Complete QCD Corrections to Hadronic $Z$--Decays in Order ${\alpha }_s^4$

Abstract

Corrections of order ${\alpha }_s^4$ for the axial singlet contributions for the decay rates of the $Z$-boson into hadrons are evaluated in the limit of the heavy top quark mass. Combined with recently finished $\mathcal{O}({\alpha }_s^4)$ calculations of the non-signlet corrections, the new results directly lead us to the first {\em complete} $\mathcal{O}({\alpha }_s^4)$ prediction for the total hadronic decay rate of the Z-boson. The new $\mathcal{O}({\alpha }_s^4)$ term in Z-decay rate lead to a significant stabilization of the perturbative series, to a reduction of the theory uncertainty in the strong coupling constant ${\alpha }_s$, as extracted from these measurements, and to a small shift of the central value.