Enhanced electromagnetic correction to the rare $B$-meson decay $B_{s,d}\to {\mu }^{+}{\mu }^{-}$

Abstract

We investigate electromagnetic corrections to the rare $B$-meson leptonic decay $B_{s,d}\to {\mu }^{+}{\mu }^{-}$ from scales below the bottom-quark mass $m_b$. Contrary to QCD effects, which are entirely contained in the $B$-meson decay constant, we find that virtual photon exchange can probe the $B$-meson structure, resulting in a "non-local annihilation" effect. We find that this effect gives rise to a dynamical enhancement by a power of $m_b/{\Lambda }_{\mathrm{QCD}}$ and by large logarithms. The impact of this novel effect on the branching ratio of $B_{s,d}\to {\mu }^{+}{\mu }^{-}$ is about $1\%$, of the order of the previously estimated non-parametric theoretical uncertainty, and four times the size of previous estimates of next-to-leading order QED effects due to residual scale dependence. We update the Standard Model prediction to $\overline{\mathcal{B}}(B_s\to {\mu }^{+}{\mu }^{-}{)}_{\mathrm{SM}}=(3.57\pm 0.17)\cdot 10^{-9}$.