Charged and Neutral $\bar B_{u,d,s} \to \gamma$ Form Factors from Light Cone Sum Rules at NLO

Abstract

We present the first analytic ${\cal O}(\alpha_s)$-computation at twist-$1$,$2$ of the $\bar B_{u,d,s} \to \gamma$ form factors within the framework of sum rules on the light-cone. These form factors describe the charged decay $\bar B_u \to \gamma \ell^- \bar{\nu}$, contribute to the flavour changing neutral currents $\bar B_{d,s} \to \gamma \ell^+ \ell^-$ and serve as inputs to more complicated processes. We provide a fit in terms of a $z$-expansion with correlation matrix and extrapolate the form factors to the kinematic endpoint by using the $g_{BB^*\gamma}$ couplings as a constraint. Analytic results are available in terms of multiple polylogarithms as ancillary files. We give binned predictions for the $\bar B_u \to \gamma \ell^- \bar{\nu}$ branching ratio along with the associated correlation matrix. By comparing with three SCET-computations we extract the inverse moment $B$-meson distribution amplitude parameter $\lambda_B = 360 (110) \text{MeV}$. The uncertainty thereof could be improved by a more dedicated analysis. In passing, we extend the photon distribution amplitude to include quark mass corrections with a prescription for the magnetic vacuum susceptibility, $\chi_q$, compatible with the twist-expansion. The values $\chi_q = 3.21 (15) \text{GeV}^{-2}$ and $\chi_s = 3.79 (17) \text{GeV}^{-2}$ are obtained.