Improved factorization for threshold resummation in heavy quark to heavy quark decays

We consider the resummation of soft-gluon effects in heavy quark to heavy quark decays, namely the processes Q(1) ? Q(2) + (non QCD partons), where Q(1) and Q(2) are two different heavy quarks. We construct a new factorization scheme for threshold resummed spectra, which allows us to consistently evaluate the distribution of the final hadron invariant mass m(X) in all the kinematic regions, i.e. when m(X) is smaller, of the same order, or larger than the mass of the final quark Q(2). A dependence of the improved Coefficient function on the threshold variable is introduced, which can however be relegated to a small interval of this variable by means of the so-called Partition of Unity. We explicitly apply our improved scheme to the b ? X-s + ? decay at next-to leading logarithmic accuracy.