%ttp93_23
%Title:  Top Quarks at a Linear Collider
%Author:  Johann H. K\"uhn
%Published:  <EM>Proceedings of the 2d International Workshop on Physics and Experiments with Linear e+e- Colliders, April 1993, F. Harris, ed., pp. 72--94.</EM>
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\begin{document}
\hfill TTP93--23\\
\mbox{}
\hfill  June  1993   \\   %                 \hfill \today \\
\vspace{0.5cm}
\begin{center}%***********************************
 {\tenbf TOP QUARKS AT A LINEAR COLLIDER%
\footnote{\normalsize
Talk presented at the Workshop
{\em Physics and Experiments with Linear $e^+e^-$ Colliders},
Waikoloa, Hawaii, April 1993, to be published in the proceedings.}
 \\}
\vglue 0.9cm
{\tenrm JOHANN H. K\"UHN%
\footnote{\normalsize Supported by BMFT Contract 055KA94P.}
\\}
\baselineskip=13pt
{\tenit Institut f\"ur Theoretische Teilchenphysik\\}
\baselineskip=12pt
{\tenit Universit\"at Karlsruhe\\}
\baselineskip=12pt
{\tenit D-76128 Karlsruhe, Germany\\}
\vglue 0.7cm
{\tenrm ABSTRACT}%
\end{center}%****************************************

%\vglue 0.3cm
\begin{center}
\begin{minipage}{12.5cm}
{\rightskip=3pc\leftskip=3pc\tenrm\baselineskip=12pt%
\narrower\smallskip\noindent%
Theoretical results for top quark physics at a linear electron positron
collider will be presented. The predictions for the
top quark decay rate and distributions are described, including
the influence of radiative corrections and of extensions of the
Standard Model. The production cross section for top quarks
through
electron positron annihilation and photon photon fusion is large
throughout, which  may allow an accurate determination of the top
magnetic moment as well as interesting measurement of CP violating
observables. The cross section in the threshold region as well as the
momentum distribution of top quarks and their decay products will lead
to an accurate determination of the top quark mass, the interquark
potential, and eventually even of the Yukawa coupling to the Higgs
boson. It is shown that higher order corrections through
final state interaction and relativistic corrections are small and
sufficiently under control.
\smallskip
}
\end{minipage}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%   MAIN TEXT
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%\newpage
\sec{1}{Introduction}
The ever-increasing energies of electron
positron colliders from PETRA, PEP, and TRISTAN up to LEP as well
as those of the hadron colliders $Sp\bar pS$ and TEVATRON have led to
an impressive increase of the lower bound of the top quark mass.
The limits from LEP (44.5 GeV) and from the absence of
$W\to t+b$
as inferred from
the $W$ decay rate (54 GeV) are derived in a fairly
model-independent way.  The limits from the TEVATRON on the other
hand depend on the dominance of the $W\to t+b$
decay mode and hence are
applicable in the context of the Minimal Standard Model (MSM)
only.  The mass bounds of 108 GeV from CDF and 103 GeV from D0
which have been quoted at this meeting \cite{li} are derived on the
basis of production cross sections which in both cases are based on
next-to-leading-order calculations \cite{nlo}.  However, the D0 limit
includes recent higher-order contributions \cite{giele}
which raise the
cross section and increase the limit by about 5 GeV.

Precision measurements of electroweak parameters exploit the
quadratic top mass dependence \cite{veltman}
of the $\rho$-parameter, resulting  in \cite{schailering}
$m_t=155+17-19{\rm GeV}$
with an additional error of 20 GeV
from the Higgs mass range between 60
GeV and 1~TeV with a central value at 200 GeV.  Future top
searches at the TEVATRON with increased statistics and efficiency
will able be to cover this interval to a large extent \cite{li}.

Top masses above 54 GeV and below the Tevatron limit are still
possible in extensions of the MSM consistent with the present
data, notably in the two Higgs doublet model (THDM) where the
decay mode $t\to H+b$ may dominate for $m_t$ below  90 GeV.
However, this window will be closed by LEP200.

The discovery of the top quark will not only complete the fermionic
spectrum of the Standard Model.  The determination of its mass
will provide important input for all calculations, leaving the
Higgs mass as only free parameter which could then be inferred
from precise measurements planned at LEP in the near future.
Once discovered, the study of the top quark and
its decay modes will constitute an important part of the physics
program at the TEVATRON, LHC, SSC, and the future linear collider%
\footnote{%
For recent reviews on top and toponium in $e^+e^-$ collisions see
\cite{KZ,KZburas,kuhndesy,PIKdesy,zerfin,fujiifin}.}.

Assuming that the top quark has been already discovered in
hadronic collisions, electron positron experiments will serve to
determine its mass with high precision, and to explore its gauge
coupling to the $W$ and $Z$ bosons.
The Yukawa coupling to the Higgs boson
which drives the large top mass should be explored and might
provide important information on the mechanism of fermion mass
generation.  A variety of novel QCD studies is at hand as a
consequence of its rapid weak
decay which intercepts the hadronization
process.  Close to threshold one may explore the QCD potential,
and eventually the Yukawa potential induced by Higgs exchange.
Decay modes of the top which are predicted in extensions of the
Standard Model
(SM) could be searched, for example decays into a charged Higgs
boson plus a bottom quark or
into the supersymmetric partner of the
top plus a photino, or one may even search for FCNC decays.
Also new couplings can be
explored with sufficient statistics, for example an anomalous
magnetic moment or CP violating coupling at the $Zt\bar t$
or the $Wt\bar t$ vertex.

Section 2 of this review will be concerned with the present
status of the calculations for the decay of top quarks.
In a first step the total
decay rates, QCD and electroweak corrections will be treated and
potential deviations in extensions of the SM discussed.  The most
recent calculations of the spectrum of $W$s and leptons from top
decays will be summarized.  The prediction for $t\bar t$ production far
above threshold are collected in section 3.  These include the
calculation of the total cross section, search techniques for CP
violation, the interplay between top decay and hadronization, and
a brief comment on $t\bar t$ production in photon-photon and electron-%
photon collisions.  Section 4 is devoted to the threshold region
and the remnants of toponium.  This includes a discussion of the
total cross section and of momentum distributions, of theoretical
ambiguities in the top mass determination, and of relativistic
corrections.

\secsubsec{2}{Top decays}{2.1}{Qualitative aspects}
%
Top quark physics is to a large extent dictated
by the large decay rate (Fig.\ref{figwidth}),
which  quickly reaches its asymptotic form
\begin{equation}
\Gamma \left(t \to b+W^+\right) \to 175{\rm MeV}
\cdot \left(\frac{m_t}{
m_W}\right)^3
\label{eq:3}
\end{equation}
Three steps in the increase of the decay rate affect the
qualitative behavior of top mesons \cite{bigi}:\\
\pictleft{8cm}{8.5cm}{6.9cm}{figwidth}%
{Solid line: The width of the top quark in the SM, including
QCD and electroweak corrections and the finite $W$ width.
Dashed line: Born and narrow width approximation.}
\noindent
{\em i)}  Once $m_t$ is above 70 GeV the hyperfine splitting between
$T$ and $T^*$ cannot be resolved anymore:
\begin{displaymath}
m_{T^*} - m_{T} \propto 1/m_t < 1 {\rm MeV} < \Gamma_t
\end{displaymath}
The angular distribution of all decay products follow the
predictions derived for a spin 1/2 $t$-quark and may serve to
analyze the top spin.

\noindent
{\em ii)}  Above 100 GeV the width exceeds 100 MeV, the characteristic
scale for the formation of top mesons.  No $T$ or $T^*$ mesons can be
formed.

\noindent
{\em iii)}  Toponium states are severely affected once
$m_t$ is above 130 GeV, leading to a toponium width
$2\Gamma_t\ge0.88 {\rm GeV}$
comparable to the $1S$-$2S$ mass difference.
A complicated interplay between binding forces and the rapid decay
is predicted.

\pictleft{9.5cm}{5.5cm}{5.4cm}{fig:f}%
{QCD correction function $f$ for the top quark
decay rate.  Solid line:  exact form; dashes:  $m_b = \Gamma_b=0$.}
\subsec{2.2}{Radiative Corrections}
These quantitative considerations are supplemented by precise
calculations including QCD and electroweak corrections to the
decay rate and the spectra.  A reduction of $\Gamma_t$ by
a relative amount $\delta^{(1)}_{QCD}$ of about 8\% is
induced by QCD \cite{jezrate1,jezrate2,jez}.
QCD corrections including the effects from the nonvanishing $\Gamma_W$
and $m_b$ are also contained in these papers.
They are conventionally cast into the form
\begin{equation}
\Gamma = \Gamma _{Born} \left( 1 -
\frac{2}{3} \frac{\alpha _s}{\pi} f(m_t,m_W,m_b,\Gamma_W) \right)
\label{eq:20}
\end{equation}
In the narrow width approximation and for vanishing bottom mass a
particularly simple formula has been derived
for the correction function $f$:
\begin{eqnarray}
f & = &  \left[ \pi ^2 + 2 Li_2 (y) - 2Li_2 (1-y)\right] \nonumber \\
 & & + \Bigl[4y(1-y-2y^2) \ln y + 2(1-y)^2 (5+4y) \ln(1-y)
  \nonumber \\
 & & - (1-y)(5+9y-6y^2) \Bigr] \Bigm/ 2(1-y)^2 (1+2y)
\label{eq:21}
\end{eqnarray}
where $y=m^2_W / m^2_t$.
The quality of this approximation can judged from Fig.\ref{fig:f}.

Potentially large electroweak corrections could be expected from
the enhanced Yukawa couplings, providing terms $\propto m_t^2/m_W^2$.
However, as
shown in \cite{Margolis} large accidental cancelations among these
leading terms render the weak corrections fairly small.  This
result has been confirmed and strengthened by the full
calculation performed in \cite{Denner,Soni}.
These electroweak corrections increase the rate by a relative amount
$\delta_{ew}$ of about 1.5\%.


The finite width of the $W$ leads to a reduction $\delta_{\Gamma_W}$
of about the same magnitude \cite{jezrate2}.
The individual corrections and the
overall predictions are displayed in Table 1 for a few
characteristic masses.

Several sources of theoretical uncertainties remain.  Not yet
calculated ${\cal O} (\alpha_s^2)$-terms
can be estimated to contribute about 1\%,
the influence of the uncertainty in $m_t$ (estimated to .5
GeV) will induce another 1\%.  With an estimated experimental
resolution of about 5 to 10\% at best at a future linear collider,
these theoretical uncertainties will not pose a serious problem in the
foreseeable future.

\begin{table}[h]
\caption{Top width as a function of top mass and the comparison of
the different approximations.}
\begin{center}
\begin{tabular}{|r|c|c|c|c|c|c|c|c|c|} \hline
$m_t\ $  & $\alpha_s(m_{t})$ & $\Gamma^{Born}_{nw}$ &
$\delta^{(0)}_{\Gamma_W}$& $\delta^{(1)}_{QCD}$ &$\delta^{}_{ew}$
&$\Gamma^{}_{t}$\\
{\scriptsize(GeV)} &  & {\scriptsize(GeV)} & {\scriptsize(\%)} &
{\scriptsize(\%)} &
{\scriptsize(\%)} &
{\scriptsize(GeV)} \\
\hline
 110.0& .115& .1955& -1.44 &-7.83&  1.20& .1796\\
 150.0& .110& .8852& -1.69 &-8.47&  1.57& .8087\\
 190.0& .106& 2.059& -1.39 &-8.47&  1.73& 1.890\\
\hline
\end{tabular}
\end{center}
\end{table}
\subsec{2.3}{Beyond the Standard Model}
An important issue will be the search for new decay modes
expected in extensions of the Standard Model.  \\
{\em Higgs decays:} The impact of the two Higgs doublet model (THDM)
on radiative corrections to $t\to W+b$ is fairly small
\cite{denhoang} and similar results are
expected for the minimal supersymmetric SM (MSSM).  However, new
decay modes with sizable rates are possible for appropriate mass
combinations, for example $t\to b+ H^+$ or $t\to \tilde t +
\tilde\gamma$. Large branching ratios are
conceivable in the most general THDM for the decay into the
charged Higgs boson which may even dominate if the ratio between
the two vacuum expectation values $v_1/v_2=\tan\beta$
is chosen sufficiently
small, corresponding to an enhanced $tbH$ Yukawa coupling above its
"natural" value of $m_t\sqrt{\sqrt{2}G_F}$  (Fig.\ref{figHiggs}).
\pictleft{8cm}{6.5cm}{6.8cm}{figHiggs}%
{Branching ratios for the decay $t\to bW^+$ and $t\to bH^+$ in
two Higgs doublet models \cite{Higgsbr}.}
For the "natural" range of $\tan\beta$
between 1 and $m_b/m_t$
and assuming a top mass of 150 GeV one predicts
branching ratios from a few up to 25\%.  These considerations can
be made more quantitative in the context of the MSSM \cite{Borzumati}.
Assuming that SUSY particle masses are outside the mass range
that will be covered in experiments in the near future,
one finds a fairly restricted range of $\tan\beta$  and $m_H$
consistent with all SUSY mass relations (Fig.\ref{figBorz}).
\pictup{15cm}{6.5cm}{figBorz}%
{Allowed values of $\tan\beta$ versus $m_t$ for relatively
loose (left, ``Set I'')  and for strong (right, ``Set II'')
assumptions on the limits on SUSY particle masses \cite{Borzumati}.}
\pictup{9cm}{8cm}{figBorz2}%
{Ratio of Higgs- versus $W$-decay for the same input as Fig.
\ref{figBorz}, ``Set I''.}
A characteristic range of branching ratios for a given value of $m_t$,
corresponding to the same values of $\tan\beta$ as in Fig.\ref{figBorz},
is shown in Fig.\ref{figBorz2}.
QCD corrections to the Higgs decay mode
\cite{Czarn} as well as electroweak corrections \cite{higgscor}
are available in the literature. \\
{\em SUSY decays:}
Decays into the stop quark $\tilde t$ and the lightest neutralino are
possible if the masses of both decay products are sufficiently
small.  The branching ratio for this channel is governed by the
ratio of electromagnetic versus Yukawa coupling and the
additional phase space suppression
\begin{equation}
\frac{\Gamma(t\to\tilde t\tilde\gamma)}{\Gamma(t\to b W)}\approx
\left(\frac{4\pi\alpha}{\sqrt{2} G_F m_t^2}\right)\times
\mbox{phase space ratio}
\end{equation}
and hence below 10\%.
The possible mass values for $m_t$, $m_{\tilde t}$ and $m_{\tilde
\gamma}$ are again related in the MSSM \cite{Borzumati}.

Perspectives for the experimental searches for these decay modes
are discussed in \cite{Venturi,PIKhaw}.
The $t\to Hb$ decay should not be
considered the prime opportunity for an $e^+e^-$ collider to discover
the charged Higgs boson, since pair production may proceed at a
lower threshold.  However, it allows to measure the strength and
chirality of the tbH coupling through the study of the rate and
of the angular distribution\cite{KZ} of the Higgs from the decay of a
polarized top quark.\\
{\em FCNC decays:}  FCNC decays into a charmed quark plus a gluon,
photon, $Z$ or Higgs boson are entirely negligible in the
context of the SM, with branching ratios of $10^{-7}$ down to $10^{-12}$
\cite{Eilam}.  The perspectives are more favorable if conventional
quarks mix with new quarks which might exist in SU(2)
representations different from $I=\frac{1}{2}$.
Typical examples are quarks
in vector representations, iso singlet quarks, or mirror quarks.
Large FCNC induced rates are predicted \cite{Fritzsch,Buchm}
with branching rations ranging from $10^{-3}$ up to $10^{-2}$.
To suppress FCNC transitions in the THDM
to a tolerable rate the structure of couplings
has to be adjusted carefully, and is normally chosen
to completely inhibit $t\to cH$ decays in Born approximation.
However, as argued in \cite{Hou},
this need only be the case for light quarks.  Nondiagonal Yukawa
couplings $\propto \sqrt{m_i m_j}/v$
are well compatible with present experiments and
would lead to branching ratios for $t\to cH$ up to 1\%.


\subsec{2.4}{Distributions}
The lowest-order prediction for the energy and angular
distribution of leptons from a polarized top quark in the narrow
width approximation and with $m_b=0$ is extremely simple and
factorizes into an energy and an angular dependent term
\begin{equation}
\frac{dN}{d\cos\theta dx}\propto x(1-x) (1+\cos\theta)
\end{equation}
The lepton angular distribution is therefore an ideal tool to
analyze the polarization of top quarks which is induced either
through longitudinal beam polarization or - naturally - through
$\gamma Z$ interference.  QCD corrections, including mass effects and
the nonvanishing $W$ width, are available in the literature
\cite{distcor}. Their
impact on the lepton spectrum is quite important, the angular
dependence, however, is hardly affected.  Predictions for the $W$
energy distribution are also available \cite{czarn,junger}.
The simple $\delta$-function like
energy distribution is smeared by the $W$ Breit Wigner distribution
and by QCD corrections as demonstrated in Fig.\ref{fwen}.
\pictleft{9.5cm}{5.5cm}{5.4cm}{fwen}%
{Distribution of the W energy for $m_t = 120$~GeV
without (dashed) and with (solid curve) QCD corrections \cite{czarn}.}

\secsubsec{3}{Open top production in  $e^+e^-$ collisions}%
{3.1}{Standard Model predictions}
The cross section for $e^+e^-\to t\bar t$ is
fairly large throughout.  It starts with a step-function-like
increase to $R\approx0.7$
at threshold as a consequence of the Coulomb
enhancement $\propto1/v$
and increases to a value of $R\approx 2$ for energies
above $2m_t+100{\rm GeV}$ (Fig.\ref{rfig}).
Electroweak radiative corrections to top production at threshold and
in the continuum have been calculated in \cite{GKKS,benhol}.
Relative to the $G_F$ parametrised Born approximation they decrease
the cross section by -6\% to -9\%, if $m_t$ is varied between 100 and
200 GeV, $m_H$ between 40 and 1000 GeV and $E_{cm}$ fixed to 500 GeV.
The influence of
Higgs exchange on the threshold behavior will be discussed in more
detail below.

The production rate for $t\bar  t$
and the number of events (for $\int {\cal L} dt=10 {\rm fb}^{-1}$) are
contrasted with the potential backgrounds $f\bar f$, $WW$ and $ZZ$
in Table 2.
Detailed simulations demonstrate \cite{PIKdesy}
that a signal-to-noise ratio of 10
can be achieved with an efficiency of 30\%.  The
top mass can be determined with an estimated uncertainty of about
0.5 GeV. More details on the experimental prospects can be found
in \cite{PIKhaw}.

The large top decay rate allows for a variety of novel
QCD studies. The rapid decay intercepts the evolution of the
string of soft hadrons between $t$ and $\bar t$ \cite{sjostr}.
For an extremely large top width of more than 2 GeV
corresponding to $m_t$ close to 200 GeV the string is actually
spanned between the $b$ and the $\bar b$
such that the multiplicity depends on the
invariant mass of the $b\bar b$ system.  Perturbative soft gluon
radiation is similarly affected and cut off for particular kinematic
configurations \cite{soft}.  This effect is particularly pronounced
for $e^+e^-$ energies in the TeV range \footnote{This field is reviewed
in more detail in \cite{PIKhaw}.}.
\pictup{10.1cm}{4cm}{rfig}%
{Cross section for $t\bar{t}$ production,
including resonances, QCD corrections
and initial state radiation in units of $\sigma_{point}$.}
\begin{table}[htbp]
\caption[]{\label{Tbackgr}
Cross sections in {\it picobarn}
for the $t\bar t$ signal and for the main
background processes. When $E_{cm}$ is close to the threshold (first
3 columns), the $t\bar t$  cross section is
taken as 0.7 times the ``point cross section''. Far above threshold,
the cross sections are those calculated by PYTHIA. All cross sections
are obtained with the inclusion of
initial state radiation \cite{PIKdesy}.}
\begin{center}
\begin{tabular}{||c||c|c|c|c|c||}
   \hline  \hline
Top mass (GeV)          & 120 & 150 & 180 & 150 & 150 \\
$E_{c.m.}$ ($GeV$)  & 250 & 310 & 370 & 400 & 500 \\
   \hline \hline
$t\bar t$    & 0.972 & 0.632 & 0.444 & 0.821 & 0.656 \\
$f \bar f$ & 58.6 & 36.6 & 25.6 & 21.9 & 16.6 \\
$W^+W^-$  & 15.2 & 12.5 & 10.3 & 9.46 & 7.44 \\
$Z^0Z^0$ & 1.25 & 0.983 & 0.792 & 0.719 & 0.572\\
   \hline \hline
\end{tabular}
\end{center}
\end{table}

\subsec{3.2}{Form factors and CP violation}
The analysis of top decay
products may serve to measure a variety of form factors governing
$t\bar t$ production, notably a possible anomalous magnetic moment, and
top decay.  The physics motivation for such an analysis and the
experimental perspectives are reviewed in \cite{PIKhaw,Peskin}.  It has
to be remembered, however, that also hadronic interactions induce
a nonvanishing anomalous magnetic moment.
In the static limit the familiar $g-2=F_2(q^2=0)
=\frac{\alpha_s}{\pi}$ is
predicted, for positive $q^2=s$ one expects a more complicated
energy dependence with an enhancement towards the threshold.
A determination of the anomalous magnetic moment
to better than 1\% is thus overwhelmed by
uncertainties from perturbative QCD.

Above threshold $F_2$ develops
an imaginary part which is responsible for a "T-odd" effect, the
polarization of top quarks perpendicular to the production plane
\begin{equation}
P_\perp\propto \beta\frac{\alpha_s}{\pi}\frac{m_t}{\sqrt s}\sin\theta
\end{equation}
which could be observed through an excess of leptons above (or
below) the production plane \cite{Kane,Reiter,Arens}.
This induced ``T-odd'' effect which is expected in the SM must be
distinguished from true CP violation.  The latter can be
established experimentally only through the comparison of $t$ and
$\bar t$ decay products.
As far as CP violation in production amplitudes
is concerned, the basic parameters are the electric dipole moment
$d^\gamma$ which gives rise to an effective $\gamma t\bar t$ coupling
$d^\gamma\sigma_{\mu\nu} k^\nu\gamma_5$ and a
similar $Z t \bar t$ coupling proportional to $d^Z$.
Through their
interference with the SM amplitude they induce nonvanishing
expectation values for CP odd observables, for instance
\begin{equation}
\langle(\hat e\times\hat t)(\vec s_+ - \vec s_-)\rangle=
\sqrt{s}\left(c^\gamma \mbox{Re} \frac{d^\gamma}{e} + c^Z\mbox{Re}
         \frac{d^Z}{e}\right)
\end{equation}
Here $\hat e$ denotes the direction of
the electron beam, $\hat t$ the direction of the outgoing top momentum
and $\vec s_+$ and $\vec s_-$ the $t$ and $\bar t$ spins.
Exploiting the pronounced
correlation between spin and (anti-)lepton direction $\ell_\pm$
one expects a nonvanishing expectation value of
\begin{equation}
A_1=\langle(\hat e\times \hat t)\,(\hat\ell_+ - \ell_-)\rangle
\neq 0
\end{equation}
The simulation of this decay chain leads to an estimate for the
``sensitivity coefficients'' $c_\gamma$ and $c_Z$.
$c_\gamma$ varies between 0.02 and
0.08 from threshold to 500 GeV and $c_Z$ between 0.08 and 0.24
correspondingly \cite{Bernr}.

Similar considerations are applicable for the imaginary parts of
$d_\gamma$ and $d_Z$,
which can be studied through the correlation
$A_2 =\langle\hat t(\hat\ell_+ - \hat\ell_-)\rangle$ or for
the decay amplitude.
The predicted effects are entirely negligible in the context of the SM.
\pictup{15cm}{6cm}{figBern}%
{Signal-to-noise ratios $A_1/\Delta{\cal O_1}$ (left)
and $A_2/\Delta{\cal O_2}$ (right)
for $m_t=150$~GeV, $m_{\phi_1}=60$~GeV and
$\gamma_{CP}=2$. (From \cite{Bernr})}

Apart from the strategy to simply postulate ad hoc
a nonvanishing electric dipole moment one may construct
and investigate models specifically tailored for large CP
violation.
Typical examples are the THDM with explicitly CP breaking
parameters, the Weinberg model with three Higgs doublets and
spontaneous CP breaking, or SUSY models with CP violating
$t\tilde t \tilde g$
interactions, incorporating a relative phase between right- and
lefthanded couplings.  Asymmetries up to one percent may arise for
specific assignments of masses and coupling constants, in
particular for light Higgs bosons, or $m_{\tilde t}$ close to $m_t$
and for
energies close to the $t\bar t$ production threshold \cite{Bernr}.
The ratio between
the shift in the central value $A$ of the distribution of the observable
$\cal O$ and the width of this distribution $\Delta \cal O$
characterizes the statistical significance of a signal, and is
denoted by ``signal''/noise in Fig.\ref{figBern}.
The predictions for this
ratio  based on specific assignments of the masses and
coupling constants in the THDM
are shown in Fig.\ref{figBern} for the two observables
discussed above.  Similar considerations can be found in
\cite{Yuan,Atwood}, where the possibilities are discussed to analyze
also $b$ and $\bar b$ jets from top decay or the transverse
polarization of tau leptons from the semileptonic top decay.

\subsec{3.3}{Yukawa couplings.}
The determination of the $t\bar t H$ coupling
constitutes one of the central aims of top quark studies at a
linear collider.  Apart from the influence of Higgs exchange
in the threshold region two
reactions have been suggested which are sensitive to the Yukawa
coupling:  for small Higgs mass the radiation of a real Higgs boson
\cite{Dj}
\begin{equation}
e^+e^-\to t\bar t H
\end{equation}
for a heavy
Higgs boson the reaction\cite{Hagi}
\begin{equation}
e^+e^-\to t\bar t Z
\end{equation}
which receives important
contributions from an intermediate Higgs boson.  Cross sections
which are typically in the range of several femtobarn require
large luminosities and optimal background rejection \cite{PIKhaw}.

\subsec{3.4}{Top production in photon photon collisions.}
Energetic electron
beams can be efficiently converted into well collimated
energetic photon beams by colliding the former with a pulse of
low energy photons from a high power laser source.  Adjusting the
photon and electron helicities appropriately, one may enhance
$\gamma\gamma$
collisions with fairly well defined cm energies and with a dominantly
helicity zero configuration.  This leads to a large enhancement
of $t\bar t$ production in the threshold region.  QCD corrections lead
to a further increase close to threshold by the factor
$(1+\pi\alpha_s/v)$.
The full evaluation of QCD corrections and of the impact of polarized
laser and electron beams was performed in \cite{KMS,KZZ} and related
consideration can be found in \cite{Eboli,Gabbiani,Brodski}.
QCD corrections are fairly important as a consequence of the large
threshold enhancement. This is demonstrated in Fig.\ref{figgamm}.
\pictup{10.5cm}{5.7cm}{figgamm}%
{Effective cross section for the production of $t\bar t$ in
photon-photon collisions after folding with the  $\gamma\gamma$
luminosity. (Laser energy $\omega_0= 1.26\,{rm eV}$, $m_t=150 {\rm
GeV}$.) Solid (dashed) line, with (without) QCD corrections
\cite{KMS}.}
Production of single top quarks through the reaction
\begin{equation}
e\gamma\to\nu\bar t b
\end{equation}
was also considered at this conference \cite{Yehudai}.
It is induced by the
charged current and could therefore be exploited for a
determination of the CKM element $V_{tb}$.



\sec{4}{Cross section in the threshold region}%
A lot of effort has been invested during the
past years in the theoretical study of the total cross section
and of momentum distributions in the threshold region.  For a
stable top quark one would expect a series of narrow toponium
bound states.  As a consequence of the large decay rate of the
top quark only remnants of these resonances can be observed which
merge into a fairly smooth cross section.  Nevertheless a number
of important topics can be investigated in this region:
{\em i)} The top mass can be determined with high precision.
{\em ii)} The QCD interquark potential can be probed down to very small
distances.
{\em iii)} The production of a large number of polarized top quarks
practically at rest allows the study of formfactors and of
conventional and
potentially new decay modes under simple kinematic conditions.
{\em iv)} Under favorable circumstances, namely a light Higgs and a
heavy top, one may become sensitive toward the Yukawa coupling.
%
\subsec{4.1}{The total cross section}
%
As stated before, one faces a series of many overlapping
resonances, once $m_t$ lies above 120 GeV.  A convenient technique
is proposed in \cite{khoze} to perform this summation.  The sum over
individual Breit Wigner resonances multiplied by the wave
function at the origin is easily transformed into the imaginary
part of the Green's function.
\begin{equation}
       \sum_{n} \left| \psi_n (0) \right|^2
       \frac{\Gamma_t}{(E_n - E)^2 + \Gamma^2_t} =
       \sum_{n} \mbox{Im} \frac{\psi^{\*}_n (0) \psi^*_n (0)}
       {E_n - E - i \Gamma_t}=
       \mbox{Im} G(0,0,E+i\Gamma)
\end{equation}

The Green's function can be obtained as the solution of the
inhomogeneous Schr\"odinger equation with a $\delta$ function like
source term, located at $\vec x=0$ and with an oscillatory time
dependence $e^{-iEt}$ that has been factored out.
$G$ can be calculated analytically for a pure Coulomb potential and
numerically \cite{peskin} for the more realistic QCD potential, whose
short distance behavior includes the proper running of $\alpha_s$.

The height of the peak cross section, the exact location
of the threshold and the relative momentum of
the top quarks (see below) depend sensitively on the actual value of
$\Lambda_{\overline{MS}}$ or $\alpha_s$ (Fig.\ref{figalph}).
Resonance and continuum cross section join smoothly \cite{GKZ}
as demonstrated in Fig.\ref{joinfig}
\pictup{9.5cm}{5.5cm}{figalph}%
{Production cross section in the threshold region
without initial state correction for different
values of $\Lambda_{\overline MS}$.}
\pictup{9.5cm}{5.5cm}{joinfig}%
{Comparison of ``resonance'' calculation and QCD corrected continuum
formula in the transition region for $m_t=150$~GeV.}

The total cross section is affected by initial state radiation
that is accounted for by the standard radiator function.
Electroweak corrections affect the cross section by an
overall factor that can be calculated independently of the
potential and other QCD effects as a consequence of the short
distance nature of $W$ and $Z$ exchange.
Only the exchange of a light
Higgs boson with large couplings may exhibit a more complicated
interplay with the potential.  The leading correction is in this
case written as a factor
\begin{displaymath}
1+\frac{\sqrt{2} G_F m_t^2}{4\pi}\frac{m_t}{m_H}
\end{displaymath}
and can be traced to the Coulomb-like singularity from an
instantaneous Yukawa potential.
\begin{equation}\label{Higgspot}
V(r)=-\frac{\sqrt{2} G_F m_t^2}{4\pi}\frac{e^{-r m_H}}{r}
\end{equation}

For realistic masses $m_t\approx150{\rm GeV}$ and $m_h> 60 GeV$
retardation effects must be incorporated which are in this
case as important as the $1/m_t$ term \cite{guth}.
A combined treatment incorporating both the instantaneous
potential and the retardation effects is given in \cite{tobe}.
It amounts to subtracting the $m_t/m_H$ term from the perturbative one
loop calculation and incorporating instead the instantaneous
potential (\ref{Higgspot})  into the Schr\"odinger equation.
\pictuph{10.5cm}{6.7cm}{mom}{%
Green's function $|p{\cal G}(p,E+i\Gamma_t)|^2$ for
$m_t = 120$  and  $E= -2.3$  GeV
(1s peak) -- solid, $E = 0$ --
dotted, and $E = 2$ GeV -- dashed line.}
\subsec{4.2}{Momentum distributions}
The momentum distribution of
top quarks from an individual resonance can be calculated from
the Fourier transform of the wave functions.
\begin{equation}
\frac{dN}{d\vec p} \bigg|_{E=E_n} = |\tilde\psi_n(\vec p)|^2
\end{equation}
giving rise to the following differential cross section for a single
resonance
\begin{equation}
{d\sigma_n\over d\vec{p}} \left(\vec p,E\right) =
{24\pi^2\alpha^2Q_t^2\over sm_t^2}
|\psi_n(\vec{r}=0)|^2
{\Gamma_t\over(E-E_n)^2+\Gamma_t^2}
|\tilde\psi_n(\vec{p})|^2
 \label{eq:md2}
\end{equation}
%
For a series of overlapping resonances, the momentum distribution
of top quarks and the decay products is again conveniently
evaluated with the help of Green's functions: \cite{JKT,Sumino,jezt}
\begin{equation}
{d\sigma\over d\vec{p}} \left(\vec p,E\right) =
{3\alpha^2Q_t^2\over \pi sm_t^2}
\Gamma_t\left|{\cal G}(\vec{p},E+i\Gamma_t)\right|^2
\label{eq:md3}
\end{equation}
$G(\vec p,E=i\Gamma)$ can be calculated
numerically for complex energies and an
arbitrary QCD potential by solving the Lipmann-Schwinger equation
\begin{equation}
{\cal G}(\vec{p},E+i\Gamma_t) =
{\cal G}_0(\vec{p},E+i\Gamma_t) +
{\cal G}_0(\vec{p},E+i\Gamma_t)
\int {d\vec{q}\over(2\pi)^3}
\tilde V(\vec{p}-\vec{q})
{\cal G}(\vec{q},E+i\Gamma_t)
\label{eq:md5}
\end{equation}
where $\tilde V(\vec{p})$ is the potential in momentum space.
The free Hamiltonian that is used to define the Green's function
${\cal G}_0$ includes the width.
Hence, for the $t-\bar t$ system
\begin{equation}
{\cal G}_0(\vec{p},E+i\Gamma_t)
= {1\over E - {p^2\over m_t} + i\Gamma_t}
\label{eq:md6}
\end{equation}
\pictuph{15cm}{8cm}{momav}{{\em Left:}
Energy dependence of $\langle p\rangle$, the
average t quark momentum
for $\alpha_s = 0.13$ (dotted)
0.12 (dashed) and 0.11 (solid) line for
$m_t=120$ GeV  and
$\Gamma_t=0.3 {\rm GeV}$.\\
{\em Right:} Average momentum for $\alpha_s=0.125$ and
different top masses.}
The resulting momentum distribution is shown in Fig.\ref{mom} for a
characteristic set of energies.  The average momentum as a
function of energy (Fig.\ref{momav}) depends on $\alpha_s$ for
negative energies, that is in
the binding region.  Above threshold the momentum simply reflects
the kinematic behavior $p=\sqrt{m_t E}$
and can be used to measure $m_t$ independently from $\alpha_s$.
The sensitivity on $\alpha_s$ in the resoncance region is clearly
visible in the energy distribution of the  $W$'s from top decay
as shown in Fig.\ref{figW}.
\pictup{12cm}{9cm}{figW}%
{$W$ energy distribution for $m_t=150$~GeV at $E=-2.7$~GeV
and different values of $\alpha_s(m_Z)$.}

At this point a brief discussion is in order on the relation
between the top masses determined on the basis of different
potentials.  The perturbative two loop QCD
potential has been calculated in momentum space and is fixed
unambigously for sufficiently large $Q^2$:
\begin{equation}
V\left( Q^2 , \alpha _{\overline{MS}} (Q^2)\right) = - \frac{16\pi}{3}
 \frac{\alpha _{\overline{MS}} (Q^2)}{Q^2}
  \left[ 1 + \left( \frac{31}{3} -
  \frac{10}{9} n_f \right)
   \frac{\alpha _{\overline{MS}}(Q^2)}{4\pi} \right]
\label{eq:49}
\end{equation}
with
\begin{eqnarray}
\frac{\alpha_{\overline{MS}}(Q^2)}{4\pi } &=&
\frac{1}{b_0 \log
 \left( Q^2/\Lambda^2_{\overline{MS}}
 \right)}
 \left[    1 - \frac{b_1}{b^2_0}
          \frac   {\log\log
          \left(  Q^2 / \Lambda^2_{\overline{MS}} \right) }
                  {\log\left( Q^2 / \Lambda^2_{\overline{MS}}
          \right) }
 \right]\quad
\label{eq:50}\\
b_0 &=& 11 - \frac{2}{3}n_f, \qquad
b_1 = 102 - \frac{38}{3}n_f
\nonumber
\end{eqnarray}

Power law suppressed terms
cannot be calculated in this approach.  To evaluate the Fourier
transform of this function in order to calculate the potential in
coordinate space the small $Q^2$ behavior has to be specified in
an ad hoc manner.  Different assumptions will lead to the same
short distance behavior.  The potentials will, however, differ
with respect to their long distance behavior.  In \cite{khoze} it
has been argued convincingly that the long distance tail is cut off
by the large top width.  However, an additive constant in coordinate
space can be induced by the small momentum part of $\tilde V(Q^2)$.
This additional term would lead to a shift in the $t\bar t$ threshold,
which in turn can be reabsorbed by a corresponding shift in $m_t$.
These different assumptions are reflected in differences between
the predictions of \cite{peskin,Sumino,JKT}
for the precise location of the $t\bar t$
threshold for identical values of $\alpha_s$ and $m_t$ and in
differences
in the $\alpha_s$ dependence of the momentum distributions for fixed
$m_t$ and energy (see also \cite{Martinez}).
All these differences can be attributed to the
freedom in the additive constant discussed before.  The same additive
constant appears in $b\bar b$ spectroscopy, such that the mass
difference between top and bottom is independent from these
considerations.

A specific choice for the potential has been adopted in \cite{jezt}.
Different values of $\alpha_s$ correspond to a different behavior at
short distances, but are tuned to the same long distance
behavior.  This feature is reflected in the energy dependence of
the average top momentum $\langle p\rangle$ for fixed $m_t$ and
different $\alpha_s$
which can be viewed in close analogy to the predictions based on a pure
Coulomb behavior.  A fairly pronounced decrease of $\langle p\rangle$
with increasing $\alpha_s$
for energies below the nominal threshold is evident (Fig.\ref{momav}a).
This can be compensated by an increase in $m_t$ \cite{Martinez}.
For positive energies, however, one observes the kinematic behavior
of the momentum
$\langle p\rangle \approx \sqrt{m_t E}$
independent of $\alpha_s$ and the correlation can be disentangled.
\subsec{4.3}{Forward-backward asymmetry}
Additional information can be drawn from the interference between
$S$ and $P$ wave contributions close to threshold which can be
observed through the forward-backward asymmetry of top quarks
\cite{MurSu}.  This asymmetry can be observed if one assumes that $t$
and $\bar t$ decay products can be recombined to yield the direction
of the original quark.  In the threshold region a marked
difference between narrow and wide resonances is predicted.
Narrow S and P wave resonances do not overlap and individually
each gives rise to isotropic top distributions.  As a consequence
of the large width the interference between these two waves may
become important and an asymmetry of order 10\% is predicted.  The
detailed structure depends on $\alpha_s$, $m_t$, and $\Gamma_t$.
Two examples are given in Fig.\ref{figSu}.
\pictup{11.8cm}{16cm}{figSu}%
{The total cross section for $t\bar t$ production (left axis)
and the forward backward asymmetry (right axis) for different
$\alpha_s$ and $m_t$  \cite{MurSu}.}
\subsec{4.4}{Rescattering corrections and bound state effects.}
Up to this
point, final state interactions and the reduction of phase space
by the $t\bar t$ binding have been neglected.  However, recently also
corrections linear in $\alpha_s$ as well as those of order
$\alpha_s^2$ have been considered.
The $b$ quark originating from $t$ decay is influenced
by the potential of the ``spectator'' $\bar t$, and this gluon exchange
will generally lead to a slight shift of the b quark momentum
\cite{Sumcor}
\begin{equation}\label{Sumeq}
\delta\frac{{\rm d}\sigma}{{\rm d} \vec p}\propto \alpha_s
\int\frac{{\rm d}\vec q}{(2\pi)^3}\frac{1}{|\vec p - \vec q|^3}
{\rm Im}[G^*(\vec q) G(\vec p)]
\end{equation}
This correction vanishes in the narrow width approximation, that is
for $\Gamma_t\to0$.  It leads to a shift of the typical momentum by
about 1 GeV as shown in Fig.\ref{figSummom}.
\pictup{11.2cm}{8.5cm}{figSummom}%
{Effect of final state interactions on the momentum distribution
of $t$ quarks \cite{Sumcor}.}
This corresponds to a shift in $m_t$ by
about 100 MeV.  From (\ref{Sumeq}) it is evident that the total cross
section is unaffected in first order of $\alpha_s$,
a result independently derived in \cite{Yakov}.

In second order one anticipates effects from relativistic
corrections, from the reduction of the phase space through the
binding energy and from the Coulomb wave function of the $b$ quark.
Individually these effects are large.  For the sake of argument,
let us adopt a pure Coulomb potential and a binding energy of
\mbox{-2.5 GeV.}
From the virial theorem one derives a potential energy
of -5 GeV.  The phase space of the quark decaying first is
therefore reduced by this same amount.  Assuming $m_t = 150$~GeV
one would arrive at a reduction of $\Gamma_t$ by about 10\%.  A full
calculation of all ${\cal O}(\alpha_s^2)$
effects is not available at present and one
has to resort to models and analogies \cite{Sumino,jezt,jez}.  For
example, it has been shown \cite{uberall}
that the decay rate of a muon bound in the field
of a nucleus is given by
\begin{equation}
\Gamma=\Gamma_{free}[1-5(Z\alpha)^2] [1+5(Z\alpha)^2] [1-(Z\alpha)^2/2]
\end{equation}
where the first correction factor
originates from the phase space suppression, the second from the
Coulomb enhancement, and the third from time dilatation.
Thus there is no first order correction to the total rate from
rescattering in the nucleus potential, similar to the $t\bar t$ case
discussed above. The second order contributions evidently compensate to
a large extent. In a
model calculation where these features are implemented \cite{jezt}
through a momentum dependent width, it is found that the total
cross section as well as the momentum distribution are hardly
affected.  However, even if an extreme suppression through the
momentum dependent width is incorporated \cite{Sumino} the shape of
the total cross section (but not its normalization) and the peak
of the momentum distribution remain nearly unchanged.  Since
these are the key ingredients for a combined determination of $m_t$
and $\alpha_s$, the basic conclusions remain valid even in this
extreme case.\\
{\em In summary:}  An $e^+e^-$
collider will lead to precise determination of
the top mass.  Top studies will provide an exciting laboratory
for QCD.  The determination of the $Ht\bar t$ Yukawa coupling will be of
prime importance.  The $Wtb$ gauge coupling can be measured and
limits on anomalous couplings can be derived.  Novel decay modes
could be searched and studied in a clean environment.\\
{\em Acknowledgements:}
The author gratefully acknowledges the collaboration with
P. Igo-Kemenes,
M. Je\.zabek,
T. Teubner and
P. Zerwas
on a variety of aspects of top quark physics discussed in this
review. Inportant additional material was provided by the members of the
European Top Quark working group W. Bernreuther, F. Borzumati,
M. Martinez, R. Miquel, S. Orteu, O. Podobrin and A. Venturi
as well as by K. Fujii, M. Peskin and Y. Sumino and by all those
who have contributed to the Session on Top Quarks.







\begin{thebibliography}{99}
\tenrm
\sloppy
\raggedright
%\input latjour
\def\app#1#2#3{{\it Act. Phys. Pol. }{\bf B #1} (#2) #3}
\def\apa#1#2#3{{\it Act. Phys. Austr.}{\bf#1} (#2) #3}
\def\lhc{Proc. LHC Workshop, CERN 90-10}
\def\npb#1#2#3{{\it Nucl. Phys. }{\bf B #1} (#2) #3}
\def\plb#1#2#3{{\it Phys. Lett. }{\bf B #1} (#2) #3}
\def\prd#1#2#3{{\it Phys. Rev. }{\bf D #1} (#2) #3}
\def\pR#1#2#3{{\it Phys. Rev. }{\bf #1} (#2) #3}
\def\prl#1#2#3{{\it Phys. Rev. Lett. }{\bf #1} (#2) #3}
\def\prc#1#2#3{{\it Phys. Reports }{\bf #1} (#2) #3}
\def\cpc#1#2#3{{\it Comp. Phys. Commun. }{\bf #1} (#2) #3}
\def\nim#1#2#3{{\it Nucl. Inst. Meth. }{\bf #1} (#2) #3}
\def\pr#1#2#3{{\it Phys. Reports }{\bf #1} (#2) #3}
\def\sovnp#1#2#3{{\it Sov. J. Nucl. Phys. }{\bf #1} (#2) #3}
\def\jl#1#2#3{{\it JETP Lett. }{\bf #1} (#2) #3}
\def\jet#1#2#3{{\it JETP Lett. }{\bf #1} (#2) #3}
\def\zpc#1#2#3{{\it Z. Phys. }{\bf C #1} (#2) #3}
\def\ptp#1#2#3{{\it Prog.~Theor.~Phys.~}{\bf #1} (#2) #3}
\def\nca#1#2#3{{\it Nouvo~Cim.~}{\bf #1A} (#2) #3}
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