The Yukawa Coupling in Three Dimensions

TL;DR

This paper investigates Yukawa couplings in three-dimensional models, highlighting differences in renormalization properties between Euclidean and Minkowski spaces and exploring conditions for asymptotic freedom.

Contribution

It analyzes the renormalization and asymptotic freedom of Yukawa couplings in various 3D models, including supersymmetric and multi-fermion scenarios, emphasizing space-time signature effects.

Findings

Yukawa couplings are asymptotically free in certain 3D Euclidean models.

In Minkowski space, asymptotic freedom is generally lost due to Dirac matrix properties.

Different fermion-scalar coupling configurations exhibit distinct renormalization behaviors.

Abstract

We consider several renormalizable, scale free models in three space-time dimensions which involve scalar and spinor fields. The Yukawa couplings are bilinear in both the spinor and scalar fields and the potential is of sixth order in the scalar field. In a model with a single scalar field and a complex Fermion field in three Euclidean dimensions, the couplings in the theory are both asymptotically free. This property is not retained in 2+1 dimensional Minkowski space, as we illustrate by considering a renormalizable scale-free supersymmetric model. This is on account of the different properties of the Dirac matrices in Euclidean and Minkowski space. We also examine a model in 2+1 dimensional Minkowski space in which two species of Fermions, associated with the two unitarily inequivalent representations of the $2 \times 2$ Dirac matrices, couple in two different ways to two distinct scalar fields. There are two types of Yukawa couplings in this model, and either one or the other of them can be asymptotically free (but not both simultaneously).