\subsection{Perspectives for \epem experiments}
The investigation of top quarks constitutes one of the
physics goals for a future linear \epem collider
\cite{eacco,PIK,Zersaar,Igo}.
With an $R$ value
between $\sim0.7$ in the region very close to threshold and about 2
in the continuum (Fig.\ref{fig:3.5}) one expects $5\,000$ to
$10\,000$ \ttbn events for an integrated luminosity of
$10^{40}cm^{-2}$. This large counting rate must be extracted from the
background which consists mainly of $W$-pair and $Z$-pair production.
(Table \ref{Tbackgr1}).
\subsubsection{$t$ quark selection}
Selection of \ttbn final states on the basis of topology,
charged particle multiplicity and missing energy (Fig.\ref{Fsign1})
leads to a drastic reduction of
the background with still acceptable loss in efficiency.
\begin{figure}
\vspace{17cm}
\caption{\label{Fsign1} \em
Characteristic distributions for \ttbn signal events
(shaded histograms) and background events (white histograms). the
normalisation of the distributions is arbitary. (a) Thrust, (b) charged
particle multiplicity and (c) missing energy distributions.}
\end{figure}
Signal to
background ratios from 3 to 7 can be reached with these simple methods
with tagging efficiencies between 20\% and 30\%. The selection of events
with leptons aims at the prominent subsample with semileptonic
$t\rightarrow b W(\rightarrow\ell\nu)$ decays. As discussed before
these events are particularly useful for the analysis of
the $tbW$ coupling. The additional suppression from the semileptonic
branching ratio may
%\clearpage\noindent
\begin{table}
\caption[]{\label{Tbackgr1}
{\it Cross sections in {\it picobarn}
for the \ttbn signal and for the main
background processes. The \ttbn cross section is
taken as 0.7 times the ``point cross section" $R$.
All cross sections
are obtained with the inclusion of
initial state radiation (from\cite{PIK}).}}
\begin{center}
\begin{tabular}{||c||c||}
\hline \hline
Top mass (GeV) & 180 \\
$E_{c.m.}$ ($GeV$) & 370 \\
\hline \hline
\ttb & 0.444 \\
$f \bar f$ & 25.6 \\
$W^+W^-$& 10.3 \\
$Z^0Z^0$ & 0.792 \\
\hline \hline
\end{tabular}
\end{center}
\end{table}
pay off in terms of a pure event
sample. Selecting $b$ jets through their content of energetic leptons
or through the displaced decay vertex of $B$ mesons one may achieve up
to 50\% efficiency with a S/B ratio of up to 5. In the subsequent
considerations, taken from \cite{PIK}, a conservative \ttbn tagging
efficiency of 25\% and a S/B ratio of 5 were assumed.
%
\subsubsection{Threshold scan}
As stated already in the introduction two important goals of top quark
physics can be anticipated in the context of the SM: The determination
of the top quark mass through the measurement of $t$ quark decay
products and the threshold scan which will provide information on $m_t$,
and at the same time on the interaction between $t$ and $\bar{t}$
through the QCD potential and eventually through Higgs exchange.
The extraction of the rapidly varying cross section depends critically
on the beam energy spread and on the smearing through beamstrahlung.
A large variety of machine designs can be found in the literature which
differ drastically in luminosity and energy smearing.
Even in the most favorable (TESLA) design the toponium peak is almost
completely smeared out. There remains, however, a marked step at
threshold. This indicates that the determination of $m_t$ will be
possible to fairly high accuracy from the shape of the cross section;
the normalisation might carry information on the strenght of the QCD
potential. The extraction of more subtle parameters like the
Yukawa coupling or the top quark decay width will presumably be difficult,
assuming of course that they are not too different from their
standard model values.
\begin{figure}[ht]
\vspace*{7cm} % was 12cm
%\begin{center}
%\epsfig{file=top_exp_thr.eps,width=10cm,angle=89.5}
%\end{center}
\caption[]{\label{Fsim1} \em
Sensitivity of the exitation curve, convoluted by the
luminosity spectrum of the TESLA collider design, to the top quark mass,
at $m_t=180 GeV$. The full
lines correspond to the nominal value of $m_t$, the dashed (doted)
lines to values which differ from their nominal value by $\pm200 MeV$,
($\pm400 MeV$). The `` eperimental'' points correspond to a measuring
time of $10^6$ seconds each.}
\end{figure}
This expectation is borne out by a more detailed simulation
(Fig.\ref{Fsim1}), which suggests that a one sigma deviation of one to
several hundred MeV can be achieved.
Allowing for a realistic uncertainty in $\alpha_s$ and hence in the QDC
potential an error of 400 MeV to 600 MeV is expected. Using the total
cross section, the momentum distribution and the forward backward
asymmetry simultaneously \cite{Igo,Comas},
this can be further reduced to even less
than 100 MeV (Fig.~\ref{fig:comas});
%\begin{figure}[htbp]
%\begin{center}
% \leavevmode
% \vspace*{8cm}
%\end{center}
%\caption{Top momentum distribution at $\protect\sqrt{s}=360$
% GeV. Solid line: Born cross section; dashed line: with initial state
% radiation (ISR); dotted line: with ISR and beam effects, computed
% with the parameters of the TESLA accelerator design (from
% \protect\cite{Comas}).}
%\caption{\em Results of the $\chi^2$ fit using the total cross section and
% the momentum distribution (from \protect\cite{Comas}).}
%\label{fig:comas}
%\end{figure}
more than an
order of magnitude superior to the expectations from jet analysis at an
hadron collider.
\subsubsection{$t$ quark reconstruction}
The study of the excitation curve is complemented by the reconstruction
of top quark decay products, leading to an independent mass
determination. This may arise from the analysis of the $W$ momentum in
semileptonic \ttbn events, a method particularly suited for the
threshold region. At energies significantly higher than $2m_t$ the
invariant mass per hemisphere is a sensitive measure of $m_t$.
Identifying jets and combining them to reconstruct the top quark in a
constrained fit seems at present to be the most promising technique,
leading to a determination of the order of 1 GeV.
Combining the results of the different techniques the systematic
experimental errors should be well under control \cite{PIK}.
To evaluate the
influence of fragmentation
\newpage
\begin{figure}[p]
%\begin{center}
% \leavevmode
\vspace*{8cm}
%\end{center}
%\caption{Top momentum distribution at $\protect\sqrt{s}=360$
% GeV. Solid line: Born cross section; dashed line: with initial state
% radiation (ISR); dotted line: with ISR and beam effects, computed
% with the parameters of the TESLA accelerator design (from
% \protect\cite{Comas}).}
\caption{\em Results of the $\chi^2$ fit using the total cross section and
the momentum distribution (from \protect\cite{Comas}).}
\label{fig:comas}
\end{figure}
\begin{figure}[hbt]
\vspace{8cm}
\caption{\label{FHiggsde}
\em ``Explored'' regions (below the curves) for different values of
$\int{\cal L}$ in the $M_{H^\pm}$ vs. $\tan\beta$ plane (99.73\%
C.L.) (from \protect\cite{Venturi}).}
\end{figure}
\clearpage
\noindent
and to relate this kinematic top mass to the
one determined in the threshold scan seem to be the limiting factors at
present.
%\clearpage\noindent
%\begin{figure}
%\vspace{8cm}
%\caption{\label{FHiggsde}
%\em ``Explored'' regions (below the curves) for different values of
%$\int{\cal L}$ in the $M_{H^\pm}$ vs. $\tan\beta$ plane (99.73\%
%C.L.) (from \protect\cite{Venturi}).}
%\end{figure}
\subsubsection{Anomalous top quark decays}
The search for anomalous top quark decays is facilitated by the
favourable signal to background ratio predicted for electron positron
annihilation \cite{PIK,Venturi}.
These decays could be an important element in the study of
physics beyond the standard model.
\underline{The decay mode $t\rightarrow H^+ b$}
is predicted with a sizeable branching ratio in a two Higgs doublett
model. It would manifest itself by an excess of $\tau$ leptons
(discussed before) as a consequence of the large
branching ratio for $H\rightarrow \tau\nu$.
Searching for $\tau$ ``jets'' with a single charged prong and posssible
additional neutral particles leads to a total $\tau$ efficiency of 30\%.
This allows the detection of the anomalous decay $t\rightarrow H^+ b$
with a statistical significance of at least 3 standard deviations for
a large part of the parameter space, in particular large
$\tan\beta$ in a data sample of 10~fb$^{-1}$, as
exemplified in Fig.\ref{FHiggsde}.
%\begin{figure}
%\vspace{10cm}
%\caption{\label{FHiggsde}
%\em ``Explored'' regions (below the curves) for different values of
%$\int{\cal L}$ in the $M_{H^\pm}$ vs. $\tan\beta$ plane (99.73\%
%C.L.) (from \protect\cite{Venturi}).}
%\end{figure}
With the masses of supersymmetric particles being only loosely
constrained\\
\underline{the decay mode $t\rightarrow \tilde t\tilde\gamma$}
followed by the decay chain
$\tilde t\rightarrow b\tilde W\rightarrow b W\tilde\gamma$
would provide a remarkable laboratory for
supersymmetry. An exploratory study \cite{PIK,Venturi}
indicates that this channel, if
kinematically accessible, may be detected and studied
with a a reasonable investment of luminosity.