The double radiative $\bar{B}\to X_s\gamma \gamma$ decay at $O(\alpha_{s})$ in QCD

Abstract

In these proceedings, we briefly review the individual interference contribution of the electromagnetic dipole operator $\mathcal{O}_{7}$ to the double differential decay width $d\Gamma_{77}/(ds_1\, ds_2)$ for the process $\bar{B} \to X_s \gamma \gamma$ at $O(\alpha_s)$ in QCD, which is based on our previous work. We define two kinematical variables $s_1$ and $s_2$ 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. While the (renormalized) virtual corrections are worked out exactly for a certain range of $s_1$ and $s_2$, we retained in the gluon bremsstrahlung process only the leading power w.r.t. the (normalized) hadronic mass $s_3=(p_b-q_1-q_2)^2/m_b^2$ in the underlying triple differential decay width $d\Gamma_{77}/(ds_1 ds_2 ds_3)$. We found that the double differential decay width, based on this approximation, is free of infrared- and collinear singularities when summing up the virtual- and real-radiation corrections, while this was not the case when keeping all powers in $s_3$ in the gluon bremsstrahlung process due to the configurations allowing collinear photon emission from the (massless) $s$-quark. Lastly, we compare our analytical results with those obtained in a recently extended work, where a non-zero strange quark mass was introduced to regulate the collinear photon configurations.