\chapter{Top quarks in $e^+e^-$ annihilation}

A variety of reactions is conceivable for top quark production
at an electron positron collider.  Characteristic Feynman
diagrams are shown in
Fig.\ref{fig:3.1}.
% the resulting cross sections in Fig.\ref{fig:3.2}.
$e^+ e^-$ annihilation through the virtual photon and
$Z$ (Fig.\ref{fig:3.1}a)
dominates and constitutes the reaction of interest for the currently
envisaged energy region.
\begin{figure}
\vspace{10cm}
\caption{Feynman diagrams for
 $t\bar{t}$ or $t\bar{b}$ production.}

\label{fig:3.1}
\end{figure}

In addition one may also consider \cite{fusi} a variety of
gauge boson fusion reactions (Fig.\ref{fig:3.1}b-d) that
are in close analogy to $\gamma \gamma$
fusion into hadrons at $e^+e^-$ machines of lower energy.
Specifically these are single top production,
\begin{equation}
e^+e^- \to \bar{\nu} e^- t \bar{b}
\label{eq:26}
\end{equation}
or its charge conjugate and top pair production through neutral or charged
gauge boson fusion
\begin{eqnarray}
e^+e^- & \to & e^+e^- t \bar{t} \nonumber\\
e^+e^- & \to & \bar{\nu} \nu t \bar{t}
\label{eq:27}
\end{eqnarray}

The experimental observation of these reactions would allow
to determine the coupling of top quarks to gauge bosons,
in particular also to longitudinal $W$ bosons and $Z$ bosons,
in the space-like region and eventually at large momentum transfers.
This would constitute a nontrivial test of the mechanism
of spontaneous symmetry breaking.

The various cross sections increase with energy in close analogy
to $\gamma \gamma$ reactions, and eventually even exceed $e^+e^-$
annihilation rates.  However, at energies accessible in the
foreseeable future these reactions are completely
negligible:  for an integrated luminosity of $10^{40}$ cm$^{-2}$,
at $E_{cm} = 500~GeV$ and for $m_t = 150~GeV$ one expects about
one $e^+e^ - t\bar{t}$ event (still dominated by
$\gamma \gamma$ fusion). At that same energy the cross sections for
$e^+ \nu \bar{t} b + c.c.$ and $\nu \bar{\nu} t \bar{t}$ final states
are still one to two orders of magnitude smaller.

Another interesting class of reactions is $e^+e^-$ annihilation
into heavy quarks in association with gauge or Higgs bosons:
\begin{eqnarray}
e^+ e^- & \to & t \bar{t} Z \\
%\label{eq:28}
e^+ e^- & \to & t \bar{b} W^- \\
%\label{eq:29}
e^+ e^- & \to & t \bar{t} H \\
%\label{eq:30}
e^+ e^- & \to & t \bar{b} H^-
\label{eq:31}
\end{eqnarray}

Two amplitudes contribute to the first reaction \cite{hagi}:
The $t \bar{t}$ system may be produced through a virtual Higgs
boson which by itself was radiated from a $Z$ (Fig.\ref{fig:3.2}).
The corresponding amplitude dominates the rate and
provides a direct measurement of the Yukawa coupling.
The radiation of longitudinal $Z$'s from the quark line
in principle also carries information on the symmetry
breaking mechanism of the theory.
\begin{figure}
\vspace{50mm}
\caption{\label{fig:3.2}
Amplitudes relevant for
$e^+e^- \to t\bar{t}Z$ and for $e^+e^- \to t\bar{t}H$.}
\end{figure}
The transverse part of the $t \bar{t} Z$ coupling,
i.e. the gauge part, can
be measured directly through the cross section or
various asymmetries in
$e^+e^- \to t \bar{t}$.  The longitudinal part, however,
could only be isolated
with $t \bar{t} Z$ final states.  For an integrated
luminosity of $10^{40}$ cm$^{-2}$
one expects only about 40 events (see sect. 3.1.7)
and it is therefore not
clear whether these can be filtered from the huge background
and eventually used for a detailed analysis.

Light Higgs bosons may be produced in conjunction with
$t \bar{t}$ \cite{djo}. They are radiated either from
the virtual $Z$ with an amplitude that is present
also for massless fermions or directly from heavy quarks as a consequence of
the large Yukawa coupling (Fig.\ref{fig:3.2}).
The latter dominates by far and may therefore be tested
specifically with heavy quark final states.  The predictions for the
rate will be discussed in sect.~3.1.7.  Depending on the mass of the
Higgs and the top 
quark, the reaction could perhaps be detected with an integrated luminosity
of $10^{40}cm^{-2}$.

Top quark production in $\gamma \gamma$ collisions is conceivable at a
``Compton collider''.  It requires special experimental provisions for the
conversion of electron beams into well-focused beams of energetic photons
through rescattering of laser light.  A detailed discussion can be
found in \cite{DESY}. 

Chapter 3 will be entirely devoted to $t\bar t $ 
production in $e^+e^-$ annihilation.  Section 3.1 will be 
concerned with the energy region far above threshold --- with electroweak 
aspects as well as with specific aspects of top hadronisation. The emphasis 
of section 3.2 will be on the threshold region which is governed by the 
interplay between bound state formation and the rapid top decay.



\section{Top production above threshold}
\subsection{Born predictions}
From the preceding discussion it is evident that the bulk of top studies
at an $e^+ e^-$ collider will rely on quarks produced in $e^+ e^-$ annihilation
through the virtual $\gamma$ and Z, with a production cross section of the
order of $\sigma _{point}$.  For quarks tagged at an angle $\vartheta $, the
differential cross section in Born approximation is a binomial in $\cos \vartheta $
\begin{equation}
\frac{d\sigma}{d \cos\vartheta} = \frac{3}{8} \left( 1 + \cos^2 \vartheta \right) \sigma _U + \frac{3}{4} \sin^2 \vartheta \sigma _L + \frac{3}{4} \cos \vartheta \sigma _F
\label{eq:32}
\end{equation}
$U$ and $L$ denote the contributions of unpolarized and longitudinally polarized
gauge bosons along the $\vartheta$ axis, and $F$ denotes the difference
between right and left polarizations.  The total cross section is the sum
of $U$ and $L$,
\begin{equation}
\sigma = \sigma _U + \sigma _L
\label{eq:33}
\end{equation}
the forward/backward asymmetry is given by the ratio
\begin{equation}
A^{FB} = \frac{3}{4} \frac{\sigma _F}{\sigma}
\label{eq:34}
\end{equation}

The $\sigma^i$ can be expressed in terms of the cross sections for the
massless case in Born approximation,
\begin{eqnarray}
\sigma ^U_B & = & \beta \sigma^{VV} + \beta ^3 \sigma ^{AA} \nonumber \\
\sigma ^L_B & = & \frac{1}{2} \left( 1 - \beta ^2 \right) \beta \sigma^{VV} \nonumber \\
\sigma ^F_B & = & \beta ^2 \sigma^{VA}
\label{eq:35}
\end{eqnarray}
with
\begin{eqnarray}
\sigma ^{VV} & = & \frac{4 \pi \alpha^2 (s) e^2_e e^2_Q}{s} \nonumber \\
& & + \frac{G_F (s)}{\sqrt{2}} e_e e_Q (\upsilon _e + \rho a_e )
\upsilon _Q \frac{m^2_Z \left( s-m^2_Z \right)}
{\left( s-m^2_Z \right)^2 + \left(\frac{s}{m_Z}
\Gamma _Z \right)^2} \nonumber \\
& & + \frac{G^2_F}{32 \pi} \left( \upsilon ^2_e
+ a^2_e + 2 \rho \upsilon _e a_e \right) \upsilon ^2_Q \frac{m^4_Z s}
{\left( s-m^2_Z \right)^2
+ \left(\frac{s}{m_Z} \Gamma _Z \right)^2} \nonumber \\
\sigma ^{AA} & = & \frac{G^2_F}{32 \pi}
\left( \upsilon ^2_e + a^2_e + 2 \rho \upsilon _e a_e \right)
a^2_Q \frac{m^4_Z s}{\left( s-m^2_Z \right)^2
+ \left(\frac{s}{m_Z} \Gamma _Z \right)^2} \nonumber \\
\sigma ^{VA} & = & \frac{G_F \alpha (s)}{\sqrt{2}} e_e
(\rho \upsilon _e + a_e )e_Q a_Q \frac{m^2_Z
\left( s-m^2_Z \right)}{\left( s-m^2_Z \right)^2
+ \left(\frac{s}{m_Z} \Gamma _Z \right)^2} \nonumber\\
& & + \frac{G^2_F}{16 \pi} \left( 2 \upsilon _e a_e
+ \rho (\upsilon ^2_e + a^2_e)\right) \upsilon_Q a_Q
\frac{m^4_Z s}{\left( s-m^2_Z\right)^2
+ \left(\frac{s}{m_Z}\Gamma _Z \right)^2}
\label{eq:36}
\end{eqnarray}

The fermion couplings are given by
\begin{eqnarray}
\upsilon_f = 2I^f_3 - 4e_f \sin ^2 \theta _w \quad , \qquad a_f = 2I^f_3
\label{eq:37}
\end{eqnarray}
and the possibility of longitudinal electron polarization
($\rho = -1;+1;0$ for
right\-handed; lefthanded; unpolarized electrons) has been included.
Alternatively one may replace $G_F m^2_Z$ by
\begin{equation}
G_F m^2_Z = \frac{\pi \alpha (s)}{\sqrt{2} \sin^2 \theta_W \cos^2 \theta _W}
\label{eq:38}
\end{equation}
With $\sin^2\theta_W$ ($\approx$0.23)
interpreted as $\sin^2 \theta_{eff}$ \cite{con}
this formula accommodates the leading logarithms from the running coupling
constant as well as the quadratic top mass terms in the threshold region.  

\subsection{Radiative corrections}

\underline{QCD corrections}
to this formula are available for arbitrary $m^2 / s$ up
to first order in $\alpha _s$:
\begin{eqnarray}
\sigma & = & \frac{(3 - \beta ^2)}{2} \beta \sigma^{VV} \left( 1 + \frac{4}{3} \frac{\alpha _s}{\pi} K_V \right) \nonumber \\
& & + \beta ^3 \sigma ^{AA} \left( 1 + \frac{4}{3} \frac{\alpha _s}{\pi} K_A \right)
\label{eq:40}
\end{eqnarray}
The exact result \cite{exac} for $K_{V,A}$ can be found in \cite{KueZer}.
These  QCD enhancement factors are
well approximated by \cite{appr}
\begin{eqnarray}
K_V & = & 1 + \frac{4}{3} \alpha_s
\left[ \frac{\pi}{2\beta} - \frac{3+\beta}{4}
\left( \frac{\pi}{2} - \frac{3}{4\pi} \right) \right] \nonumber \\
K_A & = & 1 + \frac{4}{3} \alpha_s
\left[ \frac{\pi}{2\beta} -
\left(\frac{19}{10} - \frac{22}{5}\beta +\frac{7}{2}\beta^2\right)
\left( \frac{\pi}{2} - \frac{3}{4\pi} \right) \right] \nonumber \\
\alpha_s & = & \frac{12\pi}{25 \log (4p^2_t / \Lambda ^2)}
\label{eq:43}
\end{eqnarray}
The next to leading order corrections 
to $K_V$ were calculated only recently \cite{HKT,CKS}.
The scale in $\alpha _s$ chosen
above was guessed on the basis of general
arguments \cite{guesk1} which were confirmed by the forementioned
complete calculations.
\par\noindent
For small $\beta$ these factors develop the familiar Couloumb
enhancement $\sim{2\pi\alpha_s\over 3\beta}$, compensating the
phase space $\sim \beta$.  This leads to a nonvanishing cross
section which smoothly joins the resonance region.  Details of
this transition will be treated in section 3.2.

To prepare this discussion, let us briefly study the limit of applicability 
of fixed order perturbation theory.  The leading terms in the perturbative 
expansion close to threshold are obtained from Sommerfeld's rescattering 
formula $\left(x\equiv {4\over 3}{\pi\alpha\over \beta}\right)$
\be
K_V^{\rm Som} = {x\over 1 - e^{-x}} &=& 1 +{x\over 2} + {B_1 x^2\over 2!} 
- {B_2 x^4\over 4!}+  {B_3 x^6\over 6!}\pm\ldots
\nonumber \\
&=& 1+{x\over 2} + { x^2\over 12} - {x^4\over 720} + {x^6\over 
5040}\pm\ldots
\ee
with $B_i$ being the Bernoulli numbers.  At first glance one might require 
$x\le 1$ for the perturbative expansion to be valid.  However, 
significantly larger values of $x$ are acceptable. The full  Sommerfeld 
factor $K_V^{\rm Som}$ is remarkably well approximated by the first three 
terms of the series for surprisingly large $x$ (only 6\% deviation for 
$x=4!$).  For top quarks this corresponds to $\beta\approx 0.13-0.14$ 
and hence down to about 3 GeV above the nominal threshold.  Upon closer 
inspection one also observes that the formula given in
eq.~(\ref{eq:43}) (a result  
of order $\alpha_s$) coincides numerically well with the correction factor 
$K_V^{\rm Som} \left(1-{16\over 3}{\alpha_s\over \pi}\right)$ which 
incorporates rescattering and hard gluon vertex corrections.  The results 
presented in these lectures are based on the Born predictions plus ${\cal 
O} (\alpha_s)$ corrections.


\underline{Initial state radiation}
has an important influence on the magnitude
of the cross section.  $\sigma(s_{eff})$ is folded with
the Bonneau Martin structure function, supplemented by the
summation of large logarithms.  A convenient formula for
the non-singlet structure function in the leading logarithmic
approximation has been obtained in \cite{jez},
which is a natural extension of a formula
proposed in \cite{jad}.
%
This leads to a significant suppression by about a factor
\begin{equation}
\left( \frac{\delta W}{m_t} \right) ^{ {2\alpha\over \pi}\left(
\ln{m_t^2\over m_e^2}-1\right)}
 \approx 0.5-0.6
\label{eq:47}
\end{equation}
with $\delta W = 1-5~GeV$ in the resonance
and threshold region. The correction factor increases rapidly
with energy, but stays below 0.9 in
the full range under consideration (Fig.\ref{fig:3.5}).



\underline{Electroweak corrections}
to the production cross section
in the continuum have been studied in \cite{been1}.  Apart from
a small region close to threshold they are negative.
Relative to the $G_F$ parametrized Born approximation they
decrease the cross section by -6.3\% to -9.3\%, if $m_t$ is
varied between 100 and 200~GeV, $m_H$ between 42 and 1000~GeV,
and $E_{cm}$ fixed at 500~GeV.  QCD and electroweak
corrections are thus of equal importance (Fig.~\ref{fig:3.6}).

Close to threshold and for relatively small Higgs boson masses a rapid 
increase of these corretions is observed (Fig.~\ref{fig:3.7}) which
can be attributed  
to the attractive Yukawa potential induced by light Higgs boson exchange.  
Several GeV above threshold, and for $m_H$ around or below 100 GeV it is 
more appropriate to split these corrections into hard and soft exchange and 
incorporate the latter in an instantaneous Yukawa potential \cite{JKsim}.

\begin{figure}[ht]
  \begin{center}
    \leavevmode
    \epsfxsize=12.cm
    \epsffile[65 220 410 450]{fig_3.3.ps}
    \hfill
\caption{Cross section for $t\bar{t}$ production,
including resonances, QCD corrections
and initial state radiation in units of $\sigma_{point}$.}
\label{fig:3.5}
  \end{center}
\end{figure}
\begin{figure}
\vspace{80mm}
\caption[]{Genuine electroweak
corrections to top production in $e^+e^-$ annihilation.
From \cite{been1}.}
\label{fig:3.6}
\end{figure}
\begin{figure}
\vspace{80mm}
\caption{Relative size (in percent) of electroweak corrections in the
threshold region for $m_t=200$ GeV and different Higgs masses (from
\protect\cite{BH}).}
\label{fig:3.7}
\end{figure}

\subsubsection{Longitudinal polarization}

It should be mentioned that linear colliders might well operate to
a large extent with polarized (electron) beams.  The cross section for
this case can be derived from (\ref{eq:36}).  For top quarks the resulting
right/left asymmetry
\begin{equation}
A_{LR} =  (\sigma _L - \sigma _R) / (\sigma _L + \sigma _R )
\label{eq:48}
\end{equation}
is sizable (Fig.\ref{fig:RL}) and amounts to about $-\,0.4$,
\begin{figure}[ht]
  \begin{center}
    \leavevmode
    \epsfxsize=12.cm
    \epsffile[85 280 510 565]{fig_3.6.ps}
    \hfill
\caption{Right/left asymmetry as function
of $E_{cm}$ for $m_t = 150~GeV$.}
\label{fig:RL}
  \end{center}
\end{figure}
reducing the production cross section with righthanded electrons.
However, selection of righthanded electron beams decreases the $W$ pair
cross section even stronger, thereby enhancing the top quark
signal even before cuts are applied.
Electroweak corrections to $A_{LR}$ in the threshold region have been
calculated in \cite{GKKS}.

\subsection{Top quark fragmentation}

The experimental analysis of charm and bottom fragmentation functions has
clearly demonstrated that heavy quark fragmentation is hard in
contrast to the fragmentation of light
quarks. This is a consequence of the inertia of heavy particles, the momentum
of which is not altered much if a light quark $\bar q$ is attached to
the heavy quark $Q$ in the fragmentation process
to form a bound state $(Q\bar q)$, see e.g.\ \cite{fr1}. At the same
time soft infrared gluon radiation is damped if the color source is
heavy.

For $m_t \ge 100$ GeV the strong fragmentation process and
the weak decay mechanism are intimately intertwined \cite{fr3}. The
lifetime 
$\tau_* < \Lambda^{-1}$ becomes so short that the mesonic $(t\bar q)$
and baryonic $(tqq)$ bound states cannot be built--up anymore.
Depending on the initial top quark energy, even remnants of the $t$
quark jet may not form anymore \cite{bigi2}. Hadrons can be created in
the string stretched between the $t$ and the $\bar t$ only if the
quarks are separated by about 1 fermi before they decay. If the flight
path $\gamma \tau_*$ is less than $1/2$ fm,
the length of  the $t-\bar t$ string is too short to form
hadrons and jets cannot develop anymore along the flight direction
of the top quarks.  For $m_t\approx 180$ GeV  top quark energies above
1 TeV are required to allow nonperturbative strings between $t$ and
$\bar t$.  
``Early'' nonperturbative production of particles from the string between 
$t$ and $\bar t$ is thus absent for all realistic experimental 
configurations.  ``Late'' production from the $b$ and $\bar b $ jets 
produced in top decays dominates.  
\begin{figure}
\vspace{155mm}
\caption{Rapidity distributions of ``early'' particles (full) and
``late'' ones (dashed) for three different top masses: 90 GeV in a)
100 GeV in b) and 120 GeV in c).  (From \protect\cite{fr3}.)}
\label{fig:SjZ}
\end{figure}
This is illustrated in Fig.~\ref{fig:SjZ}.  Early 
production dominates for $m_t = 90$ GeV, late production for $m_t=120$ GeV 
and {\it a forteriori} for the actual value around 180 GeV. 


The perturbative radiation of soft gluons, too, is interrupted by the
$t$ quark decay \cite{fr2}.
The angular distribution $(\Theta)$ and the
energy distribution $(\omega)$ of the radiated gluons is
approximately given by
\begin{equation}
dP_g = \frac{4\alpha_s}{3\pi} \frac{\Theta^2 d\Theta^2}{\left[
\Theta^2 + \frac{1}{\gamma^2} \right]^2 + \left[ \frac{\Gamma}{\gamma
\omega} \right]^2 } \frac{d\omega}{\omega}
\end{equation}
\noindent for a short--lived radiation source accelerated to $\gamma =
E_t/m_t$. The gluons accumulate on the surface of a cone with
half--aperture $\Theta_c \sim \gamma^{-1}$ for a long--lived $t$, but
$\sim \gamma^{-1} \sqrt{\gamma\Gamma/\omega}$ if the particle decays
quickly. The energy spectrum rises from zero to a maximum at $\omega
\sim \gamma\Gamma$ before falling off $\sim 1/\omega$ for large $\omega$,
if the width is greater than the confinement scale $\Lambda$.

The impact of the finite width on the angular distribution of gluon 
radiation will be visible if $\omega \le \Gamma_t E_t/m_t$. For a linear 
collider with c.m.~energy of 2 TeV gluon jets with energies of 10 GeV and 
below would be affected \cite{KOS}.  The radiation pattern  is shown in
\begin{figure}
\vspace{10cm}
\caption{Sensitivity of the soft gluon distribution to the top width
for back-to-backk $t$ and $b$: $dN/d\cos(\theta_g)$ for $M$ = 140 GeV,
$\theta_b=180^\circ$, and $\Gamma$ = 0, 0.7, 5, 20 GeV, and $\infty$
as marked. $W$ = 1000 GeV. (From \protect\cite{KOS}.)}
\label{fig:KhozeOrrSt}
\end{figure}
Fig.~\ref{fig:KhozeOrrSt}  for $m_t = 140$  GeV and  
$\sqrt{s} = 600 $ GeV, with $\Gamma_t$ tuned to different values in order 
to demonstrate the sensitivity of such a measurement.  The impact of the 
conversion of gluons to hadrons has been ignored in this study.  The 
picture is, further, complicated by the interference between radiation from 
top production and decay --- a phenomen characteristic for unstable 
particles. These phenomena allow to probe the time evolution of 
hadronisation in a unique manner.  Their understanding is a necessary 
prerequisite for any top mass measurement through a kinematic analysis of 
$b$-jet-$W$ final states.

%%%%%%%%%%%%%Nur in Review%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Static $t$ parameters}
%
Because of the large $t$ mass, deviations from the Standard
Model may manifest themselves in the top quark sector first.
Examples in which the large mass is crucial are provided by
multi--Higgs doublet models, models of dynamical symmetry breaking and
compositeness. These effects can globally be described by form factors
parametrizing the electroweak $t\bar t$ production current $(a =
\gamma,Z)$ and the weak $(t,b)$ decay
current $(a=-)$ \cite{st1,BerOve},
\begin{equation}
j_\mu^a \sim F_{1L}^a \gamma_\mu P_L + F_{1R}^a \gamma_\mu P_R +
\frac{i\sigma_{\mu\nu} q_\nu}{2m_t} [ F_{2L}^a P_L + F_{2R}^a P_R]
\end{equation}
\noindent [$P_{L,R}$ project on the
left and right chirality components of the wave
functions.] In the \sm, $F_{1L}^- = 1$ while all other $F^-_i$ vanish;
$F_{1L}^\gamma = F_{1R}^\gamma = 1$ and $F_{2L}^\gamma = F_{2R}^\gamma
= 0$, analogously for the $Z$ current. ${\cal{CP}}$ invariance
requires $F_{2L}^{\gamma,Z} = F_{2R}^{\gamma,Z}$ in the $t\bar t$
production current, and equal phases for $F_{1L}^-$ and $F_{2R}^-$
{\it etc.}\ in the decay current. The static values of the form factors
$F_2^{\gamma,Z}$ are the anomalous magnetic and electric dipole moments
of the top quark.

The form factors are determined experimentally by measuring the
angular distribution of the $t\bar t$ decay products, $e^+e^- \ra
t\bar t, t \ra b W^+, W^+ \ra f \bar f' $ {\it etc.}
This requires the top quark to be treated as a free particle,
the polarization of which not being affected by non--perturbative
hadronic binding effects. This assumption is justified by the short
lifetime of the top quark as discussed earlier. Details of the general
helicity analysis can be found in the literature \cite{st1,st2}.


\subsection{Normal polarization of the top quarks}
%
A non--zero component of the $t$ polarization vector that is
normal to the production plane can be generated only by the
interference between complex helicity flip and non--flip
amplitudes. Such relative phases can arise from ${\cal{CP}}$ violation
but also from higher order loop corrections due to gluon exchange in
the final state \cite{kuehn2,st4,st1}
or electroweak corrections involving
Higgs and gauge bosons \cite{st1}. The QCD induced normal polarization
is generally less than 5\%, the electroweak normal polarization is
smaller still.
[By contrast, longitudinal and transverse polarization components within the
$t\bar{t}$ production plane are generated already at the tree level of
the electroweak interactions and they are large in general; see
\cite{kuehn2} for the discussion of details.]

\subsection{Angular correlations of $t\bar t$ decay products}
As stated in the previous chapter, top quarks are produced through the
virtual photon and $Z$. In the threshold region they are polarized to a
degree
\begin{equation}
P_t= A_{RL}\approx - 0.4
\end{equation}
Assuming for the distribution of leptons from the decay of polarized
top quarks
\begin{equation}
\frac{dN}{dx\,d\cos\theta}=f(x)+ g(x) P_t\cos\theta
\end{equation}
(with $g(x)=f(x)$ in the \sm, see eq.\ref{eq:15})
the angular distribution 
allows to test for the chirality of the $tb$
current. Implicitly it was assumed that
hadronization does not affect
the top spin degrees of freedom \cite{acta,jk41}. This assumption can
be tested independently through the study of correlations between $t$ and
$\bar t$ decay products. In the threshold region the spins are
correlated
$\propto (1+\frac{1}{3} \vec{s_+}\cdot\vec{s_-})$. This leads to the
following correlated $\ell_+\ell_-$ distribution:
\begin{equation}
\frac{dN}{dx_+dx_-d\cos\theta_{+-}}= f_+(x_+)f_-(x_-) +
                      \frac{1}{3} g_+(x_+)g_-(x_-)\cos\theta_{+-}
\end{equation}
where $f_+=f_-$ and $g_+= -g_-$. $\theta_{+-}$ denotes the angle between
$\ell_+$ and $\ell_-$. After averaging the lepton energies,
\begin{equation}
\frac{dN}{d\cos\theta_{+-}}=1+\frac{1}{3} h_+ h_- \cos\theta_{+-}
\end{equation}
Note that the coefficient of the correlation term is $-h^2_+/3$ and
hence always negative (assuming CP conservation). Since
$|h_+|\leq1$ it ranges between 0 and $-1/3$. This limiting value is
assumed in the \sm. A detailed discussion with illustrative examples is
given in \cite{PIK}.


\subsection{Testing the Yukawa Coupling}

With its relatively large Yukawa coupling $g_Y=\sqrt{2} m_t/v\approx
1$  
the top quark 
is uniquely suited to test one of the basic ingredients of the Standard 
Model, the coupling between top quarks and the Higgs boson.  The 
verification of this crucial prediction would confirm the mechanism for the 
generation of fermion masses and hence complete the measurement and 
analysis of basic couplings.  Alternatively, any deviation would provide 
unambiguous proof for new physics.

Different strategies are at hand at an \epem collider which are closely 
tied to the cms energy available and to the mass of the Higgs boson.  For 
relatively light bosons a variety of possibilities appear to be promising: 
vertex corrections affect the cross section for $t\bar t$ production in the 
threshold region.  For a collider in its early stage with an energy around 
500 GeV it may well be the only option available and will be discussed more 
thoroughly in section 3.2.  For higher energies, say around  1 TeV, a 
promising choice is the radiation of Higgs 
bosons from $t\bar t$ (see Fig.~\ref{fig:3.2}) 
with a cross section around 1 fb 
\begin{figure}[htb]
\psfig{figure=djouadi.ps,angle=-90,width=60mm,bbllx=60pt,bblly=60pt,%
bburx=510pt,bbury=340pt}
\caption{The cross section $\sigma(e^+e^- \to t\bar t H)$ (from
\protect\cite{djo}).} 
\label{fig:sigma}
\end{figure}
(Fig.~\ref{fig:sigma}) \cite{djo}. Alternatively one 
may analyse the top quark final states in conjunction with a $Z$ boson.  
This reaction receives important contributions from the left one of
the diagrams depicted in 
Fig.~\ref{fig:3.2}
if the Higgs mass happens to be relatively close to 2$m_t$
\cite{hagi}.
  Again, for 
simple kinematical reasons, high energies are crucial for the reaction to 
be accessible (Fig.~\ref{fig:Hagi}). 
\begin{figure}
\vspace{15cm}
\caption{The Higgs mass ($m_H$) dependences of the total cross
sections of $e^+e^- \to t\bar t Z$ for various top quark masses $m_t$.
The c.m.~energy $\protect\sqrt{s}$ is set to be 500 GeV (a) and 1 TeV
(b). (From \protect\cite{hagi}.)}
\label{fig:Hagi}
\end{figure}


\section{Threshold behaviour}
The previous section dealt with top quark production sufficiently far
above threshold for the reaction to be well described by the Born
cross section, modified slightly by QCD and electroweak corrections.
This is in contrast to the situation in the threshold region, where
QCD plays an important role and controls the cross section.  Strong forces
modify the Born prediction. They compensate the phase space suppression and
enhance the production rate significantly, leading to a step function like
behaviour at threshold.  The large top decay rate also plays an important
role. Quarkonium resonances cease to exist and merge into a structureless
excitation curve which joins smoothly with the continuum prediction above
the nominal threshold.

This sharply rising cross section allows to study top quarks in a
particularly clean environment and with large rates. The following physics
questions can be addressed:
\begin{itemize}
\item The QCD potential can be scrutinized at short distances, with the
non perturbative tail cut off by the top decay. As a result
$\Lambda_{QCD}$ or $\alpha_s$ could be determined accurately.
\item The top quark mass can be measured with a precision which is only
limited by the theoretical understanding of the excitation curve, but in
any case better than 500 MeV.
\item Top quarks are strongly polarized (about 40\%) even for unpolarized
beams; and longitudinal beam polarisation will enhance this value even
further. Detailed studies of top decays, in particular of the $V-A$ structure
of the $tbW$ coupling are therefore feasible.
\item The interquark potential is --- slightly --- modified by the Yukawa
potential induced by Higgs exchange.  The excitation curve and the top
quark momentum distribution may therefore lead to an indirect measurement
of the Yukawa coupling.
\item The large number of top quarks in combination with the constrained
kinematics at threshold, could facilitate the search for new decay modes
expected in extensions of the SM.
 \end{itemize}
With this motivation in mind the following points will be discussed: After
a brief review of qualitative features of threshold production (section
3.2.1), the present status of our theoretical understanding of the total
cross section will be presented in section 3.2.2.

The momentum distribution of top quarks and their decay products offers an
alternative and complementary route to probe the interquark potential, as
shown in section 3.2.3.  Spin effects and angular distributions are
sensitive towards the small $P$-wave contribution induced by the axial part
of the neutral current.  The theoretical framework and the resulting
predictions are collected in section 3.2.4.  Rescattering, relativistic
corrections and other terms of order $\alpha_s^2$ will be touched upon in
section 3.2.5.

\subsection{Introductory remarks}

For a qualitative understanding it is illustrative to compare the different 
scales which govern top production close to threshold.  The quarks are 
produced at a scale comparable to their Compton wave length
\be
d_{\rm prod} \sim 1/m_t
\ee
Electroweak vertex corrections do not alter this behaviour significantly, 
since $Z$- or $W$-boson exchange proceeds at a distance $\approx 1/m_Z$ 
which is still short compared to scales characteristic for the bound state 
dynamics. For the QCD potential 
\be
V_{\rm QCD} = -{4\over 3}{\alpha_s\over r}
\ee
one anticipates an effective coupling constant ${4\over 3}\alpha_s \approx 
0.2$, if $\alpha_s$ is evaluated at the scale of the Bohr momentum
\be
k_B \approx    {4\over 3}\alpha_s {m_t\over 2} \approx 20 GeV
\ee
The resulting Bohr radius 
\be
r_{\rm Bohr}= 1/k_B
\ee
is small compared to hadronic scales. The binding energy of the 1S level 
\be
E_B =  \left( {4\over 3}\alpha_s\right)^2 {m_t\over 4}\approx 2 
GeV
\ee
and, quite generally, the separation between different resonances, is 
smaller than the decay rate 
\be
2\Gamma_t \approx 3 GeV
\ee
whence all resonances will merge and join smoothly with the continuum.  

The coupling strength $\kappa$ of the Yukawa potential 
\be
V_Y = -{\kappa\over r}e^{-m_Hr}\quad {\rm with } \quad 
\kappa = \sqrt{2}G_F {m_t^2\over 4\pi} = 0.042
\ee
is comparable to the QCD coupling $4/3\alpha_s=0.2$ in magnitude. The 
exponential damping, however, with a cutoff $1/m_H \ll r_{\rm Bohr}$ and a 
lower limit $m_H > 65 $ GeV, reduces the impact of the Yukawa potential 
quite drastically.  (The  situation may be different in multi-Higgs-models: 
the couplings could be enhanced and, even more important, the Higgs might be 
lighter!)  Furthermore, the nonrelativistic treatment is no longer adequate 
and retardation effects must be taken into consideration.

The large top quark width plays a crucial role for the threshold behaviour, 
which is best understood in the framework of (nonrelativistic) Green's 
function techniques. The production of $t\bar t$ from a pointlike source 
(actually of extension $1/m_t$) at $x'$ with frequency $\omega = E$ is 
characterized by the time dependent Green's function  $G(\vec r,\vec r',t)$ 
which is a solution of the time dependent Schr\"odinger equation with a 
pointlike source term
\be
(i\partial_t-H) G(\vec r,\vec r',t) =  \delta(\vec r - \vec r') e^{-iEt}
\ee
with
\be
H={p^2\over 2m} + V(\vec r)
\ee
In the problem at hand $m=m_t/2$ is the reduced mass, $\vec r$ the relative 
distance between $t$ and $\bar t$, and the width $\Gamma=2\Gamma_t$.  The 
location of the source is at the origin
 $\vec r'=0$ by convention, and the second argument 
of $G$  will be suppressed, $G(\vec r,t)\equiv G(\vec r,0,t)$.  For a 
qualitative discussion one may ignore the potential and obtains for a
stable quark
\be
G(\vec r,t)&=& -{m\over 2\pi r} e^{i\sqrt{2mE}r} e^{-iEt}
\nonumber\\
\widetilde G(\vec p,t) &=& {1\over E-{p^2\over 2m}} e^{-iEt}
\label{eq:Green}
\ee
The corresponding current is flowing in radial direction from the source
\be
\vec j = {i\over 2m} \left( G\vec\nabla G^* -G^*\vec\nabla G \right)
=\vec e_r {m^2v\over 4\pi^2r^2}
\ee
with a constant flux through a sphere around the origin, reflecting the 
conservation of probability.

The width $\Gamma = 2\Gamma_t$ is introduced in the Schr\"odinger equation 
through the replacement
\be
H\to H-i\Gamma/2
\ee
and, consequently, through the substitution $E\to E+i\Gamma/2$ in 
eq.~(\ref{eq:Green}). The exponential damping of the flux in radial 
direction
\be
\vec j= \vec e_r {m^2 v\over 4 \pi^2 r^2} e^{i(\sqrt{2m(E+i\Gamma/2)}
-\sqrt{2m(E-i\Gamma/2)})r}
\ee
is most easily interpreted in two limiting cases.  For $E\ll \Gamma$ the 
decrease 
\be
\vec j\sim  {\vec e_r\over r^2} e^{-\sqrt{2m\Gamma}r}
\ee
is solely driven by the large width, with a cutoff $x_c^{-1} = 
\sqrt{2m\Gamma} \approx 2.4$ GeV for realistic parameters.  For $E\gg 
\Gamma$, on the other hand, the current decreases like
\be
\vec j\sim  {\vec e_r\over r^2} e^{-\Gamma r/v}
\ee
In this latter case top quarks may travel appreciable distances, up to 
$x_{\rm cut} \approx v/\Gamma$. However, for realisitic beam energies
they hardly propagate beyond the perturbative region.

%\subsection{The total cross section}

\subsubsection{Predictions for  the Coulomb potential}
The large top decay rate restricts the range of sensitivity to the short 
distance part of the potential
\be
\widetilde V(q) = -{16\pi\over 3} {\alpha_V(q)\over q^2}
\ee
which is approximately Coulombic, with a logarithmic variation of 
$\alpha_V$. Most of the qualitative features of top quark threshold physics 
can be understood even on the basis of the results for constant $\alpha_V$, 
which are available in analytical form. 

A remarkable feature of heavy quark production is the sharp rise of the 
cross section at threshold, a consequence of the attractive Coulomb force.  
The step function joins smoothly with the smeared resonances. 

Let us try to quantify this aspect with the help of simple nonrelativistic 
quantum mechanics ($\alpha={4\over 3}\alpha_s$, $m=m_t/2$).  The narrow 
resonances below the nominal threshold ($E$=0) are located at $E_n = 
-E_{Ryd}/n^2$ with $E_{Ryd}=k_B^2/(2m)=\alpha^2m/2$;

The production amplitude from a pointlike source with frequency $\omega=E$ 
is proportional to
\be
\langle \vec r=0 | \psi_n\rangle = \psi_n(0),
\ee
the rate correspondingly to
\be
R\sim |\psi_n(0)|^2\delta(E-E_n)
\ee
The wave function at the origin decreases with the third power of the 
radial quantum number,
\be
 |\psi_n(0)|^2 \sim 1/n^3
\ee
their spacing becomes increasingly dense
\be
\Delta E = E_{n+1}-E_n = E_{Ryd} \left({1\over n^2} -  {1\over 
(n+1)^2} \right)\sim E_{Ryd} {2\over n^3}
\ee
such that their average contribution to the cross section approaches a 
constant value (Fig.~\ref{fig:S18}).
  Above threshold one has to project the state $\langle \vec 
r=0|$ onto the Coulomb wave functions in the continuum $\psi_E(\vec r)$.
These replace the plane waves which are appropriate for the case where
final state interaction is absent.

The production amplitude is thus governed by
\be
\langle\vec r=0| \psi_E\rangle = \psi_E(0)
\ee
and the rate $\sim |\psi_E(0)|^2$. The threshold phase space factor $v$ 
is thus compensated by the $1/v$ singularity in $ |\psi_E(0)|^2$, and the 
cross section approaches a constant value for $E\to 0$. 
\begin{figure}[htb]
%\psfig{figure=resonance.ps,width=100mm,bbllx=70pt,bblly=135pt,%
%bburx=515pt,bbury=550pt}
\psfig{figure=resonance.ps,width=120mm,bbllx=0pt,bblly=270pt,%
bburx=450pt,bbury=550pt}
    \caption{Schematic representation of resonances (wiht the area of
the boxes adjusted to represent the weights of the respective delta
functions) and the continuum cross section.}
\label{fig:S18}
\end{figure}

The explicit calculation yields 
\be 
R(E) = {9\pi\over 2\alpha^2} \Gamma_e \delta(E-E_n) = 3Q_t^2{3\over 2} 
\sum_n {k_B^2\over m^2} {1\over n^3} \pi \delta\left(E-E_{Ryd}{1\over 
n^2}\right)
\label{eq:RE}
\ee
   for energies below threshold, and
\be
R(E)=3Q^2_t {3\over 2} \beta {x\over 1-e^{-x}} 
\label{eq:347}
\ee
with $x=k_B/k = \pi\alpha/\beta$
for energies above threshold.\footnote{For a textbook discussion of Coulomb 
scattering states and a derivation of this result see 
e.g.~\cite{Landau,Messiah}}

The perturbative expansion breaks down in the limit $\beta\to 0$.  The 
first term of this formal series
\be
\beta   {x\over 1-e^{-x}}  = \beta \left( {x\over 2}-\ldots\right)
\ee
underestimates the exact result by a factor two.



Eq.~(\ref{eq:RE}) allows to connect the formalism based on narrow individual 
resonances with a formulation which is tailored to the situation at hand, 
namely wide overlapping resonances which merge into a smooth continuum.


Instead of summing 
the contributions from a large number of high radial excitations
one may directly calculate the imaginary part of the Greens
function for complex energy
\begin{eqnarray}
\sigma (e^+e^- \to t\bar{t})
      = \frac{24\pi^2\alpha^2}{s} \frac{\rho_v(s)}{m^2_t}
       \left( 1 - \frac{16}{3} \frac{\alpha_s}{\pi} \right)
       \sum_{n} \left| \psi_n (0) \right|^2
       \frac{\Gamma_t}{(E_n - E)^2 + \Gamma^2_t} \nonumber\\
      = \frac{24\pi^2\alpha^2}{s} \frac{\rho_v(s)}{m^2_t}
       \left( 1 - \frac{16}{3} \frac{\alpha_s}{\pi} \right)
       \sum_{n} Im \frac{\psi^{\*}_n (0) \psi^*_n (0)}
       {E_n - E - i \Gamma_t} \nonumber\\
      = -\frac{24\pi^2\alpha^2}{s} \frac{\rho_v(s)}{m^2_t}
       \left( 1 - \frac{16}{3} \frac{\alpha_s}{\pi} \right)
       Im G(0,0,E+i\Gamma)
\label{eq:60}
\end{eqnarray}
The factor  $\rho_v(s)$  incorporates the contributions from the
intermediate photon and $Z$ and is given by 
\begin{eqnarray}
\rho^{Born}_v(s) = \left| e_t e_e + \frac{1}{y^2}
                \frac{\upsilon_t \upsilon_e M^2_\theta}
                     {s - M^2_Z + iM_Z \Gamma_Z} \right|^2
              + \left| \frac{1}{y^2}
                \frac{\upsilon_t a_e M^2_G}
                     {s - M^2_Z + iM_Z \Gamma_Z} \right|
\label{eq:58}
\end{eqnarray}
\[ \upsilon_f = 2I_{3f} - 4e_f \qquad
   a_f = 2I_{3f} \qquad
   y = 16 \sin^2 \theta_W \cos^2\theta_W \]
($\alpha=\alpha_{\rm eff}=1/128$ has been adopted in the numerical
   evaluation. Radiative corrections to this formula have been discussed in
\cite{GKKS}.)

The problem can be solved in closed analytical form for an exact
Coulomb potential \cite{fk} 
\be
{\rm Im}G_{E+i\Gamma_t} (0,0) &=& -{m_t^2\over 4\pi}
\left[ {k_2\over m_t} + {2k_B\over m_t}\arctan {k_2\over k_1}
\right.
\nonumber \\
&&\left. 
 +\sum_{n=1}^\infty {2k_B^2 \over m_t^2\pi^4} 
{\Gamma_t k_B n + k_2 \left(n^2
\sqrt{E^2+\Gamma_t^2}+{k_B^2\over m_t}\right)
\over
\left( E+{k_B^2\over m_tn^2}\right)^2+\Gamma_t^2}\right]
\nonumber\\
k_{1,2} &=& \left[
m_t\left( \sqrt{E^2+\Gamma_t^2}\mp E\right)/2\right]^{1/2},
\nonumber\\
k_B&=& {2\over 3} \alpha_s m_t.
\ee



To arrive at a realistic prediction of the total (and, in sect 3.2.3
of the differential) cross section the Coulomb potential must be
replaced by the realistic QCD potential.



\subsection{The QCD potential}

On the basis of earlier conceptual work in \cite{apple,sus} the
asymptotic behaviour of the static potential has been derived in
\cite{fis,bil}.  In momentum space the potential reads in the
$\overline{MS}$ subtraction scheme
\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}
The renormalization scale $\mu ^2$ has been chosen as $Q^2$, and $n_f$
refers to the number of massless quarks.  Employing standard arguments based
on the renormalization group, the $Q^2$ expansion of $\alpha _{\overline{MS}}(Q^2)$
is given by
% eq typed by chetyrkin
\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}

The leading behaviour of the potential at small distances
$[\gamma_E=0.5772\cdots]$
\begin{eqnarray}
V(r) & = & \frac{16\pi}{(33-2n_f)r \log 1 / (\Lambda _{\overline{MS}}r)^2}
\nonumber \\
& & \left[ 1 - \frac{b_1}{b^2_0}
               \frac{\log\log 1 / (\Lambda _{\overline{MS}}r)^2}
               {\log 1 / (\Lambda _{\overline{MS}}r)^2}
             + \frac{\left( \frac{31}{3} - \frac{10}{9} n_f \right)
               / b_0 + 2 \gamma_E }
               {\log 1 / (\Lambda _{\overline{MS}}r)^2 } + \cdots \right]
\label{eq:51}
\end{eqnarray}
is thus directly given in terms of the QCD scale parameter $\Lambda$.
The exploration of $V(r)$ for small distances could thus lead to a
direct determination of $\Lambda$.  For quark masses above 50-100~GeV the
ground state properties become independent of the potential in the
nonperturbative region.  As discussed in the previous section the
large decay rate acts as a cutoff and the predictions are fairly
insensitive to the actual regularisation.  However, an additional
constant which can be traded against a shift in $m_t$ must be
carefully calibrated.

In practice one connects the theoretically predicted short distance part
smoothly with the empirically determined potential above $\sim$0.1~fermi.
The asymptotic form given in (\ref{eq:49})
is based on the assumption that $n_f$ species of light quarks, taken
as massless, contribute to the vacuum polarization, and heavier ones
are ignored.  The value of $\Lambda_{\overline{MS}}$ in (\ref{eq:50})
must be properly related to $\Lambda_{\overline{MS}}$ as determined
from other experiments with a different number of effective light
flavors \cite{bern,marc}.  For the momentum range of around 15~GeV explored
by the $t\bar{t}$ system $n_f=$5 seems adequate. 

In the subsequent discussion the Green's function will be calculated in 
momentum space with the help of the Lippman-Schwinger integral
equation \cite{JKT}.  The 
representation of the QCD potential in momentum space with the large $Q^2$ 
behaviour given by eqs.~\ref{eq:49} and \ref{eq:50} will be employed.
The intermediate  
and small momentum dependence will be based on Richardson's potential. This 
choice allows to vary $\alpha_{\overline{MS}}(M_Z^2)$ (or equivalently 
$\Lambda_{\overline{MS}}$) between 0.11 and 0.13, while maintaining a 
smooth $Q^2$ dependence of $\alpha_V(Q^2)$ (Fig.~\ref{fig:AEFF}).
\begin{figure}[htb]
  \begin{center}
    \leavevmode
    \epsfxsize=60mm
    \epsffile[130 600 390 610]{fig_3.12.ps}
\vspace*{110mm}
    \hfill
\caption{$\alpha_{\rm eff}$ for different values of $\alpha_s(M_Z)$:
solid: 0.12, dashed: 0.11, dashed-dotted: 0.13, dotted: 0.10 and 0.14.}
\label{fig:AEFF}
  \end{center}
\end{figure}
  An additive constant 
in coordinate space (corresponding to a $\delta$-function in momentum 
space)  is adjusted to fix $V(r=1 GeV^{-1})=-1/4$ GeV for arbitrary  
$\alpha_{\overline{MS}}$. This constraint avoids the unmotivated and 
uncontroled variation of the long distance part of $V(r)$ with a change in 
$\alpha_{\overline{MS}}$. The potential in coordinate space is shown in 
Fig.~\ref{fig:COOR}.

\begin{figure}[ht]
  \begin{center}
    \leavevmode
    \epsfxsize=12.cm
    \epsffile[70 135 515 710]{fig_3.13.ps}
    \hfill
\caption{QCD potential in the position space $V_{JKT}$
 for different values of $\alpha_s(M_Z)$:
solid: 0.12, dashed: 0.11, dashed-dotted: 0.13, dotted: 0.10 and 0.14.}
\label{fig:COOR}
  \end{center}
\end{figure}

%\begin{figure}[htb]
%%\vspace*{100mm}
%\psfig{figure=fig_3.13.ps,width=120mm,bbllx=70pt,bblly=135pt,%
%bburx=515pt,bbury=710pt}
%\caption{QCD potential in the position space $V_{JKT}$
% for different values of $\alpha_s(M_Z)$:
%solid: 0.12, dashed: 0.11, dashed-dotted: 0.13, dotted: 0.10 and 0.14.}
%\label{fig:COOR}
%\end{figure}
