B_d,s → γℓ ℓ ¯ decay with an energetic photon

We calculate the differential branching fraction, lepton forward-backward asymmetry and direct CP asymmetry for B_d,s → γℓ 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 B_d,s → γℓ ℓ ¯ decay shares features of the charged-current decay B_u → γℓ ν¯ _ℓ., and the FCNC decays B → 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^(∗)ℓ ℓ ¯ , four-quark and dipole operators contribute to the B_d,s → γℓ 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.