The structure function of semi-inclusive heavy flavour decays in field theory

We consider the decay of a heavy flavour into an inclusive hadronic state X of invariant mass m(X) small with respect to its energy E-X, m(X) much less than E-X. The electron spectrum and the hadronic mass distribution in semileptonic b --> u decays, or the photon spectrum in b --> s gamma decays, all require, close to their endpoints, a control over this region. This region is affected both by non-perturbative phenomena related to the Fermi motion of the heavy quark and by perturbative soft gluon radiation in the final state (Sudakov form factor). Fermi motion can be described by the shape function f(m*), which represents the distribution of the effective mass m* of the heavy quark at disintegration time. We perform a factorization with a simple technique in order to consistently separate perturbative from non-perturbative effects. We find that the shape function, contrary to naive expectations, is not a physical distribution, as it is affected by substantial regularization scheme effects, controlling even the leading, double-logarithmic term. It factorizes, however, the bulk of non-perturbative effects in lattice-like regularizations. Some non-perturbative effects are present in the coefficient function even at leading twist, but they are expected to be suppressed on physical grounds. Finally, we clarify a controversial factor of 2 in the evolution kernel of the shape function.