Universality of low-energy scattering in 2+1 dimensions

Abstract

For any relativistic quantum field theory in 2+1 dimensions, with no zero mass particles, and satisfying the standard axioms, we establish a remarkable low-energy theorem. The $S$-wave phase shift, ${\ensuremath{\delta}}_{0}(k),$ $k$ being the c.m. momentum, vanishes as either ${\ensuremath{\delta}}_{0}\ensuremath{\rightarrow}c/\mathrm{ln}(k/m)$or ${\ensuremath{\delta}}_{0}\ensuremath{\rightarrow}{O(k}^{2})$ as $\stackrel{\ensuremath{\rightarrow}}{k}0.$ The constant $c$ is universal and $c=\ensuremath{\pi}/2.$ This result follows only from the rigorously established analyticity and unitarity properties for 2-particle scattering. This kind of universality was first noted in non-relativistic potential scattering, albeit with an incomplete proof which missed, among other things, an exceptional class of potentials where ${\ensuremath{\delta}}_{0}(k)$ is ${O(k}^{2})$ near $k=0.$ We treat the potential scattering case with full generality and rigor, and explicitly define the exceptional class. Finally, we look at perturbation theory in ${\ensuremath{\varphi}}_{3}^{4}$ and study its relation to our non-perturbative result. The remarkable fact here is that in $n$-th order the perturbative amplitude diverges like $(\mathrm{ln}{k)}^{n}$ as $\stackrel{\ensuremath{\rightarrow}}{k}0,$ while the full amplitude vanishes as $(\mathrm{ln}{k)}^{\ensuremath{-}1}.$ We show how these two facts can be reconciled.