$B_{d,s}\to\gamma\ell\bar{\ell}$ decay with an energetic photon

Abstract

We calculate the differential branching fraction, lepton forward-backward asymmetry and direct CP asymmetry for $B_{d,s}\to\gamma\ell\bar{\ell}$ decays with an energetic photon. We employ factorization methods, which result in rigorous next-to-leading order predictions 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 $q^2$. The $B_{d,s}\to\gamma\ell\bar{\ell}$ decay shares features of the charged-current decay $B_u \to \gamma\ell\bar\nu_\ell$, and the FCNC decays $B\to K^{(*)}\ell\bar\ell$. 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 $B\to K^{(*)}\ell\bar\ell$, four-quark and dipole operators contribute to the $B_{d,s}\to\gamma\ell\bar{\ell}$ decay in an essential way, limiting the calculation to $q^2\lesssim 6\,$GeV$^2$ below the charmonium resonances in the lepton invariant mass spectrum. A detailed analysis of the main observables and theoretical uncertainties is presented.