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

TL;DR

The paper extends the hexagon bootstrap to seven points, introducing the heptagon function framework built from a finite 42-letter cluster alphabet on Gr(4,7) and identifying a unique weight-6 MHV symbol that satisfies the last-entry and collinear constraints while remaining dihedral and parity-symmetric. Remarkably, its 7||6 collinear limit recovers the three-loop six-particle MHV symbol, implying it is the symbol of the seven-particle three-loop MHV amplitude and suggesting strong predictive power for n-gon bootstrap beyond n=6. The authors develop and compare two constructive schemes for integrable words, implement a computational pipeline using integer-linear-algebra and LLL reductions, and demonstrate that the collinear and symmetry constraints tightly fix the higher-weight structure, with far fewer free parameters than expected. They also explore weight-2 behavior across general n-gons, revealing tight constraints on possible last-entry functions and offering insights into the scalability and limitations of the bootstrap program. Overall, the work provides a compelling glimpse into a powerful, principled approach to determining higher-point amplitudes in planar SYM and motivates future explorations to higher loops and larger n.

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.