Planar Six-Point Feynman Integrals for Four-Dimensional Gauge Theories

TL;DR

This paper computes all relevant planar two-loop six-point Feynman integrals in massless gauge theories, introducing a differential-equations approach that simplifies calculations by exploiting four-dimensional kinematic constraints.

Contribution

It formulates a novel differential-equations method tailored for four-dimensional kinematics, reducing the integral basis needed for two-loop six-point calculations in gauge theories.

Findings

Reduced the integral basis to 8 propagators due to four-dimensional constraints.

Constructed a pure basis with canonical differential equations for these integrals.

Facilitated numerical solutions for complex two-loop scattering integrals.

Abstract

We compute all planar two-loop six-point Feynman integrals entering scattering observables in massless gauge theories such as QCD. A central result of this paper is the formulation of the differential-equations method under the algebraic constraints stemming from four-dimensional kinematics, which in this case leaves only 8 independent scales. We show that these constraints imply that one must compute topologies with only up to 8 propagators, instead of the expected 9. This leads to the decoupling of entire classes of integrals that do not contribute to scattering amplitudes in four dimensional gauge theories. We construct a pure basis and derive their canonical differential equations, of which we discuss the numerical solution. This work marks an important step towards the calculation of massless $2\to 4$ scattering processes at two loops.