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In the paper, we study the properties of the $Z$-boson hadronic decay widthby using the $\mathcal{O}(\alpha_s^4)$-order quantum chromodynamics (QCD)corrections with the help of the principle of maximum conformality (PMC). Byusing the PMC single-scale approach, we obtain an accurate renormalizationscale-and-scheme independent perturbative QCD (pQCD) correction for the$Z$-boson hadronic decay width, which is independent to any choice ofrenormalization scale. After applying the PMC, a more convergent pQCD serieshas been obtained; and the contributions from the unknown$\mathcal{O}(\alpha_s^5)$-order terms are highly suppressed, e.g.conservatively, we have $\Delta \Gamma_{\rm Z}^{\rm had}|^{{\calO}(\alpha_s^5)}_{\rm PMC}\simeq \pm 0.004$ MeV. In combination with the knownelectro-weak (EW) corrections, QED corrections, EW-QCD mixed corrections, andQED-QCD mixed corrections, our final prediction of the hadronic $Z$ decay widthis $\Gamma_{\rm Z}^{\rm had}=1744.439^{+1.390}_{-1.433}$ MeV, which agrees withthe PDG global fit of experimental measurements, $1744.4\pm 2.0$ MeV.

european physical journal. c, particles and fields

$Z$-boson hadronic decay width up to $\mathcal{O}(\alpha_s^4)$-order QCD  corrections using the single-scale approach of the principle of maximum  conformality