Center-Symmetric Effective Theory for High-Tempera

arXiv:0801.1566v2[hep-ph]6May2008PreprinttypesetinJHEPstyle-HYPERVERSIONarXiv.org/0801.1566[hep-ph]CERN-PH-TH/2008-007HIP-2008-01/THTUW-08-01Center-SymmetricEffectiveTheoryforHigh-TemperatureSU(2)Yang-MillsTheoryPh.deForcrand12∗,A.Kurkela3†andA.Vuorinen24‡1Institutf¨urTheoretischePhysik,ETHZ¨urich,CH-8093Z¨urich,Switzerland2CERN,PhysicsDepartment,THUnit,CH-1211Geneva23,Switzerland3TheoreticalPhysicsDivision,DepartmentofPhysicalSciences,P.O.Box64,FI-00014UniversityofHelsinki,Finland4Institutf¨urTheoretischePhysik,TechnischeUniversit¨atWien,WiednerHauptstr.8-10,A-1040Vienna,AustriaAbstract:Weconstructandstudyadimensionallyreducedeffectivetheoryforhigh-temperatureSU(2)Yang-Millstheorythatrespectsallthesymmetriesoftheunderlyingtheory.Ourmainmotivationistostudywhetherthecorrecttreatmentofthecentersymmetrycanhelpextendtheapplicabilityofthedimensionalreductionproceduretowardstheconfinementtransition.Afterperformingperturbativematchingtothefulltheoryatasymptoticallyhightemperatures,wemapthephasediagramoftheeffectivetheoryusingnon-perturbativelatticesimulations.Wefindthatatlowertemperaturethetheoryundergoesasecondorderconfiningphasetransition,incompleteanalogywiththefulltheory,whichisadirectconsequenceofhavingincorporatedthecentersymmetry.Keywords:ThermalFieldTheory,QCD,LatticeFieldTheory.∗[e-mail]†[e-mail]‡[e-mail]ffectivetheory32.1Degreesoffreedom32.2TheLagrangian42.3Theeffectivepotential52.4Minimizingthepotential63.Parametermatchingathightemperature83.1ReductiontoEQCD83.2Domainwallproperties93.3Leadingorderparameters94.Beyondperturbationtheory104.1Numericalsimulations104.2Phasediagramandthematchingofr134.3Discussion155.Conclusions16A.Calculationoftheeffectivepotential18B.Effectivetheorydomainwall19C.Setupforthenumericalsimulations211.IntroductionInthestudyofhightemperaturegaugetheoryinthermalequilibrium,aparticularlyusefulapproachhasturnedouttobethatofdimensionalreduction[1].There,onedescribesthesystemviaad−1dimensionalstaticeffectivetheorybuiltusingthefactthat,athightemperature,thenon-staticfieldmodesdecouplequiteefficientlyfromthedynamicsoflengthscaleslargerthanorequaltotheinverseDebyemass.Thishasledtoseveralimportantadvances,includinge.g.newresultsforthespatialstringtensionandvariouscorrelationlengths[2],aswellanefficientreorganizationoftheweakcouplingexpansionoftheQCDpressure[3].Forthelatterquantity,dimensionalreductionhasevenprovidedaframeworkforextendingtheexpansiontothefullg6order,whichiswherethepressure–1–obtainsitsfirstnon-perturbativecontributions—atopicthathasrecentlyreceivedmuchattention[4].Despitetheplethoraofresultsderivedutilizingdimensionalreductioninperturbativesetups,thereisassuchnoreasontoviewthemethodasonlyatoolforweakcouplingexpansions.Afterall,itiswell-knownthatatleastamodestseparationbetweenthehard(2πT)andthesoftandultrasoftscales(mD,mmag)existsallthewaydowntothephasetransitionregion.Aminimalextensionofthecurrentperturbativeapproacheswouldbeanon-perturbative,numericaldeterminationofthecouplingsofthedimensionallyreducedtheoryasafunctionoftemperature,asattemptedinRef.[5].Anothermoreradical—andalsomorepromising—approachistoattempttocurethelackofcentersymmetrywithintheusualthree-dimensionaleffectivetheory,EQCD,asitistherestorationofthissymmetrythatdrivesthephasetransitiontotheconfiningphaseinpureSU(N)Yang-Millstheory.Recently,aneweffectivetheoryforhightemperatureSU(3)Yang-MillstheoryhasbeenproposedthathasthevirtueofrespectingtheZ(3)centersymmetryoftheoriginaltheory[6].Byconstruction,itcurestheunphysicalpropertiesofEQCDthatinvalidateitsuseclosetothedeconfinementtransitionregion,suchasthefactthatthephysicallyrelevantphaseofEQCDisinfactonlymetastable[7].ThenewtheoryisthusexpectedtobevalidoverawidertemperaturerangethanEQCD,anditsproperties,includingthedetailedstructureofitsphasediagram,arecurrentlyundernon-perturbativestudy.Thelatticesimulationsofthenewtheoryare,however,technicallyquitedemandingduetothehighdimensionalityoftherespectiveparameterspace[8].Totestthegeneralideabehindtheconstructionofdimensionallyreducedcentersym-metrictheories,ouraiminthispaperistoperformananalysisofhightemperatureSU(2)Yang-MillstheoryanditscorrespondingZ(2)invarianteffectivetheory.ThephysicalsetuphereishighlyanalogoustothatoftheSU(3)case,buthasimportantsimplificationsduetothelesscomplicatedstructureofthegaugegroup.Insteadofhavingtointroducetennewheavy(andstrictlyspeakingunphysical)degreesoffreedomtoensurethatasuper-renormalizabletheorywiththecorrectsymmetriescanbeconstructed,itwillthistimebesufficienttoconsideronlyoneextrarealscalarfield,whosemassisthentheonlyadjustableparameterinthetheory.Thismakesamarkeddifferencefromthepointofviewofthelatticesimulations.Acompeting—orrathercomplementary—approachtothebuildingofcenter-symmetriceffectivetheoriesforthermalSU(N)gaugetheoryistosettherequirementofrenormalizabilityaside,butratherconcentrateonthevariouskindsofmodelsonecanconstructforthefulltheoryWilsonlinenearTc[9,10].AnimportantstepinthisdirectionwastakeninRef.[11],whereane