Factorization and Sudakov Resummation in Leptonic Radiative $B$ Decay -- A Reappraisal

The $B$-meson light-cone distribution amplitude is an important non-perturbative quantity arising in the factorization of the amplitudes for many exclusive decays of $B$ mesons, such as $B^-\to\gamma\,\ell^-\bar\nu$. We reconsider the renormalization-group (RG) equation satisfied by this function and present its solution at next-to-leading order (NLO) in RG-improved perturbation theory in Laplace space and, for the first time, in momentum space and the so-called diagonal (or dual) space. Since the information needed to describe the $B$ decay processes at leading order in $\Lambda_{\rm QCD}/m_b$ is most directly contained in the distribution amplitude in Laplace space evaluated near the origin, we propose an unbiased parameterization of this object in terms of a small set of uncorrelated hadronic parameters. Using recent results on the three-loop anomalous dimension for heavy-light current operators, we derive an expression for the convolution integral appearing in the $B^-\to\gamma\,\ell^-\bar\nu$ factorization formula that is explicitly scale independent, and we evaluate this formula at (approximate) NNLO.