Exact Renormalization Group in Large $N$

TL;DR

This paper uses the exact renormalization group to analyze the four-dimensional O(N) linear sigma model at large N, revealing issues with the continuum limit and how a finite cutoff resolves them, with explicit formulas involving Lambert W.

Contribution

It provides a detailed application of the exact renormalization group to the large N sigma model, including explicit effective potential formulas and insights into cutoff dependence.

Findings

Naive continuum limit leads to tachyon and unbounded potential.

Restoring a finite cutoff resolves instabilities.

Explicit effective potential expressed via Lambert W function.

Abstract

We apply the exact renormalization group formalism to compute the effective action and potential of the four dimensional O$(N)$ linear sigma model in large $N$. With a finite momentum cutoff in place, the model is well defined. In the naive continuum limit where the cutoff is taken to infinity, the effective action suffers from a tachyon, and the effective potential is unbounded from below and develops a negative imaginary part. These problems disappear once a small enough cutoff is restored. The effective potential of the naive continuum limit, obtained by analytic continuation, is given explicitly in terms of the Lambert $W$ function.