Threshold resummed spectra in B -> X(u) l nu decays in NLO. III

We resum to next-to-leading order (NLO) the distribution in the light-cone momentum p+=EX-|p⃗X| and the spectrum in the electron energy Ee in the semileptonic decays B→Xulν, where EX and p⃗X are the total energy and three-momentum of the final hadron state Xu respectively. By expanding our formulae, we obtain the coefficients of all the infrared logarithms αSnLk at O(αS2) and at O(αS3), with the exception of the αS3L coefficient. We explicitly show that the relation between these semileptonic spectra and the hadron mass distribution in the radiative decay B→Xsγ is not a purely short-distance one. By comparing our O(αS2nf) 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 (nf 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.