdss 2336 - 0107 in 2002 january , using the echelle spectrograph and imager ( esi ; epps & miller 1998 ) on the keck ii telescope . the seeing was again 06 . the esi has a dispersion of 0.15 to 0.3 per pixel over a wavelength range of 4000 to 10500 , and the 1 slit used for these observations projects to 6.5 pixels . the 900 s exposure was obtained at an airmass of 1.33 , with the slit aligned at the position angle of the components , 958 . the spectra show that component a is a standard quasar with broad emission lines while b is found to be a bal quasar , totally lacking any emission features ( figure 2 ) . the shape of the overall spectral energy distributions have been corrected to agree with the photometry from the sdss images . the redshift of component a is @xmath14 from , and @xmath15 from ] ; as ] can be contaminated by , ] , and emission , we adopt the redshift . the redshift of component b is difficult to measure accurately because of the lack of emission features and the broad nature of the absorption . if we assume that the absorption feature at 6418.8 , which is somewhat broad and flat - bottomed , is due to the red half the the doublet at 2803.5 , the redshift is 1.2895 . there is also a narrow absorption line at 6525.0 which , ascribed to 2853.0 , gives @xmath16 , consistent with and perhaps more accurate than the estimate . adopting this as the redshift of component b yields an apparent velocity difference between a and b of only @xmath17 . component b also has absorption features consistent with ca h & k at a redshift of 1.2843 ( figure 2 ) . the esi spectrum of component a reveals a @xmath18 intervening absorption system at @xmath19 , perhaps due to one of the faint galaxies which appear in the @xmath8-band image ( figure 1 ) . there is also absorption from what may be 2515 and 2722 . the brighter galaxy ( g1 ) is 125 to the east of a while the fainter ( g2 ) is 230 to the west , corresponding to physical impact parameters of 9.4 and 17.4 kpc , respectively . also weakly detected in the spectrum of component a is [ ] in emission at a wavelength corresponding to @xmath20 ( figure 2 ) , within 400 of the emission features . this feature is in a region between bright sky lines and also appears on a shorter esi exposure of sdss 2336 - 0107 taken two nights previous to the data presented here , so its reality is beyond doubt . though noisy , the [ ] feature does appear to be broadened by a few hundred , well in excess of the instrumental resolution . combining the redshifts with the sdss and nirc photometry , the absolute magnitudes of the two components can be computed in the quasar restframe as @xmath21 and @xmath22 . while not exceptionally bright , both components are luminous enough to qualify as bona fide quasars . the possibility that sdss 2336 - 0107 is gravitationally lensed is immediately ruled out by the strikingly different spectra ( figure 2 ) and the short time delay between the two paths . the analytic formula of witt , mao , & keeton ( 2000 ) yields time delay estimates of a few months to a year for any reasonable lensing geometry . the nearly identical redshifts and tiny velocity difference , however , makes it a definite binary quasar . for our adopted cosmology , the apparent separation of 167 translates to a physical separation of @xmath23 kpc , making sdss 2336 - 0107 the smallest separation binary ever found in ground - based imaging and the second smallest ever . the smallest is lbqs 0103 - 2753 ( junkkarinen et al . 2001 ) , a 03 ( 2.3 kpc ) separation binary found serendipitously during an ultraviolet spectroscopy survey of bal systems using hst and stis . in lbqs 0103 - 2753 , the brighter component is a standard high ionization bal ( hibal ) quasar . the two binary quasars with the smallest physical separations both contain a bal member . although this could be simply small number statistics , it is natural to speculate that the bal features are somehow induced by the small distances between the agn . mass transfer onto one of the black holes would naturally result in a tidally interacting pair of galaxies where the agns had reached relatively small separations of less than a galactic radius . this is further evidence supporting the view that bal features are a short - lived evolutionary phase during the life of a quasar ( becker et al . 2000 ; gregg et al . 2000 ) , rather than the result of viewing an agn along a particular line of sight . additional support of this argument comes from canalizo & stockton ( 2001 ) who show that all four of the known low ionization bal ( lobal ) quasars with @xmath24 are in systems which exhibit tidal tails or evidence of merging , suggesting a strong link between these phenomena and the presence of bal spectral features . the orientation picture can not so easily account for the occurrence of bal features in the two smallest separation binary quasars or all four low redshift lobal examples , except as by chance . if the merger / accretion origin of bals is correct , the material directly responsible for the bal spectral features is almost certainly within tens of parsecs of the agn and not within reach of ground - based imaging resolution , yet mergers and tidal debris will occur over kiloparsec scales . it is then somewhat counter - intuitive that it is the standard quasar component ( a ) of sdss j2336 - 0107 which is sandwiched by the galaxies g1 and g2 in the nirc image ( figure 1 ) and which exhibits the narrow [ ] in its spectrum ( figure 2 ) . the @xmath8 image hints at a light bridge between g1 and component a , perhaps a real physical connection . the weak [ ] feature of component a is certainly consistent with this picture , indicating some level of star formation , quite expected for an intensely interacting system . some or all of the [ ] could come from the material in the region between g1 and a ; the 1 esi slit would have entirely missed g1 , though . the intervening system in the spectrum of a ( figure 2 ) could arise from g2 at an intervening location ; direct spectroscopy of the galaxies in this field are required to sort out the physical relationships . the @xmath8 seeing and s / n are not good enough to explore the region directly between the two components more fully , but space - based imaging would reveal details of the expected tidal interactions which may be driving the bal behavior . if the two quasars are in the nuclei of two large galaxies separated by only @xmath23 kpc , tidal effects should be strongly pronounced . figure 3 plots the photometry of the two components , @xmath10 through @xmath8 , highlighting the contrasting color difference . component b , the bal , is in fact brighter in the redder bands , by @xmath25 magnitude in @xmath9 and @xmath8 . for comparison , we plot photometry synthesized from the sdss quasar composite ( vanden berk , et al . 2001 ) after redshifting to z=1.2855 . the synthetic magnitudes have been adjusted so that @xmath11 equals that of component a. the excess flux in @xmath26 in the composite may be due to host starlight contribution in the low redshift quasar spectra which contribute to the composite in this wavelength region . the even larger excess in the bal component is probably a combination of starlight and `` back warming '' from dust extinction . there is evidence that felobals are heavily reddened ( hall et al . 1997 ; najita , dey , & brotherton 2000 ) and may be associated with young galaxies and star formation ( egami et al . 1996 ) . perhaps the most intriguing finding is that component b is an felobal , a quasar with low ioniziation broad absorption from plus broad and narrow absorption from metastable states of . in figure 4 , the spectrum of component b is compared to two other felobals , j1556 + 3517 ( becker et al . 1997 ) and the original of the class , q0059 - 2735 ( hazard et al . the spectrum of sdss 2336 - 0107 b exhibits strong at @xmath27 and 2750 . the broad absorption depresses the spectrum blueward to at least 2630 , or -18,000 in the quasar restframe . while not quite as red as j1556 + 3517 , sdss 2336 - 0107 b has essentially no emission lines whatsoever . there is now one felobal quasar ( sdss 2336 - 0107 ) known among the 15 binary qso systems and one felobal ( j1004 + 1229 ; lacy et al . 2002 ) among the @xmath28 known gravitationally lensed quasars . these statistics argue strongly that felobals must be much more common than their presently cataloged population suggests . lacy et al . ( 2002 ) discuss the details of red quasar luminosity functions and lensing probabilities , but not with regard to felobals in particular . with 1 felobal in @xmath28 lensed quasars and one among 40 quasars in @xmath29 binary systems , the simplest explanation is that from 1% to a few percent of quasars are felobals , or alternatively , that the felobals phase lasts for a similar percentage of the total quasar lifetime . the very red spectral energy distribution and weak or completely absent broad emission lines ( figure 1 ) makes these objects extremely difficult to find , especially at redshifts @xmath30 where the absorption edge moves into the @xmath9 band . an ir - selected sample of quasars is needed to better gauge the frequency of felobal numbers , but evidence is mounting ( becker et al . 1997 ; lacy et al . 2002 ; hall et al . 2002 ) that they are much more common than suspected even just a few years ago when the rather blue example of q0059 - 2735 was the only one known ( hazard et al . a deep imaging survey of bal quasars and a control sample of non - bal quasars , from the ground or space , is needed to document the frequency of each in systems that show tidal interactions , which would be a more comprehensive version of the work done by canalizo & stockton ( 2001 ) . if bal - ness is explained simply by viewing an ordinary quasar along a line of sight which skims the `` dusty torus '' ( weymann et al . 1991 ) , then the two populations should appear with equal frequency in merging or interacting systems . if bal features are produced by conditions created in mergers , and orientation plays less or no role , then they will be more common in chaotic systems with tidal tails and other signs of host galaxy interactions . dynamical modeling of the tidal effects may help constrain the timescales of various quasar phases . felobals may be a common feature of the agn landscape , especially if there are seyfert - type luminosity class analogs to the brighter objects now being unearthed . if felobals are indeed common , then infrared sky surveys which reach deeper than 2mass , such as those being undertaken with sirtf , will turn up numerous examples along with other extreme bal phenomena in quasar and galaxy luminosity objects . the felobal - like hawaii 167 ( cowie et al . 1994 ) was discovered in just such a survey . such surveys are needed to provide more accurate estimates of the extremely red quasar population , especially at redshifts above 2.5 where such objects are essentially invisible in the optical . we thank mark lacy for helpful discussions . the authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of mauna kea has always had within the indigenous hawaiian community . we are most fortunate to have the opportunity to conduct observations from this mountain . part of the work reported here was done at the institute of geophysics and planetary physics , under the auspices of the u.s . department of energy by lawrence livermore national laboratory under contract no . w-7405-eng-48 . funding for the creation and distribution of the sdss archive has been provided by the alfred p. sloan foundation , the participating institutions , the national aeronautics and space administration , the national science foundation , the u.s . department of energy , the japanese monbukagakusho , and the max planck society . the sdss web site is http://www.sdss.org/. llllllrrrrrrl component a & @xmath31 & @xmath32 & 0.114 & 1.2853 & 19.75 & 19.65 & 19.35 & 19.26 & 19.12 & 18.19 & 17.22 & -23.77 + component b & @xmath33 & @xmath34 & 0.114 & 1.2871 & @xmath35 & 21.37 & 19.71 & 18.94 & 18.68 & 17.24 & 16.23 & -24.72 +
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in an ongoing infrared imaging survey of quasars at keck observatory , we have discovered that the @xmath0 quasar sdss j233646.2 - 010732.6 comprises two point sources with a separation of 167 .
resolved spectra show that one component is a standard quasar with a blue continuum and broad emission lines ; the other is a broad absorption line ( bal ) quasar , specifically , a bal qso with prominent absorption from and metastable , making it a member of the `` felobal '' class .
the number of known felobals has recently grown dramatically from a single example to more than a dozen , including a gravitationally lensed example and the binary member presented here , suggesting that this formerly rare object may be fairly common .
additionally , the presence of this bal quasar in a relatively small separation binary adds to the growing evidence that the bal phenomenon is not due to viewing a normal quasar at a specific orientation , but rather that it is an evolutionary phase in the life of many , if not all , quasars , and is particularly associated with conditions found in interacting systems .
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we study scaling limits of _ internal diffusion limited aggregation _ ( `` internal dla '' ) , a growth model introduced in @xcite . in internal dla , one inductively constructs an * occupied set * @xmath8 for each time @xmath9 as follows : begin with @xmath10 and @xmath11 , and let @xmath12 be the union of @xmath13 and the first place a random walk from the origin hits @xmath14 . the purpose of this paper is to study the growing family of sets @xmath13 . following the pioneering work of @xcite , it is by now well known that , for large @xmath1 , the set @xmath13 approximates an origin - centered euclidean lattice ball @xmath15 ( where @xmath16 is such that @xmath17 has volume @xmath1 ) . the authors recently showed that this is true in a fairly strong sense @xcite : the maximal distance from a point where @xmath18 is non - zero to @xmath19 is a.s . @xmath2 if @xmath3 and @xmath4 if @xmath5 . in fact , if @xmath20 is large enough , the probability that this maximal distance exceeds @xmath21 ( or @xmath22 when @xmath5 ) decays faster than any fixed ( negative ) power of @xmath1 . some of these results are obtained by different methods in @xcite . this paper will ask what happens if , instead of considering the maximal distance from @xmath19 at time @xmath1 , we consider the `` average error '' at time @xmath1 ( allowing inner and outer errors to cancel each other out ) . it turns out that in a distributional `` average fluctuation '' sense , the set @xmath13 deviates from @xmath17 by only a constant number of lattice spaces when @xmath23 and by an even smaller amount when @xmath5 . appropriately normalized , the fluctuations of @xmath13 , taken over time and space , define a distribution on @xmath24 that converges in law to a variant of the gaussian free field ( gff ) : a random distribution on @xmath24 that we will call the * augmented gaussian free field*. ( it can be constructed by defining the gff in spherical coordinates and replacing variances associated to spherical harmonics of degree @xmath25 by variances associated to spherical harmonics of degree @xmath26 ; see [ ss.augmentedgff ] . ) the `` augmentation '' appears to be related to a damping effect produced by the mean curvature of the sphere ( as discussed below ) . , with particles started uniformly on @xmath27 . though we do not prove this here , we expect the cluster boundaries to be approximately flat cross - sections of the cylinder , and we expect the fluctuations to scale to the _ ordinary _ gff on the half cylinder as @xmath28 . ] to our knowledge , no central limit theorem of this kind has been previously conjectured in either the physics or the mathematics literature . the appearance of the gff and its `` augmented '' variants is a particular surprise . ( it implies that internal dla fluctuations although very small have long - range correlations and that , up to the curvature - related augmentation , the fluctuations in the direction transverse to the boundary of the cluster are of a similar nature to those in the tangential directions . ) nonetheless , the heuristic idea is easy to explain . before we state the central limit theorems precisely ( [ ss.twostatement ] and [ ss.generalstatement ] ) , let us explain the intuition behind them . write a point @xmath29 in polar coordinates as @xmath30 for @xmath31 and @xmath32 on the unit sphere . suppose that at each time @xmath1 the boundary of @xmath13 is approximately parameterized by @xmath33 for a function @xmath34 defined on the unit sphere . write @xmath35 where @xmath36 is the volume of the unit ball in @xmath24 . the @xmath37 term measures the deviation from circularity of the cluster @xmath13 in the direction @xmath32 . how do we expect @xmath38 to evolve in time ? to a first approximation , the angle at which a random walk exits @xmath13 is a uniform point on the unit sphere . if we run many such random walks , we obtain a sort of poisson point process on the sphere , which has a scaling limit given by space - time white noise on the sphere . however , there is a smoothing effect ( familiar to those who have studied the continuum analog of internal dla : the famous hele - shaw model for fluid insertion , see the reference text @xcite ) coming from the fact that places where @xmath38 is small are more likely to be hit by the random walks , hence more likely to grow in time . there is also secondary damping effect coming from the mean curvature of the sphere , which implies that even if ( after a certain time ) particles began to hit all angles with equal probability , the magnitude of @xmath38 would shrink as @xmath1 increased and the existing fluctuations were averaged over larger spheres . the white noise should correspond to adding independent brownian noise terms to the spherical fourier modes of @xmath38 . the rate of smoothing / damping in time should be approximately given by @xmath39 for some linear operator @xmath40 mapping the space of functions on the unit sphere to itself . since the random walks approximate brownian motion ( which is rotationally invariant ) , we would expect @xmath40 to commute with orthogonal rotations , and hence have spherical harmonics as eigenfunctions . with the right normalization and parameterization , it is therefore natural to expect the spherical fourier modes of @xmath38 to evolve as independent brownian motions subject to linear `` restoration forces '' ( a.k.a . ornstein - uhlenbeck processes ) where the magnitude of the restoration force depends on the degree of the corresponding spherical harmonic . it turns out that the restriction of the ( ordinary or augmented ) gff on @xmath24 to a centered volume @xmath1 sphere evolves in time @xmath1 in a similar way . of course , as stated above , the `` spherical fourier modes of @xmath38 '' have not really been defined ( since the boundary of @xmath13 is complicated and generally _ can not _ be parameterized by @xmath41 ) . in the coming sections , we will define related quantities that ( in some sense ) encode these spherical fourier modes and are easy to work with . these quantities are the martingales obtained by summing discrete harmonic polynomials over the cluster @xmath13 . the heuristic just described provides intuitive interpretations of the results given below . theorem [ t.fluctuations ] , for instance , identifies the weak limit as @xmath42 of the internal dla fluctuations from circularity at a fixed time @xmath1 : the limit is the two - dimensional augmented gaussian free field restricted to the unit circle @xmath43 , which can be interpreted in a distributional sense as the random fourier series @xmath44\ ] ] where @xmath45 for @xmath46 and @xmath47 for @xmath48 are independent standard gaussians . the ordinary two - dimensional gff restricted to the unit circle is similar , except that @xmath49 is replaced by @xmath50 . the series unlike its counterpart for the one - dimensional gaussian free field , which is a variant of brownian bridge is a.s . divergent , which is why we use the dual formulation explained in [ ss.generalstatement ] . the dual formulation of amounts to a central limit theorem , saying that for each @xmath48 the real and imaginary parts of @xmath51 converge in law as @xmath52 to normal random variables with variance @xmath53 ( and that @xmath54 and @xmath55 are asymptotically uncorrelated for @xmath56 ) . see @xcite for numerical data on the moments @xmath55 in large simulations . before we set about formulating our central limit theorems precisely , we mention a previously overlooked fact . suppose that we run internal dla in continuous time by adding particles at poisson random times instead of at integer times : this process we will denote by @xmath57 ( or often just @xmath58 ) where @xmath59 is the counting function for a poisson point process in the interval @xmath60 $ ] ( so @xmath59 is poisson distributed with mean @xmath1 ) . we then view the entire history of the idla growth process as a ( random ) function on @xmath61 , which takes the value @xmath62 or @xmath63 on the pair @xmath64 accordingly as @xmath65 or @xmath66 . write @xmath67 for the set of functions @xmath68 such that @xmath69 whenever @xmath70 , endowed with the coordinate - wise partial ordering . let @xmath71 be the distribution of @xmath72 , viewed as a probability measure on @xmath67 . [ t.fkg ] _ ( fkg inequality ) _ for any two increasing functions @xmath73 , the random variables @xmath74 and @xmath75 are nonnegatively correlated . one example of an increasing function is the total number @xmath76 of occupied sites in a fixed subset @xmath77 at a fixed time @xmath1 . one example of a decreasing function is the smallest @xmath1 for which all of the points in @xmath78 are occupied . intuitively , theorem [ t.fkg ] means that on the event that one point is absorbed at a late time , it is conditionally more likely for all other points to be absorbed late . the fkg inequality is an important feature of the discrete and continuous gaussian free fields @xcite , so it is interesting ( and reassuring ) that it appears in internal dla at the discrete level . note that sampling a continuous time internal dla cluster at time @xmath1 is equivalent to first sampling a poisson random variable @xmath79 with expectation @xmath1 and then sampling an ordinary internal dla cluster of size @xmath79 . ( by the central limit theorem , @xmath80 has order @xmath81 with high probability . ) although using continuous time amounts to only a modest time reparameterization ( chosen independently of everything else ) it is aesthetically natural . our use of `` white noise '' in the heuristic of the previous section implicitly assumed continuous time . ( otherwise the total integral of @xmath38 would be deterministic , so the noise would have to be conditioned to have mean zero at each time . ) [ cols="^,^ " , ] the restriction to harmonic @xmath82 ( as opposed to a more general test function @xmath83 ) seems to be necessary in large dimensions because otherwise the derivative of the test function along @xmath43 appears to have a non - trivial effect on ( see [ ss.fixedtimeproof ] ) . this is because has a lot of positive mass just outside of the unit sphere and a lot of negative mass just inside the unit sphere . it may be possible to formulate a version of theorem [ t.highdconvergence ] ( involving some modification of the `` mean shape '' described by @xmath84 ) that uses test functions that depend only on @xmath32 in a neighborhood of the sphere ( instead of using only harmonic test functions ) , but we will not address this point here . deciding whether theorem [ t.lateness ] as stated extends to higher dimensions requires some number theoretic understanding of the extent to which the discrepancies between @xmath84 and @xmath85 ( as well as the errors that come from replacing a @xmath86 with a smooth test function @xmath83 ) average out when one integrates over a range of times . we will not address these points here either . we may write a general vector in @xmath24 as @xmath30 where @xmath87 and @xmath88 . we write the laplacian in spherical coordinates as @xmath89 a polynomial @xmath90 $ ] is called _ harmonic _ if @xmath91 is the zero polynomial . let @xmath92 denote the space of all homogenous harmonic polynomials in @xmath93 $ ] of degree @xmath25 , and let @xmath94 denote the space of functions on @xmath95 obtained by restriction from @xmath92 . if @xmath96 , then we can write @xmath97 for a function @xmath98 , and setting to zero at @xmath99 yields @xmath100 i.e. , @xmath101 is an eigenfunction of @xmath102 with eigenvalue @xmath103 . note that continues to be zero if we replace @xmath25 with the negative number @xmath104 , since the expression @xmath105 is unchanged by replacing @xmath25 with @xmath106 . thus , @xmath107 is also harmonic on @xmath108 . now , suppose that @xmath101 is normalized so that @xmath109 by scaling , the integral of @xmath110 over @xmath111 is thus given by @xmath112 . the @xmath113 norm on all of @xmath114 is then given by @xmath115 a standard identity states that the dirichlet energy of @xmath101 , as a function on @xmath95 , is given by the @xmath113 inner product @xmath116 . the square of @xmath117 is given by the square of its component along @xmath95 plus the square of its radial component . we thus find that the dirichlet energy of @xmath118 on @xmath114 is given by @xmath119 now suppose that we fix the value of @xmath118 on @xmath111 as above but harmonically extend it outside of @xmath114 by writing @xmath120 for @xmath121 . then the dirichlet energy of @xmath118 outside of @xmath114 can be computed as @xmath122 which simplifies to @xmath123 combining the inside and outside computations in the case @xmath124 , we find that the harmonic extension @xmath125 of the function given by @xmath101 on @xmath95 has dirichlet energy @xmath126 . if we decompose the gff into an orthonormal basis that includes this @xmath125 , we find that the component of @xmath125 is a centered gaussian with variance @xmath127 . if we replace @xmath125 with the harmonic extension of @xmath128 ( defined on @xmath111 ) , then by scaling the corresponding variance becomes @xmath129 . now in the augmented gff the variance is instead given by , which amounts to replacing @xmath127 with @xmath130 . considering the component of @xmath128 in a basis expansion the space of functions on @xmath111 requires us to divide by @xmath131 ( to account for the scaling of @xmath118 ) and by @xmath132 ( to account for the larger integration area ) , so that we again obtain a variance of @xmath133 for the augmented gff , versus @xmath129 for the gff . in some ways , the augmented gff is very similar to the ordinary gff : when we restrict attention to an origin - centered annulus , it is possible to construct independent gaussian random distributions @xmath134 , @xmath135 , and @xmath136 such that @xmath134 has the law of a constant multiple of the gff , @xmath137 has the law of the augmented gff , and @xmath138 has the law of the ordinary gff . in light of theorem [ t.fluctuations ] , the following implies that ( up to absolute continuity ) the scaling limit of fixed - time @xmath13 fluctuations can be described by the gff itself . [ p.abscont ] when @xmath3 , the law @xmath139 of the restriction of the gff to the unit circle ( modulo additive constant ) is absolutely continuous w.r.t . the law @xmath140 of the restriction of the augmented gff restricted to the unit circle . the relative entropy of a gaussian of density @xmath141 with respect to a gaussian of density @xmath142 is given by @xmath143 note that @xmath144 , and in particular @xmath145 . thus the relative entropy of a centered gaussian of variance @xmath62 with respect to a centered gaussian of variance @xmath146 is @xmath147 . this implies that the relative entropy of @xmath140 with respect to @xmath139 restricted to the @xmath148th component @xmath149 is @xmath150 . the same holds for the relative entropy of @xmath139 with respect to @xmath140 . because the @xmath149 are independent in both @xmath140 and @xmath139 , the relative entropy of one of @xmath140 and @xmath139 with respect to the other is the sum of the relative entropies of the individual components , and this sum is finite . we recall that increasing functions of a poisson point process are non - negatively correlated @xcite . ( this is easily derived from the more well known statement @xcite that increasing functions of independent bernoulli random variables are non - negatively correlated . ) let @xmath140 be the simple random walk probability measure on the space @xmath151 of walks @xmath152 beginning at the origin . then the randomness for internal dla is given by a rate - one poisson point process on @xmath153 where @xmath139 is lebesgue measure on @xmath154 . a realization of this process is a random collection of points in @xmath155 . it is easy to see ( for example , using the abelian property of internal dla discovered by diaconis and fulton @xcite ) that adding an additional point @xmath156 increases the value of @xmath57 for all times @xmath1 . the @xmath57 are hence increasing functions of the poisson point process , and are non - negatively correlated . since @xmath157 and @xmath158 are increasing functions of the @xmath57 , they are also increasing functions of the point process and are thus non - negatively correlated . let @xmath159 be a polynomial that is harmonic on @xmath24 , that is @xmath160 in this section we give a recipe for constructing a polynomial @xmath161 that closely approximates @xmath82 and is discrete harmonic on @xmath0 , that is , @xmath162 where @xmath163 is the symmetric second difference in direction @xmath164 . the construction described below is nearly the same as the one given by lovsz in @xcite , except that we have tweaked it in order to obtain a smaller error term : if @xmath82 has degree @xmath165 , then @xmath166 has degree at most @xmath167 instead of @xmath168 . discrete harmonic polynomials have been studied classically , primarily in two variables : see for example duffin @xcite , who gives a construction based on discrete contour integration . consider the linear map @xmath169 \to \r[x_1,\ldots , x_d]\ ] ] defined on monomials by @xmath170 where we define @xmath171 [ discretepolynomial ] if @xmath172 $ ] is a polynomial of degree @xmath165 that is harmonic on @xmath24 , then the polynomial @xmath173 is discrete harmonic on @xmath0 , and @xmath166 is a polynomial of degree at most @xmath167 . an easy calculation shows that @xmath174 from which we see that @xmath175 = \xi [ \frac{\partial^2}{\partial x_i^2 } \psi ] .\ ] ] if @xmath82 is harmonic , then the right side vanishes when summed over @xmath176 , which shows that @xmath177 $ ] is discrete harmonic . note that @xmath178 is even for @xmath165 even and odd for @xmath165 odd . in particular , @xmath179 has degree at most @xmath167 , which implies that @xmath180 has degree at most @xmath167 . to obtain a discrete harmonic polynomial @xmath86 on the lattice @xmath181 , we set @xmath182 where @xmath165 is the degree of @xmath82 . for each fixed @xmath82 with @xmath183 the process @xmath184 is a martingale in @xmath1 . each time a new particle is added , we can imagine that it performs brownian motion on the grid ( instead of a simple random walk ) , which turns @xmath185 into a continuous martingale , as in @xcite . by the martingale representation theorem ( see ( * ? ? ? * theorem v.1.6 ) ) , we can write @xmath186 , where @xmath187 is a standard brownian motion and @xmath188 is the quadratic variation of @xmath185 on the interval @xmath60 $ ] . to show that @xmath189 converges in law as @xmath28 to a gaussian with variance @xmath190 , it suffices to show that for fixed @xmath1 the random variable @xmath191 converges in law to @xmath192 . by standard riemann integration and the @xmath13 fluctuation bounds in @xcite ( the weaker bounds of @xcite would also suffice here ) we know that @xmath193 converges in law to @xmath194 as @xmath28 . thus it suffices to show that @xmath195 converges in law to zero . this expression is actually a martingale in @xmath1 . its expected square is the sum of the expectations of the squares of its increments , each of which is @xmath196 . the overall expected square of is thus @xmath197 , which indeed tends to zero . recall and note that if we replace @xmath1 with @xmath59 , this does not change the convergence in law of @xmath191 when @xmath183 . however , when @xmath198 , it introduces an asymptotically independent source or randomness which scales to a gaussian of variance @xmath199 ( simply by the central limit theorem for the poisson point process ) , and hence remains correct in this case . similarly , suppose we are given @xmath200 and functions @xmath201 . the same argument as above , using the martingale in @xmath1 , @xmath202 implies that @xmath203 converges in law to a gaussian with variance @xmath204 the theorem now follows from a standard fact about gaussian random variables on a finite dimensional vector spaces ( proved using characteristic functions ) : namely , a sequence of random variables on a vector space converges in law to a multivariate gaussian if and only if all of the one - dimensional projections converge . the law of @xmath205 is determined by the fact that it is a centered gaussian with covariance given by . recall that @xmath13 for @xmath206 denotes the discrete - time idla cluster with exactly @xmath1 sites , and @xmath207 for @xmath208 denotes the continuous - time cluster whose cardinality is poisson - distributed with mean @xmath1 . define @xmath209 and @xmath210 fix @xmath211 , and consider a test function of the form @xmath212 where the @xmath213 are smooth functions supported in an interval @xmath214 . we will assume , furthermore , that @xmath215 is real - valued . that is , the complex numbers @xmath213 satisfy @xmath216 it follows from lemma [ dual ] below ( with @xmath220 , @xmath221 ) , that theorem [ gffdisk0 ] can be interpreted as saying that @xmath222 tends weakly to the gaussian random distribution associated to the hilbert space @xmath223 with norm @xmath224\frac{dr}{r}\ ] ] where @xmath225 and @xmath226 . ( the subscript @xmath227 means nonradial : @xmath223 is the orthogonal complement of radial functions in the sobolev space @xmath228 . ) [ dual ] let @xmath229 and let @xmath82 be a real valued function on @xmath230 . denote @xmath231 where the supremum is over real - valued @xmath118 , compactly supported and subject to the constraint @xmath232 then @xmath233 in the case @xmath234 , replace @xmath118 in @xmath235 with @xmath236 change order of integration and apply the cauchy - schwarz inequality . for the case @xmath237 , multiply by the appropriate factors @xmath238 and @xmath239 to deduce this from the case @xmath234 . theorem [ gffdisk1 ] is a restatement of theorem [ t.lateness ] . ( as in the proof of theorem [ t.highdconvergence ] , the convergence in law of all one - dimensional projections to the appropriate normal random variables implies the corresponding result for the joint distribution of any finite collection of such projections . ) theorem [ gffdisk1 ] says that @xmath245 tends weakly to a gaussian distribution for the hilbert space @xmath228 with the norm @xmath246\frac{dr}{r}.\ ] ] by way of comparison , the usual gaussian free field is the one associated to the dirichlet norm @xmath247\frac{dr}{r}.\ ] ] comparing these two norms , we see that the second term in @xmath248 has an additional @xmath249 , hence our choice of the term `` augmented gaussian free field . '' as derived in [ ss.augmentedgff ] , this @xmath249 results in a smaller variance @xmath250 in each spherical mode of degree @xmath25 of the augmented gff , as compared to @xmath251 for the usual gff . the surface area of the sphere is implicit in the normalization , and is accounted for here in the factors @xmath252 above . let @xmath254 , and for @xmath48 let @xmath255 , where @xmath256\ ] ] is the discrete harmonic polynomial associated to @xmath257 as described in [ s.polynomials ] . the sequence @xmath258 begins @xmath259 for instance , to compute @xmath260 , we expand @xmath261\ ] ] and apply @xmath262 to each monomial , obtaining @xmath263\ ] ] which simplifies to @xmath264 . one readily checks that this defines a discrete harmonic function on @xmath265 . ( in fact , @xmath266 is itself discrete harmonic , but @xmath267 is not for @xmath268 . ) to define @xmath258 for negative @xmath165 , we set @xmath269 . part ( a ) of this lemma was proved by van der corput in the 1920s ( see @xcite , theorem 87 p. 484 ) . part ( b ) follows from the same method , as proved below . part ( c ) follows from part ( b ) and the stronger estimate of lemma [ discretepolynomial ] , @xmath280 for @xmath281 ( and @xmath282 ) . we prove part ( b ) in all dimensions . let @xmath283 be a harmonic polynomial on @xmath24 of homogeneous of degree @xmath165 . normalize so that @xmath284 where @xmath285 is the unit ball . in this discussion @xmath165 will be fixed and the constants are allowed to depend on @xmath165 . consider @xmath294 a smooth , radial function on @xmath24 with integral @xmath62 supported in the unit ball . then define @xmath295 characteristic function of the unit ball . denote @xmath296 then @xmath297 this is because @xmath298 is nonzero only in the annulus of width @xmath299 around @xmath300 in which ( by the van der corput bound ) there are @xmath301 lattice points . the poisson summation formula implies @xmath302*\hat p_k(\xi)/r^k\ ] ] in the sense of distributions . the fourier transform of a polynomial is a derivative of the delta function , @xmath303 . because @xmath286 and @xmath178 is harmonic , its average with repect to any radial function is zero . this is expressed in the dual variable as the fact that when @xmath304 , @xmath305 ) = 0\ ] ] so we our sum equals @xmath306*\hat p_k ( \xi)/r^k\ ] ] next look at @xmath307 @xmath308 all the terms in which fewer derivatives fall on @xmath309 and more fall on @xmath310 give much smaller expressions : the factor @xmath311 corresponding to each such differentiation is replaced by an @xmath312 . the asymptotics of this oscillatory integral above are well known . for any fixed polynomial @xmath313 they are of the same order of magnitude as for @xmath314 , namely @xmath315 this is proved by the method of stationary phase and can also be derived from well known asymptotics of bessel functions . denote @xmath320 applying the formula above for @xmath321 , @xmath322 to estimate the error term @xmath323 , note first that the coefficients @xmath213 are supported in a fixed annulus , the integrand above is supported in the range @xmath324 . furthermore , by @xcite , there is an absolute constant @xmath20 such that for all sufficiently large @xmath311 and all @xmath1 in this range , the difference @xmath325 is supported on the set of @xmath326 such that @xmath327 . thus @xmath328 moreover , lemma [ discrete - error ] applies and @xmath329 next , lemma [ vandercorput](a ) says ( since @xmath330 ) @xmath331 thus replacing @xmath82 by @xmath332 gives an additional error of size at most @xmath333 in all , @xmath334 for @xmath335 , consider the process @xmath336 note that @xmath337 as @xmath338 . note also that lemma [ vandercorput](c ) implies @xmath339 because @xmath258 are discrete harmonic and @xmath340 for all @xmath341 , @xmath342 is a martingale . it remains to show that @xmath343 in law . as outlined below , this will follow from the martingale central limit theorem ( see , e.g. , @xcite or @xcite ) . for sufficiently large @xmath311 , the difference @xmath344 is nonzero only for @xmath345 in the range @xmath346 ; and @xmath347 . we now show that this implies @xmath348 so that the martingale central limit theorem applies . to prove , observe that @xmath349 where @xmath350 is the @xmath351th point of @xmath13 . then @xmath352 implies @xmath353 , and hence @xmath354 recalling that @xmath355 unless @xmath356 , we have @xmath357 which confirms . because @xmath13 fills the lattice @xmath265 as @xmath358 , we have @xmath359 we prove in three steps : replace @xmath360 by @xmath267 ( or @xmath361 if @xmath362 ) ; replace the lower limit @xmath363 by @xmath364 ; replace the sum of @xmath350 over lattice sites with the integral with respect to lebesgue measure in the complex @xmath350-plane . we begin the proof of by noting that the error term introduced by replacing @xmath258 with @xmath267 is @xmath365 in the integral this is majorized by @xmath366 since there are @xmath367 such terms , this change contributes order @xmath368 to the sum . next , we change the lower limit from @xmath363 to @xmath369 . since @xmath370 , the integral inside @xmath371 is changed by @xmath372 thus the change in the whole expression is majorized by the order of the cross term @xmath373 again there are @xmath374 terms in the sum over @xmath350 , so the sum of the errors is @xmath375 . lastly , we replace the value at each site @xmath376 by the integral @xmath377 where @xmath378 is the unit square centered at @xmath376 and @xmath379 . because the square has area @xmath62 , the term in the lattice sum is the same as this integral with @xmath380 replaced by @xmath376 at each occurrence . since @xmath381 , @xmath382 after we divide by @xmath383 , the order of error is @xmath384 . adding all the errors contributes at most order @xmath384 to the sum . we must also take into account the change in the lower limit of the integral , @xmath385 is replaced by @xmath386 . since @xmath381 , @xmath387 recall that in the previous step we previously changed the lower limit by @xmath388 . thus by the same argument , this smaller change gives rise to an error of order @xmath384 in the sum over @xmath376 . the proof of is now reduced to evaluating @xmath389 integrating in @xmath32 and changing variables from @xmath7 to @xmath390 , @xmath391 then change variables from @xmath1 to to @xmath392 to obtain @xmath393 this ends the proof of theorem [ gffdisk0 ] . the proof of theorem [ gffdisk1 ] follows the same idea . we replace @xmath13 by the poisson time region @xmath394 ( for @xmath395 ) , and we need to find the limit as @xmath217 of @xmath396 the error terms in the estimation showing this quantity is within @xmath397 of @xmath398 are nearly the same as in the previous proof . we describe briefly the differences . the difference between poisson time and ordinary counting is @xmath399 . it follows that for @xmath400 , @xmath401 as in the previous proof for @xmath363 . further errors are also controlled since we then have the estimate analogous to the one above for @xmath13 , namely @xmath402 we consider the continuous time martingale @xmath403 instead of using the martingale central limit theorem , we use the martingale representation theorem . this says that the martingale @xmath404 when reparameterized by its quadratic variation has the same law as brownian motion . we must show that almost surely the quadratic variation of @xmath185 on @xmath405 is @xmath406 . @xmath407 integrating with respect to @xmath345 gives the quadratic variation @xmath406 after a suitable change of variable as in the previous proof . theorem [ t.fluctuations ] follows almost immediately from the @xmath3 case of theorem [ t.highdconvergence ] and the estimates above . consider @xmath408 where @xmath409 is as in . what happens if we replace @xmath83 with a function @xmath410 that is discrete harmonic on the rescaled mesh @xmath411 within a @xmath412 neighborhood of @xmath413 ? clearly , if @xmath83 is smooth , we will have @xmath414 . since there are at most @xmath415 non - zero terms in , the discrepancy in @xmath416 which tends to zero as long as @xmath417 . the fact that replacing @xmath418 with @xmath409 has a negligible effect follows from the above estimates when @xmath3 . this may also hold when @xmath419 , but we will not prove it here . instead we remark that theorem [ t.fluctuations ] holds in three dimensions provided that we replace with , and that the theorem as stated probably fails in higher dimensions even if we make a such a replacement . the reason is that is positive at points slightly outside of @xmath420 ( or outside of the support of @xmath84 ) and negative at points slightly inside . if we replace a discrete harmonic polynomial @xmath82 with a function that agrees with @xmath82 on @xmath413 but has a different derivative along portions of @xmath43 , this may produce a non - trivial effect ( by the discussion above ) when @xmath421 . r. durrett , _ probability : theory and examples _ , 2nd ed . , 1995 . c. m. fortuin , p. w. kasteleyn and j. ginibre , correlation inequalities on some partially ordered sets , _ comm . * 22 * : 89103 , 1971 . a. ivi , e. krtzel , m. khleitner , and w.g . nowak , lattice points in large regions and related arithmetic functions : recent developments in a very classic topic , _ elementare und analytische zahlentheorie _ , schr . wiss . ges . johann wolfgang goethe univ . frankfurt am main , 20 , 89128 , 2006 . d. jerison , l. levine and s. sheffield , internal dla : slides and audio . _ midrasha on probability and geometry : the mathematics of oded schramm . _ http://iasmac31.as.huji.ac.il:8080/groups/midrasha_14/weblog/855d7/images/bfd65.mov , 2009 . d. jerison , l. levine and s. sheffield , logarithmic fluctuations for internal dla . d. jerison , l. levine and s. sheffield , internal dla in higher dimensions . http://arxiv.org/abs/1012.3453[arxiv:1012.3453 ] g. lawler , _ intersections of random walks _ , birkhuser , 1996 . l. levine and y. peres , strong spherical asymptotics for rotor - router aggregation and the divisible sandpile , _ potential anal . _ * 30 * ( 2009 ) , 127 . http://arxiv.org/abs/0704.0688[arxiv:0704.0688 ] l. lovsz , discrete analytic functions : an exposition , _ surveys in differential geometry _ * ix*:241273 , 2004 .
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in previous works , we showed that the internal dla cluster on @xmath0 with @xmath1 particles is almost surely spherical up to a maximal error of @xmath2 if @xmath3 and @xmath4 if @xmath5 .
this paper addresses `` average error '' : in a certain sense , the average deviation of internal dla from its mean shape is of _ constant _ order when @xmath3 and of order @xmath6 ( for a radius @xmath7 cluster ) in general .
appropriately normalized , the fluctuations ( taken over time and space ) scale to a variant of the gaussian free field .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
describing the laws of physics in terms of underlying symmetries has always been a powerful tool . lie algebras and lie superalgebras are central in particle physics , and the space - time symmetries can be obtained by an inn - wigner contraction of certain lie ( super)algebras . @xmath0lie algebras @xcite , a possible extension of lie ( super)algebras , have been considered some times ago as the natural structure underlying fractional supersymmetry ( fsusy ) @xcite ( one possible extension of supersymmetry ) . in this contribution we show how one can construct many examples of finite dimensional @xmath0lie algebras from lie ( super)algebras and finite - dimensional fsusy extensions of the poincar algebra are obtained by inn - wigner contraction of certain @xmath0lie algebras . the natural mathematical structure , generalizing the concept of lie superalgebras and relevant for the algebraic description of fractional supersymmetry was introduced in @xcite and called an @xmath0lie algebra . we do not want to go into the detailed definition of this structure here and will only recall the basic points , useful for our purpose . more details can be found in @xcite . let @xmath5 be a positive integer and @xmath6 . we consider now a complex vector space @xmath7 which has an automorphism @xmath8 satisfying @xmath9 . we set @xmath10 , @xmath11 and @xmath12 ( @xmath13 is the eigenspace corresponding to the eigenvalue @xmath14 of @xmath8 ) . hence , @xmath15 we say that @xmath7 is an @xmath0lie algebra if : 1 . @xmath16 , the zero graded part of @xmath7 , is a lie algebra . @xmath17 @xmath18 , the @xmath19 graded part of @xmath7 , is a representation of @xmath16 . 3 . there are symmetric multilinear @xmath20equivariant maps @xmath21 where @xmath22 denotes the @xmath0fold symmetric product of @xmath23 . in other words , we assume that some of the elements of the lie algebra @xmath16 can be expressed as @xmath0th order symmetric products of `` more fundamental generators '' . 4 . the generators of @xmath7 are assumed to satisfy jacobi identities ( @xmath24 , @xmath25 , @xmath11 ) : @xmath26,b_3\right ] + \left[\left[b_2,b_3\right],b_1\right ] + \left[\left[b_3,b_1\right],b_2\right ] = 0 , \nonumber \\ \left[\left[b_1,b_2\right],a_3\right ] + \left[\left[b_2,a_3\right],b_1\right ] + \left[\left[a_3,b_1\right],b_2\right ] = 0,\nonumber \\ \left[b,\left\{a_1,\dots , a_f\right\}\right ] = \left\{\left[b , a_1 \right],\dots , a_f\right\ } + \cdots + \left\{a_1,\dots,\left[b , a_f\right ] \right\ } , \nonumber \\ \sum\limits_{i=1}^{f+1 } \left [ a_i,\left\{a_1,\dots , a_{i-1 } , a_{i+1},\dots , a_{f+1}\right\ } \right ] = 0 . \label{rausch : eq : jac}\end{aligned}\ ] ] the first three identities are consequences of the previously defined properties but the fourth is an extra constraint . more details ( unitarity , representations , _ etc . _ ) can be found in @xcite . let us first note that no relation between different graded sectors is postulated . secondly , the sub - space @xmath27 @xmath28 is itself an @xmath0lie algebra . from now on , @xmath0lie algebras of the types @xmath29 will be considered . most of the examples of @xmath0lie algebras are infinite dimensional ( see _ e.g. _ @xcite ) . however in @xcite an inductive theorem to construct finite - dimensional @xmath0lie algebras was proven : + * theorem 1 * _ let @xmath30 be a lie algebra and @xmath31 a representation of @xmath30 such that _ \(i ) @xmath32 is an @xmath0lie algebra of order @xmath33;. in this case the notion of graded @xmath34lie algebra has to be introduced @xcite . @xmath35 , is a graded @xmath34lie algebra if ( i ) @xmath36 a lie algebra and @xmath31 is a representation of @xmath36 isomorphic to the adjoint representation , ( ii ) there is a @xmath37 equivariant map @xmath38 such that @xmath39 + \left[f_2 , \mu(f_1 ) \right ] = 0 , f_1,f_2 \in \ { g}_1 $ ] . ] \(ii ) @xmath40 admits a @xmath41equivariant symmetric form @xmath42 of order @xmath43 . then @xmath44 admits an @xmath0lie algebra structure of order @xmath45 , which we call the @xmath0lie algebra induced from @xmath46 and @xmath42 . + by hypothesis , there exist @xmath47equivariant maps @xmath48 and @xmath49 . now , consider @xmath50 defined by @xmath51 where @xmath52 and @xmath53 is the group of permutations on @xmath54 elements . by construction , this is a @xmath47equivariant map from @xmath55 , thus the three first jacobi identities are satisfied . the last jacobi identity , is more difficult to check and is a consequence of the corresponding identity for the @xmath0lie algebra @xmath46 and a factorisation property ( see @xcite for more details ) . an interesting consequence of the theorem of the previous section is that it enables us to construct an @xmath0lie algebras associated to _ any _ lie ( super)algebras . consider the graded @xmath34lie algebra @xmath56 where @xmath57 is a lie algebra , @xmath58 is the adjoint representation of @xmath57 and @xmath59 is the identity . let @xmath60 be a basis of @xmath57 , and @xmath61 the corresponding basis of @xmath58 . the graded @xmath34lie algebra structure on @xmath7 is then : @xmath62 = f_{ab}^{\ \ \ c } j_c , \qquad \left[j_a , a_b \right ] = f_{ab}^{\ \ \ c } a_c , \qquad \mu(a_a)= j_a,\end{aligned}\ ] ] where @xmath63 are the structure constants of @xmath57 , the second ingredient to construct an @xmath0lie algebra is to define a symmetric invariant form on @xmath64 . but on @xmath64 , the adjoint representation of @xmath65 , the invariant symmetric forms are well known and correspond to the casimir operators @xcite . then , considering a casimir operator of order @xmath66 of @xmath67 , we can induce the structure of an @xmath0lie algebra of order @xmath68 on @xmath69 . one can give explicit formulae for the bracket of these @xmath0lie algebras as follows . let @xmath70 be a casimir operator of order @xmath66 ( for @xmath71 , the killing form @xmath72 is a primitive casimir of order two ) . then , the @xmath0bracket of the @xmath0lie algebra is @xmath73 for the killing form this gives @xmath74 if @xmath75 , the @xmath0lie algebra of order three induced from the killing form is the @xmath0lie algebra of @xcite . the construction of @xmath0lie algebras associated to lie superalgebras is more involved . we just give here a simple example ( for more details see @xcite ) : the @xmath0lie algebra of order @xmath76 @xmath77 induced from the ( i ) lie superalgebra @xmath78 and ( ii ) the quadratic form @xmath79 , where @xmath8 is the invariant symplectic form on @xmath80 and @xmath81 the invariant symplectic form on @xmath82 . let @xmath83 be a basis of @xmath84 and @xmath85 be a basis of @xmath86 . let @xmath87 be a basis of @xmath88 . then the four brackets of @xmath7 take the following form @xmath89 it is interesting to notice that this @xmath0lie algebra admits a simple matrix representation @xcite : @xmath90 and @xmath91 . it is well known that supersymmetric extensions of the poincar algebra can be obtained by inn - wigner contraction of certain lie superalgebras . in fact , one can also obtain some fsusy extensions of the poincar algebra by inn - wigner contraction of certain @xmath0lie algebras as we now show with one example @xcite . let @xmath92 be the real @xmath0lie algebra of order three induced from the real graded @xmath34lie algebra @xmath93 and the killing form on @xmath94 ( see eq . [ eq:3-lie ] ) . using vector indices of @xmath95 coming from the inclusion @xmath96 , the bosonic part of @xmath97 is generated by @xmath98 , with @xmath99 and the graded part by @xmath100 . letting @xmath101 after the inn - wigner contraction , @xmath102{\lambda } } q_{\mu \nu } , & j_{4 \mu } \to \frac{1}{\sqrt[3]{\lambda } } q_{\mu } , \end{array}\end{aligned}\ ] ] one sees that @xmath103 and @xmath104 generate the @xmath105 poincar algebra and that @xmath106 are the fractional supercharges in respectively the adjoint and vector representations of @xmath107 . this @xmath0lie algebra of order three is therefore a non - trivial extension of the poincar algebra where translations are cubes of more fundamental generators . the subspace generated by @xmath108 is also an @xmath0lie algebra of order three extending the poincar algebra in which the trilinear symmetric brackets have the simple form : @xmath109 where @xmath110 is the minkowski metric . in this paper a sketch of the construction of @xmath0lie algebras associated to lie ( super)algebras were given . more complete results , such as a criteria for simplicity , representation theory , matrix realizations _ etc . _ , was given in @xcite . 9 rausch de traubenberg m and slupinski m. j. 2000 j. math . phys 41 4556 - 4571 [ hep - th/9904126 ] . rausch de traubenberg m and slupinski m. j. 2002 _ proceedings of institute of mathematics of nas of ukraine _ , p 548 - 554 , vol . 43 , editors a.g . nikitin , v.m . boyko and r.o . popovych , kyiv , institute of mathematics [ arxiv : hep - th/0110020 ] . rausch de traubenberg m and slupinski m. j. 2002 _ finite - dimensional lie algebras of order @xmath5 _ , arxiv : hep - th/0205113 , to appear in j. math . durand s 1993 mod . lett a 8 23232334 [ hep - th/9305130 ] . rausch de traubenberg m and slupinski m. j. 1997 mod . a 12 3051 - 3066 [ hep - th/9609203 ] . rausch de traubenberg m 1998 hep - th/9802141 ( habilitation thesis , in french ) . chevalley c and eilenberg s 1948 trans . 63 85 - 124 . ahmedov h , yildiz a and ucan y 2001 j. phys . a 34 6413 - 6424 [ math.rt/0012058 ] .
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@xmath0lie algebras are natural generalisations of lie algebras ( @xmath1 ) and lie superalgebras ( @xmath2 ) .
we give finite dimensional examples of @xmath0lie algebras obtained by an inductive process from lie algebras and lie superalgebras .
matrix realizations of the @xmath0lie algebras constructed in this way from @xmath3 are given .
we obtain a non - trivial extension of the poincar algebra by an inn - wigner contraction of a certain @xmath0lie algebras with @xmath4 .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
diseases spread over networks . the spreading dynamics are closely related to the structure of networks . for this reason network epidemiology has turned into of the most vibrant subdisciplines of complex network studies . @xcite a topic of great practical importance within network epidemiology is the vaccination problem : how should a population be vaccinated to most efficiently prevent a disease to turn into an epidemic ? for economic reasons it is often not possible to vaccinate the whole population . some vaccines have severe side effects and for this reason one may also want to keep number of vaccinated individuals low . so if cheap vaccines , free of side effects , does not exist ; then having an efficient vaccination strategy is essential for saving both money and life . if all ties within the population is known , then the target persons for vaccination can be identified using sophisticated global strategies ( cf . @xcite ) ; but this is hardly possible for nation - wide ( or larger ) vaccination campaigns . in a seminal paper cohen _ et al . _ @xcite suggested a vaccination strategy that only requires a person to estimate which other persons he , or she , gets close enough to for the disease to spread to i.e . , to name the `` neighbors '' in the network over which the disease spreads . for network with a skewed distribution of degree ( number of neighbors ) the strategy to vaccinate a neighbor of a randomly chosen person is much more efficient than a random vaccination . in this work we assume that each individual knows a little bit more about his , or her , neighborhood than just the names of the neighbors : we also assume that an individual can guess the degree of the neighbors and the ties from one neighbor to another . this assumption is not very unrealistic people are believed to have a good understanding of their social surroundings ( this is , for example , part of the explanation for the `` navigability '' of social networks ) @xcite . finding the optimal set of vaccinees is closely related to the attack vulnerability problem @xcite . the major difference is the dynamic system that is confined to the network disease spreading for the vaccination problem and information flow for the attack vulnerability problem . to be able to protect the network efficiently one needs to know the worst case attacking scenario . large scale network attacks are , presumably , based on local ( rather than global ) network information . so , a grave scenario would be in the network was attacked with the same strategy that is most efficient for vaccination . we will use the vaccination problem as the framework for our discussion , but the results applies for network attack as well . in our discussion we will use two measures of network structure : the _ clustering coefficient _ @xmath0 of the network defined as the ratio of triangles with respect to connected triples normalized to the interval @xmath1 $ ] . @xcite if @xmath2 there is a maximal number of triangles ( given a set of connected triples ) ; if @xmath3 the graph has no triangles . we also measure the degree - degree correlations through the _ assortative mixing coefficient _ defined as @xcite @xmath4 where @xmath5 is the degree of the @xmath6th argument of an edge in a list of the edges , and @xmath7 denotes average over that edge - list . we let @xmath8 denote the number of vertices and @xmath9 the number of edges . we will test the vaccination strategies we propose on both real - world and model networks . the first real - world network is a scientific collaboration network @xcite . the vertices of this network are scientists who have uploaded manuscripts to the preprint repository arxiv.org . an edge between two authors means that they have coauthored a preprint . we also study two small real - world social networks : one constructed from an observational study of friendships in a karate club , another based on an interview survey among prisoners . the edges of these small networks are , probably , more relevant for disease spreading than the arxiv network , but may suffer from finite size effects . the three model networks are : 1 . the holme - kim ( hk ) model @xcite that produces networks with a power - law degree distribution and tunable clustering . basically , it is a barabsi - albert ( ba ) type growth model based on preferential attachment @xcite just as the ba model it has one parameter @xmath10 controlling the average degree and one ( additional ) parameter @xmath11 $ ] controlling the clustering . we will use @xmath12 and @xmath13 giving the maximal clustering for the given @xmath8 and @xmath9 . 2 . the networked seceder model , modeling social networks with a community structure and exponentially decaying degree distributions @xcite . briefly , it works by sequentially updating the vertices by , for each vertex @xmath14 , rewiring all @xmath14 s edges to the neighborhood of a peripheral vertex . with a probability @xmath15 an edge of @xmath14 can be rewired to a random vertex ( so @xmath15 controls the degree of community structure ) . we use the parameter values @xmath16 , @xmath17 and @xmath18 iterations on an erds - rnyi network @xcite . the watts - strogatz ( ws ) model @xcite generates networks with exponentially decaying degree distributions and tunable clustering . the ws model starts from the vertices on a circular topology with edges between vertices separated by 1 to @xmath19 steps on the circle . then one goes through the edges and rewire one side of them to randomly selected vertices with a probability @xmath20 . we use @xmath21 and @xmath22 . .statistics of the networks . note that the arxiv , prison and seceder model networks are not connected the largest connected components contains @xmath23 , @xmath24 and @xmath25 nodes respectively . [ cols="<,<,<,<,<",options="header " , ] now we turn to the definition of the strategies . we assume a fraction @xmath26 of the population is to be vaccinated . as a reference we consider random vaccination ( rnd , equivalent to site percolation ) . we use the above mentioned _ neighbor vaccination _ ( rnb)to vaccinate the neighbor of randomly chosen vertices and the trivial improvement @xcite if knowledge about the neighbors degrees are included : pick a vertex at random and vaccinate one ( randomly chosen ) of its highest - degree neighbors ( we call it deg ) . to avoid overvaccination of a neighborhood one can consider to vaccinate neighbors of a vertex @xmath14 with a maximal number of edges out of @xmath14 s neighborhood ( out ) . for all strategies except rnd we also consider `` chained '' versions were one , instead of vaccinating a neighbor of a randomly chosen vertex , vaccinates a neighbor of the vertex vaccinated in the previous time step ( if all neighbors are vaccinated a neighbor of a random vertex is chosen instead ) . for the acronyms of the chained versions a suffix `` c '' is added . the results of this paper are presented in three sections : first we study how the number of vertices in the largest connected subgraph @xmath27 depends on the fraction @xmath26 of vaccinated vertices . then we show that the conclusions from @xmath27 also hold for dynamical simulations of disease spreading . to interpret the results we also investigate @xmath27 for a fixed @xmath26 as a function of the clustering and assortative mixing coefficients . as a static efficiency measure we consider the size of the average largest connected component of susceptible ( non - vaccinated ) vertices , @xmath27 . we average over @xmath28 runs of the vaccination procedures . the model networks are also averaged over @xmath29 network realizations . ( smaller or larger @xmath30 and @xmath31 does not make any qualitative difference . ) in fig . [ fig : s1 ] we display @xmath27 as a function of @xmath26 . for all except the ws model network the deg and out ( chained and unchained versions ) form the most efficient set of strategies . within this group the order of efficiency varies : for the arxiv network the out strategy is more than twice as efficient as any other for @xmath32 . for the hk and seceder model networks the chained strategies are considerably more efficient than the unchained ones . we note that the difference between the chained and unchained versions of out and deg is bigger than between out and deg ( or outc and degc ) . out do converge to deg in the limit of vanishing @xmath0 but all networks we test have rather high clustering . another interesting observation is that even if the degree distribution is narrow , such as for the seceder model of fig . [ fig : s1](e ) ( where @xmath33 ) the more elaborate strategies are much more efficient than random vaccination . this is especially clear for higher @xmath26 which suggests that the structural change of the network of susceptible vertices during the vaccination procedure is an important factor for the overall efficiency . for the ws model network the chained algorithms are performing poorer than random vaccination . this is in contrast to all other networks . we conclude that epidemiology related results regarding the ws model networks should be cautiously generalized to real - world systems . static measures of vaccination efficiency are potential over - simplifications there is a chance that the interplay between disease dynamics and the underlying network structure has a significant role . to motivate the use of @xmath27 we also investigate the sis and sir models @xcite on vaccinated networks . in the sis model a vertex goes from `` susceptible '' ( s ) to `` infected '' ( i ) and back to s. in the sir model is just the same , except that an infected vertex goes to the `` removed '' ( r ) state and remain there . the probability to go from @xmath34 to @xmath35 ( per contact ) is zero for vaccinated vertices and @xmath36 for the rest . the i state lasts @xmath37 time steps . we use synchronous updating and one randomly chosen initially infected person . the disease dynamics are averaged @xmath38 times for all @xmath28 runs of the vaccination schemes . in fig . [ fig : dyn](a ) we plot the average number of individuals that at least once have been infected during an outbreak @xmath39i.e . , until there are no i - vertices left , or ( for sis ) has reached an endemic state ( defined in the simulations as when there are no susceptible vertices that have not had the disease at least once)for the arxiv network . other networks and simulation parameters give qualitatively similar results . qualitatively , the large picture from the @xmath27 calculations remains the chained and unchained deg and out strategies are very efficient , and the chained versions are more efficient than the unchained . a difference is that the unchained rnb also performs rather well . quantitatively , the differences between the strategies are huge , this is a result of the threshold behaviors of the sis and sir models @xcite . the conclusion of fig . [ fig : dyn ] ( and similar plots for other networks ) is that the order of the strategies efficiencies are largely the same as concluded from the @xmath40-curves . but if high resolution is required , the measurement of network fragility has to be specific for the studied system . to gain some insight how the network structure govern the relative efficiencies of the strategies we measure @xmath41 for varying assortative mixing and clustering coefficients . the results hold for other small @xmath26 values . we keep the size and degree sequence constant to the values of the arxiv network . to perform this sampling we rewire pairs of edges @xmath42 and @xmath43 to @xmath44 and @xmath45 ( unless this would introduce a self - edge or multiple edges ) . to ensure that the @xmath46 rewiring realizations are independent we start with rewiring @xmath47 pairs of edges . then we go through pairs of edges randomly and execute only changes that makes the current @xmath15 or @xmath0 closer to their target values . when the value of @xmath15 or @xmath0 are within @xmath48 of the target value the iteration is braked . the results seen in fig . [ fig : rew ] shows that , just as before the out and deg strategies , chained or unchained , are most efficient throughout the parameter space . the unchained versions are most efficient for @xmath49 . an explanation is that , for high @xmath15 , the chained versions will effectively only vaccinate the high - connected vertices ( that are grouped together for very high @xmath15 ) and leave chains of low - degree vertices unvaccinated . the @xmath0-dependence plotted in fig . [ fig : rew](b ) shows that the unchained versions outperform the chained versions for @xmath50 . this is possibly a result of that the chains , for combinatorial reasons , get stuck in one part of the network . it is not an effect of biased degree - degree correlations since if the rewiring procedure is conditioned to a fixed @xmath15 fig . [ fig : rew](b ) remains essentially unaltered . we note that the structure of the original arxiv network differs from the rewired networks . for example , at @xmath51 of fig . [ fig : s1](a ) the out is 22% more efficient than outc , but in fig . [ fig : rew ] the out and outc curves differ very little . for the rnb strategy the chained version is better than the unchained throughout the range of @xmath15 and @xmath0 values . to summarize , we have investigated strategies for vaccination and network attack that are based only on the knowledge of the neighborhood information that humans arguably possess and utilize . both static and dynamical measures of efficiency are studied . for most networks , regardless of the number of vaccinated vertices , the most efficient strategies are to choose a vertex @xmath14 and vaccinate a neighbor of @xmath14 with highest degree ( deg ) , or the neighbor of @xmath14 with most links out of @xmath14 s neighborhood ( out ) . @xmath14 can be picked either as the lastly vaccinated vertex ( chained selection ) or at random ( unchained selection ) . for real - world networks the chained versions tend to outperform the unchained ones , whereas this situation is reversed for the three types of model networks we study . we investigate the relative efficiency of chained and unchained strategies further by sampling random networks with a fixed degree sequence and varying assortative mixing and clustering coefficients . we find that the unchained strategies are preferable for networks with a very high clustering or strong positive assortative mixing ( larger values than in seen in real - world networks ) . in ref . @xcite the authors propose the strategy to vaccinate a random neighbor of a randomly selected vertex . this strategy ( rnb ) requires less information of the neighborhood than deg and out do . thus the practical procedure gets simpler : one only has to ask a person `` name a person you meet regularly '' rather than `` name the acquaintance of yours who meet most people you are not acquainted with regularly '' ( for out ) . ( `` meet with regularly '' should be replaced with some phrase signifying a high risk of infection transfer for the pathogen in question . ) on the other hand , if the information of the neighborhoods is incomplete deg and out will , effectively , be reduced to rnb ( and thus not perform worse than rnb ) . to epitomize , choosing the people to vaccinate in the right way will save a tremendous amount of vaccine and side - effect cases . the best strategy can only be selected by considering both the structure of the network the pathogen spreads over , and the disease dynamics . if nothing of this is known the outc strategy our recommendation it is better , or not much worse , than the best strategy in most cases . together with degc , outc is most efficient for low clustering and assortative mixing coefficients , which is the region of parameter space for sexually transmitted diseases the most interesting case for network based vaccination schemes ( due to the well - definedness of sexual networks ) . the author is grateful for comments from m. rosvall and acknowledges support from the swedish research council through contract no.2002 - 4135 .
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we study how a fraction of a population should be vaccinated to most efficiently stop epidemics .
we argue that only local information ( about the neighborhood of specific vertices ) is usable in practice , and hence we consider only local vaccination strategies .
the efficiency of the vaccination strategies is investigated with both static and dynamical measures . among other things
we find that the most efficient strategy for many real - world situations is to iteratively vaccinate the neighbor of the previous vaccinee that has most links out of the neighborhood .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
diffusive particle acceleration at non - relativistic shock fronts is an extensively studied phenomenon . detailed discussions of the current status of the investigations can be found in some excellent reviews ( drury 1983 ; blandford & eichler 1987 ; berezhko & krimsky 1988 ; jones & ellison 1991 ; malkov & drury 2001 ) . while much is by now well understood , some issues are still subjects of much debate , for the theoretical and phenomenological implications that they may have . one of the most important of these is the reaction of the accelerated particles on the shock : the violation of the _ test particle approximation _ occurs when the acceleration process becomes sufficiently efficient that the pressure of the accelerated particles is comparable with the incoming gas kinetic pressure . both the spectrum of the particles and the structure of the shock are changed by this phenomenon , which is therefore intrinsically nonlinear . at present there are three viable approaches to determine the reaction of the particles upon the shock : one is based on the ever - improving numerical simulations ( jones & ellison 1991 ; bell 1987 ; ellison , mbius & paschmann 1990 ; ellison , baring & jones 1995 , 1996 ; kang & jones 1997 ; kang , jones & gieseler 2002 ; kang & jones 2005 ) that allow one to achieve a self - consistent treatment of several effects . the second approach is based on the so - called two - fluid model , and treats cosmic rays as a relativistic second fluid . this class of models was proposed and discussed in ( drury & vlk 1980 , 1981 ; drury , axford & summers 1982 ; axford , leer & mckenzie 1982 ; duffy , drury & vlk 1994 ) . these models allow one to obtain the thermodynamics of the modified shocks , but do not provide information about the spectrum of accelerated particles . the third approach is semi - analytical and may be very helpful to understand the physics of the nonlinear effects in a way that sometimes is difficult to achieve through simulations , due to their intrinsic complexity and limitations in including very different spatial scales . blandford ( 1980 ) proposed a perturbative approach in which the pressure of accelerated particles was treated as a small perturbation . by construction this method provides the description of the reaction only for weakly modified shocks . alternative approaches were proposed by eichler ( 1984 ) , ellison & eichler ( 1984 ) , eichler ( 1985 ) and ellison & eichler ( 1985 ) , based on the assumption that the diffusion of the particles is sufficiently energy dependent that different parts of the fluid are affected by particles with different energies . the way the calculations are carried out implies a sort of separate solution of the transport equation for subrelativistic and relativistic particles , so that the two spectra must be somehow connected at @xmath0 _ a posteriori_. in ( berezhko , yelshin & ksenofontov 1994 ; berezhko , ksenofontov & yelshin 1995 ; berezhko 1996 ) the effects of the non - linear reaction of accelerated particles on the maximum achievable energy were investigated , together with the effects of geometry . the maximum energy of the particles accelerated in supernova remnants in the presence of large acceleration efficiencies was also studied by ptuskin & zirakashvili ( 2003a , b ) . the need for a _ practical _ solution of the acceleration problem in the non - linear regime was recognized by berezhko & ellison ( 1999 ) , where a simple analytical broken - power - law approximation of the non - linear spectra was presented . recently , some promising analytical solutions of the problem of non - linear shock acceleration have appeared in the literature ( malkov 1997 ; malkov , diamond & vlk 2000 ; blasi 2002 , 2004 ) . blasi ( 2004 ) considered for the first time the important effect of seed pre - existing particles in the acceleration region ( the linear theory of this phenomenon was first studied by bell ( 1978 ) ) . in a recent work by kang & jones ( 2005 ) the seed particles were included in numerical simulations of the acceleration process . numerical simulations have been instrumental to identify the dramatic effects of the particles reaction : they showed that even when the fraction of particles injected from the thermal gas is relatively small , the energy channelled into these few particles can be an appreciable part of the kinetic energy of the unshocked fluid , making the test particle approach unsuitable . the most visible effects of the reaction of the accelerated particles on the shock appear in the spectrum of the accelerated particles , which shows a peculiar hardening at the highest energies . the analytical approaches reproduce well the basic features arising from nonlinear effects in shock acceleration . there is an important point which is still lacking in the calculations of the non - linear particle acceleration at shock waves , namely the possible amplification of the background magnetic field , found in the numerical simulations by lucek & bell ( 2000 , 2000a ) and bell & lucek ( 2001 ) and recently described by bell ( 2004 ) . this effect is still ignored in all calculations of the reaction of cosmic rays on the shock structure . we will not include this effect in the present paper . nonlinear effects in shock acceleration of thermal particles result in the appearance of multiple solutions in certain regions of the parameter space . this phenomenon is very general and was found in both the two - fluid models ( drury & vlk 1980 , 1981 ) and in the kinetic models ( malkov 1997 ; malkov et al . 2001 ; blasi 2004 ) . monte carlo approaches do not show multiple solutions . this behaviour resembles that of critical systems , with a bifurcation occurring when some threshold is reached in a given order parameter . in the case of shock acceleration , it is not easy to find a way of discriminating among the multiple solutions when they appear . in ( mond & drury 1998 ) , a two fluid approach has been used to demonstrate that when three solutions appear , the one with intermediate efficiency for particle acceleration is unstable to corrugations in the shock structure and emission of acoustic waves . plausibility arguments may be put forward to justify that the system made of the shock plus the accelerated particles may sit at the critical point ( see for instance the paper by malkov , diamond & vlk ( 2000 ) ) , but we are not aware of any real proof that this is what happens . the physical parameters that play a role in this approach to criticality are the maximum momentum achievable by the particles in the acceleration process , the mach number of the shock , and the injection efficiency , namely the fraction of thermal particles crossing the shock that are accelerated to nonthermal energies . the last of them , the injection efficiency , hides a crucial physics problem by itself , and plays an important role in establishing the level of shock modification . this efficiency parameter in reality is determined by the microphysics of the shock and should not be a free parameter of the problem . unfortunately , our poor knowledge of such microphysics , in particular for collisionless shocks , does not allow us to establish a clear and universal connection between the injection efficiency and the macroscopic shock properties . put aside the possibility to have a fully self - consistent picture of this phenomenon , one can try to achieve a phenomenological description of it . kang , jones & gieseler ( 2002 ) introduced a sort of weight function to determine a return probability of particles in the downstream fluid to the upstream fluid , as a function of particle momentum . only sufficiently suprathermal particles can jump back to the upstream region and therefore take part in the acceleration process . here we adopt an injection recipe which is similar to the _ thermal leakage _ model of kang et al . ( 2002 ) ( see also previous papers by malkov ( 1998 ) and by gieseler et al . ( 2000 ) ) and implement it in the semi - analytical approach of blasi ( 2002 , 2004 ) . we investigate then the phenomenon of multiple solutions and show that the injection recipe dramatically reduces the appearance of these situations . we also study in some detail the efficiency for particle acceleration as a function of the mach number of the shock and the maximum momentum of the accelerated particles . the paper is structured as follows : in section [ sec : nonlin ] we briefly describe the method proposed by blasi ( 2002 ) for the calculation of the spectum and pressure of particles accelerated at a modified shock . we describe the appearance of multiple solutions in section [ sec : multiplesolutions ] , and the comparison with the method of malkov ( 1997 ) in section [ sec : comparisonmalkov ] . in section [ sec : injectionrecipe ] we introduce a recipe for the injection of particles from the thermal pool . this recipe is then used in section [ sec : singlesolutions ] to show how the regions of parameter space where multiple solutions appear are drastically reduced by the self - regulated injection . in section [ sec : escape ] we discuss the efficiency of particle acceleration at modified shocks , and stress the role of escape of particles from upstream infinity . the consequences of the cosmic ray modification on the shock heating are investigated in section [ sec : heating ] . we conclude in section [ sec : conclusions ] . in this section we describe the method proposed by blasi ( 2002 , 2004 ) for the calculation of the spectrum and pressure of the particles accelerated at a shock surface , when the reaction of the particles is taken into account . no seed particles are included here . the equation that describes the diffusive transport of particles in one dimension is @xmath1 - u \frac{\partial f ( x , p)}{\partial x } + \ ] ] @xmath2 where we assumed stationarity ( @xmath3 ) . the @xmath4 axis is oriented from upstream to downstream , as in fig . the pressure of the accelerated particles slows down the fluid upstream before it crosses the shock surface , therefore the gas velocity at upstream infinity , @xmath5 , is different from @xmath6 , the fluid speed just upstream of the shock . the injection term is taken in the form @xmath7 . as a first step , we integrate eq . [ eq : trans ] around @xmath8 , from @xmath9 to @xmath10 , denoted in fig . 1 as points `` 1 '' and `` 2 '' respectively , so that the following equation can be written : @xmath11_2 - \left [ d \frac{\partial f}{\partial x}\right]_1 + \frac{1}{3 } p \frac{d f_0}{d p } ( u_2 - u_1 ) + q_0(p)= 0,\ ] ] where @xmath6 ( @xmath12 ) is the fluid speed immediately upstream ( downstream ) of the shock and @xmath13 is the particle distribution function at the shock location . by requiring that the distribution function downstream is independent of the spatial coordinate ( homogeneity ) , we obtain @xmath14_2=0 $ ] , so that the boundary condition at the shock can be rewritten as @xmath11_1 = \frac{1}{3 } p \frac{d f_0}{d p } ( u_2 - u_1 ) + q_0(p ) . \label{eq : boundaryshock}\ ] ] we can now perform the integration of eq . [ eq : trans ] from @xmath15 to @xmath9 ( point `` 1 '' ) , in order to take into account the boundary condition at upstream infinity . using eq . [ eq : boundaryshock ] we obtain @xmath16 @xmath17 we introduce the quantity @xmath18 defined as @xmath19 whose physical meaning is instrumental to understand the nonlinear reaction of particles . the function @xmath18 is the average fluid velocity experienced by particles with momentum @xmath20 while diffusing upstream away from the shock surface . in other words , the effect of the average is that , instead of a constant speed @xmath6 upstream , a particle with momentum @xmath20 experiences a spatially variable speed , due to the pressure of the accelerated particles . since the diffusion coefficient is in general @xmath20-dependent , particles with different energies _ feel _ a different compression coefficient , higher at higher energies if , as expected , the diffusion coefficient is an increasing function of momentum ( see ( blasi 2004 ) for further details on the meaning of the quantity @xmath18 ) . with the introduction of @xmath18 , eq . [ eq : step ] becomes @xmath21 + q_0(p ) = 0 , \label{eq : step1}\ ] ] where we used the fact that @xmath22.\ ] ] the solution of eq . [ eq : step1 ] for a monochromatic injection at momentum @xmath23 is @xmath24 @xmath25\right\}=\ ] ] @xmath26 @xmath27\right\}. \label{eq : inje}\ ] ] here we used @xmath28 , with @xmath29 the gas density immediately upstream ( @xmath9 ) and @xmath30 the fraction of the particles crossing the shock which take part in the acceleration process . here we introduced the two quantities @xmath31 and @xmath32 , which are respectively the compression factor at the gas subshock and the total compression factor between upstream infinity and downstream . for a modified shock , @xmath33 can attain values much larger than @xmath34 and more in general , much larger than @xmath35 , which is the maximum value achievable for an ordinary strong non - relativistic shock . the increase of the total compression factor compared with the prediction for an ordinary shock is responsible for the peculiar flattening of the spectra of accelerated particles that represents a feature of nonlinear effects in shock acceleration . in terms of @xmath34 and @xmath33 the density immediately upstream is @xmath36 . in eq . [ eq : inje ] we can introduce a dimensionless quantity @xmath37 so that @xmath38 @xmath39 the nonlinearity of the problem reflects in the fact that @xmath40 is in turn a function of @xmath13 as it is clear from the definition of @xmath18 . in order to solve the problem we need to write the equations for the thermodynamics of the system including the gas , the cosmic rays accelerated from the thermal pool and the shock itself . the velocity , density and thermodynamic properties of the fluid can be determined by the mass and momentum conservation equations , with the inclusion of the pressure of the accelerated particles . we write these equations between a point far upstream ( @xmath15 ) , where the fluid velocity is @xmath5 and the density is @xmath41 , and the point where the fluid velocity is @xmath18 ( density @xmath42 ) . the index @xmath20 denotes quantities measured at the point where the fluid velocity is @xmath18 , namely at the point @xmath43 that can be reached only by particles with momentum @xmath44 ( this is clearly an approximation , but as shown in section [ sec : comparisonmalkov ] it provides a good agreement with other calculations where this approximation is not used ) . the mass conservation implies : @xmath45 conservation of momentum reads : @xmath46 where @xmath47 is the gas pressure at the point @xmath48 and @xmath49 is the pressure of accelerated particles at the same point ( we use the symbol @xmath50 to mean _ cosmic rays _ , in the sense of _ accelerated particles _ ) . the mass and momentum escaping fluxes in the form of accelerated particles have reasonably been neglected . note that at this point the equation for energy conservation has not been used . our basic assumption , similar to that used in ( eichler 1984 ) , is that the diffusion is @xmath20-dependent and more specifically that the diffusion coefficient @xmath51 is an increasing function of @xmath20 . therefore the typical distance that a particle with momentum @xmath20 travels away from the shock is approximately @xmath52 , larger for high energy particles than for lower energy particles increases with @xmath20 faster than @xmath18 does , therefore @xmath53 is a monotonically increasing function of @xmath20 . ] . as a consequence , at each given point @xmath43 only particles with momentum larger than @xmath20 are able to affect appreciably the fluid . strictly speaking the validity of the assumption depends on how strongly the diffusion coefficient depends on the momentum @xmath20 ( see section [ sec : comparisonmalkov ] ) . since only particles with momentum @xmath44 can reach the point @xmath48 , we can write @xmath54 where @xmath55 is the velocity of particles with momentum @xmath20 , @xmath56 is the maximum momentum achievable in the specific situation under investigation . from eq . [ eq : pressure ] we can see that there is a maximum distance , corresponding to the propagation of particles with momentum @xmath56 such that at larger distances the fluid is unaffected by the accelerated particles and @xmath57 . the equation for momentum conservation is : @xmath58 + \frac{1}{\rho_0 u_0 ^ 2 } \frac{d p_{cr}}{dp } = 0.\ ] ] using the definition of @xmath59 and multiplying by @xmath20 , this equation becomes @xmath60 = \frac{4\pi}{3 \rho_0 u_0 ^ 2 } p^4 v(p ) f_0(p ) , \label{eq : eqtosolve}\ ] ] where @xmath13 is known once @xmath40 is known . [ eq : eqtosolve ] is therefore an integral - differential nonlinear equation for @xmath40 . the solution of this equation also provides the spectrum of the accelerated particles . the last missing piece is the connection between @xmath34 and @xmath33 , the two compression factors appearing in eq . [ eq : inje1 ] . the compression factor at the gas shock around @xmath8 can be written in terms of the mach number @xmath61 of the gas immediately upstream through the well known expression @xmath62 on the other hand , if the upstream gas evolution is adiabatic , then the mach number at @xmath9 can be written in terms of the mach number of the fluid at upstream infinity @xmath63 as @xmath64 so that from the expression for @xmath34 we obtain @xmath65^{\frac{1}{\gamma_g+1}}. \label{eq : rsub_rtot}\ ] ] now that an expression between @xmath34 and @xmath33 has been found , eq . [ eq : eqtosolve ] basically is an equation for @xmath34 , with the boundary condition that @xmath66 . finding the value of @xmath34 ( and the corresponding value for @xmath33 ) such that @xmath66 also provides the whole function @xmath40 and , through eq . [ eq : inje1 ] , the distribution function @xmath67 . if the reaction of the accelerated particles is small , the _ test particle _ solution is recovered . in the problem described in the previous section there are several independent parameters . while the mach number of the shock and the maximum momentum of the particles are fixed by the physical conditions in the environment , the injection momentum and the acceleration efficiency are free parameters . the procedure to be followed to determine the solution was defined in ( blasi 2002 ) : the basic problem is to find the value of @xmath34 ( and therefore of @xmath33 ) for which @xmath66 . in fig . 2 we plot @xmath68 as a function of @xmath33 , for @xmath69 , @xmath70 and @xmath71 in the left panel and @xmath72 in the right panel ( @xmath73 here is the mass of protons ) . the parameter @xmath30 was taken @xmath74 in the left panel and @xmath75 in the right panel . the different curves refer to different choices of the mach number at upstream infinity . the physical solutions are those corresponding to the intersection points with the horizontal line @xmath66 , so that multiple solutions occur for those values of the parameters for which there is more than one intersection with @xmath66 . these solutions are all physically acceptable , as far as the conservation of mass , momentum and energy are concerned . [ fig : sol ] it can be seen from both panels in fig . 2 that for low values of the mach number , only one solution is found . this solution may be significantly far from the quasi - linear solution . indeed , for @xmath76 the solution corresponds to @xmath77 , instead of the usual @xmath78 solution expected in the linear regime . lower values of the mach number are required to fully recover the linear solution . when the mach number is increased , there is a threshold value for which three solutions appear , one of which is the quasi - linear solution . for very large values of the mach number the solution becomes one again , and it coincides with the quasi - linear solution . [ fig : multi2 ] in fig . 3 we show the appearance of the multiple solutions for the case @xmath69 , @xmath70 and @xmath72 with mach number @xmath79 ( @xmath71 and @xmath79 ) in the left ( right ) panel . the curves here are obtained by changing the value of @xmath30 . the same comments we made for fig . 2 apply here as well : low values of @xmath30 correspond to weakly modified shocks , while for increasingly larger efficiencies multiple solutions appear . the solution becomes one again in the limit of large efficiencies , and it always corresponds to strongly modified shocks . the problem of multiple solutions is not peculiar of the kinetic approaches to the non - linear theories of particle acceleration at shock waves . the same phenomenon was in fact found initially in two - fluid models ( drury & vlk 1980 , 1981 ) , where however no information on the spectrum of the accelerated particles and on the injection efficiency was available . multiple solutions were also found by malkov ( 1997 ) and malkov et al . ( 2000 ) , in the context of a semi - analytical kinetic approach . aside from the technical differencies between that method and the one proposed by blasi ( 2002 , 2004 ) , the main difference is in the fact that the former requires the knowledge of the exact expression for the diffusion coefficient as a function of the momentum of the particles , while the latter only requires that such diffusion coefficient is an increasing function of the particle momentum . while the first approach may provide us with an _ exact _ solution to the problem , the second is in fact more practical , in the sense of providing an approximate solution even in those cases , the majority , in which no detailed information on the diffusion properties of the fluid is available . the solution provided in ( blasi 2002 , 2004 ) is particularly accurate when the diffusion coefficient is bohm - like , @xmath80 , expected in the case of saturated self - generation of waves in the vicinity of the shock surface ( lagage & cesarsky 1983 ) . we will now discuss the quantitative comparison between the results of malkov ( 1997 ) and those of blasi ( 2002 , 2004 ) , by considering a single situation in which multiple solutions are predicted ( in both approaches ) , and determining the spectra and compression factors in both methods . we start with briefly summarizing the approach of malkov ( 1997 ) . the following flow potential is introduced there : @xmath81 which is used as a new independent spatial variable to replace @xmath4 . using the flow potential , it is possible to define an integral transformation of the flow profile as follows : @xmath82 \frac{du}{dx } dx,\ ] ] where @xmath83 is the spectral index of the particle distribution function and @xmath51 is the diffusion coefficient , which is assumed to be independent of the position . an integral equation for @xmath84 can be derived by applying eq . [ inttra ] to the @xmath4derivative of the euler equation ( malkov 1997 ; malkov et al . 2000 ) : @xmath85 @xmath86^{-1 } \times\ ] ] @xmath87,\ ] ] where @xmath88 is an injection parameter defined as @xmath89 and related to the compression factor by the following equation ( malkov 1997 ) : @xmath90 @xmath91\bigg\}^{-1}. \label{mal2}\ ] ] here @xmath92 is the compression factor in the shock precursor . [ eq : rsub_rtot ] , [ mal1 ] and [ mal2 ] form a closed system that can be solved numerically . before showing the results it is worth noticing that the injection parameters @xmath30 and @xmath88 adopted in the two approaches are defined in two non equivalent ways . however , the relation between @xmath30 and @xmath88 can easily be found by using eqs . [ eq : inje1 ] and [ efficiency ] and can be written as : @xmath93 one may notice that this relation contains the compression factors @xmath33 and @xmath34 , which are what we are searching for . this fact implies that three solutions characterized by the same value of @xmath30 may correspond to three distinct values of @xmath88 . in order to compare the results of the two different approaches we consider a shock having mach number @xmath94 and temperature at upstream infinity @xmath95 . we set the value of the injection and maximum momenta equal to @xmath96 and @xmath97 respectively and we adopt an efficiency @xmath98 . using the approach proposed by blasi ( 2002 ) , we find three solutions , characterized by the values of the compression factor @xmath99 . the last solution is the quasi - linear one , in which the precursor is very weak . we adopt now these three values for the precursor compression factor to solve the system of equations given by eq . [ eq : rsub_rtot ] , [ mal1 ] , and [ mal2 ] ) for different choices of the diffusion coefficient . in fig . [ fig : malk_vel ] we plot the velocity profiles for the three solutions , as derived with the method of blasi ( 2002 , 2004 ) and detailed above ( solid line ) . in the figure @xmath37 with @xmath18 defined through eq . [ eq : up ] for the method of blasi ( 2002 , 2004 ) . it is easy to show that @xmath40 is related to the @xmath84 through the relation @xmath100 . the dotted and dashed lines are the results obtained with the calculation of malkov ( 1997 ) with a bohm and kolmogorov diffusion respectively . for bohm diffusion the two approaches give very similar results . for kolmogorov diffusion the differencies are larger , as expected . in fig . [ fig : malk_spec ] we plot the spectra of the accelerated particles , as obtained in this paper ( solid line ) and for a bohm ( dotted line ) and kolmogorov ( dashed line ) diffusion coefficient , as derived by carrying out the calculation of malkov ( 1997 ) . we recall that from the theoretical point of view the bohm diffusion is in fact what should be expected in the proximity of a shock if the turbulence necessary for the acceleration is strong and generated by the same cosmic rays that are being accelerated ( lagage & cesarsky 1983 ) . in this perspective , we look at the results illustrated in this section as very encouraging in using the approach presented in blasi ( 2002 , 2004 ) , since it is simple and at the same time accurate in reproducing the major physical aspects of particle acceleration at cosmic ray modified shocks . the presence of multiple solutions is typical of many non - linear problems and should not be surprising from the mathematical point of view . in terms of physical understanding however , multiple solutions may be disturbing . the typical situation that takes place in nature when multiple solutions appear in the description of other non linear systems is that ( at least ) one of the solutions is unstable and the system _ falls _ in a stable solution when perturbed . the stable solutions are the only ones that are physically meaningful . some attempts to investigate the stability of cosmic ray modified shock waves have been made by mond & drury ( 1998 ) and toptygin ( 1999 ) , but all of them refer to the two - fluid models . a step forward is being carried out by blasi & vietri ( in preparation ) in the context of kinetic models . in addition to the stability , another issue that enters the physical description of our problem is the identification of possible processes that determine some type of backreaction on the system . it may be expected that when some types of processes of self - regulation are included , the phenomenon of multiple solutions is reduced . in this section we investigate the type of reaction that takes place when a self - consistent , though simple , recipe for the injection of particles from the thermal pool is adopted . this recipe is similar to that proposed by kang , jones & gieseler ( 2002 ) in terms of the underlying physical interpretation of the injection , but probably simpler in its implementation . for non - relativistic shocks , the distribution of particles downstream is quasi - isotropic , so that the flux of particles crossing the shock surface from downstream to upstream can be written as @xmath101 , \label{eq : flux1}\ ] ] where @xmath55 is the velocity of particles with momentum @xmath20 and @xmath102 is the shock speed in the frame comoving with the downstream fluid . the term @xmath103 is the component along the direction perpendicular to the shock surface of the velocity of particles with momentum @xmath20 moving in the direction @xmath104 . it follows that the flux of particles moving tangent to the shock surface ( namely with @xmath105 ) is zero . we recall that , having in mind collisionless shocks , the typical thickness of the shock , @xmath106 , is the collision length associated with the magnetic interactions that give rise to the formation of the discontinuity . useless to say that these interactions are all but well known , and at present the best we can do is to attempt a phenomenological approach to take them into account , without having to deal with their detailed physical understanding . it is however worth recalling that many attempts have been made to tackle the problem of injection at a more fundamental level ( malkov 1998 ; malkov & vlk 1995 ; malkov & vlk 1998 ) . here , we consider the reasonable situation in which @xmath107 , where @xmath108 is the larmor radius of the particles in the downstream fluid that carry most of the thermal energy , namely those with momentum @xmath109 ( @xmath110 here is the momentum of the particles in the thermal peak of the maxwellian distribution in the downstream plasma , having temperature @xmath111 ) . we stress here the important point that the temperature of the downstream gas ( and therefore @xmath112 ) is determined by the shock strength , which in the presence of accelerated particles , is affected by the pressure of the non - thermal component . in particular , the higher the efficiency of the shock as a particle accelerator , the weaker its efficiency in terms of heating of the background plasma ( see section [ sec : heating ] ) . for collisionless shocks , it is not clear whether the downstream plasma can actually be thermalized and the distribution function be a maxwellian . on the other hand , it is generally assumed that this is the case , so that in the following we consider the case in which the bulk of the background plasma is thermal and has a maxwellian spectrum at temperature @xmath113 given by the generalized rankine - hugoniot relations in the presence of accelerated particles ( see section [ sec : nonlin ] ) . for modified shocks , the points discussed above apply to the so - called subshock , where the injection of particles from the thermal pool is expected to take place . we recall that for strongly modified shocks the subshock is weak , and rather inefficient in the heating of the background plasma . [ fig : structure ] from eq . [ eq : flux1 ] we get : @xmath114 where we assumed that the temperature downstream implies non - relativistic motion of the quasi - thermal particles ( @xmath115 ) . in eq . [ eq : flux2 ] we write the minimum momentum in terms of a parameter @xmath116 , such that @xmath117 . with this formalism , the particles that can cross the shock surface are those that satisfy the condition : @xmath118 the parameter @xmath116 defines the thickness of the shock in units of the gyration radius of the bulk of the thermal particles . our recipe for injection is pictorially illustrated in fig . 6 : thermal particles have a pathlength smaller than the shock thickness and can not cross the shock surface , being advected downstream before the crossing occurs . only particles with momentum sufficiently larger than the thermal momentum of the downstream particles can actually return upstream and be accelerated . in the following we will neglect the fluid speed @xmath102 compared with @xmath55 , which is a good approximation if the injected particles are sufficiently more energetic than the thermal particles . this is done only to make the interpretation of the result simpler , but there is no technical difficulty in keeping the dependence of the results on @xmath102 . we introduce an effective injection momentum @xmath119 defined by the equation : @xmath120 which in terms of dimensionless quantities , with @xmath121 reads : @xmath122 it is easy to show that @xmath123 for @xmath124 ( half a larmor rotation of the particles with momentum @xmath125 inside the thickness of the shock ) and @xmath126 for @xmath127 ( one full larmor rotation of the particles with momentum @xmath125 inside the _ thickness _ of the shock . the fraction of particles at momentum @xmath128 times larger than the thermal one is @xmath129 for @xmath130 and @xmath131 for @xmath132 . the actual values of @xmath128 are expected to be somewhat higher if the effect of advection with the downstream fluid is not neglected . the sharp decrease in the fraction of _ leaking _ particles that may take part in the acceleration process is due to the exponential behaviour of the maxwellian at large momenta . although the fraction of particles in the maxwellian that get accelerated only depends on the parameter @xmath128 which in turn is expected to keep the information about the microscopic structure of the shock , the absolute number of and energy carried by these particles depend on the temperature of the downstream gas , which is an output of our calculation . this simple argument serves as an explanation of the physical reason why there is a nonlinear reaction on the system due to injection . if the parameter @xmath128 is assumed to be determined by the microphysics of the shock , and if we adopt our simple recipe to describe such microphysics , then the shock thickness is easily estimated once the temperature of the downstream gas is known , and the latter can be calculated from the modified rankine - hugoniot relations . the parameter @xmath30 in eq . [ eq : inje1 ] is no longer a free parameter , being related in a unique way to the parameter @xmath128 and to the physical conditions at the subshock . the condition that fixes @xmath30 is that the total number of particles in the non - thermal spectrum equals the number of particles in the maxwellian at momenta larger than @xmath23 . due to the very strong dependence of the spectrum on the momentum for both the maxwellian and the power law at low momenta , the condition described above is very close to require the continuity of the distribution function , namely that @xmath133 . in the following we adopt this condition for the calculations . this can be shown to imply the following expression for @xmath30 : @xmath134 we recall that the compression factor at the subshock , @xmath34 , approaches unity when the shock becomes cosmic ray dominated . this makes evident how the backreaction discussed above works : when the shock becomes increasingly more modified , the efficiency @xmath30 tends to decrease , limiting the amount of energy that can be channelled in the non - thermal component . although the recipe provided here is certainly far from representing the complexity of the reality of injection of particles from the thermal pool , it may be considered as a useful attempt to include the main physical aspects of this phenomenon . in this section we describe the role played by the injection recipe discussed above for the appearance of multiple solutions . it can be expected that the phenomenon is somewhat reduced because , as discussed in the previous section , the injection provides an efficient backreaction mechanism on the shock as a particle accelerator . indeed we find that the appearance of multiple solutions is drastically reduced , and that the phenomenon still exists only in regions of the parameter space which are very narrow and of limited physical interest . in the quantitative calculations we use the value @xmath135 for the injection parameter , as suggested by the simple estimate in section [ sec : injectionrecipe ] and as suggested also in the numerical work of kang & jones ( 1995 ) . the dependence of the effect on the value of @xmath128 is discussed below . in fig . [ fig : single ] we illustrate the dramatic change in the physical picture by plotting @xmath68 as a function of @xmath33 for @xmath135 and adopting the same values for the parameters as those used in obtaining fig . the efficiency @xmath30 is now calculated according with the recipe described in the previous section . it can be seen very clearly that when the mach number of the shock is changed , there is a single solution ( compare with fig . 2 where multiple solutions where found for the same values of the parameters , but without thermal leakage ) . [ fig : parspace ] the appearance of multiple solutions can be investigated in the whole parameter space , in order to define the regions where the phenomenon appears , when it does . in fig . 8 we highlight the regions where there are multiple solutions ( dark regions ) in a plane @xmath136 , for different values of the mach number of the shock . in most cases the dark regions are very narrow and cover a region of values of @xmath128 which is rather high ( small efficiency ) . in fig . [ fig : transition ] we plot the value of @xmath33 as a function of @xmath128 for @xmath137 , @xmath138 and @xmath139 from left to right . the line is continuous when there are no multiple solutions and dashed when multiple solutions appear . the dashed regions are , as stressed above , rather narrow . for instance , for @xmath140 there are multiple solutions only for @xmath141 . any small perturbation of the system that changes the values of @xmath128 at the percent level implies that the system shifts to one of the single solutions if it is sitting in the intermediate solution before the perturbation . the sharp transition between the strongly modified solution and the quasi - linear solution when @xmath128 is increased suggests that the intermediate solution may be unstable , though a formal demonstration can not be provided here . in order to make sure that this is the case , a careful analysis of the stability is required , and will be presented in a forthcoming publication ( blasi & vietri , in preparation ) . on the other hand , a previous study , carried out in the context of the two - fluid models , showed that when multiple solutions are present , the solution with intermediate efficiency is in fact unstable to corrugations of the shock surface ( mond and drury 1998 ) . it is rather remarkable that the kinetic model of blasi ( 2002 , 2004 ) does not require explicitely the use of the equation for energy flux conservation . however , once the solution of the kinetic problem has been found , the equation for conservation of the energy flux provides very useful information , as we show below . the equation can be written in the following form : @xmath142 @xmath143 where @xmath144 is the flux of particles escaping at the maximum momentum from the upstream section of the fluid ( berezhko & ellison , 1999 ) . notice that this term is usually neglected in the linear approach to particle acceleration at shock waves because the spectra are steep enough that , in most cases , we can neglect the flux of particles leaving the system at the maximum momentum . the fact that particles leave the system make the upstream fluid behave as a radiative fluid , and makes it more compressible . this is a crucial consequence of particle acceleration at modified shocks , and is shown here to be a natural consequence of energy conservation . [ fig : escape ] in eq . [ eq : energy ] we can divide all terms by @xmath145 and calculate the normalized escaping flux : @xmath146 @xmath147 from momentum conservation at the subshock we also have : @xmath148 so that the escaping flux only depends upon the _ environment _ parameters ( for instance the mach number at upstream infinity ) and the compression parameter @xmath34 which is part of the solution . note also that the adiabatic index for cosmic rays , @xmath149 , is here calculated self - consistently as : @xmath150 where @xmath151 is the energy density in the form of accelerated particles and @xmath152 is the kinetic energy of a particle with momentum @xmath20 . it can be easily seen that @xmath153 when the energy budget is dominated by the particles with @xmath154 ( namely for strongly modified shocks ) and @xmath155 for weakly modified shocks . in eq . [ eq : energy_norm ] the term @xmath156 is clearly the fraction of flux which is advected downstream with the fluid . in fig . 10 we plot the escaping flux ( @xmath157 ) , the advected flux ( @xmath158 ) and the sum of the two ( @xmath159 ) normalized to the incoming flux @xmath145 , as functions of the mach number at upstream infinity @xmath63 . here we used @xmath160 , and @xmath135 , while the maximum momentum has been chosen as @xmath161 in the left panel and @xmath162 in the right panel . several comments are in order : * at low mach numbers the escaping flux is inessential , as one would expect for a weakly modified shock . we recall that the escaping flux is due to the particles with momentum @xmath56 leaving the system from upstream infinity . for a weakly modified shock at low mach number the spectrum is steeper than @xmath163 , so that the energy carried by the highest energy particles is a small fraction of the total . * at large mach numbers the shock becomes increasingly more cosmic ray dominated , and for the cases at hand the total efficiency gets very close to unity , meaning that the shock behaves as an extremely efficient accelerator . at mach numbers around 4 on the other hand the total efficiency is around @xmath164 for @xmath161 and @xmath165 for @xmath162 , dropping fast below mach number 4 . clearly the efficiency would be higher in this region for lower values of the parameter @xmath128 . * despite the fact that the total efficiency of the shock as a particle accelerator is close to unity at large mach numbers , the fraction of the incoming energy which is actually advected toward downstream infinity is only @xmath166 at @xmath167 for @xmath161 . most of the enegy flux in this case is in fact in the form of energy escaping from upstream infinity at the highest momentum @xmath56 . for @xmath162 the normalized advected flux roughly saturates at @xmath168 and is comparable with the escape flux at the same mach number . for a distant observer these escaping particles would have a spectrum close to a delta function around @xmath56 . energy conservation has the natural consequence that a smaller fraction of the kinetic energy of the fluid is converted into thermal energy of the downstream plasma in cosmic ray modified shocks , compared with the case of ordinary shocks . the reduction of the heating at nonlinear shock waves is fully confirmed by our calculation in the context of the injection recipe introduced in section [ sec : injectionrecipe ] . in fig . [ fig : jump ] we plot the temperature jump between downstream infinity ( at temperature @xmath111 ) and upstream infinity ( at temperature @xmath169 ) . the thick solid line is the jump as predicted by the standard rankine - hugoniot relations without cosmic rays . the other lines represent the temperature jump at cosmic ray modified shocks with @xmath170 ( thin solid line ) , @xmath171 ( dashed line ) , @xmath172 ( dotted line ) and @xmath173 ( dash - dotted line ) . such a drastic reduction of the downstream temperature is expected to reflect directly in the thermal emission of the downstream gas in those environments in which collisions are relevant . note that for strongly modified shocks the compression factor between upstream infinity and downstream are much larger than for ordinary shocks , so that the downstream turns out to be denser but colder than in the linear case . the missing energy ends up in the form of accelerated particles . the effect of suppression of the heating in cosmic ray modified shocks also appears in the spectra of the particles ( thermal plus non - thermal ) in the shock vicinity . in fig . [ fig : spec ] we show these spectra ( including the maxwellian thermal bump ) for @xmath160 , @xmath135 and @xmath174 . the vertical dashed line shows the position of the thermal peak as expected in the absence of accelerated particles . in fact this position should depend on the mach number , but the dependence is very weak for large mach numbers . the positions of the thermal peaks clearly show the effect of cooler downstream gases for modified shocks . at the same time , the effect is accompanied by increasingly more modified spectra of accelerated particles , with most of the energy pushed toward the highest momenta . the efficiency of the first order fermi acceleration at shock fronts depends in a crucial way upon details of the mechanism that determines the injection of a small fraction of the particles from the thermal pool to the _ acceleration box_. in reality the processes of formation of a collisionless shock wave , of plasma heating and particle acceleration are expected to be all parts of the same problem , though on different spatial scales . we hide our lack of knowledge of the microphysics of the shock structure in a simple recipe for injection , in which the particles that take part in the acceleration process are those that have momentum larger by a factor @xmath128 than the momentum of the thermal particles in the downstream fluid . this is motivated by the fact that for collisionless shocks the thickness of the shock surface is determined by the gyromotion of the bulk of the thermal particles . we estimated that @xmath175 . this recipe implies that the fraction of particles that get accelerated is rather small ( @xmath176 ) . we implemented this recipe in a calculation of the non - linear reaction of cosmic rays on the shock structure proposed by blasi ( 2002 , 2004 ) . similarly to other models , also this approach shows the appearance of multiple solutions , for a wide choice of the parameters of the problem . when the simple model of particle injection is used , this phenomenon is drastically reduced : the multiple solutions disappear for most of the parameter space , and when they appear they look as narrow strips in the parameter space , at the boundary between unmodified and modified shocks . we argued that this result suggests that the narrow regions may signal the transition between two stable solutions , although this needs further confirmation through detailed analyses of the stability of the solutions . this interpretation seems to be supported in part by the calculations of mond & drury ( 1998 ) , that showed that when three solutions are present , the intermediate one is indeed unstable for small corrugations of the shock front . this calculations was however performed in the context of a two - fluid model , while an investigation of the stability for kinetic models is still lacking . we find that the phenomenology of the particle acceleration at modified shocks is characterized by three main features : * the modification of the shock increases with the mach number of the fluid . for low mach numbers the quasi - linear solution is recovered , but departures from it are evident already at relatively low mach numbers . the modification of the spectra manifests itself with a hardening at high momenta and a softening at low momenta . the @xmath177 shows a characteristic dip at intermediate momenta , typically around @xmath178 ( for very large values of @xmath56 , the dip can be found at even larger momenta , which is of interest for the acceleration of ultra - high energy cosmic rays ) . * the total efficiency for particle acceleration saturates at large mach numbers at a number of order unity . however , as shown in fig . 10 , the largest fraction of the energy is not advected downstream but rather escapes from upstream infinity at the maximum momentum . this effect was also discussed in the context of the simple model by berezhko & ellison ( 1999 ) . * the high efficiency for particle acceleration reflects in a reduced ability of cosmic ray modified shocks in the heating of the background plasma . this effect is at the very basis of the backreaction introduced by the injection recipe on the acceleration process , and determines the saturation of the total efficiency for large mach numbers . the heating suppression is shown in fig . [ fig : jump ] and in fig . [ fig : spec ] . pb is grateful to e. amato , l.oc . drury , d. ellison and m. vietri for many useful discussions . the research of pb was partially funded though cofin-2004 at the arcetri astrophysical observatory . sg gratefully acknowledges support from the alexander von humboldt foundation . he also thanks stephane barland for useful discussions on nonlinear systems .
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the dynamical reaction of the particles accelerated at a shock front by the first order fermi process can be determined within kinetic models that account for both the hydrodynamics of the shocked fluid and the transport of the accelerated particles .
these models predict the appearance of multiple solutions , all physically allowed .
we discuss here the role of injection in selecting the _ real _ solution , in the framework of a simple phenomenological recipe , which is a variation of what is sometimes referred to as _ thermal leakage_. in this context we show that multiple solutions basically disappear and when they are present they are limited to rather peculiar values of the parameters .
we also provide a quantitative calculation of the efficiency of particle acceleration at cosmic ray modified shocks and we identify the fraction of energy which is advected downstream and that of particles escaping the system from upstream infinity at the maximum momentum .
the consequences of efficient particle acceleration for shock heating are also discussed .
cosmic rays ; high energy ; origin ; acceleration
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most of the neutron star low mass x ray binaries can be divided into two classes , z and atoll sources , based on the correlated behavior of their timing properties at low frequencies ( @xmath0 hz ) and their x ray spectral properties @xcite . both classes show quasi - periodic oscillations ( qpos ) with frequencies ranging from a few hundred hz to more than 1000 hz ( kilohertz qpos ) . the low frequency part of the power spectra of both classes is usually dominated by a similar broad band limited noise component , but it is unclear if these are physically the same component @xcite . in addition to the band limited noise both classes show several quasi - periodic oscillations below 200 hz . in the z sources these are named after the branch of the z track in the color color diagram ( see below ) where they mostly occur : horizontal ( hbos ) , normal ( nbos ) and flaring branch oscillations ( fbos ) . the hbo shows a sub and a second harmonic at @xmath1 and @xmath2 times the frequency of the main peak , and sometimes a peak at @xmath3 times the frequency of the hbo @xcite . obviously , the most straightforward interpretation of this is that the sub hbo is the fundamental frequency , with second , third and fourth harmonics all observed @xcite . below 200 hz the atoll sources show several lorentzian components ( see e.g. * ? ? ? note that these components are called lorentzian and not qpo ; this is because these features are sometimes too broad ( fwhm @xmath4 centroid frequency/2 ) to be classified as a qpo . in the atoll sources all components become broader as their characteristic frequency decreases @xcite . it has been suggested that the hbo in the z and the low frequency lorentzian in the atoll sources are the same physical components @xcite . the energy spectrum of neutron star low mass x ray binaries can be usefully parametrized through the use of color color or color intensity diagrams , where a color is the ratio of counts in two different energy bands . as the energy spectrum of a source changes , it moves through these diagrams . the timing properties of both the z and the atoll sources depend on the position of the source in the color color diagram . the z sources move fast through the color color diagram and draw up a z track within hours to days ( see e.g. gx 340 + 0 ; * ? ? ? * ) whereas the typical atoll sources move slowly through the color color diagram and draw up a c shaped track within weeks to months ( see e.g. 4u 172834 ; * ? ? ? * ) , although the `` banana '' part of the track , a curved branch to the bottom and right hand sides of the diagram , is traced out as fast as in the z sources . the slow motion in the island part of the diagram , to the left hand and top sides , combined with observational windowing , tends to lead to the formation of isolated patches of data points , which is the origin of the term `` island state '' @xcite . whereas the continuum power spectra of the banana state are dominated by a power law component at low frequencies with perhaps a weak band limited noise component ( which becomes stronger as the source approaches the island state ) , the island state power spectra are dominated by the band limited noise . as the source moves away from the banana into the island state the count rate drops , the x ray spectrum becomes harder and the band limited noise becomes stronger while its characteristic frequency decreases . in the most extreme island states ( 4u 160852 ; @xcite ; @xcite ; 4u 0614 + 09 ; @xcite ; @xcite ; 4u 172834 ; @xcite ; @xcite ; 4u 170544 ; @xcite ; @xcite ; @xcite ; aql x1 ; @xcite ; ks 1731260 ; @xcite ) the source is faint and hard and the band limited noise is strong ; this is the state in which neutron stars are most similar to bhcs in the low hard state @xcite . because of the low count rates and slow motion through the color color diagram the precise properties of the extreme island state have been hard to ascertain , although observations of 4u 160852 with tenma @xcite indicated the existence of an extended branch in this state that was traced out over an interval of weeks . recently , @xcite and @xcite used large data sets from the rossi x ray timing explorer ( rxte ) to study the color color diagrams of several of the z and atoll sources , which contain interesting additional information about , in particular , the nature of the extreme island state in the transient atoll sources 4u 160852 , 4u 170544 and aql x1 . they suggested that the atoll sources trace out similar three branch patterns as the z sources , with the extreme island state cast in the role of z source horizontal branch . however , these authors did not address the timing properties of the sources they studied . in this paper , we continue our work on the correlated x ray spectral and timing behavior of z and atoll sources using rxte @xcite with an analysis of 4u 160852 . in these previous analyses , we have found strong correlations between the behavior of the timing features and the position of the source in the color color diagram in both z and atoll sources . 4u 160852 is a transient source that shows outbursts with a recurrence time varying between 80 days and several years @xcite . frequency qpos were discovered in the `` island '' state of 4u 160852 with the ginga satellite @xcite and kilohertz qpos were discovered with rxte @xcite . the source was included in the samples studied by @xcite and @xcite and was one of the sources that was reported to show z like behavior in the color color diagram . by connecting the timing with the energy spectral properties we can test whether 4u 160852 indeed behaves as a z source . if this is true one would expect the power spectral properties to change smoothly along the z track , as is the case in the z sources . we find that this is not the case . in [ sec.disc ] we also perform a more general comparison of frequencies observed in the z and the atoll sources . parallel tracks in color color and color intensity diagrams were first observed in the z sources @xcite and later also in the atoll sources 4u 163653 @xcite , 4u 1735 - 44 @xcite and 4u 0614 + 09 @xcite . recently @xcite reported further parallel tracks in the color intensity diagrams of several atoll sources . the parallel track phenomenon in the plot of lower kilohertz qpo frequency versus intensity was first observed in 4u 0614 + 09 @xcite and 4u 160852 @xcite and has since been observed in several more atoll sources ( see e.g. * ? ? ? it has been suggested that the parallel tracks in the plot of intensity versus frequency of the lower kilohertz qpo and the parallel tracks in the color intensity diagrams might be the same phenomena @xcite . van der klis ( 2001 ) has proposed a possible explanation for this parallel track phenomenon in terms of a filtered response of part of the x ray emission to changes in the mass accretion rate through the disk . the frequency versus intensity parallel track phenomenon is particularly clear in 4u 160852 @xcite . in [ sec.tracks ] we investigate the relation between these frequency versus intensity parallel tracks and the parallel tracks in the color intensity diagrams . in this analysis we use all available public data from 1996 march 3 to 2000 may 24 from rxte s proportional counter array ( pca ; for more instrument information see * ? ? ? the data are divided into observations that consist of one to several satellite orbits . we exclude data for which the angle of the source above the earth limb is less than 10 degrees or for which the pointing offset is greater then 0.02 degrees . in our data set we found 7 type i x ray bursts and we exclude those ( @xmath5 s before and @xmath6 s after the onset of each burst ) from our analysis . we use the 16s time resolution standard 2 mode to calculate the colors . for each of the five pca detectors ( pcus ) we calculate a hard color , defined as the count rate in the energy band [email protected] kev divided by the rate in the energy band [email protected] kev , and a soft color , defined as the count rate in the energy band [email protected] kev divided by the rate in the energy band [email protected] kev . per detector we also calculate the intensity , the count rate in the energy band [email protected] kev . to obtain the count rates in these exact energy ranges we interpolate linearly between count rates in the pcu channels . we calculate the colors and intensity for each time interval of 16 s. we subtract the background contribution in each band using the standard background model for the pca version 2.1e . in order to correct for the changes in effective area between the different gain epochs and for the gain drifts within those epochs as well as the differences in effective area between the pcus themselves we used the method introduced by @xcite : for each pcu we calculate the colors of the crab , which can be supposed to be constant in its colors , in the same manner as for 4u 160852 . we then average the 16 s crab colors and intensity per pcu for each day . in figure [ fig.crab ] we can see the clear differences in the soft color trends of crab between the five pcus caused by the effects mentioned above . for each pcu we divide the 16 s color and intensity values obtained for 4u 160852 by the corresponding crab values that are closest in time but within the same gain epoch . we then average the colors and intensity over all pcus . if we multiply the soft color by 2.36 , the hard color by 0.56 and the intensity by 2400 c / s / pcu we approximately recover the observed , uncorrected values for 4u 160852 ( quoted values are averages of the crab colors during the observations over all five pcus ) . to improve statistics we rebin the 16 s color and intensity points to 256 s and exclude all data for which the resulting relative errors are larger than 5% . this led to a loss of about 40 ks of data at count rates below @xmath8 crab . for most of these data neither timing nor colors were of sufficient quality to determine the source state . starting may 12 , 2000 , the propane layer on pcu0 , which functions as an anti coincidence shield for charged particles , was lost . however , as all our data after may 12 , 2000 ( @xmath9 ks ) was excluded because the relative color errors were larger than 5% this was not a issue in our analysis . for the fourier timing analysis we use the 122@xmath10s time resolution event and single bit modes and the 0.95@xmath10s time resolution good xenon modes . we only use data for which all energy channels are available . this led to a loss of about 9 ks of data in the lower banana state . in some observations there was data overflow in the timing modes due to excessive count rates from the source ; we excluded those data . this led to a loss of about 27 ks of data at the highest intensity level ( see also below ) . the power spectra were constructed using data segments of 256 s and 1/8192 s time bins such that the lowest available frequency is 1/256 hz and the nyquist frequency 4096 hz ; the normalization of @xcite was used . to get a first impression of the timing properties at the different dates and positions in the color color diagram , the power spectra were combined per observation . the resulting power spectra were then converted to squared fractional rms . we subtracted a constant poisson noise level estimated between 2000 and 4000 hz where neither noise nor qpos are known to be present . as fit function we use the multi lorentzian function ; a sum of lorentzian components @xcite plus a power law to fit the vlfn ( see * ? ? ? we only include those lorentzians in the fit whose significance based on the error in the power integrated from 0 to @xmath11 is above 3.0 @xmath12 . two to five lorentzian components were generally needed for a good fit . for the lorentzian that is used to fit the band limited noise we fixed the centroid frequency to zero . we plot the power spectra in the power times frequency representation ( @xmath13 ; e.g. * ? ? ? * ; * ? ? ? * ) , where the power spectral density is multiplied by its fourier frequency . for a multi lorentzian fit function this representation helps to visualize a characteristic frequency , @xmath14 , namely , the frequency where each lorentzian component contributes most of its variance per logarithmic frequency interval : in @xmath13 the lorentzian s maximum occurs at @xmath14 ( @xmath15 , where @xmath16 is the centroid and @xmath17 the hwhm of the lorentzian ; * ? ? ? we represent the lorentzian relative width by @xmath18 defined as @xmath19 . in this section we try to get an idea of the timing properties of the source as it moves through the color color diagram . we step through the data in chronological order and look at the timing properties ( per observation , see [ sec.obs ] ) and the position of the source in the color color diagram . in this way we select continuous time intervals for which the power spectrum and the position in the color color diagram remain similar . for each continuous time interval we construct a representative power spectrum by adding up all observations within that interval . to get a first idea of the timing properties we made initial fits to the representative power spectra of the continuous time intervals using the multi lorentzian fit function described in [ sec.obs ] . in most cases the power spectral features present could be directly identified with known components seen in other atoll sources by comparing with the power spectral features of 4u 0614 + 09 and 4u 172834 ( using figures 1 , 2 and 3 of * ? ? ? a more thorough fitting and description of the power spectral features will be presented in [ sec.comb_ps ] where we obtain optimal statistics by constructing representative intervals for all our data by adding up the continuous time intervals that have similar positions in the color color diagram and show similar power spectra . the data on 4u 160852 can be usefully divided into 3 segments . the first segment ranges from 1996 march 3 to december 28 ( the decay of the 1996 outburst , see * ? ? ? * ) , the second segment from 1998 february 3 to september 29 ( the 1998 outburst , see * ? ? ? * ) and the third segment from 2000 march 6 to may 10 ( persistent data ) . in practice most data were available for the second segment ( the 1998 outburst ) so we will present the results for the second segment first . these results can then serve as a template for the rest of the data . in figure [ fig.cc_deel2 ] we show as black points the lightcurve and color color diagram for the second segment of the data ( the 1998 outburst ) . the grey points in the lower frame represent the overall color color diagram including all the data . intensity and colors are normalized to the crab ( see [ sec.obs ] ) . the numbers 917 ( 18 are reserved for the first segment of the lightcurve ) indicate the continuous time intervals for which the power spectra and the position in the color color diagram remain similar ( see [ sec.timingtour ] ) . the data for which no power spectra could be computed due to data overflow ( see also [ sec.obs ] ) are indicated by the larger crosses at the peak of the outburst and in interval 9 of the color color diagram . in table [ tbl.timingtour ] we present the duration of each interval , the time until the next interval and the 216 kev intensity during each interval . we identify several power spectral features and classify the continuous time intervals into color intervals a j ; these will be discussed in [ sec.comb_ps ] where we put all information together . for the 1998 outburst we find 7 different classes , i.e. , the source returns twice to similar positions in the color color diagram where it displays similar power spectral shapes . each class is marked with a different symbol in the color color diagram of figure [ fig.cc_deel2 ] . for the 1998 outburst we confirm the result of @xcite that the color color diagram shows the classical atoll shape ( see * ? ? ? the characteristic frequencies of most of the power spectral components increase along the track starting at the open triangles and ending at the crosses in figure [ fig.cc_deel2 ] ( for a more detailed discussion of the power spectral behavior with respect to the position in the color color diagram , see [ sec.move_cc ] ) . in figure [ fig.cc_deel1 ] we show as black points the lightcurve and color color diagram for the first segment of the data ( the decay of the 1996 outburst ) . the grey points represent the overall color color diagram including all the data . the numbers 18 again indicate continuous time intervals for which the power spectra and the position in the color color diagram remain similar . the symbols and entries in table [ tbl.timingtour ] are as described in [ sec.segment2 ] . we find one class additional to those observed during intervals 917 . this class is composed of continuous time intervals 2 , 4 and 6 , fills up the region between the open diamonds and the filled stars in the color color diagram , and is marked with open stars in figure [ fig.cc_deel1 ] . note further that part of the open diamonds here are in a slightly different position in the color color diagram , at a higher hard color and a lower soft color , than where they were during the second segment of the lightcurve . this is accompanied by a lower intensity . if we sort the classes by characteristic frequency as measured in the power spectra , the characteristic frequencies change along an `` @xmath20shaped '' track in the color color diagram ( see also [ sec.move_cc ] ) . in the third segment of the data the source count rates were low and in most observations one or more detectors were switched off . this , together with the short durations of the observations ( only @xmath21 s per observation ) led to bad statistics . in most cases it was impossible to identify any power spectral features , therefore the classification for this segment of the data was solely done on position in the color color diagram . if any power spectral features were detected they were always consistent with this color based classification . as before , in figure [ fig.cc_deel3 ] black points represent the lightcurve and color color diagram for the third segment of the data and grey points the overall color color diagram . the numbers 1830 now indicate continuous time intervals for which the position in the color color diagram remained similar , and the symbols mark the classification , here based on the position in the color color diagram only ; entries in table [ tbl.timingtour ] are as described in [ sec.segment2 ] . note that the source is in the class marked with the crosses for only the second time during continuous time interval 21 , but now at much lower intensity ( 0.03 crab ) than it was the first time in continuous time interval 9 ( 0.241.30 crab ) at the peak of the 1998 outburst . to improve the statistics we average the power spectra in each of the 8 classes . in the class marked with the filled circles the lower kilohertz qpo is extremely narrow and varies in frequency over several hundreds of hz . this leads to multiple peaks in the power spectrum . therefore we split up this class into three parts depending on lower kilohertz qpo frequency ; the first with the lower kilohertz qpo ranging from 540 to 640 hz , the second with the lower kilohertz qpo ranging from 640 to 710 hz and the third with the lower kilohertz qpo ranging from 710 to 900 hz . a finer division would have compromised the statistics at low frequencies . the observations in this class where no lower ( or upper ) kilohertz qpo was detected were added based on position in the color color diagram . in figure [ fig.cc_int ] we show the resulting 10 intervals in the color color diagram and in figure [ fig.hid_sid_int ] we show the corresponding hard color and soft color vs. intensity diagrams . note that the data for which no power spectra could be computed due to data overflow ( see [ sec.obs ] and [ sec.segment2 ] ) are not included in figures [ fig.cc_int ] and [ fig.hid_sid_int ] . we mark the different classes in order of increasing characteristic frequencies from a to j. we use letters here to avoid confusion with the numbered continuous time intervals of [ sec.timingtour ] . we fit each interval with the multi lorentzian fit function described in 2 . for the intervals where the kilohertz qpos have sufficiently high frequencies not to interfere with the low frequency features and vice versa , we fit the kilohertz qpos between 500 and 2048 hz and then fix the kilohertz qpo parameters when we fit the whole power spectra . this is for computational reasons only ; the results are the same as those obtained with all parameters free . if the @xmath18 value of a lorentzian becomes negative in the fit , we fix it to zero . no significantly negative q s occured . the fits have a @xmath22/dof below 1.4 for intervals a g . for interval h the @xmath22/dof is high ( 5.43 ) ; this is caused by the motion of the lower kilohertz qpo from 710 to 900 hz ( see above ) . for intervals i and j the @xmath22/dof are 2.5 ( dof = 100 ) and 1.8 ( dof = 97 ) . this is caused by deviations of the vlfn from the power law used to fit it ( see appendix [ sec.detailed_ps ] ) . we now specify the terminology that we will use for the various power spectral components . typical power spectral components observed for the atoll sources are described for the two sources 4u 0614 + 09 and 4u 172834 in @xcite . as in @xcite we call the upper kilohertz qpo l@xmath23 and its characteristic frequency @xmath24 . the lower kilohertz qpo has been linked to a broad bump at @xmath9 to @xmath25 hz found in the low luminosity bursters 1e 17243045 , gs 182624 and slx 1735269 @xcite and in the atoll source 4u 0614 + 09 at its lowest characteristic frequencies @xcite . although we emphasize that this identification is very tentative ( see [ comp : timing : atoll ] ) we will , as was done in @xcite , call both the lower kilohertz qpo and this 1025 hz bump ( which do not occur simultaneously ) , l@xmath26 ( characteristic frequency @xmath27 ) . no standard terminology yet existed for the hectohertz lorentzian @xcite ; here we will call it l@xmath28 ( characteristic frequency @xmath29 ) . the behavior of the band limited noise and qpos in the 0.150 hz range in the atoll sources 4u 172834 and 4u 0614 + 09 is complex ( see also * ? ? ? * ; * ? ? ? * ) , we describe it as a function of @xmath24 or position in the color color diagram ( see figure [ fig.freq_freq ] ) . first , the characteristic frequency of the band limited noise increases from @xmath30 to @xmath31 hz . in this phase the band limited noise is broad and usually fitted with a zero centered lorentzian or a broken power law . then at @xmath31 hz this noise component appears to `` transform '' into a narrow qpo ( called very low frequency lorentzian in * ? ? ? * ) whose frequency smoothly continues to increase up to @xmath32 hz , while what appears to be another band limited noise component appears at lower characteristic frequencies . it is unclear exactly how these three components are related . here , to remain true to the naming scheme of @xcite we call the band limited noise component that becomes a qpo as well as the qpo it becomes l@xmath33 ( characteristic frequency @xmath34 ) and the `` new '' broad band limited noise appearing at lower frequency l@xmath35 ( characteristic frequency @xmath36 ) , but please note the uncertainties in the interpretation underlying this nomenclature . finally , in the banana branch of 4u 172834 and 4u 0614 + 09 ( where usually no kilohertz qpos are detected ) a broad lorentzian is also present at @xmath37 hz for which it is unclear whether it is l@xmath33 , l@xmath35 or a new component ; here we shall list it as l@xmath35 . the atoll sources 4u 172834 and 4u 0614 + 09 both additionally show a @xmath38 hz lorentzian at frequencies above l@xmath33 , called the low frequency lorentzian by @xcite . at high characteristic frequencies ( @xmath39 hz ) this lorentzian appears as a narrow qpo . at low characteristic frequencies ( @xmath40 hz ) this lorentzian is broad and can be identified with the component in the low luminosity bursters which @xcite call l@xmath41 ( hump ) . for this reason , we also call this low frequency lorentzian l@xmath41 ( characteristic frequency @xmath42 ) . in some observations the low luminosity bursters show a narrow low frequency qpo ( called l@xmath43 in * ? ? ? * ) simultaneously with l@xmath41 . the @xmath14 of l@xmath43 is slightly lower than that of l@xmath41 . for 1e 17243045 the lorentzian centroid frequencies of l@xmath43 and l@xmath41 coincided but this was not the case for gs 182624 @xcite . l@xmath43 was not observed in 4u 172834 or 4u 0614 + 09 and we also do not detect it in 4u 160852 . a detailed description of the behaviour of the power spectral components of 4u 160852 is given in appendix [ sec.detailed_ps ] . in figure [ fig.powspec_1608 ] we show the averaged power spectra and best fit functions of the intervals a j . the fit parameters for the low frequency part of the power spectra are listed in table [ tbl.lowfitpar ] , the fit parameters for the high frequency part are listed in table [ tbl.highfitpar ] . we obtain 95% confidence upper limits for the fractional rms amplitude of l@xmath28 in intervals b , c and e and of l@xmath41 in intervals b , c and e using @xmath44 , fixing @xmath18 to 0.2 and allowing @xmath29 to run between 100 and 200 hz . for setting upper limits to l@xmath41 we fix @xmath18 to 1.5 , 3.0 and 3.5 and let @xmath42 run between 2232 hz , 3444 hz and 4250 hz respectively ( the values expected from the results for 4u 172834 and 4u 0614 + 09 ) . in figure [ fig.freq_freq ] we plot the characteristic frequencies of 4u 160852 versus @xmath24 , together with the results of @xcite for 4u 172834 and 4u 0614 + 09 . the black points mark the results for 4u 160852 , the grey points the results for 4u 172834 and 4u 0614 + 09 . the different symbols indicate different power spectral components . the results for 4u 160852 mostly fall on the relations established for 4u 172834 and 4u 0614 + 09 . also the @xmath18 values of the various components versus @xmath24 are similar to those for 4u 172834 and 4u 0614 + 09 . in figure [ fig.rms_all ] we plot the rms fractional amplitude of all components versus @xmath24 . the general trends in this plot for 4u 160852 are similar to those of 4u 172834 and 4u 0614 + 09 @xcite , but there is an offset in rms between the relations for the three sources @xcite . for the two kilohertz qpos the rms fractional amplitudes in 4u 160852 are similar to those in 4u 0614 + 09 but larger than in 4u 172834 @xcite . for all other features ; l@xmath28 , l@xmath41 , l@xmath33 , l@xmath35 and at low frequencies l@xmath26 the rms fractional amplitudes for 4u 0614 + 09 are the largest followed by 4u 172834 whereas the rms fractional amplitudes for 4u 160852 are the lowest . as we describe in appendix [ sec.detailed_ps ] in more detail , both the relations in figures [ fig.freq_freq ] and [ fig.rms_all ] as well as a direct comparison with the power spectra of 4u 172834 and 4u 0614 + 09 ( figures 1 and 2 in * ? ? ? * ) allow us to identify the power spectral components of 4u 160852 in the different intervals within the identification scheme described above . now that we have described and identified all power spectral components in 4u 160852 , we can link the timing properties ( tables [ tbl.lowfitpar ] and [ tbl.highfitpar ] ) to the position in the color color diagram . we can also link the timing properties and x ray spectral properties to the classical island and banana states described by @xcite . intervals a d are occurrences of the island state ( strong broadband noise with low @xmath14 , no vlfn ) in which intervals a c can be classified as the extreme island state . the extreme island state shows l@xmath33 , l@xmath41 , l@xmath26 and l@xmath23 all at low characteristic frequencies . the power spectral components are all broad and strong . interval e forms a transition between the island state and the lower banana , it still has relatively strong broad band noise but also shows a pair of narrow kilohertz qpos that are typical for what @xcite called the lower left banana state . so , intervals f h are all lower left banana state . here the vlfn appears , the kilohertz qpos are double , l@xmath33 transforms from a band limited noise component into a qpo and l@xmath35 appears ( see also [ sec.comb_ps ] ) . intervals i and most of interval j are also still in what @xcite called the lower banana state based on their position in the color color diagram ; only the upper right part in the color color diagram of interval j is in the upper banana state based on color color position . the vlfn is strong and the broad band noise is weak in these intervals . intervals i and j show l@xmath23 at the highest frequencies , l@xmath35 , and a strong vlfn component . the characteristic frequencies increase in order a j and form an `` @xmath20shaped '' track in the color color diagram . recently , @xcite and @xcite studied the color color diagrams of several of the z and atoll sources , including 4u 160852 , and suggested that the atoll sources trace out similar three branch patterns as the z sources . we observe the same shape of the color color diagram for 4u 160852 as @xcite and @xcite did ( fig . [ fig.cc_int ] ) . interval c in figure [ fig.cc_int ] represents a deviation from the classical atoll shape . according to the interpretation of @xcite and @xcite the source would have to move in the z track order d c in figure [ fig.cc_int ] . interval c would then correspond to the horizontal branch of the z sources . we observe a transition from d to c and back to d with gaps of three days during the decay of the 1996 outburst ( fig . [ fig.cc_deel1 ] ) . we also observe a transition from e to c and back to e with gaps of only one day during the decay of the 1998 outburst ( fig . [ fig.cc_deel2 ] ) . the source is first in e for an interval of @xmath45 day during which the hard color increases by @xmath46 ( so @xmath47 day@xmath48 ) , then there is a gap of 0.9 day after which the source appears in c with an increase in hard color of @xmath30 ( if the source moved directly from e to c @xmath49 day@xmath48 ) . then in c there is an interval of @xmath50 day where the hard color increases by @xmath46 ( @xmath49 day@xmath48 ) . so this is consistent with the source moving directly from e to c at approximately constant speed ( probably through d ) and not through a or b as required if 4u 160852 behaved as a z source . in c there is a gap of @xmath51 day after which the source appears again in c but at a slightly ( @xmath52 ) lower hard color . for an @xmath45 day interval the source stays in c while the hard color decreases by @xmath52 ( @xmath53 day@xmath48 ) . then there is a gap of @xmath54 day and after that the source has returned to e with a hard color of @xmath30 less ( @xmath55 day@xmath48 ) this is consistent with the source moving directly back from c to e , again at constant speed , and not through a or b. transitions from and to states a and b had gaps of 7 and more days , so it is impossible to draw conclusions from these . the characteristic frequencies of the timing features decrease in the order d b a ( appendix [ sec.detailed_ps ] ) . this is also not consistent with the idea that 4u 1608 - 52 behaves as a z source , as in z sources the characteristic frequencies of the timing features increase along the z starting at the horizontal branch ( i.e. , this would predict frequencies decreasing in the order dbac ) . one might say that 4u 160852 draws up an approximate `` @xmath20 '' shape in the color color diagram along which the characteristic frequencies of the timing features change smoothly . parallel tracks in the intensity versus lower kilohertz qpo frequency in 4u 160852 were discovered by @xcite , and extensively studied by @xcite and @xcite . the hard color and soft color vs. intensity diagrams of 4u 160852 ( see fig . [ fig.hid_sid_int ] ) and those of several other atoll sources also show narrow parallel tracks @xcite . based on the appearance of the diagrams , it has been suggested that there is a relation between these two types of parallel tracks @xcite . based on our new analysis it is now possible to directly link the parallel tracks in the color intensity diagrams to those in the qpo frequency intensity diagrams using the frequency of the lower kilohertz qpo as obtained by @xcite . similarly to @xcite and @xcite we only include data for which both kilohertz qpos are detected simultaneously and therefore the lower kilohertz qpo is identified unambiguously . we rebin our 16 s colors in such a way that they match the 64448 s data intervals of @xcite . in figure [ fig.mendez_rate ] we plot the frequency of the lower kilohertz qpo and the hard and soft color versus the 2.016.0 kev intensity . the alternating black / grey symbols represent the parallel tracks in the intensity versus the frequency diagram ; the tracks contain data that are continuous in time , with only the @xmath56 s gaps due to earth occultations . note that the parallel track phenomenon in the lower kilohertz qpo frequency vs. intensity diagram is only observed in a small region of the color intensity diagrams ( see figure [ fig.hid_sid_int ] ) where the lower kilohertz qpo is strong and narrow enough to be accurately traced on 64448 s timescales . with this analysis , we can investigate how the parallel tracks in the frequency vs. intensity diagram relate to those in the color intensity diagrams . interestingly , it turns out that the frequency intensity tracks can not , as previously thought , be identified with the narrow vertical tracks visible in the color intensity diagrams of figures [ fig.hid_sid_int ] and [ fig.mendez_rate ] . each of these latter tracks corresponds to a single satellite orbit . the parallel tracks in the frequency vs. intensity diagram are only identifiable as ( rather fuzzy ) tracks in the color intensity diagrams after several satellite orbits have elapsed and thus turn out to be composed of several of those narrow vertical tracks . the parallel tracks in the frequency versus intensity diagram could also identified in the hard color intensity diagram of 4u 163653 , where two banana tracks shifted by 20 % in intensity were observed in the hard color intensity diagram @xcite , and exactly the same 20 % shift was observed between the corresponding two parallel lines in the frequency versus intensity diagram . in 4u 160852 we do not observe complete banana tracks shifted in intensity as in 4u 163653 . the drift in x ray flux in 4u 160852 already occurs on timescales of several days @xcite , too short for a complete track to form , where in 4u 163653 the source can stay on one track for several months @xcite . these fast changes in intensity in 4u 160852 are due to its transient character , where 4u 163653 is a persistent source for which changes in intensity occur much more gradually . note , that within each of the two parallel lines in the frequency versus intensity diagram of 4u 163653 there is no correlation observed between frequency and intensity . the narrow vertical tracks in the color intensity diagrams are caused by variations in color that for the most part can be explained by the scatter due to counting statistics . the parallel tracks in the frequency versus intensity diagram , although composed of data covering several consecutive satellite orbits , already show up within individual orbits due to a short term correlation between the lower kilohertz qpo frequency and intensity in combination with small errors in these quantities but persist over several consecutive orbits . the correlation between colors and intensity ( or kilohertz qpo frequency ) in individual orbits is veiled by the limited counting statistics , so in the color intensity diagrams the parallel qpo tracks can only be identified on the longer timescales of several satellite orbits . to further illustrate this point in figure [ fig.mendez_sc ] we plot soft color versus the frequency of the lower kilohertz qpo ; lines connect the points of individual satellite orbits , and for clarity four representative individual orbits are highlighted in black . on orbit time scales , the lower kilohertz qpo frequency and soft color seem uncorrelated due to the counting statistics errors on the color , whereas on a longer time scale a clear correlation emerges . we note that for some parallel tracks in figure [ fig.hid_sid_int ] there is a correlation between color and intensity on short timescales , especially in interval j. in those cases the counting statistics are better due to the higher count rates and thus the parallel track phenomenon in the color diagrams is more similar to that in the kilohertz qpo frequency versus intensity diagram @xcite . however , in our interval j there is no timing feature present that is trackable on short timescales . in conclusion we can now identify the parallel qpo tracks in the color intensity diagrams . bad counting statistics that shows up in the form of the narrow nearly vertical parallel tracks in the color intensity diagrams cause the @xmath57color correlation to be veiled on short timescales ( less than hours ) so in the color intensity diagrams the parallel qpo tracks can only be found for the longer timescales of several satellite orbits . we have studied the color diagrams and the power spectral behavior of the atoll source 4u 160852 . we found that the timing behavior of 4u 160852 is almost identical to that of the other atoll sources 4u 172834 and 4u 0614 + 09 . if we plot the characteristic frequencies of the timing features versus the characteristic frequency of the upper kilohertz qpo , together with the results of @xcite for 4u 172834 and 4u 0614 + 09 , the three sources follow the same relations ( see fig . [ fig.freq_freq ] ) . also the behavior of the @xmath18 value is the same for the three sources . the general trends in rms fractional amplitude for 4u 160852 are also similar to those of 4u 172834 and 4u 0614 + 09 , but there is an offset between the relations for the three sources ( see fig . [ fig.rms_all ] ) . we connected the timing behavior with the position of the source in the color color diagram and found that the timing behavior is not consistent with the idea that 4u 1608 - 52 traces out a three - branched z shape in the color - color diagram along which the timing properties vary gradually as is the case in in z sources . instead , the power spectral properties change along an `` @xmath20shaped '' track . finally , our measurements for the colors together with the precise measurements of lower kilohertz qpo frequency of @xcite gave us an opportunity to link the parallel tracks in the intensity versus color diagrams with the parallel tracks in the intensity versus lower kilohertz qpo frequency diagrams . we found that the parallel tracks in the frequency versus intensity diagram can be found back in the color intensity diagrams as fuzzy structures ; they should not be confused with the narrow vertical parallel tracks visible in the intensity versus color diagrams which are mostly due to the errors in the colors . the power spectral properties in 4u 160852 change along an `` @xmath20shaped '' track . the question now arises whether the `` @xmath20shaped '' track we find in the color color diagram of 4u 160852 is universal for atoll sources . we can compare our results with those of @xcite who studied the x ray color and timing properties of a well sampled state transition of 4u 170544 where the source moves from the lower banana to the extreme island state and back . this transition was also included in @xcite who studied a larger dataset of 4u 170544 . in the color color diagram the source moves from the lower banana , to the left of the extreme island state as the count rate decreases . then both the soft color and the count rate increase , whereas the hard color remains approximately constant , and the source traces out a horizontal track in the color color diagram . then , while the count rate continues to increase , the source moves back from the right of the extreme island state to the lower banana @xcite . in the whole horizontal track the power spectra remained the same @xcite . in this extreme island state 4u 170544 did not reach characteristic frequencies as low as those in interval a of 4u 160852 ( it is very similar to interval b ) . if we look at the overall light and color curves of 4u 170544 presented in @xcite we see that apart from the extreme island state just described , which took place in the forty days around mjd 51230 , there is another extreme island state , with much sparser sampling , in which a higher hard color is reached . if we take a quick look at the timing of this mjd 51380 data we find that when the source reaches this higher hard color , the source shows a power spectrum very similar to that of interval a of 4u 160852 . as for the mjd 51230 extreme island state , during this mjd 51380 extreme island state the count rate and the soft color increase simultaneously and in the color color diagram another horizontal track is drawn up above the mjd 51230 one . returning now to 4u 160852 we note that in the state transition of 4u 160852 during the 1998 outburst ( see figure [ fig.cc_deel2 ] ) we observe the same phenomenon of a horizontal track being traced out in the extreme island state . the source is in interval c ( continuous time interval 12 in fig . [ fig.cc_deel2 ] ) for two observations , the first of which has a count rate about a factor two higher than the second . this higher count rate is accompanied by a 5 % higher soft color in the first observation . the power spectrum is the same in both observations . so , like the case of 4u 170544 , the change in count rate is accompanied by a correlated change in soft color which traces out a small almost ( the hard color changes slightly , see above ) horizontal track in the color color diagram ( continuous time interval 12 in fig . [ fig.cc_deel2 ] ) . a possible explanation for this behavior of atoll sources in the extreme island state is the so called `` secular motion '' in the color color diagram . this phenomenon was first observed in z sources , in which the z shaped track in the color color and color intensity diagrams is traced out within several hours up to a day . on longer timescales the whole z track shifts both in soft color and count rate ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the timing properties remain mostly unaffected by these shifts and are primarily determined by the position along the z track ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? . the same phenomenon has been observed in the banana state of the atoll source 4u 163653 @xcite . the horizontal tracks in the extreme island state of the atoll sources may be entirely caused by a secular motion , similar to that in the z sources . because the sources remain in one particular island state ( similar timing properties and hard color ) for a long time ( weeks to months ) , the slow process of secular motion has time to draw up a horizontal branch . in this horizontal branch the timing remains similar , as is the case for z source secular motion . as noted by @xcite , this aspect of the behavior can be explained by the scenario that @xcite proposed to explain the parallel track phenomenon in the intensity versus lower kilohertz qpo frequency diagram . it seems that the z shape in the color color diagram of some of the atoll sources is caused by transitions between the banana and the extreme island states that occur at different soft color . figure [ fig.schema ] provides a schematic of what in our interpretation occurs in these sources . several different horizontally extended extreme island branches appear above each other in the color color diagram at different hard color values . these are traced out at different epochs . to first order only the hard color determines the timing properties . within each extreme island state the soft color changes in correlation with intensity . during state transitions the extreme island state is entered or left at a soft color value depending on intensity . a possible explanation for the fact that the timing remains similar while the intensity ( and thus soft color ) changes and that the state transitions can occur at different intensities ( soft colors ) is the scenario of @xcite where the truncation radius of the disk , which determines the timing properties and therefore the state , is not set by the accretion rate through the disk but by the ratio of this accretion rate over its long term average ( c.f . so , if a source shows a state transition to an extreme island state at a minimum intensity which increases after that ( as happened in the mjd 51230 state transition of 4u 170544 * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , the source will enter the extreme island state at a lower soft color and move to a higher soft color as the intensity increases . if a state transition to an extreme island state takes place during a decay after an outburst of a transient source , we would expect that after the extreme island state is entered at a particular soft color , the source then moves to a lower soft color as it fades ( this and the reverse state transition from an extreme island to the banana state during the outburst rise were observed in aql x1 , see * ? ? ? * ) . in our interpretation , the shape we observe in the color color diagram of 4u 160852 is caused by observations of the source in extreme island states at different intensities and therefore at different soft colors . extreme island states a and b are observed at higher intensities than extreme island state c , causing c to have a lower soft color . contrary to the case in 4u 170544 and aql x1 we do not observe a large change in soft color ( or intensity ) within each extreme island state and it is unclear whether this is because the source was not observed sufficiently long or dense enough in these extreme island states , or whether this is a property of 4u 160852 . the three atoll sources , 4u 160852 , 4u 160852 and 4u 0614 + 09 , we have studied up to now ( * ? ? ? * this paper ) show a similar and very distinct timing behaviour ( figures [ fig.freq_freq ] and [ fig.rms_all ] ) . 4u 160852 differs from 4u 0614 + 09 and 4u 172834 in that 4u 160852 is a transient source whose intensity changes by about 2 orders of magnitude where this is only about 1 order of magnitude for 4u 0614 + 09 and 4u 172834 . although 4u 160852 has a much larger range in intensity ( @xmath58 crab in the 2.016.0 kev range ) than 4u 0614 + 09 ( @xmath59 crab ) , 4u 160852 and 4u 0614 + 09 show a similar range in characteristic frequencies for the different power spectral components . for 4u 160852 @xmath24 ranges from 216 to 1061 hz ( excluding intervals i and j for which the identification of @xmath24 is uncertain see appendix [ sec.detailed_ps ] ) . this range in @xmath24 is reached with a minimum intensity range of @xmath60 crab ( see fig . [ fig.hid_sid_int ] ) . for 4u 0614 + 09 a similar @xmath24 range of 233 to 1067 hz is reached with a minimum intensity range of @xmath61 crab . so , both sources reach similar frequency ranges within a similar change in intensity . 4u 172834 has not yet shown a similar range in characteristic frequencies as 4u 0614 + 09 and 4u 160852 ( until now @xmath62 hz ; * ? ? ? * ; * ? ? ? also no 2.016.0 kev intensities were published for the intervals used in @xcite and @xcite . there are also some differences in the timing behaviour ; in particular in l@xmath33 when it appears as a qpo . as a function of @xmath24 , @xmath63 is slightly higher in 4u 160852 than in 4u 172834 and 4u 0614 + 09 but it covers the same range . the @xmath18 values for l@xmath33 here are similar for the three sources . the relation between @xmath63 and @xmath24 in figure [ fig.freq_freq ] has a turnover at 30 hz in 4u 0614 + 09 @xcite ; there might be a turnover at 45 hz in 4u 172834 , but in 4u 160852 no turnover has been observed . the plot of rms fractional amplitude of all components versus @xmath24 ( fig . [ fig.rms_all ] ) shows an offset between the relations for the three atoll sources 4u 160852 , 4u 0614 + 09 , and 4u 172834 . this offset is different for the low frequency features , l@xmath28 , l@xmath41 , l@xmath33 , l@xmath35 , and at low frequencies l@xmath26 , than for the two kilohertz qpos ( see also [ sec.comb_ps ] ) . it seems that there are two groups , one containing the high and the other the low frequency features . the difference between the two groups might be due to geometrical effects . the different offsets in figure [ fig.rms_all ] hint towards the lorentzian at @xmath64 hz , found in the lowest frequency states of 4u 160852 ( interval a ) and 4u 0614 + 09 , being l@xmath23 ( see appendix [ sec.detailed_ps ] ) , whereas it could have been identified as either l@xmath23 or l@xmath28 based on its frequency ( see appendix [ sec.detailed_ps ] ) . the identification of the @xmath64 hz lorentzian is important because the power spectra of 4u 160852 and 4u 0614 + 09 at the lowest inferred mass accretion rate closely resemble those of the low luminosity bursters 1e 17243045 , gs 182624 , and slx 1735269 @xcite and the millisecond x ray pulsar sax j1808.4 - 3658 @xcite . these sources all seem to be atoll sources that are only observed at low mass accretion rate . if the @xmath64 hz lorentzian is the upper kilohertz qpo then the high frequency peaks in those sources are probably also upper kilohertz qpos . note that although in 4u 160852 l@xmath41 already becomes undetectable at @xmath65 hz whereas this happens at @xmath66 hz in 4u 172834 and 4u 0614 + 09 , this is not a significant difference as the upper limits on the fractional rms of l@xmath41 in 4u 160852 are consistent with this component still being present at those frequencies . the broad l@xmath26 component found in intervals a and b was previously found in 4u 160852 with ginga by @xcite and has also been identified in the atoll source 4u 0614 + 09 @xcite . this component has also been found in the low luminosity bursters 1e 17243045 , gs 182624 , and slx 1735269 @xcite . the lorentzian has been tentatively identified with the lower kilohertz qpo based on extrapolations of frequency frequency relations ( see * ? ? ? * ; * ? ? ? * ; * ? ? ? the rms fractional amplitude of the broad l@xmath26 is much higher than that of the lower kilohertz qpo ( see fig . [ fig.rms_all ] ) , however , a similar increase in rms is visible in that figure for l@xmath23 . but note that l@xmath23 is detected in all intervals where l@xmath26 is present as the lower kilohertz qpo in intervals h e then dissapears in intervals d and c and only shows up again as the broad lorentzian in intervals b and a. interval b of 4u 160852 shows a narrow qpo with a characteristic frequency ( 2.458 hz ) between those of l@xmath33 and l@xmath67 . this qpo is probably the same as the one discovered in the island state of 4u 160852 by @xcite : their power spectrum of the 1989 august 2526 interval is very similar to our interval b , it has l@xmath33 with a characteristic frequency of @xmath68 hz , l@xmath41 at @xmath69 hz and a narrow qpo at @xmath70 hz ( see figure 4 of * ? ? ? luminosity bursters 1e 17243045 , gs 182624 @xcite as well as the bhcs gx 3394 @xcite and cyg x1 @xcite also show narrow qpos with a characteristic frequency between those of l@xmath33 and that of l@xmath41 . for 1e 17243045 the centroid frequency of the narrow qpo coincides with the centroid frequency of l@xmath41 , but for 4u 160852 this is not the case ; here the centroid frequency of the narrow qpo ( @xmath71 hz ) is close to half the centroid frequency of l@xmath41 ( @xmath72 hz ) . no similar narrow lorentzians were fitted by @xcite for 4u 172834 or 4u 0614 + 09 , but 4u 172834 does show narrow residuals between the characteristic frequency of l@xmath33 and that of l@xmath41 . in figure [ fig.narrowqpo ] we plot @xmath42 versus the @xmath14 of these narrow qpos . for cyg x1 many narrow qpos were fitted by @xcite ; we only plot those qpos that have a characteristic frequency between @xmath63 and @xmath42 . the points of the low luminosity bursters and most of those of gx 3394 line up in figure [ fig.narrowqpo ] . these qpos were all labeled as l@xmath43 by @xcite . the two points of gx 3394 that deviate from the line are from two observations where gx 3394 showed two simultaneous qpos ; the highest frequency qpo falls on the relation in figure [ fig.narrowqpo ] while the lowest frequency qpo falls below it . if we fit the points of the low luminosity bursters and gx 3394 with a power law our result as well as those of @xcite for 4u 160852 fall below this relation . so we can not identify this narrow qpo in 4u 160852 with l@xmath43 . in the two observations where gx 3394 showed two narrow simultaneous qpos , the characteristic frequencies of the lowest frequency qpo as well as the results for cyg x1 also fall well below the power law . it might be that these qpos and those of 4u 160852 are related . these points fall close to a line that indicates half the low luminosity burster relation which is also plotted in figure [ fig.narrowqpo ] . note that if we use centroid frequency instead of @xmath14 the relations in figure [ fig.narrowqpo ] worsen . in addition to comparing the color color diagram tracks and the associated power spectra , we can also compare the timing properties of z sources with those of atoll sources by making similar plots of @xmath14 of the different power spectral components versus @xmath24 as we did for the atoll sources 4u 160852 , 4u 0614 + 09 and , 4u 172834 in figure [ fig.freq_freq ] . in figure [ fig.atoll_z ] we do this for the z sources gx 5 - 1 @xcite , gx 340 + 0 @xcite , gx 17 + 2 @xcite and cyg x2 @xcite . we include the lower kilohertz qpo , the low frequency noise ( lfn ) , the horizontal branch oscillations ( hbo ) , and the harmonic and sub harmonic of the hbo , and plot these versus @xmath24 . the grey symbols are the results of the atoll sources also displayed in figure [ fig.freq_freq ] , the black symbols represent the z sources . note that the broad band noise in these z sources was not fitted with a zero centered lorentzian but with a cutoff power law , @xmath73 , for gx 5 - 1 , gx 340 + 0 , and gx 17 + 2 and with a smooth broken power law , @xmath74^{-1}$ ] , for cyg x2 . this leads to expressions for @xmath14 , defined as the frequency of maximum power density in @xmath13 , of @xmath75 for the cutoff power law , and @xmath76^{1/\beta } \nu_{b}$ ] for the smooth broken power law . from figure [ fig.atoll_z ] we confirm the identification of the hbo in the z sources with l@xmath41 in the atoll sources made by @xcite . however , the suggestions that either the lfn @xcite or the sub hbo @xcite in the z sources might be similar to the classical broad band noise in the island state of the atoll sources seems not to be supported by the relation of the characteristic frequencies of these components with @xmath24 . at @xmath77 hz the band limited noise component in the l@xmath33l@xmath23 relation is replaced by a qpo ( see [ sec.comb_ps ] ) and the band limited noise component l@xmath35 appears and follows a new relation with @xmath24 . the lfn points fall below the l@xmath33l@xmath23 relation but seem to line up with l@xmath35 . the sub hbo points fall above the l@xmath33l@xmath23 relation until @xmath77 hz where the qpo takes over from the band limited noise component ( see above ) . based on this comparison of frequencies , the lfn might still be associated with l@xmath35 and the sub hbo might be related with l@xmath33 when it is a qpo , although for 4u 0614 + 09 and 4u 172834 no harmonic relation was present between l@xmath41 and l@xmath33 when it is a qpo @xcite . note also that the lfn in gx 17 + 2 follows a different relation compared to that in the other z sources . other differences between gx 17 + 2 and the other z sources in figure [ fig.atoll_z ] are that the lfn in gx 17 + 2 is peaked , the harmonic of the hbo is relatively strong and it shows a flaring branch oscillation ( fbo ) whereas the other sources show a flat lfn , a relatively weak harmonic of the hbo and no fbo @xcite . we note that none of these differences with the other z sources serve to make gx 17 + 2 more similar to the other atoll sources . the results presented in sections [ sec.comb_ps ] , appendix [ sec.detailed_ps ] , and this section show that the low frequency part of the power spectra of both the atoll and the z sources behaves in a very complex manner . for the atoll sources many different lorentzian components appear , disappear or change from broad lorentzians into narrow qpos . although this behavior is complex , the power spectral components of the three atoll sources for which we have performed a multi lorentzian timing study show remarkable similarities . by using frequency vs. frequency plots such as the one in figure [ fig.freq_freq ] , and by comparing the power spectra of different sources directly with each other it is possible to identify all these components within a single classification . we do this here for 4u 160852 ( section [ sec.comb_ps ] ) . note however , that sometimes additional narrow features become significant in the power spectra ( see [ comp : timing : atoll ] ) . as another example , a recent rxte observation of 4u 160852 during an outburst in 2001 , shows a power spectrum ( figure [ fig.nupnu_2001 ] ) very similar to that of interval b ( fig . [ fig.powspec_1608 ] ) , except that instead of the broad peak at @xmath78 hz , a narrow ( @xmath79 ) qpo with an rms fractional amplitude of @xmath80 appears at 30 hz . also several , marginally significant , features are visible between 2 and 5 hz . if we compare the low frequency features of the atoll sources with those found in the z sources , in a frequency vs. frequency plot we see that only the hbo of the z sources can be unambiguously identified with the l@xmath67 of the atoll sources . the frequency vs. @xmath24 relations of the other low frequency features partly overlap in figure [ fig.freq_freq ] but are not identical , and the relation of the characteristic frequency of the lfn with @xmath24 is not even the same for all z sources . it appears that for some components the frequency vs. @xmath24 relations are not universal . there might be small physical differences between the atoll sources and z sources ( and within the z sources themselves ) that affect these relations . so , either these features are different phenomena in different types of sources , or something else ( e.g. neutron star mass , magnetic field strength ) affects the relations . if this is so , this hidden parameter does not change much from source to source , or the relations do not depend strongly on this parameter as the relations are similar , as the relations are very similar within each source type . three different characteristic frequencies can be used to test the frequency relations predicted by the several qpo models ; @xmath14 , @xmath16 and @xmath81 . @xmath14 is the frequency at which a lorentzian contributes most of its variance per log frequency @xcite and is the one used in this paper . @xmath16 is the centroid frequency of the lorentzian and can be obtained from our fit parameters @xmath14 and @xmath18 as @xmath82 . according to @xcite the @xmath16 of the lorentzian is shifted with respect to the eigenfrequency of the oscillation due to damping . this shift depends on the damping rate which can be estimated from the width of the lorentzian @xcite . we can calculate this eigenfrequency , @xmath81 , from @xmath14 and @xmath18 as @xmath83 . in this section we use our fit parameter @xmath14 unless stated otherwise . the transition layer model ( tlm * ? * ; * ? ? ? * ) associates the lower kilohertz qpo with the keplerian frequency of the inner disk edge ( @xmath84 ) . between the neutron star and the keplerian disk a transition layer is present . in the tlm the upper kilohertz qpo is produced by radial oscillations of a blob thrown out of the transition layer into a magnetosphere . this radial eigenmode or hybrid frequency ( @xmath85 ) relates to @xmath84 as @xmath86 , where @xmath87 is the rotational frequency of the star s magnetosphere near the equatorial plane . this implies that @xmath85 should always exceed @xmath88 . for 4u 160852 @xcite found that @xmath87 is always larger than 300 hz and thus @xmath89 hz should apply . for 4u 160852 we find that when there is a pair of kilohertz qpos present ( intervals e h ) the upper kilohertz qpo frequency ranges from 830 to 1062 hz . but the 200682 hz single kilohertz qpo of intervals a d , which we identify as the upper kilohertz qpo based on frequency color @xcite , frequency frequency ( fig . [ fig.freq_freq ] ) and rms frequency ( see * ? ? ? * and fig . [ fig.rms_all ] ) correlations , fall well below the 600 hz predicted by the tlm model . note that neither the use of @xmath16 , nor @xmath81 , shifts these frequencies above the 600 hz limit . if these single kilohertz qpos were not the upper , but the lower kilohertz qpos , this would lead to the unlikely scenario that the presence of an upper kilohertz qpo would lead to a completely different power spectrum for a similar lower kilohertz qpo frequency ( compare , e.g. , c with e or d with g in fig . [ fig.powspec_1608 ] ) . the @xmath42 vs. @xmath27 relation for neutron stars and bhcs ( see @xcite for the relation in the @xmath16 representation , and @xcite for the relation in the @xmath14 representation ) was recently extended towards lower frequencies by including 17 white dwarf sources ( see * ? ? ? * ; * ? ? ? * and references therein ) . the points of 4u 0614 + 09 , 4u 172834 , and 4u 160852 fall on the @xmath42 vs. @xmath27 relation ; only the point from interval b of 4u 160852 deviates . the @xmath42 vs. @xmath27 relation can be fitted with a power law with an index close to 1 @xcite . the tlm explains this by assuming that @xmath27 is the keplerian frequency of the inner disk ( see above ) , and @xmath42 represents the frequency of magnetoacoustic oscillations , @xmath90 , in the disk transition layer . the relation between @xmath42 and @xmath27 is then a result of a global relation between @xmath90 of the transition layer and @xmath84 at the adjustment radius @xcite . for each individual source the tlm predicts an index for the @xmath90 vs. @xmath84 relation that is steeper than one and should be studied separately @xcite . note , however , that the atoll sources cover a large range of the @xmath42 vs. @xmath27 relation , as there are points around @xmath91 hz and around @xmath92 hz . while 4u 160852 only contributes points around @xmath91 hz and 4u 172834 only around @xmath92 hz , 4u 0614 + 09 contributes both . this suggests that the @xcite relation is not just a global relation between different sources as suggested by @xcite but can also be found within individual sources . note also that although the tlm explains the @xcite relation as above in @xcite , the hbos of the z sources , which are part of @xcite relation , are explained differently , namely as the vertical eigenmode of a blob rotating with a keplerian frequency thrown into the magnetosphere ( see above ) in other tlm papers ( e.g. * ? ? ? * ) . in the sonic point beat frequency model ( spbfm ; * ? ? ? * ) the upper kilohertz qpo represents the keplerian frequency at the inner disk edge . the lower kilohertz qpo then arises from a beat of the keplerian frequency at the inner disk edge with the neutron star spin frequency . this led to an early prediction that @xmath93 , the frequency difference between the upper and the lower kilohertz qpo , is equal to the spin frequency and should therefore be constant . observations of several sources , among which 4u 172834 @xcite and 4u 160852 @xcite , showed a significant decrease in @xmath93 when the upper kilohertz qpo frequency increased . further refinements of the spbfm by @xcite could explain this decrease in @xmath93 . neither the use of @xmath14 nor @xmath81 instead of @xmath16 makes @xmath93 constant . note that our measurements of the high frequency kilohertz qpos ( @xmath94 hz ) are not the best to use here as the addition of a lot of data leads to an artificial broadening of the lorentzian ( see [ sec.comb_ps ] ) . therefore here we used the precise measurements of these high frequency kilohertz qpos made by @xcite for 4u 172834 and by @xcite for 4u 160852 to calculate @xmath14 and @xmath81 . for both sources @xmath93 increases by less than 1.5 % if @xmath14 is used instead of @xmath16 and by less than 3.1 % if @xmath81 is used instead of @xmath16 . note that , just as for the tlm , the low frequencies we find for the upper kilohertz qpo ( see above ) are a problem for the spbfm . in the spbfm the lower bound on the upper kilohertz qpo frequency is set by the keplerian frequency at the maximal radius where the radiation coming from the neutron star surface can still remove sufficient angular momentum from the gas in the disk so that it falls supersonically to the neutron star surface @xcite . this radius is about @xmath95 , where @xmath96 is the radius of the marginally stable orbit . so , to reach @xmath97 hz the mass of the neutron star has to exceed @xmath98 @xmath99 . the relativistic precession model ( rpm ; * ? ? * ; * ? ? ? * ) assumes that the upper kilohertz qpo represents the keplerian frequency of the inner disk . the lower kilohertz qpo frequency represents the periastron precession frequency of the accretion disk , which is assumed to contain slightly elliptical orbits . this leads to the prediction that the frequency difference ( @xmath93 ) between the two kilohertz qpos decreases both at low and high kilohertz qpo frequencies ( see figure 1 of * ? ? ? * ) . if the broad l@xmath26 at low frequencies ( intervals a and b of 4u 160852 and interval 1 of 4u 0614 + 09 in * ? ? ? * ) is the lower kilohertz qpo ( but see [ comp : timing : atoll ] ) , then this predicted decrease in @xmath93 at low kilohertz qpo frequencies is observed in 4u 0614 + 09 and 4u 160852 . note that the point of interval b of 4u 160852 falls above the curves of figure 1 in @xcite by @xmath100 . more complicated modelling within the framework of the rpm as is done for sco x1 in @xcite is beyond the scopes of this paper . the rpm also predicts a low frequency qpo at the lense thirring precession frequency of @xmath101 hz @xcite , where @xmath102 is the moment of inertia in units of @xmath103 g @xmath104 , @xmath105 is the mass of the neutron star in @xmath99 , and @xmath106 is the neutron star spin frequency . we searched our data for a power law with an index of 2 as predicted by the rpm by fitting the frequency vs. @xmath24 relations for the low frequency features ( fig . [ fig.freq_freq ] ) for the three sources 4u 172834 , 4u 0614 + 09 and 4u 160852 with power laws . we used @xmath14 , @xmath16 , and @xmath81 for the characteristic frequencies ( see above ) . we find that the @xmath42 vs. @xmath24 relation has an index that is the closest to the 2 predicted by the rpm . if we use @xmath14 or @xmath81 for the frequency the relations clearly deviate from a power law . the points fall below the fitted power law at low and high frequencies and above at intermediate frequencies , leading to very large @xmath22/dof of 4605/21 ( @xmath14 ) and 7179/21 ( @xmath81 ) . if we use @xmath16 for the frequency we find that the @xmath42 vs. @xmath24 relation can be described by a power law of the form @xmath107 ( see figure [ fig.powerlawfit ] ) . the @xmath22/dof of this fit is still high ( 373/20 ) but this is not due to systematic deviations from a power law . the averaged relative scatter around the fitted power law is only 6 % on the @xmath42 axis and only 3 % on the @xmath24 axis . the high @xmath22/dof is probably a result of not including any systematic errors when determining the errors of the power spectral fit parameters ( see * ? ? ? note , that if we use @xmath16 for the frequency we exclude those points that have @xmath108 ( from fits with a zero centered lorentzian ) . the result that the @xmath42 vs. @xmath24 relation can be described by a power law with an index close to 2 was found previously for a more limited frequency range ( @xmath62 hz ) in 4u 172834 ( index 2.11 ; * ? ? ? * ) and 4u 0614 + 09 ( index 2.46 ; * ? ? ? it is remarkable that now that we have extended the @xmath42 vs. @xmath24 relation towards @xmath109 hz , the relation can still be described by a power law and that the power law index is found to be even closer to 2 . we can get an indication of the spin frequency by looking at the burst oscillation frequency which is 363 hz for 4u 172834 @xcite and 620 hz for 4u 160852 @xcite . if these two frequencies are the spin frequency of the neutron star in each case ( the most likely scenario , see * ? ? ? * ) , then it is difficult to understand why the @xmath42 vs. @xmath24 relations of these sources coincide . if we assume that we see the spin frequency in 4u 172834 and twice the spin frequency in 4u 160852 , we find an average spin frequency of about 335 hz for these sources . for the @xmath16 representation fit , this leads to @xmath110 which is too large for proposed equations of state ( the acceptable range is @xmath111 ; * ? ? ? if we assume that we see the spin frequency in 4u 160852 and half the spin frequency in 4u 172834 , the average spin frequency is about 670 hz for these sources . this leads to @xmath112 which is acceptable . recently , @xcite studied the x ray energy spectra of 4u 160852 as a function of position in the color color diagram . they find that , similar to other atoll sources , the spectra of the island state ( our intervals a , b and c ) of 4u 160852 are dominated by a hard power law spectrum . there is also a soft component present which in their view is probably due to the neutron star surface rather than the accretion disk . the banana state ( our intervals e j ) spectrum is soft and it is uncertain whether the soft component is due to the neutron star surface or the disk . to explain the spectral behavior of 4u 160852 @xcite use a scenario where in the island state the disk is far from the surface of the neutron star . the large radius of the inner disk prevents the disk from being observed directly in the pca spectra . in the island / banana state transition the inner accretion flow , which was geometrically thick and optically thin in the island state , collapses into a geometrically thin and optically thick disk . the inner edge of the disk is now close to the neutron star , and it is observed as the soft component and increases the soft color . from the diskbb model , @xcite estimate an inner disk radius in the banana state of @xmath113 km . the inner accretion flow becomes more optically thick lowering the hard color . we can compare this scenario with the results of our timing study . if we assume that @xmath24 represents the keplerian frequency at the inner disk edge ( e.g. * ? ? ? * ) we find , assuming a neutron star mass of 1.4 m@xmath114 ( for a different neutron mass the results change by a factor ( m@xmath115 , which is 1.2 for a 2.4 m@xmath114 neutron star ) , that the inner disk terminates at @xmath116 km for the island state ( intervals a , b and c ) . for the banana state ( intervals e j ) we find an inner disk radius of @xmath117 km and decreasing towards higher soft color . indeed we find that the inner disk radius is closer to the neutron star in the banana than in the island state , but the values in both the banana and the island state are close to the @xmath113 km estimated by @xcite for the banana state . so , the difference in radius between island and banana state seems too small to make the disk disappear from the pca spectrum in the island state as proposed in the scenario of @xcite described above . if we assume that instead of @xmath24 , @xmath118 represents the keplerian frequency at the inner disk edge ( e.g. * ? ? ? * ) , this would lead to much larger differences in inner disk radius between the island ( @xmath119 km ) and the banana state ( @xmath120 km ) , so this interpretation could be consistent with the @xcite scenario . the identification of @xmath27 as the keplerian frequency at the inner disk edge is also suggested by the recently proposed extension of the pbk relation @xcite towards lower frequencies by including 17 white dwarf sources ( see * ? ? ? * and references therein ) . for the white dwarf sources a qpo is plotted versus the dwarf nova oscillation , dno , the dnos are thought to occur at the keplerian frequency of the inner disk edge ( * ? ? ? * and references therein ) . the behavior in the soft color vs. intensity diagram of 4u 160852 and 4u 170544 in their extreme island states can also be explained within the scenario of @xcite . let us assume that @xmath118 represents the keplerian frequency of the inner disk edge . in the extreme island states the disk is far out at 200 km and can not be seen in the pca energy spectrum ( see above ) , the soft color should therefore only depend on the spectral properties of the neutron star . if we look at the extreme island states a and b of 4u 160852 and 4u 170544 in the soft color versus intensity diagram , there seems to be a one to one relation between soft color and intensity ( see fig . [ fig.hid_sid_int ] for 4u 160852 ) . we can fit this relation with a power law with an index of about 0.1 . we propose that in the extreme island state as the accretion rate ( intensity ) increases , the neutron star surface gets more heated and the soft color increases . outside the extreme island states the soft color is determined by the spectral properties of both the neutron star surface and the disk and its dependence on accretion rate is therefore more complicated . the main conclusions of this paper can be summarized as follows : * the timing behavior of 4u 160852 is almost identical to that of the atoll sources 4u 172834 and 4u 0614 + 09 , and can be described in terms of a well defined set of components ( l@xmath23 , l@xmath26 , l@xmath28 , l@xmath41 , l@xmath33 , l@xmath35 ) which all vary in characteristic frequency together ( except for l@xmath28 for which @xmath14 is constant ) , sharing the same frequency correlations . * the timing behavior is not consistent with the idea that 4u 1608 - 52 traces out a three - branched z shape in the color - color diagram along which the timing properties vary gradually as in z sources . instead , the power spectral properties change smoothly along an `` @xmath20shaped '' track , which is probably composed of three , partially observed , extended extreme island branches . * 4u 160852 , at its lowest frequencies , shows a power spectrum almost identical to that of the low luminosity bursters 1e 17243045 , gs 182624 , and slx 1735269 , and the millisecond x ray pulsar sax j1808.4 - 3658 . the low luminosity bursters and the millisecond x ray pulsar all seem to be atoll sources at low mass accretion rate . a similar conclusion was reached previously based on 4u 0614 + 09 @xcite . * the high frequency peak at about 200 hz in interval a of 4u 160852 is more likely to be the upper kilohertz qpo than the hectohertz lorentzian . this means that the corresponding @xmath121 hz peaks in 4u 0614 + 09 , 1e 17243045 , gs 182624 , slx 1735269 and sax j1808.4 - 3658 also most likely represent the upper kilohertz qpo . * the low frequency part of the power spectra of both the atoll and the z sources behaves in a very complex manner . nevertheless , for the atoll sources it is possible to identify all these components within a single classification . if we compare the low frequency features of the atoll sources with those found in the z sources we find that the hbo of the z sources can be identified with l@xmath41 of the atoll sources . based upon the frequency relations with the upper kilohertz qpo neither the lfn nor the sub harmonic of the hbo of the z sources can straightforwardly be identified with the band limited noise of the atoll sources . * the parallel tracks in the frequency versus intensity diagram can be identified in the intensity versus color diagrams . however , the statistical spread in the colors , which shows up as the narrow nearly vertical parallel tracks in the color intensity diagrams , cause the @xmath57color correlation to be veiled on short timescales ( less than hours ) so in the color intensity diagrams parallel tracks similar to the parallel qpo tracks can only be identified on longer timescales . * we have tested the transition layer model , the sonic point beat frequency model , and the relativistic precession model using our results for the three atoll sources 4u 0614 + 09 , 4u 172834 , and 4u 160852 . neither the transition layer model nor the sonic point beat frequency model can explain the upper kilohertz qpo frequency range we find in these sources . the @xmath42 vs. @xmath24 can be described by a power law with an index of 2 as predicted by the relativistic precession model , however it is very likely that the burst oscillation frequencies found for 4u 172834 ( 363 hz ) , and 4u 160852 ( 620 hz ) represent the spin frequency of the neutron star in each case . then it is difficult to understand why the @xmath42 vs. @xmath24 relations of these sources coincide . * our timing and color results are consistent with the scenario proposed by @xcite to explain the results of their x ray energy spectral study of 4u 160852 , only if @xmath118 , and not @xmath24 represents the keplerian frequency of the inner disk edge . in this scenario the disk is close to the neutron star in the banana state , and far from the neutron star in the island state . the large radius of the inner disk in the island state prevents the disk from being observed directly in the pca spectra . this work was supported by nwo spinoza grant 080 to e.p.j . van den heuvel , by the netherlands organization for scientific research ( nwo ) , and by the netherlands research school for astronomy ( nova ) . this research has made use of data obtained through the high energy astrophysics science archive research center online service , provided by the nasa / goddard space flight center . we would like to thank peter jonker for providing us with tables for the gx 51 and gx 340 + 0 data . we now provide a detailed description of the components found in the combined power spectra of 4u 160852 and compare them to previous results on 4u 172834 , 4u 0614 + 09 of @xcite and the low luminosity bursters discussed by @xcite . the combined power spectrum of * interval a * ( fig . [ fig.powspec_1608 ] ) is very similar to interval 1 of 4u 0614 + 09 ( see figure 2 in * ? ? ? it shows l@xmath33 and l@xmath41 at the lowest characteristic frequencies observed in atoll sources . figure [ fig.freq_freq ] confirms the identification of these components . as was the case in 4u 0614 + 09 , interval a shows l@xmath26 at @xmath31 hz and a broad lorentzian at @xmath64 hz . the lorentzian at @xmath64 hz found in 4u 160852 ( interval a ) and 4u 0614 + 09 can be identified as either l@xmath23 or l@xmath28 based on its frequency . however , this lorentzian has a similar rms fractional amplitude for both 4u 160852 and 4u 0614 + 09 . the upper kilohertz qpos of 4u 160852 and 4u 0614 + 09 always have a similar fractional rms , whereas for l@xmath28 the fractional rms is always about 5 percentage points less in 4u 160852 than in 4u 0614 + 09 ( see fig . [ fig.rms_all ] ) . this hints towards this component being l@xmath23 . * interval b * is similar to interval a , it shows l@xmath33 , l@xmath41 , l@xmath26 and l@xmath23 , but all at higher characteristic frequencies . interval b also shows a narrow qpo at 2.458 hz which will be discussed in [ comp : timing : atoll ] . intervals a and b both show a component designated l@xmath26 , at @xmath31 and @xmath122 hz respectively . the broad l@xmath26 at these low frequencies can be tentatively identified with the lower kilohertz qpo based on extrapolations of frequency frequency relations ( see * ? ? ? * ; * ? ? ? * ; * ? ? ? . however , this lorentzian might also be an unrelated feature ( see [ comp : timing : atoll ] ) . * interval c * shows only three components , l@xmath33 , l@xmath41 and l@xmath23 again all at higher frequencies than in the previous interval . * interval d * shows l@xmath33 , l@xmath28 and l@xmath23 . compared to interval c , l@xmath41 is no longer present and l@xmath28 has appeared . based on the similarities with 4u 172834 and 4u 0614 + 09 , l@xmath41 would be expected to be more narrow than in intervals a c ( expected @xmath18 about 12 ) and have a characteristic frequency of about 30 hz . figure [ fig.powspec_1608 ] shows a non significant narrow peak at @xmath123 hz in interval d which can probably be attributed to l@xmath41 . as noted above , l@xmath28 , a broad lorentzian with a constant @xmath14 of @xmath124 hz , appears in interval d for the first time . in interval a l@xmath28 would lie on top of l@xmath23 ( see above ) and in intervals b and c l@xmath28 would lie in the flank of l@xmath23 . therefore it is hard to detect l@xmath28 in intervals a c . note that there is some excess power at the expected location of 150 hz in interval c. in interval d l@xmath23 has moved above this frequency range so that l@xmath28 is more easily detectable . both the frequency and the fractional rms of l@xmath33 in interval d are above the relations in figures [ fig.freq_freq ] and [ fig.rms_all ] . including the little peak at @xmath123 hz in the fit hardly affects l@xmath33 . these deviations from the relations in figures [ fig.freq_freq ] and [ fig.rms_all ] are most likely due to frequency shifts inside interval d , which covers a large range in the color color diagram ( see fig . [ fig.cc_int ] ) . splitting this interval further up would compromise the statistics . * interval e * shows l@xmath33 , l@xmath35 and a pair of kilohertz qpos . here the character of l@xmath33 has changed from a band limited noise component ( fitted with a zero centered lorentzian ) to a qpo . l@xmath28 is visible in the power spectrum but is not significant ( 2.4 @xmath12 single trial ) in this interval . because leaving it out of the fit makes l@xmath35 fit power up to hundreds of hz leading to an overestimated @xmath36 and rms fractional amplitude , we include a l@xmath28 at 150 hz ( the average value in 4u 172834 and 4u 0614 + 09 ) with @xmath125 ( see above ) in the fit . in table [ tbl.highfitpar ] and in figure [ fig.rms_all ] we only show the upper limit for this l@xmath28 . the characteristic frequency of l@xmath35 in interval e is still high compared to the general relation between @xmath36 and @xmath24 ( see fig . [ fig.freq_freq ] ) . this is similar to what was observed in 4u 172834 where in this range there are also two points where @xmath36 is significantly above the relation . these deviations all occur close to the l@xmath33 `` transformation '' and when the l@xmath33 qpo is still weak compared to the band limited noise component l@xmath35 ( see fig . [ fig.rms_all ] ) , so one possible explanation is that in these transition power spectra a narrow and a broad l@xmath33 are merged together . the power spectra of * intervals f , g and h * all show very low frequency noise ( vlfn ) , and l@xmath33 , l@xmath35 , l@xmath28 and the pair of kilohertz qpos all increasing in frequency . * intervals i and j * both show vlfn , l@xmath28 and a broad kilohertz qpo . it is unclear whether the kilohertz peak represents the upper , the lower or even a blend of both kilohertz peaks . there were no other power spectral features present which could be used to identify this component with the help of the correlations of figure [ fig.freq_freq ] . if these peaks , with characteristic frequencies of about 1050 and 1250 hz , would be the lower kilohertz qpo they would be among the highest frequency lower kilohertz qpos found in any source up to date ( see e.g. * ? ? ? so to be slightly conservative and also because it is more convenient , as in figures [ fig.freq_freq ] and [ fig.rms_all ] we use the characteristic frequency of l@xmath23 to plot against , we list these kilohertz qpos as l@xmath23 . the @xmath18 values for the kilohertz qpos in intervals i and j are much lower than those in 4u 172834 and 4u 0614 + 09 . this is probably an artificial broadening due to the movement of the kilohertz qpo during the intervals . a finer subdivision of intervals i and j does not lead to observing l@xmath23 and l@xmath26 separately . note that no kilohertz qpos were found in the individual observations of these intervals @xcite . interval j shows an additional broad bump which is also seen in 4u 172834 ( interval 18 and 19 of figure 1 in * ? ? ? * ) and 4u 0614 + 09 ( interval 9 of figure 2 in * ? ? ? * ) ; this lorentzian can be identified as either l@xmath33 , l@xmath35 or a new type of lorentzian . here we list it as l@xmath35 . the vlfn in 4u 160852 shows a similar behavior to that in other atoll sources . the power law index is about constant around 1.5 , with the exception of interval f where it is 2.5 . the rms fractional amplitude generally increases with position in the color color diagram . in intervals i and j the vlfn clearly deviates from just a power law . this leads to high @xmath22/dof values of 2.5 and 1.8 . adding lorentzians in interval i and j to fit the deviations leads to significantly better @xmath22/dof of 1.3 and 1.2 . the @xmath14 of the lorentzian in interval i is then 0.02 hz and in interval j 0.3 hz in interval i the lorentzian does not only fit the deviation but almost the entire vlfn . the inclusion of the lorentzians lowers the rms fractional amplitude of the power law to 1.59 % for interval i and 2.27 % for interval j. + 1 & 3 & 3 & 0.160.25 & h + 2 & 1 & 3 & 0.05 & d + 3 & 1 & 3 & 0.040.05 & i + 4 & 1 & 3 & 0.020.03 & d + 5 & 1 & 4 & 0.02 & c + 6 & 1 & 16 & 0.030.04 & d + 7 & 116 & 84 & 0.030.04 & b + 8 & 64 & 402 & 0.010.02 & c + + 9 & 43 & 5 & 0.241.20 & j + 10 & 6 & 1 & 0.080.21 & fgh + 11 & 1 & 1 & 0.05 & e + 12 & 2 & 1 & 0.020.04 & c + 13 & 1 & 1 & 0.03 & e + 14 & 5 & 2 & 0.050.10 & i + 15 & 1 & 7 & 0.030.04 & fg + 16 & 7 & 38 & 0.040.05 & b + 17 & 119 & 523 & 0.030.05 & a + + 18 & 1 & 4 & 0.060.07 & i + 19 & 1 & 2 & 0.03 & d + 20 & 1 & 7 & 0.01 & c + 21 & 1 & 3 & 0.03 & j + 22 & 1 & 9 & 0.01 & c + 23 & 1 & 12 & 0.02 & i + 24 & 1 & 4 & 0.01 & c + 25 & 1 & 4 & 0.020.03 & g + 26 & 1 & 4 & 0.02 & c + 27 & 1 & 4 & 0.070.08 & f + 28 & 6 & 2 & 0.030.11 & i + 29 & 1 & 2 & 0.01 & d + 30 & 1 & & 0.01 & c + lcccccccccccccc a & & & [email protected] & 0 ( fixed ) & [email protected] & & & & [email protected] & [email protected] & [email protected] + b & & & [email protected] & 0 ( fixed ) & [email protected] & & & & [email protected] & [email protected] & [email protected] + c & & & [email protected] & 0 ( fixed ) & [email protected] & & & & [email protected] & [email protected] & [email protected] + d & & & [email protected] & 0 ( fixed ) & [email protected] & & & & & & @xmath127 + e & & & [email protected] & 3.8@xmath128 & 4.46@xmath129 & [email protected] & 0 ( fixed ) & [email protected] & & & @xmath130 + f & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 0 ( fixed ) & [email protected] & & & @xmath131 + g & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 10.6@xmath132 & 0 ( fixed ) & 2.44@xmath133 & & & + h & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 15.4@xmath134 & 0 ( fixed ) & [email protected] & & & + i & [email protected] & [email protected] & & & & & & & & & + j & [email protected] & [email protected] & & & & [email protected] & [email protected] & [email protected] & & & + lccccccccc a & & & & [email protected] & [email protected] & [email protected] & 216@xmath12634 & 0 ( fixed ) & [email protected] + b & & & @xmath135 & [email protected] & 0 ( fixed ) & [email protected] & 309@xmath12613 & [email protected] & [email protected] + c & & & @xmath136 & & & & 474@xmath12621 & [email protected] & [email protected] + d & 189@xmath12623 & [email protected] & [email protected] & & & & [email protected] & [email protected] & [email protected] + e & & & @xmath137 & 531@xmath138 & 6.1@xmath139 & [email protected] & [email protected] & [email protected] & [email protected] + f & [email protected] & [email protected] & [email protected] & [email protected] & 19@xmath140 & 8.3@xmath141 & [email protected] & [email protected] & [email protected] + g & 157@xmath12618 & [email protected] & [email protected] & [email protected] & 16.8@xmath142 & 9.91@xmath143 & [email protected] & [email protected] & [email protected] + h & [email protected] & 3.0@xmath144 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] + i & [email protected] & [email protected] & [email protected] & & & & 1046@xmath12645 & [email protected] & [email protected] + j & 123@xmath12625 & [email protected] & [email protected] & & & & 1252@xmath12695 & [email protected] & [email protected] +
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we have studied the atoll source 4u 160852 using a large data set obtained with the rossi x - ray timing explorer .
we find that the timing properties of 4u 160852 are almost exactly identical to those of the atoll sources 4u 0614 + 09 and 4u 172834 despite the fact that contrary to these sources 4u 160852 is a transient covering two orders of magnitude in luminosity .
the frequencies of the variability components of these three sources follow a universal scheme when plotted versus the frequency of the upper kilohertz qpo , suggesting a very similar accretion flow configuration .
if we plot the z sources on this scheme only the lower kilohertz qpo and hbo follow identical relations . using the mutual relations between the frequencies of the variability components we tested several models ; the transition layer model ,
the sonic point beat frequency model , and the relativistic precession model .
none of these models described the data satisfactory .
recently , it has been suggested that the atoll sources ( among them 4u 160852 ) trace out similar three
branch patterns as the z sources in the color color diagram .
we have studied the relation between the power spectral properties and the position of 4u 160852 in the color color diagram and conclude that the timing behavior is not consistent with the idea that 4u 1608 - 52 traces out a three - branched z shape in the color - color diagram along which the timing properties vary gradually , as z sources do .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
an electron incident on a superconductor from a normal metal , with an energy smaller than the superconducting energy gap , can not propagate into the superconductor and thus should be perfectly reflected . however , andreev discovered a mechanism for transmission , in which an electron may form a cooper pair with another electron and be transmitted across the superconductor . as a consequence of charge conservation a hole must be left behind , which , as a result of momentum conservation , should propagate in a direction opposite to that of the incident electron . this process is termed andreev reflection @xcite . apart from providing a confirmation for the existence of cooper pairs and superconductor energy gaps @xcite , this process may also have applications in spintronics . it has been suggested that point contact andreev reflection can be used to probe spin polarization of ferromagnets by fabricating ferromagnet - superconductor nanojunctions @xcite . materials - specific modelling of such experiments , however , is complex and so far it has been somehow unsatisfactory . for instance tight - binding based scattering theory @xcite and green s functions theory @xcite calculations found poor fits to the experimental data for ferromagnet - superconductor junctions , while produced excellent fitting to normal metal - superconductor junctions results . based on this observation , xia and co - workers suggested that there may be an interaction between the ferromagnet and superconductor which is not accounted for in the blonder - tinkham - klapwijk ( btk ) model @xcite . consequently , the simple interpretation and two - parameter btk model fitting of experimental data to extract the spin polarization of various ferromagnets , was also called into question . more recently , chen , tesanovic and chien proposed a unified model for andreev reflection at a ferromagnet - superconductor interface @xcite . this is based on a partially polarized current , where the andreev reflection is limited by minority states and the excess majority carriers provide an evanescent contribution . however , this model has also been called into doubt by eschrig and co - workers @xcite . in particular , they pointed out that the additional evanescent component is introduced in an _ ad - hoc _ manner , and that the resulting wavefunction violates charge conservation . so , the debate about the correct model to describe andreev reflection at a ferromagnet - superconductor junction seems far from being settled . among other mesoscopic systems , andreev reflection has also been measured in carbon nanotubes ( cnts ) @xcite . there has been a theoretical study of normal metal - molecule - superconductor junction from density functional theory based transport calculations @xcite . in this study it was shown that the presence of side groups in the molecule can lead to fano resonances in andreev reflection spectra . topological insulators , a very recent and exciting development in condensed matter physics , have also been shown to be characterized by perfect andreev reflection @xcite . wang and co - authors have recently suggested performing a self - consistent calculation of the scattering potential to study andreev reflection at normal metal - superconductor junctions @xcite . they calculated the conductance for carbon chains sandwiched between a normal and a superconducting al electrode and found different values depending on whether or not the calculation was carried out self - consistent over the hartree and exchange - correlation potential . however , the theoretical justification for such a self - consistent procedure is at present not clear . in particular , it is difficult to argue that the variational principle , which underpins the hohenberg - kohn theorems , is still obeyed when a pairing energy is added _ by hand _ to the kohn - sham potential . in principle a rigorous self - consistent treatment should use the superconducting version of density functional theory @xcite , which probably remains computationally too expensive for calculating the interfaces needed to address a scattering problem . given such theoretical landscape and the fact that a non self - consistent approach to density functional theory based transport calculations has shown excellent agreement to experimental results for normal metal - superconductor junctions , we follow this methodology in the present work . in this paper , we study andreev reflection in normal - superconductor junctions , including all - metal junctions and carbon nanotubes sandwiched between normal and superconducting electrodes . we take into account the atomistic details of the junction by using density functional theory to obtain the underlying electronic structure , and then employ an extended btk model to solve the normal - superconductor scattering problem . our transverse momentum resolved calculations allow identifying the contributions to conductance from different parts of the brillouin zone . we also study the variation of conductance as a function of an applied potential difference between the electrodes for various normal metal - superconductor junctions , by performing approximate finite bias calculations . after this introduction , the rest of our paper is organized as follows : in section [ formulation ] we summarize the extended btk model and beenakker s formula , which we employ in this work . in the subsequent section [ results ] , we present our results for cu - pb , co - pb and au - al junctions , as well as al - cnt - al junctions . we also include the computational details in each of these subsections . finally , we conclude and summarize our findings in section [ conclusions ] . for the sake of completeness , here we briefly summarize the extended btk model @xcite that we use to study andreev reflection at a normal metal - superconductor interface . following refs . [ ] , we begin with the bogoliubov - de gennes equation @xmath0 where @xmath1 is the single particle hamiltonian for majority ( @xmath2 ) and minority ( @xmath3 ) spins , @xmath4 is the pairing potential and @xmath5 and @xmath6 are respectively the electron and hole wavefunctions . the energy @xmath7 sets the reference to the fermi energy , @xmath8 . we follow the approach of beenakker consisting in inserting a layer of superconductor in its normal state between the metal - superconductor interface . this ensures that at the fictitious normal metal - superconductor interface the only scattering process is andreev scattering . , indicated by the arrow . self - energies are used to simulate the effect of semi - infinite leads attached to the edge of the scattering region . ] other scattering processes are accounted for at the junction between the normal metal the and superconductor in its normal state . at this interface the scattering matrix can be written as @xmath9 here the superscripts @xmath10 and @xmath11 denote the right- and left - going states and the subscripts @xmath12 and @xmath13 refer to the normal and fictitious normal metal regions , respectively . the normal state scattering matrix reads @xmath14 now at the fictitious normal metal - superconductor interface @xmath15 where the factor @xmath16 is @xmath17 , \quad |\varepsilon|<\delta \nonumber \\ & = & \frac{1}{\delta}[\varepsilon-\mathrm{sign}(\varepsilon)\sqrt{\varepsilon^{2}-\delta^{2 } } ] , \quad |\varepsilon|>\delta\:.\end{aligned}\ ] ] the states in the normal metal are given by @xmath18 then the reflection coefficients for the complete system are @xmath19 and @xmath20 finally the conductance of the system is given by @xmath21 the implicit assumptions in the above derivation are that the superconducting order parameter is switched on abruptly as a step function ( i.e. , there are no proximity effects ) and the order parameter is much smaller than the fermi energy ( the so - called andreev approximation ) . a great simplification occurs if one considers scattering at fermi energy , namely @xmath22 , and the presence of time reversal symmetry , i.e. , the normal metal is not a ferromagnet . the above expression for conductance reduces to @xmath23 where the eigenvalues of the transmission matrix product @xmath24 are @xmath25 . this is the beenakker s formula @xcite . notice that all the dependence on the superconductor pairing has dropped out and the conductance depends on the normal state transmission eigenvalues . in this case superconductivity enters implicitly in the form of a boundary condition . in our first - principles transport code smeagol @xcite , we construct the full scattering matrix and then use the expressions in equations ( [ ree ] ) and ( [ rhe ] ) to evaluate the conductance from equation ( [ gnsfull ] ) . for the special case of @xmath26 , we construct the transmission matrix , @xmath24 . it is then straightforward to obtain its eigenvalues by numerical diagonalization . these are then interted into the beenakker s formula [ equation ( [ gnsbeenakker ] ) ] to obtain @xmath27 , while a direct summation of the eigenvalues yields @xmath28 . to compute the current , @xmath29 , at a bias @xmath30 , we use @xmath27 from equation ( [ gnsfull ] ) and calculate @xmath31g(\varepsilon)\:,\ ] ] and the finite bias conductance is evaluated from @xmath32 here @xmath33 can either be the normal state conductance , @xmath34 , or the normal metal - superconductor conductance , @xmath27 , and @xmath35 is the fermi function . we begin by presenting our results for cu - pb junctions , which have also been investigated experimentally in the past @xcite . we choose cu @xmath36 and @xmath37 and pb @xmath38 and @xmath39 as valence electrons and the effect of other core electrons are described by troullier - martins norm - conserving pseudopotentials . the local density approximation with the ceperley - alder parametrization was employed for the exchange - correlation functional . we choose an energy cutoff of 400 rydberg for the real space mesh , and a double-@xmath40 polarized basis set . the lattice constants of cu ( @xmath41 ) and pb ( @xmath42 ) are quite different , however a matching is obtained by rotating cu unit cell by a @xmath43 angle . in this geometry a small strain ( @xmath44% ) exists on both cu ( compressive ) and pb ( tensile ) . for the self - consistent calculation we use a @xmath45 in plane monkhorst - pack grid , while transport quantities are evaluated over a much denser @xmath46 @xmath47 grid . , between cu and pb . note that @xmath48 remains positive for all the distances investigated here . ] plane ( orthogonal to transport direction , @xmath49 ) periodic boundary conditions are employed . ] the scattering region for a cu - pb junction is shown in fig . [ cu - pb - setup ] . we use periodic boundary conditions in the plane orthogonal to the transport direction , and open boundary conditions along the direction of transport . we plot the available channels for both electrodes resolved over the brillouin zone ( bz ) at the fermi energy in fig . [ cu - pb - kp](a ) . for the left electrode ( cu ) four channels are available in quadrants centered at the edge of the bz , with a residual region around the zone center in which either three or two channels are available . for the right electrode ( pb ) around the bz center there exists a rectangular region with three open channels , while at the bz corners there are small pockets of reduced available channels , which even drop down to zero . the normal conductance , @xmath34 , is large over almost the entire bz , along with small pockets of lower transmission at the edges of the bz , which are inherited from the reduced channel pockets in the pb electrode , as shown in fig . [ cu - pb - kp](b ) . another small conductance pocket is present at the zone center , which originates from the distribution of open channels across the bz in the cu electrode . the overall conductance remains largely unchanged as the cu - pb distance is increased from @xmath50 1.5 to 3.0 . next we show the normal metal - superconductor junction conductance , @xmath27 , in fig . [ cu - pb - kp](c ) . at @xmath50 1.5 , the pockets of small conductance at the zone edges are more prominent , as compared to @xmath34 . moreover , the region around @xmath51 , with reduced conductance is also larger . on increasing the distance to 2 , these low conductance pockets shrink in size and the overall conductance increases . at larger distances , a broader region of low conductance develops and this reduces overall @xmath27 . in table [ cu - pb - tab ] , we provide the @xmath47-averaged value of the conductance above ( @xmath34 ) and below ( @xmath27 ) the pb superconducting temperature . for both quantities a maximum is obtained at @xmath50 2 . we also tabulate the ratio @xmath52 , which is the quantity expressing the zero - bias suppression due to andreev reflection . for a single channel btk model describing an ideal interface this ratio is exactly two , however when one takes into account the band structure mismatch and the underlying electronic structure of the electrodes a much lower value for this ratio can be obtained . for the cu - pb equilibrium distance ( @xmath53 ) we find @xmath52 close to 1.4 , which is in excellent agreement with the experimental value of 1.38 reported in ref . . , between the two constituents . ] .cu - pb junction : normal conductance , @xmath34 , normal - superconductor conductance , @xmath27 , and their ratio at different cu / pb distances . [ cols="^,^,^,^,^ " , ] finally , we study the variation of normalized conductance as a function of an applied bias , which is plotted in fig . [ al - cnt - bias ] . at low bias , for al - cnt(3,0)-al junction the normal metal - superconductor conductance is greater than the normal conductance . interestingly , the situation is reversed at a voltage of 0.1 mv . in contrast , for al - cnt(4,0)-al junction , the normalized conductance remains negative for voltages less than the superconducting gap . in conclusion , we have studied andreev reflection in normal - superconductor junctions using density functional theory based transport calculations . this approach allowed us to include the atomistic details of the junction electronic structure in the extended blonder - tinkham - klapwijk model . we studied au - al and cu - pb all metal junctions and calculated the normal and normal - superconductor conductances for different separations of the two materials at the interface . our transverse momentum resolved analysis has allowed us to identify contributions to these quantities from different parts of the brillouin zone . we found that the conductances for junctions in the superconducting state follows a similar @xmath47-point dependence as the normal state conductance . in other words , andreev reflection is higher in brillouin zone regions , where transmission is also high . we have also investigated co - pb ferromagnet - superconductor junctions . in this case , while at zero bias , our results satisfactorily match the experimental reports , a discrepancy was revealed at a finite bias , particularly at voltages close to the superconductor gap . this could possibly be attributed to stray magnetic fields from the ferromagnet or to proximity effects , both causes which are not included in the extended blonder - tinkham - klapwijk model . we further studied andreev reflection from carbon nanotubes sandwiched between normal metal and superconducting electrodes and found @xmath52 ratios to lie on opposite sides of unity for @xmath54 ( higher than one ) and @xmath55 ( lesser than one ) carbon nanotubes . this highlights the sensitivity of such calculations to details and the need for a truly atomistic theory for tackling this problem . concerning the potential outlook for future studies , our work provides a stepping stone for analyzing with first - principles methods the experimental setups needed to investigate and detect majorana fermions . these particles , which are their own anti - particles , are expected to play a crucial role in topological quantum computing and have recently garnered significant attention in the condensed matter community . after several theoretical proposals , signatures of this particle were found experimentally in large spin - orbit nanowires in proximity with superconductors @xcite . however , a number of issues remain unresolved and important questions need to be answered to confirm that indeed majoranas were observed . our implementation of the phenomenology of andreev reflection in a first - principles approach can be quite useful to study such a setup , in particular , by taking into account the underlying electronic structure of the nanowires . when combined with the order-@xmath56 implementation of our smeagol code @xcite , which allows us treating thousands of atoms , it opens the opportunity of recreating theoretically the aforementioned experiments in an _ ab inito _ manner , which till now have been modelled empirically . an is financially supported by irish research council under the embark initiative . ir and ss acknowledge additional support by kaust ( acrab project ) . the computational resources have been provided by trinity centre for high performance computing .
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we study andreev reflection in normal metal - superconductor junctions by using an extended blonder - tinkham - klapwijk model combined with transport calculations based on density functional theory . starting from a parameter - free description of the underlying electronic structure , we perform a detailed investigation of normal metal - superconductor junctions , as the separation between the superconductor and the normal metal is varied .
the results are interpreted by means of transverse momentum resolved calculations , which allow us to examine the contributions arising from different regions of the brillouin zone .
furthermore we investigate the effect of a voltage bias on the normal metal - superconductor conductance spectra .
finally , we consider andreev reflection in carbon nanotubes sandwiched between normal and superconducting electrodes .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the current observed neutrino mixing @xcite suggests around the maximal 2 - 3 mixing angle and zero 1 - 3 mixing angle : @xmath1 , @xmath2 . in such a symmetric limit where both @xmath3 and @xmath4 vanish , the resulting @xmath5 effective majorana neutrino mass matrix forms in the flavor basis as @xcite ( ccc x & c & c + c & a & b + c & b & a ) . [ m - z2 ] this matrix has an exact symmetric form under a @xmath6 symmetry , i.e. the 2 - 3 ( @xmath7-@xmath8 ) permutation , and is diagonalized by the unitary matrix : u_z_2 = ( ccc _ 12 & -_12 & 0 + _ 12/ & _ 12/ & -1/ + _ 12/ & _ 12/ & 1/ ) , with remaining the solar mixing angle @xmath9 arbitrary and the entry @xmath10 of ( [ m - z2 ] ) is determined as x = a + b + . now we well know two special values for @xmath9 which give typical mixing matrices ; one is bimaximal and the other is tri - bimaximal mixing matrices . in a limit of bimaximal mixing @xcite where @xmath11 , resulting mns matrix forms u_bm = ( ccc 1/ & -1/ & 0 + 1/2 & 1/2 & -1/ + 1/2 & 1/2 & 1/ ) , with x = a + b. for the case @xmath12 with so - called tri - bimaximal mixing which is proposed by harrison , perkins and scott , then we have the hps type matrix @xcite : u_hps = ( ccc 2/ & 1/ & 0 + -1/ & 1/ & -1/ + -1/ & 1/ & 1/ ) , where @xmath9 is fixed by as well x = a + b + c is also derived . note that the tri - bimaximal structure is consistent with current experimental data , where @xmath9 is not maximal . so far , the discrete symmetry @xmath0 is successfully applied for leptons . namely , the tri - bimaximal mixing pattern can be realized naturally in a number of specific models @xcite . however , it is not easy to have small quark mixing angles in naive and straight way . in such applications the generic prediction @xcite is that @xmath13 , the quark mixing matrix becomes just the unit matrix . starting with @xmath14 , the realistic small quark mixing angles can be generated by extending interactions beyond those of the standard model , such as in supersymmetry @xcite or breaking @xmath0 symmetry explicitly @xcite . our study is aimed to obtain realistic quark masses and mixing angles entirely within the @xmath0 context @xcite . it is worthy of mention that the other types of unified models for quarks and leptons with the @xmath0 symmetry @xcite and models which predict tri - bimaximal mixing matrix with @xmath15 symmetry @xcite have been also studied . @xmath0 is the symmetry group of the tetrahedron and the finite groups of the even permutation of four objects . it has twelve elements which are derived into four equivalence classes : [ @xmath16 : ( 1234 ) , [ @xmath17 : ( 2143 ) , ( 3412 ) , ( 4321 ) , [ @xmath18 : ( 1342 ) , ( 4213 ) , ( 2431 ) , ( 3124 ) and [ @xmath19 : ( 1423 ) , ( 3241 ) , ( 4132 ) , ( 2314 ) , corresponding to its four irreducible representations we call three one - dimensional representations ( singlets ) as @xmath20 , @xmath21 , @xmath22 , and one three - dimensional representation ( triplet ) as @xmath23 , respectively . the @xmath0 is the smallest discrete group which includes the three - dimensional irreducible representation . the presence of the three - dimensional irreducible representation might be ideal for describing three families of quarks and leptons . the character table of four representations is shown in table [ table : ct ] . here @xmath24 is the order of each element , @xmath25 is the number of elements and the complex number @xmath26 is the cube root of unity : = ( 2 i/3 ) = - + i , 1 + + ^2 = 0 . .character table of @xmath0 . [ table : ct ] [ cols="^,^,^,^,^,^,^",options="header " , ] in minimal su(5 ) , there is just one @xmath27 representation of higgs bosons , yielding thus only two invariants , i.e. @xmath28 ( that is for up quark mass matrix ) and @xmath29 ( for charged lepton and down quark mass matrices ) . the second invariant implies @xmath30 at the unification scale which is phenomenologically desirable . we follow the usual strategy of using both @xmath31 and @xmath32 representations of higgs bosons , so that one linear combination couples to only leptons , and the other only to quarks . both transform as @xmath23 under @xmath0 . there are also two @xmath27 representations transforming as @xmath21 and @xmath22 under @xmath0 which couple only to up type quarks and @xmath22 , other patterns can be assigned with different prediction . ] . with this @xmath0 assignment for higgs doublets , the relevant yukawa couplings linking @xmath33 with @xmath34 are given by & & h_1 d_1 ( d^c_1 ^0_d1 + d^c_2 ^0_d2 + d^c_3 ^0_d3 ) + h_2 d_2 ( d^c_1 ^0_d1 + d^c_2 ^0_d2 + ^2 d^c_3 ^0_d3 ) + & & + h_3 d_3 ( d^c_1 ^0_d1 + ^2 d^c_2 ^ 0_d2 + d^c_3 ^0_d3 ) , resulting in the @xmath5 down quark mass matrix : m_d = ( ccc h_1 & 0 & 0 + 0 & h_2 & 0 + 0 & 0 & h_3 ) ( ccc 1 & 1 & 1 + 1 & & ^2 + 1 & ^2 & ) ( ccc v_1 & 0 & 0 + 0 & v_2 & 0 + 0 & 0 & v_3 ) , where @xmath35 , @xmath36 are three independent yukawa couplings , and @xmath37 are the vacuum expectation values of @xmath38 . to get quark mass hierarchy , @xmath39 should be satisfied contrary to the lepton sector ( @xmath40 ) . by contrast , the higgs doublets transforming as @xmath21 and @xmath22 linking @xmath41 with @xmath42 give , as following the manner of eq . ( [ 1x1x1 ] ) , the @xmath5 symmetric up quark mass matrix : m_u = ( ccc 0 & _ 2 & _ 3 + _ 2 & m_2 & 0 + _ 3 & 0 & m_3 ) , where @xmath43 , @xmath44 come from @xmath45 and @xmath46 , @xmath47 from @xmath48 and three of them can be taken real in general . in the limit @xmath49 and @xmath50 , we obtain the three eigenvalues of @xmath51 as m_t |m_3| , m_c |m_2| , m_u |+ | , with mixing angles v_uc , v_ut , v_ct 0 . in the down sector , we note that m_dm_d^= ( ccc y |h_1|^2 & z^h_1 h_2^ & z h_1 h_3^ + z h_1^h_2 & y |h_2|^2 & z^h_2 h_3^ + z^h_1^h_3 & z h_2^h_3 & y |h_3|^2 ) , where y & = & |v_1|^2 + |v_2|^2 + |v_3|^2 , + z & = & |v_1|^2 + |v_2|^2 + ^2 |v_3|^2 . its eigenvalue @xmath52 satisfies the equation & & ^3 - y(|h_1|^2 + |h_2|^2 + |h_3|^2)^2 - ( y^3+z^3+z^3 -3y|z|^2)|h_1|^2|h_2|^2|h_3|^2 & & + ( y-|z|^2)(|h_1|^2|h_2|^2 + |h_1|^2|h_3|^2 + |h_2|^2|h_3|^2)= 0 . if @xmath53 as well as assumed in the charged lepton case , then @xmath54 , @xmath55 and three eigenvalues are simply @xmath56 . we choose them instead to be different , but we still assume @xmath57 . in that case , we find m_b^2 & & y|h_3|^2 , + m_s^2 & & ( ) |h_2|^2 , + m_d^2 & & ( ) |h_1|^2 , and the mixing angles are given by v_sb & & ( ) , + v_db & & ( ) , + v_ds & & ( ) , thereby requiring the condition | | | | . using current experimental values for the left - hand side , we see that quark mixing in the down sector alone can not explain the observed quark mixing matrix @xmath13 . taking into account @xmath58 , we then have v_ckm = v_u^v_d . hence v_us & & v_ds - v_uc ( ) - , [ vus ] + v_cb & & v_sb ( ) , [ vcb ] + v_ub & & v_db - v_uc v_sb - v_ut ( ) - ( ) - . [ vub ] it is noted that our up and down quark mass matrices are restricted by our choice of @xmath0 representations to have only five independent parameters each . in the up sector , we have three real and one complex parameters ( for example we choose here @xmath43 , @xmath46 and @xmath47 to be real with @xmath44 complex ) . the five independent parameters can be chosen as the three up quark masses , one mixing angle and one phase . in the down sector , the yukawa couplings @xmath59 can all be chosen real , @xmath60 is just an overall scale , and @xmath61 is complex . the five independent parameters can be chosen as the three down quark masses and two mixing angles . now we have ten parameters in these two matrices except their overall normalizations or magnitudes of yukawa couplings . since we also have ten observables for six quark masses , three angles and one phase , it may appear that a fit is not so remarkable . however , the forms of the mass matrices are very restrictive , and it is by no means trivial to obtain a good fit . indeed , we find that @xmath62 is strongly correlated with the cp phase @xmath63 which is one of angles of the unitary triangle . if we were to fit just the six masses and the three angles , the structure of our mass matrices would allow only a very narrow range of values for @xmath63 at each value of @xmath64 . this means that future more precise determinations of these two parameters will be a decisive test of this model . cp violation is also predicted in our model . the jarlskog invariant @xcite is given by j_cp ( ) ( 1 + ) , [ jcp ] which is mainly comes from the down sector and the up sector only affects as correction terms . in order to fit the ten observables ( six quark masses , three ckm mixing angles and one cp phase ) , ten parameters of our model have been generated numerically . we choose the parameter sets which are allowed by the experimental data . we show the prediction of @xmath64 verses @xmath63 in fig . [ f1 ] , with the following nine experimental inputs @xcite : & & m_u = 0.9 ~2.9 ( mev ) , m_c = 530 ~680 ( mev ) , m_t = 168 ~180 ( gev ) , & & m_d = 1.8 ~5.3 ( mev ) , m_s = 35 ~100 ( mev ) , m_b = 2.8 ~3 ( gev ) , & & |v_us| = 0.221 ~0.227 , |v_cb| = 0.039 ~0.044 , j_cp = ( 2.75 ~3.35)10 ^ -5 , & & [ input1 ] which are given at the electroweak scale . we see that the experimental allowed region of @xmath63 ( @xmath65 radian at @xmath66 c.l . ) @xcite corresponds to @xmath64 in the range @xmath67 , which is consistent with the experimental value of @xmath68 . thus our model is able to reproduce realistically the experimental data of quark masses and the ckm matrix . plane , where the value of @xmath63 is expressed in radians . the horizontal and vertical lines denote experimental bounds at @xmath66 c.l . ( [ input1 ] ) . ] precisely measured heavy quark masses and ckm matrix elements are expected in future experiments and precise light quark masses are expected in future lattice evaluations . if the allowed regions of the current data shown in eq . ( [ input1 ] ) are reduced , the correlation between @xmath64 and @xmath63 will become stronger . we show in fig . [ f2 ] the case where the experimental data are restricted to some very narrow ranges about their central values : & & m_u = 1.4 ~1.5 ( mev ) , m_c = 600 ~610 ( mev ) , m_t = 172 ~176 ( gev ) , & & m_d = 3.4 ~3.6 ( mev ) , m_s = 60 ~70 ( mev ) , m_b = 2.85 ~2.95 ( gev ) , & & & & [ input2 ] here we use the tighter constraints on the mass ratios of light quarks , i.e. @xmath69 and @xmath70 , consistent with the well - known successful low - energy sum rules @xcite : clearly , future more precise determinations of @xmath64 and @xmath63 will be a sensitive test of our model . plane , where input data are restricted in the narrower regions shown in eq ( [ input2 ] ) . ] a comment is in order . our quark mass matrices are in principle given at the su(5 ) unification scale . however , the @xmath0 flavor symmetry is spontaneously broken at the electroweak scale . therefore , the forms of our mass matrices are not changed except for the magnitudes of the yukawa couplings between the unification and electroweak scales . hence , our numerical analyses are presented at the electroweak scale . we should also comment on the hierarchy of @xmath36 and @xmath37 . the order of @xmath36 are fixed by the quark mixing ( [ vus ] ) , ( [ vcb ] ) . the ratios of h_1/h_2 ( 0.22 ) , h_2/h_3 ^2 , are required by @xmath71 and @xmath72 , respectively . once @xmath36 are fixed , quark masses determine the hierarchy of @xmath37 as follows : v_1/v_3 ^2 , v_2/v_3 ~^1/2 . these hierarchies of @xmath36 and @xmath37 are also consistent with the magnitude of @xmath73 given in eq . ( [ jcp ] ) . the @xmath0 family symmetry which has been successful in understanding the mixing pattern of neutrinos ( tri - bimaximal mixing ) is applied to quarks , motivated by the quark - lepton assignments of su(5 ) . realistic quark masses and mixing angles are obtained entirely with the @xmath0 context , in good agreement with data . in particular , we find a strong correlation between @xmath62 and the cp phase @xmath63 , thus a decisive future test of this model can be allowed . it is one of a powerful guideline to find the constraints from neutrinos to grand unification models . discrete symmetries are suitable to decode the flavor problem : they can accommodate maximal 2 - 3 mixing and explain zero 1 - 3 mixing . moreover they can be the origin of texture zeros or equalities . i would like to thank y. koide for his great hospitality and organization of this stimulating workshop in shizuoka . i am also grateful to e. ma and m. tanimoto for their collaboration . 99 m. maltoni , t. schwetz , m.a . tortola and j.w.f . valle , new j. phys . * 6 * ( 2004 ) 122 . fogli , e. lisi , a. marrone and a. palazzo , prog . part . * 57 * ( 2006 ) 742 . see for example , w. grimus , a.s . joshipura , s. kaneko , l. lavoura , h. sawanaka and m. tanimoto , nucl . b * 713 * ( 2005 ) 151 , and references therein . barger , s. pakvasa , t.j . weiler and k. whisnant , phys . b * 437 * ( 1998 ) 107 . harrison , d.h . perkins and w.g . scott , phys . b * 530 * ( 2002 ) 167 ; p.f . harrison and w.g . scott , phys . b * 535 * ( 2002 ) 163 . see also , z.z . xing , phys . b * 533 * ( 2002 ) 85 ; x.g . he and a. zee , phys . b * 560 * ( 2003 ) 87 . e. ma and g. rajasekaran , phys . d * 64 * ( 2001 ) 113012 . babu , e. ma and j.w.f . valle , phys . b * 552 * ( 2003 ) 207 . babu , b. dutta and r.n . mohapatra , phys . rev . d * 60 * ( 1999 ) 095004 . e. ma , mod . a * 17 * ( 2002 ) 627 ; x.g . he , y.y . keum and r.r . volkas , jhep * 0604 * ( 2006 ) 039 . see for example , m. hirsch , j.c . romao , s. skadhauge , j.w.f . valle and a. villanova del moral , phys . d * 69 * ( 2004 ) 093006 ; e. ma , phys . d * 70 * ( 2004 ) 031901r ; phys . d * 72 * ( 2005 ) 037301 ; mod . lett . a * 20 * ( 2005 ) 2601 ; phys . d * 73 * ( 2006 ) 057304 ; g. altarelli and f. feruglio , nucl . b * 720 * ( 2005 ) 64 ; nucl . b * 741 * ( 2006 ) 215 ; nucl . b * 724 * ( 2005 ) 423 ; k.s . babu and x.g . he , hep - ph/0507217 ; a. zee , phys . b * 630 * ( 2005 ) 58 ; b. adhikary , b. brahmachari , a. ghosal , e. ma and m.k . parida , phys . b * 638 * ( 2006 ) 345 ; l. lavoura and h. kuhbock , mod . a * 22 * ( 2007 ) 181 ; g. altarelli , f. feruglio and y. lin , hep - ph/0610165 . e. ma , h. sawanaka and m. tanimoto , phys . b * 641 * ( 2006 ) 301 . e. ma , mod . a * 21 * ( 2006 ) 2931 . s.f . king and m. malinsky , phys . b * 645 * ( 2007 ) 351 . w. grimus and l. lavoura , jhep * 0601 * ( 2006 ) 018 . n. haba , a. watanabe and k. yoshioka , phys . * 97 * ( 2006 ) 041601 . mohapatra , s. nasri and h.b . yu , phys . b * 639 * ( 2006 ) 318 . mohapatra and a.y . smirnov , ann . nucl . part . * 56 * ( 2006 ) 569 . g. altarelli , hep - ph/0611117 . h. fritzsch and z.z . xing , prog . * 45 * ( 2000 ) 1 . particle data group , w.m . yao _ et al . _ , j. phys . g * 33 * ( 2006 ) 1 . ckm fitter group , j. charles , hep - ph/0606046 . c. jarlskog , phys . * 55 * ( 1985 ) 1039 . h. leutwyler , phys . b * 378 * ( 1996 ) 313 ; hep - ph/9609467 .
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realistic quark masses and mixing angles are obtained applying the successful @xmath0 family symmetry for leptons , motivated by the quark - lepton assignments of su(5 ) .
the @xmath0 symmetry is suitable to give tri - bimaximal neutrino mixing matrix which is consistent with current experimental data .
we study new scenario for the quark sector with the @xmath0 symmetry .
* quark and lepton mass matrices + with @xmath0 family symmetry * + +
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You are an expert at summarizing long articles. Proceed to summarize the following text:
observations show that the two point correlation function of the galaxies is an approximate power law over a range of scales . this result , known for decades now , defies a simple first - principle " explanation because of the complexity of the physical processes involved . in the conventional big - bang cosmology the dominant contribution to the energy density of the universe is in the form of nonbaryonic dark matter and the visible galaxies ( made of baryons ) form due to complex processes of cooling and fragmentation within the dark matter halos . it is possible to model non linear gravitational clustering of _ dark matter halos _ by a scaling ansatz ( see eg . @xcite ) and show that a power law correlation is expected ( in a spatial and temporal interval ) for scale invariant initial conditions . this result arises from the fact that newtonian gravitational dynamics in a @xmath1 universe does not have any preferred scale which is of interest to cosmology . but to translate this result to baryonic structures is not easy and at present the only explanation for the power law correlation of galaxies arises from numerical simulations . there are , however , alternate ( and less popular ) models for galaxy formation in quasi - steady state cosmology ( qssc ) in which the process is addressed without invoking gravitational instability explicitly @xcite . it has been claimed , based on numerical simulations , that these models also lead to power law correlation function for galaxy distribution . this result is of intrinsic interest even to conventional cosmologists who may not accept the alternate model because it provides a scenario to probe the relationship between ( i ) the conventional scenario of galaxy formation via gravitational instability ( accepted by most of the working cosmologists ) and ( ii ) the power law correlation function for the galaxies . if a completely different model can lead to power law correlation function , then it is clear that observations of galaxy correlation function , by itself , can not be used as a discriminator between the theories . since the `` rule '' used for creating galaxies in qssc is simple and explicit , it is indeed possible to study this process completely analytically . we provide such an analysis in this paper and show that the model is intrinsically unstable . however , the galaxies produced by this rule _ will _ exhibit a power law correlation function for a range of intermediate time scales . the rest of the paper is arranged as follows . in the next section , we provide an analytic description of the model for galaxy formation in the qssc , which was used in @xcite . this is done in terms of discrete time steps to parallel the previous work . in section 3 we discuss the continuum limit of this model that lends itself to a completely tractable analysis . in section 4 we describe a very general class of processes which obey the same equations as described here and a broader class of applications . the last section gives the conclusions . a non standard model for galaxy formation in qssc is built along the following lines : consider a set of points in the 3-dimensional space which represents the location of galaxies at a given instant of time . we now generate a set of new galaxies near each one of the galaxies ( `` non - local self - replication '' ) by a physical process which should quite minimally involve some extra negative energy field . the key ingredient of the nonstandard cosmological models is the existence of some `` creation field '' so that galaxies could generate new galaxies nearby ( see @xcite ) . let the probability for any given new galaxy to be located at a distance @xmath2 from an old galaxy be @xmath3 . this process will increase the number of galaxies as the universe evolves . in qssc , a balance between creation and quasi - steady state structure is maintained by rescaling the size of the universe in each cycle . mathematically , this is taken into account by rescaling each of the 3 dimensions by a factor @xmath4 thereby increasing the volume of available space . ( if we create _ one _ new galaxy near an old one with a probability @xmath5 , then we will be doubling the number of galaxies during the self replication . the rescaling should be such that the volume doubles giving @xmath6 . for future convenience , we shall keep @xmath7 as an arbitrary constant . ) we now select a subset of galaxies in the central region such that the total number of galaxies remain the same . this step renormalizes the process back to the original situation so that the process can now be repeated with the new subset of galaxies . in the specific case of number of galaxies doubling per cycle , we should take the central region containing half the volume . it is obvious that such a process of galaxy formation will lead to correlations between the galaxies since new galaxies are created close to the old ones with a probability @xmath5 . the key question is how the correlation function scales with the distance scale . we will now provide a mathematical analysis of this problem . the evolutionary rule described above can be stated mathematically in the form : @xmath8 \label{threeq}\end{aligned}\ ] ] where @xmath9 are constants with @xmath10 and @xmath11 ; @xmath5 is a probability function normalized to unity for integration over all @xmath2 . we shall also assume that @xmath12 is normalized in such a way that its integral over all @xmath13 is unity . this gives us the conditions @xmath14 it is obvious that equation ( [ threeq ] ) preserves these conditions under evolution because of the explicit normalization chosen on the right hand side . in the context of the quasi steady state cosmological model , @xmath12 will be the ratio between the number density of galaxies and the mean density . the normalization in ( [ twoint ] ) preserves the quasi steady state condition between different cycles under the simultaneous action of matter creation and expansion . while equation ( [ threeq ] ) was motivated from a particular model for galaxy formation , the rest of the analysis only uses this equation and is independent of the assumptions which go into it . in section 4 we will comment on a wider class of phenomena which could be modeled by such an equation . let us now consider the solutions to equation ( [ threeq ] ) . linearity in @xmath12 suggests switching to the fourier space variables @xmath15 with @xmath16 equation ( [ threeq ] ) now reduces to a simple form @xmath17 \label{threek}\ ] ] where we have denoted by the same symbol @xmath18 the fourier transform of the probability function . given the form of this probability function , this equation iteratively determines the evolution of @xmath19 . while the equation is fairly simple in structure , it is not easy to find its general solution . note that the conservation condition ( [ twoint ] ) which demands @xmath20 for all @xmath21 is satisfied in this case because @xmath22 for a normalized probability distribution . this also shows that solutions of the form @xmath23 , with some constant @xmath24 , are unacceptable for @xmath25 because they will violate the normalization condition . further , since we have the freedom to choose the initial condition @xmath26 , any general solution to equation ( [ threek ] ) must contain one arbitrary function ; it is difficult to obtain such a _ general _ solution . a special class of solutions to equation ( [ threek ] ) will correspond to a steady state such that @xmath27 . this function satisfies the equation @xmath28.\label{twof}\ ] ] two trivial solutions to this equation correspond to @xmath29 and , in fact , we shall see later that the most generic initial conditions will drive the system to either of these two limits by our process . the only nontrivial solution which exists can be found by the iteration in the form @xmath30 \left [ 1 + \lambda w ( { { \bf k } \over \bar\mu } ) \right]}\nonumber \\ & = & f_s \left ( { { \bf k } \over \bar \mu^n}\right ) \prod^n _ { n=0 } { ( 1 + \lambda ) \over 1 + \lambda w ( { \bf k}/ \bar\mu^n ) } . \end{aligned}\ ] ] taking the logarithms , followed by the limit @xmath31 , and using @xmath32 we get the result @xmath33 . \label{lnsum}\ ] ] to study the properties of the solution we note that @xmath34 for all @xmath35 . this is true for a wide class of probability distributions with some characteristic scale @xmath36 , and @xmath37 usually decreases for @xmath38 . we will also assume that @xmath37 depends only on the magnitude of @xmath39 because of the statistical isotropy of the process ( though our results are easily generalizable to other cases ) . it follows that each of the denominators in ( [ lnsum ] ) is close to unity for all @xmath40 making all the terms negligibly small for @xmath40 . so we need to sum the series in ( [ lnsum ] ) only up to @xmath41 $ ] with the integer value , lower than the bound , taken for @xmath42 . the result of the sum depends on the form of @xmath43 at large @xmath44 . the simplest case corresponds to assuming the @xmath43 vanishes for @xmath38 which is exact if we take @xmath45 where @xmath46 for @xmath47 and zero otherwise . then the asymptotic solution is given by @xmath48 ( the equivalence of the two forms follows from simple algebra and definition of @xmath42 . ) this is a power law solution with the index determined essentially by the two parameters of the problem . before proceeding further , let us consider the effect of the normalization condition ( [ twoint ] ) more closely . this is of some interest because one might feel that such a normalization is unnecessarily restrictive and a wider class of phenomena can be modeled by relaxing this condition . it turns out , however , that our qualitative considerations are not affected by this constraint . relaxing this normalisation can be done most conveniently by replacing the factor @xmath49 in the denominator of the right hand side of ( [ threeq ] ) by @xmath50 where @xmath51 is a constant different from @xmath52 . this will change equation ( [ threek ] ) to the form : @xmath53 . \label{newthreek}\ ] ] setting @xmath54 and using @xmath55 , we get @xmath56 since the integral over all space of @xmath57 is just @xmath58 we find that @xmath59 this shows how the total number of galaxies changes with time if @xmath60 . for example , a cosmological model in which the mean densities of galaxies decreases with time can be modeled with @xmath61 and interpreting @xmath12 as the number density of galaxies . in this case , equation ( [ newthreek ] ) admits solutions of the form @xmath23 with @xmath62 this equation has the same form as ( [ twof ] ) with the factor @xmath49 replaced by @xmath63 so that the solution is @xmath64 taking the logarithms , followed by the limit @xmath31 , we get the result @xmath65 + \ln a(0 ) . \label{newlnsum}\ ] ] the convergence of the product in the right hand side ( [ fora ] ) is now more tricky . if @xmath66 becomes close to unity for sufficiently large @xmath21 , then we will pick up a factor @xmath67 in each of the terms . unless this factor is unity , the product in ( [ fora ] ) will either diverge or vanish . thus we get the condition @xmath68 for convergence thereby making equation ( [ fora ] ) identical to ( [ twof ] ) . thus , when the normalization condition in ( [ twoint ] ) is relaxed , we again get the same @xmath69dependence as in ( [ twolns ] ) with an extra time ( @xmath21 ) dependence of the form @xmath70 which takes into account the condition ( [ forq ] ) . this clearly shows that nothing changes qualitatively as far as the spatial dependence is concerned by relaxing ( [ twoint ] ) . we thus have three possible steady state solutions @xmath71 $ ] _ none _ of which incorporates _ arbitrary _ initial conditions . obviously if the system is started at any of these solutions , it will stay in it without any evolution . the question arises as to whether any of them acts as a fixed point for the system evolving from a _ nontrivial _ initial condition or even random poisson initial conditions for which @xmath72 . while this question is difficult to analyze in the discrete model , it can be answered using a much more detailed description of the system in the continuum limit . in the continuum limit we need to study the system at two infinitesimally separated moments in time @xmath73 and @xmath74 and obtain a partial differential equation for the evolution of @xmath75 or equivalently for @xmath76 . we will also need to change the parameters @xmath52 and @xmath7 to @xmath77 and @xmath78 respectively for consistency ; this has the effect of making the _ rate _ of creation and _ rate _ of stretching finite , as it should . equation ( [ twof ] ) now becomes @xmath79 . \label{newsix}\ ] ] we have assumed that @xmath43 depends only on @xmath80 making @xmath19 also depend only on @xmath80 . expanding the equation retaining up to linear terms in @xmath81 , and using the result @xmath82 we get the final partial differential equation satisfied by @xmath19 to be @xmath83 the general solution to this equation is straightforward to obtain ; we find that @xmath84\ ] ] where @xmath85 is an arbitrary function of its argument with the condition @xmath86 . [ this condition incorporates our normalization condition ( [ twoint ] ) . this condition can be relaxed in exactly the same manner as in the discrete case ; however , as we shall see later , our conclusions do not change . ] this function - in turn - can be expressed in terms of the initial condition for the problem @xmath87 , which is assumed to be known . doing this we can write the solution in the form @xmath88 \right ] . \label{dqq}\ ] ] we shall now study the properties of the solution . let us first consider the simple case in which @xmath43 is given by equation ( [ onew ] ) . in this case the solution is found to be @xmath89 at any finite @xmath73 , there is a range of @xmath44 values for which the power spectrum ( which is proportional to @xmath90 ) is a power law with the index @xmath91 ) . but note that as @xmath92 the solution decays exponentially at all scales for a wide class of initial conditions ( including poisson distribution with @xmath93 ) . more generally , if @xmath94 , then the solution in the three ranges go as @xmath95 \cr k^{\beta } \exp \left [ \mu ( \beta - { \lambda \over \mu } ) t \right].\cr}\ ] ] for generic values of the parameters @xmath96 _ all _ the solutions tend to either zero or infinity at late times . the only exception is if we choose the initial spectrum with @xmath97 . then , for @xmath98 , we get a pure power law @xmath99 . this is precisely the solution we found in the discrete case [ see equation ( [ twolns ] ) ] , since in the continuum limit , we can set @xmath100 it must be emphasised that only a very special choice can lead to a nontrivial steady state solution . even with this special choice we get a solution which has _ no _ time dependence at all . it is , however , possible to have a power law solution that is valid for a large range of @xmath44 at any finite @xmath73 but with amplitude decreasing exponentially . equation ( [ dqq ] ) can be explicitly integrated for several cases of @xmath43 . a particularly simple case is the one with @xmath101 in this case the solution is given by @xmath102^{{\lambda \over 2 \mu}}.\ ] ] as @xmath103 , the solution goes to @xmath104 which is again in conformity with the results obtained above , thereby showing that they were not an artifact of the sharp cutoff assumed in equation ( [ onew ] ) . finally , we briefly mention the results in the continuum limit when the normalisation condition ( [ twoint ] ) is relaxed . in this case , the factor @xmath49 in the denominator on the right hand side of equation ( [ newsix ] ) will be replaced by a new constant @xmath50 . this changes equation ( [ partialone ] ) and the solution ( [ dqq ] ) to @xmath105 @xmath106 \right ] . \label{newdqq}\ ] ] writing the factor inside the integral as @xmath107 it is trivial to see that the solution in this case is the same as the solution with @xmath108 multiplied by the factor @xmath109 $ ] . that is , @xmath110 . \label{newfff}\ ] ] this is exactly what one would expect based on our results in the discrete limit in which we found that the solution gets multiplied by a factor in ( [ extrafactor ] ) . in the continuum limit , @xmath111 and @xmath52 gets replaced by @xmath112 and @xmath113 and @xmath21 becomes @xmath114 . in the limit of @xmath115 the extra factor in ( [ extrafactor ] ) becomes @xmath116 \nonumber \\ & = & \exp [ -(\lambda_1 - \lambda ) t ] \end{aligned}\ ] ] which is precisely the extra factor in ( [ newfff ] ) . thus , even in the continuum limit , our qualitative conclusions do not change when the condition ( [ twoint ] ) is relaxed . while the discussion above was modeled and motivated by galaxy formation in qssc , the process described here has a much broader range of applicability . the fact that our results were obtained for a model that does not involve gravity explicitly ( for example , neither newtonian gravitational constant nor the fact that gravitational force varies as a power law is used in the rule " mentioned above ) , suggests that results of the above kind could be quite general . in fact , there exists several natural phenomena that are described by power law correlations ( see eg . @xcite ) . more often than not , such a correlation function seem to arise in a manner that does not depend critically on the details of the underlying dynamical model . it would be interesting to see whether one can provide a mathematical model with a minimal set of assumptions which can reproduce the power law correlation . one such minimal set of assumptions can be extracted from the above analysis and we can show that models based on these assumptions will have a generic behaviour . consider a dynamical process in which some physical quantity @xmath117 evolves in time in a manner which depends on its value non locally . such an evolution can be treated in the discrete version in terms of a rule which allows one to compute @xmath118 in terms of @xmath57 where @xmath21 represents the discretised version of time with , say , @xmath119 and @xmath120 representing a convenient time interval . for example , the amount of bacteria in a culture , trees in an orchard , buildings in a city or the number of galaxies in some region of the universe could be studied by such prescription . to be more specific , we shall consider processes of the following kind : we start with a set of points in the d - dimensional space which represents the location of bacteria or galaxies or trees , say . we now generate a set of new points near each one of the points ( `` non - local self - replication '' ) . ( the bacteria creating new bacteria nearby or trees generating new trees nearby or even cities leading to the formation of new cities nearby seem reasonable . ) let the probability for any given new point to be located at a distance @xmath2 from an old point is @xmath121 . next , we rescale each of the @xmath122 dimensions by a factor @xmath4 thereby increasing the volume of available space . ( this is useful in the case of growing bacterial culture or a jungle of trees or cities in order to avoid boundary effects which will limit the process . in the cosmological model mentioned above , this is a natural consequence of the expansion of the universe . ) finally we select a subset of particles in the central region such that the total number of particles remains the same . this step renormalizes the process back to the original situation so that the process can now be repeated with the new subset of points . this evolutionary rule can be stated mathematically in the form : @xmath123 \label{geneq}\end{aligned}\ ] ] where @xmath9 are constants with @xmath10 and @xmath11 ; @xmath122 is the dimension of the space in which the vector @xmath13 lives , and @xmath5 is a probability function normalized to unity for integration over all @xmath2 . this is same as equation ( [ threeq ] ) with @xmath124 replaced by @xmath125 . we shall also assume that @xmath12 is normalized in such a way that its integral over all @xmath13 is unity . as discussed above , this can be easily relaxed if required . it must be stressed that we can consider equation ( [ geneq ] ) as the basic postulate of this analysis rather than any physical model described in the last paragraph ( involving bacteria , trees , cities , galaxies .... ) . in particular : ( i ) we need not restrict to any specific form of @xmath5 or a choice for dimension @xmath122 ; ( ii ) it is also not necessary to identify the vector @xmath13 with the position vector in real space . the equation ( [ threeq ] ) can also describe quite effectively the power transfer in fourier space when the vector @xmath13 is actually identified with a fourier space vector . there are phenomena , like fluid turbulence , in which our analysis can be applied by using the power spectrum as the basic variable in @xmath39 space . the essential postulate will then be that power at nearby wave numbers is generated with a given probability . this could provide a tool for attacking a wide class of nonlinear phenomena . two point correlations functions which exhibit power law behaviour are very prevalent in nature . the analysis here suggests that two ingredients which we have called ( i ) non local self replication and ( ii ) rescaling can lead to such correlation functions fairly generically . the first of these two ingredients ( non local self replication ) allows similar entities to be created at nearby locations in some space . this could be as varied as bacterium creating new bacterium nearby by cell division , trees creating trees nearby by seeding or galaxies creating fresh galaxies because of the existence of a creation field . if we take the space to be the fourier domain , then transfer of power from a given wave number @xmath44 to nearby wave numbers will also constitute such a process . this variety shows that the ingredient ( i ) is fairly generic and natural . an immediate consequence of self replication as defined here is the development of correlations . this is because we have tacitly assumed that the members of the second generation are preferentially ( in the probabilistic sense ) produced near the original parents . this will naturally lead to the second generation to be correlated with the first , spatially . if the process takes place in the fourier domain , then the correlation will arise in the space of wave numbers and the consequent relationship in real space will be more complicated ; but even in this case correlation will definitely be established . ( of course , in principle one can also introduce negative correlation in the model by choosing the probability distribution to be an increasing function of distance near the origin . ) in the context of conventional galaxy formation due to gravitational instability , the development of correlation is more indirect and dynamical while the process described here is direct and kinematical . ultimately , the kinematics of the model , encoded in the probability distribution function @xmath126 , need to be connected to an underlying model ( say , the creation field in the case of qssc ) in order to provide the dynamical basis . while this is the basic paradigm in physics , it must be stressed that it suffers from two well known difficulties : ( a ) if the physical process is sufficiently complicated ( fluid turbulence , galaxy formation ... ) , it just may not be possible to provide such a dynamical underpinning . ( b ) modeling different processes separately could lead one to miss the existence of a very general description for widely different classes of phenomena . much of the initial attraction for fractals in the description of natural phenomena originated from its promise to provide such a unifying perspective . the mathematical formalism developed here should be viewed against such a backdrop . the second ingredient that we have used ( rescaling ) essentially serves to renormalise the scales and is , in fact , very similar to ideas used in the theory of renormalization group . once self replication takes place , the system has become denser on the whole and in order to concentrate on the intrinsic correlation , it is necessary to renormalise the system back to the original state . in all the examples which we have it is always possible to link this rescaling with a tangible physical process . the combination of these two ingredients very nicely and naturally leads to a system with higher level of correlation at each time step . in fact , at least in the case of galaxy formation , the combined effect of these two phenomena is very similar to actual gravitational attraction between the particles . the key result of the analysis is that processes with these two ingredients are inherently unstable in the sense that the correlation function either grows without bound or decays to zero ( with the system becoming more and more dilute ) as @xmath127 . once again such an instability is reminiscent of similar phenomena seen in self gravitating systems though we have not used any gravitational dynamics . there is one special initial condition which leads to a static solution but as we discussed in the text , this is rather too special . the interest in these two systems lies in the intermediate time scale during which it could exhibit a power law correlation function very generically . once again the situation is similar to galaxy formation due to gravitational instability in which the observed power law correlation function will exist _ only _ during a limited temporal and spatial window in the numerical simulations . even in the case of fractals , it is generally known that one has to introduce cut - off in spatial and temporal scales in order to maintain power law correlation functions . in conclusion , it is interesting how a fairly simple mathematical model could lead to an approximate description of a wide class of phenomena . only further investigations for specific contexts will determine whether such a unified treatment is of some value or whether these ideas are destined to remain as mathematical curiosities . i thank j.v . narlikar for comments on an earlier version of the manuscript . i appreciate the detailed and constructive comments from an anonymous referee which significantly improved the presentation of the ideas . padmanabhan t. , 1996 , mnras , 278 , l29 [ astro - ph-9508124 ] . nayeri , a. engineer , s. narlikar , j. v. hoyle , f. , 1999 , ap j , 525 , 10 . mandelbrot , b. b. , 1983 , fractal geometry of nature , ( w.h . freeman , new york ) .
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a large class of evolutionary processes can be modeled by a rule that involves self - replication of some physical quantity with a non local rescaling .
we show that a class of such models are exactly solvable in the discrete as well as continuum limit and can represent several physical situations as varied from the formation of galaxies in some cosmological models to growth of bacterial cultures .
this class of models , in general , has no steady state solution and evolve unstably as @xmath0 for generic initial conditions
. they can however exhibit ( unstable ) power law correlation function in the continuum limit , for an intermediate range of times and length scales .
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magnetic fields are ubiquitous . all astrophysical objects are known to have magnetic fields of different magnitudes , e.g . , 1 gauss at the stellar scale to @xmath2 gauss at the galactic scale @xcite . the origin of such fields ( _ primordial field _ ) is not very clear - there are several competing theories which attempt to describe this @xcite . however , a finite magnetic field in any physical system undergoes a temporal decay due to the finite conductivity of the medium . so , for steady magnetic fields to occur in astrophysical bodies , there has to be a mechanism of regeneration of the magnetic fields , which takes place due to the dynamo process @xcite . most astrophysical bodies are thought to have _ fast _ dynamo operating within themselves ( there are exception to this , e.g. , the moon , venus and mars in our solar system ) resulting into exponential growth of the magnetic fields . this mechanism requires a turbulent velocity background @xcite [ though non - turbulent velocity fields too can make a seed ( initial ) magnetic field to grow ( for details see @xcite ) , we will not consider such cases here ] . since the dynamo equation , in the linear approximation ( see below ) gives unbounded exponentially growing solutions for the long wavelength ( large scale ) part of the magnetic fields , it is linearly unstable in the low wavenumber limit . however , one does not see a perpetual growth of magnetic fields in the core of the earth or in the sun . for example , geomagnetic fields ( @xmath3 1 gauss ) are known to be stable for about @xmath4 years @xcite . thus , the physically realisable solutions of the dynamo equations can not be unstable in the long time limit . it is now believed that the non - linear feedback due the lorentz force term in the navier - stokes equation is responsible for the saturation of the magnetic field growth ( see , e.g. , @xcite ) . the study of this problem has already been the subject of previous work by many groups . for example pouquet , frisch and lorat @xcite studied the connections between the dynamo process and the inverse cascade of magnetic and kinetic energies within a eddy damped quasi - normal markovian approximation . moffatt @xcite has examined the back reactions due to the lorentz force for magnetic prandtl number @xmath5 by linearising the equations of motion of three - dimensional ( @xmath6 ) magnetohydrodynamics ( mhd ) . vainshtein and cattaneo @xcite discussed several nonlinear restrictions on the generations of magnetic fields . et al _ @xcite discussed nonlinear @xmath0-effects within a two - scale approach . rogachevskii and kleeorin @xcite studied the effects of an anisotropic background turbulence on the dynamo process . brandenburg examined non - linear @xmath0-effects in numerical simulation of helical mhd turbulence @xcite . in particular , he examined the dependences of dynamo growth and the saturation field on the magnetic prandtl number @xmath7 ( the ratio of the magnetic- to the kinetic- viscosities ) . bhattacharjee and yuan @xcite studied the problem in a two - scale approach by linearising the equations of motion . dynamo mechanism has two competing processes at work : amplification of the magnetic field by the dynamo process and ohmic dissipation due to finite resistivity of the medium concerned . which one among these two effects will dominate depends on the case in study . in some specific models , however , one can analyze this completely . a good example of such models is the kraichnan - kazantzev dynamo @xcite where the velicity field is assumed to be gaussian - distributed , delta - correlated in time and the magnetic field is governed by the induction equation @xcite . in this model the statistics of the velocity field is taken to be parity invariant so that the @xmath0-effect is ruled out . the main results from this model include i)the existence of dynamo in the infinite magnetic reynolds number limit for a particular choice of the variance of the velocity distribution @xcite and ii)the existence of a critical magnetic reynolds number only above which dynamo growth is possible @xcite . however , not much is known about this when invariance due to parity is broken and when the velocity field is not temporally delta - correlated . in a recent simulations @xcite the authors found , in a model simulation for the solar convection zone , a monotonic increase of the horizontal @xmath0-effect with rotation . et al _ showed , in numerical simulations , that unless magnetic hyperviscosity is less than a critical value , magnetic fields did not grow @xcite , confirming the existence of a critical magnetic reynolds number ( @xmath8 ) . our studies generalize the existing results . in this paper we use a minimal model of @xmath0-effect ( see below ) to study dynamo with @xmath0-effect to calculate the @xmath0 coefficient for arbitrary correlations and viscosities , and ask the following questions : 1 . do the turbulent dynamo growth and the saturation processes require any turbulent background ? or do they function with arbitrary parity - breaking and fluctuating velocity and initial magnetic field correlations ? for the kinetic and magnetic energies and cascades of appropriate quantities ; if there is no mean magnetic field then the energy spectra is expected to be k41-type - see ref.@xcite . ] . 2 . what is the _ hydrodynamic limit _ ( long wavelength limit ) of the dynamo problem ? by this we ask how the magnetic field correlations scale in the infra red limit during the initial - growth regime . 3 . can arbitrarily large magnetic viscosity prevent dynamo growth ? in other words , is there a critical magnetic reynolds number @xmath8 above which the dynamo growth sets in ? to study the above mentioned questions we employ a diagrammatic perturbation theory , which has been highly successful in the contexts of critical dynamics @xcite , driven systems @xcite , etc . this can be easily extended to higher orders in perturbation expansion and is very suitable for handling continuous kinetic and magnetic spectra , unlike the two - scale approximation . this was first used to study stationary , homogeneous and isotropic mhd in ref.@xcite . we use this method to study non - stationary statistical states ( dynamo growth ) which facilitates studies on the hydrodynamic limit of the dynamo problem in a renormalisation group framework . we use diagrammatic perturbation theory to calculate expressions for the @xmath0 coefficiants for arbitrary background velocity and initial magnetic field correlations and magnetic prandtl number @xmath7 for both the early growth and the late time saturation . with our expressions for @xmath0 we examine the three issues mentioned above . we investigate these for arbitrary correlations and magnetic prandtl number @xmath7 with no approximations other than the existence a perturbation theory . our principal results are : * we calculate the @xmath0-coefficients for arbitrary correlations and viscosities . * we examine the hydrodynamic limit in the kinematic regime and predict the existence of a critical @xmath8 or rotation above which dynamo growth will occur for certain correlations with infra red singularity . in our all our studies , we do not assume any variance for the velocity field . instead , we use the navier - stokes equation to describe the dynamics of the velocity field . this allows us to use a renormalisation group framework to study the hydrodynamic limit . the first question that we investigate is phenomenologically very important because different systems may have different velocity and initial magnetic field spectra . therefore , it is important to understand the dependence of the dynamo on these spectra . we explicitly demonstrate that the nonlinear feedback of the magnetic fields on the velocity fields in the form of the lorentz force stabilises the growth for arbitrary velocity and initial magnetic field correlations . this demonstrates that the basic features of the dynamo mechanism are qualitatively independent of the velocity and magnetic field spectra and , essentially , are properties of the @xmath6mhd equations . details ( e.g. , the values of the @xmath0-coefficients ) of course , depend upon the actual forms of the spectra . our renormalization group analysis indicates that dynamo growth takes place only if the ekman number @xmath9 ( for a given @xmath8 ) when the velocity and the initial magnetic field spectra are sufficiently singular in the long wavelength limit . the structure of this paper is as follows : in sec.[dyn ] we discuss the general dynamo mechanism within the standard linear approximation for arbitrary velocity and initial magnetic field correlations and viscosities . in sec.[halt ] we show that beyond the linear approximation non - linear effects lead to the eventual saturation of magnetic field growth for arbitrary background kinetic energy and initial magnetic energy spectra , and viscosities . we elucidate how different background kinetic energy and initial magnetic energy spectra affect the values of the @xmath0-coefficients . in sec.[renor ] we analyze the initial dynamo growth in a renormalization group framework . we show that for sufficiently singular velocity and magnetic field spectra the ekman number must be @xmath10 1/2 for the magnetic fields to grow . for velocity and magnetic field spectra which go to zero in the long wavelength limit there are no such restrictions . in sec.[summ ] we present our conclusions . in the kinematic approximation @xcite , i.e. , in the early - time regime , when the magnetic energy is much smaller than the kinetic energy ( @xmath11 , where @xmath12 and @xmath13 are the velocity and magnetic fields respectively ) the lorentz force term of the navier stokes equation is neglected . in that weak magnetic field limit , which is reasonable at an early time , the time evolution problem for the magnetic fields is a linear problem as the induction equation @xcite is linear in magnetic fields @xmath14 : @xmath15 where @xmath16 is the magnetic viscosity . the velocity field is governed by the navier - stokes equation @xcite ( in the absence of the lorentz force ) @xmath17 here @xmath18 is the fluid viscosity , @xmath19 an external forcing function , @xmath20 the pressure and @xmath21 the density of the fluid . we take @xmath19 to be a zero mean , gaussian stochastic force with a specified variance ( see below ) . in a two - scale @xcite approach one can then write an _ effective _ equation for @xmath22 , the long - wavelength part of the magnetic fields @xcite : @xmath23 where the _ electromotive force _ @xmath24 . @xmath25 is the large scale component of the velocity field @xmath26 . an _ operator product expansion _ ( ope ) is shown to hold @xcite which provides a gradient expansion in terms of @xmath27 for the product @xmath24 @xcite @xmath28 for homogenous and isotropic flows ( @xmath29 ) eq.([ope ] ) gives , @xmath30 which is the standard turbulent dynamo equation . here @xmath16 now is the _ effective _ magnetic viscosity which includes both the microscopic magnetic viscosity and the turbulent diffusion , represented by @xmath31 in eq.([ope ] ) . @xmath0 depends upon the statistics of the velocity field ( or , equivalently , the correlations of @xmath19 ) . retaining only the @xmath0 -term and dropping all others from the rhs of eq.([dyna ] ) , the equations for the cartesian components of @xmath27 become ( we neglect the dissipative terms proportional to @xmath32 as we are interested only in the long wavelength properties ) + @xmath33 the eigenvalues of the matrix is @xmath34 . thus depending on the sign of the product @xmath35 , one mode grows and the other decays . the third mode is unphysical , because the corresponding eigenfunction is proportional to @xmath36 and hence in conflict with @xmath37 . since growth rate is proportional to @xmath38 and dissipation is proportional to @xmath32 , large scale fields continue to grow leading to long wavelength instability . thus in the long time limit effectively only the growing mode remains . growth rate @xmath0 is a pseudo - scalar quantity , i.e. , under parity transformation @xmath39 , @xmath40 @xcite . since @xmath0 depends upon the statistical properties of the velocity field , its statistics should not be parity invariant . this can happen in a rotating frame , where the angular velocity explicitly breaks reflection invariance . the navier - stokes ( ns ) ( including the lorentz force ) and the induction equation in an inertial frame in @xmath41 space take the form @xmath42 and @xmath43 here , @xmath44 and @xmath45 are the fourier transforms of @xmath46 and @xmath47 respectively , @xmath48 is the projection operator , which appears due to the divergence - free conditions on the velocity and magnetic fields ( we consider incompressible fluid for simplicity ) . equations ( [ nsk ] ) and ( [ indk ] ) have to be supplemented by appropriate correlations of @xmath49 and initial conditions on @xmath50 . we choose @xmath51 and @xmath52 to have zero mean and to be gaussian distributed with the following variances : @xmath53 @xmath54 where @xmath55 and @xmath56 are some functions of @xmath57 ( to be specified later ) . in a rotating frame with a rotation velocity @xmath58 the eq.([nsk ] ) takes the form @xmath59 whereas eq.([indk ] ) has the same form in the rotating frame . @xmath60 is the coriolis force . the centrifugal force @xmath61 is a part of the _ effective pressure_=@xmath62 which does not contribute to the dynamics of incompressible flows . the bare propagator @xmath63 ( obtained from the linearized version of eq.([nskr ] ) ) of @xmath64 + @xmath65 + such that @xmath66 where @xmath26 is the column vector @xmath67 + one can verify that with the form of the bare propagator given above , an odd - parity part in the velocity auto - correlator @xmath68 appears which is proprotional to the rotation @xmath69 . notice that @xmath70 is different from @xmath71 - this is just the consequence of the fact that @xmath69 distinguishes the @xmath72-direction as a preferred direction in space , making the system anisotropic . however for frequencies @xmath73 or length scales @xmath74 ( here @xmath72 is the dynamical exponent ) isotropy is restored . in that regime , to @xmath75 the role of the global rotation is to introduce a non - zero odd - parity part in @xmath76 proportional to @xmath69 . this can be also seen by noting that in the inertial frame the correlation @xmath77 is of the form @xmath78 [ cf . eq.([varf ] ) ] where @xmath79 is a scalar function of @xmath57 and hence in the rotating frame the correlator is proportional to @xmath80 where @xmath81 s are appropriate rotation matrices ( we have suppressed the indices ) . similarly , initial magnetic field correlations , given by eq.([varb ] ) transforms accordingly in the rotating frame . since rotation matrices act on @xmath82 and eq.([varb ] ) in the same way , magnetic field auto - correlator @xmath83 has an odd parity part in the rotating frame with the same sign as the odd parity part in the velocity correlator . thus the effects of rotation can be _ modeled _ ( to the lowest order ) by introducing parity breaking parts in eqs.([varf ] ) and ( [ varb])@xcite @xmath84 in conjunction with the eqs.([nsk ] ) and ( [ indk ] ) , where @xmath85 is the totally antisymmetric tensor in @xmath6 . this way of modeling rotation effects is , of course , only approximate , but suffices for our purposes as this explicitly incorporates parity breaking . one can , however , construct experimental set ups @xcite which are described correctly by eqs.([varbi ] ) . the parity breaking parts in the noise correlations or initial conditions ensure that the velocity and the initial magnetic field correlators have non - zero odd parity parts , as would happen in a rotating frame . an important dimensionless number is the _ ekman number _ @xmath86 which can be related to @xmath87 by equating the parity braking parts of the velocity correlator calculated from ( linearized ) eq.([nskr ] ) and eq.([varf ] ) with that from eqs . ( [ nsk ] ) and ( [ varbi ] ) . this gives @xmath88 . now , one may ask what is the relative sign between @xmath87 and @xmath89 ? since the parity breaking parts of the correlators of the velocity and the magnetic fields have same sign and are proportional to @xmath90 respectively , @xmath87 and @xmath89 must have same sign . as already noted , introduction of parity breaking terms in the force / initial correlations is well - known in the literature , we , nevertheless , give the analysis in details in order to emphasise on the fact that fluid and magnetic helicities must have the same sign . furthermore , for a complete description of the effects of rotation , in addition to the coriolis force , a forcing with a preferred direction is also required . we , however , do not include all these details as introduction of parity - breaking correlations is sufficient for our purposes . in this sense , this can be thought of as a _ reduced _ or a _ minimal model _ for dynamo . one may note that a nonzero kinetic helicity is required for the @xmath0-effect as the @xmath0-coefficient is proportional to the kinetic helocity . even though a global rotation explicity breaks the parity invariance of the system under space reversal , rotation alone is not enough to yield a non - zero helicity . this is because the helicity is pseudo - scalar and , therefore , can be constructed only out of an axial vector ( here , rotation @xmath91 ) and a polar vector . in typical astrophysical settings , the latter one could be provided by , say , a density inhomogeneity . even though this is not contained in eq . ( [ nsk ] ) , our minimal model , nevertheless , produces a finite helicity due to the helical nature of the forcing function . thus , our minimal model is able to capture both the breakdown of parity due to the rotation and the generation of helicity due to the rotation and any other preferred direction . in the kinematic approximation , which neglects the lorentz force term of the navier - stokes equation , the time evolution of the magnetic fields follows from the linear induction equation ( [ indeq ] ) . we assume , for the convenience of calculations , that the velocity field ( @xmath26 ) has reached a statistical steady state . this is acceptable as long as the loss due to the transfer of kinetic energy to the magnetic modes by the dynamo process is compensated by the external drive . in the kinematic ( i.e. , linear ) approximation , we work with the eqs.([nsk ] ) ( without the lorentz force ) and ( [ indk ] ) . we choose @xmath92 to be a zero - mean , gaussian random field with correlations @xmath93 our initial conditions for the magnetic fields are @xmath94 since we are interested to investigate the dynamo process with arbitrary statistics for the velocity and magnetic fields we work with arbitrary @xmath95 and @xmath96 . for k41-type spectra , we require @xcite @xmath97 and @xmath98 . these choices ensure that under spatial rescaling @xmath99 , @xmath100 which is the kolmogorov scaling @xcite . note that both the force correlations in the eq.([nsk ] ) and the initial conditions on eq.([indk ] ) have parts that are parity breaking , in conformity with our previous discussions we now calculate the @xmath0-term . we use an iterative perturbative method which is very similar to and discussed in details in ref.@xcite . in this method , terms in each order of the perturbation series can be represented by appropriate feynman diagrams @xcite . even though , for simplicity , we confine ourselves to the lowest order in the perturbation theory ( represented by the _ tree level diagrams _ ) , which is sufficient for our purposes , higher order calculations represented by higher order digrams can be done in a straight forward manner . below we give the expression for @xmath0 in the kinematic approximation ( which we call the ` direct ' term - responsible for growth ) in the lowest order of the perturbation theory ( see fig.[diag]a ) : @xmath101 from which one can read the @xmath0-term : @xmath102\ ] ] giving @xmath103 for large @xmath104 . the suffix @xmath105 refers to _ growth _ or the _ direct _ term , as opposed to _ feedback _ which we discuss in the next sec.[halt ] . the growth term is proportional to @xmath38 and diffusive decay proportional to @xmath32 . the angular brackets represent averaging over the noise and initial - condition ensembles . when the magnetic fields become strong , it is no longer justified to neglect the feedback of the magnetic fields in the form of the lorentz force . so we need to work with the _ full _ eqs.([nsk ] ) and ( [ indk ] ) . the ideas of ope as elucidated in sec.[dyn ] are still valid for the full non - linear problem . but the value of @xmath0 is expected to change from its value in the linear problem . in presence of the lorentz force there is an additional contribution to @xmath0 ( fig.1b ) . to evaluate that , we follow a diagrammatic perturbation approach similar to that described in the previous section . here also we restrict ourselves to the lowest order only ( i.e. , the tree level diagrams ) though extension to higher orders is straight forward . we obtain @xmath106 which gives ( @xmath107 refers to feedback ) @xmath108 which , after some simplifications , yields , @xmath109 where @xmath110\tilde{d}_2(q)$ ] is a growing function of time for small wavenumbers . as before , angular brackets refer to averaging over noise and initial - condition ensembles . thus @xmath111 grows in time . since , at any finite time @xmath104 , when the non - linear feedback on the velocity field due to the lorentz force is nolonger negligible , both @xmath112 and @xmath111 are non - zero and we get @xmath113}},\nonumber \\ \alpha_f&=&{2s_3\over 3}{4\over 15}\int { d^3q\over ( 2\pi)^3}{\tilde{d}_2(q , t)q^2 \over { |(\alpha_d+\alpha_f)q|-2\mu q^2 } } , \label{divexp}\end{aligned}\ ] ] with @xmath114\tilde { d}_2(q ) . \label{selfd2}\ ] ] equations ( [ divexp ] ) and ( [ selfd2 ] ) are to be solved self - consistently @xcite . thus the net growth rate is proportional to @xmath115 for the mode @xmath116 . the expressions ( [ divexp ] ) have apparent divergences at finite @xmath117 ; so in perturbative calculations one should treat the @xmath0-terms as perturbations which remove these divergences . this problem is akin to that in kuramoto - shivashinsky equation for flame front propagation @xcite . so the expressions for @xmath112 and @xmath111 are @xmath118 which do not have any finite wavevector singularity . expressions ( [ finexp ] ) are obtained , as mentioned before , by truncating the perturbation series at the tree level . extensions to higher orders are straight forward . illustrative examples of higher order diagrams have been shown in fig.[oneloop ] . let us now consider various @xmath57 dependences of @xmath119 and @xmath96 . when the background velocity field is driven by the navier - stokes equation with a conserved noise ( thermal noise ) one requires that @xmath120 , giving @xmath121 . if we assume similar @xmath57-dependences for @xmath122 then we require @xmath123 constant and @xmath124 . these choices yield @xmath125\over 2\mu |q| } , \label{alphathermal}\end{aligned}\ ] ] which remain finite even if the system size diverges . a fully developed turbulent state , characterised by k41 energy spectra , is generated by @xmath126 and @xmath127 . in addition if we assume that the initial magnetic fields correlation also have k41 scaling then @xmath128 and @xmath129 . if one starts with a k41-type initial correlations for the magnetic fields , then at a later time the scale dependence for the magnetic field correlations are likely to remain same ; only the amplitudes grow . notice that the spectra diverge as wavevector @xmath130 , i.e. , as the system size diverges . this is a typical characteristic of fully developed turbulence . for such a system we find @xmath131 the notable difference between the expressions eqs.([alphathermal ] ) and ( [ alphak41 ] ) for the @xmath0 coefficients is that the @xmath0 coefficients diverge with the system size if the energy spectra are singular in the infra red limit ( as in for fully developed turbulence ) . these divergences are reminiscent of the divergences that appear in critical dynamics @xcite which are handled by renormalisation group methods . in general , at early times ( small @xmath111 ) , @xmath111 increases exponentially in time . the growth rate of @xmath111 decreases with time . since @xmath112 and @xmath111 have different signs , @xmath132 ) and eq.([alphak41 ] ) suggest that the early - time growth and late time saturation of magnetic fields take place for different types of background velocity correlations and initial magnetic field correlations . therefore dynamo instability and its saturation are rather intrinsic properties of the @xmath6mhd equations with broken reflection invariance . one may also note that for @xmath133-type of correlations ( singular in the infrared limit ) one has forward cascade of kinetic energy @xcite : this is because energy is fed into the system mostly in the large scale ( i.e. , for small @xmath57 ) whereas , dissipation acts primarily in the small scales ( large @xmath57 ) , resulting into a cascade of energy from the large- to small- scales . on the other hand , for correlations smooth in the infra red limit , there is no such cascade . these results indicate that the existence of the dynamo mechanism does not require any special background velocity field spectrum , though the value of the @xmath0-coefficient depends upon it . our results also suggest that these processes may take place for varying magnetic prandtl number @xmath134 . the above analysis crucially depends on the fact that @xmath111 and @xmath112 have opposite signs , which , in turn , imply that @xmath87 and @xmath89 have same signs . we have already seen that in a physically realisable situation where parity is broken entirely due to the global rotation , @xmath87 and @xmath89 indeed have the same sign . in the _ first order smoothing approximation _ @xcite in the kinematic limit , to calculate @xmath135 one considers only the induction equation as @xmath26 is supposed to be given . however when one goes beyond the kinematic approximation , one has to consider the navier - stokes equation as well . thus in the first - order smoothing approximation one writes the equations for the fluctuations @xmath26 and @xmath14 as ( to the first order ) @xmath136 and @xmath137 where the ellipsis refer to all other terms in the navier - stokes equation and @xmath138 and @xmath139 are the large scale ( _ mean field _ ) part of the velocity and magnetic fields @xcite . with these we can write @xmath140 here the ellipsis refer to non-@xmath0 terms in the expansion of @xmath141 ( see eq.(4 ) ) . thus for isotropic situations @xmath142 $ ] where @xmath143 is a correlation time . thus @xmath0 is proportional to the difference in the fluid and magnetic _ torsalities_@xcite , ( fluid helicity being the same as fluid torsality and magnetic helicity being proportional to magnetic torsality ) a result obtained in @xcite using other methods and approximations . note that eqs . ( 21 ) and ( [ finexp ] ) are very similar to but not exactly the one that were obtained in @xcite ( in our notations @xmath87 is proportional to fluid torsality ( or fluid helicity ) and @xmath89 is proportional to magnetic torsality ) . we ascribe this difference to the essential difference between a two - scale approach and our dirgrammatic perturbation theory which , we believe is more suitable for handling continuous kinetic and magnetic spectra . we have seen that in eqs.([alphak41 ] ) the @xmath0-coefficients diverge in the hydrodynamic ( @xmath130 ) limit which calls for a renormalisation group ( rg ) analysis as a natural extension of our diagrammatic perturbative calculations . in fully developed @xmath6mhd , in the steady state , correlation and response functions exhibit dynamical scaling with the _ dynamic exponent _ @xmath144 @xcite ( for a different approach see @xcite ) , which means renormalised viscosities ( kinetic as well as magnetic ) diverge @xmath145 for a wavenumber @xmath57 belonging to the inertial range . even for decaying mhd with initial k41-type correlations this turns out to be true @xcite where equal time correlations exhibit dynamical scaling with @xmath144 . the question is , what it is in the initial transient of dynamo growth ( @xmath146 saturation time ) . we examine this in a renormalization group framework . since we are interested in the early growth , we neglect the lorentz force and work with eq.([indk ] ) inconjunction with the initial magnetic field correlations and noise correlation given by eq.([varbi ] ) . as before , we assume a statistical steady state for the velocity field . it is well - known that correlations @xmath147 exhibit scaling form @xmath148 where @xmath149 and @xmath72 are the spatial scaling and dynamical exponents respectively @xcite where @xmath150 is a scaling function . the galilean invariance of the mhd equations constraints these exponents to obey the relation @xmath151 @xcite . in addition to that , for fully developed turbulence due to non - renormalization of the noise - correlators [ cf . eq.([varf ] ) ] the exponents are fully determined : @xmath152 , which means the renormalised fluid viscosity diverges as @xmath153 in the limit wavevector @xmath130 . during early growth , equal - time magnetic field correlations @xmath154 are expected to exhibit a scaling form @xmath155 ( @xmath146 saturation time ) where @xmath156 and @xmath157 are the magnetic spatial scaling and dynamical exponents respectively , and @xmath158 is a scaling function . similar conditions arising from the galilean invariance and non - renormalization of the initial k41-like magnetic field spectrum determines @xmath159 and @xmath160 . we perform a renormalization group analysis following @xcite . as mentioned earlier , the @xmath0-term is treated as a perturbation . in a renormalisation - group transformation , one integrates out a shell of modes @xmath161 , and and simultaneously rescales length scales , time intervals and fields through @xmath162 . this has the effect that the nonlinearities are affected only by naive rescaling ( this , a consequence of the galilean invariance of the @xmath6mhd equations , essentially implies that the diagrammatic corrections to the nonlinearties vanish in the long wavelength limit ) . the variances eq.([varf ] ) , which diverge at low wavenumbers remain unrenormalised and thus affected only by rescaling . there are however fluctuations corrections to @xmath16 and @xmath112 which we evaluate at the lowest order . the resulting rg flow equations for @xmath16 and @xmath112 , obtained in a one - loop calculation are @xmath163,\\ \label{flow1 } { d\alpha_d\over dl}&=&\alpha_d[z_b-1+a_2{\tilde{d}_1\over \alpha_d \nu ( \nu+\mu)\lambda^{3 } } ] , \label{flow2}\end{aligned}\ ] ] where @xmath164 are numerical constants . equations ( [ flow1 ] ) and ( [ flow2 ] ) are similar to those presented in ref.@xcite [ eqs . ( 10.13 ) and ( 10.14 ) ] but not exactly same . the differences arise mainly ( apart from some detail technical differences in the perturbation theories involed ) from the fact that in ref.@xcite the expressions for the @xmath0-coefficients were derived for a given variance of the velocity field . in contrast , we use the navier - stokes equation , driven by a stochastic force of given variance , in place of a given velocity variance . by substituting the value of the exponents in eqs.([flow1 ] ) and ( [ flow2 ] ) we find _ renormalized _ ( i.e. , wavevector dependent ) @xmath165 in the hydrodynamic ( @xmath130 ) limit . thus in that limit , the effective dynamo equation takes the form @xmath166 where the ellipsis refer to non - linear terms and @xmath167 refers to the growing mode . thus , in the hydrodynamic limit , there is growth of the magnetic fields only if @xmath168 . this can happen only if the renormalised magnetic viscosity is less than a critical value , set by @xmath112 , i.e. , the kinetic helicity . in terms of the ekman number @xmath169 this condition means @xmath1 for anti dynamo , i.e. , no growth , equivalently @xmath9 for growth of the magnetic fields . this can be achieved in two ways , namely by increasing rotation , keeping the magnetic viscosity ( or the magnetic reynolds number ) constant , or decreasing the magnetic viscosity ( i.e. , increasing the magentic reynolds number ) for a constant rotation . this conclusions are in good agreement with the numerical results of ref.@xcite . since renormalised magnetic viscosity increases with its bare ( microscopic ) value , it suggests that bare magnetic viscosity must be less than a critical value for growth to be possible . thus our rg results qualitatively explain the numerical results of kida _ et al _ @xcite who they found that unless magnetic hyperviscosity was less than a critical value there was no growth ( it can be easily argued that a hypermagnetic viscosity gives rise to a magnetic viscosity in the longer scale and hence their result in effect imposes a critical value of the magnetic viscosity ) . in our model @xmath0-effect is proportional to @xmath170 which in turn is proportional to the global rotation frequency . hence our results suggest that @xmath0-effect is likely to grow with increasing rotational speed which is in agreement with the results of ref.@xcite . on the other hand , if the background velocity and the initial magnetic field correlators do not have an infra red singularity ( i.e. , when the correlators @xmath171 ) there is no fluctuation correction to the magnetic viscosity and to the @xmath0-coefficient resulting in the fact that the growth term ( @xmath172 ) dominates over the dissipation ( @xmath173 ) for sufficiently small wavenumber @xmath57 , leading to growth even for arbitrarily large magnetic viscosity . therefore , there is no critical @xmath8 . thus the effects of the infrared divergences that appear in the expressions for the @xmath0-coefficients [ eq . ( [ alphak41 ] ) ] are quite significant : they indicate , as for the driven diffusive nonequilibrium systems with diverging kinetic coefficients in the hydrodynamic limit @xcite , divergence of time - scales in the hydrodynamic limit . since , the @xmath0-term in eq . ( [ indk ] ) is proportional to wavenumber @xmath57 , the time - scale of growth of the mode with wavenumber @xmath57 is @xmath174 . this remains true , even in the hydrodynamic limit , for the case when there is no divergence in the @xmath0-coefficients . in contrast , when the @xmath0-coefficient diverge in the infra red limit , the growth rate changes _ qualitatively _ from its linear dependence on wavenumber @xmath57 in the hydrodynamic limit . for example , with the the background velocity correlations and the initial magnetic field correlations given by eq . ( [ varf ] ) , the @xmath0 coefficients diverge as @xmath175 in the long wavelength limit . hence , the effective growth rate is changed to @xmath176 . a full self - consistent calculation ( when feedback due to the lorentz force can not be neglected ) for the @xmath0-coefficients require simultaneous solutions of the self - consistent expressions for magnetic prandtl number , magnetic- to kinetic- energy ratio and the @xmath0-coefficients which can be handled in our scheme of calculations . the self - consistent solutions are expected to be influenced by the degree of crosscorrelations between the velocity and magnetic fields @xcite . so far , we have assumed that both @xmath177 and @xmath119 have the same infra red singularity ( @xmath178 and @xmath179 ) . this need not be the case always . however , if @xmath119 is non - singular then @xmath112 does not diverge . as a result , the growth rate is just @xmath180 even in the hydrodynamic ( long wavelength ) limit . effective dissipation , however , will still be @xmath181 and thus it will dominate over @xmath182 growth . therefore , there will be no growth in the hydrodynamic limit . thus , our analyses suggest that in any fully developed turbulent system with @xmath0-effect , helicity spectrum ( given by @xmath119 ) should be as singular as the kinetic energy spectrum ( given by @xmath177 ) . in conclusions , we have calculated expressions for the @xmath0-coefficients in a diagrammatic perturbation theory on a minimal model for arbitrary background velocity and initial magnetic field correlations , and fluid and magnetic viscosities . we show that the parity breaking parts of the velocity and magnetic field variances must have the same sign , which is the case in any physical system . we explicitly show that the processes of early growth and late - time saturations may take place independent of any special velocity and initial magnetic field correlations . even though our explicit calculations were done by using simple initial conditions for the calculational convenience , the results that we obtain are general enough and it is apparent that the feedback mechanism is qualitatively independent of the details of the initial conditions and force correlations . one may note that for one of the force / initial correlations there is no kinetic energy cascade in the conventional sense but we still find dynamo action . it is quite reasonable to expect that our results should be valid for more realistic initial conditions also . in effect we have explicitly demonstrated the robustness and generality of the dynamo mechanism and that the dynamo mechanism is an intrinsic property of the @xmath6mhd equations . we have also shown , within our rg analysis , that the magnetic viscosity should be less than a critical value for growth of magnetic fields a result which was previouly observed in numerical simulations . we conclude the existence of a critical ekman number for k41-type correlations : we find growth only when @xmath9 , confirming recent numerical results . this is easily understood in our framework . the issue of divergent effective viscosities in the inertial range assumes importance as it may help to overcome some of the non - linear restrictions as discussed by vainshtein and cattaneo @xcite . a system of magnetohydrodynamic turbulence in a rotating frame , after the saturation time ( i.e. , after which there is no net growth of the magnetic fields ) belongs to the universality class of usual three - dimensional magnetohydrodynamic turbulence in a laboratory . this can be seen easily in both the lab and the rotating frames ; the critical exponents characterising the correlation functions can be calculated exactly by using the galilean invariance and noise - nonrenormalisation conditions @xcite . an important question , which remains open for further investigations , is the multiscaling properties of the velocity and the magnetic field structure functions at various stages of the growth of the magnetic fields . in what concerns an experimental observation of our results , one should add that even though it is not easy to verify our results in an experimental set up , numerical simulations of eqs.([nsk ] ) and ( [ indk ] ) with the variances ( [ varbi ] ) with different @xmath57-dependences can be performed to check these results . the author wishes to thank j. k. bhattacharjee for drawing his attention to this problem , and r. pandit , j. santos and the anonymous referee for many fruitful comments and suggestions . the author thanks the alexander von humboldt foundation , germany for financial support . h.k moffatt , _ magnetic field generation in elctrically conducting fluids _ , cambridge university press , cambridge ( 1978 ) . p. p. kronberg , _ rep . phys . _ * 57 * , 325(1994 ) ; 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we analyze the effects of the background velocity and the initial magnetic field correlations , and viscosities on the turbulent dynamo and the @xmath0-effect . we calculate the @xmath0-coefficients for arbitrary magnetic and fluid viscosities , background velocity and the initial magnetic field correlations .
we explicitly demonstrate that the general features of the initial growth and late - time saturation of the magnetic fields due to the non - linear feedback are qualitatively independent of these correlations .
we also examine the hydrodynamic limit of the magnetic field growth in a renormalization group framework and discuss the possibilities of suppression of the dynamo growth below a critical rotation .
we demonstrate that for kolmogorov- ( k41 ) type of spectra the _ ekman number _
@xmath1 for dynamo growth to occur .
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cooperation has played a fundamental role in the early evolution of our societies@xcite and continues playing a major role still nowadays . from the individual level , where we cooperate with our romantic partner , friends , and co - workers in order to handle our individual problems , up to the global level where countries cooperate with other countries in order to handle global problems , our entire life is based on cooperation . given its importance , it is not surprising that cooperation has inspired an enormous amount of research across all biological and social sciences , spanning from theoretical accounts @xcite to experimental studies @xcite and numerical simulations@xcite . since the resolution of many pressing global issues , such as global climate change and depletion of natural resources , requires cooperation among many actors , one of the most relevant questions about cooperation regards the effect of the size of the group on cooperative behavior . indeed , since the influential work by olson @xcite , scholars have recognized that the size of a group can have an effect on cooperative decision - making . however , the nature of this effect remains one of the most mysterious areas in the literature , with some scholars arguing that it is negative @xcite , others that it is positive @xcite , and yet others that it is ambiguous @xcite or non - significant @xcite . interestingly , the majority of field experiments seem to agree on yet another possibility , that is , that group size has a curvilinear effect on cooperative behavior , according to which intermediate - size groups cooperate more than smaller groups and more than larger groups @xcite . the emergence of a curvilinear effect of the group size on cooperation in real life situations is also supported by data concerning academic research , which in fact support the hypothesis that research quality of a research group is optimized for medium - sized groups @xcite . here we aim at shedding light on this debate , by providing evidence that a single parameter can be responsible for all the different and apparently contradictory effects that have been reported in the literature . specifically , we show that the effect of the size of the group on cooperative decision - making depends critically on a parameter taking into account different ways in which the notion of cooperation itself can be defined when there are more than two agents . indeed , while in case of only two agents a cooperator can be simply defined as a person willing to pay a cost @xmath0 to give a greater benefit @xmath1 to the other person @xcite , the same definition , when transferred to situations where there are more than two agents , is subject to multiple interpretations . if cooperation , from the point of view of the cooperator , means paying a cost @xmath0 to create a benefit @xmath1 , what does it mean from the point of view of the _ other _ player__s _ _ ? does @xmath1 get earned by each of the other players or does it get shared among all other players , or none of them ? in other words , what is the marginal return for cooperation ? of course , there is no general answer and , in fact , previous studies have considered different possibilities . for instance , in the standard public goods game it is assumed that @xmath1 gets earned by each player ( including the cooperator ) ; instead , in the n - person prisoner s dilemma ( as defined in @xcite ) it is assumed that @xmath1 gets shared among all players ; yet , the volunteer s dilemma @xcite and its variants using critical mass @xcite rest somehow in between : one or more cooperators are needed to generate a benefit that gets earned by each player , but , after the critical mass is reached , new cooperators do not generate any more benefit ; finally , it has been pointed out @xcite that a number of realistic situations can be characterized by a marginal return which increases linearly for early contributions and then decelerates , reflecting the natural decrease of marginal returns that occurs when output limits are approached . in order to take into account this variety of possibilities , we consider a class of _ social dilemmas _ parametrized by a function @xmath2 describing the marginal return for cooperation when @xmath3 people cooperate in a group of size @xmath4 . more precisely , our _ general public goods game _ is the n - person game in which n people have to simultaneously decide whether to cooperate ( c ) or defect ( d ) . in presence of a total of @xmath3 cooperators , the payoff of a cooperator is defined as @xmath5 ( @xmath6 represents the cost of cooperation ) and the payoff of a defector is defined as @xmath7 . in order to have a social dilemma ( i.e. , a tension between individual benefit and the benefit of the group as a whole ) we require that : * full cooperation pays more than full defection , that is , @xmath8 , for all @xmath4 ; * defecting is individually optimal , regardless of the number of cooperators , that is , for all @xmath9 , one has @xmath10 . the aim of this paper is to provide further evidence that the function @xmath11 might be responsible for the confusion in the literature about group size effect on cooperation . in particular , we focus on the situation , inspired from realistic scenarios , in which the natural output limits of the public good imply that @xmath7 increases fast for small @xmath3 s and then stabilizes . indeed , in our previous work @xcite , we have shown that the size of the group has a positive effect on cooperation in the standard public goods game and has a negative effect on cooperation in the n - person prisoner s dilemma . a reinterpretation of these results is that , if @xmath12 increases linearly with @xmath4 ( standard public goods game ) , then the size of the group has a positive effect on cooperation ; and , if @xmath12 is constant with @xmath4 ( n - person prisoner s dilemma ) , then the size of the group has a negative effect on cooperation . this reinterpretation suggests that , in the more realistic situations in which the benefit for full cooperation increases fast for early contributions and then decelerates once the output limits of the public good are approached , we may observe a curvilinear effect of the group size , according to which intermediate - size groups cooperate more than smaller groups and more than larger groups . to test this hypothesis , we have conducted a lab experiment using a general public goods game with a piecewise function @xmath11 , which increases linearly up to a certain number of cooperators , after which it remains constant . while it is likely that realistic scenarios would be better described by a smoother function , this is a good approximation of all those situations in which the natural output limits of a public good imply that the increase in the marginal return for cooperation tends to zero as the number of contributors grows very large . the upside of choosing a piecewise function @xmath11 is that , in this way , we could present the instructions of the experiment in a very simple way , thus minimizing random noise due to participants not understanding the decision problem at hand ( see method ) . our results support indeed the hypothesis of a curvilinear effect of the size of the group on cooperative decision - making . taken together with our previous work @xcite , our findings thus ( i ) shed light on the confusion regarding the group size effect on cooperation , by pointing out that different values of a single parameter might give rise to qualitatively different group size effects , including positive , negative , and even curvilinear ; and ( ii ) they help fill the gap between lab experiments and field experiments . indeed , while lab experiments use either the standard public goods game or the n - person prisoner s dilemma , _ real _ public goods game are mostly characterized by a marginal return of cooperation that increases fast for early contributions and then approaches a constant function as the number of cooperators grows very large - and our results provide evidence that these three situations give rise to three different group size effects . we have recruited participants through the online labour market amazon mechanical turk ( amt ) @xcite . after entering their turkid , participants were directed to the following instruction screen . _ welcome to this hit . _ _ this hit will take about 5 minutes and you will earn 20c for participating . _ _ this hit consists of a decision problem followed by a few demographic questions . _ _ you can earn an additional bonus depending on the decisions that you and the participants in your cohort will make . _ _ we will tell you the exact number of participants in your cohort later . _ _ each one of you will have to decide to join either group a or group b. _ _ your bonus depends on the group you decide to join and on the size of the two groups , a and b , as follows : _ * _ if the size of group a is 0 ( that is , everybody chooses to join group b ) , then everybody gets 10c _ * _ if the size of group a is 1 , then the person in group a gets 5c and each person in group b gets 15c _ * _ if the size of group a is 2 , then each person in group a gets 10c and each person in group b gets 20c _ * _ if the size of group a is 3 , then each person in group a gets 15c and each person in group b gets 25c _ * _ if the size of group a is 4 , then each person in group a gets 20c and each person in group b gets 30c _ * _ and so on , up to 10 : if the size of group a is 10 , then each person in group a gets 50c and each person in group b gets 60c _ * _ however , if the size of group a is larger than 10 , then , independently of the size of the two groups , each person in group a will still get 50c and each person in group b will still get 60c . _ after reading the instructions , participants were randomly assigned to one of 12 conditions , differing only on the size of the cohort ( @xmath13 ) . for instance , the decision screen for the participants in the condition where the size of the cohort is 3 was : _ you are part of a cohort of 3 participants . _ _ which group do you want to join ? _ by using appropriate buttons , participants could select either group a or group b. we opted for not asking any comprehension questions . we made this choice for two reasons . first , with the current design , it is impossible to ask general comprehension questions such as `` what is the strategy that benefits the group as a whole '' , since this strategy depends on the strategy played by the other players . second , we did not want to ask particular questions about the payoff structure since this may anchor the participants reasoning on the examples presented . of course , a downside of our choice is that we could not avoid random noise . however , as it will be discussed in the results section , random noise can not be responsible for our findings . instead , our results would have been even cleaner , if we had not had random noise , since the initial increase of cooperation and its subsequent decline would have been more pronounced ( see results section for more details ) . after making their decisions , participants were asked to fill a standard demographic questionnaire ( in which we asked for their age , gender , and level of education ) , after which they received the `` survey code '' needed to claim their payment . after collecting all the results , bonuses were computed and paid on top of the participation fee , that was $ 0.20 . in case the number of participants in a particular condition was not divisible by the size of the cohort ( it is virtually impossible , in amt experiments , to decide the exact number of participants playing a particular condition ) , in order to compute the bonus of the remaining people we formed an additional cohort where these people where grouped with a random choice of people for which the bonus had been already computed . additionally , we anticipate that only 98 subjects participated in the condition with n=100 . this does not generate deception in the computation of the bonuses since the payoff structure of the game does not depend on @xmath4 ( as long as @xmath14 ) . as a consequence of these observations , no deception was used in our experiment . according to the dutch legislation , this is a non - wmo study , that is ( i ) it does not involve medical research and ( ii ) participants are not asked to follow rules of behavior . see http://www.ccmo.nl / attachments / files / wmo- engelse-vertaling-29-7-2013-afkomstig-van-vws.pdf , section 1 , article 1b , for an english translation of the medical research act . thus ( see http://www.ccmo.nl / en / non - wmo- research ) the only legislations which apply are the agreement on medical treatment act , from the dutch civil code ( book 7 , title 7 , section 5 ) , and the personal data protection act ( a link to which can be found in the previous webpage ) . the current study conforms to both . in particular , anonymity was preserved because amt `` requesters '' ( i.e. , the experimenters ) have access only to the so - called turkid of a participant , an anonymous i d that amt assigns to a subject when he or she registers to amt . additionally , as demographic questions we only asked for age , gender , and level of education . a total of 1.195 _ distinct _ subjects located in the us participated in our experiment . _ distinct _ subjects means that , in case two or more subjects were characterized by either the same turkid or the same ip address , we kept only the first decision made by the corresponding participant and eliminated the rest . these multiple identities represent usually a minor problem in amt experiments ( only 2% of the participants in the current dataset ) . participants were distributed across conditions as follows : 101 participants played with @xmath15 , 99 with @xmath16 , 102 with @xmath17 , 101 with @xmath18 , 98 with @xmath19 , 103 with @xmath20 , 97 with @xmath21 , 99 with @xmath22 , 97 with @xmath23 , 101 with @xmath24 , 99 with @xmath25 , 98 with @xmath26 . 1 summarizes the main result . the rate of cooperation , that is the proportion of people opting for joining group a , first increases as the size of the group increases from @xmath15 to @xmath18 , then it starts decreasing . the figure suggests that the relation between the size of the group and the rate of cooperation is _ not _ quadratic : while the initial increase of cooperation is relatively fast , the subsequent decrease of cooperation seems extremely slow . this is confirmed by linear regression predicting rate of cooperation as a function of @xmath4 and @xmath27 , which shows that neither the coefficient of @xmath4 nor that of @xmath27 are significant ( @xmath28 , resp . ) . for this reason we use a more flexible econometric model than the quadratic model , consisting of two linear regressions , one with a positive slope ( for small @xmath4 s ) and the other one with a negative slope ( for large @xmath4 s ) . as a switching point , we use the @xmath18 , corresponding to the size of the group which reached maximum cooperation . doing so , we find that both the initial increase of cooperation and its subsequent decline are highly significant ( from @xmath15 to @xmath18 : coeff @xmath29 , @xmath30 ; from @xmath18 to @xmath26 : coeff @xmath31 , @xmath32 ) . to @xmath18 : coeff @xmath29 , @xmath30 ; from @xmath18 to @xmath26 : coeff @xmath31 , @xmath32)._,title="fig : " ] [ fig : intermediate ] we conclude by observing that not only random noise can not explain our results , but , without random noise , the effect would have been even stronger . indeed , first we observe that there is no a priori worry that random noise would interact with any condition and so we can assume that it is randomly distributed across conditions . then we observe that subtracting a binary distribution with average @xmath33 from a binary distribution with average @xmath34 , one would obtain a distribution with average @xmath35 . similarly , subtracting a binary distribution with average @xmath33 from a binary distribution with average @xmath36 one would obtain a distribution with average @xmath37 . thus , if the @xmath38 s are the averages that we have found ( containing random noise ) and the @xmath39 s are the _ true _ averages ( without random noise ) , the previous inequalities allow us to conclude that the initial increase of cooperation and its following decrease would have been stronger in absence of random noise . here we have reported on a lab experiment providing evidence that the size of a group can have a curvilinear effect on cooperation in one - shot social dilemmas , with intermediate - size groups cooperating more than smaller groups and more than larger groups . joining the current results with those of a previously published study of us @xcite , we can conclude that group size can have qualitatively different effects on cooperation , ranging from positive , to negative and curvilinear , depending on the particular decision problem at hand . interestingly , our findings suggest that different group size effects might be ultimately due to different values of a single parameter , the number @xmath12 , describing the benefit for full cooperation . if @xmath12 is constant in @xmath4 , then group size has a negative effect on cooperation ; if @xmath12 increases linearly with @xmath4 , then group size has a positive effect on cooperation ; in the _ middle _ , all sorts of things may a priori happen . in particular , in the realistic situation in which @xmath12 is a piecewise function that increases linearly with @xmath4 up to a certain @xmath40 and then remains constant , then group size has a curvilinear effect , according to which intermediate - size groups cooperate more than smaller groups and more than larger groups . see table 1 . .summary of the different group size effects on cooperation depending on how the benefit for full cooperation varies as a function of the group size . [ cols="<,^,^",options="header " , ] to the best of our knowledge , ours is the first study reporting a curvilinear effect of the group size on cooperation in an experiment conducted in the ideal setting of a lab , in which confounding factors are minimized . previous studies reporting a qualitatively similar effect @xcite used field experiments , in which it is difficult to isolate the effect of the group size from possibly confounding effects . in our case , the only possibly confounding factor is random noise due to a proportion of people that may have not understood the rules of the decision problem . as we have shown , our results can not be driven by random noise and , in fact , the curvilinear effect would have been even stronger , without random noise . moreover , since our experimental design was inspired by a tentative to mimic all those _ real _ public goods games in which the natural output limits of the public good imply that the increase of the marginal return for cooperation , when the number of cooperators diverges , tends to zero , our results might explain the apparent contradiction that field experiments tend to converge on the fact that the effect of the group size is curvilinear , while lab experiments tend to converge on either of the two linear effects . our contribution is also conceptual , since we have provided evidence that a single parameter might be responsible for different group size effects : the parameter @xmath12 , describing the way the benefit for full cooperation varies as a function of the size of the group . of course , we do not pretend to say that this is the only ultimate explanation of why different group size effects have been reported in experimental studies . in particular , in real - life situations , which are typically repeated and in which communication among players is allowed , other factors , such as within - group enforcement , may favor the emergence of a curvilinear effect of the group size on cooperation , as highlighted in @xcite . if anything , our results provide evidence that the curvilinear effect on cooperation goes beyond contingent factors and can be found also in the ideal setting of a lab experiment using one - shot anonymous games . we believe that this is a relevant contribution in light of possible applications of our work . indeed , the difference between @xmath12 and the total cost of full cooperation @xmath41 can be interpreted has the incentive that an institution needs to pay to the contributors in order to make them cooperate . since institutions are interested in minimizing their costs and , at the same time , maximizing the number of cooperators , it is crucial to understand what is the `` lowest '' @xmath11 such that the resulting effect of the group size on cooperation is positive . this seems to be an non - trivial question . for instance , does @xmath42 give rise to a positive effect or is it still curvilinear or even negative ? the technical difficulty here is that it is hard to design an experiment to test people s behavior in these situations , since one can not expect that an average person would understand the rules of the game when presented using a logarithmic functions . in terms of economic models , our results are consistent with utilitarian models such as the charness & rabin model @xcite and the novel cooperative equilibrium model @xcite . both these models indeed predict that , in our experiment , cooperation initially ( i.e. , for @xmath43 ) increases with @xmath4 ( see @xcite for the details ) , and then starts decreasing . this behavioral transition follows from the simple observation that free riding when there are more than 10 cooperators costs zero to each of the other players and benefits the free - rider . thus , cooperation in larger groups is not supported by utilitarian models , which then predict a decrease in cooperative behavior whose speed depends on the particular parameters of the model , such as the extent to which people care about the group payoff versus their individual payoff , and people s beliefs about the behavior of the other players . thus our results add to the growing body of literature showing that utilitarian models are qualitatively good descriptors of cooperative behavior in social dilemmas . however , we note that while theoretical models predict that the rate of cooperation should start decreasing at @xmath17 , our results show that the rate of cooperation for @xmath18 is marginally significantly higher than the rate of cooperation for @xmath17 ( rank sum , @xmath44 ) . although ours is a between - subjects experiment , this finding seems to hint at the fact that there is a proportion of subjects who would defect for @xmath17 and cooperate for @xmath18 . this is not easy to explain : why should a subject cooperate with @xmath18 and defect with @xmath17 ? one possibility is that there is a proportion of `` inverse conditional cooperators '' , who cooperate only if a small percentage of people cooperate : if these subjects believe that the rate of cooperation decreases quickly after @xmath17 , they would be more motivated to cooperate for @xmath18 than for @xmath17 . another possibility , of course , is that this discrepancy is just a false positive . in any case , unfortunately our experiment is not powerful enough to detect the reason of this discrepancy between theoretical predictions and experimental results and thus we leave this interesting question for future research . v.c . is supported by the dutch research organization ( nwo ) grant no . this material is based upon work supported by the national science foundation under grant no . 0932078000 while the first author was in residence at the mathematical science research institute in berkeley , california , during the spring 2015 semester . 1 kaplan h , gurven m. the natural history of human food sharing and cooperation : a review and a new multi - individual approach to the negotiation of norms . in : gintis h , bowles s , boyd r , fehr e , editors . moral sentiments and material interests : the foundations of cooperation in economic life . cambridge , ma : mit press ; 2005 . tomasello m. a natural history of human thinking . cambridge , ma : harvard university press ; 2014 . trivers r. the evolution of reciprocal altruism . q rev biol . 1971 ; 46 : 35 - 57 . axelrod r , hamilton wd . the evolution of cooperation . science . 1981 ; 211 : 1390 - 1396 . fehr e , fischbacher u. the nature of human altruism . nature . 2003 ; 425 : 785 - 791 . five rules for the evolution of cooperation . science . 2006 ; 314 , 1560 - 1563 . perc m , szolnoki a. coevolutionary games - a mini review . biosystems 2010 ; 99 : 109 - 125 . press wh , dyson fj . iterated prisoner s dilemma contains strategies that dominate any evolutionary opponent . proc natl acad sci usa . 2012 ; 109 : 10409 - 10413 . perc m , gmez - gardees j , szolnoki a , flora lm , moreno y. evolutionary dynamics of group interactions on structured populations : a review . j roy soc interface . 2013 ; 10 : 20120997 . capraro v. a model of human cooperation in social dilemmas . plos one 2013 ; 8 : e72427 . hilbe c , nowak ma , sigmund k. the evolution of extortion in iterated prisoners dilemma games . proc natl acad sci usa . 2013 ; 110 : 6913 - 6918 . rand dg , nowak ma . human cooperation . trends cogn sci . 2013 ; 17 : 413 - 425 . capraro v , halpern jy . translucent players : explaining cooperative behavior in social dilemmas . 2014 . available : http://ssrn.com/abstract=2509678 . andreoni j. why free ride ? 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in a world in which many pressing global issues require large scale cooperation , understanding the group size effect on cooperative behavior is a topic of central importance . yet
, the nature of this effect remains largely unknown , with lab experiments insisting that it is either positive or negative or null , and field experiments suggesting that it is instead curvilinear .
here we shed light on this apparent contradiction by considering a novel class of public goods games inspired to the realistic scenario in which the natural output limits of the public good imply that the benefit of cooperation increases fast for early contributions and then decelerates .
we report on a large lab experiment providing evidence that , in this case , group size has a curvilinear effect on cooperation , according to which intermediate - size groups cooperate more than smaller groups and more than larger groups . in doing so ,
our findings help fill the gap between lab experiments and field experiments and suggest concrete ways to promote large scale cooperation among people .
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planetary nebulae ( pne ) form around stars of low and intermediate mass ( @xmath1 m@xmath2 ) towards the end of their evolution . the nebula originates from the interaction between a fairly dense ( @xmath310@xmath4 m@xmath2 yr@xmath5 ) ` slow wind ' ( @xmath310 km s@xmath5 ) lost by the star as a red giant on the asymptotic giant branch ( agb ) with a tenuous ( @xmath310@xmath6 m@xmath2 yr@xmath5 ) ` fast wind ' ( @xmath31000 km s@xmath5 ) that follows after the agb phase . the nebula material becomes ionized once the effective temperature of the central star , which evolves into a white dwarf , exceeds 25000 k ( for a review see @xcite ) . ngc 6302 ( pn [email protected] ) is a bipolar pn . its complex and clumpy morphology shown in h@xmath0 and [ nii ] _ hubble space telescope _ ( _ hst _ ) images @xcite can be roughly approximated by a bipolar shape with the two main lobes extending in the east - west direction . it presents a highly pinched waist which is characteristic for butterfly - shaped bipolar pne @xcite . the central star of ngc 6302 was directly detected for the first time by @xcite . it was partially obscured by a dense equatorial lane @xcite and molecular material observed in co emission @xcite that is tracing an expanding torus oriented in the north - south direction , approximately perpendicular to the optical nebula axis . the distance to this nebula was not known accurately . @xcite gave a firm lower limit of [email protected] kpc based on a radio expansion proper - motion ( pm ) measurement of the nebular core , and estimate a distance of [email protected] kpc from measurements of pressure - broadening of radio recombination lines . lately , @xcite determined an unambiguous distance of [email protected] kpc , derived from expansion pms of 15 knots located at the northwestern lobe of the nebula . these pms were measured comparing two ground - based images with a time separation of @xmath351 yr , indicating a hubble - type expansion . hubble - type outflows are characterised by an expansion velocity that increases linearly with the distance to the central star . this type of outflow has been observed also in other objects , i.e. , the nebula around the symbiotic system hen 2 - 104 @xcite , ngc 6537 @xcite , mz 3 @xcite , see @xcite for other examples . recently , szyszka et al . ( 2011 ) measured the expansion pms in ngc 6302 , comparing two _ hst _ images in [ nii ] separated by 9.4 yr . the velocity field follows a hubble law in agreement with the previous results of @xcite . the pm vectors present a pattern mostly radial pointing back to the central source with a position close to the central star detected by @xcite . their results show that the lobes of ngc 6302 were ejected during a brief event 2250@xmath835 yr ago ( in agreement with the result of @xcite ) , and they find evidence for a subsequent acceleration that increased the velocity of the inner regions by 9.2 km s@xmath5 , possibly related to the onset of the ionization . in this work , we investigate whether or not the morphological and kinematical characteristics of ngc 6302 can result from the interaction between an isotropic fast wind with a toroidally - shaped slow wind . we have also considered a clumpy structure for the slow wind , based on observations of agb shells , which look like clumpy and filamentary shell structures @xcite . the paper is organised as follows . in section 2 , we describe the basic assumptions and the initial conditions used in the numerical simulations . in section 3 , we present the results including synthetic h@xmath0 emission maps , simulated pms of nebular knots , and their comparison with the observations . finally , in section 4 , we discuss the implications of our results and give our conclusions . our simulations are based on the generalisation of the interacting stellar winds model ( gisw ) in which , an isotropic fast wind launched by a star expands into a previously ejected toroidally - shaped slow wind @xcite . we assumed that the slow wind has a density distribution with a high contrast between the equator and the pole , which is described by the following equation ( given by @xcite ) @xmath9 with @xmath10,\ ] ] where @xmath11 is the distance from the central star and @xmath12 is the polar angle ( @xmath13 at the pole , and @xmath14 at the equator ) . the parameter @xmath0 determines the ratio between the value of the density at the equator and that at the pole , while @xmath15 determines the shape of the variation , and therefore that of the slow wind ( see @xcite ) . the value of @xmath16 can be calculated from the mass - loss rate @xmath17 as : @xmath18 where @xmath19 is the constant terminal velocity of the agb wind and @xmath20 is the radius of the region where the stellar wind is imposed . the 3d numerical simulations were performed with the hydrodynamical code @xcite , which integrates the gas dynamical equations with a second - order accurate scheme ( in time and space ) using the ` flux - vector splitting ' method of @xcite on a binary adaptive grid . a rate equation for neutral hydrogen is integrated together with the gas dynamics equations to include the radiative losses through a parametrized cooling function that depends on the density , temperature and hydrogen ionization fraction @xcite . to model the ngc 6302 nebula , we have used a computational domain of @xmath21 cm along the @xmath22- , @xmath23- , and @xmath24-axes , respectively . five refinement levels were allowed in the adaptive cartesian grid , achieving a resolution of @xmath25 cm at the finest level . for the initial condition we have filled the computational domain with the density distribution obtained from equation ( 1)-(3 ) with @xmath17 , @xmath19 , and @xmath20 equal to @xmath26 m@xmath2 yr@xmath5 , @xmath27 , and @xmath28 cm , respectively . we have used @xmath29 and @xmath30 in order to reproduce the observed morphology . to simulate a ` clumpy ' circumstellar medium ( csm ) , we have modulated the density by a fractal structure with a spectral index of @xmath31 , which is consistent with a turbulent interstellar medium ( see @xcite ) . this clumpy density is imposed on the initial condition and has fluctuations on the order of @xmath32 of the mean density value . in this csm , we start an isotropic and fast stellar wind with mass injection @xmath33 of @xmath34 m@xmath2 yr@xmath5 , and a velocity @xmath35 of @xmath36 . the 3d hydrodynamical simulations of two interacting winds were carried out with the setup given in section 2 . the fast wind generates a shock wave which propagates into the surrounding csm . a global bipolar morphology is generated and a similar size to the observed one is achieved after a time of 2000 yr , which is in agreement with the dynamical age given by szyszka et al . ( 2011 ) . from the density and temperature distribution given by our 3d hydrodynamical simulations , we can perform synthetic h@xmath0 maps . in fig . [ fha ] , we show the synthetic h@xmath0 emission maps obtained for the @xmath37 and @xmath38 projections ( left and right panels of fig . [ fha ] , respectively ) , using an angle @xmath39 . this angle @xmath40 is the angle between the nebula axis ( the @xmath24 axis of the computational domain ) and the plane of the sky , in agreement with previous observational results @xcite . [ fha ] shows a bipolar morphology for both projections , with a waist of @xmath41 cm , which has a similar size to the observed one ( @xmath42 if a distance of 1.17 kpc is considered ) . furthermore , both panels display a filamentary and clumpy structure , which is a consequence of the interaction between the fast wind and the clumpy slow wind . emission maps obtained at an integration time of 2000 yr and for the @xmath37 and @xmath38 projections ( left and right panel , respectively ) . the angle @xmath40 is the angle between the nebula axis ( the @xmath43 axis of the computational domain ) and the plane of the sky . the vertical and horizontal axes are in units of 10@xmath44 cm . the horizontal colour bar gives the h@xmath0 flux in units of erg s@xmath5 @xmath45 sr@xmath5.,width=309 ] to calculate the pms of the nebular knots we have re - started our simulation at an integration time of 2000 yr , and leave it to evolve by 10 additional yr . then , we obtain the pms of the knots found in the simulation using a cross - correlation method . we have used the `` pm mapping '' technique described by @xcite , in which , the pms from pair of images are derived by defining boxes including emitting knots and carrying out a cross - correlation function of the emission within the boxes . the pm is obtained from a fit to the peak of the cross - correlation function . this method has proven to be better than carrying out direct fits ( e.g. , a gaussian or a two - dimensional paraboloid ) to the observed emission features , because the cross - correlation functions , which are integrals of the emission within the chosen boxes , have higher signal - to - noise ratios than the images . the cross - correlation boxes have a size of 30@xmath4630 pixels equivalent to 10@xmath4610 arcsec@xmath47 . for a given box of size @xmath48 , we first check whether or not the condition @xmath49 is satisfied in at least one pixel within the central inner box of size @xmath50 ( see appendix of @xcite ) . here @xmath51 is the h@xmath0 flux of the image and @xmath52 is set equal to @xmath34 erg s@xmath5 @xmath45 sr@xmath5 . if this condition is met at least for a single pixel in each of the two epochs that are being analysed , the cross - correlation function ( within the @xmath48-size box in the two images ) and the pm are computed . [ fhapm ] shows the @xmath38 projection of the synthetic h@xmath0 map overlaid by white arrows which represent pm vectors calculated with the `` pm mapping '' technique . interestingly , the pms show deviations from the radial direction mostly for filamentary features in the outer regions of the outflow lobes . for a spatially extended filamentary structure , the pm perpendicular to the locus of the emission is well determined . while the component of the pm of the flow along the filament is highly uncertain , resulting in a large dispersion of the pm vectors determined for filamentary features . pm map of ngc 6302 . the nebula axis is oriented at @xmath53 with respect to the plane of the sky . the vertical and horizontal axes have the same dimensions as in fig . [ fha ] . the vertical colour bar gives the h@xmath0 flux in units of erg s@xmath5 @xmath45 sr@xmath5 . the length of each arrow indicates the relative pm of the h@xmath0 emission knots computed in 10@xmath4610 arcsec@xmath47 boxes . the arrow at bottom left corresponds to a value of 200 km s@xmath5.,width=309 ] fig . [ figpms ] shows pms vs. distance to the central star plots for the top lobe ( left panel ) and bottom lobe ( right panel ) . from this figure , we note that the more distant filaments and knots move faster than those located near the source . however , the pms obtained from our model do not resemble the behaviour given by a hubble - type expansion , such as found in the ngc 6302 observations ( szyszka et al . 2011 ) . the hubble - type expansion is represented in both panels of fig . [ figpms ] by dashed lines . we can see from the fig . [ figpms ] that many of the knots far from the source have velocities comparable to the observations , however we see a majority with velocities below the hubble - type expansion . at the same time , the knots that are closer to the source ( @xmath54 ) , have a remarkably constant pm , well below the observations . this result suggest that an additional acceleration mechanism is needed , particularly near the central star , which is consistent with the findings of szyszka et al . a possible candidate for this acceleration mechanism is the ionizing flux from the central star , which would photoevaporate the clumps and push them through a rocket effect ( see * ? ? ? this , and any other acceleration mechanism is not included in our models , and will be pursued in a following work . we have carried out 3d hydrodynamical simulations of a two interacting - winds model , in order to explain both the global morphology and the tangential velocity of the pn ngc 6302 . we reproduce the morphology and size of the nebula after an integration time of 2000 yr , which is similar to the dynamical time estimate given by szyszka et al . we note the that it was necessary to use a high equator to pole density contrast of @xmath31000 to obtain the observed morphology . a pm study was done on the synthetic h@xmath0 images during a time span of 10 yr similar to the time span between observed images . the magnitude of the pms as well as the separation from the central star are quantitatively similar , although the obtained pms do not follow a hubble - type expansion as the observations indicate . this suggests that an additional acceleration mechanism is acting , in particularly near the central star where the pm discrepancy is larger . a possible candidate for this mechanism is the rocket effect from the photoevaporation of the clumps , due to the radiation of the central star . we would like to dedicate this paper to the memory of our dear colleague yolanda gmez , who sadly passed away in 2012 . she was also involved in this project . lu and pb are grateful to john meaburn for his comments related to a very early draft version of this paper . we are thankful to our referee , myfanwy lloyd , for her valuable comments on the manuscript . lu acknowledges support from grant pe9 - 1160 of the greek general secretariat for research and technology in the framework of the program support of postdoctoral researchers . pfv , acr , jc , and ae acknowledge support from the conacyt grants 167611 , 167625 , and unam dgapa grant ig100214 . 99 balick b. , frank a. , 2002 , ara&a , 40 , 439 corradi r. l. m. , 2004 , aspc , 313 , 148 corradi r. l. m. , livio m. , balick b. , munari u. , schwarz h. e. , 2001 , apj , 553 , 211 corradi r. l. m. , schwarz h. e. , 1993 , a&a , 269 , 462 cox n. l. j. , et al . , 2012 , a&a , 537 , a35 dinh - v - trung , bujarrabal v. , castro - carrizo a. , lim j. , kwok s. , 2008 , apj , 673 , 934 esquivel a. , lazarian a. , 2005 , apj , 631 , 320 esquivel a. , lazarian a. , pogosyan d. , cho j. , 2003 , mnras , 342 , 325 esquivel a. , raga a. c. , 2007 , mnras , 377 , 383 gmez y. , rodrguez l. f. , moran j. m. , garay g. , 1989 , apj , 345 , 862 icke v. , preston h. l. , balick b. , 1989 , aj , 97 , 462 matsuura m. , zijlstra a. a. , molster f. j. , waters l. b. f. m. , nomura h. , sahai r. , hoare m. g. , 2005 , mnras , 359 , 383 meaburn j. , lloyd m. , vaytet n. m. h. , lpez j. a. , 2008 , mnras , 385 , 269 mellema g. , 1995 , mnras , 277 , 173 mellema g. , frank a. , 1995 , mnras , 273 , 401 ossenkopf v. , esquivel a. , lazarian a. , stutzki j. , 2006 , a&a , 452 , 223 peretto n. , fuller g. , zijlstra a. , patel n. , 2007 , a&a , 473 , 207 raga a. c. , navarro - gonzlez r. , villagrn - muniz m. , 2000 , rmxaa , 36 , 67 raga a. c. , noriega - crespo a. , carey s. j. , arce h. g. , 2013 , aj , 145 , 28 raga a. c. , reipurth b. , 2004 , rmxaa , 40 , 15 redman m. p. , oconnor j. a. , holloway a. j. , bryce m. , meaburn j. , 2000 , mnras , 312 , l23 steffen w. , koning n. , esquivel a. , garca - segura g. , garca - daz m. t. , lpez j. a. , magnor m. , 2013 , mnras , 436 , 470 szyszka c. , walsh j. r. , zijlstra a. a. , tsamis y. g. , 2009 , apj , 707 , l32 szyszka c. , zijlstra a. a. , walsh j. r. , 2011 , mnras , 416 , 715 van leer b. , 1982 , lecture notes in physics , 170 , 507
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we present 3d hydrodynamical simulations of an isotropic fast wind interacting with a previously ejected toroidally - shaped slow wind in order to model both the observed morphology and the kinematics of the planetary nebula ( pn ) .
this source , also known as the butterfly nebula , presents one of the most complex morphologies ever observed in pne . from our numerical simulations , we have obtained an intensity map for the h@xmath0 emission to make a comparison with the _ hubble space telescope _ ( _ hst _ ) observations of this object .
we have also carried out a proper motion ( pm ) study from our numerical results , in order to compare with previous observational studies .
we have found that the two interacting stellar wind model reproduces well the morphology of , and while the pm in the models are similar to the observations , our results suggest that an acceleration mechanism is needed to explain the hubble - type expansion found in _ hst _ observations .
[ firstpage ] methods : numerical planetary nebulae : general planetary nebulae : individual : ngc 6302
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the spectacular hickson compact group 31 ( hickson 1982 ) shows a wide range of indicators of galaxy interaction and merging : tidal tails , irregular morphology , complex kinematics , vigourous star bursting ( e.g. rubin et al . 1990 ) and possible formation of tidal dwarf galaxies ( e.g. hunberger et al . all the objects belonging to the group are embedded in a common large hi envelope ( williams et al . the group is formed by members a+c , b , e , f , g , q ( rubin et al . 1990 ) . two scenarios have been put forward to explain the nature of the central system a+c : it is either two systems that are about to merge ( e.g. rubin et al . 1990 ) or a single interacting galaxy ( richer et al . 2003 ) . in this letter we use our new fabry - perot maps and deep imaging from gemini - n to revisit the important problem of the merger nature of the central object of the group , a+c , and , in addition , we investigate the internal kinematics of the tidal dwarf galaxies , in an attempt to identify if they are self gravitating objects or not . we adopt a distance of 54.8 mpc , from the redshift z=0.0137 ( hickson et al . 1992 ) and using h@xmath3=75 km s@xmath4 mpc@xmath4 , hence 1 " @xmath5 0.27 kpc . observations were carried out with the fabry - perot instrument cigale ( gach et al . 2002 ) attached to the eso 3.6 m telescope in august 2000 . interferograms were obtained with a high order ( p=1938 ) fabry - perot scanning interferometer , giving a free spectral range of @xmath6 with a _ finesse = 24 _ leading to a spectral resolution of @xmath7 . the pixel size is 0.405 arcsec ; the total exposure time was 72 min ( 6 cycles of 12 min each , 48 scanning steps per cycle ) and the fwhm of the interference filter centered around 6651 @xmath8 was 20 @xmath8 . the velocity sampling was 3 @xmath9 @xmath10 and the relative velocity accuracy is @xmath11 over the whole field where the s / n is greater than 3 . reduction of the data cubes were performed using the cigale / adhocw software ( boulesteix , 2002 ) . the data reduction procedure has been described e.g. in blais - ouellette et al . ( 1999 ) and garrido et al . for the adopted distance of the group , one pixel corresponds to @xmath5 0.11 kpc . in august 2003 , we obtained two images with gmos at gemini - n in g and r with exposure times of 1200 and 900 seconds respectively . these images have a pixel size of 0.14 `` and typical seeing of 0.75 '' . a color map of the group is presented in fig . 1a . it shows a wealth of star forming regions and the large extent of the optical diffuse light which envelopes the group . for the first time , regions @xmath2 and @xmath0 , to the east of complex a+c , are seen in great detail and depth . d show the h@xmath12 monochromatic map of the group , the velocity map and several zoom panels showing the typical velocity profiles in selected regions of the group . the velocity field was corrected from free spectral range ambiguity using previous kinematic observations . at a first look , galaxies a and c appear to be a single kinematic entity , as their velocity fields show no discontinuity . under this assumption , we can infer a rough mass for the a+c complex , within a radius of @xmath5 18 arcsec ( @xmath5 4.9 kpc ) of @xmath13 . the following parameters and assumptions were used for this mass determination : a maximum rotational velocity of 70@xmath14 ( only from the nw side of the system given that the se side is too disturbed ) , a kinematic inclination and position angle of 51@xmath15 and 130@xmath16 degrees respectively and the assumption that the measured motions are due to disk rotation . nevertheless , multiple profiles ( see fig . 1d ) , evident almost everywhere in the main body of a+c , strongly suggests that a+c is not a single entity and therefore this rough determination provides only an order of magnitude for the mass . 2 shows the velocity gradients of the objects @xmath2 , @xmath0 , e and f , situated around the pair a+c ( see fig 1a ) and which have been thought to be candidate tidal dwarf galaxies ( hunsberger et al . 1996 , iglesias - pramo & vlchez 2001 and richer et al . 2003 ) . the curves are not corrected for inclination and the central position and velocity were chosen such that the curves were as symmetric as possible , with both sides matching , when possible . the object with the highest velocity gradient is e , with ordered velocities which range from 3950 @xmath17@xmath4 to 4000 @xmath17@xmath4 . object @xmath0 also shows ordered motion , with velocities going from 4125 to 4175 @xmath17@xmath4 . surprisingly , object f , thought to be the best tidal dwarf galaxy candidate in the group , shows a completely flat rotation curve , as does also the smaller object to the northeast of a+c , fragment @xmath2 . in addition , @xmath2 and f have low gaseous velocity dispersions . these results will be discussed in the next section . we obtained the map of the gaseous velocity dispersion at each pixel of the image ( amram et al . 2004 , in preparation ) assuming the profiles are well represented by a single gaussian . the value for the velocity dispersion ranges from 10 to 30 @xmath17@xmath4 throughout the group . in particular , objects @xmath2 , @xmath0 , e and f show typical velocity dispersions of 15 @xmath17@xmath4 , which in some isolated regions can reach up to 25 @xmath17@xmath4 . the highest values for the velocity dispersion lie in the overlapping region between a and c , mainly due to double components . it is noticeable that these highest values do not match the most intense star forming regions everywhere in the galaxies but particularly where disk a and c overlap , implying that the line broadening and the multiple components are not directly linked to star formation triggered by interaction with another galaxy but specifically by the merging of a+c . hickson & menon ( 1985 ) reached a similar conclusion , analyzing a radio continuum map of the group . they found that the 20 cm peak of emission comes from the overlapping regions of a and c , indicating additional evidence of recent excessive starburst activity in this region . richer et al . ( 2003 ) also presented fabry - perot velocity maps of hcg 31 . there is fairly good general agreement between their velocity field and ours . the spectral resolution of our maps is nevertheless six times higher ( 7900 vs 45900 ) and the detection limit several magnitudes fainter . several authors supported that a and c are separate entities in an ongoing merging phase ( e.g. vorontsov - velyaminov and arhipova , 1963 ; rubin et al . , 1990 ) while richer et al . ( 2003 ) supported the scenario that a+c is a single interacting spiral galaxy . the new piece of evidence in support of the merging scenario reported in this paper is the presence of double velocity components throughout the system a+c . in fact , our higher resolution velocity field shows kinematic structures which are not naturally explained by a single disk but by the merging of two disks . although sc galaxies are , in general , transparent objects ( bosma 1995 ) , moderate amounts of molecular gas ( as traced by the co ) and cold dust may make some regions opaque . the co emission is weak in hcg 31 but the brightest co peak occurs in the overlapping region between galaxies a and c ( yun et al . 1997 ) where the broader h@xmath12 profiles are observed . if the co belongs to the foreground galaxy , multiple components are observed in regions optically thick and then could not be observed if disk a and c are two separate galaxies seen in projection , i.e. chance alignment . multiple gaseous components are observed in the same disk plane when they are not in equilibrium ( e.g. stlin et al . this occurs when two different entities merge , or when the feedback gas due to star formation interacts with the ism . we observe in hcg 31 the signature of both mechanisms , the second one being probably a consequence of the first one . the general pattern of the velocity field of hcg31 a+c is somewhat similar to that of ngc 4038/9 ( the antennae , amram et al . , 1992 ) in which a continuity in the isovelocities between both galaxies is also observed in the overlapping region . the merging stage of the antennae is slightly less advanced than that for a+c . in the antennae , the two galaxies are clearly separate entities and their bodies , which are not yet overlapping , each display an increasing velocity gradient , which is roughly parallel and run from the ne to the sw ( amram et al . it is likely that when the disks of ngc 4038 and of ngc 4039 overlap , the velocity field will also present total continuity , as observed in a+c . galaxy c ( mrk 1089 ) has been classified as a double nucleus markarian galaxy , the two nuclei being separated by 3.4 arcsec ( mazzarella & boroson , 1993 ) and it is difficult to explain the existence of the two nuclei without invoking a merging scenario , as shown by numerical simulations ( e.g. barnes & hernquist 1992 ) . to reproduce the double line profile in ngc 4848 , vollmer et al ( 2001 ) have used numerical simulations . they interpret them as the consequence of infalling gas which collides with the ism within the galaxy . this gives rise to an enhanced star formation observed in the h@xmath12 and in the 20 cm continuum map . hcg 31 is most probably a group in an early phase of merger , growing through slow and continuous acquisition of galaxies from the associated environment . moreover , several evidences for interaction with the other galaxy components of the group , namely b , q and g ( e.g. the group is completely embedded in a large hi envelope showing a local maximum on g to the sw and another one around q to the ne ) indicate that the complex a+c is most probably accreting the surrounding galaxies . several papers in the past ( hunsberger et al . 1996 , johnson and conti 2000 , richer et al . 2003 and iglesias - pramo & vlchez 2001 ) have mentionned the possibility that tidal dwarf galaxies in hcg 31 were formed . the best candidates are objects e and f , for which metallicities were measured and they were determined to be similar to that of the complex a+c , suggesting a tidal origin for these objects ( richer et al . , 2003 ) . as shown in fig . 2 , we detect ordered motions only for objects @xmath0 and e and not for @xmath2 and f. it might be suspected that the internal velocity motions measured in @xmath0 and e could be due to streaming motions in incipient tidal tails in formation ( see fig . this is , however , not the case because these objects are counterrotating with respect to the main body of a+c . the discontinuity of the isovelocities can be clearly seen from fig . 1c : the velocities go from high ( northeast ) to low ( southwest ) in a+c , towards object e. then , along the body of e they go in the opposite sense . similarly , for object @xmath0 , it presents counterrotation with respect to its immediate neighbor to the west : galaxy a. these objects may fall back onto their progenitor . in fact , from their velocity differences with respect to the a+c complex ( + 115 @xmath17@xmath4 and -60 @xmath17@xmath4 respectively ) and from their relative projected distances ( 6.7 kpc and 5.4 kpc respectively ) and assuming a total mass for the a+c complex of @xmath13 , we could determine that these two objects will indeed , most probably , fall back onto a+c . the same is true for object f , which although more distant from a+c , has a very small radial velocity difference of @xmath5 60 @xmath17@xmath4 . we note , however , that given the fact that we measure radial velocities ( and not the velocity component in the plane of the sky ) , the observed internal velocities of the tidal fragments are lower limits . region f , which was previously thought to be the best example of a tidal dwarf galaxy in hcg 31 , is indeed part of the main merger , following the same kinematic pattern of the parent galaxy and presenting no rotation nor significant internal velocity dispersion . region f was found to have low or inexistent old stellar population by johnson and conti ( 2000 ) . if there is no old stellar population the velocity dispersion of the gas is mainly indicating the dynamics of the cloud . the range of values derived for the internal velocity dispersion of f , 15 - 25 @xmath17@xmath4 , is too close to the natural turbulence of the gas and/or the expanding velocity due to starburst winds . the lack of rotation and the continuity of the kinematics between the main body of the merger and object f suggest it is simply a tidal debris , although , considering its projected distance from a+c ( 16 kpc ) and the large amount of fuel available in the whole area of the merger ( 2.1 @xmath18 of hi gas ) , it could perhaps develop into a tidal dwarf galaxy in the future , by accretion of infalling material . there is also the possibility that regions @xmath2 and f present no velocity gradient due to their rotation pattern be along the line of sight . although this could be a possibility for the smaller and rounder region @xmath2 , it is less likely the case for region f , given its elongated morphology . our two main results are : \1 ) we measure multiple kinematic components throughout the body of a+c which we interpret as a strong indication that this complex is an ongoing merger . the double photometric nucleus has been identified in several previous images of hcg 31 including the spectacular hst image published by johnson and conti ( 2000 ) and in fig . the double kinematic component is shown here for the first time in fig . \2 ) f and @xmath2 present flat rotation velocity profiles and insignificant velocity dispersions . in contrast , @xmath0 and e are structures counterrotating with respect to the parent a+c complex . we conclude that hcg 31 is a merging group which is probably going to soon end up as a field elliptical galaxy . it would be very valuable to compare our new , high spectral resolution maps to simulations of groups to investigate which interaction and merger parameters fit these data . the authors thank olivier boissin for help during the observations , jorge iglesias - pramo for kindly providing the calibrated h@xmath12 image and to acknowledge financial support from the french - brazilian pics program . cmdo , ls and esc would like to thank the brazilian pronex program , fapesp , cnpq . cmdo deeply acknowledges the funding and hospitality of the mpe institut in garching , where this work was finalized . cc and oh acknowledge support from fqrnt , qubec and nserc , canada . amram p. , marcelin m. , boulesteix j. et al 1992 , a&a , 266 , 106 amram p. , mendes de oliveira c. , boulesteix j. et al . 1998 , a&a , 330 , 881 blais - ouellette s. ; carignan c. , amram p. et al , 1999 , aj , 118 , 2123 boulesteix j. , 2002 , adhocw red.package , www.oamp.fr/adhoc/ bosma a. , 1995 in nato advanced science inst . c , 469 , 317 gach j .- l . , hernandez o. , boulesteix j. et al . 2002 , pasp , 114 , 1043 garrido o. , marcelin m. , amram p. et al , 2002 , a&a , 387 , 821 hickson p. 1982 , apj , 255 , 382 hickson p. , mendes de oliveira c. , huchra j.p . 1992 , apj 399,353 hickson p. & menon t.k . 1985 , apj 296 , 60 hunsberger s.d . , charlton j.c . & zaritsky d. 1996 , apj 462 , 50 iglesias - pramo j. & vlchez j.m . 2001 , apj 550 , 204 johnson k.e . & conti p.s . 2000 , aj 119 , 2146 stlin g. , amram p. , bergvall n. et al . 2001 , a&a 374 , 800 richer m.g . , georgiev l. , rosado m. et al . , 2003 , a&a , 397 , 99 rubin v.c . , hunter d.a . , ford w.k.jr . 1990 , apj , 365 , 86 vollmer b. , braine j. , balkowski c. et al . , 2001 , a&a,374,824 vorontsov - velyaminov b.a . & arhipova , trudy g. astron . sht . , 33 , 1 , 1963 williams b.a . , mcmahon p.m. , van gorkom j.h . , 1991 , aj , 101 , 1957 yun m.s . , verdes - montenegro l. , del olmo a. et al . , 1997 , apj , 475l,21 .
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we have obtained high spectral resolution ( r = 45900 ) fabry - perot velocity maps of the hickson compact group hcg 31 in order to revisit the important problem of the merger nature of the central object a+c and to derive the internal kinematics of the candidate tidal dwarf galaxies in this group .
our main findings are : ( 1 ) double kinematic components are present throughout the main body of a+c , which strongly suggests that this complex is an ongoing merger ( 2 ) regions @xmath0 and e , to the east and south of complex a+c , present rotation patterns with velocity amplitudes of @xmath1 and they counterrotate with respect to a+c , ( 3 ) region f , which was previously thought to be the best example of a tidal dwarf galaxy in hcg 31 , presents no rotation and negligible internal velocity dispersion , as is also the case for region @xmath2 .
hcg 31 presents an undergoing merger in its center ( a+c ) and it is likely that it has suffered additional perturbations due to interactions with the nearby galaxies b , g and q.
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collective features in the microscopic dynamics often lead to the emergence of surprising and unexpected effects in the evolution of a physical system at the macroscopic level @xcite . the collective cooling of many two - level particles to very low temperatures is discussed here as an example of such a macroscopic manifestation of microscopic collective behavior @xcite . it is shown that the collective behavior of a large number of particles can produce much higher cooling rates than they could be achieved by means of individual cooling based on the spontaneous decay of the individual particles @xcite . as in laser sideband cooling techniques for single two - level atoms @xcite , we consider an experimental setup , where red - detuned laser fields increase the excitation of the particles , thereby continuously reducing the number of phonons . afterwards , the phonon energy can be removed constantly from the system . this requires energy dissipation and yields an overall decrease of the von neumann entropy in the setup @xcite . one possible decay channel is spontaneous emission from the excited states of the particles . during such a photon emission , a particle returns most likely into its ground state without regaining the phonon energy lost in the excitation process . the net result is a _ conversion _ of the phonons , originally existing in the setup in the form of thermal energy , into photons escaping the system . here we are interested in realizing much higher cooling rates than could be achieved with the help of the above described spontaneous decay of individual particles . this is possible , when the time evolution of the system remains restricted onto a highly symmetric and strongly coupling subspace of states throughout the whole cooling process . maximum cooling is obtained when the particles exhibit _ cooperative _ behavior in the excitation step as well as in the de - excitation step . to achieve this we assume as in ref . @xcite , that the outward energy dissipation is conducted by an optical cavity . as shown in fig . [ setup ] , the particles should be placed inside a resonant optical cavity with a relatively large rate for the leakage of photons through the cavity mirrors . the particles can then transfer their excitation collectively into the resonator field , from where the energy leaves the system without affecting the number of phonons in the setup and without changing the symmetry of the involved states . [ cols="^ " , ] collective behavior of the system requires furthermore that the rabi frequency @xmath2 of the laser field for the cooling of a vibrational mode with frequency @xmath3 and the cavity coupling constant @xmath4 are for all particles ( practically ) the same . initially , the particles should all be prepared in their ground state ( in the large @xmath0 limit fluctuations can be neglected ) . as shown below , the collective states of the assembly of @xmath0 particles then experience a very strong coupling to the laser field as well as to the cavity mode @xcite . as a consequence , the number of phonons decreases exponentially with a with as large as @xmath1 times the single - particle coupling constants , which can be much higher than previously predicted cooling rates in comparable setups @xcite . the proposed cooling scheme might be used to cool a large number of particles very efficiently . so it should be applicable to the preparation of bose einstein condensates . currently , these experiments mainly use evaporative cooling @xcite which systematically removes those atoms with a relatively high temperature from the trap . consequently , only a small percentage of the initially trapped atoms is finally included in the condensate . if one could instead cool all the atoms efficiently , yet at the same time avoid the loss of particles , it should become easier to experiment with large condensates . cooling is also crucial for ion trap quantum computing , where the achievable gate operation times can depend primarily on the efficiency of the cooling of a common vibrational mode @xcite . first cavity - cooling experiments involving many particles and observing enhanced cooling rates have already been performed @xcite . in this paper , we consider some algebraic features of the dynamics ruling the collective cooling of many particles and further clarify some aspects of the underlying mechanisms , which have already been studied in ref . a remarkable feature is , for example , the presence of a relatively large and negative coherence @xmath5 . instead of solving the time evolution of the system explicitly , we avoid certain approximations made in ref . @xcite by referring to the heisenberg picture . the system considered in this paper and its collective dynamics may be a paradigmatic example for other applications of physical interest , like the evolution of a system undergoing a continuous phase transition , yet preserving some specific features during such an evolution . the cavity coupling constant @xmath4 is for all particles the same , when the particles distribute themselves in the antinodes of the resonator field , as discussed in ref . @xcite ( see fig . [ setup](a ) ) . alternatively , a ring resonator could be used , as proposed in refs . @xcite . in the following we consider a collection of @xmath0 two - level particles ( atoms , ions or molecules ) with ground states @xmath6 and excited states @xmath7 ( see fig . [ setup](b ) ) . the setup should be operated in a parameter regime , where @xmath8 here @xmath9 is the decay rate of a single photon in the cavity mode , @xmath10 is the spontaneous decay rate of a particle in the excited state and @xmath11 denotes the lamb - dicke parameter characterizing the steepness of the trap @xcite . as in ref . @xcite , we discuss in this paper the two extreme cases , namely the cooling of common vibrational modes and the cooling of the individual phonon modes in the absence of common vibrational modes . in the following , @xmath12 denotes the annihilation operator for a phonon in the common vibrational mode with frequency @xmath3 , while @xmath13 is the annihilation operator for the cavity photons and @xmath14 is the lowering operator for particle @xmath15 . using the rotating wave approximation @xcite and going over to the interaction picture with respect to the interaction - free hamiltonian , the time evolution of the system can be described by the hamiltonian @xmath16 one way to simplify this hamiltonian is to introduce the effective phonon annihilation operator @xmath17= 1 \ , , \end{aligned}\ ] ] which allows to write the hamiltonian ( [ hi0 ] ) as @xmath18 we now consider the operator @xmath19 for the annihilation of the phonons of mode @xmath3 of particle @xmath15 . again we assume that the corresponding laser rabi frequencies @xmath2 are for all particles the same . then the hamiltonian of the system equals in the interaction picture and in the rotating wave approximation @xmath20 proceeding as above , introducing the effective phonon annihilation operator @xmath21= 1\end{aligned}\ ] ] and using eq . ( [ par ] ) , the hamiltonian ( [ h0 ] ) becomes @xmath22 although this hamiltonian has some similarities with the hamiltonian ( [ hi ] ) , it describes a physically different situation . instead of coupling to a set of common vibrational modes , each particle sees his own set of phonons . spontaneous emission is described in the following by the master equation @xcite @xmath23 + \kappa \ , \big(c \rho c^\dagger - { \textstyle{1 \over 2 } } c^\dagger c \rho - { \textstyle{1 \over 2 } } \rho c^\dagger c \big ) \nonumber \\ & & + \gamma \sum_i \big(\sigma_i \rho \sigma_i^\dagger - { \textstyle{1 \over 2 } } \sigma_i^\dagger \sigma_i \rho - { \textstyle{1 \over 2 } } \rho \sigma_i^\dagger \sigma_i \big)\end{aligned}\ ] ] with @xmath24 being @xmath25 or @xmath26 , respectively , @xmath9 being the decay rate of a photon in the cavity and @xmath10 being the decay rate of the particle excited state . from this equation one can easily see that the dissipation of cavity photons reduces the energy in the system without affecting the state of the particles . cavity decay therefore does not disturb the collective behavior of the system . however , this does not apply to the emission of photons from the particles , which can negatively interfere with the collective cooling process . in case of the cooling of common vibrational modes , the setup consists effectively of _ two _ different subsystems . one is the two - level particle system . as an effect of the emergence of the cooperative behavior , the collection of particles manifests itself as a _ bosonic _ system . we will see below that a further consequence of this is the transition to a strong coupling regime , which in turn implies a much shorter time scale for the system evolution . the other subsystem is the bosonic system of phonons and photons with a continuous _ conversion _ of phonons into photons . we will see that it is convenient to consider a boson mode which is a superposition of them . this is analogous to the field - atom polariton of hopfield @xcite . the physical implication of the familiar commutator relations ( [ e2 ] ) is that the particles behave no longer like _ individual fermions_. instead , the time evolution generates the excitation of bosonic modes , namely collective dipole waves , with @xmath39 denoting the creation and annihilation operators of the associated quanta obeying the usual commutation relation ( [ e2 ] ) . as a consequence , the collection of the single two - level particles manifests itself as a _ bosonic _ system . in other words , the ladder of equally - spaced dicke states approximates to a weakly - excited harmonic oscillator . using the operators ( [ col ] ) , the hamiltonian ( [ hi ] ) for the cooling of _ common _ vibrational modes can simply be written as @xmath41 where we have introduced the notation @xmath42 from this one sees that the time evolution of the system is mainly governed by the parameters @xmath43 , @xmath44 and @xmath9 , which scale as @xmath1 . we thus have , as a consequence of the emergence of the particle collective behavior , the transition to the strong coupling regime , @xmath45 and @xmath46 . the evolution of the system happens no longer on the time scale given by the parameters @xmath47 and @xmath4 but on a much shorter time scale defined by @xmath48 and @xmath49 . another factor that contributes significantly to the described cooling mechanism is the _ interface _ or _ ambivalent _ role played by the particles with respect to the phonons and the photons in the setup . the only difference between the particle - phonon and particle - photon interaction is the difference of the coupling constants , given by @xmath50 and @xmath49 , respectively . in some sense , the particles act as an engine transforming phonon energy into photon energy . this last one is then dissipated outward through the coupling with the cavity which allows energy leakage . however , in the absence of the leakage of photons through the cavity , the inverse transformation , photons to phonons , is also possible in principle . in such a situation , interesting interference and coherence effects arise , which we analyze below in detail . indeed , a closer look at eq . ( [ xxx ] ) reveals that the hamiltonian for the cooling of common vibrational modes can alternatively be written as @xmath51 with @xmath52=1 \ , . ~~\end{aligned}\ ] ] instead of interacting with the phonons and photons separately , the particles see the boson mode with annihilation operator @xmath53 and number operator @xmath54\end{aligned}\ ] ] with @xmath55 physically , the creation of bosons corresponding to @xmath56 does not only affect the number of phonons and the number of photons in the system . inevitably , it also creates a coherence between the @xmath57 and the @xmath13 subsystem . this coherence @xmath5 provides a symmetric " channel for the phonon - photon energy transformation . however , the leakage of energy outside the cavity perturbs such a symmetric phonon - photon balancing due to the @xmath5 action . the system reacts by subsequent adjustments , trying to recover its lost balance . crucial to such a re - adjustment mechanism is the difference in the time scales . the time scale for reaching the quasi - stationary state is of the order @xmath58 , while the time scale for spontaneous decay is , as we see in the next section , of order @xmath59 . the time evolution of the system turns out to be highly non - linear since the second and higher order derivatives of the physical observables are much larger than their first order derivatives . the system reaches a quasi - stationary state on a time scale of the order @xmath58 . the formation of this local equilibrium is solely governed by the effect of the hamiltonian @xmath60 . let us therefore first consider the situation , where we can neglect spontaneous emission and @xmath61 and @xmath62 . in this case , the time evolution of the system , governed by the hamiltonian ( [ yyy ] ) , results in a redistribution of population between the bosonic subsystem described by @xmath63 and the particles described by @xmath64 . to analyse this process we introduce the operators @xmath65 with the familiar @xmath31 commutators @xmath66 = { \rm i } \epsilon_{ijk } \ , l_k \ , .\end{aligned}\ ] ] with this notation , the hamiltonian ( [ yyy ] ) becomes @xmath67 and this , formally , simply generates a rotation around the 1-axis . in such an algebraic picture , we can immediately conclude that the angular momentum @xmath68 and the total angular momentum @xmath69\end{aligned}\ ] ] are conserved during the time evolution of the system under the considered conditions . especially , the conservation of @xmath70 implies the conservation of the total number of bosons , as accounted for by the number operator @xmath71 . when the number of particles in the excited state @xmath7 remains small compared to @xmath0 , the commutator relations ( [ e2 ] ) hold . in such a case , we can assume that the expectation value of @xmath72 remains small . neglecting @xmath64 in eq . ( [ comp ] ) implies @xmath73 using the definition ( [ a ] ) , this conservation law translates into @xmath74 which coincides with eq . ( 16 ) in ref . @xcite and describes the continual balance between the rate of change of the expectation value of @xmath5 and the rate of change of the total number of phonons and photons in the system . to show that eq . ( [ cons ] ) does not violate the conservation of the number of particles in the setup , we remark that the system obeys a second conservation law . considering again the heisenberg picture , we indeed find @xmath75 in the presence of many particles , with most of them remaining in their ground state , the expectation value of @xmath64 remains negligible . this implies @xmath76 and , consequently , also @xmath77 @xcite . inserting this into eq . ( [ energy ] ) , we obtain the particle number conservation law @xmath78 despite being a coherence , the expectation value of @xmath5 acts like a population . this allows the system to obey conservation of the particle number @xmath63 as well as conservation of @xmath79 by balancing the coherence @xmath5 accordingly . in the following , we study the conversion of phonons into cavity photons and creation of a non - zero coherence @xmath5 in more detail . in the absence of spontaneous emission , the system reaches after a very short time a stationary state with constant expectation values for @xmath80 , @xmath81 and @xmath5 . to calculate the corresponding values of the cavity photon number and coherence @xmath5 as a function of the phonon number , we use again the heisenberg picture , and obtain the second order differential equations @xmath82 these equations imply that the first order derivatives of the operators @xmath80 , @xmath81 and @xmath5 change on a time scale of the order @xmath58 , which is , for large @xmath0 , much faster than the time scale on which the time evolution of @xmath83 , @xmath81 and @xmath5 takes place . since we are only interested in the time dependence of the phonon number @xmath80 , we can safely assume that the first order derivatives @xmath84 , @xmath85 and @xmath86 adapt adiabatically to the state of the system . setting the right hand sides of the differential equations ( [ sec ] ) equal to zero , we find @xmath87 moreover , eq . ( [ m ] ) implies , due to the conservation laws ( [ cons ] ) and ( [ cons2 ] ) , that @xmath88 this means that , after a short time of order @xmath58 , the system reaches a quasi - stationary state with a constant ratio between the expectation value of the coherence @xmath5 and the number of photons inside the cavity , respectively , compared to the total number of phonons in the system . to a very good approximation , these ratios remains constant throughout the whole cooling process . in the absence of spontaneous emission , at most a redistribution of phonon energy can occur in the system @xcite . efficient cooling and the irreversible removal of energy from the system requires dissipation . we therefore now consider the effect of dissipation in more detail . using the master equation ( [ rho ] ) we find @xmath89 the leakage of photons through the cavity mirrors with decay rate @xmath9 not only decreases the number of photons in the cavity mode but also affects the size of @xmath5 . in the previous subsection , we have seen that the system reaches a stationary state on a time scale of the order @xmath58 with @xmath81 and @xmath5 being a multiple of @xmath80 . using eq . ( [ trop ] ) , we find @xmath90 + { x \over y } \ , \big [ { \textstyle { 1 \over 2 } } \kappa \ , k_3 + { \textstyle { { \rm d } \over { \rm d}t } } k_3 \big ] \ , . \nonumber \\ & & \end{aligned}\ ] ] this shows that on the time scale we are interested in , namely a time scale of the order @xmath59 , both , the operator @xmath81 and the presence of a negative valued operator @xmath5 , provide effective cooling channels in the system . inserting the results ( [ m ] ) and ( [ wednesday ] ) into eq . ( [ now ] ) we obtain indeed @xmath91 if @xmath92 denotes the phonon number expectation value @xmath93 and @xmath94 , we finally obtain @xmath95 \ , .\end{aligned}\ ] ] this describes exponential cooling of the atomic sample with a rate that can be as large as @xmath49 and @xmath96 ( cf . ( [ par ] ) ) . the result ( [ rate ] ) coincides with eq . ( 17 ) in ref . @xcite and is in good agreement with an exact numerical solution of the time evolution of the system ( see fig . [ num](a ) ) . spontaneous emission from excited atomic levels with decay rate @xmath10 is too slow to contribute to the cooling of the system , since most of the particles remain in their ground states and @xmath97 . finally , we remark that the cooling process we describe here is not simply a redistribution of phonons into cavity photons and @xmath5 , which then decay into the environment with the spontaneous decay rates @xmath9 and @xmath98 , respectively . instead , the leakage of photons through the cavity mirrors disturbs the equilibrium expressed by eqs . ( [ cons ] ) and ( [ cons2 ] ) perturbing the otherwise conserved quantities @xmath99 the coupling to the environment causes a _ dynamical response _ in the system , which generates a transition from a state with a fixed ratio of phonons to cavity photons ( cf . ( [ m ] ) ) into a state with no phonons and photons in the setup . in atom - cavity systems , the time evolution of the system is , on a very short time scale , usually dominated by spontaneous emission . however , in the presence of many particles , the evolution is primarily governed by the hamiltonian @xmath25 , which drives the system into a quasi - stationary state characterised by the preserved quantities @xmath100 and @xmath101 within a time of the order @xmath58 . the presence of the spontaneous decay rate @xmath9 , which scales as @xmath1 ( cf . ( [ par ] ) ) , disturbs this stationary state and causes the system to continuously assume new values for @xmath100 and @xmath101 . using eqs . ( [ m ] ) and ( [ ooo ] ) , we find @xmath102 while the relative size of the photon number @xmath103 and the coherence @xmath104 with respect to the number of phonons @xmath92 remain constant , their sum , as accounted for by @xmath100 and @xmath101 decrease exponentially in time due to the presence of dissipation , namely leakage of photons through the cavity mirrors . this is typical for a system undergoing a phase transition . there are many similarities but also many differences between the cooling of the common and the cooling of the individual phonon modes of particles . for example , one can not simplify the hamiltonian @xmath26 using the collective lowering operator @xmath105 , as we did in eq . ( [ xxx ] ) . however , there is still a high symmetry in the system and all particles are treated in exactly the same way . to take this into account we introduce the vector operators @xmath106 with the usual scalar product , such that , for example , @xmath107 this notation allows us to write the hamiltonian ( [ h ] ) as @xmath108 which is of a similar form as the hamiltonian @xmath25 in eq . ( [ xxx ] ) . in analogy to section [ aaa ] , we proceed by introducing an effective annihilation operator @xmath109 for each particle @xmath15 with @xmath110=1 \ , . ~~\end{aligned}\ ] ] then the hamiltonian ( [ xxxx ] ) can be written as @xmath111 each particle couples individually to a new type of bosons , which are a superposition of a single phonon and a cavity photon . the number operator accounting for all bosons @xmath109 equals @xmath112 \ , , \end{aligned}\ ] ] where we defined the coherence @xmath113 while @xmath114 counts the total number of phonons in the system , the expectation value of @xmath115 . again , the coherence @xmath5 describes a certain `` symmetry '' between phonons and photons in the setup and accounts for a continual conversion of the two types of bosons into each other . neglecting spontaneous emission , i.e. assuming @xmath61 and @xmath62 , and using the hamiltonian ( [ xxxx ] ) , we find @xmath116 in the derivation of these equations , we neglected the operators @xmath117 and @xmath118 with @xmath119 since there are no interactions between particles and the phonons of other particles . from eq . ( [ trop2 ] ) we see that there are , as before , two conserved quantities in the system , namely @xmath120 one is associated with the total number of bosonic particles with annihilation operators @xmath109 , the other one counts the total number of phonons and photons in the setup and @xmath121 from the formal equivalence of the hamiltonians ( [ yyy ] ) and ( [ yyyy ] ) pointed out in the previous subsection , one might have had expected that the expectation value @xmath122 is preserved in the time evolution of the system . however , this would only be the case if @xmath109 and @xmath123 commute with each other for @xmath119 and does not apply here . to calculate the distribution of phonons , cavity photons and coherence @xmath5 in the system , which builds up in the absence of spontaneous emission , we now consider the second derivatives of the operators @xmath114 , @xmath124 and @xmath5 in the heisenberg picture . neglecting again the population in the excited states , as accounted for by @xmath125 , we obtain @xmath126 setting the right hand side of these equations equal to zero and taking the conservation laws ( [ lee ] ) in the absence of dissipation into account , we find that the system possesses a stationary state with @xmath127 in contrast to the results in section [ fast ] , this equation describes a state with only a relatively small number of photons in the cavity mode compared to the total number of phonons in the setup . however , as we see in the next subsection , the presence of a negative coherence @xmath5 of the same order of magnitude as @xmath114 provides a cooling channel , which is sufficient to obtain a cooling rate of the same order of magnitude as the cavity decay rate @xmath9 . as before , the effective removal of phonons from the setup requires leakage of photons through the cavity mirrors . to take spontaneous emission into account we consider again the master equation ( [ rho ] ) and find @xmath128 which implies @xmath129 \nonumber \\ & & + { x \over y } \ , \big [ { \textstyle { 1 \over 2 } } \kappa \ , k_3 + { \textstyle { { \rm d } \over { \rm d}t } } k_3 \big ] \ , .\end{aligned}\ ] ] as already mentioned above , the number of photons in the cavity remains relatively small and does not contribute considerably to the cooling process . neglecting @xmath130 and using the results from the previous subsection , we obtain @xmath131 and @xmath132 \ , , \end{aligned}\ ] ] where @xmath133 and @xmath134 now describe the total number of phonons in the system , as accounted for by @xmath135 at time @xmath136 and @xmath137 , respectively . as in the case of the cooling of common vibrational modes , cooling rates of the same order of magnitude as the cavity decay rate @xmath9 can be obtained , which can be as large as @xmath1 times the single particle coupling constants . however , there are some differences with respect to the case considered in section [ comm ] . in the present case , achieving such a high cooling rate and cooling the system to very low temperatures first requires the build up of a reasonably large coherence @xmath5 , which does not exist in the absence of any laser driving . suppose , there are no initial correlations between the particles and their motional degrees of freedom . then @xmath138 ( cf . ( [ trop3 ] ) ) . moreover , the second order derivative of @xmath5 increases only on a time scale of order one ( cf . ( [ sec2 ] ) ) . initially , the system is far away from the quasi - stationary state described in the previous subsection . however , once @xmath5 reaches its equilibrium , the collective cooling process can begin . the coherence @xmath5 needs no longer to be established ; the system only has to adapt to the small changes of @xmath100 and @xmath101 caused by the leakage of photons through the cavity mirrors . for a numerical solution of the time evolution of the system see fig . [ num](b ) . the emergence of collective dynamics in a system of a large number of particles manifests itself in some macroscopic features in the system behavior . as an example , we considered the problem of fast and efficient cooling of an assembly of @xmath0 two - level particles trapped inside a leaky optical cavity . results obtained here confirm those derived in ref . the particles are excited by red - detuned laser fields . when the coupling constants are for all particles the same , a collective behavior emerges and the cooling rate can be as large as @xmath1 times the single - particle coupling constants . the generation of cooperative behavior of the @xmath0 particles is crucial in the excitation step as well as in the de - excitation step . the collective states of the assembly of @xmath0 particles then experience a very strong coupling to the laser field as well as to the cavity mode . we have considered the case of particles sharing a common phonon mode and the case of individual phonon modes for each particle . the two cases are similar and in both cases the collective cooling rate is very fast as compared to the single - particle cooling rate . the two cases are , however , different for the behavior of the coherence @xmath5 : in the individual mode case , high cooling rates are achieved by first building up a reasonably large coherence @xmath5 , which does not exist in the absence of any laser driving . initially , the system is far away from the quasi - stationary state , where the phonon and the photon populations are balanced through the @xmath5 action . only when @xmath5 reaches its equilibrium , the collective cooling process can begin and the system can adapt to the small changes of @xmath100 and @xmath101 caused by the leakage of photons through the cavity mirrors . in the common mode case , there is no need for the pre - cooling " phase for a build up of the coherence @xmath5 . the system considered in this paper may be a paradigmatic example for other applications of physical interest , such as systems undergoing continuous phase transitions , yet preserving some specific features during their evolution . systems presenting such a behavior might be of interest as well in biology . this work was supported in part by the european union , coslab ( esf program ) , infn , infm and the uk engineering and physical sciences research council . acknowledges funding as a james ellis university research fellow from the royal society and the gchq . 00 p. w. anderson , science , * 177 * , ( 1972 ) 393 . a. beige , p. l. knight , and g. vitiello , _ cooling many particles at once _ , quant - 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in a recent paper [ beige , knight , and vitiello , quant - ph/0404160 ] , we showed that a large number @xmath0 of particles can be cooled very efficiently using a bichromatic interaction .
the particles should be excited by red - detuned laser fields while coupling to the quantized field mode inside a resonant and leaky optical cavity .
when the coupling constants are for all particles the same , a collective behavior can be generated and the cooling rate can be as large as @xmath1 times the single - particle coupling constants . here
we study the algebraic structure of the dynamics and the origin of the collective cooling process in more detail .
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the flavor - changing - neutral - current ( fcnc ) transitions @xmath4 provide potentially stringent tests of standard model ( sm ) in flavor physics and are not allowed at tree level but are induced by the glashow - iliopoulos - miani ( gim ) amplitudes @xcite at the loop level in the sm . in addition , these are also suppressed in sm due to their dependence on the weak mixing angles of the quark - flavor rotation matrix @xmath5 the cabibbo - kobayashi - maskawa ( ckm ) matrix @xcite . these two circumstances make the fcnc decays relatively rare and hence important for the study of physics beyond the sm commonly known as new physics . the experimental observation of inclusive @xcite and exclusive @xcite decays , @xmath6 and @xmath7 , has prompted a lot of theoretical interest on rare @xmath8 meson decays . though the inclusive decays are theoretically better understood but are extremely difficult to be measured in a hadron mechine , such as the lhc , which is the only collider , except for a super-@xmath8 factory , that could provide enough luminosity for the precise study of the decay distribution of such rare processes . in contrast , the exclusive decays are easy to detect experimentally but are challanging to calculate theoretically and the difficulty lies in describing the hadronic structure , which provides the main uncertainty in the predictions of exclusive rare decays . in exclusive @xmath9 decays the long - distance effects in the meson transition amplitude of the effective hamiltonian are encoded in the meson transition form factors which are the scalar functions of the square of momentum transfer and are model dependent quantites . many exclusive @xmath10@xcite , @xmath11@xcite , @xmath12@xcite , @xmath13@xcite processes based on @xmath14 have been studied in literature and many fameworks have been applied to the description of meson transition form factors : like constituent quark models , qcd sum rules , lattice qcd , approaches based on heavy quark symmetry and analytical constraints . rare @xmath8 decay modes also provide imporatant ways to look for physics beyond the sm . there are various extensions of the sm in the literature , but the models with extra dimensions are of viable interest as they provide a unified framework for gravity and other interactions . in this way they give some hints on the hierarchy problem and a connection with string theory . among different models of extra dimensions , which differ from one another depending on the number of extra dimensions , the most interesting are the scenario with universal extra dimensions . in these ued models all the sm fields are allowed to propagate in the extra dimensions and compactification of extra dimension leads to the appearance of kaluza - klein ( kk ) partners of the sm fields in the four dimensional description of higher dimensional theory , together with kk modes without corresponding sm partners . the appelquist , cheng and dobrescu ( acd ) model @xcite with one universal extra dimension ( ued ) is very attractive because it has only one free parameter with respect to the sm and that is the inverse of compactification radius @xmath2 @xcite . by analyzing the signature of extra dimenions in the different processes , one can get bounds to the size of extra dimensions which are different in different models . these bounds are accessible for the processes already known at the particle accelerators or within the reach of planned future facilities . in case of ued these bounds are more sever and constraints from tevatron run i allow to put the bound @xmath15 gev @xcite . rare @xmath8 decays can also be used to constraint the acd scenario and in this regard buras and collaborators have already done some work . in addition to the effective hamiltonian they have calculated for @xmath16 decays and also investigated the impact of ued on the @xmath17 mixing as well as on the ckm unitarity triangle @xcite . due to the availability of precise data on the decays @xmath18 , colangelo et al . have studied these decays in acd model by calculating the branching ratio and forward - backward asymmetry for the decay @xmath19 . we will study the rare semileptonic decay modes , @xmath20on the same footing as @xmath21 because both are induced by the same quark level transitions , i.e. @xmath22 . we compare results of forward backward asymmetry for @xmath23 using our form factors with those obtained by colangelo @xmath24 @xcite . the comparision shows clear distinction as shown in fig 4 . these decays may provide us step forward towards the study of existance of new physics beyond the sm and therefore deserve serious attention , both theoretically and experimentally . the paper is organized as follows . in section 2 we will briefly introduce the acd model . section 3 deals with the study of effective hamiltonian and the corresponding matrix elements for @xmath20 decay . now the new physics manifest in these decays in two different ways , either through new operators in the in the effective hamiltonian which are absent in the sm or through new contributions to the wilson coefficients @xcite . in acd no new operator appears at tree level and therefore the new physics comes only through the wilson coefficients which are calculated in literature @xcite and we will summarize them in the same section . finally , in section 4 we will calculate the decay rate and forward - backward asymmetry and summarize our results . in our usual universe we have 3 spatial @xmath251 temporal dimensions and if an extra dimension exists and is compactified , fields living in all dimensions would menifest themselves in the @xmath26 space by the appearence of kaluza - klein excitations . the most pertinent question is whether ordinary fields propagate or not in all extra dimensions . one obvious possibilty is the propagation of gravity in whole ordinary plus extra dimensional universe , the `` bulk '' . contrary to this there are the models with universal extra dimensions ( ued ) in which all the fields propagate in all available dimensions @xcite and appelquist , cheng and dobrescu model belongs to one of ued scenarios @xcite this model is the minimal extension of the sm in @xmath27 dimensions , and in literature a simple case @xmath28 is considered @xcite . the topology for this extra dimension is orbifold @xmath29 , and the coordinate @xmath30 runs from @xmath31 to @xmath32 , where @xmath2 is the the compactification radius . the kaluza - klein ( kk ) mode expension of the fields are determined from the boundary conditions at two fixed points @xmath33 and @xmath34 on the orbifold . under parity transformation @xmath35 @xmath36 the fields may be even or odd . even fields have their correspondent in the @xmath37 dimensional sm and their zero mode in the kk mode expansion can be interpreted as the ordionary sm field . the odd fields do not have their correspondent in the sm and therefore do not have zero mode in the kk expansion . the significant features of the acd model are : * the compactification radius @xmath2 is the only free parameter with respect to sm * no tree level contribution of kk modes in low energy processes ( at scale @xmath38 ) and no production of single kk excitation in ordinary particle interactions is a consequence of conservation of kk parity . the detailed description of acd model is provided in @xcite ; here we summarize main features of its construction from @xcite . * gauge group * as acd model is the minimal extension of sm therefore the gauge bosons associated with the gauge group @xmath39 are @xmath40 , @xmath41,@xmath42,@xmath43,@xmath44,@xmath45 and @xmath46 , and the gauge couplings are @xmath47 and @xmath48 ( the hat on the coupling constant refers to the extra dimension ) . the charged bosons are @xmath49 and the mixing of @xmath50 and @xmath46 give rise to the fields @xmath51 and @xmath52 as they do in the sm . the relations for the mixing angles are : @xmath53 the weingberg angle remains the same as in the sm , due to the relationship between five and four dimensional constants . the gluons which are the gauge bosons associated to @xmath54 are @xmath55 . * higgs sector and mixing between higgs fields and gauge bosons * the higgs doublet can be written as : @xmath56 with @xmath57 . now only field @xmath58 has a zero mode , and we assign vacuum expectation value @xmath59 to such mode , so that @xmath60 . @xmath61 is the the sm higgs field , and the relation between expectation values in five and four dimension is : @xmath62 . the goldstone fields @xmath63 , @xmath64 aries due to the mixing of charged @xmath65 and @xmath66 , as well as neutral fields @xmath67 . these goldstone modes are then used to give masses to the @xmath68 and @xmath69 , and @xmath70 , @xmath71 , new physical scalars . * yukawa terms * in sm , yukawa coupling of the higgs field to the fermion provides the fermion mass terms . the diagonalization of such terms leads to the introduction of the ckm matrix . in order to have chiral fermions in acd model , the left and right - handed components of the given spinor can not be simultaneously even under @xmath72 . this makes the acd model to be the minimal flavor violation model , since there are no new opeators beyond those present in the sm and no new phase beyond the ckm phase and the unitarity triangle remains the same as in sm @xcite . in order to have 4-d mass eigenstates of higher kk levels , a further mixing is introduced among the left - handed doublet and right - handed singlet of each flavor @xmath73 . the mixing angle is such that @xmath74 giving mass @xmath75 , so that it is negligible for all flavors except the top @xcite . integrating over the fifth - dimension @xmath76 gives the four - dimensional lagrangian : @xmath77 which describes : ( i ) zero modes corresponding to the sm fields , ( ii ) their massive kk excitations , ( iii ) kk excitations without zero modes which do not corresponds to any field in sm . feynman rules used in the further calculation are given in ref . at quark level the decay @xmath20 is same like @xmath78 as discussed by ali _ et al_.@xcite , i.e.@xmath79 and it can be described by effective hamiltonian obtained by integrating out the top quark and @xmath80 bosons @xmath81 where @xmath82@xmath83 are four local quark operators and @xmath84 are wilson co - effeicents calculated in naive dimensional regularization ( ndr ) scheme @xcite . one can write the above hamiltonian in the following free quark decay amplitude @xmath85 \left [ \bar{\ell}\gamma ^{\mu } \ell \right ] \\ + c_{10}\left [ \bar{s}\gamma _ { \mu } lb\right ] \left [ \bar{\ell}\gamma ^{\mu } \gamma ^{5}\ell \right ] \\ -2\hat{m}_{b}c_{7}^{eff}\left [ \bar{s}i\sigma _ { \mu \nu } \frac{\hat{q}^{\nu } % } { \hat{s}}rb\right ] \left [ \bar{\ell}\gamma ^{\mu } \ell \right]% \end{array } \right\ } \nonumber \\ & & \label{2.2}\end{aligned}\ ] ] with @xmath86 , @xmath87 which is just the momentum transfer from heavy to light meson . the amplitude given in eq . ( [ 2.2 ] ) contains long distance effects encoded in the form factors and short distance effects that are hidden in wilson coefficients . these wilson coeffients have been computed at next - to - next leading order ( nnlo ) in the sm @xcite . specifically for exclusive decays , the effective coefficient @xmath88 can be written as @xmath89 where the perturbatively calculated result of @xmath90 is @xcite @xmath91 here the hat denote the normalization in term of @xmath8 meson mass . for the explicit expressions of @xmath92 s and numerical values of the wilson coefficients appearing in eq . ( [ reson - expr ] ) we refer to @xcite . in acd model the new physics comes through the wilson coefficients . buras et al . have computed the above coefficients at nlo in acd model including the effects of kk modes @xcite ; we use these results to study @xmath93 decay . as it has already been mentioned that acd model is the minimal extension of sm with only one extra dimension and it has no extra operator other than the sm , therefore , the whole contribution from all the kk states is in the wilson coefficients , i.e. now they depend on the additional acd parameter , the inverse of compactification radius @xmath2 . at large value of @xmath94 the sm phenomenology should be recovered , since the new states , being more and more massive , decoupled from the low - energy theory . our objective is to calculate the decay rate and forward - backward asymmetry for @xmath0 using the lower bound on @xmath94 provided by colangelo et al . for @xmath78 decay @xcite . in acd model , the wilson coefficients are modified and they contain the contribution from new particles which are not present in the sm and comes as an intermediate state in penguin and box diagrams . thus , these coefficients can be expressed in terms of the functions @xmath95 , @xmath96 , which generalize the corresponding smfunction @xmath97 according to : @xmath98 with @xmath99 and @xmath100 @xcite . the relevant diagrams are @xmath101 penguins , @xmath102 penguins , gluon penguins , @xmath102 magnetic penguins , chormomagnetic penguins@xmath103and the corresponding functions are @xmath104 , @xmath105 , @xmath106 , @xmath107 and @xmath108 respectively . these functions can be found in @xcite but to make the paper self contained , we collect here the formulae needed for our analysis . @xmath109 in place of @xmath110 one defines an effective coefficient @xmath111 which is renormalization scheme independent @xcite : @xmath112 where @xmath113 and @xmath114 the superscript @xmath115 stays for leading @xmath116 approximation . furthermore : @xmath117 the functions @xmath118 and @xmath119 are given be eq . ( [ wilson3 ] ) with @xmath120 @xmath121 @xmath122 @xmath123 following @xcite one gets the expressions for the sum over @xmath124@xmath125\coth ( \pi m_{w}r\sqrt{y } ) \nonumber \\ & & + \frac{(-2+x_{t})x_{t}(1 + 3x_{t})}{6(x_{t}-1)^{4}}j(r,-\frac{1}{2 } ) \nonumber \\ & & -\frac{1}{6(x_{t}-1)^{4}}% [ x_{t}(1 + 3x_{t})-(-2 + 3x_{t})(1+(-10+x_{t})x_{t})]j(r,\frac{1}{2 } ) \nonumber \\ & & + \frac{1}{6(x_{t}-1)^{4}}[(-2 + 3x_{t})(3+x_{t})-(1+(-10+x_{t})x_{t})]j(r,% \frac{3}{2 } ) \nonumber \\ & & -\frac{(3+x_{t})}{6(x_{t}-1)^{4}}j(r,\frac{5}{2 } ) ] \label{wilson8}\end{aligned}\ ] ] @xmath126 \nonumber \\ & & -\frac{x_{t}(1 + 3x_{t})}{(x_{t}-1)^{4}}j(r,-\frac{1}{2 } ) \nonumber \\ & & + \frac{1}{(x_{t}-1)^{4}}[x_{t}(1 + 3x_{t})-(1+(-10+x_{t})x_{t})]j(r,\frac{1}{% 2 } ) \nonumber \\ & & -\frac{1}{(x_{t}-1)^{4}}[(3+x_{t})-(1+(-10+x_{t})x_{t})]j(r,\frac{3}{2 } ) \nonumber \\ & & + \frac{(3+x_{t})}{(x_{t}-1)^{4}}j(r,\frac{5}{2 } ) ] \label{wilson9}\end{aligned}\ ] ] where @xmath127 . \label{wilson10}\ ] ] @xmath128 in the acd model and in the ndr scheme one has @xmath129 where @xmath130 $ ] and the last term is numerically negligible . besides @xmath131 with @xmath132 \nonumber \\ z_{0}(x_{t } ) & = & \frac{18x_{t}^{4}-163x_{t}^{3}+259x_{t}^{2}-108x_{t}}{% 144(x_{t}-1)^{3 } } \nonumber \\ & & + [ \frac{32x_{t}^{4}-38x_{t}^{3}+15x_{t}^{2}-18x_{t}}{72(x_{t}-1)^{4}}-% \frac{1}{9}]\ln x_{t } \label{wilson13}\end{aligned}\ ] ] @xmath133 \label{wilson14}\ ] ] and @xmath134 \label{wilson15}\ ] ] @xmath135 @xmath136 is @xmath137 independent and is given by @xmath138 the normalization scale is fixed to @xmath139 gev . wilson coefficients give the short distance effects where as the long distance effects involve the matrix elements of the operators in eq . ( 2.2 ) between the @xmath8 and @xmath1 mesons . using standard parameterization in terms of the form factors we have @xcite : @xmath140 \label{matrix-1 } \\ \left\langle k_{1}(k,\varepsilon ) \left| a_{\mu } \right| b(p)\right\rangle & = & \frac{2i\epsilon _ { \mu \nu \alpha \beta } } { m_{b}+m_{k_{1}}}\varepsilon ^{\ast \nu } p^{\alpha } k^{\beta } a(s ) \label{matrix2}\end{aligned}\ ] ] where @xmath141 and @xmath142 are the vector and axial vector currents respectively and @xmath143 is the polarization vector for the final state axial vector meson . the relationship between different form factors which also ensures that there is no kinematical singularity in the matrix element at @xmath144 is @xmath145 in addition to the above form factors there are also some penguin form factors which are : @xmath146 f_{2}(s ) \nonumber \\ & & + ( \varepsilon ^{\ast } \cdot q)\left [ q_{\mu } -\frac{s}{% m_{b}^{2}-m_{k_{1}}^{2}}(p+k)_{\mu } \right ] f_{3}(s ) \nonumber \\ & & \label{matrix-5 } \\ \left\langle k_{1}(k,\varepsilon ) \left| \bar{s}i\sigma _ { \mu \nu } q^{\nu } \gamma _ { 5}b\right| b(p)\right\rangle & = & -i\epsilon _ { \mu \nu \alpha \beta } \varepsilon ^{\ast \nu } p^{\alpha } k^{\beta } f_{1}(s ) \label{matrix-6}\end{aligned}\ ] ] with @xmath147 form factors are the non - perturbative quantities and are the scalar function of the square of momentum transfer . different models are used to calculate these form factors . the form factors we use here in the analysis of physical variables like decay rate and forward - backward asymmetry have been calculated using ward identities . the detailed calculation and their expressions are given in ref . @xcite and can be summarized as : @xmath148 with @xmath149 the corresponding values for @xmath150 form factors at @xmath144 are given by @xmath151 in this section we define the decay rate distribution which we shall use for the phenomenological analysis . following the notation from ref.@xcite we can write from eq . ( [ 2.2 ] ) @xmath152 \label{4.1}\ ] ] where @xmath153 the definition of different momenta involved are defined in reference10 , where the auxiliary functions are @xmath154 \nonumber \\ c\left ( \hat{s}\right ) & = & \frac{1}{\left ( 1-\hat{m}_{k_{1}}^{2}\right ) } % \left\ { c_{9}^{eff}(\hat{s})v_{2}(\hat{s})+2\hat{m}_{b}c_{7}^{eff}\left [ f_{3}(\hat{s})+\frac{1-\hat{m}_{k_{1}}^{2}}{\hat{s}}f_{2}(\hat{s})\right ] \right\ } \nonumber \\ d(\hat{s } ) & = & \frac{1}{\hat{s}}\left [ \begin{array}{c } \left ( c_{9}^{eff}(\hat{s})(1+\hat{m}_{k_{1}})v_{1}(\hat{s})-(1-\hat{m}% _ { k_{1}})v_{2}(\hat{s})-2\hat{m}_{k_{1}}v_{0}(\hat{s})\right ) \\ -2\hat{m}_{b}c_{7}^{eff}f_{3}(\hat{s})% \end{array } \right ] \nonumber \\ e(\hat{s } ) & = & -\frac{2a(\hat{s})}{1+\hat{m}_{k_{1}}}c_{10 } \nonumber \\ f(\hat{s } ) & = & \left ( 1+\hat{m}_{k_{1}}\right ) c_{10}v_{1}(\hat{s } ) \nonumber \\ g(\hat{s } ) & = & \frac{1}{1+\hat{m}_{k_{1}}}c_{10}v_{2}(\hat{s } ) \nonumber \\ h(\hat{s } ) & = & \frac{1}{\hat{s}}\left [ c_{10}(\hat{s})(1+\hat{m}% _ { k_{1}})v_{1}(\hat{s})-(1-\hat{m}_{k_{1}})v_{2}(\hat{s})-2\hat{m}% _ { k_{1}}v_{0}(\hat{s})\right ] . \label{4.4}\end{aligned}\ ] ] considering the final state lepton as muon the branching ratio for @xmath155 is calculated in ref . @xcite and its numerical value is @xmath156 the above value of branching ratio is for the case if one does not include @xmath157 in eq . ( [ reson - expr ] ) . the error in the value reflects the uncertainty from the form factors , and due to the variation of input parameters like ckm matrix elements , decay constant of @xmath8 meson and masses as defined in table i. by including @xmath159 the central value of branching ratio reduces to @xmath160it is already mentioned that in acd model there is no new opeartor beyond the sm and new physics will come only through the wilson coefficients . to see this , the differential branching ratio against @xmath161 is plotted in fig . 1 using the central values of input parameteres . one can see that their is significant enhancement in the decay rate due to kk contribution for @xmath162 gev whereas the value is shifted towards the sm at large value of @xmath163 . the enhancemen is prominent in the low value of @xmath161 but such effects are obscured by the uncertainites involved in different parameters like , form factors , ckm matrix elements etc . the numerical value at these two different values of @xmath94 is @xmath164the effects of ued becomes more clear if we look for the fb asymmetry in the dilepton angular distribution because it depends upon the wilson coefficients . it is known that in sm , due to the opposite sign of the @xmath165 and @xmath166 , the forward - backward asymmetry passes from its zero position and has very weak dependence on form factors and uncertainities in input parameteres . the differential forward - backward asymmetry for @xmath167 reads as follows@xcite @xmath168 \label{4.6}\]]where @xmath169the variable @xmath170 corresponds to @xmath171 , the angle between the momentum of the @xmath8 meson and the positively charged lepton in the dilepton c.m . system frame . in sm the zero - point of forward - backward asymmetry for @xmath155 is calculated by paracha et al . 24 and it lies at @xmath172 @xmath173 . they have shown that the due to the uncertainities in the form factors zero position of forward - backward asymmetry @xmath174 deviate slightly from the central value in the low @xmath83 region where as in the large @xmath83 region these deviations are highly suppressed and zero of the forward - backward asymmetry became insensitive to these uncertainities and therefore we do not include them while analysing the above decay in ued model . to see the new physics effects due to extra dimension , the differential forward - backward asymmetry with @xmath161 is plotted in fig . it can be seen that the zero position of forward - backward asymmetry @xmath174 shifts towards the left in acd model with single universal extra dimension and this shifting is more clear for @xmath175 gev . in future , when we have some data on these decays , this sensitivity of the zero position to the compactification parameter , will be used to constrain @xmath176 the method of calculation of the form factors for @xmath0 decay descried in @xcite can be straighforwardly used to calculate the form factors for @xmath78 . now after calculating these form factors for @xmath177 we have plotted the forward - backward asymmetry with @xmath161 in fig . we believe that it provides a useful comparsion , if one compares the effect of our form factors to the zero of @xmath174 with the others like @xcite and references therein . again the zero of @xmath174 is shifted towards the left in acd model with single universal extra dimension and this shifting is more clear for @xmath175 gev . this paper deals with the study of semileptonic decay @xmath178 in acd model with single universal extra dimension which is strong contender to study physics beyond sm and has received a lot of interest in the literature . we studied the dependence of the physical observables like decay rate and zero position of forward - backward asymmetry on the inverse of compactification radious @xmath94 . the value of the branching ratio is found larger then the corresponding sm value . the zero postion of the fb asymmetry is very sensitive to @xmath94 and it is seen that it shifts significantly to the left . the shifting is large at @xmath175 gev and approaches to the sm value if we increase the value of @xmath94 . the future experiments , where mora data is expected , will put stringent constraints on the compactification radius and also give us some deep understanding of @xmath8-physics . t. m. aliev , m. k. cakmak and m. savci , nucl . b 607 , 305 ( 2001 ) [ arxiv : hep - ph/0009133 ] ; 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phys . rev . * d65 * , 074004 ( 2002 ) ; phys . rev . * d66 * , 034009 ( 2002 ) ; h. m. asatrian , k. bieri , c. greub and a. hovhannisyan , phys . rev . * d66 * , 094013 ( 2002 ) ; a. ghinculov , t. hurth , g. isidori and y. p. yao , nucl . phys . * b648 * , 254 ( 2003 ) ; a. ghinculov , t. hurth , g. isidori and y. p. yao , nucl . phys . * b685 * , 351 ( 2004 ) ; c. bobeth , p. gambino , m. gorbahn and u. haisch , jhep * 0404 * , 071 ( 2004 ) . 1 ) : the differential branching ratio as a function of @xmath161 is plotted using the form factors defined in eq . ( [ form - factors ] ) . the solid line denotes the sm result , dashed - dotted line is for @xmath175 gev and dashed line is for @xmath179 gev . all the input parameters are taken at their central values . 2 ) : the differential forward - backward ( fb ) asymmetry as a function of @xmath180 is plotted using the form factors defined in eq . ( [ form - factors ] ) . the solid line denotes the sm result , dashed - dotted line is for @xmath175 gev and dashed line is for @xmath179 gev . all the input parameters are taken at their central values . 3 ) : the differential forward - backward ( fb ) asymmetry for @xmath181 as a function of @xmath161 is plotted using the form factors defined in eq . ( [ form - factors ] ) with obvious replacements for @xmath182 . the solid line denotes the sm result , dashed line is for @xmath162 gev and long - dashed line is for @xmath179 gev . all the input parameters are taken at their central values . 4 ) : comparision of the differential forward - backward ( fb ) asymmetry for @xmath23 in standard model ( sm ) as a function of @xmath180 is plotted using the form factors defined in eq.([form - factors ] ) vs form factors given in colangelo @xmath24 @xcite . the solid line denotes the colangelo result and dashed line denotes our result .
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decay rate and forward - backward asymmetries in @xmath0 , @xmath1 is the axial vector meson , are calculated in the universal extra dimension ( ued ) model .
the dependence of these physical quantities on the compactification radius @xmath2 , the only unknown paramter in ued model , is studied and it is shown that zero of forward - backward asymmetry is sensitive to the ued model , therefore they can be very useful tool to establish new physics predicted by the ued model .
this work is briefly extended to @xmath3
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the nature of galactic formaldehyde ( h@xmath0co ) masers is a growing mystery . while hundreds of galactic oh , h@xmath0o , and ch@xmath6oh masers are known , only five galactic star - forming regions have associated h@xmath0co maser emission . to date , this emission is seen only in the @xmath7 transition at 6 cm wavelength . shortly after the discovery of the first h@xmath0co maser in ngc 7538 ( downes & wilson 1974 ; forster et al . 1980 ) , a radiative pumping model was proposed ( boland and de jong 1981 ) . the h@xmath0co masers discovered subsequently did not meet the conditions required for this mechanism ( gardner et al . 1986 ; mehringer , goss , & palmer 1994 , hereafter mgp94 ; hoffman et al . 2003 ; hereafter h03 ) . thus , twenty - five years after the discovery of the first h@xmath0co maser , these sources remain rare and the excitation mechanism remains unknown . sgr b2 , the northernmost component of the extended sgr b radio source , is located within a few hundred pc of the galactic center ( reid et al . ( the distance to the galactic center is assumed to be 8.5 kpc in this paper . ) sgr b2 is comprised of three main star - forming complexes designated north ( n ) , middle or main ( m ) , and south ( s ) , and many smaller hii regions . the h@xmath0co masers occur throughout sgr b2 , shown in figure [ fig1 ] . the heating mechanisms and complex chemistry of the region are subjects of ongoing study ( e.g. , gaume & claussen 1990 ; goicoechea et al . 2004 ) . sgr b2 contains nine individual h@xmath0co maser regions , several of which have multiple velocity components . all of the masers are unresolved at 1 angular resolution , except for maser c which mgp94 suggest consists of several masers blended within the beam . these regions are near hii regions distributed over the @xmath83.6 arcmin@xmath9 complex ( mgp94 ) . whiteoak and gardner ( 1983 ) and mgp94 designated the maser regions with letters ( fig . [ fig1 ] ) . the h@xmath0co masers are observed over the velocity range @xmath10 km s@xmath11@xmath12 km s@xmath11 , while other species , such as h@xmath0o masers , are observed over a larger range @xmath13 km s@xmath11@xmath14 km s@xmath11 ( kobayashi et al . 1989 ; mcgrath , de pree , and goss 2004 ) . of the nine maser regions observed in sgr b2 by mgp94 , the g maser was shown to be time variable , at least quadrupling in intensity over 10 yr . ( similarly , h03 found one ngc 7538 feature to triple in intensity over @xmath15 yr . ) as initially noted by whiteoak and gardner ( 1983 ) , all of the sgr b2 h@xmath0co masers lie close to oh , h@xmath0o , ch@xmath6oh , and nh@xmath6 masers . for most of the masers in mgp94 , the separation to an oh maser was less than 0.05 pc . recent successes in search techniques for new masers ( araya et al . 2004 , 2005 , 2006a ) and in high - resolution observational techniques ( h03 ) promise to provide empirical constraints for the development of a realistic model for the galactic h@xmath0co maser emission . the necessary steps in compiling an empirical picture of the h@xmath0co emission in sgr b2 are ( 1 ) detailed imaging of the masers in order to quantify the intrinsic properties of the emission ( e.g. , brightness temperature ) , ( 2 ) assessment of intensity variability in the masers , and ( 3 ) precise astrometry for elucidating spatial relationships between the h@xmath0co masers and more common masers ( oh , h@xmath0o , ch@xmath6oh ) . in this paper , we present new observations of the h@xmath0co masers in sgr b2 using the the very long baseline array ( vlba ) and very large array ( vla ) of the nrao . we observed the a and d h@xmath0co masers in sgr b2 using the ten antennas of the vlba and the 27 antennas of the vla as an 11-station vlbi array . parameters of the observations are summarized in table [ tbl - tab1 ] . the total observing time was approximately 8.0 hours alternating between the a and d pointing positions which are separated by approximately 1 . at each pointing position there is a useful correlated field of view of approximately 1 ( e.g. , bridle & schwab 1999 ) . to observe each of the nine h@xmath0co masers in sgr b2 optimally would have required observations at nine pointing centers . because of signal - to - noise considerations , we observed only the two masers measured to be most intense by mgp94 . the baseline lengths of the vlba+y27 array range from 52 km to 8611 km ; the array is not sensitive to angular scales larger than 025 . the antennas have right- ( r ) and left- ( l ) circularly polarized feeds from which rr , ll , rl , & lr cross - correlations were formed . the visibilities were integrated for 8.4 s. the amplitude scale is set using online system temperature monitoring and _ a priori _ antenna gain measurements . the station delays were determined from observations of j1733 - 130 . the maser observations were phase referenced to j1745 - 283 ( called w56 in bower et al . because the properties of this source limit our observations , we discuss it in some detail . the absolute position uncertainty of this source is 12 milliarcseconds ( hereafter : mas ) ( reid et al . 1999 ) . bower et al . find that j1745 - 283 is probably the core of an extragalactic jet source and that its apparent size at 5 ghz ( 30 mas ) is determined by scatter - broadening . we also observed j1745 - 283 on 22 november 2002 using only the ten vlba stations ( a snapshot observation with a bandwidth of 4 mhz in both polarizations ) . the 2002 observation yields a deconvolved angular size of @xmath16 mas and an integrated flux density of 28 mjy ; the 2003 observation yields a size of @xmath17 mas and a flux density of 31 mjy . ( the flux density determined by y27 [ the vla alone ] during the 2003 observations was 161 mjy . ) bower et al . report an angular size @xmath18 mas and peak intensity 89 mjy beam@xmath11 from vlba+y1 observations . therefore , the flux density must have decreased significantly between 1999 and 2002 . such variability would not be unusual for an inverted spectrum source like j1745 - 283 ( e.g. , urry & padovani 1995 ) . our imaging of j1745 - 283 with the vlba makes use of only the inner - most stations ( vla , pie town , los alamos , fort davis , kitt peak , and owens valley ) because the source is resolved on longer baselines . some additional resolution for the maser observations was gained by self - calibration ; but , because only the shorter baselines in the vlba data were absolutely calibrated in phase , the position registration accuracy of the resulting vlba images is approximately 15 mas . in summary , we could not make use of the full potential resolution of the vlba when using this phase referencing calibrator , and no other suitable nearby source is known . we detected masers toward the a and d regions . the f maser is also within the correlated field of view near the a maser , but was not detected with the current sensitivity . the image and spectrum of the a maser from the vlbi data is shown in figure [ vlba - a ] . because the radio continuum of sgr b2 is fully resolved by the vlba , no continuum subtraction was necessary . the positions , center velocities ( @xmath19 ) , linewidths ( @xmath20 ) , peak flux densities ( @xmath21 ) , and deconvolved major and minor axes and position angles are summarized in table [ tbl - vlba ] . in parentheses following each entry are the 1-@xmath22 errors . the deconvolved sizes of the vlbi images of the a and d masers are @xmath23 mas . at the distance of sgr b2 , this size corresponds to a linear diameter of approximately 80 au , comparable to the sizes ( 30 to 130 au ) observed for the h@xmath0co masers in ngc 7538 and g29.96 - 0.02 using the vlba ( h03 ) . the brightness temperatures of the sgr b2 a and d masers in the current vlbi data are @xmath24 k and @xmath25 k , respectively , similar to the @xmath26 k observed for other h@xmath0co masers ( h03 ) . no significant linear or circular polarization is detected in either maser ( @xmath27 20% for the strongest maser [ a]).gipsy/ ) and the astronomical image processing system ( aips ) software package ( http://www.nrao.edu/aips/ ) . ] because of the known variability of h@xmath0co masers ( forster et al . 1985 ; h03 ) , the sgr b2 masers were observed in 2005 with the vla in order to access any possible time - variability . parameters of the observations are summarized in table [ tbl - tab1 ] . the array configuration available was dnc for an observation period of approximately 7 hours . two pairs of rr and ll bands were recorded centered at the expected velocities of the a and the d masers . of the nine maser regions described by mgp94 , eight of the sources were detected . six velocity components in the regions observed by mgp94 lie outside the velocity range of the current observations . no new h@xmath0co maser regions were discovered . the 2005 data have inferior angular resolution ( 10 versus 1 ) but improved spectral resolution ( 0.19 km s@xmath11 versus 1.5 km s@xmath11 ) compared with the 1993 vla observations of mgp94 . comparison of the current data to the mgp94 data is uncertain due to ( 1 ) the severe blending of the strong h@xmath0co absorption with the nearby masers and ( 2 ) the insufficient velocity resolution of the mgp94 data , which did not spectrally resolve the lines . variability of the flux density of the masers is apparent and is discussed in [ disc - vary ] . the 2005 results are summarized in table [ tbl - vla ] and a spectrum from these data ( region e ) is shown in figure [ vla - e ] . the c maser region is not discussed in this paper due to confusion of the maser spectra with nearby absorption and continuum emission . with the current 10 angular resolution of the vla data , as with the 1 resolution of mgp94 , none of the masers are spatially resolved . in [ disc - vary ] , we also compare the current data with the 1983 vla observations of gardner et al . the velocity resolution of the 1983 data is 0.76 km s@xmath11 and the angular resolution is 125 . although we resolve many of the maser line profiles in velocity for the first time with the 2005 vla data , most of the linewidths presented in table [ tbl - vla ] agree with the values measured by gardner et al . ( 1986 ) . no circular polarization is detected in any of the h@xmath0co masers . this corresponds to a 3-@xmath22 upper limit of approximately 4% circular polarization for the strongest maser ( a ) . images of radio sources are angularly broadened by scattering by the ionized component of the interstellar medium . in the direction of the galactic center , this problem becomes severe ( e.g. , lazio & cordes 1998 ) . as discussed in 2.1 , j1745 - 283 , is an extragalactic point - like source whose image is broadened to @xmath830 mas at 6 cm wavelength by scattering ( bower et al . 2001 ) . in this section we discuss the extent to which this scattering medium affects the current vlbi observations of the h@xmath0co masers in sgr b2 . the observed size for j1745 - 283 , @xmath17 mas , is larger than the observed deconvolved size for the a maser , @xmath28 mas . this result may be expected even if both sources are intrinsically point - like because the proximity of the maser to the scattering medium results in reduced angular broadening ( e.g. , rickett 1990 ) . nevertheless , as discussed below , we expect significant angular broadening in the images of the masers . therefore , the deconvolved angular sizes in table [ tbl - vlba ] are upper limits to the intrinsic sizes and the brightness temperatures are lower limits . gwinn et al . ( 1988 ) quantified the scattered sizes of h@xmath0o and oh masers in sgr b2 , finding a @xmath29 dependence for the broadening . from their determinations of minimum angular sizes of 0.3 mas for 1.35-cm h@xmath0o masers and 100 mas for 18-cm oh masers , we expect a minimum apparent size of approximately 9 mas for the 6.2-cm h@xmath0co masers . the deconvolved angular sizes of the masers in table [ tbl - vlba ] are in agreement with this expectation . in comparing the current vla observations with earlier vla data , both the differing angular resolution and spectral resolution must be considered . as discussed in [ vla - obs ] , both the differences in array configuration and in correlator setup were significant . to compensate for the difference in velocity resolution , in table [ tbl - fd ] we tabulate the velocity - integrated flux density for observations made in 1983 ( gardner et al . 1986 ) , 1993 ( mgp94 ) , and 2003 and 2005 ( this paper ) . however , it is impossible to compensate for emission outside of the velocity range covered in the current observations and for the confusion of features caused by the lower angular resolution . ( the notes in the final column of table [ tbl - vla ] summarize limitations of the current data from these causes . ) the error values in table [ tbl - fd ] are dominated by systematic uncertainties rather than by thermal noise in most cases . we find that the a maser has approximately doubled in velocity - integrated flux density since 1993 , while b , g , and h increased between 1983 and 1993 , and subsequently decreased . the velocity - integrated flux densities from regions d , e , f , and i have not changed significantly since 1993 . for the a and d h@xmath0co masers in sgr b2 , a comparison of the current vlba+y27 data with past and current vla data shows two major differences : ( 1 ) only a fraction of the flux density seen with the vla is observed in the vlbi data ( @xmath870% for the a maser ; @xmath840% for the d maser ) ; and ( 2 ) the velocity widths in the vlbi data are more narrow than in the vla data ( @xmath880% for the a maser ; @xmath850% for the d maser ) . in this section we discuss a schematic model for the structure of the emission which addresses both the observed angular distribution and velocity widths . the flux density detected by the vla but resolved by vlbi baselines must lie at angular scales between @xmath30 mas ( the resolution of the shortest vlba+y27 interferometer spacings ) and @xmath31 mas ( the masers are unresolved in the mgp94 observations with 1 resolution ) , while the flux density observed with the vlbi is emitted by a source @xmath810 mas . these limits suggest a core - halo source morphology . a similar morphology was proposed by h03 for the h@xmath0co masers in ngc 7538 and g29.96 - 0.02 . a model consisting of two coincident circular gaussian components with different angular sizes can reproduce the observed results . for the a maser , the flux density observed with the vla is about 2000 mjy . to determine the decrease in flux density with increasing projected baseline length , we use the highest signal - to - noise baselines only : those between y27 and other antennas . the average flux densities observed on baselines with y27 are : 1400 mjy at 0.71 m@xmath32 ( to pie town , average projected spacing @xmath844 km ) denotes million wavelengths ] ; 1300 mjy at 3 m@xmath32 ; and 1200 mjy at 5 m@xmath32 . these data are fit by a two component gaussian model with a 600 mjy in a 300 mas halo and 1400 mjy in a 10 mas core . the brightness temperature of the halo is @xmath33 k ; for the 10 mas core component , @xmath34 k ( table [ tbl - vlba ] , fig . [ vlba - a ] ) . a more precise model is justified only after higher signal to noise measurements . it is reasonable to assume that the background radiation is relatively uniform on 300 mas angular scales because the continuum emission is well resolved on the vlba baselines . therefore , the difference in brightness temperatures between the core and halo components may be attributed to differences in maser gain . we estimate that the halo has a gain of approximately 10 ( in the exponential amplification regime ) and the core has a gain of approximately 2000 ( in the saturated regime ) . because the observed angular sizes are upper limits for the intrinsic sizes , the derived brightness temperatures are lower limits for the intrinsic brightness temperatures ; therefore , the actual gains may exceed the above values . the gain of the maser medium is dependent upon four factors : the pathlength , the density of h@xmath0co , the level inversion , and the velocity coherence ( see 4.2 of h03 ) . therefore , one or more of these factors must be significantly different between the core and halo components . we suggest that the narrower line widths arising from the core component indicate that the velocity coherence is enhanced in the core region compared to the halo , yielding a higher gain . similar arguments may be applied to the d maser , but the resulting ranges of size and @xmath35 are not well constrained because of the higher noise level on the vla - pie town baseline . additional maser species in other star - forming regions excited oh ( palmer , goss , & devine 2003 ) , oh in supernova remnants ( hoffman et al.2005 ) , as well as other h@xmath0co masers ( h03 ) exhibit a similar narrowing of line widths between vla and vlba angular scales . h@xmath0co absorption toward sgr b2 has been studied extensively with 10 @xmath36 20 resolution both by martn - pintado et al . ( 1990 ) and mehringer et al . martn - pintado et al . ( 1990 ) noted that the absorption is dominated by three velocity components ( @xmath19= 55 , 64 , and 80 km s@xmath11 ) , each with @xmath37 . linewidths of these absorption features ranged from 9 26 km s@xmath11 , i.e. much greater than that of the maser features ( @xmath38 km s@xmath11 ; see table [ tbl - vla ] ) . mehringer et al . ( 1995 ) provide optical depth profiles toward selected positions . all but maser e lies within 30 of one of these positions displayed . ( maser e lies at a position with no radio continuum . ) it is striking that the maser velocities , except for maser i ( which is more than a beamwidth away from the position of a displayed profile ) , do not occur in velocity ranges with large h@xmath0co optical depths . masers a and f occur in a velocity range with @xmath39=1 , but this velocity range is a rather sharp minimum in the optical depth profile at this position . therefore , we conclude that the gas containing the h@xmath0co masers is distinct from the bulk of the h@xmath0co containing gas observed in absorption . insight into why the maser containing gas may be distinct is provided by ( 3,3 ) and ( 4,4 ) nh@xmath6 observations made with the vla by martn - pintado et al . ( 1999 ) . with their 3 resolution , 80 - 90% of the nh@xmath6 emission is resolved out , and only small angular scale features remain . among these features are a number of rings and arcs , most naturally interpreted as complete or partial shells with linear sizes @xmath82 pc . the shells are hot ( t@xmath40 50 70 k ) , and the h@xmath0 densities derived for them are typically a factor of 10 greater than those derived from the h@xmath0co absorption studies of martn - pintado et al . h@xmath0co masers c , d , e , g , and h fall within the field imaged by martn - pintado et al . as these authors note , all of the h@xmath0co masers occur at positions on hot nh@xmath6 shells . the higher temperature ( t@xmath41 100 k ) region of apparent interaction between shells a and b contains the closely spaced d , g , and h masers . martn - pintado et al . ( 1999 ) propose that the pumping mechanism for the h@xmath0co masers as well as that for the nh@xmath6 and class ii ch@xmath6oh masers must depend on the physical conditions in the hot shells . in the search for additional constraints on the h@xmath0co emission environment , we examine the possible association of h@xmath0co masers with other molecular emission for which physical conditions are better understood . in table [ tbl - others ] we present a summary of possible associations with h@xmath0o masers observed by mcgrath , goss , and de pree ( 2004 ) , ch@xmath6oh masers observed by houghton and whiteoak ( 1995 ) , and oh masers observed by argon , reid , and menten ( 2000 ) . the excitation conditions for the different species are mutually exclusive , but if the masers exist in a shocked region as proposed by martn - pintado et al . ( 1999 ) for sgr b2 ( and for ngc 7538 by h03 ) , a rapid change in densities , temperatures , and velocity fields over a small linear distance is to be expected . of the nh@xmath6 masers in sgr b2 with interferometically determined positions , only masers m1 and m6 lie similarly near an h@xmath0co maser ( e ) , and both differ in velocity by @xmath4210 km s@xmath11 from the h@xmath0co maser . the existence of h@xmath0o and ch@xmath6oh masers near the h@xmath0co f maser is significant because previously the f maser had no known maser or continuum associations ( mgp94 ) . the h@xmath0co maser in ngc 7538 ( h03 ) and many of those reported in section 3.2 varied in intensity with timescales from years to decades . h03 noted a common variability of some of the h@xmath0co and h@xmath0o masers in ngc 7538 . this was further confirmed by long - term observations by lekht et al.(2003 , 2004b ) . similarly in sgr b2 , the long - term monitoring of the h@xmath0o masers presented by lekht et al . ( 2004a , 2004c ) may indicate common variability of h@xmath0o masers with the h@xmath0co a maser . however , variablity of an h@xmath0co maser in iras 18566 + 0408 was recently seen to occur on much more rapid timescale ( araya et al . we present vlba+y27 images of the sgr b2 a and d h@xmath0co masers . the measured sizes ( @xmath4 au ) and brightness temperatures ( @xmath5 k ) are comparable to those found in other vlbi studies of h@xmath0co masers . however , about half of the flux density from these regions is resolved out with the vlba data . a comparison between vla and vlba observations shows that the missing flux density exhibits a broader linewidth than the emission from the compact vlbi source . we demonstrate quantitatively the applicability of a core - halo model for these masers . h@xmath0o and ch@xmath6oh masers discovered near the h@xmath0co masers may indicate associations among the species , suggestive of a related origin . the association of h@xmath0co masers in sgr b2 with hot nh@xmath6 shells proposed by martn - pintado et al . ( 1999 ) , together with the arguments for shock excitation of the maser region in ngc 7538 in h03 , provide an encouraging stepping stone toward a solution of the problem of the excitation of these poorly understood masers . we are indebted to an anonymous referee for a very careful reading of this paper and for directing our attention to two very important references . araya , e. , hofner , p. , linz , h. , sewilo , m. , watson , c. , churchwell , e. , olmi , l. , & kurtz , s. 2004 , , 154 , 579 araya , e. , hofner , p. , kurtz , s. , linz , h. , olmi , l. , sewilo , m. , watson , c. , churchwell , e. 2005 , , 618 , 339 araya , e. , hofner , p. , goss , w. m. , kurtz , s. , linz , h. , & olmi , l. 2006a , , 643 , l33 araya , e. , hofner , p. , sewilo , m. , linz , h. , kurtz , s. , olmi , l. , watson , c. , & churchwell , e. 2006b , ( in press ) argon , a. l. , reid , m. j. , & menten , k. m. 2000 , , 129 , 159 boland , w. & de jong , t. 1981 , , 98 , 149 bower , g. c. , backer , d. c. , sramek , r. a. 2001 , , 558 , 127 bridle , a. h. & schwab , f. r. 1999 , in synthesis imaging in radio astronomy ii . edited by g. b. taylor , c. l. carilli , and r. a. perley . asp conference series , 180 , 371 downes , d. & wilson , t. l. 1974 , , 191 , l77 forster , j. r. , goss , w. m. , wilson , t. l. , downes , d. , & dickel , h. r. 1980 , , 84 , l1 forster , j. r. , goss , w. m. , gardner , f. f. , & stewart , r. t. 1985 , , 216 , 35p gardner , f. f. , whiteoak , j. b. , forster , j. r. , & pankonin , v. , 1986 , , 218 , 385 gaume , r. a. & claussen , m. j. 1990 , , 351 , 538 goicoechea , j. r. , rodrguez - fernndez , n. j. , cernicharo , j. 2004 , , 600 , 214 gwinn , c. r. , moran , j. m. , reid , m. j. , schneps , m. h. 1988 , , 330 , 817 hoffman , i. m. , goss , w. m. , palmer , p. , & richards , a. m. s. 2003 , , 598 , 1061 ( h03 ) hoffman , i. m. , goss , w. m. , brogan , c. l. , claussen , m. j. 2005 , , 620 , 257 houghton , s. & whiteoak , j. b. 1995 , , 273 , 1033 kobayashi , h. , ishiguro , m. , chikada , y. , ukita , n. , morita , k .- i . , okumura , s. k. , kasuga , t. , & kawabe , r. 1989 , , 41 , 141 lazio , t. j. w. & cordes , j. m. 1998 , , 505 , 715 lekht , e. e. , munitsyn , v. a. , & tolmachev , a. m. 2003 , astronomy reports , 47 , 838 lekht , e. e. , ramrez , o. , tolmachev , a. m. , & berulis , i. i. 2004 , astronomy reports , 48 , 171 lekht , e. e. , munitsyn , v. a. , & tolmachev , a. m. 2004 , astronomy reports , 48 , 200 lekht , e. e. , ramrez , o. , & tolmachev , a. m. 2004 , astronomy reports , 48 , 965 martn - pintado , j. , de vicente , p. , wilson , t. l. , & johnston , k. j. 1990 , , 236 , 193 martn - pintado , j. , gaume , r. a. , rodrguez - fernndez , n. , de vicente , p. , & wilson , t. l. 1999 , , 519 , 667 mcgrath , e. j. , goss , w. m. , & de pree , c. g. 2004 , , 155 , 577 mehringer , d. m. , goss , w. m. , & palmer , p. 1994 , , 434 , 237 ( mgp94 ) mehringer , d. m. , palmer , p. , & goss , w. m. 1995 , , 97 , 497 palmer , p. , goss , w. m. , & devine , k. e. 2003 , , 599 , 324 reid , m. j. , schnepps , m. h. , moran , j. m. , gwinn , c. r. , genzel , r. , downes , d. , & rnnng , b. 1988 , , 330 , 809 reid , m. j. , readhead , a. c. s. , vermeulen , r. c. , & treuhaft , r. n. 1999 , , 524 , 816 rickett , b. j. 1990 , , 28 , 561 urry , c. m. , padovani , p. 1995 , , 107 , 803 whiteoak , j. b. & gardner , f. f. 1983 , , 205 , 27p l c c observing date(s ) & 2003 may 17 & 22 & 2005 october 11 , 13 , & 14 + position ( j2000.0 ) & 17 47 20.0463 , -28 23 46.587 & 17 47 19.96 , -28 22 59.8 + & 17 47 19.8562 , -28 22 12.900 & + synthesized beam & ( a ) @xmath43 mas , p.a.= @xmath44 & @xmath45 , p.a.= @xmath46 + & ( d ) @xmath47 mas , p.a.= @xmath44 & + flux density calibrator & & 3c286 + phase calibrator & j1745 - 283 & j1751 - 253 + bandpass calibrator & j1733 - 130 & j1733 - 130 + rest frequency & 4829.6569 mhz & 4829.6590 mhz + number channels & 128 & 63 + channel spacing & 1.953 khz & 3.052 khz + velocity resolution & 0.24 km s@xmath11 & 0.19 km s@xmath11 + center velocity & 50.6 km s@xmath11 & 75.6 km s@xmath11 & 51.4 km s@xmath11 & 74.6 km s@xmath11 + total velocity range & @xmath487.5 km s@xmath11 & @xmath485.5 km s@xmath11 + typical noise per channel & 35 mjy beam@xmath11 & 5 mjy beam@xmath11 + c r@ r@ [email protected] r@ r@ [email protected] c c c c c c a & 17&47&19&856(1 ) & @xmath4928&22&12&99(2 ) & 75.33(1 ) & 0.59(2 ) & 645(15 ) & 13(3 ) & 5(2 ) & 110(20 ) + d & & & 20&047(2 ) & & 23&46&59(2 ) & 50.07(2 ) & 0.36(5 ) & 160(20 ) & 10(4 ) & 8(4 ) & 5(40 ) + c r@ r@ [email protected] r@ r@ [email protected] c c c c a & 17&47&19&94 & @xmath4928&22&13&0 & 75.31(7 ) & 0.71(2 ) & 1900 & u + b & & & 20&04 & & 22&40&6 & 51.0(2 ) & 0.4(2 ) & 80 & u + d & & & 20&01 & & 23&47&2 & 50.1(1 ) & 0.75(5 ) & 510 & + & & & & & & & 53.9(1 ) & 0.88(9 ) & 320 & + e & & & 18&64 & & 24&24&5 & 49.1(1 ) & 0.94(4 ) & 230 & f + & & & & & & & 51.5(1 ) & 0.60(6 ) & 120 & f + f & & & 19&61 & & 22&13&5 & 76.3(2 ) & 0.3(1 ) & 40 & u + g & & & 19&57 & & 23&49&9 & 48.7(3 ) & 0.8(2 ) & 120 & u + h & & & 20&43 & & 23&46&7 & 70.6(3 ) & 0.3(2 ) & 50 & u + & & & & & & & 73.1(1 ) & 0.9(5 ) & 50 & u + & & & & & & & 74.6(1 ) & 0.8(3 ) & 70 & u , n + i & & & 24&72 & & 21&43&0 & 70.9(1 ) & 0.92(6 ) & 60 & + c c c c c a & 720@xmath4815 & 850@xmath4815 & 640@xmath4850 & 1900@xmath48400 + b & 80@xmath4815 & 150@xmath4810 & & 20@xmath4810 + d & 700@xmath4825 & 1400@xmath4830 & 55@xmath4820 & 1200@xmath48300 + e & 230@xmath4880 & 350@xmath4820 & & 330@xmath4830 + f & @xmath50 & 70@xmath4815 & & 40@xmath51 + g & @xmath52 & 320@xmath4815 & & 75@xmath53 + h & 140@xmath4860 & 350@xmath4815 & & 35@xmath4820 + i & @xmath50 & 100@xmath4815 & & 75@xmath4820 + c c c c a & 0.6 & 5000 & 0.1 + & 0.1 & 850 & 18.8 + f & 0.2 & 1700 & 9.9 + h & 0.7 & 5500 & 6.6 + & 0.7 & 5500 & 11.9 + b & 0.7 & 6000 & + d & 0.8 & 7400 & + e & 0.5 & 4200 & + f & 4.5 & 38000 & + i & 0.5 & 4200 & + a & 0.6 & 5000 & 2.5 + b & 0.6 & 5000 & 1.2 + c & 0.15 & 130 & 0.2 + & 0.05 & 40 & 0.2 + & 0.06 & 50 & 0.2 + & 0.18 & 150 & 0.2 + d & 0.07 & 60 & 0.9 + & 0.10 & 850 & 1.1 + h & 0.21 & 180 & 0.1 + & 0.24 & 200 & 2.0 +
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observations of two of the formaldehyde ( h@xmath0co ) masers ( a and d ) in sgr b2 using the vlba+y27 ( resolution @xmath1 001 ) and the vla ( resolution @xmath2 ) are presented .
the vlba observations show compact sources ( @xmath3 milliarcseconds , @xmath4 au ) with brightness temperatures @xmath5 k. the maser sources are partially resolved in the vlba observations .
the flux densities in the vlba observations are about 1/2 those of the vla ; and , the linewidths are about 2/3 of the vla values . the applicability of a core - halo model for the emission distribution is demonstrated .
comparison with earlier h@xmath0co absorption observations and with ammonia ( nh@xmath6 )
observations suggests that h@xmath0co masers form in shocked gas .
comparison of the integrated flux densities in current vla observations with those in previous observations indicates that ( 1 ) most of the masers have varied in the past 20 years , and ( 2 ) intensity variations are typically less than a factor of two compared to the 20-year mean .
no significant linear or circular polarization is detected with either instrument .
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low - dimensional strongly correlated electron systems have attracted great attention in the last two decades . the reason dates back to anderson s proposal @xcite that the @xmath5 version of the hubbard model might carry the basic mechanisms underlying the high - tc superconductivity observed in cuo@xmath0 compounds . despite that this remains an open issue , the above suggestion fertilized intensive investigations on many related fundamental topics , such us itinerant electron magnetism , mott metal - insulator transitions and quantum critical phenomena . amongst several features of interest , we mention the possibility of realization of spiral @xcite , nagaoka @xcite and resonating - valence - bond ( rvb ) states @xcite , spatially separated phases @xcite and luttinger liquid behavior @xcite , which may present strong deviations from the landau fermi liquid theory . in this work , we report numerical results of the hubbard model on the doped ab@xmath0 chain away from half filling , which show that its special unit cell topology greatly enriches the phase diagram found in the doped standard linear chain . in fact , all features mentioned above are shown to be associated with well defined ground state ( gs ) phases of this doped chain . doped ab@xmath0-hubbard chains were previously studied through hartree - fock , quantum monte carlo and exact diagonalization ( ed ) techniques both in the weak and strong coupling limits @xcite , including also the @xmath5 model @xcite using density matrix renormalization group ( dmrg ) and recurrent variational anstzes , and the infinite - u limit @xcite using ed . in particular , these chains represent an alternative route to reaching two - dimensional quantum physics from one - dimensional systems @xcite . at half filling the ab@xmath0-hubbard chain exhibits a quantum ferrimagnetic gs @xcite , whose magnetic excitations have been studied in detail both in the weak and strong coupling limits @xcite , and in the light of the quantum heisenberg model @xcite . further studies have considered the anisotropic @xcite and isotropic @xcite critical behavior of the ab@xmath0-quantum - heisenberg model , including its spherical version @xcite , and the statistical mechanics of the ab@xmath0-classical - heisenberg model @xcite . on the experimental side , the ab@xmath0 chain topology is of relevance to the understanding of the physics of some low - dimensional strongly correlated electronic systems . one class is the line of trimer clusters present in fosfates with formula a@xmath6cu@xmath6(po@xmath7)@xmath7 , where a = ca @xcite , sr @xcite and pb @xcite . the trimers have three cu@xmath8 ( @xmath9 ) paramagnetic ions antiferromagnetically coupled . although the superexchange intertrimer interaction is much weaker than the intratrimer coupling , it proves sufficient to turn them bulk ferrimagnets . another quasi - one - dimensional inorganic material closely associated with the ferrimagnetic phase of the ab@xmath0 chain is the nicu bimetallic chain @xcite . these compounds display alternating ni@xmath8 ( @xmath10 ) and cu@xmath8 ( @xmath9 ) ions connected through suitable ligands in a line ; and are modeled by the alternating spin-@xmath11/spin-1 antiferromagnetic heisenberg chain @xcite . we also would like to mention a more recently synthesized organic ferrimagnetic compound consisting of three @xmath9 paramagnetic radicals @xcite in its magnetic unit cell , as well as possible connections with the physics of the oxocuprates @xcite . this paper is organized as follows : in sec . ii we introduce the model system and the numerical techniques used to calculate several quantities suitable to characterize the occurrence of distinct phases as function of doping and coulomb coupling . in sec . iii , we discuss spiral and nagaoka states at low hole doping , whose magnetic properties are shown to exhibit very interesting features in the weak and infinite - u limit , respectively . in sec . iv we show that for higher hole doping the system phase separates , before reaching a mott insulating phase of short - range rvb states at @xmath2 . in sec . v we discuss several features of the crossover region , which takes place before the luttinger liquid behavior observed for @xmath12 . finally , in sec . vi we present a summary and some conclusions concerning the reported results . the ab@xmath0 chain is a bipartite lattice with three sites ( named a , b@xmath13 and b@xmath0 ) per unit cell , as illustrated in fig . [ disp](a ) . the hubbard hamiltonian for a lattice with @xmath14 unit cells and @xmath15 sites reads : @xmath16+u\sum_{i=1}^nn_{i\uparrow}n_{i\downarrow},\ ] ] where @xmath17 and @xmath18 are the creation operators of an electron with spin @xmath19 at site a and in a bonding state between sites @xmath20 and @xmath21 of the cell @xmath22 , respectively , @xmath23 is the hopping amplitude and @xmath24 is the coulomb coupling . for @xmath25 , double occupancy is completely excluded and the hamiltonian takes the form : @xmath26p_g,\ ] ] where @xmath27 is the gutzwiller projector operator . the model is invariant under the interchange of the @xmath28 sites of the same cell , a symmetry that implies in a well defined local parity ( @xmath29 ) for the gs wave function . as a result , in computing some quantities we find it convenient to use the _ effective linear chain _ ( elc ) generated by the map illustrated in figs . [ disp](a ) and [ disp](b ) , i. e. , any quantity @xmath30 associated with a @xmath28 site at cell @xmath22 of the elc is given by @xmath31 . this mapping does not change the physical content of the gs and excited states , being used only to expose in a more clear fashion some properties of these states . in the tight - binding description ( @xmath32 ) this model presents three bands @xcite : one flat with @xmath14 odd parity states [ antibonding orbitals , @xmath33 ) ] and energy @xmath34 ; and two dispersive branches , @xmath35 with @xmath36 , @xmath37 , built from a sites and bonding ( even parity ) orbitals , as shown in fig . [ disp](c ) . at half filling ( @xmath38 , where @xmath39 is the number of electrons ) the gs total spin @xmath40 is degenerate , with @xmath40 ranging from the minimum value ( @xmath41 or @xmath42 ) to @xmath43 , where @xmath44 is the number of sites in the a ( b ) sublattice . as proved by lieb @xcite the coulomb repulsion lifts this huge degeneracy and selects the @xmath45 ground state for any finite @xmath24 , giving rise to a ferrimagnetic gs @xcite . on the other hand , for @xmath25 , one hole ( @xmath46 ) and periodic boundary conditions ( bc s ) , the system satisfies the requirements of nagaoka s theorem for saturated ferromagnetism @xcite . for nagaoka ferromagnetism and lieb ferrimagnetism the gs is homogeneous in parity with @xmath47 for any cell @xmath22 . due to this symmetry , the spectrum of the ab@xmath0 chain in the heisenberg limit ( @xmath48 , @xmath38 ) at the sector @xmath49 is identical to that of the alternating heisenberg spin-@xmath11/spin-1 chain @xcite . here we focus on the effect of hole doping , @xmath50 , both in the weak coupling and the infinite - u limit , using exact diagonalization ( ed ) through the lanczos algorithm for closed bc s and dmrg for open bc s @xcite . in the ed procedure , the bc s are such to minimize the energy , except for @xmath51 and @xmath52 [ fig . [ pdez](c ) ] in which the bc s ( periodic or antiperiodic ) are such that the fermi wave vector @xmath53 in the thermodynamic limit is included in the set of wave vectors for the finite system @xcite . we used finite size dmrg for open chains with a sites in its extrema , keeping 364 to 546 states per block in the last sweep . the maximum discarded weight in the last sweep was typically @xmath54 , except for odd phases and @xmath51 , where the discarded weight was @xmath55 . in the dmrg calculations we treated @xmath20 and @xmath21 as a composite site with 9 states for @xmath25 and 16 states for @xmath51 however , by considering the parity symmetry , we can decompose this supersite into the two possible symmetry sectors @xmath56 and @xmath57 . within this scheme , we have considered all parity symmetry sectors of the form @xmath58 , with @xmath59 contiguous cells of odd parity in one side of the open chain and @xmath60 contiguous cells of even parity in the other . in addition , we have verified the stability of this phase separation against the formation of a mixed phase composed of smaller domains . the energy is studied as function of @xmath59 for increasing number of states kept per block in order to localize the value of @xmath59 for which the energy is minimum , as shown in figs . [ pdez](a ) and [ pdez](b ) . the phase - separated boundaries are thus determined by the limiting dopings for which an inhomogeneous phase ( non - uniform parities ) is observed . we have also developed a simple variational approach for @xmath25 and @xmath52 , which is explained in detail in appendix a. the results calculated using this approach are shown in figs . [ pdez](d ) and [ sqsefl](c ) . in fig . [ pdez](c ) ( @xmath51 ) and fig . [ pdez](d ) ( @xmath25 ) we present the average parity , @xmath61 as function of doping , computed using the above - mentioned methods . in both regimes , we observe the occurrence of an homogeneous phase near half filling with @xmath49 . for higher doping , i. e. , @xmath62 [ @xmath63 and @xmath64 $ ] the system phase separates in one region with odd parity cells and the other with even ones . for @xmath65 the gs is homogeneous with @xmath66 . in order to present an overview of the conducting properties of the @xmath67 chain phases in the infinite - u limit , we display in fig . [ pdez](e ) the quantity @xcite @xmath68 calculated in the elc using ed , where @xmath69 , @xmath70 , @xmath71 is the electron density at site @xmath72 and @xmath73 is such that @xmath74 , with @xmath75 and @xmath73 co - primes . the phase of @xmath76 corresponds to the gs expectation value of the position operator , while its modulus defines the localization length ; in an insulator , @xmath77 , as @xmath78 , while in a conductor , @xmath79 , for closed boundary conditions @xcite . the increase of @xmath80 with system size for @xmath81 , as well as in the phase separated region , are evidences of insulating phases at these dopings . these conclusions will be better fundamented by studying the drude weight using ed and the charge gap for larger systems with dmrg . in figs . [ sqsefl](a ) and [ sqsefl](b ) we display the magnetic structure factor @xmath82 calculated at @xmath83 and @xmath51 using dmrg for the elc . first , notice the presence of peaks at @xmath84 and @xmath85 revealing the ferrimagnetic order at half filling . these peaks sustain up to two holes ( @xmath86 ) ; however , it is not clear whether the ferrimagnetic phase is robust against doping in the thermodynamic limit . indeed , by increasing the hole doping , spiral peaks at @xmath1-dependent positions appear near @xmath84 and @xmath85 . the analysis of the charge gap , @xmath87 suggests that these states are metallic , in opposition to the mott insulating ferrimagnetic state at @xmath88 . it is worth mentioning that the occurrence of spiral phases in oxocuprates has been a challenging and topical subject @xcite . in fig . [ sqsefl](c ) we present the gs total spin as function of doping for @xmath25 . for @xmath89 itinerant saturated ferromagnetism due to hole kinematics ( nagaoka mechanism ) is observed . it is interesting to notice that our estimate for the upper hole density ( @xmath90 ) beyond which nagaoka ferromagnetism is unstable is in very good agreement with similar predictions for ladders @xcite and the square lattice @xcite . we have also considered the presence of an aharonov - bohn flux @xmath91 for a closed chain through the gauge transformation : @xmath92 with @xmath93 . the flux variation is equivalent to a change in the boundary condition : @xmath94 represents periodic and @xmath95 antiperiodic boundary conditions . in fig . [ d0scn](a ) we present the dependence of the energy gap @xmath96 between the lowest energy state for a flux @xmath91 and that for saturated ferromagnetism ( @xmath97 ) as function of @xmath98 at @xmath99 . we have identified many level crossings in this curve . in fact , as the flux increases from @xmath100 , the total spin decreases from the maximum value , @xmath101 , to the minimum value @xmath102 ( @xmath9 ) for @xmath39 even ( odd ) , a behavior also observed in the square lattice @xcite . notice that @xmath103 tends to saturation with system size , indicating that the level spacings decrease with @xmath104 . these results suggest that the thermodynamic gs displays spontaneously su(2 ) symmetry breaking as a result of an ergodic combination of infinitely many states ( @xmath105 ) , including the singlet spiral state @xcite . in figs . [ d0scn](b ) and [ d0scn](c ) we present the spin correlation function between cell spins @xmath106 and the magnetic structure factor @xmath107 as function of distance @xmath22 and wave vector @xmath108 , @xmath109 , respectively . as we can observe , the saturated ferromagnetic and the spiral singlet states are adiabatically connected , such that all states contributing to the thermodynamic gs exhibit long - range ordering . in particular , as the flux increases from @xmath100 the peak of @xmath110 at @xmath84 ( saturated ferromagnetism ) steadily decreases , while the spiral state peak at @xmath111 increases . we noted also that the charge structure factor @xmath112 where @xmath113 and @xmath114 is the electron occupation number at cell @xmath22 , is not affected by the flux variation and displays a peak at @xmath115 [ fig . [ d0scn](d ) ] , where @xmath53 is the tight - binding _ spinless _ fermi wave vector @xcite , with @xmath116 , @xmath117 . in the phase - separated regime the charge compressibility diverges following the linear dependence of the energy with doping . in figs . [ cszb1b2 ] and [ nhs ] we present some properties of the gs in this regime calculated through dmrg for the elc . first we notice that all these properties clearly exhibit some modulation on the same sublattice in the _ metallic _ odd parity region due to charge itinerancy . in particular , this modulation is stronger in the @xmath51 spiral phase as evidenced by the correlation function @xmath118 shown in fig . [ cszb1b2](a ) , but also noticed in the itinerant nagaoka phase ( @xmath25 ) as manifested by the site magnetization @xmath119 shown in fig . [ cszb1b2](c ) . on the other hand , in the _ insulating _ even parity phase a flat behavior is observed , except for boundary and interface effects . these paramagnetic phases [ see figs . [ cszb1b2](b ) and [ cszb1b2](d ) ] are characterized by strong singlet correlations between spins at sites @xmath20 and @xmath21 at the same cell , i. e. , @xmath120 @xmath121 for @xmath51 @xmath122 , as shown in figs . [ cszb1b2](b ) and [ cszb1b2](d ) . in contrast , in the metallic phase this correlation varies very little with @xmath24 and indicates robust triplet correlations , i. e. , @xmath123 @xmath124 for @xmath51 @xmath122 . notice that in the absence of hole hopping , even when restricted to a cell as in the insulating phase , the value of @xmath125 in a singlet ( triplet ) state should be @xmath126 ( @xmath127 ) . the hole density @xmath128 is shown in figs . [ nhs](a ) and [ nhs](b ) . in the odd parity metallic phase , holes do not occupy antibonding orbitals , whereas in the even parity insulating phase these orbitals are accessible for them . therefore , in the first case the hole densities at sites @xmath129 and @xmath130 are very similar . this may also occur in the second case if double occupancy is excluded ( @xmath25 ) . an illustration of the phase - separated regime for @xmath25 is shown in fig . [ nhs](c ) . in this coupling limit , unsaturated ferromagnetism was suggested to occur in ladders @xcite and the square lattice @xcite as an intermediate phase between saturated ferromagnetism and paramagnetism as function of doping . however , in the context of the @xmath5 model the situation is more complex and predictions of phase separation , both for ladders @xcite and the square lattice @xcite , and stripe formation for the square lattice @xcite have been reported . at @xmath2 , i. e. , one hole per @xmath129 site for open bc s using dmrg @xcite , the gs has even parity and is fully dominated by the mott insulating phase ( even parity ) illustrated in fig . [ nhs](c ) for @xmath25 . the charge gap @xmath131 , where @xmath132/\delta n_e$ ] , @xmath133 ( @xmath134 ) , and @xmath135 , must be calculated with care . first , notice that adding electrons to @xmath2 places the system in the phase - separated ( inhomogeneous ) region where the chemical potential @xmath136 is flat . indeed , by comparing results using dmrg and ed calculations for @xmath25 , for which @xmath137 presents little finite size corrections [ fig . [ dsdcsz13](a ) ] , we concluded that boundary effects are minimized by taking @xmath138 and placing the symmetry inverted cells at the chain center . we thus find [ fig . [ dsdcsz13](a ) ] @xmath139 ( @xmath140 ) for @xmath51 ( @xmath141 ) . this problem is absent in the case of hole doping since the phase is homogeneous . the extrapolated spin gap , @xmath142 characterized by symmetry inversion of a cell at the chain center , is also shown in fig . [ dsdcsz13](a ) for @xmath51 ( @xmath143 ) and @xmath25 ( @xmath144 ) , with the spin gap at @xmath25 presenting little finite size dependence . it is a quite massive excitation with the magnon localized at the odd symmetry cell , mostly at the b sites , as shown in fig . [ dsdcsz13](b ) . in this context , sierra _ et al . _ @xcite found @xmath145 using the @xmath5 model ( @xmath146 ) for @xmath147 , i. e. , @xmath148 . we have confirmed this result by studying the @xmath24 dependence of @xmath149 using ed . in fig . [ cor13 ] we show that the spin correlation functions at @xmath2 , calculated using dmrg , present a fast decay and can be fitted with the exponential form @xmath150}$ ] , where @xmath151 is the correlation length , @xmath22 is the cell index in the elc and @xmath152 denotes the central cell of the system . this behavior is expected from the presence of a finite spin gap . the values of @xmath151 for the correlations @xmath153 , @xmath154 and @xmath155 are @xmath156 0.4 ( 2.2 ) , 0.25 ( 0.45 ) and 0.39 ( 0.75 ) , respectively , for @xmath25 ( @xmath51 ) , with @xmath152 denoting the central cell . thus , except for the correlation @xmath153 at @xmath51 , the correlation length is extremely short with spins correlated only within a cell . further , the calculated bulk values of @xmath157 at @xmath2 are in very good agreement with those in the even phase of the separated region shown in figs . [ cszb1b2](b ) and [ cszb1b2](d ) . the above results support a short - range - rvb ( sr - rvb ) @xcite state for the gs at @xmath2 , as illustrated in fig . [ cor13](d ) . in this context , sierra _ et al . _ @xcite reached similar conclusions using the @xmath5 model on the ab@xmath0 chain , while giesekus has proved @xcite that a sr - rvb state is the gs of a non - bipartite lattice with the same local symmetry but a different hopping pattern . we now focus on the behavior of the system for @xmath158 by considering a chain with closed boundary conditions and @xmath159 for @xmath25 using ed . the first noticeable feature is the behavior of the spin correlation functions after doping the @xmath2 gs with two holes . the value of @xmath160 ( where @xmath161 denotes an arbitrary cell ) changes from -0.41 to -0.28 . this variation can be understood by considering that the two holes added to the system break two singlet bonds and reside predominately at b sites . in this picture the correlation function would amounts to @xmath162 , which is close to -0.28 . furthermore , the spin correlation functions shown in fig . [ cor13dopcross](a ) evidence the formation of long ranged bonds between electrons on @xmath28 sites , while the other correlations remain short ranged , as in the @xmath2 ground state . this fact indicates that the electrons picked from the sr - rvb by hole doping are antiferromagnetically coupled and delocalized through the system , as illustrated in fig . [ cor13dopcross](b ) . in order to describe the system behavior for finite dopings , we display in fig . [ cor13dopcross](c ) the correlation function @xmath163 and electronic densities as function of @xmath1 . notice that for @xmath164 the electronic density at @xmath129 sites is almost fixed , while that at @xmath28 sites are monotonically depopulated . as a consequence , @xmath163 continuously vanishes as the doping increases . moreover , in fig . [ cor13dopcross](d ) we show the relevant nearest - neighbor spin correlation functions . these correlations display quite different magnitudes at @xmath2 , but their values approach each other for @xmath12 . we thus consider the doping interval @xmath164 as a _ crossover region _ , where doping starts to build the luttinger liquid which is fully established for @xmath12 . we have also calculated the charge compressibility @xmath165 through @xmath166,\label{ll2}\ ] ] where @xmath167 is the volume and @xmath168 is the electronic density ; the charge excitation velocity @xmath169 with @xmath170 and @xmath171 the system length ; and the drude weight @xmath172_{\phi_{min}},\label{ll4}\ ] ] where @xmath173 is the flux value that minimizes the energy @xcite . in an insulating phase these quantities satisfy the limits below @xmath174 while for a metal @xmath175 , @xmath176 and @xmath177 are finite . as shown in fig . [ csiud ] , at @xmath2 , @xmath175 and @xmath176 increases , while @xmath177 decreases with system size for both @xmath51 and @xmath25 , although the insulating character is better evidenced for @xmath25 due to its sizable charge gap , as shown in fig . [ dsdcsz13](a ) . at the other commensurate density , @xmath178 , we can see the signals of an insulating phase for @xmath25 , while for @xmath51 we does not observe any especial behavior . in order to clarify this point , we have used dmrg to study the size dependence of the charge gap for larger systems at this doping . for a finite open chain , the occupation of two holes per cell tends to @xmath178 in the thermodynamic limit . in fig . [ dsdc23](a ) , we can clearly observe that for @xmath25 the system is in a mott insulating phase with @xmath179 ; however , the gap for @xmath51 is extremely small . in order to better understand the u - dependence of this gap , we have also calculated @xmath137 for intermediate values of @xmath24 , as also shown in fig . [ dsdc23](a ) . in the inset of fig . [ dsdc23](a ) we have fit @xmath180 using an expression similar to the limiting behavior of the charge gap as @xmath181 of the lieb - wu solution for a linear chain at half filling @xcite : @xmath182 , in which @xmath183 and @xmath184 are fitting parameters . notice , however , that contrary to the lieb - wu solution @xcite , @xmath137 saturates to a finite value ( @xmath185 ) for @xmath25 . on the other hand , similarly to the linear chain at half filling @xcite , the data shown in fig . [ dsdc23](b ) indicates the absence of spin gap at @xmath178 in the thermodynamic limit for both @xmath51 and @xmath25 . in the luttinger model , it is well known @xcite that @xmath175 , @xmath176 and @xmath177 are related through @xmath186 with @xmath187 where @xmath188 is the exponent governing the decay of the correlation functions . in order to probe the doped region for which the lower energy spectrum of the @xmath67 chain can be mapped onto the luttinger model , we consider the ratio @xmath189 which must be equal to one if the system is in the ll universality class @xcite . since the @xmath67 chain is not strictly one - dimensional , care must be taken with the length scales ( @xmath167 and @xmath171 ) in eqs . ( [ ll2 ] ) , ( [ ll3 ] ) and ( [ ll4 ] ) . for @xmath32 , the orbitals at sites @xmath129 and bonding orbitals at sites @xmath28 are translationally equivalent and both build the dispersive branches shown in fig . [ disp](c ) . in this case , the system can be mapped onto a tight - binding linear chain with @xmath190 sites and a rescaled hopping parameter , @xmath191 , with @xmath192 . in order that eq . ( [ ll5 ] ) matches this result for @xmath193 , we must choose @xmath194 with @xmath195 ; or , likewise , @xmath196 and the dispersions as written in eq . ( [ disp ] ) . in both cases @xmath197 , with @xmath198 . consider , for example , the former option . for @xmath32 , the charge excitation velocity is equal to the fermi velocity @xmath199 , which can be easily calculated as @xmath200 on the other hand , substituting the gs energy , @xmath201 into the continuous version of eq . ( [ ll2 ] ) , we obtain , @xmath202 using now eqs . ( [ ll6 ] ) and ( [ ll7 ] ) in eq . ( [ ll5 ] ) we find , as expected , @xmath192 . we now turn to the interacting case using ed . as shown in fig . [ ratiokr](a ) the ll character is quite clear for @xmath12 , while for @xmath164 we identify the crossover region . the ed results for @xmath203 are presented in fig . [ ratiokr](b ) . notice that @xmath188 is close to 1 ( non - interacting fermions ) for @xmath51 ; while , @xmath188 is close to @xmath42 ( non - interacting _ spinless _ fermions ) for @xmath25 @xcite . in order to check these results , we used dmrg to calculate the elc spin correlation function @xmath204 whose asymptotic behavior should match that for the luttinger model @xcite : @xmath205^{1/2}}{l^{1+k_\rho}}. \label{cll}\ ] ] in eq . ( [ dmrgcorr ] ) we have considered an average over all possible pairs of sites separated by the same distance @xmath22 , a procedure that reduces open boundary effects . in figs . [ cor23dop](a ) and [ cor23dop](b ) we show @xmath206 calculated at @xmath207 for @xmath51 and @xmath25 , respectively . also shown are the fittings to @xmath206 using @xmath208 with @xmath197 and @xmath188 taken from the results shown in fig . [ ratiokr](b ) after linear interpolation : @xmath209 ( @xmath51 ) and @xmath210 ( @xmath25 ) . motivated by a compromise between large values of @xmath22 and minimum boundary effects , we have considered intermediate values of @xmath22 in the fitting , which is quite good for both values of @xmath24 . we thus conclude that the luttinger model correctly describes the low energy physics of the @xmath67 chain for @xmath12 . in summary , the numerical results presented here have clearly evidenced the rich phase diagram exhibited by the hubbard model on the doped ab@xmath0 chain both for @xmath51 and in the infinite - u limit . we have shown that at the commensurate dopings @xmath2 and @xmath211 the system display insulating phases , although for @xmath51 the charge gap @xmath137 is very small at @xmath178 , with indications that @xmath137 present an essential singularity as @xmath181 . for @xmath51 and @xmath212 the gs exhibit a ferrimagnetic phase reminiscent of the undoped regime , while for @xmath213 incommensurate magnetic correlations are observed . for @xmath25 and @xmath88 the gs total spin is degenerate , whereas for @xmath214 hole itinerancy ( nagaoka mechanism ) sets a fully polarized gs . in this case , we have also observed the presence of an extensive number of low - lying levels with total spin ranging from the minimum value to @xmath215 and level spacing decaying with system size as @xmath104 . for higher doping , the system phase separates into coexisting metallic and insulating phases for @xmath216 ( with @xmath217 and @xmath218 ) . the insulating state presents a finite spin gap and fully fills the system at @xmath2 , which is well described by a short - ranged - rvb state . finally , a crossover region is observed for @xmath219 , while a luttinger liquid behavior is explicitly characterized for @xmath12 . in closing , we would like to stress that the above - reported results might also stimulate further experimental and theoretical investigations on quasi - one - dimensional compounds displaying complex unit cell structures @xcite . we acknowledge useful discussions with a. l. malvezzi and m. h. oliveira . this work was supported by cnpq , finep , facepe and capes ( brazilian agencies ) . in the metallic saturated ferromagnetic region ( parity symmetry -1 ) the energy as function of doping is known to have a non - interacting spinless fermion behavior : @xmath220 where @xmath221 , @xmath222 , and @xmath223 is the linear size of the system . on the other hand , in the insulating paramagnetic phase ( sr - rvb states with even parity symmetry ) at @xmath2 ( one hole per cell ) @xmath224 and the energy per cell @xmath225 is almost independent of the system linear size and can be estimated either by using ed or dmrg : @xmath226 let us now consider a phase - separated regime in which a paramagnetic phase with size @xmath227 coexists with a ferromagnetic one with size @xmath223 , so the energy per cell reads @xmath228 it is convenient to write @xmath229 as @xmath230 where @xmath231 , @xmath232 , @xmath233 and @xmath234 . using the above notation we rewrite eq . ( [ var1 ] ) in the form below @xmath235\label{var4}\ ] ] here we should notice the presence of a singularity at @xmath236 for any finite value of @xmath237 ( see fig . [ var ] ) . however , the region of physical values of @xmath59 is defined by @xmath238 @xmath239 in fig . [ var ] we present @xmath240 for @xmath241 , in which the physical region is @xmath242 and can be found by eq . ( [ var_2 ] ) , with a minimum in @xmath240 for @xmath243 . the value of @xmath59 which minimizes the energy for a given @xmath1 , @xmath244 , satisfies the equation @xmath245_\delta=0 $ ] , which can be written as @xmath246 where @xmath247 the roots of eq . ( [ var5 ] ) are numerically calculated and conduct to @xmath248 we thus conclude that @xmath217 , which is in very good agreement with ed and dmrg calculations . the magnetization is null at the even phase and maximum at the odd one . we can thus derive the following expression for the gs total spin per unit cell : @xmath249\end{aligned}\ ] ] the dependence of the average parity @xmath75 on @xmath1 can also be easily written as @xmath250 finally , using eq . ( [ var11 ] ) for @xmath251 , the above results for @xmath75 and @xmath40 are plotted in figs . 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we present an extensive numerical study of the hubbard model on the doped ab@xmath0 chain , both in the weak coupling and the infinite - u limit . due to the special unit cell topology
, this system displays a rich variety of phases as function of hole doping ( @xmath1 ) away from half - filling . near half - filling ,
spiral states develop in the weak coupling regime , while nagaoka itinerant ferromagnetism is observed in the infinite - u limit . for higher doping the system
phase - separates before reaching a mott insulating phase of short - range rvb states at @xmath2 .
moreover , for @xmath3 we observe a crossover , which anticipates the luttinger liquid behavior for @xmath4 .
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entanglement of formation ( eof)@xcite and relative entropy of entanglement ( ree)@xcite are two major entanglement monotones for bipartite systems . for pure states @xmath0 the eof @xmath1 is defined as a von neumann entropy of its subsystem @xmath2 . on the contrary , ree is defined as minimum value of the relative entropy with separable states ; @xmath3 where @xmath4 is a set of separable states , it is called `` distance entanglement measure '' . another example of the distance entanglement measure is a geometric entanglement measure defined as @xmath5 , where @xmath6 is a maximal overlap of a given state @xmath7 with the nearest product state@xcite . ] . it was shown in ref.@xcite that @xmath8 is a upper bound of the distillable entanglement@xcite . the separable state @xmath9 , which yields a minimum value of the relative entropy is called the closest separable state ( css ) of @xmath10 . surprising fact , at least for us , is that although definitions of eof and ree are completely different , they are exactly same for all pure states@xcite . this fact may indicate that they are related to each other although the exact connection is not revealed yet . the main purpose of this paper is to explore the veiled connection between eof and ree . for mixed states @xmath10 eof is defined via a convex - roof method@xcite ; @xmath11 where the minimum is taken over all possible pure - state decompositions with @xmath12 and @xmath13 . the ensemble that gives the minimum value in eq.([two3 ] ) is called the optimal decomposition of the mixed state @xmath10 . thus , the main task for analytic calculation of eof is derivation of an optimal decomposition of the given mixture . few years ago , the procedure for construction of the optimal decomposition was derived@xcite in the two - qubit system , the simplest bipartite system , by making use of the time - reversal operation of spin-1/2 particles appropriately . in these references the relation @xmath14 is used , where @xmath15 is a binary entropy function @xmath16 and @xmath17 is called the concurrence . this procedure , usually called wootters procedure , was re - examined in ref.@xcite in terms of antilinearity . introduction of antilinearity in quantum information theory makes it possible to derive concurrence - based entanglement monotones for tripartite@xcite and multipartite systems@xcite . due to the discovery of the closed formula for eof in the two - qubit system , eof is recently applied not only to quantum information theory but also to many scientific fields such as life science@xcite . while eof is used in various areas of science , ree is not because of its calculational difficulty . in order to obtain ree analytically for given mixed state @xmath10 one should derive its css , but still we do nt know how to derive css@xcite even in the two - qubit system except very rare cases@xcite . in ref.@xcite ree for bell - diagonal , generalized vedral - plenio@xcite , and generalized horodecki states@xcite were derived analytically through pure geometric arguments@xcite . due to the notorious difficulty some people try to solve the ree problem conversely . let @xmath9 be a two - qubit boundary states in the convex set of the separable states . in ref.@xcite authors derived entangled states , whose css are @xmath9 . this converse procedure is extended to the qudit system@xcite and is generalized as convex optimization problems@xcite . however , as emphasized in ref.@xcite still it is difficult to find a css @xmath9 of given entangled state @xmath10 although the converse procedure may provide some useful information on the css@xcite . in this paper we will try to find a css for given entangled two - qubit state without relying on the converse procedure . as commented , eof and ree are identical for bipartite pure states although they are defined differently . this means that they are somehow related to each other . if this connection is unveiled , probably we can find css for arbitrary two - qubit mixed states because we already know how to compute eof through wootters procedure . to explore this issue is original motivation of this paper . we will show in the following that ree of many mixed symmetric states can be analytically obtained from eof if one follows the following procedure : 1 . for entangled two - qubit state @xmath10 let @xmath18 be an optimal decomposition for calculation of eof . since @xmath19 are pure states , it is possible to obtain their css @xmath20 . thus , it is straight to derive a separable mixture @xmath21 . if @xmath22 is a boundary state in the convex set of separable states , the procedure is terminated with @xmath23 . if @xmath22 is not a boundary state , we consider @xmath24 . by requiring that @xmath25 is a boundary state , one can fix @xmath26 , _ say _ @xmath27 . then we identify @xmath28 . this procedure is schematically represented in fig . 1 . in order to examine the validity of the procedure we have to apply the procedure to the mixed states whose ree are already known . thus , we will choose the bell - diagonal , generalized vedral - plenio and generalized horodecki states , whose ree were computed in ref.@xcite through different methods . also , we will apply the procedure to the less symmetric mixed states such as vedral - plenio - type and horodecki - type states whose ree were computed in ref.@xcite by making use of the the converse procedure introduced in ref.@xcite . the paper is organized as follows . in section ii we show that the procedure generates the correct css for bell - diagonal states . in section iii and section iv we show that the procedure generates the correct css for generalized vedral - plenio and generalized horodecki states , respectively . in section v we consider two less symmetric states , vedral - plenio - type and horodecki - type states . it is shown that while the procedure generates a correct css for the former , it does not give a correct one for the latter . in section vi a brief conclusion is given . in appendix we prove that eof and ree are identical for all pure states by making use of the schmidt decomposition . the schmidt bases derived in this appendix are used in the main body of this paper . in this section we will show that the procedure mentioned above solves the ree problem of the bell - diagonal states : @xmath29 where @xmath30 , and @xmath31 the css and ree of @xmath32 were obtained in many literatures@xcite through various different methods . if , for convenience , @xmath33 , the css and ree of @xmath32 are @xmath34 now , we will show that the procedure we suggested also yields the same result . following wootters procedure , one can show that the optimal decomposition of @xmath32 for @xmath35 , @xmath32 is a separable state . ] is @xmath36 where @xmath37 and @xmath38 all @xmath39 have the same concurrence @xmath40 and , hence , the same @xmath41 ( defined in eq . ( [ def-2 - 2 ] ) ) as @xmath42 the schmidt bases of @xmath43 can be explicitly derived by following the procedure of appendix a and the result is @xmath44 \nonumber \\ & & { \lvert 1_a \rangle } = \frac{-1}{n_- } \left [ \left(\sqrt{1 - \lambda_3 } - \sqrt{\lambda_4}\right ) { \lvert 0 \rangle } - \left(\sqrt{\lambda_1 } - i \sqrt{\lambda_2}\right ) { \lvert 1 \rangle } \right ] \\ \nonumber & & { \lvert 0_b \rangle } = \frac{1}{n_+ } \left [ \left(\sqrt{\lambda_1 } + i \sqrt{\lambda_2}\right ) { \lvert 0 \rangle } + \left(\sqrt{1 - \lambda_3 } + \sqrt{\lambda_4}\right ) { \lvert 1 \rangle } \right ] \nonumber \\ & & { \lvert 1_b \rangle } = \frac{1}{n_- } \left [ \left(\sqrt{\lambda_1 } + i \sqrt{\lambda_2}\right ) { \lvert 0 \rangle } - \left(\sqrt{1 - \lambda_3 } - \sqrt{\lambda_4}\right ) { \lvert 1 \rangle } \right ] , \nonumber\end{aligned}\ ] ] where the normalization constants @xmath45 are @xmath46 thus the css of @xmath43 , say @xmath47 , can be straightforwardly computed by making use of eq . ( [ result-2 ] ) ; @xmath48 where @xmath49 similarly , one can derive the schmidt bases for other @xmath50 and the corresponding css @xmath20 . then , one can show that the separable state @xmath51 with @xmath52 for all @xmath53 is @xmath54 this is a boundary state in the convex set of the separable states , because the minimal eigenvalue of its partial transposition , _ say _ @xmath55 , is zero . thus , the procedure mentioned in the introduction is terminated with identifying @xmath23 . in fact , it is easy to show that @xmath22 is exactly the same with @xmath56 in eq . ( [ bd-3 ] ) . thus , the procedure we suggested correctly derives the css of the bell - diagonal states . in this section we will derive the css of the generalized vedral - plenio ( gvp ) state defined as @xmath57 by following the procedure mentioned above . in fact the css and ree of the gvp were explicitly derived in ref.@xcite using a geometric argument , which are @xmath58 where @xmath59.\ ] ] now , we define @xmath60 and @xmath61 $ ] . we also define the unnormalized states @xmath62 , where @xmath63 are eigenstates of @xmath64 ; @xmath65 \\ \nonumber & & { \lvert \lambda_{- } \rangle } = \frac{1}{n } \left[\lambda_1 { \lvert 01 \rangle } - \left ( \sqrt{\lambda_1 ^ 2 + \left(\lambda_2 - \lambda_3\right)^2 } + ( \lambda_2 - \lambda_3 ) \right ) { \lvert 10 \rangle } \right].\end{aligned}\ ] ] in eq . ( [ vp-5 ] ) @xmath66 is a normalization constant given by @xmath67 then , following ref.@xcite , the optimal decomposition of @xmath64 for eof is @xmath68 , where @xmath69 and @xmath70 \left ( { \lvert v_+ \rangle } + i { \lvert v_- \rangle } \right ) \\ \nonumber & & { \lvert \psi_2^{vp } \rangle } = \frac{-i}{\omega } \left [ 2b + i \left\{\sqrt{(a - c)^2 + 4 b^2 } - ( a - c ) \right\ } \right ] \left ( { \lvert v_+ \rangle } - i { \lvert v_- \rangle } \right).\end{aligned}\ ] ] following appendix a one can derive the css for @xmath71 directly . then , one can realize that @xmath72 and @xmath73 have the same css , which is identical with @xmath74 . thus , the procedure also gives a correct css for the gvp states . in this section we will show that the procedure also generates the correct css for the generalized horodecki states @xmath75 with @xmath76 and @xmath77 , @xmath78 becomes a separable state . ] . the css and ree of @xmath79 were derived in ref.@xcite using a geometrical argument and the results are @xmath80 following ref.@xcite one can straightforwardly construct the optimal decomposition of @xmath79 for eof , which is @xmath81 , where @xmath82 and @xmath83 in order to treat @xmath84 as an unified manner let us consider @xmath85 . then , @xmath41 defined in eq . ( [ def-2 - 2 ] ) is @xmath86 where @xmath87 . since @xmath41 is independent of @xmath88 , this fact indicates that @xmath41 of @xmath84 are equal to eq.([gh-4 ] ) for all @xmath53 . following appendix a , it is straightforward to show that the schmidt bases of @xmath89 are @xmath90 } } { \lvert 0 \rangle } + \sqrt{\frac{r - \left(\sqrt{\lambda_2 } - \sqrt{\lambda_3 } \right)}{2 r } } e^{-i \theta } { \lvert 1 \rangle } \nonumber \\ & & { \lvert 1_a \rangle } = -\sqrt{\frac{\lambda_1}{r \left[r + \left(\sqrt{\lambda_2 } - \sqrt{\lambda_3 } \right ) \right ] } } { \lvert 0 \rangle } + \sqrt{\frac{r + \left(\sqrt{\lambda_2 } - \sqrt{\lambda_3 } \right)}{2 r } } e^{-i \theta } { \lvert 1 \rangle } \\ \nonumber & & \hspace{3.0 cm } { \lvert 0_b \rangle } = e^{i \theta } { \lvert 0_a \rangle } \hspace{1.0 cm } { \lvert 1_b \rangle } = -e^{i \theta } { \lvert 1_a \rangle}.\end{aligned}\ ] ] then the css @xmath91 of @xmath89 is @xmath92 where @xmath93 \\ \nonumber & & { \cal b } = \frac{\sqrt{2 \lambda_1}}{4 r^2 } \left[2 \sqrt{\lambda_3 } + \left(\sqrt{\lambda_2 } + \sqrt{\lambda_3 } \right)\left(\lambda_1 - 2 \sqrt{\lambda_2 \lambda_3}\right ) \right].\end{aligned}\ ] ] thus , the css @xmath20 of @xmath84 can be obtained by letting @xmath94 , @xmath95 , @xmath96 , respectively . then , @xmath97 with @xmath98 reduces @xmath99 however , @xmath22 is not a boundary state in the convex set of the separable states , because the minimum eigenvalue of @xmath55 is positive . thus , we define @xmath100 the condition that the minimum eigenvalue of @xmath101 is zero fixes @xmath102 as @xmath103 inserting eq.([gh-10 ] ) into @xmath9 , one can show that @xmath9 reduces to @xmath104 . thus , our procedure gives a correct css for the generalized horodecki states . in the previous sections we have shown that the procedure generates the correct css and ree for various symmetric states such as bell - diagonal , gvp , and generalized horodecki states . in this section we will apply the procedure to the less symmetric states . the first quantum state we consider is @xmath105 where @xmath106 and @xmath107 . of course , if @xmath108 , @xmath109 , and @xmath110 , @xmath111 reduces to @xmath64 in eq . ( [ vp-1 ] ) . thus , we call @xmath111 as vedral - plenio - type state . in order to apply the procedure to @xmath111 we introduce @xmath112 \hspace{1.0 cm } \lambda_2 = \frac{1}{2 } \left[\left ( a_2 + a_3\right ) - r \right ] \\ \nonumber & & { \lvert \lambda_1 \rangle } = \cos \theta { \lvert 01 \rangle } + \sin \theta { \lvert 10 \rangle } \hspace{1.0 cm } { \lvert \lambda_2 \rangle } = \sin \theta { \lvert 01 \rangle } - \cos \theta { \lvert 10 \rangle}.\end{aligned}\ ] ] applying ref.@xcite , it is possible to derive the optimal decomposition of @xmath111 for eof ; @xmath113 , where @xmath114 \hspace{1.0 cm } p_2 = \frac{1}{2 } \left[1 - \frac{a_2 - a_3}{\sqrt{1 - 4 d^2 } } \right]\ ] ] and @xmath115 \\ \nonumber & & { \lvert w_2 \rangle } = \frac{1}{{\mathcal y}_- } \left [ \left(\sqrt{\xi_+ \eta_- } - \sqrt{\xi_- \eta_+ } \right ) \sqrt{\lambda_1 } { \lvert \lambda_1 \rangle } - \left(\sqrt{\xi_+ \eta_+ } + \sqrt{\xi_- \eta_- } \right ) \sqrt{\lambda_2 } { \lvert \lambda_2 \rangle } \right].\end{aligned}\ ] ] in eq . ( [ vpt-4 ] ) @xmath116 , @xmath117 , and @xmath118 are @xmath119.\end{aligned}\ ] ] following appendix a , one can derive the css @xmath47 and @xmath120 of @xmath121 and @xmath122 after long and tedious calculation . the final results are @xmath123 ^ 2 { \lvert 01 \rangle } { \langle 01 \lvert } \nonumber \\ & + & \left[\frac{\sin \theta \sqrt{\lambda_1 } \left(\sqrt{\xi_+ \eta_+ } + \sqrt{\xi_- \eta_- } \right ) - \cos \theta \sqrt{\lambda_2 } \left(\sqrt{\xi_+ \eta_- } - \sqrt{\xi_- \eta_+ } \right ) } { { \mathcal y}_+}\right]^2 { \lvert 10 \rangle } { \langle 10 \lvert } \\ \nonumber \sigma_2&= & \left[\frac{\cos \theta \sqrt{\lambda_1 } \left(\sqrt{\xi_+ \eta_- } - \sqrt{\xi_- \eta_+ } \right ) - \sin \theta \sqrt{\lambda_2 } \left(\sqrt{\xi_+ \eta_+ } + \sqrt{\xi_- \eta_- } \right ) } { { \mathcal y}_-}\right]^2 { \lvert 01 \rangle } { \langle 01 \lvert } \\ \nonumber & + & \left[\frac{\sin \theta \sqrt{\lambda_1 } \left(\sqrt{\xi_+ \eta_- } - \sqrt{\xi_- \eta_+ } \right ) + \cos \theta \sqrt{\lambda_2 } \left(\sqrt{\xi_+ \eta_+ } + \sqrt{\xi_- \eta_- } \right ) } { { \mathcal y}_-}\right]^2 { \lvert 10 \rangle } { \langle 10 \lvert}.\end{aligned}\ ] ] then , @xmath124 simply reduces to @xmath125 this is manifestly boundary state in the convex set of separable states . thus , the procedure states that @xmath22 is a css of @xmath111 . this is exactly the same with theorem @xmath126 of ref.@xcite . the second less symmetric quantum state we consider is @xmath127 where @xmath128 and @xmath129 . if @xmath130 , @xmath131 , and @xmath132 , @xmath133 reduces to @xmath78 in eq . ( [ gh-1 ] ) . thus , we call @xmath133 as horodecki - type state . applying ref.@xcite , one can derive the optimal decomposition of @xmath133 for eof as @xmath134 , where @xmath52 for all @xmath53 and @xmath135 in order to consider @xmath136 all together , we define @xmath137 for @xmath138 the schmidt bases are @xmath139 \nonumber \\ & & { \lvert 1_a \rangle } = \frac{1}{2 { \mathcal z}_- } \bigg [ \sqrt{2 } \left ( \sqrt{a - d } \sqrt{1 + { \mathcal c } } - \sqrt{a + d } \sqrt{1 - { \mathcal c } } \right ) { \lvert 0 \rangle } \\ \nonumber & & \hspace{2.0 cm } + e^{-i \theta } \left\ { \left(\sqrt{a_1 } + \sqrt{a_4 } \right ) \sqrt{1 + { \mathcal c } } + \left(\sqrt{a_1 } - \sqrt{a_4 } \right ) \sqrt{1 - { \mathcal c } } \right\ } { \lvert 1 \rangle } \bigg ] \\ \nonumber & & { \lvert 0_b \rangle } = \frac{1}{2 { \mathcal z}_+ } \bigg [ \sqrt{2 } e^{i \theta}\left\ { \sqrt{a + d } \left(\sqrt{a_1 } + \sqrt{a_4 } \right ) + \sqrt{a - d } \left(\sqrt{a_1 } - \sqrt{a_4 } \right ) \right\ } { \lvert 0 \rangle } \\ \nonumber & & \hspace{4.0 cm } + \left\ { - \left(a_1 - a_4 \right ) + 2 \sqrt{a^2 - d^2 } + \sqrt{1 - { \mathcal c}^2 } \right\ } { \lvert 1 \rangle}\bigg ] \\ \nonumber & & { \lvert 1_b \rangle } = \frac{1}{2 { \mathcal z}_- } \bigg [ \sqrt{2 } e^{i \theta}\left\ { \sqrt{a + d } \left(\sqrt{a_1 } + \sqrt{a_4 } \right ) + \sqrt{a - d } \left(\sqrt{a_1 } - \sqrt{a_4 } \right ) \right\ } { \lvert 0 \rangle } \\ \nonumber & & \hspace{4.0 cm } + \left\ { - \left(a_1 - a_4 \right ) + 2 \sqrt{a^2 - d^2 } - \sqrt{1 - { \mathcal c}^2 } \right\ } { \lvert 1 \rangle}\bigg],\end{aligned}\ ] ] where @xmath140 and @xmath141.\ ] ] thus , the css @xmath142of @xmath138 is @xmath143 similarly , it is straightforward to derive the css @xmath144 of @xmath145 . then , one can show @xmath146 \nonumber \\ & & \hspace{2.0 cm } = \left ( \begin{array}{cccc } a_1 & 0 & 0 & 0 \\ 0 & a & d & 0 \\ 0 & d & a & 0 \\ 0 & 0 & 0 & a_4 \end{array } \right)\end{aligned}\ ] ] where @xmath147 \nonumber \\ & & a_4 = \frac{1}{4 ( 1 - { \mathcal c}^2 ) } \bigg [ ( 1 + { \mathcal c } ) \left(\sqrt{a_1 } + \sqrt{a_4 } \right)^2 + ( 1 - { \mathcal c } ) \left(\sqrt{a_1 } - \sqrt{a_4 } \right)^2 \nonumber \\ & & \hspace{9.0 cm } - 2 ( 1 - { \mathcal c}^2 ) \left(a_1 - a_4 \right ) \bigg ] \\ \nonumber & & a = \frac{1}{2 ( 1 - { \mathcal c}^2 ) } \left [ ( 1 + { \mathcal c } ) \left(a - d \right ) + ( 1 - { \mathcal c } ) \left(a + d \right ) \right ] \\ \nonumber & & d = \frac{2 a \sqrt{a_1 a_4 } + d \left ( a_1 + a_4 \right)}{1 - { \mathcal c}^2}.\end{aligned}\ ] ] one can show that if @xmath130 , @xmath131 , and @xmath132 , @xmath148 reduces to eq . ( [ gh-8 ] ) . since @xmath148 is not a boundary state in the set of separable states , we define @xmath149 then , the css condition of @xmath150 is @xmath151 \left[x \left(a_4 - a_4 \right ) + a_4 \right ] = \left[x ( d - d ) + d \right]^2.\ ] ] in the horodecki state limit eq . ( [ ht-10 ] ) gives a solution ( [ gh-10 ] ) . using @xmath152 and @xmath153 where @xmath154 the solution of @xmath102 , _ say _ @xmath155 , can be obtained by solving the quadratic equation ( [ ht-10 ] ) . inserting @xmath155 in eq . ( [ ht-9 ] ) , one can compute @xmath150 explicitly , which is a candidate of css for @xmath133 . the css of @xmath133 was derived in the theorem @xmath156 of ref.@xcite by using the converse procedure introduced in ref.@xcite . the explicit form of the css is @xmath157 where @xmath158 \\ \nonumber & & r_4 = \frac{1}{f } \left[2a_4 ( a_2 + a_4 ) ( a_1 + a_2 + a_4 ) + d^2 ( a_1 - a_4 ) + \delta \right ] \\ \nonumber & & r = \frac{1}{f } \left [ 2(a_1 + a_2 ) ( a_2 + a_4 ) ( a_1 + a_2 + a_4 ) - d^2 ( a_1 + 2 a_2 + a_4 ) - \delta \right]\end{aligned}\ ] ] and @xmath159 . in eq . ( [ th-2 - 3 ] ) @xmath160 and @xmath161 are @xmath162 our candidate @xmath163 does not coincide with the correct css @xmath164 . thus , the procedure does not give a correct ree for @xmath133 , although it gives correct ree for bell - diagonal , gvp , generalized horodecki , and vedral - plenio - type states . in this paper we examine the possibility for deriving the closed formula for ree in two - qubit system without relying on the converse procedure discussed in ref.@xcite . since ree and eof are identical for all pure states in spite of their different definitions , we think they should have some connection somehow . in this context we suggest a procedure , where ree can be computed from eof . the procedure gives correct ree for many symmetric states such as bell - diagonal , gvp , and generalized horodecki states . it also generates a correct ree for less symmetric states such as @xmath111 . however , the procedure failed to produce a correct ree for the less symmetric states @xmath133 . this means our procedure is still incomplete for deriving the closed formula of ree . we think still the connection between eof and ree is not fully revealed . if this connection is sufficiently understood in the future , probably the closed formula for ree can be derived . we hope to explore this issue in the future . * acknowledgement * : this research was supported by the basic science research program through the national research foundation of korea(nrf ) funded by the ministry of education , science and technology(2011 - 0011971 ) . 99 c. h. bennett , d. p. divincenzo , j. a. smokin and w. k. wootters , _ mixed - state entanglement and quantum error correction _ , phys . rev . * a 54 * ( 1996 ) 3824 [ quant - ph/9604024 ] . v. vedral , m. b. plenio , m. a. rippin and p. l. knight , _ quantifying entanglement _ * 78 * ( 1997 ) 2275 [ quant - ph/9702027 ] . v. vedral and m. b. plenio , _ entanglement measures and purification procedures _ , phys . rev . * a 57 * ( 1998 ) 1619 [ quant - ph/9707035 ] . a. shimony , _ degree of entanglement _ , in d. m. greenberg and a. zeilinger ( eds . ) , fundamental problems in quantum theory : a conference held in honor of j. a. wheeler , ann . n. y. acad . * 755 * ( 1995 ) 675 ; h. barnum and n. linden , _ monotones and invariants for multi - particle quantum states _ , j. phys . a : math . gen . * 34 * , ( 2001 ) 6787 [ quant - ph/0103155 ] ; t .- c . wei and p. m. goldbart , _ geometric measure of entanglement and application to bipartite and multipartite quantum states _ , phys . rev . * a 68 * ( 2003 ) 042307 [ quant - ph/0307219 ] . a. uhlmann , _ fidelity and concurrence of conjugate states _ , phys . rev . * a 62 * ( 2000 ) 032307 [ quant - ph/9909060 ] . s. hill and w. k. wootters , _ entanglement of a pair of quantum bits _ , * 78 * ( 1997 ) 5022 [ quant - ph/9703041 . w. k. wootters , _ entanglement of formation of an arbitrary state of two qubits _ , phys . lett . * 80 * ( 1998 ) 2245 [ quant - ph/9709029 ] . v. coffman , j. kundu and w. k. wootters , _ distributed entanglement _ , phys . rev . * a 61 * ( 2000 ) 052306 [ quant - ph/9907047 ] . a. osterloh and j. siewert , _ constructing @xmath66-qubit entanglement monotones from antilinear operators _ , phys . rev . * a 72 * ( 2005 ) 012337 [ quant - ph/0410102 ] ; d. . dokovi and a. osterloh , _ on polynomial invariants of several qubits _ , j. math . * 50 * ( 2009 ) 033509 [ arxiv:0804.1661 ( quant - ph ) ] . m. sarovar , a. ishizaki , g. r. fleming , k. b. whaley , _ quantum entanglement in photosynthetic light harvesting complexes _ , nature physics , * 6*(2010 ) 462 [ arxiv:0905.3787 ( quant - ph ) ] and references therein . o. krueger and r. f. werner , _ some open problems in quantum information theory _ , quant - ph/0504166 . r. horodecki and m. horodecki , _ information - theoretic aspects of inseparability of mixed states _ , phys . rev . * a 54 * , ( 1996 ) 1838 [ quant - ph/9607007 ] . h. kim , m. r. hwang , e. jung and d. k. park , _ difficulties in analytic computation for relative entropy of entanglement _ , phys . * a 81 * ( 2010 ) 052325 [ arxiv:1002.4695 ( quant - ph ) ] . d. k. park , relative entropy of entanglement for two - qubit state with @xmath165-directional bloch vectors , int . * 8 * ( 2010 ) 869 [ arxiv:1005.4777 ( quant - ph ) ] . m. horodecki , p. horodecki , and r. horodecki , in _ quantum information : an introduction to basic theoretical concepts and experiments _ , edited by g. alber _ ( springer , berlin , 2001 ) , p. 151 . a. miranowicz and s. ishizaka , _ closed formula for the relative entropy of entanglement _ , phys . rev . * a78 * ( 2008 ) 032310 [ arxiv:0805.3134 ( quant - ph ) ] . s. friedland and g gour , _ closed formula for the relative entropy of entanglement in all dimensions _ , j. math . * 52 * ( 2011 ) 052201 [ arxiv:1007.4544 ( quant - ph ) ] . m. w. girard , g. gour , and s. friedland , _ on convex optimization problems in quantum information theory _ , arxiv:1402.0034 ( quant - ph ) . in this section we will show that ree and eof are identical for two - qubit pure states . this fact was already proven in theorem @xmath166 of ref.@xcite . we will prove this again more directly , because explicit schmidt bases are used in the main body of the paper . let us consider a general two - qubit pure state @xmath167 with @xmath168 . then , its concurrence is @xmath169 . now , we define @xmath170 where @xmath171 \hspace{1.0 cm } { \cal n}_{\pm}^2 = |\alpha_1^ * \alpha_2 + \alpha_3^ * \alpha_4|^2 + |\lambda_{\pm } - ( |\alpha_1|^2 + |\alpha_3|^2)|^2.\ ] ] now , we consider @xmath172 matrix @xmath173 , whose components @xmath174 are @xmath175 then schmidt bases for each party are defined as @xmath176 where @xmath177 using eq . ( [ schmidt-1 ] ) , one can show straightforwardly that @xmath178 reduces to @xmath179 . thus , its css @xmath9 are simply expressed in terms of the schmidt bases as @xmath180 applying eq . ( [ ree-1 - 1 ] ) , one can show easily @xmath181 , which is exactly the same with eof .
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it is well - known that entanglement of formation ( eof ) and relative entropy of entanglement ( ree ) are exactly identical for all two - qubit pure states even though their definitions are completely different .
we think this fact implies that there is a veiled connection between eof and ree . in this context , we suggest a procedure , which enables us to compute ree from eof without relying on the converse procedure .
it is shown that the procedure yields correct ree for many symmetric mixed states such as bell - diagonal , generalized vedral - plenino , and generalized horodecki states .
it also gives a correct ree for less symmetric vedral - plenio - type state .
however , it is shown that the procedure does not provide correct ree for arbitrary mixed states .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
magnetic fields are a major agent in the ism and also control the density and propagation of cosmic rays . the radio infrared correlation indicates that turbulent fields are strongest in star - forming regions . magnetic fields and cosmic rays can provide the pressure to drive galactic outflows . outflows from starburst galaxies in the early universe may have magnetized the intergalactic medium . in spite of our increasing knowledge of cosmic magnetic fields , many important questions , especially their origin , strength in intergalactic space , first occurrence in young galaxies and their dynamical importance for galaxy evolution remain unanswered . the most promising mechanism to sustain magnetic fields in the interstellar medium of galaxies is the _ dynamo_. in young galaxies a small - scale dynamo probably amplified seed fields from the protogalactic phase to the energy density level of turbulence within less than @xmath2 yr ( schleicher et al . @xcite ) . to explain the generation of large - scale fields in galaxies , the mean - field dynamo has been developed ( beck et al . it is based on turbulence , differential rotation and helical gas flows , driven by supernova explosions ( gressel et al . @xcite ) and cosmic rays ( hanasz et al . the mean - field dynamo in galaxy disks predicts that within a few @xmath2 yr large - scale regular fields are excited from the seed fields ( arshakian et al . @xcite ) , forming patterns ( `` modes '' ) with different azimuthal symmetries in the disk and vertical symmetries in the halo . most of what we know about galactic magnetic fields comes through the detection of radio waves . the intensity of _ synchrotron emission _ is a measure of the number density of cosmic - ray electrons and of the strength of the total magnetic field component in the sky plane . the assumption of energy equipartition between the total cosmic rays and total magnetic fields allows us to calculate the total magnetic field strength from the synchrotron intensity ( beck & krause @xcite ) . polarized emission originates from ordered fields . as polarization `` vectors '' are ambiguous by @xmath3 , they can not distinguish _ regular ( coherent ) fields _ , defined to have a constant direction within the telescope beam , from _ anisotropic fields _ , which are generated from turbulent fields by compressing or shearing gas flows , so that their direction frequently reverses perpendicular to the flow direction . unpolarized synchrotron emission indicates _ turbulent fields _ which have random directions . the intrinsic degree of linear polarization of synchrotron emission is about 75% . the observed degree of polarization is smaller due to the contribution of unpolarized thermal emission , by _ faraday depolarization _ along the line of sight and across the beam ( sokoloff et al . @xcite ) and by geometrical depolarization due to variations of the field orientation within the beam . at radio wavelengths of a few centimeters and below , the orientation of the observed b vector is parallel to the field orientation , so that the magnetic patterns of many galaxies can be mapped directly ( e.g. beck @xcite ) . at longer wavelengths , the observed polarization vector is rotated in a magnetized thermal plasma by _ faraday rotation_. the rotation angle increases with the square of the wavelength @xmath4 and with the _ rotation measure ( rm ) _ , which is the integral of the plasma density and the strength of the component of the field along the line of sight . as the rotation angle is sensitive to the sign of the field direction , only regular fields give rise to faraday rotation , while anisotropic and random fields do not . the gravitational potential of strongly barred galaxies drives noncircular orbits and gas inflow . in many barred galaxies a circumnuclear ring is formed . numerical models show that gas streamlines are deflected in the bar region along shock fronts , behind which the cold gas is compressed in a fast shearing flow ( athanassoula @xcite ) . the compression regions traced by massive dust lanes develop along the edge of the bar that is leading with respect to the galaxy s rotation because the gas rotates faster than the bar pattern . 20 galaxies with large bars were observed with the very large array ( vla ) and with the australia telescope compact array ( atca ) ( beck et al . the total radio luminosity ( a measure of the total magnetic field strength ) is strongest in galaxies with high far - infrared luminosity ( indicating high star - formation activity ) , a result similar to that in non - barred galaxies . the average radio intensity , radio luminosity and star - formation activity all correlate with the relative bar length . polarized emission was detected in 17 of the 20 barred galaxies . ngc 1097 ( fig . [ fig : n1097 ] ) is one of the nearest barred galaxies and hosts a huge bar of about 16kpc length . the total and polarized radio intensities are strongest in the downstream region of the dust lanes ( southeast of the center ) by compression of turbulent fields in the bar s shock . the pattern of field lines in ngc 1097 is similar to that of the gas streamlines as obtained in numerical simulations ( athanassoula @xcite ) . this suggests that the ordered magnetic field is aligned with the flow and amplified by shear . remarkably , the optical image of ngc 1097 shows dust filaments in the upstream region which are aligned with the ordered field ( fig . [ fig : n1097 ] ) . between the region upstream of the southern bar and the downstream region the field lines smoothly change their orientation by almost @xmath5 . the ordered field is also hardly compressed , probably coupled to the diffuse gas and strong enough to affect its flow ( beck et al . @xcite ) . the polarization pattern in barred galaxies can be used as a tracer of shearing gas flows in the sky plane and complements spectroscopic measurements of radial velocities . barred galaxies can create a gas reservoir and starburst in the central kiloparsec , fuelling an agn needs other processes ( e.g. davies et al . the central regions of barred galaxies are often sites of ongoing intense star formation and strong magnetic fields . bright radio emission has been observed in several galaxies with the following equipartition strengths of the total magnetic fields ( beck et al . @xcite ) : * ngc 1097 : @xmath6@xmath7 g in the circumnuclear ring , 50@xmath7 g in the ring knots * ngc 1365 : @xmath8@xmath7 g in the dust lanes ( no ring ) * ngc 1672 : @xmath9@xmath7 g in the ring knots * ngc 7552 : @xmath10@xmath7 g in the ring note that these values are lower limits because the energy densities of cosmic - ray protons and electrons in starburst regions are reduced by various losses ( thompson et al . this is supported by an investigation of the @xmath11-ray and radio emission from the starburst galaxies m 82 and ngc 253 which suggests that most of the radio emission is from secondaries undergoing strong bremsstrahlung and ionization losses ( lacki et al . @xcite ) . one of the key problems of agn physics is how to fuel the nucleus with gas from the central region , which needs outwards transport of angular momentum ( davies et al . magnetic fields can help here . the circumnuclear ring of ngc 1097 is bright in the optical , ir and radio spectral ranges . it has a diameter of about 1.5kpc and an active nucleus in its center ( fig . [ fig : n1097_center ] ) . the nonthermal and weakly polarized radio emission is a signature of strong turbulent magnetic fields ( beck et al . @xcite ) . magnetic stress in the differentially rotating circumnuclear ring can drive mass inflow ( balbus & hawley @xcite ) at a rate of : @xmath12 where @xmath13 is the surface mass density , @xmath14 its angular rotation velocity and @xmath15 is the magnetic stress tensor . @xmath16 and @xmath17 denote the radial and azimuthal field components . its dominant component can be written in terms of the alfvn velocity @xmath18 as @xmath19 . this yields : @xmath20 where @xmath21 is the scale height of the gas , @xmath22 the strength of the turbulent field and @xmath23 that of the ordered field . the correlation between @xmath24 and @xmath25 is generated by shear from differential rotation . for ngc 1097 , @xmath26pc , @xmath27km / s at 1kpc radius , @xmath2850@xmath7 g and @xmath2910@xmath7 g give an inflow rate of several @xmath30/yr , which is sufficient to fuel the activity of the nucleus of this galaxy ( beck et al . this mechanism can be expected to operate in many galaxies . equipartition between the energy densities of cosmic rays and magnetic fields yields : @xmath31 where @xmath32 is the total field strength , @xmath33 is the radio synchrotron intensity which is proportional to the infrared intensity @xmath34 ( the radio - infrared correlation ) and hence to the star - formation rate @xmath35 . the relation between the bolometric luminosity of agns @xmath36 and the @xmath35 of their hosts ( e.g. trakhtenbrot & netzer @xcite ) can be explained by this scenario . the relation between the accretion rate and sfr ( eq . 3 ) should be tested with a sample of galaxies . the ordered field in the ring of ngc 1097 has a spiral pattern and extends towards the nucleus . its pitch angle agrees with that of the spiral filaments seen in dust and h@xmath37 ( prieto et al . @xcite , davies et al . @xcite ) , suggesting gas inflow along the magnetic field lines . radio emission is a tracer of star formation also in distant galaxies . the radio infrared correlation holds to redshifts of at least @xmath38 ( murphy @xcite ) , indicating strong magnetic fields in young galaxies , probably generated by the turbulent dynamo mechanism ( schleicher et al . infrared observations indicate that star formation in the hosts of the most luminous agns peaked at @xmath38 and decreased at later epochs ( serjeant et al . @xcite ) , which is consistent with the evolution of the accretion rate onto supermassive black holes ( trakhtenbrot et al . the strong magnetic fields in young galaxies may serve as the link between star formation and accretion . ngc 253 is a prototypical starburst galaxy , hosting a starburst nucleus ( brunthaler et al . its high inclination of @xmath39 allows observing the extraplanar emission . the bright radio halo ( fig . [ fig : n253 ] ) is probably a result of a galactic wind with a speed of about 300km / s ( heesen et al . @xcite ) . radio observations of the inner starburst region with high resolution revealed four filaments with widths of less than 40pc and lengths of at least 500pc ( fig . [ fig : n253_center ] ) . these are interpreted as the boundaries of the outflow cone of hot gas interacting with the halo gas ( heesen et al . the equipartition strength of the total field in the filaments is at least 40@xmath7 g , that in the central region @xmath1 g . faraday rotation measures ( rm ) between 3 cm and 6 cm wavelengths are consistent with an axisymmetric , even - parity , spiral magnetic field in the disk at radii of more than 2kpc , while the field in the inner disk is anisotropic ( heesen et al . @xcite ) . near the se minor axis , in the region of the x - ray outflow cone ( fig . [ fig : n253_x ] ) , rm jumps between + 300radm@xmath40 and -300radm@xmath40 . this is interpreted as the faraday rotation of the polarized emission from the background disk by the helical field in the walls of the outflow extending to at least 1.2kpc from the center ( fig . [ fig : n253_model ] ) . only the inner part of this helical field structure is seen in emission ( fig . [ fig : n253_center ] ) . the northern outflow cone is located behind the disk , so that its x - ray emission is mostly absorbed and can not faraday - rotate the radio emission . the galaxy ngc 3079 hosts an energetic wind - driven outflow which extends 3kpc above and below the plane ( duric & seaquist @xcite ) and emits strong and highly polarized synchrotron emission , in contrast to ngc 253 . radio - bright nuclear outflows seem to be rare . magnetic fields in the central regions of galaxies are dynamically important . magnetic stress drives gas inflow from the circumnuclear regions towards the nuclei . this process connects the star formation activity to the accretion rate and could be important for the formation of supermassive black holes . ordered fields as observed by polarized radio emission may trace the direction of gas in- and outflows . results have been obtained so far only for radio - bright and nearby galaxies . a better insight into the interaction between gas and magnetic fields , especially near galactic nuclei , needs higher resolution and higher sensitivity . the _ evla _ has largely improved the sensitivity for radio continuum observations . the _ evla _ project _ changes _ ( pi : judith irwin ) will search for outflows in a large sample of edge - on galaxies . the goal for the next decade is the _ square kilometre array ( ska ) _ with a collecting area of about 100x that of present - day telescopes . the ska will open a new era in the observation of cosmic magnetic fields ( beck @xcite ) . starburst galaxies will be observable out to redshifts of @xmath38 ( murphy @xcite ) .
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radio continuum observations of barred galaxies revealed strong magnetic fields of @xmath0 g in the circumnuclear starbursts .
such fields are dynamically important and give rise to magnetic stress that causes inflow of gas towards the center at a rate of several solar masses per year , possibly along the spiral field seen in radio polarization and as optical dust lanes .
this may solve the long - standing question of how to feed active nuclei , and explain the relation between the bolometric luminosity of agn nuclei and the star - formation rate of their hosts .
the strong magnetic fields generated in young galaxies may serve as the link between star formation and accretion onto supermassive black holes .
magnetic fields of @xmath1 g strength were measured in the central region of the almost edge - on starburst galaxy ngc 253 .
four filaments emerging from the inner disk delineate the boundaries of the central outflow cone of hot gas .
strong faraday rotation of the polarized emission from the background disk indicates a large - scale helical field in the outflow walls .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the workload management task ( work package 1 , or wp1 ) @xcite of the eu datagrid project @xcite ( also known , and referred to in the following text , as edg ) is mandated to define and implement a suitable architecture for distributed scheduling and resource management in the grid environment . during the first year and a half of the project ( 2001 - 2002 ) , and following a technology evaluation process , edg wp1 defined , implemented and deployed a set of services that integrate existing components , mostly from the condor @xcite and globus @xcite projects . this was described in more detail at chep 2001 @xcite . in a nutshell , the core job submission component of condorg ( @xcite ) , talking to computing resources ( known in datagrid as computing elements , or ces ) via the globus gram protocol , is fundamentally complemented by : * a job requirement matchmaking engine ( called the _ resource broker _ , or rb ) , matching job requests to computing resource status coming from the information system and resolving data requirements against the replicated file management services provided by edg wp2 . * a job logging and book - keeping service ( lb ) , where a job state machine is kept current based on events generated during the job lifetime , and the job status is made available to the submitting user . the lb events are generated with some redundancy to cover various cases of loss . * a stable user api ( command line , c++ and java ) for access to the system . job descriptions are expressed throughout the system using the condor classified ad language , where appropriate conventions were established to express requirement and ranking conditions on computing and storage element info , and to express data requirements . more details on the structure and evolution of these services and the necessary integration scaffolding can be found in various edg public deliverable documents . this paper focuses on how the experience of the first year of operation of the wp1 services on the edg testbed was interpreted , digested , and how a few design _ principles _ were learned ( possibly the hard way ) from the design and implementation shortcomings of the first release of wp1 software . these principles were applied to design and implement the second major release of wp1 software , that is described in another chep 2003 paper ( @xcite ) . to illustrate the logical path that leads to at least some of these principles , we start by exploring the available techniques to model the behaviour and throughput of the integrated workload management system , and identify two factors that significantly complicate the system analysis . the workload management system provided by edg - wp1 is designed to rely as much as possible on existing technology . while this has the obvious advantages of limiting effort duplication and facilitating the compatibility among different projects , it also significantly complicates troubleshooting across the various layers of software supplied by different providers , and in general the understanding of the integrated system . also , where negotiations with external software providers could nt reach an agreement within the edg deadlines , some of the interfaces and communication paths in the system had to be adapted to fit the existing external software incarnations . to get a useful high - level picture of the integrated workload management system , beyond all these practical constraints , we can model it as a queuing system , where job requests traverse a network of queues , and the service stations " connected to each queue represent one of the various processing steps in the job life - cycle . a few of these steps are exemplified in figure [ fig - netqueue ] . establishing the scale factors for each service in the wp1 system ( e.g. : how many users can a single matchmaking / job submission station serve , how many requests per unit time can a top - level access point to the information system serve , what is the sustained job throughput that can be achieved through the workload management chain , etc . ) is one of the fundamental premises for the correct design of the system . one could expect to obtain this knowledge either by applying queuing theory to this network model ( this requires obtaining a formal representation of all the components , their service time profiles and their interconnections ) or by measuring the service times and by identifying where long queues are likely to build up when a realistic " request load is injected in the system . this information could in principle also be used to identify the areas of the system where improvement is needed ( sometimes collectively called _ bottlenecks _ ) . experience with the wp1 software integration showed that both of these approaches are impractical for either dimensioning the system or ( possibly even more important ) for identifying the trouble areas that affect the system throughput . we identified two non - linear factors that definitely work against the predictive power of queuing theory in this case , and require extra care even to apply straightforward reasoning when bottlenecks are to be identified to improve system throughput . these are the consequence of common programming practice ( and are therefore easy to be found in the software components that we build or are integrating ) and are described in the following section . one of the most common ( and most frustrating , both to developers and to end users ) experiences in troubleshooting the wp1 workload management system on the edg testbed has been the fact that often , perceived _ improvements _ to the system ( sometimes even simple bug fixes ) result in a _ decrease _ in the system stability , or reliability ( fraction of requests that complete successfully ) . the cause is often closely related to the known fact that removing a bottleneck , in any flow system , can cause an overflow downstream , possibly close to the _ next _ bottleneck . the complicating factor is that there are at least two characteristics that could ( and possibly still can ) be found in many elements of our integrated workload management queuing network , that can cause problems to appear even very far from the area of the network where an _ improvement _ is being attempted : * _ queues of job requested can form where they can impact on the system load . _ + different techniques can be chosen or needed to pass job requests around . sometimes a socket connection is needed , sometimes sequential request processing ( one request at a time in the system ) is required for some reason , and multiple processes / threads may be used to handle individual requests . having a number of tasks ( processes / threads ) wait for a socket queue or a sequential processing slot is one way to queue " requests that definitely generates much extra work for the process scheduler , and can cause any other process served by the same scheduler to be allocated less and less time . queues that are unnecessarily scanned while waiting for some other condition to allow the processing of their element can also impact on the system load , especially if the queue elements are associated to significant amounts of allocated dynamic memory . * _ some system components can enforce hard timeouts and cause anomalies in the job flow . _ + when handling the access ( typically via socket connections ) to various distributed services , provisions typically need to be made to handle all possible failure modes . reasonably " long timeouts are sometimes chosen to handle failures that are perceived to be very unlikely by developers ( failure to establish communication to a local service , for instance ) . this kind of failures , however , can easily materialise when the system resources are exhausted under a stress test or load peak . figure [ fig - fmode ] illustrates how these two effects can conspire to frustrate a genuine effort to remove what seems the limiting bottleneck in the system ( the example in the figure does nor refer to any real case or component ) : removing the bottleneck ( 1 ) causes a request queue to build up at the next station ( 2 ) , and this interferes via the system load to cause hard timeouts and job failures elsewhere ( 3 ) . this example is used to rationalise some of the unexpected reactions that , in many cases , were found while working on the wp1 integrated system . the experience on practical troubleshooting cases similar to this one , while bringing an understanding of the difficulties inherent in building distributed systems , also drove us to formulate some of the principles that are presented in the next section . the attempts at getting a deeper understanding of the edg - wp1 workload management system and their failures led us to formulate a few design principles and to apply them to the second major software release . here are the principles that descend from the paradigm example described in section [ sec - fmode ] : 1 . * queues of various kinds of requests for processing should be allowed to form where they have a minimal and understood impact on system resources . * + queues that get ` filled ' in the form of multiple threads or processes , or that allocate significant amounts of system memory should be avoided , as they not only adversely impact system performance , but also generate inter - dependencies and complicate troubleshooting . * limits should always be placed on dynamically allocated objects , threads and/or subprocesses . * + this is a consequence of the previous point : every dynamic resource that gets allocated should have a tunable system - wide limit that gets enforced . * special care needs to be taken around the pipeline areas where serial handling of requests is needed . * + the impact of any contention for system resources becomes more evident near areas of the queuing system that require the acquisition of system - wide locks . so far we concentrated on a specific attempt at modeling and understanding the workload management system that led to an increased attention to the usage of shared resources . there were other specific practical issues that emerged during the deployment and troubleshooting of the system and that led to the awareness of some fundamental design or implementation mistake that was made . here is a short list , where the fundamental principle that should correct the fundamental mistake that was made is listed : 1 . * communication among services should always be reliable : * * always applying double - commit and rollback for network communications . * going through the filesystem for local communications . + in general , forms of communication that do nt allow for data or messages to be lost in a broken pipe lead to easier recovery from system or process crashes . where network communication is necessary , database - like techniques have to be used . every process , object or entity related to the job lifecycle should have another process , object or entity in charge of its well - being . * + automatic fault recovery can only happen if every entity is held accountable and accounted for . * information repositories should be minimized ( with a clear identification of authoritative information ) . * + many of the software components that were integrated in the edg - wp1 solution are stateful and include local repositories for request information , in the form of local queues , state files , database back - ends . only one site with authoritative information about requests has to be identified and kept . monolithic , long - lived processes should be avoided . * + dynamic memory programming , using languages and techniques that require explicit release of dynamically allocated objects , can lead to leaks of memory , descriptors and other resources . experimental , r&d code can take time to leak - proof , so it should possibly not be linked to system components that are long - lived , as it can accelerate system resource starvation . short - lived , easy - to - recover components are a clean and very practical workaround in this case more thought should be devoted to efficiently and correctly recovering a service rather than to starting and running it . * + this is again a consequence of the previous point : the capability to quickly recover from failures or interruption helps in assuring that system components ` can ' be short - lived , either by design or by accident . edg - wp1 has been distributing jobs over the edg testbed in a continuous fashion for one and a half years now , with a software solution where existing grid technology was integrated wherever possible . the experience of understanding the direct and indirect interplay of the service components could not be reduced to a simple _ scalability _ evaluation . this because understanding and removing _ bottlenecks _ is significantly complicated by non - linear and non - continuous effects in the system . in this process , few principles that apply to the very complex practice of distributed systems operations were learned the hard way ( i.e. not by just reading some good book on the subject ) . edg - wp1 tried to incorporate these principles in its second major software release that will shortly face deployment in the edg testbed . j. frey , t. tannenbaum , i. foster , m. livny , s. tuecke , `` condor - g : a computation management agent for multi - institutional grids '' , _ proceedings of the tenth ieee symposium on high performance distributed computing ( hpdc10 ) _ , 2001 datagrid wp1 members ( c. anglano _ et al . _ ) , * integrating grid tools to build a computing resource broker : activities of datagrid wp1 * " presented at the chep 2001 conference , beijing ( p. 708 in the proceedings )
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application users have now been experiencing for about a year with the standardized resource brokering services provided by the workload management package of the eu datagrid project ( wp1 ) .
understanding , shaping and pushing the limits of the system has provided valuable feedback on both its design and implementation .
a digest of the lessons , and better practices " , that were learned , and that were applied towards the second major release of the software , is given .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
in 1979 siemens and rasmussen formulated a model describing the hadron production in ne + na f reactions at the beam energy of 800 mev per nucleon @xcite . the physical picture behind the model was that the fast hydrodynamic expansion of the produced hadronic matter leads to a sudden decoupling of hadrons and freezing of their momentum distributions , which retain their thermal character ( although modified by the collective expansion effects ) until the observation point . in their own words , siemens and rasmussen described the collision process as follows : `` central collisions of heavy nuclei at kinetic energies of a few hundred mev per nucleon produce fireballs of hot , dense nuclear matter ; such fireballs explode , producing blast waves of nucleons and pions '' . in this way , with ref . @xcite , the concept of the blast waves of hadrons and the blast - wave model itself entered the field of relativistic heavy - ion collisions . although the model of siemens and rasmussen was motivated by an earlier hydrodynamic calculation by bondorf , garpman , and zimanyi @xcite , the results presented in ref . @xcite were not obtained by solving the hydrodynamic equations but followed from the specific assumptions on the freeze - out conditions . the most important ingredient of the model was the spherically symmetric expansion of the shells of matter with constant radial velocity . with an additional assumption about the times when such shells disintegrate into freely streaming hadrons ( this point will be discussed in a greater detail in sect . [ sect : rad ] ) siemens and rasmussen obtained the formula for the momentum distribution of the emitted hadrons @xcite @xmath0 . \label{sr1}\ ] ] in eq . ( [ sr1 ] ) @xmath1 is a normalization factor , @xmath2 denotes the hadron energy , @xmath3 is the temperature of the fireball ( the same for all fluid shells ) , and @xmath4 is the lorentz gamma factor with @xmath5 denoting the radial collective velocity ( radial flow ) . a dimensionless parameter @xmath6 is defined by the equation @xmath7 small values of @xmath5 ( and @xmath6 ) correspond to small expansion rate and , as expected , a simple boltzmann factor is obtained from eq . ( [ sr1 ] ) in the limit @xmath8 , @xmath9 the fits to the data based on the formula ( [ sr1 ] ) gave @xmath3 = 44 mev and @xmath5 = 0.373 . interestingly , the value of the radial flow @xmath5 turned out to be quite large suggesting the strong collective behavior . this was an unexpected feature summarized by the authors with the statement : `` monte carlo studies suggest that ne + na f system is too small for multiple collisions to be very important , thus , this evidence for a blast feature may be an indication that pion exchange is enhanced , and the effective nucleon mean free path shortened in dense nuclear matter '' . below we shall analyze the formal steps leading to eq . ( [ sr1 ] ) . our starting point is the expression defining the momentum distribution of particles as the integral of the phase - space distribution function @xmath10 over the freeze - out hypersurface @xmath11 , i.e. , the renowned cooper - frye formula @xcite , @xmath12 the three - dimensional element of the freeze - out hypersurface in eq . ( [ cf1 ] ) may be obtained from the formula @xmath13 where @xmath14 is the levi - civita tensor and @xmath15 are the three independent coordinates introduced to parameterize the hypersurface . we note that for systems in local thermodynamic equilibrium we have @xmath16 where the function @xmath17 is the equilibrium distribution function @xmath18^{-1}. \label{eq}\ ] ] here the case @xmath19 corresponds to the fermi - dirac ( bose - einstein ) statistics , and the limit @xmath20 yields the classical ( boltzmann ) statistics . for a static fireball one finds @xmath21 and eq . ( [ cf2 ] ) is reduced to the formula @xmath22 where @xmath23 is the volume of the system . ( [ cf3 ] ) agrees with eq . ( [ sr2 ] ) in the classical limit if the normalization constant @xmath1 is taken as @xmath24 + for spherically symmetric freeze - outs it is convenient to introduce the following parameterization of the space - time points on the freeze - out hypersurface @xcite @xmath25 the freeze - out hypersurface is completely defined if a curve , i.e. , the mapping @xmath26 in the @xmath27 space is given . this curve defines the ( freeze - out ) times when the hadrons in the shells of radius @xmath28 stop to interact , see fig . [ fig : tr ] . the range of @xmath29 may be always restricted to the interval : @xmath30 . the three coordinates : @xmath31 , \theta \in [ 0,\pi]$ ] , and @xmath32 $ ] play the role of the variables @xmath15 appearing in eq . ( [ d3sigma ] ) . hence , the element of the spherically symmetric hypersurface has the form @xmath33 where the prime denotes the derivatives taken with respect to @xmath29 . besides the spherically symmetric hypersurface we introduce the spherically symmetric ( hydrodynamic ) flow @xmath34 where @xmath35 is the lorentz factor , @xmath36 . in a similar way the four - momentum of a hadron is parameterized as @xmath37 , \label{pmurad}\ ] ] and we find the two useful expressions : @xmath38 @xmath39 we note that the spherical symmetry allows us to restrict our considerations to the special case @xmath40 . in the case of the boltzmann statistics , with the help of eqs . ( [ cf1 ] ) , ( [ purad ] ) and ( [ sigmaprad ] ) , we obtain the following form of the momentum distribution @xmath41 r^2(\zeta ) d\zeta . \label{dnd3prad1}\ ] ] here @xmath42 and @xmath43 are functions of @xmath29 , and the parameter @xmath6 is defined by eq . ( [ a ] ) . the thermodynamic parameters @xmath3 and @xmath44 may also depend on @xmath29 . to proceed further we need to make certain assumptions about the @xmath29-dependence of these quantities . in particular , to obtain the model of siemens and rasmussen we assume that the thermodynamic parameters as well as the transverse flow velocity are constant @xmath45 moreover , we should assume that the freeze - out curve in the @xmath46 space satisfies the condition @xmath47 in this case we obtain the formula @xmath48 \int\limits_0 ^ 1 r^2(\zeta ) { dr \over d\zeta } d\zeta . \label{dnd3prad2}\ ] ] equation ( [ dnd3prad2 ] ) coincides with eq . ( [ sr1 ] ) if we use eq . ( [ z ] ) and make the following identification @xmath49 note that the quantity @xmath50 does not necessarily denote the maximum value of the radius of the system , see the dotted line on the left - hand - side of fig . [ fig : tr ] . an interesting and perhaps unexpected feature of the model proposed by siemens and rasmussen is the relation between the times and positions of the freeze - out points , see eq . ( [ dtvdr ] ) illustrated on the right - hand - side part of fig . [ fig : tr ] . ( [ dtvdr ] ) indicates that the fluid elements which are further away from the center freeze - out later . moreover , taking into account eq . ( [ dtvdr ] ) in the formula ( [ d3sigmarad ] ) we find that the four - vector describing the hypersurface is parallel to the four - vector describing the flow , compare eqs . ( [ d3sigmarad ] ) and ( [ umurad1 ] ) giving @xmath51 in this case . as we shall see the same features are assumed in the single - freeze - out model @xcite . it is worth to emphasize that in the hydrodynamic approach the @xmath27 freeze - out curves contain the space - like and time - like parts . the treatment of the space - like parts leads to conceptual problems since particles emitted from such regions of the hypersurface enter again the system and the hydrodynamic description of such regions ( combined with the use of the cooper - frye formula ) is inadeqate . recently much work has been done to develop a consistent description of the freeze - out process from the space - like parts @xcite . however , very often only a quantitative argument is presented @xcite that the contributions from the space - like parts are small and may be neglected compared to the contributions from the time - like regions . the choice of siemens and rasmussen seems to have anticipated such arguments . the model presented above is appropriate for the low - energy scattering processes where the two nuclei completely merge at the initial stage of the collision and further expansion of the system is , to large extent , isotropic . at higher energies such a picture is not valid anymore and , following the famous paper by bjorken @xcite , the boost - invariant and cylindrically symmetric models have been introduced to describe the collisons . moreover , the assumption about the cylindrical symmetry is valid only for the most central data . ] . the boost - invariance ( symmetry with respect to the lorentz transformations ) may be incorporated in the hydrodynamic equations , kinetic equations , and also in the modeling of the freeze - out process . in the latter case , the appropriate formalism was developed by schnedermann , sollfrank , and heinz @xcite . the ansatz for the boost - invariant , cylindrically symmetric freeze - out hypersurface has the form @xmath52 here , the parameter @xmath53 is the space - time rapidity . at @xmath54 the longitudinal coordinate @xmath55 is also zero and the variable @xmath56 coincides with the time coordinate @xmath43 . similarly to the spherical expansion discussed in sect . [ sect : rad ] , the boost - invariant freeze - out hypersurface is completely defined if the freeze - out curve @xmath57 is given . this curve defines the freeze - out times of the cylindrical shells with the radius @xmath58 . because of the boost - invariance it is enough to define this curve at @xmath59 , since for finite values of @xmath55 the freeze - out points may be obtained by the lorentz transformation . the volume element of the freeze - out hypersurface is obtained from eq . ( [ d3sigma ] ) , @xmath60 similarly to eq . ( [ xmubinv ] ) the boost - invariant four - velocity field has the structure @xmath61 we note that the longitudinal flow is simply @xmath62 ( as in the one - dimensional bjorken model ) , whereas the transverse flow is @xmath63 . with the standard parameterization of the particle four - momentum in terms of rapidity @xmath64 and the transverse mass @xmath65 , @xmath66 we find @xmath67 and @xmath68 \rho(\zeta){\tilde \tau}(\zeta ) d\zeta d\alpha_\parallel d\phi . \label{sigmapbinv}\ ] ] for the boltzmann statistics , with @xmath69 , the cooper - frye formalism gives the following momentum distribution @xmath70 \nonumber \\ & & \times \exp\left[-\beta m_\perp \hbox{cosh}(\alpha_\perp ) \hbox{cosh}(\alpha_\parallel - y)+ \beta p_\perp \hbox{sinh}(\alpha_\perp ) \cos(\phi-\varphi ) \right ] . \label{bmdn1}\end{aligned}\ ] ] the form of eq . ( [ bmdn1 ] ) shows explicitly that the distribution @xmath71 is independent of @xmath64 and @xmath72 , in accordance with our assumptions of the boost - invariance and cylindrical symmetry . the integrals over @xmath53 and @xmath73 in eq . ( [ bmdn1 ] ) are analytic and lead to the bessel functions @xmath74 and @xmath75 , @xmath76 i_0 \left[\beta p_\perp \hbox{sinh}(\alpha_\perp)\right ] \int\limits_0 ^ 1 d\zeta \,\ , \rho(\zeta ) { \tilde \tau}(\zeta ) { d\rho \over d\zeta } \nonumber \\ & & - { e^{\beta \mu } \over 2 \pi^2 } p_\perp k_0\left[\beta m_\perp \hbox{cosh}(\alpha_\perp ) \right ] i_1 \left[\beta p_\perp \hbox{sinh}(\alpha_\perp)\right ] \int\limits_0 ^ 1 d\zeta \,\ , \rho(\zeta ) { \tilde \tau}(\zeta ) { d{\tilde \tau } \over d\zeta}. \nonumber \\ \label{kandi}\end{aligned}\ ] ] in the spirit of the blast - wave model of siemens and rasmussen we have assumed here that the radial velocity is constant , @xmath77 , otherwise the bessel functions should be kept under the integral over @xmath29 . in order to achieve the simplest possible form of the model , the common practice is to neglect the second line of eq . ( [ kandi ] ) . this procedure means that one assumes implicitly the freeze - out condition @xmath78 . in this case the boost - invariant blast - wave model is reduced to the formula @xmath79 i_0 \left[\beta p_\perp \hbox{sinh}(\alpha_\perp)\right ] . \label{k1i0a}\ ] ] where the constant has absorbed the factor @xmath80 . ( [ k1i0a ] ) forms the basis of numerous phenomenological analyses of the transverse - momentum spectra measured at the sps and rhic @xcite energies . the main drawback of the formalism outlined above is that it neglects the effect of the decays of hadronic resonances . such an approach may be justified at lower energies but should be improved at the relativistic energies where most of the light particles are produced in the decays of heavier resonance states . the expressions giving the rapidity and transverse momentum spectra of particles originating from two- and three - body decays of the resonances with a specified momentum distribution were worked out by sollfrank , koch , and heinz @xcite . their formulae may also be used to account for the feeding from the resonances in the blast - wave model , as already proposed in ref . @xcite . in other words , we wish to stress that the choice of the freeze - out hypersurface and of the flow profile are elements _ completely independent _ of the treatment of the resonances . both are important with the latter being the basic ingredient in the calculation of particle abundances and the key to success of thermal models . at the sps and rhic energies it is important to include not only the decays of the most common resonances such as @xmath81 or @xmath82 , but also of much heavier states . although their contributions are supressed by the boltzmann factor , their number increases strongly with the mass @xcite , hence their role can be easily underestimated . the effects of sequential decays of heavy resonances were first realized in statistical analyses of the ratios of hadronic abundances / multiplicities ( for recent results see @xcite ) which showed that the statistical models give a very good description of the data , provided most of the hadrons appearing in the particle data tables are included in the calculations . in order to discuss the role of the sequential decays of the resonances it is convenient to start with a general formalism giving the lorentz - invariant phase - space density of the measured particles @xcite @xmath83 . \nonumber \\ \label{ornik1}\end{aligned}\ ] ] here the indices @xmath84 label hadrons in one chain of the sequential decays . the first resonance is produced on the freeze - out hypersurface and has the label @xmath85 . the final hadron has the label @xmath86 , for more details see @xcite . the function @xmath87 is the probability distribution for a resonance with momentum @xmath88 to produce a particle with momentum @xmath89 in a two - body decay @xmath90 the function @xmath87 satisfies the normalization condition @xmath91 where @xmath92 is the branching ratio for a given decay channel and @xmath93 is the momentum ( energy ) of the emitted particle in the resonance s rest frame ( a generalization to three - body decays is straightforward and explained in refs . @xcite ) . integration of eq . ( [ ornik1 ] ) over all space - time positions gives the formula for the momentum distribution @xmath94 . \nonumber \\ \label{ornik2}\end{aligned}\ ] ] equation ( [ ornik2 ] ) serves as the starting point to prove that for constant values of the thermodynamic parameters on the freeze - out hypersurface the ratios in the full phase - space ( @xmath95 ) are the same as in the local fluid elements . in this way , a connection between the measured ratios and the local thermodynamic parameters is obtained @xcite . one may also check that for the boost - invariant systems it is enough to consider the ratios at any value of the rapidity to infer the values of the thermodynamic parameters @xcite . the experimental rhic data show , however , that the rapidity distributions are of gaussian shape and the thermodynamic parameters vary with rapidity ( the measured @xmath96 ratio depends on @xmath64 ) , hence , the system created at rhic is , strictly speaking , not boost - invariant . in this situation the relation between the measured ratios and thermodynamic parameters is not obvious . fortunately , the rhic data show also a rather flat rapidity distribution and constant ratios in the rapidity range @xmath97 @xcite . in this region ( the central part of the broad gaussian ) the system to a good approximation may be treated as boost - invariant and the standard analysis of the ratios may be performed to obtain the thermodynamic parameters at @xmath98 . the analysis of the ratios of hadron multiplicities measured at rhic gives a typical temperature of 170 mev . on the other hand , the analysis of the spectra based on eq . ( [ k1i0a ] ) gives a lower temperature of about 100 - 140 mev . such a situation was observed already at the sps energies , which motivated the introduction of the concept of two different freeze - outs . certainly , if the spectra contain important contributions from high lying states , the value of @xmath3 obtained from the blast - wave formula fitted to the spectra can not be interpreted as the temperature of the system in the precise thermodynamic sense . first , the contributions from the resonances ( feeding mostly the low - momentum region ) should be subtracted from the spectra of light hadrons , giving the insight to the properties of the primordial particles . using other words , we may argue that the calculation of the ratios should include the same number of the resonances as the corresponding calculation of the spectra . an example of such a calculation is the single - freeze - out model formulated in refs . @xcite . in this model the decays of the resonances as well as the transverse flow change the spectra of the primordial particles in such a way that it is possible to describe well the spectra and the ratios with a single value of the temperature . the basic effect here is that the hadronic decays lead to effective cooling of the spectra . similarly to the original blast - wave models discussed above , the single freeze - out model assumes a certain form of the freeze - out hypersurface in the minkowski space . in this case it is defined by the constant value of the proper time @xmath99 the transverse size of the system is defined by the parameter @xmath100 , @xmath101 and the velocity field at freeze - out is taken in the hubble - like form ) - ( [ hubflow ] ) with hydrodynamic calculation see ref . ] @xmath102 the natural parameterization of the freeze - out hypersurface has the form @xmath103 which may be considered as the special case of the formula ( [ xmubinv ] ) . ( [ d3sigma ] ) leads to the following expression defining the volume element @xmath104 a very important feature of the choice ( [ tau ] ) - ( [ hubflow ] ) is that the volume element is proportional to the four - velocity field . this feature holds also in the model of siemens and rasmussen . in this case the treatment of the resonance is very much facilitated . in particular , eq . ( [ ornik2 ] ) may be rewriten in the form @xmath105 \nonumber \\ & = & \int d\sigma \left ( x_{n}\right ) p_{1}\,\cdot u\left ( x_{n}\right ) \,f_{1}\left [ p_{1}\cdot u\left ( x_{n}\right ) \right ] , \label{ornik3}\end{aligned}\ ] ] where we have introduced the notation @xmath106 \label{trsim } \nonumber \\ & & = \int \frac{d^{3}p_{i}}{e_{p_{i}}}b\left ( p_{i},p_{i-1}\right ) p_{i}\,\cdot u\left ( x_{n}\right ) \,f_{i}\left [ p_{i}\cdot u\left ( x_{n}\right ) \right ] . \label{acta1}\end{aligned}\ ] ] in the local rest frame , the iterative procedure defined by eq . ( [ acta1 ] ) becomes a simple one - dimensional integral transform @xmath107 where @xmath108 . ( [ acta1 ] ) and ( [ acta2 ] ) allow us to deal with a very large number of decays in the very efficient way , very similar to that used in the calculation of the hadron abundances . the model described above may be generalized to the non boost - invariant version in the minimal way by the modification of the system boundaries . introducing a dependence of the transverse size on the longitudinal coordinate @xmath55 ( or @xmath53 ) , we break explicitly the assumption of the boost - invariance . at the same time , however , the local properties of the hypersurface and flow remain unchanged allowing us to treat the resonances in the same simple way as described in the previous section . since the measured rapidity distributions are approximately gaussian , it is natural to start with the gaussian ansatz for the dependence of the transverse size on the parameter @xmath53 and restrict the region of the integration over @xmath109 to the interval @xmath110.\ ] ] the original boost - invariant version is recovered in the limit @xmath111 . using the values of the thermodynamic parameters obtained from the boost - invariant version of the model applied to the hadronic ratios measured at midrapidity ( @xmath3 = 165.6 mev and @xmath112 mev ) , we are left with three extra parameters ( @xmath113 , @xmath114 , and @xmath82 ) , which should be fitted to the @xmath115-spectra collected at different values of the rapidity . the result of such a fit to the available brahms data on @xmath116 , @xmath117 , @xmath118 , and @xmath119 production are shown in fig . [ fig : brpt ] . the optimal values of the parameters found in the fit are : @xmath120 fm , @xmath121 , and @xmath122 . one can see that the model reproduces the data very well in a wide range of the transverse - momentum and rapidity . in fig . [ fig : bry ] we show the model rapidity distributions compared to the data . small discrepancies ( of about 10% ) between the model and the data may be seen for the pions at @xmath98 . note that the comparison of the rapidity distributions in fig . [ fig : bry ] is done with a linear scale ; definitely , small discrepancies may be expected for a such simplified description of the freeze - out . it should be emphasized that the non - boost invariant version of the model presented above is not capable of describing correctly the @xmath96 ratio . in the present framework this requires an introduction of the rapidity dependence of the baryon chemical potential . we have discussed several parameterizations of the freeze - out conditions in the relativistic heavy - ion collisions . we have argued that the single freeze - out model used to describe the rhic data is a natural development of the blast - wave models worked out , among others , by siemens , rasmussen , heinz , schnedermann , and sollfrank . the main advantage of the single - freeze - out model is that it includes all well established resonance decays , allowing us to treat the chemical and thermal freeze - out as essentially one phenomenon . in this respect , the single - freeze - out model is very similar to the original blast - wave model . further similarities concern the shape of the freeze - out hypersurface ( only the time - like parts are considered ) and the strict use of the cooper - frye formula . due to the limited space , we have not discussed here the variety of models where , instead of the cooper - fry formula , the so - called emission functions are introduced and modeled . an example of such an a approach is the buda - lund model @xcite .
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popular parameterizations of the freeze - out conditions in relativistic heavy - ion collisions are discussed .
similarities and differences between the blast - wave model and the single - freeze - out model , both used recently to interpret the rhic data , are outlined .
a non - boost - invariant extension of the single - freeze - out model is proposed and applied to describe the recent brahms data .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
presently planned @xmath0 linear collider ( lc ) projects will operate at an initial center of mass system ( cms ) energy of about 500 gev , with upgrades to higher energies designed in from the start . the tev class colliders tesla @xcite and nlc / jlc @xcite target 800 gev and 1 - 1.5 tev , respectively , as their maximum cms energies . increasing the energy further would require either a change in acceleration technology or an extension in accelerator length beyond the presently foreseen 30 - 40 km @xcite . this would also increase the number of active elements , which will likely decrease the overall efficiency of such a facility . the nature of the new physics which will hopefully be discovered and studied at the lhc and a tev class lc will determine the necessity and importance of exploring the multi - tev range with a precision machine such as an @xmath2 collider . this paper summarizes the work of the e3 subgroup 2 on multi - tev colliders of the snowmass 2001 workshop ` the future of particle physics ' . based on our knowledge today , the case for the multi - tev collider rests on the following physics scenarios : ( 200,1 ) ( 25,-357)_presented at the aps / dpf / dpb summer study on the future of particle physics ( snowmass 2001 ) , _ ( 120,-369)_30 june - 21 july 2001 , snowmass , colorado , usa _ * the study of the higgs + for a light higgs , a multi - tev @xmath2 collider can access with high precision the triple higgs coupling , providing experimenters with the opportunity to measure the higgs potential . the large event statistics will allow physicists to measure rare higgs decays such as @xmath3 . for heavy higgses , predicted by e.g. supersymmetric models , the range for discovery and measurement will be extended for masses up to and beyond 1 tev . * supersymmetry + in many susy scenarios only a subset of the new sparticles will be light enough to be produced directly at a tev class lc . some of the heavier sparticles will be discovered at the lhc , but a multi - tev lc will be needed to complete the spectrum and to precisely measure the heavy sparticles properties ( flavor , mass , width , couplings ) . furthermore , polarized beams will help disentangle mixing parameters and aid cp studies . ultimately we _ will _ need to measure all sparticles as precisely as possible to fully pin down and test the underlying theory . * new resonances + many alternative theories and models for new physics predict new heavy resonances with masses larger than 1 tev . if these new resonances ( e.g. new gauge bosons , kaluza - klein resonances , or @xmath4 resonances ) have masses in the 1 tev - 5 tev range , a multi - tev collider becomes a particle factory , similar to lep for the @xmath5 . the new particles can be produced directly and their properties can be accurately determined . * no new particles + if _ no _ new particles are observed directly , apart from perhaps one light higgs particle , then a multi - tev collider will probe new physics indirectly ( extra dimensions , @xmath6 , contact interactions ) at scales in the range of 30 - 400 tev via precision measurements . * unexpected phenomena + this is probably the most exciting of all : perhaps nature has chosen a road as yet not explored ( extensively ) by our imagination . recent examples of new ideas are string quantum gravity effects , non - commutative effects , black hole formation , nylons , and split fermions . .event rates for several processes in the multi - tev range , for 1 ab@xmath7 integrated luminosity . [ cols="^,^,^ " , ] [ tab : sum ] linear @xmath0 colliders operating in the multi - tev energy range are likely to be based on the clic two - beam acceleration concept . to achieve a large luminosity , such an accelerator would need to operate in the high beamstrahlung region , rendering experimentation at such a collider more challenging . studies so far indicate that this is not a substantial handicap , and the precision physics expected from an @xmath0 collider will be possible . the two - beam accelerator technology is not yet available today for use at a large scale collider . r&d on this technology will continue until 2006 at least , after which if no bad surprises emerge one can plan for a full technical design of such a collider . from the physics program side , a multi - tev collider has a large potential to push back the high energy horizon further , up to scales of 1000 tev , where if the higgs is light new physics can no longer hide from experiment . if no new scale is found by then we have to revise our understanding of nature . a multi - tev collider with high luminosity can be used for precision measurements in the higgs sector . it can precisely measure the masses and couplings of heavy sparticles , thereby completing the susy spectrum . if extra dimensions or even black holes pop up in the multi - tev range , such a collider will be a precision instrument to study quantum gravity in the laboratory . the physics reach , as envisioned today , for a multi - tev collider is summarized in table [ tab : sum ] . in short a collider with @xmath8 3 - 5 tev is expected to break new grounds , beyond the lhc and a tev class lc . , desy 2001 - 011 . , slac - r-571 . jlc-1 , jlc group , s matsumoto et al . , kek report 92 - 16 . p. burrows and r. patterson , _ lc expandability and upgradability _ , these proceedings .. m. battaglia , hep - ph/0103338 . the clic study team , _ a 3 tev e@xmath9e@xmath10 linear collider based on clic technology _ , cern 2000 - 008 . braun et al . , proc . of 18th international conference on high energy accelerators ( heacc2001 ) , 26 - 30 march 2001 , tsukuba , japan , and clic note 473 ; see also http://geschonk.home.cern.ch/geschonk/ r. corsini et al . , cern / ps 2001 - 030 ( ae ) , and 2001 particle accelerator conference ( pac2001 ) , chicago , illinois , usa , june 18 - 22 , 2001 d. asner et al . , hep - ex/0111056 v. telnov , _ photon colliders at multi - tev energies _ , these proceeedings . r. assmann and f. zimmerman , _ polarization at clic _ , these proceeedings . g.guignard , _ the clic study _ , these proceedings . d. schulte , _ machine - detector interface at clic _ , these proceedings + d. schulte , cern - ps-99 - 066 . r. settles , _ detector requirements at multi - tev lc _ , these proceedings . m. battaglia , in proceedings of the lcws2000 workshop , p831 . m. pohl and h.j . schreiber , _ simdet - a parametric monte carlo for a tesla detector _ , desy 99 - 030 . m. battaglia , adaption of the delphi pvec package . m. battaglia , e. boos , and w. yao , _ studying the higgs potential at the linear collider _ , hep - ph/0111276 . m. battaglia , a.de roeck , _ determination of the muon yukawa coupling at high energy @xmath0 linear colliders _ , these proceedings , hep - ph/0111307 . j. f gunion ( 1997 ) , hep - ph/9703203 . t. plehn and d. rainwater , phys . lett . * b520 * , 108 ( 2001 ) , hep - ph/017180 . marco battaglia et al . , hep - ph/0106204 . m. battaglia , a. kiiskinen and a. ferrari,_study of charged higgs bosons _ , these proceedings , hep - ex/0112015 . g. blair , w. porod , p.m. zerwas phys . rev . * d63 * ( 2001 ) 017703 , hep - ph/0007107 . g. moortgat - pick , _ physics opportunities with polarized @xmath11 and @xmath12 beams at a linear collider _ , these proceedings . g. wilson in proceedings of the lcws2000 workshop , p485 . m. battaglia , private communication . m. klasen , talk at the workshop on gamma - gamma colliders , chicago 2001 . a. ferrari , _ study of majorana neutrinos at clic _ , these proceedings . m. battaglia , s. riemann , s. de curtis and d. dominici , _ probing new scales at e+e- linear collider _ , these proceedings ; + battaglia et al . , hep - ph/0101114 . e. eichten , k lane and m. peskin , phys . lett * 50 * ( 1983 ) 811 ; h. kroha , phys . rev . * d46 * ( 1992 ) 58 . t. hambye and k. riesselmann , phys . * d55 * ( 1997 ) 7255 , hep - ph/9708416 . t. barklow , private communication . d. barger et al . , phys rev . * d52 * ( 1995 ) 3815 , phys rev . * d55 * ( 1997 ) 142 . m. butterworth , b.e . cox and j.r . forshaw , in preparation . a. dobado et al . , phys rev * d62 * ( 2000 ) 05501 . a. de roeck , _ ww scattering at clic _ , these proceedings . t. rizzo , hep - ph/0108235 . n. arkani - hamed , s. dimopoulos , and g. dvali , b429 263 1998 , and d59 086004 1999 ; i. antoniadis , n. arkani - hamed , s. dimopoulos , and g. dvali , b436 257 1998 . see , for example , i. antoniadis , b246 377 1990 ; i. antoniadis , c. munoz and m. quiros , b397 515 1993 ; i. antoniadis and k. benalki , b326 69 1994 and a15 4237 2000 ; i. antoniadis , k. benalki and m. quiros , b331 313 1994 . l. randall and r. sundrum , 83 3370 1999 . for an introduction to add phenomenology , see g.f . giudice , r. rattazzi and j.d . wells , b544 3 1999 ; t. han , j.d . lykken and r. zhang , phys . rev . * d59 * , 105006 ( 1999 ) , e.a . mirabelli , m. perelstein and m.e . peskin , phys . lett . * 82 * , 2236 ( 1999 ) ; j.l . hewett , phys . rev . lett . * 82 * , 4765 ( 1999 ) ; t.g . rizzo , d60 115010 1999 . m. battaglia , a. de roeck and t. rizzo , _ graviton production at clic _ , these proceedings , hep - ph/0112169 . h. davousdiasl and t. rizzo , hep - ph/0104199 . j hewett , f. petriello , and t. rizzo , hep - ph/0010354 . j. lykken , talk at lcws2000 . n. arkani - hamed , y. grossman , m. schmaltz , phys . * d61 * ( 2000 ) 115004 , hep - ph/9909411 . r. barbieri , l. j. hall , y. nomura , phys . * d63 * ( 2001 ) 105007 , hep - ph/0011311 .
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the physics at an @xmath0 linear collider with a center of mass energy of 3 - 5 tev is reviewed .
the following topics are covered : experimental environment , higgs physics , supersymmetry , fermion pair - production , @xmath1 scattering , extra dimensions , non - commutative theories , and black hole production .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
attempts to geometrical unification of gravity with other interactions , using higher dimensions other than our conventional @xmath1 space time , began shortly after invention of the special relativity ( * sr * ) . nordstrm was the first who built a unified theory on the base of extra dimensions @xcite . tight connection between sr and electrodynamics , namely the lorentz transformation , led kaluza @xcite and klein @xcite to establish @xmath0 versions of general relativity ( * gr * ) in which electrodynamics rises from the extra fifth dimension . since then , considerable amount of works have been focused on this idea either using different mechanism for compactification of extra dimension or generalizing it to non compact scenarios ( see e.g. ref . @xcite ) such as brane world theories @xcite , space time matter or induced matter ( * i m * ) theories @xcite and references therein . the latter theories are based on the campbell magaard theorem which asserts that any analytical @xmath6dimensional riemannian manifold can locally be embedded in an @xmath7dimensional ricci flat riemannian manifold @xcite . this theorem is of great importance for establishing @xmath1 field equations with matter sources locally to be embedded in @xmath0 field equations without _ priori _ introducing matter sources . indeed , the matter sources of @xmath1 space times can be viewed as a manifestation of extra dimensions . this is actually the core of i m theory which employs gr as the underlying theory . on the other hand , jordan @xcite attempted to embed a curved @xmath1 space time in a flat @xmath0 space time and introduced a new kind of gravitational theory , known as the scalar tensor theory . following his idea , brans and dicke @xcite invented an attractive version of the scalar tensor theory , an alternative to gr , in which the weak equivalence principle is saved and a non minimally scalar field couples to curvature . the advantage of this theory is that it is more machian than gr , though mismatching with the solar system observations is claimed as its weakness @xcite . however , the solar system constraint is a generic difficulty in the context of the scalar tensor theories @xcite , and it does not necessarily denote that the evolution of the universe , at all scales , should be close to gr , in which there are some debates on its tests on cosmic scales @xcite . although it is sometimes desirable to have a higher dimensional energy momentum tensor or a scalar field , for example in compactification of extra curved dimensions @xcite , but the most preference of higher dimensional theories is to obtain macroscopic @xmath1 matter from pure geometry . in this approach , some features of a @xmath0 vacuum brans dicke ( * bd * ) theory based on the idea of i m theory have recently been demonstrated @xcite , in where the role of gr as fundamental underlying theory has been replaced by the bd theory of gravitation . actually , it has been shown that @xmath0 vacuum bd equations , when reduced to four dimensions , lead to a modified version of the @xmath1 brans dicke theory which includes an induced potential . whereas in the literature , in order to obtain accelerating universes , inclusion of such potentials has been considered in _ priori _ by hand . a few applications and a @xmath8dimensional version of this approach have been performed @xcite . though , in refs . @xcite , it has also been claimed that their procedure provides explicit definitions for the effective matter and induced potential . besides , some misleading statements and equations have been asserted in ref . @xcite , and hence we have re derived the procedure in section @xmath9 . actually , the reduction procedure of a @xmath0 analogue of the bd theory , with matter content , on every hypersurface orthogonal to an extra cyclic dimension ( recovering a modified bd theory described by a 4metric coupled to two scalar fields ) has previously been performed in the literature @xcite . however , the key point of i m theories are based on not introducing matter sources in @xmath0 space times . in addition , recent measurements of anisotropies in the microwave background suggest that our ordinary @xmath1 universe should be spatially flat @xcite , and the observations of type ia supernovas indicate that the universe is in an accelerating expansion phase @xcite . hence , the universe should mainly be filled with a dark energy or a quintessence which makes it to expand with acceleration @xcite . then after an intensive amount of work has been performed in the literature to explain the acceleration of the universe . in this work , we explore the friedmann robertson walker ( * frw * ) type cosmology of a @xmath0 vacuum bd theory and obtain solutions and related conditions . this model has extra terms , such as a scalar field and scale factor of fifth dimension , which make it capable to present accelerated universes beside decelerated ones . in the next section , we give a brief review of the induced modified bd theory from a @xmath0 vacuum space time to rederive the induced energy momentum tensor , as has been introduced in ref . @xcite , for our purpose to employ the energy density and pressure . in section @xmath10 , we consider a generalized frw metric in the @xmath0 space time and specify frw cosmological equations and employ the weak energy condition ( * wec * ) to obtain the energy density and pressure conditions . then , we probe two special cases of a constant scale factor of the fifth dimension and a constant scalar field . in section @xmath11 , we proceed to exhibit that @xmath0 vacuum bd equations , employing the generalized frw metric , are equivalent , in general , to the corresponding vacuum @xmath1 ones . this equivalency can be viewed as the main point within this work which distinguishes it from refs . @xcite . in section @xmath12 , we find exact solutions for flat geometries and proceed to get solutions fulfilling the wec while being compatible with the recent observational measurements . we also provide a few tables and figures for a better view of acceptable range of parameters . finally , conclusions are presented in the last section . following the idea of i m theories @xcite , one can replace gr by the bd theory of gravitation as the underlying theory @xcite . for this purpose , the action of @xmath0 brans dicke theory can analogously be written in the jordan frame as @xmath13=\int\sqrt{|{}^{_{(5)}}g| } \left ( \phi \ ^{^{(5)}}\!r-\frac{\omega}{\phi}g^{_{ab}}\phi_{,_{a}}\phi_{,_{b}}+ 16\pi l_{m } \right ) d^{5}x\ , , \ ] ] where @xmath14 , the capital latin indices run from zero to four , @xmath15 is a positive scalar field that describes gravitational coupling in five dimensions , @xmath16 is @xmath0 ricci scalar , @xmath17 is the determinant of @xmath0 metric @xmath18 , @xmath19 represents the matter lagrangian and @xmath20 is a dimensionless coupling constant . the field equations obtained from action ( 1 ) are @xmath21 and @xmath22 where @xmath23 , @xmath24 is @xmath0 einstein tensor , @xmath25 is @xmath0 energy momentum tensor , @xmath26 . also , in order to have a non ghost scalar field in the conformally related einstein frame , i.e. a field with a positive kinetic energy term in that frame , the bd coupling constant must be @xmath27 @xcite . as explained in the introduction , we propose to consider a @xmath0 vacuum state , i.e. @xmath28 , where equations ( 2 ) and ( 3 ) read @xmath29 and ) for later on convenient . ] @xmath30 for cosmological purposes one usually restricts attention to @xmath0 metrics of the form , in local coordinates @xmath31 , @xmath32 where @xmath33 represents the fifth coordinate , the greek indices run from zero to three and @xmath34 . it should be noted that this ansatz is restrictive , but one limits oneself to it for reasons of simplicity . assuming the @xmath0 space time is foliated by a family of hypersurfaces , @xmath35 , defined by fixed values of the fifth coordinate , then the metric intrinsic to every generic hypersurface , e.g. @xmath36 , can be obtained when restricting the line element ( [ 6 ] ) to displacements confined to it . thus , the induced metric on the hypersurface @xmath37 can have the form @xmath38 in such a way that the usual @xmath1 space time metric , @xmath39 , can be recovered . hence , equation ( [ 4 ] ) on the hypersurface @xmath37 can be written as @xmath40,\ ] ] where @xmath41 is an induced energy momentum tensor of the effective @xmath1 modified bd theory , which is defined as @xmath42 with @xmath43 \bigg \}\end{aligned}\ ] ] and @xmath44-\frac{1}{2}g'_{\alpha\beta}\phi ' \bigg \}.\ ] ] also , the induced potential has been defined in the formal identification as @xcite @xmath45\equiv -\epsilon \frac{\omega}{b^{2}}\frac{\phi^{'2}}{\phi } \big |_{_{\sigma_{0 } } } , \ ] ] where the prime denotes derivative with respect to the fifth coordinate . such an identification has been claimed @xcite to be valid depending on metric background and considering separable scalar fields . however , this definition is different from what has been used in ref . @xcite . reduction of equation ( [ 5 ] ) on the hypersurface @xmath37 gives @xmath46-\frac{b_{,\mu}}{b}\phi^{,\mu}\ , , \ ] ] which after manipulation resembles the other field equation of a modified bd theory in four dimensions with induced potential . the definition @xmath41 and equation ( [ freduction ] ) are all we need for our purpose in this work and an interested reader can consult refs . @xcite for further details . in the next section we assume a generalized frw metric in a vacuum @xmath0 universe to find its cosmological implications . for a @xmath0 universe with an extra space like dimension in addition to the three usual spatially homogenous and isotropic ones , metric ( [ 6 ] ) can be written as @xmath47+b^2(t , y)dy^2\,,\ ] ] that can be considered as a generalized frw solution . the scalar field @xmath15 and the scale factors @xmath48 and @xmath49 , in general , are functions of @xmath50 and @xmath33 . however , for simplicity and physical plausibility , we assume the extra dimension is cyclic , i.e. the hypersurface orthogonal space like is a killing vector field in the underlying @xmath0 space time @xcite . hence , all fields are functions of the cosmic time only , and definition ( [ 6.3 ] ) makes the induced potential vanishes . in this case , we will show that such a universe can have accelerating and decelerating solutions . note that , the functionality of the scale factor @xmath49 on @xmath33 , either can be eliminated by transforming to a new extra coordinate if @xmath49 is a separable function , and or makes no changes in the following equations if @xmath49 is the only field that depends on @xmath33 . besides , in the compactified extra dimension scenarios , all fields are fourier expanded around @xmath51 , and henceforth one can have terms independent of @xmath33 to be observable , i.e. physics would thus be effectively independent of compactified fifth dimension @xcite . considering metric ( [ 7 ] ) , equations ( [ 4 ] ) and ( [ 5 ] ) result in cosmological equations @xmath52 @xmath53 @xmath54 and @xmath55 which are not independent equations and where @xmath56 , @xmath57 and @xmath58 . by employing relation ( [ 6.4 ] ) , one can interpret the right hand side of equations ( [ 8 ] ) and ( [ 9 ] ) as energy density and pressure of the induced effective perfect fluid , i.e. @xmath59 and @xmath60 where @xmath61 or @xmath9 or @xmath10 without summation on it . the latter equality in ( [ 11.2 ] ) comes from equation ( [ 22 ] ) which will be derived in the next section . therefor , the equation of state is @xmath62 the usual matter in our universe has a positive energy density , this basically has been demanded by the wec , in which time like observers must obtain positive energy densities . actually , the complete wec is @xcite @xmath63 now , let us consider that the scale factor of the fifth dimension and the scalar field are not constant values , i.e. @xmath64 and @xmath65 . then , by applying conditions ( [ 11.3 ] ) into relations ( [ 11.1 ] ) and ( [ 11.2 ] ) , one gets @xmath66 or @xmath67 where we also have assumed expanding universes , i.e. @xmath68 . using conditions ( [ 11.4 ] ) and ( [ 11.5 ] ) in relation ( [ 11.6 ] ) gives @xmath69 or @xmath70 in where the effective dust matter can be achieved when @xmath71 goes to negative or positive infinity , respectively . in section @xmath12 , we explore characteristic of the corresponding universes for the above results . meanwhile , in the following , we consider two special cases of a constant scale factor of the fifth dimension and a constant scalar field . + + * constant scale factor of fifth dimension * + when @xmath49 is a constant , equations ( [ 8])([11 ] ) reduce to @xmath72 these are exactly the ordinary vacuum bd equations in @xmath1 space time , with @xmath73 , as expected . + + * constant scalar field * + when @xmath15 is a constant , action ( [ 1 ] ) reduces to a @xmath0 einstein gravitational theory that has been considered in ref . @xcite in general situation ( i.e. the extra dimension is not cyclic ) . in this case , equations ( [ 8])([11 ] ) become @xmath74 and , the usual frw equations are equipped with @xmath75 , which refers to a radiation like dominated universe for any kind of geometry without a _ priori _ assumption that the scale factor of the fifth dimension is proportional to the inverse of the usual scale factor , i.e. @xmath76 . actually , the radiation like result is expected . for where there is no dependency on the extra dimension , the usual four dimensional part of metric ( [ 7 ] ) and the third equation ( [ 15 ] ) give a wave equation for the scale factor of fifth dimension . hence , definitions ( [ 6.5 ] ) and ( [ 6.6 ] ) yield a traceless induced energy momentum tensor , as mentioned in ref.@xcite . exact solution of the second equation of ( [ 15 ] ) is @xmath77 substituting solution ( [ 19 ] ) into the first or third equation of ( [ 15 ] ) gives @xmath78 where @xmath79 and @xmath80 are constants of integration , and we have assumed that @xmath1 space time has originated from a big bang . for a closed geometry , solution ( [ 19 ] ) admits @xmath81 and predicts a big crunch at @xmath82 for the usual spatial coordinates while the fifth dimension tends to infinite size and is always real , for the maximum value of the usual scale factor is @xmath83 . but , a flat geometry expands for ever and accepts @xmath81 . an open geometry also expands for ever and admits @xmath84 . in this case , @xmath85 results in @xmath86 and @xmath87 . time evolution of scale factors correspond to closed , flat and open geometries have been illustrated in fig . @xmath88 with constant values of @xmath89 and @xmath90 as an example . in the next two sections , we again consider a more general situation in which the scale factor of the fifth dimension and the scalar field are not constants . [ cols="^,^,^ " , ] analogous to the approach of i m theories , one can consider the bd gravity as the underlying theory . hence , extra geometrical terms , coming from the fifth dimension , are regarded as an induced matter and induced potential . we have followed , with some corrections , the procedure of ref . @xcite for introducing the induced potential and have employed a generalized frw type solution for a @xmath0 vacuum bd theory . hence , the scalar field and scale factors of the @xmath0 metric can , in general , be functions of the cosmic time and the extra dimension . however , for simplicity , we have assumed the scalar field and scale factors to be only functions of the cosmic time , where this makes the induced potential , by its definition , vanishes . we then have revealed that in general situations , in which the scale factor of the fifth dimension and scalar field are not constants , the @xmath0 equations , for any kind of geometry , admit a power law relation between the scalar field and scale factor of the fifth dimension . hence , the procedure exhibits that @xmath0 vacuum frw like equations are equivalent , in general , to the corresponding @xmath1 vacuum ones with the same spatial scale factor but a new ( or modified ) scalar field and a new coupling constant . this equivalency can be viewed as the distinguished point of this work from refs . indeed , through investigating the @xmath0 vacuum frw like equations , we have shown that its equivalent @xmath1 vacuum equations admit accelerated scale factors , contrary to what one may have expected from a vacuum space time . conclusions of the complete investigation of the induced @xmath1 equations are as follows . following our investigations for cosmological implications , we have shown that for the special case of a constant scale factor of the fifth dimension , the @xmath0 vacuum frw like equations reduce to the corresponding equations of the usual @xmath1 vacuum bd theory , as expected . in the special case of a constant scalar field , the action reduces to a @xmath0 einstein gravitational theory and the equations reduce to the usual frw equations with a typical radiation dominated universe . for this situation , we also have obtained dynamics of scale factors of the ordinary and extra dimensions for any kind of geometry without any _ priori _ assumption among them . solutions predict a limited life time for closed geometries and unlimited one for flat and open geometries . a typical time evolutions of scale factors correspond to closed , flat and open geometries have been illustrated in fig . @xmath88 . then , we have focused on spatially flat geometries and have obtained exact solutions of scale factors and scalar field . solutions are found to be in the form of power law and exponential ones in the cosmic time . we also have employed the wec for the induced matter of the @xmath1 modified bd gravity , that gives two conditions ( [ 49.1 ] ) and ( [ 49.2 ] ) . we then have pursued properties of these solutions and have indicated mathematically and physically acceptable ranges of them , and the results have been presented in a few tables and figures . all types of solutions fulfill the wecs in different ranges , where the exponential solutions are more restricted . the solutions fulfilling the wec ( [ 49.1 ] ) have negative pressures , but the figures illustrate that for the power law results there are decelerating solutions beside accelerating ones . for this condition , both @xmath91 and @xmath92 decrease with the cosmic time , but the extra dimension grows . on the other hand , the solutions satisfying the wec ( [ 49.2 ] ) have positive pressures , where the power law results accept accelerating solutions in addition to decelerating ones . for this condition , again decreasing energy density and pressure with the time can occur for some solutions , however all with shrinking extra dimension . the homogeneity between the extra dimension and the usual spatial dimensions , i.e. @xmath93 , can take place in the solutions , but for the power law ones the wecs exclude it . by considering non ghost scalar fields and appealing the recent observational measurements , the solutions have been more restricted . actually , we have illustrated that the accelerating power law solutions , which satisfy the wec and have non ghost scalar fields , are compatible with the recent observations in ranges @xmath3 for the bd coupling constant and @xmath4 for dependence of the fifth dimension scale factor with the usual scale factor . these ranges also fulfill the condition @xmath5 which prevents ghost scalar fields in the equivalent @xmath1 vacuum bd equations . incidentally , this range is more restricted than the one obtained in ref . @xcite , i.e. @xmath94 , where the difference may have been caused by the distinct definition of the induced potential in two approaches of ref . @xcite and ref . however , we should remind that it has also been shown 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we follow the approach of induced matter theory for a five dimensional ( @xmath0 ) vacuum brans dicke theory and introduce induced
matter and induced potential in four dimensional ( @xmath1 ) hypersurfaces , and then employ a generalized frw type solution . we confine ourselves to the scalar field and scale factors be functions of the cosmic time .
this makes the induced potential , by its definition , vanishes , but the model is capable to expose variety of states for the universe . in general situations , in which the scale factor of the fifth dimension and scalar field are not constants ,
the @xmath0 equations , for any kind of geometry , admit a power
law relation between the scalar field and scale factor of the fifth dimension . hence
, the procedure exhibits that @xmath0 vacuum frw like equations are equivalent , in general , to the corresponding @xmath1 vacuum ones with the same spatial scale factor but a new scalar field and a new coupling constant , @xmath2 .
we show that the @xmath0 vacuum frw like equations , or its equivalent @xmath1 vacuum ones , admit accelerated solutions . for a constant scalar field ,
the equations reduce to the usual frw equations with a typical radiation dominated universe .
for this situation , we obtain dynamics of scale factors of the ordinary and extra dimensions for any kind of geometry without any _ priori _ assumption among them . for non constant scalar fields and spatially flat geometries , solutions are found to be in the form of power law and exponential ones .
we also employ the weak energy condition for the induced matter , that gives two constraints with negative or positive pressures .
all types of solutions fulfill the weak energy condition in different ranges .
the power
law solutions with either negative or positive pressures admit both decelerating and accelerating ones .
some solutions accept a shrinking extra dimension . by considering non ghost scalar fields and appealing the recent observational measurements ,
the solutions are more restricted .
we illustrate that the accelerating power law solutions , which satisfy the weak energy condition and have non ghost scalar fields , are compatible with the recent observations in ranges @xmath3 for the coupling constant and @xmath4 for dependence of the fifth dimension scale factor with the usual scale factor .
these ranges also fulfill the condition @xmath5 which prevents ghost scalar fields in the equivalent @xmath1 vacuum brans dicke equations .
the results are presented in a few tables and figures .
-2.7 cm keywords : brans dicke theory ; induced matter theory ; frw cosmology .
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the ultrarelativistic heavy - ion collider facilities like rhic at bnl and lhc at cern strive to produce a new form of hot qcd matter . the experiments show @xcite that it has very intricate properties and presents a big challenge especially for theoretical understanding . while above the ( pseudo)critical temperature @xmath5 mev this matter is often called the quark - gluon plasma ( qgp ) , it can not be a perturbatively interacting quark - gluon gas ( as widely expected before rhic results @xcite ) until significantly higher temperatures @xmath6 . instead , the interactions and correlations in the hot qcd matter are still strong ( e.g. , see refs . @xcite ) so that its more recent and more precise name is _ strongly coupled qgp _ ( sqgp ) @xcite . one of its peculiarities seems to be that strong correlations in the form of quark - antiquark ( @xmath7 ) bound states and resonances still exist @xcite in the sqgp well above @xmath8 . in the old qgp paradigm , even deeply bound charmonium ( @xmath9 ) states such as @xmath10 and @xmath11 were expected to unbind at @xmath12 , but lattice qcd simulations of mesonic correlators now indicate they persist till around @xmath13 @xcite or even above @xcite . similar indications for light - quark mesonic bound states are also accumulating from lattice qcd @xcite and from other methods @xcite . this agrees well with the findings on the lattice ( e.g. , see ref . @xcite for a review ) that for realistic explicit chiral symmetry breaking ( chsb ) , i.e. , for the physical values of the current quark masses , the transition between the hadron phase and the phase dominated by quarks and gluons , is not an abrupt , singular phase transition but a smooth , analytic crossover around the _ pseudo_critical temperature @xmath8 . it is thus not too surprising that a clear experimental signal of , e.g. , deconfinement , is still hard to find and identify unambiguously . the most compelling signal for production of a new form of qcd matter , i.e. , sqgp , would be a restoration - in hot and/or dense matter - of the symmetries of the qcd lagrangian which are broken in the vacuum . one of them is the [ su@xmath14(@xmath15 ) flavor ] chiral symmetry , whose dynamical breaking results in light , ( almost-)goldstone pseudoscalar ( @xmath16 ) mesons namely the octet @xmath17 , as we consider all three light - quark flavors , @xmath18 . the second one is the u@xmath14(1 ) symmetry . its breaking by the non - abelian axial adler - bell - jackiw anomaly ( ` gluon anomaly ' for short ) makes the remaining pseudoscalar meson of the light - quark sector , the @xmath0 , much heavier , preventing its appearance as the ninth ( almost-)goldstone boson of dynamical chiral symmetry breaking ( dchsb ) in qcd . the first experimental signature of a partial restoration of the u@xmath14(1 ) symmetry seems to have been found in the @xmath19 gev central au+au reactions at rhic . namely , csrg _ et al . _ @xcite analyzed combined data of phenix @xcite and star @xcite collaborations very robustly , through six popular models for hadron multiplicities , and found that at 99.9% confidence level , the @xmath0 mass , which in the vacuum is @xmath20 mev , is reduced by at least 200 mev inside the fireball . it is the sign of the disappearing contribution of the gluon axial anomaly to the @xmath0 mass , which would drop to a value readily understood together with the ( flavor - symmetry - broken ) octet of @xmath21 ( @xmath22 ) pseudoscalar mesons . this is the issue of the return of the prodigal goldstone boson " predicted @xcite as a signal of the u@xmath14(1 ) symmetry restoration . another related but less obvious issue to which we want to draw attention , concerns the status , at @xmath3 , of the famous witten - veneziano relation ( wvr ) @xcite @xmath23 between the @xmath0 , @xmath24 and @xmath25-meson masses @xmath26 , pion decay constant @xmath27 , and yang - mills ( ym ) topological susceptibility @xmath28 . wvr was obtained in the limit of large number of colors @xmath29 @xcite . it is well satisfied at @xmath4 for @xmath28 obtained by lattice calculations ( e.g. , @xcite ) . nevertheless , the @xmath1-dependence of @xmath28 is such @xcite that the straightforward extension of eq . ( [ wittenvenez ] ) to @xmath3 @xcite , i.e. , replacement of all quantities unless their @xmath1-dependence is specifically indicated in formulas or in the text . ] therein by their respective @xmath1-dependent versions @xmath30 , @xmath31 , @xmath32 , @xmath33 _ and _ @xmath34 , leads to a conflict with experiment @xcite . since this extension of eq . ( [ wittenvenez ] ) to @xmath3 was studied in ref . @xcite before the pertinent experimental analysis @xcite , one of the purposes of this paper is to revisit the implications of the results of ref . @xcite for wvr at @xmath3 , and demonstrate explicitly that they are practically model - independent . the other , more important purpose is to propose a mechanism which can enable wvr to agree with experiment at @xmath3 . both issues pointed out before eq . ( [ wittenvenez ] ) and around it , are best understood in a model - independent way if one starts from the chiral limit of vanishing current quark masses ( @xmath35 ) for all three light flavors , @xmath36 . then not only pions and kaons are massless , but is also @xmath24 , which is then ( since the situation is also su(3)-flavor - symmetric ) a purely su(3)-octet state , @xmath37 . in contrast , @xmath0 is then purely singlet , @xmath38 ; since the divergence of the singlet axial quark current @xmath39 is nonvanishing even for @xmath35 due to the gluon anomaly , the @xmath0 mass squared receives the anomalous contribution @xmath40 ( @xmath41 in the notation of ref . @xcite ) which is nonvanishing even in the chiral limit : @xmath42 however , @xmath43 and @xmath44 are known accurately decay constant " @xmath44 is , strictly speaking , not a well - defined quantity , as _ two _ @xmath0 decay constants are actually needed : the singlet one , @xmath45 , and the octet one , @xmath46 ; _ e.g. _ , see an extensive review @xcite or the short appendix of ref . @xcite . ] only in the large @xmath29 limit . there , in the leading order in @xmath47 , @xmath48 is given by the ym ( i.e. , pure glue " ) topological susceptibility @xmath28 times @xmath49 @xcite , and the @xmath0 decay constant " @xmath44 is the same as @xmath50 @xcite . thus , keeping only the leading order in @xmath47 , the last equality is wvr in the chiral limit . the consequences of eq . ( [ wv_chlim ] ) remain qualitatively the same realistically away from the chiral limit . this will soon become clear on the basis of , e.g. , eq . ( [ m2_prop_m ] ) below . namely , due to dchsb in qcd , for relatively light current quark masses @xmath51 @xmath52 , the @xmath21 bound - state pseudoscalar meson masses ( including the _ nonanomalous _ parts of the @xmath0 and @xmath24 masses ) behave as @xmath53 the pseudoscalar mesons ( including @xmath0 ) thus obtain relatively light nonanomalous contributions @xmath54 to their masses @xmath55 , allowing them to reach the empirical values . that is , instead of the eight strictly massless goldstone bosons , @xmath56 and @xmath24 are relatively light almost - goldstones . among them , in the limit of isospin symmetry ( @xmath57 ) , only @xmath24 now receives also the gluon - anomaly contribution since the explicit su(3 ) flavor breaking between the nonstrange ( @xmath58 ) @xmath59-quarks and @xmath60-quarks causes the mixing between the isoscalars @xmath24 and @xmath0 . for @xmath61 , eq . ( [ wv_chlim ] ) is replaced by the usual wvr ( [ wittenvenez ] ) containing also the nonanomalous contributions to meson masses . nevertheless , these contributions largely cancel due to the approximate su(3 ) flavor symmetry and to dchsb [ i.e. , eq . ( [ m2_prop_m ] ) ] . this can be seen assuming the usual su(3 ) @xmath7 content of the pseudoscalar meson nonet with well - defined isospin opens the possibility of studying the interesting scenario of maximal isospin violation at high @xmath1 @xcite , but as the effects of the small difference between @xmath62 and @xmath63 are not important for the present considerations , we stick to the isospin limit throughout the present paper . ] quantum numbers , in particular the isoscalar @xmath64 octet and singlet etas , @xmath65 , @xmath66 , whose mixing yields the physical particles @xmath24 and @xmath0 . since the _ nonanomalous _ parts of the @xmath67 and @xmath68 masses squared , @xmath69 and @xmath70 , are respectively @xmath71 and @xmath72 ( see , e.g. , ref . @xcite ) , and since @xmath73 , the nonanomalous parts of the @xmath24 and @xmath0 masses are canceled by @xmath74 in wvr ( [ wittenvenez ] ) . another way of seeing this is expressing the nonanomalous parts of @xmath75 by eq . ( [ m2_prop_m ] ) . thus again @xmath76 , showing again that already wvr s chiral - limit - nonvanishing part ( [ wv_chlim ] ) reveals the essence of the influence of the gluon anomaly on the masses in the @xmath0-@xmath24 complex . this is important also for the presently pertinent finite-@xmath1 context because thanks to this , below it will be shown model - independently that wvr ( [ wittenvenez ] ) containing the ym topological susceptibility @xmath28 implies @xmath1-dependence of @xmath0 mass in conflict with the recent experimental results @xcite . namely , the gluon - anomaly contribution ( [ wv_chlim ] ) is established at @xmath4 but it is not expected to persist at high temperatures . ultimately , @xmath0 should also become a massless goldstone boson at sufficiently high @xmath1 , where @xmath77 . however , according to wvr , @xmath78 falls only for @xmath1 where @xmath79 does not fall faster than @xmath80 , as stressed in ref . @xcite . the wvr s chiral - limit version ( [ wv_chlim ] ) manifestly points out the ratio @xmath81 as crucial for the anomalous @xmath0 mass , but the above discussion shows that this remains essentially the same away from the chiral limit . in the present context , it is important for practical calculations to go realistically away from the chiral limit , in which the chiral restoration is a sharp phase transition at its critical temperature @xmath2 where the chiral - limit pion decay constant vanishes very steeply , i.e. , as steeply as the chiral quark condensate . in contrast , for realistic explicit chsb , i.e. , @xmath62 and @xmath63 of several mev , this transition is a _ smooth crossover _ ( e.g. , see ref . @xcite ) . for the pion decay constant , this implies that @xmath33 still falls relatively steeply around _ pseudo_critical temperature @xmath2 , but less so than in the chiral case , and even remains finite , enabling the usage of wvr ( [ wittenvenez ] ) for the temperatures across the chiral and u@xmath14(1 ) symmetry restorations . wvr is very remarkable because it connects two different theories : qcd with quarks and its pure - gauge , ym counterpart . the latter , however , has much higher characteristic temperatures than qcd with quarks : the melting temperature " @xmath82 where @xmath34 starts to decrease appreciably was found on lattice to be , for example , @xmath83 mev @xcite or even higher , @xmath84 mev @xcite . in contrast , the pseudocritical temperatures for the chiral and deconfinement transitions in the full qcd are lower than @xmath82 by some 100 mev or more ( e.g. , see ref . @xcite ) due to the presence of the quark degrees of freedom . this difference in characteristic temperatures , in conjunction with @xmath34 in wvrs ( [ wittenvenez ] ) and ( [ wv_chlim ] ) would imply that the ( partial ) restoration of the u@xmath14(1 ) symmetry ( understood as the disappearance of the anomalous @xmath85 mass ) should happen well after the restoration of the chiral symmetry . but , this contradicts the rhic experimental observations of the reduced @xmath0 mass @xcite _ if _ wvrs ( [ wittenvenez ] ) , ( [ wv_chlim ] ) hold unchanged also close to the qcd chiral restoration temperature @xmath2 , around which @xmath33 decreases still relatively steeply of the unphysical @xmath86 pseudoscalar with @xmath33 in fig . [ chi1757_qq_f ] . ] @xcite for realistic explicit chsb , thus leading to the increase of @xmath87 and consequently also of @xmath88 . there is still more to the relatively high resistance of @xmath34 to temperature : not only does it start falling at rather high @xmath82 , but @xmath34 found on the lattice is falling with @xmath1 _ relatively _ slowly . in some of the applications in the past ( e.g. , see refs . @xcite ) , it was customary to simply rescale a temperature characterizing the pure - gauge , ym sector to a value characterizing qcd with quarks . ( for example , refs . @xcite rescaled @xmath89 mev found by ref . @xcite to 150 mev ) . however , even if we rescale the critical temperature for melting of the topological susceptibility @xmath34 from @xmath82 down to @xmath2 , the value of @xmath87 still increases a lot @xcite for the pertinent temperature interval starting already below @xmath2 . this happens because @xmath34 falls with @xmath1 more slowly than @xmath79 . ( it was found @xcite that the rescaling of @xmath82 would have to be totally unrealistic , to less than 70% of @xmath2 , in order to achieve sufficiently fast drop of the anomalous contribution that would allow the observed enhancement in the @xmath0 multiplicity . ) these wvr - induced enhancements of the @xmath0 mass for @xmath90 were first noticed in ref . this reference used a concrete dynamical model ( with an effective , rank-2 separable interaction , convenient for computations at @xmath91 ) @xcite of low energy , nonperturbative qcd to obtain mesons as @xmath21 bound states in dyson - schwinger ( ds ) approach @xcite , which is a bound - state approach with the correct chiral behavior ( [ m2_prop_m ] ) of qcd . nevertheless , this concrete dynamical ds model was used in ref . @xcite to get concrete values for only the nonanomalous parts of the meson masses , but was essentially _ not _ used to get model predictions for the mass contributions from the gluon anomaly , in particular @xmath34 . on the contrary , the anomalous mass contribution was included , in the spirit of @xmath47 expansion , through wvr ( [ wittenvenez ] ) . thus , the @xmath1-evolution of the @xmath0-@xmath24 complex in ref . @xcite was not dominated by dynamical model details , but by wvr , i.e. , the ratio @xmath87 . admittedly , @xmath33 was also calculated within this model , causing some _ quantitative _ model dependence of the anomalous mass in wvr , but this can not change the qualitative observations of ref . @xcite on the @xmath0 mass enhancement . namely , our model @xmath33 , depicted as the dash - dotted curve in fig . [ chi1757_qq_f ] , obviously has the right crossover features @xcite . it also agrees qualitatively with @xmath33 s calculated in other realistic dynamical models @xcite . various modifications were tried in ref . @xcite but could not reduce much the @xmath0 mass enhancement caused by this ratio , let alone bring about the significant @xmath0 mass reduction found in the rhic experiments @xcite . , of @xmath92 , @xmath93 , @xmath27 and @xmath94 , i.e. , the @xmath95-dependences of the quantities entering in the anomalous contributions to various masses in the @xmath0-@xmath24 complex see eq . ( [ metaprime ] ) and formulas below it . the solid curve depicts @xmath92 for @xmath96 in eq . ( [ chitildechicond ] ) , and the short - dashed curve is @xmath92 for @xmath97 . at @xmath4 , the both @xmath98 s are equal to @xmath99 , the weighted average @xcite of various lattice results for @xmath28 . the dotted ( red ) curve depicts @xmath100 , the dash - dotted ( blue ) curve is @xmath27 , and the long - dashed ( blue ) curve is @xmath94 . ( colors online . ) , width=302 ] one must therefore conclude that either wvr breaks down as soon as @xmath1 approaches @xmath2 , or that the @xmath1-dependence of its anomalous contribution is different from the pure - gauge @xmath34 . we will show that the latter alternative is possible , since wvr can be reconciled with experiment thanks to the existence of another relation which , similarly to wvr , connects the ym theory with full qcd . namely , using large-@xmath29 arguments , leutwyler and smilga derived @xcite , at @xmath4 , @xmath101 the relation ( in our notation ) between the ym topological susceptibility @xmath28 , and the full - qcd topological susceptibility @xmath102 , the _ chiral - limit _ quark condensate @xmath103 , and @xmath104 , the harmonic average of @xmath15 current quark masses @xmath51 . that is , @xmath104 is @xmath15 times the reduced mass . in the present case of @xmath18 , @xmath105 , so that @xmath106 equation ( [ chitilde ] ) is a remarkable relation between the two pertinent theories . for example , in the limit of all very heavy quarks ( @xmath107 ) , it correctly leads to the result that @xmath28 is equal to the value of the topological susceptibility in _ quenched _ qcd , @xmath108 . this holds because @xmath102 is by definition the vacuum expectation value of a gluonic operator , so that the absence of quark loops would leave only the pure - gauge , ym contribution . however , the leutwyler - smilga relation ( [ chitilde ] ) also holds in the opposite ( and presently pertinent ) limit of light quarks . this limit still presents a problem for getting the full - qcd topological susceptibility @xmath102 on the lattice @xcite , but we can use the light - quark - sector result @xcite @xmath109 where @xmath110 stands for corrections of higher orders in small @xmath51 , and thus of small magnitude . the leading term is positive ( as @xmath111 ) , but @xmath110 is negative , since eq . ( [ chitilde ] ) shows that @xmath112 . although small , @xmath110 should not be neglected , since @xmath113 would imply , through eq . ( [ chitilde ] ) , that @xmath114 . instead , its value ( at @xmath4 ) is fixed by eq . ( [ chitilde ] ) : @xmath115 all this starting from eq . ( [ chitilde ] ) has so far been at @xmath4 . if the left- and right - hand side of eq . ( [ chitilde ] ) are extended to @xmath3 , it is obvious that the equality can not hold at arbitrary temperature @xmath3 . the relation ( [ chitilde ] ) must break down somewhere close to the ( pseudo)critical temperatures of full qcd ( @xmath116 ) since the pure - gauge quantity @xmath28 is much more temperature - resistant than the right - hand side , abbreviated as @xmath117 . the quantity @xmath117 , which may be called the effective susceptibility , consists of the full - qcd quantities @xmath102 and @xmath103 , the quantities of full qcd with quarks , characterized by @xmath2 , just as @xmath33 . as @xmath118 , the chiral quark condensate @xmath119 drops faster than the other dchsb parameter in the present problem , namely @xmath33 for realistically small explicit chsb . ( see fig . [ chi1757_qq_f ] for the results of the dynamical model adopted here from ref . @xcite , and , e.g. , refs . @xcite for analogous results of different ds models ) . thus , the troublesome mismatch in @xmath1-dependences of @xmath33 and the pure - gauge quantity @xmath34 , which causes the conflict of the temperature - extended wvr with experiment at @xmath120 , is expected to disappear if @xmath34 is replaced by @xmath121 , the temperature - extended effective susceptibility . the successful zero - temperature wvr ( [ wittenvenez ] ) is , however , retained , since @xmath122 at @xmath4 . extending eq . ( [ chi_small_m ] ) to @xmath3 is something of a guesswork as there is no guidance from the lattice for @xmath123 [ unlike @xmath34 ] . admittedly , the leading term is straightforward as it is plausible that its @xmath1-dependence will simply be that of @xmath119 . nevertheless , for the correction term @xmath110 such a plausible assumption about the form of @xmath1-dependence can not be made and eq . ( [ cat0 ] ) , which relates ym and qcd quantities , only gives its value at @xmath4 . we will therefore explore the @xmath1-dependence of the anomalous masses using the following ansatz for the @xmath91 generalization of eq . ( [ chi_small_m ] ) : @xmath125^{\delta } \ , , \label{chi_small_mt}\ ] ] where the correction - term @xmath1-dependence is parametrized through the power @xmath126 of the presently fastest - vanishing ( as @xmath127 ) chiral order parameter @xmath119 . the @xmath91 extension ( [ chi_small_mt ] ) of the light - quark @xmath102 , eq . ( [ chi_small_m ] ) , leads to the @xmath91 extension of @xmath117 : @xmath128^{\delta } \right ) . \label{chitildechicond}\ ] ] we now use @xmath129 in wvr instead of @xmath34 used by ref . @xcite . this gives us the temperature dependences of the masses in the @xmath24-@xmath0 complex , such as those in fig . [ m_pchi1757 ] illustrating the cases @xmath96 and @xmath97 . it is clear that @xmath129 ( [ chitildechicond ] ) blows up as @xmath127 if the correction term there vanishes faster than @xmath119 _ squared_. thus , varying @xmath126 between 0 and 2 covers the cases from the @xmath1-independent correction term , to ( already experimentally excluded ) enhanced anomalous masses for @xmath126 noticeably above 1 , to even sharper mass blow - ups for @xmath130 when @xmath127 . on the other hand , it does not seem natural that the correction term vanishes faster than the fastest - vanishing order parameter @xmath119 . indeed , already for the same rate of vanishing of the both terms ( @xmath97 ) , one can notice in fig . [ m_pchi1757 ] the start of the precursors of the blow - up of various masses in the @xmath0-@xmath24 complex as @xmath127 although these small mass bumps are still experimentally acceptable . thus , in fig . [ m_pchi1757 ] we depict the @xmath97 case , and @xmath96 ( @xmath1-independent correction term ) as the other acceptable extreme . since they turn out to be not only qualitatively , but also quantitatively so similar that the present era experiments can not discriminate between them , there is no need to present any ` in - between results ' , for @xmath131 . next we turn to completing the explanation how the above - mentioned results in fig . [ m_pchi1757 ] were obtained . , of the pseudoscalar meson masses for two @xmath129 , namely eq . ( [ chitildechicond ] ) with @xmath96 ( upper panel ) and with @xmath97 ( lower panel ) . the meaning of all symbols is the same on the both panels : the masses of @xmath0 and @xmath24 are , respectively , the upper and lower solid curve , those of the pion and nonanomalous @xmath132 pseudoscalar are , respectively , the lower and upper dash - dotted curve , @xmath133 and @xmath134 are , respectively , the short - dashed ( red ) and long - dashed ( red ) curve , @xmath135 is the medium - dashed ( blue ) , and @xmath136 is the dotted ( blue ) curve . ( colors online . ) the straight line @xmath137 is twice the lowest matsubara frequency.,width=302 ] , of the pseudoscalar meson masses for two @xmath129 , namely eq . ( [ chitildechicond ] ) with @xmath96 ( upper panel ) and with @xmath97 ( lower panel ) . the meaning of all symbols is the same on the both panels : the masses of @xmath0 and @xmath24 are , respectively , the upper and lower solid curve , those of the pion and nonanomalous @xmath132 pseudoscalar are , respectively , the lower and upper dash - dotted curve , @xmath133 and @xmath134 are , respectively , the short - dashed ( red ) and long - dashed ( red ) curve , @xmath135 is the medium - dashed ( blue ) , and @xmath136 is the dotted ( blue ) curve . ( colors online . ) the straight line @xmath137 is twice the lowest matsubara frequency.,width=302 ] using @xmath129 in wvr instead of @xmath34 used by ref . @xcite , does not change anything at @xmath4 , where @xmath138 , which remains an excellent approximation even well beyond @xmath4 . nevertheless , this changes drastically as @xmath1 approaches @xmath2 . for @xmath90 , the behavior of @xmath129 is dominated by the @xmath1-dependence of the chiral condensate , tying the restoration of the u@xmath14(1 ) symmetry to the chiral symmetry restoration . as for the nonanomalous contributions to the meson masses , we use the same ds model ( and parameter values ) as in ref . @xcite , since it includes both dchsb and correct qcd chiral behavior as well as realistic explicit chsb . that is , all nonanomalous results ( @xmath139 , as well as @xmath140 and @xmath94 , the mass and decay constant of the unphysical @xmath132 pseudoscalar meson ) in the present paper are , for all @xmath1 , taken over from ref . we used this same model also for computing the chiral quark condensate @xmath103 , including its @xmath1-dependence displayed in fig . [ chi1757_qq_f ] . this defines completely how the results displayed in fig . [ m_pchi1757 ] were generated . for details , see ref . @xcite ( and ref . @xcite for @xmath133 and @xmath134 ) . here we list only the formulas which , in conjunction with fig . [ chi1757_qq_f ] , enable the reader to understand easily the @xmath1-dependences of the masses in fig . [ m_pchi1757 ] : the theoretical @xmath0 and @xmath24 mass eigenvalues are @xmath141 , \quad \label{metaprime } \\ \label{meta } m_{\eta}^2(t ) & = & \frac{1}{2 } \left [ m_{\eta_{ns}}^2(t ) + m_{\eta_{s}}^2(t ) - \delta_{\eta \eta'}(t ) \right ] , \quad\end{aligned}\ ] ] where @xmath142 ^ 2 + 8 \beta^2 x^2}$ ] , @xmath143 @xmath144 @xmath145 @xmath146 in all expressions after eq . ( [ meta ] ) , the @xmath1-dependence is understood . in both cases considered for the topological susceptibility ( [ chi_small_mt ] ) [ @xmath96 , i.e. , the constant correction term , and @xmath97 , i.e. , the strong @xmath1-dependence @xmath147 of both the leading and correction terms in @xmath123 ] , the results are consistent with the experimental findings on the decrease of the @xmath0 mass of csrg _ et al . in the light of the recent experimental results on the @xmath0 multiplicity in heavy - ion collisions @xcite , we revisited the earlier theoretical work @xcite concerning the thermal behavior of the @xmath0-@xmath24 complex following from wvr straightforwardly extended to @xmath3 . we have confirmed the results of ref . @xcite on wvr where the ratio @xmath81 dominates the @xmath1-dependence , and clarified that these results are practically model - independent . it is important to note the difference between our approach and those that attempt to give model predictions for topological susceptibility , such as refs . @xcite . by contrast , in refs . @xcite and here , as well as earlier works @xcite at @xmath4 , a ds dynamical model is used ( as far as masses are concerned ) to obtain only the nonanomalous part of the light pseudoscalar meson masses ( where the model dependence is however dominated by their almost - goldstone character ) , while the anomalous part of the masses in the @xmath0-@xmath24 complex is , through wvr , dictated by @xmath149 . in this ratio , @xmath33 is admittedly model - dependent in quantitative sense , but other realistic models yield qualitatively similar crossover behaviors @xcite of @xmath33 for @xmath150 , as exemplified by our fig . [ chi1757_qq_f ] , and fig . 2 in ref . @xcite , and by fig . 6 in ref . @xcite . such @xmath33 behaviors are also in agreement with the @xmath1-dependence expected of the dchsb order parameter on general grounds : a pronounced fall - off around @xmath2 but exhibiting , in agreement with lattice @xcite , a smooth crossover pattern for nonvanishing explicit chsb , a crossover which gets slower with growing @xmath51 [ e.g. , compare @xmath33 with @xmath151 in fig . [ chi1757_qq_f ] ] . in contrast to the qcd topological susceptibility @xmath102 , the ym topological susceptibility and even its @xmath1-dependence @xmath34 , including its melting " temperature @xmath152 , can be extracted @xcite reasonably reliably from the lattice @xcite . thus , it was not modeled in ref . hence our assertion that the results of ref . @xcite unavoidably imply that the straightforward extension of wvr to @xmath3 is falsified by experiment @xcite , especially if one recalls that even the sizeable @xmath1-rescaling @xcite @xmath153 was among the attempts to control the @xmath0 mass enhancement @xcite . nevertheless , we have also shown that there is a plausible way to avoid these problems of the straightforward , naive extension of wvr to @xmath3 , and this is the main result of the present paper . thanks to the existence of another relation , eq . ( [ chitilde ] ) , connecting the ym quantity @xmath28 with qcd quantities @xmath102 and @xmath154 , it is possible to define a quantity , @xmath117 , which can meaningfully replace @xmath34 in finite-@xmath1 wvr . it remains practically equal to @xmath28 up to some 70% of @xmath2 , but beyond this , it changes following the @xmath1-dependence of @xmath155 . in this way , the successful zero - temperature wvr is retained , but the ( partial ) restoration of u@xmath14(1 ) symmetry [ understood as the disappearing contribution of the gluon anomaly to the @xmath0 ( @xmath67 ) mass ] is naturally tied to the restoration of the su@xmath14(3 ) flavor chiral symmetry and to its characteristic temperature @xmath2 , instead of @xmath152 . it is very pleasing that this fits in nicely with the recent _ ab initio _ theoretical analysis using functional methods @xcite , which finds that the anomalous breaking of u@xmath14(1 ) symmetry is related to dchsb ( and confinement ) in a self - consistent manner , so that one can not have one of these phenomena without the other . of course , the most important thing is that this version of the finite-@xmath1 wvr , obtained by @xmath156 , is consistent with experiment @xcite for all reasonable strengths of @xmath1-dependence [ @xmath157 in eq . ( [ chi_small_mt ] ) ] . namely , the both tablets in fig . [ m_pchi1757 ] show , first , that @xmath0 mass close to @xmath2 suffers the drop of more than 200 mev with respect to its vacuum value . this satisfies the minimal experimental requirement abundantly . second , fig . [ m_pchi1757 ] shows an even larger drop of the @xmath67 mass , to some 400 mev , close to the best " value of the in - medium @xmath0 mass ( 340 mev , albeit with large errors ) obtained by csrg _ et al . this should be noted because the @xmath67 mass inside the fireball is possibly even more relevant . namely , although it is , strictly speaking , not a physical meson , @xmath67 is the state with the @xmath7 content closest to the @xmath7 content of the physical @xmath0 _ in the vacuum_. thus , among the isoscalar @xmath7 states inside the fireball , @xmath67 has the largest projection on , and thus the largest amplitude to evolve , by fireball dissipation , into an @xmath0 in the vacuum . d. h. , d. kl . and s. b. were supported through the project no . 0119 - 0982930 - 1016 , and d. ke . through the project no . 098 - 0982887 - 2872 of the ministry of science , education and sports of croatia . d. h. and d. kl . acknowledge discussions with r. alkofer . the support by compstar network is also acknowledged . e. v. shuryak and i. zahed , phys * d70 * , 054507 ( 2004 ) , [ hep - ph/0403127 ] . m. a. stephanov , pos * lat2006 * , 024 ( 2006 ) [ arxiv : hep - lat/0701002 ] . d. b. blaschke and k. a. bugaev , fizika * b13 * , 491 ( 2004 ) , [ nucl - th/0311021 ] . s. datta , f. karsch , p. petreczky and i. wetzorke , nucl . . suppl . * 119 * , 487 ( 2003 ) , [ hep - lat/0208012 ] . m. asakawa and t. hatsuda , phys . lett . * 92 * , 012001 ( 2004 ) , [ hep - lat/0308034 ] . e. shuryak , nucl . phys . * a774 * , 387 ( 2006 ) , [ hep - ph/0510123 ] . f. karsch _ et al . _ , nucl . phys . * a715 * , 701 ( 2003 ) , [ hep - ph/0209028 ] . p. maris , c. d. roberts , s. m. schmidt and p. c. tandy , phys . c * 63 * , 025202 ( 2001 ) [ arxiv : nucl - th/0001064 ] . m. mannarelli and r. rapp , phys . rev . * c72 * , 064905 ( 2005 ) , [ hep - ph/0505080 ] . b. lucini , m. teper and u. wenger , nucl . phys . * b715 * , 461 ( 2005 ) , [ hep - lat/0401028 ] . l. del debbio , l. giusti and c. pica , phys . rev . lett . * 94 * , 032003 ( 2005 ) , [ hep - th/0407052 ] . b. alles , m. delia and a. di giacomo , phys . * d71 * , 034503 ( 2005 ) , [ hep - lat/0411035 ] . s. drr , z. fodor , c. hoelbling and t. kurth , jhep * 0704 * , 055 ( 2007 ) [ arxiv : hep - lat/0612021 ] .
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we discuss and propose the minimal generalization of the witten - veneziano relation to finite temperatures , prompted by star and phenix experimental results on the multiplicity of @xmath0 mesons . after explaining why these results show that the zero - temperature witten - veneziano relation can not be straightforwardly extended to temperatures @xmath1 too close to the chiral restoration temperature @xmath2 and beyond , we find the quantity which should replace , at @xmath3 , the yang - mills topological susceptibility appearing in the @xmath4 witten - veneziano relation , in order to avoid the conflict with experiment at @xmath3 .
this is illustrated through concrete @xmath1-dependences of pseudoscalar meson masses in a chirally well - behaved , dyson - schwinger approach , but our results and conclusions are of a more general nature and , essentially , model - independent .
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one of the main interests in hypernuclear physics is to investigate how an addition of @xmath0 particle influences the properties of atomic nuclei . due to the absence of pauli s principle between nucleon and @xmath0 particle , it is believed that a @xmath0 hyperon can be treated as an _ impurity _ to probe deep interior of the nuclear medium . with the presence of hyperon as an impurity , some bulk properties of nuclei such as the shape and collective motions may be changed @xcite . indeed , the shrinkage of @xmath5li , with respect to @xmath6li , has been observed experimentally by measuring the b(e2 ) value from the @xmath7 state to the @xmath8 state of @xmath5li @xcite . as the shape of nuclei plays a decisive role in determining their properties such as quadrupole moment and radius , mean - field model calculations have been performed in recent years to investigate the change of nuclear shape due to the addition of a @xmath0 hyperon . deformed skyrme - hartree - fock(shf ) studies in ref . @xcite have shown that the deformation parameter of the hypernuclei which they studied is slightly smaller ( within the same sign ) than that of the corresponding core nuclei , and thus no significant effect of @xmath0 hyperon on nuclear deformation was found . on the other hand , we have performed a relativistic mean field ( rmf ) study and found that the nuclear deformation of the core nuclei completely disappears for @xmath9c and @xmath10si hypernuclei due to the addition of @xmath0 particle @xcite , although we have obtained similar results to the shf calculations in ref . @xcite for many other hypernuclei . in ref . @xcite , we have compared between the shf and rmf approaches and have shown that the different results with respect to nuclear deformation between the two approaches is due to the fact that the rmf yields a somewhat stronger polarization effect of @xmath0 hyperon as compared to the shf approach . we have also shown that the disappearance of the deformation realizes also in the shf approach if the energy difference between the optimum deformation and the spherical configuration is less than about 1 mev @xcite . all of these mean - field calculations have assumed axial symmetric deformation . although many nuclei are considered to have axially symmetric shape , the triaxial degree of freedom plays an important role in transitional nuclei , nuclei with shape coexistence , and nuclei with gamma soft deformation @xcite . in particular , we mention that recent studies with the constrained hartree - fock - bogoliubov plus local quasi - particle random phase approximation ( chfb+lqrpa ) method @xcite as well as the rmf plus generator coordinate method ( rmf+gcm ) @xcite have revealed an important role of triaxiality in large amplitude collective motion in sd - shell nuclei . the aim of this paper is to extend the previous mean - field studies on deformation of hypernuclei by taking into account the triaxial degree of freedom , that is , by including both the @xmath11 and @xmath4 deformations in order to investigate the effect of @xmath0 hyperon in the full ( @xmath11,@xmath4 ) deformation plane . we particularly study the potential energy surface ( pes ) in ( @xmath11,@xmath4 ) deformation plane with the shf method for carbon hypernuclei as well as some sd - shell hypernuclei . notice that the shape evolution of the c isotopes in the full ( @xmath1 ) plane has been studied by zhang _ _ using the shf method @xcite . we extend the work of zhang _ et al . _ by introducing a hyperon degree of freedom . the paper is organised as follows . in sec . ii , we briefly summarise the skyrme - hartree - fock method for hypernuclei . in sec . iii , we present the results for the potential energy surface for the c isotopes as well as for sd - shell nuclei , @xmath2si and @xmath3 mg . we also discuss the softness of the energy surface along the @xmath4 deformation . in sec . iv , we summarize the paper . the self - consistent mean field approach provides a useful means to study the ground state properties of hypernuclei . the core polarization effect , that is , the change of properties of a core nucleus due to an addition of @xmath0 particle such as a change of total energy and radius can be automatically taken into account with this method @xcite . the self - consistent non - relativistic mean field calculations with a skyrme - type @xmath0-nucleon(@xmath0n ) interaction have been performed by rayet in refs . the relativistic mean field approach has also been applied to hypernuclei in _ e.g. , _ refs . it has been pointed out that the neutron drip line is extended by the addition of @xmath0 hyperon @xcite . in the present paper , we employ the skyrme - type @xmath0n interaction and perform mean - field calculations by extending the computer code ev8 @xcite to @xmath0 hypernuclei . the code solves the hartree - fock equations by discretizing individual single - particle wave functions on a three - dimensional cartesian mesh . the pairing correlation is taken into account in the bcs approximation . with this method , both axial and triaxial quadrupole deformations can be automatically described . the code ev8 has been applied to the study of shape transition and deformation of several nuclei @xcite in the ( @xmath1 ) deformation plane . the skyrme - type @xmath0n interaction is given in complete analogy with the nuclear skyrme interaction @xcite . the skyrme part of the total hypernuclear energy thus reads @xmath12,\ ] ] where @xmath13(*r * ) is the standard nuclear hamiltonian density based on the skyrme interaction . e.g. , _ refs . @xcite for its explicit form . @xmath14(*r * ) is the hyperon hamiltonian density given in terms of the lambda and nucleon densities as ( with a correction for the coefficient of the @xmath15 term ) @xcite @xmath16 here , @xmath17 , @xmath18 , and @xmath19 are the particle density , the kinetic energy density , and the spin density of the @xmath0 hyperon . these are expressed using the single - particle wave function @xmath20 for the @xmath0 particle . @xmath21 , @xmath22 , and @xmath23 are the total densities for the nucleons . @xmath24 , and @xmath25 are the skyrme parameters for the @xmath0n interaction . the hartree - fock equations for the single - particle wave functions are obtained by taking variation of the energy @xmath26 . the equation for the nucleons reads , @xmath27\phi _ q = e_q\phi _ q , \label{hf - nu}\ ] ] where @xmath28 refers to protons or neutrons , while that for the @xmath0 particle reads @xmath29\phi _ { \lambda } = e_{\lambda}\phi _ { \lambda}.\ ] ] here , @xmath30 and @xmath31 are the single - particle energies and the effective mass for the hyperon is given by @xmath32 @xmath33 is the single - particle potential originating from the nucleon - nucleon skyrme interaction @xcite . @xmath34 and @xmath35 are the single - particle potentials originating from the @xmath0n interaction . these are expressed as @xcite @xmath36 and @xmath37 the pairing correlations among the nucleons are treated in the bcs approximation . for the pairing interaction , we employ a zero - range density - dependent pairing force @xcite , @xmath38 where @xmath39 is the spin - exchange operator , @xmath40 @xmath41 , and @xmath42 . in this paper , we mainly use the sgii interaction @xcite for the nn interaction , while the set no.@xmath43 in ref . @xcite for the @xmath0n interaction . the latter interaction was constructed by fitting to the binding energy of @xmath44o , yielding the well depth for a @xmath0 particle , @xmath45= 29.38 mev , in infinite nuclear matter . notice that the spin - orbit strength @xmath25 in the @xmath46 interaction is zero for the set no . 1 @xcite . for the pairing interaction , we follow ref . @xcite to use @xmath47 mev@xmath48@xmath49 for both protons and neutrons for carbon hypernuclei , while we follow ref . @xcite to use @xmath50 mev@xmath48@xmath49 for calculations of sd - shell hypernuclei . a smooth pairing energy cutoff of 5 mev around the fermi level is used @xcite . we assume that the @xmath0 particle occupies the lowest single - particle state . since the primary purpose of this paper is to draw the potential energy surfaces ( pes ) of @xmath0 hypernuclei as a function of @xmath11 and @xmath4 deformation parameters , the isoscalar quadrupole constraint is imposed on the total energy . we relate the deformation parameter @xmath11 for hypernuclei with the total quadrupole moment @xmath51 using the equation @xmath52 where @xmath53 is the mass number of the core nucleus and @xmath54 fm is the radius of the hypernucleus . c and ( b ) @xmath55c in the ( @xmath1 ) deformation plane obtained with the sgii parameter set . each contour line is separated by 0.07mev . the triangles indicate the absolute minima in the pes . , title="fig : " ] + c and ( b ) @xmath55c in the ( @xmath1 ) deformation plane obtained with the sgii parameter set . each contour line is separated by 0.07mev . the triangles indicate the absolute minima in the pes . , title="fig : " ] + c and @xmath55c along the axially symmetric deformation corresponding to fig . the energy surface for @xmath55c is shifted by a constant amount as indicated in the figure . ] we now numerically solve the hartree - fock equations and discuss the deformation properties of hypernuclei in the ( @xmath1 ) deformation plane . we first investigate the shape of carbon hypernuclei from @xmath55c to @xmath56c . to this end , we follow refs . @xcite and reduce the strength of the spin - orbit interaction by a factor of 0.6 in the skyrme functional . this prescription was introduced in order to reproduce an oblate ground state of @xmath57c @xcite . c and @xmath9c . each contour line is separated by 0.2mev.,title="fig : " ] c and @xmath9c . each contour line is separated by 0.2mev.,title="fig : " ] c and @xmath9c . ] figure [ 10c ] shows the potential energy surfaces for @xmath58c and @xmath55c so obtained . the triangles in the figure indicate the ground state minimum in the energy surface . the energy curve along the axially symmetric deformation is also shown in fig . [ 10caxial ] as a function of the quadrupole deformation parameter @xmath11 . along the axially symmetric configuration , there are deep energy minima in both sides of prolate and oblate configurations of @xmath58c ( that is , the shape coexistence ) , having a very small energy difference of less than 40 kev between them . the ground state corresponds to the prolate configuration with @xmath11=0.35 . however , the energy surface is almost flat along the triaxial deformation @xmath4 as one can see in fig . [ 10c ] ( a ) in the ( @xmath1 ) deformation plane . with the addition of a @xmath0 hyperon , the ground state configuration moves from prolate to oblate , although the energy surface is so flat along the @xmath4 degree of freedom ( see fig . [ 10c](b ) ) that the ground state configuration may not be well defined in the mean field approximation . we have confirmed that this feature remains the same even if we use the skyrme @xmath0n interaction set no.2 and no . 5 @xcite , instead of no . c and @xmath59c . each contour line is separated by 0.15mev , title="fig : " ] c and @xmath59c . each contour line is separated by 0.15mev , title="fig : " ] for the optimum values of the @xmath11 deformation parameters shown in fig . [ the energy surface for @xmath59c is shifted by a constant amount as indicated in the figure . ] [ 12c ] ( a ) and [ 12caxial ] show the energy surface for @xmath57c in the @xmath60 deformation plane and that along the @xmath61 line , respectively . for this nucleus , the skyrme - hartree - fock method with the reduced spin - orbit interaction yields a deep oblate minimum . in this case , the addition of a @xmath0 particle shows little effect on the energy surface , although the deformation is slightly smaller than its core nucleus ( see figs . [ 12c](b ) and [ 12caxial ] ) . this is similar to the result for @xmath62ne discussed in ref.@xcite with the rmf method . for the case of @xmath63c , it has a triaxial minimum at @xmath64 and @xmath65 at an energy of @xmath660.154mev with respect to the prolate configuration ( see fig . [ 18c ] ( a ) ) . we find that @xmath59c has a ground state configuration with similar values of @xmath11 and @xmath4 deformation parameters to those of the core nucleus , @xmath63c , as indicated in fig . [ 18c ] ( b ) . the triaxial minimum is again shallow with an energy difference of 0.077 mev between the optimum deformation and the prolate configuration . in fig . [ 18c - g ] , we plot the energy curve as a function of the triaxial deformation parameter @xmath4 for the optimum values of the @xmath11 deformation parameter . with the addition of a @xmath0 particle to @xmath63c , one sees that the energy curve becomes slightly softer ( compare also figs . [ 18c](a ) and [ 18c](b ) ) . c and @xmath63c nuclei in the @xmath60 deformation plane . ] the energy difference between @xmath59c and @xmath63c at each deformation , that is , @xmath67 , in the @xmath60 deformation plane is plotted in fig . it clearly shows that the addition of a @xmath0 particle prefers the spherical configuration even if the core nucleus has a deformed minimum . we have confirmed that this is the case for the other carbon isotopes as well . notice that the @xmath0 particle slightly prefers the prolate configuration for a fixed value of @xmath11 , causing the slightly softer energy curve towards the prolate configuration shown in fig . this originates from the fact that the overlap between the deformed nuclear density and a spherical @xmath0 density is maximum at the prolate configuration , as we discuss in the appendix . the results of our calculations are summarised in table i , together with the results for the other carbon isotopes . si and @xmath10si . each contour line is separated by 0.6 mev.,title="fig : " ] si and @xmath10si . each contour line is separated by 0.6 mev.,title="fig : " ] mg and @xmath68 mg . each contour line is separated by 0.5 mev.,title="fig : " ] mg and @xmath68 mg . each contour line is separated by 0.5 mev.,title="fig : " ] let us next discuss the deformation energy surfaces of @xmath2si , and @xmath3 mg nuclei in the ( @xmath1 ) deformation plane . for these nuclei , we use the original strength for the spin - orbit interaction . we first show the potential energy surface of @xmath69si and @xmath70 mg in figs . [ 28si](a ) and fig . [ 24mg](a ) , respectively . the energy surface for @xmath69si shows a deep oblate minimum , while that for @xmath70 mg shows a deep prolate minimum . notice that n = z=14 is an oblate magic number . the energy curves for @xmath70 mg as a function of @xmath11 with @xmath71 , and of @xmath4 with @xmath72 are also plotted in in figs . [ 24axial](a ) and [ 24axial](b ) , respectively . mg and @xmath68 mg along the axially symmetric deformation corresponding to fig . ( b ) the energy curve as a function of the triaxial deformation parameter @xmath4 for the optimum values of the @xmath11 deformation parameters shown in fig . the energy surfaces for @xmath68 mg are shifted by a constant amount as indicated in the figures.,title="fig : " ] mg and @xmath68 mg along the axially symmetric deformation corresponding to fig . ( b ) the energy curve as a function of the triaxial deformation parameter @xmath4 for the optimum values of the @xmath11 deformation parameters shown in fig . the energy surfaces for @xmath68 mg are shifted by a constant amount as indicated in the figures.,title="fig : " ] for these nuclei , as the potential minimum is deep , the addition of a @xmath0 particle does not change significantly the shape of the potential energy surface , as shown in figs . [ 28si](b ) , [ 24mg](b),and [ 24axial ] . as shown in the previous subsection , the energy gain due to the additional @xmath0 particle appears mainly around the spherical configuration . notice , however , that the @xmath0 particle makes the energy curve slightly softer along the triaxial degree of freedom , @xmath4 . si and @xmath73si . each contour line is separated by 0.4 mev.,title="fig : " ] si and @xmath73si . each contour line is separated by 0.4 mev.,title="fig : " ] mg and @xmath73 mg . each contour line is separated by 0.4 mev.,title="fig : " ] mg and @xmath73 mg . each contour line is separated by 0.4 mev.,title="fig : " ] mg and @xmath73 mg along the axially symmetric deformation corresponding to fig . ( b ) the energy curve as a function of the triaxial deformation parameter @xmath4 for the optimum values of the @xmath11 deformation parameters shown in fig . the energy surfaces for @xmath73 mg are shifted by a constant amount as indicated in the figures.,title="fig : " ] mg and @xmath73 mg along the axially symmetric deformation corresponding to fig . ( b ) the energy curve as a function of the triaxial deformation parameter @xmath4 for the optimum values of the @xmath11 deformation parameters shown in fig . the energy surfaces for @xmath73 mg are shifted by a constant amount as indicated in the figures.,title="fig : " ] we next discuss the @xmath73 mg and @xmath74si nuclei . @xmath75 mg has z=12 and n=14 , that is , the protons favour a prolate configuration while the neutrons favour an oblate configuration . as a competition of these two opposite effects , the structure of @xmath75 mg may not be trivial @xcite . @xmath75si is the mirror nucleus of @xmath75 mg , and the deformation properties are expected to be similar to @xmath75 mg . figures [ 26si](a ) and fig . [ 26mg](a ) show the potential energy surfaces for @xmath75si and @xmath75 mg , respectively . indeed , the two energy surfaces are similar to each other , and show an oblate minimum with a considerably flat surface along the @xmath4 direction . the energy difference between the oblate and the prolate configurations is 0.12 mev for @xmath75si and 0.39 mev for @xmath75 mg . the energy curve for @xmath75 mg as a function of @xmath11 with @xmath71 , and of @xmath4 with @xmath76 are plotted in in figs . [ 26axial](a ) and [ 26axial](b ) , respectively . the energy curves for @xmath75si are qualitatively similar , and are not shown . as in @xmath70 mg and @xmath69si , the addition of a @xmath0 hyperon does not significantly alter the potential energy surface of these nuclei , although it somewhat softens the energy surface along the @xmath4 direction ( see figs . [ 26si](b ) , [ 26mg](b ) , and [ 26axial ] ) . we again find that the additional @xmath0 particle favours the spherical configuration . the calculations presented in this subsection are performed with the sgii set of the skyrme interaction . we have repeated the same calculations with another skyrme parameter , siii , and have found that the results are qualitatively the same as the results obtained with sgii . for a long time , the @xmath70 mg nucleus has been considered to be a candidate of nuclei with a triaxial shape , because of the low - lying second 2@xmath77state in the rotational spectrum @xcite . that is , the experimental spectrum of @xmath70 mg has been interpreted as consisting of a @xmath78=0 ground state rotational band and a @xmath78=2 rotational band built upon a @xmath4 vibrational state at 4.23 mev @xcite . the previous mean - field calculations for the ground state of @xmath70 mg using rmf @xcite and shf @xcite have shown an axially symmetric prolate ground state . recently , it has been pointed out that the angular momentum projection is essential in order to reproduce the triaxial ground state of @xmath70mg@xcite . the 3-dimensional angular momentum projection plus generator coordinate method ( 3damp+gcm ) calculations have shown a good agreement with the experimental data on low - spin states of @xmath70 mg @xcite . in order to see the effect of @xmath0 hyperon on @xmath4 vibration , we compute the second derivative of the energy curve with respect to @xmath4 around the minimum , @xmath79 . that is , when we approximate the energy curve around the minimum as , @xmath80 the second derivative @xmath81 provides information on the frequency @xmath82 for the @xmath4 vibration if the vibrational moment of inertia @xmath83 is known . by numerically taking the second derivative , we obtain @xmath84 = 0.961 and @xmath85 = 0.988 . that is , the addition of a @xmath0 particle makes the @xmath4 vibration softer if the change in the moment of inertia @xmath83 is negligible . although the @xmath0 particle softens the energy curve both for @xmath68 mg and @xmath73 mg , the mechanism is somewhat different between the two nuclei . for the @xmath73 mg nucleus , the prolate configuration decreases more energy as compared to the oblate configuration ( see fig . 13 ( b ) ) , because the prolate configuration has a larger overlap between the @xmath0 particle and the nucleon densities , as is discussed in the appendix . as a consequence , the curvature of the energy curve at the oblate minimum becomes smaller with the addition of a @xmath0 particle . in contrast , the effect of the smaller value of @xmath11 is more significant in @xmath68 mg . that is , the energy curve along the axially symmetric configuration is considerably steep for @xmath86 ( see fig . 10 ( a ) ) , and even a small change in @xmath11 induces a significant energy change at the oblate configuration . therefore , for @xmath68 mg , the energy decreases more at the oblate side as compared to the prolate side for a fixed value of @xmath11 ( see fig . 10 ( b ) ) , leading to the softer gamma - vibration . notice that the absolute value of @xmath11 is relatviely small for @xmath75 mg , lying in a `` flat '' region , and this effect is much less important in the @xmath73 mg nucleus . [ cols="^,^,^,^,^,^,^,^,^,^",options="header " , ] we have investigated the shape of @xmath0 hypernuclei in the ( @xmath1 ) deformation plane with the skyrme - hartree - fock + bcs approach . in contrast to the previous mean - field studies , we have taken into account the triaxial deformation using a 3-d cartesian mesh method . we have studied the potential energy surface for the carbon hypernuclei from @xmath55c to @xmath56c as well as sd - shell hypernuclei @xmath73si , @xmath10si , @xmath68 mg and @xmath73 mg . the potential energy surface for @xmath58c , @xmath75 mg , and @xmath75si is characterized by a flat surface along the @xmath4 degree of freedom connecting the prolate and the oblate configurations . we have found that the addition of a @xmath0 particle makes the energy surface slightly softer along the triaxial degree of freedom , although the gross feature of the energy surface remains similar to the energy surface for the corresponding core nuclei . in refs . @xcite , we have argued that the influence of the addition of a @xmath0 particle is stronger in the relativistic mean - field approach as compared to the non - relativistic skyrme - hartree - fock approach employed in this paper . this implies that the softening of the energy curve for the @xmath4-vibration may be larger than that estimated in this paper , if we employ the rmf approach instead of the shf approach . it would be an interesting future subject to carry out three - dimensional calculations for hypernuclei with rmf in order to confirm whether it is the case . there would be many ways to improve our calculations presented in this paper . firstly , as the angular momentum projection is shown to be essential to yield the triaxial shape of @xmath70 mg @xcite , it may be important to carry out the angular momentum projection on top of the mean - field energy surface in order to discuss the effect of @xmath0 particle on the shape of hypernuclei . secondly , for nuclei with a flat energy surface along the @xmath4 direction , the generator coordinate method may be required . in particular , it will provide a more quantitative estimate for the energy change of the @xmath4 vibrational state due to the addition of @xmath0 particle . in any case , the mean - field calculations presented in this paper provide a good starting point for these calculations . it will be an interesting subject to experimentally measure the deformation properties and collective motions of hypernuclei . a discussion has been started for a future experiment of @xmath4-ray spectroscopy of sd - shell hypernuclei at the new generation experimental facilities , _ e.g. , _ the j - parc facility @xcite . the change of deformation will be well investigated if excitation energies in a rotational band and the b(e2 ) values can be measured experimentally . we thank h. tamura , y. zhang , n. hinohara and a. ono for useful discussions . this work was supported by the japanese ministry of education , culture , sports , science and technology by grant - in - aid for scientific research under the program number 22540262 . particle densites as a function of the triaxiality @xmath4 for a fixed value of @xmath87 . the @xmath0-particle density is assumed to be a spherical gaussian function , while a deformed woods - saxon shape is considered for the nucleon density . ] in this appendix , we discuss the overlap between @xmath0 particle and nucleon density distributions using simple parametrizations for the density distributions . since we consider that a @xmath0 particle occupies the lowest single - particle state , we assume that the @xmath0 particle density is almost spherical and is given by , @xmath88 on the other hand , for the nucleon density , we assume that it is given by a deformed woods - saxon form , that is , @xmath89},\ ] ] where @xmath90.\end{aligned}\ ] ] here , the radius @xmath91 is determined for each @xmath11 and @xmath4 in order to satifsy the volume conservation condition , that is , @xmath92},\ ] ] is independent of @xmath11 and @xmath4 . figure 14 shows the overlap between @xmath93 and @xmath94 , that is , @xmath95 as a function of the triaxiality @xmath4 for a fixed value of @xmath87 . to this end , we use @xmath96 fm , and @xmath97=0.55 fm . the value of @xmath98 is fixed so that the volume integral of @xmath94 is 24 . we use @xmath99=1.565 fm , which corresponds to the harmonic oscillator with a frequency of @xmath100 mev . as one can see , the overlap is the largest for the prolate configuration , although the variation is small with respect to @xmath4 .
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we study the shape of @xmath0 hypernuclei in the full ( @xmath1 ) deformation plane , including both axially symmetric and triaxial quadrupole deformations . to this end
, we use the constrained skyrme hartree - fock+bcs method on the three - dimensional cartesian mesh .
the potential energy surface is analyzed for carbon hypernuclei as well as for sd - shell hypernuclei such as @xmath2si and @xmath3 mg .
we show that the potential energy surface in the ( @xmath1 ) plane is similar to each other between the hypernuclei and the corresponding core nuclei , although the addition of @xmath0 hyperon makes the energy surface somewhat softer along the @xmath4 direction .
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cometary globules are believed to be molecular cloud condensations , which are so dense that they are not disrupted when an region expands into the molecular cloud(s ) surrounding it . such globules are always bright rimmed and often have a cometary or tear shaped form , because the stellar wind and ionizing radiation from the central o stars ablate away the low density gas on the side facing the o stars and sweep away dust towards the tail . cometary globules are thought to form from `` elephant trunks '' @xcite , although for compact cores surrounded by low density gas this phase can be quite short . the rosette nebula is a good example of an region , which shows an abundance of both cometary globules and elephant trunk structures @xcite . although elephant trunks and cometary globules have been known for a long time , the fact that these sites may be forming stars was realized only in the last few decades . the first clear confirmation that stars form in cometary globules was a result of the identification of bernes 135 in the cometary globule cg1 , one of the large cometary globules in the gum nebula , as a pre - main - sequence ( pms ) star @xcite . the discovery of molecular outflows associated with cold iras sources without optical counterparts in three bright - rimmed globules @xcite , definitely confirmed that stars form in elephant trunks and cometary globules . probably the most well - known example of star forming in the tip of elephant trunks is seen in the hubble poster , `` the pillars of creation '' , in the eagle nebula ( m16 ) @xcite . in this paper we examine star formation in a more nearby cometary globule , ori i-2 , located in the large region ic434 . this is one of the three globules , in which @xcite discovered a molecular outflow . by using _ irac and mips images , 2mass data , deep r and h@xmath1 images from the nordic optical telescope ( not ) , and scuba sub - millimeter imaging we can identify all young stars in and near the globule and provide more accurate information on their physical characteristics . @xcite were the first to notice this globule in the large region ic434 and named it orii-2 . the region ic434 is illuminated by the trapezium like system @xmath3 ori with at least five early type stars @xcite . the two hottest members , a and b ( o9.5v and b0.5v ) form a binary system , which allows for an accurate distance determination . @xcite derived a distance of 334 pc assuming that they form a binary , although he also considered the possibility that it is a hierarchical triple system , in which case the distance would be @xmath4 385 pc . @xcite , in their review of distance estimates to the @xmath3 orionis cluster , quote a slightly larger distance , 420 @xmath5 30 pc . the morphology of orii-2 , with the bright rim facing @xmath3 ori and the tail aligned in the direction of the star suggests that the globule is at the same distance or slightly in front of @xmath3 ori . in this paper we adopt the distance to orii-2 as 380 pc , favoring the distance estimate by @xcite . @xcite included orii-2 in their molecular line survey of small bok globules and found it to be the densest , the most compact , and the warmest , 18 k , of all the globules in their sample . @xcite discovered a bipolar molecular outflow in the globule . they found that the outflow had a linear extent of 0.34 pc , a dynamical age of 5 @xmath2 10@xmath6 yr , and was centered on a low luminosity ( l = 11 ) cold iras source , iras @xmath7 , without any optical counterpart . @xcite did a more detailed study of orii-2 in co , @xmath8co and c@xmath9o j = @xmath10 , hcn j = @xmath11 , cs j = @xmath12 and cs j = @xmath13 with higher spatial resolution and sensitivity than @xcite and @xcite . they confirmed that the molecular outflow is bipolar and centered on the iras source , iras @xmath7 , but found it to be more compact ( 0.22 pc ) than what @xcite estimated from their low - spatial resolution observations . by examining the palomar sky survey prints and doing ccd imaging in r , i , and narrowband h@xmath1 and [ ] , they showed that the bright rim is a diffuse region and not a reflection nebula . they identified a faint star , star a , as a possible counterpart to the iras source . they found that the gas temperature of the shock heated gas in the bright rim is @xmath4 25 k , while the gas temperature in the opaque ( a@xmath14 @xmath4 20@xmath15 ) core of the globule is @xmath4 12 k , similar to that of normal isolated globules . they determined the total mass of the globule to be @xmath4 8 , with the opaque core having a mass of @xmath4 2.3 ( corrected to a distance of 380 pc ) . orii-2 was included in the catalogue of remnant molecular clouds in the ori ob1 association by @xcite as cloud number 40a . they also identified three fainter cometary globules `` behind '' the tail of orii-2 ( 40a ) , which they called 40b , c , and d. @xcite , who did a photographic and ccd imaging survey of the l 1630 and l 1641 molecular cloud regions in orion , identified a chain of hh objects , hh289 , to the east of ori i-2 and a tube like feature ( cavity ) protruding out on the western side of the globule , both apparently excited by the embedded iras source . their observations show that the iras source drives a giant outflow with a projected size of @xmath16 pc to the east . a good overview of orii-2 and other globules in the vicinity of @xmath3 orionis can be found in @xcite . we have extracted irac and mips ( 24 & 70 ) observations from the spitzer space observatory archive ( program i d 30050 : star formation in bright rimmed clouds by fazio et al . ) . the irac data were taken in the high dynamic readout ( hdr ) mode using a single aor ( astronomical observation request ) with a five - point dither pattern . we have processed both the short ( 0.6 sec ) and the long ( 12 sec ) integration basic calibrated data ( bcd ) frames in each channel using the artifact mitigation software developed by sean carey and created mosaics using mopex . these irac observations go far deeper than any previous observations of orii-2 . in the short wavelength bands we can detect point - sources down to 30 40 @xmath17jy and are slightly less sensitive in the long wavelength bands , but in regions without nebulosity we can still detect point - sources down to @xmath4 60 @xmath17jy . we have also created mosaics of the mips 24 and 70 bcds using mopex . both data sets are of excellent quality . the 70 @xmath17m - image , due to the lower angular resolution , shows only one bright point - like source embedded in the globule . the image also shows strong emission from the hot ionized rim facing @xmath3 ori and fainter emission on the western side of the globule . we have carried out multiframe psf photometry using the ssc - developed tool apex on all the _ spitzer _ irac images and on the mips images . the 3.6 and 4.5 irac images show signs of saturation on bright stars . we have used combination of automated routines and eye - inspection to detect sources and extract photometry of these sources from the irac and mips images . for sources , which apex failed to detect at one or several wavelengths , we have used the user list option in apex to supply the coordinates for the source to extract it and perform photometry on it . this enabled us to derive photometry for every source , which we could visually identify on any image . in the irac images we have detected 118 , 122 , 41 and 32 sources respectively at 3.6 , 4.5 , 5.8 and 8.0 . since the field of view of the two irac cameras do not completely overlap , we end up with a total of 125 mid - infrared sources . we detect five of them in the 24 mips image , but only one in the 70 image . we have crosscorrelated the sources detected in the irac and mips bands , and the 2mass point source catalogue . we have used the following association radii : 1 for the irac images , 25 for the mips 24 image and 2 for 2mass data . table [ tab_mirsrc ] gives the coordinates of the 125 sources identified in orii-2 together with their @xmath0-band and 2mass magnitudes , _ spitzer _ irac and mips flux densities and a preliminary classification based on selected color - color plots . in table [ tab_mirsrc ] we have given them the prefix orii-2 , but throughout the paper we simply refer to them as mir - nn , where nn is the number of the source . out of the 125 sources 51 were found to have 2mass counterparts . all the sources detected in the mips 24 image were also detected in the four irac bands . we find that around 50% of the sources in the list have been detected only at 3.6 and 4.5 and have no counterparts in 2mass as well , although many of them are seen in the deep @xmath0-band image . the optical ccd images presented here were obtained on october 26 2007 using the 2.56 m nordic optical telescope ( not ) located at el observatorio del roque de los muchachos on the island of la palma in the canary islands . we used the andalucia faint object spectrometer and camera ( alfosc ) , which uses a thinned loral 2048 @xmath2 2048 ccd array with 15 @xmath17 m pixels giving a field of view of 6.5 @xmath2 6.5 arcmin@xmath18 with 0.187 arcseconds pixel@xmath19 . for these observations we obtained two images with 300 seconds exposure time in bessel r ( 6500 ) separated by offsets of @xmath2090,+90 from the center of the globule in right ascension and declination , and a short 30 second exposure centered on the globule . we also observed the globule in narrow - band h@xmath1 . the narrow band h@xmath1 filter was a circular filter giving a more restricted field of view and centered on 6564 with a bandwidth of 33 . here we obtained two images , each with 300 second exposure and separated by 60 in right ascension . the observing conditions were excellent with an average seeing of @xmath4 075 . the images were reduced with the starlink program suite ccdpack . the transformation to the world coordinate system ( wcs ) was done using the starlink program gaia and we used more than twenty 2mass stars as astrometric reference stars . the 2mass stars have very good astrometric accuracy and the fit to these reference stars indicate that the astrometry of the ccd images have an accuracy of @xmath21 01 for both the r and the narrow - band h@xmath1 image compared to the 2mass reference frame . the images of the photometric standard stars were unavailable , hence the @xmath0-band image was calibrated using the 30 second exposure and seven stars in the 14 - 15@xmath15 range from the usno - a2.0 catalogue . we estimate the photometric calibration accuracy to be @xmath4 0.1 - 0.2@xmath15 . this calibration was transferred to the h@xmath1 image using a somewhat larger set of overlapping stars with approximately neutral colors judged from 2mass j magnitudes . the h@xmath1 image is estimated to have a photometric accuracy no better than 0.3@xmath15 . the r band image has a limiting magnitude of @xmath4 23.5@xmath15 and stars brighter than 15@xmath15 are partially saturated . we did aperture photometry in r and h@xmath1 of all stars , which coincided with an irac source to within 1 . we present only the result of the r band photometry in table [ tab_mirsrc ] . the h@xmath1 is virtually identical , except for one star , see below . we detect all 2mass sources in r band as well as 26 irac sources without 2mass detection . comparison of the r and h@xmath1 photometry revealed only one h@xmath1 emission line star . mir-52 has an h@xmath1 magnitude of @xmath4 18.5 , while it is @xmath4 19.9 in r ( table [ tab_mirsrc ] ) . the excess seen in h@xmath1 , 1.4@xmath15 , is much larger than our photometric uncertainty . therefore the star is definitely an h@xmath1 emission line star . mir-52 has a strong ir excess in the mips 24 band , so that we identify it as a class ii object based on irac and mips color color diagrams , see section [ sec_ysoclass ] . the 850 @xmath17 m and 450 @xmath17 m continuum observations were obtained with bolometer array scuba on jcmt , mauna kea , hawaii . scuba @xcite has 37 bolometers in the long and 91 in the short wavelength array separated by approximately two beam widths in a hexagonal pattern . the field of view of both arrays is @xmath42.3 . both arrays can be used simultaneously by means of a dichroic beamsplitter . the scuba observations reported in this paper were all obtained in jiggle - map mode @xcite . on october 17 , 1997 we obtained two separate five integration maps with a chop throw of 100 in azimuth ; which unfortunately was not sufficient to completely chop outside of the globule . on december 17 , 1997 we therefore used a 120 chop with a fixed position angle of 80 measured from north ( equatorial reference frame ) . this time we obtained three maps , each with small position offsets ( dither ) with three , eight , and five integrations . the total integration time from both nights was therefore 3328 seconds . the sky conditions were good on both nights . on october 17 the atmospheric opacity measured with the caltech submillimeter observatory ( cso ) taumeter , cso @xmath22 was @xmath4 0.07 , and on december 17 @xmath23 was 0.04 - 0.05 . the dust emission from the globule is too extended to be observed in jiggle map mode and the emission from the cold cloud core is therefore likely to be underestimated , especially at 450 @xmath17 m . any strong emission in the off source positions was carefully blanked out in the data reduction stage . it should not affect the morphology or photometry of the strong , compact source , which dominates the emission both at 850 and 450 @xmath17 m . pointing corrections were derived from observations of the blazar 0528 + 134 on october 17 , and the blazar 0529 + 075 on december 17 . the secondary calibrators crl618 and hl tau were used for flux density calibration . we estimate the calibration accuracy to be @xmath4 10% at 850 and @xmath4 20% at 450 . the half power beam width ( hpbw ) was estimated from observations of uranus obtained in the early part of each night and was found to be @xmath4 145 15 at 850 @xmath17 m and @xmath4 8 85 at 450 @xmath17 m . the data were reduced in a standard way using surf @xcite and starlink imaging software , i.e. , flat fielded , extinction corrected , sky subtracted , despiked , and calibrated the images in jy beam@xmath19 . each data set was corrected for any drift in pointing between successive pointing observations and the data were added together to determine the most likely submillimeter position at 850 . once we had derived a basic 850 astrometric image , we made gaussian fits of the compact sub - millimeter source in each data set and derived small additional ra and dec corrections to each scan ( shift and add ) to sharpen the final image to this position . since there is a small mis - alignment between the 850 and 450 arrays , we first corrected the 450 images for any pointing drifts and then did shift and add using the position derived at 850 . we estimate the astrometric accuracy to be @xmath24 2 . the final coadd was done by noise - weighting the data in order to minimize the noise in the final images . the rms of the 450 image is @xmath4 0.50 mjy beam@xmath19 and @xmath4 10 mjy beam@xmath19 for the 850 image . all the maps were converted to fits - files and exported to miriad @xcite for further analysis . in order to correct for the error lobe contribution , especially at 450 , we have deconvolved all the maps using clean with the same beam model used by @xcite . since the hpbw varies slightly from night to night , this model beam is not ideal , but it is the best we can do . the actual beam size is probably somewhat more extended . figure [ fig_scuba ] shows the 450 and 850 scuba images of orii-2 . these images show a single bright source , located at the head of the globule beyond the optically bright rim . this source , smm1 , is coincident with mir-54 , which is the only source seen in the mips 70 image . in order to derive the position and size of smm1 we have fitted a two component elliptical gaussian using the task imfit in miriad , one for the sub - mm source , and the other for the surrounding cloud . the fit to the broader component is mainly to provide a good subtraction of the extended emission , and is not to estimate the flux density of the surrounding cloud . the sub - millimeter position is @xmath1(2000.0 ) = 05@xmath25 38@xmath15 05098 , @xmath26(2000.0 ) = @xmath27 01 45 114 . the integrated flux densities are 0.54 @xmath5 0.08 jy and 3.36 @xmath5 0.75 jy for 850 and 450 , respectively . the fitted size , 179@xmath2 154 , should be considered an upper limit , since the true beam sizes were probably somewhat larger than the model beams used for deconvolving the images . since orii-2 lies within the @xmath3 orionis cluster ( age @xmath4 3 myr ) , we expect to detect some young stars belonging to the cluster in the 5 @xmath2 5 field that we are analyzing . stars belonging to the @xmath3 orionis cluster should be preferentially located outside the globule and generally appear older than stars that have recently formed in the globule . figure [ fig_composite ] present three - color images of orii-2 using irac 3.6 , 4.5 and 8 in the _ left _ panel and the irac 3.6 and 8 bands combined with the mips 24 channel in the _ right _ panel . both images show that the front side of the globule , which is dominated by strong polyaromatic hydrocarbon emission ( pah - emission ) is far from smooth . this may in part be due to rayleigh - taylor instabilities at the ionization front , but it is also due to outflows from the young stars which have recently formed in the cloud . the outflow from mir-54 ( iras 05355 - 0146 ) , the bright red star in the irac / mips color image , is clearly seen as a cavity like structure to the east of the star and a bluish nebulosity on the western side of the star . this outflow is very prominent in the h@xmath1 image ( figure [ fig_halpha_rband ] ) , where it shows up as a limb brightened cavity , extending @xmath4 50 from the western edge of the globule and @xmath4 1.5 from the star exciting it . another outflow just south of the h@xmath1 outflow and approximately aligned with it , can be seen in the irac images terminating at a blue star ( unrelated to the outflow ) . this outflow is powered by another young star , mir-52 , @xmath4 20 to the south of mir-54 . to the east both outflows overlap . the western side of the globule is very sharp with an ionized rim . to the east the the boundary between the globule on the surrounding region is more diffuse . the most opaque part of the globule is just north or northwest of the iras source . the bright star located @xmath4 43 away from mir-54 in the color image is sao 132389 ( hd 37389 ) . this star , identified as mir-73 in table [ tab_mirsrc ] , is detected even at 24 , although it is an unreddened foreground a0 star with an @xmath28 . the star has a radial velocity of @xmath29 km s@xmath19 , which is very different from the radial velocity ( 12 ) of orii-2 @xcite . most of the faint stars outside the globule are field stars , although a few are young stars belonging to the @xmath3-orionis cluster . the irac and mips data are ideal for detecting young stellar objects ; especially in obscured regions like in orii-2 , which has a visual extinction exceeding 20@xmath15 . outside the globule , where the extinction is low , search for brown dwarfs , for which the spectral energy distribution ( sed ) peaks at j , can be most efficiently performed using near - infrared data . however , we go deeper than 2mass in the 3.6 and 4.5 irac bands . the same is true even for the not broadband r filter . there is therefore no particular advantage in using the 2mass data alone . all 2mass sources in our field have also been detected by irac , at least in the two short wavelength bands . furthermore , at the distance of the @xmath3 orionis cluster only the brightest m dwarfs are barely detectable with 2mass . since we can see several galaxies in the deep r band image , we first analyzed the irac data to eliminate any galaxies , which could have mir colors similar to a yso . @xcite demonstrated that ( a ) normal star - forming galaxies and narrow - line agns with increasing 5.8 and 8.0 and ( b ) broad - line agns with red , nonstellar seds , result in colors which are very similar to bona - fide ysos . it is therefore necessary to inspect deep irac images for contamination due to the extragalactic sources . @xcite have extensively discussed the criteria for identifying such extragalactic objects in the irac color - color diagrams . if we use the criteria given in the appendix of their paper , we identify mir-16 , 25 , 88 , 89 , 93 , and 114 as extragalactic . however , inspection of these sources in the r - band ccd image , which was taken in excellent seeing conditions , suggests that most of them are stellar , with the exception of mir-93 . this source is clearly extended , and therefore it is an extragalactic source . for mir-88 , which is close to the detection limit in the r - band , we get no help from the r band image , it could be stellar , or it could be extragalactic . mir-89 , which is not detected in the r band , was also identified as an extragalactic object . however , not only irac , but also the mips colors suggest that mir-89 is a class 0/i object . this classification is also supported by its location in the eastern side of the bright rim . we therefore only positively identify mir-93 as an extragalactic object . figure [ fig_iracmipscol ] presents color - color diagrams of 2mass , irac and mips 24 sources detected in the orii-2 field . we have used several criteria ( shown as dashed lines and boxes in figure [ fig_iracmipscol ] ) to identify potential pms stars using these color - color diagrams . the most stringent classification scheme uses the irac ( [ 3.6][5.8 ] ) colors and the [ 8][24 ] irac and mips color . at 24 the reddening due to extinction is small and the photospheric colors are very close to zero for all spectral types @xcite . therefore the [ 8][24 ] color is very sensitive to infrared excess , but of course not all young stars bright are enough to be detected at 24 . using this color - color diagram we find four sources with infrared excess ( figure [ fig_iracmipscol ] ( _ left _ ) ) . we identify mir-54 and mir-89 as class 0/i sources , while mir-41 and mir-52 are in the class ii regime . figure [ fig_iracmipscol ] ( _ middle _ ) presents the [ 3.6][4.5 ] vs [ 5.8]-[8.0 ] color - color plot for the sources detected in all the irac bands . sources with the colors of stellar photospheres are centered at ( [ 3.6][4.5],[5.8][8.0])=(0,0 ) and include foreground and background stars as well as diskless ( class iii ) pre - main sequence stars . the box outlined in figure [ fig_iracmipscol ] ( _ middle _ ) , defines the location of class ii objects @xcite , i.e. sources whose colors can be explained by young , low - mass stars surrounded by disks . @xcite have shown from their observations of young stars in the taurus - auriga complex that class 0/i protostars require [ 3.6][4.5]@xmath30 and [ 5.8][8.0]@xmath30 . while mir-89 clearly falls in the class 0/i region , mir-54 does not . it is well above the horizontal line corresponding to [ 3.6][4.5]=0.7 , but the color [ 5.8][8.0 ] is slightly less than 0.7 . since the demarcating lines primarily serve as a guidance and mir-54 is securely identified as a class 0/i object based on the irac / mips color - color plot , we identify it as a class 0/i object here as well . in this color - color diagram five stars end up in the class ii color regime : mir-23 ( marginally ) , mir-25 , mir-41 , mir-51 , and mir-88 . mir-51 , which shows a large excess in the [ 5.8][8.0 ] color is a class ii / i source , i.e. a flat spectrum source . we did not pick up mir-52 , because the [ 3.6][4.5 ] color is only 0.12 , yet it is the only strong h@xmath1 emission line star in our sample ( see section [ not ] ) and definitely a pms star . it has , however , a strong infrared excess longward of 8 , which is why it was picked up in the [ 3.6][5.8 ] vs. [ 8][24 ] color - color diagram . @xcite also investigated color - color diagrams using the shortest irac bands , which have the highest sensitivity and which are least affected by pah emission with near - ir colors taken from the 2mass catalogue . we have similarly used the photometry for the sources presented in table [ tab_mirsrc ] which have 2mass counterparts to plot the @xmath31[3.6 ] versus [ 3.6][4.5 ] color - color diagram ( figure [ fig_iracmipscol ] _ right _ ) . we identify 12 potential pms stars , i.e. stars which are located outside the box demarcating stellar sources . these sources are mir-10 , 29 , 41 , 47 , 49 , 51 , 52 , 61 , 63 , 78 , 93 , and 117 . if we account for the large errors in faint 2mass sources we find at least one more source , mir-17 , which appears to have a clear excess in the 5.8 filter . for both mir-10 and 78 the 2mass @xmath31 fluxes have large uncertainties . it is therefore desirable to have additional evidence of ir excess in order to justify their classification as pms objects . while mir-10 shows a clear excess at 5.8 , mir-78 is not detected beyond 4.5 and it s sed is completely stellar with no sign of an ir excess . thus we do not consider mir-78 to be a pms object . since mir-93 was found to be an extragalactic source , we are left with 10 potential pms stars . of these we have already identified mir-41 , 49 , 51 and 52 as pms stars based on mir color - color plots . the two class 0/i sources , mir-54 and 89 , and mir-88 , a potential class ii source , were not detected in @xmath31 , and hence do not appear in this color - color diagram . neither do the sources mir-23 , and 25 , both of which were classified as pms objects in the irac color - color diagram , show @xmath31[3.6 ] or [ 3.6][4.5 ] colors expected for class ii objects . the @xmath31[3.6 ] color is 0.3 for mir-23 , which would put it in the class ii regime , while it is @xmath4 0 for mir-25 . the latter , however , does have a significant color excess at 8 . not all of the stars identified in the 2mass / irac color color diagram are likely to be pms stars , they could just be heavily reddened background stars . mir-29 , 47 , and 61 are all located in the western or southern rim of the globule , and have very blue irac colors . they are all almost certainly heavily reddened background stars . the same is true for mir-63 which is seen toward the opaque part of the globule . we identify mir-63 with the star that @xcite labeled b , and proposed might be a stellar counterpart to the iras source . however , the [ 3.6 ] [ 4.5 ] color for mir-63 is extremely blue , @xmath272.4@xmath15 , which shows that the red near - ir colors are simply due to extinction . the same is true for mir-117 , which is located east of the globule in a relatively low extinction area . for this star the @xmath32[3.6 ] color excess is rather marginal and there is no excess in the irac bands . it appears to be a reddened background star . mir-10 lies outside the globule in a region of low extinction and is very red in the [ 4.5 ] [ 5.8 ] color band , and could therefore be a pms star , although it was not detected at 8 . in all we therefore identify 11 potential pms objects based on 2mass , irac and mips colors . we classify mir-54 and 89 as class 0/i stars , while mir-10 , 17 , 23 , 25 , 41 , 49 , 51 , 52 , and 88 are class ii objects . we also note that mir-104 , which lies just outside the southern rim of the globule , was only detected at 5.8 and 8 , suggesting that it has an infrared excess . it is therefore a potential pms object . among these pms objects mir-10 , 17 , and 41 are located well south of the globule , while mir-25 is west of the cometary globule . if all of these four sources are pms stars , they most likely belong to the young @xmath3-orionis cluster and are unrelated to orii-2 . all the rest could potentially have formed in the globule . in section [ sed_modeling ] we investigate the seds of these candidate pms objects in more detail to see whether they are young stars , and if so , what their physical properties are . @xcite identified a parsec scale herbig - haro outflow , hh289 , to the east of the globule by analyzing images from the aat / ukst h@xmath1 survey , broadband r ( ivn ) plates and narrowband ccd images in h@xmath1 + [ ] . their hh objects 2 5 , i.e. knots c f , are all outside the area we imaged in r and h@xmath1 . we do detect their knot b in our h@xmath1 image , but not knot a. knot b appears very elongated and is located in the cavity wall from the red - shifted outflow from mir-54 . @xcite s feature 1 , which coincides with the outflow cavity emerging to the sw out of the globule is also readily visible in the h@xmath1 image ( figure [ fig_halpha_rband ] ) . although we see no trace of knot a in either r or h@xmath1 , it seems to coincide with the bluish nebulosity west of mir-54 , which is very prominent in the irac three color image ( figure [ fig_composite ] ) . in the 3.6 and 4.5 images the mir-54 nebulosity appears to be the inner part of an outflow cavity which is not visible in the mid - ir outside the bright western rim of the globule . however , this outflow lobe is seen more prominently as a limb - brightened h@xmath1 cavity extending out another 50 from the western edge of the globule . the total length of the outflow is therefore @xmath4 1.5 , 0.17 pc . since this outflow extends into `` empty '' space , i.e. the low density region ic434 , it is possible that the outflow extends even further , but becomes so diffuse that it is no longer visible . it is probably the counterpart to the large herbig - haro flow , but since the density of the globule is much lower on the eastern side of the globule , it has been able to expand more freely . however , in the irac images one can also see an outflow cavity from mir-52 protruding out of the globule . this outflow is approximately parallel with the mir-54 outflow . it is brighter on the southern side of the cavity and has an apparent length of @xmath4 50 from mir-52 . to the east the two outflows appear as a large evacuated hole ( cylinder ) and can not be separated . either star could therefore drive the large hh outflow . however , we also detect a herbig - haro like nebulosity with three knots or condensations in the r - band image @xmath4 140 east of mir-54 . a blow - up of this nebulosity is shown in figure [ fig_halpha_rband ] , although it is not seen at all in the h@xmath1 image . normally hh objects are bright in h@xmath1 . if this is an hh object , the emission must completely be dominated by [ sii ] emission , i.e. it has to be an extreme low excitation hh object . it is not detected by 2mass , nor is it seen in the irac images , which go deeper than 2mass . it is , however , seen faintly in @xcite s h@xmath1 + [ sii ] image , which proves that it is real and not just an instrumental artifact . in fact it appears quite strong in @xcite s 2.12 molecular hydrogen image , while the nebulosity is completely absent on their ivn plate . this confirms that there is no continuum at the position of the nebulosity . the only conclusion we can draw , is that this is a rather low excitation hh object , which is completely dominated by line emission from [ sii ] and which is also seen in vibrationally excited h@xmath33 at 2.12 . the coordinates of the three hh knots , which are here labeled hh992 , are given in table [ tab_hhobjects ] . this hh object is not a part of the large herbig - haro flow , hh289 , seen by @xcite . the most likely exciting source of the hh992 complex is mir-89 , which is @xmath475 west of the hh objects . mir-89 was identified as a class 0/i object in the color - color diagrams , but sed modeling ( section [ sed_modeling ] ) suggest that it is a class ii object . in the 3.6 and 4.5 images one can see a limb - brightened fan - shaped nebulosity emerging to the west from the star . this nebulosity may trace an outflow into the globule , which is roughly aligned ( p.a . @xmath4 -95 ) with the hh objects on the opposite side of the star . in order to interpret the observed seds for the sources identified to be pms and better characterize these sources we have explored the archive of two - dimensional ( 2d ) axisymmetric radiative transfer models of protostars calculated for a large range of protostellar masses , accretion rates , disk masses and disk orientations created by @xcite . this archive also provides a linear regression tool which can select all model seds that fit the observed sed better than a specified @xmath34 . each sed is characterized by a set of model parameters , such as stellar mass , temperature , and age , envelope accretion rate , disk mass , and envelope inner radius . we have used this online tool to generate models , which fit the observed seds for the 11 candidate pms objects . seds of six of the objects could be adequately fit with accretion disk models , the other five were better fit with stellar atmospheric models , reddened by foreground extinction . we restricted the sed fitting tool to explore only distances between 350 and 450 pc . below we present and briefly discuss the best fit models . figure [ fig_sedfits ] shows results of detailed modeling of the observed sed in the mid - infrared ( and fir and sub - mm for mir-54 ) for all the candidate pms objects . a major criticism against the use of these models has been the non - uniqueness of the solutions obtained from the model library . however we find that for four out of the six pms objects , _ viz . , mir-51 , 52 , 54 and 89 _ the best fit models are either distinctively better in reproducing all the observed flux densities than the next few models or that the distances for other possible models do not match the distance to orii-2 . for mir-23 we find 14 models with @xmath34@xmath35 and for mir-41 we find four models with @xmath34@xmath36 , which appear plausible . for mir-23 we find two independent models for which the inclination is varied to generate the 14 models . for each of these two sources we have obtained the average values and variances of the fitted parameters by taking a weighted average of the apparently acceptable models in a manner similar to @xcite . table [ tab_sedfits ] presents the parameters corresponding to the best fit models for the six sources which we fitted with accretion disk models . @xcite presented a classification scheme which is essentially analogous to the class scheme , but refers to the actual evolutionary stage of the object based on its physical properties like disk mass and envelope accretion rate rather than the slope of its near / mid - ir sed . according to the @xcite classification scheme , mir-54 , is a stage 0/i object , while all the others are stage ii objects . at a first glance it is surprising to find that mir-89 , which we classified as a young class 0/i object both in the irac and irac - mips color - color diagrams , is now identified as an evolved stage ii object . the sed model shows that it only has a minuscular envelope and there is no sign of envelope accretion . all reasonable sed models suggest that it is a heavily extincted ( @xmath37 50@xmath15 ) , very low - mass star . this is the sole reason why it appears as a class 0/i object in the color - color diagrams , because the observed magnitudes are not corrected for extinction in these diagrams . the star still has a rather massive disk compared to the stellar mass , @xmath38@xmath4 0.02 , and a moderately high disk accretion rate of 3.2 10@xmath39 yr@xmath19 . it is located in the bright rim on the eastern edge of the cometary globule orii-2 , and appears to drive an outflow seen as a fan - shaped nebulosity west of the star , and as the herbig - haro objects , hh992 , seen @xmath465 , 0.12 pc to the east of the star . mir-51 , which we identified as a class i / ii object based on color - color diagrams , appears more similar to mir-52 ( class ii object ) and if anything slightly older , not younger . both appear to be young stage ii objects ( @xmath40/@xmath41 10@xmath42 yr@xmath19 and @xmath43/ @xmath41 10@xmath44 ) and are embedded inside the globule . this is consistent with mir-52 having strong h@xmath1 emission and powering an outflow . the sed models place mir-23 and 41 as late stage ii objects . they have no measurable envelope and very low disk accretion rates ( table [ tab_sedfits ] ) . the models also suggest no or low foreground extinction . these results of modeling together with the location of these two objects in the region suggest that these objects are therefore more evolved pms stars belonging to the @xmath3-orionis cluster , although we can not exclude that they could have formed in the globule at an earlier epoch . we could not obtain any believable dust / envelope models for mir-10 , 17 , 25 , and 88 , which are all outside the globule in regions of low or moderate extinction , or for mir-49 , which is located in the most opaque part of the globule . we therefore fitted these objects with pure stellar atmospheric models reddened by foreground extinction . these fits are not very accurate , since we have no optical data , other than the r magnitudes . furthermore , mir-10 , 17 and 49 were not detected at 8 , while mir-88 was not detected by 2mass . we have however used upper limits from these bands to better constrain the fits . * mir-10 : * at wavelengths shortward of 4.5 , the observed sed is fitted reasonably well by a stellar photosphere = 5250 k , log@xmath45z / h@xmath46$]=-0.5 , log@xmath45g@xmath46 $ ] = 0.0 and = 1.45@xmath15 . however , the star shows an excess at 5.8 , which suggests that it could be a transitional disk object . * mir-17 : * similar to mir-10 the 2mass and short wavelength irac data for this star can be explained by reddening . even though it also shows an ir excess at 5.8 , when compared with the sed due to stellar photosphere = 3500 k , log@xmath45z / h@xmath46$]=-0.5 , log@xmath45g@xmath46 $ ] = 5.0 and @xmath4 2@xmath15 , the 8 upper limit suggests that there is no excess at 8 . we therefore dismiss mir-17 as a pms star . * mir-25 : * a stellar model with @xmath47 = 3500 k , log @xmath45z / h@xmath46$]=-2.5 , log@xmath45g@xmath46 $ ] = 5.0 , and = 0@xmath15 gives a reasonable fit to the observed data . there might be a small excess at 8 , which could continue to longer wavelengths . based on the available data , we therefore retain mir-25 as a candidate pms star , although the evidence for dust excess in this star is rather marginal . * mir-49 : * the sed of this source is well reproduced by a stellar photosphere with @xmath47 = 3500 k and = 16@xmath15 . mir-49 therefore appears to be a heavily reddened background star . * mir-88 : * the stellar model with = 3500 k , log@xmath45z / h@xmath46$]=-0.5 , log@xmath45g@xmath46 $ ] = 0.0 = 6.1@xmath15 reproduces the observed sed significantly better than the accretion disk models and there is no evidence for excess emission in the irac bands . it is therefore a reddened background star . most of the stars , which could not be fit with accretion disk models , appear to be reddened background stars , except perhaps for mir-10 and 25 , both of which could have some long wavelength excess . it is possible that they could be transitional disk objects belonging to the @xmath4 3 myr @xmath3 orionis cluster . transitional disk objects are stars , for which the inner disk has been cleared out , see e.g. @xcite , and which therefore do not show any excess until 6 or 8 . for mir-54 the models completely fail to reproduce the observed sub - millimeter flux densities , irrespective of the values of any of the input parameters . we discuss mir-54 in more detail in section [ sect_mir-54 ] . based on our sed modeling we conclusively identify six pms stars . all the pms stars in the orii-2 region appear to be low luminosity stars of spectral types k and m. the pms stars outside the globule have very low disk masses and disk accretion rates , 10@xmath48 to a few times 10@xmath49 yr@xmath19 ( table [ tab_sedfits ] ) . this is not surprising since these stars are most likely members of the @xmath3-orionis cluster , which has an age of @xmath4 3 myr @xcite . mir-54 is the mid - infrared counterpart to iras 05355 - 0146 , which was detected in all four iras bands . mir-54 is within 13 of the nominal iras position . it is the only source in orii-2 detected at 24 and 70 as well as in the sub - millimeter . the mips 70 flux density for mir-54 is @xmath50 jy . the iras flux densities are much higher than what we observe with mips . this is to be expected , since the large iras beam will include emission from the hot bright rim as well . the luminosity for the iras source , i.e. mir-54 , has therefore been severely over estimated in the past . we can obtain a more precise estimate of the bolometric luminosity by doing a greybody fit to the mips and scuba data . this is illustrated in figure [ fig_greyfit ] where we show a two - component greybody fit to the observed flux densities between 24 and 850 . based on the observed size of the source in the mid - infrared and sub - mm , we have assumed @xmath51 to be 10and 13 for the warm and the cold components , respectively . the dust temperatures ( ) , dust emissivity indices ( @xmath52 ) and masses calculated for the best fit model are presented in figure [ fig_greyfit ] . the total dust and mass gas , @xmath53 was derived from the fitted and @xmath52 using @xmath54 where @xmath55 is the distance in @xmath45kpc@xmath46 $ ] , s@xmath56 is the total flux ( in jy ) at frequency @xmath57 and m@xmath58 is given in @xmath45@xmath46 $ ] . based on the greybody fitting we obtain dust temperatures of 23 k and 67 k and @xmath52 of 0.9 and 0.8 for the cold and the warm components . between 24 and sub - mm wavelengths , the observed sed of the source is reasonably well fit using this simple two - temperature greybody model and it is consistent with the observed 1.3 mm flux density . however we could not derive a single radiative transfer based model , which reproduces the entire sed starting from 3.6 to the sub - mm wavebands ( figure [ fig_sedfits ] ) . the model is particularly poor at the longer wavelengths . however , both the greybody model and the radiation transfer based models give a luminosity of 1.31.8 and mass of 0.190.24 . mir-54 is therefore a deeply embedded ( not detected in the near - ir ) low - mass , very young object . it excites an h@xmath33o maser @xcite . it is most likely the exciting source for the large parsec scale outflow , hh289 . inside the globule this is seen as a low velocity molecular outflow @xcite . the outflow is also visible in the irac images ( section [ sec_hh992 ] ) . the sed modeling predicts @xmath40/@xmath4 2 10@xmath59 yr@xmath19 , which satisfies the criterion ( @xmath40/@xmath41 10@xmath60 yr@xmath19 ) used by @xcite to identify stage 0/i objects . since mir-54 powers such a large outflow , it is probably not a class 0 ( or stage 0 ) protostar , but rather a deeply embedded , somewhat more evolved class i protostar . we find clear evidence for triggered star formation in the cometary globule orii-2 , with four young stars embedded in the front side of the globule . the pms object furthest away from the bright rim and approximately at the center of the globule , mir-54 , is a young class i or stage i protostar . it is most likely responsible for the large parsec scale herbig - haro outflow , with its counter flow seen as a bright h@xmath1 outflow breaking out on the western side of the globule . however , mir-52 , the heavily obscured h@xmath1 emission line star @xmath4 20 south of mir-54 , also appears to drive an outflow , which is approximately aligned with the outflow from mir-54 . to the east both outflows overlap , and either star could therefore be the exciting star for the large hh outflow . mir-89 , which is located in the bright rim , is a more evolved class ii object , which most likely excites the hh object , hh992 , which we discovered in this study . hh992 lies outside the globule , @xmath4 65 east of the star and south of the large herbig - haro outflow , hh289 . hh992 breaks up into three knots , and must be rather low excitation , since it is not seen in the h@xmath1 image . another candidate young star , mir-104 , which is close to the globule and near mir-89 ( figure [ fig_halpha_rband ] ) may also have formed in the globule . however , mir-104 was only detected at 5.8 and 8 and we therefore have insufficient information to determine whether it really is a pms object . if it is , it could have formed inside the globule before the gas was ablated away by the strong uv radiation from @xmath3 ori . we identify two infrared excess stars , one of which is far to the south , @xmath42 from the globule and the other is seen in projection against the bright rim in the tail region of the globule . both these sources have zero or very little foreground extinction and are probably unrelated to the globule , although we can not exclude the possibility that they could have formed in the globule in an earlier epoch of star formation . we also find two other candidate pms stars in the field , which may have weak mid - ir excesses . all these stars appear older , i.e. they are pre - transitional or transitional disk stars , which now are part of @xmath3-orionis cluster , which has an age of @xmath4 3 myr . models of cometary globules can successfully explain their morphology and suggest that they form stars through radiation driven implosion ( rdi ) @xcite . this was already qualitatively discussed by @xcite , who showed that cometary globules are active sites of star formation . in these models cometary globules are denser cores in the molecular cloud surrounding an region , which get exposed to the uv radiation from the central ob association , when the region expands into it . the uv flux of the ob association ionizes the external layers of the core facing the ob association and an ionization front is formed at the front surface . the gas is heated during the ionization and the temperature increases . the increased pressure due to the hot ionized gas drives an isothermal shock into the cloud , which compresses the neutral gas in the core . at the same time , the ionized and heated gas at the surface of the core flows radially away from the surface and forms an evaporating layer surrounding the surface of the core . due to the curvature of the front surface of the core the shock waves preceding the ionization front converge and collide at their focus , causing the interior of core to be compressed to much higher densities . this causes the central part of the core to become gravitationally unstable , triggering the formation of a star . the observations of @xcite of the isolated cometary globule brc37 in the region ic1396 appears to match this model rather well . their observations show a string of pms objects along the symmetry axis of the globule , with the oldest being in front of the globule , and the youngest object still embedded in the head of the globule , suggesting sequential formation of young stars as the ionization front proceeds towards the tail of the globule . for some of the other cometary globules in the same region , like ic1396n ( brc38 ) , this scenario is less clear . @xcite found an elongated clustering of x - ray sources aligned with the globule , with the youngest stars still embedded inside it , consistent with the rdi model for triggered star formation . however @xcite , who studied the same cometary globule using deep j , h , k@xmath61 , and narrowband h@xmath33 2.12 imaging and previously published mm imaging did not find any color or age gradient in the south - north direction , i.e. in the direction of the globule . they therefore conclude that not all star formation in this globule can be explained in terms of triggering . we can think of two possible explanations for why ic1396n does not appear to show the same type of age progression and sequential triggering as seen in brc37 . if the x - ray sources in front of the globule are late class ii objects , i.e. transitional disk like objects , or class iii objects , they will show no color excess in the near - ir and therefore appear as field stars . secondly , ic1396n is not a well defined isolated cometary globule with a simple head / tail structure , suggesting that it could consist of more than one cometary globule . orii-2 , the cometary globule studied in this paper , does appear more similar to brc38 . it shows clear evidence for triggered star formation , with the youngest object , mir-54 , being furthest away from the front side of the globule . the two other young class ii objects are definitely older and closer to the bright rim and the ionizing star , @xmath3 orionis . all three stars are also close to the symmetry axis of the globule , as predicted by the rdi model . mir-89 and possibly mir-104 , which both lie on the eastern side of the globule , do not appear to have formed by focussed compression along the symmetry axis of the globule . other processes , like rayleigh - taylor instabilities in the bright rim , may also play a role and form low - mass stars like mir-89 . a similar situation is seen in the horsehead , a bright rimmed elephant trunk structure east of @xmath3 orionis , which also shows some evidence for sequential star formation triggered by the expanding region @xcite . these young pms stars are also low - mass stars , which appear to have formed in the bright rim by rayleigh - taylor instabilities , since hardly any of them lie close to the symmetry axis of the horsehead . several , if not all of the young stars in orii-2 drive outflows . it is interesting to note that all of the outflows are roughly perpendicular to the symmetry axis of the globule . it maybe a coincidence , but other cometary globules with embedded protostars , like brc37 , show a similar behavior . if this is generally true , it would provide additional support to the focussed rdi model , which predicts that the gas inside a cometary globule will be compressed in a ridge along the symmetry axis of the globule . when density condensations , i.e. cores in this ridge become gravitationally unstable and collapse , the collapse would preferentially be towards the symmetry axis , therefore aligning the protoplanetary disk along the symmetry axis . since outflows are driven by disks , one would therefore see outflows which are perpendicular to the symmetry axis of the cometary globule . orii-2 is thus a low mass star forming cometary globule . the star formation in this region appears to be triggered by a combination of rdi and rayleigh - taylor instabilities in the rim of the globule . in the deep r - band images for the first time we detect hh 992 , which based on its non - detection in the image is most likely an extremely low excitation hh object . follow - up optical and infrared spectroscopic observations to understand the true nature of hh 992 as well as mir-52 , the sole emission star in the field , are needed . we find that many of the young stars drive outflows which are aligned perpendicular to the symmetry axis of the globule . higher spatial resolution imaging of the outflows in orii-2 in near - infrared and millimeter would provide structural details of the outflows and lead to correct identification of the sources exciting these outflows . based on observations made with the nordic optical telescope , operated on the island of la palma jointly by denmark , finland , iceland , norway , and sweden , in the spanish observatorio del roque de los 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& [email protected] & 14.00@xmath5 0.03 & 13.82@xmath5 0.06 & [email protected] & 0.71 @xmath5 0.01 & 0.48 @xmath5 0.01 & 0.32 @xmath5 0.02 & + mir-2 & 5:37:55.41 & -1:45:22.1 & 20.3 & & & & [email protected] & 0.13 @xmath5 0.00 & & & + mir-3 & 5:37:55.43 & -1:43:16.4 & 16.3 & [email protected] & 14.15@xmath5 0.04 & 14.14@xmath5 0.06 & [email protected] & 0.44 @xmath5 0.01 & 0.30 @xmath5 0.01 & 0.17 @xmath5 0.02 & + mir-4 & 5:37:55.47 & -1:47:24.7 & 17.2 & [email protected] & 14.05@xmath5 0.04 & 13.85@xmath5 0.05 & [email protected] & 0.59 @xmath5 0.01 & 0.43 @xmath5 0.03 & & + mir-5 & 5:37:55.55 & -1:46:04.4 & 22.1 & & & & [email protected] & 0.07 @xmath5 0.00 & & & + mir-6 & 5:37:55.93 & -1:45:56.4 & 21.2 & & & & [email protected] & 0.05 @xmath5 0.00 & & & + mir-7 & 5:37:55.94 & -1:42:33.4 & 17.3 & [email protected] & 14.13@xmath5 0.04 & 13.85@xmath5 0.05 & & 0.62 @xmath5 0.01 & & & + mir-8 & 5:37:56.30 & -1:42:55.4 & 15.2 & [email protected] & 11.81@xmath5 0.02 & 11.63@xmath5 0.02 & [email protected] & 4.81 @xmath5 0.02 & 3.03 @xmath5 0.03 & 1.78 @xmath5 0.02 & + mir-9 & 5:37:56.39 & -1:45:15.2 & 23.3 & & & & [email protected] & 0.06 @xmath5 0.00 & & & + mir-10 & 5:37:57.04 & -1:47:40.9 & 18.2 & [email protected] & 16.18@xmath5 0.15 & 16.81@xmath5 & [email protected] & 0.08 @xmath5 0.00 & 0.10 @xmath5 0.01 & & + mir-11 & 5:37:57.28 & -1:44:10.5 & & & & & & & & 0.17 @xmath5 0.02 & + mir-12 & 5:37:57.80 & -1:44:29.5 & 14.0 & [email protected] & 12.38@xmath5 0.02 & 12.37@xmath5 0.03 & [email protected] & 2.32 @xmath5 0.01 & 1.49 @xmath5 0.02 & 0.86 @xmath5 0.02 & + mir-13 & 5:37:57.99 & -1:47:27.4 & & & & & [email protected] & 0.04 @xmath5 0.00 & & & + mir-14 & 5:37:58.01 & -1:45:23.4 & 22.2 & & & & [email protected] & 0.05 @xmath5 0.00 & & & + mir-15 & 5:37:58.12 & -1:44:40.7 & 17.9 & [email protected] & 16.27@xmath5 0.16 & 14.96@xmath5 & [email protected] & 0.07 @xmath5 0.00 & & & + mir-16 & 5:37:58.29 & -1:42:37.4 & 19.3 & & & & & 0.13 @xmath5 0.00 & & 0.29 @xmath5 0.02 & + mir-17 & 5:37:58.38 & -1:47:30.3 & 20.0 & [email protected] & 16.31@xmath5 0.16 & 15.50@xmath5 0.22 & [email protected] & 0.12 @xmath5 0.00 & 0.14 @xmath5 0.01 & & + mir-18 & 5:37:58.64 & -1:47:30.6 & 21.5 & & & & [email protected] & 0.10 @xmath5 0.00 & & & + mir-19 & 5:37:58.77 & -1:44:56.3 & 18.1 & [email protected] & 16.34@xmath5 0.17 & 15.95@xmath5 & [email protected] & 0.07 @xmath5 0.00 & & & + mir-20 & 5:37:58.78 & -1:45:50.0 & 15.6 & [email protected] & 12.40@xmath5 0.02 & 12.19@xmath5 0.02 & [email protected] & 2.70 @xmath5 0.01 & 1.95 @xmath5 0.02 & 1.06 @xmath5 0.02 & + mir-21 & 5:37:58.86 & -1:47:28.1 & 23.1 & & & & [email protected] & 0.07 @xmath5 0.00 & & & + mir-22 & 5:37:59.07 & -1:44:14.2 & & & & & & & & 0.14 @xmath5 0.02 & + mir-23 & 5:37:59.13 & -1:43:54.2 & 16.9 & [email protected] & 13.77@xmath5 0.04 & 13.63@xmath5 0.05 & [email protected] & 1.16 @xmath5 0.10 & 1.04 @xmath5 0.02 & 1.50 @xmath5 0.02 & + mir-24 & 5:37:59.24 & -1:46:35.5 & & & & & [email protected] & 0.03 @xmath5 0.00 & & & + mir-25 & 5:37:59.35 & -1:45:17.9 & 16.9 & [email protected] & 14.70@xmath5 0.04 & 14.57@xmath5 0.07 & [email protected] & 0.26 @xmath5 0.01 & 0.16 @xmath5 0.01 & 0.16 @xmath5 0.02 & + mir-26 & 5:37:59.36 & -1:44:32.1 & 20.5 & [email protected] & 16.44@xmath5 0.18 & 15.54@xmath5 0.22 & [email protected] & 0.11 @xmath5 0.00 & 0.09 @xmath5 0.01 & & + mir-27 & 5:37:59.49 & -1:45:52.2 & 19.0 & [email protected] & 15.57@xmath5 0.10 & 15.25@xmath5 0.16 & [email protected] & 0.15 @xmath5 0.00 & 0.10 @xmath5 0.01 & & + mir-28 & 5:37:59.53 & -1:46:49.2 & & & & & [email protected] & 0.08 @xmath5 0.00 & & & + mir-29 & 5:38:00.00 & -1:44:30.6 & 18.3 & [email protected] & 15.24@xmath5 0.07 & 15.23@xmath5 0.17 & [email protected] & 0.19 @xmath5 0.00 & & & + mir-30 & 5:38:00.04 & -1:46:22.5 & 18.6 & & & & [email protected] & 0.06 @xmath5 0.00 & & & + mir-31 & 5:38:00.44 & -1:42:59.4 & 19.5 & & & & [email protected] & 0.05 @xmath5 0.00 & & & + mir-32 & 5:38:00.90 & -1:46:44.9 & & & & & [email protected] & 0.04 @xmath5 0.00 & & & + mir-33 & 5:38:01.03 & -1:44:33.7 & & & & & [email protected] & 0.11 @xmath5 0.00 & & & + mir-34 & 5:38:01.06 & -1:47:14.1 & 16.4 & [email protected] & 14.30@xmath5 0.04 & 14.46@xmath5 0.07 & [email protected] & 0.31 @xmath5 0.00 & 0.24 @xmath5 0.01 & 0.15 @xmath5 0.02 & + mir-35 & 5:38:01.07 & -1:44:50.8 & 22.5 & & & & [email protected] & 0.07 @xmath5 0.00 & & & + mir-36 & 5:38:01.07 & -1:43:51.1 & & & & & [email protected] & 0.05 @xmath5 0.00 & & & + mir-37 & 5:38:01.60 & -1:44:32.5 & & & & & [email protected] & 0.04 @xmath5 0.00 & & & + mir-38 & 5:38:01.83 & -1:45:50.1 & 15.1 & [email protected] & 11.82@xmath5 0.02 & 11.62@xmath5 0.03 & [email protected] & 4.40 @xmath5 0.02 & 4.07 @xmath5 0.03 & 3.86 @xmath5 0.03 & + mir-39 & 5:38:02.10 & -1:44:10.9 & 23.2 & & & & [email protected] & 0.04 @xmath5 0.00 & & & + mir-40 & 5:38:02.24 & -1:47:30.1 & & & & & [email protected] & 0.05 @xmath5 0.00 & & & + mir-41 & 5:38:02.33 & -1:47:40.4 & 19.1 & [email protected] & 13.49@xmath5 0.03 & 13.10@xmath5 0.03 & [email protected] & 3.21 @xmath5 0.30 & 2.88 @xmath5 0.03 & 2.55 @xmath5 0.06 & 2.40 @xmath5 0.05 + mir-42 & 5:38:02.35 & -1:43:47.3 & & & & & [email protected] & 0.05 @xmath5 0.00 & & & + mir-43 & 5:38:02.36 & -1:44:01.8 & 22.7 & & & & [email protected] & 0.03 @xmath5 0.00 & & & + mir-44 & 5:38:02.87 & -1:42:36.7 & 17.4 & [email protected] & 12.58@xmath5 0.02 & 12.23@xmath5 0.03 & & 3.34 @xmath5 0.02 & 2.13 @xmath5 0.04 & 1.26 @xmath5 0.03 & + mir-45 & 5:38:03.42 & -1:45:15.1 & & & & & [email protected] & 0.04 @xmath5 0.00 & & & + mir-46 & 5:38:03.57 & -1:44:15.6 & & & & & [email protected] & 0.06 @xmath5 0.00 & & & + mir-47 & 5:38:03.87 & -1:45:57.5 & 17.6 & [email protected] & 15.40@xmath5 0.08 & 15.40@xmath5 0.17 & [email protected] & 0.16 @xmath5 0.00 & & & + mir-48 & 5:38:04.10 & -1:46:25.1 & 18.5 & & & & [email protected] & 0.05 @xmath5 0.00 & & & + mir-49 & 5:38:04.33 & -1:44:37.9 & & 17.27@xmath5 & 16.68@xmath5 0.26 & 15.20@xmath5 0.16 & [email protected] & 0.54 @xmath5 0.01 & 0.43 @xmath5 0.01 & & + mir-50 & 5:38:04.43 & -1:45:50.3 & & & & & [email protected] & 0.09 @xmath5 0.00 & 0.76 @xmath5 0.01 & & + mir-51 & 5:38:04.56 & -1:45:53.0 & 17.3 & [email protected] & 14.72@xmath5 0.04 & 14.64@xmath5 0.10 & [email protected] & 0.76 @xmath5 0.10 & 1.67 @xmath5 0.37 & 2.72 @xmath5 0.03 & + mir-52 & 5:38:04.78 & -1:45:32.2 & 19.9 & [email protected] & 12.73@xmath5 0.04 & 12.16@xmath5 0.03 & [email protected] & 5.13 @xmath5 0.12 & 4.00 @xmath5 0.38 & 2.91 @xmath5 0.03 & 11.37 @xmath5 0.05 + mir-53 & 5:38:05.10 & -1:45:41.9 & 22.8 & & & & [email protected] & 0.02 @xmath5 0.00 & & & + mir-54 & 5:38:05.15 & -1:45:12.4 & & & & & [email protected] & 4.97 @xmath5 0.04 & 15.73 @xmath5 0.06 & 13.63 @xmath5 0.06 & 231.90 @xmath5 0.44 + mir-55 & 5:38:05.19 & -1:46:18.9 & 20.7 & & & & [email protected] & 0.06 @xmath5 0.00 & & & + mir-56 & 5:38:05.40 & -1:47:05.8 & 22.5 & & & & [email protected] & 0.06 @xmath5 0.00 & & & + mir-57 & 5:38:05.50 & -1:42:43.1 & 21.7 & & & & & 0.10 @xmath5 0.00 & & & + mir-58 & 5:38:05.83 & -1:42:36.3 & 14.1 & [email protected] & 11.94@xmath5 0.02 & 11.84@xmath5 0.03 & [email protected] & 3.47 @xmath5 0.02 & 2.26 @xmath5 0.03 & 1.28 @xmath5 0.02 & + mir-59 & 5:38:05.86 & -1:47:04.1 & 19.1 & [email protected] & 16.26@xmath5 0.16 & 15.70@xmath5 0.23 & [email protected] & 0.08 @xmath5 0.00 & & & + mir-60 & 5:38:05.90 & -1:43:05.7 & 19.8 & & & & [email protected] & 0.05 @xmath5 0.00 & & & + mir-61 & 5:38:06.06 & -1:46:00.2 & 18.2 & [email protected] & 15.34@xmath5 0.07 & 15.06@xmath5 0.13 & [email protected] & 0.20 @xmath5 0.00 & & & + mir-62 & 5:38:06.26 & -1:43:59.6 & & & & & [email protected] & 0.04 @xmath5 0.00 & & & + mir-63 & 5:38:06.38 & -1:45:02.1 & 20.2 & [email protected] & 16.23@xmath5 0.15 & 15.40@xmath5 0.18 & [email protected] & 0.02 @xmath5 0.00 & & & + mir-64 & 5:38:06.38 & -1:46:36.4 & 16.6 & [email protected] & 13.24@xmath5 0.02 & 12.99@xmath5 0.03 & [email protected] & 1.30 @xmath5 0.01 & 0.93 @xmath5 0.02 & 0.47 @xmath5 0.02 & + mir-65 & 5:38:06.65 & -1:46:55.5 & 20.8 & & & & [email protected] & 0.07 @xmath5 0.00 & & & + mir-66 & 5:38:06.77 & -1:47:14.8 & 21.6 & & & & [email protected] & 0.05 @xmath5 0.00 & & & + mir-67 & 5:38:06.84 & -1:47:03.8 & & & & & [email protected] & 0.03 @xmath5 0.00 & & & + mir-68 & 5:38:06.99 & -1:43:04.0 & 20.1 & [email protected] & 16.07@xmath5 0.14 & 15.77@xmath5 0.26 & [email protected] & 0.10 @xmath5 0.00 & & & + mir-69 & 5:38:07.25 & -1:45:59.7 & 20.7 & & & & [email protected] & 0.06 @xmath5 0.00 & & & + mir-70 & 5:38:07.27 & -1:47:13.0 & 23.3 & & & & [email protected] & 0.04 @xmath5 0.00 & & & + mir-71 & 5:38:07.64 & -1:42:46.8 & 18.5 & [email protected] & 16.15@xmath5 0.14 & 15.86@xmath5 & [email protected] & 0.07 @xmath5 0.00 & & & + mir-72 & 5:38:07.65 & -1:44:17.9 & 16.2 & [email protected] & 13.01@xmath5 0.03 & 12.76@xmath5 0.04 & [email protected] & 1.78 @xmath5 0.01 & 1.25 @xmath5 0.02 & 0.65 @xmath5 0.02 & + mir-73 & 5:38:08.01 & -1:45:07.7 & s & [email protected] & 8.55@xmath5 0.03 & 8.57@xmath5 0.02 & [email protected] & 59.6 @xmath5 0.2 & 41.4 @xmath5 0.1 & 21.8 @xmath5 0.1 & 6.32 @xmath5 0.05 + mir-74 & 5:38:08.11 & -1:42:43.5 & 20.1 & [email protected] & 16.20@xmath5 0.15 & 15.47@xmath5 0.19 & [email protected] & 0.11 @xmath5 0.00 & 0.04 @xmath5 0.01 & & + mir-75 & 5:38:08.19 & -1:43:16.0 & 16.3 & [email protected] & 14.27@xmath5 0.04 & 14.08@xmath5 0.06 & [email protected] & 0.45 @xmath5 0.01 & 0.30 @xmath5 0.01 & 0.12 @xmath5 0.02 & + mir-76 & 5:38:08.68 & -1:43:01.1 & 18.2 & [email protected] & 15.16@xmath5 0.06 & 14.80@xmath5 0.10 & [email protected] & 0.22 @xmath5 0.00 & 0.08 @xmath5 0.01 & & + mir-77 & 5:38:08.88 & -1:43:28.3 & 22.6 & & & & [email protected] & 0.03 @xmath5 0.00 & & & + mir-78 & 5:38:08.92 & -1:44:48.2 & 18.2 & [email protected] & 16.11@xmath5 0.14 & 16.81@xmath5 & [email protected] & 0.07 @xmath5 0.00 & & & + mir-79 & 5:38:09.14 & -1:45:06.7 & & & & & [email protected] & 0.05 @xmath5 0.00 & & & + mir-80 & 5:38:09.40 & -1:44:20.7 & & & & & [email protected] & 0.13 @xmath5 0.00 & & & + mir-81 & 5:38:09.66 & -1:47:16.8 & 17.0 & [email protected] & 14.64@xmath5 0.04 & 14.41@xmath5 0.08 & [email protected] & 0.33 @xmath5 0.00 & 0.27 @xmath5 0.01 & 0.14 @xmath5 0.02 & + mir-82 & 5:38:09.69 & -1:44:51.0 & & & & & [email protected] & 0.04 @xmath5 0.00 & & & + mir-83 & 5:38:09.78 & -1:44:42.6 & 19.4 & & & & [email protected] & 0.05 @xmath5 0.00 & & & + mir-84 & 5:38:09.93 & -1:45:47.5 & 16.2 & [email protected] & 14.19@xmath5 0.03 & 14.12@xmath5 0.06 & [email protected] & 0.44 @xmath5 0.01 & 0.30 @xmath5 0.01 & 0.21 @xmath5 0.02 & + mir-85 & 5:38:09.96 & -1:45:53.0 & & & & & [email protected] & 0.02 @xmath5 0.00 & & & + mir-86 & 5:38:09.97 & -1:47:00.6 & 20.3 & & & & [email protected] & 0.07 @xmath5 0.00 & & & + mir-87 & 5:38:10.01 & -1:43:35.6 & 17.8 & [email protected] & 15.94@xmath5 0.13 & 15.75@xmath5 0.25 & [email protected] & 0.10 @xmath5 0.00 & & & + mir-88 & 5:38:10.02 & -1:45:38.1 & 23.5 & & & & [email protected] & 0.35 @xmath5 0.10 & 0.12 @xmath5 0.01 & 0.15 @xmath5 0.02 & + mir-89 & 5:38:10.16 & -1:45:15.6 & & & & & [email protected] & 0.58 @xmath5 0.10 & 0.73 @xmath5 0.01 & 2.91 @xmath5 0.02 & 21.40 @xmath5 0.05 + mir-90 & 5:38:10.30 & -1:44:15.4 & & & & & [email protected] & 0.06 @xmath5 0.00 & & & + mir-91 & 5:38:10.31 & -1:43:44.8 & & & & & [email protected] & 0.07 @xmath5 0.00 & & & + mir-92 & 5:38:10.35 & -1:45:59.4 & & & & & [email protected] & 0.02 @xmath5 0.00 & & & + mir-93 & 5:38:10.41 & -1:45:26.4 & 19.2 & [email protected] & 15.13@xmath5 0.08 & 14.62@xmath5 0.11 & [email protected] & 0.35 @xmath5 0.00 & 0.29 @xmath5 0.01 & 0.51 @xmath5 0.02 & + mir-94 & 5:38:10.52 & -1:43:13.7 & 18.6 & [email protected] & 16.03@xmath5 0.14 & 15.54@xmath5 & [email protected] & 0.08 @xmath5 0.00 & & & + mir-95 & 5:38:10.69 & -1:44:07.4 & & & & & [email protected] & 0.06 @xmath5 0.00 & & & + mir-96 & 5:38:10.70 & -1:45:03.9 & 23.7 & & & & [email protected] & 0.05 @xmath5 0.00 & & & + mir-97 & 5:38:10.79 & -1:44:11.6 & 22.9 & & & & [email protected] & 0.06 @xmath5 0.00 & & & + mir-98 & 5:38:10.80 & -1:44:44.5 & 19.2 & & & & [email protected] & 0.04 @xmath5 0.00 & & & + mir-99 & 5:38:10.85 & -1:46:57.6 & & & & & [email protected] & 0.07 @xmath5 0.00 & & & + mir-100 & 5:38:10.85 & -1:46:52.1 & & & & & [email protected] & 0.05 @xmath5 0.00 & & & + mir-101 & 5:38:10.98 & -1:44:03.1 & & & & & [email protected] & 0.06 @xmath5 0.00 & & & + mir-102 & 5:38:11.08 & -1:46:09.5 & & & & & [email protected] & 0.04 @xmath5 0.00 & & & + mir-103 & 5:38:11.21 & -1:45:35.0 & 16.9 & [email protected] & 15.39@xmath5 0.08 & 15.15@xmath5 0.15 & [email protected] & 0.17 @xmath5 0.00 & 0.07 @xmath5 0.01 & & + mir-104 & 5:38:11.66 & -1:44:56.5 & & & & & & & 0.10 @xmath5 0.01 & 0.30 @xmath5 0.02 & + mir-105 & 5:38:11.71 & -1:46:57.4 & & & & & [email protected] & 0.05 @xmath5 0.00 & & & + mir-106 & 5:38:11.71 & -1:44:48.1 & 23.1 & & & & [email protected] & 0.04 @xmath5 0.00 & & & + mir-107 & 5:38:11.81 & -1:43:01.1 & 17.6 & [email protected] & 14.01@xmath5 0.04 & 13.80@xmath5 0.05 & [email protected] & 0.74 @xmath5 0.01 & 0.45 @xmath5 0.01 & 0.25 @xmath5 0.02 & + mir-108 & 5:38:11.97 & -1:46:38.8 & & & & & [email protected] & 0.12 @xmath5 0.00 & & & + mir-109 & 5:38:11.99 & -1:45:44.5 & 21.2 & & & & [email protected] & 0.03 @xmath5 0.00 & & & + mir-110 & 5:38:12.01 & -1:45:06.5 & 18.2 & [email protected] & 16.46@xmath5 0.19 & 15.72@xmath5 0.23 & [email protected] & 0.05 @xmath5 0.00 & & & + mir-111 & 5:38:12.04 & -1:43:36.6 & & & & & [email protected] & 0.04 @xmath5 0.00 & & & + mir-112 & 5:38:12.16 & -1:46:56.6 & 23.7 & & & & [email protected] & 0.06 @xmath5 0.00 & & & + mir-113 & 5:38:12.44 & -1:46:53.8 & & & & & [email protected] & 0.08 @xmath5 0.00 & & & + mir-114 & 5:38:12.62 & -1:47:07.5 & 17.5 & [email protected] & 15.29@xmath5 0.08 & 14.95@xmath5 0.12 & [email protected] & 0.20 @xmath5 0.00 & 0.16 @xmath5 0.01 & 0.13 @xmath5 0.02 & + mir-115 & 5:38:13.32 & -1:47:17.5 & 17.4 & [email protected] & 15.17@xmath5 0.09 & 14.97@xmath5 0.13 & [email protected] & 0.21 @xmath5 0.00 & 0.14 @xmath5 0.01 & & + mir-116 & 5:38:13.37 & -1:42:41.9 & 13.8 & [email protected] & 12.10@xmath5 0.03 & 12.02@xmath5 0.03 & [email protected] & 3.14 @xmath5 0.01 & 1.93 @xmath5 0.02 & 1.10 @xmath5 0.02 & + mir-117 & 5:38:13.57 & -1:43:36.1 & 16.4 & [email protected] & 14.67@xmath5 0.06 & 14.98@xmath5 0.15 & [email protected] & 0.30 @xmath5 0.00 & 0.14 @xmath5 0.01 & & + mir-118 & 5:38:13.59 & -1:44:27.8 & & & & & [email protected] & 0.04 @xmath5 0.00 & & & + mir-119 & 5:38:13.69 & -1:45:15.3 & 19.7 & & & & [email protected] & 0.07 @xmath5 0.00 & & & + mir-120 & 5:38:14.08 & -1:43:50.9 & 14.2 & [email protected] & 11.63@xmath5 0.02 & 11.51@xmath5 0.03 & [email protected] & 4.90 @xmath5 0.02 & 3.21 @xmath5 0.03 & 1.84 @xmath5 0.02 & + mir-121 & 5:38:14.68 & -1:44:53.1 & 23.5 & & & & [email protected] & 0.07 @xmath5 0.00 & & & + mir-122 & 5:38:14.90 & -1:46:07.9 & & & & & [email protected] & 0.07 @xmath5 0.00 & & & + mir-123 & 5:38:14.92 & -1:47:04.1 & 18.4 & [email protected] & 16.52@xmath5 0.21 & 16.35@xmath5 & [email protected] & 0.06 @xmath5 0.00 & & & + mir-124 & 5:38:15.01 & -1:45:58.6 & 20.4 & & & & [email protected] & 0.10 @xmath5 0.00 & & & + mir-125 & 5:38:15.45 & -1:45:56.1 & 20.4 & & & & [email protected] & 0.09 @xmath5 0.00 & & & + + crrrrrrrr mir-23 & 3110@xmath518 & [email protected] & 1.7(-4)@xmath55.3(-5 ) & 2.9(-6)@xmath59.2(-7 ) & 0.0 & 1.9(-11)@xmath56.2(-13 ) & [email protected] & [email protected] + mir-41 & 3750@xmath5184 & [email protected] & 5.0(-3)@xmath53(-3 ) & 2.0(-6)@xmath51.8(-6 ) & 0.0 & 1.4(-8)@xmath57.1(-9 ) & [email protected] & [email protected] + mir-51 & 4500 & 1.56 & 2.0(-4 ) & 3.3(-4 ) & 1.9(-7 ) & 8.2(-10 ) & 6.5 & 1.9 + mir-52 & 2980 & 0.14 & 5.2(-5 ) & 9.7(-4 ) & 1.0(-7 ) & 3.1(-11 ) & 0.24 & 4.8 + mir-54 & 3380 & 0.30 & 2.5(-3 ) & 1.9(-1 ) & 5.9(-6 ) & 1.1(-8 ) & 1.36&12 + mir-89 & 3370 & 0.24 & 4.5(-3 ) & 2.1(-9 ) & 0.0 & 3.3(-8 ) & 0.3 & 50 +
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we investigate the young stellar population in and near the cometary globule orii-2 .
the analysis is based on deep nordic optical telescope @xmath0-band and h@xmath1 images , jcmt scuba 450 and 850 images combined with near - infrared 2mass photometry and mid - infrared archival _
spitzer _
images obtained with the irac ( 3.6 , 4.5 , 5.8 and 8 ) , and mips ( 24 and 70 ) instruments .
we identify a total of 125 sources within the 5@xmath25 region imaged by irac . of these sources 87
are detected in the @xmath0-band image and 51 are detected in the 2mass survey .
the detailed physical properties of the sources are explored using a combination of near / mid - infrared color - color diagrams , greybody fitting of seds and an online sed fitting tool that uses a library of 2d radiation transfer based accretion models of young stellar objects with disks .
orii-2 shows clear evidence of triggered star formation with four young low luminosity pre - main sequence stars embedded in the globule .
at least two , possibly as many as four , additional low - mass pms objects , were discovered in the field which are probably part of the young @xmath3 orionis cluster . among the pms stars which have formed in the globule
, mir-54 is a young , deeply embedded class 0/i object , mir-51 and 52 are young class ii sources , while mir-89 is a more evolved , heavily extincted class ii object with its apparent colors mimicking a class 0/i object .
the class 0/i object mir-54 coincides with a previously known iras source and is a strong sub - millimeter source .
it is most likely the source for the molecular outflow and the large parsec scale herbig - haro flow .
however the nearby class ii source , mir-52 , which is strong a h@xmath1 emission line star , also appears to drive an outflow approximately aligned with the outflow from mir-54 , and because of the proximity of the two outflows , either star could contribute .
mir-89 appears to excite a low excitation hh object , hh 992 , discovered for the first time in this study .
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in a recent communication m.m . block and f. halzen @xcite ( hereafter referred to as bh ) have presented some critical comments on our analysis @xcite ( hereafter fms ) . some points raised by these authors have already been addressed and discussed in our subsequent work @xcite ( hereafter fms2 ) , available as arxiv since august 16 , 2012 . in this note we reply to the bh criticisms directed to fms , recalling also some aspects of interest presented in fms2 . first , to facilitate the discussion and the reference to each part of the bh commentaries , some explanations on the organization of this reply are in order . excluding the acknowledgments and references , bh arxiv comprises four pages and the effective criticisms to fms correspond to approximately one page . all the rest of the manuscript ( three pages ) largely overlap with their previous work @xcite ( as advised in the arxiv admin note " @xcite ) . we shall focus on this 25 % material , in our section [ s2 ] . although not characterized as criticisms , the rest of the bh reproduces their former work on the subject , as a kind of lesson to be learned . in this respect , a discussion on some aspects of the fms analysis and a brief commentary on the bh former work are presented in section [ s3 ] . our conclusions and final remarks are the contents of section [ s4 ] . the content of the criticisms to fms , presented in bh , can be divided in three blocks , one block referring to the @xmath0 information ( page 1 in bh ) , another block referring to statistical probabilities ( page 2 ) and the last one to predictions at 7 tev ( page 2 ) . in what follows , each block will be treated as a subsection , in which we first situate and summarize the commentary , or quote it explicitly , and then present our reply . _ - commentary _ the first effective criticism appears in page 1 , fourth paragraph of the section _ introduction_. it concerns the fact that in fms the @xmath0 information was not used in global fits with the total cross section data . according to them , a statement alluded to ( but _ not _ carried out ) in appendix ... " . they also add : ... in their appendix , they give a rather cumbersome evaluation using their variant 3 model , to _ separately evaluate _ @xmath0 .... " _ - our reply _ in fms , the analysis has been based only on the @xmath2 data ( without the inclusion of the @xmath0 information ) for the six reasons explained there , which we consider as six facts . however , addressing the comments by one of the _ three referees _ in the submission to the braz . , we have included appendix a in a revised version . in this appendix we have shown that , even in the case of the largest values of the exponent @xmath3 ( method 1 , v3 and method 2 , v5 ) , the _ predictions _ for @xmath4 are in agreement with the experimental information . to connect @xmath5 and @xmath4 in an analytical way , we have used singly - subtracted derivative dispersion relations in the operational form introduced by kang and nicolescu @xcite ( also discussed in @xcite ) . in particular we have obtained an _ extended _ analytical result for the case of @xmath3 as a _ real _ parameter ( equations a7 and a8 in appendix a of fms ) . in respect to the effect of the @xmath0 information in data reductions , we have stated at the end of appendix a ( the references that follows concern fms ) : finally , we recall that in simultaneous fit to @xmath2 and @xmath0 the subtraction constant affects both the low- and high - energy regions [ 47,48 ] . that is a consequence of the strong correlation among the subtraction constant and all the other physical free fit parameters . we plan to discuss this consequence and other aspects of the fit procedures in a forthcoming paper . " also , in the last paragraph of our conclusions ( third sentence ) we added : these are aspects that we expect to consider in a future work , since they may provide information that is complementary to the results here presented . " in fact , in the fms2 we have extended the fms analysis in several aspects as referred to and outlined in the introduction of fms2 . in special , not only individual but also novel simultaneous fits to total cross section and @xmath0 information have been developed , leading to solutions with @xmath3 greater than 2 , _ despite the constraint involved_. the origin and role of the subtraction constant have also been discussed in some detail . therefore , we see no sense in depreciating statements like alluded but not carried out " or they give a rather cumbersome evaluation " . the second criticism in bh appears in page 2 , section _ statistical probabilities _ and involves five paragraphs , four with criticisms ( left column ) and the final one with the conclusions ( right column ) . the main point here concerns the use of the integrated probability @xmath6 to punctually analyze the reliability of the fit results . we shall treat each paragraph separately and in sequence . however , before that , some aspects on our fit procedures and on statistical analysis demand a few comments for further reference . @xmath7 introductory remarks - _ on the fit procedures _ the parametrization for the total cross section used in fms , introduced by amaldi _ @xcite , reads @xmath8^{-b_1 } + \ , \tau \ , a_2\ , \left[\frac{s}{s_l}\right]^{-b_2 } + \alpha\ , + \beta\ , \ln^{\gamma } \left(\frac{s}{s_h}\right),\end{aligned}\ ] ] [ eq1 ] where @xmath9 = -1 ( + 1 ) for @xmath1 ( @xmath10 ) scattering and @xmath11 = 1 gev@xmath12 is fixed . the dependence is linear in four parameters ( @xmath13 , @xmath14 , @xmath15 and @xmath16 ) and nonlinear in the other four ( @xmath17 , @xmath18 , @xmath3 and @xmath19 ) . as stated by bevington and robinson @xcite ( section 8.2 searching parameter space ) and also quoted recently in @xcite , _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ fitting nonlinear functions to data samples sometimes seems to be more an art than a science . " _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ nonlinear data reductions are not a trivial task . they demand a methodology for the choice of the initial ( feedback ) values of the free parameters . our strategy has been to test a _ grid _ of different ( physical ) feedback values for the free fit parameters so as to check the stability of the results , as shortly recalled in what follows . in both cases , fms and fms2 , we have considered as feedback the results already found by the particle data group ( pdg ) , which uses the standard compete collaboration highest - rank parametrization ( @xmath3 = 2 , a fixed parameter ) @xcite . in fms we used the 2010 pdg edition @xcite and in fms2 the updated result from the 2012 pdg edition @xcite . although applied to only a _ subset _ of the dataset analyzed in pdg , we understand that with these conservative choices " ( @xmath3 = 2 ) , we start with reasonable stable solutions ( already found by the pdg ) . with this input we are then able to investigate possible _ departures from this solution _ in the case of @xmath3 as a free parameter ( including or not the totem datum @xcite , as done in fms and including this point in all fits , as done in fms2 ) . _ in addition _ , in order to investigate the effect of the feedback values in the fit results , we have considered another distinct choices in both fms and fms2 and 3 different versions in each case of fms ( referred to as six variants and a direct fit ) . as explicitly quoted in fms and fms2 , our data reductions have been carried out with the objects of the class tminuit of the root framework @xcite . the statistical interpretations of the fit results , as well as aspects related to the error matrix , correlation matrix and analytical error propagation , have been based on the bevington - robinson book @xcite . for further reference , we recall that the error ( or covariance ) matrix provides the variances ( diagonal ) and covariances ( out of diagonal ) associated with each free fit parameter . the symmetric correlation matrix gives a measure of the correlation between each pair of free parameter through a coefficient with numerical limits @xmath20 1 ( full correlation ) and 0 ( no correlation ) . in the minimization program @xcite a confidence level of one standard deviation was adopted in all fits ( up = 1 ) . in each test of fit , successive running of the migrad have been considered ( up to 200 calls in fms and up to 2,000 calls in fms2 ) , until full convergence has been reached , with the smallest fcn ( @xmath21 ) and edm ( estimated distance to minimum ) , specifically edm @xmath22 ( adequate for the one sigma cl ) . in addition , the error in the parameters should not exceed the central value . among _ several different tests _ , we have _ selected _ , under the above criteria , the seven variants presented in fms . these variants are related to two different choices for the input values for all fits , denoted method 1 and method 2 in fms . for further reference and clarity , we summarize and situate below the structure of the grid considered in fms ( v stands for variant and df for direct fit , also a variant ) . method 1 . initial feedback values from pdg 2010 ( table 1 , second column ) : @xmath23 method 2 . initial feedback values distinct from method 1 ( table 3 , second column ) : @xmath24 in the six variants ( including the df ) we have investigated the effects of fixing or not the three fundamental parameters directly related with the energy dependence , namely @xmath17 , @xmath18 and @xmath3 . for the reggeon intercepts we have tested either _ ad hoc _ fixed values 1/2 or fixed ( central ) values from spectroscopic data ( chew - frautschi plots ) . we stress that these results constitute final solutions , selected under the above mentioned criteria . therefore our strategy in fms ( and fms2 ) did not involve extremely detailed use of different routines to possibly reach an absolute minimum . our point has been to investigate a _ grid _ of reasonable physical choices for feedbacks and variants for two ensembles , obtaining solutions through standard running of the minuit . despite of these possible limitations , we have found several reasonable consistent solutions with @xmath3 greater than 2 and that has been the only essential point raised in fms ( and developed also in fms2 ) . - _ on statistics _ in fms and fms2 , following the pdg procedure , the dataset include statistical and _ systematic errors _ , added in quadrature . in our opinion , the inclusion of systematic errors puts certain limits in a _ full _ statistical interpretation of the fit results . in fact , the @xmath21 test for goodness of fit is based on the assumption of a _ gaussian error distribution _ @xcite . although statistical uncertainties are considered to follow this distribution , that , certainly , is not the case for systematic uncertainties , which are equally probable quantities . therefore , we understand that a _ full _ statistical interpretation of data reductions including systematic uncertainties has a somewhat limited validity , specially in what concerns integrated probability ( due to the inclusion of equally probable quantities ) . in the fms and fms2 , the corresponding dof ( @xmath25 ) and @xmath26 for each fit has been displayed only to shown that they constitute reasonable ( acceptable ) statistical results . the condition of reduced @xmath21 closest to 1.0 has been one of the criteria used to select a given result , but not the only one . attempts to a full statistical interpretation of the results , mainly in terms of integrated probabilities , may lead to questionable conclusions , as discussed in what follows . at last , it is important to note that the focus in fms does not concern comparison among models ( or variants ) , but between two ensembles , with or without the totem datum . also , when we refer to _ statistically consistent results _ for @xmath3 = 2 or @xmath3 above 2 , we mean that _ the corresponding numerical result for @xmath3 is consistent within their uncertainty_. let us now treat each paragraph from section _ statistical probabilities _ in bh . we shall adopt here their notation ( lower index 1 for the @xmath27 = 1.8 tev ensemble and lower index 2 for the @xmath27 = 7 tev ensemble ) . @xmath7 first paragraph - commentary bh discuss our results in table 1 of fms ( @xmath27 = 1.8 tev ) , comparing the data reductions that can be summarized as follows : df@xmath28 : @xmath29 @xmath25 = 156 , @xmath29 @xmath26 = 0.931 , @xmath29 @xmath30 = 0.721,@xmath29 case of @xmath3 = 2 ( fixed ) @xmath31v1 : @xmath29 @xmath25 = 155 , @xmath29 @xmath26 = 0.937 , @xmath29 @xmath32 = 0.701 , @xmath29 case of @xmath3 free according to block and halzen : ... we get the somewhat strange result that fms have a _ ... better , somewhat more reliable _ fit when they fix the value of @xmath3 at 2 , ... , than they allow it to float , suggesting perhaps that the _ true _ minimum @xmath21 was not achieved in their minimization process . " they state in the last sentence : in any event , fms concluded that the value @xmath3 = 2 was correct for the energy interval 5 @xmath33 1,800 gev . " - our reply : the v1 result has been obtained from df@xmath28 as feedback . based on bevington and robinson @xcite ( chapter 11 ) and in our above comments on statistics , for @xmath34 155/156 we do not think that @xmath35 0.72 @xmath36 @xmath37 0.70 strictly implies in a better fit , namely that df@xmath28 could be more reliable " than v1 . in fact , let us compare the values of the free parameters in both fits in table 1 of fms ( third and fourth columns ) . note that the central values of the parameters are identical up to 3 figures , except for @xmath16 ( 0.264 and 0.263 ) and @xmath19 ( 12.0 and 12.2 ) . all the parameters in both fits are consistent within their errors , leading to the conclusion that both results are , effectively , equivalent ( the corresponding curves in figures 1 and 2 of fms overlap ) . we understand that in this nonlinear fit , based on the same dataset ( @xmath27 = 1.8 tev ) , to let free _ only one parameter _ ( @xmath3 ) does not allow the punctual statistical interpretation by block and halzen ( we shall return to this point in the discussion on the fourth paragraph ) . the very small differences in the central values of the parameters , and those in the errors , are associated with the _ correlations among all the fit parameters _ , resulting , in this particular case , in a small decrease of the probability when @xmath3 is let free . the correlation coefficients associated with both fits are displayed in table [ t1 ] : df@xmath28 above the diagonal of the table and v1 below the diagonal ( we shall return to this point in what follows ) . perhaps there can be some small differences in reaching a true minimum , but , in our opinion , that does not invalidate our results and interpretation . .correlation coefficients from the ( symmetric ) correlation matrices @xcite associated with df@xmath28 and v1@xmath28 results . the off - diagonal coefficients from df@xmath28 are displayed above the diagonal of the table ( not filled ) and those from v@xmath28 , below that diagonal . [ cols="^,^,^,^,^,^,^,^,^,^ " , ] [ t3 ] from figure 1 , the _ predictions _ by the compete 2002 and pdg 2010 are in agreement with the totem result , but not the data reduction from the pdg 2012 , which includes this point in the updated dataset : the curve and uncertainty region lie below the totem lower uncertainty bar , _ corroborating _ , therefore , _ the results and conclusions previously presented in fms_. note also that these results ( figure 1 ) corroborate another conclusion in fms , relating the totem point with the highest cosmic - ray estimates for the @xmath1 total cross section ( fly s eye collaboration @xcite and pierre auger collaboration @xcite ) . in fact , in what concerns our fit results with the amaldi _ _ parametrization ( and in the particular case of figure 1 with @xmath3 = 2 , namely the compete highest - rank result ) , there is no agreement among these three points : curves in consistency with the totem datum lie above the central values of the cosmic - ray estimations and the same is true in the inverse sense . based on all these facts , we understand that , once included in the dataset , it is , at least , not obvious the 7 tev totem result can be described by a standard @xmath38 $ ] leading dependence and that was the essential discovery " in fms . from our analysis , this effect is related with the fundamental correlation between @xmath3 and the scale factor @xmath19 ( and also with the subtraction constant @xmath39 in case of simultaneous fits to @xmath2 and @xmath0 data : see appendix a in fms2 on the correlation matrices ) . note , from table [ t3 ] , that in the compete case , @xmath40 34 gev@xmath12 and in the pdg 2012 edition , @xmath40 16 gev@xmath12 ( below , therefore , @xmath41 25 gev@xmath12 ) : the effect of these differences can be seen in fig . 1 . in particular , _ once included in the dataset _ , it is not expected the 7 tev totem datum might be described for fixed @xmath3 = 2 and @xmath42 , as is the case in bh . the totem collaboration has already obtained three new high - precision measurements of the total cross section at 7 tev , through different methods and techniques @xcite . all the measurements are consistent within their uncertainties and therefore confirm the first result they have obtained ( which has been used in fms and fms2 ) . in this respect , we can advance that the inclusion of these three points in our dataset leads also to solutions with @xmath3 greater than two @xcite , corroborating the conclusions in fms . however , the asymptotic ratio between elastic / total - cross - section , discussed in fms2 , is still under investigation . a luminosity - independent measurement at 8 tev has been also reported , indicating @xmath43 101.6 @xmath20 2.9 mb @xcite . this value lies above the prediction in fms2 , namely 98.7 @xmath20 1.0 mb , for @xmath3 = 2.346 @xmath20 0.013 and @xmath19 = 0.383 @xmath20 0.041 gev@xmath12 @xcite . at this point , we could conjecture ( if not speculating ) on the implication of a possible increase of @xmath2 faster than @xmath44 . in contrast with an effective violation of the froissart - martin bound , a fast rise of the total cross section might also be associated with some local effect at the lhc energy region , so that , asymptotically , the bound remains valid . a faster - than - squared - logarithm rise points also to the possibility of a power - like behavior @xcite , which has always been and important and representative approach ( see , for example , the unitarized model @xcite ) . these conjectures are not in disagreement with the recent theoretical arguments by azimov @xcite . here we present some critical comments on the bh parametrization , the aspen model and the bh results for the total , elastic and inelastic cross sections . the bh analytical parametrization for the total cross sections , used in global fits to @xmath2 and @xmath0-values from @xmath1 ( + ) and @xmath10 ( - ) scattering , reads @xcite @xmath45^{\mu - 1 } \ , \pm \ , \delta \ , \left[\frac{\nu}{m}\right]^{\alpha - 1 } + c_0\ , + c_1\ , \ln\left(\frac{\nu}{m}\right)\ , + c_2\ , \ln^2 \left(\frac{\nu}{m}\right),\end{aligned}\ ] ] [ eq2 ] where @xmath25 is the laboratory energy , @xmath46 is the proton mass and , in terms of the c.m . energy , @xmath47 . comparison with the parametrization used in fms , eq . ( 1 ) , shows that , for @xmath3 = 2 ( fixed ) , both forms have the same _ analytical structure_. however , the striking difference between fms and bh approaches concerns the number of fixed and free parameters and mainly , in the bh case , the way the parameters are fixed , as discussed in what follows . in the fms analysis , except for a dimensional notation choice , namely @xmath11 = 1 gev@xmath12 , all the 8 parameters involved have been treated as free or fixed in the 6 variants considered . that is , there is no _ ad hoc _ fixed parameters , except for particular variant tests . in all cases , the uncertainties in the free parameters have been explicitly given . moreover , by letting free different parameters in a _ nonlinear _ fit , we are able to investigate all the correlations involved , the variances , covariances and , as a consequence , the global uncertainties in all fitted and predicted quantities , as done in fms2 . in the bh approach , besides the arbitrary fixed energy scale ( corresponding to @xmath48 at both low and high energy regions ) , among the 7 parameters in eq . ( 2 ) , 5 are fixed ( without uncertainties ) and only 2 are free in data reductions . specifically , from table iii in @xcite : - fixed parameters : @xmath49 = 31.10 mb , @xmath50 = 0.5 , @xmath51 = - 28.56 mb , @xmath15 = 0.415 and @xmath52 = 37.32 mb ; - fitted parameters : @xmath53 = - 1.440 @xmath20 0.070 mb and @xmath54 = 0.2817 @xmath20 0.0064 mb . as a consequence , the parametrization is linear in any reduction to @xmath2 data , leading to unique solution @xcite . that , obviously , contrasts with eq . ( 1 ) and with the strategies in fms and fms2 , as commented above . in bh the reggeons intercepts are fixed , corresponding to @xmath17 = 0.5 and @xmath18 = 0.585 in eq . ( 1 ) , which is not in agreement with the spectroscopic data ( chew - frautschi plots ) and scattering fit results , as obtained by several authors @xcite . since this assumption permeated the intermediate and low energy region , in our opinion , it puts some limits on the reliability of a formal connection with the finite energy sum rules at low energy . the fixed parameters do not allow the study of correlations and their effects in the fitted and , most importantly , in the predicted quantities . we shall return to this point in the next subsection on the aspen model . we also note that , although ( 1 ) and ( 2 ) have the same analytical structure , the bh and fms high energy formulations for @xmath3 = 2 are not equivalent , even in the cases ( variants ) with @xmath17 and @xmath18 fixed . in fact , in fms all the other parameters are free and the fit extends _ simultaneously _ to both low- and high - energy data ( @xmath55 and @xmath56 contributions in fms ) . as a consequence there is strong correlations among the parameters from both @xmath55 and @xmath56 ( see , for example , the v4@xmath28 coefficients in table ii in the case of @xmath13 and @xmath19 , @xmath15 and @xmath16 ) . since that is not the case in bh formulation ( @xmath49 and @xmath51 , corresponding to @xmath13 and @xmath14 , are fixed ) , we see no correspondence between the high energy formulations , as stated in bh ( after eq . ( 7 ) in that paper ) . in the bh approach the main hypothesis concerns the imposing that the fits to the high energy data smoothly join the cross section and energy dependence obtained by averaging the resonances at low energy " @xcite . in the fms analysis , on the contrary , we have tried to identify possible high - energy effects that may be unrelated to the trends of the lower - energy data ... " @xcite . therefore , the assumptions , approaches and strategies in fms and bh are completely different . we see no reason for the comparative discussion presented in bh . in the aspen model @xcite two fundamental quantities , the mass scale @xmath57 and the coupling constant @xmath58 , are unknown parameters , fixed to _ ad hoc _ values of 600 mev and 0.5 , respectively in order to obtain best fits in data analysis . the @xmath59 parameter , from the gluon structure function , is also fixed at 0.05 . the two fundamental parameters , @xmath57 and @xmath58 , have been reinterpreted by luna _ et al . _ in the context of a dynamical gluon mass ( dgm ) approach @xcite . that has allowed connections with nonperturbative qcd , as expected in the soft sector represented by the elastic scattering processes . in the dgm approach , two essential parameters are the dynamical gluon mass scale @xmath60 and the soft pomeron intercept @xmath59 . more recently and most important for our purposes , fagundes , luna , menon and natale ( flmn ) have developed a detailed analysis on the influence in the evaluated quantities associated with physical intervals for the @xmath60 and @xmath59 parameters @xcite . moreover , in a similar way as done recently by achilli _ @xcite , flmn have established bounds and uncertainty regions in all evaluated quantities ( fitted and predicted ) , in accordance with the relevant physical intervals for @xmath60 and @xmath59 . the main conclusion is that the uncertainty regions play a crucial role in the energy region above that used in the data reductions . in other words , the relevant intervals for each parameters affect substantially the high - energy predictions and they can not be fixed at _ ad hoc _ values , without a clear physical justification . the above conclusion , expressed in @xcite , puts serious limits on the predictions of phenomenological models constructed on the basis of fixed parameters , whose numerical values do not have an explicit justification and/or whose _ consequences in the evaluated quantities are not investigated or even discussed_. in our opinion , these limitations are present in the foundations of both the bh analytical parametrization and the aspen model . in order to connect their global analytical ( empirical ) fits to @xmath2 and @xmath0 data with the inelastic cross section , @xmath61 , block and halzen use the predictions from the aspen model , in a kind of hybrid approach ( semi - empirical or perhaps semi - phenomenological ) @xcite . the model prediction is parametrized by an analytic expression ( eq . ( 8) in @xcite ) , with fixed mass and power parameters , without any reference to the uncertainties in the fit parameters . with this hybrid approach , from the model evaluation of @xmath61 and the fit result for @xmath2 they infer that @xmath62 a result statistically consistent with the black - disc limit . as commented in fms2 , we understand that the least ambiguous way to estimate the inelastic cross section is through the s - channel unitarity , @xmath61 = @xmath2 - @xmath63 , as has been done by the totem collaboration @xcite . that avoids the model dependence involved in direct estimation of @xmath61 , due to single and double diffraction contributions . in this respect , it may be interesting to note the prediction in bh for the inelastic cross secion at 7 tev ( fig . 3 in @xcite ) : the uncertainty region barely reaches the lower error bar in the totem result . moreover , for the total cross section , the central value of the prediction reaches the low extreme bar of the totem result ( and the uncertainty region , above the central value , lies through the lower error bar of the totem point ) . it should be also noted that as a one - channel eikonal formalism , the aspen model does not take explicit account of the inelastic diffractive contributions . in what concerns the bh total cross section prediction at 57 tev and the recent estimation of this quantity by the pierre auger collaboration @xcite , we stress a peculiar statement in @xcite ( also present in @xcite ) : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ in particular , the agreement with the new highest energy ( 57 tev ) experimental measurement of both @xmath2 and @xmath61 is striking . " _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ first , certainly these are not _ experimental measurements _ , due to the strong model dependence involved , as already discussed by several authors and also in @xcite . they constitute estimations of these quantities because they are based on extrapolations from models tested only in lower energies . moreover , recall that the estimation for the @xmath1 total cross section at 57 tev reads @xcite @xmath64\ , \mathrm{mb } $ ] . as discussed in fms2 ( and also in our section ii.b ) , systematic and , here , theoretical ( glauber ) uncertainties are equally probable quantities ( do not follow a gaussian distribution ) . that means the above central value is equally likely to lie in any place limited by the corresponding nonstatistical uncertainties , namely around @xmath20 23 mb . therefore , we see no physical meaning in a statement referring to a striking agreement " . at last , we understand that the bh analysis and the aspen model represent important contributions in the investigation of the high - energy elastic hadron scattering . however , given the model character and _ ad hoc _ assumptions involved we see no conclusive evidence that the bh results constitute an unique and exclusive solution for the rise of the total cross section at high energies . in this respect , a reanalysis by block and halzen , including in their dataset the four high - precision totem measurements and presenting their linear fit prediction at higher energies , might be instructive . in fms we have presented a study on the rise of the total hadronic cross section , with focus on the recent 7 tev totem result . the analysis was based on a specific class of analytical parameterization , eq . since the effects of all parameters involved have been considered , we were faced with a nonlinear data reduction , which constitutes a non - trivial problem . our strategy in fms ( and fms2 ) has been to investigate a grid of different physical choices for feedbacks and the corresponding solutions . in both fms and fms2 , beyond a second distinct possibility , we have considered conservative " choices for the initial values , namely results previously obtained by the pdg with @xmath3 = 2 . the data reductions have been developed through standard running of the minuit , namely the default minuit error analysis . perhaps , in the fits presented in fms a true minimum had not been reached in some cases . we understand that this does not invalidate the _ general and global conclusions _ from our grid strategy " approach ( summarized in section ii and , the corresponding results , in fig . 8 of fms ) . anyway , further analysis , looking for optimizations in the use of the minuit code , including also tests with other computational tools for nonlinear fits ( as the subroutine minos @xcite ) , are certainly important and we intend to implement that . the lack of unique solutions in our nonlinear fit procedures has been referred to in fms and fms2 . we have never claimed to have obtained _ unique or absolute solutions_. in particular , we have concluded in fms ( the equations refer to that paper ) , _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ from our data reductions through parametrization ( 3 - 5 ) to @xmath1 and @xmath10 scattering above 5 gev , including the 7 tev totem result , we conclude that the total hadronic cross section may rise faster than @xmath44 at high energies . " _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ our results suggest that the energy dependence of the hadronic total cross section at high energies still constitutes an open problem . " _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ and in fms2 we have stressed , _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ we also emphasize that our results represent possible consistent statistical solutions for the behavior of the total cross section , but do not correspond to unique solutions . " _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ it is important to note that at the highest energies , once treated as free fit parameters , the exponent @xmath3 and the scale factor @xmath19 are correlated in nonlinear data reductions . in the lack of formal and/or theoretical information on the value of this scale factor and without the _ canonical assumption _ @xmath3 = 2 , different solutions , in agreement with the experimental data , can be obtained for the leading @xmath65 contribution . therefore , we understand that the analytical description of the rise of the total hadronic cross section at high energies still constitutes an open problem , demanding further investigations . new high - precision data to be published and to be obtained by the totem collaboration at 8 - 14 tev are expected to shed light on the subject . at last , based on both the @xmath66 " by block and halzen and this @xmath67 " , we see no evidence that invalidate the analysis , results , conclusions and the honest statements expressed in fms . we are thankful to j.r.t . de mello neto for useful discussions . research supported by fapesp ( contracts nos . 11/15016 - 4 , 11/00505 - 0 , 09/50180 - 0 ) . azimov , froissart bounds for amplitudes and cross sections at high energies , proc . xlv winter school of pnpi , edited by v.t . kim ( pnpi , st . petersburg , russia , 2011 ) pp . 20 - 26 , arxiv : 1204.0984 [ hep - ph ] . fagundes , m.j . menon and p.v.r.g . silva , forward amplitude analysis and the recent 7 tev totem measurements , presented at xxiv reunio de trabalho sobre interaes hadrnicas , centro brasileiro de pesquisas fsicas ( cbpf ) , 3 - 4 december 2012 , rio de janeiro , brazil ( slides available at http://www.ifi.unicamp.br/@xmath69menon/24-rtih ) .
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a reply to the above mentioned commentary by m.m . block and f. halzen on our quoted work is presented .
we answer to each point raised by these authors and argument that our data reductions , strategies and methodology are adequate to the nonlinear - fit - problem in focus . in order to exemplify some arguments , additional information from our subsequent analysis
is referred to . a brief commentary on the recent results by block and halzen is also presented .
we understand that this reply gives support to the results and conclusions presented in our quoted work .
pacs : 13.85.-t hadron - induced high- and super - high - energy interactions , 13.85.lg total cross sections , 11.10.jj asymptotic problems and properties * table of contents * i. introduction \ii .
reply to the criticisms ii.a on the @xmath0 information ii.b statistical probabilities ii.c prediction of the 7 tev total @xmath1 cross section \iii .
discussion iii.a on the fms analysis and results iii.b brief commentary on bh analysis and results \iv .
conclusions and final remarks
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this paper is the second in a series investigating the evolved - star populations in nearby globular clusters . with the large - field ccd imagers now available it is possible to measure nearly complete samples of giant stars in clusters , and at the same time measure stars faint enough that we can normalize the luminosity functions ( lfs ) to the unevolved main sequence . because the lfs for evolved stars directly probe the timescales and fuel consumed in the different phases of stellar evolution , they provide a stringent test of the models for the evolution of low - mass stars . these models are the basis for our use of globular clusters to set a lower limit to the age of the universe and are a fundamental tool in the interpretation of the integrated spectra and colors of elliptical galaxies . the subject of this study is m30 ( ngc 7099 = c2137 - 174 ) , a relatively nearby cluster ( @xmath8 kpc ; peterson 1993 ) at high galactic latitude ( @xmath9 = @xmath10468 ) . m30 has a high central density ( @xmath11 ) , a moderate total mass ( @xmath12 ; pryor & meylan 1993 ) , and is at the metal - poor end of the cluster [ fe / h ] distribution . it is one of approximately 10% of clusters that have cusps at the core of their surface brightness profiles , and it also has one of the largest radial color gradients of any cluster ( stetson 1991b ) . previous studies of the lf for stars in metal - poor clusters have uncovered unexpected features . in a lf formed from the combination of ccd - based observations of the clusters m68 ( ngc 4590 = c1236 - 264 ) , ngc 6397 ( c1736 - 536 ) , and m92 ( ngc 6341 = c1715 + 432 ) , stetson ( 1991a ) found an excess of stars on the subgiant branch ( sgb ) just above the main - sequence turnoff ( msto ) . bolte ( 1994 ) and bergbusch ( 1996 ) both observed m30 using a mosaic of small - field ccd images and found an excess of sgb stars . ( the sgb is defined here as the transitional region between the main - sequence turnoff and the base of the red giant branch at the point of maximum curvature . ) another unexpected observation involving lfs is a mismatch between theoretical predictions and the observed size of the `` jump '' dividing the main sequence ( ms ) and the red giant branch ( rgb ) . when normalized to the ms , there is an excess of observed rgb stars compared to models ( stetson 1991a , bergbusch & vandenberg 1992 , bolte 1994 , bergbusch 1996 ) , although this has been disputed by deglinnocenti , weiss , & leone ( 1997 ) . these results might be explained by the action of core rotation ( vandenberg , larson , & depropris 1998 ) , or perhaps ( as discussed later ) we are witnessing the results of deep mixing and the delivery of fresh fuel into the hydrogen shell - burning regions . langer & hoffman ( 1995 ) suggested that , if the abundance patterns of light elements seen in bright cluster giants ( e.g. kraft 1994 ) are due to deep mixing , hydrogen - rich envelope material is almost certainly mixed into the hydrogen burning shell ( prolonging the giant phase of evolution ) , and some of the helium produced is returned to the envelope . because of the potential importance of such non - standard physics in stars , and because of the caveats associated with earlier lf studies , the most productive next step is to derive better lfs in a number of galactic globular clusters ( ggcs ) . in the next section , we describe our observations of the cluster . in 3 , we discuss the features observed in the color - magnitude diagram , describe the method of computing the luminosity functions , and present the results of artificial star experiments . in 4 , we discuss the constraints that can be put on the global parameters of the cluster metallicity , distance , and age . the method of data reduction is described in appendix a. the data used in deriving the @xmath13- and @xmath14-band lfs of m30 were taken on july 7/8 , 1994 at the cerro tololo inter - american observatory ( ctio ) 4 m telescope . in all , six exposures of 120 s , one exposure of 60 s and two exposures of 10 s were made in @xmath13 , and six exposures of 120 s , one exposure of 60 s , and one exposure of 10 s were made in @xmath14 . all frames were taken using the 2048 @xmath15 2048 pixel `` tek # 4 '' ccd chip , which has a sampling of about 044 per pixel , and a field 15@xmath16 on a side . these exposures were reduced individually for the purpose of constructing the color - magnitude diagram . in performing artificial star experiments and deriving the lf , the three best - seeing images in both @xmath13- and @xmath14-bands were combined into master long - exposure images . the frames were centered approximately 2@xmath16 east of the cluster center , in order to avoid a bright field star nearby . the night of the 4 m observations was not photometric . in order to set the observations on a standard photometric system , we used observations made at the ctio 1.5 m telescope on one photometric night ( october 18/19 , 1996 ) . the detector used was the `` tek # 5 '' 2048 @xmath15 2048 ccd , having a field of about 148 on a side . landolt ( 1992 ) standard star observations were used to calibrate a secondary field that overlapped the 4 m field . on that night , 10 s and 120 s exposures were taken in each band , along with exposures of 27 standards in 7 landolt fields . a sample of 118 stars having @xmath17 and @xmath18 was calibrated as secondary standards in this way . the field was centered approximately @xmath19 south of the cluster center . during the same run on the 1.5 m telescope , frames were taken of m30 on the non - photometric night of october 16/17 . five additional exposures were taken in each band ( 20 s , 200s , and 3@xmath15600s in v , and 15 s , 180 s , and 3@xmath15600s in @xmath14 ) . the details of the data reduction and calibration are described in appendix a. in figure [ m30cmd ] , we plot the total @xmath0 sample of 25279 stars ( upper panel ) and a sample that has been restricted in projected radius to @xmath20 from the cluster center ( lower panel ) . the inner radius was chosen in accord with the restriction placed on stars to be used later in the lfs , while the outer radius restriction was chosen to exclude regions that were affected by field star contamination . fiducial points for the ms and lower rgb of the clusters were determined by finding the mode of the color distribution of the points in magnitude bins . the fiducial line on the upper rgb was determined by finding the mean color of the stars in magnitude bins . once a mean was determined , stars falling more than @xmath21 from the fiducial point were discarded ( so as to eliminate agb and hb stars , as well as blends and poorly measured stars ) , and the mean redetermined . this procedure was iterated until the star list did not change between iterations . at the tip of the rgb and on the agb , the positions of individual stars were included as fiducial points if they appeared to be continuations of the mean fiducial line . the fiducial line for the hb was obtained by determining mean points in magnitude bins for the blue tail , and in color bins for the horizontal part of the branch . no smoothing has been applied . table [ fidtab ] lists the fiducial lines for our samples , as well as the number of stars used in computing each point . the procedure used to correct the `` observed '' lf back to the `` true '' lf is described in detail in sandquist et al . ( 1996 ) . as in that paper , we carried out artificial star tests on only four frames : a long exposure frame ( composed of the average of the three best - seeing images ) and a short exposure frame in both @xmath13 and @xmath14 band . in 11 runs , 20965 artificial stars were processed . allframe s coordinate transformations were found to be unable to follow the nonuniform spatial distortions introduced by the 4 m field corrector . to avoid this problem , we reduced all of the frames through allstar as usual , derived a master detected star list for each filter , and rereduced the frames in allstar with the improved positions . this procedure improved the overall quality of the photometry ( as judged by the scatter around the fiducial lines of the cluster ) , as allframe normally does . in figures [ delta ] [ fsfig ] we plot our computed values for median magnitude biases @xmath22 , median external error @xmath23 , and completeness probability @xmath24 as a function of magnitude and radius . there is little variation in most of the quantities until the innermost radial region ( @xmath25 20 , where crowding of stellar images is worst ) is reached . one change we have made since our first study was in the error estimation . the uncertainty in the incompleteness factor @xmath26 was previously found by simultaneously varying @xmath27 , and @xmath28 in such a way as to cause the maximum change in @xmath26 away from our best value . the magnitude of this change was used as the error estimate . we have improved this , following a suggestion by bergbusch ( 1996 ) , by estimating the error by varying @xmath29 and @xmath28 individually and adding the resulting error estimates in quadrature . using this information , we eliminated stars from consideration for the lf if they fell far enough away from the nearest point on the fiducial line of the cluster . the `` distance '' was defined in terms of difference in magnitude and color divided by their respective external errors , and then added in quadrature . so , stars were eliminated if @xmath30 . on the upper rgb , where contamination by the agb could be a factor , we adjusted the error cutoff by hand until we were sure the agb stars were being eliminated , but not at the expense of the rgb stars . the luminosity functions are listed in tables [ vlftab ] and [ ilftab ] . totals of 14772 and 14507 stars were used in creating the @xmath13- and @xmath14-band lfs . stars on the ms and sgb were only included if they fell more than 20 from the center of the cluster . this reduced the significance of crowding effects on the photometry . stars with magnitudes @xmath31 ( or @xmath32 ) were included in the determination of the lf to within @xmath33 of the center of the cluster . this was done to get a better indication of the `` global '' lf for the red giants since mass segregation is expected to occur in m30 , which would affect the rgb star counts taken from a small range of radii . most previous studies have made estimates of e(@xmath34 ) . we will assume e@xmath35 ( cardelli , clayton , & mathis 1989 ) for the rest of the discussion and refer to the reddening in e@xmath36 . m30 is situated well out of the galactic plane ( @xmath37 ) and the reddening is likely small ( e@xmath38 ) . reddening maps of burstein & heiles ( 1982 ) indicate that e@xmath39 . reed , hesser & shawl ( 1988 ) derive a _ negative _ reddening based on the comparison of m30 s integrated color and spectral type . zinn ( 1980 ) gets a value of 0.01 from integrated - light measurements . reed , hesser , & shawl s data indicates that m30 appears to have the bluest intrinsic colors of all of the globular clusters they examined . however , m30 is the cluster that shows the strongest color gradient of any ggc and reddening values measured from integrated colors or spectra will depend on the range of radii over which the observations are made . given the sense of the gradient , bluer towards the center , it seems likely that these reddening measurements will be biased to lower values . dickens ( 1972 ) and richer , fahlman , & vandenberg ( 1988 ; hereafter rfv ) independently derived significantly higher values , e(@xmath40 , based on the ubv colors of blue hb stars in m30 . on the other hand , @xmath41-band photometry is notoriously difficult to calibrate and the precision of reddening estimates from color - color plots is @xmath42 mag even in the best cases . neither the dickens nor the rfv study appears to have had a photometric calibration good enough to warrant error bars less than this . differential cmd comparisons with m92 ( vandenberg , bolte , & stetson 1990 ; hereafter vbs ) and m68 ( figure [ m30vsm68 ] ) imply e@xmath43 to 0.06 . the recent reddening maps based on iras and cobe measurements of far - ir flux from infrared cirrus suggest e(@xmath44 . we can make our own estimate using using sarajedini s ( 1994 ) simultaneous reddening and metallicity method ( for @xmath45 and @xmath46 , where the quoted errors allow some room for calibration errors ) . we find e@xmath47 . the errors were derived from monte carlo tests with the quoted errors on @xmath48 and @xmath49 . most of the reddening estimates we have thus far indicate a relatively high value e@xmath50 . the compilation of zinn & west ( 1984 ) has [ fe / h ] @xmath51 and numerous studies since have determined values between @xmath52 and @xmath53 ( geisler , minniti , & clari 1992 ; minniti et al . 1993 ; carretta & gratton 1996 ) . from the simultaneous reddening and metallicity method above , we find [ fe / h]@xmath54 . anticipating later discussion of the level of the hb , we examine the effects of an anomalously high on the simultaneous reddening and metallicity method . a high value can come about due to a high helium abundance , whether primordial or the result of a `` deep mixing '' scenario ( langer & hoffman 1995 ) . if the `` true '' value ( in the absence of helium enrichment ) is fainter by 0.10 mag , then we would have @xmath55 , and calculate [ fe / h ] @xmath56 and e@xmath57 . recently reid ( 1997 ) , gratton et al . ( 1997 ) and pont et al . ( 1998 ) have used subdwarf parallaxes measured with the _ hipparcos _ satellite ( esa 1997 ) to redetermine the distance moduli to several of the best observed clusters . the general result of these studies is to increase cluster distance moduli by 0.2 to 0.4 mag , implying high luminosities for the horizontal branches ( as bright as m@xmath58 mag for the [ fe / h]@xmath59 clusters ) . for m30 specifically , gratton et al . find ( m - m)@xmath60 ( for e@xmath61 ) although this is based on only three subdwarfs . we repeat this exercise for m30 with our data and a larger set of subdwarfs . this method is sensitive to uncertainties in the color of the unevolved main - sequence , which result from zero - point errors in the photometric calibration , the reddening uncertainties already mentioned , and uncertainties in the placement of the main - sequence fiducial line . our data for m30 are not optimum for using subdwarf fitting to measure the distance , primarily because of the uncertainty in the reddening , but also because we suffer from each of the other problems to a small degree . nevertheless , we selected subdwarfs that satisfy the following criteria : parallaxes from the _ hipparcos _ mission having relative errors @xmath62 , metal abundances from the study of gratton , carretta , & castelli ( 1996 ) , and @xmath0 photometry . the restriction on parallax error was chosen to minimize the effect of bias corrections . ( as a result , lutz - kelker corrections only change our derived distance moduli by 0.01 mag . ) the gratton et al . metallicity scale was chosen for its homogeneity and because it minimizes the possibility of systematic abundance errors with respect to carretta & gratton globular cluster metallicities . when available , we used @xmath0 photometry for the subdwarfs tabulated in mandushev et al . ( 1996 ) . in the remaining cases , we followed their procedure of combining literature values from the following sources : carney & aaronson ( 1979 ) , carney ( 1980 , 1983b ) , and ryan ( 1989 , 1992 ) . the studies involving carney all used the johnson @xmath14 filter , so we applied the transformation from carney ( 1983a ) to convert them to the cousins system . known spectroscopic binaries were excluded , and , for the cases where it has been measured , any reddening of the subdwarfs ( at most a very small amount ) has been subtracted . our sample of subdwarfs and metal - poor subgiants is shown in table [ subdtab ] . we used the subdwarfs to estimate the distance to m30 in two different ways . first , to simultaneously estimate the distance and e@xmath36 of m30 we created a grid of chi - square - like sums . the m30 main sequence between @xmath63 was represented by a third - order polynomial . for ranges of m - m and e@xmath64 ) , the minimum distance of each subdwarf from the main - sequence polynomial was calculated . this distance was normalized by the combined errors in the subdwarfs colors and magnitudes and the main - sequence fiducial uncertainties in color and magnitude ( assumed to be 0.04 mag in color and 0.05 mag in @xmath13 magnitude ) . our `` @xmath65 '' sum is : @xmath66 we will refer to this as the chi - square sum although it does not match the usual definition of chi - square and it is not normalized in a way to give true confidence intervals . because the slope of the main - sequence changes with @xmath13 , there is a different weighting given to @xmath67 and @xmath68 ) for each subdwarf . the minimum chi - square values are for m@xmath10m@xmath69 and e@xmath70 , although the reddening in particular is poorly constrained . our second approach was to fit the m30 main sequence to the subdwarfs using only the distance modulus as a variable for two e(@xmath71 ) values : 0.02 and 0.06 . table [ dmtab ] shows the changes in this value for different subsets of our sample and for different input data . ( the quoted errors include contributions from the scatter of values in the subdwarf fit , a cluster reddening error of 0.02 mag , and the absolute cluster metallicity error of 0.2 dex . ) clearly , reddening uncertainty is dominant in the total uncertainty . two fits at different reddening are shown in figure [ subdvred ] . if we use the value of metal content given by zinn & west ( [ fe / h ] @xmath72 ) along with the carney et al . ( 1994 ) abundance scale for the subdwarfs ( which has roughly the same zero point as zinn & west ) , the distance modulus is increased by only a few hundredths of a magnitude . restricting the sample to only metal - poor ( [ fe / h]@xmath73 ) subdwarfs also does not significantly change the distance modulus . considering the number of distance modulus measurements for m30 in the literature , it is best to try to compare using a common reddening of e@xmath74 ( and using @xmath75 to correct values ) . from the two methods we presented here we have 14.6 and 14.65 . from the pre - hipparcos ground - based measurements of bolte ( 1987 ) and rfv , we have 14.62 and 14.68 ( the rfv value must also be corrected for their lower assumed metallicity for m30 ) . for the hipparcos - based distances of reid ( 1997 ) and gratton et al . ( 1997 ) , we find 14.75 and 14.87 ( although gratton et al . use only three subdwarfs in their fit ) . it is clear that our distance moduli are consistent with the ground - based measurements , and over 0.1 mag smaller than the other hipparcos - based measurements . neverthless , the different distance measures are in good general agreement , at least for a fixed reddening value . although it is not crucial for the conclusions that follow , we will adopt ( m - m)@xmath76 for the rest of the paper . from figure [ subdvred ] , it is clear that the smaller reddening and distance modulus values are more consistent with the model isochrones and , for our preferred larger reddening value of e@xmath3 , the models do not match the shape of the m30 fiducial above the turnoff . the ratio @xmath77 , where @xmath78 is the number of rgb stars brighter than the luminosity level of the hb , is the traditional quantity used to estimate the helium abundance of stars in globular clusters . dickens ( 1972 ) first noted that m30 s value for `` r '' was unusually large and alcaino & wamsteker ( 1982 ) claimed a significant gradient in this ratio in the sense of a small value at the center of the cluster increasing to one of the largest measured in any cluster at large radii . we have sufficient numbers of stars on the red and blue sides of the instability strip to define the level of the hb . from five stars near the blue edge of the instability strip ( with colors @xmath79 ) , we calculate an average magnitude @xmath80 . from six stars on the red side of the instability strip , we find @xmath81 . all stars used in these averages were found at radii greater than 10 from the center of the cluster , so that the photometry should be quite accurate . because the estimate of the hb magnitude is crucial in later arguments , we examine the topic further . our blue side estimate is consistent with those of bolte ( 1987 ) and bergbusch ( 1996 ) , even with the different calibrations of our respective datasets . because it is possible that the red hb stars in m30 are evolved , it is wise to check this possibility before simply interpolating between the two sides . the number of stars ( 8 with projected radius @xmath82 10 , compared to 93 on the blue hb ) at @xmath83 is roughly consistent with timescales for stars evolving from the blue hb , as the evolutionary tracks tend to parallel each other closely through this part of the cmd as they move toward the agb . thus , we have chosen to look at other clusters of similar metallicity with large , well - studied rr lyrae populations to compute a magnitude correction to go from the red edge of the blue hb into the middle of the instability strip . for the clusters m15 ( bingham et al . 1984 ) , m68 ( walker 1994 ) , and m92 ( carney et al . 1992 ) we find agreement on a correction of @xmath84 mag . thus , we have @xmath85 . to define the rgb star sample for the @xmath5 method , we need to establish the relative bolometric magnitudes of the hb and rgb stars . we have used the hb models of dorman ( 1992 ) in conjunction with the isochrones of bergbusch & vandenberg ( 1992 ; hereafter bv92 ) . the stellar models involved in these studies were computed with a consistent set of physics and compositions . although the composition used is somewhat out - of - date , the _ differential _ bolometric corrections should be fine . the corrections as a function of [ fe / h ] can be approximated by : @xmath86 } + 0.229 \mbox{[m / h]}^{2 } + 0.034 \mbox{[m / h]}^{3 } .\ ] ] because @xmath87-element enhancements influence the position of the hb and rgb in the cmd like a change in [ fe / h ] ( salaris , chieffi , & straniero 1993 ) , they must be taken into account when computing [ m / h ] . for m30 , we find that @xmath88 . we have used [ m / h ] @xmath89 ( correcting [ fe / h ] by 0.21 dex for [ @xmath87/fe]@xmath90 ) , although for this range of metallicities , changes in metallicity have a very small effect on @xmath91 . because of contamination and blending problems toward the center of the cluster , we restrict our samples of rgb and hb stars to @xmath92 . with this choice , we find @xmath93 . this makes m30 s @xmath5 the highest of any of the clusters examined . in table [ clusttab ] , we present @xmath5 values that have been derived from published photometry for the clusters similar in [ fe / h ] to m30 m68 ( walker 1994 ) , m53 ( cuffey 1965 ) , ngc 5053 ( sarajedini & milone 1995 ) , ngc 5466 ( buonanno et al . 1984 ) , and m15 ( buonanno et al . 1983 ) along with several more metal - rich clusters . photometry exists for the central @xmath94 of m30 from the hubble space telescope ( yanny et al . 1994 ; hereafter ygsb ) . by merging their list with ours and eliminating common stars , we have created a master list of hb and rgb stars that completely covers the cluster out to 70 , with portions included out to about @xmath95 . the data for the full sample is presented in table [ poptab ] . using this sample , we find a global value for @xmath5 of @xmath96 . ( even if our value of @xmath48 is too bright , and if we use the magnitude of the red edge of the blue hb , we find @xmath97 still a high value in a relative sense . ) this global @xmath5 value for m30 is on firm ground , because there is no place for the bright stars to hide . the photometry is easily good enough to distinguish between the agb and rgb star samples , so this is not a source of uncertainty . ( in fact , m30 may be deficient in agb stars as well as rgb stars . ) the use of the lower buzzoni et al . ( 1983 ) value @xmath98 for the differential bolometric correction would increase the high @xmath5 value . approximately 30 additional rgb stars would have to be included to bring m30 s value in line with that of other clusters a 24% change in the sample size . there are a few potential explanations for a _ global _ depletion of bright giants in this cluster . first , because the ratio @xmath5 is a helium abundance indicator , the abnormally high value could indicate a higher - than - average helium abundance in m30 stars more luminous than the hb . this does not necessarily imply high y for lower luminosity stars , since a deep - mixing mechanism would also be expected to dredge up freshly produced helium ( sweigart 1997 ) . second , the environment within the cluster may affect the stellar populations by a mechanism that truncates rgb evolution and/or produces additional hb stars . we now consider the arguments for the two sides . using the buzzoni et al . ( 1983 ) calibration and our m30 @xmath5 value , we find @xmath99 . the value derived using @xmath48 of the blue edge of the instability strip gives @xmath100 . that @xmath48 is definitely a faint limit , making @xmath101 a lower limit . in any case , the @xmath5 value for m30 is significantly higher than those for other clusters when the revised differential bolometric corrections are used . [ the low value of @xmath102 for the other clusters is discussed in sandquist ( 1998 ) . ] one check we can make is to examine other helium indicators to see if they also indicate a high abundance relative to other clusters . caputo , cayrel , & cayrel de strobel ( 1983 ) introduced two indicators : @xmath103 ( the mass - luminosity relation for rr lyrae stars of ab type ) and @xmath6 ( the magnitude difference between the hb and the point on the ms where the dereddened @xmath34 color is 0.7 ) . while m30 s rrab variables have not been studied to the extent necessary to compute @xmath103 , we can compute a @xmath6 value from the @xmath104 photometry of bolte ( 1987 ) and rfv . assuming for both that e@xmath105 , we find @xmath106 and @xmath107 respectively , where the primary contributions to the error are the uncertainty in the reddening and the small number of stars used to define the hb magnitude . ( the lower rfv value can be traced to a fainter hb magnitude relative to bolte s data . ) we have computed comparison values for the clusters m68 ( mcclure et al . 1987 , walker 1994 ) , and m15 ( durrell & harris 1993 ) , as summarized in table [ clusttab ] . the @xmath6 values for the other clusters agree with the theoretical value of 6.30 for [ m / h ] @xmath108 ( [ fe / h]@xmath109 ) . we can directly compare our data with @xmath0 for other clusters if we redefine @xmath6 ( hereafter , @xmath110 ) by choosing the ms point to have @xmath111 . from isochrones this is approximately equivalent to @xmath112 . from our fiducial line , we find @xmath113 for e@xmath105 . @xmath0 data exist for m92 ( johnson & bolte 1998 ) and m68 ( walker 1994 ) . we find that @xmath114 ( e@xmath115 ) and @xmath116 ( e@xmath117 ) , respectively . the m92 value relies essentially on one star for the hb magnitude , so the @xmath118 value is uncertain . for @xmath110 and @xmath118 we see evidence ( though not overwhelming ) that the m30 value is high compared to other clusters . the values above indicate that m30 s helium abundance is high by about 0.02 for [ m / h]@xmath119 if the helium is primordial , or 0.03 if the helium enrichment only affects the level of the hb , as in the deep mixing scenario . ( note that a lower value for the reddening would bring the @xmath6 values into consistency with the other clusters . ) a high helium abundance would tend to make the hb distribution bluer on the whole . according to fusi pecci et al . ( 1993 ) , the color of the peak of the hb star distribution in m30 is one of the bluest , but other clusters are rather close ( m53 is bluer , m15 has approximately the same peak color , and m92 and ngc 5466 are slightly redder ) . because m30 has a core of high stellar density , we consider the possibility that this environment has influenced the populations of evolved stars in the cluster . m30 has one of the most robustly determined color gradients ( approximately linear in @xmath120 : @xmath121 mag dex@xmath122 ; piotto , king , & djorgovski 1988 ) of all the globular clusters in the galaxy . the sense of the color gradient is such that the integrated colors become bluer towards the cluster center . in m30 and other post - core - collapse clusters it has been suggested that the color gradient is due to a decrease in the ratio of rgb - to - bhb stars resulting from stellar interactions in the dense cluster cores . although djorgovski & piotto ( 1993 ) claim the color gradient measured in m30 is due to a deficit of rgb stars in the inner few tens of arcseconds , burgarella & buat ( 1996 ) show that the gradient ( which extends to radii @xmath123 ) is not due to differences in the spatial distribution in the evolved - star populations or in the blue stragglers . our results are consistent with this latter claim . buonanno et al . ( 1988 ) claimed to have detected a radial variation in the ratio @xmath124 on the basis of a smaller sample of stars . from our sample , we find @xmath5 ranges from about @xmath125 in the inner @xmath33 to @xmath126 for @xmath127 to @xmath128 for stars with projected radius @xmath129 . there may be marginal evidence for a difference between the outer annulus , and the inner two , but over the range of radii for which the bluer - inward color gradient has been observed , there is no evidence of a trend in the bright populations . the sense of the difference between the outer and inner populations is in any case opposite to that required to make the color gradient . the cumulative radial distribution ( figure [ crd ] ) also shows no strong trends in radius , contrary to claims in other studies with smaller samples ( buonanno et al . 1988 , piotto et al . 1988 ) , but in agreement with studies of the core ( yanny et al . 1994 , burgarella & buat 1996 ) . a kolmogorov - smirnoff test indicates a 33% chance that the two samples are drawn from the same distribution . the inhomogeneity of the rgb sample seems to be responsible for this noncommittal probability . we conclude that despite the color gradient in m30 , and the apparently ripe conditions for interactions to alter the stellar populations , environment - based processes are not responsible for the high r value we measure . based on the color - difference method , vbs claimed that the most metal - poor clusters , including m30 , are coeval at the level of 1 gyr . our comparison ( figure [ m30vsm68 ] ) of the fiducial lines of m30 and m68 ( walker 1994 ) in the neighborhood of the sgb indicates that the ages of m30 and m68 ( assuming similar main - sequence y and [ @xmath87/fe ] ) are nearly identical . with our uniform calibration of msto and evolved stars in m30 , we can determine with good precision the other commonly applied age estimator @xmath7 . in computing the values presented in table [ clusttab ] , we have attempted to use the studies with the largest samples having uniform photometry from the level of the hb to below the to . we were able to derive values for the clusters m68 ( walker 1994 ) , m53 and ngc 5053 ( heasley & christian 1991 ) , m92 ( bolte & roman 1998 ) , and m15 ( durrell & harris 1993 ) . although the clusters all have blue hb morphologies , it has not been necessary to make corrections to find the `` true '' hb level : either the hb is populated on both sides of the instability strip ( m53 , ngc 5053 ) or there are a number of well - measured rr lyrae stars ( m15 , m68 , m92 ) . from our photometry of m30 , we find @xmath132 , and @xmath133 . while m15 , m53 , m68 , m92 , and ngc 5053 have @xmath130 values that agree to within the errors ( and also agree with the values derived for clusters of higher metallicity ) , m30 has a value about 0.15 mag higher . this is in disagreement with values given in the extensive tabulation of chaboyer , demarque , & sarajedini ( 1996 ) . the values for m30 and m92 in particular have been put on firmer ground here since consistent photometry exists from the hb to the to . using the more robust v(bto ) ( the apparent magnitude of a point 0.05 mag redder than the turnoff ; chaboyer et al 1996 ) , we can compare @xmath134 values for m30 and m68 , which also has @xmath0 data . we find @xmath135 for m30 , and @xmath136 for m68 . there are two plausible ways to explain the apparent 0.15 mag excess in @xmath130 for m30 relative to other clusters . first , m30 could be older by @xmath137 gyr . this conclusion would be in conflict with that inferred by vbs based on the color - difference method . ( this is perhaps the first case for which the relative age indicators @xmath130 and the subgiant - branch color extent give significantly different answers . ) alternatively , m30 stars could have a larger initial helium abundance by approximately 0.027 . if the higher helium abundance is restricted to the cluster hb stars , as would be the case in a deep - mixing scenario , the increase required is approximately 0.045 . ( the agreement between the m30 and m68 cmds everywhere but on the hb would argue against a difference in the initial helium abundances in the two clusters , as would beliefs about galactic chemical evolution . ) the size of the potential helium enhancement is close to what was inferred earlier from the helium indicators @xmath6 and @xmath5 for m30 . in the following , we will be using a combination of oxygen - enhanced ( bv92 ) and @xmath87-element enhanced ( vandenberg 1997 ) theoretical lfs to interpret the data . the current state of knowledge indicates that all of the @xmath87 elements have enhancements ( pilachowski , olszewski , & odell 1983 ; gratton , quarta , & ortolani 1986 ; sneden et al . the available evidence also suggests that the oxygen enhancement remains constant , at least for [ fe / h ] @xmath138 ( e.g. suntzeff 1993 , carney 1996 ) . on the rgb , stellar evolution is insensitive to the oxygen abundance because the luminosity evolution is driven almost entirely by the helium core mass ( refsdal & weigert 1970 ) , while the color is primarily determined by h@xmath139 opacity . oxygen has a relatively high ionization potential , and hence does not contribute electrons to the opacity . the fainter one goes on the ms , the more insensitive the evolution is to the oxygen abundance because of the same opacity effect , and because @xmath140 chain reactions are dominant over cno cycle reactions in influencing the luminosity . as a result , the different distribution of heavy elements causes negligible differences in the theoretical lfs on the rgb and lower ms ( see figure 17 of sandquist et al . 1996 ) . it is primarily the turnoff region that is affected by changes in the oxygen abundance , because cno cycle reactions begin to become important , and because oxygen ionization regions are close enough to the surface to influence surface temperatures . increased oxygen abundance increases the envelope opacity , creating redder models . increased cno cycle activity causes a star to adjust to accommodate the increased luminosity by reducing the temperature and density of the hydrogen burning regions , which results in a net _ decrease _ in the luminosity of the turnoff and sgb relative to solar - ratio models . thus , the sgb `` jump '' moves in magnitude in the lf . in the cmd , it also changes slightly in slope , but for metal - poor clusters like m30 , this does not cause a significant change in the shape of the sgb jump in the lf . an examination of bv92 models indicates that the @xmath13-band lf is not very age - sensitive for this range of metallicities . it is most sensitive on the sgb and then , as found in sandquist et al . ( 1996 ) , only when the sgb is nearly horizontal in the cmd . in @xmath13 band for a cluster as metal - poor as m30 , the sgb has a relatively large slope , and so only a large systematic age error will influence the fit . in light of the _ hipparcos _ parallax data , this possibility should be considered , since derived distance moduli indicate brighter to magnitudes ( and thus , younger ages ) for metal - poor clusters . figure [ vlfages ] shows a comparison of the @xmath13-band lf with theoretical lfs for different ages , using an apparent distance modulus of @xmath141 , as derived from one fit to the _ hipparcos _ subdwarf sample . previous studies of m30 s @xmath13-band lf ( piotto et al . 1987 , bolte 1994 , bergbusch 1996 ) uncovered two unusual features in comparisons with theoretical models : an excess of faint red giants relative to main - sequence stars , and an excess of subgiant stars . our photometry goes fainter on the ms , allowing us to verify that the normalization of the theoretical models has not been made in an `` abnormal '' section , while our wide field allows us to measure the largest sample of red giants in the cluster to date . figure [ vlfcomp ] shows a comparison of the studies , with magnitude shifts according to measured zero - point differences . in large part there is excellent agreement . our lf is significantly below bergbusch s at his faint end , most likely due to underestimated incompleteness corrections in his study . at the bright end of the rgb ( @xmath142 ) , our lf points are also below most of bergbusch s . however , we observed a larger number of giants , and our bins are larger , making our points more significant statistically . in the following subsections , we discuss the main features in the lfs . there is a apparently a considerable excess of stars on the rgb for @xmath143 ( we will refer to this as the `` lower rgb '' ) when compared to the models normalized to the unevolved main sequence . to judge the reality of the excess , we need to accurately normalize the theoretical lfs in the horizontal and vertical directions , and choose the photometry subsample to maximize the statistical significance . the horizontal normalization can be accomplished by shifting the theoretical lf in magnitude so that the to matches that of the observational lf ( stetson 1991a ) . in the vertical direction , we have normalized to the ms in a range of magnitudes where there are large numbers of measured stars , and where incompleteness is a relatively small consideration . the mass function controls how well the normalized theoretical lf is fit to the ms portion in the present example . as shown in figure [ vlffig ] , fits using small values for the power - law mass function exponent @xmath144 indicate that the relative numbers of stars on the rgb and lower ms can be matched by canonical stellar evolution models . with such a choice though , bins with @xmath145 are not well - modeled . we find that the lf can be modeled from near the faint limit of our survey to the base of the rgb if we use a higher value for @xmath144 . this alleviates the depression in the star counts in this magnitude range seen by bolte ( 1994 ) . however , we are still left with an excess of rgb stars relative to ms stars in the range @xmath146 . the effects of mass segregation have been previously observed within m30 in the form of a variation of the local mass function exponent @xmath147 with radius ( rfv , bolte 1989 , piotto et al . 1990 , sosin 1997 ) . as a result , the best comparison that can be made would be between theory and a faint sample restricted to the outskirts of the cluster . the models of pryor , smith , & mcclure ( 1986 ) , as well as observational studies , indicate that restricting the sample to stars more than 20 core radii from the center should minimize the effects of mass segregation on faint end of the lf . figure [ vlfrgbms ] shows the lf we computed for this purpose . the presence of the rgb - ms discrepancy in this case suggests that the problem is not related to the dynamical effects on the mass function , at least in the outskirts . we can get good overall agreement with the shape of the lf on the ms , but there is a relative excess of rgb stars , and the sgb region is not well fit . the sgb comparison is insensitive to age and metallicity using this method of matching the msto . helium abundance , however , has a larger effect ( stetson 1991a ) . because the rgb stars in m30 become more populous relative to hb stars ( and presumably ms stars ) towards the center , we expect that the cluster core would show a larger discrepancy . as with the sgb excess , this effect has only been observed in metal - poor clusters ( in other words , not in the lfs of ngc 288 or m5 ) . to add stars to the canonical number at a point in the red giant lf , one must either increase the hydrogen content of the mass being fed into the hydrogen - burning shell , or reduce the density or temperature of the burning shell . one possibility for the excess stars on the lower rgb is that we are seeing the effects of deep mixing , which brings hydrogen - rich envelope material into the energy generating shell . if this kind of mixing occurred on the lower rgb , it could eliminate the rgb bump by erasing the chemical discontinuity left by a surface convection zone . alternately , vandenberg , larson , & depropris ( 1998 ) have examined the effects of rotation on rgb evolution . they found that core rotation can expand the outer portions of the stellar core enough to cause a reduction of the shell temperature . this results in a decrease in the rate of evolution for rgb stars , and hence leads to an increase in the number of stars per luminosity bin . this is in the correct direction to explain the rgb excess . this rotation could be related to deep mixing scenarios that are required to explain abundance anomalies in rgb stars ( e.g. shetrone 1996 ) most notably a decline in the @xmath148c/@xmath149c ratio relative to theoretical predictions , and na - o and al - o anti - correlations ( as surface material is mixed into regions where o is being converted to n in the cno cycle ) . [ note that rotation _ can not _ explain subgiant branch excesses because the burning region in core - burning stages is too small to contain a significant amount of angular momentum ( vandenberg 1995 ) . even if rotation does affect the structure of the star outside the core , and thereby changes the core temperature , this would not produce isothermalization that would lead to sgb excesses . ] the rotation and mixing pictures ( with the assumption that mixing is somehow based on internal rotation ) receive some support from observations of rotation in hb stars of some clusters . there is definite evidence of stellar rotation in blue hb stars in ngc 288 , m3 , and m13 ( peterson , rood , & crocker 1995 ) . m13 has the fastest rotators , with stars falling into two groups : some with @xmath150 km s@xmath122 , and some with @xmath151 km s@xmath122 . m3 has a @xmath152 distribution consistent with @xmath153 km s@xmath122 , while ngc 288 s stars are consistent with @xmath154 km s@xmath122 . cohen & mccarthy ( 1997 ) also found projected rotation rates between 15 and 40 km s@xmath122 for five blue hb stars in m92 . the presence of stellar rotation on the hb implies that angular momentum may have been stored during the rgb phase in a rapidly rotating core , avoiding loss of angular momentum through the stellar wind . ( such mass loss is needed to be able to create hb stars of appropriate masses to match observed cluster hb morphologies . ) if this is true , it would be particularly interesting to compare lfs for m3 and m13 to look for the effects of rotation , and perhaps even different levels of rotation . further stellar rotation measurements for m30 , m68 , and m92 would also be helpful in examining rotation as a cause of ms - rgb discrepancy in the combined lf . figure [ clfhbs ] presents the cumulative lf ( clf ) for the cluster . in this graph we have included rgb stars from @xmath155 to @xmath156 from the center of the cluster . the rgb bump is typically identified from a break in slope in the cumulative lf . at this point , the shell - burning source begins consuming material of constant , lower helium content ( in other words , the shell reaches what was formerly the base of the convection zone at its maximum extent fusi pecci et al . fusi pecci et al . examined clusters over a range of metallicities , and found a linear relation between @xmath157 and [ fe / h ] , as predicted by theory . by combining cmds for three of the most metal - poor clusters ( m15 , m92 , and ngc 5466 ) , they found @xmath158 . in addition , for ngc 6397 , the most metal - poor cluster for which they were able to find the bump , they found @xmath159 . as shown in figure [ clfhbs ] , we have examined data for m68 ( walker 1994 ) , a cluster of nearly the same metallicity as m30 , in order to get a better idea of where the bump should be . there is a clear indication of a slope break for m68 : @xmath160 , or @xmath161 , for m68 . that result shows that the continuation of the fusi pecci et al . relation to lower metallicity appears correct . we have chosen to shift m30 and m68 so that their mstos align because of the evidence that m30 s hb may be anomalously bright ( see [ dist ] ) . the comparison reveals that there may be a feature at the same position as in m68 , although we do not see significant signs of slope change in the clf at the position of the feature . we find that a few bins on the sgb ( @xmath162 ) show an excess of stars relative to the theoretical predictions for the best fitting models , confirming the result of bolte ( 1994 ) . in figure [ vlfsgb ] , we plot the lf with a radius cut closer to the cluster center so as to get better statistics on the sgb . as figure [ vkept ] shows , there is little scatter in the vicinity of the sgb in the cmd that would tend to wash out or contribute to the observed excess at @xmath163 . the excess is based on a single point having a significance of @xmath164 , where the error in almost entirely due to poisson statistics . bolte ( 1994 ) states the significance of the bump as @xmath165 , and it appears to occupy two lf bins in his figure 7 . the significance of his result is probably smaller than that because of the difficulty in determining the position of the `` jump '' ( @xmath166 mag brighter than the msto in @xmath13 ) in his lf . an examination of the @xmath14-band lf in figure [ ilf ] shows the presence of a deviation at the same position in the cmd . this is important because the slope of the sgb is steeper in an @xmath167 cmd than in a @xmath168 cmd . as a result , the bump can no longer be ascribed to a feature caused by the exact slope : it must be the result of an increase in the number of stars congregating near a point on the cluster s fiducial line . at the analogous position in the @xmath14-band lf , there are two bins with excesses of @xmath169 and @xmath170 compared to theory , for a combined significance of @xmath171 . the appearance of the subgiant branch excess in both the @xmath13- and @xmath14-band lfs indicates that the cause must be due to an excess of stars ( rather than being caused by the exact slope of the sgb thus eliminating the exact metallicity , helium content , and oxygen abundance as causes ) . so , the sgb excess has marginal significance in our lfs . in order to more definitively determine the reality of the feature , photometry reaching into the center of this cluster will be needed . the observation of this feature in the combined lf of m68 , m92 , and ngc 6397 ( stetson 1991a ) lends more credence to the phenomenon , but more investigation is necessary . sandquist et al . ( 1996 ) found that there was no evidence for an sgb excess in the lf of the more metal rich cluster m5 ( which has a good @xmath14-band lf for easy comparison with figure [ ilf ] ) . bergbusch ( 1993 ) saw no evidence of an excess in his @xmath13-band lf of ngc 288 . these pieces of evidence seem to indicate that any cause must only be effective at low metallicities . there is , however , a general lack of useful lf data covering the sgb for globular clusters with metallicities between m5 and m30 , or more metal rich than m5 . if the feature is real , there are at least two potential means of creating such an excess : a fluctuation in the initial mass function , and an unknown physical process isothermalizing the stellar core of turnoff mass stars . the excess in stetson s m68-m92-ngc 6397 lf makes mass function fluctuations less likely . a star can be forced to pause on the sgb , but still burning hydrogen in its core , if isothermality is imposed on a large portion of the core ( faulkner & swenson 1993 ) . if such a process occurred in a large enough fraction of the stars in a cluster , a sgb excess could be created in the lf . a way to create such an excess is to invoke a process that increases the efficiency of energy transfer over a large portion of the core . for this to happen , the mean free path of the transporting particle must be large . no such particle has been identified to date . \1 . determinations of the reddening for m30 disagree at a @xmath172 mag level and cases can be made for values ranging from 0.03 to 0.07 in e@xmath64 ) . this uncertainty is the main factor preventing a more accurate determination of the distance modulus . by fitting subdwarfs with _ parallax data to the @xmath0 fiducial line , we find satisfactory fits for @xmath173 ; e(@xmath71 ) pairs ranging from 14.87 ; 0.06 to 14.65 ; 0.02 with the statistical errors of around 0.12 mag ( all for the case [ fe / h ] @xmath174 ) . when shifted to a common reddening , we find our distance modulus is consistent with ground - based estimates , and at least 0.1 mag smaller than other hipparcos - based estimates . m30 has a larger @xmath5 value [ @xmath175 than any of the other metal - poor clusters for which this quantity has been measured . this quantity is usually used as a helium indicator and our measured @xmath5 value suggests a helium abundance @xmath176 larger than the mean of the other metal - poor clusters . m30 s value for the helium indicator @xmath6 is also relatively high although for the case of e@xmath177 , it is consistent with the other metal - poor clusters . if there is a helium abundance enhancement in m30 , it is probably not an initial abundance difference since galactic chemical evolution and the similarity of the m30 and m68 fiducial lines ( see next point ) argue against it . the @xmath130 value for m30 is demonstrably large relative to clusters of similar metallicity . the m30 fiducial line ( except for the hb ) overlies that of m68 ( see figure [ m30vsm68 ] ) and m92 ( vbs ) very closely , indicating that m30 probably has the same age as these two clusters . we suggest that the hb luminosity in m30 is high due to a larger - than - average y for the m30 hb stars . the lfs of the cluster show definite evidence for an excess of rgb stars relative to ms stars , and marginally significant ( @xmath178 ) evidence for an excess of sgb stars , as compared with theory . the sgb feature has slightly higher significance in the @xmath14 band . the possibility remains that these anomalies are only present in low - metallicity clusters . stellar rotation is a possible explanation for the excess number of rgb stars relative to ms stars . alternatively , the excess giants could be a signpost for mixing events on the lower rgb in which fresh hydrogen is mixing into the energy generation region . this could also be identified as the source of the envelope y - enrichment we infer from the hb and brighter rgb stars . we do not find an obvious rgb bump in m30 , in spite of the size of our rgb sample . using the cumulative lf , we have detected the bump in the metal - poor cluster m68 with a @xmath179 value that agrees with the linear trend with [ fe / h ] found by fusi pecci et al . ( 1990 ) . it is possible that points 2 5 are all related to the deep - mixing events inferred for some globulars based on the surface abundances of elements that participate in the energy generation cycles . the hypothesis that we are seeing the effects of the mixing of hydrogen - rich material into the energy - generation regions and helium - rich material out into the stellar envelope can qualitatively explain all of these ( 2 through 5 ) observations . if this hypothesis is correct then we predict that detailed abundance studies of the bright giants in m30 should show the characteristic patterns of deep mixing low oxygen and carbon abundances accompanied by high nitrogen , aluminum , and sodium . we would especially like to thank d. vandenberg for providing us with theoretical @xmath87-enhanced isochrones and luminosity functions prior to publication and p. stetson for the use of his excellent software . it is a pleasure to thank p. guhathakurta , z. webster , and r. rood for useful conversations . this research has made use of the simbad database , operated at cds , strasbourg , france . m.b . is happy to acknowledge support from nsf grant ast 94 - 20204 . electronic copies of the listing of the photometry are available on request to the first author . aperture photometry was performed using the program daophot ii ( stetson 1987 ) . using these data , growth curves were constructed for each frame using daogrow ( stetson 1990 ) in order to extrapolate from the flux measurements over a circular area of finite radius to the total flux observable for the star . the aperture magnitudes and the known standard system magnitudes of landolt ( 1992 ) were then used to derive coefficients for the transformation equations : @xmath180 @xmath181 where @xmath182 and @xmath183 are observed aperture photometry magnitudes , @xmath13 and @xmath14 are the standard system magnitudes , and @xmath184 is the airmass . the primary standard stars covered a color range @xmath185 , completely encompassing the color range of the cluster sample . the coefficients for the transformation equations are given in table [ coeftab ] . the residuals for the sample of 25 stars are shown in figure [ prime ] , and the average residuals are given in table [ restab ] . ( in this and all subsequent comparisons , the residuals are calculated in the sense of ours theirs . ) we chose 118 relatively bright and isolated stars in the m30 field observed with the 1.5 m telescope during the photometric night to be `` secondary standards '' . the stars selected were required to be unsaturated , brighter than the turnoff , in relatively uncrowded regions of the images , and close to the apparent fiducial line of the cluster ( since this acts as an additional check on the accuracy of the photometry ) . once the list was finalized , all other stars were subtracted from the frames and aperture photometry was obtained . the colors for these standards cover the range @xmath186 and @xmath187 both the ctio 4 m and 1.5 m data for m30 were reduced using the standard suite of programs developed by peter stetson ( daophot / allstar ; stetson 1987 , 1989 ) , and following the procedures in sandquist et al . ( 1996 ) . we used the secondary standards established with the 1.5 m observations on the single photometric night to determine the coefficients in the transformation equations for all of the 1.5 m profile fitting photometry : @xmath188 @xmath189 where @xmath182 and @xmath183 are the instrumental magnitudes from the profile fitting , @xmath13 and @xmath14 are the standard values from the aperture photometry , and @xmath190 is an index referring to individual frames . the coefficients of the color terms are given in table [ coeftab ] , the average residuals and standard deviation of the residuals for the comparison of the profile fitting and aperture photometry are given in table [ restab ] , and individual star residuals are shown in figure [ aptopsf ] . in the next step , we chose to calibrate the 4 m profile fitting photometry to the 1.5 m profile fitting photometry rather than the aperture photometry of the secondary standards . this was done primarily to ensure that all of our profile fitting was on the same system over as large a range of magnitudes as possible . the m30 frames taken at the 1.5 m telescope on the one photometric night did not go particularly deep , while the 4 m photometry had few unsaturated observations of the brighter stars in the cluster . the data taken on the non - photometric nights at the 1.5 m telescope did , however , cover a range of magnitudes similar to that of the 4 m data . we selected a sample of 248 stars found in both fields at least 300 pixels away from the cluster center . these stars were used to determine the transformation coefficients for the equations : @xmath191 @xmath192 where @xmath182 and @xmath183 are the instrumental magnitudes from the 4 m observations , and @xmath13 and @xmath14 are the standard values from the 1.5 m observations . the coefficients of the color terms are given in table [ coeftab ] , while the residuals of the comparison of the photometry for the 4 m and 1.5 m measurements of the secondary standards are shown in figure [ secres ] . for the final calibration , we used the transformation equations for the 1.5 m and 4 m profile - fitting data . all of the profile - fitting photometry from both telescopes was combined with weights equal to the inverse square of the internal measurement errors in order to determine our standard - system magnitude and color values . because of the large sky coverage of the ctio frames , most other surveys of m30 overlap the program area at least partially . table [ restab ] provides a summary of the zero - point offsets for comparisons with these studies . we would particularly like to point out that there is considerable difference among them , highlighting the importance of the calibration . the fields used by bolte ( 1987 ) , rfv , and samus et al . ( 1995 ) are completely included on all frames . a comparison with the photometry of bolte is given in figure [ boltecomp ] . in figure [ alccomp ] , we show the comparison with the study of m30 by samus et al . ( we do this partly because it involves the same filter bands , and partly because the residuals are the lowest on average ( although the scatter in star - to - star residuals is large ) . as a note , comparisons with the most recent study ( bergbusch 1996 ) show no signs of color trends in the residuals except within about a magnitude of the tip of the giant branch . cccccc & 17.275 & 0.820 & 29 23.000 & 1.293 & 303 & 17.216 & 0.824 & 28 22.800 & 1.293 & 399 & 17.077 & 0.833 & 20 22.600 & 1.209 & 556 & 16.925 & 0.840 & 23 22.400 & 1.147 & 771 & 16.781 & 0.847 & 14 22.200 & 1.115 & 918 & 16.618 & 0.854 & 38 22.000 & 1.052 & 1048 & 16.391 & 0.863 & 35 21.925 & 1.035 & 827 & 16.140 & 0.880 & 40 21.775 & 1.002 & 811 & 15.868 & 0.901 & 25 21.625 & 0.967 & 850 & 15.621 & 0.913 & 37 21.475 & 0.928 & 896 & 15.367 & 0.927 & 18 21.325 & 0.890 & 934 & 15.127 & 0.953 & 14 21.175 & 0.869 & 907 & 14.877 & 0.969 & 14 21.025 & 0.837 & 910 & 14.662 & 0.990 & 11 20.875 & 0.810 & 965 & 14.395 & 1.013 & 6 20.725 & 0.779 & 908 & 14.117 & 1.052 & 8 20.575 & 0.748 & 876 & 13.924 & 1.068 & 6 20.425 & 0.719 & 856 & 13.604 & 1.115 & 3 20.275 & 0.699 & 843 & 13.319 & 1.143 & 3 20.125 & 0.675 & 740 & 13.125 & 1.178 & 7 19.975 & 0.672 & 765 & 12.830 & 1.223 & 3 19.825 & 0.646 & 694 & 12.634 & 1.260 & 3 19.675 & 0.634 & 620 & 12.384 & 1.329 & 1 19.525 & 0.622 & 566 & 12.016 & 1.481 & 3 19.375 & 0.605 & 519 & 19.225 & 0.588 & 444 & 15.821 & -0.016 & 6 19.000 & 0.576 & 527 & 15.584 & 0.022 & 6 18.800 & 0.568 & 466 & 15.386 & 0.084 & 4 18.600 & 0.558 & 371 & 15.302 & 0.124 & 3 18.400 & 0.575 & 305 & 15.187 & 0.169 & 2 18.240 & 0.595 & 272 & 15.083 & 0.306 & 5 18.094 & 0.645 & 95 & 14.900 & 0.701 & 6 17.968 & 0.695 & 53 & 17.884 & 0.745 & 56 & 14.184 & 0.970 & 4 17.800 & 0.764 & 101 & 14.354 & 0.924 & 1 17.575 & 0.793 & 57 & 14.483 & 0.914 & 1 17.425 & 0.809 & 44 & 14.564 & 0.894 & 1 [ fidtab ] cccccccccc 12.543 & 0.766 & 0.212 & 13 & 1.000 & 18.855 & 191.524 & 14.548 & 184 & 0.960 13.522 & 1.028 & 0.265 & 15 & 1.000 & 19.005 & 240.516 & 16.396 & 229 & 0.952 14.275 & 1.957 & 0.449 & 19 & 1.000 & 19.155 & 276.068 & 17.539 & 261 & 0.945 14.727 & 2.885 & 0.771 & 14 & 1.001 & 19.305 & 285.390 & 17.865 & 267 & 0.934 15.027 & 4.539 & 0.968 & 22 & 1.001 & 19.455 & 332.817 & 19.475 & 306 & 0.918 15.328 & 6.614 & 1.169 & 32 & 0.998 & 19.605 & 361.638 & 20.207 & 331 & 0.915 15.554 & 7.433 & 1.752 & 18 & 1.001 & 19.756 & 386.727 & 20.903 & 351 & 0.907 15.704 & 5.785 & 1.546 & 14 & 1.000 & 19.906 & 451.675 & 22.748 & 403 & 0.892 15.854 & 8.263 & 1.848 & 20 & 1.000 & 20.056 & 475.933 & 23.433 & 419 & 0.880 16.004 & 9.912 & 2.023 & 24 & 1.000 & 20.206 & 556.464 & 25.362 & 488 & 0.877 16.154 & 8.259 & 1.847 & 20 & 1.000 & 20.356 & 574.396 & 25.920 & 498 & 0.867 16.304 & 7.433 & 1.752 & 18 & 1.001 & 20.506 & 632.285 & 27.495 & 537 & 0.850 16.455 & 11.985 & 2.226 & 29 & 1.000 & 20.656 & 664.107 & 28.509 & 555 & 0.837 16.605 & 11.161 & 2.148 & 27 & 1.000 & 20.805 & 698.896 & 29.602 & 575 & 0.824 16.755 & 12.813 & 2.302 & 31 & 1.000 & 20.955 & 733.417 & 30.831 & 592 & 0.809 16.905 & 16.977 & 4.117 & 17 & 1.001 & 21.105 & 781.575 & 32.377 & 622 & 0.798 17.055 & 12.982 & 3.601 & 13 & 1.001 & 21.254 & 751.962 & 32.336 & 588 & 0.784 17.205 & 17.971 & 4.236 & 18 & 1.001 & 21.404 & 805.608 & 33.886 & 620 & 0.774 17.355 & 25.019 & 5.004 & 25 & 0.998 & 21.553 & 779.031 & 34.422 & 580 & 0.749 17.505 & 16.997 & 4.122 & 17 & 0.999 & 21.702 & 903.370 & 38.618 & 648 & 0.722 17.655 & 34.968 & 5.912 & 35 & 1.000 & 21.851 & 922.898 & 40.324 & 643 & 0.702 17.806 & 50.947 & 7.140 & 51 & 1.000 & 22.000 & 986.270 & 44.897 & 643 & 0.658 17.956 & 63.013 & 7.972 & 63 & 1.000 & 22.148 & 998.885 & 46.770 & 622 & 0.631 18.105 & 102.573 & 10.276 & 102 & 0.997 & 22.296 & 1036.098 & 48.374 & 601 & 0.584 18.255 & 117.435 & 11.112 & 116 & 0.990 & 22.445 & 1134.865 & 55.105 & 567 & 0.505 18.405 & 123.294 & 11.517 & 121 & 0.983 & 22.593 & 1350.132 & 71.038 & 501 & 0.378 18.555 & 159.431 & 13.317 & 155 & 0.973 & 22.740 & 3956.208 & 404.959 & 500 & 0.129 18.705 & 191.698 & 14.697 & 185 & 0.964 & & & & & [ vlftab ] cccccccccc 11.149 & 0.590 & 0.170 & 12 & 1.000 & 17.467 & 76.966 & 8.715 & 78 & 1.000 12.225 & 0.850 & 0.236 & 13 & 1.000 & 17.619 & 111.136 & 10.508 & 112 & 0.999 12.993 & 1.281 & 0.355 & 13 & 1.000 & 17.770 & 104.511 & 10.283 & 104 & 0.992 13.452 & 2.373 & 0.685 & 12 & 1.000 & 17.921 & 131.605 & 11.657 & 130 & 0.985 13.756 & 2.779 & 0.743 & 14 & 1.000 & 18.071 & 188.224 & 14.172 & 183 & 0.972 14.060 & 3.586 & 0.845 & 18 & 1.000 & 18.221 & 185.096 & 14.125 & 178 & 0.964 14.287 & 4.359 & 1.315 & 11 & 1.001 & 18.370 & 253.828 & 16.588 & 243 & 0.959 14.439 & 6.772 & 1.643 & 17 & 0.997 & 18.520 & 290.213 & 17.909 & 273 & 0.943 14.591 & 7.986 & 1.786 & 20 & 1.001 & 18.670 & 302.430 & 18.308 & 282 & 0.936 14.742 & 5.598 & 1.496 & 14 & 1.000 & 18.819 & 345.452 & 19.861 & 313 & 0.911 14.893 & 5.998 & 1.549 & 15 & 1.000 & 18.968 & 421.574 & 21.974 & 380 & 0.905 15.044 & 6.780 & 1.645 & 17 & 1.000 & 19.117 & 465.390 & 23.205 & 414 & 0.894 15.195 & 9.954 & 1.991 & 25 & 1.000 & 19.266 & 506.015 & 24.377 & 441 & 0.881 15.347 & 5.985 & 1.546 & 15 & 1.000 & 19.415 & 613.201 & 27.068 & 526 & 0.860 15.498 & 9.583 & 1.956 & 24 & 1.000 & 19.564 & 693.000 & 29.054 & 583 & 0.853 15.649 & 8.810 & 1.879 & 22 & 1.000 & 19.712 & 777.236 & 31.180 & 637 & 0.832 15.799 & 8.811 & 1.879 & 22 & 1.000 & 19.859 & 820.648 & 32.433 & 657 & 0.819 15.950 & 12.407 & 2.229 & 31 & 1.000 & 20.006 & 833.922 & 33.018 & 656 & 0.807 16.101 & 11.995 & 2.190 & 30 & 1.001 & 20.152 & 961.583 & 36.406 & 730 & 0.779 16.252 & 14.400 & 2.400 & 36 & 1.000 & 20.298 & 1024.388 & 38.171 & 758 & 0.762 16.403 & 14.773 & 2.429 & 37 & 1.000 & 20.443 & 1042.008 & 39.215 & 755 & 0.748 16.554 & 21.175 & 2.909 & 53 & 0.999 & 20.587 & 1104.607 & 42.537 & 743 & 0.708 16.705 & 21.109 & 2.900 & 53 & 1.000 & 20.730 & 1145.276 & 43.849 & 759 & 0.695 16.857 & 25.699 & 5.040 & 26 & 1.001 & 20.873 & 1314.175 & 49.073 & 833 & 0.666 17.008 & 26.650 & 5.129 & 27 & 1.000 & 21.013 & 1385.530 & 55.311 & 777 & 0.610 17.161 & 41.972 & 6.400 & 43 & 1.000 & 21.155 & 1525.717 & 70.014 & 738 & 0.501 17.315 & 46.912 & 6.771 & 48 & 1.000 & & & & & [ ilftab ] lcccccccl 14594 & 8.04 & 0.66 & 0.02585 & 0.044 & @xmath193 & @xmath194 & 0.66 & hd19445 18915 & 8.51 & 1.01 & 0.05414 & 0.020 & @xmath195 & @xmath196 & 0.99 & hd25329 24316 & 9.43 & 0.65 & 0.01455 & 0.069 & @xmath197 & @xmath198 & 0.61 & hd34328 38541 & 8.27 & 0.77 & 0.03529 & 0.029 & @xmath199 & @xmath200 & 0.75 & hd64090 40778 & 9.73 & 0.60 & 0.01036 & 0.142 & @xmath201 & @xmath202 & 0.57 & bd+54 1216 53070 & 8.22 & 0.63 & 0.01923 & 0.059 & @xmath203 & @xmath204 & 0.59 & hd94028 57939 & 6.44 & 0.89 & 0.10921 & 0.007 & @xmath205 & @xmath206 & 0.84 & hd103095 60632 & 9.66 & 0.63 & 0.01095 & 0.118 & @xmath207 & @xmath208 & 0.60 & hd108177 74234 & 9.46 & 1.01 & 0.03368 & 0.050 & @xmath209 & @xmath210 & 0.99 & hd134440 74235 & 9.08 & 0.92 & 0.03414 & 0.040 & @xmath209 & @xmath211 & 0.90 & hd134439 98020 & 8.83 & 0.75 & 0.02532 & 0.046 & @xmath212 & @xmath213 & 0.70 & hd188510 100568 & 8.66 & 0.67 & 0.02288 & 0.054 & @xmath214 & @xmath215 & 0.60 & hd193901 100792 & 8.35 & 0.63 & 0.01794 & 0.069 & @xmath216 & @xmath217 & 0.55 & hd194598 104659 & 7.37 & 0.66 & 0.02826 & 0.036 & @xmath218 & @xmath219 & 0.56 & hd201891 3026 & 9.25 & 0.64 & 0.00957 & 0.144 & @xmath220 & @xmath221 & 0.54 & hd3567 33221 & 9.07 & 0.63 & 0.00911 & 0.111 & @xmath222 & @xmath223 & 0.53 & cpd-33 3337 48152 & 8.33 & 0.55 & 0.01244 & 0.085 & @xmath224 & @xmath225 & 0.55 & hd84937 55790 & 9.07 & 0.63 & 0.01099 & 0.135 & @xmath226 & @xmath227 & 0.55 & hd99383 68464 & 8.73 & 0.64 & 0.00977 & 0.135 & @xmath228 & @xmath229 & 0.60 & hd122196 76976 & 7.22 & 0.69 & 0.01744 & 0.056 & @xmath230 & @xmath231 & 0.77 & hd140283 [ subdtab ] cccccc @xmath232 & 0.06 & @xmath233 & * @xmath234 * & @xmath235 & @xmath236 @xmath232 & 0.02 & @xmath237 & * @xmath2 * & @xmath238 & @xmath239 @xmath240 & 0.06 & @xmath233 & @xmath241 & @xmath242 & @xmath243 @xmath240 & 0.02 & @xmath244 & @xmath245 & @xmath246 & @xmath247 [ dmtab ] lccccc ngc 104 ( 47 tuc ) & @xmath248 & @xmath249 & @xmath250 & @xmath251 & @xmath214 ngc 5904 ( m5 ) & @xmath252 & @xmath253 & @xmath254 & @xmath255 & @xmath256 ngc 5272 ( m3 ) & @xmath257 & @xmath258 & @xmath259 & @xmath260 & @xmath261 ngc 4590 ( m68 ) & @xmath262 & @xmath263 & @xmath264 & @xmath265 & @xmath266 ngc 5024 ( m53 ) & @xmath267 & @xmath268 & & @xmath269 & @xmath270 ngc 5053 & @xmath271 & @xmath272 & & @xmath273 & @xmath274 ngc 5466 & @xmath275 & @xmath276 & & & @xmath277 ngc 6341 ( m92 ) & @xmath278 & @xmath279 & & @xmath280 & @xmath281 ngc 6397 & @xmath232 & @xmath282 & & @xmath283 & @xmath284 ( 0.69 ) ngc 7078 ( m15 ) & @xmath285 & @xmath286 & @xmath287 & @xmath288 & @xmath289 ngc 7099 ( m30 ) & @xmath240 & @xmath96 & @xmath290 & @xmath291 & @xmath292 [ clusttab ] lcccccc @xmath293 & 57 & 2 : & 1 & 60 & 3 & 46 & @xmath125 & @xmath294 & @xmath295 & @xmath296 & @xmath297 & @xmath298 & 59 & 4 & 2 & 65 & 4 & 48 & @xmath126 & @xmath299 & @xmath300 & @xmath301 & @xmath302 & @xmath303 & 65 & 3 & 9 & 77 & 4 & 42 & @xmath128 & @xmath304 & @xmath305 & @xmath296 & @xmath306 & total & 181 & 9 & 12 & 202 & 11 & 136 & @xmath96 & @xmath307 & @xmath308 & @xmath309 & @xmath310 & ygsb & 51 & 1 : & 1 & 53 & 1 & 34 & @xmath311 & & & & & [ poptab ] ccccc @xmath13 & @xmath312 & @xmath313 & @xmath314 & 1 & & & @xmath315 & 2 & & & @xmath316 & 3 @xmath14 & @xmath317 & @xmath318 & @xmath319 & 1 @xmath13 & & & @xmath320 & 1 @xmath14 & & & @xmath321 & 1 @xmath13 & & & @xmath322 & 1 & & & @xmath323 & 2 @xmath14 & & & @xmath324 & 1 & & & @xmath325 & 2 [ coeftab ] ccccccccc 1.5 m & landolt & 0.0005 & 0.0006 & @xmath326 & 0.0119 & @xmath327 & 0.0010 & 27 psf & aperture & @xmath327 & 0.0204 & @xmath328 & 0.0198 & @xmath329 & 0.0252 & 118 4 m & 1.5 m & @xmath330 & 0.0447 & @xmath331 & 0.0635 & @xmath332 & 0.0515 & 248 4 m + 1.5 m & bolte 1987 ( s ) & @xmath333 & 0.1067 & & & 59 4 m + 1.5 m & bolte 1987 ( l ) & @xmath334 & 0.0928 & & & 401 4 m + 1.5 m & rfv & @xmath335 & 0.1419 & & & 1374 4 m + 1.5 m & samus et al . 1995 & @xmath336 & 0.1449 & @xmath337 & 0.1771 & @xmath338 & 0.1139 & 255 4 m + 1.5 m & bergbusch 1996 & @xmath339 & 0.0611 & & & 316 [ restab ]
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we present new @xmath0 photometry for the halo globular cluster m30 ( ngc 7099 = c2137 - 174 ) , and compute luminosity functions ( lfs ) in both bands for samples of about 15,000 hydrogen - burning stars from near the tip of the red giant branch ( rgb ) to over four magnitudes below the main - sequence ( ms ) turnoff . we confirm previously observed features of the lf that are at odds with canonical theoretical predictions : an excess of stars on subgiant branch ( sgb ) approximately 0.4 mag above the turnoff and an excess number of rgb stars relative to ms stars . based on subdwarfs with _
hipparcos_-measured parallaxes , we compute apparent distance moduli of @xmath1 and @xmath2 for reddenings of e@xmath3 and 0.02 respectively . the implied luminosity for the horizontal branch ( hb ) at these distances is @xmath4 and 0.37 mag .
the two helium indicators we have been able to measure ( @xmath5 and @xmath6 ) both indicate that m30 s helium content is high relative to other clusters of similar metallicity .
m30 has a larger value for the parameter @xmath7 than any of the other similarly metal - poor clusters for which this quantity can be reliably measured .
this suggests that m30 has either a larger age or higher helium content than all of the other metal - poor clusters examined .
the color - difference method for measuring relative ages indicates that m30 is coeval with the metal - poor clusters m68 and m92 .
= -0.5 in epsf
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You are an expert at summarizing long articles. Proceed to summarize the following text:
a central open question in classical fluid dynamics is whether the incompressible three - dimensional euler equations with smooth initial conditions develop a singularity after a finite time . a key result was established in the late eighties by beale , kato and majda ( bkm ) . the bkm theorem @xcite states that blowup ( if it takes place ) requires the time - integral of the supremum of the vorticity to become infinite ( see the review by bardos and titi @xcite ) . many studies have been performed using the bkm result to monitor the growth of the vorticity supremum in numerical simulations in order to conclude yes or no regarding the question of whether a finite - time singularity might develop . the answer is somewhat mixed , see _ e.g. _ references @xcite and the recent review by gibbon @xcite . other conditional theoretical results , going beyond the bkm theorem , were obtained in a pioneering paper by constantin , fefferman and majda @xcite . they showed that the evolution of the direction of vorticity posed geometric constraints on potentially singular solutions for the 3d euler equation @xcite . this point of view was further developed by deng , hou and yu in references @xcite and @xcite . an alternative way to extract insights on the singularity problem from numerical simulations is the so - called analyticity strip method @xcite . in this method the time is considered as a real variable and the space - coordinates are considered as complex variables . the so - called `` width of the analyticity strip '' @xmath5 is defined as the imaginary part of the complex - space singularity of the velocity field nearest to the real space . the idea is to monitor @xmath1 as a function of time @xmath6 . this method uses the rigorous result @xcite that a real - space singularity of the euler equations occurring at time @xmath7 must be preceded by a non - zero @xmath1 that vanishes at @xmath7 . using spectral methods @xcite , @xmath1 is obtained directly from the high - wavenumber exponential fall off of the spatial fourier transform of the solution @xcite . this method effectively provides a `` distance to the singularity '' given by @xmath1 @xcite , which can not be obtained from the general bkm theorem . note that the bkm theorem is more robust than the analyticity - strip method in the sense that it applies to velocity fields that do not need to be analytic . however , in the present paper we will concentrate on initial conditions that are analytic . in this case , there is a well - known result that states : _ in three dimensions with periodic boundary conditions and analytic initial conditions , analyticity is preserved as long as the velocity is continuously differentiable _ ( @xmath8 ) _ in the real domain _ @xcite . the bkm theorem allows for a strengthening of this result : analyticity is actually preserved as long as the vorticity is finite @xcite . the analyticity - strip method has been applied to probe the euler singularity problem using a standard periodic ( and analytical ) initial data : the so - called taylor - green ( tg ) vortex @xcite . we now give a short review of what is already known about the tg dynamics . numerical simulations of the tg flow were performed with resolution increasing over the years , as more computing power became available . it was found that except for very short times and for as long as @xmath1 can be reliably measured , it displays almost perfect exponential decrease . simulations performed in @xmath9 on a grid of @xmath10 points obtained @xmath11 ( for @xmath6 up to @xmath12 ) @xcite . this behavior was confirmed in @xmath13 at resolution @xmath14 @xcite . more than @xmath15 years after the first study , simulations performed on a grid of @xmath16 points yielded @xmath17 ( for @xmath6 up to @xmath18 ) @xcite . if these results could be safely extrapolated to later times then the taylor - green vortex would never develop a real singularity @xcite . the present paper has two main goals . one is to report on and analyze new simulations of the tg vortex that are performed at resolution @xmath0 . these new simulations show , for the first time , a well - resolved change of regime , leading to a faster decay of @xmath1 happening at a time where preliminary @xmath3 visualizations show the collision of vortex sheets . that was reported in mhd for the so - called imtg initial data at resolution @xmath16 in reference @xcite . ] the second goal of this paper is to answer the following question , motivated by the new behavior of the tg vortex : how fast does the analyticity - strip width have to decrease to zero in order to sustain a finite - time singularity , consistent with the bkm theorem ? to the best of our knowledge , this question has not been formulated previously . to answer this question we introduce a new bound of the supremum norm of vorticity in terms of the energy spectrum . we then use this bound to combine the bkm theorem with the analyticity - strip method . this new bound is sharper than usual bounds . we show that a finite - time blowup exists only if the analyticity - strip width goes to zero sufficiently fast at the singularity time . if a power - law behavior is assumed for @xmath1 then its exponent must be greater than some critical value . in other words , we provide a powerful test that can potentially rule out the existence of a finite - time singularity in a given numerical solution of euler equations . we apply this test to the data from the latest @xmath0 taylor - green numerical simulation in order to see if the change of behavior in @xmath1 can be consistent with a singularity . the paper is organized as follows : section [ sec : theo ] is devoted to the basic definitions , symmetries and numerical method related to the inviscid taylor - green vortex . in sec . [ sec : numerics_classical ] , the new high - resolution taylor - green results are presented and are analyzed classically in terms of analyticity - strip method and bkm . in sec . [ sec : as_bkm ] , the analyticity - strip method and bkm theorem are bridged together . the section starts with heuristic arguments that are next formalized in a mathematical framework of definitions , hypotheses and theorems . in sec . [ sec : newanal ] , our new theoretical results are used to analyze the behavior of the decrement . section [ sec : conclusion ] is our conclusion . the generalization to non tg - symmetric periodic flows of the results presented in sec . [ sec : as_bkm ] are described in an appendix . let us consider the 3d incompressible euler equations for the velocity field @xmath19 defined for @xmath20 and in a time interval @xmath21 : @xmath22 the taylor - green ( tg ) flow @xcite is defined by the @xmath23-periodic initial data @xmath24 , where @xmath25 the periodicity of @xmath26 allows us to define the ( standard ) fourier representation @xmath27 the kinetic energy spectrum @xmath28 is defined as the sum over spherical shells @xmath29 and the total energy @xmath30 is independent of time because @xmath26 satisfies the 3d euler equations ( [ eq : euler ] ) . a number of the symmetries of @xmath31 are compatible with the equation of motions . they are , first , rotational symmetries of angle @xmath32 around the axis @xmath33 and @xmath33 ; and of angle @xmath34 around the axis @xmath35 . a second set of symmetries corresponds to planes of mirror symmetry : @xmath36 , @xmath37 and @xmath38 . on the symmetry planes , the velocity @xmath31 and the vorticity @xmath39 are ( respectively ) parallel and perpendicular to these planes that form the sides of the so - called impermeable box which confines the flow . it is demonstrated in reference @xcite that these symmetries imply that the fourier expansions coefficients of the velocity field in eq . @xmath40 vanishes unless @xmath41 are either all even or all odd integers . this fact can be used in a standard way @xcite to reduce memory storage and speed up computations . the euler equations are solved numerically using standard @xcite pseudo - spectral methods with resolution @xmath42 . time marching is done with a second - order runge - kutta finite - difference scheme . the solutions are dealiased by suppressing , at each time step , the modes for which at least one wave - vector component exceeds two - thirds of the maximum wave - number @xmath43 ( thus a @xmath0 run is truncated at @xmath44 ) . the simulations reported in this paper were performed using a special purpose symmetric parallel code developed from that described in @xcite . the workload for a timestep is ( roughly ) twice that of a general periodic code running at a quarter of the resolution . specifically , at a given computational cost , the ratio of the largest to the smallest scale available to a computation with enforced taylor - green symmetries is enhanced by a factor of @xmath45 in linear resolution . this leads to a factor of @xmath46 savings in total computational time and memory usage . the code is based on fftw and a hybrid mpi - openmp scheme derived from that described in @xcite . the runs were performed on the idris bluegene / p machine . at resolution @xmath0 we used @xmath47 mpi processes , each process spawning @xmath45 openmp threads . ( see eq . ) at @xmath48 and b ) maximum of vorticity @xmath49 . results from runs performed at different resolutions are displayed together : @xmath50 ( brown triangles ) , @xmath51 ( blue squares ) , @xmath16 ( green diamonds ) and @xmath0 ( red circles).,height=377 ] runs were performed at resolutions @xmath50 , @xmath51 , @xmath16 and @xmath52 . the behavior of the energy spectra and the spatial maximum of the norm of the vorticity @xmath53 are presented in fig . [ fig : energy_spectra_maxvort ] . visualization of tg vorticity @xmath54 at resolution @xmath0 : a ) full impermeable box @xmath55 , @xmath56 and @xmath57 at @xmath58 . zooms over the subbox marked near @xmath59 , @xmath60 are displayed in b ) at @xmath61 , in c ) at @xmath58 and in d ) at @xmath62.,height=359 ] it is apparent in fig . [ fig : energy_spectra_maxvort](a ) that resolution - dependent even - odd oscillations are present , at certain times , on the tg energy spectrum . note that this behavior is produced when the tail of the spectrum rises above the round - off error @xmath63 . this phenomenon can be explained in terms of a _ resonance _ @xcite , along the lines developed in reference @xcite . in practice we will deal with this problem by averaging the spectrum over shells of width @xmath64 . apart from this it can be seen that spectra computed using different resolutions are in good agreement for all times . in contrast , it is visible in fig . [ fig : energy_spectra_maxvort](b ) that the maximum of vorticity @xmath49 computed at different resolutions are in agreement only up to some resolution - dependent time ( see the inset ) . the fact that @xmath49 at a given time @xmath65 decreases if one truncates the higher wavenumbers of the velocity field ( see fig . [ fig : energy_spectra_maxvort](b ) ) strongly suggests that @xmath49 has significant contributions coming from high - wavenumbers modes . this forms the basis of the heuristic argument presented below in sec . [ subsec : heur ] . figure [ fig : vort_3d_viz ] shows @xmath3 visualizations ( using the vapor software ) of the high vorticity regions in the impermeable box , corresponding to the @xmath0 run at late times . a thin vortex sheet is apparent in fig.[fig : vort_3d_viz](a ) on the vertical faces @xmath66 , @xmath32 and @xmath67 , @xmath32 of the impermeable box . the emergence of this thin vortex sheet is well understood by simple dynamical arguments about the flow on the faces of the impermeable box that were first given in reference @xcite . we now briefly review these arguments . the initial vortex on the bottom face is first forced by centrifugal action to spiral outwards toward the edges and then up the side faces . a corresponding outflow on the top face and downflow from the top edges onto the side faces leads to a convergence of fluid near the horizontal centreline of each side face , from where it is forced back into the centre of the box and subsequently back to the top and bottom faces . the vorticity on the side faces is efficiently produced in the zone of convergence , and builds up rapidly into a vortex sheet ( see figs . 1 and 2 of reference @xcite and fig . 8 of reference @xcite ) . while these considerations explain the presence of the thin vortex sheet in fig.[fig : vort_3d_viz](a ) , the dynamics presented in fig.[fig : vort_3d_viz](b - d ) also involves the collision of vortex sheets happening near the edge @xmath59 , close to @xmath60 . note that , as stated above in sec . [ subsec : symm ] , the vortex lines are perpendicular to the faces of the impermeable box . thus , because the collision takes place near an edge , the corresponding vortex lines must be highly curved , with strong variations of the direction of vorticity . the geometric constraints on potential singularities posed by the evolution of the direction of vorticity developed in references @xcite could be applied to the situation described in fig . [ fig : vort_3d_viz ] . however , such an analysis goes beyond the bkm theorem and involves extensive post - processing of very large datasets . this task is thus left for further work and we concentrate here on simple bkm diagnostics for the vorticity supremum and analyticity strip analysis of energy spectra . the analyticity - strip method @xcite is based on the fact that when the velocity field is analytic in space the energy spectrum satisfies @xmath68 in the asymptotic ` ultraviolet region ' @xmath69 with a proportionality factor that may contain an algebraic decay in @xmath70 a multiplicative function of time and , depending on the complexity of the physical flow , even an oscillatory ( in @xmath71 ) modulation @xcite . ( red markers ) ; times and fit intervals are indicated in the legend . ] the basic idea is thus to assume that @xmath28 can be well approximated by a function of the form @xmath72 in some wave numbers interval between @xmath73 and @xmath74 ( the maximum wavenumber permitted by the numerical resolution @xmath42 ) . the common procedure to determine @xmath75 is to perform a least - square fit at each time @xmath6 on the logarithm of the energy spectrum @xmath28 , using the functional form @xmath76 the error on the fit interval @xmath77 , @xmath78 is minimized by solving the equations @xmath79 , @xmath80 and @xmath81 . note that these equations are linear in the parameters @xmath82 , @xmath83 and @xmath84 the transient oscillations of the energy spectrum observed at the highest wavenumbers ( see above fig . [ fig : energy_spectra_maxvort](a ) are eliminated by averaging the tg spectrum on shells of width @xmath64 before performing the fit @xcite . we present in fig . [ fig : fit_comp ] , examples of tg energy spectra fitted in such a way on the intervals @xmath85 , where @xmath86 denotes the beginning of round off noise . it is apparent that the fits are globally of a good quality . the time evolution of the fit parameters @xmath87 , @xmath88 and @xmath89 computed at different resolutions are displayed in fig . [ fig : fit_evolution ] . the measure of the fit parameters is reliable as long as @xmath1 remains larger than a few mesh sizes , a condition required for the smallest scales to be accurately resolved and spectral convergence ensured . thus the dimensionless quantity @xmath90 is a measure of spectral convergence . it is conventional @xcite to define a ` reliability time ' @xmath91 by the condition @xmath92 and to say that the numerical simulation is reliable for times @xmath93 . this reliability time can be extended only by increasing the spatial resolution available for the simulation , so the more computer power is available the larger is the reliability time . the resolution - dependent reliability condition is marked by the horizontal lines in fig . [ fig : fit_evolution](c ) . the exponential law @xmath94 that was previously reported at resolution @xmath16 in reference @xcite is also indicated in fig . [ fig : fit_evolution](c ) by a dashed black line . it is thus apparent that our lower - resolution results well reproduce the previous computations that were discussed above in sec . [ sec : intro ] ( see text preceding references @xcite ) . in table [ tab : table_rel ] , the reliability time obtained from the fit parameter @xmath88 of fig . [ fig : fit_evolution ] is compared with the reliability time stemming from the exponential behavior . .reliability time deduced from the exponential behavior compared with the reliability time obtained from the fit parameter @xmath88 of fig . [ fig : fit_evolution ] . [ cols="^,^,^",options="header " , ] the results for exponent and predicted singular time of table [ tab : table_dels ] have to be read carefully . because of the local @xmath95-point method used to derive them from the data in table [ tab : table_int ] , they use the values of @xmath88 at @xmath96 , the last one being marginally reliable ( see sec.[sec : numefits ] ) . in fact , they amount to linear @xmath97-point extrapolation of the data in fig . [ fig : delta1 ] ( see the inset ) : @xmath7 is the intersection of the straight line extrapolation with the time axis and @xmath98 is the inverse of the slope . one can guess that there is room for a power - law type of behavior , with exponent @xmath99 if we consider the data at @xmath100 and @xmath101 if we include the data at @xmath102 . we now use corollary 11 ( see sec . [ sec : as_bkm ] ) to test if these estimates of power - law are consistent with the hypothesis of finite - time singularity . there , the product @xmath103 must be greater than or equal to one if finite - time singularity is to be expected . with the conservative estimate @xmath104 obtained by inspection of fig . [ fig : fit_evolution](b ) ( or equivalently using the values of @xmath89 in table [ tab : table_int ] ) , we obtain that @xmath105 for the data at @xmath106 and @xmath58 , but @xmath107 for the data at @xmath102 . these results are insensitive to the fit interval , see table [ tab : table_dels ] . therefore , if the latest data is considered , corollary 11 can not be used to negate the validity of the hypothesis of finite - time singularity . however , there is no sign that the data values of @xmath98 and @xmath7 in table [ tab : table_dels ] are settling down into constants , corresponding to a simple power - law behavior . another piece of analysis consists of comparing the singular time predicted from the data for the decrement @xmath1 with the singular time predicted from the direct data for the vorticity supremum norm . they seem both to be close to @xmath108 ( compare table [ tab : table_dels ] to table [ tab : table_omegasup ] ) . in this context , we should perhaps mention feynman s rule : `` never trust the data point furthest to the right '' , a comment attributed to richard feynman , saying basically that he would never trust the last points on an experimental graph , because if the people taking data could have gone beyond that , they would have . higher - resolution simulations are clearly needed to investigate whether the new regime is genuinely a power law and not simply a crossover to a faster exponential decay . our conclusion for this section is thus similar to that of sec . [ subsec : fit_methods_omegas ] : although our late - time reliable data for @xmath1 shows @xmath109 and is therefore not inconsistent with our corollary 11 , clear power - law behavior of @xmath1 is not achieved . in summary , we presented simulations of the taylor - green vortex with resolutions up to @xmath0 . we used the analyticity strip method to analyze the energy spectrum . we found that , around @xmath110 , a ( well - resolved up to @xmath111 ) change of regime is taking place , leading to a faster decay of the width of the analyticity strip @xmath1 . in the same time - interval , preliminary @xmath3 visualizations displayed a collision of vortex sheets . applying the bkm criterium to the growth of the maximum of the vorticity on the time - interval @xmath2 we found that the occurrence of a singularity around @xmath112 was not ruled out but that higher - resolution simulations were needed to confirm a clear power - law behavior for @xmath113 . we introduced a new sharp bound for the supremum norm of the vorticity in terms of the energy spectrum . this bound allowed us to combine the bkm theorem with the analyticity - strip method and to show that a finite - time blowup can exist only if @xmath1 vanishes sufficiently fast . applying this new test to our highest - resolution numerical simulation we found that the behavior of @xmath1 is not inconsistent with a singularity . however , due to the rather short time interval on which @xmath1 is both well - resolved and behaving as a power - law , higher - resolution studies are needed to investigate whether the new regime is genuinely a power law and not simply a crossover to a faster exponential decay . let us finally remark that our formal assumptions of section [ subsec : main_results ] are motivated and to some extent justified by the fact that , in systems that are known to lead to finite - time singularity , the analogous of the working hypothesis ( [ eq : fit_bound ] ) is verified . for the analogy to apply , a version of the bkm theorem must be available . this is the case of the @xmath73-d inviscid burgers equation for a real scalar field @xmath114 defined on the torus : @xmath115 , \,\forall \ , t \in [ 0,t_*),\ ] ] which admits a bkm - type of theorem @xcite , with singularity time @xmath7 defined by @xmath116 . in the 1-d case , the analogous of our bound is @xmath117 using the simple trigonometric initial data @xmath118 , the energy spectrum can be expressed in terms of bessel functions that admit simple asymptotic expansions . it is straightforward to show ( see @xcite for details ) that , for @xmath119 , one has the large-@xmath71 asymptotic expansion @xmath120 with @xmath121 while , at @xmath122 , @xmath123 in fact , the @xmath124 power law appears already before @xmath7 ( see the remark following eq . ( 3 - 10 ) of reference @xcite ) . it is easy to check that the analytical solution admits , for all @xmath71 and for all @xmath6 sufficiently close to @xmath7 , a working hypothesis ( [ eq : fit_bound ] ) of the form @xmath125 with analytically - obtainable functions @xmath126 and @xmath127 with @xmath128 . the analogous of corollary [ cor : beta ] gives the inequality @xmath129 which is saturated by the analytically - obtained exponents @xmath130 , @xmath128 . we acknowledge useful scientific discussions with annick pouquet , uriel frisch and giorgio krstulovic who also helped us with the visualizations of fig . [ fig : vort_3d_viz ] . the computations were carried out at idris ( cnrs ) . support for this work was provided by ucd seed funding projects sf304 and sf564 , and ircset ulysses project `` singularities in three - dimensional euler equations : simulations and geometry '' . here we provide the generalization to non tg - symmetric periodic flows of the results presented in section [ subsec : main_results ] . definition [ defn : spectrum ] and the working hypothesis ( hypothesis [ hypo : working ] ) are modified slightly in the general case . accordingly , the new bounds leading to lemma [ lem : main ] and theorem [ thm : main ] need to be modified slightly to accommodate the general case . the crucial derived relations between @xmath131 and @xmath132 in lemma [ lem : strong ] and corollaries [ cor : finite - time - sing ] and [ cor : beta ] will apply directly to the general periodic case and will not be discussed . the main technical difference is that the new bounds presented in section [ subsec : main_results ] apply for a flow with tg symmetries ( see section [ subsec : symm ] ) which imply that only modes with even - even - even and odd - odd - odd wavenumber components are populated . the general periodic case does not follow this restriction , which slightly modifies the bounds . we will assume , to simplify matters , that the so - called zero - mode of the velocity field is identically zero : @xmath133 + notice that all remaining wave numbers are populated . this means that all sums involving the scalar @xmath71 in equations ( [ eq : ineq_1 ] ) and ( [ eq : ineq_2_tyg ] ) will start effectively from @xmath134 also , because modes with mixed even - odd wavenumber components are allowed , the definitions of @xmath135 in lemma 2 and constant @xmath136 in equation ( [ eq : ineq_2_tyg ] ) must be replaced by more appropriate quantities . therefore , the corresponding general periodic versions of lemma [ lem : main ] ( equation ( [ eq : ineq_1 ] ) ) and practical bound ( equation ( [ eq : ineq_2_tyg ] ) ) are : + * lemma [ lem : main] ( general periodic version of lemma [ lem : main ] ) . * _ let @xmath137 be a velocity field with energy spectrum defined by equation ( [ eq : spectrum ] ) and let @xmath138 be its vorticity , defined on the periodicity domain @xmath139 ^ 3.$ ] then the following inequality is verified for all times @xmath21 when the sum in the rhs is defined , and independently of any evolution equation that @xmath140 might satisfy : _ @xmath141 _ where @xmath142 is the number of lattice points in a spherical shell of width 1 and radius @xmath143 . _ + * practical bound , general case . * @xmath144 where @xmath145 . we can easily check that the bounds for taylor - green , equations ( [ eq : ineq_1 ] ) and ( [ eq : ineq_2_tyg ] ) , are sharper ( by a factor close to 2 ) to their respective general bounds , equations ( [ eq : ineq_1_gp ] ) and ( [ eq : ineq_2_gp ] ) . finally , theorem [ thm : main ] is replaced by + * theorem [ thm : main]. * _ let a solution of the 3d euler equations satisfy the working hypothesis ( [ eq : fit_bound ] ) with @xmath146 included . then the maximal regularity time @xmath7 of the solution must satisfy _ @xmath147
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numerical simulations of the incompressible euler equations are performed using the taylor - green vortex initial conditions and resolutions up to @xmath0 .
the results are analyzed in terms of the classical analyticity strip method and beale , kato and majda ( bkm ) theorem .
a well - resolved acceleration of the time - decay of the width of the analyticity strip @xmath1 is observed at the highest resolution for @xmath2 while preliminary @xmath3 visualizations show the collision of vortex sheets .
the bkm criterium on the power - law growth of supremum of the vorticity , applied on the same time - interval , is not inconsistent with the occurrence of a singularity around @xmath4 .
these new findings lead us to investigate how fast the analyticity strip width needs to decrease to zero in order to sustain a finite - time singularity consistent with the bkm theorem .
a new simple bound of the supremum norm of vorticity in terms of the energy spectrum is introduced and used to combine the bkm theorem with the analyticity - strip method .
it is shown that a finite - time blowup can exist only if @xmath1 vanishes sufficiently fast at the singularity time .
in particular , if a power law is assumed for @xmath1 then its exponent must be greater than some critical value , thus providing a new test that is applied to our @xmath0 taylor - green numerical simulation .
our main conclusion is that the numerical results are not inconsistent with a singularity but that higher - resolution studies are needed to extend the time - interval on which a well - resolved power - law behavior of @xmath1 takes place , and check whether the new regime is genuine and not simply a crossover to a faster exponential decay .
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jets have been observed around a large variety of astrophysical objects , such as young stellar objects ( ysos ) , accreting white dwarfs , x - ray binaries ( xrbs ) , and active galactic nuclei ( agn ) , and are also thought to drive gamma - ray bursts ( grbs ) . in ysos jets facilitate angular momentum transport , allowing the central star to accrete more matter , and likely play a similar role in accreting black hole systems , where in agn they are thought to also affect the evolution of their host galaxy ( e.g. * ? ? ? while there are enormous differences between the types and scales of objects around which jets can occur , the origins of jets seem to be remarkably similar , requiring the basic ingredients of infalling / collapsing , rotating matter , and magnetic fields . despite these seemingly simple initial conditions , there are currently many outstanding problems in our understanding of jets , from their creation to their matter content and internal physics . one important facet of jets observationally is their hallmark synchrotron emission dominating the radio bands in particular , extending up to at least the near infrared ( nir ) in xrbs @xcite . at higher frequencies the typical power - law spectral energy distributions ( seds ) are generally interpreted as optically thin synchrotron emission from accelerated particles , both in agn @xcite and in xrbs @xcite . an effective way to accelerate radiating particles into a power - law distribution is via diffusive shock acceleration ( e.g. * ? ? ? * ; * ? ? ? * ) off scattering centers in turbulent plasma flows . once initiated , this process must to be distributed throughout the flow to account for the lack of spectral aging over vast distances along the jets . but where does the acceleration itself begin , and what triggers it ? there is increasing evidence that the start of this region is offset from the central compact object . for example in the jet of the agn m87 , recent vlbi observations show the synchrotron emission starting at a region offset by @xmath0 from the core @xcite . similarly , the start of power - law acceleration in compact jets would be indicated by a transition from optically thick emission , with a flat / inverted spectral index , to an optically thin power - law at a distinct location in the sed . such a break has been observed directly so far only in one source , the galactic xrb gx 339 - 4 , in the nir @xcite , during the `` hard '' or `` nonthermally dominated '' accretion state associated with compact jet formation ( see state definitions in , e.g. , * ? ? ? because of the stratification of emission regions in compact jets ( e.g. * ? ? ? * ) , the lower the frequency where this turnover occurs , the further the location along the jet where particle acceleration starts . in xrbs , a break in the nir corresponds to an offset of @xmath1 @xmath2 , and models of the broadband data of most black hole xrbs in the hard state so far seem to require such a break . the fact that the start of the acceleration region seems to occur at roughly the same location in several systems could be indicative of a critical point occurring in a magnetohydrodynamical ( mhd ) flow , particularly the magnetosonic modified fast point ( mfp ) . at the mfp the collimating magnetic field lines turn inwards towards the jet axis , potentially leading to recollimation shocks , while at the same time the flow becomes causally disconnected so shocks can occur without disrupting the flow upstream . such a shock region thus would occur at a fixed location in the flow , closely connected to the mfp , and would be an ideal location for particle acceleration to begin . we wish to investigate the feasiblity of this premiss in this paper . because of the complexity involved in relativistic mhd including strong gravity , many groups are using the results of simulations to study the formation and development of jets . these simulations often show the development of a steady outflow in which the magnetic field is remarkably self - similar and axisymmetric near the launch point ( see , e.g. , fig . 2 and fig . 11 in * ? ? ? however to study specifically the development of critical points in the flow and their dependence on external boundary conditions , current mhd simulations either do not extend far enough from the black hole , or if they do , they are too computationally expensive . assuming that the resulting flows retain a self - similar structure , at least when gravity does not dominate as indicated in the simulations , we adopt the formalism developed by ( * ? ? ? * hereafter vk03 ) . by assuming axisymmetry and a self - similar field line geometry , vk03 reduce the exact equations of special relativistic mhd to a one - dimensional problem . although vk03 focused on jets in grbs , this treatment is also applicable to other mhd jets such as in agn and microquasars . in an earlier self - similar treatment that was non - relativistic , @xcite presented a solution where the flow crosses the mfp at a finite height above the disk . in vk03 , however , they only found relativistic solutions with an mfp occurring at infinity ( meaning that the flow asymptotically approaches a perfect cylindrical geometry ) . in this paper we extend the study of vk03 and derive new solutions in which the relativistic flow crosses the mfp at a finite location above the disk . in 2 we describe the vk03 model and our method for exploring the full parameter space of solutions . in 3 we present the first relativistic solutions that pass through an mfp . in 4 we discuss our results , and the dependence of the mfp location on the model parameters . we also describe how this work sets the stage for further development to connect the flow to regions near the disk where gravity can no longer be ignored . 5 contains our conclusions . during the acceleration and collimation of jets , magnetic fields are thought to efficiently extract rotational energy from either the compact object @xcite or the accretion disk @xcite . the latter models are in the newtonian limit , with the matter considered cold , meaning there is negligible thermal pressure causing bulk acceleration to non - relativistic velocities . these cold , non - relativistic solutions were generalized to the relativistic regime by @xcite , allowing the bulk velocity to attain relativistic speeds . vk03 further extended the solutions to include the `` hot '' regime , allowing the random motions of the particles to become relativistic and thus the jets to be hydrodynamically accelerated even at the base where temperatures are high . it is this last scenario that we base this work upon . starting from the equations of time - dependent special relativistic mhd , we make the following assumptions to render them more tractable : ideal mhd , no gravitational field or external force ( and thus self - similarity holds ) , axisymmetry , a zero azimuthal electric field and time independence . following the terminology of vk03 , after scaling the equations to make them non - dimensional , we are left with two coupled differential equations ( equations and in the appendix ) . combining these two coupled differential equations , we obtain a single equation for @xmath3 ( equation , with @xmath4 the alfvnic mach number and @xmath5 the angle of the point on the field line with the axis of symmetry ) which acts like a `` wind equation '' , much akin to the wind equation of the parker solar wind model @xcite . this wind equation , along with the other algebraic equations ( see appendix ) , can be solved for the velocity , magnetic and electric field strength , density and pressure along a field line . due to the self - similar assumption , once solutions are obtained for one field line , all other field lines can be obtained by simple scaling . an example of this self - similarity and the meaning of some of the parameters used can be found in figure [ geometry ] . instead of the single critical ( or sonic ) point of the parker solar wind model , due to the inclusion of magnetic fields , the obtained wind equation has three locations where the denominator crosses zero . starting from the accretion disk , these are the modified slow point ( msp ) , the alfvn point , and the mfp . the msp and mfp are also called the slow and fast magnetosonic separatrix surfaces . the alfvn point is the location where the relativistic collimation speed of the flow towards the axis ( @xmath6 ) is given by @xmath7 where @xmath8 is the lorentz factor , @xmath9 the strength of the magnetic field , @xmath10 the cylindrical radius in units of the light cylinder radius , @xmath11 the baryon rest - mass density , and @xmath12 the specific relativistic enthalpy ( the variables are described in more detail in [ variables ] ) . the denominator of @xmath3 with the alfvn point divided out , can be expressed as @xmath13 \nonumber \\ & + \frac{u_\mathrm{s}^2}{c^2 } \frac{b_\theta^2 ( 1 - x^2)}{4 \pi \rho_0 \xi c^2},\end{aligned}\ ] ] with @xmath14 @xmath15 the velocity of light , @xmath16 the strength of the electric field , and @xmath17 the polytropic index . the msp and mfp are , by definition , the locations where @xmath18 ( vk03 ) . at every critical point , the numerator of @xmath3 should also pass through zero to ensure a smooth crossing . this translates into a regularity condition at the critical points and fixes the value of a free parameter . even though the msp should be crossed smoothly to obtain a solution that describes the entire jet from the accretion disk to the termination point , gravitational effects can not be ignored at the msp . since the equations do not include gravity ( as it is not compatible with the self - similarity assumption in relativistic flow ) , we do not try to fit for the msp . therefore we fit two critical points , and correspondingly two parameters are fixed , in our approach described below @xmath19 for the alfvn point and @xmath20 for the mfp . the physical importance of the mfp is that it is the location where not even the fastest signals can travel upstream anymore , meaning anything downstream from the mfp is causally disconnected from the region upstream . if , for example , a shock were to exist beyond the mfp , it could not disrupt the flow leading to that shock , allowing it to be a permanent feature . as mentioned above , at the mfp there is a component of the velocity heading towards the axis . this can lead to a collimation shock shortly beyond the mfp , causing the magnetic energy to be converted to particle energy and the jet to become kinetically dominated . another possibility is that the flow remains magnetically dominated , and , after reaching a minimum radius , bounces back , retaining an ordered magnetic field ( see , e.g. , * ? ? ? this is not in conflict with the statement in vk03 that the only physically acceptable case in the super - alfvnic regime is for the flow to become asymptotically cylindrical , as this statement only applies to solutions with @xmath21 . the prescription we are following from vk03 has 9 free parameters that determine the solution , whose effects are described below . following the same notation , a roman subscript a signifies the value of a variable at the alfvn point and an italic _ a _ denotes a value with respect to the poloidal magnetic flux function . the exponent @xmath22 determines the current distribution . a value @xmath21 corresponds to the current - carrying regime , with higher values of f ensuring faster collimation , but if @xmath23 we are in the return - current regime . the restriction on @xmath22 is that it can not be negative : . although we consider @xmath22 to be a free parameters , for this paper we chose to keep it fixed at 0.75 . the adiabatic index @xmath17 can have values of @xmath24 for relativistic and @xmath25 for non - relativistic solutions . @xmath26 gives the angle where the alfvn point is located with respect to the axis of symmetry . it is limited by the value of @xmath27 : . @xmath27 gives the poloidal slope of the field line at the alfvn point with respect to the accretion disk . this in turn is limited by @xmath26 : . @xmath28 is the radius squared of the alfvn point in terms of the light cylinder radius . for @xmath29 the solution becomes more force - free . the allowed values are . @xmath20 is the magnetization parameter in the monopole solution of @xcite and is related to the mass - to - magnetic flux ratio . the constraint is . @xmath30 is the dimensionless adiabatic coefficient and is constant along a field line . for a large value of @xmath30 , the specific relativistic enthalpy of the matter is high , for @xmath31 equation shows @xmath32 and the flow is cold . therefore . @xmath19 is the derivative of @xmath33 with respect to the polar angle @xmath5 at the alfvn point . accelerating flow implies . @xmath34 , with reference magnetic field @xmath35 and reference length @xmath36 , is the scaling of the solution , relating the dimensionless values to physical dimensions . we do not yet apply our solutions to specific black hole systems , so this parameter is not used here but it will be important for future applications of our solutions . the smooth crossing of the alfvn point is ensured by calculating @xmath19 from the corresponding regularity condition , given by equation . in the same way we determine @xmath20 by crossing the mfp , although this parameter sometimes can have two values ( see figure [ 3d ] ) . this leaves @xmath22 , @xmath17 , @xmath26 , @xmath27 , @xmath28 , and @xmath30 ( and @xmath34 ) to satisfy the boundary conditions at the source and , indirectly , at the end of the jet . there are also parameters derived from the above values : @xmath37 determines the total energy - to - mass - flux ratio ( @xmath38 ) and is conserved along a field line . this parameter is determined from equation and . @xmath39 is the value of the magnetization function @xmath40 , defined as the poynting - to - matter energy flux , at the alfvn point . this parameter is determined from equation . as @xmath37 can not be negative , from equation follows . and finally we describe the other variables used in the equations : @xmath41 determines the specific ( per baryon mass ) relativistic enthalpy ( @xmath12 ) . if @xmath42 we have cold , pressureless matter . @xmath41 will drop from the high temperatures at the base of the jet as matter is mainly accelerated hydrodynamically and at some point above the disk drop down to 1 . from this point on all acceleration is magnetic . this variable is determined from equation . @xmath4 is the alfvnic mach number , the velocity of the flow in terms of the alfvn velocity . @xmath43 is the radius in terms of @xmath44 and is therefore equal to 1 at the alfvn point . to find solutions with an mfp we obtained expressions for @xmath45 , given by equation , and for @xmath46 by combining the derivative of the bernoulli equation with the transfield equation using the determinant method . because @xmath46 is very unstable near the alfvn point , as the numerator has a first order zero point there and the denominator a second order one , we reverted to the bernoulli equation , given by , to determine @xmath47 . to start off the integration from the alfvn point we specify @xmath22 , @xmath17 , @xmath26 , @xmath28 , @xmath27 , @xmath30 and an initial guess for @xmath20 . we determine @xmath19 from the alfvn regularity condition , equation . integrating outward from the alfvn point , we determine whether the numerator or denominator crosses zero first and adjust @xmath20 accordingly until both cross at the same time . we then proceed to explore the range of solutions which cross both the alfvn point and mfp ( see figure [ solutionc ] ) . for plotting purposes , we have divided out the factor @xmath48 in both the numerator and denominator . as discussed above , because there is no gravity in the model , we do not try explicitly to cross the modified slow point ( msp ) . gravitational effects should play a large role close to the black hole , and the self - similar equations can not predict accurately where the msp is located . to begin our exploration of parameter space , we chose a solution from @xcite known to have an mfp ( specified below their figure 4 ) with parameters @xmath49 , @xmath50 , @xmath51 , @xmath52 , @xmath53 , @xmath54 , @xmath55 , @xmath56 and @xmath57 . by comparing terms in @xcite and vk03 , it is possible to translate these parameters to the parameters used in vk03 . because we are using the relativistic equations from vk03 , the parameters of our first solution ( see table [ parametertable ] ) differ slightly from the corresponding parameters above and we vary @xmath20 to obtain a critical solution again . from this solution we were able to traverse parameter space , while allowing only critical solutions with an mfp . by increasing @xmath28 we were able to obtain higher velocities of the jet . after achieving relativistic velocities , we focused on finding solutions with higher values of @xmath30 , as the solutions so far were cold . but for a fixed value of @xmath28 there is a maximum value of @xmath30 that produces critical solutions crossing the mfp . this is due to the fact that the collection of all solutions form a surface in the multidimensional parameter space , and we had reached a maximum for @xmath30 for the fixed values of the other parameters ( see figure [ 3d ] ) . the surfaces of valid solutions have roughly the same appearance for the explored range of @xmath26 and @xmath28 . to describe the effect of @xmath26 and @xmath28 on the solutions , we approximate the graph as a cone with the base in the @xmath58-plane and with the maximally allowed value of @xmath30 as its height . if we increase @xmath44 the base of the cone shrinks while moving to higher @xmath20 , and the height decreases . the latter limits the maximum value that @xmath28 can be increased to . if we decrease @xmath26 , the height increases , indirectly allowing higher values for @xmath28 . the area of the base becomes bigger and shifts to higher @xmath27 and @xmath20 , with the upper surface becoming steeper . due to the shape of the surface , it is possible to increase any two parameters of @xmath28 , @xmath30 and @xmath20 at the expense of the third . by extending our search to three parameters ( @xmath27 , @xmath30 and @xmath20 ) , we were able to move around this point and continue increasing @xmath30 . this revealed a multidimensional surface that is double - valued in @xmath20 . an example of this surface is shown in figure [ 3d ] , which also includes our most relativistic solution presented below , solution @xmath15 . the numerator , denominator , and acceleration of @xmath33 in this latter solution are also plotted in figure [ solutionc ] . in this section we present the various solutions crossing the mfp , from the first one that we found to one with relativistic temperature and bulk flow we sought , while describing the features particular to a certain solution . the parameters of our solutions are given in table [ parametertable ] and the main properties in figure [ overview ] . as we are not yet applying our solutions to specific black hole systems , we do not use the scaling parameters @xmath34 and @xmath36 . [ cols="<,^,^,^,^,^,^,^,^,^,^ " , ] + note.the values for the first six parameters ( @xmath22 through @xmath27 ) are exact , for the last four ( @xmath20 through @xmath37 ) they are rounded off . [ parametertable ] this solution is the closest to the non - relativistic parameter values given in @xcite that conforms to the alfvn regularity condition ( arc ) , which is the transfield equation at the alfvn point . the solution crosses the mfp at @xmath60 or @xmath61 ( @xmath62 ) . the alfvn point is located at @xmath63 or @xmath64 . the top left panel of figure [ overview ] gives the meridional projection of the magnetic field lines . the jet overcollimates after a maximum radius of almost 16 times the alfvn radius at @xmath65 or @xmath66 , shortly before the mfp . the second left panel of figure [ overview ] shows the flow is cold throughout ( @xmath67 ) due to the very small value for @xmath30 and low @xmath28 . as @xmath68 at the alfvn point , equation gives a value for @xmath69 very close to 1 . the energy of the matter @xmath41 ( including the dominant rest mass energy ) is much higher than the energy in the magnetic field throughout . therefore , even though the magnetic acceleration is efficient , the jet is not accelerated to relativistic velocities ( @xmath70 ) . the third center panel shows the `` causal connection '' opening angle @xmath71 and the opening half - angle of the outflow , which goes from @xmath64 to a few degrees overcollimation . although the causal connection opening angle has little importance for non - relativistic flows , as it remains very close to @xmath72 since @xmath73 , it is shown for completeness . after our first solution , we increased the velocity of our jet by increasing @xmath28 . as the top center panel in figure [ overview ] shows , after a long period where the field line remains almost parabolic , the jet in this solution overcollimates as well . this is caused by magnetic hoop stresses and may allow a shock region to develop beyond the mfp . the alfvn point is again located at @xmath63 and the mfp at @xmath75 or @xmath76 ( @xmath77 ) . the second center panel shows the lorentz factor and the enthalpy of the flow . the flow here also is seen to be cold ( @xmath78 ) , meaning the jet is mainly magnetically accelerated , which is again due to the small value of @xmath30 . the poynting - to - mass flux ratio ( @xmath79 ) decreases , showing magnetic energy being transferred into kinetic energy , with the flow reaching a lorentz factor of 2.8 . the bottom center panel shows the squares of the light cylinder radius , @xmath80 , and the alfvnic mach number , @xmath4 . when @xmath81 the light surface is reached , which is the radius where the field circular velocity reaches the speed of light . after having achieved a relativistic solution for the cold plasma case , we would like to find solutions with an increased flow temperature . to do so requires increasing the value of @xmath41 . as @xmath33 is given by @xmath82 at the alfvn point , equation shows that increasing @xmath28 and/or @xmath30 has the desired effect . unfortunately , for larger @xmath28 the maximum value of @xmath30 decreases . by choosing a lower value for @xmath26 the attainable values for @xmath28 and @xmath30 are increased , leading to a warmer flow . this solution is shown in the third column of figure [ overview ] . the alfvn point is located at @xmath83 or @xmath84 and the mfp at @xmath85 or @xmath86 ( @xmath87 ) . near the beginning of the flow @xmath88 . the lorentz factor of the flow at the mfp is 8.3 , which is mainly due to the high initial poynting flux . it can be seen in the second right panel of figure [ overview ] that @xmath41 always drops to 1 and from the fourth right panel that @xmath33 always dominates @xmath89 near the mfp . as the lorentz factor is given by @xmath90 the final lorentz factor is approximately @xmath37 . this means that while the jets may start as poynting flux - dominated , eventually they convert most of their poynting flux and become kinetic energy - dominated . since we are interested in the location of the mfp , we would like to know how it depends on the model parameters . as it is challenging to sample the full parameter space , we will focus on the region around the solution closest to observed jets , solution @xmath15 . by allowing all parameters to vary and looking at the effect this has on the location where the mfp occurs , we can draw the following conclusions : the mfp moves outward ( smaller @xmath5 ) when the alfvn point occurs at a smaller angle , ( lower @xmath26 ) , when the temperature at the base of the flow is increased ( higher @xmath30 ) , when the flow at the alfvn point is already close to collimation ( higher @xmath27 ) , or when the alfvn point moves closer to the light cylinder radius , making the flow more force - free ( higher @xmath44 ) . we have succeeded in obtaining new , solutions for a relativistic , magnetized flow that smoothly crosses the mfp at a finite height ( @xmath91 ) above the system equator . these solutions suggest that it should be possible to construct better , more mhd - consistent jet models where the location of the acceleration region is determined _ a priori _ from the physical boundary conditions . so far none of the solutions derived remains poynting flux - dominated up to the mfp , which is probably due to the relatively small value of @xmath20 found so far . increasing @xmath28 and especially decreasing @xmath26 will allow higher values of @xmath20 to be used . the same can also be done at the expense of @xmath30 . we also have described some of the relations between the different parameters ( see [ variables ] ) and the effect they have on each other . having a more force - free solution ( higher @xmath28 ) decreases the allowed range of temperatures ( @xmath30 ) and collimation at the alfven point ( @xmath27 ) , but at the same time allows a higher value of @xmath20 that provides a critical solution . keeping all other parameters fixed , there is a maximum value of @xmath28 for which a solution is possible at all . this maximum may be increased by moving the alfvn point closer to the disk ( smaller @xmath26 ) . this change has the additional effects of allowing a broader range for the collimation angle at the alfvn point ( @xmath27 ) while at the same time shifting this range towards higher collimation . it also allows for a higher temperature of the flow ( @xmath30 ) or a higher magnetic field strengths ( @xmath20 ) . any pair of parameters @xmath28 , @xmath30 and @xmath20 may be increased at the expense of the third . two parameters not varied so far are @xmath22 and @xmath17 . higher values for @xmath22 should ensure faster collimation and might therefore be very important for the exact location of the mfp . similarly , we may want to explore the @xmath92 case for jets with extremely relativistic temperature . however , for the weaker jets in agn and xrbs that we plan to target , the radiating particle distributions are generally thought to peak at mildly relativistic energies . if the start of the particle acceleration region in steady jets is indeed associated with the magnetosonic fast critical point in the bulk flow , then our results support the conclusion that such a region could occur at a fairly stable location about the launch point . all of the solutions found overcollimate shortly before the mfp , as would be expected for the initiation of shock development . by starting with a solution that crosses the mfp in the non - relativistic case of @xcite , we were able to extend the solution through the multidimensional parameter space towards relativistic velocities and temperatures , while retaining the critical point . this feature sets our results apart from the work of vk03 , whose formulism we adopted , in that they are immediately applicable to observed compact jet sources with an optically thick - to - thin break in the synchrotron spectrum , such as hard state xrbs and weakly accreting agn . our most promising solution is a jet outflow with mildly relativistic temperature , and lorentz factor of @xmath93 , also appropriate for the steady jets in both xrbs as well as agn . it is clear that a wide range of parameter space is left still unexplored , that can be exploited for matching physical boundary conditions appropriate to known astrophysical sources . however , before a radiative model can be constructed around the dynamical `` backbone '' provided by the solutions presented here , a prescription for including gravity must be included , to extend these solutions through the msp and allow connection with a physical model of the accretion flow / corona . we are currently working on matching these necessarily non - self similar solutions to those presented here , which will be presented in a separate work . once this solution is in place , we will have a much more physically consistent model ( compared to , e.g. , * ? ? ? * ) to use in the fitting of data from accreting black holes across the mass scale , which show compact , steady jets . as there seems to be no shortage of possible solutions , we are confident that we can match physical boundary conditions with critical solutions . p. polko and s. markoff gratefully acknowledge support from a netherlands organization for scientific research ( nwo ) vidi fellowship . in addition , s. markoff is grateful for support from the european community s seventh framework program ( fp7/2007 - 2013 ) under grant agreement number itn 215212 `` black hole universe '' . part of the research described in this paper was carried out at the jet propulsion laboratory , california institute of technology , under a contract with the national aeronautics and space administration . we would like to thank the anonymous referee for helpful comments that improved this manuscript . here we list the equations we have used for reference . see [ variables ] for a description of the parameters and variables . the transfield equation is given by @xmath98 & = ( f-1 ) \frac{x_{\mathrm{a}}^4 \mu^2 x^2}{f^2 \sigma_{\mathrm{m}}^2 } \left ( \frac{1 - g^2}{1 - m^2 - x^2 } \right)^2 \nonumber \\ & - \sin^2(\theta ) \frac{m^2 + f x^2 - f + 1}{\cos^2(\psi + \theta ) } \nonumber \\ & - \frac{x_{\mathrm{a}}^4 \mu^2 x^2}{f^2 \sigma_{\mathrm{m}}^2 m^2 } \left ( \frac{g^2 - m^2 - x^2}{1 - m^2 - x^2 } \right)^2 \nonumber \\ & + 2 \frac{\gamma - 1}{\gamma } \frac{f - 2}{f^2 \sigma_{\mathrm{m}}^2 } \frac{\xi ( \xi - 1 ) x^4}{m^2 } \label{transfield}\end{aligned}\ ] ] the magnetization function @xmath40 and the fractions at the alfvn point are given by @xmath99 we obtain @xmath100 by inserting these relations into the bernoulli equation . @xmath101 ^ 2 } \left[x_{\mathrm{a}}^2 \xi_{\mathrm{a}}^2 + \frac{f^2 \sigma_{\mathrm{m}}^2 \left(1 - x_{\mathrm{a}}^2 \right)^2 \sin^2(\theta_{\mathrm{a}})}{x_{\mathrm{a}}^2 \cos^2(\theta_{\mathrm{a}}+ \psi_{\mathrm{a } } ) } \right ] \label{mu2}\ ] ] the alfvn regularity condition is obtained by substituting these relations into the transfield equation . @xmath102 \bigg\ } \nonumber \\ = \left [ x_{\mathrm{a}}^2 - \sigma_{\mathrm{a}}(1 - x_{\mathrm{a}}^2 ) \right]^2 - ( f - 1 ) \sigma_{\mathrm{a}}^2 ( 1 - x_{\mathrm{a}}^2 ) - 2 \frac{\gamma - 1}{\gamma } ( f - 2 ) \frac{\xi_{\mathrm{a}}- 1}{\xi_{\mathrm{a } } } \left\ { x_{\mathrm{a}}^2 - \left [ x_{\mathrm{a}}^2 - \sigma_{\mathrm{a}}(1 - x_{\mathrm{a}}^2 ) \right]^2 \right\ } \label{arc}\end{aligned}\ ] ] [ windequation ] @xmath103 where @xmath104 } \right ) \bigg\ } \\ b_1 & = - 2 f^2 \sigma_{\mathrm{m}}^2 m^4 \sin^2(\theta ) \tan(\psi + \theta ) g^2 ( 1 - m^2 - x^2)^2 \\ c_1 & = x^4 \cos^2(\psi + \theta ) \mu^2 \frac{{\mathrm{d}}g^2}{{\mathrm{d}}\theta } \left [ - \left ( 1 - m^2 - x_{\mathrm{a}}^2 \right)^2 + 2 x_{\mathrm{a}}^2 \left ( g^2 - m^2 -x^2 \right ) ( 1 - x_{\mathrm{a}}^2 ) \right ] \nonumber \\ & \qquad + \frac{{\mathrm{d}}g^2}{{\mathrm{d}}\theta } ( 1 - m^2 -x^2 ) \left ( 1- m^2 - 3 x^2 \right ) \left [ \xi^2 x^4 \cos^2(\psi + \theta ) + f^2 \sigma_{\mathrm{m}}^2 m^4 \sin^2(\theta ) \right ] \nonumber \\ & \qquad + 2 f^2 \sigma_{\mathrm{m}}^2 m^4 \sin^2(\theta ) g^2 ( 1 - m^2 - x^2)^2 \left [ \frac{\cos(\theta)}{\sin(\theta ) } - \frac{1}{g^2 } \frac{{\mathrm{d}}g^2}{{\mathrm{d}}\theta } + \tan(\psi + \theta ) \right ] \\ a_2 & = -\sin^2(\theta ) \tan(\psi + \theta ) \\ b_2 & = \sin^2(\theta)\frac{1 - m^2 - x^2}{\cos^2(\psi + \theta ) } \\ c_2 & = -\sin^2(\theta)\frac{1- m^2 - x^2}{\cos^2(\psi + \theta ) } + \sin^2(\theta ) \tan(\psi + \theta ) \left [ x_{\mathrm{a}}^2 \frac{{\mathrm{d}}g^2}{{\mathrm{d}}\theta } + ( 1 - m^2 - x^2 ) \frac{1}{g } \frac{{\mathrm{d}}g}{{\mathrm{d}}\theta } \right ] \nonumber \\ & \qquad + ( f-1 ) \frac{x_{\mathrm{a}}^4 \mu^2 x^2}{f^2 \sigma_{\mathrm{m}}^2 } \left ( \frac{1 - g^2}{1 - m^2 - x^2 } \right)^2 - \sin^2(\theta ) \frac{m^2 + f x^2 - f + 1}{\cos^2(\psi + \theta ) } \nonumber \\ & \qquad - \frac{x_{\mathrm{a}}^4 \mu^2 x^2}{f^2 \sigma_{\mathrm{m}}^2 m^2 } \left ( \frac{g^2 - m^2 - x^2}{1 - m^2 - x^2 } \right)^2 + 2 \frac{\gamma - 1}{\gamma } \frac{f - 2}{f^2 \sigma_{\mathrm{m}}^2 } \frac{\xi ( \xi - 1 ) x^4}{m^2}\end{aligned}\ ] ]
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observations of relativistic jets from black holes systems suggest that particle acceleration often occurs at fixed locations within the flow .
these sites could be associated with critical points that allow the formation of standing shock regions , such as the magnetosonic modified fast point . using the self - similar formulation of special relativistic magnetohydrodynamics by @xcite
, we derive a new class of flow solutions that are both relativistic and cross the modified fast point at a finite height .
our solutions span a range of lorentz factors up to at least 10 , appropriate for most jets in x - ray binaries and active galactic nuclei , and a range in injected particle internal energy . a broad range of solutions exists , which will allow the eventual matching of these scale - free models to physical boundary conditions in the analysis of observed sources .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
it is well known that quantum key distribution is one of the most interesting subjects in quantum information science , which was pioneered by c.bennett and g.brassard in 1984[1 ] . in the original paper of bennett , single photon communication was employed as implementation of quantum key distribution . however , despite that it is not essential in great idea of bennett , many researchers employed single photon communication scheme to realize bb84 , b92[2 ] . because of the difficulties of single photon communication in practical sense , it was discussed whether one can realize a secure key distribution guaranteed by quantum nature based on light wave communication or not . in 1998 , h.p.yuen and a.kim[3 ] proposed another scheme for key distribution based on communication theory(signal detection theory ) . this scheme corresponds to an implementation of secret key sharing which was information theoretically predicted by maurer[4 ] , et al . however , yuen s idea was found independently from maurer s discussion . in the first paper of yuen - kim[3 ] , they showed that if noises of eve(eavesdropper ) and bob(receiver ) are statistically independent , secure key distribution can be realized even if they are classical noises , in which they employed a modification of b92 protocol[2 ] . following yk s first paper , a simple experimental demonstration of yk protocol based on classical noise was reported[5 ] , and recently yk scheme with 1 gbps and 10 km long fiber system based on quantum shot noise was demonstrated[6 ] . however , these schemes are not unconditional secure . that is , ability of signal detection of eve can be superior to that of bob . as a result , an interesting question arises `` is it possible to create a system with current technology that could provide a communication in which always bob s error probability is superior to that of eve ? '' in proceedings paper of qcm and c 2002 , yuen and his coworker reported that yk protocol can be unconditional secure , even if one uses conventional optical communication system[7 ] . this is interesting result for engineer , and will open a new trend of quantum cryptography . in this report , we simulate practical feature of yuen - kim protocol for quantum key distribution with unconditional secure , and propose a scheme to implement them using our former experimental setup[6 ] . a fundamental concept of yuen - kim protocol follows the next remark . + * remark * : _ if there are statistically independent noises between eve and bob , there exist a secure key distribution based on communication . _ + they emphasized that the essential point of security of the key distribution is detectability of signals . this is quite different with the principle of bb-84 , et al which are followed by no cloning theorem . that is , bb-84 and others employ a principle of disturbance of quantum states to give a guarantee of security , but yk protocol employs a principle of communication theory . it was clarified that this scheme can be realized as a modification of b-92 . however , this scheme allows us use of classical noise , and it can not provide unconditional secure . then , yuen and his coworker showed that yk scheme is to be unconditional secure in which a fundamental theorem in quantum detection theory was used for his proof of security as follows . + * theorem * : ( helstrom - holevo - yuen ) + _ signals with non commuting density operators can not be distinguished without error . _ + so if we assign non commuting density operators for bit signals 1 and 0 , then one can not distinguish without error . when the error is 1/2 based on quantum noise , there is no way to distinguish them . so we would like to make such a situation on process between alice and eve . to do so , a new version of yk scheme was given as follows : _ _ * the sender(alice ) uses an explicit key(a short key:@xmath0 , expanded into a long key:@xmath1 by use of a stream cipher ) to modulate the parameters of a multimode coherent state . * state @xmath2 is prepared . bit encoding can be represented as follows : @xmath3 where @xmath4 . * alice uses the running key @xmath1 to specify a basis from a set of m uniformly distributed two - mode coherent state . * the message @xmath5 is encoded as @xmath6 . this mapping of the stream of bits is the key to be shared by alice and bob . because of his knowledge @xmath1 , bob can demodulate from @xmath6 to @xmath5 . here , let us introduce the original discussion on the security . the ciphering angle @xmath7 could have @xmath8 in general as discrete or continuous variable determined by distribution of keys . a ciphered two mode state may be @xmath9 the corresponding density operator for all possible choices of @xmath8 is @xmath10 , where @xmath11 or @xmath12 . the problem is to find the minimum error probability that eve can achieve in bit determination . to find the optimum detection process for discrimination between @xmath13 and @xmath14 is the problem of quantum detection theory . the solution is given by[8 ] @xmath15 as an example of encoding to create @xmath16 which is the error probability of eve , yuen et al suggested certain modulation scheme . in that case , closest values of a given @xmath8 can be associated with distinct bits from the bit at position @xmath8 , and two closest neighboring states represent distinct bits which means a set of base state . in this scheme , they assumed that one chooses a set of basis state(keying state for 1 and 0 ) for bits without overlap . the error probability for density operators @xmath13 and @xmath14 becomes 1/2 , when number of a set of basis state increases . asymptotic property of the error probability depends on the amplitude of coherent state[7][9 ] . original scheme of yk protocol in the above can be realized by practical devices . to apply them to fiber communication system , we would like to realize them by intensity modulation / direct detection scheme . if one does not want to get perfect yk scheme , one can more simplify the implementation of yk protocol . from a fundamental principle in quantum detection theory , we can construct non - commuting density operators from sets based on non - orthogonal states when one does not allow overlap of the selection of a set of basis state for 1 and 0 . on the other hand , when we allow overlap for selection of a set of basis state , one can use orthogonal state to construct the same density operators for 1 and 0 . that is , @xmath17 . however , in this case , unknown factor for eve is only an initial short key , and a stream of bits that eve observed is perfectly the same as those of alice and bob , though eve can not estimate the bits at that time . this gives still insecure situation . so , here , we employ a combination of non - orthogonality and overlap selection in order to reduce the number of basis sets . let us assume that the maximum amplitude is fixed as @xmath18 . we divide it into 2 m . so we have m sets of basis state@xmath19 . total set of basis state is given as shown in fig.1 . each set of basis state is used for @xmath20 , and @xmath21 , depending on initial keys . @xmath22 so the density operators for 1 and 0 for eve are @xmath23 for the sets of @xmath24 , @xmath25,@xmath26 , let us assign 0 and 1 by the same way as eqs(4),(5 ) . in this case , eve can not get key information , but she can try to know the information of quantum states used for bit transmission . so this is the problem for discrimination of 2 m pure states . the error probability is given by @xmath27 although we have many results for calculation of optimum detection problems[10][11][12 ] , to solve this problem is still difficult at present time , because the set of states does not have complete symmetric structure . so we here give the lower bound and tight upper bound . the lower bound is given by the minimum error probability:@xmath28 for signal set @xmath29 which are neighboring states . it is given as follows : @xmath30 } \right)\ ] ] the upper bound is given by applying square root measurement for 2 m pure states . the numerical properties are shown in fig.2-(a ) . thus if m increases , then her error for information on quantum states increases . in this case , pure guessing corresponds to @xmath31 . the error probability of bob , however , is independent of the number of set of basis state , and it is given as follows : @xmath32 we emphasize that eve can not get key information in this stage , because the information for 1 and 0 are modulated by the way of eqs(4),(5 ) . furthermore , this scheme can send 2 m bits by m sets of basis state . let us apply the original scheme such that m bits are sent by m sets of basis state . in this case , eve will try to get key information , so the density operators for eve become mixed states @xmath13 , @xmath14 consisting of set of states which send 1 and 0 , respectively . the numerical properties are shown in fig.2-(b ) . both schemes have almost same security , but the latter can only send m bits by m sets of basis state . in other word , the number of sets is reduced to 1/2 in the former scheme . in implementing yk protocol by conventional fiber communication system , we use here our proposed system . figure 3 shows the experimental setup . the laser diode serves as 1.3@xmath33 m light source . a pattern generator provides a signal pulse string to send keys . a modulator which selects basis state follows a driver of laser diode . the selector gives selection of amplitude and assignment of 1 and 0 , and is controlled by initial keys . the laser driver is driven by output signals of modulator . the optical divider corresponds to eve . the case 1 is a type of opaque " , and the case 2 is a type of translucent " . the channel consists of 10 km fiber and att . we can change the distance equivalently from 10 km to 200 km by att . the speed of pulse generator to drive laser diode is 311mbps , 622mbps , and 1.2gbps . the detector of bob is ingaas pin photo loaded by 50@xmath34 register , and it is connected to an error probability counter which can apply to 12 gbps . the dark current is 7@xmath35 and the minimum received power of our system is about -30 dbm . in this system , the problem for degree of security is only power advantage of eve which will be set in near transmitter(alice ) . when the eavesdropping is opaque , the error probability of bob increases drastically , and the error probability counter shows almost 1/2 , which means that the error of eve is also 1/2 . in this case , problem of communication distance is not so important . we can detect the existence of eve in any distance of channel . when the eavesdropping is translucent , eve has to take only few power(@xmath36 from the main stream of bits sequence in order to avoid the power level disturbance . in this case , the error of bob does not increase . as a result , alice and bob can not detect the existence of eve . the secure communication distance depends on the error probabilities of bob and eve . let @xmath37 be transparency of channel from alice to bob . the detectability for bob in this experiment setup depends on the signal distance(amplitude difference between two states as basis state ) : @xmath38 for @xmath39 , and that of eve depends on the signal distance : @xmath40 . here we assume that @xmath41 , and the total loss is 20db which corresponds to 100 km . since our receiver requires about -30 dbm , the transmitter is -10 dbm . when m increases , sufficiently the error of eve increases . we examined a simulation of yk protocol based on intensity modulation / direct detection fiber communication system , and showed a design of implementation of secure system based on our experimental setup which was used to demonstrate the first version of implementation of yk protocol . we will soon report complete demonstration in experiment by the above system .
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in this report , we simulate practical feature of yuen - kim protocol for quantum key distribution with unconditional secure . in order to demonstrate them experimentally by intensity modulation / direct detection(imdd ) optical fiber communication system , we use simplified encoding scheme to guarantee security for key information(1 or 0 ) . that is , pairwise m - ary intensity modulation scheme is employed .
furthermore , we give an experimental implementation of yk protocol based on imdd .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
there has long been considerable circumstantial evidence that at least some luminous , low - redshift qsos are the result of strong interactions or mergers of galaxies ( _ e.g. , _ * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; see * ? ? ? * for a review ) . however , a concrete suggestion for an evolutionary scenario for such objects was lacking until @xcite showed that ultraluminous infrared galaxies ( uligs ) , virtually all of which are compelling examples of ongoing mergers , had bolometric luminosities and space densities similar to those of qsos . these similarities suggested the possibility that uligs are dust - enshrouded qsos which , after blowing away the dust , become classical qsos . if this hypothesis is correct , one should be able to observe examples of objects that are at intermediate stages of this evolutionary sequence . we are conducting a study of a sample of low - redshift objects that may be in such a transitionary state . these objects are recognized as bona - fide qsos and are found at an intermediate position in a far infrared ( fir ) color - color diagram between the regions occupied by typical qsos and uligs ( see fig . [ firplot ] ) . fir color color diagrams have been used as tools to detect and discriminate different types of activity in the nuclear and circumnuclear regions of galaxies . different kinds of objects such as qso / seyfert , starbursts , and powerful ir galaxies , occupy fairly well defined regions in the diagram ( see , _ e.g. , _ * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . with deep imaging and spectroscopic observations of the host galaxies , we are attempting to construct interaction histories for each of these `` transition '' objects . if strong interactions triggered the qso activity and induced starbursts , one might expect both events to occur roughly simultaneously , since both are plausibly dependent on gas flows to the inner regions . thus , we are placing these objects on an age sequence by measuring the time elapsed since the last major starburst event . this age sequence along with interaction histories can help us answer the question of whether the intermediate position of these objects is indicative of evolution from the ulig to the classical qso population , or whether it simply indicates a range of characteristics in qsos . our sample is drawn from the @xcite , @xcite , and @xcite samples of _ infrared astronomical satellite _ ( _ iras _ ) objects , and it consists of those objects which have : ( 1 ) a luminosity above the cutoff defined for quasars by @xcite , _ @xmath3 for @xmath4 kms@xmath5 mpc@xmath5 ( or @xmath6 for @xmath7 kms@xmath5 mpc@xmath5 ) , ( 2 ) a redshift @xmath8 , ( 3 ) a declination @xmath9 , ( 4 ) firm _ iras _ detections at @xmath10 m , @xmath11 m , and @xmath12 m , and ( 5 ) a position in the fir color color diagram which is intermediate between the ulig and qso loci ( fig . [ firplot ] ) . although mrk231 just misses the luminosity threshold given above , its active nucleus is known to suffer heavy extinction ( see [ mrk231 ] ) , apart from which it would clearly be a member of the sample . we know of no other objects satisfying the other criteria for which this is true . we have therefore chosen to include it for the present , although it may be approriate to exclude it from some of the analyses of the whole sample , which will be presented in a subsequent paper . so far , we have presented results for two of the nine objects in the sample : 3c48 ( * ? ? ? * hereafter cs2000 ) , an ongoing merger near the peak of starburst activity ; and pg1700 + 518 ( * ? ? ? * , hereafter cs97 and scc98 ; see also ; * ? ? ? * ) , where a tidally disturbed companion with a dominant 85 myr old post - starburst population may be in the process of merging with the host galaxy . in this paper we present the results for three additional objects : mrk1014 , iras07598 + 651 , and mrk231 . we assume @xmath7 km s@xmath5 mpc@xmath5 and @xmath13 throughout this paper , so that the projected physical length subtended by 1 is 2.43 kpc for mrk1014 , 2.26 kpc for iras07598 + 651 , and 0.77 kpc for mrk231 . spectroscopic observations for the three objects were carried out using the low - resolution imaging spectrometer ( lris ; * ? ? ? * ) on the keck ii telescope . for iras07598 + 6508 , we used a 600 groove mm@xmath5 grating blazed at 5000 yielding a dispersion of 1.28 pixel@xmath5 . for mrk231 and mrk1014 , we used a 300 groove mm@xmath5 grating blazed at 5000 with a dispersion of 2.44 pixel@xmath5 . the slit was 1 wide , projecting to @xmath145 pixels on the tektronix 2048@xmath152048 ccd . we obtained two or three exposures for each slit position , dithering along the slit between exposures . table [ journal ] shows a complete journal of observations , with specification of the slit positions , and total integration times . the spectra were reduced with iraf , using standard reduction procedures . after subtracting bias , dividing by a normalized halogen lamp flat - field frame and removing sky lines , we rectified the two - dimensional spectra and placed them on a wavelength scale using the least - mean - squares fit of cubic spline segments to identified lines in a hg - kr - ne lamp . we calibrated the spectra using spectrophotometric standards from @xcite observed with the slit at the parallactic angle . the distortions in the spatial coordinate were removed with the iraf _ apextract _ routines . for each slit position , we had two or three individual frames ; we averaged the spatially corrected spectra using the iraf task _ scombine_. we then corrected the spectra for galactic extinction , using the values given by @xcite . since we were aiming to observe the youngest populations in the host galaxies of these objects , we chose the slit positions based on previously obtained color maps of the host galaxies . we obtained imaging data for the three objects with the university of hawaii 2.2 m telescope as specified in table [ imaging ] . observations with the @xmath16 filter ( centered at 3410 with a bandpass of 320 ) sample the spectral energy distribution ( sed ) of galaxies on the short side of both the 4000 break and the balmer limit . this region is very sensitive to the age of the stellar population , and will be brightest in regions of very recent star formation ( ages @xmath17 myr ) , as well as regions with scattered qso continuum . observations in @xmath18band sample the region redwards of the balmer limit , where @xmath14a - type stellar populations are expected to peak , while longer wavelength optical and near infrared images will map the distribution of late - type stellar populations . the @xmath19 color maps will then highlight the regions of most recent star formation while @xmath20 will point to the somewhat older ( @xmath21 myr ) populations . thus our slit positions generally cover the regions brightest in these color maps . in general , the spectra from each slit position were subdivided into regions corresponding to some of the main features observed in the optical ground - based images , and one - dimensional spectra were extracted by summing pixels corresponding to the regions of interest . the spectra of regions close to the qso nucleus were contaminated by scattered qso light . the scattered light was removed by subtracting from each region a version of the quasar nuclear spectrum , scaled to match the broad - line flux . in the case of mrk1014 , none of our slit positions went through the qso , so we obtained a separate 200 s exposure of the nucleus . spectra from those slit positions which actually went through the qso nucleus suffered from strong light scattering within the spectrograph , particularly in the spectral region around h@xmath22 . therefore we were unable to obtain spectra of regions closer than @xmath23 from the qso nucleus for these slit positions . wfpc2 images of the three objects were obtained from the _ hst _ data archive . we used three 600 s wfc2 images of mrk1014 in the f675w filter , one 400 s and two 600 s pc1 images of iras07598 + 6508 in the f702w filter , and two 1100 s pc1 images of mrk231 in the f439w filter . most cosmic rays were removed by subtracting a median image from each of the individual frames , then thresholding the difference at a 3@xmath24 level , setting points above this threshold to the median of the difference image . pixels near the position of the peak of the qso were excluded from this process . the corrected difference image was then added back to the median image , giving a corrected version of the original image . the few cosmic rays within the relevant region that escaped this process were removed manually with the iraf task _ imedit_. in the case of mrk231 , where only two images were available , we first subtracted one from the other , and proceeded as above . the procedure was repeated , interchanging the images . all the corrected images for each object were then averaged . we use @xcite isochrone synthesis models to fit the spectra of the host galaxies . we will see in the following sections that the three objects which are the subject of this paper show strong evidence of having undergone some major tidal interaction . as we have described in our previous work ( cs97 , cs2000 ) , spectra of the host galaxies of such objects show features from both young ( _ e.g. , _ strong balmer lines ) and old ( _ e.g. , _ absorption ) stellar populations , and can usually be fitted satisfactorily by a two - component model . this model includes an old underlying stellar population , presumably the stellar component present prior to interaction , and a younger instantaneous burst model , presumably produced as a result of the interaction . a population with no age dispersion will be a reasonable approximation of the actual starburst as long as the period during which the star formation rate was greatly enhanced is short compared to the age of the population itself . we have also noted previously ( scc98 ) that the age of the superposed starburst is remarkably robust with respect to the different assumptions about the nature of the older stellar component . thus , we select a reasonable old underlying population ( with certain variations as described in each section below ) and assume that the same underlying population is present everywhere in the host galaxy . to this population we add instantaneous burst , @xcite initial mass function , solar metallicity models of various ages . we then perform a @xmath25 fit to the data to determine the scaling of each component and the age of the most recent starburst . the errors in the starburst ages that we quote are estimated by noting the youngest and oldest best fits for which @xmath26 changes by 15% with respect to the minimum value ( see cs2000 for details ) . in some cases , stellar absorption features ( most often the balmer lines ) were contaminated by emission coming from the extended narrow emission line region around the qso . in some cases , we subtracted a scaled synthetic spectrum of the recombination lines assuming case b. however , in calculating @xmath26 for the model fitting , we generally excluded those lines that suffered most from contamination . all spectra are displayed as observed ( _ i.e. , _ without line subtraction ) , unless otherwise specified . as we discussed in cs2000 , because of our limited spatial resolution ( generally @xmath27 ) and projection along the line of sight , we are likely observing the integrated spectrum of several starbursts of different ages , and the age we determine will be somewhat older than the youngest starbursts . therefore the ages we report should be regarded as upper limits to the most recent episodes of star formation along the line of sight . in objects with recent starbursts , the effect of reddening by dust is an obvious concern . however , studies of low - redshift agns and uligs at millimeter and submillimeter wavelengths indicate that dust is generally heavily concentrated within @xmath28 kpc of the nucleus @xcite . in addition , have found ( scc98 ) that even in a case where the optical ir spectral index is strongly affected by dust , the stellar ages from spectral features in the rest - frame 32005200 region remain fairly robust . this relative insensitivity to dust can be attributed to the fact that we are largely dealing with dust that is intermixed with the stars and that we preferentially observe regions with low extinction and , thus , low reddening . even in regions where dust along the line of sight is significant , the reddening in our observed bandpass is likely to be largely compensated by blue light scattered into our line of sight . mrk1014 ( @xmath29 ) is a luminous ( m@xmath30 ) , infrared loud ( _ e.g. , _ * ? ? ? * ) radio - quiet qso which shows a luminous host galaxy . the host galaxy has two large `` spiral - like arms '' ( @xcite ; see fig . [ mrk1014mos ] and fig . [ mrk1014hst ] ) . its spectrum indicates a mixture of old and young stars @xcite . recently , @xcite , in a spectroscopic survey of 26 rlqs , rqqs , and radio galaxies , observed the host galaxy of mrk1014 with the mayall 4 m telescope at kitt peak national observatory and with the 4.2 m william herschel telescope at la palma . they modeled the spectra and determined an age of 12 gyr for the host galaxy of mrk1014 . their approach is exactly opposite to ours ( see [ modeling ] ) : they fix the age of a possible young single starburst population to 0.1 gyr and let that of a second , older single starburst population ( what we would call the `` underlying population '' ) vary . their reason for including the 0.1 gyr population is to account for `` the spectral shape of the blue light '' which they attribute to either a recent burst of star formation or contamination of the slit by scattered light from the qso nucleus . their results will be discussed further in [ smrk1014spec ] . the host galaxy of mrk1014 shows stellar absorption features with redshifts remarkably close to those of the qso broad and narrow emission lines ( @xmath31 , as measured from our spectrum ) . we have obtained and modeled spectra of different regions in the host galaxy of mrk1014 , and we shall refer to them according to their label in fig . [ mrk1014slit ] . we have chosen a 10 gyr old population with an exponentially declining star formation rate with an e - folding time of 3 gyr as an old underlying population . this model fits reasonably well the spectrum of a galaxy 60 west - southwest of the qso at the same redshift and only slightly smaller than the `` bulge '' ( see below ) of mrk1014 . ( this galaxy was in our slit while obtaining the 200 s exposure of the qso ) . the assumption that the host galaxy ( or parent galaxies ) of mrk1014 had a similar star formation history to this galaxy need not be accurate as the precise model we use as the pre - existing population makes little difference in the age determination of the starburst population ( scc98 ) . for comparison , we have tried using a generic elliptical galaxy spectrum , and a model with a longer e - folding time ( _ i.e. , _ 5 gyr ) as underlying populations in the modeling of the spectra in mrk1014 . we obtain the same starburst ages , though slightly different flux contributions from the old population , regardless of the model used . figure [ mrk1014mos ] includes a @xmath32 color map of mrk1014 . this image emphasizes those regions with a steeper blue continuum spectrum , peaking just redwards of the balmer limit . these regions are concentrated mainly along the north edge of the tail ( regions @xmath33 and @xmath34 in fig . [ mrk1014slit ] ) , in a clump on the east end of the tail ( @xmath35 ) , directly east of the nucleus ( not covered by our slits ) , and southwest of the qso nucleus ( @xmath36 ) . spectroscopy of these regions confirms that they are indeed the youngest stellar populations we find in the host , with ages ranging from 180 myr in region @xmath36 to 290 myr in region @xmath37 . figure [ mrk1014young ] shows the spectrum of region @xmath33 , with the best @xmath26 fit of the model to the data superposed , and the relative contributions of the 200 myr starburst and the old underlying population . the error in these ages is typically @xmath38 myr . figure [ mrk1014hst ] shows compact knots at the positions of regions @xmath33 through @xmath37 , and planetary camera _ hst _ images @xcite show a larger and very bright blue knot at the position of region @xmath36 . even though there are knots at the positions of @xmath33 and @xmath34 , it is evident from colors and spectroscopy that there is recent star formation all along the north edge of the tail ( _ i.e. , _ between @xmath33 and @xmath34 ) . other regions of the host galaxy appear redder in the @xmath32 color map in fig . [ mrk1014mos ] . these regions , sampled by @xmath39 and @xmath40 , appear to be dominated by an older , @xmath28 gyr population , and to have very little , if any , contribution from the old underlying population ( fig . [ mrk1014older ] , top panel ) . it is not entirely clear whether these are truly intermediate age populations , or if they are simply , as in the case of region @xmath41 , dominated by an old underlying population with a very small contribution from a younger population like those found in regions @xmath33 through @xmath36 . we attempted to fit models with the latter characteristics to the observed spectra . the resulting fits are reasonable , with minimum values of @xmath26 10% and 25% larger than that obtained for a dominant intermediate - age population for regions @xmath39 and @xmath40 respectively . the bottom panel of fig . [ mrk1014older ] shows the spectrum of region @xmath39 with the best fit to the data of the sum of an old underlying population and a 250 myr instantaneous burst population . while a reasonable fit , it does show significant discrepancies in the region around the 4000 break and the @xmath42 line . the potential presence of this intermediate age population suggests that a better fit for the younger populations might be achieved by adding a third component that accounts for this intermediate age component . however the flux of the young starburst population in those regions is so dominant ( typically contributing 80% of the total flux at rest - frame 5000 ) , that a third component makes a negligible difference to the fit . region @xmath43 shows very strong emission , which almost certainly comes from gas ionized by the qso rather than from star - forming regions . the equivalent widths of the emission lines here are greater than those of the emission lines elsewhere in the galaxy by at least a factor of 5 , and at least twice those of the qso nucleus for [ ] . the emission line ratios indicate a power - law ionizing continuum @xcite . this region is clearly seen as a large , discrete knot in fig . 2@xmath37 of @xcite . the gas has an approaching velocity of @xmath44 km s@xmath5 with respect to the stellar absorptions in that region . the underlying spectrum is similar to that of region @xmath41 . region @xmath41 also shows emission lines with approaching velocities of @xmath45 km s@xmath5 with respect to the stellar features , but this region shows an additional , weaker emission component clearly visible in [ ] , [ ] , and h@xmath46 , blueshifted by @xmath47 km s@xmath5 with respect to the stronger component . the stellar population here seems to be dominated by an old population with a small fraction of the flux coming from younger stars . we have added spectra from two different slit positions at region @xmath48 to improve the signal to noise in this very faint region . this is part of the long extension on the west side clearly seen in the high contrast image in fig . [ mrk1014mos ] . we find a continuum with a red sed and a clear 4000 break at a redshift close to , but slightly larger than that of the main body of the galaxy . region @xmath49 , which is very bright in the @xmath50 image in fig . [ mrk1014mos ] , would appear to be an extension of the east tail . however , its spectrum shows that it is a background galaxy at @xmath51 . likewise , region @xmath52 on the west tail is a galaxy at @xmath53 . a third galaxy se of the qso ( @xmath54 , @xmath55 ) has a similar redshift as well ( @xmath56 ) , so there seems to be a group or cluster of galaxies at this redshift . the 200 s exposure spectrum of the elongated object 92 west - southwest of the qso shows narrow emission lines at the same redshift as the qso superposed on a red continuum . 2@xmath37 of @xcite shows that there is strong emission just north of this object coming from extended gas ionized by the qso and not necessarily associated with the object . the spectrum is too noisy to distinguish stellar features , so it is unclear whether this object is interacting with the system or if it is a chance projection . the near edge - on galaxy @xmath57west of the qso , on the other hand , shows [ ] emission confined to the galaxy at @xmath58 in a university of hawaii 2.2 m telescope spectrum ( canalizo 2000 , unpublished ) , as suggested by @xcite from their [ ] imaging . how do our results compare to those of @xcite ? as mentioned above , nolan et al . use a fixed 0.1 gyr starburst population . they determine that this population makes up 1.1% of the total luminous mass along the line of sight , and that the rest of the flux is well characterized by a 12 gyr instantaneous burst . in contrast , we find several regions of recent ( @xmath59 gyr ) star formation where the star forming mass typically amounts to 12% , and sometimes up to 30% , of the total luminous mass along the line of sight . one of our slit positions is very similar to one used by nolan et al.(see fig . 2 in @xcite ) , but our slit is narrower and slightly closer to the nucleus . as a way to compare , we added the flux along the slit as nolan et al . seem to have done , including the background galaxy @xmath49 , which is also in their slit . even if we add up all the flux along this slit ( subtracting the qso light , which amounted to 5% of the total flux ) , we still find that the star forming regions make up 8% of the total luminous mass along the line of sight . obviously , the difference in ages will lead to a smaller percentage for their choice of parameters . so , we tried fixing our parameters to match theirs ( _ i.e. , _ 12 gyr + 0.1 gyr populations ) , and this yields a 5% by mass for the young population , but the fit is much inferior to the ones discussed in this section . their slit position may have fortuitously missed the major star forming regions , thus leading to this smaller percentage . we have previously cautioned ( cs2000 ) that different slit positions can lead to different age determinations , and we emphasize the importance of carefully selecting slit positions if one wishes ( as we do ) to find the major starburst regions . it is also far more difficult to obtain a reliable age for an older population in the presence of a contaminating younger population than the reverse . images of mrk1014 show a very prominent tail extending to the northeast , reminiscent of the tidal tail of 3c48 ( cs2000 ) . like the tidal tail in 3c48 , this tail has a number of small clumps ( see _ hst _ image in fig . [ mrk1014hst ] ) of star formation , which are commonly found in merging systems . the bulk of the bright portion of the tail appears to be dominated by an intermediate age population ( @xmath141 gyr ) with a very small contribution , if any , from an old underlying population ; alternatively , it could be dominated by an older population , with some flux scattered from the bright star - forming regions ( @xmath33 through @xmath37 ) or fainter small regions distributed along the host . however , the north edge of the tail appears as a very sharp feature in the short wavelength images , and contains stellar populations which are as young as those of the clumps . as we find no redshift variations along this edge , it would appear that we are observing the tail nearly face - on , and that this is the leading edge where the material has been compressed , thus producing star formation . this , again , is similar to the blue leading edge observed in 3c48 ( fig . 2@xmath36 in cs2000 ) , presumably sharper in mrk1014 because of the lower inclination angle . both the ground - based and the _ hst _ images of mrk1014 ( figs . [ mrk1014mos ] and [ mrk1014hst ] ) show a long , low - surface - brightness extension of the bright tail on the east of the nucleus and arching towards the south ( more evident after the removal of the background galaxy , @xmath49 ) as well a very extended faint secondary tail , rotationally symmetric to the bright ( primary ) tail . each tail extends for as much as @xmath60 or @xmath14100 kpc ( note the inset in fig . [ mrk1014mos ] showing the well - known local interacting system m51/ngc5195 at the same scale ) . spectra of the secondary tail west of the nucleus indicates that the tail is made up by older stars , and this is consistent with this tail being visible in our @xmath61 ( not shown ) and @xmath62 images . no bright clumps of star formation are evident along the secondary tail ; this absence is not unusual as tidal dwarf formation appears not to be a ubiquitous process in mergers @xcite . the _ hst _ image of the nucleus ( see inset in fig . [ mrk1014hst ] ) shows a small extension on the south side which follows the direction of the secondary tidal tail . a similar extension is seen in _ hst nicmos _ images by @xcite . there appears to be a `` bulge '' or enhanced brightness area elongated roughly along the axis connecting the beginning of both tails ( _ i.e. , _ se ) , with a half radius of @xmath63 kpc . assuming a projected velocity of 300 km s@xmath5 on the plane of the sky , the dynamical age for the tails is @xmath64 myr . the tails are then older than every major post - starburst knot they contain , consistent with the idea that the latter were formed after the tails were first launched . at the same time , the tails are dynamically younger than the bulk of the stars that form them , though perhaps only slightly so if the stellar populations observed in @xmath39 and @xmath40 are truly @xmath28 gyr old , instead of being @xmath65 gyr with a small admixture of younger stars . mrk1014 shows some striking similarities to 3c48 ( cs2000 ) : the morphology of the ( primary ) tidal tail , the clumps of star formation along the tail as well as its blue leading edge , the relation of the starburst ages to the dynamical age of the tails ( both ages @xmath66 myr younger in 3c48 ) , and the clumpy extended emission line region @xcite with high velocity ( @xmath67 km s@xmath5 ) components . as with 3c48 , the data strongly suggest that mrk1014 is the result of a merger of two galaxies of comparable size , both of which were disks in this case . the starbursts in the main body of the host of mrk1014 , however , appear to be less intense and less widespread than those of 3c48 . if the intermediate age ( 1 gyr ) population we see in regions @xmath39 and @xmath40 is real , it may be the relic of a starburst ignited at an initial passage of the two interacting galaxies . indeed , we know of another system ( unj1025@xmath680040 ; * ? ? ? * ) where the interacting galaxies have a difference in starburst ages of this order , possibly coincident with their orbital period . if this were the case for mrk1014 , one might expect most of the star formation at the present to be very strongly concentrated towards the nucleus , as most of the gas would have been driven towards the center starting @xmath28 gyr ago . the luminous , @xmath69 , radio - quiet qso iras07598 + 6508 was first detected by _ iras _ , identified as an agn candidate by @xcite , and spectroscopically identified as a qso by @xcite . the optical spectrum of the qso is dominated by extremely strong emission @xcite , and the uv spectrum shows low- and high - ionization broad absorption lines ( bal ) extending to blueshifts of 5200 to 22000 km s@xmath70 @xcite . figures [ ir0759mos ] and [ ir0759hst ] show , respectively , ground - based and _ hst _ images of iras07598 + 6508 ( see * ? ? ? * for a psf - subtracted version of the _ hst _ image ) . these images show two clumpy regions @xmath71 south and southeast of the qso , bright in the psf subtracted @xmath16 image , but barely visible in @xmath61-band images ( not shown ) . the great number of knots in these regions , presumably ob associations , already argues for recent star formation . our slit positions cover these two regions , labeled @xmath33 and @xmath35 in fig . [ ir0759slit ] , as well as some of the fainter emission surrounding the nucleus . spectra of these regions are shown in fig . [ ir0759spec ] . regions @xmath33 and @xmath35 are fit by single starburst models of ages @xmath72 and @xmath73 myr respectively . the light from these starbursts dominates the spectra as we do not find any significant contribution from an older component . although single burst models fit the data better than two - component models , there are still some discrepancies in the fit . the observed spectra show an excess with respect to the model in the region between 3900 and 4100 , apparently because the observed continuum is steeper in this region . to test whether the discrepancy could be an indication of scattered qso light , we subtracted qso spectra with several different scalings from the stellar spectra , but we were unable to obtain better fits . in contrast to @xmath33 and @xmath35 , the spectra of the regions closer to the nucleus ( labeled @xmath37 and @xmath36 in fig . [ ir0759slit ] ) show an older stellar population ( fig . [ ir0759spec ] , bottom panel ) . the seds of these spectra are much redder and there is a clear , though not very prominent , 4000 break . we detect this old population from @xmath74 to @xmath75 north of the qso and from @xmath74 to @xmath76 south of the qso . we were unsuccessful in attempting to subtract the qso scattered light closer to the nucleus because of the strong light scattering along the slit in these regions . therefore we are unable to determine whether the populations closest to the nucleus are as old as those of @xmath37 and @xmath36 or if there might be a younger starburst concentrated around the nucleus . regions @xmath33 and @xmath35 have a slightly higher redshift than the fainter emission around the qso : @xmath77 and @xmath78 , compared to @xmath79 for the regions closer to the nucleus , but the difference is barely significant . measuring a precise redshift for the qso is difficult because of the strong emission contaminating the broad emission lines , and the absence of narrow emission lines . values in the literature include @xmath80 as measured from co @xcite and @xmath81 as measured from broad balmer emission lines @xcite ; both are within 180 km s@xmath5 ( and within the errors ) of the values we measure for the host galaxy . iras07598 + 6508 has some limited extended narrow emission as seen in the [ ] image in fig . [ ir0759mos ] . the ionized gas seems to be correlated with the stellar component : the velocity difference between emission and absorption lines in regions where both are present is never larger than 50 km @xmath82 ( notice the profile of h@xmath46 in fig . [ ir0759spec ] ; emission is visible slightly redshifted with respect to the absorption ) . region @xmath34 shows the strongest emission and appears brightest in the [ ] image . the emission line spectrum has an underlying blue continuum with stellar features . the 2-dimensional spectrum shows that this region is broken up into two discrete clumps a little over 1 each , having some small scale velocity structure . the line flux ratios indicate that the south clump has lower ionization , and it may be an region rather than gas ionized by the qso . a second , fainter component with a velocity gradient from 0 to @xmath83 km @xmath82 apparently originating from the north clump extends 1 towards the south . about 11 north of the qso , we find another emission region ( labeled @xmath40 ) which is visible in the [ ] image , but not in any of the broad band images . no stellar continuum is evident from the spectrum , either . this region is also broken into two discrete , somewhat larger clumps , but in this case the clump closer to the nucleus has lower ionization . both clumps are at a lower redshift than those of region @xmath34 , _ i.e. , _ @xmath84 _ vs. _ @xmath85 in @xmath34 . our deep @xmath50 image ( fig . [ ir0759mos ] ) shows a tidal tail extending from north to the east and arching towards the south of the nucleus for @xmath86 or @xmath87 kpc . the dynamical age of this feature ( assuming a projected velocity of 300 km @xmath82 ) is @xmath88 myr , again older than the starbursts ages found for this object . we have a very faint spectrum of this tail at region @xmath39 . the sed of the spectrum is similar to that of @xmath37 and @xmath36 , but the spectrum is too noisy to show stellar features . an [ ] @xmath89 emission line is confined to this region ( as seen in the 2-d spectrum ) at a redshift of @xmath90 , which is probably close to that of the stellar component as in other regions . this tidal tail is strongly suggestive of a merger . the fact that only one tail is evident may indicate that we are seeing the result of a merger of a spiral with an elliptical galaxy . this configuration , however , could also result from the merger of two spiral galaxies , one of which is counter rotating to the relative orbit . in mergers with such geometry , the gas and stars in the counter - rotating disk are only slightly perturbed and are not pulled into tidal bridges and tails @xcite . it is also possible that a second tail may not be evident because of projection effects . regions @xmath33 and @xmath35 may be part of the host galaxy with enhanced surface brightness due to the recent star formation , or they may be remnants of companion galaxies which have strongly interacted with the host galaxy . in either case it is clear that these regions are tidally disturbed and they will likely be completely mixed with the host within a few crossing times . the two galaxies @xmath91 south of the qso are likely to be companion galaxies since they are bright in our [ ] image ( fig . [ ir0759mos ] ) , but we do not have a spectrum to confirm this . the colors for the se galaxy indicate that it may have recent star formation , if at the same redshift as iras07598 + 6508 . we find two additional objects with the same redshift as iras07598 + 6508 that happened to fall in our slits : a faint emission line object @xmath92 south - southwest of the qso , and a bright galaxy with an absorption and emission line spectrum @xmath93 southwest of the qso . mrk231 ( @xmath94 ) , often classified as a seyfert 1 galaxy , is slightly below the luminosity cutoff for qsos defined by @xcite . however , the nucleus is heavily reddened , with an estimated @xmath95 of foreground extinction ( * ? ? ? * ; * ? ? ? * ; see also * ? ? ? * ) , so it would be well above this threshold if it were unobscured . the central plateau of the host galaxy is off center with respect to the nucleus , and the presence of tails to the north and south , as well as a low - surface - brightness extension to the east clearly indicate a recent merger ( fig . [ mrk231mos ] ) . closer to the nucleus , _ hst _ imaging shows a large number of stellar associations indicating recent star formation ( @xcite ; see fig . [ mrk231hst ] ) . as in the case of iras07598 + 651 , the spectrum of the qso is dominated by strong emission @xcite and shows a peculiar low - ionization bal system with velocities up to @xmath96 km s@xmath5 @xcite . previous spectroscopy of the host galaxy indicates the presence of a young stellar population @xcite . we have obtained and modeled spectra of several regions in the host galaxy as labeled in fig . [ mrk231slit ] . in regions @xmath49 and @xmath43 , we find no evidence for a significant young starburst population . a 10 gyr - old population with an exponentially decreasing sfr and an e - folding time of 5 gyr fits this spectrum quite well , as shown in the bottom panel of fig . [ mrk231spec ] . therefore we use this model as an underlying population to fit the rest of the spectra in the host galaxy . unfortunately , because of the poor sensitivity of lris shortwards of 4000 , and the fact that mrk231 has a lower redshift ( @xmath97 ) than the other two objects , we were unable to obtain the near - uv portion of the spectrum , which is the most helpful region in determining the ages of post - starburst populations . we are left then with the region on the long side of 3800 , where more than one combination of models can give similar fits to the spectra . indeed , we found degenerate fits in some regions of the host galaxy , particularly in those regions with the youngest populations and those where the absorption lines were heavily contaminated by emission so that we could not use balmer lines to discriminate between models . in such regions we obtained two @xmath26 minima , generally with one corresponding to a model with a small contribution ( _ i.e. , _ small percentage of total mass along the line of sight ) from a very young starburst , and the other to a model with a large contribution from a somewhat older starburst . we illustrate the problem in fig . [ degenerate ] , where we have plotted two models resulting from a 4 myr and a 42 myr old populations contributing , respectively , 1% and 12% of the total luminous mass . the models are nearly identical in the spectral region redwards of 3800 , and both are good fits to the data ( see inset in fig . [ degenerate ] ) with @xmath26 values for these models differing only by 10% . the models , however , diverge quickly at shorter wavelengths . in order to discriminate between the two `` best fits '' to the data , we have measured photometry of the different regions ( @xmath33 through @xmath49 as indicated in fig . [ mrk231slit ] ) from our ground - based optical images . we have been careful to measure fluxes only within the areas limited by the slits , at similar resolutions , and subtracting scattered light from the qso where necessary . the solid circles in fig . [ degenerate ] indicate the photometry of region @xmath33 . clearly , the older model ( red line ) is a better fit to the data . it is important to note , however , that if there is considerable reddening by dust along the lines of sight to these regions in the host galaxy , the flux values of @xmath16 will be depressed ; therefore the photometric points can only help us to determine upper limits to the ages of the stellar populations . for consistency , the age errors quoted in this section are as defined in [ modeling ] , and do not take into account additional constraints placed by the photometry . region @xmath33 corresponds to the west side of the arc - shaped structure ( the horseshoe ; * ? ? ? * ) , @xmath63 south of the qso nucleus . the _ hst _ f702w image in fig . [ mrk231hst ] shows that region @xmath33 is formed by multiple knots , much like those of regions @xmath33 and @xmath35 in iras07598 + 6508 . however this structure , unlike those of iras07598 + 6508 , has a remarkably similar morphology from the near uv to the near ir as shown in the top right panel of fig . [ mrk231mos ] . as discussed above , we determine an age of 42 ( + 22 , -17 ) myr . this age is younger than the 225 myr estimated from @xmath98 colors of this region by @xcite . while several authors have noted the blue color of the `` horseshoe '' ( _ e.g. , _ * * ; * ? ? ? * ) , we find a region that is relatively much brighter at shorter wavelengths 16south of the qso , labeled @xmath35 in fig . [ mrk231slit ] . this region appears as a bright blue extended blob dominating the @xmath19 color map in fig . [ mrk231mos ] . the spectrum has a very steep blue continuum , and very strong emission lines at the same redshift as the absorption lines , likely from an hii region @xcite . the top panel of fig . [ mrk231spec ] shows the spectrum of region @xmath35 with the balmer emission lines subtracted as described in [ modeling ] . we find an age for this region of 5 myr . as such a young starburst age is comparable to the expected duration of a typical individual starburst , and we are likely observing a collection of several such starbursts , it is virtually impossible to distinguish between continuous star formation and instantaneous bursts . the age we find is , therefore , simply indicative of ongoing star formation . as in region @xmath33 , we find degeneracy in the modeling of this spectrum , with a second @xmath26 minimum at 100 myr . the near - uv continuum of the older model , however , falls @xmath99 below the @xmath16 photometric point . neither the deep optical ground - based images nor the _ hst _ image show evidence for stellar associations in region @xmath35 . this is somewhat surprising since , although the starburst population contributes only 1% to the luminous mass along the line of sight , its flux amounts to 42% of the total flux at 5000 (rest wavelength ) , and even more at shorter wavelengths . regions @xmath34 and @xmath36 appear as single clumps in the images and have ages of 140 ( + 80 , @xmath6870 ) myr and 180 ( + 60 , @xmath6880 ) myr respectively . the starburst populations dominate the spectra in these regions , contributing @xmath100% of the total flux at rest frame 5000 and up to 36% of the total luminous mass along the line of sight . regions @xmath37 , @xmath39 ( fig . [ mrk231spec ] ) , @xmath40 , and @xmath41 all show very similar spectra , as expected from the color maps ( fig . [ mrk231mos ] ) , with ages between 300 and 360 myr . the starbursts in these regions typically contribute only @xmath1450% to the total flux and @xmath101% of the total luminous mass , except for region @xmath37 , for which these values are very similar to those of regions @xmath34 and @xmath36 . we detect very weak continuum from the north tail ( region @xmath52 ) at the @xmath102 level . the continuum is slightly blue , and there is a hint of @xmath34 at the redshift of the host galaxy . this region corresponds to the blue region in the @xmath103 color map , just before the condensation at the end of the north tail . it is possible that this region contains a knot of star formation like those found in the tail of mrk1014 . @xcite reports the presence of an extended bal in the west side of the host galaxy . the only place where we see evidence for this feature is in region @xmath48 . the spectrum of @xmath48 , uncorrected for qso scattered light , shows a wide absorption feature ( @xmath104 km s@xmath5 fwhm compared to @xmath105 km s@xmath5 for the qso bal ) as well as narrow absorption and emission ( see below ) at the redshift of the stellar component . after correcting for qso contamination , the feature appears narrower , but still at @xmath1066400 km s@xmath5 with respect to the emission line . the hypothetical bal feature is , however , much weaker than that of the qso , and it could be an artifact of an imperfect subtraction of the qso light . while the evidence for extended bal is not strong in our data , @xcite find an `` excess flux '' extending to the blue side of [ ] @xmath1076548 , which they interpret as a broad blueshifted component of h@xmath22 or h@xmath22 + [ ] indicative of an outflow with velocities up to 1500 km s@xmath5 . we observe this excess most clearly in regions @xmath41 and @xmath40 extending to even larger velocities ( @xmath108 km s@xmath5 ) . @xcite find some level of excess light in the blue wing of the h@xmath22 line in every low - ionization bal qso in their sample and suggest that this excess flux could be from the bal material itself . the narrow emission line shows a peculiar behavior . we observe narrow emission in regions @xmath41 and @xmath48 on the red side of the absorption line , forming what looks like a p - cygni profile . the only other emission lines evident in region @xmath41 are [ ] @xmath1076583 , and weak [ ] @xmath1096717,6731 , but these lines are slightly _ blueshifted _ with respect to the absorption lines . in region @xmath40 the emission line disappears , but the absorption line becomes very weak as well , so it is possible that the emission line is blueshifted into the absorption line ; [ ] , however , is at a higher redshift than in region @xmath41 . thus , the very low ionization gas appears to be decoupled from the moderately low ionization gas in these regions . @xcite find a `` green '' ( _ i.e. , _ visible only in their @xmath110 image ) , jet - like feature extending from near the nucleus to the northeast . they suggest it `` may be line emission in [ ] , perhaps tracing ionizing radiation originating in the nucleus '' . our slit goes through the western side of where this feature would be ( see fig . 2 in * ) , and region @xmath48 should sample the brightest part of the ridge leading to the bright knot . however , we find only weak [ ] in the spectrum of this region , certainly much weaker than in other regions we sample . therefore it is unlikely that this `` green '' feature , if real , is due to ionized gas . images of mrk231 show a greatly disturbed host galaxy both in large and small scales ( fig . [ mrk231mos ] ) . at large scales , two symmetric tails extend on the east side of the nucleus for @xmath111 or @xmath112 kpc each , and there is some very extended low surface brightness material east of the north tail . the nucleus is off - center on a bright `` plateau '' @xcite 20 kpc across that shows complex morphology , including linear jet - like structures ( region @xmath40 ) , and curved tail - like structures ( region @xmath37 ) . closer to the nucleus ( fig . [ mrk231hst ] ) we find the `` horseshoe '' 3 kpc to the south with numerous clumps of star formation , and arm - like features spiraling around the nucleus which connect to the outer tidal tails @xcite this morphology is indicative of a merger between two disk galaxies of similar mass . while we are unable to obtain stellar spectra within the central kpc ( @xmath113 ) of mrk231 , @xcite have found from uv spectropolarimetry of the nuclear region that high polarization in the optical falls off quite rapidly shortward of @xmath114 ; this effect is most easily interpreted as a dilution of the polarized component by o and b stars from an ongoing starburst @xcite . co and radio observations also show evidence for a centrally concentrated starburst @xcite . it is possible that the knots along the arm - like features around the nucleus have ages similar to those of region @xmath33 , if not younger . we find a relatively wide range of starburst ages around the host galaxy . if , once again , we assume a projected velocity of 300 km s@xmath5 , the tidal tails have a dynamical age of 110 myr . mrk231 is unusual in that there seems to be a significant post - starburst population somewhat older ( 300360 myr ) than the tails , which may indicate that wide spread star formation was ignited prior to the stages of final merger . numerical simulations of mergers @xcite predict such a scenario when the merging galaxies lack a significant bulge to stabilize them against an early dissipation . in cs2000 we noted that the apparent correlation between bulge mass and black - hole mass @xcite suggests a possible correlation between qso luminosity and the delay in star formation activity . thus , the early starburst activity in mrk231 may indicate that the host galaxy did not have a substantial bulge , which would in turn imply a less massive black - hole that may result into a less luminous agn . although we do in fact observe a relatively less luminous agn in mrk231 , these connections are highly speculative , and they are based on relations @xcite and models @xcite that still have many uncertainties . mrk1014 , iras07598 + 6508 , and mrk231 show many similarities . we already knew that these three objects , while being optically or ir selected qsos , were also part of the ulig family . in this study we have found additional properties which they share and which may be related to their intermediate position in the fir diagram . while mrk1014 and mrk231 have long been known to have tails , we have shown that iras 07598 + 6508 also has at least one tidal tail . we have found evidence that all three objects have undergone a strong interaction and are now in the final stages of mergers . the morphology of the hosts ( highly perturbed galaxies with tidal tails and destroyed disks ) as well as the extent of the starbursts indicate that these are all major mergers between galaxies of comparable mass rather than accretion events of low mass dwarf companions @xcite . all three objects show spectra typical of e+a galaxies @xcite , that is , spectra characterized by the simultaneous presence of strong balmer absorption lines indicative of a young stellar population and features from an older population such as , and the absence of strong emission lines typical of star forming galaxies ( we reiterate that the emission lines seen in our spectra generally come from extended gas ionized by the qso rather than from star forming regions , with a few exceptions ) . e+a galaxies are frequently linked to galaxy - galaxy mergers and interactions ( _ e.g. , _ * ? ? ? * ) . in the case of objects with e+a spectra undergoing tidal interactions , the young superposed population is clearly related to the interaction . the three objects discussed in this paper , along with 3c48 ( cs2000 ) and pg1700 + 518 ( cs97 ) , all have interaction - induced starbursts . in addition to the similarities noted in the host galaxies of the three objects , there are also some common characteristics in the spectra of the active nuclei themselves . all three objects show strong emission , with two of these objects ( iras07598 + 6508 and mrk231 ) being `` extreme '' emitters ( _ i.e. , _ showing ratios of @xmath115(feii @xmath116)/@xmath115(h@xmath46 ) @xmath117 2 ; @xcite ) . the latter two objects , like pg1700 + 518 ( cs97 ; scc98 ) , are also low - ionization bal qsos and have weak or absent narrow emission lines . how do the ages of these starbursts relate to the merger / interaction stage ? the dynamical ages for tidal features in these objects are uncertain because of projection effects . however , our rough estimates indicate that in two cases , mrk1014 and iras07598 + 6508 , the peak of the starburst occurred after the tidal tails were launched . this requires some mechanism to stabilize the gas contents of the galaxies against bursting in star formation until the later stages of the merger ; numerical simulations indicate that a significant bulge in the host can provide this mechanism @xcite . mrk231 , on the other hand , shows starburst ages which indicate that much of the star formation activity may have started before the last stages of the merger ; hence , one or both of the merging galaxies may have lacked a significant bulge . whenever we have found a significant range of starburst ages in the host galaxies of qsos , we have also found evidence that the youngest major starburst regions are preferentially concentrated towards the center of the galaxy . in 3c48 ( cs2000 ) we found the clearest example of this , with starburst ages becoming progressively younger and more dominating as we approached the galaxy / qso nucleus . mrk231 shows older starbursts in the `` plateau '' extending @xmath118 kpc around the nucleus , with some of the youngest populations only 3 kpc from the nucleus , and possibly even younger populations in the central kpc of the galaxy . mrk1014 shows stronger relatively recent star formation activity along the tail than any of the other objects . however , the starburst region we observe within 2 kpc of the nucleus is far more massive , luminous , and larger @xcite than the starburst regions along the tail , and has the youngest age found in the host . the interaction histories of the five objects discussed so far clearly favor a strong connection between interactions and vigorous bursts of star formation . since the gas flows towards the inner regions can not only trigger star formation but also serve as fuel to the qso , one might expect the age of the qso activity to be closely related to the age of the initial starbursts in the central regions of the galaxy . however , observations of central starbursts are in every case hampered by the presence of the qso ; even if this region could be observed , the spectrum would likely be dominated by continuing recent starburst activity and not by a starburst that was coincident with the onset of the qso activity . furthermore , both starburst and qso activity may be episodic . all of this is to say that there is an unavoidable intrinsic uncertainty in using starburst ages to place qsos in an evolutionary sequence . we defer the detailed discussion of a possible age sequence to a subsequent paper where we will present the results of the four remaining objects in our sample ( _ i.e. , _ iras00275@xmath682859 , iras04505@xmath682958 , izw1 , and pg1543 + 489 ) . we emphasize , though , that many of our general conclusions from the objects discussed here , as well as from our observations of 3c48 ( cs2000 ) , have an interest that is quite independent of any attempt to use some sort of starburst age as a proxy for a qso age . ( 1 ) the confirmation that these are all starburst or post - starburst objects and that they all show obvious tidal tails validates their close connection with other uligs , virtually all of which are mergers or strongly interacting pairs . while there has long been strong _ circumstantial _ evidence that a large fraction of the qso population has resulted from triggering of the qso activity by interactions and mergers , we now have much more direct evidence for this mechanism for at least one subclass of qsos . ( 2 ) both the spatial distribution and the time history of star formation in a qso host galaxy give clues to nature of the galaxies that have participated in the merger . while more sophisticated models of star formation during interactions will be necessary to exploit these data fully , we already have some hints in terms of the enhanced star formation along the leading edge of the tails in 3c48 and mrk1014 , and in the relative ages of the tail structures and the star - forming regions contained within them . ( 3 ) the youth of the stellar populations in these objects reinforces previous suggestions connecting strong emission and low - ionization bal features with the relatively recent triggering of qso activity . we will discuss these connections in detail in the paper presenting the observations of the remaining four objects in our sample . we thank gerbs bauer , scott dahm , and susan ridgway for assisting in some of the observations , and bill vacca for helpful discussions about imfs . we also thank the referee , dean hines , for his very prompt review of the paper and his suggestions , which helped us improve both its content and its presentation . this paper was partly written while both authors were visitors at the research school of astronomy and astrophysics of the australian national university , and we thank both the director , jeremy mould , and the staff there for their hospitality . this research has made use of the nasa / ipac extragalactic database ( ned ) which is operated by the jet propulsion laboratory , california institute of technology , under contract with the national aeronautics and space administration . this research was partially supported by nsf under grant ast95 - 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ph/0005122 ] taniguchi , t. , kawara , k. , nishida , m. , tamura , s. , nishida , m. t. 1988 , , 95 , 1378 taylor , g. b. , silver , c. s. , ulvestad , j. s. , & carilli , c. l. 1999 , , 519 , 185 toomre , a. , & toomre , j. 1972 , , 178 , 623 veilleux , s. & osterbrock , d. e. 1987 , , 63 , 295 zabludoff , a. i. et al . 1996 , , 466 , 104 lclccc mrk1014 & 103.0 & 4.0 n & 2.44 & 3600 & 97 sep 13 + mrk1014 & 87.1 & 6.9 n & 2.44 & 3600 & 98 sep 01 + mrk1014 & 148.6 & 2.2 w & 2.44 & 2400 & 98 sep 01 + iras07598 + 6508 & 243.6 & 6.9 s & 1.28 & 3600 & 96 oct 13 + iras07598 + 6508 & 190.0 & 0.0 & 1.28 & 3600 & 96 oct 14 + mrk231 & 6.0 & 0.0 & 2.44 & 1800 & 97 jun 12 + mrk231 & 83.0 & 16.0 s & 2.44 & 3600 & 98 mar 21 + mrk231 & 30.0 & 7.0 w & 2.44 & 2400 & 98 mar 21 + llcccc mrk1014 & orbit ccd & @xmath16 & 8@xmath151200 & 0.138 & 97 nov 01 + mrk1014 & orbit ccd & @xmath119 & 5@xmath15300 & 0.138 & 97 oct 31 + mrk1014 & tek1024 ccd & @xmath50 & 5@xmath15300 & 0.222 & 91 nov 13 + mrk1014 & quirc & @xmath61 & 29@xmath15300 & 0.061 & 99 oct 29 + mrk1014 & nicmos-3 & @xmath62 & 39@xmath1570 & 0.374 & 90 oct 28 + iras07598 + 6508&loral ccd&@xmath16 & 8@xmath151200 & 0.138 & 97 mar 09 + iras07598 + 6508&tek1024 ccd&@xmath50 & 15@xmath15300 & 0.222 & 92 mar 01 + iras07598 + 6508&tek1024 ccd&[oiii ] & 7@xmath151560 & 0.222 & 92 feb 27 + iras07598 + 6508 & quirc & @xmath61 & 22@xmath15120 & 0.061 & 99 oct 30 + mrk231 & loral ccd & @xmath16 & 5@xmath151200 & 0.138 & 97 mar 08 + mrk231 & loral ccd & @xmath120 & 18@xmath15200 & 0.138 & 97 mar 09 + mrk231 & loral ccd & @xmath121 & 6@xmath15300 & 0.138 & 97 mar 08 + mrk231 & quirc & @xmath61 & 69@xmath1520 & 0.061 & 99 apr 04 +
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we present deep spectroscopic and imaging data of the host galaxies of mrk1014 , iras07598 + 6508 , and mrk231 .
these objects form part of both the qso and the ultraluminous infrared galaxy ( ulig ) families , and may represent a transition stage in an evolutionary scenario .
our imaging shows that all three objects have highly perturbed hosts with tidal tails and destroyed disks , and appear to be in the final stages of major mergers .
the host galaxies of the three objects have spectra typical of e+a galaxies , showing simultaneously features from an old and a young stellar component .
we model spectra from different regions of the host galaxies using @xcite spectral synthesis models using two component models including an old underlying population and recent superposed starbursts .
mrk1014 has intense star formation concentrated in a large knot @xmath0 kpc from the nucleus , along the leading edge of the tidal tail , and in several knots scattered around the host .
the starburst ages in these regions range from 180 to 290 myr .
iras 07598 + 6508 has multiple knots of star formation concentrated in two regions within 16 kpc of the qso nucleus , with ages ranging from 30 to 70 myr ; the host galaxy shows an older population in other regions .
mrk231 shows a wider range of starburst ages , ranging from 42 myr in the arc 3 kpc south of the nucleus , to @xmath1 myr spread on a `` plateau '' @xmath2 kpc across around the nucleus , as well as a uv bright region 12 kpc south of the nucleus , which is apparently a region of currently active star formation .
our results indicate a strong connection between interactions and vigorous bursts of star formation in these objects .
we propose that the starburst ages found are indicative of young ages for the qso activity .
the young starburst ages found are also consistent with the intermediate position of these objects in the far infrared color - color diagram .
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the theory of games provides a general structure within which both cooperation and competition among independent entities may be modeled , and provides powerful tools for analyzing these models . applications of this theory have fundamental importance in many areas of science . this paper considers games in which the players may exchange and process quantum information . we focus on competitive games , and within this context the types of games we consider are very general . for instance , they allow multiple rounds of interaction among the players involved , and place no restrictions on players strategies beyond those imposed by the theory of quantum information . while classical games can be viewed as a special case of quantum games , it is important to stress that there are fundamental differences between general quantum games and classical games . for example , the two most standard representations of classical games , namely the _ normal form _ and _ extensive form _ representations , are not directly applicable to general quantum games . this is due to the nature of quantum information , which admits a continuum of pure ( meaning extremal ) strategies , imposes bounds on players knowledge due to the uncertainty principle , and precludes the representation of general computational processes as trees . in light of such issues , it is necessary to give special consideration to the incorporation of quantum information into the theory of games . a general theory of quantum games has the potential to be useful in many situations that arise in quantum cryptography , computational complexity , communication complexity , and distributed computation . this potential is the primary motivation for the work presented in this paper , which we view as a first step in the development of a general theory of quantum games . the following facts are among those proved in this paper : * every multiple round quantum strategy can be faithfully represented by a single positive semidefinite operator acting only on the tensor product of the input and output spaces of the given player . this representation is a generalization of the choi - jamiokowski representation of super - operators . the set of all operators that arise in this way is precisely characterized by the set of positive semidefinite operators that satisfy a simple collection of linear constraints . * if a multiple round quantum strategy calls for one or more measurements then its representation consists of one operator for each of the possible measurement outcomes . the probability of any given pair of measurement outcomes for two interacting strategies is given by the inner product of their associated operators . * the maximum probability with which a given strategy can be forced to output a particular result is the _ minimum _ value of @xmath1 for which the positive semidefinite operator corresponding to the given measurement result is bounded above ( with respect to the lwner partial order ) by the representation of a valid strategy multiplied by @xmath1 . we give the following applications of these facts : * a new and conceptually simple proof of kitaev s bound for strong coin - flipping , which states that every quantum strong coin - flipping protocol allows a bias of at least @xmath2 . * the exact characterization @xmath0 of the class of problems having quantum refereed games ( i.e. , quantum interactive proof systems with two competing provers ) . this establishes that quantum and classical refereed games are equivalent in terms of expressive power : @xmath3 . it is appropriate for us to comment on the relationship between the present paper and a fairly large collection of papers written on a topic that has been called _ quantum game theory_. meyer s _ pq penny flip _ game @xcite is a well - known example of a game in the category these papers consider . the work of eisert , _ et al . _ @xcite is also commonly cited in this area . some controversy exists over the interpretations drawn in some quantum these papers see , for instance , refs . @xcite . a key difference between our work and previous work on quantum game theory is that our focus is on multiple - round interactions . understanding the actions available to players that have quantum memory is therefore critical to our work , and to our knowledge has not been previously considered in the context of quantum game theory . a second major difference is that , in most of the previous quantum game theory papers we are aware of , the focus is on rather specific examples of classical games and on identifying differences that arise when so - called quantum variants of these games are considered . as a possible consequence , it may arguably be said that none of the results proved in these papers has had sufficient generality to be applicable to any other studies in quantum information . in contrast , our interest is not on specific examples of games , but rather on the development of a general theory that holds for all games . it remains to be seen to what extent our work will be applied , but the applications that we provide suggest that it may have interesting uses in other areas of quantum information and computation . a different context in which games arise in quantum information theory is that of _ nonlocal games _ @xcite , which include _ pseudo - telepathy games _ @xcite as a special case . these are cooperative games of incomplete information that model situations that arise in the study of multiple - prover interactive proof systems , and provide a framework for studying bell inequalities and the notion of nonlocality that arises in quantum physics . while such games can be described within the general setting we consider , we have not yet found an application of the methods of the present paper to this type of game . possibly there is some potential for further development of our work to shed light on some of the difficult questions in this area . this section gives a brief overview of various quantum information - theoretic notions that will be needed for the remainder of the paper . we assume the reader has familiarity with quantum information theory , and intend only that this overview will serve to establish our notation and highlight the main concepts that we will need . readers not familiar with quantum information are referred to the books of nielsen and chuang @xcite and kitaev , shen and vyalyi @xcite . when we speak of the vector space associated with a given quantum system , we are referring to some complex euclidean space ( by which we mean a finite - dimensional inner product space over the complex numbers ) . such spaces will be denoted by capital script letters such as @xmath4 , @xmath5 , and @xmath6 . we always assume that an orthonormal _ standard basis _ of any such space has been chosen , and with respect to this basis elements of these spaces are associated with column vectors , and linear mappings from one space to another are associated with matrices in the usual way . we will often be concerned with finite sequences @xmath7 of complex euclidean spaces . we then define @xmath8 for nonnegative integers @xmath9 , and define @xmath10 for @xmath11 . it is convenient to define various sets of linear mappings between given complex euclidean spaces @xmath4 and @xmath5 as follows . let @xmath12 denote the space of all linear mappings ( or _ operators _ ) from @xmath4 to @xmath5 , and write @xmath13 as shorthand for @xmath14 . we write @xmath15 to denote the set of hermitian operators acting on @xmath4 , @xmath16 to denote the set of all positive semidefinite operators acting on @xmath4 , and @xmath17 to denote the set of all density operators on @xmath4 ( meaning positive semidefinite operators having trace equal to 1 . ) an operator @xmath18 is a _ linear isometry _ if @xmath19 . the existence of a linear isometry in @xmath12 of course requires that @xmath20 , and if @xmath21 then any linear isometry @xmath18 is unitary . we let @xmath22 denote the set of all linear isometries from @xmath4 to @xmath5 . the operator @xmath23 denotes the identity operator on @xmath4 . transposition of operators is always taken with respect to standard bases . the hilbert - schmidt inner product on @xmath13 is defined by @xmath24 for all @xmath25 . for given operators @xmath26 , the notation @xmath27 means that @xmath28 . this relation is sometimes called the _ lwner partial order _ on @xmath15 . when we refer to _ measurements _ , we mean povm - type measurements . formally , a measurement on a complex euclidean space @xmath4 is described by a collection of positive semidefinite operators @xmath29 satisfying the constraint @xmath30 here @xmath31 is a finite , non - empty set of _ measurement outcomes_. if a state represented by the density operator @xmath32 is measured with respect to such a measurement , each outcome @xmath33 results with probability @xmath34 . a _ super - operator _ is a linear mapping of the form @xmath35 where @xmath4 and @xmath5 are complex euclidean spaces . a super - operator of this form is said to be _ positive _ if @xmath36 for every choice of @xmath37 , and is _ completely positive _ if @xmath38 is positive for every choice of a complex euclidean space @xmath6 . the super - operator @xmath39 is said to be _ admissible _ if it is completely positive and preserves trace : @xmath40 for all @xmath41 . admissible super - operators represent discrete - time changes in quantum systems that can , in an idealized sense , be physically realized . the choi - jamiokowski representation @xcite of super - operators is as follows . suppose that @xmath42 be a given super - operator and let @xmath43 be the standard basis of @xmath4 . then the _ choi - jamiokowski representation _ of @xmath39 is the operator @xmath44 it holds that @xmath39 is completely positive if and only if @xmath45 is positive semidefinite , and that @xmath39 is trace - preserving if and only if @xmath46 . for two complex euclidean spaces @xmath4 and @xmath5 , we define a linear mapping @xmath47 by extending by linearity the action @xmath48 on standard basis states . we make extensive use of this mapping in some of our proofs , as it is very convenient in a variety of situations . let us now state some identities involving the @xmath49 mapping , each of which can be verified by a straightforward calculation . [ prop : identities ] the following hold : 5 mm for any choice of @xmath50 , @xmath51 , and @xmath52 for which the product @xmath53 makes sense we have @xmath54 [ item : trace ] for any choice of @xmath55 we have @xmath56 for any choice of @xmath57 we have @xmath58 [ item : choi ] let @xmath59 and suppose @xmath60 is given by @xmath61 for all @xmath41 . then @xmath62 for any non - empty set @xmath63 of hermitian operators , the _ polar _ of @xmath64 is defined as @xmath65 and the _ support _ and _ gauge _ functions of @xmath64 are defined as follows : @xmath66 these functions are partial functions in general , but it is typical to view them as total functions from @xmath15 to @xmath67 in the natural way . for any set @xmath64 of positive semidefinite operators , we denote @xmath68 [ prop : polar ] let @xmath4 be a complex euclidean space and let @xmath64 and @xmath69 be non - empty subsets of @xmath15 . then the following facts hold : 1 . [ item : polar - contain ] if @xmath70 then @xmath71 . [ item : polar - pos ] if @xmath72 for each @xmath37 then @xmath73 . [ item : polar - polar ] if @xmath64 is closed , convex , and contains the origin , then the same is true of @xmath74 . in this case we have @xmath75 , @xmath76 the first two items in the above proposition are elementary , and a proof of the third may be found in rockafellar @xcite . in this section we define the notions of a quantum _ strategy _ and the _ choi - jamiokowski representation _ of quantum strategy . the remainder of the paper is concerned with the study of these objects and their interactions . we begin with our definition for quantum strategies , which we will simply call _ strategies _ given that the focus of the paper is on the quantum setting . [ def : strategy ] let @xmath77 and let @xmath78 and @xmath79 be complex euclidean spaces . @xmath80-turn non - measuring strategy _ having input spaces @xmath78 and output spaces @xmath79 consists of : 5 mm complex euclidean spaces @xmath81 , which will be called _ memory spaces _ , and an @xmath80-tuple of admissible mappings @xmath82 having the form @xmath83 an _ @xmath80-turn measuring strategy _ consists of items 1 and 2 above , as well as : 5 mm a measurement @xmath84 on the last memory space @xmath85 . we will use the term _ @xmath80-turn strategy _ to refer to either a measuring or non - measuring @xmath80-turn strategy . figure [ fig : strategy ] illustrates an @xmath80-turn non - measuring strategy . ( 23734,2939)(-600,1700 ) ( 18612,4212)(19812,4212)(19812,3012)(18612,3012)(18612,4212 ) ( 1812,4212)(3012,4212)(3012,3012)(1812,3012)(1812,4212 ) ( 5412,4212)(6612,4212)(6612,3012)(5412,3012)(5412,4212 ) ( 9012,4212)(10212,4212)(10212,3012)(9012,3012)(9012,4212 ) ( 2412,3612)(0,0)@xmath86 ( 6012,3612)(0,0)@xmath87 ( 9612,3612)(0,0)@xmath88 ( 19212,3612)(0,0)@xmath89 ( 16512,3612)(18312,3612 ) ( 18072.000,3552.000)(18312.000,3612.000)(18072.000,3672.000 ) ( 18072.000,3552.000 ) ( 17712,1512)(18912,2712 ) ( 18784.721,2499.868)(18912.000,2712.000)(18699.868,2584.721 ) ( 18784.721,2499.868 ) ( 19512,2712)(20712,1512 ) ( 20499.868,1639.279)(20712.000,1512.000)(20584.721,1724.132 ) ( 20499.868,1639.279 ) ( 912,1512)(2112,2712 ) ( 1984.721,2499.868)(2112.000,2712.000)(1899.868,2584.721 ) ( 1984.721,2499.868 ) ( 2712,2712)(3912,1512 ) ( 3699.868,1639.279)(3912.000,1512.000)(3784.721,1724.132 ) ( 3699.868,1639.279 ) ( 4512,1512)(5712,2712 ) ( 5584.721,2499.868)(5712.000,2712.000)(5499.868,2584.721 ) ( 5584.721,2499.868 ) ( 6312,2712)(7512,1512 ) ( 7299.868,1639.279)(7512.000,1512.000)(7384.721,1724.132 ) ( 7299.868,1639.279 ) ( 8112,1512)(9312,2712 ) ( 9184.721,2499.868)(9312.000,2712.000)(9099.868,2584.721 ) ( 9184.721,2499.868 ) ( 9912,2712)(11112,1512 ) ( 10899.868,1639.279)(11112.000,1512.000)(10984.721,1724.132 ) ( 10899.868,1639.279 ) ( 3312,3612)(5112,3612 ) ( 4872.000,3552.000)(5112.000,3612.000)(4872.000,3672.000 ) ( 4872.000,3552.000 ) ( 6912,3612)(8712,3612 ) ( 8472.000,3552.000)(8712.000,3612.000)(8472.000,3672.000 ) ( 8472.000,3552.000 ) ( 10512,3612)(12312,3612 ) ( 12072.000,3552.000)(12312.000,3612.000)(12072.000,3672.000 ) ( 12072.000,3552.000 ) ( 20112,3612)(21912,3612 ) ( 21672.000,3552.000)(21912.000,3612.000)(21672.000,3672.000 ) ( 21672.000,3552.000 ) ( 15012,3612 ) ( 15012,3612 ) ( 13212,3612 ) ( 13212,3612 ) ( 14112,3612 ) ( 14112,3612 ) ( 15912,3612 ) ( 15912,3612 ) ( 1112,2112)(0,0)[b]@xmath90 ( 4712,2112)(0,0)[b]@xmath91 ( 8312,2112)(0,0)[b]@xmath92 ( 17912,2112)(0,0)[b]@xmath93 ( 3662,2112)(0,0)[b]@xmath94 ( 7262,2112)(0,0)[b]@xmath95 ( 10862,2112)(0,0)[b]@xmath96 ( 20612,2112)(0,0)[b]@xmath97 ( 4212,4012)(0,0)@xmath98 ( 7862,4012)(0,0)@xmath99 ( 11462,4012)(0,0)@xmath100 ( 17462,4012)(0,0)@xmath101 ( 21012,4012)(0,0)@xmath85 although there is no restriction on the dimension of the memory spaces in a quantum strategy , it is established in the proof of theorem [ theorem : characterization ] that every measuring strategy is equivalent to one in which @xmath102 for each @xmath103 . we also note that our definition of strategies allows the possibility that any of the input or output spaces is equal to @xmath104 , which corresponds to an empty message . one can therefore view simple actions such as the preparation of a quantum state or performing a measurement without producing a quantum output as special cases of strategies . when we say that an @xmath80-turn strategy is described by _ linear isometries _ @xmath105 , it is meant that the admissible super - operators @xmath106 defining the strategy are given by @xmath107 for @xmath108 . notice that , when it is convenient , there is no loss of generality in restricting ones attention to strategies described by linear isometries in this way . this is because every admissible super - operator can be expressed as a mapping @xmath109 for some linear isometry @xmath50 , followed by the partial trace over some `` garbage '' space that represents a tensor factor of the space to which @xmath50 maps . by including the necessary `` garbage '' spaces as tensor factors of the memory spaces , and therefore not tracing them out , there can be no change in the action of the strategy on the input and output spaces . along similar lines , there is no loss of generality in assuming that a given measuring strategy s measurement is projective . a given @xmath80-turn strategy expects to interact with something that provides the inputs corresponding to @xmath78 and accepts the strategy s outputs corresponding to @xmath79 . let us define an @xmath80-turn _ co - strategy _ to be the sort of object that a strategy interfaces with in the most natural way . let @xmath77 and let @xmath78 and @xmath79 be complex euclidean spaces . the spaces @xmath78 are viewed as the input spaces of some @xmath80-turn strategy while @xmath79 are to be viewed as its output spaces . @xmath80-turn non - measuring co - strategy _ to these spaces consists of : 5 mm complex euclidean _ memory _ spaces @xmath110 , a density operator @xmath111 , and an @xmath80-tuple of admissible mappings @xmath112 having the form @xmath113 an _ @xmath80-turn measuring co - strategy _ consists of items 1 , 2 and 3 above , as well as : 5 mm a measurement @xmath114 on the last memory space @xmath115 . as for strategies , we use the term _ @xmath80-turn co - strategy _ to refer to either a measuring or non - measuring @xmath80-turn co - strategy . figure [ fig : interaction ] represents the interaction between an @xmath80-turn strategy and co - strategy . ( 23734,4539)(-600,200 ) ( 20412,1212)(21612,1212)(21612,12)(20412,12)(20412,1212 ) ( 18612,4212)(19812,4212)(19812,3012)(18612,3012)(18612,4212 ) ( 1812,4212)(3012,4212)(3012,3012)(1812,3012)(1812,4212 ) ( 3612,1212)(4812,1212)(4812,12)(3612,12)(3612,1212 ) ( 5412,4212)(6612,4212)(6612,3012)(5412,3012)(5412,4212 ) ( 7212,1212)(8412,1212)(8412,12)(7212,12)(7212,1212 ) ( 9012,4212)(10212,4212)(10212,3012)(9012,3012)(9012,4212 ) ( 10812,1212)(12012,1212)(12012,12)(10812,12)(10812,1212 ) ( 12,1212)(1212,1212)(1212,12)(12,12)(12,1212 ) ( 612,612)(0,0)@xmath116 ( 4212,612)(0,0)@xmath117 ( 7812,612)(0,0)@xmath118 ( 11412,612)(0,0)@xmath119 ( 21012,612)(0,0)@xmath120 ( 2412,3612)(0,0)@xmath86 ( 6012,3612)(0,0)@xmath87 ( 9612,3612)(0,0)@xmath88 ( 19212,3612)(0,0)@xmath89 ( 16512,3612)(18312,3612 ) ( 18072.000,3552.000)(18312.000,3612.000)(18072.000,3672.000 ) ( 18072.000,3552.000 ) ( 18312,612)(20112,612 ) ( 19872.000,552.000)(20112.000,612.000)(19872.000,672.000 ) ( 19872.000,552.000 ) ( 17712,1512)(18912,2712 ) ( 18784.721,2499.868)(18912.000,2712.000)(18699.868,2584.721 ) ( 18784.721,2499.868 ) ( 19512,2712)(20712,1512 ) ( 20499.868,1639.279)(20712.000,1512.000)(20584.721,1724.132 ) ( 20499.868,1639.279 ) ( 912,1512)(2112,2712 ) ( 1984.721,2499.868)(2112.000,2712.000)(1899.868,2584.721 ) ( 1984.721,2499.868 ) ( 2712,2712)(3912,1512 ) ( 3699.868,1639.279)(3912.000,1512.000)(3784.721,1724.132 ) ( 3699.868,1639.279 ) ( 4512,1512)(5712,2712 ) ( 5584.721,2499.868)(5712.000,2712.000)(5499.868,2584.721 ) ( 5584.721,2499.868 ) ( 6312,2712)(7512,1512 ) ( 7299.868,1639.279)(7512.000,1512.000)(7384.721,1724.132 ) ( 7299.868,1639.279 ) ( 8112,1512)(9312,2712 ) ( 9184.721,2499.868)(9312.000,2712.000)(9099.868,2584.721 ) ( 9184.721,2499.868 ) ( 9912,2712)(11112,1512 ) ( 10899.868,1639.279)(11112.000,1512.000)(10984.721,1724.132 ) ( 10899.868,1639.279 ) ( 3312,3612)(5112,3612 ) ( 4872.000,3552.000)(5112.000,3612.000)(4872.000,3672.000 ) ( 4872.000,3552.000 ) ( 6912,3612)(8712,3612 ) ( 8472.000,3552.000)(8712.000,3612.000)(8472.000,3672.000 ) ( 8472.000,3552.000 ) ( 1512,612)(3312,612 ) ( 3072.000,552.000)(3312.000,612.000)(3072.000,672.000 ) ( 3072.000,552.000 ) ( 5112,612)(6912,612 ) ( 6672.000,552.000)(6912.000,612.000)(6672.000,672.000 ) ( 6672.000,552.000 ) ( 8712,612)(10512,612 ) ( 10272.000,552.000)(10512.000,612.000)(10272.000,672.000 ) ( 10272.000,552.000 ) ( 10512,3612)(12312,3612 ) ( 12072.000,3552.000)(12312.000,3612.000)(12072.000,3672.000 ) ( 12072.000,3552.000 ) ( 12312,612)(14112,612 ) ( 13872.000,552.000)(14112.000,612.000)(13872.000,672.000 ) ( 13872.000,552.000 ) ( 11712,1512)(12912,2712 ) ( 12784.721,2499.868)(12912.000,2712.000)(12699.868,2584.721 ) ( 12784.721,2499.868 ) ( 20112,3612)(21912,3612 ) ( 21672.000,3552.000)(21912.000,3612.000)(21672.000,3672.000 ) ( 21672.000,3552.000 ) ( 21912,612)(23712,612 ) ( 23472.000,552.000)(23712.000,612.000)(23472.000,672.000 ) ( 23472.000,552.000 ) ( 16812,612 ) ( 16812,612 ) ( 15012,3612 ) ( 15012,3612 ) ( 13212,3612 ) ( 13212,3612 ) ( 14112,3612 ) ( 14112,3612 ) ( 15912,3612 ) ( 15912,3612 ) ( 15012,612 ) ( 15012,612 ) ( 15912,612 ) ( 15912,612 ) ( 17712,612 ) ( 17712,612 ) ( 1112,2112)(0,0)[b]@xmath90 ( 4712,2112)(0,0)[b]@xmath91 ( 8312,2112)(0,0)[b]@xmath92 ( 11962,2112)(0,0)[b]@xmath121 ( 17912,2112)(0,0)[b]@xmath93 ( 3662,2112)(0,0)[b]@xmath94 ( 7262,2112)(0,0)[b]@xmath95 ( 10862,2112)(0,0)[b]@xmath96 ( 20612,2112)(0,0)[b]@xmath97 ( 4212,4012)(0,0)@xmath98 ( 7862,4012)(0,0)@xmath99 ( 11462,4012)(0,0)@xmath100 ( 17462,4012)(0,0)@xmath101 ( 2412,212)(0,0)@xmath122 ( 6012,212)(0,0)@xmath123 ( 9612,212)(0,0)@xmath124 ( 13212,212)(0,0)@xmath125 ( 19212,212)(0,0)@xmath126 ( 21012,4012)(0,0)@xmath85 ( 22662,212)(0,0)@xmath115 we have arbitrarily defined strategies and co - strategies in such a way that the co - strategy sends the first message . this accounts for the inevitable asymmetry in the definitions . while it is possible to view any @xmath80-turn co - strategy as being an @xmath127-turn strategy having input spaces @xmath128 and output spaces @xmath129 , it will be convenient for our purposes to view strategies and co - strategies as being distinct types of objects . similar to strategies , there will be no loss of generality in assuming that the initial state @xmath116 of a co - strategy is pure , that each of the admissible super - operators @xmath130 takes the form @xmath131 for some linear isometry @xmath132 , and , in the case of measuring co - strategies , that the measurement @xmath133 is a projective measurement . an @xmath80-turn strategy and co - strategy are _ compatible _ if they agree on the spaces @xmath78 and @xmath79 . by the _ output _ of a compatible strategy and co - strategy , assuming at least one of them is measuring , we mean the result of the measurement or measurements performed after the interaction between the strategies takes place . in particular , if both the strategy and co - strategy make measurements , then each output @xmath134 results with probability @xmath135 the definitions of strategies and co - strategies given above are natural from an operational point of view , in the sense that they clearly describe the actions of the players that they model . in some situations , however , representing a strategy ( or co - strategy ) in terms of a sequence of admissible super - operators is inconvenient . we now describe a different way to represent strategies that is based on the choi - jamiokowski representation of super - operators . let us first extend this representation to @xmath80-turn non - measuring strategies . to do this , we associate with the strategy described by @xmath82 a single admissible super - operator @xmath136 this is the super - operator that takes a given input @xmath137 and feeds the portions of this state corresponding to the input spaces @xmath78 into the network pictured in figure [ fig : strategy ] , one piece at a time . the memory space @xmath85 is then traced out , leaving some element @xmath138 . such a map is depicted in figure [ fig : choijam ] for the case @xmath139 . the choi - jamiokowski representation of the strategy described by @xmath82 is then simply defined as the choi - jamiokowski representation @xmath140 of the super - operator @xmath141 we have just defined . an alternate expression for the choi - jamiokowski representation of a strategy exists in the case that it is described by linear isometries @xmath142 . specifically , its representation is given by @xmath143 where @xmath144 is defined by the product @xmath145 ( 6999,2697)(0,-10 ) ( 1587,1137)(2262,1137)(2262,12)(1587,12)(1587,1137 ) ( 3162,1812)(3837,1812)(3837,687)(3162,687)(3162,1812 ) ( 4737,2487)(5412,2487)(5412,1362)(4737,1362)(4737,2487 ) ( 1925,575)(0,0)@xmath86 ( 3500,1250)(0,0)@xmath87 ( 5075,1925)(0,0)@xmath88 ( 6350,2262)(0,0)[l]traced out ( 687,1587)(0,0)[r ] @xmath146{0mm}{0mm}\right.$ ] ( 6312,912)(0,0)[l ] @xmath147{0mm}{0mm}\right\}\xi(\xi)$ ] ( 2362,912)(3062,912 ) ( 2942.000,882.000)(3062.000,912.000 ) ( 2942.000,942.000)(2942.000,882.000 ) ( 787,912)(1487,912 ) ( 1367.000,882.000)(1487.000,912.000 ) ( 1367.000,942.000)(1367.000,882.000 ) ( 3937,1587)(4637,1587 ) ( 4517.000,1557.000)(4637.000,1587.000 ) ( 4517.000,1617.000)(4517.000,1557.000 ) ( 5512,2262)(6212,2262 ) ( 6092.000,2232.000)(6212.000,2262.000 ) ( 6092.000,2292.000)(6092.000,2232.000 ) ( 5512,1587)(6212,1587 ) ( 6092.000,1557.000)(6212.000,1587.000 ) ( 6092.000,1617.000)(6092.000,1557.000 ) ( 2362,237)(6212,237 ) ( 6092.000,207.000)(6212.000,237.000 ) ( 6092.000,267.000)(6092.000,207.000 ) ( 787,1587)(3062,1587 ) ( 2942.000,1557.000)(3062.000,1587.000 ) ( 2942.000,1617.000)(2942.000,1557.000 ) ( 3937,912)(6212,912 ) ( 6092.000,882.000)(6212.000,912.000 ) ( 6092.000,942.000)(6092.000,882.000 ) ( 787,2262)(4637,2262 ) ( 4517.000,2232.000)(4637.000,2262.000 ) ( 4517.000,2292.000)(4517.000,2232.000 ) ( 5862,2487)(0,0)@xmath100 ( 5862,1812)(0,0)@xmath96 ( 5862,1137)(0,0)@xmath95 ( 5862,462)(0,0)@xmath94 ( 1137,2487)(0,0)@xmath92 ( 1137,1812)(0,0)@xmath91 ( 1137,1137)(0,0)@xmath90 ( 4287,1812)(0,0)@xmath99 ( 2712,1137)(0,0)@xmath98 next we consider measuring strategies . assume that an @xmath80-turn measuring strategy is given , where the measurement is described by @xmath148 for some finite , non - empty set @xmath31 of measurement outcomes . in this case we first associate with the strategy a collection of super - operators @xmath149 , each having the form @xmath150 the definition of each super - operator @xmath151 is precisely as in the non - measuring case , except that the partial trace over @xmath85 is replaced by the mapping @xmath152 notice that @xmath153 where @xmath141 is the mapping defined as in the non - measuring case . each super - operators @xmath151 is completely positive but generally is not trace - preserving . the choi - jamiokowski representation of the measuring strategy described by @xmath82 and @xmath154 is defined as @xmath155 . in the situation where a measuring strategy is described by linear isometries @xmath105 and a measurement @xmath154 , its choi - jamiokowski representation is given by @xmath156 for @xmath157 where @xmath158 for @xmath50 as defined above . it is not difficult to prove that for given input spaces @xmath78 and output spaces @xmath79 , a collection @xmath156 of operators is the choi - jamiokowski representation of some @xmath80-turn measuring strategy if and only if @xmath159 is the representation of an @xmath80-turn non - measuring strategy over the same spaces . finally , we define the choi - jamiokowski representation of measuring and non - measuring co - strategies in precisely the same way as for strategies , except that for technical reasons the resulting operators are transposed with respect to the standard basis . ( this essentially allows us to eliminate one transposition from almost every subsequent equation in this paper involving representations of co - strategies . ) specifically , a given @xmath80-turn co - strategy is viewed as an @xmath127-turn strategy by including an empty first and last message as discussed previously . this strategy s choi - jamiokowski representation is defined as above . the operators comprising this strategy representation are then transposed with respect to the standard basis to obtain the choi - jamiokowski representation of the co - strategy . as we work almost exclusively with the choi - jamiokowski representation of strategies and co - strategies hereafter , we will typically use the term _ representation _ rather than _ choi - jamiokowski representation _ for brevity . the applications of choi - jamiokowski representations of strategies given in this paper rely upon three key properties of these representations , stated below as theorems [ theorem : interaction - output - probability ] , [ theorem : characterization ] , and [ theorem : gauge ] . this section is devoted to establishing these properties . the first property provides a simple formula for the probability of a given output of an interaction between a strategy and a co - strategy . [ theorem : interaction - output - probability ] let @xmath156 be the representation of a strategy and let @xmath160 be the representation of a compatible co - strategy . for each pair @xmath161 of measurement outcomes , we have that the output of an interaction between the given strategy and co - strategy is @xmath162 with probability @xmath163 . let us fix a strategy and co - strategy whose representations are @xmath156 and @xmath164 , respectively . without loss of generality , the strategy is described by linear isometries @xmath105 and a projective measurement @xmath165 , while the co - strategy is described by a pure initial state @xmath166 , linear isometries @xmath167 and a projective measurement @xmath168 . for each output pair @xmath134 , define @xmath169 as follows : @xmath170 the probability of the outcome @xmath162 is @xmath171 for each pair @xmath162 . now , making use of the @xmath49 mapping , we may express @xmath172 in a different way : @xmath173 where @xmath174 the probability of outcome @xmath162 is therefore @xmath175 \op{vec}(i_{\y_{1\ldots n}\otimes\x_{1\ldots n}})\\ & = \tr\left [ \left(\tr_{\z_n } x_a x_a^{\ast}\right ) \left(\tr_{\w_n } y_b y_b^{\ast}\right)^{\t } \right ] \\ & = \ip{q_a}{r_b } \end{aligned}\ ] ] as required . let us denote by @xmath176 the set of all representations of @xmath80-turn strategies having input spaces @xmath78 and output spaces @xmath79 . we may abbreviate this set as @xmath177 or @xmath178 whenever the spaces or number of turns is clear from the context . similarly , we let @xmath179 denote the set of all representations of co - strategies for the same spaces . it will be convenient to define @xmath180 . the second property of strategy representations that we prove provides a characterization of @xmath181 in terms of linear constraints on @xmath182 . [ theorem : characterization ] let @xmath77 , let @xmath78 and @xmath79 be complex euclidean spaces , and let @xmath183 . then @xmath184 if and only if @xmath185 for @xmath186 . first assume that @xmath187 , which implies that there exist memory spaces @xmath81 and admissible super - operators @xmath188 that comprise a strategy whose representation is @xmath189 . let @xmath190 be the super - operator associated with this strategy as described in section [ sec : strategies ] , and let @xmath191 be the super - operator associated with the @xmath192-turn strategy obtained by terminating the strategy described by @xmath82 after @xmath193 turns . we have @xmath194 and so @xmath195 for @xmath196 as required . now assume that @xmath183 satisfies @xmath197 for some choice of @xmath186 . our goal is to prove that @xmath198 . this will be proved by induction on @xmath80 . in fact , it will be easier to prove a somewhat stronger statement , which is that the assumptions imply that there exists a strategy whose representation is @xmath189 that ( i ) is described by linear isometries @xmath105 , and ( ii ) satisfies @xmath199 . if @xmath200 , there is nothing new to prove : it is well - known that if @xmath201 satisfies @xmath202 , then @xmath203 for some admissible super - operator @xmath204 . any such super - operator can be expressed as @xmath205 for @xmath206 and some choice of a linear isometry @xmath207 . the 1-turn strategy we require is therefore the strategy described by @xmath208 . now assume that @xmath209 . by the induction hypothesis , there exist spaces @xmath210 and linear isometries @xmath211 with @xmath212 such that @xmath213 for @xmath214 defined as @xmath215 as required , we let @xmath85 be a complex euclidean space with dimension equal to the rank of @xmath189 , and let @xmath216 be any operator satisfying @xmath217 such a choice of @xmath51 must exist given that the dimension of @xmath85 is large enough to admit a purification of @xmath189 . note that @xmath218 next , let @xmath219 be a complex euclidean space with dimension equal to that of @xmath93 , and let @xmath220 be an arbitrary unitary operator . we have @xmath221 and therefore @xmath222 at this point we have identified two purifications of @xmath223 . we will use the isometric equivalence of purifications to define an isometry @xmath224 that will complete the proof . specifically , because @xmath225 has the minimal dimension required to admit a purification of @xmath223 , it follows that there must exist a linear isometry @xmath226 such that @xmath227 this equation may equivalently be written @xmath228 we now define @xmath229 . this is a linear isometry from @xmath230 to @xmath231 that satisfies @xmath232 this implies that the strategy described by @xmath105 has representation @xmath233 and therefore completes the proof . theorem [ theorem : characterization ] is equivalent to the following statement : an operator @xmath183 is the representation of some @xmath80-turn strategy with input spaces @xmath78 and output spaces @xmath79 if and only if there exist operators @xmath234 , where @xmath235 , such that the following linear constraints are satisfied : @xmath236 each operator @xmath237 in of course uniquely determined by the representation @xmath189 , and represents the strategy obtained by terminating any strategy represented by @xmath189 after @xmath238 turns . theorem [ theorem : characterization ] also gives a characterization of @xmath80-turn co - strategies , as stated in the following corollary . [ cor : characterization ] let @xmath77 , let @xmath78 and @xmath79 be complex euclidean spaces , and let @xmath183 . then @xmath239 if and only if @xmath240 for @xmath241 satisfying @xmath242 the fact that @xmath181 and @xmath243 are bounded and characterized by the positive semidefinite constraint together with finite collections of linear constraints yields the following corollary . let @xmath77 , let @xmath78 and @xmath79 be complex euclidean spaces . then the sets @xmath181 and @xmath243 are compact and convex . just as @xmath181 consists of all representations of non - measuring strategies , the set @xmath244 consists of all elements of representations of measuring strategies . in other words , for any @xmath80-turn measuring strategy representation @xmath245 , it holds that @xmath246 for each @xmath33 . moreover , for each operator @xmath189 there exists an @xmath80-turn measuring strategy @xmath156 of which @xmath189 is an element if and only if @xmath247 . the set @xmath248 has similar analogous properties . the final property of strategy representations that we will prove concerns the maximum probability with which some interacting co - strategy can force a given measuring strategy to output a given outcome . [ theorem : gauge ] let @xmath77 , let @xmath78 and @xmath79 be complex euclidean spaces , and let @xmath249 represent an @xmath80-turn measuring strategy with input spaces @xmath78 and output spaces @xmath79 . then for each @xmath33 , the maximum probability with which this strategy can be forced to output @xmath250 , maximized over all choices of compatible co - strategies , is given by @xmath251\,:\ , q_a \leq p r\;\ ; \text{for some}\;r\in{\mathcal{s}}_n(\x_{1\ldots n},\y_{1\ldots n})\}.\ ] ] an analogous result holds when @xmath156 is a measuring co - strategy . the remainder of the present section is devoted to a proof of this theorem . [ lemma : polar - technical ] let @xmath219 and @xmath252 be complex euclidean spaces and let @xmath253 be any closed , convex set that contains the origin . then for @xmath254 we have @xmath255 the assumption @xmath256 implies that @xmath257 for every @xmath258 , and therefore @xmath259 . consider any choice of @xmath260 , and note that @xmath261 for every choice of @xmath262 . if it is the case that @xmath263 then we have @xmath264 for all @xmath265 , and therefore @xmath266 . on the other hand , if @xmath266 then @xmath264 for all @xmath265 . it follows from the fact that @xmath189 is positive semidefinite that @xmath267 for all @xmath268 , and therefore @xmath269 . [ lemma : switch - polar ] let @xmath219 be a complex euclidean space , let @xmath270 be non - empty closed and convex sets , and suppose @xmath271 then @xmath272 let @xmath273 as @xmath274 for every @xmath275 , it follows that @xmath276 . clearly @xmath277 , and therefore @xmath278 . thus , @xmath279 on the other hand , we have that every @xmath280 is contained in @xmath281 , implying that @xmath282 for all @xmath283 . as @xmath284 , this implies that @xmath267 for @xmath285 . consequently , @xmath286 . thus @xmath287 , and so @xmath288 as required . [ lemma : polar ] let @xmath77 and let @xmath78 and @xmath79 be complex euclidean spaces . then for all @xmath289 we have 5 mm @xmath290 if and only if @xmath291 for some choice of @xmath292 . @xmath293 if and only if @xmath291 for some choice of @xmath187 . the proof is by induction on @xmath80 . as @xmath294 $ ] and @xmath295 $ ] , the lemma holds for the case @xmath296 . now suppose that @xmath77 . the two items in the statement of the lemma are equivalent by lemma [ lemma : switch - polar ] , so it will suffice to prove the first . define @xmath297 as @xmath298 by lemma [ lemma : polar - technical ] we have @xmath299 also define @xmath300 as @xmath301 again applying lemma [ lemma : polar - technical ] , we obtain @xmath302 now , by theorem [ theorem : characterization ] we have @xmath303 , and so @xmath304 . by the induction hypothesis we have @xmath305 and therefore @xmath306 by corollary [ cor : characterization ] we have that @xmath307 which completes the proof . let @xmath308 $ ] denote the maximum probability with which @xmath156 can be forced to output @xmath250 in an interaction with some compatible co - strategy . it follows from theorem [ theorem : interaction - output - probability ] that @xmath309 . using lemma [ lemma : polar ] , along with the fact that @xmath310 is positive semidefinite , we have @xmath311 which completes the proof . quantum coin - flipping protocols aim to solve the following problem : two parties , alice and bob , at separate locations , want to flip a coin but do not trust one another . a quantum coin - flipping protocol with bias @xmath312 is an interaction between two ( honest ) strategies @xmath50 and @xmath51 , both having output sets @xmath313 , that satisfies two properties : 1 . the interaction between the honest parties @xmath50 and @xmath51 produces the same outcome @xmath314 for both players , with probability @xmath315 for each outcome . ( neither player outputs `` abort '' when both are honest . ) 2 . if one of the parties does not follow the protocol but the other does , neither of the outcomes @xmath314 is output by the honest player with probability greater than @xmath316 . protocols that satisfy these conditions are generally referred to as _ strong _ coin - flipping protocols , because they require that a cheater can not bias an honest player s outcome toward either result 0 or 1 . ( in contrast , _ weak _ protocols assume that alice desires outcome 0 and bob desires outcome 1 , and only require that cheaters can not bias the outcome toward their desired outcome . ) kitaev @xcite proved that no strong quantum coin - flipping protocol can have bias smaller than @xmath2 , meaning that one cheating party can always force a given outcome on an honest party with probability at least @xmath317 . kitaev did not publish this proof , but it appears in refs . @xcite . here we give a different proof based on the results of the previous section . suppose @xmath318 is the representation of honest - alice s strategy and @xmath319 is the representation of honest - bob s co - strategy in some coin - flipping protocol . these strategies may involve any fixed number of rounds of interaction . the first condition above implies @xmath320 now , fix @xmath314 , and let @xmath1 be the maximum probability that a cheating bob can force honest - alice to output @xmath321 . obviously we have @xmath322 . theorem [ theorem : gauge ] implies that there must exist a strategy @xmath189 for alice such that @xmath323 . if a cheating alice plays this strategy @xmath189 , then honest - bob outputs @xmath321 with probability @xmath324 given that @xmath325 for all @xmath326 , we have that either honest - alice or honest - bob can be convinced to output @xmath321 with probability at least @xmath317 . this proof makes clear the limitations of strong coin - flipping protocols : the inability of bob to force alice to output @xmath321 directly implies that alice can herself bias the outcome toward @xmath321 . weak coin - flipping does not directly face this same limitation . currently the best bound known on the bias of weak quantum coin - flipping protocols , due to ambainis @xcite , is that @xmath327 rounds of communication are necessary to achieve a bias of @xmath312 . the best weak quantum coin - flipping protocol currently known , which is due to mochon @xcite , achieves bias approximately 0.192 ( which surpasses the barrier @xmath328 on strong quantum coin - flipping ) . next , we define quantum refereed games . it will be noted that von neumann s min - max theorem for zero - sum quantum refereed games follows from the facts we have proved about representations of strategies together with well - known generalizations of the classical min - max theorem . although it is completely expected that the min - max theorem should hold for quantum games , it has not been previously noted in the general case with which we are concerned . ( lee and johnson @xcite proved this fact in the one - round case . ) this discussion will also be helpful for the application to interactive proof systems with competing provers that follows . let us first define specifically what is meant by a zero - sum quantum refereed game . such a game is played between two players , alice and bob , and is arbitrated by a referee . the referee s output after interacting with alice and bob for some fixed number of rounds determines their pay - offs . [ def : referee ] an _ @xmath80-turn referee _ is an @xmath80-turn measuring co - strategy @xmath329 whose input spaces @xmath78 and output spaces @xmath79 take the form @xmath330 for complex euclidean spaces @xmath331 , @xmath332 , @xmath333 and @xmath334 , for @xmath335 . an _ @xmath80-turn quantum refereed game _ consists of an @xmath80-turn referee along with functions @xmath336 defined on the referee s set of measurement outcomes , representing alice s payoff and bob s payoff for each output @xmath33 . such a game is a _ zero - sum _ quantum refereed game if @xmath337 for all @xmath33 . the referee s actions in a quantum refereed game are completely determined by its representation @xmath329 . during each turn , the referee simultaneously sends a message to alice and a message bob , and a response is expected from each player . the spaces @xmath331 and @xmath332 correspond to the messages sent by the referee during turn number @xmath238 , while @xmath333 and @xmath334 correspond to their responses . after @xmath80 turns , the referee produces an output @xmath338 . a refereed quantum game does not in itself place any restrictions on the strategies available to alice and bob . for example , alice and bob might utilize a strategy that allows quantum communication , they might share entanglement but be forbidden from communicating , or might even be forbidden to share entanglement . specific characteristics of a given game , such as its nash equilibria , obviously depend on such restrictions in general . the focus of the remainder of the paper is on the comparatively simple setting of zero - sum quantum refereed games . in this case , we assume that alice and bob do not communicate or share entanglement before the interaction takes place . more precisely , we assume that alice and bob play independent strategies represented by @xmath339 and @xmath340 , respectively . the combined actions of alice and bob are therefore together described by the operator @xmath341 . it is a completely natural assumption that alice and bob play independent strategies in a zero - sum quantum refereed game , given that it can not simultaneously be to both players advantage to communicate directly with one another or to initially share an entangled state . this should not be confused with the possibility that entanglement among the players and referee might exist at various points in the game , or that the referee might choose to pass information from one player to the other . these possibilities are not disallowed when alice and bob s joint strategy is represented by @xmath342 . now , assume that a zero - sum quantum refereed game is given , and that alice and bob play independent strategies @xmath50 and @xmath51 as just discussed . let us write @xmath343 and define @xmath344 alice s expected pay - off is then given by @xmath345 while bob s expected pay - off is @xmath346 . now , @xmath347 is a real - valued bilinear function in @xmath50 and @xmath51 . because the operators @xmath50 and @xmath51 are drawn from compact , convex sets @xmath348 and @xmath349 respectively , we have that @xmath350 this is the min - max theorem for zero - sum quantum games . note that the above expression does not immediately follow from von neumann s original min - max theorem , but follows from an early generalization due to j. ville @xcite and several subsequent generalizations such as the well - known min - max theorem of ky fan @xcite . the real number represented by the two sides of this equation is called the _ value _ of the given game . classical interactive proof systems with competing provers have been studied by several authors , including feige , shamir , and tennenholts @xcite , feige and shamir @xcite , feigenbaum , koller , and shor @xcite , and feige and kilian @xcite . a quantum variant of this model is defined by allowing the verifier to exchange quantum information with the provers @xcite . in both cases these interactive proof systems are generalizations of single prover interactive proof systems @xcite . interactive proof systems with two competing provers are naturally modeled by zero - sum refereed games . to highlight this connection we will refer to the verifier as the referee and the two provers as alice and bob . the referee is assumed to be computationally bounded while alice and bob are computationally unrestricted . alice , bob , and the referee receive a common input string @xmath351 , and an interaction follows . after the interaction takes place , the referee decides that either alice wins or bob wins . a language or promise problem @xmath352 is said to have a classical refereed game if there exists a referee , described by a polynomial - time randomized computation , such that : ( i ) for every input @xmath353 , there is a strategy for alice that wins with probability at least @xmath354 against every strategy of bob , and ( ii ) for every input @xmath355 , there is a strategy for bob that wins with probability at least @xmath354 against every strategy of alice . the class of promise problems having classical refereed games is denoted @xmath356 . it is known that @xmath356 is equal to @xmath357 . the work of koller and megiddo @xcite implies @xmath358 , while feige and kilian @xcite proved the reverse containment . let us note that zero - sum classical refereed games , and therefore the class @xmath356 , are unaffected by the assumption that alice and bob may play quantum strategies , assuming the referee remains classical . this assumes of course that the classical referee is modeled properly within the setting of quantum information , which requires that any quantum information that it touches immediately loses coherence . equivalently , the referee effectively measures all messages sent to it by alice and bob with respect to the standard basis before any further processing takes place . as there also can not be a mutual advantage to alice and bob to correlate their strategies using shared entanglement , there is no advantage to alice or bob to play a quantum strategy against a classical referee . this is not the case in the cooperative setting , because there alice and bob might use a shared entangled state to their advantage @xcite . quantum interactive proof systems with competing provers are defined in a similar way to the classical case , except that the referee s actions are described by polynomial - time generated quantum circuits and the referee may exchange quantum information with alice and bob . the complexity class of all promise problems having quantum refereed games is denoted @xmath359 . the containment @xmath360 follows from @xmath361 . it was previously known that @xmath362 @xcite , and we will improve this to @xmath363 . this establishes the characterization @xmath0 , and implies that quantum and classical refereed games are equivalent with respect to their expressive power . in the refereed game associated with a competing prover quantum interactive proof system , the referee declares either alice or bob to be the winner . specifically , the referee outputs one of two possible values @xmath364 , with @xmath250 meaning that alice wins and @xmath321 meaning that bob wins . by setting @xmath365 and @xmath366 , we obtain a quantum refereed game whose value is the maximum probability with which alice can win . we will show that this optimal winning probability for alice can be efficiently approximated : it is the value of a semidefinite programming problem whose size is polynomial in the total dimension of the input and output spaces of the referee . it follows from the polynomial - time solvability of semidefinite programming problems @xcite that @xmath363 . we will use similar notation to the previous subsection : for a fixed input @xmath367 , the referee is represented by operators @xmath368 , and assuming @xmath80 is the number of turns for this referee we let the input and output spaces to alice be denoted by @xmath369 while @xmath370 denote the input and output spaces to bob . the assumption that the referee is described by polynomial - time generated quantum circuits implies that the entries in the matrix representations of @xmath371 and @xmath372 with respect to the standard basis can be approximated to very high precision in exponential time . now , given any strategy @xmath339 for alice , let us define @xmath373 the functions @xmath374 and @xmath375 are linear and extend to uniquely defined super - operators . under the assumption that alice plays the strategy represented by @xmath339 and bob plays the strategy represented by @xmath340 , we have that the referee outputs @xmath250 with probability @xmath376 and outputs @xmath321 with probability @xmath377 . one may think of @xmath378 as being the co - strategy that results from `` hard - wiring '' alice s strategy represented by @xmath50 into the referee . now , alice s goal is to minimize the maximum probability with which bob can win . for a given strategy @xmath50 for alice , the maximum probability with which bob can win is @xmath379 which , by theorem [ theorem : gauge ] , is given by @xmath380 the following optimization problem therefore determines the maximum probability @xmath1 for bob to win , minimized over all strategies for alice : @xmath381 this optimization problem can be expressed in terms of linear and semidefinite constraints as follows : @xmath382 & \tr_{\c_k}(a_k ) = a_{k-1 } \otimes i_{\a_k } \quad & ( 2\leq k\leq n),\\ & \tr_{\c_1}(a_1 ) = i_{\a_1},\\[2 mm ] & q_k = p_k \otimes i_{\d_k } \quad & ( 1\leq k\leq n),\\ & \tr_{\b_k}(p_k ) = q_{k-1 } \quad & ( 2\leq k\leq n),\\[2 mm ] & a_k \in \pos{\c_{1\ldots k}\otimes\a_{1\ldots k } } \quad & ( 1\leq k\leq n ) , \\ & q_k \in \pos{\d_{1\ldots k}\otimes\b_{1\ldots k } } \quad & ( 1\leq k\leq n ) , \\ & p_k \in \pos{\d_{1\ldots k-1}\otimes\b_{1\ldots k}}\quad & ( 1\leq k\leq n).\end{aligned}\ ] ] quantum interactive proofs with competing provers . in _ proceedings of the 22nd symposium on theoretical aspects of computer science _ ( 2005 ) , vol . 3404 of _ lecture notes in computer science _ , springer , pp .
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we study properties of _ quantum strategies _ , which are complete specifications of a given party s actions in any multiple - round interaction involving the exchange of quantum information with one or more other parties .
in particular , we focus on a representation of quantum strategies that generalizes the choi - jamiokowski representation of quantum operations .
this new representation associates with each strategy a positive semidefinite operator acting only on the tensor product of its input and output spaces .
various facts about such representations are established , and two applications are discussed : the first is a new and conceptually simple proof of kitaev s lower bound for strong coin - flipping , and the second is a proof of the exact characterization @xmath0 of the class of problems having quantum refereed games .
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consider @xmath2 a subset of a group @xmath3 . the elements of @xmath2 constitute the vertices of the commuting graph @xmath4 , in which @xmath5 are joined by an edge whenever @xmath6 . if @xmath2 is a set of involutions then we call @xmath4 a commuting involution graph . commuting graphs have been studied by many authors in various contexts . for example fischer @xcite studied commuting involution graphs for the case when @xmath2 is a conjugacy class of involutions , in his work on 3-transposition groups . segev and seitz looked in @xcite at commuting graphs for finite simple groups @xmath3 where @xmath2 consists of the non - identity elements of @xmath3 . more recently giudici and pope @xcite gave some results on bounding the diameters of commuting graphs of finite groups . commuting graphs for elements of order 3 have been considered in @xcite ; there have also been many papers dealing with the case where @xmath2 consists of all non - identity elements of a given group such as for example @xcite . + in @xcite , bates et al looked into the commuting involution graph @xmath4 where @xmath2 is a conjugacy class of involutions in @xmath3 and @xmath3 is @xmath7 . the remaining finite coxeter groups were analysed in @xcite . commuting involution graphs in affine coxeter groups of type @xmath8 have been considered in @xcite . in this article , we investigate the commuting involution graphs @xmath9 for type @xmath10 . + for the rest of this article , let @xmath11 be an affine weyl group of type @xmath12 , for some @xmath13 , writing @xmath3 when @xmath14 is not specified , and let @xmath2 be a conjugacy class of involutions of @xmath3 . we write @xmath15 for the diameter of @xmath4 when @xmath4 is a connected graph , in other words the maximum distance @xmath16 between any @xmath5 in the graph . since conjugation by any group element induces a graph automorphism , we can determine the diameter by fixing any @xmath17 , and then @xmath18 . our main result is the following . [ summary ] let @xmath2 be a conjugacy class of involutions in the group @xmath11 of type @xmath12 . if the commuting graph @xmath4 is connected , then its diameter is at most @xmath1 . the proof of theorem [ summary ] is broken into several cases depending on the type of the conjugacy class . to describe this further , and in order to state our results about connectedness , we need to describe a parameterisation of the involution conjugacy class . we will explain in section 2 that involutions in @xmath3 can be written as ` labelled permutations ' ( notation [ notation ] ) . these are permutations expressed as products of disjoint cycles in which every cycle has a sign and an integer written above it . for example in @xmath19 one of the involutions is @xmath20 . a cycle is positive if it has a plus sign , and negative if it has a minus sign . the _ labelled cycle type _ of an involution @xmath21 will be a quadruple @xmath22 where @xmath23 is the number of transpositions , @xmath24 is the number of negative @xmath25-cycles with an even number above them , @xmath26 is the number of negative 1-cycles with an odd number above them , and @xmath27 is the number of positive 1-cycles . for the example in @xmath19 , its labelled cycle type is @xmath28 . + we will show ( theorem [ conjclass ] ) that involution conjugacy classes in @xmath3 are parametrised by labelled cycle type . we may now give the conditions under which the graph @xmath4 is connected . [ main ] let @xmath2 be a conjugacy class of involutions in @xmath11 ( where @xmath29 ) with labelled cycle type @xmath22 . then @xmath4 is disconnected in each of the following cases . @xmath30 and @xmath31 ; @xmath32 , @xmath31 and either @xmath33 or @xmath34 ; @xmath32 and @xmath35 ; @xmath36 and @xmath37 ; @xmath38 , @xmath37 and @xmath39 . in all other cases , @xmath4 is connected . we note that the bound on diameter given in theorem [ summary ] is best possible . for example we have verified that when @xmath40 , @xmath37 and @xmath41 , the commuting involution graph has diameter 10 . in section [ sec2 ] we will establish notation and describe the conjugacy classes of involutions in @xmath3 . section 3 is dedicated to proving the main theorems . in section 4 we give examples of selected commuting involution graphs . for the remainder of this paper @xmath42 is a fixed involution with @xmath2 its conjugacy class in the affine irreducible coxeter group @xmath3 . our first job is to establish what involutions in @xmath3 look like . + let @xmath43 be a finite weyl group with root system @xmath44 and @xmath45 the set of coroots . the affine weyl group @xmath46 is the semidirect product of @xmath43 with translation group @xmath47 of the coroot lattice @xmath48 of @xmath43 . see , for example , ( * ? ? ? * chapter 4 ) for more detail . for @xmath49 and @xmath50 we have @xmath51 for the rest of the paper @xmath52 will denote a coxeter group of type @xmath53 , and set @xmath54 . we write , respectively , @xmath43 and @xmath3 whenever it is not necessary to specify @xmath14 . we may take the roots of @xmath43 to be @xmath55 and @xmath56 , for @xmath57 , where @xmath58 is the standard orthonormal basis for @xmath59 . the coroots are then @xmath60 and @xmath56 . therefore in this case @xmath61 we may view the elements of @xmath43 as signed permutations ; they act on @xmath59 by permuting the subscripts of basis vectors and changing their signs . to obtain a signed permutation we write a permutation in sym(@xmath14 ) ( including 1-cycles ) , add a plus sign or a minus sign above each @xmath62 , and say @xmath62 is positive or negative accordingly . we adopt the convention of reading the sign first ; that is , if @xmath63 , then @xmath64 , @xmath65 and @xmath66 . + expressing @xmath21 as a product of disjoint cycles , we say that a cycle @xmath67 of @xmath21 is _ positive _ if there is an even number of minus signs above its elements , and _ negative _ if the number of minus signs is odd . for example , @xmath68 is a negative cycle , whereas @xmath69 is positive . it is straightforward to check that an involution of @xmath43 only has 1-cycles ( positive or negative ) and positive 2-cycles . by the definition of group multiplication in @xmath11 , we see that the element @xmath70 of @xmath3 is an involution precisely when @xmath71 . this allows us to characterise the involutions in @xmath11 . [ invn ] a non - identity element @xmath70 of @xmath11 is an involution if and only if @xmath21 , when expressed as a product of disjoint signed cycles , has the form @xmath72 for some @xmath73 and @xmath27 ; and , writing @xmath74 , we have @xmath75 when @xmath21 contains @xmath76 , @xmath77 when @xmath21 contains @xmath76 and @xmath78 for @xmath79 . if @xmath80 is an involution , then @xmath81 . thus @xmath21 is an involution of @xmath43 , and so its cycles are all either 1-cycles or positive 2-cycles . thus @xmath21 has the form given in the statement of the lemma . write @xmath82 . we must have @xmath83 for all @xmath84 . for @xmath85 , we have @xmath86 . thus @xmath75 . for @xmath87 , we have @xmath88 and @xmath89 . so @xmath77 . for @xmath90 we have @xmath91 , so there is no restriction on @xmath92 . for @xmath93 we have @xmath94 , forcing @xmath78 . these are the necessary conditions on @xmath21 and @xmath95 ; it is clear that they are also sufficient . [ notation ] from now on we will employ a shorthand for writing involutions @xmath96 of @xmath3 : if @xmath74 , then above each signed number @xmath62 in the expression of @xmath21 as a product of disjoint signed cycles , we will write @xmath97 . however for transpositions @xmath98 of @xmath21 , where the number above @xmath42 determines the number above @xmath99 as described in lemma [ invn ] , we write @xmath100 for @xmath101 and @xmath102 for @xmath103 . we will call this the _ labelled cycle form _ of @xmath42 . where it is helpful , we adopt the convention that cycles @xmath104 are omitted , as these fix both @xmath42 and @xmath105 . let @xmath42 be an involution in @xmath11 . the _ labelled cycle type _ of @xmath42 is the tuple @xmath106 , where @xmath23 is the number of transpositions , @xmath24 is the number of negative 1-cycles with an even number above them , @xmath26 is the number of negative 1-cycles with an odd number above them , and @xmath27 is the number of positive 1-cycles ( fixed points ) , in the labelled cycle form of @xmath42 . for example , the labelled cycle type of @xmath107 is @xmath108 . + having characterised the involutions , we must now determine the conjugacy classes . a well - known result , due to richardson , gives a description of involution conjugacy classes in coxeter groups . [ equiv ] let @xmath43 be an arbitrary coxeter group , with @xmath109 the set of fundamental reflections . we say that two subsets @xmath110 and @xmath111 of @xmath109 are @xmath43-equivalent if there exists @xmath112 such that @xmath113 . in the next result , we use the notation @xmath114 for the longest element of a finite standard parabolic subgroup @xmath115 . [ richardson ] let @xmath43 be an arbitrary coxeter group , with @xmath109 the set of fundamental reflections . let @xmath116 be an involution . then there exists @xmath117 such that @xmath118 is central in @xmath115 , and @xmath119 is conjugate to @xmath118 . in addition , for @xmath120 , @xmath118 is conjugate to @xmath121 if and only if @xmath110 and @xmath111 are @xmath43-equivalent . the coxeter graphs of @xmath122 and @xmath123 , @xmath124 are as follows . + 1.00 mm ( 58.00,20)(-10,0 ) ( 10,10.80)(1,0)12 ( 10,9.00)(1,0)12 ( 10,10.00 ) ( 22,10.00 ) ( 34,10.00 ) ( 10,14)(0,0)[cc]@xmath125 ( 22,14)(0,0)[cc]@xmath126 ( 32,14)(0,0)[cc]@xmath127 ( -10,10)(0,0)[lc]@xmath128 ( 22,10.80)(1,0)12 ( 22,9.00)(1,0)12 1.00 mm ( 58.00,14)(-30,0 ) ( 10,10.80)(1,0)12 ( 10,9.00)(1,0)12 ( 22,10.00)(1,0)8 ( 50,10.00)(1,0)8 ( 58,10.80)(1,0)12 ( 58,9.02)(1,0)12 ( 58.00,10.00 ) ( 10,10.00 ) ( 22,10.00 ) ( 70,10.00 ) ( 10,14)(0,0)[cc]@xmath125 ( 22,14)(0,0)[cc]@xmath126 ( 58,14)(0,0)[cc]@xmath129 ( 70,14)(0,0)[cc]@xmath130 ( -20,10)(0,0)[lc ] @xmath131 ( 35.00,10.00)(1,0)0.4 ( 45.00,10.00)(1,0)0.4 ( 40.00,10.00)(1,0)0.4 we may set @xmath132 , @xmath133 for @xmath134 , and @xmath135 . + it is well known that in the finite coxeter group @xmath43 of type @xmath53 , elements are conjugate if and only if they have the same signed cycle type . in particular two involutions are conjugate when they have the same number of transpositions , the same number of negative 1-cycles and the same number of positive 1-cycles . + in @xmath3 ( which is of type @xmath12 ) , the element @xmath136 is conjugate to @xmath137 via some @xmath138 if and only if : @xmath139 [ conjclass ] involutions in @xmath3 are conjugate if and only if they have the same labelled cycle type . in particular , every involution is conjugate to exactly one element @xmath140 of the form @xmath141 let @xmath142 be an involution of @xmath3 . by theorem [ richardson ] , @xmath142 is conjugate to @xmath118 for some finite standard parabolic subgroup @xmath115 of @xmath3 in which @xmath118 is central . therefore the connected components of the coxeter graph for @xmath115 are of types @xmath143 or @xmath144 for some @xmath62 ( including , by a slight abuse of notation , @xmath145 , where we have connected components with just the vertex @xmath146 or @xmath130 ) . thus @xmath147 for some @xmath148 with @xmath149 , where @xmath111 is a subset of @xmath150 no two elements of which are adjacent vertices in the coxeter graph . by conjugation in @xmath151 ( which after all is isomorphic to the symmetric group @xmath152 ) , we can assume that for some @xmath148 and @xmath23 with @xmath153 we have @xmath154 this gives that @xmath142 is conjugate to @xmath118 , where @xmath155 let @xmath156 , where @size10@mathfonts @xmath157 from equation we see that @xmath158 therefore , by setting @xmath159 we see that each involution in @xmath3 is conjugate to at least one element of the required form . + now consider an involution @xmath142 in @xmath3 with labelled cycle type @xmath22 , and suppose @xmath160 for some simple reflection @xmath161 . write @xmath162 and @xmath163 . by equation , @xmath164 is conjugate to @xmath21 in the underlying weyl group @xmath43 . hence @xmath164 and @xmath21 have the same number of transpositions , negative 1-cycles and positive 1-cycles as each other . in other words , the labelled cycle type of @xmath165 is @xmath166 for some @xmath167 satisfying @xmath168 . now , from equation again , @xmath169 is a labelled 1-cycle of @xmath142 if and only if @xmath170 is a labelled 1-cycle of @xmath165 , where @xmath171 and @xmath172 . in particular this means @xmath173 is a signed cycle of @xmath164 , so that @xmath174 . now @xmath175 . so @xmath176 . therefore @xmath177 . hence @xmath178 and so @xmath142 and @xmath165 have the same labelled cycle type . in particular @xmath142 is conjugate to at most one element @xmath42 of the form stated in the theorem . conversely , any two involutions of the same labelled cycle type @xmath22 are both conjugate to @xmath179 , and hence to each other . thus conjugacy is parameterised by labelled cycle type , and the set of elements @xmath180 contains exactly one representative of each conjugacy class of involutions in @xmath11 . we next prove three preliminary lemmas which will be used repeatedly in the proofs in section [ sec3 ] . [ 1cycle ] let @xmath181 and @xmath182 . then @xmath183 commutes with @xmath184 for all @xmath185 , whereas @xmath183 commutes with @xmath186 if and only if @xmath187 . for convenience , we assume without loss of generality that @xmath188 . now @xmath189 , so @xmath190 commutes with @xmath191 . in the case of two negative 1-cycles we have @xmath192 and @xmath193 , so @xmath190 commutes with @xmath194 if and only if @xmath195 . [ 2cycle ] let @xmath196 be distinct elements of @xmath197 and let @xmath185 , @xmath198 and @xmath199 be integers . @xmath200 and @xmath201 commute if and only if @xmath187 , and @xmath202 and @xmath203 commute if and only if @xmath187 . but @xmath200 and @xmath203 commute for all @xmath185 and @xmath198 . @xmath204 and @xmath205 commute for all @xmath185 , but there is no value of @xmath198 or @xmath185 for which @xmath204 and @xmath206 or @xmath207 commute . @xmath208 and @xmath209 commute if and only if @xmath210 , whereas @xmath202 and @xmath209 commute if and only if @xmath211 . we lose no generality by assuming , for ease of notation , that @xmath188 , @xmath212 and @xmath213 . recall that @xmath214 and @xmath215 are shorthand for @xmath216 and @xmath217 respectively . involutions commute precisely when their product is an involution ( or the identity ) , which we can check using lemma [ invn ] . we calculate @xmath218 this means @xmath214 and @xmath219 commute if and only if @xmath187 . we also have @xmath220 so again @xmath215 and @xmath221 commute if and only if @xmath187 . finally a similar calculation shows that @xmath222 , which is an involution for all values of @xmath185 and @xmath198 , so @xmath214 and @xmath221 always commute . certainly @xmath223 and @xmath224 commute for all @xmath185 ; after all , @xmath224 is just the identity element when @xmath213 . moreover , already in the underlying group @xmath43 of type @xmath225 we observe that @xmath226 does not commute with @xmath227 or @xmath228 , so there is no value of @xmath198 or @xmath185 for which @xmath223 and @xmath229 or @xmath230 commute . we have @xmath231 ; this is an involution if and only if @xmath232 . thus @xmath233 and @xmath234 commute if and only if @xmath210 . similarly @xmath235 ; this is an involution if and only if @xmath236 . therefore @xmath237 and @xmath238 commute if and only if @xmath211 . [ doubletrans ] let @xmath239 , @xmath240 , @xmath241 , @xmath242 , @xmath243 and @xmath244 , for distinct @xmath245 in @xmath197 and integers @xmath246 . then @xmath247 commutes with @xmath248 if and only if @xmath249 ; @xmath247 does not commute with @xmath250 ; @xmath247 commutes with @xmath251 if and only if @xmath252 ; @xmath253 commutes with @xmath250 if and only if @xmath254 ; @xmath253 does not commute with @xmath251 ; @xmath255 commutes with @xmath251 if and only if @xmath256 . we lose no generality by assuming , for ease of notation , that @xmath36 and @xmath257 @xmath258 and @xmath259 . hence @xmath260 now @xmath247 and @xmath248 commute if and only if @xmath261 is an involution . this occurs if and only if @xmath262 and @xmath263 . rearranging gives @xmath254 , as required for part ( i ) . since @xmath264 and @xmath265 do not commute , it is impossible for @xmath247 to commute with @xmath250 . we have @xmath258 and @xmath266 , and @xmath267 now @xmath247 and @xmath251 commute if and only if @xmath268 is an involution . from the above calculation this occurs if and only if @xmath269 and @xmath270 . that is , if and only if @xmath271 . we calculate @xmath272 . thus @xmath253 commutes with @xmath250 if and only if @xmath273 . since @xmath274 and @xmath275 do not commute , @xmath253 can not commute with @xmath251 . this part follows from the fact that @xmath276 . we end this section by stating the relevant results from @xcite about commuting involution graphs in weyl groups of type @xmath277 . [ finite ] suppose that @xmath43 is of type @xmath277 , and let @xmath278 set @xmath279 and @xmath280 . then the following hold . if @xmath30 , then @xmath4 is a complete graph . if @xmath281 , then @xmath282 . if @xmath283 and @xmath284 , then @xmath4 is disconnected . if @xmath285 and @xmath286 , then @xmath287 . if @xmath288 , @xmath37 and @xmath289 then @xmath290 . if @xmath288 , @xmath37 and @xmath291 then @xmath292 . finally if @xmath36 , @xmath37 and @xmath289 then @xmath4 is disconnected we begin by looking at connectedness . for an element @xmath293 in a conjugacy class @xmath2 of @xmath3 , define @xmath294 . then let @xmath295 be the conjugacy class of @xmath296 in @xmath43 . clearly if @xmath297 , then @xmath298 . [ easy ] suppose @xmath297 . if @xmath299 , then @xmath300 . if @xmath301 is disconnected , then @xmath302 is disconnected . the result follows immediately from the observation that is @xmath119 commutes with @xmath303 in @xmath3 , then @xmath296 commutes with @xmath304 in @xmath43 . we can now prove theorem [ main ] ( which gives necessary and sufficient conditions for @xmath4 to be disconnected ) in one direction . the proof in the other direction will arise from bounding the diameters of graphs not shown in theorem [ mainhalf ] to be disconnected . [ mainhalf ] let @xmath2 be a conjugacy class of involutions in @xmath11 ( where @xmath29 ) with labelled cycle type @xmath22 . then @xmath4 is disconnected in each of the following cases . @xmath30 and @xmath31 ; @xmath32 , @xmath31 and either @xmath33 or @xmath34 ; @xmath32 and @xmath35 ; @xmath36 and @xmath37 ; @xmath38 , @xmath37 and @xmath39 . let @xmath2 be a conjugacy class of involutions in @xmath11 , with labelled cycle type @xmath305 , and @xmath306 as defined in theorem [ conjclass ] . we deal with each case in turn . suppose @xmath31 and @xmath30 . then any @xmath142 in @xmath2 is of the form @xmath307 for some @xmath308 ( where @xmath24 of the @xmath308 are even and @xmath26 are odd ) . by lemma [ 1cycle ] , @xmath142 does not commute with any other element of @xmath2 . so in fact @xmath4 is completely disconnected in this case . suppose @xmath31 and @xmath309 ( the case @xmath33 is similar ) . then @xmath310 suppose @xmath311 such that @xmath42 commutes with @xmath99 . consider the cycle of @xmath99 that contains @xmath14 . this must be a negative 1-cycle or a 2-cycle , because @xmath99 has the same labelled cycle type as @xmath42 . if it is a 2-cycle , then @xmath99 can not commute with @xmath42 , by lemma [ 2cycle](iii ) . therefore it is a negative 1-cycle @xmath312 where @xmath185 is odd , and then by lemma [ 1cycle ] , @xmath99 contains @xmath313 . the same argument shows that any element @xmath314 of @xmath2 which commutes with @xmath99 must also contain @xmath313 , and inductively all elements in the connected component of @xmath4 containing @xmath42 must contain @xmath313 . therefore @xmath4 is disconnected . suppose @xmath284 and @xmath35 . if @xmath31 then @xmath4 is disconnected by ( ii ) . so we can assume @xmath315 . if either of @xmath24 or @xmath26 is zero , then theorem [ finite](iii ) and lemma [ easy ] imply that @xmath4 is disconnected . it remains to consider the case @xmath316 . here , @xmath317 and @xmath318 suppose @xmath311 such that @xmath42 commutes with @xmath99 , and suppose @xmath319 is a 1-cycle of @xmath99 . if @xmath320 then as @xmath42 contains a transposition @xmath321 for some @xmath322 , lemma [ 2cycle ] ( ii ) and ( iii ) show that the only way @xmath42 and @xmath99 could commute is if @xmath99 contained @xmath323 or @xmath324 where @xmath325 , contradicting our assumptions about the labelled cycle type of @xmath99 . therefore the elements appearing in 1-cycles of @xmath99 are @xmath326 , @xmath327 and @xmath14 . inductively this holds for all elements in the connected component of @xmath4 containing @xmath42 . therefore @xmath4 is disconnected . suppose @xmath36 and @xmath37 . if @xmath35 then @xmath4 is disconnected by ( iii ) . if this does nt happen , then one of @xmath24 , @xmath26 or @xmath27 is 2 . by theorem [ finite](v ) and lemma [ easy ] , @xmath4 is again disconnected . suppose @xmath38 , @xmath37 and @xmath39 . for any @xmath142 in @xmath2 , we have @xmath328 , where @xmath329 , @xmath198 and @xmath330 are even , and @xmath331 are odd . associate a set @xmath332 to @xmath142 . given that , by lemma [ 2cycle ] , a transposition can only commute with a pair of negative 1-cycles if either both cycles are odd or both are even , and also with reference to lemma [ 1cycle ] , we see that if @xmath165 in @xmath2 commutes with @xmath142 , then @xmath333 . therefore , for example , @xmath334 and @xmath335 are not connected in @xmath4 . our first result bounding diameters is when @xmath30 . [ disc ] if @xmath30 and @xmath336 , then @xmath337 in all other cases @xmath338 . in this case we have @xmath339 let @xmath340 . then for appropriate @xmath341 and @xmath342 we have that @xmath343 . since @xmath344 , exactly @xmath24 of the @xmath308 must be even . if @xmath345 , that is , @xmath346 , then @xmath119 commutes with an element @xmath303 in @xmath2 which contains the cycles @xmath347 . now @xmath303 certainly commutes with @xmath42 , and so @xmath348 . + now we suppose that @xmath349 , and that one of @xmath24 and @xmath26 is zero . without loss of generality , we may choose @xmath350 . then , writing @xmath351 , we have @xmath352 . since conjugation by elements of centralizer of @xmath42 preserves distance in @xmath4 , without loss of generality we may take @xmath119 to be of the following form for some integer @xmath84 with @xmath353 and even integers @xmath308 : @xmath354 now consider the following sequence , where @xmath355 and @xmath356 , @xmath357 , @xmath358 are arbitrary even integers . @xmath359 it is clear that @xmath360 commutes with @xmath361 for @xmath362 . moreover @xmath119 commutes with @xmath363 , and @xmath364 commutes with @xmath42 . if @xmath27 divides @xmath365 , then @xmath366 , which implies that @xmath367 . if @xmath27 does not divide @xmath365 , then @xmath368 . the diameter of the graph in each case does equal this bound because at each stage of a path from @xmath119 to @xmath42 we can add at most @xmath27 to the number of correct negative 1-cycles , but this requires the element being considered to share no fixed points with @xmath42 . hence , for example the element @xmath369 must be distance at least @xmath370 from @xmath42 . + the remaining case is when @xmath24 and @xmath26 are both nonzero , and @xmath349 . + for any @xmath371 define @xmath372 to be the number of ` correct ' negative 1-cycles in @xmath142 . that is , cycles @xmath373 where @xmath374 or @xmath375 where @xmath376 . thus , for example , @xmath377 . we say that other negative 1-cycles are ` incorrect ' . let @xmath344 . then @xmath119 commutes with an element @xmath378 whose positive 1-cycles are @xmath379 . we will describe a sequence @xmath380 where at each stage @xmath381 is an element of @xmath2 such that @xmath382 and the positive 1-cycles of @xmath381 are @xmath383 for some @xmath384 where @xmath385 and @xmath386 , with @xmath387 . moreover , for @xmath388 , @xmath381 will commute with @xmath389 . + observe that the positive 1-cycles of @xmath378 have the required form , and @xmath390 . assume that we have @xmath391 and let the positive 1-cycles of @xmath381 be @xmath383 . to form @xmath392 we look for incorrect cycles @xmath393 of @xmath381 , where each @xmath394 is even , each @xmath395 is odd , @xmath396 and @xmath397 . if such cycles can be found , then @xmath381 commutes with @xmath392 where @xmath392 is given by replacing the cycles @xmath398 of @xmath381 with @xmath399 , and leaving all other cycles unchanged . it is clear that @xmath392 commutes with @xmath381 ; moreover @xmath400 , has the appropriate positive 1-cycles and @xmath401 . this sequence can continue until we have some @xmath381 with cycles @xmath383 but ( without loss of generality ) fewer than @xmath84 incorrect cycles @xmath402 where @xmath185 is even and @xmath403 . suppose there are exactly @xmath404 such cycles ( with @xmath405 ) . so @xmath381 has the cycles @xmath406 . since @xmath407 for each @xmath408 there must be @xmath409 where @xmath410 for each @xmath411 , and even numbers @xmath394 such that @xmath381 has the cycles @xmath412 . similarly @xmath381 has cycles @xmath413 for odd @xmath198 , some @xmath414 , @xmath397 and @xmath415 . we now define @xmath165 to be @xmath416 with @xmath417 replaced with @xmath418 now @xmath381 commutes with @xmath165 , and @xmath419 . we observe that @xmath420 . notice that every @xmath421 with @xmath376 is either a fixed point of @xmath165 or appears in a cycle @xmath422 with @xmath198 odd . every incorrect even negative 1-cycle of @xmath165 features a fixed point of @xmath42 , so for some @xmath423 , and conjugating @xmath165 by a suitable element of the centralizer of @xmath42 if necessary , we can assume that the even negative 1-cycles of @xmath165 are @xmath424 now @xmath165 has at least @xmath425 correct negative 1-cycles . if we ignore the even negative 1-cycles and the correct odd negative 1-cycles , then @xmath426 cycles remain ( including @xmath27 fixed points ) . we can use the result for @xmath427 on this remaining part of @xmath165 to see that @xmath165 is distance at most @xmath428 from the element @xmath429 of @xmath2 given by @xmath430 now @xmath429 commutes with @xmath42 , and so @xmath431 . if @xmath432 and @xmath433 , then in fact @xmath434 so @xmath435 . if @xmath436 then again @xmath435 . if @xmath437 then we have @xmath438 . to give a lower bound on the diameter consider @xmath439 to create @xmath27 additional correct negative 1-cycles at each stage of a path from @xmath119 to @xmath42 one requires each fixed point to be a point not fixed by @xmath42 ; moreover in this case completion of the process for , say , the even negative 1-cycles requires the recreation of at least one fixed point between @xmath440 and @xmath14 and hence fewer than @xmath27 correct negative 1-cycles being created at the next stage . thus when @xmath315 we have @xmath441 , and when @xmath437 we have @xmath442 . this completes the proof . from now on , assume that @xmath284 . then we can take @xmath443 where @xmath444 . [ m>0k=0 ] suppose @xmath445 . if @xmath446 , then @xmath447 . if @xmath448 and @xmath449 then @xmath450 . let @xmath371 . we can write @xmath451 and @xmath452 , where @xmath453 are conjugate elements of the underlying weyl group @xmath43 . define an element @xmath454 , by @xmath455 , where @xmath456 is the unique central involution of @xmath43 . the result is that every minus sign in @xmath303 corresponds to a plus sign in @xmath457 , and every plus sign corresponds to a minus sign . since @xmath281 , @xmath303 and @xmath457 are conjugate in @xmath43 . so , by theorem [ finite](i ) , @xmath458 when @xmath459 and @xmath460 otherwise . now @xmath461 and @xmath165 commutes with @xmath142 by lemmas [ 1cycle ] and [ 2cycle ] . the result follows immediately . [ mnonzerol=0 ] suppose @xmath462 , @xmath31 , one of @xmath24 and @xmath26 is zero , and either @xmath463 or @xmath464 ( or both ) . then @xmath4 is connected with diameter at most @xmath14 . if @xmath371 and @xmath465 then @xmath37 and the transposition of @xmath142 is @xmath466 for some @xmath185 . let @xmath371 . without loss of generality @xmath350 . we start with the case @xmath37 . we will show that if the transposition of @xmath142 is @xmath466 for some @xmath185 , then @xmath467 . otherwise @xmath468 . the graph for @xmath288 ( and hence @xmath37 ) is figure [ l0 ] . we can see from this graph that the inductive hypothesis holds for @xmath288 , so assume @xmath469 . suppose the transposition of @xmath142 contains some @xmath470 with @xmath471 . then by lemma [ 2cycle](iii ) @xmath142 commutes with some @xmath472 such that @xmath165 has the 1-cycle @xmath373 and such that the transposition of @xmath165 is not @xmath466 . ignoring the cycle @xmath373 we can work within @xmath473 , to see that inductively @xmath474 . hence @xmath468 . if the transposition of @xmath142 is @xmath466 then @xmath142 certainly commutes with an element of @xmath2 which does not have this transposition . so @xmath467 as required . + now we assume @xmath475 and proceed by induction on @xmath24 to show that @xmath476 . suppose @xmath477 . then @xmath142 is distance at most 2 from an element @xmath165 of @xmath2 which has the transposition @xmath478 . to see this , note that if both @xmath327 and @xmath14 appear in transpositions of @xmath142 , or if both appear in 1-cycles of @xmath142 , then lemma [ 2cycle ] or lemma [ doubletrans ] , as appropriate , implies that @xmath142 commutes with some @xmath479 in @xmath2 which contains a transposition of the form @xmath480 for some @xmath185 . then @xmath479 commutes with @xmath165 . if on the other hand , @xmath142 contains ( for example ) the 1-cycle @xmath481 and @xmath327 appears in a transposition @xmath482 for some @xmath483 less than @xmath327 , then @xmath142 commutes with @xmath484 in @xmath2 containing the transpositions @xmath485 and @xmath486 for some @xmath185 and @xmath487 . lemma [ doubletrans ] now implies that @xmath142 commutes with an appropriate @xmath165 , in particular one containing the transpositions @xmath488 and @xmath489 . now @xmath165 in turn commutes with some @xmath429 in @xmath2 with the 1-cycles @xmath490 and @xmath491 . if we ignore these cycles and work in @xmath492 , then table [ tableg4 ] implies that when @xmath38 @xmath493 , and when @xmath494 proposition [ m>0k=0 ] tells us that @xmath495 . therefore @xmath476 . finally , suppose @xmath496 and @xmath497 . suppose there is some transposition of @xmath142 containing an element @xmath470 with @xmath498 . then by lemma [ 2cycle](iii ) @xmath142 commutes with some @xmath472 such that @xmath165 has the 1-cycle @xmath373 . by induction @xmath474 . hence @xmath468 . the final possibility is that the elements of the transpositions of @xmath142 are @xmath499 . since @xmath463 we can use lemma [ doubletrans ] to show that @xmath142 commutes with some @xmath165 in @xmath2 containing the transposition @xmath500 . working in @xmath501 ( using the case @xmath37 and induction on @xmath23 ) we see that @xmath474 . hence @xmath468 , which completes the proof of proposition [ mnonzerol=0 ] . [ kois2 ] suppose @xmath37 , @xmath31 , @xmath502 , @xmath503 and @xmath371 . then @xmath302 is connected with diameter at most @xmath504 . let @xmath371 . the distance of @xmath142 from @xmath42 will largely hinge on the whereabouts of @xmath14 and @xmath327 . but first we deal with the case where the transposition of @xmath142 is @xmath505 . here @xmath142 commutes with an element @xmath165 of @xmath2 having cycles @xmath506 with @xmath198 and @xmath330 odd . using proposition [ mnonzerol=0 ] on the remaining cycles of @xmath165 , we see that @xmath165 is distance at most @xmath326 from the element @xmath99 of @xmath2 whose cycles are the same as @xmath42 except that we have @xmath507 instead of @xmath508 . clearly @xmath509 . hence @xmath510 . this also shows that any element with odd negative 1-cycles @xmath506 is distance at most @xmath14 from @xmath42 . assume from now on that the transposition of @xmath142 is not @xmath505 , and that its odd negative 1-cycles are not @xmath506 . + if the transposition of @xmath142 is @xmath511 then @xmath142 is distance 2 from some @xmath472 with cycles @xmath512 . ignoring these cycles we use proposition [ mnonzerol=0 ] in @xmath492 to see that @xmath474 if the transposition of @xmath165 is @xmath505 and @xmath513 otherwise thus @xmath467 . if @xmath142 has cycles @xmath514 where @xmath515 , then @xmath142 is distance 3 from some @xmath165 in @xmath2 with cycles @xmath512 . thus @xmath510 if the transposition of @xmath142 is @xmath505 and @xmath467 otherwise . if one of @xmath327 and @xmath14 , say @xmath14 , appears in the transposition of @xmath142 and the other appears in an even negative 1-cycle @xmath516 , then @xmath142 commutes with some @xmath517 with the cycles @xmath518 . then @xmath479 is distance 2 from some @xmath472 with the cycles @xmath512 , and such that the transposition of @xmath165 is not @xmath519 for any @xmath21 . by proposition [ mnonzerol=0 ] again , @xmath467 . if @xmath142 contains @xmath520 where @xmath185 is even , @xmath198 is odd , @xmath21 is an integer and @xmath521 , then @xmath142 commutes with an element of @xmath2 containing @xmath522 , which commutes with an element of @xmath2 containing @xmath523 , which commutes with an element of @xmath2 containing @xmath524 , which finally commutes with an element @xmath165 of @xmath2 containing @xmath525 , such that the transposition of @xmath165 is the same as the transposition of @xmath142 , namely not @xmath526 . by proposition [ mnonzerol=0 ] , hence @xmath510 . the final case to consider is where @xmath14 is contained in the transposition of @xmath142 and @xmath327 is in an odd negative 1-cycle . so @xmath142 contains @xmath527 for some @xmath196 and integers @xmath528 with @xmath198 odd . if @xmath529 then subject to appropriate conjugation we can set @xmath530 . using proposition [ mnonzerol=0 ] in @xmath531 we see that @xmath142 is distance at most @xmath532 from the element @xmath165 where @xmath533 . then @xmath165 commutes with @xmath534 , which commutes with @xmath535 which is distance 2 from @xmath42 . thus @xmath510 . the last case is where without loss of generality @xmath536 and we can assume @xmath470 is 2 or 5 . let @xmath537 be the other element of @xmath538 . then @xmath142 is distance 2 from some @xmath479 in @xmath2 containing @xmath539 where @xmath540 is determined by @xmath142 but we may choose @xmath164 arbitrarily , and @xmath541 are integers with @xmath199 odd . now let @xmath542 . then @xmath543 and @xmath544 . what is @xmath545 ? if @xmath546 then @xmath547 . set @xmath548 . now working in @xmath549 we see from proposition [ mnonzerol=0 ] that @xmath550 . if @xmath551 then @xmath552 . this time set @xmath553 . then @xmath479 commutes with @xmath554 for some @xmath555 , which commutes with @xmath556 which commutes with @xmath165 , so @xmath557 . hence in all cases @xmath510 , which completes the proof of lemma [ kois2 ] . [ connectm=1l=0 ] suppose @xmath558 , @xmath462 , @xmath31 and @xmath24 and @xmath26 are both at least 2 . then @xmath4 is connected with diameter at most @xmath1 . if @xmath496 , then @xmath559 . assume that @xmath560 . suppose first that @xmath37 and let @xmath371 . we use induction on @xmath26 to show that if the transposition of @xmath142 is not @xmath561 , then @xmath510 . otherwise @xmath562 . if @xmath502 the result holds by lemma [ kois2 ] . if @xmath563 and the transposition of @xmath142 is not @xmath561 , then @xmath142 contains cycles @xmath564 where @xmath471 , @xmath565 , @xmath566 is a cycle of @xmath42 and @xmath567 and @xmath21 are all congruent modulo 2 . now @xmath142 commutes with some @xmath517 containing the cycles @xmath568 for appropriate @xmath569 . if we ignore @xmath566 and work in @xmath570 , then inductively @xmath571 . hence @xmath510 . suppose the transposition of @xmath142 is @xmath561 . then @xmath142 commutes with some @xmath165 in @xmath2 that does not contain this transposition , and we have seen that @xmath572 . hence @xmath562 . this completes the case @xmath37 . + if @xmath463 and @xmath502 then it is easy to see that @xmath142 is distance at most 2 from an element of @xmath2 containing the transposition @xmath573 . thus @xmath142 is distance at most 3 from an element @xmath165 of @xmath2 containing @xmath574 . ignoring these 1-cycles we may work in @xmath492 and apply proposition [ mnonzerol=0 ] to see that @xmath575 . hence @xmath467 . + now suppose @xmath576 and @xmath563 . if @xmath142 has a transposition containing an element @xmath470 with @xmath498 , then @xmath142 commutes with an element @xmath165 containing @xmath373 or @xmath577 ( choose whichever of these is a cycle of @xmath42 ) . then we can ignore this cycle and work in @xmath578 . inductively , using the base case @xmath502 , we see that @xmath579 . . finally we deal with the case that every transposition of @xmath142 is of the form @xmath580 where @xmath581 . because @xmath582 , it must be the case that @xmath142 contains cycles : @xmath583 where @xmath584 , @xmath585 , @xmath586 , @xmath587 and @xmath588 . then @xmath142 is distance 2 from an element @xmath165 with the cycles @xmath589 and @xmath590 . now , working inductively in @xmath591 we see that @xmath474 . hence @xmath467 . [ n=5 ] if @xmath592 , @xmath37 , @xmath315 and @xmath477 , then @xmath593 . let @xmath371 . if the transposition is @xmath594 then using table [ tableg3 ] for @xmath595 we see that @xmath596 . suppose the transposition of @xmath142 is @xmath597 where @xmath598 , and let @xmath599 be the remaining element of @xmath600 . then @xmath142 commutes with @xmath601 for some even integers @xmath185 and @xmath602 . now @xmath479 commutes with @xmath603 , which commutes with @xmath42 . so @xmath596 . the remaining cases are ( interchanging 1 and 2 if necessary ) when @xmath604 or @xmath605 for appropriate @xmath606 and @xmath602 . the following is a path of length at most 5 from either of these to @xmath42:@xmath607 @xmath142 , @xmath608 , @xmath609 , @xmath610 , @xmath611 , @xmath42 . hence in all cases @xmath612 , which completes the proof . we observe , because we will need it for lemma [ n=6 ] later , that the proof of lemma [ n=5 ] shows that @xmath613 in all cases except where ( modulo interchanging 1 and 2 , or 3 and 4 ) the transposition of @xmath142 is @xmath614 . [ m>0l>0 ] suppose @xmath462 , @xmath615 and @xmath616 . then @xmath302 is connected with diameter at most @xmath14 . suppose @xmath14 is minimal such that @xmath302 is a counterexample , and let @xmath371 such that @xmath617 . by lemma [ n=5 ] we can assume @xmath469 . if @xmath449 then @xmath371 commutes with some @xmath472 containing @xmath618 . ignoring this 1-cycle we can work in @xmath619 to find a path to @xmath42 , which inductively is of length at most @xmath327 , which implies @xmath467 , contrary to our choice of @xmath142 . hence @xmath432 + if elements @xmath470 and @xmath421 lying between @xmath620 and @xmath621 are contained in transpositions of @xmath142 , then @xmath142 commutes with some @xmath517 having the transposition @xmath597 for some @xmath185 ( if @xmath37 then we can set @xmath622 ) . if we ignore this transposition of @xmath479 we can work in @xmath623 , which is either the case @xmath315 with a smaller @xmath23 , so inductively the graph has diameter at most @xmath326 , or ( if @xmath37 ) we can use theorem [ disc ] , in which case the graph has diameter @xmath327 . in either case , we see that @xmath142 is distance at most @xmath327 from the element @xmath99 of @xmath2 whose cycles are the same as @xmath42 except that @xmath99 has @xmath624 instead of @xmath625 . since @xmath99 commutes with @xmath42 we have @xmath467 . the same argument holds if @xmath626 . if @xmath627 then similar reasoning shows again @xmath142 is distance at most @xmath327 from an element @xmath99 containing @xmath628 that commutes with @xmath42 . so @xmath467 . thus none of these pairs @xmath196 exist in @xmath142 . this implies @xmath629 . moreover if @xmath630 then @xmath631 and without loss of generality the transpositions of @xmath142 contain @xmath632 and @xmath14 . + if there is some @xmath421 in a transposition of @xmath142 with @xmath633 and if @xmath634 , then @xmath142 commutes with some @xmath472 containing @xmath375 . inductively we can work in @xmath635 to see that @xmath579 . hence @xmath467 . similarly , as long as either @xmath497 or @xmath634 ( or both ) , if there is some @xmath470 in a transposition of @xmath142 with @xmath636 then inductively @xmath467 . this means that if @xmath630 then @xmath477 and @xmath309 . + suppose that @xmath630 , @xmath477 and @xmath309 , so that @xmath637 . we have observed that the transpositions of @xmath142 must contain @xmath638 and @xmath639 . if 6 is not contained in an even negative 1-cycle of @xmath142 , then @xmath142 commutes with some @xmath165 containing a transposition @xmath597 for some @xmath185 , where @xmath640 . as at the start of this proof , more explicitly , inductively @xmath165 is distance at most 6 from an element @xmath99 containing @xmath642 that commutes with @xmath42 . hence @xmath643 , a contradiction . thus @xmath37 . + if @xmath142 contains @xmath644 where @xmath645 then @xmath142 commutes with some @xmath517 containing @xmath577 . so , using the result for @xmath646 , we get @xmath647 . thus @xmath467 . the same reasoning holds if @xmath497 and @xmath142 contains @xmath644 for some @xmath470 where @xmath636 . suppose first that @xmath648 . since @xmath469 we must have @xmath634 or @xmath350 and @xmath503 . thus the transposition of @xmath142 can not contain any @xmath470 with @xmath649 . hence without loss of generality @xmath142 contains @xmath650 . suppose that @xmath142 contains @xmath651 where @xmath652 mod @xmath653 and either @xmath654 or @xmath655 . then , @xmath142 commutes with some @xmath472 where @xmath165 contains @xmath656 for some @xmath185 . now @xmath42 commutes with some element @xmath429 containing @xmath657 . using theorem [ disc ] on @xmath658 ( that is , removing @xmath659 , from @xmath165 and @xmath429 ) we see that @xmath660 . therefore , @xmath467 . finally , we are reduced to the possibility that @xmath142 contains @xmath661 and any pair @xmath662 where @xmath663 and @xmath664 satisfies @xmath665 . since @xmath286 and @xmath648 , the only way this can occur is when @xmath666 and @xmath552 . conjugating by an element of the centraliser of @xmath42 if necessary , we can assume that @xmath667 where @xmath21 and @xmath555 are even , and @xmath198 , @xmath330 are odd . the following is a path from @xmath142 to @xmath42 in the graph : @xmath142 , @xmath668 , @xmath669 , @xmath670 , @xmath671 , @xmath672 , @xmath42 . so @xmath673 , another contradiction . + the final case to consider is where @xmath37 and @xmath309 . assume first that @xmath674 . suppose that @xmath142 contains @xmath675 with @xmath567 both even . then @xmath142 commutes with some @xmath472 containing @xmath676 for some @xmath199 . as observed at the start of this proof , @xmath579 . hence @xmath467 . now suppose @xmath142 does not contain @xmath677 . then there is a set @xmath678 containing 1 , 2 and @xmath327 such that @xmath679 where @xmath185 and @xmath602 are both even and @xmath198 is odd . by replacing @xmath142 with a conjugate under the centraliser of @xmath42 , we can further assume that @xmath680 for some @xmath421 . let @xmath681 . now , using the case @xmath682 on @xmath683 we see that @xmath684 . next we apply theorem [ disc ] to @xmath685 and @xmath686 ( noting that here there are no odd negative 1-cycles ) to see that @xmath687 . hence @xmath688 , another contradiction . + the remaining possibility is that @xmath38 , @xmath37 , @xmath689 . but lemma [ n=6 ] immediately after this proof shows that this graph has diameter 6 , which is the final contradiction completing the proof of theorem [ m>0l>0 ] . [ n=6 ] if @xmath38 , @xmath37 , @xmath477 and @xmath309 then @xmath690 . we have @xmath691 . let @xmath371 . suppose @xmath142 contains @xmath692 for some @xmath21 . by theorem [ disc ] , the graph of @xmath693 in @xmath694 has diameter 5 . hence @xmath612 . if @xmath142 contains @xmath695 where @xmath567 are even , then @xmath142 commutes with some @xmath479 containing @xmath692 for some @xmath21 , and we have just seen that @xmath696 . hence @xmath673 . + we next consider the cases where 1 and 2 are in different types of 1-cycle of @xmath142 . because interchanging 1 and 2 does not affect the distance of @xmath142 from @xmath42 , there are just three cases to consider here : @xmath142 contains @xmath697 , @xmath698 or @xmath699 , where @xmath185 is even and @xmath198 is odd . in the first case we have @xmath700 where @xmath701 , @xmath567 are even and @xmath198 is odd . here @xmath142 commutes with @xmath702 , which commutes with the element @xmath99 given by @xmath703 . glancing at table [ tableg4 ] we observe that in @xmath694 elements at distance 4 or 5 from @xmath693 require at least one even label to be nonzero . so @xmath704 . hence @xmath612 . for the second case , where @xmath142 contains @xmath698 , note that @xmath142 commutes with some @xmath479 containing @xmath697 . hence , using the first case we get @xmath673 . the third case is where @xmath142 contains @xmath699 . here , @xmath142 commutes with some @xmath479 of the form @xmath705 . the following is a path from @xmath479 to @xmath42 : @xmath706 so @xmath673 . + it remains to take care of the possibility that exactly one of 1 and 2 lies in the transposition of @xmath142 . here , without loss of generality , we can assume @xmath142 contains @xmath707 for @xmath471 . if @xmath142 contains @xmath708 for @xmath198 odd , or @xmath709 , then @xmath142 commutes with some @xmath479 containing @xmath697 , for even @xmath185 , and as shown earlier in this proof , @xmath696 . hence @xmath673 . we may thus assume that @xmath710 . if @xmath711 then @xmath142 commutes with @xmath479 given by @xmath712 . by theorem [ n=5 ] applied to @xmath713 we see that @xmath696 and so @xmath673 . so we can assume that @xmath714 . now @xmath142 is distance 4 from the element @xmath99 given by @xmath715 , as shown by the following path . @xmath716 if @xmath717 , then @xmath718 so @xmath612 . if @xmath719 , then @xmath720 , so @xmath673 . if @xmath721 , then @xmath720 so @xmath673 . since interchanging 3 and 4 does not affect the distance from @xmath42 , there is only one case left to deal with : @xmath722 and @xmath723 . so @xmath724 . but @xmath142 is the same distance from @xmath42 as @xmath725 , and @xmath165 commutes with @xmath429 given by @xmath726 for appropriate @xmath727 and @xmath555 . by the ` @xmath717 ' case above , @xmath728 . therefore @xmath729 . this completes the proof . [ [ proof - of - theorems - summary - and - main ] ] proof of theorems [ summary ] and [ main ] + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + suppose @xmath30 . if @xmath31 then @xmath4 is disconnected by theorem [ mainhalf](i ) . otherwise @xmath730 by theorem [ disc ] . suppose @xmath37 and @xmath31 . if one of @xmath24 and @xmath26 is 1 or the largest of @xmath24 and @xmath26 is 2 , then @xmath4 is disconnected by theorem [ mainhalf ] ( ii ) , ( iv ) and ( v ) . otherwise @xmath731 by propositions [ m>0k=0 ] , [ mnonzerol=0 ] , lemma [ kois2 ] and theorem [ connectm=1l=0 ] . suppose @xmath576 and @xmath31 . if either of @xmath24 or @xmath26 is 1 , then @xmath4 is disconnected by theorem [ mainhalf](ii ) . otherwise @xmath559 by propositions [ m>0k=0 ] , [ mnonzerol=0 ] and theorem [ connectm=1l=0 ] . finally suppose @xmath462 and @xmath732 . if @xmath733 then @xmath4 is disconnected by theorem [ mainhalf](iii ) . otherwise @xmath559 by lemma [ n=5 ] , theorem [ m>0l>0 ] and lemma [ n=6 ] . we first summarise the information on commuting involution graphs for @xmath734 , @xmath735 and @xmath736 . for fixed @xmath17 , the @xmath737 disc @xmath738 is the set of elements of @xmath2 which are distance @xmath62 from @xmath42 . since a length preserving graph automorphism interchanges classes with @xmath560 and those with @xmath739 , we list here only those classes with @xmath560 . in describing the elements we omit positive 1-cycles . in @xmath734 there are two connected graphs , both of diameter 2 . if @xmath740 then @xmath741 with the remaining elements of @xmath2 comprising the second disc . if @xmath742 then @xmath743 , with all other elements of @xmath2 lying in the second disc . tables [ tableg3 ] and [ tableg4 ] give , for each connected graph and each disc , a list of orbit representatives under the action of @xmath744 . we use @xmath567 and so on to represent arbitrary even numbers , with @xmath745 $ ] being an arbitrary nonzero even number . we use @xmath746 and so on for arbitrary odd numbers , with @xmath747 $ ] being any odd number other than 1 . finally @xmath21 , @xmath555 and so on will be arbitrary integers with @xmath748 $ ] an arbitrary nonzero integer . figure [ l0 ] is the collapsed adjacency graph for @xmath750 in @xmath751 , which has diameter 5 . in the graph @xmath21 is an arbitrary integer , @xmath748 $ ] is an arbitrary non - zero integer and @xmath567 and @xmath727 are arbitrary even integers . for simplicity we have only included shortest paths that is , we have omitted edges between nodes in the same disc . 99 f. ali , m. salman , and s. huang . _ on the commuting graph of dihedral group _ , communications in algebra vol . 44 , iss . 6 ( 2016 ) 23892401 . a. nawawi and p. rowley . _ on commuting graphs for elements of order 3 in symmetric groups _ , electron . j. combin . 22 ( 2015 ) , no . 1 , paper 1.21 , 12 pp . b. fischer . _ finite group generated by 3-transpostions _ , i.invent . math.13 ( 1971 ) , 232 - 246 . c. bates , d. bundy , s. perkins and p. rowley . _ commuting involution graphs for symmetric groups _ , j. algebra , 266(1)(2003 ) , 133 - 153 . c. bates , d. bundy , s. perkins and p. rowley . _ commuting involution graphs for finite coxeter groups _ , j. group theory 6(2003),461 - 476 . m. giudici and a. pope . _ on bounding the diameter of the commuting graph of a group _ , j. group theory , 17 ( 1)(2014 ) , 131 - 149 . _ reflection groups and coxeter groups _ , cambridge studies in advanced mathematics , 29 ( 1990 ) . s. perkins . _ commuting involution graphs in the affine weyl group @xmath8 _ , arch . math . 86 ( 2006 ) , no . 1,16 - 25 . _ conjugacy classes of involutions in coxeter groups _ , bull . * 26 * ( 1982 ) , 115 . y. segev and g.m . _ anisotropic groups of type @xmath752 and the commuting graph of finite simple groups _ , pacific j. math . 202(2002 ) , 125226 .
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in this article we consider commuting graphs of involution conjugacy classes in the affine weyl group of type @xmath0 .
we show that where the graph is connected the diameter is at most @xmath1 .
+ msc(2000 ) : 20f55 , 05c25 , 20d60 .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
discovery of charge migration in dna molecules has opened new avenues to investigate various possibilities ranging from its role in the dna oxidative damage and repair @xcite to application of dna in nanoelectronic device developments @xcite . in fact , dna - based molecular electronic devices are expected to operate within the picoseconds range @xcite that can exceed the potential of the present solid state devices . quite expectedly , the dna molecule has become a subject of intense research activities both theoretically @xcite and experimentally @xcite . from all these studies of charge migration in the dna molecule reported as yet , it is clear that there are two mechanisms for transfer of charge depending on the dna structure and transfer parameters : a superexchange charge transfer and the incoherent hopping @xcite . the charge migration leads to the geometry changes in the nucleotides and the surrounding environment , which significantly contribute to the charge migration process . due to the interaction of the @xmath0 orbitals of the nearest neighbor duplexes and insignificant ip difference between them , hole can be distributed over several sites in the ( a - t)@xmath1 and ( g - c)@xmath1 oligomers . this significantly changes the magnitudes of the geometry relaxation of the nucleobases inner - sphere component and environment contribution outer - sphere component . however , the investigation of the transfer parameters , such as orbital overlapping @xcite and activation energy for charge migration i.e. the ip and the reorganization energy @xcite , have been performed mostly for the nucleobases or / and base pairs . the main purpose of our work is to estimate the electronic coupling between the two nearest nucleobases , their charge distribution and inner - sphere reorganization energy , when they are placed within the ( a - t)@xmath1 and ( g - c)@xmath1 oligomer duplexes . all these computations have been performed using accurate quantum - chemical methods . the relatively small reaction free energy in the dna molecule makes the dna hole transfer mechanism qualitatively different from that in most proteins @xcite . the electron transfer in the dna molecule was found to be strongly dependent on the details of the donor and acceptor energies and deviation of their geometries @xcite . the charge transfer in a dna molecule occurs due to the overlapping between the @xmath0-electrons of the carbon and the nitrogen atoms that forms the @xmath4 orbitals between the parallel nucleobases . charge migration in the molecular systems with weakly interacting donors and acceptors , such as between the base pairs in the dna molecule , is described by the standard high - temperature nonadiabatic electron - transfer rate @xmath5 where @xmath6 is the electronic donor - acceptor matrix element , and fc is the franck - condon factor . the electronic donor - acceptor matrix element @xmath6 is defined by the coupling of the orbitals of the donor and the acceptor and depends on the structure of the dna molecule . for the ( a - t)@xmath1 and ( g - c)@xmath1 oligomers the simple expressions for the deviation of the electronic coupling on the sequence number @xmath2 have been generated @xcite . according to these expressions , the value of the electronic coupling decreases with elongation of the oligomers @xcite . in sect . iii a , we simulate the electronic coupling of the nucleobases within the ( a - t)@xmath1 and ( g - c)@xmath1 oligomers using the quantum chemistry methods with the jaguar 6.5 program @xcite . according to the koopmans theorem , the electronic coupling can be estimated as half of the adiabatic state splitting between the homo and the homo-1 of the closed shell neutral system , determined in a hartree - fock self - consistent field . therefore , the rhf/6 - 31@xmath7g@xmath8 have been applied for the electronic coupling calculations . the 6 - 31@xmath7g@xmath8 basis set is appropriate for our purposes . previous investigations indicated that any further extension has little influence on the electron coupling @xcite . the geometries of the separated dna base pairs have been optimized with the rhf/6 - 31@xmath7g@xmath8 bases and in the following , the optimized geometries of the base pairs have been stacked with a twist angle 36@xmath9 and a distance of 3.38 . the stacking of the preliminary optimized geometries allows us to consider the same nucleobases to be ` in resonance ' @xcite within the structures of the ( a - t)@xmath1 and ( g - c)@xmath1 oligomers . the fc factor deals with the influence of the vibronic interaction on the charge propagation and can be expressed as @xmath10 where @xmath11 is the free energy of the reaction , and @xmath12 is the reorganization energy . the interaction of the molecule with the solvent environment is included in the outer - sphere reorganization energy @xmath13 , while the relaxation of the acceptor , the donor and the molecular bridge geometries are included in the inner - sphere reorganization energy @xmath14 . for adding one electron to the positive ion and ( b ) the reorganization energy @xmath15 for removing one electron from the neutral geometry , where @xmath16 and @xmath17 are the hole donor and the hole acceptor . ] the inner - sphere reorganization energy accounts for the low - frequency inter - molecular modes and can be estimated within the quantum chemical approach as @xcite @xmath18 where @xmath19 is the energy of the neutral state in a neutral geometry , @xmath20 is the energy of the neutral state in an ionic geometry , @xmath21 is the energy of the ionic state in an ionic geometry , and @xmath22 is the energy of the ionic state in a neutral geometry . the reorganization energy @xmath15 is the energy to remove an electron from the hole acceptor @xmath17 , while the reorganization energy @xmath23 is the energy to add an electron to the hole donor @xmath24 . the scheme for calculation of the reorganization energy is presented in figure [ fig : fig1 ] . clearly , the vibronic interactions stabilize the geometry of the donor and the acceptor from a non - equilibrium state ( @xmath25 ) to the equilibrium state ( @xmath26 ) . the vertical ionization potential is determined as @xmath27 and differ from the adiabatic @xmath28 by the inner - sphere reorganization energy . the inner - sphere reorganization energy has been evaluated within the unrestricted becke3p86/6 - 311@xmath7g@xmath8 approximation of the dft method . the dft theory was found to be reasonable for this purpose based on a comparison of the results of ref . these results show that the dft theory predicts the magnitude of the inner - sphere reorganization energy with a minimum error when compared to the experimental data @xcite . furthermore , we have also tested the application of the hf method and the dft theory for the vertical ionization potential ( vip ) calculations and have found significant qualitative and quantitative disagreement of the hf with the experimental data @xcite , while the becke3p86 approximation is appropriate for this purpose . at first we consider the system of two stacked duplexes . the results for the highest occupied base orbital ( hobo ) are presented in table [ tab : table2 ] . in the case when the pyrimidine / pyrimidine and purine / purine bases are stacked in one strands , the hobos of the adenine and guanine bases have lowest energy in comparison to the pyrimidine / purine configurations . for the ( a - t)@xmath29 and ( g - c)@xmath29 oligomers the hobos are delocalized over the two intrastrand nucleobases , and therefore , it produces a significant coupling between the @xmath0 orbitals of the stacked pyrimidine / pyrimidine and purine / purine bases . for oligomers where the pyrimidine and the purine bases are stacked in the same strand ( a - t / t - a , g - c / c - g , a - t / c - c ) or in the mixed structures ( a - t / g - c and g - c / a - t ) , for some cases the @xmath0 orbitals are delocalized , but electronic coupling is weak . for others the @xmath0 orbitals are localized mostly on one nucleobase , and we can consider the weak intrastrand and interstrand coupling between the nucleobases as well . the low interstrand coupling for the cases a - t / t - a and g - c / c - g has been observed experimentally @xcite . [ cols="^,^,^,^,^,^,^,^ " , ] further , the hydrogen bonds are the channels for charge transfer between the nucleobases . in the oxidized state the hydrogen bonds participate in the charge transfer between the nucleobases to bring the pairs from the nonequilibrum state , where the charge is localized only on the nucleobase with a lower ip , to the equilibrium state , where the charge is spread over the base pair @xcite . we consider the a - t and g - c base pairs as a single state for the following calculations of @xmath14 . the stacking of the base pairs into the ( g - c)@xmath1 and ( a - t)@xmath1 oligomers leads to a decrease of the inner - sphere reorganization energy @xmath30 and a decrease of the vip as well . the results are presented in figure [ fig : fig2 ] , where the decrease of @xmath14 is seen to occur due to the contribution of the rotation and translation of the base pairs relative to each other and to the spreading of the charge between the pairs . according to our data , with elongation of the ( a - t)@xmath1 and ( g - c)@xmath1 oligomers the twist of the base pairs mostly contributes to the decrease of the geometry relaxation of each nucleobase and in a reduction of the @xmath30 . the decrease of the energies of the adiabatic ip ( see fig . [ fig : fig5 ] ) and the inner - sphere reorganization energy ( see fig . [ fig : fig2 ] ) provide the decrease of the vip , which is the sum of above two components . and the vip values versus the number of pairs in a dna duplex oligomers ( a - t)@xmath1 and ( g - c)@xmath1 performed with ub3p86/6 - 311@xmath7g@xmath8 ] as we mentioned above , @xmath30 depends on the charge distribution over the chain . the electrostatic potential distribution in the ( g - c)@xmath1 and ( a - t)@xmath1 oligomers and respectively the residence of the homo in the oligomer centers provides the localization of the charge on the central guanines and adenines in the oxidized state . we have calculated the charge distribution as the difference between oxidized @xmath32 and neutral states @xmath33 with mulliken population analysis @xcite . in the ( g - c)@xmath1 and ( a - t)@xmath1 sequences the charge is distributed along the chain and is characterized by the low charge density at the dna molecule sides . for example , the density of the atomic partial charge localized on the @xmath2=1 site is lower than that at the chain center by 0.06 coul for the ( a - t)@xmath1 and by 0.25 coul for the ( g - c)@xmath1 sequences . for the ( g - c)@xmath34 sequence our results are in agreement with the data in ref . @xcite . therefore , the charge accumulation in the oligomer centers in the oxidized state produces the maximum geometry relaxation in the center of the dna chain . we have performed an estimation of the geometry relaxation of the separated base pairs @xmath35 within the optimized geometries of the ( a - t)@xmath1 and ( g - c)@xmath1 oligomers , where @xmath36 . the simulation results of @xmath35 for @xmath2=3 and @xmath2=5 are presented in figure [ fig : fig3 ] . clearly , for the ( g - c)@xmath1 sequences the difference of the structure relaxation at the sides of the chain and in the center is significant than that for the ( a - t)@xmath1 sequences . the behavior of these curves repeats primarily the charge distribution in the ( a - t)@xmath1 and ( g - c)@xmath1 sequences . corresponding to the single base pairs within the ( a - t)@xmath1 and the ( g - c)@xmath1 oligomers , where @xmath37 and @xmath2=5 are calculated with ub3p86/6 - 311@xmath7g@xmath8 . ] the difference between the inner - sphere reorganization energy of the ( a - t)@xmath1 and ( g - c)@xmath1 oligomers should provides the larger magnitude of the vibrational coupling constant for the g - c pairs than that for the a - t pairs , and larger for the guanine than that for the adenine ( see table [ tab : table1 ] ) . we have performed accurate quantum - chemical calculations to determine the electron coupling and the inner - sphere reorganization energy for the ( a - t)@xmath1 and ( g - c)@xmath1 dna oligomers , where @xmath36 . the electronic coupling between the two neighbor nucleobases within the same strand decreases exponentially with increasing of the base pairs number @xmath2 participating in the chain formation . the @xmath384 is the sequence number required for an accurate evaluation of the electron coupling in the dna molecule . the orbital distribution in oligomers with the hobo residing on the central nucleobase have been found to be the main reason for charge accumulation on the base pair located close to the chain center . the charge distribution in the chain determines degree of the the geometry relaxation of the base pair during the oxidation process in dependence on their location within the oligomer . therefore , the base pairs in the chain center have stronger geometry distortion during the oxidation process . such results are in good agreement with the theory of polaron formation in the dna molecule , where the maximum structure distortion occurs in the polaron center @xcite . the authors would like to thank dr . e.b . starikov for useful discussion . this work has been supported by the canada research chair program and a canadian foundation for innovation ( cfi ) grant . k.b . beckman , b.n . ames , _ j. biol . chem . _ * 1997 * , _ 272 _ , 19633 - 19636 ; 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v.m . apalkov , t. chakraborty , _ phys . * 2006 * , _ 72 _ , 161102 ( r ) . f.d . lewis , r.s . kalgutkar , y. wu , x. liu , j. liu , r. t. hayes , s. e. miller , m. r. wasielewskia , _ j. am . _ * 2000 * , _ 122 _ , 12346 - 12351 . g. b. schuster , _ acc . chem . _ * 2000 * , _ 33 _ , 253 - 260 . s. hess , m. gotz , w.b . davise , m.e . michel - beyerle , _ j. am . soc . _ * 2001 * _ 123 _ , 10046 - 10055 . oneill , j.k . soc . _ * 2004 * , _ 126 _ , 11471 - 11483 . k. senthilkumar , f. c. grozema , c. f. guerra , f. m. bilkelhaupt , f. d. lewis , y. a. berlin , m. a. ratner , l. d. a. siebbeles , _ soc . _ * 2005 * , _ 127 _ , 14894 - 14903 . tavernier , m.d . fayer , _ j. phys b _ * 2000 * , _ 104 _ , 11541 - 11550 . g.s.m . tong , i.v . kurnikov , d.n . beratan , _ j. phys b _ * 2002 * , _ 106 _ , 2381 - 2392 . a. harriman , _ angew . ed . _ * 1999 * , _ 38 _ , 945 - 949 . j. olofsson , s. larsson , _ j. chem b _ * 2001 * , _ 105 _ , 10398 - 10406 . hush , a.s . cheung , _ chem . _ * 1975 * , _ 34 _ , 11 - 13 . starikov , _ philosophical magazine _ * 2005 * , _ 85 _ , 3435 - 3462 . jaguar , version 6.5 . llc : new york , ny , 2005 . marcus , _ j. phys . chem . _ * 1994 * , _ 98 _ , 7170 - 7174 . sevilla , b. besler , a.o . colson , _ j. phys . chem . _ * 1995 * , _ 99 _ , 1060 - 1063 . orlov , a.n . smirnov , y.m . varshavsky , _ tetrahedron lett . _ * 1976 * , _ 48 _ , 4377 - 4378 . oneill , j.k . barton , _ pnas _ * 2002 * , _ 99 _ , 16543 - 16550 . q.zhu , p.r . lebreton , _ _ * 2000 * , _ 122 _ , 12824 - 12834 . baker , d. sept , s. joseph , m.j . holst , j.a . mccammon , _ proc . usa _ * 2001 * , _ 98 _ 10037 - 10041 . besler , k.m . merz , p.a . kollman , _ j. comput . chem . _ * 1990 * , _ 11 _ , 431 - 439 . guerra , f.m . bickelhaupt , j. g. snijders , e. j. baerends , _ chem . j. _ * 1999 * , _ 5 _ , 3581 - 3595 . mulliken , _ * 1955 * , _ 23 _ , 1833 - 1840 . s.s . alexandre , e. artacho , j.m . soler , h. chacham , _ phys . lett . _ * 2003 * , _ 91 _ , 108105 . p. maniadis , g. kalosakas , k.o . rasmussen , a.r . bishop , _ phys . b _ * 2003 * , _ 68 _ , 174304 .
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we report on our calculations of the inner - sphere reorganization energy and the interaction of the @xmath0 orbitals within dna oligomers .
the exponential decrease of the electronic coupling between the highest and second highest occupied base orbitals of the intrastrand nucleobases in the ( a - t)@xmath1 and ( g - c)@xmath1 oligomers have been found with an increase of the sequence number @xmath2 in the dna structure . we conclude that for realistic estimation of the electronic coupling values between the nucleobases within the dna molecule , a dna chain containing at least four base pairs is required .
we estimate the geometry relaxation of the base pairs within the ( a - t)@xmath1 and ( g - c)@xmath1 oligomers ( @xmath3 ) due to their oxidation .
the decrease of the inner - sphere reorganization energy with elongation of the oligomer structure participating in the oxidation process have been observed .
the maximum degree of geometry relaxation of the nucleobase structures and correspondingly the higher charge density in the oxidized state are found to be located close to the oligomer center .
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in recent years , diamond as a possible material for particle detectors has been the subject of considerable interest @xcite . significant progress in the techniques to produce synthetical diamond films of very high quality has been achieved by means of the chemical vapour deposition method ( cvd ) . a number of commercial manufactures of cvd diamond films @xcite and research institutes @xcite have made systematic studies of the properties of this material feasible . the main advantage of the material compared to other semiconductor detector materials is its radiation hardness , which has recently been demonstrated to neutron fluences of up to @xmath0@xcite . the radiation hardness of the material is of strong interest for the detector development at projected experiments where high radiation levels are expected due to the increasing luminosity and energy , as e.g. the experiments at the large hadron collider at cern . the main problem of using cvd diamond as a detector material are the charge collection properties , since an application as a detector for ionising particles requires that the material response is homogeneous throughout the volume and that a sufficient fraction of the produced charge is collected . general studies of the cvd diamond growth conditions for detector applications with a large sample are presented in @xcite . in this paper we present different studies of the charge collection properties of cvd diamond films . we investigate both the bulk properties of the material and the behaviour of charge produced close to the surface . one of the aspects which make cvd diamond films an attractive material for the detector development is its very high specific resistivity of @xmath1 . this allows a very simple construction of a detector as a solid state ionisation chamber ( see fig . [ fig - detector ] ) . by contacting the material on opposite sides and applying a sufficient potential across the contacts , charges produced in the bulk of the material by an incident charged particle start to drift towards the electrodes and induce mirror charges . the ratio of the amount of charge measured , @xmath2 , to the amount of charge produced , @xmath3 , is the collection efficiency @xmath4.\ ] ] the efficiency is below 100% , if trap- or recombination - sites in the material hinder the charge carriers to reach the electrodes , as is the case for diamond . the mean free path length @xmath5 , also called schubweg @xcite , for each type of charge carriers ( @xmath6 for electrons and holes ) can be expressed as the product of the mobility @xmath7 , the electric field strength @xmath8 and the life - time @xmath9 , as @xmath10 the combined mean free paths of electrons and holes , @xmath11 is the mean distance a hole and an electron can separate . this distance is referred to as the collection distance @xcite . the collection distance and the efficiency @xmath12 are related according to ramo s theorem @xcite by @xmath13 with @xmath14 being the detector thickness . the above discussion holds for uniform ionisation within the detector volume and @xmath15 . if the charge is created in the vicinity of one electrode ( @xmath16 , with @xmath17 the distance of the charge to the electrode ) the van hecht - equation @xcite relates @xmath12 and @xmath5 as @xmath18 where @xmath19 denotes the schubweg for electrons and holes respectively , depen- + ding on the direction of the electrical field . throughout this paper , we will use the quantity @xmath12 because the definition of @xmath12 involves fewer assumptions than the charge collection distance . however , the values are given in collection distance as well where appropriate . systematic studies of the dependence of @xmath12 on growth parameters have shown that high values of @xmath12 are reached for films produced with slow growth rates @xcite . until now charge collection efficiencies of up to 38% @xcite have been reported . it is known that the exposure of diamond to ionising radiation and uv - light can substantially increase @xmath12 . this phenomenon is called priming @xcite . the passivation of traps by occupation with free charge carriers is the reason for the observed behaviour . the same mechanism leads to a deterioriation of the signal as for example seen when irradiating diamond with alpha particles or protons at energies corresponding to stopping ranges of a few micron . the deterioriation is caused by the build - up of space charge and the resulting compensation of the applied field @xcite . another important issue is the homogeneity of @xmath12 . it is known that the collection distance ( which relates to @xmath12 by eq . [ eq - cd ] ) varies with the film thickness , as low values are typically observed on the substrate side and high values on the growth side . this is due to the fact that polycrystalline cvd diamond films consist of several columnar micro - crystallites . typically the crystallites are very small at the beginning of the growth process and become larger as the film thickness increases @xcite . a linear model has been proposed to describe the collection distance as a function of the film thickness @xcite . for an application as a particle detector , it is important that the average @xmath12 is constant over the active detector area . while highly energetic minimum ionising particles yield a rather homogeneous response ( see section [ s - mips ] ) , it has been shown that this is not the case for particles depositing their charge in a thin surface layer @xcite . thin films for which the stopping range of the ionising particle is of the order of the film thickness seem to exhibit a more homogeneous response @xcite . we investigate the charge collection properties of cvd diamonds by four different methods . the response of cvd diamond to homogeneous ionisation densities has been measured with beta particles from a source . the ionisation density produced by beta particles above @xmath20 is similar to the ionisation density produced by minimum ionising particles ( mips ) , which are to be detected in the application as a vertex detector in a high energy particle physics experiment . by fitting the pulse height distribution , we obtained information about the homogeneity of @xmath12 . to investigate the priming effect on the cvd diamond samples , @xmath12 has been measured with beta particles while the samples were under continuous exposure of ionising radiation . in a second experiment , the diamond has been scanned with a beam of 10 kev photons . the photons ionise via the photo effect and produce when averaged over many events a similar ionisation density distribution as mips . with this method the local variations of @xmath12 have been measured with a resolution of about @xmath21 . the results are compared with the data obtained with mips . in contrast , the other two methods described in this paper use radiation which produces charge carriers only in a shallow layer below the surface . hence , these methods are not measurements of the bulk properties as are the first two . since all the produced charge is deposited in a surface layer of the detector , they are more sensitive to polarisation effects , as will be discussed in section [ s - conclusion ] . one method uses alpha particles from an source . various pulse height spectra have been recorded at different electrical field strengths and polarities in order to extract information about @xmath12 . the other method uses protons with an energy of 2 mev and allows to do a spatially resolved measurement . the recorded pulse spectra are mapped onto the diamond surface . the spatial information has been used to compare the data to sem ( scanning - electron - microscopy ) pictures and to data obtained with the 10 kev photon beam described above . for the studies presented here , two cvd diamond samples produced by different manufacturers ( sample a from norton @xcite and sample b from fhg @xcite ) have been used . the sample a was initially grown to a thickness of @xmath22 after which @xmath23 were removed from the substrate side @xcite . the growth side has been polished to reduce the surface roughness , which has been measured to be smaller than @xmath24 after polishing @xcite . the grain - size on the growth side is in the order of @xmath21 . no information is available about the grain size on the substrate side . the sample size is @xmath25 . sample b has been grown to a thickness of @xmath26 and is not polished . the grains have a typical size of @xmath27 on the growth side . from sem we estimate a surface roughness of below @xmath28 . the sample has a circular shape and a diameter of @xmath29 . pole figure measurements @xcite showed no preferred orientation for sample a and a slight ( 111)-texture for sample b. in order to provide electrical contacts , both samples have been metallised on growth and substrate side with layers of ti ( 50 nm ) , pt ( 30 nm ) and au ( 60 nm ) , followed by an annealing step and treatment in oxygen plasma to reduce surface conductivity @xcite . the quality of the contacts has been checked by measuring the current - voltage ( i - v ) characteristic in darkness at room temperature . all i - v - curves are symmetric , i.e. independent of polarity , indicating , that front and back - contact are of the same type . the specific resistivities and some of the properties of the samples are listed in table [ tab - samples ] . .[tab - samples]properties of the investigated samples . @xmath30 stands for growth rate , @xmath14 for film thickness and @xmath31 for the specific resistivity . dcaj stands for _ direct current arc jet process _ and mwp for _ micro wave plasma process_. [ cols="<,<,<,<,<,<,<,<,<",options="header " , ] within the accuracy of the matching of the sem - picture and the @xmath12 map we assigned the highlighted area shown in fig . [ fig - agathe - sem ] to this hot spot . the size of the highlighted area includes the estimated error on the matching . it shows that the hot spot sits most likely on the cloven structure as indicated in the picture . we also scanned the substrate side of the diamond . the results of the substrate side gave a similar picture . again the edges of the metallisation showed significant higher counting rates and hot spots appeared also on the substrate side . however , the population with hot spots was denser on the substrate side ( spot density of about @xmath32 ) and they were smaller in dimension ( typically @xmath33 to @xmath24 ) than on the growth side . this can be explained by the fact that the average grain size is smaller on the substrate side than on the growth side @xcite . furthermore , the existence of hot spots on the substrate side indicates that the surface roughness is not responsible for this phenomenon , since the substrate side was polished . we tried to correlate @xmath12 maps of the growth side and the substrate side . if hot spots would appear at the same locations on the growth and on the substrate side it could indicate that continuous crystal columns of high collection efficiency are present at this locations . however , we could not observe a spatial correlation between the hot spot from the substrate and the growth side . this indicates that the lattice orientation or the crystal quality of the columnar structured crystallites are not a sufficient condition for a hot spot . the instability of some hot spots indicate a close link to polarisation phenomena as proposed by @xcite . the result indicates that the crystallite orientation , quality and size are not sufficient conditions to produce a hot spot . the results favour the interpretation of a complex polarisation phenomenon at grain boundaries @xcite . this view is supported by the spatially resolved measurement of @xmath12 with 10 kev photons , which ionise uniformly in the bulk and suppress polarisation . no enhancement of @xmath12 was seen with photons at the hot spot locations found with protons . we have presented four experiments using different types of ionising radiation to investigate the charge collection properties of cvd diamond . in particular , we have used beta - particles of up to @xmath34 , a narrow beam of @xmath35 photons from a synchrotron source , @xmath36 alpha - particles and a @xmath37 proton micro - beam . the samples show an increase of the charge collection efficiency , @xmath12 , when irradiated with beta - particles . the radiation dose needed to achieve saturation of this so called priming effect is different by a factor 15 between the two investigated samples . the relative improvements of @xmath12 ( @xmath38 for sample a and @xmath39 for sample b ) are comparable . to obtain information about the lateral distribution of @xmath12 , the signal spectrum recorded with beta particles was fitted with a smeared vavilov - distribution which results in an intrinsic broadening of @xmath12 of @xmath40 for sample a ( see section [ s - mips ] ) . this value is in good agreement with a direct measurement of the width and the spatial distribution of @xmath12 with a beam of @xmath35 photons , which measures for the same sample @xmath41 in contrast , the response behaviour of the cvd diamond samples to mono - energetic alpha - particle and protons with stopping ranges of a few microns indicates a much higher value for @xmath42 . the particles produce a very broad pulse distribution with an exponential fall - off towards large pulse - heights . a strong polarisation build - up decreases the average value of @xmath12 and the counting rate during irradiation . this confirms observations reported in @xcite . the spatially resolved measurement of @xmath12 with a proton micro - beam reveals single spots ( hot spots ) in the material with a high response embedded in passive areas , which is in agreement with observation made in @xcite . the shape of the hot spots and the correlation with sem - pictures indicates that hot spots are bound to the crystallite structure . since hot spots appear on the growth and on the polished substrate side we conclude that hot spots are neither caused by distinct surface topologies , nor require a certain crystallite size . the most likely explanation seems to be that particular configurations of grain boundaries in the bulk , which are most likely to possess a high trap density @xcite , influence the polarisation field in the neighbourhood of single crystallites . the unstable behaviour of some hot spots and the good response on metallisation edges corroborates this assumption . as a result of this proposed complex and non - uniform polarisation , @xmath12 as seen by low energy particles is inhomogeneous . this is supported by the measurements with alpha - particles . the pulse distributions obtained with alpha - particles are compatible with the proton micro - beam measurements . furthermore , the hot spots seen with the proton micro - beam do not show significantly higher values of @xmath12 compared to a @xmath35 photon beam . this suggests that the existence of hot spots is bound not only to a good crystal quality in this particular spot , but also to distinct features of the polarisation field in this area . the maps of @xmath12 obtained with a proton micro - beam are highly distorted by the local polarisation fields . thus this method is not suitable to determine maps of @xmath12 for mips . the observed increase of @xmath12 under irradiation with mips and the decrease of @xmath12 observed under irradiation with protons and alpha particles is not contradictory . in the former case traps are filled throughout the bulk thus increasing @xmath12 due to the decrease of active trap levels . however , because of the rather homogeneous distribution of the trapped charge no effective polarisation field is produced . in the latter case the majority of the charge is deposited in a shallow layer . in the electric field the charge carriers are separated and trapped and thus create a strong polarisation field . the polarisation is dominating over the increase of @xmath12 from trap passivation . the applicability of cvd diamond as a particle detector depends on the radiation to be detected . for radiation which deposits charge inhomogeneously as in the case of alpha - particles , whose stopping range is much less than the film thickness , @xmath12 varies strongly and the stability is very poor . for homogeneous ionisation typically produced by mips or synchrotron - radiation at photon energies with @xmath43 , @xmath12 is stable if the detector is in its primed state , and a gaussian - like distribution is observed . thus , we conclude that cvd diamond has the potential of being used as a stable detector for homogeneous ionisation density distributions . however the relatively high value of @xmath42 might limit the achievable resolution of a position sensitive device . we would like to thank prof . h. c. g. lindstrm and dr . e. fretwurst from the ii . institut fr experimentalphysik of the univeritt hamburg for their support of the measurements with alpha particles and of the i - v - curves , dr . e. fretwurst and dr . m. niecke for their support of the measurements with a proton micro - beam , and dr . t. wroblewski from hasylab for his support of the measurements at the synchrotron facility . we would also like to thank a. bluhm and dr . l. schfer from the fraunhofer institut fr schicht- und oberflchentechnik , braunschweig , for the preparation of sample b and m. zeitler of the univesitt augsburg for the pole figure measurements .
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the charge collection properties of cvd diamond have been investigated with ionising radiation . in this study two cvd diamond samples , prepared with electrical contacts have been used as solid state ionisation chambers .
the diamonds have been studied with beta particles and 10 kev photons , providing a homogeneous ionisation density and with protons and alpha particles which are absorbed in a thin surface layer . for the latter case a strong decrease of the signal as function of time is observed , which is attributed to polarisation effects inside the diamond .
spatially resolved measurements with protons show a large variation of the charge collection efficiency , whereas for photons and minimum ionising particles the response is much more uniform and in the order of 18% .
these results indicate that the applicability of cvd diamond as a position sensitive particle detector depends on the ionisation type and appears to be promising for homogeneous ionisation densities as provided by relativistic charged particles .
epsf ps .
ps , , , , and cvd diamond , charge collection efficiency . 29.40
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You are an expert at summarizing long articles. Proceed to summarize the following text:
dendrimers are an important class of artificial macromolecules with a treelike structure , which are synthesized by repeating units in a hierarchical self - similar fashion around a central core @xcite . their unique structural features make them promising candidates for a broad range of potential applications such as light harvesting antennae @xcite and molecular amplifiers @xcite , the latter of which can be used as efficient platforms for drug delivery @xcite . in view of their practical significance , thus far , dendrimers have received extensive attention within the scientific community @xcite . in the context of light harvesting by dendrimers , it is the large number of absorbing elements at the periphery and an efficient transfer of the absorbed energy to the center that make dendrimers work as antennas @xcite . generally , light harvesting can be described as a trapping process with a ( fluorescent ) trap located at the center . a primary quantity related to trapping is average trapping time ( att ) , which is the average of mean first - passage time ( mfpt ) @xcite to the target over all starting nodes , where mfpt from a node to the trap is the expected time steps needed for a walker starting off from this node to visit the trap for the first time . att is a quantitative indicator measuring the trapping efficiency , which has been much studied for diverse complex systems @xcite . _ _ because of the theoretical and practical relevance , trapping in dendrimers has also been devoted to concerted efforts . the problem was first addressed in refs . @xcite , where the mfpt from a peripheral node to the central node was computed analytically . in refs . @xcite , mfpt from any node to the central node , as well as the att to the center , were deduced . these works unveiled the effect of structure on the mfpt and att to the central node . note that most previous works on trapping dendrimers based on discrete time random walks focused on the unbiased random walk . however , sometimes it is more suitable to describe some particular problems by a biased random walk than the unbiased one @xcite , since the transition probability depends on not only network topologies but also other properties relevant to the diffusion dynamics @xcite . among numerous biased random walks , maximal entropy random walk ( merw ) @xcite , maximizes the entropy of paths , has been studied recently @xcite . in merw , the transition probability incorporates the node centrality measured by the eigenvector associated with the largest eigenvalue of adjacent matrix of the graph where the dynamical process takes place . this local transition rate leads to substantial effects on the diffusion behaviors such as stationary distribution @xcite and relaxation time @xcite . the principle of entropy maximization has found some applications , including optimal sampling algorithm @xcite and demographic stability of population @xcite . very recently , as a powerful tool , merw has been fruitfully applied in the analysis of complex networks @xcite . at present , it is still of theoretical and practical interest to explore possible applications of merw on other areas . in this paper , we study the trapping problem in cayley trees as models of dendrimers @xcite , based on merw that incorporates the centrality of nodes of the macromolecular graphs . we concentrate on a particular case with a perfect trap placed at the central node , for which we derive an analytical exact expression for att . we then provide an upper bound for att , whose leading scaling behaves with the system size @xmath1 as @xmath2 , a scaling much smaller than that corresponding to unbiased random walk in cayley trees , for which the att scales linearly with @xmath1 . theses results show that for trapping process in dendrimers with a trap fixed at the center , merw is considerably more efficient in comparison to unbiased random walk . in this section , we introduce the construction and some relevant features of dendrimers modeled by cayley trees , which are built in an iterative way . the particular construction process allows for analytically determining the properties of cayley trees and deriving exact solutions for diverse dynamical processes on large but finite structures . cayley trees after @xmath3 iterations ( generations ) , denoted by @xmath4 ( @xmath5 , @xmath6 ) are constructed as follows @xcite . at the initial generation ( @xmath7 ) , @xmath8 contains only a central node ; at @xmath9 , @xmath10 new nodes are generated and linked to the central node to form @xmath11 . these @xmath10 new single - degree nodes constitute the peripheral nodes of @xmath11 . for any @xmath12 , @xmath4 is obtained from @xmath13 : for each peripheral node of @xmath13 , @xmath14 new nodes are created and are connected to the peripheral node . figure [ cayley ] illustrates the structure for a specific dendrimer @xmath15 . let @xmath16 denote the number of nodes in @xmath17 , which are created at @xmath18th generation . then , we can verify that @xmath19 thus , the total number of nodes in @xmath4 is @xmath20 . ] the special construction of cayley trees also makes it possible to analytically determine the eigenvalues and their associated eigenvectors @xcite . let @xmath21_{n \times n}$ ] denote the adjacency matrix of network @xmath4 , in which @xmath22 if nodes @xmath18 and @xmath23 are linked to each other , @xmath24 otherwise . although for a general graph , it is often difficult to determine the eigenvalues and eigenvectors of its adjacency matrix , for @xmath4 the problem can be settled . note that @xmath4 has @xmath1 eigenvalues . we represent these @xmath1 eigenvalues as @xmath25 , @xmath26 , @xmath27 , @xmath28 , and @xmath29 , respectively . every eigenvalue @xmath30 takes the form @xmath31 @xmath32 where @xmath33 . in the sequel , what we are concerned with is the largest eigenvalue and its corresponding eigenvector of @xmath34 , which are denoted by @xmath35 and @xmath36 , respectively . the largest eigenvalue @xmath35 can be expressed as @xmath37 where @xmath38 with regard to the eigenvector @xmath36 , since all its entries @xmath39 for nodes in a given generation @xmath18 are identical , it can be written as : @xmath40 in order to express the entry @xmath41 , we introduce the following quantity : @xmath42 for @xmath43 . it is easy to verify @xmath44}{\sin\theta}\,.\ ] ] from eq . ( [ b2 ] ) , we have @xmath45 in the following text , we choose @xmath46}$ ] . thus , @xmath47}}\,.\ ] ] the selection of @xmath48 ensures the proper normalization of every entry of eigenvector @xmath36 : @xmath49=1\,.\ ] ] after introducing the construction and related properties of cayley trees @xmath4 , in this section , we study merw in dendrimers with a perfect trap fixed at the central node . our goal is to unveil the influence of maximal entropy random walk on the efficiency of trapping performed on this important family of polymer networks . the merw considered here is a discrete time random walk @xcite . at each discrete time step , the walker jumps from its current position @xmath18 to any of its neighboring nodes @xmath23 with probability @xmath50 let @xmath51 denote the set of neighbors of a node @xmath18 . then , for an arbitrary node @xmath18 , @xmath52 fulfills the condition @xmath53 , since @xmath54 the stationary distribution for merw on @xmath4 is @xcite @xmath55 the main subject we focus on here is trapping problem in @xmath4 described by merw with a deep trap located at the innermost node . and the main quantity we are interested in is the att to the trap . for convenience of description , we label the central node of @xmath17 by @xmath56 , and consecutively label all other nodes as @xmath57 , and @xmath1 . we use @xmath58 to represent the trapping time of node @xmath18 , which is defined as the expected time for a walker starting off from node @xmath18 to reach the trap in @xmath17 for the first time . then , the att denoted by @xmath59 is the average of @xmath58 over all non - trap nodes distributed uniformly in @xmath17 , that is , s @xmath60 note that merw in any connected binary network can be represented as generic random walk in a corresponding weighted network @xcite . for @xmath17 , the generalized adjacency matrix ( weight matrix ) @xmath61_{n \times n}$ ] corresponding to the weighted networks , denoted by @xmath62 , associated with merw is defined as follows . the entries @xmath63 if nodes @xmath18 and @xmath23 are adjacent in @xmath17 , and the element @xmath64 if nodes @xmath18 and @xmath23 are not directly connected by an edge in @xmath17 . in weighted networks @xmath62 , the strength @xcite of a node @xmath18 is defined by @xmath65 , which is @xmath18th nonzero entry of the diagonal strength matrix @xmath66 , and the laplacian matrix of @xmath62 is defined to be @xmath67 . for generic random walk in @xmath62 , the transition probability for the walker from current state @xmath18 to one of its neighboring nodes @xmath23 is @xmath68 , which is the same as that of merw in @xmath17 . therefore , the stationary distribution for generic random walk in @xmath62 is also the same as that corresponding to merw in @xmath4 . according to recently obtained result about trapping in weighted networks @xcite , the explicit expression for att @xmath59 in eq . ( [ d4 ] ) can be expressed as @xmath69 in eq . ( [ d5 ] ) , @xmath70 is the sum of strengths over all nodes in @xmath62 , namely @xmath71 ; @xmath72 are the @xmath1 eigenvalues of @xmath73 , rearranged as @xmath74 , and @xmath75 are the corresponding mutually orthogonal eigenvectors of unit length , where @xmath76 . it should be mentioned that although the expression for att @xmath59 provided by eq . ( [ d5 ] ) seems compact , it requires computing the eigenvalues and eigenvectors of matrix @xmath73 . since the network size @xmath1 grows exponentially with @xmath3 as shown in eq . ( [ c2 ] ) , for moderately large @xmath3 , the computation of spectra demands an impractically large computational effort . moreover , by using eq . ( [ d5 ] ) it is very hard and even impossible to obtain useful information about the dependence of the leading behavior of @xmath59 on the network size @xmath1 . it is thus of significant practical importance to seek other approaches for determining the att @xmath59 . fortunately , the specific architecture of cayley trees allows for analytical calculation of @xmath59 and evaluation of its dominant scaling . according to the structure of dendrimers , all the nodes in @xmath17 can be classified into @xmath77 levels . the central node is at level @xmath78 , and nodes born at the first generation @xmath18 are at level @xmath56 , and so forth . by symmetry , the trapping time for nodes at the same level is identical . in the case without confusion , we use @xmath58 to denote the trapping time for a node at @xmath18th level . then , the following relations hold : @xmath79+p_i^{\rm{down}}[1+f_{i+1}(n)],&{0<i < n},\\ 1+f_{i-1}(n),&{i = n}. \end{cases}\ ] ] in eq . ( [ c10 ] ) , @xmath80 represents the transition probability for a walker hopping from a node at level @xmath18 to its ( unique ) father node at level @xmath81 , while @xmath82 denotes the transition probability from any node at level @xmath18 to its @xmath14 child nodes at level @xmath83 . for unbiased random walk in @xmath4 @xcite , @xmath84 and @xmath85 , obeying relation @xmath86 however , as will be shown below , for merw in @xmath4 , @xmath80 and @xmath82 are different from those corresponding to unbiased random walk , although @xmath86 also holds . and the disparity for transition probability between these two kinds of random walks leads to quite different scalings for att . by definition of transition probability for merw in @xmath4 , see eq . ( [ d1 ] ) , @xmath80 and @xmath82 for merw in @xmath4 are @xmath87 and @xmath88 respectively . considering above - obtained results , @xmath80 and @xmath82 can be further recast as @xmath89 } { \sin\theta}}{(m-1)^{(n - i)/2}\frac{\sin[(n - i+1)\theta]}{\sin\theta } } \nonumber \\ & = & \frac{1}{2\cos\theta}\frac{\sin[(n - i+2)\theta]}{\sin[(n - i+1)\theta ] } \nonumber\\ & = & \frac{\sin[(n - i+2)\theta]}{\sin[(n - i+2)\theta]+\sin[(n - i)\theta]}\end{aligned}\ ] ] and @xmath90 } { \sin\theta}}{(m-1)^{(n - i)/2}\frac{\sin[(n - i+1)\theta]}{\sin\theta } } \nonumber\\ & = & \frac{1}{2\cos\theta}\frac{\sin[(n - i)\theta]}{\sin[(n - i+1)\theta ] } \nonumber \\ & = & \frac{\sin[(n - i)\theta]}{\sin[(n - i+2)\theta]+\sin[(n - i)\theta]}\,,\end{aligned}\ ] ] both of which evidently fulfil the relation @xmath86 , implying that our computation for @xmath80 and @xmath82 is correct . using eqs . ( [ c16 ] ) and ( [ c18 ] ) , eq . ( [ c10 ] ) becomes @xmath91[1+f_{i-1}(n)]}{\sin[(n - i+2)\theta]+\sin[(n - i)\theta ] } \\ + \frac{\sin[(n - i)\theta][1+f_{i+1}(n)]}{\sin[(n - i+2)\theta]+\sin[(n - i)\theta ] } , & { 0<i < n } , \\ 1+f_{i-1}(n),&{i = n}.\\ \end{cases}\ ] ] therefore , for @xmath92 , we have @xmath93+\sin[(n - i)\theta]\}}f_i(n)\nonumber \\ & = & \sin[(n - i+2)\theta][1+f_{i-1}(n)]\nonumber \\ & & + \sin[(n - i)\theta][1+f_{i+1}(n)]\,,\end{aligned}\ ] ] which can be rewritten as @xmath94[f_i(n)-f_{i-1}(n)]\nonumber \\ & = & \sin[(n - i+2)\theta]+\sin[(n - i)\theta ] \nonumber \\ & & + \sin[(n - i)\theta][f_{i+1}(n)-f_i(n)]\,.\end{aligned}\ ] ] in order to determine the quantity @xmath95 , we define @xmath96 then , @xmath97}+\frac{\sin{i\theta}}{\sin[(i+2)\theta]}b_{i-1}(n)\ ] ] holds for all @xmath92 . note that @xmath98-f_{n-1}(n ) \nonumber \\ & = & 1\,,\end{aligned}\ ] ] using which eq . ( [ c23 ] ) can be solved to yield @xmath99}{\sin[(i+2-k)\theta]}+1.\ ] ] making use of eqs . ( [ c22 ] ) and ( [ c25 ] ) , @xmath58 can be evaluated as @xmath100}{\sin[(n - j+2-l)\theta]}+1\bigg]\,,\end{aligned}\ ] ] where @xmath101 was used . equation ( [ c26 ] ) provides a closed form expression for trapping time for an arbitrary node . inserting eq . ( [ c26 ] ) into eq . ( [ d4 ] ) , we can obtain the closed - form formula of att @xmath95 for merw in @xmath4 with a single trap positioned at the central node , which reads @xmath102}{n-1 } \nonumber \\ & = & \frac{\displaystyle{\sum_{i=1}^n}\left\{m(m-1)^{i-1}\displaystyle{\sum_{j=1}^i}\left[2\displaystyle{\sum_{k=0}^{n - j-1}}\displaystyle{\prod_{l=0}^k}\frac{\sin[(n - j - l)\theta]}{\sin[(n - j+2-l)\theta ] } + 1\right]\right\}}{\frac{m[(m-1)^n-1]}{m-2}}\nonumber \\ & = & \frac{m-2}{(m-1)^n-1}\sum_{i=1}^n \left\{(m-1)^{i-1}\sum_{j=1}^i \left[2\sum_{k=0}^{n - j-1}\prod_{l=0}^k\frac{\sin[(n - j - l)\theta]}{\sin[(n - j+2-l)\theta]}+1 \right]\right\}\,.\end{aligned}\ ] ] in fig . [ trapping ] , we present results about the att @xmath95 to the central node for merw in @xmath4 . the results are obtained by using eqs . ( [ d5 ] ) and ( [ c27 ] ) , respectively . figure [ trapping ] shows that the results generated by eq . ( [ d5 ] ) and eq . ( [ c27 ] ) agree with each other , confirming our theoretical results for @xmath95 provided by eq . ( [ c27 ] ) . ( color online ) the att @xmath103 to the central node corresponding to merw in @xmath4 for various @xmath10 and @xmath3 . the hollow symbols represent the results generated by eq . ( [ d5 ] ) , while the solid symbols stand for the analytical results given by eq . ( [ c27 ] ) . ] although the expression for the att @xmath95 given in eq . ( [ c27 ] ) is exact , it is rather lengthy and awkward , from which we can not see obvious dependence of @xmath95 on the network size @xmath1 . however , we will show below that , when the networks are large enough , from eq . ( [ c27 ] ) one can derive an upper bound for the leading scaling of @xmath95 in terms of the network size @xmath1 . first , let us examine the product term @xmath104}{\sin[(n - j+2-l)\theta]}\ ] ] in eq . ( [ c27 ] ) . seemingly , there are @xmath105 multipliers in both the denominator and numerator in eq . ( [ c28 ] ) . in fact , for any @xmath106 we can decrease the number of multipliers from @xmath105 to two by simplifying eq . ( [ c28 ] ) as follows : @xmath107}{\sin[(n - j+2-l)\theta ] } \nonumber \\ & = & \frac{\sin[(n - j)\theta]\sin[(n - j-1)\theta]\cdots\sin[(n - j - k)\theta]}{\sin[(n - j+2)\theta]\sin[(n - j+1)\theta]\cdots\sin[(n - j+2-k)\theta ] } \nonumber \\ & = & \frac{\sin[(n - j+1-k)\theta]\sin[(n - j - k)\theta]}{\sin[(n - j+2)\theta]\sin[(n - j+1)\theta]}\,.\end{aligned}\ ] ] since @xmath108 , for any @xmath109 ( @xmath110 ) and @xmath111 ( @xmath112 ) , every term in both the denominator and numerator of eq . ( [ c29 ] ) has the form @xmath113 , where @xmath114 is an integer between @xmath56 and @xmath77 . thus , @xmath115 and @xmath116 . according to eq . ( [ b0 ] ) , we have @xmath117 where @xmath118 is a certain positive constant . note that if constant @xmath118 is chosen properly , the following relation also holds : @xmath119-[(n+1)(m-2)]}{(m-2)n+2m-2 } \nonumber \\ & > & \frac{1}{cn}\,.\end{aligned}\ ] ] on the other hand , when @xmath3 is large enough , @xmath120 . then , each term @xmath121 $ ] in the numerator of eq . ( [ c28 ] ) satisfies @xmath122>\sin \frac{1}{cn } = \frac{1}{cn}$ ] . thus , @xmath123}{\sin[(n - j+2-l)\theta]}\le \frac{1}{\frac{1}{cn } } \frac{1}{\frac{1}{cn } } \approx d\,n^2,\ ] ] where @xmath124 is another positive constant . substituting the result in eq . ( [ c32 ] ) into eq . ( [ c27 ] ) , we obtain @xmath125\bigg\ } \nonumber \\ & \le&\frac{m-2}{(m-1)^n-1}\sum_{i=1}^n(m-1)^{i-1}dn^4 \nonumber \\ & \le&\frac{m-2}{(m-1)^n-1}\frac{(m-1)^n-1}{m-2}dn^4 \nonumber \\ & \le&d\,n^4\,.\end{aligned}\ ] ] note that , eq . ( [ c2 ] ) enables us to represent @xmath3 in terms of the system size @xmath1 as @xmath126-\ln m}{\ln(m-1)}\,.\ ] ] hence , the upper bound for @xmath103 in eq . ( [ c33 ] ) can be expressed in the following form : @xmath127-\ln m}{\ln(m-1)}\right]^4 \nonumber \\ & \le & f(\ln n)^4\end{aligned}\ ] ] where @xmath128 is a certain positive constant . equation ( [ c35 ] ) provides an upper bound of dominant behavior of @xmath103 for merw in massive networks @xmath4 with the immobile trap located at the central node , which shows that the efficiency of trapping in dendrimers described by merw is very high . it is in sharp contrast with the att for unbiased random walk in the same networks , where the leading term of att increases lineally with the system size @xcite . thus , merw is instrumental in improving the efficiency of trapping process taking place in dendrimers . based on maximal entropy random walk ( merw ) we have performed an analytical research on trapping in dendrimers with a single trap placed at the central node . we have derived an explicit expression for att as an indicator of the trapping efficiency , grounded on which we have deduced an upper bound for the leading scaling of att . the obtained upper bound scales with the network size @xmath1 as @xmath2 , and is considerably lower than the att for trapping in dendrimers based on unbiased random walk , which grows lineally with the network size @xmath1 . this means that in dendrimers merw can dramatically lessen the att to the central node , making the diffusion process highly efficient . this theoretical work might prove useful to address light harvesting by dendrimers , as well as various other dynamical processes occurring in nanoscale systems . j. gmez - gardees and v. latora , phys . e * 78 * , 065102(r ) ( 2008 ) . w. parry , trans . soc . * 112 * , 55 ( 1964 ) . z. burda , j. duda , j. m. luck , and b. waclaw , phys . lett . * * 1**02 , 160602 ( 2009 ) .
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we use maximal entropy random walk ( merw ) to study the trapping problem in dendrimers modeled by cayley trees with a deep trap fixed at the central node .
we derive an explicit expression for the mean first passage time from any node to the trap , as well as an exact formula for the average trapping time ( att ) , which is the average of the source - to - trap mean first passage time over all non - trap starting nodes .
based on the obtained closed - form solution for att , we further deduce an upper bound for the leading behavior of att , which is the fourth power of @xmath0 , where @xmath1 is the system size .
this upper bound is much smaller than the att of trapping depicted by unbiased random walk in cayley trees , the leading scaling of which is a linear function of @xmath1 .
these results show that merw can substantially enhance the efficiency of trapping performed in dendrimers .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the theoretical expression for the angular two - point correlation function @xmath3 can be derived from its spatial counterpart @xmath4 by projection via the relativistic limber equation ( peebles 1980 ) : @xmath5 ^ 2 } , \label{eq : limber}\end{aligned}\ ] ] where @xmath6 is the comoving radial coordinate , @xmath7 ( for a flat universe and in the small angle approximation ) , and @xmath8 is the number of objects within the shell ( @xmath9 ) . + the mass - mass correlation function @xmath10 to be inserted in eq.([eq : limber ] ) has been obtained following the work by @xcite ( see also @xcite and @xcite ) , which provides an analytical way to derive the trend of @xmath10 both in the linear and non - linear regime . note that @xmath10 _ only _ depends on the underlying cosmology , which we fix by adopting @xmath11 , @xmath12 , @xmath13 and a cobe - normalized value of @xmath14 . the relevant properties of scuba galaxies are included in the redshift distribution of sources @xmath8 , and in the bias factor @xmath15 . the effective bias factor @xmath16 of all the dark matter haloes with masses greater than some threshold @xmath17 is then obtained by integrating the quantity @xmath18 ( whose expression has been taken from @xcite ) - representing the bias of individual haloes of mass @xmath19 - opportunely weighted by the number density @xmath20 of scuba sources : @xmath21 note that , as @xmath22 can be thought as the fraction of haloes hosting a galaxy in the process of forming stars , its expression can also be written as @xmath23 , where @xmath24 is the mass spectrum of haloes with masses between @xmath19 and @xmath25 ( @xcite ) , @xmath26 is the duration of the star - formation burst and @xmath27 is the life - time of the haloes in which these objects reside ( see @xcite ) . according to @xcite , sources showing up in the scuba counts can be broadly divided into three categories : low - mass ( masses in the range @xmath28 , duration of the star formation burst @xmath29 gyr , and typical fluxes @xmath30 mjy ) , intermediate - mass ( @xmath31@xmath32 and @xmath33 gyr ) and high - mass ( @xmath34 , @xmath35 gyr , dominating the counts at fluxes @xmath36 mjy ) . note that by @xmath37 we denote the mass in stars at completion of the star formation process . + in order to evaluate the bias factor in eq.([eq : bias ] ) we then consider two extreme cases for the ratio between the mass in stars and the mass of the host dark halo : @xmath38 and @xmath39 . @xmath40 roughly corresponds to the ratio @xmath41 between total and baryon density , where we adopted for the latter quantity the standard value from primordial nucleosynthesis ; this corresponds to having assumed all the baryons to be locked into stars and , as a consequence , has to be considered as a conservative lower limit . @xmath38 is instead related to @xmath42 , @xmath43 being the present mass density in visible stars : the likely value is expected to be @xmath44 % . armed with the above results we can then evaluate the two - point correlation function in eq . ( [ eq : limber ] ) for different @xmath45 ratios and different flux cuts . figure 1 presents our predictions for @xmath3 , respectively for a flux cut of 50 ( solid line ) , 10 ( dashed line ) and 1 ( dotted line ) mjy . higher curves of each kind correspond to the case @xmath38 , while lower curves refer to @xmath40 . + the highest clustering amplitude is found for the brightest sources ( @xmath46 mjy ) . this is because they are associated to the most massive dark halos and are therefore highly biased tracers of the dark matter distribution . in addition , according to @xcite , they have a rather narrow redshift distribution so that the dilution of the clustering signal is minimum . the very sharp drop of all the curves at @xmath47 is due to the absence of nearby objects . this reflects the notion that the actively star - forming phase in spheroids is completed at @xmath48 . + note that , since the clustering amplitude strongly depends on the quantity @xmath45 , measurements of the angular correlation function @xmath3 are in principle able to discriminate amongst different models of scuba galaxies and in particular to determine both their star - formation rate , via the amount of baryonic mass actively partaking the process of star formation , and the duration of the star - formation burst . a first attempt to measure the angular correlation function of @xmath49 mjy scuba sources has been recently presented by @xcite . although such measurements are dominated by noise due to small - number statistics , it is nevertheless interesting to note that as illustrated by the left - hand panel of figure 2 our model ( with a preference for the @xmath38 case ) shows full consistency with the data ( kindly provided by s. scott ) . another possible way to obtain some information on the nature of scuba sources via their clustering properties is provided by the predictions of @xcite for lyman - break galaxies to be the low- to intermediate - mass tail of primeval spheroidal galaxies , with @xmath50 gyr and a star - formation rate ranging from a few to a hundred @xmath51 . in figure 2 ( right - hand panel ) we then plotted the predicted @xmath3 for those sources with @xmath52 mjy ( corresponding to @xmath53 ) , expected to be found within the redshift range @xmath54 , covered by the original ( steidel et al . , 1996 ) sample . as in the former case , the higher curve is for @xmath38 , the lower one for @xmath40 . the data points show the @xcite measurements . even though large errors once again affect the observational findings , it is nevertheless clear that the predicted trend for @xmath3 can correctly reproduce the data for high ( @xmath55 ) values of the @xmath2 ratio . this result is consistent with the predictions by @xcite and implies a well defined relationship between scuba galaxies and lbgs . furthermore , it also confirms the expectations for a small fraction ( on the order of a few percent ) of the total mass to be confined into stars . finally , it is also worth noticing that our predictions are in agreement with the strong clustering of eros recently detected by @xcite , since we expect ( @xcite ) these objects to be the direct descendants of scuba galaxies , and therefore to exhibit the same clustering properties . an issue intimately connected with the analysis of galaxy clustering is the study of the contribution of unresolved sources ( i.e. sources with fluxes fainter than some detection limit @xmath56 ) to the background intensity . its general expression is given by : @xmath57 ( see e.g. @xcite ) , where @xmath58 denotes the differential number counts , @xmath59 and @xmath60 are respectively the maximum and minimum local luminosity of the sources , @xmath61 is the k - correction , @xmath62 is the redshift when the sources begin to shine , @xmath63 is the redshift at which a source of luminosity @xmath64 is seen with a flux equal to the detection limit @xmath56 , @xmath65 is the luminosity function ( i.e. the comoving number density of sources per unit @xmath66 ) , and @xmath6 is the comoving radial coordinate . the intensity fluctuation @xmath67 due to inhomogeneities in the space distribution of unresolved sources is then given by eq . ( [ eq : i ] ) , with the quantity @xmath65 replaced by @xmath68 . it is easily shown that the angular correlation of such intensity fluctuations @xmath69 , where @xmath70 and @xmath71 define two positions on the sky separated by an angle @xmath72 , can be expressed as the sum of two terms @xmath73 and @xmath74 , the first one due to poisson noise ( i.e. fluctuations given by randomly distributed objects ) , and the second one owing to source clustering . it is possible to show ( @xcite ) that , in the case of highly clustered sources , the poissonian term @xmath73 is negligible with respect to the one due to clustering . in the following we therefore only concentrate on temperature fluctuations caused by the @xmath74 term ( hereafter simply called @xmath75 ) . by making use of the quantities previously defined and of eq . ( [ eq : i ] ) , the clustering term @xmath75 takes the form : @xmath76 where the effective volume emissivity @xmath77 is expressed as : @xmath78}\!\!\!\!\ ! \phi(l , z)\ ; k(l , z)\ ; l\;d{\rm log}l \label{eq : jeff}\end{aligned}\ ] ] . @xmath79 in eq.([eq : cth ] ) has been evaluated separately for the three cases of low- , intermediate- and high - mass objects , by plugging in eq . ( [ eq : jeff ] ) the appropriate expressions for the luminosity function . the total contribution of clustering to intensity fluctuations has then been derived by adding up all the values of @xmath79 obtained for the different mass intervals and by also taking into account the cross - correlation terms between objects of different masses , according to the expression @xmath80 where the indexes @xmath81,@xmath82 stand for high , intermediate and low masses . + note that , the quantity @xmath83 in eq.([eq : cth ] ) should indeed be read as @xmath84 , where @xmath85 is the maximum halo mass corresponding to the maximum visible bulge mass ( i.e. upper limit for the mass locked into stars ) , which corresponds in eq.([eq : bias ] ) to a replacement in the upper limit of the integrals of @xmath86 with @xmath85 . the angular power spectrum of the intensity fluctuations can then be obtained via ( see @xcite ) : @xmath87 ^ 2{\rm exp}\left(- \frac{h\nu}{k_b t}\right)/\left(\frac{h\nu}{k_b t } \right)^2,\end{aligned}\ ] ] which relates intensity and temperature fluctuations . figure [ fig : dt ] shows the predicted values for the quantity @xmath88 ( in units of k ) at respectively 353 ghz ( @xmath89 m left - hand panel ) and 545 ghz ( @xmath90 m right - hand panel ) the central frequencies of two of the channels of the high frequency instrument ( hfi ) of the esa s planck mission as a function of the multipole @xmath91 up to @xmath92 . results are plotted for two different values of the source detection limit ( @xmath93 and 10 mjy for the 353 ghz case and @xmath94 and 45 mjy for the 545 ghz case ) and the usual two values of @xmath2 . also shown , for comparison , is the power spectrum of primary ( cmb ) anisotropies ( solid line ) predicted for the cosmology specified in the caption , computed with the cmbfast code developed by @xcite . at @xmath89 m , our model predicts fluctuations of amplitude due to clustering comparable to ( and possibly even larger than ) those obtained for primary cmb anisotropies at @xmath95 . this is because most of the clustering signal comes from massive galaxies with fluxes @xmath96 mjy , which lie at substantial redshifts and are therefore highly biased tracers of the underlying mass distribution . also the strongly negative k - correction increases their contribution to the effective volume emissivity [ eq . ( [ eq : jeff ] ) ] and therefore to the fluctuations . the @xmath90 m case is even more striking , since the contribution from clustering is expected to be more than an order of magnitude greater than the one originating from primordial fluctuations , regardless of the flux detection limit . + this implies that important information on the clustering properties of faint sub - mm galaxies ( and hence on physical properties such as their mass and/or the amount of baryons involved in the star - formation process ) will reside in the planck maps at frequencies greater than 353 ghz where , however , the dominant signal is expected to come from interstellar dust emission . in order to show this effect , in figure 3 we have also plotted the expected contribution from galactic dust emission averaged all over the sky ( upper dashed - dotted curves ) . this was derived from the iras maps at @xmath97 m , rescaled at the frequencies under exam by assuming a grey - body spectrum with @xmath98 k. as already anticipated , this signal appears to be the dominant one in both the 353 and 545 ghz planck channels . nevertheless , it is still possible to extract information on the nature of sub - mm galaxies if one restricts the analysis to high galactic latitude regions ( i.e. @xmath99 , lower dashed - dotted curves in figure 3 ) , which are the least affected by galactic dust emission . in fact , as it can be seen from figure 3 , the dust contribution in this region becomes less important than the one due to the clustering of unresolved sources for @xmath100 . daddi , e. , cimatti , a. , pozzetti , l. , hoekstra , h. , rttgering , h.j.a . , renzini , a. , zamorani , g. , mannucci , f. 2000 , a&a 361 , 535 de zotti , g. , franceschini , a. , toffolatti , l. , mazzei , p. , danese , l. , 1996 , ap . lett.comm . 35 , 289 giavalisco , m. , steidel , c.c . , adelberger , k.l . , dickinson , m.e . , pettini , m. , kellogg , m. 1998 , apj 503 , 543 granato , g.l . , silva , l. , monaco , p. , panuzzo , p. , salucci , p. , de zotti , g. , danese , l. 2001 , mnras 324,757 jing , y.p . 1998 , apj 503 , l9 magliocchetti , m. , bagla , j. , maddox , s.j . , lahav , o. 2000 , mnras 314 , 546 magliocchetti , m. , moscardini , l. , panuzzo , p. , granato , g.l . , de zotti , g. , danese , l. 2001 , mnras , accepted , astro - ph/0102464 martini , p. & weinberg , d.h . , 2001 , apj 547 , 12 moscardini l. , coles p. , lucchin f. , matarrese s. , 1998 , mnras , 299 , 95 peacock , j.a . & dodds s.j . , 1996 , mnras 267 , 1020 peebles , p.j.e . 1980 , the large - scale structure of the universe , princeton university press seljak , u. , zaldarriaga , m. 1996 , apj 469 , 437 . scott , s.e . , et al . , astro - ph/0107446 sheth , r.k . & tormen , g. 1999 , mnras 308 , 119 smail , i. , ivison , r.j . , blain , a.w . 1997 , apj 490 , l5
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the clustering properties of scuba - selected galaxies are investigated within the framework of a unifying scheme relating the formation of qsos and spheroids .
the theoretical angular correlation function is derived for different bias functions , corresponding to different values of the ratio @xmath0 between the mass of the dark halos hosting such galaxies and the mass in stars produced at the end of the major star - formation burst .
scuba sources are predicted to be strongly clustered , with a clustering strength increasing with mass .
comparisons with the best available measurements show better fits for @xmath1 . the model can also account for the clustering of lyman - break galaxies , seen as the optical counterpart of low- to intermediate - mass primeval spheroidal galaxies .
best agreement is once again obtained for high values of the @xmath2 ratio .
we also discuss implications for small scale fluctuations observed at different wavelengths by forthcoming experiments such as the planck mission aimed at mapping the cosmic microwave background ( cmb ) .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
during the last 70 years , significant progress has been achieved in the study of the physics in the interior of the sun and stars @xcite . this advance has been possible due to the great development of several key fields of experimental and theoretical physics , such as statistical physics , magneto - hydrodynamics , particle physics , and nuclear physics , among others . astronomers are now able to describe with high - precision the physics that takes place inside stars like the sun , as well as many other classes of stars , with masses different than that of the sun , and in quite different stages of stellar evolution . in general , the physics of stars is well understood since the first moments of their formation , up to the most advanced stages of stellar evolution , including the formation of highly compact objects , like white dwarfs , neutron stars and black holes . the progress in the understanding of the physical principles operating inside stars was accompanied and challenged by important developments in the observational fields of astrophysics , such as astrometry , photometry and high - resolution spectroscopy , as well as the new fields of helioseismology and asteroseismology , which create powerful tools to probe the interior of stars . the recent past has shown that the prosperity of modern stellar physics was possible due to the powerful partnership built between theoretical physics and astrophysics . in this new phase of stellar physics , the large amount of data made available by several observational projects permits to use the sun and stars as a tool to challenge our knowledge about fundamental physics,@xcite and in doing so it opens new branches of research , such as gravitation tests and probes of the existence of new particles . among other applications , to validate the new gravitational theories proposed as an alternative to general relativity @xcite , and to probe the existence of dark matter ( dm ) inside stars to investigate the dm problem @xcite . currently , the interest of studying the interaction of dm with stars is twofold : on one hand to identify which type of particles dm is made of , and , on the other hand , to understand the physical mechanisms by which dm contributes to the formation of stars . in the former , stars are used as a complementary tool to test dm candidates , being in that way an alternative method to test the candidates proposed by modern theories of particle physics , or alternatively , to test candidates detected by experiments of direct or indirect dm searches . in the latter , the aim is to explore how dm contributes for the structure formation in the universe , comprising galaxies and the first generation of stars , not only by locally changing the gravitational field where stars are formed , but also to explore how the interaction of dm with baryons changes the evolution of stars . our universe is constituted by 5% of _ baryonic matter _ , a type of matter in which we have become great experts during the last two centuries , by developing several branches of physics ; 27% is constituted by _ dark matter _ , which plays a major role in the formation of structure in the universe , but its fundamental characteristics are yet poorly understood ; and another 68% of the total energy density of the universe is usually refereed to as _ dark energy _ , which physical origin is even more uncertain @xcite . although the basic properties of dm are not known , namely , which type of particles is dm made of , there is strong evidence of its existence , both from astrophysical and cosmological observations , as well as from numerical simulations @xcite . among other direct evidence of the existence of a gravitational field caused by the presence of dm , we make reference to the velocity of galaxies in clusters , the rotation curves of galaxies , the cosmic microwave background anisotropies , the velocity dispersions of dwarf spheroidal galaxies and the inference of the dm by gravitational lensing . all these observational and theoretical results suggest that most of the formation of structure in our universe can only be explained by the presence of a gravitational field caused by the presence of a new type of particles that must be non - baryonic and cold @xcite , such as the particles belonging to the group of the wimps ( for weakly interacting massive particles ) . therefore , it is no surprise that with such an amount of observational evidence for the existence of dm , a large effort is being devoted to theoretical work and experimentation in several branches of astrophysics , cosmology and particle physics , with the intent of discovering such fundamental particles . if dm particles exists , then in the near future we should expect to detect such particles in the large hadron collider at cern or in other direct detection experiments . another possibility is the confirmation of the existence of dm by the indirect detection of dm by - products , like the production of high - energy neutrinos or gamma rays caused by the annihilation of dm pairs . in the last 30 years , several classes of particles have been proposed as dm candidates , among others , axions , wimps , asymmetric dm particles , and other more exotic types of matter . at present , two groups of particles merit special attention , because they sum up most of the critical properties necessary to be the ideal dm particle . first , the well known weakly interacting massive particles ( wimps ) , which interact gravitationally with other particles and have weak interaction with baryons . wimps are among the most popular dm candidates . such class of particles occur in several extensions of the standard model of particle physics , like super - symmetric ( susy ) models.@xcite in such models , the lightest susy particle , the neutralino , a stable particle with a self - annihilation cross section of the order of the weak - scale interaction , is the most suitable candidate for dm . the second type of candidates are known as asymmetric dm particles,@xcite which like wimps have interactions with baryons at the weak - scale , even if they do not self - annihilate inside compact objects.@xcite unlike wimps , these particles carry a conserved charge analogous to the baryon number asymmetry . as a consequence , dm becomes asymmetric , _ i.e. _ , there is an unbalanced amount of particles and antiparticles , introducing an asymmetric parameter in the dm sector identical to baryon - anti - baryon asymmetry parameter , the so - called baryonic asymmetry . furthermore , these particles are expected to have a mass of the order of a few gev @xcite . in recent years , several underground experiments have been built to search for direct signatures of the interactions between dm particles and a baryons @xcite . most of such dm searches have not detected any dm signal . this is the case of experiments like xenon10/100,@xcite picasso,@xcite simple,@xcite and lux.@xcite however , in disagreement with such results are the positive detections of dama / libra,@xcite cogent,@xcite cresst - ii @xcite and cdms ii @xcite experiments , which all found evidence of events that can be credited to dm particles with similar properties . the former two experiments report evidence of an annual modulation in the differential event rate , which is explained as a consequence of the motion of the earth around the sun , which in turn moves through a cloud of dm particles @xcite . nevertheless , these results remain utterly controversial and further experimental work must be done in order to converge on a plausible answer about these detections . a possible solution to accommodate all experimental results comes from a new theoretical interpretation of the interaction of dm particles with baryons in the detectors . the simpler interpretation , used so far to analyse the data , is to assign the annual modulation to a collision of a dm particle with nucleons inside the detector . in such scenario , the dm particle is estimated to have a mass of the order of a few gev ( likely between @xmath0 and @xmath1 gev ) , and a dm - nucleon scattering cross - section of the order of @xmath2 for spin - independent ( si ) interactions or @xmath3 for spin - dependent ( sd ) interactions on protons . dama and cogent experiments show very similar positive detections , yet the dama experiment is favourable to a larger scattering cross - section than the cogent experiment . nevertheless , this problem is resolved by several theoretical solutions that have been proposed to overcome the inconsistency between the different experimental results @xcite . under such interpretations the data obtained by the different experiments can be reconciled . among the various possible theoretical explanations , one of the more appealing suggestions is the possibility that the dm particle couples unequally to the protons and neutrons of the collision nuclei because of isospin violation . usually , the dm particle is considered to couple equally with protons and neutrons . accordingly , the si scattering cross - section of heavy elements scales with @xmath4 , where @xmath5 is the atomic number of the nucleus . if the dm particles couple differently to protons and neutrons , that leads to a quite distinct interpretation of direct dm searches , leading to the reconciliation of almost all the experiments @xcite . this nuclear mechanism is known as isospin coupling violation @xcite . in such cases , the proton scattering cross section for the si interaction increases by @xmath6 to @xmath7 relatively to the usual interpretation of the experimental results , leading to an effective si scattering cross - section with values between @xmath8 and @xmath9.@xcite . other models propose momentum and velocity - dependent interactions to explain the data.@xcite we point out here that these types of dm particles can modify the internal properties of stars like the sun by conducting energy very efficiently,@xcite leading to a quite different flux of solar neutrinos @xcite and helioseismology data @xcite . dark matter impacts the evolution of a star by means of two mechanisms : through the change of the transport of energy @xcite and by the creation of an additional source of energy.@xcite the former mechanism can even be important for stars in our galactic neighbourhood , like the sun @xcite and other main - sequence stars.@xcite the latter mechanism is more pronounced in environments with a very high dm density , several million times the dm density of the solar neighbourhood . this type of scenario occurs in stellar populations located in the centre of galaxies , including the milk way,@xcite or during the formation of the first generation of stars in the primordial universe.@xcite on the other hand , the gravitational influence of the dm mass accumulated in the interior of stars similar to the sun is several tens of orders of magnitude smaller than the star s total mass , so it can be neglected . most dm particles cross the stellar interior without undergoing any type of interaction , yet a few of them scatter off nuclei losing part of their kinetic energy . in some cases , the loss of energy will result in the particle being captured by the star , _ i.e. _ , the particle can no more escape the star s gravitational field . as expected , a star with a large mass has a large gravitational field , and consequently captures a larger amount of dm during its evolution . the capture rate of dm particles by a star is inversely proportional to the mass and the dispersion velocity of the dm particle , but proportional to the local dm density of the halo and the scattering cross - section off baryons @xcite . two leading parameters define the scattering of the dm particles with the nuclei of the stellar isotopes : the sd scattering cross - section that is only relevant for hydrogen ; and the si scattering cross - section that defines the interaction of the dm particles with the heavy nuclei . the values of the scattering cross sections used in the calculations with stellar codes are usually the maximum allowed by the null results of direct detection experiments , or those in agreement with the values suggested by the interpretation of positive experimental results . however , it is worth noticing that , if the value of the si scattering cross - section is larger than a hundreth of the sd scattering cross - section , then the capture of dm particles is dominated by collisions with heavy nuclei , rather than by collisions with hydrogen @xcite . the scenario that is usually considered corresponds to a dm halo with a density in the solar neighbourhood of @xmath10 , constituted by particles with a mass of a few gev and with a maxwellian velocity distribution with a dispersion of @xmath11 . the interactions of dm particles with stars have been implemented in a stellar code @xcite that explicitly follows the capture rate of the dm particles by the different chemical elements present inside stars , some of them changing in isotopic abundance during the star s evolution . capture rates of dm were first calculated by ref . in the case of the sun , by ref . for generic massive bodies , and by ref . for main sequence stars . presently , the capture rate is computed numerically from the integral expression of ref . implemented as indicated in ref . . a detailed discussion about the dependence of the capture on the properties of dm particles , as well as the impact that the uncertainties in the dm and stellar parameters have on the capture rate by the sun and other stars is presented in ref the accumulation of dm inside a star leads to the formation of a very small dm core , with a radius of the order of the few percent of the stellar radius . the impact on the stellar properties comes from the additional energy transport mechanism provided by dm conduction , or by the dm self - annihilation . in both cases , the evolution of the star is only affected when the changes produced by the dm particles compete with the classical transport or production of energy mechanisms . mostly , only low - mass stars can be influenced , and the impact of dm becomes insignificant for more massive stars . the efficiency of the energy transport provided by dm particles depends on the average distance travelled by particles between consecutive collisions , _ i.e. _ , the mean free path of the dm particle . if the mean free path is short compared with the dm scale height , then the stellar plasma and the dm are in local thermal equilibrium . alternatively , if the mean free path is large , the successive collisions are widely separated , so that the energy transfer proceeds within the knudsen regime @xcite . the energy generation rate due to pair annihilation of dm particles is more significant in dm halos of high density . in such cases , the dm in the stellar core provides an extra source of energy . it follows that every pair of captured dm particles annihilates , being converted into additional energy for the star . without loss of generality , all the products of dm annihilation , except neutrinos , are assumed to interact with the stellar plasma , so these new particles have a very short mean free paths in the star s core and rapidly reach the thermal equilibrium . the process can be very efficient : for each annihilation pair most of the energy is converted in thermal energy and only a small fraction is lost in the form of high - energy neutrinos that escape the star s gravitational field . recent simulations of the self - annihilation of two neutralinos ( which are majorana particles in most susy models ) have shown that the energy loss in a star as the sun could be as low as 10% of the total energy produced by the annihilation of the dm pair @xcite . [ cols="^,^ " , ] in other scenarios , the dm annihilation provides stars with an extra source of energy that can dramatically change their evolution path.@xcite however , environmental dm densities millions of times greater than the local dm density are needed for these important modifications to occur . these huge dm densities are only expected in very particular locations in the local universe , such as the galactic center or the dwarf spheroidal galaxies , and in the early stages of the universe . the study of the impacts of dm in stars in environments with high dm densities is subjected to important difficulties from the observational side and larger uncertainties from the theoretical side . the most important ones are those on the stellar mass and velocity , as well as on the density of dm around the star , which strongly depends on the model . on the other hand , in these cases the potential effects of dm on low - mass stars can be dramatic : the creation of an unexpected convective core that may be detected with asteroseismology ( see figure [ fig - seism_isoc ] left),@xcite the evolution through a different path in the hertzsprung - russell diagram at a lower evolutionary speed,@xcite and the changes in the global properties of a whole cluster of stars ( see figure [ fig - seism_isoc ] right).@xcite similarly , the first generation of stars may have also been strongly influenced by the presence of dm @xcite . in the last fifteen years , the use of the sun and stars as cosmological tools to probe and test the dm particle candidates has become a regular procedure in dm research . this is testified by the arrival of several groups working in such a new research field @xcite , as well as by a significant increase on the number of publications in this subject . the increase of our knowledge of the physics of the solar interior made possible by the solar neutrinos as well as by helioseismolgy will allow accurate tests of the existence of different dm particle candidates , and also the extension of such studies to other fields of fundamental physics , such as alternative theories of gravitation . presently , with the large amount of data made available by observational asteroseismology , mostly by the spacial missions kepler@xcite and corot , the pulsation spectrum has been identified and measured in more than ten thousand stars . some of these stars are identical to the sun , but most of them have quite different masses and they are in distinct phases of stellar evolution . with this large and diverse set of stars it should be possible to further constrain the properties of the dm particles . the approach highlighted here provides a complementary contribution to the multidisciplinary effort of dm research . only a collaborative work across the several fields involved will allow the scientific community to produce fruitful results , and in doing so to overcome one of the most exciting and difficult problems of modern astrophysics , particle physics and cosmology , which is the discovery of the constituent particle(s ) of dm . j.c . acknowledges the support from the alexander von humboldt foundation . i.l . acknowledges the support from the fundao para a cincia e tecnologia .
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during the last century , with the development of modern physics in such diverse fields as thermodynamics , statistical physics , and nuclear and particle physics , the basic principles of the evolution of stars have been successfully well understood . nowadays , a precise diagnostic of the stellar interiors is possible with the new fields of helioseismology and astroseismology .
even the measurement of solar neutrino fluxes , once a problem in particle physics , is now a powerful probe of the core of the sun .
these tools have allowed the use of stars to test new physics , in particular the properties of the hypothetical particles that constitute the dark matter of the universe . here
we present recent results obtained using this approach .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the missions of @xmath8factories under constructions are ( i ) to test the cp violation in the standard model ( sm ) _ la _ kobayashi - maskawa scheme @xcite , and ( ii ) to find out any new flavor violation and especially new source of cp violation beyond the km phase in the sm with three generations . the latter is well motivated by the fact that the km phase in the sm may not be enough to generate the baryon number asymmetry in the universe . in terms of physics view point , the second mission seems more exciting one , since it could uncover a veil beyond the sm and provide an ingredient that is necessary to explain baryon number asymmetry of the universe . then , one has to seek for a possible signal of new physics in rare decays of @xmath8mesons and cp violation therein . one could choose his / her own favorite models to work out the consequences of such model to the physics issues that could be investigated at b factories . or one could work in the effective field theory framework , in a manner as much as model - independent as possible . in the following , we choose the second avenue to study the possible signals of new physics that could be studied in detail at @xmath9 factories . then we give explicit examples ( that satisfy our assumptions made in the model independent analysis ) in supersymmetric ( susy ) models with gluino - mediated @xmath10 transition . if one considers the sm as an effective field theory ( eft ) of more fundamental theories below the scale @xmath11 , the new physics effects will manifest themselves in higher dimensional operators ( dim @xmath12 \geq 5 $ ] ) that are invariant under the sm gauge group . several groups have made a list of dimension-5 and dimension-6 operators in the last decade @xcite . assuming the lepton and baryon number conservations , there are about 80 operators that are independent with each other . it would be formidable to consider all of such operators at once , even if we are interested in their effects in @xmath9 physics . however , if we restrict to @xmath13 , only two operators become relevant : @xmath14 after the electroweak ( ew ) symmetry breaking ( @xmath15 is the higgs vacuum expectation value ) . here s are dimensionless coefficients . thus the above operators can be recasted into the following form : @xmath17,\ ] ] where @xmath18 in the wolfenstein parametrization @xcite ) and @xmath19 the operator @xmath20 is obtained from @xmath21 by the exchange @xmath22 . similarly one can expect a new physics contribution to @xmath23 : @xmath24,\ ] ] where @xmath25 and @xmath26 is obtained from @xmath27 by the exchange @xmath28 . these two processes @xmath29 and @xmath23 are unique in the sense that they are described in terms of only two independent operators @xmath30 and @xmath31 whatever new physics there are . this fact makes it easy to study these decays in a model indepdent manner @xcite . the sm predictions for the @xmath32 at the @xmath33 scale are ( in the limit @xmath34 ) @xmath35 note that @xmath36 in the sm is suppressed compared to @xmath37 by @xmath38 , because @xmath39 boson couples only to the left - handed fermions . such terms proportional to @xmath40 will be neglected in our work by setting @xmath34 whenever they appear . on the other hand , this chirality suppression needs not be true in the presence of new physics such as left - right symmetric ( lr ) model or in a certain class of supersymmetric models with specific flavor symmetries . such new physics contributions can be parametrized in terms of four complex parameters , @xmath41 where @xmath42 are new complex numbers , whose phases parametrize the effects of the new sources of cp violation beyond the km phase in the sm . the sm case corresponds to @xmath43 and @xmath44 . it is convenient to define the ratio @xmath45 as following : @xmath46 in many interesting cases , this parameter @xmath45 is real @xcite as assumed in this work . implications of new physics contributions to @xmath23 have been discussed by various group in conjunction with the possible solutions for the discrepancies between theoretical expectations and the data on the semileptonic branching ratio of and the missing charms in @xmath9 meson decays , and the unexpectedly large branching ratio for @xmath47 . it has been advocated that @xmath48 $ ] can solve these problems simultaneously @xcite . however , this claim is now being challenged by the new measurement @xmath49 cl ) @xcite . in this work , we impose this new experimental data , rather than assume that the @xmath50 is large enough to solve the aformentioned puzzles in @xmath9 decays . in the presence of new physics contributions to @xmath13 , there should be also generic new physics contributions to @xmath51 through electromagnetic penguin diagrams . this effect will modify the wilson coefficient @xmath52 of the dim-6 local operator @xmath53 : @xmath54 , \ ] ] where @xmath55 in the sm , the wilson coefficients @xmath56 s are given by @xmath57 let us parametrize the new physics contribution to @xmath58 in terms of @xmath59 ( or @xmath60 ) as following : @xmath61 since we assume that the new physics modifies only @xmath62 and @xmath23 , we have @xmath63 . penguin contribution to @xmath0 is supressed relative to the photonic penguin by a factor of @xmath64 , and thus neglected in this work . ] there is no model - independent relation between @xmath65 and @xmath59 , although they are generate by the same feynman diagrams for @xmath66 . in sec . iv , we will encounter examples for both @xmath67 and @xmath68 in general susy models with gluino - mediated flavor changing neutral current ( fcnc ) . in principle , there are many more dim-6 local operators that might contribute to @xmath0 @xcite . in the presence of so many new parameters , it is difficult to figure out which operators are induced by new physics , since we are afforded only a few physical observables , such as @xmath69 and the tau polarization asymmetry @xmath70 in @xmath6 . therefore , it would be more meaningful to consider the simpler case before we take into account the most general case to figure out which operators are significantly affected by new physics . up to now , we considered @xmath71 and @xmath72 relevant to @xmath73 , assuming new physics significantly contributes to @xmath1 and @xmath2 through dim-5 operators , eqs . ( 2)(5 ) . in doing so , five more complex numbers ( @xmath74 ) have been introduced . if we further assume that the new physics does not induce new operators that are absent in the sm , we can drop @xmath75 by setting @xmath76 , thereby reducing the number of new parameters characterizing new physics effects into three complex numbers @xmath77 s ( or , equivalently @xmath78 and @xmath79 . still the number of new parameters are larger than the physical observables at our disposal . however , in many interesting cases ( and especially susy models with gluino - mediated @xmath80 transition that is to be described in sec . iv ) , it turns out that both @xmath45 and @xmath60 are real . therefore , we will assume that both @xmath45 and @xmath60 are real hereafter , and we are end up with 4 real parameters , which we choose to be @xmath81 and @xmath60 . then we can overconstrain these parameters from the following observables : * the branching ratio for @xmath82 relative to the sm prediction @xmath83 * the direct cp violation in @xmath82 ( @xmath84 * the branching ratio for @xmath85 relative to the sm prediction @xmath86 * the branching ratio for @xmath87 relative to the sm prediction @xmath88 * the forward - backward asymmetry in @xmath87 ( @xmath89 ) * the tau polarization asymmetry in @xmath90 ( @xmath91 ) at this point , it is timely to recall that there have been several works on the model - indepedent determination of the wilson coefficients , @xmath92 from @xmath93 and the kinematic distributions in @xmath94 @xcite @xcite . our work is different from these previous works in a few aspects . first of all , we include the possibility that there is a new physics contribution to @xmath95 with a new cp violating phase ( i m @xmath96 ) . this necessarily calls for studying the direct cp violation in @xmath82 as advocated by kagan and neubert @xcite , and invalidates the most previous works on the model - independent determination of @xmath97 s . secondly we include the recent experimental constraint on @xmath98 , instead assuming that it can be large enough to solve the semileptonic branching ratio problem in @xmath9 decays . finally , we assume that the new physics does not introduce any new operators with chiralities different from those in the sm , and simply modifies the wilson coefficients of @xmath99 . thus our analysis does not consider the left - right symmetric extension of the sm . this paper is organized as follows . in sec . ii , we give basic formulae for the relevant physical observables such as @xmath93 , @xmath100 , etc . as functions of four real parameters , @xmath101 and @xmath60 . in sec . iii , we present the model - independent numerical analysis for both @xmath102 and @xmath103 cases . we show the possible ranges of @xmath104 etc . , when we impose the experimental data on @xmath93 and @xmath105 . in sec . iv , we discuss explicit susy models with gluino - mediated fcnc that enjoy the several assumptions we make in this work . the results of this work are summarized in sec . in the sm , the branching ratios for @xmath82 and @xmath107 are obtained including the @xmath108 corrections and the nonperturbative effects of @xmath109quark s fermi motion inside the @xmath9 meson . relegating the details to the recent works by kagan and neubert @xcite , we show the final expressions that will be used in the following : @xmath110 + r_2 ( \chi ) [ \\ r_{g } \equiv { b(b\rightarrow x_{sg } ) \over b_{\rm sm } ( b\rightarrow x_{sg } ) } = 1 + r_3 ( \chi ) [ { \rm re } ( \xi_7 ) - 1 ] + r_4 ( \chi ) [ @xmath111 gev for the case of @xmath93 ) , the functions @xmath112 s can be approximated by @xcite @xmath113 the recent cleo data @xmath114 and the sm predictions on these decays ( @xmath115 ) imply that @xmath116 cp violation in the inclusive @xmath82 ( @xmath117 gev ) is characterized by cp asymmetry , , whereas those at the @xmath33 scale are written as @xmath118 explicitly . ] @xmath119 - 9.52 ~{\rm i m } [ c_8 c_7^ * ] + 0.10 ~{\rm i m } [ c_2 c_8^ * ] \right\}~(\%)~ , \nonumber \\ & = & \frac{a_1 ( \chi ) ~{\rm im}(\xi_7 ) } { a_2 ( \chi ) + a_3 ( \chi)~ | \xi_7 |^2 + a_4 ( \chi ) ~{\rm re } ( \xi_7 ) } ~(\%)~,\end{aligned}\ ] ] where @xmath120 now let us consider the decay @xmath121 , which occurs through the electroweak penguin diagrams and the box diagrams in the sm . if there is a new physics beyond the sm , there would be generically dim-6 operators with chiralities different from @xmath122 shown above through the electroweak penguin diagrams and the box diagrams . considered effects of such new operators ( 10 operators ) on the branching ratio and the forward - backward asymmetry ( @xmath123 ) in @xmath124 @xcite . in our opinion , it would be more meaningful to consider the effects of modified @xmath32 on the decay @xmath121 , since they are generically given by dim.-5 local operators . especially the effects of @xmath95 is enhanced by @xmath125 factor in the low @xmath126 region ( see the third line of eq . ( [ eq : bsll ] ) below ) . in any rate , we assume that the new physics does not introduce new operators with chiralities different from those in the sm , so that we assume that the new physics affects the @xmath0 only through modification of the @xmath1 . therefore , the wilson coefficients @xmath127 may change ( with a new cp - violating phase ) , and @xmath128 will not be affected at all in our case . the differential branching ratio for @xmath129 is given by @xcite @xmath130 where all the wilson coefficients are evaluated at @xmath131 by the renormalization group equations , @xmath132 , the function @xmath133 is the phase space factor for the semileptonic @xmath134 decays , and the function @xmath135 defined as @xmath136\ ] ] is the qcd correction factor thereof . the effective wilson coefficient @xmath137 is defined as @xmath138 where @xmath139 s are the wilson coefficients at @xmath131 in the leading logarithmic approximation : @xmath140 with @xmath141 , and the numbers @xmath142 s and @xmath143 s are given in table xxvii in ref . the functions @xmath144 s and @xmath145 are @xcite @xmath146 , \nonumber \\ \alpha_2 ( x , y , z ) & = & \left [ - 2 x^2 + x ( 1+y ) + ( 1-y)^2 \right ] + { 2 z \over x}~\left [ 4 x^2 - 5 ( 1 + y ) x + ( 1 - y ) ^2 \right ] \nonumber \\ \alpha_3 ( x , y , z ) & = & \left ( 1 + { 2 z \over x } \right)~\left [ - ( 1 + y ) x^2 - ( 1 + 14 y + y^2 ) x + 2 ( 1 + y ) ( 1 - y)^2 \right ] , \nonumber \\ \alpha_4 ( x , y , z ) & = & \left ( 1 + { 2 z \over x } \right)~\left [ ( 1 - y ) ^2 - ( 1 + y ) x \right ] , \nonumber \\ \omega ( \hat{s } ) & = & \sqrt { \left [ \hat{s } - ( 1 + \hat{m_s } ) ^2 \right]~ \left [ \hat{s } - ( 1 - \hat{m_s } ) ^2 \right ] } , \end{aligned}\ ] ] with @xmath147 . and @xmath148 is given by @xcite @xmath149 the function @xmath150 represents the @xmath108 corrections of the matrix elements , whose explicit form can be found at ref . the new physics contributions can induce @xmath151 through @xmath152 . this will modifies the wilson coefficients @xmath153 s , whose effects can be seen in the direct cp violation in the @xmath9 decay amplitude . however these will not affect @xmath13 and @xmath0 at the order we are working on . for the realistic prediction , one also has to include the long distance contribution through @xmath154 followed by ( @xmath155 . this can be taken into account by adding to the perturbative @xmath150 the resonance contributions @xcite : @xmath156 with @xmath157 . to avoid the large contributions from the @xmath158 and @xmath159 resonances , we consider the following two regions : the low @xmath160 region , @xmath161 gev@xmath162 for @xmath163 case , and the high @xmath160 region , @xmath164 for @xmath165 . using these informations , it is straightforward to evaluate @xmath100 : @xmath166 % + b_2 ( \chi ) { \rm i m } ( \xi_7 ) \nonumber \\ % & + & b_3 ( \chi ) [ | \xi_7 |^2 - 1 ] , \end{aligned}\ ] ] for the decay @xmath167 , @xmath168 with @xmath169 for the decay @xmath90 , @xmath170 with @xmath171 another interesting observable at b factories is the forward - backward asymmetry of the lepton in the center of mass frame of the lepton pair : @xmath172 ~d^2b / d\hat{s } d\cos\theta } \over { [ \int_{0}^1 d(\cos\theta ) + \int_{-1}^0 d(\cos\theta ) ] ~d^2b / d\hat{s } d\cos\theta } } \nonumber \\ & = & - { 3 \omega(\hat{s } ) \sqrt{1 - 4 \hat{m_l}^2 / \hat{s } } ~c_{10 } { \rm re}~\left\ { \hat{s } [ c_9 + y ( \hat{s } ) ] + 2 c_7 \right\ } \over \left\ { | c_9 + y(\hat{s } ) |^2 \alpha_1 + c_{10}^2 \alpha_2 + ( 4/\hat{s } ) c_7 ^ 2 \alpha_3 + 12 \alpha_4 { \rm re } c_7 [ c_9 + y(\hat{s } ) ] \right\ } } , \end{aligned}\ ] ] where @xmath173 is the angle between the positively charged lepton and the @xmath9 flight direction in the rest frame of the dilepton system . for the decay @xmath167 , the integrated forward - backward asymmetry is given by @xmath174 where @xmath175 for the decay @xmath90 , @xmath176 where @xmath177 the last observable we discuss is the tau polarization asymmetry @xmath178 in @xmath90 defined as @xcite @xmath179 \over \left\ { | c_9 + y(\hat{s } ) |^2 \alpha_1 + c_{10}^2 \alpha_2 + ( 4/\hat{s } ) c_7 ^ 2 \alpha_3 + 12 \alpha_4 { \rm re } c_7 [ c_9 + y(\hat{s } ) ] \right\ } } \quad .\end{aligned}\ ] ] the integrated tau polarization asymmetry @xmath180 can be expressed as @xmath181 where @xmath182 since @xmath9 decays into the tau pair probes high @xmath183 gev ) region , the observable @xmath70 is sensitive to the deviation of @xmath58 from their sm values which dominates the @xmath184 at high @xmath126 region . now we are ready to do a model - independent analysis using the formulae obtained in the previous section . there are two different cases depending on @xmath67 or not . in principle , any new physics contributing to magnetic form factor in @xmath29 may affect the electric form factor as well . therefore one would expect generically @xmath103 . however this needs not be necessarily true as discussed in the next section ( the case ( i ) ) . so we discuss @xmath67 and @xmath103 separately in this section . our strategy is the following : impose the experimental data on @xmath93 and @xmath98 : * * e1 * : @xmath185 as in ref . @xcite * * e2 * : @xmath186 @xcite for given @xmath45 and @xmath60 , these constraints ( e1 ) and ( e2 ) determine the allowed region in the complex @xmath65 plane . then , in the allowed @xmath65 plane , one can calculate other physical observables , @xmath187 and @xmath70 . because the number of observables are greater than the number of unknown parameters ( one complex number @xmath65 and two real numbers @xmath45 and @xmath60 ) , one can overconstrain these 4 real parameters . if there is no consistent solution , there would be a few possibilities : @xmath45 and/or @xmath60 may be complex , @xmath128 is modified by new physics effects , or one has to enlarge the operator basis by including operators with different chiralities from those in the sm , as in ref . @xcite . let us first consider the case with @xmath67 . in fig . [ fig_ch0 ] , we show the scattered plots of various observables as functions of @xmath93 for @xmath188 . the sm case is denoted by a square , possible values in our model are represented by dots , whereas the filled circles represent the case where there is no new cp violating phase , namely @xmath189 , but @xmath190 . implications of these figures are clear . for example , the cp asymmetry in @xmath29 can not be larger than @xmath191 if @xmath192 , and @xmath193 can be anywhere between 0.98 to 2.2 . for comparison , let us discuss the minimal sugra model with universal soft mass terms at gut scale , in which typical values of @xmath45 and @xmath60 are @xmath194 and @xmath195 respectively @xcite . therefore , the predictions in the minimal sugra model are very close to the dots in fig . [ fig_ch0 ] . namely , in the sugra case , there are two bands for the possible @xmath193 for a given @xmath93 , whereas in our case , @xmath193 can be anywhere in between becasue of the presence of a new cp - violating phase given by @xmath196 . in figs . [ fig_ch5 ] and [ fig_ch-5 ] , we show similar plots for @xmath197 and @xmath198 , respectively . this choice of @xmath45 covers a large class of new physics as discussed in ref . implications of these figures are almost the same as fig . 1 , except that there is now rather strong constraint from @xmath85 ( e2 ) . in this case we can have larger direct cp violation in @xmath82 upto @xmath199 . also the ( e2 ) constraint removes substantial parts of available @xmath200 and @xmath201 as shown in fig . [ fig_ch5 ] ( @xmath197 ) , compared to fig . [ fig_ch0 ] where the constraint ( e2 ) was not imposed . this effect is much more prominent for negative @xmath45 as shown in fig . [ fig_ch-5 ] ( @xmath202 ) . for example , the @xmath203 correlation is almost identical to the case with vanishing new phase @xmath204 . from figs . [ fig_ch0][fig_ch-5 ] , it is clear that the existence of a new cp violating phase not only can generate a large cp asymmetry in @xmath205 , but can it also induce quite a lot deviations of various observables in @xmath0 for @xmath206 and @xmath207 . for @xmath188 and @xmath197 , deviations of the observables @xmath208 from their sm values can be large enough that they can be clearly observed at future b factories , whereas deviations of other observables @xmath209 and @xmath70 from their sm values are rather small that it would be very difficult to measure them . for @xmath202 , only @xmath210 and @xmath211 shows substantial deviations from the sm values because of the ( e2 ) constraint again . if the experimental data on @xmath212 and @xmath180 show large deviations from their sm values , it would indicate that @xmath45 and/or @xmath60 are complex , or some new physics contributes to @xmath128 ( with a possibly new cp violating phase ) , and/or even generates @xmath213 and possibly other dimension-6 @xmath214 operators with different chiralities from @xmath122 in the sm . the nonvanishing @xmath60 does not affect @xmath13 and @xmath23 so that the allowed region in the complex @xmath65 plane remains the same as before , for a given @xmath45 . however , it does change the observables related with @xmath121 , and we show them in fig . [ fig_ch5pr03 ] for @xmath215 , where we chose @xmath216 that is typical in the gluino - mediated susy models considered in the next section . the @xmath217 dependence on @xmath93 differ from those in fig . [ fig_ch5pr03 ] , and the possible deviations of these observables from their sm values are smaller if @xmath218 . if there is no new cp violating phase , the differences are so tiny that one may not be able to distinguish two cases in practice . the message of this model - independent study is that the previous methods @xcite-@xcite has to be enlarged to include a new observable @xmath210 that could be sensitive to a new cp violating phase . in the presence of such a new phase , simple correlations among various observables in @xmath3 and @xmath87 ( namely , correlations among @xmath219 with ( @xmath220 ) and @xmath70 that are represented as thick dots in figs . 1 - 4 simply disappear , and there is no more apparent correlations among these observables . still one can perform a global analysis as before using the formulae given in the previous sections , including the observable @xmath210 . this will provide additional information and one can overconstrain four real parameters , @xmath221 and @xmath60 . if there is no consistent solution for these four real parameters , one has to consider the possibility that @xmath45 and/or @xmath60 are also complex . in this case one may be able to determine the wilson coefficients , if one can measure all the observables related with @xmath222 . this task will be possible , only after b factories accumulate the data for the first several years . or one might have to consider the modified @xmath128 and new operators that are not possible in the sm . in the previous section , we presented model - independent analysis of physics related with @xmath127 assuming there is a new cp - violating phase and both @xmath45 and @xmath60 are real . in this section , we wish to present specific models that satisfy such assumptions . let us consider the fcnc in generalized susy models , in which squark mass matrices are nondiagonal in the basis where fermion mass matrices are diagonal . in this case , there can be a potentially important contributions to the fcnc processes and cp violation that arise from flavor changing @xmath223 vertices @xcite . the sources of susy fcnc are the nondiagonal @xmath224 and @xmath225 . different susy breaking models have different patterns / hierarchies for the flavor mixings in the squark mass matrices . since we study the new physics contributions to @xmath127 , the wilson coefficients of the operators already present in the sm in this work , we will consider only two cases : ( i ) the @xmath226 mixing dominating and ( ii ) the @xmath227 mixing dominating cases . there are some models in the literature which fall into these two categories . as discussed below , the case ( i ) does not contribute to @xmath52 so that @xmath228 ( or , @xmath102 ) . on the other hand , the case ( ii ) contributes to @xmath52 as well as to @xmath65 and @xmath229 . also , there would be generically other contributions from @xmath230 and @xmath231 loops . if these loop effects are competent with the gluino - mediated loop effects we consider in the following , then our assumption that both @xmath45 and @xmath60 are real would not be true any longer . in the following , we assume that these ( susy ) electroweak loops are indeed negligible compared to the gluino mediated fcnc loop amplitudes . the latter is enhanced by @xmath232 , as usually assumed . however there is a suppression factor in the latter case , the mixing angle in the squark sector given by @xmath233 ( or , @xmath234 in the mass insertion approximation ) . also the heavy squark - gluino loops will be suppressed compared to the charged higgs - top , chargino - stop and neutralino - down squarks , unless all the susy particles have similar masses so that squark and gluinos are not too heavy . so one has to keep in mind that our assumption may break down for too small mixing angle in the squark sector or too heavy squark / gluino . with this caveat in mind , new physics contributions considered here depend on only one new phase so that @xmath45 and @xmath60 are real , as assumed in the previous section . in order to estimate the @xmath77 in the generalized susy models with gluino - mediated fcnc , we consider both the vertex mixing ( vm ) method and the mass insertion approximation ( mia ) . the latter approximation is good , when squarks are almost degenarate . the corresponding expressions can be obtained from the former expressions by taking a suitable expansion in @xmath235 , where @xmath236 is a suitable average mass of almost degenerate squarks . on the other hand , in the scenario in which the susy fcnc and susy cp problem are solved by decoupling of the ( nearly degenerate ) first two generation squarks such as in the effective susy models , there is a large hierarchy between the first two and the third squarks so that the mia is no longer a good approximation . in such case , we have to resort to the vm method . the full expressions for the wilson coefficients @xmath127 due to the fcnc gluino exchange diagrams are @xcite @xmath237 , \nonumber \\ c_{8}^{susy } ( m_w ) & = & -{\pi \alpha_s \over \sqrt{2 } g_f m_{\tilde{g}}^2 \lambda_t}~\sum_{i=1}^6 ~x_i ( \gamma_{gl}^{d \dagger } ) _ { 2i } \nonumber \\ & \times & \left [ ( \gamma_{gl}^d ) _ { i3 } \left\ { 3 f_1 ( x_i ) + { 1\over 3 } f_2 ( x_i ) \right\ } + ( \gamma_{gr}^d ) _ { i3 } { m_{\tilde{g } } \over m_b } \left\ { 3 f_3 ( x_i ) + { 1 \over 3 } f_4 ( x_i ) \right\ } \right ] , \nonumber \\ c_{9}^{susy } ( m_w ) & = & { 16 \pi \alpha_s \over 9 \sqrt{2 } g_f m_{\tilde{g}}^2 \lambda_t } ~\sum_{i=1}^6 ~x_i ( \gamma_{gl}^{d \dagger } ) _ { 2i } ( \gamma_{gl}^d ) _ { i3 } f_6 ( x_i ) , \end{aligned}\ ] ] where @xmath238 . @xmath233 and @xmath239 determine the @xmath240 vertices as follows : @xmath241 d_{j\alpha } \tilde{d}_{i\beta}^ { * } , \ ] ] with @xmath242 and @xmath243 . they are related with the mixing matrix elements diagonalizing the down - squark mass matrix via @xmath244 , @xmath245 , with the following identification : @xmath246 the functions @xmath247 s are given by @xcite @xmath248 the corresponding expressions in the mia is obtained from the above expressions by making a taylor expansion around @xmath249 as follows : @xmath250 and using the unitarity condition for @xmath251 . this way one can recover the results in ref . @xcite . for completeness , we present the resulting expressions below : @xmath252 , \nonumber \\ c_{8}^{susy } ( m_w ) & = & { \pi \alpha_s \over \sqrt{2 } g_f \tilde{m}^2 \lambda_t}~\left [ \left ( \delta_{23}^d \right)_{ll } \left ( { 1\over 3 } m_3 ( x ) + 3 m_4 ( x ) \right ) + \left ( \delta_{23}^d \right)_{lr } { m_{\tilde{g } } \over m_b } \left ( { 1\over 3 } m_1 ( x ) + 3 m_2 ( x ) \right ) \right ] , \nonumber \\ c_{9}^{susy } ( m_w ) & = & { 16 \pi \alpha_s \over 9 \sqrt{2 } g_f \tilde{m}^2 \lambda_t } ~\left ( \delta_{23}^d \right)_{ll } p_1 ( x ) .\end{aligned}\ ] ] the functions @xmath253 and @xmath254 are defined as @xmath255 in order to estimate the @xmath78 and @xmath60 , we assume that the @xmath256 mixing is the same order of the corresponding ckm matrix element with an unknown new phase @xmath257 . for example , @xmath258 for both cases ( i ) and ( ii ) , and similarly for @xmath259 . then it is clear that @xmath45 and @xmath60 are real in the mia both in the cases ( i ) and ( ii ) . in case of the vm approximation , the relevant model is the effective susy model where only the third family squark can be lighter than @xmath260 tev so that @xmath261 and we may keep only terms proportional to @xmath262 in the summation over @xmath263 in eqs . ( then , the @xmath45 and @xmath60 are real again , as assumed in the previous section . finally , in the following subsection , we will consider only two observables @xmath210 and @xmath193 for simplicity among several observables considered in the previous section . these two observables will be sufficient for us to find out the generic features considered in the previous section in the specific susy models with gluino mediated fcnc . let us first discuss the case ( i ) : @xmath226 insertion . since the flavor changing @xmath226 mixing terms are not generated by susy breaking in the limit of vanishing yukawa couplings , they are proportional to the corresponding yukawa couplings . therefore , the mass insertion approximation is always appropriate , and we consider the @xmath226 insertion only in the mia . from eqs . ( 44 ) , one gets @xmath264 note that @xmath45 and @xmath60 are functions of @xmath265 only , whereas @xmath266 depends on @xmath267 and also on @xmath268 . therefore , for a fixed @xmath265 and assuming @xmath269 , one can calculate the @xmath210 as a function of @xmath236 and @xmath270 with the constraints ( e1 ) and ( e2 ) . the result is that only @xmath271 is consistent with the constraints ( e1 ) and ( e2 ) . as @xmath265 increases , the contribution to @xmath93 and/or @xmath272 get(s ) too large . for @xmath273 and @xmath274 , the allowed range of @xmath275 and @xmath193 as functions of @xmath270 are shown in fig . [ fig_lr03 ] and fig . [ fig_lr08 ] , respectively , along with the constant @xmath236 contours . in the present case where the mia is appropriate , one also has to take into account the constraints on squark masses from cdf ( @xmath276 gev for @xmath277 ) @xcite and d0 ( @xmath278 gev for @xmath277 ) @xcite . for @xmath279 , one can read off the allowed mass range for the squark mass from the @xmath280 exclusion plot @xcite . roughly speaking , @xmath281 gev for @xmath273 and @xmath282 gev for @xmath283 . [ fig_lr03 ] ( a ) ( @xmath273 for which @xmath284 ) indicates that the direct cp asymmetry @xmath285 is in the range @xmath286 for the squark mass @xmath287 gev and the new cp violating phase @xmath288 . this aysmmetry is probably too small to be observed . but for the same range of @xmath236 and @xmath270 , the @xmath193 can be as large as 2.1 ( see fig . [ fig_lr03 ] ( b ) ) . so @xmath289 is more sensitive to the @xmath226 mixing than the direct cp asymmetry in @xmath13 if @xmath273 . from fig . [ fig_lr08 ] ( a ) ( @xmath274 for which @xmath290 and @xmath67 ) , the @xmath275 is in the range @xmath291 for @xmath292 . it seems that there is a definite lower bound to the @xmath275 , but this is an artifact due to our choice of @xmath293 tev . for heavier @xmath236 it vanishes very slowly ( see fig . [ fig_conv ] ( a ) and the following paragraph ) . however if all the squarks ( including the third family squarks ) are heavier than @xmath294 tev , the motivation for the low energy susy is lost , since the fine tuning problem is reintroduced . therefore we think that the condition @xmath295 tev is a reasonable requirement in the scenarios for the soft susy breakings where the mia is valid . with this caveat , the predicted values for @xmath210 are within reach of the b factories . the impact on @xmath193 is less striking than the @xmath273 case , but there is still a modest enhancement upto 1.44 of @xmath193 over its sm value which may be also detectable at b factories . one interesting feature of the @xmath226 mixing case is that the observables we show in figs . [ fig_lr03 ] and [ fig_lr08 ] can probe the effects of very heavy squark masses @xmath296 gev for @xmath297 . moreover , the heavier squarks can generate larger @xmath210 , which may be in conflict with the naive expectation based on the decoupling of heavy particles in susy models . however , this is just an artifact of our requirement @xmath298 tev , as described at the end of the previous paragraph . this is because we have fixed @xmath265 , since the heavier squark mass @xmath236 for a fixed @xmath265 implies the heavier gluino mass @xmath299 . therefore the @xmath65 decreases rather slowly as @xmath236 increases with a fixed @xmath265 because of the @xmath299 factor in the numerator of the second term . in fig . [ fig_conv ] ( a ) and ( b ) , we plot the direct cp asymmetry @xmath210 and @xmath193 as functions of @xmath236 for @xmath274 . we fixed @xmath300 and @xmath301 . if @xmath270 changes its sign , the direct cp asymmetry @xmath210 also changes its sign . we observe that @xmath210 is maximized around @xmath302 tev or so . the effects of heavy squarks decouple very slowly for @xmath210 in the @xmath226 mixing case . on the contrary , the effect on @xmath193 is larger for the lighter squark mass as usual . next let us consider the case ( ii ) : @xmath227 insertion . in this case , the susy breaking terms are the main source of the flavor changing @xmath227 mixing , which are not related with the yukawa couplings in principle . therefore , the mia may not be always valid , depending on the superparticle spectra . for example , a class of models @xcite,@xcite falls into this case where the @xmath227 mixing dominates . these models @xcite @xcite predict that @xmath256 mixing is order of @xmath303 . the mass spectra of the down - squarks in the model @xcite are nearly degenerate , whereas in the model @xcite only the @xmath304 , gauginos and the lightest neutral higgs are relatively light compared to @xmath305 tev . therefore , one can use the mia for the first models @xcite , whereas one has to use the vertex mixing for the second model @xcite . below , we will consider the mia case first . in the mass insertion approximation , @xmath306 in this case we consider two different choices for @xmath307 in order to compare our results with other existing literatures : @xmath308 @xcite and @xmath309 @xcite . as before , one imposes the experimental informations on @xmath93 and @xmath272 , and gets the allowed regions for @xmath210 and @xmath193 for a given phase @xmath270 , as well as the direct search limit on the squark mass from cdf and d0 . for @xmath310 and 3.0 , the @xmath311 and @xmath312 , respectively . therefore , the overall features of various observables will be close to fig . [ fig_ch5pr03 ] , except that @xmath65 should be fixed to some definite value . constraints are imposed . however there may be some visible deviation in @xmath211 as inferred from fig . [ fig_ch5pr03 ] . if @xmath315 as assumed in ref . @xcite , then one expects that @xmath275 can be as large as @xmath316 to @xmath317 for @xmath318 for @xmath319 , although the @xmath193 does not change very much from its sm value ( figs . [ fig_px03 ] and [ fig_px1 ] ) . if @xmath265 gets larger , the @xmath210 gets smaller and eventually becomes undetectable ( _ e.g. _ , @xmath320 for @xmath321 , if we impose @xmath282 gev ) . in ref . @xcite , it was noted that this new cp violating phase could result in the cp violation in the decay amplitudes for @xmath322 at the level of @xmath323 of the sm amplitude depending on the squark mass . there would be some intrinsic theoretical uncertainties in such estimates of nonleptonic exclusive @xmath9 decays . on the contrary , the direct cp violation in @xmath29 can provide independent informations on @xmath307 with less theoretical uncertainties , since we are dealing with the inclusive decay rate . in any rate the observable @xmath210 can play an important role in probing a new cp violating phase in @xmath9 decays if the condition @xmath315 is met . the vertex mixing case can be obtained from eq . ( 40 ) by following identifications : @xmath324 in the @xmath227 mixing case we consider here , one has @xmath325 . also , we assume that @xmath326 with a new phases of @xmath294 . this assumption is motivated by a recent model by kaplan _ @xcite , which is a susy model of flavor based on the single @xmath327 generating the fermion spectra as well as communicating susy breaking to the visible sector . in this model , only the third generation squarks are lighter than @xmath305 tev , and the 1st and the 2nd generation squarks simply decouple . therefore , we can keep only the third family squarks ( @xmath328 ) in the sum over the squark mass eigenstates , since others are all heavier than @xmath294 tev and/or the relations ( 40),(49 ) hold . after one imposes the experimental informations on @xmath93 and @xmath272 , one gets the allowed regions for @xmath275 and @xmath193 for a given phase @xmath270 , as shown in figs . 10 ( a)-(c ) for @xmath329 and 3.0 . the corresponding values of @xmath330 are @xmath331 and @xmath332 , respectively . in fig . 10 , we superposed the contours for three different values of @xmath333 gev . in the case only the third generation squarks are light , the strongest bound on the lighter stop comes from lep experiments @xcite , and @xmath334 gev is not excluded yet by lep experiment . therefore , one expects that @xmath210 can be as large as 6 @xmath335 to 12 @xmath335 for @xmath336 radian for @xmath329 and 3.0 respectively , although the @xmath193 does change very little : @xmath337 . however , this large @xmath210 quickly diminishes as @xmath338 gets heavier , and @xmath339 for @xmath340 gev . therefore , it is very difficult to see the effects of the @xmath227 insertion in the effective susy models ( for which the vm method is valid ) as well as the case of almost degenarate squarks ( for which mia is valid ) from @xmath193 . in other words , the @xmath227 insertion can generate a large direct cp violation in @xmath341 if there is a new cp violating phase associated with the squark mass matrix , @xmath342 , whereas there can be no significant change in @xmath193 compared with the sm case . also the deviation from the sm diminish very quickly as stop gets heavier . practically speaking , it would be impossible to notice the new physics effects if @xmath282 gev unless @xmath343 , for which new physics signal can be visible for heavier stop until @xmath344 gev if @xmath265 is not too large . this is in contrast to the @xmath226 mixing case , for which the new physics effects can increase @xmath193 up to 2.15 , and the @xmath210 can be as large as 11 @xmath335 for fairly large stop , @xmath345 gev and @xmath346 ( see fig . although @xmath210 and @xmath193 are not sensitive to the @xmath227 insertion for @xmath347 , there is another observable which is complementary to our observables : namely , cp violating lepton asymmetry in b decays discussed in ref . @xcite . for larger @xmath348 , direct cp violations in nonleptonic @xmath9 decays through @xmath349 penguin operators can provide additional informations @xcite . again , different channels are sensitive to different types of new physics , and it will be helpful to study as many modes as possible in order to find out new physics signals at b factories . in conclusion , we considered the possible new physics effects on the @xmath0 through the modified @xmath13 vertex . the cp violation in @xmath341 can be very different from the sm expectation ( @xmath350 ) , and the branching ratio and @xmath351 in @xmath0 can be affected by the new physics contributing to @xmath13 . in particular , the usual model - independent extraction of the wilson coefficients @xmath97 may be useless in the presence of new physics that modifies the @xmath32 with a new cp - violating phase ( namely , i m @xmath96 ) . therefore , not only is the cp asymmetry in @xmath341 a sensitive probe of new physics that might be discovered at @xmath8factories , but also it is indispensable for the model - independent analysis of @xmath0 . search for @xmath210 is clearly warranted at @xmath8factories . we also considered specific models which satisfy our assumptions made in the model - independent analysis : namely , generalized susy models with gluino - mediated fcnc . in the case of @xmath226 mixing , @xmath193 can be enhanced compared to the sm value . also the direct asymmetry @xmath210 can be as large as @xmath291 for @xmath346 and @xmath302 tev . in this case , the direct asymmetry @xmath210 is sensitive to the heavy squark masses , since the decoupling occurs very slowly , beyond @xmath352 tev ( see fig . 7 ( a ) ) . also there is a lower bound on @xmath210 since all the squarks can not be simultaneously heavier than @xmath294 tev . this is quite an intersting feature of the @xmath226 mixing scenario . in the @xmath227 mixing case , there is no observable effects both for @xmath210 and @xmath193 if @xmath353 . but there can be an appreciable amount of @xmath210 upto @xmath317 , if @xmath354 in the mia . in the @xmath227 mixing with the vm approximation , one may be observable @xmath210 upto @xmath355 depending on @xmath265 and the new cp violating phase @xmath270 for @xmath356 which is the typical values in the model by kaplan _ _ @xcite . our study is also complimentary to other previous works , _ e.g. _ , the dilepton asymmetry considered by randall and su @xcite , and the cp asymmetry in the decay amplitudes for nonleptonic @xmath9 decays considered by ciuchini _ it is very important to measure various kinds of cp asymmetries at b factories , especially those cp asymmetries which ( almost ) vanish in the sm like the direct asymmetry in @xmath13 and dilepton asymmetry , in order to probe new cp violating phase(s ) that may be necessary for us to understand the baryon number asymmetry of the universe . different channels may be sensitive to different parameter values in new physics , and thus can provide indepedent informations on new physics . while we were preparing this manuscript , we received a preprint by chua _ @xcite considering the cp - violation in @xmath13 in supersymmetric models . it somewhat overlaps with sec . iv of our present work . but they did not consider the @xmath23 constraint , and get somewhat larger @xmath275 asymmetry than our work . y.k . and p.k . acknowldge the hospitality of korea institute for advanced study ( kias ) where a part of this work has been done . this work is supported in part by kosef contract no . 971 - 0201 - 002 - 2 , by kosef through center for theoretical physics at seoul national university , by the ministry of education through the basic science research institute , contract no . bsri-98 - 2418 , the german - korean scientific exchange programme dfg-446-kor-113/72/0 ( pk ) , and the kaist center for theoretical physics and chemistry ( pk , yk ) . m. kobayashi and k. maskawa , prof . . phys . * 49 * , 652 ( 1973 ) . burges and h.j . schnitzer , nucl . * b 228 * , 464 ( 1983 ) ; c.n . leung , s.t . love and s. rao , z. phys.*c 31 * , 433 ( 1986 ) ; w. buchmller and d. wyler , nucl . * b 268 * , 621 ( 1986 ) . a. j. buras and r. fleischer , _ quark mixing , cp violation and rare decays after the top quark discovery _ , in _ heavy flavors ii _ , buras and m. lindner , world scientific , 1998 , p.65 . l. wolfenstein , phys . lett . * 51 * , 1945 ( 1983 ) . a. kagan and m. neubert , cern - 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we consider possible new physics contributions to @xmath0 assuming the new physics modifies ( chromo)magnetic and electric form factors in @xmath1 and @xmath2 with the same chirality structure as in the standard model . parametrizing the new physics effects on @xmath1 and @xmath2 in terms of four real parameters ,
one finds that there are enough region of parameter space in which the measured branching ratio for @xmath3 can be accomodated , and the predicted cp violation effect could be as large as @xmath4 .
moreover , the branching ratio and the forward - backward asymmetry of a lepton in @xmath5 and the tau polarization asymmetry in @xmath6 can be deviated from the sm predictions by a factor of @xmath7 , which can be accessible at b factories .
we also discuss these observables in a specific class of supersymmetric models with gluino - mediated flavor changing neutral current ( fcnc ) .
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a central goal of the gaia mission is to teach us how the galaxy functions and how it was assembled . we can only claim to understand the structure of the galaxy when we have a dynamical model galaxy that reproduces the data . therefore the construction of a satisfactory dynamical model is in a sense a primary goal of the gaia mission , for this model will encapsulate the understanding of galactic structure that we have gleaned from gaia . preliminary working models that are precursors of the final model will also be essential tools as we endeavour to make astrophysical sense of the gaia catalogue . consequently , before launch we need to develop a model - building capability , and with it produce dynamical models that reflect fairly fully our current state of knowledge . the modern era of galaxy models started in 1980 , when the first version of the bahcall - soneira model appeared @xcite . this model broke new ground by assuming that the galaxy is built up of components like those seen in external galaxies . earlier work had centred on attempts to infer three - dimensional stellar densities by directly inverting the observed star counts . however , the solutions to the star - count equations are excessively sensitive to errors in the assumed obscuration and the measured magnitudes , so in practice it is essential to use the assumption that our galaxy is similar to external galaxies to choose between the infinity of statistically equivalent solutions to the star - count equations . bahcall & soneira showed that a model inspired by data for external galaxies that had only a dozen or so free parameters could reproduce the available star counts . @xcite did not consider kinematic data , but @xcite updated the classical work on mass models by fitting largely kinematic data to a mass model that comprised a series of components like those seen in external galaxies . these data included the oort constants , the tangent - velocity curve , the escape velocity at the sun and the surface density of the disk near the sun . @xcite were the first to fit both kinematic and star - count data to a model of the galaxy that was inspired by observations of external galaxies . they broke the disk down into seven sub - populations by age . then they assumed that motion perpendicular to the plane is perfectly decoupled from motion within the plane , and further assumed that as regards vertical motion , each subpopulation is an isothermal component , with the velocity dispersion determined by the observationally determined age - velocity dispersion relation of disk stars . each sub - population was assumed to form a disk of given functional form , and the thickness of the disk was determined from the approximate formula @xmath0/\sigma^2\}$ ] , where @xmath1 is an estimate of the overall galactic potential . once the thicknesses of the sub - disks have been determined , the mass of the bulge and the parameters of the dark halo were adjusted to ensure continued satisfaction of the constraints on the rotation curve @xmath2 . then the overall potential is recalculated , and the disk thicknesses were redetermined in the new potential . this cycle was continued until changes between iterations were small . the procedure was repeated several times , each time with a different dark - matter disk arbitrarily superposed on the observed stellar disks . the geometry and mass of this disk were fixed during the interations of the potential . star counts were used to discriminate between these dark - matter disks ; it turned out that the best fit to the star counts was obtained with negligible mass in the dark - matter disk . although in its essentials the current ` besanon model ' @xcite is unchanged from the original one , many refinements and extensions to have been made . in particular , the current model fits near ir star counts and predicts proper motions and radial velocities . it has a triaxial bulge and a warped , flaring disk . its big weakness is the assumption of constant velocity dispersions and streaming velocities in the bulge and the stellar halo , and the neglect of the non - axisymmetric component of the galaxy s gravitational field . a consensus that ours is a barred galaxy formed in the early 1990s @xcite and models of the bulge / bar started to appear soon after . @xcite and @xcite modelled the luminosity density that is implied by the ir data from the cobe mission , while @xcite and @xcite used extensions of schwarzschild s ( 1979 ) modelling technique to produce dynamical models of the bar that predicted proper motions in addition to being compatible with the cobe data . there was an urgent need for such models to understand the data produced by searches for microlensing events in fields near the galactic centre . the interplay between these data and galaxy models makes rather a confusing story because it has proved hard to estimate the errors on the optical depth to microlensing in a given field . the recent work of the basel group @xcite and the microlensing collaborations @xcite seems at last to have produced a reasonably coherent picture . @xcite fit a model to structures that are seen in the @xmath3 diagrams that one constructs from spectral - line observations of hi and co. the model is based on hydrodynamical simulations of the flow of gas in the gravitational potential of a density model that was fitted to the cobe data @xcite . they show that structures observed in the @xmath3 plane can be reproduced if three conditions are fulfilled : ( a ) the pattern speed of the bar is assigned a value that is consistent with the one obtained by @xcite from local stellar kinematics ; ( b ) there are four spiral arms ( two weak , two strong ) and they rotate at a much lower pattern speed ; ( c ) virtually all the mass inside the sun is assigned to the stars rather than a dark halo . @xcite go on to construct a stellar - dynamical model that reproduces the luminosity density inferred by @xcite . the model , which has no free parameters , reproduces both ( a ) the stellar kinematics in windows on the bulge , and ( b ) the microlensing event timescale distribution determined by the macho collaboration @xcite . the magnitude of the microlensing optical depth towards bulge fields is still controversial , but the latest results agree extremely well with the values predicted by bissantz & gerhard : in units of @xmath4 , the eros collaboration report optical depth @xmath5 at @xmath6 @xcite while bissantz & gerhard predicted @xmath7 at this location ; the macho collaboration report @xmath8 at @xmath9 @xcite , while bissantz & gerhard predicted @xmath10 at this location . thus there is now a body of evidence to suggest that the galaxy s mass is dominated by stars that can be traced by ir light rather than by invisible objects such as wimps , and that dynamical galaxy models can successfully integrate data from the entire spectrum of observational probes of the milky way . since 1980 there has been a steady increase in the extent to which galaxy models are dynamical . a model must predict stellar velocities if it is to confront proper - motion and radial velocity data , or predict microlensing timescale distributions , and it needs to predict the time - dependent , non - axisymmetric gravitatinal potential in order to confront spectra - line data for hi and co. some progress can be made by adopting characteristic velocity dispersions for different stellar populations , but this is a very poor expedient for several reasons . ( a ) without a dynamical model , we do not know how the orientation of the velocity ellipsoid changes from place to place . ( b ) it is not expected that any population will have gaussian velocity distributions , and a dynamical model is needed to predict how the distributions depart from gaussianity . ( c ) an arbitrarily chosen set of velocity distributions at different locations for a given component are guaranteed to be dynamically inconsistent . therefore it is imperative that we move to fully dynamical galaxy models . the question is simply , what technology is most promising in this connection ? the market for models of external galaxies is currently dominated by models of the type pioneered by @xcite . one guesses the galactic potential and calculates a few thousand judiciously chosen orbits in it , keeping a record of how each orbit contributes to the observables , such as the space density , surface brightness , mean - streaming velocity , or velocity dispersion at a grid of points that covers the galaxy . then one uses linear or quadratic programming to find non - negative weights @xmath11 for each orbit in the library such that the observations are well fitted by a model in which a fraction @xmath11 of the total mass is on the @xmath12th orbit . schwarzschild s technique has been used to construct spherical , axisymmetric and triaxial galaxy models that fit a variety of observational constraints . thus it is a tried - and - tested technology of great flexibility . it does have significant drawbacks , however . first the choice of initial conditions from which to calculate orbits is at once important and obscure , especially when the potential has a complex geometry , as the galactic potential has . second , different investigators will choose different initial conditions and therefore obtain different orbits even when using the same potential . so there is no straightforward way of comparing the distributon functions of their models . third , the method is computationally very intensive because large numbers of phase - space locations have to be stored for each of orbit . finally , predictions of the model are subject to discreteness noise that is larger than one might naively suppose because orbital densities tend to be cusped ( and formally singular ) at their edges and there is no natural procedure for smoothing out these singularities . in oxford over a number of years we developed a technique in which orbits are not obtained as the time sequence that results from integration of the equations of motion , but as images under a canonical map of an orbital torus of the isochrone potential . each orbit is specified by its actions @xmath13 and is represented by the coefficients @xmath14 that define the function @xmath15 that generates the map . once the @xmath16 have been determined , analytic expressions are available for @xmath17 and @xmath18 , so one can readily determine the velocity at which the orbit would pass through any given location . since orbits are labelled by actions , which define a true mapping of phase space , it is straightforward to construct an orbit library by systematically sampling phase space at the nodes of a regular grid of actions @xmath19 . moreover , a good approximation to an arbitrary orbit can be obtained by interpolating the @xmath20 . if the orbit library is generated by torus mapping , it is easy to determine the distributon function from the weights . when the orbit weigts are normalized such that @xmath21 , and the distribution function is normalized such that @xmath22 , then @xmath23 if the action - space gid is regular with spacing @xmath24 , we can obtain an equivalent smoothed distribution function by replacing @xmath25 by @xmath26 if @xmath13 lies within a cube of side @xmath24 centred on @xmath27 , and zero otherwise . different modellers can easily compare their smoothed distribution functions . finally , with torus mapping many fewer numbers need to be stored for each orbit just the @xmath16 rather than thousands of phase - space locations @xmath28 . the drawbacks of torus mapping are these . first , it requires complex special - purpose software , whereas orbit integration is trivial . second , it has to date only been demonstrated for systems that have two degrees of freedom , such as an axisymmetric potential @xcite , or a planar bar @xcite . finally , orbits are in a fictitious integrable hamiltonian @xcite rather than in the , probably non - integrable , potential of interest . i return to this point below . in both the schwarzschild and torus modelling strategies one starts by calculating an orbit library , and the weights of orbits are determined only after this step is complete . @xcite suggested an alternative stratey , in which the weights are determined simultaneously with the integration of the orbits . combining these two steps reduces the large overhead involved in storing large numbers of phase - space coordinates for individual orbits . moreover , with the syer tremaine technique the potential does not have to be fixed , but can be allowed to evolve in time , for example through the usual self - consistency condition of an n - body simulation . to describe the syer tremaine algorithm we need to define some notation . let @xmath29 denote an arbitrary point in phase space . then each observable @xmath30 is defined by a kernel @xmath31 through @xmath32 for example , if @xmath33 is the density at some point @xmath34 , then @xmath35 would be @xmath36 . in an orbit model we take @xmath37 to be of the form @xmath38 and the integral in the last equation becomes a sum over orbits : @xmath39 if we simultaneously integrate a large number of orbits in a common potential @xmath1 ( which might be the time - dependent potential that is obtained by assigning each particle a mass @xmath40 ) , then through equation ( [ discy ] ) each observable becomes a function of time . let @xmath33 be the required value of this observable , then syer & tremaine adjust the value of the weight of the @xmath12th orbit at a rate @xmath41\over z_\alpha } \left({y_\alpha\over y_\alpha}-1\right).\ ] ] here the positive numbers @xmath42 are chosen judiciously to stress the importance of satisfying particular constraints , and can be increased to slow the rate at which the weights are adjusted . the numerator @xmath43 $ ] ensures that a discrepancy between @xmath44 and @xmath33 impacts @xmath11 only in so far as the orbit contributes to @xmath30 . the right side starts with a minus sign to ensure that @xmath11 is decreased if @xmath45 and the orbit tends to increase @xmath30 . @xcite have recently demonstrated the value of the syer & tremaine algorithm by using it to construct a dynamical model of the inner galaxy in the pre - determined potential of @xcite . n - body simulations have been enormously important for the development of our understanding of galactic dynamics . to date they have been of rather limited use in modelling specific galaxies , because the structure of an n - body model has been determined in an obscure way by the initial conditions from which it is started . in fact , a major motivation for developing other modelling techniques has been the requirement for initial conditions that will lead to n - body models that have a specified structure ( e.g * ? ? ? * ) . nothwithstanding this difficulty , @xcite was able to find an n - body model that qualitatively fits observations of the inner galaxy . it will be interesting to see whether the syer tremaine algorithm can be used to refine a model like that of fux until it matches all observational constraints . when trying to understand something that is complex , it is best to proceed through a hierarchy of abstractions : first we paint a broad - bruish picture that ignores many details . then we look at areas in which our first picture clearly conflicts with reality , and understand the reasons for this conflict . armed with this understanding , we refine our model to eliminate these conflicts . then we turn to the most important remaining areas of disagreement between our model and reality , and so on . the process terminates when we feel that we have nothing new or important to learn from residual mismatches between theory and measurement . this logic is nicely illustrated by the dynamics of the solar system . we start from the model in which all planets move on kepler ellipses around the sun . then we consider the effect on planets such as the earth of jupiter s gravitational field . to this point we have probably assumed that all bodies lie in the ecliptic , and now we might consider the non - zero inclinations of orbits . one by one we introduce disturbances caused by the masses of the other planets . then we might introduce corrections to the equations of motion from general relativity , followed by consideration of effects that arise because planets and moons are not point particles , but spinning non - spherical bodies . as we proceed through this hierarchy of models , our orbits will proceed from periodic , to quasi - periodic to chaotic . models that we ultimately reject as oversimplified will reveal structure that was previously unsuspected , such as bands of unoccupied semi - major axes in the asteroid belt . the chaos that we will ultimately have to confront will be understood in terms of resonances between the orbits we considered in the previous level of abstraction . the impact of hipparcos on our understanding of the dynamics of the solar neighbourhood gives us a flavour of the complexity we will have to confront in the gaia catalogue . when the density of stars in the @xmath46 plane was determined @xcite , it was found to be remarkably lumpy , and the lumps contained old stars as well as young , so they could not be just dissolving associations , as the classical interpretation of star streams supposed . now that the radial velocities of the hipparcos survey stars are available , it has become clear that the hyades - pleiades and sirius moving groups are very heterogeous as regards age @xcite . evidently these structures do not reflect the patchy nature of star formation , but have a dynamical origin . they are probably generated by transient spiral structure @xcite , so they reflect departures of the galaxy from both axisymmetry and time - independence . such structures will be most readily understod by perturbing a steady - state , axisymmetric galaxy model . a model based on torus mapping is uniquely well suited to such a study because its orbits are inherently quasi - periodic structures with known angle - action coordinates . consequently , we have everything we need to use the powerful techniques of canonical perturbation theory . even in the absence of departures from axisymmetry or time - variation in the potential , resonances between the three characteristic frequencies of a quasi - periodic orbit can deform the orbital structure from that encountered in analytically integrable potentials . important examples of this phenomenon are encountered in the dynamics of triaxial elliptical galaxies , where resonant ` boxlets ' almost entirely replace box orbits when the potential is realistically cuspy @xcite , and in the dynamics of disk galaxies , where the 1:1 resonance between radial and vertical oscillations probably trapped significant numbers of thick - disk stars as the mass of the thin disk built up @xcite . @xcite has shown that such families of resonant orbits may be very successfully modelled by applying perturbation theory to orbits obtained by torus mapping . if the resonant family is exceptionally large , one may prefer to obtain its orbits directly by torus mapping @xcite rather than through perturbation theory . figures [ fig1 ] and [ fig2 ] show examples of each approach to a resonant family . both figures show surfaces of section for motion in a planar bar . in figure [ fig1 ] a relatively weak resonance is successfuly handled through perturbation theory , while in figure [ fig2 ] a more powerful resonance that induces significant chaos is handled by directly mapping isochrone orbits into the resonant region . these examples demonstrate that if we obtain orbits by torus mapping , we will be able to discover what the galaxy would look like in the absence of any particular resonant family or chaotic region , so we will be able to ascribe particular features in the data to particular resonances and chaotic zones . this facility will make the modelling process more instructive than it would be if we adopted a simple orbit - based technique . a dynamical model galaxy will consist of a gravitational potential @xmath47 together with distribution functions @xmath48 for each of several stellar populations . each distribution function may be represented by a set of orbital weights @xmath11 , and the populations will consist of probability distributions in mass @xmath49 , metallicity @xmath50 and age @xmath51 that a star picked from the population has the specified characteristics . thus a galaxy model will contain an extremely large number of parameters , and fitting these to the data will be a formidable task . since so much of the galaxy will be hidden from gaia by dust , interpretation of the gaia catalogue will require a knowledge of the three - dimensional distribution of dust . such a model can be developed by the classical method of comparing measured colours with the intrinsic colours of stars of known spectral type and distance . at large distances from the sun , even gaia s small parallax errors will give rise to significantly uncertain distances , and these uncertainties will be an important limitation on the reliability of any dust model that one builds in this way . dynamical modelling offers the opportunity to refine our dust model because newton s laws of motion enable us to predict the luminosity density in obscured regions from the densities and velocities that we see elsewhere , and hence to detect obscuration without using colour data . moreover , they require that the luminosity distributions of hot components are intrinsically smooth , so fluctuations in the star counts of these populations at high spatial frequencies must arise from small scale structure in the obscuring dust . therefore , we should solve for the distribution of dust at the same time as we are solving for the potential and the orbital weights . in principle one would like to fit a galaxy model to the data by predicting from the model the probability density @xmath52 of detecting a star at given values of the catalogue variables , such as celestial coordinates @xmath53 , parallax @xmath54 , and proper motons @xmath55 , and then evaluating the likelihood @xmath56 , where the product runs over stars in the catalogue and @xmath57 with @xmath58 the measured values and @xmath59 the associated uncertainties . unfortunately , it is likely to prove difficult to obtain the required probability density @xmath60 from an orbit - based model , and we will be obliged to compare the real catalogue to a pseudo - catalogue derived from the current model . moreover , standard optimization algorithms are unlikely to find the global maximum in @xmath61 without significant astrophysical input from the modeller . in any event , evaluating @xmath62 for each of @xmath63 observed stars is a formidable computational problem . devising efficient ways of fitting models to the data clearly requires much more thought . fine structure in the galaxy s phase space may provide crucial assistance in fitting a model to the data . two inevitable sources of fine structure are ( a ) resonances , and ( b ) tidal streams . resonances will sometimes be marked by a sharp increase in the density of stars , as a consequence of resonant trapping , while other resonances show a deficit of stars . suppose the data seem to require an enhanced density of stars at some point in action space and you suspect that the enhancement is caused by a particular resonance . by virtue of errors in the adopted potential @xmath1 , the frequencies will not actually be in resonance at the centre of the enhancement . by appropriate modification of @xmath1 it will be straightforward to bring the frequencies into resonance . by reducing the errors in the estimated actions of orbits , a successful update of @xmath1 will probably enhance the overdensity around the resonance . in fact , one might use the visibility of density enhancements to adjust @xmath1 very much as the visibility of stellar images is used with adaptive optics to configure the telescope optics . a tidal stream is a population of stars that are on very similar orbits the actions of the stars are narrowly distributed around the actions of the orbit on which the dwarf galaxy or globular cluster was captured . consequently , in action space a tidal stream has higher contrast than it does in real space , where the stars diverging angle variables gradually spread the stars over the sky . errors in @xmath1 will tend to disperse a tidal stream in action space , so again @xmath1 can be tuned by making the tidal stream as sharp a feature as possible . dynamical galaxy models have a central role to play in attaining gaia s core goal of determining the structure and unravelling the history of the milky way . even though people have been building galaxy models for over half a century , we are still only beginning to construct fully dynamical models , and we are very far from being able to build multi - component dynamical models of the type that the gaia will require . at least three potentially viable galaxy - modelling technologies can be identified . one has been extensively used to model external galaxies , one has the distinction of having been used to build the currently leading galaxy model , while the third technology is the least developed but potentially the most powerful . at this point we would be wise to pursue all three technologies . once constructed , a model needs to be confronted with the data . on account of the important roles in this confrontation that will be played by obscuration and parallax errors , there is no doubt in my mind that we need to project the models into the space of gaia s catalogue variables @xmath64 . this projection is simple in principle , but will be computationally intensive in practice . the third and final task is to change the model to make it fit the data better . this task is going to be extremely hard , and it is not clear at this point what strategy we should adopt when addressing it . it seems possible that features in the action - space density of stars associated with resonances and tidal streams will help us to home in on the correct potential . there is much to do and it is time we started doing it if we want to have a reasonably complete box of tools in hand when the first data arrive in 20122013 . the overall task is almost certainly too large for a single institution to complete on its own , and the final galaxy - modelling machinery ought to be at the disposal of the wider community than the dynamics community since it will be required to evaluate the observable implications of any change in the characteristics or kinematics of stars or interstellar matter throughout the galaxy . therefore , we should approach the problem of building galaxy models as an aspect of infrastructure work for gaia , rather than mere science exploitation . i hope that in the course of the next year interested parties will enter into discussions about how we might divide up the work , and define interface standards that will enable the efforts of different groups to be combined in different combinations . it is to be hoped that these discussions lead before long to successful applications to funding bodies for the resources that will be required to put the necessary infrastructure in place by 2012 .
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techniques for the construction of dynamical galaxy models should be considered essential infrastructure that should be put in place before gaia flies .
three possible modelling techniques are discussed .
although one of these seems to have significantly more potential than the other two , at this stage work should be done on all three .
a major effort is needed to decide how to make a model consistent with a catalogue such as that which gaia will produce . given the complexity of the problem , it is argued that a hierarchy of models should be constructed , of ever increasing complexity and quality of fit to the data .
the potential that resonances and tidal streams have to indicate how a model should be refined is briefly discussed .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
in many applications and implementations of quantum information processing , one has to compare two different quantum states . in this context , the quantum fidelity @xcite is a very useful tool to measure the `` closeness '' between two states in the hilbert space of a quantum system . for two arbitrary states @xmath0 and @xmath1 , it is defined as @xmath2 for any pair of pure states @xmath3 and @xmath4 , the quantum fidelity reduces to their ( squared ) overlap , @xmath5 . although the fidelity does not define a metric on the state space , it is the core ingredient for several of them , like for instance the bures distance @xmath6^{1/2}$ ] @xcite . the fidelity is also widely used to define various entanglement monotones . the bures distance @xmath7 is itself such an example @xcite . for multipartite pure states , the geometric measure of entanglement @xmath8 with @xmath9 is another example that exploits the maximal fidelity between the state @xmath3 to characterize and the set @xmath10 of all fully separable states @xmath11 @xcite . the question arises as to how the supremum in eq . ( [ fs ] ) can be computed . in general , it is known to be an np - hard task @xcite . for multipartite states @xmath12 that are symmetric with respect to the permutations of the parties it has been shown @xcite that this supremum is realized among the symmetric separable states @xmath13 only , @xmath14 in fact it can even be proven that for three or more particles the state maximizing the overlap in the definition of @xmath15 is necessarily symmetric @xcite . this nice property considerably simplifies the calculation of the geometric measure of entanglement for symmetric states . the maximization of the fidelity over sets other than the separable states @xmath10 has proven to be very useful for discrimination strategies of inequivalent classes of multipartite entangled states with witnesses @xcite or other methods @xcite . in this case , the maximization is typically to be performed on sets of states equivalent through either local unitary operations ( lu ) or stochastic local operations assisted by classical communication ( slocc ) . one needs to evaluate the maximal fidelity @xmath16 with @xmath17 any considered lu or slocc class of states . mathematically , these classes are defined as follows : the lu equivalence class of a pure state @xmath18 is given by all states of the form @xmath19 , where the @xmath20 are unitary matrices acting on the @xmath21-th party . the slocc equivalence class of @xmath22 is given by normalized states of the form @xmath23 , where the @xmath24 are invertible matrices @xcite . the lu and slocc equivalence classes of states never coincide , except for the fully separable states that are all both lu and slocc equivalent . in eq . ( [ fc ] ) , if @xmath17 contains symmetric states , the question naturally arises whether the simplification given by eq . ( [ fss ] ) for the particular case of the slocc ( and lu ) class @xmath10 of separable states generalizes similarly . in other words , do we have for any symmetric state @xmath12 and any lu or slocc classes @xmath17 containing symmetric states @xmath25 this paper provides answers to this question for multiqubit systems . first , in the case of lu classes , the answer is positive and this is formally proven in sec . [ luclasssec ] . second , for the case of slocc classes , the answer is surprisingly negative and spectacular violations of eq . ( [ fcc ] ) will be given in sec . [ sloccclasssec ] . in sec . [ conclusionsection ] we summarize and discuss further open problems . when considering lu equivalence classes @xmath17 and multiqubit systems in eq . ( [ fcc ] ) and since any two lu - equivalent symmetric states can be transformed into each other with the same local unitary acting on each party @xcite , the question can be rephrased as follows : do we have , for any @xmath26-qubit symmetric states @xmath12 and @xmath27 , @xmath28 where @xmath29 is the group of unitary matrices of dimension @xmath30 ? since only the absolute value of the overlap matters , we can choose the phases of the @xmath20 as we like . so , it suffices to take matrices with determinant @xmath31 and consider @xmath32 with @xmath33 the group of unitary matrices of determinant 1 . the question ( [ supslu2 ] ) can thus be rephrased as @xmath34 to tackle this problem , we note that an arbitrary @xmath33 matrix can be written as @xmath35 we then define the function @xmath36 with @xmath37 where @xmath38 and @xmath39 are considered as _ real _ hilbert spaces . the function @xmath40 is symmetric with respect to the permutations of the variables , because @xmath12 and @xmath27 are symmetric states . furthermore , it is @xmath41-multilinear in the coefficients @xmath42 , that is , @xmath43 the multilinearity is only ensured for real @xmath44 and @xmath45 and this is the reason why we have to consider the _ hilbert spaces @xmath46 and @xmath47 . under these conditions , hrmander s theorem 4 of ref . @xcite and its extension to the case of real hilbert spaces of any dimension @xcite can be applied and we have @xmath48 where @xmath49 . because of the @xmath41-multilinearity of @xmath40 , @xmath50 with @xmath51 and thus @xmath52 equation ( [ horm ] ) then yields @xmath53 that is to say , @xmath54 so , the answer to question posed in eq . ( [ supslsu2 ] ) and in eq . ( [ supslu2 ] ) is clearly positive . this finishes the proof . now we turn to the case when slocc classes are considered in eq . ( [ fcc ] ) . since any two slocc - equivalent symmetric states can be transformed into each other with the same invertible local operation ( ilo ) acting on each party @xcite , the question addressed here for multiqubit systems is the following : do we have , for any @xmath26-qubit symmetric states @xmath12 and @xmath27 , @xmath55 where @xmath56 is the group of invertible matrices of dimension @xmath30 ? here , the expression to be maximized contains a normalization constant that also depends on the state @xmath27 . contrary to the lu case , the left - hand side term of eq . ( [ supslocc ] ) can not be cast in a multilinear form divided by a product of norms like the left - hand side term of eq . ( [ horm ] ) . hrmander s theorem therefore can not be exploited to tentatively prove eq . ( [ supslocc ] ) . actually , the counterexamples to this equation identified hereafter prove that any attempt in this direction would be doomed to failure . as the first observation , for three - qubit systems , extensive numerical simulations showed no violation of eq . ( [ fcc ] ) . similarly , numerical simulations for @xmath26 up to 8 gave indications that eq . ( [ fcc ] ) seems also to hold for the classes of states slocc - equivalent to the @xmath31-excitation dicke states @xmath57 @xcite , hereafter denoted by the classes @xmath58 . these classes gather both symmetric and nonsymmetric states . when restricted to the symmetric subspace , they merely identify to the @xmath59 families of symmetric states of ref . this brings us to conjecture that the equality @xmath60 actually holds for any @xmath26 and any symmetric state @xmath12 . the generalization of eq . ( [ conj ] ) to arbitrary slocc classes containing symmetric states , however , is not correct . spectacular violations are obtained when considering the classes of states slocc - equivalent to the @xmath21-excitation dicke states @xmath61 @xcite , hereafter denoted by the classes @xmath62 , for @xmath63 and @xmath64 . all these classes gather both symmetric and nonsymmetric states . when restricted to the symmetric subspace , they identify to the @xmath65 families of symmetric states of ref . @xcite . for the values of @xmath26 and @xmath21 considered , there are symmetric states @xmath12 for which @xmath66 to prove this result , we first consider the state @xmath67 . for all aforementioned @xmath26 and @xmath21 , @xmath68 @xcite and one gets ( see appendix [ apa ] for a detailed calculation ) @xmath69}}_{y \in [ -1,1 ] } f(x , x',y)\ ] ] with @xmath70^{j}}.\ ] ] equation ( [ rsnk ] ) can not be reduced analytically . it can , however , be straightforwardly evaluated numerically . we illustrate it in fig . [ fsdn1dnmkk ] for @xmath71 and @xmath72 . the figure clearly shows that @xmath73 remains significantly below one with an asymptotic behavior as @xmath26 tends to infinity for fixed @xmath21 . for fixed @xmath26 , the considered fidelities decrease with increasing @xmath21 . the largest ones are obtained for @xmath74 with an asymptotic value for large @xmath26 of @xmath75 . by contrast , surprisingly we have @xmath76 which means that the dicke state @xmath57 can be approached as closely as desired by non - symmetric slocc inequivalent @xmath62 states . we thus clearly have @xmath77 and this is a neat violation of eq . ( [ fcc ] ) . to prove eq . ( [ maxrnk ] ) , we define for any @xmath78 the non - symmetric @xmath62 state @xmath79 with @xmath80 the @xmath26-qubit non - symmetric local operation @xmath81 where , in the computational basis @xmath82 , @xmath83 the state @xmath84 belongs to the @xmath62 slocc class since for any @xmath78 @xmath85 is an invertible local operation : @xmath86 the state @xmath84 is also non - symmetric since a non - symmetric ilo acting on @xmath61 always yields a non - symmetric state for @xmath87 @xcite . a detailed calculation then gives , for any @xmath78 @xcite , ( see appendix [ apb ] ) @xmath88 where @xmath89 and @xmath90 we have @xmath91 and thus @xmath92 this implies that @xmath93 and this proves eq . ( [ maxrnk ] ) . these results show that for any @xmath63 and @xmath94 , a non - symmetric ilo of the form @xmath95 can transform the dicke state @xmath61 into a non - symmetric state located as closely as desired to the dicke state @xmath57 , even though @xmath61 and @xmath57 are slocc inequivalent . this result can not be achieved when only symmetric slocc operations are considered . as an aside and out of curiosity , it is interesting to study the inverse operation @xmath96 acting on the @xmath57 state . while one obviously has @xmath97 the state @xmath98 reads , up to a normalization constant , @xmath99 , \nonumber\end{aligned}\ ] ] this state is totally independent of @xmath100 and differs significantly from @xmath61 . it has only components onto states with at least @xmath101 excitations . therefore , except for @xmath102 , @xmath103 has no overlap with @xmath61 . for @xmath102 , the fidelity with the state @xmath61 amounts only to @xmath104 . @xmath105 \hline \hline 4 & 2 & 0.5 & 7 & 2 & 0.457 \\ 5 & 2 & 0.477 & & 3 & 0.383 \\ 6 & 2 & 0.465 & 8 & 2 & 0.451 \\ & 3 & 0.4 & & 3 & 0.372 \\ & & & & 4 & 0.344\\ \hline \end{array}\ ] ] equation ( [ maxrnk ] ) can even be generalized . any state @xmath106 of the @xmath58 slocc class satisfies @xmath107 indeed , for such states an ilo @xmath108 can be found such that @xmath109 we then define for any @xmath78 the @xmath62 states @xmath110 with @xmath111 as defined by eq . ( [ psink2 ] ) . we get trivially , up to a normalization constant , @xmath112 and thus @xmath113 it follows that @xmath114 and this implies eq . ( [ genn1 ] ) . this result shows that all states of the @xmath58 slocc class can be approached as closely as desired by non - symmetric states of any of the @xmath62 slocc classes , for @xmath115 . topologically , this means that the @xmath58 slocc class of states lies at the boundary of the non - symmetric side of any of the @xmath62 classes . this result sheds an additional light on the general topology of the slocc classes of multiqubit systems , whose restriction on the only symmetric subspace was established in ref . @xcite . the converse of the previous result is by far not true : the states of the @xmath62 slocc classes can not be approached as closely as desired by @xmath58 states , even if they are non - symmetric . as an example , a detailed calculation yields ( see appendix [ apa ] ) @xmath116^{n - k-1 } \left[\tilde{k } + ( 1 - 2\tilde{k } ) \tilde{k}_r \right ] \ ] ] with @xmath117 and @xmath118 and extensive numerical simulations [ see our conjecture in eq . ( [ conj ] ) ] showed that this should also correspond to @xmath119 . we exemplify some values of @xmath120 in table [ tabi ] . it is noteworthy to mention that these fidelities decrease with increasing @xmath26 and @xmath21 . for fixed @xmath21 , we have @xmath121 explicitly , for @xmath122 , this limit reads @xmath123 , @xmath124 , @xmath125 , @xmath126 , @xmath127 , @xmath128 , and @xmath129 , respectively . all these results clearly show that while the @xmath57 state can be approached as closely as desired by non - symmetric states that are slocc equivalent to any of the @xmath61 states ( @xmath64 ) , none of these latter can be closely approached by any state slocc equivalent to the dicke state @xmath57 . in this paper we have analyzed the maximization of the quantum fidelity between symmetric multiqubit states and sets of lu- or slocc - equivalent states that contain symmetric states . we have shown that the open question in eq . ( [ fcc ] ) admits a positive answer when lu classes of qubit states are considered , while the answer is negative when turning to slocc classes of states . in the case of lu sets , the positive answer simplifies considerably the calculation of the desired maximal overlap . for slocc classes of states , we have shown significant violations of eq . ( [ fcc ] ) where for some states @xmath12 and classes @xmath17 the fidelity @xmath130 takes the maximal possible value 1 , while the symmetric restriction @xmath131 has only significantly much lower values . this is in particular the case when considering any states @xmath12 of the @xmath58 slocc class in combination with any of the @xmath62 slocc classes , for @xmath63 and @xmath64 . finally , extensive numerical simulations have also lead us to conjecture that eq . ( [ fcc ] ) seems to be correct when the considered slocc class @xmath17 identifies to @xmath58 , whatever the state @xmath12 . there are several directions in which our work can be generalized . first , concerning lu equivalence classes , it would be highly desirable to prove our result also for higher - dimensional systems and not only for qubits . in our proof , we made use of the simple parametrization of @xmath33 matrices , which is not so simple in higher dimensional systems . second , for slocc equivalence classes it would be very useful to find out under which additional conditions the optimization over symmetric states is enough . based on numerical evidence we identified some examples where this seems to be the case , but so far no clear understanding has been reached . from a more general perspective , our work presents examples where symmetries can help to solve optimization problems related to the numerical range @xcite . understanding further the role of symmetry in such problems is clearly a challenging task , nevertheless it will have a significant impact on various problems in quantum information theory . a.n . acknowledges a fria grant and the belgian f.r.s .- fnrs for financial support . o.g . acknowledges financial support from the fqxi fund ( silicon valley community foundation ) , the dfg , and the erc ( consolidator grant no . 683107/tempoq ) . tb acknowledges financial support from the belgian f.r.s .- fnrs through iisn grant no . 4.4512.08 . @xmath136}}_{y \in [ -1,1 ] } \frac{\sum_{j=0}^{k ' } \left[(2 - \delta_{j,0 } ) \sum_{j ' = j}^{k ' } c_{j'}(x , x ' ) c_{j'-j}(x , x')\right ] t_j(y)}{\sum_{j=0}^{k ' } c_{k'}^j c_{n - k'}^j \left [ x^2 x'^2 + ( 1-x^2)(1-x'^2 ) + 2 y x x ' \sqrt{(1-x^2)(1-x'^2)})\right]^{j}},\ ] ] where @xmath137 is the binomial coefficient between @xmath138 and @xmath139 , with the usual convention @xmath140 if @xmath141 or @xmath142 , @xmath143 denotes the kronecker delta , @xmath144 is the @xmath145-th degree chebyshev polynomial of the first kind , and , for @xmath146 , @xmath147 in particular , for @xmath148 , one gets the well known result @xcite @xmath149 } x^{2k } ( 1-x^2)^{n - k } \nonumber \\ & = c_n^k \tilde{k}^k \left(1-\tilde{k}\right)^{n - k},\end{aligned}\ ] ] with @xmath117 the fractional excitation of the dicke state @xmath61 . for @xmath150 , one gets @xmath151}}_{y \in [ -1,1 ] } \frac{c_0(x , x')^2 + c_1(x , x')^2 + 2 c_0(x , x ' ) c_1(x , x ' ) y}{1 + ( n-1)\left[x^2 x'^2 + ( 1-x^2)(1-x'^2 ) + 2 y x x ' \sqrt{(1-x^2)(1-x'^2)}\right ] } \nonumber \\ & = c_n^k \tilde{k}_r^{k-1 } \left ( 1-\tilde{k}_r \right)^{n - k-1 } \left[\tilde{k } + ( 1 - 2\tilde{k } ) \tilde{k}_r \right],\end{aligned}\ ] ] with @xmath152 for @xmath153 , eq . ( [ rsnkkp ] ) can not be reduced analytically and it must be evaluated numerically . we noticed that for all tested cases the supremum was systematically obtained for @xmath154 . to prove eq . ( [ rsnkkp ] ) , we first observe that in the computational basis the dicke state @xmath61 merely reads @xmath155 , where the sum runs over all multiqubit states with any @xmath21 qubits in the @xmath156 state and the remaining @xmath157 qubits in the @xmath158 state . we then note that any symmetric state of the @xmath159 slocc class , i.e. , states of the @xmath160 family , can be written in the form @xcite @xmath161 , where @xmath162 is a normalization constant and the sum runs over all multiqubit states with any @xmath163 qubits in an @xmath164 state and the remaining @xmath165 qubits in a distinct @xmath166 state . writing the single qubit states @xmath164 and @xmath167 in the form @xmath168 and @xmath169 ( @xmath170 , \phi,\phi ' \in [ 0,2\pi[$ ] ) , respectively , yields @xmath171 with @xmath172 , @xmath173 and @xmath174 . we then get @xmath175 } } _ { \delta \phi \in [ 0,2\pi [ } \mathcal{n}^2 c_n^k \left|\sum_{j=0}^{k ' } c_j(x , x ' ) e^{i j \delta \phi}\right|^2,\ ] ] with @xmath176 as given by eq . ( [ cjxxp ] ) . finally , using the identity @xmath177 and setting @xmath178 yields eq . ( [ rsnkkp ] ) . we then define the unnormalized dicke states @xmath182 , which satisfy @xmath183 . \label{eq : decomposition_gd}\end{aligned}\ ] ] inserting eq . ( [ bnkeps ] ) in eq . ( [ eq : decomposition_gd ] ) and observing that for any @xmath26 and @xmath21 the symmetric state @xmath184 reads in the dicke state basis @xmath185 yields straightforwardly @xmath186 with @xmath187 , @xmath188 , and @xmath189 as given by eq . ( [ ajbjcj ] ) and @xmath190 . in the sum over @xmath145 in eq . ( [ gnkeps ] ) , the first term @xmath191 merely yields the state @xmath192 , which is nothing but the dicke state @xmath57 [ see eq . ( [ unkrec ] ) ] . the rest of the sum from @xmath193 to @xmath194 is by definition the state @xmath195 [ eq . ( [ psieps ] ) ] . we thus get @xmath196 from which eq . ( [ psink2 ] ) immediately follows for any @xmath78 . for @xmath197 , @xmath198 and the normalized state @xmath199 is not defined . c. d. cenci , d. w. lyons , s. n. walck , arxiv:1011.5229 ; _ theory of quantum computation , communication and cryptography _ , edited by d. bacon , m. martin - delgado , and m. roetteler , lecture notes in computer science , vol . 6745 ( springer , berlin , 2014 ) , p. 198 . we recall that the @xmath21-excitation dicke states ( @xmath200 ) are defined as @xmath201 , where @xmath202 denotes the binomial coefficient @xmath203 , the multiqubit states in the sum contain @xmath21 qubits in state @xmath156 , and @xmath204 denotes all permutations of the qubits leading to different terms in the sum . all @xmath205 dicke states ( @xmath200 ) are symmetric and they form an orthonormal basis in the symmetric subspace of the multiqubit system . for @xmath206 , where @xmath207 denotes the floor function , the dicke states @xmath61 are slocc inequivalent between each others @xcite . in contrast , the @xmath208 and @xmath61 states are lu equivalent . although the @xmath26-qubit local operation @xmath80 remains defined for @xmath197 , though not invertible in this case , the normalized state @xmath199 is in contrast not defined in this case since @xmath209 ( see appendix [ apb ] ) .
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for two symmetric quantum states one may be interested in maximizing the overlap under local operations applied to one of them .
the question arises whether the maximal overlap can be obtained by applying the same local operation to each party .
we show that for two symmetric multiqubit states and local unitary transformations this is the case ; the maximal overlap can be reached by applying the same unitary matrix everywhere . for local invertible operations ( stochastic local operations assisted by classical communication equivalence ) , however , we present counterexamples , demonstrating that considering the same operation everywhere is not enough .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the field of cold atoms has expanded dramatically over the last 15 years . it has now reached a stage where experimentalists are capable of designing the form of the confining potential . going to extreme aspect ratios , conditions of quasi - one- and quasi - two - dimensional behavior have been achieved . in other experiments , it has also become possible to design toroidal trapping potentials @xcite , in which persistent currents have been observed @xcite . recent theoretical studies have examined bose - einstein condensates in one - dimensional annular traps . for example , quantum - tunneling - related effects in vertically @xcite and concentrically @xcite coupled double - ring traps were investigated . also , the rotational properties of a mixture of two distinguishable bose gases that are confined in a single ring were addressed @xcite . one of the basic points of the above studies is the fact that the ability to design traps , control and manipulate the atoms with a very high accuracy , may allow the investigation of novel quantum phenomena , like quantum phase transitions , for example . in the present work we consider a mixture of two distinguishable bose gases @xcite which interact via an effectively - repulsive contact potential , and are confined in a two - dimensional concentric double - ring - like trap , as shown in fig.[fig1 ] . using the mean - field approximation , we investigate two main questions : first , we identify the various phases in the ground state of the system , varying the interaction strength between the atoms . in the trapping potential that we consider , we observe that the two gases separate radially via discontinuous transitions ; in this case , each gas resides in one of the two minima of the trapping potential , preserving the circular symmetry of the trapping potential . we also observe the expected azimuthal ( and continuous ) phase separation between the two gases in each potential minimum @xcite . a similar effect has also been studied in the case of a single ring @xcite ; see also @xcite . the second main question that we examine are the rotational properties of this system , including its response to some rotational frequency of the trap @xmath0 , as well as the stability of the persistent currents for variable couplings , and variable relative populations of the two components . the expectation value of the angular momentum of the system as a function of @xmath0 shows an interesting structure , reflecting the various phase transitions that take place with increasing @xmath0 . regarding the ( meta)stability of the currents , it is remarkable that for equal populations between the two components the vast majority of the coupling strengths that we have examined yield metastable states , except for a very small range where all the coupling strengths are exactly or nearly equal . it is worth mentioning that analogous single and concentric ring geometries have been addressed in semiconductor heterostructures , both theoretically and experimentally , see e.g. @xcite , and also @xcite for reviews on the subject . in these systems , the applied external magnetic field plays the same role as the trap rotation in the present problem and allows the investigation of , e.g. , electron localization effects and persistent electron currents in field - free regions . in what follows we first describe in sec.ii our model . in sec.iii we present the results for the ground state of the system , identifying the states where the species coexist , or separate , either radially or azimuthally . in sec.iv we examine the rotational properties for a fixed rotational frequency of the trap , and the ( meta)stability of the persistent currents . we study the stability as a function of the coupling between the atoms , as well as of the ratio of the populations of the two components . finally , in sec.v we present a summary and our conclusions . we consider two distinguishable kinds of bosonic atoms , labelled as @xmath1 and @xmath2 , which are trapped in a two - dimensional potential of the form @xmath3 where @xmath4 is the usual radial coordinate in cylindrical coordinates and @xmath5 is the atom mass , assumed to be equal for the two components . the two ( overlapping ) parabolae in @xmath6 with frequencies @xmath7 and @xmath8 are centered at the positions with @xmath9 and @xmath10 , giving rise to the potential plotted in fig.[fig1 ] @xcite . of eq.([trpot ] ) , where @xmath11 , @xmath12 , and @xmath13.,width=264 ] in our calculations we consider @xmath11 and @xmath14 , where @xmath15^{1/2}$ ] is the oscillator length corresponding to @xmath16 , and finally @xmath13 . in the outer ring the potential is more tight , @xmath17 , in order to compensate for the fact that @xmath18 , i.e. , to make the product of the width " of each annulus times the radius of each annulus to be comparable to each other . ( this is typically also the case in the studies on electrons in quantum rings , in semiconductor heterostructures that we mentioned above . ) to simplify the discussion , we also assume that there is a very tight trapping potential along the @xmath19 axis ( omitted in the potential above ) , which completely freezes out the degrees of freedom of the gases along this direction . with this assumption , our problem becomes effectively two - dimensional , with the tight dimension entering only implicitly through the parameters @xmath20 in the hamiltonian of eq.([ham ] ) . with the usual assumption of a contact interatomic potential , the hamiltonian becomes @xmath21 here @xmath22 , with @xmath23 being the state of lowest energy of the potential along the @xmath19 axis and @xmath24 being the @xmath25wave scattering lengths for zero - energy elastic atom - atom collisions . the coupled gross - pitaevskii - like equations for the order parameters of the two components @xmath26 and @xmath27 , resulting from the above hamiltonian , are @xmath28 in the above equations @xmath29 , and @xmath30 . also , @xmath31 , @xmath32 , @xmath33 , and @xmath34 are the chemical potentials . in what follows , we consider repulsive interactions only , @xmath35 , @xmath36 , and also assume that @xmath37 . the method that we adopt to solve eqs.([gpe ] ) is a fourth - order split - step fourier method within an imaginary - time propagation approach @xcite . we start with a reasonable initial state for the two components and propagate it in imaginary time , making sure that we proceed a sufficiently large number of time steps , which guarantee that we have reached a steady state . we start with the ground state of the system . there are three energy scales in the problem , namely the single - particle energy that is set by the trap , the intra - atomic interaction energy , and the inter - atomic interaction energy . for weak interactions , the energy is dominated by the single - particle term and the ground state is determined by the minimization of this term . on the other hand , for strong interactions , for a given ratio of the two populations @xmath38 , the actual symmetry of the ground state results from the competition between the intra - and inter - species coupling strengths ; the former favors the maximum possible spread of the two gases within the system , whereas the latter favors the minimization of their spatial overlap . the most pronounced difference of this problem as compared to the case where there is only one potential minimum i.e. , a single annulus is the existence of a phase where each component resides in only one of the two potential minima , thus separating radially . in addition , we have also observed the expected azimuthal phase separation when both species occupy the same potential minimum @xcite . in fig.[fig2 ] we illustrate the phase diagram showing the symmetry of the ground - state density distribution of the two gases as @xmath39 and @xmath40 are varied , for @xmath41 . since @xmath42 and @xmath41 , the results are symmetric when the two components are interchanged . the axes in the phase diagram of fig.[fig2 ] may also be considered to represent the values of the scattering lengths @xmath43 and @xmath44 ( scaled appropriately ) . we have found three different phases : ( i ) coexistence of the two species ( squares , red color ) , ( ii ) azimuthal separation ( triangles , blue color ) , and ( iii ) radial separation ( circles , green color ) , see fig.[fig3 ] . the difference between solid and empty symbols in fig.[fig2 ] refers to the stability of the persistent currents and is explained in the following section . -versus-@xmath40 plane , which shows the symmetry of the ground - state density distribution of the two components . squares , triangles and circles correspond to phase coexistence , azimuthal phase separation , and radial phase separation , respectively . the solid ( empty ) symbols denote points where the currents are metastable ( unstable).,width=302 ] as seen in fig.[fig2 ] , the phase boundary between phase coexistence and separation of the two components is , to a very good approximation , a straight line given by ( setting @xmath45 ) @xmath46 as we have found by fitting numerically our data . certain limiting cases in the phase diagram of fig.2 may be analyzed and understood easily . in the case @xmath47 and @xmath48 , the two gases do not interact with each other . in order to minimize their energy , they distribute homogeneously along the rings and coexist . in the other limiting case @xmath49 , for @xmath39 smaller than @xmath50 , the two components also coexist ; however , for larger values than 2 , they separate azimuthally . although in this phase the kinetic energy increases , the interaction energy is lowered due to the repulsion between the two species , and the azimuthal symmetry - breaking persists with increasing @xmath39 . it is also instructive to understand the internal structure of the phase diagram . as one moves vertically , i.e. , for a fixed value of @xmath39 ( being sufficiently large , such that the components separate azimuthally ) , for small enough @xmath40 the two components occupy mainly the inner ring since the repulsion between the particles can not compensate for the stronger confinement of the outer ring . as @xmath40 increases , the outer ring becomes progressively more occupied . when the inter - component repulsion @xmath39 becomes large enough , the gases minimize their energy by separating radially . however , if @xmath40 increases further , the inner ring becomes too small to host one of the species entirely . azimuthal phase separation takes place again , now with both gases being largely spread within the whole system . eventually , for even larger values of @xmath40 , the dominant term in the energy is the intra - atomic interaction , and thus the two components coexist . as anticipated before , we illustrate this effect in fig.[fig3 ] , where we plot the densities of the cases with @xmath51 , and ( a ) @xmath52 , ( b ) @xmath53 , ( c ) @xmath54 , and ( d ) @xmath55 , corresponding to azimuthal phase separation , radial phase separation , azimuthal phase separation , and phase coexistence , respectively . it is also of interest to investigate the nature of the phase transitions occurring in the system . as one crosses the boundary from coexistence to azimuthal phase separation , the two components decrease continuously their overlap , developing sharper profiles as the repulsion increases . this transition is thus continuous ( second order ) , as it is also the case in purely one - dimensional single rings @xcite . in the corresponding energy surface , the minimum ( which determines the ground state ) moves continuously as one crosses the phase boundary . on the contrary , the transitions involving radial separation are discontinuous ( first order ) , indicating that two local minima in the energy surface compete and that the system jumps abruptly from the one state to the other . of the two components , for @xmath51 and ( a ) @xmath52 , ( b ) @xmath53 , ( c ) @xmath54 , and ( d ) @xmath55 , corresponding to azimuthal phase separation , radial phase separation , azimuthal phase separation , and phase coexistence , respectively . the peaks in the densities correspond to the minima of the inner and the outer parabolae of the confining potential.,width=340 ] the phase transitions that we described above also have a clear influence on the rotational properties of the system . we start by examining the response of the system to some finite rotational frequency @xmath56 of the trap . in the following we determine the total angular momentum per particle @xmath57 as a function of @xmath0 , @xmath58 . following the usual procedure , we minimize the energy of the system in the rotating frame , i.e. , we minimize @xmath59 , where @xmath60 is the total energy . the result of this calculation is shown in fig.[fig4 ] , where we have set @xmath61 , @xmath62 , and @xmath63 . the typical values of @xmath0 are rather small because we have scaled it with @xmath7 . however , at least in the case of a purely one - dimensional ring potential , the scale for the typical @xmath0 is on the order of the frequency @xmath64 of the corresponding kinetic energy @xmath65 @xcite , where @xmath66 is the radius of the ring . in atomic units , @xmath67 , @xmath68 , while @xmath69 ( for @xmath66 , say , equal to 5 ) . this introduces a factor @xmath70 . it is also instructive to comment on the behavior of the gas as @xmath0 increases and see the connection of the function @xmath71 with the density distribution of the two components . for @xmath72 the two species are separated radially and the total angular momentum is zero . for @xmath73 , the density of the two components breaks its azimuthal symmetry discontinuously , and the angular momentum jumps abruptly to a finite value . beyond this value of @xmath74 the angular momentum increases linearly with @xmath0 and is carried by both components , which undergo solid - body rotation . this is an expected result due to the azimuthal symmetry - breaking of the density , as we have confirmed by studying the phases of the two order parameters . when @xmath75 , a new plateau appears , which corresponds to an angular momentum per particle equal to two , and the two species separate radially . as one can see from the plot , there is a sequence between plateaus at integer values of @xmath76 and straight lines with a positive slope , which are separated by abrupt jumps . this sequence persists up to @xmath77 . beyond this value of @xmath0 there is no longer radial separation of the two components due to the large centrifugal force , which forces both gases to occupy the two potential minima and therefore to separate azimuthally . similar discontinuous transitions in the function @xmath71 occur in a single - component weakly - interacting bose - einstein condensate that rotates in a harmonic trap @xcite , and are associated with discontinuous transitions between phases of different symmetries of the single - particle density distribution of the gas . in the present problem , the corresponding different symmetries are the ones where the components separate radially , or azimuthally . furthermore , the jumps in the @xmath71 plot are consistent with the fact that for the parameters considered , the system supports persistent currents . in other words , had this function been continuous , then metastability would have not been possible ( for a discussion of this effect we refer to ref.@xcite ) . from the above observations we see that the general picture that emerges by considering a finite rotation of the trap resembles the one where the couplings are varied , with a series of phase transitions between radial and azimuthal separation . , as a function of the angular frequency of rotation of the trap @xmath0 , which results from the minimization of the energy in the rotating frame.,width=340 ] another interesting question is the possible existence of ( meta)stable currents . in physical terms , we investigate the energetic ( meta)stability of current - carrying states . in the case that there is an energy barrier that separates a current - carrying state from the ground state , in the presence of some dissipative mechanism ( such as a thermal cloud , for example ) such a state does not decay , and therefore the system supports persistent currents . we examine two separate aspects of this problem . firstly , we consider the points of the phase diagram shown in fig.[fig2 ] , for a fixed population of the two gases . secondly , we fix the interaction strengths and vary the ratio @xmath38 . in both cases we examine the imaginary - time evolution of initial states with some finite , nonzero expectation value of the angular momentum , in the absence of any external rotation of the trap . for a given set of parameters , the existence of a converged final state with a nonzero expectation value of @xmath76 implies that the associated current is metastable . we have examined all the points that are shown in the phase diagram of fig.[fig2 ] . those corresponding to a final state with a nonzero expectation value of the angular momentum are represented in fig.2 with `` solid '' symbols , while the ones that decay to the non - rotating ground state are represented with `` open '' symbols . clearly , the vast majority of states correspond to metastable currents , except those close to the diagonal @xmath78 , as well as those with sufficiently small @xmath40 ( for all values of @xmath39 ) . the obtained results show that the angular momentum of the metastable states is always an integer multiple of the particle population , which implies that the associated densities are necessarily circularly symmetric . this is consistent with the statement that circular symmetry is a necessary though not sufficient condition for the ( meta)stability of the currents , as otherwise the circulation may escape from the gas ( since in this case there is no barrier separating the rotating state from the non - rotating one @xcite ) . let us now study the effect of a variable relative population between the two gases on the ( meta)stability of the currents . we thus fix the interaction strengths , as well as the population of the one component @xmath79 , and study this question starting from the case with @xmath80 , all the way up to the limit @xmath81 , with an initial state that has some angular momentum in component @xmath2 . since @xmath82 is fixed , the above procedure corresponds physically to keeping the scattering lengths fixed . as mentioned above , the ( meta)stability of the persistent currents depends on the competition between azimuthal phase separation and circular symmetry of the density distribution of the two components . thus , when @xmath80 , the @xmath2 component spreads within the whole system for any value of @xmath40 and the currents can be metastable , provided that the coupling @xmath40 is sufficiently large . if the population of the component @xmath1 becomes nonzero but is small enough , it acts only as a weak perturbation , and azimuthal symmetry is still preserved . however , beyond a critical ratio @xmath38 , the two species separate azimuthally and the currents are no longer metastable . this critical value depends on the actual intra- and inter - component interaction strengths . a further increase of @xmath83 with respect to @xmath82 drives the system to a phase of radial separation , and metastability is recovered . finally , in the limit where @xmath83 becomes too large , this component can not fit into only one of the two rings . this leads again to azimuthal phase separation of the gases , which is preserved even in the limit @xmath81 . as a result , the stability of the currents is lost again . we show in fig.[fig5 ] the densities for the case @xmath54 , and @xmath51 and for various values of the ratio @xmath38 , in order to illustrate the mentioned sequence of phase transitions . in particular , the results correspond to @xmath84 , @xmath85 , @xmath86 , and @xmath87 , with the first and the third cases corresponding to the metastable states . according to our simulations , changing the values of the interaction strengths modifies the sizes of the `` windows '' in @xmath83 that separate the different phases , but yields qualitatively similar results . and @xmath88 with @xmath54 , and @xmath89 of the component @xmath1 ( dark blue ) and @xmath2 ( light yellow ) , for ( a ) @xmath84 , ( b ) @xmath90 , ( c ) @xmath86 , and ( d ) @xmath87 . in the cases ( a ) and ( c ) the states support metastable currents , while in ( b ) and ( d ) they do not . the peaks in the densities correspond to the minima of the inner and the outer parabolae of the confining potential.,width=302 ] as shown in the present study , a coupled system of two distinguishable bose gases that interact with an effectively - repulsive contact potential and are loaded in a concentric double annular trap reveals a series of phase transitions and the existence of metastable currents . for weak interactions , when the chemical potential is much smaller than the barrier that separates the two potential minima , the gases are confined in the inner ring , with the width of their transverse profile being smaller than the radius of the ring . thus , their motion is ( at least ) close to being quasi - one - dimensional . on the other hand , as the couplings increase , the barrier plays a decreasingly important role , and their transverse width becomes comparable to the radius of the ring(s ) . such a trapping potential interpolates up to some extent between one - dimensional and two - dimensional motion , depending on the strength of the coupling between the atoms . the interplay between this effect and the strength of the inter- and intra - atomic couplings gives rise to interesting phase transitions in the ground state of the system , including three different geometries : phase coexistence , radial phase separation , and azimuthal phase separation . an interesting feature of the system considered is its response to some finite rotational frequency of the trap . the basic picture resembles very much the one where the couplings are varied , inducing axial and/or radial phase separation of the two components . the robustness in the ( meta)stability of the currents that we found for the vast majority of the points in the phase diagram of fig.[fig2 ] is another interesting aspect of this study . metastability is not found for the cases when the couplings are all nearly equal or exactly equal to each other , as well as for the cases with small @xmath40 , independently of @xmath39 . according to ref.@xcite , a necessary ( but not sufficient ) condition for metastability is that the trapping potential does not increase monotonically from the center of the trap . the present results suggest that multiple variations in the monotonicity of the trapping potential enhance the stability of the currents . last but not least , the discontinuous phase transitions we have found both in the ground - state phase diagram when the two gases separate radially as the couplings are varied , as well as in the response of the system when the rotation of the trap is varied , imply that hysteresis should show up as the coupling / rotational frequency increases / decreases . we thank s. bargi and k. krkkinen for useful discussions . this work was financed by the swedish research council . the collaboration is part of the nordforsk nordic network `` coherent quantum gases - from cold atoms to condensed matter '' . b. szafran and f. m. peeters , phys . b * 72 * , 155316 ( 2005 ) ; j. m. escartn , f. malet , a. emperador , and m. pi , phys . rev . b * 79 * , 245317 ( 2009 ) ; t. mano , t. kuroda , s. sanguinetti , t. ochiai , t. tateno , j. kim , t. noda , m. kawabe , k. sakoda , g. kido , and n. koguchi , nanoletters * 5 * , 425 ( 2005 ) ; s. viefers , p. s. deo , s. m. reimann , m. manninen , and m. koskinen , phys . rev . b * 62 * , 10668 ( 2000 ) ; m. manninen , m. koskinen , s. m. reimann , and b. mottelson , eur . phys . j. d * 16 * , 381 .
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a two - component bose - einstein condensate confined in an axially - symmetric potential with two local minima , resembling two concentric annular traps , is investigated .
the system shows a number of phase transitions that result from the competition between phase coexistence , and radial / azimuthal phase separation .
the ground - state phase diagram , as well as the rotational properties , including the ( meta)stability of currents in this system , are analysed .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
compact binary star systems are likely to be an important source of gravitational waves for the broadband laser interferometric detectors now under construction @xcite , as they are the best understood of the various types of postulated gravity wave sources in the detectable frequency band and their waves should carry a large amount of information . within our own galaxy , there are three known neutron star binaries whose orbits will decay completely under the influence of gravitational radiation reaction within less than one hubble time , and it is almost certain that there are many more as yet undiscovered . current estimates of the rate of neutron star / neutron star ( ns / ns ) binary coalescences @xcite based on these ( very few ) known systems project an event rate of three per year within a distance of roughly 200 mpc ; and estimates based on the evolution of progenitor , main - sequence binaries suggest a distance of as small as roughly 70 mpc for three events per year . these distances correspond to a signal strength which is within the target sensitivities of the ligo and virgo interferometers @xcite . however , to find the signals within the noisy ligo / virgo data will require a careful filtering of the interferometer outputs . because the predicted signal strengths lie so close to the level of the noise , it will be necessary to filter the interferometer data streams in order to detect the inspiral events against the background of spurious events generated by random noise . the gravitational waveform generated by an inspiraling compact binary has been calculated using a combination of post - newtonian and post - minkowskian expansions to post@xmath9-newtonian order by the consortium of blanchet , damour , iyer , will , and wiseman @xcite , and will be calculated to post@xmath10-newtonian order long before the ligo and virgo interferometers come on - line ( c. 2000 ) . because the functional form of the expected signal is so well - known , it is an ideal candidate for matched filtering , a venerable and widely known technique which is laid out in detail elsewhere @xcite and briefly summarized here : the matched filtering strategy is to compute a cross - correlation between the interferometer output and a template waveform , weighted inversely by the noise spectrum of the detector . the signal - to - noise ratio is defined as the value of the cross - correlation of the template with a particular stretch of data divided by the rms value of the cross - correlation of the template with pure detector noise . if the signal - to - noise ratio exceeds a certain threshold , which is set primarily to control the rate of false alarms due to fluctuations of the noise , a detection is registered . if the functional form of the template is identical to that of the signal , the mean signal - to - noise ratio in the presence of a signal is the highest possible for any linear data processing technique , which is why matched filtering is also known as optimal filtering . in practice , however , the template waveforms will differ somewhat from the signals . true gravitational - wave signals from inspiraling binaries will be exact solutions to the einstein equations for two bodies of non - negligible mass , while the templates used to search for these signals will be , at best , finite - order approximations to the exact solutions . also , true signals will be characterized by many parameters ( e.g. the masses of the two objects , their spins , the eccentricity and orientation of the orbit ... ) , some of which might be neglected in construction of the search templates . thus , the true signals will lie somewhat outside the submanifold formed by the search templates in the full manifold of all possible detector outputs ( see fig . [ fig : manifold ] ) . apostolatos @xcite has defined the `` fitting factor '' @xmath11 to quantitatively describe the closeness of the true signals to the template manifold in terms of the reduction of the signal - to - noise ratio due to cross - correlating a signal lying outside the manifold with all the templates lying inside the manifold . if the fitting factor of a template family is unity , the signal lies in the template manifold . if the fitting factor is less than unity , the signal lies outside the manifold , and the fitting factor represents the cross - correlation between the signal and the template nearest it in the template manifold . even if the signal were to lie within the template manifold , it would not in general correspond to any of the actual templates used to search the data . the parameters describing the search templates ( masses , spins , etc . ) can vary continuously throughout a finite range of values . the set of templates characterized by the continuously varying parameters is of course infinite , so the interferometer output must be cross - correlated with a finite subset of the templates whose parameter values vary in discrete steps from one template to the next . this subset ( the `` discrete template family '' ) has measure zero on the manifold of the full set of possible templates ( the `` continuous template family '' ) , so the template which most closely matches a signal will generally lie in between members of the discrete template family ( again , see fig . [ fig : manifold ] ) . the mismatch between the signal and the nearest of the discrete templates will cause some reduction in the signal - to - noise ratio and therefore in the observed event rate , as some signals which would lie above the threshold if cross - correlated with a perfectly matched filter are driven below the threshold by the mismatch . thus the spacing between members of the discrete template family must be chosen so as to render acceptable the loss in event rate , without requiring a prohibitive amount of computing power to numerically perform the cross - correlations of the data stream with all of the discrete templates . the high computational demands of a laser interferometric detector may in fact make it desirable to perform a _ hierarchical search_. in a hierarchical search , each stretch of data is first filtered by a set of templates which rather sparsely populates the manifold , and stretches which fail to exceed a relatively low signal - to - noise threshold are discarded . the surviving stretches of data are filtered by a larger set of templates which more densely populates the manifold , and are subjected to a higher threshold . the spareseness of the first - pass template set insures that most of the data need only be filtered by a small number of templates , while the high threshold of the final pass reduces the false alarm rate to an acceptable level . theoretical foundations for choosing the discrete set of templates from the continuous family were laid by sathyaprakash and dhurandhar for the case of white noise in ref . , and for ( colored ) power - recycling interferometer noise in ref . . both papers used a simplified ( so - called `` newtonian '' ) version of the waveform which can be characterized by a single parameter , the binary s `` chirp mass '' @xmath12 . recently , sathyaprakash @xcite began consideration of an improved , `` post - newtonian '' set of templates characterized by two mass parameters . he found that , by a judicious choice of the two parameters , the spacing between templates can be made constant in both dimensions of the intrinsic parameter space . sathyaprakash s parameters also make it obvious ( by producing a very large spacing in one of the dimensions ) that a two - parameter set of templates can be constructed which , if it does not populate the manifold too densely , need not be much more numerous than the one - parameter set of templates used in refs . . in this paper i shall recast the s - d formalism in geometric language which , i believe , simplifies and clarifies the key ideas . i shall also generalize the s - d formalism to an arbitrary spectrum of detector noise and to a set of template shapes characterized by more than one parameter . this is necessary because , as apostolatos @xcite has shown , no one - parameter set of templates can be used to filter a post - newtonian signal without causing an unacceptably large loss of signal - to - noise ratio . in one respect , my analysis will be more specialized than that of the s - d formalism . my geometric analysis requires that the templates of the discrete set be spaced very finely in order that certain analytical approximations may be made , while the numerical methods of sathyaprakash and dhurandhar are valid even for a large spacing between templates ( as would be the case in the early stages of a hierarchical search ) . the small spacing approximation is justified on the grounds that at some point , even in a hierarchical search , the data must be filtered by many closely spaced templates in order to detect a reasonable fraction ( of order unity ) of the binary inspirals occurring in the universe within range of the ligo / virgo network . the rest of this paper is organized as follows : in sec . [ sec : formalism ] , i develop my generalized , geometric variant of the s - d formalism . i then apply this formalism to the general problem of detection of gravitational waves from inspiraling binaries , and develop general formulas for choosing a discrete template family from a given continuous template family . in sec . [ sec:1pn ] , i detail an example of the use of my formalism , choosing discrete templates from a continuous template family which describes nonspinning , circularized binaries to post@xmath1-newtonian order in the evolution of the waveform s phase . i also estimate the computing power required for a single - pass ( non - hierarchical ) search using this discrete template family , and compare to the previous work of sathyaprakash @xcite . finally , in sec . [ sec : conclusion ] , i summarize my results and suggest future directions for research on the choice of discrete search templates . in this section , a geometric , multiparameter variant of the s - d formalism is developed . unless otherwise stated , the following conventions and definitions are assumed : following cutler and flanagan , we define the inner product between two functions of time @xmath13 and @xmath14 ( which may be templates or interferometer output ) as @xmath15 .\end{aligned}\ ] ] here @xmath16 is the fourier transform of @xmath13 , @xmath17 and @xmath18 is the detector s noise spectrum , defined below . the interferometer output @xmath19 consists of noise @xmath20 plus a signal @xmath21 , where @xmath22 is a dimensionless , time - independent amplitude and @xmath23 is normalized such that @xmath24 . thus , @xmath22 describes the strength of a signal and @xmath23 describes its shape . waveform templates are denoted by @xmath25 , where @xmath26 is the vector of `` intrinsic '' or `` dynamical '' parameters characterizing the template shape and @xmath27 is the vector of `` extrinsic '' or `` kinematical '' parameters describing the offsets of the waveform s endpoint . examples of intrinsic parameters @xmath28 are the masses and spins of the two objects in a compact binary ; examples of extrinsic parameters @xmath29 are the time of a compact binary s final coalescence @xmath30 and the phase of the waveform at coalescence @xmath31 . templates are assumed to be normalized such that @xmath32 for all @xmath27 and @xmath26 . expectation values of various quantities over an infinite ensemble of realizations of the noise are denoted by @xmath33 $ ] . the interferometer s strain spectral noise density @xmath18 is the one - sided spectral density , defined by @xmath34 = \frac{1}{2}\delta ( f_1 - f_2 ) s_h(f_1)\ ] ] for positive frequencies and undefined for negative frequencies . the noise is assumed to have a gaussian probability distribution . newton s gravitational constant @xmath35 and the speed of light @xmath36 are set equal to one . in developing our formalism , we begin by defining the signal - to - noise ratio . for any single template @xmath37 of unit norm , the cross - correlation with pure noise @xmath38 is a random variable with mean zero and variance unity ( cf . ii.b . of ref . , wherein it is shown that @xmath39 = \langle a | b\rangle$ ] ) . the signal - to - noise ratio of a given stretch of data @xmath19 , after filtering by @xmath37 , is defined to be @xmath40 this ratio is the statistic which is compared to a predetermined threshold to decide if a signal is present . if the template @xmath41 is the same as the signal @xmath42 , it optimally filters the signal , and the corresponding ( mean ) optimal signal - to - noise ratio is @xmath43 & = & { \rm e}[\langle n+{\cal a}u | u\rangle ] \nonumber \\ & = & { \cal a}.\end{aligned}\ ] ] if the template @xmath41 used to filter the data is not exactly the same as the signal @xmath42 , the mean signal - to - noise ratio is decreased somewhat from its optimal value : @xmath44 & = & { \rm e}[\langle n+{\cal a}s | u\rangle ] \nonumber \\ & = & { \cal a}\langle s | u\rangle.\end{aligned}\ ] ] the inner product @xmath45 , which is bounded between zero and one , is the fraction of the optimal @xmath46 $ ] retained in the mismatched filtering case , and as such is a logical measure of the effectiveness of the template @xmath41 in searching for the signal shape @xmath42 . now suppose that we search for the signal with a family of templates specified by an extrinsic parameter vector @xmath27 and an intrinsic parameter vector @xmath26 . let us denote the values of the parameters of the actual templates by @xmath47 . for example , @xmath48 might be the value of the time @xmath30 of coalescence for the @xmath49th template in the family , and @xmath50 might be the phase of the @xmath49th template waveform at coalescence . the search entails computing , via fast fourier transforms ( fft s ) , all the inner products @xmath51 for @xmath52 in these numerical computations , the key distinction between the extrinsic parameters @xmath27 and the intrinsic parameters @xmath26 is this : one explores the whole range of values of @xmath27 very quickly , automatically , and efficiently for a fixed value of @xmath26 ; but one must do these explorations separately for each of the @xmath53 . in this sense , dealing with the extrinsic parameters is far easier and more automatic than dealing with the intrinsic ones . as an example ( for further detail see sec . 16.2.2 of schutz @xcite ) , for a given stretch of data one explores _ all _ values of the time of coalescence @xmath54 of a compact binary simultaneously ( for fixed values of the other template parameters ) via a single fft . if we write the fourier transform ( for notational simplicity ) as a continuous integral rather than a discrete sum , we get @xmath55 the discrete fft yields the discrete analog of the function of @xmath30 as shown above , an array of numbers containing the values of the fourier transform for all values of @xmath30 . because , for fixed @xmath53 , the extrinsic parameters @xmath27 are dealt with so simply and quickly in the search , throughout this paper we shall focus primarily on a template family s intrinsic parameters @xmath26 , which govern the shape of the template . correspondingly , we shall adopt the following quantity as our measure of the effectiveness with which a particular template shape i.e . a particular vector @xmath53 of the intrinsic parameters matches the incoming signal : @xmath56{c}{\textstyle\max}\\{\scriptstyle\mu}\end{array } \langle s | u(\bbox{\mu},\bbox{\lambda}_{(k ) } ) \rangle .\ ] ] here the maximization is over all continuously varying values of the extrinsic parameters . then the logical measure of the effectiveness of the entire discrete family of templates in searching for the signal shape is @xmath57{c}{\textstyle\max}\\{\scriptstyle k}\end{array } \bigl [ \begin{array}[t]{c}{\textstyle\max}\\{\scriptstyle\mu}\end{array } \langle s | u(\bbox{\mu},\bbox{\lambda}_{(k ) } ) \rangle \bigr ] , \ ] ] which is simply ( [ sort of match ] ) maximized over all the discrete template shapes . in order to focus on the issue of discretization of the template parameters rather than on the inadequacy of the continuous template family , let us assume that the signal shape @xmath42 is identical to some template . the discussion of the preceding paragraphs suggests that in discussing the discretization of the template parameters we will want to make use of the _ match _ between two templates @xmath58 and @xmath59 which we will define as @xmath60{c}{\textstyle\max}\\{\scriptstyle\mu,\delta\mu}\end{array } \langle u(\bbox{\mu},\bbox{\lambda } ) | u(\bbox{\mu}+\delta\bbox{\mu},\bbox{\lambda}+\delta\bbox{\lambda } ) \rangle .\ ] ] this quantity , which is known in the theory of hypothesis testing as the _ ambiguity function , _ is the fraction of the optimal signal - to - noise ratio obtained by using a template with intrinsic parameters @xmath26 to filter a signal identical in shape to a template with intrinsic parameters @xmath61 . using the match ( [ match definition ] ) it is possible to quantify our intuitive notion of how `` close '' two template shapes are to each other . since the match clearly has a maximum value of unity at @xmath62 , we can expand in a power series about @xmath62 to obtain @xmath63 this suggests the definition of a metric @xmath64 so that the _ mismatch _ @xmath65 between two nearby templates is equal to the square of the proper distance between them : @xmath66 having defined a metric on the intrinsic parameter space , we can now use it to calculate the spacing of the discrete template family required to retain a given fraction of the ideal event rate . schematically , we can think of the templates as forming a lattice in the @xmath67-dimensional intrinsic parameter space whose unit cell is an @xmath67-dimensional hypercube with sides of proper length @xmath68 . the worst possible case ( lowest @xmath69 $ ] ) occurs if the point @xmath70 describing the signal is exactly in the middle of one of the hypercubes . if the templates are closely spaced , i.e. @xmath71 , such a signal has a squared proper distance @xmath72 from the templates at the corners of the hypercube . we define the _ minimal match _ @xmath73 to be the match between the signal and the nearest templates in this worst possible case , i.e. the fraction of the optimal signal - to - noise ratio retained by a discrete template family when the signal falls exactly `` in between '' the nearest templates . this minimal match is the same quantity that dhurandhar and sathyaprakash in ref . denote as @xmath74 ; but since it is the central quantity governing template spacing it deserves some recognition in the form of its own name . our choice of name closely parallels the term `` fitting factor '' @xmath11 , which apostolatos introduced in ref . @xcite to measure the similarity between actual signals and a continuous template family . the minimal match , which is chosen by the experimenter based upon what he or she considers to be an acceptable loss of ideal event rate , will determine our choice of spacing of the discrete template parameters and therefore the number of discrete templates in the family . more specifically , the experimenter will choose some desired value of the minimal match @xmath73 ; and then will achieve this @xmath73 by selecting the templates to reside at the corners of hypercubes with edge @xmath68 given by @xmath75 the number of templates in the resulting discrete template family will be the proper volume of parameter space divided by the proper volume per template @xmath76 , i.e. @xmath77 the formalism above applies to the detection of any set of signals which have a functional form that depends on a set of parameters which varies continuously over some range . we now develop a more explicit formula for the metric , given an analytical approximation to the ligo noise curve and a particular class of inspiraling binary signals . we approximate the `` initial '' and `` advanced '' benchmark ligo noise curves by the following analytical fit to fig . 7 of ref . @xcite : @xmath78\},&f > f_s\\ \infty,&f < f_s , \end{array } \right.\ ] ] where @xmath79 is the `` knee frequency '' or frequency at which the interferometer is most sensitive ( which is determined by the reflectivities of the mirrors and is set by the experimenters to the frequency where photon shot noise begins to dominate the spectrum ) and @xmath80 is a constant whose value is not important for our purposes . this spectrum describes photon shot noise in the `` standard recycling '' configuration of the interferometer ( second term ) superposed on thermal noise in the suspension of the test masses ( first term ) , and it approximates seismic noise by setting @xmath81 infinite at frequencies below the `` seismic - cutoff frequency '' @xmath82 . throughout the rest of this paper , the `` first ligo noise curve '' will refer to ( [ ligo noise curve ] ) with @xmath82 = 40 hz and @xmath79 = 200 hz , and the `` advanced ligo noise curve '' will refer to ( [ ligo noise curve ] ) with @xmath82 = 10 hz and @xmath79 = 70 hz . these numbers are chosen to closely fit fig . 7 of ref . @xcite for the first ligo interferometers and for the advanced ligo benchmark . in this paper , when various quantities ( such as the number of discrete templates ) are given including a scaling with @xmath79 , this indicates how the quantity changes while @xmath79 is varied but @xmath83 is held fixed . at this point it is useful to define the moments of the noise curve ( [ ligo noise curve ] ) , following poisson and will , as [ moment definition ] @xmath84 the upper limit of integration @xmath85 denotes the coalescence frequency or high - frequency cutoff of whatever template we are dealing with , which very roughly corresponds to the last stable circular orbit of a test particle in a non - spinning black hole s schwarzschild geometry . for both first and advanced ligo noise curves , the majority of inspiraling binary search templates will occupy regions of parameter space for which @xmath85 is many times @xmath79 . because we will always be dealing with @xmath86 for @xmath87 , and because the noise term in the denominator of the integrand in eq . ( [ moment definition ] ) rises as @xmath88 for @xmath89 , we can simplify later calculations by approximating @xmath90 in the definition of the moments . to illustrate the metric formalism , we shall use templates based on a somewhat simplified version of the post - newtonian expansion . since the inner product ( [ inner product definition ] ) has negligible contributions from frequencies at which the integrand oscillates rapidly , it is far more important to get the phase of @xmath91 right than it is to get the amplitude dependence . therefore , we adopt templates based on the `` restricted '' post - newtonian approximation in which one discards all multipolar components except the quadrupole , but keeps fairly accurate track of the quadrupole component s phase ( for more details see secs . ii.c . and iii.a . of ref applying the stationary phase approximation to that quadrupolar waveform , we obtain @xmath92,\ ] ] up to a multiplicative constant which is set by the condition @xmath93 @xcite . the function @xmath94 , describing the phase evolution in ( [ template definition ] ) , is currently known to post@xmath9-newtonian order for the case of two nonspinning point masses in a circular orbit about each other as @xmath95 \end{array}\ ] ] ( cf . ( 3.6 ) of ref . ) . here the mass parameters have been chosen to be @xmath96 , the total mass of the system , and @xmath97 , the ratio of the reduced mass to the total mass . the actual amplitude @xmath22 of a waveform is proportional to @xmath98 , where @xmath99 is the distance to the source ; and this lets us find the relation between minimal match and event rate which we will need in order to wisely choose the minimal match . assuming that compact binaries are uniformly distributed throughout space on large distance scales , this means that the rate of events with a given set of intrinsic parameters and with an amplitude greater than @xmath22 is proportional to @xmath100 . setting a signal - to - noise threshold @xmath101 is equivalent to setting a maximum distance @xmath102 to which sources with a given set of intrinsic parameters can be detected . thus , if we could search for signals with the entire continuous template family , we would expect the observed event rate to scale as @xmath103 . this ideal event rate is an upper limit on what we can expect with a real , discrete template family . we can obtain a lower bound on the observed event rate by considering what happens if all signals conspire to have parameters lying exactly in between the nearest search templates . in this case , all events will have reduced signal - to - noise ratios of @xmath73 times the optimal signal - to - noise ratio @xmath22 . this is naively equivalent to optimally filtering with a threshold of @xmath104 , so a pessimistic guess is @xmath105 for a fixed rate of false alarms . in real life , @xmath101 is affected by the total number of discrete templates and by the minimal match of the discrete template family . this can be seen by the fact that the signal - to - noise ratio , @xmath106{c}{\textstyle\max}\\{\scriptstyle k}\end{array } \bigl [ \begin{array}[t]{c}{\textstyle\max}\\{\scriptstyle\mu}\end{array } \langle o | u(\bbox{\mu},\bbox{\lambda}_{(k ) } ) \rangle \bigr ] , \ ] ] is the maximum of a number of random variables . the covariance matrix of these variables will be determined by the minimal match , and will itself determine the probability distribution of @xmath107 ( in the absence of a signal ) which is used to set the threshold @xmath101 in order to keep the false alarm rate below a certain level . however , since these effects are fairly small at high signal - to - noise ratios ( such as those to be used by ligo ) and the issue of choosing thresholds is a problem worthy of its own paper @xcite , we will use ( [ er vs mm ] ) for the rest of this paper . for this two - parameter template family , the formula for the match ( [ match definition ] ) can be simplified somewhat by explicitly performing the maximization over the extrinsic parameters @xmath27 and @xmath108 . since the integrand in the inner product ( [ inner product definition ] ) depends on @xmath27 and @xmath108 as @xmath109 $ ] , there is no dependence on @xmath27 but only on @xmath108 . maximizing over @xmath110 is easy : instead of taking the real part of the integral in the inner product ( [ inner product definition ] ) , we take the absolute value . to maximize over @xmath111 we go back a step . let us define @xmath112 , and consider the @xmath113-dimensional space formed by @xmath114 and @xmath115 . we expand the inner product between adjacent templates to quadratic order in @xmath116 to get a preliminary metric @xmath117 , where the greek indices range from @xmath118 to @xmath67 ( latin indices range from @xmath119 to @xmath67 ) : @xmath120\right| } { \int_0^{\infty}df\frac{\displaystyle f^{-7/3}}{\displaystyle s_h(f)}}\right\ } \right ] { \atop\delta\lambda^\alpha = 0}.\ ] ] here @xmath121 @xcite . we define the moment functional @xmath122 such that , for a function @xmath123 , @xmath124 = \frac{1}{i(7)}\int_{f_s / f_0}^{f_c / f_0 } dx\frac{x^{-7/3}}{s_h(x~f_0)}a(x),\ ] ] and thus @xmath125 = \sum_n a_n j(7 - 3n).\ ] ] we also define the quantities @xmath126 such that @xmath127 where the derivative is evaluated at @xmath128 and @xmath129 is the part of @xmath94 in eq . ( [ template definition ] ) that is frequency dependent ( any non - frequency - dependent , additive parts of @xmath94 are removed when we take the absolute value in the maximized inner product ) . evaluation of the derivative in eq . ( [ premetric - def ] ) then shows that , in the limit @xmath130 , @xmath131 - { \cal j}[\psi_\alpha]{\cal j}[\psi_\beta]\right).\ ] ] finally , we minimize @xmath132 with respect to @xmath111 ( i.e. , we project @xmath117 onto the subspace orthogonal to @xmath30 ) and thereby obtain the following expression for the metric of our continuous template family : @xmath133 by taking the square root of the determinant of this metric and plugging it into eq . ( [ cal n definition ] ) , we can compute the number of templates @xmath0 that we need in our discrete family as a function of our desired minimal match @xmath73 , or equivalently of the loss of ideal event rate . although the phase of the inspiraling binary signal has recently been calculated to post@xmath9-newtonian order @xcite , it is useful to calculate the number of templates that would be required in a universe where the waveforms evolve only to post@xmath1-newtonian order and all binaries are composed of nonspinning objects in circular orbits . there are several reasons for this exercise . 1 . apostolatos @xcite has shown that amplitude modulation of the waveform due to spin effects is important in an inspiraling binary search only for a few extremal combinations of parameters , and also that ( at higher post - newtonian order ) templates without spin - related phase modulation can match phase modulated signals almost as well as can templates that include spin parameters . therefore the bulk of the final set of templates actually used when the detectors come on - line will not need to include the extra spin parameters , and we may ignore them in this preliminary work . we assume circular orbits because gravitational radiation reaction circularizes most eccentric orbits on a timescale much less than the lifetime of a compact binary @xcite . the phase of the templates is truncated at post@xmath1-newtonian order for simplicity . although apostolatos has demonstrated in ref . @xcite that post@xmath1-newtonian templates will not have a large enough fitting factor to be useful , consideration of such a set is a first step toward obtaining an adequate set of templates and it is a particularly important step since the metric coefficients will turn out to be constant over the template manifold . having chosen as the continuous template family the set of post@xmath1-newtonian , circular , spinless binary waveforms , we must now choose the discrete templates from within this continuous family . the first step is to calculate the coefficients of the metric on the two - dimensional dynamical parameter space . it is convenient to change the mass parameterization from the variables @xmath134 to the sathyaprakash variables @xcite @xmath135 note that @xmath136 and @xmath137 are simply the newtonian and post@xmath1-newtonian contributions to the time it takes for the carrier gravitational wave frequency to evolve from @xmath79 to infinity . the advantage of these variables is that the metric coefficients in @xmath138 coordinates are constant ( in the limit @xmath139 ) for all templates . this is because the phase of the waveform @xmath91 is linear in the sathyaprakash variables , and so the integral in the definition of the match ( [ match definition ] ) depends only on the displacement @xmath140 between the templates , not on the location @xmath138 of the templates in the dynamical parameter space . the dynamical parameter - dependent part of the templates phase is given by [ eq . ( [ 2pn psi ] ) truncated to first post - newtonian order and reexpressed in terms of @xmath138 using eqs . ( [ def : tau1 ] ) and ( [ def : tau2 ] ) ] @xmath141 and it is easy to read off @xmath142 and @xmath143 [ eq . ( [ psi - def ] ) ] as the coefficients of @xmath136 and @xmath137 @xcite . by inserting these @xmath144 into eq . ( [ series functional ] ) , the relevant moment functionals can be expressed in terms of the moments of the noise : @xmath145 & = & 2\pi f_0~j(4 ) , \\ { \cal j}[\psi_1 ] & = & 2\pi f_0~\frac{3}{5}j(12 ) , \\ { \cal j}[\psi_2 ] & = & 2\pi f_0~j(10 ) , \\ { \cal j}[\psi_0 ^ 2 ] & = & ( 2\pi f_0)^2~j(1 ) , \\ { \cal j}[\psi_0\psi_1 ] & = & ( 2\pi f_0)^2~\frac{3}{5}j(9 ) , \\ { \cal j}[\psi_0\psi_2 ] & = & ( 2\pi f_0)^2~j(7 ) , \\ { \cal j}[\psi_1 ^ 2 ] & = & ( 2\pi f_0)^2~\frac{9}{25}j(17 ) , \\ { \cal j}[\psi_1\psi_2 ] & = & ( 2\pi f_0)^2~\frac{3}{5}j(15 ) , \\ { \cal j}[\psi_2 ^ 2 ] & = & ( 2\pi f_0)^2~j(13 ) . \end{array}\ ] ] we can compute the needed moments of the noise by numerically evaluating the integrals ( [ moment definition ] ) . by setting the upper limit of integration to infinity , i.e. by approximating @xmath146 as infinite for all templates under consideration , we find that the moments have the constant values given in table [ tab : j s ] ; and therefore the moment functionals ( [ 1pn functionals ] ) have the constant values given in table [ tab : cal j s ] . inserting these values into eqs . ( [ premetric ] ) and ( [ metric conversion ] ) yields , for the coordinates @xmath147 , the 3-metric and 2-metric @xmath148 @xmath149 for the first ligo noise curve , and @xmath150 @xmath151 for the advanced ligo noise curve ( where the dots denote terms obtained by symmetry ) . we shall also estimate the errors in the metric coefficients due to the approximation @xmath152 : the moment integrals defined in eq . ( [ moment definition ] ) can be rewritten as @xmath153 where the first integral is the expression used in the above metric coefficients and the second is the correction due to finite @xmath146 . the second integral can be expanded to lowest order in @xmath154 as @xmath155 and from this the errors in the moments ( and therefore in the metric coefficients ) due to approximating @xmath85 as infinite are estimated to be less than or of order ten percent for the first ligo interferometers and one percent for the advanced ligo interferometers over most of the relevant volume of parameter space . since the two - parameter , post@xmath1-newtonian continuous template family is known to be inadequate for the task of searching for real binaries , these errors are small enough to justify our use of the @xmath156 approximation in this exploratory analysis . since the metric coefficients are constant in this analysis , the formula for the required number of templates [ eq . ( [ cal n definition ] ) ] reduces to @xmath157 the square root of the determinant of the metric is given by @xmath158 for the advanced ligo noise curve and by @xmath159 for the initial ligo noise curve , so once we have decided on the range of parameters we deem astrophysically reasonable we will have a formula for @xmath0 as a function of @xmath73 . the most straightforward belief to cherish about neutron stars is that they all come with masses greater than a certain minimum @xmath3 , which might be set to @xmath160 ( based on the minimum mass that any neutron star can have ) or @xmath161 ( based on the observed masses of neutron stars in binary pulsar systems @xcite ) . in terms of the variables @xmath134 the constraint @xmath162 and @xmath163 is easily expressed as @xmath164 but in terms of the sathyaprakash variables [ eqs . ( [ def : tau1 ] ) and ( [ def : tau2 ] ) ] the expression becomes rather unwieldy to write down . however , see fig . [ fig : volume ] for a plot of the allowed region in @xmath138 coordinates . for this reason we have found it convenient to use a monte carlo integration routine @xcite to evaluate the coordinate volume integral @xmath165 . the monte carlo approach becomes especially attractive when evaluating the proper volume integral @xmath166 for cases where the integrand is allowed to vary and in fact may itself have to be evaluated numerically , as will be the case for a post@xmath9-newtonian set of templates . the integral has numerical values of @xmath167 and @xmath168 seconds@xmath9 for initial and advanced ligo interferometer parameters , respectively , assuming a @xmath3 of @xmath160 and arbitrarily large @xmath169 . the integral can be shown ( numerically ) to scale roughly as @xmath170 ( independent of @xmath79 ) and as @xmath171 for @xmath3 ranging from 0.2 to 1.0 solar masses ( the dependence on @xmath169 is negligible for any value greater than a few solar masses ) . inserting the above numbers into eq . ( [ flat cal n ] ) , we find that @xmath172 for the first ligo noise curve and @xmath173 for the advanced ligo noise curve . the fiducial value of @xmath73 has been chosen as 0.97 to correspond to an event rate of roughly 90 percent of the ideal event rate [ cf . ( [ er vs mm ] ) ] . in terms of the template - spacing - induced fractional loss @xmath2 of event rate , the number of templates required is @xmath174 for the first ligo noise curve and @xmath175 for the advanced ligo noise curve . with the aid of the metric coefficients given in eqs . ( [ 1st 1pn metric ] ) and ( [ adv 1pn metric ] ) , it is a simple task to select the locations of the templates and the spacing between them . because the metric coefficients form a constant @xmath176 matrix , we can easily find the eigenvectors @xmath177 and @xmath178 of @xmath179 and use them as axes to lay out a grid of templates . the numerical values are @xmath180 for the first ligo noise curve and @xmath181 for the advanced ligo noise curve . the infinitesimal proper distance is given in terms of the eigen - coordinates as @xmath182 , where @xmath183 and @xmath184 are the eigenvalues of the metric . therefore we simply use eq . ( [ mm vs dl ] ) to obtain the template spacings @xmath185 we find that the eigenvalues of the metrics ( [ 1st 1pn metric ] ) and ( [ adv 1pn metric ] ) are @xmath186 times 0.721 and 0.00427 ( first ligo ) , and @xmath186 times 1.25 and 0.00984 ( advanced ligo ) . therefore the template spacings are given by @xmath187 for the first ligo noise curve and by @xmath188 for the advanced ligo noise curve . figure [ fig : contours ] shows the locations of some possible templates superposed on a contour plot of the match with the template in the center of the graph . drawing on the previous work of schutz @xcite concerning the mechanics of fast - fourier - transforming the data , we can estimate the cpu power required to process the interferometer output on - line through a single - pass ( non - hierarchical ) search involving @xmath0 templates . although the data will be sampled at a rather high rate ( tens of khz ) , frequencies above some upper limit @xmath189 can be thrown away ( in fourier transforming the data ) with only negligible effects on the signal - to - noise ratio . this lowers the effective frequency of sampling to @xmath190 ( the factor of two is needed so that the nyquist frequency is @xmath191 ) , and thereby considerably reduces the task of performing the inner - product integrals . if the length of the array of numbers required to store a template is @xmath192 and that required to store a given stretch of data is @xmath193 , the number of floating point operations required to process that data stretch through @xmath0 filters is @xmath194 [ cf . ( 16.37 ) of schutz @xcite , with the fractional overlap between data segments @xmath195 chosen as roughly @xmath196 . actually , @xmath192 varies from filter to filter , but most of the search templates occupy regions of parameter space where the mass is very low and thus the storage size of the template , @xmath197 is very large @xcite . the longest filter is the one computed for two stars of mass @xmath3 , so by inserting @xmath198 and @xmath199 into eq . ( [ def : tau1 ] ) and combining with eq . ( [ storage size ] ) , we find that we can make a somewhat pessimistic estimate of the computational cost by using @xmath200 the required cpu power @xmath6 for an on - line search is obtained by dividing eq . ( [ n operations ] ) by the total duration of the data set , @xmath201 to find that @xmath202 combining eq . ( [ cpu power ] ) with eqs . ( [ 1st cal n ] ) and ( [ adv cal n ] ) gives us @xmath203 for an on - line search by the first ligo interferometers and @xmath204 for the advanced ligo interferometers . although the estimates in the paragraph above are not to be believed beyond a factor of order unity , the magnitude of the numbers shows that a hierarchical search strategy may be desirable to keep the computing power requirements at a reasonable level for non - supercomputing facilities . that is , the data would first be filtered through a more widely spaced ( low minimal match ) set of templates with a relatively low signal - to - noise threshold , and only the segments which exceed this preliminary threshold would be analyzed with the finely spaced ( high minimal match ) templates . the metric - based formalism of this paper only holds for the finely spaced set of templates used in the final stage of the hierarchical search ; the template spacing used in the earlier stages of the search will have to be chosen using more complex methods such as those of sathyaprakash and dhurandhar . the only previous analysis of the problem of choosing the discrete search templates from the two - parameter , restricted post@xmath1-newtonian continuous template family is that of sathyaprakash @xcite , in which he found that the entire volume of parameter space corresponding to @xmath205 could be covered by a set of templates which vary only in @xmath206thereby reducing the effective dimensionality of the mass parameter space to one . this implied a value of @xmath0 similar to that obtained in the one - parameter ( newtonian template ) analysis of dhurandhar and sathyaprakash in ref . . it is not possible to fairly compare my value for @xmath0 to the values given by dhurandhar and sathyaprakash in table ii of ref . due to our differing assumptions concerning the sources and the desirable level of the minimal match therefore i will compare the assumptions . dhurandhar and sathyaprakash typically consider a minimal match of 0.8 or 0.9 rather than 0.97 , this would lead to a loss of thirty to fifty percent of the ideal event rate [ cf . ( [ er vs mm ] ) ] . if the current `` best estimates '' of inspiraling binary event rates @xcite are correct , the ideal event rate for ligo and virgo will not be more than about one hundred per year even when operating at the `` advanced interferometer '' noise levels , and the loss of up to half of these events would be unacceptable . from eqs . ( [ 1st cal n ] ) and ( [ adv cal n ] ) it can be seen that the dependence of @xmath0 on @xmath3 is the most important factor influencing the number of templates . the two - parameter analysis of sathyaprakash @xcite uses a value for @xmath3 of @xmath207 , which is based on the statistics of ( electromagnetically- ) known binary pulsars . however , because there is no known , firm physical mechanism that prevents neutron stars from forming with masses between 0.2 and 1 @xmath208 , ligo and virgo should use a discrete template family with @xmath209 . after all , laser interferometer gravitational wave detectors are expected to bring us information about astronomical objects as yet unknown . during the final stages of completion of this manuscript , a new preprint by balasubramanian , sathyaprakash , and dhurandhar appeared in the xxx.lanl.gov archive @xcite , applying differential geometry to the problem of detecting compact binary inspiral events and extracting source parameters from them . the preprint applies the tools of differential geometry primarily to the problem of parameter measurement rather than that of signal detection , and so does not develop the geometrical formalism as far as is done in sec . [ sec : formalism ] of this paper . the metric constructed in ref . @xcite is identical to the information matrix which was suggested for use in the construction of a closely - spaced discrete template family in the authors previous work . while this is quite useful for parameter measurement , it neglects maximization over kinematical parameters and thus is not very useful for the construction of search templates . also , the assumptions about @xmath73 and @xmath3 are no different from those made in previous work up to and including ref . @xcite , and so the result for @xmath0 is no different . the main difference between the results of ref . @xcite and previous analyses by the same authors and therefore the most important part of the preprint as far as the detection problem is concerned is the introduction of the possibility of choosing search templates to lie outside the manifold of the continuous template family . using an _ ad hoc _ example , the authors show that such a placement of templates can result in a spacing roughly double that between discrete templates chosen from the manifold formed by the continuous template family . my analysis in this paper does not consider this possibility , but the formalism of sec . [ sec : formalism ] can easily be extended to investigate this problem in the future . this paper has presented a method for semi - analytically calculating the number of templates required to detect gravity waves from inspiraling binaries with ligo as a function of the fraction of event rate lost due to the discrete spacing of the templates in the binary parameter space . this method , based on differential geometry , emphasizes that ultimately a finer template spacing is required than has previously been taken as typical in the literature , in order to retain a reasonable fraction of the event rate . this paper details the first calculation of this kind that uses post - newtonian templates and a noise curve which takes into account the coloration of noise in the detector due to both standard recycling photon shot noise and thermal noise in the suspension of the test masses . the result is that it is possible to search the data for binaries containing objects more massive than @xmath160 thoroughly enough to lose only @xmath210 10 percent of the ideal event rate without requiring a quantum leap in computing technology . the computational cost of such a search , conducted on - line using a single pass through the data , is roughly 20 gflops for the first ligo interferometers ( ca . 2000 ) and 270 gflops for the advanced ligo interferometers ( some years later ) . this is feasible ( or very nearly feasible ) even for a present - day supercomputing facility , but a hierarchical search strategy ( using as its first stage a widely - spaced set of templates similar to that analyzed by sathyaprakash @xcite ) may be desirable to reduce the cost . a thorough investigation of hierarchical search strategies is in order : how should the threshold and the minimal match of the first stage be set in order to minimize the cpu power required while keeping the false alarm and false dismissal rates at acceptable levels ? how would non - gaussian noise statistics affect the first stage threshold and minimal match ? would a hierarchical search benefit by using more than two stages ? how is the threshold affected by the minimal match when the approximation of high signal - to - noise ratio can no longer be made ? the formalism of this paper should be applied to choose discrete templates from a better continuous template family than the one considered here . the best two - parameter templates will be based on the highest post - newtonian order computations that have been performed for circularized , spinless binaries , augmented perhaps by terms of still higher order from the theory of perturbations of schwarzschild or kerr spacetime . i plan to soon apply my geometric formalism to the post@xmath211-newtonian templates which are currently the best available . the areas of parameter space where spins can not be neglected ( noted by apostolatos in ref . @xcite ) must also be investigated , and the inclusion of an orbital eccentricity parameter should be considered . there are several more issues which i plan to address using my formalism or some extension of it . an analysis needs to be made for the case when the signal is not identical to some member of the continuous template family ( i.e. the fitting factor is not equal to one ) ; and the result of such an analysis should be used to set definite goals for both the fitting factor and the minimal match in terms of event rate . the effect of non - quadrupolar harmonics of the gravitational wave on the construction of search templates should be considered . these harmonics have been ignored in all previous analyses of detection and even of parameter measurement , but they may have a noticeable effect when a very high minimal match is desired . finally , a systematic investigation of the optimal choice of search templates outside the continuous template family is in order . this problem has been addressed in a preliminary way in ref . @xcite , but is deserving of further scrutiny . my thanks to theocharis apostolatos and eanna flanagan for helping me get started . i am most indebted to kip thorne for his guidance and his patience in reviewing the manuscript . this work was supported in part by my nsf graduate fellowship and in part by nsf grant phy-9424337 . a. abramovici , w. e. althouse , r. w. p. drever , y. grsel , s. kawamura , f. j. raab , d. shoemaker , l. sievers , r. e. spero , k. s. thorne , r. e. vogt , r. weiss , s. e. whitcomb , and m. e. zucker , science * 256 * , 325 ( 1992 ) . c. bradaschia , e. calloni , m. cobal , r. del fasbro , a. di virgilio , a. giazotto , l. e. holloway , h. kautzky , b. michelozzi , v. montelatici , d. pascuello , and w. velloso , in _ gravitation 1990 _ , proceedings of the banff summer institute , ed . r. mann and p. wesson ( world scientific , singapore , 1991 ) . see , e.g. , n. wiener , _ the extrapolation , interpolation and smoothing of stationary time series with engineering applications _ ( wiley , new york , 1949 ) or l. a. wainstein and v. d. zubakov , _ extraction of signals from noise _ ( prentice - hall , london , 1962 ) . note that the @xmath31 used here is the coalescence phase of the _ waveform_. in the restricted post - newtonian approximation , this is twice the coalescence phase of the _ orbit _ @xmath212 , which is used by cutler and flanagan in ref . . this @xmath113-dimensional metric @xmath117 is related to the fisher information matrix , which was used by dhurandhar and sathyaprakash in the appendix of ref . to compute the template spacing analytically in the small - spacing limit . however , in taking the absolute value to maximize over @xmath110 , we have lost some dependence on the dynamical ( intrinsic ) template parameters , so our @xmath213-metric is not even the projection of the information matrix onto the subspace perpendicular to @xmath31 . this lets us tolerate a greater mismatch between the templates dynamical parameters than was possible before maximization , thereby reducing the total number of templates required . the @xmath144 given here are different from those given in eq . ( 13a-3 ) of ref . @xcite because sathyaprakash does not use @xmath30 and @xmath31 as kinematical parameters , but rather the time and phase at which the quadrupole part of the waveform reaches a frequency of @xmath79 . numerical integrations were performed with the aid of mathematica [ s. wolfram , _ mathematica : a system for doing mathematics by computer _ ( addison - wesley , redwood city , california , 1988 ) ] and with a monte carlo code based on numerical recipes [ w. h. press , s. a. teukolsky , w. t. vetterling , and b. p. flannery , _ numerical recipes : the art of scientific computing _ ( cambridge university press , cambridge , england , 1992 ) ] . the storage size of the template is equal to the effective sampling frequency @xmath190 times the duration of the template waveform . in the restricted post - newtonian approximation , the duration of a binary chirp waveform template is equal to the time it takes for the gravitational wave frequency to sweep from the seismic frequency @xmath82 up to infinity . since @xmath214 is the dominant ( newtonian ) contribution to the time it takes the gravitational wave frequency to sweep from @xmath79 up to infinity , the duration of a waveform template is roughly @xmath215 . thus we obtain eq . ( [ storage size ] ) . r. balasubramanian , b. s. sathyaprakash , and s. v. dhurandhar , _ gravitational waves from coalescing binaries : detection strategies and monte carlo estimation of parameters _ , submitted to phys . d ( los alamos xxx.lanl.gov archive preprint gr - qc/9508011 ) .
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gravitational waves from inspiraling , compact binaries will be searched for in the output of the ligo / virgo interferometric network by the method of `` matched filtering''i.e . , by correlating the noisy output of each interferometer with a set of theoretical waveform templates .
these search templates will be a discrete subset of a continuous , multiparameter family , each of which approximates a possible signal .
the search might be performed _ hierarchically _ , with a first pass through the data using a low threshold and a coarsely - spaced , few - parameter template set , followed by a second pass on threshold - exceeding data segments , with a higher threshold and a more finely spaced template set that might have a larger number of parameters .
alternatively , the search might involve a single pass through the data using the larger threshold and finer template set .
this paper extends and generalizes the sathyaprakash - dhurandhar ( s - d ) formalism for choosing the discrete , finely - spaced template set used in the final ( or sole ) pass through the data , based on the analysis of a single interferometer .
the s - d formalism is rephrased in geometric language by introducing a metric on the continuous template space from which the discrete template set is drawn .
this template metric is used to compute the loss of signal - to - noise ratio and reduction of event rate which result from the coarseness of the template grid .
correspondingly , the template spacing and total number @xmath0 of templates are expressed , via the metric , as functions of the reduction in event rate .
the theory is developed for a template family of arbitrary dimensionality ( whereas the original s - d formalism was restricted to a single nontrivial dimension ) .
the theory is then applied to a simple post@xmath1-newtonian template family with two nontrivial dimensions . for this family
, the number of templates @xmath0 in the finely - spaced grid is related to the spacing - induced fractional loss @xmath2 of event rate and to the minimum mass @xmath3 of the least massive star in the binaries for which one searches by @xmath4 for the first ligo interferometers and by @xmath5 for advanced ligo interferometers .
this is several orders of magnitude greater than one might have expected based on sathyaprakash s discovery of a near degeneracy in the parameter space , the discrepancy being due to this paper s lower choice of @xmath3 and more stringent choice of @xmath2 .
the computational power @xmath6 required to process the steady stream of incoming data from a single interferometer through the closely - spaced set of templates is given in floating - point operations per second by @xmath7 for the first ligo interferometers and by @xmath8 for advanced ligo interferometers .
this will be within the capabilities of ligo - era computers , but a hierarchical search may still be desirable to reduce the required computing power .
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the notion of an @xmath0-algebra ( _ i.e. _ algebra over the sphere spectrum ) is well - known in homotopy theory ( _ cf . _ _ e.g. _ @xcite ) : in the categorical form used in this paper and implemented by the concept of a discrete @xmath1-ring , it was formalized in the late 90 s . @xmath0-algebras are also intimately related to the theory of brave new rings ( first introduced in the 80 s ) and of functors with smash products ( fsp ) for spectra in algebraic topology ( _ cf . _ @xcite ) . the goal of this article is to explain how the implementation of the notion of an @xmath0-algebra in arithmetic , in terms of segal s @xmath1-rings , succeeds to unify several constructions pursued in recent times aimed to define the notion of absolute algebra " . in particular , we refer to the development , for applications in number theory and algebraic geometry , of a suitable framework apt to provide a rigorous meaning to the process of taking the limit of geometry over finite fields @xmath2 as @xmath3 . in our previous work we have met and used at least three possible categories suitable to handle this unification : namely the category @xmath4 of monods as in @xcite , the category @xmath5 of hyperrings of @xcite and finally the category @xmath6 of semirings as in @xcite.in @xcite , n. durov developed a geometry over @xmath7 intended for arakelov theory applications by implementing monads as generalizations of classical rings . in his work certain combinatorial structures replace , at the archimedean place(s ) , the geometric constructions classically performed , at a finite prime ideal , in the process of reduction modulo that ideal . in the present article we argue that all the relevant constructions in _ op.cit . _ can be naturally subsumed by the well - known theory of @xmath0-algebras and segal s @xmath1-sets - space is needed in homotopy theory and is simply that of a simplicial @xmath1-set . ] in homotopy theory . this theory is taken up as the natural groundwork in the recent book @xcite . in proposition [ assembly ] we prove that the assembly map of @xcite provides a functorial way to associate an @xmath0-algebra to a monad on pointed sets . while in the context of @xcite the tensor product @xmath8 produces an uninteresting output isomorphic to @xmath9 , in proposition [ tensq ] we show that the same tensor square , re - understood in the theory of @xmath0-algebras , provides a highly non - trivial object . as explained in @xcite , the category of pointed @xmath1-sets is a symmetric closed monoidal category and the theory of generalized schemes of b. ten and m. vaqui in @xcite applies directly to this category while it could not be implemented in the category of endofunctors under composition of @xcite which is not symmetric . we endow the arakelov compactification @xmath10 of @xmath11 with a natural structure of a sheaf @xmath12 of @xmath0-algebras and each arakelov divisor provides a natural sheaf of modules over that structure sheaf . moreover , this new structure of @xmath10 over @xmath0 endorses a one parameter group of weakly invertible sheaves whose tensor product rule is the same as the composition rule of the frobenius correspondences over the arithmetic site @xcite . the fundamental advantage of having unified the various attempts done in recent times to provide a suitable definition of absolute algebra " by means of the well established concept of @xmath0-algebra is that this latter notion is at the root of the theory of topological cyclic homology which can be understood as cyclic homology over the absolute base @xmath0 . segal s @xmath1-spaces form a model for stable homotopy theory and they may be equivalently viewed as simplicial objects in the category of @xmath1-sets ( _ cf._@xcite ) . note that general @xmath1-spaces in _ op.cit . _ are not assumed to be special or very special , and it is only by dropping the very special condition that one can use the day product . moreover , the prolongation of a @xmath1-space to a spectrum is a purely formal construction and does not need the special / very special hypotheses . the very special condition only appears for fibrant " objects in the appropriate quillen model category . thus , the stable homotopy category , understood using general @xmath1-spaces , is properly viewed as the derived category of the algebraic category of @xmath1-sets , _ i.e. _ of @xmath0-modules . topological cyclic homology then appears as the direct analogue of ordinary cyclic homology of rings when one replaces the category of abelian groups by the category of @xmath1-sets . in particular it is now available to understand the new structure of @xmath10 using its structure sheaf @xmath12 and the modules . our original motivation for using cyclic homology in the arithmetic context arose from the following two results:@xmath13 in @xcite we showed that cyclic homology ( a fundamental tool in noncommutative geometry ) determines the correct infinite dimensional ( co)homological theory for arithmetic varieties to recast the archimedean local factors of serre as regularized determinants . the key operator in this context is the generator of the @xmath14-operations in cyclic theory.@xmath15 l. hesselholt and i. madsen have proven that the de rham - witt complex , an essential ingredient of crystalline cohomology , arises naturally when one studies the topological cyclic homology of smooth algebras over a perfect field of finite characteristic ( _ cf . _ _ e.g. _ @xcite ) . our long term objective is to use cyclic homology over the absolute base @xmath0 and the arithmetic site defined in @xcite to obtain a global interpretation of @xmath16-functions of arithmetic varieties . before recalling the definition of a @xmath1-set we comment on its conceptual yet simple meaning . a @xmath1-set is the most embracing generalization of the datum provided on a set by a commutative addition with a zero element . the commutativity of the addition is encoded by the fact that the output of a ( finite ) sum depends only upon the corresponding finite set ( with multiplicity ) . a preliminary definition of a @xmath1-set is given by a covariant functor @xmath17 from the category @xmath18 of finite sets to the category @xmath19 of sets , mapping a finite set @xmath20 to the set @xmath21 of all possible strings of elements indexed by @xmath20 which can be added . the action of @xmath17 on morphisms encodes the addition , while its covariance embodies the associativity of the operation . one introduces a base point @xmath22 in the labelling of the sums and a base point @xmath23 to implement the presence of the @xmath24-element ( neutral for the sum ) . thus @xmath17 is best described by a pointed covariant functor from the category @xmath25 of finite pointed sets to the category @xmath26 of pointed sets . note that while the category of covariant functors @xmath27 is a topos the category of pointed covariant functors @xmath28 is no longer such and admits an initial object which is also final , in direct analogy with the category of abelian groups . the theory reviewed in chapter ii of @xcite is that of segal s @xmath1-spaces , _ i.e. _ of simplicial @xmath1-sets . the category of @xmath1-spaces admits a natural structure of model category ( _ cf.__op.cit . _ definition 2.2.1.5 ) allowing one to do homotopical algebra . the transition from @xmath1-sets to @xmath1-spaces parallels the transition from abelian groups to chain complexes of abelian groups in positive degrees and the constructions of topological hochschild and cyclic homology then become parallel to the construction of their algebraic ancestors . for each integer @xmath29 , one introduces the pointed finite set @xmath30 , where @xmath24 is the base point . let @xmath31 be the small , full subcategory of @xmath25 whose objects are the sets @xmath32 s , for @xmath29 . the morphisms in the category are maps between pointed finite sets , preserving the base points . the notion of a discrete @xmath1-space , _ i.e. _ of a @xmath1-set suffices for our applications and at the same time it is intimately related to the topos @xmath33 of covariant functors @xmath34 . the small category @xmath35 is a pointed category _ i.e. _ it admits a ( unique ) initial and final object , namely the object @xmath36 formed by the base point alone . in general , if @xmath37 is a pointed category one defines a @xmath1-object of @xmath37 to be a covariant functor @xmath38 preserving the base point . [ defngamset ] a @xmath1-set @xmath17 is a functor @xmath39 between pointed categories . the morphisms @xmath40 between two @xmath1-sets are natural transformations of functors . the category @xmath41 of @xmath1-sets is a symmetric , closed , monoidal category ( _ cf . _ @xcite , chapter ii ) . the monoidal structure is given by the smash product of @xmath1-sets which is an example of a day product : we shall review this product in [ salgsect ] . the closed structure property is shown in @xcite ( _ cf . _ also @xcite theorem 2.1.2.4 : replace the category @xmath42 with @xmath43 ) . the internal @xmath44 structure is given by @xmath45 where @xmath46 . for two pointed sets @xmath47 their smash product is defined by collapsing the set @xmath48 to @xmath49 in the product @xmath50 . our next task is to explain how the closed monoidal category @xmath41 encompasses several attempts to model a notion of an absolute algebra " . let @xmath51 be a commutative monod denoted additively and with a @xmath24 element . one defines a functor @xmath52 by setting @xmath53 in the last formula @xmath54 and the sum over the empty set is the @xmath24 element . moreover the base point of @xmath55 is @xmath56 , @xmath57 . the formula is meaningful in view of the commutativity and associativity of the monod @xmath51 . one easily checks that the maps @xmath58 with @xmath59 do not create any problem for the functoriality . the functor @xmath60 is in fact a particular case of the covariant functor @xmath61 , @xmath62 defined by the formula @xmath63 . more precisely , @xmath60 is obtained by restricting to @xmath25 and by subsequently dividing using the equivalence relation @xmath64 by assuming that @xmath65 preserves the base point and if @xmath66 , one gets @xmath67 next , we recall the notion of an @xmath0-algebra as given in @xcite ( definition 2.1.4.1 ) . this requires to define first the smash product of two @xmath1-sets , _ i.e. _ of two pointed functors @xmath68 , @xmath69 . the definition of the smash product @xmath70 is dictated by the internal @xmath44 structure of and the adjunction formula @xmath71 thus the smash product @xmath70 is such that for any @xmath1-set @xmath72 , a morphism @xmath73 is simply described by a natural transformation of bifunctors @xmath74 _ i.e. _ by maps of pointed sets , natural in both objects @xmath47 of @xmath35 @xmath75 the evaluation of the @xmath1-set @xmath70 on an object @xmath76 of @xmath25 is given by the following colimit @xmath77 where for any morphisms @xmath78 and @xmath79 in @xmath25 one uses the morphism @xmath80 provided that @xmath81 , with @xmath82 . thus , with the exception of the base point , a point of @xmath83 is represented by the data @xmath84 given by a pair of objects @xmath47 of @xmath25 , a map @xmath85 and a pair of non - base points @xmath86 , @xmath87 . moreover , notice the following implication @xmath88 the colimit is in general not filtered . the specialization of definition 2.1.4.1 . of @xcite to the case of @xmath1-sets yields the following [ defnsalg ] an @xmath0-algebra @xmath89 is a @xmath1-set @xmath90 endowed with an associative multiplication @xmath91 and a unit @xmath92 , where @xmath93 is the canonical inclusion functor . since @xmath0 is the inclusion functor , the obvious identity map @xmath94 defines the product @xmath95 in @xmath0 which is clearly associative . by construction , any @xmath0-algebra is an algebra over @xmath0 . this elementary categorical object can be naturally associated to @xmath7 _ i.e. _ to the most basic algebraic structure underlying the absolute ( geometric ) point . a morphism @xmath96 of @xmath0-algebras is a morphism of the underlying @xmath1-sets which is compatible with the unit and with the product . this latter condition is equivalent to the commutativity of the following diagram for any objects @xmath47 of @xmath35 @xmath97^{\mu_{\ca } } \ar[rr]^{\rho_x\wedge \rho_y } & & \ar[d]^{\mu_\cb } \cb(x)\wedge \cb(y ) \\ \ca(x\wedge y ) \ar[rr]_{\rho_{x\wedge y } } & & \cb(x\wedge y ) } \ ] ] we let @xmath98 be the category of @xmath0-algebras . it follows from _ op.cit . _ ( _ cf . _ 2.1.4.1.6 ) that , given an @xmath0-algebra @xmath99 and an integer @xmath100 , one obtains an @xmath0-algebra of @xmath101 matrices by endowing the @xmath1-set @xmath102 with the natural multiplication of matrices having only one non - zero entry in each column . there is moreover a straightforward notion of module over an @xmath0-algebra ( _ cf . _ _ definition 2.1.5.1 ) : an @xmath0-module being just a @xmath1-set . we begin this section by reviewing an easy construction of an @xmath0-algebra derived from a functor from the category of ( not necessarily commutative ) multiplicative monods with a unit and a zero element , to the category of @xmath0-algebras . this construction is described in example 2.1.4.3 , 2 . of @xcite , where the monod @xmath51 is not assumed to have a @xmath24 element and the obtained @xmath0-algebra is called spherical monod algebra . let @xmath51 be a multiplicative monod with a multiplicative unit and a zero element @xmath24 . we define the covariant functor @xmath103 with @xmath104 viewed as the base point and with maps @xmath105 , for @xmath65 . [ mono2sss ] let @xmath51 be a multiplicative monod with a unit and a zero element . then the product in @xmath51 endows @xmath106 with a structure of an @xmath0-algebra . the product in @xmath51 , viewed as a map @xmath107 determines the following map , natural in both objects @xmath20 , @xmath108 of @xmath25 @xmath109 the multiplicative unit @xmath110 determines a natural transformation @xmath111 . this construction endows @xmath106 with a structure of an @xmath0-algebra.the multiplicative monod @xmath112 ( frequently denoted by @xmath7 ) determines , in this framework , the canonical inclusion functor @xmath0 : _ i.e. _ @xmath113 . conversely , given an @xmath0-algebra @xmath114 one obtains , using the product @xmath115 and the unit @xmath116 , a canonical structure of multiplicative monod ( with a base point @xmath24 and a multiplicative identity @xmath117 ) on the set @xmath118 . [ monomono ] let @xmath99 be an @xmath0-algebra , @xmath51 a monod and @xmath119 a morphism of monods , then the following map defines a morphism of @xmath0-algebras @xmath120 the map @xmath121 gives the adjunction @xmath122 . the unit @xmath116 for @xmath99 defines a natural transformation of functors compatible with the product . since the product @xmath123 is natural in @xmath20 and @xmath108 , this shows that by taking @xmath124 , the morphism @xmath125 defines a natural transformation @xmath126 . the associativity of @xmath127 shows that @xmath125 is multiplicative . the required adjunction then follows using .note that the counit of the adjunction gives a canonical morphism @xmath128 . a semiring @xmath129 is a set endowed with two binary operations : @xmath130 and @xmath131 . the addition @xmath130 defines on @xmath129 the structure of an additive , commutative mono " i d with neutral element @xmath132 ; the multiplication @xmath133 is left and right distributive with respect to the addition and defines on @xmath129 the structure of a multiplicative mono " i d with neutral element @xmath117 and absorbing element @xmath24 . we let @xmath134 be the category of semirings . an important construction of an @xmath0-algebra is provided by the following result ( _ cf._@xcite ) [ sssalg ] let @xmath129 be a semiring , then the functor @xmath135 , @xmath136 is naturally endowed with a structure of an @xmath0-algebra . the additive structure on @xmath129 defines the @xmath1-set @xmath137 using . we first describe the product and the unit @xmath138 . to define the product we introduce the following map , natural in both objects @xmath47 of @xmath35 @xmath139 the naturality of the above operation follows from the bilinearity of the product in @xmath129 . the unit @xmath138 is defined as follows @xmath140 one obtains in this way a natural transformation since there is at most one non - zero value in the sum @xmath141 which computes @xmath142 . moreover , a non - zero value occurs exactly when @xmath143 , which shows that @xmath144 . notice that we have defined both the product and the transformation @xmath138 without using the additive group structure of a ring : in fact the semiring structure suffices . finally , the axioms of @xmath0-algebras are checked in the same way as for rings ( _ cf._@xcite example 2.1.4.3 for details ) . [ sssalg1 ] the functor @xmath145 from semirings to @xmath0-algebras is fully faithful . one needs to check that , for two semirings @xmath146 the natural map @xmath147 is bijective . as a set , one obtains the bijection @xmath148 by using the map defined by @xmath149 , @xmath150 , which maps the base point to @xmath151 . the product in @xmath152 is recovered by the map @xmath153 , using the fact that @xmath154.notice that for a pointed functor @xmath155 of the form @xmath156 , with @xmath152 a semiring , one derives the special property that given two elements @xmath157 there exists a _ unique _ element @xmath158 whose images by the maps @xmath159 of the form @xmath160 are given by @xmath161 , @xmath162 . one then obtains @xmath163 this shows that a morphism of functors @xmath164 determines , by restriction to @xmath165 , a homomorphism @xmath166 of semirings . the uniqueness of this homomorphism is clear using the bijection @xmath167 . the equality @xmath168 follows from the naturality of @xmath169 which implies that the projections @xmath170 , @xmath171 , fulfill @xmath172 the formula for the addition in @xmath173 retains a meaning also in the case where the above special property , for a pointed functor @xmath155 is relaxed by dropping the condition that the solution to @xmath174 , @xmath175 is unique . in this case , one can still define the following generalized addition _ i.e. _ the hyper - operation which associates to a pair @xmath176 a subset of @xmath177 @xmath178 this fact suggests that one can associate an @xmath0-algebra to a hyperring . this construction will be described in more details in section [ secthyp ] . the apparent simplicity of the smash product of two @xmath1-sets in the day product hides in fact a significant difficulty of the concrete computation of the colimit defined in . in this section we consider the specific example of the @xmath0-algebra @xmath179 , where @xmath180 is the smallest semiring of characteristic one ( idempotent ) and compute explicitly the smash product @xmath181 . we start by producing an explicit description of the @xmath0-algebra @xmath179 . [ sssalg1 ] the @xmath0-algebra @xmath179 is canonically isomorphic to the functor @xmath182 which associates to an object @xmath20 of @xmath35 the set @xmath183 of subsets of @xmath20 containing the base point . the functoriality is given by the direct image @xmath184 . the smash product is provided by the map @xmath185 which associates to the pair @xmath186 the smash product @xmath187 . the unit @xmath188 is given by the natural transformation @xmath189 defined by the map @xmath190 . a map @xmath191 is specified by the subset @xmath192 . one associates to @xmath193 the subset @xmath194 . the formula @xmath195 which defines the functoriality shows that it corresponds to the direct image @xmath184 . the last two statements are straightforward to check using the product in @xmath196 . [ f2]it is interesting to compare the structures of the @xmath0-algebras @xmath179 and @xmath197 because , when evaluated on an object @xmath20 of @xmath35 , they both yield the same set . @xmath198 is indeed equal to the set @xmath183 of subsets of @xmath20 containing the base point : this equality is provided by associating to @xmath199 the subset @xmath194 . the products and the unit maps are the same in both constructions since the product in @xmath200 is the same as that in @xmath196 . the difference between the two constructions becomes finally visible by analyzing the functoriality property of the maps . the simple rule @xmath184 holding in @xmath179 is replaced , in the case of @xmath197 by adding the further condition that on the set @xmath201 one only retains the points @xmath202 such that the cardinality of the preimage @xmath203 is an odd number . it follows from the colimit definition that @xmath205 is the set @xmath206 of connected components of the following category @xmath207 . the objects of @xmath207 are 4-tuples @xmath208 , where @xmath47 are objects of @xmath25 , @xmath209 is a morphism in that category and @xmath210 . a morphism @xmath211 in @xmath207 is given by a pair of pointed maps @xmath78 , @xmath79 such that @xmath212 and @xmath213 . notice that the set @xmath183 of subsets of @xmath20 containing the base point can be equivalently viewed as the set of all subsets of @xmath214 . thus @xmath215 , @xmath216 , is determined by a pair @xmath186 of non - empty subsets , @xmath217 , @xmath218 . one can thus encode an object @xmath208 of @xmath207 such that @xmath219 by a 5-tuple @xmath220 where @xmath217 , @xmath218 are non - empty sets . a morphism @xmath211 of such 5-tuples is then given by a pair of pointed maps @xmath221 such that @xmath222 for @xmath223 , a morphism @xmath211 is given by a pair of pointed maps @xmath221 such that @xmath224 let @xmath208 be an object of @xmath207 then the _ support _ of @xmath225 is empty if @xmath226 and for @xmath216 , @xmath227 one lets @xmath228 an object @xmath225 of @xmath207 is _ degenerate _ iff @xmath229 . in particular one sees that any object @xmath230 is degenerate . [ assoc0 ] let @xmath231 be a morphism in @xmath207 then @xmath232 assume first that @xmath223 . then by the restriction of @xmath233 to @xmath234 is equal to @xmath24 . thus @xmath229 . assume now that @xmath235 . for @xmath236 , one has @xmath237 , hence @xmath238 , @xmath239 and @xmath240 . conversely , let @xmath241 , then since @xmath242 there exist @xmath243 , @xmath244 , @xmath245 , @xmath246 . one then has @xmath247 since @xmath248 . this shows that @xmath249.let @xmath250 be the object of @xmath207 where @xmath20 and @xmath108 are reduced to the base point . one has [ assoc05 ] an object of @xmath207 is degenerate if and only if it belongs to the connected component of @xmath251 . the condition @xmath229 defines a connected family of objects by lemma [ assoc0 ] . thus it is enough to show that if @xmath229 then @xmath225 is in the same component as @xmath251 . the inclusions @xmath252 give in general a morphism in @xmath207 @xmath253 if @xmath229 the restriction of @xmath233 to @xmath234 is equal to @xmath24 . this shows that the unique pair of maps @xmath254 , @xmath255 gives a morphism in @xmath207 @xmath256 so that @xmath257 and @xmath251 are in the same component . to determine an explicit description of the ( pointed ) set @xmath204 we introduce the following terminology [ kmultigraph ] @xmath13 a @xmath258-_relation _ is a triple @xmath259 where @xmath17 and @xmath72 are non - empty finite sets and @xmath260 is a map of sets such that no line or column of the corresponding matrix is identically @xmath24 there exists @xmath261 such that @xmath262 and symmetrically @xmath263 , @xmath264 . ] . @xmath15 a morphism @xmath265 between two k - relations @xmath259 and @xmath266 is a pair of surjective maps @xmath267 , @xmath268 such that @xmath269 . @xmath270 a @xmath258-relation @xmath259 is said to be reduced when no line is repeated and no column is repeated . we let @xmath271 be the category of @xmath258-relations . the next lemma produces a retraction of the full subcategory of @xmath207 whose objects are non - degenerate , on @xmath271 . [ assoc]@xmath13 let @xmath257 be a non - degenerate object of @xmath207 . set @xmath272 , @xmath273 where @xmath274 and @xmath275 are the projections . then the triple @xmath276 defines a k - relation . @xmath15 a morphism @xmath277 in @xmath207 induces by restriction a morphism @xmath278 of k - relations . @xmath270 for @xmath279 a k - relation , let @xmath280 , where @xmath281 is the unique extension of @xmath233 as a pointed map . then @xmath282 extends to a functor from the category @xmath271 of @xmath258-relations to @xmath207 and one has @xmath283 . @xmath284 the objects @xmath225 and @xmath285 belong to the same connected component of @xmath207 . @xmath13 notice that @xmath286 is a subset of @xmath287 whose projections are @xmath288 and @xmath289 , thus the restriction of @xmath290 to @xmath291 defines a k - relation . let @xmath231 be a morphism in @xmath207 between non - degenerate objects . one has a commutative diagram @xmath292^{p_a } \ar[rr]^{f\times g } & & \ar[d]^{p_{a ' } } \supp(\alpha ' ) \\ f \ar[rr]_{f } & & f ' } \ ] ] where both @xmath293 and @xmath294 are surjective . this shows that @xmath295 and similarly that @xmath296 . thus the restrictions of @xmath297 and @xmath298 define surjections @xmath267 and @xmath268 , where @xmath272 , @xmath273 . the restrictions @xmath299 , @xmath300 fulfill the equality @xmath301 and one thus obtains a morphism @xmath302 of k - relations.@xmath270 let @xmath303 be a morphism of k - relations , we extend both @xmath304 to maps of pointed sets @xmath305 , @xmath306 . then one easily checks that one obtains a morphism @xmath307 ( _ cf . _ ) . one has by construction @xmath283 . @xmath284 by the object @xmath257 is equivalent to @xmath308 . we now prove that @xmath309 is equivalent to @xmath310 , where @xmath311 , @xmath312 , and @xmath313 . let @xmath314 act as the identity on @xmath315 and map the elements of @xmath316 to the base point . we define @xmath317 in a similar manner . then by construction one has @xmath318 . moreover for @xmath319 one has @xmath320 since both sides vanish unless @xmath321 and they agree on @xmath291 . this shows that @xmath225 and @xmath285 belong to the same connected component . every k - relation @xmath259 admits a canonical reduction @xmath322 . this is obtained by dividing @xmath17 and @xmath72 by the following equivalence relations @xmath323 by construction , the value @xmath324 only depends upon the classes @xmath325 of @xmath326 and thus the canonical reduction map @xmath327 defines a morphism of k - relations . more precisely [ assoc17]@xmath13 the reduction map defines an endofunctor @xmath328 of the category of @xmath258-relations which determines a retraction on the full subcategory of reduced @xmath258-relations.@xmath15 let @xmath329 be a morphism of @xmath258-relations , with @xmath330 reduced , then @xmath297 and @xmath298 are bijections and @xmath331 is an isomorphism . @xmath13 let @xmath332 be a morphism of @xmath258-relations . we show that there exists a unique morphism @xmath333 making the following diagram commutative @xmath334^{r_c } \ar[rr]^{(f , g ) } & & \ar[d]^{r_{c ' } } c ' \\ r(c ) \ar[rr]_{(r(f),r(g ) ) } & & r(c ' ) } \ ] ] indeed , for @xmath335 one has @xmath336 , since using the equality @xmath212 and the surjectivity of @xmath298 one derives @xmath337 similarly for @xmath338 one has @xmath339 . thus @xmath297 and @xmath298 induce maps @xmath340 and @xmath341 on the reductions making the diagram commutative . the surjectivity of @xmath297 and @xmath298 implies the surjectivity of the induced maps @xmath340 and @xmath341 . @xmath15 let @xmath332 be a morphism of @xmath258-relations , with @xmath330 reduced . then for @xmath342 let @xmath343 such that @xmath344 . one has @xmath345 which shows that @xmath297 is injective . thus both @xmath297 and @xmath298 are bijections and @xmath331 is an isomorphism . let @xmath346 be the set of isomorphism classes of reduced @xmath258-relations , and let @xmath347 be the set obtained by adjoining a base point . we extend the association @xmath348 to a functor @xmath349 by letting a morphism @xmath350 act on elements of @xmath347 as follows @xmath351 where the right hand side reduces to the base point when @xmath352 is degenerate . [ wedge2 ] @xmath13 for @xmath353 , the map @xmath354 induces a canonical bijection of sets @xmath355 @xmath15 the isomorphism @xmath356 induces an equivalence of functors @xmath357 . @xmath13 by construction , @xmath205 is the union of its base point with the set @xmath206 of connected components of the category @xmath207 . it follows from lemma [ assoc ] that the category @xmath271 of @xmath258-relations is a retraction of the category of non - degenerate objects of @xmath207 . thus using lemma [ assoc05 ] one obtains @xmath358 . moreover by lemma [ assoc17 ] the reduction @xmath359 gives a bijection @xmath360 . @xmath15 this follows since by construction of @xmath181 one has @xmath361 which can be applied to any representative @xmath362 of a given class in @xmath206 . [ pp1 ] @xmath13 for @xmath363 , let @xmath364 be the graph of the identity map on the set with @xmath365-elements . then the @xmath117-relations @xmath364 belong to distinct connected components of the category @xmath366 and they define distinct elements of @xmath367 . @xmath15 the action of the cyclic group @xmath368 on @xmath367 is the transposition acting on isomorphism classes of reduced @xmath117-relations . both its set of fixed points and its complement are infinite . @xmath13 the statement follows from the fact that the @xmath117-relations @xmath364 are reduced and pairwise non - isomorphic.@xmath15 the action of the transposition @xmath369 replaces a reduced 1-relation @xmath225 by its transpose @xmath370 . the fixed points are given by the reduced 1-relations such that @xmath225 is isomorphic to @xmath370 ; all the @xmath364 as in @xmath13 are thus fixed points . the relations between finite sets of different cardinality determine infinitely many non fixed points . @xmath13 in general , the natural map @xmath371 is _ not _ surjective . this can be easily seen by choosing @xmath372 , since @xmath373 is finite unlike @xmath367 . @xmath15 a square reduced relation is not necessarily symmetric . for instance , among the @xmath374 reduced @xmath117-relations of figure [ krelations9 ] , only the following reduced 1-relation and its transposed @xmath375 which have equal number ( 3 ) of lines and columns are _ not symmetric _ , since the number of ( non - zero ) elements in the lines are respectively @xmath376 and @xmath377 which can not be matched by any pair of permutations . -relations @xmath378[krelations9 ] ] -algebras[levels ] ] the definition of [ hyperpart ] which gives back the classical addition on @xmath129 for the @xmath0-algebra @xmath137 associated to a semiring @xmath129 is multivalued for a general @xmath0-algebra @xmath379 . the extension of ring theory when the additive group structure is replaced by a multivalued addition has been investigated by m. krasner in his theory of hyperrings and hyperfields . in our recent research we have encountered these types of generalized algebraic structures in the following two important cases ( _ cf._@xcite ) : @xmath380 the adle class space of a global field @xmath17 is naturally a hyperring and it contains the krasner hyperfield @xmath381 as a sub - hyperalgebra precisely because one divides the ring of the adles of @xmath17 by the non - zero elements @xmath382 of the field @xmath17 . @xmath380 the dequantization process described by the `` universal perfection '' at the archimedean place of a number field yields a natural hyperfield structure @xmath383 on the set of real numbers and a natural hyperfield structure @xmath384 in the complex case . the structure of a hyperfield is more rigid than that of a semifield since the operation @xmath385 is present in the hyper - context while it is missing for semirings . this fact makes the requirement that non - zero elements are invertible more restrictive for hyperfields than for semifields . a natural construction of hyperrings is achieved by simply dividing an ordinary ( commutative ) ring @xmath152 with a subgroup @xmath386 of the group of invertible elements of @xmath152 . in fact , in corollary 3.10 of @xcite we proved that any ( commutative ) hyperring extension of the krasner hyperfield @xmath381 ( @xmath387 ) , without zero divisors and of dimension @xmath388 is of the form @xmath389 , where @xmath152 is a commutative ring and @xmath390 is the group of non - zero elements of a subfield @xmath315 . moreover , theorem 3.13 of _ op.cit . _ shows that a morphism of hyperrings of the above form lifts ( under mild non - degeneracy conditions ) to a morphism of the pairs @xmath391 . in this section we provide the construction of the @xmath0-algebra associated to a hyperring of the form @xmath389 , where @xmath152 is a commutative ring and @xmath386 is a subgroup . notice that the structure needed to define _ uniquely _ the @xmath0-algebra @xmath392 is provided by the pair @xmath393 and that this datum is more precise than simply assigning the hyperring @xmath389 . [ sssalg3 ] let @xmath152 be a commutative ring and @xmath386 be a subgroup of the group of invertible elements of @xmath152 . for each object @xmath20 of @xmath25 , let @xmath394 be the quotient of @xmath395 by the following equivalence relation @xmath396 then , the functor @xmath397 defines an @xmath0-algebra and the quotient map @xmath398 is a morphism of @xmath0-algebras . we first check the functoriality of the construction , _ i.e. _ that for @xmath65 in @xmath25 , the map @xmath399 respects the equivalence relation @xmath400 . for @xmath401 , @xmath402 one has @xmath403 since these equations hold for the same @xmath298 and for all @xmath401 , one derives that @xmath404 . the above equivalence relation is compatible with the product : in fact using the commutativity of @xmath152 , one has @xmath405 the unit @xmath406 defines by composition a natural transformation @xmath407 : @xmath408 it follows easily from the construction that the quotient map @xmath398 is a morphism of @xmath0-algebras . [ sssalgback ] let @xmath152 be a commutative ring , let @xmath386 be a subgroup of the group of invertible elements of @xmath152 and @xmath392 the @xmath0-algebra defined in proposition [ sssalg3 ] . then , the set @xmath409 endowed with the hyper - addition defined in and the product @xmath410 induced by the @xmath0-algebra structure , is canonically isomorphic to the hyperring @xmath389 . by using the bijection @xmath411 , @xmath412 which maps the base point to @xmath151 one derives the bijection of sets @xmath413 . the product in @xmath409 is recovered by the map @xmath414 using the relation @xmath154 . it is the same as the product in @xmath389 . by using and , the hyper - addition is described , for @xmath415 as follows @xmath416 for @xmath417 and @xmath418 let @xmath419 with @xmath420 and @xmath421 . then the coset @xmath422 belongs to @xmath423 and all elements of @xmath423 are of this form . in this way one recovers the hyper - addition on @xmath389 . given an @xmath0-algebra @xmath99 a necessary condition for @xmath99 being of the above type @xmath392 is that the equations @xmath174 , @xmath175 always admit solutions . the lack of uniqueness of solutions is directly related to the multivalued nature of the addition . the construction of the @xmath0-algebras @xmath392 given in proposition [ sssalg3 ] applies in particular when one uses the notion of @xmath424-model introduced in @xcite . we recall that given a hyperfield @xmath425 , a @xmath424-_model _ of @xmath425 is by definition a triple @xmath426 where @xmath424 is a field@xmath427 is a homomorphism of hyperfields@xmath428 is a multiplicative section of @xmath169 . because @xmath428 is a multiplicative section of @xmath169 , the map @xmath169 is surjective and identifies the multiplicative monod @xmath425 with the quotient of @xmath424 by the multiplicative subgroup @xmath429 . since @xmath427 is a homomorphism of hyperfields , one has @xmath430 for any @xmath431 . thus @xmath169 defines a morphism of hyperfields @xmath432.a morphism @xmath433 of @xmath424-models is a field homomorphism @xmath434 such that @xmath435 and @xmath436 . we recall that a @xmath424-model of @xmath425 is said to be _ universal _ if it is an initial object in the category of @xmath424-models of @xmath425 . when such universal model exists one easily sees that it is unique up to canonical isomorphism : we denote it by @xmath437 . in this case it is thus natural to associate to @xmath425 the @xmath0-algebra @xmath438 of its universal @xmath424-model @xmath439 , where @xmath440 . [ shyper ] let @xmath441 , ( @xmath442 ) be the hyperfield of signs . the associated @xmath424-model is @xmath443 where the morphism @xmath444 is given by the sign of rational numbers ( _ cf._@xcite ) . the corresponding @xmath0-algebra is then @xmath445 . one has @xmath446 and @xmath447 is the set of half lines @xmath16 from the origin in the rational plane @xmath448 ( including the degenerate case @xmath449 ) . the maps @xmath450 and @xmath451 ( _ cf . _ ) are the projections onto the two axes . the map @xmath452 ( _ cf . _ ) is the projection on the main diagonal . it follows that the hyper - operation gives back the hyperfield structure on @xmath453 . one may wonder how to relate the @xmath0-algebra @xmath445 with @xmath179 . to this end , one first considers the @xmath0-subalgebra @xmath454 which is defined using the sub - semiring @xmath455 . the subset @xmath456 corresponds to the collection of half lines @xmath16 through the origin in @xmath457 which belong to the first quadrant @xmath458 . one then defines a morphism of @xmath0-algebras @xmath459 by using the morphism of semirings @xmath460 and one also notes that @xmath461 induces a morphism of @xmath0-algebras @xmath462 . one can thus describe the relation between @xmath99 and @xmath179 by the following map @xmath463 we expect that a similar diagram holds more generally when passing from a semifield of characteristic @xmath117 to a hyperfield and its @xmath424-model . -algebras.[levels2 ] ] in this section we briefly discuss the concept of levels for @xmath0-algebras using which one may derive an explicit approximation of an @xmath0-algebra @xmath114 . the first two levels of such approximation are obtained by restricting the functor @xmath99 to sets with @xmath464 elements to define the level @xmath117 , and to sets with @xmath465 elements to produce the level @xmath466 . in this way , we obtain * level @xmath117 * the theory of multiplicative monods * level @xmath466 * the theory of hyperrings with partially defined addition . in [ sectlevel1 ] we have seen that at level @xmath117 the @xmath0-algebra @xmath99 is well approximated by the @xmath0-subalgebra corresponding to the monod @xmath467 . next we discuss the compatibility of the additive structure given in with the morphism of @xmath0-algebras @xmath468 . for a general mono " i d @xmath51 , implementing to @xmath106 with the same notations used there , one gets @xmath469 one has : @xmath470 , while the elements of the set @xmath471 are the base point @xmath24 and the pairs @xmath472 for @xmath69 , @xmath473 , @xmath474 . one also has @xmath475 and @xmath476 . thus the only rule related to the addition that is retained at level @xmath117 simply states that the base point @xmath24 plays the role of the neutral element : @xmath477 this rule is of course preserved by the morphism of @xmath0-algebras @xmath468 . in [ fromrtohyp ] we have seen that the notion of an @xmath0-algebra is compatible with the operation of quotient by a subgroup of the multiplicative monod associated to level @xmath117 . in the following [ sectnorm ] we shall describe how to associate a sub-@xmath0-algebra to a sub - multiplicative seminorm on a semiring . by combining these two constructions one provides a good approximation to the description of level @xmath466 of an @xmath0-algebra . this viewpoint shows that , in general , the operation defined by yields a partially defined hyper - addition so that the approximation of level @xmath466 is by hyperrings with partially defined addition . the theory of @xmath0-algebras contains however all ( non - negative ) levels , even though only for the first level we have an easy explicit description . this section is motivated by the theory developed in @xcite aiming to determine an `` absolute '' , algebraic foundation underlying arakelov geometry : in particular , we refer to the theory of monads which defines the groundwork of _ op.cit . _ first , in [ sectsemi ] we explain how the basic construction of _ op.cit . _ with the seminorms can be adapted in the framework of @xmath0-algebras . then , in [ sectstructsh ] we describe a natural structure sheaf @xmath12 of @xmath0-algebras associated to the arakelov compactification @xmath10 of @xmath478 . as a topological space , @xmath10 is obtained from the zariski topological space @xmath478 by adding a new point @xmath479 treated as an additional closed point . this compactification is the same topological space as the space denoted by @xmath480 in _ op.cit . _ ( _ cf . _ [ durov1 ] ) . we show that the arakelov divisors @xmath481 on @xmath10 give rise to sheaves of @xmath12-modules and that the smash product rule for these modules parallels the composition rule of frobenius correspondences introduced in @xcite . let @xmath129 be a semiring . by a sub - multiplicative seminorm on @xmath129 we mean a map @xmath482 , @xmath483 , such that @xmath484 , @xmath485 and the following inequalities hold @xmath486 the basic construction of @xcite adapts directly as follows [ sssalg2 ] @xmath13 let @xmath129 be a semiring and @xmath487 a sub - multiplicative seminorm on @xmath129 . then the following set - up defines an @xmath0-subalgebra @xmath488 @xmath489 @xmath15 let @xmath490 be an @xmath129-module and @xmath491 a seminorm on @xmath490 such that @xmath492 , @xmath493 , @xmath494 . then for any real @xmath495 the following structure defines a module @xmath496 over @xmath497 @xmath498 where @xmath499 is defined as in . @xmath13 the sub - multiplicative seminorm on @xmath129 fulfills . in particular , the triangle inequality shows that , using the formula @xmath500 , one derives @xmath501 thus , for any morphism @xmath502 , the map @xmath503 restricts to a map @xmath504 . finally , the stability under product is derived from the sub - multiplicativity of the seminorm as follows @xmath505 @xmath15 the proof of @xmath13 shows that @xmath496 is a @xmath1-set . it also shows that one obtains a map of sets , natural in both objects @xmath20 , @xmath108 of @xmath35 @xmath506 this gives the required @xmath497-module structure @xmath507 on @xmath496 . [ ss1]@xmath13 the only multiplicative seminorm on the semifield @xmath196 is given by : @xmath484 , @xmath485 . in this case one obtains @xmath508 since the condition restricts the functor @xmath509 of lemma [ sssalg1 ] to the range of the unit map @xmath510.@xmath15 with the notations of , one also defines a submodule @xmath511 by setting @xmath512 we let @xmath10 be the set @xmath513 endowed with the topology whose non - empty open sets @xmath514 are the complements of finite sets @xmath17 of primes , with @xmath17 possibly containing @xmath479 . in particular all non - empty open sets contain the generic point @xmath515 . for any open set @xmath516 the subset @xmath517 is open . the structure sheaf @xmath518 is canonically a subsheaf of the constant sheaf @xmath519 : it associates to the complement of a finite set @xmath17 of primes the ring @xmath520 of fractions with denominators only involving primes in @xmath17 . we view this ring as a subring of @xmath519 . we shall construct the structure sheaf @xmath12 as a subsheaf of the constant sheaf of @xmath0-algebras @xmath521 . first , we use the functor @xmath522 to transform @xmath518 into a sheaf of sub-@xmath0-algebras @xmath523 . then , we extend this sheaf at the archimedean place @xmath524 by associating to any open set @xmath514 containing @xmath479 the @xmath0-subalgebra of @xmath521 given by @xmath525 where @xmath526 , @xmath527 is the usual absolute value on @xmath519 which induces a multiplicative seminorm on @xmath528 so that makes sense using .we recall that an arakelov divisor on @xmath10 is given by a pair @xmath529 where @xmath530 is a divisor on @xmath478 and @xmath531 is a real number @xmath532 . one uses the notation @xmath533 . the principal divisor @xmath534 associated to a rational number @xmath535 is @xmath536 . to a divisor @xmath530 on @xmath478 corresponds a sheaf @xmath537 of modules over @xmath518 and this sheaf is canonically a subsheaf of the constant sheaf @xmath519 . [ propspecz ] @xmath13 the definition together with the inclusions @xmath538 and the restriction maps define a sheaf @xmath539 of @xmath0-algebras over @xmath540 . @xmath15 let @xmath481 be an arakelov divisor on @xmath540 . the following defines a sheaf @xmath541 of @xmath539-modules over @xmath540 extending the sheaf @xmath537 on @xmath11 @xmath542 the sheaf @xmath537 on @xmath11 is viewed as a subsheaf of the constant sheaf @xmath519 which gives a meaning to @xmath527 . @xmath13 by construction the @xmath0-algebras @xmath543 are naturally @xmath0-subalgebras of @xmath521 . moreover the restriction map @xmath544 for @xmath545 corresponds to the inclusion as @xmath0-subalgebras of @xmath521 . these maps are injective and compatible with the product . to show that one has a sheaf of @xmath0-algebras it is enough to show that for each @xmath29 , the ( pointed ) sets @xmath546 form a sheaf of sets . given a covering @xmath547 , the elements @xmath548 of @xmath549 which agree on the intersections @xmath550 correspond to the same element of @xmath551 . moreover this element belongs to @xmath546 since this means that @xmath552 for all @xmath553 , @xmath554 and , if @xmath555 , that @xmath556 which is true since @xmath557 for some @xmath558 . @xmath15 let @xmath514 be a non - empty open set containing @xmath479 . we apply proposition [ sssalg2 ] with @xmath559 and the module @xmath560 : these are naturally subsets of @xmath519 with the inherited ring and module structures . thus the norms induced by the usual norm on @xmath519 satisfy the hypothesis of proposition [ sssalg2 ] and one obtains using the same argument as in @xmath13 , a sheaf @xmath541 of @xmath539-modules over @xmath10 . note that at level @xmath117 , one has , for @xmath555 , @xmath561 where the positivity of a divisor is defined pointwise . let @xmath99 be an @xmath0-algebra , @xmath51 a right @xmath99-module and @xmath562 a left @xmath99-module . by definition ( _ cf . _ @xcite definition 2.1.5.3 ) the smash product @xmath563 is the coequalizer @xmath564 where the maps come from the two actions . we shall use the following two properties of the smash product : the first states that @xmath565 and the second asserts that for filtered colimits of @xmath99-modules one has @xmath566 these properties can be shown by applying , for any @xmath0-module @xmath567 , the isomorphism @xmath568 , where @xmath569 is the internal one . likewise for the tensor product of sheaves modules over sheaves of rings , the smash product @xmath570 is defined as the sheaf associated to the presheaf @xmath571 a similar argument as in proposition [ propspecz ] then applies showing that one has a natural morphism of @xmath539-modules @xmath572 next , we prove that the smash product rule for the sheaves @xmath541 , when taken over the structure sheaf @xmath539 is similar to the law of composition of frobenius correspondences over the arithmetic site of @xcite . let @xmath533 and @xmath573 . then the corresponding conditions at the level @xmath117 ( of the @xmath0-algebras ) determine the restrictions @xmath574 and @xmath575 . the additive law , asserting that @xmath576 is an isomorphism , means here that the map @xmath577 is surjective , and fails to be so when both @xmath578 and @xmath579 are irrational while their product is rational . the sheaf @xmath580 , which plays the role of the infinitesimal deformation of the identity correspondence in @xcite , is defined as follows ( using ) @xmath581 [ proppic ] @xmath13 the multiplication rule @xmath582 holds for the @xmath539-modules @xmath541 and @xmath583 except when @xmath578 and @xmath579 are irrationals while their product is rational . in that case one has @xmath584 @xmath15 two arakelov divisors @xmath585 are equivalent modulo principal divisors if and only if the sheaves @xmath541 and @xmath583 are isomorphic as abstract sheaves of @xmath539-modules . @xmath13 one checks that @xmath576 is an isomorphism , when both @xmath578 and @xmath579 are rational : this is verified using and the fact that in that case the modules are locally trivial . one then obtains in the case @xmath579 is rational using an increasing sequence @xmath586 such that @xmath587 $ ] for a fixed prime @xmath588 . on the open set @xmath589 one gets that @xmath590 where @xmath591 is defined by replacing the component at @xmath479 of @xmath481 by @xmath592 . one then applies to obtain . finally , when @xmath579 is also irrational one uses an increasing sequence @xmath593 such that @xmath594 $ ] together with to obtain @xmath13 . @xmath15 for any @xmath535 , one has a canonical isomorphism @xmath595 of sheaves of @xmath539-modules over @xmath10 which is defined by multiplication by @xmath596 . more precisely , the module @xmath560 maps to @xmath597 by letting @xmath598 . one also has @xmath599 this shows that if @xmath585 are equivalent modulo principal divisors , then the sheaves @xmath541 and @xmath583 are isomorphic as abstract sheaves of @xmath539-modules . conversely , an isomorphism of abstract sheaves of @xmath539-modules induces an isomorphism on the set of global sections @xmath600 . thus one obtains an invariant of the isomorphism class of abstract sheaf by setting @xmath601 where @xmath602 is the arakelov divisor with vanishing finite part . one also has the identification @xmath603 thus @xmath604 using the multiplication rule @xmath13 one gets that @xmath605 . since every arakelov divisor is equivalent , modulo principal divisors , to a @xmath606 one gets @xmath15 . [ picrem]to handle the non - invertibility of the sheaf @xmath541 for irrational values of @xmath607 in one can use @xmath580 ( _ cf . _ ) rather than @xmath539 as the structure sheaf . the obtained @xmath0-algebras @xmath608 are no longer unital but one still retains the equality @xmath609 . one defines in a similar manner the sheaves @xmath610 , using the strict inequality . the multiplication rule is now @xmath611 this implies that the group of isomorphism classes of invertible sheaves of @xmath580-modules contains at least the quotient of the idele class group of @xmath519 by the maximal compact subgroup @xmath612 . thus this new feature bypasses the defect of the construction of @xcite mentioned in the open question 4 . 6 of @xcite . [ cyclicrem]the above developments suggest that one should investigate the ( topological ) cyclic homology of @xmath540 , viewed as a topological space endowed with a sheaf of @xmath0-algebras and apply the topological hochschild theory to treat the cohomology of the modules @xmath541 . [ ss12]proposition [ sssalg2 ] , combined with the general theory of schemes developed by b. ten and m. vaqui in @xcite for a symmetric closed monoidal category , provides the tools to perform the same constructions as in @xcite ( reviewed briefly below in [ durov1 ] ) , in the framework of @xmath0-algebras . we shall not pursue this venue , rather we will now explain , using the assembly map of @xcite , a conceptual way to pass from the set - up of monads to the framework of @xmath0-algebras . the main fact highlighted in this section is that the assembly map of @xcite determines a functorial way to associate an @xmath0-algebra to a monad on @xmath43 . we derive as a consequence that several main objects introduced in @xcite can be naturally incorporated in the context of @xmath0-algebras . we start off in [ durov1 ] by reviewing some basic structures defined in _ op.cit . _ in [ sectassemb ] we recall the definition of the assembly map which plays a fundamental role , in proposition [ assembly ] , to define the functor that associates an @xmath0-algebra to a monad . finally , proposition [ tensq ] of [ sectalgmon ] shows that the smash product @xmath0-algebra @xmath613 is ( non - trivial and ) not isomorphic to @xmath614 . this result is in sharp contrast with the statement of @xcite that @xmath615 ( _ cf . _ 5.1.22 p. 226 ) . a monad on @xmath19 is defined by an endofunctor @xmath616 together with two natural transformations @xmath617 and @xmath618 which are required to fulfill certain coherence conditions ( @xcite ) . the required associativity of the product @xmath127 is encoded by the commutativity of the following diagram , for any object @xmath20 of @xmath19 @xmath619^{\mu_{\sigma(x ) } } \ar[rr]^{\sigma(\mu(x ) ) } & & \ar[d]^{\mu_{x } } \sigma^2(x ) \\ \sigma^2(x ) \ar[rr]_{\mu_x}&&\sigma(x ) } \ ] ] there is a similar compatibility requirement for the unit transformation @xmath620 . this set - up corresponds precisely to the notion of a monod in the monoidal category @xmath621 of endofunctors under composition , where @xmath622 is the identity endofunctor . next , we review briefly some of the constructions of @xcite . let @xmath129 be a semiring , then in _ op.cit . _ one encodes @xmath129 as the endofunctor of @xmath129-linearization @xmath623 . it associates to a set @xmath20 the set @xmath624 of finite , formal linear combinations of points of @xmath20 , namely of finite formal sums @xmath625 , where @xmath626 and @xmath627 . one uses the addition in @xmath129 to simplify : @xmath628 . to a map of sets @xmath65 one associates the transformation @xmath629 the multiplication in @xmath129 is encoded by the elementary fact that a linear combination of finite linear combinations is still a finite linear combination , thus one derives @xmath630 this construction defines a natural transformation @xmath631 which fulfills the algebraic rules of a monad . in fact , the rules describing a monad allow one to impose certain restrictions on the possible linear combinations which one wants to consider . this fits well , in particular , with the structure of the convex sets so that one may for instance define the monad @xmath632 by considering as the ring @xmath129 the field @xmath633 of real numbers and by restricting to consider only the linear combinations @xmath634 satisfying the convexity condition @xmath635 . one can eventually replace @xmath633 by @xmath519 and in this case one obtains the monad @xmath636 . one may combine the monad @xmath636 with the ring @xmath9 as follows . the first step takes an integer @xmath562 and considers the localized ring @xmath637 $ ] and the intersection @xmath638 . this is simply the monad @xmath639 of finite linear combinations @xmath634 with @xmath640 and @xmath635 . by computing its prime spectrum one finds that @xmath641 . one then reinstalls the missing primes _ i.e. _ the set @xmath642 , by gluing @xmath643 with @xmath644 on the large open set that they have in common . one obtains in this way the set @xmath645 finally , one eliminates the integer @xmath562 by taking a suitable projective limit under divisibility . the resulting space is the projective limit space @xmath646 as a topological space it has the same topology ( zariski ) as that of @xmath644 with simply one further closed point ( _ i.e. _ @xmath479 ) added . at the remaining non - archimedean primes this space fulfills the same topological properties as @xmath644 . one nice feature of this construction is that the local algebraic structure at @xmath479 is modeled by @xmath636 . as explained in 2.2.1.1 of @xcite , a @xmath1-set @xmath647 automatically extends , using filtered colimits over the finite subsets @xmath648 , to a pointed endofunctor @xmath649 let @xmath129 be a ( commutative ) semiring and @xmath137 be the associated @xmath0-algebra as in lemma [ sssalg ] . then the extension @xmath650 is , except for the nuance given by the implementation of the base point in the construction , the same as the endofunctor @xmath651 of [ durov1 ] . next , we review the definition of the assembly map in the simple case of @xmath1-sets : we refer to @xcite for more details . let @xmath652 be a @xmath1-set , and @xmath653 its extension to a pointed endofunctor as in . one derives a map natural in the two pointed sets @xmath47 ( _ cf . _ 2.2.1.2 of @xcite ) @xmath654 which is obtained as the family , indexed by the elements @xmath655 , of maps @xmath656 with @xmath657 similarly , we let @xmath647 be a second @xmath1-set , and @xmath658 its extension to a pointed endofunctor of @xmath43 . one then constructs a map natural in @xmath20 and @xmath108 as follows @xmath659 where @xmath660 and @xmath661 . by restricting to finite pointed sets @xmath47 , _ i.e. _ to objects of @xmath662 , one obtains a natural transformation of bi - functors and hence a morphism of @xmath1-sets @xmath663 @xmath13 the functor @xmath670 coincides with the endofunctor @xmath671 . for a pointed set @xmath76 , we denote the elements of the set @xmath672 as formal sums @xmath673 , where only finitely many non - zero terms occur in the sum , and the coefficient of the base point @xmath251 is irrelevant , _ i.e. _ the term @xmath674 drops out . let @xmath675 and @xmath676 , then one has @xmath677 the image of @xmath678 by @xmath679 is described by @xmath680 and one finally obtains @xmath681 by applying one has @xmath682 any element @xmath683 is represented by @xmath684 with @xmath85 and @xmath685 as above . its image is described , using , by @xmath686 for @xmath32 an object of @xmath35 , an element of @xmath687 is of the form @xmath688 where @xmath689 is a finite set of indices . let @xmath690 and @xmath691 . let @xmath692 and @xmath693 . let @xmath209 be given by @xmath694 if @xmath695 and @xmath696 , @xmath697 . with this choice for @xmath698 , one then obtains @xmath699 @xmath15 for @xmath667 , the element of @xmath204 associated to the @xmath258-relation @xmath259 corresponds to @xmath698 where @xmath700 , @xmath701 , @xmath702 @xmath703 , @xmath704 @xmath263 . thus becomes @xmath705 . notice that @xmath15 of proposition [ assembly0 ] implies that the assembly map is _ not _ injective for @xmath667 , since by corollary [ pp1 ] one knows that @xmath204 is infinite and countable for any @xmath706 , while @xmath707 remains finite for any choice of @xmath258 . the constructions of @xcite for monads reviewed in [ durov1 ] can be easily reproduced in the context of @xmath1-sets and yield the @xmath0-algebras : @xmath708 and @xmath709 , where the multiplicative seminorms are _ resp . , _ the usual absolute value on @xmath633 and its restriction to @xmath519 . this fact suggests the existence of a functor that associates to a monad on @xmath43 an @xmath0-algebra . we first briefly comment on the use of the category of pointed sets in place of the category of sets as in @xcite . one has a pair of adjoint functors @xmath710 where @xmath711 is the functor @xmath712 of implementation of a base point . it is left adjoint to the forgetful functor @xmath713 . the unit of the adjunction is the natural transformation @xmath714 which maps the set @xmath20 to its copy in the disjoint union @xmath715 . the counit @xmath716 is the natural transformation which is the identity on @xmath717 while it maps the extra base point to the base point of @xmath20 . let @xmath522 be a pointed endofunctor of @xmath43 , the composition @xmath718 is then an endofunctor of @xmath19 . moreover given a natural transformation @xmath719 of endofunctors of @xmath43 , one obtains a natural transformation of endofunctors of @xmath19 , @xmath720 as follows @xmath721 this construction determines a natural correspondence between monads on @xmath43 and monads on @xmath19 . note in particular that the monad on @xmath19 given by @xmath722 , where @xmath723 is the identity endofunctor is the monad called @xmath7 in @xcite . moreover , for any semiring @xmath129 , with @xmath670 being the extension of @xmath137 to an endofunctor of @xmath43 as in , one checks that the natural monad structure on @xmath670 corresponds to the monad @xmath724 , _ i.e. _ that @xmath725 . [ assembly ] let @xmath726 be a pointed algebraic monad on @xmath43 , then the restriction @xmath99 of @xmath726 to @xmath662 defines an @xmath0-algebra with the product @xmath727 defined by the composite of the assembly map and the product @xmath617 of the monad @xmath726 . one uses the fact stated in remark 2.19 of @xcite that the assembly map makes the identity functor on @xmath1-sets a lax monoidal functor from the monoidal category @xmath728 to the monoidal category @xmath729 . more explicitly , the composition @xmath727 becomes a morphism of @xmath1-sets @xmath730 . the compatibility of the assembly map with the associativity shows that the commutative diagram gives the commutativity of the following diagram of maps of @xmath1-sets @xmath731^{m\wedge \id } \ar[rr]^{\id \wedge m } & & \ar[d]^{\mu\circ \alpha_{\sigma,\ca } } \ca\wedge \ca \\ \ca\wedge \ca \ar[rr]_{\mu\circ \alpha_{\sigma,\ca } } & & \ca } \ ] ] the compatibility with the unit is handled in a similar manner . the assembly map makes the identity functor on @xmath1-sets a lax monoidal functor from the monoidal category @xmath728 to the monoidal category @xmath729 . thus given an algebraic monad @xmath726 on @xmath43 one can associate to a ( left or right ) algebraic module @xmath490 on @xmath726 in the monoidal category @xmath728 a ( left or right ) module over the @xmath0-algebra @xmath99 of proposition [ assembly ] . the underlying @xmath1-set is unchanged and the action of @xmath99 is obtained using the assembly map as in proposition [ assembly ] . besides the notion of ( left or right ) algebraic module @xmath490 on @xmath726 in the sense of the monoidal category @xmath728 as discussed in 4.7 of @xcite , a simpler category @xmath733 of modules over a monad is also introduced and used in 3.3.5 of _ op.cit . _ a module @xmath490 on a monad @xmath726 on @xmath43 in this sense is simply a pointed set @xmath490 together with a map of pointed sets @xmath734 fulfilling the two conditions : @xmath735 . this notion of a module over a monad defined in @xcite implies that modules over @xmath724 correspond exactly to ordinary @xmath129-modules ( _ cf . _ _ or @xcite just before proposition 2 , p. 11 ) . in particular , by taking the monad @xmath736 ( which is denoted by @xmath737 in @xcite ) , one finds that any set @xmath51 is a module in the unique manner provided by the map @xmath738 . this fact continues to hold if one considers the monad which corresponds to @xmath7 ( _ cf . _ @xcite 4.4 p. 24 ) . then , one finds that the modules over @xmath7 are just pointed sets . in the framework of @xmath0-algebras , an @xmath129-module @xmath490 over a ring @xmath129 gives rise to an @xmath137 module @xmath499 but a general module over @xmath137 is not always of this form ( _ cf . _ @xcite remark 2.1.5.2 , 4 ) . this nuance between @xmath129-modules and @xmath137-modules luckily disappears at the homotopy level . since this is a crucial fact , we shall comment on it here below : we refer to @xcite and @xcite for the full treatment . let @xmath258 be a _ commutative _ @xmath0-algebra , then one defines the smash product @xmath739 of two @xmath258-modules as the coequalizer ( _ cf . _ @xcite , definition 2.1.5.3 ) @xmath740 where the two maps are given by the actions of @xmath258 on @xmath51 and @xmath562 . one obtains in this way the symmetric closed monoidal category of @xmath258-modules and a corresponding notion of @xmath258-algebra . in order to perform homotopy theory , one passes to the associated categories of simplicial objects . the fundamental fact ( _ cf . @xcite , theorem 4.1 ) is that the homotopy category of @xmath614-algebras , obtained using the simplicial version of @xmath614-algebras , is equivalent to the homotopy category of ordinary simplicial rings . this result continues to hold if one replaces @xmath9 by any commutative ring @xmath129 and simplicial rings by simplicial @xmath129-algebras . this shows that , in the framework of @xmath0-algebras , nothing is lost by using the more relaxed notion of @xmath137-module in place of the more restricted notion based on monads . in @xcite ( _ cf . _ 5.1.22 p. 226 ) one gets the equality @xmath615 which is disappointing for the development of the analogy with the geometric theory over functions fields . the following result shows the different behavior of the corresponding statement over @xmath0 . assume that there exists an isomorphism of @xmath0-algebra @xmath741 . it is then unique since , by proposition [ sssalg1 ] , the @xmath0-algebra @xmath614 has no non - trivial automorphism . furthermore , by proposition [ assembly0 ] the assembly map @xmath742 is surjective and this fact implies that the composition @xmath743 is also surjective . let @xmath744 be the natural transformation as in , then the composite @xmath745 is a morphism of @xmath1-sets : @xmath746 . since @xmath744 is surjective , one then derives that the composite @xmath746 is also surjective , thus coming from a surjective endomorphism ( _ i.e. _ an automorphism ) of the abelian group @xmath9 ( here we refer to the proof of proposition [ sssalg1 ] ) . these facts would imply that @xmath747 is also injective , hence an isomorphism @xmath748 . the set @xmath749 is @xmath750 with @xmath24 as the base point . the maps @xmath751 of act as follows @xmath752 thus @xmath753 is naturally isomorphic to the @xmath1-set which assigns to @xmath32 the laurent polynomials in @xmath258-variables modulo constants @xmath754^{\otimes k}/\z$ ] . the three maps @xmath751 of determine the following maps @xmath755\to \z[t]$ ] @xmath756 in particular , we see that the map @xmath757 given by the pair of maps @xmath758 is not injective since one can add to @xmath759 any multiple of @xmath760 without affecting @xmath761 and @xmath762 . for the @xmath1-set @xmath614 the map @xmath763 given by the pair of maps @xmath764 is bijective , thus the @xmath1-set @xmath765 can not be isomorphic to @xmath614 and this determines a contradiction . [ homtop]when working at the secondary level of topological spectra , it is a well known fact that the smash product of the eilenberg maclane spectra @xmath766 is not isomorphic to @xmath767 . in fact , the corresponding homotopy groups are not the same . indeed , for any cofibrant spectrum @xmath20 one has the equality @xmath768 , where @xmath769 denotes the spectrum homology with integral coefficients . applying this fact to @xmath770 , one interprets @xmath771 as the integral homology of the eilenberg - mac lane spectrum @xmath767 . this homology is known to be finite in positive degrees but not trivial ( _ cf . _ theorem 3.5 of @xcite ) . this argument however , can not be applied directly to provide an alternative proof of proposition [ tensq ] since then one would need to compare the spectra @xmath772 and @xmath773 and this comparison is usually done by using a cofibrant replacement @xmath774 of the @xmath1-set @xmath614 . a. connes , c. consani , _ characteristic one , entropy and the absolute point _ , `` noncommutative geometry , arithmetic , and related topics '' , the twenty - first meeting of the japan - u.s . mathematics institute , baltimore 2009 , jhup ( 2012 ) , pp 75139 . a. connes , c. consani , _ the universal thickening of the field of real numbers _ , to appear on advances in the theory of numbers thirteenth conference of the canadian number theory association ( 2014 ) ; arxiv:1202.4377 [ math.nt ] ( 2012 ) .
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* abstract * we show that the basic categorical concept of an @xmath0-algebra as derived from the theory of segal s @xmath1-sets provides a unifying description of several constructions attempting to model an algebraic geometry over the absolute point .
it merges , in particular , the approaches using monods , semirings and hyperrings as well as the development by means of monads and generalized rings in arakelov geometry .
the assembly map determines a functorial way to associate an @xmath0-algebra to a monad on pointed sets .
the notion of an @xmath0-algebra is very familiar in algebraic topology where it also provides a suitable groundwork to the definition of topological cyclic homology .
the main contribution of this paper is to point out its relevance and unifying role in arithmetic , in relation with the development of an algebraic geometry over symmetric closed monoidal categories .
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let @xmath2 be a measurable space , @xmath3 , @xmath4 and let @xmath5 be nonatomic finite measures defined on the same @xmath6algebra @xmath7 . let @xmath8 stand for the set of all measurable partitions @xmath9 of @xmath10 ( @xmath11 for all @xmath12 , @xmath13 , @xmath14 for all @xmath15 ) . let @xmath16 denote the @xmath17-dimensional simplex . for this definition and the many others taken from convex analysis , we refer to @xcite . a partition @xmath18 is said to be @xmath0optimal , for @xmath19 , if @xmath20 this problem has a consolidated interpretation in economics . @xmath10 is a non - homogeneous , infinitely divisible good to be distributed among @xmath1 agents with idiosyncratic preferences , represented by the measures . a partition @xmath21 describes a possible division of the cake , with slice @xmath22 given to agent @xmath12 . a satisfactory compromise between the conflicting interests of the agents , each having a relative claim @xmath23 , @xmath12 , over the cake , is given by the @xmath0optimal partition . it can be shown that the proposed solution coincides with the kalai - smorodinski solution for bargaining problems ( see kalai and smorodinski @xcite and kalai @xcite ) . when @xmath5 are all probability measures , i.e .. @xmath24 for all @xmath12 , the claim vector @xmath25 describes a situation of perfect parity among agents . the necessity to consider finite measures stems from game theoretic extensions of the models , such as the one given in dallaglio et al . @xcite . when all the @xmath26 are probability measures , dubins and spanier @xcite showed that if @xmath27 for some @xmath28 , then @xmath29 . this bound was improved , together with the definition of an upper bound by elton et al . a further improvement for the lower bound was given by legut @xcite . the aim of the present work is to provide further refinements for both bounds . we consider the same geometrical setting employed by legut @xcite , i.e. the partition range , also known as individual pieces set ( ips ) ( see barbanel @xcite for a thorough review of its properties ) , defined as @xmath30 let us consider some of its features . the set @xmath31 is compact and convex ( see lyapunov @xcite ) . the supremum in is therefore attained . moreover @xmath32 so , the vector @xmath33 is the intersection between the pareto frontier of @xmath31 and the ray @xmath34 . to find both bounds , legut locates the solution of the maxsum problem @xmath35 on the partition range . then , he finds the convex hull of this point with the corner points @xmath36 ( @xmath37 is placed on the @xmath38-th coordinate ) to find a lower bound , and uses a separating hyperplane argument to find the upper bound . we keep the same framework , but consider the solutions of several maxsum problems with weighted coordinates to find better approximations . fix @xmath39 and consider @xmath40 let @xmath41 be a non - negative finite - valued measure with respect to which each @xmath26 is absolutely continuous ( for instance we may consider @xmath42 ) . then , by the radon - nikodym theorem for each @xmath43 @xmath44 where @xmath45 is the radon - nikodym derivative of @xmath26 with respect to @xmath46 finding a solution for is rather straightforward : ( see ( * ? ? ? * theorem 2 ) , ( * ? ? ? * theorem 2 ) ( * ? ? ? * proposition 4.3 ) ) let @xmath47 and let @xmath48 be an @xmath49partition of @xmath10 . if @xmath50 then @xmath51 is optimal for . given @xmath47 , an _ efficient value vector ( evv ) _ with respect to @xmath52 , @xmath53 is defined by @xmath54 the evv @xmath55 is a point where the hyperplane @xmath56 touches the partition range @xmath57 so @xmath55 lies on the pareto border of @xmath58 as we will see later only one evv is enough to assure a lower bound , we give a general result for the case where several evvs have already been computed . we derive this approximation result through a convex combination of these easily computable points in @xmath57 which lie close to @xmath59 [ teo_main ] consider @xmath60 linearly independent vectors @xmath61 where @xmath62 , @xmath63 is the evv associated to @xmath64 , @xmath65 . let @xmath66 be an @xmath67 matrix and denote as @xmath68 an @xmath69 submatrix of @xmath70 such that @xmath71 a. @xmath72 + if and only if @xmath73 where @xmath74 is the @xmath69 matrix obtained by replacing the @xmath75th column of @xmath76 with @xmath77 , obtained from @xmath78 by selecting the elements corresponding to the rows in @xmath76 . moreover , @xmath79 if and only if @xmath80 b. for any choice of @xmath81 , @xmath82 moreover , if holds , then @xmath83_{ij } } \leq v^{\alpha}\ ] ] where @xmath84_{ij}$ ] is the @xmath85-th element of @xmath86 . to prove @xmath87 , suppose holds . we show that @xmath88 , and therefore that holds , by verifying that the following system of linear equations in the variables @xmath89 , @xmath90 @xmath91 has a unique solution @xmath92 with @xmath93 , for @xmath94 . first of all , @xmath95 implies @xmath96 for at least an @xmath97 , otherwise all the evvs would lie on the same hyperplane , contradicting the linear independence of such vectors . this fact and imply that the coefficient matrix has rank @xmath98 and its unique solution can be obtained by deleting the @xmath99 equations corresponding to the rows not in @xmath76 . denote each column of @xmath76 as @xmath100 , @xmath94 and denote as @xmath101 , the vector obtained from @xmath78 by selecting the same components as each @xmath102 by cramer s rule we have for each @xmath94 , @xmath103 @xmath104 since by either a determinant is null or it has the same sign of the other determinants . if holds , then @xmath105 for every @xmath94 and holds . conversely , each row of @xmath70 not in @xmath76 is a linear combination of the rows in @xmath76 . therefore , each point of @xmath106 is identified by a vector @xmath107 whose components correspond to the rows in @xmath76 , while the other components are obtained by means of the same linear combinations that yield the rows of @xmath70 outside @xmath76 . let @xmath108 denote the @xmath109 matrix obtained from the matrix @xmath76 without @xmath110 , @xmath111 . consider a hyperplane @xmath112 in @xmath106 through the origin and @xmath113 evvs @xmath114 where @xmath115 . if @xmath116 , when we separate the subspace through @xmath112 , for all @xmath115 either the ray @xmath117 is coplanar to @xmath112 , i.e. , @xmath118 or @xmath119 and @xmath110 lie in the same half - space , i.e. , @xmath120 moving the first column to the @xmath121-th position in all the matrices above , we get . in case holds , only the inequalities in are feasible and holds . to prove @xmath122 , consider , for any @xmath12 , the hyperplane that intersects the ray @xmath117 at the point @xmath123 , with @xmath124 since @xmath31 is convex , the intersection point is not internal to @xmath31 . so , @xmath125 for @xmath126 , and , therefore , @xmath127 . we get the lower bound for @xmath128 as solution in @xmath89 of the system . by cramer s rule , @xmath129_{ij } } , \end{aligned}\ ] ] where @xmath130 is the @xmath85-th minor of @xmath76 . the second equality derives by suitable exchanges of rows and columns in the denominator matrix : in fact , swapping the first rows and columns of the matrix leaves the determinant unaltered . the last equality derives by dividing each row @xmath38 of @xmath76 by @xmath23 , @xmath94 . finally , by we have @xmath131 . the above result shows that whenever @xmath132 , then @xmath133 therefore , the corresponding evv @xmath134 is irrelevant to the formulation of the lower bound and can be discarded . we will therefore keep only those evv that satisfy and will denote them as the _ supporting evvs _ for the lower bound . in the case @xmath135 we have [ cor_mequalsn ] suppose that there are @xmath1 vectors @xmath136 where @xmath137 , @xmath12 , is the evv associated to @xmath64 , @xmath65 . if @xmath138 , @xmath139 and , for all @xmath12 @xmath140 where @xmath141 is the @xmath142 matrix obtained by replacing @xmath134 with @xmath78 in @xmath70 , then @xmath143_{ij } } \leq v^{\alpha } \leq \min_{i \in n } \cfrac{\sum_{j \in n } \beta_{ij } u_{ij}}{\sum_{j \in n } \beta_{ij } \alpha_j}\ ] ] where @xmath144_{ij}$ ] is the @xmath85-th element of @xmath145 . we next consider two further corollaries that provide bounds in case only one evv is available . the first one works with an evv associated to an arbitrary vector @xmath146 [ oneevv ] ( ( * ? ? ? * proposition 3.4 ) ) let @xmath147 be finite measures and let @xmath148 be the evv corresponding to @xmath47 such that @xmath149 then , @xmath150 } \leq v^{\alpha } \leq \cfrac{\sum_{i \in n } \beta_i u_i}{\sum_{i \in n } \alpha_i u_i}.\ ] ] consider the corner points of the partition range @xmath151 where @xmath37 is placed on the @xmath38-th coordinate ( @xmath12 ) , and the matrix @xmath152 , where @xmath153 occupies the @xmath121-th position . now @xmath154 and , for all @xmath155 , @xmath156 which is positive by . therefore , @xmath70 satisfies the hypotheses of corollary [ cor_mequalsn ] . since @xmath70 has inverse @xmath157 the following lower bound is guaranteed for @xmath128 : @xmath158 } .\ ] ] the upper bound is a direct consequence of theorem [ teo_main ] . in case all measures @xmath26 , @xmath159 are normalized to one and the only evv considered is the one corresponding to @xmath160 , we obtain legut s result . [ legut_result ] ( ( * ? ? ? * theorem 3 ) ) let @xmath147 be probability measures and let @xmath148 be the evv corresponding to @xmath160 . let @xmath115 be such that holds . then , @xmath161 where @xmath162 . simply apply corollary [ oneevv ] with @xmath24 , for all @xmath12 and @xmath160 . then @xmath163 where @xmath164 . finally , by theorem [ teo_main ] we have @xmath165 it is important to notice that the lower bound provided by theorem [ teo_main ] certainly improves on legut s lower bound only when one of the evvs forming the matrix @xmath70 is the one associated to @xmath166 we consider a @xmath167 $ ] good that has to be divided among three agents with equal claims , @xmath168 and preferences given as density functions of probability measures @xmath169 \ ; , \ ] ] @xmath170 being the density function of a @xmath171 distribution . the preferences of the players are not concentrated ( following definition 12.9 in barbanel @xcite ) and therefore there is only one evv associated to each @xmath172 ( cfr . @xcite , theorem 12.12 ) . agent 1 : tiny dashing ; agent 2 : large dashing ; agent 3 : continuous line.,scaledwidth=60.0% ] the evv corresponding to @xmath173 is @xmath174 . consequently , the bounds provided by legut are @xmath175 consider now two other evvs @xmath176 corresponding to @xmath177 and @xmath178 , respectively . the matrix @xmath179 satisfies the hypotheses of theorem [ teo_main ] and the improved bounds are @xmath180 the bounds for @xmath128 depend on the choice of the evvs that satisfy the hypotheses of theorem [ teo_main ] . any new evv yields a new term in the upper bound . since we consider the minimum of these terms , this addition is never harmful . improving the lower bound is a more delicate task , since we should modify the set of supporting evvs for the lower bound , i.e. those evvs that include the ray @xmath117 in their convex hull . when we examine a new evv we should verify whether replacing an evv in the old set will bring to an improvement . a brute force method would require us to verify whether conditions are verified with the new evv in place of @xmath181 only in this case we have a guarantee that the new evv will not make the bound worst . then , we should verify again , with the new evv replacing one of the @xmath182 evvs , in order to find the evv from the old set to replace . however , in the following proposition we propose a more efficient condition for improving the bounds , by which we simultaneously verify that the new evv belongs to the convex hull of the @xmath182 evvs and detect the vector to replace . for any couple @xmath183 denote as @xmath184 the @xmath109 matrix obtained from @xmath76 by replacing column @xmath121 with @xmath185 and by deleting column @xmath186 . [ teo_imp ] let @xmath187 be @xmath188 evvs with the last @xmath182 vectors satisfying conditions and . if there exists @xmath189 such that @xmath190 then @xmath191 moreover , if all the inequalities in are strict then in both and the vectors belong to the relative interior of the respective cones . before proving the individual statements , we sketch a geometric interpretation for condition . as in theorem [ teo_main ] we restrict our analysis to the subspace @xmath106 . for any @xmath192 , the hyperplanes @xmath193 should separate @xmath194 and @xmath78 ( strictly if all the inequalities in are strict ) in the subspace @xmath106 . to prove , argue by contradiction and suppose @xmath195 then , for any @xmath196 there must exist a @xmath197 such that the hyperplane @xmath198 passes through all the evv ( including @xmath185 ) but @xmath194 and @xmath199 , and supports @xmath200 therefore , @xmath78 and @xmath194 belong to the same strict halfspace defined by @xmath198 , contradicting . to show the existence of such hyperplane consider the hyperplane @xmath201 in @xmath106 passing through @xmath81 and denote with @xmath202 the intersection of @xmath203 with such hyperplane . restricting our attention to the points in @xmath201 , the vectors @xmath81 form a simplicial polyhedron with @xmath204 . there must thus exist a @xmath197 such that the @xmath205-dimensional hyperplane @xmath206 in @xmath207 passing through @xmath208 and @xmath209 , supports @xmath210 and contains @xmath194 ( and @xmath78 ) in one of its strict halfspaces ( see appendix . ) if we now consider the hyperplane in @xmath106 passing through the origin and @xmath206 we obtain the required hyperplane @xmath211 to prove we need some preliminary results . first of all , under , @xmath212 otherwise , @xmath185 would be coplanar to @xmath112 and any hyperplane @xmath198 , @xmath197 , would coincide with it . in such case the separating conditions would not hold . moreover , there must exist some other @xmath213 for which @xmath214 otherwise , @xmath185 would coincide with @xmath194 , making the result trivial . we also derive an equivalent condition for . let @xmath215 be the @xmath69 matrix obtained from @xmath76 by replacing vectors @xmath194 and @xmath199 by some other vectors , say @xmath216 and @xmath217 , respectively . if we move the first column to the @xmath186-th position , becomes @xmath218 switching positions @xmath121 and @xmath186 in the second matrix we get @xmath219 and therefore @xmath220 from , and from part @xmath87 of the present theorem , we derive @xmath221 and therefore , yields @xmath222 condition and theorem [ teo_main ] allow us to conclude that @xmath223 . regarding the last statement of the theorem , we have already shown . moreover , if hold with a strict inequality sign for any @xmath197 , then @xmath224 for the same @xmath186 and @xmath225 . similarly , would hold with strict inequality signs and @xmath226 . if holds , we get not only that @xmath117 intersects the convex hull of the @xmath182 evvs @xmath227 , but also that the ray @xmath203 intersects the convex hull of the @xmath182 evvs @xmath228 . we can therefore replace @xmath194 with @xmath185 in the set of supporting evvs for the lower bound . if the test fails for each @xmath189 , we discard @xmath185 we keep the current lower bound ( with its supporting evvs ) . in case holds with an equality sign for some @xmath186 , conditions and together imply @xmath229 . therefore , we could discard @xmath199 from the set of supporting evvs for the lower bound . we consider a list of 1000 random vectors in @xmath230 and , starting from the identity matrix , we iteratively pick each vector in the list . if this satisfies condition , then the matrix @xmath70 is updated . the update occurs 9 times and the resulting evvs which generate the matrix @xmath70 are @xmath231 corresponding , respectively , to @xmath232 correspondingly , the bounds shrink to @xmath233 the previous example shows that updating the matrix @xmath70 of evvs through a random selection of the new candidates is rather inefficient , since it takes more than 100 new random vectors , on average , to find a valid replacement for vectors in @xmath70 . a more efficient way method picks the candidate evvs through some accurate choice of the corresponding values of @xmath52 . in @xcite a subgradient method is considered to find the value of @xmath128 up to any specified level of precision . in that algorithm , legut s lower bound is used , but this can be replaced by the lower bound suggested by theorem [ teo_main ] . considering the improved subgradient algorithm , we obtain the following sharper bounds @xmath234 after 27 iterations of the algorithm in which , at each repetion , a new evv is considered . the authors would like to thank vincenzo acciaro and paola cellini for their precious help . the proof of in theorem [ teo_imp ] is based on the following lemma . this is probably known and too trivial to appear in a published version of the present work . however , we could not find an explicit reference to cite it . therefore , we state and prove the result in this appendix consider @xmath1 affinely independent points @xmath235 in @xmath236 and @xmath237 . for each @xmath115 there must exist a @xmath197 such that the hyperplane passing through @xmath185 and @xmath209 supports @xmath210 and has @xmath194 in one of its strict halfspaces . fix now @xmath115 and consider @xmath112 , the hyperplane passing through the points @xmath242 . also denote as @xmath243 the intersection between @xmath112 and the line joining @xmath185 and @xmath194 . clearly @xmath244 . therefore , for any @xmath245 , the hyperplane @xmath246 in @xmath112 passing through @xmath247 will strictly separate @xmath199 and @xmath243 . consequently , @xmath243 should simultaneously lie in the halfspace of @xmath246 not containing @xmath248 , and in the cone formed by the other hyperplanes @xmath249 , @xmath250 and not containing @xmath248 . a contradiction .
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we provide a two - sided inequality for the @xmath0optimal partition value of a measurable space according to @xmath1 nonatomic finite measures .
the result extends and often improves legut ( 1988 ) since the bounds are obtained considering several partitions that maximize the weighted sum of the partition values with varying weights , instead of a single one .
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liquid crystals are characterized by large thermal fluctuations in their local orientational order arising from collective alignment of the long axis of their constituent molecules @xcite . due to such soft anisotropy , liquid crystals tend to respond easily to external forces . confining geometries such as thin films , on which most applications of liquid crystals are based , change the fluctuation spectrum . this can cause not only structural changes @xcite but also leads to fluctuation - induced effective forces between the substrates @xcite , also known as thermodynamic casimir effect . in correlated fluids such as liquid crystals this casimir force exhibits a universal power - law decay as a function of the separation between the substrates @xcite . however , this behavior is modified in the presence of other characteristic scales in the system @xcite . in the case that the substrates are laterally modulated , discrete lateral modes of thermal fluctuations are also excited . under such conditions , in addition to the forces acting perpendicularly to the substrates , effective lateral forces arise @xcite with potentially interesting technological applications . we study the influence of anchoring conditions , which vary periodically in one lateral direction , on the fluctuations of a uniformly ordered nematic liquid crystal . obviously , the periodicity @xmath0 of the substrate pattern gives rise to an oscillatory behavior for the lateral force as a function of the lateral shift @xmath1 between the substrates . for small inhomogeneities the lateral force is proportional to @xmath2 . ( the analysis of nonperiodic patterns would provide an understanding of nematic phases exposed to chemically disordered substrates . ) the present study actually extends our previous work @xcite where we considered the case in which only one of two confining substrates exhibits a chemical pattern so that there are no lateral forces . here , in addition to the fluctuation - induced lateral forces , we calculate the lateral force between the patterned substrates across the vacuum , i.e. , the background van der waals force acting parallel to the substrates . this background force is generated by the necessary chemical modulations providing the laterally varying anchoring strengths . in sec . [ ii ] our model and the theoretical formalism are specified . in sec . [ iii ] the fluctuation - induced lateral force is obtained . in sec . [ iv ] we calculate the lateral van der waals force between the patterned substrates . the results for the fluctuation - induced normal force are presented in sec . [ v ] and finally sec . [ vi ] summarizes our results . we consider a nematic liquid crystal confined by two flat but chemically patterned substrates at a separation @xmath3 . the patterns on the both substrates consist of the same periodic stripes of anchoring energies per area @xmath4 and @xmath5 along the @xmath6 direction but shifted relative to each other by the length @xmath1 ( see fig . 1 ) the substrates are translationally invariant in the y direction . the stripes are considered to vary with respect to the strength of homeotropic anchoring so that the mean orientation of the director @xmath7 is spatially homogeneous but the thermal fluctuations vary laterally giving rise to effective lateral forces . and @xmath5 with the widths @xmath8 and @xmath9 , respectively . the wavelength of the periodicity is denoted as @xmath10 and the lateral shift between the origins of the patterns on the top and the bottom substrate is denoted by @xmath1 . anchoring at both boundaries is homeotropic everywhere so that the thermal average of the director field @xmath11 is spatially homogeneous . ] based on the bulk structural frank free energy @xcite given by @xmath12=\frac{1}{2}\int_{v}{\rm d}^3x\big[k_1(\mbox{\boldmath $ \nabla$}\cdot{\bf n})^2 + k_2({\bf n}\cdot \mbox{\boldmath $ \nabla$}\times{\bf n})^2 \nonumber\\+k_3({\bf n}\times \mbox{\boldmath $ \nabla$}\times{\bf n})^2\big ] , \label{eq0}\end{aligned}\ ] ] where @xmath13 is the nematic volume , @xmath14 , @xmath15 , and @xmath16 are the splay , the twist , and the bend elastic constants , respectively , the free energy of gaussian fluctuations in the one - constant approximation reads @xmath17=\sum_{i=1}^{2}{k\over 2}\int_{v } { \rm d}^3x\left[\nabla \nu_{i}({\bf x},z)\right]^{2 } , \label{eqe}\ ] ] where @xmath18 , @xmath19 , is either of the two independent components of the fluctuating part @xmath20 of the director @xmath7 , @xmath21 is the effective elastic constant , and @xmath22 are the lateral components of the cartesian coordinates @xmath23 . to describe the interaction of the liquid crystal and the substrates , we employ the rapini - papoular surface free energy given by @xmath24=-{1\over 2}\int_{s } { \rm d}^2x \,w^{z=0}({\bf x})\,({\bf n}\cdot \hat{{\bf z}})^2-\nonumber\\{1\over 2}\int_{s } { \rm d}^2x \,w^{z = d}({\bf x})\,({\bf n}\cdot \hat{{\bf z}})^2\ = f_s^{z=0}+f_s^{z = d } \label{eq3}\end{aligned}\ ] ] where @xmath25 is the anchoring energy per area @xmath26 and @xmath27 is the unit vector in z - direction . here $ ] at the lower substrate located at @xmath29 and @xmath30 $ ] at the upper substrate located at @xmath31 where @xmath32 and @xmath33 describe the stripe modulations at the lower and the upper substrates , respectively , @xmath34 is the heaviside step function , and @xmath0 is the periodicity . the stripes have the same width @xmath35 and are separated by sharp chemical steps . the functions @xmath36 and @xmath37 equal one at the regions characterized by @xmath4 and zero elsewhere at the lower and the upper substrates , respectively . thus the surface free energy of the gaussian fluctuations given by @xmath38=f_{s}[\nu_{1}]+f_{s}[\nu_{2}]$ ] reads @xmath39={1\over 2}\big[w_a\int_{s}{\rm d}^2x\ , \big[\nu ( { \bf x},z=0)\big]^2 a(x ) \nonumber\\ + w_b\int_{s}{\rm d}^2x\ , \big[\nu ( { \bf x},z=0)\big]^2 [ 1-a(x)]\big]\ , \label{eq3}\end{aligned}\ ] ] at the lower substrate and @xmath40={1\over 2}\big[w_a\int_{s}{\rm d}^2x\ , \big[\nu ( { \bf x},z = d)\big]^2 b(x ) \nonumber\\ + w_b\int_{s}{\rm d}^2x\ , \big[\nu ( { \bf x},z = d)\big]^2 [ 1-b(x)]\big]\ , \label{eq3}\end{aligned}\ ] ] at the upper substrate . minimization of total free energy @xmath41 + f_{\rm s}^{z=0}[\nu_1,\nu_2]+f_{\rm s}^{z = d}[\nu_1,\nu_2]$ ] leads to two boundary conditions : @xmath42=0 , \ ; z=0,\label{eq : bca}\end{aligned}\ ] ] @xmath43=0 , \ ; z = d,\label{eq : bcb}\end{aligned}\ ] ] where @xmath44 is either @xmath45 or @xmath46 . the normalized [ see , c.f . , after eq . ( [ eq9 ] ) ] partition function @xmath47 of the fluctuating fields @xmath48 , @xmath49 , subject to the boundary conditions given by eqs . ( [ eq : bca ] ) and ( [ eq : bcb ] ) can be calculated within the path integral approach ( see ref . @xcite and references therein ) . the boundary conditions act as constraints which can be implemented by delta functions . they , in turn , can be written as integral representations by introducing two auxiliary fields localized at @xmath29 and @xmath31 , respectively @xcite . after performing the corresponding gaussian integrals over @xmath18 , @xmath49 , the path integral reduces to a gaussian functional integral over the auxiliary fields with a matrix kernel @xmath50 , so that obtaining the result for @xmath47 reduces to calculating @xmath51 . for the geometry considered here , the matrix @xmath50 is found to have the following matrix elements @xmath52 : @xmath53 \big[1+{\lambda_b-\lambda_a\over \lambda_a}a(x')\big]\nonumber\\ & + & { \lambda_b ( \lambda_b-\lambda_a)\over \lambda_a } [ a(x)-a(x')]\partial_z-\lambda^2_b\partial^2_z\big\ } g({\bf x}-{\bf x'},z - z{'})\big |_{z = z{'}=0},\nonumber\\ & m_{12}&({\bf x},{\bf x}{'})= \big[1+{{\lambda_b-\lambda_a}\over \lambda_a}[a(x')+b(x ) ] -2\lambda_b\partial_{z'}-{{\lambda_{b}(\lambda_b-\lambda_a)}\over \lambda_a}[a(x')+b(x)]\partial_{z'}\nonumber\\ & + & \big({{\lambda_b-\lambda_a}\over \lambda_a}\big)^{2}a(x ' ) b(x)+\lambda_b^{2}\partial^{2}_{z'})\big ] g({\bf x}-{\bf x'},z - z{'})\big |_{z = d , z{'}=0},\nonumber\\ & m_{21}&({\bf x},{\bf x}{'})= \big[1+{{\lambda_b-\lambda_a}\over \lambda_a}[a(x)+b(x ' ) ] -2\lambda_b\partial_{z}-{{\lambda_{b}(\lambda_b-\lambda_a)}\over \lambda_a}[a(x)+b(x')]\partial_{z}\nonumber\\ & + & \big({{\lambda_b-\lambda_a}\over \lambda_a}\big)^{2}a(x ) b(x')+\lambda_b^{2}\partial^{2}_{z})\big ] g({\bf x}-{\bf x'},z - z{'})\big |_{z{'}=d , z=0},\nonumber\\ & m_{22}&({\bf x},{\bf x}{'})=\big\{\big[1+{\lambda_b-\lambda_a\over \lambda_a}b(x)\big ] \big[1+{\lambda_b-\lambda_a\over \lambda_a}b(x')\big]\nonumber\\ & + & { \lambda_b ( \lambda_b-\lambda_a)\over \lambda_a } [ b(x')-b(x)]\partial_z-\lambda^2_b\partial^2_z\big\ } g({\bf x}-{\bf x'},z - z{'})\big |_{z = z{'}=d},\nonumber\\ \label{eq9}\end{aligned}\ ] ] where @xmath54 is the so - called extrapolation length and @xmath55 is the two - point correlation function of the scalar field @xmath18 in the bulk with its statistical weight given by @xmath56 where the normalizing factor @xmath57 is the bulk partition function and @xmath58 is the thermal energy . in terms of the partition function @xmath47 , the free energy is given by @xmath59 . we note that normalizing @xmath47 by @xmath57 amounts to subtracting the bulk free energy , so that the free energy @xmath60 includes only the surface free energy and the finite - size contribution . the surface free energy depends neither on @xmath3 nor on @xmath1 , so the fluctuation - induced force @xmath61 reads @xmath62 where @xmath63 is either @xmath64 or @xmath65 corresponding to lateral or normal displacements giving rise to lateral or normal forces , respectively . the matrix kernel @xmath50 is a functional of the patterning function @xmath36 on the substrates and therefore calculation of the inverse of @xmath50 is nontrivial . however , in systems with in - plane symmetries one may proceed by a lateral fourier transformation with respect to the lateral coordinates @xmath22 . in ref . @xcite , it is shown how the electrodynamic casimir force can be calculated for a periodically modulated substrate ( see also refs . @xcite ) . in this reference , the lateral periodicity is used to transform the matrix @xmath50 to a block - diagonal form in fourier space @xmath66 in which @xmath67 . similarly , also here the matrix elements of the block @xmath68 with @xmath69 , @xmath70 , are given by @xmath71 for @xmath72 , where @xmath73 is the lateral extension of the system in the @xmath6 direction and the @xmath74 are @xmath75 matrices providing the following decomposition of the matrix @xmath50 : @xmath76 however , it is interesting to note that the matrix @xmath77 can also be represented , using a more direct derivation than in ref . @xcite , in a form in which @xmath50 is diagonal @xcite . in view of the discrete lateral periodicity along the @xmath6 direction , it is suitable to express @xmath6 and @xmath78 as @xmath79 with @xmath80 and @xmath81 , @xmath82 . since @xmath36 is periodic with wavelength @xmath0 it follows that @xmath83 and @xmath84 depends on @xmath85 and @xmath86 only via the difference @xmath87 . this property and translational invariance along the @xmath88 direction imply the fourier decomposition @xmath89 where @xmath90 is diagonal . furthermore with @xmath81 , @xmath91 one can form @xmath92 as expected , the different representations of @xmath50 in terms of the matrices @xmath93 or @xmath94 ( @xmath93 is the fourier transform of @xmath95 and @xmath94 is the fourier transform of @xmath96 ) do not change the final result for the force [ eq . ( [ force ] ) ] which is given by @xcite @xmath97 here @xmath98 denotes the partial trace with respect to the indices @xmath99 of the infinite - dimensional matrix @xmath94 ( or @xmath93 ) and we have taken into account the contribution of both fluctuating components of the director field . in the following we continue with the block - diagonal form of the matrix kernel @xmath50 since in this representation the patterning functions @xmath100 and @xmath101 are somehow simpler than their counterparts in the diagonal form of @xmath50 . accordingly , the matrices @xmath102 [ eq . ( [ eq2 ] ) ] are given by @xmath103 \phi_{m}^{ba}(d)&\phi_{m}^{bb}(0 ) \end{array } \right ) + \delta_{m,0}\left ( \begin{array}{cc } { z_{a}^2\over 2p}-{\lambda_b^{2}p\over 2 } & { ( 1-\lambda_{b}p)^{2}\over 2p}e^{-pd } \\[3 mm ] \ { ( 1-\lambda_{b}p)^{2}\over 2p}e^{-pd } & { z_{b}^2\over 2p}-{\lambda_b^{2}p\over 2 } \end{array } \right ) \nonumber\\ & + & { \lambda_b-\lambda_a\over 2\lambda_a}\delta_{m,0 } \left ( \begin{array}{cc } 0 & \big[a_0\big({z_{b}^c\over p}-\lambda_b\big)+b_0\big({z_{a}^c\over p}-\lambda_b\big)\big]e^{-pd } \\[3 mm ] \big[a_0\big({z_{b}^c\over p}-\lambda_b\big)+b_0\big({z_{a}^c\over p}-\lambda_b\big)\big]e^{-pd}&0 \end{array } \right ) \end{aligned}\ ] ] for @xmath104 even , and @xmath105 b_m\big({z_{a}\over p}-\lambda_b\big)e^{-pd}+a_m\big({z_{b}\over p_m}-\lambda_b\big)e^{-p_{m}d } & ~~ b_{m}z_{b}\big ( { 1\over p}+{1\over p_m}\big ) \end{array } \right ) \ ] ] for @xmath104 odd , with @xmath106 , @xmath107 , @xmath108 , @xmath109 , @xmath110 , @xmath111 , @xmath112 , and @xmath113 where the prime at the summation sign indicates that in the sum the terms with even @xmath114 are excluded . in the limit @xmath115 we find that the contributions from the elements @xmath116 [ eqs . ( [ eqb ] ) and ( [ eq2 ] ) ] to the force decrease rapidly with increasing absolute values of @xmath117 , so that the expression for the force [ eq . ( [ tforce ] ) ] converges already at small orders of @xmath50 with @xmath118 . taking into account only the elements @xmath116 [ eq . ( [ eqb ] ) ] for @xmath119 , the asymptotic behavior of the fluctuation - induced lateral force @xmath120 in the limit @xmath121 is given by @xmath122 with the dimensionless function @xmath123 the lateral force oscillates as function of @xmath1 reflecting the underlying lateral periodic pattern and its magnitude decays exponentially as function of @xmath124 , because @xmath125 . for arbitrary values of @xmath3 , we evaluate the force in eq . ( [ tforce ] ) numerically . although the matrix @xmath94 [ eq . ( [ eqb ] ) ] is infinite - dimensional , the value of the force saturates at some finite values for @xmath99 . our numerical results for the fluctuation - induced lateral force as function of the shift @xmath1 for arbitrary strength of the contrast @xmath126 are shown in fig . the lateral force acts against the increase of the lateral displacement @xmath1 in the interval @xmath127 by being a restoring force and acts favorably with the increase of @xmath1 in the interval @xmath128 $ ] by being a pulling force . therefore the force is antisymmetric with respect to @xmath129 . upon approaching the maximum misalignment , i.e. , @xmath129 , the restoring force vanishes . this implies that the interaction free energy @xmath130 has its maximum at @xmath129 where the opposing parts of the substrates face each other and attains its minimum at @xmath131 . the force is maximal at @xmath132 and @xmath133 where @xmath134 exhibits its strongest dependence on @xmath1 ( fig . [ fig00 ] ) . in fig . [ fig3d ] we show the decay of @xmath135 as function of @xmath3 . asymptotically , @xmath136 vanishes as @xmath137 for @xmath138 , @xmath139 , and @xmath132 . we note that due to the assumption @xmath140 interchanging @xmath141 and @xmath142 leaves the system unchanged [ fig . [ fig0 ] ] . thus the force must be identical for @xmath143 . while in eqs . ( [ osci ] ) and ( [ osci1 ] ) this symmetry is explicitly valid up to the second order in @xmath144 ) , the numerical results respect this symmetry fully . this provides a very useful check of the numerical calculations . in units of @xmath145 between two periodically patterned substrates at distance @xmath3 as function of the shift @xmath1 in units of the periodicity @xmath0 for @xmath146 , and @xmath147 ( see fig . [ fig0 ] ) . the anchoring on the stripes in terms of the extrapolation lengths is taken to be @xmath138 and @xmath139 . @xmath148 is antisymmetric around @xmath129 . there is no restoring force if the misalignment is maximal , i.e. , at @xmath129 . @xmath149 means that the plates are pulled back towards the preferred alignment at @xmath131 ; for @xmath150 the plates are pulled forward towards preferred alignment at @xmath151 . ] in units of @xmath152 between two periodically patterned substrates at distance @xmath3 as function of the shift @xmath1 in units of the periodicity @xmath0 for @xmath153 , and @xmath147 and @xmath138 , @xmath139 . for all values of @xmath124 the inflection points of the potential are at @xmath132 and @xmath133 . ] in units of @xmath145 between two periodically patterned substrates as function of the film thickness @xmath3 in units of the periodicity @xmath0 . the anchoring on the stripes in terms of the extrapolation lengths is taken to be @xmath138 , @xmath139 , and the lateral shift between the patterns is @xmath132 . asymptotically , @xmath154 for @xmath132 vanishes as @xmath155 ( dashed line ) . the variation of @xmath135 for small values of @xmath124 is shown in the inset for @xmath138 ( circles ) and @xmath156 ( triangles ) . the other system parameters remain the same . as expected the absolute value of the amplitude of the lateral force increases upon increasing the contrast @xmath157 . ] endowing the substrates with the envisaged stripe patterns requires corresponding chemical patterns which in turn involve at least two different species providing the chemical contrast . these species do not only interact ( differently ) with the nematic liquid crystal in the vicinity of the substrate , giving rise to two different extrapolation lengths @xmath141 and @xmath142 , but also interact across the liquid crystal with each other via dispersion forces . the later interaction provides a lateral force as well which adds to the fluctuation - induced lateral force . note that such a lateral force due to direct interactions is the same if the two patterned substrates are separated by vacuum or by a nematic liquid crystal , as long as the _ mean _ nematic order is not affected by the stripe patterns . in order to estimate these direct interactions between the patterned substrates , we consider each substrate to be covered by a monolayer whose chemical composition varies periodically , alternating between a- and b - particles . we neglect non - additivity aspects of the dispersion forces and consider pairwise interactions between the patterned monolayers at @xmath29 and @xmath31 . since we consider @xmath3 to be large compared with the diameters of the @xmath158 and @xmath94 particles , we can disregard that particles forming the monolayers occupy discrete lattice sites . as pair potentials between the two species we take lennard - jones potentials @xmath159 , ~ i , j = a , b\ ] ] where a(b)-particles give rise to the extrapolation length @xmath160 . since in the present context @xmath161 , for the lateral force only the attractive part of the pair potentials matters this leads to the following expression for the van der waals potential energy between the two monolayers : @xmath162 ^ 3 } \label{23}\ ] ] with @xmath163\nonumber\\ + e_{bb}[1-a(x_1)][1-b(x_2;\delta)]\end{aligned}\ ] ] and @xmath164 where @xmath165 is the areal number density of @xmath166-particles in the monolayer forming the stripe @xmath160 . carrying out the integration over @xmath167 and @xmath168 in eq . ( [ 23 ] ) , one obtains @xmath169^{5/2 } } , \label{pot}\end{aligned}\ ] ] where @xmath73 is the lateral extension of the system both in the @xmath6 and @xmath88 directions . from this the lateral van der waals force @xmath170 can be calculated : @xmath171 with the scaling function @xmath172^{7/2 } } \label{eq28}\end{aligned}\ ] ] where @xmath173 . the force and the potential as function of @xmath174 are shown in fig . [ vdw ] . the comparison between figs . [ fig3 ] and [ vdw ] ( a ) reveals that the lateral van der waals force is practically constant over a wide range of shift values and varies steeply around the positions of maximum and minimum misalignment while the fluctuation - induced force varies more smoothly across all shift values . apart from that , the qualitative features are the same for both forces . ( [ eq27 ] ) and ( [ eq28 ] ) ] of the lateral force induced by direct van der waals interactions between the chemically patterned monolayers covering the two substrates at distance @xmath3 as function of the shift @xmath1 in units of the periodicity @xmath0 of the pattern for @xmath175 and @xmath176 . ( b ) corresponding lateral potential @xmath177.$ ] ] in order to estimate the lateral van der waals force we assume that the particles are closely packed within the monolayer @xcite , so that the areal number density is @xmath178^{-2}/\sqrt{3}$ ] where @xmath179 ( @xmath180 ) is the radius of the @xmath158 ( @xmath94 ) particles forming a triangular lattice . according to table i in ref . @xcite typical values are @xmath181 and @xmath182 . for @xmath183 one has @xmath184 . according to fig . [ vdw ] ( a ) for @xmath185 this implies @xmath186 for the contribution @xmath187 to the actual lateral van der waals force @xmath188 proportional to @xmath173 . thus for a suitably chosen constant @xmath189 , without compromising the goal of achieving @xmath190 ( as used in our calculations ) , the lateral van der waals force can be quite smaller than @xmath191 per area @xmath192 from fig . [ fig3 ] one finds , for the same system parameters @xmath0 and @xmath3 considered above and at room temperature , for the fluctuation - induced lateral force @xmath193 . thus the background lateral van der waals force tends to be stronger than the nematic fluctuation induced force . however , for a suitably chosen chemical contrast of the particles forming the chemical stripes , it appears to be possible to determine the fluctuation - induced lateral force by measuring the shear force once with and once without the nematic liquid between the patterned substrates . the ratio of the two forces for @xmath175 , @xmath138 , @xmath139 , and @xmath194 as function of @xmath1 is shown in fig . [ fig56 ] . it appears that the fluctuation - induced lateral force becomes more prominent around @xmath195 . from figs . [ fig3 ] and [ vdw ] ( a ) one notices that both @xmath196 and @xmath197 vanish linearly at @xmath198 [ as @xmath199 and @xmath200 at @xmath129 , respectively , for @xmath175 , @xmath201 , @xmath183 , and @xmath202 and since the slope of @xmath203 is much larger than the corresponding slope of @xmath135 , the ratio @xmath204 at @xmath205 is small . from fig . [ fig56 ] one should not draw the conclusion that the lateral fluctuation induced force is at most half a percent of the corresponding lateral van der waals background force . this ratio is inversely proportional to @xmath206 . figure [ fig56 ] corresponds to a parameter choice for which @xmath189 is estimated by an individual @xmath207 and not by the actual contrast expressed by e , which vanishes for @xmath208 . accordingly , for suitable choices of @xmath158 and @xmath94 , @xmath189 can be significantly smaller than the @xmath207 used in fig . [ fig56 ] which then leads to a significantly larger ratio . and the lateral van der waals force @xmath188 as function of the shift @xmath1 in units of the periodicity @xmath0 for @xmath175 , @xmath138 , @xmath139 , @xmath209 , and at @xmath210 . @xmath135 is more prominent around @xmath132 and @xmath133 . ] the effect of a periodic anchoring at one substrate , with the second substrate being homogeneous , on the fluctuation - induced normal force was studied in ref . it turned out that for the description of the normal force the single patterned substrate can be replaced by a uniform substrate with an effective anchoring strength , i.e. , the force is given by the force found between two uniform substrates characterized by their effective anchoring . depending on the model parameters , the normal force is either repulsive or attractive corresponding to an effective similar - dissimilar or an effective similar - similar boundary condition , respectively . in the present case of two patterned substrates , we have calculated the normal force numerically [ eq . ( [ tforce ] ) ] . figure [ fig1 ] shows the dependence of the normal force @xmath211 on @xmath3 . in units of @xmath145 between two periodically patterned substrates as function of the film thickness @xmath3 in units of the periodicity @xmath0 . the anchoring on the stripes in terms of the extrapolation lengths is taken to be @xmath139 and @xmath138 ( full line ) , @xmath212 ( dashed line ) . for @xmath213 as shown here the anchoring is weak but finite @xcite at both substrates and the force is attractive . as expected the absolute value of the amplitude of the force decreases upon decreasing the extrapolation length . the force depends very weakly on the shift @xmath1 . here @xmath174 is set to @xmath214 . ] for @xmath215 as shown here , the force is attractive and decays monotonically as function of @xmath3 . in this regime , the anchoring is weak at both substrates so that the effect of the periodicity is not visible . we note that in this case the fluctuation - induced normal force [ fig . [ fig1 ] ] is about @xmath216 times larger than the fluctuation - induced lateral force [ fig . [ fig3d ] ] for @xmath217 . figure [ fig8 ] , however , shows the behavior of @xmath218 as function of @xmath1 in the regime of strong but finite anchoring @xmath219 . in this case the normal force is oscillatory and attractive , and its magnitude is comparable with the fluctuation - induced lateral force [ fig . [ fig9 ] ] . in units of @xmath145 between two periodically patterned substrates at distance @xmath220 as function of the shift @xmath1 in units of the periodicity @xmath0 . the anchoring on the stripes in terms of the extrapolation lengths is taken to be @xmath221 and @xmath222 ( triangles ) , @xmath223 ( circles ) . for @xmath224 as shown here the anchoring is strong but finite at both substrates and the force is attractive . as expected the absolute value of the amplitude of the force decreases upon increasing the extrapolation length @xcite . the force oscillates as function of the shift @xmath1 and for complete misalignment , i.e. , @xmath129 the attraction is weakest . ] divided by fluctuation - induced lateral force @xmath148 as function of the contrast @xmath126 in units of the periodicity @xmath0 for @xmath221 , @xmath220 , corresponding to strong but finite anchoring , and @xmath132 . @xmath211 and @xmath225 are of comparable size . ] we have calculated the fluctuation - induced forces acting on two substrates chemically modulated with period @xmath0 and confining a nematic film of thickness @xmath3 ( fig . [ fig0 ] ) . the substrates are characterized by homeotropic anchoring with alternating extrapolation lengths @xmath141 and @xmath142 . we have studied the shear force as function of the lateral shift @xmath1 between the patterns on the substrates and of their separation @xmath3 . for @xmath226 and @xmath227 , the lateral force sinusoidally oscillates as a function of @xmath174 and decays exponentially with @xmath124 [ eq . ( [ osci ] ) and fig . [ fig3d ] ] . for stronger contrasts , the lateral force and its corresponding potential have been evaluated numerically [ figs . [ fig3 ] and [ fig00 ] ] . it turns out that for a suitably chosen chemical contrast the fluctuation - induced lateral force is comparable [ fig . [ fig56 ] ] with the background lateral van der waals force between the corresponding monolayers on the top and bottom substrates forming the chemical heterogeneity [ fig . [ vdw ] ] . in order to complete the picture of the forces in the presence of the patterned substrates we have also calculated numerically the fluctuation - induced normal forces [ figs . [ fig1 ] and [ fig8 ] ] and found them to be comparable with the fluctuation - induced lateral force in the case of strong anchoring [ fig . [ fig9 ] ] . patterning at small length scales gives rise to rich interfacial phenomena . surface modulations open the possibility of controlling the morphology of wetting films and generate structural phase transitions which are central to the behavior of structural forces induced by distortions of the liquid crystal order parameter . in such cases , in addition to the fluctuation - induced forces , the substrates are subject to liquid - crystalline _ elastic _ forces @xcite . such elastic forces scale with k [ eq . ( [ eqe ] ) ] and are for large @xmath3 larger than the fluctuation - induced forces which scale with @xmath228 . since for small @xmath3 the elastic forces scale as @xmath229 @xcite but the fluctuation - induced force as @xmath230 @xcite , the latter can , however , even dominate for small @xmath3 . mean field _ director contributions are completely eliminated for such model parameters and boundary conditions for which the director structure is uniform . in ref . @xcite it has been demonstrated that this uniformity can indeed occur for suitable combinations of model parameters , even in cases of competing planar and homeotropic anchoring conditions , which have not been considered here . capillary forces due to capillary condensation and the formation of bridge phases @xcite are other sources for the structural forces in the vicinity of the nematic - isotropic phase transition . if the fluid is confined to very narrow slits , patterning may give rise to capillary bridges of different liquid crystalline order . under such conditions , it would be interesting to study the stress under the shear strains by shifting the lateral substrate structures out of phase @xcite which might give rise to rather strong lateral forces . lateral forces can technologically be used to align the parallel substrate structures . for instance , for those ranges of the model parameters for which the liquid crystalline lateral forces are significant , the liquid crystal can be filled into the slit pore to align the substrate structure and then be removed . p. g. de gennes and j. prost , _ the physics of liquid crystals _ ( clarendon , oxford , 1993 ) . s. kondrat , a. poniewierski , and l. harnau , eur . j. e * 10 * , 163 ( 2003 ) . a. poniewierski and s. kondrat , j. mol . liq . * 112 * , 61 ( 2004 ) . a. ajdari , l. peliti , and j. prost , phys . lett . * 66 * , 1481 ( 1991 ) . p. zihrel , r. podgornik , and s. umer , chem . . lett . * 295 * , 99 ( 1998 ) . p. ziherl , f. karimi pour haddadan , r. podgornik , and s. umer , phys . e * 61 * , 5361 ( 2000 ) . f. karimi pour haddadan , d. w. allender , and s. umer , phys . e * 64 * , 061701 ( 2001 ) . p. ziherl and i. muevi , liq . cryst . * 28 * , 1057 ( 2001 ) . t. emig , a. hanke , r. golestanian , and m. kardar , phys . rev . a * 67 * , 022114 ( 2003 ) . r. bscher and t. emig , preprint , cond - mat/0412766 ( 2004 ) . f. karimi , f. schlesener , and s. dietrich , phys . e * 70 * , 041701 ( 2004 ) . h. li and m. kardar , phys . lett . * 67 * , 3275 ( 1991 ) . r. golestanian and m. kardar , phys . a * 58 * , 1713 ( 1998 ) . t. emig , europhys . 62 * , 466 ( 2003 ) . r. bscher and t. emig , phys . a * 69 * , 062101 ( 2004 ) . a. hanke , private communication . p. m. chaikin and t. c. lubensky , _ principles of condensed matter physics _ ( cambridge university press , cambridge , 1995 ) . t. getta and s. dietrich , phys . e * 47 * , 1856 ( 1993 ) . s. kondrat , a. poniewierski , and l. harnau , liq . cryst . * 32 * , 95 ( 2005 ) . k. koevar , a. bortnik , i. muevic , and s. umer , phys . . lett . * 86 * , 5914 , ( 2001 ) . d. andrienko , p. patrcio , and o. i. vinogradova , j. chem . phys . * 121 * , 4414 ( 2004 ) . h. stark , j - i . fukuda , and h. yokoyama , phys . * 92 * , 205502 ( 2004 ) . h. bock and m. schoen , j. phys : condens . matter * 12 * , 1545 ( 2000 ) .
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we consider a nematic liquid crystal confined by two parallel flat substrates whose anchoring conditions vary periodically in one lateral direction . within the gaussian approximation
, we study the effective forces between the patterned substrates induced by the thermal fluctuations of the nematic director . the shear force oscillates as function of the lateral shift between the patterns on the lower and the upper substrates .
we compare the strength of this fluctuation - induced lateral force with the lateral van der waals force arising from chemically structured adsorbed monolayers .
the fluctuation - induced force in normal direction is either repulsive or attractive , depending on the model parameters .
[ present address : ] institute for studies in theoretical physics and mathematics ( ipm ) , school of physics , po box 19395 - 5531 , tehran , iran
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understanding the source of quantum advantage in quantum computation is a long - standing issue in quantum information science . previous researches have shown that certain quantum computation is ` classical ' , for the reason that it is efficiently simulateable by classical computers . one example is any computation performed just by local operations and classical communication ( locc ) @xcite without using any entangled resources . all models of quantum computation outperforming classical counterparts use entanglement resources ( such as measurement - based quantum computation @xcite ) or some kind of non - locc operation . non - locc operations are called ` global ' operations . the source of quantum speedup must be due to the properties of the global operations . in this paper , we refer to the properties exclusive to global operations as _ globalness _ of quantum operations . it is also known that not all global operations result in quantum speedup for quantum computation . there must be a specific globalness that differentiates the quantum operations leading to quantum speedup from those do not . the difference may be due to more than one kind of globalness , but even this is not clear at this point . for this reason , having a good understanding of the globalness of quantum operations is important . in this paper , we try to understand the simplest case of the global operations , namely , bipartite unitary operations . to investigate globalness of unitary operations , it is important to clarify what kind of states is given as inputs of the unitary operations . we want to evaluate the globalness that does not depend on a choice of a particular input state . by introducing the concept of _ pieces of quantum information _ , we analyze characterizations of unitary operations for two pieces of quantum information represented by arbitrary unknown states , in terms of _ delocalization power _ @xcite and _ entanglement cost _ @xcite . we compare these characterizations with another characterization , _ entangling power _ of global operations @xcite , which characterizes the globalness of unitary operations acting on a set of known states . then we extend our analysis of globalness in terms of the delocalization power in two ways by introducing new locc tasks . one of the tasks is _ locc one - piece relocalization _ for _ one piece _ of delocalized quantum information that corresponds to the case when a part of input state is unknown and arbitrary but the other part can be chosen from a set of known state . the other task is _ locc one - piece relocation _ for two pieces of delocalized quantum information , which evaluates the ability of the unitary operation to relocate one of the two pieces of quantum information from one hilbert space to another by adding locc to the unitary operation . the rest of the paper is organized as following . in section [ overview ] , we introduce the concept of pieces of quantum information and present an overview on the three characterizations . we summarize the comparison of different aspects of the globalness of bipartite unitary operations presented in the previous works in section [ comparison ] . we extend the analysis of the delocalization power in sections [ fixed_input ] and [ relocation ] . in section [ fixed_input ] , we show the result on locc one - piece relocalization for one piece of delocalized quantum information . in section [ relocation ] , we analyze locc one - piece relocation of two pieces of quantum information . finally , in section [ conclusion ] , we present our conclusion . first , we define _ a piece of quantum information _ for a @xmath0-dimensional quantum system , or _ qudit _ , whose hilbert space is denoted by @xmath1 . if a pure quantum state of @xmath2 qudits @xmath3 is given by @xmath4 where @xmath5 is a fixed set of normalized and mutually orthogonal states in @xmath6 and the coefficients @xmath7 are arbitrary and unknown except for the normalization @xmath8 , the unknown state @xmath9 is said to represent _ one piece _ of quantum information for a qudit . in the formalism presented above , a piece of quantum information for a single qudit can be stored in an @xmath2-qudit system using an arbitrary set of orthonormal states , @xmath5 . any such set of states would form a _ logical _ qudit space , but in a special case satisfying @xmath10 for all @xmath11 , where the set of states @xmath12 forms an orthonormal basis of @xmath13 and @xmath14 is independent of @xmath15 , the piece of quantum information is stored in a _ physical _ qudit . hence it is possible to _ assign _ one physical qudit for each piece of quantum information . using this formalism , now we provide the formal definition of one piece of _ localized _ quantum information for a qudit . we label the qudits of an @xmath2-qudit system from @xmath16 to @xmath17 and denote the hilbert space of qudit @xmath18 by @xmath19 . the hilbert space of @xmath17 qudits _ excluding _ a certain qudit @xmath18 will be denoted by @xmath20 . we will also assume that two different pieces of quantum information in the same system are assigned to different physical qudits . for @xmath21 , a piece of quantum information represented by an unknown @xmath2-qudit state @xmath9 is said to be _ localized _ at _ an assigned _ hilbert space @xmath19 , or simply localized when there is no fear of confusion , if it is represented in the form @xmath22 where @xmath23 is any basis of the hilbert space of the assigned qudit ( _ i.e. _ , @xmath19 ) , @xmath24 is an @xmath25-qudit state determined independently of the set of coefficients @xmath26 , and @xmath26 are arbitrary coefficients satisfying the normalization condition @xmath8 . note that the global phase factor of the coefficients is not a physical quantity , so we take the global phase equivalence . there are @xmath27 complex degrees of freedom in total for one piece of quantum information . for @xmath28 , since @xmath29 is the minimal hilbert space to store one piece of quantum information for a qudit , one piece of quantum information has to be localized in @xmath30 . we define the concept of _ delocalized _ quantum information , which is the complement of localized quantum information , and also the concept of _ delocalization _ of quantum information . if a piece of quantum information is _ not _ localized , then it is said to be _ delocalized_. the task of delocalizing quantum information is called _ delocalization_. next , we consider two - qudit states , where the state of each qudit represents one piece of localized quantum information . we denote the two hilbert spaces of the qudits by @xmath31 and @xmath32 . the two pieces of localized quantum information can be represented by a tensor product state @xmath33 , where the subscripts of the kets denote the assignment of the hilbert spaces of the qudits , @xmath34 and @xmath35 . we define delocalization for two pieces of quantum information as the following . two pieces of quantum information are said to be _ delocalized _ , if the state representing the two pieces of quantum information _ can not _ be written by @xmath36 , where @xmath37 on @xmath38 and @xmath39 on @xmath40 are arbitrary local unitary operations but independent of @xmath41 and @xmath42 . we again note that we have already assigned a hilbert space for each piece of quantum information , so the state @xmath43 represents delocalized two pieces of quantum information out from the assigned hilbert spaces . now we investigate the effects of a global unitary operation @xmath44 applied on two pieces of localized quantum information @xmath33 . if the unitary operation @xmath44 is not a tensor product of two local unitary operations on @xmath38 and @xmath40 , @xmath44 always transforms each piece of localized quantum information to delocalized quantum information . in this paper , we say that the unitary operations have _ delocalization power _ , which in a sense is the ` strength ' of delocalization of quantum information due to the unitary operations . how pieces of quantum information are delocalized is determined only by the set of orthonormal states representing the quantum information , which in turn is determined by the unitary operation used for the delocalization . therefore , the globalness of unitary operations can be studied by understanding how a unitary operation delocalizes pieces of quantum information . later , we argue that certain pieces of quantum information are ` more ' delocalized than others . the difference in the level of delocalization can only have come from the difference in the globalness of the unitary operation , namely , the delocalization power . hence , we can classify the delocalization power by analyzing the level of the delocalization that each unitary operation achieves . to define and classify the level of delocalization , we introduce the following locc task , _ locc one - piece relocalization _ , that aims to localize just one of the two pieces of delocalized quantum information by sacrificing the other piece of quantum information in @xmath45 . we denote the set of density operators on the hilbert space @xmath13 by @xmath46 . _ locc one - piece relocalization _ of qudit @xmath47 for two pieces of quantum information delocalized by a global unitary operation @xmath44 is a task to find an locc - implementable completely positive trace preserving ( cptp ) map @xmath48 satisfying @xmath49 = { { \left \vert}\psi_b { \right \rangle}}_b { { \left \langle}\psi_b { \right \vert}}\ ] ] for any @xmath34 and @xmath35 . we characterize the delocalization power of global unitary operations in terms of their ability to allow locc one - piece relocalization on two pieces of quantum information delocalized by the global unitary operations . we define the order of the delocalization power of two global unitary operations @xmath44 and @xmath50 on two pieces of quantum information by the following . if locc one - piece relocalization of two pieces of delocalized quantum information is possible for a unitary operation @xmath44 , but not possible for another unitary operation @xmath50 , the order of the delocalization power of @xmath44 is defined to be smaller than that of @xmath50 in terms of locc one - piece relocalization . another way to quantify the globalness of a unitary operation applied on quantum information is to evaluate how much extra global resource is required on top of locc operations to implement the unitary operation on two pieces of quantum information . the minimum amount of entanglement required to deterministically implement a given global operation is unique , based on the fact that entanglement can not be generated by locc . we define an locc task , _ entanglement assisted deterministic locc implementation _ and then define _ entanglement cost _ of the unitary operation on quantum information in terms of this locc task . _ entanglement assisted deterministic locc implementation _ of a global unitary operation @xmath44 on two pieces of localized quantum information @xmath51 using a fixed bipartite resource state @xmath52 is a task of finding an locc - implementable cptp map @xmath53 satisfying @xmath54 for any @xmath55 and @xmath56 . _ entanglement cost _ of a unitary operation @xmath44 on two pieces of quantum information is given by the minimum amount of entanglement of the resource state @xmath57 required to perform entanglement assisted deterministic locc implementation of @xmath44 on two pieces of localized quantum information . the entanglement cost of unitary operations can be regarded as the minimum entanglement cost for delocalizing two pieces of quantum information . thus this is another way to characterize the globalness of unitary operations applied on quantum information . we can also quantify the globalness of a unitary operation by evaluating its ability of entanglement generation in place of entanglement cost , similarly to the pair of the definitions for evaluating the globalness of quantum states , namely , distillable entanglement and entanglement cost . however , the amount of entanglement generated by a unitary operation strongly depends on the choice of input states , therefore it is difficult to define a quantity in terms of quantum information , namely , unknown states . instead , _ entangling power _ of a global operation @xcite is defined by optimizing the generated amount of entanglement over a set of known input states . the entangling power of a bipartite unitary operation @xmath44 ( denoted by @xmath58 ) is defined as the maximum amount of entanglement generated between the bipartite cut by applying @xmath44 on a known state , _ i.e. _ , @xmath59 where @xmath60 is an entanglement measure of choice and @xmath61 is chosen among the given set of states . in the previous section , we introduced three different characterizations for the globalness of bipartite unitary operations : delocalization power , entanglement cost and entangling power . in this section , we summarize known results on the three characterizations and investigate whether the globalness characterized by each method is same to or different from that of by the others . [ theorem : qq1pr ] locc one - piece relocalization for two pieces of quantum information of qudits delocalized by a unitary operation @xmath44 is possible if and only if @xmath44 is a locally unitary equivalent to a controlled - unitary operation @xmath62 , where @xmath63 forms an orthonormal basis for one of the subsystems and @xmath64 is a set of unitary operations on the other subsystem . @xcite the characterization of globalness based on the delocalization power reveals that there are two classes of globalness for bipartite unitary operations , one class is a local unitary equivalent of a controlled - unitary operations , and the other class is all the rest of global unitary operations . for any given two - qubit controlled - unitary operation , its entanglement cost for entanglement assisted deterministic locc implementation is 1 ebit when the schmidt number , the number of non - zero schmidt coefficients , of the resource state is 2 . @xcite for other two - qubit unitary operations , the entanglement cost of the swap operation @xmath65 , of which action is given by @xmath66 for any @xmath67 and @xmath68 , is easily shown to be 2 ebit by considering the situation where the two input qubits are parts of maximally entangled states . however , for more general operations , it is not easy to evaluate the minimum entanglement cost of entanglement assisted locc implementation , therefore , it is still unknown . the formulation of entangling power depends on the set of allowed input states and the measure of entanglement . entangling power is usually difficult to calculate because it involves two optimizations . one is the maximization over all possible input states ( usually taken to be separable or product states ) . the other is the calculation of the amount of generated entanglement according to the chosen entanglement measure . even when the quantum operation is restricted to bipartite unitary operations , the exact value is obtained for only limited cases @xcite . for example , it is known that the cnot operation @xmath69 ( @xmath70 where @xmath71 and @xmath72 is given by the pauli matrix @xmath73 ) has the entangling power of 1 ebit and the swap operation @xmath65 has the entangling power of 2 ebit when we allow to use ancilla qubits . nevertheless , we can make a relatively generic statement about entangling power if the entanglement measure is continuous . the statement is as follows . the identity operation clearly generates no entanglement at all , hence its entangling power should be zero . invoking a continuity argument , there should be a set of operations in the neighborhood of the identity operation such that their entangling power is arbitrarily small . however , when we evaluate the globalness in terms of the delocalization power and entanglement cost , a fundamental difference arrises . by using these two characterizations , all two - qubit controlled - unitary operations of the form @xmath74 where @xmath75 is a single qubit unitary operation , belong to the same class of globalness irrelevant to their entangling power . thus , two - qubit controlled - unitary operations and their local unitary equivalent operations belong to a distinct class from the the class of identity operation , even when the local unitary operations are close to the identity ( @xmath76 ) , in contrast to the characterization in terms of entangling power . on the other hand , for general _ known _ bipartite pure qudit states , the locc convertibility condition between two states is known . [ majorizationtheorem ] a bipartite state @xmath77 can be transformed to another state @xmath78 by using only locc , if and only if the schmidt coefficients of the @xmath77 is majorized by those of @xmath78 @xcite . since the locc conversion protocol depends on the choice of states @xmath77 and @xmath78 ( when locc conversion from @xmath77 to @xmath78 is possible ) , it is essential that these states are known . by taking @xmath79 and @xmath80 and using theorem [ majorizationtheorem ] , we can see that if we have a resource state of which schmidt coefficients are equal to the those of @xmath78 , it is possible to obtain @xmath81 by locc . for two - qubit unitary operations , which can only create an entangled state with schmidt number 2 , the majorization condition is equivalent to the comparison of the amount of entanglement . thus , we have the following corollary . the entanglement cost of the resource state for entanglement assisted deterministic locc implementation of unitary operations on a _ given known _ state @xmath33 is given by the the amount of entanglement of @xmath82 . this corollary gives a justification to define the entanglement cost for an entanglement assisted deterministic locc implementation on a set of known states by finding the largest minimum entanglement cost to perform @xmath44 over the set of input states . in this case , the entanglement cost for a set of known states coincides to the entangling power of @xmath44 . the results in this section indicate that there are several aspects of globalness in quantum operations . it is particularly important to clarify the types of input states , known states or unknown states representing pieces of quantum information , for analyzing globalness of unitary operations , since they lead to a fundamental difference . in the classification of globalness of unitary operations in terms of the delocalization power presented in the previous sections , we analyzed global properties of two pieces of delocalized quantum information . that is , we analyzed the globalness of unitary operations totally independent of input states . on the other hand , in section [ delocalizationpwer ] , we also defined _ one piece _ of delocalized quantum information . this situation corresponds to the case where one of the two input qudits is in an arbitrary and unknown state , but that of the other qudit is in a _ known _ state , and we can choose the most suitable state for performing tasks . in this section , we extend our analysis on globalness of unitary operations in terms of delocalization power to the case for one piece of quantum information . we define the task of locc one - piece relocalization of _ one piece _ of delocalized quantum information . _ locc one - piece relocalization _ of the qudit @xmath47 for _ one piece _ of quantum information delocalized by a global unitary operation @xmath44 is a task to find an locc - implementable cptp map @xmath48 and a state @xmath83 satisfying @xmath84 = { { \left \vert}\psi_b { \right \rangle}}_b { { \left \langle}\psi_b { \right \vert}}\ ] ] for any @xmath35 . we show the following lemma . the global unitary operations that allow locc one - piece relocalization for one piece of delocalized quantum information is in a strictly wider class of global unitary operations than that allows locc one - piece relocalization for two pieces of delocalized quantum information . to prove this lemma , we present an example of two - qubit unitary operations , @xmath85 , where locc one - piece relocalization is impossible for delocalized two pieces of quantum information , but it becomes possible if one of the qubits is promised to be in a particular pure state . let us take the computational basis , which is an orthonormal basis of the composite hilbert space @xmath45 given by @xmath86 , where @xmath87 and @xmath88 are orthonormal base for @xmath38 and @xmath40 , respectively . the matrix representation of @xmath89 in the computational basis is given by @xmath90 first , we show that locc one - piece relocalization of the qubit @xmath47 for one piece of quantum information delocalized by @xmath85 is possible by presenting that @xmath85 can be simulated by a locally unitary equivalent operation to a controlled - unitary operation if qubit @xmath91 is set to a particular state . we set the state @xmath92 to be @xmath93 . it is easy to check that for an arbitrary @xmath42 , @xmath94 where @xmath69 is a controlled - not operation and @xmath95 denotes the single - qubit hadamard operation represented in the computational basis by @xmath96 ( the same calculation can be done using the stabilizer formalism @xcite by exploiting the fact that @xmath89 is a clifford operation . ) thus the action of @xmath85 can be simulated by @xmath97 , a locally unitary equivalent operation to the controlled - not operation , when we fix one of the qubits to be in the state @xmath98 . from theorem [ theorem : qq1pr ] , any operation which is locally unitary equivalent to a controlled - unitary operation is locc one - piece relocalizable for two pieces of delocalized quantum information . note that , if an locc protocol relocalizes two pieces of delocalized quantum information , the same protocol must also relocalize one piece of delocalized quantum information . therefore , @xmath85 is locc one - piece relocalizable for one piece of delocalized quantum information . next , we show that @xmath89 itself is not locally unitary equivalent to controlled - unitary operations , therefore it is _ not _ locc one - piece relocalizable for two pieces of delocalized quantum information . to show this , we analyze the cartan numbers for two - qubit unitary operations . it is known that any two - qubit unitary operator on @xmath45 has the following cartan decomposition @xcite , @xmath99 \cdot v_a \otimes v_b,\ ] ] by taking appropriate local unitary operations @xmath37 , @xmath100 on @xmath38 and @xmath39 , @xmath101 on @xmath40 , coefficients @xmath102 ( @xmath103 ) , where @xmath73 , @xmath104 and @xmath105 denote the pauli matrices and each subscript of the pauli matrix indicates the corresponding hilbert space . in this decomposition , the nonlocal component of the unitary operation is represented by the set of coefficients @xmath106 . in this paper , we refer @xmath107 to be a cartan coefficient , and the number of non - zero cartan coefficients to be the cartan number . the cartan number of a unitary operation can not be changed by local unitary operations , and two unitary operations with different cartan numbers can not be locally unitary equivalent to each other @xcite . the cartan decomposition of @xmath89 is given by @xmath108 \cdot v_a \otimes v_b\ ] ] by using appropriate local unitary operators @xmath37 , @xmath100 , @xmath39 and @xmath101 @xcite . thus the cartan number of @xmath89 is 2 . on the other hand , the cartan decomposition of a controlled phase operation @xmath109 where the single - qubit phase operation @xmath110 is defined by @xmath111 is given by @xmath112.\ ] ] thus , the cartan number of the controlled - phase operation is 1 . it is also known that any controlled - unitary operations @xmath113 is locally unitary equivalent to @xmath114 , therefore , the cartan number of the operations that are locally unitary equivalent to controlled - unitary operations is also 1 . therefore , @xmath89 can not be locally unitary equivalent to the controlled - unitary operations . note that some unitary operations remain locc one - piece _ _ un__relocalizable even for one piece of delocalized quantum information . such an example is the swap operation @xmath115 . by performing @xmath115 , even if one of the input qubit states is fixed to a particular known state , a piece of quantum information represented by the other qubit s unknown input state completely moves out from the original hilbert space and stored in the other hilbert space . this phenomena is an example of what we call a _ relocation _ of a piece of quantum information . once this relocation happens , it is not possible to relocalize the piece of quantum information back to the original hilbert space by locc alone . it requires 1 ebit of entanglement to relocalize the one piece of relocated quantum information on top of locc by using quantum teleportation @xcite . in the previous section , we briefly introduced the concept of _ relocation _ of a piece of quantum information . but actually , @xmath115 provides relocation of both two pieces of quantum information . @xmath115 is the only unitary operation that has the delocalization power strong enough to relocate two pieces of quantum information simultaneously without any additional operation or resource . a wider class of unitary operations , namely , the local unitary equivalents of @xmath115 , also relocates two pieces of quantum information , if local operations are allowed as an extra operation . to define and classify delocalization power of unitary operations in terms of relocation , we further relax the condition of the additional operations to locc and investigate locc relocatability of one of the two pieces of quantum information delocalized by unitary operations . _ locc one - piece relocation _ from @xmath38 to @xmath40 for two pieces of quantum information delocalized by a unitary operation @xmath44 is the task to find an locc - implementable cptp map @xmath116 satisfying @xmath117 = { { \left \vert}\psi_a { \right \rangle}}_b { { \left \langle}\psi_a { \right \vert}}\ ] ] for any @xmath118 and @xmath119 . this is a task similar to locc one - piece relocalization for two pieces of quantum information , in the sense by sacrificing one of two pieces of quantum information , we obtain one piece of localized quantum information . the difference between these tasks is the location of the piece of localized quantum information . we define the order of the delocalization power of two global unitary operations @xmath44 and @xmath50 on two pieces of quantum information in terms of locc one - piece relocation by the following . if locc one - piece relocation of two pieces of delocalized quantum information is possible for a unitary operation @xmath44 , but not possible for another unitary operation @xmath50 , the order of the delocalization power of @xmath44 is defined to be larger than that of @xmath50 in terms of locc one - piece relocation . note that for locc one - piece relocation , feasibility of the task implies more delocalization power , whereas for locc one - piece relocalization , feasibility of the task implies less delocalization power . as the first step to classify the delocalization power of global unitary operations in terms of locc one - piece relocation , we show that where the locally unitary equivalent class of controlled - unitary operations lies in this classification . if two pieces of quantum information are delocalized by an operation locally unitary equivalent to controlled - unitary operations , locc one - piece relocation is not possible . to prove this lemma , we employ the formulation of locc using accumulated operators @xcite . in the following subsections , we first summarize the formulation , and then show the proof by contradiction . we adopt the standard formulation of locc @xcite . in a two - party scenario , alice and bob perform one local measurement operation in turns while exchanging the outcome of each measurement operation by classical communication . the measurement operation at a particular turn is chosen according to all the outcomes by the both parties up to that turn , where the choice is made following a protocol agreed beforehand by the parties . strictly speaking , we may consider locc protocols which can not be expressed in this form . these protocols , however , can always be substituted by the protocols in this standard form . each local quantum operation can be described as a generalized measurement , which is represented by a set of operators @xmath120 satisfying the completeness relation @xmath121 there exists one operator for each outcome in the measurement , which is denoted by the superscript @xmath122 . we add a subscript to the outcome index , for example @xmath123 , to specify to which measurement operation the index belongs . in this notation , @xmath123 belongs to the @xmath18-th measurement operation in the sequence . we use @xmath124 to denote the set of measurement outcomes of the first @xmath18 measurement operations in the sequence . the @xmath125-th measurement operation is a function of @xmath126 and we denote the set of operators describing this measurement operation by @xmath127 let us denote alice s measurement operations by @xmath128 and bob s by @xmath129 . we set @xmath130 and @xmath131 note that @xmath128 is an operator on @xmath38 and @xmath129 is on @xmath40 . when @xmath2-th turn is alice s turn then @xmath132-th turn is bob s turn , which implies that alice does not perform any operation during this @xmath132-th turn . in this case , we set alice s measurement operation to the identity operation , _ i.e. _ , @xmath133 if this @xmath132-th turn happens to be bob s , then his measurement operation is set to the identity operation . the effect of the measurement operations accumulates as an locc protocol proceeds . the accumulated effect up to a particular turn is expressed by the product of all the measurement operators corresponding to all the measurement outcomes obtained up to that point . given a particular sequence of measurement outcomes @xmath134 , we represent the accumulated effect corresponding to this sequence by an _ accumulated operator _ @xmath135 defined by @xmath136 bob s accumulated operator will be denoted by @xmath137 defined by a similar way of @xmath135 . let us focus on two - qubit controlled - unitary operations for simplicity . the following argument can be extended to arbitrary two-_qudit _ controlled - unitary operations . we prove by contradiction that locc one - piece relocation of two pieces of quantum information delocalized by any controlled - unitary operation is impossible . now , consider the following scenario where alice has an extra ancilla qubit , whose hilbert space is denoted by @xmath138 . let @xmath139 denote a maximally entangled state between alice s input qubit and the ancilla qubit defined by @xmath140 suppose that there is an locc one - piece relocation protocol for the given controlled - unitary operation @xmath113 defined by eq . ( [ c - u ] ) . let alice set her input qubit and the ancilla in the state of @xmath139 while bob s input remains arbitrary . alice and bob perform @xmath113 and the locc protocol to complete the relocation of one piece of quantum information from @xmath38 to @xmath40 . then alice s ancilla qubit and bob s input qubit are in the state of @xmath141 which implies that @xmath142 holds for an arbitrary @xmath143 . using the accumulated operator representation of @xmath144 , we have @xmath145 = { { \left \vert}\phi { \right \rangle}}_{ab}{{\left \langle}\phi { \right \vert}}.\ ] ] we modify the locc protocol @xmath144 by adding an extra measurement operation by alice described by @xmath146 , just after alice s final measurement . we denote this modified protocol by @xmath147 . direct substitution reveals that @xmath148\\ + { \mathrm{tr}}_a [ \sum_{{\vec{r}_{n } } } ( { { \left \vert}0 { \right \rangle}}_a{{\left \langle}1 { \right \vert}}a^{{\vec{r}_{n } } } \otimes b^{{\vec{r}_{n } } } ) ( c_u { { \left \vert}\phi { \right \rangle}}_{aa}{{\left \langle}\phi { \right \vert } } \otimes { { \left \vert}\psi_b { \right \rangle}}{{\left \langle}\psi_b { \right \vert } } c_u^\dag ) ( { { \left \vert}0 { \right \rangle}}_a{{\left \langle}1 { \right \vert}}a^{{\vec{r}_{n } } } \otimes b^{{\vec{r}_{n}}})^\dag ] . \end{gathered}\ ] ] since the partial trace @xmath149 is taken and the additional measurement introduced for the protocol @xmath147 acts only on @xmath38 , we have @xmath150 thus we obtain @xmath151 because the right hand side is a pure state , it must be true that @xmath152\\ = p^{{\vec{r}_{n}},k,\psi_b } { { \left \vert}\phi { \right \rangle}}_{ab}{{\left \langle}\phi { \right \vert}}\end{gathered}\ ] ] for all @xmath134 and @xmath153 , where @xmath154 is a positive coefficient normalized by @xmath155 since eq . ( [ relocation_perbranch ] ) holds for any @xmath143 , we can replace @xmath68 by a completely mixed state @xmath156 , and obtain @xmath157\\ = \frac{1}{2 } { \mathrm{tr}}_a [ ( { { \left \vert}0 { \right \rangle}}_a{{\left \langle}k { \right \vert}}a^{{\vec{r}_{n } } } \otimes b^{{\vec{r}_{n } } } ) ( c_u { { \left \vert}\phi { \right \rangle}}_{aa}{{\left \langle}\phi { \right \vert } } \otimes { { \left \vert}0 { \right \rangle}}{{\left \langle}0 { \right \vert } } c_u^\dag ) ( { { \left \vert}0 { \right \rangle}}_a{{\left \langle}k { \right \vert}}a^{{\vec{r}_{n } } } \otimes b^{{\vec{r}_{n}}})^\dag]\\ \qquad + \frac{1}{2 } { \mathrm{tr}}_a [ ( { { \left \vert}0 { \right \rangle}}_a{{\left \langle}k { \right \vert}}a^{{\vec{r}_{n } } } \otimes b^{{\vec{r}_{n } } } ) ( c_u { { \left \vert}\phi { \right \rangle}}_{aa}{{\left \langle}\phi { \right \vert } } \otimes { { \left \vert}1 { \right \rangle}}{{\left \langle}1 { \right \vert } } c_u^\dag ) ( { { \left \vert}0 { \right \rangle}}_a{{\left \langle}k { \right \vert}}a^{{\vec{r}_{n } } } \otimes b^{{\vec{r}_{n}}})^\dag]\\ = ( p^{{\vec{r}_{n}},k,0 } + p^{{\vec{r}_{n}},k,1 } ) { { \left \vert}\phi { \right \rangle}}_{ab}{{\left \langle}\phi { \right \vert}}.\end{gathered}\ ] ] note that @xmath158 acts only on alice s input qubit . taking the partial trace over alice s _ ancilla _ qubit @xmath159 , eq . ( [ relocation_oni ] ) gives @xmath160 = \frac{(p^{{\vec{r}_{n}},k,0 } + p^{{\vec{r}_{n}},k,1})}{2 } { \mathbb{i}},\ ] ] where we have used the relation @xmath161 noting that the identity operator commutes with any unitary operators , after performing the partial trace @xmath149 , we have @xmath162 this equation guarantees that bob s accumulated operator @xmath137 for each sequence of measurement outcomes @xmath134 is proportional to a unitary operator , _ i.e. _ , @xmath163 where the coefficient @xmath164 is set to satisfy @xmath165 for any linear operator @xmath166 on @xmath38 and the maximally entangled state given by @xmath139 , @xmath167 , where @xmath168 denotes the transpose of @xmath166 in the computational basis , holds . let @xmath169 denote a set of operators forming a basis for the operators on @xmath170 ( where @xmath171 ) . that is , for any @xmath166 on @xmath170 , there exists a set of complex numbers @xmath172 such that @xmath173 ( an example of such a basis is the set of pauli operators and the identity operator , if the hilbert space in question has the dimension of 2 . ) with this basis , @xmath113 on @xmath174 can be expressed as a linear combination of @xmath175 , namely , @xmath176 where @xmath177 denotes the coefficient of @xmath175 . let us choose @xmath178 to satisfy @xmath179 in the computational basis and define @xmath180 on @xmath181 by @xmath182 under these conventions , we have @xmath183 comparing this equation to eq . ( [ relocation_perbranch ] ) , it must be that @xmath184 which is equivalent to @xmath185 let an ancilla state ( not necessarily normalized ) @xmath186 be defined by @xmath187 by substituting eq . ( [ bisunitary ] ) into eq . ( [ relocation_vectorform ] ) , we conclude that @xmath188 holds for all @xmath68 . the right hand side is collinear to @xmath189 for all @xmath68 . on the other hand , because @xmath190 is invertible , the left hand side returns linearly independent vectors when @xmath191 are chosen linearly independently . this , however , is a contradiction proving that the assumption that locc one - piece relocalization is possible for two pieces of quantum information delocalized by the controlled - unitary operations @xmath113 must not hold . this proof strongly depends on the fact that bob s input state is kept arbitrary , namely , we considered the situation of delocalized two pieces of quantum information . indeed , if we are allowed to choose bob s input state , one - piece relocation is possible for certain controlled - unitary operations . an example is the controlled - not operation on two qubits . in this paper , we first introduced the concept of pieces of quantum information and reviewed three different characterizations of the globalness of bipartite unitary operations , which were delocalization power , entanglement cost , and entangling power . the first two characterizations are on the globalness of the unitary operations on two pieces of quantum information represented by unknown states , and the last one is on the globalness of the unitary operations on a set of known states . we showed the fundamental difference between these two types of globalness of the unitary operations . next , we extended our analysis on characterization in terms of the delocalization power by introducing a new locc task , locc one - piece relocalization of _ one piece _ of quantum information delocalized by a unitary operation . we showed that there are unitary operations which belong to a higher globalness class in terms of the delocalization power than the local unitary equivalents of controlled - unitary operations , and such operations can be further divided into two subclasses depending on the possibility of this task . we also introduced another new task , locc one - piece relocation of two pieces of delocalized quantum information . we proved that locc one - piece relocation is impossible for any controlled - unitary operations . this confirms that the local unitary equivalents of controlled - unitary operations , which are locc one - piece relocalizable , belong to a class of global operations with relatively weak globalness also in terms of locc relocation of quantum information . in our analysis of the locc tasks , we focused on the locc tasks that transform two pieces of quantum information within the two - qudit hilbert space . this is because our main purpose is to investigate the delocalization power of two - qudit unitary operations . but in general , we can investigate more general properties of delocalized pieces of quantum information by considering locc tasks that transforms @xmath2 pieces of quantum information delocalized in an @xmath192-qudit subspace of a totally @xmath193-qudit hilbert space ( @xmath194 and @xmath195 ) to @xmath196 pieces of quantum information delocalized in an @xmath197-qudit subspace ( @xmath198 and @xmath199 ) , by limiting the allowed operations to be locc for a certain devision of the total hilbert space . self - teleportation @xcite can be interpreted as a special case of this generalized locc task for @xmath200 , @xmath201 and @xmath202 . by denoting the total hilbert space by @xmath203 , only locc is allowed between the division of @xmath38 and @xmath204 in this case . it is shown that asymptotically , @xmath18-copies of any delocalized two pieces of quantum information in @xmath45 can be approximately ` relocated ' to @xmath204 . the error probability of this relocation drops exponentially with the number of copies @xmath18 as long as the two pieces of quantum information are delocalized , namely , not in a product state . in our analysis of the delocalization power , we characterized the order of delocalization power of unitary operations by their ability allowing the locc tasks . for more quantitative analysis of the unitary operations that do not allow the locc tasks , it is important to analyze the entanglement cost of the corresponding entanglement assisted versions of the locc relocalization / relocation tasks . quantum state merging @xcite can be interpreted as evaluating the entanglement cost for performing entanglement assisted approximate generalized locc task for @xmath205 , @xmath206 and @xmath207 ( @xmath208 ) , where only locc is allowed between the division of @xmath38 and @xmath204 and no operation is allowed on @xmath209 . this is an entanglement assisted locc task to achieve relocation of three pieces of quantum information delocalized in @xmath210 to @xmath211 . in the asymptotic limit , it is shown that the entanglement cost coincides with the quantum conditional entropy , which provides an operational interpretation of the quantum conditional entropy . to understand information theoretical meanings of our locc tasks , it is interesting to analyze asymptotic settings of our locc and entanglement assisted locc tasks . we leave these investigations for future works . bennett , c. h. , brassard , g. , crpeau , c. , jozsa , r. and peres , w.k . ( 1993 ) teleporting an unknown quantum state via dual classical and einstein - podolsky - rosen channels . lett . _ * 70 * , 18951899 . nielsen , m. a. , dawson , c.m . , dodd , j. l. , gilchrist , a. , mortimer , d. , osborne , t.j . , bremner , m. j. , harrow , a. w. and hines , a. ( 2003 ) quantum dynamics as a physical resource . a _ * 67 * , 052301 .
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we compare three different characterizations of the globalness of bipartite unitary operations , namely , delocalization power , entanglement cost , and entangling power , to investigate global properties of unitary operations .
we show that the globalness of the same unitary operation depends on whether input states are given by unknown states representing pieces of quantum information , or a set of known states for the characterization .
we extend our analysis on the delocalization power in two ways .
first , we show that the delocalization power differs whether the global operation is applied on one piece or two pieces of quantum information .
second , by introducing a new task called locc one - piece relocation , we prove that the controlled - unitary operations do not have the delocalization power strong enough to relocate one of two pieces of quantum information by adding locc .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
quantum pumping is a transport phenomenon originally proposed by thouless@xcite and first realized by switkes _ et al_.@xcite it proposed that directed current can be induced by time - dependent modulation of external and internal parameters without bias in a quantum phase coherent nanoscale conductor . theoretical and experimental research of quantum pumping has become a very important and active direction in mesoscopic physics . it is also significant in the field of quantum dynamic theory . by scale of the modulation frequency the quantum pump can be categorized into the adiabatic and nonadiabatic ones , with the former frequency much smaller than the characteristic tunneling times and vice the latter@xcite . adiabatic quantum pumping can be described by berry phase of the scattering matrix accumulated during the cyclic modulation in the parameter space@xcite and also by the energy quanta absorption and emission processes equivalent to the nearest sideband approximation@xcite . nonadiabatic quantum pumping can be described by the floquet scattering scheme picturing quantities of interest in terms of sideband formation@xcite . the non - equilibrium green s function@xcite , equation of motion@xcite , galileo transformation@xcite , and etc . also show physics of nonadiabatic quantum pumping from different views . adiabatic and nonadiabatic quantum pumping has been investigated in various mesoscopic systems , such as nanowire@xcite , mesoscopic rings@xcite , quantum - dot structures@xcite , spin - orbit coupled conductors@xcite , magnetic tunnel junction@xcite , graphene@xcite , and superconductor junction with majorana fermions@xcite , etc . current fluctuations are present in almost all kinds of conductors including dynamical transport systems . the shot noise is the quantum contribution in the current fluctuation produced by the quantum coherence of charge carriers , which can give rich physical information in mesoscopic transport systems and is more significant in nanoscale quantum devices than in the traditional non - quantum devices@xcite . although intensive work has been done on the bias driven shot noise in various mesoscopic conductors@xcite , adiabatic pumped noise is also extensively investigated@xcite , and the general scattering theory for nonadiabatic pumped shot noise@xcite is derived , the specific pumped shot noise properties in different quantum transport systems are less covered . they represent the underlying physics of different materials and devices , some of which is beyond conductance information . generally a transport approach covers the shot noise as well as the conductance . the floquet scattering matrix approach was developed for nonadiabatic noise properties as detailed by moskalets _ et al_.@xcite general expressions for the pumped current , heat flow , and shot noise are derived for adiabatically and non - adiabatically driven quantum pumps . this approach stresses the existence of sidebands of electrons passing the time - dependent scatterer and these sidebands are connected to the currents and noise directly . recently , park and ahn@xcite derived an expression for the admittance and the current noise for a driven nanocapacitor in terms of the floquet scattering matrix and obtained a non - equilibrium fluctuation dissipation relation . the scattering matrix renormalized by interaction has been used by devillard _ et al_.@xcite to study the effect of weak electron - electron interaction on the noise . under the geometric framework , there have been beautiful mathematical descriptions from the current to the noise@xcite . in this work , we focus on the non - adiabatic quantum pump driven by a single oscillating potential well , in which fano resonance is predicted in the floquet transmission spectrum when one of the floquet levels matches the quasibound level of the static potential well@xcite . in this case , the pumped current vanishes due to time and spatial reversal symmetry and the fano resonance is unavailable in the current measurement . however , energy and information is transfused into the pump from exterior by instantaneous transport within a driving cycle . in the pumping process , virtual or temporary transmission within a cyclic period generates considerable noise . supposing the resonance feature can be characterized in the correlation , we investigated its noise and heat flow properties . we consider a one - dimensional width-@xmath0 time - dependent potential - well sketched in fig . the time - dependent potential , which oscillates with frequency @xmath1 and is located between @xmath2 and @xmath3 , has the form @xmath4 the time - dependent hamiltonian of the electrons can be expressed as @xmath5 @xmath6 is the effective mass of electrons and our discussion is based on single electron approximation and coherent tunneling . firstly , we consider the quasibound states within the one - dimensional static quantum well . when the electron is confined in the well with its energy @xmath7 , the wave function can be written as @xmath8 where @xmath9 and @xmath10 . continuity equations of the wave function and its derivative at @xmath11 and @xmath12 are @xmath13 solvability of these equations gives rise to the secular equation @xmath14 roots of @xmath15 for this equation are the quasibound state energies . it can be obtained numerically that the only quasibound state for the considered well configuration is at the energy@xcite of @xmath16 mev . we use the floquet scattering theory to investigate the quantum pumping properties of the oscillating quantum well@xcite . wave functions in the three scattering regimes can be written as @xmath17 in the left and right free regions , the incident and outgoing electron waves consist of infinite number of sidebands , as shown in fig @xmath18 is the eigenenergy of the @xmath19-th order floquet state with the fermi energy @xmath20 and @xmath21 is the corresponding wave vector . @xmath19 are integers varying from @xmath22 to @xmath23 in an ideal exactness . while @xmath24 , @xmath25 is imaginary meaning an evanescent mode , the current for this channel vanishes . @xmath26 and @xmath27 are the probability amplitudes corresponding to those flowing out of and flowing into the left / right leads , respectively . here we did not use flux normalization for algebra simplicity of continuity conditions and its justification is elaborated in the appendix . the floquet scattering matrix can be constructed by relations between @xmath26 and @xmath27 . for example , if we set @xmath28 , @xmath29 corresponds to the reflection amplitude from the left lead to the left lead in the @xmath19-order floquet channel , and so on . @xmath30 is wavevector in the middle oscillating quantum well region . @xmath31 is the first kind of bessel function deriving from @xmath32 , which only exists in the oscillating region . @xmath33 and @xmath34 are the wave function amplitudes in the oscillating region and present only in the continuity equation solving processes . by solving equations of the boundary conditions at interfaces @xmath35 and @xmath36 which must hold for all time , the floquet scattering matrix without flux normalization can be obtained by matrix algebra . it connects different floquet modes as @xmath37 with @xmath38 and @xmath39 column vectors made up of @xmath26 and @xmath27 of all @xmath40s . @xmath41 and @xmath42 are the transmission and reflection amplitudes incoming from the @xmath43-th floquet channel and going into the @xmath19-th floquet channel . considering the real current flux , the floquet scattering matrix elements are @xmath44 transmission of evanescent modes would vanish as only the real part of the outgoing wave vectors is considered . from the scattering matrix , the total transmission probability is defined as @xmath45 we are interested in the element @xmath46 of the floquet scattering matrix given in eq . ( [ fluxnormalizedscatteringmatrix ] ) . it measures the scattering amplitude of the electron incident through lead @xmath47 with energy @xmath15 and leaving through lead @xmath48 with energy @xmath49 . by interacting with the oscillating potential the electron absorbs or loses energy quanta of @xmath50 , with its final energy @xmath51 . for the non - adiabatic quantum pump , the floquet scattering matrix is sensitive to the spatial symmetry of the potential . if the system is present of perturbations broken the spatial symmetry or time - reversal symmetry it can pump a dc current . with only one oscillating potential , both the spatial symmetry and time - reversal symmetry is present in the device , thus no pumped current exists with @xmath52 the heat flow is carried by the non - equilibrium particles , which occurs in the process of scattering and the direction of heat flow is defined as from the oscillating potential to the reservoirs as @xmath53 . \label{eq9}\ ] ] here @xmath54 is equilibrium fermi distribution function . @xmath55 is the chemical potential , which is the same in all reservoirs at zero bias . we also assume all the reservoirs have the same temperature . the problem of current noise is closely connected with the matrix elements of @xmath46 . for a phase - coherent conductor the noise is sensitive to the quantum - mechanical interference effects . we can describe the correlation function of the current as@xcite @xmath56 where @xmath57 , and @xmath58 is the quantum - mechanical current operator in the lead @xmath48 , which can be expressed as@xcite @xmath59 } e^{i\left ( { e - e ' } \right)t/\hbar } , \label{eq11}\ ] ] with @xmath60 and @xmath61 annihilation operators of the incident and outgoing electrons to the driven potential and @xmath62 from eqs . ( [ eq1 ] ) to ( [ bsa ] ) the pumped shot noise and pumped heat flow noise can be expressed in terms of the floquet scattering matrix as@xcite @xmath63 ^ 2 } } } } } , \label{eq2}\ ] ] @xmath64 ^ 2 , \label{eq3}\ ] ] with @xmath65 describing the quantum - mechanical exchange during scattering of electrons with energy @xmath66 incident from leads @xmath67 and outgoing to the leads @xmath68 with energy @xmath69 , respectively . current flux conservation secures that @xmath70 and @xmath71 . we consider one of the four and label @xmath72 as @xmath73 , @xmath74 as @xmath75 , and @xmath76 . to magnify the resonance spectrum , we also considered the derivatives of the noise and heat flow over the fermi energy with @xmath77 in this paper we have adopted the floquet scattering matrix approach to investigate the pumped effect of phase coherent mesoscopic systems of noninteracting electrons . the floquet scattering matrix describes existence of sidebands of electrons entering and exiting the pump . the nonequilibrium electrons generated by the pump carry heat from the oscillating potential to the reservoirs and transfer charge between the two reservoirs . only the first sidebands @xmath78 are excited if the oscillating amplitude is small . the total transmission probability @xmath79 as a function of the incident energy is shown in fig . 2 . in all of the numerical consideration @xmath80 is set to be 1 mev . we take into account different floquet sidebands both above and below the incident energy , with @xmath81 . @xmath82 cutoff is used , with its precision satisfactory for the small driving amplitude . in the quantum well there exists a quasibound state , when the first order floquet sideband overlaps with the quasibound level , a fano " resonance occurs ( also confer fig . 1 ) , which was discovered in ref . . due to time and spatial reversal symmetry , no charge current is generated by a single oscillating quantum well . it is known that when pumped charge current is zero , the pumped shot noise can be considerably large due to virtual transmission processes@xcite . we suppose that the nonequilibrium transmission properties can be recorded in the shot noise spectrum and the fano " resonance can thus be observed . we calculated the pumped current noise , heat noise and heat flow driven by the nonadiabatic oscillating quantum well using the floquet scattering scheme with eqs . ( [ eq9 ] ) , ( [ eq2 ] ) , and ( [ eq3 ] ) . their variation as a function of the fermi energy was depicted in fig . 3 . the pumped current noise , heat noise and heat flow increases with the fermi energy when more energy channels contribute to the transport for larger fermi energies . for a small driving amplitude in our case , most of the transmission comes from the original incident level and the two first order floquet sidebands ( @xmath83 ) . when these bands completely go out of the quantum well with @xmath84 , a decrease occurs in the noise spectrum . noise is an effect of correlation , concrete or virtual . when all the active floquet bands are out of the quantum well , the transmitting electron sees " no structure in the conductor therefore ballistic transport governs giving rise to the shot noise decrease . an inflection could be found at the fermi energy @xmath85 mev corresponding to the fano " resonance in the total transmission . the magnification of the inflection point is shown in the insets . since the transport properties are an accumulating effect of all energy channels , contribution of a single energy channel of the resonance is limited . to magnify the fano " resonance of the nonadiabatic quantum pump , we calculated the differentials of the pumped charge and heat noise to the fermi energy and dramatic resonance pattern could be found . differentials of the pumped current noise , heat noise , and heat flow as a function of the fermi energy are shown in fig . 4 . these curves have a sharp dip followed by a peak at @xmath86 mev , demonstrating an asymmetric fano " resonance , which ensures that there exists a quasibound state ( the energy of the quasibound state is @xmath87 mev ) in the deep quantum well@xcite . electrons in the propagating states can emit photons and drop into the quasibound state and bounce back before exiting the well , thus contributing to the transport ( see fig . 1 ) . the heat flow also shows a sharp peak at the resonance fermi energy . the pumped noise properties can be interpreted as follows . at the resonant fermi energy , transport process and the electron - electron correlation achieve maximal strength . the nonequilibrium electrons created by the oscillating scatterer move in different directions carried the heat flow into the electron reservoirs of the two sides . the differential shot noise demonstrates peaks corresponding to the fano " resonance , as a result of the virtual motion of nonequilibrium electrons . we considered the noise properties of a nonadiabatic quantum pump driven by an oscillating potential well . due to time and space reversal symmetry no dc charge current can be produced . due to virtual transmission of the electrons , heat current is not zero with its direction from the conductor into the leads at both reservoirs . to experimentally observe the fano " resonance found in the transmission@xcite , we investigated the heat current and shot noise of the charge and heat current . sharp fano "- shape resonance was found . the differential current noise , heat noise , and heat flow demonstrate peak structure from the interaction of electrons with the oscillating potential when one of the floquet sideband matches the quasibound state . electrons in incident channel can drop into the quasibound state by emitting photons . similarly , electron in the bound state can also absorb photons and bounce back into the floquet channels . thus a fano " resonance occurred . the resonance position of the fermi energy is then governed by the energy of the static quasibound state . the authors acknowledge enlightening discussions with professor wen - ji deng . this project was supported by the national natural science foundation of china ( no . 11004063 ) and the fundamental research funds for the central universities , scut ( no . 2014zg0044 ) . the current operator in the left lead ( far from the sample ) can be expressed in a standard way@xcite , @xmath88 } , \label{acurrent}\ ] ] where the field operator @xmath89 is defined as @xmath90}^{{1 \mathord{\left/ { \vphantom { 1 2 } } \right . \kern-\nulldelimiterspace } 2}}}}}\left [ { { { \hat a}_l}\left ( { { e_n } } \right){e^{i{k_{n}}x } } + { { \hat b}_l}\left ( { { e_n } } \right){e^ { - i{k_{n}}x } } } \right ] } . \label{afieldoperator}\ ] ] to avoid tediousness we consider single transverse channel and multiple energy channels , the latter of which is necessary for the nonadiabatic dynamic process . @xmath91 and @xmath92 are defined identical to the main text . it can be seen that the field operator is naturally flux normalized with @xmath93 in the denominator . substituting eq . ( [ afieldoperator ] ) into eq . ( [ acurrent ] ) , and using the relations , @xmath94 and @xmath95 we could have @xmath96 } \right . } } \\ \times \left [ { { e^{i\left ( { { k_{lm } } - { k_{ln } } } \right)x}}\hat a_l^\dag \left ( { { e_n } } \right){{\hat a}_l}\left ( { { e_m } } \right ) - { e^ { - i\left ( { { k_{lm } } - { k_{ln } } } \right)x}}\hat b_l^\dag \left ( { { e_n } } \right){{\hat b}_l}\left ( { { e_m } } \right ) } \right ] \\ \left . { - \left [ { { { \left ( { \frac{{{k_{lm}}}}{{{k_{ln } } } } } \right)}^{{1 \mathord{\left/ { \vphantom { 1 2 } } \right . \kern-\nulldelimiterspace } 2 } } } - { { \left ( { \frac{{{k_{ln}}}}{{{k_{lm } } } } } \right)}^{{1 \mathord{\left/ { \vphantom { 1 2 } } \right . \kern-\nulldelimiterspace } 2 } } } } \right]\left [ { { e^ { - i\left ( { { k_{lm } } + { k_{ln } } } \right)x}}\hat a_l^\dag \left ( { { e_n } } \right){{\hat b}_l}\left ( { { e_m } } \right ) - { e^{i\left ( { { k_{lm } } + { k_{ln } } } \right)x}}\hat b_l^\dag \left ( { { e_n } } \right){{\hat a}_l}\left ( { { e_m } } \right ) } \right ] } \right\ } . \\ \end{array}\ ] ] we consider the dc current driven by nonadiabatic periodic parameter variation . the time - averaged current can be calculated in an arbitrary pumping cycle as @xmath97 where @xmath98 means the quantum and statistical average of the current operator . with the integral @xmath99 we could obtain @xmath100 } } , \label{acurrentwithn}\ ] ] here it should be noted that the floquet channels occur only in the scattering process and the original and final states are energy conserved with the energy of @xmath15 . then , we have @xmath101 } . \label{aaveragedcurrent}\ ] ] the outgoing operators can be expressed in terms of the incoming ones by the floquet scattering matrix @xmath102 for reservoirs at equilibrium , we have @xmath103 substituting eqs . ( [ adefinesmatrix ] ) and ( [ afermidistribution ] ) into the averaged current ( [ aaveragedcurrent ] ) , we can obtain @xmath104 } , \ ] ] which reproduced the result of ref . . unitarity of the floquet scattering matrix naturally follows from the current conservation . the current flowing from the left lead is identical to that flowing into the right lead @xmath105 ( positive direction of the current is defined as flowing from the reservoir to the scatterer ) . from eq . ( [ acurrentwithn ] ) , we have @xmath106 } = \sum\limits_n { \left [ { \hat b_r^\dag \left ( { { e_n } } \right){{\hat b}_r}\left ( { { e_n } } \right ) - \hat a_r^\dag \left ( { { e_n } } \right){{\hat a}_r}\left ( { { e_n } } \right ) } \right ] } .\ ] ] by eq . ( [ adefinesmatrix ] ) , it follows that @xmath107 which could be written into the matrix form as @xmath108 with corresponding elements @xmath109 and @xmath110 directly from eq . ( [ asmatrixunitarity ] ) follows the unitarity relation @xmath111 all elements of the floquet scattering matrix are only well defined for incident and outgoing propagating modes . in deducing the floquet scattering matrix from continuity conditions , it is more convenient to use the wave function without flux normalization as in eq . ( [ wavefunction ] ) . by considering the real current flux , we do the transform of eq . ( [ fluxnormalizedscatteringmatrix ] ) to obtain the floquet scattering matrix . we could reproduce previous derivations by defining @xmath112 and @xmath113 with @xmath114 defined in eq . ( [ eq6 ] ) and directly obtainable from continuity relations . . there is a quasibound state in the potential well with the binding energy @xmath115 . energy is infused into the system by ac modulation of the potential well . fano resonance occurs when one of the floquet sideband overlaps with the quasibound state . equilibrium well depth is @xmath116 and its variation in time has the form of @xmath117.,width=377,height=264 ] as a function of the incident energy@xcite . driving amplitude @xmath118 mev , static well depth @xmath119 mev , well width @xmath120 , and energy quanta of the driving frequency @xmath121 mev . a resonance occurs at @xmath85 mev . , width=453,height=377 ] , ( b ) heat flow shot noise @xmath75 , and ( c ) heat flow @xmath122 , as functions of the fermi energy . their units are obtained by substituting @xmath123 mev into the energy and absorbing additional @xmath124 into the data . an inflection occurs at @xmath85 mev corresponding to the resonance in transmission . insets are the zoom - in of the inflection point . , width=415,height=377 ] , heat flow noise @xmath125 , and the heat flow @xmath126 , as functions of the fermi energy . their units are obtained by substituting @xmath123 mev into the energy and absorbing additional @xmath124 into the data . sharp resonance could be seen at the inflection in fig . 3 with @xmath85 mev . , width=529,height=377 ]
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we use the floquet scattering theory to study the correlation properties of the nonadiabatic pumped dc current and heat flow through a time - dependent quantum well .
electrons can transit through the quasibound state to the oscillator induced floquet states leading to resonant tunneling effect .
virtual electron scattering processes can produce pumped heat flow , pumped shot noise and pumped heat flow noise , with presence of time and spatial reversal symmetry . when one of the floquet levels matches the quasibound level there strikes a fano " resonance .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the so - called _ adiabatic piston _ is a long known problem in classical thermodynamics , which can be stated as follows @xcite . an isolated cylinder of length @xmath0 , containing a gas , is divided by an adiabatic wall ( no internal degrees of freedom ) , _ the piston _ , into two compartments ( fig . [ fig:1 ] ) . the initial condition is prepared in the following way : the piston is kept fixed by a clamp at a given position @xmath1 ; the gases in the left ( @xmath2 ) and right ( @xmath3 ) compartments are in equilibrium defined by their pressure , temperature and volume : @xmath4 . by assuming that the two gases are perfect and composed by @xmath5 molecules with equal masses @xmath6 , the gas state equation @xmath7 holds in both chambers ( where the boltzmann constant is set to unity by rescaling the temperatures ) . being the piston adiabatic , the two subsystems are in equilibrium even if @xmath8 . at @xmath9 the clamp is removed and the piston is free to move without friction with the cylinder . the nontrivial question is to predict the system evolution and the final position of the piston and of the thermodynamic quantities . and @xmath3 indicate the left and right compartments , respectively . ] at the beginning of last century , the above setup was used as experimental device for measuring the ratio @xmath10 of the specific heat of gases @xcite , that is linked to the period of the piston oscillations . renewed interest on the problem has lead to recent experiments @xcite . meanwhile , several attempts were made to predict the final equilibrium state by using the laws of thermodynamics only , ending in controversial answers . a naive application of the first two laws of thermodynamics lead to the ( wrong ) conclusion that the equilibrium conditions is @xmath11 . a more careful treatment @xcite shows that the correct answer is @xmath12 . however , such a condition says nothing about the final position of the piston and gas temperatures , which remain undetermined . therefore , equilibrium thermodynamics can not predict the final state . to shed light on the problem one has to cope with the non - equilibrium process that occurs after the clamp removal . from a microscopic point of view the adiabatic piston problem for ideal gases ( non - interacting particles ) can be described in terms of a one dimensional model , where the piston is a heavy particle of mass @xmath13 much larger that the mass @xmath6 of the gas molecules , which collide elastically with the piston . as argued by feynman , the system first converges toward a state of mechanical equilibrium with @xmath14 , consistently with the thermodynamic prediction . then , the pressure fluctuations , which are asymmetric because @xmath15 , very slowly drive the system toward thermal equilibrium @xmath16 @xcite . in this way the final position of the piston and the thermodynamic quantities are determined . more recently , the problem was the subject of renewed attention , mainly stimulated by the talk of lieb at the statistical physics conference in 1998 @xcite , and by the connection of this problem with the physics of mesoscopic systems @xcite and brownian motors @xcite . among the first attempts to understand quantitatively the time evolution of the adiabatic piston we mention crosignani et al . @xcite who introduced a set of ordinary differential equations for the macroscopic observables . however , this model was only able to account for the position of the piston in the state of mechanical equilibrium and not for the final thermodynamic one . remaining in the framework of ideal gases , a systematic investigation in statistical mechanics terms , together with numerical simulations , has been carried on in the last decade by gruber and coworkers @xcite . in these works the problem has been examined in several limits ( see ref . @xcite for a review ) . in particular , in the thermodynamic limit taken by letting the system size @xmath0 and the piston mass @xmath13 to go to infinity by holding fixed the ratios @xmath17 and @xmath18 , it has been shown that the system evolution can be reduced to a set of ordinary differential equations for the macroscopic observables ( i.e. the gas left / right temperatures , and the moments of the piston velocity ) . these equations were obtained by using the liouville and boltzmann equations . within such an approach it is possible to control the deviations from maxwell - boltzmann distribution for the gas velocities , observed in the simulations , and a whole hierarchy of equations can be written for all moments of the piston velocity . remarkably , these equations describe not only the reaching of mechanical equilibrium , which comes from the treatment at zero order in @xmath19 @xcite , but also the final equilibrium state , which comes from the first order terms in @xmath19 @xcite . these analytical results have been shown by the same authors to be in agreement with numerical simulations of the ideal gas piston problem . though it remains open the problem of a detailed description of the early stage of the dynamics in which the presence of shock waves has an important impact on the dynamics . some recent attempts in this direction can be found in ref . @xcite . when the initial pressures are different , the system phenomenology can be described as follows @xcite . in a first stage , the piston oscillates driven by the pressure difference . these oscillations are then damped till the `` mechanical equilibrium '' state , @xmath20 but @xmath21 , is reached . then , as argued by feynman , it follows a regime controlled by the asymmetry in the fluctuations felt by the left / right walls . this phase is characterized by a very slow approach to the thermodynamic equilibrium , @xmath20 and @xmath22 , with the piston position fluctuating around the middle of the cylinder . in the oscillatory phase , both experiments @xcite , numerical computations and analytical arguments @xcite have shown the existence of two different regimes : _ weak and strong damping _ , the relevant parameter being @xmath23 . for @xmath24 the adiabatic oscillations of the piston are weakly damped , while for @xmath25 they are over - damped , @xmath26 being @xmath27 @xcite . still in the context of ideal gases , it is worth mentioning some recent approaches based on dynamical systems theory that have been developed by chernov , lebowitz and sinai @xcite . in this context , also the case of gases starting from non - equilibrium conditions has been considered @xcite . clearly , the ultimate goal would be to quantitatively understand the behavior of the system for an interacting gas , but this seems to be still too ambitious . indeed only very few studies analyzed the case of gas composed by interacting particles @xcite . in this paper , we consider a limiting case which has the advantage of being more tractable while displaying most of the non - trivial features of the problem . the basic idea of our approach is to assume that the gases in the two compartments are composed of interacting molecules and thus characterized by a relaxation time toward the equilibrium state . our main hypothesis is that this time is very short compared with all the other characteristic times of the system . in particular , we require that any fluctuation away from equilibrium ( which is characterized by homogeneously distributed gas molecules with a maxwell - boltzmann velocity statistics ) induced by the collision with the piston is re - adsorbed before the new collision with the piston walls . physically speaking , the efficient re - adsorption of the fluctuations means that the ( mechanical ) work done by the piston is immediately converted into heat ; an obvious consequence is that shock waves are ruled out . for the sake of simplicity , we also assume that the gases follow the perfect gas law . these hypothesis make the problem tractable while retaining the basic phenomenology of the original problem . although a microscopic model of a gas able to fulfill the above requirements may sound rather artificial , at a practical level such a `` microscopic model '' can be easily implemented on a computer . the basic idea is to start with an equilibrium configuration with temperatures @xmath28 for the gases , and then reinitialize the gas molecules as soon as one particle collides with the piston . the temperatures are recomputed after the collision and used for extracting a new configuration of the gas molecules . the procedure is then repeated . in the following we shall call such a model _ randomized gas_. even though , no actual interaction among the particles is actually considered , one can think that the re - generation of the gas configuration from an equilibrium one ( but with the new temperature ) is the result of such `` unresolved '' interactions . with the above assumptions for the gas , we will derive a set of ordinary differential equations for the time evolution of the macroscopic quantities describing the state of the system . indeed the fact that the gas is always homogeneous and following the maxwell - boltzmann distribution allows us to compute the joint probability density function that , in a given state of the system , the first colliding gas particle hits the piston in a time @xmath29 and with a velocity @xmath30 . then , by averaging over this joint distribution the energy and momentum exchange due to the collisions with the piston , we derive the evolution of the macroscopic observables . the minimal set of variables required to have a closed set of equations is made up of the gas temperatures , the mean piston position , the first and second moment of the piston velocity . the second moment is required for accounting the piston fluctuations which , as argued by feynman , are crucial for recovering the correct thermodynamic equilibrium @xcite . the equations are derived perturbatively up to the first order in @xmath19 . as we will see , though with a different approach and assumption , these equations are very similar to those derived by gruber and coworkers @xcite , in particular at the zeroth order in @xmath19 they are identical . we compare then the evolution of the system obtained by simulations of the ideal and randomized gas . in particular , the agreement in the first ( mechanical ) regime is quantitatively perfect in the case of the randomized gas . while in the second regime , which is dominated by the fluctuations , the agreement seems to be only qualitative . somehow surprisingly , we found that , in this regime , a better quantitative agreement seems to be possible disregarding some @xmath31 terms . however , with such terms excluded the equipartition of energy at equilibrium is violated by the piston . some hints to explain these findings could come by higher orders terms in the expansion . unfortunately , the computation of the higher order terms is very cumbersome . since the newest aspects of our work is in the proposed derivation and in the introduction of the randomized gas model , we present in this paper the all approach up to first order in @xmath19 . the paper is organized as follows . in sect . [ sec:1 ] we present our approach based on the collision statistics and derive the equations for the macroscopic observables . in sect . [ sec:2 ] we compare the results of the model with those obtained by simulations . discussions and conclusions can be found in section [ sec:3 ] . in order to avoid long appendices , the technical material , with the detailed derivation and all the formulas needed to make explicit the equations , is presented as _ electronic supplementary material _ the underlying idea of our approach is to derive a set of deterministic dynamical equations for the `` macroscopic '' variables describing the evolution of the thermodynamic state of the system under the assumption that , at any time , the gases in both chambers are perfect and at equilibrium . in other words the gases are able to instantaneously dissipate the fluctuations induced by the collisions with the piston . thus a maxwell - boltzmann equilibrium state holds always in both compartments but , in general , with different temperatures and volumes . while the above hypothesis define the macroscopic state of the gas , for the piston the problem is more subtle . we would like to describe its motion on times longer than the single collisions , that is to average its instantaneous position and velocity @xmath32 over the collisions so to obtain a deterministic ( macroscopic ) trajectory defined by the average position @xmath33 and velocity @xmath34 the symbol @xmath35}$ ] denotes the average over the collisions . as discussed in the introduction , it is crucial to account also for the fluctuations of the piston velocity . for this reason , the second moment of the piston velocity @xmath36 is included in the description . in the thermodynamic limit we will consider , one can argue that the fluctuation of the piston position can be safely ignored . this means that in the following we will consider the piston position as a deterministic quantity and we shall use only the mean piston position @xmath37 . given the piston position , the gas is characterized by the temperature @xmath28 , and volume @xmath38 , with @xmath39 and @xmath40 ( we assume a 1d geometry for the sake of simplicity ) . being perfect gases , the pressures are given by the equation of state @xmath41 . in the sequel we show how to derive a set of differential equations for the evolution of @xmath37 , @xmath42 , @xmath36 and @xmath28 . in order to keep the presentation as simple as possible , here we shall sketch how the equations can be derived and the averages performed skipping all the algebra of the computation , which is detailed in @xcite . for the formal derivation of the deterministic equations , we only need the above discussed assumptions and a microscopic ingredient : the elastic collision rules @xmath43 primes denote postcollisional velocities , and @xmath30 the colliding gas particle velocity . the quantities we are interested in are the time derivatives of the macroscopic observables @xmath44 where we set the boltzmann constant @xmath45 . the time derivatives should be computed starting from the collision rules as the averages @xmath46\rangle_{l , r}= \overline{[\dots]}_{l , r}/\overline{\delta t}$ ] suggest , @xmath47 being the mean collisions time ( for a more precise and operative definition see sect [ subsec : average ] ) . the subscripts @xmath48 denote averages performed over the collisions with particles residing on the left / right compartments . it is useful to introduce @xmath49 which evolves as @xmath50 where we used eqs . ( [ eq : vx ] ) and ( [ eq : sigma ] ) . it should be noted that , at this level , the piston is completely described by @xmath42 and @xmath51 . this amounts to assume that its velocity distribution is gaussian @xmath52 since at the initial time @xmath53 , one starts with @xmath54 and @xmath55 , the above probability distribution is initially a @xmath56 function . plugging ( [ eqnurto ] ) into ( [ eq : vx]-[eq : temp ] ) , we obtain @xmath57 \label{eq1:sigma } \\ \frac{{\rm d}t_{l}}{{\rm d}t } & = & \frac{4mm}{(m\!+\!m)^2}\left[m\langle \sigma_v^2\rangle_l+m\langle v_x^2\rangle_l \right.\nonumber\\ & -&\left.(m\!-\!m)\langle v v \rangle_l -m\langle v^2\rangle_l \right ] \,,\label{eq : templ } \\ \frac{{\rm d}t_{r}}{{\rm d}t}&=&\frac{4mm}{(m\!+\!m)^2}\left[m\langle \sigma_v^2\rangle_r+m\langle v_x^2 \rangle_r\right.\nonumber\\ & -&\left.(m\!-\!m)\langle v v \rangle_r -m\langle v^2\rangle_r \right ] \,.\label{eq : tempr}\end{aligned}\ ] ] the equation for @xmath58 ( [ eq1:sigma ] ) is obtained from ( [ eq : dsigdt ] ) and ( [ eq : sigma ] ) by using the collision rules ( [ eqnurto ] ) . in ( [ eq1:vx ] ) and ( [ eq1:sigma ] ) , the prefactor @xmath59 appears as a result of a time rescaling , that sets the time unit to the average collision time , which is order @xmath60 . said differently , the change of the gas temperatures due to the collision with the piston is order @xmath60 . notice that , for reasons that will become clear in the following , the average of the type @xmath61 is different from @xmath62 . we anticipate that this difference is not due to a breakdown of molecular chaos hypothesis ( as one may naively think ) but to the fact that the collision statistics depends on the instantaneous value of the piston velocity . more explicitly , @xmath62 represents only the zeroth order term of @xmath61 , and a term coming from the fact that @xmath63 is a fluctuating quantity will also appear . notice also that the above equations conserve the total ( gas plus piston ) energy @xmath64 the consistent ( first order in @xmath19 ) equations can then be obtained from ( [ eq1:vx]-[eq : tempr ] ) by expanding the various prefactors , performing the limit @xmath65 and suitably expanding around it . since the procedure is delicate , we proceed step by step . first of all we have to specify the limiting procedure , that as explained in @xcite can be done in different ways . we are interested in the limit @xmath66 in which we keep fixed @xmath67 and the nondimensional mass ratio @xmath68 . we have now to expand around this limit retaining all terms which are first order in @xmath19 ( and consequently at the first order in @xmath60 ) . aiming to make explicit the zeroth and first order terms we formally write the averages as : @xmath69\rangle=\langle [ \dots]\rangle^{(0)}+\langle [ \dots]\rangle^{(1)}\,,\ ] ] where the first and second terms on the r.h.s are the zeroth and first order terms of the expansion . how to explicitly perform such an expansion will be explained in the following subsections . we warn the reader that for maintaining the notation as compact and explicit as possible in the following we adopt the convection to indicate with @xmath46\rangle^{(1)}$ ] all averages which are @xmath31 irregardless if this comes from the expansion of the average or from the averaged quantity . for instance , by direct inspection of eq . ( [ eq1:sigma ] ) at equilibrium , one easily realizes that @xmath58 is @xmath31 . therefore , we shall always indicate its average with @xmath70 . finally , notice also that all the terms involving powers of @xmath71 vanish at the zeroth order . keeping in mind these simplifications , the ( expanded ) equations become : @xmath72 \label{eq2:vx } \\ \strut{\hspace{-.1truecm}}\frac{{\rm d}\sigma^2_v}{{\rm d}t}\!\ ! & = & \ ! \!4r\left [ -\langle \sigma^2_v \rangle^{(1 ) } + { \frac{m}{m}}\langle ( v - v_x)^2 \rangle^{(0 ) } + \langle ( v - v_x)v \rangle^{(1)}+\langle ( v - v_x)v_x \rangle^{(1)}\right ] \label{eq2:sigma } \\ \strut{\hspace{-.6truecm}}\frac{{\rm d}t_l}{{\rm d}t}\!\!&= & \!\!4m\ ! \left\{\ ! \langle v_x(v_x\!-\!v)\rangle_l^{(0 ) } \!+\ ! \left[\ ! \langle v_x(v_x\!-\!v)\rangle_l^{(1 ) } \!\!\!-\!2{\frac{m}{m}}\langle v_x^2 \rangle_l^{(0 ) } \!\!+\ ! 3{\frac{m}{m}}\langle v_xv\rangle^{(0)}_l \!\!+\!\langle \sigma_v^2\rangle^{(1)}_l \right.\right . -\left.\left.\langle ( v\!-\!v_x)v\rangle^{(1)}_l \!\!-\!{\frac{m}{m}}\langle v^2\rangle^{(0)}_l \!\right]\!\right\ } \label{eq2:tl } \\ \strut{\hspace{-.6truecm}}\frac{{\rm d}t_r}{{\rm d}t}\!\!&=&\!\ ! 4m\ ! \left\{\ ! \langle v_x(v_x\!-\!v)\rangle_r^{(0 ) } \!+\ ! \left[\ ! \langle v_x(v_x\!-\!v)\rangle_r^{(1 ) } \!\!\!-\!2{\frac{m}{m}}\langle v_x^2 \rangle_r^{(0 ) } \!+\ ! 3{\frac{m}{m}}\langle v_xv\rangle^{(0)}_r \!\!+\!\langle \sigma_v^2\rangle^{(1)}_r \right.\right . -\left.\left.\langle ( v\!-\!v_x)v\rangle^{(1)}_r \!\!-\!{\frac{m}{m}}\langle v^2\rangle^{(0)}_r \!\right]\!\right\ } \,.\label{eq2:tr}\end{aligned}\ ] ] before sketching the way the above averages can be computed ( see sect . [ subsec : average ] and @xcite ) we briefly discuss some properties of the above equations . the first observation is that eqs . ( [ eq2:vx]-[eq2:tr ] ) ensure the energy conservation ( [ eq : energy_conservation ] ) both at the zeroth and first order , meaning that the expansion is consistent . notice also that the relative importance of the various terms is not the same at all times . as the system evolves , their relative weights change corresponding to the different stages of the evolution , briefly summarized in the introduction and detailed in the following . for example , consider @xmath58 . at the beginning @xmath73 , then it grows until it reaches its equilibrium value . consequently , the terms which involve the velocity fluctuations are not important at the beginning . while they become @xmath31 and play a crucial role in the final stage of the system evolution . the opposite is true for the terms involving the average drift @xmath42 , which is close to zero in the second ( brownian ) stage of the evolution , and large in the first ( mechanical ) part of the system evolution . among the terms involving the piston velocity fluctuations , we should mention a special role played by those in which it does not ( explicitly ) appear @xmath58 . these are the terms of the form @xmath74 and @xmath75 , as we will see they will be both proportional to @xmath58 and , as anticipated , find their origin in the way the fluctuations of @xmath63 affect the collision statistics ( see next subsections and the supplementary material @xcite ) . however , a closer inspection shows that @xmath76 is very small at all times . in the first stage the fluctuations are negligible while in the second stage the average drift is very small . though we retained this term in the equations and in the numerical simulations , one can show that they can be removed without problem . differently the terms @xmath77 are very important in the final stage of the evolution and , as discussed below , for obtaining the correct equilibrium state . we are still left with performing the infinite volume limit . we anticipate here that all the terms appearing in the averages are proportional to either @xmath78 , when coming from a left - chamber average , or @xmath79 when coming from a right - chamber average . there is no other dependence on @xmath37 or @xmath0 in equations ( [ eq : x]-[eq : temp ] ) and , consequently , eqs . ( [ eq2:vx]-[eq2:tr ] ) . this implies that one can simply introduce the hydrodynamic time @xmath80 and the rescaled @xmath37 coordinate @xmath81 . with this rescaling , @xmath0 does not appear anymore in the equations and there is no need to perform the limit , meaning that there are no corrections to the results due to finite size effects . for a simpler comparison with the simulations , we will keep writing in the following the equations for finite values of @xmath0 ; the corresponding expressions in the hydrodynamic time can be simply obtained with the above substitution , that in practice corresponds to set @xmath82 . let us start a closer inspection of the equations starting from the zeroth order terms , i.e. assuming @xmath84 . the velocity fluctuations of the piston are ignored ( meaning the motion of the piston is purely deterministic ) and the final equilibrium position depends on the initial conditions . the result is anyway nontrivial . as discussed in @xcite , having considered the dynamics and not only the thermostatics ( which only tells us the equality of pressures ) , we can now determine the mechanical equilibrium position . in the sequel , we shall show that at the zeroth order in @xmath19 we obtain , by using a different approach , the same equations of gruber and coworkers @xcite , and following them we sketch how the mechanical equilibrium point can be computed . for obtaining the equations at the @xmath83 order , we need to set @xmath85 and to ignore all averages indicated with the superscript @xmath86 in ( [ eq2:vx ] ) , ( [ eq2:tl ] ) and ( [ eq2:tr ] ) . in other words we only need the averages @xmath87 \label{eq : vmed}\\ \langle v \rangle_r^{(0 ) } & = & -\frac{t_r}{2(l - x ) m } \left[1+\mathrm{erf}\left(v_x\sqrt{\frac{m}{2t_r}}\ , \right)\right]\nonumber \,,\end{aligned}\ ] ] with @xmath88 , and the averages @xmath89\label{eq : vxmed}\\ \langle v_x\rangle^{(0)}_{r}\!\!\!\ ! & = & \!\ ! \frac{v_x}{l\!-\!x}\!\ ! \left[\sqrt{\frac{t_r}{2 \pi m } } e^{-\frac{m v_x^2}{2t_r}}\!+\ ! \frac{v_x}{2}\!+\!\frac{v_x}{2}\mathrm{erf}\!\left(v_x\sqrt{\frac{m}{2t_r}}\,\right)\!\right]\nonumber\,.\end{aligned}\ ] ] see the supplements @xcite for the derivation of the above expressions . assuming @xmath90 ( which is reasonable for realistic values of the physical parameters ) , and expanding ( [ eq : vmed ] ) and ( [ eq : vxmed ] ) in @xmath42 , the equations ( [ eq2:vx ] ) and ( [ eq2:tl]-[eq2:tr ] ) reads @xmath91 where for the friction coefficients it holds @xmath92 ensuring energy conservation ( [ eq : energy_conservation ] ) at the @xmath83 order ; from ( [ eq : vmed]-[eq : vxmed ] ) , at the lowest order in @xmath42 one has : @xmath93 we remind that @xmath39 and @xmath40 . note that apparently , in the limit @xmath94 , @xmath95 , this is not the case if the hydrodynamic rescaling is properly applied . indeed , taking the hydrodynamic limit the above expression remains unchanged , keeping in mind that in this case the `` hydrodynamic volumes '' are @xmath96 and @xmath97 . in particular , the damping coefficients go to a finite value also in the hydrodynamic limit . it is worth remarking , that the linearized equations ( [ eq : vxlin]-[eq : trlin ] ) coincide with those derived in ref . @xcite with a different method . in the absence of friction , one can easily see that they describe a purely adiabatic transformation of a one - dimensional perfect ( mono - atomic ) gas . indeed the first term on the r.h.s . of eqs . ( [ eq : vxlin ] ) is simply the pressure difference on the two sides of the piston , while the first term of eq . ( [ eq : tllin ] ) and ( [ eq : trlin ] ) can be obtained by differentiating with respect to time the equation of an iso - entropic process , namely @xmath98 where @xmath99 is the specific heats ratio and the initial conditions fix @xmath100 . in the absence of the friction terms this would give rise to periodic oscillations of the piston . as discussed in @xcite , the friction terms are responsible for the irreversible evolution toward a state of mechanical equilibrium for which @xmath101 where the tilde indicates the mechanical equilibrium quantities . notice that in this framework irreversibility naturally emerges as a result of the averaging over the collisions @xcite . ( [ eq : equalpress ] ) is also the result of thermostatics , but it is not enough to determine @xmath102 and @xmath103 . indeed , the full dynamics given by ( [ eq : vxlin]-[eq : trlin ] ) is needed to predict such mechanical equilibrium point , as shown in the sequel where we briefly summarize the results first derived in gruber et al . first , notice that ( [ eq : equalpress ] ) together with ( [ eq : energy_conservation ] ) tell us that @xmath104 , with @xmath105 . so that the equilibrium pressure is @xmath106 . second , defining @xmath107 and by using ( [ eq : vxlin]-[eq : trlin ] ) one can easily see that @xmath108 , which if @xmath109 means that @xmath110 is conserved . @xmath111 provides the missing condition to determine the mechanical equilibrium . the resulting equation for the equilibrium point is therefore @xcite : @xmath112 which should be solved for @xmath102 after plugging @xmath113 and @xmath114 . when the first order ( in @xmath19 ) terms are retained , the terms in @xmath51 ( among which , as discussed above , we have also to consider the terms @xmath116 and @xmath117 ) allows for energy exchange among the two compartments mediated by the fluctuation of the piston . of course , such terms start to play a role once the mechanical regime ( described by the @xmath83 order terms ) is finished , i.e. when the fluctuations of the piston become relevant . this regime driven by the fluctuations results from the expansions in @xmath19 and @xmath60 which , as it happens commonly in brownian motor - like systems @xcite , are intertwined and add new ( sometimes unexpected ) features to the dynamics . in particular , in this case one can show that eq . ( [ eq2:vx]-[eq2:tr ] ) evolve toward a nontrivial stable fixed point corresponding to the thermodynamic equilibrium , i.e. : @xmath118 the last equality can not be explicitly seen from ( [ eq2:sigma ] ) , which simply states that at equilibrium @xmath119 . as we discussed , @xmath120 and with the explicit computation at equilibrium @xcite one can see that @xmath121 , which is a pleasant result since it is in agreement with the condition of equipartition of energy . notice that eq . ( [ fixedpoint ] ) suggests to interpret @xmath122 as the temperature of the piston . we conclude this subsection mentioning that the above equations are similar with the ones obtained by gruber , pache and lesne @xcite . due to the very long expressions involved in the equations at the first order , we could not decipher whether they are exactly equal . at the end of next section we shall discuss the possible source of differences . however , we stress that in these two works a different approach and different assumptions were made on the gases . in particular , gruber , pache and lesne derived the equations from an expansion of the boltzmann and liouvulle equations . in some sense , our and their different assumptions can be seen as two different ways to close the hierarchy of equations to the second order and one should expect the phenomenology of the two equations to be , at least , qualitatively similar . section [ sec:2 ] is devoted to compare the evolution of the macroscopic observables obtained by integrating ( [ eq2:vx]-[eq2:tr ] ) with that of the microscopic model . in the following subsection we detail the procedure by which the averages can be computed . in order to finalize our program we have now to make explicit the averages in ( [ eq2:vx]-[eq2:tr ] ) . let us start by making explicit the formal expression of averages such as @xmath123\rangle_{l , r}$ ] , which should be interpreted as follows . denote with @xmath124 the probability of having a left / right collision in a time @xmath29 with a velocity @xmath30 for the r / l - particle conditioned to a realization in which the piston has velocity @xmath63 , and indicate with @xmath125 their sum , i.e. @xmath126 , which is normalized to 1 . then performing the average @xmath127 of a generic function @xmath128 means : @xmath129 where @xmath130 is the mean collision time . let us now derive @xmath131 and @xmath132 . in particular , we shall compute them in the thermodynamic limit by explicitly considering the terms order @xmath60 and consequently @xmath19 . these terms are those entering the averages we indicated with @xmath46 \rangle^{(1)}_{l , r}$ ] . we start by the equilibrium distribution of the gases , which is uniform in the particle positions @xmath133 and maxwell - boltzmann for the velocities @xmath30 : @xmath134 that , under our assumptions , describe the gases at all times . note that the above distributions depend ( parametrically ) on the dynamical variables @xmath37 and @xmath28 . from the equilibrium joint distributions ( [ eq : equilibrio ] ) we can derive the probability density @xmath135 that a particle on the left / right collides in a time @xmath29 given its velocity @xmath30 and the macroscopic state of the system defined by the temperatures , the piston position @xmath37 and velocity @xmath63 and the equation of state . of course , most of the weight to such an hitting probability comes from particles that are close to the piston and that have a large ( negative ) relative velocity with it . these particles are far from the bulk of the gas and in this derivation we assume that we can use the evolution of free particles to compute their hitting time with the piston . within such an assumption , it is easy to realize that the probability @xmath136 is simply obtained through a change of variables from ( [ eq : equilibrio ] ) @xmath137 @xmath138 being the unitary step function . note that in the second expression we ignored the second @xmath138 function . this is justified by the fact that @xmath29 indicates the time between two consecutive collisions , which for @xmath139 is always much shorter than the time needed for a particle to travel along a whole chamber . further , considering only positive times , the functions @xmath140 do not depend on @xmath29 . we shall then use the compact notation @xmath141 . notice also that we use the instantaneous piston velocity @xmath63 and not its average @xmath42 , while we ignore the fluctuations in position . moreover , we neglected possible correlations between the velocities and positions , which amounts to implicitly assume molecular chaos . given @xmath135 , the joint probability of having an impact of a particle with @xmath30 in a time @xmath29 is : @xmath142 the probabilities @xmath143 that a left / right particle collides in a time @xmath144 is then given by @xmath145 which we rewrite as @xmath146 with @xmath147 in the following we will use the shorthand notation @xmath148 and @xmath149 . considering that we have @xmath59 particles on both the left and right , the probability densities that one of them on the left / right impacts the piston in @xmath37 with a velocity @xmath30 are given by @xmath150^{n\!-\!1}\!\left[1\!-\!f_r(t_{m})\right]^n\!f_l(v|v ) \nonumber\\ & & \strut\hspace{-1.05truecm}g_r(t_{m},\!v)\!=\!n\!\left[1\!-\!f_r(t_{m})\right]^{n\!-\!1}\!\left[1\!-\!f_l(t_{m})\right]^n \!f_r(v|v ) . \label{step1}\end{aligned}\ ] ] we can now perform the limit @xmath151 and @xmath152 holding @xmath153 fixed . noticing that @xmath154 and expanding @xmath155^{n}\!=\ ! \exp[n\ln(1\!-\!f_{l , r})]\approx \exp[n(-f_{l , r}-f_{l.r}^2/2)]$ ] and retaining terms only up to @xmath60 , we find @xmath156 \!\tilde{g}_{l , r}(\tau , v|v ) \label{step3}\end{aligned}\ ] ] with @xmath157 by recalling that @xmath158 , and aiming to retain only the first order @xmath19 terms ( [ step3 ] ) can be rewritten as @xmath159 where we substituted @xmath160 in place of @xmath161 and @xmath42 in place of @xmath63 in the part which is already at the first order in @xmath19 . however another contribution to the @xmath162 term comes from the expansion of the first term in ( [ eq : gfinal ] ) . we need now to further expand ( [ eq : gfinal ] ) in @xmath63 around @xmath42 , this can be accomplished by taylor expanding @xmath163 @xmath164 noticing that @xmath165 and that @xmath166 we can write , for example , the ( expanded ) average collision time : @xmath167.\nonumber\end{aligned}\ ] ] finally , we can write the correct expansion of ( [ eq : averages ] ) at the first order in @xmath19 , by plugging all the expanded terms in ( [ eq : averages ] ) to have the zero and first order terms of the average of a generic observable , @xmath168 . the result is : @xmath169\ , a\!+\!\left[\ ! { \frac{m}{m}}\frac{h_{l , r}}{r(h_l+h_r)}\!-\!\sigma_v^2\right.\nonumber\\ & & \strut{\hspace{-.4 cm } } \left.\frac{(h_l+h_r)^2 } { 2 } \int \!\!\!\!\int\!\ ! \mathrm{d}t\ , \mathrm{d}v\ , t\ , \partial^2_v\tilde{g}(\tau,\!v|v_x)\right ] \int\!\!\ ! \mathrm{d}v \ , f_{l , r}(v|v_x ) a \label{eq : av1}\end{aligned}\ ] ] which can be used to compute all the averages in ( [ eq2:vx]-[eq2:tr ] ) . however we mention that there are exceptions to the above recipe . for instance , the average of @xmath170 , whose result is @xmath171 and not @xmath172 . this is due to the convention adopted for the average : remember that we chose to write the superscript @xmath86 also when the @xmath19 order comes from the averaged quantity and not from the expansion of the collision distribution , like in this case . notice that in the first term of the r.h.s of ( [ eq : av1 ] ) , as for the average collision time , we expanded @xmath163 with the aid of ( [ eq : expansion ] ) . depending on the observable @xmath173 , which may have or not a linear term in @xmath174 , also the first derivative of @xmath163 may appear . in this section we compare the evolution of the macroscopic observables given by ( [ eq2:vx]-[eq2:tr ] ) with numerical simulations of the microscopic model . we consider two kinds of microscopic simulations : the ideal and the randomized gas . the latter is meant to fulfill the assumptions we made on the gas in deriving the equations . let us now better clarify how this is realized . from a computational point of view , it is very easy to realize the randomized gas , the idea is to let the gases relax in an artificial manner through a randomization procedure . more precisely , the simulations are performed in the following way : we generate an equilibrium configuration of the system , corresponding to given values of the macroscopic observables . then the system is let to evolve up to the first collision of the piston _ without _ interactions among the gas particles . after the collision , the energy of the gas containing the colliding particle changes ; the fast relaxation of the system is mimicked by updating the gas temperature ( corresponding to the new energy ) and redrawing an equilibrium configuration of the gases corresponding to the new temperatures ( and volumes ) . then the process is iterated . notice that in this way one keeps track of all the observables except the piston velocity fluctuation , which needs some kind of average to be defined / measured . collisions are evaluated , as for the ideal gas , with an event driven algorithm . in figure [ fig : evolution ] we show the evolution of the piston position by numerical simulations of the ideal and randomized gas for two different values of @xmath23 ( namely @xmath175 ( top ) and @xmath176 ( bottom ) . as discussed in the introduction ( see @xcite for a more detailed treatment ) these two choices correspond to the case of strongly and weakly damped oscillations , respectively . in both cases one has two regimes : mechanical and brownian . the former is characterized by damped oscillations of the piston , which in the strongly damped case are very few or nonexistent . for the ideal gas , as discussed by gruber et al . @xcite the detailed damping of the oscillations depends on the presence of shock waves , which are absent in the randomized gas . in fact , the latter is damped much more efficiently than the former . as shown in the bottom panel , for @xmath176 the evolution is weakly damped and many oscillations are observable . in this case , the piston is very slow and the gases perform quasi - adiabatic oscillations , which are damped more and more mildly as @xmath177 . again , also in this case the damping appear to be much more efficient in the randomized case . ) the dotted line indicates the adiabatic phase @xmath178 , while the solid straight line the isobar that characterizes the final phase ( brownian motor like regime ) . the rightmost arrow indicates the initial state of the gas in left compartment @xmath179 @xmath180 which evolves ( red dots ) toward the final equilibrium in the middle . the leftmost arrow indicate the initial state of the gas in left compartment @xmath179 @xmath181 which evolves ( blue dots ) toward the final equilibrium in the middle . the simulation is done with the ideal gas , for which the initial oscillations are much more evident that for the randomized one . , scaledwidth=49.0% ] the brownian motor - like regime @xcite , occurs when the oscillations are completed damped , i.e. the mechanical equilibrium is realized with approximately equal pressures @xmath182 , which differ only for terms @xmath31 . from now on , both the ideal and randomized gas remain in a state of marginal equilibrium approximately along the isobar @xmath183 , which is the prediction of thermodynamics ( shown in fig . [ fig : tv ] for the ideal gas only ) . the brownian motor - like mechanism is responsible for heat transfer from the warmer to the colder chamber mediated _ via _ the piston fluctuations @xcite . this stage occurs on a long timescale ( proportional to @xmath13 ) . for realistic values of the various parameters , one can realize that the timescales necessary for reaching the thermodynamic equilibrium state are enormous . it is therefore very difficult to observe this regime in experiments , and well controlled numerical simulations are mandatory . in the final state the two chambers have the same temperatures and the piston fluctuates reaching equipartition with the same temperature . it is worth noticing that for the ideal gas , since the molecules interact only through the collisions with the piston , the probability distribution of the velocities of the gas molecules may ( and actually do ) deviate sensibly from the maxwell - boltzmann distribution that is recovered only after the reaching of the final equilibrium state @xcite . by construction , this problem is not present in the randomized gas which is forced to remain maxwell - boltzmann . in the following we shall compare the evolution of the macroscopic equations with the simulations . we start the comparison by considering the oscillatory regime in the weakly damped case . here we know from previous studies @xcite that the model is able to correctly predict the period of the oscillations . the period of the oscillations in the model can be estimated from the linearized dynamics at the zeroth order in @xmath19 , eqs . ( [ eq : vxlin]-[eq : trlin ] ) . as discussed in sect . [ sec : expandedequations ] , the oscillations will occur around a mechanical equilibrium position @xmath102 defined by eq . ( [ eq : eqpoint ] ) . then to recover the period of the oscillations it is enough to linearize ( [ eq : vxlin]-[eq : trlin ] ) around the mechanical equilibrium state defined by @xmath102 , @xmath103 and @xmath184 . linearizing ( [ eq : tllin]-[eq : trlin ] ) one can easily recognize that the equation define an iso - entropic process , i.e. ( [ eq : adiabatic ] ) , meaning that @xmath185 and @xmath186^{c_p / c_v-1}$ ] , which plugged in ( [ eq : vxlin ] ) and expanding in @xmath187 lead to the equation of a damped oscillator : @xmath188 where we recall that @xmath99 and @xmath106 ( see sec . [ sec : mecheq ] ) . since the friction coefficient does not modify the period one immediately gets : @xmath189 this formula is at the basis of the measurement of the specific heat ratio in experiments @xcite . , @xmath190 , @xmath191 , @xmath192 , @xmath193 , @xmath194 corresponding to @xmath195 . ( top ) evolution of the piston position . ( bottom ) evolution of the temperatures . blue open squares refer to the ideal gas and red filled circles to the randomized gas , the solid line is the model ( [ eq2:vx]-[eq2:tr ] ) . ] , @xmath190 , @xmath191 , @xmath192 , @xmath193 , @xmath194 corresponding to @xmath195 . ( top ) evolution of the piston position . ( bottom ) evolution of the temperatures . blue open squares refer to the ideal gas and red filled circles to the randomized gas , the solid line is the model ( [ eq2:vx]-[eq2:tr ] ) . ] in fig . [ fig : oscill - x ] , we compare the ideal gas simulations with that of the randomized gas and the numerical integration of the determinist equations ( [ eq2:vx]-[eq2:tr ] ) . as one can see , the ideal and randomized gas have the same period , while the damping is different , and the model is in perfect agreement with the randomized gas simulations . integrating the equation at the zeroth order , we find @xmath196 which agrees with the predicted value ( [ eq : eqpoint ] ) and eq . ( [ eq : ruch ] ) predicts @xmath197 which is in very good agreement with the the measured period in both the randomized and ideal gas . ( top ) with superimposed the thick black line superimposing on randomized gas data ( blue empty circles ) is obtained by the numerical integration of ( [ eq2:vx]-[eq2:tr ] ) . the agreement of the model with the randomized gas is perfect also for the temperatures ( not shown ) . ] we mention that in ref . @xcite a detailed study of the period of the oscillations was reported . since our linearized equations coincide with those of ref . @xcite , we shall not repeat here this study . it is however interesting to compare the first stage of the evolution of the ideal and randomized gas with the model in the case of strong damping . in fig . [ fig : strong ] we show the leftmost plot of fig . [ fig : evolution ] ( top ) , as one can see although the model is unable to reproduce the oscillation of the ideal gas , its evolution coincides with that of the randomized gas . as discussed in ref.@xcite , estimating the decay rate of the oscillations for the ideal gas is a nontrivial task , since it requires a detailed study of the dissipation mechanisms of the shock waves created by the piston motion . these shock waves survive for a long time in the ideal gas , while they lifetime is expected to be shorter in the presence of interactions , which should be able to decrease their coherence . the randomized gas represents a sort of limiting case of interaction in which the shock waves are completely absent . most likely the absence of shock waves is at the origin of the faster damping of the oscillations in the randomized gas and of the very good agreement between its evolution and the one obtained from the macroscopic equations . it would be interesting to investigate the transition between weak and strong damping in the randomized model . this is far from the aim of the present paper , however the above results suggest that the critical value of @xmath23 for the transition will probably be smaller ( but of the same order ) of that of the ideal gas which is order unity . as the mechanical equilibrium is reached @xmath182 , the system slowly evolves , along an approximate isobar , driven by fluctuations toward the thermodynamical equilibrium defined by eq . ( [ fixedpoint ] ) , which is the ( stable ) fixed point of the ordinary differential equations ( [ eq2:vx]-[eq2:tr ] ) . , @xmath190 and @xmath198 with @xmath199 and @xmath200 with @xmath201 : red triangles refer to the randomized gas and the blue squares to the ideal gas , respectively . the dashed line is the prediction of the macroscopic equation , the solid line ( which perfectly superimposes on the randomized gas data ) is explained in the text . the simulation data are obtained by performing an average over about @xmath202 realizations to reduce the fluctuations . ] we shall now compare the evolution given by the macroscopic equations with that of the ideal and randomized gases in the brownian regime . to minimize the possible differences between the ideal and randomized gases we performed a simulation which starts in a mechanical equilibrium state having the gas molecules distributed according to the maxwell - boltzmann distribution . for the ideal gas , this might be not the typical situation : usually when the system arrives to the mechanical equilibrium from a non - equilibrium state , it may be strongly non - maxwellian @xcite . as exemplified in fig . [ fig : relax ] , where we show the deviation from the final equilibrium of the piston position , @xmath203 , the relaxation is exponential . the randomized gas relaxes faster than the ideal one , likely because the latter develops slightly non - maxwellian distribution ( we observed that the difference in the relaxation time tends to diminish as the number of particles is increased ) . note that the simulation has been performed with @xmath204 because the time scale for reaching the equilibrium is controlled by @xmath13 ; with @xmath201 , as here , working with @xmath205 would have implied the necessity to reach too large time scales to study the relaxation . as shown in the figure , the macroscopic equations ( dashed line ) predict a relaxation slower than both the randomized and ideal gases . this came out as a surprise for us , because we were encouraged by the very good agreement in the mechanical regime , discussed in the previous section . with the aim of understanding such a difference we examined all the terms appearing in the macroscopic equations and we realized that the mismatch in the relaxation was due to the terms @xmath206 that , as discussed in sect . [ sec : brown ] , are those which ensures the equipartition of energy at equilibrium , i.e. the fact that @xmath207 . in particular , eliminating such terms from ( [ eq2:vx]-[eq2:tr ] ) ( note that this is not affecting the conservation of energy ) , we found a perfect agreement between the macroscopic equations , now modified , and the randomized gas , as shown by the solid line in the figure . nevertheless , with such a modification the energy equipartition is not verified anymore and at equilibrium @xmath208 . ( bottom ) now plotted in log - lin scale . the ( red ) symbols refer to the randomized gas simulations , the dashed curve to the evolution predicted by the macroscopic equations and the solid curve ( which perfectly superimposes on the simulation data ) is explained in the text . ( bottom ) same as ( top ) but for the evolution of the gas temperatures @xmath209 and @xmath210 as in label . the two curved which start from zero show the evolution of the piston temperature @xmath211 as obtained from the macroscopic equations ( dashed curve which saturates to @xmath212 and as explained in the text the solid one which saturates ad @xmath213 . ] ( bottom ) now plotted in log - lin scale . the ( red ) symbols refer to the randomized gas simulations , the dashed curve to the evolution predicted by the macroscopic equations and the solid curve ( which perfectly superimposes on the simulation data ) is explained in the text . ( bottom ) same as ( top ) but for the evolution of the gas temperatures @xmath209 and @xmath210 as in label . the two curved which start from zero show the evolution of the piston temperature @xmath211 as obtained from the macroscopic equations ( dashed curve which saturates to @xmath212 and as explained in the text the solid one which saturates ad @xmath213 . ] one may think that the agreement is incidental , and so we did a more severe test . in fig . [ fig : relax2 ] we show the evolution of the piston position ( top ) and of the gas temperatures ( bottom ) with the simulations of the randomized gas previously shown in fig . [ fig : evolution](bottom ) . as one can see , the modified ( solid lines ) and original ( dashed lines ) macroscopic equations generate , a part from @xmath58 , two indistinguishable dynamics for @xmath37 and @xmath28 up to times for which the mechanical equilibrium is reached . actually for such times the dynamics is essentially given by the zeroth order equations ( [ eq : vxlin]-[eq : trlin ] ) which are the same for both the original and modified equations : this explain their behavior in the mechanical regime . as one can see the main difference in the dynamics is that retaining the terms @xmath214 leads the system to stay for a longer time interval in a state of approximate mechanical equilibrium and thus to a slower relaxation to equilibrium . the evolution of the piston temperature @xmath215 shown in fig . [ fig : relax2 ] ( bottom ) clearly show the difference in the two dynamics , and in particular the breaking of equipartition at equilibrium for the modified dynamics . it should be stressed that the behaviors shown in the above figures is not related to the peculiar parameters choice as it has been verified in other simulations ( not shown ) . the picture emerging from this comparison is that the model we introduced goes to the correct equilibrium state and respects the basic physics principles as conservation of energy and its equipartition . on the other hand , the relaxation is slower than the one of the microscopic model it should correspond to ( i.e. the randomized gas ) . as shown in the above figures the terms responsible for this slowing down are the ones coming from the fluctuation of the piston velocity in the expansion of the function @xmath125 . at present we dot not have a definite understanding either of this difference in the dynamics of the macroscopic equations and of the randomized gas , or of the very good agreement between the modified equations and the randomized gas . we suspect that the contribution to the relaxation time of these terms might be counterbalanced by the resummation of higher order terms in the @xmath19 expansion . however , carrying the analysis to higher order terms is very demanding due to the proliferation of terms in the expansion , and therefore we shall not discuss it here . we conclude this section by mentioning that the equations derived by gruber and coworkers @xcite predict a reasonable relaxation time and equipartition at the same time . we remark that going to higher order in their approach requires to include into the description also higher order moments of the piston velocity , namely @xmath216 ; while within our approach the description remains at the level of the second order moment because the statistics is constrained to remain gaussian ( maxwell - boltzmann ) for the gases and the piston . this may be the origin of the difference between our and their derivation . their equations at the first order might likely correspond to a resummation of higher order terms in our approach , meaning that their closure with the second moment of the velocity is exact while ours remains only approximate . in the framework of kinetic theory , we derived a set of deterministic equations describing the evolution of the macroscopic variables in the adiabatic piston problem . our basic assumptions are that at each time the gases in the two compartments are perfect , spatially homogeneous , and described by the maxwell - boltzmann statistics . thus , at the level of simulations , a ( randomized ) gas model has been introduced with aim to have microscopic model respecting such assumptions . we obtained a set of five ordinary differential equations for the variables that describe the macroscopic state of the system , namely the mean position of the piston , the average velocity of the piston , the temperatures of the gases in the two compartments and the second moment of the piston velocity . the equations are derived up to the first order in @xmath19 . at the zeroth order they describe a deterministic piston characterized by a velocity distribution collapsed on the mean , namely @xmath217 . this is enough to solve the problem of finding the final state of mechanical equilibrium and the result coincides with that derived in ref . @xcite by using a different approach . at the first order the fluctuations of the piston velocity , now assumed to be gaussian , allow for recovering the correct final thermodynamic equilibrium . although the evolution of the macroscopic observables provided by this set of equations is in good qualitative agreement with simulations of the randomized gas , we found some quantitative discrepancy for the relaxation timescales . apart from the performance in comparing with the simulations , we would like to stress the conceptual aspects of the method we developed . it allows for a transparent description of the macroscopic dynamics of a nontrivial non - equilibrium problem , similarly to how the perfect gas law can be derived from the microscopic collisions by using elementary kinetic theory . we thanks e. caglioti , f. cecconi , b. crosignani , p. di porto , a. lesne , g.p . morriss , and c. van den broeck for useful discussions , remarks and correspondence . this work has been partially supported by prin2003 `` complex systems and many body problems '' , and prin2005 `` statistical mechanics of complex systems '' by miur . s.p . has benefited from a mec - mius joint program ( italy - spain integral actions ) .
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a simplified version of a classical problem in thermodynamics the adiabatic piston is discussed in the framework of kinetic theory .
we consider the limit of gases whose relaxation time is extremely fast so that the gases contained on the left and right chambers of the piston are always in equilibrium ( that is the molecules are uniformly distributed and their velocities obey the maxwell - boltzmann distribution ) after any collision with the piston . then by using kinetic theory we derive the collision statistics from which we obtain a set of ordinary differential equations for the evolution of the macroscopic observables ( namely the piston average velocity and position , the velocity variance and the temperatures of the two compartments ) .
the dynamics of these equations is compared with simulations of an ideal gas and a microscopic model of gas settled to verify the assumptions used in the derivation .
we show that the equations predict an evolution for the macroscopic variables which catches the basic features of the problem .
the results here presented recover those derived , using a different approach , by gruber , pache and lesne in _
j. stat .
phys .
_ * 108 * , 669 ( 2002 ) and * 112 * , 1177 ( 2003 ) .
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this paper considers ( multivariate ) generating functions of the form : _ n=1^ _ g [ n ] ( g ) s^e(g ) , [ eq : genfn ] where @xmath2 indicates a subclass of graphs . @xmath3 is the number of edges a graph @xmath4 . @xmath5 $ ] indicates that the graph has vertex set @xmath6 $ ] and @xmath7 is a specified positive graph weight . the exponents of the variables @xmath8 and @xmath9 indicate the size of the vertex set , respectively , the number of edges . when evaluating at @xmath10 , there are some remarkable cancellations , leading , in some cases , to simple formul for the coefficients . this paper gives combinatorial explanations for the class of two - connected graphs in particular . two - connected graphs are those graphs for which we can remove any vertex and its incident edges and the resulting graph remains connected . there are four cases of used in mayer s theory of cluster and virial expansions , depending on the class of graphs considered and the weights . the sum is either over connected graphs , denoted by @xmath11 , or two - connected graphs , denoted by @xmath12 . the weights are either those for a discrete hard core gas , often referred to as the one - particle hard core gas , or for a continuum one - dimensional hard core gas , also named the tonks gas . for the discrete gas , the goal is to count the number of graphs ; for the continuum model , the coefficients are given by the volume of a polytope associated with the graph @xmath4 . we write a graph @xmath4 as the ordered pair of its vertex set and edge set as @xmath13 . we define the polytope corresponding to the graph @xmath4 as : _ g : = \{()_[2,n]^n-1 |_i-_j|<1 \{i , j } e(g ) } [ eq : graphpolytope ] , with @xmath14 . we use the notation @xmath15}:=(x_2 , \cdots , x_n)$ ] . mayer , in @xcite , established important connections between weighted graph generating functions and expansions in statistical mechanics . these connections are also presented in the framework of combinatorial species of structure in the work of ducharme , labelle and leroux @xcite , leroux and kaouche @xcite and faris @xcite . the results of mayer are that the weighted sum over connected graphs gives the pressure as a function of activity and the weighted sum over two - connected graphs is related to the virial expansion of pressure expanded in terms of density . the two formul are : @xmath16 } w(g ) \\ \beta p(\rho ) & = \rho - \sum\limits_{n=2}^{\infty}(n-1 ) \frac{\rho^n}{n ! } \sum\limits_{g \in \mathcal{b}[n]}w(g ) , \end{aligned}\ ] ] where @xmath17 is the graph weight specified by the particular model . the answers for the four cases are given by the formul for the connected graph discrete case : _ g [ n](-1)^e(g)= ( -1)^n-1(n-1 ) ! and for the connected graph continuum case : _ g [ n](-1)^e(g)(_g ) = ( -1)^n-1n^n-1 . there are also formul for the two - connected discrete case : _ g [ n](-1)^e(g)=-(n-2 ) ! [ eq : virial1phc ] and the two - connected graph continuum case : _ g [ n](-1)^e(g)(_g)=-n(n-2 ) ! [ eq : virialtg ] .for the discrete cases the results are straightforward computations . for the continuum case , derivations are given in @xcite . the statistical mechanical background is explained in full detail in section [ sec : models ] . it is tempting to try and find a simple combinatorial interpretation that explains the cancellations in a direct way . this was posed as a challenge in the paper of ducharme , labelle and leroux @xcite . in the connected graph cases , this was done by bernardi @xcite . the approach was to use an involution that exhibits the result of the almost perfect cancellation as a contribution from the fixed points of the involution . the fixed points were identified as increasing trees in the discrete case and rooted trees in the continuum case . the purpose of this paper is to present similar derivations for the two - connected graph cases . as always this is considerably more complicated . the concept of using an involution to understand the cancellations is natural . recall the formula that , for any finite non - empty set @xmath18 , we have : _ s x(-1)^|s|=0 . in order to prove this , we show we have the same number of sets with even cardinality as we do of odd cardinality . one approach is to pair sets differing by one element . this pairing idea is captured by the involution . in this example , the involution is defined by first fixing a singleton subset of @xmath18 , say @xmath19 , and taking the symmetric difference @xmath20 . if we consider a fixed vertex set @xmath5 $ ] for a graph , then a graph @xmath4 is determined precisely by its edge set @xmath21 , which are subsets of the collection of unordered pairs in @xmath5 $ ] , denoted @xmath5^{(2)}$ ] . we can also use this symmetric difference operation on the edge set for graphs . an important complication is that we consider particular subsets for which taking the symmetric difference with a fixed edge will not suffice , since the removal or addition of the edge may take us outside of the prescribed collection of subsets . we need to find an efficient way of choosing an edge based on the graph we are considering so that we obtain a pairing that will not take us outside of the prescribed collection . in section [ sec : results ] , we present the combinatorial structures that give the interpretations of the cancellations in the two - connected case . sections [ sec:1phcinv ] and [ sec : chcginv ] give the proofs of the one particle hard core and the tonks gas case respectively . in the latter , the decomposition of polytopes into unimodular simplices attributed to lass is given so that it may be proved as an extension of the previous case . we provide an interpretation why @xmath22 should appear as the number of edges in section [ sec : structure ] . from the perspective of statistical mechanics , the motivation for understanding such cancellations is to be able to adapt the understanding to models where more complicated weights are used . indeed , the key idea is to emulate what is done for the connections between connected graphs and trees and understand how to modify these in this context . the first parallel to draw is that the involution of bernardi fits within a general concept of externally and internally active elements of a set with a matroid structure as given by bjrner and sokal @xcite . the idea to emphasise here is that this allows the set of connected graphs to be partitioned into subsets , indexed by trees . when we consider graphs with the partial order defined by bond inclusion , the blocks in this partition are boolean . that is , each set has a tree @xmath23 as minimal graph and a corresponding maximal graph @xmath24 , all graphs with edge set @xmath25 such that @xmath26 are included in the set in the partition . this form of a partition lends itself well to performing estimates on the cluster coefficients . this was actually realised earlier by penrose @xcite in the specific case of connected graphs . understanding this partition into boolean subsets also gives rise to an alternative involution . it is intriguing to realise that the general construction does not include the penrose construction as a subcase . these ideas are addressed in section [ sec : extend ] . this combinatorial understanding is also closely linked to the tree - graph identities of brydges battle and federbush @xcite , for which a symmetric version is provided by abdesselam and rivasseau @xcite and a matroid generalisation by faris @xcite . these identities allow estimations to be made on these coefficients , since we may express the sum over connected graphs as a sum over trees with modified weights . a greater goal is to extend these to partially ordered sets where a matroid structure may not be present . interest in providing such bounds on the virial expansion coefficients has recently been renewed with the papers by pulvirenti and tsagkarogiannis @xcite and morais and procacci @xcite , which use the canonical ensemble as a method of achieving bounds . the paper by jansen @xcite suggests that at high temperatures the radius of convergence should be improved : actual improvements on the bounds of lebowitz and penrose @xcite have been proposed recently @xcite . in a classical gas system of @xmath0 indistinguishable interacting particles in a vessel @xmath27 with only two - body interactions and no external potential , we may write the hamiltonian as : h(,)=_i=1^n + _ 1 i < j n(q_i , q_j ) , where @xmath28 represents the generalised coordinates and @xmath29 the conjugate momenta . the canonical partition function of the gas model is : z(,,n)=_i=1^n ( _ ^dq_i _ ^d ^dp_i ) ( - h ) . integrating out the gaussian integrals for the momenta , we obtain a factor @xmath30 , where @xmath31 is the thermal wavelength . the partition function is therefore : z ( , , n)= _ i=1^n ( _ ^dq_i ) _ 1 i < j n(-(q_i , q_j ) ) [ eq : canonicalpfinvol ] . the mayer trick @xcite , allows us to rewrite the canonical partition function in terms of weighted graphs . the first stage is to define the mayer @xmath32-function : f(q_i , q_j):=(-(q_i , q_j))-1 .we realise that the product of exponentials in may be rewritten as : _ 1 i < j n ( -(q_i , q_j))=_1 i < j n(1+f(q_i , q_j))=_g [ n]_(i , j ) e(g)f(q_i , q_j ) , where @xmath33 $ ] is the set of simple graphs ( no multiple edges or loops ) on @xmath0 points . we write a graph @xmath34 , where @xmath35^{(2)}$ ] is the edge set and @xmath36 $ ] is the vertex set . this motivates the graph weight : w(g)=_i=1^n ( _ ^dq_i ) _ ( k , l ) e(g)f(q_k , q_l ) . we can therefore write the partition function as : z ( , , n)=_g [ n]w(g ) . in order to obtain the grand canonical partition function we sum : ( , , z)=_n=0^ z^n^n z ( , , n ) , where @xmath37 the activity and @xmath38 is the chemical potential . in terms of graphs , we write this as : ( , , z)=_n=0^_g [ n]w(g ) = : _ the pressure is defined to be : p= _ || ( , , z ) .if we define the new weight @xmath39 , then the pressure function can be written in terms of connected graphs : p = _ w(z)=_n=1^ _ g [ n]w(g ) [ eq : pressureconnectedinv ] .this is the content of mayer s first theorem @xcite and is explained in the paper @xcite . the density @xmath40 is : = z p = ^_w(z ) , where @xmath41 denotes a rooted connected graph . from mayer s second theorem @xcite or by the dissymmetry theorem @xcite , we are able to obtain a series expansion for pressure in terms of density , in which the coefficients are , up to a prefactor , the @xmath42-weighted two - connected graphs . p = - _ n=2^ _ g [ n]w(g ) [ eq : pressure2connectedinv].one may also consult the book by mccoy @xcite for an explanation of the derivation of these two theorems . the potential for a one - particle hard core gas is : ( q_i , q_j)= , so that @xmath43 and @xmath44 . the grand canonical partition function is : ( z)=1+z . the statistical mechanical relationships give pressure and density as : @xmath45 we may invert , to obtain : z = and substitute for @xmath8 in , to obtain : p = - ( 1- ) . the two series expansions derived from statistical mechanics are : @xmath46 if we compare these two power series with and respectively , using the graph weight @xmath47 , where @xmath3 is the number of edges in graph @xmath4 , we obtain : @xmath48}(-1)^{e(g)}&=(-1)^{n-1}(n-1 ) ! \\ \sum\limits_{g \in \mathcal{b}[n]}(-1)^{e(g)}&=-(n-2 ) ! .\end{aligned}\ ] ] for a continuum hard core gas in one dimension with diameter @xmath49 , the potential is : ( q_i , q_j)= & |q_i - q_j|<1 + 0 & . the exponential and mayer @xmath32-functions are : @xmath50 we therefore have the graph weight : w(g)=(-1)^e(g)_^n-1 _ \{i , j } e(g)(|x_i - x_j|<1 ) x_2 x_n , where @xmath14 and @xmath51 is the indicator function . in @xcite , this is interpreted as a the volume of a convex polytope @xmath52 in @xmath53 . the polytope is defined by : _ g = \ { ( ) _ [ 2,n ] ^n-1 |x_i - x_j|<1 \{i , j } e(g ) x_1=0 } . we use the notation @xmath54=\{2 , 3 , \cdots , n\}$ ] and @xmath15}=(x_2 , \cdots x_n)$ ] . hence the graph weight may be written as : w(g)=(-1)^e(g)(_g ) . the derivation of the cluster and virial expansions , using statistical mechanics , are more difficult in this case , but they are done in @xcite and we achieve : @xmath55 where @xmath56 is the lambert @xmath57-function . if we compare these to the results of mayer s first and second theorems , and , we obtain the combinatorial relationships : @xmath48}(-1)^{e(g)}{\mathop{\mathrm{vol}}}(\pi_g ) & = ( -1)^{n-1}n^{n-1 } \\ \sum\limits_{g \in \mathcal{b}[n]}(-1)^{e(g)}{\mathop{\mathrm{vol}}}(\pi_g ) & = -n(n-2 ) ! .\end{aligned}\ ] ] the results of this article are the combinatorial interpretations of the cancellations in the alternating sums of weighted two - connected graphs . [ thm:1phc ] the difference of two - connected graphs with an even number of edges and an odd number of edges is given by the following formula : _ g [ n](-1)^e(g)=-(n-2)!. this is proved through an involution @xmath58 , given in section [ sec:1phcinv ] , which effectively pairs graphs differing by only one edge , leaving some small collection of graphs fixed , which give the @xmath59 factor . the fixed graphs are formed from an increasing tree on the vertex set @xmath60 $ ] with the vertex @xmath0 adjacent to every other vetex . the number of increasing trees on @xmath60 $ ] is @xmath59 . the tree has @xmath61 edges and we add @xmath62 edges from the vertex labelled @xmath0 to achieve @xmath22 edges . this gives the definite minus sign and the combinatorial factor . an increasing tree is a labelled tree on which the sequence of vertex labels along all paths from the vertex labelled @xmath49 to the leaves form increasing sequences . an example of such a graph is shown in figure [ fig : increasingtree ] . [ thm : chcg ] when we add the polytope weights to the alternating graph sum , we achieve the following identity : _ g [ n](-1)^e(g)(_g)=-n(n-2 ) ! . this is proved through a collection of involutions @xmath63 . the index @xmath64 is related to the partition of the polytopes into areas of equal volume attributed to lass in @xcite . the meaning of @xmath64 is explained in subsection [ subsec : lasspolytope ] . the fixed points of these involutions occur only when @xmath64 is of the form @xmath65 , meaning that any edge is possible . there are precisely @xmath0 possibilities of these sequences , which corresponds to the @xmath0 positions of the last zero . the particular @xmath64 provides a bijection @xmath66 \to [ n]$ ] on which the fixed graphs correspond to an increasing tree ( given by the order @xmath67 if and only if @xmath68 ) on the labels @xmath69 . this is paired with every edge from @xmath70 to the vertices @xmath69 . the number of these increasing trees on @xmath62 vertices is @xmath59 and hence we obtain the factor @xmath71 . we notice that these graphs are on @xmath22 edges as above , which provides the minus sign . the two - connected case is necessarily more complicated than the connected case . first of all , minimal two - connected graphs do not all have the same number of edges for a fixed number of vertices as trees ( minimal connected graphs ) do . simply removing edges appropriately down to a minimal graph can not provide a combinatorial understanding as there will still be sign differences to take care of . furthermore , the sign of the factor is constant - the number of edges must always be odd for whatever value of @xmath0 we take . as indicated in the introduction , the proof of theorem [ thm:1phc ] is done through an involution . to explain how the involution @xmath58 provides the combinatorial factor through the number of fixed points , we use the manipulations of bernardi @xcite , where we know that the involution either adds an edge , removes an edge or leaves the graph fixed . we have that : _ g [ n](-1)^e(g)=_g [ n](-1)^e((g ) ) , since @xmath58 is a bijection . the sum of these is therefore : @xmath72}(-1)^{e(g)}&=\sum\limits_{g \in \mathcal{b}[n]}((-1)^{e(g)}+(-1)^{e(\psi(g ) ) } ) \notag \\ & = 2\sum\limits_{g \in \mathcal{b}[n ] \vert \psi(g)=g}(-1)^{e(g ) } . \end{aligned}\ ] ] the fixed points of the involution thus give us the combinatorial factor . this section describes the involution and proves it does what is required . for graphs , the analogous operation to symmetric difference explained in the introduction is the operation @xmath73 . @xmath74 is the graph @xmath75 . the specific task of the proof of both identities is to identify for each graph a unique edge that we can add or remove . this has to be done in a consistent and efficient manner . consistent in the sense that if we identify @xmath76 as the unique edge in @xmath4 , then we want @xmath77 so that @xmath58 is an involution . it needs to be efficient in the sense that the only graphs it leaves fixed are those that provide the combinatorial factor relevant for the alternating sum . we do not want further cancellations to consider . in each graph @xmath4 , we consider the vertex labelled @xmath0 . when the vertex @xmath0 is adjacent to every vertex , we realise that the collection of two - connected graphs with this property may be identified with the collection of connected graphs on the vertex set @xmath60 $ ] . bernardi @xcite has already provided an involution on this set that we can use in this case to obtain cancellations , since they will all come with the same prefactor @xmath78 from the @xmath62 edges from the vertex labelled @xmath0 . we thus firstly introduce the involution of bernardi and make rigorous the connection between connected graphs on @xmath60 $ ] and the particular subset of two - connected graphs where @xmath0 is adjacent to every other vertex . for those graphs where the vertex labelled @xmath0 is not adjacent to every other vetex , we may use the two - connected property of the graph to find an edge suitable for the involution . this is done through using a corollary due to whitney of menger s theorem and introducing a definition of permissible edges . we emphasise how these combine to give a complete involution and that the only contributions arise from the bernardi involution . firstly , we define the neighbourhood of a vertex @xmath79 in a graph @xmath4 . for a graph @xmath4 and a vertex @xmath79 , we define the neighbourhood of @xmath79 in @xmath4 as @xmath80 . we define the _ lexicographic order _ on edges @xmath81^{(2)}$ ] by : \{i , j } < \{k , l } \{i , j } < \{k , l } + \{i , j } = \{k , l } \{i , j } < \{k , l } . for a subset @xmath82 of a totally ordered set , we define @xmath83 . for a graph @xmath84 and an edge @xmath85 , we define @xmath86 with respect to the lexicographic order above . we give here bernardi s involution on connected graphs , since it used for the two - connected graph version . we write it for the vertex set @xmath60 $ ] as this is the form in which it will be used . an edge @xmath87^{(2)}$ ] is externally active for the graph @xmath88 $ ] , if there is a path in @xmath89 between the endpoints of @xmath85 . if a connected graph @xmath4 has an externally active edge , we define @xmath90 to be the maximal such edge . the involution @xmath91 \to \mathcal{c}[n-1]$ ] defined by bernardi @xcite , is given by : _ b : g g _ g & g + g & . the result of the involution is the following lemma . under the involution @xmath92 , only increasing trees are kept fixed . we introduce the following notation to simplify the formulation of the connection between two - connected graphs with vertex set @xmath5 $ ] , where the vertex labelled @xmath0 is adjacent to every other vertex , and connected graphs with vertex set @xmath60 $ ] . * for a graph @xmath84 , we denote by @xmath93 , the graph @xmath94 . * we denote the subset of two - connected graphs on vertex set @xmath5 $ ] with vertex @xmath0 adjacent to all other vertices by @xmath95 $ ] . we define the mapping @xmath96 \to \mathcal{c}[n-1]$ ] , by @xmath97 . we emphasise that removing a vertex and its incident edges from a two - connected graph leaves a connected graph and so defining the codomain of @xmath98 as @xmath99 $ ] is fine . the map @xmath96 \to \mathcal{c}[n-1]$ ] is a bijection . firstly it is injective . if @xmath100 , this means @xmath101^{(2)}=e(h ) \cap [ n-1]^{(2)}$ ] and since @xmath4 and @xmath102 $ ] , the remaining elements of @xmath21 and @xmath103 , namely @xmath104\}$ ] , are the same and so @xmath105 . this is surjective , since for any connected graph on @xmath60 $ ] , if we add the vertex labelled @xmath0 and all edges @xmath106 such that @xmath107 $ ] , the resulting graph is two - connected . if we consider removing any vertex @xmath108 from this new graph we see that every vertex is connected to every other vertex via @xmath0 . if @xmath0 is removed then it is connected by definition and hence it is two - connected . we define the inverse map of @xmath98 to be @xmath38 . a path is an alternating sequence of vertices and edges in a graph @xmath109 , which begins and ends with a vertex . the edges are written in terms of the preceding and following vertices : @xmath110 . two paths @xmath109 and @xmath111 are internally disjoint if the only common vertices or edges are the endpoints @xmath112 and @xmath113 . for an edge @xmath114 the endpoints are defined as the vertices @xmath79 and @xmath115 . the following result in the case @xmath116 is used to find an edge in each graph where @xmath117 $ ] , by using the fact that we have two internally disjoint paths between @xmath0 and some @xmath118 \setminus n_g(n)$ ] . this is a classical theorem of whitney @xcite based upon menger s theorem . [ thm : whitney ] a graph @xmath4 is @xmath119-connected if and only if every pair of vertices is connected by @xmath119 internally disjoint paths . we introduce the notion of permissible edges as those edges which have both endpoints in the neighbourhood of a vertex @xmath79 and can easily be understood as a chord in the graph @xmath4 , when we neglect any edges in @xmath120 . we will focus on the case when @xmath121 . given a ( two - connected ) graph @xmath4 and a vertex @xmath122 , such that @xmath123 , we define an edge @xmath81^{(2)}$ ] to be @xmath124-permissible if the following condition holds : @xmath125 there exists an @xmath126 , such that we have two vertex disjoint paths @xmath127 , intersecting each once in @xmath128 . the intersection vertices are the endpoints of @xmath85 . if a two - connected graph @xmath4 with @xmath36 $ ] has a @xmath129-permissible edge , then we denote the largest such edge in lexicographical ordering by @xmath130 . for every @xmath131 \setminus \mathcal{b}^{\delta}[n]$ ] , we have a @xmath129-permissible edge . we know that @xmath132 \setminus n_g(n ) \neq \emptyset$ ] , because we are outside of @xmath95 $ ] . if we choose some @xmath133 , then we know by theorem [ thm : whitney ] we have two internally disjoint paths between the vertices labelled @xmath134 and @xmath0 . both paths must hit @xmath135 at some point . when they first hit @xmath135 , then they could go straight to @xmath0 and so each path need only intersect @xmath135 in one place . this provides us with a permissible edge and so @xmath130 is well defined for every @xmath131 \setminus \mathcal{b}^{\delta}[n]$ ] . we define the involution @xmath136 \to \mathcal{b}[n]$ ] through bernardi s involution @xmath92 and the permissible edge concept . * if @xmath137 $ ] , we consider the graph @xmath138 . this is a connected graph and we may apply bernardi s involution to this subgraph and retain the vertex @xmath0 and its incident edges . + this can be written as @xmath139}:=\mu \circ \psi_b \circ \zeta$ ] . * if @xmath117 $ ] , then we define the involution @xmath140 . the first point to emphasise is that due to the bijection between @xmath95 $ ] and @xmath99 $ ] , we are able to obtain cancellations for these graphs in the same way as bernardi . we are left with increasing trees on the set @xmath60 $ ] and the vertex @xmath0 adjacent to every other vertex . we still need to prove that @xmath58 is indeed an involution . @xmath58 is an involution and its image is contained within @xmath141 $ ] . the fact this is true for @xmath142}$ ] follows from the proof of bernardi . if an edge is permissible , we note that it is a chord in a cycle within the graph @xmath4 . if we add an edge to a two - connected graph it remains two - connected . we prove below that if we remove a chord from a two - connected graph , then it remains two - connected . we denote the chord we are considering by @xmath143 , the original graph by @xmath4 and the graph @xmath144 by @xmath145 . we prove @xmath145 is two - connected by considering the effect of removing a vertex from @xmath145 . there are two cases : * @xmath146 and @xmath147 are connected as they are the same graphs as @xmath148 and @xmath149 respectively , which are connected since @xmath4 is two - connected . * if we consider another vertex @xmath119 . we assume for contradiction that the graph @xmath150 is not connected . we know @xmath151 is connected and the only difference is that we have the additional edge @xmath152 . this would then imply that @xmath79 and @xmath115 are in different connected components in @xmath150 . we know that @xmath79 and @xmath115 appear in a cycle in @xmath145 . this means if we remove one vertex then we still have a path between @xmath79 and @xmath115 , hence we obtain a contradiction unless @xmath150 is connected . the collection of permissible edges depends only on edges within @xmath132 \setminus n_g(n)$ ] , between @xmath82 and @xmath135 and edges involving @xmath0 . adding or removing a permissible edge does not change the available edges on which one can make the two internally disjoint paths . hence the collection of permissible edges for @xmath4 and @xmath153 are the same . this means that the largest elements in each set are the same i.e. @xmath154 . therefore it is an involution . hence , @xmath58 is an involution and has only fixed points in the set @xmath95 $ ] . the fixed points are those given by bernardi as increasing trees on the vertex set @xmath60 $ ] with @xmath0 adjacent to all vertices in @xmath60 $ ] . in order to deal with the polytope volume weights , we decompose the polytopes into simplices . this first appeared in @xcite and is used in @xcite to prove the connected graph case . this splitting of polytopes into unimodular simplices is attributed to lass . this subsection explains how this splitting of the polytopes into simplices is used to construct the involution for the continuum case . these ideas are important in reducing the case of the tonks gas to the one particle hard core model . the key idea is to split @xmath53 into @xmath155-simplices of equal volume . we then realise that a polytope either fully contains a simplex , intersects only on the boundary of the simplex or is disjoint from the simplex . the sum is then reorganised so that we may sum over each simplex on the outside and then undertake the alternating sum on the restricted set of graphs whose associated polytopes contain the simplex considered . consider @xmath15 } \in \mathbb{r}^{n-1}$ ] and let @xmath156 be the integer part of @xmath157 and @xmath158 be the fractional part such that @xmath159 . let @xmath160 \to [ 2,n]$ ] be a bijection . we may define the simplex @xmath161 , by the set of @xmath162 with integer part @xmath64 and whose fractional parts satisfy : @xmath163 . this simplex has volume @xmath164 . the condition @xmath165 is equivalent to @xmath166 . we therefore have that @xmath167 if and only if for all @xmath168 , we have that @xmath169 with @xmath170 and @xmath171 . for any graph @xmath172 $ ] , the value @xmath173 counts the pairs @xmath174 and @xmath175 such that @xmath176 is a subpolytope of @xmath52 . we may rearrange the sums over connected or two - connected graphs of the graph weights by first casting the sum as a sum over the pairs @xmath177 and symmetrising the weight over isomorphic graphs . the symmetrisation procedure can be understood by considering a permutation @xmath178 of @xmath54 $ ] and defining for any vector @xmath179 , @xmath180 . for any graph @xmath4 with labels in @xmath5 $ ] , the graph @xmath181 is the graph , with the same vertex set and satisfies @xmath182 . @xmath167 if and only if @xmath183 for any permutation @xmath178 of @xmath54 $ ] . this equivalence can be elucidated by rewriting @xmath184 and @xmath185 . this allows us to rewrite the latter statement as : @xmath186 . this implies , for the entries in vector @xmath187 , that @xmath188 , @xmath189 . since @xmath190 , we may rewrite this as : @xmath191 @xmath192 . we make the identification that @xmath193 to see that we get precisely the statement that @xmath194 . we let @xmath2 denote either @xmath11 or @xmath12 and then we rewrite : @xmath195 \\ \pi(\mathbf{h } , \sigma ) \subset \pi_g}}(-1)^{e(g ) } & = \sum\limits_{\substack{\mathbf{h } \in \mathbb{z}^{n-1 } \ , g \in \mathcal{h}[n ] \\ \pi(\sigma^{-1}(\mathbf{h}),\text{id } ) \subset \pi_{\sigma(g)}}}(-1)^{e(g ) } \notag \\ & = \sum\limits_{\substack { \mathbf{h } \in \mathbb{z}^{n-1 } \ , g \in \mathcal{h}[n ] \\ \pi(\mathbf{h } , \text{id } ) \subset \pi_g}}(-1)^{e(\sigma^{-1}(g ) ) } \notag \\ & = \sum\limits_{\substack{\mathbf{h } \in \mathbb{z}^{n-1 } \ , g \in \mathcal{h}[n ] \\ \pi(\mathbf{h } , \text{id } ) \subset \pi_g}}(-1)^{e(g ) } \end{aligned}\ ] ] we may therefore , understand the weight as : @xmath196}w(g)&=\sum\limits_{g \in \mathcal{h}[n]}(-1)^{e(g)}{\mathop{\mathrm{vol}}}(\pi_g ) = \frac{1}{(n-1)!}\sum\limits_{\substack { \mathbf{h } \in \mathbb{z}^{n-1 } \sigma \in s_{n-1 } \\ \text{such that } \pi(\mathbf{h},\sigma ) \subset \pi_g}}(-1)^{e(g ) } \notag \\ & = \sum\limits_{\substack{\mathbf{h } \in \mathbb{z}^{n-1 } \ , g \in \mathcal{h}[n ] \\ \pi(\mathbf{h},\text{id } ) \subset \pi_g}}(-1)^{e(g ) } \end{aligned}\ ] ] we define the _ centroid _ of the vector @xmath64 , by @xmath197 , where @xmath198 and @xmath199 . we define @xmath200 as the graph on @xmath5 $ ] where the edges are all pairs @xmath114 such that @xmath201 . we define @xmath202:=\{g \in \mathcal{h}[n ] \vert e(g ) \cap e(k_{\mathbf{h}})=e(g ) \}$ ] where @xmath2 can be replaced by @xmath11 or @xmath203 . the final sum indicates that we need to count pairs @xmath64 and @xmath4 such that @xmath204 . that is that the centroid @xmath205 , since @xmath206 is in the interior of @xmath207 . this can be recast as : for @xmath205 , we require that : \{i , j } e(g ) _ [ n](-1)^e(g ) we can thus consider the total sum as first a sum over the subset of graphs @xmath202 $ ] for each @xmath64 and add the results . this leads to considering separate @xmath208 \to \mathcal{b}_{\mathbf{h}}[n]$ ] which are involutions and finding their fixed points . we define an involution @xmath209 for each @xmath174 on the set @xmath210 $ ] of two connected graphs , which are compatible with the vector @xmath64 . we note that , by the definition of @xmath210 $ ] , edges with @xmath211 are forbidden . we call an edge @xmath212 such that @xmath213 allowed . in order to make the connection with the proof in the discrete case , we indicate a bijection @xmath214 related to the particular @xmath64 that provides a suitable relabelling of the vertices to allow for an efficient application of the lemmas of section [ sec : chcginv ] to a relabelled graph . we reframe the consequences of these lemmas in the context of the allowed edges . it is important to check that an edge we may want to add or remove by the prescription in section [ sec : chcginv ] is allowed within the specific collection of graphs @xmath210 $ ] . it is then proved that when we have a non empty set of forbidden edges , all terms cancel . in the case when the set of forbidden edges is empty , we obtain the exact values taken by @xmath64 and everything reduces to the discrete gas case with a relabelling . we have a definite order on the entries of @xmath206 , since each entry has a different fractional part . we define a re - ordering of the set @xmath5 $ ] , through a bijection @xmath215 \to [ n]$ ] . this re - ordering is defined through the order for the entries of @xmath206 : @xmath216 . the re - ordered lexicographic order on edges is given by : \{_(i),_(j ) } < \{_(k),_(l ) } \{i , j } < \{k , l } + \{i , j } = \{k , l } \{i , j } < \{k , l } . instead of considering @xmath129-permissible edges , we consider @xmath217-permissible edges since it makes the formulation of the involution easier . [ lem : permisallowed ] all edges @xmath218 are allowed . we realise that @xmath219 we have that @xmath220 and so for every pair @xmath221 , @xmath213 and hence the edge is allowed in @xmath210 $ ] . all @xmath217-permissible edges are allowed . if @xmath222)$ ] , then @xmath210=\mathcal{b}[n]$ ] and @xmath64 is of the form of an initial sequence of zeroes with remaining entries @xmath223 . by lemma [ lem : permisallowed ] , all edges in @xmath224)^{(2)}$ ] are allowed . the edges @xmath225 for all @xmath226)$ ] are already in the graph and so can not be forbidden . hence every edge is allowed and so @xmath210=\mathcal{b}[n]$ ] . since @xmath199 , this means @xmath227 for all @xmath79 . we also note that if @xmath228 , then @xmath229 for all @xmath230 . this arises from the fact that the entries of @xmath64 are restricted to @xmath231 . for a negative entry we will have @xmath232 , which is not within distance @xmath49 of the value @xmath233 for any @xmath230 this means that @xmath64 is of the special form of an initial sequence of zeroes with the remaining entries @xmath223 . we define the set @xmath234 $ ] as the collection of two - connected graphs where @xmath235 is adjacent to all other vertices . we have the corresponding maps @xmath236 and @xmath237 between @xmath234 $ ] and @xmath238 \setminus \{\xi_{\mathbf{h}}(n)\}]$ ] , which are the same as in section [ sec : chcginv ] , except we are removing the vertex @xmath235 instead of @xmath0 . formally , we can write these bijections as a conjugation with @xmath214 , when interpreted as its action on graphs . in this case : @xmath239 we define the modified bernardi involution @xmath240 as in section [ sec : chcginv ] , except @xmath89 is interpreted in the sense of the re - ordered lexicographic ordering and for @xmath90 to be maximal externally active edge we use this ordering too . this can also be simply written using the graphical label conjugation : _ b , : = _ _ b _ ^-1 the largest ( using the re - ordered lexicographic order ) @xmath241-permissible edge is denoted by @xmath242 . we define @xmath209 as the involution on @xmath210 $ ] , defined by : * if @xmath222)$ ] , then we may use a modified version of bernardi , since all edges are possible in @xmath243 . ^_[n ] : = _ _ b , _ * otherwise , we have a permissible edge and can perform the involution @xmath244 . @xmath209 retains the property of being an involution on two - connected graphs as in section [ sec : chcginv ] . we are thus left with only those graphs that have @xmath222)$ ] and are increasing with respect to the re - ordered lexicographic order . the only @xmath64 vectors that contribute are those with an initial sequence of zeros followed by @xmath223s . there are @xmath0 possibilities of these sequences , since the final @xmath245 can appear in any of the entries @xmath246 . the permutation @xmath214 related to the @xmath64-vector with @xmath247 for @xmath248 and @xmath249 for @xmath250 takes the special form : _ : s n. we observe that the entry @xmath251 has the smallest value so @xmath252 . we then note that the following entries are negative and are in increasing order . the preceding entries are also in increasing order but are positive . hence we have @xmath253 for @xmath254 and @xmath255 for @xmath256 , which can be written in the form in the lemma for brevity . hence we have a precise collection of two - connected graphs . we have the examples from section [ sec : chcginv ] with these linear relabellings . for @xmath257 , the fixed graphs are all distinct . we indicate that there are no labelled graph automorphisms of the form of @xmath214 described above for the increasing trees on @xmath60 $ ] with @xmath0 adjacent to every vertex . the first observation is that @xmath214 has no fixed vertex labels . we know the degree of the vertex labelled @xmath0 is @xmath62 . if we were to have an automorphism with no fixed labels , then we require another vertex of the same degree to send @xmath0 to . this means we need a vertex in the increasing tree adjacent to all other vertices in the increasing tree . when a tree has at least three vertices , only one vertex can be adjacent to the rest , since if we have two vertices adjacent to all vertices we have them adjacent to each other and some third vertex . this creates a @xmath258-cycle contradicting the fact a tree is acyclic . furthermore , in this increasing tree , this vertex can only be the vertex labelled @xmath49 or @xmath259 . for any @xmath260 $ ] , @xmath119 can not be attached to both @xmath49 and @xmath259 , or else we will have a @xmath258-cycle , as we always have the edge @xmath261 . we therefore require that the graph automorphism exchanges the labels of the two vertices . the automorphisms are translations and since @xmath262 or @xmath263 , we have to translate by @xmath49 or @xmath259 , but then the vertex labelled @xmath49 or @xmath259 would not be relabelled as @xmath0 as we would require . hence @xmath214 is not an automorphism for any of the prescribed graphs and so the collection of these graphs for @xmath257 are all distinct . in this section , we indicate how the structure of two - connected graphs indicates the importance of graphs with @xmath22 edges . firstly , we explain some preliminary concepts about block cutpoint trees and then use these to explain why minimal two - connected graphs , that is a two - connected graph , such that the removal of an edge renders the graph no longer two - connected , on @xmath0 vertices have at most @xmath264 edges . in this section , we introduce the notion of a block cut - point tree and state a result relating the number of vertices in the individual blocks to the number of vertices in the whole graph . we use the notation @xmath265 to denote the collection of trees . * an _ articulation point _ in a connected graph is a vertex , which when it and its incident edges are removed , renders the graph disconnected . a synonym that is frequently used is a _ cutpoint_. * a _ two - connected graph _ is a connected graph without articulation points . * a _ block _ is a maximal two - connected subgraph of a connected graph . maximal in terms of edges and vertices it includes . the block cutpoint tree ( bc - tree ) associated to a connected graph @xmath4 is a ( bipartite ) graph where the vertices represent the articulation points and the blocks in a connected graph . an edge , between an articulation point and a block , is present in this graph , when an articulation point is contained in a block . it is a tree , since if there were a cycle in this graph then the cycle itself would have been a block . an example of a block cutpoint tree is shown in figure [ fig : bctreethesis ] . to define the centre of a tree formally , we define first the _ eccentricity _ @xmath266 of a vertex @xmath267 as the minimal graph distance of @xmath267 to a leaf . this may be formally written as @xmath268 , where @xmath269 indicates the hamming or graph distance in the tree . the centre of a tree is the collection of vertices at which the maximum eccentricity is attained . this can either be two neighbouring vertices or a single vertex . in the former case , we often call the edge between the vertices the centre of the tree . one can apply the function @xmath270 , which for any given tree , removes all leaves and the edges incident to the leaves . formally , we can write this as : f:(v(),e ( ) ) ( v ( ) l , e ( ) ( l v ( ) ) ) , where @xmath271 , the collection of leaves . repeated application of @xmath32 , gives a sequence of trees , @xmath272 which becomes constant either when we have a single vertex or the empty graph . in the case of the single vertex , this is the centre of the tree . for the empty graph , the penultimate step will have been two vertices and an edge . this edge or the pair of vertices is defined as the centre . a bc - tree is bipartite with all leaves in one set ( the blocks ) . it therefore has a unique centre , since the eccentricity of the articulation points will be odd and the eccentricity of the blocks will be even so two neighbours can not have the same maximum eccentricity . since we have a unique vertex at the centre of the bc - tree , we may define a digraph arising from the bc - tree , where the edge is oriented to point away from the centre . an example is displayed in figure [ fig : digraph ] . [ lem : blockdecomp ] if we decompose a connected graph on @xmath0 vertices into its block structure and let @xmath273 index the collection of blocks and @xmath274 be the sequence of block sizes , then we have the following equality : _ i i(k_i-1)=n-1 [ eq : bctreeidentity ] . the key idea is to indicate what vertex we omit inside each block on the left hand side of . the digraph gives an ( essentially ) unique prescription of the missing vertex in each block and in which block an articulation point is counted . the digraph comprises of two types of directed edge @xmath275 and @xmath276 , where @xmath134 indicates an articulation point and @xmath277 a block . the arrow points from the first entry to the second entry . since there is a unique path from the centre to every other vertex , every vertex has precisely one edge in which they are the second entry . there are two key cases : * * the centre is an articulation point * + for a block , @xmath277 , the unique vertex we neglect on the left hand side of is the articulation point , @xmath134 , where @xmath276 is the directed edge in the digraph . + every articulation point , @xmath278 , except the centre appears in an edge @xmath279 , for which it is the second entry , meaning it is enumerated in the left hand side of in precisely one block . the central articulation point is the only neglected vertex , which gives the right hand side of . * * the centre is a block * + in this case every block , except the centre , can be given the prescription as for the first case . for the central block , we can choose precisely one of its neighbours to neglect . all articulation points in this case have an edge in which they are the second entry and so are counted , excepting the articulation point identified by the central block . therefore , we have . to understand why the two - connected graphs on @xmath0 vertices with @xmath22 edges play a special role , we first indicate that two - connected graphs with at least this number of edges can not be minimal . given a graph @xmath4 on the vertex set @xmath5 $ ] , we denote by @xmath280 , the degree of the vertex labelled @xmath49 . two - connected graphs on @xmath0 vertices with @xmath281 edges are not minimal , that is they necessarily have a chord . this is done by induction on the number of vertices @xmath0 . the cases @xmath282 are vacuous and one can see from the examples in figure [ fig:5edges4vertices ] that this holds when @xmath283 . the connected graph @xmath284 may be decomposed into its bc - tree . each block with @xmath285 vertices in the tree has to have @xmath286 edges or else we have a smaller graph which has a chord by induction . we note here that blocks of size @xmath259 or @xmath258 need to be treated separately . we let @xmath287 denote the size of the @xmath79th block not of size @xmath259 or @xmath258 and @xmath288 and @xmath289 denote the number of blocks of size @xmath259 and @xmath258 respectively . we have from lemma [ lem : blockdecomp ] : _ i(l_i-1)+b_2 + 2b_3=n-2 the total number of edges in @xmath290 must then not exceed : _ i2(l_i-1)-2b_4+b_2 + 3b_3 2n-4 - b_2-b_3 - 2b_4 where @xmath291 indicates the number of blocks with more than four vertices . we know that @xmath292 and so we obtain the inequality : d_1 1+b_2+b_3 + 2b_4 1 + if we have only one block , then we either have two neighbours of @xmath49 and can apply induction to this block , as it will be a two - connected graph on @xmath62 vertices and at least @xmath293 edges . if we have at least three neighbours of @xmath49 in a block , say @xmath278 , @xmath294 and @xmath295 , then we may find a path @xmath296 . this follows from theorem [ thm : whitney ] , since we have two internally disjoint paths between @xmath278 and @xmath294 and between @xmath294 and @xmath295 . if we go along one of the paths between @xmath278 and @xmath294 until we first hit one of the two paths between @xmath294 and @xmath295 , from here we follow the path towards @xmath294 and then take the disjoint path to @xmath295 , this is then a path between @xmath278 and @xmath295 that goes via @xmath294 and does not self - intersect . in this case @xmath297 forms a chord . the final case is if we have at least two blocks and at most two neighbours of @xmath49 in a block . then we have a block with two neighbours of @xmath49 call them @xmath278 and @xmath294 and we have a third neighbour of @xmath49 , @xmath295 in some other block . let @xmath298 be the articulation point of the block containing @xmath278 and @xmath294 closest to @xmath295 . we have a path from @xmath298 to @xmath295 outside of this block since it is a connected graph . we are also able to construct a path @xmath299 since they are all in one block . concatenating these paths gives again a path @xmath296 from which we determine @xmath297 is a chord . it is also possible to construct a graph with @xmath0 vertices and @xmath264 edges that is minimally two - connected , as shown in figure [ fig : minimally2conn ] . the number of edges being @xmath22 marks some transition in the possibility of being minimal . edges and @xmath0 vertices which is minimally two - connected ] edges and @xmath0 vertices which is minimally two - connected ] in this section , we convey the connection between involutions and partition schemes for connected graphs and how the latter is used to give estimations of the coefficients in the expansions . this is used as motivation to consider whether the two - connected graph involution may have such a connection . the paper @xcite presents the notion of the partition in the sense of penrose and gives the general idea of a partition . we define a partial order of @xmath300 $ ] by bond inclusion : @xmath301 @xmath302 @xmath303 . for @xmath304 , we define the set @xmath305=\{k\vert \ , g \leq k \leq h\}$ ] the penrose construction partitions the set of connected graphs into subsets of the form @xmath306 $ ] , where @xmath307 \to \mathcal{c}[n]$ ] . many different constructions can be used to achieve an @xmath308 . penrose gave one explicit example in @xcite . a partition scheme for the set of connected graphs @xmath300 $ ] is any map @xmath309 \to \mathcal{c}[n]$ ] @xmath310 , such that : * @xmath311 and * @xmath300 $ ] is the disjoint union of the sets @xmath312 $ ] for @xmath313 $ ] . the penrose scheme is as follows : for any vertex @xmath79 of @xmath313 $ ] , we denote by @xmath314 the tree distance between the vertices @xmath79 and @xmath49 . we let @xmath315 be the predecessor of @xmath79 i.e. @xmath316 and @xmath317 . we associate to @xmath23 , the graph @xmath318 found by adding ( only once ) to @xmath23 all edges @xmath319^{(2)}$ ] such that either : * @xmath320 edges between vertices at same generation * @xmath321 and @xmath322 edges between vertices one generation away for a partition scheme @xmath308 , denote by @xmath323 \vert \ , r(\tau)=\tau \}$ ] the set of @xmath308-trees . in particular , @xmath324 is the set of penrose trees . the following proposition emphasises where the boolean partition offers advantages to providing estimations . in models where we have soft repulsion ( a positive potential ) , the mayer @xmath32-function satisfies @xmath325 . using a partition scheme , we have the bound : _ g [ n]_e e(g)f_e _ e e()|f_e| for any numbers @xmath326^{(2)}}$ ] , we have : @xmath48 } \prod\limits_{e \in e(g ) } f_e & = \sum\limits_{\tau \in \mathfrak{a}[n ] } \prod\limits_{e \in e(\tau)}f_e \sum\limits_{f \subset e(r(\tau ) ) \setminus e(\tau ) } \prod\limits_{e \in f } f_e \notag \\ & = \sum\limits_{\tau \in \mathfrak{a}[n]}\prod\limits_{e \in e(\tau)}f_e \prod\limits_{e \in e(r(\tau))\setminus e(\tau)}(1+f_e ) \label{eq : penroserewrite } \end{aligned}\ ] ] when we take the absolute value of the right hand side , we may use the triangle inequality and bound the second product in by @xmath49 . in the hardcore case , the second product in is zero unless @xmath327 , giving that the fixed points of this @xmath308 function also give a combinatorial interpretation of the cancellations.the alternative combinatorial interpretation of fixed points provided by penrose trees is that , considering the tree as being rooted at @xmath49 , we are required to have precisely one vertex in each generation . this necessarily gives a linear tree . we have to determine the positions of @xmath328 $ ] , which are defined uniquely by their distance from @xmath49 , which corresponds to a bijection , @xmath329 \to [ n-1]$ ] , giving the @xmath330 factor . to define the penrose involution arising from the penrose construction , we make the following definition of a penrose active edge . for a graph @xmath4 , we define the hamming distance between vertices labelled @xmath79 and @xmath115 as @xmath331 which is the length of the shortest path between @xmath79 and @xmath115 . an edge @xmath114 is called _ penrose active _ for @xmath4 if , either : * @xmath332 or * @xmath333 and @xmath334 such that @xmath335 with @xmath336 . we let @xmath337 be the greatest penrose active edge for @xmath4 in lexicographic order . the mapping : _ : g g e_,g^ & g + g & is an involution on connected graphs . we first prove that @xmath338 . the two graphs @xmath4 and @xmath339 differ only on an edge @xmath340 , where @xmath341 . throughout this proof in the case where we have equality , we assume without loss of generality that @xmath342 . for any @xmath119 , we consider the distance from the vertex labelled @xmath119 to @xmath49 in both graphs . this is defined through the shortest path from @xmath49 to @xmath119 . we indicate that for any path between @xmath49 and @xmath119 containing the edge @xmath114 we can find a path of the same or shorter length that does not contain this edge . if @xmath343 , then considering a path from @xmath49 to @xmath119 up to this edge , we realise that the shortest length the path up to this edge can be is @xmath344 , but we know that there is a shorter path to this endpoint because @xmath343 and so we can replace this initial path with a shorter path . we are left with the case @xmath345 . we know from property @xmath346 that there is some @xmath315 such that @xmath336 and @xmath347 is an edge in both graphs . therefore if the initial segment of a path includes the edge @xmath114 , then the shortest this can be is @xmath348 . if the initial segment ends at @xmath115 rather than @xmath79 then we know we have a shorter path to @xmath115 that we can replace this initial segment by . otherwise it ends at @xmath79 . we know that we have a path of length @xmath349 to @xmath315 on which we can attach the edge @xmath350 to construct a new path of the same length but not using this edge . we now have that condition @xmath351 for penrose active edges is the same in both graphs , since the graph distance is the same . we now indicate that an edge satisfies condition @xmath346 independent of the presence of @xmath114 . we realise if @xmath114 was added or removed satisfying @xmath351 then it has no effect on an edge satisfying @xmath346 , since @xmath346 depends on edges between generations . therefore , we consider that @xmath114 satisfies @xmath346 . since @xmath333 , we have an @xmath322 such that @xmath347 is an edge in both graphs and @xmath336 . this means that if we use @xmath115 to invoke applying condition @xmath346 for an edge to be penrose active , then we can invoke it in both cases by using @xmath315 . we can also go the other way and find a bernardi construction to provide an appropriate partition . the map @xmath309 \to \mathcal{c}[n]$ ] , which adds to @xmath23 all externally active edges for the given tree graph @xmath23 is the appropriate partition scheme . this is explained in the context of matroids below . in the work of bjrner and sokal @xcite , it is explained that for a matroid @xmath352 , where we give a total order to the underlying set @xmath353 , we may find a partition of the collection of subsets of @xmath354 according to the matroid structure . we introduce below some key definitions for matroids to introduce this connection , which can be found in the book of oxley @xcite and the work of faris @xcite . a matroid @xmath352 on the ground set @xmath355 is defined by a collection of independent subsets , denoted @xmath356 . these subsets must satisfy the following three axioms : 1 . @xmath357 ( non empty ) 2 . if @xmath358 and @xmath359 then @xmath360 ( downward closed ) 3 . if @xmath358 and @xmath361 and @xmath362 , then there exists @xmath363 with @xmath364 ( augmentation property ) for a graphical matroid , the ground set is @xmath5^{(2)}$ ] . we define the independent sets as forests or acyclic graphs . a maximal independent set @xmath365 is called a basis . the set of bases is denoted @xmath366 . the maximal independent sets for a graphical matroid are therefore trees . the rank of a matroid @xmath352 , @xmath367 is the cardinality of a basis element . all bases have the same cardinality and so the rank is well defined . a matroid can be defined by its set of bases , since @xmath368 if and only if @xmath369 , for some @xmath370 . given a matroid , @xmath352 , consider @xmath371 . there is a matroid @xmath372 , which is the restriction of @xmath352 to @xmath18 . it has ground set @xmath18 and @xmath373 . for @xmath371 , the rank of @xmath18 , @xmath374 is the rank of the matroid @xmath372 or alternatively the cardinality of the largest independent subset of @xmath18 . we note that @xmath375 if and only if @xmath18 is independent , so the rank function completely determines the matroid . the dual of a matroid is defined on the same ground set , but has a dual rank function rk@xmath376 , defined by : ^(a):=|a|-(e)+(e a ) let @xmath12 be the set of bases for @xmath25 . the dual basis set is then @xmath377 . we fix a total order on @xmath25 in the following . let @xmath378 . an element @xmath379 is _ externally active _ on @xmath277 if @xmath85 is dependent on the list of elements of @xmath277 larger than it . we let @xmath380 be the set of externally active elements . an element @xmath381 is _ internally active _ on @xmath277 , if in the dual matroid @xmath85 is externally active on the complement @xmath382 . we denote by @xmath383 the set of internally active elements . for @xmath384 , we define @xmath385=\{a \vert r \subseteq a \subseteq s\}$ ] . @xmath386 can be written as the disjoint union : 2^e=_b [ b(b),b ( b ) ] [ eq : c1 ] . for the case of the graphical matroid , we recall that the bases are the collection of trees . if we use the lexicographical order on the edges , then an edge is externally active for a tree @xmath23 in this sense , if and only if it is externally active in the sense of bernardi @xcite . this is due to the fact that all independent sets are forests and so a set of edges is dependent if it creates a cycle . we emphasise that for connected graphs , internally active edges play no role , since trees are minimally connected graphs . this therefore gives , when we intersect each set with connected graphs : [ n]=_[n ] [ , r ( ) ] , where @xmath24 has edge set @xmath387 . we note that the penrose construction does not fit in the construction given above . in figure [ fig : penrosenoorder ] , we see that we would add the dashed edge in each case . in order to do this , we can not have a consistent ordering on the edges @xmath388 , @xmath389 and @xmath390 . the motivation of emphasising this connection is to understand if a similar connection may be drawn for two - connected graphs as the important context of the result . the main conclusion is that we are able to identify combinatorially the cancellations in the alternating sums of weighted two - connected graphs . the combinatorial factor arises from increasing trees on the subset @xmath60 $ ] of the vertices , with the vertex @xmath0 adjacent to every other vertex . there are modified versions for this in the case of the polytope , where we have the isomorphic graph structures , differing only through a relabelling in the form @xmath391 for all @xmath392 $ ] . the key outlook for the work contained in this paper is to modify the set up explained in section [ sec : extend ] towards two - connected graphs so that we obtain a helpful resummation of the graphs amenable to suitable estimation , which is important for the virial expansion . the parallel that is useful to draw here is that for the cluster expansion , we have the increasing and cayley trees as the combinatorial objects representing the two cases above . it has been shown by groeneveld @xcite that these examples provide the extreme cases for positive potentials and an adaptation is available for stable potentials . * acknowledgements . * the author would like to thank the anonymous referees for constructive comments in improving this article . the work for this paper has been funded by epsrc grant ep / g056390/1 and sfb tr12 . the author acknowledges helpful discussions with d. brydges , r. koteck , d. ueltschi and s. jansen for particular discussions relating to this paper . a. bjrner , homology and shellability of matroids and geometric lattices in _ matroid applications _ , n. white ( ed . ) encyclopdia of mathematics and its applications , vol . cambridge university press , ( 1992 )
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mayer s second theorem in the context of a classical gas model allows us to write the coefficients of the virial expansion of pressure in terms of weighted two - connected graphs .
labelle , leroux and ducharme studied the graph weights arising from the one - dimensional hardcore gas model and noticed that the sum of these weights over all two - connected graphs with @xmath0 vertices is @xmath1 .
this paper addresses the question of achieving a purely combinatorial proof of this observation .
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let @xmath0 be a compact oriented connected surface of genus @xmath1 with one boundary component . _ homology cylinders _ over @xmath0 , each of which consists of a homology cobordism @xmath2 between two copies of @xmath0 and markings of both sides of the boundary of @xmath2 , appeared in the context of the theory of finite type invariants for 3-manifolds ( see goussarov @xcite , habiro @xcite , garoufalidis - levine @xcite and levine @xcite ) , and play an important role in a systematic study of the set of 3-manifolds . in our previous paper @xcite , we observed their relationship to knot theory by introducing _ homologically fibered knots_. the set @xmath3 of isomorphism classes of homology cylinders over @xmath0 becomes a monoid by the natural stacking operation . it is known that the monoid @xmath3 contains the mapping class group @xmath4 of @xmath0 as the group of units ( see example [ ex : mgtocg ] ) . moreover , many techniques and invariants to study @xmath4 such as johnson homomorphisms and the magnus representation can be extended to @xmath3 ( see @xcite , @xcite ) . by using them , we can observe that @xmath3 and @xmath4 hold many properties in common . taking account of the similarity between @xmath3 and @xmath4 , we now pay our attention to abelian quotients of them . it is known that @xmath4 is a perfect group for @xmath5 , namely it has no non - trivial abelian quotients . in this paper , however , we will show that the opposite holds for @xmath3 . the outline of this paper is as follows . after introducing homology cylinders in section [ section : cylinder ] , we see that @xmath3 has a big abelian quotient arising from the reducibility of a homology cylinder as a 3-manifold ( theorem [ prop : notfg1 ] ) . however this fact seems not to be suitable for our purpose of comparing @xmath4 and @xmath3 because all homology cylinders coming from @xmath4 have irreducible underlying 3-manifolds . therefore we shall introduce the submonoid @xmath6 of @xmath3 consisting of _ irreducible homology cylinders_. the main result is theorem [ thm : notfg2 ] that @xmath6 has a big abelian quotient originating in a quite different context from that of @xmath3 mentioned above . in fact , we prove it in section [ sec : proof ] as an application of sutured floer homology theory . sutured floer homology was defined first in @xcite by juhsz , then an alternative definition was given in @xcite by ni . it is a variant of heegaard floer homology theory defined by ozsvth and szab ( here we only refer to @xcite , which contains the results we use later , for details ) . in the last section , we discuss our results from the viewpoint of homology cobordisms of homology cylinders . the authors would like to thank dr . motoo tange for giving them a lecture about sutured floer homology theory . they also would like to thank the referee for his / her careful reading of the manuscript and valuable comments . the final publication is available at www.springerlink.com . we first recall the definition of homology cylinders , following garoufalidis - levine @xcite and levine @xcite . a _ homology cylinder _ @xmath7 _ over _ @xmath8 consists of a compact oriented 3-manifold @xmath2 with two embeddings @xmath9 such that : 1 . @xmath10 is orientation - preserving and @xmath11 is orientation - reversing ; 2 . @xmath12 and @xmath13 ; 3 . @xmath14 ; and 4 . @xmath15 are isomorphisms . two homology cylinders @xmath16 and @xmath17 over @xmath0 are said to be _ isomorphic _ if there exists an orientation - preserving diffeomorphism @xmath18 satisfying @xmath19 and @xmath20 . we denote by @xmath21 the set of all isomorphism classes of homology cylinders over @xmath0 . we define a product operation on @xmath3 by @xmath22 for @xmath16 , @xmath23 , so that @xmath21 becomes a monoid with the identity element @xmath24 , \mathrm{id } \times 1 , \mathrm{id } \times 0)$ ] . [ ex : mgtocg ] for each diffeomorphism @xmath25 of @xmath8 which fixes @xmath26 pointwise , we can construct a homology cylinder by setting @xmath27 , \mathrm{id } \times 1 , \varphi \times 0),\ ] ] where collars of @xmath28 and @xmath29 are stretched half - way along @xmath30 $ ] . it is easily checked that the isomorphism class of @xmath24 , \mathrm{id } \times 1 , \varphi \times 0)$ ] depends only on the ( boundary fixing ) isotopy class of @xmath25 and that this construction gives a monoid homomorphism from the mapping class group @xmath31 of @xmath8 to @xmath21 . in fact , it is an injective homomorphism ( see garoufalidis - levine ( * ? ? ? * section 2.4 ) , levine ( * ? ? ? * section 2.1 ) and also ( * ? ? ? * proposition 2.3 ) ) . it is known that the image of this homomorphism coincides with the group of units of @xmath21 ( see habiro - massuyeau ( * ? ? ? * proposition 2.4 ) for example ) . as seen in example [ ex : mgtocg ] , we may regard @xmath3 as an enlargement of @xmath4 . we recall here that @xmath4 is a perfect group for @xmath5 ( see korkmaz s survey @xcite and papers listed there for details ) . in comparing the structures of @xmath3 and @xmath4 , it seems interesting to discuss abelian quotients of @xmath3 . note that we need to be careful when mentioning abelian quotients of @xmath3 since it is not a group but a monoid . we avoid this by considering its universal group @xmath32 and its abelian quotients . recall that for every monoid @xmath33 , there exist ( uniquely up to isomorphism ) a group @xmath34 together with a monoid homomorphism @xmath35 satisfying the following : for every monoid homomorphism @xmath36 to a group @xmath37 , there exists a unique group homomorphism @xmath38 such that @xmath39 . one of the possible constructions of @xmath34 is to regard a monoid presentation of @xmath33 as a group presentation . in our discussion below , the case where @xmath40 is exceptional . we can check that @xmath41 is isomorphic to the monoid @xmath42 of all ( integral ) homology 3-spheres whose product is given by connected sums . indeed , an isomorphism @xmath43 is given by assigning to each homology 3-sphere @xmath44 the homology cylinder @xmath45)\sharp x , { \mathop{\mathrm{id}}\nolimits}\times 1 , { \mathop{\mathrm{id}}\nolimits}\times 0)$ ] with the inverse homomorphism given by taking _ closures _ ( see example [ ex : knot ] ) . consequently , @xmath41 is an abelian monoid which is not finitely generated . we begin our main argument by the following observations . [ lem : incompressible ] for each @xmath46 with @xmath47 , the surfaces @xmath48 and @xmath49 are incompressible in @xmath2 . it suffices to show that @xmath50 is injective . since @xmath51 induces an isomorphism on homology , it follows from stallings theorem ( * ? ? ? * theorem 3.4 ) that @xmath52 induces isomorphisms on all stages of nilpotent quotients . combining it with the fact that @xmath53 is free , in particular residually nilpotent , we see that @xmath54 is injective . [ prop : notfg1 ] the monoid @xmath21 is not finitely generated , for every @xmath55 . in fact , the abelianization of @xmath32 has infinite rank . the case where @xmath40 is as mentioned above . we now assume @xmath47 . for each homology cylinder @xmath46 , the underlying 3-manifold @xmath2 has a prime decomposition of the form @xmath56 where @xmath57 is the unique prime factor containing @xmath58 and @xmath59 are homology 3-spheres . using this decomposition , we can define the _ forgetting _ map @xmath60 by @xmath61 the uniqueness of the prime decomposition of a 3-manifold shows that @xmath62 is well - defined . we now claim that @xmath62 is a surjective monoid homomorphism . let @xmath63 @xmath64 . we decompose @xmath2 into @xmath65 , where @xmath57 is the prime factor containing @xmath58 and @xmath66 is a homology 3-sphere . similarly , we have @xmath67 . by lemma [ lem : incompressible ] , the underlying 3-manifold of the product @xmath68 has a decomposition @xmath69 such that @xmath70 is the prime factor containing the boundary . this shows that @xmath62 is a monoid homomorphism . the surjectivity of @xmath62 follows from the existence of a section @xmath71 defined by @xmath72 ) \sharp x , { \mathop{\mathrm{id}}\nolimits}\times 1 , { \mathop{\mathrm{id}}\nolimits}\times 0)$ ] . the result follows from the fact that @xmath42 satisfies the conditions mentioned in the statement . proposition [ prop : notfg1 ] says that the monoid @xmath21 has a different property about its abelian quotients from the mapping class group @xmath31 , which arises from the reducibility of the underlying 3-manifold of a homology cylinder . however the underlying 3-manifolds of homology cylinders obtained from @xmath31 are all product @xmath73 $ ] and , in particular , irreducible . therefore it seems reasonable to consider the following subset of @xmath21 . [ def : irred ] a homology cylinder @xmath16 is said to be _ irreducible _ if the underlying 3-manifold @xmath2 is irreducible . we denote by @xmath74 the subset of @xmath21 consisting of all irreducible homology cylinders . note that @xmath74 is a submonoid of @xmath21 , for @xmath75 . in particular , @xmath76 is the trivial monoid . note also that we have an injective monoid homomorphism @xmath77 . the following is the main result of this paper , whose proof will be given in the next section . [ thm : notfg2 ] the monoid @xmath74 is not finitely generated , for every @xmath47 . in fact , the abelianization of @xmath78 has infinite rank . our proof of theorem [ thm : notfg2 ] will be obtained as an application of sutured floer homology theory due to juhsz @xcite and ni @xcite . for each homology cylinder @xmath46 , we have a natural decomposition @xmath79 of @xmath58 . such a decomposition defines a _ sutured manifold _ @xmath80 with the suture @xmath81 . sutured manifolds were originally defined by gabai @xcite , to which we refer for details . [ ex : knot ] for a knot @xmath82 in a closed oriented connected 3-manifold @xmath83 with a seifert surface @xmath33 , let @xmath2 be the manifold obtained from the knot exterior @xmath84 by cutting open along @xmath33 and @xmath85 the annulus @xmath86 . then , @xmath80 is called the _ complementary sutured manifold for _ @xmath33 . the core curve of @xmath85 is denoted by @xmath87 , and called the _ suture_. the suture @xmath87 and @xmath82 are parallel . on the other hand , for each @xmath46 ( or , more generally , each marked cobordism of @xmath0 ) , we have a closed oriented connected @xmath88-manifold @xmath89 called the _ closure _ and a knot @xmath90 with a seifert surface @xmath91 in @xmath92 . sutured floer homology is an invariant of _ balanced _ sutured manifolds ( see juhsz ( * ? ? * definition 2.11 ) ) , where all of the sutured manifolds mentioned above satisfy this condition . it assigns a finitely generated abelian group @xmath93 to each balanced sutured manifold @xmath80 . we rely on papers of juhsz @xcite and ni @xcite for the definition and fundamental properties of sutured floer homology , and concentrate on using this theory . juhsz ( * ? ? ? * theorem 1.5 ) showed that @xmath94 holds for any sutured manifold @xmath80 mentioned in example [ ex : knot ] , where the right hand side is the genus @xmath95 part of the knot floer homology of the knot @xmath87 in @xmath92 with the seifert surface @xmath33 of genus @xmath95 . for each homology cylinder @xmath46 , we put @xmath96 with @xmath97 . [ prop : nontrivial ] @xmath98 contains @xmath99 for any @xmath46 . we first assume that @xmath2 is irreducible . by juhsz ( * theorem 1.4 ) , all we have to do is to check that @xmath16 gives a _ taut _ sutured manifold . that is , * @xmath2 is irreducible ; and * @xmath48 and @xmath49 are incompressible and thurston norm minimizing in their homology classes in @xmath100 with @xmath97 . the condition ( i ) is automatic and the first half of ( ii ) follows from lemma [ lem : incompressible ] . for the latter half of ( ii ) , it suffices to show that @xmath101 is thurston norm minimizing . suppose that we have a proper embedding of a surface @xmath102 representing @xmath103 \in h_2 ( m,\gamma)$ ] with smaller norm than that of @xmath101 . we may assume that @xmath102 does not have any closed component , for such a component comes from @xmath104 and hence removing it does not change the class @xmath105 \in h_2 ( m,\gamma)$ ] nor increase the norm . next , consider the intersection @xmath106 of @xmath102 and the annulus @xmath85 . generically , it consists of oriented circles , each of which is an essential simple loop in @xmath85 or bounds a disk in @xmath85 . for an innermost circle bounding a disk , we attach this disk to @xmath102 and move it away from @xmath58 by an isotopy . repeating this , we can eliminate all circles bounding disks in @xmath106 , so that the intersection consists of parallel copies of the essential simple loop in @xmath85 . if @xmath106 is disconnected , we can find an annulus in @xmath85 bounding ( with coherent orientations ) two adjacent components of @xmath106 . then we attach this annulus to @xmath102 and move it away from @xmath58 . ( when a closed component is produced , remove it . ) repeating this , @xmath106 becomes connected . note that the above procedure does not change @xmath105 \in h_2 ( m,\gamma)$ ] nor increase the norm since disks and annuli in @xmath85 are trivial in @xmath107 . consequently , we may assume that @xmath102 is a connected surface with one boundary component . now suppose that we have a proper embedding @xmath108 with @xmath109 and satisfying @xmath110=[i_+({\ensuremath{\sigma_{g,1 } } } ) ] \in h_2(m,\gamma)$ ] . we take a basis @xmath111 of @xmath112 such that @xmath113 \in [ \pi_1(m),\pi_1(m)]\ ] ] under suitable orientations . here we set the basepoint on @xmath87 . by stallings ( * lemma 3.1 ) , we see that @xmath114}{[\pi_1 ( { \ensuremath{\sigma_{g,1}}}),[\pi_1 ( { \ensuremath{\sigma_{g,1}}}),\pi_1({\ensuremath{\sigma_{g,1 } } } ) ] ] } \longrightarrow \frac{[\pi_1 ( m),\pi_1(m)]}{[\pi_1 ( m),[\pi_1 ( m),\pi_1(m)]]}\ ] ] is an isomorphism , and we pull back ( [ eq : boundary ] ) to @xmath115}{[\pi_1 ( { \ensuremath{\sigma_{g,1}}}),[\pi_1 ( { \ensuremath{\sigma_{g,1}}}),\pi_1({\ensuremath{\sigma_{g,1}}})]]}$ ] , which is known to be isomorphic to @xmath116 . then we obtain an equality @xmath117 ) \wedge i_+^{-1}([\delta_{2i } ] ) \in \wedge^2 ( h_1({\ensuremath{\sigma_{g,1}}})),\ ] ] where @xmath118 \in h_1(m)$ ] denotes the homology class of @xmath119 . on the other hand , we have @xmath120 , the symplectic form , for any symplectic basis @xmath121 of @xmath122 . define an endomorphism of @xmath122 by @xmath123 ) , \ \ y_i \mapsto i_+^{-1}([\delta_{2i } ] ) \quad \mbox{for \ $ 1 \le i \le h$},\\ & x_j , y_j \mapsto 0 \quad \mbox{for \ $ h+1 \le j \le g$}. \end{aligned}\ ] ] by definition , this endomorphism is not injective , but preserves the symplectic form . however , such an endomorphism does not exist since the symplectic form embodies the intersection form on @xmath124 , which is nondegenerate , a contradiction . therefore @xmath101 is thurston norm minimizing and we finish the proof when @xmath2 is irreducible . when @xmath2 is not irreducible , we take a prime decomposition @xmath125 as in section [ section : cylinder ] , where @xmath59 are all homology 3-spheres . then we obtain the conclusion by an argument similar to the proof of ( * ? ? ? * corollary 8.3 ) using the connected sum formula ( * ? ? ? * proposition 9.15 ) . by formulas of juhsz ( * proposition 8.6 ) and ni ( * ? ? ? * theorem 4.1 , 4.5 ) together with the fact ( [ eq : sfh - hfk ] ) , we have @xmath126 for @xmath16 , @xmath23 . hence by taking the rank of @xmath127 , we obtain a monoid homomorphism @xmath128 defined by @xmath129 where @xmath130 is the monoid of positive integers whose product is given by multiplication . we call @xmath131 the _ rank homomorphism_. note that the restriction of @xmath131 to @xmath4 is trivial since every element of @xmath4 has its inverse . by the uniqueness of the prime decomposition of an integer , we can decompose @xmath131 into prime factors @xmath132 where @xmath133 is a copy of @xmath134 , the monoid of non - negative integers whose product is given by sums , corresponding to the power of the prime number @xmath135 . we now restrict the above homomorphisms to @xmath6 . [ prop : independent ] for @xmath47 , the set @xmath136 contains infinitely many non - trivial homomorphisms that are linearly independent as elements of @xmath137 . to prove this proposition , we have to consider the images of the homomorphisms @xmath138 . more specifically , we need many homology cylinders whose ranks of @xmath127 are known . we now use _ homologically fibered knots _ defined in the previous paper @xcite to construct such homology cylinders . recall that a knot @xmath82 in @xmath139 is said to be _ homologically fibered _ if @xmath82 satisfies the following two conditions : * the degree of the normalized alexander polynomial @xmath140 of @xmath82 is @xmath141 ; and * @xmath142 , where @xmath143 is the genus of @xmath82 and @xmath140 is normalized so that its lowest degree is @xmath144 . in ( * theorem 3.4 ) , we showed that @xmath82 is homologically fibered if and only if @xmath82 has a seifert surface @xmath33 whose complementary sutured manifold is a homology product . here , a homology product means a homology cylinder without markings . note that crowell and trotter observed in @xcite this essentially . ( see also @xcite . ) we also showed that if @xmath82 is homologically fibered , then any minimal genus seifert surface gives a homology product . for a homologically fibered knot @xmath82 of genus @xmath1 , we obtain an irreducible homology cylinder @xmath145 by fixing a pair of markings of the boundary of the complementary sutured manifold @xmath146 for a minimal genus seifert surface . by ( [ eq : sfh - hfk ] ) , @xmath147 holds for such a homology cylinder . we first give a proof of the case where @xmath148 . we now consider pretzel knots @xmath149 with @xmath150 . as depicted in figure [ pretzel1 ] , each of such knots has a genus @xmath151 seifert surface . it is well known that the normalized alexander polynomial of @xmath149 is given by @xmath152 using this formula , we see that the sequence @xmath153 consists of homologically fibered knots of genus @xmath151 since @xmath154 . moreover , a computation due to ozsvth - szab ( * ? ? ? * latter formula of theorem 1.3 ) gives @xmath155 from which we see that @xmath156 . we now analyze the sequence @xmath157 of positive integers . the first supplement to quadratic reciprocity says that there exists a positive integer @xmath158 such that @xmath159 for any odd prime @xmath135 satisfying @xmath160 . in this case , set @xmath161 then , if @xmath158 is odd , we have @xmath162 and if @xmath158 is even , we also have @xmath163 hence we can conclude that @xmath138 is non - trivial if @xmath160 . by dirichlet s theorem on arithmetic progressions , there exist infinitely many such prime numbers . for a homology cylinder @xmath164 , let @xmath165 take a sequence of homology cylinders @xmath166 such that @xmath167 . then we can see that @xmath168 are linearly independent by evaluating them on the @xmath169 s . this concludes the proof of the case where @xmath148 . let @xmath170 be a homologically fibered knot of genus @xmath171 obtained from @xmath172 by taking connected sums with @xmath173-tuples of trefoils . by ( * ? * theorem 1.1 ) and ( * ? ? ? * corollary 8.8 ) together with the fact that the trefoil is a fibered knot of genus @xmath151 , so that @xmath174 , we have @xmath175 therefore , the cases where @xmath176 follow from the same argument as above . suppose @xmath6 was finitely generated . then except for finitely many primes , the homomorphisms @xmath138 are trivial on any finite set of generators , and hence on whole @xmath6 . this contradicts proposition [ prop : independent ] and we have proved the first half of our claim . the latter half follows from the construction that uses infinitely many homomorphisms whose targets are abelian . in @xcite , garoufalidis - levine introduced _ homology cobordisms _ of homology cylinders , which give an equivalence relation of homology cylinders . we finish this paper by two observations concerning our results ( proposition [ prop : notfg1 ] and theorem [ thm : notfg2 ] ) from the viewpoint of this equivalence relation . we denote by @xmath182 the quotient set of @xmath21 with respect to the equivalence relation of homology cobordism . the monoid structure of @xmath21 induces a group structure of @xmath182 . it is known that @xmath4 can be embedded in @xmath183 ( see ( * ? ? ? * section 2.4 ) , ( * ? ? ? * section 2.1 ) ) . one important problem is to determine whether @xmath183 is perfect or not . has @xmath184 as an abelian quotient , for every @xmath47 . ] in fact , no non - trivial abelian quotients of @xmath182 are known at present . we now observe that it is difficult to give an answer to this problem by using the homomorphisms used in this paper . first we consider the forgetting homomorphism @xmath185 discussed in section [ section : cylinder ] . it follows from myers result ( * ? ? ? * theorem 3.2 ) that every homology cylinder in @xmath3 with @xmath47 is homology cobordant to an irreducible one , whose image by @xmath62 is trivial by definition . hence if @xmath189 factors through @xmath183 for a monoid homomorphism @xmath190 , then @xmath191 must be trivial . next we consider the rank homomorphisms @xmath192 discussed in section [ sec : proof ] . it induces a group homomorphism @xmath193 on universal groups . note that the quotient group of @xmath194 by homology cobordism relation is also @xmath183 as mentioned in the proof of theorem [ thm : ab_quot1 ] . s. garoufalidis , j. levine , _ tree - level invariants of three - manifolds , massey products and the johnson homomorphism _ , graphs and patterns in mathematics and theorical physics , proc . pure math . 73 ( 2005 ) , 173205 .
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a homology cylinder over a surface consists of a homology cobordism between two copies of the surface and markings of its boundary .
the set of isomorphism classes of homology cylinders over a fixed surface has a natural monoid structure and it is known that this monoid can be seen as an enlargement of the mapping class group of the surface .
we now focus on abelian quotients of this monoid .
we show that both the monoid of all homology cylinders and that of irreducible homology cylinders are not finitely generated and moreover they have big abelian quotients .
these properties contrast with the fact that the mapping class group is perfect in general . the proof is given by applying sutured floer homology theory to homologically fibered knots studied in a previous paper .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
after the discovery @xcite of @xmath10 followed by the other experiments @xcite , identifying the properties of the particle is one of the central problems in hadron physics . while the isospin of @xmath11 is likely to be zero @xcite , the spin and the parity and the origin of its tiny width still remain open questions @xcite . in spite of many theoretical studies on @xmath11 @xcite , the nature of this exotic particle , including the very existence of the particle , is still controversial . among theoretical approaches , the lattice qcd calculation is considered as one of the most reliable _ ab initio _ methods for studying the properties of hadronic states , which has been very successful in reproducing the non - exotic hadron mass spectra @xcite . up to now , several lattice qcd studies have been reported , which aim to look for pentaquarks in various different ways . however , the conclusions are unfortunately contradictory with each other . on one hand , the authors in refs . @xcite conclude that the parity of @xmath11 is likely to be negative , while in ref . @xcite the state with the similar mass to @xmath11 in the positive parity channel is reported . in refs . @xcite , the absence of @xmath11 is suggested . one of the difficulties in the spectroscopy calculation with lattice qcd arises from the fact that the hadron masses suffer from systematic errors due to the discretization , the chiral extrapolation , the quenching effect , the finite volume effect and the contaminations from higher excited - states . the difficulty specific to the present problem is that the signal of @xmath11 is embedded in the discrete spectrum of nk scattering states in finite volume . in order to verify the existence of a resonance state , one needs to isolate the first few low energy states including the lowest nk scattering state , identify a resonance state and study its volume dependence which can distinguish itself from other scattering states . therefore , ideally one should extract multistates from a high statistics unquenched calculation for several different physical volumes , where both the continuum and the chiral limits are taken . however , due to enormous computational costs , so far there are no lattice qcd study which performs all these steps . in the present situation where even the very existence of the resonance state is theoretically in dispute , the primary task is to provide evidences which distinguish the candidate resonance state from a scattering state . as long as other systematic errors only affect the numerical values of the masses but not the characteristic evidences of the resonance state , they may be neglected . even so , the isolation of the first few low energy states and the study of the volume dependence is a minimum requisite . therefore , at this stage as a first step towards a more complete analysis , we propose to focus only on analyses using rather heavy quarks on coarse quenched lattices but with a good statistics . by such a strategy , we can afford taking several different lattice volumes with thousands of gauge configurations so that the careful separation of states and the studies of volume dependence are possible . although giving well controlled continuum and chiral extrapolations may be important , we simply assume that the contents in spectra would not be drastically changed , although there are some cases where level crossings of resonance states occur as the quark masses decrease @xcite . even with such a compromise , we can hopefully learn about the existence and much of the qualitative properties of @xmath11 . in this paper , we study @xmath8 channel in quenched lattice qcd to search for possible resonance states . we adopt two independent operators with @xmath12 and @xmath13 and diagonalize the @xmath7 correlation matrices by the variational method for all the combinations of lattice sizes and quark masses to extract the 2nd - lowest state slightly above the nk threshold in this channel . after the careful separation of the states , we investigate the volume dependence of the energy as well as the spectral weight @xcite of each state so that we can distinguish the resonance state from simple scattering states . the paper is organized as follows . we present the formalism used in the analysis in sec . [ formalism ] and show the simulation conditions in sec . [ setup ] . the process of the analysis is shown in sec . [ negative1]-[positive ] are devoted to the interpretation of the obtained results and the verification of the existence of a resonance state , as well as some checks on the consistency and the reliability of the obtained data . in sec . [ discussion ] , we discuss the operator dependence of the results and compare our results with the previous works . we finally summarize the paper in sec . [ summary ] . in appendix , we show the result of another trial to estimate the volume dependences of the spectral weights , which requires no multi - exponential fit . as @xmath11 lies above the nk threshold , any hadron correlators , which have the @xmath11 signal , also contain the discrete - level nk scattering states in a finite volume lattice . in order to isolate the resonance state from the scattering states , one needs to extract _ at least _ two states before anything else . since a double - exponential fit of a single correlator becomes numerically ambiguous , we adopt the variational method using correlation matrices constructed from independent operators @xcite . a set of independent operators , \{@xmath14 } for sinks and\{@xmath15 } for sources , is needed to construct correlation matrices @xmath16 , which can be decomposed into the sum over the energy eigenstates @xmath17 as @xmath18 with the general matrices which depend on the operators as @xmath19 and the diagonal matrix @xmath20 from the product @xmath21 we can extract the energies \{@xmath22 } as the logarithm of eigenvalues \{@xmath23 } of the matrix @xmath24 . while there are @xmath25 independent operators for the correlation matrix , the number of the intermediate states @xmath17 which effectively contribute to this matrix may differ from @xmath25 in general . let us call this number as @xmath26 . if @xmath26 is larger than @xmath25 , the higher excited - states are non negligible and their contaminations give rise to a @xmath27-dependence of eigenvalues as \{@xmath28}. if on the other hand @xmath26 is smaller than @xmath25 , @xmath29 becomes non - invertible so that the extracted energies become numerically fairly unstable and we can not extract all the @xmath25 eigenvalues . in order to have a reliable extraction of states , we therefore need to find an appropriate window of @xmath27 ( @xmath30 ) so that @xmath31 . ( of course , even in the case when @xmath32 , we can extract @xmath26 eigenvalues with the reduced @xmath33 correlation matrices . ) the stability of \{@xmath28 } against @xmath27 is expected in this @xmath27 range and we can obtain @xmath25 eigenenergies \{@xmath22 } ( @xmath34 ) by fitting the eigenvalues @xmath28 as @xmath35 in @xmath30 . since finding the stability of the energies against @xmath27 in noisy data may suffer from uncontrollable biases , the result could be quite subjective . in order to avoid such biases , one should impose some concrete criteria to judge the stability as will be explained in later sections and select only those data which satisfy the criteria . after the separation of the states , we need to distinguish a possible resonance state from nk scattering states by the volume dependence of each state . it is expected that the energies of resonance states have small volume dependence , while the energies of nk scattering states are expected to scale as @xmath36 according to the relative momentum @xmath37 between n and k on a finite periodic lattice , provided that the nk interaction is weak and negligible which is indeed the case for the leading order in chiral perturbation theory . although the variational method is powerful for extracting the energy spectrum , one can obtain only part of the information on the spectral weights @xmath38 . in order to extract the spectral weights , we also perform constrained double exponential fits using the energies from the variational method as inputs . we carry out simulations on four different sizes of lattices , @xmath2 , @xmath3 , @xmath4 and @xmath5 with 2900 , 2900 , 1950 and 950 gauge configurations using the standard plaquette ( wilson ) gauge action at @xmath39 and the wilson quark action . the hopping parameters for the quarks are @xmath40=@xmath41 , @xmath42 , @xmath43 , @xmath44 and @xmath45 , which correspond to the current quark masses @xmath46 , @xmath47 , @xmath48 , @xmath49 and @xmath50 , respectively in the unit of mev @xcite . the lattice spacing @xmath51 from the sommer scale is set to be 0.17 fm , which implies the physical lattice sizes are @xmath52 @xmath53 , @xmath54 @xmath53 , @xmath55 @xmath53 and @xmath56 @xmath53 . we adopt the following two operators used in ref . @xcite for the interpolating operators at the sink @xmath57 ; @xmath58 \{u_e(x)[{\overline{s_e}(x)\gamma_5d_c(x ) } ] -(u\leftrightarrow d)\ } , \label{theta1}\end{aligned}\ ] ] which is expected to have a larger overlap with @xmath11 state , and @xmath59 \{u_c(x)[{\overline{s_e}(x)\gamma_5d_e(x ) } ] -(u\leftrightarrow d)\ } , \label{theta2}\end{aligned}\ ] ] which we expect to have larger overlaps with nk scattering states . here , the dirac fields @xmath60 , @xmath61 and @xmath62 are up , down and strange quark fields , respectively and the roman alphabets \{a , b , c , e } denote color indices . for measuring the energy spectrum , the two operators at the source @xmath63 are chosen to be @xmath64 and @xmath65 defined using spatially spread quark fields @xmath66 with the coulomb gauge : @xmath67\{u_e(x_3)[{\overline{s_e}(x_4 ) \gamma_5 d_c(x_5 ) } ] -(u\leftrightarrow d)\},\end{aligned}\ ] ] and @xmath68\{u_c(x_3)[{\overline{s_e}(x_4 ) \gamma_5 d_e(x_5 ) } ] -(u\leftrightarrow d)\}.\end{aligned}\ ] ] the above operators give a @xmath7 correlation matrix in the channel with the quantum number of @xmath8 . we note here that the baryonic correlators have the spinor indices , which we omit in the paper , and they contain the propagations of both the positive and negative parity particles . for the parity projection , we simply multiply the correlators by @xmath69 and extract the contributions proportional to @xmath70 from the negative - parity and positive - parity particles , respectively . we fix the source operator @xmath71 on @xmath72 plane to reduce the effect of the dirichlet boundary on @xmath73 plane @xcite . we adopt the operators @xmath74 as sink operators , which is summed over all space to project out the zero - momentum states . we finally calculate @xmath75 using two independent operators , we can extract the first _ two _ states , namely the lowest and the next - lowest states . the lowest state is considered to be the `` lowest '' nk scattering state . in order to extract a possible resonance state with controlled systematic errors , we need to choose the physical volume of the lattice in an appropriate range . if we choose @xmath76 to be too large , the resonance state becomes heavier than the 2nd - lowest nk scattering state whose energy is naively expected to scale as @xmath77 according to the spatial lattice extent @xmath76 . in this case we need to extract the 3rd state using a @xmath78 correlation matrix , which requires more computational time . the energy difference between the lowest and the next - lowest nk scattering states @xmath79 , for example , ranges from 180 mev to 860 mev in @xmath80 . taking into account that @xmath11 lies about 100 mev above the nk threshold , we take 3.5 fm as the upper limit of @xmath76 . on the other hand , if we choose @xmath76 too small , unwanted finite - volume artifacts from the finite sizes of particles become non - negligible . it is however difficult to estimate the lower limit of @xmath76 , because the finite - volume effect is rather uncontrollable . we shall take the spatial extents @xmath81 at @xmath82 as a trial . we take periodic boundary conditions in all directions for the gauge field , whereas we impose periodic boundary conditions on the spatial directions and the dirichlet boundary condition on the temporal direction for the quark field in order to avoid possible contaminations from those propagating beyond the boundary at @xmath73 in ( anti)periodic boundary conditions . since the source of possible contaminations is peculiar to the pentaquark and has not been properly noticed in previous studies , it is worthwhile to dwell on this problem for a moment . let us denote the correlators in the pentaquark channel with periodic / antiperiodic boundary conditions and dirichlet boundary conditions as @xmath83 and @xmath84 , respectively . inserting complete set of states these correlators read @xmath85 where the states @xmath86 are the eigenstates with energies @xmath87 , @xmath88 , @xmath89 , @xmath90 respectively . @xmath91 is the state which corresponds to the dirichlet boundary condition and @xmath92 is the factor which represents the @xmath93 sign with antiperiodic boundary condition . the factor @xmath92 is equal to @xmath94 when @xmath95 contains an even(odd ) number of valence quarks . it should be noted that the up , down , and strange quark numbers @xmath96 for the state @xmath97 are restricted to zero since all the quark fields @xmath98 are set to zero in dirichlet boundary condition , while there is no restriction in the quark sector for the states @xmath95 which appear in periodic / antiperiodic boundary conditions . this means that the following states can contribute in the correlators @xmath99 where @xmath100 and @xmath101 are the antiparticle states of @xmath102 and @xmath103 , respectively . therefore , the correlators have the contributions in the long range limit , @xmath104 in fig . [ hadronic_corr ] , we give a schematic picture of the contributions to the correlators . the first two terms in eq.([cpap ] ) are the contributions from the five quark states as given in diagram ( a ) . the third , the fourth and the fifth terms in eq.([cpap ] ) which correspond to diagrams ( c),(d ) and ( b ) are hadronic contributions which propagate beyond the boundary . schematic figure for the explanation on the possible contaminations of the particles propagating over the temporal boundary . @xmath105 and @xmath106 are the interpolating operators and their arguments are the distances from the source point . @xmath105 is a generic hadronic operator , which creates and annihilates the particles which can not decay in quenched qcd . the wave lines represent the propagations of states . resonance states like @xmath11 are represented by `` 5q '' in the figure , as well as nk scattering states . _ the five - quark state can dissociate into forward - propagating nucleon ( kaon ) and backward - propagating kaon ( nucleon ) . _ ] as a result , the correlation @xmath107 inevitably contains unwanted contributions such as @xmath108 in this case , the effective mass plot approaches @xmath109 below the nk threshold as @xmath27 is increased . on the other hand , the contributions corresponding to diagrams ( c),(d ) and ( b ) do not exist with dirichlet boundary conditions ( eq.([cd ] ) ) . therefore , we find that it would be safest to impose the dirichlet boundary condition on the temporal direction , since no quark can go over the boundary on t=0 in the temporal direction . although the boundary is transparent for the particles composed only by gluons ; _ i.e. _ glueballs , due to the periodicity of the gauge action , it would be however safe to neglect these gluonic particles going beyond the boundary since these particles are rather heavy . then , the correlation @xmath110 mainly contains only such terms ( ( a ) in fig . [ hadronic_corr ] ) as @xmath111 with @xmath112 the weight factor and @xmath113 the eigenenergy of @xmath114-th state in positive / negative parity channel , respectively . one sees that one can now apply the prescription mentioned in the last section . one may wonder if these contaminations can be discarded with the parity projection of the correlators by taking linear combinations with periodic and antiperiodic boundary conditions . this method indeed works for ordinary three quark states where one can single out one of the two contributions diagram ( x ) and ( y ) in fig . [ hadronic_corr ] . however even if one takes such linear combinations , one can not make the contributions from diagram ( c ) seen in eq.([cpap ] ) cancel out as opposed to the contributions from diagram ( b ) and ( d ) . it is because of the fact that the factor @xmath92 for the contribution ( c ) is always equal to @xmath115 . ( we note here that we can avoid these contaminations using the `` averaged quark propagator '' @xcite . ) some of the previous lattice qcd studies on @xmath11 adopted a parity projection method using the combination with periodic and antiperiodic boundary conditions @xcite . we stress that one should in principle be careful whether the result is free from the contamination owing to the boundary condition which is peculiar to the pentaquark and can mimic a fake plateau in the propagator . after obtaining the energy spectrum , we carry out a study of the spectral weight for @xmath40=@xmath44 . introducing two smeared operators @xmath116 , @xmath117 we compute the following correlators @xmath118 from which we extract the spectral weights using a constrained double exponential fit . the details will be explained in sec . [ negative2 - 3 ] . the `` effective mass '' plot @xmath119 as the function of @xmath27 , the separation between source operators and sink operators , in @xmath120 channel with the hopping parameters @xmath121 on @xmath2 , @xmath3 , @xmath4,@xmath5 lattice at @xmath82 . the stability of each @xmath119 against @xmath27 means the smallness of the unwanted higher excited - state contaminations . the solid line and the dashed lines represent the central value and the error of the fitted masses @xmath122 and @xmath123 . , title="fig : " ] the `` effective mass '' plot @xmath119 as the function of @xmath27 , the separation between source operators and sink operators , in @xmath120 channel with the hopping parameters @xmath121 on @xmath2 , @xmath3 , @xmath4,@xmath5 lattice at @xmath82 . the stability of each @xmath119 against @xmath27 means the smallness of the unwanted higher excited - state contaminations . the solid line and the dashed lines represent the central value and the error of the fitted masses @xmath122 and @xmath123 . , title="fig : " ] + the `` effective mass '' plot @xmath119 as the function of @xmath27 , the separation between source operators and sink operators , in @xmath120 channel with the hopping parameters @xmath121 on @xmath2 , @xmath3 , @xmath4,@xmath5 lattice at @xmath82 . the stability of each @xmath119 against @xmath27 means the smallness of the unwanted higher excited - state contaminations . the solid line and the dashed lines represent the central value and the error of the fitted masses @xmath122 and @xmath123 . , title="fig : " ] the `` effective mass '' plot @xmath119 as the function of @xmath27 , the separation between source operators and sink operators , in @xmath120 channel with the hopping parameters @xmath121 on @xmath2 , @xmath3 , @xmath4,@xmath5 lattice at @xmath82 . the stability of each @xmath119 against @xmath27 means the smallness of the unwanted higher excited - state contaminations . the solid line and the dashed lines represent the central value and the error of the fitted masses @xmath122 and @xmath123 . , title="fig : " ] the `` effective mass '' plot @xmath119 as the function of @xmath27 , the separation between source operators and sink operators , in @xmath120 channel with the hopping parameters @xmath124 on @xmath2 , @xmath3 , @xmath4,@xmath5 lattice at @xmath82 . , title="fig : " ] the `` effective mass '' plot @xmath119 as the function of @xmath27 , the separation between source operators and sink operators , in @xmath120 channel with the hopping parameters @xmath124 on @xmath2 , @xmath3 , @xmath4,@xmath5 lattice at @xmath82 . , title="fig : " ] + the `` effective mass '' plot @xmath119 as the function of @xmath27 , the separation between source operators and sink operators , in @xmath120 channel with the hopping parameters @xmath124 on @xmath2 , @xmath3 , @xmath4,@xmath5 lattice at @xmath82 . , title="fig : " ] the `` effective mass '' plot @xmath119 as the function of @xmath27 , the separation between source operators and sink operators , in @xmath120 channel with the hopping parameters @xmath124 on @xmath2 , @xmath3 , @xmath4,@xmath5 lattice at @xmath82 . , title="fig : " ] the `` effective mass '' plot @xmath119 as the function of @xmath27 , the separation between source operators and sink operators , in @xmath120 channel with the hopping parameters @xmath124 on @xmath2 , @xmath3 , @xmath4,@xmath5 lattice at @xmath82 . , title="fig : " ] the `` effective mass '' plot @xmath119 as the function of @xmath27 , the separation between source operators and sink operators , in @xmath120 channel with the hopping parameters @xmath124 on @xmath2 , @xmath3 , @xmath4,@xmath5 lattice at @xmath82 . , title="fig : " ] + the `` effective mass '' plot @xmath119 as the function of @xmath27 , the separation between source operators and sink operators , in @xmath120 channel with the hopping parameters @xmath124 on @xmath2 , @xmath3 , @xmath4,@xmath5 lattice at @xmath82 . , title="fig : " ] the `` effective mass '' plot @xmath119 as the function of @xmath27 , the separation between source operators and sink operators , in @xmath120 channel with the hopping parameters @xmath124 on @xmath2 , @xmath3 , @xmath4,@xmath5 lattice at @xmath82 . , title="fig : " ] the `` effective mass '' plot @xmath119 as the function of @xmath27 , the separation between source operators and sink operators , in @xmath120 channel with the hopping parameters @xmath125 on @xmath2 , @xmath3 , @xmath4,@xmath5 lattice at @xmath82 . , title="fig : " ] the `` effective mass '' plot @xmath119 as the function of @xmath27 , the separation between source operators and sink operators , in @xmath120 channel with the hopping parameters @xmath125 on @xmath2 , @xmath3 , @xmath4,@xmath5 lattice at @xmath82 . , title="fig : " ] + the `` effective mass '' plot @xmath119 as the function of @xmath27 , the separation between source operators and sink operators , in @xmath120 channel with the hopping parameters @xmath125 on @xmath2 , @xmath3 , @xmath4,@xmath5 lattice at @xmath82 . , title="fig : " ] the `` effective mass '' plot @xmath119 as the function of @xmath27 , the separation between source operators and sink operators , in @xmath120 channel with the hopping parameters @xmath125 on @xmath2 , @xmath3 , @xmath4,@xmath5 lattice at @xmath82 . , title="fig : " ] the `` effective mass '' plot @xmath119 as the function of @xmath27 , the separation between source operators and sink operators , in @xmath120 channel with the hopping parameters @xmath125 on @xmath2 , @xmath3 , @xmath4,@xmath5 lattice at @xmath82 . , title="fig : " ] the `` effective mass '' plot @xmath119 as the function of @xmath27 , the separation between source operators and sink operators , in @xmath120 channel with the hopping parameters @xmath125 on @xmath2 , @xmath3 , @xmath4,@xmath5 lattice at @xmath82 . , title="fig : " ] + the `` effective mass '' plot @xmath119 as the function of @xmath27 , the separation between source operators and sink operators , in @xmath120 channel with the hopping parameters @xmath125 on @xmath2 , @xmath3 , @xmath4,@xmath5 lattice at @xmath82 . , title="fig : " ] the `` effective mass '' plot @xmath119 as the function of @xmath27 , the separation between source operators and sink operators , in @xmath120 channel with the hopping parameters @xmath125 on @xmath2 , @xmath3 , @xmath4,@xmath5 lattice at @xmath82 . , title="fig : " ] before obtaining the energies of the lowest state and the 2nd - lowest state , there are only a few simple steps . first , we calculate the @xmath7 correlation matrix @xmath126 defined in eq.([correlation_matrix ] ) and obtain the `` energies '' \{@xmath119 } as the logarithm of eigenvalues \{@xmath28 } of the matrix product @xmath24 . after finding the @xmath27 range ( @xmath30 ) , where \{@xmath119 } are stable against @xmath27 , we can extract the energies @xmath22 by the least @xmath127-squared fit of the data as @xmath128 in @xmath30 . since the volume dependence of the energy is crucial to judge whether the state is a resonance or not , a great care must be paid in extracting the energy . therefore , the systematic error by the contaminations from higher excited - states should be avoided by a careful choice of fitting ranges . for this reason , we impose the following criteria for the reliable extraction of the energies . although this set of the criteria is nothing more than just one possible choice , we believe it is important to impose some concrete criteria for the fit so that we can reduce the human bias for the fit , though not completely . 1 . the effective mass plot should have `` plateau '' for both the lowest and the 2nd - lowest states simultaneously in a fit range [ @xmath129 , @xmath130 , where the length @xmath131 should be larger than or equal to 3 ( @xmath132 ) . 2 . in the plateau region , the signal for the lowest and the 2nd - lowest states should be distinguishable , so that the gap between the central values of the lowest and the 2nd - lowest energies should be larger than their errors . the fitted energies should be stable against the choice of the fit range ; i.e. the results of the fit with @xmath133 time slices and with @xmath134 time slices should be consistent within statistical errors for both the lowest and the 2nd - lowest states . 4 . the lowest state energy obtained by the diagonalization method using the @xmath135 correlation matrix should be consistent with the value from a single exponential fit for a sufficiently large @xmath136 . if the fit does not satisfy the above conditions , we discard the result since either the data in the fit range may be contaminated by higher excited - states or the 2nd - lowest - state signal is too noisy for a reliable fit . figs [ effective_mass ] show the `` effective mass '' plot @xmath119 for the heaviest combination of quarks @xmath121 . as is mentioned in sec . [ formalism ] , we need to find the @xmath27 region ( @xmath30 ) where each @xmath119 shows a plateau . in the case of @xmath4 lattice in fig . [ effective_mass ] , for example , we choose the fit range of @xmath137 and @xmath138 ( @xmath139 ) . notice that the source operators are put on the time slice with @xmath140 . the plateau in this region satisfies the above criteria so that we consider the fit @xmath122 and @xmath123 for the range @xmath141 as being reliable . the situation is similar for the cases of @xmath3 and @xmath5 lattices . on the other hand , in the case of @xmath2 lattice we do not find a plateau region satisfying the above criteria . [ meff221],[meff112],[meff331],[meff111 ] shows the `` effective mass '' plots for the combinations with smaller quark masses . we find that the signal is noisier for the lighter quarks and the fit with the smaller volumes @xmath2 and @xmath3 lattices do not satisfy the criteria . now we show the lattice qcd results of the lowest state in @xmath12 and @xmath142 channel . the filled circles in fig . [ negativegses ] show the lowest - state energies @xmath122 in @xmath12 and @xmath142 channel on four different volumes . here the horizontal axis denotes the lattice extent @xmath76 in the lattice unit and the vertical axis is the energy of the state . the lower line denotes the simple sum @xmath143 of the nucleon mass @xmath144 and kaon mass @xmath145 obtained with the largest lattice . though @xmath144 and @xmath145 are slightly affected by finite volume effects , the deviation of @xmath146 from @xmath147 is about a few % ( table [ results ] . ) . we therefore simply use @xmath147 as a guideline . at a glance , we find that the energy of this state takes almost constant value against the volume variation and coincides with the simple sum @xmath143 . we can therefore conclude that the lowest state in @xmath12 and @xmath142 channel is the nk scattering state with the relative momentum @xmath148 . the good agreement with the sum @xmath143 implies the weakness of the interaction between n and k. in fact , the scattering length in the @xmath12 channel is known to be tiny ( @xmath149 fm ) from compilations of hadron scattering experiments @xcite , whereas the current algebra prediction from pcac with su(3 ) symmetry predicts that the scattering length @xmath150 . we here compare our data with the previous lattice qcd studies , which were performed with almost the same conditions as ours , in order to confirm the reliability of our data . the well - known hadron masses @xmath151,@xmath152 and @xmath153 listed in table [ results ] can be compared with the values in ref . our data are consistent with those in ref . the lowest nk scattering state in @xmath9 channel is carefully investigated in ref . @xcite with almost the same parameters as our present study . it is worth comparing our data with them . for the complete check of our data , we re - extract the lowest state by the ordinary single - exponential fit of the correlator as @xmath154 in the large - t region , and compare them with the present lattice data @xmath122 obtained by the multi - exponential method as well as the data in ref . @xcite . in table [ results ] , we list the data of the lowest state @xmath155 obtained by the single - exponential fit . they almost coincide with the present data @xmath122 extracted by the multi - exponential method with about 1% deviations , which may be considered as the slightly remaining contaminations of the higher excited - state . in ref . @xcite , the authors extracted the energy difference @xmath156 with the hopping parameters @xmath157 using @xmath158 lattice . we therefore compare our data @xmath156 obtained with the hopping parameters @xmath159 on @xmath4 lattice . the energy difference @xmath160 in our study is found to be @xmath161 , which is consistent with the value of @xmath162 in ref . @xcite taking into account that this error includes only statistical one . it is now confirmed that the lowest state extracted using the multi - exponential method is consistent with the previous works and that our data and method are reliable enough to investigate the 2nd - lowest state in this channel . the @xmath9 state is one of the candidates for @xmath10 . since @xmath11 is located above the nk threshold , it would appear as an excited state in this channel . we show the lattice data of the 2nd - lowest state in this channel . in order to distinguish a possible resonance state from nk scattering states , we investigate the volume dependence of both the energy and the spectral weight of each state . it is expected that the energies of resonance states have small volume dependence , while the energies of nk scattering states are expected to scale as @xmath36 according to the relative momentum @xmath37 between n and k on a finite periodic lattice , provided that the nk interaction is weak and negligible . we can take advantage of the above difference for the discrimination . a possible candidate for the volume dependence of the energies of nk scattering states is the simple formula as @xmath163 with the relative momentum @xmath37 between n and k in finite periodic lattices , which is justified on the assumption that nucleon and kaon are point particles and that the interaction between them is negligible . in practice , there may be some corrections to the volume dependence of @xmath164 . we therefore estimate here three possible corrections ; the existence of the nk interaction , the application of the momenta on a finite discretized lattice and the estimation of the implicit finite - size effects . there can be small hadronic interactions between nucleon and kaon , which may lead to correction to naively expected energy spectrum @xmath164 of the nk scattering states . using l " uscher formula @xcite , one can relate the scattering phase shift to the energy shift from @xmath164 on finite lattices . for example , in the case when a system belongs to the representation @xmath165 of cubic groups , which is relevant in the present case , the relation between the phase shift and the possible momentum spectra is @xmath166 here @xmath167 is the zeta function defined as @xmath168 with the eigenenergy @xmath169 on a finite lattice . we have simply omitted the corrections from the partial waves with angular momenta higher than the next smallest one ( @xmath170 ) . although our current quark masses are heavier than those of the real quarks , we use the empirical values of the phase shift in nk scattering in ref.@xcite , by simply neglecting the quark mass dependence . the correction using the empirical values results in at most a few % larger energy than the simple formula @xmath164 within the volume range under consideration ; the energies are slightly increased by the weak repulsive force between nucleon and kaon . one may claim that one has to adopt momenta on a finite discretized lattice : @xmath171 for kaon and @xmath172 for nucleon , respectively . this correction turns out to be within only a few % lower energy than @xmath164 , although it is not certain whether this correction is meaningful or not for composite particles like nucleon or kaon . we find that these corrections lead to at most a few % deviations from @xmath164 . we then neglect these corrections for simplicity in the following discussion and use the simple form @xmath164 . so far , we have neglected the implicit finite - size effects in @xmath173 , other than the explicit ones due to the lattice momenta @xmath174 . some smart readers may suspect that the dispersions @xmath175 and @xmath176 may be affected by the uncontrollable finite - size effects due to the finite sizes of n and k , and no longer valid . in order to make sure of the small implicit artifacts especially with @xmath177 , which we are mainly interested in , we also calculate the sum @xmath178 of energies of nucleon @xmath179 and kaon @xmath180 with the smallest non - zero lattice momentum @xmath181 . we extract @xmath182 and @xmath183 from the correlators @xmath184 and @xmath185 . these results are denoted by the open squares in fig . [ negativegses ] as the sum @xmath178 . the upper lines in fig . [ negativegses ] show @xmath186 . the deviations of @xmath178 from @xmath187 are very small , which implies the smallness of the implicit finite - size artifacts . therefore , provided that the interaction between nucleon and kaon is weak , which we assume throughout the present analysis , the naive expectation for the 2nd - lowest nk scattering states denoted by the upper line in fig . [ negativegses ] would be able to follow the energies of the 2nd - lowest nk scattering states even on the @xmath188 lattices in our setup . the black ( gray ) filled - squares denote the lattice qcd data of the 2nd - lowest state in @xmath9 channel extracted with @xmath134 ( @xmath133 ) data plotted against the lattice extent @xmath76 . the filled circles represent the lattice qcd data @xmath122 of the lowest state in @xmath9 channel . the open symbols are the sum @xmath178 of energies of nucleon @xmath179 and kaon @xmath180 with the smallest lattice momentum @xmath181 . the upper line represents @xmath189 with @xmath190 the smallest relative momentum on the lattice . the lower line represent the simple sum @xmath143 of the masses of nucleon @xmath144 and kaon @xmath145 . we adopt the central values of @xmath144 and @xmath145 obtained on the largest lattice to draw the two lines . , title="fig : " ] + the black ( gray ) filled - squares denote the lattice qcd data of the 2nd - lowest state in @xmath9 channel extracted with @xmath134 ( @xmath133 ) data plotted against the lattice extent @xmath76 . the filled circles represent the lattice qcd data @xmath122 of the lowest state in @xmath9 channel . the open symbols are the sum @xmath178 of energies of nucleon @xmath179 and kaon @xmath180 with the smallest lattice momentum @xmath181 . the upper line represents @xmath189 with @xmath190 the smallest relative momentum on the lattice . the lower line represent the simple sum @xmath143 of the masses of nucleon @xmath144 and kaon @xmath145 . we adopt the central values of @xmath144 and @xmath145 obtained on the largest lattice to draw the two lines . , title="fig : " ] the black ( gray ) filled - squares denote the lattice qcd data of the 2nd - lowest state in @xmath9 channel extracted with @xmath134 ( @xmath133 ) data plotted against the lattice extent @xmath76 . the filled circles represent the lattice qcd data @xmath122 of the lowest state in @xmath9 channel . the open symbols are the sum @xmath178 of energies of nucleon @xmath179 and kaon @xmath180 with the smallest lattice momentum @xmath181 . the upper line represents @xmath189 with @xmath190 the smallest relative momentum on the lattice . the lower line represent the simple sum @xmath143 of the masses of nucleon @xmath144 and kaon @xmath145 . we adopt the central values of @xmath144 and @xmath145 obtained on the largest lattice to draw the two lines . , title="fig : " ] + the black ( gray ) filled - squares denote the lattice qcd data of the 2nd - lowest state in @xmath9 channel extracted with @xmath134 ( @xmath133 ) data plotted against the lattice extent @xmath76 . the filled circles represent the lattice qcd data @xmath122 of the lowest state in @xmath9 channel . the open symbols are the sum @xmath178 of energies of nucleon @xmath179 and kaon @xmath180 with the smallest lattice momentum @xmath181 . the upper line represents @xmath189 with @xmath190 the smallest relative momentum on the lattice . the lower line represent the simple sum @xmath143 of the masses of nucleon @xmath144 and kaon @xmath145 . we adopt the central values of @xmath144 and @xmath145 obtained on the largest lattice to draw the two lines . , title="fig : " ] the black ( gray ) filled - squares denote the lattice qcd data of the 2nd - lowest state in @xmath9 channel extracted with @xmath134 ( @xmath133 ) data plotted against the lattice extent @xmath76 . the filled circles represent the lattice qcd data @xmath122 of the lowest state in @xmath9 channel . the open symbols are the sum @xmath178 of energies of nucleon @xmath179 and kaon @xmath180 with the smallest lattice momentum @xmath181 . the upper line represents @xmath189 with @xmath190 the smallest relative momentum on the lattice . the lower line represent the simple sum @xmath143 of the masses of nucleon @xmath144 and kaon @xmath145 . we adopt the central values of @xmath144 and @xmath145 obtained on the largest lattice to draw the two lines . , title="fig : " ] we compare the lattice data @xmath123 with the expected behaviors @xmath187 for the 2nd - lowest nk scattering states . the filled squares in fig . [ negativegses ] denote @xmath123 , the 2nd - lowest - state energies in this channel . the black and gray symbols are the lattice data obtained by the fits with @xmath134 and @xmath133 time slices , respectively ( see the criterion.3 in the sec . [ data ] ) . the upper line shows the expected energy - dependence on @xmath191 of the 2nd - lowest nk scattering state @xmath186 estimated with the next - smallest relative momentum between n and k , and with the masses @xmath145 and @xmath144 extracted on the @xmath192 lattices . although the lattice qcd data @xmath123 and the expected lines @xmath193 almost coincide with each other on the @xmath192 lattices , which one may take as the characteristics of the 2nd - lowest scattering state , the data @xmath123 do not follow @xmath193 in the smaller lattices . ( at the smallest lattices with @xmath188 ( 1.4 fm ) in the physical unit , some results apparently coincide with each other again . however we consider that the volume with @xmath194 fm is too small for the pentaquarks ; it is difficult to tell which is the origin of the coincidence , uncontrollable finite volume effects of the pentaquarks or expected volume dependence of the 2nd - lowest nk scattering state . ) especially when the quarks are heavy , composite particles will be rather compact and we expect smaller finite volume effects besides those arising from the lattice momenta @xmath195 . moreover the statistical errors are also well controlled for the heavy quarks . thus , the significant deviations in @xmath196 fm with the combination of the heavy quarks , such as @xmath197 , are reliable and the obtained states are difficult to explain as the nk scattering states . therefore one can understand this behavior with the view that this state is a resonance state rather than a scattering state . in fact , while the data with the lighter quarks have rather strong volume dependences which can be considered to arise due to the finite size of a resonance state , the lattice data exhibit almost no volume dependence with the combination of the heavy quarks especially in @xmath196 fm , which can be regarded as the characteristic of resonance states . for further confirmation , we investigate the volume dependence of the spectral weight @xcite . as mentioned in sec . [ formalism ] , the correlation function @xmath198 can be expanded as @xmath199 . the spectral weight of the @xmath114-th state is defined as the coefficient @xmath200 corresponding to the overlap of the operator @xmath201 with the @xmath114-th excited - state . the normalization conditions of the field @xmath202 and the states @xmath203 give rise to the volume dependence of the weight factors @xmath200 in accordance with the types of the operators @xmath201 . for example , in the case when a correlation function is constructed from a point - source and a zero - momentum point - sink , as @xmath204 , the weight factor @xmath200 takes an almost constant value if @xmath203 is the resonance state where the wave function is localized . if the state @xmath203 is a two - particle state , the situation is more complicated . nevertheless if there is almost no interaction between the two particles , the weight factor is expected to be proportional to @xmath205 . in the case when a source is a wall operator @xmath206 as taken in this work , a definite volume dependence of @xmath200 is not known . therefore , we re - examine the lowest state and the 2nd - lowest state in @xmath9 channel using the locally - smeared source @xmath207 with @xmath208 , which we introduce to partially enhance the ground - state overlap . since smeared operators , whose typical sizes are much smaller than the total volume , can be regarded as local operators , we can discriminate the states using the locally - smeared operators as in the case of point operators . ( we also investigated the weight factor using the point source . the results are consistent with those obtained using the locally - smeared one , but are rather noisy . ) we adopt the hopping parameter @xmath197 and additionally employ @xmath209 lattice for this aim . the spectral weight factors defined in sec . [ negative2 - 3 ] are plotted against the lattice volume @xmath191 . the left figure shows @xmath210 for the lowest state in @xmath9 channel and the right figure shows @xmath211 for the 2nd - lowest state . in the case when the weight factor @xmath200 for the @xmath114-th state @xmath203 in a point - point correlator shows no volume dependence , @xmath203 is likely to be a resonance state . on the contrary , when the @xmath114-th state @xmath203 is a two - particle state , @xmath200 scales according to @xmath212 . , title="fig : " ] the spectral weight factors defined in sec . [ negative2 - 3 ] are plotted against the lattice volume @xmath191 . the left figure shows @xmath210 for the lowest state in @xmath9 channel and the right figure shows @xmath211 for the 2nd - lowest state . in the case when the weight factor @xmath200 for the @xmath114-th state @xmath203 in a point - point correlator shows no volume dependence , @xmath203 is likely to be a resonance state . on the contrary , when the @xmath114-th state @xmath203 is a two - particle state , @xmath200 scales according to @xmath212 . , title="fig : " ] we extract @xmath210 and @xmath211 using the two - exponential fit as @xmath213 . the fit with four free parameters @xmath210 , @xmath211 , @xmath214 and @xmath215 is however unstable and therefore we fix the exponents using the obtained values @xmath122 and @xmath123 . the weight factors @xmath210 and @xmath211 are then obtained through the two - parameter fit as @xmath216 in as large @xmath136 range ( @xmath217 ) as possible in order to avoid the contaminations of the higher excited - states than the 2nd - excited state ( 3rd - lowest state ) , which will bring about the instability of the fitted parameters . the fluctuations of @xmath122 and @xmath123 are taken into account through the jackknife error estimation . [ weightfactor ] includes all the results with the various fit range as ( @xmath129,@xmath218)= ( 16,18),(17,19),(18,20 ) to see the fit - range dependence . though the results have some fit - range dependences , the global behaviors are almost the same among the three . the left figure in fig . [ weightfactor ] shows the weight factor @xmath210 of the lowest state in @xmath9 channel against the lattice volume @xmath191 . we find that @xmath210 decreases as @xmath191 increases and that the dependence on @xmath191 is consistent with @xmath205 , which is expected in the case of two - particle states . it is again confirmed that the lowest state in this channel is the nk scattering state with the relative momentum @xmath148 . next , we plot the weight factor @xmath211 of the 2nd lowest state in the right figure . in this figure , almost no volume dependence against @xmath191 is found , which is the characteristic of the state in which the relative wave function is localized . ( in appendix , we try another prescription to estimate the volume dependences of the spectral weights , which requires no multi - exponential fit . ) this result can be considered as one of the evidences of a resonance state lying slightly above the nk threshold . to summarize this section , the volume dependence analysis of the eigenenergies and the weight factors of the 2nd - lowest state in @xmath9 channel suggests the existence of a resonance state . although there remain the statistical errors and the possible finite - volume artifacts , the data can be consistently accounted for assuming the 2nd - lowest state to be different from ordinary scattering states . if the 2nd - lowest state were an ordinary scattering state , one had to assume a large systematic errors for heavier quarks which is hard to understand consistently the lattice qcd data in the @xmath219 channel are plotted against the lattice extent @xmath76 . the solid line denotes the simple sum @xmath220 of the masses of the lowest - state negative - parity nucleon @xmath221 and kaon @xmath145 obtained with the largest lattice . , title="fig : " ] the lattice qcd data in the @xmath219 channel are plotted against the lattice extent @xmath76 . the solid line denotes the simple sum @xmath220 of the masses of the lowest - state negative - parity nucleon @xmath221 and kaon @xmath145 obtained with the largest lattice . , title="fig : " ] the lattice qcd data in the @xmath219 channel are plotted against the lattice extent @xmath76 . the solid line denotes the simple sum @xmath220 of the masses of the lowest - state negative - parity nucleon @xmath221 and kaon @xmath145 obtained with the largest lattice . , title="fig : " ] the lattice qcd data in the @xmath219 channel are plotted against the lattice extent @xmath76 . the solid line denotes the simple sum @xmath220 of the masses of the lowest - state negative - parity nucleon @xmath221 and kaon @xmath145 obtained with the largest lattice . , title="fig : " ] the lattice qcd data in the @xmath219 channel are plotted against the lattice extent @xmath76 . the solid line denotes the simple sum @xmath220 of the masses of the lowest - state negative - parity nucleon @xmath221 and kaon @xmath145 obtained with the largest lattice . , title="fig : " ] in the same way as @xmath9 channel , we have attempted to diagonalize the correlation matrix in @xmath219 channel using the wall - sources @xmath71 and the zero - momentum point - sinks @xmath222 . in this channel , the diagonalization is rather unstable and we find only one state except for tiny contributions of possible other states . we plot the lattice data in fig . [ positivess ] . one finds that they have almost no volume dependence and that they coincide with the solid line which represents the simple sum @xmath220 of @xmath221 and @xmath145 , with @xmath221 the mass of the ground state of the _ negative - parity _ nucleon . from this fact , the state we observe is concluded to be the @xmath223-@xmath224 scattering state with the relative momentum @xmath148 . it may sound strange because the p - wave state of n and k with the relative momentum @xmath190 should be lighter than the @xmath223-@xmath224 scattering state with the relative momentum @xmath148 ; this lighter state is missing in our analysis . this failure would be due to the wall - like operator @xmath225 . the fact that the wall operator @xmath225 is constructed by the spatially spread quark fields @xmath226 with zero momentum may leads to the large overlaps with the nk scattering state with zero relative momentum . the relation between operators and overlap coefficients is an interesting problem and is to be explored in detail for further studies . anyway , the strong dependence on the choice of operators suggests that it is needed to try various types of operators before giving the final conclusion . before closing this section , we show the spectral weight @xmath227 obtained by the fit using the form @xmath228 in fig . [ weightfactor2 ] . the spectral weight factor of the extracted state in @xmath219 channel with the hopping parameters @xmath197 is plotted against the lattice volume @xmath191 . we note here that the @xmath229-like volume dependence _ is not _ always the characteristics of scattering states when we do not adopt point - point correlators as is the present case . to conclude from this dependence , we need to determine the precise volume dependences of the weight factors in wall - point correlators . ] although one sees the @xmath205-like volume dependence in fig . [ weightfactor2 ] , one can conclude nothing only from this behavior unless the precise volume dependences of the weight factors in wall - point correlators are estimated . we here mention the operator dependences in @xmath9 channel . as is seen in sec . [ positive ] , the overlap factors with states strongly depend on the choice of operators . we survey the effective masses of the five correlators ; @xmath230 , @xmath231 , @xmath232 , @xmath233 and @xmath234 . the effective mass e(t ) is defined as a ratio between correlators with the temporal separation @xmath27 and @xmath235 , @xmath236 which can be expressed in terms of the eigenenergies and spectral weights as @xmath237 a plateau in e(t ) at @xmath238 implies the ground - state dominance in the correlator . effective mass plots @xmath239 are often used to find the range where correlators show a single - exponential behavior ; the higher excited - state contributions @xmath240 are negligible in comparison with the ground - state component @xmath241 . here , @xmath242 is an interpolation operator defined as @xmath243[u_g(x)c\gamma_5d_h(x)]c\bar s_c(x)\ ] ] which has a di - quark structure similar to that proposed by jaffe and wilczek @xcite , and is also used in refs . @xcite . the effective mass plots constructed from @xmath230 , @xmath231 , @xmath232 , @xmath233 and @xmath234 , in @xmath9 channel in @xmath5 lattice at @xmath82 employing the hopping parameters ( @xmath244)=(0.1600,0.1600 ) are plotted , along with the dashed line which denotes the lowest - state energy @xmath122 . ] [ variouseffectivemass ] shows the effective mass plots constructed from @xmath230 , @xmath231 , @xmath232 , @xmath233 and @xmath234 . one can see two typical behaviors in the figure . one is the line damping from a large value to the energy @xmath122 of the lowest nk scattering state . the other is the one arising upward to @xmath122 . surprisingly , the differences of the spinor structure or the color structure among the operators are hardly reflected in the effective mass plots . the difference is enough to perform the variational method but seems insufficient for a clear change of the effective mass plots . instead , the effective mass plots seem sensitive to the spatial distribution of operators . the upper three symbols are data using the point - point correlators and the lower two symbols are those from the wall - point correlators . this means the overlap factor with each state is controlled mainly by the spatial distribution rather than the internal structure of operators , except for the overall constant . the spatially smeared operators seem to have larger overlaps with the scattering state with the relative momentum @xmath148 . ( one can find that the overlap factor of the wall operator with the observed state in @xmath219 in fig . [ weightfactor2 ] is 1000 times larger than those of point operators in fig . [ weightfactor ] . ) one often expects that the overlap with a state could be enhanced using an operator whose spinor or color structures resembles the state . we find however no such tendency in the present analysis . the insensitivity to the spinor structures may come from the fact that the kn - type operator ( @xmath245 ) and the di - quark type operator ( @xmath242 ) are directly related by a factor of @xmath246 and a fierz rearrangement @xcite . though we have no idea about the mechanism of the insensitivity to the color structure at present , this insensitivity would have some connection with the internal color - structure of @xmath11 . the upper three data slowly damp and do not reach the lowest energy @xmath122 in this @xmath27 range , which can be explained in terms of the spectral weight . as is seen in fig . [ weightfactor ] , @xmath210 is ten - times smaller than @xmath211 in the case of the point - point correlator . then , the term @xmath247 in the effective mass survives at relatively large @xmath27 . hence the effective mass needs larger @xmath27 to show a plateau at @xmath122 . the insensitivity of the overlaps to the internal structure of operators could be helpful for us : we have adopted two operators whose color and spinor structures are different from each other . although the difference is enough in @xmath9 channel , it may be insufficient in @xmath219 channel , which leads to the failure in the diagonalization . if we use operators with spatial distributions different from each other , it would be more effective in the diagonalization method . here we comment on other works previously published , especially for the pioneering works by csikor _ et al . _ @xcite and sasaki @xcite . the simulation condition for the former is rather similar to ours . csikor _ et al . _ first reported the possible pentaquark state slightly above the nk threshold in @xmath9 channel in @xcite . in ref . @xcite , they tried chiral extrapolations and taking the continuum limit at the quenched level for the possible pentaquark state . however they used the single - exponential fit analysis for the non - lowest state , namely the possible pentaquark state , for the main results . it is difficult to justify their result unless the coupling of the operators to the lowest nk state is extremely small . sasaki found a double - plateau in the effective mass plot and identified the 2nd - lowest plateau as the signal of @xmath11 . unfortunately , we does not find a double - plateau in the present analysis . the double - plateau - like behavior in effective mass plots can appear only under the extreme condition that @xmath211 is much larger than @xmath210 . @xmath211 which is ten times larger than @xmath210 seen in fig . [ weightfactor ] and the statistical fluctuations may cause the deviation of the effective mass plot from the single monotonous line . in fact , the effective mass plot very slowly approaches @xmath122 as @xmath27 increases in fig . [ variouseffectivemass ] . he extracted the mass of the next - lowest state with a single and double exponential fits . the result do not contradict with ours . @xcite reports a lattice qcd study which adopted the overlap - fermions with the exact chiral symmetry . the hybrid - boundary method was suggested in ref . @xcite and the authors tried to single out the possible resonance state . in these two studies , the absence of resonance states with a mass a few hundred mev above the nk threshold was concluded . we have not found the resonance state which coincides just with the mass of @xmath11 in the chiral limit . in this sense , the results in refs . @xcite are not inconsistent with ours . we perform chiral extrapolations for kaon , nucleon , nk threshold ( a simple sum of a kaon mass and a nucleon mass ) and the 2nd - lowest state in the @xmath9 channel . we adopt the lattice data with @xmath5 lattice , the largest lattice in our analysis . one can find in fig . [ negativegses ] that the 2nd - lowest state , which is expected to be a resonance state , is already affected by the finite volume effects for @xmath248 with the lightest combination of quarks , and we therefore adopt the largest - lattice data for safety . we can expect from this fact that the typical diameter of this resonance is about 2 fm or longer and that it is desirable to use larger lattices than , @xmath249 for the analysis of @xmath11 . in fig . [ chiralextrapolations ] , @xmath250 and @xmath123 obtained with each combination of quark masses for @xmath4 and @xmath5 lattices are plotted against @xmath251 . we assume the linear function of quark masses , @xmath252 , for nucleon and the 2nd - lowest state with @xmath253 free parameters fitted using the five lattice data . we determine the critical @xmath254 ( @xmath255 ) by @xmath256 and fix the @xmath257 so that the physical kaon mass is reproduced in the chiral limit , using the form for pseudo scalar mesons @xmath258 . the chiral - extrapolated values of @xmath145 , @xmath144 , @xmath250 and @xmath123 for @xmath5 lattice are 0.4274(12 ) , 0.7996(60 ) , 1.227(6 ) and 1.500(52 ) in the lattice unit and 0.5001(14 ) , 0.9355(70 ) , 1.436(7 ) and 1.755(61 ) in the unit of gev , respectively . we find that the results for @xmath4 lattice are consistent within errors as shown in fig . [ chiralextrapolations ] . the value of @xmath123=1755(61 ) mev in the chiral limit is significantly larger than the mass of @xmath10 in the real world . how can we interpret this deviation ? one possibility is the systematic errors from the discretization , the chiral extrapolation , or quenching . another possibility is that the observed 2nd - lowest state might be a signal of a resonance state lying higher than @xmath11 . unfortunately there is no clear explanation at this point . obviously more extensive studies on finer lattices with lighter quark masses in unquenched qcd are required . however , we can at least conclude that _ our quenched lattice calculations suggest the existence of a resonance - like state slightly above the nk threshold for the parameter region we have investigated . _ a comparison of the chiral extrapolations in @xmath9 channel on @xmath4 ( left ) and @xmath5 ( right ) lattices for the nk threshold energy @xmath250 and the 2nd - lowest state energy @xmath123 . @xmath250 and @xmath123 are plotted against @xmath259 . the filled circles ( triangles ) denote the energies of the nk threshold ( 2nd - lowest state ) in the chiral limit . @xmath257 is fixed to be @xmath260 so that the physical kaon mass is reproduced in the chiral limit . , title="fig : " ] a comparison of the chiral extrapolations in @xmath9 channel on @xmath4 ( left ) and @xmath5 ( right ) lattices for the nk threshold energy @xmath250 and the 2nd - lowest state energy @xmath123 . @xmath250 and @xmath123 are plotted against @xmath259 . the filled circles ( triangles ) denote the energies of the nk threshold ( 2nd - lowest state ) in the chiral limit . @xmath257 is fixed to be @xmath260 so that the physical kaon mass is reproduced in the chiral limit . , title="fig : " ] we have performed the lattice qcd study of the @xmath261 states on @xmath2 , @xmath3 , @xmath4 and @xmath5 lattices at @xmath6=5.7 at the quenched level with the standard plaquette gauge action and wilson quark action . to avoid the possible contaminations originating from the ( anti)periodic boundary condition , which are peculiar to the pentaquark and have not been properly noticed in previous studies , we have adopted the dirichlet boundary condition in the temporal direction for the quark field . with the aim to separate states clearly , we have adopted two independent operators with @xmath12 and @xmath262 so that we can construct a @xmath7 correlation matrix . from the correlation matrix of the operators , we have successfully obtained the energies of the lowest state and the 2nd - lowest state in the @xmath9 channel . the volume dependence of the energies and spectral weight factors show that the 2nd - lowest state in this channel is likely to be a resonance state located slightly above the nk threshold and that the lowest state is the nk scattering state with the relative momentum @xmath148 . as for the @xmath219 channel , we have observed only one state in the present analysis , which is likely to be a @xmath263 scattering state of the ground state of the negative - parity nucleon @xmath223 and kaon with the relative momentum @xmath148 . we have also investigated the overlaps using five independent operators . as a result , we have found that the overlaps seem to be insensitive to the spinor and color structure of operators while the overlaps are mainly controlled by the spatial distributions of operators , at least for a few low - lying state in this analysis . for the diagonalization method , it may be more effective to vary the spatial distributions rather than the internal structures . the volume dependence of @xmath123 suggests that this resonance - like state in the @xmath9 channel is a rather spread object with the radius of about 1 fm or more . the possibility of a resonance state lying in @xmath9 channel is desired to be confirmed by other theoretical studies , such as quark models , qcd sum rules , string models and so on @xcite . unfortunately , four quarks _ uudd _ and one antiquark @xmath264 in @xmath265 state can hardly reproduce the unusually narrow width of @xmath11 so far , while the obtained mass in @xmath142 channel could be assigned to the observed resonance state @xcite . hence , @xmath266 or @xmath267 states are favored to reproduce the width in terms of the quark model . however , there are many unknown problems left so far such as the internal structure of multi - quark hadrons @xcite or the dynamics of the string / flux - tubes @xcite . the discovery of @xmath11 gives us many challenges in the hadron physics and more detailed theoretical study including the lattice qcd studies are awaited . for further analyses , a variational analysis using the @xmath78 correlation matrix or larger matrices will be desirable . the observation of wave functions will be also useful to distinguish a resonance state from scattering states and to investigate the internal structures of hadrons . we can use the lattice qcd calculations in order to estimate the decay width @xcite and to study the flux - tube dynamics @xcite , which should give useful inputs for model calculations . we acknowledge the yukawa institute for theoretical physics at kyoto university , where this work was initiated from the discussions during the yitp - w-03 - 21 workshop on `` multi - quark hadrons : four , five and more ? '' . t. t. t. thanks dr . f. x. lee for the useful advice . t. u. and t. o. thank dr . t. yamazaki for the fruitful discussion . t. t. t. and t. u. were supported by the japan society for the promotion of science ( jsps ) for young scientists . t. o. and t. k. are supported by grant - in - aid for scientific research from the ministry of education , culture , sports , science and technology of japan ( nos . 13135213,16028210 , 16540243 ) and ( nos . 14540263 ) , respectively . this work is also partially supported by the 21st century for center of excellence program . the lattice qcd monte carlo calculations have been performed on nec - sx5 at osaka university and on hitachi - sr8000 at kek . after the completion of this paper , refs . @xcite which also study the pentaquark state with lattice qcd have appeared on the preprint server . 0 t. nakano _ et al . _ [ leps collaboration ] , phys . rev . lett . * 91 * , 012002 ( 2003 ) [ arxiv : hep - ex/0301020 ] . v. v. barmin _ et al . _ [ diana collaboration ] , phys . atom . nucl . * 66 * , 1715 ( 2003 ) [ yad . fiz . * 66 * , 1763 ( 2003 ) ] [ arxiv : hep - ex/0304040 ] . s. stepanyan _ et al . _ [ clas collaboration ] , phys . lett . * 91 * , 252001 ( 2003 ) [ arxiv : hep - ex/0307018 ] . j. barth _ et al . _ [ saphir collaboration ] , phys . b * 572 * , 127 ( 2003 ) . s. chekanov _ et al . _ [ zeus collaboration ] , phys . b * 591 * , 7 ( 2004 ) [ arxiv : hep - ex/0403051 ] . a. e. asratyan , a. g. dolgolenko and m. a. kubantsev , phys . nucl . * 67 * , 682 ( 2004 ) [ yad . * 67 * , 704 ( 2004 ) ] [ arxiv : hep - ex/0309042 ] . a. airapetian _ et al . _ [ hermes collaboration ] , phys . b * 585 * , 213 ( 2004 ) [ arxiv : hep - ex/0312044 ] . m. abdel - bary _ et al . _ [ cosy - tof collaboration ] , phys . b * 595 * , 127 ( 2004 ) [ arxiv : hep - ex/0403011 ] . v. kubarovsky _ et al . _ [ clas collaboration ] , phys . * 92 * , 032001 ( 2004 ) [ erratum - ibid . * 92 * , 049902 ( 2004 ) ] [ arxiv : hep - ex/0311046 ] . m. oka , prog . . phys . * 112 * , 1 ( 2004 ) and references therein [ arxiv : hep - ph/0406211 ] . r. l. jaffe , phys . rept . * 409 * , 1 ( 2005 ) [ nucl . suppl . * 142 * , 343 ( 2005 ) ] [ arxiv : hep - ph/0409065 ] . d. diakonov , v. petrov and m. polyakov , z. phys . * a 359 * , 305 ( 1997 ) . m. praszalowicz , phys . b * 575 * , 234 ( 2003 ) [ arxiv : hep - ph/0308114 ] . stancu and d. o. riska , phys * b575 * , 242 ( 2003 ) [ arxiv : hep - ph/0307010 ] ; 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( @xmath268,@xmath257)=(0.1650,0.1650 ) .[results ] the pion masses @xmath269 , kaon masses @xmath145 , nucleon masses @xmath144 , masses of the ground state of negative - parity nucleon @xmath221 , energies of the lowest state @xmath122 , energies of the 2nd - lowest state @xmath123 , energies of the lowest state @xmath155 ( obtained by single - exponential fit ) in the @xmath9 channel are listed . the energies of the obtained state @xmath270 in the @xmath219 channel are also listed . @xmath268 and @xmath257 are the hopping parameters for @xmath271 quarks and @xmath272 quark respectively . [ cols="^,^,^,^,^,^,^,^,^",options="header " , ] in this appendix , we make another trial to estimate volume dependences of weight factors in @xmath9 channel . as seen in sec . [ negative2 - 3 ] , we have extracted the weight factors using double - exponential fit , which is however rather unstable and we have therefore fixed the exponents . we here discuss the possibility of methods without any multi - exponential fits . let us again consider @xmath273 correlation matrices constructed by @xmath274-sink @xmath275-source and @xmath274-sink @xmath6-source correlators . here @xmath276 denote the types of operators , such as `` point '' or `` wall '' or `` smear '' and so on . the notations are the same as those in sec . [ formalism ] . the @xmath277 and @xmath278 correlation matrices are described as @xmath279 with @xmath273 matrices ( @xmath280 ) and ( @xmath281 ) being possible higher excited - state contaminations . we hereby consider two quantities ; @xmath282^t { \cal c}_{ij}^{\rm \gamma\alpha}(t)$ ] defined using one type of the correlation matrix and @xmath283^{-1}{\cal c}_{ij}^{\rm \gamma\beta}(t)$ ] , which with large @xmath27 lead to @xmath284^t { \cal c}_{ij}^{\rm \gamma\alpha}(t ) = c^{\gamma\dagger } c^\alpha + { \cal f}(d(t))+ ... .\ ] ] and @xmath285^{-1}{\cal c}_{ij}^{\rm \gamma\beta}(t ) = ( c^{\alpha})^{-1}c^\beta + { \cal f}'(d(t))+ ... , \ ] ] respectively . here @xmath286 and @xmath287 are terms including @xmath273 diagonal matrix @xmath288 . then , each component of @xmath282^t { \cal c}_{ij}^{\rm \gamma\alpha}(t)$ ] and @xmath283^{-1}{\cal c}_{ij}^{\rm \gamma\beta}(t)$ ] gets stable and shows a plateau in large @xmath27 region , where @xmath286 and @xmath287 are negligible . next , we relate these quantities to spectral weights . for this aim , we simply take the determinants . in the case when the correlation matrices are @xmath7 matrices , the determinant @xmath289 is explicitly written as @xmath290 , and the determinant @xmath291 is expressed as @xmath292 . the term @xmath293 ( @xmath294 ) denotes the product of the overlaps of the @xmath6(@xmath275)-type operator with the lowest state and the 2nd - lowest state . on the other hand , @xmath295 ( @xmath296 ) corresponds to the spectral weight for the lowest ( 2nd - lowest ) state in the @xmath6-@xmath274 correlator in terms of a volume dependence . let us consider the several cases when ( @xmath276)=\{w(wall ) , s(smeared ) , p(point)}. the term @xmath297 behaves showing the same volume dependence as the product of the spectral weights for the lowest and the 2nd - lowest state in the smeared - point correlator , which _ should be _ @xmath298 _ if _ the lowest state is a scattering state and the 2nd - lowest state is a resonance state . the left panel in fig . [ app1 ] represents @xmath299^t { \cal c}_{ij}^{\rm ps}(t)\right)$ ] on each volume . however , @xmath299^t { \cal c}_{ij}^{\rm ps}(t)\right)$ ] on each volume , which approaches @xmath297 with large @xmath27 , has relatively large errors and fluctuations with no clear plateau and we fail to extract @xmath297 . this would be due to the smallness of the signals in smeared - point correlators . meanwhile , @xmath300^t { \cal c}_{ij}^{\rm pw}(t)\right)$ ] shown in the middle panel in fig . [ app1 ] and @xmath301^{-1}{\cal c}_{ij}^{\rm ps}(t)\right)$ ] shown in the right panel in fig . [ app1 ] , which approach @xmath302 and @xmath303 respectively , show relatively clear plateaus . therefore we extract @xmath302 and @xmath303 by the fits @xmath304^t { \cal c}_{ij}^{\rm pw}(t)\right)$ ] and @xmath305^{-1}{\cal c}_{ij}^{\rm ps}(t)\right)$ ] in the @xmath27 range where they show plateaus and we finally obtain @xmath297 as @xmath306 . in fig . [ app2 ] , we show @xmath297 obtained by the prescription shown above . the solid line denotes the best - fit curve by @xmath307 and the dashed line does the best - fit curve by @xmath308 . the best fit parameters are @xmath309 and @xmath310 , and @xmath311 is 1.72 and 7.13 respectively . the volume dependence of @xmath297 seems not to be inconsistent with @xmath205 . if we know the precise volume dependence of overlaps of wall operators , we may be discriminate the states using @xmath302 or @xmath303 . ^t { \cal c}_{ij}^{\rm ps}(t)\right)$ ] as the function of @xmath27 on each volume . the middle figure is the plot of @xmath300^t { \cal c}_{ij}^{\rm pw}(t)\right)$ ] against @xmath27 . @xmath312^{-1}{\cal c}_{ij}^{\rm ps}(t)\right)$ ] is plotted in the right figure . @xmath313^t { \cal c}_{ij}^{\rm ps}(t)\right)$ ] , @xmath300^t { \cal c}_{ij}^{\rm pw}(t)\right)$ ] and @xmath301^{-1}{\cal c}_{ij}^{\rm ps}(t)\right)$ ] show plateaus in the large @xmath27 region and coincide with @xmath297 , @xmath302 and @xmath303 respectively . [ app1],title="fig : " ] ^t { \cal c}_{ij}^{\rm ps}(t)\right)$ ] as the function of @xmath27 on each volume . the middle figure is the plot of @xmath300^t { \cal c}_{ij}^{\rm pw}(t)\right)$ ] against @xmath27 . @xmath312^{-1}{\cal c}_{ij}^{\rm ps}(t)\right)$ ] is plotted in the right figure . @xmath313^t { \cal c}_{ij}^{\rm ps}(t)\right)$ ] , @xmath300^t { \cal c}_{ij}^{\rm pw}(t)\right)$ ] and @xmath301^{-1}{\cal c}_{ij}^{\rm ps}(t)\right)$ ] show plateaus in the large @xmath27 region and coincide with @xmath297 , @xmath302 and @xmath303 respectively . [ app1],title="fig : " ] ^t { \cal c}_{ij}^{\rm ps}(t)\right)$ ] as the function of @xmath27 on each volume . the middle figure is the plot of @xmath300^t { \cal c}_{ij}^{\rm pw}(t)\right)$ ] against @xmath27 . @xmath312^{-1}{\cal c}_{ij}^{\rm ps}(t)\right)$ ] is plotted in the right figure . @xmath313^t { \cal c}_{ij}^{\rm ps}(t)\right)$ ] , @xmath300^t { \cal c}_{ij}^{\rm pw}(t)\right)$ ] and @xmath301^{-1}{\cal c}_{ij}^{\rm ps}(t)\right)$ ] show plateaus in the large @xmath27 region and coincide with @xmath297 , @xmath302 and @xmath303 respectively . [ app1],title="fig : " ] is plotted as the function of the lattice volume @xmath191 . the solid line denotes the best - fit curve by @xmath307 and the dashed line does the best - fit curve by @xmath308 . the best fit parameters are @xmath309 and @xmath310 , and @xmath311 is 1.72 and 7.13 respectively . these data behaves consistently in accordance with @xmath212 .
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we study spin @xmath0 hadronic states in quenched lattice qcd to search for a possible @xmath1 pentaquark resonance .
simulations are carried out on @xmath2 , @xmath3 , @xmath4 and @xmath5 lattices at @xmath6=5.7 at the quenched level with the standard plaquette gauge action and the wilson quark action .
we adopt a dirichlet boundary condition in the time direction for the quark to circumvent the possible contaminations due to the ( anti)periodic boundary condition for the quark field , which are peculiar to the pentaquark . by diagonalizing the @xmath7 correlation matrices constructed from two independent operators with the quantum numbers @xmath8 ,
we successfully obtain the energies of the lowest state and the 2nd - lowest state in this channel .
the analysis of the volume dependence of the energies and spectral weight factors indicates that a resonance state is likely to exist slightly above the nk threshold in @xmath9 channel .
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phase transition in a given physico - chemical system is characterized by parameters like the range of the microscopic interactions , the space dimensionality @xmath1 and the dimensionality of the order parameter , often referred to the spin dimensionality @xmath2 . there are features whose qualitative nature is determined by the universality class to which the system belongs . short - range interactions , double and super - exchange nearest - neighbor type , classical and quantum spins @xmath2 in @xmath1-dimensional systems have been studied @xcite . double - exchange ( de ) interaction or indirect exchange , is the source of a variety of magnetic behavior in transition metal and rare - earth compounds@xcite . the origin of de lies in the intra - atomic coupling of the spin of itinerant electrons with localized spins @xmath3 . this coupling favors a ferromagnetic ( f ) background of local spins and may lead to interesting transport properties such as colossal magnetoresistance . this mechanism has been widely used in the context of manganites @xcite . this f tendency is expected to be frustrated by anti - ferromagnetic ( af ) inter - atomic super - exchange ( se ) interactions between localized spins @xmath3 as first discussed by de gennes@xcite who conjectured the existence of canted states.in spite of recent interesting advances , our knowledge of magnetic ordering resulting from this competition is still incomplete . although it may look academic , the one - dimensional ( 1d ) version of this model is very illustrative and helpful in building an unifying picture . on the other hand , the number of pertinent real 1d systems as the nickelate one - dimensional metal oxide carrier - doped compound @xmath0@xcite is increasing . haldane gap @xmath4 has been observed for the parental compound @xmath5 @xmath6 ( s=1 ) from susceptibility and neutron scattering measurements . in these compounds , carriers are essentially constrained to move parallel to @xmath7 chains and a spin - glass - like behavior was found at very low temperature @xmath8 for typical dopings @xmath9 , @xmath10 and @xmath11 . at high temperature curie - like behavior of the magnetic susceptibility was found . the question is how physical properties change by introducing @xmath12 holes in the system . in the doped case the itineracy of doped electrons or holes plays an important role taken into account by the double - exchange mechanism . recently , it has been shown that three - leg ladders in the oxyborate system fe@xmath13bo@xmath14 may provide evidence for the existence of spin and charge ordering resulting from such a competition@xcite . naturally , the strength of the magnetic interactions depends significantly on the conduction electron band filling , @xmath15 . at low conduction electron density , f polarons have been found for localized @xmath16 quantum spins batista19982000 . island phases , periodic arrangement of f polarons coupled anti - ferromagnetically , have been clearly identified at commensurate fillings both for quantum spins in one dimension @xcite and for classical spins in onekoshibae1999 and two dimensions @xcite . phase separation between hole - undoped antiferromagnetic and hole - rich ferromagnetic domains has been obtained in the ferromagnetic kondo model @xcite . phase separation and small ferromagnetic polarons have been also identified for localized @xmath17 quantum spins @xcite . in addition to the expected f - af phase separation appearing for small super - exchange coupling , a new phase separation between small polarons ordered ( one electron within two or three sites ) and af regions for larger se coupling was found @xcite . these phase separations are degenerate with phases where the polarons can be ordered or not giving a natural response to the instability at the fermi energy and to an infinite compressibility as well . wigner crystallization and spin - glass - like behavior were also obtained and could explain the spin - glass - like behavior observed in the nickelate 1d doped compound @xmath0 @xcite . in this paper , we present a study of the parallel static magnetic susceptibility in an ising - like exchange model . short - range spin - spin correlations are also presented . our results are compared with the curie - like behavior observed at high temperature in the nickelate one - dimensional compound @xmath0 @xcite . the paper is organized as follows . in section ii a brief description of the model is given . in section iii , results and a discussion are presented . finally , our results are summarized in section iv . the de hamiltonian is originally of the form , @xmath18where @xmath19 are the fermions creation ( annihilation ) operators of the conduction electrons at site @xmath20 , @xmath21 is the hopping parameter and @xmath22 is the electronic conduction band spin operator . in the second term , @xmath23 is the hund s exchange coupling . here , hund s exchange coupling is an intra - atomic exchange coupling between the spins of conduction electrons @xmath24 and the spin of localized electrons @xmath3 . this hamiltonian simplifies in the strong coupling limit @xmath25 , a limit commonly called itself the de model . in this strong coupling limit itinerant electrons are now either parallel or anti - parallel to local spins and are thus spinless . the complete one dimensional de+se hamiltonian becomes , @xmath26 @xmath27 is the relative angle between localized spins at sites @xmath20 , @xmath28 defined with respect to a z - axis taken as the spin quantization axis of the itinerant electrons . the super - exchange coupling is an anti - ferromagnetic inter - atomic exchange coupling between localized spins @xmath3 . this coupling is given in the second term of the former equation . here @xmath29 is the super - exchange interaction energy . an ising - like model with itinerant electrons will be considered in this paper , i. e. @xmath30 and @xmath31 or @xmath32 . for itinerant electrons ( holes ) an electron ( hole)-single approximation will be used . the nickelate one - dimensional parental compound @xmath33 , is basically formed of quasi one - dimensional chains of @xmath6 . @xmath34 and @xmath35 are two relevant @xmath6 orbitals in this system . @xmath35 is basically localized while @xmath34 has finite overlap with @xmath36 orbital of the o @xcite . so , to make contact with the nickelate one - dimensional compound @xmath0 , @xmath37 localized s=1/2 spins in the @xmath35 orbital will be considered . on the other hand itinerant electrons @xmath38 or holes @xmath12 will be placed in the @xmath34 orbital . the role of these electrons ( holes ) within the parental compound @xmath5 , will be considered by the de mechanism . within our ising - like model there is an electron - hole symmetry . exact parallel static magnetic susceptibility @xmath39 and short - range spin - spin correlations are presented using a standard canonical ensemble . to obtain @xmath39 within the electron ( hole)-single approximation is necessary to calculate eigenvalues of the following matrix @xmath40 where @xmath41 in the former equation first term is super - exchange interaction and the second one is the zeeman coupling of the localized background of s=1/2 spins . third term is the coupling between the magnetic moment @xmath42 of the itinerant electron and the magnetic field @xmath43 . a magnetic field was introduced to calculate @xmath39 . @xmath44 with eigenvalues of equation ( [ hamil ] ) is easy to obtain partition function @xmath45 in the canonical ensemble within the electron - single approximation @xmath46 for one ( @xmath20 ) , two ( @xmath20 and @xmath47 ) , three ( @xmath20 , @xmath47 and @xmath48 ) and ( @xmath49 ) itinerant electrons respectively . @xmath50 being @xmath51 boltzmann constant and @xmath52 temperature@xmath53 magnetic susceptibility is related with partition function as @xmath54 mean value of all operators can be related to partition function i. e. @xmath55 @xmath56 on the other hand , the phenomenological ising - like model was proposed because of our previous results using classical localized spins lead basically to an ising - like model @xcite . high temperature @xmath39 will be compared with experimental results of the nickelate one - dimensional compound @xmath0 @xcite . in this section , phase diagram , parallel static magnetic susceptibility ms and short - range spin - spin correlations are presented for a particular open linear chain of @xmath57 sites . in the thermodynamic limit , phase diagram is shown in figure [ fig1cl ] . this phase diagram is similar to our previous one using classical localized spins ( s=3 ) @xcite . phase separation between ferromagnetic ( f ) @xmath58 and anti - ferromagnetic ( af ) @xmath59 phases is found for low super - exchange interaction energy . on the other hand phase separation between p2 @xmath60 and p3 @xmath61 phases and the af phase was obtained for high @xmath62 . because of the scalar @xmath63 spin character used in this paper canted cp3 , cp2 and t phases are not obtained in this paper @xcite . the af phase observed at @xmath64 was previously studied for an ising ( s=1 ) and classical ( s=3 ) model respectively in references @xcite . vs super - exchange interaction energy @xmath62 phase diagram . ] vs temperature @xmath65 for @xmath66 i.e. one itinerant electron and a typical value of the super - exchange interaction energy @xmath67 . curie - weiss like behavior at high temperature limit can be observed . solid line represents @xmath68 limit . ] but for two itinerant electrons @xmath69 . ] but for three itinerant electrons @xmath70 . ] figures [ fig2cl ] , [ fig3cl ] and [ fig4cl ] show the inverse of the magnetic susceptibility vs temperature for one , two and three itinerant electrons respectively . solid lines in those figures represent high temperature @xmath68 limit . curie - weiss behavior can be easily observed in those figures as @xmath71 being @xmath72 curie constant and @xmath73 curie - weiss like temperature . ( @xmath74 @xmath75 ) , ( @xmath76 @xmath77 ) and ( @xmath78 @xmath79 ) for one , two and three itinerant electrons respectively . curie constant can be rigorously extracted for the former limit @xmath25 and @xmath80 . for this goal it is considered @xmath37 localized spins and @xmath81 itinerant electrons . because of @xmath25 limit hilbert space is reduced . so there are @xmath81 and @xmath82 free particles with @xmath83 and @xmath84 energies respectively . being @xmath43 the magnetic field . the former gives @xmath85 . curie constant is identified like @xmath86 . it gives @xmath87 , @xmath88 and @xmath89 for one , two and three itinerant electrons respectively i. e. @xmath90 and @xmath91 . these values are very close to those obtained in figures [ fig2cl]-[fig4cl ] . now , we can use our curie constant 1 + 3x to make contact with results of the nickelate one - dimensional compound @xmath0 . @xmath92 ( s=1/2 ) for curie constant was proposed by kojima _ @xcite kojima _ et al . _ proposed that each ca - atom introduces three @xmath16 spins . they studied hole dopings @xmath93 and @xmath94 . in our case these itinerant holes correspond to @xmath95 and @xmath96 itinerant electrons studied here . it means curie constant ( 1 + 3x ) as @xmath97 and @xmath98 or simply @xmath99 if we introduce holes as kojima . @xmath5 corresponds to @xmath100 or ne = n electrons coupled with n localized spins s=1/2 by an infinite hund s coupling . on the other hand , @xmath101 is exactly n localized spins s=1/2 with @xmath102 . so , the effect to introduce holes in our itinerant electron system is to reduce curie constant . for low temperature the model proposed by kojima _ et al . _ is very close to our p3+af phase separation . on the other hand , curie - weiss like temperature @xmath73 decreases as itinerant electron density increases . itinerant electrons are responsible for the former f behavior because of our de interaction . short range spin - spin correlations @xmath103 at zero magnetic field can be observed in figures [ fig5cl]-[fig7cl ] for a typical value of @xmath67 and four different temperatures @xmath104 solid circles , cross , large open circles and plus symbols respectively were used . to obtain these short range correlations negative in - site @xmath105 energies were used to pin one , two and three polarons in the linear chain as can be observed in figures [ fig5cl]-[fig7cl ] respectively . these negative in - site energies can be related with impurities in our linear chain . for low temperature can be clearly seen polarons of three sites in an af background . similar polarons were found in reference @xcite by using quantum s=3/2 core spins . this phase with disordered polarons is degenerated to our p3+af phase separation . it means ordered polarons of three sites in an af background . at high temperature @xmath106 polarons disperse and a very low correlation is observed . and @xmath67 . solid circles , cross , large open circles and plus symbols respresent four different temperatures @xmath107 and @xmath108 respectively . ] but for two itinerant electrons @xmath69 . ] but for three itinerant electrons @xmath70 . ] in the same way , figures [ fig8cl]-[fig10cl ] show short range spin - spin correlations @xmath103 for another typical value of @xmath109 and three different temperatures @xmath110 and @xmath111 . in this case only one in - site @xmath105 energy was utilized to pin de f phase as can be seen in figures [ fig8cl]-[fig10cl ] . for low temperature f - af phase separation can be observed . the f phase increases as the itinerant electron density @xmath38 increases , see figures [ fig8cl]-[fig10cl ] . the former is because of de interaction . at high temperature the f phase disperses and a very low correlation is observed . and @xmath109 . solid circles , cross and large open circles respresent three different temperatures @xmath110 and @xmath111 respectively . ] but for two itinerant electrons @xmath69 . ] but for three itinerant electrons @xmath70 . ] and @xmath67 . solid circles , cross , plus and open circles respresent four different temperatures @xmath112 and @xmath113 respectively . fitting solid lines are also shown in the same figure . ] but for an itinerant electron density of @xmath114 . ] but for a chain of @xmath115 sites , one itinerant electron @xmath69 is presented . ] it is tempting to apply our results to the magnetic properties of the hole doped @xmath0 . doing so raises the question of the relation between quantum spins and classical spins cases . it is clear that some properties are specific to the quantum character of the spins , in particular the haldane gap occurring in heisenberg @xmath116 chains , as in the case of un - doped @xmath33 . however , in the doped case the itineracy of doped electrons or holes plays an important role taken into account by the double - exchange mechanism . the essential behavior of the spin correlations in the quantum level is similar in the classical case . for the commensurate filling @xmath117 the polaronic phase @xmath118 reference @xcite is qualitatively similar to the quantum @xmath16 case . we have calculated magnetic susceptibility for typical values of the conduction electron density to make contact with experiments@xcite . the inverse of magnetic susceptibility ( @xmath119 ) vs @xmath52 presents a complicated behavior as described in the former lines . at high temperature curie - weiss behavior was obtained . as shown , curie constant is basically t - j independent . our ising - like results give @xmath120 or @xmath99 . et al . _ from experimental results proposed @xmath121 ( s=1/2 ) . in our case we remove electrons from an s=1 system @xmath5 . in the case of kojima , holes are added . in this case our ising - like model may be can be related with experimental results . curie - weiss temperature @xmath73 is t - j dependent and can be related with curie - like behavior observed in this compound @xcite . it is important to mention that the contribution related to the haldane gap in @xmath116 spin chains decreases exponentially with decreasing temperature and becomes negligible at low temperature @xmath122 @xcite . it is difficult to identify the different contributions to the magnetic susceptibility in such a complex magnetic ground state . of course , our comparison with the experimental results becomes irrelevant below the spin - glass transition identified to be @xmath123 . finite size effects are taken into account to show that our @xmath57 sites are of relevance . inverse of magnetic susceptibility vs inverse of n sites for different temperatures are shown in figures [ fig11cl ] and [ fig12cl ] for an itinerant electron density of @xmath124 and @xmath114 respectively . fitting solid lines @xmath125 with an error of @xmath126 , 95 per cent of confidence levels are shown in the same figures . as can be seen in the same figures an error of @xmath127 is obtained if @xmath57 sites are taken into account . the @xmath80 limit , that is n - site independent , is also compared with these thermodynamic limits , giving an error of @xmath128 @xcite . finite size effects for a heisenberg and an ising model ( without itinerant electrons @xmath64 ) were studied in reference @xcite . as can be seen in that reference , magnetic susceptibility is almost n - site independent at high temperature limit . in our model , because of itinerant electrons , both high and low temperature limits lead to the same qualitative behavior . it is also presented , in figure [ fig13cl ] , short - range spin - spin correlations for one itinerant electron and @xmath115 sites @xmath129 and @xmath67 . these results can be compared with results shown in figure [ fig6cl ] for @xmath57 sites and two itinerant electrons . the same spin - spin correlations behavior can be observed . magnetic phase diagram for classical localized spins and an exchange model , as used in this paper , is compared with the thermodynamic limit in reference @xcite . as can be observed in that reference , the same magnetic phases were obtained . of course that because of our exact results very long systems can not be studied easily because of a huge cpu time used . in this work , we presented exact parallel static magnetic susceptibility calculations and short - range spin - spin correlations of an equivalent ising - like de+se model using large hund s coupling . magnetic susceptibility was calculated in a region where p3-af and f - af phase separation can be found . at high temperature curie - weiss behavior and a very low correlated system were obtained . curie constant is basically t - j independent and could be related with the curie - like behavior observed in the nickelate one - dimensional compound @xmath0 . finite size effects were considered to show the relevance of our finite @xmath57 system . it is important to mention that another itinerant electronic densities and super - exchange couplings were considered . almost the same errors were obtained and the same qualitative behavior in magnetic susceptibility and spin - spin correlations .
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spin - polarons are obtained using an ising - like exchange model consisting of double and super - exchange interactions in low dimensional systems . at zero temperature , a new phase separation between small magnetic polarons , one conduction electron self - trapped in a magnetic domain of two or three sites , and the anti - ferromagnetic phase was previously reported . on the other hand
the important effect of temperature was missed .
temperature diminishes boltzmann probability allowing excited states in the system .
static magnetic susceptibility and short - range spin - spin correlations at zero magnetic field were calculated to explore the spin - polaron formation . at high temperature curie - weiss behavior
is obtained and compared with the curie - like behavior observed in the nickelate one - dimensional compound @xmath0 .
exchange and super - exchange interactions , classical spin models , phase separation 75.30.et,75.10.hk,64.75.+g
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the existence of homo and heteroclinic cycles in systems with symmetry is no longer a surprising feature . there are several examples of cycles arising in differential equations symmetric under the action of a specific compact lie group . several definitions of heteroclinic cycles and networks have been given in the literature . these objects are associated with intermittent dynamics and used to model stop - and - go behaviour in various applications . throughout the present article , we use the following definition valid for a finite dimensional system of ordinary equations ( ode ) : a _ heteroclinic cycle _ is a finite collection of equilibria @xmath0 of the ode together with a set of heteroclinic connections @xmath1 where @xmath2 is a solution of the ode such that : @xmath3 and @xmath4 . when @xmath5 , we say that the set @xmath6 is a _ homoclinic cycle_. a _ homoclinic network _ is a connected finite union of homoclinic cycles associated to the same equilibrium point . complex behaviour near a homo / heteroclinic network is often connected to the occurence of _ switching_. there are different types of switching , leading to increasingly complex behaviour near the network : * _ switching at a node _ @xcite characterised by existence of initial conditions , near an incoming connection to that node , whose trajectory follows any of the possible outgoing connections at the same node . incoming connection does not predetermine the outgoing choice at the node . * _ switching along a heteroclinic connection _ @xcite , which extends the notion of switching at a node to initial conditions whose trajectories follow a prescribed homo / heteroclinic connection . * _ infinite switching _ @xcite , which ensures that any sequence of connections in the network is a possible path near the network . this is different from _ random switching _ in which trajectories shadow the network in a non - controllable way @xcite . the absence of _ switching along a connection _ prevents _ infinite switching _ and , therefore , chaotic behaviour near the network . the term _ switching _ has also been used to describe simpler dynamics where there is one change in the choices observed in trajectories . this is the case described in @xcite . in this case , the network consists of two cycles and trajectories are allowed to change from a neighbourhood of one cycle to a neighbourhood of the other cycle . this change is referred to as switching , although it is a very weak example of this phenomenon . in @xcite , the expression _ railroad switching _ is used in relation to switching at a node . complex behaviour near a network can also arise from the presence of noise - induced switching , see @xcite . we do not address the presence of noise in this note . the authors of @xcite find a form of complicated switching ( possibly not infinite ) leading to regular and irregular cycling near a network . there are several examples in the literature where the existence of infinite switching leads to chaotic behaviour near the network , see @xcite . all the networks considered by these authors have at least one invariant saddle at which the linearized vector field has complex eigenvalues . in general , infinite switching is related with the existence of suspended horseshoes in its neighbourhood . see for example the works @xcite . in these articles , the authors proved the existence of infinitely many initial conditions that realize a given forward infinite path . these solutions lie on the sequence of suspended horseshoes that accumulate on the network . so , infinite switching seems to be connected with the existence of suspended horseshoes . the natural question is : * ( q1 ) * are there examples of homo / heteroclinic networks exhibiting infinite switching and without suspended horseshoes around it ? the main goal of this note is to answer this question . we will exhibit a class of vector fields whose flow has a homoclinic network exhibiting infinite switching and without suspended horseshoes around it . the example is based on the most famous and rich examples in the dynamical systems theory the shilnikov model of a homoclinic cycle to a saddle - focus with negative saddle value @xcite . although we deal with the classic shilnikov homoclinic loop , as far as we know , the combination _ @xmath7-symmetry _ and _ negative saddle - value _ of the saddle - focus is new . this note provides an example of a simple attracting network exhibiting sensitivity with respect to initial conditions . let @xmath8 be a compact three - dimensional manifold possibly without boundary and let @xmath9 the banach space of @xmath10 vector fields on @xmath8 endowed with the @xmath10 whitney topology with @xmath11 . consider a vector field @xmath12 defining a system @xmath13 and denote by @xmath14 , with @xmath15 , the associated flow ( with initial condition @xmath16 ) . in this paper , we will be focused on an equilibrium @xmath17 of ( [ general ] ) such that its spectrum ( _ i.e. _ the eigenvalues of @xmath18 ) consists of one pair of non - real complex numbers with negative real part and one positive real eigenvalue . this is what we call a _ saddle - focus_. a _ homoclinic connection _ associated to @xmath17 is a trajectory biasymptotic to @xmath17 in forward and backward times . homoclinic network _ associated to @xmath17 consists of the equilibrium and a finite union of homoclinic connections associated to o. let @xmath19 be a homoclinic network associated to @xmath17 with @xmath20 . [ def2 ] if @xmath21 , a finite path of order @xmath22 on @xmath23 is a sequence @xmath24 of homoclinic connections , where @xmath25 is an arbitrary map . we use the notation @xmath26 for this type of finite path . for an infinite path , take @xmath27 . let @xmath28 be a neighbourhood of the network @xmath23 and let @xmath29 be a neighbourhood of @xmath17 . for each homoclinic connection @xmath30 in @xmath23 , consider a point @xmath31 and a neighbourhood @xmath32 of @xmath33 . the collection of neighbourhoods @xmath34 should be pairwise disjoint . given neighbourhoods as above , we say that the trajectory of a point @xmath35 follows a finite path @xmath36 , if there exist two monotonically increasing sequences of times @xmath37 and @xmath38 such that for all @xmath39 , we have @xmath40 and : 1 . @xmath41 for all @xmath42t_{1},t_{k+1}[$ ] ; 2 . @xmath43 for all @xmath44 and @xmath45 for all @xmath39 ; 3 . for all @xmath46 there exists a proper subinterval @xmath47z_{j},z_{j+1}[$ ] such that , given @xmath42z_{j},z_{j+1}[$ ] , @xmath48 if and only if @xmath49 . the notion of a trajectory following an infinite path can be stated similarly . along the paper , when we refer to points that follow a path , we mean that their trajectories do it . based in @xcite , we define : there is : [ def1 ] 1 . _ finite switching _ near @xmath23 if for each finite path and for each neighbourhood @xmath28 there is a trajectory in @xmath28 that follows it and 2 . _ infinite switching _ ( or simply _ switching _ ) near @xmath23 by requiring that for each infinite path and for each neighbourhood @xmath28 there is a trajectory in @xmath28 that follows it . an infinite path on @xmath23 can be considered as a pseudo - orbit of ( [ general ] ) with infinitely many discontinuities . switching near @xmath23 means that any pseudo - orbit in @xmath23 can be realized . in @xcite , using connectivity matrices , the authors gave an equivalent definition of switching , emphasising the possibility of coding all trajectories that remain in a given neighbourhood of the network in both finite and infinite times . our object of study is the dynamics around a special type of homoclinic network , for which we give a rigorous description here . specifically we consider a family of vector fields in @xmath9 , @xmath11 , with a flow given by the unique solution @xmath50 of ( [ general ] ) satisfying the following hypotheses : 1 . the point @xmath17 is a saddle - focus equilibrium where the eigenvalues of @xmath51 are @xmath52 and @xmath53 , where @xmath54 and @xmath55 . 2 . there is a trajectory @xmath56 biasymptotic to @xmath17 . the vector field is @xmath7-symmetric under the action of @xmath57 . hypotheses * ( h2 ) * and * ( h3 ) * imply the presence of an additional trajectory , say @xmath58 . thus @xmath59 is a homoclinic network ; in particular @xmath60 in definition [ def2 ] . we address the reader to @xcite for more details about equations with symmetry . there are several papers in the literature dealing with the case where the inequality @xmath55 fails , all of them dealing with the complexity of solutions in a neighbourhood of @xmath23 , namely a sequence of suspended horseshoes accumulating on the network . finitely many of these horseshoes survive under the addition of generic perturbing terms . our main result says that although @xmath23 is attracting ( the statistical limit set associated to @xmath23 is the point @xmath17 ) , the approach to the network is chaotic . [ main ] for a vector field @xmath12 whose flow satisfies * ( h1)**(h3 ) * , the following conditions hold : * there are no suspended horseshoes in the neighbourhood of @xmath23 ; * the network @xmath23 is asymptotically stable , in the sense that all trajectories starting in a small open neighbourhood of @xmath23 are attracted to the network ; * there is infinite switching near @xmath23 , realized by infinitely many initial conditions . the proof of * ( a ) * and * ( b ) * may be found in the works @xcite . see also @xcite . the proof of * ( c ) * is addressed in [ sec6 ] of the present note . in particular , in a small neighbourhood of @xmath23 , @xmath61 , if @xmath62 is @xmath63-close to @xmath64 , the set of non - wandering trajectories of @xmath64 in @xmath61 consists of @xmath17 and one or two attracting limit cycles . using theorem [ main ] , we conclude that the transient dynamics should visit the ghost of the homoclinic cycles ( in any prescribed order ) before falling on the basins of attractions of the periodic solutions . in [ localdyn ] we linearize the vector field around the saddle - focus , obtaining an isolating block around it ; this section is concerned with introducing the notation for the proof of switching . in [ global ] , we obtain a geometrical description of the way the flow transforms a segment of initial conditions across the stable manifold of @xmath17 . this curve is wrapped around the isolating block and accumulates on the unstable manifold of @xmath17 ( @xmath65-lemma for flows ) and , in particular , on the next connection . the local stable manifold of @xmath17 crosses infinitely many times the previous curve . the geometric setting is explored in [ sec6 ] to obtain intervals of the segment that are mapped by the flow into curves next to @xmath17 in a position similar to the first one . this allows to establish the recurrence neeeded for infinite switching . for any infinite sequence of homoclinic connections , say : @xmath66 without using perturbation theory , we find infinitely many trajectories that visits the neighbourhoods of these connections in the same sequence . throughout this note , we have endeavoured to make a self contained exposition bringing together all topics related to the proofs . we have stated short lemmas and we have drawn illustrative figures to make the paper easily readable . the behaviour of the vector field @xmath64 in the neighbourhood of the network @xmath23 is given , up to topological equivalence , by the linear part of @xmath64 in the neighbourhood of @xmath17 and by the transition map between two discs transversal to the flow in those neighbourhoods . in this section , we choose coordinates in the neighbourhood of @xmath17 in order to put @xmath64 in the canonical form and we assume that the transition map is linear . the main point is the application of samovol s theorem @xcite to @xmath67linearize the flow around @xmath17 , and to introduce cylindrical coordinates around the equilibrium . there are no @xmath67-resonances here . we use neighbourhoods with boundary transverse to the linearized flow . since @xmath17 is hyperbolic , by samovol s theorem @xcite , the vector field @xmath64 is @xmath68conjugate to its linear part in a @xmath69-small open neighbouhood around @xmath17 , @xmath70 . we choose cylindrical coordinates @xmath71 near @xmath17 so that the linearized vector field can be written as : @xmath72 . for @xmath73 , on a segment @xmath74 there are infinitely many subsegments that are mapped by @xmath75 into @xmath76 , each one containing a point mapped into @xmath77 . the small sub - segments contain smaller ones that are mapped into by @xmath78 into @xmath79 and the process may be continued forming a nested sequence . , height=264 ] after a linear rescaling of the local variables , we consider a cylindrical neighbourhood of @xmath17 of radius @xmath80 and height @xmath81 that we denote by @xmath82 see figure [ neigh_vw ] . their boundaries consist of three components : the cylinder wall parametrized by @xmath83 and @xmath84 with the usual cover @xmath85 and two disks ( top and bottom ) . we take polar coverings of these disks @xmath86 where @xmath87 , @xmath88 and @xmath89 . by convention , the intersection points of @xmath23 with the wall of the cylinder has @xmath90 and @xmath91 angular coordinate . the set of points in the cylindrical wall with positive ( resp . negative ) second coordinate is denoted by @xmath92 . ( resp @xmath93 ) . the similar holds for @xmath94 . as depicted in figure [ neigh_vw ] , the cylinder wall of @xmath82 is denoted by @xmath95 . note that @xmath77 corresponds to the circle @xmath96 . the top and the bottom of the cylinder are simply denoted by @xmath94 . the boundary of @xmath82 can be written as the disjoint union : @xmath97 where @xmath98 is the part of @xmath99 where the flow is not transverse . it follows by the above construction that : let @xmath100 . solutions starting : 1 . at @xmath95 go inside the cylinder @xmath82 in positive time ; 2 . at @xmath94 go outside the cylinder @xmath82 in positive time ; 3 . at @xmath101 leave the cylindrical neighbourhood @xmath82 at @xmath102 . if @xmath103 , let @xmath104 be the time of flight through @xmath82 of the trajectory whose initial condition is @xmath105 . it only depends on @xmath106 and is given explicitly by @xmath107 in particular @xmath108 . now , we obtain the expression of the local map that sends points in the boundary where the flow goes in , into points in the boundary where the flows goes out . the local map @xmath109 near @xmath17 is given by @xmath110 where @xmath111 is the _ saddle index _ of @xmath17 . observe that if @xmath112 is fixed , then @xmath113 the same expression holds for the local map from the other connected component of @xmath114 to @xmath115 ( where @xmath116 ) with the exception that the first coordinate of @xmath117 changes its sign . in @xcite , the author obtains precise asymptotic expansions for the local map @xmath118 . in the present article , we omit high order terms because they are not needed to our purposes . let @xmath119 . by the _ tubular flow theorem _ @xcite , solutions starting near @xmath120 follow one of the connections in @xmath23 . we may then define the transition map @xmath121 by flow box fashion and the return map to @xmath95 : @xmath122 hereafter , we concentrate our attention on initial conditions that do not escape from @xmath61 ; otherwise take a smaller subset in @xmath123 where the returm map is well defined . the explicit expression for the return map is highly nonlinear since the distortion near the hyperbolic saddle - foci is tremendous . for our purposes , it is not needed . the coordinates and notations of [ localdyn ] will be used to study the geometry of the local dynamics near the saddle - focus . this is the main goal of the present section but first we introduce the concept of a _ segment _ on @xmath95 . let @xmath100 . a _ segment _ @xmath124 : 1 . [ segment w ] _ on _ @xmath79 is a smooth regular parametrized curve @xmath125\rightarrow \sigma^{in}_j$ ] that meets @xmath126 at the point @xmath127 and such that , writing @xmath128 , both @xmath129 and @xmath130 are monotonic and bounded functions of @xmath131 . the definition of _ segment _ may be relaxed : the components do not need to be monotonic for all @xmath132 $ ] . we use the assumption of monotonicity to simplify the arguments . let @xmath133 , @xmath134 be a disc centered at @xmath135 and @xmath136 a line passing through @xmath137 . 1 . a _ spiral _ on @xmath134 around the point @xmath137 is a smooth curve @xmath138 satisfying @xmath139 and such that if @xmath140 is its expression in polar coordinates around @xmath137 then : 1 . the map @xmath141 is bounded by two monotonically decreasing maps converging to zero as @xmath142 ; 2 . the map @xmath143 is monotonic for some unbounded subinterval of @xmath144 and 3 . a _ double spiral _ on @xmath134 around the point @xmath137 is the union of two spirals accumulating on @xmath137 and a curve connecting the other end points . 3 . given a _ spiral _ @xmath146 on @xmath134 around the point @xmath137 , an _ half circle _ bounded by @xmath136 is a connected component of @xmath147 . the next result characterizes the local dynamics near the saddle - focus . [ lemma3 ] let @xmath148 . a _ segment _ @xmath124 on @xmath79 is mapped by @xmath149 into a spiral on @xmath150 accumulating on the point defined by @xmath151 . the proof will be done for @xmath152 . the other case is analogous . let @xmath74 be a segment on @xmath153 . write @xmath154 , where : * @xmath132 $ ] , * @xmath155 is an increasing map as function of @xmath131 and * @xmath156 . the function @xmath149 maps the segment @xmath157 into the curve defined by : @xmath158= \left[(y^\star(s))^\delta , -\frac{\alpha}{e}\ln map @xmath159 is a spiral on @xmath160 accumulating on the point defined by @xmath161 because @xmath162 and @xmath163 are monotonic ( see remark [ rem1 ] ) and @xmath164 [ rem1 ] let @xmath148 . the coordinates @xmath165 may be chosen so as to make the map @xmath166 increasing or decreasing , according to our convenience . from now on , we omit the dependence of @xmath167 on @xmath131 to simplify the notation . in this section we put together the information about the return map . first note that if @xmath168 , we denote by @xmath169 and @xmath170 its topological closure and topological interior , respectively . in what follows we remove the point @xmath171 from the @xmath172 because the return map is not defined at this point . from now on , let us fix @xmath61 , a small neighbourhood of @xmath23 . figure [ neigh_vw ] illustrates the main idea of the following proof . the next result shows that there are infinitely many points in @xmath173 which are mapped under @xmath75 into @xmath77 and that separate segments of initial conditions that follow the different connections @xmath56 and @xmath174 . [ first lemma ] let @xmath175 be a rectangle in @xmath95 centered at one point of @xmath176 . for any segment @xmath177 \rightarrow \mathcal{r}\cap \sigma^{in}_+$ ] , there is a family of intervals of the type @xmath178 $ ] such that for all @xmath179 , we have : as already said , we concentrate our attention on initial conditions that do not escape from @xmath61 . by lemma [ lemma3 ] , the image of @xmath183 under @xmath184 is a spiral accumulating on @xmath185 . in its turn , this spiral is mapped by @xmath186 into another spiral in @xmath95 accumulating on one point of @xmath176 . the curve @xmath77 cuts transversely this spiral into infinitely many points . let @xmath187 the points for which @xmath188 . by construction , it is easy to see that if @xmath189 either @xmath190 or @xmath191 . suppose , without loss of generality , that the first case holds . then , by continuity of @xmath75 , for @xmath192 we get @xmath191 . [ second lemma ] let @xmath175 be a rectangle in @xmath95 centered at one point of @xmath176 and let @xmath193 be as in lemma [ first lemma ] . then for sufficiently large @xmath194 , there are @xmath195 $ ] such that for all @xmath196 , we have : our starting point is the half - circle @xmath201 , @xmath179 , bounded by @xmath77 . since the image under @xmath75 of each half circle can be seen as the image of two connected segments , using lemma [ lemma3 ] the set @xmath202 is a double spiral accumulating on @xmath203 , which is mapped under @xmath186 into a double spiral accumulating on @xmath176 . the line @xmath77 intersects this double spiral infinitely many times . let @xmath204 the sequence of points such that @xmath197 . for a given @xmath205 , the arguments used before may be used to conclude that there exists a sequence @xmath206 such that @xmath199)$ ] is an half - circle in @xmath92 bounded by @xmath77 and @xmath207)$ ] is an half - circle in @xmath93 bounded by @xmath77 . in the previous sections we have proved that a segment cutting transversely the stable manifold of @xmath17 contains subsegments @xmath193 that are mapped into new segments cutting transversely the stable manifold of @xmath17 . starting with a segment @xmath74 on @xmath153 , we may obtain , recursively , nested compact subsets containing initial conditions that follow any prescribed sequence of connections . let @xmath208 . we say that the path @xmath209 of order @xmath22 on the homoclinic network @xmath23 is inside the path @xmath210 of order @xmath211 if @xmath212 for all @xmath213 . we denote this relation by @xmath214 . given a path of order @xmath215 , @xmath216 , we want to find trajectories that follow it . let us fix a segment @xmath217 given by lemma [ first lemma ] and set that all initial conditions in @xmath218 follow the connection @xmath219 . take a closed subset @xmath220 of @xmath217 . by construction , all initial conditions starting in @xmath220 follow the connection @xmath219 . the set @xmath221 is an half circle cutting transversely @xmath222 infinitely many times . by lemma [ second lemma ] , one can obtain again sequences of points in @xmath223 , where a similar result to that in lemma [ first lemma ] can be stated for @xmath78 instead of @xmath75 . take a closed subset @xmath224 of @xmath225 . by construction , all initial conditions starting in @xmath224 follow the path @xmath226 . a recursive argument allows the construction of a compact set @xmath227 of initial conditions whose trajectories follow @xmath26 . [ infinite2 ] we first need to introduce some extra terminology . given a path of order @xmath215 , @xmath228 we denote by @xmath229 the compact set @xmath230 obtained in the proof of proposition [ theoremfiniteswitching ] and we say that @xmath229 is an admissible set . recall that all points in @xmath229 correspond to inifinitely many solutions following @xmath26 . fix an infinite path @xmath232 , with @xmath233 . for each @xmath234 define the finite path @xmath235 . taking into account remark [ admissible ] it follows that there exists an infinite sequence of admissible sets @xmath236 such that @xmath237 for all @xmath21 . since the sequence of compact sets @xmath236 is nested , @xmath238 . any initial condition in @xmath239 gives a trajectory which follows @xmath240 . the different solutions are distinguished by the number of revolutions around the isolating block of @xmath17 @xcite . again by construction we find trajectories realising the required switching arbitrarily close to @xmath23 . based on theorem [ main ] , the answer to the question * ( q1 ) * is _ yes_. at this point , other questions arise , namely : _ are there other examples of heteroclinic networks where infinite switching holds without suspended horseshoes emerging in its neighbourhood ? _
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in general , infinite switching behaviour near networks is associated with the existence of suspended horseshoes .
trajectories that realize switching lie within these transitive sets . in this note , revisiting the equivariant shilnikov scenario , we describe an attracting homoclinic network exhibiting forward switching and without suspended horseshoes in its neighbourhood .
thus we provide an example of an asymptotically stable network exhibiting sensitive dependence on initial conditions . en gnral , le comportement de commutation autour des rseaux est associ lexistence de fers cheval en suspension .
trajectoires qui ralisent la commutation se situent dans ces ensembles transitifs .
dans cette note , visitant le scnario classique de shilnikov avec une symtrie , nous dcrivons un rseau homocline qui prsente commutation nayant pas des fers cheval en suspension autour de lui .
plus prcisement nous donnons un exemple dun rseau asymptotiquement stable avec sensible dpendance des conditions initiales .
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molecular gas is one of the essential components in galaxies because it is closely related to star formation , which is a fundamental process of galaxy evolution . thus the observational study of molecular gas is indispensable to understand both star formation in galaxies and galaxy evolution . however , the most abundant constituent in molecular gas , h@xmath15 , can not emit any electro - magnetic wave in cold molecular gas with typical temperature of @xmath16 10 k due to the lack of a permanent dipole moment . instead , rotational transition lines of @xmath1co , the second abundant molecule , have been used as a tracer of molecular gas . for example , some extensive @xmath1co surveys of external galaxies , which consist of single pointings toward central regions and some mappings along the major axis , have been reported ( e.g. , @xcite ; @xcite ; @xcite ) . these studies provided new findings about global properties of galaxies , such as excitation condition of molecular gas in galaxy centers and radial distributions of molecular gas across galaxy disks . in order to understand the relationship between molecular gas and star formation in galaxies further , spatially resolved @xmath1co maps covering whole galaxy disks are necessary because star formation rates ( sfrs ) are often different between galaxy centers and disks . in particular , single - dish observations are essential to measure @xmath17 molecular gas content in the observing beam from dense component to diffuse one avoiding the missing flux ( e.g. , @xcite ) . so far , two major surveys of wide - area @xmath1co mapping toward nearby galaxies are performed using multi - beam receivers mounted on large single - dish telescopes . one is the @xmath1co(@xmath0 ) mapping survey of 40 nearby spiral galaxies performed with the nobeyama radio observatory ( nro ) 45-m telescope in the position - switch mode ( @xcite , hereafter k07 ) . their @xmath1co(@xmath0 ) maps cover most of the optical disks of galaxies at an angular resolution of 15@xmath18 , and clearly show two - dimensional distributions of molecular gas in galaxies . k07 found that the degree of the central concentration of molecular gas is higher in barred spiral galaxies than in non - barred spiral galaxies . in addition , they found a correlation between the degree of central concentration and the bar strength adopted from @xcite ; i.e. , galaxies with stronger bar tend to exhibit a higher central concentration . this correlation suggests that stronger bars accumulate molecular gas toward central regions more efficiently , which may contribute the onset of intense star formation at galaxy centers ( i.e. , higher sfrs than disks ) . using the @xmath1co(@xmath0 ) data , @xcite investigated the physical properties of molecular gas in the barred spiral galaxy maffei 2 . they found that molecular gas in the bar ridge regions may be gravitationally unbound , which suggests that molecular gas is hard to become dense , and to form stars in the bar . the other survey is the heterodyne receiver array co line extragalactic survey performed with the iram 30-m telescope @xcite . they observed @xmath1co(@xmath7 ) emission over the full optical disks of 48 nearby galaxies at an angular resolution of 13@xmath18 , and found that the @xmath1co(@xmath7)/@xmath1co(@xmath0 ) line intensity ratio ( hereafter @xmath8 ) typically ranges from 0.6 to 1.0 with the averaged value of 0.8 . in addition , @xcite examined a quantitative relationship between surface densities of molecular gas and sfrs for 30 nearby galaxies at a spatial resolution of 1 kpc using the @xmath1co(@xmath7 ) data . they found a first - order linear correspondence between surface densities of molecular gas and sfrs but also found second - order systematic variations ; i.e. , the apparent molecular gas depletion time , which is defined by the ratio of the surface density of molecular gas to that of sfr , becomes shorter with the decrease in stellar mass , metallicity , and dust - to - gas ratio . they suggest that this can be explained by a co - to - h@xmath15 conversion factor ( @xmath19 ) that depends on dust shielding . however , such global co maps of galaxies have raised a new question ; the cause of the spatial variation in star formation efficiencies ( sfes ) defined as sfrs per unit gas mass . it is reported that sfes differ not only among galaxies ( e.g. , @xcite ) but also within locations / regions in a galaxy ( e.g. , @xcite ) ; i.e. , higher sfes are often observed in galaxy mergers rather than normal spiral galaxies and also observed in the nuclear star forming region rather than in galaxy disks . some observational studies based on hcn emission , an excellent dense gas tracer , suggest that sfes increase with the increase in molecular gas density ( or dense gas fraction ) in galaxies ( e.g. , @xcite ; @xcite ; @xcite ; @xcite ) , but the cause of the spatial variation in sfes is still an open question because hcn emission in galaxy disks is too weak to obtain its map except for some gas - rich spiral galaxies ( e.g. , m 51 ; @xcite ; @xcite ) . instead , isotopes of co molecule are promising probes of molecular gas density . in particular , @xmath2co(@xmath0 ) is thought to be optically thin and thus trace denser molecular gas ( @xmath20 ) rather than @xmath1co(@xmath0 ) , which is optically thick and traces relatively diffuse molecular gas ( @xmath21 ) . therefore , the relative intensity between @xmath2co(@xmath0 ) and @xmath1co(@xmath0 ) is sensitive to physical properties of molecular gas . for example , spatial variations in @xmath2co(@xmath0)/@xmath1co(@xmath0 ) intensity ratios ( hereafter @xmath9 ) were observed in nearby galaxy disks ( e.g. , @xcite ; @xcite ; @xcite ) . such variations in @xmath9 , typically ranging from 0.05 to 0.20 , are interpreted as the variation in molecular gas density ; i.e. , @xmath9 increases with the increase in molecular gas density . however , some observations suggest that @xmath9 in central regions of nearby galaxies are lower than those in disk regions ( e.g. , @xcite ; @xcite ; @xcite ; @xcite ) although central regions of galaxies often show intense star formation activities , suggesting higher molecular gas density . the cause of the low @xmath9 in central regions is thought to be high temperature of molecular gas due to the heating by uv radiation from a lot of young massive stars . such a degeneracy between density and temperature of molecular gas in a line ratio can be solved using two ( or more ) molecular line ratios with a theoretical calculation on the excitation of molecular gas such as the large velocity gradient ( lvg ) model @xcite . for example , the density and kinetic temperature of giant molecular clouds ( gmcs ) were determined using @xmath9 and @xmath1co(@xmath22)/@xmath1co(@xmath0 ) ratio for large magellanic cloud @xcite and m 33 @xcite , and also determined using @xmath9 and @xmath8 for the spiral arm of m 51 @xcite . this method to determine molecular gas density is useful to investigate the cause of the variation in sfes . thus the dependence of sfes on molecular gas density should be investigated for various galaxies at high angular resolution based on multiple line ratios including @xmath9 . in this paper , we investigate the relationship between sfe and molecular gas density within a nearby barred spiral galaxy ngc 2903 using an archival @xmath1co(@xmath7 ) map combined with @xmath1co(@xmath0 ) and @xmath2co(@xmath0 ) maps which are newly obtained by the co multi - line imaging of nearby galaxies ( coming ) project with the nro 45-m telescope . ngc 2903 is a gas - rich galaxy exhibiting bright nuclear star formation ( e.g. , @xcite ; @xcite ; @xcite ; @xcite ) . the distance to ngc 2903 is estimated to be 8.9 mpc @xcite ; thus the effective angular resolution of 20@xmath18 for the on - the - fly ( otf ) mapping with the nro 45-m corresponds to 870 pc . this enables us to resolve major structures within ngc 2903 , such as the center , bar , and spiral arms although its inclination of @xcite is not so small . in addition , ngc 2903 is rich in archival multi - wavelength data set ; i.e. , not only the @xmath1co(@xmath7 ) map to examine @xmath8 but also h@xmath23 and infrared images to calculate sfrs are available . thus this galaxy is a preferable target to examine the cause of the variation in sfe in terms of molecular gas density . basic parameters of ngc 2903 are summarized in table 1 . the structure of this paper is as follows : we describe the overview of the coming project and explain the detail of the co observations and data reduction for ngc 2903 in section 2 . then , we show results of observations ; i.e. , spectra and velocity - integrated intensity maps of @xmath1co(@xmath0 ) and @xmath2co(@xmath0 ) emission in section 3 . we obtain averaged spectra of @xmath1co(@xmath0 ) , @xmath2co(@xmath0 ) , and @xmath1co(@xmath7 ) emission for nine representative regions , and measure averaged @xmath9 and @xmath8 for each region in section 4.1 . we determine molecular gas density and kinetic temperature for the center , bar , bar - ends , and spiral arms using @xmath9 and @xmath8 based on the lvg approximation in section 4.2 . finally , we investigate the cause of the variation in sfe by examining the dependence of sfe on molecular gas density and kinetic temperature . coming is a project to map @xmath0 emission of @xmath1co , @xmath2co , and c@xmath3o molecules simultaneously for 70% area of optical disks of 238 galaxies using the four - beam receiver system on 45-m telescope ( forest ; @xcite ) at nro . the main purposes of the coming are to characterize properties of molecular gas as sequence of , hubble types , dynamical structures , central concentrations , and star formation activities , as well as surrounding environments of galaxies . more detailed information on coming project including the current status of the survey will be reported in the forthcoming paper ( sorai et al . in preparation ) . @xmath1co(@xmath0 ) , @xmath2co(@xmath0 ) , and c@xmath3o(@xmath0 ) emission observations of ngc 2903 were performed using the nro 45-m telescope from april to may , 2015 , employing the otf mapping mode . the observed area is about @xmath4 , which corresponds to @xmath24 kpc at the distance of 8.9 mpc , as indicated in figure 1 . the total time for the observations was 13 hrs . we used a new @xmath25 focal - plane dual - polarization sideband - separating sis mixer receiver for the single side band ( ssb ) operation , forest , which provides 8 intermediate frequency ( if ) paths ( i.e. , 4 beam @xmath5 2 polarization ) independently . owing to the wide if range of 4 to 12 ghz , we could simultaneously observe @xmath1co(@xmath0 ) emission at 115 ghz ( if = 10 ghz ) and @xmath2co(@xmath0 ) and c@xmath3o(@xmath0 ) emission at 110 ghz ( if = 5 ghz ) when the frequency of the local oscillator was set to 105 ghz . the backend was an fx - type correlator system , sam45 , which consists of 16 arrays with 4096 spectral channels each . we set the frequency coverage and resolution for each array of 2 ghz and 488.24 khz , which gives velocity coverage and resolution of 5220 km s@xmath26 and 1.27 km s@xmath26 at 115 ghz , and those of 5450 km s@xmath26 and 1.33 km s@xmath26 at 110 ghz . we assigned 8 of 16 arrays to 115 ghz band ( i.e. , if = 9 11 ghz for @xmath1co(@xmath0 ) emission ) and other 8 arrays to 110 ghz band ( i.e. , if = 4 6 ghz for @xmath2co(@xmath0 ) and c@xmath3o(@xmath0 ) emission ) . the half - power beam widths of the 45-m with the forest were @xmath27 at 115 ghz and @xmath28 at 110 ghz , respectively . the system noise temperatures were 300 500 k at 115 ghz and 200 250 k at 110 ghz during the observing run . we performed the otf mapping along the major and minor axes of the galaxy disk whose position angle was ( k07 ) . the separation between the scan rows was set to @xmath29 , and the spatial sampling interval was @xmath30 applying a dump time of 0.1 second and a scanning speed of @xmath31 s@xmath26 . the data sets scanned along two orthogonal axes were co - added by the basket - weave method @xcite to remove any effects of scanning noise . in order to check the absolute pointing accuracy every hour , we observed an sio maser source , w - cnc , using a 43 ghz band receiver . it was better than @xmath32 ( peak - to - peak ) throughout the observations . in addition , we observed @xmath1co(@xmath0 ) and @xmath2co(@xmath0 ) emission of w 3 and irc+10216 every day to obtain the scaling factors for converting the observed antenna temperature to the main beam temperature for each if . note that these scaling factors not only correct the main - beam efficiency ( @xmath33 ) of the 45-m antenna but also compensate the decrease in line intensity due to the incompleteness of the image rejection for ssb receiver ( e.g. , @xcite ) . the absolute error of the temperature scale for each co spectrum was about @xmath34% , mainly due to variations in @xmath33 and the image rejection ratio of the forest . data reduction was made using the software package nostar , which comprises tools for otf data analysis , developed by nro @xcite . we excluded bad spectra , which includes strong baseline undulation and spurious lines , from the raw data . then , linear baselines were subtracted , and the raw data were regridded to 6@xmath35 per pixel with an effective angular resolution of approximately 20@xmath35 ( or 870 pc ) . we binned the adjacent spectral channels to a velocity resolution of 10 km s@xmath26 for the spectra . finally , we created three - dimensional data cubes in @xmath1co(@xmath0 ) , @xmath2co(@xmath0 ) , and c@xmath3o(@xmath0 ) emission . the resultant r.m.s . noise levels ( 1 @xmath36 ) were 60 mk , 39 mk , and 40 mk for @xmath1co(@xmath0 ) , @xmath2co(@xmath0 ) , and c@xmath3o(@xmath0 ) , respectively . figure 2 shows @xmath1co(@xmath0 ) spectra of the whole optical disk in ngc 2903 . as is the case in earlier studies ( e.g. , @xcite , k07 , @xcite ) , strong @xmath1co(@xmath0 ) emission , whose peak temperature was @xmath16 0.6 k , was found at the center and significant @xmath1co(@xmath0 ) emission was detected in the bar ( @xmath16 0.3 k ) , bar - ends ( 0.4 0.5 k ) , and spiral arms ( 0.1 0.3 k ) . we calculate velocity - integrated @xmath1co(@xmath0 ) intensities ( @xmath37 ) from the spectra . in order to obtain more accurate line intensities ( in other words , to minimalize the effects of the noise and the undulation of baseline for weak line ) , we defined the `` line channels '' , which are successive velocity channels where significant emission exists , in advance for each pixel as described below . in order to define the `` line channels '' , we utilized @xmath1co(@xmath7 ) data ( @xcite ) , which was regridded and convolved to 20@xmath35 to match our @xmath1co(@xmath0 ) spectra . since the 1 @xmath36 r.m.s . of @xmath1co(@xmath7 ) data of 6 mk at 20@xmath35 and 10 km s@xmath26 resolutions was 10 times better than that of our @xmath1co(@xmath0 ) data , the @xmath1co(@xmath7 ) spectra are appropriate for the decision of `` line channels '' in each pixel . we first identified a velocity channel exhibiting the peak @xmath1co(@xmath7 ) temperature and defined the channel as the `` co peak channel '' for each pixel . then , successive channels whose @xmath1co(@xmath7 ) emission consistently exceeds 1 @xmath36 including the `` co peak channel '' are defined as `` line channels '' . finally , we calculated @xmath37 for the specified `` line channels '' in each pixel . figure 3 shows the @xmath37 map of ngc 2903 . the strongest @xmath37 of 92 k km s@xmath26 is observed at the center , and the secondary peak of 55 k km s@xmath26 is at the northern bar - end . the total molecular gas mass for the observed area in ngc 2903 is estimated to ( @xmath38 ) @xmath39 @xmath40 under the assumptions of the constant @xmath19 of @xmath41 @xmath42 ( k km s@xmath26)@xmath26 @xcite over the disk and the uncertainty of 20% in brightness temperature scape of co line . this value is consistent with @xmath43 @xmath40 obtained by k07 , which is recalculated using the same distance and @xmath19 . we also compare @xmath37 obtained by coming with those obtained by k07 to confirm the validity of our @xmath1co(@xmath0 ) data . we examined the pixel - by - pixel comparison for the two @xmath37 maps at the same angular resolution of 20@xmath18 as shown in figure 4 , and confirmed that both @xmath37 are well correlated with each other . the median and the standard deviation in @xmath37 are 12.9 k km s@xmath26 and 12.4 k km s@xmath26 for coming dataset , and those are 15.2 k km s@xmath26 and 12.6 k km s@xmath26 for k07 dataset . figure 5(a ) shows the global @xmath2co(@xmath0 ) spectra , which are overlaid by @xmath1co(@xmath0 ) spectra for comparison . in addition , figure 5(b ) , ( c ) , and ( d ) show the magnified @xmath2co(@xmath0 ) spectra at the northern bar - end , the center , and the southern bar - end , respectively . we found significant @xmath2co(@xmath0 ) emission at the center and both bar - ends . however , we could not detect any significant c@xmath3o(@xmath0 ) emission . we calculated the velocity - integrated @xmath2co(@xmath0 ) intensities ( @xmath44 ) . as is the case of @xmath1co(@xmath0 ) , we utilized the `` line channels '' defined by @xmath1co(@xmath7 ) spectra . figure 6 shows a spatial distribution of @xmath44 in pseudo - color overlaid by @xmath37 in contour . the global distribution of @xmath2co(@xmath0 ) is similar to @xmath1co(@xmath0 ) ; several peaks whose @xmath44 exceeds 5 k km s@xmath26 are observed at the center , bar - ends , and in spiral arms . intensity ratios of two ( or more ) molecular lines provide important clues to estimate physical properties of molecular gas , such as density and temperature . we examined the spatial variations in line intensity ratios among @xmath1co(@xmath0 ) , @xmath2co(@xmath0 ) , and @xmath1co(@xmath7 ) emission . figure 7 shows spatial distributions of @xmath8 and @xmath9 over the disk of ngc 2903 . in these maps , we displayed pixels with each line intensity exceeding 2 @xmath36 ( 5 k km s@xmath26 , 3 k km s@xmath26 , and 0.5 k km s@xmath26 for @xmath1co(@xmath0 ) , @xmath2co(@xmath0 ) , and @xmath1co(@xmath7 ) , respectively ) . we found some local peaks of @xmath8 ( @xmath16 1.0 ) near the center and at the downstream side of the northern spiral arm , whereas lower @xmath8 ( @xmath16 0.5 0.6 ) was observed in the bar . the spatial distribution of @xmath9 is unclear due to poor signal - to - noise ( s / n ) ratio of @xmath2co(@xmath0 ) emission . as described in section 3.3 , the spatial distribution of @xmath9 seems noisy and unclear due to the poor s / n although we could obtain the spatial distribution of @xmath44 . in order to improve the s / n of weak emission such as @xmath2co(@xmath0 ) , the stacking analysis of co spectra with the velocity - axis alignment seems a promising method . the stacking technique for co spectra in external galaxies are originally demonstrated by schruba et al . ( 2011 , 2012 ) . since the observed velocities of each position within a galaxy are different due to its rotation , a simple stacking causes a smearing of spectrum . in order to overcome such difficulty , @xcite demonstrated the velocity - axis alignment of co spectra in different regions in a galaxy disk according to mean h velocity . they stacked velocity - axis aligned co spectra , and successfully confirmed very weak @xmath1co(@xmath7 ) emission ( @xmath45 1 k km s@xmath26 ) with high significance in h - dominated outer - disk regions of nearby spiral galaxies . in addition , @xcite applied this stacking technique to perform the sensitive search for weak @xmath1co(@xmath7 ) emission in dwarf galaxies . furthermore , @xcite applied the stacking technique to @xmath2co(@xmath0 ) emission in the optical disk of the nearby barred spiral galaxy ngc 3627 . by the stacking with velocity - axis alignment based on mean @xmath1co(@xmath0 ) velocity , they obtained high s / n @xmath2co(@xmath0 ) spectra which are improved by a factor of up to 3.2 compared to the normal ( without velocity - axis alignment ) stacking analysis . these earlier studies clearly suggest that the stacking analysis is very useful to detect weak molecular line . in this section , we employ the same stacking technique as @xcite to improve the s / n of @xmath2co(@xmath0 ) emission and to obtain more accurate line ratios . based on our @xmath37 image ( figure 3 ) , we have separated ngc 2903 into nine regions according to its major structures ; i.e. , center , northern bar , southern bar , northern bar - end , southern bar - end , northern arm , southern arm , inter - arm , and outer - disk . the left panel of figure 8 shows the separation of each region overlaid by the grey - scale map of @xmath37 . for each region , we stacked @xmath1co(@xmath0 ) , @xmath2co(@xmath0 ) , and @xmath1co(@xmath7 ) spectra with velocity - axis alignment based on the intensity - weighted mean velocity field calculated from our @xmath1co(@xmath0 ) data ( right panel of figure 8) . we successfully obtained the stacked co spectra as shown in figure 9 . the s / n of each co emission is dramatically improved , and thus we could confirm the significant @xmath2co(@xmath0 ) emission for all the regions . we found the difference in the line shape of stacked co spectra according to regions . in particular , the stacked @xmath1co spectra in the bar show flat peak over the velocity width of 100 150 km s@xmath26 . this is presumably due to the rapid velocity change in the bar , which makes the velocity - axis alignment difficult . we summarize the averaged line intensities and line ratios for each region in table 2 . we found that the averaged @xmath8 shows the highest value of 0.92 at the center , and a moderate value of 0.7 0.8 at both bar - ends and in the northern arm . a slightly lower @xmath8 of 0.6 0.7 is observed in the bar , southern arm , inter - arm , and outer - disk . such a variation in @xmath8 ranging from 0.6 to 1.0 in ngc 2903 is quite consistent with those observed in nearby galaxies ( e.g. , @xcite ) . however , the highest @xmath9 of 0.19 is observed not at the center but in the northern arm . the central @xmath9 of 0.11 is similar to those in other regions ( 0.08 0.13 ) except for the northern arm and outer - disk ( @xmath16 0.04 ) . the typical @xmath9 of @xmath16 0.1 is frequently observed in nearby galaxies ( e.g. , @xcite ; @xcite ) , but slightly higher than the averaged @xmath9 in representative regions of ngc 3627 , 0.04 0.09 @xcite . using @xmath8 and @xmath9 , we derive averaged physical properties of molecular gas , its density ( @xmath10 ) and kinetic temperature ( @xmath11 ) , in seven regions ( center , northern bar , southern bar , northern bar - end , southern bar - end , northern arm , and southern arm ) of ngc 2903 based on the lvg approximation . some assumptions are required to perform the lvg calculation ; the molecular abundances @xmath46(@xmath1co ) = [ @xmath1co]/[h@xmath15 ] , [ @xmath1co]/[@xmath2co ] , and the velocity gradient @xmath48 . firstly , we fix the @xmath46(@xmath1co ) of @xmath49 and @xmath48 of 1.0 km s@xmath26 pc@xmath26 ; i.e. , @xmath1co abundance per unit velocity gradient @xmath46(@xmath1co)/(@xmath48 ) was assumed to @xmath49 ( km s@xmath26 pc@xmath26)@xmath26 . this is the same as the assumed @xmath46(@xmath1co)/(@xmath48 ) for gmcs in m 33 @xcite . then , we determine the [ @xmath1co]/[@xmath2co ] abundance ratio to be assumed in this study by considering earlier studies . @xcite found a systematic gradient in the @xmath1c/@xmath2c isotopic ratio across in our galaxy ; from @xmath16 30 in the inner part at 5 kpc to @xmath16 70 at 12 kpc with a galactic center value of 24 . for external galaxies , the reported @xmath1c/@xmath2c isotopic ratios in their central regions are 40 for ngc 253 @xcite , 50 for ngc 4945 @xcite , @xmath50 for m 82 and @xmath51 for ic 342 @xcite . @xcite reported a higher [ @xmath1co]/[@xmath2co ] abundance ratio of 50 75 in the central region of m 82 . @xcite also reported a higher @xmath1c/@xmath2c isotopic ratios of @xmath52 50 100 in the central regions of m 82 and ngc 253 . in summary , reported @xmath1c/@xmath2c isotopic ( and [ @xmath1co]/[@xmath2co ] abundance ) ratios in nearby galaxy centers ( 30 100 ) are typically higher than that in the inner 5 kpc of our galaxy ( 24 30 ) , but the cause of such discrepancies in @xmath1c/@xmath2c and [ @xmath1co]/[@xmath2co ] between our galaxy and external galaxies is still unresolved . here , we assumed an intermediate [ @xmath1co]/[@xmath2co ] abundance ratio of 50 in ngc 2903 without any gradient across its disk for our lvg calculation . note that we perform an additional lvg calculation for the center of ngc 2903 assuming the [ @xmath1co]/[@xmath2co ] abundance ratios of 30 and 70 to evaluate the effect of the variation in the assumed [ @xmath1co]/[@xmath2co ] abundance ratio on results of lvg calculation . figure 10 shows results of lvg calculation for each region in ngc 2903 . the thin line indicates a curve of constant @xmath8 as functions of @xmath10 and @xmath11 , and the thick line indicates that of constant @xmath9 . we can determine both @xmath10 and @xmath11 at the point where two curves intersect each other . under the assumption of [ @xmath1co]/[@xmath2co ] abundance ratio of 50 , the derived @xmath10 ranges from @xmath161000 @xmath13 ( in the disk ; i.e. , bar , bar - ends , and spiral arms ) to 3700 @xmath13 ( at the center ) and the derived @xmath11 ranges from 10 k ( in spiral arms ) to 30 k ( at the center ) . note that both @xmath10 and @xmath11 vary depending on the assumption of [ @xmath1co]/[@xmath2co ] abundance ratio ; at the center of ngc 2903 , the abundance ratio of 30 yields lower @xmath10 of 1800 @xmath13 and higher @xmath11 of 38 k , whereas the abundance ratio of 70 yields higher @xmath10 of 5900 @xmath13 and intermediate @xmath11 of 29 k. it seems that @xmath10 is proportional to [ @xmath1co]/[@xmath2co ] abundance ratio . this trend of @xmath10 can be naturally explained if we consider the optical depth of @xmath1co and @xmath2co emission . @xmath1co is always optically thick and thus its emission emerges from the diffuse envelope of dense gas clouds , while @xmath2co emission emerges from further within these dense gas clouds due to its lower abundance . since the increase in the assumed [ @xmath1co]/[@xmath2co ] abundance ratio means that @xmath2co becomes more optically thin , @xmath2co emission emerged from deeper within the dense gas clouds and thus it probes denser gas . derived physical properties , @xmath10 and @xmath11 , are summarized in table 3 . we compare the derived @xmath10 and @xmath11 in ngc 2903 with those determined in other external galaxies . @xcite determined @xmath10 and @xmath11 for gmcs associated with the giant h region ngc 604 in m 33 at a spatial resolution of 100 pc using three molecular lines , @xmath1co(@xmath0 ) , @xmath2co(@xmath0 ) , and @xmath1co(@xmath22 ) , based on the lvg approximation . the derived @xmath10 and @xmath11 are 800 2500 @xmath13 and 20 30 k , respectively , which are similar to our study for ngc 2903 in spite of the difference in the spatial resolution . however , @xcite obtained different physical properties for gmcs in spiral arms of m 51 . they performed the lvg analysis using @xmath9 and @xmath8 at a spatial resolution of 120 180 pc . for the case of constant @xmath48 = 1.0 km s@xmath26 pc@xmath26 , the derived @xmath11 ranges from 10 to 50 k , which is similar to our study for ngc 2903 , whereas @xmath10 ranges from 100 to 400 @xmath13 , which is 5 10 times lower than that in the disk of ngc 2903 in spite that the values of @xmath8 and @xmath9 in m 51 are not so different from those in ngc 2903 . this is presumably due to the differences in assumed @xmath46(@xmath1co ) and [ @xmath1co]/[@xmath2co ] abundance ratio . the authors assumed @xmath46(@xmath1co ) of 8.0 @xmath53 , which is higher than that assumed in our study , and a lower [ @xmath1co]/[@xmath2co ] abundance ratio of 30 . under the lvg approximation with the assumption of @xmath46(@xmath1co ) of 8.0 @xmath53 , we found that the derived @xmath10 is typically @xmath163 times lower than that with assumption of @xmath46(@xmath1co ) of 1.0 @xmath53 . physically , high @xmath46(@xmath1co ) means abundant @xmath1co molecules among molecular gas . in this condition , the optical depth of @xmath1co line also increases , and thus the photon - trapping effect in molecular clouds becomes effective . since this effect contributes the excitation of @xmath1co molecule , an effective critical density of @xmath1co line decreases . in other words , since the @xmath1co is easily excited to upper @xmath54 levels even in low molecular gas density , @xmath10 at a given @xmath8 decreases . as a result , lvg analysis with the assumption of @xmath46(@xmath1co ) of 8.0 @xmath53 yields lower @xmath10 . in addition , the low [ @xmath1co]/[@xmath2co ] abundance ratio of 30 causes the decrease in the derived molecular gas density as described above . therefore , the difference in the derived @xmath10 between ngc 2903 and m 51 can be explained by the difference in the assumed @xmath46(@xmath1co ) and the [ @xmath1co]/[@xmath2co ] abundance ratio . as described in section 1 , sfes often differ between galaxy centers and disks . since ngc 2903 has a bright star forming region at the center , its sfe is expected to be higher than those in other regions . here , we calculate sfes for seven regions where averaged physical properties of molecular gas are obtained , and compare sfe with @xmath10 and @xmath11 in each region to examine what parameter controls sfe in galaxies . sfe is expressed using the surface density of sfr ( @xmath55 ) and that of molecular hydrogen ( @xmath56 ) as follows : @xmath57= \left ( \frac{\sigma_{\rm sfr}}{m_{\odot}\,{\rm yr^{-1}\,pc^{-2 } } } \right ) { \displaystyle \biggl/ } \left ( \frac{\sigma_{\rm h_2}}{m_{\odot}\,{\rm pc^{-2 } } } \right ) \end{aligned}\ ] ] we calculated extinction - corrected sfrs from a linear combination of h@xmath23 and @xmath58/mips 24 @xmath59 luminosities using a following formula @xcite : @xmath60 where @xmath61 and @xmath62 mean h@xmath23 and 24 @xmath59 luminosities , respectively . @xmath63 is the inclination of for ngc 2903 and @xmath64 is the covered area for each region ( in the unit of pc@xmath65 ) . we used archival continuum - subtracted h@xmath23 and 24 @xmath59 images of ngc 2903 obtained by @xcite and the local volume legacy survey project @xcite , respectively . in addition , we calculated @xmath56 using @xmath37 as follows : @xmath66 & = & 2.89 \times { \rm cos } \ i \left ( \frac{i_{\rm 12co(1 - 0)}}{{\rm k\,\,km\,\,s^{-1 } } } \right ) \times \left\ { \frac{x_{\rm co}}{1.8 \times 10^{20}\,{\rm cm}^{-2}\,({\rm k\,\,km\,\,s^{-1}})^{-1 } } \right\ } .\end{aligned}\ ] ] here , we adopted a constant @xmath19 value of @xmath41 @xmath42 ( k km s@xmath26)@xmath26 @xcite . we found that sfe at the center , @xmath67 yr@xmath26 , is 2 4 times higher than those in other regions . calculated sfes are listed in table 3 . we examined the dependence of sfe on @xmath10 and @xmath11 as shown in figure 11 . we found that sfe positively correlates with both @xmath10 and @xmath11 . however , the trend of these correlations might change because it is possible that variations in the [ @xmath1co]/[@xmath2co ] abundance ratio and @xmath19 affect the estimate of @xmath10 , @xmath11 , and sfe . in fact , both the [ @xmath1co]/[@xmath2co ] abundance ratio and @xmath19 often differ between galaxy centers and disks . therefore , we examine how variations in the [ @xmath1co]/[@xmath2co ] abundance ratio and @xmath19 alter the estimate of @xmath10 , @xmath11 , and sfe at the center of ngc 2903 . we first consider the effect of the variation in [ @xmath1co]/[@xmath2co ] abundance ratio on the estimate of @xmath10 and @xmath11 . as described in section 4.2.1 , it is reported that the @xmath1c/@xmath2c abundance ratio in our galaxy increases with the galactocentric radius @xcite . thus we examine the case of the lower [ @xmath1co]/[@xmath2co ] abundance ratio at the center of ngc 2903 . if we adopt the [ @xmath1co]/[@xmath2co ] abundance ratio of 30 at the center , @xmath10 and @xmath11 are estimated to be 1800 @xmath13 and 38 k , respectively . this @xmath10 value is slightly lower than that in the northern arm , whereas the positive correlation between sfe and @xmath10 is not destroyed . similarly , the @xmath11 of 38 k does not destroy the positive correlation between sfe and @xmath11 . next , we consider the effect of the variation in @xmath19 on the estimate of sfe . in central regions of disk galaxies , @xmath19 drops ( i.e. , co emission becomes luminous at a given gas mass ) by a factor of 2 3 or more ( e.g. , @xcite ; @xcite ) , including the galactic center ( e.g. , @xcite ; @xcite ) . such a trend is presumably applicable to ngc 2903 considering the relationship between @xmath19 and metallicity , 12 + log(o / h ) . in general , @xmath19 decreases with the increase in metallicity because the co abundance should be proportional to the carbon and oxygen abundances ( e.g. , @xcite ; @xcite ) . in addition , it is reported that metallicity decreases with the galactocentric distance in ngc 2903 ( e.g. , @xcite ; @xcite ) . these observational facts suggest a smaller @xmath19 by a factor of 1.5 2 at the center than in the disk of ngc 2903 , which yields a smaller gas mass , providing a higher sfe than the present one shown in table 3 and figure 11 . however , even if a higher sfe by a factor of 2 is adopted for the center , the global trend of the correlations shown in figure 11 does not change so much because the original sfe at the center is already the highest in ngc 2903 . therefore , we concluded that variations in the [ @xmath1co]/[@xmath2co ] abundance ratio and @xmath19 do @xmath68 affect the correlations of sfe with @xmath10 and @xmath11 in ngc 2903 . note that the smaller @xmath19 corresponds to the larger @xmath46(@xmath1co ) at the center of ngc 2903 , but the larger @xmath48 is also suggested because the typical velocity width at the center ( 250 300 km s@xmath26 ) is wider than those in other regions ( 150 200 km s@xmath26 ) due to the rapid rotation of molecular gas near the galaxy center . thus we consider that @xmath69co)/(@xmath48 ) itself does not differ between the center and the disk in ngc 2903 even if the @xmath69co ) at the center is larger than that in the disk . finally , we examine the correlation coefficient for the least - square power - law fit @xmath14 between sfe and @xmath10 and that between sfe and @xmath11 shown in figure 11 . we found that the former is 0.50 and the latter is 0.08 . the significant correlation between sfe and @xmath10 with @xmath14 of 0.50 suggests that molecular gas density governs the spatial variations in sfe . this speculation is well consistent with earlier studies based on hcn emission ( e.g. , @xcite ; @xcite ; @xcite ; @xcite ) . in order to confirm whether such a relationship between sfe and @xmath10 is applicable to other galaxies or not , we will perform further analysis toward other coming sample galaxies , considering variations in the [ @xmath1co]/[@xmath2co ] abundance ratio , @xmath19 , and @xmath69co)/(@xmath48 ) , in forthcoming papers . we have performed the simultaneous mappings of @xmath0 emission of @xmath1co , @xmath2co , and c@xmath3o molecules toward the whole disk ( @xmath4 or @xmath24 kpc at the distance of 8.9 mpc ) of the nearby barred spiral galaxy ngc 2903 with the nro 45-m telescope equipped with the forest at an effective angular resolution of @xmath6 ( or 870 pc ) . a summary of this work is as follows . 1 . we detected @xmath1co(@xmath0 ) emission over the disk of ngc 2903 . in addition , significant @xmath2co(@xmath0 ) emission was found at the center and bar - ends , whereas we could not detect any significant c@xmath3o(@xmath0 ) emission . 2 . in order to improve the s / n of co emission and to measure @xmath8 and @xmath9 with high significance , we performed the stacking analysis for our @xmath1co(@xmath0 ) , @xmath2co(@xmath0 ) , and archival @xmath1co(@xmath7 ) spectra with velocity - axis alignment in nine representative regions ( i.e. , center , northern bar , southern bar , northern bar - end , southern bar - end , northern arm , southern arm , inter - arm , and outer - disk ) of ngc 2903 . we successfully obtained the stacked co spectra with highly improved s / n , and thus we could confirm the significant @xmath2co(@xmath0 ) emission for all the regions . we examined the averaged @xmath8 and @xmath9 for nine regions , and found that the averaged @xmath8 shows the highest value of 0.92 at the center , and moderate or lower values of 0.6 0.8 are observed in the disk . however , the highest @xmath9 of 0.19 is observed not at the center but in the northern arm . the central @xmath9 of 0.11 is similar to those in other regions ( 0.08 0.13 ) except for the northern arm and outer - disk ( @xmath16 0.04 ) . 4 . we determined @xmath10 and @xmath11 of molecular gas using @xmath8 and @xmath9 based on the lvg approximation . under the assumption of [ @xmath1co]/[@xmath2co ] abundance ratio of 50 , the derived @xmath10 ranges from @xmath161000 @xmath13 ( in the bar , bar - ends , and spiral arms ) to 3700 @xmath13 ( at the center ) and the derived @xmath11 ranges from 10 k ( in the bar and spiral arms ) to 30 k ( at the center ) . we examined the dependence of sfe on @xmath10 and @xmath11 of molecular gas , and found the positive correlation between sfe and @xmath10 with the correlation coefficient for the least - square power - law fit @xmath14 of 0.50 . this suggests that molecular gas density governs the spatial variations in sfe . we thank the referee for invaluable comments , which significantly improved the manuscript . we are indebted to the nro staff for the commissioning and operation of the 45-m telescope and their continuous efforts to improve the performance of the instruments . this work is based on observations at nro , which is a branch of the national astronomical observatory of japan , national institutes of natural sciences . this research has made use of the nasa / ipac extragalactic database , which is operated by the jet propulsion laboratory , california institute of technology , under contract with the national aeronautics and space administration . co(@xmath0 ) emission in ngc 2903 . the grid spacing was set to 12@xmath18 in order to display the spectrum in each pixel clearly . the temperature scale of spectra is indicated by the small box inserted in the lower left corner . , width=302 ] in ngc 2903 obtained by coming and those obtained by k07 . the vertical and horizontal lines indicate the 2 @xmath36 of @xmath37 for coming and k07 data , respectively . the diagonal solid line indicates the ratio of @xmath37 by coming to those by k07 of unity , and the dashed lines indicate the ratio of 0.5 and 2.0 . both @xmath37 are well correlated with each other . , width=302 ] co(@xmath0 ) emission in ngc 2903 . for comparison , spectra of @xmath1co(@xmath0 ) emission multiplied by 0.5 are overlaid in red line . as well as figure 2 , the grid spacing was set to 12@xmath18 . the temperature scale of spectra is indicated by the small box inserted in the lower left corner . ( b ) magnified spectra with the grid spacing of 6@xmath18 at the northern bar - end . ( c ) same as ( b ) , but at the center . ( d ) same as ( b ) , but at the southern bar - end . , width=642 ] ( left ) and @xmath9 ( right ) superposed on the contour map of @xmath37 of ngc 2903 . the contour levels of @xmath37 are the same as figure 3 . there are some local peaks of @xmath8 ( @xmath16 1.0 ) near the center and at the downstream side of the northern spiral arm , whereas lower @xmath8 ( @xmath16 0.5 0.6 ) was observed in the bar . the spatial distribution of @xmath9 is unclear due to poor s / n of @xmath2co(@xmath0 ) emission . , width=642 ] of ngc 2903 . the green frame indicates the center , the purple and orange indicate the northern and southern bars , the blue and red indicate the northern and southern bar - ends , the cyan and magenta indicate the northern and southern arms , the yellow indicates the inter - arm , and the grey indicates the outer - disk . ( right ) intensity - weighted mean velocity field calculated from @xmath1co(@xmath0 ) data superposed on the contour map of @xmath37 . the contour levels of @xmath37 are the same as figure 3 . , width=642 ] co(@xmath0 ) emission multiplied by 0.5 , the red indicates @xmath1co(@xmath7 ) emission multiplied by 0.5 , and the blue indicates @xmath2co(@xmath0 ) emission . the s / n of each co emission is dramatically improved , and thus a significant @xmath2co(@xmath0 ) emission is confirmed for all the regions . , width=642 ] ( thin line ) and @xmath9 ( thick line ) as functions of molecular gas density @xmath10 and kinetic temperature @xmath11 . @xmath1co fractional abundance per unit velocity gradient @xmath69co)/(@xmath48 ) was assumed to be @xmath49 ( km s@xmath26 pc@xmath26)@xmath26 . the [ @xmath1co]/[@xmath2co ] abundance ratio was assumed to be a fixed value of 50 for the bar , bar - ends , and spiral arms , but three different [ @xmath1co]/[@xmath2co ] abundance ratios of 30 , 50 , and 70 were assumed for the center . dashed lines indicate @xmath70 error of each line ratio . , width=642 ] lc + morphological type@xmath71 & sab(rs)bc + map center@xmath72 : & + right ascension ( j2000.0 ) & + declination ( j2000.0 ) & + distance@xmath73 & 8.9 mpc + linear scale & 43 pc arcsec@xmath26 + inclination@xmath74 & + + lccccc + & @xmath37 & @xmath75 & @xmath44 & @xmath8 & @xmath9 + & [ k km s@xmath26 ] & [ k km s@xmath26 ] & [ k km s@xmath26 ] & & + + center & @xmath77 & @xmath78 & @xmath79 & @xmath80 & @xmath81 + northern bar & @xmath82 & @xmath83 & @xmath84 & @xmath85 & @xmath86 + southern bar & @xmath87 & @xmath88 & @xmath89 & @xmath90 & @xmath91 + northern bar - end & @xmath92 & @xmath93 & @xmath94 & @xmath95 & @xmath96 + southern bar - end & @xmath97 & @xmath98 & @xmath99 & @xmath95 & @xmath86 + northern arm & @xmath100 & @xmath101 & @xmath102 & @xmath103 & @xmath104 + southern arm & @xmath105 & @xmath106 & @xmath107 & @xmath108 & @xmath109 + inter - arm & @xmath110 & @xmath111 & @xmath112 & @xmath108 & @xmath96 + outer - disk & @xmath113 & @xmath114 & @xmath115 & @xmath116 & @xmath117 + + lcccc + region & [ @xmath1co]/[@xmath2co ] & @xmath10 & @xmath11 & sfe + & & [ @xmath118 @xmath13 ] & [ k ] & [ @xmath120 yr@xmath26 ] + + center & 50 & @xmath121 & @xmath122 & @xmath123 + northern bar & 50 & @xmath124 & @xmath125 & @xmath126 + southern bar & 50 & @xmath127 & @xmath128 & @xmath129 + northern bar - end & 50 & @xmath130 & @xmath131 & @xmath132 + southern bar - end & 50 & @xmath112 & @xmath133 & @xmath134 + northern arm & 50 & @xmath135 & @xmath136 & @xmath137 + southern arm & 50 & @xmath138 & @xmath139 & @xmath140 + center & 30 & @xmath141 & @xmath142 & @xmath123 + center & 70 & @xmath143 & @xmath144 & @xmath123 + +
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we present simultaneous mappings of @xmath0 emission of @xmath1co , @xmath2co , and c@xmath3o molecules toward the whole disk ( @xmath4 or 20.8 kpc @xmath5 13.0 kpc ) of the nearby barred spiral galaxy ngc 2903 with the nobeyama radio observatory 45-m telescope at an effective angular resolution of @xmath6 ( or 870 pc ) .
we detected @xmath1co(@xmath0 ) emission over the disk of ngc 2903 .
in addition , significant @xmath2co(@xmath0 ) emission was found at the center and bar - ends , whereas we could not detect any significant c@xmath3o(@xmath0 ) emission . in order to improve the signal - to - noise ratio of co emission and to obtain accurate line ratios of @xmath1co(@xmath7)/@xmath1co(@xmath0 ) ( @xmath8 ) and @xmath2co(@xmath0)/@xmath1co(@xmath0 ) ( @xmath9 ) , we performed the stacking analysis for our @xmath1co(@xmath0 ) , @xmath2co(@xmath0 ) , and archival @xmath1co(@xmath7 ) spectra with velocity - axis alignment in nine representative regions of ngc 2903 .
we successfully obtained the stacked spectra of the three co lines , and could measure averaged @xmath8 and @xmath9 with high significance for all the regions .
we found that both @xmath8 and @xmath9 differ according to the regions , which reflects the difference in the physical properties of molecular gas ; i.e. , density ( @xmath10 ) and kinetic temperature ( @xmath11 ) .
we determined @xmath10 and @xmath11 using @xmath8 and @xmath9 based on the large velocity gradient approximation .
the derived @xmath10 ranges from @xmath12 @xmath13 ( in the bar , bar - ends , and spiral arms ) to 3700 @xmath13 ( at the center ) and the derived @xmath11 ranges from 10 k ( in the bar and spiral arms ) to 30 k ( at the center ) .
we examined the dependence of star formation efficiencies ( sfes ) on @xmath10 and @xmath11 , and found the positive correlation between sfe and @xmath10 with the correlation coefficient for the least - square power - law fit @xmath14 of 0.50 .
this suggests that molecular gas density governs the spatial variations in sfes .
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the science of nanostructures is one of the fields of growing interest in materials science . nanotubes , in particular , are of both fundamental and technological importance ; being quasi - one - dimensional ( 1d ) structures , they possess a number of exceptional properties . while the peculiar electronic structure metallic versus semiconducting behavior of carbon nanotubes depends sensitively on the diameter and the chirality , i.e. , on the way the graphene sheet is wrapped up into a tube @xcite , boron nitride ( bn ) tubes display a more uniform behavior with a wide band - gap ( larger than 4 ev ) , almost independent of diameter and chirality @xcite . the potential applications of nanotubes in nanodevices are numerous @xcite : super - tough nanotube fibres @xcite , gas sensors @xcite , field effect displays @xcite , and electromechanical devices @xcite , to name only a few . one of the most spectacular examples is the realization of field effect transistors both with carbon nanotubes @xcite and with bn - nanotubes @xcite . since carbon and bn nanotubes are routinely produced in gram quantities , the challenge now consists in having fast experimental tools for characterization of nanotube samples and , if possible , single isolated nanotubes . optical spectroscopic techniques provide us with this tool @xcite . the final goal is to find a unique mapping of the measured electronic and vibrational properties onto the tube indices @xmath0 . to that end , the electronic structure and dielectric properties of the tubes are two key areas to study . one possible spectroscopic method is optical absorption spectroscopy with direct excitations from occupied to unoccupied states . for carbon tubes , the energy difference @xmath1 between corresponding occupied and unoccupied van hove singularities ( vhss ) in the 1-dimensional electronic density of states ( dos ) is approximately inversely proportional to the tube diameter @xmath2 . in resonant raman spectroscopy @xcite and scanning tunneling spectroscopy @xcite , this scaling is employed for the determination of tube diameters ( and @xmath0 indexes in resonant raman @xcite ) . a recent example is the fluorescence spectroscopy of single carbon tubes in aqueous solution , where @xmath3 is probed through the frequency of the excitation laser and @xmath4 is probed simultaneously through the frequency of the emitted fluorescent light @xcite . also optical absorption spectroscopy of nanotubes in aqueous solution @xcite allows for the spectral resolution of peaks that can be associated with distinct types of nanotubes . for the distance between the first vhss in semiconducting carbon tubes , a simple @xmath5-electron tight - binding fit yields the relation @xmath6 , where @xmath7 is the distance between nearest - neighbor carbon atoms . the value for the hopping matrix element @xmath8 varies between 2.4 ev and 2.9 ev , depending on the experimental context in which it is used . this fact is a clear indication that the above relation gives only qualitative and not quantitative information on the tube diameter and/or chirality . furthermore , for small - diameter tubes the band structure completely changes with respect to the band structure of large diameter tubes , including a reordering of the vhss in the density of states and displaying fine structure beyond the first and second vhss @xcite . this structure sensitively depends on the tube indices and may be probed by optical absorption spectroscopy over a wider energy range , possibly extending into the uv regime . the spectroscopical characterization of macroscopic tube samples is made difficult by the fact that in the bulk solid and even in bundles the tubes are not perfectly aligned and do not have a well defined diameter and helicity , rendering , in the case of carbon tubes , a random mixture of semiconducting and metallic tubes @xcite . additionally , the role of intertube interaction in the spectra needs to be addressed since the tubes are close packed and could interact with each other via long - range forces induced by the excitations . only very recently optical absorption spectra were reported for aligned single - wall carbon tubes of a very narrow diameter distribution ( [email protected] ) grown inside the channels of a zeolite matrix @xcite . geometric arguments predict three possible tubular helicities for the range of diameters around 4 : the armchair ( 3,3 ) , the zig - zag ( 5,0 ) and the chiral ( 4,2 ) configuration . therefore , this particular case serves as an important case study where a direct comparison between experimental data and theoretical calculations becomes possible . indeed in ref . @xcite we have shown the relevance of a first - principles calculation for these small - diameter carbon nanotubes by reproducing the polarization dependence of the measured optical spectra . at present , the dielectric response of tubes in the frequency range of the electronic interband transitions and the collective excitations ( plasmons ) of the valence electrons ( up to 50 ev ) is not well understood since previous studies focused on the low - energy regime and excitations @xcite . for higher frequencies , up to now , the majority of the theoretical studies of the response , besides model calculations @xcite , are mostly limited to summing over independent band - to - band transitions obtained within the semi - empirical tight - binding method @xcite or the density - functional theory ( dft ) framework @xcite , or to calculations of the joint density of states ( jdos ) @xcite which use the bandstructure with no explicit evaluation of the transition matrix elements . as we will show below , these approximations are not sufficient for a full interpretation of the experiments . this shortcoming is not due to the quality of the bandstructure calculation itself but instead due to the neglect of the induced microscopic components in the response to the external field , the local field effects ( lfe ) @xcite . these effects strongly modify the total response for certain polarizations of the external perturbation . also , induced exchange and correlation ( xc ) components obtained beyond the random - phase approximation ( rpa ) might contribute @xcite . therefore , important questions concerning the electron interaction , excitations and screening still remain unanswered . our approach here is to determine the spectra by _ ab - initio _ calculations incorporating important ingredients of the electron response as in previous works @xcite . the present paper is organized as follows : after an exposition of the theoretical framework in section [ theory ] , we present in section [ tubeabs ] _ ab initio _ calculations of optical spectra of small single - wall carbon and boron nitride nanotubes . we identify the influence of crystal lfe , xc effects and intertube interaction in the spectra . we also point out the similarities and differences between carbon and bn structures which are related to similarities and differences in the respective electronic band structures . a certain number of comparisons with experiment allows us to verify that the chosen _ ab initio _ approach is an important improvement and well suited for the description of the spectra of these systems . in section [ sheetabs ] , we analyse the spectra of the building blocks of the tubes , i.e. the graphene and hexagonal bn sheets ( layers ) including the optical spectrum of graphite . this gives information about the the interaction between objects ( sheet - sheet ) in the excited state , even when these objects can be considered to be isolated in the ground state . we also discuss to which extent the dielectric response of the tubes can be understood in terms of the response of the sheets . this will help to understand whether some aspects of the response of the tubes are _ inherent _ to the sheets , and if so , they could also be observed in other systems of more practical interest , e.g. samples comprising a mixture of tubes with different diameters and orientation or multiwall tubes . in section [ eelsec ] , we present results for electron energy loss in order to have some additional validation from existing experimental data , and because a comparison between optical and loss spectra allows a deeper discussion and understanding of interaction effects . finally , we conclude with section [ finsec ] , with an overall discussion of the results . _ ab initio _ calculations in the framework of density functional theory ( dft ) have yielded high - quality results for a large variety of systems : from molecules to periodic solids and structural defects @xcite . these results are however mostly limited to quantities related to the electronic _ ground state _ , whereas additional phenomena that occur in the excited state are not correctly described @xcite . in particular , the self - consistency between the total perturbing potential and the charge response induces hartree and exchange - correlation potentials that have to be dealt with . the former give rise to the so - called local field effects , whereas the latter can lead , for example , to excitonic effects . today , in the solid state _ ab initio _ framework two main approaches can include both lfe and xc effects @xcite and can be therefore suitable for the description of electronic excitations in nanostructures . first , green s functions approaches within many - body perturbation theory : here one adds self - energy corrections to the dft kohn - sham bandstructure and the electron - hole interaction is included via the solution of the bethe - salpeter equation @xcite . this approach has given excellent results for various bulk and finite systems , but it is computationally very cumbersome and not ready yet to be applied systematically to more complex systems . the time - dependent hartree - fock ( tdhf ) method represents a certain level of approximation within this approach , where correlation ( i.e. screening ) in the self energy is neglected . lfe , on the other hand , are still retained . tdhf has been employed , up to now in a semi - empirical way , to calculate optical spectra , including excitons , of carbon nanotubes @xcite and also for the 4 - diameter ones @xcite . however , the obtained absorption peak assignments in the latter case were in disagreement with predictions based on the operative dipole selection rules for the specific tubular space groups @xcite . this suggests that in such systems with important metallic character a neglect of screening may lead to problems . the second way to calculate spectra is given by the time - dependent dft ( tddft ) @xcite , where all manybody effects are embodied in the exchange - correlation potential and kernel . it is in practice an approximate way but always treats the variations of the hartree potential exactly ( right as in both the green s functions and tdhf approaches ) . this approach , using the ( static ) adiabatic local density approximation ( tdlda ) for the xc effects , or even completely neglecting the density variations of the xc potential ( rpa , on the basis of an lda bandstructure ) has been applied successfully to many finite and infinite systems @xcite . in particular , excellent results for the loss spectra of graphite have been obtained using this approximation @xcite . this is the approach we follow in the present work . as a first step in our tddft approach , we determined the electronic ground state of the systems . the kohn - sham single - particle equations were solved self - consistently in the lda for exchange and correlation @xcite . for the description of the valence - core interaction we have used norm - conserving pseudopotentials which were generated from free - atom all - electron calculations @xcite . for the `` isolated '' geometries of the tubes and of the sheets we calculated periodic arrays with a large distance between the building blocks in a supercell geometry , in order to minimize the mutual interaction . the crystalline valence - electron wavefunctions were expanded using a plane - wave basis set ( with an energy cutoff of 62 rydberg for carbon systems , and 50 rydberg for bn systems ) . a part of the calculations was carried out using the abinit code @xcite . the next step is the linear response calculation of the independent particle polarizability @xmath10 @xcite . it involves a sum over excitations from occupied bands to unoccupied bands : @xmath11 , \label{chi0}\end{aligned}\ ] ] where @xmath12 means that the indices @xmath13 and @xmath14 of the first term are exchanged . the result is checked for convergence with respect to the number of bands @xcite and the discrete sampling of * k*-points within the first brillouin zone . within tdlda , the full polarizability @xmath15 is connected to @xmath16 via@xcite @xmath17 where v@xmath18 is the bare coulomb interaction and @xmath19 , the so - called exchange - correlation kernel , is the functional derivative of the lda exchange - correlation potential with respect to the electron density . by setting @xmath19 to zero , exchange and correlation effects in the electron response are neglected and one obtains the random - phase approximation ( rpa ) . we have carried out our calculations in the rpa and also in the tdlda for certain cases . the inverse dielectric function for a periodic system and momentum transfer @xmath20 is obtained from : @xmath21 with @xmath20 in the first brillouin zone and @xmath22 , @xmath23 are reciprocal lattice vectors . the absorption spectrum is obtained as the imaginary part of the macroscopic dielectric function @xmath24 whereas the loss function for a transferred momentum @xmath25 is given by @xmath26 $ ] . in inhomogeneous systems , e.g. periodic solids , clusters and structural imperfections the inhomogeneity in the electron response gives rise to local field effects ( lfe ) and the @xmath27 can not be considered as being purely a diagonal matrix . therefore , its off - diagonal elements have to be included in the matrix inversion . making the approximation @xmath28 corresponds to neglecting the inhomogeneity of the response , i.e. to neglecting the lfe . when these effects are neglected and , moreover , all transition matrix elements in @xmath16 are supposed to be constant , one arrives at the widely used approximation that the absorption spectrum is proportional to the jdos , i.e. proportional to the sum over interband transitions from occupied ( @xmath29 ) to empty ( @xmath30 ) states over the brillouin zone points @xmath31 , @xmath32 . let us first look at the optical absorption spectra of small - diameter carbon nanotubes ( all three possible helicities ) , for which experimental results have recently become available . the calculations of the spectra @xcite were done for the tubes arranged in a hexagonal lattice with an intertube distance ( distance between tube walls ) , equal to @xmath33 5.5 which leads to nearly isolated tubes . additionally , for the ( 3,3 ) armchair ones we repeated the calculations for a solid with a smaller intertube distance , @xmath33 3.2 , which is close to the interlayer distance in graphite ( @xmath34 3.4 ) . the optical absorption spectra for the small diameter tubes are diplayed in fig . [ fig1 ] ( as well as jdos curves for two cases ) . in the upper panel of the figure , the jdos divided by the square of the excitation energy , for the ( 3,3 ) nearly isolated ( thin line ) and interacting ( in the solid ) ( thick line ) tubes are shown . it can be seen that after an initial very steep decrease ( @xmath35 1 ev ) , in both jdos curves there is a gradual increase starting from 2 ev . in the case of isolated tubes a sequence of pronounced peak structures up to 5 ev is observed : this can be explained from the occurrence of direct interband transitions between the van hove singularities of the density of states ( dos ) . these peaks are smeared out in the jdos of the solid where the tubes are strongly interacting @xcite . in the lower panels of fig . [ fig1 ] the calculated absorption spectra for light polarizations perpendicular and parallel to the tube axis are displayed ( as well as the experimental data plotted in the inset ) . the dashed lines denote results in the rpa neglecting lfe , continuous lines including lfe . for parallel polarization both lfe and lda - xc effects were found to be negligible and , therefore , in this case only the rpa results without lfe are presented . the fact that both these effects turned out to be negligible can explain why the peak positions ( a , b and c ) predicted here for the parallel polarization match very well the ones found in two recent dft - rpa studies @xcite , which completely neglected lfe and xc effects in the response . our calculated peak positions ( a , b and c ) in fig . [ fig1 ] are in good agreement ( to within 0.2 ev ) with the experimentally observed peak structures for the parallel polarization ( inset ) . it is also important to note that the present results support the same peak - to - helicity assignment as in previous works @xcite . namely , the three observed peaks a , b and c are due to the ( 5,0 ) , ( 4,2 ) and ( 3,3 ) tubes , respectively . such an assignment is also in accordance with the dipole selection rules for these tubular space groups @xcite . concerning the effect of intertube interaction in the absorption we found that it is rather small for this polarization : tube - tube interaction leads only to a broadening of the main absorption peak ( thick solid curve for the ( 3,3 ) tubes ) . this can be explained from an increase of the energy range of the possible interband transitions brought about by the interaction . all the discussion of the results up to now seems to indicate that lfe may not be needed . nonetheless , for light polarizations perpendicular to the tube axis due to the presence of depolarization lfe play an important role which can not be ignored . more specifically , for this polarization the experimental spectrum displays vanishing intensity for frequencies up to 4 ev @xcite ( see inset in fig . [ fig1 ] , dotted curve ) . clearly , neither the jdos calculation nor the calculated spectrum without lfe can capture this effect . instead , they both predict a series of peaks from 2 to 5 ev . the reason that rpa - without - lfe fails here is due to the depolarization effect @xcite which is created by the induced polarization charges . the depolarization is only accounted for if lfe are included : as it can be seen in fig . [ fig1 ] ( second panel ; continuous curve ) _ lfe suppress the low - energy absorption peaks _ and render the tubes almost transparent below 5 ev in agreement with the experiment . it should not come , therefore , as a surprise that the recent dft - rpa studies @xcite ( which did not consider lfe ) did not reproduce this transparency for the perpendicular polarization . the tdlda result , displayed as the dotted curve in the second panel , turned out to be qualitatively similar to the rpa - with - lfe result : again the low - energy absorption peaks are suppressed . this shows that the main effect comes from fluctuations of the hartree- , and not from those of the xc - potential ( as also obtained for the case of graphite @xcite ) . lda - xc effects only cause a small ( 0.3 ev ) redshift of the remaining absorption peak to 5.5 ev , with respect to rpa - with - lfe . this is a characteristic behavior of finite systems @xcite . still there is an open question about the role of electron - hole interaction in the case of the small band - gap ( 4,2 ) tube @xcite , however the good agreement with experiment indicates that its contribution should be small for the parallel polarization . lfe also have a similar drastic impact for the system of isolated zig - zag ( 5,0 ) tubes ( not shown ) . the manner according to which lfe operate may be understood as follows : for perpendicular polarization the tubes form an assembly of almost isolated , but highly polarizable , objects . an applied external field induces hence a local , i.e. microscopic , response - the lfe - which strongly weakens the total perturbation ( i.e. it is a `` depolarization '' ) . the macroscopic response to this weak perturbation is only very moderate , because the electrons are localized on the tubes . this is totally different from the screening in a bulk metal or small - gap semiconductor , where even a very weak total perturbation still leads to a strong response at low frequencies . for polarization parallel to the tube axis , the situation resembles rather this latter case , something which explains the absence of lfe for this polarization . on the other hand for the perpendicular polarization when the tubes are interacting in the solid ( third panel ) the depolarization is much weaker : the tubes start to absorb ( they are no longer transparent ) because the electronic states start to delocalize and the system is now more similar to a bulk metal . this different behavior of the response depending on the intertube distance leads to a very important consequence : it suggests that _ the inter - tube interaction can be detected experimentally _ in a qualitative study of the absorption spectra for perpendicular light polarizations . in fig . [ bntubesfig ] we present optical absorption spectra for three different bn nanotubes . first , we compare for the case of the bn(3,3 ) tube the spatially averaged spectrum with the joint density of states ( jdos ) , divided by the square of the transition energy . if local field efffects are neglected , most peaks of the jdos are visible in the averaged absorption spectra while some peaks are suppressed due to small or vanishing oscillator strength in eq . ( [ chi0 ] ) . this demonstrates that the fine structure in the spectra is not an artifact of low k - point sampling but is due to the presence of van - hove singularities in the 1-dimensional density of states of the tubes . proper inclusion of lfe leads to a smoothing of the spectrum . however , some fine structure survives and may be discernible in high - resolution optical absorption experiments . the spectrum with perpendicular polarization demonstrates that for bn nanotubes , depolarization effects play a similarly important role as for carbon tubes : neglecting lfe , the onset of absorption would be around 4.5 ev . lfe , however , lead to a redistribution of the oscillator strength to higher energies and render the tubes almost transparent up to 8 ev . the band gap of bn nanotubes in dft - lda is 4 ev and is only weakly dependent on radius and chirality @xcite . accordingly , the absorption spectra for polarization parallel to the tube axis are very similar for the ( 3,3 ) , ( 5,5 ) , and ( 6,0 ) tubes which all display a strong absorption peak around 5 ev and a second high peak around 14 ev . we expect these structures to be stable also for tubes with larger diameters . only the fine structure of the spectra ( e.g. , the absorption peak at 9 ev for the ( 3,3 ) tube ) depends on the details of the band - structure and varies for the different diameters and chiralities @xcite . for the ( 6,0 ) tube , the first high absorption peak is split and the dominant peak shifted towards lower energy . this is a curvature effect which leads to a reduction of the band - gap for small zigzag bn tubes . in the next section we will see to which extent the above findings for carbon and bn nanotubes can be understood by an analysis based on results for graphene and bn sheets and graphite . hexagonal graphite has the aba bernal stacking sequence of the graphene sheets . in the present work we assumed the experimental lattice parameter a@xmath36 and ( c / a)@xmath36 ratio ( 2.46 and 2.73 , respectively @xcite ) . the calculated rpa optical absorption spectra @xcite , with and without lfe , are shown in fig . [ graphabs ] for in - plane light polarization ( * e * @xmath37 * c * ) and in fig . [ fig4 ] ( a ) for polarization parallel to the * c * axis ( * e * @xmath38 * c * ) . the in - plane spectrum is dominated by a very intense peak structure at low frequencies ( up to 5 ev ) and also another peak structure of broader frequency range which sets in beyond 10 ev and has a pronounced peak at 14 ev . the origin of these peak structures is due to @xmath39 and @xmath40 interband transitions , respectively , according to the earlier interpretations by bassani and paravicini @xcite who assumed a two - dimensional approximation no interaction between the graphene sheets and the operative dipole selection rules for this polarization . our calculations of the oscillator strength for specific transitions between bands in the brillouin zone ( bz ) are in agreement with their interpretation . lfe are found to be nearly negligible for this polarization . this is not surprising since for in - plane polarizations graphite is homogeneous in the long - wavelength limit ( @xmath41 ) . the general aspects of the spectrum peak positions , their intensity and lineshape are in close agreement with the existing experimental results @xcite and the previous all - electron calculation of ahuja et al . @xcite who neglected lfe . * c*. the star symbol denotes the unoccupied electron states.,width=302 ] the absorption spectrum of graphite for the light polarization parallel to the * c * axis is shown in fig . [ fig4 ] ( a ) . it is characterized by a weak intensity in the low frequency range ( 0 - 5 ev ) and important peaks in the frequency range beyond 10 ev . for this polarization the bandstructure does not play the exclusive direct role in defining the absorption spectrum . now lfe are very important . when lfe are not considered the peak positions for this polarization are at 11 and 14 ev as in the earlier dft - rpa calculation @xcite . however , when lfe are included transitions are mixed and the absorption spectrum is appreciably modified . the main effect of local fields is to shift the oscillator strength at 10 - 15 ev to higher frequencies . they decrease the intensity of the 11 ev peak and are responsible for the appearance of the 16 ev peak in the spectrum . the latter peak is seen in experiments as a shoulder suggesting that the inclusion of lfe is necessary . * c * , for graphite and the graphene - sheet geometries with 2 ( c / a)@xmath36 and 3 ( c / a)@xmath36.,width=302 ] the non - zero oscillator strength found below 5 ev is attributed to the inter - layer interaction which is present in the solid . it is also observed experimentally . the dipole selection rules @xcite for an isolated graphene sheet ( i.e. complete two - dimensionality ) lead to vanishing matrix elements and oscillator strength at this frequency range . lfe do not have any influence on the lower part of the spectrum ( less than 10 ev ) . the existing experimental evidence is not conclusive for the dielectric function in the 11 ev frequency region . the frequency - dependent im[@xmath42 obtained from electron energy loss data @xcite displays a very sharp and intense peak ( im[@xmath42=10 ) at 11 ev . on the other hand , on the basis of optical measurements @xcite the observed maximum at this frequency is of considerably smaller intensity . the earlier interpretations @xcite were based on semi - empirical tight - binding bandstructure calculations in the two - dimensional approximation ( i.e. isolated graphene layers ) and they predicted peak structure between 13.5 and 16.5 ev . it is therefore also important to understand the effect of the inter - layer interaction in the optical response and if this interaction is primarily responsible for the occurrence of the intense peak at 11 ev . for this purpose , we progressively increased the inter - layer spacing ( doublying and tripling the ( c / a)@xmath36 ratio ) . this yields stackings of graphene layers in the unit cell with much weaker mutual interaction . the absorption spectra for these graphene - sheet geometries are displayed in fig . [ fig4 ] ( b , c ) . the first observation is that the oscillator strength vanishes completely in the frequency region below 10 ev in complete accordance with the predictions based on the dipole - selection rules for an isolated graphene layer . therefore , the double inter - layer spacing leads to non - interacting graphene layers as far as the rpa absorption spectrum ( at this frequency range ) is concerned . when lfe are neglected , the peak structure in the 10 - 15 ev range stays intact , now with a smaller intensity due to the larger volume . with increasing interlayer separation , lfe become progressively more important . the shift of oscillator strength induced by lfe is so big that the absorption peaks at 10 - 15 ev are almost completely suppressed . these findings demonstrate that both inter - layer interaction and lfe influence considerably the intensity of the absorption peak at 11 ev . qualitatively , the depolarization effects found in the tubes can hence be explained by the local field effects observed in the graphitic response perpendicular to the graphene sheets . likewise , the effect of intertube interaction in the spectra for polarizations perpendicular to the tube axis is also consistent with the increased propensity of lfe to shift oscillator strength at higher frequencies when the intersheet distance progressively increases ( see fig . [ fig4 ] ) . we conclude this section with a comparative presentation of the absorption spectra of the graphene and bn sheets using the same inter - sheet distance for both cases , shown in fig . [ sheetsfig ] . also we make comparisons with the spectra of the corresponding tubes . the spectrum of the graphene sheet for the in - plane polarization resembles closely the in - plane polarization spectrum of graphite ( fig . [ graphabs ] ) ( except for a scaling due to the change in the volume ) , confirming once more , as in the case of the tubes for the polarization parallel to the tube axis , that the position of the absorption peaks and their lineshape is only weakly influenced by the distance and inter - sheet interaction in this case . the main difference between the graphene and bn - sheet spectra for the in - plane polarization is the complete absence of any feature below 4 ev in the bn spectrum . this is clearly related to the 4 ev lda - band gap of bn . the out - of - plane polarization spectra of the graphene and bn sheets are remarkably similar . most importantly , both display a transparency up to 10 ev and both demonstrate a strong depolarization effect with shift of oscillator strength to higher energies @xcite . we compare now the calculated sheet - absorption spectra for in - plane polarization with the corresponding tube - absorption spectra for polarization parallel to the tube axis ( fig . [ fig1 ] for carbon and fig . [ bntubesfig ] for bn ) . in the carbon case , the strong absorption feature of the sheet between 0 and 4 ev maps onto one or several absorption peaks of the tubes in this energy range . the details of this mapping are , however , sensitively dependent on the diameter and the chirality of the tubes , since dipole selection rules play a strong role in these highly symmetric systems . in particular , for tubes with larger diameter than the ones calculated in this article , the distance @xmath4 of the first van hove singularities strongly depends on whether the tube is metallic or semiconducting . therefore , calculations on graphite or graphene alone will not be sufficient to predict absorption spectra of small - diameter carbon nanotubes . for bn , in contrast , the first high absorption peak at 5.5 ev maps directly onto a corresponding peak in the tubes . also the high energy absorption feature around 14 ev is very similar for the bn sheet and the bn tubes . due to the large band gap of bn , the comparison between sheet and tube spectra is much more favourable for bn than for carbon tubes . on the other hand , in the case of carbon , the extrapolation of results from graphite and graphene to the tubes is much more straightforward for the electron energy loss spectra as discussed in the following section . having seen the strong interaction effects that occur in optical spectra for a polarization perpendicular to the planes or tube axis , it is interesting to make an excursion to a different type of spectroscopy , namely electron energy loss ( eels ) @xcite . although also in this case one can measure , like in the absorption experiment , a momentum transfer close to zero , there is a substantial difference in the definition of the response function : in the absorption measurement , one detects the response to the total macroscopic field , whereas in eels the response to the external field is reported . therefore , as pointed out above , absorption is linked to the imaginary part of the macroscopic dielectric function , but eels is linked to the imaginary part of the inverse of the latter . mathematically this translates into the fact that the crucial response function for eels is governed by equation ( [ eq : chi ] ) , whereas in the case of absorption one can use slightly different quantity : the macroscopic dielectric function can in fact be rewritten as @xmath43 where v@xmath44 is the long - range component of the coulomb potential , and it has a @xmath45 divergence for vanishing @xmath46 and the quantity @xmath47 obeys an equation similar to ( [ eq : chi ] ) : @xmath48 , but setting the @xmath49 component of @xmath50 to zero ( @xmath51 ) @xcite . therefore , this seemingly tiny difference is responsible for the difference between , e.g. , the position of the main absorption peaks and that of the valence plasmons in solids , and one can also expect that it will be crucial when interaction effects on the spectra are discussed . in particular , eels spectra should show , due to the presence of this long - range term , stronger interaction effects than absorption spectra . this is in fact the case , as we will illustrate in the following for the total @xmath52 plasmon in graphite . this plasmon represents the collective excitation mode of all the valence electrons in graphite . before discussing the results we stress here that the tubes can not be considered as completely isolated objects in this calculation of the loss function with the 5.5 intertube distance we refer to the tubes in the latter geometry as _ distant_. fig . [ ctubeels ] shows the rpa loss function , @xmath53 , for the ( 3,3 ) tube , in the range 15 - 35 ev and for a vanishing momentum transfer * q * parallel to the tube axis . for this orientation lfe are negligible . a strong shift of the @xmath52 plasmon from 22 to 28 ev due to intertube interactions in the solid can be seen @xcite . the magnitude of this shift reveals a strong dependence of the plasmon position upon the intertube distance ( hence the average valence electron density ) essentially following the plasmon - frequency dependence in the case of the homogeneous electron gas @xcite . this shows that the tubes respond as homogeneous and highly polarizable objects for parallel * q * in the long - wavelength limit ( @xmath54 ) . a direct consequence would then be that the atomic arrangement , orientation of bonds and helicity may play a secondary role in the response in this frequency and * q * range . therefore , the result for the @xmath52 plasmon ( shown in fig . [ ctubeels ] ) would be representative of either of the three tubes since all of them being of nearly the same diameter possess the same average electron density . this is indeed the case as it can be seen in the inset of fig . [ ctubeels ] where an almost indistinguishable @xmath52 plasmon was also obtained for the ( 5,0 ) tube . hence , in contrast to an optical absorption experiment , small - q loss measurements of _ the @xmath52 plasmon can not determine tubular helicities _ for a given tube diameter . the governing factor for the @xmath52 plasmon must be traced to the in - layer graphitic response as it can be seen in fig . [ ctubeels ] where the loss function for graphite and graphene is also shown ( for in - layer * q * orientation ) at comparable ( to the tubes ) average electron densities ( to within 10 % ) @xcite ( dashed curves in fig . [ ctubeels ] ) . this shows that the loss function of the tubes for parallel * q * in this frequency range is governed by the average - density - dependent part of the in - layer graphitic response . similar plasmon shifts , therefore , can also be expected in other carbon systems with graphene - based structural blocks e.g. multiwalled tubes . the present results outline the significance of the _ @xmath52 plasmon as a key measurable spectroscopic quantity _ which could gauge the _ intertube distances and interactions _ in real samples of carbon nanotubes . in order to gain a more complete understanding of the in - layer graphitic response at small q s and how the latter is influenced by the inter - layer interaction we determined both the loss and dielectric function for various graphene - like geometries i.e. varying the interlayer spacing or , equivalently , the ( c / a)@xmath36 ratio . in these calculations we also looked at the lower - frequency @xmath5 plasmon . bearing in mind the discussion in the previous paragraph , these calculations could then serve as benchmarks for predicting the position of the @xmath52 plasmon in nanotubes as a function of the intertube distances for low q s . ) for graphite and the graphene - like geometries with multiple ( c / a)@xmath36 ratios . only the results without lfe are presented since lfe are negligible in this q range.,width=302 ] the loss and dielectric function for small in - plane * q * ( 0.22 @xmath55 ) is shown in fig . [ grapheels ] for graphite and the graphene geometries with multiple ( c / a)@xmath36 ratios . it can be seen that the peak positions of both plasmons have shifted to lower frequencies when the inter - layer separation is increased . this effect is very pronounced for the @xmath52 plasmon position . these results reaffirm that the total ( @xmath52 ) plasmon is extremely sensitive to the inter - layer interaction for small in - plane q s . this can be explained as follows : with increasing ( c / a)@xmath36 , i.e. interlayer spacing , the system becomes an assembly of nearly isolated graphene sheets . this leads to a decrease of screening ( re[@xmath56 \to 1 $ ] ; see fig . [ grapheels ] ) with the effect that @xmath57 \to im[\varepsilon_{m}]$ ] , namely a coincidence of the loss and absorption functions . since for the in - plane polarization the positions of the absorption peaks do not change with increasing ( c / a)@xmath36 ( see im[@xmath42 in fig . [ grapheels ] ) , then the loss function undergoes important changes , in particular the @xmath52 plasmon . the latter is displaced at a much faster rate to lower frequencies towards the 14 ev peak of the absorption spectrum whereas the @xmath5 plasmon peak is rather insensitive to ( c / a)@xmath36 since it is already located very close to the 0 - 5 ev peak structure of the @xmath39 transitions in @xmath58 $ ] . at present , existing measurements @xcite of the loss spectra of samples of single - wall carbon nanotubes have given a @xmath52 plasmon in the frequency range 2124 ev for small momentum transfer q. our predicted frequency of the @xmath52 plasmon for the case of the distant tubes is within this frequency range ( fig.[ctubeels ] ) . however , a direct comparison of the present results for the loss function of ( 3,3 ) and ( 5,0 ) tubes with the measured loss data is not straightforward . the difficulty stems from two factors , tube diameter and alignment , which have a competing effect on the @xmath52 plasmon position : a ) bulk samples of single - wall tube material possess a mean diameter of 14 @xcite , which is considerably larger to the range of 4 studied in the present work . assuming a common intertube spacing , then larger diameters would give rise to a displacement of the @xmath52 plasmon towards lower frequencies since the diametrically - opposed wall parts of the same tube are facing each other at larger distances , i.e. a situation resembling a stacking of graphene sheets with larger ( c / a)@xmath36 ratios ( see fig . [ grapheels ] ) . for instance , the 5 ( c / a)@xmath36 ratio corresponds to an intersheet separation of 16.8 and the corresponding @xmath52 plasmon peak is at 16 ev . b ) the alignment of the tubes in the samples is not perfect ; therefore it is to be expected that also out - of - plane excitations of graphitic origin will contribute to the response . these excitations @xcite should tend to produce a more diffuse shape for the loss spectrum , heavily dampening the @xmath52 plasmon and shifting the observed peak towards higher frequencies . the type of response described just above can be clearly seen in fig . [ ctubeels2 ] which shows the loss function of the distant ( 3,3 ) tubes for * q * orientation perpendicular to the tube axis . for this orientation in - plane as well as out - of - plane graphene excitations contribute to the tube response . the latter cause the diffuse shape of the loss function ( see inset of fig . [ ctubeels2 ] ) . lfe are now very important and the peak position of the @xmath52 plasmon is at 28 ev . ) of orientation perpendicular to the tube axis . in the inset the loss function of graphene @xcite for 2 ( c / a)@xmath36 and for a similar * q * of orientation perpendicular to the sheet is displayed . continuous and dashed curves denote results obtained with and without lfe , respectively.,width=302 ] for completeness , we show in fig . [ fig9 ] also the calculated eels spectra for small bn nanotubes with momentum transfer @xmath59 along the tube axis . as in the case of carbon tubes , two main features are clearly pronounced : the @xmath5-plasmon at 6 - 7 ev and the high energy collective oscillation ( @xmath60 plasmon ) around 20 ev . the exact position of the peaks depends on the radius and chirality of the tubes and on the intertube distance . in order to compare with experimental eels - spectra on multi - wall bn tubes @xcite , an extrapolation to larger - diameter tubes is needed and the inter - wall interactions have to be taken into account . along the direction of the tube axis . the spectra are calculated with an inter - tube distance of 7.4 . since the spectra with and without lfe are almost indistinguishable , only the calculation without lfe is shown.,width=302 ] in this closing section we should like first to comment on the validity of rpa and tdlda for the description of the optical spectra of the tubes . it is well known that rpa and tdlda often give excellent results for _ loss _ spectra @xcite . however , in this work we have seen that also very good _ absorption _ spectra were obtained for the carbon tubes , even at the rpa level , despite the fact that both rpa and tdlda are known to fail badly in the description of absorption spectra of many bulk materials ( silicon and argon being two representative cases ) @xcite . our explanation in this regard is the following : for the light polarization parallel to the tube axis , the ensuing screening is significant and , therefore , xc effects are damped . for the perpendicular polarization and for the larger inter - wall distance ( @xmath33 5.5 ) the tubes behave essentially like isolated systems , where strong cancellations are known to occur between self - energy corrections and the electron - hole interaction , i.e. between xc effects ( see ref.@xcite ) . the experimental precision is then not high enough ( also due to the almost vanished absorption intensity in the relevant frequency range ) to discern to which extent xc effects should be better described by approximations beyond the tdlda . in view of these considerations , we can be confident regarding the quality of the calculations . on the other hand , the results for the strongly interacting tubes ( smaller intertube distance ) should be regarded as qualitative since the system becomes then more similar to a solid where the cancellations may be more incomplete , and since no direct comparison to experiment is possible , at present , in this case . still , excitonic effects are not included in the calculations . they could play a role in both carbon and bn nanotubes , leading to redistribution of oscillator strength and/or appearence of new peaks in the bandgap ( bound excitons ) . they might be the reason for the anomalous @xmath61 ratio @xcite measured recently for carbon tubes @xcite . we expect a stronger deviation of measured optical spectra from theoretical ones in the case of bn nanotubes . for this large band gap material , two effects will most likely play an important role : quasi - particle corrections will widen the band gap as in the case of hexagonal bn , where a gw calculation has demonstrated a band - gap increase from 4 ev to 5.5 ev @xcite . excitonic effects , in contrast , will lead to isolated states in the band gap or to an overall reduction of the band - gap . to which extent quasi - particle corrections and excitonic effects cancel each other is presently not clear . work along these lines is in progress @xcite . for the role of quasi - particle corrections and excitonic effects in carbon tubes , we refer the reader to ref . @xcite . in conclusion , the present _ ab initio _ calculations of the optical absorption of small - diameter carbon and bn nanotubes give good agreement with the available experimental data . the inclusion of local field effects in the response to a perturbation with perpendicular polarization is necessary for a proper description of the depolarization effect leading to a suppression of the low - energy absorption peaks for both types of tubes . this suppression can also explain recent findings in near - field raman microscopy of single - wall carbon nanotubes @xcite . the proper analysis of the polarization dependence of the absorption cross section is very important in order to describe the surface enhanced raman scattering experiments in isolated carbon nanotubes @xcite . in carbon tubes the position of the first absorption peak strongly varies with the tube indices while in bn tubes the first peak is determined by the band gap of bn and is therefore mostly independent of @xmath0 . for the bn tubes some of the fine - structure which distinguishes tubes of different chirality is only visible in the uv region which gives rise to the hope that this energy regime will be probed in the future . the intertube interaction was also found to be very important . for the carbon tubes this interaction is the governing factor which determines the position of the higher - frequency @xmath52 plasmon . this plasmon , hence , may prove to be a very useful spectroscopic quantity probing intertube interactions and distances in real samples . finally , the corresponding results for the layered constituents graphene and bn sheets revealed that some aspects of the tubular dielectric response can be explained even at a quantitative level from the in - layer and interlayer response of the sheets . this work was supported by the european community research training networks nanophase ( hprn - ct-2000 - 00167 ) and comelcan ( hprn - ct-2000 - 00128 ) , by spanish mcyt(mat2001 - 0946 ) and university of the basque country ( 9/upv 00206.215 - 13639/2001 ) . the computer time was granted by idris ( project no . 544 ) , dipc and cepba ( barcelona ) . the authors also gratefully acknowledge fruitful discussions with thomas pichler and nathalie vast . for an overview over the numerous works on resonant raman spectroscopy of carbon nanotubes see , e.g. , r. saito and h. kataura in _ carbon nanotubes : synthesis , structure , properties , and applications _ , m.s . dresselhaus , g. dresselhaus , and ph . avouris ( editors ) , springer verlag ( 2001 ) . l. wirtz , v. olevano , a.g . marinopoulos , l. reining and a. rubio , in proceedings of _ electronic properties of novel materials : xviith international winterschool _ , ed . h. kuzmany , j. fink , m. mehring and s. roth , world scientific , singapore ( 2003 ) ; and unpublished results . de heer _ et al . _ , science * 268 * , 845 ( 1995 ) ; t. pichler , _ et al . . lett . * 80 * , 4729 ( 1998 ) ; h. kataura _ et al . _ synth . met . * 103 * , 2555 ( 1999 ) ; m. ichida _ jpn * 68 * , 3131 ( 1999 ) ; x. liu _ b * 66 * , 045411 ( 2002 ) . c.l . kane and e.j . mele , phys . lett . * 78 * , 1932 ( 1997 ) ; r. egger and a.o . gogolin , _ ibid . _ , * 79 * , 5082 ( 1997 ) ; c.l . kane , l. balents and m.p.a . fisher , _ ibid . _ , * 79 * , 5086 ( 1997 ) ; r. egger , _ ibid . _ , * 83 * , 5547 ( 1999 ) . lin and k.w.k . shung , phys . b * 50 * , 17744 ( 1994 ) ; s. tasaki , _ et al . _ , _ ibid . _ * 57 * , 9301 ( 1998 ) ; m.f . lin , _ et al . _ , _ ibid . _ * 61 * , 14114 ( 2000 ) ; f.l . shyu and m.f . lin , _ ibid . _ * 62 * , 8508 ( 2000 ) . see for example , w. kohn , rev . * 71 * , 1253 ( 1999 ) ; c. fiolhais , f. nogueira , m. marques ( eds . ) , _ a primer in density functional theory _ ( springer , berlin , 2003 ) ; reviews in modern quantum chemistry : a celebration of the contributions of r.g . parr , k.d . sen , ed(s ) , ( world scientific , 2002 ) ; and references therein . l. hedin and s. lundqvist , in _ solid state physics _ , edited by f. seitz , d. turnbull and h. ehrenreich ( academic , new york , 1969 ) , vol . 23 , p. 1 ; m.s . hybertsen and s.g . louie , phys . lett . * 55 * , 1418 ( 1985 ) ; phys . b * 34 * , 5390 ( 1986 ) ; r.w . godby , m. schlter and l.j . sham , phys . lett . * 56 * , 2415 ( 1986 ) ; phys . b * 37 * , 10159 ( 1988 ) . @xmath10 is expanded in reciprocal lattice vectors @xmath62 and depends on @xmath63 and on the momentum transfer @xmath20 . it describes the change of the density @xmath64 in response to the change of the total potential @xmath65 ( which in turn is composed of the hartree , the exchange - correlation and the external potentials ) : @xmath66 . typically , for the systems and excitation energies treated in this work , it was sufficient to compute bands up to @xmath67 , where @xmath68 is the number of occupied bands . in certain cases , especially for evaluating the loss function , we included more unoccupied bands in the summations . s. g. louie _ et al . _ , ( to be published ) ; and in proceedings of _ electronic properties of novel materials : xviith international winterschool _ , ed . h. kuzmany , j. fink , m. mehring and s. roth , world scientific , singapore ( 2003 ) . calculations are performed for a periodic array of nanotubes with an intertube distance of 14 a.u . the absolute value of @xmath69 scales with the dimension of the employed super - cell , but we are interested only in the relative absorption cross section @xmath70 in arbitrary units . the quasi - one dimensional brillouin zones of the tube are sampled by 20 irreducible * k*-points along the tube axis . a.g . marinopoulos , l. reining , a. rubio and v. olevano ( to be submitted ) . we have used up to 16464 * k*-points for the brillouin - zone sums , and 121 * g*-vectors in the matrices . our results are fully converged with respect to the brillouin zone sampling for frequencies above 1 ev . a proper evaluation of the spectra at lower frequencies for this semi - metal would require an improved treatment of intraband transitions , and is beyond the scope of the present work . model calculations for a linear array of carbon tubes also show an important shift of the higher - frequency plasmon due to intertube interaction : g. gumbs and g.r . aizin , phys . rev . b * 65 * , 195407 ( 2002 ) .
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we present results for the optical absorption spectra of small - diameter single - wall carbon and boron nitride nanotubes obtained by _ ab initio _ calculations in the framework of time - dependent density functional theory .
we compare the results with those obtained for the corresponding layered structures , i.e. the graphene and hexagonal bn sheets .
in particular , we focus on the role of depolarization effects , anisotropies and interactions in the excited states .
we show that already the random phase approximation reproduces well the main features of the spectra when crystal local field effects are correctly included , and discuss to which extent the calculations can be further simplified by extrapolating results obtained for the layered systems to results expected for the tubes .
the present results are relevant for the interpretation of data obtained by recent experimental tools for nanotube characterization such as optical and fluorescence spectroscopies as well as polarized resonant raman scattering spectroscopy .
we also address electron energy loss spectra in the small - q momentum transfer limit . in this case , the interlayer and intertube interactions play an enhanced role with respect to optical spectroscopy .
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the theoretical description of multiphase flows is essentially based on analyzing the reponse of a cloud of dispersed particles of different size ranges in a fluid . these particles constitute dynamic phases and hence a multiphase flow . a widely used multiphase system is a bubble column which is a reactor where a discontinuous gas phase in the form of bubbles , moves relative to a continuous phase . bubble columns have a wide range of applications in chemical industries , biotechnology or in nuclear reactors @xcite , @xcite , @xcite , @xcite , nucreac . the transient behavior is important at the start - up of these systems and its analysis is essential in order to characterize the dynamic performance of the columns . among the phenomena that occur in these systems void wave propagation mechanisms are of great importance since many transient and steady states are controlled by the propagation of these waves and , in this sense among others , the dynamic characterization of multiphase flows is essential for the prevention of instabilities . the ( in)stability of bubbly flows which are characterized by almost uniformly sized bubbles , is usually described in terms of the propagation properties of void fraction and pressure disturbances caused by natural or imposed fluctuations of the rate of air supply @xcite , @xcite , @xcite . bubble size , rise velocity , size distribution and liquid and bubble velocity profile have a direct bearing on the performance of bubble columns . however , most of the time the dispersion devices deliver a dispersed phase with a given size distribution . the importance of the size distributions is only scarcely evaluated , most of the time by direct empirical trials and its influence on the global behavior has still to be studied . actually , to our knowledge and from the theoretical point of view , it has not been yet well established whether the stability of the motion of a swarm of bubbles is different for monodispersed or polydispersed bubble flows . the main objective of this work is to investigate the effects that size polydispersity might produce on the stability of a bubble column . we shall introduce the effect of polidispersity through the drag force in the hydrodynamic equations , using a method based on statistical concepts and on a point - force approximation @xcite . as we shall see below , the corrections on the drag force factor , @xmath1 , due to polydispersity depend only on the first three moments of a given particle size distribution and they also have an effect on several properties of kinematic waves . in particular , we found that size polydispersity enhances the stability of void waves by a factor which varies between @xmath2 as a function of frequency and for a particular type of bubble column . in this way our model predicts effects that might be verified experimentally but this , however , remain to be assessed . to this end the paper is organized as follows . in section 2 we briefly review a hydrodynamical model for bubbly fluids introduced by biesheuvel and gorissen @xcite . next , in section 3 we consider a dispersion of spherical air bubbles of different radi in water and we calculate the effect of size polidispersity on the gain factor , mean bubble velocity , kinematic wave velocities as a function of void fraction , for different wave frequencies . in this section we summarize the main ideas and steps behind the hydrodynamical model for bubbly fluids introduced in ref . the equations of motion for a swarm of bubbles in a bubble column have been derived in the literature by using standard methods of kinetic theory to average over an ensamble or realizations of the flow @xcite , @xcite . in ref . @xcite a dispersion of equally sized air bubbles in a water column where the bubbles are small enough to remain spherical through the whole system , is considered . they assumed that the air can be taken as an incompressible fluid where no mass transfer is allowed between the bubbles and the water , which is assumed to be an incompressible newtonian liquid . the conservation equation for the mean number density of the gas bubbles , @xmath3 , and the conservation equation for the mean bubble momentum , @xmath4 ( kelvin impulse ) , were obtained for this system @xcite , @xmath5 @xmath6 + \mathbf{\nabla } _ { \mathbf{x}}\cdot \left [ n\left ( \frac{4}{3}\pi a^{3}\rho _ { g}\mathbf{v+i}_{l}\right ) \right ] \nonumber \\ & & -\mathbf{\nabla } _ { \mathbf{x}}\cdot \left ( \mathbb{t}_{g}+\mathbb{t}% _ { l}\right ) \nonumber \\ & = & n\mathbf{f}_{d}+n\frac{4}{3}\pi a^{3}(\rho _ { l}-\rho _ { g})\mathbf{g . } \label{2}\end{aligned}\]]@xmath7 is the fluid impulse , @xmath8 and @xmath9 are the fluid stresses ; @xmath10 is the drag force exerted by the fluid on the bubble and @xmath11 stands for the gravity field ; @xmath12 @xmath13 denote , respectively , the mass densities of water and air . @xmath14 stands for the liquid s viscosity . in order to describe the flow parameters of the bubble swarm , eqs . ( [ 1 ] ) and ( [ 2 ] ) should be expressed in terms of the volume fraction of bubbles ( or void fraction ) @xmath15 and their velocity field @xmath16 . following ref . @xcite we assume that the uniform flow of bubbles is along the axial direction of the column with a mean axial rise velocity @xmath17 , therefore , @xmath18 . the effect of hydrodynamic interactions between the bubbles on the mean frictional force may be represented by introducing a function @xmath19 into @xmath17 in the form @xmath20 , @xcite . the magnitude of the terminal velocity , @xmath21 , of a single bubble of radius @xmath22 in a stagnant liquid is given by @xcite @xmath23 , where @xmath24 @xmath25 is the drag force factor and experiments suggest that wallis , @xmath26 . the mean fluid impulse is modelled by @xmath27where @xmath28 takes into account the effect of the hydrodynamic interactions . according to ref . @xcite an expression for @xmath29 that renders reliable results up to large values of @xmath30 is @xmath31 . since in a nonuniform bubbly flow the stress @xmath32 @xmath33 play the role of an effective pressure , they also assume that the _ kinetic _ contribution , @xmath34 , is proportional to the effective density of the bubbles , @xmath35 , and to the mean square of their velocity fluctuations @xmath36 , @xcite . here @xmath37 stands for the limit of closest packaging of a set of spheres and is close to the value 0.62 . thus , @xmath38 . furthermore , if the non - uniformity is the main cause of an additional transfer of bubble momentum and fluid impulse associated with stress , biesheuvel and gorissen @xcite postulate that such a contribution to the stress should be given by the force @xmath39 . therefore , taking into account both contributions to the stress , @xmath40 , where @xmath41 is the one dimensional nonuniform flow velocity and @xmath42 is an effective viscosity . on the other hand , the mean frictional force is enhanced by an effective diffusive flux of bubbles due to their fluctuating motion . this effect is similar to an steady drag force acting upon each one of the bubbles and proportional to the mean number density gradient . therefore , using ( [ 2 ] ) this force is represented by @xmath43 $ ] . substitution of the above expressions into eqs . ( [ 1 ] ) and ( [ 2 ] ) leads to the following closed set of one - dimensional equations of motion for the bubbly flow in a zero volume flux reference frame , @xmath44 @xmath45 + \frac{% \partial } { \partial z}\left [ \rho _ { ef}(\varepsilon ) v^{2}\right ] -\frac{% \partial } { \partial z}\mathbb{t } \nonumber \\ & = & -c_{d}\varepsilon f_{0}\left ( v+\frac{\mu _ { e}(\varepsilon ) } { \varepsilon \rho _ { ef}}\frac{\partial \varepsilon } { \partial z}\right ) -\varepsilon \left ( \rho _ { g}-\rho _ { l}\right ) g. \label{15}\end{aligned}\ ] ] these equations may be rewritten in a laboratory reference frame by considering the mean axial velocity of the dispersion , @xmath46 , defined by @xmath47 . here @xmath48 and @xmath49 are the mean bubble and fluid axial velocity in the laboratory reference frame . note that due to the incompressibility of both , liquid and gas , @xmath46 is only a function of time . therefore @xmath50 and a galileo transformation of eqs . ( [ 14 ] ) and ( [ 15 ] ) gives @xmath51 @xmath52 \nonumber \\ & & + \frac{\partial } { \partial z}\left [ \varepsilon \left ( \rho _ { g}u_{g}+% \frac{1}{2}\rho _ { l}m_{0}\left ( u_{g}-u\right ) \right ) u_{g}\right ] \nonumber \\ & & -\frac{\partial } { \partial z}\left ( -p_{e}+\mu _ { e}\frac{\partial u_{g}}{% \partial z}\right ) -\varepsilon \rho _ { g}\frac{\partial u}{\partial t } \nonumber \\ & = & -c_{d}\varepsilon f_{0}\left ( \left ( u_{g}-u\right ) + \frac{\mu _ { e}(\varepsilon ) } { \varepsilon \rho _ { ef}}\frac{\partial \varepsilon } { % \partial z}\right ) -\varepsilon \left ( \rho _ { g}-\rho _ { l}\right ) g , \label{18}\end{aligned}\]]together with the incompressibility condition @xmath53 consider a quiscent equilibrium state of the dispersion described by @xmath54 . the deviations from this state will be denoted by @xmath55 and @xmath56 . linearization of eqs . ( [ 17 ] ) - ( [ 19 ] ) around the reference state yields the wave - hierarchy equation@xmath57 \nonumber \\ & & = -\left [ \left ( \frac{\partial } { \partial t}+c_{0}\frac{\partial } { % \partial z}\right ) \delta \varepsilon -\nu _ { \varepsilon } \frac{\partial ^{2}\delta \varepsilon } { \partial z^{2}}\right ] \label{20}\end{aligned}\]]with lower and higher - order wave velocities given by @xmath58 and @xmath59 ^{1/2}. \label{22}\]]here @xmath60 and @xmath61 $ ] . the primes ( @xmath62 ) denote derivatives with respect to @xmath15 and evaluated at the unperturbed state @xmath54 . for relatively low radial frequencies the wave propagation is described by a linearized burgers / korteweg - de vries equation@xmath63 \frac{\partial ^{2}\delta \varepsilon } { \partial z^{2}}+\tau _ { \varepsilon } \nu _ { \varepsilon } ( u_{g_{0}}-c_{0})\frac{\partial ^{3}\delta \varepsilon } { % \partial z^{3 } } , \label{25}\]]with a solution @xmath64 where @xmath65 is the frequency of the void wave and @xmath66 -% \frac{\nu _ { \varepsilon } \omega ^{2}\left ( u_{g}-c_{0}\right ) } { c_{0}^{3}}% \left [ \tau _ { \varepsilon } ( c^{+}-c_{0})(c_{0}-c^{-})+\delta _ { \varepsilon } % \right ] . \label{27}\]]in terms of these quantities the so called gain factor , @xmath67 $ ] , where @xmath68 denotes the real part and @xmath69 the distance between two impedance probes in the experiments to measure @xmath70 @xcite . the method developed by tam @xcite uses the concept of randomness of the bubble cloud and derives equations describing the average properties of the fluid motion . these averages are taken over a statistical ensemble of particle configurations . a slow viscous flow past a large collection of spheres of a given size distribution is considered to derive a particle drag formula free from empirical assumptions . the result essentially replaces the disturbance produced by a sphere in low reynolds number flow , by that of a point force located at the centre of the sphere . the correction drag force factor is given by @xmath71 c_{d } , \label{28}\]]where @xmath72 ^{1/2}}{(1 - 3c)}. \label{29}\ ] ] @xmath73 are the moments of the size distribution @xmath74 and @xmath75 . since the terminal velocity of a bubble depends on @xmath76 , it is reasonable to assume that in the polydispersed case @xmath77 should be replaced by @xmath78 . substitution of this asumption into eqs . ( [ 17 ] ) - ( [ 19 ] ) , carrying out the linearization procedure described in the last section and using the explicit expressions of @xmath79 , one can show that these quantities scale as @xmath80 . if these polydispersed quantities are substituted into eq . ( [ 27 ] ) , one obtains an expression for the polydispersed gain factor @xmath81 $ ] . to compare the monodisperesed and polydispersed results on the gain factor , mean bubble velocity , kinematic wave velocities as a function of void fraction , we used the the following parameter values for an air - water bubble column , @xmath82 @xmath83 , @xmath84 @xmath85 , @xmath86 @xmath87 @xmath88 @xmath89 , @xmath90 @xmath89 , @xmath91 @xmath87 @xmath92 , @xmath93 . in fig . 1 we plot @xmath70 and @xmath94 vs. @xmath15 for different frequencies @xmath95 and for a log - normal distribution @xmath74 with average and dispersion @xmath96 , @xmath97 , respectively . , @xmath94 vs. @xmath15 for waves with frequencies of @xmath98 , @xmath99 and @xmath100 . the liquid is stagnant and the parameter values are those given in section 4 . ] note that for values @xmath101 @xmath102 the attenuation rate drops significantly for for instance , the per cent difference defined by @xmath103 @xmath104 . for the range both cases . for instance , for a frequency of @xmath105 this difference ranges from @xmath106 per cent , whereas for a frequency of @xmath100 it varies in the interval @xmath107 per cent . this means that stability is larger in about 23 per cent for the latter case , a change that is significant in bubble reactors @xcite . the quantities @xmath108 and @xmath109 are plotted as functions of @xmath15 in fig . the curve for @xmath110 is always between that for @xmath111 and @xmath112 . according to the whitham stability criterion @xcite , when @xmath113 @xmath114 the uniform flow is unstable . this occurs for both distributions , however , for the monodispersed case it occurs for @xmath115 , whereas for the polydispersed case the system is stable up to a larger value of the void fraction , e.g. @xmath116 . and @xmath117 as functions of the void fraction @xmath15 for the same parameter values as in fig . 1 . ] summarizing , in this work we have analyzed the effects of size polydispersity in several features of the void fraction waves and their stability properties . we found that the presence of a size distribution reinforces the stability of the waves , as shown in figs . 1 and 2 . furthermore , the per cent difference may be quantified by estimating @xmath103 @xmath118 , @xmath119 amounts to a maximum percentual difference of @xmath120 . it is convenient to emphasize once again , that the hydrodynamic model used in this work @xcite is idealized in many aspects . for instance , compressibility and hydrodynamic interactions between bubbles and with the boundaries , have not been taken into account . however , given the complexity of these effects and of the system itself , the simple dimensional model proposed by biesheuvel and gorissen seems to be a good first step in modeling the complex behavior of a bubble column . it also ilustrates how some of the methodology and concepts of kinetic theory and statistical mechanics may be used to deal with complex phenomena in engineering systems . we should also mention that in this work we have assumed an initial polydisperse size distribution and the coalescence of bubbles has not been considered @xcite . however , this remains to be assessed . some other important effects remain to be considered as well , like the bubble - bubble interaction mechanisms.nevertheless the approach followed here by including the influence of the distribution through the drag effects , considering a mean field approach , is an attempt to set a first framework for the bubble size distribution incorporation to further studies . our results reinforce this point of view in the sense that a description of a bubble column based on the concept of randomness of a bubble cloud and average properties of the fluid motion , may be a useful approach that has not been exploited in engineering systems . y. mercadier , contribution ltude des propagations de perturbations de taux de vide dans les coulements disphasiques eau - air bulles . thsis , universit scientifique et medicale et institut national polytechnique des grenoble , france , 1981
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the occurrence of swarms of small bubbles in a variety of industrial systems enhances their performance .
however , the effects that size polydispersity may produce on the stability of kinematic waves , the gain factor , mean bubble velocity , kinematic and dynamic wave velocities is , to our knowledge , not yet well established .
we found that size polydispersity enhances the stability of a bubble column by a factor of about @xmath0 as a function of frequency and for a particular type of bubble column . in this way our modelpredicts effects that might be verified experimentally but this , however , remain to be assessed .
our results reinforce the point of view advocated in this work in the sense that a description of a bubble column based on the concept of randomness of a bubble cloud and average properties of the fluid motion , may be a useful approach that has not been exploited in engineering systems .
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