All three-loop four-point correlators of half-BPS operators in planar N=4 SYM

TL;DR

The paper addresses the perturbative computation of planar four-point correlators of half-BPS operators of arbitrary weights in ${\cal N}=4$ SYM up to three loops. It develops an integrand-based bootstrap that leverages superconformal symmetry, singularity constraints, and planarity, with a powerful light-cone OPE relation between correlators of shifted weights to fix almost all coefficients; the result expresses all correlators in terms of a finite basis of one-, two-, and three-loop planar conformal integrals. A key finding is the uniformity: only 9 independent functions at two loops and 55 at three loops are needed, reflecting a degeneracy generalisation observed at lower loops and compatible with integrability predictions for OPE structure constants. The work provides nontrivial checks against integrability-based predictions and hints at deeper integrable structures governing multipoint correlators in planar ${\cal N}=4$ SYM, with potential extensions to higher loops and more points and connections to amplitudes/Wilson loops. The results also establish averaging rules and consistency checks that reinforce the planarity-driven bootstrap as a viable route to understanding nontrivial dynamical data in the theory.

Abstract

We obtain the planar correlation function of four half-BPS operators of arbitrary weights, up to three loops. Our method exploits only elementary properties of the integrand of the planar correlator, such as its symmetries and singularity structure. This allows us to write down a general ansatz for the integrand. The coefficients in the ansatz are fixed by means of a powerful light-cone OPE relation between correlators with different weights. Our result is formulated in terms of a limited number of functions built from known one-, two- and three-loop conformal integrals. These results are useful for checking recent integrability predictions for the OPE structure constants.