Analytic Properties of the QCD Running Coupling Constant and $\tau$ Decay

A non-perturbative expansion method which gives a well-defined analytic continuation of the running coupling constant from the spacelike to the timelike region is applied to the inclusive semileptonic decay of the $\tau$–lepton. The method allows us to evaluate $R_{\tau }$ by integration over the non-perturbative physical region, rather than by using Cauchy’s theorem, and hence to incorporate threshold effects in a very direct way. Within our framework the difference between the effective coupling constants in the timelike and spacelike domains can be substantial and is not simply a matter of the standard ${\pi }^2$ terms.