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.