Mixing and Decay Constants of Pseudoscalar Mesons

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

MixingandDecayConstantsofPseudoscalarMesons

Th.Feldmann ,P.Kroll

FachbereichPhysik,Universit¨atWuppertal,D-42097Wuppertal,Germany

B.Stech

Institutf¨urTheoretischePhysik,Universit¨atHeidelberg,D-69120Heidelberg,Germany

(Copyright1998byTheAmericanPhysicalSociety.)

Weproposeanewη–η′mixingschemewherewestartfromthequark avorbasisandassume

thatthedecayconstantsinthatbasisfollowthepatternofparticlestatemixing.Onexploiting

thedivergencesoftheaxialvectorcurrents–whichembodytheaxialvectoranomaly–allbasic

parametersare xedto rstorderof avorsymmetrybreaking.Thatapproachnaturallyleadsto

amassmatrix,quadraticinthemasses,withspeci edelements.Wealsotestourmixingscheme

againstexperimentanddeterminecorrectionstothe rstordervaluesofthebasicparametersfrom

phenomenology.Finally,wegeneralizethemixingschemetoincludetheηc.Againthedivergences

oftheaxialvectorcurrents xthemassmatrixand,hence,mixinganglesandthecharmcontentof

theηandη′.

PACS:14.40.Aq,11.40.Ha,11.30.Hv

arXiv:hep-ph/9802409v2 20 Jul 1998I.INTRODUCTIONη–η′mixingisasubjectofconsiderableinterestthathasbeenexaminedinmanyinvestigations,seee.g.[1–4]andreferencestherein.Asitiswell-known[5],theU(1)Aanomalyplaysadecisiverole.Fortheoctet-singletmixingangleθofthesepseudoscalarmesonsvaluesintherangeof 10 to 23 havebeenobtaineddependingondetailsoftheanalysis,seee.g.[3].Thephenomenologicalanalysisofteninvolvesdecayprocesseswhere,besidesstatemixing,alsoweakdecayconstantsappear.Thedecayconstantsarede nedbyii 0|Jµ5|P(p) = fPpµ(i=8,1;P=η,η′),(1.1)81whereJµ5denotestheSU(3)FoctetandJµ5theSU(3)Fsingletaxial-vectorcurrent,respectively.Frequently,itisassumedthatthedecayconstantsfollowthepatternofstatemixing:8=f8cosθ,fη8fη′=f8sinθ,1= f1sinθ,fη1fηf1cosθ.′=(1.2)However,recenttheoretical[6]aswellasphenomenological[7]investigationshaveshownthatEq.(1.2)cannotbecorrect.Adoptingthenewandgeneralparametrization[6]

8=f8cosθ8,fη8fη′=f8sinθ8,1= f1sinθ1,fη1fηf1cosθ1.′=(1.3)

θ1andθ8turnedouttodi erconsiderably.Thephenomenologicalanalysis[7],whichinvolvedthecombinedanalysisofthetwo-photondecaywidthsoftheηandη′,theηγandη′γtransitionformfactorsandtheadditionalconstraintfromtheradiativeJ/ψdecays,allowedtodeterminethefourquantitiesoccuringinEq.(1.3).Mostimportantly,thevaluesobtainedsatisfytheconstraints[6]fromchiralperturbationtheory(ChPT).

Theappearanceofthefourparametersf8,f1,θ8,θ1raisesanewtheproblemoftheirmutualrelationsandtheirconnectionwiththemixingangleoftheparticlestates.Thisangleisnecessarilyasingleonesincemixingwithhigherstates–theηcforinstance–canbeneglectedatthisstage.TherelationofthisanglewiththefourparametersinEq.(1.3)doesnotneedtobesimple:inapartonlanguage[8]thedecayconstantsarecontrolledbyspeci cFock

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′′

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

Transformingthenon-strangeandstrange

axial-vector

currentsintooctetandsingletcurrents,onecanalsoconnectthedecayconstantsde nedinEq.(1.1)tofqandfs

81 fηfη=U(φ)FU (θideal)(2.4)81fη′fη′

withtheresult

f8=

f1=

andthus 2fs/fq),2fq/fs)

√(2.5)tan(θ1 θ8)=

q-s

4π .GG(2.7)

itsdual;midenotethecurrentquarkmasses.Thevacuum–mesonGdenotesthegluon eldstrengthtensorandG

transitionmatrixelementsoftheaxialvectorcurrentdivergencesaregivenbytheproductofthesquareofthemeson2mass,MP,andtheappropriatedecayconstant.Forinstance,

s2s 0| µJµ5|η =Mηfη.(2.8)

Themassfactors,whichnecessarilyappearquadraticallyhere,canbeconsideredastheelementsoftheparticlemassmatrix 2 M0ηM2=.(2.9)20Mη′

iWiththehelpofEq.(2.3)thematrixelementsof µJµ5(i=q,s)canthenbeidenti edasthoseofthematrixproduct

M2U(φ)F.Transformingittothequark avorbasisandsolvingforthemassmatrix

2M2qs=U(φ)MU(φ),(2.10)

oneeasily nds

M2qs= m2qq+

2√fq 0|4παsfs |ηs GG 0|1αsm2ss+4π |ηs GG(2.11)

whereweusetheabbreviations

√m2qq=fq¯iγ5d|ηq ; 0|muu¯iγ5u+mddm2ss=2

2 0|αs 0|αsfs

.(2.13)

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

Forlateruseitisalsoconvenienttointroducetheabbreviation

a2=αs1 0|2fq

2

y=22(Mη′ Mη)sin2φ,.(2.15)(2.16)2a2

ThecombinationofEqs.(2.13)and(2.16)providesaninterestingrelationbetweenfq/fsandthemixingangleφ.Alsoworthnotingistherelationbetweenφandθ8thatisobtainedbycombiningEq.(2.5)withEqs.(2.13)and(2.16)

cotθ8= 2Mη′

2 f2.2fKπ(2.19)

NotethatV-spinconsiderationsprovidealinearrelationbetweenthedecayconstantsfs,fπandfKwhich,totheconsideredorderof avorsymmetrybreaking,canbereplacedbytheabovequadraticrelation.AscanbeseenfromEq.(2.6),thesetheoreticalresultsforfqandfsleadtoasubstantialdi erencebetweenθ1andθ8.OnlyinthestrictSU(3)Flimitwherefq=fs(or,equivalently,fK=fπorf8=f1)onewouldhaveθ1=θ8=θ,withθbeingtheoctet-singletmixingangle.AccordingtoLeutwyler[6],ChPTprovidestworelations(upto1/Nccorrections)amongthedecayconstants

22 1fKf8=3

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

powercompletely1.Inthenextsectionwewillconfrontourapproachwithexperiment.Wedeterminethemixingangleφandthebasicdecayconstantsfqandfsphenomenologicallyandlookforconsistencyandfordeviationsfromthe rstorderofSU(3)Fbreaking.

III.THEPHENOMENOLOGICALVALUESOFφ,Fq,Fs

Severalpossibilitiestoextractthevalueofthemixinganglefromexperimenthavebeendiscussedintheliterature,see,forinstance,[1,3,15,16].Wecanpro tfromthesepapersbyproperlyadaptingthemtotheqsmixingscheme.Wenotethatinthephenomenologicalanalyses[1,3,15,16]someadditionalsimplifyingassumptionshadtobemade.Thus,forinstance,OZI-suppressedcontributionsormassdependenciesofformfactorsandcouplingconstantsareignored.Westartby rstdiscussingprocesseswhichareindependent(orinsensitive)tothedecayconstantsandallowaoneparameter toftheparticlestatemixingangleφ.

ThedecayJ/ψ→Pρ:WeconsidertheratioofthedecaywidthsΓ[J/ψ→η′ρ]andΓ[J/ψ→ηρ].IntheseprocessesG-parityisnotconserved,theyproceedthroughavirtualphoton(seeFig.1a).Contributionsfromtheisospin-violatingpartofQCDaresupposedlyverysmallascanbeinferredfromthesmallnessoftheJ/ψ→φπwidthandwillbeneglected.Thecalculationofthedecaywidthsrequirestheknowledgeoftheρ–Ptransition2formfactorsatmomentumtransferq2=MJ/ψ.Onaccountofthe avorcontentoftheρmeson,thistransitionformfactoronlyprobestheηqcomponentsoftheηandη′ifOZI-suppressedcontributionsareneglected.Hence,

Fρη(q2)=cosφFρηq(q2),

Fρη′(q2)=sinφFρηq(q2).

andtherefore

Γ[J/ψ→η′ρ]

kηρ

where

222kPV=MJ/ψ[1 (MP+MV)/MJ/ψ]/2.(3.1) 3(3.2)(3.3)

Fromtheexperimentalvalue0.54±0.11forthisratioofdecaywidths[17]weobtainφ=39.9 ±2.9 .Almostthesamevalueforφhasbeenfoundinananalysisofallisospin-1J/ψ→PVdecays(includingpions)[16].Aglobal ttoallJ/ψ→PVdecaymodesonthebasisofaparticularmodel,yieldsφ=37.8 ±1.7 [16].Becauseofitsmodeldependencewewillnotusethelatterresultinevaluatingtheaverageofthemixingangle.SinceinthederivationofEq.(3.2)onlythemixingangleoftheparticlestatesenters,onemayfreelytransformfrom

sbasistotheoctet-singletoneaswasdone,forinstance,in[16].Nevertheless,thesimplerelationtheq

betweentheratioofdecaywidthsandthemixingangle,independentofthedynamics,isanadvantageoftheqsbasisusedhere.Intheoctet-singletmixingschemeonewouldhavetodealwithalinearcombinationoftwoaprioridi erentformfactors.Wewillpro tfromthisadvantagealsointhefollowing veprocesses.

Thedecaysη′→ργandρ→ηγ:Thetransitionmatrixelementscontrollingtheseprocessescanbedecomposedcovariantly[18]

( )νλσ γP(pP)|T|ρ(pρ) = egρPγ µνλσpµpρερ,Pεγ(3.4)

leadingtothefollowingexpressionsforthedecaywidths

23Γ[η′→ργ]=αgρη′γkρ,Γ[ρ→ηγ]=α

Usingthephenomenologicalparametersfoundin[7],andtranslatingthemintothequark avorbasiswithadmissionoftwomixingangles,φqandφs,de nedinanalogytoEq.(1.3),one ndsfq=1.09fπ;fs=1.38fπ;φq=39.4 andφs=38.5 .ThefactthatthetwoanglesnearlycoincidegivedirectsupporttothevalidityofEq.(1.4).1

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

where

kf=Mi[1 Mf2/Mi2]/2,(3.6)

beingthe3-momentaofthe nalstatemesonf.

M

i

denotes

the

mass

of

the

decaying

meson.

αisthe nestructureconstant.

Using

state

mixing(2.1),one nds

gρη′γ

Γ[a2→ηπ]=tan2φ kη′π

σ(π p→ηn)2=tan2φ(s MP)(3.9)

Thetwoexperimentsleadtoφ=36.5 ±1.4 [21]and39.3 ±1.2 [22].Sincethetworesultsarenotfully¯.consistentwitheachotherwewilldoubletheerrorsintheevaluationoftheaveragedvalueφ

Annihilationprocessespp¯→PM(M=π0,η,ω):TheCrystalBarrelCollaboration[23]measuredtheratiosforannihilationintoηMandη′Mandquotedavalueofφ=37.4 ±1.8 forthemixingangle.However,sincetheexperimentwascarriedthroughatlowenergies,theresultforφisrathersensitivetophasespacefactorsandtothemomentumdependenceoftheannihilationamplitudes.Wethereforediscardthatvalueofφinthedeterminationoftheaveragedmixinganglealthoughitwillturnouttobeconsistentwithit.

ThedecayJ/ψ→Pγ:Accordingto[24,15]thephotonisemittedbythecquarkswhichthenannihilateintolighterquarkpairsthroughthee ectoftheanomaly.Thus,thecreationofthecorrespondinglightmesonsiscontrolledbythematrixelement 0|αs

Γ[J/ψ→ηγ]=tan2φ4Mη′

kη 3=cot2θ8 kη′

Onemayextendthisanalysistotheωandφcases.Ignoringthesmalle ectduetotheω φmixing,onederivesgωη′γ/gωηγ tanφandgφη′γ/gφηγ cotφ,respectively.Fromthemeasuredvalues/bounds[17]weobtainφ 37 ±8 andφ>21 ,respectively.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

Fromthemeasuredvalue

[17]RJ/ψ=5

.

±

.6themixingangleφbecomes39.0 ±1.6 .Obviously,Eq.(3.10)isnotequivalenttothenaivesingletdominancepredictionforwhichthefactorcotθ8wouldhavetobereplacedbycotθ.AswelearnedinSect.II,θ8markedlydi ersfromtheoctet-singletmixingangle.

FIG.1.a)ElectromagneticcontributiontoJ/ψ→ρη,η′.b)PoleansatzfortheDs→P νsemi-leptonicdecay.

Aweightedaverageoftheabovesevenhigh-lightedvaluesyields

2,Cs=1/9)

Γ[η→γγ]=9α2

fq

16π3

andsolvingforfqandfs,wearriveat

fq=3Cqα

sinφΓ[η→3γγ]/Mη3Mη′ Cssinφ 2 Cqsinφfs,(3.13)+sinφ Γ[η′→3γγ]/Mη′4π3/2 1.(3.14)

WeevaluateEq.(3.14)withthemixingangleaccordingtoEq.(3.11)andthefollowingexperimentalvaluesforthedecaywidthsΓ[η→γγ]=(0.51±0.026)keVandΓ[η′→γγ]=(4.26±0.19)keV[17].Thevalueof0.324±0.046keV,obtainedfromthePrimako productionmeasurementofη→γγ,isnotincluded.Itturnsoutthatfsisnotwelldeterminedthisway,itacquiresaratherlargeerrorfs=(1.42±0.16)fπ.Wethereforeevaluatefsalsofromfqandthephenomenologicalvalueoftheratioyandformtheweightedaverageofbothvaluesto ndamoreprecisevalueforfs.Bythismeansweobtain

fq=(1.07±0.02)fπ,

fs=(1.34±0.06)fπ.(3.15)

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

Thesevaluesforthebasicdecayconstantsdi erfromthetheoreticalvalues(2.19)onlymildly.Since,withintheerrors,boththevaluesoffsdeterminedhereagreewitheachother,theexperimentalvaluesof

the

two-photondecaywidthsarewellreproducedbytheparameterset(3.11),(3.15).

Asanimmediatetestoftheparameters(3.11)and(3.15)wecomputethePγtransitionformfactorsalongthelinesdescribedindetailin[7].We ndexcellentagreementbetweentheoryandexperiment[14].Thenewresultsarepracticallyindistinguishablefromthe tperformedin[7](χ2/d.o.f.is28/34ascomparedto26/33in[7]).TheformfactoranalysisisbasedonapartonFockstatedecompositionofthephysicalmesons.ThewavefunctionsofthevalenceFockstates,providingtheleadingcontributiontotheformfactoraboveQ2=1GeV2,areassumedtohavetheasymptoticform.Thevaluesofthesewavefunctionsattheoriginofcon gurationspacearerelatedtothedecayconstants[7].

AcomparisonbetweenthetheoreticalandphenomenologicalvaluesofthemixingparametersismadeinTableI.Ascanbenoticedthereisnosubstantialdeviationbetweenbothsetofvalues,i.e.higherorder1/Nccorrections,absorbedinthephenomenologicalvalues,seemtobereasonablysmall.InTableIIwelistthevaluesoftheparametersde nedinEq.(1.3),i.e.intheparametrizationintroducedbyLeutwyler[6],asobtainedfromvarioussources.Thetheoreticalvaluesoff8,f1,θ8arecomputedfromthedecayconstantsgiveninEq.(2.19)andthetheoreticalmixinganglelistedinTableIwhilethephenomenologicalvaluesfollowfromEqs.(3.11)and(3.15).Ascanbeseentheresultsobtainedfromtheanalysesperformedinthisworkandin[6,7]agreeratherwellwitheachother.Theconventionalanalyses,e.g.[3,15],arenotincludedinthetablebecausethedi erencebetweenθ8andθ1isnotconsidered.

IV.GENERALIZINGTOη–η′–ηcMIXING

Fromtheprevioussectionswelearnedthatourcentralassumption(1.4)combinedwiththedivergencesoftheaxialvectorcurrentsleadstoavarietyofinterestingpredictionswhichcomparewellwithexperiment.Thereasonforthissuccessislikelytheratherlargedi erencebetweenthecurrentmassesofthestrangeandtheup/downquarks.Sincethecharmquarkmassisevenheavierthanthestrangeone,itistemptingtogeneralizetotheqs–c

q–scbasisreads(i,j=q,s,c),

222M2qsc=U(φ,θy,θc)diag(Mη,Mη′,Mηc)U(φ,θy,θc).(4.3)

Ontheotherhand,generalizingEq.(2.11)andusingtheabbreviations(2.12),(2.13)and(2.14)introducedinSect.II,wemaywritethemassmatrixasfollows

√ 2 222ay2amqq00√2 .+M2=0m0(4.4)yqscss200mcc2a2yza2z2a2

OnexploitingagainthedivergencesoftheaxialvectorcurrentsandthepropertiesofthemassmatrixanumberofconsequencesfollowsfromwhichallnewparametersappearinginEq.(4.4)canbe xed

z=fq/fc,θy=θ8,

222=m2Mηcc+zac(4.5)

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

and

θc= z 2Mηc(4.6)

witha2asgiveninEq.(2.15).UsingthephenomenologicalparametervaluesquotedinTableI,we ndthefollowing2=m2numericalresults:z=0.35±0.03;θc= 1.0 ±0.1 .Sincez2a2=0.03GeV2wehaveMηcctoaverygoodc′approximation.Thecharmdecayconstantsoftheηandtheηtakethevalues

cfη= (2.4±0.2)MeVcfη′= (6.3±0.6)MeV(4.7)

Theirvaluesareinroughagreementwiththeresultspresentedin[13,27,28]butindramaticcon ictwiththevaluescquotedin[10,11].fη′lieswellwithintheboundestimatedin[7].Ouranalysissupportstheconclusionsdrawnin[13]thatthecharmcontentoftheη′isnotthesolutionfortheabnormallylargeB→Kη′decaywidth,theexplanationofwhichremainsanopenproblem.

Usingtheabovevaluesforthemixinganglesφ,θyandθc,we ndforthequarkcontentofthephysicalmesons

|η =0.77|ηq 0.63|ηs 0.006|ηc0 |η′ =0.63|ηq +0.77|ηs 0.016|ηc0 |ηc =0.015|ηq +0.008|ηs +|ηc0 .(4.8)

Thecharmadmixturestotheηandη′aresomewhatsmallerthanestimatedin[1]butslightlylargerthanquotedin

[28].Apossibletestfortheηc0contentisprovidedbytheradiativeJ/ψdecays.ForthedecaysJ/ψ→ηγ,η′γweusedalreadytheactionofthegluonsasdescribedbythematrixelementsoftheanomaly(notethatθ8=θy).Sincetheη′hastheηc0contentθccosθ8whiletheηc0contentofηcispracticallyone,weexpect

3 αs 3′ 0|Γ[J/ψ→ηγ]√=(4.9)kηckηc

Theexperimentalnumberforthisratio,0.33±0.10[17],givesusanother–admittedlylessreliable–determinationofthecharmadmixtureinη′.Theresult|θccosθ8|=0.014±0.002isingoodagreementwiththenumbercontainedinEq.(4.8).

V.SUMMARY

Inthedescriptionofη-η′mixingthereare veparametersinvolved,themixingangleoftheparticlestatesandfourdecayconstants.Motivatedbytheobservationofnearlyidealmixinginvectorandtensorparticleswetakeasourbasisthestatesaccordingtotheirquark avorcompositions.Ourcentralassumptionisthenthatinthisparticularbasisthemixingofthedecayconstantsfollowsthatofstatemixing.Thisnewmixingschemeisveryrestrictive.It xesthestructureofthemassmatrixandpredictsthemixingangleandthefourdecayconstantsupto rstorderin avorsymmetrybreaking:

i)Thefourdecayconstantsareimmediatelyreducedtotwoconstantsfqandfsandasingleangleφthatisidenticaltothestatemixingangleanddescribesthedeviationfromidealmixing.

ii)Thedivergencesoftheaxialvectorcurrentsprovideuswithamassmatrixquadraticintheparticlemasseswitho diagonalelementsentirelydeterminedbytheanomaly.Theoldproblemofquadraticversuslinearmassmatriceshasfounditsanswer.

iii)ingthisresultthemixingangleφcanbecalculatedfromfq/fs.iv)SU(3)Frelations xfqandfsintermsoffπandfKto rstorderin avorsymmetrybreaking,and xthose22partsofthemassmatrixwhichcontainthecurrentquarkmassesintermsofMπandMK.Thedecayconstantsobtainedthiswayobeytherequirementsofchiralperturbationtheorywhichareknownuptoorder1/Nccorrections.v)Withtheseingredientsandbyusingtheknownmassesofthephysicalstatesthemassmatrixisover-determined.Althoughsizeablecorrectionstothe avorsymmetryresultscouldhavebeenexpected,theresultingparameter-freede-terminationofthemixingangleandthedecayconstantsisinreasonableagreementwithapreviousphenomenologicalanalysiswithunconstrainedparameters.

vi)Weperformedanewanalysisanddeterminedphenomenologicallyφandfqandfsfromseveralindependentexperiments.Allresultswereconsistentwitheachother.Thus,theweightedaveragevalueforthemixingangleis

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

ratherprecise:weobtainedφ=39.3 ±1 whichgivesasingle-octetmixingangleofθ= 15.4 .Fortheangleθ8whichisresponsiblefortheη,η′ratioinradiativeJ/ψdecayswefoundavalueof 21.2 .Thevaluesforfqandforfsdi erfromthetheoreticalpredictions(to rstorderof avorsymmetrybreaking)onlymildly.

vii)Itisstraightforwardtogeneralizethenewmixingschemetoincludethemixingwiththeηcwhichisofparticularrecentinterest.HerethedecayconstantfcenterswhichwetakeequaltofJ/ψ.Withthisingredientthec

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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

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