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 $d{\Gamma }_{77}/(d{s}_{1}d{s}_2)$ 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 $d{\Gamma }_{77}/(d{s}_{1}d{s}_{2}d{s}_3)$. 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.

Comments: 4 pages, 2 figures, presented at the IVth International Conference on Hadron Physics, 1-5 July 2014, Canakkale, Turkey. arXiv admin note: substantial text overlap with arXiv:1207.2397