One-loop hexagon integral to higher orders in the dimensional regulator

TL;DR

This paper analytically computes the one-loop hexagon integral in dimensional regularization, identifying its function alphabet and providing integral representations, thereby facilitating higher-order two-loop amplitude calculations in QCD.

Contribution

It introduces a detailed analysis of the one-loop hexagon integral, including its function alphabet and integral representations, advancing the tools for two-loop amplitude computations.

Findings

Identified the function alphabet for the one-loop hexagon integral.

Provided integral representations up to weight four.

Analyzed differences between dimensional regularization schemes.

Abstract

The state-of-the-art in current two-loop QCD amplitude calculations is at five-particle scattering. Computing two-loop six-particle processes requires knowledge of the corresponding one-loop amplitudes to higher orders in the dimensional regulator. In this paper we compute analytically the one-loop hexagon integral via differential equations. In particular we identify its function alphabet for general $D$-dimensional external states. We also provide integral representations for all one-loop integrals up to weight four. With this, the one-loop integral basis is ready for two-loop amplitude applications. We also study in detail the difference between the conventional dimensional regularization and the four-dimensional helicity scheme at the level of the master integrals and their function space.