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

Chris L Lin,Carlos R Ordóñez

doi:10.1155/2016/2809290

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 anO(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 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.

Highlights

Fujikawa showed that, within the path integral formalism, all anomalies are the result of noninvariance of the measure under symmetry transformations [1,2,3]
It is known that the quantum effective action preserves the symmetries of the classical action, provided that the measure is invariant under the symmetry transformations [4]
There should be a relationship between Fujikawa’s method and the noninvariant terms of the quantum effective action. We investigate this relationship in the context of O(N), λφ4 theory, by comparing, term by term, the Taylor expansion of the Fujikawa determinant with all diagrams in the 1-loop expansion of the quantum effective potential

Summary

Introduction

Within the path integral formalism, all anomalies are the result of noninvariance of the measure under symmetry transformations [1,2,3]. In this paper we attempt to explore the connection between certain terms in the effective potential when it is expanded by the number of vertices and certain terms in the Jacobian of Fujikawa’s method when it is Taylor expanded, thereby clarifying the statement that putting the quadratic part of the effective action in the regulating function captures the 1-loop effects. In the sixth section, we apply Noether’s theorem to the effective action and compare it to anomalous scale-breaking of the classical action

Fujikawa’s Method

Field Method

Fujikawa Calculation

Equivalence of Fujikawa with Background Field Calculation

Noether’s Theorem and Dimensional Transmutation

Conclusion