Two-point functions of chiral operators in {\cal N}=4 SYM at order g^4

TL;DR

The paper tests the nonrenormalization of two-point functions for chiral primaries in N=4 SYM by computing <Tr Φ^3><Tr ¯Φ^3> at order g^4 for SU(N). Using N=1 superspace, dimensional regularization, and the method of uniqueness, the authors show all perturbative corrections cancel for all N, not only at g^2 but also at g^4, including nonplanar diagrams. This provides a nontrivial direct check of conjectured all-orders nonrenormalization and supports the AdS/CFT expectation that these correlators remain unrenormalized. The approach also establishes a framework for analyzing higher-k chiral primaries and other extremal correlators within the same formalism.

Abstract

We compute two-point functions of chiral operators Tr Φ^3 in {\cal N}=4 SU(N) supersymmetric Yang-Mills theory to the order g^4 in perturbation theory. We perform explicit calculations using {\cal N}=1 superspace techniques and find that perturbative corrections to the correlators vanish for all N. While at order g^2 the cancellations can be ascribed to the nonrenormalization theorem valid for correlators of operators in the same multiplet as the stress tensor, at order g^4 this argument no longer applies and the actual cancellation occurs in a highly nontrivial way. Our result is obtained in complete generality, without the need of additional conjectures or assumptions. It gives further support to the belief that such correlators are not renormalized to all orders in g and to all orders in N.