Renormalization group flow of Weyl invariant dilaton gravity

Any theory can be made Weyl invariant by introducing a dilaton. It is shown here how to construct renormalization group equations for gravity that maintain this property. Explicit calculations are given only in the simplest approximation, namely for the one-loop beta functions of a dilaton conformally coupled to a dynamical metric, but the results have wider validity. This formalism could be used to define the meaning of a theory with a position-dependent cutoff: it is equivalent to a theory with a constant cutoff but a conformally related metric.