Anomalies and symmetries of the regularized action

TL;DR

This paper explores the symmetries of the regularized scalar field action in 1+1 dimensions, revealing a non-local symmetry that extends Weyl invariance and offers a new perspective on the conformal anomaly.

Contribution

It introduces a non-local symmetry of the regularized action that generalizes Weyl symmetry, providing a novel interpretation of the conformal anomaly via the Jacobian.

Findings

The regularized action exhibits a non-local symmetry involving the regulator field.

The Jacobian of this symmetry accounts for the conformal anomaly.

In the limit of infinite regulator mass, the symmetry reduces to standard Weyl transformations.

Abstract

We show that the Pauli-Villars regularized action for a scalar field in a gravitational background in 1+1 dimensions has, for any value of the cutoff M, a symmetry which involves non-local transformations of the regulator field plus (local) Weyl transformations of the metric tensor. These transformations, an extension to the regularized action of the usual Weyl symmetry transformations of the classical action, lead to a new interpretation of the conformal anomaly in terms of the (non-anomalous) Jacobian for this symmetry. Moreover, the Jacobian is automatically regularized, and yields the correct result when the masses of the regulators tend to infinity. In this limit the transformations, which are non-local in a scale of 1/M, become the usual Weyl transformation of the metric. We also present the example of the chiral anomaly in 1+1 dimensions.