Mixing and Decay Constants of Pseudoscalar Mesons(2)

We propose a new eta-eta' mixing scheme where we start from the quark flavor basis and assume that the decay constants in that basis follow the pattern of particle state mixing. On exploiting the divergences of the axial vector currents - which embody the

statewavefunctionsatzerospatialseparationofthequarkswhilestatemixingreferstothemixingintheoverallwavefunctions.

Inthisworkweexpressηandη′aslinearcombinationsoforthogonalstatesηqandηswhichcanbegenerated

u+d2andsbytheaxialvectorcurrentswiththe avorstructureq

qands

2.Wewilldemonstratethattheproperuse

ofthisquark avorbasisprovidesfornewinsightsandsuccessfulpredictions.Wepointoutthatweemploy xed(momentum-independent)basisstates.Thus,ourstatemixingangleismomentumindependentandwell-de nedalsoinanyotherbasisobtainedbyanorthogonaltransformation.Thisdi ersfromotherpossibleapproachesinwhichmomentumdependentmassmatricesareintroduced(see,forinstance,[9]).Thedecayconstants,ontheotherhand,willingeneraldependonq2,i.e.theparticlestatesandmasses,andwillthusrequireaparametrizationbytwodi erentmixinganglesasin(1.3).Theseanglesdependonthebasiswhichisusedforthede nitionofthedecayconstants.Asdescribedbelow,thebasicassumptionwhichwewilluseinthispaperisthatthedecayconstantsfollowthestatemixingifandonlyiftheyarede nedwithrespecttothequarkbasis.Inthiscircumstancethetwoanglesforthedecayconstantsobtainedinthisbasisandthecorrespondingstatemixinganglecoincideandarethusmomentum-independent.

De ningnowdecayconstantsanalogoustoEq.(1.1)butwithi=q,sanddenotingtheη–η′mixinganglethatdescribesthedeviationfromidealmixing,byφ,wepropose

qfη=fqcosφ,qfη′=fqsinφ,sfη= fssinφ,

sfη′=fscosφ.(1.4)

Thatthedecayconstantsinthequark avorbasisfollowinthiswaythepatternofparticlestatemixingisourcentralassumption.Itisequivalenttotherequirementthatthecontributionfq(fs)tothedecayconstantsobtainedfromtheηq(ηs)componentsofthewavefunctionsisindependentofthemesoninvolved.Thisassumptionappearsplausiblebutwehavenorigorousjusti cationforitandhavetotestit.ItiscertainlyrestrictiveaswillbeshowninSect.II: rst,itreducesthenumberofparametersagaintothree.Secondly,byinvokingthedivergencesofthecurrents,theangleφisconnectedtofq/fs.Finally, avorsymmetry xesfqandfsto rstorderofSU(3)Fbreaking,leavingus–tothisorder–withnofreeparameter.Massmixingofthepseudoscalarmesonsisalsodiscussedinthissection.NumerousphenomenologicalchecksarepossibleandperformedinSect.III.WedeterminephenomenologicalvaluesforthethreeparametersfromthedataandcheckforconsistencywithChPTandtheearlierdetermination[7]ofthefourquantitiesf8,f1,θ8,θ1.OurschemewillthenbegeneralizedtoincludetheηcinSect.IV.Thegeneralizedapproachallowstocestimatethequarkcontentofthethreepseudoscalarmesons,themixinganglesandthecharmdecayconstantsfηcandfη′whichattractedmuchinterestinthecurrentdiscussion[10–13]oftheratherlargebranchingratiofortheprocessB→Kη′asmeasuredbyCLEO[14].OursummaryispresentedinSect.V.

II.THEqsMIXINGSCHEME

Thetwostatesηqandηsarerelatedtothephysicalstatesbythetransformation

ηqη=U(φ),′ηηs

whereUisaunitarymatrixde nedby

U(α)= cosα

sinα sinαcosα .(2.1)(2.2)

Weassumethatthephysicalstatesareorthogonal,i.e.thatmixingwithheavierpseudoscalarmesons(e.g.theηc)canbeignored,see,however,Sect.IV.Westressthataslongasstate-mixingisconsidered,onemayfreelytransformfromoneorthogonalbasistotheother.Forexample,thestandardoctet-singletmixingangleisgivenbyθ=φ θideal.AccordingtoourcentralassumptiondescribedinSect.1,wetake

q s fηfηfq0=U(φ)F,F=.(2.3)qsfηfη0fs′′