Hidden symmetry of four-point correlation functions and amplitudes in N=4 SYM

TL;DR

This paper reveals a hidden symmetry in four-point correlation functions in N=4 SYM, enabling the prediction and determination of multi-loop integrands and their consistency with known operator dimensions, advancing understanding of superconformal theories.

Contribution

It uncovers a novel symmetry in four-point functions, allowing all-loop integrand predictions and explicit three-loop results in N=4 SYM, and extends techniques to higher loops.

Findings

Identified complete symmetry under exchange of external and internal points.

Predicted three-loop integrand up to four constants using symmetry.

Fully determined three-loop integrand in planar limit and verified with OPE.

Abstract

We study the four-point correlation function of stress-tensor supermultiplets in N=4 SYM using the method of Lagrangian insertions. We argue that, as a corollary of N=4 superconformal symmetry, the resulting all-loop integrand possesses an unexpected complete symmetry under the exchange of the four external and all the internal (integration) points. This alone allows us to predict the integrand of the three-loop correlation function up to four undetermined constants. Further, exploiting the conjectured amplitude/correlation function duality, we are able to fully determine the three-loop integrand in the planar limit. We perform an independent check of this result by verifying that it is consistent with the operator product expansion, in particular that it correctly reproduces the three-loop anomalous dimension of the Konishi operator. As a byproduct of our study, we also obtain the three-point function of two half-BPS operators and one Konishi operator at three-loop level. We use the same technique to work out a compact form for the four-loop four-point integrand and discuss the generalisation to higher loops.