Resummation analysis of the $\tau$ decay width using the four-loop $\beta$-function

Abstract

We extract the strong coupling constant ${\alpha }_s(m_{\tau }^2)$ from the semileptonic $\tau$ decay width taking into account resummation effects from the running of the strong coupling constant. In the $\overline{\mathrm{MS}}$ scheme the result reads ${\alpha }_s=0.375\pm 0.007$ to third order and ${\alpha }_s=0.378\pm 0.007$ to fourth order in the $\beta$-function, respectively, where we use the recently computed four-loop coefficient ${\beta }_3$. These values for the coupling constant have to be compared with the value ${\alpha }_s=0.354\pm 0.005$ derived from a third order analysis of $\tau$ 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.

12 pages, LaTeX, more detailed explanation of the analysis done for the determination of the convergence radius, one PostScript figure added

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