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\subsubsection{The strong coupling constant}
An accurate determination of $\alpha_s$ may be obtained from the precise experimental measurement of the inclusive hadronic Z decay rate. The theoretical prediction for such an inclusive observable is known with ${\rm N_3LO}$ QCD corrections~\cite{Baikov_Chetyrkin_Kuhn_2008, Chetyrkin_Kuhn_Rittinger_2012}, with strongly suppressed non-perturbative effects. Some caveat is in order since Electroweak corrections can in principle be sensitive to the particle content of the
electroweak Electroweak theory. The extraction of $\alpha_{\rm s}$ may
therefore not be completely free of model dependence of
electroweak Electroweak nature. A good way around this caveat is to constraint radiative correction effects with other Electroweak measurements at the Z pole or elsewhere. In the case at stake here, the hadronic partial width is sensitive to new physics through the ‘oblique’ electroweak corrections known as $\epsilon_1 (\equiv \Delta\rho)$ and $\epsilon_3$, and through the vertex correction $\delta_{\rm b}$ to the $ {\rm Z} \to \bbbar$ partial width. The $\Delta \rho$ sensitivity cancels when taking the ratio $R_\ell$ with the leptonic partial width, and the $\epsilon_3$ corrections can be strongly constrained by the determination of $\sintw$ from leptonic asymmetries or from $\ALR$. The b-vertex contribution can be constrained by the direct extraction of $R_{\rm b}$ hence is not expected to be a limitation.
The ratio $ R_\ell$ has been used for the determination of $\alpha_s$ at LEP. Up to a few years ago, when only NNLO QCD predictions were available, and the Higgs boson mass was still unknown, this measurement was translated to~\cite{Bethke_2004}
\begin{equation}