Conformal symmetry and the chiral anomaly

Two-loop contributions to the anomalous correlation function $〈J_{\mu }{\mathrm{}\left(x\right)J}_{\nu }{\mathrm{}\left(y\right)J}_{\rho }\mathrm{}\left(z\right)\mathrm{}〉$ of three chiral currents are calculated by a method based on the conformal properties of massless field theories. The method was previously applied to virtual photon diagrams in quantum electrodynamics, and it is extended here to diagrams with scalars and chiral spinors in the Abelian Higgs model and in the $\mathrm{SU}\mathrm{}\left(3\right)\mathrm{}\times \mathrm{SU}\mathrm{}\left(2\right)\mathrm{}\times \mathrm{U}\mathrm{}\left(1\right)\mathrm{}$ standard model. In each case there are nonvanishing contributions to the gauge current correlator from self-energy insertions, vertex insertions, and nonplanar diagrams, but their sum exactly vanishes. The two-loop contribution to the anomaly, therefore, also vanishes, in agreement with the Adler-Bardeen theorem. An application of the method to the correlator $〈R_{\mu }{\mathrm{}\left(x\right)R}_{\nu }{\mathrm{}\left(y\right)K}_{\rho }\mathrm{}\left(z\right)\mathrm{}〉$ of the $R$ and Konishi axial vector currents in supersymmetric gauge theories which was reported by Anselmi et al. is discussed here. The net two-loop contribution to this correlator also vanishes.