\section{Introduction}
\label{sec:Introduction}

The design study of the Future Circular Colliders (FCC) in a 100-km ring in the Geneva area has started at CERN at the beginning of 2014, as an option for post-LHC particle accelerators. The study has an emphasis on proton-proton and electron-positron high-energy frontier machines~\cite{FCCWebSite}. In the current plans, the first step of the FCC physics programme would exploit a high-luminosity ${\rm e^+e^-}$ collider called FCC-ee, with centre-of-mass energies ranging from below the Z pole to the ${\rm t\bar t}$ threshold and beyond. A first look at the physics case of the FCC-ee can be found in Ref.~\cite{Bicer_2014}.

In this first look, the focus regarding top-quark physics was on precision measurements of the top-quark mass, width, and Yukawa coupling through a scan of the ${\rm t\bar t}$ production threshold, with $\sqrt{s}$ comprised between~340 and~350\,GeV. The expected precision on the top-quark mass was in turn used, together with the outstanding precisions on the Z peak observables and on the W mass, in a global electroweak fit to set constraints on weakly-coupled new physics up to a scale of 100\,TeV. Although not studied in the first look, measurements of the top-quark electroweak couplings are of interest, as new physics might also show up via significant deviations of these couplings with respect to their standard-model predictions. Theories in which the top quark and the Higgs boson are composite lead to such deviations. The inclusion of a direct measurement of the ttZ coupling in the global electroweak fit is therefore likely to further constrain these theories. 

It has been claimed that both a centre-of-mass energy well beyond the top-quark pair production threshold and a large longitudinal polarization of the incoming electron and positron beams are crucially needed to independently access the tt$\gamma$ and the ttZ couplings for both chirality states of the top quark. In Ref.~\cite{Baer_2013}, it is shown that the measurements of the total event rate and the forward-backward asymmetry of the top quark, with 500\,${\rm fb}^{-1}$ at $\sqrt{s}=500$\,GeV and with beam polarizations of ${\cal P} = \pm 0.8$, ${\cal P}^\prime = \mp 0.3$, allow for this distinction. 

The aforementioned claim is revisited in the present study. The sensitivity to the top-quark electroweak couplings is estimated here with an optimal-observable analysis of the lepton angular and energy distributions of over a million events from ${\rm t\bar t}$ production at the FCC-ee, in the $\ell \nu {\rm q \bar q b \bar b}$ final states (with $\ell = {\rm e}$ or $\mu$), without incoming beam polarization and with a centre-of-mass energy not significantly above the  ${\rm t\bar t}$ production threshold. 

Such a sensitivity can be understood from the fact that the top-quark polarization arising from its coupling to the Z is maximally transferred to the final state particles via the weak top-quark decay ${\rm t \to W b}$ with a 100\% branching fraction: the lack of initial polarization is compensated by the presence of substantial final state polarization, and by a larger integrated luminosity. A similar situation was encountered at LEP, where the measurement of the total rate of ${\rm Z} \to \tau^+\tau^-$ events and of the tau polarization was sufficient to determine the tau couplings to the Z, regardless of initial state polarization~\cite{Altarelli:1989hv,Schael_2006}.

This letter is organized as follows. First, the reader is briefly reminded of the theoretical framework. Next, the statistical analysis of the optimal observables is described, and realistic estimates for the top-quark electroweak coupling sensitivities are obtained as a function of the centre-of-mass energy at the FCC-ee. Finally, the results are discussed, and prospects for further improvements are given. 
