NLL QCD contribution of the Electromagnetic Dipole operator to B-> X_s gamma gamma

TL;DR

This paper computes the NLL QCD corrections to the double differential decay width of B -> X_s gamma gamma from the electromagnetic dipole operator, providing analytical results and addressing infrared and collinear singularities.

Contribution

It presents the first calculation of O(alpha_s) corrections to the double differential decay width for B -> X_s gamma gamma involving the electromagnetic dipole operator, with an analytical approach and singularity analysis.

Findings

Corrections are free of infrared and collinear singularities when using the leading power approximation.

Analytical expressions for the corrected decay width are obtained.

Uncanceled singularities appear when all powers of s_3 are retained, indicating the need for non-perturbative inputs.

Abstract

We calculate the set of O(alpha_s) corrections to the double differential decay width dGamma_77/(ds_1*ds_2) for the process B -> X_s gamma gamma originating from diagrams involving the electromagnetic dipole operator O_7. 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. 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 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 dGamma_77/(ds_1*ds_2*ds_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 bremstrahlung corrections contains uncanceled 1/epsilon singularities (which are due to collinear photon emission from the s-quark) and other concepts, which go beyond perturbation theory, like parton fragmentation functions of a quark or a gluon into a photon, are needed which is beyond the scope of our paper.