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

We prove that, in (2+1) dimensions, the S-wave phase shift, $ δ_0(k)$, k being the c.m. momentum, vanishes as either $δ_0 \to {c\over \ln (k/m)} or δ_0 \to O(k^2)$ as $k\to 0$. The constant $c$ is universal and $c=π/2$. 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 $ϕ_3^4$ 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.