TOY MODELS OF NONPERTURBATIVE ASYMPTOTIC FREEDOM IN 6-DIMENSIONAL PHI(3) THEORY

We study idealizations of the full nonlinear Schwinger-Dyson equations for the asymptotically free phi(3) theory in six dimensions in its me tastable vacuum. We begin with the cubic nonlinearity and go on to all -order nonlinearities that contain instanton effects. In an asymptotic ally free theory the relevant Schwinger-Dyson equations are homogeneou s and ultraviolet finite and perturbative methods fail from the outset . We show how our toy models of the cubic Schwinger-Dyson equations co ntain the usual diseases of perturbation theory in the massless limit (e.g., factorially divergent beta functions, singular Borel-transform kernels associated with infrared renormalons) and show how these model s yield specific mechanisms for removing such singularities when there is a mass gap. The solutions to these homogeneous equations, in spite of being ultraviolet finite, still depend on an undetermined paramete r equivalent to the perturbative renormalization scale mu. In the all- order nonlinear equation we show how to recover the usual renormalizat ion-group-improved instant on effects and associated factorial diverge nces.