Split-Helicity Tree Amplitudes and Flag Cluster Algebras

TL;DR

This paper investigates the cluster adjacency properties of split-helicity tree-level gluon amplitudes, proposing a conjecture that their poles satisfy certain permutation-based cluster adjacency conditions, supported by multiple case checks.

Contribution

It introduces a conjecture relating split-helicity amplitudes' poles to cluster adjacency via arc permutations, expanding the understanding of scattering amplitudes and cluster algebras.

Findings

Confirmed cluster adjacency in several cases

Proposed a general conjecture for all split-helicity amplitudes

Connected scattering amplitudes with flag cluster algebra structures

Abstract

Recent work has uncovered a connection between the symbol letters of general massless scattering and (permutations of) cluster variables of partial flag varieties. In this paper we explore the cluster adjacency of tree-level gluon amplitudes, specifically focusing on split-helicity amplitudes which can be written in closed form in terms of zigzag diagrams. We check in several cases, and conjecture in general, that the poles in each term satisfy cluster adjacency under a set of permutations that is built from arc permutations of the corresponding zigzag.