Scaling and Chiral Ward Identities and the Axial-Vector Current Anomaly

We combine chiral and "scaling" Ward identities in an analysis of the axial-vector current anomaly. This leads to a proof of the Adler-Bardeen theorem. The "scaling" Ward identities we used may be properly called "response equations"; they are slight generalizations of the Callan-Symanzik equation. Our discussion is given for a class of theories which includes the $\sigma$ model and massive vector-boson theory.