Relationship between Fujikawa's Method and the Background Field Method for 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.

9 pages (this version is the published version)

References in corpus (5)

• Effective Action, Conformal Anomaly and the Issue of Quadratic Divergences
• Anomalous Commutator Algebra for Conformal Quantum Mechanics
• Path-Integral Fujikawa's Approach to Anomalous Virial Theorems and Equations of State for Systems with $SO(2,1)$ Symmetry
• Path-Integral Derivation of the Non-relativistic Scale Anomaly
• Virial Theorem for Non-relativistic Quantum Fields in D Spatial Dimensions

Cited by in corpus (3)

• Virial expansion for the Tan contact and Beth-Uhlenbeck formula from 2D SO(2,1) anomalies
• Bose and Fermi Statistics and the Regularization of the Nonrelativistic Jacobian for the Scale Anomaly
• Quantum anomaly and thermodynamics of one-dimensional fermions with antisymmetric two-body interactions