Numerical evaluation of two-loop QCD helicity amplitudes for $gg\to t \bar{t} g$ at leading colour

TL;DR

This paper provides the first numerical benchmark of two-loop QCD helicity amplitudes for top-quark pair production with a jet, using a novel special function basis to handle elliptic integrals and achieve analytical pole cancellation.

Contribution

It introduces a new method for expressing master integrals with special functions, enabling precise numerical evaluation of complex two-loop amplitudes in QCD.

Findings

First benchmark evaluation of two-loop finite remainders for $gg\to t\bar{t}g$

Development of a new special function basis for elliptic integrals

Analytical cancellation of infrared and ultraviolet poles

Abstract

We present the first benchmark evaluation of the two-loop finite remainders for the production of a top-quark pair in association with a jet at hadron colliders in the gluon channel. We work in the leading colour approximation, and perform the numerical evaluation in the physical phase space. To achieve this result, we develop a new method for expressing the master integrals in terms of a (over-complete) basis of special functions that enables the infrared and ultraviolet poles to be cancelled analytically despite the presence of elliptic Feynman integrals. The special function basis makes it manifest that the elliptic functions appear solely in the finite remainder, and can be evaluated numerically through generalised series expansions. The helicity amplitudes are constructed using four dimensional projectors combined with finite-field techniques to perform integration-by-parts reduction, mapping to special functions and Laurent expansion in the dimensional regularisation parameter.