Toy Models of Non-perturbative Asymptotic Freedom in ${\varphi }_6^3$

Abstract

We study idealizations of the full nonlinear Schwinger-Dyson equations for the asymptotically free theory of ${\varphi }^3$ in six dimensions in its meta-stable vacuum. We begin with the cubic nonlinearity and go on to all-order nonlinearities which contain instanton effects. 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. In the all-order nonlinear equation we show how to recover the usual renormalization-group-improved instanton effects and associated factorial divergences.

30 pages (REVTeX), 9 figures in uuencoded compressed Postscript

References in corpus (3)

• Scale Setting in QCD and the Momentum Flow in Feynman Diagrams
• Principal Value Resummation
• Nonperturbative Renormalon Structure of Infrared Unstable Theories

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• Asymptotic behavior and critical coupling in the scalar Yukawa model from Schwinger-Dyson equations
• Solution of coupled vertex and propagator Dyson-Schwinger equations in the scalar Munczek-Nemirovsky model
• Non-perturbative solution of metastable scalar models
• A Finite Energy-Momentum Tensor for the ${\varphi }^3$ theory in $6$ dimensions