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

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