Abstract :
We perform factorization of the most general distribution in semileptonic B→Xu decays and we resum the threshold logarithms to next-to-leading order. From this (triple-differential) distribution, any other distribution is obtained by integration. As an application of our method, we derive simple analytical expressions for a few distributions, resummed to leading approximation. It is shown that the shape function can be directly determined by measuring the distribution in mX2/EX2, not in mX2/mB2. We compute the resummed hadron energy spectrum, which has a “Sudakov shoulder”, and we show how the distribution in the singular region is related to the shape function. We also present an improved formula for the photon spectrum in B→Xsγ, which includes soft-gluon resummation and non-leading operators in the effective Hamiltonian. We explicitly show that the same non-perturbative function — namely, the shape function — controls the non-perturbative effects in all the distributions in the semileptonic and in the rare decay.
