Factorization of heavy-to-light form factors in soft-colline(20)

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

Figure5:In nitesetsofFeynmangraphswithattachmentsofsoftgluonsto

collinearquarksandvice-versa.Integratingouttheintermediatehard-collinear

propagatorsleadstotheWilsonlinesYs andWc,respectively,see(37).

diagramsresultin6

¯c1 gAψ/s

1← in Ds

in+Dc ¯cY , ξs(37)1 1gn+Acqs Wcqs.

Inpracticeusingthe xedgauge(33)inintermediatestepsandrestoringthegaugesym-metryvia(35)ismoree cient,inparticularwhenkeepingtrackofWilsonlinesarisingfrommulti-gluonverticesinthenon-Abeliantheory.

3.2E ectivecurrentattreelevel

¯ΓQ,whereWenowconsidertherepresentationofheavy-to-lighttransitioncurrentsJ=ψ

ΓisaDiracmatrix,inSCET(c,s).Weareonlyconcernedwithtreediagramsinthissubsection.

Inthefollowingweintroducedi erent eldsforhard-collinearandcollinearmodes.Weworkwiththegaugen+Ac=n+Ahc=n As=0andpresentthegauge-invariantresultonlyattheend.Integratingouthardintermediateheavy-quarkpropagatorsintreediagrams,weobtainthecurrentinSCET(hc,c,s)intheform

J(x)=e

where

ψ=ξc+ηc+ξhc+ηhc+qs

1=ξc+ξhc+qs

Q=

6 imbvx ¯ΓQ(x),ψ (38)2((iD/⊥+m)(ξc+ξhc)+(gA/⊥c+gA/⊥hc)qs), 1+iD/sn vn/ Wedonotwriteoutthe+i prescriptiononthepropagators,whichreads+i for1/(in+ )and i for1/(in ).Thisfollows,becausetheinternalhard-collinearpropagatorsarealwaysspace-like,with(p′ l)2 n ln+p′<0,wheren+p′>0describesanoutgoingcollinearmomentum,andn l>0anincomingsoftmomentum.Henceinpositionspace1/(in+ in +i )is1/(in+ +i )1/(in i ).