Jumpstarting the all-loop S-matrix of planar N=4 super Yang-Mills

TL;DR

This paper derives a set of differential equations based on Yangian symmetry that uniquely determine the all-loop S-matrix of planar N=4 super Yang-Mills theory, enabling systematic computation of scattering amplitudes.

Contribution

It introduces a novel set of first-order differential equations for the all-loop S-matrix, derived from Yangian symmetry, and demonstrates their application to specific multi-loop amplitude calculations.

Findings

Reproduces known two-loop NMHV hexagon results

Fixes coefficients in three-loop MHV hexagon Ansatz

Derives simplified equations for 2D subspace amplitudes

Abstract

We derive a set of first-order differential equations obeyed by the S-matrix of planar maximally supersymmetric Yang-Mills theory. The equations, based on the Yangian symmetry of the theory, involve only finite and regulator-independent quantities and uniquely determine the all-loop S-matrix. When expanded in powers of the coupling they give derivatives of amplitudes as single integrals over lower-loop, higher-point amplitudes/Wilson loops. We outline a derivation for the equations using the Operator Product Expansion for Wilson loops. We apply them on a few examples at two- and three-loops, reproducing a recent result on the two-loop NMHV hexagon and fixing previously undermined coefficients in a recent Ansatz for the three-loop MHV hexagon. In addition, we consider amplitudes restricted to a two-dimensional subspace of Minkowski space and derive a particularly simple set of partially closed equations in that case.