Beyond the measurement of $R_\ell$ at the Z pole, another interesting possibility for the $\alpha_{\rm s}$ determination is to use the W hadronic width as measured from W-pair events at and above 161 GeV. The quantity of interest is the branching ratio $B_{\rm had}= \Gamma_{\rm W \to hadrons}/\Gamma^{\rm tot}_{\rm W}$, which can be extracted by measuring the fractions of WW events to the fully leptonic, semi-leptonic and fully hadronic final states: 
\begin{eqnarray}
{\rm BR}({\rm W^+W^-} \to \ell^+ \nu \ell’^- \bar{\nu}) & = & (1-B_{\rm had})^2,\\
{\rm BR}({\rm W^+W^-} \to \ell^+ \nu {\rm q\bar{q}^\prime}) & = & (1-B_{\rm had})\times B_{\rm had},\\
{\rm BR}({\rm W^+W^-} \to {\rm q\bar{q}^\prime q^{\prime\prime}\bar{q}^{\prime\prime\prime}}) & = & B_{\rm had}^2.
\end{eqnarray}


The LEP2 data taken at centre-of-mass energies ranging from 183 to 209 GeV led to $B_{\rm had} = 67.41 \pm 0.27$~\cite{1302.3415}, a measurement with a 0.4\% relative precision. This measurement was limited by WW event statistics of about $4\times 10^4$ events. With over $2 \times 10^8$ W pairs expected at TLEP at $\sqrt{s} =$ 161, 240 and 350~GeV, it may therefore be possible to reduce the relative uncertainty on $B_{\rm had}$ by a factor $\sim 70$, down to $5\times 10^{-5}$, and thus the absolute uncertainty on $\alpha_{\rm s}$ to $\pm 0.00015$. 

This measurement is both competitive with and complementary to that performed with the Z hadronic width, because the sensitivity to Electroweak effects is completely different in $B_{\rm had}$ and in $R_\ell$. In particular, the coupling of the W to pairs of quarks and leptons is straightforwardly given by the CKM matrix elements with little sensitivity to any new particles.\\
\textbf{\textit{A reasonable target for the measurement of $\boldsymbol{\alpha_s (m^2_W)}$ with the runs at and above 161~GeV with TLEP is therefore a precision better than 0.0002. When combined with the measurement at the Z pole, a precision of 0.0001 is within reach for $\boldsymbol{\alpha_s (m^2_Z)}$.}}
