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, δ₀(k), k being the c.m. momentum, vanishes as either δ₀→c/ln(k/m)or δ₀→O(k²) as →k0. The constant c is universal and c=π/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 δ₀(k) is O(k²) 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 φ₃⁴ and study its relation to our non-perturbative result. The remarkable fact here is that in n-th order the perturbative amplitude diverges like (lnk)ⁿ as →k0, while the full amplitude vanishes as (lnk)^-1. We show how these two facts can be reconciled.