Differential Regularization of a Non-relativistic Anyon Model

Abstract

Differential regularization is applied to a field theory of a non-relativistic charged boson field $ϕ$ with $λ(ϕ{}^{*} ϕ)^2$ self-interaction and coupling to a statistics-changing $U(1)$ Chern-Simons gauge field. Renormalized configuration-space amplitudes for all diagrams contributing to the $ϕ{}^{*} ϕ{}^{*} ϕϕ$ 4-point function, which is the only primitively divergent Green's function, are obtained up to 3-loop order. The renormalization group equations are explicitly checked, and the scheme dependence of the $β$-function is investigated. If the renormalization scheme is fixed to agree with a previous 1-loop calculation, the 2- and 3-loop contributions to $β(λ,e)$ vanish, and $β(λ,e)$ itself vanishes when the ``self-dual'' condition relating $λ$ to the gauge coupling $e$ is imposed.