Outer automorphisms are sufficient conditions for RG fixed points

We point out that the existence of an outer automorphism (Out) is a sufficient condition for the existence of a fixed hyperplane (fixed point, separatrix) in the renormalization group (RG) flow of a Quantum Field Theory (QFT). The corresponding RG fixed hyperplane is determined by a symmetry argument and can be computed without resorting to perturbation theory. This provides the mathematical underpinning of 't Hooft’s technical naturalness argument, and results in a systematic way to derive non-perturbative all-order constraints on the RG beta functions. If an Out exists, the symmetry of the fully coupled system of beta functions is larger than the symmetry of the action. We also stress the importance of including goofy transformations in these considerations.