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

TL;DR

This paper demonstrates that low-energy scattering in (2+1) dimensions exhibits universal behavior with the S-wave phase shift vanishing logarithmically or quadratically as momentum approaches zero, supported by multiple theoretical frameworks.

Contribution

It establishes the universality of low-energy scattering behavior in (2+1) dimensions across different theoretical approaches, including Schrödinger equation, field theory, and perturbation theory.

Findings

The phase shift $\, o rac{ ext{constant}}{\, ext{log}(k/m)}$ as $k o 0$

The phase shift $\, o O(k^2)$ as $k o 0$

Perturbative amplitudes diverge like $( ext{log} k)^n$, but the full amplitude vanishes as $( ext{log} k)^{-1}$.

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 $\delta_0 \to {c\over \ln (k/m)} or \delta_0 \to O(k^2)$ as $k\to 0$. The constant $c$ is universal and $c=\pi/2$. This result is established first in the framework of the Schr\"odinger 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 $\phi_3^4$ 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.