Structure of infrared singularities of gauge-theory amplitudes at three and four loops

A bstract The infrared divergences of massless n -parton scattering amplitudes can be derived from the anomalous dimension of n -jet operators in soft-collinear effective theory. Up to three-loop order, the latter has been shown to have a very simple structure: it contains pairwise color-dipole interactions among the external partons, governed by the cusp anomalous dimension and a logarithm of the kinematic invariants s ij , plus a possible three-loop correlation involving four particles, which is described by a yet unknown function of conformal cross ratios of kinematic invariants. This function is constrained by two-particle collinear limits and by the known behavior of amplitudes in the high-energy limit. We construct a class of relatively simple functions satisfying these constraints. We also extend the analysis to four-loop order, finding that three additional four-particle correlations and a single five-particle correlation appear, which again are governed by functions of conformal cross ratios. Our results suggest that the dipole conjecture, which states that only two-particle color-dipole correlations appear in the anomalous dimension, may need to be generalized. We present a weaker form of the conjecture, stating that to all orders in perturbation theory corrections to the dipole formula are governed by functions of conformal cross ratios, and are suppressed relative to the dipole term. If true, this conjecture implies that the cusp anomalous dimension obeys Casimir scaling to all orders in perturbation theory.