Factorization of heavy-to-light form factors in soft-colline(8)

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

Theintegralcanbedonewithstandardmethods,butitwillbeusefultoobtainanintermediateresult,whereonlythen kintegrationisperformed.Thevariablen+p′isrelatedtotheenergyofthe“photon”,son+p′>0.Wethenclosethecontourinthelower

2halfplaneandpickupthepoleat( k⊥+m2 i )/n+kfor0<n+k<n+p′.(Inour

2conventionk⊥isnegative.)Thisgives

Ic=

= 12p′1

δ·l δ n+p′0dn+kn+k ln δ µe2γE dd 2k⊥2k⊥ m2+ln2p′·lm2

Hereweperformthen+kintegral rst.Assumingn l>0,weclosethecontourinthe

2lowerhalfplaneandpickupthepoleat( k⊥+m2 i )/n kfor0<n k<n l.This

gives

Is= 1

n k

1

m2 [(k l)2]1+δ[k2 m2][ n+p′n k].(11)n lπd/2 1 ν2δn l= Γ( )m2

δΓ(δ+ )Γ(1 2δ 2 ) 221+δ k⊥)