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The soft emission factor is a central ingredient in the factorization ofgeneric $n$-particle gauge theory amplitudes with one soft gluon in theexternal state. We present the complete two-loop soft factor, capturing theleading power behavior in the soft-gluon momentum. At two loops, the colorstructure and the kinematic dependence of the soft factor become nontrivial asthe soft gluon can couple to three hard partons for the first time (tripoleterms). 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 consistentwith the behavior of the recently computed symbol of the two-loop five-particleamplitude in $\mathcal{N}=4$ super-Yang-Mills theory. In the limit where theoutgoing soft gluon is also collinear with an incoming hard parton, potentiallydangerous factorization-violating terms can arise.

Soft gluon emission at two loops in full color