{\bf Discussion}

The above results are obtained under the assumption that the gauge-invariance-violating form factor ($F_{1A}^\gamma$) and the CP-violating form factors ($F_{2A}^{\gamma,Z}$) vanish, to allow for a one-to-one and straightforward comparison with Ref.~\cite{Baer_2013}. From an experimental point-of-view, however, there is no a-priori reason why these form factors could not be extracted from the measurements of the lepton angular and energy distributions. The present study is therefore extended, with $2.4\,{\rm ab}^{-1}$ at $\sqrt{s} = 365$\,GeV, to the following two configurations by relaxing the constraints on $F_{1A}^\gamma$, $F_{2A}^\gamma$ and $F_{2A}^Z$: either the four form factors $F_{1V,A}^X$ are varied simultaneously while the four $F_{2V,A}^X$ are fixed to their standard model values, or vice-versa. 

In the first configuration, it turns out that relaxing the constraint on $F^\gamma_{1A}$ does not sizeably change the precision on the other three $F^X_{1V,A}$ form factors, as shown in Table~\ref{tab:f1}. A per-cent accuracy is also obtained on $F^\gamma_{1A}$.
\begin{table}[htbp]
\begin{center}
\caption{\label{tab:f1} Precision on the four $F_{1V,A}^X$ expected with $2.4\,{\rm ab}^{-1}$ at $\sqrt{s} = 365$\,GeV at the FCC-ee, if $F_{1A}^\gamma$ is fixed to its standard model value (first row) or if this constraint is relaxed (second raw). The precisions expected with $500\,{\rm fb}^{-1}$ at $\sqrt{s} = 500$\,GeV are indicated in the third row.}
\begin{tabular}{|l|l|l|l|l|}
\hline Precision on & $F_{1V}^\gamma$ & $F_{1V}^Z$ & $F_{1A}^\gamma$ & $F_{1A}^Z$ \\
\hline\hline Only three $F_{1V,A}^X$ &     $1.2\, 10^{-3}$     &    $2.9\, 10^{-3}$     &   $0.0\, 10^{-2}$    &   $2.2\, 10^{-2}$ \\
\hline All four $F_{1V,A}^X$  & $1.2\, 10^{-3}$ & $3.0\, 10^{-3}$ & $1.3\, 10^{-2}$ & $2.4\, 10^{-2}$ \\ 
\hline $\sqrt{s} = 500$\,GeV  & $5.5\, 10^{-3}$ & $1.5\, 10^{-2}$ & $1.0\, 10^{-2}$ & $2.2\, 10^{-2}$ \\ 
\hline
\end{tabular}
\end{center}
\end{table}

The situation with the $F_{2V,A}^X$ form factors in the second configuration is even clearer. Indeed, the distributions related to $F_{2A}^\gamma$ and $F_{2A}^Z$ form factors are CP odd, while those related to $F_{2V}^\gamma$ and $F_{2V}^Z$ are CP even. With vanishing correlation coefficients, the two pairs of form factors can therefore be determined independently from each other. The precisions on $F_{2V}^\gamma$ and $F_{2V}^Z$, expected with $2.4\,{\rm fb}^{-1}$ at $\sqrt{s} = 365$\,GeV at the FCC-ee, are thus unchanged with respect to Fig.~\ref{fig:baer} when the constraint on $F_{2A}^\gamma$ and $F_{2A}^Z$ is relaxed, and amount to $8.1\, 10^{-4}$ and $2.3\, 10^{-3}$ respectively. With $500\,{\rm fb}^{-1}$ at $\sqrt{s} = 500$\,GeV, the precisions would be $2.5\, 10^{-3}$ and $8.3\, 10^{-3}$ respectively, as also shown in Fig.~\ref{fig:variation}.

The accuracy of the CP-violating form factors with the sole lepton angle and energy distributions is moderately constraining ($1.4\, 10^{-1}$ on $F_{2A}^\gamma$ and $8.2\, 10^{-1}$ on $F_{2A}^Z$) because of the important correlation between the two distributions $f_D^\gamma$ and $f_D^Z$, well visible in Fig.~\ref{fig:distributions}. A relevant precision of $1.7\, 10^{-2}$, however, is reached on the linear combination $F_{2A}^\gamma + 0.17 \times F_{2A}^Z$ with $2.4\,{\rm ab}^{-1}$ at $\sqrt{s} = 365$\,GeV, reduced to $0.9\, 10^{-2}$ with $500\,{\rm fb}^{-1}$ at $\sqrt{s} = 500$\,GeV. A reduction of the correlation between these two form factors requires the analysis of additional observables, beyond the scope of the present study.

Similarly, when all eight parameters are considered simultaneously, the lepton angle and energy distributions are no longer sufficient to avoid large correlations between form factors. The same observation was made in Refs.~\cite{Baer_2013} and~\cite{Amjad_2013} with the four observables chosen for the analysis at 500\,GeV and with incoming beam polarizations. A generator-level exercise with more observables in the fully leptonic final state has been recently attempted in Ref.~\cite{ledib_2015}, released after the present study. In this exercise, an optimal-observable analysis of the matrix element squared is carried out with thirteen different observables (the top quark direction, the $\ell^+$ and $\ell^-$ angles and energies, the ${\rm b}$ and ${\rm \bar b}$ angles and energies, and the invariant masses of the top quarks and W bosons), with unambiguous identification and reconstruction under the assumption of a perfect detector. With these additional variables, the few degeneracies between form factors are indeed removed, but the conclusion is identical to that of this paper: the incoming beam polarizations are not essential in the process. 

A similar analysis could be undertaken for semi-leptonic final states at $\sqrt{s} = 365$\,GeV, in order to determine all eight form factors simultaneously with the ultimate accuracy, but the assumption of a perfect detector cannot be expected to give fully reliable results when the jets and the missing energy from the top decays are to be included, as acknowledged in Ref.~\cite{ledib_2015}. Such an analysis will be carried out when a complete simulation and reconstruction in a realistic detector becomes available for the FCC-ee study.  
