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

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