  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
O(1/N_c^2) 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.
