\section{Top quarks in hadron colliders}
Searching for new quarks and exploring their properties
has been a most important task at hadron colliders in the past.
The Tevatron proton-antiproton collider is the only machine in which
top quarks have been produced so far.
The high energy collider LHC is needed to
provide the large number of events [order $10^7$] for a detailed study
of the top quark properties.
The main production mechanisms
for top quarks in proton-antiproton collisions, Fig.\ref{Fhaddia},
are gluon-gluon and quark-antiquark fusion \cite{gluec1}
$$
gg \;\;\mbox{and} \;\;q\overline{q} \rightarrow t\overline{t}.
$$
$W$--gluon fusion \cite{wille1}
$$ Wg \rightarrow t\overline{b} $$
is an interesting weak process that will allow us to measure the $tbW$ coupling.
\begin{figure}[htb]
\vspace{8cm}
\caption[dummi]{\label{Fhaddia}\em The main production
mechanisms for top quarks in \ppb\ and $pp$ colliders
[generic diagrams].}
\end{figure}
\subsection{Hadronic fusion mechanism}
The dominant Born terms of the \underline{total cross section}
for top--pair production 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 $pp$ cross
sections then follow by averaging the partonic cross sections over the
$q\overline{q}$ and $gg$ luminosities in $pp$ collisions,
\beq
\sigma \left( pp \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
Since the parton luminosities rise steeply for $\tau \ra 0$, the
$\ttb$ production cross sections, increase
dramatically with rising energy.
Higher-order QCD corrections \cite{nason1}, \cite{beena1}
include loop corrections
to the Born terms and $2 \ra 3$ contributions like $gg \ra \ttb g$,
$\qqb \ra \ttb g$ etc. 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}.
The subdominant $2 \ra 3$ contributions add up to less
than $10 \%$ of the dominant lowest-order results.
It is also instructive to study separate, physically distinct components of
the $\alpha_s^3$ results
\cite{meng1}. Whereas the initial (final) state bremsstrahlung (ISGB)
processes, illustrated for the gluon initiated reactions
in Fig.\ref{Fhaddi2}, dominate
around threshold [$\sqrt{\shatn}\ge 2m_t$ or $p_t$
of the $\ttb$
pair is just a few $m_t$.
The top production is central,
$< y > = 0$
with a spread $\Delta y = \pm 2$,
the rapidity values where the cross section has fallen to $1/2$ of
its maximum. The distribution of the transverse momenta has its
maximum value at $p^{max}_t \sim m_t / 2$.
\subsection{$Wg$ Fusion to $t \bar{b}$}
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,
$\sigma \rightarrow G_F^2 m_W^2 / 2\pi$. Folding this subprocess with the
quark-gluon luminosities, the fall-off of the total cross section
$\sigma
(pp \rightarrow t\overline{b})$ is less steep than for the QCD fusion
processes. The $Wg$ fusion process is
suppressed at masses of order
175 GeV by about a factor five: $\sigma (Wg \to tb) \sim 100 \, pb$.
More than 10$^6$ top quarks will nevertheless be produced by this mechanism
at an integrated luminosity of $\int\cal L$ = 10$^4
pb^{-1}$.\hfill
A close inspection of the diagrams in Fig.\ref{Fhaddi2}
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. The cross section is reduced by
an order of magnitude with respect to gluon fusion/quark--antiquark
annihilation.
\subsection{The discovery of the top quark and mass measurements}
After first indications one year earlier, the experimental proof for the
discovery of top quarks was delivered by the CDF collaboration \cite{cdf1}
and the D0 collaboration early in 1995.
Top signals can be
classified according to the number of isolated and non--isolated
charged leptons in the final state.
The CDF collaboration found 6 dilepton candidates with a
background of $1.3 \pm 0.3$ events; moreover, they also found
37 events with tagged $b$ quarks (see Fig.\ref{Fchrquigg}), containing
a $W$ boson and at least 3 jets. Similarly, the D0 collaboration
isolated 17 top candidate events with a background of $3.8 \pm 0.6$ events.
The reconstructed mass from CDF is shown in Fig.\ref{Freconstrmass}. Combining
the two measurements from CDF and D0 the present value of the top mass is
given \cite{cquigg} by
$$
m_t = 175.5 \pm 5.1 GeV
$$
i.e. the $t$ mass is known within an error of 3 percent. The production rate,
Fig.\ref{FCDF/D0}, coincides with the theoretical expectation for a top mass
of $\sim 175$ GeV.
\begin{figure}[phbt]
\vspace{7.5cm}
\caption[dummi]{\label{Fchrquigg}\em Candidate event for top--antitop
production at the Tevatron from the CDF Collaboration, adopted
from Ref.~\cite{cquigg}.
Both top quarks decay
into a $W$--boson plus a bottom quark. The $W^+$ decays to
$e^+$ plus an invisible neutrino, and the $W^-$ decays into a quark
and antiquark that show up as jets of hadrons.}
\end{figure}
\begin{figure}[phbt]
\vspace{8.5cm}
\caption[dummi]{\label{Freconstrmass}\em
Reconstructed mass distributions for the $b$--tagged $ W + \ge 4-$jet
events (solid), adopted from Ref.~\cite{cquigg}.
Also shown are the background shape (dotted) and the sum
of background plus $t \overline{t}$ Monte Carlo for $M_{top} = 175
\makebox { GeV}/c^2$ (dashed), with the background constrained to the
calculated value, $6.9^{+2/5}_{-1/9}$ events. The inset shows the likelihood
fit used to determine the top mass. [CDF \cite{cdf1}].}
\end{figure}
\begin{figure}[hbt]
\vspace{7cm}
\caption[dummi]{\label{FCDF/D0}\em Combined CDF and D0 cross--section
and mass measurements, compared with the theoretical expection for
($p \overline{p} \to t \overline{t}$) as a function of top mass, derived
in Ref.\cite{laen}.}
\end{figure}
Several methods can be used to measure the top--quark mass in proton--(anti)
proton collisions at the Tevatron and LHC quite accurately: (i) $jjj$
recoil--mass measurements in 1--lepton events; (ii) The most precise
measurement of the top mass can be performed by
evaluating the
$\left(l^+ l^- \right)$
invariant mass in trilepton events
where $l^{'}$ and $l^+$ are isolated while $l^-$ belongs to a $b$
jet \cite{foley1}. Since
$\langle m \left( b l^+ \right) \rangle $
$\propto m_t$, with the coefficient determined by the invariant
$l^+l^-$ mass
dis\-tri\-bu\-ti\-on
and the
$b \ra l^-$
fragmentation function, this method is the least background-prone
determination of the top mass \cite{fayar1}. Simulations for the LHC
have shown that a precision of $\delta m_t = \pm 5 \, \mbox{GeV} $
can eventually be achieved. This number does not only match the accuracy
from electroweak radiative corrections but it improves the precision
by at least a factor of 2. If finally all methods are combined, a
precision of $\delta m_t \approx$ 2 to 3 GeV may be achieved.
\newpage