Relationship between Fujikawa's Method and the Background Field Method for the Scale Anomaly

TL;DR

This paper demonstrates the equivalence between Fujikawa's method and the background field method in calculating the scale anomaly for an $O(N)$ scalar field theory, clarifying their conceptual relationship.

Contribution

It establishes a formal connection between two different approaches to scale anomaly calculation, linking Fujikawa's method with the diagrammatic background field approach.

Findings

Fujikawa's method corresponds to vacuum diagrams in 1-loop expansion.

Anomaly causes breakdown of Ward identities and Noether's theorem.

The two methods are equivalent in describing the scale anomaly.

Abstract

We show the equivalence between Fujikawa's method for calculating the scale anomaly and the diagrammatic approach to calculating the effective potential via the background field method, for an $O(N)$ symmetric scalar field theory. Fujikawa's method leads to a sum of terms, each one superficially in one-to-one correspondence with a vacuum diagram of the 1-loop expansion. From the viewpoint of the classical action, the anomaly results in a breakdown of the Ward identities due to a scale-dependence of the couplings, whereas in terms of the effective action, the anomaly is the result of the breakdown of Noether's theorem due to explicit symmetry breaking terms of the effective potential.