%Title: Top Quarks
%Author: J.H. K\"uhn
%Published:  <EM>Acta Phys. Pol.</EM><B> B26 </B><EM>(1995) 711--729.</EM>
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\noindent{\bf REFERENCES}\\
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\begin{document}
\begin{center}%***********************************
 {\tenbf TOP QUARKS
\footnote{Work supported by BMFT contract 0056KA93P}\\}%
\vglue 0.9cm
{\tenrm JOHANN H. K\"UHN\\}
\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}%****************************************

\vspace{-7cm}
\begin{flushright}
TTP95-11
\end{flushright}
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\begin{center}
\begin{minipage}{12.5cm}
{\rightskip=3pc\leftskip=3pc\tenrm\baselineskip=12pt%
\narrower\smallskip\noindent%
Recent theoretical results on the production and decay of top quarks are 
presented. The implications of the new experimental results from the
TEVATRON are briefly discussed. Predictions for the top quark decay 
rate and distributions are described, including the influence of QCD and
electroweak radiative corrections.
Top production at an $e^+e^-$ collider is discussed with 
emphasis towards the threshold region.
The polarization of top quarks in the threshold region
is calculated with techniques based on Green's functions for $S$ and 
$P$ waves.
\smallskip
}
\end{minipage}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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 
for a long time only to
a steady increase of the lower bound of the top quark mass.
The direct limits from LEP of about 45 GeV and the indirect limit
from the absence of $W\to t+b$ as inferred from
the $W$ decay rate (62 GeV) were derived in a fairly
model-independent way. 
With increasing luminosity at the TEVATRON the limit had increased
steadily until the spring of 1994 when first evidence for the top quark
was presented by the CDF collaboration \cite{cdf94}.
Finally, after collecting a data sample corresponding to 67 pb$^{-1}$,
the top quark has been observed by both collaborations CDF \cite{cdf95}
as well as \Dnull\ \cite{dzero} with a mass of about 
176 GeV and 199 GeV respectively.

These results are nicely consistent with 
precision measurements of electroweak parameters which exploit the
quadratic top mass dependence \cite{veltman}
of the $\rho$-parameter, resulting  in \cite{schailecern}
$m_t=178\pm 11\pm 19 {\rm GeV}$ where the second error originates from 
the variation of 
the Higgs mass between 60
GeV and 1~TeV with a central value at 300 GeV.

The discovery of the top quark has completed the fermionic
spectrum of the Standard Model.  The precise 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.
The study of the top quark and
its decay modes will thus constitute an important part of the physics
program at the TEVATRON, the LHC, and the future linear collider.

After the discovery of the top quark in
hadronic collisions and the subsequent exploratory studies, 
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 measured 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.

After a brief discussion of the recent discovery of the top quark
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.  
Calculations of the spectrum of $W$s and leptons from top
decays will be summarized.  The prediction for $t\bar t$ production
in $e^+e^-$ annihilation is summarized in section 3.
Emphasis is put on 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. Recent results concerning the polarization of top quarks
in the threshold region will be treated in section 4.

\secsubsec{2}{Mass, decay rate and distributions}{2.1}{Top quark mass}
%
After collecting a large amount of data in $p \bar{p}$ collisions at
an energy of 1.8 TeV, corresponding to 67 pb$^{-1}$ for the CDF 
experiment and 50 pb$^{-1}$ for \Dnull, 
both collaborations have reported the 
observation of the top quark, tagged in $t \bar{t}$ events where one
quark decays semileptonically, the other hadronically or also
semileptonically.
With a production cross section of $6.8^{+3.6}_{-2.4}$ pb as measured
by CDF and $6.4\pm 2.2$ pb from \Dnull\ theory and experiment are 
in fair agreement for a mass around 180 GeV as demonstrated in 
Fig.\ref{dzerofig} (taken from \cite{dzero}).
%
% [figure 3 aus \Dnull]
%
\bild{95}{405}{530}{720}{8cm}{dzerofig}
	{\Dnull\ measured $t\bar t$ production cross section
	(solid line with one standard deviation error band)
	as a function of assumed top quark mass.
	Also shown is the theoretical cross section curve
	(dashed line).}{dnull3.ps}

On the other hand the sample of reconstructed top events leads to 
a mass of $m_t = 176\pm 8\pm 10 {\rm GeV}$ at CDF (Fig.\ref{cdffig})
and $m_t = 199^{+19}_{-21}\pm 22 {\rm GeV}$ at \Dnull\ nicely 
consistent with the cross section measurements.
%
% [figure 3 aus CDF]
%
\bild{130}{235}{520}{650}{8cm}{cdffig}
	{Reconstructed mass distribution for the $b$-tagged
	$W + \ge 4$-jet events (solid).
	Also shown are the background shape (dotted) and the sum of
	background plus $t\bar{t}$ Monte Carlo for 
	$M_{top} = 175 {\rm GeV/c^2}$ (dashed), with the background
	constrained to the calculated value, $6.9^{+2.5}_{-1.9}$
	events. The inset shows the likelihood fit used to
	determine the top mass.}{cdf3.ps}

This important result not only provides the final quark required for 
consistency of the SM. The agreement of the mass measurements with
the predictions based on electroweak radiative corrections of
$178\pm 11\pm 19 {\rm GeV}$ \cite{schailecern} confirms
the importance of quantum corrections in the electroweak theory.
A reduction of the uncertainty in the mass determination down to about
$\pm 1 {\rm GeV}$ as expected from experiments at a linear electron
positron collider would allow to fully exploit the precision expected
in electroweak observables in the near future (Fig.\ref{schailefig}).
\bild{55}{160}{585}{700}{8cm}{schailefig}
	{Constraints on the Higgs mass for a central value of 
	150 GeV from final data of LEP I, the direct $W$-mass
	measurements and the left-right $e^+e^-$ polarization
	asymmetry, if the mass is determined externally with large
	or small error \cite{schailecern}.}{schaile.ps}

However, also a modest $\delta m_t = \pm 5 {\rm GeV}$ which could be
achieved at the LHC would lead to important restrictions on the mass
of the Higgs boson in the Standard Model.

The discovery has immediate implications for top decays:
If the experimentally measured and the calculated cross section are
in good agreement, then the decay mode $t\rightarrow b + W$ must be 
dominant and any more exotic channel like $t\rightarrow bH$
or decays envisaged in SUSY models like 
$t\rightarrow \tilde{t} \widetilde{\gamma}$ can only appear as small 
admixtures.
This justifies to study the top decay mode $t\rightarrow bW$,
including all refinements of QCD and electroweak corrections.

\subsec{2.2}{Top decays}
%
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.}
\bild{115}{220}{465}{585}{10cm}{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.}{jkplot3.ps}

\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} \ll \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. Hence, for the actual value of $m_t\approx 180 {\rm GeV}$
energy and angular distributions of all decay products can be described
to a good approximation by those resulting of the decay of 
``free'' spin $1/2$ quarks.

%\pictleft{9.5cm}{5.5cm}{5.4cm}{fig:f}%
%{QCD correction function $f$ for the top quark
%decay rate.  Solid line:  exact form; dashes:  $m_b = \Gamma_b=0$.}

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}.
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 $y=m^2_W / m^2_t$.
This approximation describes the exact correction function to better
than 1\% for $m_t$ above 160 GeV.
In fact, even the asymptotic result 
$f(0) = {2\over 3} \pi^2 - {5\over 2} \approx 4.08$
provides an adequate approximation which is valid to about 10\% accuracy.
In general $\mu^2 \approx m_t^2$ is adopted for the characteristic
scale of $\alpha_s(\mu^2)$. Recently, however, arguments based on 
partial calculations have been given \cite{voloshin} that the 
effective scale should be reduced significantly down to
$\mu = 0.28 m_t$ which would lead to a significant decrease
of the rate.

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 cancellations 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 1-2\%,
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.
At a hadron collider no measurement of the decay rate can be performed.

\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
 170.0& .108& 1.405& -1.52 &-8.49&  1.67& 1.287\\
 180.0& .107& 1.714& -1.45 &-8.48&  1.70& 1.572\\
 190.0& .106& 2.059& -1.39 &-8.47&  1.73& 1.890\\
\hline
\end{tabular}
\end{center}
\end{table}

An important issue will be the search for new decay modes
expected in extensions of the Standard Model.  

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 $m_t\sqrt{\sqrt{2}G_F}$. 
However, this choice is theoretically unattractive and with the top
quark discovery at the TEVATRON already excluded.
For the ``natural'' range of $\tan\beta$
between 1 and $m_t/m_b$ 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}.

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, isosinglet quarks, or mirror quarks.

\subsec{2.3}{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.  
Charged lepton angular distributions are therefore an ideal tool to
determine the spin of top quarks. Neutrino distributions, on the other
hand, are particularly sensitive towards small admixtures of righthanded
currents in the $tbW$ coupling \cite{JezK}.
Predictions for the $W$
energy distribution are also available \cite{czarn,junger}.

\secsubsec{3}{Production in  $e^+e^-$ collisions}%
{3.1}{Far above threshold}
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 $G_F$ parametrized 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 has been discussed in more
detail in \cite{jk_plb316}.

The production cross section for $t\bar  t$ has to be
contrasted with the potential backgrounds $f\bar f$, $WW$ and $ZZ$.
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. 

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}.
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.

%\pictup{10.1cm}{4cm}{rfig}%
%{Cross section for $t\bar{t}$ production,
%including resonances, QCD corrections
%and initial state radiation in units of $\sigma_{point}$.}
\bild{85}{240}{370}{410}{10cm}{rfig}
	{Cross section for $t\bar{t}$ production,
	including resonances, QCD corrections
	and initial state radiation in units of $\sigma_{point}$.}
	{tpicop4i.ps}	

{\em 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.
Given favorable mass assignments these reactions might be
experimentally accessible.

\subsec{3.2}{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 form factors 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.

As stated before, one faces a series of many overlapping
resonances, once $m_t$ lies above 120 GeV.  A convenient technique
to perform this summation was proposed in \cite{khoze}.  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 QCD corrected continuum cross section 
join smoothly \cite{GKZ}.
%\pictup{9.5cm}{5.5cm}{figalph}%
%{Production cross section in the threshold region
%without initial state correction for different
%values of $\Lambda_{\overline {MS}}$.}
\bild{100}{300}{460}{515}{10cm}{figalph}
	{Production cross section in the threshold region
	without initial state correction for different
	values of $\Lambda_{\overline {MS}}$.}
	{tpiclms3.ps}

%\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.}
\bild{35}{210}{535}{600}{10cm}{mom}
	{Green's function $|p{\cal G}(p,E+i\Gamma_t)|^2$ 
	for $m_t = 180$  and  $E= -2.9$  GeV (1s peak) -- solid, $E = 0$ --
	dotted, and $E = 2$ GeV -- dashed line.}{gg.ps}
%
\subsec{3.3}{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 Lippmann-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}
The resulting momentum distribution is shown in Fig.\ref{mom} for a
characteristic set of energies.  

\subsec{3.4}{The Potential}
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
unambiguously 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.
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.

\subsec{3.5}{Relativistic corrections}
It has been  argued in \cite{JKT,jezt} and demonstrated rigorously in
\cite{Yakov} that ${\cal O} (\alpha_s$) corrections to the 
total cross section are absent. 

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{JKT,Sumino,jezt}. 
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. These considerations have recently been confirmed in a more
formal approach \cite{KumMoed}.

\sec{4}{Polarization near threshold \cite{HJKT}}
The calculations of top polarization well above the threshold
for $t\bar t$ production were published long ago \cite{krz}.
Recently also the angular distributions of top quarks \cite{MurSu}
and their polarization \cite{HJKT} has been investigated in the 
threshold region which deserves a special study. This polarization
is experimentally accessible as a consequence of the strong
analyzing power of leptons from top decay.

Due to restricted phase space the amplitude is
dominantly $S$ wave and the electron and positron
polarizations are directly transferred to the top quark.
For a quantitative study
this simple picture
has to be extended and the modifications
originating from $S-P$ wave interference
should be taken into account.
As a consequence of final state interaction between
$t$ and $\bar t$, which leads to the familiar
QCD potential, the parton model prediction
where these terms are simply proportional to
$\beta=\sqrt{1-4m_t^2/s}$ has to be modified.
The interference terms are therefore calculated
from numerical solutions of Lippmann-Schwinger
equations.

We describe the longitudinal polarization of the $e^+e^-$ system
in its center-of-mass frame as a function of the variable
\begin{equation}
\chi = {P_{e^+}-P_{e^-}\over 1 - P_{e^+}P_{e^-}}
\end{equation}
where $P_{{e^\pm}}$ denote the polarizations of $e^\pm$
with respect to the directions of $e^+$ and $e^-$ beams,
respectively.
A righthanded system of coordinates is defined through the
triplet of orthogonal unit vectors: $\hat n_{\bot}$, $\hat n_{_N}$
and $\hat n_{^\|}$ where
$\hat n_{^\|}$ points in the direction of $e^-$ beam,
$\hat n_{_N}\sim \vec p_{e^-}\times \vec p_t$  is normal
to the production plane and
$\hat n_{\bot}=\hat n_{_N}\times\hat n_{^\|}$.
In the absence of phases from final state interaction,
which can be induced by higher orders in $\alpha_s$ and will be
considered elsewhere, the top quark polarization vector
is in the production plane and $P_{^\|}$ and $P_{\bot}$ denote
its longitudinal and transverse components.
The definition of $P_{^\|}$ and $P_{\bot}$ with respect
to the beam direction is convenient for the treatment
close to threshold and differs from the definition
of \cite{krz} where the quantities have been defined with respect
to the direction of flight of the top quark.
The angle $\vartheta$
denotes the angle between $e^-$ and the top quark.

\noindent
In the threshold region the top quark is nonrelativistic (with
velocity $\beta=p/m_t\sim\alpha_s$)
and the kinetic energy of the $t\bar t$ system
$E=\sqrt{s}-2m_t$ is of order ${\cal O}(\beta^2)$.
As a consequence of the $t\bar t$ interaction the top quarks
will exhibit a momentum spread (Fermi motion) even for fixed
energy $E$. Also the polarization and the angular
distribution will depend on both $E$ and $p$.
Retaining only the terms up to
${\cal O}(\beta)$ one derives the following expressions for
the components of the polarization vector, as functions
of $E$, $p$ and $\vartheta$
\begin{eqnarray}
{\cal P}_{^\|}(p,E,\vartheta) &=& C^0_{^\|}(\chi)
+ C^1_{^\|}(\chi)\varphi(p,E)\cos\vartheta
\label{eq:pol2}\\
{\cal P}_\bot(p,E,\vartheta) &=&C_\bot(\chi)\varphi(p,E)\sin\vartheta
\label{eq:pol3}
\end{eqnarray}
and, after the integration over top quark momentum $p$,
\begin{eqnarray}
P_{^\|}(E,\vartheta)  &=& C^0_{^\|}(\chi)
+ C^1_{^\|}(\chi)\Phi(E)\cos\vartheta
\label{eq:2a}\\
P_\bot(E,\vartheta)  &=&C_\bot(\chi)\Phi(E)\sin\vartheta
\label{eq:3a}
\end{eqnarray}
The coefficients $C^0_{^\|}(\chi)$, $C^1_{^\|}(\chi)$
and $C_\bot(\chi)$ are given in \cite{HJKT}. They depend on the 
polarization $\chi$, the
electroweak coupling constants, the $Z$ mass and the center-of-mass
energy $\sqrt{s}\approx 2m_t$. They are plotted in
Fig.\ref{fig-CLchi} for $m_t=174$~GeV,
$\sin^2\theta_W = 0.2317$ and $M_Z = 91.1888\; GeV$.
$C^0_{^\|}(\chi)$ and $C^1_{^\|}(\chi)$
are shown in Fig.\ref{fig-CLchi}a as the solid and the dashed
lines, respectively, and $C_\bot(\chi)$ as the solid line in
Fig.\ref{fig-CLchi}b.
\bild{30}{295}{530}{515}{12cm}{fig-CLchi}
	{Coefficient functions: a) $C^0_{^\|}(\chi)$ -- solid line
	and $C^1_{^\|}(\chi)$ -- dashed line, b) $C_\bot(\chi)$ -- solid line
	and $C_{FB}(\chi)$ -- dashed line.}{figpol.ps}
%
The functions $\varphi(p,E)$  and $\Phi(E)$
in eqs.(\ref{eq:pol2})--(\ref{eq:3a}) replace the simple
factor $\beta$ in a calculation for free, noninteracting quarks.
They describe the complicated dynamics of the
$t\bar t$ system near threshold which in particular
includes effects of the would-be toponium resonances
and Coulomb enhancement. Nevertheless, it is possible to calculate
these functions using the Green function method.
The same functions $\varphi(p,E)$ and $\Phi(E)$ also govern the
forward backward asymmetry 
\begin{equation}
A_{FB}(E) = C_{FB}(\chi)\Phi(E)
\end{equation}
which has been first obtained in \cite{MurSu} for $\chi=0$.

To evaluate the functions $\varphi(p,E)$ and
$\Phi(E)$ the Green function method
is adequate which
has become a standard tool for studying
$e^+e^-$ annihilation in the threshold region
\cite{khoze,peskin,Sumino,JKT}.
In \cite{HJKT} the momentum space approach has been adopted.
The Lippmann-Schwinger equation was solved numerically
for the $S$-wave and $P$-wave Green functions
\begin{eqnarray}
G(p,E) &=&
G_0(p,E) +
G_0(p,E)
\int {{\rm d}^3q\over(2\pi)^3}
\tilde V\left(\,|\,\vec{p}-\vec{q}\,|\,\right)
G(q,E)
\label{eq:LS1}\\
F(p,E) &=&
G_0(p,E) +
G_0(p,E)
\int {{\rm d}^3q\over(2\pi)^3}
{{\vec p}\cdot{\vec q}\over p^2}
\tilde V\left(\,|\,\vec{p}-\vec{q}\,|\,\right)
F(q,E)
\label{eq:LS2}
\end{eqnarray}
where again 
\begin{equation}
G_0(p,E) = \left(\,
E- {p^2/ m_t}+ {\rm i}\Gamma_t\,\right)^{-1} 
\label{eq:8}
\end{equation}
In the nonrelativistic approximation the momentum distribution
of the top quark is dominated by the $S$-wave contribution and
it is proportional to
\begin{equation}
{\cal D}_{S-S}(p,E)=p^2 \left|\, G\,\right|^2 
\end{equation}

Contributions of ${\cal O}(\beta)$ to
${\cal P}_{^\|}$ and ${\cal P}_{\bot}$
as well as to the forward-backward asymmetry ${\cal A}_{FB}$
arise from the interference of $S$- and $P$- waves
and are therefore proportional to
\begin{equation}
{\cal D}_{S-P}(p,E)= p^3\, {\cal R}e \left( \,G\,F^*\,\right)/m_t
\end{equation}
This distribution exhibits a significantly wider spread in momentum,
a feature naturally explained by the larger momentum
of $P$-states. The angular distribution,
the forward-backward asymmetry, and the components
(\ref{eq:pol2}) and (\ref{eq:pol3}) of the polarization vector
for fixed $E$ and $p$ are then governed by the ratio
\begin{equation}
\varphi(p,E)\, =\,  {
\left( 1- {4\alpha_s\over 3\pi} \right)\, {\cal D}_{S-P}(p,E)
\over
\left( 1- {8\alpha_s\over 3\pi} \right)
{\cal D}_{S-S}(p,E)  } .
\end{equation}
The factors
$\left( 1- {4\alpha_s\over 3\pi} \right)$
and
$\left( 1- {8\alpha_s\over 3\pi} \right)$
originate from transversal gluon exchange  \cite{KZ}.
In a similar way the function
\begin{equation}
\Phi(E) = {
\left( 1- {4\alpha_s\over 3\pi} \right)
\int_0^{p_{m}}\,{\rm d}p\, {\cal D}_{S-P}(p,E)
\over
\left( 1- {8\alpha_s\over 3\pi} \right)
\int_0^{p_{m}}\,{\rm d}p\, {\cal D}_{S-S}(p,E) }
\label{eq:Phi}
\end{equation}
describes the integrated quantities.
The upper limit
$p_{m}$ has been introduced in order to cut off
the logarithmic divergence of the numerator.
In experimental analyzes the
contributions of very large intrinsic momenta
will be automatically suppressed
by the separation of $t\bar t$ events from the background.
In \cite{HJKT} $p_{m}= m_t/3$ was adopted.
%
\bild{25}{310}{520}{515}{12cm}{fig-FiR}
	{Energy dependence in the threshold region
	of: a) $\Phi(E)$ and
	b) $R=\sigma(e^+e^-\to t\bar t\,)/\sigma(e^+e^-\to \mu^+\mu^-)$
	for $m_t=174$~GeV and $\alpha_s(m_Z)=0.11$, $0.12$ and $0.13$
	-- dashed, solid and dotted lines, respectively.}{phisig.ps}
	{phisig.ps}
The function
$\Phi(E)$ is plotted in Fig.\ref{fig-FiR}a for $m_t=174$~GeV.
For a comparison in
Fig.\ref{fig-FiR}b the annihilation cross section
$\sigma(e^+e^-\to t\bar t\,)$
is shown in units of
$\sigma(e^+e^-\to \mu^+\mu^-)$.
The increase of $\Phi(E)$ with $E$ resembles qualitatively the
behavior $\beta\approx \sqrt{E/m_t}$ expected for noninteracting
quarks. Linear terms in $\beta$ amount to typically 0.1 and could,
therefore, be accessible experimentally, whereas relativistic
corrections $\sim\beta^2$ are evidently  small.

\par\noindent
Combining Figs.(\ref{fig-CLchi})--(\ref{fig-FiR})
the predictions for the polarizations
and the angular distributions can be discussed easily.
The coefficient $C^0_{^\|}(\chi)$
governs the leading term for the longitudinal polarization.
For $\chi=\pm 1$ the top quark polarization is simply given by
the polarization of the incoming electrons, resulting in
$C^0_{^\|}(\chi)=\mp 1$ and
$C^1_{^\|}(\chi)=0$, and consequently in a nearly completely
polarized sample of top quarks, a strong argument for top
production with polarized beams. The deviation from the $\beta=0$
prediction for the longitudinal top polarization is maximal
at $\chi\approx -0.4$, where
$C^0_{^\|}(\chi)$ happens to vanish, but remains generally small.
The coefficient
$C_\bot(\chi)$ is small for unpolarized beams. For
$\chi= -1$ however, it amounts to 0.65 which leads to sizable
polarization of top quarks perpendicular to the beam,
if they are produced at large angles.
Since leptons from top decays determine its spin with maximal analyzing
power, these polarizations may well be measured at a future linear
collider.

\vspace{.3cm}
\noindent
{\bf\underline{Acknowledgements:}}\\[.2cm]
The contributions of A. Czarnecki, R. Harlander and T. Teubner to the
TOPics presented in this review are gratefully acknowledged.
Special thanks, however, go to Marek Je\.zabek for a long lasting
and fruitful collaboration. Without the \TeX nical help of
R. Harlander the work would not have been finished in time.

\begin{thebibliography}{99}
\tenrm
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\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}
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\end{document}




