A Symbol of Uniqueness: The Cluster Bootstrap for the 3-Loop MHV Heptagon

TL;DR

This paper identifies a unique mathematical symbol related to seven-particle scattering amplitudes in super-Yang-Mills theory, linking it to known six-particle results and suggesting a powerful new bootstrap approach for higher particle numbers.

Contribution

It introduces a systematic study of heptagon function symbols, discovering a unique, symmetric symbol that connects six- and seven-particle amplitudes without prior assumptions.

Findings

Found a unique weight-six symbol satisfying specific physical conditions.

Linked the seven-particle symbol to the six-particle three-loop amplitude.

Suggested the n-gon bootstrap's potential for n>6 amplitudes.

Abstract

Seven-particle scattering amplitudes in planar super-Yang-Mills theory are believed to belong to a special class of generalised polylogarithm functions called heptagon functions. These are functions with physical branch cuts whose symbols may be written in terms of the 42 cluster A-coordinates on Gr(4,7). Motivated by the success of the hexagon bootstrap programme for constructing six-particle amplitudes we initiate the systematic study of the symbols of heptagon functions. We find that there is exactly one such symbol of weight six which satisfies the MHV last-entry condition and is finite in the $7 \parallel 6$ collinear limit. This unique symbol is both dihedral and parity-symmetric, and remarkably its collinear limit is exactly the symbol of the three-loop six-particle MHV amplitude, although none of these properties were assumed a priori. It must therefore be the symbol of the three-loop seven-particle MHV amplitude. The simplicity of its construction suggests that the n-gon bootstrap may be surprisingly powerful for n>6.