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

Abstract

We consider the structure-dependent amplitude of the decay $B_s \to l^+ l^- γ$ $(l=e,μ)$ 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 $< γ| \bar{s} γ_μ(1 \mp γ_5) b | \bar{B}_s>$ and $< γ| \bar{s} σ_{μν} (1 \mp γ_5) b | \bar{B}_s>$ are taken to have the universal form $f_V \approx f_A \approx f_T \approx f_{B_s} M_{B_s} R_s / (3 E_γ)$ 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 $A(x_γ)$ and the photon energy spectrum $d Γ/ d x_γ(x_γ= 2 E_γ/M_{B_s})$. The decay rates are calculated in terms of the decay rate of $B_s \to γγ$. The branching ratios are estimated to be $Br(B_s \to e^+ e^- γ) = 2.0 \times 10^{-8}$ and $Br(B_s \to μ^+ μ^- γ) = 1.2 \times 10^{-8}$, somewhat higher than earlier estimates.