Two-loop Octagons, Algebraic Letters and $\bar{Q}$ Equations

TL;DR

This paper computes the symbol of the two-loop eight-point NMHV amplitude in planar ${ m N}=4$ SYM, revealing a complex alphabet with rational and algebraic letters, and demonstrates the use of $ar{Q}$ equations for such calculations.

Contribution

It introduces a method to compute two-loop amplitudes with algebraic letters using $ar{Q}$ equations, providing explicit results for the octagon case and insights into all-multiplicity predictions.

Findings

The symbol alphabet includes 180 rational and 18 algebraic letters.

Explicit two-loop octagon NMHV amplitude result obtained.

Insights into all-loop predictions for final entries.

Abstract

We compute the symbol of the first two-loop amplitudes in planar ${\cal N}=4$ SYM with algebraic letters, the eight-point NMHV amplitude (or the dual octagon Wilson loops). We show how to apply $\bar{Q}$ equations for computing the differential of two-loop $n$-point NMHV amplitudes and present the result for n=8 explicitly. The symbol alphabet for octagon consists of 180 independent rational letters and 18 algebraic ones involving Gram-determinant square roots. We comment on all-loop predictions for final entries and aspects of the result valid for all multiplicities.