Dispersive Techniques for $\alpha_s$, $R_{had}$ and Instability of the Perturbative Vacuum

Abstract

Recent dispersive techniques developed by us are applied to discuss three problems: 1. A long standing discrepan-cy between the measurements of $R(s)$ for $\sqrt{s} = (5\div 7.5)GeV$ by Crystal Ball and MARK I has been analyzed and its consequences analyzed for the number of contributing quarks. 2. Noting that the perturbative $\alpha_s$ has the wrong analyticity, analytic models consistent with asymptotic freedom (AF) and confinement have been constructed and applied to discuss $\tau$ decay. 3. It is shown that AF leads to a wrong sign for $\im\big(\alpha(s)\big)$ which signals an instability of the perturbative QCD vacuum.