Charge Asymmetry and Photon Energy Spectrum in the Decay $B_s\to l^{+}{l}^{-}\gamma$

We consider the structure-dependent amplitude of the decay $B_s \to l^+ l^- \gamma$ $(l=e,\mu )$ in a model based on the effective Hamiltonian for $b \bar{s} \to l^+ l^-$containingtheWilsoncoefficients$C_7,C_9$and$C_{10}$. The form factors characterising the matrix elements $< \gamma | \bar{s} \gamma_\mu (1 \mp \gamma_5) b | \bar{B}s>$and$< \gamma | \bar{s} \sigma{\mu\nu} (1 \mp \gamma_5) b | \bar{B}s>$ are taken to have the universal form $f_V \approx f_A \approx f_T \approx f{B_s} M_{B_s} R_s / (3 E_\gamma)$suggestedbyrecentworkinQCD,where$R_s$ is a parameter related to the light cone wave function of the $B_s$ meson. Simple expressions are obtained for the charge asymmetry $A(x_{\gamma })$ and the photon energy spectrum $d\mathrm{\Gamma }\mathrm{/}d{x}_{\gamma }(x_{\gamma }=2E_{\gamma }\mathrm{/}{M}_{B_s})$. The decay rates are calculated in terms of the decay rate of $B_s\to \gamma \gamma$. The branching ratios are estimated to be $Br(B_s\to e^{+}{e}^{-}\gamma )=2.0\times {10}^{-8}$ and $Br(B_s\to {\mu }^{+}{\mu }^{-}\gamma )=1.2\times {10}^{-8}$, somewhat higher than earlier estimates.