QCD factorization for the \(B \rightarrow \gamma \ell \nu _{\ell }\) decay beyond leading power

The radiative leptonic \(B \rightarrow \gamma \ell \nu _{\ell }\) 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 \(\Lambda _\textrm{QCD}/m_b\). 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 \(B \rightarrow \gamma \ell \nu _{\ell }\) 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.