Generalities

The phenomenon of anomaly plays an important role in quantum field theory: in many cases it determines whether or not a theory is self-consistent and can be realized in the physical world and, therefore, allows one to select the acceptable physical theories. In a given theory anomalies are often related to the appearance of new quantum numbers (topological quantum numbers; see Chapter 4), result in the emergence of mass scales, determine the spectrum of physical states. So, despite its name, anomalies are a normal and significant attribute of any quantum field theory.

The term “anomaly” has the following meaning: Let the classical action of the theory obey some symmetry, i.e. let it be invariant under certain transformations. If this symmetry is violated by quantum corrections, such a phenomenon is called an “anomaly.” (Reviews of anomalies are given in [1]–[4].) There are two types of anomalies – internal and external. In the first case, the gauge invariance of the classical Lagrangian is destroyed at the quantum level. The theory becomes nonrenormalizable and cannot be considered as a self-consistent theory. The standard method to solve this problem is a special choice of fields in the Lagrangian in such a way as to cancel all internal anomalies. (Such an approach is used in the standard model of electroweak interaction – the Glashow–Illiopoulos–Maiani mechanism.) An external anomaly corresponds to violation of symmetry of interaction with external sources, not related to gauge symmetry of the theory. Such anomalies arise in QCD and are considered below. There are two kinds of anomalies in QCD: axial (chiral) anomaly and scale anomaly.