Factorization of heavy-to-light form factors in soft-colline(5)

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

InSection3weturnourattentiontotherepresentationoftheheavy-to-lightcurrentinthee ectivetheorywiththehard-collinearscaleremoved.Weintegrateoutthehard-collinearmodesintreegraphsbysolvingtheclassical eldequationsforthehard-collinearquarkandgluon elds.Despitethecomplexbranchingsoftherelevanttrees,thesolu-tioncanbefoundbychoosingaspecialgaugeforthecalculationandreconstructingthe¯ completeresultthroughgaugeinvariance.The rsttermwithanon-vanishing π|...|B

3/2matrixelementintheexpansionofthecurrentisλ3suppressed,whichexplainsthe1/mbsuppressionofheavy-to-lightformfactorsatlargerecoil.Thecalculationalsoshowsthat

thee ectiveoperatorishighlynon-local,implyingconvolutionsintwolight-likedirections.Theconvolutionsaredivergent,asexpected,butwealso ndthatquark-antiquark-gluonamplitudesintheBmesonandinthelightmesoncontributeatleadingpower,whichisanewfeatureofheavy-to-lighttransitions.Attheendofthissectionwebrie ydis-cusshard-collinearquantumcorrectionstodeterminethegeneralformofoperatorsandshort-distancekernelsinthee ectivetheoryofsoftandcollinearmodes.

Theexistenceofdivergentconvolutionssignalsthatheavy-to-lightformfactorsdonotfactorizestraightforwardly.InSection4wereturntothefactorizationformula(1),andshowthatitisindeedvalidtoallordersinthestrongcouplingandtoleadingpowerin1/mb.Weshallde netheuniversalformfactorξπasamatrixelementinSCETbeforeintegratingouthard-collineare ects.Wethenshowthatthetermsnotcontainedinthisde nitionfactorizeintoconvolutionsoflight-conedistributionamplitudeswithconvergentintegralsafterintegratingouthard-collinearmodes.Theproofofconvergencereliesonpowercounting,boostinvariance,andthecorrespondenceofcollinearandsoftendpointdivergencesthroughsoft-collinearfactorization.WeconcludeinSection5.

2Thescalar“photon”vertex

Thepurposeofthissectionistodemonstratebytheexampleofaspeci cFeynmaninte-gralthatthedistinctionofhard-collinearandcollinearmodeshasatechnicalmeaning.Weshallalsoseehowthefactorizationofcollinearandsoftmodesintroduces“endpointsingu-larities”inthelongitudinalintegrations,andhowthesingularitiesarerelatedtocancelinthesumofallterms.Finally,wesketchhowthediagrammaticresultwouldbeinterpretedinthecontextofe ective eldtheory.

Weconsiderthescalarintegral

I= [dk]1

i µ2eγE(2π)d=µe2 γEddk