The Two-Loop Six-Gluon MHV Amplitude in Maximally Supersymmetric Yang-Mills Theory

TL;DR

This paper computes the two-loop six-gluon MHV amplitude in N=4 super-Yang-Mills theory, confirming the need for a remainder function and supporting the Wilson loop/amplitude duality at six points.

Contribution

It provides a numerical evaluation of the two-loop six-gluon MHV amplitude and confirms the amplitude/Wilson loop duality beyond strong coupling and Regge limits.

Findings

The ABDK/BDS ansatz requires an additive remainder function.

Wilson loop and amplitude remainders match within numerical precision.

Results support a duality between Wilson loops and MHV amplitudes at six points.

Abstract

We give a representation of the parity-even part of the planar two-loop six-gluon MHV amplitude of N=4 super-Yang-Mills theory, in terms of loop-momentum integrals with simple dual conformal properties. We evaluate the integrals numerically in order to test directly the ABDK/BDS all-loop ansatz for planar MHV amplitudes. We find that the ansatz requires an additive remainder function, in accord with previous indications from strong-coupling and Regge limits. The planar six-gluon amplitude can also be compared with the hexagonal Wilson loop computed by Drummond, Henn, Korchemsky and Sokatchev in arXiv:0803.1466 [hep-th]. After accounting for differing singularities and other constants independent of the kinematics, we find that the Wilson loop and MHV-amplitude remainders are identical, to within our numerical precision. This result provides non-trivial confirmation of a proposed n-point equivalence between Wilson loops and planar MHV amplitudes, and suggests that an additional mechanism besides dual conformal symmetry fixes their form at six points and beyond.