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

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 $m_b/\Lambda_{\rm QCD}$ 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 $\overline{{\cal B}}(B_s \to \mu^+\mu^-)_{\rm SM} = (3.57 \pm 0.17) \cdot 10^{-9}$.