Wilson Loops @ 3-Loops in Special Kinematics

TL;DR

This paper derives a compact 3-loop expression for the octagon MHV amplitude/Wilson loop in planar N=4 SYM within special 2D kinematics, utilizing symmetry properties and assumptions about polylogarithmic functions.

Contribution

It provides the first compact 3-loop formula for the octagon amplitude/Wilson loop in special kinematics, expressed in terms of unfixed coefficients and symmetry considerations.

Findings

Expression involves 7 unfixed coefficients.

Utilizes symmetry and soft/collinear limits.

Uplifted to 10-point case.

Abstract

We obtain a compact expression for the octagon MHV amplitude / Wilson loop at 3 loops in planar N=4 SYM and in special 2d kinematics in terms of 7 unfixed coefficients. We do this by making use of the cyclic and parity symmetry of the amplitude/Wilson loop and its behaviour in the soft/collinear limits as well as in the leading term in the expansion away from this limit. We also make a natural and quite general assumption about the functional form of the result, namely that it should consist of weight 6 polylogarithms whose symbol consists of basic cross-ratios only (and not functions thereof). We also describe the uplift of this result to 10 points.