Three-loop corrections to the soft anomalous dimension in multi-leg scattering

TL;DR

This paper calculates the three-loop corrections to the soft anomalous dimension in multi-leg scattering amplitudes, revealing non-dipole effects and their dependence on kinematic variables, with implications for factorization and high-energy limits.

Contribution

It provides the first complete three-loop correction to the soft anomalous dimension, including non-dipole terms and their kinematic dependence, expressed via harmonic polylogarithms.

Findings

Non-dipole corrections appear at three loops for three or more partons.

Kinematic dependence arises only through conformally-invariant cross ratios.

The non-dipole correction reduces to a constant in collinear limits, preserving factorization.

Abstract

We present the three-loop result for the soft anomalous dimension governing long-distance singularities of multi-leg gauge-theory scattering amplitudes of massless partons. We compute all contributing webs involving semi-infinite Wilson lines at three loops and obtain the complete three-loop correction to the dipole formula. We find that non-dipole corrections appear already for three coloured partons, where the correction is a constant without kinematic dependence. Kinematic dependence appears only through conformally-invariant cross ratios for four coloured partons or more, and the result can be expressed in terms of single-valued harmonic polylogarithms of weight five. While the non-dipole three-loop term does not vanish in two-particle collinear limits, its contribution to the splitting amplitude anomalous dimension reduces to a constant, and it only depends on the colour charges of the collinear pair, thereby preserving strict collinear factorization properties. Finally we verify that our result is consistent with expectations from the Regge limit.