Complete function space for planar two-loop six-particle scattering amplitudes

TL;DR

This paper derives a complete system of differential equations and solutions for all planar two-loop six-particle master integrals, enabling efficient evaluation of six-particle scattering amplitudes in four dimensions.

Contribution

It provides the full analytic solution space, including the function alphabet and iterated integrals, for all relevant master integrals in two-loop six-particle scattering.

Findings

Derived canonical differential equations for all master integrals.

Provided explicit solutions and numerical implementations.

Validated solutions against numerical Feynman integral evaluations.

Abstract

We derive the full system of canonical differential equations for all planar two-loop massless six-particle master integrals, and determine analytically the boundary conditions. This fully specifies the solutions, which may be written as Chen iterated integrals. We argue that this is sufficient information for evaluating any scattering amplitude in four dimensions up to the finite part. We support this claim by reducing, for the most complicated integral topologies, integrals with typical Yang-Mills numerators. We use the analytic solutions to the differential equations, together with dihedral symmetry, to provide the full solution space relevant for two-loop six-particle computations. This includes the relevant function alphabet, as well as the independent set of iterated integrals up to weight four. We also provide the answer for all master integrals in terms of iterated integrals that can be readily evaluated numerically. As a proof of concept, we provide a numerical implementation that evaluates the integrals in part of the Euclidean region, and validate this against numerical evaluation of the Feynman integrals. Our result removes the bottleneck of Feynman integral evaluation, paving the way to future analytic evaluations of six-particle scattering amplitudes.