Differential regularization of a nonrelativistic anyon model

Differential regularization is applied to a field theory of a nonrelativistic charged boson field $\phi$ with $\lambda {\left({\phi }^{*}\phi \right)}^2$ self-interaction and coupling to a statistics-changing U(1) Chern-Simons gauge field. Renormalized configuration-space amplitudes for all diagrams contributing to the ${\phi }^{*}{\phi }^{*}\phi \phi$ four-point function, which is the only primitively divergent Green's function, are obtained up to three-loop order. The renormalization group equations are explicitly checked, and the scheme dependence of the $\beta$ function is investigated. If the renormalization scheme is fixed to agree with a previous one-loop calculation, the two- and three-loop contributions to $\beta (\lambda ,e)$ vanish, and $\beta (\lambda ,e)$ itself vanishes when the "self-dual" condition relating $\lambda$ to the gauge coupling $e$ is imposed.