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${}_s$ + γ decay at next-to-leading logarithmic accuracy.