Constructing the correlation function of four stress-tensor multiplets and the four-particle amplitude in N=4 SYM

TL;DR

This paper constructs the integrand of a four-point stress-tensor correlator in N=4 SYM using symmetry principles, graph theory, and divergence analysis, achieving fixed results up to six loops and confirming duality with four-particle amplitudes.

Contribution

It introduces a symmetry-based, diagram-free method for constructing correlation function integrands applicable to both planar and non-planar sectors in N=4 SYM.

Findings

Planar integrand fixed up to six loops

No non-planar corrections at three loops

Correlation function matches four-particle amplitude

Abstract

We present a construction of the integrand of the correlation function of four stress-tensor multiplets in N=4 SYM at weak coupling. It does not rely on Feynman diagrams and makes use of the recently discovered symmetry of the integrand under permutations of external and integration points. This symmetry holds for any gauge group, so it can be used to predict the integrand both in the planar and non-planar sectors. We demonstrate the great efficiency of graph-theoretical tools in the systematic study of the possible permutation symmetric integrands. We formulate a general ansatz for the correlation function as a linear combination of all relevant graph topologies, with arbitrary coefficients. Powerful restrictions on the coefficients come from the analysis of the logarithmic divergences of the correlation function in two singular regimes: Euclidean short-distance and Minkowski light-cone limits. We demonstrate that the planar integrand is completely fixed by the procedure up to six loops and probably beyond. In the non-planar sector, we show the absence of non-planar corrections at three loops and we reduce the freedom at four loops to just four constants. Finally, the correlation function/amplitude duality allows us to show the complete agreement of our results with the four-particle planar amplitude in N=4 SYM.