Universality of low-energy scattering in (2+1) dimensions

Abstract

We prove that, in (2+1) dimensions, the S-wave phase shift, ${\delta }_0(k)$, k being the c.m. momentum, vanishes as either ${\delta }_0\to \frac{c}{\ln (k/m)}or{\delta }_0\to O(k^2)$ as $k\to 0$. The constant $c$ is universal and $c=\pi /2$. This result is established first in the framework of the Schrödinger equation for a large class of potentials, second for a massive field theory from proved analyticity and unitarity, and, finally, we look at perturbation theory in ${\varphi }_3^4$ and study its relation to our non-perturbative result. The remarkable fact here is that in n-th order the perturbative amplitude diverges like $(\ln k{)}^n$ as $k\to 0$, while the full amplitude vanishes as $(\ln k{)}^{-1}$. We show how these two facts can be reconciled.

23 pages, Latex