Resummation analysis of the τ decay width using the four - loop β - function

We extract the strong coupling constant α s (m 2 τ) from the semileptonic τ decay width taking into account resummation effects from the running of the strong coupling constant. In the MS scheme the result reads α s = 0.375 ± 0.007 to third order and α s = 0.378 ± 0.007 to fourth order in the β-function, respectively, where we use the recently computed four-loop coefficient β 3. These values for the coupling constant have to be compared with the value α s = 0.354 ± 0.005 derived from a third order analysis of τ decays. We determine the exact value of the convergence radius of the perturbation series by analyzing the singularity structure of the complex coupling constant plane.