Abstract
We construct renormalization group equations for the effective potential in the leading logarithmic approximation. These equations are valid for arbitrary \(SO(N)\) -symmetric scalar field theories, including nonrenormalizable ones, in four dimensions on a curved background with nonminimal coupling. The solutions to these equations represent the sum of leading logarithms in all orders of perturbation theory for two contributions to the effective potential: one that is linear in curvature and another that corresponds to the effective potential on a flat background. In the renormalizable case, the obtained equations reduce to the standard renormalization group equations. Finally, we consider possible cosmological applications of the solutions to the renormalization group equations.
