Non-factorization and the Decays B into Jpsi + K()

arXiv:hep-ph/9409261v1 8 Sep 1994WM-94-110Non-factorizationandtheDecaysB→J/ψ+K(∗)

CarlE.CarlsonPhysicsDepartment,CollegeofWilliamandMary,Williamsburg,VA23187

J.MilanaPhysicsDepartment,UniversityofMaryland,CollegePark,MD20742(February1,2008)

Abstract

Manyknownmodels,whichgenerallyuseafactorizationhypothesis,giveapooraccountofthedecaysB→J/ψ+K(∗).Usuallythereisafreeoverallfactor,whichisfittothedata,sothattestsofthemodelsrelyuponratios.ThemodelstendtogivetoomuchK∗comparedtoKandtoomuchtransversepolarizationcomparedtolongitudinal.Ourmicroscopiccalculations,whichuseperturbativeQCD,dowellforbothratios.Amicroscopiccalculationallowsustoseehowwellfactorization,heavyquarksymmetry,andotherfeaturesofvariousmodelsareworking.Inthepresentcase,agreementwiththeexperimentalratiosisdependentuponabreakdownoffactorizationforoneoftheamplitudes.

PACSnumbers:13.20.He,12.15.-y,12.38.Bx

TypesetusingREVTEX1I.FACTORIZATIONANDDATAGourdin,Kamal,andPham[1]andAleksanetal.[2]pointoutthatmanyknownmod-els[3–6],allofwhichusethefactorizationhypothesis,giveapooraccountofthedecaysB→J/ψ+K(∗).Inmostmodelsthereisanoverallfactor,generallycalleda2[3],whichisfittothedata,sothattestsofthemodelsrelyuponratiosofK∗andKdecays,andratiosoflongitudinalandtranversepolarizationintheJ/ψ+K∗decays.ThemodelstendtogivetoomuchK∗comparedtoKandtoomuchtransversepolarizationcomparedtolongitudinal.Ourmicroscopiccalculations,whichuseperturbativeQCD,dowellforbothratios.Al-thoughtheresultshavebeenpublishedinsomedetail[7–9],thecharmoniumBdecaysdeservesomefurtherthoughtbecauseofthepresentinterestinthem,andwewillattempttomakeself-containedatleastthequalitativepartsofourpresentremarks.Amicroscopiccalculationallowsustoseehowwellfactorization,heavyquarksymmetry,andotherfea-turesofvariousmodelsareworking.Inthepresentcase,wefindaseriousbreakdownoffactorizationforoneoftheamplitudes.Moreexplicitly,theratiosunderstudyandtheirexperimentalvaluesare[10],

R≡Br(B→J/ψ+K∗)

Γ󰀈K∗=󰀇0.80±0.08±0.05CLEO[10]0.66±0.10±0.10CDF[11](2)

Ourownresultsforthetworatiosare1.76and0.65,respectively(usingTableIVof[7]).FactorizationimpliesthatthedecaysdependuponasetofformfactorsforacurrentconnectingBtoK(∗).Asabenchmark—yes,weknowtheK(∗)islight—therelationsthatheavyquarksymmetry[12–14]impliesamongtheformfactorsleadto

R=m2B+4m2J/ψ

Γ󰀈K∗=m2

Bcontributionsareabouthalfthesizeandoppositeinsigntothefactorizableones,whichhasroughlytheeffectofturningthe“4”intoa“1”inthepreviousequations,andgivingdecentagreementwiththe(ΓLL/Γ)K∗data.Also,surprisinglyinthiscontext,wefindtheheavyquarksymmetrysymmetrypredic-tionsfortheformfactorsofthefactorizablepartsoftheamplitudeworksurprisinglywell.Onemightexpectsignificantdifferencesduetoanonperturbativecause,namelythatthewavefunctionsordistributionamplitudesoftheKandK∗aredifferent.AwavefunctiondifferenceattheoriginisshownbydatathatgivesunequaldecayconstantsfortheKandK∗,andtheshapesofthetwowavefunctionsarealsodifferent.WeusethedistributionamplitudesforKandK∗workedoutfromQCDsumrulesbyChernyak,Zhitnisky,andZhitnitsky[15].Theupshotisthattheformfactorsrelativetotheheavyquarksymmetrypredictionsaregood,andthatsmallcorrectionsandnonfactorizablecontributionskeepthetworatiosfromjustbeinginversesofeachother.Somedetailswillnowcome.

II.MOREDETAILEDDISCUSSIONFactorizationWeshouldstatewhatfactorizationmeansinthecontextofB→J/ψ+K(∗).TherelevantpartoftheeffectiveHamiltoniandensityis

Heff=GF

2VcbVcs¯sγµLc¯cγµLb,(5)

whereγµL=γµ(1−γ5)andthematrixelementwewantis,generically,M=󰀌X,ψ|Heff|B󰀍,(6)whereψ=ηc,J/ψ,....Thefactorizationhypothesisisthatthecharmedquarkswhicharecreatedgointotheψand,exceptfortheweakinteractionvertices,areunconnectedtootherquarksintheprocess.Wealsoassumethattheoutgoingcharmedquarksintheψhavesmalltransversemomentumrelativetothedirectionoftheψ.Ifthefactorizationhypothesisisvalid,onecanshow

M=−1√Non-factorizationInFig.1,parts(a)and(d)correspondtothefactorizablecontributions,inthepresentcontext,andparts(b)and(c)tothenonfactorizableones.Uponfirstview,itiseasytobelievethatthenonfactorizablecontributionsaresmall.ThegluoncouplestotwooppositelycoloredquarksthatarenearlyatthesamepointbecauseoftheW-exchange.Indeed,thelargestpartsofdiagrams(b)and(c)canceleachotherandthesubleadingO(qG),whereqGisthegluonmomentum,termsgivethesurvivingresult.ThepiecesofFigs.1(b)and(c)fromoneweakvertex,throughtheJ/ψ(whosepolarizationvectorisξ)includingthegluonemmissionvertex(γν),andtotheotherweakvertex,havenumeratorsthatsumto

4mJ/ψ(1+γ5)(ξγνqG−qGγνξ)(1−γ5),(10)whichdoesgotozeroforgluonsoflongwavelength,orqGgoingtozero.(ThenumeratorsofFigs.1(a)and(d)donotgotozerointhesamelimit.)However,thegluonmomentumisnotsosmall;infactwearguethatitislargeenoughthataperturbativecalculationisplausiblyvalid.Itsuppliesthemomentumtransferneededbythelightquark,whichisoforder¯ΛB,thepartofthemassoftheBmesoncarriedbythelightquark,whichisabout500MeVorafewtimesΛQCD.However,forB→J/ψ+KandthelongitudinalpartofB→J/ψ+K∗,thereisfurthercancellationbetweenthesubleadingpartsofthetwononfactorizablediagrams.Incontrast,theyaddforthetransversedecay,sothisnonfactorizableamplitudecangetlarge.WhilethetransverseB→J/ψ+K∗doesrequirechiralityviolations,theensuingsuppressionisofO(mJ/ψ/mB),whichisnotadecisivefactor.