Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry

TL;DR

This paper advances the understanding of singularities in scattering amplitudes for eight- and nine-particle processes in super Yang-Mills theory by leveraging cluster algebras and tropical geometry, predicting new symbol letters.

Contribution

It introduces a refined mathematical framework for predicting amplitude singularities at higher multiplicities using tropical Grassmannians and cluster algebra sequences.

Findings

Identifies 3,078 rational and 2,349 square-root letters for nine-particle amplitudes.

Shows the only additional letters from scattering diagrams are related to four-mass boxes.

Recovers the alphabet from a 2-loop NMHV symbol calculation.

Abstract

We further exploit the relation between tropical Grassmannians and $\operatorname{Gr}(4,n)$ cluster algebras in order to make and refine predictions for the singularities of scattering amplitudes in $\mathcal{N}=4$ planar super Yang-Mills theory at higher multiplicity $n\ge 8$. As a mathematical foundation that provides access to square-root symbol letters in principle for any $n$, we analyse infinite mutation sequences in cluster algebras with general coefficients. First specialising our analysis to the eight-particle amplitude, and comparing it with a recent, closely related approach based on scattering diagrams, we find that the only additional letters the latter provides are the two square roots associated to the four-mass box. In combination with a tropical rule for selecting a finite subset of variables of the infinite $\operatorname{Gr}(4,9)$ cluster algebra, we then apply our results to obtain a collection of $3,078$ rational and $2,349$ square-root letters expected to appear in the nine-particle amplitude. In particular these contain the alphabet found in an explicit 2-loop NMHV symbol calculation at this multiplicity.