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.

Highlights

  • Introduction and summary of the resultsIn the present paper we continue the investigation of the four-point correlation function of the stress-tensor multiplet in N = 4 super-Yang-Mills (SYM) theory initiated in [1]

  • We show that the four-point correlation function is uniquely fixed at four loops in the planar sector only, whereas in the non-planar sector the obtained expression depends on four arbitrary constants

  • We study four-point correlation functions of the simplest representative of the class of half-BPS operators in the N = 4 SYM theory

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Summary

Introduction and summary of the results

Using the permutation symmetry and some input from the duality with the amplitude, the precise form of the planar correlation function in terms of a set of conformal fourpoint integrals was established at three and four loops This was done without even mentioning Feynman diagrams, whose number and complexity render a conventional perturbative calculation at this level very hard. We use the duality relation between the correlation function and scattering amplitude to obtain the expression for the planar four-particle amplitude up to six loops and we observe perfect agreement with the known results. Some technical details of our calculation are described in four appendices

Symmetries of four-point correlation functions
Conformal symmetry and OPE constraints
Permutation symmetry
Graph-theoretical interpretation
Duality with the planar four-particle scattering amplitude
Rung rule
Correlation function to two loops
The one-loop integrand
Logarithmic singularities
The two-loop integrand
Relation to the two-loop amplitude
Correlation function at three loops
The three-loop integrand
Fixing the coefficients from the logarithmic singularities
Relation to the three-loop amplitude
Rung rule at three loops
Correlation function at four loops
Planar sector
Rung rule at four loops
Relation to the four-loop amplitude
Non-planar sector
Five loops
Six loops
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