The fate of non-polynomial interactions in scalar field theory

TL;DR

This paper provides an exact renormalization group analysis showing that non-polynomial interactions in scalar field theory do not generate new universality classes and can be re-expressed in terms of polynomial couplings, clarifying their role.

Contribution

It proves non-polynomial eigen-perturbations do not lead to new interactions or universality classes in scalar field theory at the fixed point.

Findings

Non-polynomial eigen-perturbations do not enlarge the universality class.

These perturbations can be re-expressed in terms of polynomial couplings.

The RG flow of these perturbations does not originate from the fixed point.

Abstract

We present an exact RG (renormalization group) 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 non-polynomial eigen-perturbations found in the LPA (local potential approximation) at the linearised level, do not lead to new interactions, \textit{i.e.} enlarge the universality class, neither in the LPA or treated exactly. Non-perturbatively, their RG flow does not emanate from the fixed point. For the equivalent Wilsonian effective action they can be re-expressed 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 behaviour is fixed by these perturbations.