Continuum extrapolation of finite temperature meson correlation functions in quenched lattice QCD

0.21 MB·English

by  Anthony Francis

Continuum extrapolation of finite temperature
1 meson correlation functions in quenched lattice
1
0 QCD
2

n
a
J

8
2

] AnthonyFrancis∗
t FakultätfürPhysik,UniversitätBielefeld,D-33615Bielefeld,Germany
a
l E-mail: [email protected]
-
p
e Frithjof Karsch†
h
PhysicsDepartment,BrookhavenNationalLaboratory,Upton,NY11973,USA
[
E-mail: [email protected]

1
v
1 We explore the continuumlimit a 0 of meson correlationfunctionsat finite temperature. In
7 →
detail we analyze finite volume and lattice cut-off effects in view of possible consequencesfor
5
5 continuumphysics. Weperformcalculationsonquenchedgaugeconfigurationsusingtheclover
1. improvedWilsonfermionaction.WepresentanddiscusssimulationsonisotropicNs3 16lattices
×
0 withNs =32,48,64,128and1283 Nt latticeswithNt =16,24,32,48correspondingtolattice
×
1 spacingsintherangeof0.01fm<a<0.031fmatT 1.45T. Continuumlimitextrapolations
c
1 ≃
ofvectormesonandpseudoscal∼ar co∼rrelatorsare performedandtheirlargedistanceexpansion
:
v
in terms of thermal moments is introduced. We discuss consequences of this analysis for the
i
X calculationoftheelectricalconductivityoftheQGPatthistemperature.
r
a
TheXXVIIIInternationalSymposiumonLatticeFieldTheory,Lattice2010
June14-19,2010
Villasimius,Italy
Postercontribution.
∗
†Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
ContinuumextrapolationoffinitetemperaturequenchedlQCD A.Francis,F.Karsch
1. Introduction
The heavy-ion experiments at RHIC and soon LHC are probing deeper and deeper into the
high temperature region of QCD. As such more and more experimental results are presented that
requireamorecompleteknowledgeoffinitetemperatureQCD.NaturallyoneturnstolatticeQCD
togivesomeanswerstothequestions arising. Howeversomequestions ofinterest arenotoriously
difficult to answer from a lattice perspective. This is in particular the case for the analysis of
spectralpropertiesofhadroncorrelationfunctions. Althoughherethemaininterestisinextracting
information on hadronic spectral functions at low frequencies, this cannot be achieved without
good control over the large frequency region. Correlation functions thus need to be controlled at
short as well as large distances. High accuracy data on large lattices at several values of the cut-
off are needed to control finite volume and lattice cut-off effects. Only then a reliable continuum
extrapolation becomespossible whichisaprerequisite forextracting dependable physicsresults.
In this combined talk and poster contribution we present results from a lattice analysis of
meson correlation functions at finite temperature [1]. We emphasize here our systematic analysis
ofthefinitevolume and cut-off dependence ofthermal mesoncorrelation functions. Thisanalysis
leadstotheconclusionthatweindeedcantakethecontinuumlimitforthevectorandpseudoscalar
current correlation functions in a large Euclidean time interval. We also calculate several thermal
moments of hadron spectral functions to better explore the spectral properties at low frequencies.
Weclosewithadiscussion oftheconstraints arisingfromouranalysisforthedetermination ofthe
electrical conductivity inthequarkgluonplasmaatfinitetemperature [2,3].
2. MesonCorrelationFunctions
OurkeyobservableofinterestistheEuclideancorrelation functionofagivenparticlecurrent,
Jn q¯(t ,~x)gn q(t ,~x), (2.1)
≡
where choosing the appropriate gamma matrix we obtain the vector particle channels for gn =gm
where m =0,...,4andthepseudoscalarforgn =g5.
In this work we analyze the Euclidean temporal two-point correlation function for the above
channels atfixedmomentum,
Gmn (t ,~p)=(cid:229) Gmn (t ,~x)ei~p·~x, (2.2)
~x
where
Gmn (t ,~x)= Jm (t ,~x)Jn†(0,~0) , (2.3)
h i
while we denote thermal expectation values with ... . In this work we will set the momentum to
h i
zero,~p=0,andsuppressthesecondargument inthehadronic correlation functions.
Athightemperaturesthespectrumofmesonresonancesismoreandmorechangedbythermal
effects; thewidth of resonances will broaden and the onset of thecontinuum in the spectrum may
2
ContinuumextrapolationoffinitetemperaturequenchedlQCD A.Francis,F.Karsch
shift. Asaresulttheinterest intheanalysis oftheabovecurrent-current correlation function shifts
totheirrepresentation intermsofaspectralfunction r mn [4]:
¥ dw cosh(w (t 1/2T))
Gmn (t ,T)=Z0 2p r mn (w ,T) sinh(w −/2T) . (2.4)
Fromhereonwedenotewithmn iithesumoverthespace-likecomponentsofthevectorspectral
≡
function r , while the full vector spectral function is given by r r +r . We also use the
ii V 00 ii
≡
notation PS for mn 55. Note that the correlation function G (t ,T) is connected to the net
00
≡
number of quarks (q q¯) in a given flavor channel, i.e., n (t )= d3xJ (t ,~x). As the net quark
0 0
−
number is conserved, it does not depend on time, n (t )=n . ThuRs the corresponding correlation
0 0
function isconstantinEuclideantimeandthespectral functionissimplygivenby,
r (w )= 2pc wd (w ), (2.5)
00 q
−
withthequarknumbersusceptibility
c = d3x J (t ,~x)J†(0,~0) . (2.6)
q −Z h 0 0 i
Consequently the time-like correlation function obeys the relation G (t ,T) c T, which im-
00 q
≡−
mediately leadstoanexactrelation betweenG (t ,T)andG (t ,T):
ii V
G (t ,T)=c T+G (t ,T). (2.7)
ii q V
Inthefreefieldlimitthespectralfunctionsareseentoincreasequadraticallyforlargefrequen-
ciesw . Oneobtains:
3
r (w ) = 2pc wd (w )+ w 2tanh(w /4T) (2.8)
ii q 2p
3
r (w ) = r (w )+r (w )= w 2tanh(w /4T) (2.9)
V 00 ii 2p
3
r (w ) = w 2tanh(w /4T) (2.10)
PS 4p
Atfinitetemperaturetheexactcancellationofthed -functionsinther (w )spectralfunctionisonly
V
approximate, asr (w )continuestobeproportionaltoad -functionduetotheconnectionwithnet
00
quark number, while the d -function in r (w ) is subject to thermal effects. As a consequence this
ii
contribution issmearedoutintoaBreit-Wigner shapedpeak[5]:
w G /2 3
r (w ) r BW(w )=2c c +(1+k ) w 2tanh(w /4T), (2.11)
ii → ii q BWw 2+(G /2)2 ·2p
wherek parametrizescorrectionstothehighfrequencybehaviorofthefreefieldlimit. Thissmear-
ing inthe low frequency region isdirectly related tothe appearance offinite transport coefficients
inthethermalmedium,e.g. theelectrical conductivity,
s C r (w )
em ii
= lim , (2.12)
T 6 w 0 w T
→
whereC isthesumoverthesquaredchargesofthecontributing quarkflavors. Thepseudoscalar
em
spectralfunctionontheotherhanddoesnotpersecontainanadditionallowfrequencycontribution;
aconnection withtransportphenomena isnotexpected.
3
ContinuumextrapolationoffinitetemperaturequenchedlQCD A.Francis,F.Karsch
3. Thermal Moments ofthe CorrelationFunction
A useful set of observables that helps to characterize the structure of spectral functions, is
given by “thermal moments” of the spectral function at vanishing momentum. At a given order
wedefinethesemomentsastheTaylorexpansioncoefficientsofthecorrelation functionexpanded
around themidpoint oftheEuclideantimeinterval1,i.e.,aroundt T =1/2,
1 dG (t T) 1 ¥ dw w 2n r (w )
(2n) H H
G = = (3.1)
H (2n)! d(t T)2n (cid:12)(cid:12)tT=1/2 (2n)!Z0 2p (cid:16)T (cid:17) sinh(w /2T)
(cid:12)
and (cid:12)
¥ 2n
G (t T)= (cid:229) G(2n) 1 t T . (3.2)
H H (cid:18)2− (cid:19)
n=0
The infinite temperature, free field limit of the correlation function can be calculated analyti-
cally. Formasslessquarksoneobtains[6]:
1+cos2(2pt T) cos(2pt T)
Gfree(t T)=2Gfree(t T) = 6T3 p (1 2t T) +2 (3.3)
V PS (cid:18) − sin3(2pt T) sin2(2pt T)(cid:19)
Gfree(t T) = T3+Gfree(t T) (3.4)
ii V
Usingtheseresultsthefirstthreenon-vanishing momentsarethengivenby
2
G(0),free = 2G(0),free= G(0),free=2T3, (3.5)
V PS 3 ii
28p 2
G(2),free = 2G(2),free= T3, (3.6)
H PS 5
124p 4
G(4),free = =2G(4),free= T3 . (3.7)
H PS 21
In our lattice simulation we will also analyze ratios of moments. In the free field limit they are
givenby
G(2),free 14p 2
R(2,0) = V =R(2,0) = 27.635 (3.8)
V,free G(0),free PS,free 5 ≃
V
2
(2,0) (2,0)
R = R 18.423 (3.9)
ii,free 3 V,free≃
155p 2
(4,2)
R = 10.407 , (3.10)
H,free 147 ≃
which can be obtained from the ratio of the mid-point subtracted correlation functions and the
corresponding freefieldvalues:
D (t T) G (t T) G(0) G(2) 1
H = H − H = H 1+(R(4,2) R(4,2) )( t T)2+... .(3.11)
D free(t T) Gfree(t T) G(0),free G(2),free(cid:18) H − H,free 2− (cid:19)
H H − H H
Notethatthedistinction betweenH =iiorH =V isunnecessary whenevaluating D (t T),asdif-
H
ferences inhtecorrelators, whicharisefromcontributions ofd -functions inthespectral functions,
areeliminated insubtracted correlators.
1Asthespectralfunctionsaswellastheintegrationkernelinthespectralrepresentationofthecorrelatorsareodd
functionsofthefrequency,alloddmomentsvanish.
4
ContinuumextrapolationoffinitetemperaturequenchedlQCD A.Francis,F.Karsch
Nt Ns #conf b a[fm] T/Tc cSW k mMS/T[m =2GeV] ZV ZPS
48 128 451 7.793 0.010 1.43 1.3104 0.13340 0.1117(2) 0.861 0.79
32 128 255 7.457 0.015 1.45 1.3389 0.13390 0.0989(4) 0.851 0.78
24 128 340 7.192 0.021 1.42 1.3673 0.13431 0.1062(2) 0.842 0.76
156 0.13440 0.02367(5)
16 128 191 6.872 0.031 1.46 1.4125 0.13495 0.02429(5) 0.829 0.74
64 191
48 229
32 251
Table1: SimulationparametersforthegenerationofgaugefieldconfigurationsonlatticesofsizeNs3 Nt.
×
4. SimulationParameters and Results
4.1 ComputationalDetails
Ournumerical results areobtained from quenched QCDgauge fieldconfigurations generated
withthestandard SU(3)singleplaquette Wilsongaugeaction[7]. Usinganover-relaxed heatbath
algorithm configurations were generated on lattices of size Ns3 Nt , where 32 <Ns < 128 and
×
Nt =16, 24, 32and 48, withaseparation of500 updates perconfiguration. Correlation functions
andplaquetteexpectationvaluescalculatedontheseconfigurationshavebeencheckedandareseen
tobeuncorrelated. Detailsonthestatistics collected aregiveninTab.1.
Allcalculationspresentedherehavebeenperformedatasingletemperaturevalue,T 1.45T .
c
≃
Thegaugecouplings havebeenadjusted accordingly forthedifferent temporallatticesizes. Todo
so we extrapolate known results for the critical coupling b c(Nt ) and the square root of the string
tension √s [8]. This is achieved by fitting T/√s using the ansatz suggested in [9] within the
range of b [5.6,6.5] and extrapolating to our relevant region of b [6.8,7.8]. All simulation
∈ ∈
parameters aregiveninTab.1.
In the fermion sector we employ the clover improved Wilson action with non-perturbatively
chosen clover coefficient c [10] and determine the correlation functions using an even-odd de-
SW
composed CG-algorithm. The hopping parameter k is chosen to be close to its critical value [11]
andtuned togiveapproximately constant quarkmassesforfouroftheexamined lattices. Herethe
quarkmassesareestimatedusingtheaxialWardidentity(AWI)tocalculatetheAWIcurrentquark
massm forthedifferentvaluesofcut-off. Wherebyweusethenon-perturbatively improvedax-
AWI
ialvectorcurrentwithcoefficientc notedin[12]. Inthenextstepwecalculatetherenormalization
A
groupinvariantmassm andrescaletothecommonlyquotedMS-schemeatthescalem =2GeV.
RGI
TheresultsarealsolistedinTab.1.
4.2 NumericalResults
Duetothesubtletyoftheeffectsinquestionitisnotveryilluminatingtoexaminethecorrelator
directly. Toillustratethisweshowthecorrelationfunctionsofthevectorandpseudoscalarcurrents
together with their analytically obtained free continuum and free discretized, lattice counterparts
in Fig. 1. Clearly the exponential decay of the correlation function dominates and obscures all
5
ContinuumextrapolationoffinitetemperaturequenchedlQCD A.Francis,F.Karsch
111111111000000000000000000000000000000000000 111111111000000000000000000000000000000000000
GGGGGGGGGVVVVVVVVV(((((((((tttttttttTTTTTTTTT)))))))))/////////TTTTTTTTT333333333 11111 11111.....11111.....2222255555 GVfree,lat/GVfree GGGGGGGGGPPPPPPPPPSSSSSSSSS(((((((((tttttttttTTTTTTTTT)))))))))/////////TTTTTTTTT333333333 11111 .....1111111111.....5555522222 GPfreSe,lat/GPfreSe
111111111000000000000000000000000000 11111.....11111 111111111000000000000000000000000000 11111.....11111
11111.....0000055555 11111.....0000055555
111111111000000000000000000 11111 tT 111111111000000000000000000 11111 tT
00000 00000.....11111 00000.....22222 00000.....33333 00000.....44444 00000.....55555 00000 00000.....11111 00000.....22222 00000.....33333 00000.....44444 00000.....55555
111111111000000000 f111111111111111111111111re222222222222222222222222e888888888888888888888888 333333333333333333333333cxxxxxxxxxxxxxxxxxxxxxxxxo111111111222323234234234n666666666444242428428428t 111111111000000000 111111111111111111111111222222222222222222222222888888888888888888888888333333333333333333333333xxxxxxxxxxxxxxxxxxxxxxxx111111111222323234234234666666666444242428428428
free cont
tttttttttTTTTTTTTT tttttttttTTTTTTTTT
111111111 111111111
000000000 000000000.........111111111 000000000.........222222222 000000000.........333333333 000000000.........444444444 000000000.........555555555 000000000 000000000.........111111111 000000000.........222222222 000000000.........333333333 000000000.........444444444 000000000.........555555555
Figure1:Thevector(left)andpseudoscalar(right)correlationfunctionstogetherwiththeirfreecontinuum
(magentalines)andfreelattice(opensymbols)counterparts. (Blow-ups): Theratiooffreelatticeandfree
continuumcorrelationfunctions.
otherinteresting physicsfeatures. Tocircumventthiseffectitisusefultolookatcertainratiosthat
largely canceltheexponential partoftheEuclideantimedependence ofthecorrelators.
One of these ratios is that of the correlation function itself divided by the free correlation
function giveninEq.3.3,
G (t ,T)
R (t ,T)= H . (4.1)
H Gfree(t ,T)
H
The immediate advantage of a ratio as that given in Eq. 4.1 is that it gives a direct handle on the
deviation of the correlation function calculated at finite and infinite temperature, respectively. As
thecontinuumaswellaslatticefreecorrelationfunctionsareknownanalytically,thisalsoprovides
insight into the importance of lattice cut-off effects as function of Euclidean time. The insertions
in Fig. 1 show the ratio of the free lattice and the free continuum correlation functions. They, of
course, show that cut-off effects are largest at small distances. Moreover, it becomes obvious that
withincreasingcut-offvalues(largerNt )theonsetofcut-offeffectsisshiftedtosmallerdistances.
Asweemploy thelocal current correlations introduced inEq.2.1allresults havetoberenor-
malizedmultiplicatively,
Jnlat =(2k ZH)q¯(t ,~x)gn q(t ,~x), (4.2)
where the renormalization constants have been determined non-perturbatively for the vector [11]
andperturbativelyuptotwo-looporderforthepseudoscalarchannel[13]. TheyarelistedinTab.1.
Inthecaseofthevector correlation functions theabovestatement holds truealsoforthetime-like
component G (t ,T) because the local lattice current is not conserved at non-zero lattice spac-
00
ing. Consequently the time-like correlation function G (t ,T) may not be strictly t -independent.
00
However,asallvectorcurrent-currentcorrelationfunctionsaresubjecttothesamerenormalization
constants arescalingofthecorrelation functions G (t ,T)andG (t ,T)oreventheratioR (t ,T)
V ii H
byG (t ,T)yieldsaquantity independent ofrenormalization.
00
6
ContinuumextrapolationoffinitetemperaturequenchedlQCD A.Francis,F.Karsch
Nt 16 24 32 48 ¥
c /T2 0.882(10) 0.895(16) 0.890(14) 0.895(8) 0.897(3)
q
Table2: Quarknumbersusceptibility(c q/T2)calculatedonlatticesofsize1283 Nt . Thequarknumber
×
susceptibilitieshavebeenrenormalizedusingtherenormalizationconstantslistedinTab.1.Thelastcolumn
givestheresultofacontinuumextrapolationtakingintoaccountcut-offerrorsofO(a2).
1.2 1.2
-G00(tT)/T3 -G00(tT)/T3
1.15 1.15
1.1 1283x16 1.1 1283x16
643x16 1283x24
1.05 483x16 1.05 1283x32
323x16 1283x48
1 1
0.95 0.95
0.9 0.9
0.85 0.85
tT tT
0.8 0.8
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.2 0.4 0.6 0.8 1
Figure2: Thetime-likecomponentofthevectorspectralfunction,G (t T)/T3,calculatedatT 1.45T.
00 c
≃
ThelefthandpartofthefigureshowsthevolumedependenceofG00(t T)/T3 forNt =16and32 Ns
≤ ≤
128. Therighthandfigureshowsthecut-offdependenceofG00(t T)/T3 forNs =128and16 Nt 48.
≤ ≤
4.3 FiniteVolumeandLatticeEffects
4.3.1 QuarkNumberSusceptibility
As mentioned above the time-like component of the vector correlation function, G (t T),
00
will be used for rescaling in the following, so it is the first channel to be tested for finite volume
and cut-off effects. We find these effects to be small, as can be seen from Fig. 2, where we plot
G (t T)/T3. The quark number susceptibility may then be extracted from the long distance
00
−
behavior ofthecorrelator, c /T2= G (t T 0.5)/T3.
q 00
− ≃
OntheleftofFig.2weshowtheresultsfortheNt =16latticewithfixedcut-offwhilevarying
thespatialextent. ExceptforaspectratioNs /Nt =2allresultsagreewithinstatisticalerrorsatthe
level of 1%. On the right the cut-off dependence at fixed spatial extent Ns =128 is shown. Here
toosystematic effectsinthequarknumbersusceptibility areseentobesmall.
Note even though we implement the non-conserved local current of the time-like vector cor-
relation function, we obtain results that are t -independent to a very high degree. The deviations
at small distances t T might be understood as lattice effects. The results for the quark number
susceptibility aresummarizedinTab.2.
4.3.2 VectorandPseudoScalarCorrelation Functions
Turning tothevector andpseudo scalar correlation functions westress thatthelarge rangeof
available spatial lattice sizes at fixed cut-off, with aspect ratios ranging between 2 Ns /Nt 8,
≤ ≤
allowstoquantifyfinitevolumeeffects. Moreover,thelargetemporalextentofmaximumNt =48
7
ContinuumextrapolationoffinitetemperaturequenchedlQCD A.Francis,F.Karsch
1.35 GV(tT)/GVfree(tT) 2.6 GPS(tT)/GPfreSe(tT)
1283x16
1.3 643x16 2.4
483x16
1.25 1238233xx2146 2.2
1283x24-2
1283x32 2
1.2 1283x48 1.8 1624833xx1166
1.15 483x16
1.6 323x16
1283x24
1.1 1.4 1218238x32x43-22
1.05 1.2 1283x48
tT tT
1 1
0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5
Figure 3: The vector (left) and pseudo scalar (right) correlation functions, calculated on lattices of size
Ns3 Nt atT 1.45Tc,where“1283 24 2“denotesthelowerquarkmassonthislattice. Theresultsare
× ≃ × −
normalizedusingthecorrespondingfreecorrelationfunctions.Shownaredatafort T >1/Nt only.
on an isotropic (!) lattice reduces the lattice spacing to 0.01fm at T 1.45T . The variation of
c
≃
Nt by up to a factor of three allows to control lattice cut-off effects. In the following it will be
shownthatfinitevolumeeffectsinthecorrelationfunctionsarewellundercontrolandacontrolled
extrapolation tothecontinuumlimitisindeedpossibleinalargeEuclideantimeinterval. Notethat
that our current analysis improves on systematic errors that were present in earlier calculations of
thevectorspectralfunction performedbyemploying thesamediscretization scheme[14].
In Fig. 3 we show results for the vector and pseudo scalar correlation functions using the
ratios introduced in Eq. 4.1. On the left hand side the ratio for H =V and on the right hand side
for H =PS is shown for all available lattice sizes. Data sets with fixed spatial size at Ns =128
while varying the cut-off Nt shown in black. Data sets with fixed cut-off (Nt =16) and varying
volume are shown in color. For one value of the cut-off (Nt =24) we performed calculations for
twodifferent values ofthequark masses. Wefindthat finitequark masseffects aresmalland well
within 2%. Fromthe fixedcut-off (colored) Nt =16results inboth plots finitevolume effects for
t T 0.3areseentoremainwithinafewpercent evenforthelargest Euclideantimeseparation at
≥
t T =0.5. As a consequence these results show that finite volume effects are under control. For
t T <0.3intheleftH =V plotitisimmediatelyapparent thatcut-offeffectsarelargeintheratio.
In the pseudo scalar case the situation concerning cut-off effects is not immediately evident.
The ratio shown in Fig. 3(right) shows large deviations from the free field behavior even at short
distances. Atall distances the correlator thus seems to be controlled by large non-perturbative ef-
fects. Moreover, theanalysis ofcut-off effectsisobscured bythefactthatdatahavebeenrescaled
byusingrenormalizationconstants,whichareknownonlyperturbatively. Thisintroducesunknown
uncertainties. Toeliminateatleasttheseuncertainties weshowinFig.4thepseudo scalarcorrela-
tionfunctionnormalizedbythepseudoscalarcorrelationfunctionatt T =0.5. Aswefocusonthe
cut-off dependence weonly show equal quark mass Ns =128 results. Theleft hand figure shows
the pseudo scalar correlator normalized by free continuum correlator and in the right hand figure
the free lattice correlation function has been used. From the left hand plot the cut-off dependence
becomes immediately evident and we conclude that, similarly to the vector case, cut-off effects
above t T =0.3 are small and increase withdecreasing t T. Eventhough their effect becomes ap-
8
ContinuumextrapolationoffinitetemperaturequenchedlQCD A.Francis,F.Karsch
1 GPS(tT)/[GPS(1/2)GPfreSe(tT)] 1 GPS(tT)/[GPS(1/2)GPfreSe,lat(tT)]
0.9 0.9
0.8 0.8
1283x16 1283x16
0.7 11228833xx2342 0.7 11228833xx2342
1283x48 1283x48
0.6 0.6
0.5 0.5
tT tT
0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5
Figure 4: The pseudo scalar correlationfunction normalizedby the free continuum(left) and free lattice
(right)correlationfunctions,bothadditionallyrescaledbythecorrelatoratt T =1/2,G (t T =0.5).
PS
parent fort T 0.3, theydonotdominate thebehavior ofthecorrelation function asinthevector
≤
channel. ActuallytherighthandsideofFig.4indicatesthatthet -dependenceofthecut-offeffects
issimilar tothatofthefreelattice correlation functions, asanycut-off effectishardly visible also
fort T 0.3.
≤
4.4 ContinuumExtrapolation
Theresultsshownintheprevioussectionindicatethatthecontinuumlimitcanbetakenforthe
vectorcorrelationfunctionsatleastfort T>0.25. TodosoweuseaquadraticansatzinaT =1/Nt
to fit ratios of the vector correlator and th∼e corresponding free field values. We normalize these
ratios using the quark number susceptibility c /T2 and perform fits at fixed temporal extent t T.
q
Ascut-off effects are large on the Nt =16 lattice wewillonly use data from the Nt =24, 32, 48
lattices, forreference wewillhoweverincludetheformerinourfigures.
InFig.5weshowresultsofthisextrapolationin1/Nt2fort T =1/2, 7/16, 3/8, 5/16and1/4
whereweusedthefreecontinuumaswellasthefreelatticecorrelationfunctionsfornormalization.
Notethatweperformedextrapolations forallEuclidean timest T available ontheNt =48lattice.
Whereverthesmallerlatticesfailtohaveacorresponding pointint T weinterpolate usingaspline
construction. SubsequentlytheerrorsarethencalculatedusingaJackknife-method. InFig.5thisis
thecasefort T =7/16and5/16. Thefigurerevealsthatthecontinuum limitcanbecleanlytaken
andconsistent resultsareobtained byusingthefreecontinuum andthefreelatticenormalizations,
respectively.
In the left hand part of Fig. 6 we show the results of the extrapolation in the vector channel
as described above. Note that the largest deviation from the free correlation function occurs at
t T =1/2. Infact,theestablished bendingofthevectorcorrelationfunctioniscrucialforaquanti-
tativedescription ofthelowfrequency regionofthevectorspectral function intermsoftheansatz
suggested in Eq. 2.11. The short distance part of the correlation function obviously can be well
describedbythefreespectralfunctionincludingthecorrectionfactor(1+k )asalsohasbeendone
inEq.2.11.
IntherighthandpartofFig.6thecorresponding resultforthepseudoscalarcorrelation func-
tion is shown. Here it is not possible to suppress the renormalization effects using suitable ratios
9
ContinuumextrapolationoffinitetemperaturequenchedlQCD A.Francis,F.Karsch
1.7
T2GV(tT)/[c qGVfree(tT)]
1.6
1.5
1.4
1.3
1.2
1/Nt2
1.1
0 0.001 0.002 0.003 0.004
Figure 5: The ratio R (t T) normalized by the quark number susceptibility using the free continuum
V
(full symbols) and lattice (open symbols) correlation functions over 1/Nt2. Shown are the results of
t T = 1/2,7/16,3/8,5/16 and 1/4, for legibility the results have been offset by 0.1 respectively. For
t T =7/16and5/16splineinterpolationswereusedtoestimate thecorrespondingresultsonthe Nt =24
lattice.
2.8
1.45 T2GV(tT)/[c qGVfree(tT)] 1283x16 2.6 GPS(tT)/GPfreSe(tT)
1.4 11228833xx2342
1283x48 2.4
1.35 Extrapolation
Fit 2.2
1.3
2
1.25 1.8 1283x16
1283x24
1.2 1.6 11228833xx3428
Extrapolation
1.15 1.4
tT tT
1.1 1.2
0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5
Figure6: (left): TheratioR (t T)normalizedbythequarknumbersusceptibilityusingthefreecontinuum
V
correlation functions over t T and its corresponding continuum extrapolation. (right): The unnormalized
R (t T) including its continuum extrapolation. In both cases the extrapolation was done as described,
PS
fillinginsplineinterpolationswhennecessary.Thesolidcurveinthelefthandfigureshowsthefitdiscussed
insection5.
of correlation functions. Theextrapolation necessarily also includes this ambiguity. Asthe corre-
lator normalized by its value at the midpoint was found to be almost cut-off independent and as
finitevolumeeffectswereseentobesmallrenormalization effectsdominatetheuncertainty ofthe
extrapolation.
4.5 CurvatureoftheVectorandPseudoScalarCorrelation Functions
As discussed in section 3, thermal moments give additional insight into the spectral repre-
sentation of hadronic correlation functions. Theyare especially interesting asthey are obtained at
the largest Euclidean timeseparation wherethecorrelation functions aremostsensitive tothelow
frequency region of the spectral function. In particular, the lower orders of the thermal moments
10