deletions | additions       diff --git a/aphas.tex b/aphas.tex  index 3e84142..a694267 100644  --- a/aphas.tex  +++ b/aphas.tex 
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{\ }^{+0.0002\ (m_{\rm top} = 180 {\rm GeV}/c^2)}_{-0.0002\ (m_{\rm top} = 170 {\rm GeV}/c^2)} \pm 0.0002\ ({\rm th}) = 0.1226^{+0.0058}_{−0.0038}  \end{equation}  Now that the uncertainty due to the Higgs boson mass dependence is no longer relevant, that the uncertainty due to the top-quark mass dependence is negligible, and the pQCD scale uncertainty from the latest {\rm N_3LO} calculations~\cite{doi:10.1103/PhysRevLett.101.012002} calculations~\cite{Baikov_Chetyrkin_Kuhn_2008}  [6, 7] has dropped to 0.0002, this method should allow access to high precision on $\alpha_s$. The LEP measurement $R_\ell$ = 20.767±0.025 [4] is mainly limited by lepton statistics. With $10{12} Z$ events expected from TLEP and assuming the selection efficiency uncertainties scale with statistics, one might expect a reduction of the uncertainty by a factor of 250. At this level of precision, we will have to consider many subtle systematic uncertainties and a detailed analysis would be necessary. The hadronic partial width is sensitive to new physics through the ‘oblique’ electroweak corrections known as $\Delta \rho \equiv \epsilon_1$ and $\epsilon_3$ and through the vertex correction $\delta_b$ to the $ Z\rightarrow b bar{b} $ partial width. The $\Delta \rho$ sensitivity cancels when taking the ratio with the leptonic partial width, and the $\epsilon_3$ corrections can be strongly constrained by the determination of $\sin^2{\vartheta_w^{eff}}$ from leptonic asymmetries or $A_{LR}$. The b-vertex contribution can be constrained by the direct extraction of $R_b = Z\rightarrow b bar{b}/(Z \rightarrow hadrons$ so this is not expected to be a limitation. The TLEP target goal of the measurement of $ R_\ell$ with a relative precision of $< 10^{-5}$ corresponds to a measurement of $\alpha_s (M^2_Z)$ to a precision better than $10{-4}$,