B d,s → $$ \gamma \mathrm{\ell}\overline{\mathrm{\ell}} $$ decay with an energetic photon

We calculate the differential branching fraction, lepton forward-backward asymmetry and direct CP asymmetry for Bd,s → $$ \gamma \mathrm{\ell}\overline{\mathrm{\ell}} $$ γ ℓ ℓ ¯ decays with an energetic photon. We employ factorization methods, which result in rigorous next-to-leading order predic- tions in the strong coupling at leading power in the large-energy/heavy-quark expansion, together with estimates of power corrections and a resonance parameterization of sub- leading power form factors in the region of small lepton invariant mass q2. The Bd,s → $$ \gamma \mathrm{\ell}\overline{\mathrm{\ell}} $$ γ ℓ ℓ ¯ decay shares features of the charged-current decay Bu → $$ \gamma \mathrm{\ell}{\overline{\nu}}_{\mathrm{\ell}} $$ γ ℓ ν ¯ ℓ ., and the FCNC decays B → $$ {K}^{\left(\ast \right)}\mathrm{\ell}\overline{\mathrm{\ell}} $$ K ∗ ℓ ℓ ¯ . As in the former, the leading-power decay rates can be expressed in terms of the B-meson light-cone distribution amplitude and short-distance factors. However, similar to B →$$ {K}^{\left(\ast \right)}\mathrm{\ell}\overline{\mathrm{\ell}} $$ K ∗ ℓ ℓ ¯ , four-quark and dipole operators contribute to the Bd,s → $$ \gamma \mathrm{\ell}\overline{\mathrm{\ell}} $$ γ ℓ ℓ ¯ decay in an essential way, limiting the calculation to q2 ≲ 6 GeV2 below the charmonium resonances in the lepton invariant mass spectrum. A detailed analysis of the main observables and theoretical uncertainties is presented.