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(0)(k), k being the c.m. momentum, vanishes as either delta(0) --> c/ln(k/m)or delta(0) --> O(k(2)) as k --> 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(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 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 --> 0 while the full amplitude vanishes as (lnk)(-1). We show how these two facts can be reconciled. [S0556-2821(98)02714-3].