The one-loop six-dimensional hexagon integral and its relation to MHV amplitudes in N=4 SYM

TL;DR

This paper derives an explicit analytic expression for a six-dimensional one-loop hexagon integral in N=4 SYM, revealing its deep connection to two-loop amplitude integrals and providing tools for simplifying related calculations.

Contribution

It presents a new analytic formula for the one-loop scalar hexagon integral and uncovers its differential relations to known amplitude integrals in N=4 SYM.

Findings

Analytic formula for the hexagon integral in terms of polylogarithms.

Demonstrates differential relations linking the hexagon integral to amplitude integrals.

Introduces kinematic variables that simplify the analysis of these integrals.

Abstract

We provide an analytic formula for the (rescaled) one-loop scalar hexagon integral $\tilde\Phi_6$ with all external legs massless, in terms of classical polylogarithms. We show that this integral is closely connected to two integrals appearing in one- and two-loop amplitudes in planar $\\mathcal{N}=4$ super-Yang-Mills theory, $\Omega^{(1)}$ and $\Omega^{(2)}$. The derivative of $\Omega^{(2)}$ with respect to one of the conformal invariants yields $\tilde\Phi_6$, while another first-order differential operator applied to $\tilde\Phi_6$ yields $\Omega^{(1)}$. We also introduce some kinematic variables that rationalize the arguments of the polylogarithms, making it easy to verify the latter differential equation. We also give a further example of a six-dimensional integral relevant for amplitudes in $\\mathcal{N}=4$ super-Yang-Mills.