The one-loop six-dimensional hexagon integral with three massive corners

TL;DR

This paper analytically computes a complex six-dimensional hexagon integral with three massive corners, expressing it in terms of conformally invariant cross-ratios and special polylogarithmic functions, aiding amplitude calculations in super Yang-Mills theory.

Contribution

It introduces a novel method combining differential equations and symbol reconstruction to evaluate a six-dimensional hexagon integral with three masses, advancing amplitude computations.

Findings

Explicit expression for the hexagon integral in terms of polylogarithms.

Methodology for reconstructing functions from their symbols.

Potential applications to two-loop scattering amplitude analysis.

Abstract

We compute the six-dimensional hexagon integral with three non-adjacent external masses analytically. After a simple rescaling, it is given by a function of six dual conformally invariant cross-ratios. The result can be expressed as a sum of 24 terms involving only one basic function, which is a simple linear combination of logarithms, dilogarithms, and trilogarithms of uniform degree three transcendentality. Our method uses differential equations to determine the symbol of the function, and an algorithm to reconstruct the latter from its symbol. It is known that six-dimensional hexagon integrals are closely related to scattering amplitudes in N=4 super Yang-Mills theory, and we therefore expect our result to be helpful for understanding the structure of scattering amplitudes in this theory, in particular at two loops.