Soft Theorem to Three Loops in QCD and ${\cal N} = 4$ Super Yang-Mills Theory

The soft theorem states that scattering amplitude in gauge theory with a soft gauge-boson emission can be factorized into a hard scattering amplitude and a soft factor. In this paper, we present calculations of the soft factor for processes involving two hard colored partons, up to three loops in QCD. To accomplish this, we developed a systematic method for recursively calculating relevant Feynman integrals using the Feynman-Parameter representation. Our results constitute an important ingredient for the subtraction of infrared singularities at N$^4$LO in perturbative QCD. Using the principle of leading transcendentality between QCD and ${\cal N}=4$ super Yang-Mills theory, we determine the soft factor in the latter case to three loops with full-color dependence. As a by-product, we also obtain the finite constant $f_2^{(3)}$ in the Bern-Dixon-Smirnov ansatz analytically, which was previously known numerically only.