Some conservation laws, to be valid in classical theory, cease to be fulfilled in quantum field theory (QFT) at the right inclusion of quantum effects. These phenomena are called anomalies. Their origin is connected with ultraviolet divergences in QFT. Even in renormalizable theories, the presence of divergences causes us to apply regularization or cut-off in real calculations. In some cases, regularization is impossibly carried out so that all symmetry demands of the initial classical field theory will be simultaneously satisfied. Moreover, the regularization used may violate the initial symmetry and even after it will be removed in the end of calculations, violation tracks may be left. As a case in point, we consider QED. Here, we have the vector current conservation law:

∂xµ Jµ(x) = 0, (6.1)

where Jµ(x) = eψ(x)γµψ(x). Along with this current, one may examine the axial-vector current: Jµ5(x) = eψ(x)γµγ5ψ(x). Thanks to the Dirac equation, the divergence of the axial-vector current is given by the expression

∂xµ Jµ5(x) = 2iemψ(x)γ5ψ(x) ≡ 2mJ5(x), (6.2)

where a quantity J5 is called a chiral density. From this equation it follows that, in the limit of a zero electron mass, the axial-vector current is conserved. This represents the consequence of chiral symmetry. However, the more accurate consideration demonstrates that this conclusion fails. Really, in the definition of the axial-vector current, the product of anticommuting operators ψ(x) and ψ(x) taken in one and the same point x is contained. Such a product needs extension of the definition (regularization). If the regularization is carried out so that the vector current conservation law is not violated,∗ then it turns out that the right expression for the divergence of the axial-vector current becomes the view:

∂xµ Jµ5(x) = 2mJ5(x)− e

16pi2 µνλσF

µν(x)Fλσ(x), (6.3)

Particle and

where Fµν is the electromagnetic field tensor. Thus, the axial-vector current is not conserved even in the limit of a massless electron. This phenomenon is named an axial anomaly. It was discovered by Schwinger in 1951 [56] under analysis of the decay pi0 → γγ. In 1969 Bell and Jackiw [57] found out that violation of the chiral symmetry is a source of this anomaly. The violation, in its turn, is caused by introducing an regularization that is needed to obtain consequences resulting from the neutral axial-vector current conservation for one-loop Feynman diagrams. In gauge theories Ward identities are associated with current conservation laws. Therefore, in theories with fermions, axial Ward identities are not automatically true. In QED chiral anomalies were discovered by Adler [58] under the investigation of axial-vector Ward identities. Afterwards, these anomalies are called AdlerBell-Jackiw (ABJ) ones. The appearance of the ABJ anomalies happen because particular one-loop diagrams give anomalous terms to prevent the reproduction of Ward identities in higher orders of the perturbation theory. In QED and QCD, the vector interaction of the type

gV JµW µ,

whereWµ is a gauge field, Jµ ∼ ψγµψ is a vector current of Fermi matter fields and internal symmetry indices have been omitted, is the unique type of interaction between matter and gauge fields. Just due to the vector current conservation law, the Ward identity for a vertex function is fulfilled. Thus, for example, the graph shown in Fig. 6.1 is associated with the amplitude

A =< p′|gV JµWµ|p > and we safely get

(p′ − p)µJµ = qµJµ = 0. However, in the presence of an axial-vector coupling between matter and gauge fields, as


The expansion of the vector coupling between the gauge field and matter fields with an accuracy of e3.