Factorization of heavy-to-light form factors in soft-colline(15)

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

therenormalizationoftheconventionallight-conedistributionamplitudes,istheendpointdivergenceofthen+kintegralin(10).Theassociatedν-dependencecancelsagainsttheν-dependenceofthesoftcontribution.Ingeneral,thedistributionamplitudesmaythemselvesdependontheadditionalregularization,andhencedi erfromthedistributionamplitudesthatappearinthehard-scatteringformalism.

Tosummarizethisdiscussion,wedistinguishtwostepsoffactorization.Inthe rststep,weintegrateoutthelarge-virtualityhard-collinearmodes,andrepresenttheresultintermsofoperatorsofsoftandcollinear elds.Theseoperatorswillbenon-local,re ectingthefactthatthehard-scatteringkernelsappearinconvolutionsratherthanasmultiplicativefactors.Thesecondfactorizationstepreferstotheseparationofsoftandcollinearmodeswithinthee ectivetheoryofsoftandcollinearmodes.Inourexample,thephotoncouplesonlytocollinearlines,andtheq¯bstatecouplesonlytosoftlines,sowewouldexpectthee ectivetheorymatrixelementstofactorizeintoamatrixelementofcollinear eldsbetweenthephotonandthevacuum,andamatrixelementofsoft eldsbetweenthevacuumandtheq¯bstate.Ifthiswerethecase,theprocesswouldfactorizeintoS T Φ.ThefactorizationscaledependenceofthesoftfactorSandofthecollinearfactorΦwouldcancelseparatelywiththatofthehard-scatteringkernelT,butthesoftandcollinearfactorswouldbeunrelated.Theendpointdivergencespreventsuchacompletefactorization.Forourtoyexamplewe ndinsteadafactorizationformulathattakestheschematicform

γ|J|q¯b =(C0+C1) φq¯b+C2 φq¯bg ν+

The rsttermontheright-handsiderepresentsadirectphotoncontribution;inthethirdtermthepartonicstructureofthephotonisresolved.Thesquarebracketsindicatetheadditionalscale-dependenceintroducedbytheendpointdivergences,whichconnectthesecondwiththethirdterm.Ifthescaleνischosensuchthatthethirdtermcontainsnolargelogarithmrelatedtotheendpointdivergence,wecaninterpretitasaendpoint-subtractedhard-scatteringcontributiontotheq¯b→γtransition.Forourtoyintegral,

2(10,13)showthatthiscorrespondstotakingνoforder2p′·l.Thecorrespondingendpointlogarithmthenresidesinthesecondterm,whichwemaycallthe“softoverlap”contribution

(sinceasoftlineconnectstheinitialstatewiththephotonasseenfromthemiddlecolumnofFigure3).Thetwotermsarerelatedviatheirν-dependence,suchthatthesumisindependentoftheimplementationofsoft-collinearor“endpoint”factorization.AsimilarstructureisexpectedfortheB→πformfactor[12]. φγqq¯ C3 ν φq¯b.(21)

3Heavy-to-lighttransitionsinSCET(c,s)

¯ΓQisob-Thee ectivetheoryrepresentationoftheheavy-to-lighttransitioncurrentsψ

tainedintwosteps: rstthehardmodesareintegratedout,andthecurrentisdescribedinsoft-collineare ectivetheoryincludinghard-collinearmodes.Weshalldenotethisthe-orybySCET(hc,c,s)(alsocalledSCETIintheliterature[15]).Thisstep,inwhichitisnotnecessarytodistinguishhard-collinearandcollinear,hasalreadybeendiscussedin