Cluster adjacency properties of scattering amplitudes

TL;DR

The paper proposes a cluster-algebraic framework for the analytic structure of planar $\mathcal{N}=4$ SYM amplitudes, positing a cluster adjacency rule that constrains consecutive symbol letters beyond the alphabet and is anchored in $\mathrm{Gr}(4,n)$ cluster algebras. It demonstrates the conjecture using heptagon amplitudes, deriving admissible symbol structures at weight up to six and computing a new three-loop heptagon symbol that is consistent with the adjacency constraints. By analyzing neighbor sets, integrability, and differential relations, the authors show that cluster structure tightly governs both discontinuities and derivatives, with Steinmann relations recovered as a special case. The work provides explicit predictions for which letter pairs can appear consecutively, supports the adjacency principle with explicit three-loop data, and outlines broader implications for MHV/NMHV amplitudes and higher-point functions expressed as cluster polylogarithms.

Abstract

We conjecture a new set of analytic relations for scattering amplitudes in planar N=4 super Yang-Mills theory. They generalise the Steinmann relations and are expressed in terms of the cluster algebras associated to Gr(4,n). In terms of the symbol, they dictate which letters can appear consecutively. We study heptagon amplitudes and integrals in detail and present symbols for previously unknown integrals at two and three loops which support our conjecture.

Figures (3)

• Figure 1: Seven-point, three-loop, massless integral.
• Figure 2: The seven-point, two-loop integrals $I^{(2)}$ and $\tilde{I}^{(2)}$.
• Figure 3: The one-loop hexagon integrals $I^{(1)}$ and $\tilde{I}^{(1)}$.