Analytic Properties of the QCD Running Coupling Constant and $τ$ 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 $τ$--lepton. The method allows us to evaluate $R_τ$ 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 $π^2$ terms.
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