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 . 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. . 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 $$ comprised between 340 and 350GeV. 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 100TeV. 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 ttγ and the ttZ couplings for both chirality states of the top quark. In Ref. , 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 $=500$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 μ), 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 . 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.

INTRODUCTION The Higgs boson with mass around 125 GeV recently discovered by the ATLAS and CMS experiments  at the LHC is found to have properties compatible with the Standard Model predictions , as shown for example in Fig. [fig:ellis] . Coupled with the absence of any other indication so far for new physics at the LHC, be it either through precision measurements or via direct searches, this fundamental observation seems to push the energy scale of any physics beyond the Standard Model above several hundred GeV. The higher-energy LHC run, which is expected to start in 2015 at $ \sim 13$-14 TeV, will extend the sensitivity to new physics to 1 TeV or more. Fundamental discoveries may therefore be made in this energy range by 2017-2018. Independently of the outcome of this higher-energy run, however, there must be new phenomena, albeit at unknown energy scales, as shown by the evidence for non-baryonic dark matter, the cosmological baryon-antibaryon asymmetry and non-zero neutrino masses, which are all evidence for physics beyond the Standard Model. In addition to the high-luminosity upgrade of the LHC, new particle accelerators will be instrumental to understand the physics underlying these observations.

When the measurements from the FCC-ee become available, an improved determination of the standard-model “input” parameters will be needed to fully exploit the new precision data towards either constraining or fitting the parameters of beyond-the-standard-model theories. Among these input parameters is the electromagnetic coupling constant estimated at the Z mass scale, $(m^2_{\rm Z})$. The measurement of the muon forward-backward asymmetry at the FCC-ee, just below and just above the Z pole, can be used to make a direct determination of $(m^2_{\rm Z})$ with an accuracy deemed adequate for an optimal use of the FCC-ee precision data.