Factorization of heavy-to-light form factors in soft-colline(2)

Heavy-to-light transition form factors at large recoil energy of the light meson have been conjectured to obey a factorization formula, where the set of form factors is reduced to a smaller number of universal form factors up to hard-scattering corrections

1Introduction

ThepurposeofthispaperistodevelopfurtherthetheoryofexclusiveBdecaystolightenergeticmesons.Wearespeci callyinterestedin“hard-spectatorscattering”,i.e.scatter-¯meson.ThesoftexternalquarkingmechanismsthatinvolvethelightantiquarkintheB

lineisoneofthecrucialdi erencestothestandardsituationofhardexclusiveprocesses

[1]involvingonlylighthadrons,whereallexternallinescarrylargemomentum.Aconse-quenceofthisdi erenceisthatexclusiveBmesondecaysinvolvetwohardscales,m2bandmbΛ,whereΛisoforderofthestronginteractionscale.

Hard-spectatorscatteringisanimportantingredientinQCDfactorizationfornon-leptonicBdecaystocharmless nalstates[2,3],andisevenmoreimportantintheso-calledPQCDapproach[4].Abetterunderstandingofspectatorinteractionsisneededtojustifythefactorizationhypothesesofthetwoapproachestoallordersinperturbationtheoryandtoleadingorderintheheavy-quarklimit.However,evenapparentlysimplerprocessessuchasthesemi-leptonicdecayB→πlνatlargemomentumtransfertothepionarecurrentlynotcompletelyunderstood.TheoneexceptionisB→γlν,whichhasreceivedmuchattentionrecently[5,6,7,8].AfactorizationformulaoftheschematicformF=T φB,wherethestar-productdenotesconvolutionofahard-scatteringkernelwiththeBmesonlight-conedistributionamplitude,hasnowbeenshowntobevalidtoallordersinperturbationtheory[7,8].InthispaperweconsiderB→πlν(heavy-to-lightformfactors)atlargepionenergy.Asummaryoftheresultshasalreadybeengivenin[9].AstraightforwardextensionoftheresultsforB→γlνdecaytotheB→πformfactorsrelevanttosemi-leptonicdecaysfails.Ifonewritestheformfactorsasφπ T φBinanalogytotheπ→πtransitionformfactoratlargemomentumtransfer,one ndsthattheconvolutionintegralsdonotconvergeattheirendpoints.Inotherwords,theformfactorsreceiveleadingcontributionsfrommomentumcon gurationswheresomepartonsinthepionappeartohavesmallmomentum[10,11].In[12]thefactorizationformula

Fi=Ciξπ+φB Ti φπ(1)

hasbeenconjecturedforthethreeLorentzinvariantB→πformfactorsandshowntobevalidatorderαs.TheadditionaltermCiξπinvolvesashort-distancecoe cientandasingle“softformfactor”ξπ,whichobeysthelarge-recoilsymmetries[13].FactorizationfortheB→πformfactorsismorecomplicatedthanforboth,theπ→πandB→Dformfactors.Atlargemomentumtransfersoftinteractionscancelintheπ→πtransitionatleadingpower.Theremaininghardandcollineare ectsarefactoredintoconvolutionsasinthesecondtermontheright-handsideof(1).Whenbothmesonsareheavy,suchasinB→D,collineare ectsareirrelevant.Theremaininghardandsoftinteractionsfactorintoashort-distancecoe cientandtheIsgur-Wiseformfactor[14]similartothe rsttermontheright-handsideof(1).TheB→πformfactor,however,involveshard,collinearandsofte ects.Furthermore,duetothepresenceofseveralscalesonemustdistinguishshort-andlong-distancecollineare ects,aswediscussinmoredetailbelow.

Aseparationofallthesee ectsandanoperatorde nitionofthevariousshort-andlong-distancequantitiesin(1)toallordersinαsandtoleadingorderin1/mbhasnot