A non-renormalization theorem for 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.