Analytic two-loop amplitudes for $q\bar{q}\to \gamma \gamma$ and $gg \to \gamma \gamma$ mediated by a heavy-quark loop

TL;DR

This paper presents an analytic approach to compute two-loop scattering amplitudes involving heavy-quark loops for photon pair production, utilizing elliptic integrals and differential equations to improve accuracy and efficiency.

Contribution

It introduces a canonical basis for elliptic Feynman integrals and derives explicit expressions, advancing the analytic computation of complex two-loop amplitudes.

Findings

Derived a canonical basis for elliptic integrals

Achieved fast, reliable numerical evaluations across phase space

Uncovered cancellations in the physical amplitude expressions

Abstract

We address the analytic computation of the two-loop scattering amplitudes for the production of two photons in parton-parton scattering, mediated by loops of heavy quarks. Due to the presence of integrals of elliptic type, both partonic channels have been previously computed using semi-numerical methods. In this paper, leveraging new advances in the theory of differential equations for elliptic Feynman integrals, we derive a canonical basis for all integrals involved and compute them in terms of independent iterated integrals over elliptic and polylogarithmic differential forms. We use this representation to showcase interesting cancellations in the physical expressions for the scattering amplitudes. Furthermore, we address their numerical evaluation by producing series expansion representations for the whole amplitudes, which we demonstrate to be fast and numerically reliable across a large region of the phase space.