A Cluster Bootstrap for Two-Loop MHV Amplitudes

TL;DR

This work develops a cluster-analytic bootstrap for two-loop MHV amplitudes in planar $\mathcal{N}=4$ SYM, arguing that the most intricate part of the weight-4 coproduct, $\delta(R_n^{(2)})|_{\Lambda^2 B_2}$, is fixed by a simple cluster-structure constraint together with dihedral symmetry, analytic structure, supersymmetry, and well-defined collinear limits. The authors construct a $\{v\}_2 \wedge \{z\}_2$ basis, where $v$ encodes first-entry (analytic) constraints and $z$ encodes last-entry (SUSY) constraints, and show that the coproduct can be determined for all $n$ via an explicit all-$n formula (eq. (['eq:npt'])), validated against symbol data up to $n=13$. They explicitly demonstrate the bootstrap at $n=7$ and $n=8$, fixing coefficients through collinear limits and symmetry, and conjecture a general all-$n$ result, which suggests a deeper geometric structure (Poisson brackets vanish among contributing pairs) and hints at further extensions to higher-loop or non-MHV amplitudes. The work also notes limitations in capturing $\mathrm{B}_3 \otimes \mathbb{C}^*$ and $\mathrm{Li}_4$ data within the current ansatz, pointing to future directions in the cluster-bootstrap program.

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 $Λ^2 B_2$ 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.