$B_{d,s}\to \gamma \ell \bar{\ell }$ 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 $B\to K^{(\ast )}\ell \bar{\ell }$. 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 $B\to K^{(\ast )}\ell \bar{\ell }$, 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.

50 pages, LaTeX; v2: Corrected sign typos in (3.48) and (3.52), results not affected

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