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

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

1998-05-07

arxiv.org/abs/hep-th/9805036v1

Phys.Rev.D58:025014,1998

Physics

High Energy Physics

High Energy Physics - Theory

23 pages, Latex

Scientific paper

10.1103/PhysRevD.58.025014

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.

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