Dynamical symmetry breaking as a bootstrap

Abstract

This paper is concerned with the bootstrap nature of dynamical symmetry breaking and with the infrared origins of the mass of the electron. We present a general calculational procedure for handling the situation in which a composite operator $\overline{\Psi }\Psi$ acquires a dynamical vacuum expectation value. We apply our procedure to finite quantum electrodynamics to show how the infrared divergences of the theory self-consistently cause $〈\overline{\Psi }\Psi 〉$ to become nonzero nonperturbatively so that it can provide a scale for a purely dynamical electron mass. Since no Goldstone boson need accompany this spontaneous breakdown, the electron mass bootstraps itself about the ${\gamma }_5$ degenerate vacuum. The mechanism also yields a new eigenvalue condition for the fine-structure constant. We discuss the deep interplay between the ultraviolet and the infrared, a characteristic feature of dynamical symmetry breaking. We use this interplay to show how to extend the Wilson operator-product expansion to the situation in which there is a degenerate vacuum. We discuss the possibility that the infrared structure of the weak interaction provides a dynamical origin for the Gell-Mann, Oakes, and Renner Hamiltonian. We indicate briefly the possibility that anomalous dimensions can soften a 4-Fermi interaction sufficiently to make it renormalizable.