Wilson Loops @ 3-Loops in Special Kinematics

TL;DR

The paper advances the analytic understanding of Wilson loops / MHV amplitudes in planar N=4 SYM by focusing on a 3-loop octagon in AdS3-like special kinematics. It exploits cyclic and parity symmetry, soft/collinear limits, and a symbol-based ansatz restricted to cross-ratio variables to derive a compact, polylogarithmic expression with 7 unfixed coefficients, and demonstrates a systematic uplift to the decagon. The approach relies on the integrability constraint and near-collinear OPE insights to constrain the functional form, enabling reconstruction without direct multi-loop integrals. The results illustrate the power of symbol methods for constraining high-loop amplitudes and set a roadmap for extending to higher polygons.

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.