Theory of Inclusive B Decays

arXiv:hep-ph/9702310v112Feb1997CERN-TH/97-19hep-ph/9702310TheoryofInclusiveBDecaysMatthiasNeubertTheoryDivision,CERN,CH-1211Geneva23,SwitzerlandAbstractWepresentthetheoryofinclusivedecaysofhadronscontainingaheavyquarkanddiscussitsmostimportantapplicationstothedecaysofBmesons.Wealsoreviewthetheoreticalunderstandingofthehadronicparametersλ1andλ2(orμ2πandμ2G)enteringtheheavy-quarkexpansion.Invitedtalkpresentedatthe4thKEKTopicalConferenceonFlavourPhysicsKEK,Japan,29-31October1996CERN-TH/97-19February19971TheoryofInclusiveBDecaysMatthiasNeubertTheoryDivision,CERNCH-1211Geneva23,SwitzerlandWepresentthetheoryofinclusivedecaysofhadronscontainingaheavyquarkanddiscussitsmostimportantapplicationstothedecaysofBmesons.Wealsoreviewthetheoreticalunderstandingofthehadronicparametersλ1andλ2(orμ2πandμ2G)enteringtheheavy-quarkexpansion.1.INTRODUCTIONHadronicboundstatesofaheavyquarkwithlightconstituents(quarks,antiquarksandgluons)arecharacterizedbyalargeseparationofmassscales:theheavy-quarkmassmQismuchlargerthanthemassscaleΛQCDassociatedwiththelightdegreesoffreedom.Equivalently,theComp-tonwavelengthoftheheavyquark(λQ∼1/mQ)ismuchsmallerthanthesizeofthehadroncon-tainingit(Rhad∼1/ΛQCD).Insuchasitua-tion,itisappropriatetoseparatethephysicsas-sociatedwiththesetwoscales,sothatalldepen-denceontheheavy-quarkmassbecomesexplicit.Theframeworkinwhichtoperformthissepara-tionistheheavy-quark(or1/mQ)expansion[1]–[8],whichisaspecificrealizationoftheoperatorproductexpansion(OPE)[9,10].Thereare(atleast)twoimportantreasonswhyitisdesirabletoseparateshort-andlong-distancephenomena:Atechnicalreasonisthatafterthisseparationwecancalculateabigportionoftherelevantphysics(i.e.allshort-distanceeffects)usingperturbationtheoryandrenormalization-grouptechniques;inparticular,inthiswayweareabletocontrolalllogarithmicdependenceontheheavy-quarkmass.Animportantphysicalreasonisthat,aftertheshort-distancephysicshasbeenseparated,thelong-distancephysicsmaysimplifyduetotherealizationofapproximatesymmetries,whichrelatethelong-distancepropertiesofmanyobservablestoasmallnumberofhadronicmatrixelements.Thesecondpointisparticularlyexcit-ing,sinceitallowsustomakestatementsbeyondtherangeofapplicabilityofperturbationtheory.Inourcase,anapproximatespin–flavoursym-metryisrealizedinsystemsinwhichasingleheavyquarkinteractswithlightdegreesoffree-dombytheexchangeofsoftgluons[11]–[15].Theoriginofthissymmetryisnotdifficulttounder-stand.Inaheavyhadron,theheavyquarkissurroundedbyamostcomplicated,stronglyinter-actingcloudoflightquarks,antiquarksandglu-ons.However,thefactthattheComptonwave-lengthoftheheavyquarkismuchsmallerthanthesizeofthehadronleadstosimplifications.Toresolvethequantumnumbersoftheheavyquarkwouldrequireahardprobe;thesoftglu-onsexchangedbetweentheheavyquarkandthelightconstituentscanonlyresolvedistancesmuchlargerthan1/mQ.Therefore,thelightdegreesoffreedomareblindtotheflavour(mass)andspinorientationoftheheavyquark.Theyexperienceonlyitscolourfield,whichextendsoverlargedis-tancesbecauseofconfinement.Intherestframeoftheheavyquark,itisinfactonlytheelectriccolourfieldthatisimportant;relativisticeffectssuchascolourmagnetismvanishasmQ→∞.Sincetheheavy-quarkspinparticipatesininter-actionsonlythroughsuchrelativisticeffects,itdecouples.Itfollowsthat,inthelimitmQ→∞,hadronicsystemswhichdifferonlyintheflavourorspinquantumnumbersoftheirheavyquarkhavethesameconfigurationoftheirlightdegreesoffreedom.Althoughthisobservationstilldoesnotallowustocalculatewhatthisconfigurationis,itprovidesrelationsbetweenthepropertiesofsuchparticlesastheheavymesonsB,D,B∗andD∗,ortheheavybaryonsΛbandΛc.2Heavy-quarksymmetryisanapproximatesym-metry,andcorrectionsarisesincethequarkmassesarenotinfinitelyheavy.Nevertheless,re-sultsderivedonthebasisofheavy-quarksymme-tryaremodel-independentconsequencesofQCDinawell-definedlimit.Thesymmetry-breakingcorrectionscan,atleastinprinciple,bestudiedinasystematicway.Aconvenientframeworkforanalysingthesecorrectionsisgivenbytheheavy-quarkeffectivetheory(HQET)[16],whichpro-videsasystematicexpansionaroundthelimitmQ→∞.IntheHQET,aheavyquarkin-sideahadronmovingwithvelocityvisdescribedbyavelocity-dependentfieldhvsubjecttotheconstraint/vhv=hv.Thisfieldisrelatedtotheoriginalheavy-quarkfieldbyaphaseredef-inition,sothatitcarriesthe“residualmomen-tum”k=pQ−mQv,whichcharacterizesthein-teractionsoftheheavyquarkwithgluons.TheeffectiveLagrangianoftheHQETis[16]–[19]Leff=¯hviv·Dhv+Ckin2mQ¯hv(iD⊥)2hv+Cmaggs4mQ¯hvσμνGμνhv+O(1/m2Q),(1)whereDμ⊥=Dμ−(v·D)vμcontainsthecom-ponentsofthegauge-covariantderivativeorthog-onaltothevelocity,andgsGμν=i[Dμ,Dν]isthegluonfield-strengthtensor.TheleadingtermintheeffectiveLagrangian,whichgivesrisetotheFeynmanrulesoftheHQET,isinvariantunderaglobalSU(2nh)spin–flavoursymmetrygroup,wherenhisthenumberofheavy-quarkflavours.Thissymmetryisexplicitlybrokenbythehigher-dimensionaloperatorsarisingatorder1/mQ,whoseoriginismosttransparentintherestframeoftheheavyhadron:thefirstoper-atorcorrespondsto(minus)thekineticenergyresultingfromthemotionoftheheavyquarkin-sidethehadron(intherestframe,(iD⊥)2istheoperatorfor−k2)