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.Comment: 9 pages (this version is the published version
