High-precision $\alpha_S$ from W and Z hadronic decays

The extraction of the QCD coupling $\alpha_s$ from the comparison of experimental data on inclusive W and Z bosons hadronic decays to state-of-the-art perturbative QCD calculations is reviewed. The relatively small amount of W data from $e^+e^-\to \rm{W}^+ \rm{W}^-$ collisions at LEP leads today to an non-competitive extraction of the strong coupling at the Z mass from the measured $\rm{R_W}$ ratio of hadronic-to-leptonic branching fractions, $\alpha_s(m_Z) = 0.117 \pm 0.042_{\rm{exp}} \pm 0.004_{\rm{th}} \pm 0.001_{\rm{par}}$ with a $\sim 35\%$ propagated experimental uncertainty. Analysis of the much more abundant hadronic results at the Z pole leads to $\alpha_s(m_Z) = 0.1203 \pm 0.0030$, with a $2.5\%$ uncertainty by combining three different pseudo-observables (ratio of hadronic-to-leptonic widths $\rm{R_Z}$, hadronic peak cross section $\sigma^{\rm{had}}_Z$, and total width $\Gamma^{\rm{tot}}_Z$). An $\alpha_s$ determination with per mille uncertainty requires high-statistics W and Z bosons data samples at future $e^+e-$ colliders, such as the FCC-ee, combined with even higher precision (N$^4$LO) pQCD calculations.