\section{Sensitivity to the top-quark electroweak couplings}
\label{sec:sensitivities}

The aforementioned covariance matrices assume a perfect event reconstruction, an event selection efficiency of 100\%, a $4\pi$ detector acceptance, and the absence of background processes. While these hypotheses would not be utterly unrealistic at an ${\rm e^+ e^-}$ collider, a more conservative approach is in order to render the present estimates credible and reliable. 

{\bf Event reconstruction} 

The only reconstructed quantities used for the determination of the covariance matrices are the lepton direction and the lepton energy (or momentum). Both quantities can be reconstructed with more than adequate precision, as was the case with the detectors built for the LEP collider. The numerical evaluation of the integrals in Eqs.~\ref{eq:firstInt} to~\ref{eq:lastInt} are however performed with 50 bins in $x$ and $\cos\theta$. This procedure corresponds to conservatively assuming a lepton energy resolution of 1\,GeV and a lepton angular resolution of 20\,mrad, figures vastly exceeded by LEP detectors.

{\bf Event selection and particle identification}

The event selection relies on the presence of an energetic isolated lepton and two energetic b-quark jets in the final state, accompanied by either two light-quark jets or an additional lepton. At $\sqrt{s} = 365$\,GeV, the lepton momentum can take values between 13.5 and 120\,GeV/$c$, a range in which an identification efficiency of 80\% can be conservatively assumed, with a negligible fake rate.  Similarly, the b-quark jet energies can take values between 49 and 94\,GeV, for which b-tagging algorithms are very efficient and pure, especially with two b jets in the final state. A very conservative b-tagging efficiency of 60\% is assumed here. To emulate these efficiencies, all terms of Eq.~\ref{eq:optimal}, hence all covariance matrix elements, are multiplied by $0.6 \times 0.8 = 0.48$. 

{\bf Detector acceptance}

The polar-angle coverage of a typical detector at ${\rm e^+ e^-}$ colliders is usually assumed to be from 10 to 170 degrees. To be conservative, the leptons are assumed here to be detected only for $ | \cos\theta |< 0.9$, {\it i.e.}, in a range from 26 to 154 degrees. This effect is emulated by evaluating the integrals of Eqs.~\ref{eq:firstInt} to~\ref{eq:lastInt} between $\cos\theta_{\rm min} = -0.9$ and  $\cos\theta_{\rm max} = 0.9$. Given the large value of the minimum lepton energy, the integration bounds over $x$ are left untouched. 

{\bf Background processes}

The major background identified in Ref.~\cite{Amjad_2013} (which Ref.~\cite{Baer_2013} is based upon) is the single-top production in association with a W boson and a b quark, through WW$^\ast$ production, as it leads to the same final state as the top-quark pair production. The corresponding cross section~\cite{Boos_2001} increases fast with the centre-of-mass energy, and critically depends on the incoming beam polarization. At $\sqrt{s} = 500$\,GeV, the single-top production cross section can reach up to 20\% of the top-pair production cross section in the final state with an electron or a positron and in the ${\rm e^-_L e^+_R}$ initial polarization configuration. Yet, this background has not been included in the top-quark electroweak coupling study of Ref.~\cite{Baer_2013}. At $\sqrt{s} = 365$\,GeV and with unpolarized beams, however, the single-top cross section in the same final state amounts to about 0.1\% of the pair production cross section. It was therefore ignored for the first estimate of precisions given below.. 

{\bf Other experimental uncertainties}

A number of other experimental uncertainties are listed in Ref.~\cite{Amjad_2013}, such as those affecting the measurement of the beam polarization (which enters crucially the cross section measurement); the effects of beamstrahlung; or the ambiguous top-quark reconstruction (which enters crucially the forward-backward asymmetry measurement). These uncertainties apply neither to the FCC-ee, where beamstrahlung effects are negligible and no beam polarization needs to be measured, nor to the present study, as the top-quark direction needs not be reconstructed. The experimental uncertainties affecting the lepton energy and angular distributions can be safely neglected, given the conservative assumptions on the resolutions. The total event rate, needed for the present study, requires a precise luminosity determination, a measurement that can be controlled to a fraction of a per mil, hence neglected here. 

{\bf Theoretical uncertainties}

The dominant systematic uncertainty is of theoretical nature. The total event rate indeed requires an accurate prediction of the total cross section for top pair production. The precision of this prediction is inferred to be at the level of a few per mil in Ref.~\cite{Amjad_2013} for $\sqrt{s} = 500$\,GeV. A similar precision can be expected at smaller centre-of-mass energy as long as it is reasonably above the production threshold.  

{\bf Integrated luminosity profile}

The target luminosities at the FCC-ee are displayed in Fig.~\ref{fig:lumi}~\cite{FCCeeWebSite} as a function of the centre-of-mass energy, together with the target luminosities of other ${\rm e^+ e^-}$ colliders under study in the world. At $\sqrt{s} = 350$\, GeV, a luminosity of $7.2 \times 10^{34}\, {\rm cm}^{-2}{\rm s}^{-1}$ is expected to be democratically distributed to four interaction regions, leading to an integrated luminosity of $3.6\, {\rm ab}^{-1}$ over a period of five years. About $1{\rm ab}^{-1}$ ought to be kept for threshold measurements (leading to a statistical precision on the top mass of about 15\,MeV), and the rest can be used to perform measurements above the production threshold. 