Test of the Running of ${\alpha }_s$ in $\tau$ Decays

Abstract

The $\tau$ decay rate into hadrons of invariant mass smaller than $\sqrt{s_0}\gg {\Lambda }_{\mathrm{QCD}}$ can be calculated in QCD assuming global quark--hadron duality. It is shown that this assumption holds for $s_0>0.7$~GeV${}^2$. From measurements of the hadronic mass distribution, the running coupling constant ${\alpha }_s(s_0)$ is extracted in the range 0.7~GeV${}^2<s_0<m_{\tau }^2$. At $s_0=m_{\tau }^2$, the result is ${\alpha }_s(m_{\tau }^2)=0.329\pm 0.030$. The running of ${\alpha }_s$ is in good agreement with the QCD prediction.

Comments: 9 pages, 3 figures appended; shortened version with new figures, to appear in Physical Review Letters (April 1996)