Factorization of heavy-to-light form factors in soft-colline(9)

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

Thereisasingularityfork⊥→∞foranyn k.Thepoleatδ=0isanendpointdivergencefromn k→0foranyk⊥.Thisimpliesthatn+kbecomeslargefor xedk⊥～λ2,andhencethe“quark”withmomentumkbecomescollinear.Inthesoftregionthetransversemomentumandlongitudinalmomentumintegralsdonotfactorize,andthereisalsoadivergencewhenk⊥→∞andn k→0simultaneously,whichcorrespondstothedoublepoleinthehard-collinearintegral.

Sincewedidnotregularizethehard-collinearcontributionanalytically,thecorrectprocedureistoexpand rstinδandthenin .Infactthepoleinδcancelswiththecollinearcontributionbeforeexpandingin .However,performingbothexpansionstocomparewith(10)weobtain

Is= 1

δ lnm2

lnm2

2+1

µ2 1

µ2+5π2

2p′·l 1

ln2p′·lµ2lnm2

2ln2m2

12 .(14)

Finallyaddingtothisthehard-collinearcontribution(7),thesingularityin alsocancels,andweobtain ′2p·l1,(15)ln2Ic+Is+Ihc= 23

inagreementwiththeexpansion(5)ofthefullintegral.Weconcludethatingeneralhard-collinear,collinearandsoftmomentumregionsmustbeconsidered.Inthescalarintegral

(3)allthreeregionscontributealreadytotheleadingtermintheexpansion.Semi-hardmodeswithscaling(λ,λ,λ)arenotneededinthiscalculation,sincethecorrespondingintegralsarescaleless.

InQCDthephoton-vertexintegralcontainsanumeratorproportionalton kwhichsuppressesthecollinearregionbyafactorofλ2relativetothehard-collinearandsoftregion.Forthisreasonitissu cienttoconsideronlyhard-collinearandsoftcon gurationsinthefactorizationtheoremforB→γlνatleadingpowerin1/mb,ashasbeendonein

[6,7,8].Hard-collinearmodesareperturbativeandcanbeintegratedout,resultinginhard-scatteringkernels.Softandcollinearmodeshavevirtualityλ4～Λ2,andcannotbetreatedinperturbationtheory.The1/mbsuppressionofthecollinearcontributioninQCDimpliesthatthehadronicstructureofthephotonisasub-leadinge ectinB→γlνdecay.

2.2O -shellregularization

Thescalarintegral(10)hasrecentlybeendiscussedin[25],howeverwithm=0andtheexternalcollinearandsoftlineso -shell,l2≡ L2=n+ln l～λ4,and(p′)2≡ (P′)2=n+p′n p′～λ4.Itisinstructivetodiscussthedi erencetothecaseabove.