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

TL;DR

This paper analyzes the $O(\alpha_s)$ QCD corrections to the double radiative decay of B mesons, focusing on interference effects of the electromagnetic dipole operator and ensuring the finiteness of the decay width calculations.

Contribution

It provides a detailed calculation of the interference contribution to the decay width at NLO, including virtual and real corrections, with a focus on infrared and collinear singularity cancellation.

Findings

Double differential decay width is free of infrared and collinear singularities after summing corrections.

Leading power approximation in gluon bremsstrahlung simplifies calculations and avoids singularities.

Comparison with extended work including strange quark mass regulation shows consistency.

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.