A non-renormalization theorem for conformal anomalies

TL;DR

This paper proves a non-renormalization theorem for conformal anomalies in operators with zero anomalous dimensions, with implications for correlation functions in superconformal theories and their invariance under renormalization.

Contribution

It establishes a non-renormalization theorem for conformal anomaly coefficients and clarifies conditions under which correlation functions remain unrenormalized in superconformal theories.

Findings

Conformal anomaly coefficients for certain operators do not renormalize.

2- and 3-point functions of chiral primaries in N=4 SYM are protected from renormalization.

The non-renormalization depends on the behavior of contact terms and a generalized Adler-Bardeen theorem.

Abstract

We provide a non-renormalization theorem for the coefficients of the conformal anomaly associated with operators with vanishing anomalous dimensions. Such operators include conserved currents and chiral operators in superconformal field theories. We illustrate the theorem by computing the conformal anomaly of 2-point functions both by a computation in the conformal field theory and via the adS/CFT correspondence. Our results imply that 2- and 3-point functions of chiral primary operators in N=4 SU(N) SYM will not renormalize provided that a ``generalized Adler-Bardeen theorem'' holds. We further show that recent arguments connecting the non-renormalizability of the above mentioned correlation functions to a bonus U(1)_Y symmetry are incomplete due to possible U(1)_Y violating contact terms. The tree level contribution to the contact terms may be set to zero by considering appropriately normalized operators. Non-renormalizability of the above mentioned correlation functions, however, will follow only if these contact terms saturate by free fields.