# Six-Gluon Amplitudes in Planar ${\cal N}=4$ Super-Yang-Mills Theory at Six and Seven Loops

## TL;DR

This paper computes six-particle scattering amplitudes in planar ${\cal N}=4$ super-Yang-Mills theory up to seven loops, using advanced mathematical principles to identify and analyze their structure and behavior in various limits.

## Contribution

It extends the calculation of six-particle amplitudes to seven loops, employing the extended Steinmann relations and cosmic Galois coaction, and introduces new methods for amplitude determination at high loop orders.

## Key findings

- Amplitudes are uniquely determined up to five loops using known limits.
- Beyond five loops, additional data from the Pentagon OPE is required.
- Amplitudes show remarkable stability across different loop orders.

## Abstract

We compute the six-particle maximally-helicity-violating (MHV) and next-to-MHV (NMHV) amplitudes in planar maximally supersymmetric Yang-Mills theory through seven loops and six loops, respectively, as an application of the extended Steinmann relations and using the cosmic Galois coaction principle. Starting from a minimal space of functions constructed using these principles, we identify the amplitude by matching its symmetries and predicted behavior in various kinematic limits. Through five loops, the MHV and NMHV amplitudes are uniquely determined using only the multi-Regge and leading collinear limits. Beyond five loops, the MHV amplitude requires additional data from the kinematic expansion around the collinear limit, which we obtain from the Pentagon Operator Product Expansion, and in particular from its single-gluon bound state contribution. We study the MHV amplitude in the self-crossing limit, where its singular terms agree with previous predictions. Analyzing and plotting the amplitudes along various kinematical lines, we continue to find remarkable stability between loop orders.