NNLO QCD corrections to the weak radiative $B$-meson decay with exact dependence on $m_c$

Abstract

The inclusive weak radiative $B$-meson decay provides important constraints on many popular extensions of the Standard Model. Some of the numerically relevant Next-to-Next-to-Leading-Order QCD corrections to its branching ratio have so far been evaluated using an interpolation in the charm quark mass between the $m_c=0$ and $m_c \gg m_b$ limits. In the current paper, we present their determination for the physical value of $m_c$, which required calculating hundreds of four-loop propagator diagrams with unitarity cuts and two mass scales. Our final result for the CP- and isospin-averaged branching ratio reads ${\mathcal B}_{s \gamma}^{\rm SM} = (3.54 \pm 0.14)\times 10^{-4}$ for $E_\gamma>1.6\,{\rm GeV}$ in the decaying meson rest frame. It is in good agreement with the current Particle Data Group average ${\mathcal B}_{s \gamma}^{\rm exp} = (3.49 \pm 0.19)\times 10^{-4}$.