An Introduction to Anomalies

Abstract

These lectures are dedicated to the study of the recent progress and implications of anomalies in quantum field theory. In this introduction we would like to recapitulate some of the highlights in the history of the subject. In its original form,1 one considers a triangle diagram with two vector currents and an axial vector current. Requiring Bose symmetry and vector current conservation in the vector channels, one finds that the axial vector current is not conserved, leading to the breakdown of chiral symmetry in the presence of external gauge fields. The existence of this anomaly led to an understanding of π0 decay, and later on to the resolution of the U(1) problem2 in QCD. These anomalies correspond to the breakdown of global axial symmetries, and their existence does not jeopardize unitarity or renormalizability. More dangerous anomalies appear whenever chiral currents are coupled to gauge fields. For example, in four dimensions we can consider V-A currents coupled to gauge fields as in the standard Weinberg-Salam model, and compute the same triangle diagram with V-A currents on each vertex.3 Again one finds an anomaly, and unless the anomaly cancels after summing over all the fermion species, the theory will not be gauge invariant, implying a loss of renormalizability. If we recall the Feynman rules for non-Abelian gauge theories coupled to fermions in some representation Ta of the gauge group G, the anomaly for gauge currents is proportional to a purely group theoretic factor times a Feynman integral.