Quark Pair Production in the Chiral Phase Transition(2)

(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

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