Factorization of heavy-to-light form factors in soft-colline(6)

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

¯→γlν.Γdenotestheweakb→udecayFigure1:PhotonvertexcorrectiontoB

vertex.Weconsiderthecorrespondingvertexintegralwithalllinessimpli edto

scalarpropagatorsandallvertexfactorssetto1.

TheintegralIisultravioletandinfrared nite,butdimensionalregularizationwillbeneededtoconstructtheexpansion.The+i prescriptionforthepropagatorsisunderstood.

Theexternalmomentaofthevertexsubgraphare:acollinear“photon”momentum′p=(n+p′,p′⊥,n p′)～(1,0,0)withp′2=0;asoft“lightquark”momentuml～(λ2,λ2,λ2)withl2=m2;ahard-collinear“lightquark”momentump′ l～(1,λ2,λ2)withvirtualityλ2.Thetwoinvariantsare2p′·l～λ2andm2～λ4,soImustbeafunctionofthesmalldimensionlessratiom2/(2p′·l).Astraightforwardcalculationgives

I=1

m2 π2

2p′·l 1

2p′·l+π2

whichvanishindimensionalregularization,sincetheonlypossibleinvariantp′2=0.Thisisnotsurprising,becausethereisnoexternalinvariantoforder1.Proceedingfordi erent

[k2]a[ 2p′·k]b×polynomial,(6)