Quantum properties of a non-Abelian gauge invarian

arXiv:hep-th/0605288v23Oct2006LTH-703Quantumpropertiesofanon-AbeliangaugeinvariantactionwithamassparameterM.A.L.Capria,∗D.Dudalb,†J.A.Graceyc,‡V.E.R.Lemesa,§R.F.Sobreiroa,¶S.P.Sorellaa,∗∗andH.Verscheldeb††aUERJ-UniversidadedoEstadodoRiodeJaneiroRuaS˜aoFranciscoXavier524,20550-013Maracan˜aRiodeJaneiro,BrasilbGhentUniversityDepartmentofMathematicalPhysicsandAstronomyKrijgslaan281-S9B-9000Gent,BelgiumcTheoreticalPhysicsDivisionDepartmentofMathematicalSciencesUniversityofLiverpoolP.O.Box147,Liverpool,L693BX,UnitedKingdomWecontinuethestudyofalocal,gaugeinvariantYang-Millsactioncontainingamassparameter,whichweconstructedinapreviouspaperstartingfromthenonlocalgaugeinvariantmassdimensiontwooperatorFμν(D2)−1Fμν.Wereturnbrieflytotherenormalizabilityofthemodel,whichcanbeproventoallordersofperturbationtheorybyembeddingitinamoregeneralmodelwithalargersymmetrycontent.WepointouttheexistenceofanilpotentBRSTsymmetry.Althoughouractioncontainsextra(anti)commutingtensorfieldsandcouplingconstants,weprovethatourmodelinthelimitofvanishingmassisequivalentwithordinarymasslessYang-Millstheories.ThefulltheoryisrenormalizedexplicitlyattwoloopsintheMSschemeandalltherenormalizationgroupfunctionsarepresented.Weendwithsomecommentsonthepotentialrelevanceofthisgaugemodelfortheissueofadynamicalgluonmassgeneration.I.INTRODUCTION.Yang-Millsgaugetheories,withquantumchromodynamics(QCD)modelingthestronginteractionbetweenelementarypar-ticlesasoneofthekeyexamples,arequitewellunderstoodatveryhighenergies.Inthisenergyregion,asymptoticfreedom[1,2,3,4]setsin,whichinturnensuresthatthecouplingconstantg2issmallenoughtomakeaperturbativeexpansioninpowersofg2possible.TheelementaryQCDexcitationsarethegluonsandquarks.Ourcurrentunderstandingofnon-Abeliangaugetheoriesisstillincompleteintheinfraredregion.Atlowerenergies,theinteractiongrowsstronger,preventingtheuseofstandardperturbationtheorytoobtainrelativelyacceptableresults.Nonper-turbativeaspectsofthetheorycomeintoplay.Themostnotable,yettoberigourouslyprovennonperturbativephenomenon,isthefactthattheelementarygluonandquarkexcitationsnolongerbelongtothephysicalspectrum,beingconfinedintocolorlessstatessuchasglueballs,mesonsandbaryons.Awidelyusedstrategytoparametrizecertainnonperturbativeeffectsofthetheoryamountstotheintroductionofsocalledcondensates,whicharetheexpectationvaluesofcertainoperatorsinthevacuum.Furthermore,onecanemploytheoperatorproductexpansion(OPE)(viz.shortdistanceexpansion)whichcanbeappliedtolocaloperators,inordertorelatetheassociatedcondensatestononperturbativepowercorrectionswhich,inturn,giveadditionalinformationnexttotheperturbativelycalculablecontributions.Asweareconsideringagaugetheory,thesecondensatesshouldbegaugeinvariantiftheyaretoenterphysicalobservables.Thisputsratherstrongrestrictionsonthepossiblecondensates,theoneswithlowestdimensionalityarethedimensionthreequarkcondensatehψψiandthedimensionfourgluoncondensateF2μν.Thereisavarietyofmethodstoobtainestimatesof∗Electronicaddress:[e-mail]†Electronicaddress:[e-mail]‡Electronicaddress:[e-mail]§Electronicaddress:[e-mail]¶Electronicaddress:[e-mail]∗∗Electronicaddress:[e-mail];WorksupportedbyFAPERJ,Fundac¸˜aodeAmparo`aPesquisadoEstadodoRiodeJaneiro,undertheprogramCientistadoNossoEstado,E-26/151.947/2004.††Electronicaddress:[e-mail]2thesecondensates,suchasthephenomologicalapproachbasedontheSVZsumrules,[5]forarecentoverview,theuseoflatticemethods,[6],aswellastheuseofinstantoncalculus,[7].Gaugecondensatesarenecessarilynonperturbativeinnature,asgaugetheoriesdonotcontainamasstermintheactionduetotherequirementofgaugeinvariance.However,throughnonperturbativeeffects,anontrivialvaluefore.g.F2μνcanarise.In[8],itwasalreadyarguedthatalsogaugevariantcondensatescouldinfluencegaugevariantquantitiessuchasthegluonpropagator.Inparticular,thedimensiontwogluoncondensateA2μhasreceivedmuchattentionintheLandaugauge[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34]overthepastfewyears.AnOPEargumentbasedonlatticesimulationshasprovidedevidencethatthiscondensatecouldaccountforquadraticpowercorrectionsoftheform∼1Q2,reportedintherunningofthecouplingconstantaswellasinthegluonpropagator[11,13,14,24,27,31,33].ThisnonvanishingcondensateA2μgivesrisetoadynamicallygeneratedgluonmass,[11,12,16,19,20].Theappearanceofmassparametersinthegluontwopointfunctionisacommonfeatureoftheexpressionsemployedtofitthenumericaldataobtainedfromlatticesimulations,[35,36,37].Letusmentionthatagluonmasshasbeenfoundtobeusefulalsointhephenomenologicalcontext,[38,39].ThelocaloperatorA2μintheLandaugaugehaswitnessedarenewedinterestduetotherecentworks[9,10],asthequantityA2min≡minU∈SU(N)1VTZd4xDAUμ
2E,(1)whichisgaugeinvariantduetotheminimizationalongthegaugeorbits,couldbephysicallyrelevant.Infact,asshownin[9,10]inthecaseofcompactthree-dimensionalQED,thequantityA2minseemstobeusefulinordertodetectthepresenceofnontrivialfieldconfigurationslikemonopoles.OneshouldnoticethattheoperatorA2minishighlynonloca