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.Comment: 14 pages; v2: improved discussion and corrected several statements
thanks to referee, to appear in pr