Next-to-leading logarithmic QCD contribution of the electromagnetic dipole operator toB¯→Xsγγ

Abstract

We calculate the set of $O({\ensuremath{\alpha}}_{s})$ corrections to the double differential decay width $d{\ensuremath{\Gamma}}_{77}/(d{s}_{1}d{s}_{2})$ for the process $\overline{B}\ensuremath{\rightarrow}{X}_{s}\ensuremath{\gamma}\ensuremath{\gamma}$ originating from diagrams involving the electromagnetic dipole operator ${\mathcal{O}}_{7}$. The kinematical variables ${s}_{1}$ and ${s}_{2}$ are defined as ${s}_{i}=({p}_{b}\ensuremath{-}{q}_{i}{)}^{2}/{m}_{b}^{2}$, where ${p}_{b}$, ${q}_{1}$, and ${q}_{2}$ are the momenta of the $b$ quark and two photons. While the (renormalized) virtual corrections are worked exactly for a certain range of ${s}_{1}$ and ${s}_{2}$, we retain in the gluon bremsstrahlung process only the leading power with respect to the (normalized) hadronic mass ${s}_{3}=({p}_{b}\ensuremath{-}{q}_{1}\ensuremath{-}{q}_{2}{)}^{2}/{m}_{b}^{2}$ in the underlying triple differential decay width $d{\ensuremath{\Gamma}}_{77}/(d{s}_{1}d{s}_{2}d{s}_{3})$. The double differential decay width, based on this approximation, is free of infrared and collinear singularities when combining virtual and bremsstrahlung corrections. The corresponding results are obtained analytically. When retaining all powers in ${s}_{3}$, the sum of virtual and bremsstrahlung corrections contains uncanceled $1/ϵ$ singularities (which are due to collinear photon emission from the $s$ quark), and other concepts, which go beyond perturbation theory, such as parton fragmentation functions of a quark or a gluon into a photon, are needed which is beyond the scope of our paper.