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

TL;DR

The paper derives an all-loop, first-order differential equation for the S-matrix of planar N=4 SYM based on Yangian symmetry, yielding regulator-independent, finite relations that uniquely determine amplitudes from lower-loop data. The key tool is the bar Q equation, formulated via a one-dimensional collinear integral over higher-point amplitudes and supported by an Operator Product Expansion analysis of null polygon Wilson loops. The authors apply the framework to reproduce known two-loop results for MHV and NMHV hexagons, fix undetermined coefficients in a three-loop MHV hexagon Ansatz, and provide evidence for all-loop validity including a dispersion-relation constraint. They also explore a two-dimensional kinematics sector where the equations simplify to partially closed forms, illustrating the method's reach and potential for higher-loop predictions.

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.

Figures (2)

• Figure 1: All-loop equation for planar $\mathcal{N}=4$ S-matrix.
• Figure 2: Fermion insertion on the Wilson loop versus kink insertion