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

Abstract The radiative leptonic $$B \rightarrow \gamma \ell \nu _{\ell }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>→</mml:mo> <mml:mi>γ</mml:mi> <mml:mi>ℓ</mml:mi> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>ℓ</mml:mi> </mml:msub> </mml:mrow> </mml:math> 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 $$\Lambda _\textrm{QCD}/m_b$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mtext>QCD</mml:mtext> </mml:msub> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>b</mml:mi> </mml:msub> </mml:mrow> </mml:math> . 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 $$B \rightarrow \gamma \ell \nu _{\ell }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>→</mml:mo> <mml:mi>γ</mml:mi> <mml:mi>ℓ</mml:mi> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>ℓ</mml:mi> </mml:msub> </mml:mrow> </mml:math> 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.