  Two-loop contributions to the anomalous correlation function
<J_mu(x)J_nu(y)J_rho(z)> 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 SU(3)xSU(2)xU(1) 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(x)R_nu(y)K_rho(z)> of the R and Konishi axial currents in
supersymmetric gauge theories which was reported in hep-th/9608125 is discussed
here. The net two-loop contribution to this correlator also vanishes.
