Three-loop universal structure constants in N=4 susy Yang-Mills theory

TL;DR

This work targets the three-loop universal structure constants for twist-two operators in $N=4$ SYM by analyzing the four-point function of stress-tensor multiplets in a double OPE limit. The authors deploy expansion-by-regions to asymptotically evaluate two unknown three-loop integrals $E$ and $H$, and then perform a conformal partial wave analysis to relate these results to twist-two conformal blocks. They show that both the anomalous dimensions and the OPE constants for twist-two operators can be written as linear combinations of harmonic sums in the exchanged spin, providing explicit coefficients up to three loops and matching known results. The findings support an integrable-structure interpretation and offer a framework for extending the approach to higher twists and loops, with potential guidance for expressing correlation functions in terms of harmonic polylogarithms.

Abstract

We present a conjecture for the normalisation of the twist two conformal partial waves in a double OPE limit of the four-point function of stress tensor multiplets in N = 4 super Yang-Mills theory up to three loops. This contains information about the structure constants in the OPE. Like the twist two anomalous dimensions our result is expressed as a linear combination of harmonic sums whose argument is the spin of the exchanged operators. To arrive at the result we derive asymptotic expansions for the twist two part of two unknown three-loop integrals using the method of expansion by regions, complemented by some intuition gained on the example of the ladder integrals up to three loops.