Analysis of the photon spectrum in inclusive B --> X$_{s}\gamma$ decays

Using a combination of the operator product and heavy quark expansions we resum the leading nonperturbative contributions to the inclusive photon spectrum in $B\to X_s\,\gamma$ decays. The shape of the spectrum is determined by a structure function, which describes the distribution of the light-cone momentum fraction of the $b$-quark inside the $B$ meson. The moments of the spectrum are proportional to forward matrix elements of higher-dimension operators. As a byproduct, we obtain the bound $\lambda_1<0$ for one of the fundamental parameters of the heavy quark effective theory. The integral over the $B\to X_s\,\gamma$ structure function is related to the shape function that governs the fall-off of the lepton spectrum close to the endpoint in $B\to X_u\,\ell\,\bar\nu$ decays. A measurement of the photon spectrum in rare $B$ decays can therefore help to get a model independent determination of $V_{ub}$.