Theory of the Quark-Gluon Plasma

arXiv:hep-ph/0107131v111Jul2001TheoryoftheQuark-GluonPlasmaJean-PaulBlaizotServicedePhysiqueTh´eorique,CEASaclay91191Gif-sur-YvetteCedex,FranceSPhT-T01/0741IntroductionInspiteofwhatthetitlemightsuggest,Ishallnottrytocoverintheselec-turesallinterestingaspectsofthetheoryofthequark-gluonplasma.Ishallratherfocusonprogressmadeinrecentyearsinunderstandingthehightem-peraturephaseofQCDbyusingweakcouplingtechniques.Suchtechniquesgofarbeyondstrictperturbationtheoryviewedasanexpansioninpowersofthegaugecoupling.Infactsuchanexpansionbecomesmeaninglessassoonasthecouplingisnotvanishinglysmall.However,weshallseethatarathersimplestructureemergesfromweakcouplingstudies,withacharacteristichierarchyofscalesanddegreesoffreedom.Theinteractionsrenormalizethepropertiesoftheseelementarydegreesoffreedom,butdoesnotdestroythesimplepictureofthehightemperaturequark-gluonplasmaasasystemofweaklyinteractingquasiparticles.Asweshallseeattheendoftheselectures,thispictureissupportedbyafirstprinciplecalculationoftheentropywhichreproducesaccuratelylatticedataabove2or3timesthecriticaltemperature.Someofthematerialpresentedhereisborrowedfromtherecentreview[1],andcomplementscanalsobefoundin[2,3,4,5,6].AnotherperspectiveonsomeofthetopicsdiscussedherecanbefoundinthelecturesbyA.Rebhan.Theoutlineofthelecturesisthefollowing.InordertogetafirstroughpictureofthephasediagramofhadronicmatterIusethebagmodeltodescribethequark-hadronphasetransition:thisexercisewillgiveussomefamiliaritywiththethermodynamicsofmassless,non-interacting,particles.ThenIbrieflyrecallsometechniquesofquantumfieldtheoryatfinitetem-peratureneededtotreattheinteractions[7,8,9,10,11,12],andintroducetheconceptofeffectivetheoryinasimplecaseofascalarfield.ThenIproceedtoananalysisofthevariousimportantscalesanddegreesoffreedomofthequark-gluonplasmaandfocusontheeffectivetheoryforthecollectivemodeswhichdevelopattheparticularmomentumscalegT,wheregisthegaugecouplingandTthetemperature.Apowerfultechniquetoconstructtheef-fectivetheoryisbasedonkineticequationswhichgovernthedynamicsoftheharddegreesoffreedom.Someofthecollectivephenomenathataredescribedbythiseffectivetheoryarebrieflymentioned.ThenIturntothecalculationoftheentropyandshowhowtheinformationcodedintheeffectivetheorycanbeexploitedin(approximately)self-consistentcalculations[13,14,15].2Jean-PaulBlaizot2Thequark-hadrontransitioninthebagmodel.Thephasediagramofdensehadronicmatterhastheexpectedshapeindi-catedinFig.1.Thereisalowdensity,lowtemperatureregion,correspondingtotheworldofordinaryhadrons,andahighdensity,hightemperatureregion,wherethedominantdegreesoffreedomarequarksandgluons.Theprecisedeterminationofthetransitionlinerequireselaboratenonperturbativetech-niques,suchasthoseoflatticegaugetheories(seethelecturesbyF.Karsch).Butonecangetroughordersofmagnitudeforthetransitiontemperatureanddensityusingasimplemodeldealingmostlywithnon-interactingparticles[3,5].μTQuark-GluonPlasmaHadronsTcBcμFig.1.Theexpectedphasediagramofhotanddensehadronicmatterintheplane(μB,T),whereTisthetemperatureandμBthebaryonchemicalpotentialLetusfirstconsiderthetransitioninthecasewhereμB=0.Atlowtemperaturethisbaryonfreematteriscomposedofthelightestmesons,i.e.mostlythepions.Atsufficientlyhightemperatureoneshouldalsotakeintoaccountheaviermesons,butinthepresentdiscussionthisisaninessentialcomplication.Weshallevenmakeafurtherapproximationbytreatingthepionasamasslessparticle.Atveryhightemperature,weshallconsiderthathadronicmatteriscomposedonlyofquarksandantiquarks(inequalnum-bers),andgluons,formingaquark-gluonplasma.Inboththehightempera-tureandthelowtemperaturephases,interactionsareneglected(exceptforthebagconstanttobeintroducedbelow).Thedescriptionofthetransitionwillthereforebedominatedbyentropyconsiderations,i.e.bycountingthedegreesoffreedom.TheenergydensityεandthepressurePofagasofmasslesspionsaregivenby:ε=3·π230T4,P=3·π290T4,(1)wherethefactors3accountforthe3typesofpions(π+,π−,π0).Theenergydensityandpressureofthequark-gluonplasmaaregivenbysimilarformulae:ε=37·π230T4+B,Theoryofthequark-gluonplasma3P=37·π290T4−B,(2)where37=2×8+78×2×2×2×3istheeffectivenumberofdegreesoffreedomofgluons(8colors,2spinstates)andquarks(3colors,2spins,2flavors,qand¯q).ThequantityB,whichisaddedtotheenergydensity,andsubtractedfromthepressure,summarizesinteractioneffectswhichareresponsibleforachangeinthevacuumstructurebetweenthelowtemperatureandthehightemperaturephases.Itwasintroducedfirstinthe“bagmodel”ofhadronstructureasarestoringforceneededtoequilibratethepressuregeneratedbythekineticenergyofthequarksinsidethebag[16].Roughly,theenergyofthebagisE(R)=4π3R3B+CR,(3)whereC/Risthekineticenergyofmasslessquarks.MinimizingwithrespecttoR,onefindsthattheenergyatequilibriumisE(R0)=4BV0,whereV0=4πR30/3istheequilibriumvolume.ForaprotonwithE0≈1GeVandR0≈0.7fm,onefindsE0/V0≃0.7GeV/fm3,whichcorrespondstoa“bagconstant”B≈175MeV/fm3,orB1/4≈192MeV.Wecannowcomparethetwophasesasafunctionofthetemperature.Fig.2showshowPvariesasafunctionofT4.OneseesthatthereexistsatransitiontemperatureTc=4517π21/4B1/4≈0.72B1/4,(4)beyondwhichthequark-gluonplasmaisthermody