Charming-loop contribution to $B_s\to \gamma \gamma$ decay

We present a detailed theoretical study of nonfactorizable contributions of the charm-quark loop to the amplitude of the $B_s\to \gamma \gamma$ decay. This contribution involves the $B$-meson three-particle Bethe-Salpeter amplitude, $⟨0\mathrm{\mid }\bar{s}(y)G_{\mu \nu }(x)b(0)\mathrm{\mid }{\bar{B}}_s(p)⟩$, for which we take into account constraints from analyticity and continuity. The charming-loop contribution of interest may be described as a correction to the Wilson coefficient $C_{7\gamma }$, $C_{7\gamma }\to C_{7\gamma }(1+\delta C_{7\gamma })$. We calculate an explicit dependence of $\delta C_{7\gamma }$ on the parameter ${\lambda }_{B_s}$. Taking into account all theoretical uncertainties, $\delta C_{7\gamma }$ may be predicted with better than 10% accuracy for any given value of ${\lambda }_{B_s}$. For our benchmark point ${\lambda }_{B_s}=0.45$ GeV, we obtain $\delta C_{7\gamma }=0.045\pm 0.004$. Presently, ${\lambda }_{B_s}$ is not known with high accuracy, but its value is expected to lie in the range $0.3\le {\lambda }_{B_s}(\mathrm{G}\mathrm{e}\mathrm{V})\le 0.6$. The corresponding range of $\delta C_{7\gamma }$ is found to be $0.02\le \delta C_{7\gamma }\le 0.1$. One therefore expects the correction given by charming loops at the level of at least a few percent.