Tricritical point of lattice QCD with Wilson quarks at finit

First principle study of QCD at finite temperature $T$ and chemical potential $\mu$ is essential for understanding a wide range of phenomena from heavy-ion collisions to cosmology and neutron stars. However, in the presence of finite density, the critical

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002 voN 61 3v9105040/tal-ep:hviXraTricriticalpointoflatticeQCDwithWilsonquarksat nitetemperatureanddensity

Xiang-QianLuo

CCAST(WorldLaboratory),P.O.Box8730,Beijing100080,China

DepartmentofPhysics,Zhongshan(SunYat-Sen)University,Guangzhou510275,China

(February1,2008)FirstprinciplestudyofQCDat nitetemperatureTandchemicalpotentialµisessentialforunderstandingawiderangeofphenomenafromheavy-ioncollisionstocosmologyandneutronstars.However,inthepresenceof nitedensity,thecriticalbehaviorlatticegaugetheorywithoutspeciesdoubling,isunknown.Atstrongcoupling,weexaminethephasestructureonthe(µ,T)plane,usingHamiltonianlatticeQCDwithWilsonfermions.Atricriticalpointisfound,separatingthe rstandsecondorderchiralphasetransitions.Suchatricriticalpointat niteThasnotbeenfoundinpreviousworkintheHamiltonianformalismwithKogut-Susskindfermionsornaivefermions.12.38.Gc,11.10.Wx,11.15.Ha,12.38.Mh

PublishedinPhys.Rev.D70,091504(RapidCommun.)(2004).

I.INTRODUCTION

InLagrangianformulationofSU(3)LGTat niteµ,complexactionspoilsnumericalsimulationswithimpor-Oneofthemostchallengingissuesinparticlephysicstancesampling.TherecentyearshaveseenenormousistostudyQCDinextremeconditions.Precisedetermi-e orts[4–6]onsolvingthecomplexactionproblem,andnationoftheQCDphasediagramontemperatureTandsomeveryinterestinginformation[7]onthephasedia-chemicalpotentialµplanewillprovidevaluableinfor-gramforQCDwithKSfermionsatlargeTandsmallmationforquark-gluonplasma(QGP)andneutronstarµhasbeenobtained.Nevertheless,topreciselylocatephysics.

thetricriticalpointandcriticallineatlargeµisstillForQCDwithtwomasslessquarks,severalapproxi-anextremelydi culttask.Onehasalsotoresolvethemationmodels(e.g.linearsigmamodel,Nambu-Jona-contradictionbetweenMonteCarlo(MC)simulationsatLasinomodel,randommatrixmodel,statisticalboot-intermediatecouplingandstrongcouplinganalysisfor1strap)[1]suggesttheexistenceofatricriticalpointonstaggered avor(correspondingto4 avorsinthecontin-the(µ,T)planeseparatingthe rstordertransitionlineuum):MCdata[6]indicatethatthechiralphasetran-atlowerTandlargerµ,andthesecondordertransitionsitionatµ=0andsome niteTCisof rstorder,lineathigherTandsmallerµ.Therehasbeenapro-whilestrongcouplinganalysis[8,9]favorsthesecondor-posal[2]forexperimentalsearchforthetricriticalpoint,dertransition;TheauthorofRef.[9]studiesthephaseviaevent-by-event uctuationsinheavy-ioncollisions.diagramonthe(µ,T)plane,anddiscoversatricriticalLatticegaugetheory(LGT),proposedbyWilson[3]point,whileinMCsimulation[6],onlyalineof rstor-isthemostreliablenon-perturbativeapproachtoQCD,derphasetransitionisfound.Itmightwellbethattheybasedon rstprinciples.Unfortunately,itsu ersprob-belongtodi erentuniversalityclasses.

lemslikethecomplexactionat niteµandspeciesdou-QCDatlargeµisofparticularimportanceforneu-blingwithnaivefermions.

tronstarorquarkstarphysics.Hamiltonianformula-Kogut-Susskind(KS)’sapproachtolatticefermionstionofLGTdoesn’tencounterthenotorious“complexthinsthedegreesoffreedom,andpreservestheremnantactionproblem”.Recently,weproposedaHamiltonianofchiralsymmetry,butitdoesn’tcompletelysolvetheapproachtoLGTwithnaivefermionsat niteµ[10,11],speciesdoublingproblem;Itbreaksthe avorsymme-andextendedittoWilsonfermions[12].Thechiralphasetryaswell.Wilson’sapproachtolatticefermions[3]transitionatT=0andsome niteµCwasfoundtobehasbeenextensivelyusedinhadronspectrumcalcula-of rstorder.InRefs.[13,14],theauthorsstudiedthetionsaswellasinQCDat nitetemperature;ItavoidsphasediagramofQCDwithKSfermionsfor2and4 a-thespeciesdoublingandpreservesthe avorsymmetry,vorsandnaivefermionsfor4 avors.Alineofsecondbutitexplicitlybreaksthechiralsymmetry,oneoftheorderphasetransitionwasfoundinbothcases,butnomostimportantsymmetriesoftheoriginaltheory;Non-tricriticalpointwasfoundatany niteT.

perturbative ne-tuningofthebarefermionmasshastoInthispaper,westudythephasediagramusingHamil-bedone,inordertode nethechirallimit.

tonianlatticeQCDwithWilsonfermions.Atstrongcou-

First principle study of QCD at finite temperature $T$ and chemical potential $\mu$ is essential for understanding a wide range of phenomena from heavy-ion collisions to cosmology and neutron stars. However, in the presence of finite density, the critical

plingandinthenon-perturbativelyde nedchirallimit,we ndatricriticalpointonthe(µ,T)plane.Therestofthepaperisorganizedasfollows.InSec.II,wederivethee ectiveHamiltonianat niteTandµ.InSec.III,weanalyzetheQCDphasediagram.Theresultsaresum-marizedinSec.IV.

II.EFFECTIVEHAMILTONIANATTHE

STRONGCOUPLING

A.Theµ=0case

AccordingtoEq.(3)andfollowingtheprocedureinSec.IIA,theroleoftheHamiltonianatstrongcouplingisnowplayedby

µHeff=Heff µN.

(5)

ThevacuumenergyistheexpectationvalueofH µN

initsgroundstate| ,andalsotheexpectationvalueofµHeffinitsgroundstate| eff ,givenby

µ

E = |H µN| = eff|Heff| eff .

(6)

WebeginwithQCDHamiltonianwithWilson

fermionsatµ=0on1dimensionalcontinuumtimeandd=3dimensionalspatialdiscretizedlattice,

¯(x)ψ(x)H=Mψ

x

C.ResultsforlargeNc

+

1

d xj=1

AsshowninRef.[12],underthemean- eldapproxima-tion,i.e.,byWick-contractingapairoffermion eldsin

thefourfermiontermsinHeff,onecanobtainabilinearHamiltonianinleadingorderof1/Nc

µµ¯(x)ψ(x)ψHeff～HMFA=A

x

2a

1

αα

Ej(x)Ej(x)

+(B µ)

whereA=M

a4aNc

Kd

v

2

2

x

ψ (x)ψ(x)+C,

(7)

a

,(2)

m,a,randgarerespectivelythebarefermionmass,

spatiallatticespacing,Wilsonparameter,andbarecou-plingconstant.U(x,k)isthegaugelinkvariableatsite

.Theconventionγ k= γkisused.xanddirectionk

αEj(x)isthecolor-electric eldatsitexanddirectionjandUpistheproductofgaugelinkvariablesaroundanelementaryspatialplaquette.

For1/g2<<1,onecanintegrateoutthegauge eldsandderivethee ectiveHamiltonianHeff,consistingoftermswithtwofermionsandfourfermions[15,16].

B.TheT=0andµ=0case

(1+r)v (1 r)¯v

22

NsNf.(8)

Thecoe cientAplaystheroleofdynamicalmassof

¯quark.v¯andv arerespectivelytheexpectationofψψ

andψ ψin| eff ,pidedbyNsandNf,i.e.,thetotalnumberoflatticesitesandthenumberof avors.

K=

1

Inthecontinuum,thegrandcanonicalpartitionfunc-tionofQCDat niteTandµis

Z=Tre β(H µN),

β=(kBT) 1,

(3)

aNc

v¯.

(10)

wherekBistheBoltzmannconstantandNisparticle

numberoperator

N=ψ (x)ψ(x).(4)

x

Inthechirallimit,thereareonlytwofreeparameters

left:randµinEq.(7).Thevacuumenergyis2

First principle study of QCD at finite temperature $T$ and chemical potential $\mu$ is essential for understanding a wide range of phenomena from heavy-ion collisions to cosmology and neutron stars. However, in the presence of finite density, the critical

¯ 1) E =2NcNfNsMchiral(n+n

a

2 Kd

n+n¯2+1 2¯n+

Kdr2

′2TC

1

′2TC

1

1

,

1+3r2

(15)

nlnn+(1 n)ln(1 n)

a

+n¯lnn¯+(1 n¯)ln(1 n¯),

whichisdepictedbythedottedlineforr=1inFig.4.

Inthelowerandleftcorner,thechiralcondensateanddynamicalmassofquarkarenon-zero.Intheotherside,theyvanishidentically.

′

Belowsome niteT3,thesituationisdi erent.Thereisa rstorderchiralphasetransitionline

2

µ′C=1+2r.

(16)

(13)

istheentropy.

Oncenandn¯areknown,thechiralcondensateandquarknumberdensityintheleadingorderof1/Nccandirectlybeobtained

¯ |xψ(x)ψ(x)| ¯ = ψψ

v

2NcNfNs

1→

′

fromsome niteT3downtoT′=0.thisisillustratedbythesolidlineforr=1inFig.4.Chiralcondensateanddynamicalmassofquarkjumpfromnon-zeroforµ′<µ′C

′′

tozeroforµ>µC.Thisisconsistentwiththedataob-tainedbyminimizingΦ,asshowninFigs.1,2and3.µ′Cislargerthanthedynamicalmassoffermionatzerotem-perature.Thereason,asexplainedindetailsinRef.[12],isduetothefactthatWilsonfermionsbreakexplicitlythechiralsymmetry.Thisseemscounter-intuitive,sincefromathermodynamicalpointofview,atransitionisex-pectedwhenthechemicalpotentialequalsthedynamicalfermionmass.Itwouldbeinterestingtoseewhetherthedi erencedisappearsinthecontinuumlimit.

ThepointswhentwolinesdescribedbyEq.(15)and

′′′′′′

Eq.(16)joinare(µ′3,T3)and(µ3,T).(µ3,T3)isthetri-criticalpoint,whiletheotheroneathightemperatureisjustacriticalpointonthesecondorderphasetransitionline.TableIgivesthelocationofthetricriticalpointfor

′

variousr.Forr=1,we nd(µ′3,T3)=(3,0.4498),i.e.thecircleinFig.4.Thephasestructureforanyr=0isqualitativelythesame.Notethatthesystemalongthe

+

lineµ′=µ′3+0experiencesaninvertedbehavior:For

′

T′∈[0,T3),thesystemisinthechiral-symmetricphase;

′

WhileforT∈(T3,T′′),thesystementersintothechiral-brokenphase;ForT′>T′′,thesystemisagaininthechiral-symmetricphase.ThevalueofT′′isalsogiveninTab.I.Suchabehaviorexistsevenfornaivefermions,thoughthereisnotricriticalpointat niteTwhenr=0.Figure5showstheresultsforthechiralcondensateandquarknumberdensity(14)asafunctionµ′forT′=1andr=1,abovethetricriticalpoint;Theresultsin-dicatethereisasecondorderchiralphasetransitionatµ′=3.1770.Figure6showsthoseforT′=0.25andr=1,belowthetricriticalpoint;Thereisa rstorderchiralphasetransitionatµ′=3.Figure7showsthechiralcondensateforµ′=0andr=1;ThereisasecondorderchiralphasetransitionatT′=2.TheseresultsareobtainedbylocatingtheminimumofΦ,andthecriticalpointisconsistentwiththepredictionofEqs.(15)and(16).

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First principle study of QCD at finite temperature $T$ and chemical potential $\mu$ is essential for understanding a wide range of phenomena from heavy-ion collisions to cosmology and neutron stars. However, in the presence of finite density, the critical

IV.DISCUSSIONS

Intheprecedingsections,wehaveinvestigatedtheQCDphasediagraminHamiltonianlatticeformulationwithWilsonfermions.Atthestrongcoupling,we ndatricriticalpointonthe(µ,T)plane,whichhasnotbeenfoundinpreviousworkintheHamiltonianfor-malismwithKogut-Susskindfermionsornaivefermions.Our ndingsimplythatonthe(µ,T)phase,thephasestructureofQCDwithWilsonfermions(withoutspeciesdoubling)mightbequalitativelydi erentfromnaiveorKogut-Susskindfermions(withspeciesdoubling).Fur-therdetailedlatticestudywillbeveryimportantforun-derstandingtheQCDphasediagram.

r=1,µ′=0,T′=0.4

2NcNfNs

-1.5-2.0-2.5-3.0

0.20.4

n0.60.8

10.80.6¯0.4n

0.2

10

ACKNOWLEDGMENTS

FIG.1.3Dplotofthegrandthermodynamicpotentialasafunctionofnandn¯atr=1,µ′=0,andT′=0.4,wherethesystemisinthechiral-broken

phase.

IthankV.Azcoiti,Y.Fang,S.Guo,S.Katz,liena,andM.Lombardoforusefuldiscussions.ThisworkissupportedbytheKeyProjectofNationalSci-enceFoundation(10235040),andNationalandGuang-dongMinistriesofEducation.

Φ

r=1,µ′=3.5,T′=0.4

2NcNfNs

-6.6-6.8-7.0

0.20.4

n0.60.8

10.80.6¯0.4n

0.2

10

FIG.3.ThesameasFig.1,butforr=1,µ′=3.5,andT′=0.4,wherethesystemisinthechiral-symmetricphase.

4

First principle study of QCD at finite temperature $T$ and chemical potential $\mu$ is essential for understanding a wide range of phenomena from heavy-ion collisions to cosmology and neutron stars. However, in the presence of finite density, the critical

r=1

2.52

-<ψψ>

1.5T′

10.50

r=1,T′=0.25

00.511.5

2

µ′

2.533.54

µ′

FIG.4.Phasediagram.Thesolidanddottedlinesstandrespectivelyforthe rstandsecondordertransitions.Thecircleisthetricriticalpoint.

r0.0

1.02

0.2

1.18

0.4

1.50

0.6

1.98

0.8

2.62

1.0

′T30.0000

1.2

0.3504

0.0087

0.4269

0.0489

2Nc

10.80.60.40.200

0.5

1

1.5

2

2.5

0.5771

0.1336

0.7979

0.2670

1.0867

0.4498

FIG.7.ChiralcondensateasafunctionofT′atr=1andµ′=0.Eq.(15)predictsasecondorderchiralphase

′

transitionatTC=2.

1

0.90.80.70.60.50.40.30.20.10

2Nc

00.511.522.533.54

FIG.5.Chiralcondensateandquarknumberdensityasafunctionofµ′atr=1andT′=1.Eq.(15)predictsasecondorderchiralphasetransitionatµ′C=3.1770.

5

nq

¯

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