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

Abstract

We extract the strong coupling constant $α_s(m_τ^2)$ from the semileptonic $τ$ decay width taking into account resummation effects from the running of the strong coupling constant. In the $\overline{\rm MS}$ scheme the result reads $α_s=0.375\pm 0.007$ to third order and $α_s=0.378\pm 0.007$ 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 $α_s=0.354\pm 0.005$ 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.