A non-renormalization theorem for conformal anomalies

TL;DR

The paper proves a non-renormalization theorem for conformal anomaly coefficients associated with operators of zero anomalous dimension and validates it by explicit 2-point function calculations in both conformal field theory and its AdS/CFT dual. It argues that, under SUSY, the ratio of anomaly coefficients to corresponding 2- and 3-point normalizations remains invariant under renormalization, provided a generalized Adler-Bardeen saturation holds for the R-current. By performing detailed CFT and AdS computations of the 2-point anomaly, the work demonstrates agreement between weak and strong coupling descriptions, supporting the non-renormalization idea. It also scrutinizes the role of a bonus U(1)_Y symmetry in N=4 SYM, showing that contact terms can invalidate naive symmetry-based arguments unless saturated by free-field contributions. The results have implications for the non-renormalization of chiral primary correlators in N=4 SYM and offer a framework for testing AdS/CFT further through conformal anomalies.

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.