Real Time Correlators in Hot (2+1)d QCD

arXiv:hep-th/9705026v15May1997USITP-97-04SUNY-NTG-97-17April1997RealTimeCorrelatorsinHot(2+1)dQCDT.H.Hansson1,J.WirstamInstituteofTheoreticalPhysicsUniversityofStockholmBox6730,S-11385Stockholm,SwedenI.Zahed2DepartmentofPhysicsSUNYatStonyBrookStonyBrook,NewYork,11794,USAABSTRACTWeusedimensionalreductiontechniquestorelaterealtimefiniteTcorrelationfunc-tionsin(2+1)dimensionalQCDtoboundstateparametersinageneralized’tHooftmodelwithaninfinitenumberofheavyquarkandadjointscalarfields.WhilestaticsusceptibilitiesandcorrelationfunctionsoftheDeTartypecanbecalculatedusingonlythelight(static)gluonicmodes,thedynamicalcorrelatorsrequiretheinclusionoftheheavymodes.InparticularwedemonstratethattheleadingTperturbativeresultcanbeunderstoodintermsoftheboundstatesofthe2dmodelandthatconsistencyrequiresboundstatetrajectoriescomposedofbothquarksandadjointscalars.Wealsoproposeanon-perturbativeexpressionforthedynamicalDeTarcorrelatorsatsmallspatialmo-menta.1SupportedbytheSwedishNaturalScienceResearchCouncil.2SupportedbytheUSDepartmentofEnergyunderGrantNo.DE-FG02-88ER40388.1.IntroductionAbout10yearsago,DeTar[1]conjecturedthatthefiniteTexcitationspectrumofQCDwasmuchricherthanpreviouslyexpected-excitationswithwavelengthslongerthanthemagneticscale1/g2Tarecolorsinglets,attheelectricscale1/gTtherearecoloredplasmonsandatthehardscale1/Tcoloredquarksandgluons.Healsoproposedtotestthisconjecturebystudyingcolorsingletcurrent-currentcorrelatorsofthetypeGαα(~x)=1β2*Zβ0dτψ(~x,τ)Γαψ(~x,τ)Zβ0dτ′ψ(~0,τ′)Γαψ(~0,τ′)+,(1.1)whereh···iistheaverageovertheGibbsensemble.Latticemeasurementsofthese“De-Tarcorrelators”inEuclideanspace,relevantforstudyingscreeningeffectsinthefinitetemperaturetheory,suggestthatthescreeningmassesareabout2πTforthelightvectorandpseudovectorexcitations,andsmallerforthescalarandpseudoscalar.Someyearsago,itwassuggested[2,3]thatthelatticeresultsonthescreeninglengthsandthespatialextensionofthestaticcorrelatorsathightemperaturecouldbeunder-stoodintermsofadimensionallyreducedtheorywhichisknowntoconfineinthehightemperaturelimit.ItwasshownthatthestaticcorrelationfunctionscanbecalculatedfromaneffectivetheoryinvolvingnonrelativisticquarkswithcoloredCoulombinterac-tions.Thescreeninglengthswerefoundtoasymptotethefreequarkvalues,withspatiallycorrelatedquarksinthetransversedirectionsatalltemperatures.Theseideashavebeenfurtherdevelopedinseveralpapers,notablythoseofHuangandLissia[4,5],whohavestudiedstaticpropertiesathightemperatureinQCDandintheGross-Neveumodel.InspiteofourincreasedunderstandingofthestaticaspectsofhightemperatureQCD,DeTar’smainpoint,namelythatthehightemperaturephaseofQCDisconfinedinthesensethatalllowenergyexcitationsarecolorsinglets,isstillneitherprovennordisproven.ThemaindifficultyisthatalthoughtherearemanysophisticatedandefficienttechniquesforcalculatingEuclideancorrelationfunctions,therearenogoodmethods,neitheranalyticalnornumerical,toobtaintherealtime,finiteTcorrelators.InprinciplethesecanbeobtainedbyanalyticallycontinuingtheEuclideanfunctions,asisdone2routinelyinperturbationtheory,butonthelatticethisisverydifficultwithouthavingaratherdetailedaprioriknowledgeabouttheanalyticstructureofthefiniteTspectralfunctions(forsomealternativeattemptsinQCDsee[6]andinsimplemodelsystemssee[7]).Thereareseverallinesofattackonthisproblem.Themostdirectisperturbationtheory,wheretherearewelldevelopedmethodsforcalculatingrealtimecorrelators[8],butsinceQCDisknowntobenonperturbativeatlargedistancesevenatinfiniteT,thismethodisoflimitedvalue.Anotherwayistomakesomespecificassumptionaboutthespectralfunctions,asforinstancepoledominancewithTdependentmasses,andthenusesum-ruletechniquestochecktheconsistencyoftheassumptionsandestimatespectralparametersintermsofcondensatevalues.ThismethodislikelytobegoodattemperatureswellbelowTc[9,10,11,12],buthasalsobeenattemptedatlargeT[13].Thelastapproachistoattemptexact,oratleastcontrolled,analyticalcalculationsinsimplemodels,withthehopethatsomeofthefeaturescanbegeneralizedtorealisticcases.Thepresentpaperisdevotedtoastudyofthiskindusing2+1dimensionalQCDasamodeltheory.ThemainideaofourcalculationistoexpresstherealtimefiniteTcorrelatorsin2+1dQCDintermsofspectralparameters(boundstatemasses,formfactors)ofageneralized’tHooftmodel,whichisobtainedsimplybyrewritingthetheoryin2dlanguage.Thisisofcourseonlyadvantageousiftheheavy(i.e.nonstatic)modescanbetotally,ortoalargeextent,ignoredinthehighTlimit.Thisisknowntobethecaseforstaticobservables,andweshallarguethatforzerotwo-momentum,~q=~0,thisistruealsoforcertaindynamicalcorrelationfunctionsinthesensethatonlytheheavymodesofthequarkfieldshavetoberetained.Thisclaimisbasedonacomparisonbetweenthehightemperaturelimitofperturbationtheoryinthe2+1dmodelandthegeneralized’tHooftmodel.Thiscomparisonalsoshowsthatfornonzero~q,theheavyscalars(originatingfromthenon-staticgluonmodes)mustberetained,whichimpliesthatthetwo-dimensionalmethodisoflimitedvalue.However,inthiscasewecanshowthatconsistencyrequires3theexistenceoflineartrajectoriesofboundstatescontainingheavyadjointscalarsinadditiontoquarksandantiquarks.Weshallbrieflydiscuss