
\subsection{Realistic predictions for $\sigma_{t\bar t}$.}

For a realistic QCD potential the Green's function can only be calculated 
with numerical methods. An elegant algorithm for a solution in coordinate 
space has been suggested in \cite{pesk}. As a consequence of the 
optical theorem (see also eqn.~(\ref{eq:60})) only the imaginary part
of 
 $G(\vec r=0,\vec r'
=0,E+i\Gamma_t)$  
is needed to predict the total cross section.  The differential equation 
for the Green's function
\be
\left[ \left(E+i\Gamma_t\right)-\left( -{\nabla^2\over m_t} + V(\vec 
r)\right)\right] G(\vec r, \vec r'=0,E+i\Gamma_t) = \delta(\vec r)
\ee
is solved in a way which provides direct access to   Im$G(\vec r=0,\vec r' 
=0,E+i\Gamma_t)$ without the need to calculate the full $\vec r$ dependence. 
Alternatively, in \cite{JKT,TJ} the Green's function in momentum space was 
obtained from the Lippmann-Schwinger equation
\be
G(\vec p,E+i\Gamma_t)& =& G_0(\vec p,E+i\Gamma_t)  + G_0(\vec p,E+i\Gamma_t)  
\nonumber \\
&&\times \int{d\vec q\over (2\pi)^3} \widetilde V(\vec p - \vec q)
 G(\vec q,E+i\Gamma_t)  \nonumber\\
 G_0(\vec p,E+i\Gamma_t)  &=& {1\over E-p^2/m_t+i\Gamma_t}
\label{eq:355}
\ee
The total cross section is in this case obtained from the integral over the 
differential distribution
\be
{d\sigma\over d^3p} = {3\alpha^2\over \pi s m_t^2} \rho_v(s) \Gamma_t 
|G(\vec p,E+i\Gamma_t)|^2
\ee
This second formulation is particularly suited to introducing a momentum 
and energy dependent width $\Gamma(p,E)$ which allows to incorporate the 
phase space suppression and certain $\alpha_s^2$ rescattering corrections 
to be discussed below in section 3.2.6.

It is well known that the coupling of the virtual photon to the quarkonium 
boundstate is modified by ``hard'' gluon exchange.  The vertex correction 
to the vector current produces an additional factor $\left(1-{16\over 
3}{\alpha_s\over \pi}\right)$ for the quarkonium decay rate into
{\epem} 
through the virtual photon or $Z$.  This factor can be calculated by 
separating the gluon exchange \cite{Buchm} correction to the vertex into 
the instanteneous potential piece and a remainder which  is attributed to 
gluons with high virtualities of order $m_t$.  A similar approach has been 
developed in \cite{JKsim} for Higgs exchange.  The vertex correction is 
again decomposed into a part which is given by the instantaneous Yukawa 
potential
\be
V_{Yuk}(r) = -\kappa {e^{-m_Hr}\over r}
\ee
with $\kappa=\sqrt{2} Gm_t^2/4\pi$ and a remainder which is dominated by 
highly virtual Higgs exchange.  The rapid increase of the correction in the  
threshold region (cf.~sect. 3.1.2) is driven by the potential; the 
remainder, the hard vertex correction, is fairly energy independent.  The
total cross section is thus sensitive to the top mass, the width (which in 
the SM is uniquely determined by $m_t$), the strong coupling constant 
$\alpha_s$ and the mass of the Higgs boson.  This dependence is illustrated 
in Figs.~\ref{fig:310}-\ref{fig:313}.  Apart from the trivial shift
of the threshold due to a  
change in $m_t$ the shape of $\sigma$ is affected by the rapidly increasing 
width of the top quark which amounts to 0.81 GeV, 1.57 GeV and 2.24 GeV for 
$m_t =$ 150 GeV, 180 GeV, and 200 GeV respectively. This is demonstrated in 
Fig.~\ref{fig:310}: A fairly 
\begin{figure}[ht]
  \begin{center}
    \leavevmode
    \epsfxsize=12.cm
    \epsffile[110 265 465 560]{fig_3.10.ps}
    \hfill
\caption{Total cross section as function of $E=\protect\sqrt{s}-2m_t$
for three values of the top quark mass.}
\label{fig:310}
  \end{center}
\end{figure}
\begin{figure}[ht]
  \begin{center}
    \leavevmode
    \epsfxsize=12.cm
    \epsffile[110 265 465 560]{fig_3.11.ps}
    \hfill
\caption{Total cross section as function of $E=\protect\sqrt{s}-2m_t$
for three values of $\alpha_s$. }
\label{fig:311}
  \end{center}
\end{figure}
\begin{figure}[ht]
  \begin{center}
    \leavevmode
    \epsfxsize=12.cm
    \epsffile[110 280 465 560]{fig_3.16.ps}
    \hfill
\caption{Comparison between the predicted cross section for constant
(dashed and dotted lines) and running (solid lines) $\alpha_s$.}
\label{fig:312}
  \end{center}
\end{figure}
\begin{figure}[ht]
  \begin{center}
    \leavevmode
    \epsfxsize=10.cm
%    \epsffile[110 140 465 595]{fig_3.17.ps}
    \epsffile[110 260 465 595]{fig_3.17.ps}
    \hfill
\caption{Total cross section as function of $E=\protect\sqrt{s}-2m_t$
for different values of the Higgs mass.}
\label{fig:313}
  \end{center}
\end{figure}
pronounced 1S peak is still visible for $m_t$ = 150 GeV, for $m_t=200$ GeV, 
however, only a smooth shoulder is predicted.  The behaviour is 
qualitatively very similar, if we keep $m_t$ fixed say at 180 GeV and 
decrease or increase $\Gamma_t$ be the corresponding amount.  The shape of 
the cross section will therefore allow to determine the width of the top 
quark.  A qualitatively very different response is observed towards a 
change in $\alpha_s$ (Fig.~\ref{fig:311}). The binding energy increases with 
$\alpha_s$, the aparent threshold is thus lowered (This is the reason for 
the strong correlation between $\alpha_s$ and $m_t$ in the experimental 
analysis based on $\sigma_{tot}$ only \cite{Fujii,Igo}.) and the height of the 
``would-be resonance'' is increased.  Even several GeV above threshold one 
observes a slight increase of the cross section with $\alpha_s$, a 
consequence of the enhanced attraction between $t$ and $\bar t$
(cf.~eq.~\ref{eq:347}).   The 
impact of the running of $\alpha_s$ on the shape of the cross section is 
evident from Fig.~3.15.  The full QCD prediction with running
$\alpha_s$  (for  
$\alpha_{\overline{MS}}(M_Z^2)$ = 0.125) is compared to the prediction for 
a Coulomb potential with $\alpha_s$ fixed. It is impossible to describe the 
height of the peak and the continuum above with the same value of 
$\alpha_s$, even allowing for an
arbitrary additive constant $V_0$.  The influence of a variation in $m_H$ 
is shown in Fig.~\ref{fig:313}.  Cross section measurements with a
precision better  
than 10\% will become sensitive to the effect of a light Higgs boson. 

Up to this point the amplitude induced by virtual $Z$ and $\gamma$ are 
included in Born approximation only.  Electroweak corrections and 
initial state radiation are neglected.  A detailed discussion of 
electroweak corrections to the cross section and the left right asymmetry 
in the context of the SM can be found in \cite{GKKS}.     The corresponding 
discussion for the two-Higgs-doublet model is presented in \cite{GuthK}. In 
this model one might encounter enhanced Yukawa couplings which would 
amplify the effect under discussion. 

Initial state radiation leads to a fairly drastic distortion of the shape of 
the cross section, in particular to a smearing of any pronounced structure. 
This is illustrated in Fig.~\ref{fig:initstate}
\begin{figure}
\psfig{figure=isr.ps,width=130mm,bbllx=40pt,bblly=270pt,%
bburx=450pt,bbury=550pt}
\caption{Comparison of the $t\bar t$ production cross section without
(dashed) and with (solid line) initial state radiation.}
\label{fig:initstate}
\end{figure}
where the predictions with and without initial state radiation are compared 
for otherwise identical parameters.

Beamstrahlung and the energy spread of the beam lead to a further smearing 
of the apparent cross section.  These accelerator dependent issues are 
treated in more detail in \cite{Fujii}. (For a related discussion see also 
\cite{Igo}.)


\subsection{Momentum distributions of top quarks}

The Green's function in momentum space and the momentum distribution of top 
quarks (and thus their decay products) are intimately related. For a narrow 
quarkonium resonance with orbital quantum number $n$ the quarks' momentum 
distribution is evidently given by the wave function in momentum space
\be
{dN\over d\vec p} = {|\widetilde \psi_n(\vec p)|^2 \over (2\pi)^3}
\ee
For $J/\psi$ or $\Upsilon$ this distribution is not directly accessible to 
experiment since these states decay through $Q\bar Q$ annihilation only.  
For toponium, however, which is dominated by single quark decay, the decay 
products carry the information of their parent momentum and hence allow for 
the reconstruction of the original quark momentum distribution \cite{JKm}.

For one individual resonance this leads to the differential $t\bar t$ 
cross section (without $Z$ contribution and transverse gluon correction).  
\be
{d \sigma_n\over d\vec p} (\vec p,E) = {3\alpha^2 Q_t^2\over
\pi sm_t^2} |\psi_n(\vec r =0)|^2 
{\Gamma_t\over (E-E_n)^2+\Gamma_t^2}  |\widetilde \psi_n(\vec p)|^2
\ee
Once $\Gamma_t$ is sufficiently large, interferences between different 
radial excitations become important and the right-hand side of this 
equation has to be replaced by the square of the Green's function 
\cite{Sumino,JKT}
\be
{d \sigma_n\over d\vec p} (\vec p,E) = 
{3\alpha^2 Q_t^2\over \pi sm_t^2}
\Gamma_t 
\left|G(\vec p,E+i\Gamma_t) \right|^2
\ee
with
\be
G(\vec p,E+i\Gamma_t) = \int d\vec r e^{i\vec p \vec r} 
G(\vec r,\vec r'=0,E+i\Gamma_t)
\ee
As discussed in sect.~3.2.2, the Green's function can be obtained in 
momentum space as a solution of the Lippmann-Schwinger equation.  For an 
energy close to the 1S peak it exhibits a  fairly smooth 
behaviour reminiscent of the 1S wave function in momentum space
(Fig.~\ref{fig:2}).
\begin{figure}
  \begin{center}
    \leavevmode
    \epsfxsize=12.cm
    \epsffile{fig_3.19a.eps}

\rule{0mm}{10mm}

\rule{-1mm}{0mm}
    \epsfxsize=12.cm
    \epsffile{fig_3.19b.eps}

\rule{0mm}{1mm}
\caption{Real (dashed) and imaginary (dotted) 
 parts of the Green's function for an
energy corresponding to the 1S peak (upper figure) and for $E=0$
(lower figure). Solid curve: $|pG(p)|^2\cdot 0.002.$}
\label{fig:2}
  \end{center}
\end{figure}
With increasing energy an oscillatory pattern of the amplitude is observed, 
and a shift towards larger momenta (Fig.~\ref{fig:2}).  These results are 
intentionally displayed for $m_t=120$ GeV, where the oscillations are still 
clearly visible, in contrast to $m_t$=180 GeV where all oscillations are 
smeared by the large width $\Gamma_t$.  The corresponding predictions for the 
distributions at $m_t = 180$ GeV are displayed in Fig.~\ref{fig:3}.   
\begin{figure}[ht]
  \begin{center}
    \leavevmode
    \epsfxsize=12.cm
    \epsffile[15 210 580 635]{fig_3.20.ps}
    \hfill
\caption{Momentum distribution of top quarks for three different cms
energies.}
\label{fig:3}
  \end{center}
\end{figure}
\begin{figure}[ht]
  \begin{center}
    \leavevmode
    \epsfxsize=12.cm
    \epsffile[15 210 580 635]{fig_3.21.ps}
    \hfill
\caption{Energy distribution of $W$'s from top quark decay for three
different cms energies.}
\label{fig:4}
  \end{center}
\end{figure}
The transition 
from a wide distribution below the nominal threshold to a narrow one with 
the location of the peak determined by trivial kinematics is clearly 
visible.  The impact on the energy distribution of the $W$'s from top decay 
is shown in Fig.~\ref{fig:4}. 

 To characterize the momentum distribution by a single parameter, one may 
either choose its peak value, or the expectation value of the modulus of 
the momentum $\langle p\rangle$, the latter being well adopted to the 
experimental analysis.  In the situation at hand the definition of 
$\langle p\rangle$ has to be introduced with some care. The free Green's 
function $G_0$ (see eq.~\ref{eq:355})
 drops $\sim p^{-2}$ for large momenta and this 
behaviour is recovered also in the presence of interaction. The expectation 
value $\langle p\rangle$ diverges logarithmically with the cutoff. In the 
narrow width approximation one finds for the leading terms
\be
\langle p\rangle &=& 
{ \int_0^{p_m} d\vec p p \left| G_0(\vec p,E+i\Gamma) \right|^2 
   \over
 \int_0^{p_m} d\vec p  \left| G_0(\vec p,E+i\Gamma) \right|^2 
}
\nonumber \\ &=&
\sqrt{m_tE}\left( 1+ {\Gamma\over E}{\ln p_m^2/
\left(\sqrt{E^2+\Gamma^2} m_t\right) \over \pi}\right)
\ee
where a cutoff $p_m$ has been introduced. As a consequence of the small 
numerical prefactor of the divergent term and its
 logarithmic cutoff dependence 
the result is fairly insensitive to the exact value of the cutoff for $p_m$ 
of order $m_t$.  Alternatively one may replace the phase space element 
$d\vec p/m_t$ by the relativistic version   $d\vec p/E = d\vec 
p/\sqrt{m_t^2+p^2}$ to obtain a convergent result. In future measurements the 
cutoff will presumably be provided by the experimental analysis.

In order to study the dependence of $\langle p\rangle$ on the strong 
coupling constant, consider for the moment the predictions for a stable 
quark.
Some
intuition and qualitative understanding can already be gained from the
predictions based on a pure Coulomb potential \cite{JKTZer}.

For a stable top quark of fixed mass the ``effective threshold''
can be associated with the location of the $1S$ resonance
$\sqrt{s_{thr}} = 2 m_t + E_{1S}$ with $E_{1S}=-E_{Ryd}= - \alpha^2m_t/4$
which decreases with increasing $\alpha$. The height of the resonance
cross
section is proportional to the square of the wave function at the origin
and hence proportional to $\alpha^3$, as long as the resonances are
reasonably well separated. In the limit of large $\Gamma_t$, i.e.~far
larger than $E_{Ryd}$, the overlapping $1S$, $2S$ $\dots$ resonances have
to fill the gaps between the peaks. Since these gaps themselves
increase proportional to $\alpha^2$, one is left in the extreme case of large
width with a cross section linear in $\alpha$. Note that this
corresponds to the behaviour of the cross section close to but slightly
above the threshold which is also proportional to $\alpha$.

For realistic top masses one thus observes a dependence
of the peak cross section linear in $\alpha$.
Since the location of the peak itself depends on $\alpha$, only the
analysis of the full shape allows to extract the relevant information.

In a next step also the momentum distribution of top quarks
has to be
exploited to obtain further information. The discussion is again
particularly simple for the Coulomb potential $V(r)=-\alpha/r$ and
provides a nice exercise in nonrelativistic quantum mechanics.
The average momentum,
in units of the Bohr momentum $\alpha m_t/2$, can be written in terms of
a function $f(\epsilon)$ which depends only on one variable $\epsilon=
E/E_{Ryd}$ if
the energy $E=\sqrt{s}- 2m_t$ is measured in terms of the Rydberg
energy.

\newcommand{\pav}{\langle p\rangle}
\begin{equation}\label{paveq}
\pav = \frac{\alpha m_t}{2} f(\epsilon)
\end{equation}
For positive arguments  the function $f$ can be derived
from obvious kinematical considerations.

\begin{equation}
f(\epsilon)=\sqrt{\epsilon} \qquad {\rm for} \qquad \epsilon \ge0
\end{equation}

For the discrete negative
arguments $\epsilon_n=-1/n^2$, corresponding to the
locations of the bound states,
the radial wave functions in momentum space are given 
in terms  of the Gegenbauer polynomials $C^m_n$

\begin{equation}
\psi(\vec p\,)= \frac{16\pi n^{3/2}}{(1+n^2 p^2)^2} C^1_{n-1}
\left(\frac{n^2 p^2 -1} {n^2 p^2 + 1} \right) Y_0^0\left(\theta,\varphi\right)
\end{equation}
with
\begin{equation}
\int\frac {d\vec p}{(2\pi)^3} |\psi(\vec p\,)|^2 = 1\ .
\end{equation}
Using the explicit forms of $C^m_n$
\begin{equation}
C^1_0(z)=1,\qquad C^1_1(z)=2z,\qquad C^1_2(z)=4z^2-1
\end{equation}
one obtains through straightforward calculation

\begin{equation}
f(-1)=\frac{8}{3\pi},\qquad f(-1/4)=\frac{16}{15\pi},
\qquad f(-1/9)=\frac{24}{35\pi}\ .
\end{equation}
For arbitrary $n$ one derives the general result
\begin{equation}
f\left(-\frac{1}{n^2}\right)=\frac{8n}{(2n-1)(2n+1)\pi}
\label{eq8}
\end{equation}
with the asymptotic behaviour
\begin{equation}\label{asym}
f\left(-\frac{1}{n^2}\right)\to \frac{2}{n \pi}\ .
\end{equation}

This is in accord with the result expected from classical mechanics:
For the average momentum of a particle on a closed orbit
in the Coulomb potential one derives
\begin{equation}
\langle p^{2\beta }\rangle=\left(\frac{\alpha m_t}{2}\right)^{2\beta }
\left(\frac{-E}{E_{Ryd}}\right)^\beta 
\frac{1}{2\pi}
\int_0^{2\pi} d\xi
\frac{(1-e^2\cos^2\xi)^\beta}{(1-e\cos\xi)^{2\beta -1} }\ .
\end{equation}
Quantum mechanical orbits with angular momentum zero and
high radial quantum numbers correspond to classical motions
with excentricity $e=1$ (i.e.~straight lines). 
In this limiting case the classical
expectation value is easily evaluated, and for $\beta=1/2$ one finds
agreement with the quantum mechanical result.
For small negative energies one
therefore obtains the behaviour $f(\epsilon) = 2\sqrt{-\epsilon}/\pi$.
Significantly below threshold, however, the average momentum
obtained from the Green's function
 increases
more rapidly with decreasing energy and between the $1S$ and the $2S$
state one observes an approximately
linear dependence on the energy.

From these considerations the dependence of the average momentum
on $\alpha$ (with $E$ fixed) is easily understandable, in particular
the seemingly surprising observation that well below threshold
$\pav$ decreases with
increasing $\alpha$.
From (\ref{paveq}) one derives for
a shift in $\alpha$ (keeping the energy $E$ fixed) the following
shift in $\pav$
\begin{equation}
\frac{\delta\pav}{\pav}=
\left(1-2\frac{f'(\epsilon)}{f(\epsilon)} \epsilon\right)
\frac{\delta\alpha}{\alpha}\ .
\end{equation}
Above threshold as well as close to but below threshold $f\propto
\sqrt{|\epsilon|}$.
Hence $\epsilon f'/f = 1/2$ and the average momentum remains
unaffected.
 The location of the minimum is thus an ideal place to fix the mass of
 the top quark.
Significantly below threshold, however, $\epsilon f'/f \approx 1$
and the factor in front of $\delta\alpha/\alpha$ becomes negative.
This explains the decrease of $\pav$  with increasing $\alpha$.

\begin{figure}
\epsffile{tavmom_coul.ps}
\vspace*{-15mm}
\caption{a) Average momentum as a function of $E$ for different values
of $\alpha_s$. The markers show the results of the analytical
calculation at $1S$, $2S$, $3S$ energies.
b) Comparison with the analytical result for discrete
energies and with the square--root dependence close to threshold.}
\label{fig1}
\end{figure}

These results are illustrated in Fig.\ref{fig1}.
In Fig.\ref{fig1}b we demonstrate that $\pav$ as evaluated with the
program for the Green function (solid line) coincides perfectly
well with the values calculated from the analytical formula on
resonance, indicated
by the triangles. The prediction from classical mechanics, namely
$\pav \propto \sqrt{|\epsilon|}$ is shown by the dotted line
and agrees nicely for positive and negative energies. In Fig.\ref{fig1}a
$\alpha_s$ is increased from 0.20 to 0.24 and $\pav$ changes in accord
with the previous discussion.

For definiteness we have
chosen $m=m_t/2=60$ GeV for the reduced mass and
$\alpha=4\alpha_s/3$ with $\alpha_s$ varying between 0.20 and 0.24.
The curves demonstrate the decrease of $\pav$ by
about 10\%  for the corresponding increase in $\alpha$. The triangles
mark the locations of the resonances and the expectation values
for the momentum as calculated from (\ref{eq8}).

\begin{figure}
\epsffile{tavmom_qcd.ps}
\vspace*{-15mm}
\caption{Energy dependence of $\langle p\rangle$, the
average t quark momentum
for $\alpha_s = 0.13$ (dotted)
$0.12$ (dashed) and $0.11$ (solid) line for
$m_t=120$ GeV. a) $\Gamma_t=0.03$ GeV and b) $\Gamma_t=0.3$ GeV.}
\label{fig2}
\end{figure}

The qualitative behaviour remains unchanged for realistic QCD
potentials  corresponding to different values of $\alpha_s(M_Z)$.
Qualitatively the same behaviour is observed as in Fig.\ref{fig1}.
In Fig.\ref{fig2}a the top quark width has been set to an artificially
small value of $0.03$ GeV, in Fig.\ref{fig2}b the realistic value of
$0.3$ GeV has been adopted. The finite width leads to an additional
contribution to the momentum of order $\sqrt{\Gamma m}$.

An important feature is evident from Fig.\ref{fig2}: The momentum
calculated for
positive energy is nearly independent from $\alpha_s$ and reflects
merely the kinematic behaviour, just as in the case of the Coulomb
potential. This is characteristic for the choice of a potential \cite{TJ}
where the large distance behaviour is fixed by phenomenology and
decoupled from the short distance value of $\alpha_s$.

The different assumptions on the long distance behaviour are reflected
in differences between
the predictions of \cite{peskin,Sumino,JKT}
for the precise location of the $t\bar t$
threshold for identical values of $\alpha_s$ and $m_t$ and in
differences
in the $\alpha_s$ dependence of the momentum distributions for fixed
$m_t$ and energy (see also \cite{Igo}).
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.


In Fig.~\ref{fig:s30} the predictions for $\langle p \rangle$  vs.~energy 
are presented for the case of a realistic QCD potential, assuming
$m_t=120/150/180$ GeV. The strong rise of $\langle p \rangle$ as a
consequence of the strong increase of $\Gamma_t$ is clearly visible. 
\begin{figure}
  \begin{center}
    \leavevmode
    \epsfxsize=12.cm
    \epsffile[15 135 580 705]{fig_3.24.ps}
    \hfill
\caption{Energy dependence of the average top quark momentum for $m_t$
= 120/150/180 GeV. Triangles, stars and circles correspond to $\langle
p_t \rangle$ for S-states with $\Gamma_t=0$. }
\label{fig:s30}
  \end{center}
\end{figure}


\subsection{Angular distributions and polarization}

Close to threshold the production amplitude is dominantly $S$-wave which 
leads to an isotropic angular distribution.  The spin of top quarks  
is alligned with the beam direction, with a degree of polarization 
determined by the electroweak couplings, the beam polarization and the mass 
of the top quark, but independent of the production dynamics, in particular 
of the potential. 

 Small, but nevertheless experimentally accessible corrections do arise 
from the small admixture of $P$-wave contributions and from rescattering of 
the top quark decay products. Let us concentrate for the moment on the 
first mechanism.  $P$-wave amplitudes are proportional to the top quark 
momentum.  For stable noninteracting particles the momentum vanishes at 
threshold.  However, as discussed in the previous section the expectation 
value of the quark momentum is nonzero for all energies ---  a consequence 
of the large top decay rate and the uncertainty principle.   Technically the 
$P$-wave contribution is calculated with the help of the Green's function 
technique. The generalization of the Lippman-Schwinger equation
(\ref{eq:355})  from $S$- 
to $P$-waves reads as follows
\be
{\cal F} (\rmp,E) &=& G_0(p,E) + G_0(p,E)
\int {d^3 k \over (2 \pi)^3}
        {\bfp\cdot \bfk \over \bfp^2}
        V(\bfp - \bfk) {\cal F} (k,E) 
\ee

It is then straightforward to calculate the
differential momentum distribution and the polarization of
top quarks produced in electron positron annihilation.
Let us recall the following conventions for the fermion
couplings
\begin{equation}
v_f = 2 I^3_f - 4 q_f \sin^2\theta_{\rm W} , \qquad  a_f = 2 I^3_f .
\end{equation}
$P_\pm$ denotes the longitudinal electron/positron
polarization and
$\chi=(P_+-P_-)/(1-P_+P_-)$
can be interpreted as effective longitudinal polarization of
the virtual intermediate photon or $Z$ boson.
The following abbreviations will be useful below:
\begin{eqnarray}
a_1 &=& q_e^2 q_t^2 + (v_e^2 + a_e^2) v_t^2 d^2 +
        2 q_e q_t v_e v_t d \nonumber \\
a_2 &=& 2 v_e a_e v_t^2 d^2 + 2 q_e q_t a_e v_t d \nonumber \\
a_3 &=& 4 v_e a_e v_t a_t d^2 + 2 q_e q_t a_e a_t d \label{coupl}\\
a_4 &=& 2 (v_e^2 + a_e^2) v_t a_t d^2 + 2 q_e q_t v_e a_t d \nonumber\\
d &=& {1\over 16 \sin^2\theta_{\rm W}\cos^2\theta_{\rm W}}\,{s\over s - M_Z^2}.
    \nonumber
\end{eqnarray}
The differential cross section, summed over polarizations of quarks
and including $S$-wave and $S$-$P$--interference contributions,
is thus given by
\begin{eqnarray}
{d^3\sigma \over dp^3} &=&
{3 \alpha^2 \Gamma_t \over 4 \pi m_t^4} (1-P_+P_-)
\left[ { (a_1 + \chi a_2)
        \left(1-{16 \alpha_{\rm s} \over 3 \pi} \right)
        \left|G(\rmp,E)\right|^2 + }\right. \nonumber\\
& & %\mbox{\hspace{2.cm}}
\left. {+(a_3+\chi a_4)
\left( 1-{12\alpha_{\rm s} \over 3 \pi} \right)
{\rmp \over m_t} \Re \left(\,G(\rmp,E) F^*(\rmp,E)\,\right)\,
\cos\vartheta} \right]  \label{dsig_d3p} .
\end{eqnarray}
The vertex corrections from hard gluon exchange for $S$-wave
and $P$-wave  amplitudes are included in this
formula. It leads to the following forward-backward asymmetry
\cite{MurSum2,hjkt}
\begin{equation}\label{afb}
{\cal A}_{\rm FB}(\rmp,E) = C_{\rm FB}(\chi)\, \varphi_{\rm _R}(\rmp,E),
\end{equation}
with
\begin{equation}
   C_{\rm FB}(\chi) = {1 \over 2}\, {a_3
    + \chi a_4 \over a_1 + \chi a_2} ,
\end{equation}
$\varphi_{\rm _R} = \Re\,\varphi$,  and
\begin{equation}\label{phi}
\varphi(\rmp,E) =
{(1-{4 \alpha_{\rm s}/3 \pi})\over (1-{8 \alpha_{\rm s}/3 \pi})}\,
        {\rmp \over m_t}\,
        {F^* \!(\rmp,E) \over G^* \!(\rmp,E)}  .
\end{equation}
This result is still differential in the top quark momentum.
Replacing $\varphi(\rmp,E)$ by
\begin{equation}\label{cap_phi}
\Phi(E) =
{(1-{4 \alpha_{\rm s}/3 \pi})\over (1-{8 \alpha_{\rm s}/3 \pi})}\,
{\int_0^{\rmp_m} d\rmp\,
{\rmp^3 \over m_t}\, F^*(\rmp,E)G(\rmp,E) \over
\int_0^{\rmp_m} d\rmp \, \rmp^2 \left|G(\rmp,E)\right|^2}  .
\end{equation}
one obtains the integrated forward-backward
asymmetry again.
Again, the cutoff $\rmp_m$ must be introduced to eliminate the
logarithmic divergence of the integral.

\begin{figure}[ht]
\begin{flushleft}
\leavevmode
\epsfxsize=8cm
\epsffile[100 370 500 520]{fig3.ps}\\[-1.5cm]
\hfill
\parbox{6.cm}{\small
\caption[]{\label{dyn.ps}\sloppy Definition of the spin directions.
        The normal component ${\bf s}_{\rm N}$ points out of the plane.}}
\end{flushleft}
\end{figure}

{\em Polarization \cite{hjkt,ttp48}}\\
To describe top quark polarization in the threshold region it
is convenient to align the reference system with the beam
direction (Fig.~\ref{dyn.ps}) and to define
\begin{eqnarray}
{\bf s}_{\|} = {\bf n}_{e^-}, \quad
{\bf s}_{\rm N} = {{\bf n}_{e^-} \times {\bf n}_t \over
        |{\bf n}_{e^-} \times {\bf n}_t|},               \label{basis}
\quad
{\bf s}_\bot = {\bf s}_{\rm N} \times {\bf s}_{\|} . 
\end{eqnarray}
%
In the limit of small $\beta$ the quark spin is essentially
aligned with the beam direction apart from small
corrections
proportional to $\beta$, which depend on the production
angle. A system of reference with ${\bf s}_\|$ defined with respect
to the top quark momentum \cite{krz}
is convenient in the high energy limit but evidently becomes
less convenient close to threshold.

Including the QCD potential one obtains for the three components
of the polarization
\begin{eqnarray}
{\cal P}_\|(\bfp,E,\chi) &=& C_\|^0(\chi)
+ C_\|^1(\chi)\, \varphi_{\rm _R}(\rmp,E)\,\cos\vartheta\,
 \label{thr_long}\\
{\cal P}_\bot(\bfp,E,\chi) &=& C_\bot(\chi)\,
\varphi_{\rm _R}(\rmp,E)\,
\sin\vartheta\,
\label{thr_perp}\\
{\cal P}_{\rm N}(\bfp,E,\chi) &=& C_{\rm N}(\chi)
\varphi_{\rm _I}(\rmp,E)
\sin\vartheta\,
        \label{thr_norm} ,
\end{eqnarray}\\
\parbox{75.ex}{
\begin{eqnarray*}
& &\hspace{5.ex}C_\|^0 (\chi) =
-{a_2 + \chi a_1 \over a_1 + \chi a_2} ,\hspace{6.9ex}
  C_\|^1 (\chi) = \left( 1-\chi^2 \right) {a_2 a_3 - a_1 a_4   \over
        \left(a_1 + \chi a_2 \right)^2} ,\\
& &\hspace{5.ex}C_\bot(\chi)  = -{1\over 2} \,
{a_4 + \chi a_3 \over a_1 + \chi a_2} ,
    \qquad C_{\rm N}(\chi) =-{1 \over 2}\, {a_3
    + \chi a_4 \over a_1 + \chi a_2}\, =\, - C_{\rm FB}(\chi) ,
\end{eqnarray*}}
\hfill
\parbox{5.ex}{
\begin{eqnarray} \label{coefs} \end{eqnarray} }
with $\varphi_{\rm _I} = \Im\,\varphi$, and $\varphi(\rmp,E)$ as defined
in (\ref{phi}).  The momentum integrated quantities are obtained by
the replacement $\varphi(\rmp,E) \to \Phi(E)$. The case of
non-interacting stable quarks is recovered by the replacement
$\Phi\to\beta$, an obvious consequence of (\ref{cap_phi}).
\par\noindent
Let us emphasize the main qualitative features of the result:
\begin{itemize}
\item Top quarks in the threshold region are highly polarized.  Even
  for unpolarized beams the longitudinal polarization amounts to about
  $-0.41$ and reaches $\pm1$ for fully polarized electron beams. This
  later feature is of purely kinematical origin and independent of the
  structure of top quark couplings.  Precision studies of polarized
  top decays are therefore feasible.
\item Corrections to this idealized picture arise from the small
  admixture of $P$-waves. The transverse and the normal components of
  the polarization are of order 10\%. The angular dependent part of
  the parallel polarization is even more suppressed.  Moreover, as a
  consequence of the angular dependence its contribution vanishes upon
  angular integration.
\item The QCD dynamics is solely contained in the functions $\varphi$
  or $\Phi$ which is the same for the angular distribution and the
  various components of the polarization. (However, this
  ``universality'' is affected by the rescattering corrections.)
  These functions which evidently
  depend on QCD dynamics can thus be studied in a variety of ways.
\item The relative importance of $P$-waves increases with energy,
  $\Phi\sim\sqrt{E/m_t}$.  This is expected from the close analogy
  between $\Phi_{\rm R}=\Re\,\Phi$ and $\beta$. 
  In fact, the order of magnitude of
  the various components of the polarization above, but close to
  threshold, can be estimated by replacing $\Phi_{\rm R}\to \rmp/m_t$.
\end{itemize}
The $C_i$ are displayed in Fig.~\ref{pol_coefs.ps} as functions of the
variable $\chi$ ($\sin^2\!\theta_{\rm W}= 0.2317$, $m_t=180$ GeV).
As discussed before, $C_\|^0$ assumes its maximal value $\pm 1$ for
$\chi=\mp 1$ and the coefficient $C_\|^1$ is small throughout. The
coefficient $C_\bot$ varies between $+0.7$ and $-0.5$ whereas $C_{\rm
  N}$ is typically around $-0.5$.  The dynamical factors $\Phi$ are
around $0.1$ or larger, such that the $P$-wave induced effects should
be observable experimentally.
\begin{figure}[ht]
 \begin{center}
  \leavevmode
  \epsfxsize=14cm
  \epsffile[40 290 535 525]{fig4.ps}
  \caption[]{\label{pol_coefs.ps}\sloppy
        The coefficients (\ref{coefs}) for $\sqrt{s}/2 = m_t=180$ GeV.}
 \end{center}
\end{figure}
\par\noindent
The normal component of the polarization which is proportional to
$\varphi_{\rm _I}$ has been predicted for stable quarks in the framework of
perturbative QCD \cite{dev,krz}. In the threshold region the phase can be
traced to the $t\bar t$ rescattering by the QCD potential. For 
stable quarks, assuming a pure Coulomb potential $V=-4\alpha_{\rm s}/3r$, 
the nonrelativistic problem can be solved analytically \cite{FKotsky} and
one finds
\begin{eqnarray}
\lim_{\Gamma_t \rightarrow 0 \atop E\rightarrow \bfp^2/m_t} 
        \left(E - {\bfp^2\over m_t} + i\Gamma_t \right) G(\rmp,E) &=&
        \exp\left(\pi {\bar k} \over 2 \rmp\right)\,
        \Gamma(1+{i{\bar k}/\rmp}) \label{lim1_gt0}\\
\lim_{\Gamma_t \rightarrow 0 \atop E\rightarrow \bfp^2/m_t} 
        \left(E - {\bfp^2\over m_t} + i\Gamma_t \right) F(\rmp,E) &=&
        \left(1 - i{{\bar k} \over \rmp}\right) 
        \exp\left(\pi {\bar k} \over 2 \rmp\right)\,
        \Gamma(1+{i{\bar k}/\rmp}) \label{lim2_gt0},\ \ \ \ \
\end{eqnarray}
with ${\bar k} = 2m_t\alpha_s/3$
and hence
\begin{eqnarray}
\varphi_{\rm _I}(\rmp,E) &\rightarrow&
{2\over 3}\alpha_{\rm s}{1-4\alpha_{\rm s}/3\pi\over
        1-8\alpha_{\rm s}/3\pi} \label{phi_gt0}\\
\Phi_{\rm I}(E) &\rightarrow&
{2\over 3}\alpha_{\rm s}{1-4\alpha_{\rm s}/3\pi\over
        1-8\alpha_{\rm s}/3\pi} . \label{cap_phi_gt0}
\end{eqnarray}
The component of the polarization normal to
the production plane is thus
approximately independent of $E$ and essentially measures the strong
coupling constant. In fact one can argue that this is a unique way to
get a handle on the scattering of heavy quarks through the QCD
potential.

Predictions for real and imaginary parts of the function $\varphi$ are
displayed in Fig.~\ref{phi.ps} for four different energies.  
\begin{figure}
 \begin{center}
  \leavevmode
  \epsfxsize=135mm
  \epsffile[40 185 535 630]{fig7.ps}
  \caption[]{\label{phi.ps}\sloppy 
        Real (solid) and imaginary (dashed) part of the function
        $\varphi(\rmp,E)$ for $m_t = 180$ GeV, $\alpha_{\rm s}=0.125$ 
        and four
        different energies. The dotted line shows the free particle case
        $\Re\,\varphi = \mbox{p}/m_t$ (from \protect\cite{ttp48}).}
 \end{center}
\end{figure}

The momentum integrated functions $\Phi(E)$ are shown in
Fig.~\ref{alph_dep.ps}.  From this figure, in combination with
Fig.~\ref{pol_coefs.ps}, it is clear that the contribution of $P$-wave
amplitudes to the quark polarization will amount to 10\% at most and
by construction vanishes upon angular integration.  
\begin{figure}
 \begin{center}
  \leavevmode
  \epsfxsize=135mm
  \epsffile[70 100 510 735]{fig13.ps}
  \caption[]{\label{alph_dep.ps}\sloppy Real and imaginary part of $\Phi(E)$
        for three different values of $\alpha_{\rm s}$ (from
\protect\cite{ttp48}). }
 \end{center}
\end{figure}



\subsection{Rescattering}
For a particle with a very small decay rate production and decay amplitudes 
can be clearly separated.  This is fairly evident from the space-time 
picture of such a sequence.  Prior to its decay the particle travels away 
from the production point and any coherence is lost between the two 
reactions. The situation is different for the case under discussion, an 
unstable top quark which decays within the range of interaction between $t$ 
and $\bar t$.  In such a situation the decay products from $t$ are still 
affected by the force originating from $\bar t$ and vice versa 
(Fig.~\ref{tbbar}).
\begin{figure}[h]\begin{center}
\begin{tabular}{cc}
  \epsfxsize 63mm \mbox{\epsffile[25 570 200 650]{fig5a.ps}} &
  \epsfxsize 63mm \mbox{\epsffile[25 570 200 650]{fig5b.ps}} \\
  a) & b)
\end{tabular}\end{center}
\caption{\label{tbbar}Lowest order rescattering diagrams.}
\end{figure}

In ref.~\cite{MelYak1,sumph} it has been demonstrated that the total 
cross section remains unaffected by rescattering in order $\alpha_s$.  This 
result had been anticipated in \cite{JKT} on the basis of earlier work 
which considered the decay rate of a muon bound in the strong field of a 
nucleus \cite{Huff}.  In contrast momentum and angular distributions 
\cite{MurSum2,sumph,FMY94} as well as the top quark polarization 
\cite{ttp48} are affected by rescattering.  For example the momentum 
distribution has to be corrected by a factor $(1+\psi_1(p,E)) $ with 
\be
   \psi_1(\rmp,E) = 2\,\Im\int\!\frac{d^3k}{(2\pi)^3}V(|{\bf k}-{\bf p}|)
        \frac{G(\rmk,E)}{G(\rmp,E)}\frac{\arctan{\frac{|{\bf k}-{\bf p}|}
        {\Gamma_t}}}{|{\bf k}-{\bf p}|} \\
\ee
\begin{figure}
\begin{tabular}{cc}
  \epsfxsize 70mm \mbox{\epsffile[0 0 567 454]{fig8a.ps}} & 
  \epsfxsize 70mm \mbox{\epsffile[0 0 567 454]{fig8b.ps}} \\  
    a) E=-3GeV & b) E=0GeV \\
  \epsfxsize 70mm \mbox{\epsffile[0 0 567 454]{fig8c.ps}} & 
  \epsfxsize 70mm \mbox{\epsffile[0 0 567 454]{fig8d.ps}} \\
    c) E=2GeV  & d) E=5GeV
\end{tabular}
\caption{Modification of the momentum distribution through rescattering.
Dashed line: no rescattering corrections included; Solid line: rescattering
contribution with full potential included; dotted line: rescattering
contribution with pure Coulomb potential and $\alpha_{\rm s}=0.187$
included (from \protect\cite{ttp48}).
\label{rescdsig}}
\end{figure}
The distribution is shifted towards smaller momenta by about 5\% 
(Fig.~\ref{rescdsig}), an effect that could become relevant in precision 
experiments.  The influence on the forward-backward asymmetry and the 
polarization is even more pronounced
\cite{ttp48}, as far as the 
$S$-$P$-wave interference terms are concerned which are  thus
intrinsically of  
order $\beta$.  A detailed discussion of these effects is beyond the scope 
of these lectures and can be found in \cite{ttp48}.

\subsection{Relativistic corrections}
In  ${\cal O}(\alpha^2_s)$ 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 the
argument, let us adopt a pure Coulomb potential and a binding energy
of -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 = 180 $ GeV one would
arrive at a reduction of $\Gamma_t$ by about 10\%.  A full calculation
of all ${\cal O}(\alpha^2_s)$ effects is not available at present and
one has to resort to models and analogies \cite{JKT,Sumino,TJ}. For
example, it has been shown \cite{Huff,Uber} that the decay rate of a
muon bound in the field of a nucleus is given by
\be
\Gamma = \Gamma_{\rm free} 
\left[1-5(Z\alpha)^2\right]
\left[1+5(Z\alpha)^2\right]
\left[1-{(Z\alpha)^2\over 2}\right],
\ee
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{TJ} 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{MKum}.




