 
\section{Top quarks in the $e^+e^-$ continuum}

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{\em Feynman diagrams for
 $t\bar{t}$ or $t\bar{b}$ production in $e^+ e^-$ colliders.}

\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 $5\cdot 10^{40} cm^{-2}$,
at $E_{cm} = 500~GeV$ and for $m_t = 150~GeV$ one expects about
five $e^+e^ - t\bar{t}$ 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}
\em 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 $10^{41} cm^{-2}$ at $E_{CM}=1$ TeV
one expects only about 400 events (Fig.\ref{fig:F3.10})
and it is therefore not
clear whether these can be filtered from the huge background
and eventually used for a detailed analysis.

\begin{figure}
\vspace{13.5cm}
\caption{\em 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:F3.10}
\end{figure}


\begin{figure}[htb]
\psfig{figure=djouadi.ps,angle=-90,width=60mm,bbllx=60pt,bblly=60pt,%
bburx=510pt,bbury=340pt}
\caption{\em The cross section $\sigma(e^+e^- \to t\bar t H)$ (from
\protect\cite{djo}).} 
\label{fig:F3.9}
\end{figure}


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.  
The latter dominates by far and may therefore
be tested specifically with heavy quark final states.  The predictions
for the rate are shown in Fig.\ref{fig:F3.9}.  Depending on the mass of
the Higgs and the top quark, the reaction could perhaps be detected
with an integrated luminosity
of $10^{41}cm^{-2}$ \cite{Marti}.
%



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.  This will be discussed in a separate
part at the end of this section.

\subsection{$e^+ e^- \to t \bar{t}$}
\subsubsection{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
 \cite{jersa1} 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.
In the subsequent
nu\-me\-rical examples $G_F m^2_Z$ has been replaced 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.  For
$\alpha (s)$ we adopt \cite{seidel}
\begin{eqnarray}
1-\frac{\alpha}{\alpha(s)} & = & { \sum_\ell}
\frac{\alpha}{3\pi}\left(\ln\frac{s}{m^2_\ell}
-\frac{5}{3}\right)\nonumber \\
& & + 0.00165 + 0.003 \ln( 1 + s/1~GeV^2)
\label{eq:39}
\end{eqnarray}

\subsubsection{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 cast in the following form
\begin{eqnarray}
K_V & = & \frac{1}{\beta} \left[ A + \frac{P_V}{\left(1- \beta^2 / 3
\right) } \log \frac{1+\beta}{1-\beta} + \frac{Q_V}
{\left(1-\beta^2/3 \right) } \right] \nonumber \\
K_A & = & \frac{1}{\beta} \left[ A + \frac{P_A}{\beta ^2}
\log \frac{1+\beta}{1-\beta} + \frac{Q_A}{\beta ^2} \right]
\label{eq:41}
\end{eqnarray}

\begin{eqnarray}
A & = & (1+\beta^2) 
\left[
Li_2 \left[\left( \frac{1-\beta}{1+\beta}\right)^2\right]
+2Li_2 \left( \frac{1-\beta}{1+\beta}\right)
+\log{1+\beta\over 1-\beta}  \log{(1+\beta)^3\over 8\beta^2}
\right] 
\nonumber \\
& & + 3\beta \log \frac{1-\beta^2}{4\beta} -
\beta \log \beta \nonumber \\
P_V & = & \frac{33}{24} + \frac{22}{24} \beta ^2
- \frac{7}{24} \beta ^4 \nonumber \\
P_A & = & \frac{21}{32} + \frac{59}{32} \beta ^2
+ \frac{19}{32} \beta ^4 - \frac{3}{32} \beta ^6 \nonumber \\
Q_V & = & \frac{5}{4} \beta - \frac{3}{4} \beta ^3 \nonumber \\
Q_A & = & - \frac{21}{16} \beta + \frac{30}{16} \beta ^3
+ \frac{3}{16} \beta ^5
\label{eq:42}
\end{eqnarray}

%It is
QCD enhancement factors are
well approximated by \cite{appr}
\begin{eqnarray}
R_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 \\
R_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}

Next to leading order corrections have been calculated recently for
vector as well as axial current induced parts. Four contributions have
to be considered separately in both cases:

i) A purely gluonic correction proportional $C_F^2$, denoted $R_A$,
since it is equaly present in an abelian theory like QED. The
threshold behaviour $\sim 1/\beta^2$ is a consequence of the Coulomb
singularity and has to be absorbed in the Sommerfeld rescattering
correction. 
ii) The corresponding nonabelian part proportional $C_FC_A$ is
characteristic for the nonabelian group structure. For small $\beta$
it is enhanced $\sim \ln\beta/\beta$; a consequence of the large QCD
coupling close to threshold. These two parts have been evaluated
through semianalytical methods \cite{CKS,CHKST,CKS2}. The remaining
two contributions iii) and iv) with a gluon split into a light or
massive quark pair with a colour structure $\sim C_F Tn_l$ and $C_FT$
are available in analytical and semianalytical form
\cite{HKT}. Top production in the continuum, from $2m_t$ plus
about 10 GeV upwards is thus well under control. 

For small $\beta$ the leading (abelian) corrections can be obtained
from Sommerfeld's rescattering formula ($x\equiv {4\over
  3}{\pi\alpha_s\over \beta}$)
\begin{equation}
  K_V^{Som} = {x\over 1-e^{-x}}=1+{x\over 2}
+{x^2\over 12}
-{x^4\over 720}+\ldots
\end{equation}
To arrive at a reliable prediction close to threshold these terms have
to be resummed. For $\beta$ above $\sim 0.2$, corresponding to  an
energy roughly 10 GeV above the threshold, the fixed order calculation
in NLO approximation is adequate, and for the following discussion
even eq.~\ref{eq:43} provides a valid approximation.

\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. 
This leads to a significant suppression by about a factor
\begin{equation}
\left( \frac{\delta W}{m_t} \right) ^\beta \approx 0.5-0.6
\label{eq:47}
\end{equation}
with $\beta ={2\alpha\over \pi}\left(\ln{m_t^2\over m_e^2}-1\right)
=0.12$ and $\delta W = 1-5~GeV$ 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}).

\begin{figure}[ht]
  \begin{center}
    \leavevmode
    \epsfxsize=12.cm
    \epsffile[65 220 410 450]{fig_3.3.ps}
    \hfill
\caption{\em 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}

\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 about 7\%, for $m_H$ 
 between 100 and 1000~GeV,
and $E_{cm}$ fixed at 500~GeV. For even larger energies the
corrections decrease the $t\bar t$ cross section even further. This is
partially compensated by Higgs and $Z$ boson radiation. 
QCD and electroweak
corrections are thus of equal importance.

\begin{figure}
\vspace{10.5cm}
\caption[]{\em Genuine electroweak
corrections to top production in $e^+e^-$ annihilation.
From \cite{been1}.}
\label{fig:3.6}
\end{figure}

Close to threshold and for relatively small Higgs boson masses a rapid 
increase of these corretions is observed  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}.

\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 and amounts to about $-\,0.4$,
%\begin{figure}
%\vspace{80mm}
%\caption{Right/left asymmetry as function
%of $E_{cm}$ for $m_t = 150~GeV$.}
%\label{fig:3.7}
%\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}



For $m_t \sim 175$ GeV, the strong--interaction fragmentation process and
the weak decay mechanism are intimately intertwined. The lifetime
$\tau_* < \Lambda^{-1}$ is so short that the mesonic $(t\bar q)$
and baryonic $(tqq)$ bound states cannot be built--up any more.
Depending on the initial top quark energy, even remnants of the $t$
quark jet may not form any more \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 $\gamma \tau_* < 1/2$ fm, i.e.
\begin{equation}
E < \frac{1}{2} \frac{m_t^4}{m_W^3}
\end{equation}
\noindent the length of  the $t-\bar t$ string is too short to form
hadrons and jets cannot develop any more along the flight direction
of the top quarks.

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 $\Lambda$.

Top quarks are decelerated only by the early radiation of hard,
non--collinear gluons. The non--zero $t$ quark width $\Gamma$
acts as an infrared cut--off for gluon radiation.

Neglecting the rare hard non--collinear gluon radiation, the flux tube
forms between the $b$ and $\bar b$ quarks after the decay of the top
quarks, $e^+e^- \ra t\bar t \ra b \bar b W^+W^-$,
if the top mass is so large
 that
no $t$ jet remnants are left--over. As a result, the light hadrons
created in the flux tube, will be boosted into the $b\bar b$
hemisphere while the region opposite to the $b\bar b$ hemisphere
will
be depleted of particles \cite{fr3}. This is demonstrated
in Fig.\ref{Fazim} where
the distribution of the azimuthal angles of the final particles in the
plane perpendicular to the $t\bar t$ direction is shown; the $W$
bosons are assumed to decay leptonically in this event sample and the
rapidity of the final particles is restricted to $|y| < 1$, with $B$
decay products removed. The distribution of particles created  in the
$b\bar b$ string [``1--jet'']
is clearly distinct from the distribution where the
$b$ and $\bar b$ flux lines are connected to the $t$ and $\bar t$
fragments and three jets develop [``3--jet''].

\begin{figure}[htb]
\vspace{11cm}

\caption[]{\em Distributions of the azimuthal angles of particles with
$|y| < 1$ [$B$ decay products removed], created in the $b\bar b$ flux
tube (full curve) and in final states in which the $b$ and $\bar b$
flux lines have been connected with $t$ and $\bar t$ fragments (dashed
curve);  $\sqrt{s} = 500$ GeV, Ref.\cite{fr3}. }
\label{Fazim}
\end{figure}
%%%%%%%%%%%%%Nur in Review%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ifnum\wg=1
\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,st2,st3},
\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.} The
corresponding
helicity amplitudes have been given in Refs.\cite{st1,st2} at the quark
level. 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. While the general
helicity analysis can be found in the literature \cite{st1,st2}, we
shall focus here on a few physically interesting specific examples.

\subsubsection{Anomalous magnetic dipole moments of the top quark}
If the electrons in $e^+e^- \ra t\bar t $ are left--handedly
polarized, the top quarks are preferentially produced as left--handed
particles in the forward direction and only a small fraction is
produced as right--handed particles in the backward direction. As a
consequence of this \sm\  prediction, the backward direction is most
sensitive to small anomalous magnetic moments of the top quarks. This
is demonstrated quantitatively in Fig.\ref{Fheldis}
where the angular distribution
of the top quarks is broken down to the various helicity
contributions. It is apparent that the anomalous magnetic moments can
be bounded to a few percent through the measurement of the angular
dependence of the $t$ quark cross section.  Also in a recent study
\cite{schmitt} it is demonstrated that both production and decay
distributions are sensitive probes of new physics, manisfest in
production or decay vertices.

\begin{figure}[hbt]
\vspace{11cm}

\caption[]{\em Angular distributions of the $t$ production cross sections
for fixed $t$ and $\bar t$ helicities in polarized $e^-_Le^+_R$
annihilation. The parameter $\delta$ measures the anomalous magnetic
$(\gamma,Z)$
dipole moment of the top quark. From Ref.\cite{st2}.}
\label{Fheldis}
\end{figure}
%
\subsubsection{Electric dipole moments of the top quark}
A term $\sim F_2 \sigma_{\mu\nu} q_\nu \gamma_5$ can only be
generated in a ${\cal{CP}}$ non--invariant electromagnetic or $Z$
interaction. The static limit $d^{\gamma,Z}$ of $F_2$ is the electric
dipole moment of the top quark. Independently of the ${\cal{CP}}$
properties of the $t \ra bW$ decay amplitude, such a contribution
would be signalled by a non--zero expectation value of the
${\cal{CP}}$ odd momentum tensor \cite{st3}
\begin{equation}
T_{ij} = (q_+ - q_-)_i \frac{(q_+ \times q_-)_j}{|q_+ \times q_-|}
\end{equation}
\noindent The greatest sensitivity is achieved by choosing the unit
momentum vectors of the charged $W$--decay leptons for $q_{\pm}$
\begin{equation}
\langle T_{ij} \rangle = \frac{\sqrt{s}}{e} \left[ c_\gamma d^\gamma +
c_Z d^Z \right] \cdot \mbox{diag}\left[-\frac{1}{6}, -\frac{1}{6}, +\frac{1}{3}
 \right]
\end{equation}
\noindent where the coefficient $c_\gamma$, for instance, falls rapidly from
zero at threshold to $\sim -0.4$ at $\sqrt{s} = 500$ GeV for $m_t =
150$ GeV. Correlations between $q_+ \times q_-$ and the initial
$e^+e^-$ beam direction can be exploited to disentangle $d^\gamma$
from $d^Z$. Sensitivity limits to the electric dipole moments are
listed in the following table for  a top mass $m_t = 150$ GeV and the
integrated luminosity $\int{\cal{L}} = 10 \mbox{ fb}^{-1}$:
\begin{center}
\begin{tabular}{c|c}
$\sqrt{s}$ & $d^{\gamma,Z}$ \\
\hline
310 GeV & $< 7.0 \times 10^{-18}$ e cm \\
500 GeV & $< 2.8 \times 10^{-18}$ e cm
\end{tabular}
\end{center}

Other correlations, like $q_{e^+} [ q_{e^+} \times q_b -q_{e^-} \times
q_{\bar b}]$, and energy asymmetries can be defined that are
sensitive
to ${\cal{CP}}$ violation in the decay current \cite{st3} (see also
Ref.\cite{st1}).
%
\subsubsection{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 2\%, 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.]

The polarization of the top quark is transmitted through the decay
process to the polarization of the $W$ boson. Since the
$b$ quark is always left--handedly polarized,  the
%\clearpage\noindent
$W$ emitted parallel
to the $t$ polarization vector, for instance,
must be longitudinally polarized, while
it is left--handedly polarized if emitted in the opposite direction.
The $W$ polarization reflects itself in the angular distribution of
the charged decay lepton. As a result of the preceding discussion,
the lepton $l^+$ in the decay chain $t \ra b W^+ \ra l^+ \nu_l$ will
the be emitted preferentially parallel to the $t$ spin while the emission
into the opposite direction is forbidden \cite{stre},
\begin{equation}
d\Gamma(\vec{s}_t) = d\Gamma_0 \cdot \mbox{$\frac{1}{2}$} (1 +
\cos\theta_{l^+})
\end{equation}
\noindent This property is approximately preserved if radiative QCD
corrections are taken into account \cite{czar2}.
%

\subsubsection{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
%\clearpage\noindent
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 $-1/3$. The limiting  value
$-1/3$  is
assumed in the \sm. A detailed disussion with illustrative examples is
given in \cite{PIK}.

%\begin{figure}
%\vspace{55mm}
%\caption[]{\em Photon energy spectrum for
%different combinations of helicities of photons
%from the laser source and the electron beam
%($E_{\gamma} = 1.26~eV$, $E_e = 250~GeV$) for
%collinear configuration.}
%\label{fig:3.8}
%\end{figure}

%\begin{figure}
%\vspace{55mm}
%\caption[]{\em $\gamma \gamma$ luminosities for
%unpolarized initial state (dashed) and for both laser
%beams and both electron beams polarized (solid).}
%\label{fig:3.9}
%\end{figure}

%\begin{figure}
%\vspace{55mm}
%\caption{\em Decomposition of the $\gamma \gamma$
%luminosity into the non-flip (solid), single-flip (dashed),
%and double-flip (dotted) contribution for $h_{\gamma \gamma}=0$ in
%the initial state and polarized beams.}
%\label{fig:3.10}
%\end{figure}

%\newpage

%\begin{figure}
%\vspace{75mm}
%\caption{\em Cross section for $\gamma \gamma \to t\bar{t}$
%with polarized (solid, dashed) and unpolarized (dotted) photons.}
%\label{fig:3.11}
%\end{figure}

%\begin{figure}
%\vspace{85mm}
%\caption{\em Cross section for $t\bar{t}$ production
%in units of $\sigma_{point}$ after folding with the $\gamma \gamma$
%luminosity.  Solid:  $2\lambda P = 2\tilde{\lambda}\tilde{P} = -1$.
%Dashed:  $2\lambda P = 2\lambda \tilde{P} = 0$.}
%\label{fig:3.12}
%\end{figure}




\fi%static  t parameters nur in review
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\subsection{Top quark production in $\gamma \gamma$ collisions}

It has been repeatedly advocated \cite{ginz} to convert an electron
collider into a photon collider in order to have access with large
luminosities to $C\!=\!+1$ final states.  $\gamma \gamma$ fusion into a
Higgs boson is one interesting possibility \cite{rich}, $W$ pair
production leading to a variety of tests of the $W$ magnetic moment is
another one \cite{schr}.  Here we will deal with $t \bar{t}$ production,
which on the one hand could be considered to be a background to the
other reactions, but could on the other hand allow the study of
top quark couplings to real photons, a subject interesting
in its own right.

Adjusting the helicities of incoming electron ($P_e$) and photon
($P_{\gamma}$) from the laser source appropriately
($2P_e P_{\gamma} = -1$), most of
the electron energy is transferred to the scattered photon
(Fig.\ref{fig:3.8}).
The helicity of the scattered photon is dominantly flipped.
Therefore, adjusting the two laser beams in an opposite
helicity configuration (and the electron beams accordingly)
one obtains a $\gamma \gamma$ configuration
peaked at large energies (Fig.\ref{fig:3.9}) and dominantly in a
state of helicity zero (Fig.\ref{fig:3.10}).
Since the $t\bar{t}$ cross section close to threshold is
dominated by the helicity zero configuration
(Fig.\ref{fig:3.11}), one finds a large top cross section in
the threshold region even after folding with the
$\gamma \gamma$ luminosity for appropriately
polarized electron and laser beams (Fig.\ref{fig:3.12}).
For details see \cite{steeg}.


\begin{figure}
\vspace{75mm}
\caption[]{\em Photon energy spectrum for
different combinations of helicities of photons
from the laser source and the electron beam
($E_{\gamma} = 1.26~eV$, $E_e = 250~GeV$) for
collinear configuration.}
\label{fig:3.8}
\end{figure}

\begin{figure}
\vspace{75mm}
\caption[]{\em $\gamma \gamma$ luminosities for
unpolarized initial state (dashed) and for both laser
beams and both electron beams polarized (solid).}
\label{fig:3.9}
\end{figure}

\begin{figure}
\vspace{75mm}
\caption{\em Decomposition of the $\gamma \gamma$
luminosity into the non-flip (solid), single-flip (dashed),
and double-flip (dotted) contribution for $h_{\gamma \gamma}=0$ in
the initial state and polarized beams.}
\label{fig:3.10}
\end{figure}

\begin{figure}
\vspace{95mm}
\caption{\em Cross section for $\gamma \gamma \to t\bar{t}$
with polarized (solid, dashed) and unpolarized (dotted) photons.}
\label{fig:3.11}
\end{figure}

\begin{figure}
\vspace{10.4cm}
\caption{\em Cross section for $t\bar{t}$ production
in units of $\sigma_{point}$ after folding with the $\gamma \gamma$
luminosity.  Solid:  $2\lambda P = 2\tilde{\lambda}\tilde{P} = -1$.
Dashed:  $2\lambda P = 2\lambda \tilde{P} = 0$.}
\label{fig:3.12}
\end{figure}




















