Simplicity of Polygon Wilson Loops in N=4 SYM

TL;DR

This paper investigates the properties of polygonal Wilson loops in N=4 super Yang-Mills theory, providing evidence for a universal remainder function at weak and strong coupling and analyzing its behavior for large polygons.

Contribution

It presents numerical evidence that the remainder function for polygon Wilson loops is universal across coupling regimes and explores its linear growth for large n-gons.

Findings

Remainder function is consistent at weak and strong coupling up to a constant.

Remainder function grows linearly with the number of sides n at large n.

Universality suggests similar behavior for MHV amplitudes across coupling regimes.

Abstract

Wilson loops with lightlike polygonal contours have been conjectured to be equivalent to MHV scattering amplitudes in N=4 super Yang-Mills. We compute such Wilson loops for special polygonal contours at two loops in perturbation theory. Specifically, we concentrate on the remainder function R, obtained by subtracting the known ABDK/BDS ansatz from the Wilson loop. First, we consider a particular two-dimensional eight-point kinematics studied at strong coupling by Alday and Maldacena. We find numerical evidence that R is the same at weak and at strong coupling, up to an overall, coupling-dependent constant. This suggests a universality of the remainder function at strong and weak coupling for generic null polygonal Wilson loops, and therefore for arbitrary MHV amplitudes in N=4 super Yang-Mills. We analyse the consequences of this statement. We further consider regular n-gons, and find that the remainder function is linear in n at large n through numerical computations performed up to n=30. This reproduces a general feature of the corresponding strong-coupling result.