Ultraviolet and infrared aspects of the axial anomaly I

This is the first part of a brief review of some perturbative aspects of the Adler-Bell-Jackiw axial anomaly, described in terms of ultraviolet and infrared behaviour of the famous VVA triangle graph. Apart from a general overview of the diversified role played by the anomaly in quantum field theory and particle physics, we present here an elementary introduction to the subject of the anomaly, which hopefully should be comprehensible to an uninitiated reader with only a basic background in quantum field theory. In this article we stress the ultraviolet aspects of the anomaly and the topics covered here are the following: Vector and axial-vector Ward identities for the VVA triangle graph, the anomaly and several ways to derive it, namely the symmetric momentum cut-off and shifting the integration variables in linearly divergent integrals, the Adler-Rosenberg argument, the Pauli-Villars method and dimensional regularization.