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

TL;DR

This paper calculates the three-loop soft factor in QCD and ${ m extbf{N}}=4$ super Yang-Mills theory, providing key ingredients for high-order infrared divergence subtraction and revealing new constants analytically.

Contribution

It introduces a systematic recursive method for three-loop soft factor calculations in QCD and extends the results to ${ m extbf{N}}=4$ SYM using transcendentality principles, also deriving an unknown constant analytically.

Findings

Computed the three-loop soft factor in QCD for two hard partons.

Extended the soft factor calculation to ${ m extbf{N}}=4$ SYM using transcendentality.

Analytically derived the finite constant $f_2^{(3)}$ in the BDS ansatz.

Abstract

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.