Three-loop octagons and n-gons in maximally supersymmetric Yang-Mills theory

TL;DR

This paper explores simplified structures of scattering amplitudes in planar N=4 supersymmetric Yang-Mills theory within two-dimensional kinematics, providing new formulas for complex multi-loop, multi-point amplitudes and proposing their extension to general n-point cases.

Contribution

It introduces a compact formula for one-loop N^2MHV amplitudes and computes the complete two-loop NMHV and three-loop MHV octagons, advancing understanding of multi-loop amplitudes.

Findings

Simplified structures in two-dimensional kinematics

Explicit formulas for two-loop NMHV and three-loop MHV octagons

Conjecture for extending octagon results to general n-point amplitudes

Abstract

We study the S-matrix of planar $\mathcal{N}=4$ supersymmetric Yang-Mills theory when external momenta are restricted to a two-dimensional subspace of Minkowski space. We find significant simplifications and new, interesting structures for tree and loop amplitudes in two-dimensional kinematics; in particular, the higher-point amplitudes we consider can be obtained from those with lowest-points by a collinear uplifting. Based on a compact formula for one-loop N${}^2$MHV amplitudes, we use an equation proposed previously to compute, for the first time, the complete two-loop NMHV and three-loop MHV octagons, which we conjecture to uplift to give the full $n$-point amplitudes up to simpler logarithmic terms or dilogarithmic terms.