Nonrenormalization Theorems

Abstract

In Chap. 4, we have seen that the BRS invariance of gauge theories may be broken at the quantum level by the gauge anomaly, which would render the theory inconsistent. However our purely algebraic approach, based on the cohomology in the space of local field polynomials, gives only a potential anomaly. It does not yield the value of its coefficient r (c.f. (4.55)), which is needed in order to know if the anomaly is actually present. This coefficient r is a power series in the coupling constants or in ħ, and a priori receives contributions from Feynman diagrams of all orders. But a remarkable nonrenormalization theorem due to Adler and Bardeen [118] states that the gauge anomaly is definitively absent if the coefficient r vanishes at the first order in ħ. This means that, for checking the consistency of a physical model, it is sufficient to examine the one-loop contribution to the anomaly.