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ThenatureofthethermalphasetransitionwithWilsonquarksAZPH-TH/93-29UUHEP93/4UCSBTH-93-18IUHET262ClaudeBernardDepartmentofPhysics,WashingtonUniversity,St.Louis,MO63130,USAThomasA.DeGrandandA.HasenfratzPhysicsDepartment,UniversityofColorado,Boulder,CO80309,USACarletonDeTarPhysicsDepartment,UniversityofUtah,SaltLakeCity,UT84112,USAStevenGottliebDepartmentofPhysics,IndianaUniversity,Bloomington,IN47405,USAandDepartmentofPhysics,Bldg.510A,BrookhavenNationalLaboratoryUpton,LongIsland,NY11973,USALeoKarkkainenandD.ToussaintDepartmentofPhysics,UniversityofArizona,Tucson,AZ85721,USAR.L.SugarDepartmentofPhysics,UniversityofCalifornia,SantaBarbara,CA93106,USA(October21,1993)1

AbstractWedescribeaseriesofsimulationsofhightemperatureQCDwithtwoa-vorsofWilsonquarksaimedatclarifyingthenatureofthehightemperaturephasefoundincurrentsimulations.Mostofourworkiswithfourtimeslices,althoughweincludesomerunswithsixandeighttimeslicesforcomparison.InadditiontotheusualthermodynamicobservableswestudythequarkmassdenedbythedivergenceoftheaxialcurrentandthequarkpropagatorintheLandaugauge.WendthatthesharpnessoftheNt=4thermaltransitionhasamaximumaround=0:19and6=g2=4:8.12.38.Gc,11.15.HaTypesetusingREVTEX2

I.INTRODUCTIONLatticesimulationsareanimportantsourceofinformationonthebehaviorofquantumchromodynamicsathightemperature.MostworkhasbeendonewithKogut-Susskindquarksbecauseoftheexactremnantofchiralsymmetry.SincetheexactchiralsymmetryofKogut-SusskindquarksisaU(1)symmetry,thereissomequestionabouthowwelltheresultsreproducetherealworldwithitsSU(2)chiralsymmetry.Inthecontinuumlimitthecompletechiralsymmetryisrestored.However,inthecontinuumlimittheresultsshouldbeindependentoftheregularizationusedforthequarks.TotestthisitisimportanttostudyhightemperatureQCDwiththeothercommonformoflatticequarks,theWilsonquarks.TherstsimulationsofhightemperatureQCDwithtwoavorsofWilsonquarksrevealedapotentialproblem|forthevaluesof6=g2forwhichmostlowtemperaturesimulationsweredone,4:56=g25:7,thehightemperaturetransitionoccursatavalueofquarkhoppingparameterforwhichthepionmassmeasuredatzerotemperatureisquitelarge[1,2].Inotherwords,itisdiculttondasetofparametersforwhichthetemperatureisthecriticaltemperatureandthequarkmassissmall.Furtherworkconrmedthatthepionmassislargeatthedeconnementtransitionforthisrangeof6=g2[3,4].(Arecentstudyhasconcludedthatforfourtimeslicesthechirallimitisreachedataverysmallvalueof6=g23:9[5]. )ScreeningmassesforcolorsingletsourcesshowanapproachtoparitydoublinginthehightemperaturephasesimilartowhatisseenwithKogut-Susskindquarks[2,3].Also,measurementsofthepionmassshowashallowminimumatthehightemperaturetransition[6].PrevioussimulationswithWilsonfermionshavelocatedt,thevalueofthehoppingparameteratwhchthehightemperaturecrossoverorphasetransitionoccurs,asafunctionof6=g2forNt=4and6.Thecriticalvalueofthehoppingparameter,c,forwhichthepionmassvanishesatzerotemperaturehasbeenlocatedwithsomewhatlessprecision[1,2,6,7,3,4].Somemeasurementsofhadronmassesbeencarriedoutonzerotemperaturelatticesforvaluesofand6=g2closetothetcurve,allowingonetosetascaleforthetemperature,andtoestimatecinthevacinityofthethermaltransition[3,4,8].InmorerecentworkatNt=6wehaveobservedcoexistenceofthelowandhightem-peraturephasesoverlongsimulationtimes,andwehaveextendedtheseobservationsinthepresentproject.ThechangeintheplaquetteacrossthetransitionismuchlargerthanforthehightemperaturetransitionwithKogut-Susskindquarks[4].Thisunexplainedbehavior,aswellasworkbyHasenfratzandDeGrandontheeectofheavyquarks[9],hasledustoextendourwork.ThispaperreportsonaseriesofsimulationswithWilsonquarksathightemperature,inwhichwehavestudiedanumberofindicatorsforthenatureofthephases.Using834lattices,wehaveextendedearlierstudiesofthelocationofthethermaltransitionorcrossoverto=0:20,0:21and0:22.Intherangesof6=g2andthathavebeenstudiedearlier,wehavedoneextensivesimulationson88204lattices,withadditionalworkon1236,122246and82208lattices.Foronevalueof6=g2wemadeaseriesofrunson66204latticestomakesurethattheeectsweseearenotduetothespatialsizeofthelattice.Inthisreportwewillconcentrateontheresultswithfourtimeslices.SimulationswithNt=6and8arestillunderwayandwillbedescribedlater.3

Weseeanumberofinexplicableeects.Atlargeandsmallthecrossoverfromtheconnedphasetothehightemperaturephaseissmooth.Beginningat(;)=(5:1;:16)andextendingdowntoabout(;)=(4:51;:20)thecrossoverbecomesabrupt,thoughprobablynotrstorder.Arapidcrossoverisseenintheplaquette,realpartofthePolyakovloop, ,theentropy,andthequarkmassderivedfromtheaxialcurrent.For<4:5,>0:20thetransitiononceagainbecomesverysmooth.Section2discussesthequantitieswemeasured,andsection3summarizesthesimulationsandtheresults.Conclusionsareinsection4.II.MEASUREDQUANTITIESInoursimulationswehavemeasuredtheexpectationvaluesofthePolyakovloop,thespace-spaceandspace-timeplaquettes,thechiralcondensate ,theentropy,screeningmassesformesonsources,thequarkmassdenedbythedivergenceoftheaxialcurrent,andquarkpropagatorsinLandaugauge.TheexpectationvalueofthePolyakovloop,hPi,issimplyinterpretedasexp(Fq=T),whereFqisthefreeenergyofastatictestquark.WithdynamicalquarkshPiisalwaysnonzero,butitincreasesdramaticallyatthehightemperaturetransition.Wealsomeasuredthespace-spaceandspace-timeplaquettes,h2ssiandh2sti.Inournormalizationtheseareequaltothreeonacompletelyorderedlattice.Theenergy,pressure,entropyand withWilsonquarksareobtainedbydierentiatingthepartitionfunctionwithrespecttothetemporalsize,thespatialsize,andthequarkmass,respectively.DetailsaregiveninappendixA.Westudytheentropytolowestorderingand ,usingtheformulae:D E=1N3sNt4Nf2ReTr1My(1)andDsga4E1aNt=13N3sNtXx*@Sg@t+13@Sg@x+13@Sg@y+13@Sg@z+=1N3sNtXx* 8g2+4(CC)! (2st2ss)+(2)andDsfa4E1aNt=fa4+pfa4=2N3sNtNf2Re*Tr1My =D013Xi=Di!+(3)wheresgandsfarethegluonandfermionentropies,respectively.Wemeasuredscreeningmassesformesonsourceswithquantumnumbersofthe,,anda1.Thesemeasurementsareastandardhadronspectrumcalculation,exceptthatthe4

propagationisinthezdirection.Weusedawallsourcecoveringtheentirez=0sliceofthelattice,withthegaugexedtoaspatialCoulombgaugewhichmaximizesthetracesofthex,yandtdirectionlinks.Afterblockingvetotenmeasurementstogethertominimizetheautocorrelations,wetallthepropagatorstoasingleexponentialusingthefullcovariancematrixofthepropagatorelements.Aquarkmasscanbedenedfromthedivergenceoftheaxialcurrent[10,11].Thebasicrelationisacurrentalgebrarelationrh 5 (0) 5 (x)i=2mqh 5 (0) 5 (x)i(4)Ifwesumoverx,y,tslices,andmeasuredistanceinthezdirection,thisbecomes:@@zXx;y;th 5 (0) 53 (x)i=2mqXx;y;th 5 (0) 5 (x)i(5)WedenePS(z)asthepioncorrelatorwithapointsink:PS(z)=DW(0) 5 (z)E(6)andA(z)astheaxialcurrentcorrelator:A(z)=DW(0) 5z (z)E(7)whereW(0)isthewallsourceatz=0.AtlongdistancesbothPS(z)andA(z)willfalloasexp(mz).Thereforeweperformasimultaneousttothetwopropagatorsonalatticeperiodicinthezdirectionusingthreeparameters,C,mandmq,PS(z)=Csinh(m)[exp(mz)+exp(m(Nzz))](8)A(z)=C2mq[exp(mz)exp(m(Nzz))](9)Thefactorofsinh(m)inEq.(8)comesfromusingthelatticedierencef(z+1)f(z1)forthederivativeinEq. (5).NotethatPS(z)isperiodicinzwhileA(z)isantiperiodic.Weusethepointlikeaxialcurrent (z)5 (z)ratherthanapointsplitcurrent.Thesearequarkmassesinlatticeunits;toconverttocontinuumquarkmassesrequresalattice-to-continuumrenormalization.SeeRef.[17]foradiscussionofthispoint.ThequarkpropagatorintheLandaugaugewasalsomeasured.ThispropagatorhasbeenstudiedwithKogut-SusskindquarksinRef.

[12].Wechoseasourceconstantintheydirectionandafunctioninx,zandtwithonlytherealpartoftherstDiraccomponentnon-zero(intheWeylbasisweuse).Becauseofthe-functionallpossiblemomentainx,zandtdirectionswereexcited.TodistinguishamongthedierentmomentaweperformedaFouriertransformofthepropagatorinxandtdirections(takingintoaccountthatithastohaveoddfrequenciesintdirection).Thisgivesthepropagationofthequarkinthezdirectionasfunctionofkxandkt,i.e.thedispersionrelationofthescreeningpropagator.Inordertokeeptheamountofgenerateddataatareasonablelevel,thepropagatorwassaved5

onlyforon-axismomentumvaluesofkxandkt.Thisenabledustomeasuretheon-axisdispersionrelationofthequarkscreeningmass,inparticularthescreeningmassdierenceofthequarkandlightdoublers.Theformtowhichthespatialpropagatoristtedisusuallymotivatedbytheformofthefreepropagator.Wesupposethatatlargedistances,eachseparatemomentumcomponentofthespatialpropagatorresemblesthecorrespondingfreequarkform,butwithitsownrenormalizedquarkmass,orinthiscaseofWilsonfermions,witharenormalized.InmomentumspacethefreeWilsonpropagatorisG(k)=[12Pcos(p)]i2Psin(p)[12Pcos(p)]2+42Psin2(p):(10)WithourchoiceofthesourcetherstDiraccomponentofthepropagator,G1,isreal:G1=[12Pcos(p)][12Pcos(p)]2+42Psin2(p):(11)Then,fornon-zerozvaluesG(p1;p2;z;p0)=LzXk=1exp[i2kz=Lz]G(p1;p2;k;p0)=Lz"(16+4A)24(B+1)8(16+4A)2#"cosh[ma(zLz=2)]sinh(ma)sinh(maLz=2)#;(12)whereA=X=1;2;0sin2[p=2];(13)B=X=1;2;0sin2[p];(14)andma=2sinh10@vuut42B+(18+4A)28(16+4A)1A:(15)Atzeromomentum(onalowtemperaturelatticewherethelowestMatsubarafrequencyisclosetozero)thisrelationturnsintoma=ln162:(16)Themassvanisheswhen!1=8asexpected.Invertingthisforgives=12exp[ma]+6;(17)which,forsmallmassesreducestothenaiverelation,6

=12ma+8(18)thatoneexpectslookingatthetermsoftheLagrangian.Forlargelatticesthelowestdoublermassbecomesmadoubler=lim!1=8ln122=ln[3]=1 :09861:(19)ForKogut-Susskindfermions[12]thefreepropagatorturnsouttobeasumoftwoterms,havingpartswithanalternatingsigninz-direction.ForWilsonfermions,withourchoiceofthesource,thepropagatorisasingleexponential,orhyperboliccosine,onanitelattice.Furthermore,thesignofG1atk=0changesatc=1=8.Therefore,measuringthesignofthepropagatorcanbeusedasanindicatorofwhetheriseectivelygreaterorlessthanc.OnecaninferfromEqs.13{15thattheonlyeectofnitespatiallatticesizeisthediscretisationofthemomenta.Foragivenmomentumalllatticesizesgivethesamevalueofthescreeningmass.Forasmallerlattice,therangeofallowedmomentaismorerestricted,ofcourse.FIG.1.ThespatialscreeningmassatdierentspatialmomentaforfreeWilsonfermionsasafunctionofforan82204lattice.Atc=1=8thehighermassesareforhighermomenta.Tobespecicletuslookatwhathappenswithourlatticesize:82204.Thisisshowningure1.Atcthelowestmomentumscreeningmassisatitsminimum.Ifoneincreasesthescreeningmassesstarttoconvergetoasinglevalueclosetooneat=0:152.Atthispointthedispersionrelationisat.Thesignofthepropagatorwiththissourcedependsonthemomentum.Generally,thevalueofatwhichthesignchangesincreaseswiththemomentum.Forzeromomentum7

itoccursatc;forthesmallestnonzeromomentuminourlatticesizeittakesplaceat=(2+p2p3)=11=0:1374.Theamplitudeforthedoublerdoesnotchangesigninthiskapparange.Forourlatticesize,insertingtheappropriatemomentatoEq. (15)oneobtainsthefollowingquarkscreeningmassesatc=1=8):k=(0;0;0;=4)ma=0:6610k=(=4;0;0;=4)ma=0:8906k=(=2;0;0;=4)ma=1:1171k=(3=4;0;0;=4)ma=1:2149k=(;0;0;=4)ma=1:2411(20)Forpurelytemporalmomentathefreeeldscreeningmassisk=(0;0;0;=4);ma=0:6610k=(0;0;0;3=4);ma=1:0711(21)Hence,thetemporaldoubler'sscreeningmassissmallerthanthatofthespatialdoubler.III.SIMULATIONSANDRESULTSNtNx;y6=g2traj.ignoredtaccept484.90.1806501000.020.88484.90.1813201000.020.94484.90.1828101000.020.90484.90.1825824(c)1000.020.86484.90.1825780(h)1000.020.88484.90.183624(h)1000.020.90484.90.183810(c)6000.020.87484.90.1846101000.020.93485.00.1735401000.0250.86485.00.1755001000.020.92485.00.1774001000.020.90485.00.1784741000.020.94485.00.1806301000.020.92485.00.1822561000.020.86685.00.175240800.020.94685.00.180360600.01670.87885.00.1753501000.01670.91TABLEI.Tableofrunsatxed6=g2withvarying.\(h)"and\(c)"indicatehotandcoldstarts.SimulationswererunontheInteliPSC/860andParagon,andthenCUBE-2attheSanDiegoSupercomputerCenter,ontheThinkingMachinesCorporationCM5attheNational8

NtNx;y6=g2traj.ignoredtaccept485.10.16511201000.0250.89485.10.16726601000.0250.89485.10.1694601000.0250.87485.10.1707001000.0250.90485.10.1719801000.0250.89485.10.17233801000.0250.87485.10.1735001000.0250.85485.10.1754601000.0250.86485.10.17718001000.0250.88485.10.1796601000.0250.88465.10.1698602000.03330.82465.10.17010001000.03330.85465.10.17113601000.03330.81465.10.17211201000.03330.83465.10.1757201000.03330.81885.10.1675121000.0250.82885.10.1732791000.020.87885.10.1774401000.020.66485.30.15524001000.03330.78485.30.1576601000.03330.76485.30.15812391000.03330.89485.30.1596601000.03330.89485.30.16017771000.0250.88485.30.1614801000.0250.88485.30.1624801000.0250.83485.30.1636801000.0250.87485.30.1644601000.0250.90485.30.1657201000.0250.88485.30.1666601000.0250.87485.30.1679121000.0250.86485.30.1685401000.0250.894y85.30.1688401000.0250.80485.30.1693801000.0250.86485.30.1703801000.0250.79485.30.1724401000.0250.876125.30.1553201000.01770.886125.30.160552600.01770.916125.30.1656662160.01770.856125.30.16614034000.01770.846125.30.1677603020.01770.846125.30.1686032000.01770.85TABLEII.Tableofrunsatxed6=g2withvarying.\(h)"and\(c)"indicatehotandcoldstarts.MostoftheNt=4runswereon82204lattices.Therunindicatedwithayat=0:168,6=g2=5:3wasdoneona82404lattice.9

NtNx;y;z6=g2traj.ignoredtaccept484.750.195781000.0142860.912(13)484.7550.1915047500.0142860.960(6)484.76(c)0.198375000.0142860.948(11)484.76(h)0.193621000.0142860.962(12)484.320.20156500.020.83(4)484.360.20172500.020.83(3)484.400.20188500.020.79(3)484.440.20152500.020.77(4)484.480.20368500.020.69(3)484.500.202441000.0142860.84(3)484.520.208411500.0142860.773(16)484.540.205661500.0142860.72(2)484.560.2013242000.0142860.941(7)484.600.20365500.0142860.937(14)484.640.20244500.020.959(14)484.100.2174500.010.79(8)484.200.21267500.0050.90(2)484.260.215861000.0050.85(2)484.280.214781000.005!0.00250.82(2)484.300.214541000.0050.904(16)484.320.21227500.0050.94(2)484.340.21259500.0071430.943(15)484.360.212811000.0025!0.0071430.960(14)484.400.21197500.0025!0.010.95(2)484.440.21249500.007143!0.010.977(9)484.500.21120500.05!0.010.94(3)483.800.2259150.0010.93(4)483.900.2256300.002!0.00040.69(9)483.960.2239250.002!0.00050.36(13)484.000.22119800.004!0.0020.50(8)484.040.22161500.0040.71(4)484.060.22119400.0033330.78(5)484.100.22234500.0050.86(3)484.200.2290500.0071430.98(3)484.300.2258500.005!0.0071431.00(0)484.400.2290500.010.95(3)484.500.22122500.01!0.020.96(2)TABLEIII.Tableofrunsatxedwithvarying6=g2.\(h)"\(c)"indicatehotandcoldstarts.Theacceptancerategivestheaverageoverallrunsinthesamplekeptformeasurement,whetherornotdtwaschangingduringtheruns.10

CenterforSupercomputingApplications,andonaclusterofRS6000workstationsattheUniversityofUtah.WeusedthehybridMonteCarloalgorithmwithtwoavorsofdynamicalquarksinalloursimulations[13].TheparametersofourrunsarelistedintablesI,IIandIII.Forthe82204runsweusedtrajectorieswithalengthofoneunitofsimulationtimeandmademeasurementsaftereverysecondtrajectory.ThestepsizefortheserunsrangedinthenormalizationofRef.[14]from0.033forthelargest6=g2andsmallestto0:02attheotherextreme.Acceptanceratesfortheserunsrangefrom70%to90%,withanaverageoveralltherunsof87%.Forcomputationofthefermionforceintheupdatingandthepropagatorsinthemeasurementsweusedtheconjugategradientalgorithmwitheven-oddILUpreconditioning[15].Theconjugategradientresidual,denedas~My~Mxb=jbjwhere~Misthepreconditionedmatrix,bisthesourcevectorandxisthesolutionvector,was106.Runsweremadeat6=g2=5:3,5:1,5:0and4:9withNt=4.At6=g2=5:3wealsomadeaseriesofrunswithNt=6.At6=g2=5:1werantwopointswithNt=8andat6=g2=5:0twopointswithNt=6.Wealsoranaseriesofsimulationsat6=g2=5:1on62204latticestoverifythatthespatialsizeofthelatticewasnotseriouslyaectingourresults.At6=g2=5:3aseriesofshortrunson634latticeswasmadeforverylarge.Forreferenceweshowaphasediagramfortherelevantrangeofand6=g2inFig.2.Previousworkshowedthatasincreasedfrom0.16to0.19alongtheNt=4hightem-peraturecrossoverlinethepionmassdecreased,suggestingacloserapproachtothehightemperaturetransitioninthechirallimit[3].MorerecentworkbyIwasakietal.,beginningfromthe6=g2=0limit,suggestedthatahightemperaturetransitionforzeroquarkmassmightbefoundat0:225[16].Wehavedoneaseriesofrunson834latticesinwhichwevaried6=g2at=0:20,0:21and0:22toextendthepreviouswork.Asexpected,thenumberofconjugategradientiterationsrequiredintheupdatingincreasesasincreasesinthisrange,andthesizeofthepossibleupdatingtimestepdecreases.Thustheserunshavelimitedstatistics.InFig.3weshowtheplaquetteandPolyakovloopasafunctionof6=g2forthevariousvaluesof.Noticethatthetransitionappearstobesharpestat0:19,becomingsmootherforlargerandsmaller.Eveninthosecaseswherethetransitionisveryabrupt,wedonotseethesortsofmetastabilityandtunnelingcharacteristicofstrongrstordertransitions.Wedondcaseswhereequilibrationtakesalongtime.Theworstcasewasintherunat6=g2=4:9and=0:1825.Inthiscasewehaveplottedtwopoints,fromhotandcoldstarts.ThesepointsaremarkedbyarrowsinFig.3.However,thesetworunseventuallyconvergedtosimilarvalues,lyinginbetweenthevaluesintheearlypartsoftheruns.ThetimehistoryofthePolyakovloopinthesetworunsisshowninFig.4.Wenowexaminethe82204runsinmoredetail.Figure5showstherealpartofthePolyakovloopasafunctionofforthedierentvaluesof6=g2.For6=g2=5:3wealsoincludevaluesforNt=6toshowhowthetransitionpointmovesasNtincreases.Forallofthesevaluesof6=g2weseetheexpectedsharpincreaseinthePolyakovloopatavaluesoflessthanc,wherecisthevalueatwhichthesquaredpionmassvanishesonazerotemperaturelattice.Weestimatecatthesevaluesof6=g2frompublishedvaluesofcinRefs. [7]and[4]andarecentmeasurementat6=g2=5:3bytheHEMCGCgroup:c(5:3)=0:16794[17].Fromaquadraticttothesevalues,shownbyalineinFig.2,wendc(6=g2)=0:1687(2)at5:3,0:1795(4)at5:1,0:1861(12)at5:0and0:1941(40)at4:9.Althoughnotaphysicalquantity,thenumberofconjugategradientiterationsusedin11

FIG.2.Phasediagramshowingestimatesforthehightemperaturetransitionandc.CirclesrepresentthehightemperaturetransitionorcrossoverforNt=4,squaresthehightemperaturetransitionforNt=6anddiamondsthezerotemperaturec.PreviousworkincludedinthisgureisfromRefs. [2],[6],[7],[3]and[4].Weshowerrorbarswheretheyareknown.Forseriesofrunsdoneatxedtheerrorbarsarevertical,whileforseriesdoneatxed6=g2thebarsarehorizontal.Pointscomingfromthisworkareshowninheaviersymbols.ThesolidlinesaretstotforNt=4andtocusedininterpolatingandextrapolating.12

FIG.3.TheplaquetteandPolyakovloopasafunctionof6=g2forvariousvaluesof.ThediamondsarepreviousresultsofRef. [3]for=0:12,0:14,0:16,0:17,0:18and0:19.For=0:12and0:14datafromlongrunsaswellassomedatafromshortrunscollectedwhilegeneratinghysteresisloopsisshown.Theoctagonsat=0:20,0:21and0:22arenewresultsfrom834lattices.Thesquarescomefromrunson82204lattices.Theserunsweredoneatxedvaluesof6=g2withvarying.Theyhavebeenmappedontothisgurebyttingthe6=g2t,tline(withatshownasalineinFig.2),andmovingthepointsinthe,6=g2planeparalleltothisline.Specically,weplotthepointsat6=g2effective=6=g2run@(6=g2t)@t( runt) .Thetfortat6=g2=5:3,5:1,5:0and4:9is0.1579,0.1713,0.1772and0.1827respectively.13

FIG.4.TimehistoryoftherealpartofthePolyakovloopforrunswithhotandcoldstartsat6=g2=4:9and=0:1825.solving~My~Mx=bindicateshowsingular~Misontheaverage.ThisquantityhasbeenusedasaprobeofthephysicsinRef. [16]InFig.6weshowtheaveragenumberofconjugategradientiterationsusedinanupdatingstep,wherealinearextrapolationofthelasttwotimestepswasusedtoproduceastartingguessforthesolutionvector.For6=g2=5:3andNt=4thereisverylittleeectonthenumberofiterationsatt.As6=g2isdecreasedforNt=4thereisanincreasinglysharppeakinthenumberofiterationsatc.NoticealsothesharppeakintheNt=6resultsfor6=g2=5:3.Figure7showstheaverageplaquetteintheseruns.Ournormalizationissuchthattheplaquetteisthreeforalatticeofunitmatrices.Theplaquettealsoshowsasharpriseasthehightemperaturecrossoverispassed.Noticethatfor6=g2=5:3wehaveresultsforNt=4and6showingthatthisincreaseisinfactduetothetimesizeofthelattice,orthetemperature.Thechiralcondensate islessusefulforWilsonquarksthanforKogut-Susskindquarks,sinceitdoesnotgotozerointhehightemperaturephasewithoutdicultsubtractions.NeverthelessweplotitinFig.8.Thereisacleardropin asthehightemperaturetransitioniscrossed.Thisdropincreasesdramaticallyas6=g2decreases.Perhapsthemostphysicallyrelevantobservableistheentropy.InFig.9weplotTsinunitsofa4.Togiveanideaofthenormalizationofthisgraph,foreightgluonsinfreeeldtheoryonan88204latticethegluonentropywouldbeTsglue;free=0:040a4,whilefortwoavorsoffreeWilsonquarksat=c=0:125theentropywouldbeTsquark;free=0:125a4.Theeectsofthelatticespacingandspatialsizeareverylargehere;inthecontinuumwithinnitespatialextentthesenumbersare0:027and0:036respectively.Strangely,whenwedividetheentropyintogaugeandfermionpartsasinEqs.2and3we14

FIG.5.ExpectationvalueofthePolyakovloopasafunctionofforthevariousvaluesof6=g2.ResultsareshownforNt=4for6=g2=5:3,5:1,5:0and4:9(octagons).For6=g2=5:3wealsoshowresultsforNt=6(diamonds).Thecrossesalongthe6=g2=5:1lineareresultsona62244latticeat6=g2=5:1,toshowthatthespatialsizeofthelatticeisnotgreatlyaectingtheresults.Thedottedsymbolsextendingthe6=g2=5:3lineareshortrunsona634lattice,showingthatthebehaviorissmoothouttoverylarge.Theverticallinesmarkthezerotemperaturecfor6=g2=5:2,5:1and5:0respectively.(For6=g2=4:9,c(T=0)0:194. )15

FIG.6.Conjugategradientiterationsforupdatingstep,asafunctionofforthevariousvaluesof6=g2.AgainthediamondsareNt=6resultsat6=g2=5:3.FIG.7.Expectationvalueoftheplaquetteasafunctionofforthevariousvaluesof6=g2.HereweincludedvaluesforlargerNttoemphasizetheeectofthetemperature.Thedottedsymbolsfor6=g2=5:3areshortrunsona634latticeextendingtofarbeyondthezerotemperaturecshownbytheverticalline.16

FIG.8.Expectationvalueof asafunctionofforthevariousvaluesof6=g2.Thedottedsymbolsfor6=g2=5:3areshortrunsona634latticeextendingtofarbeyondthezerotemperaturecshownbytheverticalline.FIG.9.Entropy(actuallyTsa4)asafunctionofforthevariousvaluesof6=g2.Againweshowthe62244resultsforcomparison.17

ndthatthegaugeentropyiscomparabletothefermionentropyinsteadofmuchsmalleraswouldbethecasewithfreeeldsonalatticeofthissize.FIG.10.Quarkmassfromtheaxialcurrentasafunctionofforthevariousvaluesof6=g2.Pointsmarkedwithquestionmarksindicaterunswherewewereunabletogetconsistenttsasafunctionofdistance.Theplussignsonthemq=0linearethezerotemperaturecfor6=g2=5:3,5:1and5:0.ThequarkmassdenedbythedivergenceoftheaxialpionpropagatorisplottedinFig.10.WhenthisquarkmasswassmallwehadgreatdicultyingettinggoodtstotheformsinEqs.8and9.Thisisexpected,becausewhenthequarkmassissmalltheamplitudeforthepropagatorA(z)isverysmall.Additionally,thereisatendencyfortheeectivequarkmass,orthequarkmasscomingfromatoverashortdistancerange,toincreasewithdistancefromthesource.Incaseswherewewereunabletogetatwithasatisfactory2orwherethequarkmasswasnotconvincinglyindependentofdistance,weplotthepointwithaquestionmarkinFig.10.Topursuethisfurtherweranoneofthedicultpoints,6=g2=5:3and=0:168,ona82404lattice,allowingustomeasuretheratioouttoadistanceoftwenty.Figure11summarizestheresults.InthisgureweshowtheeectivepionmassobtainedfromPS(z)andA(z)byttingtwotwosuccessivedistances,andthequarkmassobtainedfromsimultaneouslyttingbothpropagatorsatthetwosuccessivedistances(aonedegreeoffreedomt).Unfortunately,inallothercasesthelatticewasonlytwentysiteslongandwehavetodrawconclusionsfromdistanceslessthanten.InFig.10weseethatwhentheNt=4latticeentersthehightemperatureregimethepointlikeaxialcurrentquarkmassnolongeragreeswiththelowtemperaturelattices(Nt=6and8).Theplussesatmq=0inFig.10areestimatesforthezerotemperaturec.Theaxialcurrentquarkmassesgothroughzeroatlessthanthezerotemperaturec.Whentheaxialcurrentquarkmassvanishes,thesystemisinthehightemperature18

FIG.11.PioneectivescreeningmassesfromPS(z)(circles)andfromA(z)(squares),andtheeectivequarkscreeningmassfromtheirratio.Theresultsarefroman82404latticewith6=g2=5:3and=0:168.phasefor>5:0,whileat=4:9ktapearstocoincidewiththepointwheretheaxialcurrentquarkmassvanishes,withinexperimentaluncertainty.Notehowever,thatthepionscreeningmassintheconnementphaseisstillnonzeroatthetransitionpointat=4:9.InFig.12weshowthesquaredpionscreeningmassesintheseruns.Againweseeanincreasinglysharpdipattas6=g2decreasesandincreases.Theappearanceofthecuspat=5:1coincideswiththebeginningoftheregionwherethetransitionisabrupt.Screeningmassesforthe,,anda1mesonsareshowninFigs.13,14and15.Inallcasesweseethescreeningmassescomingtogetherasthehightemperaturetransitioniscrossed.However,wedonotseeanyindicationthattheora1splittingsinthehightemperatureregimearedecreasingas6=g2decreases.Althoughthesmallerpionmassesinthecoldregimesuggestthatchiralsymmetryisbeingapproachedaswemovetowardsmaller6=g2alongthetline,wedonotseethistrendinthehightemperaturescreeningmasses.Alsonoticethattherearenonzerosplittingsbetweentheparitypartnersatthepointswheretheaxialcurrentquarkmassiszero.Thusthevanishingofthisquarkmassisnotanindicatorforcompletechiralsymmetryrestorationinthesystem.ToinvestigatethecontributionsofthedoublerstothermodynamicquantitiessuchastheentropywemeasuredtheeectivemassesfromthequarkpropagatorinLandaugaugeatafewvaluesofand6=g2.Wendthatttingthequarkscreeningpropagatorsismoredicultthanttingthemesonpropagators.Inpartthisisbecausethequarkpropagatorsuctuatemorefromcongurationtoconguration.Therealsoseemstobeasystematictrendtowardlargereectivequarkmassesatlargerdistances.Withthesecaveats,themassesofthephysicalquarkandthelightestdoublersaregiveninTableIV.Thetswere19

FIG.12.Pionscreeningmasssquaredasafunctionofforthevariousvaluesof6=g2.Again,thecirclesareforNt=4,thediamondsforNt=6andthecrossesforNt=8.TheburstsarezerotemperaturepionmassesfromtheHEMCGCcollaborationat6=g2=5:3.FIG.13.Mesonscreeningmassesfor6=g2=5:30.ThepointsconnectedbysolidlinesareforNt=4andthepointsconnectedbydashedlinesforNt=6.20

FIG.14.Mesonscreeningmassesfor6=g2=5:10atNt=4.FIG.15.Mesonscreeningmassesfor6=g2=4:90atNt=4.Thetwopointsat=0:1825arefromcoldandhotstarts.21

selectedbychoosingthelargesttrangethatgivesanacceptablecondencelevel.TherangesandcondencelevelsarealsogiveninTableIV.Ntsignma(0;4)ma(;4)ma(0;34)masmatqrange40.1655.10+1.13(4)2.3(4)1.8(2)1.1(4)0.7(2)0.473-1040.1675.10+1.13(4)2.4(5)1.8(3)1.3(5)0.6(3)0.143-1040.1725.10+0.97(16)1.5(3)1.5(2)0.5(3)0.6(4)0.434-1040.1775.10{1.05(9)1.28(15)1.33(11)0.23(19)0.27(15)0.694-840.1555.30+1.06(2)1.63(13)1.65(11)0.57(13)0.60(11)0.683-940.1605.30+0.89(4)1.38(9)1.22(5)0.49(9)0.34(7)0.843-1040.1675.30{0.92(6)1.51(9)1.36(7)0.59(11)0.44(10)0.573-9TABLEIV.Thescreeningmassesforthequarkandthelightestdoublers.mas(t)isthedierenceofthespatial(temporal)doublerscreeningmasstothequarkscreeningmassatthelowestmomenta.ThesignisforG(kt==4).Thetsweredonesimultaneouslytoallthreepropagatorstakingintoaccountthecrosscorrelations.Thecondencelevelqandtherangeofeachtisalsodisplayed.IV.CONCLUSIONSThemostnaiveexpectationregardingthethermodynamicsoftwoavorsofWilsonquarksatxedNtisthattherewouldbealineinthe;planeatwhichaconnement-deconnementtransitionoccurs,thatthetransitionwouldbesmooth(crossoverorsecondorder),thatthepionmasswouldsmoothlydecreasealongthatline,andthatatsomepoint,possiblycorrespondingtothepointwherethetransitionlinecrossedthezerotemperaturecline,thepionmasswouldgotozero.Atthatpointonewouldhaveanitetemper-atureconnement-deconnementorchirallyrestoringtransitionanalogoustothatseeninstaggeredfermions.Simplearguments[9]wouldputthispointaround=5:0atNt=4.Thesenaiveexpectationsarenotborneoutbythedata.Thechirallimitisreachedataverysmallvalueifitisreachedatall.However,near=5:0Nt=4Wilsonthermodynamicsdisplaysanumberoffeatureswhichhavenoanalogsinstaggeredfermionsystems.Thetransitionbecomesverysharp,thoughnotrstorderasfaraswecantell.Acuspinthepionscreeningmassappearsasonecrossesfromtheconnedtothedeconnedphase.TheaxialvectorquarkmassbecomesstronglyNtdependentatthispointandforsmallNtdoesnotgotozeroatitszerotemperaturevalue(atxed).Thesharptransitionpersistsdownto=4:5;=0:20orso,atwhichpointitisonceagainbecomessmooth.Asfaraswecantell,thezerotemperature=cpointplaysnoroleinanyNt=4eectswehaveobserved.Itistemptingtospeculatethatthecrossoverlineinthe,6=g2planeisclosetosomephaseboundarywherethetransitionissteepest.WearecurrentlyexploringthisregionwithNt=6,wherepreliminaryresultsindicateachangeinthenatureofthehightemperaturetransitionaroundthisvalueof.Indicatorsforthenatureofthehightemperaturephasegiveasomewhatmixedpicture.Itisclearfromthemesonscreeningmassesandfrom thatchiralsymmetryisatleast22

partiallyrestoredathightemperature.Whiletheaxialcurrentquarkmassgoestozerotheanda1splittingsinthescreeningmassesremainnonzero.QuarkpropagatorsintheLandaugaugesuggestalargeconstituentquarkmassatthetransition,atleastfor6=g2=5:3and5:1.Thisisconsistentwithearlierwork[3]whereattheNt=4crossoverpointnearthese(;)valuesthezerotemperaturepionwasfoundtobequiteheavy.Noticethattheseriesofrunsat6=g2=5:3extendstosignicantlylargerthanthezerotemperaturec,andthereisnonoticeableeectonanyofthemeasuredquantitieswhenthisciscrossed.(Infact,wehavedoneshortrunson834latticesforaslargeas0.19at6=g2=5:3andseennoeects. )Thesignofthepropagatorofthezeromomentumquark,shownintableIV,isconsistentwiththesignoftheaxialcurrentquarkmass.Bothofthesequantitiesarebehavinginthewayonewouldexpectinafreeeldtheoryat>c.ACKNOWLEDGMENTSThesecalculationswerecarriedoutontheiPSC/860,theParagonandthenCUBE2attheSanDiegoSupercomputerCenter,ontheCM5attheNationalCenterforSuper-computingApplications,on15IBM/RS6000workstationsinthePhysicsDepartmentattheUniversityofUtah,onanIBM/RS6000clusterattheUtahSupercomputingInstituteandonourlocalworkstations.Wearegratefultothestasofthesecentersfortheirhelp.WealsothankTonyAndersonandReshmaLalofIntelScienticComputersfortheirhelpwiththeParagon.WewouldliketothankAkiraUkawaandFrithjofKarschforhelpfuldiscussions.SeveraloftheauthorshaveenjoyedthehospitalityoftheInstituteforNuclearTheory,theInstituteforTheoreticalPhysicsandtheUCSBPhysicsdepartment,wherepartsofthisworkweredone.ThisresearchwassupportedinpartbyDepartmentofEnergygrantsDE-2FG02{91ER{40628,DE-AC02{84ER{40125,DE-AC02{86ER{40253,DE-FG02{85ER{40213,DE-FG03{90ER{40546,DE-FG02{91ER{40661,andNationalScienceFoundationgrantsNSF{PHY90{08482,NSF{PHY93{09458,NSF{PHY91{16964,NSF{PHY89{04035andNSF{PHY91{01853.APPENDIXExpressionsfortheenergy,pressureand arefoundbydierentiatingthepartitionfunctionwithrespectto1=T,volumeandquarkmass,respectively.First,wewritetheactionwithadjustablelatticespacingsinalldirections.Introducingdimensionlessparameters,wewritethelatticespacinginthedirectionasa=a.Clearlythisisredundant,sincewehaveveparameters,aandtospecifyfourlatticespacings,butitisconvenientandsymmetric.IntheconventionalnotationofKarsch,=i=t,whereallthespatial'sarethesame.Whenwearedonetakingderivatives,allthewillbesettoone.ThepartitionfunctionisZ=Z[dU]eSg+Sf(22)wherethegaugeactionis23

Sg=XxX>2g2xyzt222(23)where2istheplaquetteintheplanenormalizedtothreeforunitmatrices.Weallowadierentgaugecouplinggineachplane.ThefermionactionisSf=nf2TrlogMyM(24)whereM=1X=D(25)where=D=(1+)U(x)y;x+^+(1)Uy(x^)y;x^(26)Theinthecoecientof=Dtakescareofthedimensionalscalingoftherstderivative.NoticethatwehavemadeasomewhatarbitrarychoiceinMwhenwescaledtheirrelevantsecondderivativepartwithinthesamewaythatwescaledtherstderivativepart.TheinthecoecientmustbeadjustedtogetcorrelationfunctionstobeEuclideaninvariant.ItsroleissimilartotheKarschcoecentsCandCinthegaugeaction.PresumablyhasapowerseriesexpansioningjustasCandC.Onceagainwehavemoreparametersthanweneed:fourandforfourdirections.Thisparameterizationisconvenientbecauseitincludesthecustomaryand,later,casparameters.Wecanxtheambiguitywellenoughforourpurposesbyrequiringthat=1whenalltheareequal.Inotherwords,ifalldirectionsarescaledbythesamefactortheonlythingthatchangesis.Letcbethevalueofatwhichthepionmassandquarkmassvanish,atleastonaninnitelattice.Followingfreeeldtheory,weintroduceaquarkmass2ma=11c(27)sothatM=11c+2ma 2ma+1cX=D!(28)Here1cwillbeafunctionofthecouplingsgandthescalefactors.Infreeeldtheory,1c=2P1.(=1infreeeldtheory. )Wendtheenergy,pressureand bydierentiatingthepartitionfunction:=1V@logZ@jV;mconstant(29)p=1@logZ@Vj;mconstant(30) =1V@logZ@mj;Vconstant(31)24

HereVisthevolume,V=a3QiNii=a3N3s3s.istheinversetemperature,=aNtt.Heretheenergyandpressurederivativesaretakenwithmconstant,ratherthanwithconstant.Thisisbecausecdependsonthe,sothatifwedistortthelatticewhileholdingxed,thequarkmass,andeveryphysicalmass,willvarysharply.Wechangethetemperatureandvolumeusing@@=1Nta@@t(32)and@@V=13N3sa32s@@s(33)Alternatively,itmaybeeasiertovarythevolumebyvaryingonlyoneofthespatiallatticespacings@@V=1N3sa3xy@@z(34)Thegaugeenergyandpressurearestandard[18{20].FollowingRef.[18],wedenetwoderivatives.C=@g2@(35)whereisnotoneoforandC=@g2@(36)whereisoneofor.Becausestretchingboththetimeandspacedirectionsisequivalenttochangingthelatticespacing,CandCarerelatedtothebetafunction. X@@!g2ab=2C2C=@g2@log(a)(37)ThecontributionstotheenergyandpressurefromSgandSfadd.Doingthedierenti-ation,andthensettingthetoone,thegluonenergyisga4=1N3sNtXx*6g2(2st2ss)+6C2ss+6C2st+(38)Here2ssand2starethespace-spaceandspace-timeplaquettes,againnormalizedtothreeforalatticeofunitmatrices.Forthegluonpressure,wendpga4=1N3sNtXx*2g2(2st2ss)2C(2ss+22st)2C(22ss+2st)+(39)25

Wealsoconsiderthelinearcombination+p,theentropy.sga41aNt=ga4+pga4=1N3sNtXx* 8g2+4(CC)!(2st2ss)+(40)TheentropyisobviouslyzeroatT=0.Justasthegaugecouplingsvarywiththelatticespacings,1candvarywiththelatticespacingsaswetrytoholdmxed.Thereisanexplicitdependenceof1conmplusadependenceof1cong,wheregisvaryingwiththe.Nowitclearlydoesn'tmatterwhichdirectionwestretchthelattice,since1cisdenedontheinnitelattice,so@1c@t=@1c@x... :(41)Therefore@1c@sj=1=3@1c@tj=1(42)Therearetwoindependentderivativesofthe,analogoustoCandC.DeneB=@@(43)andB=@@;6=(44)Tocompute wecanjustsetandtooneatthebeginning.a3 =1N3sNtNf2*@@maTrlogMyM+=1N3sNtNf2*Tr1MyM @My@maM+My@M@ma!+=1N3sNtNf2*Tr 1My@My@ma+1M@M@ma!+(45)Thetwopartsarecomplexconjugates,sokeeponlyoneandtaketwicetherealpart.UsingEq.

(28)a3 =1N3sNtNf22Re*Tr1My 2(1c+2ma)2 2ma+1cX=Dy!+2 11c+2ma! !+=1N3sNt4Nf2ReTr1My1(46)26

whereinthelaststepweused1=1c+2ma=.Lookingatthederivationshowsthatthe1inEq.(46)comesfromdierentiatingthe1=1c+2maoutsidetheparenthesesinEq. (28).HadwetakenthefermionmatrixtobeM= 2ma+1cX=D!

(47)thistermwouldbeabsent.SincethislatterformisclosertotheusualcontinuumLagrangian,weprefer =1N3sNt4Nf2ReTr1My(48)asourexpressionfor .Nowforthefermionenergy,dierentiateEq. (25)andthensetandtoone:fa4=1N3sNtNf2*2ReTr1My@My@t+=2N3sNtNf2Re1My"1(1c+2ma)2 2ma+1cX=Dy!

@1c@t(49)+11c+2ma =Dy0X@@t=Dy+@1c@t!#=2N3sNtNf2Re*@1c@t11My1My =Dy0B=Dy0XiB=Dyi!+(50)whereinthelaststepweused1=1c+2ma=.Forthefermionpressure:pfa4=13N3sNtNf2*2ReTr1My@My@s+=23N3sNtNf2Re1My"1(1c+2ma)2 2ma+1cX=Dy! @1c@s(51)+11c+2ma Xi=DiX@@s=Dy+@1c@s!#=23N3sNtNf2Re*@1c@s11My+1My 3B=Dy0+(2B+B)Xi=DyiXi=Dyi!+(52)Justasfor ,the1terminfandpfcomesfromdierentiatingtheoverallfactorof1.Itcanbeincludedornot,asdesired.Itwillcancelwhenthenitepartsoftheenergyandpressurearecalculatedbysubtractingthezerotemperatureresultfromthenonzerotemperatureresult.However,the1My@1c@ttermwillnotcancelout,since istemperaturedependent.InpracticehTr1=MyiisfairlyclosetohTr1i=43,sonumericallyitmaybebesttoleavethe1in.Thenwewoulduse27

11My==DyMy(53)toexpresstheenergyandpressurejustintermsoftheexpectationvaluesofthespatialandtemporalcomponentsof=Dy.Muchofthedicultycancelsoutifwelookattheentropy,orsumofenergyandpressure:sfa41aNt=fa4+pfa4=2N3sNtNf2Re*Tr1My(1+(BB)) =D013Xi=Di!+(54)Therelation@1c@s=3@1c@t(55)resultedinallthederivativetermscancelling.(Remember,by@=@swemean@=@x+@=@y+@=@z|varyallthespatialitogether.)Asusual,theentropyisobviouslyzeroatT=0,where=Di==D0.Sincenozerotemperaturesubtractionisrequiredfortheentropy,thetermsinvolvingBandBwillbehigherordering2thanthe\1"term,andwehaveneglectedtheminEq.(3).Obviously,thebigproblemingettingtheenergyandpressureseparatelyistond@1c@andBandB.FromEuclideaninvariance,@1c@isindependentof.Thevariable1cdependsontheintwoways.First,thereisan\explicit"dependence.Fromexaminingthefermionmatrix,Eq.(25),weseethatifallthearescaledtogetherwithgheldxed,cisproportionalto.Thus@1c@explicit=1c4(56)Secondly,thereisan\implicit"dependenceof1concomingfromthefactthatcdependsong2,andweadjusttheg2asweadjustthe.Again,Euclideaninvariancesays@1c@implicit=14@1c@log(a);(57)soonlythebetafunctionappears. @1c@implicit=14@1c@log(a)=14@1c@6=g2@6=g2@log(a)(58)Wecouldproceedbyestimating@1c=@6=g2fromourdataatvariousvaluesof6=g2,orfromcorrelationsofthehadronpropagatorswiththeplaquette.Similarly,wecouldtakeabetafunctioneitherfromperturbationtheoryorfromsomesetoflatticesimulations.Wewillnotsolvethisproblemhere,sowewillonlyquotetheentropyratherthantheenergyandpressureseparately.28

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