Brans-Dicke theory in the local potential approximation

We study the Brans-Dicke theory with arbitrary potential within a functional renormalization group framework. Motivated by the asymptotic safety scenario of quantum gravity and by the well-known relation between f(R) gravity and Brans-Dicke theory at the classical level, we concentrate our analysis on the fixed-point equation for the potential in four dimensions and with Brans-Dicke parameter . For two different choices of gauge, we study the resulting equations by examining both the local and global properties of the solutions, by means of analytical and numerical methods. As a result of our analysis we do not find any nontrivial fixed point in one gauge, but we find a continuum of fixed points in the other one. We interpret such an inconsistency as a result of the restriction to , and thus we suggest that it indicates a failure of the equivalence between f(R) gravity and Brans-Dicke theory at the quantum level.