Using recently developed methods for the evaluation of five-loop amplitudes in perturbative QCD, corrections of order ${\alpha }_s^4$ for the nonsinglet part of the cross section for electron-positron annihilation into hadrons and for the decay rates of the $Z$ boson and the $\tau$ lepton into hadrons are evaluated. The new terms lead to a significant stabilization of the perturbative series, to a reduction of the theory uncertainly in the strong coupling constant ${\alpha }_s$, as extracted from these measurements, and to a small shift of the central value, moving the two central values closer together. The agreement between two values of ${\alpha }_s$ measured at vastly different energies constitutes a striking test of asymptotic freedom. Combining the results from $Z$ and $\tau$ decays we find ${\alpha }_s(M_Z)=0.1198\pm 0.0015$ as one of the most precise and presently only result for the strong coupling constant in order ${\alpha }_s^4$.

• Received 11 January 2008

DOI:https://doi.org/10.1103/PhysRevLett.101.012002