YUKAWA COUPLING IN 3 DIMENSIONS

We consider several renormalizable, scale free models in three space-t ime dimensions which involve scalar and spinor fields. The Yukawa coup lings are bilinear in both the spinor and scalar fields and the potent ial is of sixth order in the scalar field. In a model with a single sc alar field and a complex fermion field in three Euclidean dimensions, the couplings in the theory are both asymptotically free. This propert y is not retained in (2 + 1)-dimensional Minkowski space, as we illust rate by considering a renormalizable scale-free supersymmetric model. This is on account of the different properties of the Dirac matrices i n Euclidean and Minkowski space. We also examine a model in (2 + 1)-di mensional Minkowski space in which two species of fermions, associated with the two unitarily inequivalent representations of the 2 x 2 Dira c matrices, couple in two different ways to two distinct scalar fields . There are two types of Yukawa couplings in this model, and either on e or the other of them can be asymptotically free (but not both simult aneously).