\section{Theoretical framework}
\label{sec:theory}

The top-quark couplings to the photon and the Z can be parameterized in several ways. In Ref.~\cite{Baer_2013}, for example, the analysis makes use of the usual form factors denoted $F_1$, $F_2$, defined in the following expression (with $X = \gamma, Z$):
\begin{equation}
\Gamma_\mu^{ttX} = -ie \left\{ \gamma_\mu \left( F_{1V}^X +\gamma_5 F_{1A}^X \right) + {\sigma_{\mu\nu} \over 2 m_{\rm t}} (p_t + p_{\bar t})^\nu \left( i F_{2V}^X + \gamma_5 F_{2A}^X \right)\right\},
\end{equation}
with, in the standard model, vanishing $F_2$s and
\begin{eqnarray}
F_{1V}^{\gamma} = -{2\over3} \ & , & F_{1V}^Z = {1\over 4\sin\theta_W\cos\theta_W} \left(1-{8\over3}\sin^2\theta_W\right) \ , \\ F_{1A}^{\gamma} = 0 \ & , & F_{1A}^Z = {1 \over 4\sin\theta_W\cos\theta_W} \ .
\end{eqnarray}
The sensitivities are expressed therein in terms of $\tilde{F}_1$, $\tilde{F}_2$ defined as
\begin{equation}
\tilde{F}_{1V}^X = -({F}_{1V}^X+{F}_{2V}^X)\ , \ \tilde{F}_{2V}^X = {F}_{2V}^X\ , \ \tilde{F}_{1A}^X = -{F}_{1A}^X\ , \ \tilde{F}_{2A}^X = -i{F}_{2A}^X\ .
\end{equation}

On the other hand, the optimal-observable statistical analysis presented in the next section, based on Ref.~\cite{Grzadkowski_2000}, uses the following $A,B,C,D$ parameterization (with $v = \gamma, Z$): 

\begin{equation}
\Gamma^\mu_{ttv} = {g\over 2} \left[ \gamma^\mu \left\{ (A_v+\delta A_v) - \gamma_5 (B_v+\delta B_v) \right\} + {(p_t -p_{\bar t})^\mu\over 2 m_{\rm t}}  \left( \delta C_v - \delta D_v \gamma_5 \right) \right],
\end{equation}
which easily relates to the previous parameterization with
\begin{eqnarray}
A_v+\delta A_v = - 2i\sin\theta_W \left( F_{1V}^X + F_{2V}^X \right) & \ ,  & B_v+\delta B_v = - 2i\sin\theta_W F_{1A}^X \ , \\  \delta C_v = -2i\sin\theta_W F_{2V}^X & \ , & \delta D_v = -2\sin\theta_W F_{2A}^X \ . 
\end{eqnarray}
The expected sensitivities on the anomalous top-quark couplings can be derived in any of these parameterizations. Although originally derived with that of Ref.~\cite{Grzadkowski_2000}, the final estimates presented in this study, however, use the parameterization of Ref.~\cite{Baer_2013}, for an easy comparison. For the same reason, \cross{although it is not needed,} the same restrictions as in Ref.~\cite{Baer_2013} are applied here: only the six CP conserving form factors are considered (i.e., the two $F_{2A}^X$ are both assumed to vanish), and either the four form factors $F_{1V,A}^X$ are varied simultaneously while the two $F_{2V}^X$ are fixed to their standard model values, or vice-versa. A careful reading of Ref.~\cite{Baer_2013} shows that the form factor $F_{1A}^\gamma$ was also kept to its standard model value, as a non-zero value would lead to gauge-invariance violation. It is straightforward to show that, under these restrictions, the three parameterizations lead to the same sensitivities on $F_i$, $\tilde{F}_i$ and $A,B,C,D$ (with a multiplicative factor $2\sin\theta_W \sim 0.96$ for the latter set). 

The tree-level angular and energy distributions of the lepton arising from the ${\rm t \bar t}$ semi-leptonic decays are known analytically as a function of the incoming beam polarizations and the centre-of-mass energy~\cite{Grzadkowski_2000}: 
\begin{equation}
{{\rm d}^2\sigma \over {\rm d}x {\rm d}\cos\theta} = {3\pi\beta\alpha^2(s) \over 2s} B_\ell S_\ell(x,\cos\theta),
\end{equation}
where $\beta$ is the top velocity, $s$ is the centre-of-mass energy squared, $\alpha(s)$ is the QED running coupling constant, and $B_\ell$ is the fraction of ${\rm t\bar t}$ events with at least one top quark decaying to either ${\rm e}\nu_{\rm e}{\rm b}$ or $\mu\nu_\mu{\rm b}$ (about 44\%). As the non-standard form factors $\delta(A,B,C,D)_v \equiv \delta_i$ are supposedly small, only the terms linear in $\delta_i$ are kept: 
\begin{equation}
\label{eq:optimal}
S(x,\theta) = S^0(x,\theta) + \sum_{i=1}^8 \delta_i f_i(x,\cos\theta),
\end{equation}
where $x$ and $\theta$ are the lepton (reduced) energy and  polar angle, respectively, and $S^0$ is the standard-model contribution. The eight distributions $f_{A,B,C,D}^{\gamma, Z}(x,\cos\theta) \equiv f_i(x,\cos\theta)$ and the standard-model contribution $S^0(x,\cos\theta)$ are shown for $\ell^-$ in Fig.~\ref{fig:distributions} at $\sqrt{s} = 365$\,GeV, with no incoming beam polarization. 