The Calculation of Vacuum Properties from the Glob

arXiv:nucl-th/0201001v313Apr2003TheCalculationofVacuumPropertiesfromtheGlobalColorSymmetryModelHong-shiZong1,2,Jia-lunPing3,Hong-tingYang1,Xiao-fuL¨u2,4,andFanWang11DepartmentofPhysics,NanjingUniversity,Nanjing210093,P.R.China2CCAST(WorldLaboratory),P.O.Box8730,Beijing100080,P.R.China3Departmentofphysics,NanjingNormalUniversity,Nanjing210097,P.R.Chinaand4DepartmentofPhysics,SichuanUniversity,Chengdu610064,P.R.ChinaAbstractAmodifiedmethodforcalculatingthenon-perturbativequarkvacuumcondensatesfromtheglobalcolorsymmetrymodelisderived.Withinthisapproachitisshownthatthevacuumcon-densatesarefreeofultravioletdivergencewhichisdifferentfromthepreviousstudies.Asaspecialcasethetwoquarkcondensateh¯qqiandthemixedquarkgluoncondensategh¯qGμνσμνqiarecal-culated.AcomparisonwiththeresultsofothernonperturbativeQCDapproachesisgiven.Key-words:Non-perturbativemethodsinQCD,GCM,Vacuumcondensate.E-mail:[e-mail]-perturbativestructureoftheQCDvacuumischaracterizedbyvariouscon-densates:suchasthetwoquarkcondensateh¯qqi,themixedquark-gluoncondensategh¯qGμνσμνqi,thefourquarkcondensateh¯qΓq¯qΓqi,etc.Thesecondensatesareessentialfordescribingthephysicsofthestronginteraction[1,2].Therehasbeengrowinginterestindescribingthepropertiesofhadronsinnuclearmatterintermsofthein-mediumquarkandgluoncondensates,whichareshiftedfromtheirvacuumvalues[3-8].SincetheGlobalColorSymmetryModel(GCM)[9]providesanonperturbativeframeworkwhichadmitsasimulta-neousstudyofthespontaneouschiralsymmetrybreakingandconfinement,itisusefulinexploringthephasetransitionatnonzerotemperatureanddensity.BeforewegeneralizeGCMfromzerotemperatureanddensitytononzerotemperatureanddensity,itisneces-sarytocheckwhetherGCMcanprovideagooddescriptionofvacuumpropertiesatzerotemperatureanddensity.Itistheaimofthispapertoconsiderquarkvacuumcondensatesingeneralandinparticu-larthetwoquarkcondensateandthemixedquark-gluoncondensateintheframeworkoftheglobalcolorsymmetrymodel.AlthoughGCMviolateslocalcolorSU(3)Cgaugeinvarianceandrenormalizability,itprovidesaverysuccessfuldescriptionofvariousnonperturbativeas-pectsofstronginteractionphysicsandhadronicphenomenaatlowenergies.Theseinclude,forinstance,quarkconfinement[10],UA(1)breaking,andtheη−η′masssplitting[11],lowenergychiraldynamicsofGoldstonebosons(π,K,η)[12-18],mesonformfactors[19],heavy-lightmesons[20],systemsatfinitetemperate[21],orsoliton[22]andFadeev[23]descriptionsofthenucleon.Thefirstestimationsofthetwoquarkandthemixedquark-gluoncondensatesinGCMwasmadeinRef.[24],wherethetwoquarkcondensateh¯qqiitselfissystematicallysmallerthanthe“standard”valueof−(250MeV)3[1]Here,wepresentamodifiedmethodforcalculatingthequarkvacuumcondensateintheframeworkofGCM.Sincetheu,dcurrentquarkmassissmall,weconsidertheGCMgeneratingfunctionalformasslessquarks,i.e.,thechirallimit,inEuclideanspaceZ[¯η,η]=ZD¯qDqDAexp−SGCM+Zd4x(¯ηq+¯qη),(1)whereSGCM=Zd4x¯q(x)γ·∂−igλaC2γ·Aa(x)#q(x)+Zd4xd4y12Aaμ(x)hDabμν(x−y)i−1Abν(y),2withDabμν(x−y)denotingthegluontwo-pointgreenfunction.BecausetheformofDabμν(x−y)intheinfraredregionisunknown,oneoftentreatstheDabμν(x−y)astheGCMinputparameter,which,aswewilldiscusslater,ischosentoreproducecertainaspectsoflowenergyhadronicproperties.ForconveniencewewillusetheFeynmanlikegaugeDabμν(x−y)=δμνδabD(x−y)fromnowon.PerformingthefunctionalintegraloverDAinEq.(1)(seeEq.(27)below),weobtaintheGCMgeneratingfunctionalZ[¯η,η]=ZD¯qDqen−Rd4x¯q(x)γ·∂q(x)−Rd4xd4yg22jaμ(x)D(x−y)jaμ(y)+Rd4x(¯ηq+¯qη)o,(2)herejaμ(x)denotesthecoloroctetvectorcurrentjaμ(x)=¯q(x)γμλaC2q(x).IntroducinganauxiliarybilocalfieldBθ(x,y)asinRef.[9],thegeneratingfunctionalofGCMcanbewrittenasZ[¯η,η]=ZD¯qDqDBθ(x,y)exp−S[¯q,q,Bθ(x,y)]+Zd4x(¯ηq+¯qη),(3)whereS[¯q,q,Bθ(x,y)]=ZZd4xd4y¯q(x)G−1[x,y;[Bθ]]q(y)+Bθ(x,y)Bθ(y,x)2g2D(x−y)#,withG−1[x,y;[Bθ]]=γ·∂xδ(4)(x−y)+12ΛθBθ(x,y),(4)wherethematricesΛθ=DaFbCcisdeterminedbyFierztransformationinDirac,flavorandcolorspace,andaregivenbyΛθ=12(1D,iγ5,i√2γμ,i√2γμγ5)⊗(1√31F,1√2λaF)⊗(431C,i√3λaC).PerformingthefunctionalintegraloverD¯qandDqinEq.(3),weobtaintheGCMgener-atingfunctionalZ[¯η,η]=ZDBθ(x,y)exp(−S[¯η,η,Bθ(x,y)]),(5)whereS[¯η,η,Bθ(x,y)]=−Trln/∂δ(x−y)+12ΛθBθ(x,y)+ZZd4xd4yBθ(x,y)Bθ(y,x)2g2D(x−y)+¯η(x)G(x,y;[Bθ])η(y)#.(6)3Thesaddle-pointoftheactionisdefinedasδS[¯η,η,Bθ(x,y)]/δBθ(x,y)η=¯η=0=0andisgivenbyBθ0(x−y)=g2D(x−y)tr[ΛθG0(x−y)],(7)whereG0standsforG[Bθ0]andthetraceinEq.(7)istobetakeninDiracandcolorspace,whereastheflavortracehasbeenseparatedout.Wewillcalculatethevacuumcondensatesfromthesaddle-pointexpansion,thatis,wewillworkatthemeanfieldlevel.ThisisconsistentwiththelargeNClimitinthequarkfieldsforagivenmodelgluontwo-pointfunction.Inthemeanfieldapproximation,thefieldBθ(x−y)issubstitutedbytheirvacuumBθ0(x−y).Underthisapproximation,thedressedquarkpropagatorG(x−y)≡G0(x−y)inGCMhasthedecompositionG−1(p)≡iγ·p+Σ(p)≡iγ·pA(p2)+B(p2)(8)withtheself-energydressingofthequarksΣ(p)isdefinedas:Σ(p)≡12ΛθBθ0(p)=Zd4xeip·x12ΛθBθ0(x)=iγ·p[A(p2)−1]+B(p2),(9)wheretheselfenergyfunctionsA(p2)andB(p2)aredeterminedbytherainbowDyson-Schwingerequation[9][A(p2)−1]p2=83