Fate of nonpolynomial interactions in scalar field theory

Abstract

We present an exact renormalization group (RG) analysis of $O(N)$-invariant scalar field theory about the Gaussian fixed point. We prove a series of statements that, taken together, show that the nonpolynomial eigenperturbations found in the local potential approximation at the linearized level do not lead to new interactions, i.e. enlarge the universality class, in the local potential approximation nor treated exactly. Nonperturbatively, their RG flow does not emanate from the fixed point. For the equivalent Wilsonian effective action, they can be reexpressed in terms of the usual couplings to polynomial interactions, which can furthermore be tuned to be as small as desired for all finite RG time. For the infrared cutoff Legendre effective action, this can also be done for the infrared evolution. We explain why this is nevertheless consistent with the fact that the large field behavior is fixed by these perturbations.