The Three-loop MHV Octagon from $\bar{Q}$ equations

TL;DR

This paper computes the three-loop MHV octagon amplitude in planar N=4 SYM using $ar{Q}$ equations, revealing its symbol alphabet with rational and algebraic letters, advancing understanding of amplitude structures.

Contribution

It introduces a novel computation of the three-loop MHV octagon amplitude's symbol using $ar{Q}$ equations, including algebraic letters, which was not previously achieved.

Findings

The symbol contains 204 rational letters.

Shares 18 algebraic letters with two-loop NMHV amplitude.

First three-loop MHV octagon amplitude with algebraic letters computed.

Abstract

The $\bar{Q}$ equations, rooted in the dual superconformal anomalies, are a powerful tool for computing amplitudes in planar $\mathcal{N}=4$ supersymmetric Yang-Mills theory. By using the $\bar{Q}$ equations, we compute the symbol of the first MHV amplitude with algebraic letters -- the three-loop 8-point amplitude (or the octagon remainder function) -- in this theory. The symbol alphabet for this amplitude consists of 204 independent rational letters and shares the same 18 algebraic letters with the two-loop 8-point NMHV amplitude.