Threshold Resummation in B --> X_c l nu_l Decays

We compute the QCD form factor resumming threshold logarithms in B →Xc + l + νl decays to next-to-leading logarithmic approximation. We present an interpolation formula including soft as well as collinear effects softened by the non-vanishing charm mass which, in the N-space, reads: σN = exp 1 0 dy [(1−y)N−1 −1]{ 1 y Q2y2 y+ρ Q2y2 1+ρ A[ρ;αS(k2)] dk2 k2 + 1 yD[αS( Q2y2 1+ρ )]+ ( 1 y − 1 y+ρ )Δ[αS( Q2y2 y+ρ )] + 1 y+ρ B[αS( Q2y2 y+ρ )]}, where y = (m2 Xc −m2c )/(Q2 −m2c ), ρ = m2c /(Q2 −m2c ) and Q= EXc +| pXc |. The function A(ρ;αS) describes soft radiation collinearly enhanced off the charm quark and reduces to the standard double-logarithmic function A(αS) in the massless limit ρ →0. The function B(αS) describes hard collinear emission off the charm quark, being the standard jet function. D(αS) and Δ(αS) describe soft radiation not collinearly enhanced off the beauty and the charm quark respectively.