Asymptotic Freedom of Elastic Strings and Barriers

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 $\phi_{\rm max}$.