Renormalization group flow in scalar-tensor theories: I

We study the renormalization group flow in a class of scalar-tensor theories involving at most two derivatives of the fields. We show in general that minimal coupling is self consistent, in the sense that when the scalar self couplings are switched off, their beta functions also vanish. Complete, explicit beta functions that could be applied to a variety of cosmological models are given in a five parameter truncation of the theory in $d=4$. In any dimension $d>2$ we find that the flow has only a "Gaussian Matter" fixed point, where all scalar self interactions vanish but Newton's constant and the cosmological constant are nontrivial. The properties of these fixed points can be studied algebraically to some extent. In $d=3$ we also find a gravitationally dressed version of the Wilson-Fisher fixed point, but it seems to have unphysical properties. These findings are in accordance with the hypothesis that these theories are asymptotically safe.