Quark Pair Production in the Chiral Phase Transition

The production of quarks and antiquarks during a sudden restoration of chiral symmetry, as it might occur in very energetic heavy ion collisions, is considered. If gluons are already present they can assist additively to the overall production: real gluons

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aQuarkPairProductionInTheChiralPhaseTransitionCarstenGreinerDepartmentofPhysics,DukeUniversity,DurhamNC27708-0305(February7,2008)AbstractTheproductionofquarksandantiquarksduringasuddenrestorationofchiralsymmetry,asitmightoccurinveryenergeticheavyioncollisions,isconsidered.Ifgluonsarealreadypresenttheycanassistadditivelytotheoverallproduction:realgluonsarepartiallydecayingduringsuchachiraltransition.Thetotalnumberofproducedquarksiscalculatedandfoundtobequitesizeable.Itisspeculatedthatsuchaphenomenoncouldgiverisetoasigni cantcontributiontotheoverallquarkpaircreationinthepreequilibriumstageoftheheavyioncollision.25.75.+r,11.10.Qr,12.38.Bx,24.85.+p

TypesetusingREVTEX

The production of quarks and antiquarks during a sudden restoration of chiral symmetry, as it might occur in very energetic heavy ion collisions, is considered. If gluons are already present they can assist additively to the overall production: real gluons

OneprimarygoalintheupcomingrelativisticheavyioncollisionexperimentsattheBrookhavenRelativisticHeavyIonCollider(RHIC)andtheCERNLargeHadronCollider(LHC)isthetemporaryformationandpotentialobservationofanewformofmatter,thequarkgluonplasma,adecon nedphaseofQCD[1].Itisexpectedthatthisphaseisalsoaccompaniedbyarestorationofchiralsymmetry[2].Athighenoughtemperaturesthemas-siveandcon nedquasiparticles,theconstituentquarks,becomebare,undressedquarkswithcurrentquarkmassesbeingmuchlighterthantheconstituentquarkmasses.Suchaphysicalpictureisrealizedinmodeldescriptionsoflow-energy,e ectiveQCDlagrangiansliketheNambuandJona-Lasinio(NJL)model[3,4]:Atlargeenoughcouplingchiralsymmetryofthevacuumisspontaneouslybrokenandexhibitsanonvanishingscalarquarkcondensate q¯q (250MeV)3.AsimilarbehaviourisexperiencedinlatticeQCDcalculations[5].Thescalarcondensateattributesascalarsel nteractiontothefermionpropagatorresultinginthepresenceofconstituentquarksatlowenergies,inlinewithexperiencefromhadronicspectroscopy.Forhighertemperatures,however,thecondensatebecomesweakerandvan-ishesatsu cientlylargetemperatures(ontheorderofT～200MeV)[6,3],whichisalsofoundinrecentQCDcalculations[7].

Relativisticheavyioncollisionso ertheonlypossiblewaytogainmoreinsightintothephasestructureofnuclearmatteratsuchhightemperatures(anddensities).However,fromtheonsetthereactionduringanysuchacollisionproceedsfaro equilibrium,sothatathermalquarkgluonplasmamaybereachedonlyaftersome nitetime.Accordingtotheresultsofthepartoncascademodel[8]thequarkswillreach,ifatall,achemicalsaturationonlyeventuallyinacentralrelativisticheavyioncollisionatcolliderenergieswhereasthegluonssaturateonasigni cantlysmallertimescale(inabouthalfafm/c)[9,10].This‘hotglue’scenariowasproposedbyShuryak[9].Thereasonforthisparticularbehaviourliesinthefactthatthequarksareonlyproducedinsecondorderbygluonannihilationandtherateforthisprocessissmallcomparedtothecorrespondingoneforgluonproduction.Thedecayofasinglegluonintoaquarkandantiquarkis,ofcourse,kinematicallynotallowed,aslongasallpartonsaretreatedase ectiveon-shellparticles.Inthisworkwewanttoproposethat

The production of quarks and antiquarks during a sudden restoration of chiral symmetry, as it might occur in very energetic heavy ion collisions, is considered. If gluons are already present they can assist additively to the overall production: real gluons

theoccurrenceofachiralphasetransitioncanhavesomeprofoundin uenceonthequarkpairproduction,iftherestorationwillhappenonaveryshorttimescale.Theideaisthatifindeedadecon nedstate(equilibratedornot)willbeformedinthereactionofaheavyioncollision,the‘undressing’ofthequarksshouldalreadyhaveoccurred.Beingprobablystillfarfromequilibriumthischiralrestorationmaythushappenratherspontaneously.Weassumethatthescalarquarkcondensateσ～ q¯q wouldmeltonatimescalesmallerthananyscalesetbythedynamics.Thedecayofthevacuuminducesaspontaneouscreationofquarkandantiquarkpairs.Duringthechiralrestorationtransitionthequarksarenolongeron-shellparticles,theirspectralfunctioncontainsawideo -shellspectrum,thuspermittingalsothedecayofasinglegluonintoaquark-antiquarkpair.Thedecayofboththevacuumandasubstantialfractionofenergeticgluonswouldacceleratethechemicalsaturationofthequarkphasespace.

Todescribethee ectonquarkpairproductionweconstructasimplemodelforfermionicmatter.Denotingthetimeofchiralrestorationatt=0,thelightquarkmassesthenarefunctionsoftime mc

Weassumethatnofurtherresidualinteractionsactamongthequarks.Whileansatz(1)

mayappearquitearti cial,itcapturestheessentialspiritofwhatonewouldnaivelycall‘atransitiontothephasewithrestoredchiralsymmetry’.TheconstituentvacuumbecomesunstablesothatduringtherestorationofthemassesoccupiedstatesofthenegativeDiraccontinuumarepartiallytransferredintothepositivecontinuum.TheamplitudeforthisisobtainedinastraightforwardwaywithintheIn/Outformalism.AparticlewithmomentumpsittinginthenegativeDiracseaisdescribedbytheincoming(constituent)wavefunctionψ↓(In)m(t)= ～350MeVfort≤0.(1)mb～10MeVfort>0≡ p↓= p↓(x,t)(thearrow‘↓’denotesaDirac-stateinthenegativecontinuum,

(In)(+)(c)while‘↑’denotesoneinthepositivecontinuum).Afterthechangeofthevacuumstructurethiswavefunctiongoesoverintoψ↓→ p↓=αp p↓ βp p↑,wherestateswiththelabel(+)(b)(b)

(b)denotetheoutgoing(bare)wavefunctions.Thenumberofproducedquarksisobtainedby

The production of quarks and antiquarks during a sudden restoration of chiral symmetry, as it might occur in very energetic heavy ion collisions, is considered. If gluons are already present they can assist additively to the overall production: real gluons

theprojectionsquaredontheappropriateoutgoingstatesde ningtheasymptoticparticles[11]andtakesherethesimpleform

(Out) (Out) (Out)Np= 0In| bpbp|0In

In) (In) 2 (=βpβp 0In|d pd p|0In =|βp|

=1

(b)(c)EpEp .(2)

Thenumberofproducedquarksisdepictedin g.1.Itisfoundthatthespectrumresemblesthedistributionofasaturatedhotquark-antiquarkplasmawithatemperaturebetween200and300MeV,althoughthedistributiondoesnotdropo exponentially,butasapowerlawforlargemomenta,Np≈(mc mb)2/(4p2).Thetotalnumberofproducedquarksishencelinearlydivergentandshouldberegulatedbyatypicalmomentumcuto ΛCasintheNJLmodel[3]whichcorrespondstoaseparationbetweenthelowandhighmomentumdegreesoffreedom.ForΛC 1GeVthetotalproducedquarkdensityisnq～1.5 2fm 3whichagainissimilartothequarkdensitiesinaplasmafortemperaturesbetween200and300MeV.Inadditionwehavealsoshownin g.1thesituationforabarequarkmassof150MeV(mc=550MeV),atypicalvalueforthestrangequark.Althoughtheproductionissuppressedforsmallmomenta,thetotalintegratedquarkdensityisonlyminorlya ected(ns～0.65 0.8fm 3).

Themechanismforthismultiparticleproductioncanbeunderstoodasfollows:Inthecurrentquarkpicture,whathappensisthatthe‘hadronic’vacuumstate|0In correspondstoacomplexcoherentsuperpositionofcurrentquarkstateswhichrapidlygooutofphase.Thetime tneededforthevacuumtorestorechiralsymmetryisexpectedtobepropor-tionaltotheinverseofthetypicalnonpertubativeQCDenergyscaleorthechiralsymmetry

1 1breakingscaleΛ QCD～ΛC～0.2fm/c.Asimilarestimateisbasedonthekineticequili-

brationtimeofthegluonsatcolliderenergiespredictedtobeoftheorder0.3fm/c[9,10].Asmallbut niteduration tsuppressesthepairproductionatsu cientlyhighmomenta,becausethewavefunctionofthequarkresolvesthesmoothnessofthetransition.Toquantifythiswesubstituteforthetimedependentmass(1)

The production of quarks and antiquarks during a sudden restoration of chiral symmetry, as it might occur in very energetic heavy ion collisions, is considered. If gluons are already present they can assist additively to the overall production: real gluons

m(t)= mc+mb

2sgn(t)1 e 2|t|/τ (3)

andproposeanontrivialansatzforthewavefunction

ψp(x,t)=eiα(t)cos (t)U1 ·pσ Ve iα(t)sin (t) (ε(t) p·x)h¯,(4)

whereα, andεarerealfunctionintime.InsertingtheansatzintothetimedependentDiracequationyieldsthefollowingcoupledsetofdi erentialequations

ε˙=1

sin cos h¯α˙= m (t)+pcos(2α)

h¯ ˙=psin(2α).1cos (5)

Withtheappropriateinitialconditionforthestateinthenegativecontinuumtheseequationscanbeintegratednumericallyandthenprojectedontheoutgoingparticlestate.In g.2thespectrumofparticlesproducedina‘smooth’transitionaredepictedforvariouschoicesofτ.Thetotaldurationofthetransitionisapproximately t≈1.5τ.Asexpected,theparticlenumberatlargermomentadependssensitivelyonthechoiceofτ.However,forτ<0.2fm/c(andthus t<0.3fm/c)onlythehighmomentumyieldisa ected.Weconcludethatiffordynamicalreasons,asestimatedabove,thetransitionfromthebrokentotheunbrokenchiralphaseoccursrapidlyenough,thechangeoftheunderlyingvacuumstructureisaccompaniedbyasubstantialnonperturbativeproductionofquarksandantiquarks.

Theadditionalpossibilityofgluondecaysuggestsaninteresting eldtheoreticalproblemandcanbephrasedasfollows:Whatisthenumberofquarks(andantiquarks)producedina rstorderdecayofalreadypresentgluonspropagatinginanexternaltimedependentscalarbackground eld?Weassumenowthatgluonsinteractperturbativelywiththequarks,althoughtheyareinpartresponsibleforthedynamicquarkmass.Theformulationwehavecarriedoutbyusingreal-timeGreenfunctions[12,13]isappropriatefornon-equilibriumstudies.Thenumberofproducedquarksiscontainedinthe‘<’-componentofthecomplete

The production of quarks and antiquarks during a sudden restoration of chiral symmetry, as it might occur in very energetic heavy ion collisions, is considered. If gluons are already present they can assist additively to the overall production: real gluons

one-particleGreenfunctionandisgivenbytheprojectionontheoutgoingstateinthedistantfuture

(Out) (Out) (Out) ( ∞,∞) Np= (in)|UbpbpU(∞, ∞)| (in)

=t1=t2→∞lim dx1dx233 (b) (b)( i) p↑(1)G<(1,2)(γ0 p↑(2)) .(6)

Forourpurposewehavetospecifythefermionicinitialconditions.ForthisthezerothorderGreenfunctionG0isdeterminedbythefullsetofwavefunctionsde nedinrespecttotheincomingvacuumstateandevolvingnonperturbativelyintimeaccordingtothetimedependentscalarbackgroundHamiltonian(1),i.e.G<0(1,2)=+i

G>0(1,2)= i p1

p1 p1↓(1) ¯p1↓(2) p1↑(1) ¯p1↑(2).(+)(+)(+)(+) (7)

Aperturbativeexpansionisdiagrammaticallystraightforward.The rstorderdecayiscontainedinthelowestorderself-energyinsertionΣforthequarks,wheretheaveragedistribution nk oftherealgluonsentersexplicitly.Thenumberofproducedparticlescannowbecalculated[14].ByusingsomegeneralpropertiesoftheselfenergyandtheGreenfunctions,thechangeinthequarkoccupationnumberinducedbytheself-energyinsertionreads

(Out) Np=2·t1=t2→∞lim dx1dx2

>33[Σ(3,4)G<0(4,2) dx3dx4 ( i) p↑(1)Gret0(1,3)θ(t3 t4) Σ<44(3,4)G>0(4,2)] (b) (b)(γ0 p↑(2)) .(8)

Theexpression(8)fortheamountofproduced(orscattered)particlesresemblesthetypi-calformusedinkinetictransporttheories[12,13];inparticularthetwotermsintheinnerbracketsuggesta‘Gain’and‘Loss’contribution.Tworeasons,however,makefurtheranal-ysiscumbersome.The rst,purelytechnicaloneliesinthefactthattheGreenfunctionsdependonbothtimeargumentsseparatelyduetothetimedependentexternal eld.Thesecondonestemsfromtheoccurrenceofanumberofinfraredsingularities.Besidesthepairproductiontheaboveexpressionalsoincorporatestheabsorptionoremissionofgluons

The production of quarks and antiquarks during a sudden restoration of chiral symmetry, as it might occur in very energetic heavy ion collisions, is considered. If gluons are already present they can assist additively to the overall production: real gluons

forthequarksproducedinlowestorderinthevacuumdecay(2).Aseparationofthepairproductionfromtheseprocessesisnotpossiblebecauseofquantummechanicalinterference.Intheendonehastoshowthatthesesingularitiescanceleachother,astheyindeeddo[14].Finally,(8)canbesplitintothreecontributions[14],

(Out)‘GAIN‘LOSS‘MASSHIFT, Np= Np+ Np+ Np′′′(9)

wherethe‘Gain’andthe‘Loss’termbotharenotsimplythesquaredofsomeparticularamplitude.However,asanumericalinvestigationshows,the rstismostlypositivewhilethesecondisnegative.Theinterpretationstemsfromthefactthatforexamplethe‘Gain’partcontainsquarksscatteredintothemomentumstatepeitherbyemissionorabsorptionofagluonaswellasthedecayofagluonintoaquarkandantiquark.Theexoticprocessofthesimultaneouscreationofaquark-antiquarkpairandagluonalsocontributes.The‘Loss’termcontainstheoppositeprocesses.Thecancellationoftheinfraredsingularitiesamongbothtermsiseasilyunderstoodwithinthisinterpretation.Thedesignationofa‘masshift’partforthelastcontributionisbasedontheknownfactthatinanequilibratedplasmathequarksbecomedressedanddynamicallymassivebytheinteractionwiththethermalgluons

[15].Suchamasshiftshouldenterasacorrectiontothezerothordervacuumdecay(2)[14].Anultravioletdivergencearisingfrominteractionswiththeperturbativegluonicvacuumisinterpretedasmassrenormalization.Thevacuumtermscanexplicitlybeseparatedandshouldberegardedasthe rstordervacuumcorrectiontothespontaneousdecay.However,wearemainlyinterestedintheoveralle ectduetothepresenceofrealgluons.Inthefollowingnumericalstudythesevacuumcontributionsarethereforediscarded.

Thedistributionofgluons,atthetimewheretherestorationofchiralsymmetrytakesplace,isprobablyfarfromlocalthermalorkineticequilibrium.Inparticularhighenergeticgluonsparalleltothebeamaxismaybepresent.However,forourpresentpurposeitisconvenientandsu cienttoparametrizetheincominggluonsbyathermaldistributionwithalargetemperature(intherangeof700to1000MeV)whichincludessuchhighmomenta.TheresultsforachosentemperatureofTg=700MeVareshownin g.3.TheQCD

The production of quarks and antiquarks during a sudden restoration of chiral symmetry, as it might occur in very energetic heavy ion collisions, is considered. If gluons are already present they can assist additively to the overall production: real gluons

couplingconstantαs=g2/(4π)istakentobe0.3.Forlowmomentathequarkdistributionislargeandpositiveandexceedsunitybythreeordersofmagnitude.Itthenquicklyturnsandremainsnegativeuptomomentaof400MeV.Atlargermomentathedistributionresemblesanotfullysaturatedthermalquarkdistributionofnearlythesametemperatureasthegluons.Thedistribution,however,dropsfasteratstillhighermomentawhicharenotshownhere.Thelowmomentumbehaviourisobviouslyunphysicalandshowsthatpertubationtheoryisnotapplicablehere.Ifweintegratethecalculateddistributionovermomentumspace,itturnsoutthatthebehaviouratlowmomentumisirrelevant.Themajoramountofparticlesiscreatedatlargermomenta.Anintegrationuptostillonlymoderatemomentap 1200MeVexplicitlydemonstratesthatabout12×0.36=4.3quarksperfm3areproducedbytheinitiateddecayofgluons(integratinguptomomenta 2500MeVthenumberchangesto12×0.84=10.1quarksperfm3)!Furthernumericalinvestigationshowsthattheimportanceofthise ectincreaseswithahighertemperatureemployedforthegluons.Itturnsoutthatthenearlycollineargluonswithhighmomentaareresponsibleforthedecay[14].Thissuggeststhatsuchaphenomenonshouldbeofparticularimportanceintheearlystagesofthegluonevolutionwhenthechiralsymmetryrestorationtakesplace.Thee ect,however,issmallifwetakeabaremassmbof150MeV.Thedistributionisshiftedtowardshighermomenta.Integratinguptomomenta 2500MeVthenumberofproducedquarksisevaluatedas0.34fm 3.Consequently,comparedtothelightquarks,thenumberofstrangequarksproducedbygluondecaywouldbestronglysuppressed.

Tosummarize,theoccurrenceofarapidchiralrestorationofthevacuumcanleadtosomeparticularnewandinterestingphenomenainveryenergeticrelativisticheavyioncollisions.Inthisletterweinvestigatedthepairproductionofquarkswithinasimplemodelillustratingsuchadynamicaltransition.Thepresenceofrealgluonsassisttheoverallproduction:Duringthechangeofthevacuumstructurethedirectdecayofagluonintoaquarkandantiquarkpairiskinematicallyallowed.Accordingtoour ndingindeedasigni cantamountofquarksareproducedduringthisprocess.Suchaphenomenoncouldgiverisetoanonperturbativedynamicalcreationofquarksandantiquarks(aswellas

The production of quarks and antiquarks during a sudden restoration of chiral symmetry, as it might occur in very energetic heavy ion collisions, is considered. If gluons are already present they can assist additively to the overall production: real gluons

entropy)intheveryearlystagesoftheheavyionreaction.Itistemptingtospeculatethatthetotalnumberofproducedquarksduringsuchatransitionandduringthefurtherevolutionsu cessothatnotonlythegluonicdegreesoffreedomaresaturated,butalsothefermionic.The‘hotglue’scenariocouldturnouttobeahot‘quarkgluonplasma’afterall.

ACKNOWLEDGMENTS

TheauthorwantstothanktheAlexandervonHumboldtStiftungforitssupportwithaFeodorLynenscholarship.ThisworkwasalsosupportedinpartbytheU.S.DepartmentofEnergy(grantDE-FG05-90ER40592).TheauthoristhankfulforthevariousandstimulatingdiscussionswithProf.B.M¨uller.DiscussionswithA.Sch¨aferandS.Mrowczynskiareappreciated.

The production of quarks and antiquarks during a sudden restoration of chiral symmetry, as it might occur in very energetic heavy ion collisions, is considered. If gluons are already present they can assist additively to the overall production: real gluons

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The production of quarks and antiquarks during a sudden restoration of chiral symmetry, as it might occur in very energetic heavy ion collisions, is considered. If gluons are already present they can assist additively to the overall production: real gluons

FIGURES

FIG.1.Thespontaneousproductionofquarksisshownformc=350MeV,mb 0MeV(a)andformc=550MeV,mb=150MeV(b).Toguidetheeyeathermaldistributionofmasslessquarksisalsodrawn(foratemperatureofT=200MeV(c)andT=300MeV(d)).

FIG.2.Theproductionofquarksisshownfora‘smooth’transitionintimeofthevacuum(comparetext).Theparametersgiveninthe gurearetobereadfromthelefttotheright.mc=350MeVandmbistakentobenearlyzero.

FIG.3.The rstordercorrectionsontheparticledistributionduetothepresenceofrealgluonsaredepicted.Thegluonsaretakentobedistributedwithatemperatureof700MeV.(a)correspondstotheparametersetmc=350MeV,mb=10MeV,(b)tothesameasin g.1.Foracomparisonathermaldistributionofmasslessquarkswithatemperatureof700MeVisalsodrawn(c).

The production of quarks and antiquarks during a sudden restoration of chiral symmetry, as it might occur in very energetic heavy ion collisions, is considered. If gluons are already present they can assist additively to the overall production: real gluons

This figure "fig1-1.png" is available in "png" format from:/ps/hep-ph/9406261v1

The production of quarks and antiquarks during a sudden restoration of chiral symmetry, as it might occur in very energetic heavy ion collisions, is considered. If gluons are already present they can assist additively to the overall production: real gluons

This figure "fig1-2.png" is available in "png" format from:/ps/hep-ph/9406261v1

The production of quarks and antiquarks during a sudden restoration of chiral symmetry, as it might occur in very energetic heavy ion collisions, is considered. If gluons are already present they can assist additively to the overall production: real gluons

This figure "fig1-3.png" is available in "png" format from:/ps/hep-ph/9406261v1

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