Differential Regularization of a Non-relativistic Anyon Model

TL;DR

This paper applies differential regularization to a non-relativistic anyon field theory, calculating renormalized amplitudes up to three loops and analyzing the renormalization group behavior, especially under self-dual conditions.

Contribution

It provides a detailed 3-loop renormalization of a non-relativistic anyon model using differential regularization, including explicit RG equations and scheme dependence analysis.

Findings

2- and 3-loop contributions to the beta function vanish under certain conditions.

The beta function itself vanishes when the self-dual condition is imposed.

Explicit 3-loop renormalized amplitudes for the 4-point function are obtained.

Abstract

Differential regularization is applied to a field theory of a non-relativistic charged boson field $\phi$ with $\lambda (\phi {}^{*} \phi)^2$ self-interaction and coupling to a statistics-changing $U(1)$ Chern-Simons gauge field. Renormalized configuration-space amplitudes for all diagrams contributing to the $\phi {}^{*} \phi {}^{*} \phi \phi$ 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 $\beta$-function is investigated. If the renormalization scheme is fixed to agree with a previous 1-loop calculation, the 2- and 3-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.