Two-Loop Polygon Wilson Loops in N=4 SYM

TL;DR

This paper calculates two-loop corrections to polygon Wilson loops in N=4 SYM, confirming the need for a remainder function beyond the BDS ansatz and verifying dual conformal invariance for various n-gons.

Contribution

It provides the first numerical computation of two-loop polygon Wilson loops for arbitrary n, demonstrating the structure of remainder functions and their properties.

Findings

Confirmed the necessity of a remainder function for six-point amplitudes.

Numerically computed remainder functions for n=7,8 and verified dual conformal invariance.

Showed that the collinear limits match the expected factorization properties.

Abstract

We compute for the first time the two-loop corrections to arbitrary n-gon lightlike Wilson loops in N=4 supersymmetric Yang-Mills theory, using efficient numerical methods. The calculation is motivated by the remarkable agreement between the finite part of planar six-point MHV amplitudes and hexagon Wilson loops which has been observed at two loops. At n=6 we confirm that the ABDK/BDS ansatz must be corrected by adding a remainder function, which depends only on conformally invariant ratios of kinematic variables. We numerically compute remainder functions for n=7,8 and verify dual conformal invariance. Furthermore, we study simple and multiple collinear limits of the Wilson loop remainder functions and demonstrate that they have precisely the form required by the collinear factorisation of the corresponding two-loop n-point amplitudes. The number of distinct diagram topologies contributing to the n-gon Wilson loops does not increase with n, and there is a fixed number of "master integrals", which we have computed. Thus we have essentially computed general polygon Wilson loops, and if the correspondence with amplitudes continues to hold, all planar n-point two-loop MHV amplitudes in the N=4 theory.