Two dimensional non-linear sigma models as a limit of the linear sigma models

TL;DR

This paper demonstrates how the two-dimensional O(N) non-linear sigma model can be derived as a strong coupling limit of the linear sigma model, establishing a precise relation between the two in the context of renormalized theories.

Contribution

It provides a detailed method to obtain the non-linear sigma model from the linear model in two dimensions, clarifying the coupling dependence needed for finiteness.

Findings

The non-linear sigma model emerges as a strong coupling limit of the linear model.

A specific coupling dependence of the squared mass parameter ensures finite physical mass.

The relation applies to renormalized theories, not just regularized ones.

Abstract

We show how to obtain the O(N) non-linear sigma model in two dimensions as a strong coupling limit of the corresponding linear sigma model. In taking the strong coupling limit, the squared mass parameter must be given a specific coupling dependence that assures the finiteness of the physical mass scale. The relation discussed in this paper, which applies to the renormalized theories as opposed to the regularized theories, is an example of a general relation between the linear and non-linear models in two and three dimensions.