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We report on further progress in understanding soft singularities of masslessgauge theory scattering amplitudes. Recently, a set of equations was derivedbased on Sudakov factorization, constraining the soft anomalous dimensionmatrix of multi-leg scattering amplitudes to any loop order, and relating it tothe cusp anomalous dimension. The minimal solution to these equations was shownto be a sum over color dipoles. Here we explore potential contributions to thesoft anomalous dimension that go beyond the sum-over-dipoles formula. Suchcontributions are constrained by factorization and invariance under rescalingof parton momenta to be functions of conformally invariant cross ratios.Therefore, they must correlate the color and kinematic degrees of freedom of atleast four hard partons, corresponding to gluon webs that connect four eikonallines, which first appear at three loops. We analyze potential contributions,combining all available constraints, including Bose symmetry, the expecteddegree of transcendentality, and the singularity structure in the limit wheretwo hard partons become collinear. We find that if the kinematic dependence issolely through products of logarithms of cross ratios, then at three loopsthere is a unique function that is consistent with all available constraints.If polylogarithms are allowed to appear as well, then at least two additionalstructures are consistent with the available constraints.

On soft singularities at three loops and beyond