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

TL;DR

This paper provides a detailed theoretical analysis of charm-quark loop effects on the $B_s o \gamma\,\gamma$ decay amplitude, quantifying their correction to the Wilson coefficient with high precision.

Contribution

It introduces a novel calculation of the charm-loop contribution involving the $B$-meson three-particle Bethe-Salpeter amplitude, improving the understanding of nonfactorizable effects.

Findings

Charming-loop correction $\delta C_{7\gamma}$ is predicted with better than 10 ext{ extperthousand} accuracy.

For $\lambda_{B_s}=0.45$ GeV, $\delta C_{7\gamma}=0.045 ext{ extpm}0.004$.

Range of $\delta C_{7\gamma}$ is 0.02 to 0.1 for $\lambda_{B_s}$ between 0.3 and 0.6 GeV.

Abstract

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, $\langle 0|\bar s(y)G_{\mu\nu}(x)b(0)|\bar B_s(p)\rangle$, 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 $0.3\le \lambda_{B_s}({\rm GeV})\le 0.6$. 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.