\section{Threshold behaviour}
The previous section dealt with top quark production sufficiently far
above threshold for the reaction to be well described by the Born
cross section, modified slightly by QCD and electroweak corrections.
This is in contrast to the situation in the threshold region, where
QCD plays an important role and controls the cross section. Strong forces
modify the Born prediction. They compensate the phase space suppression and
enhance the production rate significantly, leading to a step function like
behaviour at threshold. The large top decay rate also plays an important
role. Quarkonium resonances cease to exist and merge into a structureless
excitation curve which joins smoothly with the continuum prediction above
the nominal threshold.
This sharply rising cross section allows to study top quarks in a
particularly clean environment and with large rates. The following physics
questions can be addressed:
\vspace*{2mm}
\noindent
-- The QCD potential can be scrutinized at short distances, with the
non perturbative tail cut off by the top decay. As a result
$\Lambda_{QCD}$ or $\alpha_s$ could be determined accurately.\\
-- The top quark mass can be measured with a precision which is only
limited by the theoretical understanding of the excitation curve, but in
any case better than 500 MeV.\\
-- Top quarks are strongly polarized (about 40\%) even for unpolarized
beams; and longitudinal beam polarisation will enhance this value even
further. Detailed studies of top decays, in particular of the $V-A$ structure
of the $tbW$ coupling are therefore feasible.\\
-- The interquark potential is -- slightly -- modified by the Yukawa
potential induced by Higgs exchange. The excitation curve and the top
quark momentum distribution may therefore lead to an indirect measurement
of the Yukawa coupling.\\
-- The large number of top quarks in combination with the constrained
kinematics at threshold could facilitate the search for new decay modes
expected in extensions of the SM.
\vspace*{2mm}
\noindent
With this motivation in mind the following points will be discussed: After
a brief review of qualitative features of threshold production (section
\ref{ss:intr}),
the present status of our theoretical understanding of the total
cross section will be presented in section \ref{ss:pot}.
The momentum distribution of top quarks and their decay products
offers an alternative and complementary route to probe the interquark
potential, as shown in section \ref{ss:real}. Spin effects and
angular distributions are sensitive towards the small $P$-wave
contribution induced by the axial part of the neutral current. The
theoretical framework and the resulting predictions are collected in
section \ref{ss:mom}. Rescattering, relativistic corrections and
other terms of order $\alpha_s^2$ will be touched upon in section
\ref{ss:angu}.
\subsection{Introductory remarks}
\label{ss:intr}
For a qualitative understanding it is illustrative to compare the different
scales which govern top production close to threshold. The quarks are
produced at a scale comparable to their Compton wave length
\be
d_{\rm prod} \sim 1/m_t
\ee
Electroweak vertex corrections do not alter this behaviour significantly,
since $Z$- or $W$-boson exchange proceeds at a distance $\approx 1/m_Z$
which is still short compared to scales characteristic for the bound state
dynamics. For the QCD potential
\be
V_{\rm QCD} = -{4\over 3}{\alpha_s\over r}
\ee
one anticipates an effective coupling constant ${4\over 3}\alpha_s \approx
0.2$, if $\alpha_s$ is evaluated at the scale of the Bohr momentum
\be
k_B \approx {4\over 3}\alpha_s {m_t\over 2} \approx 20 GeV
\ee
The resulting Bohr radius
\be
r_{\rm Bohr}= 1/k_B
\ee
is small compared to hadronic scales. The binding energy of the 1S level
\be
E_B = \left( {4\over 3}\alpha_s\right)^2 {m_t\over 4}\approx 2
GeV
\ee
and, quite generally, the separation between different resonances, is
smaller than the decay rate
\be
2\Gamma_t \approx 3 GeV
\ee
whence all resonances will merge and join smoothly with the continuum.
The coupling strength $\kappa$ of the Yukawa potential
\be
V_Y = -{\kappa\over r}e^{-m_Hr}\quad {\rm with } \quad
\kappa = \sqrt{2}G_F {m_t^2\over 4\pi} = 0.042
\ee
is comparable to the QCD coupling $4/3\alpha_s=0.2$ in magnitude. The
exponential damping, however, with a cutoff $1/m_H \ll r_{\rm Bohr}$ and a
lower limit $m_H > 65 $ GeV, reduces the impact of the Yukawa potential
quite drastically. (The situation may be different in multi-Higgs-models:
the couplings could be enhanced and, even more important, the Higgs might be
lighter!) Furthermore, the nonrelativistic treatment is no longer adequate
and retardation effects must be taken into consideration.
The large top quark width plays a crucial role for the threshold behaviour.
The long distance part of the potential becomes irrelevant with a
cutoff corresponding to momenta $x_0^{-1}\sim \sqrt{2m_t\Gamma_t}\sim
24$ GeV,and the $t\bar t$ system can be described entirely withing the
framework of perturbative QCD.
Instead of summing
the contributions from a large number of high radial excitations
one may directly calculate the imaginary part of the Greens
function for complex energy
\begin{eqnarray}
\sigma (e^+e^- \to t\bar{t})
= \frac{24\pi^2\alpha^2}{s} \frac{\rho_v(s)}{m^2_t}
\left( 1 - \frac{16}{3} \frac{\alpha_s}{\pi} \right)
\sum_{n} \left| \psi_n (0) \right|^2
\frac{\Gamma_t}{(E_n - E)^2 + \Gamma^2_t} \nonumber\\
= \frac{24\pi^2\alpha^2}{s} \frac{\rho_v(s)}{m^2_t}
\left( 1 - \frac{16}{3} \frac{\alpha_s}{\pi} \right)
\sum_{n} Im \frac{\psi^{\*}_n (0) \psi^*_n (0)}
{E_n - E - i \Gamma_t} \nonumber\\
= -\frac{24\pi^2\alpha^2}{s} \frac{\rho_v(s)}{m^2_t}
\left( 1 - \frac{16}{3} \frac{\alpha_s}{\pi} \right)
Im G(0,0,E+i\Gamma)
\label{eq:60}
\end{eqnarray}
The factor $\rho_v(s)$ incorporates the contributions from the
intermediate photon and $Z$ and is given by
\begin{eqnarray}
\rho^{Born}_v(s) = \left| e_t e_e + \frac{1}{y^2}
\frac{\upsilon_t \upsilon_e M^2_\theta}
{s - M^2_Z + iM_Z \Gamma_Z} \right|^2
+ \left| \frac{1}{y^2}
\frac{\upsilon_t a_e M^2_G}
{s - M^2_Z + iM_Z \Gamma_Z} \right|
\label{eq:58}
\end{eqnarray}
\[ \upsilon_f = 2I_{3f} - 4e_f\sin^2\theta_W \qquad
a_f = 2I_{3f} \qquad
y = 16 \sin^2 \theta_W \cos^2\theta_W \]
($\alpha=\alpha_{\rm eff}=1/128$ has been adopted in the numerical
evaluation. Radiative corrections to this formula have been discussed in
\cite{GKKS}.)
The problem can be solved in closed analytical form for an exact
Coulomb potential \cite{fk};
to arrive at a realistic prediction of the total (and, in sect \ref{ss:real}
of the differential) cross section a realistic QCD potential must be
used.
\subsection{The QCD potential}
\label{ss:pot}
On the basis of earlier conceptual work in \cite{apple,sus} the
asymptotic behaviour of the static potential has been derived in
NLO \cite{fis,bil} and even NNLO \cite{MPeter}.
In momentum space the potential reads in the
$\overline{MS}$ subtraction scheme
\begin{eqnarray}
\lefteqn{ 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.} && \nonumber \\
& & \left.+\left( \frac{4343}{18}+66\zeta_3 + 54\pi^2 - \frac{9}{4}\pi^4
- \Big(\frac{1229}{27}+\frac{52}{3}\zeta_3\Big) n_f
+ \frac{100}{81} n_f^2 \right)
\left(\frac{\alpha _{\overline{MS}}(Q^2)}{4\pi}\right)^2
\right]
\nonumber \\
&& { }
\label{eq:49}
\end{eqnarray}
The renormalization scale $\mu ^2$ has been chosen as $Q^2$, and $n_f$
refers to the number of massless quarks.
For the momentum range above 20~GeV explored
by the $t\bar{t}$ system $n_f=$5 seems adequate.
In practice one connects the theoretically predicted short distance part
smoothly with the empirically determined potential above
$\sim$0.1~fermi (Fig.~\ref{fig:AEFF}, \ref{fig:COOR})
The
large decay rate acts as a cutoff and the predictions are fairly
insensitive to the actual regularisation of the potential for small
$Q^2$.
However, an additional
constant which can be traded against a shift in $m_t$ must be
carefully calibrated.
\begin{figure}[htb]
\begin{center}
\leavevmode
\epsfxsize=60mm
\epsffile[130 600 390 610]{fig_3.12.ps}
\vspace*{110mm}
\hfill
\caption{\em $\alpha_{\rm eff}$ for different values of $\alpha_s(M_Z)$:
solid: 0.12, dashed: 0.11, dashed-dotted: 0.13, dotted: 0.10 and 0.14;
\cite{pedag}.}
\label{fig:AEFF}
\end{center}
\end{figure}
\begin{figure} %[ht]
\begin{center}
\leavevmode
\epsfxsize=12.cm
\epsffile[70 135 515 710]{fig_3.13.ps}
\hfill
\caption{\em QCD potential in the position space $V_{JKT}$
for different values of $\alpha_s(M_Z)$:
solid: 0.12, dashed: 0.11, dashed-dotted: 0.13, dotted: 0.10 and 0.14;
\cite{pedag}.}
\label{fig:COOR}
\end{center}
\end{figure}
\subsection{Realistic predictions for $\sigma_{t\bar t}$.}
\label{ss:real}
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 \ref{sec:res}.
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.
The dependence on the top quark
mass is illustrated in Fig.~\ref{fig:310}.
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.
A fairly
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:310}). 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 $\bar t$.
A variation in $m_H$ for $m_H$ above $\sim
65$ GeV affects mainly the normalization of the cross section.
Measurements with a
precision better
than 10\% will become sensitive to the effect of a relatively
light Higgs boson.
\begin{figure} %[ht]
\begin{center}
\leavevmode
\vspace*{-2cm}
\hspace*{7mm}
\psfig{figure=sigma.ps,width=100mm,bbllx=110pt,bblly=280pt,%
bburx=470pt,bbury=550pt}
\hspace*{.5cm}
\psfig{figure=dsgdp349.ps,width=100mm,bbllx=105pt,bblly=280pt,%
bburx=470pt,bbury=550pt}
\end{center}
\caption[]{\it
Upper part: The cross section for the production of top quarks near
the threshold. Demonstrated is the sensitivity of the cross section
to the value of the top mass and the QCD coupling (normalized at the
$Z$ mass). Lower part: The momentum spectrum of the top quarks near
the threshold for a fixed total c.m. energy. The momentum depends
strongly on the top mass, yet less on the QCD coupling.
Refs.\protect\cite{JKT}. \protect\label{fig:310}}
\end{figure}
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.
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{Igo,Comas}.
\subsection{Momentum distributions of top quarks}
\label{ss:mom}
The Green's function in momentum space and the momentum distribution of top
quarks (and thus their decay products) are intimately related
\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.~\ref{ss:pot}, 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=10.cm % was 12
\epsffile{fig_3.19a.eps}
\rule{0mm}{10mm}
\rule{-1mm}{0mm}
\epsfxsize=11.cm % was 12
\epsffile{fig_3.19b.eps}
\rule{0mm}{1mm}
\caption{\em 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: $|pG(p)|^2\cdot 0.002$; \cite{JKT}.}
\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=11.cm
\epsffile[15 210 580 635]{fig_3.20.ps}
\hfill
\caption{\em Momentum distribution of top quarks for three different cms
energies.}
\label{fig:3}
\end{center}
\end{figure}
\begin{figure}[ht]
\begin{center}
\leavevmode
\epsfxsize=11.cm
\epsffile[15 210 580 635]{fig_3.21.ps}
\hfill
\caption{\em 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}.
The dependence on $\alpha_s$ and $m_t$ can be seen in Fig.~\ref{fig:310}
\cite{pedag}.
\subsection{Angular distributions and polarization}
\label{ss:angu}
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 \cite{hjkt} (for a related discussion in
coordinate space see \cite{MurSum2})
\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.
The following abbreviations will be useful below:
\begin{eqnarray}
v_f &=& 2 I^3_f - 4 q_f \sin^2\theta_{\rm W} , \qquad a_f = 2 I^3_f .
\nonumber \\
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}
$P_-$ ($P_+$) 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 differential cross section, summed over polarizations of quarks
and including $S$-wave and $S$-$P$--interference contributions,
is thus given by \cite{MurSum2,hjkt}
\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
\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.
A 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 \em Definition of the spin directions.
The normal component ${\bf s}_{\rm N}$ points out of the plane.}}
\end{flushleft}
\end{figure}
\noindent
{\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.
Including the QCD potential one obtains for the three components
of the polarization \cite{hjkt}
\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 $\Phi_{\rm R}\to \rmp/m_t$
(Fig.~\ref{alph_dep.ps}).
\item The imaginary part of $\Phi$ approaches a constant value $\approx
{2\over 3}\alpha_s$ (Fig.~\ref{alph_dep.ps}). 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.
\end{itemize}
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}
\begin{center}
\leavevmode
\epsfxsize=120mm
\epsffile[70 100 510 735]{fig13.ps}
\caption[]{\label{alph_dep.ps}\sloppy \em 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}
\label{sec:res}
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}\em 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 between $t$ and $\bar b$ and
between $\bar t$ and $b$ (rescattering between $b$ and $\bar b$ can be
ignored \cite{ttp48}). 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}} \\
& \\
{\it a) E = -3 GeV} & {\it b) E = 0 GeV} \\
& \\
& \\
\epsfxsize 70mm \mbox{\epsffile[0 0 567 454]{fig8c.ps}} &
\epsfxsize 70mm \mbox{\epsffile[0 0 567 454]{fig8d.ps}} \\
& \\
{\it c) E = 2 GeV} & {\it d) E = 5 GeV}\\
& \\
& \\
\end{tabular}
\caption{\em 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
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. A full calculation
of all relativistic
${\cal O}(\alpha^2_s)$ 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}.