Toy models of nonperturbative asymptotic freedom in six-dimensional phi3 theory.
We study idealizations of the full nonlinear Schwinger-Dyson equations for the asymptotically free {phi}{sup 3} theory in six dimensions in its metastable vacuum. We begin with the cubic nonlinearity and go on to all-order nonlinearities that contain instanton effects. In an asymptotically free theory the relevant Schwinger-Dyson equations are homogeneous and ultraviolet finite and perturbative methods fail from the outset. We show how our toy models of the cubic Schwinger-Dyson equations contain 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 models 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 parameter equivalent to the perturbative renormalization scale {mu}. In the all-order nonlinear equation we show how to recover the usual renormalization-group-improved instanton effects and associated factorial divergences.