SU(2) Yang-Mills Theory in Savvidy Background at F

arXiv:hep-th/0609090v113Sep2006SU(2)Yang-MillsTheoryinSavvidyBackgroundatFiniteTemperatureandChemicalPotentialR.Parthasarathy1andAlokKumar2TheInstituteofMathematicalSciencesC.P.T.Road,TaramaniPostChennai600113,India.AbstractTheone-loopeffectiveenergydensityofapureSU(2)Yang-MillstheoryintheSavvidybackground,atfinitetemperatureandchemicalpotentialisexaminedwithemphasisontheunstablemodes.Afteridentifyingthestableandunstablemodes,thestablemodesaretreatedinthequadraticapproxi-mation.Fortheunstablemodes,thefullexpansionincludingthecubicandthequartictermsinthefluctuationsisused.Thefunctionalintegralsfortheunstablemodesareevaluatedandaddedtotheresultsforthestablemodes.Theresultingenergydensityisfoundtobereal,coincidingwiththerealpartoftheenergydensityinthequadraticapproximationofearlierstudy.Thereisnownoimaginarypart.Numericalresultsarepresentedforthevariationoftheenergydensitywithtemperatureforvariouschoicesofthechemicalpotential.Keywords:Savvidyvacuum;SU(2)Yang-Mills;backgroundchromomag-netic;finitetemperature;chemicalpotential;unstablemodes;quartictermsinfluctuations.1e-mail:[e-mail]2e-mail:[e-mail]teofSU(2)Yang-Millstheoryinacovariantlyconstantchromomagneticfieldinthethirdcolordirection(F312=H)hadbeenstudiedbySavvidy[1].Theimportantresultobtainedwastheone-loopeffectiveenergydensitywaslowerthantheperturbativevacuumandthetheoryhasagluoncondensate,|FaμνFμνa|6=0.However,NielsonandOlesen[2]pointedoutthattheone-loopeffectiveenergydensityhadanimaginarypartduetothelowestLandaulevel.Variousstudiesweredevotedtothisaspect[3].Inrecenttimes,gaugeinvarianceargumentstoexcludetheimaginarypart[4]andastablevacuumbydynamicallygeneratingmasstermfortheoff-diagonalgluons[5]havebeenproposed.Inallthesestudies,onlythetermsquadraticinthefluctuationsarekept,throwingoutthecubicandquartictermsinthefluctuations.Asearlyas1983,theSavvidyvacuumwithintheframeworkofthebackgroundfieldmethodwasexaminedbyFlory[6]andlaterbyKay[7]treatingtheunstablemodesincludingthecubicandquartictermsandthepossibilityofobtainingrealeffectiveenergydensitywaspointedout.ThepresentauthorswithKay[8]haveexplicitlyshownatzerotemperature,theresultingenergydensity,whenthecubicandthequartictermsinthefluctuationsareincludedfortheunstablemodes,hasnoimaginarypartandcoincidedwiththerealpartoftheearliercalculations.Briefly,thestablemodesaretreatedinthequadraticapproximationandfortheunstablemodes,thefunctionalintegralisevaluatedkeepingthecubicandquartictermsalongwiththequadraticterm.Thisresult(Eqn.30of[8])hasnoimaginarypartandwhenaddedtotheresultofthestablemodes,givestheeffectiveenergydensitywhichisreal.Thecalculationsleadingtothisresultaregaugeinvariant[8].TheSavvidyvacuumatfinitetemperaturehasbeenstudiedbymanyauthors[9-15].Inallthesestudies,thequadraticapproximationhasbeenused.Asaresult,theeffectiveenergydensityinvolvedanimaginarypart,dependentonthetemperature.Thisisseriousasitpersistsathightem-peratures.IthasbeenobservedbyMeisingerandOgilvie[15]thatwiththeintroductionofanon-trivialPolyakovloop,specifiedbyφ,itispossibletostabilizethevacuum,ifβ√gHφ2π−β√gH,whereβ=1/kT,Histhechromomagneticbackgroundinthethirdcolordirection,asforthisrangeofφ,theimaginarypartbecomeszero.However,theimaginarypartisnon-zero2attheglobalminimum.So,intheunderstandingofthefinitetemperaturebehaviour,theimaginarypartseverelyinhibitstheprogress.Itisthepurposeofthisworktoextendourmethod[8]atzerotem-perature,tofinitetemperaturewithchemicalpotential,byseparatingtheunstablemodesandincludingthecubicandquartictermsinthesemodesintheevaluationofthefunctionalintegralandaddingtothecontributionofthestablemodesinthequadraticapproximation.Themotivationistogetrealeffectiveenergydensity.Anotherpurposeistoresolveadiscrepancyintheanalyticalexpressionsfortheone-loopenergydensitybetween[14]and[15].ThediscrepancyisaninterchangeofJ1andY1andalsoarelativesignbe-tweenthetwoK1functionsintheenergydensity.Ourresultsshowthatthefinitepartoftheeffectiveenergydensityisreal.Thereisnoimaginarypart.Therealenergydensitycoincideswiththerealpartof[15]whichthereforeresolvesthediscrepancyalludedaboveinfavourof[15].Thenextsectiondevelopstheformalismfortheevaluationoftheeffectiveenergydensity.Thestableandtheunstablemodesareseparatelytreated.Section.IIIprovidesthedetailsofhandlingtheunstablemodesincludingthecubicandquartictermsintheexpansion.Theresultsareaddedtothecontributionofthestablemodesandthefullexpressionfortheeffectiveenergydensityisexhibited.SectionIVcontainsthedetailsofthenumericalevaluationoftheeffectiveenerdydensityforvarioustemperatures.II.EFFECTIVEENERGYDENSITYINTHEBACKGROUNDFIELDMETHOD(BFM)TheEuclideanfunctionalintegralforanSU(2)pureYMtheoryisZ=Z[dAaμ]eS,(1)whereS=Zd4x{−14FaμνFμνa},(2)andFaμν=∂μAaν−∂νAaμ+gǫabcAbμAcν.(3)3ExpandingAaμ=¯Aaμ+aaμwith¯Aaμastheclassicalbackgroundfieldsatisfyingtheequationofmotion¯Dabμ¯Fbμν=0,with¯Dabμ=∂μδab+gǫacb¯Acμasthebackgroundcovariantderivative,¯Faμνissameas(3)with¯Aaμandusingthebackgroundgauge¯Dabμabμ=0,(4)wehaveZ=Z[daaμ]eS′,(5)withS′=Zd4x−14¯Faμν¯Faμν+12aaμΘacμνacν+gǫacd(¯Daeνaeμ)