Universality of low-energy scattering in 2+1 dimensions

Abstract

For any relativistic quantum field theory in 2ϩ1 dimensions, with no zero mass particles, and satisfying the standard axioms, we establish a remarkable low-energy theorem. The S-wave phase shift, ␦ 0 (k), k being the c.m. momentum, vanishes as eitherThe constant c is universal and c ϭ/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 ␦ 0 (k) is O(k 2 ) near kϭ0. We treat the potential scattering case with full generality and rigor, and explicitly define the exceptional class. Finally, we look at perturbation theory in 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) Ϫ1 . We show how these two facts can be reconciled.

The Virial Expansion of a Dilute Bose Gas in Two Dimensions

“…The Virial expansion developed in this paper is complementary. As the quasi-two dimensional gas of trapped atoms is also experimentally feasible, the result reported in this paper may be brought to a direct comparison with the measurements As is well-known, a perturbative treatment of a dilute Bose gas in two dimensions suffer from two difficulties: 1) The scattering amplitude vanishes in the zero energy limit and the Born expansion breaks down for a large number of potentials [10] [11].…”

“…(19) was proved rigorously by Chan, Khuri, Martin and Wu [11] for a general class of potentials that fall off faster than 1 r 2 ln r for r → ∞ . They also proved that for the same class of potentials, the correction to the corresponding function f 0 (r) of is of the order of k 2 .…”

“…This way the s-wave channel dominant over all other partial wave channels to all orders of a low energy expansion, different from three dimensions. The universality found in [11] is highlighted in the low temperature thermodynamics of the 2D system. In what follows, we shall suppress the subscript "0" for the s-wave and introduce a running coupling constant at the scale of the thermal wavelength,…”

The absolute definition of the phase-shift in potential scattering

“…This corresponds to the S-wave Schrödinger equation in two space dimensions, and is interesting to study 10 .…”

Electron-electron bound states in Maxwell-Chern-Simons-Proca QED $\scriptstyle \mathsf {}$ 3

“…The Kato condition [2] establishes the finiteness of the number of bound states, in D = 1 + 2, associated to a certain potential V, and can be used as a criterion for determining the character confining or condensating of the potential. The fact the logarithmic potential to be confining (according to the Kato criterion) indicates it does not lead to bound states, becoming clear the need of a finite range, screened interaction.…”