On Truncations of the Exact Renormalization Group

TL;DR

This paper studies the behavior of truncations in the Exact Renormalization Group for scalar field theories, revealing convergence issues and the presence of spurious solutions in approximations of the Wilson fixed point.

Contribution

It systematically analyzes the effects of truncating the ERG equations and explains the convergence and spurious solutions through the analytic properties of untruncated solutions.

Findings

Truncations initially converge to the untruncated solution

Beyond a certain order, truncations cease to converge

Spurious solutions are prevalent and hard to reject

Abstract

We investigate the Exact Renormalization Group (ERG) description of ($Z_2$ invariant) one-component scalar field theory, in the approximation in which all momentum dependence is discarded in the effective vertices. In this context we show how one can perform a systematic search for non-perturbative continuum limits without making any assumption about the form of the lagrangian. Concentrating on the non-perturbative three dimensional Wilson fixed point, we then show that the sequence of truncations $n=2,3,\dots$, obtained by expanding about the field $\varphi=0$ and discarding all powers $\varphi^{2n+2}$ and higher, yields solutions that at first converge to the answer obtained without truncation, but then cease to further converge beyond a certain point. No completely reliable method exists to reject the many spurious solutions that are also found. These properties are explained in terms of the analytic behaviour of the untruncated solutions -- which we describe in some detail.