Consequences of gauging the Weyl symmetry and the two-dimensional conformal anomaly

We discuss the generalization of the local renormalization group approach to theories in which Weyl symmetry is gauged. These theories naturally correspond to scale-invariant - rather than conformal-invariant - models in the flat-space limit. We argue that this generalization can be of use when discussing the issue of scale vs conformal invariance in quantum and statistical field theories. The application of Wess-Zumino consistency conditions constrains the form of the Weyl anomaly and the beta functions in a nonperturbative way. In this work, we concentrate on two-dimensional models including also the contributions of the boundary. Our findings suggest that the renormalization group flow between scale-invariant theories differs from the one between conformal theories because of the presence of a new charge that appears in the anomaly. It does not seem to be possible to find a general scheme for which the new charge is zero, unless the theory is conformal in flat space. Two illustrative examples involving flat space's conformal- and scale-invariant models that do not allow for a naive application of the standard local treatment are given.