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within the hierarchical model of the universe , galaxy mergers are thought to be the main formation channel for the build up of massive elliptical galaxies .
these , and fainter , galaxies are known to host a supermassive black hole ( smbh ) at their centre ( magorrian et al .
1998 ; richstone et al . 1998 ; ferrarese & ford 2005 , and references therein ) .
it has therefore been hypothesized that the depleted cores in core - srsic galaxies are due to the action of coalescing black hole binaries which are produced in major , dissipationless ( gas free or `` dry '' ) mergers of galaxies ( e.g. , begelman et al . 1980 ; ebisuzaki et al .
1991 ; milosavljevi & merritt 2001 ; merritt 2006 ) .
indeed , observations have found binary smbhs at kpc separation ( e.g. , ngc 6240 , komossa et al .
2003 ; arp 299 , ballo et al . 2004 ; 0402 + 379 , rodriguez et al . 2006 ;
mrk 463 , bianchi et al .
2008 ) , and slowly more are being found at closer separations ( e.g. , burke - spolaor 2011 ; ju et al .
2013 ; liu et al .
in such a scenario , three - body interactions involving stars and the smbh binary would decay the smbh binary orbit via the slingshot ejection of stars from the centres of the `` dry '' galaxy merger remnant , naturally creating the observed central stellar light ( mass ) deficits in giant elliptical galaxies and bulges ( e.g. , king & minkowski 1966 ; king 1978 ; binney & mamon 1982 ; lauer 1985 ; kormendy 1985 ; crane et al . 1993 ; ferrarese et al .
1994 , 2006 ; lauer et al . 1995 ; byun et al . 1996
; faber et al . 1997 ; carollo et al .
. numerical simulations targeting the evolution of massive black hole binaries have predicted that the central stellar mass deficit , @xmath10 , of a core - srsic galaxy , i.e. , the ejected stellar mass , scales with the mass of the smbh binary and the number of ( equivalent ) major `` dry '' mergers that the galaxy has experienced ( e.g. , milosavljevi & merritt 2001 ; merritt 2006 ) .
other theoretical studies have proposed enhanced mass deficits as large as 5 @xmath15 as a result of additional stellar ejections from repetitive core passages of `` recoiled '' smbhs ( e.g. , boylan - kolchin et al . 2004 ; gualandris & merritt 2008 ) , or due to the actions of multiple smbhs from merging galaxies ( kulkarni & loeb 2012 ) . tight scaling relations involving the structural parameters of both core - srsic and srsic galaxies have been shown to exist ( e.g. , graham 2013 and references therein ) . while these correlations can yield clues to the processes of galaxy formation and evolution ,
the reliability of this approach depends on the robustness of the modelling employed for deriving the structural parameters . although the parameters of the nuker model ( grillmair et al .
1994 ; kormendy et al . 1994 ; lauer et al .
1995 ) are known to vary with the fitted radial extent of the light profile , due to the fact that the straight outer power - law profile of the nuker model fails to capture the curved ( srsic ) surface brightness profiles of galaxies beyond their core region ( graham et al .
2003 ; dullo & graham 2012 ) , a work around has been suggested . applying the nuker model to the light profiles of 120 `` core '' galaxies , lauer et al .
( 2007a ) noted that the nuker model break radii ( @xmath16 ) are only roughly correlated with the galaxy properties .
however , they identified a better correlated parameter @xmath17 ( the `` cusp radius '' ) as a measure of the core size ( carollo et al . 1997a ; rest et al .
dullo & graham ( 2012 ) subsequently showed that this cusp radius closely matches the break radius of the core - srsic model which we employ here ) are typically two times bigger than the core - srsic model break radii which are defined relative to the inward extrapolation of the outer srsic function . ] .
fitting this model additionally enables us to determine a galaxy s global structural parameters such as its luminosity and half light radius , and to measure the central stellar deficit relative to the outer srsic profile .
as noted by graham et al .
( 2003 ) , the issue is not only measuring the core sizes of core - srsic galaxies , but also the misclassification of coreless `` srsic '' galaxies as galaxies having partially depleted cores . using the core - srsic model ,
dullo & graham ( 2012 ) found that 18@xmath18 of their sample of 39 galaxies which were previously alleged to have depleted cores according to the nuker model were actually srsic galaxies with no cores .
although lauer ( 2012 ) subsequently reported that the core identification disagreement between the nuker model and the core - srsic model was only at the level of @xmath19 , dullo & graham ( 2013 , their appendix a.2 ) revisited and confirmed the 18% disagreement . in some cases ,
additional nuclear components show up as an excess relative to the outer srsic profile , yet these components shallow inner profile resulted in the nuker model labelling them as `` core '' galaxies .
in this paper we analyse the 26 suspected core - srsic elliptical galaxies , presenting new light profiles which cover a large radial range @xmath20 .
our sample selection , data reduction , and light profile extraction technique are discussed in section [ sec2 ] . in section [ sec3 ]
we outline our fitting analysis , provide our results , and additionally compare them to those from published works , paying attention to the issue of double , triple and higher srsic model fits . in section [ sec4 ] ,
we present updated structural parameters and scaling relations , including central and global properties , of core - srsic early - type galaxies . in section [ sec5.1 ]
we discuss the connection between the galaxy core size and the black hole mass . in section [ sec5.2 ]
, we discuss the methodology that is applied to derive the stellar mass deficits in the core - srsic early - type galaxies , and in section [ sec5.3 ] we then compare our mass deficits with past measurements .
finally , section [ sec5.4 ] discusses alternative scenarios which have been presented in the literature for generating cores in luminous galaxies .
our main conclusions are summarised in section [ sec6 ] .
we include three appendices at the end of this paper .
the first presents the core - srsic model fits for all 26 galaxies together with a review on two galaxies with complicated structures .
notes on five suspected lenticular galaxies with a bulge plus disc stellar light distribution are given in the second appendix , while the third appendix provides a comparison between this work and modelling by others of common light profiles .
in so doing we highlight a number of issues in the literature today that are important but currently poorly recognised .
@llcccc@ galaxy&type & @xmath21 & d & @xmath22 & hst + & & ( mag)&(mpc)&(km s@xmath23)&filters + ( 1)&(2)&(3)&(4)&(5)&(6 ) + + ngc 0507@xmath24 & s0 & @xmath25 & @xmath26 & 306&f555w + ngc 0584@xmath24 & e@xmath27 @xmath28 & @xmath29 & @xmath30&206&f555w + ngc 0741@xmath24 & e @xmath28 & @xmath31 & @xmath32&291&f555w + ngc 1016@xmath24 & e @xmath28 & @xmath33 & @xmath34&302&f555w + ngc 1399@xmath35 & e @xmath28 & @xmath36 & @xmath37&342&f475w / f814w + ngc 1700@xmath24 & e @xmath28 & @xmath38 & @xmath39&239&f555w + ngc 2300@xmath24 & s0 @xmath28 & @xmath40 & @xmath41&261&f555w + ngc 3379@xmath24 & e @xmath28 & @xmath42 & @xmath43 & 209&f555w + ngc 3608@xmath24 & e @xmath28 & @xmath44 & @xmath45&192&f555w + ngc 3640@xmath24 & e @xmath28 & @xmath46 & @xmath47 & 182&f555w + ngc 3706@xmath24 & e @xmath28 & @xmath48 & @xmath49 & 270&f555w + ngc 3842@xmath24 & e @xmath28 & @xmath50 & @xmath51 & 314&f555w + ngc 4073@xmath24 & cd@xmath27 @xmath28 & @xmath52 & @xmath53&275&f555w + ngc 4278@xmath35 & e @xmath28 & @xmath54 & @xmath55&237&f475w / f850lp + ngc 4291@xmath24 & e @xmath28 & @xmath56 & @xmath57 & 285&f555w + ngc 4365@xmath35 & e @xmath28 & @xmath58 & @xmath59 & 256&f475w / f850lp
+ ngc 4382@xmath35 & s0 @xmath28 & @xmath60 & @xmath61 & 179&f475w / f850lp + ngc 4406@xmath35 & e @xmath28 & @xmath62 & @xmath63 & 235&f475w / f850lp + ngc 4472@xmath35 & e@xmath27 @xmath28 & @xmath64 & @xmath65 & 294&f475w / f850lp
+ ngc 4552@xmath35 & e@xmath27 @xmath28 & @xmath66 & @xmath67&253&f475w / f850lp + ngc 4589@xmath24 & e @xmath28 & @xmath68 & @xmath69 & 224&f555w + ngc 4649@xmath35 & e @xmath28 & @xmath70 & @xmath71&335&f475w / f850lp + ngc 5061@xmath24 & e @xmath28 & @xmath72 & @xmath73 & 186&f555w + ngc 5419@xmath24 & e@xmath27 @xmath28 & @xmath74 & @xmath75&351&f555w + ngc 5557@xmath24 & e @xmath28 & @xmath76 & @xmath77&253&f555w + ngc 5813@xmath24 & s0 @xmath28 & @xmath78 & @xmath79 & 237&f555w + ngc 5982@xmath24 & e @xmath28 & @xmath48 & @xmath80 & 239&f555w + ngc 6849@xmath24 & sb0 @xmath28 & @xmath81 & @xmath82 & 209&f555w + ngc 6876@xmath24 & e @xmath28 & @xmath83 & @xmath84 & 229&f555w + ngc 7619@xmath24 & e @xmath28 & @xmath85 & @xmath86 & 323&f555w + ngc 7785@xmath24 & e @xmath28 & @xmath87 & @xmath88&255&f555w + notes.col .
( 1 ) galaxy name .
instrument : @xmath89 _ hst _ wfpc2 ; @xmath90 _ hst _ acs .
( 2 ) morphological classification from the nasa / ipac extragalactic database ( ned ) except for ngc 3706 and ngc 5813 .
we adopt an elliptical morphology for ngc 3706 based on the results in dullo & graham ( 2013 ) , while for ngc 5813 we adopt an s0 morphology based on the fitting analysis as well as the photometric profile ( section [ sec3 ] , appendices [ apppa ] and [ apppb ] ) .
the superscript @xmath91 shows elliptical galaxies which are classified as disc galaxies in the literature ( appendix [ apppb ] ) .
( 3 ) absolute _ v_-band ( galaxy or bulge ) magnitude obtained from lauer et al .
( 2007b ) .
these magnitudes are corrected for galactic extinction and @xmath92 surface brightness dimming , and adjusted using the distances from col .
the five bulge magnitude are additionally corrected for inclination and internal dust attenuation ( driver et al .
2008 , their table 1 and eqs . 1 and 2 ) .
sources : ( @xmath93 ) tonry et al .
( 2001 ) after reducing their distance moduli by 0.06 mag ( blakeslee et al .
2002 ) ; ( @xmath94 ) from ned ( 3k cmb ) .
( 5 ) central velocity dispersion from hyperleda ( paturel et al .
( 6 ) filters .
as noted above , in this paper we have targeted 26 suspected core - srsic elliptical galaxies from dullo & graham ( 2012 ) using more radially extended light profiles .
we additionally use the data for the five core - srsic galaxies ( ngc 507 , ngc 2300 , ngc 3706 , ngc 4382 and ngc 6849 ) from dullo & graham ( 2013 ) excluding ngc 3607 because of its dusty nuclear spiral which affected the recovery of the structural parameters as detailed there .
updated magnitudes , distances and velocity dispersions pertaining to this combined sample of 31 core - srsic early - type galaxies are presented in table [ tabbb1 ] .
the _ hst _ images for the 26 core - srsic elliptical galaxies , observed with the acs and/or wfpc2 cameras were taken from the public hubble legacy archive ( hla ) .
although both the acs ( plate scale of @xmath95 ) and the wfpc2 ( plate scale of @xmath96 ) cameras have comparable high spatial resolution , the acs wide field channel ( wfc ) has a larger , rhomboidal field - of - view ( fov ) of @xmath97 compared to that of the wfpc2 ccd array ( 160@xmath98@xmath99160@xmath98 l - shaped fov ) .
we therefore prefer the acs / f475w ( @xmath100 sdss @xmath101 ) and acs / f850lp ( @xmath100 sdss @xmath102 ) images when available . for ngc 1399 ,
these were not available , and we instead used the acs / f475w ( @xmath100 sdss @xmath101 ) and acs / f814w ( similar to the johnson - cousins @xmath103-band ) images . for the other galaxies where the acs images were not available we use the wfpc2 images taken in the f555w filter ( similar to the johnson - cousins v - band ) .
table [ tabbb1 ] provides the observation summaries including the programs , instruments and filters used for imaging our sample galaxies . ) .
these colour calibrations are from acs @xmath101- and @xmath102-band magnitudes into the wfpc2 @xmath104-band magnitude except for ngc 1399 . for ngc 1399 , we transform the acs @xmath101- and @xmath103-band data into wfpc2 @xmath105-band data .
our @xmath101- , @xmath102- and @xmath103-band data points each derived here from same major - axis with equal position angle and ellipticity using the iraf / ellipse fit along with the lauer et al .
( 2005 ) @xmath104- and @xmath105-band surface brightness profiles are used for creating the data points in each panel .
each panel shows the least - squares fit and the pertaining equation ( eq ) for the galaxy .
shaded regions show the 1@xmath106 uncertainty on the best fits for only one galaxy in which the error associated with the slope of the least - square fit is large . ] as in dullo & graham ( 2013 ) , we build new , composite light profiles for all the 26 core - srsic elliptical galaxies by combining the lauer et al .
( 2005 ) very inner ( @xmath107 ) deconvolved f555w ( @xmath100 @xmath104-band ) light profiles with our new ( calibrated @xmath104-band ) major - axis light profiles ( see below ) .
we have chosen to use the inner deconvolved light profile from lauer et al .
( 2005 ) so that differences in the core parameters between our works are not attributed to the use of different treatments of the psf , but rather the application of the nuker versus the core - srsic model .
the new outer profiles are extracted using the iraf / ellipse task ( jedrzejewski 1987 ) and cover @xmath108@xmath98 in radius . our data reduction steps , along with the surface brightness profile extraction procedures , are discussed in detail in sections @xmath109 and @xmath110 of dullo & graham ( 2013 ) . in order to match the acs @xmath101 and @xmath102-band data with the very inner lauer et al .
( 2005 ) deconvolved @xmath104-band data , we made color transformations for the six galaxies without @xmath104-band data ( ngc 4278 , ngc 4365 , ngc 4406 , ngc 4472 , ngc 4552 and ngc 4649 ) using eqs.[eqqiii2]@xmath111[eqqiii7 ] which were derived here by applying a least - squares fit to each galaxy s @xmath112 and @xmath113 data ( see also dullo & graham 2013 and references therein for a similar practice ) .
for ngc 1399 , lauer et al .
( 2005 ) published this galaxy s f606w ( roughly @xmath105-band ) light profile , thus we calibrate our acs @xmath101- and @xmath103-band data to the @xmath105-band light profile using eq .
[ eqqiii1 ] .
[ figiii1 ] illustrates these linear fits to the above seven sample galaxies for which the acs wide field channel ( wfc ) images are available , as listed in table [ tabbb1 ] .
@xmath114 @xmath115 @xmath116 @xmath117 @xmath118 @xmath119 @xmath120 accurate sky level subtraction is of critical importance when determining the surface brightness profile of a galaxy at large radii .
the galaxy flux is often just a few percent of the sky background values in the outermost parts .
the automatic hla reduction pipeline subtracts the sky values from the images .
thus , poor sky subtraction is a concern for galaxies which extend beyond the _ hst _ wfpc2 and acs fovs .
however , reliable sky background determination can be done for galaxies with @xmath121 mag arcsec@xmath122 diameters @xmath123 . not only do the bulk of these galaxies lie within the wfpc2 and acs fovs but their `` counts '' at the edges of the wfpc2/acs ccds are about 10% fainter than the typical _ hst _ @xmath104-band sky value @xmath124 mag arcsec@xmath122 ( lauer et al .
eight of the 31 sample galaxies ( ngc 1399 , ngc 3379 , ngc 4365 , ngc 4382 , ngc 4406 , ngc 4472 , ngc 4552 and ngc 4649 ) have major - axis diameter ( at @xmath121 mag arcsec@xmath122 ) @xmath125 ( graham et al .
1998 ; smith et al .
2000 ; trager et al .
fortunately , all these galaxies except two ( ngc 1399 and ngc 3379 ) have published composite ( acs plus ground - based ) data ( ferrarese et al .
2006 ; kormendy et al .
the ground - based data enabled these authors to better constrain the sky level .
for ngc 3379 , schombert & smith ( 2012 ) published an extended ( @xmath126 ) ground - based profile and for ngc 1399 , li et al .
( 2011 ) published a very extended ground - based light profile .
our _ hst_-derived light profiles for all eight extended galaxies are in a good agreement with those past published profiles , suggesting only a small or negligible sky subtraction error by the pipeline . for the remaining galaxies , we additionally checked the pipeline sky subtraction by measuring the sky values at the edges of the wfpc2 and acs chips , i.e. , away from the galaxy and free of contaminating sources .
as expected , the average of the median of the sky values from several 10 @xmath99 10 pixel boxes is very close to zero for most galaxies .
for a few galaxies , we find that the average values are slightly below zero even if these galaxies are well within the fov of the wfpc2 or acs .
therefore , we adjust the background level to zero .
in general , the three - parameter srsic ( 1968 ) @xmath127 model , a generalization of the de vaucouleurs ( 1948 ) @xmath128 model , is known to provide accurate representations to the stellar light distributions of both elliptical galaxies and the bulges of disc galaxies over a large luminosity and radial range ( e.g. , caon et al .
1993 ; donofrio et al .
1994 ; young & currie 1994 ; andredakis et al .
1995 ; graham et al .
however , because the @xmath128 model was very popular , even referred to by many as the @xmath128 law , saglia et al .
( 1997 ) attempted to explain the observed @xmath127 light profiles as the sum of @xmath128 models and exponential discs . while this approach had some merit , in that pressure - supported elliptical galaxies are becoming increasingly rare , and in fact many intermediate luminosity early - type galaxies possess rotating discs ( e.g. , graham et al .
1998 ; emsellem et al . 2011
; scott et al . 2014 ) , we now know that lenticular galaxies are very well described by an @xmath127 bulge plus an exponential disc which can often additionally contain a bar and/or a lens ( e.g. laurikainen et al .
2013 , and references therein ) . the srsic model s radial intensity distribution can be written as @xmath129 , \label{eqiii8}\ ] ] where @xmath130 is the intensity at the half - light radius @xmath8 .
the variable @xmath131 , for @xmath132 ( e.g.,caon et al .
1993 ) , is coupled to the srsic index @xmath94 , and ensures that the half - light radius encloses half of the total luminosity .
the luminosity of the srsic model within any radius @xmath105 is given by @xmath133 where @xmath134 is the incomplete gamma function and @xmath135 .
the review by graham & driver ( 2005 ) describes the srsic model in greater detail .
the srsic model fits the surface brightness profiles of the low- and intermediate - luminosity ( @xmath136 mag ) spheroids all the way to the very inner region .
although , additional nuclear components are often present in these galaxies and require their own model ( e.g. , graham & guzmn 2003 ; ct et al . 2006 ; den brok et al . 2014 ) . on the other hand ,
the inner light profiles of luminous ( @xmath137 mag ) spheroids deviate downward from the inward extrapolation of their outer srsic model fits ( graham et al .
2003 ; trujillo et al . 2004 ; ferrarese et al .
2006 ) . in order to describe such galaxies light distributions , graham et al .
( 2003 ) introduced the core - srsic model which is a combination of an inner power - law with an outer srsic model .
this six - parameter model is defined as @xmath138^{\gamma /\alpha } \exp \left[-b\left(\frac{r^{\alpha}+r^{\alpha}_{b}}{r_{e}^{\alpha } } \right)^{1/(\alpha n)}\right ] , \label{eqq10}\ ] ] with @xmath139 . \label{eqq11}\ ] ] @xmath140 is intensity measured at the core break radius @xmath141 , @xmath142 is the slope of the inner power - law profile , and @xmath143 controls the sharpness of the transition between the inner power - law and the outer srsic profile .
@xmath8 represents the half - light radius of the outer srsic model , and the quantity @xmath144 has the same meaning as in the srsic model ( eq . [ eqiii8 ] ) .
the total core - srsic model luminosity ( trujillo et al . 2004 ; their eq .
a19 ) is @xmath145 fig .
[ figa1 ] shows our core - srsic model fit to the underlying host galaxy , major - axis , light distributions for all 26 galaxies .
the fit residuals together with their root - mean - square ( rms ) values are given for each galaxy in appendix [ apppa ] .
we find that the light profile for one of these suspected elliptical galaxies ( ngc 5813 ; see appendices [ apppa ] and [ apppb ] ) is better described with a core - srsic bulge plus an exponential disc model , suggesting an s0 morphology as discussed later .
we note that this galaxy s large - scale disc had negligible contribution to the dullo & graham ( 2012 ) @xmath100@xmath147 light profile fit . further , in agreement with dullo & graham ( 2012 ) , we also detect additional nuclear light components ( i.e. , agn or nuclear star clusters ) on the top of the underlying core - srsic light distributions in six galaxies ( ngc 741 , ngc 4278 , ngc 4365 , ngc 4472 , ngc 4552 , and ngc 5419 )
. we account for these nuclear light excesses using a gaussian function .
dullo & graham ( 2013 ) presented the core - srsic(+exponential ) model fits , along with the fit parameters , to five galaxies ( ngc 507 , ngc 2300 , ngc 3706 , ngc 4382 and ngc 6849 ) . table [ tabbb2 ] presents the best fit structural parameters for the full ( 26 + 5=)31 galaxy sample obtained by applying our adopted models to the @xmath104-band , major - axis , light profiles probing large radial ranges ( @xmath148 ) .
in general , from appendix [ apppa ] it is apparent that the main body of luminous ellipticals can be very well described with the core - srsic model .
our fits yield a median root - mean - square ( rms ) residual of @xmath149 mag arcsec@xmath122 . out of the full galaxy sample
, only two elliptical galaxies ( ngc 4073 and ngc 6876 ) reveal complicated structures , as such their light profiles , discussed in appendix [ apppa ] , are somewhat poorly matched by the core - srsic model . ,
table 2 , @xmath104-band ) with previously published break radii measurements ( @xmath150 ) from ( i ) trujillo et al .
( 2004 , @xmath105-band , stars ) , ( ii ) lauer et al .
( 2005 , @xmath104-band , open ellipses ) , ferrarese et al .
( 2006 , g - band , blue ellipses ) , richings et al .
( 2011 , @xmath103- and @xmath151-bands , triangles ) and dullo & graham ( 2012 , @xmath104-band , red circles for ellipticals and blue disc symbols for s0s ) .
we also compare the kormendy et al .
( 2009 ) inner most srsic model fitting radii ( @xmath152 ) and our break radii for six common galaxies ( squares).the geometric - mean break radii from ferrarese et al .
( 2006 ) were converted into semi - major axis break radii using their published galaxy ellipticities at @xmath1 .
the lauer et al .
( 2005 ) break radii came from their nuker model fits , while all remaining works mentioned above applied the core - srsic model for measuring the galaxies break radii .
the break radii from each model is the radius where each model has it s maximum curvature .
i.e. where the second derivative of the model s intensity profile is a maximum .
the solid line indicates a one - to - one relation while the dashed line is the @xmath153 = 2 @xmath154 relation .
a representative error bar is shown at the bottom of the panel . ] for three sample elliptical galaxies ngc 1700 , ngc 3640 and ngc 7785 , while their light profiles are well fit by the core - srsic model with reasonable rms residuals @xmath155 mag arcsec@xmath122 , we find that their cores are unusually small for their brightnesses .
the core - srsic model yields break radii of @xmath156 and @xmath157 for ngc 1700 , ngc 3640 , and ngc 7785 , respectively .
not surprisingly , in section [ sec4 ] , it can be seen that these three questionable cores are outliers in several galaxy scaling relations involving @xmath1 , @xmath106 , @xmath4 , and @xmath8 and as such they have not been included in the regression analysis . for the interested reader ,
these galaxies are discussed in more detail in appendix [ apppqes ] .
@lllcccccccccccc@ galaxy&type&@xmath158&@xmath159 & @xmath160 & @xmath1 & @xmath161&@xmath143&@xmath94&@xmath8&@xmath8&@xmath162&@xmath163&@xmath164 + & & & & ( arcsec)&(pc)&&&&(arcsec)&(kpc)&(mag)&&(arcsec ) + ( 1)&(2)&(3)&(4)&(5)&(6)&(7)&(8)&(9)&(10)&(11)&(12)&(13)&(14 ) + + ngc 0507@xmath165 & s0&16.16 & 16.38 & 0.33 & 102 & 0.07 & 5 & 2.2 & 5.3 & 1.65 & & 21.03 & 27.69 + ngc 0584 & e@xmath27&13.81 & 14.61 & 0.21 & 21 & 0.47 & 5 & 6.6 & 112.5 & 11.25 & + ngc 0741 & e&16.83 & 17.52 & 0.76 & 267 & 0.19 & 5 & 7.4 & 53.0 & 18.60 & 22.1 + ngc 1016 & e&16.35 & 17.00 & 0.48 & 204 & 0.15 & 2 & 5.2 & 41.7 & 17.79 + ngc 1399@xmath166&e&15.45 & 16.36 & 2.30 & 202 & 0.11 & 2 & 5.6 & 36.6 & 3.22 + ngc 1700 ? & e&13.34 & 13.38 & 0.04 & 11 & 0.19 & 5 & 6.1 & 32.0 & 8.23 + ngc 2300@xmath165&s0 & 16.23 & 16.61 & 0.53 & 70 & 0.08 & 2 & 2.2 & 7.7 & 1.02 & & 20.39 & 21.08 + ngc 3379 & e&14.76 & 15.76 & 1.21 & 102 & 0.19 & 2 & 5.9 & 50.2 & 4.21 + ngc 3608 & e&14.56 & 15.14 & 0.23 & 24 & 0.29 & 5 & 6.4 & 68.7 & 7.28 + ngc 3640 ? & e&14.80 & 14.72 & 0.03 & 4 & -0.01 & 5 & 3.5 & 28.0 & 2.99 + ngc 3706@xmath165&e & 14.15 & 14.16 & 0.11 & 24 & -0.02 & 10 & 6.4 & 42.1 & 9.18 & + ngc 3842 & e&16.73 & 17.42 & 0.72 & 315 & 0.19 & 5 & 6.9 & 102.4 & 45.17 + ngc 4073 & cd@xmath27&16.51 & 16.46 & 0.22 & 90 & -0.06 & 10 & 6.1 & 141.6 & 58.61 & & + ngc 4278 & e&15.07 & 15.84 & 0.83 & 52 & 0.22 & 5 & 3.8 & 20.2 & 1.25 & 19.4 & + ngc 4291 & e&14.87 & 15.14 & 0.30 & 36 & 0.10 & 5 & 4.4 & 13.6 & 1.64 + ngc 4365 & e&16.14 & 16.50 & 1.21 & 127 & 0.00 & 2 & 4.8 & 47.3 & 4.97 & 20.2 + ngc 4382@xmath165&s0&14.82 & 15.01 & 0.27 & 24 & 0.07 & 5 & 2.7 & 11.1 & 0.99 & & 19.50 & 35.07 + ngc 4406 & e&15.86 & 15.97 & 0.76 & 61 & 0.00 & 5 & 5.5 & 145.2 & 11.62 + ngc 4472 & e@xmath27&16.18 & 16.34 & 1.21 & 108 & -0.02 & 2 & 3.0 & 48.8 & 4.34 & 22.0 + ngc 4552 & e@xmath27&14.91 & 15.03 & 0.38 & 17 & 0.03 & 10 & 4.4 & 29.7 & 1.31 & 20.6 + ngc 4589&e&14.80 & 15.33 & 0.20 & 27 & 0.30 & 5 & 5.6 & 70.8 & 9.56 + ngc 4649&e&15.75 & 16.70 & 2.51 & 241 & 0.21 & 2 & 3.6 & 62.8 & 6.02 + ngc 5061&e&13.70 & 14.09 & 0.22 & 34 & 0.16 & 5 & 8.4 & 68.44 & 10.81 + ngc 5419&e@xmath27&17.35 & 17.53 & 1.43 & 416 & -0.06 & 2 & 5.6 & 55.0 & 16.01 & 19.9 +
ngc 5557&e&15.05 & 15.46 & 0.23 & 51 & 0.19 & 5 & 4.6 & 30.2 & 6.80 + ngc 5813 & s0&16.04 & 16.11 & 0.35 & 51 & -0.10 & 2 & 2.8 & 7.1 & 1.02 & & 20.30 & 31.28 + ngc 5982&e&15.22 & 15.48 & 0.25 & 51 & 0.09 & 5 & 4.3 & 26.8 & 5.45 + ngc 6849@xmath165 & sb0&16.33 & 16.67 & 0.18 & 69 & 0.20 & 5 & 3.2 & 7.8 & 2.98 & & 20.72 & 16.93 + ngc 6876&e&17.00 & 16.98 & 0.45 & 119 & 0.00 & 10 & 5.9 & 250.0&65.8 + ngc 7619&e&15.41 & 15.93 & 0.49 & 109 & 0.16 & 5 & 7.2 & 72.2 & 16.23 + ngc 7785?&e&14.98 & 14.76 & 0.03 & 5 & 0.00 & 10 & 4.9 & 55.1 & 12.63 + notes.structural parameters from fits to the @xmath104-band major - axis surface brightness profiles ( appendix [ apppa ] ) .
the superscript + indicates that we use an @xmath105-band surface brightness profile instead of a @xmath104-band surface brightness profile for ngc 1399 .
the superscript * shows ( 4s0s and 1e ) galaxies for which the fit parameters are taken from dullo & graham ( 2013 ) .
a `` ? '' is used to indicate three galaxies with questionable core sizes .
( 1 ) galaxy name
( 2 ) adopted morphological classification .
the superscript @xmath91 shows elliptical galaxies which are classified as disc galaxies in the literature ( appendix [ apppb ] ) .
( 3)-(11 ) best - fit parameters from the core - srsic model , eq .
[ eqq10 ] . col .
( 12 ) central point source apparent magnitude .
( 13 ) disc central surface brightness . col .
( 14 ) disc scale length .
the surface brightnesses @xmath167 , @xmath168 and @xmath169 are in units of mag arcsec@xmath122 . ) with previous srsic values ( @xmath170 ) from ( i ) trujillo et al .
( 2004 , stars ) , ( ii ) ferrarese at al .
( 2006 , blue ellipses ) , richings et al .
( 2011 , triangles ) and dullo & graham ( 2012 , red circles for ellipticals and blue disc symbols for s0s ) .
68% of the data resides within -30% and + 25% of perfect agreement , and the outliers are explained and accounted for in section 3.3 . ] here we illustrate two diagrams comparing our values of @xmath1 and @xmath94 with those from similar studies in the literature .
the agreement is generally good .
we have gone to some effort to identify and explain all notable disagreements with past studies .
we have three core - srsic galaxies ( ngc 4291 , ngc 5557 , and ngc 5982 ) in common with trujillo et al .
( 2004 ) who used a radial extent of @xmath100@xmath171 .
ngc 1700 is also an overlapping galaxy but its light profile used by trujillo et al .
( 2004 ) extends from @xmath172 @xmath173 to @xmath174 , thus they did not detect the questionably small core ( @xmath175 ) that we potentially measure here . with the exception of ngc 1700 , classified as a srsic galaxy by trujillo et al .
( 2004 ) , there is an excellent agreement between our break radii and those from trujillo et al .
( 2004 ) , see fig .
[ figiii4 ] .
their srsic indices are also consistent for all four galaxies in common with our study , i.e. including ngc 1700 ( fig . [ figiii5 ] ) .
there are six core - srsic galaxies ( ngc 4365 , ngc 4382 , 4406 , ngc 4472 , ngc 4552 and 4649 ) in common with ferrarese et al .
( 2006 ) who used a radial extent of @xmath100@xmath171 .
for these galaxies , the ferrarese et al .
( 2006 ) geometric - mean , @xmath101-band , break radii were taken and converted to major - axis values .
we prefer their g - band than the z - band data as it more closely matches our @xmath104-band data .
the agreement between their break radii and our measurements are good except for three galaxies ( ngc 4382 , ngc 4552 , and ngc 4472 ) .
the most discrepant ( by more than 100% ) is the s0 galaxy ngc 4382 but this is because it was modelled with a core - srsic+exponential model by dullo & graham ( 2013 , their fig .
ferrarese et al .
( 2006 ) treated this galaxy as a single component system , and thus fit the bulge+disc light with just a core - srsic model , resulting in a systematically higher @xmath1 and @xmath94 value ( figs .
[ figiii4 ] and [ figiii5 ] , see also dullo & graham 2012 , their fig .
the core - srsic stellar light distributions of the remaining two elliptical galaxies . ]
ngc 4552 and ngc 4472 have a broad and an intermediate ( inner core)-to-(outer srsic ) transition region , respectively , which are well described by the core - srsic model @xmath176 and @xmath177 values ( fig .
[ figa1 ] ) .
this in contrast to ferrarese et al s . sharp transition ( @xmath178 ) core - srsic model fits which poorly match these two galaxies transition regions as can be seen by the systematic bump in their fit residuals . due to this
, we find a 70% discrepancy between the ferrarese et al .
( 2006 ) break radii and ours for ngc 4552 and ngc 4472 , much bigger than the @xmath179 uncertainty range of 10% quoted in dullo & graham ( 2012 ) . however , dullo & graham ( 2012 , their fig . 7 ) already discussed the source of this discrepancy for ngc 4552 . omitting the s0 galaxy ngc 4382 , the agreement between the srsic indices of ferrrarese et al .
( 2006 ) and ours is generally good .
we note that @xmath180 of the data in fig .
[ figiii5 ] have a srsic index ratio within -30% and 25% of perfect agreement .
this is in fair agreement with typical uncertainties of 25% reported for the srsic index ( e.g. , caon et al .
1993 ) and is slightly better than the allen et al .
( 2006 ) @xmath179 uncertainty range of @xmath181 36% .
it should be remembered that using different filters as well as modelling minor , major and geometric - mean axis profiles can yield different @xmath94 values for a galaxy ( e.g. , caon et al .
1993 ; ferrari et al . 2004
; kelvin et al . 2012 ) . with past measurements @xmath182 from ( i ) dullo & graham ( 2012 , filled circles and the disc symbols are for their core - srsic ellipticals and s0s , respectively ) and ( ii ) lauer et al .
( 2007a , ellipses ) .
these surface brightnesses are where the respective intensity profile models have the maximum value of their second derivative , i.e. where the curvature of the models is greatest . ]
we have six core - srsic galaxies ( ngc 3379 , ngc 3608 , ngc 4278 , ngc 4472 , ngc 4552 and ngc 5813 ) in common with richings et al .
( 2011 ) whose data extended to @xmath183 .
their break radii for these common galaxies agree with ours except for ngc 5813 .
ngc 5813 is a similar case to that of the s0 galaxy ngc 4382 noted above ; it has a core - srsic bulge+exponential disc light distribution which was modelled using only the core - srsic model by richings et al .
( 2011 ) , thus they measured larger @xmath1 and @xmath94 values .
our srsic indices agree within 25% with richings et al .
( 2011 ) for half of the six core - srsic galaxies in common , but for the remaining half ( ngc 3379 , ngc 4552 plus the s0 ngc 5813 ) there is more than a 40% discrepancy . for ngc 3379 and ngc 4552 , the origin of this discrepancy appears to be the @xmath100@xmath147 _
nic2 f160w light profiles mag arcsec@xmath122 ) major - axis diameter of @xmath100@xmath184 , extend beyond the nicmos nic2 cdd . ]
used by richings et al .
. their profiles for these two galaxies may be too limited in radius for the core - srsic model to capture the actual galaxy light distributions .
finally , we note that the elliptical galaxy ngc 5982 is also in common with richings et al .
( 2011 ) who classified it as a srsic galaxy based on their @xmath185 srsic model fit to the @xmath100@xmath147 nicmos nic2 f160w profile .
it seems that richings et al .
( 2011 ) might have missed the core with their @xmath185 srsic fit to this large elliptical galaxy with @xmath186 km s@xmath23 and @xmath187 mag .
kormendy et al . (
2009 ) adopted graham et al s .
( 2003 ) logic of defining a core as a deficit in light relative to the inward extrapolation of a spheroid s outer srsic profile , but they fit the major - axis light profiles of their core - srsic galaxies using only the srsic model .
they advocate fitting the srsic model over the radius range where it fits well by eye and distinguishing the core region in a subjective manner .
this exercise assumes no transition region between the inner core and the outer srsic profile , but actual galaxy profiles can have a broad transition region . as with the nuker model ,
the `` break radius '' is not the outermost boundary of this transition region controlled by the parameter @xmath143but the mid - point of the transition .
the outer edge of the transition region is very hard to judge by eye , and highly subjective .
we have six core - srsic galaxies ( ngc 4365 , ngc 4382 , ngc 4406 , ngc 4472 , ngc 4552 and ngc 4649 ) in common with kormendy et al . ( 2009 ) .
it is worth comparing the inner most srsic model fitting radius ( @xmath188 ) from kormendy et al .
( 2009 ) with our break radius ( @xmath1 ) for these six overlapping galaxies : ngc 4365 , ngc 4382 , ngc 4406 , ngc 4472 , ngc 4552 and ngc 4649 . as shown in fig .
[ figiii4 ] , for each core - srsic galaxy in common with kormendy et al .
( 2009 ) , their @xmath188 are much further out than ours and part of the explanation likely arises from their method of not measuring the actual `` break radii '' .
fitting the core - srsic model provides this radius , the extent of the transition region , and the central flux deficit from within the outer - edge of the transition region . figs .
[ figiii4 ] and [ figiii5a ] additionally compare our break radii and break surface brightnesses , respectively , with those from lauer et al .
( 2005 ) and dullo & graham ( 2012 ) for all 31 galaxies ( including the three with questionable core in appendix [ apppqes ] ) .
it is important to note that lauer et al .
( 2005 ) fit the nuker model to their @xmath147 galaxy light profiles , while dullo & graham ( 2012 ) re - modelled these using the core - srsic model .
our new break radii , determined from spatially extended ( @xmath20 ) light profiles , are in excellent agreement with those from dullo & graham ( 2012 ) , i.e. , within the uncertainty range , except for ngc 1700 and ngc 2300 .
ngc 1700 is an elliptical galaxy with a questionably small core mentioned earlier ( see section [ sec4 ] and section [ sec8.1.1 ] for further details ) , while for the s0 galaxy ngc 2300 , as noted in dullo & graham ( 2013 ) , the contribution of the disc light to the @xmath189 light profile modeled by dullo & graham ( 2012 ) resulted in a bigger break radius and srsic index .
given the remarkable agreement between the core - srsic break radii of dullo & graham ( 2012 ) and those from their model - independent estimates ( their fig .
11 ) , it implies that our break radii from this work ( table [ tabbb2 ] ) are also in a very good agreement with the model - independent radii where the slope of the logarithmic profile equals -1/2 ( carollo et al .
in addition , as can be seen in fig .
[ figiii5a ] , our new break surface brightnesses fully agree with those from dullo & graham ( 2012 ) . on the other hand , in line with previous core - srsic works , we find the nuker model break radii are larger ( fig . [ figiii4 ] ) .
the nuker break radii ( e.g. , lauer et al .
2005 , 2007a , b ; krajnovi et al .
2013 ) are on average two times bigger than our core - srsic break radii . in estimating larger break radii ,
the nuker model consequently estimates the associated surface brightness up to 2 mag arcsec@xmath122 fainter ( fig .
[ figiii5a ] ) .
hopkins et al .
( 2009a , b ) fit the surface brightness profiles of both srsic and core - srsic elliptical galaxies using a double srsic model .
they claimed that these galaxies outer component is an old spheroid ( with @xmath190 ) formed by the violent relaxation of pre - existing stars from a merger event while their inner component was `` excess light '' formed from a dissipative starburst produced by the same `` wet '' ( gas - rich ) merger event ( e.g. , hernquist et al .
1993 ; mihos & hernquist 1994 ) .
while this reasonable scenario sounds plausible , we point out two concerns . first , the lower luminosity early - type ( srsic ) galaxies ( @xmath191 mag ) tend to have fast - rotating , outer exponential discs ( emsellem et al .
2011 ; krajnovi et al .
2013 ) , rather than old , outer spheroid - like components .
second , the higher luminosity ( core - srsic ) elliptical galaxies have a central deficit of light rather than an excess , and are thought to be formed from dry merger events ( e.g. , faber et al .
the violent relaxation simulations yield @xmath190 ( e.g. , van albada 1982 ; mcglynn 1984 ) and therefore can not account for luminous elliptical galaxies with @xmath192 built from dry mergers .
in addition , they do not explain how low - luminosity elliptical galaxies with @xmath193 are made , nor why these galaxies follow the same @xmath194 relation as the massive ellipticals with @xmath195 .
the hopkins et al .
( 2009a , b ) galaxy sample included lenticular galaxies ( ngc 507 , ngc 1400 , ngc 2778 , ngc 4382 , ngc 4459 , ngc 4476 , ngc 5813 and ngc 4515 ) which were thought to be ellipticals and modelled using an inner srsic model plus an outer srsic model with @xmath196 .
our fits in fig .
[ figa1 ] and those in dullo & graham ( 2013 ) show that ngc 507 , ngc 4382 and ngc 5813 are core - srsic lenticular galaxies that are well described by a core - srsic bulge plus an exponential disc model with very small rms residuals of 0.027 mag arcsec@xmath122 , 0.016 mag arcsec@xmath122 and 0.015 mag arcsec@xmath122 , respectively .
laurikainen et al .
( 2010 , 2011 ) have also detected a weak nuclear bar in the unsharp mask image of ngc 507 .
further , all of our core - srsic elliptical galaxies ( table [ tabbb1 ] ) , except for ngc 1016 , ngc 3706 and ngc 4073 , are in common with hopkins et al .
( 2009b ) . as discussed in section [ sec3.2 ] , these galaxies are well described by the core - srsic model , which can be seen from the residual profiles ( fig .
[ figa1 ] ) which also have small rms residuals @xmath100 0.045 mag arcsec@xmath122 .
this can be compared to the larger rms residuals given by hopkins et al .
( 2009b , their figs
. 10 - 14 ) from their double srsic model fits for six core - srsic galaxies in common with our sample ( ngc 4365 , ngc 4382 , ngc 4406 , ngc 4472 , ngc 4552 and ngc 4649 ) .
that is , with better fits , we have shown that these galaxies have a central deficit of light , in accord with the dry merging scenario involving supermassive black holes . in stark contrast , hopkins et al .
( 2009b ) argued that these galaxies support a wet merger scenario .
similarly , dhar & williams ( 2012 ) argued that both srsic and core - srsic galaxies can be represented well by two or three `` dw '' functions which are the 2d projections of the 3d einasto density model ( dhar & williams 2010 ) .
as in hopkins et al .
( 2009a , b ) , all their sample galaxies , including the six common core - srsic galaxies ( ngc 4365 , ngc 4382 , ngc 4406 , ngc 4472 , ngc 4552 and ngc 4649 ) , are presented as having an inner `` extra light '' component that has a half - light radius @xmath8 @xmath197 kpc .
this is somewhat similar to the ground - based work by huang et al .
( 2013 ) who argued that elliptical galaxies comprise three distinct components an inner ( @xmath198 kpc ) component , a middle ( @xmath199 kpc ) component and an outer @xmath200 kpc envelope which are all represented by srsic models with @xmath201 , at odds with the traditional picture of violent relaxation producing @xmath202 profiles .
in contrast , the fits from our study show that the inner @xmath100 kpc of `` ellipticals '' ( excluding the depleted core ) are not disconnected from , but are rather the simple extensions of , their outer regions .
of course disturbed , unrelaxed galaxies , especially those with peculiar morphology , wo nt be well described by a single core - srsic model .
these particular galaxies may well appear to have multiple ( srsic ) spheroidal components .
in addition to a handful of s0 galaxies ( ic 2006 , ngc 4697 ) , a handful of barred s0 galaxies ( ic 4329 , ngc 6673 ) , and a handful of unrelaxed peculiar galaxies ( ngc 2305 , ngc 4976 ) , huang et al .
( 2013 ) included 15 cd galaxies ( ic 1633 , ic 2597 , ic 4765 , ic 4797 , ngc 596 , ngc 1172 , ngc 1339 , ngc 1427 , ngc 3087 , ngc 4696 , ngc 4786 , ngc 6909 , ngc 6958 , ngc 7192 , ngc 7796 ) in their galaxy sample .
the tell - tale signature of a fit which has failed to fully capture the curvature in the radial stellar distribution is evidenced by the pattern in the residual profile .
this can be seen in , for example , fig .
36 from huang et al .
( 2013 ) , where , from 150@xmath98 to beyond 400@xmath98 there is a systematic hump in their residual profile for eso 185-g054 .
the artificial ring in their residual image also reveals that the fit is not optimal .
this is because the 9 parameters of their three fitted srsic models have collectively managed to approximate the light profile out to a radius @xmath203 . in this instance , the use of a fourth , extended srsic model would have enabled a better fit to the outer half of the light profile , as in the case of es0 221-g026 which huang et al .
( 2013 ) fit with 4 srsic models .
however , this does not mean that the galaxy eso 185-g054 actually has 4 components , simply that if one uses enough parameters then one can better approximate the light profile . rather than applying a multitude of srsic components
, we advocate trying to establish which components are real and then applying the appropriate function , as done by , for example , lsker et al .
( 2014 ) . to continue this important point , but avoid creating too much of a distraction in the main text , in appendix [ apppc ] we provide comparisons between our core - srsic modelling of our galaxy sample and some other recent works which obtained dramatically different results for galaxies with depleted cores .
we explore several galaxy structural parameter relations for 28 core - srsic early - type galaxies with carefully acquired core - srsic parameters .
the good agreement between the structural parameters from this work and those form our initial study using @xmath100@xmath147 profiles ( dullo & graham 2012 ) suggests that the different correlations that will be shown here agree with those of dullo & graham ( 2012 ) . fig .
[ figiii6 ] shows the relation between the core - srsic break radius @xmath1 ( table [ tabbb2 ] ) and central galaxy properties including ( a ) the core - srsic model s central @xmath104-band surface brightness @xmath204 ( table [ tabbb2 ] ) , ( b ) the break surface brightness @xmath205 ( table [ tabbb2 ] ) and ( c ) the velocity dispersion @xmath106 ( table [ tabbb1 ] ) . the solid and dashed lines shown in each panel of figs . [ figiii6 ] to [ figiii9 ] are two distinct linear regression fits obtained with and without the three elliptical galaxies ( ngc 1700 , ngc 3640 and ngc 7785 ) with questionably small cores ( @xmath206 ) .
we note that all the relations given in table [ tabbb30 ] are for the galaxy data without these three questionable galaxies . using the ordinary least squares ( ols ) bisector regression from feigelson & babu ( 1992 )
, a fit to the @xmath1 and @xmath204 data gives @xmath207 , while applying the bisector fit to @xmath1 and @xmath205 yields @xmath208 , and the bisector fit to @xmath1 and @xmath106 yields @xmath209 ( table [ tabbb30 ] ) .
@llccccccccc@ relation&ols bisector fit&@xmath210 ( vertical scatter ) + + @xmath211 & @xmath212&0.24 dex + @xmath213&@xmath214&0.18 dex + @xmath215&@xmath216&0.29 dex + @xmath217&@xmath218&0.30 dex + @xmath219&@xmath220&0.43 dex + @xmath221&@xmath222&0.45 dex + @xmath223&@xmath224&0.28 + @xmath225 & @xmath226&0.80 + @xmath227&@xmath228&0.80 + @xmath229&@xmath230&1.17 + @xmath231 ( @xmath232 derived @xmath4 for 23 galaxies & @xmath233&0.27 dex + plus 8 direct @xmath4 masses ) & & + @xmath231 ( @xmath234 derived @xmath4 for 23 galaxies & @xmath235&0.27 dex + plus 8 direct @xmath4 masses ) & & + similar to fig .
[ figiii6 ] , fig .
[ figiii7 ] reveals that the core - srsic break radii @xmath1 correlate with global galaxy properties such as ( a ) the @xmath104-band absolute magnitude @xmath21 ( table [ tabbb1 ] ) and ( b ) the effective radius @xmath8 ( table [ tabbb2 ] ) .
the bisector fit gives the near - linear relation between @xmath1 and @xmath21 as @xmath236 , while the fitted relation for the @xmath1 and @xmath8 data is @xmath237 ( table [ tabbb30 ] ) .
of all the relations ( figs .
[ figiii6 ] and [ figiii7 ] , table [ tabbb30 ] ) , the weakest correlation with a pearson correlation coefficient of @xmath238 is between @xmath1 and @xmath8 , while the for the remaining relations @xmath239 .
we note that , intriguingly , the bulges seem to reveal a systematic trend in the @xmath211 , @xmath213 ( figs [ figiii6]a and [ figiii6]b ) and @xmath219 ( fig .
[ figiii7]b ) diagrams , although it is more obvious in the @xmath219 plane . for a given break radius ,
bulges appear to be compact , i.e. , @xmath240 kpc ( see also dullo & graham 2013 and graham 2013 ) and possess somewhat fainter central and break surface brightnesses .
thus , in fig [ figiii7](b ) we additionally include the ols bisector fit to the relation between @xmath1 and @xmath8 ( dotted line ) for only the elliptical galaxies , which is given by @xmath241 ( table [ tabbb30 ] ) . combining this relation with the @xmath242 relation ( table [ tabbb30 ] , fig .
[ figiii7]a ) gives @xmath243 for elliptical galaxies with @xmath244 mag .
this can be compared with the bright end of the curved @xmath9 relation given in section 5.3.1 of graham & worley ( 2008 ) .
the linear regression fit to the luminous ( @xmath245 mag ) galaxies in graham & worley ( 2008 , their fig .
11a ) has a slope of 0.9 ( see also liu et al . 2008 , bernardi et al .
2007 ) , and it is @xmath246 steeper at brighter luminosities . while the elliptical galaxies appear to follow the steeper near - linear @xmath1 @xmath247 @xmath248 relation than the @xmath1 @xmath247 @xmath249 relation for the combined ( elliptical+bulge ) sample , the vertical rms scatters for both these relations are large ( table [ tabbb30 ] ) . , but shown here are the correlations between the core - srsic break radius @xmath1 and ( a ) absolute @xmath104-band magnitude of a galaxy or a bulge for a disc galaxy ( table [ tabbb1 ] ) , and ( b ) effective ( half - light ) radius @xmath8 ( table [ tabbb2 ] ) . in fig .
[ figiii7](b ) we also include least - squares fit to the @xmath1 and @xmath8 data for just the elliptical galaxies ( dotted line ) . ] the tight correlations seen in the @xmath213 and @xmath215 diagrams ( figs .
[ figiii6]b and [ figiii6]c ) were also shown by faber et al .
( 1997 , their fig .
8) , lauer et al .
( 2007a , their figs . 4 and 6 ) and dullo & graham ( 2012 , their fig .
similar trends to the @xmath217 distribution ( fig.[figiii7]a ) can be seen in the works by faber et al .
( 1997 , their fig . 4 ) , ravindranath et al .
( 2001 , their fig .
5a , b ) , laine et al .
( 2003 , their fig .
9 ) , trujillo et al .
( 2004 , their fig .
9 ) , de ruiter et al .
( 2005 , their fig . 8) , lauer et al .
( 2007a , their figs . 5 ) and dullo & graham ( 2012 , their fig .
the slope of the @xmath250 relation @xmath251 ( table [ tabbb30 ] ) that we find here can be compared to the similar slopes 1.15 , 0.72 , @xmath252 , @xmath253 and @xmath254 published by faber et al .
( 1997 ) , laine et al .
( 2003 ) , de ruiter et al .
( 2005 ) , lauer et al .
( 2007a ) and dullo & graham ( 2012 ) , respectively .
it is worth noting that the @xmath250 relations in faber et al .
( 1997 ) , laine et al .
( 2003 ) and de ruiter et al .
( 2005 ) were derived using the nuker break radii , while lauer et al .
( 2007a ) used the `` cusp radius''the radius at which the negative logarithmic slope of the nuker model equals 0.5 .
the slopes ( not the intercepts ) of the @xmath250 relations obtained using the core - srsic model and the nuker model break radii can coincidentally be consistent because of the way the nuker model systematically overestimates the break radius in comparison with the core - srsic model .
also provided here , achieved using well constrained core - srsic fit parameters , are the @xmath213 and @xmath215 relations which are consistent with dullo & graham ( 2012 , their eqs .
7 , and 5 ) within the errors .
due to coupling of @xmath1 and @xmath205 along the light profile ( dullo & graham 2012 , their figs .
17c and 18 ) , our @xmath213 relation agrees with that of lauer et al .
( 2007a , their eq .
17 ) . as shown in fig .
[ figiii6](b ) , the core size of a galaxy ( @xmath1 ) and its surface brightness ( @xmath205 ) are closely related .
[ figiii8 ] reveals that @xmath205 is thus also tightly correlated with ( a ) the central surface brightness @xmath204 , ( b ) the velocity dispersion @xmath106 , ( c ) the spheroid absolute magnitude @xmath21 , and ( d ) the effective ( half - light ) radius @xmath8 .
the ols bisector fits are given in table [ tabbb30 ] .
the @xmath225 and @xmath255 relations ( table [ tabbb30 ] ) agree with those in dullo & graham ( 2012 , their eqs .
10 and 9 , respectively ) .
lastly , given the disagreement between the `` core '' parameters ( @xmath256 ) of the core - srsic model and the nuker model ( section [ sec3.1 ] ) , it is expected that our galaxy scaling relations ( table [ tabbb30 ] ) may differ from similar relations obtained using the nuker model .
however , as mentioned above , some of the slopes ( not the intercepts ) of these scaling relations derived from these two models can agree . moreover , the close agreement between the core - srsic break radius and the lauer et al .
( 2007a ) `` cusp radius '' ( dullo & graham 2012 ) suggests that the scaling relations based on these two core measurements would be consistent .
if galaxy core formation proceeds by the orbital decay of black hole binaries , from merging galaxies , as suggested by simulations ( e.g. , ebisuzaki et al . 1991
; merritt 2006 ) and advocated by faber et al .
( 1997 ) , then a close relation between the core size ( @xmath1 ) and the black hole mass ( @xmath4 ) of a galaxy might be expected .
given the well known @xmath257 ( ferrarese & merritt 2000 ; gebhardt et al .
2000 ) and @xmath258 ( marconi & hunt 2003 ; graham & scott 2013 ) relations , the strong @xmath215 and @xmath250 correlations in section [ sec4.1 ] hint at a tight @xmath231 relation .
this trend is observed in fig .
[ figiii9 ] , and quantified in table [ tabbb30 ] for our sample of 31 galaxies .
eight of these galaxies have direct smbh mass measurements , while the remaining smbh masses were predicted using either the graham & scott ( 2013 ) `` non - barred @xmath257 '' relation relation , which has smaller uncertainties , because it is consistent with the core - srsic @xmath232 relation . ]
@xmath259 or their @xmath260-band core - srsic @xmath258 relation which is converted here to @xmath104-band using @xmath261 ( fukugita et al . 1995 ;
faber et al .
1997 ) , to give @xmath262 following graham et al .
( 2011 , see the discussion in their section 2.1.1 ) , we assumed a 10 % uncertainty on our velocity dispersions in order to estimate the errors on the smbh masses which were predicted using the @xmath257 relation .
the predicted masses are given in table [ tabbb3 ] .
note that since the resulting @xmath231 distributions , shown in figs .
[ figiii9]a and b , are primarily driven by galaxies with predicted smbh masses , the observed trend may simply be due to the existence of the @xmath215 ( fig .
[ figiii6]c ) , @xmath250 ( fig .
[ figiii7]a ) relations and the @xmath257 , @xmath258 relations , although we find below that this is not the case .
( table [ tabbb2 ] ) and black hole mass .
the smbh masses are acquired from direct smbh mass measurements for 8 circled galaxies , while for the remaining 23 galaxies the smbh masses are predicted using either ( a ) the graham & scott ( 2013 ) `` non - barred @xmath263 '' relation ( table [ tabbb3 ] ) or ( b ) their @xmath260-band core - srsic @xmath258 relation which is converted here to the @xmath104-band using @xmath261 ( fukugita et al .
symbolic representations are as in fig .
[ figiii6 ] .
the solid lines are the least - squares fits to our core - srsic data , the shaded regions cover the corresponding 1@xmath106 uncertainties on these regression fits .
pearson correlation coefficients , @xmath264 , ( and representative error bars ) are shown at the bottom ( top ) of each panel . ]
[ figiii11 ] plots the @xmath231 relation for the eight galaxies with directly measured smbh masses .
ngc 1399 has two distinct smbh mass measurements in the literature ( houghton et al.2006 , @xmath265 ; gebhardt et al .
2007 , @xmath266 ) .
while this galaxy has a normal core ( 202 pc ) for its absolute magnitude ( @xmath267 mag ) , it appears that the @xmath268 dynamical smbh mass measurement of gebhardt et al .
( 2007 ) _ may _ be too small for its 202 pc core size .
this mass estimate makes the galaxy an obvious outlier in the @xmath231 diagram ( fig .
[ figiii11 ] ) . using the houghton et al .
( 2006 ) smbh mass for ngc 1399 , the ols bisector fit to the @xmath1 and @xmath4 data for our eight galaxies with direct smbh measurements yields @xmath269 with an rms scatter of @xmath270 dex in the log @xmath4 direction . when using the smbh mass for ngc 1399 from gebhardt et al .
( 2007 ) rather than from houghton et al .
( 2006 ) , @xmath271 with an intercept of 1.96 @xmath181 0.14 ( fig .
[ figiii11 ] , dashed line ) . while this relation is in excellent agreement with eq .
[ eqq34 ] , the scatter in this distribution is larger ( 0.35 dex in the log @xmath4 direction ) .
we note that , as shown in fig .
[ figiii11 ] , the near - linear @xmath231 relation established by these eight galaxies ( eq . [ eqq34 ] , solid line ) is consistent with the relations constructed by including the remaining sample galaxies with predicted smbh mass measurements ( figs .
[ figiii9]a and b , table [ tabbb30 ] ) .
however , the @xmath258 relation ( eq .
[ eqq34b ] , graham & scott 2013 ) appears to somewhat overpredict the smbh masses relative to the @xmath257 relation ( eq .
[ eqq34a ] , graham & scott 2013 ) for our core - srsic galaxy sample .
this can be seen from the smaller intercept of the ( @xmath258)-based @xmath231 relation , 1.75 @xmath181 0.06 ( fig .
[ figiii9]b , table [ tabbb30 ] ) , compared to the intercept of the ( @xmath257)-based @xmath231 relation , @xmath272 ( fig .
[ figiii9]a , table [ tabbb30 ] ) .
this unexpected situation has arisen because of a difference in the @xmath273 relation between the core - srsic galaxy sample in graham & scott ( 2013 ) and that used here .
we find a 2.44@xmath106 difference between the intercepts of the @xmath273 relations from these two studies , which largely explains the 3.25@xmath106 difference between the intercepts of the ( @xmath257)-based and ( @xmath274)-based @xmath231 relations ( table [ tabbb30 ] ) . to further appreciate the ( galaxy core)-(smbh mass ) connection we derive additional @xmath231 relations by combining the non - barred @xmath257 relation from graham & scott ( 2013 ) with the @xmath215 relation ( table [ tabbb30 ] ) to obtain the new @xmath231 relation @xmath275 which is in good agreement with eq .
[ eqq34 ] .
similarly , combining the @xmath276 relation ( table [ tabbb30 ] ) with the core - srsic @xmath258 relation from graham & scott ( 2013 , their table 3 @xmath260-band ) , which is converted here to the @xmath104-band using @xmath277=1.0 , gives @xmath278 although we only have eight galaxies with direct black hole mass measurements , eq .
[ eqq34 ] is consistent ( i.e. , overlapping @xmath179 uncertainties ) with the two inferred relations ( eqs .
[ eqq35 ] and [ eqq36 ] ) .
( table [ tabbb2 ] ) and black hole mass ( table [ tabbb3 ] ) for eight galaxies with dynamically determined ( i.e. , direct ) black hole mass @xmath4 measurements . for ngc 1399
we include two direct smbh mass measurements : ( i ) ( [email protected])@xmath99@xmath279 ( gebhardt et al . 2007 ) and ( ii ) ( 1.2@xmath181@xmath280)@xmath99@xmath281@xmath282 ( houghton et al . 2006 ) .
the solid line is the least - squares fit assuming the houghton et al .
( 2006 ) smbh mass for ngc 1399 , while the dashed line uses the gebhardt et al .
( 2007 ) mass , see the text for further detail .
the inner shading marks the @xmath283@xmath106 uncertainty on eq .
[ eqq34 ] , while the outer shading extends this by 0.26 dex ( the rms scatter ) in the log @xmath4 direction .
the dotted and dashed - dotted lines are the fits for the full galaxy sample shown in figs . [ figiii9](a ) and ( b ) , respectively ( table [ tabbb30 ] ) .
the pearson correlation coefficient , r , obtained when using the houghton et al .
( 2006 ) smbh mass for ngc 1399 is shown at the bottom of the panel . ]
@llcccccc@ galaxy&@xmath284&@xmath285&log ( @xmath286)&log ( @xmath287)&log ( @xmath288)&@xmath289 + & & & & & & + ( 1)&(2)&(3)&(4)&(5)&(6)&(7 ) + + ngc 0507 & 1.40[a ] & 5.5&8.34&9.08&9.24@xmath290[p]&0.69 + ngc 0584 & 1.27[b]&4.5&7.78&8.43 & 8.29@xmath290[p]&1.38 + ngc 0741 & 1.32[a]&5.0&8.96&9.66&9.12@xmath290[p]&3.43 + ngc 1016 & 1.32[a]&5.0&9.10&9.80&9.21@xmath291[p]&3.85 + ngc 1399 & 1.21[a]&4.0&9.57&9.97&8.67 @xmath292 , 9.07 @xmath293[d]&20.0 , 7.90 + ngc 1700 & 1.29[b]&4.8&7.77&8.45&8.65@xmath291[p]&0.63 + ngc 2300 & 1.33[b]&5.0&7.81&8.51&8.86@xmath291[p]&0.45 + ngc 3379 & 1.28[b]&4.6&8.63&9.29&8.60@xmath294[d]&4.90 + ngc 3608 & 1.29[b]&4.8&7.82&8.50&8.30@xmath295[d]&1.57 + ngc 3640&1.22[a]&4.0&6.35&6.96&7.99@xmath291[p]&0.09 + ngc 3706 & 1.34[a]&5.2&8.30&9.01&8.94@xmath290[p]&1.18 +
ngc 3842&1.38[a]&5.6&9.20&9.95&9.98@xmath296[d]&0.93 + ngc 4073 & 1.17[a]&3.2&8.50&9.00&8.98@xmath290[p]&1.05 + ngc 4278 & 1.26[b]&4.5&8.31&8.96&8.62@xmath290[p]&2.17 + ngc 4291 & 1.27[a]&4.5&8.15&8.81&8.52@xmath297[d]&1.94 + ngc 4365 & 1.33[b]&5.0&8.68&9.38&8.81@xmath290[p]&3.70 + ngc 4382 & 1.10[b]&2.6&7.68&8.09&7.95@xmath290[p]&1.37 + ngc 4406 & 1.25[b]&4.5&8.28&8.93&8.61@xmath290[p]&2.12 + ngc
4472&1.33[b]&5.0&8.59&9.29&9.15@xmath290[p]&1.40 + ngc 4552 & 1.29[b]&4.8&8.02&8.70&8.67@xmath298[d]&1.08 + ngc 4589 & 1.33[b]&5.0&7.50&8.22&8.49@xmath290[p]&0.54 + ngc
4649&1.34[a]&5.2&9.05&9.76&9.67@xmath299[d]&1.22 + ngc 5061 & 1.24[a]&4.2&8.64&9.27&8.05@xmath291[p]&16.7 + ngc 5419 & 1.35[a]&4.7&9.70&10.37&9.57@xmath300[p]&6.26 + ngc 5557 & 1.18[a]&3.2&8.23&8.74&8.78@xmath290[p]&0.90 + ngc 5813&1.31[b]&5.0&8.24&8.93&8.83@xmath298[d]&1.27 + ngc 5982 & 1.26[b]&4.5&8.24&8.89&8.65@xmath291[p]&1.75 + ngc 6849&1.03[a]&2.4&7.97&8.35&8.04@xmath291[p]&2.03 + ngc
6876&1.26[a]&4.5&8.52&9.17&8.55@xmath290[p]&4.25 + ngc 7619&1.35[b]&4.7&8.99&9.66&9.37@xmath290[p]&1.95 + ngc 7785&1.33[a]&5.0&6.67&7.37&8.80@xmath290[p]&0.04 + notes.col .
( 1 ) galaxy name .
( 2 ) galaxy colour : we use the lauer et al .
( 2005 ) central @xmath284 colours [ b ] when available ; otherwise the @xmath284 colours [ a ] were taken from the hyperleda database .
( 3 ) @xmath104-band stellar mass - to - light ( @xmath301 ) ratios determined using the galaxy colours ( col .
2 ) and the colour - age - metallicity-@xmath302 relation given by graham & spitler ( 2009 , their fig .
( 4 ) central luminosity deficit in terms of @xmath104-band solar luminosity .
( 5 ) central stellar mass deficit determined using col . ( 3 ) and col .
( 6 ) smbh mass .
sources : [ p ] supermassive black hole mass predicted using the graham & scott ( 2013 , their table 3 and fig .
2 ) `` non - barred @xmath232 '' relation ( and the `` barred @xmath232 '' relation for ngc 6849 ) ; [ d ] galaxies with dynamically determined smbh mass measurements taken from graham & scott ( 2013 ) . for ngc 1399 , we use two direct ( dynamically determined ) smbh mass measurements taken from gebhardt et al .
( 2007 , @xmath3034.7@xmath304@xmath99@xmath305 @xmath282 ) and houghton et al .
( 2006 , @xmath3031.2@xmath306@xmath99@xmath281@xmath282 ) and adjusted to our distance .
we use eq . 4 from graham et al .
( 2011 ) , updated according to the relation in graham & scott ( 2013 ) , as well as the @xmath106 values in table [ tabbb1 ] and assume a 10% uncertainty on @xmath106 to estimate the error on the predicted smbh mass ( see graham et al .
2011 , their section 2.1.1 ) .
( 7 ) ratio between mass deficit and black hole mass .
as mentioned before , the central stellar mass deficits of core - srsic galaxies are naturally generated through the gravitational sling - shot ejection of core stars by the inspiraling black hole binaries that that are formed in a merger remnant ( begelman et al .
1980 ; ebisuzaki et al .
a key point to note is that high - accuracy simulations ( e.g. , milosavljevi & merritt 2001 ; merritt 2006 ) predicted that multiple dissipationless mergers will have cumulative effects on core formation .
merritt ( 2006 ) found that the total stellar mass deficit , @xmath10 , after @xmath307 successive dry major mergers is @xmath308 , with @xmath4 the final smbh mass .
past studies have quantified this stellar mass deficit from the difference in luminosity , @xmath309 , between the inward extrapolation of the outer srsic profile ( of the core - srsic model ) and a sharp - transition ( graham et al .
core - srsic model ( graham 2004 ; ferrarese et al .
2006 ; hyde et al .
2008 ) . here
, we apply the same prescription for @xmath309 as in these past works but we use a smoother transition ( instead of a sharp ) core - srsic model by applying a finite @xmath143 in eq .
[ eqq10 ] ( cf . also dullo & graham 2012 , 2013 ) .
therefore , the difference in luminosity between the outer srsic model ( eq .
[ eqq9 ] ) and the core - srsic model ( eq . [ eqq37 ] ) is the central stellar luminosity deficit @xmath310 . for each galaxy
this luminosity deficit is converted into a mass deficit using the @xmath104-band stellar mass - to - light ( @xmath301 ) ratio given in table [ tabbb3 ] . in order to determine these @xmath311 ratios , the central ( if available , otherwise the galaxy ) @xmath284 colours ( table [ tabbb3 ] ) together with the colour - age - metallicity-(@xmath301 ) diagram ( graham & spitler 2009 , their fig .
a1 ) were used , assuming a 12 gyr old stellar population .
we note that the graham & spitler ( 2009 ) colour - age - metallicity-(@xmath301 ) diagram is constructed using the bruzual & charlot ( 2003 ) stellar population models and the chabrier ( 2003 ) stellar initial mass function ( imf ) .
recent works suggest that the imf may vary with velocity dispersion for early - type galaxies but there is a significant scatter in this relation ( e.g. , cappellari et al . 2012 , 2013 ;
conroy & van dokkum 2012 ; spiniello et al .
2012 ; wegner et al . 2012 ; ferreras et al . 2013 ; zaritsky et al .
galaxies with low - velocity dispersions require a kroupa ( 2001 , or a chabrier 2003 ) imf , while high - velocity dispersion galaxies ( @xmath312 km s@xmath23 ) may prefer a `` bottom - heavy '' imf having a steeper slope than that of salpeter ( 1955 ) , although cappellari et al . ( 2013 , their fig .
15 ) found a shallow ( @xmath313@xmath106 relation for their slow rotators with @xmath314 km s@xmath23 ( see also rusli et al .
2013 ; clauwens , schaye & franx 2014 ) .
once these mass - to - light issues are settled , it may be worth trying to refine the stellar mass deficits reported here . versus black hole mass ( @xmath4 ) for the 31 core - srsic galaxies listed in table [ tabbb1 ] .
the data for four s0s ( ngc 507 , ngc 2300 , ngc 4382 and ngc 6849 ) and one elliptical ( ngc 3706 ) are taken from dullo & graham ( 2013 ) .
the graham & scott ( 2013 , their table 3 ) `` non - barred @xmath315-@xmath106 '' relation was used for estimating the smbh masses of 23 galaxies , while for the remaining 8 galaxies ( enclosed in boxes ) we used their direct smbh mass measurements as given in graham & scott ( 2013 ) . for ngc 1399
, we also plot the larger black hole mass from houghton et al .
( 2006 ) adjusted for our distance of 19.4 mpc . three questionable galaxies with unusually small cores ( ngc 1700 , ngc 3640 and ngc 7785 ) are circled .
a representative error bar is shown at the bottom of the panel . ]
[ figiii12 ] plots the mass deficits that we derive against the dynamically determined or predicted smbh masses for our 31 core - srsic early - type galaxies ( table [ tabbb3 ] ) , the largest sample of core - srsic galaxies with extended light profiles that has been modelled to date .
we find the mass deficits for these galaxies are typically @xmath316 @xmath4 , in agreement with past core - srsic model estimates ( graham 2004 ; ferrarese et al.2006 ; hyde et al .
2008 ; dullo & graham 2012 , 2013 ) .
this translates to core - srsic galaxy formation through one to several successive `` dry '' major merger events , consistent with theoretical expectations ( e.g. , haehnelt & kauffmann 2002 ) .
in addition , recent observations on close major merger pairs have revealed that massive galaxies have undergone @xmath317 major mergers since @xmath318 ( e.g. , bell et al.2004 , 2006 ; bluck et al .
2012 ; man et al .
2012 ; xu et al .
the most massive galaxies , with stellar mass @xmath319 , may have experienced up to six major mergers since @xmath320 ( conselice 2007 ) . four elliptical galaxies
ngc 1399 , ngc 3640 , ngc 5061 and ngc 7785 are outliers from the main @xmath321 distribution ( fig .
[ figiii12 ] ) .
ngc 3640 and ngc 7785 are two of the three galaxies with unusually small depleted cores ( see section 4 and appendix [ apppqes ] ) .
they both have small mass deficits ( @xmath322 and @xmath323 ) for their predicted smbh masses ( @xmath324 and @xmath325 ) . the remaining galaxy with a questionable core ( ngc 1700 ) is an outlier in most central galaxy scaling relations ( figs . [ figiii6 ] , [ figiii7 ] and [ figiii8 ] ) but it has a normal mass deficit for its smbh mass .
this owes to the fact that ngc 1700 , unlike ngc 3640 and ngc 7785 , has a relatively steep outer srsic profile ( @xmath326 ) .
this larger srsic index value helps to compensate for the small core size , taking its estimated mass deficit into the normal range in fig .
[ figiii12 ] . in the case of the potentially outlying galaxy ngc 1399 ,
the discussion given in section [ sec5.1 ] explains the behavior seen here .
the smaller dynamical smbh mass determination by gebhardt et al .
( 2007 ) yields an inflated @xmath289 ratio of @xmath327 , while assuming the larger dynamical smbh mass measurement of houghton et al .
( 2006 ) implies a somewhat reasonable value of @xmath328 .
the situation with the fourth offset elliptical galaxy ngc 5061 is somewhat unclear given it is not a deviant galaxy in the other galaxy scaling relations ( section [ sec4 ] ) .
the core - srsic model fits its light profile very well with a fairly small rms residual of @xmath329 mag arcsec@xmath122 , but from these fit parameters we determine a large mass ratio @xmath330 .
its core size @xmath331 pc is a good match to its @xmath104-band absolute magnitude @xmath332 mag ( table [ tabbb1 ] ) and velocity dispersion @xmath333 km s@xmath23 ( table [ tabbb1 ] ) , but its srsic index @xmath334 ( table [ tabbb2 ] ) may be too high for the aforementioned galaxy properties . indeed , ngc 5061 has the largest srsic index from our sample , attributed to its noticeably straight surface brightness profile ( fig .
[ figa1 ] ) .
this may suggest that the envelope of this galaxy was built via several dry minor and major merging events ( hilz et al .
2013 ) . on the other hand
, it has the third smallest velocity dispersion @xmath335 km s@xmath23 ( hyperleda s mean value ) from our sample which seems to underpredict its smbh mass @xmath336 ( table [ tabbb3 ] ) . using the largest reported velocity dispersion value @xmath337 km s@xmath23 ( davies et al .
1987 ) , instead of the mean measurement , increases its predicted smbh mass roughly by a factor of 2 , i.e. , @xmath338 .
as such the associated @xmath289 ratio reduces roughly by a factor of two , to give @xmath339 .
this latter ratio is marginally consistent with the @xmath289 distribution shown in fig .
[ figiii11 ] . in summary
, it appears that both the high srsic index and the relatively low velocity dispersion of ngc 5061 _ may _ collectively act to inflate the @xmath289 ratio to 17 . versus cumulative number of `` dry '' major merger events ( @xmath307 ) for the 31 core - srsic galaxies listed in table [ tabbb1 ] .
symbolic representations are as in fig .
[ figiii12 ] .
pearson correlation coefficient , @xmath264 , ( and representative error bar ) are shown at the top ( bottom ) of the panel . ] as shown in fig .
[ figiii12b ] , excluding the three galaxies with questionably small / real cores , and using the hougton et al .
( 2006 ) smbh mass for ngc 1399 , the ols bisector regression between @xmath340 and @xmath289 ( @xmath341 @xmath342 , merritt 2006 ) gives @xmath343 in fig .
[ figiii12c ] , we explore the behavior of the @xmath344 ratio with @xmath4 .
we find that the distribution in this diagram appears largely consistent with the simulations by merritt ( 2006 , his table 2 ) . as in merritt ( 2006 ) , for the first merger ( @xmath345 ) ,
an object with a supermassive black hole mass of ( 1/3)(@xmath346 ) was added to a system having a black hole mass of ( 2/3)(@xmath346 ) .
the same black hole mass of ( 1/3)(@xmath346 ) was then added for each successive merger .
these accumulated black hole masses are plotted against @xmath342 to construct each of the three curves shown in fig .
[ figiii12c ] .
form this figure it is apparent that core - srsic galaxies with the same smbh mass ( or merger history ) can have different merger histories ( smbh masses ) .
this is consistent with the notion that the stellar mass deficits of core - srsic galaxies reflect the amount of merging as well as the masses of their smbhs .
the mean elliptical galaxy @xmath289 ratio from graham ( 2004 ) is 2.1 @xmath181 1.1 , while ferrarese et al .
( 2006 ) reported a mean @xmath347 ratio of 2.4 @xmath181 0.8 after excluding the s0 galaxy ngc 4382 from their sample .
hyde et al . ( 2008 ) found a comparable mean @xmath289 ratio of 2.3 @xmath181 0.67 for their sample . in dullo & graham ( 2012 ) , we modelled @xmath100@xmath147 light profiles and cautioned that the outer srsic parameters might be less constrained than desirable , although the srsic indices were shown to be in a fair agreement with those determined from published fits to larger radial extents .
nonetheless , we reported tentative @xmath344 ratios that were some 0.5@xmath348 . in dullo & graham ( 2013 )
we fit the extended light profiles of four core - srsic lenticular galaxies ( ngc 507 ; ngc 2300 ; 4382 and ngc 6849 ) using a core - srsic model for the bulge plus an exponential model for the disc .
one suspected s0 galaxy ngc 3706 was found to have a stellar distribution that is best described by the core - srsic model and was thus reclassified as an elliptical galaxy . using these core - srsic fit parameters we reported a robust @xmath349 for these five core - srsic galaxies ( dullo & graham 2013 , their fig .
4 ) , which are also shown here in fig .
[ figiii12 ] . to supermassive black hole mass ( @xmath4 ) as a function of @xmath350 for the 31 core - srsic galaxies listed in table [ tabbb1 ] .
the three curves are based on the simulations by merritt ( 2006 , his table 2 ) .
that is , we started with a certain ( total ) supermassive black hole mass @xmath4 for @xmath345 and a third of this black hole mass was added for each successive merger shown by a star . as such
, the cumulative black hole mass @xmath4 increases linearly with the number of merger @xmath307 , but the binary mass ratio decreases with @xmath307 .
these black hole masses were then plotted against @xmath342(@xmath341 @xmath289 , merritt 2006 ) to construct each curve , see the text for further detail . ] prior to graham ( 2004 ) who reported @xmath289 ratio of @xmath351 , previous estimates based on nuker model parameters had been an order of magnitudes larger ( e.g. , milosavljevic & merritt 2001 ; milosavljevi et al .
2002 ; ravindranath 2002 ) .
graham ( 2004 ) argued that the universe was some ten times less violent , in terms of major galaxy mergers , than previously believed .
subsequent works using nuker model parameters ( lauer et al . 2007a ; gltekin et al .
2011 ) or subjectively identifying the core from visual inspections ( kormendy et al .
2009 ) have reported mass deficits up to an order of magnitude larger than typically found here . as discussed in section [ sec3 ] and appendix [ apppc ] , this discrepancy is partly due to the contrasting core sizes measured by these distinct methodologies . using a model - independent analysis of the light profiles , hopkins & hernquist ( 2010 ) confirmed the result of graham ( 2004 ) and reported @xmath352 .
the larger mass deficits ( @xmath353 ) of kormendy & bender ( 2009 ) are also partly because they used the dynamical mass - to - light @xmath354 ratios rather than the stellar @xmath301 ratios to derive the mass deficits .
there is not much dark matter at the centre of massive spheroids ( e.g. , dekel et al .
2005 ) , and assuming the dynamical @xmath354 for the galaxy (= [ @xmath355 + @xmath356 $ ] / @xmath2 ) overpredicts the central mass deficits .
recently , kormendy & ho ( 2013 ) scaled up the smbh masses for the galaxies in kormendy & bender ( 2009 ) by about a factor of two and reported a new lower , mean ratio @xmath357 .
alternative mechanisms for the production of enhanced depleted cores in luminous galaxies has been suggested in the literature ( boylan - kolchin et al .
2004 ; gualandris & merritt 2008 ; kulkarni & loeb 2012 ) .
gualandris & merritt ( 2008 ) invoked the recurrent core passages of gravitationally kicked and `` recoiled '' ( and then oscillating about the centre ) smbh in a merger remnant to explain the formation of large stellar mass deficits that are up to 5 @xmath15 .
kulkarni & loeb ( 2012 ) also suggested that mass deficits as large as @xmath358 could be formed as a result of the action of multiple smbhs from merging galaxies .
if these processes have always occurred , then the result we found here , i.e. , @xmath359 @xmath4 for our sample galaxies implies that these galaxies are formed via just one major merger or minor merger events only , at odds with both observations of close galaxy pairs of equal mass , and theoretical expectations ( e.g. , khochfar & burkert 2003 ; bell et al .
2004 ; naab et al .
2006 ; bluck et al .
however , spheroids with @xmath360 ( i.e. , @xmath361 probably need some black hole oscillations as 8 or more major mergers seem excessive .
we find two elliptical galaxies ( ngc 1399 , ngc 5061 ) with @xmath362 , suggesting that these oversized mass deficits might be partly due to the action of their gravitationally kicked smbhs . it should be noted that several authors have considered alternative ways in which cores can be produced in luminous galaxies .
dissipationless collapses in existing dark matter haloes were invoked by nipoti et al .
( 2006 ) as a possible mechanism for forming depleted cores .
another suggested alternative was the adiabatic expansion of the core region driven by the rapid mass loss from the effects of supernova and agn feedback ( navarro et al .
1996 ; read & gilmore 2005 ; peirani , kay & silk 2008 ; martizzi et al . 2012 , 2013 ) and krajnovi et al .
( 2013 ) pointed out that this scenario is compatible with the properties of `` core slow rotators '' .
in addition , goerdt et al .
( 2010 ) proposed that the energy transferred from sinking massive objects would produce cores that are as large as 3 kpc in size .
these suggested mechanisms , however , are not without problems . for example
, it is unclear how the cores created by the simulations in nipoti et al .
( 2006 ) are guarded against infalling satellites ( which would replenish the core ) in the absence of a central smbh . also , the oversized ( @xmath363 kpc ) cores produced by the latter two mechanisms ( e.g. , goerdt et al .
2010 ; martizzi et al.2012 , 2013 ) are generally inconsistent with the typical @xmath3640.5 kpc cores observed in galaxies ( e.g. , trujillo et al .
2004 ; ferrarese et al . 2006 ; richings et al . 2011 ; dullo & graham 2012 ) . section 6.1 of dullo & graham ( 2013 ) provides further details , including the merits and weaknesses of these core formation models in the context of the observations . finally , we note that the standard lambda cold dark matter ( @xmath365cdm ) model predicts that elliptical galaxies are built via major mergers ( e.g. , kauffmann et al .
1993 ; khochfar & burkert 2005 ) . in this hierarchical picture
, the bulges of lenticular galaxies form early via major mergers while their discs grow later through gas accretion events ( e.g. , steinmetz & navarro 2002 ) . alternatively ,
the evolutionary transformation of spiral galaxies into s0 galaxies via mechanisms such as ram pressure stripping has been suggested ( e.g. , gunn & gott 1972 ) .
comparing various galaxy scaling relations , luarikainen et al .
( 2010 ) , for example , showed that the bulges of s0 galaxies are closely correlated with the bulges of bright spiral galaxies , having @xmath366(bulge ) @xmath36720 mag , than with elliptical galaxies . on the other hand , in dullo & graham ( 2013 )
, we argued that core - srsic s0s with @xmath21(bulge ) @xmath368 mag might be assembled inside - out in two stages : an earlier `` dry '' major merger process involving smbhs forms their bulge component , while the surrounding disc is subsequently formed via cold gas accretion
. the bulges of our small s0 galaxy sample tend to have @xmath369 ( dullo & graham 2013 , see also balcells et al .
2003 and laurikainen et al .
2005 ) , however , core - srsic elliptical galaxies have @xmath370 . as we mentioned in section [ sec4.1 ] , for the same core size
, these massive bulges tend to be compact ( @xmath240 kpc , fig .
[ figiii7]b ) and have somewhat fainter break , and central , surface brightnesses than the elliptical galaxies ( figs . [ figiii6]a and [ figiii6]b ) .
furthermore , the @xmath289 ratio for s0s is lower compared to the spread seen in elliptical galaxies ( fig .
[ figiii12 ] ) .
this implies that the bulges of core - srsic s0s have experienced fewer major mergers than core - srsic elliptical galaxies ( fig .
[ figiii12b ] ) .
we extracted the major - axis surface brightness profiles from 26 core - srsic early - type galaxies observed with the _ hst _ wfpc2 and acs cameras .
we additionally included five core - srsic early - type galaxies ( ngc 507 , ngc 2300 , ngc 3706 , ngc 4382 and ngc 6849 ) from dullo & graham ( 2013 ) .
this compilation represents the largest number of core - srsic galaxies modelled to large radii @xmath371 , giving the fitting functions enough radial expanse to robustly measure the galaxy stellar distribution ( see fig .
[ figiii5 ] for a comparison of the srsic indices obtained from fits using @xmath372 profiles ) .
we fit the extended surface brightness profiles of the 26 core - srsic elliptical galaxies using the core - srsic model , while light profiles of the remaining five core - srsic s0 galaxies were modelled with the core - srsic model for the bulge plus an exponential model for the disc .
we accounted for additional nuclear cluster light using the gaussian function .
our principal results are summarised as follows : \1 .
the global stellar distributions of core - srsic elliptical galaxies are robustly represented with the core - srsic model , while core - srsic lenticular galaxies are accurately described using the core - srsic bulge + exponential disc model .
these fits yield a median rms scatter of 0.045 mag arcsec@xmath122 for our sample of 31 core - srsic galaxies , and argue against excessive multi - component srsic models ( section 3.4 ) .
we provide updated core - srsic model parameters @xmath373 for 31 core - srsic early - type galaxies with spheroidal components having @xmath374 mag and @xmath375 km s@xmath23 . in general
, there is a good agreement with the parameters obtained from our earlier analysis of the publicly available , but radially limited ( @xmath376 ) surface brightness profiles given by lauer et al .
( 2005 ) .
the bulges of our core - srsic s0s are compact ( @xmath377 2 kpc ) and have @xmath378 ( dullo & graham 2013 ) , as compared to the core - srsic elliptical galaxies which typically have @xmath379 kpc and @xmath195 . \4 .
the core - srsic model break radii are in agreement with both ( i ) the previously published core - srsic break radii and ( ii ) the model - independent break radii which mark the locations where the negative logarithmic slopes of the light profiles equal 0.5 ( carollo et al .
1997a ; dullo & graham 2012 ) .
updated structural parameter relations involving both the central and global galaxy properties are provided in section [ sec4 ] .
we have found tight correlations involving the central galaxy properties @xmath1 ,
@xmath204 , @xmath205 , @xmath106 and @xmath4 ( see table [ tabbb30 ] ) .
we have also found near - linear relations between the break radius @xmath1 , and the spheroid luminosity @xmath2 and the smbh mass @xmath4 given by @xmath380 and @xmath5 .
we additionally found a near - linear relation between @xmath1 and @xmath8 such that @xmath381 but with a large scatter .
we have derived central stellar mass deficits in 31 early - type galaxies that are typically 0.5 to 4 times the host galaxy s black hole mass .
given published theoretical results , these mass deficits suggest a few dissipationless major mergers for core - srsic galaxies . \7 . as noted in dullo & graham ( 2013 ) , mass deficits in core - srsic s0s
suggest a two stage assembly : an earlier `` dry '' major merger event involving smbhs creates the bulges with depleted cores , and the disc subsequently builds up via cold gas accretion events .
the relation between the stellar mass deficit @xmath10 and the cumulative number @xmath307 of major `` dry '' mergers that the galaxy has undergone is such that @xmath10 is roughly @xmath247 @xmath382 .
the close relation between the galaxy cores and the smbhs supports the popular core depletion hypothesis where cores are thought to be created by sinking binary smbhs that eject stars away from the centres of their host galaxies .
the small cores seen in some galaxies , if real , may arise from loss cone regeneration by newly produced stars and/or recent stellar accretion events .
alternatively , small galaxy cores can be interpreted as a sign of minor mergers .
we have identified two galaxies ( ngc 1399 and ngc 5061 ) which have a high @xmath289 ratio , suggesting that their central smbh may have experienced a kick due to a gravitational - radiation recoil event leading to multiple core passages .
this research was supported under the australian research council s funding scheme ( dp110103509 and ft110100263 ) .
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 .
we acknowledge the usage of the hyperleda database ( http://leda.univ-lyon1.fr . ) .
btd is grateful for the supra scholarship offered by swinburne university of technology , and travel support from the astronomical society of australia .
figure [ figa1 ] shows core - srsic model fits to the major - axis surface brightness profiles of the 26 ( suspected elliptical ) core - srsic galaxies listed in table [ tabbb1 ] .
notes on two of these 26 galaxies ( ngc 4073 and ngc 6876 ) with complicated structures are given below , while their photometric profiles are shown in figs .
[ figb1 ] and [ figb2 ] .
ngc 4073 is a cd galaxy in the poor mkw 4 cluster ( de vacouleurs et al .
it has a double classification in laurikainen et al .
( 2011 , their table 3 ) , i.e. , sab0@xmath383 for the inner regions and e@xmath384 for the outer parts . also , this galaxy has a small bump in its light profile over @xmath385 due to a nuclear ring of stars ( lauer et al .
our models are not designed to fit stellar rings , thus we simply exclude data points that are contaminated by the nuclear ring light .
further , the core - srsic model fit to ngc 4073 shows an excess of light on top of the core - srsic light distribution over @xmath386 , creating the residual pattern seen in fig .
[ figa1 ] .
cd galaxies are known to grow via cannibalism of their neighboring , less massive cluster galaxies ( e.g. , ostriker & hausman 1977 ) .
if these accreted objects ( or at least their dense cores ) survive , they would be visible as extra light like in ngc 4073 .
however , we did not find clear evidence for such a feature in this galaxy s 2d residual image , possibly suggesting a collection of large scale disturbances in the galaxy .
[ figb1 ] shows the surface brightness and photometric profiles for ngc 4073 determined using the iraf task ellipse .
these profiles are connected with the galaxy s residual structure observed in the model fit ( fig .
[ figa1 ] ) .
the position angle profile shows an abrupt 90@xmath387 twist at around @xmath388 .
the galaxy is also highly flattened ( @xmath389 ) outside @xmath390 , while the ellipticity shows a steady drop from 0.45 ( at @xmath391 ) to 0.1 ( at @xmath392 ) accompanied by negative b@xmath393 values , then becomes nearly circular ( @xmath394 ) inside the core ( @xmath395 ) .
fisher , illingworth & franx ( 1995 ) reported ngc 4073 as having a counterrotating stellar core which they attribute to possible cannibalism events .
laurikainen et al .
( 2011 ) interpreted the excess light and the associated ellipticity trend over @xmath396 as being due to a weak inner bar . without accounting for its light excess at @xmath397 , and omitting its ring light , the core - srsic model fits the light distribution of ngc 4073 with rms residual of @xmath398 mag arcsec@xmath122 .
ngc 6876 is a dominant elliptical galaxy in the pavo group . as shown in fig .
[ figb2 ] , this galaxy is only @xmath399 from its
smaller companion , the elliptical galaxy ngc 6877 . using multi - wave observations ,
machacek et al .
( 2005 , 2009 ) showed evidence for interaction between ngc 6876 and the highly disturbed spiral galaxy ngc 6872 .
dullo & graham ( 2012 , their fig .
4 ) showed the double optical nucleus in ngc 6876 possibly associated with the dense core of a lesser galaxy or the ends of an inclined ring ( lauer et al .
2002 ) . the residual structure at @xmath400 ( fig . [ figa1 ] ) is associated with a 13@xmath387 twist in the position angle , and the change in the ellipticity and the isophote shape of this galaxy beyond its core regions .
* ngc 1700*[sec8.1.1 ] + ngc 1700 , a luminous ( @xmath401 mag ) elliptical galaxy with an estimated ( luminosity weighted ) age of 3@xmath1811 gyr ( brown et al .
2000 ) and a velocity dispersion @xmath402 km s@xmath23 , shows post merger morphological signatures such as shells , and boxy isophote at large radii @xmath403 ( franx et al .
1989 ; statler et al . 1996
; whitmore et al .
1997 ; brown et al .
2000 ; stratler & mcnamara 2002 ) .
it was also reported to be offset from the fundamental plane ( e.g. , statler et al .
1996 ; reda et al .
given this galaxy s recent ( wet ) merger history , its small core size may be because of loss cone regeneration via newly produced and/or accreted stars .
that is , a pre - existing large core in this galaxy may be partially replenished by these new stars .
we do however caution that because this core is defined by just the inner two data points of the deconvolved surface brightness profile , i.e. , within @xmath404 , its validity can be questioned , but it remains unlikely that they are any bigger than reported here .
+ * ngc 3640 * + ngc 3640 is a fast rotating ( krajnovi et al .
2013 ) early - type galaxy with @xmath405 mag and @xmath406 km s@xmath23 .
prugniel et al .
( 1988 , see also michard & prugniel 2004 ) noted that this galaxy is a merger in progress , possibly with a gas - poor disc system .
not only did they find low surface brightness structures such as shells and ripples but they also showed that the @xmath407 radial colour profile of this galaxy gets bluer towards the inner region .
hibbard & sansom ( 2003 ) however fail to detect neutral hydrogen associated with ngc 3640 .
also , tal et al . (
2009 ) recently reported this galaxy to be highly disturbed system revealing morphological peculiarities .
therefore , as in the case of ngc 1700 , this galaxy s undersized core and mass deficit can be interpreted as loss cone replenishment via new star formation and/or stellar accretion not associated with a secondary black hole ( see also krajnovi et al.2013 ) .
it is also possible that the observed core , defined by just one inner data point , may not be real . +
* ngc 7785 * + our core - srsic model fit to this elliptical galaxy light profile yields a core size of just 5 pc , which is too small for its absolute magnitude @xmath408 mag and velocity dispersion @xmath409 255 km s@xmath23 .
there is no evidence for an ongoing or a recent merging event associated with this galaxy in the literature .
thus , its small core size as well as undersized mass deficit suggest that one or a few minor ( instead of major ) dissipationless mergers might have taken place in the absence of loss cone refilling
. alternatively , the apparent core in this galaxy , also defined by one data point , may be spurious .
this section provides a review of five galaxies ( i.e. , excluding ngc 4073 already discussed above ) from our 26 suspected elliptical galaxies which were shown to have a bulge+disc stellar distribution in the literature .
this galaxy is classified as an elliptical galaxy in the third reference catalogue , rc3 ( de vacouleurs et al .
1991 ) and as an s0 by sandage & tammann ( 1981 ) .
laurikainen et al .
( 2010 , 2011 ) fit a 3-component ( bulge+bar+disc ) model to this galaxy s
@xmath100@xmath410 @xmath411-band light profile and noted that it has a large disc - like outer envelope and a weak inner bar .
however , these ( bar and disc ) components were not detected in the @xmath100@xmath412 @xmath104-band light profile that we modelled .
we do however wish to bring them to the attention of readers .
the virgo cluster galaxy ngc 4472 is classified as an elliptical galaxy in the rc3 but as an s0 by sandage & tammann ( 1981 ) .
laurikainen et al .
( 2010 , 2011 ) identified a large - scale disc in this galaxy which dominates the light at large radii ( @xmath413 . in contrast , ngc 4472 was considered to be an elliptical galaxy by kormendy et al . ( 2009 ) who fit a single srsic model to its light profile over @xmath414 .
we did not detect a disc component in our @xmath171 profile ( see also ferrarese et al .
interestingly , however , this galaxy has the lowest srsic value ( @xmath415 ) from our 26 suspected elliptical galaxies ( table 2 ) , possibly suggesting an s0 morphology .
it is classified as a slow rotator ( over its inner region ) by the atlas3d team ( cappellari et al .
2011 ) . like ngc 4472 , ngc 4552 is a member of the virgo cluster .
it is classified as an elliptical galaxy in the rc3 , but was recognized as an s0 by sandage & tammann ( 1981 ) . while kormendy et al .
( 2009 ) adopted an elliptical morphology for this galaxy , their srsic model fit to its light profile over @xmath416 shows a clear residual structure , in agreement with the s0 morphology adopted by laurikainen et al .
( 2010 , 2011 ) . however , it too is classified as a slow rotator by cappellari et al .
( 2011 ) perhaps because they only sampled the pressure - supported , bulge - dominated inner portion of the galaxy . the disc light ( e.g. , laurikainen et al .
2011 ) , dominant at large radii , does not contribute to our @xmath100@xmath171 light profile ( see also ferrarese et al .
2006 ) . as hinted at in section [ sec3.2 ] ,
although ngc 5813 is classified as an elliptical galaxy ( e1 - 2 ) in rc3 , we find that its ( @xmath100@xmath412 ) light profile is best fit using the core - srsic bulge+exponential model with a small rms residual of @xmath417 mag arcsec@xmath122 .
this suggests the galaxy may be an s0 disc galaxy , consistent with its steadily increasing ellipticity at @xmath418 as well as the kinks in the position angle and @xmath419 profiles at @xmath420 ( fig .
[ figiii3 ] ) .
the residual image of this galaxy shown in fig . [ figiii3 ] is regular , there is no evidence for a distinct morphological feature .
trujillo et al . ( 2004 ) also concluded that ngc 5813 may be an s0 galaxy which is better described using a bulge+disc model .
this however appears to disagree with the rotation curve given by efstathiou , ellis & carter ( 1982 ) .
using long slit data , these authors showed that the core ( @xmath421 ) of ngc 5813 rotates rapidly , while beyond @xmath172@xmath147 the galaxy shows little rotation : evidence for a kinematically decoupled core .
this galaxy is classified as an elliptical galaxy in the rc3 , yet it was identified as an s0 by sandage & tammann ( 1981 ) and also recently by laurikainen et al .
we fit a core - srsic bulge plus a gaussian point source model to @xmath100@xmath171 ( @xmath104-band ) light profile , while laurikainen et al .
( 2011 ) fit a three - component ( bulge+lens+outer - disc ) model to a more radially extended @xmath422-band brightness profile .
our @xmath100@xmath171 data does not probe this galaxy s large - scale disc .
also , the lens component is not seen in our @xmath104-band light profile . recently , krajnovi et al .
( 2013 ) fit the nuker model ( kormendy et al .
1994 ) to the nuclear light profiles of 135 atlas@xmath423 galaxies .
while these authors are aware that the nuker model is not a robust parametrization ( i.e. , the parameters are unstable ) , this model was used for the purpose of distinguishing core and coreless galaxies . as in the previous nuker model
works ( e.g. , lauer et al .
2005 ) they opt to use @xmath424 ( the slope of the nuker model at @xmath173 ) as a diagnostic tool for separating their galaxies as core ( @xmath425 ) , intermediate ( 0.3 @xmath426 0.5 ) and power - law ( @xmath427 ) type .
however , it has been shown ( graham et al .
2003 ; dullo & graham 2012 , their section 5.1 ) that the distance - dependent @xmath424 is not a physically robust or meaningful quantity to use .
moreover , galaxies with small srsic indices will have @xmath428 but no central deficit relative to the inward extrapolation of their outer srsic profile
. in dullo & graham ( 2012 , see also graham et al .
2003 ) , we highlighted that the nuker model is not ideal for describing central light profiles , and we revealed that seven ( @xmath429 ) of the sample galaxies with low @xmath94 and thus shallow inner profile slopes were misidentified as galaxies having depleted cores by the nuker model .
four of these seven galaxies ( ngc 4458 , ngc 4473 , ngc 4478 and ngc 5576 ) are in common with krajnovi et al .
krajnovi et al .
( 2013 ) classify three of them ( ngc 4458 , ngc 4478 and ngc 5576 ) as an intermediate type ( i.e. , coreless galaxies based on their criteria ) in agreement with dullo & graham ( 2012 ) and at odds with lauer et al . ( 2005 ) .
they , however , reminded the reader that the nuker model actually detected cores in all these galaxies but this detection may change depending on the range over which the fits were done .
it is perhaps worth clarifying that the nuker model break radii reported by lauer et al .
( 2005 ) for ngc 4458 , ngc 4478 and ngc 5576 are @xmath430 smaller than those from krajnovi et al.(2013 ) .
these differing break radii have resulted in a significant difference in the @xmath424 measurements from these two works .
for these three galaxies in question , krajnovi et al .
( 2013 ) found @xmath424 values that are @xmath431 times larger than those from lauer et al .
for the remaining common galaxy ngc 4473 , krajnovi et al .
( 2013 ) found @xmath432 and subsequently classified it as a core galaxy in agreement with lauer et al .
( 2005 ) and in contrast to ferrarese et al .
( 2006 ) , kormendy et al .
( 2009 ) and dullo & graham ( 2012 ; 2013 , their section 9.2.2 ) .
ngc 4473 does not have a central light deficit but quite the opposite : an embedded stellar disk ( e.g. , krajnovi et al .
2006 ; capellari et al .
2007 ) .
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we have carefully quantified the radial stellar distributions of the elliptical galaxies using the core - srsic model whereas for the lenticular galaxies a core - srsic bulge plus an exponential disc model gives the best representation .
we additionally caution about the use of excessive multiple srsic functions for decomposing galaxies and compare with past fits in the literature .
the structural parameters obtained from our fitted models are in general , in good agreement with our initial study using radially limited ( @xmath0 ) profiles , and are used here to update several `` central '' as well as `` global '' galaxy scaling relations .
we find near - linear relations between the break radius @xmath1 and the spheroid luminosity @xmath2 such that @xmath3 , and with the supermassive black hole mass @xmath4 such that @xmath5 .
this is internally consistent with the notion that major , dry mergers add the stellar and black hole mass in equal proportion , i.e. , @xmath6 .
in addition , we observe a linear relation @xmath7 for the core - srsic elliptical galaxies where @xmath8 is the galaxies effective half light radii which is collectively consistent with the approximately - linear , bright - end of the curved @xmath9 relation .
finally , we measure accurate stellar mass deficits @xmath10 that are in general 0.5@xmath11 @xmath4 , and we identify two galaxies ( ngc 1399 , ngc 5061 ) that , due to their high @xmath12 ratio , may have experienced oscillatory core - passage by a ( gravitational radiation)-kicked black hole .
the galaxy scaling relations and stellar mass deficits favor core - srsic galaxy formation through a few `` dry '' major merger events involving supermassive black holes such that @xmath13 , for @xmath14 .
[ firstpage ] galaxies : elliptical and lenticular , cd galaxies : fundamental parameters galaxies : nuclei galaxies : photometry galaxies : structure | 1310.5867 |
due to the high frequency of the fastest internal motions in molecular systems , the discrete time step for molecular dynamics simulations must be very small ( of the order of femtoseconds ) , while the actual span of biochemical proceses typically require the choice of relatively long total times for simulations ( e.g. , from microseconds to milliseconds for protein folding processes ) .
in addition to this , since biologically interesting molecules ( like proteins @xcite and dna @xcite ) consist of thousands of atoms , their trajectories in configuration space are esentially chaotic , and therefore reliable quantities can be obtained from the simulation only after statistical analysis @xcite . in order to cope with these two requirements , which force the computation of a large number of dynamical steps
if predictions want to be made , great efforts are being done both in hardware @xcite and in software @xcite solutions .
in fact , only in very recent times , simulations for interesting systems of hundreds of thousand of atoms in the millisecond scale are starting to become affordable , being still , as we mentioned , the main limitation of these computational techniques the large difference between the elemental time step used to integrate the equations of motion and the total time span needed to obtain useful information . in this context ,
strategies to increase the time step are very valuable .
a widely used method to this end is to constrain some of the internal degrees of freedom @xcite of a molecule ( typically bond lengths , sometimes bond angles and rarely dihedral angles . for a verlet - like integrator @xcite , stability requires the time step to be at least about five times smaller than the period of the fastest vibration in the studied system @xcite .
here is where constraints come into play . by constraining the hardest degrees of freedom ,
the fastest vibrational motions are frozen , and thus larger time steps still produce stable simulations . if constraints are imposed on bond lengths involving hydrogens , the time step can typically be increased by a factor of 2 to 3 ( from 1 fs to 2 or 3 fs ) @xcite .
constraining additional internal degrees of freedom , such as heavy atoms bond lengths and bond angles , allows even larger timesteps @xcite , but one has to be careful , since , as more and softer degrees of freedom are constrained , the more likely it is that the physical properties of the simulated system could be severely distorted @xcite .
the essential ingredient in the calculation of the forces produced by the imposition of constraints are the so - called lagrange multipliers @xcite , and their efficient numerical evaluation is therefore of the utmost importance . in this work ,
we show that the fact that many interesting biological molecules are esentially linear polymers allows to calculate the lagrange multipliers in order @xmath0 operations ( for a molecule where @xmath0 constraints are imposed ) in an exact ( up to machine precision ) , non - iterative way .
moreover , we provide a method to do so which is based in a clever ordering of the constraints indices , and in a recently introduced algorithm for solving linear banded systems @xcite .
it is worth mentioning that , in the specialized literature , this possibility has not been considered as far as we are aware ; with some works commenting that solving this kind of linear problems ( or related ones ) is costly ( but not giving further details ) @xcite , and some other works explicitly stating that such a computation must take @xmath2 @xcite or @xmath3 @xcite operations .
also , in the field of robot kinematics , many @xmath1 algorithms have been devised to deal with different aspects of constrained physical systems ( robots in this case ) @xcite , but none of them tackles the calculation of the lagrange multipliers themselves .
this work is structured as follows . in sec .
[ sec_aclm ] , we introduce the basic formalism for the calculation of constraint forces and lagrange multipliers . in sec .
[ soc ] , we explain how to index the constraints in order for the resulting linear system of equations to be banded with the minimal bandwidth ( which is essential to solve it efficiently ) .
we do this starting by very simple toy systems and building on complexity as we move forward towards the final discussion about dna and proteins ; this way of proceeding is intended to help the reader build the corresponding indexing for molecules not covered in this work . in sec . [
sec : numerical ] , we apply the introduced technique to a polyalanine peptide using the amber molecular dynamics package and we compare the relative efficiency between the calculation of the lagrange multipliers in the traditional way ( @xmath2 ) and in the new way presented here ( @xmath1 ) . finally , in sec . [ sec : conclusions ] , we summarize the main conclusions of this work and outline some possible future applications .
if holonomic , rheonomous constraints are imposed on a classical system of @xmath4 atoms , and the dalembert s principle is assumed to hold , its motion is the solution of the following system of differential equations @xcite : [ sistembasico ] @xmath5 where ( [ newton ] ) is the modified newton s second law and ( [ constr ] ) are the equations of the constraints themselves ; @xmath6 are the lagrange multipliers associated with the constraints ; @xmath7 represents the external force acting on atom @xmath8 , @xmath9 is its euclidean position , and @xmath10 colectively denote the set of all such coordinates .
we assume @xmath7 to be conservative , i.e. , to come from the gradient of a scalar potential function @xmath11 ; and @xmath12 should be regarded as the _ force of constraint _ acting on atom @xmath8 . also , in the above expression and in this whole document we will use the following notation for the different indices : * @xmath13 ( except if otherwise stated ) for atoms .
* @xmath14 ( except if otherwise stated ) for the atoms coordinates when no explicit reference to the atom index needs to be made .
* @xmath15 for constrains and the rows and columns of the associated matrices . * @xmath16 as generic indices for products and sums .
the existence of @xmath17 constraints turns a system of @xmath18 differential equations with @xmath19 unknowns into a system of @xmath20 algebraic - differential equations with @xmath20 unknowns .
the constraints equations in ( [ constr ] ) are the new equations , and the lagrange multipliers are the new unknowns whose value must be found in order to solve the system . if the functions @xmath21 are analytical , the system of equations in ( [ sistembasico ] ) is equivalent to the following one : @xmath22 in this new form , it exists a more direct path to solve for the lagrange multipliers : if we explicitly calculate the second derivative in eq .
( [ constr2 ] ) and then substitute eq .
( [ newton2 ] ) where the accelerations appear , we arrive to @xmath23 where we have implicitly defined [ pq ] @xmath24 and it becomes clear that , at each @xmath25 , the lagrange multipliers @xmath26 are actually a _ known _ function of the positions and the velocities .
we shall use the shorthand @xmath27 and , @xmath28 , @xmath29 , and @xmath30 to denote the whole @xmath0-tuples , as usual .
now , in order to obtain the lagrange multipliers @xmath31 , we just need to solve @xmath32 this is a linear system of @xmath17 equations and @xmath17 unknowns . in the following
, we will prove that the solution to it , when constraints are imposed on typical biological polymers , can be found in @xmath33 operations without the use of any iterative or truncation procedure , i.e. , in an exact way up to machine precision . to show this , first
, we will prove that the value of the vectors @xmath29 and @xmath30 can be obtained in @xmath33 operations .
then , we will show that the same is true for all the non - zero entries of matrix @xmath34 , and finally we will briefly discuss the results in @xcite , where we introduced an algorithm to solve the system in ( [ lm ] ) also in @xmath33 operations .
it is worth remarking at this point that , in this work , we will only consider constraints that hold the distance between pairs of atoms constant , i.e. , @xmath35 where @xmath36 is a constant number , and the fact that we can establish a correspondence between constrained pairs ( @xmath37 ) and the constraints indices has been explicitly indicated by the notation @xmath38 .
this can represent a constraint on : * a bond length between atoms @xmath8 and @xmath39 , * a bond angle between atoms @xmath8 , @xmath39 and @xmath40 , if both @xmath8 and @xmath39 are connected to @xmath40 through constrained bond lengths , * a principal dihedral angle involving @xmath8 , @xmath39 , @xmath40 and @xmath41 ( see @xcite for a rigorous definition of the different types of internal coordinates ) , if the bond lengths ( @xmath37 ) , ( @xmath42 ) and ( @xmath43 ) are constrained , as well as the bond angles ( @xmath44 ) and ( @xmath45 ) , * or a phase dihedral angle involving @xmath8 , @xmath39 , @xmath40 and @xmath41 if the bond lengths ( @xmath37 ) , ( @xmath42 ) and ( @xmath46 ) are constrained , as well as the bond angles ( @xmath44 ) and ( @xmath47 ) . this way to constrain degrees of freedom
is called _
triangularization_. if no triangularization is desired ( as , for example , if we want to constrain dihedral angles but not bond angles ) , different explicit expressions than those in the following paragraphs must be written down , but the basic concepts introduced here are equally valid and the main conclusions still hold . now , from eq .
( [ sigma_generica ] ) , we obtain @xmath48 inserting this into ( [ pq1 ] ) , we get a simple expression for @xmath49 @xmath50 the calculation of @xmath51 is more involved , but it also results into a simple expression : first , we remember that the indices run as @xmath14 , and @xmath52 , and we produce the following trivial relationship : @xmath53 where @xmath54 , @xmath55 and @xmath56 are the unitary vectors along the @xmath10 , @xmath57 and @xmath58 axes , respectively . therefore , much related to eq .
( [ grad_sigma ] ) , we can compute the first derivative of @xmath59 : @xmath60 \ , \end{aligned}\ ] ] and also the second derivative : @xmath61 \nonumber \\ & & \mbox { } \cdot [ ( \delta_{3\alpha-2,\nu}\hat{i } + \delta_{3\alpha-1,\nu}\hat{j } + \delta_{3\alpha,\nu}\hat{k } ) -(\delta_{3\beta-2,\nu}\hat{i } + \delta_{3\beta-1,\nu}\hat{j } + \delta_{3\beta,\nu}\hat{k } ) ] \nonumber \\ & = & 2 ( \delta_{3\alpha-2,\mu}\delta_{3\alpha-2,\nu } + \delta_{3\beta-2,\mu}\delta_{3\beta-2,\nu } -\delta_{3\alpha-2,\mu}\delta_{3\beta-2,\nu } -\delta_{3\beta-2,\mu}\delta_{3\alpha-2,\nu } \nonumber \\ & & \mbox { } + \delta_{3\alpha-1,\mu}\delta_{3\alpha-1,\nu } + \delta_{3\beta-1,\mu}\delta_{3\beta-1,\nu } -\delta_{3\alpha-1,\mu}\delta_{3\beta-1,\nu } -\delta_{3\beta-1,\mu}\delta_{3\alpha-1,\nu } \nonumber \\ & & \mbox { } + \delta_{3\alpha,\mu}\delta_{3\alpha,\nu } + \delta_{3\beta,\mu}\delta_{3\beta,\nu } -\delta_{3\alpha,\mu}\delta_{3\beta,\nu } -\delta_{3\beta,\mu}\delta_{3\alpha,\nu } ) \ .\end{aligned}\ ] ] taking this into the original expression for @xmath51 in eq .
( [ pq2 ] ) and playing with the sums and the deltas , we arrive to @xmath62 now , eqs .
( [ defo ] ) , ( [ neop ] ) and ( [ neoq ] ) can be gathered together to become @xmath63 where we can see that the calculation of @xmath64 takes always the same number of operations , independently of the number of atoms in our system , @xmath4 , and the number of constraints imposed on it , @xmath17 .
therefore , calculating the whole vector @xmath28 in eq .
( [ lm ] ) scales like @xmath17 . in order to obtain an explicit expression for the entries of the matrix @xmath34
, we now introduce eq .
( [ grad_sigma ] ) into its definition in eq .
( [ defr ] ) : @xmath65 where we have used that @xmath66 looking at this expression , we can see that a constant number of operations ( independent of @xmath4 and @xmath17 ) is required to obtain the value of every entry in @xmath34 .
the terms proportional to the kroenecker deltas imply that , as we will see later , in a typical biological polymer , the matrix @xmath34 will be sparse ( actually banded if the constraints are appropriately ordered as we describe in the following sections ) , being the number of non - zero entries actually proportional to @xmath0 .
more precisely , the entry @xmath67 will only be non - zero if the constraints @xmath68 and @xmath69 share an atom .
now , since both the vector @xmath28 and the matrix @xmath34 in eq .
( [ lm ] ) can be computed in @xmath1 operations , it only remains to be proved that the solution of the linear system of equations is also an @xmath1 process , but this is a well - known fact when the matrix defining the system is banded . in @xcite , we introduced a new algorithm to solve this kind of banded systems which is faster and more accurate than existing alternatives .
essentially , we shown that the linear system of equations @xmath70 where @xmath71 is a @xmath72 matrix , @xmath10 is the @xmath73 vector of the unknowns , @xmath74 is a given @xmath73 vector and @xmath71 is _ banded _ , i.e. , it satisfies that for known @xmath75 @xmath76 can be directly solved up to machine precision in @xmath77 operations .
this can be done using the following set of recursive equations for the auxiliary quantities @xmath78 ( see @xcite for details ) : [ coefsband ] @xmath79 if the matrix @xmath71 is symmetric ( @xmath80 ) , as it is the case with @xmath34 [ see ( [ defr ] ) ] , we can additionally save about one half of the computation time just by using @xmath81 instead of ( [ xifinal_c ] ) .
( [ xifinal_csim ] ) can be obtained from ( [ coefsband ] ) by induction , and we recommend these expressions for the @xmath82 coefficients because other valid ones ( like considering @xmath83 , @xmath84 , which involves square roots ) are computationally more expensive . in the next sections ,
we show how to index the constraints in such a way that nearby indices correspond to constraints where involved atoms are close to each other and likely participate of the same constraints .
in such a case , not only will the matrix @xmath34 in eq .
( [ lm ] ) be banded , allowing to use the method described above , but it will also have a minimal bandwidth @xmath85 , which is also an important point , since the computational cost for solving the linear system scales as @xmath86 ( when the bandwidth is constant ) .
in this section we describe how to index the constraints applied to the bond lengths and bond angles of a series of model systems and biological molecules with the already mentioned aim of minimizing the computational cost associated to the obtention of the lagrange multipliers .
the presentation begins by deliberately simple systems and proceeds to increasingly more complicated molecules with the intention that the reader is not only able to use the final results presented here , but also to devise appropriate indexings for different molecules not covered in this work .
the main idea we have to take into account , as expressed in section [ sec_aclm ] , is to use nearby numbers to index constraints containing the same atoms .
if we do so , we will obtain _ banded _
@xmath34 matrices .
further computational savings can be obtained if we are able to reduce the number of @xmath82 coefficients in eqs .
( [ coefsband ] ) to be calculated . in more detail , solving a linear system like ( [ lm ] ) where the @xmath34 is @xmath87 and banded with semi - band width ( i.e. , the number of non - zero entries neighbouring the diagonal in one row or column ) @xmath85 requires @xmath88 operations if @xmath85 is a constant .
therefore , the lower the value of @xmath85 , the smaller the number of required numerical effort .
when the semi - band width @xmath85 is not constant along the whole matrix , things are more complicated and the cost is always between @xmath89 and @xmath90 , depending on how the different rows are arranged . in general , we want to minimize the number of zero fillings in the process of gaussian elimination ( see @xcite for further details ) , which is achieved by not having zeros below non - zero entries .
this is easier to understand with an example : consider the following matrices , where @xmath91 and @xmath92 represent different non - zero values for every entry ( i.e. , not all @xmath92 , nor all @xmath91 must take the same value , and different symbols have been chosen only to highlight the main diagonal ) : @xmath93 during the gaussian elimination process that is behind ( [ coefsband ] ) , in @xmath71 , five coefficients @xmath82 above the diagonal are to be calculated , three in the first row and two in the second one , because the entries below non - zero entries become non - zero too as the elimination process advances ( this is what we have called ` zero filling ' ) .
on the other hand , in @xmath94 , which contains the same number of non - zero entries as @xmath71 , only three coefficients @xmath82 have to be calculated : two in the first row and one in the second row .
whether @xmath34 looks like @xmath71 or like @xmath94 depends on our choice of the constraints ordering .
one has also to take into account that no increase in the computational cost occurs if a series of non - zero columns is separated from the diagonal by columns containing all zeros .
i.e. , the linear systems associated to the following two matrices require the same numerical effort to be solved : @xmath95 as promised , we start by a simple model of a biomolecule : an open linear chain without any branch . in this case
, the atoms should be trivially numbered as in fig .
[ fig : lc ] ( any other arrangement would have to be justified indeed ! ) . if we only constrain bond lengths , the fact that only consecutive atoms participate of the same constraints allows us to simplify the notation with respect to eq .
( [ sigma_generica ] ) and establish the following ordering for the constraints indices : @xmath96 with @xmath97 this choice results in a tridiagonal matrix @xmath34 , whose only non - zero entries are those lying in the diagonal and its first neighbours .
this is the only case for which an exact calculation of the lagrange multipliers exists in the literature as far as we are aware @xcite .
the next step in complexity is to constrain the bond angles of the same linear chain that we discussed above .
the atoms are ordered in the same way , as in fig .
[ fig : lc ] , and the trick to generate a banded matrix @xmath34 with minimal bandwidth is to alternatively index bond length constraints with odd numbers , @xmath98 and bond angle constraints with even ones , @xmath99 where the regular pattern involving the atom indicies that participate of the same constraints has allowed again to use a lighter notation . the constraints equations in this case are @xmath100 respectively , and , if this indexing is used , @xmath34 is a banded matrix where @xmath85 is 3 and 4 in consecutive rows and columns .
therefore , the mean @xmath101 is 3.5 , and the number of @xmath82 coefficients that have to be computed per row in the gaussian elimination process is the same because the matrix contains no zeros that are filled .
a further feature of this system ( and other systems where both bond lengths and bond angles are constrained ) can be taken into account in order to reduce the computational cost of calculating lagrange multipliers in a molecular dynamics simulation : a segment of the linear chain with constrained bond lengths and bond angles is represented in fig .
[ fig : angs ] , where the dashed lines correspond to the virtual bonds between atoms that , when kept constant , implement the constraints on bond angles ( assuming that the bond lengths , depicted as solid lines , are also constrained ) . due to the fact that all these distances are constant , many of the entries of @xmath34 will remain unchanged during the molecular dynamics simulation . as an example
, we can calculate @xmath102 where we have used the law of cosines .
the right - hand side does not depend on any time - varying objects ( such as @xmath9 ) , being made of only constant quantities .
therefore , the value of @xmath103 ( and many other entries ) needs not to be recalculated in every time step , which allows to save computation time in a molecular dynamics simulation . in order to incrementally complicate the calculations
, we now turn to a linear molecule with only one atom connected to the backbone , such the one displayed in figure [ fig : branched1 ] .
the corresponding equations of constraint and the ordering in the indices that minimizes the bandwidth of the linear system are [ singlybranched ] @xmath104 where the trick this time has been to alternatively consider atoms in the backbone and atoms in the branches as we proceed along the chain .
the matrix @xmath34 of this molecule presents a semi - band width which is alternatively 2 and 1 in consecutive rows / columns , with average @xmath105 and the same number of superdiagonal @xmath82 coefficients to be computed per row . the next molecular topology we will consider is that of an alkane ( a familiy of molecules with a long tradition in the field of constraints @xcite ) , i.e. , a linear backbone with two 1-atom branches attached to each site ( see fig . [
fig : nalkane ] ) .
the ordering of the constraints that minimizes the bandwidth of the linear system for this case is @xmath106 where the trick has been in this case to alternatively constrain the bond lengths in the backbone and those connecting the branching atoms to one side or the other .
the resulting @xmath34 matrix require the calculation of 2 @xmath82 coefficients per row when solving the linear system . if we want to additionally constrain bond angles in a molecule with the topology in fig .
[ fig : branched1 ] , the following ordering is convenient : @xmath107 this ordering produces 16 non - zero entries above the diagonal per each group of 4 rows in the matrix @xmath34 when making the calculations to solve the associated linear system .
this is , we will have to calculate a mean of @xmath108 super - diagonal coefficients @xmath82 per row . when we studied the linear molecule with constrained bond lengths and bond angles , this mean was equal to @xmath109 , so including minimal branches in the linear chain makes the calculations just slightly longer . if we now want to add bond angle constraints to the bond length ones described in sec .
[ sec : doubly_branched ] for alkanes , the following ordering produces a matrix @xmath34 with a low half - band width : @xmath110 in this case , the average number of @xmath82 coefficients to be calculated per row is approximately 5.7 . if we have cycles in our molecules , the indexing of the constraints is only slightly modified with respect to the open cases in the previous sections .
for example , if we have a single - branch cyclic topology , such as the one displayed in fig .
[ fig : rings]a , the ordering of the constraints is the following : @xmath111 these equations are the same as those in [ secbl ] , plus a final constraint corresponding to the bond which closes the ring .
these constraints produce a matrix @xmath34 where only the diagonal entries , its first neighbours , and the entries in the corners ( @xmath112 and @xmath113 ) are non - zero .
in this case , the associated linear system in eq .
( [ lm ] ) can also be solved in @xmath1 operations , as we discuss in @xcite . in general , this is also valid whenever @xmath34 is a sparse matrix with only a few non - zero entries outside of its band , and therefore we can apply the technique introduced in this work to molecular topologies containing more than one cycle . the ordering of the constraints and the resulting linear systems for different cyclic species , such as the one depicted in fig .
[ fig : rings]b , can be easily constructed by the reader using the same basic ideas . as we discussed in sec .
[ sec : introduction ] , proteins are one of the most important families of molecules from the biological point of view : proteins are the nanomachines that perform most of the complex tasks that need to be done in living organisms , and therefore it is not surprising that they are involved , in one way or another , in most of the diseases that affect animals and human beings . given the efficiency and precision with which proteins carry out their missions , they are also being explored from the technological point of view .
the applications of proteins even outside the biological realm are many if we could harness their power @xcite , and molecular dynamics simulations of great complexity and scale are being done in many laboratories around the world as a tool to understand them @xcite .
proteins present two topological features that simplify the calculation of the lagrange multipliers associated to constraints imposed on their degrees of freedom : represents the first numbered atom in each residue ( the amino nitrogen ) and @xmath114 is the number of atoms in the side chain .
* b ) * indexing of the bond length constraints ; @xmath68 denotes the index of the first constraint imposed on the residue ( the n - h bond length ) and @xmath115 is the variable number of constraints imposed on the side chain.,width=302 ] * they are linear polymers , consisting of a backbone with short ( 17 atoms at most ) groups attached to it @xcite .
this produces a banded matrix @xmath34 , thus allowing the solution of the associated linear problem in @xmath1 operations . even in the case
that disulfide bridges , or any other covalent linkage that disrupts the linear topology of the molecule , exist , the solution of the problem can still be found efficiently if we recall the ideas discussed in sec .
[ ring1 ] .
* the monomers that typically make up these biological polymers , i.e. , the residues associated to the proteinogenic aminoacids , are only 20 different molecular structures .
therefore , it is convenient to write down explicitly one block of the @xmath34 matrix for each known monomer , and to build the @xmath34 matrix of any protein simply joining together the precalculated blocks associated to the corresponding residues the protein consists of . the structure of a segment of the backbone of a protein chain is depicted in fig .
[ fig : prot ] .
the green spheres represent the side chains , which are the part of the amino acid residue that can differ from one monomer to the next , and which usually consist of several atoms : from 1 atom in the case of glycine to 17 in arginine or tryptophan . in fig .
[ fig : prot]a , we present the numbering of the atoms , which will support the ordering of the constraints , and , in fig . [
fig : prot]b , the indexing of the constraints is presented for the case in which only bond lengths are constrained ( the bond lengths plus bond angles case is left as an exercise for the reader ) .
using the same ideas and notation as in the previous sections and denoting by @xmath116 the block of the matrix @xmath34 that corresponds to a given amino acid residue @xmath117 , with @xmath118 , we have that , for the monomer dettached of the rest of the chain , @xmath119 where the explicit non - zero entries are related to the constraints imposed on the backbone and @xmath120 denotes a block associated to those imposed on the bonds that belong to the different sidechains .
the dimension of this matrix is @xmath121 and the maximum possible semi - band width is 12 for the bulkiest residues . a protein s global matrix @xmath34 has to be built by joining together blocks like the one above , and adding the non - zero elements related to the imposition of constraints on bond lengths that connect one residue with the next .
these extra elements are denoted by @xmath122 and a general scheme of the final matrix is shown in fig .
[ fig : proteinmatrix ] . for a protein molecule with @xmath123 residues . in black
, we represent the potentially non - zero entries , and each large block in the diagonal is given by ( [ defr]).,width=264 ] the white regions in this scheme correspond to zero entries , and we can easily check that the matrix is banded .
in fact , if each one of the diagonal blocks is constructed conveniently , they will contain many zeros themselves and the bandwidth can be reduced further .
the size of the @xmath124 blocks will usually be much smaller than that of their neighbour diagonal blocks .
for example , in the discussed case in which we constrain all bond lengths , @xmath124 are @xmath125 ( or @xmath126 ) blocks , and the diagonal blocks size is between @xmath127 ( glycine ) and @xmath128 ( tryptophan ) .
nucleic acids are another family of very important biological molecules that can be tackled with the techniques described in this work .
dna and rna , the two subfamilies of nucleic acids , consist of linear chains made up of a finite set of monomers ( called ` bases ' ) .
this means that they share with proteins the two features mentioned in the previous section and therefore the lagrange multipliers associated to the imposition of constraints on their degrees of freedom can be efficiently computed using the same ideas .
it is worth mentioning that dna typically appears in the form of two complementary chains whose bases form hydrogen - bonds . since these bonds are much weaker than a covalent bond , imposing bond length constraints on them such as the ones in eq .
( [ sigma_generica ] ) would be too unrealistic for many practical purposes , denotes the index of the first constraint imposed on the nucleotide and @xmath115 is the variable number of constraints imposed on the bonds in the base.,width=340 ] in fig .
[ fig : dna_constrs ] , and following the same ideas as in the previous section , we propose a way to index the bond length constraints of a dna strand which produces a banded matrix @xmath34 of low bandwidth .
green spheres represent the ( many - atom ) bases ( a , c , t or g ) , and the general path to be followed for consecutive constraint indices is depicted in the upper left corner : first the sugar ring , then the base and finally the rest of the nucleotide , before proceeding to the next one in the chain .
this ordering translates into the following form for the block of @xmath34 corresponding to one single nucleotide dettached from the rest of the chain : @xmath129 where @xmath120 is the block associated to the constraints imposed on the bonds that are contained in the base , @xmath130 , @xmath131 , @xmath132 , and @xmath133 are very sparse rectangular blocks with only a few non - zero entries in them , and the form of the diagonal blocks associated to the sugar ring and backbone constraints is the following : [ eq : r11r33 ] @xmath134 analagously to the case of proteins , as many blocks as those in eq .
( [ rpartdna ] ) as nucleotides contains a given dna strand have to be joined to produce the global matrix @xmath34 of the whole molecule , together with the @xmath122 blocks associated to the constraints on the bonds that connect the different monomers . in fig .
[ fig : matrixadn ] , a scheme of this global matrix is depicted and we can appreciate that it indeed banded . the construction of the matrix @xmath34 for a rna molecule should follow the same steps and the result will be very similar . for a dna molecule with @xmath135 nucleotides . in black , we represent the potentially non - zero entries , and each large block in the diagonal is given by ( [ rpartdna]).,width=264 ]
in this section , we apply the efficient technique introduced in this work to a series of polyalanine molecules in order to calculate the lagrange multipliers when bond length constraints are imposed . we also compare our method , both in terms of accuracy and numerical efficiency , to the traditional inversion of the matrix @xmath34 without taking into account its banded structure .
we used the code avogadro @xcite to build polyalanine chains of @xmath1362 , 5 , 12 , 20 , 30 , 40 , 50 , 60 , 80 , 90 and 100 residues , and we chose their initial conformation to be approximately an alpha helix , i.e. , with the values of the ramachandran angles in the backbone @xmath137 and @xmath138 @xcite .
next , for each of these chains , we used the molecular dynamics package amber @xcite to produce the atoms positions ( @xmath10 ) , velocities ( @xmath139 ) and external forces ( @xmath140 ) needed to calculate the lagrange multipliers ( see sec .
[ sec_aclm ] ) after a short equilibration molecular dynamics simulations .
we chose to constrain all bond lengths , but our method is equally valid for any other choice , as the more common constraining only of bonds that involve hydrogens . in order to produce reasonable final conformations , we repeated the following process for each of the chains : * solvation with explicit water molecules .
* minimization of the solvent positions holding the polypeptide chain fixed ( 3,000 steps ) . * minimization of all atoms positions ( 3,000 steps ) .
* thermalization : changing the temperature from 0 k to 300 k during 10,000 molecular dynamics steps .
* stabilization : 20,000 molecular dynamics steps at a constant temperature of 300 k. * measurement of @xmath10 , @xmath139 and @xmath140 .
neutralization is not necessary , because our polyalanine chains are themselves neutral . in all calculations we used the force field described in @xcite ,
chose a cutoff for coulomb interactions of 10 and a time step equal to 0.002 ps , and impose constraints on all bond lengths as mentioned .
in the thermostated steps , we used langevin dynamics with a collision frequency of 1 ps@xmath141 .
.,width=302 ] using the information obtained and the indexing of the constraints described in this work , we constructed the matrix @xmath34 and the vector @xmath28 and proceeded to find the lagrange multipliers using eq .
( [ lm ] ) .
since ( [ lm ] ) is a linear problem , one straightforward way to solve is to use traditional gauss - jordan elimination or lu factorization @xcite .
but these methods have a drawback : they scale with the cube of the size of the system .
i.e. , if we imposed @xmath0 constraints on our system ( and therefore we needed to obtain @xmath0 lagrange multipliers ) , the number of floating point operations that these methods would require is proportional to @xmath142 .
however , as we showed in the previous sections , the fact that many biological molecules , and proteins in particular , are essentially linear , allows to index the constraints in such a way that the matrix @xmath34 in eq .
( [ lm ] ) is banded and use different techniques for solving the problem which require only @xmath1 floating point operations @xcite .
[ table : pruebas ] .comparison of numerical complexity and accuracy between a traditional gauss - jordan solver and the banded algorithm described in this work , for the calculation of the lagrange multipliers on a series of polyalanine chains as a function of their number of residues @xmath143 . [ cols="^,^,^,^,^ " , ] in fig .
[ fig : pruebas1 ] and table [ table : pruebas ] , we compare both the accuracy and the execution time of the two different methods : gauss - jordan elimination @xcite , and the banded recursive solution advocated here and made possible by the appropriate indexing of the constraints .
the calculations have been run on a mac os x laptop with a 2.26 ghz intel core 2 duo processor , and the errors were measured using the normalized deviation of @xmath144 from @xmath145 .
i.e. , if we denote by @xmath146 the solution provided by the numerical method , @xmath147 from the obtained results , we can see that both methods produce an error which is very small ( close to machine precision ) , being the accuracy of the banded algorithm advocated in this work slightly higher .
regarding the computational cost , as expected , the gauss - jordan method presents an effort that approximately scales with the cube of the number of constraints @xmath0 ( which is approximately proportional to @xmath143 ) , while the banded technique allowed by the particular structure of the matrix @xmath34 follows a rather accurate lineal scaling .
although it is typical that , when two such different behaviours meet , there exists a range of system sizes for which the method that scales more rapidly is faster and then , at a given system size , a crossover takes place and the slower scaling method becomes more efficient from there on , in this case , and according to the results obtained , the banded technique is less time - consuming for all the explored molecules , and the crossover should exist at a very small system size ( if it exists at all ) .
this is very relevant for any potential uses of the methods introduced in this work .
we have shown that , if we are dealing with typical biological polymers , whose covalent connectivity is that of essentially linear objects , the lagrange multipliers that need to be computed when @xmath0 constraints are imposed on their internal degrees of freedom ( such as bond lengths , bond angles , etc . ) can be obtained in @xmath1 steps as long as the constraints are indexed in a convenient way and banded algorithms are used to solve the associated linear system of equations .
this path has been traditionally regarded as too costly in the literature @xcite , and , therefore , our showing that it can be implemented efficiently could have profound implications in the design of future molecular dynamics algorithms . since the field of imposition of constraints in moleculary dynamics simulations is dominated by methods that cleverly achieve that the system exactly stays on the constrained subspace as the simulation proceeds by not calculating the exact lagrange multipliers , but a modification of them instead @xcite , we are aware that the application of the new techniques introduced here is not a direct one .
however , we are confident that the low cost of the new method and its close relationship with the problem of constrained dynamics could prompt many advances , some of which are already being pursued in our group . among the most promising lines , we can mention a possible improvement of the shake method @xcite by the use of the exact lagrange multipliers as a guess for the iterative procedure that constitutes its most common implementation .
also , we are studying the possibility of solving the linear problems that appear either in a different implementation of shake ( mentioned in the original work too @xcite ) or in the lincs method @xcite , and which are defined by matrices which are different from but related to the matrix @xmath34 introduced in this work , being also banded if an appropriate indexing of the constraints is used .
finally , we are exploring an extension of the ideas introduced here to the calculation not only of the lagrange multipliers but also of their time derivatives , to be used in higher order integrators than verlet .
we would like to thank giovanni ciccotti for illuminating discussions and wise advices , and claudio cavasotto and isaas lans for the help with the setting up and use of amber .
the numerical calculations have been performed at the bifi supercomputing facilities ; we thank all the staff there for the help and the technical assistance .
this work has been supported by the grants fis2009 - 13364-c02 - 01 ( micinn , spain ) , grupo de excelencia `` biocomputacin y fsica de sistemas complejos '' , e24/3 ( aragn region government , spain ) , araid and ibercaja grant for young researchers ( spain ) .
r . is supported by a jae predoc scholarship ( csic , spain ) . , r. ron o. dror , j. salmon , j. grossman , k. mackenzie , j. bank , c. young , b. batson , k. bowers , e. edmond chow , m. eastwood , d. ierardi , j. john l. klepeis , j. jeffrey s. kuskin , r. larson , k. kresten lindorff - larsen , p. maragakis , m. m.a .
, s. piana , s. yibing , and b. towles , , in proceedings of the acm / ieee conference on supercomputing ( sc09 ) , acm press , new york , ( 2009 ) . ,
t. darden , t. e. cheatham , c. simmerling , w. junmei , d. r. e. , r. luo , k. m. merz , m. a. pearlman , m. crowley , r. walker , z. wei , w. bing , s. hayik , a. roitberg , g. seabra , w. kim , f. paesani , w. xiongwu , v. brozell , s. tsui , h. gohlke , y. lijiang , t. chunhu , j. mongan , v. hornak , p. guanglei , c. beroza , d. h. mathews , c. schafmeister , w. s. ross , and p. a. kollman , , university of california : san francisco ( 2006 ) . | in order to accelerate molecular dynamics simulations , it is very common to impose holonomic constraints on their hardest degrees of freedom . in this way ,
the time step used to integrate the equations of motion can be increased , thus allowing , in principle , to reach longer total simulation times . the imposition of such constraints results in an aditional set of @xmath0 equations ( the equations of constraint ) and unknowns ( their associated lagrange multipliers ) , that must be solved in one way or another at each time step of the dynamics . in this work it is shown
that , due to the essentially linear structure of typical biological polymers , such as nucleic acids or proteins , the algebraic equations that need to be solved involve a matrix which is banded if the constraints are indexed in a clever way .
this allows to obtain the lagrange multipliers through a non - iterative procedure , which can be considered exact up to machine precision , and which takes @xmath1 operations , instead of the usual @xmath2 for generic molecular systems .
we develop the formalism , and describe the appropriate indexing for a number of model molecules and also for alkanes , proteins and dna .
finally , we provide a numerical example of the technique in a series of polyalanine peptides of different lengths using the amber molecular dynamics package . + * keywords : * constraints , lagrange multipliers , banded systems , molecular dynamics , proteins , dna + | 1103.4425 |
in @xcite , randall and sundrum ( rs ) proposed a new approach to extra dimensions for space - time to solve the hierarchy problem . in their model , it is assumed that there exists only one extra space - like dimension which is taken homeomorphic with an orbifold @xmath1 .
this orbifold has two fixed points at @xmath2 and @xmath3 . at each fixed point
, they put a 4-dimensional brane world .
one of them which is located at @xmath3 is called the visible world and is assumed we are living on it , and the other one is called hidden world . in rs method
same as the arkani - hamed , dimopoulos and dvali ( add ) approach @xcite , it is assumed that except the graviton ( and also axions ) all the standard model ( sm ) fields are confined in these two distinct worlds .
the physical laws are the same on these two worlds but the masses and the coupling constants may differ . in this model , the classical action is assumed to be : s&= & s_gravity + s_vis + s_hid , + s_gravity&= & d^4x d \{+ 2 m r } , + s_vis&= & d^4x \{l_vis -v_vis } , + s_hid&= & d^4x \{l_hid -v_hid } , where @xmath4 and @xmath5 are the 5-dimensional planck mass and the cosmological constant respectively , and @xmath6 ( @xmath7 ) is the sm or any effective lagrangian corresponding to matter and force fields except the gravity .
the @xmath8 and @xmath9 are vacuum expectations on the branes .
the classical solution of einstein equation for the mentioned action is the following metric : ds^2 = e^-2 ( ) _
dx^dx^+ r_c^2 d^2 , with ( ) = r_c and radius of @xmath10 in the orbifold @xmath11 and @xmath12 . by integrating out the fifth dimension , the coupling constant of the effective 4-dimensional action yields the 4-dimensional planck scale @xcite : m_p = m^3 (1-e^-2r_c ) .
it is found that a field on visible brane with the fundamental mass parameter @xmath13 will appear to have a physical mass @xmath14 . by taking @xmath15 ,
the observed scale hierarchy reproduces naturally by exponential factor and no additional large hierarchies arise @xcite . at this stage , it is natural to search for any observable effects of this extra fifth dimension in rs model .
many efforts have been done to probe the effects of this extra dimension in ordinary particle interactions @xcite
. there would be two kinds of gravitons in this formalism ; the first type is massless ordinary graviton , which is also confined to the 4-dimensional physical space - time , and the others are massive gravitons . in @xcite it is shown that the effects of the massless gravitons in particles interactions are in order of @xmath16 where @xmath17 is the 4-dim planck mass . however , the contributions of the massive ones are considerable and comparable with the weak scale of the standard model .
the masses of gravitons come from kaluza - klein compactification of the fifth dimension . due to non - factorization of the geometry ,
the masses of gravitons are @xmath18 , where @xmath19 s are the roots of @xmath20 , the bessel function of order one . in this paper
, we calculate the total cross - sections of scalar - scalar to photon and graviton fields @xmath21 .
this process in standard model , without producing gravitons is forbidden by energy - momentum conservation .
at the beginning , we consider the scalar electrodynamics as the effective theory of matter and forces in visible and hidden spaces .
the action of this theory is : s_vis = s_sed & = & \ { g^(_-i e a_)^(_+i e a _ ) + & -&m_^2 ^-1 4g^g^f_f _ } d^4x , where @xmath23 and @xmath24 is the inverse of the classical metric ( 1.2 ) .
the factor @xmath5 for massless graviton is equal to @xmath17 and for the massive gravitons is @xmath25 . inserting @xmath26 in eq .
( 2.1 ) and absorbing the conformal factor @xmath27 in scalar fields and their mass , one can reduce the interaction part of lagrangian up to the first order in @xmath28 to the following terms , [ lagrangian ] l_i&= & i e a^(_^-^_)+ e^2 a^a_^ + & + & _ ^_+
i e a_(_^-^ _ ) + & - & ( ^a__a_-2 ^a__a_+ _ a^_a _ ) + & + & e^2 a_a_^. as it is pointed out in ref .
@xcite , we can use @xmath29 as @xmath30 , for this theory . since we are searching for amplitude of @xmath31 , the relevant terms of hamiltonian to this interaction are , [ hamilton ] h_1&=&-i e a^(_^- ^ _ ) , + h_2&=&- _ ^ _ , + h_3&= & -i e a_(_^- ^ _ ) , + h_4&= & ( ^a__a_-2 ^a__a_+ _ a^_a _ ) , which contribute to the following diagrams , ( 0,0)(0,0 ) ( -150,-35 ) ( -100,50)(0,0 ) ( 0,0)(-100,-50 ) ( 0,0)(100,-50)510 ( 0,0)(100,50)5 ( 0,0)3 ( 0,-67)[]the feynman diagram of @xmath32 ( -60,25)[]@xmath33 ( -60,-25)[]@xmath34 ( 58,25)[]@xmath35 ( 58,-25)[]@xmath36 ( 150,-20 ) ( -100,50)(0,0 ) ( 0,0)(100,50)5 ( 0,0)3 ( 0,0)(0,-65 ) ( 0,-65)(-100,-115 ) ( 0,-65)(100,-115)510 ( 0,-65)3 ( 0,-83)[]the feynman diagram of @xmath37 ( -60,25)[]@xmath33 ( -60,-60)[]@xmath34 ( 58,25)[]@xmath35 ( 58,-60)[]@xmath36 3.5 cm ( 0,0)(0,0 ) ( -150,0 ) ( -100,50)(0,-65 ) ( 0,-65)(0,0 ) ( 0,0)(-100,-115 ) ( 0,0)3 ( 0,-65)3 ( 0,0)(100,50)5 ( 0,-65)(100,-115)510 ( 0,-83)[]the feynman diagram of @xmath38 ( -60,25)[]@xmath33 ( -60,-60)[]@xmath34 ( 58,25)[]@xmath35 ( 58,-60)[]@xmath36 ( 115,-20 ) ( -100,50)(0,0 ) ( 0,0)(-110,-50 ) ( 0,0)3 ( 105,0)3 ( 105,0)(205,50)5 ( 0,0)(100,0)510 ( 105,0)(205,-50)510 ( 25,-60)[]the feynman diagram of @xmath39 ( -60,25)[]@xmath33 ( -64,-25)[]@xmath34 ( 112,25)[]@xmath35 ( 112,-25)[]@xmath36 + denoting in - state by @xmath40 where @xmath41 and @xmath42 are the momenta of scalar and anti - scalar particles respectively and out - state by @xmath43 , where @xmath44 and @xmath45 are the momenta due to the massive graviton and photon . now , we are going to derive the @xmath46-matrix elements for the above diagrams in the tree level , & = & < k , q t e^-i_i dt p_1 , p_2 > + & = & i ( 2)^4 ^(4 ) ( q+k - p_1-p_2)m_tot , where @xmath47 is invariant amplitude which is the sum of the following amplitudes ( see figs . ) , [ ampl ] m_1 & = & - _ ( q ) e^ ( k ) ( p_1-p_2 ) _ , + m_2 & = & - ^(q ) e^ ( k ) ( p_1-k)_p_1 ( p_2-p_1+k ) _ , + m_3 & = & - ^(q ) e^ ( k ) ( p_2-k)_p_2 ( p_2-p_1-k ) _ , + m_4&= & ( ( p_1-p_2)__(q(p_1+p_2 ) ) . + & -&.(p_1-p_2)_q_((p_1+p_2))-_(p_1+p_2)_(p_1-p_2)q
. + & + & .q_(p_1+p_2)_(p_1-p_2 ) ) . in the above equation @xmath48 and @xmath49
are the polarization of photon and graviton respectively . to calculate the unpolarized cross - section
, we should make summation over these polarizations .
we have , _
pol.e_(k)e_(k)&=&f_(k ) , + _ pol._(q)_(q)&=&-g _ , in which @xmath50 for a massive graviton is @xcite , & & f_(k)=12\ { g_g_+g_g _ -g_g _ } + & + & 12\ { g_k_k_+ g_k_k_+ g_k_k_+ g_k_k _ } + & + & 2 3(1 2g_-k_k _ ) ( 1 2g_-k_k _ ) , and for a massless graviton , f_(k)=12\ { g_g_+g_g _ -23g_g_}. using the above relations , it is straightforward to calculate the cross - section in the center of mass frame of the incident particles for the massive gravitons .
the differential cross - section of the mentioned process in the center of mass frame is @xcite , [ cs ] ( d)_cm= 1 |k| |m_tot|^2 , where @xmath51 , and @xmath52 is the center of mass energy of incident particles .
the contribution of the massless graviton can be neglected , as it is pointed out in @xcite . for a massive graviton of mass @xmath53 ,
we have calculated in the appendix the terms in @xmath54 . to obtain the total cross - section , as a function of @xmath55
, one should integrate the ( [ cs ] ) over the scattering angles and sum over all massive gravitons which their masses are less than @xmath52 .
for the calculations the computer program mathematica version 3.0 was used .
the final result is the following graph , where the solid curve shows the behavior of the total cross - section versus the energy , @xmath55 . here , we take @xmath56 , @xmath57 and @xmath58 .
according to the mass formula , @xmath59 , we obtain the first four gravitons masses , @xmath60 , @xmath61 , @xmath62 and @xmath63 . for @xmath64 , only the first graviton mode contributes to the total cross - section . for @xmath65 ,
the dashed curve shows the contribution of this first mode to the @xmath66 .
the individual behavior of the other graviton modes are similar to this dashed curve which shows a monotonic increasing behavior .
this increasing behavior is expected due to the non - renormalizability of the quantum gravity .
the peaks on the solid curve show the resonance behavior according to creation of the graviton modes .
( 0,0)(0,0 ) ( 0,0 ) ( -300,0)(300,0)(10,5,-4,0,1.5 ) ( -300,-700)(300,-700)(10,5,4,0,1.5 ) ( -300,0)(-300,-700)(12,5,4,0,1.5 ) ( 300,0)(300,-700)(12,5,-4,0,1.5 ) ( -105,-252)[].8 ( -84,-252)[]1 ( -63,-252)[]1.2 ( -42,-252)[]1.4 ( -21,-252)[]1.6 ( 0,-252)[]1.8 ( 21,-252)[]2 ( 42,-252)[]2.2 ( 63,-252)[]2.4 ( 84,-252)[]2.6 ( 105,-252)[]2.8 ( 0,-270)[]@xmath55 in tev ( -130,-126)[][l]@xmath67 in pb ( -112,-245)[r]0 ( -112,-225)[r]10 ( -112,-205)[r]20 ( -112,-185)[r]30 ( -112,-164.4)[r]40 ( -112,-143.8)[r]50 ( -112,-123.2)[r]60 ( -112,-102.6)[r]70 ( -112,-82)[r]80 ( -112,-61.4)[r]90 ( -112,-40.8)[r]100 ( -112,-20.2)[r]110 ( -112,0)[r]120 ( 0,-295)[]total cross section of @xmath68 versus energy of one incident particles , with ( 0,-310)[]@xmath69 . the peaks correspond to the productions of massive gravitons .
we would like to thank a. shafiekhani for his contribution to the primary version of this article .
in this appendix , all the ten terms of the scattering diagrams @xmath70 up to the order of @xmath71 for @xmath72 in terms of the mass of graviton , @xmath53 , and enrgy , @xmath55 , have been calculated .
_ pol.|m_1|^2&= & , + _ pol.(_1 _ 2^&+&_1 ^_2)= ( ( 4e^2 - m^2 ) ( 32.c^4 - 16.e^6m^2 .. + & - & .. 16.c^3m^2 + 0.64cm^4 + ( 0.0512 + 0.04m^2 ) m^4 + c^2m^2 ( -2.56 + 2.m^2 ) + e^2 ( -64.c^3 + 96.c^2m^2 + ... + & + & ... c ( 2.56 - 16.m^2 ) m^2 + ( -2.88 - 1.m^2 ) m^4 ) + e^4 ( 32.c^2 - 64.cm^2 + m^2 ( 0.64 + 40.m^2 ) ) ) ) , + _ pol.(_1 _ 3^&+&_1 ^_3)= ( 0.000527714 ( 4e^2 - m^2 ) ( -2.c^4 + 1.e^6m^2 .. + & - & c^3m^2 + c^2 ( 0.16 - 0.125m^2 ) m^2 + 0.04cm^4 + ( -0.0032 - 0.0025m^2 ) m^4 + e^4 ( -2.c^2 - 4.cm^2 .
+ & + & ... ( -0.04 - 2.5m^2 ) m^2 ) + e^2 ( -4.c^3 - 6.c^2m^2 + c ( 0.16 - 1.m^2 ) m^2 + ( 0.18 + 0.0625m^2 ) m^4 ) ) ) , + _ pol.(_1 _
4^&+&_1 ^_4)= - - + & + & + & + & + & + & , + _ pol.|m_2|^2&= & ( 4096.e^12 + e^10 ( -163.84 - 20480.c + 2048.m^2 ) .
+ & + & e^8 ( 40960.c^2 + c ( 655.36 - 7168.m^2 ) + ( -450.56 - 256.m^2 ) m^2 ) + & + & e^4
( 20480.c^4 + c^3 ( 655.36 - 2048.m^2 ) + c^2 ( -1228.8 - 3072.m^2 ) m^2 .
+ & + & . 640.c ( -0.276264 + m^2 ) m^2 ( 0.148264 + m^2 ) - 16 .
( -3.07951 + m^2 ) m^4 ( 0.199512 + m^2 ) ) + & + & e^2 ( -4096.c^5 + c^4 ( -163.84 - 2048.m^2 ) - 512.c^2 ( -0.492029 + m^2 ) m^2 ( 0.0520294 + m^2 ) .
+ & + & 8 .
( -0.64 + m^2 ) m^4 ( 0.0626408 + m^2 ) ( 0.817359 + m^2 ) - 16.cm^4 ( 0.0812907 + m^2 ) + & & .
( 5.03871 + m^2 ) + c^3m^2 ( 327.68 + 2560.m^2 ) ) + e^6 ( -40960.c^3 - 256 .
( -0.189783 + m^2 ) m^2 .
( 0.269783 + m^2 ) + cm^2 ( 1310.72 + 1536.m^2 ) + c^2 ( -983.04 + 8192.m^2 ) ) + m^2 ( 1024.c^5 .
+ & + & c^4 ( 40.96 - 768.m^2 ) - 12.c ( -0.64 + m^2 ) m^4
( 0.426667 + m^2 ) + 32.c^2 ( -0.0849806 + m^2 ) m^2 + & & .. ( 1.20498 + m^2 ) + c^3m^2 ( -122.88 + 128.m^2 ) + 1.m^4 ( 0.16 + m^2 ) ( 0.4096 - 1.28m^2 + m^4 ) ) ) , + _ pol.(_2 _
3^&+&_2 ^_3)= + & & ( 256.e^8 + 256.c^4 - 1280.e^6m^2 + c^2 ( -20.48 - 32.m^2 ) m^2 + 1.m^4 ( 0.171487 + m^2 ) .
+ & & ( 2.38851 + m^2 ) + e^2m^2 ( 1280.c^2 + ( -40.96 - 80.m^2 ) m^2 ) + & + & . e^4 ( -512.c^2 + m^2 ( 40.96 + 864.m^2 ) ) ) , + _ pol.(_2 _
4^&+&_2 ^_4)= ( -1024.e^12 - 7.1054310 ^ -15c^4m^2 .
+ & + & 3.5527110 ^ -15c^3m^6 + c^2 ( -1.92 - 4.4408910 ^ -16m^2 ) m^6 + ( 0.0384 + 0.12m^2 ) m^8 + + & + & e^10 ( 40.96 + 2048.c + 2304.m^2 ) + e^8 ( 1.1368710 ^ -13c^2 + c ( -81.92 - 4096.m^2 ) .
( -112.64 - 640.m^2 ) m^2 ) + e^4 ( 1024.c^4 + 1024.c^3m^2 + c^2 ( -30.72 - 512.m^2 ) m^2 .
c ( -35.84 - 128.m^2 ) m^4 + 44.m^4 ( 0.0220983 + m^2 ) ( 0.210629 + m^2 ) ) + & + & e^2m^2 ( -256.c^4 - 128.c^3m^2 - 3.m^4 ( 0.106667 + m^2 ) ( 0.8 + m^2 ) + cm^4 ( 1.28 + 8.m^2 ) .
+ & + & . c^2m^2 ( 12.8 + 64.m^2 ) ) + e^6 ( -2048.c^3 - 96 .
( -0.247773 + m^2 ) m^2 ( 0.0344401 + m^2 ) .
+ & + & .. c^2 ( 40.96 + 1024.m^2 ) + cm^2 ( 143.36 + 1280.m^2 ) ) ) , + _ pol.|m_3|^2&= & ( 4096.e^12 + e^10 ( -163.84 + 20480.c .. + & + & .2048 .
m^2 ) + e^2 ( 4096.c^5 + c^4 ( -163.84 -2048.m^2 ) + c^3 ( -327.68 - 2560.m^2 ) m^2 .
+ & - & 512.c^2 ( -0.492029 + m^2 ) m^2 ( 0.0520294 + m^2 ) + 8 .
( -0.64 + m^2 ) m^4 ( 0.0626408 + m^2 ) + & & .
( 0.817359 + m^2 ) + 16.cm^4 ( 0.0812907 + m^2 ) ( 5.03871 + m^2 ) ) + e^4 ( 20480.c^4 .
+ & + & c^2 ( -1228.8 - 3072.m^2 ) m^2 - 640.c ( -0.276264 + m^2 ) m^2 ( 0.148264 + m^2 ) - 16 . + & & .
( -3.07951 + m^2 ) m^4 ( 0.199512 + m^2 ) + c^3 ( -655.36 + 2048.m^2 ) ) + e^8 ( 40960.c^2 .
+ & - & .. ( 450.56 + 256.m^2 ) m^2 + c ( -655.36 + 7168.m^2 ) ) + e^6 ( 40960.c^3 - c(1310.72 + 1536.m^2 ) m^2 .. + & - & .. 256 .
( -0.189783 + m^2 ) m^2 ( 0.269783 + m^2 ) + c^2 ( -983.04 + 8192.m^2 ) ) + m^2 ( -1024.c^5 .. + & + & c^4 ( 40.96 - 768.m^2 ) + c^3 ( 122.88 - 128.m^2 ) m^2 + 12.c ( -0.64 + m^2 ) m^4 ( 0.426667 + m^2 ) + & + & .. 32.c^2 ( -0.0849806 + m^2 ) m^2 ( 1.20498 + m^2 ) + 1.m^4 ( 0.16 + m^2 ) ( 0.4096 - 1.28m^2 +
m^4 ) ) ) , + _ pol.(_3
_ 4^&+&_3 ^_4)= ( + e^4 ( 6 + - ) .
+ & + & . e^2 ( + - ) + + + + ) + & - & ( c ( 4e^2 - m^2 ) ( -8e^6 ( 2c + 3m^2 ) + m^2 ( -4.c^3 ... + & - & .
2.c^2m^2 + c ( 0.24 - 2.25m^2 ) m^2 + 0.06m^4 ) + e^2 ( -16.c^3 + 8.c^2m^2 + c ( 0.96 - 11.m^2 ) m^2 .
+ & + & ... ( -0.48 - 1.5m^2 ) m^4 ) + e^4 ( -32.c^2 - 12.cm^2 + m^2 ( 0.96 + 12.m^2 ) ) ) ) + & - & ( ( 4e^2 - m^2 ) ( 4e^2 + m^2 ) ( -8e^6 ( 2c + 3m^2 ) .. + & + & m^2 ( -4.c^3 - 2.c^2m^2 + c ( 0.24 - 2.25m^2 ) m^2 + 0.06m^4 ) + e^2 ( -16.c^3 + 8.c^2m^2
. + & + & ... c ( 0.96 - 11.m^2 ) m^2 + ( -0.48 - 1.5m^2 ) m^4 ) + e^4 ( -32.c^2 - 12.cm^2 + m^2 ( 0.96 + 12.m^2 ) ) ) ) + + & + & ( ( -4e^2 + m^2 ) ^2 ( 32.c^4 - 16.e^6m^2 + 16.c^3m^2 .. + & - & 0.64cm^4 + ( 0.0768 - 0.6m^2 ) m^4 + c^2m^2 ( -3.2 + 18.m^2 ) + e^2 ( 64.c^3 + 96.c^2m^2 .
+ & + & ... m^4 ( -3.52 + 15.m^2 ) + cm^2 ( -2.56 + 16.m^2 ) ) + e^4 ( 32.c^2 + 64.cm^2 + m^2 ( 0.64 + 40.m^2 ) ) ) ) , + _ pol.|m_4|^2&= & + + & + & + + & + & + & - & + & + & + & + & + & - & + & - & , where @xmath73 .
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m. e. peskin , d. v. schroeder , _ an introduction to quantum field theory _ , addison - wesley , 1995 . | considering the randall - sundrum background , we calculate the total cross - section for @xmath0 in the framework of the scalar electrodynamics .
+ l 3 mm 230 mm ipm / p2000/021 + hep - ph/0005166 | hep-ph0005166 |
if cooled below a critical temperature ( @xmath0k in @xmath1he and @xmath2 in at @xmath3hehe we mean the b - phase of @xmath3he ] at saturated vapour pressure ) , liquid helium undergoes bose - einstein condensation @xcite , becoming a quantum fluid and demonstrating superfluidity ( pure inviscid flow ) . besides the lack of viscosity
, another major difference between superfluid helium and ordinary ( classical ) fluids such as water or air is that , in helium , vorticity is constrained to vortex line singularities of fixed circulation @xmath4 , where @xmath5 is planck s constant , and @xmath6 is the mass of the relevant boson ( in the most common isotope @xmath1he , @xmath7 , the mass of an atom ; in the rare isotope @xmath3he , @xmath8 , the mass of a cooper pair ) .
these vortex lines are essentially one - dimensional space curves , like the vortex lines of fluid dynamics textbooks ; for example , in @xmath1he the vortex core radius @xmath9 m is comparable to the inter atomic distance .
this quantisation of the circulation thus results in the appearance of another characteristic length scale : the mean separation between vortex lines , @xmath10 . in typical experiments
( both in @xmath1he and @xmath3he ) @xmath10 is orders of magnitude smaller than the outer scale of turbulence @xmath11 ( the scale of the largest eddies ) but is also orders of magnitudes larger than @xmath12 .
there is a growing consensus @xcite that on length scales much larger than @xmath10 the properties of superfluid turbulence are similar to those of classical turbulence if excited similarly , for example by a moving grid .
the idea is that motions at scales @xmath13 should involve at least a partial polarization @xcite of vortex lines and their organisation into vortex bundles which , at such large scales , should mimic continuous hydrodynamic eddies .
therefore one expects a classical richardson - kolmogorov energy cascade , with larger `` eddies '' breaking into smaller ones .
the spectral signature of this classical cascade is indeed observed experimentally in superfluid helium . in the absence of viscosity , in superfluid turbulence
the kinetic energy should cascade downscale without loss , until it reaches the small scales where the quantum discreteness of vorticity is important .
it is also believed that at this point the richardson - kolmogorov eddy - dominated cascade should be replaced by a second cascade which arises from the nonlinear interaction of kelvin waves ( helical perturbation of the vortex lines ) on individual vortex lines .
this kelvin wave cascade should take the energy further downscale where it is radiated away by thermal quasi particles ( phonons and rotons in @xmath1he ) .
although this scenario seems quite reasonable , crucial details are yet to be established .
our understanding of superfluid turbulence at scales of the order of @xmath10 is still at infancy stage , and what happens at scales below @xmath10 is a question of intensive debates .
the quasi - classical " region of scales , @xmath14 , is better understood , but still less than classical hydrodynamic turbulence .
the main reason is that at nonzero temperatures ( but still below the critical temperature @xmath15 ) , superfluid helium is a two - fluid system . according to the theory of landau and tisza @xcite , it consists of two inter penetrating components : the inviscid superfluid , of density @xmath16 and velocity @xmath17 ( associated to the quantum ground state ) , and the viscous normal fluid , of density @xmath18 and velocity @xmath19 ( associated to thermal excitations ) .
the normal fluid carries the entropy @xmath20 and the viscosity @xmath21 of the entire liquid . in the presence of superfluid
vortices these two components interact via a mutual friction force@xcite .
the total helium density @xmath22 is practically temperature independent , while the superfluid fraction @xmath23 is zero at @xmath24 , but rapidly increases if @xmath25 is lowered ( it becomes 50% at @xmath26 , 83% at @xmath27 and 95% at @xmath28 @xcite ) .
the normal fluid is essentially negligible below @xmath29 .
one would therefore expect classical behaviour only in the high temperature limit @xmath30 , where the normal fluid must energetically dominate the dynamics .
experiments show that this is not the case , thus raising the interesting problem of
double - fluid " turbulence which we study here .
the aim of this article is to present the current state of the art in this intriguing problem , clarify common features of turbulence in classical and quantum fluids , and highlight their differences . to achieve our aim
we shall overview and combine experimental , theoretical and numerical results in the simplest possible ( and , probably , the most fundamental ) case of homogeneous , isotropic turbulence , away from boundaries and maintained in a statistical steady state by continuous mechanical forcing .
the natural tools to study homogeneous isotropic turbulence are spectral , thus we shall consider the velocity spectrum ( also known as the energy spectrum ) and attempt to give a physical explanation for the observed phenomena .
we recall that ordinary incompressible viscous flows are described by the navier - stokes equation @xmath31 = - \frac{1}{\rho}{{{\bf{\nabla } } } p}+\nu { { { \bf{\nabla}}}^2 } { { \bf{u}}},\ ] ] and the solenoidal condition @xmath32 for the velocity field @xmath33 , where @xmath34 is the pressure , @xmath35 the density , and @xmath36 the kinematic viscosity .
the dimensionless parameter that determines the properties of hydrodynamic turbulence is the reynolds number re@xmath37 .
the reynolds number estimates the ratio of nonlinear and viscous terms in eq . at the outer length scale @xmath11 ( typically the size of a streamlined body ) , where @xmath38 is the root mean square turbulent velocity fluctuation . in fully developed turbulence ( re@xmath39 )
the @xmath11-scale eddies are unstable and give birth to smaller scale eddies , which , being unstable , generate further smaller eddies , and so on .
this process is the richardson - kolmogorov energy cascade toward eddies of scale @xmath40 , defined as the length scale at which the nonlinear and viscous forces in eq .
approximately balance each other .
@xmath40-scale eddies are stable and their energy is dissipated into heat by viscous forces .
the hallmark feature of fully developed turbulence is the coexistence of eddies of all scales from @xmath11 to @xmath41 with universal statistics ; the range of length scales @xmath42 where both energy pumping and dissipation can be ignored is called the inertial range . in the study of homogeneous turbulence
it is customary to consider the energy density per unit mass @xmath43 ( of dimensions @xmath44 ) . in the isotropic case the energy distribution between eddies of scale @xmath45
can be characterized by the one dimensional energy spectrum @xmath46 of dimensions @xmath44 ) with wavenumber defined as @xmath47 ( or as @xmath48 ) , normalized such that + @xmath49 where @xmath50 is volume . in the inviscid limit
, @xmath43 is a conserved quantity ( @xmath51 ) , thus @xmath46 satisfies the continuity equation @xmath52 where @xmath53 is the energy flux in spectral space ( of dimensions @xmath54 ) . in the stationary case ,
energy spectrum and energy flux are @xmath55independent , thus eq
. immediately dictates that the energy flux @xmath56 is @xmath57-independent . assuming that this constant @xmath56 is the only relevant characteristics of turbulence in the inertial interval and using dimensional reasoning , in 1941 kolmogorov and ( later ) obukhov suggested that the energy spectrum is @xmath58 where the ( dimensionless ) kolmogorov constant is approximately @xmath59 .
this is the celebrated kolmogorov - obukhov @xmath60 law ( ko41 ) , verified in experiments and numerical simulations of eq . ; it states in particular that in incompressible , steady , homogeneous , isotropic turbulence , the distribution of kinetic energy over the wavenumbers is @xmath61 . in the inviscid limit
the energy flux goes to smaller and smaller scales , reaching finally the interatomic scale and accumulating there . to describe this effect ,
leith @xcite suggested to replace the algebraic relation between @xmath62 and @xmath63 by the differential form : @xmath64 this approximation dimensionally coincides with eq . , but the derivative @xmath65/dk$ ] guarantees that @xmath66 if @xmath67 . the numerical factor @xmath68 , suggested in @xcite , fits the experimentally observed value @xmath69 in eq . .
a generic energy spectrum with a constant energy flux was found in @xcite as a solution to the equation @xmath70 constant : @xmath71^{2/3}.\ ] ] notice that at low @xmath57 , eq . coincides with ko41 , while for @xmath72 it describes a thermalized part of the spectrum , @xmath73 , with equipartition of energy ( shown by the solid black line at the right of in fig .
[ f:3]a , and , underneath in the same figure , by the solid red line , although the latter occurs in slightly different contexts).a , the energy flux @xmath74 is not preserved along the cascade , but continuously decreases due to dissipation and ultimately vanishes at the maximum @xmath57 . ]
we shall have also to keep in mind that although eq .
is an important result of classical turbulence theory , it presents only the very beginning of the story .
in particular , its well known @xcite that in the inertial range , the turbulent velocity field is not self similar , but shows intermittency effects which modify the ko41 scenario . in this paper
we apply these ideas to superfluid helium , explain how to overcome technical difficulties to measure the energy spectrum near absolute zero , and draw the attention to three conceptual differences between classical hydrodynamic turbulence and turbulence in superfluid @xmath1he .
the first difference is that the quantity which ( historically ) is most easily and most frequently detected in turbulent liquid helium is not the superfluid velocity but rather the vortex line density @xmath75 , defined as the superfluid vortex length per unit volume ; in most experiments ( and numerical simulations ) this volume is the entire cell ( or computational box ) which contains the helium .
this scalar quantity @xmath75 has no analogy in classical fluid mechanics and should not be confused with the vorticity , whose spectrum , in the classical ko41 scenario , scales as @xmath76 correspondingly to the @xmath77 scaling of the velocity .
notice that in a superfluid the vorticity is zero everywhere except on quantized vortex lines . in order to use as much as possible the toolkit of ideas and methods of classical hydrodynamics
, we shall define in the next sections an `` effective '' superfluid vorticity field @xmath78 ; this definition ( which indeed @xcite yields the classical @xmath76 vorticity scaling corresponding to the @xmath77 velocity scaling ) is possible on scales that exceed the mean intervortex scale @xmath10 , provided that the vortex lines contained in a fluid parcel are sufficiently polarized .
this procedure opens the way for a possible identification of `` local '' values of @xmath79 with the magnitude @xmath80 of the vector field @xmath78 .
the second difference is that liquid helium is a two fluid system , and we expect both superfluid and normal fluid to be turbulent .
this makes the problem of superfluid turbulence much richer than classical turbulence , but the analysis becomes more involved .
for example , the existence of the intermediate scale @xmath10 makes it impossible to apply arguments of scale invariance to the entire inertial interval and calls for its separation into three ranges .
the first is a `` hydrodynamic '' region of scales @xmath81 ( corresponding to @xmath82 in k - space where @xmath83 and @xmath84 ) , which is similar ( but not equal ) to the classical inertial range ; the second is a `` kelvin wave region '' @xmath85 where energy is transferred further to smaller scales in @xmath1he@xcite . ] by interacting kelvin waves ( helix - like deformations of the vortex lines ) . in the third , intermediate region @xmath86 ,
the energy flux is caused probably by vortex reconnections .
finally , the third difference is that mutual friction between normal and superfluid components leads to ( dissipative ) energy exchange between them in either direction .
studies of classical turbulence are solidly based on the navier - stokes eq . .
unfortunately , there are no well established equations of motion for @xmath1he in the presence of superfluid vortices .
we have only models at different levels of description ( for an overview see sec .
[ s : theory ] ) .
all these issues make the problem of superfluid turbulence very interesting from a fundamental view point , simultaneously creating serious problems in experimental , numerical and analytical studies . [ cols="^,^ " , ]
in the absence of superfluid vortices , landau s two - fluid equations@xcite for the superfluid and normal fluid velocities @xmath87 and @xmath88 account for all phenomena observed in he - ii at low velocities , for example second sound and thermal counterflow . in the incompressible limit ( @xmath89 , @xmath90 )
landau s equations are : [ landau ] @xmath91 & = & - { { \bf{\nabla } } } p_s/ \rho_s,\\ \label{landau2 } \big [ ( \partial \,{{\bf{u}}}{_{\text { n}}}/\partial t)+ ( { { \bf{u}}}{_{\text { n}}}\cdot { { \bf{\nabla}}}){{\bf{u}}}{_{\text { n } } } \big]&= & - { { \bf{\nabla } } } p_n
/\rho_n + \nu_n \nabla^2 { { { \bf{u}}}{_{\text { n } } } } \,,\end{aligned}\ ] ] where @xmath92 is the kinematic viscosity , and the efficient pressures @xmath93 and @xmath94 are defined by @xmath95 and @xmath96 . on physical ground
, laudau argued that the superfluid is irrotational .
the main difficulty in developing a theory of superfluid turbulence is the lack of an established set of equations of motion for he - ii in the presence of superfluid vortices .
we have only models at different levels of description . at the first , most microscopic level of description , we must account for phenomena at the length scale of the superfluid vortex core , @xmath97 .
monte carlo models of the vortex core @xcite , although realistic , are not suitable for the study of the dynamics and turbulent motion .
a practical model of a pure superfluid is the gross - pitaevskii equation ( gpe ) for a weakly - interacting bose gas @xcite : @xmath98 where @xmath99 is the complex condensate s wave function , @xmath100 the strength of the interaction between bosons , @xmath101 the chemical potential and @xmath6 the boson mass .
the condensate s density @xmath102 and velocity @xmath103 are related to @xmath104 via the madelung transformation @xmath105 , which confirms landau s intuition that the superfluid is irrotational .
it can be shown that , at length scales @xmath106 , the gpe reduces to the classical continuity equation and the ( compressible ) euler equation .
it must be stressed that , although the gpe accounts for quantum vortices , finite vortex core size ( of the order of @xmath12 ) , vortex nucleation , vortex reconnections , sound emission by accelerating vortices and kelvin waves , it is only a qualitative model of the superfluid component .
he - ii is a liquid , not a weakly - interacting gas , and the condensate is only a fraction of the superfluid density @xmath107 .
no adjustment of @xmath100 and @xmath101 can fit both the sound speed and the vortex core radius , and the dispersion relation of the uniform solution of eq .
lacks the roton s minimum which is characteristic of he - ii @xcite .
this is why , strictly speaking , we can not identify @xmath108 with @xmath107 and @xmath109 with @xmath87 .
nevertheless , when solved numerically , the gpe is a useful model of superfluid turbulence at low @xmath25 where the normal fluid fraction vanishes , and yields results which can be compared to experiments , as we shall see . far away from the vortex core at length scales larger than @xmath12 and in the zero mach number limit
, the gpe describes incompressible euler dynamics .
this is the level of description of a second practical model , the vortex filament model ( vfm ) of schwarz@xcite . at this level ( length scales @xmath110 )
we ignore the nature of the vortex core but distinguish individual vortex lines , which we describe as oriented space curves @xmath111 of infinitesimal thickness and circulation @xmath112 , parametrised by arc length @xmath113 .
their time evolution is determined by schwarz s equation [ vfm ] @xmath114 where the self - induced velocity @xmath115 is given by the biot - savart law @xcite , and the line integral extends over the vortex configuration . at nonzero temperatures the term @xmath116 accounts for the friction between the vortex lines and the normal fluid@xcite : @xmath117\ , , \quad { { \bf{u } } } { _ { \text { ns}}}= { { { \bf{u } } } { _ { \text { n}}}}-{{{\bf{u } } } { _ { \text { si}}}}\,,\ ] ] where @xmath118 is the unit tangent at @xmath119 , and @xmath120 , @xmath121 are known @xcite temperature - dependent friction coefficients . in the very low temperature limit ( @xmath122 ) , @xmath120 and @xmath121 become negligible@xcite , and we recover the classical result that each point of the vortex line is swept by the velocity field produced by the entire vortex configuration . in numerical simulations based on the vfm
, vortex lines are discretized in a lagrangian fashion , biot - savart integrals are desingularised using the vortex core radius @xmath12 , and reconnections are additional artificial _ ad - hoc _ procedures that change the way pairs of discretization points are connected .
reconnection criteria are described and discussed in ref .
@xcite ; ref .
@xcite compares gpe and vfm reconnections with each other and with experiments .
simulations at large values of vortex line density are performed using a tree algorithm@xcite which speeds up the evaluations of biot - savart integrals from @xmath123 to @xmath124 where @xmath125 is the number of discretization points .
the major drawback of the vfm is that the normal fluid @xmath126 is imposed ( either laminar or turbulent ) , therefore the back - reaction of the vortex lines on @xmath126 is not taken into account .
the reason is the computational difficulty : a self - consistent simulation would require the simultaneous integration in time of eq . for the superfluid , and of a navier - stokes equation for the normal fluid ( implemented with suitable friction forcing at vortex lines singularities ) .
such self - consistent simulations were carried out only for a single vortex ring @xcite and for the initial growth of a vortex cloud @xcite .
this limitation is likely to be particularly important at low and intermediate temperatures ( at high temperatures the normal fluid contains most of the kinetic energy , so it is less likely to be affected by the vortices ) . at a third level of description we do not distinguish individual vortex lines any longer , but rather consider fluid parcels which contain a continuum of vortices . at these length scales
@xmath13 we seek to generalise landau s equations to the presence of vortices . in laminar
flows the vortex lines ( although curved ) remain locally parallel to each other , so it is possible to define the components of a macroscopic vorticity field @xmath78 by taking a small volume larger than @xmath10 and considering the superfluid circulation in the planes parallel to the cartesian directions ( alternatively , the sum of the oriented vortex lengths in each cartesian direction ) .
we obtain the so - called hall - vinen ( or hvbk ) coarse - grained " equations @xcite : [ nse ] @xmath127 = - \frac{1}{\rho_s}{{{\bf{\nabla } } } p_s}-{{\bf{f } } } { _ { \text { ns } } } , \\ \label{nseb } & & \hskip - 0.7 cm \big[\frac{\partial \,{{\bf{u}}}{_{\text { n}}}}{\partial t}+ ( { { \bf{u}}}{_{\text { n}}}\cdot { { \bf{\nabla}}}){{\bf{u}}}{_{\text { n } } } \big]= -\frac{1}{\rho_n } { { { \bf{\nabla } } } p_n } + \nu_n \nabla^2 { { { \bf{u}}}{_{\text { n}}}}+\frac{\rho_s}{\rho } { { \bf{f } } } { _ { \text { ns } } } , \\ \label{nsec } & & \hskip - 0.7 cm { { \bf{f } } } { _ { \text { ns}}}=\alpha { \hat { { \mbox{\boldmath $ \omega$}}}_s } \times ( { { \mbox{\boldmath $ \omega$}}}_s \times { { \bf{u } } } { _ { \text { ns } } } ) + \alpha ' { \hat { { \mbox{\boldmath $ \omega$}}}_s } \times { { \bf{u } } } { _ { \text { ns}}},\end{aligned}\ ] ] where @xmath128 , @xmath129 and @xmath130 is the mutual friction force .
these equations have been used with success to predict the taylor - couette flow of he - ii , its stability @xcite and the weakly nonlinear regime @xcite . in these flows ,
the vortex lines are fully polarised and aligned in the same direction , and their density and orientation may change locally and vary as a function of position ( on length scales larger than @xmath10 ) .
the difficulty with applying the hvbk equations to turbulence is that in turbulent flows the vortex lines tend to be randomly oriented with respect to each other , so the components of @xmath131 cancel out to zero ( partially or totally ) , resulting in local vortex length ( hence energy dissipation ) without any effective superfluid vorticity . in this case
, the hvbk equations may become a poor approximation and underestimate the mutual friction coupling . nevertheless , they are a useful model of large scale superfluid motion with characteristic scale @xmath14 , particularly because ( unlike the vfm ) they are dynamically self - consistent ( normal fluid and superfluid affect each other )
. we must also keep in mind that eq .
do not have physical meaning at length scales smaller than @xmath10 . in the next sections we shall describe numerical simulations of equations as well as shell models and theoretical models based on these equations . in some models the mutual friction force
is simplified to @xmath132 where @xmath133 .
numerical simulations in the framework of all three approaches , and are shown in figs .
they clearly show ko41 scaling , in agreement with the experimental results shown in figs .
details of this simulations will be described below .
since the pioneering work of schwarz @xcite , numerical experiments have played an important role , allowing the exploration of the consequences of limited sets of physical assumptions in a controlled way , and providing some flow visualization .
numerical simulations of the gpe in a three - dimensional periodic box have been performed for decaying turbulence @xcite following an imposed arbitrary initial condition , and for forced turbulence @xcite .
besides vortex lines , the gpe describes compressible motions and sound propagation ; therefore , in order to analyse turbulent vortex lines , it is necessary to extract from the total energy of the system ( which is conserved during the evolution ) its incompressible kinetic energy part whose spectrum is relevant to our discussion . to reach a steady state , large - scale external forcing and small - scale damping
was added to the gpe @xcite .
the resulting turbulent energy spectrum , shown in fig .
[ f:3]c , agrees with the ko41 scaling ( shown by cyan dot - dashed line ) , and demonstrates bottleneck energy accumulation near the intervortex scale at zero temperature predicted earlier in @xcite and discussed in sect .
the ko41 scaling observed in gpe simulations was found to be consistent with the vfm at zero temperature @xcite and has also been observed when modelling a trapped atomic bose
einstein condensate @xcite .
the gpe can be extended to take into account finite temperature effects .
different models have been proposed @xcite .
most vfm calculations have been performed in a cubic box of size @xmath11 with periodic boundary conditions . in all relevant experiments
we expect that the normal fluid is turbulent because its reynolds number @xmath134 is large ( where @xmath135 the root mean square normal fluid velocity ) .
recent studies thus assumed the form @xcite + @xmath136 where @xmath137 , @xmath138 and @xmath139 are wave vectors and angular frequencies .
the random parameters @xmath140 , @xmath141 and @xmath138 are chosen so that the normal fluid s energy spectrum obeys ko41 scaling @xmath142 in the inertial range @xmath143
. this synthetic turbulent flow @xcite is solenoidal , time - dependent , and compares well with lagrangian statistics obtained in experiments and direct numerical simulations of the navier - stokes equation .
other vfm models included normal - fluid turbulence generated by the navier stokes equation @xcite and a vortex - tube model @xcite , but , due to limited computational resources , only a snapshot of the normal fluid , frozen in time , was used to drive the superfluid vortices . in all numerical experiments , after a transient from some initial condition , a statistical steady state of superfluid turbulence is achieved , in the form of a vortex tangle ( see fig .
[ f:1]-b ) in which the vortex line density @xmath144@xmath145 fluctuates about an average value independent of the initial condition .
recent analytical @xcite and numerical studies@xcite of the geometry of the vortex tangle reveal that the vortices are not randomly distributed , but there is a tendency to locally form bundles of co - rotating vortices , which keep forming , vanish and reform somewhere else .
these bundles associate with the kolmogorov spectrum : if turbulence is driven by a uniform normal fluid ( as in the original work of schwarz@xcite recently verified in ref .
@xcite ) , there are nor kolmogorov scaling nor bundles .
laurie et al.@xcite decomposed the vortex tangle in a polarised part ( of density @xmath146 ) and a random part ( of density @xmath147 ) , as argued by roche & barenghi @xcite , and discovered that @xmath146 is responsible for the @xmath77 scaling of the energy spectrum , and @xmath147 for the @xmath148 scaling of the vortex line density fluctuations , as suggested in ref.@xcite . from a computational viewpoint ,
the hvbk equations are similar to the navier - stokes equation .
not surprisingly , standard methods of classical turbulence have been adapted to the hvbk equations , e.g. large eddy simulations @xcite , direct numerical simulations @xcite and eddy damped quasinormal markovian simulations @xcite .
the hvbk equations are ideal to study the coupled dynamics of superfluid and normal fluid in the limit of intense turbulence at finite temperature . indeed , by ignoring the details of individual vortices and their fast dynamics , hvbk simulations
do not suffer as much as vfm and gpe simulations from the wide separation of space and time scales which characterize superfluid turbulence .
moreover , well optimized numerical solvers have been developed for navier - stokes turbulence and they can be easily adapted to the hvbk model .
thus , simulations over a wide temperature range ( @xmath149 corresponding to @xmath150 ) evidence a strong locking of superfluid and normal fluid ( @xmath151 ) at large scales , over one decade of inertial range ( @xcite ) . in particular , it was found that even if one single single fluid is forced at large scale ( the dominant one ) , both fluids still get locked very efficiently .
[ f:3]a illustrates velocity spectra generated by direct numerical simulation of the hvbk equations , while the red and blue solid lines of fig .
[ f:3]b show spectra obtained using a shell model of the same equations ( see paragraph at the end of the section ) . in both case , a clear @xmath77 spectrum is found for both fluid components , at all temperature and large scale .
information about the quantization of vortex lines is lost in the coarse graining procedure which leads to eqs . .
as discussed in sect .
8b , a quantum constrain can be re - introduced in this model by truncating superfluid phase space for @xmath152 , causing an upward trend of the low temperature velocity spectrum of fig .
[ f:3]a which can be interpreted as partial thermalization of superfluid excitations .
this procedure also leads to the prediction @xmath153
@xcite which is consistent with experiments and allows to identify the spectrum of @xmath75@xmath154 with the spectrum of the scalar field @xmath155 .
it is found that this spectrum is temperature dependent in the inertial range with a flat part at high temperature ( reminiscent of the corresponding spectrum of the magnitude of the vorticity in classical turbulence ) which contrasts the @xmath77 decreases at low temperature ( consistent with experiments @xcite ) .
essential simplification of the hvbk eqs . can be achieved with the shell - model approximation@xcite .
the complex shell velocities @xmath156 and @xmath157 mimic the statistical behaviour of the fourier components of the turbulent superfluid and normal fluid velocities at wavenumber @xmath57 . the resulting ordinary differential eqs . for @xmath158 capture important aspects of the hvbk eqs . .
because of the geometrical spacing of the shells ( @xmath159 ) , this approach allows more decades of @xmath57-space than eqns .
( see fig .
[ fig : interim ] with eight decades in @xmath57-space @xcite ) .
this extended inertial range allows detailed comparison of intermittency effects in superfluid turbulence and classical turbulence ( see sec .
in this section we discuss theoretical models of large scale ( eddy dominated ) motions at wavenumbers @xmath160 which are important at all temperatures from 0 to @xmath161 .
these motions can be tackled using the hydrodynamic hvbk eqs . , thus generalising what we know about classical turbulence .
we shall start by considering the simpler case of @xmath3he ( sect .
[ 6a ] ) , in which there is only one turbulent fluid , then move to more difficult case of @xmath1he ( sect .
[ 6b ] ) in which there are two coupled turbulent fluids , and finally discuss intermittency ( sect .
[ 6c ] ) .
the viscosity of @xmath3he is so large that , in all @xmath3he turbulence experiments , we expect the normal fluid to be at rest ( @xmath162 ) or in solid body rotation in rotating cryostat ( in which case our argument requires a slight modification ) .
liquid @xmath3he thus provides us with a simpler turbulence problem ( superfluid turbulence in the presence of linear friction against a stationary normal fluid ) than @xmath1he ( superfluid turbulence in the presence of normal fluid turbulence ) . at scales
@xmath13 , we expect eq . to be valid , provided we add a suitable model for the friction . following ref .
@xcite , we approximate @xmath130 as + @xmath163with @xmath164 . here
@xmath165 , @xmath166 is the characteristic turbulent " superfluid vorticity , estimated via the spectrum @xmath63 .
@xmath167 is the energy pumping scale . using the differential approximation for the energy spectrum , the continuity eq . in the stationary case
becomes @xmath168 + \gamma e{_{\text { s } } } ( k)=0.\ ] ] analytical solutions of eq .
are in good agreement @xcite with the results of numerical simulation of the shell model to the hydrodynamic eq . , providing us with quasi - qualitative description of turbulent energy spectra in @xmath3he over a wide region of temperatures . in @xmath1he
, we have to account for motion of the normal component , which has very low viscosity and is turbulent in the relevant experiments .
eqs . , and
( now with @xmath169 ) result in a system of energy balance equations for superfluid and normal fluid energy spectra @xmath170 and @xmath171 that generalises eq .
@xcite : here @xmath180 and @xmath181 are characteristic interaction frequencies ( or turnover frequencies ) of eddies in the normal and superfluid components .
they are related to the well known effective turbulent viscosity @xmath182 by @xmath183 . for large mutual friction or / and small @xmath57 , when @xmath184 and can be neglected , eq .
has a physically motivated solution @xmath185 corresponding to full locking @xmath186 . in this case
the sum of eq .
( multiplied by @xmath16 ) and eq .
( multiplied by @xmath187 ) yields the navier - stokes equation with effective viscosity @xmath188 .
thus , in this region of @xmath57 , one expects classical behaviour of hydrodynamic turbulence with ko41 scaling ( up to intermittency corrections discussed in sec .
for small mutual friction or / end large @xmath57 , when @xmath189 , eqs .
gives @xcite @xmath190 i.e. full decorrelation of superfluid and normal fluid velocities . in this case
normal component will have ko41scaling , @xmath191 up to the kolmogorov micro - scale @xmath192 that can be found from the condition @xmath193 , giving the well known estimate @xmath194 simultaneously , the superfluid spectrum also obeys the same ko41 scaling , @xmath195 .
moreover , since at small @xmath57 the two fluid components are locked , we expect that @xmath196 .
assuming that the @xmath60 scaling is valid up to wavenumber @xmath197 , we estimate that @xmath198 which is similar to the estimate for @xmath192 with the replacement of @xmath199 with @xmath112 . in he - ii ,
the numerical values of @xmath200 and @xmath112 are similar , thus we conclude that the viscous cutoff @xmath192 for the normal component and the quantum cutoff @xmath201 for the superfluid component are close to each other . at a given temperature ,
the decorrelation crossover wave vector @xmath202 between the two regimes described above can be found from the condition @xmath203 using the estimate @xmath204 with @xmath205 .
we obtain @xmath206 .
the quantity @xmath207 varies between 1.2 and 0.5 @xcite in the temperature range @xmath208 where the motion of the normal fluid is important .
we conclude that @xmath209 , which means that normal fluid and superfluid eddies are practically locked over the entire inertial interval
. nevertheless , dissipation due to mutual friction can not be completely ignored , leading to intermittency enhancement described next .
the first numerical study of intermittent exponents @xcite did not find any intermittent effect peculiar to superfluid turbulence neither at low temperature ( @xmath210 , @xmath211 ) nor at and high temperature ( @xmath212 , @xmath213 ) , in agreement with experiments performed at the same temperatures ( see sect . [
s : exp ] ) .
recent shell model simulations @xcite with eight decades of @xmath57-space allowed detailed comparison of classical and superfluid turbulent statistics in the intermediate temperature range corresponding to @xmath214 .
the results were the following .
for @xmath25 slightly below @xmath161 , when @xmath215 , the statistics of turbulent superfluid @xmath1he appeared similar to that of classical fluids , because the superfluid component can be neglected , see green lines in fig .
[ fig : interim ] with @xmath216 .
the same result applies to @xmath217 ( @xmath218 ) , as expected due to the inconsequential role played by the normal component , see blue lines with @xmath219 . in agreement with the previous study the intermittent scaling exponents appeared the same in classical and low - temperature superfluid turbulence (
indeed the nonlinear structure of the equation for the superfluid component is the same as of euler equation , and dissipative mechanisms are irrelevant . )
a difference between classical and superfluid intermittent behaviour in a wide ( up to three decades ) interval of scales was found in the range @xmath220 ( @xmath221 ) , as shown by red lines in fig .
[ fig : interim ] with @xmath222 .
the exponents of higher order correlation functions also deviate further from the ko41 values .
what is predicted is thus an enhancement of intermittency in superfluid turbulence compared to the classical turbulence .
superfluid ( solid lines ) and normal fluid ( dash lines ) compensated energy spectra @xmath223 ; the compensation factor is the classical energy spectrum with intermittency correction .
inset : @xmath224 for @xmath225 .
shell model simulation of the hvbk model at @xmath226 k ( green ) , 0.9 ( red ) and 0.9 ( blue ) , corresponding to @xmath227 and 0.9 respectively @xcite .
the vertical dash lines indicate @xmath228.,scaledwidth=50.0% ]
now we come to the more complicated and more intensively discussed aspect of the superfluid energy spectrum : what happens for @xmath229 , where the quantisation of vortex lines becomes important .
this range acquires great importance at low temperatures , typically below 1 k in @xmath1he , and is relevant to turbulence decay experiments .
here we shall describe only the basic ideas , avoiding the most debated details .
for @xmath230 we neglect the interaction between separate vortex lines ( besides the small regions around vortex reconnection events , which will be discussed later ) . under this reasonable assumption , at large @xmath57 superfluid turbulence can be considered as a system of kelvin waves ( helix - like deformation of vortex lines ) with different wavevectors interacting with each other on the same vortex .
the prediction that this interaction results in turbulent energy transfer toward large @xmath57 @xcite was confirmed by numerical simulations in which energy was pumped into kelvin waves at intervortex scales by vortex reconnections @xcite or simply by exciting the vortex lines @xcite . the first analytical theory of kelvin wave turbulence ( propagating along a straight vortex line and in the limit of small amplitude compared to wavelength )
was proposed by kozik and svistunov @xcite ( ks ) , who showed that the leading interaction is a six - wave scattering process ( three incoming waves and three outgoing waves ) . under the additional assumption of locality of the interaction ( that only compatible wave - vectors contribute to most of the energy transfer ) ks found that ( using the same normalisation of other hydrodynamic spectra such as eqs . ) the energy spectrum of kelvin waves is + @xmath231 + here @xmath232 or @xmath233 in typical @xmath1he and @xmath3he experiments , and @xmath234 is the energy flux in three dimensional @xmath235-space .
later lvov - nazarenko ( ln ) @xcite criticised the ks assumption of locality and concluded that the leading contribution to the energy transfer comes from a six waves scattering in which two wave vectors ( from the same side ) have wavenumbers of the order of @xmath236 .
ln concluded that the spectrum is @xmath237 this ks vs ln controversy triggered an intensive debate ( see e.g. refs @xcite ) , which is outside the scope of this article .
we only mention that the three dimensional energy spectrum @xmath238 can be related to the one dimensional amplitude spectrum @xmath239 by @xmath240 where @xmath241 is the angular frequency of a kelvin wave of wavenumber @xmath57 , @xmath242 the energy of one quantum , and @xmath243 the number of quanta ; therefore , in terms of the kelvin waves amplitude spectrum ( which is often reported in the literature and can be numerically computed ) , the two predictions are respectively @xmath244 ( ks ) and @xmath245 ( ln ) .
the two predicted exponents ( -3.40 and -3.67 ) are very close to each other ; indeed vfm simulations @xcite could not distinguish them ( probably because the numerics were not in the sufficiently weak regime of the theory in terms of ratio of amplitude to wavelength ) .
nevertheless , more recent gpe simulations by krstulovic @xcite based on long time integration of eq . and
averaged over initial conditions ( slightly deviating from a straight line ) support the ln spectrum .
at finite temperature , it was shown in ref .
@xcite that the kelvin wave spectrum is suppressed by mutual friction for @xmath246 , reaching core scale ( @xmath247 ) at @xmath248k and fully disappears at @xmath249k , when @xmath250 .
the region of the spectrum near the intervortex scale @xmath251 is difficult because both eddy - type motions and kelvin waves are important , and the discreteness of the superfluid vorticity prevents direct application of the tools of classical hydrodynamic .
nevertheless , some progress can be made : sect .
[ 8a ] presents a differential model for the @xmath252 limit @xcite , and sect .
[ 8b ] describes a complementary truncated hvbk model @xcite designed for the @xmath253 temperature range .
the description of superfluid turbulence for @xmath254 is more complicated than @xmath255 because there are no well justified theoretical approaches ( like in the problem of kelvin wave turbulence at @xmath256 ) or even commonly accepted uncontrolled closure approximations .
nevertheless , there some qualitative predictions can be tested numerically and experimentally , at least in the zero temperature limit .
comparison @xcite of the hydrodynamic spectrum with the kelvin wave spectrum at @xmath257 suggests that the one dimensional nonlinear transfer mechanisms among weakly nonlinear kelvin waves on individual vortex lines is less efficient than the three dimensional , strongly nonlinear eddy - eddy energy transfer .
the consequence is an energy cascade stagnation at the crossover between the collective eddy - dominated scales and the single vortex wave - dominated scales .
@xcite argued that the superfluid energy spectrum @xmath63 at @xmath254 should be a mixture of three dimensional hydrodynamics modes and one dimensional kelvin waves motions ; the corresponding spectra should be @xmath258e(k)\ .\ ] ] here @xmath259^{-1}\ ] ] is theblending " function which was found @xcite by calculating the energies of correlated and uncorrelated motions produced by a system of @xmath10-spaced wavy vortex lines .
the total energy flux , @xmath62 arising from hydrodynamic and kelvin - wave contributions , was modelled @xcite by dimensional reasoning in the differential approximation , similar to eq .
: for @xmath260 the energy flux is purely hydrodynamic and @xmath63 is given by eq .
, while for @xmath261 it is purely supported by kelvin waves and @xmath63 is given by eq . .
this approach leads to the ordinary differential equation @xmath262 constant , which was solved numerically .
the predicted energy spectra @xmath63 for different values of @xmath263 are shown in fig .
[ f:3]c , exhibit a bottleneck energy accumulation @xmath73 in agreement with eq . .
recently a model @xcite has been proposed that accounts for the fact that ( according to numerical evidence @xcite and analytical estimates @xcite ) small scales excitations ( @xmath264 ) , such as kelvin waves and isolated rings , are fully damped for @xmath265k .
thus , at these temperatures , the energy flux @xmath266 should be very small at scales @xmath267 .
the idea @xcite was to use the hvbk eqs . but truncating the superfluid beyond a cutoff wavenumber @xmath268 , where @xmath269 is a fitting parameter of order one .
obviously , a limitation of this model is the abruptness of the truncation ( a more refined model could incorporate a smoother closure which accounts for the dissipation associated with vortex reconnections and the difference between @xmath270 and @xmath271 ) .
direct numerical simulations of this truncated hvbk model for temperatures @xmath272 with @xmath273 confirmed the ko41 scaling of the two locked fluids in the range @xmath274 ( see fig .
[ f:3]a ) . at smaller scales ,
an intermediate ( meso ) regime @xmath275 was found that expands as the temperature is lowered @xcite .
apparently , superfluid energy , cascading from larger length scales , accumulates beyond @xmath276 . at the lowest temperatures
, this energy appears to thermalize , approaching equipartition with @xmath277 , as shown by the red curve of fig .
the process saturates when the friction coupling with the normal fluid becomes strong enough to balance the incoming energy flux @xmath278 .
in physical space , this mesoscale thermalization should manifest itself as a randomisation of the vortex tangle .
the effect is found to be strongly temperature dependent@xcite : @xmath279 .
the truncated hydrodynamic model reproduces the decreasing spectrum of the vortex line density fluctuations at small @xmath57 and reduces to the classical spectrum in the @xmath280 limit .
this accumulation of thermalized superfluid excitations at small scales and finite temperature was predicted by an earlier model developed to interpret vortex line density spectra @xcite .
we conclude that , at large hydrodynamic scales @xmath82 , the evidence for the ko41 @xmath77 scaling of the superfluid energy spectrum which arises from experiments , numerical simulations and theory ( across all models used ) is strong and consistent , and appears to be independent of temperature ( including the limit of zero temperature in the absence of the normal fluid @xcite ) .
this direct spectral evidence is also fully consistent with an indirect body of evidence arising from measurements of the kinetic energy dissipation ( @xcite ) and vortex line density decay @xcite in turbulent helium flows .
the main open issue is the existence of vortex bundles @xcite predicted by the vfm , for which there is no direct experimental observation yet .
intermittency effects , predicted by shell models @xcite , also await for experimental evidence .
what happens at mesoscales just above @xmath281 is less understood .
the differential model ( at @xmath257 , sect .
[ 8a ] ) and the truncated hvbk model ( at finite @xmath25 , sec . [ 8b ] ) , predict an upturning of the spectrum ( temperature - dependent for the latter model ) in this region of @xmath57-space .
if confirmed by the experiments and the vfm model , this would signify the striking appearance of quantum effects at scales larger than @xmath10 .
further insight could arise from better understanding of fluctuations of the vortex line density .
it is worth noticing that similar macroscopic manifestation of the singular nature of the superfluid vorticity was also predicted for the pressure spectrum @xcite . at length scales of the order of @xmath10 and less than @xmath10
the situation is even less clear .
this regime is very important at the lowest temperatures , where the kelvin waves are not damped , and energy is transferred from the eddy dominated , three
dimensional kolmogorov - richardson cascade into a kelvin wave cascade on individual vortex lines , until the wavenumber is large enough that energy is radiated as sound .
the main open issues which call for better understanding concern the cross over and more elaborated description of the bottleneck energy accumulation around @xmath10 in the wide temperature range from 0 to about @xmath161 and the role of vortex reconnections in the strong regime ( large kelvin wave amplitudes compared to wavelength ) of the cascade . at the moment
, there is much debate on these problems but no direct experimental evidence for these effects .
it is however encouraging that the most recent gpe simulations @xcite in the weak regime ( small amplitude compared to wavelength ) seem to agree with theoretical predictions .
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( 138):537 _ _ _ _ _ _ | turbulence in superfluid helium is unusual and presents a challenge to fluid dynamicists because it consists of two coupled , inter penetrating turbulent fluids : the first is inviscid with quantised vorticity , the second is viscous with continuous vorticity . despite this double nature , the observed spectra of the superfluid turbulent velocity at sufficiently large length scales are similar to those o ordinary turbulence .
we present experimental , numerical and theoretical results which explain these similarities , and illustrate the limits of our present understanding of superfluid turbulence at smaller scales . | 1306.6248 |
the problem of classification of germs of foliations in the complex plane is stated by thom @xcite .
he conjectured that the analytic type of a foliation defined in a neighborhood of a singular point is completely determined by its associated separatrix and its corresponding holonomy .
r. moussu in @xcite gave a counterexample for this statement and showed that we have to consider the holonomy representation of each irreducible component of the exceptional divisor in a desingularization instead of the economizes .
for this new version , thom s problem is proved for cuspidal type singular points @xcite , @xcite and more generally for quasi - homogeneous foliations @xcite .
however , the new statement of thom s conjecture was refuted by j.f .
mattei by computing the dimension of the space of isoholonomic deformations @xcite , @xcite : there must be other invariants for the non quasi - homogeneous foliations .
this conclusion is confirmed by the number of free coefficients in the normal forms in @xcite , @xcite and in the hamiltonian part of the normal forms of vector field in @xcite . by adding a new invariant called _ set of slidings _ this paper solves the problem of strict classification for the non - dicritical foliations whose camacho - sad indices are not rational .
here , strict classification means up to diffeomorphism tangent to identity . a germ of singular foliation @xmath0 in @xmath1 is called _ reduced _ if there exists a coordinate system in which it is defined by a @xmath2-form whose linear part is @xmath3 @xmath4 is called the _ camacho - sad index _ of @xmath0 . when @xmath5 , the origin is called a _
saddle - node _ singularity , otherwise it is called _
nondegenerate_. a theorem of a. seidenberg @xcite says that any singular foliation @xmath0 with isolated singularity admits a canonical _ desingularization_. more precisely , there is a holomorphic map @xmath6 obtained as a composition of a finite number of blowing - ups at points such that any point @xmath7 of the _ exceptional divisor _
@xmath8 is either a regular point or a reduced singularity of the strict transform @xmath9 .
an intersection of two irreducible components of @xmath10 is called a _
corner_. an irreducible component of @xmath10 is a _
dead branch _ if in this component there is a unique singularity of @xmath11 that is a corner .
a _ separatrix _ of @xmath0 is an analytical irreducible invariant curve through the origin of @xmath0 .
it is well known that any germ of singular foliation @xmath0 in @xmath1 possesses at least one separatrix @xcite .
when the number of separatrices is finite @xmath0 is _ non - dicritical_. otherwise it is called _
dicritical_. denote by @xmath12 the set of all singularities of the strict transform @xmath11 .
let @xmath13 be a non - dicritical irreducible component of the exceptional divisor @xmath10 , then @xmath14 is a leaf of @xmath11 .
let @xmath7 be a regular point in @xmath15 and @xmath16 a small analytic section through @xmath7 transverse to @xmath11 .
for any loop @xmath17 in @xmath15 based on @xmath7 there is a germ of a holomorphic return map @xmath18 which only depends on the homotopy class of @xmath17 in the fundamental group @xmath19 .
the map @xmath20 is called the _ vanishing holonomy representation _ of @xmath0 on @xmath13 .
let @xmath21 be a foliation that also admits @xmath22 as its desingularization map .
assume that @xmath23 where @xmath24 is the set of singularities of the strict transform @xmath25 .
denote by @xmath26 in @xmath27 the vanishing holonomy representation of @xmath0 .
we say that the vanishing holonomy representation of @xmath0 and @xmath21 on @xmath13 are conjugated if there exists @xmath28 such that @xmath29 .
the vanishing holonomy representation of @xmath0 and @xmath21 are called conjugated if they are conjugated on every non - dicritical irreducible component of @xmath10 .
[ no1 ] we denote by @xmath30 the set of all non - dicritical foliations @xmath0 defined on @xmath1 such that the camacho - sad index of @xmath11 at each singularity is not rational .
if @xmath0 is in @xmath30 then after desingularization all the singularities of @xmath11 are not saddle - node .
moreover , the chern class of an irreducible component of divisor , which is an integer , is equal to the sum of camacho - sad indices of the singularities in this component @xcite . therefore , every element in @xmath31 after desingularization admits no dead branch in its exceptional divisor .
let @xmath22 as in be a composition of a finite number of blowing - ups at points .
a germ of singular holomorphic foliation @xmath32 is said @xmath22-_absolutely dicritical _ if the strict transform @xmath33 is a regular foliation and the exceptional divisor @xmath34 is completely transverse to @xmath35 .
when @xmath22 is the standard blowing - up at the origin , we called @xmath32 a _ radial foliation_. at each corner @xmath36 of @xmath10 , the diffeomorphism from @xmath37 to @xmath38 that follows the leaves of @xmath35 is called the _ dulac map _ of @xmath35 at @xmath39 .
the existence of such foliations for any given @xmath22 is proved in @xcite .
in fact , in @xcite the authors showed that if in each smooth component of @xmath10 we take any two smooth curves transverse to @xmath10 then there is always an absolutely dicritical foliation admitting them as their integral curves .
we will denote by @xmath40 if at any point @xmath41 the separatrices of @xmath11 through @xmath39 are transverse to @xmath35 .
[ lem1 ] let @xmath0 be a non - dicritical foliation such that @xmath22 is its desingularization map .
then there exists a @xmath22-absolutely dicritical foliation @xmath32 satisfying @xmath40 denote by @xmath42 the strict transforms of the separatrices of @xmath0 . on each component @xmath13 of @xmath10 that does not contain any singularity of @xmath11 except the corners we take a smooth curve @xmath43 transverse to @xmath13 .
then we have the set of curves @xmath44 such that each component of @xmath10 is transverse to at least one curve @xmath45 .
denote by @xmath46 .
by @xcite , for each @xmath47 there exists a @xmath22-absolutely dicritical foliation @xmath48 defined by a @xmath2-form @xmath49 verifying that @xmath45 is transverse to @xmath50 .
choose a local chart @xmath51 at @xmath52 such that @xmath53 and @xmath54 where `` h.o.t .
'' stands for higher order term .
write down @xmath49 in the local chart @xmath55 @xmath56 because @xmath50 is transverse to @xmath10 , we have @xmath57 .
we define @xmath58 .
there always exists a vector @xmath59 such that for @xmath60 , @xmath61 denote by @xmath62 .
then , in the local chart @xmath55 , we have @xmath63 because @xmath64 and @xmath49 , @xmath65 , have the same multiplicity on each component of @xmath10 , they have the same vanishing order .
since each component of @xmath10 contains at least one point @xmath52 and the strict transform @xmath35 of the foliation @xmath32 defined by @xmath64 is transverse to @xmath10 in each neighborhood of each @xmath52 , @xmath35 is generically transverse to @xmath10 .
by @xcite , @xmath32 is absolutely dicritical and satisfies @xmath40 .
consider first a nondegenerate reduced foliation @xmath0 in @xmath1 . by @xcite , there exists a coordinate system in which @xmath0 is defined by @xmath66 where @xmath67 .
let @xmath32 be a germ regular foliation whose invariant curve through the origin ( we call it the separatrix of @xmath32 ) is transverse to the two separatrices of @xmath0 , which are denoted by @xmath68 and @xmath69 .
then we have the following lemma , whose proof is straightforward .
[ lem2 ] the tangent curve of @xmath0 and @xmath32 , denoted @xmath70 , is a smooth curve transverse to the two separatrices of @xmath0 .
moreover , if the separatrix of @xmath32 is tangent to @xmath71 then @xmath70 is tangent to @xmath72 . and @xmath32 . ]
after a standard blowing - up @xmath73 at the origin , the strict transform @xmath74 of @xmath70 is transverse to @xmath11 and cut @xmath75 at @xmath39 .
we denote by @xmath76 and @xmath77 the vanishing holonomy representation of @xmath0 .
we choose a generator @xmath17 for @xmath78 .
then @xmath73 induces @xmath79 we call @xmath80 the _ holonomy on the tangent curve @xmath70_. denote by @xmath81 and @xmath82 the projection by the leaves of @xmath32 from @xmath70 to @xmath68 and @xmath69 respectively .
the sliding of a reduced foliation @xmath0 and a regular foliation @xmath32 on @xmath68 ( resp . , @xmath69 ) is the diffeomorphism ( figure [ figure1 ] ) @xmath83 let @xmath84 be the dulac map of @xmath32 ( section [ sec1.2 ] ) . since @xmath85 , it is obvious that @xmath86 now let @xmath0 be a non - dicritical foliation such that after desingularization by the map @xmath22 all singularities of @xmath87 are nondegenerate . by lemma [ lem1 ] there exists a @xmath22-absolutely dicritical foliation @xmath88 such that @xmath89 .
[ no5 ] we denote by @xmath90 the set of all @xmath22-absolutely dicritical foliations @xmath32 satisfying the two following properties : * @xmath35 and @xmath91 have the same dulac maps at any corner of @xmath10 .
* at each singularity @xmath39 of @xmath11 , the invariant curves of @xmath35 and @xmath91 through @xmath39 are tangent ( figure [ p38 ] ) .
of @xmath92 let @xmath32 in @xmath90 and @xmath13 be an irreducible component of @xmath10 .
suppose that @xmath93 are the singularities of @xmath11 on @xmath13 .
then we denote by @xmath94 where @xmath95 is the sliding of @xmath11 and @xmath35 in a neighborhood of @xmath52 .
the sliding of @xmath0 and @xmath32 is @xmath96 where @xmath97 is the set of all irreducible components of @xmath10 .
the set of slidings of @xmath0 relative to direction @xmath88 is the set @xmath98 we will prove in corollary [ cor3 ] that @xmath99 is an invariant of @xmath0 : if @xmath0 and @xmath21 are conjugated by @xmath100 then for each @xmath32 in @xmath90 we have @xmath101 . under some conditions for @xmath0 and @xmath21 ( theorem [ thr1 ] )
, we will have @xmath102
. therefore @xmath103 .
moreover , @xmath104 is also in @xmath90 .
consequently , @xmath105 .
[ re7 ] for each singularity @xmath39 of @xmath11 that is a corner , i.e. , @xmath36 , there are two slidings @xmath106 and @xmath107 . however , by , @xmath107 is completely determined by @xmath106 and the dulac map of @xmath35 at @xmath39 .
+ this invariant is named `` sliding '' because it gives an obstruction for the construction of local conjugacy of two foliations that fixes the points in the exceptional divisor ( corollary [ cor3 ] ) .
+ the definition of @xmath99 does not depend on choosing a element @xmath88 in @xmath90 .
more precisely , if @xmath108 then @xmath109 and @xmath110 + although @xmath111 is a set of local diffeomorphisms , it is not a local invariant .
@xmath111 also contains the information of the relation of those local diffeomorphisms because all these local diffeomorphisms are defined by the holonomy projections following the global fibration @xmath32 : in some sense , any fibration @xmath112 plays the role of a global common transversal coordinate on which the slidings invariants are computed all together and at the same time .
let us clarify here the role of the sliding invariant in the problem of classification of germs of foliations .
suppose that two non - dicritical foliations @xmath0 and @xmath21 satisfy that their separatrices and their vanishing holonomies are conjugated .
moreover , after desingularization , all the camacho - sad indices of @xmath0 and @xmath21 are coincide . then after blowing - ups , @xmath0 and @xmath21 are locally conjugated in a neighborhood of their singularities .
although we have the conjugation of their vanishing holonomies , in general , we can not glue the local conjugation together .
the obstruction is that these local conjugations induce the local diffeomorphisms on the exceptional divisor which we call the slidings . in general
, there is no reason for those slidings being parts of a global diffeomorphism of the divisor .
let @xmath113 .
we say that their _ strict separatrices are tangent _ , denoted @xmath114 , if they have the same desingularization map and the same set of singularities .
moreover , at each singularity which is not a corner of the divisor the separatrices of @xmath11 and @xmath25 are tangent . if @xmath114 and @xmath88 is an absolutely dicritical foliation satisfying @xmath89 then @xmath115 and @xmath116 .
we denote by @xmath117 the set of camacho - sad indices of @xmath11 at all singularities .
we also denote by @xmath118 if at each singularity , @xmath11 and @xmath25 have the same camacho - sad index .
[ thr1 ] let @xmath0 and @xmath21 be two foliations in the class @xmath30 ( see notation [ no1 ] ) such that @xmath114 .
suppose that @xmath88 is an absolutely dicritical foliation satisfying @xmath89 .
let @xmath90 be as in notation [ no5 ] and @xmath99 , @xmath119 the corresponding sets of slidings .
then the three following statements are equivalent : 1 .
@xmath0 and @xmath21 are strictly analytically conjugated .
their vanishing holonomy representations are strictly analytically conjugated , @xmath118 and @xmath105 .
their vanishing holonomy representations are strictly analytically conjugated , @xmath118 and @xmath120 . here
, a strict conjugacy means a conjugacy tangent to identity . we will prove that the slidings of foliations are finitely determined : [ thr2 ] let @xmath0 be a non - dicritical foliation without saddle - node singularities after desingularization .
there exists a natural @xmath121 such that if there is a non - dicritical foliation @xmath21 satisfying the following conditions : 1 .
@xmath0 and @xmath21 have the same set of singularities after desingularization and at a neighborhood of each singularity , @xmath11 and @xmath25 are locally strictly analytically conjugated , 2 .
there exist @xmath122 in @xmath90 such that @xmath123 , then there exists @xmath124 such that @xmath124 is strictly conjugated with @xmath32 and @xmath125 .
here @xmath123 means @xmath126 for all @xmath127 in @xmath111 , @xmath128 in @xmath129 , where @xmath130 stands for the regular part of degree @xmath121 in the taylor expansion of @xmath127 .
+ these two theorems also give two corollaries on finite determination of the class of isoholonomic non - dicritical foliations and absolutely dicritical foliations that have the same dulac maps ( see corollary [ cor17 ] and [ cor19 ] ) .
+ this paper is organized as follows : in section 2 , local conjugacy of the pair @xmath131 will be proved .
we prove theorem [ thr1 ] in section 3 .
section 4 is devoted to prove theorem [ thr2 ] and two corollaries of finite determination of class of isoholonomic non - dicritical foliations and absolutely dicritical foliations that have the same dulac maps .
let @xmath0 , @xmath21 be two germs of nondegenerate reduced foliations in @xmath1 .
denote by @xmath68 , @xmath69 and @xmath132 , @xmath133 the separatrices of @xmath0 and @xmath21 respectively .
let @xmath32 and @xmath134 be two germs of regular foliations such that their separatrices @xmath135 and @xmath136 are transverse to the two separatrices of @xmath0 and @xmath21 respectively .
suppose that @xmath100 is a diffeomorphism conjugating @xmath131 and @xmath137 , then the restriction of @xmath100 on the tangent curves commutes with the holonomies on @xmath70 and @xmath138 of @xmath0 and @xmath21 .
the converse is also true : [ pro8 ] suppose that @xmath0 and @xmath21 have the same camacho - sad index .
if @xmath139 is a diffeomorphism commuting with the holonomies of @xmath0 and @xmath21 then @xmath140 extends to a diffeomorphism @xmath100 of @xmath1 sending @xmath131 to @xmath137 .
moreover , if we require that @xmath100 sends @xmath68 ( resp .
@xmath69 ) to @xmath132 ( resp .
@xmath133 ) then this extension is unique . by lemma [ lem1 ] , the curves @xmath68 , @xmath69 , @xmath135 , @xmath70 ( resp
. , @xmath132 , @xmath133 , @xmath136 , @xmath138 ) are four transverse smooth curves .
it is well known that there exist two radial foliations @xmath141 and @xmath142 such that @xmath68 , @xmath69 , @xmath135 , @xmath70 and @xmath132 , @xmath133 , @xmath136 , @xmath138 are the invariant curves of @xmath141 and @xmath142 respectively . after a blowing - up at the origin ,
denote by @xmath143 , @xmath144 , @xmath145 , @xmath146 ( resp . , @xmath147 , @xmath148 , @xmath149 , @xmath150 ) the intersections of strict transforms of @xmath68 , @xmath69 , @xmath135 , @xmath70 ( resp . , @xmath132 , @xmath133 , @xmath136 , @xmath138 ) with @xmath151 .
take @xmath152 in @xmath153 that sends @xmath143 , @xmath144 , @xmath145 to @xmath147 , @xmath148 , @xmath149 respectively . by lemma [ lem1 ] , the direction of @xmath70 ( resp . , @xmath138 )
is completely determined by the camacho - sad index and the direction of @xmath135 ( resp . , @xmath136 ) .
therefore , @xmath154 . using the path lifting method after a blowing - up @xcite
, @xmath140 extends to a diffeomorphism @xmath155 of @xmath1 sending @xmath156 to @xmath157 .
denote by @xmath158 . because @xmath159 sends @xmath136 and @xmath138 to @xmath135 and @xmath70 respectively , @xmath135 is also the separatrix of @xmath88 and @xmath160 .
we denote by @xmath161 the tangent curve @xmath162 .
the proof is reduced to show that there exists a diffeomorphism fixing points in @xmath161 sending @xmath131 to @xmath163 .
choose a system of coordinates @xmath164 such that @xmath88 is defined by @xmath165 and @xmath0 is defined by a @xmath2-form @xmath166 then @xmath161 is defined by @xmath167 we claim that there exist a natural @xmath168 and a holomorphic function @xmath169 such that @xmath32 is defined by @xmath170 indeed , assume that @xmath32 is defined by @xmath171 where @xmath172 is invertible .
rewrite the equation of @xmath161 as @xmath173 where @xmath174 . because @xmath175 , the maps @xmath176 and @xmath177 are diffeomorphic .
hence there exists a diffeomorphism @xmath178 such that @xmath179 this is equivalent to @xmath180 therefore , there exist a natural @xmath181 and a function @xmath169 satisfying @xmath182 such that @xmath183 because @xmath184 is a diffeomorphism , @xmath32 is also defined by @xmath185 .
let us prove @xmath168 .
we have @xmath186 because @xmath161 is defined by @xmath187 , we have @xmath188 the fact @xmath189 forces to @xmath190 and @xmath191 . this implies @xmath192 .
consequently , @xmath168 .
now let @xmath193 tangent to @xmath0 .
now we will show that there exists @xmath194 such that the diffeomorphism @xmath195x$ ] satisfies @xmath196x(x , y)=\sum_{i\geq 0}\frac{\tau^{i(n-1)}\alpha^i}{i!}\mathrm{ad}^i_{x}(x+y)=f(x , y),\ ] ] where @xmath197 is the adjoint representation .
since @xmath198 becomes @xmath199 hence , the existence of @xmath200 comes from the implicit function theorem .
+ now we will prove the uniqueness of @xmath100 .
in fact , we only need to show that if there exists a diffeomorphism @xmath201 that sends @xmath163 to itself , preserves the two separatrices of @xmath0 and fixes the points of @xmath161 then @xmath202 . since @xmath203 , @xmath201 sends every leaf of @xmath0 into itself . by @xcite
, there exists @xmath204 such that @xmath205x.\ ] ] because @xmath88 is defined by the function @xmath206 and @xmath201 fixes points in @xmath161 , we get @xmath207x = x+y.\ ] ] decompose @xmath208 into the homogeneous terms @xmath209 since @xmath210 and @xmath211 for all @xmath47 , where @xmath212 , we have @xmath213x&=\sum_{i=0}^{\infty}\frac{\beta_0^i}{i!}x+\sum_{i=0}^{\infty}\frac{\beta_0^i}{i!}((-\lambda)^iy)+h.o.t.\\ & = \exp(\beta_0)x+\exp(-\lambda\beta_0)y+h.o.t .. \end{aligned}\ ] ] so leads to @xmath214 hence , @xmath215x&=\sum_{i=0}^{\infty}\frac{\beta_0^i}{i!}x=\exp(\beta_0 ) x = x , \label{5}\\ y\circ\exp[\beta_0]x&=\sum_{i=0}^{\infty}\frac{\beta_0^i}{i!}\left((-\lambda)^i y+c_i\right)=\exp(-\lambda
\beta_0 ) y+c = y+c,\label{6}\end{aligned}\ ] ] where @xmath216 .
we claim that @xmath217x=\exp[\beta_0]x\circ\exp[\bar\beta]x.\ ] ] indeed , for any @xmath218 we have @xmath219x\circ\exp[\bar\beta]x & = \left(\sum_{i=0}^{\infty}\frac{\beta_0^i}{i!}\mathrm{ad}^i_x(h)\right)\circ\exp[\bar\beta]x = \sum_{j=0}^{\infty}\frac{\bar\beta^j}{j!}\mathrm{ad}^j_x\left(\sum_{i=0}^{\infty}\frac{\beta_0^i}{i!}\mathrm{ad}^i_x(h)\right)\\ & = \sum_{k=0}^{\infty}\sum_{i+j = k}\frac{\bar\beta^j \beta_0^i}{j!i!}\mathrm{ad}^k_x(h)=\sum_{k=0}^{\infty } \frac{(\bar\beta+\beta_0)^k}{k!}\mathrm{ad}^k_x(h)=h\circ\exp[\beta]x.\end{aligned}\ ] ] we write @xmath220 where @xmath221 . by , , we get @xmath213x&=x\circ\exp[\beta_0]x\circ \exp[\bar\beta]x+ y\circ\exp[\beta_0]x\circ\exp[\bar\beta]x\nonumber\\ & = x\circ\exp[\bar\beta]x+ ( y+c)\circ\exp[\bar\beta]x\nonumber\\ & = \sum_{i=0}^{\infty}\frac{\bar\beta^i}{i!}x+\sum_{i=0}^{\infty}\frac{\bar\beta^i}{i!}\mathrm{ad}^i_x(y+c)\nonumber\\ & = \exp(\bar\beta)x+\exp(-\lambda\bar{\beta})y+ \sum_{i=0}^{\infty}\frac{\bar\beta^i}{i!}d_i\nonumber\\ & = x\prod_{i=1}^{\infty}\exp(\beta_i)+ y\prod_{i=1}^{\infty}\exp(-\lambda\beta_i)+ \sum_{i=0}^{\infty}\frac{\bar\beta^i}{i!}d_i.\label{8}\end{aligned}\ ] ] we will prove @xmath222 by induction . from
we have @xmath223x = x(1+\beta_1)+ y(1-\lambda\beta_1)+h.o.t.\ ] ] so forces @xmath224 .
suppose that @xmath225 , we have @xmath223x = x(1+\beta_k)+ y(1-\lambda\beta_k)+h.o.t .. \ ] ] then again leads to @xmath226 and consequently @xmath227 .
this implies that @xmath228x=(x , y+c).\ ] ] finally , again gives @xmath229 .
so @xmath202 .
[ cor11 ] suppose that @xmath0 and @xmath21 are two nondegenerate reduced foliations that are analytically conjugated .
let @xmath32 and @xmath134 be two regular foliations that are transverse to the two separatrices of @xmath0 and @xmath21 respectively .
then there exists a diffeomorphism that sends @xmath131 to @xmath137 .
let @xmath201 be the conjugacy of @xmath0 and @xmath21 . denote by @xmath230 . then the restriction @xmath231 commutes with the holonomies of @xmath0 on @xmath70 and @xmath21 on @xmath232 .
moreover by the holonomy transport , the holonomies of @xmath21 on @xmath232 and on @xmath138 are conjugated .
hence , the holonomies of @xmath0 on @xmath70 and @xmath21 on @xmath138 are conjugated . by proposition [ pro8 ] there
exists a diffeomorphism that sends @xmath131 to @xmath137 by projecting on @xmath68 and @xmath69 the holonomies defined on @xmath70 and @xmath138 respectively , we can obtain [ cor3 ] if @xmath100 is a diffeomorphism conjugating @xmath131 and @xmath137 , then @xmath233 reciprocally , if @xmath234 and @xmath235 is a diffeomorphism satisfying @xmath236 then @xmath140 uniquely extends to a diffeomorphism @xmath100 of @xmath1 sending @xmath131 to @xmath137 . because @xmath100 conjugates @xmath131 and @xmath137 , the restriction @xmath237 commutes with the holonomies @xmath80 and @xmath26 of @xmath0 and @xmath21 .
denote by @xmath81 ( resp .
, @xmath238 ) the projection by the leaves of @xmath32 ( resp . , @xmath134 ) from @xmath70 ( resp . , @xmath138
) to @xmath68 ( resp . , @xmath132 )
. since @xmath100 sends @xmath131 to @xmath137 , we have @xmath239 therefore @xmath240 reciprocally , suppose @xmath235 is a diffeomorphism commuting with the slidings of @xmath0 and @xmath21 .
denote by @xmath241 then @xmath242 by proposition [ pro8 ] , @xmath243 uniquely extends to a diffeomorphism @xmath100 that sends @xmath131 to @xmath137 . in particular , if in corollary
[ cor3 ] we have @xmath244 and @xmath245 then there exists a diffeomorphism sending @xmath131 to @xmath137 and fixing points in @xmath68 .
this whole section is devoted to prove theorem [ thr1 ] .
the direction ( ( ii)@xmath246(iii ) ) is obvious .
+ ( ( i)@xmath246(ii ) ) since the camacho - sad index is an analytic invariant , it is obvious that @xmath118
. let @xmath100 be the strict conjugacy and @xmath247 be its lifting by @xmath22 .
suppose that a non - corner point @xmath7 of @xmath10 is a fixed point of @xmath248 .
then the linear map @xmath249 has two eigenvalues .
one corresponds to the direction of the divisor .
we denote by @xmath250 the other eigenvalue and define @xmath251 for each corner @xmath7 .
[ lem9 ] @xmath102 so @xmath252 is a function defined on @xmath10 and moreover @xmath253 .
denote by @xmath73 the standard blowing - up at the origin of @xmath1 @xmath254 on @xmath255 , we use the two standard chart @xmath256 and @xmath257 together with the transition functions @xmath258 , @xmath259 .
suppose that @xmath260 then in the coordinate system @xmath256 we have @xmath261 where @xmath262 .
therefore @xmath263 fixes points in @xmath255 and @xmath264 .
let @xmath39 be a non - reduced singularity of @xmath265 on @xmath255 .
we will show that @xmath266 and apply the inductive hypothesis for @xmath155 in a neighborhood of @xmath39 .
indeed , let @xmath267 be the blowing - up at @xmath39 and @xmath268 .
denote by @xmath269 and @xmath270 all invariant curves of @xmath265 and @xmath271 through @xmath39 . because every element in @xmath31 after desingularization admits no dead component in its exceptional divisor ,
@xmath272 is not a dead component .
therefore there is at least one irreducible component @xmath273 of @xmath269 that are not tangent to @xmath255 . because @xmath274 and @xmath114 , @xmath275 has an eigenvector different from the direction of @xmath255 , which is corresponding to the direction of @xmath273 .
so @xmath276 has two eigenvectors .
since both of their eigenvalues are @xmath2 , we have @xmath266 .
now let @xmath277 and denote by @xmath278 .
since @xmath102 , the strict transforms @xmath35 and @xmath279 have the same dulac maps .
moreover , because @xmath114 , at each singularity @xmath39 of @xmath11 , @xmath280 has two eigenvectors . as @xmath253 and @xmath102 we have @xmath281 .
therefore the invariant curves of @xmath35 and @xmath279 through @xmath39 are tangent .
this gives @xmath282 . because @xmath283 fixes points in @xmath10 , by corollary [ cor3 ] the identity map commutes with the slides of @xmath0 and @xmath21 .
this leads to @xmath284 .
consequently , @xmath105 .
moreover , the vanishing holonomy representation of @xmath0 and @xmath21 are conjugated by @xmath283 . since @xmath253
this conjugacy is strict .
+ ( ( iii)@xmath246(i ) ) suppose that @xmath32 , @xmath285 satisfy @xmath284 . by corollary [ cor3 ] , at each singularity @xmath52 , @xmath286 , of @xmath11 there exists a neighborhood @xmath287 of @xmath52 and a local conjugacy @xmath288 such that @xmath289 .
let @xmath290 be a neighborhood of @xmath291 such that @xmath290 does not contain any singularity of @xmath11 .
note that @xmath290 is not connected and the restriction of @xmath11 and @xmath25 on @xmath290 are regular .
the strict conjugacy of the vanishing holonomy representations can be extended by path lifting method to the conjugacy @xmath292 satisfying that the second eigenvalue function @xmath293 is identically @xmath2 .
we will show that on each intersection @xmath294 , @xmath295 and @xmath296 coincide .
denote by @xmath297 we claim that @xmath298 on @xmath299 .
let @xmath300 in @xmath301 .
denote by @xmath302 and @xmath303 the invariant curves of @xmath35 through @xmath39 and @xmath304 respectively .
as the two maps @xmath305 and @xmath306 are conjugated by the holonomy transport , we have @xmath307 .
consequently , @xmath308 is constant on @xmath309 . since @xmath310
, it follows that @xmath311 is constant on @xmath301 .
therefore , @xmath311 is constant on @xmath312 .
moreover , at the singularity @xmath52 , @xmath313 has three eigenvectors corresponding to the directions of the divisor and the directions of invariant curves of @xmath11 and @xmath35 through @xmath52 . since @xmath313 has also one eigenvalue @xmath2 corresponding to the directions of the divisor , we have @xmath314 .
this gives @xmath315 and consequently @xmath298 .
now at each point @xmath316 , the map @xmath305 commutes with the holonomy of @xmath11 around @xmath52 .
since the camacho - sad index @xmath317 of @xmath11 at @xmath52 is not rational , lemma [ lem10 ] below says that @xmath318 and so @xmath319 .
hence we can glue all @xmath295 together and the strict conjugacy we need is the projection of this diffeomorphism on @xmath1 by @xmath22 .
[ lem10 ] let @xmath320 such that @xmath321 where @xmath322 . if @xmath323 satisfying @xmath324 and @xmath325 then @xmath326 . since @xmath322 , there is a formal diffeomorphism @xmath140 such that @xmath327
denote by @xmath328 , then @xmath329 and @xmath330 .
the proof is reduced to show that @xmath331 .
suppose that @xmath332 . then @xmath333 and @xmath334 since @xmath322 , it forces @xmath335 for all @xmath336 .
hence @xmath331 .
let @xmath337 be a germ of curve at @xmath39 in a surface @xmath338 .
we denote by @xmath339 the set of all germs of singular curves having the same desingularization map and having the same singularities as @xmath337 after desingularization . here , the singularities of @xmath337 after desingularization are the singularities of the curve defined by the union of strict transform of @xmath337 and the exceptional divisor .
if @xmath337 is smooth , we denote by @xmath340 the set of all holomorphic functions on @xmath337 whose vanishing orders at @xmath39 are at least @xmath341 . [ pro11 ]
let @xmath337 be a germ of curve in @xmath1 and @xmath342 be its irreducible components .
suppose that @xmath343 is a finite composition of blowing - ups such that all the transformed curves @xmath344 are smooth .
then there exists a natural @xmath121 such that if @xmath345 , @xmath346 , then there exists @xmath347 such that @xmath348 .
moreover , the same @xmath121 can be chosen for all elements in @xmath339 .
we first consider the statement when @xmath337 is irreducible .
if @xmath337 is smooth then @xmath349 is diffeomorphic to @xmath337 .
so we can suppose that @xmath337 is singular .
denote by @xmath350 . choose a coordinate system @xmath351 in a neighborhood of @xmath39 such that @xmath352 and @xmath353 .
then @xmath354 is defined by @xmath355 where @xmath356 , @xmath357 and @xmath358 .
so we have @xmath359 therefore , there exist a natural @xmath360 and a holomorphic function @xmath169 such that @xmath361 where @xmath362 . we claim that @xmath169 is a unit . indeed ,
suppose @xmath363 and denote by @xmath364 the curve @xmath365 .
let @xmath135 be a curve defined in @xmath1 such that @xmath366 .
let @xmath367 be a reduced equation of @xmath135 . by , @xmath368 and @xmath369 .
it contradicts @xmath357
. now denote by @xmath370 which is a unit , we have @xmath371 for each @xmath372 there exists @xmath373 such that @xmath374 .
thus @xmath375 so implies that if a holomorphic function @xmath376 satisfies @xmath377 then there exists a holomorphic function @xmath378 such that @xmath379 .
consequently @xmath380 in the general case , suppose that @xmath381 is defined by @xmath382 . if @xmath383 , @xmath346 , with @xmath121 big enough , there exist @xmath384 , @xmath346 , such that @xmath385 .
we will find a holomorphic function @xmath386 such that @xmath387 for all @xmath346 .
this is reduced to show that there exists a natural @xmath388 such that the following morphism @xmath389 is surjective @xmath390 indeed , by hilbert s nullstellensatz , there exists a natural @xmath391 such that @xmath392 for all @xmath393 .
we will show that for all @xmath346 , @xmath394 the elements @xmath395 , where @xmath396 is in the @xmath397 position , are in @xmath398 and then @xmath388 can be chosen as @xmath399 .
we decompose @xmath400 where @xmath401 . by ,
there exist @xmath402 such that @xmath403 .
this implies that @xmath404 now we will show that the same @xmath121 can be chosen for all elements of @xmath339 . in the case
@xmath337 is irreducible , let @xmath405 in @xmath339 and @xmath406 be the equation of @xmath407 .
we also have @xmath408 where @xmath409 which is a unit .
consequently , holds . in the general case , it is sufficient to show that the same @xmath391 in can be chosen for all elements of @xmath339
. let @xmath410 be the smallest natural satisfying @xmath411 we claim that @xmath412 indeed , there exists @xmath413 .
let @xmath414 , @xmath415 , be a sequence of monomials such that @xmath416 , @xmath417 and either @xmath418 or @xmath419 .
since @xmath420 we have @xmath421 for all @xmath422 .
we will show that @xmath423 is independent in the vector space @xmath424 over @xmath425 .
suppose that @xmath426 suppose there exists @xmath427 .
let @xmath428 be the smallest natural such that @xmath429 .
then @xmath430 this implies that @xmath431 in @xmath432 and it is a contradiction .
now , it is well known that the intersection number @xmath433 is a topological invariant .
it means that if two curves @xmath434 and @xmath435 are topologically conjugated then @xmath436 .
consequently , @xmath391 can be chosen as @xmath437 that does nt depend on the elements of @xmath339 .
now , we will prove the finite determinacy property of the slidings of foliations .
suppose that @xmath438 , @xmath439 where @xmath440 and @xmath441 are irreducible components of @xmath70 and @xmath138 .
then the singularities of @xmath11 and @xmath25 are @xmath442 .
denote by @xmath443 the holonomy of @xmath11 on @xmath440 .
now let @xmath52 be a singularity @xmath11 .
we first suppose that @xmath52 is not a corner .
denote by @xmath13 the irreducible component of @xmath10 through @xmath52 . because @xmath11 and @xmath25 are strictly conjugated in a neighborhood of @xmath52 , by corollaries [ cor11 ] and [ cor3 ] , there exists a diffeomorphism @xmath444 in @xmath445 tangent to identity such that @xmath446 let @xmath447 ( resp .
, @xmath448 ) be the projection from @xmath440 ( resp . , @xmath441 )
to @xmath13 that follows the leaves of @xmath35 ( resp .
, @xmath279 ) .
denote by @xmath449 then @xmath450 . since @xmath451 ,
@xmath452 is tangent to identity map at order at least @xmath121 . in the case
@xmath52 is a corner , let @xmath13 be one of two irreducible components of @xmath10 through @xmath52 and define @xmath452 as above .
we also have that @xmath452 is tangent to identity map at order at least @xmath121 . [ lem17 ] suppose that there exists a diffeomorphism @xmath100 such that the lifting @xmath453 satisfies * @xmath454 , * @xmath455 , * @xmath456
. then @xmath457 satisfies @xmath125 .
let @xmath52 be a singularity @xmath11 .
in the case @xmath52 is not a corner , we denote @xmath13 , @xmath447 , @xmath448 as above .
let @xmath458 be the projection following the leaves of @xmath459 from @xmath440 to @xmath13 , then @xmath460 we have @xmath461 if @xmath52 is a corner , @xmath462 , we also have @xmath463 since @xmath454 the dulac maps of @xmath459 and @xmath279 in a neighborhood of @xmath52 are the same .
so remark [ re7 ] leads to @xmath464 now we will prove the existence of @xmath100 in lemma [ lem17 ] for @xmath121 big enough .
suppose that @xmath0 and @xmath32 are respectively defined by @xmath465 then the tangent curve @xmath466 is defined by @xmath467 let @xmath468 be a vector field tangent to @xmath161 and @xmath469 be its lifting by @xmath22 . by the implicit function theorem ,
if @xmath121 is big enough , there exists @xmath470 defined on @xmath440 such that @xmath471\left(\restr{\tilde{x}_{q}}{t_i}\right)=\phi_i.\ ] ] using proposition [ pro11 ] , by choosing @xmath121 big enough , there exists @xmath472 such that @xmath473\restr{\tilde{x}_{q}}{t_i}=\phi_i.\ ] ] for each @xmath474 , denote by @xmath475 it is easy to see that @xmath476 if and only if @xmath477 .
for each holomorphic function @xmath478 , we denote by @xmath479x_q.\ ] ] lemma [ lemma18 ] below says that there exists a holomorphic function @xmath172 such that @xmath480 satisfies lemma [ lem17 ] for @xmath121 big enough . moreover , by proposition [ pro11 ] , we can chose @xmath121 that only depends on @xmath0 .
[ lemma18 ] if @xmath121 is big enough , for all @xmath478 in @xmath481 there exists a holomorphic function @xmath172 such that @xmath482 .
we have @xmath483 similarly , @xmath484 this implies that @xmath485 denote by @xmath486 .
then @xmath487 is the tangent curve of @xmath32 and the foliation defined by the level sets of @xmath304 . since at each singularity @xmath52 of @xmath11 , the irreducible component @xmath440 of @xmath161 is transverse to @xmath35 , the irreducible components of the strict transform of @xmath487 at @xmath52 are also transverse to @xmath440 .
this implies that @xmath488 and the two curves @xmath489 and @xmath490 are topologically conjugated . by hilbert s nullstellensatz and
the proof of proposition [ pro11 ] there exists a natural @xmath388 such that @xmath491 and @xmath492 .
this implies that @xmath493 .
so if @xmath494 , by we can choose @xmath495 such that @xmath482 .
if we replace the condition @xmath11 and @xmath25 are locally strictly analytically conjugated " in theorem [ thr2 ] by the condition `` @xmath0 and @xmath21 are in @xmath31 ''
then the conclusion in theorem [ thr2 ] becomes : `` for all natural @xmath496 there exists @xmath497 such that @xmath498 '' .
indeed , in that case , because the camacho - sad indices are not rational , @xmath11 and @xmath25 are locally formally conjugated .
so we can choose @xmath243 in such that @xmath499 [ cor17 ] let @xmath500 defined by a @xmath2-form @xmath501 then there exists a natural @xmath121 such that if @xmath502 is defined by a @xmath2-form @xmath503 satisfying that @xmath504 and the vanishing holonomy representations of @xmath0 and @xmath21 are strictly analytically conjugated , then @xmath0 and @xmath21 are strictly analytically conjugated .
let @xmath505 then @xmath506 where @xmath507 is an increasing function on @xmath121 and @xmath508 when @xmath509 . by theorem [ thr2 ] if @xmath121 is big enough there exists @xmath510 such that @xmath511 . by theorem [ thr1 ] , @xmath0 and @xmath21
are strictly analytically conjugated .
this corollary is consistent with the result of j.f .
mattei in @xcite which says that the dimension of moduli space of the equisingular unfolding of a foliation is finite .
note that the vanishing holonomy representations of two foliations that are jointed by a unfolding are conjugated but the converse is not true .
[ cor19 ] let @xmath32 be a @xmath22-absolutely dicritical foliation defined by @xmath2-form @xmath501 .
there exists a natural @xmath121 such that if @xmath134 is a @xmath22-absolutely dicritical foliation defined by @xmath503 satisfying @xmath504 and the dulac maps of @xmath35 and @xmath279 are the same then @xmath32 and @xmath134 are strictly analytically conjugated .
suppose that @xmath512 where @xmath513 is an irreducible component of @xmath10 .
we take a pair of irreducible functions @xmath470 and @xmath514 for each @xmath515 , such that the curve @xmath516 and @xmath517 satisfy the following properties : * the strict transforms @xmath518 and @xmath519 cut @xmath513 at two different points @xmath52 , @xmath520 , respectively , such that none of them is a corner . *
@xmath518 , @xmath519 are smooth and transverse to the invariant curve of @xmath35 through @xmath52 , @xmath520 respectively . because @xmath521 $ ] is an infinite field extension , there exists @xmath522 such that @xmath523 now , let us consider the non - dicritical foliation @xmath0 defined by the @xmath2-form @xmath524 then @xmath0 admits @xmath22 as its desingularization map and the singularities of the strict transform @xmath11 are the corners of @xmath10 and @xmath525 , @xmath346 .
we claim that at each singularity , the camacho - sad index of @xmath11 is not rational .
indeed , denote by @xmath526 the multiplicity of @xmath527 and @xmath528 on @xmath529 . at the corner @xmath530
, we take coordinates @xmath164 such that @xmath531 . in this coordinate system , we can write @xmath532 as @xmath533 where @xmath534 is a unit and @xmath535 is a holomorphic form . so the camacho - sad index of @xmath11 at @xmath536 is @xmath537 similarly , the camacho - sad indices of @xmath11 at @xmath52 and @xmath520 , respectively , are @xmath538 now if @xmath504 then @xmath539 where @xmath540 is an increasing function on @xmath121 and @xmath508 when @xmath509 .
moreover if @xmath121 is big enough the invariant curves of @xmath35 and @xmath279 through the singularities of @xmath11 are tangent . by using theorem [ thr2 ] for @xmath541 ,
there exists a foliation @xmath124 strictly conjugated with @xmath32 such that the two couples @xmath542 and @xmath543 are strictly conjugated .
consequently , @xmath32 and @xmath134 are strictly conjugated .
this paper is based on the main part of my doctoral thesis at the institute of mathematics of toulouse , france .
i would like to thank my advisors , yohann genzmer and emmanuel paul , for having guided me so well over all these years , jean - franois mattei for useful discussions and suggestions about the problem .
l. ortiz - bobadilla ; e. rosales - gonzlez ; s. m. voronin - _ thom s problem for the orbital analytic classification of degenerate singular points of holomorphic vector fields on the plane . _
dokl . math . * 82 * no .
2 , ( 2010 ) , 759761 .
moussu , r. - _ holonomie venescente des quations diffrentielles dgneres transverses _ , singularities and dynamical systems ( irklion , 1983 ) , 161173 , north - holland math . stud .
, 103 , north - holland , amsterdam , 1985 . | by introducing a new invariant called the set of slidings , we give a complete strict classification of the class of germs of non - dicritical holomorphic foliations in the plan whose camacho - sad indices are not rational .
moreover , we will show that , in this class , the new invariant is finitely determined .
consequently , the finite determination of the class of isoholonomic non - dicritical foliations and absolutely dicritical foliations that have the same dulac maps are proved . | 1304.4805 |
let @xmath0 be a complete finite - area orientable hyperbolic surface with one cusp , and @xmath1 the space of complete geodesic rays in @xmath0 emanating from the puncture .
then , there is a natural action of the ( full ) mapping class group @xmath2 of @xmath0 on @xmath3 ( see section [ sb ] ) .
the dynamics of the action of an element of @xmath1 plays a key role in the nielsen - thurston theory for surface homeomorphisms
. it also plays a crucial role in the variation of mcshane s identity for punctured surface bundles with pseudo - anosov monodromy , established by @xcite and @xcite .
it is natural to ask what does the action of the whole group @xmath2 ( or its subgroups ) look like .
however , the authors could not find a reference which treats this natural question , though there are various references which study the action of ( subgroups of ) the mapping class groups on the projective measured lamination spaces , which are homeomorphic to higher dimensional spheres ( see for example , @xcite ) .
in particular , such an action is minimal ( cf .
@xcite ) and moreover ergodic @xcite .
the purpose of this paper is to prove that the action of @xmath2 on @xmath1 is almost everywhere wandering ( see theorem [ b1 ] for the precise meaning ) .
this forms a sharp contrast to the above result of @xcite .
we would like to thank katsuhiko matsuzaki for his helpful comments on the first version of the paper .
let @xmath4 be a complete finite - area orientable hyperbolic surface with precisely one cusp , where @xmath5 .
let @xmath6 be the space of complete geodesic rays in @xmath7 emanating from the puncture .
then @xmath1 is identified with a horocycle , @xmath8 , in the cusp .
in fact , a point of @xmath9 determines a geodesic ray in @xmath7 emanating from the puncture , or more precisely , a bi - infinite geodesic path with its positive end going out the cusp and meeting @xmath9 in the given point . any mapping class @xmath10 of @xmath7 maps each geodesic ray to another path which can be straightened out to another geodesic ray , and hence determines another point of @xmath9 .
this gives an action of the infinite cyclic group generated by @xmath11 on @xmath12 .
a rigorous construction of this action is described as follows .
choose a representative , @xmath13 , of @xmath11 , so that its lift @xmath14 to the universal cover @xmath15 is a quasi - isometry
. then @xmath14 extends to a self - homeomorphism of the closed disc @xmath16 . for a geodesic ray @xmath17 ,
let @xmath18 be the closure in @xmath16 of a lift of @xmath19 to @xmath15 .
then @xmath20 is an arc properly embedded in @xmath16 , and its endpoints determine a geodesic in @xmath15 , which project to another geodesic ray @xmath21 .
thus , we obtain an action of @xmath11 on @xmath6 , by setting @xmath22 .
the dynamics of this action plays a key role in @xcite .
however , one needs to verify that this action does not depend on the choice of a representative @xmath13 of @xmath11 . in the following ,
we settle this issue , by using the canonical boundary of a relatively hyperbolic group described in @xcite .
though we are really interested here only in the case where the group is the fundamental group of a once - punctured closed orientable surface , and the the peripheral structure is interpreted in the usual way ( as the conjugacy class of the fundamental group of a neighborhood of the puncture ) , we give a discussion in a general setting .
let @xmath23 be a non - elementary relatively hyperbolic group with a given peripheral structure @xmath24 , which is a conjugacy invariant collection of infinite subgroups of @xmath23 . by ( * ?
* definition 1 ) , @xmath23 admits a properly discontinuous isometric action on a path - metric space , @xmath25 , with the following properties . 1 .
@xmath25 is proper ( i.e. , complete and locally compact ) and gromov hyperbolic , 2 . every point of the boundary of @xmath25 is either a conical limit point or a bounded parabolic point , 3 .
the peripheral subgroups , i.e. , the elements of @xmath24 , are precisely the maximal parabolic subgroups of @xmath23 , and 4 . every peripheral subgroup is finitely generated .
it is proved in ( * ? ? ?
* theorem 9.4 ) that the gromov boundary @xmath26 is uniquely determined by @xmath27 , ( even though the quasi - isometry class of the space @xmath25 satisfying the above conditions is not uniquely determined ) .
thus the boundary @xmath28 is defined to be @xmath26 . by identifying @xmath23 with an orbit in @xmath25
, we obtain a natural topology on the disjoint union @xmath29 which is compact hausdorff , with @xmath23 discrete and @xmath30 closed .
the action of @xmath23 on itself by left multiplication extends to an action on @xmath29 by homeomorphism .
this gives us a geometrically finite convergence action of @xmath23 on @xmath30 .
let @xmath31 be the subgroup of the automorphism group , @xmath32 , of @xmath23 which respects the peripheral structure @xmath24 .
this contains the inner automorphism group , @xmath33 .
now , by the naturality of @xmath34 ( ( * ? ? ?
* theorem 9.4 ) ) , the action of @xmath31 on @xmath23 also extends to an action on @xmath29 , which is @xmath23-equivariant , i.e. , @xmath35 for every @xmath36 , @xmath37 and @xmath38 .
( in order to avoid confusion , we use @xmath39 to denote group actions , only in this place . ) under the natural epimorphism @xmath40 , this gives rise to the same action on @xmath41 as that induced by left multiplication .
the centre of @xmath23 is always finite , and for simplicity , we assume it to be trivial . in this case , we can identify @xmath23 with @xmath33 .
suppose that @xmath42 is a parabolic point .
its stabiliser , @xmath43 , in @xmath23 is a peripheral subgroup .
now @xmath44 acts properly discontinuously cocompactly on @xmath45 , so the quotient @xmath46 is compact hausdorff ( cf .
* section 6 ) ) .
let @xmath47 be the stabiliser of @xmath48 in @xmath31 .
then @xmath44 is a normal subgroup of @xmath49 , and we get an action of @xmath50 on @xmath51 .
if there is only one conjugacy class of peripheral subgroups , then the orbit @xmath52 is @xmath31-invariant , and it follows that the group @xmath49 maps isomorphically onto @xmath53 , so in this case we can naturally identify the group @xmath54 with @xmath55 .
suppose now that @xmath7 is a once - punctured closed orientable surface , with negative euler characteristic @xmath56 .
we write @xmath57 , where @xmath58 , the universal cover , and @xmath59 .
let @xmath24 be the peripheral structure of @xmath23 arising from the cusp of @xmath7 , namely @xmath24 consists of the conjugacy class of the fundamental group of a neighbourhood of the end of @xmath7 .
then @xmath60 is a relatively hyperbolic group , because if we fix a complete hyperbolic structure on @xmath7 then @xmath61 is identified with @xmath15 and the isometric action of @xmath23 on @xmath62 satisfies the conditions ( 1)(4 ) in the above , namely ( * ? ? ? * definition 1 ) .
now @xmath61 admits a natural compactification to a closed disc , @xmath63 , where @xmath64 is the dynamically defined circle at infinity .
we can identify @xmath64 with @xmath30 .
in fact , if @xmath65 is any point of @xmath61 , then identifying @xmath23 with the orbit @xmath66 , we get an identification of @xmath29 with @xmath67 . as above
we get an action of @xmath31 on @xmath64 . if @xmath68 is parabolic , then its stabiliser @xmath44 in @xmath23 is isomorphic to the infinite cyclic group @xmath69 , and we get an action of @xmath55 on the circle @xmath70 . since @xmath55 is identified with the ( full ) mapping class group , @xmath71 , of @xmath7 , we obtain a well defined action of @xmath71 on the circle @xmath51 .
we now return to the setting in the beginning of this section , where @xmath4 is endowed with a complete hyperbolic structure
. then we can identify the ( dynamically defined ) circle @xmath72 with the horocycle , @xmath9 , in the cusp , which in turn is identified with the space of geodesic rays , @xmath6 .
this gives an action of @xmath71 on @xmath6 . since the action of @xmath23 on @xmath15 satisfies the conditions ( 1)-(4 ) in the above ( i.e. , ( * ? ? ?
* definition 1 ) ) , we see that , for each mapping class @xmath11 of @xmath7 , its action on @xmath6 , defined via the straightening process presented at the beginning of this section , is identical with the action which is dynamically constructed in the above , independently from the hyperbolic structure .
thus the problem raised at the beginning of this section is settled .
in order to state the main result , we prepare some terminology .
let @xmath73 be a group acting by homeomorphism on a topological space @xmath25 .
an open subset , @xmath74 , is said to be _ wandering _ if @xmath75 for all @xmath76 .
( note that this definition is stronger than the usual definition of wandering , where it is only assumed that the number of @xmath77 such that @xmath78 is finite . )
the _ wandering domain _ , @xmath79 is the union of all wandering open sets .
its complement , @xmath80 , is the _ non - wandering set_. this is a closed @xmath73-invariant subset of @xmath25 .
note that if @xmath81 is a @xmath82-invariant open set , then @xmath83 .
if @xmath84 is a normal subgroup , we get an induced action of @xmath85 on @xmath86 .
( in practice , the action of @xmath87 on @xmath25 will be properly discontinuous . )
one checks easily that @xmath88 with equality if @xmath89 .
note that any hyperbolic structure on @xmath7 induces a euclidian metric on @xmath51 ( via the horocycle @xmath9 ) .
if one changes the hyperbolic metric , the induced euclidian metrics on @xmath51 are related by a quasisymmetry .
however , they are completely singular with respect to each other ( see @xcite ) .
( that is , there is a set which has zero measure in one structure , but full measure in the other . ) in general , this gives little control over how the hausdorff dimension of a subset can change .
we say that a subset , @xmath90 is _ small _ if it has hausdorff dimension stricty less than 1 with respect to any hyperbolic structure on @xmath7 .
now we can state our main theorem .
[ b1 ] let @xmath7 be a once - punctured closed orientable surface , with @xmath91 , and consider the action of @xmath71 on the circle @xmath51 , defined in the above .
then the non - wandering set in @xmath51 with respect to the action of @xmath71 is small .
in particular , the non - wandering set has measure 0 with respect to any hyperbolic structure , and so has empty interior .
given that two different hyperbolic structures give rise to quasisymmetically related metrics on @xmath51 , it is natural to ask if there is a more natural way to express this .
for example , is there a property of ( closed ) subsets of @xmath51 , invariant under quasisymmetry and satisfied by the non - wandering set , which implies hausdorff dimension less than 1 ( or measure 0 ) ?
let @xmath4 be a complete finite - area orientable hyperbolic surface with precisely one cusp , where @xmath5 .
thus the universal cover @xmath58 is identified with the hyperbolic plane @xmath15 .
write @xmath64 for the ideal boundary of @xmath61 , which we consider equipped with a preferred orientation .
thus @xmath23 acts on @xmath64 as a geometrically finite convergence group .
let @xmath92 be the set of parabolic points of @xmath23 .
given @xmath93 , let @xmath94 be the generator of @xmath95 which acts on @xmath96 as a translation in the positive direction .
given distinct @xmath97 , let @xmath98 \subseteq d \cup c $ ] denote the oriented geodesic from @xmath65 to @xmath99 . if @xmath37 is hyperbolic , write @xmath100 , @xmath101 respectively , for its attracting and repelling fixed points ; @xmath102 $ ] for its axis ; and @xmath103 for the oriented closed geodesic in @xmath0 corresponding to @xmath104 , i.e. , the image of @xmath105 in @xmath0 .
if @xmath97 are distinct , then @xmath98 \cap d $ ] projects to an oriented bi - infinite geodesic path , @xmath106 , in @xmath7 . if @xmath107 , then this is a proper geodesic path , with a finite number , @xmath108 , of self - intersections .
let @xmath109 , i.e. , @xmath110 consists of pairs @xmath111 of parabolic points such that @xmath112 is a proper geodesic arc .
( by an _ arc _ , we mean an embedded path . ) given @xmath93 , write @xmath113 .
pick an element @xmath114 .
then the proper arc @xmath112 intersects a sufficiently small horocycle , @xmath9 , in precisely two points .
let @xmath115 be the horocircle centred at @xmath48 which is a connected component of the inverse image of @xmath9 , and let @xmath116 be the inverse image of the two points in @xmath117 , located in this order , such that @xmath118\cap \tilde\tau $ ] and @xmath119 .
then there is a unique element @xmath120 such that @xmath121 and @xmath122\cap \tilde \tau = s_1 $ ] .
namely , @xmath122 $ ] is the closure of the lift of @xmath112 with endpoint @xmath48 which is closest to @xmath123 $ ] , among the lifts of @xmath112 with endpoint @xmath48 , with respect to the preferred orientation of @xmath117 .
( see figure [ figure1 ] . ) in the quotient surface @xmath7 , the oriented closed geodesic @xmath124 is homotopic to the simple oriented loop obtained by shortcutting the oriented arc @xmath112 by the horocyclic arc which is the image of the subarc of @xmath125 bounded by @xmath126 and @xmath127 .
thus @xmath124 is a simple closed geodesic disjoint from the proper geodesic arc @xmath128 . in particular , @xmath129\cap \alpha(g(p , q))=\emptyset$ ] .
in fact , the map @xmath130 : \delta \longrightarrow \gamma $ ] is characterised by the following properties : for all @xmath131 , we have @xmath132 , @xmath133 , and @xmath134 \cap \alpha(g(p , q ) ) = \emptyset $ ] . and @xmath135 of the hyperbolic transformations @xmath136 and @xmath137 respectively .
the blue arcs with thin arrows represent the oriented geodesic @xmath123 $ ] and its images by the infinite cyclic groups @xmath138 and @xmath139 .
the three intersection points of the blue arcs and the horocircle @xmath117 centred at @xmath48 are @xmath140 , @xmath141 and @xmath142 , from left to right .
[ figure1 ] ] write @xmath143 and @xmath144
. then the points @xmath48 , @xmath145 , @xmath146 , @xmath147 , @xmath148 , @xmath149 occur in this order around @xmath64 .
let @xmath150 , @xmath151 and @xmath152 be open intervals in @xmath64 .
thus @xmath153 , @xmath154 for all @xmath155 , and @xmath156 for all @xmath157 . in the quotient surface @xmath7
, the oriented simple closed geodesics @xmath124 and @xmath158 cut off a punctured annulus containing the geodesic arc @xmath128 , in which the simple geodesic rays @xmath159 and @xmath160 emanating from the puncture spiral to @xmath124 and @xmath158 , respectively .
thus , each of @xmath161 projects homeomorphically onto a _
gap _ in the horocircle @xmath8 , in the sense of @xcite .
in fact , each of @xmath161 is a maximal connected subset of @xmath162 consisting of points @xmath163 such that the geodesic ray @xmath164 is non - simple .
moreover , if @xmath164 is non - simple , then @xmath163 is contained in @xmath161 for some @xmath165 ( see @xcite ) .
. then we obtain the following as a consequence of ( * ? ? ?
* corollary 5 ) and @xcite ( see also ( * ? ? ? * section 5 ) ) : [ c1 ] the elements of @xmath167 are pairwise disjoint .
the complement , @xmath168 , is a cantor set of hausdorff dimension @xmath169 . here
, of course , the hausdorff dimension is taken with respect to the euclidean metric on the horocycle , @xmath9 .
up to a scale factor , this is the same as the euclidean metric in the upper - half - space model with @xmath48 at @xmath170 .
( note that we could equally well use the circular metric on the boundary , @xmath64 , induced by the poincar model , since all the transition functions are mbius , and in particular , smooth . )
write @xmath171 .
this is a closed set , whose complementary components are precisely the intervals @xmath172 for @xmath173 .
thus the set @xmath174 is characterised by the following property : a point @xmath175 belongs to @xmath174 if and only if @xmath176 and the geodesic ray @xmath164 in @xmath0 is simple . for @xmath177 , we define maps @xmath178 , @xmath179 and @xmath180 from @xmath181 to @xmath182 , @xmath183 and @xmath184 , respectively , by the following rule . if @xmath185 , then @xmath186 for some unique @xmath187 and @xmath173 .
define @xmath188 , @xmath189 , and @xmath190 or @xmath191 according to whether @xmath192 or @xmath193 .
note that the definition is symmetric under simultaneously reversing the orientation on @xmath64 and swapping @xmath194 with @xmath195 .
it should be noted that if @xmath185 , then , in the quotient surface @xmath7 , the geodesic ray @xmath196 is obtained from the non - simple geodesic ray @xmath197 by cutting a loop , homotopic to @xmath198 , and straightening the resulting piecewise geodesic ( see figure [ figure2 ] ) .
( in the quotient , we are allowing ourselves to cut out any peripheral loops that occur at the beginning . ) in particular , if @xmath199 , then both @xmath197 and @xmath200 are proper geodesic paths in @xmath7 , and their self - intersection numbers satisfy the inequality @xmath201 . and so @xmath190 . [ figure2 ] ] by repeatedly applying these maps , we associate for a given @xmath202 , a sequence @xmath203 in @xmath23 , @xmath204 in @xmath205 , and @xmath206 in @xmath182 as follows .
* step 0 . * pick a parabolic point @xmath177 , and define @xmath207 .
thus , @xmath208 is independent of @xmath202
. * step 1 . *
if @xmath209 , we stop with the 1-element sequence @xmath208 , and define @xmath203 and @xmath206 to be the empty sequence .
if @xmath210 , set @xmath211 , @xmath212 , @xmath213 , and continue to the next step .
( the sequences @xmath203 and @xmath206 begin with index @xmath214 . )
* step 2 . *
if @xmath215 , we stop with the 1-element sequences @xmath216 and @xmath217 and @xmath218-element sequence @xmath219 . if @xmath220 , set @xmath221 , @xmath222 and @xmath223 .
we continue this process , forever or until we stop .
we call the resulting sequences @xmath203 , @xmath204 and @xmath206 the _ derived sequences _ for @xmath163 .
more specifically , we call @xmath203 and @xmath204 the _ derived @xmath23-sequence _ and the _ derived @xmath224-sequence _ for @xmath163 , respectively .
[ c2 ] let @xmath225 , and let @xmath203 , @xmath204 and @xmath206 be the derived sequences for @xmath163 . then the following hold .
\(1 ) the sequences @xmath204 and @xmath206 are determined by the sequence @xmath203 by the following rule : @xmath226 where @xmath227 , and @xmath228 or @xmath195 according to whether @xmath229 or @xmath230 .
\(2 ) a point @xmath231 has the derived @xmath23-sequence beginning with @xmath232 for some @xmath233 , if and only if @xmath234 .
\(3 ) set @xmath235 . if @xmath236 , then the derived @xmath23-sequence @xmath203 is infinite .
\(4 ) if @xmath237 , then the derived @xmath23-sequence @xmath203 is finite .
( 1 ) , ( 2 ) and ( 3 ) follow directly from the definition of the derived sequences . to prove ( 4 ) ,
let @xmath65 be a point in @xmath224 .
if @xmath238 , then @xmath203 is the empty sequence .
so we may assume @xmath199 .
then by repeatedly using the observation made prior to the construction of the derived sequences , we see that the self - intersection number @xmath239 of the proper geodesic path @xmath240 is strictly decreasing .
hence @xmath241 for some @xmath242 .
this means that @xmath243 and so the derived sequences terminate at @xmath242 .
the following is an immediate consequence of lemma [ c2](2 ) .
[ c3 ] suppose that @xmath202 has derived @xmath23-sequence beginning with @xmath244 for some @xmath245 .
then there is an open set , @xmath246 , containing @xmath65 , such that if @xmath247 , then @xmath244 is also an initial segment of the derived @xmath23-sequence for @xmath99 .
recall from section [ sb ] that @xmath248 denotes the subgroup of @xmath32 preserving @xmath224 setwise and fixing @xmath93 .
[ c4 ] let @xmath249 be an element of @xmath250 with @xmath251 .
then the following holds for every point @xmath202 . if @xmath203 , @xmath204 and @xmath206 are the derived sequences for @xmath65 , then the derived sequences for @xmath252 are @xmath253 , @xmath254 and @xmath255 .
this can be proved through induction , by using the fact that the following hold for each @xmath256 .
1 . @xmath257 .
2 . for any @xmath165
, we have : 1 .
if @xmath249 is orientation - preserving , then @xmath258 , @xmath259 , @xmath260 , and @xmath261 .
2 . if @xmath249 is orientation - reversing , then @xmath262 , @xmath263 , @xmath264 , and @xmath265 .
let @xmath65 be a point in @xmath64 and @xmath266 the ( finite or infinite ) derived @xmath205-sequence for @xmath163 .
write @xmath267 for the projection of @xmath268 \cap d $ ] to @xmath7 .
this is a proper geodesic arc in @xmath7 .
we call the sequence @xmath269 the _ derived sequence of arcs _ for @xmath163 .
we say that @xmath65 is _ filling _ if the arcs @xmath270 eventually fill @xmath7 , namely , there is some @xmath157 such that @xmath271 is a union of open discs .
let @xmath272 be the subset of @xmath273 consisting of points which are filling .
in this section , we prove the following proposition .
[ d1 ] the set @xmath272 is open in @xmath64 , and its complement has hausdorff dimension strictly less than @xmath274 . in particular , @xmath272 has full measure .
we begin with some preparation .
let @xmath275 be a simple closed geodesic in @xmath7 , and let @xmath276 be the path - metric completion of the component of @xmath277 containing the cusp .
then we can identify @xmath276 as @xmath278 , where @xmath279 is a subgroup of @xmath23 containing @xmath280 , and @xmath281 is the convex hull of the limit set @xmath282 . in other words
, @xmath276 is the `` convex core '' of the hyperbolic surface @xmath283 .
note that @xmath284 and @xmath285 .
let @xmath286 be the closure of a component of @xmath287 .
this is a bi - infinite geodesic in @xmath63 .
let @xmath288 be the component of @xmath289 not containing @xmath48 .
thus , @xmath290 is an open interval in @xmath64 , which is a component of the discontinuity domain of @xmath73 .
note in particular , that @xmath291 .
[ d2 ] suppose @xmath292 , and let @xmath293 , @xmath294 and @xmath295 .
then , if @xmath296 , we have @xmath297 . in particular , @xmath298 for every @xmath299 . to simplify notation
we can assume ( via the orientation reversing symmetry of the construction ) that @xmath192 .
note that @xmath300 , so @xmath301 \subseteq h(g ) $ ] . also @xmath302 and @xmath303 .
it follows that @xmath301 $ ] , @xmath304 and @xmath286 are pairwise disjoint .
thus , @xmath290 lies in a component of @xmath305 . since @xmath192
, the four points , @xmath306 are located in @xmath64 in this cyclic order , and so @xmath307 is a component of @xmath308 .
since @xmath290 and @xmath309 share the point @xmath65 , we obtain the first assertion that @xmath297 with @xmath192 .
the second assertion follows from the first assertion and the definition of @xmath310 .
[ d3 ] suppose that @xmath311 and that the derived @xmath23-sequence @xmath203 for @xmath65 is infinite .
then there is some @xmath312 such that @xmath313 .
suppose , for contradiction , that @xmath314 for all @xmath312 .
it follows that @xmath315 for all @xmath312 , and so @xmath316 for all @xmath312 . by lemma [ d2 ]
, we have @xmath317 for all @xmath299 .
( here @xmath204 is the derived @xmath224-sequence for @xmath65 and @xmath318 . )
now , applying lemma [ d2 ] with @xmath319 in place of @xmath48 , we get that @xmath320 .
continuing inductively we get that @xmath321 for all @xmath312 . in other words ,
the derived @xmath23-sequence for @xmath99 is identical to that for @xmath65 , and so , in particular , it must be infinite .
we now get a contradiction by applying lemma [ c2](4 ) to any point @xmath322 .
if we take @xmath323 to be a standard horoball neighbourhood of the cusp , then @xmath324 for all simple closed geodesic in @xmath0 , and so we can identify @xmath323 with a neighbourhood of the cusp in any @xmath276 .
[ d4 ] there is some @xmath325 such that for each simple closed geodesic , @xmath275 , the hausdorff dimension of @xmath326 is at most @xmath327 .
this is an immediate consequence of ( * ? ?
* theorem 3.11 ) ( see also ( * ? ? ?
* theorem 1 ) ) which refines the result of @xcite , on observing that the groups @xmath328 are uniformly `` geometrically tight '' , as defined in that paper . here
, this amounts to saying that there is some fixed @xmath329 ( independent of @xmath275 ) such that the convex core , @xmath330 , is the union of @xmath323 and the @xmath331-neighbourhood of the geodesic boundary of the convex core . from the earlier discussion , we see that @xmath331 is bounded above by the diameter of @xmath332 , and so in particular , independent of @xmath275 .
let @xmath333 be the union of the limit sets @xmath334 as @xmath335 ranges over all subgroups of @xmath184 obtained from a simple closed geodesic @xmath336 in @xmath0 .
applying lemma [ d4 ] , we see that @xmath337 is a @xmath23-invariant subset of @xmath273 of hausdorff dimension strictly less than 1 .
this is because it is a countable union of the limit sets @xmath334 whose hausdorff dimensions are uniformly bounded by a constant @xmath338 .
recall the set @xmath235 defined in lemma [ c2](3 ) .
then @xmath339 is also @xmath23-invariant and has hausdorff dimension zero by theorem [ c1 ] .
[ d5 ] if @xmath340 , then @xmath163 is filling .
namely , @xmath341 .
suppose , for contradiction , that some @xmath340 is not filling .
then there must be some simple closed geodesic , @xmath275 , in @xmath7 , which is disjoint from every @xmath342 , where @xmath269 is the derived sequence of arcs for the point @xmath65 .
consider the hyperbolic surface @xmath343 and its fundamental group @xmath344 , as described at the beginning of this section . by hypothesis , @xmath345 , and
so @xmath65 lies in some component , @xmath290 , of the discontinuity domain of @xmath73 . by lemma [ d3 ]
, there must be some @xmath346 with @xmath347 .
choose the minimal such @xmath312 .
thus , @xmath348 but @xmath349 , where @xmath350 .
we have @xmath351 and @xmath352 .
( the latter assertion can be seen as follows . if @xmath353 then @xmath354 is a parabolic fixed point of @xmath73 .
since @xmath276 has a single cusp , there is an element @xmath355 such that @xmath356 . since @xmath357 , we have @xmath358 .
this implies @xmath359 , a contradiction . )
therefore @xmath268 $ ] meets @xmath360 , giving the contradiction that @xmath342 crosses @xmath275 in @xmath7 . by lemma [ d5 ] , we have @xmath361 .
since @xmath362 and @xmath337 both have hausdorff dimension strictly less than @xmath274 , the same is true of @xmath363 .
thus , we have only to show that @xmath272 is open .
pick an element @xmath364 .
then there is some @xmath157 such that @xmath271 is a union of open discs , where @xmath270 is a derived sequence of arcs for @xmath163 . by corollary [ c3 ]
, there is an open neighbourhood @xmath365 of @xmath65 in @xmath64 such that every @xmath366 shares the same initial derived @xmath184-sequence @xmath367 with @xmath65 .
thus , every @xmath366 shares the same beginning derived sequence of arcs @xmath368 with @xmath65 .
hence every @xmath366 is filling , i.e. , @xmath369 .
recall that @xmath71 is identified with @xmath370 , where @xmath47 and @xmath371 , respectively , are the stabilisers of @xmath48 in @xmath31 and @xmath23 . as described in section [ sb ] , @xmath49 acts on @xmath96 , and @xmath372 acts on the circle @xmath373 .
the wandering domain @xmath374 is equal to @xmath375 , because @xmath376 .
( see the general remark on the wandering domain given in section [ sb ] . )
we want to show that any @xmath378 has a wandering neighbourhood . by assumption ,
some initial segment , @xmath379 , of the derived sequence of arcs for @xmath65 fills @xmath7 . by corollary [ c3 ]
, there is an open neighbourhood , @xmath365 , of @xmath65 , such that for every @xmath366 , the initial segment of length @xmath242 of the derived sequence of arcs is identical with @xmath379 .
suppose that @xmath380 for some non - trivial element @xmath249 of @xmath381 .
pick a point @xmath382 and set @xmath383 . by assumption ,
the derived sequences of arcs for both @xmath65 and @xmath99 begin with @xmath384 . on the other hand , lemma [ c4 ]
implies that the derived sequence of arcs for @xmath385 is equal to the image of that for @xmath65 by @xmath249 .
hence we see that @xmath386 for all @xmath387 .
it follows by lemma [ e2 ] below , that @xmath249 is the trivial element of @xmath71 , a contradiction .
[ e2 ] let @xmath379 be a set of proper oriented arcs in @xmath7 which together fill @xmath7 .
suppose that @xmath11 is a mapping class on @xmath7 fixing the proper homotopy class of each @xmath342 .
then @xmath11 is trivial . by proposition [ d1 ]
, @xmath272 is an open set of @xmath96 whose complement has hausdorff dimension strictly less than @xmath388 .
since @xmath389 contains @xmath272 by lemma [ e1 ] , its complement in @xmath96 also has hausdorff dimension strictly less than @xmath388 .
since @xmath390 , this implies that the non - wandering set , @xmath391 , has hausdorff dimension strictly less than @xmath388 .
fix any complete finite - area hyperbolic structure on @xmath7 , and use it to identify @xmath392 with @xmath393 . construct a graph , @xmath54 , as follows .
the vertex set , @xmath394 , is the set of bi - infinite geodesics which are lifts of the arcs @xmath342 for all @xmath312 .
two arcs @xmath395 are deemed adjacent in @xmath54 if either ( 1 ) they cross ( that is , meet in @xmath393 ) , or ( 2 ) they have a common ideal point in @xmath396 , and there is no other arc in @xmath394 which separates @xmath397 and @xmath398 .
one readily checks that @xmath54 is locally finite .
moreover , the statement that the arcs @xmath342 fill @xmath7 is equivalent to the statement that @xmath54 is connected . note that @xmath399 acts on @xmath54 with finite quotient .
note also that @xmath54 can be defined formally in terms of ordered pairs of points in @xmath400 ( that is corresponding to the endpoints of the geodesics , and where crossing is interpreted as linking of pairs ) .
the action of @xmath23 on @xmath54 is then induced by the dynamically defined action of @xmath23 on @xmath401 .
now suppose that @xmath402 .
lifting some representative of @xmath11 and extending to the ideal circle gives us a homomorphism of @xmath401 , equivariant via the corresponding automorphism of @xmath23 .
suppose that @xmath11 preserves each arc @xmath342 , as in the hypotheses .
then @xmath11 induces an automorphism , @xmath403 .
given some @xmath404 , by choosing a suitable lift of @xmath11 , we can assume that @xmath405 .
we claim that this implies that @xmath13 is the identity on @xmath54 . to see this ,
first let @xmath406 be the set of vertices adjacent to @xmath397 .
this is permuted by @xmath13 .
consider the order on @xmath407 defined as follows .
let @xmath408 and @xmath409 , respectively , be the closed intervals of @xmath401 bounded by @xmath410 which lies to the right and left of @xmath397 .
orient each of @xmath408 and @xmath409 so that the initial / terminal points of @xmath397 , respectively , are those of the oriented @xmath408 and @xmath409 .
each @xmath411 determines a unique pair @xmath412 such that @xmath413 and @xmath414 are the endpoints of @xmath19 .
now we define the order @xmath415 on @xmath407 , by declaring that @xmath416 if either ( i ) @xmath417 or ( ii ) @xmath418 and @xmath419 .
this order must be respected by @xmath13 , because @xmath13 preserves the orders on @xmath408 and @xmath409 . since @xmath407 is finite , we see that @xmath420 is the identity .
the claim now follows by induction , given that @xmath54 is connected .
it now follows that the lift of @xmath11 is the identity on the set of all endpoints of elements of @xmath394 .
since this set is dense in @xmath401 , it follows that it is the identity on @xmath401 , and we deduce that @xmath11 is the trivial mapping class as required .
h. akiyoshi , h. miyachi , m. sakuma , _ variations of mcshane s identity for punctured surface groups _ : proceedings of the workshop `` spaces of kleinian groups and hyperbolic 3-manifolds '' , london math .
soc . , lecture note series * 329 * ( 2006 ) 151185 .
a. fathi , f. laudenbach , v. ponaru , et al , _
thurston s work on surfaces _ : translated from the 1979 french original by d. m. kim and d. margalit , mathematical notes , * 48 * , princeton university press , princeton , 2012 . | let @xmath0 be a complete finite - area orientable hyperbolic surface with one cusp , and let @xmath1 be the space of complete geodesic rays in @xmath0 emanating from the puncture .
then there is a natural action of the mapping class group of @xmath0 on @xmath1 .
we show that this action is almost everywhere wandering . | 1608.03009 |
traffic problems have been attracting not only engineers but also physicists @xcite . especially it has been widely accepted that the phase transition from free to congested traffic flow can be understood using methods from statistical physics @xcite . in order to study the transition in detail , we need a realistic model of traffic flow which should be minimal to clarify the underlying mechanisms . in recent years
cellular automata ( ca ) @xcite have been used extensively to study traffic flow in this context . due to their simplicity , ca models have also been applied by engineers , e.g. for the simulation of complex traffic systems with junctions and traffic signals @xcite .
many traffic ca models have been proposed so far @xcite , and among these ca , the deterministic rule-184 ca model ( r184 ) , which is one of the elementary ca classified by wolfram @xcite , is the prototype of all traffic ca models .
r184 is known to represent the minimum movement of vehicles in one lane and shows a simple phase transition from free to congested state of traffic flow . in a previous paper @xcite , using the ultra - discrete method @xcite , the burgers ca ( bca ) has been derived from the burgers equation @xmath0 which was interpreted as a macroscopic traffic model @xcite .
the bca is written using the minimum function @xmath1 by @xmath2 where @xmath3 denotes the number of vehicles at the site @xmath4 and time @xmath5 .
if we put the restriction @xmath6 , it can be easily shown that the bca is equivalent to r184 .
thus we have clarified the connection between the burgers equation and r184 , which offers better understanding of the relation between macroscopic and microscopic models .
the bca given above is considered as the _ euler _ representation of traffic flow .
as in hydrodynamics there is an another representation , called _ lagrange _ representation @xcite , which is specifically used for car - following models .
the lagrange version of the bca is given by @xcite @xmath7 where @xmath8 and @xmath9 is the position of @xmath10-th car at time @xmath5 .
note that in ( [ lagbca ] ) @xmath11 corresponds a `` perspective '' or anticipation parameter @xcite which represents the number of cars that a driver sees in front , and @xmath12 is the maximum velocity of cars .
( [ lagbca ] ) is derived from the bca mathematically by using an euler - lagrange ( el ) transformation @xcite which is a discrete version of the well - known el transformation in hydrodynamics . in this paper
, we will develop the bca ( [ lagbca ] ) to a more realistic model by introducing slow - to - start ( s2s ) effects @xcite and a driver s perspective @xmath11 .
moreover , a stochastic generalization is also considered by combining it with the nagel - schreckenberg ( ns ) model @xcite .
first , let us extend ( [ lagbca ] ) to the case @xmath13 and combine it with the s2s model . the s2s model @xcite is written in lagrange form as @xmath14 note that the inertia effect of cars is taken into account in this model
. comparing ( [ eqs2s ] ) and ( [ lagbca ] ) , we see that , in the s2s model , the velocity of a car depends not only on the present headway @xmath15 , but also on the past headway @xmath16 .
this rule has only a nontrivial effect if @xmath17 and @xmath18 , i.e. if the leading car has started to move in the previous time step . in this case
the following car is not allowed to move immediately ( s2s ) . before combining ( [ lagbca ] ) and
( [ eqs2s ] ) , it is worth pointing out that we can choose the perspective parameter as @xmath19 in the model according to observed data .
we define the size of a cell as 7.5 m and @xmath20 in our model according to the ns model .
since @xmath12 corresponds to about 100 km / hour in reality , then one time step in the ca model becomes 1.3 s. moreover , the gradient of the free line and jamming line in the fundamental diagram , which is the dependence of the traffic flow @xmath21 on density @xmath22 , is known to be about 100 km / hour and @xmath23 km / hour @xcite according to many observed data ( see fig .
[ fig1 ] ) @xcite .
these values correspond to the typical free velocity and the jam velocity , respectively . ) and jamming line ( @xmath24 ) is known to be about 100 km / hour and @xmath23 km / hour .
we also see that there is a wide scattering area near the phase transition region from free to jamming state .
, scaledwidth=80.0% ] thus , considering the fact that the positive and negative gradient of each line is given by @xmath20 and @xmath25 , respectively , in the ca model @xcite , we should choose @xmath26 in the ca model .
since only integer numbers for @xmath11 are allowed in this model , we will simply choose @xmath19 for studying the effect of the perspective of drivers .
it is noted that other possibilities , such as velocity - dependent @xmath11 or stochastic choice of @xmath11 , are also possible .
now by combining ( [ lagbca ] ) and ( [ eqs2s ] ) we propose a new lagrange model with @xmath19 which is defined by the rules listed below : let @xmath27 be the velocity of the @xmath10-th car at a time @xmath5 .
the update procedure from @xmath5 to @xmath28 is divided into five stages : 1 .
_ accerelation _ @xmath29 2 .
_ slow - to - accelerate effect _
@xmath30 3 .
_ deceleration due to other vehicles _
@xmath31 4 .
_ avoidance of collision _
@xmath32 5 .
_ vehicle movement _ @xmath33 the velocity @xmath34 is used as @xmath27 in the next time step .
( [ avoid ] ) is the condition that the @xmath10-th car does not overtake its preceding @xmath35-th car , including anticipation .
the accerelation ( [ acdet ] ) is the same as in the ns model , which is needed for a mild accerelating behaviour of cars . in the step 2 ,
we call ( [ s2a ] ) as `` slow - to - accelerate '' instead of s2s
. this is because this rule affects not only the behaviour of standing cars but also that of moving cars , which is considered to be a generalization of s2s rule .
it is not difficult to write down the new model in a single equation for general @xmath11 .
the result is @xmath36 where the last term represents the collision - free condition explained in fig .
[ fig2 ] , and @xmath37 the condition that there is no collision between the @xmath10-th and @xmath38-th cars ( @xmath39 ) is given by @xmath40 for @xmath41 ( if @xmath42 then we simply put @xmath43 ) , which is identical to the last term in ( [ lagfm ] ) .
in contrast to the ns model , the velocity of the preceeding car is taken into account in the calculation of the safe velocity in the step 4 , i.e. our model also includes anticipation effects .
-th and @xmath38-th car . ,
next , we investigate the fundamental diagram of this new hybrid model . in fig .
[ fig3 ] , we observe a complex phase transition from a free to congested state near the critical density 0.2 @xmath44 0.4 .
there are many metastable branches in the diagram , similar to our previous models in euler form @xcite or in other models with anticipation @xcite .
we also point out that there is a wide scattering area near the critical density in the observed data ( fig .
[ fig1 ] ) which may be related to these metastable branches . as we will discuss later
, these branches may account for some aspects of the scattering area observed empirically .
first , we discuss properties of the state in the metastable branches . in all cases
it consists of _ pairs of vehicles _ that move coherently with vanishing headway ( see fig . [ fig4 ] ) .
cars are represented by black squares , and the direction of the road is horizontal right and time axis is vertical down .
the corresponding velocity distributions are also given in fig .
we see that there are stopping cars which velocity are zero only in the case of the lowest branch given in the state in fig .
[ fig4 ] ( e ) . at density @xmath45 with different strength of perturbation : ( a ) very weak , ( b ) less weak , ( c ) medium , ( d ) stronger and finally ( e ) strongest perturbation .
the details of these perturbations are all explained in detail in the text .
the stationary state of these five cases correspond to a state in each metastable branch appearing in fig .
[ fig3 ] , although the branch corresponding to ( a ) does not appear in the numerical simulations with random initial conditions .
, title="fig:",scaledwidth=40.0% ] at density @xmath45 with different strength of perturbation : ( a ) very weak , ( b ) less weak , ( c ) medium , ( d ) stronger and finally ( e ) strongest perturbation .
the details of these perturbations are all explained in detail in the text .
the stationary state of these five cases correspond to a state in each metastable branch appearing in fig .
[ fig3 ] , although the branch corresponding to ( a ) does not appear in the numerical simulations with random initial conditions .
, title="fig:",scaledwidth=40.0% ] + ( a)(b ) + at density @xmath45 with different strength of perturbation : ( a ) very weak , ( b ) less weak , ( c ) medium , ( d ) stronger and finally ( e ) strongest perturbation .
the details of these perturbations are all explained in detail in the text .
the stationary state of these five cases correspond to a state in each metastable branch appearing in fig .
[ fig3 ] , although the branch corresponding to ( a ) does not appear in the numerical simulations with random initial conditions .
, title="fig:",scaledwidth=40.0% ] at density @xmath45 with different strength of perturbation : ( a ) very weak , ( b ) less weak , ( c ) medium , ( d ) stronger and finally ( e ) strongest perturbation .
the details of these perturbations are all explained in detail in the text .
the stationary state of these five cases correspond to a state in each metastable branch appearing in fig .
[ fig3 ] , although the branch corresponding to ( a ) does not appear in the numerical simulations with random initial conditions .
, title="fig:",scaledwidth=40.0% ] + ( c)(d ) + at density @xmath45 with different strength of perturbation : ( a ) very weak , ( b ) less weak , ( c ) medium , ( d ) stronger and finally ( e ) strongest perturbation .
the details of these perturbations are all explained in detail in the text .
the stationary state of these five cases correspond to a state in each metastable branch appearing in fig .
[ fig3 ] , although the branch corresponding to ( a ) does not appear in the numerical simulations with random initial conditions .
, title="fig:",scaledwidth=40.0% ] + ( e ) , corresponding to the states given in fig .
[ fig4 ] . ,
title="fig:",scaledwidth=40.0% ] , corresponding to the states given in fig .
, title="fig:",scaledwidth=40.0% ] + ( a)(b ) + , corresponding to the states given in fig .
[ fig4 ] . , title="fig:",scaledwidth=40.0% ] ,
corresponding to the states given in fig .
, title="fig:",scaledwidth=40.0% ] + ( c)(d ) + , corresponding to the states given in fig .
[ fig4 ] . , title="fig:",scaledwidth=40.0% ]
+ ( e ) next let us calculate the flow - density relation for each branch . in the metastable branches we find phase separation into a free - flow and a jamming region . in the former , pairs move with velocity @xmath46 and a headway of @xmath47 empty cells between consecutive pairs . in the jammed region ,
the velocity of the pairs is @xmath48 and the headway @xmath49 . @xmath50 and @xmath51 are the numbers of cars in the jamming cluster and the free uniform flow , respectively .
we assume @xmath51 and @xmath50 to be even so that there are @xmath52 and @xmath53 pairs , respectively
. then the total number of cars @xmath54 is given by @xmath55 and the total length of the system becomes @xmath56 .
since the average velocity is @xmath57 and density and flow of the system are given by @xmath58 and @xmath59 , we obtain the flow - density relation as @xmath60 from the stationary states in fig .
[ fig4 ] we have @xmath61 therefore the resulting equations for each branch are @xmath62 where @xmath63 , which correspond to the branches @xmath64 and @xmath65 in fig .
[ fig6 ] , respectively .
end points of the branches are found to be given by @xmath66 and @xmath67 , where @xmath68 is the point that the metastable branches intersect the free flow branch , and @xmath69 is the maximal possible density in the metastable branches .
note that all @xmath69 lie on the line @xmath70 , which is indicated as the broken line in fig .
[ fig6 ] . and @xmath71 ) and the jamming line ( @xmath65 ) in the new model .
the highest flow state is represented by @xmath72 , which is quite unstable and easy to go down to the lower flow state @xmath73 according to the magnitude of the perturbation .
, scaledwidth=60.0% ] next let us now study the stability of each metastable branch .
we mainly consider the density @xmath45 and , in particular , we will focus on the uniform flow represented by @xmath74 , which shows the highest flow given in the point @xmath72 in fig .
[ fig6 ] .
spatio - temporal patterns due to various kinds of perturbations are already seen in fig .
perturbation in this case means that some cars are shifted backwards at the initial configuration .
the initial conditions for fig .
[ fig4](a)-(e ) are given as follows : 1 .
very weak perturbation ( one car is shifted one site backwards)@xmath75 2 . weak perturbation ( one car is shifted two sites backwards)@xmath76 3 . moderate perturbation ( one car is shifted three sites backwards)@xmath77 4 . strong perturbation ( one car is shifted five sites backwards)@xmath78 5 .
strongest perturbation ( three cars are shifted backwards ) @xmath79 the stationary state of ( a ) , ( b ) , ( c ) , ( d ) and ( e ) are given by the points @xmath80 , @xmath81 , @xmath82 , @xmath83 and @xmath84 in fig .
[ fig6 ] , respectively .
that is , if the system in @xmath72 is perturbated , then the flow easily goes down to a lower branch in the course of time depending on the magnitude of the perturbation . since the density does not change due to the perturbation , we obtain @xmath85 , @xmath86 , @xmath87 , @xmath88 and @xmath89 by substituting @xmath45 into eq .
( [ metaq ] ) .
we see a jamming cluster propagating backwards in the cases of ( d ) and ( e ) in fig .
in other cases the jamming cluster propagates forward ( ( a ) and ( b ) ) or does not move ( c ) .
these facts are related to the gradient of the metastable branches which are given by @xmath90 according to eq .
( [ metaq ] ) .
finally we will combine the above model with the ns model in order to take into account the randomness of drivers .
the ns model is written in lagrange form as @xmath91 where @xmath92 with probability @xmath93 and @xmath94 with probability @xmath95 .
the last term in the mininum in ( [ eqns ] ) represents the acceleration of cars . the randomness in this model
is considered as a kind of random braking effect , which is known to be responsible for spontaneous jam formation often observed in real traffic @xcite .
we also consider random _ accerelation _ in this model which is not taken into account in the ns model .
thus a stochastic generalization of the hybrid model in the case of @xmath19 is similarly given by the following set of rules : 1 .
_ random accerelation _
@xmath96 where @xmath97 with the probability @xmath98 and @xmath99 with @xmath100 .
slow - to - accelerate effect _
@xmath101 3 .
_ deceleration due to other vehicles _ @xmath102 4 .
_ random braking _
@xmath103 where @xmath104 with the probability @xmath105 and @xmath106 with @xmath107 .
_ avoidance of collision _
@xmath108 with @xmath109 , which is an iterative equation that has to be applied until @xmath110 converges to @xmath111 .
_ vehicle movement _
@xmath112 again the velocity @xmath113 is used as @xmath27 in the next time step .
step 5 must be applied to each car iteratively until its velocity does not change any more , which ensures that this model is free from collisions .
this is the difference between the deterministic and stochastic case . in the deterministic model
it is sufficient to apply the avoidance of collision stage only once in each update , while in the stochastic case generically it has to be applied a few times in order to avoid collisions between successive cars . and @xmath19 .
upper two figures are the case of @xmath114 and @xmath115 , middle ones are @xmath116 and @xmath117 , and the bottom ones are @xmath116 and @xmath115 .
, title="fig:",scaledwidth=40.0% ] and @xmath19 .
upper two figures are the case of @xmath114 and @xmath115 , middle ones are @xmath116 and @xmath117 , and the bottom ones are @xmath116 and @xmath115 . , title="fig:",scaledwidth=40.0% ] + and @xmath19 .
upper two figures are the case of @xmath114 and @xmath115 , middle ones are @xmath116 and @xmath117 , and the bottom ones are @xmath116 and @xmath115 . , title="fig:",scaledwidth=40.0% ] and @xmath19 .
upper two figures are the case of @xmath114 and @xmath115 , middle ones are @xmath116 and @xmath117 , and the bottom ones are @xmath116 and @xmath115 . , title="fig:",scaledwidth=40.0% ] + and @xmath19 .
upper two figures are the case of @xmath114 and @xmath115 , middle ones are @xmath116 and @xmath117 , and the bottom ones are @xmath116 and @xmath115 . , title="fig:",scaledwidth=40.0% ] and @xmath19 .
upper two figures are the case of @xmath114 and @xmath115 , middle ones are @xmath116 and @xmath117 , and the bottom ones are @xmath116 and @xmath115 . , title="fig:",scaledwidth=40.0% ] the fundamental diagrams of this stochastic model for some values of @xmath98 and @xmath105 are given in fig .
the randomization effect can be considered as a sort of perturbation to the deterministic model .
hence some unstable branches seen in the deterministic case disappear in the stochastic case , especially if we consider the random braking effect as seen in fig .
random accerelation itself does not significantly destroy the metastable branches .
moreover , from the spatio - temporal pattern it is found that spontaneous jam formation is observed only if we allow random braking .
random accerelation alone is not sufficient to produce spontaneous jamming .
we also note that a wider scattering area appears if we introduce both random accerelation and braking .
in this paper we have proposed a new hybrid model of traffic flow of lagrange type which is a combination of the bca and the s2s model .
its stochastic extension is also proposed by further incorporating stochastic elements of the ns model and random accerelation .
the model shows several metastable branches around the critical density in its fundamental diagram .
the upper branches are unstable and will decrease its flow under perturbations .
it is shown that the magnitude of a perturbation determines the final value of flow in the stationary state .
moreover , introduction of stochasticity in the model makes the metastable branches dilute and hence produces a wide scattering area in the fundamental diagram .
we would like to point out that this metastable region around the phase transition density is similar to so - called synchronized flow proposed by @xcite .
our investigation shows that one possible origin of such a region is the occurance of many intermediate congested states near the critical density .
if some of them are unstable due to perturbation or randomness , then a dense scattering area near the critical density is formed around the metastable branches .
this is in some sense in between the two cases of a fundamental diagram based approach ( with unique flow - density relation ) and the so - called 3-phase model of @xcite which exhibits a full two - dimensional region of allowed states even in the deterministic limit .
this work is supported in part by a grant - in - aid from the japan ministry of education , science and culture . | a new stochastic cellular automaton ( ca ) model of traffic flow , which includes slow - to - start effects and a driver s perspective , is proposed by extending the burgers ca and the nagel - schreckenberg ca model .
the flow - density relation of this model shows multiple metastable branches near the transition density from free to congested traffic , which form a wide scattering area in the fundamental diagram .
the stability of these branches and their velocity distributions are explicitly studied by numerical simulations . | cond-mat0311194 |
puga@xmath0 appears in two crystal structures with similar electronic structures but different magnetic structures .
et al . _ noted in 1964 the appearance of the two crystal structures and identified the low - temperature ( low-@xmath3 ) form as hexagonal and isostructural with ni@xmath0sn;@xcite the following year larson _ et al .
_ identified the high - temperature ( high-@xmath3 ) form as a 12-layer rhombohedral close - packed structure with space group r@xmath4m.@xcite four decades later , magnetic measurements revealed the low-@xmath3 form to be an antiferromagnet ( @xmath5 k ) and the high-@xmath3 form to be a ferromagnet ( @xmath6 k ) , and specific heat measurements suggest for both phases an electronic structure with heavy fermion character.@xcite these characteristics place puga@xmath0 in compelling relation to other heavy fermion systems of significant interest .
the electronic specific heat coefficient @xmath7 , a measure of the electronic density of states at the fermi energy , has values ( 220 and 100 mj / mol k@xmath8 for low-@xmath3 and high-@xmath3 , respectively ) similar to the heavy fermion superconductors pucoga@xmath9 and purhga@xmath9.@xcite neither of these are magnetic experimentally , though electronic structure calculations favor antiferromagnetic order.@xcite with rather delocalized @xmath1 electrons , puga@xmath0 lies between pucoga@xmath9 and @xmath10-pu,@xcite which has more localized @xmath1 electrons but also shows no localized magnetic moments.@xcite the combination of magnetic structure and heavy fermion behavior in puga@xmath0 suggest a challenging system for electronic structure calculations .
the electronic structure of pu , many pu compounds , and some other actinide systems requires special attention to be paid to the strong @xmath1 electron correlation .
calculations with `` standard '' density functional theory ( dft ) methods , which involve limited approximate treatments of the electronic correlation , favor an antiferromagnetic structure,@xcite and some aspects are even better modeled with disordered local moments or approximations thereof.@xcite experimentally , pure pu shows no signs of magnetic moments.@xcite the breaking of spin symmetry in dft calculations delivers a static approximation of the spatial separation experienced by dynamically correlated electrons . as a result , calculations allowing a localized magnetic moment can be used to explore nonmagnetic aspects of pu and pu compounds without introducing material - dependent parameters .
the existence of magnetic structures in puga@xmath0 entangles the magnetic moments and the electronic correlation , which , along with their entanglement with the observed crystal structures , motivates this study .
the crystal structures of puga@xmath0 can be viewed as close - packed puga@xmath0 planes with different stacking sequences.@xcite the low-@xmath3 structure follows an ab sequence , as does hexagonal close packed ( hcp ) ; stacking in the high-@xmath3 structure progresses as ababcacabcbc ( with some in - plane distortions away from the perfect close - packed planar structures ) .
this layered , close - packed nature already appears in crystal structures of pure pu : the face - centered cubic structure of @xmath10-pu exhibits abc stacking , the crocker pseudostructure for @xmath11-pu follows from the @xmath11 structure s repeating two planes of a distorted hexagonal structure,@xcite and the orthorhombic structure of @xmath7-pu exhibits close - packed pu planes stacked such that pu atoms in one plane sit above bonds in the plane underneath ( giving rise to an abcd stacking pattern ) .
the close - packed puga@xmath0 planes correspond to the close - packed pu planes with ordered substitutional placing of ga .
these stacking sequences of close - packed planes ( excluding that of @xmath7-pu ) can also be written as sequences of shifts between planes : ab , bc , ca being shifts to the right ( r ) and ac , cb , ba being shifts to the left ( l ) .
abc stacking always shifts in the same direction ( rrrr ) , ab stacking alternates between the two directions ( rlrl ) , and ababcacabcbc stacking , rewritten as ( abca)(cabc)(bcab ) , reverses direction once every four planes ( rrrl ) .
the missing unique pattern with four shifts , rrll , is abcb stacking , which corresponds to double hcp ( dhcp ) , exemplified by @xmath11-la . while the r and l shifts are equivalent , a stark difference exists between a plane that links two shifts with the same direction and one that sits at a reversal in the direction .
the local environment of the atomic sites in the ideal close - packed lattices has twelve nearest neighbors in both cases .
sites in a plane between two identical shifts have the inversion symmetry , while those between two opposite shifts do not .
the lack of inversion symmetry disrupts an otherwise straight line of bonding oriented 60@xmath12 to the planes .
a natural order of the four crystal structures arises : abc stacking has no disruptions , ab stacking has disruptions in every plane , and the remaining two stacking sequences lie in between .
the work presented here applies dft to reveal the interplay between crystal structures based on these four structural patterns , a series of magnetic structures , and the resulting electronic structures .
starting from `` standard '' dft in the generalized gradient approximation ( gga ) , calculations furthermore explore the effects of adding either spin - orbit coupling or a hubbard @xmath2 ( in the gga+u method ) .
the calculations presented here set aside thermal effects , in particular those due to phonons .
preliminary calculations of the phonons and their contribution to the free energy suggest they can not make the low-@xmath3 phase more favorable in the gga to dft without specifically addressing @xmath13 electron correlation .
the dft calculations employ the vasp package.@xcite they make use of the generalized gradient approximation ( gga ) of perdew , burke , and ernzerhof.@xcite the pu(@xmath14 ) and ga(@xmath15 ) electrons are treated in the valence using a plane - wave basis and with projector - augmented wave potentials.@xcite the calculations employ methfessel - paxton smearing ( with width 0.1 ev ) , a k - point mesh of density 40 @xmath16 , and an energy cutoff of 400 ev . the self - consistent cycles are converged to within 10@xmath17 ev .
calculations aimed at improving the treatment of the the on - site coulomb repulsion between 5@xmath13 electrons use an effective hubbard parameter @xmath2 in the rotationally invariant form of dudarev _
et al._.@xcite in this form the hubbard parameter @xmath2 and the exchange parameter @xmath18 appear only in the difference @xmath19 , throughout this report the difference is referred to as @xmath2 .
calculations that include the effects of spin - orbit coupling do so in the noncollinear mode of vasp,@xcite the implementation follows the approach of kleinman and macdonald , picket , and koelling.@xcite the calculations optimize crystal structures that start as ideal close - packed planes with one pu and three ga atoms , stacked according to one of the four patterns described above .
relaxation of the structures retains the overall layered structure , but displacements within the planes make initially equivalent planes lose their exact equality .
the size of the unit cell , in particular the number of planes ( between two and twelve ) , follows from the particular pattern and the magnetic structure used to seed the calculations .
the latter either has all spins in the same direction for the ferromagnetic ( fm ) structure , or spins that switch direction every one , two , three , four , or six planes .
these arrangements define spin wave structures with wave vectors @xmath20 of magnitude @xmath21 , @xmath22 , @xmath23 , @xmath24 , @xmath25 , and , in the fm case , @xmath26 , scaled by @xmath27 , where @xmath28 represents the interplanar spacing .
additional magnetic structure within the close - packed plane affects the results , but these are not reported here , other than to note that their energy lies above that of the antiferromagnetic ( afm , @xmath29 ) state .
the results from three approaches appear in the following three subsections . sections
[ gga ] and [ ggaplussoc ] report the results of dft calculations in the gga without and with spin - orbit coupling , respectively , for the four crystal structures in a sequence of magnetic states .
section [ ggaplusu ] focuses on results of the gga+u method applied to the low-@xmath3 and cubic structures in the afm state .
table [ tab : opt ] summarizes the energies , volumes and @xmath30 ratios calculated in the three approaches for the four crystal structures in the afm state .
figure [ fig : abcafvary ] shows the interplay between the four crystal structures and the magnetic structures using the gga to dft .
all four stacking sequence patterns show a preference for the magnetic structure with the shortest spin wave length , the afm state .
the ordering of energies of the crystal structures in the afm state correlates with the order arising from the number of changes in r and l shifts mentioned in the introduction . with this magnetic structure ,
the experimentally observed low-@xmath3 phase lies highest , 117 mev / puga@xmath0 above the favored structure with abc stacking .
this cubic structure is observed for puin@xmath0 and is often considered a building block for the layered superconductors pucoga@xmath9 , cept@xmath31in@xmath32 , ce@xmath31rhin@xmath33 , etc .
the preference for this cubic structure appears only for the magnetic structure with the shortest spin wave length ; for longer spin wave lengths it lies higher than the other states ( albeit by small amounts ) . among the fm states the high-@xmath3 phase lies lowest , the slight 13 mev / puga@xmath0 difference to the low-@xmath3 phase suggests the importance of thermal effects .
( color online ) calculated dependence of energy on stacking and magnetic structure for puga@xmath0 using gga ( @xmath34 ) .
stacking denotes initial crystal structure ; upon relaxation the planes with the same letter are no longer necessarily equivalent .
stacking direction corresponds to body diagonal of the conventional aucu@xmath0 crystal structure unit cell ; the lowest energy appears for g - type antiferromagnetism ( afm(g ) ) .
dashed lines serve to guide the eye . , width=321 ]
the optimized structures agree reasonably well with experimental volumes , while the optimized @xmath30 ratios consistently lie above the experimental values .
the afm volume calculated for the low-@xmath3 structure is only 1% smaller , but the @xmath30 ratio is close to 6% larger than the experimental value ( see table [ tab : opt ] ) . the afm volume calculated for the high-@xmath3 structure is 0.25% smaller than the experimental value , and the @xmath30 ratio is close to 4% larger than the experimental value .
the fm volume calculated for the high-@xmath3 structure is 1% larger , and the @xmath30 ratio is 3% larger than the experimental value .
the distances from pu to nearest ga atoms ( located in adjacent planes ) differ by negligible amounts between the calculated and experimental high-@xmath3 structures
. the larger calculated @xmath30 ratio does affect the angle spanned by a pu atom and two ga atoms in adjacent planes , decreasing it by as much as 13% .
figure [ fig : ggaedos1](a ) compares calculated total electronic densities of states ( dos ) and suggests why the cubic structure appears more favorable in the afm state .
the low-@xmath3 , high-@xmath3 , and cubic structure differ significantly in the highest occupied states .
the low-@xmath3 and cubic structure both exhibit a single peak , but the cubic structure has it almost 0.3 ev further below the fermi level @xmath35 .
the high-@xmath3 structure exhibits a double peak centered between the other two structures . while the band energy is only one part of the total energy , this ability of the structures to push states down and away from @xmath35 corresponds to their order in total energy .
figure [ fig : ggaedos1](b ) plots the analogous comparison for the three structures in the fm state . compared to the afm state ,
the peaks appear much more similar for the three structures than in the afm state .
accordingly , the total energies for the fm state differ by smaller amounts compared to the afm state .
the peaks sit closer to @xmath35 in the fm state , concurring with the energies of the fm state lying higher than those of the afm state .
( color online ) calculated electronic densities of states ( dos ) near the fermi energy @xmath35 using the gga to dft ( @xmath34 ) for low-@xmath3 , high-@xmath3 , and aucu@xmath0 crystal structures with ( a ) afm and ( b ) fm structure .
only the dos for one spin orientation appears for afm .
, width=321 ] ( color online ) calculated electronic dos near @xmath35 projected on a pu site with @xmath13 character using the gga to dft ( @xmath34 ) for low-@xmath3 and aucu@xmath0 crystal structures with afm magnetic structure .
the plotted dos represent the majority spin on the pu site .
, width=321 ] figure [ fig : fdos1 ] shows the @xmath13 symmetry character ( projected out on a pu site ) of the electronic dos calculated for the low-@xmath3 and aucu@xmath0 crystal structures with afm magnetic structure .
the @xmath13-projected peaks correspond to the peaks in fig .
[ fig : ggaedos1](a ) .
the projected dos are identical for all sites in each case , as expected given the sites identical environments : each site has the same structural environment and nearest neighbors with opposite spin .
the structural environment differs between the two cases , the aucu@xmath0 crystal structure s inversion symmetry allows the @xmath13-projected peaks to be pushed down lower .
the less symmetric local environment in the low-@xmath3 structure makes it less atomic - like , requiring the @xmath13 electrons to hybridize more .
( color online ) calculated electronic dos near @xmath35 projected on pu sites with @xmath13 character using gga ( @xmath34 ) for low-@xmath3 and aucu@xmath0 crystal structures with magnetic structure that has spin wave length spanning twelve planes .
the plotted dos represent the majority spin on each site . in terms of geometry ,
all sites are equivalent for each crystal structure .
they differ depending on where they sit within the magnetic structure : adjacent to the spin flip ( `` edge '' ) , one layer farther in ( `` intermediate '' ) , or most distant to the spin flip ( `` center '' ) .
, width=321 ] figure [ fig : fdos2 ] compares the @xmath13 symmetry character projected out on pu sites from the electronic dos calculated for the low-@xmath3 and aucu@xmath0 crystal structures in the magnetic state with spin wave vector magnitude @xmath36 .
this choice of spin wave vector stems from the differences it reveals among the pu sites , unlike the ferromagnetic structure where all sites ( within each crystal structure ) remain equivalent . for the low-@xmath3 structure
the projected electronic dos differs only slightly between the three types of sites , a slight shift down from @xmath35 occurs closer to the edge of the magnetic subdivision .
the cubic structure shows dramatic differences between the three types of sites : all show a projected electronic dos hugging @xmath35 from below , and only the site at the edge of the magnetic subdivision appears able to spread a significant amount down several tenths of an ev . in the fm state ,
the @xmath13-projected dos on any of the sites closely resembles the @xmath13-projected dos shown here for center atoms .
( color online ) calculated dependence of energy on stacking and magnetic structure for puga@xmath0 using gga ( @xmath34 ) and including spin - orbit coupling . dashed lines serve to guide the eye .
notation follows fig .
[ fig : abcafvary ] . , width=321 ] figure [ fig : abcafvarysoc ] shows the interplay between the four crystal structures and magnetic structures using the gga to dft and including spin - orbit coupling .
the inclusion of spin - orbit coupling reduces the energy differences overall , hence figure [ fig : abcafvarysoc ] appears much like a scaled version of fig .
[ fig : abcafvary ] .
the aucu@xmath0 crystal structure in the afm state remains the most favored , in the fm state it remains the least favored .
results from calculations that include spin - orbit coupling repeat the correlation between which structure is energetically favored and its ability to push electronic states down and away from @xmath35 . with spin - orbit coupling ,
the electronic dos of the low-@xmath3 and aucu@xmath0 crystal structure differ from one another less than in fig .
[ fig : ggaedos1 ] , but the more favored aucu@xmath0 crystal structure still succeeds better at pushing electronic states to lower energies . table [ tab : opt ] shows how treating the on - site coulomb repulsion between @xmath1 electrons with a hubbard u changes the ranking of crystal structures . setting @xmath37
ev reverses the sequence in energy from the gga result ( with or without spin - orbit coupling ) : the low-@xmath3 crystal structure becomes most favored while the aucu@xmath0 crystal structure becomes the least favored .
the high-@xmath3 and `` @xmath11-la '' structures remain in between and switch their order as well .
comparison of the energies for the different crystal structures only has meaning for each value of @xmath2 individually , which is somewhat unsatisfactory since the different crystal structures would be better described with different values ( differences in the electronic specific heat coefficient @xmath7 and in the pu - ga distances in the low-@xmath3 and high-@xmath3 crystal structure suggest different degrees of @xmath1 delocalization,@xcite implying incompatible values of @xmath2 ) .
c | c c c c | c structure & 4@xmath2 ( ev ) & soc & 0 & 1 & 2 & 3 & ( @xmath34)6relative energies ( mev / puga@xmath0 ) low-@xmath3 & 0 & 0 & 0 & 0 & 0@xmath11-la " & -49 & -9 & 6 & 109 & -26 high-@xmath3 & -72 & -53 & -26 & 120 & -47 aucu@xmath0 & -117 & -67 & 2 & 168 & -62 6volumes ( @xmath38/puga@xmath0 ) low-@xmath3 & 77.37 & 78.09 & 80.00 & 81.28 & 77.99 @xmath11-la " & 77.59 & 78.58 & 79.78 & 80.84 & 77.52 high-@xmath3 & 77.50 & 78.22 & 79.68 & 80.90 & 77.50 aucu@xmath0 & 77.72 & 77.72 & 79.88 & 80.77 & 77.67 6@xmath30 ratio low-@xmath3 & 0.38 & 0.38 & 0.37 & 0.36 & 0.37 @xmath11-la " & 0.40 & 0.39 & 0.39 & 0.39 & 0.39 high-@xmath3 & 0.39 & 0.39 & 0.39 & 0.38 & 0.39 aucu@xmath0 & 0.41 & 0.41 & 0.41 & 0.41 & 0.41 figure [ fig : edosabaf1 ] plots the electronic dos for the low-@xmath3 crystal structure in the afm state calculated with the gga+u method . as @xmath2 increases , the dominant peaks , dft s rendering of the upper and lower hubbard bands,@xcite increasingly separate .
this separation pushes the occupied states down from @xmath35 more than it pushes the unoccupied states up .
the symmetry between up and down spin remains intact , and , based on site - projected dos ( not shown here ) , the equivalence among sites with the same spin remains .
figure [ fig : edosabcaf1 ] plots the electronic dos for the aucu@xmath0 crystal structure in the afm state calculated with the gga+u method .
again the increasing @xmath2 drives the dominant peaks apart , but for this crystal structure the separation occurs mainly by pushing up the unoccupied states .
the occupied states change little as @xmath2 increases from 0 ev to 1 ev .
increasing @xmath2 from 1 ev to 2 ev pushes the occupied states down .
setting @xmath37 ev breaks the symmetries of up and down spins as well as the equivalence among sites with the same spin .
( color online ) calculated electronic dos with varying hubbard @xmath2 for the low-@xmath3 crystal structure with afm magnetic structure at the experimental volume .
the dos for the two spin orientations appear as positive and negative , respectively .
, width=321 ] ( color online ) calculated electronic dos with varying hubbard @xmath2 for the cubic crystal structure with afm magnetic structure at the experimental volume .
the dos for the two spin orientations appear as positive and negative , respectively . for u=@xmath39 ev and above the symmetry between pu sites
is broken and variations of up to 0.6% appear in the site - projected charge and of up to 5% appear in the site - projected magnetic moments .
, width=321 ] comparison of figs .
[ fig : edosabaf1 ] and [ fig : edosabcaf1 ] for each value of @xmath2 correlates well with the energy differences in table [ tab : opt ] . for @xmath40
ev the aucu@xmath0 crystal structure retains the peak around 0.5 ev below @xmath35 while the low-@xmath3 crystal structure has its main peak shifted lower than for @xmath34 . for @xmath41
ev both crystal structures have shifted ( and broadened ) the peak to around 1 ev below @xmath35 . for @xmath37
ev the overall dos changes somewhat for the aucu@xmath0 crystal structure while for the low-@xmath3 crystal structure a dramatic shift downward occurs .
systematically studying the crystal and magnetic structures of puga@xmath0 reveals how they affect the electronic structure and how the three types of structure are entangled .
the key to understanding the entanglement lies in the position of the @xmath1 electron states relative to the fermi level @xmath35 in the electronic dos .
the position relative to @xmath35 is determined by both the symmetry of the crystal structure and the imposed magnetic structure .
how far the @xmath1 peak sits below @xmath35 dovetails with how favorable the system in question is in terms of calculated total energy .
magnetic structure affects the energy more strongly than crystal structure . with or without spin - orbit coupling ,
the calculations favor the afm state over the fm state for all crystal structures .
spin density waves with wave lengths between those of the fm state ( infinity ) and of the afm state ( twice the spacing between pu planes ) give total energies between the two limiting values .
pu sites neighboring a junction between up and down spins have their @xmath1 electron states farther below @xmath35 than other pu sites .
each such junction gives the sites sandwiching the junction less hybridization of @xmath1 states with neighbors on the other side of the junction . in the limiting case of afm , every site has the least hybridization because pu sites in neighboring planes have opposite spin .
calculations using standard gga result in the wrong crystal structure ( aucu@xmath0 ) having the lowest energy in the favored afm state .
the @xmath1 electron states in the cubic structure sit farther below @xmath35 than they do in the experimentally observed ni@xmath0sn crystal structure , because the inversion symmetry at sites in the cubic structure requires less hybridization between the pu @xmath1 states and other states . adding a hubbard u to treat the strong @xmath1 electron correlation results in the correct crystal structure having the lowest energy .
the @xmath2 raises and lowers the potential acting on the unoccupied and occupied @xmath1 states , respectively , but the effect of @xmath2 on the positions of the @xmath1 states relative to @xmath35 depends on how they are hybridized.@xcite increasing the value of @xmath2 proves more effective at lowering the energy of the @xmath1 electron peak for the ni@xmath0sn crystal structure , making it most favored for @xmath37 ev .
allowing localized magnetic moments to simulate correlation effects fails for puga@xmath0 .
the strong preference for the cubic crystal structure over the hexagonal crystal structure suggests the failure stems not from the actual presence of a magnetic structure ( observed in experiment ) , but from the symmetry at pu sites in the hexagonal crystal structure being much lower than in the cubic crystal structure .
the use of allowing localized magnetic moments to simulate correlation does so by permitting the @xmath1 electrons on the same pu site to occupy more orbitals that differ spatially .
the inversion symmetry present in the cubic crystal structure makes the localized magnetic moments most effective at simulating correlation effects . in the hexagonal crystal structure
the lower symmetry prevents an adequate decoupling of @xmath13 states from hybridization and their energy can not be lowered sufficiently to make the crystal structure most favorable .
these results suggest a explanation for the effectiveness of using magnetism to approximate correlation effects in @xmath10-pu .
the crystal structure of @xmath10-pu is face - centered cubic , and all sites exhibit the inversion symmetry shown here to be important in the closely - related aucu@xmath0 structure .
given the similarities , the preference for an afm state in @xmath10-pu does not surprise .
nor does the additional effectiveness of modeling correlation effects with disordered local moments astonish , since such a magnetic `` structure '' reduces also the in - plane hybridization between @xmath13 electrons on neighboring sites . analogous to the relation between @xmath10-pu and puga@xmath0 in the aucu@xmath0 crystal structure , @xmath11-pu relates to puga@xmath0 in the ni@xmath0sn crystal structure .
the crystal structures of both @xmath11-pu and the low-@xmath3 phase of puga@xmath0 are the most stable and both have an ab stacking pattern .
the @xmath11-pu crystal structure stacks distorted close - packed pu planes ; replacing three of four pu atoms with ga removes the distortion to restore the symmetry in the close packed planes of puga@xmath0 , which could relate to the stabilization of @xmath10-pu to low temperatures by adding a small amount of ga.@xcite the electronic specific heat coefficient @xmath7 differs dramatically between the low-@xmath3 phase of puga@xmath0 , where @xmath42 mj / mol k@xmath8 , and @xmath11-pu , where @xmath43 mj / mol k@xmath8 was measured.@xcite correspondingly , @xmath11-pu can be well described by standard dft methods,@xcite while the work presented here shows that the low-@xmath3 phase of puga@xmath0 requires special attention be paid to the strong @xmath1 electron correlation , and allowing spin polarization does not suffice to describe the effects of the strong correlation .
this research was supported by the los alamos national laboratory , under the auspices of the national nuclear security agency , by the u.s .
department of energy under grant no .
ldrd - dr 20120024 ( `` pu-242 : a national resource for the fundamental understanding of the 5f electrons of pu '' ) .
many thanks go to in particular eric chisolm , anders niklasson , and john wills as well as eric bauer , john joyce , and paul tobash , for helpful and encouraging discussions .
the author expresses a deep gratitude to neil henson for assistance with the andulu computational facility .
last , but not least , fond thanks go to lucia lin and anna lan for spurring alternative approaches to understanding .
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10.1103/physrevb.75.184501 [ * * , ( ) ] link:\doibase 10.1103/physrevb.70.104504 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevb.21.2630 [ * * , ( ) ] http://stacks.iop.org/0022-3719/13/i=14/a=009 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop _ _ ( , , ) @noop * * , ( ) @noop * * , ( ) | systematically studying the crystal , magnetic , and electronic structures of puga@xmath0 with density functional theory ( dft ) reveals the entanglement of the three types of structure .
magnetic structure affects the energy more strongly than crystal structure . for dft to correctly order the crystal structures in agreement with experiment
requires special treatment of the electronic correlation in the @xmath1 states , exemplified here by the gga+u approach .
the upper and lower hubbard bands change with increasing @xmath2 in very dissimilar ways for the two most different crystal structures .
the results suggest the effectiveness of using magnetic structure to simulate correlation effects in the actinides depends on both the magnetic and the crystal structure . | 1307.0762 |
a benzenoid system is determined with all the hexagons lying inside cycle @xmath0 of the hexagonal lattice .
they represent molecules called benzenoid hydrocarbons .
these graphs are also known as the hexagonal systems and form one of the most extensively studied family of chemical graphs . for fundamental properties of benzenoid systems see @xcite , while some recent results can be found in @xcite .
if we embed benzenoid systems on a surface of a cylinder and join some edges we obtain structures called open - ended single - walled carbon nanotubes also called tubulenes .
carbon nanotubes are carbon compounds with a cylindrical structure and they were first observed in 1991 @xcite . if we close a carbon nanotube with two caps composed of pentagons and hexagons , we obtain a fullerene .
more exactly , a fullerene is a molecule of carbon in the form of a hollow sphere , ellipsoid , tube , or many other shapes .
the first fullerene molecule was discovered @xmath1 years ago . in graph theory ,
a fullerene is a @xmath2-regular plane graph consisting only of pentagonal and hexagonal faces .
the overview of some results on fullerene graphs can be found in @xcite .
papers @xcite present a sample of recent investigations .
the concept of the resonance graph appears quite naturally in the study of perfect matchings of molecular graphs of hydrocabons that represent kekul ' e structures of corresponding hydrocarbon molecules .
therefore , it is not surprising that it has been independently introduced in the chemical @xcite as well as in the mathematical literature @xcite ( under the name @xmath3-transformation graph ) and then later rediscovered in @xcite .
the equivalence of the zhang - zhang polynomial of the molecular graph and the cube polynomial of its resonance graph was established for benzenoid systems @xcite , tubulenes @xcite , and fullerenes @xcite . the zhang - zhang polynomial counts clar covers with given number of hexagons , i.e.conjugated 6-cycles .
for some recent research on the zhang - zhang polynomial see @xcite the resonance energy is a theoretical quantity which is used for predicting the aromatic stability of conjugated systems . in the conjugated - circuit model ,
the resonance energy is determined with conjugated cycles of different lengths ( see @xcite ) , not only with 6-cycles . among them , only 6-cycles and 10-cycles have uniquely determined structure . therefore , we introduce the concept of the generalized zhang - zhang polynomial , which considers both of them . in this paper
we prove the equivalence of the generalized zhang - zhang polynomial of a molecular graph and the generalized cube polynomial of the corresponding resonance graph .
a _ benzenoid system _ consists of a cycle @xmath0 of the infinite hexagonal lattice together with all hexagons inside @xmath0 . a _ benzenoid graph _ is the underlying graph of a benzenoid system .
next we formally define open - ended carbon nanotubes , also called _ tubulenes _ @xcite .
choose any lattice point in the hexagonal lattice as the origin @xmath4 .
let @xmath5 and @xmath6 be the two basic lattice vectors .
choose a vector @xmath7 such that @xmath8 and @xmath9 are two integers and @xmath10 , @xmath11 .
draw two straight lines @xmath12 and @xmath13 passing through @xmath4 and @xmath14 perpendicular to @xmath15 , respectively . by rolling up the hexagonal strip between @xmath12 and @xmath13 and gluing @xmath12 and @xmath13 such
that @xmath14 and @xmath4 superimpose , we can obtain a hexagonal tessellation @xmath16 of the cylinder . @xmath12 and
@xmath13 indicate the direction of the axis of the cylinder . using the terminology of graph theory ,
a _ tubulene _
@xmath17 is defined to be the finite graph induced by all the hexagons of @xmath18 that lie between @xmath19 and @xmath20 , where @xmath19 and @xmath20 are two vertex - disjoint cycles of @xmath18 encircling the axis of the cylinder .
the vector @xmath21 is called the _ chiral vector _ of @xmath17 and the cycles @xmath19 and @xmath20 are the two open - ends of @xmath17 .
-type tubulene . ] for any tubulene @xmath17 , if its chiral vector is @xmath22 , @xmath17 will be called an @xmath23-type tubulene , see figure [ fig - nano ] .
a _ fullerene _ @xmath24 is a @xmath2-connected 3-regular plane graph such that every face is bounded by either a pentagon or a hexagon . by euler s formula
, it follows that the number of pentagonal faces of a fullerene is exactly @xmath25 .
a _ 1-factor _ of a graph @xmath24 is a spanning subgraph of @xmath24 such that every vertex has degree one .
the edge set of a 1-factor is called a _ perfect matching _ of @xmath24 , which is a set of independent edges covering all vertices of @xmath24 . in chemical literature ,
perfect matchings are known as kekul structures ( see @xcite for more details ) .
petersen s theorem states that every bridgeless @xmath2-regular graph always has a perfect matching @xcite .
therefore , a fullerene always has at least one perfect matching .
a hexagon of @xmath24 with exactly 3 edges in a perfect matching @xmath26 of @xmath24 is called a _
sextet_. let @xmath24 be a benzenoid system , a tubulene or a fullerene with a perfect matching .
the _ resonance graph _
@xmath27 is the graph whose vertices are the perfect matchings of @xmath24 , and two perfect matchings are adjacent whenever their symmetric difference forms a hexagon of @xmath24 .
the _ hypercube _ @xmath28 of dimension @xmath8 is defined in the following way : all vertices of @xmath28 are presented as @xmath8-tuples @xmath29 where @xmath30 for each @xmath31 and two vertices of @xmath28 are adjacent if the corresponding @xmath8-tuples differ in precisely one coordinate . a _ convex subgraph _ @xmath32 of a graph @xmath24 is a subgraph of @xmath24 such that every shortest path between two vertices of @xmath32 is contained in @xmath32 .
let @xmath24 be a benzenoid system , a tubulene or a fullerene .
clar cover _ is a spanning subgraph of @xmath24 such that every component of it is either @xmath33 or @xmath34 .
zhang - zhang polynomial _ of @xmath24 is defined in the following way : @xmath35 where @xmath36 is the number of clar covers of @xmath24 with @xmath37 hexagons .
a _ generalized clar cover _ is a spanning subgraph of @xmath24 such that every component of it is either @xmath33 , @xmath38 or @xmath34 .
see figure [ clar_cover ] for an example . .
] the _ generalized zhang - zhang polynomial _ of @xmath24 is defined in the following way : @xmath39 where @xmath40 is the number of generalized clar covers of @xmath24 with @xmath37 cycles @xmath41 and @xmath42 cycles @xmath38 .
note that for a graph @xmath24 number @xmath43 equals the number of vertices of @xmath27 and @xmath44 equals the number of edges of @xmath27 .
furthermore , number @xmath45 represents the number of clar covers with @xmath37 hexagons .
let @xmath32 be a graph .
the _ cube polynomial _ of @xmath32 is defined as follows : @xmath46 where @xmath47 denotes the number of induced subgraphs of @xmath32 that are isomorphic to the @xmath37-dimensional hypercube .
let @xmath24 be a graph and @xmath48 an integer .
then by @xmath49 we denote the cartesian product of @xmath50 copies of @xmath24 , i.e. @xmath51 . also , @xmath52 .
furthermore , for any @xmath53 we define @xmath54 , where @xmath55 and @xmath56 are paths on 2 and 3 vertices , respectively .
obviously , @xmath57 is the @xmath37-dimensional hypercube . moreover ,
if @xmath58 , vertices of the graph @xmath59 can be presented as @xmath60-tuples @xmath61 , where @xmath62 if @xmath63 and @xmath64 if @xmath65 . in such representation two vertices @xmath61 and
@xmath66 are adjacent if and only if there is @xmath67 such that @xmath68 and @xmath69 for any @xmath70 .
let @xmath32 be a graph .
the _ generalized cube polynomial _ of @xmath32 is defined as follows : @xmath71 where @xmath72 denotes the number of induced convex subgraphs of @xmath32 that are isomorphic to the graph @xmath59 .
in this section we prove that the generalized zhang - zhang polynomial of every benzenoid system , tubulene or fullerene equals the generalized cube polynomial of its resonance graph .
let @xmath24 be a benzenoid system , a tubulene or a fullerene with a perfect matching .
then the generalized zhang - zhang polynomial of @xmath24 equals the generalized cube polynomial of its resonance graph @xmath27 , i.e. @xmath73 [ main ] let @xmath37 and @xmath42 be nonnegative integers . for a graph @xmath24
we denote by @xmath74 the set of all generalized clar covers of @xmath24 with exactly @xmath37 cycles @xmath33 and @xmath42 cycles @xmath38 . on the other hand , consider a graph @xmath32 ; the set of induced convex subgraphs of @xmath32 that are isomorphic to a graph @xmath59 is denoted by @xmath75 .
let us define a mapping @xmath76 from the set of generalized clar covers of @xmath24 with @xmath37 cycles @xmath33 and @xmath42 cycles @xmath38 to the set of induced convex subgraphs of the resonance graph @xmath27 isomorphic to the graph @xmath59 @xmath77 in the following way : for a generalized clar cover @xmath78 consider all perfect matchings @xmath79 , @xmath80 , @xmath81 , @xmath82 of @xmath24 such that : * if cycle @xmath33 in @xmath0 , then @xmath83 for all @xmath84 , * if cycle @xmath38 of @xmath0 is composed of two hexagons , @xmath85 and @xmath86 , then @xmath87 for all @xmath84 , * each isolated edge of @xmath0 is in @xmath88 for all @xmath84 .
finally , assign @xmath89 as an induced subgraph of @xmath27 with vertices @xmath79 , @xmath80,@xmath81 , @xmath82 .
note first that in case when @xmath90 and @xmath91 generalized clar covers are the perfect matchings of @xmath24 and if @xmath0 is such a generalized clar cover then @xmath89 is a vertex of the resonance graph and the mapping is obviously bijective .
so from now on at least one of @xmath37 and @xmath42 will be positive .
we first show that @xmath76 is a well - defined function . for each generalized clar cover @xmath78 it follows that @xmath92 .
[ lema1 ] first we show that @xmath89 is isomorphic to the graph @xmath59 .
let @xmath19 , @xmath20,@xmath81,@xmath93 be the hexagons of @xmath0 and let @xmath94 be cycles @xmath38 that are in @xmath0 .
obviously , every hexagon of @xmath0 has two possible perfect matchings .
let us call these possibility 0 " and possibility 1 " .
moreover , for every cycle @xmath38 in @xmath0 we obtain tree possible perfect matchings of graph @xmath95 , which will be denoted as possibility 0 " , possibility 1 " , and possibility 2 " .
also , if cycle @xmath38 is composed of hexagons @xmath85 and @xmath86 , possibility 1 " denotes the perfect matching containing the common edge of @xmath85 and @xmath86 . for any vertex @xmath26 of @xmath89 let @xmath96 , where @xmath97 if on @xmath98 possibility @xmath50 is selected
it is obvious that @xmath99 is a bijection .
let @xmath100 for @xmath101 .
if @xmath26 and @xmath102 are adjacent in @xmath89 , then @xmath103 for a hexagon @xmath104 of some @xmath105 , where @xmath106
. therefore , @xmath107 for each @xmath70 and @xmath108 , which implies that @xmath109 and @xmath110 are adjacent in @xmath59 .
conversely , if @xmath111 and @xmath112 are adjacent in @xmath59 , it follows that @xmath26 and @xmath102 are adjacent in @xmath89 .
hence @xmath113 is an isomorphism between @xmath89 and @xmath59 . to complete the proof we have to show that @xmath89 is a convex subgraph of @xmath27
therefore , let @xmath26 and @xmath102 be two vertices of @xmath89 . obviously
, perfect matchings @xmath26 and @xmath102 can differ only in the edges of hexagons that belong to cycles of @xmath0 . therefore , any shortest path between @xmath26 and @xmath102 in @xmath27 contains perfect matchings that are vertices of @xmath89 .
it follows that @xmath89 is convex in @xmath27 .
the following lemma shows that @xmath76 is injective .
the mapping @xmath114 is injective for any integers @xmath115 .
[ lema2 ] let @xmath0 and @xmath116 be distinct generalized clar covers in @xmath74 .
if @xmath0 and @xmath116 contain the same set of cycles , then the isolated edges of @xmath0 and @xmath116 are distinct .
therefore , @xmath89 and @xmath117 are disjoint induced subgraphs of @xmath27 and thus @xmath118 . therefore , suppose that @xmath0 and @xmath116 contain different sets of cycles .
without loss of generality we can assume that there is hexagon @xmath104 such that @xmath104 has at least five edges in @xmath0 and @xmath104 has at most three edges in @xmath116 .
hence at least one edge @xmath119 of @xmath104 does not belong to @xmath116 . from the definition of the function @xmath76
, @xmath119 is thus unsaturated by those perfect matchings that correspond to the vertices in @xmath117 .
however , there obviously exists perfect matching @xmath120 such that @xmath121 . as a result , @xmath122 and @xmath123 .
the next lemma was proved in @xcite for benzenoid systems .
the same proof can be applied in the case of tubulenes or fullerenes .
@xcite [ stiri ] let @xmath24 be a benzenoid systems , a tubulene , or a fullerene with a perfect matching .
if the resonance graph @xmath27 contains a 4-cycle @xmath124 , then @xmath125 and @xmath126 are disjoint hexagons . also , we have @xmath127 and @xmath128 .
the following lemma shows that @xmath76 is surjective .
the mapping @xmath114 is surjective for any integers @xmath115 .
[ lema3 ] let @xmath115 be integers and @xmath129 .
then the vertices of @xmath130 can be identified with strings @xmath61 , where @xmath62 if @xmath63 or @xmath64 if @xmath65 , so that two vertices of @xmath130 are adjacent in @xmath130 if and only if their strings @xmath113 and @xmath131 differ in precisely one position @xmath50 , such that @xmath132 .
let @xmath133 , @xmath134 , @xmath135 , , @xmath136 be the vertices of @xmath130 .
it is obvious that @xmath137 is an edge of @xmath27 for every @xmath138 . by definition of @xmath27 , the symmetric difference of perfect matchings @xmath26 and
@xmath139 is the edge set of a hexagon of @xmath24 .
we denote this hexagon by @xmath140 and we obtain the set of hexagons @xmath141 of graph @xmath24 . if two of these hexagons were the same , for example if @xmath142 for @xmath143 and @xmath144 , then @xmath145 - a contradiction .
hence , we have the set of @xmath146 distinct hexagons . in the next claim
we show that these hexagons are pairwise disjoint .
the hexagons @xmath140 , @xmath106 , are pairwise disjoint .
let @xmath143 and @xmath144 .
let @xmath147 be a vertex of @xmath130 having exactly two @xmath148 s ( and these are in the @xmath50th and @xmath149th position ) and @xmath150 at every other position .
obviously , @xmath151 is a 4-cycle and therefore , by lemma [ stiri ] , @xmath140 and @xmath152 are disjoint hexagons .
next , we consider the vertices @xmath153 , @xmath154 , such that @xmath155 has @xmath156 in the @xmath50th position and @xmath150 in every other position . obviously , @xmath157 is the edge of @xmath27 for any @xmath154 .
let @xmath158 be the hexagon of @xmath24 corresponding to the edge @xmath157 . if @xmath154 , the hexagon @xmath158 has exactly one common edge with @xmath140 .
it is easy to see that @xmath159 ( otherwise @xmath160 ) .
therefore , suppose that @xmath140 and @xmath158 are disjoint .
since they are both sextets in the perfect matching @xmath139 , there is a vertex @xmath161 of @xmath27 , @xmath162 , which is adjacent to @xmath26 and @xmath155 . if @xmath163 , the string of @xmath161 must differ from @xmath26 for @xmath148 in exactly one position and must differ from @xmath155 for @xmath148 in exactly one position , which means @xmath164 - a contradiction
therefore , @xmath161 is not in @xmath130 .
since @xmath165 is a shortest path between @xmath26 and @xmath155 , @xmath130 is not convex subgraph of @xmath27 , which is a contradiction .
hence , @xmath140 and @xmath158 have exactly one common edge .
let @xmath154 .
then the hexagon @xmath158 is disjoint with every @xmath152 , @xmath166 .
let @xmath161 be a vertex in @xmath130 with @xmath156 in the @xmath50th position , @xmath148 in the @xmath149th position and @xmath150 in every other position .
furthermore , let @xmath167 be a vertex in @xmath130 with @xmath148 in the @xmath50th position , @xmath148 in the @xmath149th position and @xmath150 in every other position .
obviously , @xmath168 ( otherwise @xmath169 , which is a contradiction ) . since @xmath170 is a @xmath171-cycle such that @xmath158 corresponds to the edge @xmath157 and @xmath152 corresponds to the edge @xmath172 , it follows from lemma [ stiri ] that hexagons @xmath158 and @xmath152 are disjoint .
let @xmath154 .
then the hexagon @xmath158 is disjoint with every @xmath173 , @xmath174 .
define the following vertices in @xmath130 : * @xmath175 has @xmath148 in the @xmath50th position , @xmath148 in the @xmath149th position and @xmath150 in every other position , * @xmath176 has @xmath148 in the @xmath50th position , @xmath156 in the @xmath149th position and @xmath150 in every other position ,
* @xmath177 has @xmath156 in the @xmath50th position , @xmath148 in the @xmath149th position and @xmath150 in every other position , * @xmath178 has @xmath156 in the @xmath50th position , @xmath156 in the @xmath149th position and @xmath150 in every other position .
using lemma [ stiri ] we can easily see that hexagon @xmath173 corresponds to the edge @xmath179 and hexagon @xmath158 corresponds to the edge @xmath180 . since @xmath181 is a @xmath171-cycle in the resonance graph ,
lemma [ stiri ] again implies that @xmath158 and @xmath173 are disjoint and the proof is complete .
let @xmath182 , @xmath65 be a @xmath183-cycle formed by @xmath140 and @xmath158 .
moreover , let @xmath0 be a spanning subgraph of @xmath24 such that @xmath184 .
therefore , @xmath0 is a generalized clar cover with @xmath37 hexagons and @xmath42 10-cycles .
it is obvious that every edge in @xmath130 corresponds to some hexagon @xmath140 , @xmath185 or @xmath158 , @xmath65 .
therefore , @xmath186 .
since both @xmath130 and @xmath89 are induced subgraphs of the resonance graph , it follows @xmath187 .
we have proved that @xmath76 is bijective function and hence , @xmath188 .
therefore , the proof is complete .
in this final section we give an example of a benzenoid system @xmath24 and calculate the generalized zhang - zhang polynomial of @xmath24 , i.e. the generalized cube polynomial of the resonance graph of @xmath24 .
see figures [ ben_sistem ] and [ resonancni ] . .
] . ]
the polynomials are @xmath189 @xmath190 for example , the coefficient in front of @xmath191 is 3 , since there are 3 generalized clar covers in @xmath24 with two @xmath33 and one @xmath38 . on the other hand , this coefficient counts the number of induced convex subgraphs of @xmath27 isomorphic to the graph @xmath192 .
supported in part by the ministry of science of slovenia under grant @xmath193 .
chou , j .- s .
kang , h. a. witek , _ closed - form formulas for the zhang - zhang polynomials of benzenoid structures : prolate rectangles and their generalizations_. discrete appl . math . * 198 * ( 2016 ) , 101108 . d. plavi ' c , s. nikoli ' c , n. trinajsti ' c , _ the conjugated - circuit model : application to nonalternant hydrocarbons and a comparison with some other theoretical models of aromaticity_. j. mol . struct .
( theochem ) * 277 * ( 1992 ) , 213237 . | in this paper we study the resonance graphs of benzenoid systems , tubulenes , and fullerenes .
the resonance graph reflects the interactions between the kekul ' e structures of a molecule .
the equivalence of the zhang - zhang polynomial ( which counts clar covers ) of the molecular graph and the cube polynomial ( which counts hypercubes ) of its resonance graph is known for all three families of molecular graphs .
instead of considering only interactions between 6-cycles ( clar covers ) , we also consider 10-cycles , which contribute to the resonance energy of a molecule as well .
therefore , we generalize the concepts of the zhang - zhang polynomial and the cube polynomial and prove the equality of these two polynomials . | 1612.02986 |
the method of creative telescoping , also known as zeilberger s algorithm @xcite , is a powerful tool for solving problem involving definite integration and summation of hypergeometric function .
suppose we are given a certain holonomic function of two variables @xmath67 ( @xmath68 , @xmath69 ) , and it is required to prove that the summation of @xmath70 over @xmath32 equals to @xmath71 , @xmath72 the basic idea of creative telescoping algorithm is to find a linear recurrence equation for the summands @xmath73 .
this could be done by constructing a differential operator @xmath74 with coefficients being polynomials in @xmath75 , and a new function @xmath76 satisfying , @xmath77 thus @xmath78 operating on the summation @xmath79 is determined by the difference of upper bound and lower bound @xmath80 .
then we just need to check both sides of eq .
( [ eq9 ] ) satisfy recurrence equations : @xmath81 , @xmath82 , and check eq .
( [ eq9 ] ) holds for some initial conditions .
several algorithms for computing creative telescoping relations have been developed in the past @xcite .
the main programs are zeilberger s maple program and mathematica program written by peter paule and markus schorn @xcite . here
, we use the mathematical program to compute the creative telescoping relation for our problem .
in this section , we prove @xmath83 using the method of creative telescoping ( mct ) .
we use the mathematica package holonomic functions @xcite to create a recurrence relation for the summands @xmath84 in eq .
( [ eq5 ] ) , @xmath85 where @xmath86 , @xmath87 are the partial differential operator ( @xmath88 , @xmath89 ) , @xmath90 is the shift operator satisfying @xmath91 .
summing over @xmath32 leads to , @xmath92 the second term in the above equation is a telescoping series , the central terms are cancelled and only leave the last term and first term . noting that @xmath93 are zero for @xmath94 and @xmath95 , the second term in eq .
( [ eq12 ] ) equals to @xmath96 .
hence the infinite summation of return probabilities @xmath37 satisfies , @xmath97 it is easy to check @xmath98 also satisfies the above partial differential equation . combining with the initial condition @xmath83 for @xmath45 ( see eq .
( [ eq6 ] ) ) , @xmath83 holds for all @xmath2 and @xmath6 .
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arxiv:1004.3314 the holonomic package can be downloaded at + http://www.risc.uni-linz.ac.at/research/combinat/software/ | the recurrence properties of random walks can be characterized by plya number , _
i.e. _ , the probability that the walker has returned to the origin at least once . in this paper
, we consider recurrence properties for a general 1d random walk on a line , in which at each time step the walker can move to the left or right with probabilities @xmath0 and @xmath1 , or remain at the same position with probability @xmath2 ( @xmath3 ) .
we calculate plya number @xmath4 of this model and find a simple expression for @xmath4 as , @xmath5 , where @xmath6 is the absolute difference of @xmath0 and @xmath1 ( @xmath7 ) .
we prove this rigorous expression by the method of creative telescoping , and our result suggests that the walk is recurrent if and only if the left - moving probability @xmath0 equals to the right - moving probability @xmath1 .
random walk is related to the diffusion models and is a fundamental topic in discussions of markov processes .
several properties of ( classical ) random walks , including dispersal distributions , first - passage times and encounter rates , have been extensively studied .
the theory of random walk has been applied to computer science , physics , ecology , economics , and a number of other fields as a fundamental model for random processes in time @xcite .
an interesting question for random walks is whether the walker eventually returns to the starting point , which can be characterized by plya number , _
i.e. _ , the probability that the walker has returned to the origin at least once during the time evolution .
the concept of plya number was proposed by george plya , who is a mathematician and first discussed the recurrence property in classical random walks on infinite lattices in 1921 @xcite .
plya pointed out if the number equals one , then the walk is called recurrent , otherwise the walk is transient because the walker has a nonzero probability to escape @xcite . as a consequence , plya showed that for one and two dimensional infinite lattices the walks are recurrent , while for three dimension or higher dimensions the walks are transient and a unique plya number is calculated for them @xcite . recently ,
m. tefak _
et al .
_ extend the concept of plya number to characterize the recurrence properties of quantum walks @xcite .
they point out that the recurrence behavior of quantum walks is not solely determined by the dimensionality of the structure , but also depend on the topology of the walk , choice of coin operators , and the initial coin state , etc @xcite .
this suggests the plya number of random walks or quantum walks may depends on a variety of ingredients including the structural dimensionality and model parameters . in this paper
, we consider recurrence properties for a general one - dimensional random walk .
the walk starts at @xmath8 on a line and at each time step the walker moves one unit towards the left or right with probabilities @xmath0 and @xmath1 , or remain at the same position with probability @xmath2 ( @xmath3 ) .
this general random walk model has some useful application in physical or chemical problems , and some of its dynamical properties requires a further study .
previous studies of one - dimensional random walk focus on the simple symmetric case where the walker moves to left and right with equal probability ( @xmath9 ) @xcite .
for instance , plya showed that the symmetric random walk is recurrent and its plya number equals to 1 @xcite .
however , recurrence properties of this general random walk defined here are still unknown . as a consequence
, we will calculate the plya number for this general random model and discuss its recurrence properties .
we will try to derive an explicit expression for plya number , and reveal its dependence on the model parameters @xmath0 , @xmath1 and @xmath2 .
plya number of random walks can be expressed in terms of the return probability @xmath10 @xcite , _ i.e. _ , the probability for the walker returns to its original position @xmath8 at step @xmath11 , @xmath12 hence , the recurrence behavior of random walk is determined solely by the infinite summation of return probabilities .
it is evident that if the summation of return probabilities diverges the walk is recurrent ( @xmath13 ) , and if the summation converges the walk is transient ( @xmath14 ) . to calculate the plya number , it is crucial to obtain the return probabilities . in the following , we will calculate the return probabilities for our general random walk model .
the return probability @xmath10 can be obtained using the trinomial coefficients of @xmath15 .
considering an ensemble of random walks after @xmath11 steps , in which the walker has @xmath16 steps moving left , @xmath17 steps moving right and @xmath18 steps remaining at the same position , then the probability for such random walks is @xmath19 ( @xmath20 , @xmath21 ) . since the walker s position @xmath22 is only dependant on the difference of right - moving steps @xmath17 and left - moving steps @xmath16 , @xmath23 , returning to the original position @xmath8 requires @xmath24 .
therefore , the ensemble of random walks returning to @xmath8 involves sum over all possible @xmath18 subject to the constraints @xmath24 and @xmath25 . because @xmath26 is an even number , @xmath11 and @xmath18 must have the same parity . here , we suppose @xmath27 , @xmath28 for even @xmath11 and @xmath18 , and @xmath29 , @xmath30 for odd @xmath11 and @xmath18 ( @xmath31 and @xmath32 are nonnegative integers , and @xmath33 ) .
we calculate the return probability for even @xmath11 and odd @xmath11 separately . for even @xmath11 ,
the return probability is given by , @xmath34 where @xmath27 , @xmath28 , @xmath35 are used in the above equation .
analogously , for odd @xmath11 , the return probability is given by , @xmath36 the infinite summation of return probabilities @xmath37 can be determined by the sum of @xmath38 and @xmath39 , @xmath40 in order to get a simple expression for @xmath37 , we define @xmath41 , thus @xmath42 . substituting this relation into eq .
( [ eq4 ] ) , we get @xmath43 where @xmath44 is the gauss hypergeometric function .
@xmath37 can be further simplified , for the sake of clarity , we first consider the case @xmath45 .
when @xmath45 the hypergeometric function equals to 1 , @xmath37 can be simplified as , @xmath46 the last equality follows from the taylor series expansion at @xmath47 for the function @xmath48 . for @xmath49 , we find that @xmath37 also equals to @xmath50 .
this result is surprising because @xmath37 does not depend on the remaining unmoving probability @xmath2 .
this suggests that , for all @xmath2 and @xmath6 , eq . ( [ eq5 ] ) can be simplified as , @xmath51 it is difficult to simplify eq .
( [ eq5 ] ) or prove eq .
( [ eq7 ] ) using the usual mathematical methods . here ,
in the appendix , we prove this rigorous expression ( [ eq7 ] ) by the method of creative telescoping .
the method of creative telescoping @xcite is an algorithm to compute hypergeometric summation , definite integration , and prove combinatorial identity . using this method , we transfer @xmath37 to the solution of a partial differential equation ( see the proof in the appendix ) . the plya number in eq .
( [ eq1 ] ) can be written as , @xmath52 consequently , we find a simple explicit expression for plya number , which is solely determined by the absolute difference of @xmath0 and @xmath1 , @xmath7 . according to eq .
( [ eq8 ] ) , plya number @xmath4 equals to 1 for @xmath53 .
this suggests that the walk is recurrent if and only if the left - moving probability @xmath0 equals to the right - moving probability @xmath1 .
our result is consistent with previous conclusion that one - dimensional symmetric random walk ( @xmath9 ) is recurrent .
our result also indicates that the infinite summation of return probabilities @xmath37 diverges for @xmath53 and converges for @xmath54 . to verify this point , we plot the return probability @xmath10 as a function of step @xmath11 in fig .
[ fg1 ] .
we find that @xmath10 is a power - law decay as @xmath55 for @xmath53 ( see fig .
[ fg1 ] ( a ) in the log - log plot ) and exponential decay for @xmath54 ( see fig .
[ fg1 ] ( b ) , ( c ) in the log - linear plot ) .
since @xmath10 for @xmath53 decays slower than @xmath56 and decays faster than @xmath56 for @xmath54 , the infinite summation @xmath37 diverges for @xmath53 and converges otherwise .
particularly , by means of stirling s approximation @xmath57 for @xmath45 , we find an asymptotic form for the return probability in eq .
( [ eq6 ] ) : @xmath58 for even @xmath11 and @xmath59 for odd @xmath11 .
for a certain value of @xmath60 , the decay behavior of @xmath10 seems different for different values of @xmath2 ( see fig .
[ fg1 ] ( b ) , ( c ) ) .
however , the summations of @xmath10 for different @xmath2 are identical and equal to @xmath50 .
this result is some what unexpected and we provide a strict proof in the appendix . [ 0.25 ] as a function of step @xmath11 for @xmath53 ( a ) , @xmath61 ( b ) and @xmath62 ( c ) . for each value of @xmath6 ,
we plot @xmath10 vs @xmath11 for @xmath45 ( black squares ) , @xmath63 ( red dots ) and @xmath64 ( blue triangles ) . the critical decay for convergence @xmath65 are also plotted in the figure .
@xmath10 shows a power - law decay @xmath66 for @xmath53 ( see ( a ) ) , and @xmath10 exhibits exponential decay for @xmath60 ( see ( b ) and ( c ) ) .
it should be pointed out that for the case @xmath49 , @xmath10 is nonzero at all values of @xmath11 , while @xmath10 is zero at odd @xmath11 for @xmath45 .
[ fg1],title="fig : " ] in summary , we have studied recurrence properties for a general 1d random walk on a line , in which at each time step the walker can move to the left or right with probabilities @xmath0 and @xmath1 , or remain at the same position with probability @xmath2 ( @xmath3 ) .
we calculate plya number @xmath4 of this model for the first time , and find a simple explicit expression for @xmath4 as , @xmath5 , where @xmath6 is the absolute difference of @xmath0 and @xmath1 ( @xmath7 ) .
we prove this rigorous relation by the method of creative telescoping , and our result suggests that the walk is recurrent if and only if the left - moving probability @xmath0 equals to the right - moving probability @xmath1 .
we thank armin straub and dr .
koutschan for useful discussions .
this work is supported by national natural science foundation of china under project 10975057 , the new teacher foundation of soochow university under contracts q3108908 , q4108910 , and the extracurricular research foundation of undergraduates under project ky2010056a . | 1010.2014 |
let @xmath1 be an oval , i.e. , a planar , closed , regular , simple , oriented counterclockwise , @xmath2 curve , @xmath3 , parameterized by @xmath4 and such that its radius of curvature @xmath5 .
the billiard problem on @xmath1 consists in the free motion of a point particle on the planar region enclosed by @xmath1 , being reflected elastically at the impacts on the boundary .
the trajectories will be polygonals on this planar region .
since the motion is free inside the region , it is completely determined by the points of impacts at @xmath1 , and the direction of motion immediately after each reflection , defined by the angle @xmath6 between it and the oriented tangent to the boundary at the reflection point .
therefor , to each oval @xmath1 is associated a billiard map @xmath7 from the cylinder @xmath8 into itself which to each initial condition @xmath9 associates the next impact and direction : @xmath10 .
the map @xmath7 has some very well known properties ( see for instance @xcite , @xcite , @xcite , @xcite , @xcite ) : it is a @xmath11-diffeomorphism , preserving the measure @xmath12 , where @xmath13 is the arclength parameter for @xmath1 .
it is reversible with respect to the reversing symmetry @xmath14 , which satisfies @xmath15 and @xmath16 .
it is a monotone twist map .
so , the billiard map defines a discrete reversible conservative bidimensional dynamical system in the cylinder @xmath8 .
this map has no fixed points but birkhoff @xcite proved that it has periodic orbits of any period greater or equal to 2 . in this paper
we show that , under certain generic conditions , billiards on ovals have only a finite number of periodic orbits , for each period @xmath0 , all non - degenerate and at least one of them is hyperbolic .
moreover , the invariant curves of two hyperbolic points are transversal .
the generic existence , in the analytic case , of at least one nondegenerate periodic orbit for each period , was proved by kozlov and treschev @xcite .
our results generalize theirs in the sense that we work with curves at least @xmath17 , prove the finiteness and study the heteroclinic connections .
once we have a hyperbolic periodic orbit , which unstable manifold project to @xmath18 , we can construct the instability region associated to it from the closure of the invariant manifold .
the dynamical object we obtain is the same as the one constructed by le calvez in @xcite and thus shares the same dynamical properties .
billiard maps are a special kind of diffeomorphisms and @xmath19 perturbations of the map may produce diffeomorphisms which are not billiards . in order to assure that the perturbed diffeomorphism remains a billiard , we have to perturb the boundary curve instead of the map itself .
as planar rigid motions and homotheties do not change the geometrical features of the curve , they do not change the dynamical characteristics of the associated billiard map and so it is useful to work on the set of curves modulo this equivalence relation .
let @xmath20 be the set of equivalence classes of @xmath17 ovals .
given a representative @xmath1 of an equivalence class @xmath21\in{\cal c}$ ] , the normal bundle @xmath22 is @xmath19 . for @xmath23 , a tubular neighbourhood of @xmath1
is given by @xmath24 .
@xmath25\in{\cal c}$ ] is @xmath26-close to @xmath21\in{\cal c}$ ] if there exist representatives @xmath1 in @xmath21 $ ] and @xmath27 in @xmath25 $ ] such that the image of @xmath27 is in @xmath28 , and its canonical projection onto the image of @xmath1 is a diffeomorphism . as a consequence
, @xmath27 can be written as @xmath29 , with @xmath30 at least @xmath17 and periodic .
@xmath25\in{\cal c}$ ] is @xmath26-@xmath17-close to @xmath21\in{\cal c}$ ] if @xmath25 $ ] is @xmath26-close to @xmath21 $ ] and @xmath31 .
[ prop : baire ] @xmath20 is a baire space .
@xmath32 with the @xmath17-topology is a baire space . by sard
s theorem , the subset @xmath33 of immersions is open and dense in @xmath32 , and so also a baire space . for @xmath34 let @xmath35 be its radius of curvature on each @xmath36 and
let @xmath37 be the total curvature divided by @xmath38 .
let @xmath39 and @xmath40 such that @xmath41 and @xmath42 .
note that @xmath43 is exactly the set of closed , regular , simple , oriented , @xmath17-curves with positive radius of curvature .
it is clear that @xmath44 is closed and @xmath43 is open and dense in @xmath44 , so @xmath43 , the set of ovals , is baire with the @xmath17-topology .
cutting by the equivalence relation and taking the induced topology , we have that @xmath20 is a baire space . as we have showed in @xcite , by perturbing the classes of curves we get nearby diffeomorphisms in the @xmath19-topology : [ prop : difeo ] if @xmath25\in{\cal c}$ ] is @xmath26-@xmath17-close to @xmath21\in{\cal c}$ ] then there exist representatives @xmath1 in @xmath21 $ ] and @xmath27 in @xmath25 $ ] such that the associated billiard maps @xmath45 and @xmath46 are @xmath19-close . to lighten the notation , in what follows we will denote both the curve @xmath1 and the equivalence class @xmath21 $ ] by @xmath1 , unless where confusion may happen
let @xmath47 .
since the radius of curvature @xmath35 is strictly positive , @xmath1 can be reparameterized by @xmath48 , the positive oriented angle between the tangent vector @xmath49 and a fixed direction ( say , the x - axis ) .
let @xmath50 be the associated billiard map .
a point @xmath51 is @xmath52-periodic if @xmath52 is the smallest positive integer such that @xmath53 .
such a point defines an @xmath52-periodic orbit @xmath54 an @xmath52-periodic orbit is nondegenerate if each of its points @xmath55 is a nondegenerate fixed point of @xmath56 .
otherwise it is degenerate .
let @xmath57 the set of ovals @xmath58 such that for each @xmath59 divisor of @xmath0 , the associated billiard map @xmath7 has only nondegenerate @xmath52-periodic orbits .
if @xmath60 then for each @xmath59 divisor of @xmath0 , @xmath7 has only a finite number of @xmath52-periodic orbits .
let @xmath61 be an @xmath52-periodic orbit of @xmath45 .
then @xmath62 , @xmath63 , ... , @xmath64 are the vertices of a polygon inscribed in @xmath1
. let @xmath65 be its internal angles and @xmath66 if @xmath67 or @xmath68 if @xmath69 .
as @xmath70 , @xmath71 and then @xmath72 .
so there is at least one @xmath73 such that @xmath74 .
this means that if @xmath60 then each @xmath52-periodic orbit of @xmath7 , with @xmath59 divisor of @xmath0 , has at least one point in the compact cylinder @xmath75 $ ] .
since all the points of @xmath76 are nondegenerate , there is only a finite number of them on the compact cylinder and so , only a finite number of @xmath52-periodic orbits . for diffeomorphisms @xmath19 on @xmath77 , with the @xmath19-topology , having only a finite number of nondegenerate fixed points on a compact subset is an open property . taking the restriction to billiard maps , we get @xmath57 is open on @xmath78 . [ prop : open ] by birkhoff s theorem , any @xmath79 has periodic orbits of any period .
so @xmath1 will be outside @xmath57 if it has ( finitely or infinitely many ) degenerate @xmath52-periodic orbits for @xmath59 , divisor of @xmath0 .
the following lemma provides the basic tool for proving the density of @xmath57 .
[ lem : perturb ] suppose that @xmath80 is a degenerate @xmath52-periodic orbit for the billiard map @xmath7 associated to a curve @xmath79 . then there is a curve @xmath81 , @xmath17-close to @xmath1 , such that @xmath82 is a nondegenerate @xmath52-periodic orbit for the associated billiard map @xmath83 .
let @xmath84 and @xmath85 . the jacobian matrix of @xmath7 at @xmath86 is given by : @xmath87 and , for any point @xmath88 of @xmath89 , det@xmath90 .
it follows that @xmath89 is degenerate if @xmath91 , for any point of the orbit .
we have that @xmath92 where @xmath93 let us isolate the terms that depend on , for instance , @xmath94 . only @xmath95 and @xmath96 have entries with @xmath94 and @xmath97 then @xmath98\\ & = & \frac{1}{x_1}\mbox{tr}\left(a_{n-1}a_{n-2} ...
a_2b_1\right)+ \mbox{tr}\left(a_{n-1}a_{n-2} ...
a_2c_{1,0}\right)\\ & = & \frac{b_1}{x_1}+c_1\end{aligned}\ ] ] where neither @xmath99 nor @xmath100 depend on @xmath94 . if @xmath101 , let @xmath33 be an interval such that @xmath102 and @xmath103 for @xmath104 .
let @xmath105 , where @xmath30 is a @xmath17 periodic function satisfying @xmath106 if @xmath107 , @xmath108 , @xmath109 and @xmath110 small enough to guarantee that @xmath111 , the radius of curvature of @xmath112 , is strictly positive .
the perturbed curve @xmath112 and the original one @xmath1 coincide , except on a neighbourhood of @xmath113 and , at @xmath114 , they have a contact of order one .
so the polygonal trajectory that corresponds , on the billiard table , to the periodic orbit is unchanged and @xmath82 is also a @xmath52-periodic orbit for @xmath83 .
moreover @xmath115 , the radius of curvature @xmath116 if @xmath104 and @xmath117 .
then @xmath118 and we can choose @xmath30 as small as we want such that @xmath119 and @xmath82 is a nondegenerate @xmath52-periodic orbit for @xmath83 . if @xmath120 then @xmath121 with @xmath122 where @xmath123 has the same form of @xmath124 , replacing @xmath94 by @xmath125 and @xmath126 by @xmath127
. then @xmath128 where neither @xmath129 nor @xmath130 depend on @xmath94 and @xmath131 .
if @xmath132 then we can make the normal perturbation on a neighbourhood of @xmath133 as above .
if @xmath134 we continue the process until finding a @xmath135 and then making the normal perturbation at @xmath136 or to end up with all @xmath137 s @xmath138 , for @xmath139 in which case , as @xmath140 , we will have @xmath141 as @xmath142 is the perimeter of the polygonal trajectory and then different from 0 , we can perform the normal perturbation on a neighbourhood of @xmath143 as above concluding the proof of the lemma .
@xmath57 is dense on @xmath78 .
[ prop : dense ] given @xmath144 , let @xmath145 be the set of all fixed points of @xmath146 and @xmath147 be the projection on the first coordinate . as @xmath144 ,
there is a @xmath148 such that @xmath149 is a degenerate @xmath52-periodic orbit of @xmath7 for @xmath59 , divisor of @xmath0 . by lemma [ lem : perturb ]
we can find a curve @xmath150 , close to @xmath1 , such that @xmath149 is a non - degenerate @xmath52-periodic orbit of @xmath151 and so there are intervals @xmath152 such that @xmath153 is the unique point of @xmath154 in @xmath155 .
as @xmath18 is compact , we can construct after a finite number of steps , a curve @xmath156 , as close as we want to @xmath1 , such that @xmath157 and intervals @xmath158 , with @xmath159 , and such that each @xmath160 is the unique point of @xmath161 in @xmath162 and so each fiber @xmath163 contains at least one nondegenerate fixed point and maybe other degenerate or nondegenerate fixed points .
if @xmath164 has only non - degenerate fixed points , the problem is solved .
otherwise , let @xmath165 and @xmath166 be two different fixed points of @xmath164 , with @xmath165 non - degenerate and @xmath167 $ ] degenerate .
if @xmath168 then they are both non - degenerate , so suppose that they do not belong to the same orbit .
let @xmath52 and @xmath169 be the periods of the orbits , respectively
. then @xmath170 and @xmath171 . as in the proof of lemma [ lem : perturb ]
, there is an @xmath172 such that @xmath173 with @xmath174 and then a normal perturbation in the interval containing @xmath175 will produce a curve @xmath27 , @xmath17-close to @xmath112 , such that both of @xmath165 and @xmath166 are nondegenerate fixed points of @xmath176 .
to finish the proof of the proposition , we remark that @xmath177 $ ] is a closed interval .
so , after a finite number of perturbations we can construct a curve in @xmath57 , arbitrarily close to the given @xmath178 .
taking the intersection of the open and dense sets @xmath57s and remembering that @xmath20 is a baire space it follows that generically , for billiards on ovals , there is only a finite number of periodic orbits , for each period @xmath0 , and they are all non - degenerate .
let @xmath1 be a@xmath2 oval , @xmath7 the associated billiard map and @xmath179 a lift of @xmath7 .
if @xmath180 is an @xmath52-periodic orbit of @xmath7 then there is an @xmath169 , @xmath181 such that @xmath182 .
we will say that @xmath183 is an @xmath184-periodic orbit .
given @xmath52 and @xmath169 , @xmath181 , let @xmath185 @xmath180 is a non - degenerate @xmath184-periodic orbit if and only if @xmath186 , @xmath187 , @xmath188 is a non - degenerate singularity of @xmath189 . by the mackay - meiss criterion @xcite , non - degenerate minima of @xmath189 correspond to hyperbolic orbits and non - degenerate maxima to elliptic or reverse hyperbolic ones .
if @xmath60 then all the @xmath52-periodic orbits , @xmath59 divisor of @xmath0 , are nondegenerate singularities for the appropriate @xmath189 and then are hyperbolic or elliptic .
actually , for each @xmath52 and each @xmath169 such that gcd@xmath190 , the first step in the proof of birkhoff s theorem is the existence of a global minimum of @xmath189 .
so there is at least one hyperbolic @xmath52-periodic for @xmath7 .
the stability of the other periodic orbits is strongly related to the geometry of the oval @xmath1 , and they can even be all hyperbolic , like the 2-periodic orbits in the examples given in @xcite or @xcite .
let @xmath191 be one @xmath52-periodic hyperbolic orbit .
associated to each point @xmath55 there are two @xmath11 invariant curves @xmath192 and @xmath193 called , respectively , unstable and stable curves of @xmath55 .
a point @xmath194 is hetero(homo)clinic if @xmath195 for @xmath196 ( @xmath197 ) .
for the hyperbolic periodic orbits corresponding to global minima of @xmath189 , bangert s results @xcite assures the existence of hetero and homoclinic points .
other hyperbolic periodic orbits may have only homoclinic points , as the 2-periodic orbit plotted in the figure bellow .
we do nt know if , for a generic oval @xmath1 there exists a hyperbolic periodic orbit without hetero or homoclinic points .
in particular , @xmath198 is an open property for @xmath19-diffeomorphisms , and so will be open for billiards on ovals .
an heteroclinic or homoclinic point @xmath194 is called transversal ( tangent ) if the invariant stable and unstable curves meet transversally ( tangentially ) at @xmath194 .
transversal intersection of stable and unstable curves is also an open property for @xmath19-diffeomorphisms , and so it will be open for billiards on ovals . in the next lemma
we will prove that any billiard with a tangent hetero(homo)clinic point can be approached by billiards with a transversal one .
we use the techniques introduced in @xcite and used in @xcite .
[ lem : levallois ] let @xmath60 and @xmath199 be an @xmath52-periodic hyperbolic orbit such that a stable and an unstable curve , say @xmath200 and @xmath201 intersect tangentially at @xmath194
. then @xmath1 can be approximated by curves in @xmath57 such that @xmath194 is a transverse heteroclinic ( or homoclinic ) point of the associated billiard map .
since @xmath202 there are sequences @xmath203 and @xmath204 as @xmath205 .
then , there exists an interval @xmath206 such that @xmath207 , @xmath208 and @xmath209 . as
@xmath7 is a @xmath11-diffeomorphism , each @xmath210 is also a heteroclinic tangent point for every @xmath211 .
moreover , as @xmath7 has the twist property , it is not possible to the tangency at every @xmath212 to be vertical .
so we can suppose that the stable and unstable curves are local graphs over the @xmath213-axis at a neighbourhood of , for instance , @xmath194 .
so there is an interval @xmath214 containing @xmath215 such that @xmath200 and @xmath201 are given locally by the graphs of @xmath216 and @xmath217 , with @xmath218 and @xmath219 .
those graphs define two pencils of rays that focuses forward and backward at the distances ( see , for instance @xcite ) @xmath220 where @xmath221 is the radius of curvature of @xmath1 at @xmath215 .
let @xmath222 and @xmath112 be a normal perturbation @xmath223 where @xmath106 if @xmath107 , @xmath224 , @xmath225 and @xmath226 is sufficiently small in order that @xmath227 and @xmath7 and @xmath83 are @xmath19-close .
the two curves @xmath1 and @xmath112 have a contact of order 1 at @xmath215 and the radius of curvature of @xmath112 at this contact point is @xmath228 . as @xmath229 and @xmath1 differs only on @xmath230 , @xmath231 is also a @xmath52-periodic hyperbolic orbit for @xmath83 .
moreover , every trajectory not hitting @xmath232 is the same both for @xmath1 and for @xmath112 .
let @xmath233 and @xmath234 be the stable and unstable curves of @xmath235 and @xmath55 under @xmath83 .
we can choose @xmath226 sufficiently small such that both are also local graphs over @xmath33 given by @xmath236 and @xmath237 , where @xmath238 is the angular parameter of @xmath229 with @xmath239 .
the pencil of rays defined by @xmath240 will focuses backward at the distance @xmath241 and the pencil @xmath242 will focuses forward at the distance @xmath243 with @xmath244 as the contact of @xmath1 and @xmath112 on @xmath215 is of order 1 , preserving the point and the tangent , the trajectory of @xmath194 is also the same for both billiards .
then @xmath194 is a heteroclinic point for @xmath83 and @xmath245 .
as the curve is unchanged outside @xmath33 , the beam of trajectories given by @xmath240 remains the same until it hits @xmath246 , implying that @xmath247 and then @xmath248 applying the same construction for the stable curves , with @xmath249 and @xmath250 gives @xmath251 and @xmath252 implies that the invariant curves for @xmath83 will intersect transversally .
the same reasoning also works for the invariant curves of points on different hyperbolic orbits .
as we have only a countable number of hyperbolic orbits , each one with a finite number of points , we can conclude that : generically , for billiards on ovals , the invariant curves of two hyperbolic points are transversal .
we remark that we do not prove that every homo / heteroclinic orbit is transversal .
we do know that generically two invariant stable and unstable curves either do not intersect or have at least one transversal homoclinic orbit , but there can also be tangent orbits .
a closed , simple , continuous curve @xmath253 which is not homotopic to a point is called a rotational curve .
it is invariant if @xmath254 .
the phase - space of the circular billiard , for instance , is foliated by rotational invariant curves . on the other hand ,
there are billiards on ovals with no rotational invariant curves at all , as showed in @xcite .
however , for sufficiently differentiable ovals ( @xcite , @xcite ) , the twist property implies that @xmath7 has rotational invariant curves in any small neighbourhood of the boundaries @xmath255 and @xmath256 of the cylinder @xmath77 . for generic oval billiards
the rotation number of any rotational invariant curve is irrational .
a rotational invariant curve @xmath257 is a lipschitz graph over @xmath258 @xcite .
so @xmath259 , where @xmath6 is continuous and there exists @xmath260 such that @xmath261 , @xmath262 .
let @xmath263 and @xmath264 .
then @xmath265 is a homeomorphism of the circle and so its degree is @xmath266 .
suppose that the rotation number of @xmath257 is rational .
if deg@xmath267 then @xmath265 has periodic orbits , all with the same period . if deg@xmath268 then @xmath269 has periodic orbits , all with the same period . as @xmath7 is generic , there is only a finite number of periodic orbits , for each period , and they are nondegenerate .
hence there exists an @xmath0 such that @xmath270 is a homeomorphism with a finite number of fixed points , all nondegenerate .
let @xmath271 be two consecutive fixed points .
then for every @xmath272 , @xmath273 , @xmath274 as @xmath275 and @xmath276 as @xmath277 .
let @xmath23 be such that @xmath278 and let @xmath279 and @xmath280 .
clearly @xmath281 .
let us suppose that @xmath282 is an elliptic periodic point of @xmath7 .
then @xmath283 is a rotation of angle @xmath284 .
there exists @xmath285 such that @xmath286 .
as @xmath7 is at least @xmath19 then there is @xmath287 such that @xmath282 is the unique fixed point of @xmath288 in @xmath289 and @xmath290 .
but this is impossible since @xmath291
. then all periodic points in @xmath257 must be hyperbolic .
but this is also impossible in the generic case since @xmath257 will be the union of the periodic points and saddle connections .
so , the rotation number must be irrational . from the proof of the above proposition we have the following results : [ lem : irrac ] generically
, rotational invariant curves can not cross the middle segment @xmath292 .
let , as before , @xmath293 be the reversing symmetry .
clearly @xmath257 is a rotational invariant curve if and only if @xmath294 is also a rotational invariant curve .
suppose that a point @xmath295 .
then @xmath296 and , since they are invariant , @xmath297 and @xmath298 is a 2-periodic orbit in @xmath257 , which is impossible .
then , for a generic oval billiard , if @xmath299 and @xmath300 ( resp .
@xmath301 ) then @xmath302 ( resp .
@xmath303 ) for all @xmath48 or , in other words , the orbits on @xmath257 respect the order of the circle @xmath258 ( respec . reverse the order ) .
[ lem : rot ] let @xmath257 be a rotational invariant curve and @xmath304 be a hyperbolic periodic orbit , with unstable manifold @xmath305 and stable manifold @xmath306
. then @xmath307 and @xmath308 .
suppose that @xmath309 .
then @xmath310 and then @xmath311 , since @xmath257 is a continuous invariant curve . but @xmath257 has irrational rotation number and then do not contain any periodic orbit .
the argument for @xmath305 is analogous .
birkhoff @xcite called the region between two invariant rotational curves , with no other invariant rotational curves inside , an instability region . in this subsection we will characterize the instability regions for a generic oval billiard . a cylinder @xmath312 $ ] is a non - empty closed connected set such that @xmath313 and whose boundaries are two continuous rotational curves @xmath314 and @xmath315 .
note that we are not asking neither the cylinder nor the boundary curves to be invariant under @xmath45 .
[ teo : inst ] let @xmath7 be generic and @xmath304 be a hyperbolic periodic orbit such that its unstable manifold @xmath305 satisfies @xmath316 .
then the smallest cylinder containing @xmath317 is an instability region . before proving this theorem ,
let us remark that for a generic oval billiard and for each period @xmath52 , there is only a finite number of @xmath52-periodic hyperbolic orbits and at least one of them is a global minimizer of the action @xmath189 , for a suitable @xmath169 .
this implies that it has heteroclinic connections @xcite and then @xmath316 .
hence the projection hypothesis applies to at least one orbit of each period .
let @xmath318 be a hyperbolic periodic point , @xmath304 its orbit and @xmath305 the unstable manifold of @xmath304 , with @xmath316 .
first , assume that there is an invariant rotational curve @xmath259 such that @xmath304 lies in the cylinder @xmath319 bounded by @xmath257 and @xmath294 . since , by lemma [ lem : irrac ] , @xmath257 has irrational rotation number we have that @xmath320 and @xmath321 .
moreover , @xmath319 is invariant and contains @xmath317 .
let then , @xmath322 be the smallest cylinder containing @xmath317 . since @xmath323 is also a cylinder and @xmath324 , @xmath325 , and we have that @xmath326 .
so , @xmath327 is also a cylinder containing @xmath317 . and
as @xmath328 is the smallest cylinder , we must have @xmath329 .
however , @xmath45 is area - preserving , so we conclude that @xmath330 , i.e , @xmath328 is an invariant cylinder . as @xmath45 is a diffeomorphism , @xmath331 and @xmath332 . from birkhoff
s theorem it follows that @xmath333 is the union of two rotational invariant curves @xmath100 and @xmath130 .
in fact , these two curves @xmath100 and @xmath130 are contained in @xmath317 . to prove this fact ,
let us suppose that there is a point @xmath334 .
then there is an open ball @xmath43 , centered at @xmath335 such that @xmath336 .
we can then construct a new cylinder @xmath337 with @xmath338 where @xmath339 outside @xmath340 and @xmath341 otherwise .
we have then a new cylinder satisfying @xmath342 and so @xmath328 is not the smallest one , which is a contradiction .
now , as @xmath305 is the union of continuous curves , its closure contains the boundaries @xmath100 and @xmath130 and @xmath307 for every @xmath257 rotational and invariant , it follows that there is no invariant curve in @xmath343 .
we have proved that @xmath328 is an invariant cylinder , its boundaries are rotational invariant curves and there are no other rotational invariant curves inside it .
so , it is the instability region containing @xmath317 .
suppose now that there is no invariant rotational curves curves bounding @xmath305 from bellow , i.e , @xmath344 for all rotational invariant curves @xmath259 .
we can extend @xmath7 continuously to the circle @xmath255 , observing that @xmath345 .
now , any cylinder with boundaries @xmath346 and a rotational invariant curve contains @xmath317 .
the smallest one , @xmath328 , will have boundaries @xmath346 and @xmath130 , because if there were a new boundary @xmath100 then it will be a rotational invariant curve with @xmath347 .
reasoning as above we conclude that @xmath328 is the instability region that contains @xmath317 .
the case @xmath348 for all rotational invariant curves follows from the reversibility property of @xmath7 .
finally , if @xmath45 has no rotational invariant curves , the instability region containing @xmath317 is the entire cylinder @xmath349 $ ] .
obviously , we can make a similar construction and obtain the same result for the stable manifold @xmath306 .
[ prop : ilha ] let @xmath350 be the complement of @xmath317 in the instability region @xmath328
. then @xmath351 , where all the @xmath352 s are disjoint open sets , homeomorphic to discs .
dynamically , each @xmath352 returns over itself .
@xmath350 is open and is the union of its disjoint open connected components .
let @xmath353 be one of them .
@xmath353 can not divide @xmath328 into two disjoint sets because @xmath354 contains @xmath305 , which projects over @xmath258 and converges to the two rotational curves of @xmath333 .
so @xmath353 is homotopically trivial . as @xmath355 $ ] has finite area and each connected component has positive area .
so there can not exist a non countable number of them . clearly , @xmath356 . as @xmath45 is area preserving then ,
given @xmath352 , there exists a smallest @xmath357 such that @xmath358 .
but , as each @xmath352 is a connected component @xmath350 , we must have @xmath359 . in other words ,
each component @xmath352 of @xmath360 is periodic .
the above proposition suggests the definition given @xmath211 , let @xmath357 be the smallest integer such that @xmath359 .
we call @xmath352 an island of stability and @xmath357 its period
. the invariant set @xmath361 will be called an archipel .
remark that , as each island @xmath352 is invariant by @xmath362 and homeomorphic to a ball then the boundary @xmath363 is a closed connected set , with empty interior , contained in @xmath317 and invariant by @xmath362 .
an easy consequence of proposition [ prop : ilha ] is every periodic point in an island of stability has a period multiple of the period of the island .
more strongly , we have that each island of stability contains a periodic point with the same period as the island .
let @xmath352 be an island of stability with period @xmath357 .
since the restriction @xmath364 is area preserving and conjugated to a diffeomorphism of the plane which is orientation preserving , it follows from brouwer s translation theorem that @xmath364 has a fixed point .
so there exists a point @xmath365 such that @xmath366 and so a periodic orbit @xmath367 with the smallest possible period .
remark that , as billiards have no fixed points then @xmath368 and @xmath369 .
all points in an island of stability have the same rational rotation type .
[ prop : rotnum ] let , as above , @xmath352 be an island of stability with period @xmath357 . following @xcite we observe that if @xmath370 is the lift of @xmath352 to the universal cover and @xmath371 is a lift of @xmath7 , then there exists an integer @xmath169 such that @xmath372 .
since @xmath370 is bounded , this implies that that for all @xmath373 if @xmath374 denotes a lift of @xmath375 , then there exists @xmath376 , where @xmath377 is the projetion on the first factor . for the sake of completeness , we conclude by describing some well known dynamical consequences of the area preserving and the twist properties .
let @xmath7 be generic , in the sense of the previous sections , and @xmath304 be a hyperbolic periodic orbit such that its unstable manifold @xmath305 satisfies @xmath316 .
let @xmath328 be the instability region containing @xmath317 .
following the terminology of franks in @xcite we call @xmath378 the instability set . clearly , the instability set @xmath354 does not contain any moser stable periodic orbit , since they have open islands around them . for hyperbolic periodic orbits we have let @xmath379 be a hyperbolic periodic orbit in @xmath354 and let @xmath380 and @xmath381 be its invariant stable and unstable manifolds . then either @xmath382 and @xmath383 or @xmath384 , ie , the boundary of an island . moreover , for @xmath385 or @xmath386 , if @xmath387 and @xmath388 then @xmath389 ; or if @xmath384 then two branches of the invariant curves of @xmath390 and/or @xmath391 are contained in @xmath363 .
suppose that there exists @xmath392 such that @xmath393 , @xmath394 .
then by lemma 3.1 of @xcite , the branch of @xmath395 which contains @xmath375 is entirely contained in @xmath378 .
so if each branch of @xmath395 intersects @xmath378 then @xmath396 and obviously , @xmath397 .
let us now suppose that a branch @xmath257 of @xmath395 is not in @xmath378 .
then @xmath257 is contained in an island of stability @xmath352 or is outside the region of instability @xmath328 . as the @xmath398-limit of @xmath257 is @xmath399 , either @xmath384 or @xmath400 or @xmath130 , the boundary curves of @xmath328 .
but this last case is impossible in the generic case .
so @xmath384 .
but in this case , it is enough to remark that @xmath363 is an invariant closed connected set and that the only sets with these properties containing @xmath399 are the invariant manifolds to conclude that two branches of the invariant curves of @xmath390 and/or @xmath391 are contained in @xmath363 . if @xmath388 then @xmath401 and , by theorem [ teo : inst ] , @xmath389 .
concerning non - periodic orbits and as a consequence of proposition [ prop : rotnum ] we have that any aubry- mather set of irrational type contained in @xmath328 is actually contained in the instability set @xmath378 . and
as it was proven by le calvez @xcite the instability set @xmath378 contains the closure of the orbits which @xmath1 and/or @xmath402-limits are @xmath100 and @xmath130 .
so far we have obtained a rough description of the dynamics in the instability region , very similar to what hapens for generic area preserving twist maps .
it is made of the closure of an hyperbolic orbit and the union of periodic islands . in order to proceed with this description
it is necessary to address two basic related themes . the first one has to do with the islands : the existence of a finite number of them , the existence of a lower bound for the period of an island ( e.g. there are examples with no period two islands ) and the existence of connecting orbits between the boundaries of the islands .
a very interesting question is if it is possible to have an instability region without any islands .
a starting point is the instability region that contains the period two orbits . in this region ,
the reversibility symmetry of the phase space , together with geometric properties of the boundary of the billiard may allow us to obtain new results about the above questions .
* acknowledgments * : the authors thank conselho nacional de desenvolvimento cientfico ( cnpq ) and fundao de amparo a pesquisa de minas gerais , brazilian agencies , for financial support .
spc gratefully acknowledges the hospitality of laboratoire emile picard , universit paul sabatier ( toulouse iii ) , where part of this work was done , under the financial support of coordenao de aperfeioamento de pessoal de nvel superios ( capes ) .
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s.tabachnikov : _ billiards _ , s m f panorama et syntheses 1 , 1995 . | in this paper we show that , under certain generic conditions , billiards on ovals have only a finite number of periodic orbits , for each period @xmath0 , all non - degenerate and at least one of them is hyperbolic .
moreover , the invariant curves of two hyperbolic points are transversal .
we explore these properties to give some dynamical consequences specially about the dynamics in the instability regions . | 0705.0948 |
a useful formulation of gauge theories , both from the conceptual and methodological point of view , is the one in terms of gauge invariant excitations or string - like objects .
the so - called _ p - representation _
@xcite , consisting of a hilbert space of path labeled states , has been used on the lattice to perform analytical hamiltonian calculations . a cluster approximation allowed to provide qualitatively good results for the @xmath0 qed @xcite and the @xmath1 qed @xcite with staggered fermions . a description in terms of paths or strings ,
besides the general advantage of only involving gauge invariant excitations , is appealing because all the gauge invariant operators have a simple geometrical meaning when realized in the path space .
however , the computational method implemented , up to now , on a formally _ infinite _ lattice , has the serious drawback of the explosive proliferation of clusters with the order of the approximation . in order to tackle this difficulty we propose in this paper to explore the previous method implemented now on a _
finite lattice_. as a first test , we choose the simplest lattice gauge theory with dynamical fermions , the schwinger model or ( 1 + 1 ) qed .
this massless model can be exactly solved in the continuum and it is rich enough to share relevant features with 4-dimensional qcd as confinement or chiral symmetry breaking with an axial anomaly @xcite .
for this reason it has been extensively used as a laboratory to analyze the previous phenomena .
the lattice schwinger model also become a popular benchmark to test different techniques to handle dynamical fermions @xcite-@xcite .
this article is organized into four sections . in section 2
we show the formulation of the model in the p - representation .
the electric and interaction components of the hamiltonian operator are realized in this basis of electromeson " states . in section 3 ,
first , we describe the finite lattice hamiltonian approach .
second , we show the calculation of the ground - state energy , the mass gap and chiral condensate . these results are discussed in the concluding section .
the p - representation offers a gauge invariant description of physical states in terms of kets @xmath2 , where @xmath3 labels a set of connected paths @xmath4 with ends @xmath5 and @xmath6 in a lattice of spacing @xmath7 . in the continuum ,
the connection between the p - representation and the ordinary representation ( configuration " representation ) , in terms of the fermion fields @xmath8 and the gauge fields @xmath9 $ ] , can be performed considering the natural gauge invariant object constructed from them : @xmath10 where @xmath11 $ ] ( @xmath12 denote the links ) .
the immediate problem we face is that @xmath13 is not purely an object belonging to the `` configuration '' basis because it includes the canonical conjugate momentum of @xmath8 , @xmath14 .
the lattice offers a solution to this problem consisting in the decomposition of the fermionic degrees of freedom .
let us consider the hilbert space of kets @xmath15 , where @xmath16 corresponds to the @xmath17 part of the dirac spinor and @xmath18 to the @xmath19 part .
those kets are well defined in terms of `` configuration '' variables ( the canonical conjugate momenta of @xmath20 and @xmath21 are @xmath22 and @xmath23 respectively . ) then , the internal product of one of such kets with one of the path dependent representation ( characterized by a lattice path @xmath4 with ends @xmath5 and @xmath6 ) is given by @xmath24 @xmath25 where @xmath26 and @xmath27 denote a component of the spinor @xmath16 and @xmath18 respectively .
thus , it seems that the choice of staggered fermions is the natural one in order to build the lattice p - representation .
therefore , the lattice paths @xmath4 start in sites @xmath5 of a given parity and end in sites @xmath6 with opposite parity . the one spinor component at each site can be described in terms of the susskind s @xmath28 single grassmann fields @xcite .
the path creation operator @xmath29 in the space of kets @xmath30 of a path with ends @xmath5 and @xmath6 is defined as @xmath31 its adjoint operator @xmath32 acts in two possible ways @xcite : annhilating the path @xmath33 or joining two existing paths in @xmath34 one ending at @xmath5 and the other starting at @xmath6 .
the schwinger hamiltonian is given by @xmath35 @xmath36 @xmath37 @xmath38 where @xmath5 labels sites , @xmath39 the spatial links pointing along the spatial unit vector @xmath40 , @xmath41 is the electric field operator , the kets @xmath42 are eigenvectors of this operator @xmath43 where the eigenvalue @xmath44 is the number of times that the link @xmath45 appears in the set of paths @xmath3 .
the @xmath46 are displacement " operators corresponding to the quantity defined in ( [ eq : phi ] ) for the case of a one - link path i.e. @xmath47 . the realization of both hamiltonian terms in this representation is as follows @xcite : by ( [ eq : e ] ) the action of the electric hamiltonian is given by @xmath48 the interaction term @xmath49 is realized in the loop space as @xmath50 where the factor @xmath51 is 0 or @xmath52 dictated by the algebra of the operators .
the different actions of operators @xmath46 over path - states @xmath53 and their corresponding @xmath51 are schematically summarized in fig.1
. -6 mm -5 mm
our method of calculation works assuming a lattice of some fixed even number of sites @xmath54 and periodic boundary conditions ( pbc ) . starting with the zero - path state @xmath55 ( infinite coupling vacuum ) , then a collection of new states @xmath56 are generated by applying successively the non - diagonal @xmath57 interaction hamiltonian operator whose action is to add or to eliminate links to to the path @xmath58 as it was described in the previous section up to order @xmath59 .
the traslational symmetry can be exploited in order to reduce the dimension of the space tacking only one representative @xmath60 of each class of translationally equivalent paths @xmath61 . the hamiltonian matrix , with all the transitions between the different states @xmath62 ,
is then built for the scalar and vector sectors and their eigenvalues @xmath63 are numerically evaluated . in order to perform the generation and recognition of diagrams ( the elementary lattice paths ) as well as the computation of transitions between them
, we resorted to the prolog language which is very suitable to carry out the symbolic manipulations .
the calculations of the ground - state energy , mass gap and chiral order parameter were performed on lattices ranging from size @xmath64 to @xmath65 and at least up to order @xmath66 in each case .
our aim is to extrapolate these results to the continuum limit : @xmath67 , @xmath68 ( @xmath69 . )
it is clear from the plots ( figs . 2 to 5 ) that the lattice results show convergence to the expected continuum values .
this convergence is , however , non - uniform and for @xmath70 large enough the plots show deviation from the continuum values although the region of assimptotic regime becomes larger when the size is increased .
it is patent that for a fixed lattice size @xmath54 the best results for the vacuum energy and the chiral condensate are achieved for order @xmath71 .
this appears to be the order at which the finite size effects are minimized .
this is not the case with the mass gap which always gets closer to the continumm value when the order increases .
_ ground state energy _ in the continuum limit
the ground - state energy density is known exactly @xcite : @xmath72 when the order increases @xmath73 tends to a fixed value . for a fixed size @xmath54 the closer value to ( [ eq : e - exact ] ) is given by order @xmath71 . the value for size @xmath65 and
order @xmath74 at @xmath75 is @xmath76 , so the discrepancy from the exact value is less than 0.05 % .
the approximations converge with considerable rapidity .
2 shows @xmath77 for orders @xmath78 for @xmath70 ranging from 0 to 100 on a lattice of size @xmath79 .
-13 mm -10 mm in order to obtain a result in a consistent way we compute the energy for two large values of @xmath70 , for three correlative large orders and for three correlative large sizes .
then , for fixed size and order we first extrapolate to @xmath80 assuming the behaviour @xmath81 .
second , for fixed size we extrapolate to infinite order assuming exponential dependence .
finally we extrapolate to infinite size assuming exponential behaviour .
the results are given in table 1 .
the error using lattice sizes up to @xmath65 is 0.17% .
@l@rrrr & & & & + @xmath82 & @xmath83 & @xmath84 & @xmath85 & + @xmath86 & @xmath87 & @xmath88 & @xmath89 & + @xmath90 & @xmath91 & @xmath92 & @xmath93 & + @xmath94 & @xmath95 & @xmath96 & @xmath97 & @xmath98 + _ mass gap _ the mass gap for the massless continuum schwinger model can be computed exactly @xcite : @xmath99 the lattice mass gap is computed as : @xmath100 comparing our results with those of crewther and hamer @xcite obtained by a similar method , although they use a different representation ( jordan - wigner transformation ) , we find complete agreement for given values of @xmath54 and @xmath59 .
when we reach larger @xmath54 we observe that the value of the mass gap improves substantially .
for instance , in fig .
3 we show a plot of the mass gap for @xmath101 for several orders . as it can be seen in the region @xmath102 , the mass gap values decrease with the order and the size approaching the continuum result . given the non - uniformity of the convergence it is more difficult to extrapolate to the limit although values @xmath103 are obtained at the modest size of @xmath104 .
_ chiral order parameter _
an interesting quantity to compute is the vacuum expectation of the chiral condensate per - lattice - site @xmath105 , defined as @xmath106 , \label{eq : chir - cond}\ ] ] where @xmath107 is the number of lattice sites .
the corresponding operator is realized in the p - representation and thus we get for the chiral condensate : @xmath108 where @xmath109 is the number of connected paths in @xmath3 . to compute @xmath105 the @xmath110 hamiltonian is modifyed as @xmath111 , \label{eq : w'}\ ] ] where @xmath112 is an arbitrary parameter .
thus , @xmath105 is obtained in the standard way as @xmath113 the massless continuum schwinger model undergoes a breaking of chiral symmetry with @xmath114 where @xmath115 is the euler constant .
this non - zero value of the chiral condensate is one of the main efects of the axial anomaly . in fig .
4 we report the value of the chiral condensate per - lattice - site for lattice sizes ranging from @xmath64 to @xmath101 . fig .
5 shows this chiral order parameter for different lattice sizes up to order @xmath90 for each size .
notice that the results in the weak coupling region converge to the corresponding continuum value ( [ eq : chiral - cont ] ) as long as @xmath54 increases while for a fixed @xmath54 the value improves with the order @xmath59 till the value @xmath90 is reached .
our general proposal is to to show that the p - representation is a valuable and alternative computational tool for gauge theories with dynamical fermions . in particular , in this work , we wanted to test the hamiltonian approach on finite lattices . with tis aim , we chose the simplest model : ( 1 + 1 ) qed .
this also enables us to compare with the corresponding numerical simulations @xcite using the lagrangian counterpart of the p - representation or the socalled _ worldsheet formulation _ @xcite .
this comparison shows that , for this case of one spatial dimension , the hamiltonian method is less time consuming .
the results are very good and confirm the belief of hamer et al @xcite in obtaining with considerable accuracy the observables working on lattices of moderate size .
consequently , this procedure is appealing because one can run all the needed computations in small machines obtaining quite fair results . | the schwinger model is studied in a finite lattice by means of the p - representation .
the vacuum energy , mass gap and chiral condensate are evaluated showing good agreement with the expected values in the continuum limit . | hep-lat9711049 |
recently , barger _
@xcite addressed astophysical constraints on extra dimensions by computing energy loss rates from the sun , red giant stars and ( type ii ) supernovae due to possible excitation of graviton modes , @xmath1 , in the case that the extra dimensions are compactified @xcite .
the processes @xmath2 ( in the static nucleon approximation ) , and @xmath3 where considered .
they worked in the zero density approximation , varying only the temperature .
their calculation neglected plasma effects and they anticipated that this neglect should not be important because of the high power dependence on @xmath4 , the inverse of the compactification dimension . the purpose of this paper is to address the extent to which the process @xmath5 ( and the crossed process @xmath6 ) is affected by a non - zero charged particle density and the presence of both longitudinal and transverse plasmons .
our aims are , in brief , to find numerical values for the energy loss rate per @xmath0 for densities from @xmath7 to @xmath8gm/@xmath0 and temperatures from @xmath7 to @xmath9 ev , to confirm the barger _
. _ expectation of density insensitivity , to show the extent to which the systems they considered are close to the border in density at which their expectation fails , to determine the relative contributions of longitudinal and transverse plasmon processes , to address the size of the contribution of the crossed process , and , importantly , to note the ambiguities in the form of a covariant interaction between plasmons and kaluza - klein modes .
it is clear that the expectation of barger et al .
@xcite must fail at sufficiently high density , for fixed temperature , since the energy loss rate goes as an integral over the bose - einstein distribution , @xmath10 , in which the frequencies are given by a dispersion relation with an effective photon mass that grows with density .
it is essentially a numerical question as to the point at which suppression sets in and the rate ( in density ) at which it proceeds .
intuition is hampered by the fact that the natural parameter is the electron chemical potential which is not simply related to the particle density .
we give the numerical results over the temperature and density ranges cited above .
an important problem in carrying out this work is the question of the appropriate lagrangian .
the free space coupling between the electromagnetic field and gravitons can be found in textbooks , for example @xcite , and has been generalized to the case of higher dimensional kaluza - klein excitations @xcite .
however , we have not found a parallel literature for the case in which the free space photon is replaced by a plasmon satisfying a non - trivial dispersion relation .
this difficulty is addressed in section 2 .
we adopt a diagrammatic approach and also make an approximation that we test numerically . also given in section 2
is the formalism used for the numerical calculations of section 3 .
we conclude in section 4 with a brief summary .
in a medium with nonzero temperature and density , radiation satisfies the dispersion relation @xmath11 @xmath12 is the transverse or longitudinal component of the polarization tensor , @xmath13 where @xmath14 ( @xmath15 ) are the polarization vectors and the polarization tensor @xmath16 is the photon self energy in the medium .
the contribution to this self energy from fermions in the medium is calculated in the medium rest frame by adding a term @xmath17 to the usual ( vacuum ) progagator . here , @xmath18 is the temperature , @xmath19 and @xmath20 is the electron chemical potential , which is related to the electron number density @xmath21 by @xmath22= n_e\ ] ] to lowest order in the fine structure constant @xmath23 , the polarization tensor is given by @xmath24\nonumber \\[4pt ] & & \qquad\times\frac{(p\dot k)^2g^{\mu\nu}+k^2p^{\mu}p^{\nu}-p\dot k(k^{\mu}p^{\nu}+ k^{\nu}p^{\mu})}{(p\dot k)^2}\,.\end{aligned}\ ] ] it turns out that @xmath25 and @xmath26 can be approximated to within 1% for all temperatures and densities
@xcite by @xmath27+v_*^2|\vec{k}|^2-|\vec{k}|^2\ , , \\ [ 4pt ] \pi_t & = & \omega_p^2\left[1+\frac{1}{2}g(v_*^2|\vec{k}|^2/\omega^2)\right]\,,\end{aligned}\ ] ] where @xmath28 is an average value of @xmath29 for the electron ( the only fermion which contributes for stellar temperatures and densities ) .
explicitly , @xmath30 with @xmath31 given by @xmath32 where @xmath33 is the sum of the electron and positron distributions ( the square bracket in eq.([pimunu ] ) above ) .
the plasma frequency @xmath34 is given by @xmath35 and the function @xmath36 is @xmath37\,.\ ] ] it will be important below to note that , for transverse photons , @xmath38 is @xmath39 for @xmath40 and _ increases _ as @xmath41 increases , while , for longitudinal photons , @xmath42 at @xmath40 and _ decreases _ as @xmath41 increases .
integration over @xmath41 for longitudinal photons must be cut off at the point where @xmath43 becomes negative , @xmath44\,.\ ] ] we include in our numerical evaluations the renormalization constants @xmath45 although this is inconsistent with calculating only to the lowest order in @xmath23 .
it is a check on our results that they do not change significantly when the @xmath46 are set to unity .
the @xmath46 are given by @xcite @xmath47 + ( \omega^2+|\vec{k}|^2)(\omega^2-v_*^2|\vec{k}|^2 ) } \\ [ 4pt ] z_l & = & \frac{2\omega^2(\omega^2-v_*^2|\vec{k}|^2)}{3\omega_p^2-\omega^2+v_*^2|\vec{k}|^2 } \frac{\omega^2}{\omega^2-|\vec{k}|^2}\,.\end{aligned}\ ] ] the rate of graviton emission can be calculated using the lagrangian for the coupling of kaluza - klein field @xmath48 , corresponding to the mass excitation @xmath49 , to the photon energy - momentum tensor @xmath50 . neglecting gauge terms , this coupling is @xcite @xmath51 & = & \frac{\kappa}{2}\left({\cal g}_{\vec{n}}^{\mu\nu}f_{\mu}^{\la}f_{\nu\la}-\frac{1}{4 } { \cal g}_{\vec{n},\mu}^{\mu}f^{\la\ro}f_{\la\ro}\right)\,,\end{aligned}\ ] ] where @xmath52 is the electromagnetic field tensor .
we consider only the coupling of the spin-2 component of the kaluza - klein field ; the spin-0 component does not couple to photons .
the matrix element for @xmath53 obtained from eq.([lint ] ) is @xcite @xmath54 where @xmath55 with @xmath56 d_{\mu\nu,\la\ro}(k_1,k_2 ) & = & \eta_{\mu\nu}k_{1\,\ro}k_{2\,\la } - \left[\eta_{\mu\ro}k_{1\,\nu}k_{2\,\la}+\eta_{\mu\la}k_{1\,\la}k_{2\,\nu } + ( \mu\leftrightarrow\nu)\right]\,.\end{aligned}\ ] ] the sum over polarizations of the kaluza - klein state is @xcite @xmath57 with @xmath58 e_{\mu\nu } & = & \eta_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{m_{\vec{n}}^2}\,.\label{projop}\end{aligned}\ ] ] the coupling eq.([coupling ] ) is gauge invariant even if @xmath59 and @xmath60 are not zero , e.g. @xmath61 .
however , it is not conserved , @xmath62 , if @xmath59 and/or @xmath60 differs from zero .
we can not write a conserved coupling by using the energy - momentum tensor for a massive vector field because @xmath59 is not necessarily equal to @xmath60 .
this means that if we square @xmath63 of eq.([matrixel ] ) , and use eq.([spinsum ] ) , we get extra terms of the form @xmath64 or @xmath65 from the second term in eq.([projop ] ) . to have a conserved amplitude with @xmath66
, we must include all the diagrams of fig.([diags ] ) .
the feynman rules for the , height=288 ] fermion - fermion-@xmath67 coupling and the fermion - fermion - photon-@xmath67 are given in refs.@xcite , and the loops are calculated by using eq.([prop ] ) for one of the legs .
we have shown the the sum of these diagrams is gauge invariant and conserved for arbitrary @xmath59 and @xmath60 at finite temperature and density . however ,
this was done without actually evaluating the diagrams . in particular , diagram ( d ) is very tedious and we have not computed it .
instead , we have used only diagram ( a ) ( eq.([matrixel ] ) above ) but have evaluated every energy loss twice - once including the @xmath64 and @xmath65 terms and once omitting them . in every case , the results were almost identical . while this proves nothing , it does seem to indicate that performing the complete one - loop calculation would not give a substantially different answer .
the reaction rate must be summed over the kaluza - klein states , which is done by integrating over @xmath68 where @xmath69 is the number of extra dimensions .
@xmath4 is the string scale which is related to the compactification scale @xmath70 and newton s constant @xmath71 .
specifically , we use @xmath72 our definition of @xmath73 differs from that of @xcite by a factor of @xmath74 , i.e. their @xmath4 is larger by a factor @xmath75 . as a consequence , values of the energy loss per unit volume obtained from our tables
must be multiplied by 2 when comparing with barger _
et al._@xcite . for 2 particles
@xmath76 1 particle reactions there remains a delta function from phase space which identifies @xmath77 with the center of mass squared energy @xmath78 .
thus , the integral over @xmath77 , eq.([dmsq ] ) , replaces @xmath77 by @xmath78 and our results depend on @xmath69 through the factor @xmath79 .
the rate of energy loss per unit volume is given by the standard expression @xmath80 where @xmath81 denotes the cross section times the relative velocity @xcite .
the initial photons have @xmath82 , @xmath83 and can be transverse , @xmath84 , longitudinal , @xmath85 or mixed , e.g. @xmath86 .
the factor @xmath87 gives the number of spin states : @xmath88 for @xmath89 , @xmath90 or @xmath91 . for longitudinal photons
, the @xmath41 integrals are cut off at @xmath92 given by eq.([kmax ] ) .
the corresponding expression for the energy loss in the decay @xmath93 is @xmath94 & & \qquad\times\frac{(2\pi)^4\left((k_t - k_l)^2\right)^{(n-2)/2}}{\kappa^2m_s^{n+2 } } \,|{\cal m}|^2 \,,\end{aligned}\ ] ] where @xmath63 is given by eq.([matrixel ] ) with @xmath95 .
the first step in the calculations is to obtain @xmath96 from eq.([density ] ) . in doing this
we assume that the electron number density , @xmath21 is related to the mass density @xmath97 by @xmath98 where @xmath99 is the proton mass .
this is useful for comparison purposes and is a reasonable order of magnitude approximation but needs correction ( by less than an order of magnitude ) for a supernova or a neutron star .
the results of the calculation of @xmath20 are given in tables [ mu ] and [ mutilde ] for the matrix of @xmath97 and @xmath18 values : @xmath100 gm/@xmath0 and @xmath101 .
two tables are given ( @xmath102 and @xmath103 ) in order to make clear both the deviation of @xmath20 from @xmath104 ( taken as 0.51 mev ) at low temperature and its deviation from zero at high .
note the rapid variation of @xmath20 ( for the lowest densities ) at the temperature @xmath18 around @xmath105 where pair production first begins to be copious , and the slower but similar variation at higher densities @xmath97 .
the variation is slower for higher densities because the electron - positron density difference needs to have a large value .
these variations are illustrated in fig.[mu_1 ] , where , for display purposes , the lowest value of @xmath20 shown , @xmath106 , is an upper limit on the exact numbers in table [ mu ] .
is plotted for the range of density @xmath97 and temperature @xmath18 given in the text .
here , for display purposes , the lowest value of @xmath20 shown , @xmath106 , is an upper limit on the exact numbers in table [ mu].[mu_1],height=288 ] in calculating @xmath20 we used an iteration procedure and required that the output density value equal the input to better than a percent .
we now pass on to the results of calculating @xmath107 for the case of @xmath108 extra dimensions .
it was possible to evaluate the integral in eq.([dedtdv ] ) over the cosine of the angle between the two plasmons analytically so that , for all the processes under consideration , only two integrals remain in finding the energy loss rate - the integrals over the two plasmon momenta .
it should be noted that there are only four processes to consider : ( 1 ) @xmath109 , ( 2 ) @xmath110 , ( 3 ) @xmath111 , and ( 4 ) @xmath112 .
this is because , as the plasmon momentum @xmath41 increases , the effective mass of a transverse plasmon increases while the effective mass of a longitudinal plasmon decreases @xcite .
thus the _ missing _ processes , @xmath113 and @xmath114 are forbidden by energy - momentum conservation .
the assertion is clear for the first process since @xmath115 .
for the second , we note that , in the rest frame of the decaying longitudinal plasmon , @xmath116 and conservation of energy and momentum implies that the graviton mass , @xmath117 , satisfies @xmath118 where @xmath119 is the energy of the final plasmon . using the dispersion relation for longitudinal plasmons ,
it can be shown that the right side of eq.([lneqlg ] ) is less than zero for @xmath120 .
the results of the calculations are given in tables [ tt]-[tot2 ] .
these have the energy loss rates for the four processes , and for the sum , for a matrix of density and temperature values - @xmath121 to @xmath8 gm/@xmath0 for density @xmath97 and @xmath7 to @xmath122 ev for temperature @xmath18 - in both cases in factors of @xmath123 increments . in these tables , @xmath18 increases from left to right while @xmath97 increases from top to bottom .
the entries are logs to the base 10 of the energy loss rate in ergs per @xmath0-s .
note that barger _
@xcite give results per unit mass , but results per unit volume are better for our purposes since they show more clearly the way in which the zero - density approximation breaks down as the density increases .
we give the results for @xmath124 tev .
two additional tables , [ dom ] and [ frac ] , give respectively the number of the process that dominates for the reaction ( zero if the rate is zero , i.e. below @xmath125 ) and the fraction of the total represented by the dominant contribution . in fig.[tttog](@xmath126 ) , we see at a glance the effect cited in the for the process @xmath127 is plotted for the range of density @xmath97 and temperature @xmath18 given in the text .
[ tttog],height=288 ] introduction : for fixed @xmath18 the energy loss rate is independent of the density until @xmath97 increases to a point where the effective photon mass and plasmon density are sufficiently high that the rate drops exponentially . the numerical values are given in table[tt ] .
for the process @xmath128 is plotted for the range of density @xmath97 and temperature @xmath18 given in the text .
[ lltog],height=288 ] the analogous plot for @xmath112 is shown in fig.[lltog ] . here
, the the rate grows slightly for a fixed @xmath18 before dropping exponentially in the mass of the longitudinal plasmon with increasing @xmath41 .
again , numerical values are found in table[ll ] .
the sun , a red giant , and a type ii supernova are , in the ( @xmath129 ) plane , given by barger _
_ @xcite to be at ( 156 gm/@xmath0,1.3 kev ) , ( 10@xmath130 gm/@xmath0 , 8.6 kev ) , and ( 10@xmath131 gm/@xmath0 , 30 mev ) respectively .
we see from table [ tt ] that the sun is in a low density region where the zero - density approximation holds while the supernova is on the edge of a constant density region .
we also see from table 6 that the other processes contribute little in these two cases .
the red giant ( rg ) case is more interesting . in table [ tt ] , we see from the @xmath132 and @xmath133 columns that the rg is in the gentle fall off region for the former , but the steep fall off region for the latter .
examining the region between 1 and 10 kev more closely gives , with t varying in 1.0 kev increments , for the log of the tt energy loss rate : -16.7 , -5.1 , -0.91 , 1.36 , 2.85 , 3.94 , 4.78 , 5.48 , 6.07 , 6.57 .
the total energy loss rate varies as : -15.96 , -4.14 , 0.37 , 2.65 , 3.77 , 4.59 , 5.25 , 5.82 , 6.31 , 6.77 . in short , at the rg , the tt energy loss rate , 1.4 erg / g@xmath134s , is down by a factor 4.5 from its zero - density approximation @xcite , but the other processes bring the total rate up to 2.6 erg / g@xmath134s , or within about a factor 2 of the zero density result .
finally we turn to the dependence of the emission rates on the number n of ( large ) extra dimensions .
we give , in tables [ tot3 ] and [ tot4 ] , the results for the total rates for n=3 and n=4 .
one sees , in the low - density limit , the @xmath135 behavior pointed out by barger _
@xcite for the @xmath136 rate which dominates for @xmath137 in the same places as in the @xmath108 case .
the approximations of zero density and purely transverse ( @xmath126 ) photon annihilation into gravitons of @xcite must fail , for fixed temperature , at high densities .
we have computed a first estimate of finite density corrections to two - plasmon production , for both transverse and longitudinal plasmons of kaluza - klein excitations , as well as the decay process @xmath138 as a function of plasma density and temperature over a wide range of interest in both parameters .
our conclusion is that the zero - density , pure transverse approximation is satisfactory for the sun , marginal for supernovae , and fails by about a factor of 5 for red giants .
it is interesting to note that , while very little of the @xmath139 plane is occupied , astrophysical systems appear to be preferentially located relatively close to the boundaries ( in @xmath97 ) at which the transverse photon approximation begins to fail .
our calculation is approximate in that we omit most of the diagrams of fig.([diags ] ) .
however , we believe our results indicate that the full calculation of all the diagrams is unlikely to modify our conclusions .
one of us ( v. t. ) wishes to thank r. mohapatra for helpful conversations .
this work was supported in part by the national science foundation under grant phy-9802439 and by the department of energy under contract nos .
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.the entries are @xmath140 for the process ( 1 ) ( @xmath127 ) . the rows are labeled by the density @xmath97 in gm/@xmath0 , and the columns by the temperature @xmath18 in ev .
a blank entry indicates that @xmath141erg/@xmath0-s .
the number of compact dimensions is @xmath108.[tt ] [ cols="^,^,^,^,^,^,^,^,^,^,^",options="header " , ] | recently , barger _
et al . _ computed energy losses into kaluza klein modes from astrophysical plasmas in the approximation of zero density for the plasmas .
we extend their work by considering the effects of finite density for two plasmon processes .
our results show that , for fixed temperature , the energy loss rate per @xmath0 is constant up to some critical density and then falls exponentially .
this is true for transverse and longitudinal plasmons in both the direct and crossed channels over a wide range of temperature and density . a difficulty in deriving the appropriate covariant interaction energy at finite density and temperature is addressed .
we find that , for the cases considered by barger _
et al .
_ , the zero density approximation and the neglect of other plasmon processes is justified to better than an order of magnitude . | hep-ph0005099 |
this work was partially financed by cnpq , capes and fapergs , brazil . | the diffractive photoproduction of vector mesons is usually described considering the two - gluon ( pomeron ) exchange , non - diagonal parton distributions and the contribution of the real part to the cross section . in this letter
we analyze the diffractive photoproduction of the @xmath0 at hera using an alternative model , the color evaporation model ( cem ) , where the cross section is simply determined by the boson - gluon cross section and an assumption for the production of the colorless state .
we verify that , similarly to the @xmath1 case , the hera data for this process can be well described by the cem .
moreover , we propose the analyzes of the ratio @xmath2 to discriminate between the distinct approaches . the successful operation of the hera @xmath3 collider has opened a new era of experimental and theoretical investigation into diffractive vector - meson photo- and leptoproduction . on the experimental side
, the hera accelerator extends the accessible energy range by more than one order of magnitude over previous experiments .
the hera data show that the cross sections for exclusive vector - meson production rise strongly with energy when compared to fixed - target experiments , if a hard scale is present in the process . on the theoretical side ,
vector - meson production has proven to be a very interesting process in which to test the interplay between the perturbative and nonperturbative regimes of qcd ( for a review , see for example @xcite ) . while in inclusive process , as open heavy flavor production , the cross section is described in terms of a perturbative term associated to the cross section of the partonic subprocess , and a nonperturbative term represented by the parton distributions , an analogous factorization of hard and soft physics does not apply to quarkonium production rates @xcite , which constitute a small fraction of the total open flavor cross section .
this makes quarkonium physics a challenging field @xcite .
the current picture in the literature to describe the diffractive photo- and leptoproduction of vector - mesons assumes that the color singlet property of the meson is enforced at the perturbative level by the two - gluon ( pomeron ) exchange @xcite . in this approach the amplitude , in the target rest frame ,
is factorized as a sequence of events very well separated in time : ( i ) the photon fluctuates into a quark - antiquark pair ; ( ii ) the @xmath4 pair scatters on the proton target , and ( iii ) the scattered @xmath4 pair turns into a vector meson .
the interaction is mediated by the exchange of two gluons in a color - singlet state .
moreover , the two gluon exchange amplitude can be shown to be proportional to the gluon distribution @xmath5 , with @xmath6 and @xmath7 , where @xmath8 is the @xmath9 center of mass energy and @xmath10 is the invariant mass of the @xmath11 system . in the case of the production of a heavy meson
, the presence of the heavy meson mass ensures that perturbative qcd can be applied even in the photoproduction limit .
this approach has been improved , particularly in the role of the vector meson light cone wave function @xcite , describing reasonably the hera @xmath12 data . when applied to the recent hera @xmath13 data @xcite
, it only reproduces the data if new effects , which significantly contribute , are considered @xcite : ( a ) the non - diagonal parton distributions , which probe new nonperturbative information about hadrons and are a generalization of the conventional parton distributions ( for a review see ref .
@xcite ) ; and ( b ) the real part of the scattering amplitude . in ref .
@xcite , a strong correlation between the mass of the diffractively produced state and the energy dependence of the total cross section was found , implying a distinct energy dependence for the @xmath13 and @xmath12 photoproduction .
one of the main motivations for the study of diffractive photoproduction of vector mesons in the pomeron model is the possibility of obtain a sensitive probe of the behavior of the gluon distribution , due to the quadratic dependence on the gluon distribution @xcite of the cross section in this model .
an alternative view of the diffractive photoproduction process was proposed recently in ref .
@xcite , where the @xmath12 photoproduction was analyzed using the color evaporation model ( cem ) @xcite . in this case
the color singlet is not enforced at the perturbative level and the cross section for the process is given essentially by the boson - gluon cross section plus an assumption for formation of the colorless meson .
in the cem for the diffractive photoproduction the cross section is linearly proportional to the gluon distribution .
these authors have obtained a parameter free prediction which describes very well the hera @xmath12 data . in this letter
we extend the application of the cem for the recent hera @xmath13 data and verify that , using the parameters determined in @xmath14 hadroproduction , this model reasonably describes the experimental data with no need to introduce a colorless ( pomeron ) exchange at the perturbative level , as well as , non - diagonal parton distributions or the contribution of the real part of the scattering amplitude .
moreover , we propose the analyzes of the ratio @xmath15 to discriminate between the distinct approaches .
let us start from a brief review of the color evaporation model ( cem ) .
one of the main uncertainties in the quarkonium production is related to the transition from the colored state to the colorless meson .
initially , the @xmath11 pair will in general be in a color octet state .
it subsequently neutralizes its color and binds into a physical resonance . color neutralization occurs by interaction with the surrounding color field .
if we enforce that a colorless object is already present at the perturbative level then a hard interaction with the surrounding color field is assumed , and a minimal of two gluons should be exchanged with the proton ( pomeron models ) . however ,
if the colorless object is produced at the nonperturbative level , then there is not a minimal restriction in the number of gluons exchanged with the proton .
the cem provides a simple and general phenomenological approach to color neutralization ( see also ref .
@xcite ) . in cem
, quarkonium production is treated identically to open heavy quark production with exception that in the case of quarkonium , the invariant mass of the heavy quark pair is restricted to be below the open meson threshold , which is twice the mass of the lowest meson mass that can be formed with the heavy quark . for bottomonium
the upper limit on the @xmath16 mass is then @xmath17 .
moreover , the cem assumes that the quarkonium dynamics is identical to all bottomonium states , although the @xmath16 pairs are typically produced at short distances in different color , angular momentum and spin states .
the hadronization of the bottomonium states from the @xmath16 pairs is nonperturbative , usually involving the emission of one or more soft gluons .
depending on the quantum numbers of the initial @xmath16 pair and the final state bottomonium , a different matrix element is needed for the production of the bottomonium state .
the average of these nonperturbative matrix elements are combined into the universal factor @xmath18 $ ] , which is process- and kinematics - independent and describes the probability that the @xmath16 pair binds to form a quarkonium @xmath19 of given spin @xmath20 , parity @xmath21 , and a charge conjugation @xmath22 .
once @xmath23 has been fixed for each state ( @xmath13 , @xmath24 or @xmath25 ) the model successfully predicts the energy and momentum dependence @xcite .
considering the elastic @xmath0 photoproduction at hera , the cem predicts that the cross section is given by @xmath26 = f[nj^{pc } ] \,\,\overline{\sigma}[b \overline{b } ] \,\,,\end{aligned}\ ] ] where the large distance factors @xmath23 can be written in terms of the probability @xmath27 to have a color singlet pair after the soft interactions , times the fractions @xmath28 of total bottomonium carried by the different states , in a similar way as considered in @xcite , where the corresponding @xmath29 factors for charmonium are claimed to be universal .
the short distance contribution is @xmath30 = \int_{2m_b}^{2m_b } dm_{b\overline{b } } \,\,\frac{d\sigma[b\overline{b}]}{dm_{b\overline{b } } } \,\ , .
\label{cross}\end{aligned}\ ] ] here @xmath31 $ ] is the spin- and color - averaged cross section for open heavy quark production , which describes the boson - gluon fusion process @xmath32 ; @xmath33 is the invariant mass of the @xmath16 pair , @xmath34 is the bottom quark mass and @xmath17 is the @xmath35 threshold .
the differential cross section for the boson - gluon subprocess is well known ( see eq . ( 4 ) in ref .
@xcite for the expression at leading order ) .
this will contribute to elastic photoproduction , since in lo all energy of the photon is transferred to the pair .
higher order corrections will be mostly important for the inclusive and the inelastic cross section .
thus , the soft interactions have a double effect - to eliminate the color of the pair , allowing quarkonium production , and to allow elastic production with a single perturbative gluon exchange , by having soft gluon exchanges taking place in the non - perturbative part of the process . in cem ,
the effect of these soft interactions is implicit in the non - perturbative factors .
an explicit modeling for the soft gluon exchanges is done in the sci model @xcite , which is based essentially in the same general idea . in that model , by requiring either rapidity gaps or leading proton in the final state , it was obtained explicitly a good description of several diffractive processes , including diffractive and open beauty production in the tevatron @xcite .
this shows that soft gluon exchanges really might play a role in diffraction .
the cem has been used to predict the production of @xmath13 in hadron and nuclear processes @xcite , with the parameter @xmath18 $ ] determined from the quarkonium production experimental data at fixed - target energies .
a remarkable feature of the cem is that this model also accounts ( within @xmath36 factors ) for the quarkonium production at the tevatron @xcite . as the fixed target @xmath14 data
have generally given the sum of @xmath13 , @xmath24 and @xmath37 production , due to the low mass resolution to clearly separate the peaks , only the global factor @xmath38=0.044 $ ] has been determined .
this is an effective value which reflects both direct production and chain decays of higher mass states . as discussed in ref .
@xcite , isolation of the direct production cross section for each @xmath39 requires the detection of the radiated photons associated with chain decays , which is not currently available but might be possible at the lhc .
considering some assumptions related to the decay chain the following values for @xmath23 in direct @xmath39 production have been estimated @xcite : @xmath40=0.023 $ ] , @xmath41=0.02 $ ] , @xmath42=0.0074 $ ] , where the @xmath43 superscript indicates the @xmath23 for direct production . in our formalism
the @xmath23 factors can be related to the @xmath29 factors by the simple relation @xmath44 , where the @xmath27 is the probability to have the pair in a color singlet state after the soft gluon exchanges , and @xmath28 are the universal factors with give the fractions of the onium cross section carried by the different onium states .
their corresponding values are , from above , @xmath45=0.207\,\,,\,\,\rho^d[\upsilon ^{\prime } ] = 0.18\,\,,\,\,\rho^d[\upsilon ^{\prime \prime } ] = 0.066\,\,.\end{aligned}\ ] ] once the free parameters have been determined in the hadronic processes , we can use the cem to predict the @xmath13 photoproduction at hera .
this also follows the assumption used in @xcite that the production ratios of @xmath0 , @xmath46 and @xmath37 are the same as those measured in hadron - hadron collisions .
a comment is in order here . in ref .
@xcite the @xmath13 hadroproduction has been calculated considering the next - to - leading order contributions for the cross section , which implies that the factor @xmath23 determined from data does not contain perturbative contributions beyond leading order and can be considered an universal factor which describes the probability for quarkonium production .
moreover , the calculations in ref .
@xcite have used the mrs d@xmath47 parton densities as input , but as demonstrated in @xcite , where an update from the analyzes of the @xmath14 suppression for the cms detector was made , the @xmath13 data can be equally well described using the grv 94 lo @xcite parton densities .
the cross section @xmath48 $ ] is computed at leading order using the grv 94 lo @xcite parton densities with @xmath49 gev , @xmath50 gev and the renormalization and factorization scales set to @xmath51 @xcite , with @xmath52 . in fig .
[ fig1 ] we show our predictions for @xmath14 photoproduction at hera energies . both the total @xmath53 and direct @xmath13 production are presented , since the cross sections measured by the zeus and h1 collaborations did not select the direct @xmath13 state , but rather the data were integrated over an interval of the @xmath54 mass which includes at least the @xmath13 , @xmath24 and @xmath25 resonances .
we verify that the current experimental data does not allow to discriminate the distinct contributions .
we emphasize that our results are completely parameter free and that the cem model reasonably describes the scarce experimental data .
the simplicity of the cem strongly contrasts with the number of assumptions necessary in the pomeron models to describe the same set of data .
here we proposed a signature to discriminate between these models . as we have quoted above ,
the pomeron models predict a stronger growth in energy for the diffractive @xmath13 photoproduction than in @xmath12 photoproduction , due to the strong correlation between the mass of the diffractively produced state and the energy dependence of the cross section .
in contrast , in the cem model the growth of the cross section is directly determined by the gluon distribution @xmath55 , where @xmath56 is the factorization scale .
therefore , the energy dependence of the ratio @xmath57 can be used to discriminate between models . as an intermediate step to calculate the mentioned rate ,
we show in fig .
[ fig3 ] our previous calculation of @xmath58 , contrasted with the more recent hera data @xcite .
we see that these data can be reasonable explained without tuning the parameters , which were taken as the previously used in ref .
@xcite .
this gives one more clue that the cem can be used to explain elastic @xmath59 photoproduction . in fig .
[ fig4 ] we present the energy dependence of the ratio @xmath60 calculated using the cem , where @xmath61 represents the direct @xmath14 production and we have used the results from ref .
@xcite and from above to calculate @xmath58 .
our results show that this ratio is almost constant in the kinematic range of hera , in contrast to the pomeron model @xcite , where a steep rise of the ratio is predicted [ cf . ref .
@xcite , @xmath62 .
this ratio was also obtained in @xcite , where the value predicted agrees with data within errors for the mrs gluon , but underestimates the result for the steeper grv gluon . in our case , the @xmath63 ratio agrees with the lower bound observed by zeus @xcite for both mrs and grv gluon parameterizations .
the original motivation of the study of diffractive photoproduction was the possibility to extract the gluon distribution inside the proton .
however , the dependence on the gluon distribution is one of the major differences between the descriptions of the diffractive photoproduction using either the pomeron model or the cem . whereas the pomeron model has a quadratic dependence on @xmath64 , this dependence is linear in the cem .
our result demonstrates that before to extract the gluon distribution from hera @xmath0 and @xmath1 data , one should determine the correct description for this process , for example by measuring the energy behaviour of the @xmath65 ratio .
the cem describes a large range of data in hadro- and photoproduction , as shown in refs .
@xcite . using this model ,
in this letter we obtain a parameter free description of the elastic @xmath13 photoproduction at hera energies .
we verify that this simple model reasonably describes the experimental data , similarly to the pomeron models . as a distinct feature between these models ,
the cem predicts a softer energy dependence of the ratio between the @xmath13 and @xmath66 cross sections .
of course , when more precise data become available , the discrimination between these models should be possible , which will allow us to conclude whether the cem is only an alternative phenomenological model for the current energies or it contains some underlying non - perturbative dynamics which is important in both diffractive and non - diffractive quarkonium production . in the latter case ,
more theoretical studies are necessary to understand the soft interactions in the process of color neutralization . | hep-ph0110404 |
the renormalization group equations ( rges ) for the rigid couplings and soft parameters in susy gauge theories play a crucial role in applications . actually , all predictions of the mssm are based on solutions to these equations in leading and next - to - leading orders @xcite .
typically , one has three gauge couplings , one or three yukawa couplings ( for the case of low or high @xmath0 , respectively ) and a set of soft couplings .
in leading order , solutions to the rge for the gauge couplings are simple ; however , already for yukawa couplings , they are known in an analytical form only for the low @xmath0 case , where only the top coupling is left . moreover , even in this case solutions for the soft terms look rather cumbersome and difficult to explore @xcite . in a recent paper @xcite
, it has been shown that solutions to the rge for the soft couplings follow from those for the rigid ones in a straightforward way .
one takes the solution for the rigid coupling ( gauge or yukawa ) , substitute instead of the initial conditions their modified expressions @xmath1 where @xmath2 , @xmath3 and @xmath4 are the grassmannian parameters , and expand over these parameters .
this gives the solution to the rges for the soft couplings .
hereafter the following notation is used : @xmath5 where @xmath6 and @xmath7 are the gauge and yukawa couplings , respectively , and @xmath8 are the soft masses associated with each scalar field .
this procedure , however , assumes that one knows solutions to the rge for the rigid couplings in the analytic form .
for instance , in the case of the mssm in the low @xmath0 regime this allows one to get solutions for the soft couplings and masses simpler than those known in the literature ( see @xcite ) . at the same time , in many cases such solutions are unknown .
actual examples are the mssm with high @xmath0 and nmssm .
one is bound to solve the rges numerically when the number of coupled equations increases dramatically with the soft terms being included . below
we propose simple analytical formulae which give an approximate solution to the rge for yukawa couplings in an arbitrary susy theory with the accuracy of a few per cent . performing the grassmannian expansion in these approximate solutions one can get those for the soft couplings in a straightforward way . as an illustration we consider the mssm in the high @xmath0 regime .
one can immediately see that approximate solutions obtained in this way possess infrared quasi - fixed points @xcite which can be found analytically .
they appear in the limit when the initial values of the yukawa couplings are much larger than those for the gauge ones .
then , one can analytically trace how the initial conditions for the soft terms disappear from their solutions in the above mentioned limit .
the paper is organized as follows . in sect .
2 , we consider the mssm in the low @xmath0 regime , where all solutions are known analytically and describe briefly the grassmannian expansion . in sect .
3 , we present our approximate solutions for the yukawa couplings and obtain those for the soft terms
. we also present numerical illustration and compare approximate solutions with the numerical ones .
the fixed point behaviour is discussed .
section 4 contains our conclusions .
the explicit formulae for the soft couplings and masses are given in appendices .
consider the mssm in the low @xmath0 regime .
one has three gauge and one yukawa coupling .
the one - loop rg equations are @xmath9 with the initial conditions : @xmath10 , and @xmath11 .
their solutions are given by @xcite @xmath12 where @xmath13 to get solutions for the soft terms , it is enough to perform the substitution @xmath14 and @xmath15 for the initial conditions in ( [ sol ] ) and expand over @xmath16 and @xmath17 .
expanding the gauge coupling in ( [ sol ] ) up to @xmath18 one has ( hereafter we assume @xmath19 ) @xmath20 performing the same expansion for the yukawa coupling and using the relations @xmath21 one finds the well - known expression @xcite @xmath22 to get the solution for the @xmath23 term , one has to make expansion over @xmath18 and @xmath17 .
this can be done with the help of the following relations : @xmath24 as a result one has @xcite @xmath25\!.\ , \label{si}\end{aligned}\ ] ] which is much simpler than what one finds in the literature @xcite , though coinciding with it after some cumbersome algebra .
one can also write down solutions for the individual masses using the grassmannian expansion of those for the corresponding superfield propagators .
for the first two generations one has @xmath26 where @xmath27 the third generation masses get a contribution from the top yukawa coupling @xmath28 where @xmath29 is related to @xmath30 ( [ si ] ) by @xmath31 \\ & = & \frac{\sigma_0-a_0 ^ 2}{1 + 6y_0f}+\frac{(a_0-m_{1/2}6y_0(te - f))^2}{(1 + 6y_0f)^2 } -m_{1/2}^2\frac{6y_0}{1 + 6y_0f}t^2\frac{de}{dt}- \sigma_0.\end{aligned}\ ] ] with analytic solutions ( [ sol],[a],[si ] ) one can analyze asymptotic and , in particular , find infrared quasi - fixed points @xcite which correspond to @xmath32 @xmath33\!. \label{sf}\end{aligned}\ ] ] one can clearly see that the dependence on @xmath34 and @xmath35 disappears from ( [ yf])-([sf ] ) .
some residual dependence on @xmath36 is left for the soft masses and partially cancels with that of @xmath29 .
below we demonstrate how the same procedure works in the case of approximate solutions . as a realistic example we take the mssm in the high @xmath0 regime .
the one - loop rge for the yukawa couplings in this case look like @xmath37 since the exact solution is absent and might be too cumbersome , we look for an approximate one in a simple form similar to that of ( [ sol ] ) . in choosing approximate solutions we follow the idea of @xcite where
an approximate solution for @xmath38 and @xmath39 ignoring @xmath40 has been proposed .
our suggestion is to consider separate brackets for each propagator entering into the yukawa vertex .
then , one has the following expressions for the yukawa couplings : @xmath41^{1/a } [ 1 + 2by_{t0}f_t]^{1/b } [ 1 + 3cy_{t0}f_t]^{1/c } } , \quad \frac 1a \!+\ ! \frac 1b \!+\ ! \frac 1c\!=\!1
\\ y_b&=&\frac{\displaystyle y_{b0}e_b}{\displaystyle [ 1+a(y_{t0}f_t+y_{b0}f_b)]^{1/a } [ 1 + 2by_{b0}f_b]^{1/b } [ 1+c(3y_{b0}f_b+y_{\tau 0}f_\tau ) ] ^{1/c } } , \\
y_\tau & = & \frac{\displaystyle y_{\tau 0}e_\tau } { \displaystyle [ 1+a^{\prime } y_{\tau 0}f_\tau ] ^{1/a^{\prime } } [ 1 + 2b^{\prime } y_{\tau 0}f_\tau]^{1/b^{\prime } } [ 1+c(3y_{b0}f_b+y_{\tau 0}f_\tau ) ] ^{1/c}},\quad \frac1{a^{\prime } } \!+\!\frac 1{b^{\prime } } \!+\!\frac 1c\!=\!1\end{aligned}\ ] ] where the brackets correspond to the @xmath42 , @xmath43 and @xmath44 propagators , respectively . here
@xmath45 and @xmath46 are given by ( [ e ] ) and ( [ f ] ) and @xmath47 and @xmath48 have the same form but with @xmath49 and @xmath50 , respectively .
the brackets are organized so that they reproduce the contributions of particular diagrams to the corresponding anomalous dimensions .
the coefficients @xmath51 and @xmath52 are arbitrary and their precise values are not so important .
when yukawa couplings @xmath53 are small enough , one can make expansion in each bracket , and the dependence of these coefficients disappears .
however , for large couplings , which are of interest for us because of the fixed points , we have some residual dependence .
the requirement that the sum of exponents equals 1 follows from a comparison with rges .
solutions are close to the exact ones when the brackets are roughly equal to each other .
apparently , since @xmath54 and @xmath55 one can not completely satisfy this requirement .
our choice of the coefficients @xmath51 and @xmath52 is dictated mainly by simplicity . in the following
we choose them as @xmath56 this gives approximate solutions like @xmath57^{2/7}\!\ !
\left[1 + 7y_{t0}f_t\right]^{5/7 } } , \label{y1 } \\
! \frac{\displaystyle y_{b0}e_b } { \displaystyle \left[1+\frac{7}{2}(y_{t0}f_t+y_{b0}f_b)\right]^{2/7}\!\ ! \left[1 + 7y_{b0}f_b\right]^{2/7}\left[1+\frac{7}{3}(3y_{b0}f_b+y_{\tau 0}f_\tau ) \right]^{3/7 } } \label{y2 } , \\ y_\tau
\!&\approx & \!\!\frac{\displaystyle y_{\tau 0}e_\tau } { \displaystyle \left[1\!+\!\frac{21}{4}y_{\tau 0}f_\tau \right]^{4/7}\!\ !
\left[1\!+\!\frac{7}{3}(3y_{b0}f_b\!+\!y_{\tau 0}f_\tau ) \right]^{3/7 } } .
\label{y3}\,\end{aligned}\ ] ] solutions for @xmath58 and @xmath59 can be obtained by grassmannian expansion with the initial conditions @xmath60 these initial conditions correspond to the so - called universality hypothesis which we follow in our numerical illustration for simplicity .
however , one can choose arbitrary initial conditions for the soft terms when needed .
it leads to an obvious modification of the formulae
. one can get also the corresponding solutions for the individual soft masses .
this can be achieved either by grassmannian expansion of the corresponding brackets in ( [ y1])-([y3 ] ) , or by expressing the masses through the @xmath23s in an exact way .
the second way gives a slightly better agreement with numerical solutions ( see below ) .
we present the explicit expressions for the soft terms and masses in appendix a. we start by investigating the precision of approximate solutions for the yukawa couplings .
to estimate the accuracy , we introduce a relative error which is defined as @xmath61 and corresponds to the @xmath62 scale ( @xmath63 ) at the end of the integration range . the accuracy for the solutions of soft terms
is defined in the same way .
let us take at the beginning all three yukawa couplings to be equal at the gut scale and to have their common value @xmath64 in the range @xmath65 .
the upper limit is taken in order not to leave the perturbativity regime .
we find that for @xmath66 the approximation errors are less than @xmath67 for all @xmath68 s .
while for @xmath38 it remains smaller than @xmath69 over the whole range of initial values at the gut scale , for @xmath39 the error increases up to @xmath70 and for @xmath71 up to @xmath72 ( for large values of @xmath64 ) .
it is worth mentioning that for small @xmath64 ( around @xmath73 and below ) the accuracy is very good ( fractions of per cent or better ) .
consider now @xmath74 and let the top yukawa coupling vary within the limits @xmath75 in order to examine the applicability of our formulae . in this case
the accuracy it is spoilt a little bit with increasing initial values .
namely , the error for @xmath38 increases up to @xmath76 , and for @xmath39 and @xmath40 up to @xmath77 .
however , if one keeps @xmath64 in the range @xmath78 the accuracy for @xmath38 remains better than @xmath67 , and for @xmath39 and @xmath71 better than @xmath76 .
the particular case considered above seems to have the worst accuracy .
this is not surprising since our approximate formulae are supposed to work best of all when all three yukawa couplings are nearly equal .
if we keep @xmath79 and the relative ratios less than @xmath80 , we get an average error of less than @xmath81 for @xmath38 , about @xmath81 for @xmath39 and @xmath76 for @xmath71 . this statement is illustrated in fig.[errandm ] .
for each yukawa coupling we have plotted the error as a function of @xmath82 in the range @xmath83 .
the ratios are kept within the region @xmath84 and @xmath85 .
further on , we narrow the range of initial values up to @xmath86 because the errors ( defined as in ( [ defer ] ) ) come to an asymptotic value for @xmath87 and almost vanish for @xmath88 . the comparison of numerical and approximated solutions is shown in fig.[evays ] for three different sets of @xmath64 s .
the approximate solutions follow the numerical ones quite well , preserving their shape , and they have a high accuracy , especially in the case of equal yukawa couplings at the gut scale . however , as can be seen from the top of fig .
[ evays ] , one can take arbitrary initial conditions for the yukawa couplings , in particular those which are needed to fit the @xmath89 masses , and to use our approximate solutions for these purposes .
for the soft couplings , @xmath90 , we take the initial values at the gut scale to be @xmath91 and leave @xmath64s in the narrow range as above .
then , we get an accuracy of @xmath92 for @xmath93 and @xmath94 . for @xmath95
the approximation is worse when @xmath96 is taken to be negative or smaller than @xmath97 ( see fig .
[ evays ] ) , but things go better for large initial values of @xmath96 and we get an accuracy of about @xmath98 . again
it should be mentioned that this is an accuracy at the end point where @xmath99 itself is close to 0 and the accuracy defined as ( [ defer ] ) merely gives an odd hint of the precision . along the curves
the accuracy is much better . in fig.[evays ] we have plotted the behaviour of @xmath93 , @xmath94 and @xmath99 for three different initial values of @xmath96 , namely @xmath100 and for one set of @xmath64s . as for the @xmath23 s , keeping the range of parameter space for @xmath64 and @xmath96 as above , we get an accuracy of typically @xmath101 for @xmath102 ( even better for fairly equal @xmath64s ) . for @xmath103
the precision is around @xmath70 . with @xmath104
we get into the same troubles as for @xmath95 .
the approximation becomes good ( about @xmath76 ) only for a large enough ratio of @xmath105 .
the approximation errors for @xmath90 s and @xmath23 s are linked with those for @xmath68 .
if one considers only the sets of small initial values for @xmath64 ( less than @xmath73 ) , then @xmath23 s are approximated with a precision better than @xmath106 , regardless of the @xmath96 values .
the precision for @xmath23 increases with @xmath96 , but this dependence is not so striking as the one on @xmath64 . the approximate formulae for the soft masses may be derived from @xmath23 using ( [ a1])-([a7 ] ) . in this case
the approximate solutions give an accuracy of about @xmath107 for @xmath108 , @xmath109 and @xmath92 for @xmath110 .
for the higgs masses we get a good approximation ( of about @xmath81 on average ) for @xmath111 , and a satisfactory one for @xmath112 ( typically @xmath76 ) .
this accuracy is almost insensitive to the @xmath96 variation ( we took it to be in the range @xmath113 ) and on the ratio @xmath114 ( taken to be @xmath115 ) .
the slepton masses ( see fig . [ errandm ] ) are not approximated properly in an analogous way .
this is mainly due to the less accurate approximation of @xmath116 . as a concluding remark on numerical analysis
, it should be mentioned that one has a rather good approximation for small ( less than @xmath73 ) initial values of the yukawa couplings .
for larger values of @xmath68 s one has a good approximation especially in the case of unification of the yukawa couplings .
one can easily see that solutions ( [ y1])-([y3 ] ) exhibit the quasi - fixed point behaviour when the initial values @xmath117 . in this case , one can drop 1 in the denominator and the resulting expressions become independent of the initial conditions @xmath118^{2/7}\!\ ! \left[7 f_t\right]^{5/7 } } \label{yfp1},\\ y_b^{fp}&\approx&\frac{\displaystyle e_b}{\displaystyle \left[\frac{7}{2}(f_t+f_b)\right]^{2/7}\!\!\left[7f_b\right]^{2/7}\!\ ! \left[\frac{7}{3}(3f_b+ f_\tau ) \right]^{3/7 } } \label{yfp2},\\ y_\tau^{fp } & \approx & \frac{\displaystyle e_\tau } { \displaystyle \left[\frac{21}{4}f_\tau \right]^{4/7}\!\ !
\left[\frac{7}{3}(3f_b+f_\tau ) \right]^{3/7}}. \label{yfp3}\end{aligned}\ ] ] these expressions being expanded over the grassmannian variables give the quasi - fixed points for the soft terms and masses .
the explicit expressions are presented in appendix b. we see that the irqfp behaviour is sharply expressed for @xmath38 and @xmath39 ( see fig .
[ fpays ] ) , and our approximate solution describes the fixed point line well .
the same takes place for the corresponding @xmath90s and @xmath23s . for @xmath119 and @xmath120 the accuracy is worse , however , the solution is still reliable .
the soft mass terms there exhibit the same irqfp behaviour , though some residual dependence on the initial conditions is left in full analogy with the exact solutions in the low @xmath0 case .
the approximate solutions allow one to calculate the irqfp analytically .
one can see that the fixed points for the soft terms naturally follow from the grassmannian expansion of our approximate solutions ( [ yfp1])-([yfp3 ] ) and they inherit their stability properties , as has been shown in @xcite .
in particular , the behaviour of @xmath23s essentially repeats that of the yukawa couplings in agreement with @xcite .
the existence of the irqfps allows one to make predictions for the soft masses without exact knowledge of the initial conditions .
this property has been widely used ( see , for example , @xcite ) and though the irqfps give a slightly larger top mass when imposing @xmath121 unification , it is still possible to fit the quark masses within the error - bars and to make predictions for the higgs and sparticle spectrum @xcite .
this explains general interest in the irqfps .
we hope to convince the reader that the approximate solutions presented above reproduce the behaviour of the yukawa couplings with good precision in the whole integration region and for a large range of initial values .
relative accuracy is typically a few per cent and is worse only at the end of the integration region mainly due to the smallness of the quantities themselves .
moreover , we have shown how the approximate solutions for the soft terms and masses follow from those for the rigid couplings .
this demonstrates how the grassmannian expansion , advocated in @xcite , works in the case of approximate solutions as well . for illustration
we have considered universal initial conditions for the soft terms . in recent time there appeared some interest in non - universal boundary conditions .
non - universality can also be included in our formulae at the expense of changing ( [ tayl ] ) and ( [ init ] ) using the same substitution rules , see ( [ g ] ) and ( [ y ] ) . since the form of our approximate solutions
has been `` guessed '' ad hoc starting from some reasonable arguments , there is no direct way to improve them .
however , one can imagine more constructive derivation of those solutions which would allow one to make corrections .
needless to say that it is enough to construct a solution for the rigid terms .
solutions for the soft terms will follow automatically .
* acknowledgement * we would like to thank a.v.gladyshev for valuable discussions . financial support from rfbr grants # 99 - 02 - 16650 and # 96 - 15 - 96030 is kindly acknowledged .
we here present approximate expressions for the soft couplings and masses corresponding to ( [ y1])-([y3 ] ) : @xmath122 to find the individual soft masses one can formally perform integration of the rg equations and express the masses through @xmath23s solving a system of linear algebraic equations .
this gives @xmath123 the masses of squarks and sleptons of the first two generations are given by ( [ me])-([mq ] ) .
we present here the irqfps for the soft couplings and masses .
they are obtained via grassmannian expansion of ( [ yfp1])-([yfp3 ] ) .
@xmath124 ^ 2 } { ( f_t+f_b ) ^2 } + \frac 57\ \frac{(te_t - f_t)^2 } { f_t^2 } \right)\\ \sigma_b^{fp}\!&\approx & \ !
m_{1/2}^2\left(-\ ! \frac 27\!\ \frac{t^2\frac{de_t}{dt}\!+\!t^2\frac{de_b}{dt } } { ( f_t\!+\!f_b ) } \!-\!\frac 27\ !
\ \frac{t^2\frac{de_b}{dt}}{f_b}\ ! -\ !
\frac{3t^2\frac{de_b}{dt } \!+\!t^2\frac{de_\tau } { dt } } { ( 3f_b\!+\!f_\tau ) } \right . \\ & & \left
. + \frac 27 \
\frac{[(te_t - f_t)+(te_b - f_b)]^2 } { ( f_t+f_b ) ^2 } + \frac 27\ \frac{(te_b - f_b)^2 } { f_b^2 } \right .
\\ & & \left .
+ \frac 37 \ \frac{[3(te_b - f_b)+(te_\tau -f_\tau)]^2 } { ( 3f_b+ f_\tau ) ^2 } + \frac{d}{dt}\left ( \frac{t^2}{e_b}\frac{de_b}{dt}\right)\right ) \\
\sigma_\tau^{fp } \!&\approx&\ ! m_{1/2}^2\!\left ( -\frac 47\!\ \frac{t^2\frac{de_\tau } { dt } } { f_\tau } -\frac 37\!\ \frac{3t^2\frac{de_b}{dt } + t^2\frac{de_\tau}{dt } } { ( 3f_b+\!f_\tau ) } + \ ! \frac{d}{dt}\!\left ( \frac{t^2}{e_\tau } \frac{de_\tau}{dt }
\right ) \right . \\ & & \left .
+ \frac 47\
! \frac{(te_\tau -f_\tau)^2}{f_\tau^2 } + \frac 37\ \frac{[3(te_b - f_b)+(te_\tau -f_\tau)]^2}{(3f_b+f_\tau)^2 } \right)\\\end{aligned}\ ] ] 99 v. barger , m. s. berger and p. ohmann , phys.rev . *
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hep - ph/9902290 | we present simple analytical formulae which describe solutions of the rg equations for yukawa couplings in susy gauge theories with the accuracy of a few per cent . performing the grassmannian expansion in these solutions
, one finds those for all the soft couplings and masses .
the solutions clearly exhibit the fixed point behaviour which can be calculated analytically . a comparison with numerical solutions
is made . | hep-ph9906256 |
the p - wave spin - triplet charmonium states were originally observed @xcite in radiative decays of the @xmath1 soon after the discovery of the @xmath16 and @xmath1 resonances . a number of decay modes of these states have been observed and branching fractions reported @xcite .
most of the existing results are from the mark i experiment , which had a data sample of 0.33 million @xmath1 decays @xcite . because the photon capabilities of the mark i detector were limited , the detection of the photon from the @xmath17 process was not required , and one constraint kinematic fits were used to reconstruct the final states .
recently there has been a renewed interest in the p - wave charmonium states . since in lowest - order perturbative
qcd the @xmath12 and @xmath18 decay via the annihilation of their constituent @xmath19 quarks into two gluons , followed by the hadronization of the gluons into light mesons and baryons , these decays are expected to be similar to those of a bound @xmath20 state ; a detailed knowledge of the hadronic decays of the @xmath12 and @xmath18 may provide an understanding of the decay patterns of glueball states that will help in their identification .
the mass differences between the three @xmath21 states provide information on the spin - orbit and tensor interactions in non - relativistic potential models and lattice qcd calculations .
the masses of the @xmath22 and @xmath18 have been precisely determined ( to a level of @xmath23 mev ) by fermilab experiment e760 @xcite using the line shape measured in the @xmath24 formation reaction .
in contrast , the @xmath12 mass is much more poorly known ; the pdg average for @xmath25 has an uncertainty of @xmath26 mev @xcite . in this paper , we report the analyses of all - charged - track final states from @xmath27 decays , including @xmath28 , @xmath29 , @xmath30 , @xmath31 , @xmath32 and @xmath33 .
the results for @xmath27 decays into @xmath34 , @xmath35 and @xmath36 have been reported elsewhere @xcite .
we use the combined invariant mass distribution from all of the channels under study to determine the @xmath12 mass with improved precision .
a byproduct of this analysis is a determination of the mass of the @xmath37 .
this is of interest because the @xmath38 mass difference measures the strength of the hyperfine splitting term in heavy quark interactions .
however , in spite of a number of measurements , the current experimental value of @xmath39 remains ambiguous : the pdg @xcite average is based on a fit to seven measurements with poor internal consistency @xcite and the confidence level of the fit is only 0.001 . a recent measurement from e760 @xcite disagrees with the value reported by the dm2 group @xcite by almost four standard deviations .
additional measurements may help clarify the situation .
the data used for the analysis reported here were taken with the bes detector at the bepc storage ring at a center - of - mass energy corresponding to @xmath40 .
the data sample corresponds to a total of @xmath41 @xmath1 decays , as determined from the observed number of inclusive @xmath42 decays @xcite .
bes is a conventional solenoidal magnet detector that is described in detail in ref .
a four - layer central drift chamber ( cdc ) surrounding the beampipe provides trigger information .
a forty - layer cylindrical main drift chamber ( mdc ) , located radially outside the cdc , provides trajectory and energy loss ( @xmath43 ) information for charged tracks over @xmath44 of the total solid angle .
the momentum resolution is @xmath45 ( @xmath46 in @xmath47 ) , and the @xmath43 resolution for hadron tracks is @xmath48 .
an array of 48 scintillation counters surrounding the mdc measures the time - of - flight ( tof ) of charged tracks with a resolution of @xmath49 ps for hadrons .
radially outside of the tof system is a 12 radiation length thick , lead - gas barrel shower counter ( bsc ) operating in the limited streamer mode .
this device covers @xmath50 of the total solid angle and measures the energies of electrons and photons with an energy resolution of @xmath51 ( @xmath52 in gev ) .
outside the bsc is a solenoid , which provides a 0.4 tesla magnetic field over the tracking volume .
an iron flux return is instrumented with three double layers of counters that identify muons of momentum greater than 0.5 gev / c .
we use monte carlo simulated events to determine the detection efficiency ( @xmath53 ) and the mass resolution ( @xmath54 ) for each channel analyzed . the monte carlo program ( mc )
generates events of the type @xmath55 under the assumption that these processes are pure @xmath56 transitions @xcite : the photon polar angle distributions are @xmath57 , @xmath58 and @xmath59 .
multihadronic @xmath60 decays are simulated using phase space distributions . for each channel , either 10000 or 5000 events are generated , depending on the numbers of events for the corresponding mode that are observed in the data sample .
a neutral cluster is considered to be a photon candidate when the angle in the @xmath61 plane between the nearest charged track and the cluster is greater than @xmath62 , the first hit is in the beginning 6 radiation lengths , and the difference between the angle of the cluster development direction in the bsc and the photon emission direction is less than @xmath63 .
when these selection criteria are applied to kinematically selected samples of @xmath64 and @xmath65 events , fewer than 20% of the events have @xmath66 candidates , which indicates that the fake - photon rejection ability is adequate ( see fig . [ ngm ] ) .
the number of photon candidates in an event is limited to four or less .
the photon candidate with the largest energy deposit in the bsc is treated as the photon radiated from @xmath1 and used in a four - constraint kinematic fit to the hypothesis @xmath67 .
each charged track is required to be well fit to a three - dimensional helix and be in the polar angle region @xmath68 . for each track , the tof and @xmath43 measurements
are used to calculate @xmath69 values and the corresponding confidence levels to the hypotheses that the particle is a pion , kaon and proton ( @xmath70 ) .
the reliability of the confidence level assignments is verified using a sample of @xmath71 , @xmath72 and @xmath73 events , where the particle identification confidence levels ( @xmath74 ) of the tracks in different momentum ranges are found to be distributed uniformly between zero and one as expected @xcite .
typically the @xmath74 value of each track for a given decay hypothesis is required to be greater than 1% in our analysis . for all decay channels ,
the candidate events are required to satisfy the following selection criteria : 1 .
the number of charged tracks is required to be four or six with net charge zero .
the maximum number of neutral clusters in an event is eight , and the number of photon candidates remaining after the application of the photon selection is required to be four or less .
the sum of the momenta of the lowest momentum @xmath75 and @xmath76 tracks is required to be greater than 550 mev ; this removes contamination from @xmath71 events .
the @xmath69 probability for a four - constraint kinematic fit to the decay hypothesis is greater than 0.01 .
the particle identification assignment of each charged track is @xmath77 .
a combined probability of the four - constraint kinematic fit and particle identification information is used to separate @xmath78 and the different particle assignments for the @xmath79 final states .
this combined probability , @xmath80 , is defined as @xmath81 where @xmath82 is the sum of the @xmath69 values from the four - constraint kinematic fit and those from each of the four particle identification assignments , and @xmath83 is the corresponding total number of degrees of the freedom used in the @xmath69 determinations . the particle assignment with the largest @xmath80
is selected , and further cuts on the kinematic fit probability and particle identification probability are imposed .
figure [ mks - mks ] shows a scatterplot of @xmath84 vs @xmath84 invariant masses for events with a @xmath28 mass between 3.2 and 3.6 gev .
the cluster of events in the lower left - hand corner indicates the presence of a @xmath85 signal .
a fit of a gaussian function to the @xmath86 mass distribution gives a peak mass at @xmath87 mev and a width @xmath88 mev that is consistent with the mc expectation for the mass resolution .
we select @xmath89 candidates by requiring the mass of both @xmath84 combinations in the event to be within @xmath90 of the nominal @xmath91 mass .
the invariant mass distributions for the @xmath28 , @xmath29 and @xmath85 events that survive all the selection requirements are shown in figs .
[ mallpppp ] , [ mallppkk ] and [ mallksks ] .
there are peaks corresponding to the @xmath27 states in each of the plots .
( the high mass peaks in figs . [ mallpppp ] and [ mallppkk ] correspond to the @xmath1 decays to all charged tracks final states that are kinematically fit with a fake low - energy photon . ) we fit the @xmath28 , @xmath29 or @xmath85 invariant mass distribution between 3.20 and 3.65 gev with three breit - wigner resonances convoluted with gaussian mass resolution functions and a linear background shape using an unbinned maximum likelihood method . in the fit ,
the mass resolutions are fixed to their mc - determined values and the widths of the @xmath22 and @xmath18 are fixed to the pdg average values of of 0.88 and 2.00 mev @xcite , respectively .
the results of the fit are listed in table [ tab - ppkk - res ] and shown in figs .
[ mallpppp ] , [ mallppkk ] and [ mallksks ] .
table [ tab - ppkk - res ] also lists the mc - determined efficiencies and mass resolutions .
if one of the four tracks is identified as a proton or antiproton , the event is assumed to be @xmath92 .
we assign probabilities to the remaining particle assignment using the same technique that was used for @xmath29 decays ; the combination with the highest probability is selected . the @xmath30 invariant mass distribution for the selected events is shown in fig .
[ mallpppr ] . here clear signals for all three @xmath27 states are apparent .
we fit the mass spectrum using the same method described in the previous section ; the results are listed in table [ tab - pppr - res ] and shown as the smooth curve in fig .
[ mallpppr ] . for the case
where all the tracks are kaons , the contamination from @xmath94 is not an important background , and the requirement on total momentum of the lowest momentum @xmath75 and @xmath76 tracks , which is aimed at removing these events , is not used .
the @xmath31 invariant mass distribution is shown in fig .
[ mallkkkk ] .
figure [ mphi - mphi ] shows a scatterplot of @xmath35 vs @xmath35 invariant masses for the events with @xmath31 mass between 3.2 and 3.6 gev .
the concentration of events in the lower left - hand corner of the plot indicates the presence of @xmath95 final states .
a fit to the @xmath35 mass distribution with a gaussian function gives a peak mass of @xmath96 mev and a width @xmath97 mev , consistent with mc expectations .
events where the mass of two @xmath35 combinations are in the range @xmath98 gev are identified as @xmath99 candidates .
the @xmath95 mass distribution for these events is shown in fig .
[ mallphiphi ] , where there are clear signals for the @xmath12 and @xmath18 .
the @xmath31 mass and @xmath95 mass plots are fitted with three breit - wigner resonances and two breit - wigner resonances , respectively , as described previously .
the results of the fit are listed in table [ tab - kkkk - res ] and are shown as smooth curves in figs .
[ mallkkkk ] and [ mallphiphi ] .
because of the large fraction of @xmath100 intermediate events observed in the @xmath31 mode and the significant difference between the detection efficiency for phase - space events and those coming from @xmath100 decays , the detection efficiency for the @xmath12 and @xmath101 channels is a weighted average of the phase space and @xmath100 efficiency .
the detection efficiencies and mass resolutions are listed in table [ tab - kkkk - res ] .
the @xmath103 decay channels have serious potential backgrounds from @xmath104 ( including @xmath105 ) and @xmath106 final states . to eliminate these backgrounds , we exploit the feature that there is one and only one @xmath91 with a secondary vertex in real @xmath32 events .
in each event , we determine @xmath107 , the number of two charged track combinations with net charge zero and effective mass within @xmath108 mev of @xmath109 , when the tracks are assigned a pion mass . the combination with mass closest to @xmath110 is considered to be a @xmath111 candidate .
the @xmath111 vertex is defined as the point of closest approach of these two tracks ; the primary vertex is defined as the point of closest approach of the other two charged tracks in the event .
two parameters are used to identify the @xmath111 : the distance between primary vertex and secondary vertex in the @xmath61 plane , @xmath112 , and the cosine of the angle between the @xmath111 momentum vector and its vertex direction @xmath113 , which is expected to be very near unity for a real @xmath111 event . candidate @xmath114 events are selected by requiring the mass of the @xmath111 candidate determined from the track four - vectors returned by the 4c - fit to be within @xmath115 of the nominal @xmath116 mass , @xmath117 , @xmath118 mm , and @xmath119 . in the invariant mass distribution of the selected events , shown in fig .
[ mallkskp ] , only a @xmath120 signal is prominent .
the mc simulation indicates that the numbers of events in the the @xmath121 and @xmath122 mass region are consistent with residual backgrounds from @xmath104 , @xmath105 and @xmath106 final states .
we set upper limits on the branching fractions of @xmath121 and @xmath122 .
the @xmath32 invariant mass distribution between 3.20 and 3.65 gev are fitted with the procedure described above .
the mass resolutions are fixed at their mc - determined values , the width of the @xmath12 is fixed at the recent bes value of 14.3 mev @xcite and those of the @xmath22 and @xmath18 at their pdg values @xcite .
the mass of the three @xmath21 states are also fixed at their pdg @xcite values .
the fit results are listed in table [ tab - kskp - res ] and are shown as a smooth curve in fig .
[ mallkskp ] .
after the selections based on the kinematic fit and particle i d , the main background to the @xmath124 decays comes from the decay chain @xmath125 .
the requirement on the total momentum of the lowest momentum @xmath75 and @xmath76 tracks removes one third of the mc - simulated events while rejecting almost all the @xmath94 background .
the @xmath33 invariant mass distribution for the selected events is shown in fig .
[ malltpp ] , where prominent signals for all three @xmath27 states can be seen .
the smooth curve in the figure is the result of the fitting procedure described above .
the results of the fit and the mc - determined efficiencies and resolutions are listed in table [ tab - tpp - res ] .
we determine branching fractions from the relation @xmath126 where the values for @xmath127 are taken from the pdg tables @xcite . for the @xmath85 [ @xmath95 ] channel , a factor of @xmath129 [ @xmath130
is included in the denominator .
systematic errors common to all modes include the uncertainties in the total number of @xmath131 events ( 8.2% ) and the @xmath132 branching fractions ( 8.6% , 9.2% and 10.3% for @xmath121 , @xmath120 and @xmath122 , respectively ) .
other sources of systematic errors were considered .
the variation of our results for different choices of the selection criteria range from 10% for high statistics channels to 25% for those with low statistics .
the systematic errors due to the statistical precision of the mc event samples range from 2% to 5% depending on the detection efficiencies of the channels .
changes in the detection efficiency when the phase space event generator is replaced by one using possible intermediate resonant states indicate that the systematic error on the efficiency due to the unknown dynamics of the decay processes is 15% .
the variation of the numbers of observed events due to shifts of the mass resolutions and the total widths of the @xmath60 states is 7% ; that coming from changes in the shape used for the background function is less than 5% .
the total systematic error is taken as the quadrature sum of the individual errors and ranges from 25% to 35% , depending on the channel .
the branching fraction results are listed in table [ chic - result ] , where all bes results for @xmath27 branching fractions are given , including those for the two - charged track modes reported in ref .
@xcite . in each case , the first error listed is statistical and the second is systematic . for comparison , we also provide the previous world averages for those channels when they exist @xcite . our branching fractions for @xmath2 , @xmath3 , @xmath4 , @xmath5 , @xmath6 and @xmath7 ( j=0,1,2 ) are the first reported measurements for these decays .
the results for @xmath121 and @xmath4 are in agreement with the isospin prediction of the @xmath27 decays compared with the corresponding @xmath35 branching ratios . for the other decay modes , signals with large statistics
are observed and the corresponding branching fractions are determined with precisions that are significantly better than those of existing measurements .
note that our results are consistently lower than the previous measurements , sometimes by as much as a factor of two or more .
we can find no obvious explanation for these discrepancies .
we determine @xmath133 by fitting the combined invariant mass distribution of all of the channels discussed above to three resolution - broadened breit - wigner functions with the resolution fixed at the value of 13.8 mev , which is determined from fits to the @xmath22 and @xmath18 , and the total widths of the @xmath22 and @xmath18 fixed at the pdg values @xcite .
the masses of all three @xmath27 states and the total width of the @xmath12 are left as free parameters .
the results of the fit for @xmath134 ( @xmath135 mev ) and @xmath136 ( @xmath137 mev ) agree with the pdg values within errors .
the fit value for @xmath133 is @xmath138 mev , where the error is statistical .
the fit gives a total width for the @xmath12 that is in good agreement with the recently reported bes result @xcite .
figure [ etac_mass ] shows the combined invariant mass distribution for the @xmath139 , @xmath140 , @xmath141 , and @xmath32 channels in the region of the @xmath13 , where an @xmath13 signal is evident .
superimposed on the plot is a fit to the spectrum using a resolution - smeared breit - wigner line shape with a mass that is allowed to vary , a total width fixed at the pdg value of @xmath142 mev @xcite , and a fourth - order polynomial background function .
the fit gives a total of @xmath143 events in the peak and has a @xmath144 , which corresponds to a confidence level of 27.9% .
the mass value from the fit is @xmath145 mev , where the error is statistical .
( a fit with only the background function and no @xmath13 has a confidence level of 0.8% . ) the systematic error on the mass determination includes a possible uncertainty in the overall mass scale ( @xmath146 mev ) , which is determined from the rms average of the differences between the fitted values for @xmath134 and @xmath136 and their pdg values .
the systematic errors associated with uncertainties is the particle s total widths and the experimental resolutions ( @xmath147 mev for @xmath39 and less than @xmath148 mev for @xmath133 ) are added in quadrature .
the resulting masses and errors are : @xmath149 and @xmath150 the precision of our @xmath133 measurement represents a substantial improvement on the existing pdg value of @xmath151 mev @xcite .
our result for @xmath39 agrees with the dm2 group s value of @xmath152 mev @xcite and is 2.4 standard deviations below the e760 group s result of @xmath153 mev @xcite .
events of the type @xmath17 in a @xmath154 @xmath1 event sample are used to determine branching fractions for @xmath27 decays to four and six charged particle final states .
our results for @xmath32 , @xmath85 , @xmath95 , and @xmath31 are the first measurements for these decays .
the branching fractions for @xmath155 , @xmath29 , @xmath30 , and @xmath33 final states are measured with better precision and found to be consistently lower than previous measurements . @xmath133 and
@xmath39 were determined using the same data sample .
we thank the staffs of the bepc accelerator and the computing center at the institute of high energy physics , beijing , for their outstanding scientific efforts .
this project was partly supported by china postdoctoral science foundation .
the work of the bes collaboration was supported in part by the national natural science foundation of china under contract no .
19290400 and the chinese academy of sciences under contract no .
h-10 and e-01 ( ihep ) , and by the department of energy under contract nos .
de - fg03 - 92er40701 ( caltech ) , de - fg03 - 93er40788 ( colorado state university ) , de - ac03 - 76sf00515 ( slac ) , de - fg03 - 91er40679 ( uc irvine ) , de - fg03 - 94er40833 ( u hawaii ) , de - fg03 - 95er40925 ( ut dallas )
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rev . * d58 * , 092006 ( 1998 ) . in the determination of the number of @xmath1 events , the branching ratio @xmath156 ( r. m. barnett _ et al .
_ , ( particle data group ) , phys . rev . *
d54 * part i ( 1996 ) ) was used . using the @xmath157 events sample and the particle identification procedures described in the text
, we determine the branching fraction @xmath158 which is in good agreement with the world average @xcite . | hadronic decays of the p - wave spin - triplet charmonium states @xmath0 are studied using a sample of @xmath1 decays collected by the bes detector operating at the bepc storage ring . branching fractions for the decays @xmath2 , @xmath3 , @xmath4 , @xmath5 , @xmath6 and @xmath7
are measured for the first time , and those for @xmath8 , @xmath9 , @xmath10 and @xmath11 are measured with improved precision .
in addition , we determine the masses of the @xmath12 and @xmath13 to be @xmath14 mev and @xmath15 mev . 0.1 cm plus 1pt minus 1pt | hep-ex9812016 |
one of the interesting characteristics of the percolation model is the existence of universal quantities , which are independent of the microscopic qualities of the system .
critical exponents and amplitude ratios are examples of one class of universal quantities that are only dependent on the dimensionality of the system .
shape - dependent universal quantites , which depend on the shape of the boundary as well as the dimensionality of the system , have been identified in the ising model by privman and fisher @xcite , mller @xcite and kamieniarz and blte @xcite . in percolation ,
recent research has been focused on shape - dependent universal quantities such as the crossing probabilities @xcite and the excess number of clusters @xcite .
the excess number of clusters @xmath0 on a two dimensional system is defined by @xcite @xmath8 as @xmath9 , where @xmath10 is the total number of the clusters in a system of area @xmath11 , and @xmath1 is the number of clusters per unit area in an infinite system . in ( [ definition ] ) , it is assumed that the system is at criticality , and that there are no boundaries to the system .
the average density of clusters @xmath1 is a non - universal quantity , meaning that it depends on the microscopic qualities of the system and will have different values for different systems . in two - dimensional percolation , exact values of @xmath1 have been determined theoretically for bond percolation on two lattices @xcite .
also , simulations have provided precise values of @xmath1 for 2d @xcite and 3d @xcite lattice percolation . while @xmath1 is a system - dependent quantity ,
@xmath0 is a universal quantity that is independent of the type of percolation but does depend on shape of the system boundary and the dimensionality of the system .
previous work has demonstrated the shape dependence of @xmath0 for toroidal systems using @xmath12 sytems with periodic boundary conditions , for which @xmath13 has been found exactly @xcite .
for an @xmath2 system , the theoretical prediction is @xmath14 @xcite , which agrees with the numerical value @xmath15 @xcite found for site and bond percolation on square and triangular lattices . for 3d lattice percolation ,
theoretical predictions for @xmath16 ( the excess number per unit length ) do not exist ; however , the universality of @xmath16 has been confirmed using numerical values for different @xmath17 cubic systems @xcite . in the present paper , we are interested in finding @xmath0 for percolation on the surface of a sphere , another two dimensional surface with no boundaries , for which however no theoretical prediction exists . in order to do this , we use the 2d continuum `` swiss cheese '' model of percolation , for which the critical density has recently been found to high accuracy @xcite .
in addition to the spherical systems , we are interested in using this continuum model to detemine @xmath0 for @xmath2 toroidal systems and comparing it to the theoretical prediction . in order to find @xmath0 , we must also find @xmath1 , the number of clusters per unit area for the continuum percolation model .
we note that continuum percolation on the surface of a sphere ( and hypersphere ) has also been studied in a recent publication @xcite , in the context of diffusion on fractal clusters . in the following section ,
we describe the simulations that were used to model the square and spherical system
. then we present and summarize our results for the excess number of clusters and the average density of clusters in these systems .
the basic `` cluster counting '' algorithm that was used for determining the number of clusters in the @xmath2 toroidal and spherical continuum systems is identical to the algorithm we used to study similar problems in lattice percolation . however , the implementation of this algorithm was quite different for the two continuum systems .
the @xmath2 system with periodic boundary conditions was first divided into squares of unit area .
discs , whose radius @xmath18 is equal to 0.5 , were distributed into each of the unit squares in the plane using a poisson function , where the probability @xmath19 that there are @xmath20 particles in a given volume @xmath21 is given by @xmath22 here @xmath23 is the critical density of discs ( @xmath24 for 2d continuum percolation of discs @xcite ) and @xmath25 is the volume of each of the unit squares .
[ note : the critical density of discs is often referred to as @xmath1 in the continuum percolation literature , but in order to differentiate from the average density @xmath1 , we use @xmath23 here . ] the algorithms in ref.@xcite were used to generate numbers with this distribution . for each unit square , a random number @xmath20 was generated and if @xmath26 , then @xmath20 discs were placed within that square .
the @xmath27- and @xmath28- coordinates of each disc were stored in two one - dimensional arrays , which were indexed by the order that the discs were distributed ( i. e. , the first sphere placed is numbered 0 , the second is numbered 1 , @xmath29 ) . the index of the first and last disc distributed in the square were also stored in two one - dimensional pointer arrays .
after discs were placed in each unit square , a search was made for clusters , starting with the first disc placed in the first unit square .
the search checked only the neighboring eight unit squares , as opposed to the entire system , for discs .
if the distance between any two discs was less than or equal to 1 , then the two were considered to be in the same cluster .
the coordinates of each disc in the cluster were stored in two one - dimensional list ( `` growth '' ) arrays , which were indexed by the order that the discs were determined to be part of the cluster .
after the first disc was checked for overlapping neighbors , then subsequent discs on the list arrays were checked , in the order that they were placed on the list , for overlapping neighbors .
this process was continued until a cluster stopped growing ; at which time , the same search for each of the unchecked discs in the current square was performed .
after each of the discs in a square were checked , the search moved to the next square that had unchecked discs remaining .
this cluster search was continued until all discs within the system were determined to be part of a cluster .
when simulating the spherical systems , we encountered two areas of difficulty that were not present when using the @xmath2 system .
first , we were nt able to divide the surface into smaller sub - sections because of the difficulty in producing equal area sub - sections . instead , the discs were distributed on the surface of the entire sphere , using ( [ poisson ] ) to determine the number , and the @xmath30-coordinate and the cosine of the @xmath31-coordinate for each disc were stored in two one - dimensional arrays .
the entire list was searched for each neighbor check , resulting in a much slower simulation for larger spherical systems compared to the toroidal system .
we began our cluster search algorithm with the first disc that was placed on the sphere .
the `` great circle '' distance between two discs @xmath32 , whose coordinates are ( @xmath33 ) and ( @xmath34 ) , is given by @xmath35 which is the arc length between two points on the surface of a sphere of radius @xmath36 .
two discs were considered to be overlapping if @xmath37 .
the cluster search algorithm was applied to the spherical system until every disc on the sphere was checked .
the second area of difficulty we encountered was in the selection of a criterion for the critical threshold that is consistant with the criterion used for a @xmath2 system .
different choices ( density of discs , surface area coverage , etc . ) lead to different numbers of discs . as a result of this ambiguity , we considered two different methods to determine the average number of discs that were required . in general , the total number of the discs @xmath38 is defined as @xmath39 where @xmath40 is the area covered by each disc , @xmath41 is the coverage of the discs , and @xmath11 is the total area of the system . for a finite system , @xmath42 , where @xmath23 is the critical density of discs .
first , we used `` type 1 '' discs , where we chose the number of discs per unit area on the sphere to have the same value as the @xmath2 critical system .
therefore , the number of discs is @xmath43 , where @xmath36 is the radius of the sphere and @xmath24 is the critical density of discs for a flat system . for the second type of discs ( `` type 2 '' discs )
, we determined the number of discs required to keep the critical coverage the same as it was for a flat system .
the area covered by each disc is calculated by @xmath44 where @xmath36 is the radius of the sphere , @xmath31 is the angle between the horizontal axis of the sphere and a point on the surface of the sphere , and @xmath45 is the angle between the edge of the disc and the horizontal axis as shown in fig .
[ geo ] . in this case
@xmath46 , therefore , the area covered by each type 2 disc is @xmath47 the number @xmath38 of these discs that would be required to achieve the same coverage as the @xmath2 system ( @xmath48 ) was determined using ( [ coverage ] ) , with the result that @xmath49 where @xmath36 is the radius of the sphere , where @xmath50 . using these two simulations
, we were able to study planar @xmath2 systems , where @xmath51 8 , 16 , 32 , 64 , and 128 , and spherical systems with radius @xmath52 5 , 6 , 10 , and 15 .
the simulations counted the number of clusters that existed within the system .
numerous realizations ( @xmath53 for spherical systems and @xmath54 for planar systems ) were averaged over in order to calculate the density of clusters within each system .
the random numbers used in these simulations were generated by the four - tap shift - register rule @xmath55 , where @xmath56 is the exclusive - or operation @xcite .
the number of clusters @xmath10 present in a system which has area @xmath11 is expected to follow ( [ definition ] ) .
using our simulations , we were able to calculate the total number of clusters present and then determine the overall density of clusters @xmath57 for each system size , which by rearranging ( [ definition ] ) is @xmath58 , or @xmath59
\\ \label{excesssphere } & n = n_c + b/(4\pi r^2 ) + \ldots \quad[{\rm spherical\ system } ] .
\end{aligned}\ ] ] figure [ xs ] is a plot of @xmath20 vs. @xmath60 for each of the three systems . by fitting these plots with equations ( [ excessll ] ) and ( [ excesssphere ] )
, we were able to determine the values of @xmath0 and @xmath1 for these systems from the slope and the intercept , respectively , which are summarized in table [ results ] .
several observations can be made from these results .
first , the average density of clusters for the spherical systems and for the @xmath2 system is , as expected , the same ( within numerical error ) .
also , the excess number of clusters @xmath0 for the planar @xmath3 continuum system is consistent with the theoretical value ( @xmath61 ) @xcite and the simulation value ( @xmath15 ) @xcite found for the @xmath3 lattice percolation system , which further confirms the universality of this quantity . finally , the values of the excess number of clusters @xmath0 for the spherical systems are different than that found for the planar system , which is expected because the shape and topology of the boundary is different . however , there is a difference in the value of @xmath0 for the two spherical systems , which is a result of the different definitions for the critical number of discs placed on the surface of the sphere . when the total number of discs @xmath38 that are placed on the sphere in each case are compared , the difference in @xmath0 becomes more understandable : @xmath62 \nonumber\\ & m = \rho_c ( 4\pi r^{2 } ) + \rho_c ( \pi/12 ) + o(1/r^{2 } \ldots ) \quad [ { \rm type\ 2\ discs } ] .
\label{comparenumber}\end{aligned}\ ] ] wherethe second expression follows from a taylor series expansion of ( [ number3 ] ) .
the first term in the expression for type 2 discs is exactly the number of type 1 discs present on the surface of the sphere .
however , the second term shows that of the order of one more disc is present on the surface of the sphere than type 1 disc at the same ( critical ) density .
these relatively small differences in the number of discs present on the sphere can cause a significant difference in the density of clusters . in general , about the critical point , one expects the density of clusters to behave as @xcite @xmath63 where @xmath64 in two dimensions , @xmath65 , and @xmath66 . then the number of clusters @xmath10 is @xmath67 therefore , if @xmath38 changes by an amount of order 1 , then @xmath10 will also change by an amount of order 1 , which is the same order as the @xmath0 term in ( [ definition ] ) .
thus , the slight differences in the two definitions of the total number of discs @xmath38 even though they are asymptotically identical for large @xmath36 lead to non - zero differences in the value of @xmath0 .
the universality of the excess number of clusters @xmath0 has been demonstrated using 2-d and 3-d lattice percolation , but it had never been tested with a continuum model .
our result @xmath68 for the @xmath3 planar system is the same as found for the @xmath2 lattice percolation system @xcite , thus it further confirms the universality of this quantity .
the excess number of clusters had never been calculated for spherical systems . by using the swiss cheese continuum model
, we were able to study percolation on the surface of sphere .
we found that the excess number of clusters ( @xmath69 , @xmath70 ) is slightly dependent upon the definition of the critical density of discs for the spherical system .
we believe that the type 2 definition is the most reasonable as it is based upon the idea that the surface coverage is the same as the planar system , which is consistent with the fact that the surface coverage is invariant when the @xmath2 periodic system is deformed to a torus .
therefore , we propose the value @xmath71 for the spherical surface . while theoretical results exist for @xmath0 for the @xmath72 periodic system , there is no prediction for the sphere .
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.values of the critical average density of clusters @xmath1 and the excess number of clusters @xmath0 for two - dimensional `` swiss cheese '' model .
numbers in parenthesis represent the error in the last digit . [ cols="<,<,<",options="header " , ] | monte carlo simulations were performed in order to determine the excess number of clusters @xmath0 and the average density of clusters @xmath1 for the two - dimensional `` swiss cheese '' continuum percolation model on a planar @xmath2 system and on the surface of a sphere .
the excess number of clusters for the @xmath3 system was confirmed to be a universal quantity with a value @xmath4 as previously predicted and verified only for lattice percolation .
the excess number of clusters on the surface of a sphere was found to have the value @xmath5 for discs with the same coverage as the flat critical system .
finally , the average critical density of clusters was calculated for continuum systems @xmath6 . # | cond-mat0102008 |
pseudoknots and pseudolinks are knots and links about which we have incomplete information .
this incompleteness is expressed in diagrams by the appearance of _ precrossings _ that have no over or under designation . in @xcite , pseudoknots are defined as equivalence classes of knot or link diagrams ( called _ pseudodiagrams _ in @xcite ) where some crossing information may be missing .
we picture the precrossings that lack definitive over / under information , as undecorated self - intersections .
these unknown crossings can be involved in reidemeister - like moves in predictable ways .
( see figure [ rmoves ] . ) in terms of their behavior , pseudoknots act like 4-valent rigid vertex spatial graphs ( as in @xcite ) or singular knots ( as in @xcite ) .
the one important difference between pseduoknots and these other objects is entirely characterized by the pr1 move .
the pr1 move allows us to eliminate a single unknown crossing at a curl .
the reason this move is reasonable for pseudoknots is that , regardless of whether we replace the precrossing with a positive or negative classical crossing , it can be removed with an r1 move . on the other hand , vertices or singularities
are not removable in rigid vertex spatial graphs or singular knots .
while several new pseduoknot invariants have been introduced @xcite , we introduce a schema for a large collection of new invariants .
these invariants of pseudoknots , defined by replacing precrossings with tangles , can be adapted to serve as invariants for 4-valent rigid vertex spatial graphs and singular knots as well .
in @xcite , vassiliev showed how to extend an invariant @xmath0 of links to the class of singular links .
he did so by defining the _ derivative _ of an invariant as follows : @xmath1@xmath2@xmath3@xmath4 the derivative of an invariant is an invariant of singular links with a single double - point . in general the @xmath5th derivative ,
defined recursively as follows , is an invariant of a singular link with @xmath5 double - points .
note that when @xmath6 , @xmath7 is the first derivative @xmath8 , and @xmath9 .
@xmath10@xmath11@xmath12@xmath4 this idea of extending invariants by taking linear combinations of invariant values for diagrams that are related by tangle replacements can be generalized ( see @xcite ) .
for instance , we could define @xmath13 recursively as follows .
@xmath14@xmath15@xmath16@xmath17@xmath4 if we are careful about which invariants @xmath0 we use and how we choose our coefficients @xmath18 , @xmath19 , and @xmath20 , this equation can be used to define an invariant of singular links , rigid vertex spatial graphs , or pseudoknots and links .
note that any choice of coefficients that yields an invariant of singular links is also an invariant of pseudoknots if it satisfies the following additional relation .
in general , suppose @xmath0 is a polynomial or integer , real , or complex - valued invariant of knots and links that behaves nicely under and connected sum .
in particular , suppose @xmath0 obeys the following property .
@xmath21 here , @xmath22 and @xmath23 denote arbitrary pseudoknot or link diagrams , and @xmath24 indicates a connected sum .
we observe that the jones and alexander polynomials satisfy this property , among other commonly used invariants .
once a suitable link invariant has been chosen , we choose a set of oriented , rational ( 2,2)-tangles , @xmath25 .
recall that , since each @xmath26 is a rational tangle , @xmath27 satisfies the symmetries shown in figure [ tangle_sym_free ] , by the flip theorem for rational tangles @xcite .
some examples of oriented rational tangles are shown in figure [ example_t ] .
+ * vertical flip * + + * horizontal flip * + ( a)(b)(c ) + + ( d)(e)(f ) now that we have our desired link invariant and set of tangles , we define @xmath28 recursively as follows for an oriented pseudodiagram @xmath29 containing precrossing @xmath20 .
@xmath30 in this definition , @xmath31 denotes the pseudodiagram where tangle @xmath32 is inserted at precrossing @xmath20 ( respecting the orientations of the tangles and the pseudodiagram ) , and the symbol @xmath33 is a variable .
if a given pseudodiagram @xmath34 contains no precrossings , we define @xmath35 .
the restrictions on our allowable invariants and tangles guarantee that @xmath36 is an invariant of singular links and 4-valent rigid vertex spatial graphs . to guarantee invariance under _ all _ pseudo - reidemeister moves , we need to further impose a relation among our coefficients .
@xmath37 we take @xmath28 modulo a linear combination of the values of the link invariant of the denominator closures of each of our tangles to ensure that @xmath38 is invariant under pr1 , and hence , is an invariant of oriented pseudoknots and links . note that we are implicitly assuming that the value of our chosen invariant is nonzero on the denominator closure of at least one of our tangles @xmath39 . to gain an understanding of why each of the restrictions imposed above is necessary , let us prove that @xmath38 is an invariant of oriented pseudolinks .
we begin by acknowledging that , since @xmath0 is a link invariant , invariance of @xmath13 for classical reidemeister moves is immediate .
pr1 invariance is shown in figure [ pr1_pf ] .
line 1 follows from the definition of @xmath13 , line 2 follows from our required connected sum property of the link invariant @xmath0 ( which ensures that @xmath13 behaves similarly under connected sum ) , line 3 is an algebraic distribution , and line 4 is a consequence of the relation @xmath40 .
the proof of invariance under the pr2 move illustrates why our tangles @xmath26 are chosen to have the symmetries of rational tangles .
a flype together with a rational tangle symmetry is required to show invariance of @xmath13 under each oriented pr2 move , pictured in figure [ pr2_pf ] . finally , pr3 invariance is straightforward .
figure [ pr3_pf ] illustrates how @xmath13 is invariant under pr3 since a strand that lies entirely above or entirely below a tangle can be moved freely past the tangle .
this completes our proof that @xmath13 is an invariant of pseudoknots .
note that we also proved that @xmath36 is a singular link and 4-valent rigid vertex spatial graph invariant . +
* this tangle insertion method can be generalized beyond rational tangle insertion by asking that the tangles satisfy the symmetries shown in figure [ tangle_sym_free ] .
we shall take up this aspect of the construction in another paper .
+ one obvious question to ask now that we have created a schema for creating pseudoknot invariants is : how does this schema relate to known pseudoknot invariants ? one of the simplest , yet most powerful invariants of pseudoknots is the _ weighted resolution set _ , or _
were - set _ , introduced in @xcite .
the were - set of a pseudodiagram is the set @xmath41 of pairs @xmath42 of knot types @xmath43 that can be realized by some choice of crossing information for the diagram s precrossings .
the number @xmath44 is the probability that knot @xmath43 will be produced if crossing information is randomly chosen , where positive and negative crossings are both equally likely .
it was proven that the were - set is indeed a pseudoknot invariant .
how does this invariant relate to our schema ?
first , we notice that choosing crossing information for a precrossing is equivalent to inserting a basic + 1 or -1 rational tangle ( i.e. tangles ( a ) and ( b ) in figure [ example_t ] ) , so let our tangle set @xmath45 consist of these two tangles .
next , let @xmath0 be a knot invariant such that @xmath46 if @xmath47 is the unknot ( such as the jones polynomial or the alexander polynomial ) . then choosing the coefficients @xmath48 will satisfy the relation @xmath40 , since this amounts to @xmath49!{\hcross } ! { \hcap[1 ] } [ l]!{\hcap[-1 ] } } ) + \frac{1}{2}\mathcal{i}(\xygraph { ! { 0;/r1.0pc/ : } [ u(.5)]!{\htwist }
! { \hcap[1 ] } [ l]!{\hcap[-1 ] } } ) = \frac{1}{2}+\frac{1}{2}=1,\ ] ] so an invariant @xmath13 is determined by our ingredients , following the recipe above .
the invariant we have just created with our schema is equivalent to the following composition of the were - set with @xmath0 .
@xmath50 note that , if we make a different coefficient choice , we no longer recover the were - set .
for instance , making the choice @xmath51 and @xmath52 would correspond to a distinct variant of the were - set where positive crossings are chosen with probability @xmath53 and negative crossings are chosen with probability @xmath54 .
let us return to our original motivation for tangle insertion invariants : the derivative of an invariant . in our new framework , the tangles @xmath55 that are used to define the derivative are tangles ( a ) and ( b ) in figure [ example_t ] , respectively .
their denominator closures are both the unknot , @xmath47 .
the coefficient @xmath56 is 1 and @xmath57 .
notice , then , that our imposed relation @xmath40 states that @xmath58 . but clearly @xmath59 .
so , oddly enough , our original motivating example is not an instance of our scheme after all .
this is one of the reasons why the generalization is so valuable .
it allows for the creation of many new invariants of pseudoknots just as our old singular knot invariants fail to be useful .
instead , let us consider the other example we mentioned above , namely : @xmath14@xmath15@xmath16@xmath17@xmath4 here , we take tangles ( a ) , ( b ) , and ( c ) from figure [ example_t ] to form the tangle set @xmath45 , and we do nt yet specify values for our coefficients .
@xmath60!{\hcross } ! { \hcap[1 ] } [ l]!{\hcap[-1 ] } } ) + b\overline{\mathcal{i}}(\xygraph { ! { 0;/r1.0pc/ : } [ u(.5)]!{\htwist }
! { \hcap[1 ] } [ l]!{\hcap[-1 ] } } ) + c\overline{\mathcal{i}}(\xygraph { ! { 0;/r1.0pc/ : } [ u(0.5)r]!{\hcap[1 ] } ! { \hcap[-1 ] } [ ll]!{\hcap[1 ] } ! { \hcap[-1 ] } } ) = 1\ ] ] suppose we choose our link invariant @xmath0 to be the jones polynomial , @xmath61 .
we will compute @xmath61 using the bracket polynomial , @xmath62 , via the relation @xmath63 where @xmath64 is the writhe of the link @xmath23 as in @xcite , and @xmath65 is a variable .
then any choice for @xmath18 , @xmath19 , and @xmath20 satisfying @xmath66 will do , where @xmath67 is the polynomial such that @xmath68 .
in particular , we could choose @xmath69 and @xmath70 .
note that this choice is equivalent to simply starting with the singleton tangle set consisting of the 0-tangle , ( c ) .
let us apply this invariant to a pseudo - trefoil and its mirror image , shown in figure [ tref ] .
when we insert tangle ( c ) into both pseudodiagrams , we get the links shown in figure [ tref_insert ] .
the link on the left is l2a1@xmath71 and the link on the right is l2a1@xmath72 .
both links have the same bracket polynomial value , @xmath73 , since they are the same as unoriented links .
( see @xcite , p. 151
for this computation . )
but the writhe of l2a1@xmath71 is 2 while the writhe of l2a1@xmath72 is -2 .
hence , the jones polynomial of l2a1@xmath71 is @xmath74 while the jones polynomial of l2a1@xmath72 is @xmath75 thus , the two pseudoknots shown in figure [ tref ] are distinct , since their @xmath13 values are @xmath76 and @xmath77 , respectively . another interesting pair of examples , pictured in figure [ perko ] , is derived from the famous perko pair , pictured in figure [ perko_orig ] .
let us use the alexander polynomial and take our tangle set @xmath45 to be the set consisting of tangles ( a ) and ( b ) .
if we insert tangle ( b ) at both precrossings in both diagrams , we recover the perko pair itself , which were shown by perko to be equivalent . on the other hand , the knot @xmath78 is obtained by inserting tangle ( a ) at one precrossing and tangle ( b ) at the other precrossing ( regardless of the order of insertion ) in both diagrams .
finally , if tangle ( a ) is inserted at both precrossings in both diagrams as in figure [ perko_pos ] , the two resulting knots are distinct .
one knot is determined by the alexander polynomial to be the knot @xmath79 , while the other is knot @xmath80 .
hence , the invariant @xmath13 derived from the alexander polynomial and tangle set @xmath45 ( where , say , @xmath48 ) can distinguish these two pseudoknots . +
* remark . *
notice that , because our pseudoknot invariant was defined using the alexander polynomial and tangles ( a ) and ( b ) with the coefficient choice @xmath48 , the were - set together with the alexander polynomial can alternatively be used to distinguish the perko pseudoknots .
+ .,height=124 ] in our next example , we consider the pseudodiagram related to the borromean rings pictured in figure [ borr_insert ] ( on the left ) . here
, we use the tangle insertion invariant @xmath13 that is defined using the jones polynomial ( computed via the bracket , with variable @xmath65 ) and take our tangle set to be the singleton set containing the tangle ( c ) . as shown in figure [ borr_insert ] , inserting the tangle ( c ) produces the whitehead link . since this link has jones
polynomial @xmath81 @xmath13 can be used to prove that the original borromean pseudoknot is nontrivial .
note that , if we had used a linking number invariant rather than the jones polynomial to define our pseudoknot invariant , we would not have been able to detect the nontriviality of this example since the linking number of the whitehead link is 0 .
.,height=115 ] these are just a few examples to illustrate how invariants of pseudoknots can be derived from classical link invariants and tangle sets .
we intend to consider many more examples in future work .
given this framework for developing pseudoknot invariants , there are many questions that have yet to be be explored .
we provide the reader with an initial list of open questions . 1 . are there examples of pseudoknot pairs that can only be distinguished using a tangle set @xmath45 that contains more complex tangles than the basic tangles ( a ) , ( b ) , and ( c ) in figure [ example_t ] ?
for instance , can tangles ( d ) , ( e ) , or ( f ) be used to construct more powerful pseudoknot invariants ? 2 .
given two arbitrary distinct pseudoknots @xmath82 and @xmath83 , does there exist a classical link invariant @xmath0 and tangle set @xmath45 such that the corresponding pseudoknot invariant @xmath13 distinguishes @xmath82 and @xmath83 ?
3 . can our invariant schema be generalized to include invariants of other sorts , e.g. the matrix of linking numbers ?
( this can be done for the case of a single precrossing . )
4 . is there a relationship between the gauss - diagrammatic pseudoknot invariants defined in @xcite and the invariant schema presented here ? 5 .
can we determine if a given reduced pseudodiagram has the fewest number of precrossings among all equivalent diagrams ?
specifically , can our tangle insertion invariants be used for this purpose ?
l. h. kauffman and s. lambropoulou .
classifying and applying rational knots and rational tangles . _
physical knots : knotting , linking , and folding geometric objects in _
@xmath84 ( las vegas , nv , 2001 ) .
223259 , contemp .
soc . , providence , ri _ ( 2002 ) . | _ the notion of a pseudoknot is defined as an equivalence class of knot diagrams that may be missing some crossing information .
we provide here a topological invariant schema for pseudoknots and their relatives , 4-valent rigid vertex spatial graphs and singular knots , that is obtained by replacing unknown crossings or vertices by rational tangles . _ | 1603.03808 |
mass loss via stellar winds is thought to play a dominant role in the evolution of massive o - type stars , because of the loss of _ mass _ , as winds `` peel off '' the star s
outermost layers ( conti 1976 ) , as well as through the associated loss of _ angular momentum _
( e.g. langer 1998 , meynet & maeder 2002 ) .
however , during the last decade , large uncertainty has been pointed out regarding our quantitative knowledge of the mass - loss rates of massive stars , as stellar winds have been revealed to be clumped , resulting in empirical rates that have been overestimated . although it had been known for decades that o - type winds are clumped ( lupie & nordsieck 1987 , eversberg et al .
1998 ) , the severity did not appear to be fully recognized until bouret et al .
( 2005 ) and fullerton et al .
( 2006 ) claimed mass - loss reductions of factors @xmath23 - 7 and @xmath220 - 130 respectively in comparison to unclumped h@xmath3 and radio mass - loss rates ( e.g. lamers & leitherer 1993 ) .
the h@xmath3 diagnostics depends on the density squared , and are thus sensitive to clumping , whilst ultraviolet p cygni lines such as pv are _ in_sensitive to clumping as these depend linearly on the density .
the above - mentioned bouret et al . and
fullerton et al .
analyses were based on models where the wind is divided into a portion of the wind containing all the material with a volume filling factor @xmath4 ( the reciprocal of the clumping factor ) , whilst the remainder of the wind is assumed to be void .
this pure _ micro_-clumping approach is probably an oversimplification of the real situation , but it provides interesting insights into the potential mass - loss rate reductions . in reality ,
clumped winds are likely porous , with a range of clump sizes , masses , and optical depths .
_ macro_-clumping and porosity have been investigated with respect to both the spectral analyses ( e.g. oskinova et al .
2007 , sunqvist et al .
2010 , surlan et al .
2012 ) as well as the radiative driving ( muijres et al .
the upshot from these studies is that o star mass - loss rates may only be reduced by a moderate factor of @xmath23 ( repolust et al .
2004 , puls et al .
2008 ) , which would bring their clumping properties in agreement with those of wolf - rayet ( wr ) winds , for which similar moderate clumping factors have been derived ( hamann & koesterke 1998 ) .
the latter are based on the analysis of emission line wings due to electron scattering , which have the advantage that they do not depend on detailed ionization fractions and abundances of trace elements .
these moderate clumping factors would imply that massive star evolution modelling is not affected , as current state - of - the - art rotating stellar models ( e.g. georgy et al .
2011 ; brott et al .
2011 ) already employ moderately reduced rates via the theoretical relations of vink et al .
( 2000 ) . in light of the severe mass - loss reductions claimed e.g. by fullerton et al .
( 2006 ) , smith & owocki ( 2006 ) argued that the integrated mass loss from stationary stellar winds for very massive stars ( vms ) above @xmath050 may be vastly insufficient to explain their role as the progenitors of wr stars and stripped - envelope ibc supernovae . instead , smith & owocki argued that the bulk of vms mass loss is likely of an eruptive rather than a stationary nature .
in particular , they highlighted the alternative option of eruptive mass loss during the luminous blue variable ( lbv ) phase .
in view of the new porosity results , the arguments of smith & owocki ( 2006 ) however seem to have lost weight .
furthermore , quantitative estimates on the integrated amount of eruptive mass loss are hard to obtain as both the eruption frequency , and the amounts of mass lost per eruption span a wide range ( of a factor 100 ) with lbv nebular mass estimates varying from @xmath20.1 in pcygni to @xmath210 in @xmath5car , as discussed by smith & owocki ( 2006 ) .
moreover , the energies required to produce such giant mass eruptions are very high ( @xmath6erg ) , and their energy source is unknown . soker ( 2004 ) discussed that the energy and angular momentum required for @xmath5car great eruption can not be explained within a single - star scenario .
whilst stationary winds in o and wr stars are ubiquitous , it is not at all clear if lbv - type objects like @xmath5car have encountered a special evolution ( such as a merger ) or if all massive stars go through eruptive mass - loss phases . on the other hand , for the most massive main - sequence wnh stars ( crowther et al . 2010 , bestenlehner et al . 2011 ) there is both theoretical and empirical evidence for strong eddington parameter @xmath7-dependent ( see definition eq.[eq_gamma ] ) mass loss ( grfener et al .
2011 ) . for vms
the role of stationary mass loss has thus increased rather than decreased in recent years . in summary ,
the relevant roles of eruptive versus stationary mass loss seem rather uncertain and unsettled at the current time .
there are ongoing debates as to whether wind clumping reduces the mass - loss rates by moderate factors of @xmath02 - 3 , such that stellar evolution would not be affected , or by more severe factors of order @xmath210 . in the latter case ,
line - driving would become negligible and alternatives such as eruptive mass loss would need to be considered . in order to address the relative role of wind versus eruptive mass loss , it would be beneficial to be able to calibrate either one of them . at the moment
both stellar wind and eruptive mass loss could be inaccurate by factors of 10 , and possibly even more . in this letter ,
we attempt to alleviate this problem by presenting a methodology that involves a model - independent mass - loss indicator , the transition mass - loss rate @xmath1 located right at the transition from optically thin to optically thick stellar winds . martins et al .
( 2008 ) found two mass - loss relations for vms arches cluster stars , one for the of stars and one for the late - type wnh stars respectively .
the fact that wolf - rayet stars with wnh spectral classification have optical depth larger than one has already been discussed in literature ( e.g. grfener & hamann 2008 ) , and one might thus expect to witness a transition from optically thin o - type winds to optically thick wolf - rayet winds .
vink et al . ( 2011 ) discovered a sudden change in the slope of the mass - loss versus @xmath7 relation at the transition from o - type ( optically thin ) to wr - type ( optically thick ) winds .
interestingly , this transition was found to occur for a wind efficiency parameter @xmath5 @xmath8 @xmath9 @xmath10/(@xmath11 ) of order unity .
this key result from monte carlo modelling that the transition from o to wr - type mass loss coincides with @xmath12 , can also be found analytically ( sect.[sec_anal ] ) . and
the result can be utilized to `` calibrate '' wind mass loss in an almost model independent manner ( sect.[sec_arches ] ) .
netzer & elitzur ( 1993 ) and lamers & cassinelli ( 1999 ; hereafter lc99 ) give general momentum considerations for dust - driven winds ( see lc99 pages 152 - 153 ) that can also be applied to line - driven winds .
the integral form of the momentum equation contains four terms ( eq .
7.5 of lc99 ) .
because hydrostatic equilibrium is a good approximation for the subsonic part of the wind , and the gas pressure gradient is small beyond the sonic point , lc99 argue that the second and third terms are negligible compared to the first and fourth , resulting in : @xmath13 employing the mass - continuity equation @xmath14 , one obtains @xmath15 where @xmath16 denotes the sonic radius and @xmath17 the eddington factor with respect to the total flux - mean opacity @xmath18 : @xmath19 using the wind optical depth @xmath20 , one obtains @xmath21 where it is assumed that @xmath7 is significantly larger than one , and the factor @xmath22 is thus close to unity ( lc99 s second assumption ) , resulting in @xmath23 one can now derive a key condition for the wind efficiency number @xmath5 @xmath24 the key point of our letter is that we can employ the unique condition @xmath25 right at the transition from optically thin o - star winds to optically - thick wr winds , and obtain a _ model - independent _ @xmath26 . in other words , if we were to have an empirical data - set available that contains luminosity determinations for o and wr stars , we can obtain the transition mass - loss rate @xmath1 simply by considering the transition luminosity @xmath27 and the terminal velocity @xmath28 representing the transition point from o to wr stars : @xmath29 we note that this transition point can be obtained by purely spectroscopic means , _ independent _ of any assumptions regarding wind clumping . [ tab_models ] in the above analysis we made two assumptions that we wish to check with numerical tests involving sophisticated hydrodynamic wind models ( from grfener & hamann 2008 ) and simpler @xmath30-type velocity laws , commonly used in o / wr wind modelling .
the results are compiled in tab.1 .
we first confirm lc99 s first assumption through the comparison of @xmath5 , determined from @xmath26 and @xmath31 , to the approximate @xmath32 values as computed from the right - hand - side of integral in eq.[eq_lc99 ] , where @xmath33 .
evidently , the values of @xmath5 and @xmath32 agree at the few percent level , and the first lc99 approximation is verified .
second , we investigate the assumption that the term @xmath34 in eq.[eq_gaga ] is close to unity by numerical integration of @xmath35 .
we obtain a correction factor @xmath36 , which we define by @xmath37 with this definition eq.[eq_eta ] becomes @xmath38 to compute the integral numerically , we need to obtain the density @xmath39 , and the flux - mean opacity @xmath40 in the stellar wind ( @xmath17 follows from eq.([eq_gamma ] ) .
the hydrodynamic wind models of grfener & hamann ( 2008 ) have these quantities directly available .
we have performed a direct computation for the first model ( hyd ) in tab.1 for the galactic wnh star wr22 ( grfener & hamann 2008 ) .
for this model we obtain @xmath41 .
this value is lower than , but of the order of , unity .
the terminal wind speed in this model is significantly lower than the observed value for wr22 ( 980km / s vs. 1785km / s ) .
consequently , our derived @xmath36 is likely on the low side .
we expect @xmath42 to be connected to the ratio @xmath43 , and @xmath44 . here
we follow a model - independent approach , adopting @xmath30-type velocity laws .
the mean opacity @xmath45 then follows from the resulting radiative acceleration @xmath46
@xmath47 @xmath46 follows from the prescribed density @xmath39 and velocity structures @xmath48 via the equation of motion @xmath49 where we assume a grey temperature structure to compute the gas pressure @xmath50 .
we note that these results are completely independent of any assumptions regarding wind porosity , or the chemical composition of the wind material .
the _ only _ assumption that goes into these considerations is that the winds are radiatively driven .
the resulting mean opacity @xmath45 consequently captures all physical effects that could potentially affect the radiative driving .
the obtained values for the correction factor @xmath36 are summarized in tab.1 .
the first three models in tab.1 represent a consistency test with the hydrodynamic model for wr22 . using a beta law with @xmath51 , and the same @xmath31
, we obtain almost exactly the same @xmath36 as for the hydrodynamic model , justifying our @xmath30-law approach , which we employ in the following . now employing the _ observed _ and therefore likely close to correct value of @xmath31 ( and a correspondingly increased @xmath26
, we obtain @xmath52 . to get a handle on the overall behaviour of this factor @xmath36
, we computed a series of wind models for a range of stellar parameters @xmath53 and @xmath54 , with wind efficiencies around the transition region ( @xmath55 ) .
remarkably , the resulting values of @xmath36 depend _ only _ on the adopted values of @xmath56 and @xmath30 . for @xmath57
, we obtain @xmath58 0.7 , 0.6 respectively for @xmath59 , 1.0 , 1.5 , where the last value is probably most appropriate ( vink et al .
overall , we derive values of @xmath60 in the range 0.4 - 0.8 , with a mean value of 0.6 .
we note that the error on this number @xmath60 is within the uncertainty of the luminosity determinations described in the next section . for transition objects with @xmath61
we thus expect that @xmath62 , i.e. the transition between o and wr spectral types should occur at mass - loss rates of @xmath63 the fact that the correction factor is within a factor of two of our idealized approach ( eq . [ eq_transm ] ) is highly encouraging .
we stress that this number is independent of any potential model deficiencies , as we have used the observed values of @xmath31 in this analysis .
[ tab_arches ] llllllc ' '' '' star & subtype & @xmath64 & @xmath65 & @xmath31 & @xmath66 + & & [ @xmath67 & [ @xmath68 & [ @xmath69 & [ @xmath68 + ' '' '' f9 & wn8 - 9 & 6.35 & @xmath704.78 & 1800 & @xmath704.60 + f1 & wn8 - 9 & 6.30 & @xmath704.70 & 1400 & @xmath704.54 + f14 & wn8 - 9 & 6.00 & @xmath705.00 & 1400 & @xmath704.84 + b1 & wn8 - 9 & 5.95 & @xmath705.00 & 1600 & @xmath704.95 + f16 & wn8 - 9 & 5.90 & @xmath705.11 & 1400 & @xmath704.94 + ' '' '' f15 & o4 - 6if+ & 6.15 & @xmath705.10 & 2400 & @xmath704.92 & + f10 & o4 - 6if+ & 5.95 & @xmath705.30 & 1600 & @xmath704.95 & @xmath1 + ' '' '' f18 & o4 - 6i & 6.05 & @xmath705.35 & 2150 & @xmath704.98 + f21 & o4 - 6i & 5.95 & @xmath705.49 & 2200 & @xmath705.09 + f28 & o4 - 6i & 5.95 & @xmath705.70 & 2750 & @xmath705.18 + f20 & o4 - 6i & 5.90 & @xmath705.42 & 2850 & @xmath705.25 + f26 & o4 - 6i & 5.85 & @xmath705.73 & 2600 & @xmath705.26 + f32 & o4 - 6i & 5.85 & @xmath705.90 & 2400 & @xmath705.22 + f33 & o4 - 6i & 5.85 & @xmath705.73 & 2600 & @xmath705.26 + f22 & o4 - 6i & 5.80 & @xmath705.70 & 1900 & @xmath705.17 + f23 & o4 - 6i & 5.80 & @xmath705.65 & 1900 & @xmath705.17 + f29 & o4 - 6i & 5.75 & @xmath705.60 & 2900 & @xmath705.41 + f34 & o4 - 6i & 5.75 & @xmath705.77 & 1750 & @xmath705.19 + f40 & o4 - 6i & 5.75 & @xmath705.75 & 2450 & @xmath705.33 + f35 & o4 - 6i & 5.70 & @xmath705.76 & 2150 & @xmath705.33 + designations , subtypes , luminosities ( @xmath71 ) , mass - loss rates ( @xmath26 ) , and terminal wind velocities ( @xmath31 ) according to martins et al .
the 6th column indicates the mass - loss rate where @xmath5=1 . for the arches cluster ,
we obtain @xmath72 .
martins et al .
( 2008 ) analyzed 28 vms in the arches cluster , with equal numbers of o - type supergiants and nitrogen - rich wolf - rayet ( wnh ) stars ( sometimes called `` o stars on steroids '' ) . for the o - type supergiants , we expect the winds to be optically thin , whilst the wnh stars should have optically thick winds . here
we postulate that the o4 - 6if@xmath73 represent the transition point where the optical depth crosses unity . in tab.2 , we compiled a subset of 20 stars , skipping those objects with a he - enriched surface composition .
the objects are sorted with respect to their spectral subtypes , and within each subtype bin with respect to their luminosity . together with the basic stellar and wind parameters derived by martins et al .
( 2008 ) , we list the mass - loss rate for which @xmath74 if the stars would have a wind efficiency of exactly 1 .
the values listed in tab.2 show that there is a transition between o and wr spectral types .
the _ spectroscopic _
transition for spectral subtypes o4 - 6if+ occurs at @xmath75 and @xmath76 .
this is the resulting transition mass - loss rate for the arches cluster stars .
its determined value does not depend on model uncertainties involving issues such as wind clumping .
the only remaining uncertainties are due to uncertainties in the terminal velocity and the stellar luminosity @xmath71 .
the latter results from errors in the distances and reddening parameters , as well as the determination of effective temperatures from non - lte model atmospheres .
if the derived value for @xmath1 is compared with empirically determined mass - loss rates , the uncertainties in distances and reddening parameters nearly cancel , as for empirical mass - loss rates based on recombination line analyses @xmath77 , while @xmath78 . to estimate uncertainties in the effective temperature scale for o stars we can use historical values from the last four decades ( e.g. panagia et al .
1973 , martins et al . 2005 ) as an indicator for potential systematic errors in the inclusion / neglect of certain micro - physics ( line blanketing , wind effects , etc . ) , the best error estimate is @xmath210% in effective temperature , leading to potential errors in the luminosity of at most @xmath240% .
this is several factors smaller than the order - of - magnitude uncertainties in mass - loss rates due to clumping and porosity . in other words our simple equation ( eq.[eq_transm ] )
is of tremendous value in calibrating stellar wind mass loss , and assessing its role in the mass loss during the evolution of massive stars ( sect.[sec_disc ] ) . how does our determination of the transition mass - loss rate compare to other methods ?
let us first compare our transition mass - loss value to the mass - loss rates of martins et al .
martins et al .
use the non - lte cmfgen code by hillier & miller ( 1998 ) , employing a micro - clumping approach with a volume filling factor @xmath79 for their k - band analysis .
their values are in good agreement with our transition mass - loss rates for the objects at the boundary between o and wr ( see their table 2 ) .
this is unlikely to be a coincidence . according to our findings in sect.[sec_mod ] ,
we expect mass - loss rates of the order of @xmath80 for the transition objects , i.e. @xmath81 .
we also compare the transition mass - loss rate to the oft - used theoretical mass - loss relation of vink et al .
( 2000 ) , for which we find @xmath82 for an assumed stellar mass @xmath83 .
this number is within 0.2 dex from the transition mass - loss rate @xmath84 .
the comparison is hardly compromised as a result of the vink et al .
dependence on stellar mass , as for masses in the range 40 - 80@xmath85 , the vink et al .
mass - loss rate varies by at most 0.04 dex . in summary
, we have three independent mass - loss rate determinations that agree within a factor of two .
this means that our concept of the transition mass - loss rate has indeed been able to test the accuracy of current mass - loss estimates by stellar winds .
now that we have calibrated stellar wind mass loss in the high mass and luminosity regime , we assess the role of stellar wind mass loss for massive star evolution . for a 60@xmath85 star ,
the main sequence lifetime is @xmath03 myrs ( e.g. weidner & vink 2010 and references therein ) . with a stationary mass - loss rate of @xmath86 @xmath87 as derived for the transition mass - loss rate in sect.[sec_arches ] , this means such an object will lose @xmath030 , i.e. half its initial mass , already on the main sequence during core hydrogen burning .
we have not yet addressed the new concept of @xmath7 dependent mass loss , nor any additional stellar wind mass loss during the subsequent core - helium burning wr phase . in other words ,
solar - metallicity 60@xmath85 stars are expected to lose the bulk of their initial masses through stellar winds , leaving very little ( if any ) space for additional ( e.g. eruptive ) mass loss .
it is plausible that the strong - winded vms remain on the blue side of the hr diagram , without ever entering an eruptive luminous blue variable ( lbv ) or red supergiant ( rsg ) phase .
current wisdom thus suggests that solar metallicity vms likely `` evaporate '' primarily as the result of stationary wind mass loss , without the _ necessity _ of additional eruptive mass loss .
however , finding out if eruptive mass loss might play an additional role remains an interesting exercise , especially for the lower initial mass and sub - solar metallicity ranges , as their story might be expected to be different .
moreover , we know @xmath5car analogs and supernova impostors exist in external galaxies ( e.g. van dyck et al . 2005 , pastorello et al .
2010 , kochanek et al .
2012 ) .
contrary to the most massive stars , stars below @xmath240 likely evolve into the rsg , yellow super / hypergaint , and/or lbv regimes of the stellar hr diagram ( see e.g. vink 2009 ) .
we note that the vink et al .
( 2000 ) main - sequence mass - loss rates currently in use in stellar models ( e.g. brott et al .
2011 ) for lower mass `` normal '' 20 - 60@xmath85 o stars are also already reduced by a factor 2 - 3 in comparison to previous unclumped empirical rates .
there is currently no particular reason to assume they are still overestimated .
the results presented here certainly boost confidence in the mass - loss rates currently in use , although they remain uncalibrated for the lower mass regime .
one should also realize when working down the mass range , starting from our 60@xmath85 calibrator star , that the mass - loss rates drop significantly below @xmath86 @xmath87 .
its effects on stellar evolution remain significant due to the longer evolutionary timescales for lower mass objects and the fact that it is the multiplication of the mass - loss rate times the duration that is relevant .
this is especially relevant for angular momentum evolution , possibly down to stellar masses as low as 10 - 15 ( vink et al . 2010 ) . | a debate has arisen regarding the importance of stationary versus eruptive mass loss for massive star evolution .
the reason is that stellar winds have been found to be clumped , which results in the reduction of unclumped empirical mass - loss rates .
most stellar evolution models employ theoretical mass - loss rates which are _ already _ reduced by a moderate factor of @xmath02 - 3 compared to non - corrected empirical rates .
a key question is whether these reduced rates are of the correct order of magnitude , or if they should be reduced even further , which would mean that the alternative of eruptive mass loss becomes necessary .
here we introduce the transition mass - loss rate @xmath1 between o and wolf - rayet ( wr ) stars .
its novelty is that it is model independent .
all that is required is postulating the _ spectroscopic _ transition point in a given data - set , and determining the stellar luminosity , which is far less model dependent than the mass - loss rate .
the transition mass - loss rate is subsequently used to calibrate stellar wind strength by its application to the of / wnh stars in the arches cluster .
good agreement is found with two alternative modelling / theoretical results , suggesting that the rates provided by current theoretical models are of the right order of magnitude in the @xmath250 mass range .
our results do not confirm the specific _ need _ for eruptive mass loss as luminous blue variables , and current stellar evolution modelling for galactic massive stars seems sound .
mass loss through alternative mechanisms might still become necessary at lower masses , and/or metallicities , and the _ quantification _ of alternative mass loss is desirable . | 1205.0394 |
in what follows @xmath9 is a variable vector in @xmath1 ( @xmath10 ) . as usual , @xmath11:={\ensuremath{\mathbb{r}}}[x_1,\ldots , x_d]$ ] denotes the ring of polynomials in variables @xmath12 and coefficients in @xmath13 a subset @xmath0 of @xmath1 which can be represented by @xmath14 for @xmath15 $ ] ( @xmath3 ) is said to be an
_ elementary closed semi - algebraic set _ in @xmath16 clearly , the number @xmath17 from is not uniquely determined by @xmath7 let us denote by @xmath18 the minimal @xmath17 such that is fulfilled for appropriate @xmath15.$ ] analogously , a subset @xmath19 of @xmath1 which can be represented by @xmath20 for some @xmath21 $ ] ( @xmath3 ) is said to be an _ elementary open semi - algebraic set _ in @xmath16 the quantity @xmath22 associated to @xmath19 is introduced analogously to @xmath23 the system of polynomials @xmath2 from ( resp . )
is said to be a _ polynomial representation _ of @xmath0 ( resp .
@xmath19 ) . from the well - known theorem of brcker and scheiderer ( see ( * ? ? ?
* chapter 5 ) , and and the references therein ) it follows that , for @xmath0 and @xmath19 as above , the following inequalities are fulfilled : @xmath24 both of these inequalities are sharp .
it should be emphasized that all known proofs of and are highly non - constructive . the main aim of this paper is to provide constructive upper bounds for @xmath18 and @xmath22 for certain classes of @xmath0 and @xmath19 ; see also @xcite , , , @xcite , , and for previous results on this topic .
we also mention that constructive results on polynomial representations of special semi - algebraic sets are related to polynomial optimization ; see @xcite , @xcite , @xcite , @xcite , and @xcite .
let @xmath15 $ ] and let @xmath25 be non - empty .
the assumptions of our main theorems are formulated in terms of the following functionals , which depend on @xmath2 .
the functional @xmath26 determines the set of constraints defining @xmath0 which are `` active '' in @xmath27 furthermore , we define @xmath28 where @xmath29 stands for the cardinality .
the geometric meaning of @xmath30 and @xmath31 can be illustrated by the following special situation .
let @xmath0 be a @xmath32-dimensional polytope with @xmath17 facets ( see for information on polytopes ) .
then @xmath0 can be given by with all @xmath33 having degree one ( the so - called _ h - representation _ ) .
in this case @xmath30 is the maximal number of facets of @xmath0 having a common vertex and @xmath31 is the set consisting of those vertices of @xmath0 which are contained in the maximal number of facets of @xmath7 if the polytope @xmath0 is _ simple _ ( that is , each vertex of @xmath0 lies in precisely @xmath32 facets ) , then @xmath34 and @xmath31 is the set of all vertices of @xmath7 now we are ready to formulate our main results .
[ main : n+1 ] let @xmath21 $ ] , @xmath35 , and @xmath36 assume that @xmath0 is non - empty and bounded , and @xmath37 then the following inequalities are fulfilled : @xmath38 furthermore , there exists an algorithm that gets @xmath2 and returns @xmath8 polynomials @xmath39 $ ] satisfying @xmath40 and @xmath41 @xmath42 in the case when @xmath31 is finite theorem [ main : n+1 ] can be improved .
[ main : n ] let @xmath21 $ ] , @xmath35 , and @xmath36 assume that @xmath0 is non - empty and bounded , @xmath43 is finite , and @xmath44
. then the following inequalities are fulfilled : @xmath45 furthermore , there exists an algorithm that gets @xmath2 and @xmath46 and returns @xmath5 polynomials @xmath47 satisfying @xmath48 and @xmath49 @xmath42 below we discuss existing results and problems related to theorems [ main : n+1 ] and [ main : n ] .
let @xmath0 be a convex polygon in @xmath50 with @xmath17 edges , which is given by with all @xmath33 having degree one .
bernig showed that setting @xmath51 one can construct a strictly concave polynomial @xmath52 vanishing on all vertices of @xmath0 which satisfies @xmath53 ; see fig .
[ simp : polyt:2d : fig ] . as it will be seen from the proof of theorem [ main : n ] , for the case @xmath54 and @xmath0 as in theorem [ main : n ] we also set @xmath55 and choose @xmath56 in such a way that it vanishes on each point of @xmath57 and the set @xmath58 approximates @xmath0 sufficiently well ; see fig . [
fig : semi - alg:2d ] .
however , since @xmath0 from theorem [ main :
n ] is in general not convex , the construction of @xmath56 requires a different idea .
the statement of theorem [ main : n ] concerned with @xmath19 and restricted to the cases @xmath59 and @xmath60 , @xmath61 ( with slightly different assumptions on @xmath19 ) was obtained by bernig .
c0.6 mm ( 200,50 ) ( 60,2 ) ( 0,2 ) ( 150,2 ) ( 125,25 ) ( 20,25)@xmath58 ( 80,25)@xmath62 ( 170,25)@xmath0 + c0.7 mm ( 200,50 ) ( 60,2 ) ( 0,2 ) ( 150,2 ) ( 125,25 ) ( 20,25)@xmath58 ( 80,25)@xmath62 ( 170,25)@xmath0 + the study of @xmath18 for the case when @xmath0 is a polyhedron of an arbitrary dimension was initiated by grtschel and henk . in it
was noticed that @xmath63 for every @xmath32-dimensional polytope @xmath7 on the other hand , bosse , grtschel , and henk gave an upper bound for @xmath18 which is linear in @xmath32 for the case of an arbitrary @xmath32-dimensional polyhedron @xmath7 in particular , they showed that @xmath64 if @xmath0 is @xmath32-dimensional polytope . in the following conjecture was announced .
[ bgh : conj ] for every @xmath32-dimensional polytope @xmath0 in @xmath1 the equality @xmath65 holds .
@xmath42 this conjecture has recently been confirmed for all simple @xmath32-dimensional polytopes ; see .
[ avehenkthm ] let @xmath0 be a @xmath32-dimensional simple polytope then @xmath66 furthermore , there exists an algorithm that gets polynomials @xmath2 ( @xmath3 ) of degree one satisfying @xmath67 and returns @xmath32 polynomials @xmath68 satisfying @xmath69 @xmath42 elementary closed semi - algebraic sets @xmath25 with @xmath34 can be viewed as natural extensions of simple polytopes in the framework of real algebraic geometry .
thus , we can see that theorem [ avehenkthm ] is a consequence of theorem [ main : n ] .
[ cube : p : rep ] illustrates theorem [ avehenkthm ] for the case when @xmath0 is a three - dimensional cube .
this figure can also serve as an illustration of theorem [ main : n ] with the only difference that in theorem [ main : n ] the set @xmath70 does not have to be convex anymore .
c0.5 mm ( 132,160 ) ( 35,55 ) ( -15,30 ) ( 90,30 ) ( 35,120 ) ( -15,90 ) ( 90,90 ) ( 35,0 ) ( 5,10 ) ( 45,55)@xmath0 ( -5,30)@xmath71 ( 100,30)@xmath62 ( 45,120)@xmath72 ( -5,90)@xmath73 ( 100,90)@xmath74 ( 45,0)@xmath75 + while proving our main theorems we derive the following approximation results which can be of independent interest . the _ hausdorff distance _ @xmath76 is a metric defined on the space of non - empty compact subsets of @xmath1 by the equality @xmath77 see .
[ approx : thm1 ] let @xmath15,$ ] @xmath25 , and @xmath78 assume that @xmath0 is non - empty and bounded .
then there exists an algorithm that gets @xmath2 and @xmath79 and returns a polynomial @xmath80 $ ] such that @xmath81 , @xmath82 , and the hausdorff distance from @xmath0 to @xmath83 is at most @xmath84 @xmath42 [ approx : thm2 ] let @xmath15,$ ] @xmath25 , and @xmath78 assume that @xmath0 is non - empty and bounded , @xmath85 is finite , and @xmath86 then there exists an algorithm that gets @xmath2 , @xmath57 , and @xmath79 and returns a polynomial @xmath80 $ ] such that @xmath81 , @xmath82 , the hausdorff distance from @xmath0 to @xmath83 is at most @xmath87 , and @xmath88 for every @xmath89 @xmath42 we note that some further results on approximation by sublevel sets of polynomials can be found in @xcite , @xcite , and .
the paper has the following structure .
section [ prelim : sect ] contains preliminaries from real algebraic geometry . in section
[ approx : sect ] we obtain approximation results ( including theorems [ approx : thm1 ] and [ approx : thm2 ] ) .
finally , in section [ main : proofs : sect ] the proofs of theorems [ main : n+1 ] and [ main : n ] are presented . in the beginning of the proofs of theorems [ main : n+1 ] and [ main : n ] one can find the formulas defining the polynomials @xmath90 ( see and ) as well as sketches of the main arguments .
the origin and the euclidean norm in @xmath1 are denoted by @xmath91 and @xmath92 respectively .
we endow @xmath1 with its euclidean topology . by @xmath93
we denote the closed euclidean ball in @xmath1 with center at @xmath94 and radius @xmath95 the interior ( of a set ) is abbreviated by @xmath96 we also define @xmath97 where @xmath98 is the set of all natural numbers . a set @xmath99 given by @xmath100 where @xmath101 and @xmath102 $ ] , is called _
semi - algebraic_. an expression @xmath103 is called a _ first - order formula over the language of ordered fields with coefficients in @xmath104 _ if @xmath103 is a formula built with a finite number of conjunctions , disjunctions , negations , and universal or existential quantifier on variables , starting from formulas of the form @xmath105 or @xmath106 with @xmath107 $ ] ; see .
the _ free variables _ of @xmath103 are those variables , which are not quantified . a formula with no free variables
is called a _
sentence_. each sentence is is either true or false .
the following proposition is well - known ; see also and .
[ semialg : over : formula ] let @xmath103 be a first - order formula over the language of ordered fields with coefficients in @xmath104 and free variables @xmath108 then the set @xmath109 consisting of all @xmath110 for which @xmath103 is true , is semi - algebraic .
@xmath42 a real valued function @xmath111 defined on a semi - algebraic set @xmath112 is said to be a _ semi - algebraic function _ if its graph is a semi - algebraic set in @xmath113 the following theorem presents _
ojasiewicz s inequality _ ; see @xcite and .
[ loj ] let @xmath112 be non - empty , bounded , and closed semi - algebraic set in @xmath16 let @xmath114 and @xmath115 be continuous , semi - algebraic functions defined on @xmath112 and such that @xmath116 then there exist
@xmath117 and @xmath118 such that @xmath119 for every @xmath120 @xmath42 considering algorithmic questions we use the following standard settings ; see ( * ? ? ?
* chapter 8.1 ) .
it is assumed that a polynomial in @xmath11 $ ] is given by its coefficients and that a finite list of real coefficients occupies finite memory space .
furthermore , arithmetic and comparison operations over reals are assumed to be atomic , i.e. , computable in one step .
the following well - known result is relevant for the constructive part of our theorems ; see .
[ dec : problem ] let @xmath103 be a sentence over the language of ordered fields with coefficients in @xmath104 .
then there exists an algorithm that gets @xmath103 and decides whether @xmath103 is true or false .
the following proposition ( see ) presents a characterization of the convergence with respect to the hausdorff distance .
[ hausd : convergence ] a sequence @xmath121 of compact convex sets in @xmath1 converges to a compact set @xmath112 in the hausdorff distance if and only if the following conditions are fulfilled : 1 . every point of @xmath112 is a limit of a sequence @xmath122 satisfying @xmath123 for every @xmath124 2 . if @xmath125 is a strictly increasing sequence of natural numbers and @xmath126 is a convergent sequence satisfying @xmath127 ( @xmath128 ) , then @xmath129 converges to a point of @xmath112 , as @xmath130 3 . the set @xmath131 is bounded . @xmath42 let @xmath15.$ ] the following theorem states that for the case when @xmath25 is non - empty and bounded , appropriately relaxing the inequalities @xmath132 , which define @xmath0 , we get a bounded semi - algebraic set that approximates @xmath0 arbitrarily well .
let us define @xmath133 with @xmath134 and @xmath135 [ semi : approx ] let @xmath15 $ ] , @xmath25 , and @xmath78 assume that @xmath0 is non - empty and bounded .
then there exists an algorithm that gets @xmath2 and returns values @xmath134 and @xmath136 such that the following conditions are fulfilled : 1 .
[ boundedness : part ] @xmath137 is bounded for @xmath138 2 .
[ convergence : part ] @xmath139,$ ] converges to @xmath0 in the hausdorff distance , as @xmath140 @xmath42 first we show the existence of @xmath141 and @xmath142 from the assertion , and after this we show that these two quantities are constructible .
let us derive the existence of @xmath141 and @xmath142 satisfying condition [ boundedness : part ] .
since @xmath0 is bounded , after replacing @xmath0 by an appropriate homothetical copy , we may assume that @xmath143 by proposition [ semialg : over : formula ] , the function @xmath144 is semi - algebraic .
we also have @xmath145 for all @xmath146 with @xmath147 furthermore , the set @xmath137 can be expressed with the help of @xmath111 by @xmath148 for @xmath149 the function @xmath150 is positive and non - increasing . using proposition [ hausd : convergence ] it can be shown that @xmath151 is continuous .
moreover , in view of proposition [ semialg : over : formula ] , we see that @xmath151 is semi - algebraic . in the case
@xmath152 condition [ boundedness : part ] is fulfilled for @xmath153 and @xmath154 in the opposite case we have @xmath155 , as @xmath156 then @xmath157 is a continuous semi - algebraic function on @xmath158 $ ] with @xmath159 if and only if @xmath160 thus , applying theorem [ loj ] to the functions @xmath161 and @xmath162 defined on @xmath158 $ ] , we see that there exist @xmath134 and @xmath163 such that @xmath164 for every @xmath165.$ ] consequently @xmath166 for every @xmath167 the latter implies that @xmath168 and condition [ boundedness : part ] is fulfilled for @xmath141 as above and @xmath169 now we show that condition [ boundedness : part ] implies condition [ convergence : part ] .
assume that condition [ boundedness : part ] is fulfilled .
then the set @xmath137 is bounded for all @xmath170.$ ] hence @xmath171 is well defined for all @xmath170.$ ] consider an arbitrary sequence @xmath172 with @xmath173 $ ] and @xmath174 as @xmath175 using proposition [ hausd : convergence ] we can see that @xmath176 as @xmath130 consequently , condition [ convergence : part ] is fulfilled .
finally we show that @xmath142 and @xmath141 are constructible . for determination of @xmath141 one can use the following `` brute force '' procedure .
procedure : : : determination of @xmath177 input : : : @xmath178.$ ] output : : : a number @xmath134 such that for some @xmath179 the set @xmath180 is bounded . 1
. set @xmath181 2 .
for @xmath182 introduce the first - order formula @xmath183 with free variables @xmath184 3 .
test the existence of @xmath179 for which @xmath180 is bounded .
more precisely , determine whether the sentence @xmath185 is true or false ( cf .
theorem [ dec : problem ] ) .
4 . if @xmath186 is true , return @xmath141 and stop . otherwise set @xmath187 and go to step 2 . in view of the conclusions made in the proof , the above procedure terminates after a finite number of iterations .
for determination of @xmath142 we can use a similar procedure .
we start with @xmath188 and assign @xmath189 at each new iteration , terminating the cycle as long as @xmath180 is bounded .
[ bounded : extension : remark ] we wish to show theorem [ semi : approx ] can not be improved by setting @xmath190 since @xmath191 may be unbounded for all @xmath135 let us consider the following example .
let @xmath192 @xmath193 @xmath194 and @xmath195 then the set @xmath196 is bounded .
in fact , if @xmath197 then the term @xmath198 , appearing in the definition of @xmath199 , is positive .
but the remaining terms @xmath200 and @xmath201 can not vanish simultaneously .
hence , @xmath202 for every @xmath203 with @xmath197 which shows that @xmath204 furthermore , since @xmath205 we see that @xmath0 has non - empty interior ( which shows that our example is non - degenerate enough ) .
let us show that @xmath206 is unbounded for every @xmath135 for @xmath207 with @xmath208 one has @xmath209 and @xmath210 as @xmath211 ; see also fig .
[ bd : ubd : fig ] .
this implies unboundedness of @xmath212 throughout the rest of the paper we shall use the following polynomials associated to @xmath15.$ ] for @xmath134 , @xmath213 , and @xmath214 we define @xmath215 if @xmath85 is finite , we define @xmath216 where @xmath217 [ sublevel : approx : part ] let @xmath15 $ ] ,
@xmath25 , and @xmath78 assume that @xmath0 is non - empty and bounded . then for every @xmath79 , @xmath134 , @xmath218 and @xmath214 satisfying @xmath219 the polynomial @xmath220 fulfills the relations @xmath221 furthermore , there exists an algorithm that gets @xmath2 , @xmath79 , and @xmath134 and constructs @xmath222 $ ] satisfying and .
inclusions @xmath223 and @xmath224 follow from .
it remains to show the inclusion @xmath225 assume that @xmath226 then @xmath227 consequently @xmath228 or equivalently , @xmath229 . hence @xmath230 now let us discuss the constructibility of @xmath231 it suffices to show the constructibility of @xmath232 satisfying . for determination of @xmath232
we iterate starting with @xmath233 , set @xmath234 at each new step , and use , reformulated as a first - order formula , as a condition for terminating the cycle . c0.9 mm ( 80,70 ) ( 0,2 ) ( 70,10)@xmath235 ( 10,70)@xmath236 ( 36,38)@xmath0 + one can see that theorem [ approx : thm1 ] from the introduction is a direct consequence of theorem [ semi : approx ] and lemma [ sublevel : approx : part ] .
[ 07.12.10,11:53 ] let @xmath237 , @xmath25 , and @xmath238 .
assume that @xmath0 is non - empty and bounded , @xmath43 is finite , and @xmath37 then there exists an algorithm that gets @xmath239 @xmath240 @xmath134 , and @xmath79 and returns @xmath80 $ ] fulfilling the relations @xmath241 furthermore , @xmath242 can be defined by @xmath243 where @xmath244 , @xmath213 , and @xmath217 @xmath42 analogously to the proof of theorem [ semi : approx ] , we first show the existence of @xmath242 from the assertion and then we derive the constructive part of the theorem .
we fix @xmath232 and @xmath245 satisfying and and set @xmath246 let us derive the inclusions @xmath81 and @xmath82 .
first we show that @xmath247 let @xmath248 since @xmath249 for every @xmath250 the set @xmath251 is properly contained in @xmath252 .
consequently , for every @xmath250 we get @xmath253 thus , is fulfilled .
therefore we can fix @xmath254 with @xmath255 in view of and the finiteness of @xmath256 we can fix @xmath257 such that @xmath258 and @xmath259 for all @xmath260 with @xmath261 let us consider an arbitrary @xmath262 we show that , for an appropriate choice of @xmath263 and @xmath264 we have @xmath265 and the latter inequality is strict for @xmath266 _ case a : _ @xmath267 let us fix @xmath268 such that @xmath269 since @xmath250 , we have @xmath270 furthermore , due to the choice of @xmath271 equality is attained if and only if @xmath272 let @xmath273 be an arbitrary scalar satisfying @xmath274 applying theorem [ loj ] to the functions @xmath275 and @xmath276 restricted to @xmath277 , we have @xmath278 for appropriate parameters @xmath279 and @xmath280 independent of @xmath27 in view of the choice of @xmath281 we deduce @xmath282 where @xmath283 and @xmath284 we have @xmath285 in view of , for all sufficiently large @xmath263 the inequality @xmath286 is fulfilled .
assuming that holds , and taking into account , we have @xmath287 now assume that @xmath203 lies in @xmath288 then , if @xmath289 satisfies , we get @xmath290 _ case b : _ @xmath291
then @xmath292 for every @xmath293 from the definition of elementary symmetric functions and the assumptions it easily follows that @xmath294 let us choose @xmath295 with @xmath296 thus , we get the bounds @xmath297 and @xmath298 in view of , for all sufficiently large @xmath263 the inequality @xmath299 is fulfilled . assuming that is fulfilled , we obtain @xmath290 now we show the inclusion @xmath300 .
consider an arbitrary @xmath301 then @xmath302 which is equivalent to @xmath303 the latter implies that @xmath304 and therefore @xmath305 we have @xmath306 the above estimate for @xmath307 together with the estimate @xmath308 and implies that @xmath309 if @xmath289 fulfills the inequality @xmath310 since @xmath311 , is fulfilled if @xmath263 is large enough .
thus , we obtain that the inequality @xmath312 holds for all sufficiently large @xmath313 now we show the constructive part of the assertion .
we present a sketch of a possible procedure that determines @xmath314 it suffices to evaluate the parameters @xmath315 and @xmath281 involved in the definition of @xmath242 .
constructibility of @xmath232 and @xmath245 follows from lemma [ sublevel : approx : part ] .
let us apply theorem [ dec : problem ] in the same way as in the previous proofs .
determine the following parameters in the given sequence .
we can determine @xmath316 satisfying for an appropriate @xmath317 and all @xmath318 using the same idea as in the procedure for determination of @xmath141 in the proof of theorem [ semi : approx ] . a parameter @xmath281 satisfying is constructible in view of theorem [ dec : problem ] ( by means of iteration procedure which we also used in the previous proofs ) .
an appropriate @xmath289 can be easily found from inequalities , , and .
thus , for evaluation of @xmath289 we should first find the parameters @xmath319 and @xmath320 appearing in , , and .
the parameters @xmath254 , @xmath321 , and @xmath295 are determined by means of , , and .
one can see that theorem [ approx : thm2 ] from the introduction is a straightforward consequence of theorem [ semi : approx ] and theorem [ 07.12.10,11:53 ] .
the parameters @xmath322 involved in the statements of this section were computed with the help of the theorem [ dec : problem ] .
in contrast to this , in general it is not possible to compute @xmath46 exactly , since evaluation of @xmath46 would involve solving a polynomial system of equations .
this explains why in the statement of theorem [ 07.12.10,11:53 ] the set @xmath46 is taken as a part of the input . the parameters @xmath232 and @xmath281 from lemma [ sublevel : approx : part ] and theorem [ 07.12.10,11:53 ] , respectively , are upper bounds for certain polynomial programs .
in fact , by the parameter @xmath213 is a common upper bound for the optimal solutions of @xmath17 non - linear programs @xmath323 with constraints @xmath324 , @xmath325 from the proof of theorem [ 07.12.10,11:53 ] we see that @xmath281 can be any number satisfying @xmath326 hence @xmath327 is an upper bound for the optimal solution of the polynomial program @xmath328 with @xmath329 unknowns ( which are coordinates of @xmath330 and @xmath331 ) and the @xmath332 constraints @xmath333 and @xmath334 , @xmath335 the same observations apply also to the parameters @xmath254 and @xmath295 from the proof of theorem [ 07.12.10,11:53 ] , which are used for determination of @xmath289 . in this respect
we notice that upper bounds of polynomial programs can be determined using convex relaxation methods ; see @xcite , @xcite , and @xcite .
given @xmath3 , @xmath336 and @xmath337 the _ @xmath245-th elementary symmetric function _ in variables @xmath338 is defined by @xmath339 we also put @xmath340 [ berniglemma ] let @xmath337 with @xmath341 then the following statements hold : a. [ nonstrict : part ] @xmath342 if and only if @xmath343 .
b. [ strict : part ] @xmath344
if and only if @xmath345 . the necessities of both of the parts are trivial . let us prove the sufficiencies .
we introduce the polynomial @xmath346 whose roots are the the values @xmath347 by _ vieta s formulas _ , we have @xmath348 thus , if @xmath349 for every @xmath350 , then all coefficients of @xmath351 are non - negative , while the coefficient at @xmath352 is equal to one .
it follows that @xmath351 can not have strictly positive roots .
hence @xmath353 for all @xmath354 which shows the sufficiency of part [ nonstrict : part ] .
now assume that the strict inequality @xmath355 holds for every @xmath356 then @xmath357 i.e. , zero is not a root of @xmath358 and , using the sufficiency of part [ nonstrict : part ] , we arrive a the strict inequalities @xmath359 this shows the sufficiency in part [ strict : part ] .
proposition [ berniglemma ] was noticed by bernig , who derived it from _
descartes rule of signs_. our elementary proof ( slightly ) extends the arguments given in .
[ determ : n ] let @xmath15 $ ] and @xmath360 assume that @xmath0 is non - empty and bounded .
then there exists an algorithm which gets @xmath2 and returns @xmath361 since @xmath0 is bounded , we have @xmath362 we suggest the following procedure for evaluation of @xmath361 procedure : : : evaluation of @xmath30 input : : : @xmath15.$ ] output : : : @xmath30 1 .
for @xmath363 introduce the formula @xmath364 with free variables @xmath365 2 .
set @xmath366 3 .
[ iter : begin ] introduce the formula @xmath367 with free variables @xmath365 4 .
verify whether the sentence @xmath368 is true or not .
if @xmath186 is true and @xmath249 set @xmath369 and go to step [ iter : begin ] . 6 . if @xmath186 is true and @xmath370 return @xmath5 and stop .
if @xmath186 is false , set @xmath371 , return @xmath5 , and stop it is not hard to see that the above procedure terminates in a finite number of steps and returns @xmath361 as in the previous proofs , we first show the existence of @xmath372 from the assertion and then discuss the algorithmic part .
we define @xmath373 by the formula @xmath374 where @xmath214 , @xmath134 , and @xmath213 will be fixed later .
( we recall that @xmath375 is defined by . )
let us first present a brief sketch of our arguments .
it turns out that the polynomials @xmath376 which are defined with the help of elementary symmetric functions , represent @xmath0 locally , that is , @xmath0 and @xmath377 coincide in a neighborhood of @xmath7 in order to pass to the global representation , the additional polynomial @xmath378 is chosen in such a way that the sublevel set @xmath379 approximates @xmath0 sufficiently well .
given @xmath79 let us consider the set @xmath137 defined by
. by theorem [ semi : approx ] there exist @xmath134 and @xmath179 such that @xmath180 is bounded .
since @xmath380 it follows that @xmath381 for all @xmath250 and @xmath382 thus , the above strict inequalities hold also for @xmath203 in a small neighborhood of @xmath7 consequently , by theorem [ semi : approx ] , we can fix an @xmath170 $ ] such that @xmath381 for all @xmath383 and @xmath382 we define the sets @xmath384 let us consider an arbitrary @xmath262 obviously , @xmath385 for @xmath386 where all inequalities are strict if @xmath266 assume that @xmath232 and @xmath245 satisfy and . then , by lemma [ sublevel : approx : part ] , @xmath387 where the inequality is strict if @xmath266 hence @xmath388 and @xmath389 let us show the reverse inclusions .
let @xmath390 then , by the definition of @xmath391 we have @xmath392 for @xmath393 and @xmath394 but , by the choice of @xmath87 and @xmath375 , we also have @xmath392 for @xmath382 thus , @xmath392 for @xmath395 and , in view of proposition [ berniglemma]([strict : part ] ) , we have @xmath396 for @xmath335 this shows the inclusion @xmath397 the inclusion @xmath398 can shown analogously ( by means of proposition [ berniglemma]([nonstrict : part ] ) ) . finally we discuss the constructive part of the statement .
by lemma [ determ : n ] , @xmath5 is computable .
consequently , the polynomials @xmath399 are also computable , since they are arithmetic expressions in @xmath400 the computability of @xmath378 follows from directly from theorem [ semi : approx ] .
the polynomials @xmath401 will be defined by @xmath402 where @xmath403 will be fixed below .
we give a rough description of the arguments .
we start with the same remark as in the proof of theorem [ main : n+1 ] .
namely , polynomials @xmath404 with @xmath405 represent @xmath0 locally .
we shall disturb the polynomial @xmath275 by subtracting an appropriate non - negative polynomial @xmath406 which is small on @xmath0 , has high order zeros at the points of @xmath240 and is large for all points @xmath203 sufficiently far away from @xmath7 see also fig . [
fig : semi - alg:2d ] for an illustration of theorem [ main : n ] in the case @xmath407 we first show the existence of @xmath399 from the assertion . given @xmath408 let us consider the set @xmath137 defined by . by theorem [ semi : approx ] there exist @xmath134 and @xmath179 such that @xmath180 is bounded .
since @xmath380 it follows that @xmath381 for all @xmath250 and @xmath382 thus , the above strict inequalities hold also for @xmath203 in a small neighborhood of @xmath7 consequently , by theorem [ semi : approx ] , we can fix @xmath409 $ ] such that @xmath381 for all @xmath410 and @xmath382 let us borrow the notations from the statements of theorems [ semi : approx ] and [ 07.12.10,11:53 ] .
we set @xmath411 with @xmath80 $ ] as in theorem [ 07.12.10,11:53 ] .
define the semi - algebraic sets @xmath412 let us consider an arbitrary @xmath262 obviously , @xmath385 for @xmath413 where all inequalities are strict if @xmath266 furthermore , by theorem [ 07.12.10,11:53 ] we also have @xmath414 and this inequality is strict if @xmath266 thus , we get the inclusions @xmath388 and @xmath389 it remains to verify the inclusions @xmath398 and @xmath397 let us consider an arbitrary @xmath415 that is , for some @xmath416 one has @xmath417 if @xmath418 then , by the choice of @xmath419 @xmath392 for all @xmath382 but , on the other hand , by proposition [ berniglemma]([strict : part ] ) , @xmath420 for some @xmath325 hence we necessarily have @xmath421 , and we get that @xmath422 consequently @xmath423 now assume @xmath424 then , by proposition [ berniglemma]([nonstrict : part ] ) , @xmath425 for some @xmath325 but , in the same way as we showed above , we deduce that @xmath426 hence @xmath427 which means that @xmath428 if @xmath429 then , by theorem [ 07.12.10,11:53 ] , one has @xmath430 and by this @xmath431 as for the algorithmic part of the assertion , we notice that @xmath432 can be easily computed from @xmath433 the computability of @xmath56 follows from theorem [ 07.12.10,11:53 ] .
we mention that the `` combinatorial component '' of our proofs ( dealing with elementary symmetric functions ) resembles in part the proof of theorem [ avehenkthm ] . however , the crucial parts of the proofs of theorems [ main : n+1 ] and theorem [ main : n ] concerning the approximation of @xmath0 are based on different ideas .
the polynomials @xmath68 from theorem [ avehenkthm ] can be computed in a rather straightforward way ; see .
in contrast to this , the constructive parts of the proofs of theorems [ main : n+1 ] and [ main : n ] use decidability of the first order logic over reals and , by this , lead to algorithms of extremely high complexity . even though theorem [ loj ] and proposition [ berniglemma ]
were also used in , our proofs can not be viewed as extensions of the proofs from .
i am indebted to prof .
martin henk for his support during the preparation of the manuscript .
the examples in remark [ bounded : extension : remark ] arose from a discussion with prof .
claus scheiderer . c. andradas , l. brcker , and j. m. ruiz , _
constructible sets in real geometry _ , ergebnisse der mathematik und ihrer grenzgebiete ( 3 ) [ results in mathematics and related areas ( 3 ) ] , vol .
33 , springer - verlag , berlin , 1996 .
mr 98e:14056 j. bochnak , m. coste , and m .- f .
roy , _ real algebraic geometry _ , ergebnisse der mathematik und ihrer grenzgebiete ( 3 ) [ results in mathematics and related areas ( 3 ) ] , vol .
36 , springer - verlag , berlin , 1998 , translated from the 1987 french original , revised by the authors . | let @xmath0 be an elementary closed semi - algebraic set in @xmath1 , i.e. , there exist real polynomials @xmath2 ( @xmath3 ) such that @xmath4 ; in this case @xmath2 are said to represent @xmath0 .
denote by @xmath5 the maximal number of the polynomials from @xmath6 that vanish in a point of @xmath7 if @xmath0 is non - empty and bounded , we show that it is possible to construct @xmath8 polynomials representing @xmath7 furthermore , the number @xmath8 can be reduced to @xmath5 in the case when the set of points of @xmath0 in which @xmath5 polynomials from @xmath6 vanish is finite .
analogous statements are also obtained for elementary open semi - algebraic sets .
primary : 14p10 , secondary : 14q99 , 03c10 , 90c26 approximation , elementary symmetric function , ojasiewicz s inequality , polynomial optimization , semi - algebraic set , theorem of brcker and scheiderer | 0804.2134 |
the quantum mechanical description of neutrino oscillations @xcite has been the subject of much discussion and debate in the recent literature .
the ` standard ' oscillation formula @xcite , yielding an oscillation phase or @xmath1 . ] , at distance @xmath2 from the neutrino source , between neutrinos , of mass @xmath3 and @xmath4 and momentum @xmath5 , of : are used throughout . ]
@xmath6 is derived on the assumption of equal momentum and equal production times of the two neutrino mass eigenstates .
other authors have proposed , instead , equal energies @xcite or velocities @xcite at production , confirming , in both cases , the result of the standard formula . the latter reference claims , however , that the standard expression for @xmath7 should be multiplied by a factor of two in the case of the equal energy or equal momentum hypotheses when different production times are allowed for the two mass eigenstates
. however , the equal momentum , energy or velocity assumptions are all incompatible with energy - momentum conservation in the neutrino production process @xcite .
in two recent calculations @xcite a covariant formalism was used in which exact energy - momentum conservation was imposed .
these calculations used the invariant feynman propagator @xcite to describe the space - time evolution of the neutrino mass eigenstates . in ref .
@xcite a formula for the neutrino oscillation phase differing by a factor of 9.9 from eqn(1.1 ) was found for the case of pion decay at rest , and it was predicted that correlated spatial oscillations in the detection probability of neutrinos and the recoiling decay muons could be observed .
however , the author of ref .
@xcite as well as others @xcite claimed that muon oscillations would either be completely suppressed , or essentially impossible to observe .
the present paper calculates the probabilities of oscillation of neutrinos and muons produced by pions decaying both at rest and in flight , as well as the probabilities of neutrino oscillation following muon decay or @xmath8decay of a nucleus at rest .
the calculations , which are fully covariant , are based on feynman s reformulation of quantum mechanics @xcite in terms of interfering amplitudes associated with classical space - time particle trajectories . the essential interpretational formula of this approach , though motivated by the seminal paper of dirac on the lagrangian formulation of quantum mechanics @xcite , and much developed later in the work of feynman and other authors @xcite , was actually already given by heisenberg in 1930 @xcite .
the application of the path amplitude formalism to neutrino or muon oscillations is particularly staightforward , since , in the covariant formulation of quantum mechanics , energy and momentum are exactly conserved at all vertices and due to the macroscopic propagation distances of the neutrinos and muons all these particles follow essentially classical trajectories ( _ i.e. _ corresponding to the minima of the classical action ) which are rectilinear paths with constant velocities .
the essential formula of feynman s version of quantum mechanics , to be employed in the calculations presented below , is @xcite : @xmath9 where @xmath10 is the probability to observe a final state @xmath11 , given an initial state @xmath12 , and @xmath13 are ( unobserved ) intermediate quantum states . in the applications to be described in this paper , which , for simplicity , are limited to the case of the first two generations of leptons , eqn(1.2 ) specialises to : @xmath14 for the case of neutrino oscillations and @xmath15 for the case of muon oscillations .
@xmath16 is the probability to observe the charged current neutrino interaction : @xmath17 following the decay : @xmath18 , while @xmath19 is the probability to observe the decay @xmath20 , after the same decay process . in eqns.(1.3),(1.4 ) @xmath21 are neutrino mass eigenstates while @xmath22 are the corresponding recoil muon states from pion decay .
@xmath23 , @xmath24 and @xmath25 denote invariant decay amplitudes , @xmath26 is the invariant amplitude of the charged current neutrino interaction , @xmath27 is the invariant space - time propagator of particle @xmath28 between the space - time points @xmath29 and @xmath30 and @xmath31 is an invariant amplitude describing the production of the @xmath32 by the source @xmath33 and its space - time propagation to the space - time point @xmath34
. an important feature of the amplitudes appearing in eqns(1.3 ) and ( 1.4 ) is that they are completely defined in terms of the physical neutrino mass eigenstate wavefunctions @xmath35 .
this point will be further discussed in section 5 below .
the difference of the approach used in the present paper to previous calculations presented in the literature can be seen immediately on inspection of eqns(1.3 ) and ( 1.4 ) .
the initial state contributes only a multiplicative constant to the transition probabilites .
the initial state can then just as well be defined as ` pion at @xmath34 ' , rather than @xmath36 .
this is done in the calculations presented in section 2 below . ]
is a pion at space time - point @xmath34 , the final state an @xmath37 or @xmath38 produced at space - time point @xmath39 . these are unique points , for any given event and do not depend in any way on the masses of the unobserved neutrino eigenstates propagating from @xmath40 to @xmath39 in eqn(1.3 ) .
on the other hand the ( unobserved ) space - time points @xmath40 at which the neutrinos and muons are produced _ do _ depend on @xmath41 .
indeed , because of the different velocities of the propagating neutrino eigenstates , only in this case can both neutrinos and muons ( representing _ alternative _ classical histories of the decaying pion ) both arrive simultanously at the unique point @xmath39 where the neutrino interaction occurs ( eqn(1.3 ) ) or the muon decays ( eqn(1.4 ) ) .
the crucial point in the above discussion is that the decaying pion , _ via _ the different path amplitudes in eqns(1.3 ) and ( 1.4 ) , _ interferes with itself_. to modify very slightly dirac s famous statement : ` each pion then interferes only with itself .
interference between two different pions never occurs ' .
because of the different possible decay times of the pion in the two interfering path amplitudes , the pion propagators @xmath42 in eqns(1.3 ) and ( 1.4 ) above give important contributions to the interference phase . to the author s best knowledge ,
this effect has not been taken into account in any previously published calculation of neutrino oscillations .
the results found for the oscillation phase are , for pion decays at rest : @xmath43 and for pion decays in flight : @xmath44 where @xmath45 the superscripts indicate the particles whose propagators contribute to the interference phase .
also @xmath46 , @xmath47 and @xmath48 are the energies of the parent @xmath49 and the decay @xmath50 and @xmath51 and @xmath52 , @xmath53 the angles between the pion and the neutrino , muon flight directions . in eqns.(1.5 ) to ( 1.7 ) terms of order @xmath54 , @xmath55 , and higher , are neglected , and in eqns.(1.6 ) and ( 1.7 ) ultrarelativistic kinematics with @xmath56 is assumed .
formulae for the oscillation phase of neutrino oscillations following muon decays or nuclear @xmath0-decays at rest , calculated in a similar manner to eqn(1.5 ) , are given in section 3 below .
a brief comment is now made on the generality and the covariant nature of the calculations presented in this paper .
although the fundamental formula ( 1.2 ) is valid in both relativistic and non - relativistic quantum mechanics , it was developed in detail by feynman @xcite only for the non - relativistic case .
for the conditions of the calculations performed in the present paper ( propagation of particles in free space ) the invariant space - time propagator can either be derived ( for fermions ) from the dirac equation , as originally done by feynman @xcite or , more generally , from the covariant feynman path integral for an arbitary massive particle , as recently done in ref .
@xcite . in the latter case ,
the invariant propagator for any stable particle with pole mass @xmath57 , between space - time points @xmath58 and @xmath59 in free space is given by the path integral @xcite : @xmath60 \exp \left \
{ -\frac{im}{2 } \int_{x_i}^{x_f } \left ( \frac{dx}{d\tau } \cdot \frac{dx}{d\tau}+1 \right)d \tau\right \}\ ] ] where @xmath61 is the proper time of the particle . by splitting the integral over @xmath62 on the right side of eqn(1.8 ) into the product of a series of infinitesimal amplitudes corresponding to small segments , @xmath63 , gaussian integration
may be performed over the intermediate space - time points . finally , integrating over the proper time @xmath61 , the analytical form of the propagator
is found to be @xcite : @xmath64 where : @xmath65 and @xmath66 is a first order hankel function of the second kind , in agreement with ref .
@xcite . in the asymptotic region where @xmath67 , or for the propagation of on - shell particles @xcite
, the hankel function reduces to an exponential and yields the configuration space propagator @xmath68 of eqn(2.11 ) below .
it is also shown in ref .
@xcite that energy and momentum is exactly conserved in the interactions and decays of all such ` asymptotically propagating ' particles .
the use of quasi - classical particle trajectories and the requirement of exact energy - momentum conservation are crucial ingredients of the calculations presented below .
the structure of the paper is as follows . in the following section the case of neutrino or muon oscillations following pion decay at rest
is treated .
full account is taken of the momentum wave - packets of the progagating neutrinos and muons resulting from the breit - wigner amplitudes describing the distributions of the physical masses of the decaying pion and daughter muon .
the corresponding oscillation damping corrections and phase shifts are found to be very small , indicating that the quasi - classical ( constant velocity ) approximation used to describe the neutrino and muon trajectories is a very good one .
the incoherent effects , of random thermal motion of the source pion , and of finite source and detector sizes , on the oscillation probabilities and the oscillation phases , are also calculated .
these corrections are found to be small in typical experiments , but much larger than those generated by the coherent momentum wave packets . in section 3 , formulae are derived to describe neutrino oscillations following muon decay at rest or the @xmath0-decay of radioactive nuclei .
these are written down by direct analogy with those derived in the previous section for pion decay at rest . in section 4 ,
the case of neutrino and muon oscillations following pion decay in flight is treated . in this case
the two - dimensional spatial geometry of the particle trajectories must be related to the decay kinematics of the production process . due to the non - applicability of the ultrarelativistic approximation to the kinematics of the muon in the pion rest frame , the calculation , although straightforward , is rather tedious and lengthy for the case of muon oscillations , so the details are relegated to an appendix .
finally , in section 5 , the positive aspects and shortcomings of previous treatments in the literature of the quantum mechanics of neutrino and muon oscillations are discussed in comparison with the method and results of the present paper .
to understand clearly the different physical hypotheses and approximations underlying the calculation of the particle oscillation effects it is convenient to analyse a precise experiment .
this ideal experiment is , however , very similar to lnsd @xcite and karmen @xcite except that neutrinos are produced from pion , rather than muon , decay at rest .
the different space - time events that must be considered in order to construct the probability amplitudes for the case of neutrino oscillations following pion decay at rest are shown in fig.1 .
a @xmath32 passes through the counter c@xmath69 , where the time @xmath70 is recorded , and comes to rest in a thin stopping target t ( fig.1a ) ) . for simplicity ,
the case of only two neutrino mass eigenstates @xmath71 and @xmath72 of masses @xmath3 and @xmath4 ( @xmath3 @xmath73 @xmath4 ) is considered .
the pion at rest constitutes the initial state of the quantum mechanical probability amplitudes .
the final state is an @xmath74 system produced , at time @xmath75 , via the process @xmath76 at a distance @xmath2 from the decaying @xmath32 ( fig.1d ) ) .
two different physical processes may produce the observed @xmath74 final state , as shown in fig.1b ) and 1c ) , where the pion decays either at time @xmath77 into @xmath71 or at time @xmath78 into @xmath72 .
the probability amplitudes for these processes are , up to an arbitary normalisation constant , and neglecting solid angle factors in the propagators : @xmath79 note that following the conventional ` @xmath80 ' ( final , initial ) ordering of the indices of matrix elements in quantum mechanics , the path amplitude is written from right to left in order of increasing time .
this ensures also correct matching of ` bra ' and ` ket ' symbols in the amplitudes . in eqn(2.1 ) , @xmath81 , @xmath82 are ` reduced ' invariant amplitudes of the @xmath50 charged current scattering and pion decay processes , respectively , @xmath83 , @xmath84 and @xmath85 , @xmath86 are relativistic breit - wigner amplitudes , @xmath87 and @xmath88 are elements of the unitary maki - nagagawa - sakata ( mns ) matrix @xcite , @xmath89 , describing the charged current coupling of a charged lepton , @xmath90 , ( @xmath91 ) to the neutrino mass eigenstate @xmath12 .
the reduced invariant amplitudes are defined by factoring out the mns matrix element from the amplitude for the process .
for example : @xmath92 . since the purely kinematical effects of the non - vanishing neutrino masses are expected to be very small , the reduced matrix elements may be assumed to be lepton flavour independent : @xmath93 where @xmath94 denotes a massless neutrino . in eqn(2.1 ) , @xmath95 is the lorentz - invariant configuration space propagator @xcite of the pion or neutrino .
the pole masses and total decay widths of the pion and muon are denoted by @xmath96 , @xmath97 and @xmath98 , @xmath99 respectively . for simplicity ,
phase space factors accounting for different observed final states are omitted in eqn(2.1 ) and subsequent formulae . because the amplitudes and propagators in eqn(2.1 ) are calculated using relativistic quantum field theory , and the neutrinos propagate over macroscopic distances , it is a good approximation , as already discussed in the previous section , to assume exact energy - momentum conservation in the pion decay process , and that the neutrinos are on their mass shells , _ i.e. _ @xmath100 , where @xmath101 is the neutrino energy - momentum four - vector . in these circumstances
the neutrino propagators correspond to classical , rectilinear , particle trajectories .
the pion and muon are unstable particles whose physical masses @xmath102 and @xmath103 differ from the pole masses @xmath104 and @xmath98 appearing in the breit - wigner amplitudes and covariant space - time propagators in eqn(2.1 )
. the neutrino momentum @xmath105 will depend on these physical masses according to the relation : @xmath106[w_{\pi}^2-(m_i - w_{\mu(i)})^2 ] \right]^{\frac{1}{2 } } } { 2 w_{\pi}}\ ] ] note that , because the initial state pion is the same in the two path amplitudes in eqn(2.1 ) @xmath102 does not depend on the neutrino mass index @xmath12 .
however , since the pion decays resulting in the production of @xmath71 and @xmath72 are independent physical processes , the physical masses of the unobserved muons , @xmath107 , recoiling against the two neutrino mass eigenstates are not , in general , the same . in the following kinematical calculations sufficient accuracy
is achieved by retaining only quadratic terms in the neutrino masses , and terms linear in the small quantities : @xmath108 , @xmath109 .
this allows simplification of the relativistic breit - wigner amplitudes : @xmath110 developing eqn(2.2 ) up to first order in @xmath111 , @xmath112 and @xmath113 yields the relation : @xmath114 \end{aligned}\ ] ] where @xmath115 the term @xmath116 which is also included in eqn(2.4 ) gives a negligible o(@xmath117 ) contribution to the neutrino oscillation formula . for muon oscillations ,
however , it gives a term of o(@xmath111 ) in the interference term , as discussed below .
similarly , the exact formula for the neutrino energy : @xmath118 in combination with eqn(2.4 ) gives , for the neutrino velocity : @xmath119~+~o(m_i^4,\delta_{\pi}^2,\delta_i^2)\ ] ] this formula will be used below to calculate the neutrino times - of - flight : @xmath120 . from the unitarity of the mns matrix
, the elements @xmath89 may be expressed in terms of a single real angular parameter @xmath121 : @xmath122 the parts of the amplitudes requiring the most careful discussion are the invariant space - time propagators @xmath95 , as it is mainly their treatment that leads to the different result for the neutrino oscillation phase found in the present paper , as compared to those having previously appeared in the literature . in the limit of large time - like separations , the propagator may be written as @xcite : @xmath123\ ] ] @xmath95 is the amplitude for a particle , originally at a space - time point ( @xmath124 , @xmath125 ) , to be found at ( @xmath126 , @xmath127 ) and @xmath128 , @xmath129 . in the following , according to the geometry of the experiment shown in fig.1
, only one spatial coordinate will be considered ( @xmath130 ) and only the exponential factor in eqn(2.10 ) , containing the essential phase information for particle oscillations will be retained in the amplitudes .
solid angle correction factors , taken into account by the factor in large brackets in eqn(2.10 ) , are here neglected , but are easily included in the final oscillation formulae .
writing then @xmath131 = \exp[-im\delta \tau ] \equiv \exp[-i\delta \phi]\ ] ] it can be seen that the increment in phase of the propagator , @xmath132 , when the particle undergoes the space - time displacement ( @xmath133 , @xmath134 ) is a lorentz invariant quantity equal to the product of the particle mass and the increment , @xmath63 , of proper time . using the relativistic time dilatation formula : @xmath135 and also the relation , corresponding to a classical , rectilinear , particle trajectory :
@xmath136 gives , for the phase increments corresponding to the paths of the neutrinos and the pion in fig.1 : @xmath137 \\
\delta \phi_i^{\pi } & = & m_{\pi } ( t_i - t_0 ) = m_{\pi } ( t_d - t_0)-\frac{m_{\pi}l}{v_i } \nonumber \\ & = & m_{\pi } ( t_d - t_0)-m_{\pi}l\left\{1+\frac{m_i^2}{2 p_0 ^ 2}\left[1- \frac{2\delta_{\pi}}{m_{\pi}}\frac{(m_{\pi}^2+m_{\mu}^2)}{(m_{\pi}^2-m_{\mu}^2 ) } + \frac{4\delta_i m_{\mu}}{m_{\pi}^2-m_{\mu}^2}\right]\right\ } \end{aligned}\ ] ] where terms of o(@xmath117 ) and higher are neglected . to perform the integral over @xmath112 in eqn(2.16 )
it is convenient to approximate the modulus squared of the breit - wigner amplitude by a gaussian , _ via _ the substitution : @xmath145 where the width of the gaussian is chosen so that it has approximately the same full width at half maximum , @xmath146 , as the breit - wigner function .
after the substitution ( 2.21 ) , the integral over @xmath112 in eqn(2.16 ) is easily evaluated by a change of variable to ` complete the square ' in the argument of the exponential , with the result : @xmath147\ ] ] eqn(2.18 ) gives , for @xmath148 , the numerical value : @xmath149 for typical physically interesting values ( see below ) of @xmath150 ev and @xmath151 m , @xmath148 takes the value 0.48 mev@xmath152 , so that @xmath153 then , to very good accuracy , @xmath154 , independently of the neutrino mass .
it follows that for neutrino oscillations , the muon mass dependence of the amplitudes may be neglected for any physically interesting values of @xmath155 and @xmath2 . from eqns ( 2.16 ) and
( 2.22 ) the probability to observe the reactions @xmath76 at distance @xmath2 from the pion decay point and at time @xmath75 is : @xmath156\delta_{\pi}\right]\right\ } \nonumber\end{aligned}\ ] ] the time dependent exponential factors in the curly brackets of eqn(2.24 ) are easily understood . if @xmath157 then @xmath158 .
this implies that the neutrino of mass @xmath3 results from an earlier decay than the neutrino of mass @xmath4 , in order to be detected at the same time . because of the exponential decrease with time of the pion decay amplitude , the contribution to the probability of the squared amplitude for the neutrino of mass @xmath3 is larger .
the interference term resulting from the product of the decay amplitudes of the two neutrinos of different mass , has an exponential factor that is the harmonic mean of those of the squared amplitudes for each neutrino mass eigenstate , and so is also suppressed relative to the squared amplitude for the neutrino of mass @xmath3 .
the integral over the physical pion mass is readily performed by replacing the breit - wigner function by a gaussian as in eqn(2.21 ) .
this leads to an overall multiplicative constant @xmath159 and a factor : @xmath160 multiplying the interference term .
for @xmath161 and @xmath151 m the numerical value of this factor is @xmath162 .
this tiny correction is neglected in the following equations .
integrating over @xmath75 gives the average probability to observe the @xmath74 final state at distance @xmath2 : @xmath163 \cos\frac{2 m_{\pi } m_{\mu}^2 \delta m^2 } { ( m_{\pi}^2- m_{\mu}^2)^2}l\right\ } \end{aligned}\ ] ] where all kinematical quantities are expressed in terms of @xmath164 , @xmath96 and @xmath98 .
note that the minimum value of @xmath75 is @xmath165 , @xmath166 and @xmath165 for the squared amplitude terms of neutrinos of mass @xmath3 , @xmath4 and the interference term , respectively .
on integrating over @xmath75 , the squared amplitude terms give equal contributions , the larger amplitude for mass @xmath3 being exactly compensated by a smaller range of integration . the exponential damping factor in the interference term in eqn(2.26 ) is derived using the relations : @xmath167 and @xmath168 to obtain @xmath169 the damping factor arises because the difference in the times - of - flight of the two neutrino paths is limited by the pion lifetime .
it will be seen below , however , that for distances @xmath2 of practical interest for the observation of neutrino oscillations , the damping effect is tiny .
the part of the oscillation phase in eqn(2.24 ) originating from the neutrino propagators ( the term associated with the ` 1 ' within the large curved brackets ) differs by a factor two from the corresponding expression in the standard formula .
the contribution to the oscillation phase of the propagator of the decaying pion ( the term associated with @xmath170 in the large curved brackets of eqn(2.24 ) ) has not been taken into account in any published calculation known to the author of the present paper . the oscillation phase in eqn(2.26 )
is @xmath171 times larger than that given by the standard formula ( 1.1 ) . for @xmath172 m , as in the lnsd experiment , the first oscillation maximum occurs for @xmath173 . denoting by @xmath174 the phase of the cosine interference term in eqn(2.26 )
, the pion lifetime damping factor can be written as : @xmath175 so the damping effect is vanishingly small when @xmath176 .
the oscillation formula ( 2.26 ) is calculated on the assumption that the decaying pion is at rest at the precisely defined position @xmath58 .
in fact , the positive pion does not bind with the atoms of the target , but will rather undergo random thermal motion .
this has three effects : an uncertainy in the value of @xmath58 , a doppler shift of the neutrino energy and a time dilatation correction correction factor of @xmath177 in the equation ( 2.15 ) for the pion phase increment . assuming that the target is at room temperature ( t= @xmath178 k ) , the mean kinetic energy of @xmath179 correponds to a mean pion momentum of @xmath180 mev and a mean velocity of @xmath181 km / sec .
the pion will move , in one mean lifetime ( @xmath182 sec ) , a distance of 146@xmath183 .
this is negligible as compared to @xmath2 ( typically @xmath184 30 m ) and so eqn(2.26 ) requires no modification to account for this effect .
the correction factor due to the doppler effect and time dilatation is readily calculated on the assumption of a maxwell - boltzmann distribution of the pion momentum : @xmath185 here @xmath186 mev .
details of the calculation are given in appendix a. the interference term in eqn(2.26 ) is modified by a damping factor : @xmath187l\right)^2 \right\}\ ] ] while the argument of the cosine term acquires an additional phase factor : @xmath188l\ ] ] for @xmath189 , @xmath190 and @xmath191 .
if the target in which the pion stops is of thickness @xmath192 , then the effect of different stopping points of the @xmath49 ( assumed uniformly distributed ) is to multiply the interference term in ( 2.26 ) by the factor : @xmath193 if the position of the neutrino interaction point within the target has an uncertainy of @xmath194 a similar correction factor is found , with the replacement @xmath195 in eqn(2.34 ) .
the calculation of this correction factor is also described in appendix a. if the target t in which the pion comes to rest ( fig.1a ) ) is chosen to be sufficiently thin , the pion decay process may be detected by observing the recoil muon in the counter c@xmath69 at times @xmath77 or @xmath78 ( fig.1b ) or 1c ) ) .
a sufficiently accurate measurement of the times @xmath77 , @xmath78 and @xmath75 would , in principle , enable separation of the different processes in figs.1b ) and 1c ) by the observation of separated peaks in the time - of - flight distribution at @xmath196 and @xmath197 . in this case
the interference term in eqn(2.26 ) vanishes as the two alternative space - time paths leading to the neutrino interaction are distinguishable .
however , for @xmath172 m and @xmath198(ev)@xmath199 the difference in the times of flight is only @xmath200 sec , more than ten orders of magnitude smaller than can be measured with existing techniques . as discussed in section 5 below , the momentum smearing due to the doppler effect at room temperature is some eleven orders of magnitude larger than the shift due to a neutrino mass difference with @xmath198(ev)@xmath199 .
thus , even with infinitely good time resolution , separation of such neutrino mass eigenstates by time - of - flight is not possible . the ideal experiment , described above to study neutrino oscillations , is easily adapted to the case of oscillations in the decay probability of muons produced by charged pion decay at rest . as previously pointed out in ref .
@xcite , such oscillations will occur if neutrinos with different masses exist .
as before , the pion stops in the target t at time @xmath70 ( see fig.2a ) ) . at time
@xmath77 the pion decays into @xmath71 and the corresponding recoil muon ( @xmath201 ) , whose passage is recorded in the counter c@xmath202 ( fig.2b ) ) .
similarly , a decay into @xmath72 and @xmath203 may occur at time @xmath78 ( fig.2c ) ) . with a suitable choice of the times @xmath77 and @xmath78 , such that muons following the alternative paths both arrive at the same time @xmath75 at the point @xmath59 , interference occurs between the path amplitudes when muon decay occurs at the space - time point ( @xmath59 , @xmath75 ) in the detector d ( fig.2d ) ) .
the probability for two _ classical _ trajectories to arrive at _ exactly _ the same space - time point of course vanishes .
the correct way to consider the quantum mechanical calculation is rather to ask _ given that _
the muon decay occurs at the point ( @xmath204 , @xmath75 ) , does the muon recoil against @xmath71 or @xmath72 ? if these two possiblities are not distinguished by the measurement of the decay process , the corresponding probability
_ amplitudes _
( not probabilities ) must be added in the calculation of the probability of the observed decay process .
the path amplitudes corresponding to muons recoiling against neutrinos of mass @xmath3 and @xmath4 are : @xmath205 the various factors in these equations are defined , _
mutatis mutandis _ , as in eqn(2.1 ) . with the same approximations , concerning the neutrino masses and the physical pion and muon masses , as those made above
, the velocity of the muon recoiling against the neutrino mass eigenstate @xmath206 is : @xmath207\end{aligned}\ ] ] where @xmath208 comparing with eqn(2.7 ) , it can be seen that for the muon case , unlike that where neutrino interactions are observed , there are pion and muon mass dependent correction terms that are independent of the neutrino masses , implying a velocity smearing effect due to the physical pion and muon masses that is @xmath209 larger than for the case of neutrino oscillations . the phase increments corresponding to the paths of the muons and the pion in fig . 2 are , using ( 2.4 ) . ] and ( 2.12)-(2.15 ) and ( 2.36 ) : @xmath210 \end{aligned}\ ] ] @xmath211 @xmath212 @xmath213\ ] ] where @xmath214 using eqns(2.11),(2.38 ) and ( 2.39 ) to re - write the space - time propagators in eqn(2.35 ) , as well as eqn(2.3 ) for the breit - wigner amplitudes gives : @xmath215 d \delta_i~~i=1,2 \end{aligned}\ ] ] where @xmath216 @xmath217 @xmath218\ ] ] and @xmath219 where , as in eqns(2.18 ) and ( 2.19 ) , imaginary parts of order @xmath220 are neglected . making the substitution ( 2.21 ) and performing the integral over @xmath112 according to eqn(2.22 ) ,
the following formula is found for the probability for muon decay at distance @xmath2 and time @xmath75 .
@xmath221 ^ 2 |<e^+\nu_k\overline{\nu}_{l}|t|\mu^+>|^2 e^{-\frac{\gamma_{\mu}v_0^{\mu}}{\gamma_0^{\mu}}l } \nonumber \\ & & |<\nu_0 \mu^+|t_r|\pi^+>|^2 e^{-\gamma_{\pi}(t_d - t_0 ) } \frac{\gamma_{\pi}^2}{4(\delta_{\pi}^2+\frac{\gamma_{\pi}^2}{4 } ) } \nonumber \\ & & \left\{\sin^2\theta e^{\gamma_{\pi } t_{\mu(1)}^f } + \cos^2 \theta e^{\gamma_{\pi } t_{\mu(2)}^f } \right .
\nonumber \\ & & -2\sin \theta \cos \theta e^{\frac{\gamma_{\pi}}{2}(t_{\mu(1)}^f + t_{\mu(2)}^f ) } \nonumber \\ & & \left .
{ \it re } \exp i \left [ \frac{m_{\mu}^2 \delta m^2}{2 p_0 ^ 3}\left(1-\frac { e_0^{\mu}}{m_{\pi}}\right)l + [ \alpha_{\pi}^{\mu}(1)-\alpha_{\pi}^{\mu}(2)]\delta_{\pi } \right ] \right\}\end{aligned}\ ] ] where the effect of the non - zero neutrino masses are neglected in the reduced pion decay amplitude so that @xmath222 and this amplitude is a common factor in both path amplitudes .
the muon path difference yields the term associated with @xmath223 in the interference phase in eqn(2.46 ) while the pion path is associated with ` 1 ' in the large round brackets .
the numerical value of the damping factor : @xmath224 [ 1-\frac{\alpha^{\mu } \gamma_{\mu}}{3}]^2\ ] ] resulting from the integral over the physical muon mass is , for @xmath151 m , 0.774 , so , unlike for the case of neutrino oscillations , the correction is by no means negligible .
this is because , in the muon oscillation case , the leading term of @xmath225 is not proportional to the neutrino mass squared .
the non - leading terms proportional to @xmath111 have been neglected in eqn(2.43 ) .
this correction however effects only the overall normalisation of the oscillation formula , not the functional dependence on @xmath2 arising from the interference term . integrating over @xmath113 using eqns(2.21 ) and ( 2.22 ) , as well as over @xmath75 , gives the probability of muon decay , into the final state @xmath226 , at distance , @xmath2 , from the production point , where all kinematical quantities are expressed in terms of @xmath164 , @xmath96 and @xmath98 : @xmath227[1- \frac{4}{3 } \frac{\gamma_{\mu } m_{\mu } m_{\pi } } { m_{\pi}^2- m_{\mu}^2}l]^2 \nonumber \\ & & |<e^+\nu_k\overline{\nu}_{l}|t|\mu^+>|^2 \exp \left[-\frac{2 \gamma_{\mu } m_{\pi } m_{\mu } ( m_{\pi}^2- m_{\mu}^2)}{(m_{\pi}^2 + m_{\mu}^2)^3 } l \right]|<\nu_0 \mu^+|t_r|\pi^+>|^2 \nonumber \\ & & \left\{1-\sin 2 \theta\exp[-\frac{2 \gamma_{\pi } m_{\pi}^2 m_{\mu}^2 \delta m^2 } { ( m_{\pi}^2- m_{\mu}^2)^3}l ] \cos \frac{2 m_{\pi } m_{\mu}^2 \delta m^2 } { ( m_{\pi}^2- m_{\mu}^2)^2 } l \right\}\end{aligned}\ ] ] in this expression the correction due to the damping factor of the interference term : @xmath228\ ] ] arising from the integral over the physical pion mass has been neglected . for @xmath161 and @xmath151 m the numerical value of this factor is @xmath229 . denoting by @xmath230 the argument of the cosine in eqn(2.48 ) , the exponential damping factor due to the pion lifetime may be written as : @xmath231 for @xmath232 , @xmath233 $ ] so , as in the neutrino oscillation case , the pion lifetime damping of the interference term is very small .
introducing the reduced muon decay amplitude : @xmath234 the total muon decay probability is given by the incoherent sum over the four possible final states containing massive neutrinos : @xmath235 where the unitarity of the mns matrix has been used .
@xmath236 is given by the replacement of @xmath237 by @xmath238 in eqn(2.48 ) .
eqn(2.52 ) shows that the muon decay width is independent of the values of the mns matrix elements .
corrections due to time dilatation and the doppler effect are calculated in a similar way to the neutrino oscillation case with the results ( see appendix a ) : @xmath239l\right)^2 \right\}\ ] ] and @xmath240l\ ] ] as for neutrino oscillations , the corresponding corrections are very small for oscillation phases of order unity .
the phase of the cosine in the interference term is the same in neutrino and muon oscillations , as can be seen by comparing eqns(2.26 ) and ( 2.48 ) .
it follows that the target or detector size correction ( eqn(2.34 ) ) is the same in both cases .
neutrino and muon oscillations from pion decay at rest then have an identical oscillation phase for given values of @xmath164 and @xmath2 . in view of the much larger event rate that is possible , it is clearly very advantageous in this case to observe muons rather than neutrinos , since the rate of neutrino oscillation events is severely limited by the very small neutrino interaction cross section .
in fact , it it not necessary to observe muon decay , as in the example discussed above .
the oscillation formula applies equally well if the muons are observed at the distance @xmath2 using any high efficiency detector such as , for example , a scintillation counter . according to eqn(1.2 ) ,
interference between the path amplitudes must occur if the muon detection device does not discriminate muons recoiling against @xmath71 from those recoiling against @xmath72 .
the formula describing ` @xmath241 neutrino oscillations ' appearence ' . as discussed in more detail in section 5 below , ` @xmath242 ' and ` @xmath243 ' do not exist , as physical states , if neutrinos are massive and the mns matrix is non - diagonal .
it is still , however , current practice in the literature to use the symbols ` @xmath244 ' , ` @xmath245 ' and ` @xmath246 ' to refer to massive neutrinos .
this is still a useful and meaningful procedure if it is employed only to identify , in a concise manner , the type of charged current interaction by which the neutrinos are produced or detected , i.e. ` @xmath247 ' means neutrinos ( actually several , with different generation numbers ) produced together with the charged lepton @xmath248 or detected by observation of the charged lepton @xmath249 .
it should not be forgotten however that only the wavefunctions of the mass eigenstates @xmath206 occur in the amplitudes of standard model processes . ] following the decay at rest of a @xmath250 , @xmath251 is easily derived from the similar formula for @xmath32 decay at rest , ( 2.25 ) . because the neutrino momentum spectrum is continous , smearing effects due to the finite muon lifetime may be neglected from the outset . the phase increment associated with the neutrino path
is then given by eqn(2.14 ) with the replacements @xmath252 and @xmath253 , where @xmath254 is the antineutrino momentum .
the phase increment of the decaying muon is given by the same replacements in eqn(2.15 ) with , in addition , @xmath255 and @xmath256 .
the formula , analogous to eqn(2.26 ) , for the time - averaged probability to detect the process @xmath257 at a distance l from the muon decay point is then : @xmath258 \cos \left[\frac{\delta m^2 } { p_{\overline{\nu}}}\left ( \frac{m_{\mu}}{2 p_{\overline{\nu}}}-1\right)l \right ] \right\}\end{aligned}\ ] ] the standard neutrino oscillation formula , hitherto used in the analysis of all experiments , has instead the expression @xmath259 for the argument of the cosine term in eqn(3.1 ) . denoting my @xmath260
the value of @xmath164 obtained using the standard formula , and @xmath261 that obtained using the feynman path ( fp ) formula ( 3.1 ) then : @xmath262 for a typical value of @xmath263 of 45 mev , eqn(3.2 ) implies that @xmath264 .
thus the @xmath243 oscillation signal from @xmath250 decays at rest reported by the lnsd collaboration @xcite corresponding to @xmath265 ( ev)@xmath199 for @xmath266 implies @xmath267 ( ev)@xmath199 for a similar mixing angle . for the case of @xmath0-decay : @xmath268 @xmath96 in the first line of eqn(2.15 ) is replaced by @xmath269 , the total energy release in the @xmath0-decay process : @xmath270 where @xmath271 and @xmath272 are the masses of the parent and daughter nuclei . that the phase advance of an unstable state , over a time , @xmath134 , is given by @xmath273 where @xmath274 is the excitation energy of the state , is readily shown by the application of time - dependent perturbation theory to the schrdinger equation @xcite .
a more intuitive derivation of this result has also been given in ref .
@xcite . in the present case , @xmath275 . omitting the lifetime damping correction , which is about eight orders of magnitude smaller than for pion decay ,
given a typical @xmath0-decay lifetime of a few seconds , the time - averaged probabilty to detect @xmath276 via the process @xmath277 , at distance @xmath2 from the decay point is given by the formula , derived in a similar way to eqns(2.26 ) and ( 3.1 ) : @xmath278 \right\}\end{aligned}\ ] ] where @xmath279 the lifetime of the unstable nucleus @xmath280 .
until now , all experiments have used the standard expression @xmath259 for the neutrino oscillation phase .
the values of @xmath164 found should be scaled by the factor @xmath281 , suitably averaged over @xmath263 , to obtain the @xmath164 given by the feynman path formula ( 3.4 ) .
in this section , the decays in flight of a @xmath32 beam with energy @xmath282 into @xmath283 are considered .
as the analysis of the effects of the physical pion and muon masses have been shown above to give negligible corrections to the @xmath2 dependence of the oscillation formulae , for the case of decays at rest , such effects will be neglected in this discussion of in - flight decays .
the overall structure of the path amplitudes for neutrinos and muons is the same as for decays at rest ( see eqns(2.1 ) and ( 2.35 ) ) .
however , for in - flight decays , in order to calculate the interfering paths originating at different and terminating at common space - time points , the two - dimensional spatial geometry of the problem must be properly taken into account . in fig.3
a pion decays at a into the 1 mass eigenstate , the neutrino being emitted at an angle @xmath284 in the lab system relative to the pion flight direction . if @xmath285 a later pion decay into the 2 mass eigenstate at the angle @xmath286 may give a path such that both eigenstates arrive at the point b at the same time .
a neutrino interaction @xmath287 occuring at this space - time point will than be sensitive to interference between amplitudes corresponding to the paths ab and acb .
the geometry of the triangle abc and the condition that the 1 and 2 neutrino mass eigenstates arrive at b at the same time gives the following condition on their velocities : @xmath288 expanding to first order in the small quantity @xmath289 , rearranging , and neglecting terms of @xmath290 , gives : @xmath291\ ] ] where the relation : @xmath292 has been used . rearranging eqn(4.2 ) : @xmath293\ ] ] the difference in phase of the neutrino paths ab and cb is ( see eqn(2.14 ) ) : @xmath294 since the angle @xmath295 is @xmath296 , the difference between ab and cb is of the same order , and so : @xmath297 where @xmath298 , the measured neutrino energy . from the geometry of the triangle abc : @xmath299 so , to first order in @xmath295 , and using eqn(4.4 ) : @xmath300 and @xmath301 eqns(4.8 ) and ( 4.9 ) give , for the phase increment of the pion path : @xmath302 in eqn(4.10 ) , the lorentz invariant character of the propagator phase is used . setting @xmath303 and @xmath304 ,
gives for @xmath305 a prediction consistent with that obtained from eqn(2.15 ) .
eqns(4.6 ) and ( 4.10 ) give , for the total phase difference of the paths ab , acb : @xmath306\ ] ] using the expressions , valid in the ultra - relativistic ( ur ) limit where @xmath307 : @xmath308 and @xmath309 where @xmath310 is the angle between the directions of the pion and neutrino momentum vectors in the pion rest frame , eqn(4.11 ) may be rewritten as : @xmath311 for @xmath312 the oscillation phase is the same as for pion decay at rest ( see eqn(2.26 ) ) since in the latter case , @xmath313 . using eqn(4.14 ) , the probability to observe a neutrino interaction , at point b , produced by the decay product of a pion decay occuring within a region of length , @xmath314 , ( @xmath315 ) centered at the point a , in a beam of energy @xmath46 , is given by a formula analagous to eqn(2.26 ) : @xmath316 as in the case of pion decay at rest , eqn(2.26 ) , the oscillation phase differs by the factor @xmath171 from that given by the standard formula .
the derivation of the formula describing muon oscillations following pion decays in flight is very similar to that just given for neutrino oscillations .
the condition on the velocities so that the muons recoiling against the different neutrino mass eigenstates arrive at the point b ( see fig.3 ) at the same time , is given by a formula analagous to ( 4.2 ) : @xmath317 } { v_{\pi}\sin\theta_1}\delta\ ] ] the formula relating the neutrino masses to the muon velocities is , however , more difficult to derive than the corresponding relation for neutrinos , ( 4.3 ) , as the decay muons are not ultra - relativistic in the pion rest frame .
the details of this calculation are given in appendix b. the result is : @xmath318 using eqn(4.16 ) and the relation , valid to first order in @xmath295 , : @xmath319 where @xmath134 is the flight time of the pion from a to c in fig.3 ( and also the difference in the times - of - flight of the muons recoiling against the two neutrino eigenstates ) , the angle @xmath295 may be eliminated to yield : @xmath320}\ ] ] using now the kinematical relation ( see appendix b ) : @xmath321 and the expression for the phase difference of the paths ab and acb : @xmath322 together with eqn(4.19 ) , it is found , taking the ur limit , where @xmath323 , that @xmath324\ ] ] the probability of detecting a muon decay at b is then : @xmath325 |<\nu_{0 } \mu^+|t_r|\pi^+>|^2 \nonumber \\ & & \times \left\{1-\sin 2 \theta \cos \frac { 2 m_{\mu}^2 \delta m^2}{e_{\mu}^2 ( m_{\pi}^2-m_{\mu}^2)^2 ) } \left[\frac{(m_{\mu}^2 e_{\pi}- m_{\pi}^2 e_{\mu})l}{\cos\theta_1}\right ] \right\ } \end{aligned}\ ] ] where @xmath314 is defined in the same way as in eqn(4.15 ) .
the quantum mechanics of neutrino oscillations has been surveyed in recent review articles @xcite , where further extensive lists of references may be found . in this section
, the essential differences between the calculations presented in the present paper and all previous treatments in the literature of the quantum mechanics of neutrino oscillations , as cited in the above review articles , will be summarised . a critical review of the existing literature
will then be given .
hitherto , it has been assumed that the neutrino source produces a ` lepton flavour eigenstate ' that is a superposition of mass eigenstates , at some fixed time . in this paper
it is , instead , assumed following shrock @xcite that the neutrino mass eigenstates are produced incoherently in different physical processes .
this follows from the structure of the leptonic charged current in the electroweak standard model : @xmath326 only the wavefunctions of the physical neutrino mass eigenstates @xmath206 appear in this current , and hence in the initial or final states of any physical process .
a consequence is that the neutrino mass eigenstates can be produced at different times in the path amplitudes corresponding to different mass eigenstates .
it has recently been shown that experimental measurements of the decay width ratio ; @xmath327 and of the mns matrix elements are inconsistent with the production of a coherent ` lepton flavour eigenstate ' in pion decay @xcite and that the the ` equal time ' or ` equal velocity ' hypothesis resulting from this assumption underestimates , by a factor of two , the contribution of neutrino propagation to the oscillation phase @xcite .
as demonstrated above , allowing for the possibility of different production times of the neutrinos results in an important , decay process dependent , contribution to the oscillation phase from the propagator of the source particle .
the non - diagonal elements of @xmath89 in eqn(5.1 ) describe violation of lepton flavour ( or generation number ) by the weak charged current . for massless neutrinos ,
the mns matrix becomes diagonal ; lepton flavour is conserved within each generation , and the familiar ` lepton flavour eigenstates ' are give by the replacements : @xmath328 , @xmath329 , @xmath330 . only in this case
are the lepton flavour eigenstates physical , being all mass eigenstates of vanishing mass .
the standard derivation of the neutrino oscillation phase will now be considered , following the treatment of ref .
@xcite , but using the notation of the present paper .
the calculation is performed assuming an initial ` lepton lavour eigenstate of the neutrino ' that is a superposition of the mass eigenstates @xmath331 and @xmath332 : @xmath333 where @xmath334 are mass eigenstates of fixed momentum @xmath335 .
this flavour eigenstate is assumed to evolve with laboratory time , @xmath336 , according to fixed energy solutions of the non - relativistic schrdinger equation into the mixed flavour state @xmath337 : @xmath338 where @xmath339 , @xmath340 are the laboratory energies of the neutrino mass eigenstates .
the amplitude for transition into the ` electron flavour eigenstate ' : @xmath341 at time @xmath336 is then , using eqns(5.3 ) , ( 5.4 ) : @xmath342 because it is assumed that the neutrinos have the same momentum but different energies : @xmath343 and using ( 5.5 ) and ( 5.6 ) , the probability of the flavour state @xmath244 at time @xmath336 is found to be : @xmath344 \right)\ ] ] finally , since the velocity difference of the neutrino mass eigenstates is o(@xmath164 ) , then , to the same order in the oscillation phase , the replacement @xmath345 can be made in eqn(5.7 ) to yield the standard oscillation phase of eqn(1.1 ) . * the time evolution of the neutrino mass eigenstates in eqn(5.3 )
according to the schrdinger equation yields a non - lorentz - invariant phase @xmath346 , to be compared with the lorentz - invariant phase @xmath347 given in eqn(2.14 ) above .
although the two expressions agree in the non - relativistic limit @xmath348 it is clearly inappropriate to use this limit for the description of neutrino oscillation experiments .
it may be noted that the lorentz - invariant phase is robust relative to different kinematical approximations .
the same result is obtained to order @xmath349 for the phase of spatial oscillations independent of whether the neutrinos are assumed to have equal momenta or energies .
this is not true in the non - relativistic limit .
assuming equal momenta gives the standard result of eqn(1.1 ) , whereas the equal energy hypothesis results in a vanishing oscillation phase .
a contrast may be noted here with the standard treatment of neutral kaon oscillations , which follows closely the derivation in eqns(5.2 ) to ( 5.7 ) above , except that the particle phases are assumed to evolve with time according , to the lorentz invariant expression , @xmath350 $ ] , where @xmath57 is the particle mass and @xmath61 is its proper time , in agreement with eqn(2.11 ) . * as pointed out in ref .
@xcite , the different neutrino mass eigenstates do not have equal momenta as assumed in eqns(5.2 ) and ( 5.6 ) . the approximation of assuming equal momenta might be justified if the fractional change in the momentum of the neutrino due to a non - vanishing mass were much less than that of the energy . however , in the case of pion decay as is readily shown from eqns(2.4 ) and ( 2.6 ) above , the ratio of the fractional shift in momentum to that in energy is actually @xmath351 ; so , in fact , the opposite is the case . *
the derivation of eqn(5.7 ) is carried out in the abstract hilbert space of the neutrino mass eigenstates or ` lepton flavour eigenstates ' without any reference to the production or detection processes necessary for the complete description of an experiment in which ` neutrino oscillations ' may be observed . in this calculation
the ` mass ' and ` flavour ' bases are treated as physically equivalent .
however in standard model amplitudes only states of the mass basis appear .
also it has been pointed out that ` flavour momentum eigenstates ' can not be defined in a theoretically consistent manner @xcite .
their existence is , in any case , excluded by experiment for the case of pion decay @xcite .
* what are the physical meanings of @xmath336 , @xmath335 in eqn(5.3 ) ?
in this equation it is assumed that the neutrino mass eigenstates are both produced , and both detected , at the same times .
thus both have the same time - of - flight @xmath336 .
the momentum @xmath335 can not be the same for both eigenstates , as assumed in eqn(5.6 ) , if both energy and momentum are conserved in the decay process .
for any given value of the laboratory time @xmath336 the different neutrino mass eigenstates must be at different space - time positions because they have different velocities , if it is assumed that both mass eigenstates are produced at the same time .
it then follows that the different mass eigenstates can not be probed , at the same space - time point , by a neutrino interaction , whereas the latter must occur at a definite space - time point in every detection event .
in fact , there is an inconsistent treatment of the velocity of the neutrinos .
equal production times imply equal space - time velocities , whereas it is assumed thast ` kinematical velocities ' defined as @xmath352 are different for the different mass eigenstates .
* the historical development of the calculation of the neutrino oscillation phase is of some interest . the first published prediction @xcite actually obtained a phase a factor two larger than eqn(1.1 ) _ i.e. _ in agreement with the contribution from neutrino propagation found in the present paper .
this prediction was later used , for example , in ref.@xcite .
the derivation sketched above , leading to the standard result of eqn(1.1 ) was later given in ref.@xcite .
a subsequent paper @xcite by the authors of ref.@xcite , published shortly afterwards , cited both ref.@xcite and ref.@xcite , but used now the prediction of the latter paper .
no comment was made on the factor of two difference in the two calculations . in a later review article , @xcite , by the authors of
ref.@xcite a calculation similiar to that of ref.@xcite was presented in detail .
subsequently , all neutrino oscillation experiments have been analysed on the assumption of the standard oscillation phase of eqn(1.1 ) .
it may be thought that the kinematical and geometrical inconsistencies mentioned in points ( ii ) and ( iv ) above result from a too classical approach to the problem .
after all , what does it mean , in quantum mechanics , to talk about the ` position ' and ` velocity ' of a particle , in view of the heisenberg uncertainty relations @xcite ?
this point will become clear later in the present discussion , but first , following the original suggestion of ref .
@xcite , and , as done in almost all subsequent work on the quantum mechanics of neutrino oscillations , the ` wave packet ' description of the neutrino mass eigenstates will be considered . in this approach , both the ` source ' and also possibly the ` detector ' in the neutrino oscillation experiment are described by coherent spatial wave packets . here
the ` source ' wavepacket treatment in the covariant approach of ref .
@xcite will be briefly sketched . after discussing the results obtained , and comparing them with those of the present paper , the general consistency of the wave packet approach with the fundamental quantum mechanical formula ( 1.2 ) will be examined .
a further discussion of wave packets as applied to neutrino oscillations can be found in ref .
@xcite . the basic idea of the wave packet approach of ref .
@xcite is to replace the neutrino propagator ( 2.11 ) in the path amplitude by a four - dimensional convolution of the propagator with a ` source wave packet ' which , presumably , describes the space - time position of the decaying pion . for mathematical convenience
, this wave packet is taken to have a gaussian form with spatial and temporal widths @xmath353 and @xmath354 respectively .
thus , the neutrino propagator @xmath95 is replaced by @xmath355 where : @xmath356 where @xmath58 and @xmath59 are 4-vectors that specify the neutrino production and detection positions , respectively , and : @xmath357\ ] ] the integral in ( 5.8 ) was performed by the stationary phase method , yielding the result ( up to multiplicative and particle flux factors ) , and here assuming , for simplicity , one dimensional spatial geometry : @xmath358\ ] ] where @xmath359 and @xmath360 . for the case of ` @xmath361 oscillations ' , following @xmath32 decay at rest , the probability to observe flavour @xmath244 at time @xmath336 and distance @xmath204 is given by : @xmath362 performing the integral over @xmath134 and making the ultra - relativistic approximation @xmath363 yields finally , with @xmath364 : it can be seen that the oscillation phase in eqn(5.12 ) is the same as standard one of eqn(1.1 ) .
this is a consequence of the ` equal production time ' hypothesis implicit in eqn(5.10 ) , where @xmath134 does not depend on the mass eigenstate label @xmath366 .
the exponential damping factors in eqn(5.12 ) are the same as those originally found in ref .
@xcite for spatial wave packets ( @xmath367 ) .
considering now only spatial wave packets and using the property @xmath368 derived from the fourier transform of a gaussian , the two terms in the exponential damping factor may be written as : @xmath369\ ] ] and @xmath370\ ] ] the spatial damping factor @xmath371 is usually interpreted in terms of a ` coherence length ' @xcite . if @xmath372 then @xmath373 and the neutrino oscillation term is strongly suppressed .
eqn(5.15 ) expresses the condition that oscillations are only observed provided that the wave packets overlap . since @xmath374 the separation of the wave packets is @xmath375 , so that eqn(5.15 ) is equivalent to @xmath376 ( no wave packet overlap ) .
the damping factor @xmath377 is typically interpreted @xcite in terms of the ` heisenberg uncertainty principle ' .
this factor is small , unless the difference in mass of the eigenstates is much less than @xmath378 , so it is argued that only for wide momentum wave packets can neutrino oscillations be observed , whereas in the contrary case , when the mass eigenstates are distinguishable , the interference effect vanishes . in the case of pion decay
the difference in momentum of the two interfering mass eigenstates comes only from the @xmath112 term in eqn(2.4 ) , as @xmath113 , being a property of the common initial state , is the same for both eigenstates .
the neutrino momentum smearing in pion decay is then estimated from eqn(2.4 ) as : @xmath379 for @xmath380 the value of @xmath377 is found to be @xmath381 giving complete suppression of neutrino oscillations .
this prediction is in clear contradiction with the tiny damping corrections found in the path amplitude analysis in of section 2 above . the preceding discussion of the derivation of the standard formula for the oscillation phase ( 1.1 ) in terms of ` flavour eigenstates ' revealed contradictions and inconsistencies if the neutrinos are assumed to follow classical space - time trajectories .
the ` source wave packet ' treatment gives the standard result for the oscillation phase and predicts that the interference term will be more or less damped depending on the widths in space - time and momentum space of the wave packets .
so do wave packets actually play a role in the correct quantum mechanical description of neutrino oscillations , as suggested in ref.@xcite ?
are the packets actually constrained by the heisenberg uncertainty relations ?
how do the properties of the wave packet effect the possibility to observe neutrino oscillations ?
the answers to all these questions are contained in the results of the calculations presented in section 2 above .
they are now reviewed , with special emphasis on the basic assumptions made and the physical interpretation of the equations . referring again to fig.1 , in a ) a single pion comes to rest in the stopping target t. the time of its passage is recorded by the counter c@xmath69 , which thus defines the initial state as a @xmath32 at rest at time @xmath70 .
this pion , being an unstable particle , has a mass @xmath102 that is , in general , different from its most likely value which is the pole mass @xmath96 .
what are shown in fig.1b ) and fig.1c ) are two different classical histories of this _ very same pion_. in b ) it decays into the neutrino mass eigenstate @xmath71 at time @xmath77 , and in c ) it decays into the neutrino mass eigenstate @xmath72 at time @xmath78 . because these are independent classical histories , the physical masses , @xmath382 and @xmath383 , of the muons recoiling against the mass eigenstates @xmath71 and @xmath72 , respectively , are , in general , not equal . taking into account , now , exact energy - momentum conservation in the decay processes ( appropriate because of the covariant formulation used throughout ) the eigenstates @xmath71 and @xmath72 will have momenta which depend on @xmath102 and @xmath382 and @xmath102 and @xmath383 respectively .
these momenta are calculated in eqn(2.4 ) .
the distributions of @xmath102 and @xmath103 are determined by breit - wigner amplitudes ( that are the fourier transforms of the corresponding exponential decay laws ) in terms of the decay widths @xmath97 and @xmath99 respectively . in accordance with eqn(1.2 ) , only the breit - wigner amplitudes corresponding to the physical muon masses are ( coherently ) integrated over at the amplitude level .
the integral over @xmath102 ( a property of the initial state ) is performed ( incoherently ) at the level of the transition probability . because of the long lifetimes of the @xmath49 and @xmath51 , the corresponding momentum wave packets are very narrow , so that all corrections resulting from integration over the resulting momentum spectra are found to give vanishingly small corrections .
indeed , as is evident from the discussion of eqn(5.10 ) above , the width of the momentum wave packet is much smaller than the difference in the momenta of the eigenstates expected for experimentally interesting values of the neutrino mass difference ( say , @xmath384 ) .
contrary to the prediction of eqn(5.10 ) , this does not at all prevent the observation of neutrino oscillations .
this is because the oscillations result from interference between amplitudes corresponding to different propagation times , not different momenta , of the neutrino mass eigenstates .
table 1 shows contributions to the fractional smearing of the neutrino momentum from different sources : * neutrino mass difference @xmath384 ( eqn(2.4 ) ) * coherent effect due to the physical muon mass ( @xmath385 in eqn(2.4 ) ) * incoherent effect due to the physical pion mass ( @xmath386 in eqn(2.4 ) ) * incoherent doppler effect assuming @xmath387 the pion mass effect is of the same order of magnitude as the neutrino mass shift .
the muon mass effect is two orders of magnitude smaller , while the doppler effect at room temperature gives a shift eleven orders of magnitude larger than a ( 1ev)@xmath199 neutrino mass difference squared .
according to the usual interpretation of the heisenberg uncertainty principle , the neutrinos from pion decay , which , as has been shown above , correspond to very narrow momentum wave packets , would be expected to have a very large spatial uncertainty . indeed , interpreting the width of the coherent momentum wave packet generated by the spread in @xmath388 according to the the momentum - space uncertainty relation @xmath389 gives @xmath390 km .
does this accurately represent the knowledge of the position of a decay neutrino obtainable in the experiment shown in fig.1 ? without any experimental difficulty , the decay time of the pion can be measured with a precision of @xmath391sec , by detecting the decay muon .
thus , at any later time , the distance of the neutrino from the decay point is known with a precision of @xmath392 cm .
this is a factor 4@xmath393 more precise than the ` uncertainty ' given by the heisenberg relation .
it is clear that the experimental knowledge obtainable on the position of the neutrino is essentially classical , in agreement with the theoretical description in terms of classical particle trajectories in space - time . in this case
the momentum - position uncertainty relation evidently does not reflect the possible experimental knowledge of the position and momentum of the neutrino .
this is because it does not take into account the prior knowledge that the mass of the neutrino is much less than that of the pion or muon , so that its velocity is , with negligible uncertainty , c. in spite of this , a heisenberg uncertainty relation is indeed respected in the pion decay process .
the breit - wigner amplitude that determines the coherent spread of neutrino momentum is just the fourier transform of the exponential decay law of the muon .
the width parameter of the breit - wigner amplitude and the muon mean lifetime do indeed respect the _ energy - time _
uncertainty relation @xmath394 .
it is then clear , from this careful analysis of neutrino oscillations following pion decay at rest , that , in contradiction to what has been almost universally assumed until now , _
the neutrinos are not described by a coherent spatial wave packet_. there is a coherent momentum wave packet , but it is only a kinematical consequence of a breit wigner amplitude . if for mathematical convenience
, the momentum wave packet is represented by a gaussian , a conjugate ( and spurious ) gaussian spatial wave packet will be generated by fourier transformation .
indeed , in the majority of wave packet treatments that have appeared in the literature , gaussian momentum and spatial wave packets related by a fourier transform with widths satisfying the ` uncertainty relation ' @xmath395 have been introduced .
in fact , it is evident , by inspection , that the space - time wavepacket of eqn(5.10 ) does not correctly reflect the known space - time structure of the sequence of events corresponding to pion decay .
since @xmath364 , the fixed source - detector distance , the integral of @xmath134 over the range from @xmath396 to @xmath397 assumed in order to derive eqn(5.12 ) implies that the neutrino velocities @xmath398 vary also in the range : @xmath399 .
this unphysical range of integration results in an average neutrino momentum that is less than the kinematical velocity , @xmath352 , of either mass eigenstate , due to contributions to the integral from negative values of @xmath134 @xcite .
also the production time dependence is known to be exponential , not gaussian .
there are indeed ` heisenberg uncertainties ' in the production times of the neutrinos , due to the finite source lifetime , but once the neutrinos are produced their motion in space - time is well approximated , in each alternative history , by that of a free classical particle .
the above discussion shows clearly that the _ ad hoc _ gaussian wave packet introduced in eqns(5.8 ) and ( 5.9 ) does not correspond to the actual sequence of the space - time events that constitute realistic neutrino oscillation experiments .
the limitation on the detection distance @xmath2 , for observation of neutrino oscillations , given by the damping factor @xmath400 of eqn(2.30 ) is easily understood in terms of the classical particle trajectories shown in fig.1 . for a given velocity difference , the time difference @xmath401 becomes very large when both neutrinos , in the alternative classical histories , are required to arrive simultaneously at a far distant detector . because of the finite pion lifetime , however , the amplitude for pion decay at time @xmath78 is smaller than that at @xmath77 by the factor @xmath402 $ ] . integrating over all decay times results in the exponential damping factor @xmath400 of eqn(2.30 ) .
it is clear that , contrary to the damping factor @xmath371 of eqn(5.14 ) , the physical origin of the @xmath2 dependent damping factor is quite unrelated to ` wavepacket overlap ' . for a given value of @xmath401 the coherent neutrino momentum spread originating in the breit wigner amplitude for @xmath388
produces a corresponding velocity smearing that reduces the number of possible classical trajectories arriving at the detection event .
this effect is taken into account in the integral shown in eqn(2.22 ) .
the effect is shown , in section 2 above , to be much smaller than the ( already tiny ) pion lifetime damping described above , and it is neglected in eqn(2.26 ) . as described by eqn(2.25 ) , the ( incoherent ) integration over the breit wigner function containing @xmath102 gives an additional damping correction to the interference term that is also very tiny compared to that due to the pion lifetime . a final remark is now made on the physical interpretation of the damping factors ( 5.13 ) and ( 5.14 ) that have often been derived and discussed in the literature .
@xmath371 is replaced , in the path amplitude calculations , by @xmath400 and @xmath377 by the factor resulting from the coherent integration over the physical muon mass @xmath388 .
the reason for the huge suppression factor predicted by eqn(5.13 ) , and the tiny one found in the path amplitude calculation , is that , in deriving ( 5.13 ) and ( 5.14 ) , it is assumed that the neutrino eigenstates are both produced and detected at equal times .
this will only be possible if both the hypothetical ` wave packet overlap ' is appreciable ( eqn(5.14 ) ) and also the momentum smearing is sufficiently large that the time - of - flight differences due to the different neutrino masses are washed out ( eqn(5.13 ) . in the path amplitude calculation interference and hence oscillations are made possible by different _ decay times _ of the source pion , and the damping factors analagous to @xmath371 and @xmath377 turn out to give vanishingly small corrections to the oscillation term .
it may be remarked that the physical interpretation of ` neutrino oscillations ' provided by the path amplitude description is different from the conventional one in terms of ` flavour eigenstates ' . in the latter the amplitudes of different flavours in the neutrino
are supposed to vary harmonically as a function of time .
this may be done , for example , by changing the basis states , in eqn(5.3 ) above , from the mass to the flavour basis by using the inverses of eqns(5.2 ) and ( 5.4 ) . in the amplitudes for the different physical processes in the path amplitude treatment
there is , instead , no variation of the ` lepton flavour ' in the propagating neutrinos . if the mass eigenstates are represented as superpositions of `
lepton flavour eigenstates ' there is evidently no temporal variation of the lepton flavour composition within each interfering amplitude .
indeed , the neutrinos in general occupy different spatial positions at any given time , making it impossible to project out a ` flavour eigenstate ' at any time by using the inverses of eqns(5.2 ) and ( 5.4 ) . only _ in the detection process itself _ where the different neutrino histories occupy the same space time point are the ` lepton flavour eigenstates ' projected out , and the interference effect occurs that is described as ` neutrino oscillations ' . in the case of the observation of decay products of the recoil muons
no such projection on to a ` lepton flavour eigenstate ' takes place , but exactly similar interference effects are predicted to occur . as previously emphasised @xcite ,
the ` flavour oscillations ' of neutrinos , neutral kaons and b - mesons are just special examples of the universal phenomenon of quantum mechanical superposition described by eqn(1.2 ) , that also describes all the interference effects of physical optics . by far
the most widespread difference from the path amplitude treatment of the present paper is the non - respect of the basic quantum mechanical formula , ( 1.2 ) , by the introduction of wave packets to describe ` source ' and/or ` detector ' particles[7,8,10,11,30 - 46 ] . since in any practical neutrino oscillation experiment a single initial or final quantum state , as specified by eqn(1.2 ) , is not defined , but rather sets of initial and final states @xmath403 and @xmath404 determined by experimental conditions , eqn(1.2 ) may be generalised to : @xmath405 to be contrasted with the formula used in the references cited above : @xmath406 here , @xmath407 and @xmath408 are ` source ' and ` detector ' wave packets respectively . in eqn(5.17 )
the initial states @xmath409 and final states @xmath410 of eqn(5.16 ) , that correspond to different spatial positions , and also , possibly , different kinematic properties , of the source particle or detection event , are convoluted with _ ad hoc _ spatial and/or temporal ` wave packets ' , that are , in the opinion of the present writer , for the reasons given above , devoid of any physical significance . a possible reason for the widespread use of eqn(5.17 ) instead of ( 5.16 )
may be understood following a remark of the author of ref .
@xcite concerning a ` paradox ' of the complete quantum field theory calculation that takes into account , by a single invariant amplitude , production , propagation and detection of the neutrinos .
it was noticed that , if the amplitude for the complete chain of processes is considered to correspond to one big feynman diagram , then integration over the space - time coordinates of the initial and final states particles will reduce the exponential factors , containing the essential information on the interference phase , to energy - momentum conserving delta functions , and so no oscillations will be possible .
a related remark was made by the authors of ref .
@xcite who stated that , as they were assuming exact energy - momentum conservation , the integration over the space - time coordinates could be omitted .
they still , however , ( quite inconsistently , in view of the previous remark ) retained the exponential factors containing the interference phase information .
these considerations indicate a general confusion between momentum space feynman diagram calculations , where it is indeed legitimate to integrate , at the amplitude level , over the unobserved space - time positions of the initial and final state particles , and the case of neutrino oscillations , where the amplitude is defined with initial and final states corresponding to space - time positions . in the latter case ,
it is the unobserved momenta of the propagating particles that should be integrated over , as is done in eqns(2.1 ) and ( 2.35 ) above , and not the space - time positions of the ` source ' or ` detector ' particles , as in eqn(5.17 ) .
it is clear that exact energy - momentum conservation plays a crucial role in the path amplitude calculation .
this is valid only in a fully covariant theory .
still , several authors , in spite of the ultra - relativistic nature of neutrinos , used a non - relativistic theory to describe the production , propagation and detection of neutrinos @xcite . as is well known , in such ` old fashioned perturbation theory ' @xcite energy is not conserved at the level of propagators and so no precise analysis of the kinematics and the space - time configurations of the production and detection events , essential in the covariant path amplitude analysis , is possible .
even when , in some cases , the complete production , propagation and detection process of the neutrinos were described @xcite , equal neutrino energies @xcite , equal neutrino momenta @xcite or either @xcite were assumed , in contradiction with energy - momentum conservation and a consistent space - time description of the production and detection events . as follows
directly from eqn(5.6 ) ( or the similar formula , for the neutrino momentum , obtained by assuming equal neutrino energies ) , the standard formula ( 1.1 ) for the oscillation phase was obtained in all the above cited references .
as shown in reference @xcite this is a consequence of the universal equal production time ( or equal velocity ) hypothesis .
in fact , the assumptions of equal momenta , equal energies or exact energy - momentum conservation give only negligible , o(@xmath411 ) , changes in the oscillation phase @xcite .
an interesting discussion of the interplay between different kinematical assumptions ( not respecting energy - momentum conservation ) and the space - time description of the production and detection events was provided in ref .
this treatment was based on the lorentz - invariant propagator phase of eqn(2.11 ) . by assuming either equal momentum or equal energy for the propagating neutrinos , but allowing _ different times of propagation _ for the two mass eigenstates , values of @xmath412 agreeing with eqns(2.14 ) and ( 2.24 ) above
were found , _
i.e. _ differing by a factor two from the standard formula .
alteratively , assuming equal velocities , ( and hence equal propagation times ) the standard result ( 1.1 ) was obtained . in this latter case , however , the masses , momenta and energies of the neutrinos must be related , up to corrections of o(@xmath111 ) , according to : @xmath413 since the ratio of the neutrino masses may take , in general , any value , so must then the ratio of their momenta . for the case of neutrino production from pion decay at rest with @xmath414
the relation ( 5.18 ) is clearly incompatible with eqn(2.4 ) which gives : @xmath415 on the other hand , eqn(5.19 ) is clearly compatible ( up to corrections of o(@xmath164 ) ) with the ` equal momentum ' hypothesis .
there is similar compatiblity with the ` equal energy ' hypothesis .
even so , the authors of ref .
@xcite recommended the use of the equal velocity hypothesis . with the hindsight provided by the path amplitude analysis , in which the two neutrino mass eigenstates do indeed have different propagation times
, it can be seen that the kinematically consistent ` equal momentum ' and ` equal energy ' choices are good approximations and the neutrino oscillation phase , resulting from the propagation of the neutrinos alone , is indeed a factor two larger than the prediction of the standard formula . a more detailled discussion of the effects of the ` equal momentum ' ` equal energy ' and ` equal velocity ' hypotheses may be found in reference @xcite the author of ref .
@xcite included the propagator of the decaying pion in the complete production - propagation - detection amplitude ; compare eqn(8 ) of ref .
@xcite with eqn(2.1 ) above . however , no detailed space - time analysis of production and detection events was performed .
square spatial wave packets for the ` source ' and ` detector ' were convoluted at amplitude level as in eqn(5.17 ) . as the neutrinos were assumed to have a common production time , no contribution to the oscillation phase from the source particle was possible , and so the standard result for the oscillation phase was obtained . although , following ref .
@xcite , most recent studies of the quantum mechanics of neutrino oscillations have considered the complete production - propagation - detection process , some authors still use , in spite of the criticisms of ref .
@xcite , the ` flavour eigenstate ' description @xcite . in the last two of these references
a ` quantum field theory ' approach ia adopted , leading to an ` exact ' oscillation formula @xcite that does not make use of the usual ultra - relativistic approximation .
this work included neither exact kinematics , nor an analysis of the space - time structure of production and detection events . in ref .
@xcite the equal energy hypothesis was used and in ref .
@xcite the equal momentum hypothesis . in all three cases
the standard result was found for the oscillation phase in the ultra - relativistic limit , as a consequence of the assumption of equal production times .
correlated production and detection of neutrinos and muons produced in pion decay were considered in ref .
the introduction to this paper contains a valuable discussion of the universality of the ` particle oscillation ' phenomenon .
it is pointed out that this is a consequence of the general principle of amplitude superposition in quantum mechanics , and so is not a special property of the @xmath416 , @xmath417 and neutrino systems which are usually discussed in this context .
this paper used a covariant formalism that employed the ` energy representation ' of the space - time propagator . in the introduction , the important difference between eqns(5.16 ) and ( 5.17 ) was also touched upon : even so , in the amplitude for the correlated detection of the muon and neutrino ( eqn(2.10 ) of ref .
@xcite ) not only are the space - time positions of the production points of the neutrino and muon integrated over , but they are assumed to be at _ different _ space - time points .
the propagator of the decaying pion is not included in the amplitude , and although exact energy - momentum conservation is imposed , no space - time analysis of the production and decay points is performed .
correlated spatial oscillations of neutrinos and muons are predicted , though with interference phases different from the results of both the present paper and the standard formula .
pion and muon lifetime effects were mentioned in ref .
@xcite , but neither the role of the pion lifetime in enabling different propagation times for the neutrinos nor the momentum smearing , induced by the fourier - transform - related breit wigner amplitudes , were discussed . the claim of ref .
@xcite ) that correlated neutrino - muon oscillations should be observable in pion decay was questioned in ref .
the authors of the latter paper attempted to draw conclusions on the possibility , or otherwise , of particle oscillations by using ` plane waves ' , _ i.e. _ energy - momentum eigenfunctions .
as is well known , such wavefunctions are not square integrable , and so can yield no spatial information . the probability to find a particle described by such a wave function in any finite spatial volume is zero .
due to the omission of the ( infinite ) normalisation constants of the wavefunctions many of the equations in ref .
@xcite are , as previously pointed out @xcite , dimensionally incorrect .
momentum wavepackets for the decaying pion convoluted at amplitude level as in eqn(5.17 ) were also discussed in ref . @xcite .
although exact energy - momentum conservation constraints were used , it was assumed , as in ref .
@xcite , that the muons and the different neutrino mass eigenstates are both produced and detected at common points ( eqn(35 ) of ref .
the latter assumption implies equal velocities , yielding the standard neutrino oscillation phase as well as the inconsistent kinematical relation ( 5.18 ) .
the authors of ref .
@xcite concluded that : * correlated @xmath418 oscillations of the type discussed in ref .
@xcite could be observed , though with different oscillation phases .
* oscillations would not be observed if only the muon is detected * neutrino oscillations can be observed even if the muon is not detected .
conclusion ( b ) is a correct consequence of the ( incorrect ) assumption that the muons recoiling against the different neutrino mass eigenstates have the same velocity . as both muons have the same mass they will have equal proper time increments .
so according to eqn(2.12 ) the phase increments will also be equal and the interference term will vanish .
the conclusion ( c ) is in agreement with the prediction of eqn(3.22 ) of ref .
the path amplitude calculation of the present paper shows that conclusion ( b ) is no longer valid when the different possible times of propagation of the recoiling muons are taken into account .
observation of neutrino oscillations following pion decay , using a covariant formalism ( schwinger s parametric integral representation of the space - time propagator ) was considered in ref .
exact energy - momentum conservation was imposed , and integration over the pion spatial position at amplitude level , as in eqn(5.17 ) , was done .
the propagator of the pion source was included in the amplitudes , but as the different mass eigenstates were produced and detected at the same space - time points , equal propagation velocities were implicitly assumed , so that just as in refs@xcite , where the same assumption was made , the standard neutrino oscillation phase was obtained . in the conclusion of this paper the almost classical nature of the space - time trajectories followed by the neutrinos
was stressed , although this was not taken to its logical conclusion in the previous discussion , _
e.g. _ the kinematical inconsistency of the equal velocity hypothesis that requires the evidently impossible condition ( 5.18 ) to be satisfied . in a recent paper @xcite , the standard neutrino oscillation formula with oscillation phase given by eqn(1.1 )
was compared with a neutrino decoherence model . in order to take into account incertainties in the position of the source and the neutrino energy , an average was made over the quantity @xmath419 , assuming that it is distributed according to a gaussian with mean value @xmath249 and width @xmath420 .
the average was performed in an incoherent manner .
thus the calculation is closely analagous to those for the effects of target or detector length or of thermal motion of the neutrino source , presented in the appendix a of the present paper . perhaps uniquely then , in the published literature , in ref .
@xcite the effects of source position and motion are taken into account correctly , according to eqn(5.16 ) instead of eqn(5.17 ) .
however , the source of the neutrino energy uncertainty is not specified . in
as far as it is generated from source motion the calculation is , in principle , correct .
there is however also the ( typically much smaller , see table 1 above , for the case of pion decay at rest ) coherent contribution originating from the variation in the physical masses of the unstable particles produced in association with the neutrino , as discussed in detail above .
it was concluded in ref .
@xcite that the gaussian averaging procedure used gave equivalent results to the decoherence model for a suitable choice of parameters .
it is clearly of great interest to apply the calculational method developed in the present paper to the case of neutral kaon and b - meson oscillations . indeed
the use of the invariant path amplitude formalism has previously been recommended @xcite for experiments involving correlated pairs of neutral kaons . here , just a few remarks will be made on the main differences to be expected from the case of neutrino or muon oscillations .
a further discussion can be found in ref .
@xcite and a more detailed treatment will be presented elsewhere@xcite . in the case of neutrino and muon oscillations ,
the interference effect is possible as the different neutrino eigenstates can be produced at different times .
this is because the decay lifetimes of all interesting sources ( pions , muons , @xmath0-decaying nuclei ) are much longer than the time difference beween the paths corresponding to the interfering amplitudes . to see
if a similar situation holds in the case of @xmath421 oscillations , three specific examples will be considered with widely differing momenta of the neutral kaons : these correspond to neutral kaon centre - of - mass momenta of 108 mev , 750 mev and 5 gev respectively . in each case
the time difference ( @xmath426 ) of production of @xmath427 and @xmath428 mesons , in order that they arrive at the same time at a point distant @xmath429 ( where @xmath430 is the usual relativistic parameter ) from the source in the centre - of - mass frame is calculated .
exact relativistic kinematics is assumed and only leading terms in the mass difference @xmath431 are retained . taking the value of @xmath432 and the various particle masses from ref .
@xcite the following results are found for @xmath426 in the three cases : ( i ) 2.93@xmath433sec , ( ii ) 8.3@xmath434sec and ( iii ) 6.4@xmath435sec . for comparison , for neutrino oscillations following pion decay at rest , with @xmath436 and
@xmath437 m , eqn(2.29 ) gives @xmath438sec . the result ( i )
may be compared with the mean life of the @xmath439 meson of 1.5@xmath440sec @xcite .
thus the @xmath439 lifetime is a factor of about 27 larger than @xmath426 indicating that @xmath421 interference should be possible by a similar mechanism to neutrino oscillations following pion decays , _
i.e. _ without invoking velocity smearing of the neutral kaon mass eigenstates . in cases ( i ) and
( ii ) the interference effects observed will depend on the ` characteristic time ' of the non resonant ( and hence incoherent ) strong interaction process , a quantity that has , hitherto , not been susceptible to experimental investigation .
if this time is much less than , or comparable to , @xmath426 , essentially equal velocities ( and therefore appreciable velocity smearing ) of the eigenstates will be necessary for interference to occur .
since @xmath432 and @xmath441 are comparable in size , velocity smearing effects are expected to be , in any case , much larger than for neutrino oscillations following pion decay .
these effects may be roughly estimated by using the gaussian approximation ( 2.22 ) of the present paper .
the main contribution to the velocity smearing is due to the variation of the physical mass of the @xmath427 rather than those of the @xmath428 or @xmath442 . for the @xmath443 oscillation case , analagous to ( i ) above , @xmath444 , where @xmath445 mev
, the value of @xmath446 is found to be @xmath447sec , to be compared with @xmath448sec @xcite , which is a factor 3 smaller .
thus , velocity smearing effects are expected to play an important role in @xmath443 oscillations .
this is possible , since the neutral b - meson decay width ( @xmath449 mev ) , and mass difference ( @xmath450 mev ) , have similar sizes . in closing ,
it is interesting to mention two types of atomic physics experiments where interference effects similar to the conjectured ( and perhaps observed @xcite ) neutrino oscillations have aleady been clearly seen .
the first is quantum beat spectroscopy @xcite .
this type of experiment , which has previously been discussed in connection with neutrino oscillations @xcite , corresponds closely to the gedanken experiment used by heisenberg @xcite to exemplify the the fundamental law of quantum mechanics , eqn(1.2 ) .
the atoms of an atomic beam are excited by passage through a thin foil or a laser beam .
the quantum phase of an atom with excitation energy @xmath274 evolves with time according to : @xmath451 ( see the discussion after eqn(3.3 ) above ) .
if decay photons from two nearby states with excitation energies @xmath452 and @xmath453 are detected after a time interval @xmath134 ( for example by placing a photon detector beside the beam at a variable distance @xmath454 from the excitation foil ) a cosine interference term with phase : @xmath455 where @xmath456 is the average velocity of the atoms in the beam , is observed @xcite .
an atom in the beam , before excitation , corresponds to the neutrino source pion .
the excitation process corresponds to the decay of the pion .
the propagation of the two different excited states , _ alternative _ histories of the initial atom , correspond to the _ alternative _ propagation of the two neutrino mass eigenstates .
finally the dexcitation of the atoms and the detection of a _ single _ photon corresponds to the neutrino detection process .
the particular importance of this experiment for the path amplitude calculations presented in the present paper , is that it demonstrates , experimentally , the important contribution to the interference phase of the space - time propagators of excited atoms , in direct analogy to the similar contributions of unstable pions , muons and nuclei discussed above . an even closer analogy to neutrino osillations following pion decay is provided by the recently observed process of photodetachement of an electron by laser excitation : the ` photodetachment microscope ' @xcite .
a laser photon ejects the electron from , for example , an @xmath457 ion in a beam .
the photodetached electron is emitted in an s - wave ( isotropically ) and with a fixed initial energy .
it then moves in a constant , vertical , electric field that is perpendicular to the direction of the ion beam and almost parallel to the laser beam .
an upward moving electron that is decelerated by the field eventually undergoes ` reflection ' before being accelerated towards a planar position - sensitive electron detector situated below the beam and perpendicular to the electric field direction ( see fig.1 of ref .
@xcite ) . in these circumstances
, it can be shown @xcite that , just two classical electron trajectories link the production point to any point in the kinematically allowed region of the detection plane .
typical parameters for @xmath457 are @xcite : initial electron kinetic energy , 102 @xmath51ev : detector distance , 51.4 cm ; average time - of - flight , 117 ns ; difference in emission times for the electrons to arrive in spatial - temporal coincidence at the detector plane , 160 ps .
an interference pattern is generated by the phase difference between the amplitudes corresponding to the two allowed trajectories .
the phase difference , derived by performing the feynman path integral of the classical action along the classical trajectories @xcite , gives a very good description of the observed interference pattern .
the extremely close analogy between this experiment and the neutrino oscillation experiments described in sections 2 and 3 above is evident .
notice that the neutrinos , like the electrons in the photodetachement experiment , must be emitted at different times , in the alternative paths , for interference to be possible .
this is the crucial point that was not understood in all previous treatments of the quantum mechanics of neutrino oscillations .
actually , ref .
@xcite contains , in section iv , a path amplitude calculation for electrons in free space that is geometrically identical to the discussion of pion decays in flight presented in section 4 above ( compare fig.3 of the present paper with fig.3 of ref .
@xcite ) the conclusion of ref .
@xcite is that , in this case , no interference effects are possible for electrons that are mononergetic in the source rest frame . as is shown in section 4 above ,
if these electrons are replaced either by neutrinos of different masses from pion decay , or muons recoiling against such neutrinos , observable interference effects are indeed to be expected .
random thermal motion of the decaying pion in the target has two distinct physical effects on the phase of neutrino oscillations , @xmath458 ( where the first and second terms in eqn(a1 ) give the contributions of the neutrino and pion paths respectively ) : * the observed neutrino momentum , @xmath459 , is no longer equal to @xmath460 , due to the boost from the pion rest frame to the laboratory system .
( doppler effect or lorentz boost ) * the time increment of the pion path @xmath461 ( see eqn(2.16 ) ) no longer corresponds to the pion proper time .
( relativistic time dilatation ) taking into account ( 1 ) and ( 2 ) gives , for the neutrino oscillation phase : @xmath462 where : @xmath463 here , @xmath464 is the angle between the neutrino momentum vector and the pion flight direction in the pion rest frame .
developing @xmath465 and @xmath466 in terms of the small quantity @xmath467 , eqn(a2 ) may be written as : @xmath468 \cos\theta_{\nu}^{\ast } + \left(\frac{p_{\pi}}{m_{\pi}}\right)^2 \frac{\delta m^2 l}{2 p_0}\left[1-\frac{3 m_{\pi}}{2 p_0}\right ] ~~~~~~~(a3)\ ] ] performing now the average of the interference term over the isotropic distribution in @xmath469 : @xmath470 d \cos \theta_{\nu}^{\ast } \nonumber \\ & = & \frac{1}{2 } { \it re } \exp \left\{i \phi_{12}^{\nu,\pi}(0)+ \left(\frac{p_{\pi}}{m_{\pi}}\right)^2\left(i\frac{\delta m^2 l}{2 p_0 } \left[1-\frac{3 m_{\pi}}{2 p_0}\right ] \right .
. \nonumber \\ & & \left .
~~~~~~~~~~-\frac{1}{6}\left(\frac{\delta m^2 l}{p_0 } \left[1-\frac{m_{\pi}}{p_0}\right ] \right)^2\right ) \right\}~~~~~~~~~~~~~~~~~~(a4 ) \nonumber\end{aligned}\ ] ] in deriving eqn(a4 ) the following approximate formula is used : @xmath471 = \frac{\sin \alpha}{\alpha } \simeq 1-\frac{\alpha^2}{6}~~~~~~~~~~~~~~~~~~~~~~~~~(a5)\ ] ] where @xmath472 \ll 1\ ] ] the average over the maxwell - boltzmann distribution ( 2.31 ) is readily performed by ` completing the square ' in the exponential , with the result : @xmath473\right)^2 \right .
\nonumber \\ & & + \left .
i\left[\phi_{12}^{\nu,\pi}(0)+ \frac{3}{4}\left(\frac{\overline{p}_{\pi}}{m_{\pi}}\right)^2\left(\frac{\delta m^2 l}{p_0 } \left[\frac{3 m_{\pi}}{2 p_0}-1\right]\right)\right]\right\ } \nonumber \\ & \equiv & f^{\nu}(dop)\cos [ \phi_{12}^{\nu,\pi}(0)+\phi^{\nu}(dop ) ] \nonumber~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(a6)\end{aligned}\ ] ] leading to eqns(2.32 ) and ( 2.33 ) for the doppler damping factor @xmath474 and phase shift @xmath475 , respectively .
the correction for the effect of thermal motion in the case of muon oscillations @xmath475 , respectively . is performed in a similar way .
the oscillation phase : @xmath476 is modified by the lorentz boost of the muon momentum and energy , and the relativistic time dilatation of the phase increment of the pion path , to : @xmath477 where @xmath478 and @xmath479 is given by eqn(2.37 ) . developing , as above , in terms of @xmath467 , gives : @xmath480 \cos\theta_{\mu}^{\ast } + \left(\frac{p_{\pi}}{m_{\pi}}\right)^2 \frac{m_{\mu}^2 \delta m^2 l}{p_0 ^ 3 } \left[\frac{e_0^{\mu}}{2 m_{\pi}}-1\right ] ( a9)\ ] ] performing the averages over @xmath481 and @xmath482 then leads to eqns(2.53 ) and ( 2.54 ) for the damping factor @xmath483 and phase shift @xmath484 , respectively .
the effect of the finite longitudinal dimensions of the target or detector is calculated by an appropriate weighting of the interference term according to the value of the distance @xmath485 between the decay and detection points ( see fig.1 ) .
writing the interference phase as @xmath486 , and assuming a uniform distribution of decay points within the target of thickness @xmath192 : @xmath487 substituting the value of @xmath0 appropriate to neutrino oscillations yields eqn(2.34 ) . since the value of @xmath0 is the same for neutrino and muon oscillations , the same formula is also valid in the latter case .
the same correction factor , with the replacement @xmath195 describes the effect of a finite detection region of length @xmath488 : @xmath489 the first step in the derivation of eqn(4.17 ) relating @xmath490 to @xmath164 is to calculate the angle @xmath491 , in the centre - of - mass ( cm ) system of the decaying pion , corresponding to @xmath295 in the laboratory ( lab ) system ( see fig.3 ) .
it is assumed , throughout , that the pion and muon are ultra - relativistic in the latter system , so that : @xmath492 .
the lorentz transformation relating the cm and lab systems gives the relation : @xmath493 the starred quantities refer to the pion cm system .
making the substitutions : @xmath494 , @xmath495 , eqns(b1 ) may be solved to obtain , up to first order in @xmath295 , @xmath491 and @xmath164 : @xmath496 where , ( c.f .
eqn(2.37 ) : @xmath497 using eqn(2.36 ) @xmath498 may be expressed in terms of the neutrino mass difference : @xmath499 eliminating now @xmath498 between ( b2 ) and ( b4 ) gives a relation between @xmath295 , @xmath491 and @xmath164 : @xmath500 in the lab system , and in the ur limit , the difference of the velocities of the muons recoiling against the two neutrino mass eigenstates is : @xmath501~~~~~~~~~~~~~~~~~(b6)\ ] ] where @xmath48 is the muon energy in the lab system for vanishing neutrino masses . making the lorentz transformation of the muon energy from the pion cm to the lab frames , and using eqns(2.4 ) and ( 2.36 ) to retain only terms linear in @xmath164 and @xmath491 , enables eqn(b6 ) to be re - written as : @xmath502 ~~~~~~~~~~~~~~~~~~~~~~~(b7)\ ] ] where @xmath46 is the energy of the pion beam . by combining the geometrical constraint equation for the muon velocities , ( 4.16 ) with ( b5 ) and ( b7 ) the angles @xmath295 and
@xmath491 may be eliminated to yield the equation for lab frame velocity difference : @xmath503 where @xmath504 @xmath505 to simplify ( b8 ) , the quantity @xmath506 is now expressed in terms of kinematic quantities in the pion cm system . within the ur approximation used , @xmath507 so that @xmath508 writing eqn(b1 ) to first order in @xmath284 , and neglecting terms of @xmath509 : @xmath510 using eqn(b12 ) , and expressing @xmath48 in terms of pion cm quantities , eqn(b11 ) may be written as : @xmath511 expressing the rhs of ( b13 ) in terms of @xmath46 and @xmath48 , using the relation : @xmath512 gives eqn(4.20 ) of the text . on substituting ( b13 ) into the rhs of ( b10 ) , it can be seen that the factor in the large curly brackets is the same in ( b9 ) and ( b10 ) , and so cancels in the ratio @xmath513 in eqn(b8 ) .
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69 * 810 ( 2001 ) . | feynman s path amplitude formulation of quantum mechanics is used to analyse the production of charged leptons from charged current weak interaction processes . for neutrino induced reactions the interference effects predicted
are usually called ` neutrino oscillations ' .
similar effects in the detection of muons from pion decay are here termed ` muon oscillations ' .
processes considered include pion decay ( at rest and in flight ) , and muon decay and nuclear @xmath0-decay at rest . in all cases studied , a neutrino oscillation phase different from the conventionally used one is found .
a concise critical review is made of previous treatments of the quantum mechanics of neutrino and muon oscillations .
24.5 cm -5pt -5pt -50pt addtoresetequationsection * j.h.field * dpartement de physique nuclaire et corpusculaire universit de genve .
24 , quai ernest - ansermet ch-1211 genve 4 .
pacs 03.65.bz , 14.60.pq , 14.60.lm , 13.20.cz quantum mechanics , neutrino oscillations . | hep-ph0303151 |
a long outstanding challenge in modern astrophysics is the intriguing observational behavior of the galactic rotation curves , and the mass discrepancy in clusters of galaxies .
both these observations suggest the existence of a ( non or weakly interacting ) form of dark matter at galactic and extra - galactic scales . indeed , according to newton s theory of gravitation , at the boundary of the luminous matter
, the rotation curves of test particles gravitating around galaxies or galaxy clusters should show a keplerian decrease of the tangential rotational speed @xmath1 with the distance @xmath2 , so that @xmath3 , where @xmath4 is the dynamical mass within the radius @xmath2 .
however , the observational evidence indicates rather flat rotation curves @xcite .
the tangential rotational velocities @xmath1 increase near the galactic center , as expected , but then intriguingly remain approximately constant at a value of @xmath5 km / s .
therefore , observations provide a general mass profile of the form @xmath6 @xcite
. consequently , the mass around a galaxy increases linearly with the distance from the center , even at large distances , where very little or no luminous matter can be observed .
as mentioned above , the observed behavior of the galactic rotation curves is explained by assuming the existence of some dark ( invisible ) matter , distributed in a spherical halo around the galaxies , and interacting only gravitationally with ordinary matter .
the dark matter is usually described as a pressureless and cold medium .
a recently proposed model considered the possibility that dark matter is a mixture of two non - interacting perfect fluids , with different four - velocities and thermodynamic parameters . the two - fluid model can be described as an effective single anisotropic fluid , with distinct radial and tangential pressures @xcite .
in fact , many possible candidates for non - luminous dark matter have been proposed in the literature ranging from @xmath7 solar mass black holes , running through low mass stars to @xmath8 ev axions , although the most popular being the weakly interacting massive particles ( wimps ) ( for a review of the particle physics aspects of dark matter see @xcite ) .
indeed , the interaction cross section of wimps with normal baryonic matter , although practically negligible , is expected to be non - zero , and therefore there is a possibility of detecting them directly . nevertheless , despite several decades of intense experimental and observational effort , there is presently still no direct evidence of dark matter particles @xcite .
however , it is important to emphasize that the masses in galaxies and clusters of galaxies are deduced from the observed distances and velocities of the system under consideration .
these relationships are based on newton s laws of dynamics , thus the newtonian dynamical masses of galactic systems are not consistent with the observed masses .
indeed , newton s laws have proven extremely reliable in describing local phenomena , that there is an overwhelming tendency to apply them in the intermediate galactic scales .
note that the mass discrepancy is interpreted as evidence for the existence of a _ missing mass _
, i.e. , dark matter in galactic systems .
therefore , one can not _ a priori _ exclude the possibility that einstein s ( and newtonian ) gravity breaks down at the galactic or extra - galactic scales .
indeed , a very promising way to explain the recent observational data @xcite on the recent acceleration of the universe and on dark matter is to assume that at large scales einstein s general relativity , as well as its newtonian limit , breaks down , and a more general action describes the gravitational field .
several theoretical models , based on a modification of newton s law or of general relativity , including modified newtonian orbital dynamics ( mond ) @xcite , scalar fields or long range coherent fields coupled to gravity @xcite , brane world models @xcite , bose - einstein condensates @xcite , modified gravity with geometry - matter coupling @xcite , non - symmetric theories of gravity @xcite , and eddington - inspired born - infeld gravity @xcite have been used to model galactic `` dark matter '' . from a theoretical point of view a very attractive possibility is to modify the form of the einstein - hilbert lagrangian , so that such a modification could naturally explain dark matter and dark energy , without resorting to any exotic forms of matter .
the simplest extension of the einstein - hilbert action consists in modifying the geometric part of the action , through the substitution of the ricci scalar with a generic function @xmath0 .
this change in the action introduces higher order terms in the gravitational field action .
the so - called @xmath0 gravitational theories were first proposed in @xcite , and later used to find a non - singular isotropic de sitter type cosmological solution @xcite .
detailed reviews of @xmath0 theories can be found , for instance , in @xcite .
the most serious difficulty of @xmath0 theories is that , in general , these theories can not pass the standard solar system tests @xcite .
however , there exists some classes of theories that can solve this problem @xcite . using phase space analysis of the specific involved gravitational model ,
it was shown that @xmath0 theories , in general , can explain the evolution of the universe , from a matter dominated early epoch up to the present , late - time self accelerating phase @xcite .
@xmath0 type gravity theories can be generalized by including the function @xmath0 in the bulk action of the brane - world theories @xcite .
it has been shown in @xcite that this type of generalized brane world theories can describe a universe beginning with a matter - dominated era , and ending in an accelerated expanding phase .
the classical tests of this theory were considered in @xcite .
on the other hand , @xmath0 theories have also been studied in the palatini approach , where the metric and the connection are regarded as independent fields @xcite .
in fact , these approaches are certainly equivalent in the context of general relativity , i.e. , in the case of the linear einstein - hilbert action . on the other hand , for a general @xmath0 term in the action
, they seem to provide completely different theories , with very distinct field equations .
the palatini variational approach , for instance , leads to second order differential field equations , while the resulting field equations in the metric approach are fourth order coupled differential equations .
these differences also extend to the observational aspects .
all the palatini @xmath0 models aimed at explaining the cosmic speedup studied so far lead to microscopic matter instabilities , and to unacceptable features in the evolution patterns of cosmological perturbations @xcite .
hence , in order to cure some of the pathologies of the @xmath0 gravity models in both their metric and palatini formulation , a novel approach was recently proposed @xcite , that consists of adding to the einstein - hilbert lagrangian an @xmath0 term constructed within the framework of the palatini formalism . using the respective dynamically equivalent scalar - tensor representation ,
even if the scalar field is very light , the theory can pass the solar system observational constraints .
therefore the long - range scalar field is able to modify the cosmological and galactic dynamics , but leaves the solar system unaffected .
the absence of instabilities in perturbations was also verified , and explicit models , which are consistent with local tests and lead to the late - time cosmic acceleration , were also found . furthermore , the cosmological applications of the hybrid metric - palatini gravitational theory were investigated in @xcite , where specific criteria to obtain the cosmic acceleration were analyzed , and the field equations were formulated as a dynamical system . indeed , several classes of dynamical cosmological solutions , depending on the functional form of the effective scalar field potential , describing both accelerating and decelerating universes , were explicitly obtained .
the cosmological perturbation equations were also derived and applied to uncover the nature of the propagating scalar degree of freedom and the signatures these models predict in the large - scale structure .
the general conditions for wormhole solutions according to the null energy condition violation at the throat in the hybrid metric - palatini gravitational theory were also presented in @xcite .
a new approach to modified gravity which generalizes the hybrid metric - palatini gravity was introduced in @xcite .
the gravitational action is taken to depend on a general function of both the metric and palatini curvature scalars . the dynamical equivalence with a non - minimally coupled bi - scalar field gravitational theory was proved .
the evolution of the cosmological solutions in this model was studied by using dynamical systems techniques . in @xcite
a method was developed to analyse the field content of `` hybrid '' gravity theories whose actions involve both the independent palatini connection and the metric levi - civita connection , and , in particular , to determine whether the propagating degrees of freedom are ghosts or tachyons .
new types of second , fourth and sixth order derivative gravity theories were investigated , and from this analysis it follows that the metric - palatini theory is singled out as a viable class of `` hybrid '' extensions of general relativity .
in addition to this , the stability of the einstein static universe was analysed in @xcite . in the latter , by considering linear homogeneous perturbations , the stability regions of the einstein static universe were parameterized by the first and second derivatives of the scalar potential , and it was explicitly shown that a large class of stable solutions exists in the respective parameter space .
for a brief review of the hybrid metric - palatini theory , we refer the reader to @xcite .
thus , the hybrid metric - palatini theory opens up new possibilities to approach , in the same theoretical framework , the problems of both dark energy and dark matter . in @xcite ,
the generalized virial theorem in the scalar - tensor representation of the hybrid metric - palatini gravity was analysed . more specifically , taking into account the relativistic collisionless boltzmann equation , it was shown that the supplementary geometric terms in the gravitational field equations provide an effective contribution to the gravitational potential energy .
indeed , it was shown that the total virial mass is proportional to the effective mass associated with the new terms generated by the effective scalar field , and the baryonic mass .
this shows that the geometric origin in the generalized virial theorem may account for the well - known virial theorem mass discrepancy in clusters of galaxies .
in addition to this , the astrophysical applications of the model were considered and it was shown that the model predicts that the mass associated to the scalar field and its effects extend beyond the virial radius of the clusters of galaxies . in the context of the galaxy cluster velocity dispersion profiles predicted by the hybrid metric - palatini model , the generalized virial theorem can be an efficient tool in observationally testing the viability of this class of generalized gravity models .
thus , hybrid metric - palatini gravity provides an effective alternative to the dark matter paradigm of present day cosmology and astrophysics . in this latter context , it is the purpose of the present paper to investigate the possibility that the observed properties of the galactic rotation curves could be explained in the framework of hybrid metric - palatini gravity , without postulating the existence of the hypothetical dark matter . as a first step in this study ,
we obtain the expression of the tangential velocity of test particles in stable circular orbits around galaxies . since we assume that the test particles move on the geodesic lines of the space - time , their tangential velocity is determined only by the radial distance to the galactic center , and the metric , through the derivative of the @xmath9 metric component with respect to the radial coordinate .
the metric in the outer regions of the galaxy is largely shaped by the energy contained in the effective scalar field of the hybrid metric - palatini gravitational theory .
therefore the behavior of the neutral hydrogen gas clouds outside the galaxies , and their flat rotation curves , can be explained by the presence of the scalar field generated in the model . by using the weak field limit of the gravitational field equations in the hybrid metric - palatini model
we obtain the explicit form of the tangential velocity , and show that the existence of a constant velocity region is possible for some specific values of the model parameters .
since the observations on the galactic rotation curves are obtained from the doppler frequency shifts , we generalize the expression of the frequency shifts by including the effect of the scalar field .
we also consider the velocity dispersion of the stars in the galaxy , and obtain the stellar velocity dispersion as a function of the scalar field .
thus , at least in principle , all the basic parameters of the model can be obtained directly from astronomical observations .
the knowledge of the tangential velocity allows the complete determination of the functional form of the scalar field from the observational data .
the basic parameters of the model can be immediately obtained in the flat rotation curves region , which is determined by the derivative of the scalar field potential .
hence the functional form of the scalar field can be obtained exactly , within the weak field limit of the model , for the entire galactic space - time , and tested at the galactic scale .
therefore , all the physical parameters of the hybrid metric - palatini gravitational theory can be either obtained directly , or severely constrained by astronomical observations .
the present paper is organized as follows . in section
[ sec : b ] , the field equations of the hybrid metric - palatini gravity model , as well as the general fluid representation of the stress - energy tensor are presented . in section [ sect3 ] , the tangential velocity of test particles in stable circular
orbits are derived and in section [ sect4 ] , the tangential velocities of test particles in the galactic halos in the hybrid model are discussed .
the comparison of the theoretical predictions for the rotational velocity and the observational data for four low surface brightness galaxies is considered in section [ sect5 ] .
the velocity dispersion of the stars in the galaxy , representing important observational tests of the model , as well as the red and blue shifts of the electromagnetic radiation emitted by the gas clouds are also investigated .
we discuss and conclude our results in section [ sect6 ] . in this paper
, we use the landau - lifshitz @xcite sign conventions and definitions of the geometric quantities .
in this section , we briefly present the basic formalism , for self - completeness and self - consistency , and the field equations of the hybrid metric - palatini gravitational theory within the equivalent scalar - tensor representation ( we refer the reader to @xcite for more details ) , and furthermore obtain the perfect fluid form of the stress - energy tensor of the scalar field . the action for hybrid metric - palatini gravity is obtained by adding an @xmath11 term , constructed within the framework of the palatini formalism , to the metric einstein - hilbert lagrangian @xcite , and is given by @xmath12 + s_m \ , \ ] ] where @xmath13 is the gravitational coupling constant , and the scalar curvature @xmath14 , depending on both the metric and an independent dynamical connection @xmath15 , is defined as @xmath16 @xmath17 is the ricci tensor obtained from the connection @xmath15 . the hybrid metric - palatini theory may be expressed in a purely scalar - tensor representation , by the following action @xmath18 + s_m \,,\ ] ] which differs fundamentally from the @xmath19 brans - dicke theory in the coupling of the scalar to the scalar curvature , where @xmath20 is the dimensionless brans - dicke coupling constant .
the variation of this action with respect to the metric tensor gives the field equations [ einstein_phi ] g_=^2(t _
+ t_^ ( ) ) , where @xmath21 is the ordinary matter stress - energy tensor , and t_^()&= & , is the stress - energy tensor of the scalar field .
the variation of the action with respect to the scalar field gives [ variation_phi ] r - + _
^- = 0 .
moreover , one can show that the identity @xmath22 also holds , and that the scalar field @xmath23 is governed by the second - order evolution equation @xmath24=\frac{\phi\kappa^2}{3}t\,,\ ] ] with @xmath25 , which is an effective klein - gordon equation .
the stress - energy tensor of a fluid can be generally represented as t_=(+p)u_u_-pg_+q_u_+q_u_+s _ , where @xmath26 is the four - velocity of the fluid , @xmath27 and @xmath28 are the energy density and isotropic pressure , respectively , @xmath29 is the heat flux , and @xmath30 is the tensor of the anisotropic dissipative stresses .
the heat flux four - vector and the anisotropic stress tensor satisfy the conditions @xmath31 , @xmath32 , and @xmath33 , respectively .
the four - velocity is normalized so that @xmath34 , and @xmath35 . by introducing the projection tensor @xmath36 , with the properties @xmath37 , @xmath38 , and @xmath39
, the thermodynamic parameters of the fluid can be obtained from the stress - energy tensor as [ ex ] & & = u^u^t _ , p =- h^t _ , + & & q_=u^h^_t _
, s_=h^_h^_t_+ph_. in order to obtain the perfect fluid representation of the stress - energy tensor of the scalar field in hybrid metric - palatini gravity , we introduce first the four - velocity of the scalar field as u^ _ ( ) = , which satisfies the relation @xmath40 .
therefore , with the use of eqs .
( [ ex ] ) we obtain the effective energy density @xmath41 and pressure @xmath42 in the scalar field description of hybrid metric - palatini gravity as _ & = & \{^_(_^)-_^ + & & -+t - v ( ) } , p_&=&\{--_^ + & & + -t+v ( ) } , q^()=u^ _ ( ) _ u^ _ ( ) , s^()&= & + & & , where @xmath43 is the expansion of the fluid .
it is interesting to note that the fluid - equivalent stress - energy tensor in hybrid metric - palatini gravity is not of a perfect fluid form , but contains `` heat transfer '' terms , as well as an anisotropic dissipative component .
therefore , the stress - energy of the scalar field can be written in a form equivalent to a general fluid as t^()&=&(_+p_)u^_()u^_()-p_g^+ + & & q^()u^_()+q^()u^ _ ( ) + s^ ( ) , which will be useful in determining the galactic geometry in the context of the tangential velocity curves analysis outlined below .
the most direct method for studying the gravitational field inside a spiral galaxy is provided by the galactic rotation curves .
they are obtained by measuring the frequency shifts @xmath44 of the 21-cm radiation emission from the neutral hydrogen gas clouds .
the 21-cm radiation also originates from stars .
the 21-cm background from the epoch of reionization is a promising cosmological probe : line - of - sight velocity fluctuations distort redshift , so brightness fluctuations in fourier space depend upon angle , which linear theory shows can separate cosmological from astrophysical information ( for a recent review see @xcite ) . instead of using @xmath44 the resulting redshift
is presented by astronomers in terms of a velocity field @xmath1 @xcite . in the following
, we will assume that the gas clouds behave like test particles , moving in the static and spherically symmetric geometry around the galaxy . without a significant loss of generality , we assume that the gas clouds move in the galactic plane @xmath45 , so that their four - velocity is given by @xmath46 , where the overdot stands for derivation with respect to the affine parameter @xmath47 .
the static spherically symmetric metric outside the galactic baryonic mass distribution is given by the following line element @xmath48 where the metric coefficients @xmath49 and @xmath50 are functions of the radial coordinate @xmath2 only . the motion of a test particle in the gravitational field with the metric given by eq .
( [ line ] ) , is described by the lagrangian @xcite @xmath51 , \ ] ] where @xmath52 , which simplifies to @xmath53 along the galactic plane @xmath54 . from the lagrange equations it follows that we have two constants of motion , namely , the energy @xmath55 per unit mass , and the angular momentum @xmath56 per unit mass , given by @xmath57 and @xmath58 , respectively .
the normalization condition for the four - velocity @xmath59 gives @xmath60 , from which , with the use of the constants of motion , we obtain the energy of the particle as @xmath61 from eq .
( [ energy ] ) it follows that the radial motion of the test particles is analogous to that of particles in newtonian mechanics , having a velocity @xmath62 , a position dependent effective mass @xmath63 , and an energy @xmath64 , respectively .
in addition to this , the test particles move in an effective potential provided by the following relationship @xmath65 the conditions for circular orbits , namely , @xmath66 and @xmath67 lead to @xmath68 and @xmath69 respectively .
note that the spatial three - dimensional velocity is given by @xcite @xmath70 .\ ] ] for a stable circular orbit @xmath71 , and the tangential velocity of the test particle can be expressed as @xmath72 in terms of the conserved quantities , and along the galactic plane @xmath45 , the angular velocity is given by @xmath73 and taking into account eqs .
( [ cons1 ] ) and ( [ cons2 ] ) , we finally obtain the following relationship @xcite @xmath74 therefore , once the tangential velocity of test particles is known , the metric function @xmath75 outside the galaxy can be obtained as ( r)=2 .
[ metricnu ] the tangential velocity @xmath76 of gas clouds moving like test particles around the center of a galaxy is not directly measurable , but can be inferred from the redshift @xmath77 observed at spatial infinity , for which @xmath78 \left ( 1\pm v_{tg}/c\right ) /\sqrt{1-v_{tg}^{2}/c^2}$ ] @xcite . due to the non - relativistic velocities of the gas clouds , with @xmath79 , we observe that @xmath80 , as the first part of a geometric series .
the observations show that at distances large enough from the galactic center the tangential velocities assume a constant value , i.e. , @xmath81 constant @xcite . in the latter regions of the constant tangential velocities ,
( [ metricnu ] ) can be readily integrated to provide the following metric tensor component [ nu ] e^=()^2v_tg^2/c^21 + 2 ( ) , where @xmath82 is an arbitrary constant of integration . if we match the metric given by eq .
( [ nu ] ) with the schwarzschild metric on the surface of the galactic baryonic matter distribution , having a radius @xmath83 , @xmath84 , we obtain the following relationship r_=. an important physical requirement for the circular orbits of the test particle around galaxies is that they must be stable .
let @xmath85 be the radius of a circular orbit and consider a perturbation of it of the form @xmath86 , where @xmath87 @xcite .
taking expansions of @xmath88 and @xmath89 about @xmath90 , it follows from eq .
( [ energy ] ) that @xmath91 the condition for stability of the simple circular orbits requires @xmath92 @xcite .
hence , with the use of the condition @xmath93 , we obtain the condition of the stability of the orbits as @xmath94|_{r = r_0}$ ] . by taking into account eq .
( [ vtg ] ) , it immediately follows that for massive test particles whose velocities are determined by the @xmath9 component of the metric tensor only the stability condition of the circular orbits is always satisfied .
the rotation curves only determine one , namely @xmath49 , of the two unknown metric functions , @xmath49 and @xmath50 , which are required to describe the gravitational field of the galaxy .
hence , in order to determine @xmath95 we proceed to solve the gravitational field equations for the hybrid metric - palatini gravitational theory outside the baryonic matter distribution .
this allows us to take all the components of the ordinary matter stress - energy tensor as being zero . taking into account the stress - energy tensor for the equivalent scalar field representation of hybrid metric - palatini gravity the gravitational field equations describing the geometry of the galactic halo take the form [ f1b ] -e^- ( - ) + & = & ( _ + p_)u _ ( ) tu^t_()-p _ + & & + q^()tu_()t+q^()_tu_()^t+s^()t_t , [ f2b ] e^- ( + ) - & = & -(_+p_)u _ ( ) ru^r_()+p _ + & & -q^()ru_()r - q^()_ru_()^r - s^()r_r , & & e^- ( ^++- ) = - s^ ( ) _ + & & -(_+p_)u _ ( ) u^_()+p_- q^()u_()-q^()_u_()^ , where there is no summation upon the pair of indices @xmath96 . the weak field limit of the gravitational theories at the solar system level is usually obtained by using isotropic coordinates .
however , it is useful to apply schwarzschild coordinates in studying exact solutions and in the context of galactic dynamics . in the following
, we will adopt in our analysis the schwarzschild coordinate system .
we assume that the gravitational field inside the halo is weak , so that @xmath97 , which allows us to linearise the gravitational field equations retaining only terms linear in @xmath98 .
moreover , we assume that the scalar field @xmath99 is also weak , so that @xmath100 . by representing the scalar field as @xmath101 , where @xmath102 is a small perturbation around the background value @xmath103 of the field , in the first order of perturbation
, the scalar field potential @xmath104 and its derivative with respect to @xmath99 can be represented as v()=v(_0+)v(_0)+v(_0)+ .... , and v()v(_0)+ v(_0 ) , respectively . in the linear approximation
we have @xmath105 .
therefore the effective klein - gordon type equation of the scalar field , eq .
( [ eq : evol - phi ] ) takes the form [ yuk ] ( ^2 -)=0 , where a constant on the right - hand side of this equation has been absorbed into a redefinition of @xmath106 , and the following parameter has been defined for notational simplicity = . from a physical point of view
@xmath107 represents ( in natural units ) the inverse of the mass @xmath108 of the particle associated with the scalar field , @xmath109 .
the hybrid metric - palatini gravity theory can pass the solar system observational constraints even if the scalar field is very light , that is , @xmath108 is very small @xcite . within this linear approximation the stress - energy tensor of the scalar field
is given by t_^()= , where @xmath110 and @xmath111 are defined by = - v(_0 ) , = -v(_0 ) .
therefore the linearized gravitational field equations take the form [ f1 ] ( r)=+&=&^(eff ) , + [ f2 ] -+=++&=&-p_r^(eff ) , + [ f3 ] -(+)=+&=&-p_^(eff ) .
using spherical symmetry , eq .
( [ yuk ] ) takes the form r-=0 , which yields the following general solution ( r)=_0 , where @xmath112 is an integration constant . comparing this expression with the results obtained in @xcite for the weak - field limit ( taking into account the transformation from isotropic to schwarzschild coordinates )
, we find that [ psi0 ] _ 0=-_0<0 , where @xmath113 and @xmath83 are the mass and the radius of the galactic baryonic distribution , respectively .
( [ f1 ] ) can be immediately integrated to provide ( r)&=&+^r(+)^2d + & = & + r^2-(1 + ) , where @xmath114 is an integration constant .
comparing again with the results obtained in @xcite for the weak - field limit , we find that @xmath115 .
the tangential velocity of the test particles in stable circular orbits moving in the galactic halo can be derived immediately from eq .
( [ f2 ] ) , and is given by = = -r^2-r^2-r^2 , which in terms of the solutions found above becomes & = & r^2 + - + & & , where @xmath116 .
the term proportional to @xmath117 corresponds to the cosmological background , namely the de sitter geometry , and we assume that it has a negligible contribution on the tangential velocity of the test particles at the galactic level . on the surface of the baryonic matter distribution the tangential velocity must satisfy the boundary condition , which , with the use of eq .
( [ psi0 ] ) , gives the following constraint on the parameters of the model , ( 1+)(2+r_^2)+(1+r_^2)0 . in order to satisfy
the above condition would require that @xmath118 , or , equivalently , v(_0)>0 , and 2<v(_0)r_^2<3 , respectively . in the regions near the galactic baryonic matter distribution , where @xmath119 , we have @xmath120 , to a very good approximation .
hence in this region the tangential velocity can be approximated as & & -_0 + & & -(1+r_^2)r , r_br r_. if the parameters of the model satisfy the condition 2gm_b - c^2_0(r_^2 + 2)0 , the term proportional to @xmath121 becomes negligible , while for small values of @xmath112 , and @xmath122 , the term proportional to @xmath2 can also be neglected .
therefore for the tangential velocity of test particles rotating in the galactic halo we obtain -_0- , r_br r_. since according to our assumptions , @xmath123 , the coefficient @xmath124 can be approximated as @xmath125 , which provides for the rotation curve , in the constant velocity region , the following expression , r_br r_. since @xmath126 , the scalar field potential must satisfy the condition @xmath127 . in the first order of approximation , with @xmath128 , for the tangential velocity we obtain the expression + r+r^2 .
alternatively , in general we can write the tangential velocity as follows , [ vfin ] & = & r^2 + \{1+e^}. as compared to our previous results , in this representation we have @xmath129 instead of @xmath130 .
since we are working in a regime in which @xmath131 , the choice of the constants @xmath83 or @xmath113 does not seem very relevant , since it just amounts to a rescaling of @xmath106 . from now on we will also assume that @xmath132 .
from the above equation we want to find the constraints on the model parameters that arise from the expected behavior at different scales . for that purpose
, it is convenient to write the equation , equivalently , as follows : & = & + + & & ( 2+r_^2 ) e^-+ ( 1+r_^2 ) ( ) e^- . at intermediate scales , the asymptotic tangential velocity tends to a constant .
if we expand the exponential as @xmath133 , then we obtain the following three constraints on the free parameters of the model , & & a ) 1+(2+r_^2 ) 0 , + & & b ) ( 2+r_^2 ) ( 1-)c= constant , + & & c ) ( ) |c| . with increasing @xmath2 , and by assuming that the condition @xmath134 still holds , the rotation curves will decay , at very large distances from the galactic center , to the zero value .
in the present section , we will present some observational possibilities of directly checking the validity of the hybrid metric - palatini gravitational model . more specifically
, we will first compare the theoretical predictions of the model with a sample of rotation curves of low surface brightness galaxies , respectively .
then we consider the possibility of observationally determining the functional form of the scalar field @xmath135 by using the velocity dispersion of stars in galaxies , and the red and blue shifts of gas clouds moving in the galactic halo . in order to test our results
we compare the predictions of our model with the observational data on the galactic rotation curves , obtained for a sample of low surface luminosity galaxies in @xcite .
generally , in a realistic situation , a galaxy consists of a distribution of baryonic ( normal ) matter , consisting of stars of mass @xmath136 , ionized gas of mass @xmath137 , neutral hydrogen of mass @xmath138 etc . , and the `` dark matter '' of mass @xmath139 , which , in the present model , is generated by the extra contributions to the total energy - momentum , due to the contribution of the effective scalar field .
hence the total mass of the galactic baryonic matter is @xmath140 . as for the distribution of the baryonic mass
, we assume that it is concentrated into an inner core of radius @xmath141 , and that its mass profile @xmath142 can be described by the simple relation m_b ( r ) = m_b()^3 , rr_b , r_b r_c , where @xmath143 for high surface brightness galaxies ( hsb ) and @xmath144 for low surface brightness ( lsb ) and dwarf galaxies , respectively @xcite .
for @xmath145 we have @xmath146 . by representing the coefficient @xmath124 as @xmath147 , where @xmath148 , from eq .
( [ vfin ] ) we obtain the tangential velocity of the massive particles in stable galactic circular orbits as v_tg^2=v_kepl^2+v_kepl^2e^-r / r_(1++),rr_b , where @xmath149 .
hence , by also taking into account the baryonic matter contribution , we obtain the total tangential velocity of a massive test particle as [ comp ] v_tgtot^2(km / s)&=&4.3310 ^ 4 ( ) ^6 + & & + 2.7710 ^ 4 _ 0 _ 0 + & & ( 1++ ) , rr_b , where we have used the value @xmath150 . in order to compare the prediction of eq .
( [ comp ] ) with the observed rotation curves of the lsb galaxies in the following we assume @xmath127 , and we fix the numerical values of the universal parameters @xmath151 as @xmath152 . then any variability in the behavior of the rotation curves is due to the variation of the baryonic mass of the galaxy @xmath113 , and of its baryonic mass distribution in the core , described by @xmath141 . in fig .
[ fig1 ] , we have compared the predictions of eq .
( [ comp ] ) for the behavior of the rotation curves in the `` dark matter '' region for four lsb galaxies , ddo189 , ugc1281 , ugc711 , and ugc10310 , respectively @xcite .
the comparison of the predictions of the theoretical model with the observational results show that the contribution of the scalar field energy density to the tangential velocity of the test particles can explain the existence of a constant rotational velocity region around the baryonic matter , without requiring the presence of the dark matter . of course , in order to gain a better understanding of the behavior of the galactic rotation curves in the hybrid metric - palatini gravity model , the qualitative approach considered in the present section must be reconsidered by taking into account more realistic galactic baryonic matter distributions , and a much larger sample of galaxies having different morphologies . in hybrid metric - palatini gravity one can formally associate an approximate `` dark matter '' mass profile @xmath153 to the tangential velocity profile , which taking into account eq .
( [ f1 ] ) , is given by @xmath154 so that the metric tensor component @xmath95 can be written as ( r)=. the effective `` dark matter '' density profile @xmath155 is obtained as _
dm(r)==(+ ) . in order to observationally constrain @xmath139 and @xmath155
, we assume that each galaxy consists of a single , pressure - supported stellar population that is in dynamic equilibrium and traces an underlying gravitational potential , which is created due to the presence of the scalar field @xmath135 . by assuming spherical symmetry , the equivalent mass profile induced by the scalar field (
the mass profile of the `` effective dark matter '' halo ) can be obtained from the moments of the stellar distribution function via the jeans equation @xcite @xmath156 + \frac{2\rho _ { s}\left ( r\right ) \beta _ { an}(r)}{r}=-\frac{g\rho _ { s}m_{dm}(r)}{r^{2}},\]]where @xmath157 , @xmath158 , and @xmath159 describe the three - dimensional density , the radial velocity dispersion , and the orbital anisotropy of the stellar component , where @xmath160 is the tangential velocity dispersion . with the assumption of constant anisotropy , @xmath161 ,
the jeans equation can be solved to give @xmath162 as @xcite @xmath163 with the use of eq .
( [ darkmass ] ) we obtain for the stellar velocity dispersion the equation [ integral ] _ sv_r^2 & & r^-2_an_r^s^2(2-_an)_s ( s ) + & & \{_r_b^s^2 d}ds
. the `` effective dark matter '' mass profile can be related through the projection along the line of sight to two observable quantities , the projected stellar density @xmath164 , and to the stellar velocity dispersion @xmath165 , respectively , according to the relation @xcite @xmath166 given a projected stellar density model @xmath164 , one recovers the three - dimensional stellar density from @xcite @xmath167 therefore , once the stellar density profile @xmath164 , the stellar velocity dispersion @xmath168 , and the quantities @xmath124 , @xmath169 , @xmath83 and @xmath113 , determining the geometry of the space - time outside the baryonic matter distribution , are known , with the use of the integral equation eq .
( [ integral ] ) one can constrain the explicit functional form of the scalar field @xmath170 , the two free parameters of the model , @xmath112 and @xmath171 , as well as the equivalent mass and density profiles induced by the presence of the scalar field .
the simplest analytic projected density profile is the plummer profile @xcite , given by @xmath172 , where @xmath173 is the total luminosity , and @xmath174 is the projected half - light radius ( the radius of the cylinder that encloses half of the total luminosity ) .
the rotation curves of spiral galaxies are inferred from the astrophysical observations of the red and blue shifts of the radiation emitted by gas clouds moving in circular orbits on both sides of the central region in the galactic plane .
the light signal travels on null geodesics in the galactic geometry with tangent @xmath175 .
we may , without a significant loss of generality , restrict @xmath175 to lie in the equatorial plane @xmath45 , and evaluate the frequency shift for a light signal emitted from the observer @xmath176 in circular orbit in the galactic halo , and detected by the observer @xmath177 situated at infinity . the frequency shift associated to the emission and detection of the light signal from the gas cloud
is defined as @xmath178 where @xmath179 , and the index @xmath180 refers to emission ( @xmath181 ) or detection ( @xmath182 ) at the corresponding space - time point @xcite .
we can associate with light propagation two frequency shifts , corresponding to maximum and minimum values , in the same and opposite direction of motion of the emitter , respectively . from an astrophysical point of view
such shifts are frequency shifts of a receding or approaching gas cloud , respectively . in terms of the tetrads
@xmath183 , @xmath184 , @xmath185 , @xmath186 , the frequency shifts can be represented as @xcite @xmath187 /2}\left ( 1\mp v\right )
\gamma , \ ] ] where @xmath188^{1/2}$ ] , with @xmath189 the components of the particle s four velocity along the tetrad ( i.e. , the velocity measured by an eulerian observer whose world line is tangent to the static killing field ) . in eq .
( [ 60 ] ) , @xmath190 is the usual lorentz factor , and @xmath191 represents the value of @xmath192 $ ] for @xmath193 . in the case of circular orbits in the @xmath45 plane , we obtain @xmath194 /2}\frac{%
1\mp \sqrt{r \nu ^{\prime } /2 } } { \sqrt{1-r \nu ^{\prime } /2 } } .\ ] ] it is convenient to define two other quantities , @xmath195 , giving the differences in the doppler shifts for the receding and approaching gas clouds , and @xmath196 , representing the mean value of the doppler shifts , respectively @xcite .
these redshift factors are given by @xmath197 /2}\frac{\sqrt{r\nu ^{\prime } /2 } } { \sqrt{1-r \nu ^{\prime } /2 } } , \ ] ] and @xmath198 /2}}{\sqrt{1-r
\nu ^{\prime } /2 } } , \ ] ] respectively , and they can be easily connected to the astrophysical observations @xcite .
@xmath199 and @xmath200 satisfy the relation @xmath201 $ ] , and thus in principle , the metric tensor component @xmath202 $ ] can be directly determined from observations . from eq .
( [ f2 ] ) we obtain r=-(++)r^2 , and ( r)=_r_b^rdr , respectively . by substituting these expressions of the metric tensor and of its derivative in eqs .
( [ zd ] ) and ( [ za ] ) , in principle we obtain a direct observational test of the galactic geometry , of the functional form of the scalar field , and , implicitly , of the hybrid metric - palatini gravitational model .
the behavior of the galactic rotation curves , especially their constancy , and the mass deficit in clusters of galaxies , continues to pose a major challenge to present day physics .
it is essential to have a better understanding of some of the intriguing phenomena associated with them , such as their universality , the very good correlation between the amount of dark matter and the luminous matter in the galaxy , as well as the nature of the dark matter particle , if it really does exist . to explain these intriguing observations ,
the commonly adopted models are based on exotic , beyond the standard model , particle physics in the framework of newtonian gravity , or of some extensions of general relativity . in the present paper , we have considered the observational implications of the model proposed in @xcite , and proposed an alternative view to the dark matter problem , namely , the possibility that the galactic rotation curves and the mass discrepancy in galaxies can naturally be explained in gravitational models in which an @xmath11 term , constructed within the framework of the palatini formalism , is added to the metric einstein - hilbert lagrangian .
the extra - terms in the gravitational field equations , which can be described as a function of an equivalent scalar field , modify , through the metric tensor components , the equations of motion of test particles , and induce a supplementary gravitational interaction , which can account for the observed behavior of the galactic rotation curves . due to the presence of the scalar field ,
the rotation curves show a constant velocity region , which decay to zero at large distances from the galactic center , a behavior which is perfectly consistent with the observational data @xcite , and is usually attributed to the existence of dark matter . by using the weak field limit of the gravitational field equations ,
the rotation curves can be completely reconstructed as functions of the scalar field , without any supplementary assumption .
if the galactic rotation velocity profiles are known from observations , the galactic metric can be derived theoretically , and the scalar field function can be reconstructed exactly over the entire mass distribution of the galactic halo . the formalism developed in the present paper could also be extended to the case of the galaxy clusters .
the latter are cosmological structures consisting of hundreds or thousands of galaxies .
we emphasize that the analysis of the geometric properties of the galaxy clusters can also be done in weak field approximation considered in the present paper .
the comparison of the observed velocity dispersion profiles of the galaxy clusters and the velocity dispersion profiles predicted by the hybrid metric - palatini gravity model can provide a powerful method for the observational test of the theory , and for observationally discriminating between the different modified gravity theoretical models .
the nature and dynamics of the cosmological evolution can be investigated by using a variety of cosmological observations .
one of the important methods for the study of the cosmic history relies on extracting the baryon acoustic oscillations ( bao ) in the high-@xmath44 galaxy power spectrum .
the baryon acoustic oscillations imprint a characteristic scale on the galaxy distribution that acts as a standard ruler .
the origin of the bao in the matter power spectrum can be understood as the velocity fluctuation of the baryonic fluid at the decoupling time .
the characteristic scale of the baryon oscillation is determined by the sound speed and horizon at decoupling , which is a function of the total matter and baryon densities @xcite . since this scale can be measured in both the transverse and radial directions , the bao yields both the angular diameter distance , and the hubble parameter at that redshift .
therefore , the precise measurement of the bao scale from the galaxy power spectrum can impose important constraints on the cosmic expansion history
. different expansion histories in modified gravity models shifts the peak positions of oscillations relative to the @xmath203cdm model @xcite .
therefore the predicted shifts in the bao can potentially be used to distinguish between the @xmath204cdm models and modified gravity models .
thus , by using the bao analysis it can be shown that the original dvali - gabadadze - porrati model @xcite is disfavored by observations , unless the matter density parameter exceeds 0.3 @xcite .
the recently released planck satellite data @xcite , as combined with the bao measurements @xcite provide strong constraints on the modified @xmath0 gravity model @xcite . in the @xmath0 modified gravity models the lensing amplitude return to be compatible with @xmath205 at 68% confidence limit ( c.l . )
if one consider the planck or planck combined with the hubble space telescope measurement data , and even at 95% c.l . if we consider planck data combined with bao data
moreover , in the framework of the considered @xmath0 models the standard value of the lensing amplitude @xmath205 is in agreement with the planck measurements , oppositely to what happens in the @xmath203cdm scenario .
the study of the bao and of the cmb data can provide very powerful and high precision constraints in discriminating between the hybrid metric - palatini gravity model , and alternative gravity models , as well as the standard dark matter model .
weak gravitational lensing , whereby galaxy images are altered due to the gravitational field influence of the mass along the line of sight , is a powerful probe of the dark matter in cosmology , with promising results obtained in recent years @xcite . in standard general relativity
, the weak lensing distortion field provides a direct tracer of the underlying matter distribution .
however , modifications to gravity theory can conceivably alter the way that mass curves spacetime , and thus the way that null geodesics behave in a given matter distribution .
the possibilities of constraining modified gravity theories with weak lensing were considered recently in @xcite . due to its sensitivity to the growth rate of the structure
, weak lensing can be very useful to constrain modified gravity theories , and to distinguish between various modified gravity and standard dark matter models , when combined with cmb observations .
future weak lensing surveys as euclid can constrain modified gravity models , as those predicted by scalar - tensor and @xmath0 theories . since the hybrid metric - palatini gravity model can be formulated in terms of an equivalent scalar - tensor theory ,
the analysis of the future weak lensing observational data , such as those provided by the euclid mission , may provide a powerful method to observationally constrain the free parameters of this theoretical model . even that the effective scalar field of the hybrid metric - palatini gravity model provides a gravitational `` mass '' equivalent to the dark matter , due to the specific functional form and numerical values of the model parameters ,
its imprint on the weak lensing properties on a cosmological scale is different from that of the standard dark matter . in the present model
all the relevant physical quantities , including the `` dark mass '' associated to the equivalent scalar - tensor description , and which plays the role of dark matter , its corresponding density profile , as well as the scalar field and its potential , are expressed in terms of observable parameters the tangential velocity , the baryonic ( luminous ) mass , the doppler frequency shifts of test particles moving around the galaxy , and the velocity dispersions of the stars .
therefore , this opens the possibility of directly testing the modified gravitational models with palatini type @xmath11 terms added to the gravitational action by using direct astronomical and astrophysical observations at the galactic or extra - galactic scale . in this paper
we have provided some basic theoretical ideas , which , together with the virial theorem considered in @xcite , are the necessary tools for the in depth comparison of the predictions of the hybrid metric - palatini gravity model with the observational results .
we would like to thank to the anonymous referee for comments and suggestions that helped us to significantly improve our manuscript .
sc is supported by infn ( iniziative specifiche qgsky and teongrav ) .
tsk is supported by the research council of norway .
fsnl acknowledges financial support of the fundao para a cincia e tecnologia through the grants cern / fp/123615/2011 and cern / fp/123618/2011 .
gjo is supported by the spanish grant fis2011 - 29813-c02 - 02 , the consolider programme cpan ( csd2007 - 00042 ) , and the jae - doc program of the spanish research council ( csic ) .
by convention , the weak - field limit of theories of gravity in the solar system is discussed using isotropic coordinates .
however , exact solutions and galactic dynamics are usually considered in terms of the schwarzschild coordinates . since we have already obtained the form of the weak - field limit in @xcite , in order to translate those results to the current problem we just need to transform our metric from isotropic to schwarzschild coordinates .
this will give coherence to these series of papers .
what we need to do is just to compare the schwarzschild - like line element used in the study of the galactic dynamics with the isotropic results , and find the change of coordinates . + in schwarzschild coordinates the linearized line element is ds^2&=&e^dt^2-e^dr^2-r^2d^2 + & & ( 1+)dt^2-(1+)dr^2-r^2d^2 .
the comparison of the coordinates gives the relations @xmath209 the second of these equations allows to express @xmath210 and @xmath211 in terms of the schwarzschild coordinate @xmath2 .
it also allows us to find an expression for @xmath212 as follows , @xmath213 ^ 2.\ ] ] inserting this result in eq .
( [ 81 ] ) , and expanding to leading order , we find @xmath214.\ ] ] therefore , @xmath215 and @xmath216 .
taking into account the explicit form of @xmath217 and @xmath204 @xcite , @xmath218 with @xmath219 we find that @xmath220.\end{aligned}\ ] ] here we can consider some simplifications . as a first approximation we can assume @xmath221 to be very small , which is equivalent to consider scales much smaller than the cosmological horizon .
we can also assume @xmath106 small , but we keep @xmath222 non - negligible . in this case
, we find the same relation between coordinates as in gr , namely , r^2 ^
2(1 + ) , which leads to r(1+)=gm+ . with this result ,
we find @xmath223\right|_{\rho = r - gm}.\nonumber\\\end{aligned}\ ] ] note that even though at galactic scales we have @xmath224 and @xmath225 , the exponential corrections @xmath226 experience a magnification of the _ apparent _ value of @xmath106 by an exponential factor : @xmath227 , i.e. , @xmath228 .
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zhao , a. hojjati , and l. pogosian , jcap * 08 * , 029 ( 2013 ) . | generally , the dynamics of test particles around galaxies , as well as the corresponding mass deficit , is explained by postulating the existence of a hypothetical dark matter .
in fact , the behavior of the rotation curves shows the existence of a constant velocity region , near the baryonic matter distribution , followed by a quick decay at large distances . in this work
, we consider the possibility that the behavior of the rotational velocities of test particles gravitating around galaxies can be explained within the framework of the recently proposed hybrid metric - palatini gravitational theory .
the latter is constructed by modifying the metric einstein - hilbert action with an @xmath0 term in the palatini formalism .
it was shown that the theory unifies local constraints and the late - time cosmic acceleration , even if the scalar field is very light . in the intermediate galactic scale , we show explicitly that in the hybrid metric - palatini model the tangential velocity can be explicitly obtained as a function of the scalar field of the equivalent scalar - tensor description . the model predictions are compared model with a small sample of rotation curves of low surface brightness galaxies , respectively , and a good agreement between the theoretical rotation curves and the observational data is found . the possibility of constraining the form of the scalar field and the parameters of the model by using the stellar velocity dispersions is also analyzed .
furthermore , the doppler velocity shifts are also obtained in terms of the scalar field .
all the physical and geometrical quantities and the numerical parameters in the hybrid metric - palatini model can be expressed in terms of observable / measurable parameters , such as the tangential velocity , the baryonic mass of the galaxy , the doppler frequency shifts , and the stellar dispersion velocity , respectively .
therefore , the obtained results open the possibility of testing the hybrid metric - palatini gravitational models at the galactic or extra - galactic scale by using direct astronomical and astrophysical observations . + + * keywords * : modified gravity : galactic rotation curves : dark matter : | 1307.0752 |
it is generally accepted that supernova remnants ( snrs ) are the dominant source of galactic cosmic rays , at least for energies up to 3@xmath310@xmath12 ev . these objects provide a significant fraction of the mechanical energy that heats , compresses and chemically enriches the interstellar medium ( ism ) . therefore , snrs can be used to investigate global properties of the galaxy as well as the local environment where they evolve . thanks to significant advances in the angular resolution capabilities of modern x - ray observatories such as xmm-_newton _ and _ chandra _ , important progress has been made concerning the detection of new and well - known snrs ( sasaki et al .
2004 ; bocchino et al . 2005 ; combi et al .
2010a ; combi et al .
at present , 30% of the radio snrs display x - ray emission and more than a dozen of snrs were originally discovered through their x - ray emission ( e.g. , schwentker 1994 ; aschenbach 1998 ; bamba et al .
2003 ; yamaguchi et al .
2004 ) . studying the connexion between radio , infrared and x - ray emission of snrs
enables us to explore how stars end their lives and better understand the evolutionary process in this kind of fascinating sources .
the southern galactic snr g296.8 - 0.3 ( 1156 - 62 ) lies in the direction to the scutum - crux arm of our galaxy .
it was first detected at radio frequencies by large & vaughan ( 1972 ) .
initially , it was considered as a shell - type remnant despite its unusual and ill - defined shape ( shaver & goss 1970 ; goss & shaver 1970 ) .
subsequent higher - resolution radio observations at 843 mhz ( whiteoak & green 1996 ) showed a complicated multi - ringed structure , with its diffuse interior emission being brightest in its northwest side .
hi observations carried out by gaensler et al .
( 1997 ) allowed to obtain lower and upper limits on its systemic velocity in the range + 15 to + 30 km s@xmath13 .
this corresponds to a distance of [email protected] kpc .
a flux density of 7.0@xmath14 0.3 jy was measured at 1.3 ghz for the total radio structure . throughout this paper , a mean distance of 9 kpc is assumed .
it corresponds to an angular size of @xmath2 31 pc . at x - ray energies ,
an exploratory study was carried out with the rosat satellite by hwang & markert ( 1994 ) .
these authors noted marginal x - ray emission ( 4 @xmath15 detection ) near the peak of radio emission to the northwest part of the remnant .
however , the poor statistics and limited x - ray energy range of the rosat telescope did not allow to observe a clear well - defined morphology of the x - ray emission .
as part of a program addressed to study the x - ray emission of supernova remnants , in this paper we report xmm@xmath0_newton _ observations of g296.8 - 0.3 , and infrared data obtained with the spitzer space telescope in order to study the physical characteristics of the object and the surrounding ism where it evolves .
the structure of the paper is as follows : in sect .
2 , we describe the xmm-_newton _ observations , data reduction and present our x - ray analysis .
ir results are reported in sect .
3 . in sect .
4 , we discuss the implications of our results . finally in sect . 5
, we summarize our main conclusions .
g296.8 - 0.3 was observed by the xmm - newton x - ray satellite in two separate pointings .
these were performed on 2008 february 16 ( obsid 0503780301 ) and 2008 august 16 ( obsid 0550170101 ) , with the epic mos ( turner et al . 2001 ) and epic pn ( strder et al .
2001 ) cameras .
both observations have similar pointing coordinates ( @xmath17=11@[email protected] , @xmath20=@xmath062@[email protected] ; j2000 ) , and were placed at the ccd center .
the xmm-_newton _ data were calibrated and analyzed with the xmm science analysis system ( sas ) version 10.0.0 . to exclude high background activity , which can affect the observations ,
we extracted light curves of photons above 10 kev from the entire field - of - view of the cameras , and excluded time intervals with count rate higher than 3 @xmath15 above average to produce a gti file . unfortunately , the observation obsid 0503780301 was affected by a high and variable soft proton background level ( lumb et al .
2002 ) , whereas the other one obsid 0550170101 is unaffected by background fluctuations . in order to avoid contamination for high background patterns ,
hereafter our analysis concerns only the second observation .
after the time filtering , 45.2 ks of useful data for mos1 , 45.1 ks for mos2 , and 46.5 ks for the pn cameras are available for further data analysis , which is @xmath280@xmath23 of the total exposure . to create images , spectra , and light curves , we selected events with flag @xmath24 0 and pattern @xmath25 12 for mos1/2 and pattern @xmath25 4 for pn .
figure 1 ( left panel ) shows a color composite xmm-@xmath16 image g296.8 - 0.3 in three energy bands : 0.5 - 1.0 ( red ) , 1.0 - 2.0 ( green ) , and 2.0 - 5.0 kev ( blue ) , for the combined mos1/2 cameras . images were corrected for the spatial dependent exposure , and the instrumental background was also substracted . at x - ray energies
@xmath6 2.6 kev the snr is not detected .
the contour map overplotted on figure 1 is the most 843 mhz radio continuum map ( whiteoak & green 1996 ) , which can help us to identify the origin of the x - rays .
the x - ray structure of the snr is complex and covers @xmath2 30% of the total radio extent .
it shows three different components : interior diffuse emission coincident with the unusual rectangular strip ( gaensler et al .
1998 ) running through its center seen at radio frequencies ( indicated with the dashed line in fig.1 , left panel ) , a bright soft shell - like feature with an angular size of @xmath2 8 , coincident with the northwest radio shell , and at least 9 point - like sources .
all of them are numbered in fig.1 , right panel .
these point sources were detected by using the source detection meta - task ` edetect - chain ' .
sources 1,2,3,4,5,6,7 and 9 display medium and hard x - ray emission and could be background agns .
the remaining one , source 8 , displays soft and medium x - ray emission . as can be seen , source 8 is located close to the geometrical center of the radio structure ( @xmath17=11@[email protected] , @xmath20=@xmath062@[email protected] ; j2000 ) .
the source is catalogued as 2xmmi j15836.1 - 623516 in the xmm-_newton _ serendipitous source catalogue 2xmmi - dr3 ( watson et al .
2009 ) .
the interior of the radio remnant is filled with x - ray emitting material , and the spatial coincidence of x - ray and radio emission suggests that the physical conditions of the terminal shock region are very similar to those found at the outer shocks of ordinary snrs . from the x - ray image
we see enhanced x - ray emission on the northwest part of the snr , which is probably caused by density enhancements in the medium in which the shock propagates and forms a more or less continuous structure .
we notice that the x - ray emission is entirely contained within the boundaries of the radio shell .
this bright filamentary x - ray structure is centered at ( @xmath17=11@[email protected] , @xmath20=@xmath062@[email protected] ; j2000 ) and has an angular size of 8 arcmin on the plane of the sky .
the x - ray emission of the snr fades out to the southeast , lacking a clearly defined edge . in order to study the physical properties of the plasma in the remnant
, a x - ray spectrum of g298.6 - 0.3 was extracted from the epic cameras using an elliptical region ( shown in fig .
1 , right panel ) that comprises @xmath2 50@xmath23 of the central ccd of the mos cameras and spread part of 4 ccds of the pn camera . for such purpose
we use the sas task ` evselect ' with suitable parameters for the mos 1/2 and pn cameras .
the background spectrum of the snr was also taken from an elliptical region located within the central ccd of the mos cameras .
these regions were defined exclusively on the central chip to avoid having to account for chip - to - chip variations .
the spectral analysis was performed with the xspec package ( arnaud , 1996 ) .
the x - ray spectrum of the snr is shown in fig .
2a ( left panel ) .
the global spectrum has two components .
the diffuse x - ray emission of the snr ( dominant in the energy range of [email protected] kev ) , plus the contribution of point - like sources ( mainly contributing in energy range of [email protected] kev ) . in order to obtain the x - ray spectrum of the snr
, we have excluded all the point - like sources taking circular regions with a radius of 15@xmath26 . the extracted epic mos1/2 and pn spectra
were grouped with a minimum of 30 counts per spectral bin , and the @xmath27 statistics was used .
ancillary response files ( arfs ) and redistribution matrix files ( rmfs ) were calculated .
the x - ray emission of the snr peaks in the 0.53.0 kev energy range and it is clearly dominated by thermal emission .
thus , we used a pshock model affected by an absorption interstellar model ( phabs ; balucinska - church and mccammon 1992 ) to fit it .
the x - ray parameters for the best - fit of the diffuse emission are given in table 1 .
[ cols="<,<,^ " , ] normalization is defined as 10@xmath28/4@xmath29d@xmath30@xmath31 , where @xmath32 is distance in [ cm ] , n@xmath33 is the hydrogen density [ cm @xmath34 , @xmath35 is the electron density [ @xmath9 ] , and @xmath37 is the volume [ @xmath38 .
the flux in the three energy ranges 0.5 - 1.0 , 1.0 - 2.0 and 2.0 - 3.0 kev , is absorption - corrected .
values in parentheses are the single parameter 90% confidence interval .
the abundance parameter is given relative to the solar values of anders & grevesse ( 1989 ) .
since the point - like source is surrounded by diffuse x - ray emission of the snr , we extracted its spectrum from a circular region with a radius of 6 arcsec ( ee@xmath2 50% ) and grouped it with a minimun of 18 counts per spectral bin .
the background spectrum was estimated from an annular region with radii of 8 and 25 arcsec .
the spectrum of the source is shown in figure 2 ( right panel ) . in order to study the x - ray properties of this compact source we fitted its spectrum with several spectral models . however , due to the low - photons statistic , the most representative one is a simple power - law ( pl ) model that yields a neutral hydrogen absorption column @xmath1=0.55 @xmath140.1 @xmath5 , an index @xmath39=4.3 @xmath140.7 and a normalization of 6.1(@xmath142.5)@xmath310@xmath40 @xmath5 .
the absorption corrected x - ray flux is f@xmath41=9.9 ( @xmath140.1 ) @xmath310@xmath28 erg s@xmath13 @xmath5 in the [email protected] kev band .
the fit is acceptable in terms of the minimum @xmath27 ( @xmath42= 1.1 for 24 d.o.f ) .
as can be seen , this model provides a good value of @xmath27 and the value of @xmath1 is similar to the value obtained for the snr .
this fact , supports the possibility that the point source detected at the geometrical centre of the snr has a real physical connexion with g296.8 - 0.3 .
in addition , we found no significant pulsed signal with a period greater than twice the read - out time of the epic - pn camera in the ff mode ( 73.3 ms ) , which corresponds to a nyquist limit of 0.146 s.
using radio observations performed with the most radiotelescope at 0.843 ghz , and infrared spitzer - mips ( rieke et al .
2004 ) observations of the snr , we have investigated the positional correlation between all the detected emissions .
the mips basic calibrated data ( bcd ) were downloaded from the spitzer archive .
these images were re - processed with the regular mips pipeline ( version s18.7.0 ) , and then mosaicked using mopex ( version 18.3.1 ) and the standard mips 24@xmath11 m mosaic pipeline . in fig .
3 , we show the mips image at 24 @xmath11 m with the radio contours superimposed . the 24 @xmath11 m emission
is strongly correlated with the radio shells , with several faint filaments coincident on the northwest and southeast parts of the snr .
this emission is generally interpreted as thermal emission from dust grains that have been swept up and shock - heated by the supernova blast wave ( tappe et al .
the extraction regions used for computing the infrared fluxes on the shell - like boundaries are indicated in green .
the mid - infrared fluxes at 24 @xmath11 m of the northwest and southeast regions are [email protected] jy and [email protected] jy , respectively .
the x - ray emission detected on the northwest radio shell follows the infrared emission very well , which demonstrates the connection between the x - ray emitting plasma and the heated dust grains .
this result seems to indicate that the enhanced x - ray emission is caused by the expansion through a dense ism with a density gradient toward the northwest side of the remnant . under these conditions , it is possible to roughly make quantitative estimates of the swept - up ism dust mass in the northwest and southeast rims using the formulae introduced by whittet ( 2003 ) , @xmath43,\ ] ] in this equation we have adopted a density @xmath44 for silicate / graphite grains , a distance @xmath45 for the snr , a dust temperature @xmath46 @xmath2 100 k at 24@xmath11 m , and an average ratio of grain radius @xmath47 over emissivity @xmath48 of @xmath49 ( see , tappe et al .
2006 ) . using the infrared flux densities @xmath50
computed at 24@xmath11 m and the planck function @xmath51(@xmath52 ) at 24 @xmath11 m , we derived a dust mass of @xmath2 0.010 @xmath53 and @xmath2 0.006 @xmath53 for the northwest and southeast rims , respectively .
1 is an approximation assuming spherical dust grains of uniform size , composition , and in thermal equilibrium . in order to inspect if there exists some infrared counterpart to 2xmmi j115836.1 - 623516 , we show in fig .
4 a _ spitzer_/irac deep image of the central region of g296.9 - 0.3 in the 3.6 @xmath11 m band . the extraction circle with a radius of 6 arcsec , used in the x - ray spectral analysis , is overplotted .
as can be seen only one infrared source lies near the edge of the encircled region .
the position of the x - ray source is given in the 2xmm catalogue ( watson et al .
, 2009 ) with a relative precision of one arcsec . however , it is well - known from cross - correlations with other astrometric catalogues that the absolute accuracy in the position of _ xmm - newton _ sources ranges between 5 and 10 arcsec ( della ceca et al .
2004 , lpez - santiago et al .
2007 , combi et al . 2011 ) .
therefore , we can not assure if the infrared source observed near the edge of the encircle region is physically associated or not with the x - ray object .
m image of g296.8 - 0.3 with the radio contours at 843 mhz ( in white ) overlaid .
the extraction regions used for computing the infrared fluxes are indicated in green color.,width=317 ] m band .
we have overplotted the 6 arcsec extraction region used for the spectral x - ray analysis.,width=302 ]
two possible evolutionary scenarios to explain the unusual morphology observed at radio frequencies in snr [email protected] , were initially studied by gaensler et al .
these authors suggested that the biannular appearance is induced by axial symmetry in the progenitor wind , or the snr morphology resulted from the inhomogeneous ism into which it is expanding .
they conclude that the second possibility is more probable and that [email protected] seems consistent with being the remnant of a single explosion , where the unusual linear feature running north - south through the remnant may represent a low density tunnel which has been re - energized by an encounter with the sn shock .
it is clear that g296.8 - 0.3 is found in a complex area in the scutum - crux arm of our galaxy , where the ism is particularly inhomogeneous and density variations in the pre - shock medium are present .
the radio , infrared and x - ray observations of g296.8 - 0.3 here analyzed , can provide crucial information about the origin and evolution of the snr , as well as regarding the age , energetics , ambient conditions and the presence of heated dust .
with all this information in mind , we could outline a possible framework that allows to explain the characteristics of the emissions observed from the snr .
the xmm observations reveal that there is diffuse x - ray emission in the interior of the snr well - correlated with the unusual rectangular strip running through its center seen at radio frequencies , a bright soft shell - like feature coincident with the internal northwest radio shell , and several hard point - like sources ( possibly background agns ) .
moreover , the 24 @xmath11 m observations show two limb - brightened shell - like structures on the northwest and southeast parts of the snr , and faint filaments strongly correlated with the radio shells .
the emission measure ( em ) computed for the global region of the snr can allow us to estimate the corresponding density of the x - ray emitting gas . from the x - ray image
, we can roughly assume that the x - ray emission fills an ellipsoid with radii of @xmath2 3 @xmath3 4 arcmin and estimate the volume @xmath37 of the x - ray emitting plasma .
at a distance of 9 kpc , the snr defines an x - ray emitting volume @xmath54= 6.1@xmath310@xmath55 @xmath56 . based on the em determined by the spectral fitting ( see table 1 )
, we can estimate the electron density of the plasma , @xmath57 , by @xmath57=@xmath58 , which results in @xmath57@xmath2 0.18 @xmath9 . in this case , the number density of the nucleons was simply assumed to be the same as that of electrons .
the age @xmath59 is determined from the ionization timescale , @xmath60 , by @xmath59=@xmath60/@xmath57 . therefore , the elapsed time after the plasma was heated is @xmath59 @xmath2 1.0@xmath310@xmath61 yr .
this result shows that [email protected] is a middle - aged snr .
assuming that the snr is in the adiabatic ( sedov - taylor ) phase , gaensler et al .
( 1998 ) obtained several physical parameters for the object .
taking into account a kinetic energy of the initial explosion of @xmath62= 10@xmath63 erg , and that the snr expands in an ism with a density of 0.2 @xmath9 , these authors computed an age of @xmath59=(10@xmath142)@xmath310@xmath64 yr for g296.8 - 0.3 .
this value agrees very well with the age obtained by us above using the x - ray information . concerning the point - like x - ray source located close to the geometrical center of the radio structure , we see no significant variability that disfavors an accreting binary origin , a soft thermal spectrum that eliminates a background active nucleus , lack of radio counterpart , and absence of a surrounding pulsar wind nebula . at first sight
, we can see that the source displays some characteristics of the so - called cco ( see pavlov et al .
2004 , for a review ) , a new population of isolated neutron stars ( nss ) with clear differences from isolated rotation - powered pulsars and accretion - powered x - ray pulsars in close binary systems .
the nature of these objects is still unclear .
it is thought that the x - ray emission from ccos is generally due to the thermal cooling of the ns ( e.g zavlin , trumper & pavlov 1999 ) , with typical temperatures of a few 10@xmath65 k , as inferred from their thermal - like spectra .
they have x - ray luminosities ( @xmath66 ) in the range of 10@xmath67 - 10@xmath68 erg s@xmath13 and display x - ray spectra characterized by a blackbody model with temperatures ( @xmath7 ) in the range of 0.2 - 0.5 kev or a power - law model with very steep index @xmath39 ( see pavlov et al .
halpern & gotthelf ( 2010 ) have recently suggested that these objects could be weakly magnetized nss ( @xmath69 g ) , i.e. , a kind of `` anti - magnetars '' . in order to check the characteristics of 2xmmi j115836.1 - 623516 , we computed its @xmath66 and spin - down luminosity @xmath70 , to compare with other well - known ccos ( pavlov et al .
2003 ) . adopting a mean distance of 9 kpc and a total unabsorbed x - ray flux of @xmath71=9.9@xmath310@xmath28 ergs @xmath5 s@xmath13
, we obtain an unabsorbed luminosity @xmath66= 1.0@xmath310@xmath67 ergs s@xmath13 .
a rough estimate of the spin - down luminosity can be derived using the empirical formula by seward & wang ( 1988 ) , log @xmath66 ( ergs s@xmath13)= 1.39 log @xmath70 - 16.6 , which gives @xmath70= 3.9@xmath310@xmath72 ergs s@xmath13 .
@xmath66 lies within the range suggested by pavlov et al .
( 2004 ) and by halpern & gotthelf ( 2010 ) for cco objects .
the second quantity , @xmath70 , falls below the empirical threshold for generating bright wind nebulae of @xmath73 @xmath74 4@xmath310@xmath75 ergs s@xmath13 .
these results suggest that the system g296.8 - 0.3/2xmmi j115836.1 - 623516 is a thermal snr with , possibly , a nondetected ns .
plausible reasons for the nondetection of a ns are the low - photon statistic , a short rotation period or unfavorable geometrical conditions .
we have analyzed radio , infrared and x - ray data of the snr g296.8 - 0.3 to investigate the origin of the radiative process involved in the generation of the emission observed .
the diffuse x - ray emission is clearly correlated with the unusual rectangular strip running through its center seen at radio frequencies , where we found a region with significant lower density , and a bright x - ray shell - like feature at the northwest part of the snr coincident with the internal boundary of the radio shell .
the 24 @xmath11 m observations show two limb - brightened shell - like structures on the northwest and southeast parts of the snr , and faint filaments strongly correlated with the radio shells .
the spectral study confirms that the x - ray diffuse emission is thermal and the column density of the snr is high ( @[email protected]@xmath310@xmath4 @xmath5 ) supporting distant location ( @xmath769 kpc ) for the snr .
in addition , a compact x - ray source was also detected close to the geometrical center of the snr .
the object presents some characteristics of the ccos , and the neutral hydrogen absorption column @xmath1 is consistent with that of the snr .
although these results support a physical connexion with the snr , high - resolution x - ray observations carry out with the chandra satellite are necessary to better understand the nature of the x - ray source .
the authors acknowledge support by dgi of the spanish ministerio de educacin y ciencia under grants aya2010 - 21782-c03 - 03 , feder funds , plan andaluz de investigacin desarrollo e innovacin ( paidi ) of junta de andaluca as research group fqm-322 and the excellence fund fqm-5418 .
j.a.c . and j.f.a.c .
are researchers of conicet .
j.f.a.c was supported by grant pict 2007 - 02177 ( secyt ) .
j.a.c was supported by grant pict 07 - 00848 bid 1728/oc - ar ( anpcyt ) and pip 2010 - 0078 ( conicet ) .
j.l.s . acknowledges support by the spanish ministerio de innovacin y tecnologa under grant aya2008 - 06423-c03 - 03 . | we report xmm-_newton _ observations of the galactic supernova remnant [email protected] , together with complementary radio and infrared data .
the spatial and spectral properties of the x - ray emission , detected towards [email protected] , was investigated in order to explore the possible evolutionary scenarios and the physical connexion with its unusual morphology detected at radio frequencies .
[email protected] displays diffuse x - ray emission correlated with the peculiar radio morphology detected in the interior of the remnant and with the shell - like radio structure observed to the northwest side of the object .
the x - ray emission peaks in the soft / medium energy range ( 0.5 - 3.0 kev ) .
the x - ray spectral analysis confirms that the column density is high ( @[email protected]@xmath310@xmath4 @xmath5 ) which supports a distant location ( d@xmath69 kpc ) for the snr .
its x - ray spectrum can be well represented by a thermal ( pshock ) model , with @xmath7 @xmath2 0.86 kev , an ionization timescale of 6.1@xmath310@xmath8 @xmath9 s , and low abundance ( @xmath2 0.12 @xmath10 ) .
the 24 @xmath11 m observations show shell - like emission correlated with part of the northwest and southeast boundaries of the snr .
in addition a point - like x - ray source is also detected close to the geometrical center of the radio snr .
the object presents some characteristics of the so - called compact central objects ( cco ) .
its x - ray spectrum is consistent with those found at other ccos and the value of @xmath1 is consistent with that of [email protected] , which suggests a physical connexion with the snr . | 1110.1272 |
two of the most intriguing questions in particle physics are the the ewsb mechanism and the origin of fermion masses .
although the sm remains a successful theory when compared with all the available data , it lacks predictability in the higgs sector , which determines the masses of gauge bosons , as well as of fermions through _ ad hoc _ yukawa couplings .
this suggests the possibility that new physics beyond the sm might be associated with either of these questions . in general , the energy scales and dynamics behind the ewsb sector and
the fermion masses may be unrelated . in order to avoid fine - tuning ,
the scale associated with ewsb can not be much higher than a few tev , whereas the scales where light fermion masses are generated could be much higher .
if the mechanism responsible for the breaking of the electroweak symmetry involves some new strong dynamics , deviations from the sm might be observable in low energy signals even at energies much smaller than the scale of new physics @xmath0 . reaching this new frontier by direct observation of new physical states or even of tree - level effects in the couplings of sm particles ,
may require not only very large energies but also some previous knowledge of what ( and what not ) to expect .
thus , low energy measurements might be of paramount importance in planning experiments and search strategies at high energy machines . among these low energy signals are electroweak measurements such as those at lep and the tevatron .
on the other hand , processes involving flavor changing neutral currents ( fcnc ) can play a complementary role , since the fact that these processes are largely suppressed or forbidden in the sm may compensate the suppression by factors of @xmath1 ( with @xmath2 the low energy scale , e.g. @xmath3 , @xmath4 , etc . ) . here
we address the potential of rare @xmath5 and @xmath6 decays as a complement to other low energy measurements in constraining models where strong dynamics is associated to either the ewsb sector and/or the origin of fermion masses . in the absence of a completely satisfactory theory of dynamical symmetry breaking and fermion masses , it is convenient to carry out a model - independent analysis that makes maximum use of the known properties of the electroweak interactions .
this is the case with the ewsb sector , where an effective lagrangian approach allows us to parameterize the effects of the new strong dynamics in very much the same way chiral perturbation theory parameterizes low energy qcd . on the other hand , the effects from fermion mass generation
can also be addressed by a general operator analysis . however , in addition , most theories predict the existence of relatively light states ( scalars , pseudo - goldstone bosons , etc . ) which generally couple to mass in one way or another . to exemplify the effects of such states ( which can not be integrated out ) we work with a particular set of models known as topcolor - assisted technicolor ( tatc ) .
this provides a current example of how strong dynamics model building deals with the large top - quark mass and illustrates the distinct low energy phenomenology emerging from non - standard ewsb scenarios .
in the absence of a light higgs boson the symmetry breaking sector is represented by a non - renormalizable effective lagrangian corresponding to the non - linear realization of the @xmath7 model .
the essential feature is the spontaneous breaking of the global symmetry @xmath8 . to leading order the interactions involving the goldstone bosons associated with this mechanism and the gauge fields are described by the effective lagrangian @xcite @xmath9 + \frac{v^2}{4}{{\rm tr}}\left[d_\mu u^\dagger d^\mu u\right ] , \label{lo}\ ] ] where @xmath10 and @xmath11 $ ] are the the @xmath12 and @xmath13 field strengths respectively , the electroweak scale is @xmath14 gev and the goldstone bosons enter through the matrices @xmath15 .
the covariant derivative acting on @xmath16 is given by @xmath17 . to this order
there are no free parameters once the gauge boson masses are fixed .
the dependence on the dynamics underlying the strong symmetry breaking sector appears at next to leading order . to this order ,
a complete set of operators includes one operator of dimension two and nineteen operators of dimension four @xcite .
the effective lagrangian to next to leading order in the basis of ref .
@xcite is given by @xmath18 where @xmath19 is a dimension two custodial - symmetry violating term absent in the heavy higgs limit of the sm . if we restrict ourselves to cp invariant structures , there remain fifteen operators of dimension four .
the coefficients of some of these operators are constrained by low energy observables .
for instance precision electroweak observables constrain the coefficient of @xmath19 , which gives a contribution to the electroweak parameter @xmath20 .
the @xmath21 limit requires @xmath22 the combinations @xmath23 and @xmath24 contribute to the electroweak parameters @xmath25 and @xmath26 .
for instance , the constraint on @xmath25 translates into @xmath27 in addition , the coefficients @xmath28 , @xmath29 , @xmath30 and @xmath31 modify the triple gauge - boson couplings ( tgc ) and will be probed at lepii and the tevatron at the few percent level @xcite .
the remaining operators contribute to oblique corrections only to one loop and , in some cases , only starting at two loops . to the last group belong @xmath32 and @xmath33 given that their contributions to the gauge boson two - point functions only affect the longitudinal piece of the propagators .
of particular interest is the operator @xmath32 defined by @xcite @xmath34 , \label{defo11}\ ] ] with @xmath35 and the covariant derivative acting on @xmath36 defined by @xmath37 $ ] .
the equations of motion for the @xmath38 field strength imply @xcite @xmath39 where the @xmath13 current is @xmath40 , @xmath41 denote the left - handed fermion doublets .
the dominant effect appears in the quark sector due to the presence of terms proportional to @xmath42 .
after the quark fields are rotated to the mass eigenstate basis , the operator @xmath32 can be written as @xcite @xmath43 where @xmath44 , the @xmath45 are cabibbo - kobayashi - maskawa ( ckm ) matrix elements and the dots stand for terms suppressed by small fermion masses . from the above discussion
we see that the leading effects of the ewsb sector in fcnc processes are coming from the insertion of anomalous tgc vertices and four - fermion operators like ( [ ffo11 ] ) . in the rest of this section , we review the status and future impact of these constraints on the symmetry breaking sector .
the effects of the four - fermion operators in ( [ ffo11 ] ) in rare b and k decays were considered in ref .
the loop insertion will result in contributions to several fcnc processes , that are controlled by both the coefficient @xmath46 of the effective lagrangian ( [ lnlo ] ) as well as by the high energy scale @xmath0 .
to one loop , only one parameter is needed , namely @xmath47 this parameter also governs the contributions of ( [ ffo11 ] ) to other neutral processes , both flavor changing and flavor conserving .
for instance , the @xmath48 term in ( [ ffo11 ] ) gives a contribution to @xmath49 , whereas the terms like @xmath50 appear in @xmath51 mixing @xcite .
thus the measurements of @xmath52 and the rate of @xmath5 mixing ( together with all other ckm information ) can be used to derive a bound on @xmath53 .
although the bound carries some uncertainty mainly associated with ckm quantities like @xmath54 and @xmath55 , we will take it to be , approximately @xcite @xmath56 next , we use this as the allowed range for @xmath53 in order to explore the possible impact of this physics in rare @xmath5 and @xmath6 decays .
the one - loop insertion of the terms @xmath57 induces new contributions to various fcnc vertices in @xmath5 decays ( the first two terms in ( [ rareo11 ] ) ) , as well as in @xmath6 decays ( third term in ( [ rareo11 ] ) ) .
first , let us consider @xmath58 processes leading , for instance , to the inclusive @xmath59 , since this rate has been recently measured @xcite .
the one - loop insertion of the operator @xmath32 does not give a contribution to these processes given that it does not mix with the operator @xmath60 responsible for the on - shell photon amplitude .
mixing only occurs at two loops , when qcd corrections are taken into account . as a result the effect , in all @xmath58 transitions is expected to be only a few percent of the sm branching ratios @xcite . on the other hand ,
the off - shell amplitudes for photons , @xmath61 s and gluons are non - zero at one loop .
they generate contributions to processes such as @xmath62 , @xmath63 , @xmath64 ; as well as to similar rare kaon decays like @xmath65 , etc . in order to asses
the potential effects we define @xmath66 which is plotted in fig . 1 as a function of the parameter @xmath53 defined in ( [ ydef ] ) , for the allowed range of @xmath53 ( [ ybound ] ) .
analogously , we can define the ratio @xmath67 , which tracks the effects in @xmath68 decays ; whereas the contribution to gluon penguin processes such as @xmath69 is represented by the ratio @xmath70 . as it is clear from fig .
[ fig1 ] , the effects of the operator @xmath32 are very similar in all three types of @xmath5 decays .
we see that , even with the @xmath52 and @xmath71 mixing constraints , large deviations from the sm predictions for these modes are possible .
the current experimental bounds on these processes are still not binding on @xmath53 .
however , sensitivity to sm branching ratios will be reached in the next round of experiments at the various @xmath5 factories at cornell , kek , slac and fermilab .
the distinct feature of this effect is that no significant deviation is expected in @xmath72 , even when large deviations are observed in all the other modes .
the effects are very similar in rare @xmath6 decays such as @xmath73 and @xmath74 , etc . in fig .
2 we plot @xmath75 , a quantity analogous to @xmath76 in ( [ defrat ] ) .
again , large effects of up to factors of @xmath77 deviations , are allowed .
the recently reported @xcite observation of one event in @xmath73 roughly translates into @xmath78 , which is still not constraining .
although in this model - independent approach we can not , as a matter of principle , calculate the size of the coefficients @xmath79 , we can use general arguments to estimate their approximate value .
using naive dimensional analysis @xcite we have @xmath80 with the scale of new physics obeying @xmath81 .
for instance , taking @xmath82 , one would obtain @xmath83 . on the other hand , if @xmath84 , one has @xmath85 . in any case , these are meant to be order of magnitude estimates .
therefore , the experimental relevance of the effect strongly depends on details of the dynamics we are not able to compute in a model - independent fashion .
finally , we should note that rare @xmath5 and @xmath6 decays are the most sensitive signals for this effect .
this is due to the fact that four - lepton operators are suppressed by the lepton masses , and that @xmath32 does not mix quarks and leptons . imposing @xmath86 and @xmath87 conservation , the most general form of the @xmath88 ( @xmath89 ) couplings
can be written as @xcite @xmath90 with the conventional choices being @xmath91 and @xmath92 @xcite . in principle , there are six free parameters . making contact with the electroweak lagrangian ( [ lnlo ] )
, these parameters can be expressed in terms of the next - to - leading order coefficients @xcite @xmath93 and @xmath31 .
conservation of the electromagnetic charge implies @xmath94 .
furthermore , to this order in the energy expansion ( [ lnlo ] ) @xmath95 . then we are left with @xmath96 , @xmath97 and @xmath98 .
finally , when considering rare @xmath5 and @xmath6 decays , we can neglect the contribution of @xmath97 since it will be suppressed by powers of the small external momenta over @xmath99 .
thus , in this simplistic approach , there are only two parameters relevant at very low energies .
the sm predicts @xmath100 .
the effects of anomalous tgc have been previously studied in the literature @xcite . however
, this hierarchical approach to the couplings has not been the one used in the various analyses and a more comprehensive study is needed . the experiments at lep ii and the next tevatron run are going to be sensitive to deviations from the sm prediction at the @xmath101 level @xcite .
effects of this size might be also observed in rare @xmath5 and @xmath6 decays .
for instance , @xmath102 can produce enhancements in the branching ratios of @xmath103 decay modes of up to @xmath104 @xcite . in the near future ,
@xmath5 factory experiments will have sensitivity to these processes at the sm level , turning these low energy measurements into an excellent complement of direct probes of the tgc .
up to now , we have only considered the effects of the dynamics associated with the ewsb .
these are encoded in the effective lagrangian ( [ lnlo ] ) , which only involves the goldstone boson and gauge fields . additionally , it is possible that the new strong dynamics may also affect some or all fermions .
we first comment on the effective lagrangian approach for non - sm couplings of fermions to gauge bosons , and then examine the effects of a prototypical class of theories ( topcolor ) where the dynamical generation of fermion masses imply the existence of relatively light new states .
the effects of new dynamics on the couplings of fermions with the sm gauge bosons can be , in principle , also studied in an effective lagrangian approach .
for instance , if in analogy with the situation in qcd , fermion masses are dynamically generated in association with ewsb , residual interactions of fermions with goldstone bosons could be important @xcite if the @xmath105 .
thus residual , non - universal interactions of the third generation quarks with gauge bosons could carry interesting information about both the origin of the top quark mass and ewsb . in a very general parameterization
, the anomalous couplings of third generation quarks can be written as @xmath106 where the parameters @xmath107 , @xmath108 contain the residual , non - universal effects associated with the new dynamics , perhaps responsible for the large top quark mass . then ,
if we assume that the new couplings are cp conserving , there are six new parameters .
they are constrained at low energies by a variety of experimental information , mostly from electroweak precision measurements and the rate of @xmath72 .
several simplifications are usually made in order to reduce the number of free parameters .
for instance , in most of the literature , it is assumed that @xmath109 @xcite .
a stringent bound on the right - handed charged coupling is obtained from @xmath72 @xcite : @xmath110 .
the bounds obtained on a particular coupling from electroweak observables such as @xmath25 , @xmath20 , @xmath26 and @xmath52 generally strongly depend on assumptions about the other couplings .
for example , if @xmath111 , then the combination @xmath112 is strongly constrained since it contributes to @xmath20 . on the other hand , if @xmath113 , then @xmath114 @xcite since it is the only ( linear ) contribution to @xmath20 .
thus , although in general most parameters are confined to a few percent , some of them are allowed to be as large as @xmath115 under certain conditions .
this `` model - dependent '' situation requires more experimental information . a global analysis of the effects of the couplings of eqn .
( [ lfer ] ) in rare @xmath5 and @xmath6 processes such as @xmath103 , @xmath65 , etc . may help disentangle the various possible effects and perhaps will give constraints that may be of importance in interpreting data from higher energy experiments @xcite .
the description of the residual effects of strong dynamics at low energies on fermion couplings by using ( [ lfer ] ) corresponds to cases where the states associated with the new physics are heavy compared to the weak scale .
thus , integrating out the heavy states , leaves us with effective couplings which might be generated at tree level or through loops in the full theory .
however , most theories in which electroweak symmetry and/or fermion masses have a dynamical origin also contain states with masses comparable to the weak scale .
such is the case , for instance , in technicolor models where the breaking of large chiral symmetries imply the presence of pseudo - goldstone bosons with masses of at most a few hundred gev .
it is also the case in topcolor - assisted technicolor ( tatc ) models @xcite , where a top - condensation mechanism generated by the topcolor interactions is responsible for the large dynamical top quark mass , whereas technicolor breaks the electroweak symmetry giving ( most of ) the @xmath116 and @xmath61 masses .
the tatc scenario is designed to relief the problems of extended technicolor ( etc ) in generating a heavy top @xcite .
although the new gauge bosons associated with the tatc gauge group are heavier than @xmath117tev , the presence of several scalar and pseudo - scalar states with masses in the few - hundred gev range , forces us to take these into account directly in our calculations . from the point of view of their impact in low energy observables ,
the most important of these states are the top - pions @xmath118 , the triplet of goldstone bosons associated with the breaking of the top chiral symmetry .
since top condensation does not fully break the electroweak symmetry ( @xmath119 ) , after mixing with the techni - pions , there will be a triplet of physical top - pions in the spectrum , with a coupling to third generation quarks given by @xmath120 they acquire masses of a few hundred gev due to explicit etc quark mass terms .
additionally , in most models there are scalar and pseudo - scalar bound states due to the strong ( although sub - critical ) effective coupling of right - handed @xmath121-quarks .
the closer the effective couplings are from criticality , the lighter these bound states tend to be .
the spectrum and properties of these states , unlike those of top - pions , are not determined by model - independent features of the symmetry breaking pattern but depend on details of the model .
finally , in all tatc models there will be pseudo - goldstone bosons from the breaking of techni - fermion chiral symmetries .
however , their couplings to third generation quarks are reduced with respect to ( [ tpcop ] ) by @xmath122 , where @xmath123 is a small etc mass of the order of @xmath117gev .
the presence of the relatively light top - pions , as well as the additional bound states , imposes severe constraints on topcolor models due to their potential loop effects in low energy observables , most notably @xmath52 and rare @xmath5 and @xmath6 decays .
[ [ top - pion - effects - in - r_b ] ] top - pion effects in @xmath52 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + : the one - loop contributions of top - pions to the @xmath124 process were studied in ref .
there it was shown that they shift @xmath52 negatively by an amount controlled by @xmath125 and @xmath126 .
for instance , for @xmath127gev the correction is about @xmath128 for @xmath129gev , and top - pions with masses in the expected @xmath130gev range give unacceptably large deviations .
this value of the top - pion decay constant is obtained by using the pagels - stokar formula , which gives @xmath126 a logarithmic dependence on the topcolor energy scale , chosen here to be a few tev .
potentially cancelling contributions by other states , such as the scalar and pseudo - scalar bound states , topcolor vector and axial - vector mesons , etc .
, are either of the wrong sign or not large enough
. possible ways out of this constraint are : larger top - pion masses or larger values of @xmath126 .
the larger @xmath126 is , the smaller the coupling , and the top - pions are more goldstone - boson - like . for @xmath131gev , for instance , the shift of @xmath52 is well within the experimentally allowed region even for @xmath132gev .
however , in order to obtain such an enhancement in the decay constant we must either assume large corrections to the pagels - stokar expression or introduce new and exotic fermion states .
[ [ rare - b - and - k - decays ] ] rare @xmath5 and @xmath6 decays + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + : the top - pions and other scalar states , give one - loop contributions to fcnc processes .
these depend not only on @xmath126 and @xmath125 but typically also on one or more elements of the quark rotation matrices necessary to diagonalize the quark yukawa couplings .
the contributions of top - pions , as well as `` b - pions '' ( scalar and pseudo - scalar bound states in models where @xmath133 couples to the topcolor interaction ) to @xmath72 depend on @xmath134 , the @xmath135 element in the left or right down rotation matrix .
furthermore , the two contributions tend to cancel .
thus , the freedom in this model - dependent aspects of the prediction makes it possible to have quite low masses and still satisfy the bound from the experimental measurement of @xmath59 @xcite .
the situation changes drastically in @xmath103 processes , where the cancellations are much less efficient .
although experiments have not yet reached sensitivity to sm branching ratios @xcite , it will be soon achieved at both hadron and lepton @xmath5 factories .
as an example , we plot in fig .
3 the br@xmath136 as a function of the top - pion mass with no other contributions , for @xmath137gev .
the @xmath72 constraint is in this case @xmath138 .
however , one can see that , even for heavier top - pions the effect can still be a @xmath139 enhancement over the sm prediction of @xmath140 . on the other hand , in the presence of a @xmath141gev charged b - pion the curve changes little , but the @xmath72 bound is now @xmath142 . finally , to compare the potential of these fcnc transitions with the @xmath52 constraints , let us say that if we take @xmath131gev ( which avoids conflict with @xmath52 measurements ) , then the effect of a @xmath141gev top - pion in @xmath103 is still an enhancement of more than @xmath143 with respect to sm expectations .
thus , the observation of these modes will further constrain topcolor models beyond the @xmath52 bounds .
we expect similar effects due to top - pions and/or b - pions to be present in kaon processes such as @xmath73 .
we have seen that a complete , model - independent analysis of the effects of strong dynamics in rare @xmath5 and @xmath6 decays could shed light on the nature of the ewsb mechanism and the origin of fermion masses .
the signals are also likely to be important in models where relatively light scalars couple strongly to mass , like in the case of tatc . in most cases
, the next round of experiments will have sensitivity to sm branching ratios .
this will be the case , for instance , for the tevatron experiments , as well the kek and slac @xmath5 factories in the @xmath144 modes .
it will also be the situation in the next generation of kaon experiments for @xmath73 and @xmath74 .
the amount and variety of experimental information from these processes is such that suggests a parallel to the role of electroweak measurements at the @xmath61 pole as not only a constraint on new physics sources but also as guidance in the searches to be carried out at high energy machines such as the tevatron in run ii , the lhc and eventually the nlc and/or the muon collider .
it is possible to imagine a scenario where deviations from the sm in @xmath5 and/or @xmath6 decays point to a particular source , e.g. corrections to goldstone boson propagators given by @xmath32 , anomalous tgc or anomalous couplings of third generation quarks to gauge bosons as in ( [ lfer ] ) .
the nature of the deviation might dictate the road to follow at high energies .
as an example , if the source of an effect is in one the top quark couplings @xmath145 , there would be a strong case for a lepton collider running at @xmath146 threshold .
other scenarios may not be so clear , and may require a comprehensive and careful analysis of all the data to come ( including issues like hadronic uncertainties in @xmath5 decays ) .
this , however , constitutes a very well defined research program .
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we focus on models involving new strong dynamics and show that transitions involving flavor changing neutral currents ( fcnc ) play an important role in disentangling the physics in these scenarios . in a model - independent approach to the ewsb sector , the information from rare decays is complementary to precision electroweak observables in bounding the contributions to the effective lagrangian . we compare the pattern of deviations from the standard model ( sm ) that results from these sources , with the deviations associated with the mechanism for generating fermion masses . madph-98 - 1039 + | hep-ph9802232 |
the compas group was supported in part by the russian foundation for basic research grants rfbr-98 - 07 - 90381 and rfbr-01 - 07 - 90392 , k.k .
is in part supported by the u.s .
contract de - fg-02 - 91er40688-task a , e.m .
is a visiting fellow of the belgian fnrs .
we thank professor jean - eudes augustin for the hospitality at lpnhe - universit paris 6 , where part of this work was done .
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_ [ compete collaboration ] : 2001 , in the _ proceedings of the 6th workshop on non - perturbative qcd _ ,
paris , france , 5 - 9 jun 2001 [ arxiv : hep - ph/0111025 ] . | we present predictions on the total cross sections and on the ratio of the real part to the imaginary part of the elastic amplitude ( @xmath0 parameter ) for present and future @xmath1 and @xmath2 colliders , and on total cross sections for @xmath3 hadrons at cosmic - ray energies and for @xmath4 hadrons up to @xmath5 tev .
these predictions are based on a study of many possible analytic parametrisations and invoke the current hadronic dataset at @xmath6 .
the uncertainties on total cross sections , including the systematic theoretical errors , reach @xmath7 at rhic , @xmath8 at the tevatron , and @xmath9 at the lhc , whereas those on the @xmath0 parameter are respectively @xmath9 , @xmath10 , and @xmath11 . * forward observables at + rhic , the tevatron run ii and the lhc * + + j. r. cudell , v. v. ezhela , p. gauron , k. kang , yu .
v. kuyanov@xmath12 , s. b. lugovsky@xmath12 , e. martynov@xmath13 , b. nicolescu@xmath14 , e. a. razuvaev@xmath12 , n. p. tkachenko@xmath12 + + compete collaboration this report is based on ref .
@xcite , which constitutes the conclusion of an exhaustive study @xcite of analytic parametrisations of soft forward data at @xmath6 .
as explained in v. v. ezhela s contribution to these proceedings , this study has three main purposes .
first of all , it helps maintain the dataset of cross sections and @xmath0 parameters available to the community .
secondly , it enables us to decide which models are the best , and in which region of @xmath15 .
finally , and this will be the main object of this report , it enables us to make predictions based on a multitude of models , and on all available data .
the dataset of this study includes all measured total cross sections and ratios of the real part to the imaginary part of the elastic amplitude ( @xmath16 parameter ) for the scattering of @xmath17 @xmath18 , @xmath19 , @xmath20 , and total cross sections for @xmath21 , @xmath22 and @xmath23 .
compared with the 2002 review of particle properties dataset @xcite , it includes the latest zeus points@xcite on total cross sections , as well as cosmic ray measurements @xcite .
the number of points of each sub - sample of the dataset is given in table [ table1 ] .
+ + lcccc & number & db02z & db02z & db02z + sample & of points & & @xmath24cdf & @xmath24e710/e811 + total & & 0.965 & 0.964 & 0.951 + + @xmath25 & 111 & 0.84 & 0.90 & 0.90 + @xmath26 & 57@xmath2459 & 1.15 & 1.12 & 1.05 + @xmath27 & 50 & 0.71 & 0.71 & 0.71 + @xmath28 & 95 & 0.96 & 0.96 & 0.96 + @xmath29 & 40 & 0.71 & 0.71 & 0.71 + @xmath30 & 63 & 0.62 & 0.62 & 0.61 + @xmath31 & 9 & 0.38 & 0.38 & 0.38 + @xmath32 & 37 & 0.58 & 0.58 & 0.58 + @xmath33 & 38 & 0.64 & 0.64 & 0.63 + + @xmath25 & 64 & 1.83 & 1.83 & 1.80 + @xmath26 & 11 & 0.52 & 0.52 & 0.53 + @xmath27 & 8 & 1.50 & 1.52 & 1.46 + @xmath28 & 30 & 1.10 & 1.09 & 1.14 + @xmath29 & 10 & 1.07 & 1.10 & 0.98 + @xmath30 & 8 & 0.99 & 1.00 & 0.96 + + & & 0.307(10)&0.301(10)&0.327(10 ) + [ table1 ] the base of models is made of 256 different analytic parametrisations .
we can summarize their general form by quoting the form of total cross sections , from which the @xmath0 parameter is obtained via derivative dispersion relations .
the ingredients are the contribution @xmath34 of the highest meson trajectories ( @xmath0 , @xmath35 , @xmath36 and @xmath37 ) and the rising term @xmath38 for the pomeron .
@xmath39 the first term is parametrised via regge theory , and we allow the lower trajectories to be partially non - degenerate , _
i.e. _ we allow one intercept for the @xmath40 trajectories , and another one for the @xmath41 @xcite
. a further lifting of the degeneracy is certainly possible , but does not seem to modify significantly the results @xcite . hence we use @xmath42 with @xmath43 .
the contribution of these trajectories is represented by rr . as for the pomeron term
, we choose a combination of the following possibilities : @xmath44 \label{pom2}\\ { \textrm{h}}^{ab}&=&s\left[b^{ab}\ln^{2}\left({s\over s_1}\right ) + p^{ab}\right]\label{pom3}\end{aligned}\ ] ] with @xmath43 and @xmath45 to be determined by the fit .
the contribution of these terms is marked pe , pl and pl2 respectively .
note that the pole structure of the pomeron can not be directly obtained from these forms , as multiple poles at @xmath46 produce constant terms which mimic simple poles at @xmath6 .
furthermore , we have considered several possible constraints on the parameters of eqs .
( [ lower]-[pom3 ] ) : * degeneracy of the reggeon trajectories @xmath47 , noted ( rr)@xmath48 ; * universality of rising terms ( @xmath49 independent of the hadrons ) , noted l2@xmath50 , l@xmath50 and e@xmath50 @xcite ; * factorization for the residues in the case of the @xmath22 and @xmath51 cross sections . if not otherwise indicated by the subscript @xmath52 , we impose @xmath53 ; * quark counting rules @xcite to predict the @xmath54 cross section from @xmath1 , @xmath55 and @xmath56 , indicated by the subscript @xmath57 ; * johnson - treiman - freund @xcite relation for the cross section differences , noted @xmath58 .
all possible variations of eqs .
( [ lower]-[pom3 ] ) , using the above constraints , amount to 256 variants .
these variants are then fitted to the database , allowing for the minimum c.m .
energy @xmath59 of the fit to vary between 3 and 10 gev . for @xmath60 gev ,
33 variants have an overall @xmath61 if one fits only to total cross sections , whereas 21 obeyed this criterion when one includes the @xmath0 parameters in the data to be fitted to .
one can try to lower the minimum energy of the fit , and one finds that for 11 models one can extend the minimum energy of the cross section fit to 4 gev , and that of the combined fit of @xmath62 and @xmath0 to 5 gev .
several parametrisations based on triple poles ( rrpl2 ) , double poles ( rrpl ) or simple poles ( rrpe ) are kept .
the only notable candidate which seems to be ruled out is the popular simple - pole model ( rre ) @xcite .
its predictions for @xmath1 and @xmath2 nevertheless fall within our errors parameter from _ integral _ dispersion relations
see o. v. selyugin s contribution to these proceedings . ] . after this selection is made , the remaining models are ranked .
we measure some characteristics of the fits , namely : the number of parameters , the confidence level in the considered region , the size of the region where the model achieves a @xmath63 and the value of that @xmath64 , the stability of the parameters when the minimum c.m . energy is changed , their stability with respect to the inclusion of the @xmath0 data , the uniformity of the @xmath64 for different processes and quantities , and finally the quality of the correlation matrix .
all these features are important , and we have managed to measure them , introducing new statistical indicators @xcite
. the ideal fit would be the one with the least number of parameters , the biggest region of applicability , the best @xmath65 , etc .
unfortunately , a single fit does not concentrate all these virtues . as
the new indicators do not have ( yet ) a probabilistic interpretation and as all the parametrisations which fit are _ a priori _ acceptable , we choose the `` best '' model through a ranking procedure : for each feature , the models are ordered according to how well they perform .
one then sums the position of each model for each indicator , and the model with least points is preferred .
the advantage of this method , besides the fact that it automatically looks at many qualities of each fit , is that the best model is decided on the basis of automatic criteria , which do not depend on our own prejudice . following that procedure
, the triple - pole parametrisation rrp@xmath66l2@xmath50 @xcite gives the most satisfactory description of the data .
this parameterization has a universal ( u ) @xmath67 term , a non - factorizing ( nf ) constant term and non - degenerate lower trajectories .
we are now in a position to evaluate several quantities of interest for future measurements .
first of all , our best parametrisation can of course be used to predict @xmath62 and @xmath0 , with their statistical errors .
we choose for this the parameters determined for a minimum c.m .
energy @xmath68 gev . for @xmath1 and @xmath2 ,
the central value of this fit gives @xmath69 with all coefficients in mb and @xmath15 in gev@xmath12 .
we assign errors by using the full error matrix @xmath70 from the fit , and define @xmath71 with @xmath72 or @xmath0 and @xmath73 the parameters of the model .
our predictions are given in table [ table2 ] and the corresponding 1 @xmath74 region is shown as a dark band in figs .
[ fig1 ] and [ fig2 ] .
+ + .predictions for @xmath62 and @xmath0 , for @xmath75 ( at @xmath76 gev ) and for @xmath1 ( all other energies ) .
the central values and statistical errors correspond to the preferred model rrp@xmath66l2@xmath50 , fitted for @xmath68 gev .
the first systematic errors come from the consideration of two choices between cdf and e-710/e-811 @xmath77 data in the simultaneous global fits .
the second systematic error corresponds to the consideration of the 21 parametrisations compatible with existing data . [ cols="^,^,^",options="header " , ] [ table3 ] we can now , and maybe for the first time , give a reasonable estimate of the theoretical error .
the idea is to choose a less constraining minimum energy for the fits ( we take 9 gev ) , and to consider the results of the 21 models that succeed in reproducing both @xmath62 and @xmath0 .
this gives us 21 predictions with error bars .
we can then define the theoretical systematic error by taking the distance between the highest ( resp .
lowest ) values in the @xmath78 intervals with the 1 @xmath74 central region .
we give the resulting numbers as a third error in table [ table2 ] , and as the outer curves of figs .
[ fig1 ] and [ fig2 ] .
note that the systematic errors can not be added in quadrature , and that the theoretical systematic error is an absolute shift from model to model , and does not have any probabilistic interpretation .
one can see that the errors on total cross sections are of the order of 1% at rhic , @xmath8 at the tevatron and as large as @xmath9 at the lhc . at rhic ,
the systematic errors dues to the tevatron discrepancy and those due to theory are of comparable order .
the value of the cross section is constrained by the @xmath2 data , and by the fact that we allow only one @xmath41 contribution , which is well constrained by the overall fit .
very precise rhic measurements , at the level of one in a thousand could shed light on the tevatron discrepancy , and discriminate between models .
of course , the extrapolation to lhc energies presents the largest uncertainty and is dominated by systematic theoretical errors , with the double pole models ( rrpl ) giving a cross section significantly lower than the triple poles or the simple poles .
a determination of the cross section at the 5% level could rule out one half of the models or the other .
the errors on the @xmath0 parameter are much larger , reaching 10@xmath79 at rhic , 17@xmath79 at the tevatron and 26@xmath79 at the lhc .
this is due to the fact that experimental errors are bigger , hence less constraining , but this also stems from the incompatibility of some low - energy determinations of @xmath0 @xcite , and from our use of derivative dispersion relations .
although integral dispersion relations have the potential to reduce the @xmath64 , they have the inconvenient of introducing extra parameters ( because they necessitate subtractions ) .
hence it is unlikely that a different theoretical treatment can reduce the errors . on the other hand , a re - analysis of some of the data could be envisaged .
it should involve a combination of the information on total cross section with that on elastic hadronic cross sections , on electromagnetic form factors and on regge trajectories ( see v. v. ezhela s contribution to these proceedings ) .
finally , we can use the same approach to predict cross sections for cosmic photon studies .
we show the results in table [ table3 ] , where we have given only the experimental systematic error . to conclude , we believe that we have pushed the database technology to the point where it can make predictions , and decide on which models or theories are the best .
this is an example proof of the feasibility of the compete program , and of its utility .
we have given here the best possible estimates for present and future @xmath1 and @xmath2 facilities .
the central values of our fits and the corresponding statistical error give our `` best guess '' estimate .
the systematic experimental errors tell us how much this guess could be affected by incompatible data .
the theoretical systematic errors will tell us directly whether an experiment can be fitted by one of the standard analytic parametrisations , or whether it calls for new ideas . | hep-ph0212101 |
despite years of intensive research , the pairing mechanism responsible for d - wave superconductivity ( dsc ) in the high-@xmath0 cuprates remains a puzzle .
it is generally believed that the conventional phonon mechanism is inconsistent with d - wave pairing symmetry and not strong enough to explain transition temperatures higher than 100 k. most investigations in this direction have been focused on the pure electronic mechanism , but no consensus has been reached so far .
recently , accurate experiments displayed pronounced phonon and electron - lattice effects in these materials , which are manifested by a large softening and broadening of certain phonon modes in the whole doping region . in particular ,
the in - plane copper - oxygen bond - stretching phonon , apical oxygen phonon ( aop ) , and oxygen @xmath1 buckling phonon are shown to be strongly coupled to charge carriers @xcite .
moreover , photoemission - spectroscopy - resolved kink structures @xcite are probably caused by coupling of quasiparticles to phonon modes .
these findings suggest that phonons are important for the physical properties of high-@xmath0 cuprates .
various theoretical attempts have been made to understand the role of phonons in high-@xmath0 superconductivity ( htsc ) @xcite , but the answer remains unclear . in a functional renormalization group study @xcite , honerkamp _ et al .
_ found that the @xmath1 buckling phonon enhances the d - wave pairing instability in the hubbard model .
more recently , an exact diagonalization ( ed ) study of the @xmath2 model coupled to phonons shows that coupling to the buckling mode stabilizes d - wave pairing while coupling to the breathing mode favors a p - wave pairing @xcite . on the other hand , based on dynamical cluster monte carlo calculations of the hubbard model
coupled to holstein , buckling and breathing phonons , macridin _ et al .
_ found that while these phonons can indeed enhance pairing , a strong phonon - induced reduction of quasiparticle weight leads to a suppression of dsc @xcite . in this paper
, we study the effect of aop on dsc in the more realistic three - band hubbard model .
our work is motivated by recent experiments showing that the distance between apical oxygen and the @xmath3 plane @xcite , the disorder around apical oxygen @xcite , and the apical hole state @xcite have significant effects on @xmath0 .
basically , there are two routes for apical oxygen to affect htsc : one is to tune the electronic structure of @xmath3 plane , leading to a change of @xmath0 @xcite ; the other one is to directly couple apical oxygen vibrations to charge carriers on the conducting @xmath3 plane @xcite .
here , we focus on the second route and study a strongly anharmonic vibration of apical oxygen in a double - well potential , which is evidenced in the x - ray absorption spectroscopy of several typical high-@xmath0 compounds @xcite . for a strongly anharmonic motion in the double - well potential ,
the first excitation energy @xmath4 is much smaller than the ones excited to higher energy levels @xmath5 . in this case
one can take into account only the lowest quantum states @xmath6 and @xmath7 with energies @xmath8 and @xmath9 and model the low - energy motion of apical oxygen by a local two - level system represented by a pseudospin @xcite degree of freedom .
our main results , obtained by ed and constrained - path monte carlo ( cpmc ) methods , are presented in figs .
[ sum2x2](b ) , [ binde](a ) and [ pdvdu246 ] . fig .
[ binde](a ) clearly shows that the coupling to the aop induces a strong enhancement of hole binding energy , and this enhancement effect grows as the coulomb repulsion @xmath10 on the copper site is increased .
an analysis of the contribution of different energies to the hole binding energy reveals a novel potential - energy - driven pairing mechanism that involves reduction of both electronic potential energy and phonon related energy . as a combination of increasing pairing interaction and quasiparticle weight ( see fig .
[ sum2x2](b ) ) , the d - wave pairing correlations are found to be strongly enhanced by the electron - phonon ( el - ph ) coupling ( see fig . [ pdvdu246 ] ) . our paper is organized as follows : in section [ model ] , we define the hamiltonian and the physical quantities calculated and discuss the choice of model parameters . in section [ results ] , we present our numerical results and discuss the physical mechanism responsible for the aop - induced enhancement of dsc . finally , in section [ conclusions ] , we discuss in detail our main conclusions .
to model the electronic structure of @xmath3 plane and the coupling of holes to the anharmonic aop , we adopt the following hamiltonian proposed in ref .
@xcite , @xmath11 where @xmath12 , @xmath13 , and @xmath14 stand for the kinetic motion of holes , the potential energy for holes , and phonon related energy , respectively .
they are expressed in the form : @xmath15 @xmath16 and @xmath17 here , the operator @xmath18 creates a hole at a cu @xmath19 orbital and @xmath20 creates a hole in an o@xmath21 or @xmath22 orbital . @xmath23 and @xmath24 are the pseudospin operators for @xmath25 .
@xmath10 denotes the coulomb energy at the cu sites . @xmath26 and @xmath27 are the cu - o and o - o hybridizations , respectively , with the cu and o orbital phase factors included in the sign .
the charge - transfer energy is @xmath28 , i.e. the oxygen orbital energy .
@xmath29 and @xmath30 in @xmath14 denote the strength of el - ph coupling .
@xmath31 stands for the tunneling frequency of the two - level system .
@xmath32 is the vector connecting cu and its nearest - neighbor ( nn ) o. according to quantum cluster calculations @xcite , the parameters lie in the range : @xmath33 , @xmath34 , @xmath35 , and @xmath36 . in units of @xmath37 , we choose a parameter set @xmath38 and @xmath39 , while @xmath10 is varied from weak to strong coupling , including the physical value @xmath40 .
@xmath41 and @xmath42 are assumed for the results presented below , except explicitly noted otherwise .
our calculations are performed on clusters of @xmath43 , @xmath44 , @xmath45 and @xmath46 unit cells with periodic boundary conditions using the ed and cpmc methods @xcite . in the cpmc method , we follow refs . and to use the worldline representation for the pseudospins and projected the ground state @xmath47 of el - ph interacting system from a trial wave function @xmath48 represent the hole and phonon parts , respectively .
the cpmc algorithm has been checked against ed on the @xmath49 cluster , and the difference for the electronic kinetic energy , as well as for the charge and magnetic moment at the copper sites , is less than @xmath50 up to @xmath51 .
the hole binding energy is defined as : @xmath52 with @xmath53 the ground - state energy for n doped holes .
the @xmath54 pairing correlation is defined by , @xmath55 where @xmath56\\ + & [ & p^x_{\vec{r}\uparrow}p^x_{\vec{r}+\vec{\delta}\downarrow } -p^x_{\vec{r}\downarrow}p^x_{\vec{r}+\vec{\delta}\uparrow } ] \\ + & [ & p^y_{\vec{r}\uparrow}p^y_{\vec{r}+\vec{\delta}\downarrow } -p^y_{\vec{r}\downarrow}p^y_{\vec{r}+\vec{\delta}\uparrow } ] \}\end{aligned}\ ] ] with @xmath57 .
@xmath58 for @xmath59 and @xmath60 for @xmath61 .
we calculate also the vertex contribution to the correlations defined as follows : @xmath62 where @xmath63 is the bubble contribution obtained with the dressed ( interacting ) propagator @xcite .
first , we show ed results for the @xmath43 cluster with one hole doped beyond half filling .
the electronic kinetic energy @xmath64 , the peak value @xmath65 of the single - particle spectral function @xmath66 at the fermi energy ( @xmath67 ) , the charge @xmath68 and the magnetic moment @xmath69 at the copper sites are displayed in figs .
[ sum2x2](a)-[sum2x2](d ) . here , @xmath70 with @xmath71 and @xmath72 denoting the ground - state wave function and its energy for one - hole doping .
the index @xmath73 corresponds either to the @xmath54 or to the @xmath74 orbitals , and @xmath75 .
one can clearly see that @xmath76 is lowered with increasing the el - ph coupling @xmath77 at all coulomb energies .
meanwhile , an increase of @xmath65 with increasing @xmath77 indicates that the quasiparticle weight is increased by the el - ph coupling .
in contrast , previous studies of harmonic phonons in the holstein - hubbard model found that the kinetic energy of electrons is increased with increasing el - ph coupling , accompanying a reduction of quasiparticle weight @xcite . to explore the physical reasons for lowering @xmath76 ,
we switch off the coupling of apical phonon either to copper or to in - plane oxygen , i.e. , we set @xmath78 or @xmath79 in eq.([hph ] ) .
it is found that @xmath76 is lowered for the former case , but increased for the latter case .
these results demonstrate that the special coupling of apical oxygen phonon to in - plane oxygen is responsible for lowering the electronic kinetic energy .
.@xmath77 dependence of @xmath76 ( per unit cell ) , @xmath80 , @xmath81 , and nn @xmath82 spin correlation @xmath83 on the @xmath84 cluster at @xmath51 .
the number of holes @xmath85 , corresponding to a hole doping density @xmath86 .
statistical errors are in the last digit and shown in the parentheses . [
cols="<,<,<,<,<",options="header " , ] [ tab6x6 ] from fig .
[ sum2x2](c ) , we notice that the charge is transferred from copper to oxygen sites , which , in combination with a phonon - mediated retarded attraction between holes with opposite spins , results in a reduction of magnetic moment at @xmath87 sites , as shown in fig .
[ sum2x2](d ) .
similar effects of aop on @xmath76 , @xmath80 and @xmath81 are also observed for larger clusters obtained by cpmc simulations , and representative results on the @xmath45 cluster are shown in table [ tab6x6 ] . the last column in tab .
[ tab6x6 ] shows that the value of nn @xmath82 spin correlation @xmath88 becomes less negative with increasing @xmath77 , implying a suppression of antiferromagnetic ( afm ) spin correlation .
the hole binding energy @xmath89 is shown in fig .
[ binde](a ) as a function of the coulomb energy @xmath10 at different @xmath77 .
at all @xmath10 , the binding energy is decreased by switching on the el - ph coupling , signaling an enhancement of hole pairing interaction .
it is remarkable that this enhancement effect becomes stronger with increasing @xmath10 , which is particularly evident in the region @xmath90 . in order to identify the physical origin for this enhancement , the contributions to @xmath89 from the hole kinetic energy @xmath76 , the hole potential energy @xmath91 , and the phonon related energy @xmath92 are depicted in figs .
[ binde ] ( b)-(d ) , respectively . here
, @xmath93 , @xmath94 and @xmath95 have similar definitions to @xmath89 , with @xmath96 in eq.([binding ] ) replaced with @xmath76 , @xmath97 and @xmath98 , respectively .
these quantities represent the gain in the corresponding energy when the second hole is doped in the vicinity of the first one . although the kinetic energy of holes is reduced upon increasing the el - ph coupling ( see fig .
[ sum2x2](a ) ) , an increase of @xmath93 with increasing @xmath77 displayed in fig .
[ binde](b ) indicates that the kinetic energy gain for two doped holes is reduced by the el - ph coupling . as seen in fig .
[ binde](c ) and fig .
[ binde](d ) , a decrease of @xmath94 and @xmath95 with increasing @xmath77 reveals that it is the reduction of electronic potential energy and phonon related energy between two doped holes that enhances the hole binding energy @xmath89 .
in addition , the decrease of @xmath94 and @xmath95 becomes more pronounced as @xmath10 is increased , leading to a stronger enhancement of the hole binding energy in the strong correlation regime ( see fig .
[ binde](a ) ) .
[ binde2 ] displays the hole binding energy and different contributions to @xmath89 on the @xmath49 cluster with antiperiodic and mixed ( periodic in the x direction and antiperiodic in the y direction ) boundary conditions . a comparison of the results in figs .
[ binde ] and [ binde2 ] shows that although the boundary condition has strong effects on the amplitude of hole binding energy , the aop - induced enhancement of hole binding energy is qualitatively similar for different boundary conditions .
this demonstrates that our findings reflect the intrinsic effects of aop in the studied model .
based on the ed results on the small cluster , we can conclude that the coupling of aop to holes can enhance superconductivity on the @xmath3 plane .
the question arising is whether the d - wave pairing symmetry is enhanced ?
this issue can be addressed by examining the behavior of the d - wave pairing correlation @xmath99 in eq.([pairing ] ) . in figs .
[ pdvdu246](a)-(c ) we show @xmath100 as a function of @xmath101 for the @xmath45 cluster at @xmath102 , @xmath103 and @xmath104 , respectively .
the corresponding vertex contribution is also displayed in figs .
[ pdvdu246](d)-(f ) . in the weak - correlation case ( @xmath102 )
, we observe that @xmath100 is modified slightly by the el - ph coupling .
however , when @xmath105 is increased to @xmath103 and @xmath104 , the pairing correlation is enhanced at all long - range distances for @xmath106 . at @xmath107 , the average enhancement of the long - range part of @xmath108
is estimated to be about @xmath109 and @xmath110 for @xmath111 and @xmath51 , respectively .
this increasing enhancement of @xmath112 with @xmath10 is consistent with our findings for the binding energy and quasiparticle weight .
a dramatic increment of the vertex contribution with increasing @xmath77 , as shown in fig .
[ pdvdu246](e ) and fig .
[ pdvdu246](f ) , further provides strong evidence that the d - wave pairing interaction is actually increased by the el - ph coupling .
we also study the effect of the tunneling frequency @xmath31 on the pairing correlations . in the inset of fig .
[ pdvdu246 ] ( c ) , the average of long - range d - wave pairing correlation , @xmath113 where @xmath114 is the number of hole pairs with @xmath115 , is plotted as a function of @xmath31 for @xmath51 and @xmath116 .
we notice that its frequency dependence is rather weak , suggesting that the isotope effect of apical oxygen on the superconductivity is very small .
this is in good agreement with the site - selected oxygen isotope effect in @xmath117 @xcite . to illustrate the effect of hole doping on the phonon - induced enhancement of dsc , in fig .
[ pdave ] we show @xmath118 as a function of hole doping density @xmath119 at @xmath51 for the @xmath120 , @xmath44 , and @xmath46 clusters .
the combination of the results on the three clusters shows that in a wide hole doping region @xmath121 , the d - wave pairing correlation is enhanced by the el - ph coupling , and as the hole doping density is increased from underdoping to optimal doping and then to overdoping , the enhancement of @xmath118 is weakened monotonically .
the reduced enhancement of @xmath118 with larger hole doping , together with the stronger enhancement of @xmath89 and @xmath108 with increasing @xmath10 , demonstrates that strong electronic correlations and/or afm fluctuations play a crucial role in the phonon - induced enhancement of dsc .
in summary , our numerical simulations show that the hole binding energy is strongly enhanced by an aop - induced reduction of electronic potential energy and phonon related energy . as a combination of two concurring effects , i.e. the enhancement of hole pairing interaction and the increase of quasiparticle weight ,
the long - range part of d - wave pairing correlations is dramatically enhanced with increasing the el - ph coupling strength .
our results also show that strong electronic correlations and/or afm fluctuations are crucial for this phonon - induced enhancement effect .
the consistent behavior of our results on different clusters suggests that the phonon - induced enhancement of dsc could survive in the thermodynamic limit .
we thank w. hanke , f. f. assaad . , and a. s. mishchenko for enlightening discussions .
this work was supported by nsfc under grant nos .
10674043 and 10974047 .
hql acknowledges support from hkrgc 402109 .
ea was supported by the austrian science fund ( fwf ) under grant p18551-n16 .
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our results indicate that the apical oxygen phonon plays a significant role in the superconductivity of high-@xmath0 cuprates . | 1004.2740 |
in this note we deal with the uniqueness of distributional solutions to the continuity equation with a sobolev vector field and with the property of being a lagrangian solution , i.e. transported by a flow of the associated ordinary differential equation .
let us first recall the by now classical diperna - lions theory @xcite .
we fix @xmath0 and @xmath1 and we consider a vector field @xmath2;w_{\mathrm{loc}}^{1,p}({\mathbb r}^{n};{\mathbb r}^{n})\right ) \
, , \qquad \operatorname{div}{\boldsymbol b}\in l^{1 } \left ( [ 0,t];l^\infty({\mathbb r}^n ) \right ) \ , , \\ & \frac{|{\boldsymbol b}(t,{\boldsymbol x})|}{1+|{\boldsymbol x}| } \in l^{1}\left ( [ 0,t ] ; l^1({\mathbb r}^{n } ) \right ) + l^{1}\left ( [ 0,t ] ; l^\infty({\mathbb r}^{n } ) \right)\ , .
\end{aligned}\ ] ] given an initial datum @xmath3 , we consider distributional solutions to the cauchy problem for the continuity equation @xmath4 defined as usual by a formal `` integration by parts '' after testing the equation with lipschitz test functions .
given a vector field @xmath5 as in , the diperna - lions theory @xcite guarantees uniqueness of distributional solutions @xmath6 ; l^q({\mathbb r}^n ) \right)\ ] ] to the problem , where @xmath7 is the conjugate exponent of @xmath8 , that is , @xmath9 . if @xmath10 the existence of solutions in this class can be proved by an easy approximation procedure .
moreover , such unique solution is transported by the unique regular lagrangian flow associated to @xmath5 ( see definition [ d : rlf ] ) .
we remark that the theory of @xcite has been extended to vector fields with bounded variation by ambrosio @xcite .
the need for considering solutions in the class follows from the strategy of proof in @xcite , which consists in showing the renormalization property for distributional solutions . to this aim ,
the authors prove the convergence to zero of a suitable commutator , that can be rewritten as an integral expression involving essentially the product of @xmath11 and @xmath12 .
however , distributional solutions to the cauchy problem can be defined as long as the product @xmath13 \times { \mathbb r}^n)$ ] .
therefore , the theory in @xcite leaves open the question whether uniqueness holds for solutions with less integrability than .
ideally , the `` extreme '' case would be that of @xmath14 and @xmath15 , both locally in space .
our main result in this direction is the following : [ t : main ] let @xmath5 be a vector field as in , with @xmath16 .
assume in addition that @xmath17 is continuous for @xmath18-a.e .
@xmath19 $ ] , with modulus of continuity on compact sets which is uniform in time . then , given an initial datum @xmath20 , the cauchy problem for has a unique solution @xmath21\times{\mathbb r}^{n}\right ) \,.\ ] ] such unique solution is lagrangian and renormalized .
the continuity assumption on the vector field in theorem [ t : main ] is satisfied for example when @xmath22;w_{\mathrm{loc}}^{1,p}({\mathbb r}^{n})\right)$ ] with @xmath23 .
theorem [ t : main ] can be easily extended to the case where a source term or a linear term of zero order are present in the continuity equation , under suitable integrability conditions on the coefficients .
in particular , we can also deal with the transport equation @xmath24 instead of the continuity equation .
let us describe in few words the strategy of the proof of theorem [ t : main ] .
given a distributional solution @xmath25\times{\mathbb r}^{n}\right)$ ] of the cauchy problem we aim at proving that it is transported by the regular lagrangian flow @xmath26 associated to @xmath5 .
to this aim , we change variable using the flow in the distributional formulation of .
however , due to the lack of lipschitz regularity of the flow with respect to the space variable , we do not obtain yet the lagrangian formulation in distributional sense : after the change of variable we do not obtain the full class of test functions .
nevertheless some regularity of the flow `` on large sets '' is in fact available ( see theorem [ t : regularity ] ) .
this guarantees that the test function we obtain is lipschitz on a `` large flow tube '' , although with a possibly large lipschitz constant .
we need to extend this function to a globally lipschitz test function .
the key remark is that , in order to estimate the error resulting from this extension , only the lipschitz constant along the characteristics is relevant , not the global lipschitz constant .
we then implement a `` directional extension lemma '' ( lemma [ l : extension ] ) , stating that we can construct an extension which is both globally lipschitz and directionally lipschitz along the flow , and the directional lipschitz constant can be estimated quantitatively .
this allows to conclude the proof . after presenting in
[ s : prelim ] some background material , in [ s : proof ] we give a complete proof of theorem [ t : main ] , under the additional assumption [ a : srfe ] on the existence of a directional lipschitz extension . in
[ s : lemmalip ] we sketch a proof of the validity of assumption [ a : srfe ] under the continuity assumptions on the vector field in theorem [ t : main ] . a complete proof is deferred to the follow up paper @xcite .
in the non smooth context the suitable notion of flow of a vector field is that of regular lagrangian flow , introduced in the following form in @xcite : [ d : rlf ] we say that a map @xmath27^{2}\times{\mathbb r}^{n}\to{\mathbb r}^{n}$ ] is a regular lagrangian flow associated to the vector field @xmath5 if 1 .
[ item:1rlf ] for @xmath28 ^ 2 $ ] we have @xmath29 .
2 . for @xmath30-a.e .
@xmath31 the map @xmath32 satisfies the ordinary differential equation @xmath33 we notice that @xmath34 for later use we set @xmath35 and observe that by definition [ d : rlf](i ) we have @xmath36 the theory in @xcite guarantees that , given a vector field @xmath5 as in , there exists a unique regular lagrangian flow associated to it .
moreover , in @xcite the following regularity of the regular lagrangian flow has been proved : [ t : regularity ] let @xmath5 be a vector field as in and let @xmath26 be the associated regular lagrangian flow .
assume that @xmath16 .
then , for all @xmath37 and @xmath38 there exists a compact set @xmath39 such that 1 .
@xmath40 is lipschitz continuous on @xmath41 , uniformly w.r.t .
@xmath42 $ ] .
2 . @xmath43 .
the restriction to the case @xmath44 in theorem [ t : regularity ] and therefore in theorem [ t : main ] is due to the use of some harmonic analysis estimates in its proof .
we finally introduce the following concept of directional lipschitz continuity : [ d : directional ] let @xmath45 be defined on a borel set @xmath46\times{\mathbb r}^n$ ] and let @xmath47\times a\to { \mathbb r}^{n}$ ] be a borel map , where @xmath48 is a borel set .
we say that the function @xmath45 is @xmath49-directionally lipschitz continuous if for all @xmath50 $ ] and for all @xmath51 such that @xmath52 there holds @xmath53 we focus in this paper only on directional lipschitz continuity in the specific case @xmath54 , where @xmath26 is a regular lagrangian flow .
in this section we give a complete proof of theorem [ t : main ] , under the additional assumption [ a : srfe ] on the existence of a directional lipschitz extension that we introduce in step 2 here below . a proof of assumption [ a : srfe ] is sketched in
[ s : lemmalip ] below and a full proof deferred to @xcite .
[ [ step-0 . ] ] * step 0 .
* + + + + + + + + + by the linearity of the continuity equation , it is enough to prove that @xmath55 implies @xmath56 .
we do this by showing that every distributional solution @xmath12 of satisfies a lagrangian formulation . in this context
this amounts to the fact that the function @xmath57 solves in distributional sense the equation @xmath58=0 $ ] , where @xmath59 is defined in , with initial datum @xmath60 , that is @xmath61 where @xmath62 and @xmath63 , the spaces of lipschitz functions with compact support , and of essentially bounded functions with compact support , respectively .
notice that the validity of implies that @xmath64 , and thus with we obtain @xmath56 .
since @xmath65 is dense in @xmath66 with respect to the weak star topology of @xmath67 , we reduced the proof of theorem [ t : main ] to the proof of the following claim : [ c : claim ] the lagrangian formulation holds for every @xmath68 .
we fix @xmath69 we prove in the next steps that claim [ c : claim ] holds . [
[ step-1 . ] ] * step 1 .
* + + + + + + + + + fix @xmath38 and consider a compact set of the form @xmath70\times b_{r}({\boldsymbol 0})$ ] which contains the support of the function @xmath71 fixed in .
we use theorem [ t : regularity ] to find a compact subset @xmath39 on which the regular lagrangian flow @xmath40 is uniformly lipschitz continuous .
[ l : sfsdva ] on the compact flow tube @xmath72}$ ] starting from @xmath73 the function @xmath74 is lipschitz continuous and @xmath75-directionally lipschitz continuous , with @xmath76 as in .
we start by proving the @xmath75-directional lipschitz continuity .
let @xmath77 and thus by we get @xmath78 we now prove the lipschitz continuity of @xmath79 on @xmath72}$ ] . given @xmath80 one has @xmath81 when comparing two points @xmath82 and @xmath83 , for some @xmath84 , we simply define @xmath85 and we estimate @xmath86 where in the last inequality we applied @xmath87 * step 2 . *
+ + + + + + + + + we can proceed with the proof under the following assumption .
[ a : srfe ] given @xmath38 let @xmath79 be as in .
we assume that there exists @xmath88 \times { \mathbb r}^{n}\to{\mathbb r}$ ] which is an extension of @xmath79 and in addition is 1 .
lipschitz continuous , and 2 .
@xmath89-directionally lipschitz continuous , where @xmath90 does not depend on @xmath91 .
in fact , we are able to prove that assumption [ a : srfe ] holds when the vector field @xmath5 satisfies the continuity condition assumed in theorem [ t : main ] . in section [ s : lemmalip ]
we give a sketch of the proof of this fact , and we defer a complete proof to a next paper .
[ [ step-3 . ] ] * step 3 .
* + + + + + + + + + we now derive some consequences of assumption [ a : srfe ] in the @xmath92-variables .
we define @xmath93 $ and $ { \boldsymbol x}\in { \mathbb r}^n$}\ ] ] and we observe that 1 .
[ i : szvds ] @xmath94 is @xmath95-lipschitz continuous for all @xmath96 .
this follows from assumption [ a : srfe](ii ) and from the definition of directional lipschitz continuity ( definition [ d : directional ] ) .
2 . @xmath97 for every @xmath98 and every @xmath19 $ ] .
in particular , we can test @xmath58(t,{\boldsymbol y})$ ] agains @xmath99 : by the definitions in and we obtain @xmath100\,dtd{\boldsymbol y}\,.\end{aligned}\ ] ] we now apply the change of variable @xmath101 , obtaining @xmath102 \,dtd{\boldsymbol x}=0\,,\ ] ] because @xmath12 is a distributional solution of with zero initial datum .
we stress that the first equality in follows by the definition of push - forward measure because the results in @xcite establish that the regular lagrangian flow @xmath26 satisfies the absolute continuity estimate in definition [ d : rlf ] .
this is a very important brick in this disintegration strategy , and in other settings it requires to be proved ad hoc , see for instance @xcite .
[ [ step-4 . ] ] * step 4 .
* + + + + + + + + + we conclude the proof of claim [ c : claim ] , thus establishing theorem [ t : main ] under assumption [ a : srfe ] .
the main observation is that equation gives the validity of claim [ c : claim ] with the test function @xmath71 replaced by the approximation @xmath103 defined in .
therefore , we simply estimate the integral containing @xmath71 with the integral containing @xmath104 plus an error , and we only need to show that the error converges to zero as @xmath105 .
indeed , we compute as follows : @xmath106\,dtd{\boldsymbol y}\\ & = \cancel{\int_{0}^{t}\int_{k_{\varepsilon } } \frac{u}{r}\,\partial_{t}\left[\psi-\psi_{\varepsilon}\right]\,dtd{\boldsymbol y } } + \int_{0}^{t}\int _ { b_{r}({\boldsymbol 0})\setminus k_{\varepsilon } } \ !
\frac{u}{r}\,\partial_{t}\left[\psi-\psi_{\varepsilon}\right]\,dtd{\boldsymbol y}\,,\end{aligned}\ ] ] where @xmath107 is as in step 1 , and by construction @xmath108 on @xmath70 \times k_\varepsilon$ ] .
since @xmath109 is @xmath76-lipschitz continuous by definition and each @xmath94 is @xmath95-lipschitz continuous by step 3([i : szvds ] ) , we finally get @xmath110 using and the fact that the function @xmath111 in belongs to @xmath112 \times { \mathbb r}^n)$ ] .
this concludes the proof of theorem [ t : main ] under assumption [ a : srfe ] .
we finally briefly sketch the strategy of proof of the following lemma . a full proof in a more general context
is deferred to @xcite .
[ l : extension ] let @xmath5 be a vector field as in , with @xmath16
. assume in addition that @xmath17 is continuous for @xmath18-a.e .
@xmath19 $ ] , with modulus of continuity on compact sets which is uniform in time .
then assumption [ a : srfe ] holds . in the above lemma
one can as well require that @xmath113 .
we start by noticing that a function @xmath79 is @xmath75-directionally lipschitz continuous according to definition [ d : directional ] if and only if @xmath79 is @xmath76-lipschitz continuous for the following degenerate distance @xmath114 : @xmath115 moreover , we denote by @xmath116 the usual euclidean distance in @xmath70 \times { \mathbb r}^{n}$ ] .
consider a lipschitz continuous function @xmath79 defined on the compact flow tube @xmath117}$ ] of assumption [ a : srfe ] .
we remind that we assume that @xmath79 is @xmath75-directionally lipschitz continuous , and that we need to extend @xmath79 to @xmath70 \times { \mathbb r}^n$ ] in such a way that the extension is 1 .
@xmath75-directionally lipschitz continuous , i.e. @xmath95-lipschitz continuous for @xmath114 , with @xmath95 depending on @xmath76 , and 2 .
lipschitz continuous for the euclidean distance , i.e. lipschitz continuous for @xmath116 .
in other words , we need to prove a lipschitz extension theorem with respect to two non equivalent distances at the same time : to the best of our knowledge , this is a new and non trivial task .
notice that for our purposes we need that the lipschitz constant for @xmath118 only depends on @xmath76 , while we do not need a quantitative control on the lipschitz constant for @xmath119 .
we now give a rough idea of the proof of lemma [ l : extension ] .
for @xmath120 we introduce a family of distances @xmath121 , each of them equivalent to the euclidean distance @xmath119 . the distance @xmath121 penalizes with a factor @xmath122 displacements which are not along the flow . moreover , the distances @xmath123 converge to the degenerate distance @xmath114 , i.e. @xmath124 as @xmath125 . in particular , a function which is @xmath95-lipschitz continuous for @xmath121 is also @xmath95-lipschitz continuous for @xmath114 .
the key point in the proof of lemma [ l : extension ] is the fact that , when @xmath125 , the lipschitz constant of @xmath79 for @xmath121 converges to the lipschitz constant @xmath76 of @xmath79 for @xmath114 : @xmath126 using this property , we choose @xmath127 small enough so that @xmath128 is close to @xmath76 .
we extend @xmath79 by using mcshane extension theorem for the distance @xmath129 . in this way
, we get an extension which is 1 .
@xmath130-directionally lipschitz continuous , and @xmath131 is close to @xmath76 , and 2 .
lipschitz continuous for the euclidean distance @xmath116 , since @xmath129 is equivalent to @xmath119 . in the above procedure
, we are currently able to prove only assuming that the vector field @xmath5 is continuous for @xmath18-a.e . @xmath19 $ ] , with modulus of continuity on compact sets which is uniform in time .
this work was started during a visit of lc at the university of basel and carried on during a visit of gc at the university of padova as a visiting scientist .
the authors gratefully acknowledge the support and the hospitality of both institutions .
lc is a member of the gruppo nazionale per lanalisi matematica , la probabilit e le loro applicazioni ( gnampa ) of the istituto nazionale di alta matematica ( indam ) .
gc is partially supported by the erc starting grant 676675 flirt . | we deal with the uniqueness of distributional solutions to the continuity equation with a sobolev vector field and with the property of being a lagrangian solution , i.e. transported by a flow of the associated ordinary differential equation .
we work in a framework of lack of local integrability of the solution , in which the classical diperna - lions theory of uniqueness and lagrangianity of distributional solutions does not apply due to the insufficient integrability of the commutator .
we introduce a general principle to prove that a solution is lagrangian : we rely on a disintegration along the unique flow and on a new directional lipschitz extension lemma , used to construct a large class of test functions in the lagrangian distributional formulation of the continuity equation .
0.5 * rsum * 0.5*unicit et proprit lagrangienne des solutions manquant dintgrabilit de lquation de continuit . * on tudie
lunicit des solutions distributionnelles de lquation de continuit avec des champs de vecteurs sobolev et la proprit dtre une solution lagrangienne , cest - - dire une solution transporte par le flot de lquation diffrentielle ordinaire associe au champ de vecteurs . on travaille dans un cadre o les solutions considres manquent dintgrabilit locale et o on ne peut pas appliquer la thorie classique de diperna - lions dunicit des solutions distributionnelles et de la proprit dtre lagrangienne parce que on na pas assez dintgrabilit pour le commutateur .
on introduit un principe gnral pour dmontrer la proprit dtre une solution lagrangienne : notre technique se base sur une desintgration le long le flot unique et sur un lemme dextension lipschitzienne directionnelle qui nous permet de construire une vaste famille de fonction test pour la formulation distributionnelle lagrangienne de lquation de continuit . , received * * * * * ; accepted after revision + + + + + + presented by | 1608.04324 |
in this paper , we continue our general study of the _ uniform thickness property _ ( utp ) in the context of iterated torus knots that are embedded in @xmath0 with the standard tight contact structure . as stated in a previous paper , _ studying uniform thickness i _
@xcite , our goal in this study is to determine the extent to which iterated torus knot types fail to satisfy the utp , and the extent to which this failure leads to cablings that are legendrian or transversally non - simple .
motivation for this study is due to the work of etnyre and honda @xcite , who showed that the failure of the utp is a necessary condition for transversal non - simplicity in the class of iterated torus knots .
they also established that the @xmath1 torus knot fails the utp and supports a transversally non - simple cabling .
in @xcite we extended this study of the utp by establishing new necessary conditions for both the failure of the utp and transversal non - simplicity in the class of iterated torus knots ; in so doing we obtained new families of legendrian simple iterated torus knots .
the specific goal of this note is to fully answer the first motivating question of our study by providing a complete utp classification of iterated torus knots , that is , determining which iterated torus knot types satisfy the utp , and which fail the utp .
we will also address the second motivating question of our study by proving that failure of the utp for an iterated torus knot type is a sufficient condition for the existence of transversally non - simple cablings of that knot .
specifically , we have the following two theorems and corollary : [ main theorem ] let @xmath2 be an iterated torus knot , where the @xmath3 s are measured in the standard preferred framing , and @xmath4 for all @xmath5 .
then @xmath6 fails the utp if and only if @xmath7 for all @xmath5 , where @xmath8 . in the second theorem ,
@xmath9 is the euler characteristic of a minimal genus seifert surface for a knot @xmath10 : [ second theorem ] if @xmath6 is an iterated torus knot that fails the utp , then it supports infinitely many transversally non - simple cablings @xmath11 of the form @xmath12 , where @xmath13 ranges over an infinite subset of positive integers . to state our corollary to theorem [ main theorem ] ,
recall that if @xmath10 is a fibered knot , then there is an associated open book decomposition of @xmath0 that supports a contact structure , denoted @xmath14 ( see @xcite ) .
iterated torus knots are fibered knots , and hedden has shown that the subclass of iterated torus knots where each iteration is a positive cabling , i.e. @xmath7 for all @xmath5 , is precisely the subclass of iterated torus knots where @xmath15 is isotopic to @xmath16 @xcite .
we thus obtain the following corollary : an iterated torus knot @xmath6 fails the utp if and only if @xmath17 .
we make a few remarks about these theorems .
first , it will be shown that these transversally non - simple cablings will all have two legendrian isotopy classes at the same rotation number and maximal thurston - bennequin number @xmath18 , and thus they will exhibit legendrian non - simplicity at @xmath18 .
second , in the class of iterated torus knots there are certainly more transversally non - simple cablings than those in theorem [ second theorem ] , as evidenced by etnyre and honda s example of the transversally non - simple @xmath1-cabling of a @xmath1-torus knot . however , we present just the class of transversally non - simple cablings in theorem [ second theorem ] , and leave a more complete legendrian and transversal classification of iterated torus knots as an open question
. we now present a conjectural generalization of the above two theorems and corollary . to this end ,
recall that hedden has shown that for general fibered knots @xmath10 in @xmath0 , @xmath19 precisely when @xmath10 is a fibered strongly quasipositive knot @xcite ; he also shows that for these knots , the maximal self - linking number is @xmath20 @xcite .
furthermore , from the work of etnyre and van horn - morris @xcite , we know that for fibered knots @xmath10 in @xmath0 that support the standard contact structure there is a unique transversal isotopy class at @xmath21 . in the present paper , all of these ideas are brought to bear on the class of iterated torus knots , and this motivates the following conjecture concerning general fibered knots : let @xmath10 be a fibered knot in @xmath0 ; then @xmath10 fails the utp if and only if @xmath19 , and hence if and only if @xmath10 is fibered strongly quasipositive .
moreover , if a topologically non - trivial fibered knot @xmath10 fails the utp , then it supports cablings that are transversally non - simple .
we also ask the following question of non - fibered knots in @xmath0 : if @xmath10 is a non - fibered strongly quasipositive knot , does @xmath10 fail the utp and support transversally non - simple cablings ?
we will be using tools developed by giroux , kanda , and honda , and used by etnyre and honda in their work , namely convex tori and annuli , the classification of tight contact structures on solid tori and thickened tori , and the legendrian classificaton of torus knots . most of the results we use can be found in @xcite , @xcite , @xcite , or @xcite , and if we use a lemma , proposition , or theorem from one of these works , it will be specifically referenced .
the plan of the note is as follows . in
2 we recall definitions , notation , and identities used in @xcite and @xcite . in 3 we outline a strategy of proof of theorem [ main theorem ] that yields the statement of two key lemmas . in 4 and 5
we prove the first lemma . in
6 we prove the second lemma and complete the proof of theorem [ main theorem ] . in
7 we prove theorem [ second theorem ] .
_ iterated torus knots _ , as topological knot types , can be defined recursively .
let 1-iterated torus knots be simply torus knots @xmath22 with @xmath23 and @xmath24 co - prime nonzero integers , and @xmath25 . here
@xmath23 is the algebraic intersection with a longitude , and @xmath24 is the algebraic intersection with a meridian in the preferred framing for a torus representing the unknot . then for each @xmath22 torus knot , take a solid torus regular neighborhood @xmath26 ; the boundary of this is a torus , and given a framing we can describe simple closed curves on that torus as co - prime pairs @xmath27 , with @xmath28 . in this way we obtain all 2-iterated torus knots , which we represent as ordered pairs , @xmath29 .
recursively , suppose the @xmath30-iterated torus knots are defined ; we can then take regular neighborhoods of all of these , choose a framing , and form the @xmath31-iterated torus knots as ordered @xmath31-tuples @xmath32 , again with @xmath33 and @xmath34 co - prime , and @xmath35 .
for ease of notation , if we are looking at a general @xmath31-iterated torus knot type , we will refer to it as @xmath6 ; a legendrian representative will usually be written as @xmath36 .
note that we will use the letter @xmath31 both for the rotation number ( see below ) and as an index for our iterated torus knots ; context will distinguish between the two uses .
we will study iterated torus knots using two framings .
the first is the standard framing for a torus , where the meridian bounds a disc inside the solid torus , and we use the preferred longitude which bounds a surface in the complement of the solid torus .
we will refer to this framing as @xmath37 .
the second framing is a non - standard framing using a different longitude that comes from the cabling torus .
more precisely , to identify this non - standard longitude on @xmath38 , we first look at @xmath6 as it is embedded in @xmath39 . we take a small neighborhood @xmath40 such that @xmath38 intersects @xmath39 in two parallel simple closed curves . these curves are longitudes on @xmath38 in this second framing , which we will refer to as @xmath41 .
note that this @xmath41 framing is well - defined for any cabled knot type .
moreover , for purpose of calculations there is an easy way to change between the two framings , which will be reviewed below . given a simple closed curve @xmath42 on a torus , measured in some framing as having @xmath43 meridians and @xmath44 longitudes
, we will say this curve has slope of @xmath45 ; i.e. , longitudes over meridians
. therefore we will refer to the longitude in the @xmath41 framing as @xmath46 , and the longitude in the @xmath37 framing as @xmath47 .
the meridian in both framings will have slope @xmath48 .
we will also use a convention that meridians in the standard @xmath37 framing , that is , algebraic intersection with @xmath47 , will be denoted by upper - case @xmath49 s . on the other hand , meridians in the non - standard @xmath50 framing , that is , algebraic intersection with @xmath51 , will be denoted by lower - case @xmath52 s . given a curve @xmath53 on a torus @xmath54
, there is then a relationship between the framings @xmath50 and @xmath37 on @xmath54 . in terms of a change of basis ,
we get from @xmath50 to @xmath37 by multiplying on the left by the matrix @xmath55 .
given an iterated torus knot type @xmath56 where the @xmath57 s are measured in the @xmath50 framing , we define two quantities .
the two quantities are : @xmath58 note here we use a convention that @xmath59 . also , if we restrict to the first @xmath5 iterations , that is , to @xmath60 , we have an associated @xmath61 and @xmath62 .
for example , @xmath63 .
finally , from @xcite we obtain four useful identities which we will apply extensively throughout this note : @xmath64 recall that for legendrian knots embedded in @xmath0 with the standard tight contact structure , there are two classical invariants of legendrian isotopy classes , namely the thurston - bennequin number , @xmath65 , and the rotation number , @xmath31 . for a given topological knot type , if the ordered pair @xmath66 completely determines the legendrian isotopy classes , then that knot type is said to be _
legendrian simple_. for transversal knots there is one classical invariant , the self - linking number @xmath67 ; for a given topological knot type , if the value of @xmath67 completely determines the transversal isotopy classes , then that knot type is said to be _ transversally simple_. for a given topological knot type , if we plot legendrian isotopy classes at points @xmath66 , we obtain a plot of points that takes the form of a _
legendrian mountain range _ for that knot type
. we will be examining legendrian knots which are embedded in convex tori .
recall that the characteristic foliation induced by the contact structure on a convex torus can be assumed to have a standard form , where there are @xmath68 parallel _ legendrian divides _ and a one - parameter family of _ legendrian rulings_. parallel push - offs of the legendrian divides gives a family of @xmath68 _ dividing curves _ , referred to as @xmath69 . for a particular convex torus , the slope of components of @xmath69 is fixed and is called the _ boundary slope _ of any solid torus which it bounds ; however , the legendrian rulings can take on any slope other than that of the dividing curves by giroux s flexibility theorem @xcite .
a _ standard neighborhood _ of a legendrian knot @xmath70 will have two dividing curves and a boundary slope of @xmath71 .
we can now state the definition of the _ uniform thickness property _ as given by etnyre and honda @xcite . for a knot type @xmath10 ,
define the _ contact width _ of @xmath10 to be @xmath72 in this equation the @xmath73 are solid tori having representatives of @xmath10 as their cores ; slopes are measured using the preferred framing where the longitude has slope @xmath47 ; the supremum is taken over all solid tori @xmath73 representing @xmath10 where @xmath74 is convex .
a knot type @xmath10 then satisfies the utp if the following hold : * @xmath75 , where @xmath18 is the maximal thurston - bennequin number for @xmath10 . * every solid torus @xmath73 representing @xmath10 can be thickened to a standard neighborhood of a maximal @xmath65 legendrian knot . for a topological knot type @xmath10 ,
if @xmath73 is a solid torus having a representative of @xmath10 as its core and convex boundary , then @xmath73 _ fails to thicken _ if for all @xmath76 , we have @xmath77 .
if we define @xmath78 to be the twisting of the contact planes along @xmath70 with respect to the @xmath50 framing on @xmath79 , equation 2.1 in @xcite gives us : @xmath80 observe that @xmath81 is also the twisting of the contact planes with respect to the framing given by @xmath74 , and so is equal to @xmath82 times the geometric intersection number of @xmath70 with @xmath83 .
@xmath84 will denote the maximal twisting number with respect to this framing .
we also had two definitions introduced in @xcite that will be useful in this note .
_ let @xmath73 be a solid torus with convex boundary in standard form , and with @xmath85 in some framing .
if @xmath86 is the geometric intersection of the dividing set @xmath69 with a longitude ruling in that framing , then we will call @xmath87 the _ intersection boundary slope__. note that when we have an intersection boundary slope @xmath87 , then @xmath88 is the number of dividing curves . _ for @xmath89 and positive integer @xmath13 , define @xmath90 to be any solid torus representing @xmath6 with intersection boundary slope of @xmath91 , as measured in the @xmath50 framing . also define the integer @xmath92 .
_ note that @xmath90 has @xmath93 dividing curves .
note also that the above definition is only for @xmath94 .
however , we will also define @xmath95 to be a standard neighborhood of a @xmath96 representative , and thus have this as the @xmath97 case . finally , recall that if @xmath98 is a convex annulus with legendrian boundary components , then dividing curves are arcs with endpoints on either one or both of the boundary components . dividing curves that are boundary parallel are called _ bypasses _ ; an annulus with no bypasses is said to be _
standard convex_. recall that a contact structure @xmath99 on a 3-manifold @xmath100 is said to be _ overtwisted _ if there exists an embedded disc @xmath101 which is tangent to @xmath99 everywhere along @xmath102 , and a contact structure is _ tight _ if it is not overtwisted .
moreover , one can further analyze tight contact 3-manifolds @xmath103 by looking at what happens to @xmath99 when pulled back to the universal cover @xmath104 via the covering map @xmath105 .
in particular , if the pullback of @xmath99 remains tight , then @xmath103 is said to be _ universally tight_. the classification of universally tight contact structures on solid tori is known from the work of honda .
specifically , from proposition 5.1 in @xcite , we know there are exactly two universally tight contact structures on @xmath106 with boundary torus having two dividing curves and slope @xmath107 in some framing .
these are such that a convex meridional disc has boundary - parallel dividing curves that separate half - discs all of the same sign , and thus the two contact structures differ by @xmath108 .
( if @xmath109 , there is only one tight contact structure , and it is universally tight . ) also from the work of honda , we know that if @xmath99 is a contact structure which is everywhere transverse to the fibers of a circle bundle @xmath100 over a closed oriented surface @xmath110 , then @xmath99 is universally tight .
this is the content of lemma 3.9 in @xcite , and such a transverse contact structure is said to be _
horizontal_. given a legendrian knot @xmath70 , recall that there are well - defined _ positive and negative transverse push - offs _ , denoted by @xmath111 and @xmath112 , respectively .
moreover , the self - linking numbers of these transverse push - offs are given by the formula @xmath113
in this section we present a strategy of proof for theorem [ main theorem ] .
we begin with a theorem that in previous works has in effect been proved , but not stated . in this theorem @xmath10
is a knot type and @xmath114 is the @xmath53-cabling of @xmath10 .
[ utp preserved ] if @xmath10 satisfies the utp , then @xmath114 also satisfies the utp .
the case where the cabling fraction @xmath115 is the content of theorem 1.3 in @xcite . for the case where @xmath116 , the proof follows from examining the proofs of theorem 3.2 @xcite and theorem 1.1 in @xcite and observing that legendrian simplicity of @xmath10 is not needed to preserve the utp . with this theorem in mind
, we will prove theorem [ main theorem ] by way of two lemmas , one of which uses induction .
for this purpose we make the following inductive hypothesis , which from here on we will refer to as _ the inductive hypothesis_. we will need to justify its veracity for the base case of positive torus knots . + * inductive hypothesis : * let @xmath117 be an iterated torus knot , as measured in the standard @xmath37 framing .
assume that the following hold : * @xmath7 for all @xmath5 , where @xmath8 .
( thus @xmath118 for all @xmath5 as well . ) * @xmath119 . ( thus @xmath120 . )
* any solid torus @xmath121 representing @xmath6 thickens to some @xmath90 ( including @xmath95 which is a standard neighborhood of a @xmath18 representative ) . *
if @xmath121 fails to thicken then it is an @xmath90 , and it has at least @xmath93 dividing curves . * the candidate non - thickenable @xmath90 exist and actually fail to thicken for @xmath122 , where @xmath123 is some positive integer that varies according to the knot type @xmath6 .
moreover , these @xmath90 that fail to thicken have contact structures that are universally tight , with convex meridian discs containing bypasses all of the same sign .
also , a legendrian ruling preferred longitude on these @xmath124 has rotation number zero for @xmath125 .
+ our first key lemma used in proving theorem [ main theorem ] is the following , which along with the base case of positive torus knots , will show that if @xmath126 is such that @xmath7 for all @xmath5 , then @xmath6 fails the utp .
[ indhyppreserved ] suppose @xmath6 satisfies the inductive hypothesis , and @xmath11 is a cabling where @xmath127 ; then @xmath11 satisfies the inductive hypothesis .
the main idea in the argument used to prove this lemma will be that since @xmath6 satisfies the inductive hypothesis , there is an infinite collection of non - thickenable solid tori whose boundary slopes form an increasing sequence converging to @xmath128 in the @xmath50 framing ( which is @xmath47 in the @xmath37 framing ) . as a consequence , it will be shown that cabling slopes with @xmath127 in the @xmath37 framing will have a similar sequence of non - thickenable solid tori .
our second key lemma is the following , which along with theorem [ utp preserved ] and the fact that negative torus knots satisfy the utp , will show that if at least one of the @xmath129 , then @xmath6 satisfies the utp .
[ negativeutp ] suppose @xmath6 satisfies the inductive hypothesis , and @xmath11 is a cabling where @xmath130 ; then @xmath11 satisfies the utp .
our outline for the next three sections is as follows . in the next section , 4 , we establish the truth of the inductive hypothesis for the base case of positive torus knots . in
5 we prove lemma [ indhyppreserved ] , and in 6 we prove lemma [ negativeutp ] .
in this section we show that positive torus knots @xmath22 satisfy the inductive hypothesis described in 3 . from lemma 4.3 in @xcite , we know that items 1 - 4 of the inductive hypothesis are satisfied ; it remains to establish item 5 , that each solid torus candidate @xmath131 actually exists with a universally tight contact structure and the appropriate complement in @xmath0 , and that these @xmath131 indeed fail to thicken ( for all @xmath132 in this case of positive torus knots ) . to establish item 5 , we employ arguments similar to those used in @xcite for solid tori representing the @xmath1 torus knot , specifically lemmas 5.2 and 5.3 in @xcite . from lemma 4.3 in @xcite , we know that if @xmath131 fails to thicken , its complement @xmath133 must be contactomorphic to the manifold obtained by taking a neighborhood of a hopf link @xmath134 and a standard convex annulus @xmath98 joining the two neighborhoods of the hopf link , where @xmath98 has boundary components that are legendrian ruling representatives of @xmath135 .
moreover , we know that the two components of the hopf link must have @xmath65 values equal to @xmath136 and @xmath137 , respectively , for @xmath132 . we first show that the candidate @xmath131 have universally tight contact structures .
if @xmath131 fails to thicken , then its contact structure is universally tight ; moreover , for @xmath138 , a convex meridian disc contains bypasses that all bound half - discs of the same sign . also , a legendrian ruling preferred longitude on @xmath139 has rotation number zero for @xmath125 .
the lemma is immediately true for @xmath97 , so we may assume that @xmath138 . to fix notation ,
let @xmath140 be the legendrian unknot with @xmath141 and let @xmath142 be the unknot with @xmath143
. then @xmath144 thickens outward to @xmath145 ; we denote @xmath146 and @xmath147 .
since @xmath148 and @xmath149 are convex , we can take @xmath150$]-invariant neighborhoods of each ; our convention will be that the two @xmath151 will bound a thickened torus that contains the two @xmath152 .
now @xmath153 is a convex torus with dividing curves that divide the torus into two annuli , @xmath154 and @xmath155 .
we locate a ( topological ) meridian curve @xmath43 on @xmath153 that intersects each dividing curve efficiently @xmath156 times , and so that @xmath157 consists of @xmath24 arcs which intersect @xmath154 and @xmath155 at least @xmath13 times each .
we then can realize @xmath43 as a legendrian ruling using theorem 3.7 in @xcite .
we then examine a horizontal convex annulus @xmath158 in the space @xmath159 , bounded by meridian rulings on @xmath151 .
this horizontal convex annulus @xmath158 has two dividing curves that connect its two boundary components ; the other @xmath160 bypasses have endpoints on @xmath153 . by lemma 4.14 in @xcite
, we may assume that all of these bypasses are boundary compressible , meaning there are no nested bypasses .
the two dividing curves connecting the two boundary components of @xmath158 thus divide @xmath158 into two discs , one containing all bypasses of positive sign , the other disc containing all negative bypasses .
we will show that in fact all bypasses on @xmath158 must be of the same sign . to this end , let @xmath98 be the standard convex annulus with @xmath22 legendrian rulings as its boundary components on @xmath152 .
we first examine @xmath161 , which is @xmath24 arcs with endpoints on the two @xmath162 . at first glance ,
it is possible that there may be points of intersection between these @xmath24 arcs and the boundary - parallel dividing curves on @xmath158 .
however , up to a choice of contact vector field for the convex annulus @xmath158 , we may assume that all boundary - parallel dividing curves for @xmath158 are in a collar neighborhood of @xmath153 and avoid @xmath98 .
this contact vector field may also be chosen so that the two non - separating dividing curves on @xmath158 intersect @xmath98 transversely .
now @xmath163 is one of the annuli that forms @xmath139 , and the intersection of this annulus with @xmath158 will be @xmath24 arcs , which we denote as @xmath164 for @xmath165 . by the above considerations we thus have that , as a collection , the @xmath164 have support that intersects all of the @xmath160 bypasses on @xmath158 .
see figure [ fig : universallytightsetupbb ] . .
the thick gray arcs represent the intersection of @xmath98 with @xmath158 .
some of the @xmath160 bypasses are shown ; in the figure , @xmath166.,scaledwidth=55.0% ] we next perform edge - rounding for the curves of intersection of @xmath167 and the two annuli coming from @xmath168 ; after edge - rounding we obtain @xmath139 .
thus @xmath139 intersects @xmath158 in @xmath24 ( topological ) meridian curves for @xmath131 ; this set of curves , call it @xmath169 , is _ nonisolating _ in the sense of section 3.3.1 in @xcite , meaning that each curve is transverse to the dividing set of @xmath158 and every component of @xmath170 has boundary that intersects @xmath69 .
moreover , @xmath169 is also a nonisolating set of curves on @xmath139 . then by theorem 3.7 in @xcite
, we can realize the @xmath24 topological meridian curves as legendrian meridian curves for @xmath139 .
now these legendrian meridian curves may not have efficient geometric intersection with @xmath171 .
however , by the construction of @xmath139 , any holonomy of dividing curves on the two annuli coming from the two sides of @xmath172 cancels each other out .
thus we can destabilize these legendrian meridian curves on the surface @xmath139 so that they do have geometric intersection @xmath173 with @xmath171 , and we can do so _ away from the @xmath164_. these destabilizations can thus be accomplished by attachment of bypasses off of the @xmath24 convex meridian discs , but these ( attached ) bypasses will avoid the original @xmath160 bypasses along the @xmath164 .
the resulting @xmath24 convex meridian discs therefore inherit the bypasses of @xmath158 . by construction , it is possible that one of the @xmath24 convex meridian discs may inherit @xmath174 bypasses from @xmath158 ; if this is the case , however , these bypasses must all be of the same sign , and we have the desired conclusion .
so we may assume that each of the @xmath24 meridian discs intersects @xmath13 bypasses of @xmath158 .
so suppose , for contradiction , that the @xmath160 bypasses on @xmath158 have mixed sign , meaning some are negative and some are positive .
since each of the @xmath24 meridian discs is a convex meridian disc for @xmath131 , then by the classification of tight contact structures on solid tori we know that if one of the discs has a negative bypass , then all of them must ; the same is true for positive bypasses .
but since the negative bypasses on @xmath158 are grouped in succession , and since we may assume @xmath175 , this forces one of the discs to inherit only negative bypasses , contradicting the fact that it is supposed to also have positive bypasses .
thus all of the bypasses on @xmath158 must be of the same sign , as must be all of the bypasses on a convex meridian disc for @xmath131 . as a result the contact structure for @xmath131 is universally tight .
we can now calculate the rotation number for the @xmath22 ruling on @xmath144 . since @xmath159 is universally tight
, one can show that if @xmath176 is a convex seifert surface for the longitude on @xmath177 , we must have @xmath178 . by lemma 2.2 in @xcite
, we have that the @xmath22 ruling on @xmath177 has rotation number equal to @xmath179 this yields @xmath180 .
we now let @xmath101 be a meridian disc for @xmath131 and @xmath110 be a seifert surface for the preferred longitude on @xmath139 . we know
@xmath181 , and we know that the @xmath22 torus knot , which is @xmath51 on @xmath139 , is actually a @xmath182 knot on @xmath139 in the preferred framing . so using a similar equation from above , we obtain that @xmath183 .
thus @xmath184 .
we note that there are two universally tight contact structures , diffeomorphic by @xmath108 , which satisfy the conditions set by the above lemma .
we now show that these appropriate @xmath131 and associated @xmath185 actually exist in @xmath0 .
the standard tight contact structure on @xmath0 splits into a universally tight contact structure on @xmath131 and @xmath185 .
the idea is to build @xmath0 .
to begin , choose one of the above two universally tight candidates for @xmath131 .
we then claim we can join @xmath131 to itself by a standard convex annulus @xmath186 with boundary @xmath51 rulings so that @xmath187 is a ( universally tight ) thickened torus with boundary @xmath188 having associated boundary slopes of @xmath189 and @xmath190 and two dividing curves .
one way to see this is that we can think of @xmath139 as being composed of four annuli , one from @xmath191 , one from @xmath192 , and two from @xmath193 $ ] . since we are constructing the thickened torus , with a suitable choice of holonomy of @xmath186
, we can assure that the dividing curves on @xmath194 have only one longitude , and two components . since we know the twisting of @xmath51 on @xmath131 is equal to @xmath195 , a calculation shows that the dividing curves on @xmath194 must have slope @xmath190 .
but then the slopes of the dividing curves on @xmath194 and @xmath124 are determined , making the slope of dividing curves on @xmath149 equal to @xmath189 based on equation 8 in lemma 4.3 in @xcite .
now as in the proof of lemma 5.2 in @xcite , the contact structure on @xmath196 can be isotoped to be transverse to the fibers of @xmath196 , which are parallel copies of @xmath197 , while preserving the dividing set on @xmath198 .
such a horizontal contact structure is universally tight .
we then use the classification of tight contact structures on @xmath0 , solid tori , and thickened tori to conclude that any tight contact structure on @xmath199 $ ] with boundary conditions being tori with two dividing curves and slopes @xmath189 and @xmath190 glues together with standard neighborhoods of unknots with those boundary slopes to give the tight contact structure on @xmath0 .
we now show that these @xmath131 with complements @xmath185 fail to thicken .
the @xmath131 with complement @xmath185 fail to thicken . by inequality 14 in @xcite
, it suffices to show that @xmath131 does not thicken to any @xmath200 for @xmath201 .
so to this end , observe that the @xmath22 positive torus knot is a fibered knot over @xmath202 with fiber a seifert surface @xmath110 of genus @xmath203 ( see @xcite ) .
moreover , the monodromy is periodic with period @xmath204 .
thus , @xmath185 has a @xmath204-fold cover @xmath205 .
if one thinks of @xmath185 as @xmath206 $ ] modulo the relation @xmath207 for monodromy @xmath208 , then one can view @xmath209 as @xmath204 copies of @xmath206 $ ] cyclically identified via the same monodromy . now note that downstairs in @xmath185 , @xmath51 intersects any given seifert surface @xmath204 times efficiently .
it is therefore evident that we can view @xmath185 as a seifert fibered space with base space @xmath110 and two singular fibers ( the components of the hopf link ) .
the regular fibers are topological copies of @xmath51 , which itself is a legendrian ruling on @xmath139 with twisting @xmath210 . in fact
, the regular fibers can be assumed to be legendrian isotopic to the @xmath139-fibers except for small neighborhoods around the singular fibers .
we claim the pullback of the tight contact structure to @xmath209 admits an isotopy where the @xmath202 fibers are all legendrian and have twisting number @xmath210 with respect to the product framing .
this isotopy can be accomplished because in @xmath209 , the lifts of the singular fibers have tight neighborhoods with convex boundary tori which have dividing curves with one longitude and where @xmath51 has twisting @xmath210 .
thus these neighborhoods of the lifts of the singular fibers are in fact standard neighborhoods of a legendrian fiber with twisting @xmath210 ; the contact structure can then be isotoped so that every fiber inside these neighborhoods is legendrian with twisting @xmath210 .
so , if @xmath131 can be thickened to @xmath200 , then there exists a legendrian curve topologically isotopic to the regular fiber of the seifert fibered space @xmath185 with twisting number greater than @xmath210 , measured with respect to the seifert fibration . pulling back to the @xmath204-fold cover @xmath209
, we have a legendrian knot which is topologically isotopic to a fiber but has twisting greater than @xmath210 .
we will obtain a contradiction , thus proving that @xmath131 can not be thickened to @xmath200 . to obtain our contradiction
, we let @xmath211 be the projection map onto the base space .
thus the hypothesis that @xmath131 can be thickened to @xmath200 yields a knot @xmath212 which is isotopic to @xmath213 for some @xmath214 , but where @xmath215 .
thus there is a continuous isotopy @xmath216 where @xmath217 and @xmath218 .
now look at @xmath219 .
then this is a continous map , and since @xmath220 , we actually obtain @xmath221 .
this means that @xmath222 is contained inside a tight @xmath106 that is fibered by legendrian fibers with twisting @xmath210 , and is thus a solid torus neighborhood of a legendrian knot with twisting @xmath210 . by the classification of tight contact structures on solid tori ,
such a @xmath222 can not exist .
this is our contradiction .
now that we know that the base case holds for positive torus knots , we begin to prove lemma [ indhyppreserved ] for the bulk of this section we will thus have that @xmath127 , @xmath6 satisfies the inductive hypothesis , and we work to show that @xmath11 satisfies the inductive hypothesis . we will need to break the proof of lemma 3.2 into two cases , case i being where @xmath223 , and case ii being where @xmath224 .
however , before we do that , we prove two general lemmas concerning iterated cablings that begin with positive torus knots . if @xmath6 is an iterated torus knot with @xmath225 , then @xmath226 .
we use induction .
@xmath227 is evident from equation 1 above .
then inductively , @xmath228 .
we now use the above lemma to prove the following .
let @xmath6 be an iterated torus knot with @xmath225 . if @xmath229 and both @xmath230 , then @xmath231 .
we have that @xmath231 if and only if @xmath232 . but
this is true if and only if @xmath233 , which is true .
we now directly address the two different cases in two different subsections .
+ + we work through proving items 2 - 5 in the inductive hypothesis via a series of lemmas .
the following lemma begins to address item 2 . if @xmath223 , then @xmath234 .
the proof is similar to that of lemma 3.3 in @xcite .
we first claim that @xmath235 . if not , there exists a legendrian @xmath236 with @xmath237 and a solid torus @xmath121 with @xmath236 as a legendrian divide .
but then we would have a boundary slope of @xmath223 in the @xmath37 framing , which can not occur .
so since @xmath235 , any legendrian @xmath236 must be a ruling on a convex @xmath238 with slope @xmath239 in the @xmath50 framing .
but then if @xmath240 , we have that @xmath241 .
this shows that @xmath242 is achieved by a legendrian ruling on a convex torus having slope @xmath243 in the standard @xmath37 framing . finally , note that @xmath244 . with the following lemma we prove that items 3 and 4 of the inductive hypothesis hold for @xmath11 .
[ nonthickening inductive step ] if @xmath223 , let @xmath245 be a solid torus representing @xmath11 , for @xmath89 .
then @xmath245 can be thickened to an @xmath246 for some nonnegative integer @xmath247 . moreover ,
if @xmath245 fails to thicken , then it has the same boundary slope as some @xmath246 , as well as at least @xmath248 dividing curves . in this case , for the @xmath50 framing , we have either @xmath249 or @xmath250 ( the latter being relevant only if @xmath251 ) .
the proof in this case is nearly identical to the proof of lemma 4.4 in @xcite ; we will include some of the details , however , as certain particular calculations differ .
moreover , we will use modifications of this argument in case ii and thus will be able to refer to the details here .
let @xmath245 be a solid torus representing @xmath11 .
let @xmath36 be a legendrian representative of @xmath6 in @xmath252 and such that we can join @xmath253 to @xmath254 by a convex annulus @xmath255 whose boundaries are @xmath256 and @xmath51 rulings on @xmath253 and @xmath254 , respectively .
then topologically isotop @xmath257 in the complement of @xmath245 so that it maximizes @xmath65 over all such isotopies ; this will induce an ambient topological isotopy of @xmath255 , where we still can assume @xmath255 is convex .
a picture is shown in ( a ) in figure [ fig : non - thickening1b ] . in the @xmath50 framing we will have @xmath258 where @xmath259 , since @xmath260 .
now if @xmath261 , then there will be no bypasses on the @xmath253-edge of @xmath255 , since the @xmath256 ruling would be at maximal twisting .
on the other hand , if @xmath262 , then there will still be no bypasses on the @xmath253-edge of @xmath255 , since such a bypass would induce a destabilization of @xmath257 , thus increasing its @xmath65 by one see lemma 4.4 in @xcite . to satisfy the conditions of this lemma
, we are using the fact that either @xmath249 or @xmath250 .
furthermore , we can thicken @xmath245 through any bypasses on the @xmath254-edge , and thus assume @xmath255 is standard convex .
now let @xmath263 .
inductively we can thicken @xmath121 to an @xmath90 with intersection boundary slope @xmath91 where @xmath13 is minimized over all such thickenings ( if we have @xmath97 , then we will have @xmath245 thickening to a standard neighborhood of a knot at @xmath18 see the proof of theorem 1.1 in @xcite ; so we can assume @xmath138 ) .
then consider a convex annulus @xmath264 from @xmath253 to @xmath265 , such that @xmath266 is in the complement of @xmath267 and @xmath268 consists of @xmath256 rulings .
a picture is shown in ( b ) in figure [ fig : non - thickening1b ] . by an argument identical to that used in lemma 4.4 in @xcite ,
@xmath266 is standard convex ; in brief , if @xmath266 was not standard convex , either a bypass would occur on its @xmath253-edge , or @xmath13 would not be minimized , neither of which is true .
[ nonthickening1 ] is the larger solid torus in gray ; @xmath269 is the smaller solid torus in gray.,title="fig : " ] now four annuli compose the boundary of a solid torus @xmath270 containing @xmath245 : the two sides of a thickened @xmath266 ; @xmath271 ; and @xmath272 .
we can compute the intersection boundary slope of this solid torus . to this end ,
recall that @xmath258 where @xmath273 ( @xmath274 would be the @xmath84 case which we have take care of above ) . to determine @xmath275 we note that the geometric intersection of @xmath256 with @xmath69 on @xmath265 and @xmath253 must be equal , yielding the equality @xmath276 these equal quantities are greater than zero , since @xmath277 we note here that this will yield @xmath278 for the calculations below . in the meantime , however , the above equation gives @xmath279 we define the integer @xmath280 .
we now choose @xmath281 to be a curve on these two tori such that @xmath282 , and we change coordinates to a framing @xmath283 via the map @xmath284 . under this map
we obtain @xmath285 @xmath286 we then obtain in the @xmath50 framing , after edge - rounding , that the intersection boundary slope of @xmath270 is @xmath287 this shows that any @xmath245 representing @xmath11 can be thickened to one of the @xmath246 , and if @xmath245 fails to thicken , then it has the same boundary slope as some @xmath246 .
we now note that if @xmath245 fails to thicken , and if it has the minimum number of dividing curves over all such @xmath245 which fail to thicken and have the same boundary slope as @xmath246 , then @xmath245 is actually an @xmath246 , by an argument identical to that used in lemma 4.4 in @xcite . in brief , if @xmath245 fails to thicken and is at minimum number of dividing curves , then taking @xmath288 gives an @xmath90 ; one then concludes that @xmath245 is an @xmath246 .
we now finish the proof of item 2 of the inductive hypothesis . if @xmath223 , then @xmath289 .
we show that @xmath290 for any candidate @xmath246 . as a consequence ,
since any @xmath245 thickens to some @xmath246 ( including @xmath291 ) , we have , to prevent overtwisting , that @xmath289 .
now note that our intended inequality is automatically true if @xmath292 ; thus we may assume that @xmath235 .
we have that @xmath290 holds if and only if @xmath293 inductively we know that @xmath294 where @xmath295 .
this implies that @xmath296 we can now prove inequality 11 ; we begin with @xmath297 .
we have : @xmath298 we conclude this subsection by proving item 5 of the inductive hypothesis . if @xmath223 , the candidate @xmath246 exist and actually fail to thicken for @xmath299 , where @xmath300 is some positive integer
. moreover , these @xmath246 have contact structures that are universally tight and have convex meridian discs whose bypasses bound half - discs all of the same sign . also , the preferred longitude on @xmath301 has rotation number zero for @xmath302 .
we first prove that the contact structure on a candidate @xmath246 which fails to thicken is universally tight .
to see this note that from lemma 5.2 above , and the inductive hypothesis , such a candidate @xmath246 is embedded inside a @xmath90 with a universally tight contact structure .
now there is a @xmath303-fold cover of @xmath90 that maps @xmath303 lifts @xmath304 to @xmath246 , the lifts themselves each being an @xmath106 .
this cover in turn has a universal cover @xmath305 that contains @xmath303 copies of a universal cover @xmath305 of @xmath246 .
since , by the inductive hypothesis , the universal cover of @xmath90 has a tight contact structure , a tight contact structure is thus induced on the universal cover of @xmath246 . to see that a meridian disc for @xmath246 contains bypasses all of the same sign , note that this is immediate if @xmath301 has two dividing curves . for the case of @xmath68 dividing curves where @xmath306 , we argue in a similar fashion to lemma 4.1 .
specifically , since a meridian disc for @xmath90 inductively has bypasses all of the same sign , a horizontal annulus @xmath158 with boundary on @xmath124 and @xmath307 will have @xmath308 bypasses all of the same sign .
thus , as in lemma 4.1 , a meridian disc for @xmath246 will inherit @xmath309 bypasses all of the same sign . to show that the preferred longitude on @xmath301 has rotation number zero
, we use an argument similar to that used in lemma 4.1 .
we call the meridian disc for @xmath90 , @xmath310 , and the seifert surface for the preferred longitude on @xmath124 , @xmath311 .
if we then look at the @xmath312 cable on @xmath124 , we can calculate its rotation number as @xmath313 but then since this same knot is a @xmath314 cable on @xmath301 , we have that @xmath315 @xmath316 , where @xmath110 is a seifert surface for the preferred longitude on @xmath301 .
this implies that @xmath184 .
now we know inductively that there exists a @xmath123 such that if @xmath317 , then the @xmath90 exist and fail to thicken .
so suppose @xmath318 for some @xmath122 .
we will show that @xmath246 exists and fails to thicken for @xmath319 .
then @xmath300 will be the least such @xmath318 .
we take one of the two universally tight candidate @xmath246 , and as in lemma 4.2 above we construct a universally tight @xmath320 and glue in an appropriate solid torus neighborhood of a legendrian knot @xmath36 to obtain a universally tight @xmath90 , which then glues into @xmath0 inductively .
this shows that @xmath246 exists . to show that @xmath246 fails to thicken , by lemma 5.2 it suffices to show that @xmath246 does not thicken to any @xmath321 , where @xmath322 .
inductively , we can assume @xmath323 fails to thicken ; in particular , the @xmath324 that contains @xmath246 fails to thicken .
thus , if @xmath246 admits a non - trivial thickening , it must do so inside of @xmath324 .
define @xmath325 ; then @xmath100 is a seifert fibered space with one singular fiber , @xmath257 , and with regular fibers that are topologically isotopic to the legendrian copies of @xmath11 on the boundary of @xmath100 .
@xmath100 has a @xmath303-fold cover , @xmath104 , that is a @xmath303-punctured disc times @xmath202 , where the tight contact structure admits an isotopy so that all the @xmath202 fibers are legendrian with twisting @xmath326 with respect to the product framing .
we can then glue in @xmath303 standard neighborhoods of fibers with twisting @xmath326 to obtain an @xmath106 which itself is a standard neighborhood of a knot with twisting @xmath326 .
but then , if @xmath246 thickens to a @xmath321 , where @xmath322 , that means that in this cover there will be a knot isotopic to one of the fibers , but with twisting greater than @xmath326 , contradicting the classification of tight contact structures on solid tori
. + as in case i , we work through proving items 2 - 5 in the inductive hypothesis via a series of lemmas .
we begin by proving item 2 in the inductive hypothesis . if @xmath224 , then @xmath327 .
the proof is almost identical to that of step 1 in theorem 1.5 in @xcite ; we will include the details , though , since certain key aspects differ .
we first examine representatives of @xmath11 at @xmath18 .
since there exists a convex torus representing @xmath6 with legendrian divides that are @xmath256 cablings ( inside of the solid torus representing @xmath6 with @xmath328 ) we know that @xmath329 . to show that @xmath330
, we show that @xmath331 by showing that the contact width @xmath332 , since this will yield @xmath333 .
so suppose , for contradiction , that some @xmath245 has convex boundary with @xmath334 , as measured in the @xmath50 framing , and two dividing curves .
after shrinking @xmath245 if necessary , we may assume that @xmath335 is a large positive integer .
then let @xmath98 be a convex annulus from @xmath254 to itself having boundary curves with slope @xmath51 .
taking a neighborhood of @xmath336 yields a thickened torus @xmath320 with boundary tori @xmath148 and @xmath149 , arranged so that @xmath148 is inside the solid torus @xmath121 representing @xmath6 bounded by @xmath149 .
now there are no boundary parallel dividing curves on @xmath98 , for otherwise , we could pass through the bypass and increase @xmath335 to @xmath51 , yielding excessive twisting inside @xmath245 .
hence @xmath98 is in standard form , and consists of two parallel nonseparating arcs .
we now choose a new framing @xmath283 for @xmath121 where @xmath337 ; then choose @xmath338 so that @xmath339 and such that @xmath340 and @xmath341 .
as mentioned in [ eh1 ] , this is possible since @xmath342 is obtained from @xmath343 by @xmath344 right - handed dehn twists
. then note that in the @xmath41 framing , we have that @xmath345 , and @xmath346 and @xmath347 are connected by an arc in the farey tessellation of the hyperbolic disc ( see section 3.4.3 in [ h ] ) .
thus , since @xmath348 is connected by an arc to @xmath349 in the farey tessellation , we must have that @xmath350 .
thus we can thicken @xmath121 to one of the solid tori with @xmath351 which fails to thicken .
then , just as in claim 4.2 in [ eh1 ] , we have ( i ) inside @xmath320 there exists a convex torus parallel to @xmath352 with slope @xmath346 ; ( ii ) @xmath320 can thus be decomposed into two layered _ basic slices _ ; ( iii ) the tight contact structure on @xmath320 must have _ mixing of sign _ in the poincar@xmath353 duals of the relative half - euler classes for the layered basic slices ; and ( iv ) this mixing of sign can not happen inside the universally tight solid torus which fails to thicken .
this last statement is due to the proof of proposition 5.1 in @xcite , where it is shown that mixing of sign will imply an overtwisted disc in the universal cover of the solid torus .
thus we have contradicted @xmath354 .
so @xmath355 . with the following lemma we prove that items 3 and 4 of the inductive hypothesis hold for @xmath11 .
[ nonthickening inductive step 2 ] if @xmath224 , let @xmath245 be a solid torus representing @xmath11 , for @xmath89 .
then @xmath245 can be thickened to an @xmath246 for some nonnegative integer @xmath247 . moreover ,
if @xmath245 fails to thicken , then it has the same boundary slope as some @xmath246 , as well as at least @xmath248 dividing curves .
this is the case where @xmath356 but @xmath357 ; we have that @xmath331 .
we begin as we did in case i. if @xmath245 is a solid torus representing @xmath11 , as before choose @xmath36 in @xmath252 such that @xmath307 is joined to @xmath254 by an annulus @xmath255 , and with @xmath358 maximized over topological isotopies in the space @xmath252 .
now suppose @xmath258 where @xmath359 .
then inside @xmath360 is an @xmath121 with boundary slope @xmath346 .
but then we can extend @xmath255 to an annulus that has no twisting on one edge , and we can thus thicken @xmath245 so it has boundary slope @xmath51 .
moreover , since there is twisting inside @xmath360 , we can assure there are two dividing curves on the thickened @xmath245 .
so this situation yields no nontrivial solid tori @xmath245 which fail to thicken . alternatively , suppose @xmath361 .
furthermore , for the moment suppose @xmath362 .
then we can use lemma 4.4 in @xcite to conclude that there are no bypasses on the @xmath307-edge of @xmath255 , and so we can thicken @xmath245 through bypasses so that @xmath255 is standard convex .
then the calculation of the boundary slope goes through as above in lemma 5.4 , and we conclude that @xmath245 thickens to some @xmath246 .
the @xmath90 that is used for this will have @xmath363 ; note that such @xmath90 exist since @xmath364 as @xmath13 increases .
for the remaining case , suppose @xmath361 and @xmath275 is the least positive integer satisfying this inequality .
thus @xmath365 .
again look at the @xmath307-edge of @xmath255 .
we claim that this edge has no bypasses .
so , for contradiction , suppose it does
. then we can thicken @xmath360 to a solid torus where the ( efficient ) geometric intersection of @xmath256 with dividing curves is less than @xmath366 .
suppose the slope of this new solid torus is @xmath367 , where @xmath368 since @xmath275 is minimized in the complement of @xmath245 .
we do some calculations .
note first that if @xmath369 , then @xmath370 , which means @xmath371 , which implies @xmath372 , which can not happen , again since @xmath275 is minimized in the complement of @xmath245 .
thus we must have @xmath373 .
but then the geometric intersection of @xmath256 with @xmath374 is @xmath375 = p_{r+1 } + mq_{r+1}$ ] .
this is a contradiction .
thus there are no bypasses on the @xmath307-edge of @xmath255 , and we can thicken @xmath245 through any bypasses so that @xmath255 is standard convex . the calculations that show @xmath245 thickens to @xmath246 go through as above in lemma 5.4 .
this shows that any @xmath245 representing @xmath11 can be thickened to one of the @xmath246 , and if @xmath245 fails to thicken , then it has the same boundary slope as some @xmath246 .
we now show that if @xmath245 fails to thicken , and if it has the minimum number of dividing curves over all such @xmath245 which fail to thicken and have the same boundary slope as @xmath246 , then @xmath245 is actually an @xmath246 . to see this , as above
we can choose a legendrian @xmath257 that maximizes @xmath65 in the complement of @xmath245 and such that we can join @xmath253 to @xmath254 by a convex annulus @xmath255 whose boundaries are @xmath256 and @xmath51 rulings on @xmath253 and @xmath254 , respectively .
now since @xmath245 fails to thicken , we can assume that @xmath277 and that there are no bypasses on the @xmath253-edge , and in this case we have no bypasses on the @xmath254-edge since @xmath245 fails to thicken and is at minimum number of dividing curves .
as above , let @xmath376 .
we claim this @xmath267 fails to thicken the proof proceeds identically as above in lemma 5.4 , as does the proof that @xmath245 is in fact an @xmath246 .
the following proof of item 5 of the inductive hypothesis is similar to that of case i. if @xmath224 , the candidate @xmath246 exist and actually fail to thicken for @xmath299 , where @xmath300 is some positive integer
. moreover , these @xmath246 have contact structures that are universally tight and have convex meridian discs whose bypasses bound half - discs all of the same sign . also , the preferred longitude on @xmath301 has rotation number zero for @xmath302 . the proof that the contact structure on a candidate @xmath246 which fails to thicken is universally tight is identical to the argument in case i , as is the proof that their convex meridian discs have bypasses all of the same sign , as well as the proof that the rotation number of the preferred longitude is zero .
now we know inductively that there exists a @xmath123 such that if @xmath122 , then the @xmath90 exist and fail to thicken .
so suppose @xmath318 for some @xmath122 .
also assume that @xmath363 ; we know such a @xmath13 exists since @xmath364 as @xmath13 increases .
then @xmath246 exists and fails to thicken as in the argument for case i for @xmath319 , and @xmath300 will be the least such @xmath318 .
we provide below the proof of lemma 3.3 , which is really just a matter of referencing a previous proof .
this is the case where @xmath377 , we know @xmath6 satisfies the inductive hypothesis , and we wish to show that @xmath11 satisfies the utp .
the proof is identical to that of steps 1 and 2 in the proof of theorem 1.5 from @xcite , the key being that since @xmath377 , this cabling slope is shielded from any @xmath90 that fail to thicken .
we have completed the utp classification of iterated torus knots ; it now remains to show that in the class of iterated torus knots , failing the utp is a sufficient condition for supporting transversally non - simple cablings . to this end , in this section we prove theorem [ second theorem ] ; we do so by working through a series of lemmas .
these lemmas will first give us information about just a piece of the legendrian mountain range for @xmath117 where @xmath7 for all @xmath5 ; we will then use this information to obtain enough information about the legendrian mountain ranges of certain cables @xmath11 to conclude that these cables are transversally non - simple .
we will therefore not be completing the legendrian or transversal classification of these iterated torus knots .
a formula for @xmath379 is given at the end of the proof of corollary 3 in @xcite .
in the notation used in that paper , the formula is @xmath380 , since in our case all the @xmath381 as we are cabling positively at each iteration .
however , note that our @xmath382 corresponds to @xmath383 in @xcite for @xmath384 .
[ sliceofmtnrange ] suppose @xmath117 is an iterated torus knot where @xmath7 for all @xmath5
. then there exists legendrian representatives @xmath391 with @xmath392 and @xmath393 ; also , @xmath391 destabilizes .
the lemma is true for positive torus knots @xcite , so we inductively assume it is true for @xmath389 .
then look at legendrian rulings @xmath394 on standard neighborhoods of the inductive @xmath395 . in the @xmath50 framing the boundary slope of these @xmath396
is @xmath397 , and so a calculation shows that @xmath398 ; hence @xmath399 .
to calculate the rotation number of @xmath394 , we use the following formula from @xcite , where @xmath101 is a convex meridian disc for @xmath396 and @xmath110 is a seifert surface for the preferred longitude on @xmath400 : this , along with lemma [ eulerchar ] , shows us that @xmath404 is on the right - most slope of the legendrian mountain range of @xmath6 , and @xmath405 is on the left - most edge . to the former we can perform positive stabilizations to reach @xmath406 at @xmath407 and @xmath408 ; to the latter we can perform negative stabilizations to reach @xmath409 at @xmath407 and @xmath410 we know such stabilizations can be performed since @xmath411 .
so suppose @xmath6 is an iterated torus knot that fails the utp ( which is precisely when @xmath7 for all @xmath5 )
. then we know that for @xmath122 there exist non - thickenable solid tori @xmath90 having intersection boundary slopes of @xmath91 , where these slopes are measured in the @xmath50 framing . switching to the standard @xmath37 framing ,
these intersection boundary slopes are @xmath412 .
now as @xmath413 , there are infinitely many values of @xmath174 which are prime and greater than @xmath414 . as a consequence ,
there are infinitely many @xmath90 with two dividing curves . based on this observation
, we make the following definition : _ suppose @xmath117 is an iterated torus knot where @xmath7 for all @xmath5 .
let @xmath415 be a cabling of @xmath6 with @xmath50 slope @xmath91 , where @xmath416 and there is an @xmath90 with two dividing curves that fails to thicken .
_ so given @xmath6 , there are infinitely many such cabling knot types @xmath415 , all of these being cablings of the form @xmath417 as measured in the preferred framing .
the following lemma will then prove theorem [ second theorem ] .
we now look at the two universally tight non - thickenable @xmath90 that have representatives of @xmath415 as legendrian divides .
these legendrian divides have @xmath420 . to calculate rotation numbers
, we have two possibilities , depending on which boundary of the two universally tight @xmath90 the legendrian divides reside . using the formula from @xcite
, we obtain we will call the two legendrian divides corresponding to @xmath422 , @xmath423 respectively .
we can calculate the self - linking number for the negative transverse push - off of @xmath424 to be @xmath425 .
this shows that @xmath424 is on the right - most edge of the legendrian mountain range and is at @xmath18 .
similarly , @xmath426 is on the left - most edge of the legendrian mountain range and is at @xmath18 .
we now look at solid tori @xmath427 with intersection boundary slope @xmath91 , but which _ thicken _ to solid tori with intersection boundary slopes @xmath428 .
such tori @xmath429 are embedded in universally tight basic slices bounded by tori with dividing curves of slope @xmath428 and @xmath128 .
legendrian divides on such @xmath427 have @xmath430 ; to calculate possible rotation numbers for these legendrian divides , we recall the procedure used in the proof of theorem 1.5 in @xcite .
there we used a formula for the rotation numbers from @xcite , where the range of rotation numbers was given by the following ( substituting @xmath431 for @xmath432 ) : now from lemma [ sliceofmtnrange ] we know that there is an @xmath36 with @xmath434 and @xmath435 . plugging this value of the rotation number into the expression above yields @xmath436 .
we will call the legendrian divides having these rotation numbers @xmath437 , respectively .
important for our purposes is that @xmath437 have the same values of @xmath65 and @xmath31 as @xmath423 .
consider first @xmath442 .
it is in fact one of the dividing curves on @xmath124 , and is also at maximal self - linking number for @xmath415 .
similarly , @xmath443 is one of the dividing curves on @xmath429 , and is also at maximal self - linking number . now from @xcite we know that @xmath415 is a fibered knot that supports the standard contact structure , since it is an iterated torus knot obtained by cabling positively at each iteration . as a consequence , from @xcite , we also know that @xmath415 has a unique transversal isotopy class at @xmath21 . hence we know that @xmath442 and @xmath443 are transversally isotopic .
thus there is a transverse isotopy ( inducing an ambient contact isotopy ) that takes these two dividing curves on the two different tori to each other .
thus we may assume that @xmath124 and @xmath429 intersect along one component of the dividing curves ; we call this component @xmath444 .
now suppose , for contradiction , that @xmath440 is transversally isotopic to @xmath441 .
these transverse knots are represented by the other two non - intersecting dividing curves on @xmath124 and @xmath429 , respectively , and there is a transverse isotopy taking one to the other .
we claim that this transverse isotopy can be performed relative to @xmath444 .
to see this , note that associated to @xmath445 is an open book decomposition of @xmath0 , with pages being seifert surfaces @xmath110 for the knot @xmath444 .
moreover , the standard contact structure is supported by this open book decomposition .
thus the transverse isotopy taking @xmath440 to @xmath441 will induce an ambient isotopy of open book decompositions supporting the standard contact structure , all with a transversal representative of @xmath444 on the binding .
since @xmath124 is incompressible in @xmath445 , it is therefore evident that the isotopy taking @xmath440 to @xmath441 can be accomplished simply as an isotopy of @xmath124 relative to @xmath444 .
thus we may assume that after a contact isotopy of @xmath0 , @xmath124 and @xmath429 intersect along their two dividing curves , which we denote as @xmath444 and @xmath446 , and we observe that there is an isotopy ( not necessarily a contact isotopy ) of @xmath90 to @xmath427 relative to @xmath444 and @xmath446 .
we claim that as a result @xmath427 can not thicken , thus obtaining our contradiction .
we do this by noting that the isotopy of @xmath90 to @xmath427 relative to @xmath444 and @xmath446 may be accomplished by the attachment of successive bypasses . since these bypasses are attached in the complement of the two dividing curves , none of these bypass attachments can change the boundary slope .
however , they may increase or decrease the number of dividing curves . starting with @xmath447 ,
we make the following inductive hypothesis , which we will prove is maintained after bypass attachments : * @xmath448 is a convex torus which contains @xmath444 and @xmath446 , and thus has slope @xmath91 .
* @xmath448 is a boundary - parallel torus in a @xmath150$]-invariant @xmath449 $ ] with @xmath450 , where the boundary tori have two dividing curves .
* there is a contact diffeomorphism @xmath451 which takes @xmath449 $ ] to a standard @xmath452-invariant neighborhood of @xmath124 and matches up their complements .
the argument that follows is similar to lemma 6.8 in @xcite .
first note that item 1 is preserved after a bypass attachment , since such a bypass is in the complement of @xmath444 and @xmath446 , and thus can not change the slope of the dividing curves . to see that items 2 and 3 are preserved
, suppose that @xmath453 is obtained from @xmath448 by a single bypass .
since the slope was not changed , such a ( non - trivial ) bypass must either increase or decrease the number of dividing curves by 2 .
suppose first that the bypass is attached from the inside , so that @xmath454 , where @xmath73 is the solid torus bounded by @xmath448 . for convenience ,
suppose @xmath455 inside the @xmath449 $ ] satisfying items 2 and 3 of the inductive hypothesis .
then we form the new @xmath456 $ ] by taking the old @xmath456 $ ] and adjoining the thickened torus between @xmath448 and @xmath453 . now
@xmath453 bounds a solid torus @xmath457 , and , by the classification of tight contact structures on solid tori , we can factor a nonrotative outer layer which is the new @xmath458 $ ] . alternatively , if @xmath459 , then we know that @xmath457 thickens to an @xmath90 , and thus there exists a nonrotative outer layer @xmath456 $ ] for @xmath460 , where @xmath148 has two dividing curves
. thus the proof is done , for after enough bypass attachments we will obtain @xmath461 , with @xmath427 non - thickenable . but this is a contradiction , since @xmath427 does thicken . | we prove that an iterated torus knot type fails the uniform thickness property ( utp ) if and only if all of its iterations are positive cablings , which is precisely when an iterated torus knot type supports the standard contact structure .
we also show that all iterated torus knots that fail the utp support cabling knot types that are transversally non - simple .
v c u _
sl(2,c ) ps .
psl(2,c ) _ | 0909.1452 |
an intense electromagnetic field makes possible the processes which are forbidden in a vacuum such as the neutrino production of an electron
positron pair @xmath0 . the list of papers devoted to an analysis of this process and the collection of the results obtained could be found e.g. in ref . @xcite . in most cases , calculations of this kind
were made either in the crossed field approximation , or in the limit of a superstrong field much greater than the critical value of @xmath1 g , when the electrons and positrons were born in states corresponding to the ground landau level . however , there are physical situations of the so - called moderately strong magnetic field , @xmath2 is the electron mass , and @xmath3 is the elementary charge . ] , @xmath4 , when electrons and positrons mainly occupy the ground landau level , however , a noticeable fraction may be produced at the next levels .
the indicated hierarchy of physical parameters corresponds to the conditions of the kerr black hole accretion disk , regarded by experts as the most likely source of a short cosmological gamma - ray burst .
the disc is a source of copious neutrinos and anti - neutrinos , which partially annihilate above the disc and turn into @xmath5 pairs , @xmath6 .
this process was proposed and investigated in many details , see e.g. ref .
@xcite and the papers cited therein , as a possible mechanism for creating relativistic , @xmath5-dominated jets that could power observed gamma - ray bursts . in ref .
@xcite , in addition to @xmath7 annihilation , the contribution of the magnetic field - induced process @xmath0 to the neutrino energy deposition rate around the black hole was also included .
however , in calculations of the efficiency of the electron - positron plasma production by neutrino through the process @xmath0 in those physical conditions @xcite ( @xmath8 to 180 @xmath9 , @xmath10 to 25 mev ) , it should be kept in mind that approximations of both the crossed and superstrong field have a limited applicability here . we know
a limited number of papers @xcite , where the probability of neutrino - electron processes was investigated , as the sum over the landau levels of electrons ( positrons ) . in the papers
@xcite , only the neutrino - electron scattering channel in a dense magnetized plasma was studied , which was the crossed process to the considered here neutrino creation of electron - positron pairs . in the paper @xcite , also devoted to the study of the process @xmath0 , the analytical calculations were presented in a rather cumbersome form , caused by the choice of solutions of the dirac equation .
the final results for the process probability were obtained by numerical calculations for some set of landau levels occupied by electrons and positrons . in astrophysical applications , there exists probably more interesting value than the process probability , namely , the mean value of the neutrino energy loss , caused by the influence of an external magnetic field .
thus , the aim of this paper is the study of the process @xmath0 in the physical conditions of the moderately strong magnetic field , where the electrons and positrons would be born in the states corresponding to the excited landau levels , and the theoretical description would contain a relatively simple analytical formulas for the mean value of the neutrino energy loss , for a wide range of landau levels .
more details of the analysis can be found in our recent paper @xcite .
we use the standard calculation technics , see e.g. ref . @xcite .
the effective local lagrangian of the process can be written in the form @xmath11 \ , \big [ \bar \nu \gamma^{\alpha } ( 1 - \gamma_5 ) \nu \big ] \ , , \label{eq : l}\ ] ] where the electron field operators are constructed on a base of the solutions of the dirac equation in the presence of an external magnetic field .
the constants @xmath12 and @xmath13 for different neutrino types are : @xmath14 where @xmath15 is the weinberg angle .
the total probability of the process @xmath16 where the electron and the positron are created in the states corresponding to @xmath17th and @xmath18th landau levels correspondingly , is , in a general case , the sum of the probabilities of the four polarization channels : @xmath19 for each of the channels , the differential probability over the final neutrino momentum per unit time can be written as @xmath20 where @xmath21 is the total interaction time , @xmath22 is the total volume of the interaction region , @xmath23 is the @xmath24-matrix element constructed with the effective lagrangian ( [ eq : l ] ) , and the elements of the phase volume are introduced for the electron and the positron ( the magnetic field is directed along the @xmath25 axis ) : @xmath26 in the integration over the momenta of the electron and the positron , a condition arises : @xmath27 which determines the range of integration over the final neutrino momentum . here
, @xmath28 is the change of the four - vector of the neutrino momentum equal to the four - momentum of the @xmath29 pair , @xmath30 and @xmath31 are the four - momenta of the initial and final neutrinos . in turn , the condition ( [ eq : cond ] ) can be satisfied when the energy of the initial neutrino exceeds a certain threshold value . in the reference frame ,
hereafter called @xmath32 , where the momentum of the initial neutrino directed at an angle @xmath33 to the magnetic field , the threshold energy is given by : @xmath34 in fig . [
fig : open_levels ] , the landau levels of @xmath35 are shown , similarly to fig . 1 of ref
. @xcite , to be excited accoring to the condition ( [ eq : conde ] ) , at @xmath36 mev , and at @xmath37 and @xmath38 .
to be excited when @xmath39 , at @xmath40 mev , and at @xmath37 ( left ) and @xmath38 ( right ) . ] to be excited when @xmath39 , at @xmath40 mev , and at @xmath37 ( left ) and @xmath38 ( right ) . ]
it is convenient to perform further integration over the final neutrino momentum , without loss of generality , not in an arbitrary reference frame @xmath32 , but in the special frame @xmath41 , where the initial neutrino momentum is perpendicular to the magnetic field , @xmath42 .
one can then return from @xmath41 to @xmath32 by the lorentz transformation along the field ( recall that the field is invariant with respect to this transformation ) .
it is convenient to use the dimensionless cylindrical coordinates in the space of the final neutrino momentum vector @xmath43 : @xmath44 & & r = e'/e_{\mprp } = \sqrt{\rho^2 + z^2 } \ , .
\label{eq : variab}\end{aligned}\ ] ] here , @xmath45 is the energy of the initial neutrino in the frame @xmath41 , which is connected with its energy @xmath10 in an arbitrary frame @xmath32 by the relation @xmath46 .
we do not present here the set of expressions for the probability of the process @xmath16 , which can be found in the paper @xcite .
these probabilities evaluated numerically as the functions of the initial neutrino energy and of the magnetic field strength for all channels considered in ref .
@xcite , where the electron and positron are created in the lower landau levels , are in a good agreement with the results of that paper .
the probability of the @xmath0 process defines its partial contribution into the neutrino opacity of the medium .
the estimation of the neutrino mean free path with respect to this process gives the result which is too large @xcite compared with the typical size of a compact astrophysical object , e.g. the supernova remnant , where a strong magnetic field could exist .
however , a mean free path does not exhaust the neutrino physics in a medium . in astrophysical applications , we could consider the values that probably are more essential , namely , the mean values of the neutrino energy and momentum losses , caused by the influence of an external magnetic field . these values can be described by the four - vector of losses @xmath47 , @xmath48 where @xmath49 is the total differential probability of the process @xmath0 .
the zeroth component of @xmath47 is connected with the mean energy lost by a neutrino per unit time due to the process considered , @xmath50 .
the space components of the four - vector ( [ eq : q0 ] ) are similarly connected with the mean neutrino momentum loss per unit time , @xmath51 .
it should be noted that the four - vector of losses @xmath47 can be used for evaluating the integral effect of neutrinos on plasma in the conditions of not very dense plasma , e.g. of a supernova envelope , when an one - interaction approximation of a neutrino with plasma is valid @xcite . in ref .
@xcite , the formula ( 10 ) for the energy deposition rate was taken , which was calculated in the crossed field limit @xcite . by the way , the value @xmath52 defined by eq .
( 10 ) of ref .
@xcite is not the 4-vector while the value @xmath53 is .
however , in the region of the physical parameters used in ref .
@xcite ( @xmath8 to 180 @xmath9 , @xmath10 to 25 mev ) , the approximation of a crossed field is poorly applicable , as well as the approximation of a superstrong field when @xmath29 are created in the ground landau level .
the contribution of the next landau levels which can be also excited , should be taken into account .
we present here the results of our calculation of the mean neutrino energy losses caused by the process @xmath0 in a moderately strong magnetic field , i.e. in the conditions of the kerr black hole accretion disk .
we parametrize the energy deposition rate as : @xmath54 where @xmath55 , and the dependence on the initial neutrino energy and the field strength is described by the function @xmath56 .
this function calculated in ref .
@xcite in the crossed field limit had the form @xmath57 on the other hand , in the strong field limit when both electron and positron are born in the ground landau level , the function @xmath58 was also calculated in ref .
@xcite and can be presented in the form @xmath59 where @xmath60 is the modified bessel function . in conditions of moderately strong magnetic field , when the electron and the positron are created in the process @xmath16 in the @xmath17th and @xmath18th landau levels , the result has more complicated form .
it is significantly simplified when one of the particles , electron or positron , is born in the ground landau level , and if an approximation @xmath61 is used .
we obtain the contribution of the channels @xmath62 and @xmath63 to the function @xmath58 as follows : @xmath64 & & \times \left[(1 - \rho^2)^2 + 4 r^2 - 2 r ( 1+\rho^2 ) \right ] \int\limits_0^{2 \pi } \frac{{\mathrm{d}}\phi}{2 \pi } ( r-\rho \cos \phi ) \nonumber\\[2 mm ] & & \times \ , ( 1 - 2 \rho \cos \phi + \rho^2)^{n-1 } \exp \ ! \left ( - \frac{y^2 ( 1 - 2 \rho \cos \phi + \rho^2)}{2 \eta } \right ) , \label{eq : f_(0n)}\end{aligned}\ ] ] where @xmath65 in figs .
[ fig : function180][fig : function50 ] , the function @xmath58 obtained in different approximations is shown at @xmath66 .
it can be seen that the crossed field limit gives the overstated result which is in orders of magnitude greater than the sum of the contributions of lower excited landau levels . on the other hand ,
the results with @xmath29 created at the ground landau level give the main contribution to the energy deposition rate , and almost exhaust it at @xmath67 .
this would mean that the conclusion @xcite that the contribution of the process @xmath0 to the efficiency of the electron - positron plasma production by neutrino exceeds the contribution of the annihilation channel @xmath6 , and that the first process dominates the energy deposition rate , does not have a sufficient basis .
a new analysis of the efficiency of energy deposition by neutrinos through both processes , @xmath6 and @xmath0 , in a hyper - accretion disc around a black hole should be performed , with taking account of our results for the process @xmath0 presented here .
in the paper , a calculation is performed of the mean value of the neutrino energy loss due to the process of electron - positron pair production , @xmath0 , in the magnetic field of an arbitrary strength at which the electrons and positrons can be produced in the states corresponding to the excited landau levels , which could be essential in astrophysical applications .
the results obtained should be used for calculations of the efficiency of the electron - positron plasma production by neutrinos in the conditions of the kerr black hole accretion disk , regarded by experts as the most likely source of a short cosmological gamma - ray burst . in those conditions ,
the crossed field limit used in the previous calculations led to the overstated result which was in orders of magnitude greater than the sum of the lower landau levels .
the study may be also useful for further development of computational techniques for the analysis of quantum processes in an external active environment , particularly in conditions of moderately strong magnetic field , when the allowance for the contribution of only the ground landau level is insufficient .
we dedicate this paper to the blessed memory of our teacher , colleague , and friend nickolay vladimirovich mikheev , who passed away on june 19 , 2014 . for @xmath68 obtained in the crossed field limit (
dotted line ) , with @xmath29 created at the ground ( 0,0 ) landau level ( dashed line ) , and for the sum of all lower landau levels which are excited in this energy interval according to the condition ( [ eq : conde])(solid line).,scaledwidth=87.0% ] the study was performed with the support by the project no .
92 within the base part of the state assignment for the yaroslavl university scientific research , and was supported in part by the russian foundation for basic research ( project no . ) .
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_ * 319 * , 122 ( 1997 ) . | the process of neutrino production of electron positron pairs in a magnetic field of arbitrary strength , where electrons and positrons can be created in the states corresponding to excited landau levels , is analysed .
the mean value of the neutrino energy loss due to the process @xmath0 is calculated .
the result can be applied for calculating the efficiency of the electron - positron plasma production by neutrinos in the conditions of the kerr black hole accretion disc considered by experts as the most possible source of a short cosmological gamma burst .
the presented research can be also useful for further development of the calculation technic for an analysis of quantum processes in external active medium , and in part in the conditions of moderately strong magnetic field , when taking account of the ground landau level appears to be insufficient . | 1408.6985 |
in recent years , research has illuminated a link between active galactic nuclei ( agns ) and the evolution of their host galaxies .
studies have found that a galaxy s central black hole mass is correlated with the mass of the central bulge ( e.g. , * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ) . in the local universe ,
high - luminosity agns are preferentially found in early - type galaxies with young mean stellar ages @xcite . as the redshift increases , the cosmic rates of both star formation and agn activity increase @xcite , and , at high redshift , intensely star - forming submillimeter galaxies have been found to have an agn fraction of @xmath620 - 30@xmath7 @xcite .
the connection between agns and host evolution is also observed in optical colors .
studies have shown that galaxies are organized based on rest - frame color , separating into a ` red sequence ' and a ` blue cloud ' on a color - magnitude diagram ( cmd ; e.g. , * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ) . the sparsely populated ` green valley ' between these regions is thought to be a transitional area , where blue cloud galaxies are rapidly migrating onto the red sequence after the cessation of star formation @xcite .
a proposed mechanism for this quick transition is agn feedback , where the agn activity , driven by major mergers or tidal interactions , causes a truncation of star formation and leads to the color evolution onto the red sequence @xcite .
this theory is supported by a number of studies which find an association between agn activity and the green valley @xcite .
however , some studies using mass - selected samples have found a more uniform color distribution of agn hosts @xcite .
in addition , @xcite have found that @xmath8of green valley agn hosts are dust - reddened members of the blue cloud , although these results are apparently in conflict with the more recent studies of @xcite .
studies examining agns for recent merger activity have also found mixed results ( e.g. , * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
@xcite offer an alternative to major merger driven agns , where agns are instead triggered by minor mergers , mainly in red galaxies .
star formation in this scenario would only be briefly re - ignited at the cores of many of these systems and the galaxies would return to the red sequence after its cessation .
these agns would then represent a red sequence population evolving in luminosity space rather than blue cloud galaxies migrating onto the red sequence .
in support of this theory , several studies have found agn activity associated with blue early - type galaxies or red galaxies with blue cores @xcite .
additionally , @xcite propose that recycled stellar material in elliptical galaxies could fuel central starbursts and black hole accretion . without precise observations of morphologies
, the effects of these two models would appear similar .
other plausible models exist as well , some important only in certain regimes .
for example , at high redshift ( @xmath9 ) , @xcite propose that violent disk instabilities could lead to powerful nuclear activity , while @xcite present a model where lower luminosity agns could be triggered by mild disk instabilities or gas funneled through bars .
note that the properties of individual agns are widely varying , and it could be the case that any number of modes contribute to agn triggering , with different contributions in different mass and/or redshift regimes . to better understand how agn are triggered and the role they play in galaxy evolution
, it is useful to study large - scale structures ( lsss ) at high redshift .
these environments contain a large number of galaxies in the process of transitioning from actively star - forming to passive , and there is evidence that nuclear activity increases at higher redshifts @xcite . to this end
, we have examined the x - ray selected agn population within five lsss at redshifts of @xmath10 - 0.9 .
these structures are in varying states of evolution and include complex superclusters , an interacting supergroup , and isolated clusters .
each has been studied extensively by the observations of redshift evolution in large - scale environments ( orelse ) survey , which is searching for lsss in the vicinities of 20 known clusters between @xmath11 - 1.3 .
the survey has compiled extensive multi - wavelength datasets for each structure , which include multi - band optical , radio , and x - ray imaging , as well as thousands of spectroscopic redshifts @xcite . in this paper , we present analyses of x - ray point sources and the properties of the agn population within the following five lsss : the cl1604 and cl1324 superclusters at @xmath12 and @xmath13 , respectively , the cl0023 + 0423 supergroup at @xmath14 , and two x - ray selected and relatively relaxed , isolated clusters , rxj1757.3 + 6631 at @xmath15 and rxj1821.6 + 6827 at @xmath16 .
we study this sample with a combination of chandra x - ray data , optical imaging , and near - ir and optical spectroscopy . for our cosmological model ,
we assume @xmath17 , @xmath18 , and @xmath19 km s@xmath5 mpc@xmath5 .
we discuss the clusters and superclusters in our survey in section [ sec : samp ] .
observations , data reduction , and techniques are discussed in section [ sec : red ] .
the global properties of our sample are discussed in section [ sec : globchar ] .
the statistical measurements of cumulative source counts are covered in section [ sec : csc ] .
analysis of the agns is presented in section [ sec : agn ] .
in this section we describe the five structures in our sample , which are succintly summarized in table [ strsumtab ] .
the redshift boundaries used in the following analyses were determined by visually examining each structure s redshift histogram . delineating where structures end is not straightforward , with some having associated filaments or possible nearby sheets .
the redshift boundaries are chosen with the aim to include all galaxies which could be part of each overall lss .
lccccccccc cl1604 & 16 04 15 & + 43 16 24 & 0.90 & 0.84 & 0.96 & 10 & 300 - 800 & 531 & 10 + cl0023 & 00 23 51 & + 04 22 55 & 0.84 & 0.82 & 0.87 & 4 & 200 - 500 & 244 & 7 + cl1324 & 13 24 45 & + 30 34 18 & 0.76 & 0.65 & 0.79 & 10 & 200 - 900 & 393 & 6 + rxj1821 & 18 21 32.4 & + 68 27 56 & 0.82 & 0.80 & 0.84 & 1 & @xmath20 & 90 & 3 + rxj1757 & 17 57 19.4 & + 66 31 29 & 0.69 & 0.68 & 0.71 & 1 & @xmath21 & 42 & 1 + [ strsumtab ] the cl1604 supercluster at @xmath22 is one of the largest structures studied at high redshift .
it consists of at least 10 clusters and groups and spans @xmath23mpc along the line of sight and @xmath24mpc in the plane of the sky @xcite .
the massive member clusters cl1604 + 4304 and cl1604 + 4321 were first discovered in the optical survey of @xcite .
the proximity of the clusters suggested that they were components of a larger structure .
further wide field imaging has revealed 10 distinct red galaxy overdensities , suggesting the existence of a supercluster @xcite .
spectroscopic observations have confirmed four of the overdensities to be clusters with velocity dispersions in excess of 500 km s@xmath5 , while four others were confirmed to be poor clusters or groups with dispersions in the range 300 - 500 km s@xmath5 @xcite .
the two most massive clusters in cl1604 have associated diffuse x - ray emission .
cl1604 + 4304 and cl1604 + 4314 , hereafter clusters a and b , have measured bolometric x - ray luminosities of @xmath25 and @xmath26erg s@xmath27 and x - ray temperatures of @xmath28 and @xmath29kev , respectively @xcite .
while these values place cluster a on the @xmath30-@xmath31 curve for virialized clusters , cluster b is well off from it , suggesting that cluster a is relaxed while cluster b is not .
all other clusters have only an upper limit on their bolometric luminosity of @xmath32erg s@xmath27@xcite .
while many galaxies in cl1604 have substantial [ ] emission , near - infrared spectroscopy has shown that a significant portion of this emission is due to contributions from low - ionization nuclear emission - line regions ( liners ) and seyferts @xcite . also , @xcite studied 24@xmath33 m selected galaxies in and around three clusters and three groups in cl1604 using the multiband imaging photometer for _ spitzer _ ( mips ; * ? ? ?
* ) and found evidence for recent starburst activity and an infalling population .
analysis of the morphologies using the advanced camera for surveys ( acs ; * ? ? ?
* ) on the hubble space telescope ( hst ) revealed that many of these 24@xmath33 m bright galaxies were disturbed , indicating mergers and interactions were likely responsible for starburst activity .
we refer the reader to @xcite , @xcite , and @xcite , for more details on the data processing , supercluster properties , and observations .
the cl0023 + 0423 structure at @xmath34 , hereafter cl0023 , was also discovered as an overdensity in the optical survey of @xcite .
the structure was later observed by @xcite using the low - resolution imaging spectrograph ( lris ; * ? ? ?
* ) on the keck 10 m telescope , where the overdensity was resolved into two structures .
further study has shown that the structure consists of four merging galaxy groups separated by approximately 3000 km s@xmath5 in radial velocity and @xmath35 mpc on the plane of the sky @xcite .
the constituent groups have measured velocity dispersions within @xmath36mpc of 480@xmath37 , 430@xmath38 , 290@xmath39 , and 210@xmath40 km s@xmath5 @xcite .
simulations suggest that the groups will merge to form a cluster of mass @xmath41 within @xmath61 gyr @xcite .
@xcite found cl0023 to have a large blue population , with @xmath42of galaxies bluer than their red galaxy color - color cut , down to an @xmath43-band magnitude of 24.5 .
spectroscopic analysis found that @xmath44of galaxies had measurable [ ] emission , which , because of the large blue population , is most likely due to ongoing star formation for a discussion of [ ] emission . ] .
we refer the reader to @xcite and @xcite for more details on the supergroup properties and observations .
the cl1324 supercluster is a lss at @xmath45 .
the two most massive clusters in the structure , cl1324 + 3011 at @xmath46 and cl1324 + 3059 at @xmath15 , were first discovered in the optical survey of @xcite . because of the proximity of the clusters on the sky and in redshift space , this structure was chosen for the orelse survey to investigate the possible existence of structure in the field .
wide - field imaging has revealed a total of ten clusters and groups , detected through red galaxy overdensities , and , despite extensive spectroscopy , only four have been spectroscopically confirmed as constituent clusters or groups ( see gal et al .
2012 , in preparation ) .
lcc cl0023 & 7500 - 7850 & 6200 - 9150 + cl1604 & 7700 & 6385 - 9015 + cl1324 & 7200 & 5900 - 8500 + rxj1821 & 7500 - 7800 & 6200 - 9100 + rxj1757 & 7000 - 7100 & 5700 - 8400 + [ speccovtab ] cl1324 + 3011 was previously studied in @xcite and @xcite , where a velocity dispersion of @xmath47 km s@xmath5 and a temperature of @xmath48kev , using xmm - newton , were measured for the cluster . according to these measurements ,
the cluster does not fall close to the @xmath30-@xmath31 curve for virialized clusters , which would imply that it is not well relaxed .
new chandra results for cl1324 are presented in n. rumbaugh et al .
( 2012 , in preparation ) , including new x - ray temperatures for cl1324 + 3011 and cl1324 + 3059 .
in addition , we present here updated velocity dispersions for these two clusters of @xmath49 and @xmath50 km s@xmath5 , respectively .
the new measurements place cl1324 + 3011 closer to the @xmath30-t curve for virialized clusters , only offset by @xmath51 .
while the older measurements suggested the cluster was not relaxed , the new measurements are more consistent with virialization .
similarly to cl1324 + 3011 , cl1324 + 3059 is offset from the curve , but still by less than 1@xmath30 .
the photometric and spectroscopic observations of the cl1324 supercluster will be covered in full in r. r. gal et al .
( 2012 , in preparation ) .
in this paper , we present redshift histograms of the full structure and velocity dispersions for the four confirmed groups and clusters , as well as the chandra observations .
lcccccc cl1604 & 2465 & 1785 & 158(213 ) & 112(128 ) & 43(48 ) & 38(42 ) + cl0023 & 1136 & 892 & 94(133 ) & 58(72 ) & 39(49 ) & 26(32 ) + cl1324 & 1419 & 1155 & 174(217 ) & 126(133 ) & 38(40 ) & 28(30 ) + rxj1821 & 351 & 306 & 102(132 ) & 64(72 ) & 15(18 ) & 10(13 ) + rxj1757 & 549 & 421 & 87(107 ) & 57(62 ) & 18(19 ) & 9(9 ) + [ srcsum ] the x - ray - selected cluster rxj1821.6 + 6827 , hereafter rxj1821 , at @xmath52 , was the highest redshift cluster discovered in the rosat north ecliptic pole ( nep ) survey , where it is also referred to as nep5281 @xcite .
using xmm - newton data , the cluster was found to have slightly elongated diffuse x - ray emission with a measured bolometric luminosity of @xmath53 and a temperature of @xmath54kev @xcite .
the same study measured a velocity dispersion of @xmath55 km s@xmath5 using 20 cluster members . later analysis by @xcite used 40 galaxies within 1 mpc to measure a velocity dispersion of @xmath56 km s@xmath5 .
redshift histograms of rxj1821 are characteristic of a single , isolated structure , although a small kinematically associated group has been detected to the south @xcite . while the temperature and dispersion measurements place the cluster near the @xmath30-@xmath31 relation for virialized clusters , the elongated x - ray emission could be indicative of still ongoing formation of the cluster .
@xcite measured a blue fraction of only @xmath57for rxj1821 , down to a magnitude limit of @xmath58 .
they found a population dominated by massive , old ( formation epoch of @xmath59 ) galaxies , along with fainter galaxies that were quenched more recently .
they also found that @xmath60of galaxies had detectable [ ] emission .
near - ir spectroscopy of a subset of these [ ] emitting galaxies suggests that some of the emission is due to liner or agn activity @xcite .
we refer the reader to @xcite for more details on the data processing , cluster properties , and observations .
the @xmath15 cluster rxj1757.3 + 6631 , hereafter rxj1757 , was discovered as part of the rosat nep survey , where it is also identified as nep200 @xcite .
@xcite found the structure to have an x - ray luminosity of 8.6@xmath61erg s@xmath5 in the 0.5 - 2.0 kev band .
the structure is dominated by a single , large cluster . in this paper , we present a velocity dispersion , redshift histograms , and analysis of x - ray point sources for rxj1757 , none of which have been previously published .
ground - based optical imaging data were obtained with the large format camera ( lfc ; * ? ? ? * ) on the palomar 5 m telescope .
observations were taken using the sloan digital sky survey ( sdss ) @xmath62 , @xmath43 , and @xmath63 filters .
the 5 @xmath30 point source limiting magnitudes for the five fields ranged from 25.5 - 25.1 , 25.0 - 24.5 , and 23.6 - 23.3 , in the @xmath64 , @xmath65 , and @xmath66 bands , respectively .
cl1604 was also imaged using acs . the hst imaging for cl1604 consists of 17 acs pointings designed to image nine of the ten galaxy density peaks in the field .
observations were taken using the f606w and f814w bands .
these bands roughly correspond to broadband v and i , respectively .
our photometric catalog is complemented by an unprecedented amount of spectroscopic data . for this part of the study
, we used the deep imaging multi - object spectrograph ( deimos ; * ? ? ? * ) on the keck ii 10 m telescope .
in addition , cl1604 and rxj1821 have some lris coverage ( see * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
deimos has a wide field of view ( @xmath67 ) , high efficiency , and is able to position over 120 targets per slit mask , which makes the instrument ideal for establishing an extensive spectroscopic catalog .
we targeted objects down to an @xmath43-band magnitude of 24.5 .
on deimos , we used the 1200 line mm@xmath5 grating , blazed at 7500 , and 1@xmath68 slits .
these specifications create a pixel scale of 0.33 pixel@xmath5 and a fwhm resolution of @xmath69 , or 68 km s@xmath5 .
the central wavelength was varied from structure to structure and sometimes between different masks for the same field .
central wavelengths for the spectroscopic observations for the five fields and the approximate spectral coverages are displayed in table [ speccovtab ] . when more than one central wavelength was used per field ,
a range is given .
total exposure times for the observations are in the range of 1 - 4 hours per mask .
spectroscopic targets were chosen based on color and magnitude . the number of spectroscopic targets in each field is shown in table [ srcsum ] .
redshifts were determined or measured for all targets and given a quality flag value , @xmath70 , where @xmath71 indicates that we could not determine a secure redshift , @xmath72 means a redshift was obtained using features that were only marginally detected , @xmath73 means one secure and one marginal feature were used to calculate the redshift , and @xmath74 means at least two secure features were used .
those sources determined to be stars were given a flag of @xmath75 .
see @xcite for more details on quality flags and the spectral targeting method . for our analysis , redshifts with @xmath76,3,4
were deemed satisfactory , and the number of such sources in each field is shown in table [ srcsum ] .
spectroscopic data have been previously presented for the cl1604 supercluster , cl0023 , and rxj1821 as part of the orelse survey .
we present new data for each of these structures , as well as for the cl1324 supercluster and rxj1757 .
we include ten deimos masks for cl1324 , with exposure times ranging from 6635 s to 10800 s , and four deimos masks for rxj1757 , with exposure times ranging from 7200 s to 14,730 s. we include a total of 18 spectroscopic masks for cl1604 , six more than what were included in @xcite .
we include nine total masks for cl0023 , four more than in @xcite .
we include three masks for rxj1821 , one more than in @xcite .
many of the new targets were optical counterparts to x - ray sources . from our measured redshifts with @xmath77 or 4 from the new masks , we find 114 new galaxies in cl1604 within @xmath78 , 104 new galaxies in cl0023 within @xmath79 here , @xcite use @xmath80 , excluding a sheet of galaxies at @xmath81 - 0.87 . in the smaller range of @xmath80
, we add 96 new galaxies .
we use a wider redshift range in order to be consistently liberal in our structure boundaries . ] , and five new galaxies in rxj1821 within @xmath82 .
all x - ray imaging of the clusters was conducted with the advanced ccd imaging spectrometer ( acis ) of the chandra x - ray observatory , using the acis - i array ( pi : l. m. lubin ) .
this array has a @xmath83 field of view .
some of the five structures were imaged with one pointing and some with two , but every pointing had the same approximate total exposure time of 50 ks .
cl0023 , rxj1821 , and rxj1757 were each imaged with one pointing of the array .
cl1604 and cl1324 , with angular sizes in excess of 20@xmath84 , were observed with two pointings each .
for cl1604 , the two pointings are meant to cover as much of the structure as possible , and there is a small overlap ( @xmath85 arcminutes@xmath86 ) . for cl1324 , the two pointings are centered near the two largest and originally discovered clusters , cl1324 + 3011 and cl1324 + 3059 . there is an approximately 13@xmath84 gap between the north and south pointings . in this paper , we present new chandra data for cl1324 , rxj1821 , and rxj1757 .
lccc cl1604 & 1.2 & 6.12 & 22.2 + cl0023 & 2.7 & 6.37 & 22.3 + cl1324 & 1.1 & 6.16 & 22.2 + rxj1821 & 5.6 & 8.23 & 22.3 + rxj1757 & 4.0 & 6.60 & 22.2 + [ nhc2ftab ] the reduction of the data was conducted using the _ chandra _ interactive analysis of observations 4.2 software ( ciao ; * ? ? ?
each observation was filtered by energy into three bands : 0.5 - 2 kev ( soft ) , 2 - 8 kev ( hard ) , and 0.5 - 8 kev ( full ) .
data were checked for flares using _
dmextract _ and the _ chandra _ imaging and plotting system ( chips ) routine _
lc_clean_. exposure maps were created using the routine _
merge_all_. for vignetting correction , exposure maps were normalized to their maximum value , then images were divided by this normalized exposure map . to locate point sources ,
the routine _ wavdetect _ was run on each observation , without vignetting correction , using wavelet scales of @xmath87 pixels , with @xmath88 ranging from 0 to 8 .
a threshold significance of @xmath89 was used , which would imply fewer than one spurious detection per acis chip , which has dimensions of @xmath90 pixels ( @xmath91per pixel ) . however
, this assumes a uniform background , which is almost certainly not the case . to measure realistic detection significances , we instead used photometric results explained below .
point source detection was carried out on images in each of the soft , hard , and full bands separately . for cl1604 and cl1324 ,
_ wavdetect _ was run on each of the two pointings separately .
output object positions from the three different bands were cross - correlated to create one final composite list for each field .
we carried out follow up photometry on the point sources .
circular apertures containing 95@xmath7 of the flux were created for each point source using the point - spread function ( psf ) libraries in the _ chandra _ calibration database . for _ chandra _ , the psf depends on both energy and off - axis angle . for the soft and hard bands ,
respectively , we used the psf libraries for energies of 1.497 and 4.510 kev .
photometry was carried out on the vignetting - corrected images in the soft and hard bands .
background counts for each source were calculated in annuli with inner and outer radii of @xmath92 and @xmath93 , where @xmath94 is the radius of the circular aperture containing 95@xmath7 of the flux . since only 95@xmath7 of the flux
is enclosed , net counts calculated from the apertures were multiplied by 1/0.95 to recover all the counts .
full band counts were calculated by summing those from the soft and hard bands .
the results of the photometry were used to calculate detection significances for the sources in each of the three bands using @xmath95 where @xmath96 is the net photon counts from the source and @xmath97 is the background counts @xcite .
x - ray sources with significances @xmath98 were rejected as spurious . with @xmath99of accepted sources having detection significances @xmath100 , and the remaining @xmath101with detection significances between @xmath102 and @xmath103 , we expect a spurious detection rate of @xmath104 , based on a normal distribution .
due to the low number of photons observed for many sources , we opted to normalize a power - law spectral model to the net count rate of individual point sources to determine fluxes .
we assumed a photon index of @xmath105 , which is the approximate slope of the x - ray background in the hard band @xcite .
count rates were calculated by dividing net counts by the nominal exposure time at the aimpoint of the appropriate observation .
the galactic neutral hydrogen column density was calculated at the aim point of each observation using the colden tool from the _ chandra _ proposal toolkit , which uses the dataset of @xcite .
hydrogen column densities and derived net count rate to unabsorbed flux conversion factors for each field are summarized in table [ nhc2ftab ] .
conversion factors were determined separately for the different pointings of cl1604 and cl1324 , but did not differ to the three significant figures listed in the table .
llllll cl1324 + 3011 & a & 13 24 48.7 & + 30 11 48 & @xmath49 + cl1324 + 3059 & i & 13 24 50.5 & + 30 58 19 & @xmath106 + cl1324 + 3013 & b & 13 24 21.5 & + 30 13 10 & @xmath107 + cl1324 + 3025 & c & 13 24 01.8 & + 30 25 05 & @xmath108 + [ 1324clustab ] to search for agns within the individual clusters , we matched x - ray and optical sources . in order to increase our completeness ,
the input to the matching included all point sources detected by _
wavdetect _ , regardless of significance in any of the three bands .
we used the maximum likelihood ratio technique described in @xcite , which was developed by @xcite and also used by @xcite and @xcite .
our technique is similar to @xcite , but with a few key differences .
the main statistic calculated in each case is the likelihood ratio ( lr ) , which estimates the probability that a given optical source is the genuine match to a given point source relative to the arrangement of the two sources arising by chance .
the lr is given by the equation @xmath109 here , @xmath110 is the separation between objects @xmath88 and @xmath111 , @xmath112 is the positional error of object @xmath111 , and @xmath113 is the inverse of the number density of optical sources with magnitude brighter than @xmath114 .
the inclusion of the latter quantity is designed to weight against matching to fainter optical objects .
however , in our analysis , we found that this particular weighting , used by @xcite , of @xmath115 gave too much favor to bright objects , even when they were much farther from an x - ray object than a faint source .
we adjusted the weighting to @xmath116 .
we found that this change did not have a large overall effect but changed some borderline cases where a bright object with a large separation from the point had been chosen over a dimmer , much closer object .
this includes one case , which prompted the adjustment , where spectroscopy showed that an m - type star was matched instead of a probable agn , even though the m star was three times farther from the x - ray source . for each field ,
except cl1604 , @xmath117 was measured using @xmath43 magnitudes from our lfc catalogs . for cl1604 ,
acs data were also available , but these observations did not cover the entire field .
all objects were matched to the lfc catalogs .
when possible , objects were also matched using the f814w magnitude from the acs catalogs and matches to acs objects took precedence over matches to lfc objects . from the lr , we use monte carlo simulations to derive the probability that a given match is genuine .
we ran 10,000 trials for each x - ray object .
in each trial , the object s position was randomized , and the lr was calculated based on nearby optical sources .
the lr for a given x - ray object to optical counterpart pairing , @xmath118 , was compared against the distribution of lrs from the 10,000 monte carlo trials .
we calculated the reliability as : @xmath119 where @xmath120 is the number of matches , to any optical source , across all 10,000 trials for that x - ray object with lr greater than @xmath118 .
@xmath121 can be interpreted as the probability that optical source @xmath88 is the true match of x - ray source @xmath111 , in the case of only one optical candidate .
when there are multiple candidates , we used the method of @xcite to calculate the probability that optical source @xmath88 is the true match of x - ray source @xmath111 as @xmath122 the probability that no optical source is the true match is : @xmath123 where n is the total number of optical candidates and s is a normalization factor defined so that @xmath124 . for an x - ray source with a single optical counterpart ,
a match was considered genuine if @xmath125 . for x - ray sources with multiple optical counterparts ,
a genuine match was chosen if @xmath126 ( which is equivalent to @xmath125 ) and @xmath127 for any one object @xmath88 .
if the first condition was true , but the second was not , all objects with @xmath128 were considered as matches . in subsequent sections ,
only one optical counterpart was considered for each x - ray source . in almost all cases ,
the highest probability match was used . however , in several cases , spectra of the primary and secondary matches indicated that the secondary match was an agn , and thus more likely to be a genuine match .
note that our threshold is a deviation from @xcite .
they used @xmath129 instead of 0.15 .
we decided to use the more stringent threshold of 0.15 , which has been used by others @xcite , to better limit the number of false matches .
the more restrictive cutoff omitted @xmath130 sources per field .
we determined this threshold through visual inspection of potential matches .
this calibration entailed determining at what approximate level of @xmath131 most matches visually seemed spurious .
however , optical candidates above our threshold were also visually scrutinized ( @xmath132 of the total ) , and some were accepted after this inspection where we felt the matching algorithm had failed .
note that setting a threshold for genuine matches is not entirely objective , and a precedent has been set for accomplishing this with visual inspection ( e.g. , * ? ? ?
* ) . in table
[ srcsum ] , we list , for each field , the number of x - ray sources detected at @xmath1333@xmath30 ( @xmath1332@xmath30 ) in one of the three bands , as well as the number of those sources matched to optical counterparts .
the five structures in our sample span a range of evolutionary states .
they include cl0023 , whose four constituent groups are still in the process of merging to form a single cluster ; the two isolated x - ray selected clusters , rxj1757 and rxj1821 , which appear to be in a more evolved and relaxed state ; and the two superclusters , cl1604 and cl1324 .
we would like to compare the agns within our sample based on the evolutionary states of the structures to which they belong , which could shed light on how the agns in these systems are being triggered . in order to make such a comparison , we first present the global properties of the five orelse structures .
the cl1604 supercluster , cl0023 , and rxj1821 have all been studied previously as part of the orelse survey ( see section [ sec : samp ] for individual references ) , although we have gathered new data on each , as described in section [ sec : optobs ] .
while individual clusters in cl1324 have been studied , the properties of the supercluster as a whole have not . in this section
we do a preliminary exploration of this structure , which will be covered more thoroughly in an upcoming paper ( r. r. gal et al .
2012 , in preparation ) . the cluster rxj1757 was studied only as a part of the rosat nep survey @xcite , in little detail . here
, we present new redshift histograms of these last two structures derived from our orelse data .
figure [ 1324 + 1757.rshists](a ) shows all confirmed redshifts in the spatial vicinity of cl1324 .
we can see two peaks in the histogram , at @xmath134 and @xmath135 .
these peaks coincide with the two largest clusters in the structure , cl1324 + 3011 and cl1324 + 3059 . from the distribution of red galaxies
, we find ten overdensities in the supercluster , some of which can be observed in the redshift histogram .
so far , we have confirmed four clusters and groups to be constituents , shown in table [ 1324clustab ] , with coordinates , redshifts , and measured velocity dispersions given .
additional multiobject spectroscopy to confirm the nature of the other red galaxy overdensities is planned .
the redshift histograms for rxj1757 are displayed in figures [ 1324 + 1757.rshists](c ) and [ 1324 + 1757.rshists](d ) . at @xmath136 and @xmath12 , we see two peaks in the distribution . the former is the overdensity associated with rxj1757 .
when we examine the spatial distribution of the higher redshift peak , we find that the galaxies in its vicinity are distributed nearly uniformly across the field of view , implying a sheet of galaxies .
looking at the redshift distribution of confirmed galaxies within the bounds of rxj1757 ( figure [ 1324 + 1757.rshists](d ) ) , we can see the distribution is reasonably consistent with a gaussian , confirmed by a kolmogorov - smirnov ( k - s ) test at a @xmath137 level , suggesting there is no significant substructure .
however , we caution that we have a smaller sample of confirmed redshifts compared to the other fields .
figure [ cmds ] shows cmds for all five fields . all spectroscopically confirmed supercluster / cluster members are shown .
squares indicate the confirmed x - ray agns within each structure , which are analyzed in section [ sec : agn ] . the red sequence for each field is delineated by dotted lines .
red sequence fits for each field were calculated using a linear fitting and @xmath30-clipping technique .
first , a fit to a linear model , of the form @xmath138 where @xmath96 is either @xmath139 or f606w - f814w and @xmath97 is either @xmath43 or f814w , was carried out on member galaxies within a chosen magnitude and color range using a @xmath140 minimization @xcite .
the fit was initialized with a color range chosen `` by eye '' to conform to the apparent width of the red sequence of the structure .
the magnitude bounds were defined as the range where the photometric errors were small ( @xmath141 ) .
after the initial fit , colors were normalized to remove the slope .
the color distribution was then fit to a single gaussian using iterative @xmath103 clipping . at the conclusion of the algorithm ,
the boundaries of the red sequence were defined by a 3@xmath142offset from the center , except for cl1604 and cl1324 . for the two superclusters ,
the color dispersion was inflated due to the large redshift extent of these structures , and 2@xmath30 offsets were used to achieve reasonable boundaries . for every field except cl1604 , the lfc @xmath143 color magnitude diagrams are shown .
for cl1604 , acs data were available and were used in place of lfc data because of their superior precision .
note , however , that two of the agns in cl1604 are outside our acs pointings , so that our analysis using acs data only includes 8 agns in the cl1604 supercluster .
lcc cl0023 & [email protected] & [email protected] + cl1604 & [email protected] & @xmath145 + cl1324 & [email protected] & [email protected] + rxj1821 & [email protected] & [email protected] + rxj1757 & [email protected] & [email protected] + [ bftab ] while all the structures show a substantial number of galaxies on the red sequence , there are large differences in the blue populations .
qualitatively , we observe a lower blue fraction in the cmds of rxj1821 and rxj1757 than in those of cl0023 and cl1604 .
this is quantified in table [ bftab ] , where the blue fraction for all confirmed cluster / supercluster members with @xmath43 or f814w magnitudes brighter than 23.5 are displayed in the first column .
we can see that cl0023 and cl1604 have the bluest galaxy populations , while rxj1757 has the highest fraction of galaxies on the red sequence of all the fields .
since this could be due to the low completeness of spectroscopic coverage in this field , we attempted to make corrections with two approaches : ( 1 ) using our efficiency of spectroscopically confirming structure members to estimate the total number of member galaxies , and ( 2 ) correcting using measurements of the background galaxy density .
all efforts to statistically estimate the true blue fraction yielded large errors making the measurements highly uncertain .
instead , we created a blue fraction measurement metric that could be compared between fields ( see also * ? ? ? * ) .
the different fields have a varying number of spectral masks , with several different spectroscopic priorities .
however , the first several masks , excluding those designed to specifically target x - ray matched sources , have similar priorities for choosing targets .
therefore , we chose to compare blue fractions only among sources in the first two masks for each field , since these sources should represent similar populations .
we chose two masks because this is the number of masks for rxj1757 excluding those where objects matched to x - ray point sources were preferentially targeted .
we confirmed that this sampling is representative of the entire galaxy population by recreating composite spectra using only the first two spectral masks .
since there are no large differences between the average spectral features , the method should be accurate .
the results of this comparison are displayed in the second column of table [ bftab ] .
however , we did not calculate a blue fraction for cl1604 in this way , for two reasons .
first , there are a total of 24 spectroscopic masks for cl1604 , from several different telescopes . choosing which ones to include is difficult and it may be impossible to select a population congruous with any of the other fields in this manner .
second , our spectroscopy for cl1604 is relatively complete , so we are confident in the blue fraction measured down to @xmath146 . examining the results
, we can see the same color hierarchy in the structures for both methods of measuring the blue fraction , with rxj1757 having the smallest fraction and cl1604 and cl0023 the largest .
these large blue fractions , in particular those for cl1604 and cl0023 , are consistent with the butcher - oemler effect @xcite .
we also note that the isolated x - ray - selected clusters , rxj1757 and rxj1821 , are the reddest structures , suggestive of a more advanced , dynamically relaxed evolutionary state .
the color hierarchy suggests a similar ranking of galactic star formation in the five structures , which we can explore with our spectroscopic data .
lccc cl0023 & @xmath147 & @xmath148 & @xmath149 + cl1604 & @xmath150 & @xmath151 & @xmath152 + cl1324 & @xmath153 & @xmath154 & @xmath155 + rxj1821 & @xmath156 & @xmath157 & @xmath158 + rxj1757 & @xmath159 & @xmath160 & @xmath161 + [ globchartab ] using our spectroscopic data , we examine the typical star formation history of the galaxies in each structure .
we formed composite spectra by co - adding the individual spectra of galaxies within each structure , according to the method of @xcite and @xcite .
we analyze these spectra in terms of two important features relevant to star formation : the [ ] and h@xmath2 lines .
the h@xmath2 absorption is indicative of a population of a and b stars , which disappears @xmath162 gyr after the cessation of star formation within a galaxy , due to the lifetime of a stars @xcite .
if star formation is ongoing , a population of o stars , which have weaker hydrogen features , can dominate the continuum and wash out this absorption line .
infilling can also occur from balmer emission from regions .
the [ ] emission line has been used as an indicator of star formation , especially as a proxy for the h@xmath163 emission line at higher redshifts when h@xmath163 has shifted out of the optical range @xcite .
however , recent analysis using near - ir spectroscopy of sources from the cl1604 supercluster and rxj1821 has compared [ ] and h@xmath163 emission , finding that a significant portion of [ ] emission can come from liner or seyfert related processes @xcite .
these results are supported by those of @xcite , albeit using a lower redshift sample . in light of this , caution must be exercised when interpreting [ ] measurements . for an additional diagnostic ,
we measure the d@xmath164(4000 ) strength which is an indicator of mean stellar age @xcite .
figure [ oiivshd](a ) shows [ ] vs. h@xmath2 equivalent widths for the composite spectra of members of the five structures .
the dotted line represents the average spectral properties for a cluster population composed of various fractions of `` normal '' star - forming and quiescent galaxies @xcite , based on data from the two - degree field ( 2df ) galaxy redshift survey @xcite .
asterisks on this line represent a cluster population composed of ( from left to right ) 20% , 40% , 60% , 80% , and 100% star - forming galaxies . for a cluster whose average galaxy lies above this line ,
the h@xmath2 line is too strong to be produced by normal star formation , requiring some contribution from starbursting or post - starbursting galaxies .
the dashed lines enclose 95% of normal star - forming galaxies @xcite .
the shaded regions , which are based on the spectral types of @xcite and @xcite , denote the region of this phase space inhabited by ( starting from the upper right and moving counter - clockwise ) starburst ( dark blue ) , post - starburst ( green ) , quiescent ( red ) , and normal star - forming galaxies ( light blue ) .
examining the positions of the structures in our sample on this plot , we can see that rxj1757 and rxj1821 have mostly quiescent populations , with each cluster having @xmath165 normal star - forming galaxies .
because rxj1757 is offset from the 2df line , there may be some contribution from post - starburst galaxies , but the fractional contribution is low .
the stronger [ ] emission in the cl0023 and cl1604 composite spectra suggests that these structures have a higher fraction ( @xmath166 ) of continuously star - forming galaxies .
these structures have larger blue fractions than the others , so it is unlikely that the increased ew ( [ ] ) in their average spectra is due to liner processes , which are primarily associated with red sequence galaxies .
the ew(h@xmath2 ) measured from the cl0023 and cl1604 composite spectra are significantly in excess of the 2df line , suggesting a substantial contribution from starbursting or post - starburst galaxies .
cl1324 is in an intermediate range , with @xmath167 normal star - forming galaxies and an observed ew(h@xmath2 ) for its galaxy population smaller than that of cl0023 and cl1604 .
our conclusions based on the [ ] and h@xmath2 lines are supported by the corresponding d@xmath164(4000 ) measurements ( see table [ globchartab ] ) .
these results are illustrated in figure [ oiivshd](b ) , where average measurements of ew(h@xmath2 ) and d@xmath164(4000 ) are plotted for the five structures .
for comparison , we also indicate ranges of ew(h@xmath2)d@xmath164(4000 ) phase space spanned by four different bruzual & charlot @xcite models for various times after the starburst .
the four models include a single burst ( @xmath168 ) and a secondary burst of 20% , 10% , and 5% by mass which occurs 2 gyr after the initial burst .
all models are solar metallicity and are corrected for extinction using e(b - v ) @xmath169 and a @xcite extinction law .
because bruzual & charlot models only incorporate stellar light , emission infill corrections were made for all ew(h@xmath2 ) measurements using relationships between h@xmath163 and [ ] taken from @xcite and h@xmath163 and h@xmath2 from @xcite .
although these corrections are not perfect ( e.g. , see the measurement for region 1 ) , these values should be accurate to within @xmath170 .
we can see that rxj1757 and rxj1821 have the largest continuum break strengths , indicating that they possess the oldest average stellar populations of the five structures .
the average galaxy in this structure has had 1 - 3 gyr since its last starburst , according to the @xcite model .
the other structures have smaller average d@xmath164(4000 ) measurements , consistent with the results of figure [ oiivshd](a ) showing larger fractions of star - forming galaxies and younger galaxy populations . according to the @xcite model
, the average galaxy in cl1324 , cl0023 , and cl1604 has had a progressively shorter time since the last starburst .
altogether , these spectral results parallel what was found with the blue fractions , in that the reddest structures are also the ones with the most evolved stellar populations and the lowest fraction of star - forming galaxies . despite our extensive spectroscopic sample , there are too few x - ray agns in any individual cluster ( and even supercluster ) to draw statistical conclusions .
hence , we divide our structure sample into two categories .
the first contains cl0023 and cl1604 which have the highest level of ongoing star formation and starburst activity , as shown in the preceding sections .
the second category , consisting of cl1324 , rxj1757 , and rxj1821 , contains structures whose member galaxies are typically quiescent or forming stars at a lower rate .
we refer to these two categories as `` unevolved '' and `` evolved '' , respectively , as a means of describing their typical galaxy populations .
while we acknowledge that the terms `` more evolved '' and `` less evolved '' would be more appropriate , we adopt the less accurate denominations for brevity .
in addition , these terms do not necessarily imply differing levels of cluster dynamical evolution or even a clear temporal sequence from one category to the other .
we note that , while some clusters or groups may not fit well with the global characteristics of their parent supercluster ( i.e. cluster a of cl1604 ) , we can not examine all of the agns on a cluster - by - cluster basis . as we will discuss in section [ sec : spatdist ] , many of the agns in the superclusters and the supergroup are not associated with any one cluster or group .
so , when analyzing these agns , we take the parent supercluster or supergroup as a whole .
the particular segregation of our structure sample is motivated by the clear distinction between the structures shown in figure [ oiivshd ] .
the abundant star formation in the unevolved structures suggests the presence of a large gas reservoir in many of their member galaxies . we might expect that this same gas is available for agn fueling
conversely , the typical galaxy in the evolved sample has likely consumed most of the available gas in prior star formation episodes , leaving less fuel for the agns . in the following sections , we examine whether the properties of the agn sin the two categories
reflects this distinction , and what we can can learn about the relationship of star formation to agn activity in lsss .
in this section , we examine the frequency of agn activity within the five structures to determine if there are associated excesses of x - ray point sources .
x - ray point source photometry was used to calculate cumulative x - ray source number counts , @xmath171 , using the method of @xcite : @xmath172 where @xmath173 is the number of point sources with fluxes greater than @xmath174 and @xmath175 is the area of the sky in which the _ i_th point source could have been detected at a 3@xmath30 level or higher .
variance in @xmath171 was calculated using @xmath176 . to calculate @xmath175
, we used the method of @xcite , which is similar to those used in other literature @xcite .
all point sources detected by _ wavdetect _
, in all bands and without significance cuts , were removed and replaced with an estimate of the background using the ciao tool _
dmfilth_. to create a map of the background emission , these images were binned into 32@xmath68 bins . according to equation ( [ eq : detsig ] ) , the flux limit corresponding to a 3@xmath30 detection in this analysis to be consistent with previous work .
the inclusion of sources with detection significances between @xmath102 and @xmath103 has a significant effect only on fluxes where sky coverage is below @xmath177 , and the results are generally considered unreliable . ]
occuring in one of these binned pixels is given by @xmath178 where @xmath97 is the net counts in a pixel , @xmath179 is the area of a pixel , @xmath94 is the radius of the aperture enclosing 95@xmath7 of an x - ray source s flux , as described in section [ sec : datared ] , @xmath180 is the conversion factor between photon count rate and x - ray flux , also described in section [ sec : datared ] , and @xmath181 is the exposure time of the image . for a given source with flux , @xmath174 , @xmath175 is then equal to the total number of binned pixels for which @xmath174 is greater than @xmath182 , multiplied by the area of a pixel .
the cumulative source counts for all the fields in the soft band and the 2 - 10 kev band , hereafter the hard@xmath183 band , are shown in figure [ lognlogsall ] .
the latter was extrapolated , field by field , from the 2 - 8 kev band by fitting a power law spectral model with exponent @xmath184to each detected point source . for cl1604 and cl1324
, the source counts for the two pointings were combined .
also shown are cumulative source counts measurements from the chandra deep field north and south @xcite .
one chandra observation from each deep field was used ( observation i d 582 and 2232 ) .
both have exposure times of roughly 130 ks .
we re - analyzed these observations using the same reduction pipeline that we used for the orelse fields .
we used the combined source counts from these two fields ( hereafter cdf ) to estimate the blank - field counts for comparison with our data .
we can see that , in the soft band , cl0023 and rxj1757 appear to be consistent with no overdensity compared to cdf .
the other three fields are all overdense to some degree . in the range 3@xmath185 to 10@xmath186 erg s@xmath5 @xmath187 , rxj1821 , cl1324 , and
cl1604 have average overdensities of 0.5@xmath30 , 1.0@xmath30 , and 1.5 @xmath30 , respectively , with @xmath30 calculated using the cumulative source count errors from our data and those of cdf . in the hard@xmath183 band , rxj1821 , rxj1757 , and
cl0023 all have approximate overdensities of 0.5@xmath30 in the range 7@xmath188 to @xmath189 erg s@xmath5 @xmath187 .
rxj1757 appears to be consistent with no overdensity .
cl1604 and cl1324 also appear to be overdense , with average overdensities in the flux range @xmath190 to @xmath189 erg s@xmath5 @xmath187 of 1@xmath30 and 1.5 @xmath30 , respectively .
it should be noted that the results for cl0023 differ from those presented in the previous work of @xcite .
the 2 - 8 kev band cumulative source counts were unintentionally presented as the 2 - 10 kev band counts in that paper .
however , we have also found changes due to the different versions of the ciao software used between earlier papers @xcite and this paper , version 3.3 compared to 4.2 .
we believe this to be an effect of updated response functions at off - axis angles .
the net result for our data is mainly an approximate 5% increase in flux , with some very minor differences in point source detection , the latter being almost entirely below the 3@xmath30 level . in figure
[ lognlogscomp ] , co - added cumulative source counts for all five fields are shown . to accomplish this co - addition ,
all point source lists were first combined into one composite source list . for the calculation of the effective sky area weighting factors , @xmath175 ,
the combined background of every field was used . with @xmath191
, we can see that the error in @xmath171 will be reduced from its value in any individual field .
in the soft band , the composite source counts have an approximate average overdensity of 1@xmath30 in the range @xmath192 to @xmath193 erg s@xmath5 @xmath187 , which falls off to zero brighter than 10@xmath186 erg s@xmath5 @xmath187 . in the hard@xmath183 band ,
the composite counts have an approximate average overdensity of 1@xmath30 between @xmath194 and @xmath195 erg s@xmath5 @xmath187 , but an approximate underdensity of 0.5@xmath30 between @xmath196 and @xmath197 erg s@xmath5 @xmath187 .
while there is substantial variation in the five cumulative source count measurements for the individual fields , we can see from the composite measurement that , on average , the fields of the five structures studied here have a density of x - ray point sources in excess of the control field , though not significantly so .
* , hereafter c05 ) have found a positive correlation between cluster redshift and source count overdensity . to compare to this , and our earlier results of @xcite , we first fit the cumulative source counts to a power law of the form @xmath198 , using the maximum likelihood method of @xcite and @xcite .
this method fixes the dimensionless variable @xmath180 while fitting for @xmath163 .
@xmath180 is calculated by requiring consistency between the model and the data at the flux point @xmath199 .
we use @xmath200 erg s@xmath5 @xmath187 and @xmath201 erg s@xmath5 @xmath187 for @xmath202 in the soft and hard@xmath183 bands , respectively , which are the values used by @xcite .
c05 measured overdensities using the ratio of @xmath180 in a given field to @xmath180 in a reference set of five blank fields . for these fields , they found @xmath203 and @xmath204 .
we calculated @xmath180 for the five fields individually , as well as the composite of all five fields .
it should be noted that the composite measurement is not a simple average of the five fields ( see above for details on its creation ) and that significant variation in the individual @xmath180 values come from calculating @xmath180 at a single flux point . our results are shown in figure [ capplot ] for the hard@xmath183 band , where our data are overplotted on that of c05 and the linear fit to their data is shown . just as in @xcite ,
our data are consistent with their fit .
once again ( refer to earlier explanation ) , it is posssible that the issue with the version of the ciao software may have created a systematic offset from c05 s data set .
we would expect this offset to increase the value of @xmath180 for the co5 data by about 5@xmath7 . even without this correction ,
our results are still consistent with c05 within our errors .
however , we note that it is difficult to use these overdensities to interpret the actual agn activity in an individual structure , even with our large spectroscopic sample ( see section [ sec : agn ] ) .
using optical sources with redshifts with quality flags of @xmath77 or 4 and the results of our optical matching , we were able to identify x - ray sources that are members of the clusters or superclusters in our sample ( see table [ srcsum ] ) . in summary
, we found ten confirmed agns in the bounds of cl1604 , seven in cl0023 , six in cl1324 , three in rxj1821 , and one in rxj1757 .
note that these numbers include four sources , one in each structure except rxj1757 , that were detected at a @xmath2053@xmath30 ( but at a @xmath1332@xmath30 ) level in at least one of the three x - ray passbands ( see table [ agntab ] ) .
we show in the following that these low - significance detections do not bias our results .
+ + lcccccccccc cl0023 & 1 & 00 24 10.9 & + 04 29 23 & 0.823 & 5.6 & 46.7 & 52.3 & 12.2 & ... & -1.86 + cl0023 & 2 & 00 24 15.5 & + 04 23 09 & 0.829 & 7.8 & & 7.8 & 10.1 & 1.38 & 0.67 + cl0023 & 3 & 00 23 54.9 & + 04 25 24 & 0.830 & 0.5 & 5.1 & 5.6 & 2.6 & 1.00 & 0.33 + cl0023 & 4 & 00 24 09.4 & + 04 22 41 & 0.841 & 24.3 & 35.8 & 60.1 & 39.9 & 0.71 & -6.08 + cl0023 & 5 & 00 23 52.2 & + 04 22 59 & 0.844 & 24.8 & 83.8 & 108.6 & 63.7 & 0.25 & -2.85 + cl0023 & 6 & 00 23 45.6 & + 04 22 59 & 0.850 & 10.8 & 17.0 & 27.8 & 23.2 & 0.48 & -4.70 + cl0023 & 7 & 00 24 07.6 & + 04 27 26 & 0.854 & 3.1 & 3.4 & 6.5 & 3.9 & 2.45@xmath206 & -1.72 + cl1604 & 1 & 16 04 23.9 & + 43 11 26 & 0.867 & 14.5 & 22.3 & 36.7 & 27.8 & 1.23 & -4.43 + cl1604 & 2 & 16 04 25.9 & + 43 12 45 & 0.871 & 5.5 & 8.7 & 14.2 & 10.4 & 0.66 & -1.48 + cl1604 & 3 & 16 04 15.6 & + 43 10 16 & 0.900 & 17.0 & 31.4 & 48.3 & 31.2 & 1.90@xmath206 & -1.84 + cl1604 & 4 & 16 04 37.6 & + 43 08 58 & 0.900 & 0.9 & 4.4 & 5.3 & 2.4 & 2.44 & ...
+ cl1604 & 5 & 16 04 06.1 & + 43 18 07 & 0.913 & 18.3 & 29.1 & 47.4 & 19.3 & 1.08@xmath206 & -1.02 + cl1604 & 6 & 16 04 36.7 & + 43 21 41 & 0.923 & 6.3 & 18.1 & 24.4 & 9.8 & 0.36 & -3.64 + cl1604 & 7 & 16 04 01.3 & + 43 13 51 & 0.927 & 12.3 & 25.0 & 37.3 & 18.6 & 1.03 & ... + cl1604 & 8 & 16 04 05.1 & + 43 15 19 & 0.934 & 3.6 & 4.7 & 8.4 & 4.3 & 0.29 & -1.07 + cl1604 & 9 & 16 04 10.9 & + 43 21 11 & 0.935 & 0.9 & 10.7 & 11.6 & 4.0 & 1.87@xmath206 & 0.97 + cl1604 & 10 & 16 04 08.2 & + 43 17 36 & 0.937 & 5.0 & 24.9 & 30.0 & 7.1 & 0.84 & -2.47 + cl1324 & 1 & 13 24 51.4 & + 30 12 39 & 0.660 & 1.6 & 0.7 & 2.3 & 4.1 & 0.44@xmath206 & -2.85 + cl1324 & 2 & 13 24 36.4 & + 30 23 16 & 0.662 & 1.0 & 6.8 & 7.8 & 4.2 & ... & -2.66 + cl1324 & 3 & 13 25 04.5 & + 30 22 07 & 0.696 & 1.2 & 0.6 & 1.8 & 2.1 & ... & -2.53 + cl1324 & 4 & 13 24 52.9 & + 30 52 18 & 0.697 & 1.4 & & 1.4 & 3.3 & 2.60 & -1.79 + cl1324 & 5 & 13 24 52.0 & + 30 50 51 & 0.700 & & 10.9 & 10.9 & 6.2 & ... & -1.23 + cl1324 & 6 & 13 24 28.8 & + 30 53 20 & 0.778 & 1.6 & 0.8 & 2.4 & 3.0 & 2.90@xmath206 & -0.24 + rxj1821 & 1 & 18 21 07.7 & + 68 23 38 & 0.813 & 1.9 & 2.5 & 4.4 & 2.7 & 2.17 & -0.51 + rxj1821 & 2 & 18 21 23.9 & + 68 26 33 & 0.822 & 3.4 & 4.1 & 7.5 & 5.5 & 0.67 & -1.20 + rxj1821 & 3 & 18 21 27.0 & + 68 32 34 & 0.824 & 8.6 & 10.6 & 19.3 & 10.3 & 2.13 & -0.66 + rxj1757 & 1 & 17 57 25.2 & + 66 31 50 & 0.693 & 1.9 & 5.1 & 7.1 & 6.1 & 0.24 & -2.11 + [ agntab ] examining
the spatial distribution of agns located within each cluster can give insight into what processes triggered their nuclear activity .
in figures [ allspats0 ] , [ allspats ] , and [ allspats2 ] , we show the spatial distributions on the sky and redshift distributions of the five structures studied here .
the agns are marked in red , and their positions and characteristics are given in table [ agntab ] .
of particular note is the lack of agns in dense cluster centers .
indeed , we find @xmath207 of agns in cluster cores , defined as being within a projected distance of 0.5 mpc to the nearest cluster or group .
an additional @xmath208 lie on the outskirts of clusters ( projected distances between 0.5 and 1.5 mpc ) , and a majority ( @xmath209 ) of agn host galaxies lie more than 1.5 mpc in projected distance from the nearest cluster or group .
these results are consistent with previous work studying the spatial distribution of agns , in that the agns tend to be located outside of clusters or in their outskirts .
first , the distribution of agns in cl1604 , previously studied by @xcite , is consistent with the other four structures here .
our results are also consistent with those of @xcite , who found that agns in the a901/902 supercluster at @xmath210 tend to avoid the densest areas .
several other studies have also found that x - ray agns tend to reside in regions of moderate density similar to group environments , up to @xmath211 @xcite .
it is thought that regions of intermediate density , such as the outskirts of clusters , are the most conducive to galaxy - galaxy interactions because of the elevated densities , compared to the field , but relatively low velocities @xcite .
since we find more of the agns in these areas , this lends support to the theory that mergers or tidal interactions are one of the main instigators of agn activity . since only @xmath212
inhabit dense cluster cores , processes which preferentially occur in these regions , such as ram pressure stripping , are probably not responsible for triggering agns in cluster galaxies .
however , the association between agns and these regions could also be related to higher gas availability in galaxies farther from cluster cores .
although we find that most of the x - ray agns do not reside in the cluster cores , a number of studies have measured the fraction of cluster galaxies that host agn ( e.g. , * ? ? ?
therefore , we attempt to measure the agn fraction for the individual clusters within the five structures in our sample . because we are limited by the spectral completeness of our sample ,
we make a composite measurement of the most massive , well - sampled clusters : cl1604a , cl1604b , cl1604c , cl1604d , cl1324a , cl1324i , and rxj1821 .
we compare to the results of @xcite for low - redshift clusters ( @xmath213 ) , who used galaxies with @xmath214 , within approximately 2000 km s@xmath5 of the mean redshift of cluster members for each system , and within the field of view of their _ chandra _ observations , which ranged from 1.2 to 4.5 mpc in width . to approximate these criteria ,
we adopt a magnitude cutoff that roughly corresponds to @xmath215 and use galaxies within 1 mpc and @xmath216 of the cluster centers and redshifts .
we measure the combined agn fraction for the seven clusters listed above to be 0.012@xmath217 .
this is consistent with the results of @xcite , who measured @xmath218 .
we note , however , that a number of arbitrary definitions went into our measurement .
in addition , our spectroscopy of optical counterparts to x - ray sources is significantly incomplete , in contrast to @xcite . correcting for
this incompleteness contributes the largest source of error to our measurement . because of the large uncertainties , we refrain from drawing any conclusions from our measurement .
figure [ cmds ] presents cmds , which are described in section [ sec : globchar ] , for all five fields .
confirmed agn members of the structures are shown with blue squares .
while the agns in cl1324 , rxj1821 , and rxj1757 preferentially reside within the bounds of , or very close to , the red sequence , the agn hosts in cl0023 and cl1604 are more spread out .
previous work , including @xcite and a number of wide - field surveys @xcite , has found an association between agn activity in galaxies and the transition onto the red sequence ( possibly for a second time ) in the green valley .
while these galaxies could be evolving from the blue cloud onto the red sequence , it is also possible that they could have moved down off the red sequence after a tidal interaction or merger and are evolving back @xcite .
we note , however , that some studies using mass - selected samples have found that agn hosts have a color distribution more similar to that of normal galaxies @xcite .
in addition , @xcite , using a sample of galaxies with redshifts @xmath219 , have found that many green valley agn hosts are dust - reddened blue cloud members , so that agn host colors acquire the bimodality apparent in the general galaxy population .
however , @xcite have also examined the impact of extinction on the colors of agn hosts and did not find a significant impact for galaxies in the redshift range @xmath219 , although bimodality may be introduced at higher redshifts . to address this issue for our study , we are planning to implement spectral energy distribution ( sed ) fitting to evaluate the impact of extinction on the broadband colors of our sample .
preliminary results from sed fitting of the cl1604 hosts suggest that extinction levels in our sample are not as drastic as those presented by @xcite .
it is also possible that agn host colors are contaminated by the agns themselves .
however , this is unlikely because ( 1 ) almost all agn hosts in our sample have rest - frame x - ray luminosities below the quasi - stellar object ( qso ) level of @xmath220 erg s@xmath5 ( see sec [ sec : xlum ] ) and ( 2 ) @xcite found that , in cl1604 , agn hosts with blue cores did not have a rising blue continuum indicative of qso activity .
therefore , we proceed to investigate the agn association with the transition zone and to explore differences in the evolutionary states of their host galaxies in each field by examining color offsets of the agn hosts from the red sequence .
histograms of offsets from the center of the red sequence are shown in figure [ rsoffsets ] .
the first two panels present offsets in terms of color .
in the top panel , only the structures where lfc data were used are shown , which is every field except cl1604 .
the middle panel shows only cl1604 , for which we used acs colors . in the bottom panel ,
all five structures are shown , with normalized offsets . in order to compare the acs and lfc data
, we scale by the red sequence width , @xmath221 . we define @xmath221 as the distance from the center of the red sequence fit to its boundary ( see section [ trsatbp ] and figure [ cmds ] ) . on this plot , agn hosts on the red sequence
will then be located between -1.0 and 1.0 . with red sequence offsets
, we can quantitatively examine the green valley . in figure [ rsoffsets](b ) , we plot a histogram of rs offsets , measured from the acs data , for the agns in the cl1604 supercluster . for comparison , we overplot a scaled distribution of rs offsets for all spectroscopically confirmed supercluster members with acs photometry . in the scaled histogram , we can clearly see an area of reduced number density between the red sequence and blue cloud . for cl1604
, the green valley can be approximated as the region @xmath222 , where @xmath223 is the offset of an agn host from the center of the red sequence . only @xmath224of all confirmed supercluster members with acs data fall within this region .
however , five out of eight of the cl1604 agn hosts with acs data reside within it .
while it is unclear how well this definition of the green valley extends to lfc data , because of larger photometric errors , we can see in figure [ rsoffsets]c that @xmath1 of all host galaxies have @xmath225 and @xmath226 are in the range @xmath227 . while many galaxies in both the evolved and unevolved structures lie in the ` green valley ' region , the percentage of agns on the red sequence is somewhat higher in the evolved structures compared to the unevolved structures , with 30@xmath7 and @xmath177 , respectively . examining figure
[ rsoffsets ] , we can see the distribution of agn hosts in evolved structures is clustered closer to the red sequence , while in the unevolved structures , this distribution has a large tail extending into the blue cloud .
indeed , none of the agn host galaxies in the evolved structures have @xmath228c @xmath229 , whereas four of the x - ray agn hosts in the unevolved structures have red - sequence offsets below this limit .
although these results are suggestive ( and unaffected by our inclusion of the @xmath205 3@xmath30 x - ray sources ) , the two distributions are not statistically different based on the k - s test . morphological analysis by @xcite has shown that @xmath230 of the x - ray agns in cl1604 have had recent mergers or tidal interactions , which could fuel star formation through starburst events .
more recent mergers or interactions are one possible explanation for some of the color differences that we see between the agn host galaxies in the evolved and unevolved structures .
in particular , we find that nine of the agn host galaxies are members of a kinematic close pair with a relative line - of - sight velocity of @xmath231 km s@xmath5 and projected physical separation ( on the plane of the sky ) of @xmath232 h@xmath233 kpc ( e.g. , * ? ? ? * ) .
two are in cl0023 , five in cl1604 , one in cl1324 , and one in rxj1821 ( see table [ agntab ] ) .
those agn hosts in pairs include three out of the four galaxies with the largest red sequence offsets ( i.e. , the bluest ) , all of which are members of the unevolved structures . based on their @xmath63 magnitudes or measured stellar masses ( in the case of the cl1604 members ; see @xcite ) , seven out of the nine kinematic pairs have flux or mass ratios of @xmath234 , implying a major merger scenario . the differences in color and , perhaps , merger activity are likely related to the increased level of star formation and starburst activity in the unevolved compared to evolved structures ( see section [ sec : specprop ] ) . to explore the connection between the agn and star formation history
, we can use our high - resolution spectroscopy to examine the average spectral properties of their host galaxies .
we measure the average spectral properties of the agn host galaxies in the five orelse structures using three composite spectra : one comprised of six agns from cl0023 km s@xmath5 ) which would dominate a composite spectrum and was therefore not included .
while the other has a very broad line , the other features are narrow and do not dominate the composite spectrum . as a result , it was included .
] , one comprised of all ten agns from cl1604 , and one comprised of the ten agns from the combined fields of cl1324 , rxj1821 , and rxj1757 .
the last spectrum combines all the evolved structures , necessitated by the low number of agns in rxj1821 and rxj1757 compared with the other structures .
these three composite spectra are shown in figure [ csplot ] , and measurements of spectral features are listed in table [ co - addtab ] . first , we can see that the agn hosts in all three groupings have substantial [ ] emission . in cl1604 and the evolved structures ,
most of this emission is probably from the agns , rather than star formation .
six of the ten agn host galaxies in our cl1604 sample were analyzed by @xcite using the keck ii near - infrared echelle spectrograph ( nirspec ; * ? ? ?
* ) . for five out of the six targets
, @xcite found that the [ ] /h@xmath163 flux ratio was too large for a normal star - forming galaxy , which implies that agns are the dominant contributor to [ ] emission @xcite .
the cl1324 , rxj1821 , and rxj1757 structures have agns mostly near or on the red sequence .
in addition , we found in section [ sec : globchar ] that star formation is low in all three structures ( @xmath235of all galaxies are star - forming ) .
because of this low star formation rate , and because @xcite found that @xmath236 of red [ ] emitters are dominated by liner / seyfert emission , it is likely that most of the [ ] emission from the agn host galaxies in the evolved structures comes from the agns as well .
deciphering the origin of the [ ] emission in the agn host galaxies in cl0023 is not as straightforward . while some agn host galaxies in cl0023 are on the red sequence , the structure also has the bluest host galaxies in our sample
also , as discussed in section [ sec : globchar ] , there is significant star formation in the general population of cl0023 .
while this is also true of cl1604 , the nirspec results of @xcite showed most of the [ ] emission in the agn host galaxies in that structure comes from the agns .
however , we do not have any near - ir spectroscopic data for the cl0023 agns , so we must use other means to determine the emission source . the average [ ] / [ ] ratio of the cl0023 x - ray agn hosts is 0.429 , typical of type-2 agn emission , emission from metal - poor star formation , or a superposition of the two processes @xcite .
note that cl1604 and the evolved structures have values of the [ ] / [ ] ratio of 0.549 and 0.229 , respectively , also typical of type-2 agn emission . combining this result with the blue colors of the cl0023 hosts and the high fraction of star - forming galaxies , it is likely that the observed [ ] emission of the agn hosts in cl0023 is due to a combination of normal star formation and type-2 agn activity .
examination of the balmer features reveals further insight into the star formation histories of the agn host galaxies .
specifically , based on a single stellar population model , the ew(h@xmath2 ) rises quickly from zero after a starburst in a galaxy , peaks after about 300500 myr , and then declines back to approximately zero at @xmath3 gyr after the burst @xcite .
infill can complicate the interpretation of this feature , so care must be taken when measuring the line strength . for the composite spectrum of the cl0023 agn hosts ,
the equivalent width of h@xmath2 , attempting to correct for infill , is consistent with zero .
however , we observe strong emission from other balmer features , ew@xmath237 line , figure [ csplot ] was not drawn out to this range . however , approximately two - thirds of agns in the cl0023 structure do have spectral coverage for h@xmath238 , and the measurement presented here represents the average value for these galaxies . ] and ew@xmath239 , suggesting that emission infill has a significant effect on the measured ew(h@xmath2 ) .
this infill could be due to emission from regions , emission from agns or some other liner processes , or from continuum emission produced by o stars .
since we observe h@xmath238 and h@xmath240 in emission , it is unlikely that o stars are solely responsible for the observed ew(h@xmath2 ) .
the balmer emission lines observed in the cl0023 agn hosts are not broadkm s@xmath5 . ] and are quite strong . in the average type-2 agn , a large fraction of the balmer emission originates from star formation @xcite , which suggests ongoing star formation in the cl0023 hosts .
compared to cl0023 , h@xmath240 emission from the cl1604 hosts is low , with ew(h@xmath240 ) = -0.69@xmath241 , which is consistent with a lower level of star formation in the supercluster ) could not be measured for a majority of galaxies in the cl1604 host sample . ] .
balmer lines for the evolved structures are in absorption , ew(h@xmath238 ) = 0.48@xmath242 and ew(h@xmath240 ) = 1.41@xmath243 , consistent with an even lower level of star formation .
these results combined with the earlier result analyzing the average [ ] / [ ] ratios of the agn hosts strongly indicates that star formation is occurring in the cl0023 galaxies coevally with agn activity , while less star formation activity is occuring in the other structures .
lcrccc cl0023 & 6 & @xmath244 & @xmath245 & @xmath246 & @xmath247 + cl1604 & 10 & @xmath248 & @xmath249 & @xmath250 & @xmath251 + evolved sys . & 10 & @xmath252 & @xmath253 & @xmath254 & @xmath255 [ co - addtab ] the h@xmath2 equivalent widths , combined with measurements of the 4000 break , suggest that starbursts have occurred more recently in the average cl1604 agn host compared to those in the evolved structures .
larger values of d@xmath164(4000 ) indicate a more passive galaxy , with an older average stellar population , which could mean that more time has passed since the cessation of star formation @xcite .
the ew(h@xmath2 ) measured from the cl1604 composite is @xmath256 , roughly half of that from the evolved structures composite .
similarly , the cl1604 composite has a lower value for d@xmath164(4000 ) than those in the evolved structures , indicating that the average cl1604 host is more actively star - forming or has a younger stellar population on average .
this result is supported by the bluer colors of the cl1604 agn hosts compared to those in the evolved structures .
all of these results imply the average cl1604 host has had a starburst more recently than the average agn host in the evolved structures .
the agn host composite spectra for cl0023 has a particularly low d@xmath164(4000 ) measurement ( see table [ co - addtab ] ) . since we found that this composite spectra ( which excluded one broadline source ) was consistent with type-2 agns , the agns themselves should not contribute most of the blue continuum .
this points to a stellar source , particulary o and b stars .
we would then expect significant star formation in the cl0023 hosts within the last 10 - 100 myr , as indicated by the other spectral features as well .
related to the d@xmath164(4000 ) measurement , the h+h@xmath257 and the k lines also provide information on star formation . for f ,
g , and k stars , the ratio of these lines is constant , while the h+h@xmath257/ k ratio increases for a and b stars as the overall strength decreases and the h@xmath257 strength increases @xcite .
the h+h@xmath257/ k ratio is @xmath251 for cl1604 and @xmath255 for the evolved structures ( see table [ co - addtab ] ) , consistent with the evolved structures having ( on average ) older stellar populations .
we do see a decrease in the overall strengths of both lines in the average spectrum of the cl0023 host galaxies relative to the other structures ; however , we actually measure a dramatic decrease in the ratio for cl0023 ( @xmath247 ) , the opposite of what is expected from a population of a and b stars .
the most likely explanation is significant h@xmath257 emission , which would be in concert with the other observed balmer emission .
the h@xmath257 emission could be coming from some combination of agns and regions , which would support previous conclusions about the level of activity in the cl0023 hosts . altogether ,
the composite spectra of the agn hosts in all three bins of figure [ csplot ] show that the average host galaxy has ongoing star formation or has had star formation within the last @xmath162 gyr
. however , the hosts in cl1604 and the evolved structures each have , on average , less ongoing star formation than cl0023 , as evidenced by larger values of d@xmath164(4000 ) and the absence of the balmer emission that is observed in cl0023 .
these differences suggest a progression in the temporal proximity of the last starburst event , with the hosts in cl0023 having significant ongoing star formation characteristic of a current starburst , to those in cl1604 and the evolved structures each having successively more time since the last significant starburst event . we have confirmed that none of our results based on the composite spectra are changed by removing the four lowest significance ( @xmath2053 @xmath30 ) x - ray sources from our sample , with most spectral measurements remaining the same within the errors . with our analysis of the composite spectra of agn host galaxies in the different structures
, we can compare the average properties of these hosts with the average properties of all spectroscopically confirmed galaxies within the same structures . in section [ sec : globchar ] , we found that the galaxy populations in the evolved structures were largely quiescent , with little or no contribution from starburst or post - starburst galaxies .
in contrast , the populations in the unevolved structures were comprised of large fractions of star - forming galaxies , with a more significant contribution from starburst or post - starburst galaxies . when comparing these results to the average spectral properties of the agn hosts , we find that , in _ all _ cases , the average agn host galaxy has a _ younger _ stellar population than the average galaxy in the parent structure , irregardless of the evolved or unevolved classification .
this result holds even when comparing to member galaxies outside the dense cluster cores ( @xmath258mpc ) , where the vast majority of x - ray agns reside .
the most prominent difference comes from the evolved structures where their average agn host galaxy has significantly larger ew(h@xmath2 ) and smaller d@xmath164(4000 ) than the average structure member , indicative of a post - starburst galaxy with a substantial star - formation event within the last @xmath259gyr .
such galaxies make a small contribution to the overall population in the evolved structures , which is largely quiescent .
while we do observe clear differences between the spectra of the agn hosts , with those in cl0023 having significant ongoing star formation to those in the evolved structures having the most time since the last significant starburst event , these differences are not nearly as pronounced as the differences between the average galaxy in each structure .
this suggests that agn activity has a common origin associated with current or recent star - formation . in this section
, we explore the differences between the structures and agn host properties based on the x - ray luminosities of the confirmed agns .
we calculate rest - frame luminosities for x - ray point sources with known redshifts .
k - corrections were carried out using the power law spectral models for sources , with a photon index of @xmath260 , described in section [ sec : red ] .
luminosities are measured in the x - ray soft , hard , and full bands .
a histogram of full - band rest - frame luminosity , binned by evolved and unevolved structures , is shown in figure [ xlums ] .
the luminosity distributions in the soft and hard bands are similar to the one shown . in the left panel of figure [ xlums ] , we can see that the agns in the unevolved structures have higher x - ray luminosities than those in the more evolved structures .
k - s tests show that the distributions of the two bins are different at the 99@xmath7 level in each of the three bands .
this statistically - significant result is independent of our inclusion of the four low - significance ( @xmath205 3@xmath30 ) x - ray sources .
while 10 out of 17 agns in the unevolved structures have full band luminosities above @xmath261 ergs s@xmath5 , there are _ no _ agns above this limit in cl1324 , rxj1757 , or rxj1821 .
we follow a bayesian approach , using poisson statistics to calculate the likelihoods , to estimate the probability of finding no high @xmath262 agns in the evolved structures ( @xmath263 ) given the detection rate in the unevolved structures . specifically , we calculate @xmath264 . here , @xmath265 is the number of high @xmath266 agns that are spectroscopically confirmed members in the unevolved structures , and @xmath267 and @xmath268 are the total number of high @xmath266 agns that were targeted for spectroscopy in the unevolved and evolved structures , respectively .
based on this calculation , the probability of finding no high @xmath262 agns in the evolved structures is only 0.25% .
this result is likely related to the smaller fractions of blue cloud galaxies ( and overall more quiescent populations ) in cl1324 , rxj1757 , or rxj1821 and , thus , the unavailability of large gas reservoirs . at the faint end , there are four sources in the evolved structures below @xmath269 ergs s@xmath5 , all of which are members of cl1324 . however , the x - ray source counts , in all five fields , are significantly incomplete at these luminosity levels with only @xmath270 of optically matched x - ray sources below this limit , most of which are @xmath205 3@xmath30 detections
. therefore , we can not say anything definitive about the lack of faint sources in the unevolved structures ; however , if we remove the four low - luminosity sources , the difference between the luminosity distributions in the evolved and unevolved samples is still significant at a 95% level according to the k - s test .
the reasonably high significance is clearly due to the lack of high - luminosity sources in the evolved structures . in the right panel of figure [ xlums ]
, we plot the full - band rest - frame luminosity versus the red - sequence offset , scaled by rs width ( see section [ sec : hgalcolan ] ) . from this figure
, we can see that all agns with host galaxies on the red - sequence have lower x - ray luminosities , all below @xmath271 ergs s@xmath5 ( region 4 ) .
this result is not unexpected since agn activity should diminish as the host galaxy moves onto the red sequence .
we can also see that 60% of the confirmed agns lie in the green valley ( @xmath272 ; regions 2 and 3 ) .
most interestingly , in the green valley there are almost two orders of magnitude variation in x - ray luminosity , with the unevolved structures having all of the highest @xmath273 sources .
we try to decipher the origin of these variations in section [ sec : xlums ] .
we also do not detect any low @xmath266 blue agns ( region 5 ) or any high @xmath266 red agns ( region 6 ) , not necessarily unexpected given their expected gas contents .
however , it is difficult to say for certain if these null results are significant .
we do sample optically matched x - ray sources in these regions .
specifically , in the five fields , there are a total of 85 sources ( at @xmath133 3@xmath30 ) in region 5 ( @xmath274 and @xmath275 ergs s@xmath5 ) , of which we have targeted 24 ( 28% ) . here
, we estimate the x - ray luminosities by assuming that all sources in a particular field are at the mean redshift of the structure , and we choose the lower limit of @xmath276 ergs s@xmath5 so as not to be adversely affected by incompleteness ( see above ) .
based on @xcite , the x - ray luminosity function of field agns at similar redshifts in `` optically normal '' galaxies ( comparable to the vast majority of our agn hosts ) shows increasing number densities down to @xmath277 ergs s@xmath5 . as a result
, we would naively expect to detect a larger number of fainter x - ray sources in the blue cloud . using the bayesian approach described in section [ sec : xlum ] ,
the probability of finding no agn in region 5 is 3% , given our success rate of confirming cluster members for the high @xmath266 blue galaxies ( region 1 ) .
this formal probability may , in fact , be an _
upper limit _ as we would expect a higher success rate given the larger number densities at fainter x - ray luminosities .
the fact that we observe no low @xmath266 blue galaxies in our sample may suggest that , in high - density environments compared to the field , either ( 1 ) the time to reach the highest x - ray luminosities is shorter after agn turn - on or ( 2 ) the host galaxies are transformed more quickly , moving to redder colors by the time their x - ray luminosities drop to lower levels .
similarly , there are 25 sources in region 6 ( @xmath278 and @xmath279 ergs s@xmath5 ) , of which we have targeted 7 ( 28% ) . if we assume our success rate for confirming cluster members as measured from all high @xmath266 agns ( regions 1 and 2 ) , the probability of finding no agn in region 6 is 18% . although not a significant result , the conclusions for this region are , of course , more obvious as we do not expect any true red - sequence ( i.e. , non - dust reddened ) galaxies to have enough cold gas to fuel a luminous agn . to explore the origin of the variations observed in figure [ xlums ] , we examine the average spectral properties of the host galaxies within the four distinct regions .
specifically , the sample is split by x - ray luminosity at @xmath280 and red - sequence offset , scaled by rs width , to delineate regions containing high @xmath266 blue cloud ( region 1 ) , high @xmath266 ( region 2 ) and low @xmath266 ( region 3 ) green valley , and low @xmath266 red - sequence ( region 4 ) host galaxies . in figure [ specreg ] , we plot the measured ew(h@xmath2 ) versus d@xmath164(4000 ) from the spectral composites in the four regions . for comparison
, we also plot post - starburst temporal regimes derived from four @xcite models , described in section [ sec : specprop ] .
the small d@xmath281(4000 ) and h@xmath2 in emission indicate that the high @xmath266 blue hosts are coeval with the starburst or ongoing star formation . as we examine galaxies in regions 2 to 4 going from the high to low @xmath266 hosts in the green valley to the low @xmath266 hosts in the red sequence , the time since the burst gets progressively larger .
while there is some degeneracy between time since burst and burst strength , it is clear that the low @xmath266 green valley hosts are either ( 1 ) further along since the burst than their high @xmath266 counterparts or ( 2 ) had a weaker initial burst which could explain their lower x - ray luminosities as less gas would likely be funneled to the center .
our results are robust to removing the four lowest significance ( @xmath205 3@xmath30 ) x - ray sources , as well as the four x - ray sources below @xmath276 ergs s@xmath5 , where in both cases we are highly incomplete .
the most striking results from this spectral analysis are , first , that the average agn host in _ every _ region is either in the process of having a starburst or has had one within last @xmath282 gyr .
this global finding clearly demonstrates the close connection between starburst and agn activity as normal star formation does not typically produce the h@xmath2 values seen in these host galaxies .
second , we do not detect high x - ray luminosity , young ( as indicated by time since starburst ) galaxies in the evolved structures .
this result implies that the entire galaxy population in these structures ( certainly in the isolated , x - ray selected ones ) are more advanced , suggesting that the peak of gas consumption , seen in both star formation and agn activity , occurred at an earlier time .
we studied agn activity in five high - redshift clusters and superclusters in the redshift range @xmath283 . before identifying individual agns
, we analyzed the structures using the statistical measure of cumulative source counts .
we found every structure to have x - ray excesses of 0.5 - 1.5 @xmath30 with respect to the cdfn and cdfs control fields .
the method is highly dependent on the field used as a sky estimate , which makes comparing results between studies difficult .
@xcite measured cumulative source counts in a number of structures in the range @xmath284 and found a dependence on redshift . while our data are consistent within the errors with the results of c05 , our redshift range is not large enough to evaluate the relation .
we note that it is difficult to use these overdensities to interpret the actual agn activity in an individual structure , even with extensive spectroscopy , as we have attempted to do here with our large spectroscopic sample .
we recommend caution in using this technique , as its precision can easily be overestimated .
we employed a maximum likelihood technique to match x - ray sources to optical counterparts . with an extensive deimos optical spectroscopic campaign with @xmath285 targets ,
accurate redshifts have been obtained for 126 of these x - ray sources , allowing us to identify a total of 27 agns within all of the structures .
these results show that significant spectroscopy is needed to confirm even small numbers of agn members .
we find that the spatial distribution of the agn is largely consistent with previous work at lower redshift . across all five structures
, we find that agn host galaxies tend to be located away from dense cores ( within 0.5 mpc of a cluster or group center ) , with many instead located on the outskirts of clusters or poorer groups km s@xmath5 . ] .
previous studies have found similar results up to @xmath211 @xcite .
these intermediate environments are thought to be conducive to galaxy - galaxy interactions , because of the relatively high densities compared to the field and low velocity dispersions compared to cluster cores @xcite .
our results would then lend support to these interactions as the trigger of x - ray agn activity in the environments of lsss . however , these may also be regions where gas availability is higher in the member galaxies , which could lead to increased agn activity
. with optical counterparts to x - ray sources identified , we were able to analyze the color properties of agn hosts .
our analysis showed that agn host galaxies are overrepresented in the green valley . in cl1604 , only @xmath286 of all the supercluster members with acs data were within @xmath287 of the lower boundary of the red sequence , where @xmath221 is the width of the red sequence , defined in section [ sec : globchar ] .
however , five out of the eight agn hosts in our acs pointings were in this range .
in fact , @xmath1 of host galaxies in all five structures lie within @xmath221 of the lower boundary of the red sequence and @xmath226 are within @xmath287 .
our results are supported by other studies which have found an overabundance of agn activity in the green valley @xcite .
since the green valley is thought to be a transitional region for galaxies @xcite , these results suggest x - ray agns in lss are a transitional population between blue star - forming and red quiescent galaxies .
however , we note that , while our sample is magnitude - limited , several studies using mass - selected samples have found less overrepresentation of agn hosts in the green valley @xcite .
additionally , @xcite have found that @xmath8of agn hosts in the green valley are dust - reddened blue cloud galaxies , although these results conflict with the recent studies of @xcite .
our results from preliminary sed fittings to agn hosts suggest that the effect is not as drastic in our sample .
the five structures studied in this paper occupy a range of evolutionary states .
based on the [ ] and h@xmath2 features of the composite spectra for each structure , we grouped our sample into the least evolved structures ( `` unevolved '' ) , cl1604 and cl0023 , and the most evolved structures ( `` evolved '' ) , cl1324 , rxj1821 , and rxj1757 .
this distinction is based on the average stellar populations and the presence , or lack thereof , of current star formation . with these two categories
, we sought to explore differences in agn activity between structures with different galaxy populations .
we did not find any significant differences between the five structures when examining the cumulative source counts or the spatial distributions of the agn hosts .
however , the agn host galaxies in the unevolved structures were skewed more towards bluer colors , although this was not at a statistically significant level for this sample size .
we did , however , find significant differences between the subsets when examining the x - ray luminosities of the agns and the optical spectra of their hosts .
we found that agns in the unevolved structures tend to have higher full band ( 0.58 kev ) x - ray luminosities relative to those in the evolved structures at a @xmath137 level , with all of the most luminous agns ( @xmath279 erg s@xmath5 ) found in the unevolved structures . while _ all _ agn host galaxies either have on - going star formation or have had a starburst within the last @xmath282 gyr , the host galaxies in the unevolved structures are distinctly younger than those in the evolved structures , with shorter times since the last starburst as indicated by smaller average ew@xmath288 and d@xmath164(4000 ) in their composite spectra .
the average cl0023 host has current star formation , and the average cl1604 host has had a burst within the last @xmath289 myr .
we do not detect any of these young , high x - ray luminosity agns in the evolved structures , implying that the peak of both star formation and agn activity occurred at an earlier time .
we note that , regardless of whether they are members of the evolved or unevolved structures , _ all _ agn host galaxies are younger than the average galaxy in their parent population .
we also find a large ( two orders of magnitude ) variation in x - ray luminosity for agns within the green valley , while agns in the red sequence have consistently lower luminosities ( @xmath271 erg s@xmath5 ) .
as we move from the high to low @xmath266 green - valley hosts to low @xmath266 red - sequence hosts , the time since starburst gets progressively longer .
although there is some degeneracy between burst strength and time since burst , the low @xmath266 green valley hosts are either further along since the burst than their high @xmath266 counterparts or have had a weaker initial burst , which may explain their lower x - ray luminosities .
the higher agn x - ray luminosities in the unevolved structures are most likely related to their bluer colors and , hence , larger reservoirs of gas that could lead to higher levels of black hole accretion and the higher x - ray luminosities . in addition , both the x - ray and spectral results can be explained if these galaxies had more recently undergone merger - induced , or other , starburst events .
specifically , simulations and observations have found that agn activity peaks soon after maximum star formation in a starburst event ( @xmath60.1 - 0.25 gyr ) @xcite .
as the agn x - ray luminosity declines after reaching its peak , the star formation rate should be declining as well .
this could explain why the most x - ray luminous agns are in the least evolved structures .
agn host galaxies in cl0023 were also found to have significant ongoing star formation , which could mean these galaxies have had the most recent merger events , where star formation is still near its peak .
the x - ray luminosity differences between the agns in the evolved and unevolved structures could also be viewed as a transition from quasar mode emission towards radio mode emission , as defined in @xcite and @xcite , which is also related to the star formation rate .
since agn feedback deters gas from falling into the core and depletes the cold gas in a galaxy , the fuel for both star formation and further agn emission is decreased .
ultimately , the dominant fuel for black hole accretion transitions from cold gas funneled to the galactic core from the starburst event to hot halo gas , which leads to a much more quiescent state ( * ? ? ?
* ; * ? ? ?
* ; for a similar model , see also , e.g. , @xcite , @xcite ) . in our sample
, the cl0023 agns are emitting more similarly to the quasar mode , accompanied by substantial star formation .
the agn host galaxies in cl1604 and the evolved structures have lower star formation rates than those in cl0023 , while the evolved structures have agns with lower x - ray luminosities , which suggests that they are sequentially further along the track leading toward domination of radio mode agn emission .
if the agns in the evolved structures were found to be radio emitters , it would support these conclusions .
existing vla b - array observations ( at 1.4 ghz ) of these five structures are currently being analyzed to explore this connection ( c. d. fassnacht et al .
2012 , in preparation ) . altogether
, many of our results could support several potential agn triggering scenarios .
two possibilities are that : ( 1 ) agns in these structures represent a transitional population where hosts are evolving from the blue cloud onto the red sequence or ( 2 ) agns represent a population evolving mainly in mass space , where red sequence hosts have undergone episodic nuclear star formation induced by minor mergers .
additionally , many observable effects of the latter would appear similar to models in which recycled stellar material fuels central starbursts and nuclear activity in elliptical galaxies , which tend to be located on the red sequence .
in the first case , agns are triggered by mergers or strong tidal interactions which lead to a starburst .
the feedback from the agn quenches star formation , leading to a rapid evolution across the green valley and onto the red sequence @xcite . in the second case ,
red sequence galaxies undergo minor mergers , which funnel gas into the galactic core , creating a burst of nuclear star formation and fueling the agn @xcite .
alternatively , recycled stellar material could create a nuclear starburst and a central instability , leading to black hole accretion @xcite . in either case
, the galaxies could evolve into the green valley before agn feedback brought them onto the red sequence again .
once on the red sequence , there would not be a large net change in color , in contrast to the dramatic color evoltion of the first possibility . in the case of minor mergers ,
this process would mainly entail an evolution in mass . in support of agns as a transitional population , many of the host galaxies across all five structures
are located close to the red sequence , where the green valley should lie , although it is difficult to determine the location of the green valley in the four structures without precise acs data . in support of all scenarios involving significant starbursts ,
the average agn host in both the evolved structures and the cl1604 supercluster has substantial h@xmath2 absorption , which is a sign of recently quenched star formation .
this is expected for green valley galaxies evolving onto the red sequence .
previous morphological analysis of part of our sample by @xcite found that two - thirds of cl1604 host galaxies studied showed signs of recent or pending mergers or tidal interactions , which is expected in the context of both major and minor merger theories .
in addition , we find that at least nine of the 27 agn host galaxies are part of a kinematic close pair . in seven of these cases ,
the companion galaxy has a similar stellar mass or @xmath63 magnitude to the agn host . while being far from conclusive due to sampling and selection effects , these results could point to a major merger scenario .
@xcite examined eight of the agns in cl1604 and found that half of the hosts had blue cores in an otherwise red galaxy .
other studies have also found blue cores or blue early type galaxies @xcite .
these blue cores are predicted by minor merger simulations @xcite , as well as the recycled gas models of @xcite .
in addition simulations show that central black hole accretion is most highly correlated with star formation in the nucleus , as opposed to the entire galaxy @xcite . during a major merger ,
star formation peaks at later times closer to the galactic core @xcite .
this could potentially create bluer cores as well . from our data
, it seems our results are ambiguous with regards to the various scenarios .
however , with evidence supporting more than one possibility , our results could indicate a combination of different triggering mechanisms .
these different agn triggering mechanisms could also potentially explain some of the differences that we see between the evolved and unevolved structures . the second agn mode , involving episodic nuclear activity in red sequence galaxies fueled by minor mergers , or recycled gas , is expected to involve accretion rates at a much lower eddington ratio than in major merger driven agn activity @xcite .
while agn luminosity depends on both the eddington ratio and the black hole mass @xcite , lower eddington ratios will , on average , correspond to lower luminosities .
a difference in eddington ratios could potentially explain the lower x - ray luminosities that we observe in the evolved structures , if our sample of host galaxies consists of a mix of the two agn modes .
in the unevolved structures , we would expect to observe more major mergers between blue cloud galaxies due to the larger fraction of blue galaxies , fueling the brightest agns in our sample .
since all of our structures have a substantial number of red - sequence galaxies we would expect all structures to have galaxies undergoing the second mode of agn activity , whether induced by minor mergers or , perhaps , recycled gas in ellipticals .
we would then expect the x - ray luminosity distributions of the evolved and unevolved structures to look similar except for a tail of higher luminosity objects in the unevolved structures , which is roughly what we observe . while a combination of the two agn modes considered could explain our results , they could also be explained if major mergers were the primary driver , and the quasar mode is less dominant for the agns in the evolved structures , because of the larger red fraction therein . with our current data
, we still lack the ability to distinguish between the different triggering mechanisms .
breaking the degeneracy will require high - resolution imaging to examine the morphologies of the agn hosts and the colors of their cores in the structures other than cl1604 , as well as reliable stellar masses and hst data , for measuring bulge - to - disk - ratios , with which we could reliably calculate eddington ratios .
in summary , we find that most x - ray agn hosts , across all five structures , avoid the dense cluster cores , in agreement with a number of previous studies at a range of redshifts . we interpret this to mean that x - ray agn activity is preferentially triggered in intermediate - density environments , such as the outskirts of clusters .
we also find many agn host galaxies in or near the green valley , with 36@xmath7 within one red - sequence width of the lower boundary of the red sequence and 60@xmath7 within two red - sequence widths . with numerous other studies finding a similar connection , this implies that there is an association between this transitional region and agn activity .
we divided our sample of five structures into two groups : the more and less evolved structures , which we separated using composite spectra made of all of their spectroscopically - confirmed member galaxies .
we define the more evolved structures as those with member galaxies that exhibit , on average , less [ ] emission and less h@xmath290absorption , where the [ ] and h@xmath2 lines are taken as indicators of ongoing and recent star formation , respectively .
the more evolved structures also have galaxy populations with a higher red fraction than the less evolved structures .
our spectral results indicate that the agn hosts in the less evolved structures have more ongoing star formation , while those in the more evolved structures have stronger average h@xmath2 features .
stronger h@xmath2 lines are indicative of star formation within the past @xmath61 gyr , and our results indicate starbursts occurred more recently in the agn host galaxies in the less evolved structures .
however , all of the agn hosts , regardless of whether they are members of the more evolved or less evolved structures , are younger than the average galaxy in their parent population .
we also found that agns in the less evolved structures had more luminous x - ray emission .
this may be expected , since these structures contain the highest fraction of blue galaxies which are likely to have larger reservoirs of cool gas to fuel nuclear activity .
if agns were triggered more recently in the less evolved structures , as the spectral data suggests , the difference in luminosity could also be related to a transition from `` quasar mode '' emission in newly triggered agns to `` radio mode '' emission at later times as supplies of inflowing cool gas are shut off .
we consider several scenarios for agn triggering that are in agreement with our results .
agns are triggered by major mergers or tidal interactions between blue cloud galaxies and/or agns are triggered episodically in red - sequence galaxies , fueled by recycled stellar material or induced by minor mergers .
each of these scenarios could explain the association of agn hosts with the green valley .
also , the h@xmath290absorption we observe could be indicative of the quenching of star formation that drives galaxies across the green valley .
a previous study of the cl1604 agns by @xcite found that a majority of hosts had recent or pending mergers .
half of the hosts had blue cores in otherwise red galaxies , which could support the second two scenarios , although other explanations are possible .
some of our results , such as the difference in x - ray luminosities , could be explained if the agns in our sample were triggered by a mix of these modes , with a larger fraction of agns in the less evolved structures triggered by major mergers .
however , we can not distinguish between the triggering scenarios with our data , so the cause of the agn activity is still ambiguous . future work investigating agn host morphologies , examining for blue cores , and calculating eddington ratios could potentially break the degeneracy .
the authors thank phil marshall and robert lupton for useful conversations .
this work is supported by the _ chandra _ general observing program under award numbers go6 - 7114x , go7 - 8126x , go8 - 9123a , and go9 - 0139a .
in addition , we acknowledge support by the national science foundation under grant no .
the spectrographic data presented herein were obtained at the w.m .
keck observatory , which is operated as a scientific partnership among the california institute of technology , the university of california , and the national aeronautics and space administration .
the observatory was made possible by the generous financial support of the w.m . keck foundation .
as always , we thank the indigenous hawaiian community for allowing us to be guests on their sacred mountain .
we are most fortunate to be able to conduct observations from this site .
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2006 , , 648 , 281 | we use deep chandra imaging and an extensive optical spectroscopy campaign on the keck 10-m telescopes to study the properties of x - ray point sources in two isolated x - ray selected clusters , two superclusters , and one ` supergroup ' at redshifts of @xmath0 .
we first study x - ray point sources using the statistical measure of cumulative source counts , finding that the measured overdensities are consistent with previous results , but we recommend caution in overestimating the precision of the technique .
optical spectroscopy of objects matched to x - ray point sources confirms a total of 27 agns within the five structures , and we find that their host galaxies tend to be located away from dense cluster cores .
more than @xmath1 of the host galaxies are located in the ` green valley ' on a color magnitude diagram , which suggests they are a transitional population .
based on analysis of [ ] and h@xmath2 line strengths , the average spectral properties of the agn host galaxies in all structures indicate either on - going star formation or a starburst within @xmath3 gyr , and that the host galaxies are younger than the average galaxy in the parent population .
these results indicate a clear connection between starburst and nuclear activity .
we use composite spectra of the spectroscopically confirmed members in each structure ( cluster , supergroup , or supercluster ) to separate them based on a measure of the overall evolutionary state of their constituent galaxies .
we define structures as having more evolved populations if their average galaxy has lower ew ( [ ] ) and ew(h@xmath2 ) .
the agns in the more evolved structures have lower rest - frame 0.58 kev x - ray luminosities ( all below @xmath4 erg s@xmath5 ) and longer times since a starburst than those in the unevolved structures , suggesting that the peak of both star formation and agn activity has occurred at earlier times . with the wide range of evolutionary states and timeframes in the structures
, we use our results to analyze the evolution of x - ray agns and evaluate potential triggering mechanisms . | 1108.5733 |
semitauonic @xmath11 meson decays with @xmath12 @xcite transitions are sensitive to new physics ( np ) beyond the standard model ( sm ) involving non - universal coupling to heavy fermions .
one prominent candidate for np is the two higgs doublet model ( 2hdm ) @xcite , which has an additional higgs doublet and therefore introduces two neutral and two charged higgs bosons in addition to the sm higgs boson .
the charged higgs bosons may contribute to the @xmath13 process , modifying its branching fraction and decay kinematics .
exclusive semitauonic decays of the type @xmath14 have been studied by belle @xcite , babar @xcite and lhcb @xcite .
the experiments typically measure the ratios of branching fractions , @xmath15 where the denominator is the average for @xmath16 .
the ratio cancels uncertainties common to the numerator and the denominator .
these include the cabibbo - kobayashi - maskawa matrix element @xmath17 and many of the theoretical uncertainties on hadronic form factors and experimental reconstruction effects .
the current averages of the three experiments @xcite are @xmath18 and @xmath19 , which are within @xmath20 and @xmath21 @xcite of the sm predictions of @xmath22 @xcite or @xmath23 @xcite and @xmath24 @xcite , respectively . here
, @xmath25 represents the standard deviation .
in addition to @xmath26 , the polarization of the @xmath0 lepton and the @xmath27 meson is also sensitive to np @xcite .
the polarization of the @xmath0 lepton ( @xmath28 ) is defined by @xmath29 where @xmath30 denotes the decay rate of @xmath14 with a @xmath0 helicity of @xmath31 .
the sm predicts @xmath32 for @xmath33 @xcite and @xmath34 for @xmath35 @xcite .
the @xmath0 polarization is accessible in two - body hadronic @xmath0 decays with the following formulae @xcite : @xmath36 where @xmath37 , @xmath38 and @xmath39 are , respectively , the decay rate of @xmath40 and the masses of the @xmath0 lepton and the vector meson from the @xmath0 decay .
the helicity angle , @xmath41 , is the opening angle between the momentum vectors of the virtual @xmath42 boson and of the @xmath0-daughter meson in the rest frame of the @xmath0 .
the parameter @xmath43 describes the sensitivity to @xmath28 for each @xmath0-decay mode ; in particular , @xmath44 for the decay @xmath5 . in this paper , we report a new measurement of @xmath45 in the hadronic @xmath0 decay modes @xmath4 and @xmath46 .
this measurement is statistically independent of the previous belle measurements @xcite , with a different background composition .
we also report the first measurement of @xmath28 for the decay @xmath1 .
we use the full @xmath47 data sample containing @xmath48 pairs recorded with the belle detector @xcite at the asymmetric - beam - energy @xmath49 collider kekb @xcite .
the belle detector is a large - solid - angle magnetic spectrometer that consists of a silicon vertex detector ( svd ) , a 50-layer central drift chamber ( cdc ) , an array of aerogel threshold cherenkov counters ( acc ) , a barrel - like arrangement of time - of - flight scintillation counters ( tof ) , and an electromagnetic calorimeter ( ecl ) comprised of csi(tl ) crystals located inside a superconducting solenoid coil that provides a 1.5 t magnetic field .
an iron flux - return located outside of the coil is instrumented to detect @xmath50 mesons and to identify muons ( klm ) .
the detector is described in detail elsewhere @xcite .
two inner detector configurations were used .
a 2.0 cm radius beampipe and a 3-layer silicon vertex detector was used for the first sample of @xmath51 pairs , while a 1.5 cm radius beampipe , a 4-layer silicon detector and a small - cell inner drift chamber were used to record the remaining @xmath52 pairs @xcite .
the signal selection criteria and the signal and background probability density functions ( pdfs ) used in this measurement rely on the use of monte carlo ( mc ) simulation samples .
these samples are generated by the software packages evtgen @xcite and pythia @xcite ; final - state radiation is generated by photos @xcite .
detector responses are fully simulated with the belle detector simulator based on geant3 @xcite .
the signal decay @xmath1 ( signal mode ) is generated with a decay model based on the heavy quark effective theory ( hqet ) @xcite .
we use the current world - average values for the form - factor parameters @xmath53 , @xmath54 and @xmath55 @xcite , which are based on the parameterization in ref .
decays of the type @xmath56 ( normalization mode ) , which are used for the denominator of @xmath45 , are also modeled with hqet using the above form - factor values .
background from semileptonic decays to orbitally - excited charmed mesons @xmath57 , where @xmath58 denotes @xmath59 , @xmath60 , @xmath61 or @xmath62 , are generated with the isgw model @xcite with their kinematic distributions reweighted to match the dynamics predicted by the llsw model @xcite .
additionally , theoretically - predicted radial excitation states @xmath63 are assumed to fill the gap between the inclusively - measured and the sum of the exclusively - measured branching fractions of @xmath64 @xcite .
the mc sample of @xmath65 is produced with the isgw model .
the branching fractions are assigned according to their theoretical estimates @xcite .
the remaining background mc samples are comprised of mostly hadronic @xmath11 meson decays and light quark production processes ( @xmath66 ) .
the sample sizes of the signal , @xmath67 , @xmath68 , other @xmath7 , and @xmath69 processes are 40 , 40 , 400 , 10 and 5 times larger , respectively , than the full belle data sample .
we first identify events where one of the two @xmath11 mesons ( @xmath70 ) is reconstructed in one of 1149 exclusive hadronic @xmath11 decays using a hierarchical multivariate algorithm @xcite based on the neurobayes package @xcite .
more than 100 variables are used in this algorithm , including the difference @xmath71 between the energy of the reconstructed @xmath70 candidate and the kekb beam energy in the @xmath49 center - of - mass system as well as the event shape variables for suppression of @xmath72 background .
we further require the beam - energy - constrained mass of the @xmath70 candidate @xmath73 , where @xmath74 denotes the reconstructed @xmath70 three - momentum in the @xmath75 center - of - mass system , to be greater than 5.272 gev@xmath76 and @xmath77 to lie between @xmath78 and @xmath79 . if there are two or more @xmath70 candidates retained after the selection criteria , we select the one with the highest neurobayes output value , which is related to the probability that the @xmath70 candidate is correctly reconstructed .
due to limited knowledge of hadronic @xmath11 decays , the branching ratios of the @xmath70 decay modes are not perfectly modeled in the mc .
it is therefore essential to calibrate the @xmath70 reconstruction efficiency ( tagging efficiency ) with control data samples .
we determine a scale factor for each @xmath70 decay using the method described in ref .
@xcite based on events where the signal - side @xmath11 meson candidate ( @xmath80 ) is reconstructed in @xmath81 modes .
the ratio of measured to expected rates in each decay mode ranges from 0.2 to 1.4 , depending on the @xmath70 decay mode , and is 0.68 on average . after the efficiency calibration ,
the tagging efficiencies are about 0.20% for charged @xmath11 mesons and 0.15% for neutral @xmath11 mesons .
after @xmath70 selection , we form @xmath80 candidates from the remaining particles not associated with the @xmath70 candidate . charged particles used to form @xmath80 candidates are reconstructed using information from the svd and the cdc .
the tracks that are not used in @xmath82 reconstruction are required to have impact parameters to the interaction point ( ip ) of less than 0.5 cm ( 2.0 cm ) in the direction perpendicular ( parallel ) to the @xmath83 beam axis .
charged - particle types are identified by a likelihood ratio based on the response of the sub - detector systems .
identification of @xmath84 and @xmath85 candidates is done by combining measurements of specific ionization ( @xmath86 ) in the cdc , the time of flight from the ip to the tof counter and the photon yield in the acc . for @xmath0-daughter
@xmath85 candidates , an additional proton veto is required in order to reduce background from the baryonic @xmath11 decays @xmath87 .
the ecl electromagnetic shower shape , track - to - cluster matching at the inner surface of the ecl , the photon yield in the acc and the ratio of the cluster energy in the ecl to the track momentum measured with the svd and cdc are used to identify @xmath88 candidates .
muon candidates are selected based on the comparison of the projected cdc track with interactions in the klm . to form @xmath82 candidates
, we combine a pair of oppositely - charged tracks , treated as pions .
three requirements are applied : the reconstructed vertex must be detached from the ip , the momentum vector must point back to the ip , and the invariant mass must be within @xmath8930 mev/@xmath90 of the nominal @xmath82 mass @xcite , which corresponds to about 8@xmath25 .
( in this section , @xmath25 denotes the corresponding mass resolution . )
photons are reconstructed using ecl clusters not matched to charged tracks .
photon energy thresholds of 50 , 100 and 150 mev are used in the barrel , forward - endcap and backward - endcap regions , respectively , of the ecl to reject low - energy background photons , such as those originating from the @xmath49 beams , and hadronic interactions of particles with material in the detector .
neutral pions are reconstructed in the decay @xmath91 . for @xmath92 candidates from @xmath93 or @xmath94 decay
, we impose the same photon energy thresholds described above . the @xmath92 candidate s invariant mass must lie between 115 and 150 mev/@xmath90 , corresponding to about @xmath95 around the nominal @xmath92 mass @xcite . in order to reduce the number of fake @xmath92 candidates , we apply the following @xmath92 candidate - selection procedure .
the @xmath92 candidates are sorted in descending order according to the energy of the most energetic daughter .
if a given photon is the most energetic daughter for two or more candidates , they are sorted by the energy of the lower - energy daughter .
we then retain the @xmath92 candidates whose daughter photons are not shared with a higher - ranked candidate .
the remaining @xmath92 candidates are used for @xmath93 or @xmath94 reconstruction described later . for the soft @xmath92 from @xmath27 decay
, we impose a relaxed photon energy threshold of 22 mev in all ecl regions , the same requirement for the invariant mass of the two photons , and an energy - asymmetry @xmath96 less than 0.6 , where @xmath97 and @xmath98 are the energies of the high- and low - energy photon daughters in the laboratory frame .
we do not apply the above candidate - selection procedure for the soft @xmath92 candidates . after reconstructing the final - state particles and light mesons , we reconstruct the @xmath99 candidates using 15 @xmath93 decay modes : @xmath100 , @xmath101 , @xmath102 , @xmath103 , @xmath104 , @xmath105 , @xmath106 , @xmath107 , @xmath108 , @xmath109 , @xmath110 , @xmath111 , @xmath112 , @xmath113 , @xmath114 , and four @xmath27 decay modes : @xmath115 , @xmath116 , @xmath117 and @xmath118 .
the @xmath93 invariant mass requirements are optimized for each decay mode . for the @xmath119 modes in the @xmath120 candidates , the invariant masses ( @xmath121 )
are required to be within @xmath122 ( @xmath123 ) of the nominal @xmath119 meson mass @xcite for the high ( low ) signal - to - noise ratio ( snr ) modes . for @xmath124 candidates ,
the @xmath121 requirements are loosened to @xmath125 and @xmath122 for the high- and low - snr modes , respectively .
the requirements for the @xmath126 candidates are @xmath127 for the high - snr modes and
@xmath123 for the low - snr modes around the nominal @xmath126 meson mass @xcite . here , the high - snr modes are @xmath100 , @xmath102 , @xmath103 , @xmath105 , @xmath107 , @xmath108 , @xmath109 , @xmath111 ; the low - snr modes are all remaining @xmath93 modes .
we reconstruct a @xmath27 candidate by combining a @xmath93 candidate with a @xmath85 , @xmath128 or soft @xmath92 .
the @xmath27 candidates are selected based on the mass difference @xmath129 , where @xmath130 denotes the invariant mass of the @xmath27 candidate .
the @xmath115 , @xmath131 , @xmath117 and @xmath124 candidates are required to have a @xmath132 within @xmath123 , @xmath122 , @xmath122 and @xmath133 of the nominal @xmath132 . for the @xmath134 candidates ,
the @xmath94 candidate is formed from the combination of a @xmath85 and a @xmath92 with an invariant mass between 0.66 and 0.96 mev@xmath76 .
we then associate a @xmath85 or a @xmath135 candidate ( one charged lepton ) with the @xmath27 candidate to form signal ( normalization ) candidates . for the signal mode , square of the momentum transfer
, @xmath136 is required to be greater than 4 gev@xmath137 , where @xmath138 and @xmath139 denote the energy and the three - momentum specified by the subscript .
the subscripts `` @xmath49 '' , `` tag '' and `` @xmath27 '' stand for the colliding @xmath83 and @xmath140 , the @xmath70 candidate and the @xmath27 candidate , respectively . due to the kinematic constraint in @xmath35 that @xmath141 is always greater than the square of the @xmath0 mass , almost no signal events exist with @xmath141 below 4 gev@xmath137 .
finally , we require that there be no remaining charged tracks nor @xmath92 candidates ( except for soft @xmath92 ) in the event .
after the @xmath80 reconstruction procedure is completed , the probability to have multiple candidates ( the number of retained candidates ) per event is about 9% ( 1.09 ) for the charged @xmath11 mesons and 3% ( 1.03 ) for the neutral @xmath11 mesons .
most of the multiple - candidate events are due to the existence of two or more @xmath27 candidates in an event . for the @xmath120 candidates in the charged @xmath11 meson sample , about 2% of the events
are reconstructed both in the @xmath115 and @xmath131 modes .
since the latter mode has a much higher branching fraction , we assign these events to the @xmath131 sample .
the contribution of this type of multiple - candidate events is negligibly small in the @xmath142 mode .
we then select the most signal - like event as follows . for the @xmath115 events , we select the candidate with the most energetic photon associated with the @xmath119 . for the @xmath131 and @xmath117 events ,
we select the candidate with the soft @xmath92 having the invariant mass nearest the nominal @xmath92 mass . for the @xmath124 events , we select one candidate at random since the multiple - candidate probability is only @xmath143 . after the @xmath27 candidate selection ,
roughly 2% of the retained events are reconstructed both in the @xmath4 sample and the @xmath5 sample . according to the mc study ,
about 80% of such signal events are actually @xmath5 events .
we therefore assign these events to the @xmath5 sample .
distribution for the @xmath144 mc sample.,width=377 ] in order to measure @xmath28 , the @xmath145 distribution must be reconstructed .
this is challenging , as the @xmath0 momentum vector is not fully determined . instead of @xmath145 ,
we measure the cosine of the angle @xmath146 between the momenta of the @xmath0 lepton and its daughter meson in the rest frame of the virtual @xmath42 boson , @xmath147 as shown in fig .
[ fig - kinematics ] ( a ) . here ,
@xmath138 and @xmath139 denote the energy and the three - momentum of a particle specified by the subscript , where @xmath0 and @xmath148 represent the @xmath0 lepton and its daughter meson , respectively .
this angle is equivalent to @xmath41 in this frame .
the rest frame of the virtual @xmath42 is obtained from its three - momentum @xmath149 in this frame of reference , the magnitude of the @xmath0 momentum is determined only by @xmath141 since the @xmath0 lepton is emitted in the two - body decay of the static virtual @xmath42 boson .
therefore @xmath150 is calculated as @xmath151 we only accept events , for which @xmath152 . here , more than 97% of the reconstructed signal events are retained . due to limited kinematic constraints , one degree of freedom of the @xmath0 momentum direction
is not determined
. however , the cone around @xmath153 with an angle of @xmath146 , on which @xmath154 lies , is rotationally symmetric and therefore all directions on this cone are equivalent . with this in mind , as shown in fig .
[ fig - kinematics ] ( b ) , we take the new right - handed @xmath155 coordinate such that the @xmath156-axis corresponds to the direction of the @xmath153 , and set @xmath157 .
the system is boosted to the pseudo @xmath0 rest frame with @xmath158 , where the correct value of @xmath145 is obtained . as shown in fig .
[ fig - coshel ] , there are many @xmath159 background events that peak around @xmath160 , which corresponds to @xmath161 , in the @xmath4 sample .
this peak arises from low - momentum muons that do not reach the klm and are therefore misreconstructed as pions .
to mitigate this background , we only use the region @xmath162 in the fit , in which 94% ( 81% ) of signal events are contained with the sm @xmath28 of @xmath163 ( maximum @xmath28 of @xmath164 ) .
in order to separate signal and normalization events from background , we use the variable @xmath165 , the summed energy of ecl clusters not used in the reconstruction of the @xmath80 and @xmath70 candidates .
this is a useful variable for the signal extraction since the @xmath165 shape is less affected by changes in kinematics due to np .
the variable @xmath166 is additionally used for normalization events , and is defined as @xmath167 where @xmath168 and @xmath169 are the energy and the three - momentum , respectively , of the charged lepton ; the other variables in this formula are defined in eq .
[ eq - q2 ] . due to its narrow concentration near @xmath170 , this
variable is ideal for measuring normalization events . in the fit
, we use the distributions obtained from mc for the pdfs .
the @xmath165 shape for the signal component is validated using the normalization sample , which is more than 20 times larger than the signal sample . in this comparison
, we find good agreement between the data and the mc distributions . in the @xmath166 comparison for the normalization sample ,
the @xmath166 resolution in the data sample is slightly worse than in the mc sample .
we therefore broaden the @xmath166 peak width of the pdfs to match that of the data sample .
the most significant background contribution is from events with incorrectly - reconstructed @xmath27 ( denoted `` fake @xmath27 events '' ) .
since the combinatorial fake @xmath27 background processes are difficult to be modeled precisely in the mc , we compare the pdf shapes of these events in @xmath132 sideband regions .
the sideband regions 50500 mev@xmath76 , 135190 mev@xmath76 , 135190 mev@xmath76 and 140500 mev@xmath76 are chosen for @xmath115 , @xmath131 , @xmath117 and @xmath124 , respectively , while excluding about @xmath133 around the nominal @xmath132 .
these sideband regions contain 550 times more events than the signal region .
while we find good agreement of the @xmath165 shapes between the data and the mc for the signal sample , we observe a slight discrepancy in the @xmath166 distributions of the @xmath115 and @xmath131 modes for the normalization sample .
the @xmath166 discrepancy is therefore corrected based on this comparison . in both samples , since up to 20% of the yield discrepancies are observed , the fake @xmath27 yields are scaled by the yield ratios of the data to the mc in the @xmath132 sideband regions .
semileptonic decays to excited charm modes , @xmath67 and @xmath68 , comprise an important background category as they have a similar decay topology to the signal events .
in addition , background events from various types of hadronic @xmath11 decays wherein some particles are not reconstructed are significant in this analysis since there are only hadrons and two neutrinos in the final state of the signal mode .
because there are many unmeasured exclusive decay modes of @xmath67 , @xmath68 and hadronic @xmath11 decays , we determine their yields in the fit . with one exception
, we sum all the exclusive decays of these background categories into common yield parameters .
the exception is the decay to two @xmath93 mesons , such as @xmath171 and @xmath172 , since these are experimentally well measured : we fix their yields based on the world - average branching fractions @xcite .
in addition to the yield determination , the pdf shape of these background must be taken into account , as a change in the @xmath11 decay composition may modify the @xmath165 shape and thereby introduce biases in the measurement of @xmath45 and @xmath28 . if a background @xmath11 decay contains a @xmath50 in the final state , it may peak in the @xmath165 signal region .
we correct the branching fractions of the @xmath173 and @xmath174 modes in the mc using the measured values @xcite .
we do not apply branching fraction corrections for the other decays with @xmath50 since they are relatively minor .
however , we change the relative yield from 0% to 200% to estimate systematic uncertainties , as discussed in sec .
[ sec - syst ] .
.list of the calibration factors for each calibration sample , which are used to correct the amount of each hadronic @xmath11 background in the mc .
these calibration factors are obtained from the yield comparison between the data and the mc with the @xmath175 or @xmath176 distributions .
the errors on the calibration factors arise from statistics of the calibration samples . [
cols="<,^,^",options="header " , ] [ tab - calib ] other types of hadronic @xmath11 decay background often contain neutral particles such as @xmath92 or @xmath177 or pairs of charged particles .
we calibrate the amount of hadronic @xmath11 decays in the mc based on control data samples by reconstructing seven final states with the signal - side particles : @xmath178 , @xmath179 , @xmath180 , @xmath181 , @xmath182 , @xmath183 and @xmath184 .
candidate @xmath177 mesons are reconstructed using pairs of photons with an invariant mass ranging from 500 to 600 mev@xmath76 .
we then take the yield ratios between the data and the mc for @xmath185 and @xmath186 , which is the same requirement as in the signal sample , with the signal - side energy difference @xmath175 or the beam - energy - constrained mass @xmath176 of the @xmath80 candidate .
these ratios are used as yield calibration factors .
if there is no observed event in the calibration sample , we assign a 68% confidence level upper limit on the yield .
the obtained factors are summarized in table [ tab - calib ] .
additionally , we correct the branching fractions of the decays @xmath187 , @xmath188 and @xmath87 based on refs .
@xcite .
about 80% of the hadronic @xmath11 background is covered by the calibrations discussed above .
we discuss the systematic uncertainties on our observables due to the uncertainties of the calibration factors in sec .
[ sec - syst ] .
we perform an extended binned maximum likelihood fit to the @xmath165 and the @xmath166 distributions for the signal- and the normalization - candidate samples , respectively . in order to extract @xmath28
, we divide the signal sample into two @xmath145 regions : @xmath189 ( forward ) and @xmath190 ( backward ) . according to eq .
[ eq - coshel ] , the asymmetry of the number of signal events between the forward and the backward regions is proportional to @xmath28 . in the fit ,
we divide the @xmath35 component into three groups .
signal : : + correctly - reconstructed signal events , which originate from @xmath191 events reconstructed correctly as the @xmath191 sample , are categorized in this component , and are used for the determination of @xmath45 and @xmath28 . @xmath192
cross feed : : + cross - feed events where the decay @xmath5 is reconstructed in the @xmath4 mode due to the misreconstruction of one @xmath92 , or events where the decay @xmath4 is reconstructed in the @xmath5 mode by adding a random @xmath92 , comprise this component .
as these events originate from @xmath35 , they can be used in the @xmath45 determination .
they also have some sensitivity to @xmath28 ; however , @xmath145 is distorted .
the measured @xmath28 from the distorted @xmath145 distribution is mapped to the correct value of @xmath28 using mc information .
other @xmath0 cross feed : : + events from other @xmath0 decays also can contribute to the signal sample .
they originate mainly from @xmath193 with two missing @xmath92 mesons and @xmath194 with a low - momentum muon .
the fraction of these two cross - feed components are , respectively , 11% and 73% in the @xmath4 mode and 69% and 14% in the @xmath5 mode .
these modes are less sensitive to @xmath28 since the heavy @xmath195 mass makes the @xmath43 in eq .
( [ eq - alpha ] ) almost equal to 0 , while events with two neutrinos in the @xmath194 mode wash out the @xmath28 information .
the relative contribution from the three @xmath35 components are fixed using the mc simulation sample , which contains 40 times more events than the full belle data sample .
the parameterization of @xmath45 is @xmath196 where @xmath197 denotes the branching fraction of @xmath4 or @xmath5 , and @xmath198 and @xmath199 are the efficiencies for the signal and the normalization mode , respectively .
the observed yields are expressed by @xmath200 and @xmath201 for the signal in the forward ( backward ) region and the normalization , respectively .
the polarization is represented by @xmath202 due to detector effects , the extracted value deviates from the true @xmath28 .
this detector bias is taken into account with a linear function that relates the true @xmath28 to the extracted @xmath28 .
the linear function , which is called the @xmath28 correction function in this paper , is determined using several mc sets of type - ii 2hdm @xcite . in this model
, @xmath28 varies between @xmath203 and @xmath204 as a function of the theoretical parameter @xmath205 , where @xmath206 denotes the ratio of the vacuum expectation values of the two higgs doublets coupling to up - type and down - type quarks and @xmath207 is the mass of the charged higgs boson .
we then extrapolate the obtained @xmath28 correction function to @xmath208 . for the background , we have four components .
@xmath159 : : + the decay @xmath159 contaminates the signal sample due to the misassignment of the lepton as a pion .
we fix the @xmath159 yield in the signal sample from the fit to the @xmath166 distribution in the normalization sample . @xmath67 and hadronic
@xmath11 decays : : + as discussed in the previous section , we float the sum of the yields of @xmath67 and hadronic @xmath11 decays except for the well - determined two - body @xmath93 final states in the fit .
the yield parameters are independent for each sample : @xmath209 .
continuum : : + continuum events from @xmath72 process provide a minor contribution . as the size of the contribution
is only @xmath210 , we fix the yield using the mc expectation .
fake @xmath27 : : + all events containing fake @xmath27 candidates are categorized in this component .
the yield is fixed from a comparison of the data and the mc in the @xmath132 sideband regions .
@l@|@c@@c@ source & @xmath45 & @xmath28 + hadronic @xmath11 composition & @xmath211 & @xmath212 + mc statistics for each pdf shape & @xmath213 & @xmath214 + fake @xmath27 pdf shape & 3.0% & 0.010 + fake @xmath27 yield & 1.7% & 0.016 + @xmath67 & 2.1% & 0.051 + @xmath68 & 1.1% & 0.003 + @xmath159 & 2.4% & 0.008 + @xmath0 daughter and @xmath3 efficiency & 2.1% & 0.018 + mc statistics for efficiency calculation & 1.0% & 0.018 + evtgen decay model & @xmath215 & @xmath216 + fit bias & 0.3% & 0.008 + @xmath217 and @xmath218 & 0.3% & 0.002 + @xmath28 correction function & 0.1% & 0.018 + + tagging efficiency correction & 1.4% & 0.014 + @xmath27 reconstruction & 1.3% & 0.007 + @xmath93 sub - decay branching fractions & 0.7% & 0.005 + number of @xmath7 & 0.4% & 0.005 + total systematic uncertainty & @xmath219 & @xmath220 + [ tab - syst ] we estimate systematic uncertainties by varying each possible uncertainty source such as the pdf shape and the signal reconstruction efficiency with the assumption of a gaussian error , unless otherwise stated .
in several trials , we change each parameter at random , repeat the fit , and then take the mean shifts of @xmath45 and @xmath28 from all such trials as the corresponding systematic uncertainty that is enumerated in table [ tab - syst ] .
the most significant systematic uncertainty , arising from the hadronic @xmath11 decay composition , is estimated as follows .
uncertainties of each @xmath11 decay fraction in the hadronic @xmath11 decay background are taken from the experimentally - measured branching fractions or estimated from the uncertainties in the calibration factors discussed in sec .
[ sec - background ] . for components with no experimentally - measured branching fractions and not covered by the control samples , we vary their contribution continuously from @xmath221 to @xmath222 of the mc expectation and take the maximum shifts of @xmath45 and @xmath28 as the systematic uncertainties .
the limited mc sample size used in the construction of the pdfs is also a major systematic uncertainty source .
we estimate this by regenerating the pdfs for each component and each sample using a toy mc approach based on the original pdf shapes . the same number of events are generated to account for the statistical fluctuation .
the pdf shape of the fake @xmath27 component has been validated by comparing the data and the mc in the @xmath132 sideband region . however , a slight fluctuation from the decay @xmath33 , which is a peaking background in the fake @xmath27 component , may have a significant impact on the signal yield as this component has almost the same shape as the signal mode . to be conservative
, we incorporate an additional uncertainty by varying the contribution of the @xmath33 component within the current uncertainties of the experimental averages @xcite : @xmath223 for @xmath224 and @xmath225 for @xmath226 .
we take the theoretical uncertainty on the @xmath0 polarization of the @xmath33 mode into account , which is found to be 0.002 for @xmath28 and negligibly small . in addition
, we estimate a systematic uncertainty due to the small @xmath166 shape correction for the fake @xmath27 component of the normalization sample : this is 0.15% ( 0.001 ) for @xmath45 ( @xmath28 ) .
the fake @xmath27 yield , fixed using the @xmath132 sideband , has an uncertainty that arises from the statistical uncertainties of the yield scale factors .
the uncertainty of the decays @xmath67 are twofold : the indeterminate composition of each @xmath58 state and the uncertainty in the form - factor parameters used for the mc sample production .
the composition uncertainty is estimated based on uncertainties of the branching fractions : @xmath227 for @xmath228 , @xmath229 for @xmath230 , @xmath231 for @xmath232 and @xmath233 for @xmath234 . if the experimentally - measured branching fractions are not applicable , we vary the branching fractions continuously from @xmath221 to @xmath222 in the mc expectation .
we estimate an uncertainty arising from the llsw model parameters by changing the correction factors within the parameter uncertainties and obtain 0.5% and 0.016 for @xmath45 and @xmath28 , respectively .
the uncertainty due to limited knowledge of the decays @xmath68 is estimated separately by varying the branching fractions .
since there are no experimental measurements of these decays , the branching fractions are varied continuously from 0% to 200% in the mc expectation . as @xmath235 is constrained by the branching fraction of the inclusive semitauonic decay @xmath236
@xcite , which is smaller than the sum of the branching fractions @xmath237 and @xmath238 , we conclude that our assumption is sufficiently conservative . the uncertainties due to the hqet form - factor parameters in the normalization mode @xmath159 are estimated using the uncertainties in the world - average values @xcite .
in addition , the uncertainty arising from the small @xmath166 shape correction for the normalization sample is estimated as an uncertainty related to @xmath159 : 0.4% ( 0.008 ) for @xmath45 ( @xmath28 ) . the uncertainties on the reconstruction efficiencies of the @xmath0-daughter particles in the signal sample and the charged leptons in the normalization sample are also considered . here , the uncertainties on the particle identification efficiencies for @xmath85 and @xmath239 and the reconstruction efficiency for @xmath92 are measured with control samples : the @xmath240 sample for @xmath85 , the @xmath241 sample for @xmath92 and the @xmath242 for charged leptons .
the sample @xmath243 from @xmath11 decays is also used in order to account for the difference in multiplicity between two - photon events and @xmath11 decay events .
estimated uncertainties are as follows : 0.5% and 0.003 for @xmath85 , 0.6% and 0.004 for @xmath92 , 1.6% and 0.004 for charged leptons ( the first and the second values corresponding to uncertainties of @xmath45 and @xmath28 , respectively ) .
reconstruction efficiencies of the three @xmath35 components are estimated using mc .
the efficiencies have uncertainties arising from the statistics of the signal mc and are varied independently for each component . here , the uncertainty of the fraction of the three @xmath35 components are also taken into account .
the @xmath35 decay generator of evtgen , based on the hqet form factors and implemented by belle , neglects the interference between the amplitudes of left- and right - handed @xmath0 leptons .
this mis - models the decay topology and so affects the signal reconstruction efficiency .
we compare the extracted @xmath45 and @xmath28 between this model and an alternate model based on the isgw form factors , and take differences of @xmath45 and @xmath28 as systematic uncertainties .
other small uncertainties arise due to fit bias arising from the @xmath165 bin width selection , which is estimated by comparing the extracted values of @xmath45 and @xmath28 in the nominal fit ( with a bin width of 0.05 gev ) with the values obtained in a wide - bin fit ( bin width of 0.1 gev ) ; the branching fractions of the @xmath0 lepton decays ; and errors on the parameters of the @xmath28 correction function .
axis is illustrated .
the red - hatched `` @xmath0 cross feed '' combines the @xmath192 cross - feed and the other @xmath0 cross - feed components.,title="fig:",width=566 ] axis is illustrated .
the red - hatched `` @xmath0 cross feed '' combines the @xmath192 cross - feed and the other @xmath0 cross - feed components.,title="fig:",width=566 ] axis is illustrated .
the red - hatched `` @xmath0 cross feed '' combines the @xmath192 cross - feed and the other @xmath0 cross - feed components.,title="fig:",width=566 ] in addition , common uncertainty sources between the signal sample and the normalization sample are also estimated in this analysis , although they largely cancel at first order in the branching fraction ratio .
this is due to the fact that the background yields are partially fixed from the mc expectation . here ,
uncertainties on the number of @xmath7 ( 1.9% ) events , tagging efficiencies ( 4.7% ) , branching fractions of the @xmath93 decays ( 3.4% ) and @xmath27 reconstruction efficiency ( 4.8% ) are evaluated for their impact on the final measurements . for the @xmath27 reconstruction efficiency , the uncertainty originates from reconstruction efficiencies of @xmath82 , @xmath92 , @xmath84 and @xmath85 , and is therefore correlated with the efficiency uncertainty of the @xmath0-daughter particles containing @xmath85 and @xmath92 .
this correlation is taken into account in the total systematic uncertainties shown in table [ tab - syst ] .
figure [ fig - fitresult ] shows the fits to the signal and the normalization samples .
( the figures in the forward and backward regions are shown in the appendix [ app - fwdbwd ] . ) the @xmath145 distribution is shown in fig .
[ fig - fitcoshel ] . the observed
signal and normalization yields are summarized in table [ tab - obtained - yield ] .
the @xmath244-values are found to be 15% for the normalization fit and 29% for the signal fit . from the fit
, we obtain @xmath245 the signal significance is 9.7@xmath25 ( statistical error only ) or 7.1@xmath25 ( including the systematic uncertainty ) . the significance is taken from @xmath246 , where @xmath247 and @xmath248 are the likelihood with the nominal fit and the null hypothesis , respectively .
@l@|@c@@c@ sample & + + & ( signal ) & ( @xmath0 cross feed ) + @xmath249 & @xmath250 & @xmath251 + @xmath252 & @xmath253 & @xmath254 + @xmath255 & @xmath256 & @xmath257 + @xmath258 & @xmath259 & @xmath260 + + @xmath261 & + @xmath262 & + [ tab - obtained - yield ] figure [ fig - significance ] shows a comparison of our result with the theoretical prediction based on the sm @xcite in the @xmath263 plane .
the consistency of our result with the sm is @xmath264 . and @xmath265 , respectively ) with the sm prediction ( red triangle ) .
the gray region shows the average of the experimental results as of march 2016 @xcite.,width=415 ]
we report the measurement of @xmath45 with hadronic @xmath0 decay modes @xmath4 and @xmath5 and the first measurement of @xmath28 in the decay @xmath35 , using @xmath6 @xmath266 data accumulated with the belle detector .
our preliminary results are @xmath267 which is consistent with the sm prediction within 0.6@xmath25 .
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 , the national institute of informatics , and the pnnl / emsl computing group for valuable computing and sinet4 network support .
we acknowledge support from the ministry of education , culture , sports , science , and technology ( mext ) of japan , the japan society for the promotion of science ( jsps ) , and the tau - lepton physics research center of nagoya university ; the australian research council ; austrian science fund under grant no .
p 22742-n16 and p 26794-n20 ; the national natural science foundation of china under contracts no . 10575109 , no . 10775142 , no .
10875115 , no .
11175187 , no .
11475187 and no . 11575017 ; the chinese academy of science center for excellence in particle physics ; the ministry of education , youth and sports of the czech republic under contract no .
lg14034 ; the carl zeiss foundation , the deutsche forschungsgemeinschaft , the excellence cluster universe , and the volkswagenstiftung ; the department of science and technology of india ; the istituto nazionale di fisica nucleare of italy ; the wcu program of the ministry of education , national research foundation ( nrf ) of korea grants no .
2011 - 0029457 , no .
2012 - 0008143 , no . 2012r1a1a2008330 , no .
2013r1a1a3007772 , no .
2014r1a2a2a01005286 , no . 2014r1a2a2a01002734 , no .
2015r1a2a2a01003280 , no . 2015h1a2a1033649 ; the basic research lab program under nrf grant no .
krf-2011 - 0020333 , center for korean j - parc users , no .
nrf-2013k1a3a7a06056592 ; the brain korea 21-plus program and radiation science research institute ; the polish ministry of science and higher education and the national science center ; the ministry of education and science of the russian federation and the russian foundation for basic research ; the slovenian research agency ; ikerbasque , basque foundation for science and the euskal herriko unibertsitatea ( upv / ehu ) under program ufi 11/55 ( spain ) ; the swiss national science foundation ; the ministry of education and the ministry of science and technology of taiwan ; and the u.s .
department of energy and the national science foundation .
this work is supported by a grant - in - aid from mext for science research in a priority area ( `` new development of flavor physics '' ) , from jsps for creative scientific research ( `` evolution of tau - lepton physics '' ) , a grant - in - aid for scientific research ( s ) `` probing new physics with tau - lepton '' ( no .
26220706 ) and was partly supported by a grant - in - aid for jsps fellows ( no .
25.3096 ) .
we thank y. sakaki , r. watanabe and m. tanaka for their invaluable suggestions and helps .
, respectively.,title="fig:",width=566 ] , respectively.,title="fig:",width=566 ] | we report the first measurement of the @xmath0 lepton polarization in the decay @xmath1 as well as a new measurement of the ratio of the branching fractions @xmath2 , where @xmath3 denotes an electron or a muon , with the decays @xmath4 and @xmath5 .
we use the full data sample of @xmath6 @xmath7 pairs accumulated with the belle detector at the kekb electron - positron collider .
our preliminary results , @xmath8 and @xmath9 , are consistent with the theoretical predictions of the standard model within @xmath10 standard deviation . | 1608.06391 |
density functional theory ( dft ) has been the subject of remarkable developments since its original formulation by hohenberg and kohn ( hk ) @xcite . after formal improvements , extensions , and an uncountable number of applications to a wide variety of physical problems ,
this theoretical approach has become the most efficient , albeit not infallible , method of determining the electronic properties of matter from first principles @xcite .
the most important innovation of dft , which is actually at the origin of its breakthrough , is to replace the wave function by the electronic density @xmath18 as the fundamental variable of the many - body problem . in practice , density functional ( df ) calculations are largely based on the kohn - sham ( ks ) scheme that reduces the many - body @xmath19-particle problem to the solution of a set of self - consistent single - particle equations @xcite . although this transformation is formally exact , the implementations always require approximations , since the ks equations involve functional derivatives of the unknown interaction energy @xmath20 $ ] , usually expressed in terms of the exchange and correlation ( xc ) energy @xmath21 $ ] .
therefore , understanding the functional dependence of @xmath21 $ ] and improving its approximations are central to the development of df methods .
the currently most widespread _ anstze _ for @xmath21 $ ] the local density approximation ( lda ) @xcite with spin polarized @xcite and gradient corrected extensions @xcite were originally derived from exact results for the homogeneous electron gas .
it is one of the purposes of this paper to investigate the properties of the interaction - energy functional from an intrinsically inhomogeneous point of view , namely , by considering exactly solvable many - body lattice models . despite the remarkable success of the local spin density approximation ,
present dft fails systematically in accounting for phenomena where strong electron correlations play a central role , for example , in heavy - fermion materials or high-@xmath22 superconductors .
these systems are usually described by simplifying the low - energy electron dynamics using parameterized lattice models such as pariser - parr - pople,@xcite hubbard,@xcite or anderson@xcite models and related hamiltonians @xcite .
being in principle an exact theory , the limitations of the df approach have to be ascribed to the approximations used for exchange and correlation and not to the underlying hks formalism .
it would be therefore very interesting to extend the range of applicability of dft to strongly correlated systems and to characterize the properties @xmath23 in the limit of strong correlations .
studies of the xc functional on simple models should provide useful insights for future extensions to realistic hamiltonians . moreover , taking into account the demonstrated power of the df approach in _ ab initio _ calculations , one may also expect that a dft with an appropriate @xmath23 could become an efficient tool for studying many - body models , a subject of theoretical interest on its own .
several properties of dft on lattice models have been already studied in previous works@xcite .
gunnarsson and schnhammer were , to our knowledge , the first to propose a df approach on a semiconductor model in order to study the band - gap problem@xcite . in this case
the local site occupancies were treated as the basic variables .
some years later schindlmayr and godby @xcite provided a different formulation of dft on a lattice by considering as basic variables both diagonal elements @xmath24 and off - diagonal elements @xmath0 of the single - particle density matrix ( see also @xcite ) .
then derived a more general framework that unifies the two previous approaches @xcite . using levy s
constrained search method @xcite they showed that different basic variables and different @xmath25 functionals can be considered depending on the type of model or perturbation under study .
site occupations alone may be used as basic variables , if only the orbital energies are varied ( i.e. , if all hopping integrals @xmath26 are kept constant for @xmath27 ) . however
, off - diagonal elements of the single - particle density matrix must be included explicitly if the functional @xmath25 is intended to be applied to more general situations involving different values of @xmath26 , for example , the hubbard model on various lattice structures or for different interaction regimes , i.e. , different @xmath28 . in this paper
we investigate the properties of levy s interaction - energy functional @xmath25 as a function of @xmath0 by solving the constrained search minimization problem exactly . in sec .
[ sec : teo ] the basic formalism of density - matrix functional theory ( dmft ) on lattice models is recalled and the equations for determining @xmath1 $ ] are derived .
[ sec : res ] presents and discusses exact results for the correlation energy @xmath29 of the hubbard model , which is given by the difference between @xmath25 and the hartree - fock energy @xmath30 .
these are obtained , either numerically for finite clusters with different lattice structures , or from the bethe - ansatz solution for the one - dimensional chain . finally , sec .
[ sec : conc ] summarizes our conclusions and points out some relevant extensions .
in sec . [ sec : teolat ] the main results of levy s formulation of dmft are presented in a form that is appropriate for the study of model hamiltonians such as the hubbard model . here , the hopping integrals @xmath26 between sites ( or orbitals ) @xmath6 and @xmath7 play the role given in conventional dft to the external potential @xmath31 .
consequently , the single - particle density matrix @xmath0 replaces the density @xmath18 as basic variable @xcite . in sec .
[ sec : teoex ] , we derive equations that allow to determine levy s interaction - energy functional @xmath1 $ ] in terms of the ground - state energy of a many - body hamiltonian with effective hopping integrals @xmath32 that depend implicitly on @xmath0 .
we consider the many - body hamiltonian @xmath33 where @xmath34 ( @xmath35 ) is the usual creation ( annihilation ) operator for an electron with spin @xmath36 at site ( or orbital ) @xmath6 .
@xmath37 can be regarded as the second quantization of schrdinger s equation on a basis @xcite . however , in the present paper , the hopping integrals @xmath26 and the interaction matrix elements @xmath38 are taken as parameters to be varied independently .
the matrix @xmath26 defines the lattice ( e.g. , one dimensional chains , square or triangular two - dimensional lattices ) and the range of single - particle interactions ( e.g. , up to first or second neighbors ) . from the _ ab initio _ perspective @xmath26
is given by the external potential and by the choice of the basis @xcite .
@xmath38 defines the type of many - body interactions which may be repulsive ( coulomb like ) or attractive ( in order to simulate electronic pairing ) and which are usually approximated as short ranged ( e.g. , intra - atomic ) .
( [ eq : hamgen ] ) is mainly used in this section to derive general results which can then be applied to various specific models by simplifying the interactions . a particularly relevant example , to be considered in some detail in sec . [ sec : res ] , is the single - band hubbard model with nearest neighbor ( nn ) hoppings @xcite , which can be obtained from eq .
( [ eq : hamgen ] ) by setting @xmath39 for @xmath6 and @xmath7 nn s , @xmath40 otherwise , and @xmath41 @xcite .
in order to apply dmft to model hamiltonians of the form ( [ eq : hamgen ] ) we follow levy s constrained search procedure @xcite as proposed by schindlmayr and godby @xcite .
the ground - state energy is determined by minimizing the functional @xmath42 = e_k[\gamma_{ij } ] + w [ \gamma_{ij}]\ ] ] with respect to the single - particle density matrix @xmath0 .
@xmath43 $ ] is physically defined for all density matrices that can be written as @xmath44 for all @xmath6 and @xmath7 , where @xmath45 is an @xmath19-particle state . in other words , @xmath0 must derive from a physical state .
it is then said to be pure - state @xmath19-representable @xcite . the first term in eq .
( [ eq : efun ] ) is given by @xmath46 it includes all single - particle contributions and is usually regarded as the kinetic energy associated with the electronic motion in the lattice .
notice that eq .
( [ eq : ek ] ) yields the exact kinetic energy for a given @xmath0 .
there are no corrections on @xmath47 to be included in other parts of the functional as in the ks approach . the second term in eq .
( [ eq : efun ] ) is the interaction - energy functional given by @xcite @xmath48 = min \left[\frac{1}{2 } \sum _ { nmkl \atop { \sigma \sigma ' } } v_{nmkl } \ ; \langle \psi [ \gamma_{ij } ] | \ ; \hat c_{n \sigma } ^{\dagger } \hat c_{k \sigma ' } ^{\dagger } \hat c_{l \sigma ' } \hat c_{m \sigma } \ ; |\psi[\gamma_{ij}]\rangle \right ] \ ; .\ ] ] the minimization in eq .
( [ eq : w ] ) implies a search over all @xmath19-particles states @xmath49 \rangle$ ] that satisfy @xmath50 | \ ;
\sum_\sigma \hat c_{i \sigma } ^{\dagger}\hat c_{j \sigma } \ ; |\psi [ \gamma_{ij } ] \rangle = \gamma_{ij}$ ] for all @xmath6 and @xmath7 .
therefore , @xmath1 $ ] represents the minimum value of the interaction energy compatible with a given density matrix @xmath0 .
@xmath25 is usually expressed in terms of the hartree - fock energy @xmath51=\frac{1}{2}\sum_{ijkl
\atop{\sigma \sigma ' } } v_{ijkl } \left(\gamma_{ij \sigma } \gamma_{kl \sigma ' } - \delta_{\sigma\sigma ' } \gamma_{il \sigma } \gamma_{kj \sigma } \right)\ ] ] and the correlation energy @xmath3 $ ] as @xmath52 = e_{\rm hf}[\gamma_{ij } ] + e_{\rm c}[\gamma_{ij } ] \;.\ ] ] @xmath25 and @xmath29 are universal functionals of @xmath0 in the sense that they are independent of @xmath26 , i.e. , of the system under study .
they depend on the considered interactions or model , as defined by @xmath38 , on the number of electrons @xmath9 , and on the structure of the many - body hilbert space , as given by @xmath9 and the number of orbitals or sites @xmath8 .
notice that @xmath29 in eq .
( [ eq : xc ] ) does not include any exchange contributions .
given @xmath0 ( @xmath53 in nonmagnetic cases ) there is no need to approximate the exchange term , which is taken into account exactly by @xmath30 [ eq . ( [ eq : hf ] ) ] .
nevertheless , if useful in practice , it is of course possible to split @xmath25 in the hartree energy @xmath54 and the exchange and correlation energy @xmath23 is a similar way as in the ks approach . the variational principle results from the following two relations @xcite : @xmath55\ ] ] for all pure - state @xmath19-representable @xmath0 @xcite , and @xmath56 \;,\ ] ] where @xmath57 refers to the ground - state energy and @xmath58 to the ground - state single - particle density matrix
. as already pointed out in previous works @xcite , @xmath25 and @xmath29 depend in general on both diagonal elements @xmath24 and off - diagonal elements @xmath0 of the density - matrix , since the hopping integrals @xmath26 are non local in the sites
the situation is similar to the df approach proposed by gilbert for the study of non - local potentials @xmath59 as those appearing in the theory of pseudo - potentials @xcite . a formulation of dft on a lattice only in terms of @xmath24 would be possible if one would restrict oneself to a family of models with constant @xmath26 for @xmath27
however , in this case the functional @xmath60 $ ] would depend on the actual value of @xmath26 for @xmath27 @xcite .
the functional @xmath1 $ ] , valid for all lattice structures and for all types of hybridizations , can be simplified at the expense of universality if the hopping integrals are short ranged .
for example , if only nn hoppings are considered , the kinetic energy @xmath47 is independent of the density - matrix elements between sites that are not nn s . therefore , the constrained search in eq .
( [ eq : w ] ) may restricted to the @xmath49 \rangle$ ] that satisfy @xmath50 |\ ; \sum_\sigma \hat c_{i \sigma } ^{\dagger}\hat c_{j \sigma } \;|\psi [ \gamma_{ij } ] \rangle = \gamma_{ij}$ ] only for @xmath61 and for nn @xmath62 . in this way the number of variables in @xmath1 $ ] is reduced significantly rendering the interpretation of the functional dependence far simpler . while this is a great practical advantage , it also implies that @xmath25 and @xmath29 lose their universal character since the dependence on the nn @xmath0 is now different for different lattices . in sec .
[ sec : res ] results for one- , two- , and three - dimensional lattices with nn hoppings are compared in order to quantify this effect . for the applications in sec .
[ sec : res ] we shall consider the single - band hubbard model with nn hoppings , which in the usual notation is given by@xcite @xmath63 in this case the interaction energy functional reads @xmath64 = min \left [ u \sum_l \langle \psi [ \gamma_{ij } ] |\ ; \hat n_{l\uparrow } \hat n_{l\downarrow } \;|\psi [ \gamma_{ij } ] \rangle \right ] \ ; , \ ] ] where the minimization is performed with respect to all @xmath19-particle @xmath65\rangle$ ] satisfying @xmath50 | \sum_\sigma \hat c_{i \sigma } ^{\dagger}\hat c_{j \sigma } | \psi [ \gamma_{ij } ] \rangle = \gamma_{ij}$ ] for @xmath6 and @xmath7 nn s .
if the interactions are repulsive ( @xmath16 ) @xmath1 $ ] represents the minimum average number of double occupations which can be obtained for a given degree of electron delocalization , i.e. , for a given value of @xmath0 . for attractive interactions ( @xmath17 )
double occupations are favored and @xmath1 $ ] corresponds to the maximum of @xmath66 for a given @xmath0 . in order to determine @xmath3 $ ] and @xmath1 $ ] we look for the extremes of @xmath67 \ ; + \;\varepsilon \ ; \big(1 - \langle \psi |\psi \rangle \big ) \ ; \nonumber \\ & + & \sum_{i , j } \lambda_{ij } \ ; \big ( \langle\psi| \sum_\sigma \hat c^\dagger_{i\sigma } \hat c_{j \sigma } | \psi\rangle - \gamma_{ij } \big ) \end{aligned}\ ] ] with respect to @xmath68 .
lagrange multipliers @xmath69 and @xmath32 have been introduced to enforce the normalization of @xmath70 and the conditions on the representability of @xmath0 .
derivation with respect to @xmath71 , @xmath69 and @xmath32 yields the eigenvalue equations @xmath72 and the auxiliary conditions @xmath73 and @xmath74 .
the lagrange multipliers @xmath32 play the role of hopping integrals to be chosen in order that @xmath70 yields the given @xmath0 .
the pure - state representability of @xmath0 ensures that there is always a solution @xcite . in practice , however , one usually varies @xmath32 in order to scan the domain of representability of @xmath0 . for given @xmath32 , the eigenstate @xmath75 corresponding to the lowest eigenvalue of eq .
( [ eq : evgen ] ) yields the minimum @xmath1 $ ] for @xmath76 . any other @xmath70 satisfying @xmath74 would have higher @xmath69 and thus higher @xmath25 .
the subset of @xmath0 which are representable by a ground - state of eq .
( [ eq : evgen ] ) is the physically relevant one , since it necessarily includes the absolute minimum @xmath77 of @xmath43 $ ] .
nevertheless , it should be noted that pure - state representable @xmath0 may be considered that can only be represented by excited states or by linear combinations of eigenstates of eq .
( [ eq : evgen ] ) . in the later case ,
@xmath78 , and @xmath79 is an eigenstate of the interaction term with lowest eigenvalue .
examples shall be discussed in sec .
[ sec : res ] .
for the hubbard model eq .
( [ eq : evgen ] ) reduces to @xmath80 this eigenvalue problem can be solved numerically for clusters with different lattice structures and periodic boundary conditions . in this case
we expand @xmath49 \rangle$ ] in a complete set of basis states @xmath81 which have definite occupation numbers @xmath82 at all orbitals @xmath83 ( @xmath84 with @xmath85 or @xmath86 ) .
the values of @xmath82 satisfy the usual conservation of the number of electrons @xmath87 and of the @xmath88 component of the total spin @xmath89 , where @xmath90 . for not too large clusters , the lowest energy @xmath91 \rangle$ ] the ground state of eq .
( [ eq : evhub]) can be determined by sparse - matrix diagonalization procedures , for example , by using lanczos iterative method @xcite .
@xmath92 \rangle$ ] is calculated in the subspace of minimal @xmath93 since this ensures that there are no _ a priori _ restrictions on the total spin @xmath94 .
in addition , spin - projector operators may be used to study the dependence of @xmath4 on @xmath94 . for a one - dimensional ( 1d ) chain with nn hoppings @xmath95
, translational symmetry implies equal density - matrix elements @xmath0 between nn s .
therefore , one may set @xmath96 for all nn @xmath62 , and then eq . ( [ eq : evhub ] ) has the same form as the 1d hubbard model for which lieb and wu s exact solution is available @xcite . in this case
the lowest eigenvalue @xmath69 is determined following the work by shiba @xcite .
the coupled bethe - ansatz equations are solved as a function of @xmath97 , band - filling @xmath98 , and for positive and negative @xmath99 , by means of a simple iterative procedure .
in this section we present and discuss exact results for the correlation energy functional @xmath3 $ ] of the single - band hubbard hamiltonian with nearest neighbor hoppings @xcite .
given the lattice structure , @xmath8 and @xmath9 , the model is characterized by the dimensionless parameter @xmath28 which measures the competition between kinetic and interaction energies [ see eq .
( [ eq : hamhub ] ) ] .
@xmath16 corresponds to the usual intra - atomic repulsive coulomb interaction , while the attractive case ( @xmath17 ) simulates intra - atomic pairing of electrons . in fig .
[ fig : xcanft ] the correlation energy @xmath29 of the one - dimensional ( 1d ) hubbard model is shown for half - band filling ( @xmath100 ) as a function of the density - matrix element or bond order @xmath101 between nn s .
@xmath102 for all nn s @xmath6 and @xmath7 .
results are given for rings of finite length @xmath8 as well as for the infinite chain .
several general qualitative features may be identified .
first of all we observe that on bipartite lattices @xcite @xmath103 , since the sign of the nn bond order can be changed without affecting the interaction energy @xmath104 by changing the phase of the local orbitals at one of the sublattices ( @xmath105 for @xmath106 and @xmath107 unchanged for @xmath108 , where @xmath109 and @xmath110 refer to the sublattices ) .
let us recall that the domain of definition of @xmath4 is limited by the pure - state representability of @xmath0 .
the upper bound @xmath111 and the lower bound @xmath112 for @xmath101 ( @xmath113 on bipartite lattices ) are the extreme values of the bond order between nn s on a given lattice and for given @xmath8 and @xmath9 ( @xmath102 for all nn @xmath62 ) .
they represent the maximum degree of electron delocalization . @xmath111 and
@xmath112 correspond to the extremes of the kinetic energy @xmath47 [ @xmath114 , where @xmath88 is the coordination number ] and thus to the ground state of the hubbard model for @xmath115 [ @xmath111 for @xmath116 and @xmath112 for @xmath117 , see eq .
( [ eq : hamhub ] ) ] . for @xmath118
the underlying electronic state @xmath79 is usually a single slater determinant and therefore @xmath119 . in other words ,
the correlation energy vanishes as expected in the fully delocalized limit @xcite .
as @xmath120 decreases @xmath29 decreases ( @xmath121 ) since correlations can reduce the coulomb energy more and more efficiently as the electrons localize .
@xmath29 is minimum in the strongly correlated limit @xmath122 . for half - band filling this corresponds to a fully localized electronic state ( @xmath123 ) . here
, @xmath29 cancels out the hartree - fock energy @xmath30 and the coulomb energy @xmath25 vanishes ( @xmath124 ) @xcite .
the ground - state values of @xmath125 and @xmath126 for a given @xmath28 result from the competition between lowering @xmath29 by decreasing @xmath101 and lowering @xmath47 by increasing it ( @xmath116 ) .
the divergence of @xmath127 for @xmath128 is a necessary condition in order that @xmath129 for arbitrary small @xmath16 . on the other side , for small @xmath101 , we observe that @xmath130 .
this implies that for @xmath131 , @xmath132 and @xmath133 , a well known result in the heisenberg limit of the hubbard model ( @xmath134 ) @xcite . a more quantitative analysis of @xmath4 and in particular the comparison of results for different @xmath8 is complicated by the size dependence of @xmath135 and @xmath30 .
it is therefore useful to measure @xmath29 in units of the hartree - fock energy and to bring the domains of representability to a common range by considering @xmath136 as a function of @xmath137 . fig .
[ fig : nxcanft ] shows that @xmath15 has approximately the same behavior for all considered @xmath8 .
finite size effects are small except for the very small sizes .
the largest deviations from the common trend are found for @xmath138 .
here we observe a discontinuous drop of @xmath139 for @xmath140 ( @xmath141 ) which is due to the degeneracy of the single - particle spectrum .
in fact in this case two of the four electrons occupy a doubly degenerate state in the uncorrelated limit and the minimum interaction energy @xmath104 does not correspond to a single - slater - determinant state even for @xmath142 @xcite . as @xmath8 increases
@xmath15 approaches the infinite - length limit with alternations around the @xmath143 curve .
the strong similarity between @xmath15 for small @xmath8 and for @xmath143 is a remarkable result .
it suggests that good approximations for @xmath4 in extended systems could be derived from finite cluster calculations .
[ fig : xcanfl1 ] shows the band - filling dependence of @xmath4 in a 10-site 1d hubbard ring .
results are given for @xmath144 , since for @xmath145 , @xmath146 as a result of electron - hole symmetry @xcite .
although @xmath4 depends strongly on @xmath9 , several qualitative properties are shared by all band fillings : ( i ) as in the half - filled band case , the domain of representability of @xmath101 is bound by the bond orders in the uncorrelated limits . in fact , @xmath147 , where @xmath111 ( @xmath112 ) corresponds to the ground state of the @xmath115 tight - binding model for @xmath116 ( @xmath117 ) .
on bipartite lattices @xmath148 .
notice that @xmath135 increases monotonously with @xmath9 as the single - particle band is filled up .
this is an important contribution to the band - filling dependence of @xmath29 ( see fig .
[ fig : xcanfl1 ] ) .
( ii ) in the delocalized limit , @xmath119 for all the @xmath9 for which @xmath149 derives from a single slater determinant @xcite . moreover
, the divergence of @xmath127 for @xmath150 indicates that @xmath129 for arbitrary small @xmath16 , as expected from perturbation theory .
( iii ) starting from @xmath128 , @xmath4 decreases with decreasing @xmath101 reaching its lowest possible value @xmath124 for @xmath151 ( @xmath144 ) .
the same behavior is of course observed for @xmath152 .
in particular , @xmath153 also for @xmath154 . as shown in fig . [
fig : xcanfl1 ] , @xmath155 decreases rapidly with increasing @xmath9 , since @xmath30 increases quadratically with the electron density @xcite .
( iv ) on bipartite lattices @xmath156 , while on non - bipartite structures one generally has @xmath157 , since the single - particle spectrum is different for positive and negative energies .
the decrease of @xmath29 with decreasing @xmath120 shows that the reduction of the coulomb energy due to correlations is done at the expense of kinetic energy or electron delocalization , as already discussed for @xmath100 ( fig .
[ fig : xcanft ] ) .
( v ) @xmath158 for all @xmath159 ( @xmath160 for @xmath100 ) .
@xmath161 represents the largest nn bond order that can be constructed under the constraint of vanishing coulomb repulsion energy .
a lower bound for @xmath161 is given by the bond order @xmath162 in the fully - polarized ferromagnetic state ( @xmath163 ) .
this is obtained by occupying the lowest single - particle states with all electrons of the same spin ( @xmath144 ) .
therefore , @xmath162 increases with @xmath9 for @xmath164 and then decreases for @xmath165 reaching @xmath166 at half - band filling ( @xmath167 for @xmath168 ) . in this way the non - monotonous dependence of @xmath161 on @xmath9 can be explained ( see fig . [
fig : xcanfl1 ] ) .
( vi ) the correlation energy is constant and equal to @xmath169 for @xmath170 .
these values of @xmath101 can never correspond to the ground - state energy of the hubbard model , since in this range increasing @xmath101 always lowers the kinetic energy ( @xmath116 ) without increasing the coulomb repulsion ( @xmath171 ) . for @xmath172 , @xmath101
can not be represented by a ground state of eq .
( [ eq : evhub ] ) . in this range @xmath101 can be derived from a linear combination of states having minimal coulomb repulsion @xcite . in order to compare the functional dependences of the correlation energy for different band fillings ,
it is useful to scale @xmath29 in units of the hartree - fock energy and to bring the relevant domains @xmath173 of different @xmath9 to a common range . in fig .
[ fig : nxcanfl1 ] , @xmath174 is shown as a function of @xmath175 .
we observe that the results for @xmath15 are remarkably similar for all band - fillings .
the largest deviations from the common trend are found for @xmath176 . as already discussed for @xmath177 , this anomalous behavior
is related to the degeneracy of the single - particle spectrum and to the finite size of system . fig .
[ fig : nxcanfl1 ] shows that for the hubbard model the largest part of the dependence of @xmath4 on band filling comes from @xmath30 , @xmath178 and @xmath161 .
similar conclusions are derived from the results for the infinite 1d chain presented in fig .
[ fig:1dinf ] . for a given @xmath179
, @xmath15 depends weakly on @xmath180 if the carrier density is low ( @xmath181 ) , and tends to increase as we approach half - band filling [ see fig .
[ fig:1dinf](b ) ] . for high carrier densities
it become comparatively more difficult to minimize the coulomb energy for a given degree of delocalization @xmath179 .
the effect is most pronounced for @xmath182@xmath183 , i.e. , close to the uncorrelated limit .
as we approach the strongly correlated limit ( @xmath184 ) the dependence of @xmath139 on @xmath180 is very weak even for @xmath185 .
one concludes that @xmath15 is a useful basis for introducing practical approximations on more complex systems .
the correlation energy @xmath29 is a universal functional of the complete single - particle density matrix @xmath0 .
@xmath3 $ ] and @xmath1 $ ] may depend on @xmath8 and @xmath9 but are independent of @xmath26 and in particular of the lattice structure .
the functional @xmath4 considered in this paper depends by definition on the type of lattice , since the constraints imposed in the minimization only apply to nn bonds .
in order to investigate this problem we have determined @xmath4 for 2d and 3d finite clusters having @xmath186 sites and periodic boundary conditions . in fig .
[ fig : xcfdima ] we compare these results with those of the 1d @xmath187-site periodic ring . as shown in the inset figure , the qualitative behavior is in all cases very similar .
the main quantitative differences come from the domain of representability of @xmath101 , i.e. , from the values of @xmath111 and @xmath112 ( @xmath188 ) . once scaled as a function of @xmath189
, @xmath29 depends rather weakly on the lattice structure . notice that the hartree - fock energy @xmath190 is the same for all structures .
however , for the bcc structure we obtain @xmath191 , i.e. , @xmath192 , due to degeneracies in the single - particle spectrum of the considered finite cluster [ see inset fig . [ fig : xcfdima](b ) ] . in order to correct for this finite size effect
it is here more appropriate to consider @xmath193 / w(\gamma_{12}^0)$ ] .
still , the differences in @xmath139 between bcc and fcc structures appear to be more important than between square and triangular 2d lattices .
this is probably related to the degeneracies in the spectrum of the bcc cluster , as already observed for rings with @xmath194 [ figs .
[ fig : nxcanft ] and [ fig : nxcanfl1](a ) ] .
the largest changes in @xmath139 for different lattice structures are observed for intermediate degree of delocalization ( @xmath195@xmath183 , see fig .
[ fig : xcfdima ] ) .
note that there is no monotonic trend as a function of the lattice dimension .
for example , for @xmath196@xmath183 , @xmath139 first increases somewhat as we go from 1d to 2d lattices , but it then decreases coming close to the 1d curve for the 3d fcc lattice [ @xmath197 for @xmath198 . finally , it is worth noting that in the strongly correlated limit ( @xmath199 ) the results for @xmath15 are nearly the same for all considered lattice structures ( see fig .
[ fig : xcfdima ] ) .
this should be useful in order to develop simple general approximations to @xmath4 in this limit .
the attractive hubbard model describes itinerant electrons with local intra - atomic pairing ( @xmath17 ) .
the electronic correlations are very different from those found in the repulsive case discussed so far . in particular levy s interaction energy functional @xmath200
now correspond to the maximum average number of double occupation for a given @xmath0 [ see eq .
( [ eq : whub ] ) ] .
therefore , it is very interesting to investigate the properties of the correlation energy functional @xmath201 also for @xmath17 and to contrast them with the results of the previous section . in fig .
[ fig : xcsuft ] the correlation energy @xmath4 of the attractive hubbard model is given at half - band filling for various finite rings ( @xmath202 ) and for the infinite 1d chain ( @xmath134 ) .
the band - filling dependence of @xmath4 is shown in fig .
[ fig : xcsufl ] for a finite @xmath187-site ring ( @xmath203 ) . as in the repulsive case ,
@xmath204 since the domain of representability of @xmath101 is independent of the form or type of the interaction .
moreover , @xmath205 due to the electron - hole symmetry of bipartite lattices @xcite . starting from @xmath111 or @xmath112 ( @xmath206 on bipartite lattices ) , @xmath4 decreases with decreasing @xmath120 reaching the minimum @xmath207 for @xmath208 and for @xmath154 ( @xmath209 in this case ) . for @xmath9 even , @xmath210 , and for @xmath9
odd , @xmath211 , which correspond to the maximum number of electron pairs that can be formed . for @xmath9
even , the minimum @xmath212 $ ] is achieved only for a complete electron localization ( i.e , @xmath160 ) . in contrast , for odd @xmath9 a finite - size effect is observed . in this case ,
one of the electrons remains unpaired even in the limit of strong electron correlations and the minimum of @xmath29 is @xmath213[1-(n_e+1)/(2n_a)]$ ] .
moreover , non - vanishing @xmath161 are obtained as a result of the delocalization of the unpaired electron .
@xmath161 represents the maximum bond order that can be obtained when @xmath214 electron pairs are formed ( @xmath215 for @xmath216 , @xmath9 odd ) . notice that in all cases the ground state @xmath125 is found in the interval @xmath217 .
it is interesting to observe that @xmath4 can be appropriately scaled in a similar way as for @xmath16 . in fig .
[ fig : xcsufl](b ) , @xmath218 is shown as a function of the degree of delocalization @xmath219 .
@xmath15 presents a pseudo - universal behavior in the sense that it depends weakly on @xmath8 and @xmath9 .
the main deviations from the common trend are found for @xmath220 . as already discussed for @xmath16 ,
this is a consequence of degeneracies in the single - particle spectrum . in this case , the wave function corresponding to the minimum in levy s functional for @xmath221 [ eq . ( [ eq : whub ] ) ] can not be described by a single slater determinant and @xmath222 .
density - matrix functional theory has been applied to lattice hamiltonians taking the hubbard model as a particularly relevant example . in this framework
the basic variable is the single - particle density matrix @xmath0 and the key unknown is the correlation energy functional @xmath3 $ ] .
the challenge is therefore to determine @xmath3 $ ] or to provide with useful accurate approximations for it . in this paper
we presented a systematic study of the functional dependence of @xmath4 on periodic lattices , where @xmath101 is the density - matrix element between nearest neighbors ( @xmath223 for all nn @xmath62 ) . based on finite - cluster exact diagonalizations and on the bethe - ansatz solution of the 1d chain , we derived rigorous results for @xmath4 of the hubbard model as a function of the number of sites @xmath8 , band filling @xmath180 and lattice structure . a basis for applications of density - matrix functional theory to many - body lattice models is thereby provided .
the observed pseudo - universal behavior of @xmath224 as a function of @xmath225 encourages transferring @xmath15 from finite - size systems to infinite lattices or even to different lattice geometries .
in fact , the exact @xmath4 of the hubbard dimer has been recently used to infer a simple general ansatz for @xmath4 @xcite . with this approximation to @xmath4 the ground - state energies and charge - excitation gaps of 1d and 2d lattices
have been determined successfully in the whole range of @xmath28 .
further investigations , for example , by considering magnetic impurity models or more complex multiband hamiltonians , are certainly worthwhile .
langreth and m.j .
mehl , phys .
b * 28 * , 1809 ( 1983 ) ; a.d .
becke , phys .
a * 38 * , 3098 ( 1988 ) ; j.p .
perdew , k. burke and m. ernzerhof , phys .
lett . * 77 * , 3865 ( 1996 ) and references therein . given a basis set @xmath226 ,
the hopping integrals are expressed as @xmath227 \phi_j ( \vec r ) \ ; , \ ] ] and the matrix elements of the coulomb interaction as @xmath228 a single - particle density matrix @xmath0 is said to be pure - state @xmath19-representable if an @xmath19-particle state @xmath45 exists such that @xmath229 for all @xmath6 and @xmath7 .
an extension of the definition domain of @xmath43 $ ] to ensemble - representable density matrices @xmath230 is straightforward following the work by valone ( see ref .
ensemble density matrices are written as @xmath231 . in practice ,
the @xmath230 are much easier to characterize than pure - state density matrices . c. lanczos
45 * , 255 ( 1950 ) ; b.n .
parlett , _ the symmetric eigenvalue problem _ , ( prentice - hall , engelwood cliffs , nj , 1980 ) ; j.k . collum and r.a .
willoughby , _ lanczos algorithms for large symmetric eigenvalue computations _ , ( birkhauser , boston , 1985 ) , vol . i. the electron - hole transformation @xmath232 leaves the hubbard hamiltonian and eq .
( [ eq : evhub ] ) formally unchanged , except for an additive constant and a change of sign in the hopping integrals @xmath26 [ or of @xmath32 in eq .
( [ eq : evhub ] ) ] .
@xmath233 , since the form of interaction term is unaffected by changing electrons to holes , and since the density matrix for holes satisfies @xmath234 .
moreover , on bipartite lattices with only nn hoppings @xmath235 , since the sign of the nn bond order may be changed by changing the sign of @xmath236 on one of the sublattices .
a lattice is _ bipartite _ if two distinct subsets of lattice sites @xmath109 and @xmath110 can be defined such that every lattice site belongs either to a or to b , and that there is no pair of nn s belonging to the same subset .
all nn bonds ( or hoppings ) connect a site in @xmath109 with a site in @xmath110 .
in the presence of degeneracies in the single - particle spectrum of finite systems one may find that the minimum of @xmath1 $ ] does not correspond to a single slater determinant , and that @xmath237 .
such a behavior is observed , for example , in rings with @xmath238 .
this is a finite - size effect which importance decreases with increasing @xmath8 : @xmath239 , @xmath240 and @xmath241 for rings with @xmath220 , @xmath242 and @xmath187 , respectively .
@xmath243 .
in the nonmagnetic case the hartree - fock energy of the hubbard model is @xmath244 for @xmath9 even and @xmath245 $ ] for @xmath9 odd .
notice that in this model the difference between @xmath30 and the hartree energy @xmath246 is only the self interaction .
if @xmath247 , i.e. , @xmath248 , it is more appropriate to consider @xmath249 / w(\gamma_{12}^0)$ ] in order to correct for this finite size effect . in this way
@xmath15 is less sensitive to the details of the lattice structure or cluster size . using a linear combination @xmath70 between two @xmath19-particles states @xmath250 and @xmath251 satisfying @xmath252 and latexmath:[$\langle\psi_a| \sum_{\sigma } \hat c_{i \sigma}^{\dagger } \hat c_{j \sigma } @xmath94 or @xmath93 )
, one may represent the nn matrix elements @xmath254 in the interval @xmath255 , where @xmath256 and @xmath257 are the bond orders corresponding to @xmath258 and @xmath259 ( @xmath260 ) .
for example , one may take @xmath258 as a fully localized state with maximal @xmath93 ( @xmath261 , @xmath262 ) and @xmath251 as the state representing @xmath161 for @xmath263 . | a density functional theory for many - body lattice models is considered in which the single - particle density matrix @xmath0 is the basic variable .
eigenvalue equations are derived for solving levy s constrained search of the interaction energy functional @xmath1 $ ] .
@xmath1 $ ] is expressed as the sum of hartree - fock energy @xmath2 $ ] and the correlation energy @xmath3 $ ] .
exact results are obtained for @xmath4 of the hubbard model on various periodic lattices , where @xmath5 for all nearest neighbors @xmath6 and @xmath7 .
the functional dependence of @xmath4 is analyzed by varying the number of sites @xmath8 , band filling @xmath9 and lattice structure .
the infinite one - dimensional chain and one- , two- , or three - dimensional finite clusters with periodic boundary conditions are considered .
the properties of @xmath4 are discussed in the limits of weak ( @xmath10 ) and strong ( @xmath11 ) electronic correlations , and in the crossover region ( @xmath12 ) . using an appropriate scaling
we observe that @xmath13 has a pseudo - universal behavior as a function of @xmath14 .
the fact that @xmath15 depends weakly on @xmath8 , @xmath9 and lattice structure suggests that the correlation energy of extended systems could be obtained quite accurately from finite cluster calculations . finally , the behavior of @xmath4 for repulsive ( @xmath16 ) and attractive ( @xmath17 ) interactions are contrasted . 2 | cond-mat9910264 |
e. rasmusen , _ games and information : an introduction to game theory _
( blackwell pub , oxford , 2001 ) , r. b. myerson , _ game theory : analysis of conflict _ ( harvard univ . press , cambridge ma , 1997 ) . in bos ,
if the source is ideal , that is @xmath47 , there are multiple ne s with the same payoff .
( the strategy set achieving the ne s are shown in table [ tab : prob3 ] . ) however , the strategies with @xmath96 and @xmath97 , where @xmath98 and @xmath99 are in the range @xmath100 $ ] , is a focal equilibrium point .
this strategy set enables the players to converge to the ne without a concern for the choice of @xmath98 and @xmath99 . | effects of a corrupt source on the dynamics of simultaneous move strategic games are analyzed both for classical and quantum settings . the corruption rate dependent changes in the payoffs and strategies of the players are observed .
it is shown that there is a critical corruption rate at which the players lose their quantum advantage , and that the classical strategies are more robust to the corruption in the source .
moreover , it is understood that the information on the corruption rate of the source may help the players choose their optimal strategy for resolving the dilemma and increase their payoffs .
the study is carried out in two different corruption scenarios for prisoner s dilemma , samaritan s dilemma , and battle of sexes . * _ a .
introduction : _ * classical game theory has a very general scope , encompassing questions and situations that are basic to all of the social sciences @xcite .
there are three main ingredients of a game which is to be a model for real life situations @xcite : the first of these is the rational players ( decision makers ) who share a common knowledge . the second is the strategy set which contains the feasible actions the players can take , and the third one is the payoff which are given to the players as their profit or benefit when they apply a specific action from their strategy set .
when rational players interact in a game , they will not play dominated strategies , but will search for an equilibrium .
one of the important concepts in game theory is that of nash equilibrium ( ne ) in which each player s choice of action is the best response to the actions taken by the other players . in an ne , no player can increase his payoff by unilaterally changing her action .
while the existence of a unique ne makes it easier for the players to choose their action , the existence of multiple ne s avoids the sharp decision making process because the players become indifferent between them . in pure strategies , the type and the number of ne s in a game depend on the game .
however , due to von neumann there is at least one ne when the same game is played with mixed strategies @xcite .
classical game theory has been successfully tested in decision making processes encountered in real - life situations ranging from economics to international relations . by studying and applying the principles of game theory
, one can formulate effective strategies , predict the outcome of strategic situations , select or design the best game to be played , and determine competitor behavior , as well as the optimal strategy . in recent years
, there have been great efforts to apply the quantum mechanical toolbox in the design and analysis of games @xcite . as it was the same in other fields such as communication and computation , quantum mechanics introduced novel effects into game theory , too .
it has proved to have the potential to affect our way of thinking when approaching to games and game modelling . using the physical scheme proposed by eisert _
et al . _
( see fig.[fig : scheme ] ) @xcite , it has been shown in several games that the dilemma existing in the original game can be resolved by using the paradigm of quantum mechanics @xcite .
it has also been shown that when one of the players chooses quantum strategies while the other is restricted to classical ones , the player with quantum strategies can always receive better payoff if they share a maximally entangled state @xcite .
quantum systems are easily affected by their environment , and physical schemes are usually far from ideal in practical situations .
therefore , it is important to study whether the advantage of the players arising from the quantum strategies and the shared entanglement survive in the presence of noise or non - ideal components in the physical scheme . in this paper
, we consider a corrupt source and analyze its effect on the payoffs and strategies of the players .
we search answers for the following two questions : ( i ) is there a critical corruption rate above which the players can not maintain their quantum advantage if they are unaware of the action of the noise on the source , and ( ii ) how can the players adopt their actions if they have information on the corruption rate of the source . *
_ b .
eisert s scheme : _ * in this physically realizable scheme the quantum version of a two - player - two - strategy classical game can be played as follows : ( a ) a referee prepares a maximally entangled state by applying an entangling operator @xmath0 on a product state @xmath1 where @xmath2 .
the output of this entangler , which reads @xmath3 $ ] , is delivered to the players .
( b ) the players apply their actions , which are su(2 ) quantum operations locally on their qubits , and return the resultant state @xmath4 back to the referee .
operators @xmath5 and @xmath6 are restricted to two - parameter su(2 ) operators given by @xmath7 where @xmath8 and @xmath9 .
( c ) the referee , upon receiving this state , applies @xmath10 and then makes a quantum measurement @xmath11 with @xmath12 and @xmath13 .
then the average payoffs of the players become @xmath14 where @xmath15 , @xmath16 and @xmath17 are the payoffs chosen from the classical payoff matrix when the measurement result is @xmath18 , and @xmath19 corresponds to the probability of obtaining @xmath18 .
the classical version of the game can be played using the same scheme if the operations corresponding to the classical pure strategies are chosen as @xmath20 and @xmath21 . using this scheme , quantum versions of some dilemma - containing classical games , such as prisoner s dilemma ( pd ) , samaritan s dilemma ( sd ) and battle of sexes ( bos ) whose
payoffs matrices are given in fig.[fig : payoffs ] , have been studied . in these games , it has been understood that if the referee starts with the state @xmath22 generating the entangled state @xmath23 $ ] , the players can resolve their dilemma and receive the highest possible total payoff @xmath24 .
it has also been shown that the dynamics of the games changes when the referee starts with a different initial state .
for example , if the referee starts with @xmath25 in sd , four ne s emerge with the same constant payoff making a solution to the dilemma impossible @xcite .
= 8.7 cm * _ c .
corrupt source in quantum games : _ * as we have pointed out above , the initial state from which the referee prepares the entangled state is a crucial parameter in eisert s scheme . therefore , any corruption or deviation from the ideality of the source which prepares this state will change the dynamics and outcomes of the game .
consequently , the analysis of situations where the source is corrupt is necessary to shed a light in understanding the game dynamics in the presence of imperfections .
we consider the source model shown in fig .
[ fig : corrupt ] .
this model includes two identical sources constructed to prepare the states @xmath26 s which are the inputs to the entangler at each run of the game . these sources are not ideal and have a _ corruption rate _ , @xmath27 , that is , they prepare the desired state @xmath26 with probability @xmath28 while preparing the unwanted state @xmath29 with probability @xmath27
. the state prepared by these sources thus can be written as @xmath30 .
then the combined state generated and sent to the entangler becomes @xmath31 .
this results in a mixture of the four possible maximally entangled states @xmath32 , where @xmath33 and @xmath34 .
this is the state on which the players will perform their unitary operators .
* _ scenario i : _ * in this scenario , the players alice and bob are not aware of the corruption in the source .
they assume that the source is ideal and always prepares the initial state @xmath22 , and hence that the output state of the entangler is always @xmath35 .
based on this assumption , they apply the operations that is supposed to resolve their dilemma .
we have analyzed pd , sd and bos according to this scenario , and compared the payoffs of the players with respect to the corruption rate .
the payoff they receive when they stick to their quantum strategies are compared to the payoffs when they play the game classically .
we consider the classical counterparts both with and without the presence of noise in the game .
that is , the players use the same physical scheme of the quantum version with and without the corrupt source , and apply their actions by choosing their operators from the set @xmath36 .
the results of the analysis according to this scenario are depicted in figs .
[ fig : pd]-[fig : bos ] .
a remarkable result of this analysis is that with the introduction of the corrupt source , the players quantum advantage is no longer preserved if the corruption rate , @xmath27 , becomes larger than a critical corruption rate @xmath37 . at @xmath37 ,
the classical and quantum strategies produce equal payoffs .
another interesting result is the existence of a strategy @xmath38 , where the payoffs of the players become constant independent of corruption rate .
this strategy could be attractive for risk avoiding and/or paranoid players . for pd , which is a symmetric game ,
the optimal classical strategies deliver the payoffs @xmath39 for the actions @xmath40 . in the quantum version with an uncorrupt source
, the players can get the optimal payoffs @xmath41 if they adopt the strategies @xmath42 @xcite .
hence , the dilemma of the game is resolved and the players receive better payoffs than those obtained with classical strategies . however , as seen in fig .
[ fig : pd ] , the payoffs of players with classical and quantum strategies become equal to @xmath43 when @xmath44 .
if @xmath27 satisfies @xmath45 , the quantum version of the game always does better than the classical one .
otherwise , the classical game is better .
when the classical version of pd is played with a corrupt source , we find that with increasing corruption rate , while the payoffs for the quantum strategy decrease , those of the classical one increase .
that is , if @xmath46 , then the players would rather apply their classical strategies than the quantum ones .
this can be explained as follows : when the players apply classical operations , the game is played as if there is no entanglement in the scheme .
that is , players apply their classical operators @xmath21 on the state prepared by the source . if the source is ideal , @xmath47 , they operate on the @xmath48 which results in an output state @xmath49 .
referee , upon receiving this output state and making the projective measurement , delivers @xmath39 . on the other hand ,
when @xmath50 , the state from the source is @xmath49 and the output state after the players actions becomes @xmath48 . with this output state
, referee delivers them the payoffs @xmath41 .
thus , when the players apply the classical operator @xmath21 , their payoffs continuously increase from one to three with the increasing corruption rate from @xmath47 to @xmath50 . using a classical mixed strategy in the asymmetric game of sd ,
the players receive @xmath51 at the ne . in this strategy , while alice chooses from her strategies with equal probabilities , bob uses a biased randomization where he applies one of his actions , @xmath20 , with probability @xmath52 .
the most desired solution to the dilemma in the game is to obtain an ne with @xmath53 .
this is achieved when both players apply @xmath54 to @xmath35 @xcite .
the dynamics of the payoffs in this game with the corrupt source when the players stick to their operators @xmath54 and its comparison with their classical mixed strategy are depicted in fig .
[ fig : sd ] . since this game is an asymmetric one , the payoffs of the players , in general , are not equal .
however , with the corrupt source it is found that their payoffs become equal at @xmath55 and at @xmath50 , where the payoffs are @xmath56 and @xmath57 , respectively .
the critical corruption rate , @xmath58 , which denotes the transition from the quantum advantage to classical advantage regions , is the same for both players .
while for increasing @xmath27 , @xmath59 monotonously decreases from two to zero , @xmath60 reaches its minimum of @xmath61 at @xmath62 , where it starts increasing to the value of zero at @xmath50 .
it is worth noting that when the players apply their classical mixed strategies in this physical scheme , @xmath59 is always constant and independent of the corruption rate , whereas @xmath60 increases linearly as @xmath63 for @xmath64 .
the payoffs of the players are compared in three cases @xcite : _ case 1 _ : @xmath65 ( insufficient solution ) , _ case 2 _ : @xmath66 ( weak solution ) , and _ case 3 _ : @xmath67 ( strong solution ) . in the corrupt source scenario in quantum strategies , _
case 1 _ is achieved for @xmath68,_case 2 _ is achieved for @xmath69 , and finally _ case 3 _ for @xmath70 .
the remarkable result of this analysis is that although the players using quantum strategies have high potential gains , there is a large potential loss if the source is deviated from an ideal one .
the classical strategies are more robust to corruption of the source . in bos , which is an asymmetric game , the classical mixed strategies , where alice and bob apply @xmath20 with probabilities @xmath71 and @xmath72 or vice versa , the players receive equal payoffs of @xmath72 .
however , the dilemma is not solved due to the existence of two equivalent ne . on the other hand , when the physical scheme with quantum strategies is used the players can reach an ne where their payoffs become @xmath73 and @xmath74 if both players apply @xmath21 to the maximally entangled state prepared with an ideal source @xcite .
the advantage of this quantum strategy to the classical mixed strategy is that in the former @xmath24 is higher than the latter . in the presence of corruption in the source , payoffs of the players change as shown in fig .
[ fig : bos ] . with an ideal source ,
the payoffs reads @xmath75 , however for increasing corruption rate while @xmath59 decreases from two to one , @xmath60 increases from one to two . with an completely corrupt source , @xmath50 ,
the payoffs become @xmath76 .
the reason for this is the same as explained for pd .
when the quantum strategies with and without corrupt source are compared to the classical mixed strategy without noise , it is seen that the former ones always give better payoffs to the players .
however , when the source becomes noisy ( corrupt ) , classical strategies become more advantageous to quantum ones with increasing corruption rate .
the range of corruption rate where classical strategies are better than the quantum strategy if the players stick to their operations @xmath21 , are @xmath77 and @xmath78 for alice and bob , respectively . when @xmath79 , @xmath80 and these payoffs are equal to the ones received with classical mixed strategies .
while @xmath80 independently of @xmath27 for classical mixed strategies , @xmath60 and @xmath59 differ when the players stick to their quantum strategies for @xmath81 .
another interesting result for this game is that , contrary to pd and sd , the strategy @xmath38 discussed above always gives a constant payoff @xmath82 , which is better than that of the classical mixed strategy . *
_ scenario ii : _ * in this scenario , the referee knows the characteristics of the corruption in the source , and inform the players on the corruption rate .
then the question is whether the players can find a unique ne for a known source with corruption rate @xmath27 ; and if they can , does this ne resolve their dilemma in the game or not . when the corruption rate is @xmath79 ,
the state shared between the players become @xmath83 .
then independent of what action they choose , the players receive constant payoffs determined by averaging the payoff entries .strategies which lead to ne s and the corresponding payoffs for the players in prisoner s dilemma ( pd ) if they are provided the information on the corruption rate , @xmath27 , of the source which prepares in the initial product state @xmath84 .
[ tab : prob1 ] [ cols="^,^,^,^,^,^,^",options="header " , ] for the sd game . in this game , in contrast to the other two ,
although for @xmath47 there is a unique ne solving the dilemma , for @xmath50 the players can not find a unique ne .
there emerges an infinite number of different strategies with equal payoffs @xmath85 .
the players are indifferent between these strategies and can not make sharp decisions .
therefore , the dilemma of the game survives , although its nature changes .
when we look at some intermediate values for the corruption rate , we see that corruption rate affects bos and sd strongly . for example , when @xmath86 in sd , there are infinite number of strategies and ne s which have the same payoffs @xmath87 .
these ne s are achieved when the players choose their operators as @xmath88 and @xmath89 .
the same is seen in bos for @xmath90 which results in a payoff @xmath91 when the players choose @xmath92 and @xmath89 . a more detailed analysis carried out for pd with increasing @xmath27 in steps of @xmath93 in the range
@xmath94 $ ] has revealed that the players can achieve a unique ne where their payoffs and strategies depends on the corruption rate .
therefore , information on the source characteristic might help the players to reorganize their strategies . however , whether providing the players with this kind of information in a game is acceptable or not is an open question . *
_ d .
conclusion : _ * this study shows that the strategies to achieve ne s and the corresponding payoffs are strongly dependent on the corruption of the source . in a game with corrupt source ,
the quantum advantage no longer survives if the corruption rate is above a critical value .
the corruption may not only cause the emergence of multiple ne s but may cause a decrease in the player s payoff , as well , even if there is a single ne . if the players are given the characteristics of the source then they can adapt their strategies ; otherwise they can either continue their best strategy assuming that the source is ideal and take the risk of losing their quantum advantage over the classical or choose a risk - free strategy , which makes their payoff independent of the corruption rate .
however , in the case where players know the corruption rate and adjust their strategies , the problem is that for some games there emerge multiple ne s , therefore the dilemmas in those games survive .
this study reveals the importance of the source used in a quantum game .
the authors thank to dr .
j. soderholm for the critical reading of the manuscript .
they also acknowledge the insightful discussions with dr .
f. morikoshi and dr .
t. yamamoto . | quant-ph0402038 |
the magnetocaloric effect , _
i.e. _ , a temperature change induced by an adiabatic change of an external magnetic field was discovered in iron by warburg in 1881 @xcite .
this effect also has a long history in cooling applications .
for example , adiabatic demagnetization of paramagnetic salts was the first method to reach temperatures below 1k @xcite . among the many experimental but also theoretical investigations to follow , we would like to mention in particular the computation of the field variation of the entropy in a quantum spin-1/2 @xmath2 chain @xcite .
this investigation is remarkable in so far as it is one of the very first numerical computations for an interacting quantum many - body system .
the relation of the entropy of quantum spin chains in a magnetic field to magnetic cooling was also noted @xcite and even one exact computation was performed , albeit for a chain with a strong easy - axis anisotropy @xcite .
the magnetocaloric effect in quantum spin systems has recently attracted renewed attention .
the reason for this is two - fold . on the one hand ,
field - induced quantum phase transitions lead to universal responses when the applied field is varied adiabatically @xcite . on the other hand , it was observed that the magnetocaloric effect is enhanced by geometric frustration @xcite , promising improved efficiency in low - temperature cooling applications . in this context , the isotropic spin-1/2 heisenberg chain , the isotropic and anisotropic @xmath3 chain in a transverse field @xcite , and the quantum ising chain in a transverse field @xcite were also revisited .
one further example of a non - frustrated quantum spin chain where the magnetocaloric effect was computed is the ferrimagnetic spin-@xmath4-@xmath5 chain @xcite .
finally , the adiabatic cooling rate has recently been measured on the spin-1/2 heisenberg chain compound [ cu(@xmath6-c@xmath7o@xmath8)(4-aminopyridine)@xmath7(h@xmath7o)]@xmath9 @xcite and relatedly the magnetocaloric effect of the spin-1/2 isotropic heisenberg and @xmath2 chains was computed exactly @xcite . in this paper we will investigate magnetocaloric properties of quantum spin-@xmath0 heisenberg chains given by the hamiltonians @xmath10 @xmath11 is the antiferromagnetic exchange constant , @xmath12 an external magnetic field , @xmath13 the length of the chain , and @xmath14 are quantum spin-@xmath0 operators at site @xmath15
. we will use periodic boundary conditions , _
i.e. _ , @xmath16 . in section [ sec : methods ] we will first summarize the methods which we are going to use , namely exact diagonalization ( ed ) and quantum monte carlo ( qmc ) simulations using the stochastic series expansion ( sse ) framework .
then we will apply these methods to the @xmath1 chain in section [ sec : s1chain ] .
the spin @xmath1 heisenberg chain is famous for the presence of the so - called ` haldane ' gap at @xmath17 @xcite . in particular
, we will illustrate with the @xmath1 chain that one can exploit the presence of a spin gap for cooling by adiabatic magnetization at low magnetic fields .
next , we discuss the scaling with spin quantum number @xmath0 in section [ sec : sscal ] .
in particular , we will show that there is a quantum scaling regime at low temperatures close to the saturation field where one has _ linear _ scaling with @xmath0 .
finally , in section [ sec : summary ] we will conclude with a summary .
thermodynamic quantities can be computed using spectral representations . for example , the entropy can be written as @xcite @xmath18 where @xmath19 is the partition function and @xmath20 are the eigenvalues of the hamiltonian . in order to evaluate spectral representations like ( [ eq : specs ] ) , one needs to diagonalize the hamiltonian .
if this is done numerically exactly , this is called exact diagonalization ( ed ) .
first , one should perform a symmetry reduction .
we have used translations and @xmath21-conservation as well as reflections and spin inversion where appropriate .
one can then use a library routine to perform a full diagonalization .
such an approach is very much in the spirit of classic work @xcite .
the main technical differences of our computations and @xcite are : ( i ) we have exploited @xmath22-symmetry to reconstruct the @xmath23 sector from the @xmath24 sectors and the spin - inversion resolved @xmath25 sector , ( ii ) we have used improved implementations of the diagonalization routines @xcite , ( iii ) and we have substantially more powerful computers at our disposal . the combination of these factors enables us to compute full spectra for bigger systems than in the 1960s .
if we are interested only in low - energy properties , we can use iterative diagonalization algorithms like the lanczos method @xcite .
the basic lanczos algorithm @xcite for a hermitian matrix @xmath26 proceeds as follows . for a given normalized start vector @xmath27
, one defines a sequence of normalized lanczos vectors @xmath28 via the recurrence relations @xmath29 with the initial conditions @xmath30 and @xmath31 . the real coefficients @xmath32 and @xmath33 define the so - called lanczos matrices @xmath34 the crucial point of the lanczos algorithm is that the eigenvalues of @xmath35 yield a good approximation of the extremal eigenvalues of @xmath26 already for @xmath36 much smaller than the total hilbert space dimension .
the lanczos algorithm suffers from one practical problem : the recursion relations ( [ eq : lancrec ] ) are supposed to guarantee mutual orthogonality of the vectors @xmath37 , _
i.e. _ , @xmath38 for @xmath39 . however , this fails to be correct when the computations are carried out numerically , leading to undesired spurious states , called ghost states @xcite .
several strategies have been proposed @xcite in order to deal with the ghost problem during the computation of excited eigenvalues . here
we suggest another method to compute excited states with controlled accuracy and correct multiplicity .
we start by performing a fixed number @xmath36 of lanczos iterations with a given start vector @xmath27 .
next we compute the orthogonal transformation @xmath40 , @xmath41 diagonalizing the lanczos matrix ( [ eq : lancmat ] ) .
then we fix a number @xmath42 and repeat the lanczos procedure with the same start vector @xmath27 . during this second lanczos
pass we construct @xmath43 vectors @xmath44 the second lanczos pass is needed in order to avoid storing the @xmath45 lanczos vectors @xmath28 .
the vectors @xmath46 given by ( [ eq : lancapproxev ] ) yield approximations to the eigenvectors of @xmath26 .
however , due to the ghost problem , there are spurious vectors which need to be eliminated .
we perform this in two steps . first
, using the observation that the vectors @xmath46 are supposed to be mutually orthogonal , we can eliminate those @xmath46 which have big projections on the @xmath47 with @xmath48 .
second , we reorthogonalize the remaining vectors yielding @xmath49 orthonormal vectors @xmath50 . finally , we project the matrix @xmath26 onto this subspace via @xmath51 a full diagonalization of the @xmath52 matrix @xmath53 and application of the resulting basis transformation to the vectors @xmath50 yields @xmath54 _ orthogonal _ vectors @xmath55 which approximate the eigenvectors of @xmath26 . the eigenvalues @xmath56 associated to the vectors @xmath55 and
their accuracy can be estimated according to @xmath57 according to our experience , there are neither any missing nor any spurious eigenvalues among the converged ones .
it should be mentioned that there is no guarantee that the lanczos procedure yields the complete spectrum for a given start vector @xmath27 , in particular if one has not performed a complete symmetry decomposition .
nevertheless , numerical noise seems to prevent this from happening , at least for sufficiently generic start vectors @xmath27 .
for our procedure one needs to choose the number of lanczos iterations @xmath36 and the dimension of the subspace @xmath58 in advance and the whole procedure needs to be repeated if it turns out that less than the desired number of converged eigenvalues have been obtained . in practice , however , one obtains results of a comparable quality for a class of problems where the choice of @xmath36 and @xmath43 has been adjusted for one representative case . in the examples to be reported below ,
we have been able to obtain about 100 eigenvalues in a given symmetry sector with a relative accuracy of @xmath59 or better using @xmath60 lanczos iterations and @xmath61 vectors @xmath46 . since the system sizes accessible by either full diagonalization or lanczos diagonalization
are limited , it is desirable to have other methods at our disposal .
for non - frustrated spin models like the spin-@xmath0 heisenberg chain ( [ eq : ham ] ) , one can in principle use quantum monte carlo ( qmc ) simulations
. there is , however , one problem : we are particularly interested in the entropy which is usually obtained with large statistical errors from integrating a monte - carlo result for the specific heat . in this subsection
we will summarize how one can circumvent this problem . in order to obtain a qmc estimate of the temperature dependence of the entropy , we employed an extended ensemble , broad - histogram method @xcite within the stochastic series expansion ( sse ) framework @xcite .
based on this approach , thermodynamic quantities can be obtained over a broad range of temperatures from a single qmc simulation that provides estimates of the expansion coefficients @xmath62 of the system s partition function @xmath63 in a high - temperature series expansion : @xmath64 here , @xmath65 is the inverse temperature . in terms of the expansion coefficients @xmath62 ,
the internal energy is obtained using @xmath66 and the free energy from @xmath67 finally , the entropy can be calculated using @xmath68 . in the qmc simulation , estimates of the first @xmath69 coefficients
@xmath70 are obtained by performing a random walk in the expansion order @xmath36 , such as to sample efficiently all expansion orders from @xmath71 ( where @xmath72 for a spin-@xmath0 chain of @xmath13 sites is known exactly ) to @xmath73 .
this is accomplished using an extension of the wang - landau flat - histogram sampling algorithm @xcite to the quantum case , as detailed previously in refs .
knowledge of the first @xmath69 coefficients @xmath62 allows for the calculation of thermodynamic quantities from @xmath74 down to a temperature @xmath75 for a system of @xmath13 sites , where @xmath69 scales proportional to @xmath76 @xcite .
in particular , for the spin @xmath1 heisenberg chain , we were able to treat systems with up to @xmath77 sites , accessing temperatures down to @xmath78 , which required an already large value of @xmath79 .
the data shown in this contribution ( and also those in @xcite ) were obtained based on the original version of the algorithm @xcite , while recently an improved sampling strategy was proposed , based on optimizing the broad histogram quantum monte carlo ensemble @xcite . of the @xmath1 heisenberg chain as a function of temperature @xmath80 for @xmath81 .
lines are obtained by ed for @xmath82 and @xmath83 sites , symbols show qmc results for @xmath77 sites . ]
we test and compare the aforementioned numerical methods using the example of the @xmath1 heisenberg chain ( [ eq : ham ] ) with a finite magnetization .
[ figs1enth3 ] shows the entropy per site @xmath84 as a function of temperature @xmath80 for a fixed magnetic field @xmath81 . for @xmath82 ,
we have performed a full diagonalization .
the corresponding ed curve can therefore be considered as the exact result for @xmath82 .
a full determination of the spectrum is clearly out of reach already for @xmath85 . in this case
, we therefore had to perform a severe truncation to low energies .
thus , the @xmath85 ed curve is only a low - temperature approximation . in this paper ,
the precise temperature range is fixed as follows : let @xmath86 be the highest energy until which the spectrum is definitely complete
. then we restrict to temperatures @xmath87 .
this choice ensures that missing states are suppressed by a boltzmann factor @xmath88 .
this may seem a small number , but since we are discarding many states , extending truncated data to higher temperatures would yield artifacts which are clearly visible on the figure(s ) .
qmc ( denoted by sse in fig . [ figs1enth3 ] ) is able to treat the larger system size @xmath77 .
this method , however , is best suited for high temperatures .
indeed , at @xmath89 one can see deviations caused by statistical errors in the sse result of fig .
[ figs1enth3 ] which prevented us from going to lower temperatures .
finite - size effects can be observed in fig .
[ figs1enth3 ] in the @xmath82 ed curve for @xmath90 and in the @xmath85 ed result .
otherwise it is reassuring that we observe good overall agreement between all three methods .
in passing we note that , for the value of the magnetic field used in fig .
[ figs1enth3 ] , the low - energy physics of the @xmath1 chain is described by a luttinger liquid ( see ref .
@xcite and references therein ) .
it is well known that the specific heat @xmath91 of a luttinger liquid is linear in @xmath80 @xcite .
due to the relation @xmath92 and because of @xmath93 , the entropy of a luttinger liquid is identical to its specific heat and in particular also linear in @xmath80 . indeed ,
[ figs1enth3 ] is consistent with a linear behavior @xmath94 at sufficiently large @xmath13 and low @xmath80 .
heisenberg chain as obtained from ed and qmc for different chain lengths .
the corresponding values of the entropy are @xmath95 , @xmath96 , @xmath97 , @xmath98 , @xmath99 , @xmath100 ( bottom to top ) . ] in this section we will discuss the entropy of the @xmath1 heisenberg chain in more detail . first , we show results for the value of the entropy per site @xmath84 as a function of @xmath12 and @xmath80 in fig .
[ figs1isent ] .
the ed curves have been obtained by computing the entropy @xmath101 on a mesh in the @xmath12-@xmath80-plane and determining the constant entropy curves , _ i.e. _ , the isentropes from this data .
this leads to some discretization artifacts at the cusps in the the @xmath102 curves ( the ed curves at the lowest temperature ) in fig .
[ figs1isent ] .
conversely , the qmc had problems to resolve the large low - temperature entropy around the saturation field @xmath103 .
accordingly , the @xmath77 sse data points with @xmath102 are missing at @xmath104 and @xmath105 . from our qmc simulations
we can just conclude that for an @xmath77 chain these two points are in the region @xmath106 and @xmath107 for @xmath108 and @xmath109 , respectively . the ed curves with @xmath102 in fig .
[ figs1isent ] have marked finite - size wiggles .
also at higher temperatures ( entropies ) , finite - size effects can be observed .
there are , however , two regions where finite - size effects are evidently small .
one is the low - temperature regime for fields @xmath12 smaller than the haldane gap @xcite @xmath110 also in the gapped high - field regime above the saturation field @xmath111 one observes essentially no finite - size effects at any temperature .
this renders ed the method of choice at high fields @xmath112 , in particular at low temperatures , while qmc is preferable otherwise because bigger system sizes @xmath13 are accessible .
one can read off from fig .
[ figs1isent ] pronounced temperature changes in two regimes .
firstly , one can obtain cooling by adiabatic demagnetization from a high magnetic field @xmath113 as one lowers the field to the saturation field . secondly , at low temperatures , one can also cool by adiabatic magnetization from @xmath17 to @xmath114 .
this demonstrates that one can use the gap - closing at a generic field - induced quantum phase transition for cooling purposes . in the regime @xmath115 ,
the spectrum is gapless ( compare fig .
[ figs1enth3 ] and the related discussion in section [ sec : test ] ) .
accordingly , one observes only small temperature changes induced by adiabatic ( de)magnetization in this field range .
heisenberg chain at zero magnetic field @xmath17 as a function of temperature .
note that the temperature axis is scaled as @xmath116 and the entropy axis logarithmically .
the dashed line is a fit to ( [ eqentgap ] ) with @xmath117 and @xmath118 . ]
we now focus on the low - field regime .
firstly , we consider the behavior of the entropy at @xmath17 which is shown in fig .
[ figs1enth0 ] .
one observes that finite - size effects are small for @xmath119 : the @xmath82 curve is almost within the error margins of the @xmath77 qmc curve even at low temperatures .
this can be attributed to a finite correlation length @xmath120 which is evidently sufficiently smaller than @xmath121 for all temperatures at @xmath17 .
indeed , the correlation length of the @xmath1 heisenberg chain is known to be @xmath122 at @xmath17 and @xmath123 @xcite .
because of the presence of a gap @xmath124 at @xmath17 , we expect the entropy to be exponentially activated as a function of temperature .
accordingly , the low - temperature asymptotic behavior at @xmath17 should follow the form @xmath125 if we fix the gap to the known value @xmath118 @xcite , the only free parameter in the low - temperature asymptotic behavior ( [ eqentgap ] ) is the prefactor @xmath126
. a fit of the @xmath77 qmc results yields @xmath127 . the dashed line in fig . [ figs1enth0 ]
shows that the formula ( [ eqentgap ] ) describes the low - temperature regime quite well with the given parameters .
heisenberg chain at @xmath128 , corresponding to the haldane gap .
note the doubly logarithmic scale .
the dashed line shows the square - root behavior ( [ eqentsqrt ] ) with @xmath129 . ]
secondly , we consider a magnetic field exactly equal to the haldane gap @xmath128 .
[ figs1enthgap ] shows a doubly - logarithmic plot of the entropy @xmath101 as a function of temperature @xmath80 for this value of the magnetic field .
since we are now sitting at a field - induced quantum phase transition , we expect a large number of low - lying states and relatedly an infinite correlation length @xmath130 . on a finite system , these low - lying states will be pushed to higher energies . indeed , we observe pronounced finite - size effects at low temperatures in fig .
[ figs1enthgap ] . at a quantum phase transition in one dimension which preserves a @xmath131-symmetry , the entropy @xmath101 is expected @xcite to vary asymptotically as a square root of temperature : @xmath132 as before , the amplitude @xmath133 is the only free parameter .
a fit to the @xmath77 qmc data with @xmath134 yields @xmath135 . the dashed line in fig .
[ figs1enthgap ] shows that the @xmath77 qmc data is indeed consistent with the asymptotic square root ( [ eqentsqrt ] ) in the range @xmath136 .
however , because of finite - size effects , in this case we really need to go to @xmath77 to recover the asymptotic behavior .
furthermore , we need to restrict to @xmath137 since one can see from fig .
[ figs1enthgap ] that a pure power law is no longer a good description for @xmath138 .
comparison of ( [ eqentgap ] ) and ( [ eqentsqrt ] ) shows that one can reach exponentially small temperatures @xmath139 at @xmath114 by adiabatic magnetization of a spin-@xmath4 heisenberg chain from an initial temperature @xmath140 at @xmath17 . for an initial temperature @xmath141
, the final temperature @xmath139 can be quantitatively estimated from the following combination of ( [ eqentgap ] ) and ( [ eqentsqrt ] ) @xmath142 with the parameters @xmath143 , @xmath126 , and @xmath133 given above .
we will now turn to the high - field region and consider in particular scaling with the spin quantum number @xmath0 close to the saturation field .
heisenberg chains as obtained by ed for different @xmath0 .
note that a truncation of the energy spectrum has been used for @xmath144 .
the values of the entropy per site are @xmath102 , @xmath145 , @xmath97 , @xmath98 , and @xmath99 ( bottom to top ) . ]
it is straightforward to compute the excitation energy @xmath146 of a single flipped spin propagating above a ferromagnetically polarized background ( see , e.g. , @xcite ) : @xmath147 the minimum of this one - magnon dispersion is located at @xmath148 .
the saturation field @xmath149 of the spin-@xmath0 heisenberg chain ( [ eq : ham ] ) is given by the one - magnon instability @xcite .
accordingly , it can be determined by inserting the condition @xmath150 into ( [ eq : singlemagnon ] ) : @xmath151 the fact that the single - particle energy ( [ eq : singlemagnon ] ) scales linearly with @xmath0 suggests to scale all energies close to the saturation field and at low temperatures linearly with @xmath0 . in order to test this scenario ,
we have first computed the isentropes for a fixed chain length @xmath85 and @xmath152 .
the @xmath153 curves are obtained by a full diagonalization of the hamiltonian @xcite while we have performed additional ed computations for @xmath1 , @xmath154 , and @xmath155 using the truncation procedure described in section [ sec : methods ] .
[ figmad ] shows the resulting isentropes for @xmath102 , @xmath145 , @xmath97 , @xmath98 , and @xmath99 close to the saturation field ( [ hsat ] ) .
we observe that at low temperatures , scaling both @xmath12 and @xmath80 by @xmath0 leads to a nice collapse of the isentropes around and in particular above the saturation field .
( left column ) and magnetic susceptibility per site @xmath156 ( right column ) as obtained by ed for different @xmath0 .
note that a truncation of the energy spectrum has been used for @xmath144 .
the values of the magnetic field are chosen somewhat above the saturation field ( top row ) , exactly at the the saturation field ( middle row ) , and somewhat below the saturation field ( bottom row).,title="fig : " ] ( left column ) and magnetic susceptibility per site @xmath156 ( right column ) as obtained by ed for different @xmath0 .
note that a truncation of the energy spectrum has been used for @xmath144 .
the values of the magnetic field are chosen somewhat above the saturation field ( top row ) , exactly at the the saturation field ( middle row ) , and somewhat below the saturation field ( bottom row).,title="fig : " ] + ( left column ) and magnetic susceptibility per site @xmath156 ( right column ) as obtained by ed for different @xmath0 .
note that a truncation of the energy spectrum has been used for @xmath144 .
the values of the magnetic field are chosen somewhat above the saturation field ( top row ) , exactly at the the saturation field ( middle row ) , and somewhat below the saturation field ( bottom row).,title="fig : " ] ( left column ) and magnetic susceptibility per site @xmath156 ( right column ) as obtained by ed for different @xmath0 .
note that a truncation of the energy spectrum has been used for @xmath144 .
the values of the magnetic field are chosen somewhat above the saturation field ( top row ) , exactly at the the saturation field ( middle row ) , and somewhat below the saturation field ( bottom row).,title="fig : " ] + ( left column ) and magnetic susceptibility per site @xmath156 ( right column ) as obtained by ed for different @xmath0 .
note that a truncation of the energy spectrum has been used for @xmath144 .
the values of the magnetic field are chosen somewhat above the saturation field ( top row ) , exactly at the the saturation field ( middle row ) , and somewhat below the saturation field ( bottom row).,title="fig : " ] ( left column ) and magnetic susceptibility per site @xmath156 ( right column ) as obtained by ed for different @xmath0 .
note that a truncation of the energy spectrum has been used for @xmath144 .
the values of the magnetic field are chosen somewhat above the saturation field ( top row ) , exactly at the the saturation field ( middle row ) , and somewhat below the saturation field ( bottom row).,title="fig : " ] further details can be read off from the temperature scans at a fixed magnetic field shown in fig . [
fig : scalecchi ] . in this figure , we have chosen to show the specific heat @xmath91 as a representative for energy - related quantities is equivalent to @xmath101 since in the present case @xmath157 for @xmath158 . ] and the magnetic susceptibility @xmath159 as a representative for magnetic quantities .
the top row of fig .
[ fig : scalecchi ] corresponds to a magnetic field @xmath160 .
here we observe a nice collapse with @xmath0 for @xmath161 .
the middle row of fig .
[ fig : scalecchi ] corresponds to a magnetic field exactly equal to the saturation field .
here we observe a good scaling collapse for @xmath162 . finally , the bottom row of fig .
[ fig : scalecchi ] corresponds to a magnetic field @xmath163 . in this case , one expects the scaling region to be pushed to even lower temperatures .
however , for @xmath164 the finite size @xmath85 leads to artifacts in the low - temperature behavior . therefore it is difficult to make definite statements for this case . to summarize this section ,
we have provided evidence of _ linear _ scaling with @xmath0 for thermodynamic quantities of spin-@xmath0 heisenberg chains close to the saturation field .
note that this is very different from the _ quadratic _ scaling , e.g. of @xmath80 with @xmath165 @xcite or with @xmath166 @xcite , which is needed to approach the classical limit .
in this paper we have illustrated the numerical computation of thermodynamic quantities and in particular the entropy for spin-@xmath0 heisenberg chains . from a technical point of view
, we have described in section [ sec : ed ] how to perform a reliable computation of a large number of low - lying states using the lanczos method and in section [ sec : qmc ] how to compute the entropy directly by a qmc simulation . in section [ sec : s1chain ] we have then focused on the @xmath1 heisenberg chain and shown in particular that one can cool with an adiabatic magnetization process during which the haldane gap @xmath124 is closed .
many previous investigations ( e.g. , @xcite ) have focused on the saturation field .
the reason for this is that the saturation field is a field - induced quantum phase transition at a known value of the magnetic field which also gives rise to technical simplifications .
however , the scenario of quantum phase transitions @xcite is universal and not restricted to the saturation field . indeed , fig .
[ figs1enthgap ] is consistent with the same universal square - root behavior of the entropy @xmath101 at @xmath167 in the spin-1 heisenberg chain as observed previously for @xmath153 chains exactly at the saturation field @xcite .
this may also be important from an experimental point of view since a possible spin gap may be accessible by laboratory magnetic fields even if the saturation field is out of reach . in fact , cooling by adiabatic magnetization when closing a spin gap has presumably been indirectly observed in pulse - field magnetization experiments on srcu@xmath7(bo@xmath168)@xmath7 @xcite . in section [ sec : sscal ] we then moved to the saturation field and investigated scaling with the spin quantum number @xmath0 . in contradistinction to the classical scaling regime where temperature should scale quadratically with @xmath0 @xcite , there is a quantum scaling regime around the saturation field where one observes a collapse using the linearly scaled parameters @xmath169 and @xmath170 .
also in higher dimensions the one - magnon dispersion typically scales with @xmath0 . by the same arguments as in section [ sec : sscal ] ,
we therefore expect linear scaling with @xmath0 at any continuous transition to saturation .
this expectation could be tested numerically with the methods of the present paper .
we are grateful to m.e .
zhitomirsky for useful discussions .
acknowledges support by the deutsche forschungsgemeinschaft through a heisenberg fellowship ( project ho 2325/4 - 1 ) .
s.w . acknowledges hlrs stuttgart and nic jlich for allocation of computing time .
some of our numerical simulations were based on the alps libraries @xcite .
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mater . , 2007 , * 310 * , 1187 . | we compute the entropy of antiferromagnetic quantum spin-@xmath0 chains in an external magnetic field using exact diagonalization and quantum monte carlo simulations . the magnetocaloric effect , _
i.e. _ , temperature variations during adiabatic field changes , can be derived from the isentropes .
first , we focus on the example of the spin-@xmath1 chain and show that one can cool by closing the haldane gap with a magnetic field .
we then move to quantum spin-@xmath0 chains and demonstrate linear scaling with @xmath0 close to the saturation field . in passing ,
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quantum spin chains , magnetocaloric effect , entropy , exact diagonalization , quantum monte carlo 75.10.pq ; 75.30.sg ; 75.50.ee ; 02.70.-c | 0907.3736 |
as expected , the most interesting combinatorial structures are those bearing some kind of symmetry and/or regularity .
in fact , in general , high symmetry imply high regularity , but the converse does not necessarily holds .
moreover , symmetric structures suggest definitions of new structures obtained , either by relaxing the conditions of symmetry , or having the same regularity properties as the original ones . in turn
, the latter can give rise to new definitions by relaxing the mentioned symmetry conditions .
in graph theory , a good example of the above are the distance - transitive graphs , with automorphism group having orbits constituted by all vertices at a given distance . attending to their symmetry , related concepts are the vertex - symmetric , symmetric , and orbit polynomial graphs @xcite . besides , concerning regularity , distance - transitive graphs can be generalized to distance - regular graphs @xcite , distance - polynomial graphs , and degree - regular graphs @xcite . in this paper
, we introduce the concept of a quotient - polynomial graph , which could be thought of as the regular counterpart of orbit polynomial graphs . in a quotient - polynomial graph
, every vertex @xmath0 induces the same regular partition around @xmath0 , with the additional condition that all vertices of each cell are equidistant from @xmath0 .
some properties and characterizations of such graphs are studied .
for instance , all quotient - polynomial graphs are walk - regular and distance - polynomial .
our study allows us to provide a characterization of those distance - polynomial and vertex - transitive graphs which are distance - regular .
also , we show that every quotient - polynomial graph generates a ( symmetric ) association scheme . throughout this paper , @xmath1 denotes a ( connected ) graph with vertex set @xmath2 , edge set @xmath3 , and diameter @xmath4 . for every @xmath5 and @xmath6 , let @xmath7 denote the set of vertices at distance @xmath8 from @xmath0 , with @xmath9 , and let @xmath10 be the characteristic ( @xmath0-th unitary ) vector of @xmath11 .
eccentricity _ of @xmath0 , denoted by @xmath12 , is the maximum distance between @xmath0 and any other vertex @xmath13 of @xmath1 .
let @xmath14 be the @xmath8-th distance matrix , so that @xmath15 is the adjacency matrix of @xmath1 , with spectrum @xmath16 , where @xmath17 , and the superscripts @xmath18 stand for the multiplicities .
let @xmath19 , @xmath20 be the minimal idempotents representing the orthogonal projections on the @xmath21-eigenspaces .
let @xmath22 $ ] be the _ adjacency algebra _ of @xmath1 , that is , the algebra of all polynomials in @xmath23 with real coefficients .
following fiol , garriga and yebra @xcite , the @xmath24-entry of @xmath19 is referred to as the _ crossed @xmath25-@xmath26local multiplicity _ of the eigenvalue @xmath21 , and it is denoted by @xmath27 . in particular , for a regular graph on @xmath28 vertices , @xmath29 and , hence , @xmath30 for every @xmath31 . since @xmath32 , the number of walks of length @xmath33 between two vertices @xmath34 is @xmath35 in particular , the _
@xmath36-@xmath26local multiplicities _ are @xmath37 , @xmath38 , and satisfy @xmath39 and @xmath40 , @xmath38 .
a graph @xmath1 with diameter @xmath4 is called _ @xmath41-punctually walk - regular _ , for some @xmath42 , when the number of walks @xmath43 for any pair of vertices @xmath34 at distance @xmath41 only depends on @xmath33 . from the above , this means that the crossed local multiplicities @xmath27 only depend on @xmath21 and we write it as @xmath44 ( see dalf , van dam , fiol , garriga , and gorissen @xcite for more details ) .
notice that , in particular , a @xmath45-punctually walk - regular graph is the same as a walk - regular graph , a concept introduced by godsil and mckay @xcite .
a partition @xmath46 of the vertex set @xmath2 is called _ regular _ ( or _ equitable _ ) whenever for any @xmath47 , the _ intersection numbers _
@xmath48 , where @xmath49 , do not depend on the vertex @xmath0 but only on the subsets ( _ classes _ or _ cells _ ) @xmath50 and @xmath51 . in this case ,
such numbers are simply written as @xmath52 , and the @xmath53 matrix @xmath54 is referred to as the _ quotient matrix _ of @xmath23 with respect to @xmath55 .
the _ characteristic matrix _ of ( any ) partition @xmath55 is the @xmath56 matrix @xmath57 whose @xmath8-th column is the characteristic vector of @xmath50 , that is , @xmath58 if @xmath49 , and @xmath59 otherwise
. in terms of such a matrix , it is known that @xmath55 is regular if and only if there exists an @xmath53 matrix @xmath60 such that @xmath61 moreover , in this case , @xmath62 , the quotient matrix of @xmath23 with respect to @xmath55 .
then , using this it easily follows that all the eigenvalues of @xmath63 are also eigenvalues of @xmath23 . for more details , see godsil @xcite .
in this section we introduce several types of partitions bearing some regularity properties with respect to a given vertex @xmath0 .
we begin by considering those partitions where all vertices of the same class are equidistant from @xmath0 .
let @xmath1 have diameter @xmath4 . given a vertex @xmath0 , a _
( @xmath0-)distance - faithful _ partition around @xmath0 , denoted by @xmath64 , is a partition @xmath65 , with @xmath66 such that @xmath67 and , for @xmath68 , every pair of vertices @xmath69 are at the same distance from @xmath0 : @xmath70 .
thus , in particular , @xmath64 is a _ distance partition around _
@xmath0 whenever @xmath69 if and only if @xmath71 . in other words , @xmath72 for every @xmath73 and , hence , @xmath74 . for every pair of vertices @xmath31 we consider the vectors of crossed local multiplicities @xmath75 and numbers of @xmath33-walks for @xmath76 between @xmath0 and @xmath13 @xmath77 the following result is an immediate consequence of ( see , for instance , @xcite )
. given some vertices @xmath78 , we have @xmath79 if and only if @xmath80 .
then , we can define the two following equivalent concepts : a partition @xmath81 is _ @xmath36-@xmath26walk - regular _ @xmath82or _ @xmath36-@xmath26-spectrum - regular_@xmath26 around a vertex @xmath5 if , for every @xmath73 the set @xmath83 is constituted by all vertices @xmath84 with the same vector @xmath85 @xmath82or , equivalently , @xmath86 .
now we will prove that , if a partition is both regular and @xmath0-distance - faithful , then it is also @xmath0-walk - regular . before that , we have the following straightforward lemma .
[ wr->df ] every walk - regular partition @xmath64 around a vertex @xmath5 is also distance - faithful around the same vertex . by contradiction , assume that @xmath87 and @xmath88 .
then , we would have @xmath89 but @xmath90 , against the hypothesis of @xmath0-walk - regularity .
let @xmath91 be a @xmath0-distance - faithful and regular partition .
then @xmath92 defines a @xmath0-walk - regular partition @xmath93 with @xmath94 ( by the union of some sets @xmath50 , if necessary ) .
to prove that the number of walks @xmath95 , with @xmath96 , only depend on @xmath8 and @xmath33 , we use induction on @xmath33 .
the result is clear for @xmath97 .
now suppose that the result holds for some @xmath98 .
then , for a given @xmath96 , @xmath99 and , hence , @xmath100 does not depend on @xmath13 . finally ,
if there are sets @xmath101 with vertices @xmath13 having the same vector @xmath85 , we consider their union @xmath102 to form the claimed @xmath0-walk - regular partition .
to prove the converse , we need an extra hypothesis , which in fact leads to a stronger result in terms of the new concept defined below . given a vertex @xmath0 of a graph @xmath103 , the so - called _
@xmath0-local spectrum _ is constituted by those eigenvalues @xmath104 of @xmath103 such that @xmath105 ( that is , with nonzero @xmath0-local multiplicity @xmath106 ) .
moreover , these are referred to as the _ @xmath0-local eigenvalues_. let us consider the vector space @xmath107 spanned by the vectors @xmath108 , @xmath76 .
then , it is known that @xmath107 has dimension @xmath109 and basis @xmath110 ; see e.g. @xcite .
[ defi1 ] let @xmath0 be a vertex with @xmath109 distinct local eigenvalues .
let @xmath93 be a @xmath0-walk - regular partition , with @xmath111 being the characteristic vector of @xmath83 @xmath82note that @xmath112@xmath26 .
then @xmath64 is said to be _ quotient - polynomial _
whenever @xmath113 for every @xmath114 . in the following
result the walk - regular partitions that are quotient - polynomial ( and regular ) are characterized .
[ ineq&equ ] let @xmath0 be a vertex with @xmath109 distinct local eigenvalues .
let @xmath93 be a @xmath0-walk - regular partition .
then , @xmath115 with equality if and only if @xmath64 is a quotient - polynomial partition .
moreover , in this case @xmath64 is also regular .
for @xmath73 , let @xmath116 the common value of the number of @xmath33-walks from @xmath0 to every @xmath117 , and let us consider the vector space @xmath118 .
then , as @xmath119 we have that @xmath120 and , hence , @xmath121 which proves .
of course , the same conclusion can be reached by considering the common value @xmath122 of the crossed local multiplicities @xmath27 , for every @xmath117 .
then , @xmath123 ( only @xmath109 of the above equations are not trivially null ) .
if @xmath124 , we have that @xmath125 , that is , every vector @xmath111 is a linear combination of the vectors @xmath108 for @xmath126 , and @xmath64 is a quotient - polynomial partition .
conversely , if @xmath64 is quotient - polynomial , we have @xmath127 .
hence , @xmath128 which , together with ( @xmath64 is also walk - regular ) , leads to @xmath124 .
in fact , the constants of the above linear combinations , which are the coefficients of the polynomials @xmath129 , can be computed in the following way : the @xmath130 first equations in are , in matrix form , @xmath131 but the coefficient matrix @xmath132 with entries @xmath133 , for @xmath134 , is a change - of - basis matrix and , hence , it is invertible . as a consequence , for every @xmath135 , the coefficients @xmath136 , @xmath136 , , @xmath136 of @xmath137 correspond to the @xmath8-th row of @xmath138 . finally , to prove that @xmath64 is regular ,
let us choose one vertex @xmath139 in each @xmath83 , @xmath135 and consider the @xmath140 matrices @xmath63 and @xmath141 , with entries @xmath142 and @xmath143 , @xmath144 , respectively .
then , the @xmath130 equations of can be written as @xmath145 hence , the entries @xmath146 of the matrix @xmath147 do not depend on the chosen vertices @xmath148 , and the partition is regular with quotient matrix @xmath54 . all the above result can be summarized in the followin theorem .
let @xmath0 be a vertex of a graph @xmath1 .
let @xmath64 be a partition with @xmath130 classes around @xmath0 having @xmath109 distinct local eigenvalues @xmath149 .
then , the following assertions are equivalent : * the partition @xmath64 is quotient - polynomial . *
the partition @xmath64 is @xmath0-walk - regular with @xmath124 .
* there exist polynomials @xmath137 with @xmath150 , @xmath73 , such that @xmath151 .
* @xmath125 with basis @xmath152 . by theorem [ ineq&equ ] , we only need to prove @xmath153 : from @xmath154 we have that @xmath155 and hence @xmath128 . also , as @xmath156 , where @xmath157 , we get @xmath158 , where the left - hand side is a polynomial with degree at most @xmath130 .
moreover , @xmath159 , with @xmath160 , is the polynomial of minimum degree satisfying @xmath161 ( see @xcite ) .
therefore , @xmath162 and , hence , @xmath124 , @xmath125 , and @xmath163 is a basis .
in this section we study the graphs having the same ( i.e. with the same parameters ) quotient - polynomial partition around each of their vertices . with this aim ,
we now follow a global approach .
let @xmath1 be a graph with vertex set @xmath2 , @xmath164 distinct eigenvalues , and adjacency algebra @xmath165 .
then a partition @xmath166 of @xmath167 is called _ walk - regular _ whenever each @xmath168 is the set with elements @xmath169 having identical vector @xmath170 ( or @xmath85 ) .
so , from lemma [ wr->df ] , all pairs of vertices in a given @xmath168 are at the same distance , and we assume that the pairs in @xmath171 are of the form @xmath172 ( distance zero ) .
let @xmath173 , @xmath73 , the @xmath174 matrices , indexed by the vertices of @xmath1 , representing the equivalence classes @xmath168 , that is , @xmath175 let @xmath176 be an equivalence class with elements @xmath169 satisfying @xmath177 . then , @xmath178 if and only if @xmath1 is @xmath41-punctually walk - regular and , in particular , @xmath179 if and only if @xmath1 is walk - regular . from these matrices we can now define our main concept : a graph @xmath1 , with walk - regular partition @xmath166 and adjacency algebra @xmath180 , is _ quotient - polynomial _
if @xmath181 for every @xmath73 . thus , @xmath1 is quotient - polynomial if and only if there exist polynomials @xmath137 , with @xmath182 , such that @xmath183 , @xmath73 ( this inspired our definition ) .
in fact , the following result shows that this only happens when @xmath184 .
we omit its proof since it goes along the same lines of reasoning as that of theorem [ ineq&equ ] .
[ ineq&equ ] let @xmath1 be a graph as above .
let @xmath185 be a walk - regular partition .
then , @xmath186 and equality occurs if and only if @xmath1 is quotient - polynomial . since @xmath187 , the all-1 matrix ,
the sum polynomial @xmath188 is the hoffman polynomial satisfying @xmath189 .
therefore , a quotient - polynomial graph is connected and regular ( see hoffman @xcite ) .
moreover , the same reasoning used in @xcite[th .
2.4 ] to prove that every orbit polynomial graph is vertex transitive , shows that every quotient - polynomial graph is walk - regular , that is , @xmath179 . indeed , if @xmath190 , the equality @xmath191 ( @xmath180 is a commutative algebra ) leads to a contradicion because @xmath1 is connected .
thus , for any vertex @xmath0 , the _ induced partition _
@xmath64 of @xmath2 , with characteristic vectors @xmath10 , @xmath192 , , @xmath193 is quotient - polynomial since @xmath194 then , we can summarize all the above results in the ` global analogue ' of theorem [ ineq&equ ] .
[ main - theo ] let @xmath1 be a graph with vertex set @xmath2 , and @xmath164 distinct eigenvalues .
let @xmath195 be a partition of @xmath167 with @xmath196 .
then , the following assertions are equivalent : * @xmath1 is a quotient - polynomial graph . *
the partition @xmath197 is walk - regular with @xmath184 .
* there exist polynomials @xmath137 with @xmath150 , @xmath73 , such that @xmath198 . * for every vertex @xmath0 , the induced partition @xmath64 of @xmath2 is quotient - polynomial with the same polynomials @xmath137 .
* @xmath199 with basis @xmath200.@xmath201 ( notice that the first equality in @xmath202 implies @xmath184 . )
let us consider the following example of quotient - polynomial graph : the circulant graph @xmath203 has vertices @xmath204 and vertex @xmath0 is adjacent to vertices @xmath205 and @xmath206 .
then , @xmath1 is a 4-regular vertex - transitive graph with diameter @xmath207 , and spectrum ( with numbers rounded to three decimals ) @xmath208 .,title="fig : " ] -2.5 cm as shown in fig .
[ fig1 ] , the walk - regular partition around vertex @xmath45 has clases @xmath209 , @xmath210 , @xmath211 , @xmath212 , and @xmath213 .
( the corresponding intersection diagram is shown in fig .
2 . ) then the matrices @xmath132 and @xmath141 of numbers of walks from @xmath45 to a vertex of @xmath83 are @xmath214 then , from the inverse of @xmath132 we obtain the quotient polynomials : @xmath215 whereas the transpose of the intersection matrix turns out to be @xmath216 as in the case of distance - regular graph , the entries @xmath217 ( shown also in fig .
2 ) can also be calculated by using the formulas @xmath218 where we use the scalar product @xmath219 note that , since @xmath220 for @xmath221 , the quotient polynomials @xmath137 are orthogonal with respect to such a product .
( 1,3 ) circle [ radius=0.6 ] ; at ( 0.4,3.7 ) @xmath222 ; at ( 1.75,3.25 ) @xmath223 ; ( 3.5,3 ) circle [ radius=0.6 ] ; at ( 2.9,3.7 ) @xmath224 ; at ( 2.75,3.25 ) @xmath225 ; at ( 4,3.8 ) @xmath225 ; at ( 4,2.2 ) @xmath226 ; ( 5.5,1 ) circle [ radius=0.6 ] ; at ( 4.75,0.5 ) @xmath227 ; at ( 4.75,1.45 ) @xmath226 ; at ( 5.65,1.85 ) @xmath225 ; at ( 6.25,1.45 ) @xmath225 ; at ( 5.5,0.20 ) @xmath228 ; ( 5.5,5 ) circle [ radius=0.6 ] ; at ( 4.75,5.55 ) @xmath229 ; at ( 4.75,4.5 ) @xmath225 ; at ( 5.65,4.15 ) @xmath225 ; at ( 6.25,4.5 ) @xmath225 ; at ( 5.5,5.85 ) @xmath225 ; ( 7.5,3 ) circle [ radius=0.6 ] ; at ( 6.6,3 ) @xmath230 ; at ( 7,3.8 ) @xmath225 ; at ( 7,2.2 ) @xmath225 ; at ( 8.3,3 ) @xmath226 ; ( 1.6,3)(2.9,3 ) ; ( 3.93,2.57)(5.07,1.43 ) ; ( 3.93,3.43)(5.07,4.57 ) ; ( 5.5,1.6)(5.5,4.4 ) ; ( 5.93,1.43)(7.07,2.57 ) ; ( 5.93,4.57)(7.07,3.43 ) ; at ( 1,3 ) @xmath225 ; at ( 3.5,3 ) @xmath223 ; at ( 5.5,1 ) @xmath223 ; at ( 5.5,5 ) @xmath223 ; at ( 7.5,3 ) @xmath223 ; [ fig2 ]
in this section we study the relationships of quotient - polynomial graphs with other known combinatorial structures .
delsarte @xcite proved that a graph @xmath1 with @xmath164 distinct eigenvalues is distance - regular if and only if , for every @xmath38 , @xmath231 for some polynomial @xmath137 of degree @xmath8 ; see also weischel @xcite . then , as @xmath232 , the following result is clear .
inspired by the above characterization of distance - regular graphs , weichsel @xcite defined a graph @xmath1 with diameter @xmath4 to be _ distance - polynomial _ if @xmath235 .
( that is @xmath231 for every @xmath6 with no condition on the degrees of the polynomials @xmath137 . )
then any distance - regular graph is also distance - polynomial , but the converse does not hold .
for instance , any regular graph with diameter two is easily proved to be distance - polynomial ( but not necessarily strongly regular ) .
the next result gives a condition for having the equivalence .
since every distance - regular graph is distance - polynomial and has diameter @xmath236 , necessity is clear . to prove sufficiency ,
assume that @xmath1 is distance - polynomial with @xmath236 .
then @xmath237 which , as it was proved in @xcite , implies that @xmath1 is distance - regular .
let @xmath1 have the walk - regular partition @xmath238 with corresponding matrices @xmath239 for some polynomials @xmath240 , @xmath20 .
then , the polynomials @xmath241 with @xmath242 , clearly satisfy @xmath243 for @xmath6 , and the result follows .
however , the converse result does not hold , even if we require @xmath1 to have the same equitable walk - regular partition around each of its vertices .
a counterexample is the so - called ` chordal ring ' @xmath245 or prism @xmath246 shown in fig .
this ( bipartite ) graph has @xmath247 and it has the walk - regular ( and regular ) partition with intersection diagram shown in fig .
[ fig4 ] .
cccccccccccc 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 + 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 + 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 + 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 + 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 + 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 + 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 + 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 + 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 + 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 + 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 + 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 the definition of orbit polynomial graph @xmath1 , due to beezer @xcite is similar to that of quotient polynomial graph , but now the classes @xmath254 of the partition of @xmath167 are the orbits of the action of the automorphism group @xmath255 on such a set .
namely @xmath169 and @xmath256 are in the same equivalence class @xmath168 if and only if there is some @xmath257 such that @xmath258 .
then , the @xmath174 matrices , @xmath259 , representing the equivalence classes @xmath168 , are defined as in , and @xmath1 is said to be orbit polynomial whenever @xmath260 for all @xmath73 .
recall that a ( symmetric ) association scheme @xmath261 with @xmath262 classes can be defined as a set of @xmath262 graphs @xmath263 , @xmath264 , on the same vertex set @xmath2 , with adjacency matrices @xmath14 satisfying @xmath265 , with @xmath266 , and @xmath267 , for some integers @xmath268 , @xmath269 . then , following godsil [ 24 ] , we say that the graph @xmath270 is the @xmath8-th class of the scheme , and so we indistinctly use the words graph " or class " to mean the same thing .
( for more details , see @xcite . )
a @xmath262-class association scheme @xmath261 is said to be _ generated _ by one of its matrices @xmath14 ( or graph @xmath270 ) if it determines the other relations , that is , the powers @xmath271 and @xmath272 span the bose - mesner algebra of @xmath261 .
in particular , if @xmath270 is connected , then it generates the whole scheme if and only if it has @xmath164 distinct eigenvalues .
then , since a quotient polynomial graph @xmath1 is connected , theorem [ main - theo]@xmath202 yields : let @xmath1 be a graph with @xmath164 distinct eigenvalues .
then , @xmath1 is the connected generating graph of a @xmath262-class association scheme @xmath261 if and only if @xmath1 is a quotient - polynomial graph.@xmath201 if @xmath1 is quotient polynomial , it is connected and the matrices @xmath273 clearly satisfy the conditions for being an association scheme @xmath261 with @xmath262 classes .
conversely , if @xmath1 is a connected graph that generates a @xmath262-class association scheme @xmath261 , its adjacency matrix @xmath23 has @xmath164 distinct eigenvalues @xmath274 , and @xmath261 has bases @xmath275 and @xmath276 ( the minimal idempotents ) related by the equalities @xmath277 ( with @xmath278 being the eigenvalues of the scheme , @xmath279 ) .
thus , the polynomial @xmath280 $ ] satisfying @xmath281 satisfies @xmath243 for every @xmath38 , and @xmath1 is quotient polynomial by theorem [ main - theo ] .
* acknowledgments . *
the author acknowledges the useful comments and suggestions of e. garriga and j.l.a .
this research is supported by the _ ministerio de ciencia e innovacin _ ( spain ) and the _
european regional development fund _ under project mtm2011 - 28800-c02 - 01 , and the _ catalan research council _ under project 2009sgr1387 . | as a generalization of orbit - polynomial and distance - regular graphs , we introduce the concept of a quotient - polynomial graph . in these graphs
every vertex @xmath0 induces the same regular partition around @xmath0 , where all vertices of each cell are equidistant from @xmath0 .
some properties and characterizations of such graphs are studied .
for instance , all quotient - polynomial graphs are walk - regular and distance - polynomial . also , we show that every quotient - polynomial graph generates a ( symmetric ) association scheme .
_ mathematics subject classifications : _ 05e30 , 05c50 .
_ keywords : _ graph quotient ; distance - faithful partition ; walk - regular partition ; quotient - polynomial graph ; distance - regular graph ; eigenvalues ; orthogonal polynomials ; intersection numbers . | 1506.04688 |
by numerical study we find that the branch cut integrals conform to some simple scaling laws for small @xmath1 .
if we write the bc1 and bc2 integrals as @xmath197 and @xmath198 , then we find that for small @xmath1 and for @xmath199 , @xmath200 can be written in a scaling form @xmath201 but for bc2 a very different scaling law applies : @xmath202 here @xmath203 are dimensionless , universal " functions that govern the behavior of the branch cut contributions for small @xmath1 . for bc1 ,
the behavior that the scaling law gives is very simple : eq .
( [ ddvbc2note1 ] ) implies that @xmath204 , where @xmath205 is the laplace transform of the scaling function @xmath206 .
we might have expected this behavior from eq .
( [ p8_7e8_24 ] ) , from which we can read off the scaling function for @xmath128 .
in fact it appears from numerical studies that @xmath206 hardly changes as @xmath207 is varied , except for an overall scale factor ; that is , @xmath208 .
we find that the scaling function @xmath209 is peaked at @xmath210 .
so , the memory effect described above for @xmath128 persists for finite @xmath207 , but becomes smaller .
for @xmath187 the bc2 contribution , which we will describe now , becomes dominant over the bc1 one .
returning to eq .
( [ ddvbc2note2 ] ) , if we write the fourier transform of the scaling function as @xmath211 and consider its polar form @xmath212 , then we obtain @xmath213 this shows that bc2 contributes an oscillatory part to the solution , whose @xmath4
" decay is determined by the features of the scaling function @xmath214 .
a few more observations about @xmath215 ( obtained initially from numerical study ) reveal some crucial properties of the @xmath214 function : 1 ) @xmath216 ; 2 ) @xmath217 has a single maximum at @xmath218 , where @xmath172 is some constant of order unity ; 3 ) most important for the present discussion , for @xmath219 @xmath220 approaches @xmath221 , that is , @xmath222 , where @xmath223 is another real constant of order unity .
fact 3 ) implies that , for @xmath224 , @xmath225 .
that is , we conclude that at sufficiently short time ( actually for @xmath168 , so a relatively long time ) , @xmath226 as stated in the text . | within the lowest - order born approximation , we present an exact calculation of the time dynamics of the spin - boson model in the ohmic regime .
we observe non - markovian effects at zero temperature that scale with the system - bath coupling strength and cause qualitative changes in the evolution of coherence at intermediate times of order of the oscillation period .
these changes could significantly affect the performance of these systems as qubits .
in the biased case , we find a prompt loss of coherence at these intermediate times , whose decay rate is set by @xmath0 , where @xmath1 is the coupling strength to the environment .
these calculations indicate precision experimental tests that could confirm or refute the validity of the spin - boson model in a variety of systems .
novel solid state devices that can control spin degrees of freedom of individual electrons@xcite , or discrete quantum states in superconducting circuits@xcite , show promise in realizing the ideal of the completely controllable two - state quantum system , weakly coupled to its environment , that is the essential starting point for qubit operation in quantum computation . from a fundamental point of view
, these experimental successes also bring us close to embodying the ideal test of quantum coherence as envisioned by leggett many years ago@xcite , in which a simple quantum system is placed in a known initial state , is allowed to evolve for a definite time @xmath2 under the action of its own hamiltonian and under the influence of decoherence from the environment , and is then measured .
recent experiments , starting with @xcite , show that this ideal test can be implemented in practice .
the decay of quantum oscillations due to environmental decoherence is now@xcite sufficiently weak that some tens of coherent oscillations can be observed .
if quantum computation is to become a reality , it is believed@xcite that these systems will eventually need to achieve even lower levels of decoherence , such that thousands or tens of thousands of coherent oscillations could be observed .
this prospect of producing experiments with ultra - long coherence times in quantum two state systems offers a new challenge for theoretical modelling of decoherence . despite the many years of work@xcite following on leggett s initial proposals ,
there has never been a full , systematic analysis of the most popular description of these systems , the spin - boson model , in the limit of very weak coupling to the environment . in this letter
we provide an exact analysis of the weak coupling limit of the spin boson model for the ohmic heat bath , and in the low temperature limit . in this limit the born approximation ( to the self energy ) should become essentially exact , and we make no other approximations in our solutions in particular , no markov approximation is made .
as other workers have recently emphasized@xcite , understanding the details of the short - time dynamics of this model is especially crucial for the operation of these systems as qubits .
we find important , new , non - markovian effects in this regime . at lowest order in the born expansion of the self energy superoperator , the time dynamics of the model rigorously separates into a sum of strictly exponential pieces ( the usual
@xmath3 " and @xmath4 " decays of the bloch - redfield model ) plus two distinct non - exponential pieces that arise , technically speaking , from two different kinds of branch cuts in the laplace transform of the solution of the generalized master equation that we obtain .
these two contributions both have power - law forms at long times , @xmath5 , and thus formally dominate the exponentially - decaying parts .
but more interesting is that they both give new structure to the time evolution at intermediate times @xmath2 , @xmath6 ; this structure typically occurs for @xmath2 on the order of the oscillation period .
( here , @xmath7 is a high frequency cut - off of the bath modes , defining the very short time regime , @xmath8 , which is of no interest here . )
we can explain our results in the language of the double - well potential , where the two quantum states are left " and right " ( @xmath9/@xmath10 ) , the @xmath11 state is pure @xmath9 , and the system oscillates in time via tunneling from @xmath9 to @xmath10 .
the first branch - cut contribution is most important in the unbiased case ( @xmath9 and @xmath10 energies degenerate ) and it causes the system , starting immediately in the first quantum oscillation , to spend more time in the @xmath10 well , that is , the _ opposite _ well from the one the system is in initially . the second branch - cut contribution , present when the system is biased , adds to the amplitude of the coherent oscillation , but dies out after an intermediate time which scales like the inverse square root of the interaction strength @xmath1 with the bath .
this _ prompt loss of coherence _ , whose amplitude is proportional to @xmath1 , changes qualitatively the picture of the initial decay of coherence that is so important for discussions of fault - tolerant quantum computation@xcite .
we are interested in studying the time dependence of the system density matrix @xmath12 with a time - independent system hamiltonian , and in the presence of a fixed coupling to an environment .
an exact equation for @xmath13 the generalized master equation ( gme )
is@xcite @xmath14 here the kernel @xmath15 is the self energy superoperator , the system - bath hamiltonian is written @xmath16 , the liouvillian superoperator is defined by @xmath17 $ ] , @xmath18 , @xmath19 , @xmath20 is the temperature , and @xmath21 is the projection superoperator @xmath22 .
eq .
( [ p1_3e1_11_1 ] ) is written for the case @xmath23 , and the total initial state is taken to be of the form @xmath24 . since we are interested in the case of weak coupling to the bath , we will consider a systematic expansion in powers of this coupling @xmath25 in the self - energy operator @xmath15 .
retention of only the lowest order term in this expansion , giving the born approximation , is obtained@xcite by the replacement @xmath26 in eq .
( [ p1_3e1_11_2 ] ) .
we now proceed to solve the gme _ with no further approximations .
_ this distinguishes our work from previous efforts , in which various other approximations ( secular , rotating wave , markov , non - interacting blips " , short time ) are made ( see , e.g. , @xcite ) .
we will find that , in particular , avoidance of the markov approximation endows the solution with qualitatively new features .
we obtain our solution for the special case of a two - dimensional system hilbert space , and a system - bath coupling of a simple bilinear form , @xmath27 ( @xmath28 ( @xmath29 ) is an operator in the system ( bath ) space ) . in this case
the gme ( [ p1_3e1_11_1 ] ) in born approximation can be rewritten in an ordinary operator form : @xmath30-\int_0^tdt'i_\mu(t , t')\ , , \label{p2_2e2_61}\\ i_\mu(t , t')&=&i_{\mu 0}(t')+\sum_{\nu=1}^3i_{\mu\nu}(t')\langle\sigma_\nu(t - t')\rangle\ , , \label{p2_2e2_62}\\ i_{\mu\nu}(t')&=&{\rm re}~\{c(-t'){\rm tr}_s\sigma_\nu(-t')[\sigma_\mu , s]s(-t')\}\ , .\label{p2_2e2_63}\end{aligned}\ ] ] here @xmath31 are the pauli operators , @xmath32 , @xmath33 , the bath correlation function is @xmath34=c'(t)+ic''(t)$ ] , the time dependent operators are in the interaction picture , i.e. , @xmath35 , and @xmath36 and @xmath37 denote the real and imaginary parts of the bath correlator . without loss of generality
, we can take the system operators to be of the form @xmath38 and @xmath39 .
then the gme can be written in an explicit form @xmath40 . here
@xmath41 denotes the vector @xmath42 , convolution is denoted @xmath43 , and @xmath44 with @xmath45 , and kernels given by @xmath46 , @xmath47 , @xmath48 , @xmath49 , and @xmath50@xcite .
these equations can be solved in the laplace domain . defining the laplace
transform as @xmath51 , the solutions are , for the standard " initial conditions @xmath52 , @xmath53 we can give more explicit solutions if we specialize to the spin - boson model , for which @xmath54 and @xmath55 .
if the spectral function @xmath56 is defined as @xmath57 , then @xmath58 for most of the sequel we will consider the ohmic case , for which @xmath59 ( @xmath7 is an ultraviolet cutoff ) . in this case
eq .
( [ p7_2e48 ] ) becomes @xmath60 where @xmath61 is the derivative of the digamma function@xcite . for discussing the exact solution
it is instructive to understand the structure of the solution in a markov approximation .
this approximation is obtained by replacing all the kernels @xmath62 , @xmath63 , @xmath64 , @xmath65 , and @xmath66 by their values at @xmath67 . in this case
the solutions eqs .
( [ p2_11e2_543 ] ) are rational functions of @xmath68 .
then , if the poles of these rational functions are located at positions @xmath69 in the complex @xmath68 plane , with residues @xmath70 , then the inverse laplace transform can be written @xmath71 .
we indicate here that while the residues do depend on the label @xmath72 , the pole positions do not , as is suggested by the form of eqs .
( [ p2_11e2_543 ] ) .
as is well known@xcite , there are four poles : @xmath73 , @xmath74 , and @xmath75 .
the first pole describes the long - time asymptote of the solution ( stationary state ) , the second the purely exponential , @xmath3"-type decay ( relaxation ) , and the last two ( complex conjugate paired ) describe an exponentially decaying sinusoidal part , the @xmath4"-type decay of coherent oscillations .
the residues of these poles are , to lowest order in @xmath1 , @xmath76 , @xmath77 , @xmath78 , @xmath79 , @xmath80 , @xmath81 , @xmath82 , @xmath83 , and @xmath84 .
the pole positions are , again to lowest order in @xmath1 , given by @xmath85 , @xmath86 , and @xmath87 , @xmath88@xcite , @xmath89 and @xmath90 , where we have dropped terms of order @xmath91 and higher , c is the euler constant , and @xmath92 is the digamma function @xcite .
these expressions are straightforwardly derivable , and agree with the literature@xcite , except for the energy shift due to vacuum fluctuations , @xmath93 ( which contains in general @xmath94 and not @xmath95 ) .
we note that this markovian theory satisfies the expected fundamental relation @xmath96 ( korringa relation)@xcite ; also , to lowest order in @xmath1 , the asymptotic values of @xmath97 go to the boltzmann equilibrium distribution of the system , e.g. , @xmath98 , unlike , for example , the popular non - interacting blip " approximation@xcite .
we now return to the exact solution , examining it in detail at vanishing temperature @xmath99 . in this case
the laplace transform of @xmath100 in eq .
( [ ddv25oct2002 ] ) is @xmath101 where @xmath102@xcite .
there is an important feature of this correlation function that makes the markov solution qualitatively incomplete : while the sine integral si is analytic , the cosine integral ci(s ) behaves like @xmath103 for @xmath104@xcite .
this means that @xmath105 is nonanalytic at @xmath67 it has a branch point there .
thus , the exact solutions @xmath106 have extra analytic structure not present in the markov approximation , and the real - time dynamics @xmath107 has qualitatively different features in addition to the pole contributions we have just discussed . the @xmath67 branch point in @xmath105 leads the kernels @xmath108 , @xmath109 , and @xmath110 to have branch points at @xmath111 ; the kernels @xmath112 and @xmath113 have three branch points , at @xmath67 and @xmath111 .
thus , the solutions to the gme @xmath114 also have three branch points in the complex plane .
we find by numerical study that the exact solutions still have four poles as before , which , for small @xmath1 , have nearly ( but not exactly ) the same pole positions and residues as in the markov approximation .
thus , the structure of the solutions in the complex @xmath68 plane is as shown in fig .
[ fig1]a .
the locations of the branch cuts are chosen for computational convenience , as discussed shortly .
given this branch - cut structure , the inverse laplace transform ( the bromwich integral ) is evaluated by closing the contour as shown .
thus , the exact inverse laplace transform can be expressed as ( @xmath115 ) @xmath116 here @xmath117 and @xmath118 , with @xmath119 an infinitesimal positive real number .
that is , @xmath120 is an infinitesimal displacement perpendicular to the direction of branch cut @xmath121 . for the cut choices we have made , @xmath122 , @xmath123 , @xmath124 , @xmath125 , and @xmath126 .
the closed - contour integral in the expression can be written as a sum over the four poles , and so gives complex exponential contributions to the solution as in the markovian case .
the extra terms , the sum over the three branch cuts , are new and give qualitatively different features .
the contributions of the second and third branch cuts are complex conjugates of each other , so we will discuss them together as the
branch cut 2 " ( bc2 " for short ) contribution .
the first cut will be discussed as the bc1 " contribution .
the contribution of these cuts to the solution is independent of the detailed positioning of the branch cuts , so long as they are not moved across a pole ; the choice of the direction of bc1 is a computational convenience the apparently most natural choice of this cut direction , along the negative real axis , passes it essentially on top of the @xmath62 pole , making the evaluation of the branch - cut integral numerically inconvenient . as a check , we find that the results we discuss now are indeed independent of the cut direction . we have done a detailed study of these branch - cut contributions for @xmath127 .
for the unbiased spin - boson case , @xmath128 , an essentially analytic calculation can be done for all contributions . in this case
there is no bc2 contribution , @xmath129 .
the bc1 contribution can be obtained analytically to leading order in @xmath1 : @xmath130 , @xmath131\}.\label{p8_7e8_24}\ ] ] this function , plotted along with the pole contribution in fig .
[ fig1]b for the choice of parameters shown , has the following features : @xmath132 is negative for all @xmath2 , it is monotonically increasing , and its long - time behavior is @xmath133 .
also , @xmath134 .
let us survey , then , the peculiar features that this branch cut contribution introduces into the time response @xmath135 . visualizing the @xmath128 spin - boson model as a symmetric double
well system coupled to its environment , the bc1 piece being negative means that , if the system is initially in the left well , it will , in the course of coherently tunnelling back and forth , spend more time in the _ right _ well ! this effect becomes strongest at long time , much longer than @xmath4 , for in this regime the pole contributions are exponentially small , while the bc1 contribution decays like a power law .
experimentally it may be hard to see the effect in this regime ( on account of finite - temperature effects , for example ) , so it is important to note that this memory effect appears already at early times , indicating that already in the first couple of coherent oscillations , there will be an excess amplitude in the right - well excursions as compared with the left - well excursions , by an amount proportional to @xmath1 .
we believe , on the basis of a variety of evidence@xcite , that the born approximation should be reliable up to @xmath1 s of order @xmath136 ; thus , experiments that look at coherent oscillations accurately at the percent level ( which , it seems , will ultimately be necessary for performing quantum computation ) could readily see this bc1 effect .
we note several other interesting features of our solution for @xmath128 . taking into account the non - markovian effects
, we can do a more precise calculation of the pole positions and residues ( only poles 3 and 4 contribute ) .
we find , for @xmath99 , @xmath137 , where , as before @xmath138 , and the renormalization factor r is given by @xmath139 , with @xmath140 .
further , @xmath141 , with @xmath142 . these expressions are obtained as systematic expansions in the small parameters @xmath143 and @xmath144 , and they match a direct numerical evaluation of the pole positions very well up to @xmath1 s of a few percent . for the corresponding pole residues we find the simple result in leading order @xmath145 .
this would be impossible in a markovian theory , in which @xmath146 , so that @xmath147 would be exactly 1 to all orders in @xmath1 .
in fact this excess pole residue is exactly what is needed to cancel out the initial value of the bc1 contribution to @xmath135 .
we note that our results for the residues differ from the weak - coupling expressions in the literature@xcite ( we are not aware of prior reports on @xmath148 ) . for the biased model ( @xmath149 ) the bc2 contributions become nonzero ; we find that they give other peculiar non - exponential corrections to the solution @xmath135 , very different from the bc1 contribution .
we can do a nearly analytic evaluation of the bc2 contribution to eq .
( [ ddv1_6_03 ] ) : using eq .
( [ p2_11e2_543 ] ) and expanding to lowest order in @xmath1 , we find for the integrand of the sum of the @xmath150 and 3 terms of ( [ ddv1_6_03 ] ) , @xmath151 here @xmath152 , @xmath153 , where @xmath154 and @xmath155 are given by @xmath156 ( see eq . ( [ p2_11e2_544 ] ) ) and @xmath157 ( see eq . ( [ p2_11e2_545 ] ) ) . since @xmath158 for @xmath159 , it is reasonable to expect that @xmath160 will grow linearly as one passes onto the branch cut ; and , in fact , we find from numerical study that a good ansatz is @xmath161 , with @xmath162 being a weakly varying , real function of @xmath163 . with this , for @xmath164 of order @xmath165 , eq .
( [ p10_3e10_8 ] ) simplifies to @xmath166 we find that ( [ p10_4e10_13 ] ) should be valid for @xmath167 . using ( [ p10_4e10_13 ] )
we can do the branch cut integral , which gives ( for @xmath168 see appendix for an alternative approach ) , @xmath169 here the dimensionless constants @xmath170 .
since @xmath171 , these constants are independent of @xmath1 .
the last expression for @xmath172 comes from an evaluation of @xmath173 : it is directly related to the energy renormalization in the markov approximation , @xmath174 . in fig .
[ fig2 ] we show a direct numerical evaluation of @xmath175 .
one can see the decay of the oscillatory part , which is logarithmic according to eq .
( [ ddvbc2note4 ] ) . even though the decay is very non - exponential , it is reasonable to attempt to characterize this decay by a time scale .
eq .
( [ ddvbc2note4 ] ) obviously does not work at @xmath11 , since it is logarithmically divergent .
this is not surprising , since our calculation has neglected cutoff effects ( dependence on @xmath7 ) , so eq . ( [ ddvbc2note4 ] )
is not expected to be correct for @xmath8 . however , if we consider early " time to be the first half - period of the coherent oscillation , @xmath176 , then eq . ( [ ddvbc2note4 ] )
should be valid and we can use it to characterize the decay by determining the time @xmath177 at which @xmath175 decreases to half its early - time value , i.e. , @xmath178 .
we obtain @xmath179 surprisingly , @xmath180 depends non - analytically on @xmath1 .
this explains the effect that is evident in fig .
[ fig2 ] : for small @xmath1 , @xmath181 , that is , on the scale of @xmath4 , there is a very rapid loss of coherence as contributed by bc2 . this phenomenon may be called a _
prompt loss of coherence _ , as it would appear experimentally as a fast initial loss of coherence ( from 100% to @xmath182 , @xmath183 being some constant near unity ) , followed by a much slower , exponential decay of coherence on the regular @xmath4 time scale .
we give a few final remarks about the bc2 calculation .
the absolute size of the bc2 contribution reaches a maximum near the value of @xmath184 used in fig .
[ fig2 ] ; the relative size of this contribution continues to increase as @xmath185 increases , so that it eventually becomes much larger than the pole contribution ( but all contributions to @xmath135 go to zero as @xmath186 ) .
when @xmath187 , we find that , because of the prompt loss of coherence , there is a _
deficit _ in the total pole contribution , that is , @xmath188 . even in the absence of an explicit branch cut computation
, this deficit signals the prompt loss of coherence , in that it indicates that the exponentially decaying contributions to @xmath135 do no account for all the coherence near @xmath11 .
note that this is opposite to the unbiased case , where , as a result of the bc1 part , there is an _ excess _ pole contribution .
naturally , many more regimes could be studied using the present approach .
recently , there has been interest in varying both the system@xcite and bath@xcite initial conditions , as well as in varying the model of the bath density of states@xcite .
for all these circumstances , the systematic born expansion procedure we report here can be done .
it is clear on general grounds that the appearance of branch cut contributions will not be restricted to the ohmic model we have studied in detail here ; it is easy to show that for any spectral density of the form @xmath189 at low frequencies ( @xmath190 ) , @xmath191 will have a power - law dependence at long time , and thus @xmath105 will have a branch point at @xmath67 .
so , interesting non - exponential contributions to the dynamics are expected in all these cases .
our hope is that , using the present and further exact calculations of the weak - coupling behavior of the spin - boson model , a tool will be made available to permit precision experiments to test the validity of the model ( which , at present , is only phenomenologically justified ) in various physical situations of present interest in quantum information
. a fundamentally correct , experimentally verified theory of the system and its environment should ultimately be of great value in finding a satisfactory qubit for the construction of a quantum information processor .
dpdv is supported in part by the national security agency and the advanced research and development activity through army research office contract number daad19 - 01-c-0056 .
he thanks the institute for quantum information at cal tech ( supported by the national science foundation under grant .
no .
eia-0086038 ) for its hospitality during the initial stages of this work .
dl thanks the swiss nsf , nccr nanoscience , darpa and the aro .
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the constant term @xmath7 in @xmath195 cancels in the physical correlators @xmath110 and @xmath113 .
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small submicron particles are well - suited as fillers in non - conducting polymer matrices to obtain a conducting composite with a percolation threshold ( far ) below 1 % .
the low percolation threshold is due to the formation of airy aggregates of conducting particles , in which the particles are grown together by diffusion - limited cluster aggregation , creating a network with a fractal dimension around 1.7 .
@xcite these airy aggregates can be thought of as conducting spheres forming a 3-dimensional percolating network around the expected aggregate filling fraction 0.16 . as a consequence of the fractal structure within the aggregates ,
the filler fraction of the particles at the percolation point is much lower . even in case
the particles touch , the dc conductivity ( @xmath0 ) of these composites at high filling fractions turns out to be orders of magnitude lower than of the bulk material , as was recently illustrated for a particular crosslinked epoxy composite with filler particles of phthalcon-11 , @xcite co phthalocyanine crystallites of 100 nm size , and explained by purely structural arguments.@xcite when crystalline particles with a diameter of less than 10 nm instead of 100 nm are used , the small size of the particles may impose another important restriction to the maximal possible composite conductivity , which is due to the density of states ( dos ) involved in the dc conductivity through the network of particle contacts .
compared to larger crystallites , this dos can be strongly reduced by the charging energy.@xcite we show how those microscopic parameters , which govern the charge - transport process across many decades of length scales , can accurately and consistently be determined by ac ( alternating current ) dielectric spectroscopy from a few hz to infrared frequencies .
in particular we can address the parameters for mott variable - range hopping , for heterogeneity - induced enhanced ac response , for phonon- or photon - assisted nearest - neighbor hopping , and for the drude response of individual nanocrystals . due to these quantitative results
we can unambiguously determine also the role of the nanocrystal charging energy in limiting the hopping process .
we apply the method to antimony - doped tin - oxide ( ato ) crystallites of 7 nm diameter and to 100 nm sized crystallites of phthalcon-11 .
it turns out that in densely packed crystallites of ato , due to the strong influence of the charging energy on the dos , @xmath0 at room temperature is four orders of magnitude lower than the dc conductivity extrapolated from the drude plasma frequency ( @xmath1 ) of the crystallites - a result with obvious implications for the design of conducting composites .
the dielectric method is well suited for a variety of systems with restricted geometries , as we will illustrate by a short discussion of phase - change materials @xcite and granular oxides . @xcite .
for randomly placed conducting spheres in an insulating matrix , the relation between @xmath0 and the fraction @xmath2 of spheres is known from percolation theory .
@xciteabove the percolation threshold @xmath3 where @xmath4 is the percolation threshold , @xmath5 and @xmath6 is approximately equal to the conductivity of the spheres .
@xcite when the building blocks of the network are fractal aggregates instead of solid spheres , @xmath6 has to be replaced by the aggregate conductivity @xmath7 and depends on the particle conductivity and , via the non - linear relation @xmath8 with @xmath9 , on the real percolation threshold @xmath10 of the particles.@xcite the value of the exponent @xmath11 is related to the random - walk dimension and the fractal dimension @xmath12 , and is maximally @xmath13 .
this shows that on purely geometrical grounds for a network with @xmath14 , at the highest filling fraction of the aggregates @xmath0 will be three to four orders of magnitude lower than in the pure filler powder .
@xcite as remarked in the introduction , when nanosized particles are used as fillers , charging energies ( and quantum size effects ) impose a further important restriction to the maximal possible composite conductivity.@xcite this effect can be conveniently studied in densely packed powders of filler material by dielectric spectroscopy . in the ohmic regime ,
if there is a non - negligible density of states around the chemical potential , and the temperature @xmath15 is high enough that also the coulomb interaction can be neglected ( @xmath16 ) , @xmath0 will obey mott s equation for conduction via variable - range hopping ( vrh ) : @xcite @xmath17 \label{esm}\ ] ] with @xmath18 and @xmath19 where @xmath20 denotes the decay length of the electron density , @xmath21 the boltzmann constant , and @xmath22 the density of states relevant in the hopping process .
@xcite for randomly packed spheres of radius @xmath23 and spacing @xmath24 the localization length @xmath25 will be enlarged , @xcite and can be approximated by @xcite @xmath26 in the following we drop the tilde . below @xmath27 the @xmath15 dependence of the conductivity will be dominated by a soft coulomb gap , leading to so - called efros - shklovskii ( es ) vrh : @xcite @xmath28 .
\label{eses}\ ] ] in the es vrh model in the dilute limit of a large distance between the particles @xmath29 is given by @xmath30 with @xmath31 the electron charge , @xmath32 the vacuum dielectric constant , and @xmath33 the relative dielectric constant of the medium .
@xmath27 is given by @xmath34 $ ] and ( for @xmath35 ) the charging energy by @xmath36 .
for densely packed small particles , at high temperatures but still in the regime , where coulomb interactions are important ( @xmath37 ) , es vrh behavior will evolve into nearest - neighbor hopping at a temperature @xmath38 .
above @xmath38 the conduction is thermally activated with an activation energy @xmath39 of the order of the charging energy , and @xmath40 .
the experiments of yu _ et al . _
@xcite on thin films of highly monodispersed semiconducting nanocrystals of cdse of 6 nm diameter , slightly smaller than the ato crystallites discussed here , showed good agreement between the theoretical and experimental value of @xmath29 , @xmath27 and @xmath41 .
@xcite at sufficiently low frequencies the conductivity will be frequency independent and equal to its dc value , because the inhomogeneities are averaged out by the motion of the charge carriers .
the minimal length scale for homogeneity is referred to as @xmath42 , @xmath43 where @xmath44 is the density of the carriers involved in the hopping process at the border of the homogenous regime , and the onset frequency @xmath45 for the frequency dependence of @xmath46 is divided by @xmath47 , with @xmath48 the dimension of the system . @xcite at high enough frequencies , when during half a period of the oscillation of the applied field electrons can hop solely between nearest - neighbors , the major contribution to the conductivity will be due to tunnelling between localized states at neighboring sites ( the pair limit).@xcite this incoherent process can be either by phonon - assisted or photon - assisted hopping , where in the latter case the energy difference between the sites is supplied by photons instead of phonons .
@xcite the phonon - assisted contribution to the conductivity is given by @xmath49 with @xmath50 the decay length of the electronic state outside the conducting particles , @xmath22 the relevant dos at the fermi energy @xmath51 , and @xmath52 the phonon ` attempt ' frequency.@xcite this formula is valid when @xmath53 ; at higher @xmath54 , where the contribution of phonon - assisted hopping to @xmath46 becomes constant , photon - assisted processes usually take over , with a conductivity @xmath55 given by @xmath56 the energy @xmath57 in eq .
( [ esphon ] ) is in eq .
( [ esphot ] ) replaced by @xmath58 and the phonon attempt frequency @xmath52 by @xmath59 , with @xmath60 being the ` overlap ' pre - factor for the energy levels of two neighboring sites . in analogy with @xmath52 , @xmath61 can be interpreted as the attempt frequency for photon - assisted hopping .
equation ( [ esphot ] ) is only valid when @xmath62 . as in phonon - assisted hopping
, @xmath46 passes over into a plateau at high @xmath54 . at high frequencies ( for ato in the infrared regime ) the short period of the electromagnetic field will restrict the motion of the carriers to the nanocrystallite , and the dielectric response characterized by the complex relative dielectric constant @xmath63 $ ]
will be drude - like , with @xmath64 the drude plasma resonance frequency and @xmath65 the damping rate . in practice
the constant 1 has to be replaced by @xmath66 due to other contributions in this frequency regime , like the polarization of the ion cores .
@xcite the drude plasma frequency is related to the number of carriers per unit of volume @xmath67 and the effective mass @xmath68 as @xmath69 for damping rates comparable to the drude plasma frequency , the real plasma frequency ( where the dielectric constant becomes zero ) will be larger than @xmath1 .
@xmath70 is determined by the boundaries of the nanoparticle and additional ( ionized impurity ) scattering : @xmath71 where @xmath72 is the sum of the inverse size of the particle and the inverse phonon scattering length .
measurements were performed on sb - doped tin - oxide nanoparticles with [ sb]/([sn]+[sb ] ) equal to 0 , 2 , 5 , 7 , 9 , and 13 at.% .
the particles are monocrystalline and spherical with diameters close to 7 nm .
@xcite the diameter of the 7% doped crystallite is 7.1 nm .
sb is incorporated in the casserite sno@xmath73 lattice by replacing sn@xmath74 . at the doping level of 7%
, sb is mainly present as sb@xmath75 , resulting in n - type conductivity of the ato particles according to nutz _
et al._. @xcite the amount of sb@xmath76 present in the particles is negligibly small .
@xcite the followed experimental procedures for the dc conductivity and dielectric measurements are described in refs . and .
the thickness of the samples was typically a few mm .
the dc conductivity measurements were performed in the dark under helium atmosphere .
the @xmath15 dependence is given in fig .
[ svst ] , and the frequency dependence at temperatures down to 7 k in fig .
all data shown are for 7%-doped ato .
similar results were obtained at other doping levels , be it with different absolute values .
the data were taken in the ohmic regime .
for a densely packed powder of 7%-doped ato . for @xmath77 k the data can be fitted by eq .
( [ eses ] ) ( a ) , while for @xmath78 k the @xmath15 dependence is activated ( b).,width=302 ] .transport parameters obtained for ato and indium tin oxide ( ito ) ( second row ) at doping levels of @xmath79 to @xmath80 per @xmath81 .
the room - temperature dc conductivities are given in s / cm , the drude frequencies and damping rate in s@xmath82 , and the effective mass in free - electron masses .
the first and second row are obtained for films of ato @xcite and ito @xcite resp .
, the last two rows contain the data on powders of 6% doped ato particles of ntz _
et al._@xcite and our data ( labelled as pw for present work ) on samples with 7% sb doping . for a bulk material with a drude frequency of @xmath83 hz , a scattering time of @xmath84 s and a carrier mass of 0.3 @xmath85 a dc conductivity
is expected of @xmath86 s / cm [ cols="^,^,^,^,^,^,^",options="header " , ] ) .,width=302 ] the infrared ( ir ) transmittance was measured on a pellet of kbr mixed with a small amount of ato . for the ir - reflectance we used a precipitated film of ato with a thickness of about 1 mm .
the data are shown in fig .
[ trir ] . for the analysis we also used the sub - thz transmittance and phase data ( only shown in fig .
[ sfit ] ) .
.,width=302 ] in table [ numbers ] we summarize our data on densely packed 7-nm - sized ato crystallites and compare them with measurements on doped tin oxide published in the literature .
the values of @xmath1 agree within a factor 2 , while the spread in the scattering rates is larger .
in the analysis we first show the procedure to extract the parameter values from the data in the different frequency regimes and to check their consistency .
we also make a comparison to the parameter values of phthalcon-11 , for which the data are published elsewhere .
@xcite then we concentrate on the density of states ; the latter being important for the dc conductivity .
subsequently , the implications for the use of the particles as fillers in nanocomposites are discussed . regarding the @xmath15 dependence of @xmath46 ( fig . [ svst ] )
, the data can be fitted with @xmath87 ( eq .
( [ esm ] ) ) if the fit is restricted to @xmath88 k and with @xmath89 ( eq .
( [ eses ] ) ) for @xmath90 k. the @xmath87 fit gives an activation energy of @xmath86 k , while the exponent @xmath89 at low @xmath15 gives @xmath91 k. the localization length from @xmath92 , see eq .
( [ et0es ] ) , is calculated to be @xmath20 = 3 nm .
using eq .
( [ ea ] ) and @xmath93 estimated from the packing density , we find @xmath94 nm , in good agreement with the value calculated from @xmath29 .
the onset of the frequency dependence of the conductivity ( see fig . [ svsf ] ) signals that the carrier starts to feel the inhomogeneity of the underlying structure .
( [ esphon ] ) , the typical length scale @xmath42 at the onset can be found . for 7%-doped ato at 300 k , the onset frequency @xmath95 hz and
@xmath96 s / cm give a value of @xmath97 cm@xmath82 .
the linear frequency dependence of the conductivity at 7 k in the double logarithmic plot of fig .
[ svsf ] , is in agreement with phonon - assisted tunneling , see eq .
( [ esphon ] ) . in the range of 10 - 100 cm@xmath82 photon - assisted processes
take over.@xcite applying eq .
( [ esphon ] ) to the conductivity data at 293 k and taking the usual value for the phonon frequency in solids @xmath99 s@xmath82,@xcite we find @xmath100 ev@xmath101cm@xmath82 , see fig .
[ sfit ] . turning next to the high - frequency data presented in fig .
[ trir ] , we performed a simple drude analysis .
the fit ( @xmath1 = 11000 cm@xmath82 and @xmath102 = 3300 cm@xmath82 , together with a dielectric constant of 4.0 ) reproduces the main features of the increase of the transmission and the level of the reflectance ( the oscillations in the fit to the transmittance are an artefact because the effective ato film thickness of 0.005 mm is much smaller than the real thickness of the pressed kbr pellet ) . the number of carriers @xmath67 of @xmath80 @xmath103 is directly derived from the drude frequency and is slightly lower than obtained from a simple interpretation of the chemical composition .
the bulk dc conductivity calculated from the drude parameters is @xmath86 s / cm .
the fit parameters of the present samples are given in table [ numbers ] and agree well with the literature .
[ sfit ] shows the reconstructed conductivity of ato as function of frequency due to the processes discussed above .
for ato the important values for the dc conductivity can be deduced from the combination of variables that we found from the previous analysis ( i ) @xmath104 cm@xmath82 , ( ii ) @xmath105 j@xmath101m@xmath82 or @xmath106 @xmath107 for photon - assisted hopping and @xmath108 j@xmath101m@xmath82 for the phonon - fit to the data at 7 k , and ( iii ) @xmath109 @xmath103 and @xmath110 s. using ( iii ) the ` extrapolated ' dc conductivity is @xmath86 s / cm , a factor @xmath111 larger than the found value of @xmath112 s / cm .
the estimated fermi energy @xmath51 is around 2 ev , and from @xmath113 ( valid for free electrons ) we get @xmath114 ev@xmath82@xmath103 . from ( ii ) with @xmath115 of 3 nm ,
we find for @xmath116 ev@xmath82@xmath103 , a factor @xmath117 lower than @xmath118 .
note that this is an averaged density of states involved in photon assisted hopping . due to the curvature of the density of states around the chemical potential
, @xmath119 will be lower at lower energies .
for example , for the phonon - fit at 7 k @xmath120 is equal to @xmath121 ev@xmath82@xmath103 .
the values for @xmath102 and @xmath42 can be used as a consistency check .
the combination of the estimated fermi velocity of @xmath122 cm / s , with the crystallite size of 7 nm and @xmath123,@xcite predicts a surface scattering rate of @xmath124 s@xmath82 , in agreement with the found value of @xmath70 .
next , from @xmath125 ev@xmath82@xmath103 we now can estimate @xmath44 at @xmath126 k as @xmath127 @xmath103 . using ( i ) and @xmath128
we find @xmath129 0.3 @xmath130 m . in short ,
the dielectric data of ato allow a consistent picture of the conduction process . in these densely packed crystallites
the localization length is enhanced by a factor 10 and the density of states involved in the dc conductivity is more than a factor @xmath117 smaller than that in the conduction within the crystallites .
the relatively large length scale for homogeneity is indicative for the presence of aggregates . indeed , like in ketjen - black , @xcite nanoparticles of ato are known to form chemically bonded aggregates that survive the preparation stage.@xcite due to the nature of the chemical bond , the conductivities between neighboring crystallites in and outside the aggregates are expected to be only slightly different .
note that also the value of @xmath131 has to be seen as an average , as inhomogeneities in the doping of ato might be present as well.@xcite for the studied phthalcon-11 crystallites @xmath132 s@xmath82 and @xmath133 s leading to @xmath134 @xmath103 , i.e. about 1 charge per crystallite.@xcite the other values found for phthalcon-11 are : ( i ) @xmath135 cm@xmath82 , ( ii ) @xmath136 j@xmath101m@xmath82 or @xmath137 @xmath107 . in these organic crystals with such a low carrier density , the charge carriers can be seen as an electron gas with an energy scale set by @xmath57 , and @xmath119 can be estimated from @xmath138 to be @xmath139 @xmath140 .
this value of @xmath119 is the upper limit for @xmath22 and @xmath141 . from @xmath142 @xmath140
, we find a decay length @xmath20 of 3 nm , as expected from the packing .
the phthalcon-11 parameters show that the crystals are semiconducting crystals with a low number of charge carriers .
all charges participating in the conductivity within the crystal also contribute to the dc conductivity . as for ato the obtained conduction parameters for phthalcon-11 from the dielectric scans give a consistent picture . for ato , the differences between the density of states involved in the hopping process @xmath143 ev@xmath82@xmath103 and the drude conduction within the crystallites @xmath114 ev@xmath82@xmath103 are clearly significant .
the result is as anticipated from the estimated charging energy of the order of 50 mev , and shows its importance for the dc powder conductivity . for phthalcon-11
the very low number of carriers involved in the hopping process is similar to the number of carriers that determines the drude contribution in the crystallites .
since the mean size of the particles is 20 times larger than for ato , the charging energies will be of the order of 3 mev , and hence are expected to be negligible at room temperature . in polymer nanocomposites with building blocks formed by diffusion - limited cluster aggregation
, the airy structure of the particle network gives a strong reduction in conductivity of the composite compared to the filler ( for the phthalcon11/polymer composite a factor @xmath111 ) .
@xcite this effect can be compensated by using better conducting particles .
particles of ato or ito seem to be well - suited as the material is known to be very well - conducting .
in addition , ato crystallites are relatively easily obtained in sizes around 7 nm , and when properly dispersed can give polymer composites with a low percolation threshold.@xcite however , even if the filler nanoparticles in the composite touch , they will not be in better contact than in a densely packed powder .
as shown here for ato , for these small crystallites the dos involved in @xmath0 is dramatically reduced due to the shift of the energy levels away from the fermi level by coulomb charging effects . as a consequence ,
an additional four orders of magnitude in @xmath0 are lost compared to the bulk value .
other systems where size restrictions are expected to be present might be conveniently studied in a similar way . for example
several chalcogenide alloys exhibit a pronounced contrast between the optical absorption in the metastable rocksalt after the intense laser pulse and in the initial amorphous phase .
@xcite as shown by extended x - ray absorption fine structure spectroscopy ( exafs ) the resistive change after the intense laser recording pulse goes together with a crystallization process , where also small domains are inherently present .
our dielectric method might visualize to what extent the domain walls after crystallization limit the conductivity and have consequences for the band structure calculations .
if the walls become real barriers quantum size effects in the small domains will invalidate the use of periodic boundary conditions in the calculations .
also the glassy behavior in the conductance of deposited indium - tin oxide samples in the insulating regime , @xcite and of quench - condensed insulating granular metals @xcite might be further clarified by the use of our dielectric approach and analysis .
scanning the frequency will reveal the evolution of the length scales and dos involved in the relaxation processes .
by combining data of sub - thz transmission with infrared transmission and reflection we were able to explain the full frequency response of densely - packed nanosized crystallites using the parameters for mott variable - range hopping , for heterogeneity - induced enhanced ac response , for phonon- or photon - assisted nearest - neighbor hopping , and for the drude response of individual nanocrystals . for 7
nm antimony - doped tin - oxide particles the analysis unambiguously quantified the reduction of the density of states involved in the dc conduction compared to the value extrapolated from the drude response at infrared frequencies .
dielectric scans with a similar analysis will also be revealing in other systems where size limitations are expected to play a role .
it is a pleasure to acknowledge roel van de belt of nano specials ( geleen , the netherlands ) , who made the ato samples available , and matthias wuttig from the physikalisches institut of the rwth aachen university in germany for fruitful discussions about phase - change materials . this work forms part of the research program of the dutch polymer institute ( dpi ) , project dpi435 . | conducting submicron particles are well - suited as filler particles in non - conducting polymer matrices to obtain a conducting composite with a low percolation threshold . going to nanometer - sized filler particles imposes a restriction to the conductivity of the composite , due to the reduction of the density of states involved in the hopping process between the particles , compared to its value within the crystallites .
we show how those microscopic parameters that govern the charge - transport processes across many decades of length scales , can accurately and consistently be determined by a range of dielectric - spectroscopy techniques from a few hz to infrared frequencies .
the method , which is suited for a variety of systems with restricted geometries , is applied to densely packed 7-nm - sized tin - oxide crystalline particles with various degree of antimony doping and the quantitative results unambiguously show the role of the nanocrystal charging energy in limiting the hopping process . | 0801.2228 |