Soft gluon emission at two loops in full color

TL;DR

This paper computes the complete two-loop soft gluon emission factor in gauge theories, revealing complex color and kinematic structures, including tripole terms, and confirms consistency with known amplitude symbols in super-Yang-Mills theory.

Contribution

It provides the full two-loop soft factor with nontrivial color and kinematic dependence, including tripole terms, expressed via single-valued harmonic polylogarithms, advancing understanding of soft gluon emissions.

Findings

Two-loop soft factor includes tripole terms with complex color structure.

Kinematic dependence of tripole terms has maximal transcendental weight.

Results align with the symbol of two-loop five-particle amplitudes in $ ext{N}=4$ super-Yang-Mills.

Abstract

The soft emission factor is a central ingredient in the factorization of generic $n$-particle gauge theory amplitudes with one soft gluon in the external state. We present the complete two-loop soft factor, capturing the leading power behavior in the soft-gluon momentum. At two loops, the color structure and the kinematic dependence of the soft factor become nontrivial as the soft gluon can couple to three hard partons for the first time (tripole terms). The nontrivial kinematic dependence of the tripole terms is of uniform, maximal transcendental weight, and can be expressed (in a "Euclidean" region) in terms of single-valued harmonic polylogarithms. Our results are consistent with the behavior of the recently computed symbol of the two-loop five-particle amplitude in $\mathcal{N}=4$ super-Yang-Mills theory. In the limit where the outgoing soft gluon is also collinear with an incoming hard parton, potentially dangerous factorization-violating terms can arise, but they cancel after summing over colors.