On soft singularities at three loops and beyond

TL;DR

This paper investigates the structure of soft singularities in massless gauge theory scattering amplitudes at three loops, exploring potential corrections beyond the dipole formula constrained by symmetry, factorization, and transcendentality.

Contribution

It identifies possible new contributions to the soft anomalous dimension beyond the dipole sum, constrained by conformal invariance and symmetry, at three loops.

Findings

Unique function consistent with constraints if only logarithms are used

At least two additional structures possible with polylogarithms

Analysis advances understanding of multi-leg soft singularities at three loops

Abstract

We report on further progress in understanding soft singularities of massless gauge theory scattering amplitudes. Recently, a set of equations was derived based on Sudakov factorization, constraining the soft anomalous dimension matrix of multi-leg scattering amplitudes to any loop order, and relating it to the cusp anomalous dimension. The minimal solution to these equations was shown to be a sum over color dipoles. Here we explore potential contributions to the soft anomalous dimension that go beyond the sum-over-dipoles formula. Such contributions are constrained by factorization and invariance under rescaling of parton momenta to be functions of conformally invariant cross ratios. Therefore, they must correlate the color and kinematic degrees of freedom of at least four hard partons, corresponding to gluon webs that connect four eikonal lines, which first appear at three loops. We analyze potential contributions, combining all available constraints, including Bose symmetry, the expected degree of transcendentality, and the singularity structure in the limit where two hard partons become collinear. We find that if the kinematic dependence is solely through products of logarithms of cross ratios, then at three loops there is a unique function that is consistent with all available constraints. If polylogarithms are allowed to appear as well, then at least two additional structures are consistent with the available constraints.