Universality of low-energy scattering in 2+1 dimensions

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, {delta}{sub 0}(k), k being the c.m. momentum, vanishes as either {delta}{sub 0}{r_arrow}c/ln(k/m)or {delta}{sub 0}{r_arrow}O(k{sup 2}) as k{r_arrow}0. The constant c is universal and c={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 {delta}{sub 0}(k) is O(k{sup 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 {phi}{sub 3}{sup 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 (lnthinspk){sup n} as k{r_arrow}0, while the full amplitude vanishes as (lnk){sup {minus}1}. We show how these two facts can be reconciled. {copyright} {ital 1998} {ital The American Physical Society}