On the logarithmic behaviour in N=4 SYM theory

TL;DR

This work links the appearance of short-distance logarithms in four-point functions of ${\cal N}=4$ SYM to nonzero anomalous dimensions of unprotected long multiplet operators, notably the Konishi multiplet, via the OPE framework. Through a targeted four-point function of protected current-multiplet scalars, the authors perform a one-loop perturbative analysis that reproduces ${\gamma}_{\cal K}^{\text{1-loop}} = {3\over 16\pi^2} g^2 N$ and show that only Konishi contributes logarithmically in the relevant channel. They then compute the one-instanton contribution and demonstrate the absence of nonperturbative corrections to the Konishi anomalous dimension (and to protected two-point functions), while outlining the behavior of double-trace operators. The results support a picture in which logarithms encode operator content and reinforce the consistency of the AdS/SCFT framework, with multi-trace anomalous dimensions vanishing at large $N$ and Konishi potentially growing with $N$, in line with holographic expectations.

Abstract

We show that the logarithmic behaviour seen in perturbative and non perturbative contributions to Green functions of gauge-invariant composite operators in N=4 SYM with SU(N) gauge group can be consistently interpreted in terms of anomalous dimensions of unprotected operators in long multiplets of the superconformal group SU(2,2|4). In order to illustrate the point we analyse the short-distance behaviour of a particularly simple four-point Green function of the lowest scalar components of the N=4 supercurrent multiplet. Assuming the validity of the Operator Product Expansion, we are able to reproduce the known value of the one-loop anomalous dimension of the single-trace operators in the Konishi supermultiplet. We also show that it does not receive any non-perturbative contribution from the one-instanton sector. We briefly comment on double- and multi-trace operators and on the bearing of our results on the AdS/SCFT correspondence.