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

Abstract

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 $b\bar{s}\to l^{+}{l}^{-}$ containing the Wilson coefficients $C_7,C_9$ 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 $f_V\approx f_A\approx f_T\approx f_{B_s}{M}_{B_s}{R}_s/(3E_{\gamma })$ suggested by recent work in QCD, 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 $d\Gamma /d{x}_{\gamma }(x_{\gamma }=2E_{\gamma }/M_{B_s})$. 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.

Comments: sign of $f_T$ corrected; new table and figures