How tropical are seven- and eight-particle amplitudes?

TL;DR

This paper explores the application of tropical Grassmannians to seven- and eight-particle amplitudes in super Yang-Mills theory, proposing new cluster coordinate sets and analyzing their relation to amplitude structures.

Contribution

It introduces a finite set of cluster coordinates for eight-particle amplitudes and examines the applicability of tropical Grassmannians to particle scattering amplitudes.

Findings

The positive part of Tr(4,7) relates to seven-point amplitude discontinuities.

A set of 356 cluster coordinates is proposed for eight-particle amplitudes.

A triangulation approach yields a scalar amplitude with minimal spurious poles.

Abstract

We study tropical Grassmanians Tr$(k,n)$ in relation to cluster algebras, and assess their applicability to $n$-particle amplitudes for $n=7,8$. In $\mathcal{N}=4$ super Yang-Mills theory, we first show that while the totally positive part of Tr$(4,7)$ may encompass the iterated discontinuity structure of the seven-point Maximally Helicity Violating (MHV) amplitude, it is too small for the Next-to-MHV helicity configuration. Then, using Tr$(4,8)$ we propose a finite set of 356 cluster $\mathcal{A}$-coordinates expected to contain the rational symbol letters of the eight-particle MHV amplitude, and discuss how the remaining square-root letters may be obtained from limits of infinite mutation sequences. Finally, we use a triangulation of the totally positive part of Tr$(3,8)$ to obtain the associated generalised biadjoint scalar amplitude in a form containing a near-minimal amount of spurious poles.