STUDY OF QUARK PROPAGATOR SOLUTIONS TO THE DYSON-SCHWINGER EQUATION IN A CONFINING MODEL

We solve the Dyson-Schwinger equation for the quark propagator in a mo del with singular infrared behavior for the gluon propagator We requir e that the solutions, easily found in configuration space, be tempered distributions and thus have Fourier transforms. This severely limits the boundary conditions that the solutions may satisify. The sign of t he dimensionful parameter that characterizes the model gluon propagato r can be either positive or negative. If the sign is negative, we find a unique solution. It is singular at the origin in momentum space, fa lls off like 1/p(2) as p(2)-->+/1-infinity, and is truly nonperturbati ve, in that it is singular in the limit that the gluon-quark interacti on approaches zero. If the sign of the gluon propagator coefficient is positive, we find solutions that are, in a sense that we exhibit, unc onstrained linear combinations of advanced and retarded propagators. T hese solutions are singular at the origin in momentum space, fall off like 1/p(2) asympotically, exhibit ''resonantlike'' behavior at the po sition of the bare mass of the quark when the mass is large compared t o the dimensionful interaction parameter in the gluon propagator model , and smoothly approach a linear combination of free-quark advanced an d retarded two-point functions in the limit that the interaction appro aches zero. In this sense, these solutions behave in an increasingly ' 'particlelike'' manner as the quark becomes heavy. The Feynman propaga tor and the Wightman function are not tempered distributions and there fore are not acceptable solutions to the Schwinger-Dyson equation in o ur model. On this basis we advance several arguments to show that the Fourier-transformable solutions we find are consistent with quark conf inement, even though they have singularities on the real p(2) axis.