A Cluster Bootstrap for Two-Loop MHV Amplitudes

TL;DR

This paper develops a bootstrap method to determine two-loop MHV amplitudes in planar N=4 super-Yang-Mills theory, leveraging cluster algebra properties and physical constraints to produce a concise, all-n formula.

Contribution

It introduces a novel bootstrap approach that uniquely determines the complex coproduct component of the amplitude using cluster algebra and physical constraints.

Findings

Successfully derives a closed-form expression for all n-particle amplitudes.

Demonstrates the power of cluster algebra in constraining amplitude structures.

Provides insights into the mathematical structure of two-loop MHV amplitudes.

Abstract

We apply a bootstrap procedure to two-loop MHV amplitudes in planar N=4 super-Yang-Mills theory. We argue that the mathematically most complicated part (the $\Lambda^2 B_2$ coproduct component) of the n-particle amplitude is uniquely determined by a simple cluster algebra property together with a few physical constraints (dihedral symmetry, analytic structure, supersymmetry, and well-defined collinear limits). We present a concise, closed-form expression which manifests these properties for all n.