Hedgehog Bases for A_n Cluster Polylogarithms and An Application to Six-Point Amplitudes

TL;DR

This paper develops a new basis of polylogarithm functions aligned with the cluster algebra structure of N=4 Yang-Mills amplitudes, enabling more transparent expressions of complex scattering amplitudes.

Contribution

It introduces a novel construction of Goncharov polylogarithm bases based on $A_n$ cluster coordinates, facilitating the analysis of multi-loop scattering amplitudes.

Findings

New basis simplifies the expression of 2-loop 6-particle NMHV amplitude.

Manifestation of cluster structure in amplitude expressions.

Enhanced computational framework for scattering amplitudes.

Abstract

Multi-loop scattering amplitudes in N=4 Yang-Mills theory possess cluster algebra structure. In order to develop a computational framework which exploits this connection, we show how to construct bases of Goncharov polylogarithm functions, at any weight, whose symbol alphabet consists of cluster coordinates on the $A_n$ cluster algebra. Using such a basis we present a new expression for the 2-loop 6-particle NMHV amplitude which makes some of its cluster structure manifest.