\chapter{Top quarks at hadron colliders}

The search for new quarks and the exploration of their properties
has been a most important task at hadron colliders in the past.
The recent observation of top quarks with a mass of around 180 GeV at
the TEVATRON has demonstrated again the discovery power of hadron
colliders in the high energy region.  Several ten's of top quarks have
been observed up to now.  The significant increase of luminosity
toward the end of this decade will sharpen the picture.  The branching
ratios of the dominant decay modes will be determined and the
uncertainty in the top mass reduced significantly.  For
a detailed study of the top quark properties the
high energy collider LHC will  provide the required large
number of top events [order $10^7$].

The main production mechanisms
for top quarks in proton-antiproton collisions, Fig.\ref{Fhaddia},
are  quark-antiquark fusion  supplemented by a small
admixture of gluon-gluon fusion \cite{gluec1}.
$$
gg \;\;\mbox{and} \;\;q\overline{q} \rightarrow t\overline{t}
$$
Top production at the LHC is of course dominated by the second
reaction. The  $W$-gluon fusion process \cite{wille1}
$$ Wg \rightarrow t\overline{b} $$
is interesting  on its own. It is about a factor 0.1 -- 0.2 below
the dominant reaction and thus well accessible
at the high energy $pp$ colliders --- and perhaps even at the TEVATRON.

\begin{figure}[htb]
\vspace*{10mm}
\begin{tabular}{cc}
\psfig{figure=ggtt.ps,width=80mm,bbllx=0pt,bblly=300pt,%
bburx=630pt,bbury=420pt}
&
\psfig{figure=qqtt.ps,width=80mm,bbllx=0pt,bblly=300pt,%
bburx=630pt,bbury=420pt}
\end{tabular}
\\[25mm]
\hspace*{30mm}
\begin{tabular}{c}
\psfig{figure=ugdtb.ps,width=120mm,bbllx=0pt,bblly=300pt,%
bburx=630pt,bbury=420pt}
\end{tabular}
\vspace*{5mm}
   \caption[dummi]{\label{Fhaddia} The main production
   mechanisms for top quarks in \ppb\ and $pp$ colliders
   [generic diagrams].}
\end{figure}

\section{Lowest order predictions and qualitative features}
The dominant Born terms for the {\em total top cross section} in
$gg/\qqb \ra \ttb$ fusion are well-known to be of the form \cite{gluec1}
\stepcounter{equation}
$$ \label{eq29a}
   \sigma_{gg}(\hat{s})   =   \frac{4\pi\alpha^2_s}{12\hat{s}}
   \; \left[\;(1+\rho+\frac{\rho^2}{16})\;\ln\frac{1+\beta}{1-\beta} -
   \beta\;(\frac{7}{4}+\frac{31}{16}\rho)\;\right] \\
   \eqno{(\arabic{equation}a)}
$$
$$\hspace{-5.8cm}
   \sigma_{q\overline{q}}(\hat{s})   =
   \frac{8\pi\alpha^2_s}{27\hat{s}} \;\beta\;[1+\frac{\rho}{2}]
   \eqno{(\arabic{equation}b)}
$$
with $\rho$ = 4$m_t^2/\hat{s}\;$ and
$\beta = \sqrt{1 - \rho}$ being the velocity of the $t$ quarks in
the $t\overline{t}$ cm frame with invariant energy $\sqrt{\hat{s}}$.
The total $p\bar p$ cross
sections then follow by averaging the partonic cross sections over the
$q\overline{q}$ and $gg$ luminosities in $p\bar p$ (and similarly in $pp$) 
collisions,
\beq
   \sigma \left( p\bar p \ra \ttb \right) = \int^1_{4m^2_t / s}
   d\tau \; \frac{ d {\cal L} \left( gg \right) }{d \tau}
   \sigma_{gg} \left( \tau s \right) + \left[ \qqb \right]
\eeq
The relative enhancement  of the $q\bar q$ cross section by about a
factor 3, as evident from the threshold behaviour
\be
\sigma_{q\bar q} &\approx& {4\over 9} {\pi\alpha_s^2\over \hat s}\beta
\nonumber\\
\sigma_{gg} &\approx& {7\over 48} {\pi\alpha_s^2\over \hat s}\beta
\ee
has to be combined with the prominent $q\bar q$ luminosity at the TEVATRON.
\begin{figure}
\psfig{figure=lumi.ps,width=160mm,bbllx=-100pt,bblly=240pt,%
bburx=630pt,bbury=520pt}
\caption{Parton luminosities for TEVATRON and LHC energies.}
\label{Lumi}
\end{figure}
\begin{figure}
   \vspace{12.5cm}
   \caption[dummi]{\label{Fhadsig} Production cross section
   for $\ttb\ $
   pairs in $\ppb$ and $pp$ colliders: Tevatron (1.8 GeV);
   Tevatron II (3.6 TeV); LHC (16 TeV); SSC (40 TeV). Ref.\cite{reya1}.}
\label{fig213}
\end{figure}
%Einschub 2
As  shown in Fig.~\ref{Lumi}
\be
\left. {\rm d}{\cal L}_{q\bar q} : {\rm d}{\cal L}_{gg} 
\right|_{\sqrt{\hat s}=2m}
\approx
\left\{
\begin{array}{rl}
5 & {\rm TEVATRON} \\
0.1 & {\rm LHC}
\end{array}
\right.
\ee
which implies the dominance of $q\bar q$ annihilation, in contrast to
the situation at the LHC, where gluon fusion is the dominant reaction. 

A number of important features can be read off from this lowest order
result:  
\begin{itemize}
\item
Since the parton luminosities rise steeply with decreasing $\tau$, the
production cross sections increase dramatically with the energy
(Fig.~\ref{fig213}) 
\item
Structure functions and quark-antiquark luminosities in the region of
interest for the TEVATRON, i.e. for $\sqrt{\tau}$ between 0.2 and 0.4
are fairly well known from experimental measurements at lower energies
(combined with evolution equations) and collider studies.  The
predictions are therefore quite stable with respect to variations
between different sets of phenomenologically acceptable 
parton distributions.  The near tenfold increase of the  energy at the
LHC and the corresponding decrease of $x$ and $\sqrt{\tau}$
 by nearly a factor of ten
leads to the dominance of gluon-gluon fusion   and results in a
significantly 
enhanced uncertainty in the production cross section.
\item
With the cross section proportional to $\alpha_s^2$ and uncertainties
in $\alpha_s(M_Z)$ which may be stretched up to $\pm 10\%$ one might
naively expect a resulting uncertainty in the predicted cross section.
However, the increase in the parton cross section with increasing
$\alpha_s$  is, to some extent,
compensated by a decrease in the parton luminosity (with increasing
$\alpha_s$) for the kinematical
region of interest at the TEVATRON.  This compensation mechanism has
been studied in \cite{Martin95} for inclusive jet
production (Fig.~\ref{Jet}) and applies equally well for top quark 
production.
\item
At the TEVATRON the rapidity distribution is strongly dominated by central
production, $|y| \leq 1$, a consequence of the balance between the
steeply falling proton and antiproton parton distributions. 
At the LHC, however, a rapidity plateau develops gradually and the
distribution spans nearly four units in rapidity (Fig.~\ref{transpS8}).
\begin{figure}%[htb]
\begin{center}
    \leavevmode
    \epsfxsize=16cm
    \epsffile[18 180 577 819]{stirling.ps}
\end{center}
\caption{The $p\bar p$--initiated jet $E_T$ distribution at
$\protect\sqrt{s}=1.8$ TeV normalized to the prediction from partons
with $\alpha_S = 0.115$ (i.e. MRS.115).  The data are the CDF
measurements of $d^2 \sigma/dE_T d \eta$ averaged over the
rapidity interval $0.1 < |\eta| < 0.7$.  The curves
are obtained from a leading-order calculation evaluated at $\eta =
0.4$.  The data are preliminary and only the statistical errors
are shown.  The systematic errors are approximately 25\% and are
correlated between different $E_T$ points.  
(From \protect\cite{Martin95}).}
\label{Jet}
\end{figure}

%\begin{figure}
%\psfig{figure=stirling.ps,width=100mm,bbllx=0pt,bblly=0pt,%
%bburx=630pt,bbury=820pt}
%\caption{The $p\bar p$--initiated jet $E_T$ distribution at
%$\protect\sqrt{s}=1.8$ TeV normalized to the prediction from partons
%with $\alpha_S = 0.115$ (i.e. MRS.115).  The data are the CDF
%measurements of $d^2 \sigma/dE_T d \eta$ averaged over the
%rapidity interval $0.1 < |\eta| < 0.7$.  The curves
%are obtained from a leading-order calculation evaluated at $\eta =
%0.4$.  The data are preliminary and only the statistical errors
%are shown.  The systematic errors are approximately 25\% and are
%correlated between different $E_T$ points.  
%(From \protect\cite{Martin95}).}
%\label{Jet}
%\end{figure}
\begin{figure}[htb]
\vspace*{110mm}
\caption{Rapidity distribution of top quarks in $gg/q\bar q$ fusion at
$\protect\sqrt{s}=16$ TeV
\protect\cite{reya1}. }
\label{transpS8}
\end{figure}
\item
The transverse momentum distribution is relatively flat, dropping down
to half its peak value at around $p_t \approx  m_t/2$, again a
consequence of the competition between the increase of the phase space
factor $\propto \beta$ in the parton cross section and the steeply
decreasing parton luminosity (Fig.~\ref{FigS8Beenaker}). 
\begin{figure}[htb]
\vspace*{105mm}
\caption{The differential cross section for $p+\bar p\to t(\bar t) +X$
with $m_t=120$ GeV/c$^2$ and $\mu_R=Q=\protect\sqrt{m_t^2+p_t^2}$ at
$\protect\sqrt{s}=1.8$ TeV.  The cross section is shown at different
values of rapidity for (1) dashed lines: lowest order contribution
scaled by an arbitrary factor (2) solid lines: full order $\alpha_s^2$
calculation. (From \protect\cite{Been91}.)}
\label{FigS8Beenaker}
\end{figure}
At the LHC the distribution
will be even flatter and $p_t$ values around $2m_t$ are well
within reach (Fig.~\ref{fig:transverse}).
This corresponds to  CMS energies between 0.5 and 1
TeV in the parton subsystem and extremely large subenergies
 are therefore accessible.  This opens the
possibility to search for the radiation of $W$, $Z$, or Higgs bosons
in this reaction.  For high energies the suppression of the cross
section through
electroweak virtual corrections (cf.~sect.~\ref{ssec:weak})
is, at least partially, compensated by the large
logarithm $\ln \hat s/M^2_{W,Z,H}$.
\end{itemize}
\begin{figure}[htb]
\vspace{110mm}
\caption{Transverse momentum distribution of top quarks in $gg/q\bar
q$ at $\protect\sqrt{s}=16$ TeV. (From \protect\cite{reya1}.)}
\label{fig:transverse}
\end{figure}  

\newpage

\section{QCD and electroweak corrections}
The observation of top quarks has been well established during the
last year.  One of the tools to study its properties, in particular
its mass and its decay modes, is a precise experimental determination
of its production cross section and subsequent decay
in the $t\to b+W$ channel.  A large
deficiency in the comparison between theory and experiment would
signal the presence of new decay modes which escape the canonical
experimental cuts; with $t\to bH^+$ as most prominent example.
However, the early round of experiments had indicated even an excess
of top events when compared to the theoretical prediction for
$m_t\approx 180$ GeV.  This observation
 was difficult to interpret and the original
calculations were  scrutinized again by various authors.  In particular,
the resummation of leading logarithms and the influence of the Coulomb
threshold enhancement was investigated --- in the end, however, the
prediction remained fairly stable.

In these lectures we will, therefore, in a first step, present the
results from a complete NLO calculation (sect.~\ref{ssec:NLO}). This
is supplemented by a qualitative discussion
of the resummation of higher order leading logarithmic terms.  The
influence of the Coulomb enhancement is studied in section
\ref{ssec:Coul}, electroweak corrections are presented in
sect.~\ref{ssec:weak}.  Radiation of gluons may have a sizeable effect
on the aparent mass of top quarks as observed in the experiment
(sect.~\ref{ssec:exper}), with distinct differences between initial
and final state radiation. 

\subsection{Next to leading order (NLO) corrections and resummation of
large logarithms}
\label{ssec:NLO}

Higher-order QCD corrections \cite{nason1,beena1,Been91,meng1} 
include loop corrections to
the Born terms and $2\to 3$ contributions like $gg\to t\bar t g$,
$q\bar q\to t\bar t g$ etc.  For $q\bar q$ annihilation a few
characteristic diagrams are displayed in Fig.~\ref{fig:ifsr}.  
Real and virtual
\begin{figure}[h]
\begin{minipage}{16.cm}
\hspace*{40mm}
\[
\mbox{
\hspace*{15mm}
\begin{tabular}{cc}
\psfig{figure=isr_real.ps,width=40mm,bbllx=210pt,bblly=410pt,bburx=630pt,bbury=550pt} 
&\hspace*{15mm}
\psfig{figure=isr_virt.ps,width=40mm,bbllx=210pt,bblly=410pt,bburx=630pt,bbury=550pt}
\\[15mm]
\rule{-25mm}{0mm} (a) &\hspace*{-.1cm} \rule{-4mm}{0mm}(b) 
\\
\psfig{figure=fsr_real.ps,width=40mm,bbllx=210pt,bblly=410pt,bburx=630pt,bbury=550pt} 
&\hspace*{15mm}
\psfig{figure=fsr_virt.ps,width=40mm,bbllx=210pt,bblly=410pt,bburx=630pt,bbury=550pt}
\\[15mm]
\rule{-25mm}{0mm} (c) &\hspace*{-.1cm} \rule{-4mm}{0mm}(d) 
\end{tabular}}
\]
\end{minipage}
\caption{Initial and finite state radiation in the reaction $q\bar
q\to t\bar t$}
\label{fig:ifsr}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{minipage}{16.cm}
\hspace*{40mm}
\[
\mbox{
\hspace*{25mm}
\begin{tabular}{cc}
\psfig{figure=gluon_quark.ps,width=40mm,bbllx=210pt,bblly=410pt,bburx=630pt,bbury=550pt} 
&
%\hspace*{15mm}
$+\ldots$ 
\\[15mm]
\end{tabular}}
\]
\end{minipage}
\caption{Reaction $qg\to t\bar t q$}
\label{fig:quarkgluon}
\end{figure}
initial state radiation (Fig.~\ref{fig:ifsr}a,b) dominate, final state radiation
from the slow top quarks (Fig.~\ref{fig:ifsr}c) is
unimportant, 
virtual gluon exchange at the $t\bar t$ vertex (Fig.~\ref{fig:ifsr}d) 
leads to the Coulomb
enhancement and will be discussed in sect.~\ref{ssec:Coul}.  The
separation between $q\bar q$ annihilation and $qg$ reactions
(Fig.~\ref{fig:quarkgluon}) 
depends on the choice of the so-called factorisation scale $\mu^2$
which effectively enters the definition of the structure functions.  

The differential as well as the total production cross section can be
cast into the following form
\be
{\rm d}\sigma = \sum_{i,j=partons} \int {\rm d}x_1 {\rm d}x_2
F_i^p(x_1,\mu^2)
F_j^{\bar p}(x_2,\mu^2)
{\rm d}\sigma_{ij}\left( \hat s, \ldots, \mu^2\right)
\ee
The renormalization scale $\mu_R$ and the factorisation scale $\mu_F$
are in general identified, $\mu_R = \mu_F \to \mu$,
a matter of convention and convenience more than a matter of
necessity.  The parton distributions are extracted from structure
functions as measured in deep inelastic scattering, and the analysis
has to be taylored to the order of the calculation, i.e. to the NLO in
the present case.  The integrated expressions for the total cross
sections can still be cast into a simple form
\begin{equation}
   \sigma_{ij}(\hat{s},m_{t}^2,\mu^{2}) =
   \frac{\alpha_{s}^2(\mu^{2})}{m_{t}^{2}} \left[ f_{ij}^{(0)}(\rho)
   + 4 \pi \alpha_{s}(\mu^{2}) \left( f_{ij}^{(1)}(\rho) +
   \bar{f}_{ij}^{(1)}(\rho) \ln \frac{\mu^{2}}{m_{t}^{2}} \right) \right]
\end{equation}
where $\hat{s} = x_1 x_2 s$ and the dominant lowest-order contributions
$f^{(0)}_{ij} \left( \rho \right)$ are given by the parton cross sections
above;
in addition
$f^{(0)}_{gq} = f^{(0)}_{g \bar{q}} = 0$. The subleading higher-order
expressions for $f^{(1)}_{ij}$ and $\bar{f}^{(1)}_{ij}$ are given in
Refs.\cite{nason1}, \cite{beena1}. The heavy quarks are treated within the on-shell
renormalization scheme with $m_t$ being the ''physical'' mass of the
top quark. Outside the heavy quark sector, the $\overline{MS}$ scheme
has been employed. These higher-order terms have to be used in
conjunction with the running coupling $\alpha_s \left( \mu^2 \right)$
and the gluon/light-quark parton densities evolved in 2-loop evolution
equations. $\mu$ is the renormalization scale, identified here
also with the factorization scale; typical scales that have been chosen
are $\mu = m_t$ and $\sqrt{m^2_t + p^2_T}$. More technical details are
discussed in Ref.\cite{reya1}.

\begin{sloppypar}
The lowest-  and higher-order predictions are compared with each other
in Fig.\ref{Fhadsig}.
In \cite{reya1} it has been argued that the subdominant $2 \ra 3$
contributions add up to less 
than $10 \%$ of the dominant lowest-order results.
The theoretical uncertainties of the predictions for the LHC due to
different parton distributions \cite{diemo1} were estimated  about
$\pm 10 \%$ plus 
a $\pm 10 \%$ variation due to the scale ambiguity $\mu$.
The impact of the additional shift from the resummation of large logs
arising in higher orders
will be discussed below.
Note that the ''K factor'', defined formally by the higher order
corrections to the LO parton cross section, but the parton
distributions and $\alpha_s$ kept fixed, amounts to an $\approx 50 \%$
correction of the Born terms.
\end{sloppypar}

It is also instructive to study separate, physically distinct components of
the $\alpha_s^3$ results
\cite{meng1}.  The initial  state bremsstrahlung (ISGB)
processes, illustrated for the gluon initiated reactions
in Fig.\ref{Fhaddi2}, dominate
\begin{figure}
\hspace*{10mm}
\begin{tabular}{cccc}
\psfig{figure=isgb.ps,width=30mm,bbllx=210pt,bblly=410pt,%
bburx=630pt,bbury=550pt}
&
\hspace*{5mm}
\psfig{figure=isgb2.ps,width=30mm,bbllx=210pt,bblly=410pt,%
bburx=630pt,bbury=550pt}
&
\hspace*{5mm}
\psfig{figure=gs.ps,width=30mm,bbllx=210pt,bblly=410pt,%
bburx=630pt,bbury=550pt}
&
\hspace*{5mm}
\psfig{figure=fe.ps,width=30mm,bbllx=210pt,bblly=410pt,%
bburx=630pt,bbury=550pt}
\end{tabular}
\\[15mm]
\begin{tabular}{cccc}
\hspace*{12mm} ISGB &\hspace*{24mm} FSGB
 & \hspace*{26mm} GS & \hspace*{30mm} FE
\end{tabular}
\caption{Generic QCD diagrams of the dominant higher order mechanisms.}
\label{Fhaddi2}
\end{figure}
around threshold [$\sqrt{\shatn}\ge 2m_t$ or $p_t<m_t)$], the case of
relevance at the TEVATRON. 
The gluon splitting (GS)
and the flavor excitation (FE) contributions become  
increasingly important for
$\sqrt{\shatn}\gg 2m_t$, the situation  anticipated for
the LHC.


Let us concentrate in the remainder of this section on the predictions
for TEVATRON energies.  Initial state radiation reduces the effective
energy in the partonic subsystem, requiring larger initial parton
energies to reach the threshold for top pair production.  Considering
the steeply falling parton distributions $F_j(x)$ one might, therefore,
expect a reduction of $\sigma$ through NLO contributions.  However,
the same effect is operative in the very definition of $F_j$
(Fig.~\ref{fig:sei13})
\begin{figure}[h]
\begin{minipage}{16.cm}
\hspace*{40mm}
\[
\mbox{
\hspace*{15mm}
\begin{tabular}{cc}
\psfig{figure=dis1.ps,width=40mm,bbllx=210pt,bblly=410pt,bburx=630pt,bbury=550pt} 
&\hspace*{15mm}
\psfig{figure=dis2.ps,width=40mm,bbllx=210pt,bblly=410pt,bburx=630pt,bbury=550pt}
\\[15mm]
\rule{-25mm}{0mm} (a) &\hspace*{-.1cm} \rule{-4mm}{0mm}(b) 
\end{tabular}}
\]
\end{minipage}
\caption{Definition of quark distributions, including NLO.}
\label{fig:sei13}
\end{figure} 
through deep inelastic scattering, including NLO corrections.  In fact,
without this compensation mechanism the result would not even be
finite.  However, the magnitude or even the sign of the correction
cannot be guessed on an intuitive basis, and, not quite unexpected,
even the precise form of $f^{(1)}$ and $\bar f^{(1)}$ depends on the
definition of the structure functions.  
The most prominent examples are the $\overline{MS}$ scheme where
$1/\epsilon$ poles (plus $\ln(4\pi) -\gamma$) 
from collinear singularities are simply dropped [more precisely, they
are combined with the corresponding singular terms which arise in the
NLO definition of the structure function] and
finite corrections have to be applied when comparing to deep inelastic
scattering, and the DIS scheme, where $F_j$ are defined through deep
inelastic scattering to all orders.

Let us illustrate the qualitative aspects in the simpler example of NLO
contributions to the Drell Yan process.  
The dominance of initial state radiation in the corrections to $t\bar
t$ production will allow to apply the same reasoning to the case of
interest in these lectures.  
Including NLO corrections one
obtains 
\be
{ {\rm d} \sigma \over {\rm d} Q^2 }
&=& \sigma_0 \sum_{i,j} \int_0^1 { {\rm d} x_i\over x_i} 
{ {\rm d} x_j\over x_j} F_i^p(x_i,Q^2) F_j^{\bar p}(x_j,Q^2)
\cdot 
\omega_{ij}\left({\tau\over x_ix_j},\alpha_s\right)
\nonumber \\
&=& \sigma_0 \sum_{i,j} \int_\tau^1 { {\rm d} z\over z} 
{\cal F}_{ij}(\tau/z)
\cdot 
\omega_{ij}(z,\alpha_s)
\label{eq:27a}
\ee
with
\be
\sigma_0 &\equiv & {4\pi\alpha^2\over 9Q^2s}\qquad \qquad \qquad 
\tau \equiv
{Q^2\over s}
\nonumber\\
\omega^{(0)}_{q\bar q} (z,\alpha_s) &=& e_q^2\delta(1-z)
\nonumber\\
\omega^{(1)}_{q\bar q} (z,\alpha_s) &=& e_q^2{\alpha_s\over 2\pi}C_F
 \left[
2(1+z^2)\left( \ln(1-z)\over 1-z\right)_+ +3\left({1\over
1-z}\right)_+ \right.
\nonumber\\
&& \left. \qquad
+\left(1+{4\pi^2\over 3}\right)\delta(1-z)+\mbox{regular functions}
\right]
\label{eq:DY}
\ee
(The quark-gluon induced reactions will not be discussed in this
connection.) 
The plus prescription which regulates the singularity of the
distributions  at $z=1$ arises from the subtraction of collinear
singularities.  It can be understood by considering the limit 
\be
\left( { \ln^n(1-z)\over 1-z} \right)_+ \equiv \lim_{\epsilon\to 0}
\left( { \ln^n(1-z)\over 1-z} \Theta(1-\eps-z) + \delta(1-z)
{\ln^{n+1}\eps \over n+1} 
\right)
\ee
with the coefficient of the $\delta$ function adjusted such that the
integral from zero to one vanishes.

Equivalently  the plus-distribution 
can be defined through an integral with test functions
$f(z)$.  If $f(z)$ vanishes outside the interval $\langle
\tau,1\rangle$ a convenient formula which will be of use below 
reads as follows
\be
\int_\tau^1 {\rm d}z f(z)
\left( { \ln^n(1-z)\over 1-z} \right)_+ =
\int_\tau^1 {\rm d}z
{ f(z) -f(1) \over 1-z} \ln^n(1-z) - f(1)
\int_0^\tau
dz\, { \ln^n(1-z)\over 1-z}
\ee
The Born term $\omega^{(0)}$ is simply given by a
 $\delta$ function peak at $z=1$,
corresponding to the requirement that the squared energy of the
partonic system $\tau/z\times s$ and the squared mass of the muon pair
$\tau\cdot s= Q^2$ be equal.  ${\cal O} (\alpha_s)$ corrections
contribute to the $\delta$ function through vertex corrections and a
continuous part from initial state radiation extending through the
range 
\be
\tau \leq z \leq 1
\ee
The upper limit $z=1$ corresponds to the kinematic endpoint without
radiation, the requirement $\tau\leq z$ originates from the fact that
the parton luminoisities ${\cal F}_{ij}(\tau/z)$ 
vanish for $\tau/z >1$. The
regular and the subleading pieces of $\omega$
are process dependent, the leading
singularity $\sim (1+z^2)\left( { \ln(1-z)\over 1-z} \right)_+ $ is
universal (and closely related to the $q\to qg$ splitting function) and
equally present in $t\bar t$ production. 

The suppression of final state radiation in top pair production allows
to extend the analogy to the Drell Yan process and to employ
resummation techniques that were successfully developed and applied
for muon pair production \cite{Appel}.
A complete treatment of this resummation is outside the scope of these
lectures.  Nevertheless we shall try to present at least qualitative
arguments which allow to understand the origin of these large
logarithms.  (For a similar line of argument see \cite{Appel}.)
 With the energies of the partonic reaction
$\sqrt{\hat s}={\cal O}(2-4m_t)\approx 350 - 700$ GeV and the CMS energy
$\sqrt{s}=1800$ GeV of comparable magnitude it is clear that the ratio
$\hat s/s$ will not give rise to large logs.  However, large
logarithms can be traced to the interplay between the collinear
singularity in the subprocess and the rapidly falling parton
luminosity ${\cal F}_{ij}$ (cf.~eq.~(\ref{eq:27a})). This rapid decrease
leads to a reduction 
in the effective range of integration.  Let us, for the sake of
argument assume a range reduced from
\be
\tau \leq z \leq 1
\ee
to 
\be
1-\delta \leq z \leq 1
\ee
and evaluate the leading term.  For a constant luminosity ${\cal
F}(\tau/z)/z={\cal F}$ one would obtain
\be
\int_\tau^1 {\rm d}z\; 2 \left( { \ln (1-z)\over 1-z} \right)_+ 
{\cal F}
= \ln^2(1-\tau) 
{\cal F}
\ee
If the region of integration extended through the full kinematic range
and with $\tau = 4m_t^2/s=0.04$ there would be no large log. 
For the restricted range of integration, however, one finds
\be
\int_{1-\delta}^1 {\rm d}z \;2 \left( { \ln (1-z)\over 1-z} \right)_+ 
{\cal F}
= \ln^2(\delta) 
{\cal F}
\ee
For small $\delta$, corresponding in practice to steeply falling
luminosities one thus obtains large, positive (!) corrections from the
interplay between ${\cal F}_{ij}(\tau/z)$ and $\omega_{q\bar q}$. 

To arrive at a reliable prediction the leading terms of the form
$(\alpha_s\ln^2\ldots)^n$ thus have to be included.  The results are based
on alternatively momentum space or impact parameter techniques which
were originally developed for the Drell Yan process
and applied to top pair production in \cite{Laenen}. An additional
complication arises from the blow up of the coupling constant
associated with the radiation of soft gluons for $\hat s\to 4m^2_t$.
This has been interpreted in \cite{Laenen} as a breakdown of
perturbation theory.  Different regulator prescriptions have been
advertised. In \cite{Laenen}
 a cutoff $\hat s-4m^2_t >\mu_0^2$ with $\mu^2_0 \ll
4m^2_t$ was introduced to exclude a small fraction
of the phase space.

The result is fairly stable for $q\bar q$ induced reactions with
$\mu_0/m_t$ chosen between 0.05 and 0.2.  The small contribution from
gluon fusion, however, is  sensitive towards $\mu_0$ which had
to be chosen in the range between $0.2m_t$ and $0.3m_t$, a consequence
of the enhancement of radiation from gluons.
A slightly different approach (``principal value resummation'')
 has been advocated in \cite{Berger}
which circumvents the explicit $\mu_0$ dependence of the result, but
leads essentially to the same final answer (Table \ref{table:cross}).
\begin{table}[htb]
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
$m_t$ [GeV] & 175 & 180 \\ \hline \hline
$\mu_0$ (min) &4.72 & 3.86  \\ \hline
$\mu_0$ (centr) & 4.95 &  4.21\\ \hline
$\mu_0$ (max) & 5.65 & 4.78  \\ \hline
``principal value'' & 5.6 & 4.8  
\\ \hline
\end{tabular}
\end{center}
\caption{Top production cross section (in pb)
for $\protect\sqrt s$ =1800 GeV for
different values of the cutoff $\mu_0$ \protect\cite{Laenen} (first
three lines) and 
for the  ``principal value'' prescription \protect\cite{Berger}
(fourth line).
} 
\label{table:cross}
\end{table}
\begin{table}[htb]
\begin{center}
\begin{tabular}{|l|l|}
\hline
  &  $\protect\sigma$ [pb] \\ \hline
Altarelli et al. \protect\cite{nason1}  & $3.52$ (DFLM) 
\\
                                          & $4.10$ (ELHQ)
\\ \hline
Laenen et al. \protect\cite{Laenen} & 
           $ \left.\begin{array}{ll} 3.5 &\quad(\mu^2=4m^2)\\
                                   3.8  &\quad(\mu^2=m^2)\\
                                   4.05 &\quad(\mu^2=m^2/4)
                  \end{array}
\right\}$ MRSD
\\ \hline
\hline
Resummation &   \\ \hline
Laenen et al. \cite{Laenen}   &  $
            \left.\begin{array}{c} 3.86\\4.21\\4.78\end{array}
\right\} $ vary $\mu_0$
\\ \hline
Berends et al. \cite{Berends} & 4.8 central value
\\ \hline
Berger et al. \cite{Berger} & 4.8 ``principal value res.''
\\ \hline
\end{tabular}
\end{center}
%\label{table:}
\caption{History of predictions for the production cross section for
$\protect\sqrt{s} =1.8$ TeV and $m_t = 180$ GeV.}
\label{table:history}
\end{table}

The result of the improved prediction (central value) is compared to
the fixed order calculation (with $\mu^2=4m^2,m^2,m^2/4$) in
fig.~\ref{fig:seite15}. 
\begin{figure}[htb]
\vspace*{13cm}
\caption{The NLO exact cross section as a function of the top quark
  mass for  three choices of scale: 
$\mu = m/2$ (upper solid line),
$\mu = m$ (central solid line)
and $\mu = 2m$ (lower solid line), and the NLO exact cross section
plus the ${\cal O}(\alpha^4_s)$ contribution at $\mu=m$ (dashed
line) (from \protect\cite{Laenen}).
}
\label{fig:seite15}
\end{figure}
Resummation evidently increases the cross section slightly above the
previously considered range.  The history of predictions is shown in
table~\ref{table:history}, with $\sqrt{s}=1.8$ TeV and $m_t=180$ GeV
as reference values.  The table demonstrates that the spread of
predictions through a (fairly extreme) variation of structure
functions (DFLM vs. ELHQ) 
and through a variation of the renormalisation and factorisation are
comparable --- typically around $\pm 10\%$. Leading log resummation
increases the cross sections by $10-15\%$, with a sizeable sensitivity
towards the cutoff prescription.  A reduction in $m_t$ by 5 GeV leads
to an increase of $\sigma$ by about 0.8 pb. 
Theory and experiment,
with its present result of $7.6^{+2.4}_{-2.0}$ pb and $6.3\pm 2.2$ pb
from CDF and D0 respectively are thus well compatible
(Fig.~\ref{fig:seite17}).  
\begin{figure}[htb]
\vspace*{13cm}
\caption{Comparison of experimental results for the top quark mass and
production cross section with the theoretical prediction.}
\label{fig:seite17}
\end{figure}


\subsection{Threshold behaviour}
\label{ssec:Coul}

Near the $\ttb$ threshold the cross sections are affected by resonance
production and Coulomb rescattering forces \cite{bigi2}, \cite{guesk1},
\cite{fadin1}. These
corrections can be estimated in a simplified potential picture. The
driving one-gluon exchange potential is attractive if the $\ttb$ is in
color-singlet state and repulsive in a color-octet state \cite{fadin1},
\begin{eqnarray}
\sigma^{(1)}(gg\ra t\overline{t}) & = & \frac{2}{7}
       \;\sigma_B(gg\ra t\overline{t}) \mid\!\Psi^{(1)}\!\mid^2 \nonumber \\
\sigma^{(8)}(gg\ra t\overline{t}) & = & \frac{5}{7}
       \;\sigma_B(gg\ra t\overline{t}) \mid\!\Psi^{(8)}\!\mid^2 \\
\sigma^{(8)}(q\overline{q}\ra t\overline{t}) & = & \phantom{\frac{5}{7}}
       \;\sigma_B(q\overline{q}\ra t\overline{t})
       \mid\!\Psi^{(8)}\!\mid^2 \nonumber
\end{eqnarray}



with the correction factors (see fig.~\ref{fig:coul})
given in NLO by 
\be
\beta |\psi|^2 = \beta \left(
1+{\pi\alpha_s\over 2\beta_t}\left\{
\begin{array}{cc}
4/3 & \mbox{singlet} \\
-1/6 & \mbox{octet} 
\end{array}
\right.
\right)
\label{eq:star1}
\ee
The summation of the leading $\pi\alpha_s/\beta$ terms to all orders
results in the familiar Sommerfeld correction factor
\be
\beta|\psi|^2 = \beta {x\over 1-e^{-x}}
\label{eq:18a}
\ee
For $t\bar t$ in the singlet configuration
$x=x^{(1)}\equiv {4\over 3} {\pi \alpha_s\over \beta}$, for octet states
 $x=x^{(8)}\equiv -{1\over 6} {\pi \alpha_s\over \beta}$.


The Coulombic attraction thus leads to a sharp rise of the cross section
at the threshold in the singlet channel, even if no resonance can be
formed anymore, since the phase space suppression of the Born term
$\sigma_B \propto \beta_t$
is neutralized by the Coulomb enhancement of the wave function
$\left| \Psi_1 \right|^2 \propto \alpha_s / \beta$.
In the octet channel (dominant for $q\bar q$ annihilation), by
contrast, the cross sections are strongly 
reduced by the Coulombic repulsion which leads effectively to an
exponential fall-off of the cross sections
$\sigma_8 \propto \exp \left[ - \pi \alpha_s / 6 \beta_t \right]$
at the threshold \cite{fadin1}.
Due to the
averaging over parton luminosities the effects are less
spectacular in $p\bar p$ or $pp$ than in $e^+e^-$ collisions.

The enhancement and suppression factors are compared to simple phase
space $\sim \beta$ in Fig.~\ref{fig:seite20}.
\begin{figure}
\hspace*{20mm}
\begin{tabular}{cl}
\psfig{figure=Coul1.ps,width=35mm,bbllx=210pt,%
bblly=410pt,bburx=630pt,bbury=550pt} & $\sim \beta_t$
\\[12mm]
\psfig{figure=Coul2.ps,width=35mm,bbllx=210pt,%
bblly=410pt,bburx=630pt,bbury=550pt} & $\sim \beta_t\cdot
{\pi\alpha_s\over
2\beta_t} \left\{\begin{array}{ll}
4/3 & {\rm singlet}\\
-1/6 & {\rm octet}
\end{array}\right.
$
\end{tabular}
\vspace*{15mm}
\caption{Vertex corrections from gluon exchange in the threshold
region.}
\label{fig:coul}
\end{figure}
The dotted line corresponds to the phase space factor $\beta$, the
dashed line to the perturbative NLO calculation
(\ref{eq:star1}),
the solid line to the Coulomb enhancement given in eq.~(\ref{eq:18a}).
The predictions for the singlet, octet
($\equiv q\bar q$), and properly weighted gluon fusion channel are
displayed in Figs.~\ref{fig:seite20} a, b, and c, respectively.
\begin{figure}[htbp]
  \begin{center}
    \leavevmode
\psfig{figure=fig_2.14.ps,width=150mm,bbllx=0pt,bblly=115pt,%
bburx=630pt,bbury=699pt}
  \end{center}
  \caption{Threshold enhancement and suppression factors for singlet
(upper), octet (middle), and properly weighted gluon fusion (lower
figure) for $\alpha_s=0.15$. Dotted line: Born approximation; dashed
line: NLO approximation; solid line: Coulomb enhancement.}
  \label{fig:threshold}
\end{figure}
\subsection{Electroweak corrections}
\label{ssec:weak}
Another potentially important modification which is closely tied to
the Coulomb enhancement originates from vertex corrections induced by
light Higgs boson exchange.  In a simplified treatment these are
lumped into a Yukawa potential 
\be
V(r) = { \sqrt{2} G_F m_t^2\over 4\pi} {e^{-m_Hr}\over r}
\ee
resulting in a reduction $\delta E$ of the apparent threshold, with
$\delta E = -200$ MeV for $m_H = 100$ GeV as characteristic example.
The change in the normalisation by $+ 10\%$ could become
relevant for precision measurements.  The situation is quite similar
to the one discussed for $e^+e^-$ colliders in section 3.2.

Genuine electroweak contributions of
${\cal O} \left( \alpha \alpha^2_s \right)$
have been calculated to both the
$\qqb$ and $gg \ra \ttb$
subprocesses \cite{holli2}. The corrections include vertex corrections and
box diagrams built-up by vector bosons and the Higgs boson
(Fig.~\ref{fig:seite21}). 

\begin{figure}
\hspace*{20mm}
\begin{tabular}{ccc}
\psfig{figure=sei22a.ps,width=35mm,bbllx=210pt,%
bblly=410pt,bburx=630pt,bbury=550pt}
&
\hspace*{5mm}
\psfig{figure=sei22d.ps,width=35mm,bbllx=210pt,%
bblly=410pt,bburx=630pt,bbury=550pt}
&
\hspace*{5mm}
\psfig{figure=sei22f.ps,width=35mm,bbllx=210pt,%
bblly=410pt,bburx=630pt,bbury=550pt}
\\[20mm]
\psfig{figure=sei22b.ps,width=35mm,bbllx=210pt,%
bblly=410pt,bburx=630pt,bbury=550pt}
&
\hspace*{5mm}
\psfig{figure=sei22c.ps,width=35mm,bbllx=210pt,%
bblly=410pt,bburx=630pt,bbury=550pt}
&
\hspace*{5mm}
\psfig{figure=sei22e.ps,width=35mm,bbllx=210pt,%
bblly=410pt,bburx=630pt,bbury=550pt}
\end{tabular}
\vspace*{15mm}
\caption{Feynman diagrams contributing to the electroweak radiative
  corrections.  Dashed lines stand for $H$, $Z$, $\gamma$, or $W$.}
\label{fig:seite21}
\end{figure}
\begin{figure}[htbp]
  \begin{center}
    \leavevmode
    \vspace*{15cm}
  \end{center}
  \caption{Relative correction to the parton cross section for
$m_t=250$ GeV (upper figure: $q\bar q\to t\bar t$, lower figure:
$gg\to t\bar t$) (from \protect\cite{holli2}).}
  \label{fig:seite20}
\end{figure}

Except for a small region close to the production threshold, which is
dominated by the Yukawa potential, the
corrections are always negative; they can become sizeably large,
in particular
if the top is very heavy
and if the energy of the subsystem exceeds 1 TeV, not uncommon for
$t\bar t$ production at the LHC.  In this case however, the large
negative corrections are compensated by positive contributions from
real radiation of $W$, $Z$, or $H$.  The corrections for the $q\bar q$
and $gg$ subprocess as functions of the parton energies are shown in
Fig.~\ref{fig:seite20}.  The sharp increase of the corrections
close to threshold for a light Higgs is clearly visible and,
similarly, the large negative correction for large parton energies. 
 After convoluting
the cross sections of the subprocesses with the parton distributions,
a reduction of the Born
cross section at a level of a few percent is observed
(Fig.~\ref{fig:been17}). 
\begin{figure}
\vspace*{90mm}
\caption{Relative correction to the hadronic cross section for $s=(16
  $ TeV)$^2$ (from \protect\cite{holli2})} 
\label{fig:been17}
\end{figure}

\subsection{Gluon radiation}
\label{ssec:exper}

Up to this point the discussion has centered around the predictions
for inclusive top quark production.  Additional ingredients for the
experimental analysis are the detailed topological structure of the
signal, the number of jets,  the characteristics of the underlying
event, and, of course, predictions for the background.  This
information allows to adjust in an optimal way experimental cuts and
to measure the top quark through a kinematic analysis of its decay
products.  As a typical example the impact of gluon radiation on the
top mass determination has been analysed recently.  An idealized study
has been performed e.g. in \cite{Lampe}.  Radiated gluons are merged
with the $b$ jet from top decay or with the quark jet from $W\to q\bar q$ if
they are found within a cone of opening angle
\be
R=\sqrt{(\Delta \eta)^2 + (\Delta \phi)^2} < R_{cut}
\ee
with respect to $b$, $q$ or $\bar q$ and if their rapidity is below
$|\eta| < 2$. In this case the gluon is considered as top decay
product and hence contributes to its invariant mass.  If the gluon jet
falls outside the cuts, it is assigned the rest of the event.

\begin{figure}[htb]
\hspace*{25mm}
\begin{tabular}{cc}
\psfig{figure=gluon_ini.ps,width=50mm,bbllx=210pt,%
bblly=410pt,bburx=630pt,bbury=550pt}
&
\hspace*{20mm}
\psfig{figure=gluon_fin.ps,width=50mm,bbllx=210pt,%
bblly=410pt,bburx=630pt,bbury=550pt}
\end{tabular}
\\[30mm]
\hspace*{65mm}
\psfig{figure=decay.ps,width=50mm,bbllx=210pt,%
bblly=410pt,bburx=630pt,bbury=550pt}
\\
\vspace*{15mm}
\caption{Gluon radiation from top production (upper) and decay (lower
  diagrams).}
\label{fig:seite22}
\end{figure}
Gluon radiation associated with top quark production, if erroneously
associated with top decay, will thus increase the apparent $m_t$.
Radiation from top decay, if outside the forementioned cuts, will
however, decrease the measured mass of the quark.  The interplay
between the two compensating effects is displayed in
Fig.~\ref{fig:seite23}. \begin{figure}[htbp]
\begin{center}
  \leavevmode
  \vspace*{10cm}
\end{center}
\caption{Shift in the apparent top mass as a function of $R$ cut (from
  \protect\cite{Lampe}).}
\label{fig:seite23}
\end{figure}
For a realistic $R_{cut} \approx 0.6$ a reduction $\Delta m$ of around
2 GeV is predicted.

\section{Single top production}
\label{singletop}
Virtual $W$ bosons, originating from $u\to dW$ splitting, can merge
with bottom quarks from gluon splitting $g\to b\bar b$ to produce
single top quarks in association with fairly collinear $d$- and $\bar
b$- jets (Fig. \ref{fig:seite25}).
\begin{figure}
\begin{minipage}{16.cm}
\hspace*{40mm}
\[
\mbox{
\hspace*{15mm}
\begin{tabular}{c}
\psfig{figure=u_split.ps,width=40mm,bbllx=210pt,bblly=410pt,bburx=630pt,bbury=550pt}
\\[15mm]
\end{tabular}}
\]
\end{minipage}
\caption{Characteristic diagram for single top production. }
\label{fig:seite25}
\end{figure}
The interaction radius in the QCD $gg$ fusion process shrinks with
rising energy so that the cross section $\sigma(gg \rightarrow \ttbn) \sim
\alpha_s^2 / \shatn$ [mod.~log's] vanishes asymptotically. By contrast, the
interaction radius in the weak fusion process \cite{wille1} is set by the
Compton wave length of the $W$ boson and therefore asymptotically non-zero,
$\sigma \rightarrow G_F^2 m_W^2 / 2\pi$. The subprocess has to be
folded
 with the
quark-gluon luminosities
\be
\sigma(p\bar p\to t + X) = \int_\tau^1 {{\rm d}z\over z} {\cal
L}_{ug}(\tau/z)
\sigma(ug\to t+\bar b+d)
\ee
plus a similar contribution from $\bar d g\to t + \bar b + \bar u$.  
The fall-off of the total cross section
$\sigma(pp \rightarrow t\overline{b})$ is less steep than for the QCD fusion
processes. As a result, the $Wg$ fusion process
would have dominanted for large top
quark masses $\ge$ 250 GeV at the LHC (Fig.~\ref{Fhads2}).

\begin{figure}[htb]
\vspace*{5mm}
\begin{tabular}{cc}
\hspace*{25mm}
\psfig{figure=Wb.ps,width=50mm,bbllx=210pt,%
bblly=410pt,bburx=630pt,bbury=550pt}
&
\hspace*{20mm}
\psfig{figure=ut.ps,width=50mm,bbllx=210pt,%
bblly=410pt,bburx=630pt,bbury=550pt}
\\[18mm]
(a) & (b)
\end{tabular}
\vspace*{5mm}
\caption{Characteristic diagrams for top production via Compton
  scattering (a) and Drell Yan process (b)}
\label{fig:Wg}
\end{figure}


For $m_t =$ 180 GeV, the case of practical interest, single top
production is about a factor 5 below the QCD reaction.
Nevertheless, as shown in Fig.\ref{Fhads2},
about 10$^6$ top quarks will be produced at the LHC by this mechanism
at an integrated luminosity of $\int\cal L$ = 10$^4
pb^{-1}$. Also at the TEVATRON this process should be accessible with
the anticipated luminosity.
\begin{figure}[hbt]
\vspace*{115mm}
\caption{Cross sec\-ti\-ons for several
      mechanisms of top quark production \protect\cite{reya1}. Various
                   pa\-ra\-me\-tri\-za\-ti\-ons and models of the parton
          densities \protect\cite{diemo1} have been used.}
\label{Fhads2}
\end{figure}

A close inspection of the various contributions to the subprocess
$ug\to t+\bar b+d$ 
reveals immediately that the
by far dominant part of the cross section is due to  $b$ exchange, with the
b quark being near its mass shell. Since the $b$
quark is almost collinear to the incoming gluon,
this cross section is logarithmically enhanced $\sim\ln(m_t^2/m_b^2)$
over other mechanisms.
This naturally suggests to approximate
the process by the subprocess $u$ + $b$
$\stackrel{\mbox{\scriptsize W}}{\rightarrow}$ $d$ + $t$ with the $b$-quark
distribution generated perturbatively
by gluon splitting based on
massless evolution equations.
The weak cross sections can be presented in a compact form,
\begin{eqnarray}
\sigma(ub\stackrel{\mbox{\scriptsize W}}{\rightarrow}dt) & = &
\frac{G_F^2 m_W^2}{2\pi}\;\frac{(\shatn-m_t^2)^2}
{\shatn(\shatn+m_W^2-m_t^2)}
\mbox{\hspace{5.00cm} }\rightarrow\frac{G_F^2 m_W^2}{2\pi}
\nonumber \\
\sigma(d\overline{b}\stackrel{\mbox{\scriptsize W}}{\rightarrow}
u\overline{t})
&\rule[-1mm]{0mm}{10mm}= &
\frac{G_F^2 m_W^2}{2\pi}\;
\left[\;1+\frac{\shatn(2m_W^2-m_t^2)-2m_W^2m_t^2}{\shatn^2} \right.
\nonumber \\
& &    \mbox{\hspace{1.6cm}} \left.
-\;\frac{(2\shatn+2m_W^2-m_t^2)m_t^2}{\shatn^2}\;\log
\frac{\shatn+m_W^2-m_t^2}
{m_W^2}\;\right] \mbox{\hspace{0.00cm}}\rightarrow\frac{G_F^2 m_W^2}
{2\pi} \nonumber
\end{eqnarray}
and identically the same expressions for the $\cal C$-conjugate reactions.

Top quarks are created in $u+g$ collisions, anti-top quarks in
$d+g$ collisions where the absorption of a $W^-$ transforms a
$\overline{b}$ quark to a $\overline{t}$ quark. The
na$\ddot{\mbox{\i}}$ve expectation from valence quark counting
for the ratio of $t / \overline{t}$ cross sections,
$\sigma(u\rightarrow t):\sigma(d\rightarrow \overline{t})$ $\sim$ 2 : 1 is
corroborated by a detailed analysis; in fact, the ratio
turns out to be 2.1 for top quark
masses of about 150 GeV.

The remaining possibilities for single top production are Compton
scattering (Fig.~\ref{fig:Wg}(a))
\be 
g+b\to W+t
\ee
and the Drell Yan process (Fig.~\ref{fig:Wg}(b))
\be
u+\bar d \to t+ \bar b
\ee
The predicted cross sections are too small to be of practical
interest.  Single top quark production via the dominant mechanism
(Fig.~\ref{fig:seite25})  offers a unique way for a
measurement of the CKM matrix element $V_{tb}$ and thus, indirectly of
the top quark life time.  As discussed in section \ref{sect:qualit}
 $V_{tb}$ is strongly constrained to be very close to one for three 
generations --- in a four generation
model $V_{tb}$ may be quite different from these expectations. 

\section{Quarkonium production}

Both charm and bottom quarks have been discovered at hadron colliders
in the form of quarkonium resonances $J/\psi$ and $\Upsilon$ through
their distinct signals in the $\mu^+\mu^-$ channel. The search for
toponium at a hadron collider is, however, entirely useless.  The
broad ($\Gamma\sim$ 2 GeV) resonances decay with an overwhelming
probability through single quark decay and are therefore
indistinguishable from open top quarks produced close to threshold.  

The situation could be different in extensions of the SM. Decays of a
fourth generation $b^\prime$ 
\be
b^\prime \to t + W
\ee
are suppressed by small mixing angles.  Alternatively, if
$m_{b'}<m_t+m_W$, the $b'\to c+W$ mode would have to compete with loop
induced FCNC decays --- leaving ample room for narrow quarkonium
states. Another example would be the production of weak isosinglet
quarks which are predicted in Grand Unified Theories.  The decay of
these quarks would again be inhibited by small mixing angles.  

Of particular interest is the search for $\eta_{b'}$, the $^1S_0$ state
composed of $b'$ and $\bar b'$ \cite{Barger,KM,KP}. It is produced
with appreciable cross section. Its dominant decay mode
\be
\eta_{b'} \to H+Z
\ee
is enhanced by the large Yukawa coupling, governing the coupling of
the heavy quark to the Higgs and the longitudinal $Z$.  For large
$m_{b'}$ one obtains 
\be
{\Gamma(\eta_{b'}\to H+Z)\over \Gamma(\eta_{b'}\to \gamma\gamma)}
\sim
{m_{b'}^4\over M_W^4}
\ee
The branching ratios as functions of $M_\eta$ are displayed in
Fig.~\ref{fig:seite30oben}.
\begin{figure}
\vspace*{105mm}
\caption{Branching ratios of $\eta_{b'}$ for the dominant decay modes
  as functions of the bound state mass $M$.}
\label{fig:seite30oben}
\end{figure}
The complete set of QCD corrections for leading and subleading
annihilation decay modes
can be found in \cite{KP}.  They do not alter the picture
significantly.

It should be emphasized that the decay $\eta_{b'}\to H+Z$ proceeds
through the axial part of the neutral current coupling which, in turn,
is proportional to the third component of the weak isospin.  
Bound states of isosinglet quarks would, therefore, decay dominantly
into two gluon jets.

The cross section for {\em open} $b\bar b'$ production at the
LHC (with $m_{b'}=300$ GeV) amounts to about 100 pb.  The fraction of
the phase space where bound states can be formed, i.e. for relative
quark velocity $\beta<0.1$, covers around $10^{-2}$ of the relevant
region
\be
{\int_0^{0.1} {\rm d}\beta \beta 
\over
\int_0^{1} {\rm d}\beta \beta }
=10^{-2}
\ee
and indeed one predicts a production cross section somewhat less than
1 pb from a full calculation.
 
For a  detailed calculation of the production cross section a proper
treatment of the QCD potential is required to obtain a reliable
prediction for the bound state wave function at the origin.  The
structure of the 
NLO corrections for the production cross section, in particular of the
dominant terms, bears many similarities with the result for open
production and for the Drell Yan process (eq.~\ref{eq:DY}).  For gluon
fusion the partonic cross section is 
(in the $\overline{MS}$ scheme) given by \cite{Letter,Paper}
\be
\lefteqn{\hat \sigma^{gg}  =  {1\over s}{\pi^2\over 3}{R^2(0)\over M^3}
\alpha^2_{\overline{\rm MS}} (M^2)
\left\{ \rule{0mm}{5mm} \delta(1-z) \right.}
\nonumber
\\
&&\left. +{\alpha_{\overline{\rm MS}} (M^2) \over \pi}
\left\langle
\delta(1-z)\left( N_C\left(1+{\pi^2\over 12}\right)
C_F\left({\pi^2\over 4}-5\right)
-{4\over 3}T_f \ln(2) \right) +N_C F(z)\right\rangle \right\}
\nonumber\\
\label{eq:ttp9235}
\ee
where
\be
F(z)&=&\Theta(1-z)\left[
{11z^5+11z^4+13z^3+19z^2+6z-12\over 6z(1+z)^2}\right.
\nonumber\\
&&+4\left( {1\over z} +z(1-z) -2\right) \ln(1-z) +4\left(
{\ln(1-z)\over 1-z} \right)_+
\nonumber\\
&&\left.
+\left( {2(z^3-2z^2-3z-2)(z^3-z+2)z\ln(z)\over
(1+z)^3(1-z)}-3\right)
{1\over 1-z}\right]
\ee
and $z=M^2/{\hat s}$.  Both Born term and the virtual correction are
proportional to $\delta(1-z)$, the structure of the dominant term due
to gluon splitting $\sim \ln(1-z)/(1-z)$ is again
universal. Quark-gluon and quark-antiquark initiated subprocesses of
order $\alpha^3_s$ can be found in \cite{Letter,Paper}.  It may
be worth mentioning that the structure of QCD corrections to light
Higgs production \cite{lighthiggs} is nearly identical to
eq.~\ref{eq:ttp9235}. 
From Fig.~\ref{fig:prod} 
it is evident that $\eta_{b'}$ states with masses up to 1
TeV are produced at the LHC with sizeable rates.  The fairly clean
signature of the $Z+H$ decay mode might allow to discover these exotic
quarkonia and the Higgs boson at the same time.

\begin{figure}
\vspace*{105mm}
\caption{Production cross section for $\eta_{b'}$ including QCD
  corrections as function of the bound state mass $M$ for
  $\protect\sqrt{s}=16$ TeV and 40 TeV.}
\label{fig:prod} 
\end{figure}
