Threshold resummed spectra in B-> Xu l n decays in NLO (III)

We resum to next-to-leading order (NLO) the distribution in the light-cone momentum p(+)=E-X-vertical bar p(X)vertical bar and the spectrum in the electron energy E-e in the semileptonic decays B -> X(u)l nu, where E-X and p(X) are the total energy and three-momentum of the final hadron state X-u respectively. By expanding our formulae, we obtain the coefficients of all the infrared logarithms alpha L-n(S)k at O(alpha(2)(S)) and at O(alpha(3)(S)), with the exception of the alpha L-3(S) coefficient. We explicitly show that the relation between these semileptonic spectra and the hadron mass distribution in the radiative decay B -> X-s gamma is not a purely short-distance one. By comparing our O(alpha(2)(S)n(f)) result for the p(+) distribution with a recent Feynman diagram computation, we obtain an explicit and nontrivial verification of our resummation scheme at the two-loop level (n(f) is the number of massless quark flavors). We also discuss the validity of the so-called Brodsky-Lepage-Mackenzie (BLM) scheme in the evaluation of the semileptonic spectra by comparing it with our results, finding that it is a reasonable approximation. Finally, we show that the long-distance phenomena, such as Fermi motion, are expected to have a smaller effect in the electron spectrum than in the hadron mass distribution.