Nonfactorizable charming-loop contribution to FCNC $B_s\to \gamma l^+l^-$ decay

We present the first theoretical calculation of nonfactorizable charm-quark loop contributions to the $B_s\to \gamma l^+l^-$ amplitude. We calculate the relevant form factors, $H_{A,V}^{\rm NF}(k'^2,k^2)$, and provide convenient parametrizations of our results in the form of fit functions of two variables, $k'^2$ and $k^2$, applicable in the region below hadron resonances, $k'^2 < M_{J/\psi}^2$ and $k^2 < M_{\phi}^2$. We report that factorizable and nonfactorizable charm contributions to the $B_s\to\gamma l^+l^-$ amplitude have opposite signs. To compare the charm and the top contributions, it is convenient to express the NF charming loop contribution as a non-universal (i.e., dependent on the reaction) $q^2$-dependent correction $\Delta^{\rm NF}C_7(q^2)$ to the Wilson coefficient $C_7$. For the $B_s\to\gamma l^+l^-$ amplitude, the correction is found to be positive, $\Delta^{\rm NF} C_7(q^2)/C_7 > 0$.