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 Q1 → Q2 + (non QCD partons), where Q1 and Q2 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 mX in all the kinematic regions, i.e. when mX is smaller, of the same order, or larger than the mass of the final quark Q2. 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 → Xs + γ decay at next-toleading logarithmic accuracy.