Four-point functions in N=4 supersymmetric Yang-Mills theory at two loops

TL;DR

The paper computes two-loop four-point functions of gauge-invariant operators in D=4, N=4 SYM using N=2 harmonic superspace, expressing results as differential operators acting on a scalar two-loop integral. It shows that in the coincidence limit two points approach each other the leading singularity has the form $$(x^2)^{-1}\log x^2,$$ with additional purely logarithmic subleading terms, and that analytic operator correlators remain analytic at this order. By decomposing N=4 into N=2 multiplets and exploiting superconformal invariance, the authors reduce the problem to tractable one- and two-loop integrals and verify harmonic analyticity. The findings provide a concrete perturbative check of analyticity postulates and offer data for comparisons with AdS/CFT supergravity results, highlighting where perturbation theory and gravity calculations may align. The work also clarifies the structural role of the A1,A2,A3 functions in encoding the leading terms of four-point functions and sets the stage for higher-loop and holographic tests.

Abstract

Four-point functions of gauge-invariant operators in D=4, N=4 supersymmetric Yang-Mills theory are studied using N=2 harmonic superspace perturbation theory. The results are expressed in terms of differential operators acting on a scalar two loop integral. The leading singular behaviour is obtained in the limit that two of the points approach one another. We find logarithmic singularities which do not cancel out in the sum of all diagrams. It is confirmed that Green's functions of analytic operators are indeed analytic at this order in perturbation theory.