Anomalies in Ward Identities for Three-Point Functions Revisited

TL;DR

This paper revisits the calculation of Ward identities in three-point functions for a free fermion model, proposing a regulator-independent method that clarifies ambiguities and supports consistent anomaly analysis.

Contribution

It introduces a general, regulator-independent approach to analyze divergences in Ward identities, clarifying ambiguities and providing conditions for anomaly consistency.

Findings

Divergences are expressed in five basic objects independent of regulators.

Ambiguities are linked to only three combinations of divergent objects.

The approach aligns with Dimensional Regularization and Gertsein-Jackiw methods.

Abstract

A general calculational method is applied to investigate symmetry relations among divergent amplitudes in a free fermion model. A very traditional work on this subject is revisited. A systematic study of one, two and three point functions associated to scalar, pseudoscalar, vector and axial-vector densities is performed. The divergent content of the amplitudes are left in terms of five basic objects (external momentum independent). No specific assumptions about a regulator is adopted in the calculations. All ambiguities and symmetry violating terms are shown to be associated with only three combinations of the basic divergent objects. Our final results can be mapped in the corresponding Dimensional Regularization calculations (in cases where this technique could be applied) or in those of Gertsein and Jackiw which we will show in detail. The results emerging from our general approach allow us to extract, in a natural way, a set of reasonable conditions (e.g. crucial for QED consistency) that could lead us to obtain all Ward Identities satisfied. Consequently, we conclude that the traditional approach used to justify the famous triangular anomalies in perturbative calculations could be questionable. An alternative point of view, dismissed of ambiguities, which lead to a correct description of the associated phenomenology, is pointed out.