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 \({\rm t}\bar{\rm t}\) 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 \({\rm Z}\to{\rm b}\bar{\rm b}\) 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 \(\sin^{2}\theta_{\rm W}^{\rm eff}\) from leptonic asymmetries or from \(A_{\rm LR}\). 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}

Now that (i) the uncertainty due to the Higgs boson mass dependence is no longer relevant; (ii) the uncertainty due to the top-quark mass dependence is negligible; and (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.
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.