Nonrelativistic Limit of the Scalar Chern-Simons Theory and the Aharonov-Bohm Scattering

TL;DR

This paper investigates the nonrelativistic limit of a scalar Chern-Simons theory, analyzing scattering amplitudes and identifying a critical self-interaction parameter where the scattering reduces to the Aharonov-Bohm effect, including relativistic corrections.

Contribution

It introduces a nonrelativistic reduction scheme and effective Lagrangian for the scalar Chern-Simons theory, highlighting a critical self-interaction value affecting scattering behavior.

Findings

Existence of a critical self-interaction parameter for Aharonov-Bohm scattering

Calculation of two-body scattering amplitude up to order p^2/m^2

Relativistic corrections to the Aharonov-Bohm scattering identified

Abstract

We study the nonrelativistic limit of the quantum theory of a Chern-Simons field minimally coupled to a scalar field with quartic self-interaction. The renormalization of the relativistic model, in the Coulomb gauge, is discussed. We employ a procedure to calculate scattering amplitudes for low momenta that generates their $|p|/m$ expansion and separates the contributions coming from high and low energy intermediary states. The two body scattering amplitude is calculated up to order $p^2/m^2$. It is shown that the existence of a critical value of the self-interaction parameter for which the 2-particle scattering amplitude reduces to the Aharonov-Bohm one is a strictly nonrelativistic feature. The subdominant terms correspond to relativistic corrections to the Aharonov-Bohm scattering. A nonrelativistic reduction scheme and an effective nonrelativistic Lagrangian to account for the relativistic corrections are proposed.