\subsection{Reducing the theory uncertainties}

The unprecedented precision in Higgs, W, Z and top measurements at TLEP will require significant theoretical effort in a new generation of theoretical calculations in order to reap the full benefits from their interpretation, as illustrated in Fig.~\ref{fig:GFitter2}. In their absence, a few considerations are given here, based on calculations made in the context of GigaZ and MegaW studies at the ILC~\cite{Ferroglia_Sirlin_2013}. The current measurements of $m_{\rm H}$, $m_{\rm Z}$, $\alpha_{\rm em}$, $m_{\rm top}$ and $\alpha_{\rm s}$ may be used to estimate $m_{\rm W}$ and $\sintw$,
\begin{eqnarray}
m_{\rm W} & = & 80.361 \pm 0.006 \pm 0.004 \; {\rm GeV},\\
\sintw & = & 0.23152 \pm 0.00005 \pm 0.00005,
\label{eq:uncertainties}
\end{eqnarray}
where in each case the first error is the parametric uncertainty and the second is the estimated uncertainty due to higher-order Electroweak corrections. 

In both cases~\cite{1007.5232}, the dominant parametric uncertainty is due to the experimental error in the top mass, $\delta m_{\rm top} \sim 1$~GeV, responsible for $\delta m_{\rm W} \sim 6$~MeV and $\delta \sintw \sim 3 \times 10^{-5}$. A measurement of $m_{\rm top}$ with a statistical precision of 10 to 20 MeV, as discussed above, could in principle reduce these parametric uncertainties to $\delta m_{\rm W} \sim 0.1$~MeV and $\delta\sintw < 10^{-6}$, respectively. However, there is currently a theoretical uncertainty in $m_{\rm top}$ associated with non-perturbative QCD, of the order of $\sim 100$~MeV or more, which would need to be understood better. Other important parametric uncertainties are those due to $\delta m_{\rm Z}$, responsible for $\delta m_{\rm W} \sim 2.5$~MeV and $\delta \sintw \sim 1.4 \times 10^{-5}$. The projected measurement of $m_{\rm Z}$ with an error $\delta M_Z \sim 0.1$~MeV would reduce these two parametric uncertainties to $\delta m_{\rm W} \sim 0.1$~MeV and $\delta\sintw \sim 10^{-6}$ as well. Other important parametric uncertainties are those associated with $\alpha_{\rm em}(m_{\rm Z})$, which are currently $\delta m_{\rm W} \sim 1$~MeV and $\delta \sintw \sim 1.8 \times 10^{-5}$. The exploitation of the full power of TLEP would require reducing $\delta\alpha_{\rm em}(m_{\rm Z})$ by almost an order of magnitude, which will require significant improvements not only in lower-energy measurements of $\epemto$ hadrons, but also in the theoretical understanding of radiative corrections~\cite{aQEDD,aQEDC,aQEDB,aQEDA}.

These prospective reductions in the parametric errors of Eq.~\ref{eq:uncertainties} will need to be accompanied by order-of-magnitude reductions in the uncertainties associated with Electroweak corrections. This will require a new generation of Electroweak calculations to higher order in Electroweak perturbation theory, that are perhaps beyond the current state of the art, but within reach on the time scale required by TLEP.
