(O_8, O_8) contribution to $\bar{B} \to X_s \gamma \gamma$ at O(\alpha_s)

Abstract

In this analysis, we present the contribution associated with the chromomagnetic dipole operator O_8 to the double differential decay width d\Gamma/(ds_1 ds_2) for the inclusive process $\bar{B} \to X_s \gamma \gamma$. The kinematical variables s_1 and s_2 are defined as s_i=(p_b - q_i)^2/m_b^2, where p_b, q_1, q_2 are the momenta of b-quark and two photons. This contribution (taken at tree level) is of order \alpha_s, like the recently calculated QCD corrections to the contribution of the operator O_7. In order to regulate possible collinear singularities of one of the photons with the strange quark, we introduce a non zero mass m_s for the strange quark. Our results are obtained for exact m_s, which we interpret as a constituent mass being varied between 400 and 600 MeV. Numerically it turns out that the effect of the (O_8, O_8) contribution to the branching ratio of $\bar{B} \to X_s \gamma \gamma$ does not exceed +0.1 % for any kinematically allowed value of our physical cutoff parameter c, confirming the expected suppression of this contribution relative to the QCD corrections to d\Gamma_{77}/(ds_1 ds_2).