Asymptotic Freedom of Elastic Strings and Barriers

TL;DR

This paper demonstrates that a quantized elastic string constrained by an impenetrable wall exhibits asymptotic freedom, with the mass gap behavior confirmed through nonperturbative, perturbative, and 1/N expansion methods, aligning with the O(N) sigma model.

Contribution

It provides a nonperturbative proof of asymptotic freedom for elastic strings with barriers, clarifying disputed scaling behavior and connecting to the O(N) sigma model in the large N limit.

Findings

Theories are asymptotically free for N ≥ 1.

Perturbation theory and 1/N expansion confirm asymptotic freedom.

Large N limit matches the O(N) nonlinear sigma model.

Abstract

We study the problem of a quantized elastic string in the presence of an impenetrable wall. This is a two-dimensional field theory of an N-component real scalar field $\phi$ which becomes interacting through the restriction that the magnitude of $\phi$ is less than $\phi_{\rm max}$, for a spherical wall of radius $\phi_{\rm max}$. The N=1 case is a string vibrating in a plane between two straight walls. We review a simple nonperturbative argument that there is a gap in the spectrum, with asymptotically-free behavior in the coupling (which is the reciprocal of $\phi_{\rm max}$) for N greater than or equal to one. This scaling behavior of the mass gap has been disputed in some of the recent literature. We find, however, that perturbation theory and the 1/N expansion each confirms that these theories are asymptotically free. The large N limit coincides with that of the O(N) nonlinear sigma model. A theta parameter exists for the N=2 model, which describes a string confined to the interior of a cylinder of radius $\phi_{\rm max}$.