deletions | additions
diff --git a/The_above_results_are_obtained__.tex b/The_above_results_are_obtained__.tex
index ccadea1..abf6078 100644
--- a/The_above_results_are_obtained__.tex
+++ b/The_above_results_are_obtained__.tex
...
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 $2.4\,{\rm
fb}^{-1}$ 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}
\begin{center}
\caption{\label{tab:f1} Precision on the four
form factors $F_{1V,A}^X$, $F_{1V,A}^X$ expected with $2.4\,{\rm
fb}^{-1}$ 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).} raw). The precision expected with $500\,{\rm fb}^{-1}$ at $\sqrt{s} = 500$\,GeV is 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 $F_{1V,A}^X$ & $1.2\, 10^{-3}$ & $2.9\, 10^{-3}$ & $0.0\, 10^{-2}$ & $2.4\, 10^{-2}$ \\
\hline All four $F_{1V,A}^X$ & $1.2\, 10^{-3}$ & $3.0\, 10^{-3}$ & $1.3\, 10^{-2}$ & $2.6\,
10^{-3}$ \\
\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 $F_{2A}^Z$ are CP even.
The With vanishing correlation coefficients, the two pairs of form factors can therefore be determined independently from each
other, with vanishing correlation coefficients. In practice, the determination of $F_{2A}^\gamma$ other. The precisions on $F_{2V}^\gamma$ and
$F_{2A}^Z$ can be achieved $F_{2A}^Z$, expected with
the difference of the ($x, \cos\theta$) distribution of the $\ell^+$ for events in which the top decays leptonically, and the ($x, -\cos\theta$) distribution of the $\ell^-$ for events in which the anti-top decays leptonically. Such a distribution $2.4\,{\rm fb}^{-1}$ at $\sqrt{s} = 365$\,GeV at the FCC-ee, is
expected thus unchanged with respect to
be identically 0 in the standard model and for Fig.~\ref{fig:baer} when the
other six form factors (which are all CP odd), hence depends solely of constraint on $F_{2A}^\gamma$ and
$F_{2A}^Z$. Conversely, the sum of the ($x, \cos\theta$) distribution of the $\ell^+$ and the ($x, -\cos\theta$) distribution of the $\ell^-$ $F_{2A}^Z$ is
expected 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
identically zero for these two form factors, hence depends solely of $F_{2V}^\gamma$ $2.5\, 10^{-3}$ and
$F_{2V}^Z$, when all $F_{1A,V}^X$ form factors are fixed to their standard model values. $8.3\, 10^{-3}$ respectively.
The
precision expected on the $F_{2V,A}^X$ form factors is summarized in Table~\ref{tab:f2}. The accuracy of the CP-violating form factors with the sole lepton angle and energy distributions is moderately
constraining, constraining ($1.4\, 10^{-1}$ and
will gain from an analysis $8.2\, 10^{-1}$ respectively) because of
additional observables.
\begin{table}
\begin{center}
\caption{\label{tab:f2} Precision the important correlation between the two distributions $f_D^\gamma$ and $f_D^Z$, well visible in Fig.~\ref{fig:distributions}, but a relevant precision of 1.7\% is reached on the
four form factors $F_{2V,A}^X$ expected linear combination $F_{2A}^\gamma + 0.17 \times F_{2A}^Z$ with $2.4\,{\rm
fb}^{-1}$ ab}^{-1}$ at $\sqrt{s} =
365$\,GeV 365$\,GeV, reduced to 0.85\% with $500\,{\rm fb}^{-1}$ at
$\sqrt{s} = 500$\,GeV. A reduction of the
FCC-ee by relaxing correlation between these two form factors will benefit from the
constraint on $F_{2A}^{\gamma,Z}$.}
\begin{tabular}{|l|l|l|l|l|}
\hline Precision on & $F_{2V}^\gamma$ & $F_{2V}^Z$ & $F_{2A}^\gamma$ & $F_{2A}^Z$ \\
\hline\hline & $8.1\, 10^{-4}$ & $2.3\, 10^{-3}$ & $1.4\, 10^{-1}$ & $8.2\, 10^{-1}$ \\ \hline
\end{tabular}
\end{center}
\end{table}
When analysis of additional observables.
Similarly, when all eight parameters are considered simultaneously,
however, the lepton angle and energy
distributions are no longer sufficient to avoid large correlations between form factors.
It is well visible from Fig.~\ref{fig:distributions} that $f_A^\gamma$ and $f_C^\gamma$ (i.e., $\tilde{F}_{1V}^\gamma$ and $\tilde{F}_{2V}^\gamma$, on The same observation was made in Refs.~\cite{Baer_2013,Amjad_2013} with the
one hand, and $f_A^Z$ and $f_C^Z$ (i.e., $\tilde{F}_{1V}^Z$ and $\tilde{F}_{2V}^Z$, on four observables chosen for the
other, are very much correlated. It is only 500\,GeV analysis. A generator-level exercise with
an analysis of more observables in the fully
reconstructed leptonic final
state, and all the pertaining observables, that these correlations are expected to disappear. Such an exercise state has
recently been
recently attempted in Ref.~\cite{ledib_2015}, released after the present study. In this exercise,
performed at $\sqrt{s} = 500$\,GeV and with incoming beam polarization, an
optimal-obervable optimal-observable analysis of the matrix element squared
in the fully leptonic final states is carried out with thirteen different observables (the top quark direction, the lepton angles and energies, the
b-quark ${\rm b}$ and ${\rm \bar b}$ angles and energies, and the invariant masses of the top quarks and W bosons),
all unambiguously determined with unambiguous identification and reconstruction under the assumption of a perfect detector.
Their The few degeneracies between form factors are indeed removed, and the conclusion is identical to that of this paper: the incoming beam polarizations are not essential
for a determination of in the
top-quark form factors. 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
entirely fully reliable results when the jets and the missing energy from the top decays are to be included. Such an analysis will be carried out when a complete simulation and reconstruction in a realistic detector become available for the FCC-ee study.
