\subsubsection{The strong coupling constant}

The prospective TLEP precisions on the EWSB parameters call for a similar improvement of the strong coupling constant accuracy, which would otherwise become a leading systematic uncertainty in the theoretical interpretation of the TLEP measurements, and in particular in the determination of the top quark mass from the measurement of the $\ttbar$ production threshold cross section. Complementary determinations of the strong coupling constant, $\alpha_{\rm s}$, may be obtained both at the Z pole and at energies at the WW threshold and above, with similar accuracies.

The precise experimental measurement of the inclusive hadronic Z decay rate at the Z pole is sensitive to $\alpha_{\rm s}$. The theoretical prediction for such an inclusive observable is known with ${\rm N_3}{\rm LO}$ 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 theory. The extraction of $\alpha_{\rm s}$ may therefore not be completely free of model dependence of Electroweak nature. A good way around this caveat is to constrain 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 $\alpha_{\rm s}$ 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}
\scriptstyle
\alpha_{\rm s} (m^2_{\rm Z})= 0.1226 \pm 0.0038 \ ({\rm exp}) {\ }^{+0.0028\ (\mu = 2.00 m_{\rm Z})}_{-0.0005\ (\mu = 0.25 m_{\rm Z})}
{\ }^{+0.0033\ (m_{\rm H} = 900\ {\rm GeV})}_{-0.0000\ (m_{\rm H} = 100\ {\rm GeV})}
{\ }^{+0.0002\ (m_{\rm top} = 180\ {\rm GeV})}_{-0.0002\ (m_{\rm top} = 170\ {\rm GeV})} \pm 0.0002\ ({\rm th}).
\end{equation}

Now that {\it (i)} the uncertainty due to the Higgs boson mass dependence is no longer relevant; {\it (ii)} the uncertainty due to the top-quark mass dependence is negligible; and {\it (iii)} the pQCD scale uncertainty from the latest ${\rm N_3}{\rm LO}$ calculations~has dropped to 0.0002, this method potentially allows access to a high-precision measurement of $\alpha_{\rm s}$. As shown in Eq.~\ref{eq:Rl}, $R_\ell$ was measured at LEP with a relative uncertainty of 0.12\%. As mentioned in Section~\ref{sec:Rell}, this precision is expected to improve to $5 \times 10^{-5}$ with TLEP. The LEP experimental error of 0.0038 on $\alpha_s (m^2_{\rm Z})$ will scale accordingly to 0.00015 at TLEP, becoming of the same order as the theory uncertainty.\\
\textbf{\textit{A reasonable target for the measurement of $\boldsymbol{\alpha_s (m^2_Z)}$ with a run at the Z pole with TLEP is therefore a precision of 0.0002.}}


