We resum to next-to-leading order the distribution in the ratio of the invariant hadron mass $m_X$ to the total hadron energy $E_X$ and the distribution in $m_X$ in the semileptonic decays $B\to X_{u}l\nu$. By expanding our formulas, we obtain the coefficients of all the infrared logarithms at $O({\alpha }_S^2)$ and of the leading ones at $O({\alpha }_S^3)$. We explicitly show that the relation between these semileptonic spectra and the photon spectrum in the radiative decay $B\to X_s\gamma$ is not a purely short-distance one. There are long-distance effects in the semileptonic spectra which are not completely factorized by the structure function as measured in the radiative decay and have to be modeled in some way.

• Received 30 March 2006

DOI:https://doi.org/10.1103/PhysRevD.74.034005