The local potential approximation in quantum gravity

Within the context of the functional renormalization group flow of gravity, we suggest that a generic f(R) ansatz (i.e. not truncated to any specific form, polynomial or not) for the effective action plays a role analogous to the local potential approximation (LPA) in scalar field theory. In the same spirit of the LPA, we derive and study an ordinary differential equation for f(R) to be satisfied by a fixed point of the renormalization group flow. As a first step in trying to assess the existence of global solutions (i.e. true fixed point) for such equation, we investigate here the properties of its solutions by a comparison of various series expansions and numerical integrations. In particular, we study the analyticity conditions required because of the presence of fixed singularities in the equation, and we develop an expansion of the solutions for large R up to order N = 29. Studying the convergence of the fixed points of the truncated solutions with respect to N, we find a characteristic pattern for the location of the fixed points in the complex plane, with one point stemming out for its stability. Finally, we establish that if a non-Gaussian fixed point exists within the full f(R) approximation, it corresponds to an R² theory.