Quantum corrections to the effective potential in nonrenormalizable theories

For the effective potential in the leading logarithmic approximation, we construct a renormalization group equation that holds for arbitrary scalar field theories, including nonrenormalizable ones, in four dimensions. This equation reduces to the usual renormalization group equation with a one-loop beta-function in the renormalizable case. The solution of this equation sums up the leading logarithmic contributions in the field in all orders of the perturbation theory. This is a nonlinear second-order partial differential equation in general, but it can be reduced to an ordinary one in some cases. In specific examples, we propose a numerical solution of this equation and construct the effective potential in the leading logarithmic approximation. We consider two examples as an illustration: a power-law potential and a cosmological potential of the $\operatorname{tan}^2\phi$ type. The obtained equation in physically interesting cases opens up the possibility of studying the properties of the effective potential, the presence of additional minima, spontaneous symmetry breaking, stability of the ground state, etc.