

\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 $p_t<m_t)$],
the gluon splitting (GS)
and the flavor excitation (FE) contributions become equally important for
$\sqrt{\shatn}\gg 2m_t$ or $p_t\ge m_t$.
\begin{figure}[hbt]
   \vspace{4cm}
   \caption[dummi]{\label{Fhaddi2}\em Generic QCD diagrams of the
   dominant higher order mechanisms.}
\end{figure}

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 1-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}
The Coulombic attraction 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, 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.
Due to the
averaging over parton luminosities \cite{fadin1} the effects are less
spectacular in $pp$ than in $e^+e^-$ collisions.

The calculation of the production cross section for top pairs can be refined
by summing up the radiation of infrared gluons in the
initial state. These effects are large in the threshold region. Various
algorithmic techniques have been developed to approach
this problem. Either a cut--off scale (CS) is introduced
\cite{laen} at which the infrared radiation is stopped, 
or principle--value (PVR) techniques are adopted \cite{berg,cata}
to perform the resummation. Even if the method is fixed, there are
still non--negligible uncertainties due to the cut--off choice and due to
different algorithmic approaches.

In addition to the uncertainties in resummation techniques, the theoretical 
predictions depend on the error in $\alpha_s$, the specific choice
of the renormalization and factorization scales near
$\mu \sim m_t$, and to a lesser extent on the parton
distributions. The size of the uncertainties may be estimated
from the spread of the theoretical predictions which appears to be of order
20\%.

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 the electroweak vector bosons and the Higgs boson.
Except for a small region close to the production threshold, the
corrections are always negative. After convoluting
the cross sections of the subprocesses with the parton distributions,
a reduction of the Born
cross section at a level of 10\% is expected.

\begin{table}[hbt]
\begin{center}
\begin{tabular}{|c||c|c||c|}
\hline
& & &  \\
 Method &  $\sigma (p \overline{p} \to t \overline{t})$  & 
$\sigma (pp \to t \overline{t})$ & Ref. \\
 & & & \\
\hline
\hline
 & & & \\
NLO, fix ord.   & $4.20^{+0.28}_{-0.54}$ nb &  & \cite{elli}   \\
 & & & \\
NLO, resum.   & $4.94^{+0.71}_{-0.45}$ nb &  &  \cite{laen}   \\
 & & & \\
NLO, resum.   & $5.52^{+0.07}_{-0.45}$ nb &  & \cite{berg}   \\
 & & & \\
NLO, resum.   & $4.75^{+0.63}_{-0.68}$ nb & $0.77^{+0.25}_{-0.12}$ nb  & 
\cite{cata} \\
 & & & \\
\hline
\end{tabular}
\end{center}
\caption[table]{\label{tabhadtop} \em
Theoretical predictions of the cross sections for $p \overline{p} 
\rightarrow t \overline{t}$ at the Tevatron, $\sqrt{s} = 1.8$ TeV,
and for $pp \rightarrow t \overline{t}$ at the LHC,
$\sqrt{s} = 14$ TeV, for $m_t = 175$ GeV.}
\end{table}

Recent results for the cross sections $\sigma (p \overline{p} \to
t \overline{t})$ at the Tevatron and $\sigma (pp \to t \overline{t})$
at the LHC  are summarized in Table \ref{tabhadtop}. The
resummation of infrared gluon radiation increases the cross section with 
respect to the fixed order $\alpha^3_s$ prediction. With an integrated
luminosity of $10^4 pb^{-1}$ in low--luminosity runs at the 
beginning of the LHC operation, about $10^7$ top--quark pairs will be 
produced at this machine -- a solid base for the first step of 
high--precision top analyses.

The \underline{final states} in the fusion processes can be 
characterized by three observables. The average invariant energy
$< \sqrt{\hat{s}} >$
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.

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