QCD factorization for the $B\to \gamma\ell\nu_{\ell}$ decay beyond leading power

The radiative leptonic $B\to \gamma\ell\nu_{\ell}$ decay serves as an ideal platform to determine the $B$-meson inverse moment which is a fundamental nonperturbative parameter for the $B$ meson. In this paper, we explore precise QCD contributions to this decay with an energetic photon. We reproduce the next-to-next-to-leading-logarithmic resummation formula for the decay amplitude at leading power in $\Lambda_{\rm QCD}/m_b$. Employing operator identities, we calculate subleading-power contributions from the expansion of the hard-collinear propagator of the internal up quark and the heavy-quark expansion of the bottom quark. We update the contributions from the hadronic structure of the photon to the $\decay$ process with the dispersion technique. Together with other yet known power corrections, phenomenological applications including the partial branching fraction and ratio of the branching fractions of the radiative $B$ decay are investigated.