Soft theorem to three loops in QCD and \( \mathcal{N} \) = 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 N⁴LO in perturbative QCD. Using the principle of leading transcendentality between QCD and $\mathcal{N}$ \( \mathcal{N} \) = 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 $f_2^{\left(3\right)}$ \( {f}_2^{(3)} \) in the Bern-Dixon-Smirnov ansatz analytically, which was previously known numerically only.