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, $H_{A,V}^{\mathrm{N}\mathrm{F}}(k^{\mathrm{\prime }2},k^2)$, and provide convenient parametrizations of our results in the form of fit functions of two variables, $k^{\mathrm{\prime }2}$ and $k^2$, applicable in the region below hadron resonances, $k^{\mathrm{\prime }2}<M_{J\mathrm{/}\psi }^2$ 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 ${\mathrm{\Delta }}^{\mathrm{N}\mathrm{F}}{C}_7(q^2)$ to the Wilson coefficient $C_7$. For the $B_s\to \gamma l^{+}{l}^{-}$ amplitude, the correction is found to be positive, ${\mathrm{\Delta }}^{\mathrm{N}\mathrm{F}}{C}_7(q^2)\mathrm{/}{C}_7>0$.