\section{Optimal-observable statistical analysis}
\label{sec:optimal}

There are nine different functions entering Eq.~\ref{eq:optimal}, and eight form factors $\delta_i$ to be evaluated from a given sample of ${\rm t \bar t}$ events. In principle, all eight form factors and their uncertainties can therefore be determined simultaneously, under the condition that the nine functions are linearly independent. Experimentalists usually maximize numerically a global likelihood $L$ -- or equivalently, minimize the negative Log-likelihood ($-\log L$) -- with respect to all form factors:
\begin{equation}
\label{eq:likelihood}
L = {\mu^N \over N!}{\rm e}^{-\mu} \times \prod_{k=1}^N p(k),
\end{equation} 
where $N$ is the total number of ${\rm t \bar t}$ events observed in the data sample, $\mu$ is the number of events expected for the integrated luminosity ${\cal L}$ of the data sample ($\mu = \sigma_{\rm tot} \times {\cal L}$), and
\begin{equation} 
p(k)  = {1 \over \sigma_{\rm tot} } { {\rm d}^2\sigma \over {\rm d}x {\rm d}\cos\theta }(x_k,\cos\theta_k) {\rm , with \ } 
\sigma_{\rm tot} =  \int { {\rm d}^2\sigma \over {\rm d}x {\rm d}\cos\theta } {\rm d}x {\rm d}\cos\theta .
\end{equation}
The covariance matrix obtained from the numerical minimization of the negative log-likelihood is then inverted to get the uncertainties on the form factors, $\sigma(\delta_i)$. It can be shown~\cite{Davier_1993} that, in the linear form given in Eq.~\ref{eq:optimal}, this method is statistically optimal for the determination of the $\sigma(\delta_i)$.  The functions $f_i(x,\cos\theta)$ are therefore called "optimal observables". It turns out~\cite{Diehl_1994} that the covariance matrix, hence the statistical uncertainties on the form factors, can be obtained analytically in the limit of a large number of events, which is the case considered in this letter. Specifically, if the total event rate is included in the derivation of the likelihood as is the case in Eq.~\ref{eq:likelihood}, the elements of the covariance matrix $V$ are given by (${\rm d}\Omega \equiv {\rm d}x {\rm d}\cos\theta$)
\begin{equation}
\label{eq:rate}
V_{ij} = {\cal L} \int {\rm d}\Omega { f_i \times f_j \over S^0}  \ ,
\end{equation}
while if the total event rate is not included in the likelihood, namely by removing the first term of the product in Eq.~\ref{eq:likelihood}, these elements take the form
\begin{equation}
\label{eq:norate}
V_{ij} = {\cal L} \left[ \int {\rm d}\Omega {f_i \times f_j \over S^0}  -  { \int {\rm d}\Omega  f_i  \int {\rm d}\Omega  f_j \over \int {\rm d}\Omega S_0 } \right],
\end{equation}
and the uncertainty on the form factor $\delta_i$ is simply 
\begin{equation}
\sigma(\delta_i) = \sqrt{ [ V^{-1} ]_{ii} } \ .
\end{equation}
This analytical procedure is used in Ref.~\cite{Grzadkowski_2000} to determine the sensitivity to top-quark electroweak couplings in $500\,{\rm fb}^{-1}$ of ${\rm e^+ e^-}$ collisions at $\sqrt{s} = 500$\,GeV, with or without incoming beam polarization. In this article, the authors evaluate the covariance matrix with Eq.~\ref{eq:rate}, but they let the total number of events float by adding a fictitious multiplicative form factor $\delta_0$ in front of $S^0$ in Eq.~\ref{eq:optimal}, hence increase the rank of the covariance matrix from 8 to 9. It was checked that this work-around is numerically equivalent to using Eq.~\ref{eq:norate}, {\it i.e.}, to not use the total event rate in the likelihood. 

A quick survey of Fig.~\ref{fig:distributions}, however, shows that $f_A^\gamma(x,\cos\theta)$, in the top-left corner, is almost degenerate with the standard model contribution $S^0(x,\cos\theta)$, in the bottom-right corner. Letting the normalization of the standard model contribution float is therefore bound to lead to very large statistical uncertainties on all form factors, as is indeed observed in Ref.~\cite{Grzadkowski_2000}. For this reason, and as is done in Ref.~\cite{Baer_2013}, the present study includes the total event rate in the determination of the uncertainties. 

As already mentioned, it is possible to determine simultaneously all eight form factors and their uncertainties. In the first configuration of Ref.~\cite{Baer_2013}, however, only the three coefficients $F_{1V}^\gamma$, $F_{1V}^Z$ and $F_{1A}^Z$ are allowed to vary. The other five form factors are fixed to their standard model values. In this simplified situation, Eq.~\ref{eq:optimal} reads
\begin{equation}
S(x,\theta) = S^0(x,\theta) - 2i\sin\theta_W \delta F_{1V}^\gamma  f_A^\gamma - 2i\sin\theta_W \delta F_{1V}^Z f_A^Z +  -2i\sin\theta_W \delta F_{1A}^Z f_B^Z \ ,
\end{equation}
which leads to the following $3\times 3$ covariance matrix $V_1 = 4\sin^2\theta_W \times {\cal L} \times X$, with
\begin{eqnarray}
\label{eq:firstInt} X_{11} = \int {\rm d}\Omega {(f_A^\gamma)^2 \over S^0} \ , \ & {\displaystyle X_{12} = \int {\rm d}\Omega {f_A^\gamma \times f_A^Z \over S^0}} \ , \ & X_{13} = \int {\rm d}\Omega {f_A^\gamma \times f_B^Z \over S^0} \ , \\
& {\displaystyle X_{22} = \int {\rm d}\Omega {(f_A^Z)^2 \over S^0} } \ , \ & X_{23} = \int {\rm d}\Omega {f_A^Z \times f_B^Z \over S^0} \ , \\
& & X_{33} = \int {\rm d}\Omega {(f_B^Z)^2 \over S^0} \ .
\end{eqnarray}
In the second configuration of Ref.~\cite{Baer_2013}, only the two coefficients $F_{2V}^\gamma$ and $F_{2V}^Z$ are allowed to vary, which leads to the even simpler expression of Eq.~\ref{eq:optimal}:
\begin{equation}
S(x,\theta) = S^0(x,\theta) - 2i\sin\theta_W \delta F_{2V}^\gamma  (f_A^\gamma + f_C^\gamma) - 2i\sin\theta_W \delta F_{2V}^Z  (f_A^Z + f_C^Z)\ ,
\end{equation}
and the following $2\times 2$ covariance matrix $V_2 = 4\sin^2\theta_W \times {\cal L} \times Y$, with 
\begin{eqnarray}
Y_{11} = \int {\rm d}\Omega {(f_A^\gamma + f_C^\gamma)^2 \over S^0} \ , \ & {\displaystyle Y_{12} = \int {\rm d}\Omega {(f_A^\gamma + f_C^\gamma) \times (f_A^Z + f_C^Z) \over S^0}} &  \ , \\
\label{eq:lastInt} & {\displaystyle Y_{22} = \int {\rm d}\Omega {(f_A^Z + f_C^Z)^2 \over S^0} } & \ .
\end{eqnarray}
The numerical results are presented in the next section for the case of the FCC-ee. 