The symbol and alphabet of two-loop NMHV amplitudes from $\bar{Q}$ equations

TL;DR

This paper analyzes the symbol and alphabet of two-loop NMHV amplitudes in planar ${\cal N}=4$ super-Yang-Mills theory using $\bar{Q}$ equations, revealing algebraic structures, explicit formulas, and new classes of integrals.

Contribution

It introduces a first-principles method to compute the symbol alphabet of two-loop NMHV amplitudes, including algebraic letters and their patterns across multiplicities.

Findings

Derived explicit formulas for algebraic letters in the symbol.

Identified the number of independent algebraic letters related to Gram determinants.

Presented the complete symbol for the 9-point case with detailed alphabet structure.

Abstract

We study the symbol and the alphabet for two-loop NMHV amplitudes in planar ${\cal N}=4$ super-Yang-Mills from the $\bar{Q}$ equations, which provide a first-principle method for computing multi-loop amplitudes. Starting from one-loop N${}^2$MHV ratio functions, we explain in detail how to use $\bar{Q}$ equations to obtain the total differential of two-loop $n$-point NMHV amplitudes, whose symbol contains letters that are algebraic functions of kinematics for $n\geq 8$. We present explicit formula with nice patterns for the part of the symbol involving algebraic letters for all multiplicities, and we find $17-2m$ multiplicative-independent letters for a given square root of Gram determinant, with $0\leq m\leq 4$ depending on the number of particles involved in the square root. We also observe that these algebraic letters can be found as poles of one-loop four-mass leading singularities with MHV or NMHV trees. As a byproduct of our algebraic results, we find a large class of components of two-loop NMHV, which can be written as differences of two double-pentagon integrals, particularly simple and absent of square roots. As an example, we present the complete symbol for $n=9$ whose alphabet contains $59\times 9$ rational letters, in addition to the $11 \times 9$ independent algebraic ones. We also give all-loop NMHV last-entry conditions for all multiplicities.