Colour-dressed hexagon tessellations for correlation functions and non-planar corrections

Abstract

We continue the study of four-point correlation functions by the hexagon tessellation approach initiated in 1611.05436 and 1611.05577. We consider planar tree-level correlation functions in $\mathcal{N} = 4$ supersymmetric Yang-Mills theory involving two non-protected operators. We find that, in order to reproduce the field theory result, it is necessary to include $SU(N)$ colour factors in the hexagon formalism; moreover, we find that the hexagon approach as it stands is naturally tailored to the single-trace part of correlation functions, and does not account for multi-trace admixtures. We discuss how to compute correlators involving double-trace operators, as well as more general $1/N$ effects; in particular we compute the whole next-to-leading order in the large-$N$ expansion of tree-level BMN two-point functions by tessellating a torus with punctures. Finally, we turn to the issue of "wrapping", L\"uscher-like corrections. We show that $SU(N)$ colour-dressing reproduces an earlier empirical rule for incorporating single-magnon wrapping, and we provide a direct interpretation of such wrapping processes in terms of $\mathcal{N}=2$ supersymmetric Feynman diagrams.