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arXiv:hep-ph/9308262v213Aug1993IKDA93/26hep-ph/9308262OperatorProductExpansionforInclusiveSemileptonicDecaysinHeavyQuarkEffectiveFieldTheoryThomasMannelInstitutf¨urKernphysikTechnischeHochschuleDarmstadtSchlossgartenstr.9,D–6100DarmstadtGermanyFebruary1,2008AbstractInclusivesemileptonicdecaysarediscussedintheframeworkofheavyquarkeffectivefieldtheorybyemployingtheshortdistanceexpansionintheeffectivetheory.Thelowestordertermturnsouttobethepartonmodel;thehigherordertermsmayberegardedascorrectiontermstothepartonmodelresult.Thefirstnonvanishingcorrectionstothepartonmodelresultaregivenandtheleptonenergyspectrumofinclusivesemileptonicdecaysofheavymesonsiscalculated.1IntroductionHeavyquarkeffectivefieldtheory[1,2]hasturnedouttobeausefultoolforthedescriptionofheavyquarksystems.Duetoadditionalsymmetries[1]emergingintheheavyquarklimitseveralpredictionsforthedecaysofheavyhadronsmaybeobtainedwhicharecompletelymodelindependent.Inparticular,theformfactorsofweakdecaysinvolvingaheavytoheavytransitionareseverelyconstrainedbyheavyquarksymmetryandsomepredictionsmaybeobtainedevenforheavytolightdeacays[3].Theheavyquarkexpansionhasalsobeenappliedtoinclusivedecays.ThefirstdiscussionofinclusivesemileptonicdecaysinthisframeworkhasbeengivenbyChay,GeorgiandGrinsteinin[4].AlongsimilarlinesproceedstheworkofBigi,Shifman,UraltsevandVainstein[5]andofBlok,Koyrakh,ShifmanandVainshtein[6],whereinclusivenonleptonicaswellassemileptonicdecayshavebeenconsidered.Usuallytheinclusivedecayratesofheavyhadronsandtheirspectraareap-proximatedbythedecayoftheheavyquarksinapartonmodelapproximation.However,thisisamodelapproachandanestimateofitserrorisdifficult.Thisiniscontrasttothemethodsdescribedin[4,5]andalsotothepresentapprochwhichisacontrollableapproximationtoQCDcorrespondingtoexpansionsinsmallparameters.Themethodproposedin[4,5]todealwithinclusiveprocessesistouseanoper-atorproductexpansion,whichisjustifiedforheavyhadronsduetoitslargemass.Thelowestdimensionoperatorisadimensionthreeoperator;itsmatrixelementsarenormalizedduetoheavyquarksymmetriesandyieldthepartonmodelresult.Correctionsenterthroughoperatorsofdimensionfive;theirmatrixelementsareparameterstobetakenfrommeasurements.Thesecorrectionshavebeencalculatedfornonleptonicandforsemileptonicdecaysofmesonsin[5]andforthedifferentialdistributionsinsemileptonicdecaysin[6];semileptonicdecaysofbaryonshavebeenconsideredin[7].Inthepresentpaperweshallconsideronlysemileptonicdecaysandformulateaslightlydifferentapproachastheoneproposedin[5,6].AfterswitchingtoheavyquarkeffectivetheorythespacetimedependenceoftheheavyquarkfieldsinthehadronictensorisonlyduetotheresidualmomentumkoftheheavyquarkQinsidetheheavyhadronH,i.e.k=PH−mQv.InsemileptonicdecaysthereisinadditiontothescalesetbytheheavymassmQalsothemomentumtransfertotheleptonsq2.AfterswitchingtotheeffectivetheorytherelevantvariablebecomesQ2=(q−mQv)2,whichdefinesinadditiontothemassmQoftheheavyquarka1secondscaleintheproblem.InlargeportionsofthephasespaceQ2isoftheorderofm2Qandthusasimul-taneousexpansioninboth1/mQand1/Q2isappropriate.Thisistheapproachchosenin[5,6]1.However,onemayalsoconsidertheregioninphasespace,wherem2Q≫(q−mQv)2≫Λ2QCDandthusanexpansiononlyinpowersof1/Q2isnecessarywhileonlytheleadingtermofthe1/mQiskept.Inthepresentpaperweshallconsiderthelatterapproachandcomparetotheoneof[5,6].Thepaperisorganizedasfollows.Inthenextsectionwefixournotationbygivingthebasicformulaefortheinclusivedecayrates.Insection3wesetuptheoperatorproductexpansionintheeffectivetheoryanddiscussthelowestordertermandthefirstnonvanishingcorrection.Thesetupgivenheremaybeusedforbaryonicaswellasformesonicdecaysandinsection4wegiveaparametrizationforthehadronicmatrixelementsformeson-andbaryoninclusivedecaysanddiscussshortdistanceQCDcorrections.Finally,insection5wediscusstheleptonenergyspectrumininclusivesemileptonicdecaysinsomedetail.2InclusiveDecay:MatrixElementsandRatesInordertosetupthenotationweshallbrieflyrecallthebasicformulaeforthedescriptionofinclusivesemileptonicdecays.TheinclusivesemileptonicdecayofaheavyhadronH(eitheraheavymesonoraheavybaryon)intosomehadronicstateXandtwoleptonsH(PH=mHv)→X(PX)+ℓ(k)+¯νℓ(k′)(1)ismediatedbyaneffectiveHamiltonianoftheformHeff=GF√2VQq(¯Qγμ(1−γ5)q)(¯ℓγμ(1−γ5)νℓ)(2)whereQistheheavyquarkcontainedintheintheheavyhadronH2.ThedifferentialdecayratemaybewrittenintermsofthehadronictensorWμν(q,v)Wμν(q,v)=ΣZX(2π)4δ4(PH−q−PX)(3)H(v)|(¯Qγμ(1−γ5)q)|XX|(¯qγμ(1−γ5)Q)|H(v).1IthankMarkWiseforaclarifyingdicussiononthispoint.2WeshallassumethattheflavorofqmaybedeterminedfromthehadronicstateX.2Weshallconsideronlythecasewherethespinsofthefinalstateleptonsarenotmeasured;hencetheleptonictensorisgivenbyΛμν=8kμk′ν+k′μkν−gμν(kk′)+iǫμναβkαk′β(4)ThedifferentialdecayrateintherestframeoftheheavyhadronisthendΓ=G2F4mH|VQq|2Wμν(k+k′,v)Λμνd(PS)(5)whered(PS)isthephasespaceintegrald(PS)=Zfdkfdk′δ(observables)fdk=d4k(2π)3δ(k2−m2Lep)Θ(k0).(6)Thedeltafunctionδ(observables)projectsouttheobservablestobeconsidered;forthecaseoftheenergyspectrumofthechargedleptonintherestframeofthedecayingheavymesonitisδ(observables)=δ(E−vk′),sin