 

\chapter{$t$ \hspace{2mm}QUARK}

The top quark completes the fermion spectrum of the Standard Model
so that all its properties are uniquely determined when the mass
is fixed. Top quarks were elusive particles for a long time, searched
for more than a decade before they could be isolated at the Tevatron
\cite{cdf1} with a mass value \cite{cquigg}
\begin{equation}
m_t = 175.5 \pm 5.1 \makebox { GeV}
\end{equation}

This directly measured value matches very well early 
estimates which were derived from precision measurements of the electroweak 
parameters in $e^+e^-$ annihilation: $m_t = 177 \pm 19$ GeV 
\cite{cern96-183}. The coincidence of the two values is a 
triumph of field theory  allowing us to 
control the quantum fluctuation in the $W, Z$ boson masses which
are affected by virtual top quarks. 

The next major step after the discovery of the top quark 
will be the high--precision
analysis of its properties. A large number 
of top quarks [order $10^7$] will be produced at the $pp$ collider
LHC \cite{reya1}, necessary to measure the mass within a few GeV
and to search for hypothetical very rare decay modes. High energy
$e^+e^-$ linear colliders \cite{eacco} can improve the accuracy
in the top sector considerably. The precision with which the fundamental
$t$ mass, a key parameter of flavor physics, can be determined,
increases to a level of 100 MeV. Moreover, the static
electroweak parameters and  non--standard decay modes can be
analysed with high precision.


\section{Early evidence for top quarks}
\subsection{$t$ -- The ekasilicon of the Standard Model}
Several experimental results had provided quite early very strong evidence 
that the
fermion spec\-trum of the \sm\
\[
   { \left[ \nu_e    \atop    e^- \right]_L } \quad { \atop    e^-_R } \qquad
   { \left[ \nu_\mu  \atop  \mu^- \right]_L } \quad { \atop  \mu^-_R } \qquad
   { \left[ \nu_\tau \atop \tau^- \right]_L } \quad { \atop \tau^-_R }
\]
\nopagebreak
\[ \!
   { \left[ {\;    u \;    } \atop d \right]_L }
     \quad { u_R \atop d_R } \qquad
   { \left[ {\, \, c \, \, } \atop s \right]_L }
     \quad { c_R \atop s_R } \qquad
   \,
   { \left[ {\, \, t \, \, } \atop b \right]_L }
     \quad { t_R \atop b_R }
\]
must include the top quark, imprinting the same multiplet structure on
the third family as observed in 
the first two families. The evidence was deduced from
measurements of the weak isospin of the $b$ quark which had been
proved to be non-zero, $I_3 = -1/2$, thus demanding an $I_3 = +1/2$
partner in this isospin multiplet.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Absence of triangle anomalies}
A compelling theoretical argument for the existence of top quarks follows
from consistency requirements. The renormalizability of the
\sm\ demands the absence of triangle anomalies.
Triangular fermion loops built-up by an axialvector charge $I_{3A} = -I_{3L}$
combined with two electric vector charges $Q$ would spoil the renormalizability
of the gauge theory. Since the anomalies do not depend on the masses of the
fermions
circulating in the loops, it is sufficient to demand that the sum
\begin{eqnarray*}
   & & \\
   \hspace{4cm} & & \sim \sum_L I_{3A} Q^2 = - \sum_L I_3 \left[ I_3 +
     \frac{1}{2} Y \right]^2 \\
   \hspace{4cm} & & \sim \sum_L Y \sim \sum_L Q \\
   & &
\end{eqnarray*}
of all contributions be zero. Such a requirement can be translated into a
condition on the electric charges of all the left-handed fermions
\beq
   \sum_L Q = 0
\eeq
This condition is met in a complete standard family in which the
electric charges of the
lepton plus those of all color components of the up and down quarks add up
to zero,
\[
   \sum_L Q = - 1 + 3 \times \left[ \left( + \frac{2}{3} \right) + \left(
   - \frac{1}{3} \right) \right] = 0
\]
If the top quark were absent from the third family, the condition would be
violated and the \sm\ would be theoretically inconsistent.




\subsubsection{Absence of FCNC decays}
Mixing between quarks which belong to different isospin multiplets
\[
   { \left[ c \atop s^{'} \right]_L } \quad { \atop b^{'}_L }
   \hspace{2cm} {
     \! s^{'} = \: s \cos \vartheta^{'} + b \sin \vartheta^{'} \atop
     \: b^{'} =   -s \sin \vartheta^{'} + b \cos \vartheta^{'} }
\]
generates non-diagonal neutral current couplings, i.e. the breaking of the
GIM mechanism

\pagebreak

\begin{eqnarray*}
   <I_3> & = & + \frac{1}{2} \left( c_L,c_L \right)
               - \frac{1}{2} \left( s^{'}_L, s^{'}_L \right) \\
         & = & \mbox{diag.} - \frac{1}{2} \sin \vartheta^{'} \cos \vartheta^{'}
               \left( s_L, b_L \right)
               + \mbox{h.c.} \quad .
\end{eqnarray*}
The non-diagonal current induces flavor-changing neutral lepton pair decays
$b \ra s + l^+l^-$ which have been estimated to be a
substantial fraction of all semileptonic $B$ meson decays \cite{kaue1}
\beq
   \frac{ \mbox{BR} \left( B \ra l^+l^-X \right) }
        { \mbox{BR} \left( B \ra l^+ \nu_l X \right) } \ge 0.12
\eeq
This ratio is four orders of magnitude larger than a bound set recently by
the UA1 Collaboration \cite{albaj3}
\beq
   \frac{ \mbox{BR} \left( B \ra \mu^+ \mu^-X    \right) }
        { \mbox{BR} \left( B \ra \mu   \nu_\mu X \right) } <
   \frac{ 5.0 \times 10^{-5} }{0.110 \pm 0.008 }
\eeq
so that the working hypothesis of an isosinglet $b$ quark is clearly ruled out
experimentally also by this method.


\subsubsection{Partial width $\Gamma ( Z \ra \bbb )$ and
                        forward-backward asymmetry of $b$ quarks}
The isospin quantum numbers of the $b$ quarks have been
 measured directly through
the $Z$--decay width to $b$ quarks $\Gamma(Z \ra b\overline{b})$ and the
forward--backward asymmetry of $b$ quarks on the $Z$, $A_{FB}(b)$, at LEP. If
combined with $b$ data from PETRA/PEP and TRISTAN, a unique solution can be
derived for the third isospin components of the left-- and
right--handed $b$ quarks \cite{SchailZ}, Fig.\ref{FSchailZ},
\begin{eqnarray}
\ltowidth{\{ I_3^L(b) \}_{exp} = -0.490^{+0.015}_{-0.012}}{\{ I_3^R(b) \}_{exp}
 = -0.028 {\pm 0.056}  }
 & \qquad \ra \quad & I_3^L(b) = -1/2 \nonumber \\
\{ I_3^R(b) \}_{exp} = -0.028 {\pm 0.056} & \qquad \ra \quad& I_3^R(b) = 0
\nonumber
\end{eqnarray}
\noindent which coincide with the isospin assignment of the Standard Model.
\begin{figure}[hbt]

\vspace{11.5cm}

\caption[]{\label{FSchailZ}%
{\it The weak isospins $I_3^L(b)$ and $I_3^R(b)$ of the left-- and
right--handed $b$ quark components, extracted from the data on $\Gamma(Z\ra
b\overline{b})$ and $A_{FB}(b)$ at LEP, and PETRA/PEP and TRISTAN,
Ref.\cite{SchailZ}.}}
\end{figure}

\subsection{Mass estimate from radiative corrections}

%\begin{figure}[hbt]
 %  \vspace{11cm}
 %  \caption[dummi]{\label{FCDFbd}Comparison of experimental upper
 %  bounds on the
 %  $t$ production cross sections with theoretical
 %  predictions for top masses up to 91 GeV
 %  \cite{cdf1}.}
%\end{figure}

%
First indications of a very high top quark mass were derived from the rapid
$(B - \overline{B})$ oscillations observed by ARGUS \cite{albre1}.
However, due to the
uncertainties of the \calckm\ matrix element $V_{td}$ and of the
$(b \bar{d})$
wave function, no more than qualitative conclusions can be drawn from such an
analysis as the oscillation frequency $\Delta m \sim \left| V_{td} \right|^2
f^2_B m^2_t$ depends on three [unknown] parameters.

The analysis of the radiative corrections to high precision electroweak
observables, measured at LEP and elsewhere, 
 could be carried out at a quantitativ level.
Since Higgs mass effects are
weak as a result of the screening theorem \cite{veltm1},
the top mass is the central
unknown parameter in the framework of the \sm.
Combining the high precision
measurements of the $Z$ mass with $\sin^2 \theta_W$ from the $Z$ decay
rate, the forward-backward
asymmetry and LR polarization measurements, the top quark mass has been
determined up to a residual uncertainty of $\pm 19$ GeV, including
the uncertainty  through the variation of $m_H$.

The key r\^ole is played by the connection of
the $Z$ mass with $\sin^2 \theta_W$
and the low-energy couplings $\alpha, G_F$. Through radiative
corrections
\footnote{Again, the on-shell renormalization scheme will be adopted.}
this connection is strongly affected by the $t$ quark mass. Linking
the Glashow-Salam-Weinberg theory to the effective low-energy Fermi theory
of the weak interactions relates the Fermi coupling $G_F$ with the
GSW $SU(2)$ coupling $g = e / \sin \theta_W$ and the $W$ mass.
After the QED
radiative corrections are taken into account by substituting for $\alpha$
the running coupling $\alpha \left(m^2_Z \right) = \alpha / \left( 1 -
\Delta \alpha \right) = 1/128.896$ ..., the top quark mass is
introduced through
the genuine electroweak corrections, essentially the $\rho$ parameter of
the ($tb$) loop in the $W$ propagator (see also Sect. 1.1.1). 
The final relation can be cast
into the form
\beq \label{sintheta}
   \sin^2 \theta_W \cos^2 \theta_W m^2_Z = \frac{\pi \alpha}
   {\sqrt{2} G_F \left[ 1 - \Delta \alpha + 3 G_F m^2_t
   \cot^2 \theta_W / 16 \pi^2 \right] } \quad 
\eeq
Given the high precision with which $m_Z$ and $\sin^2 \theta_W$ 
have been measured, the mass of the top quark can be derived from
eq.(\ref{sintheta}). Including small corrections from loops involving
the Higgs particle, one finds \cite{cern96-183} 
\beq
   m_t = 177^{+7 +17}_{-8 -19}
%   \begin{array}{c} +7 \\ -8 \end{array}
%   \begin{array}{c} +17 \\ -19 \end{array}
    \mbox{ GeV}
\eeq
The second error results from the variation of the Higgs mass between
60 and 1000 GeV with a central value of 300 GeV. The same
analysis leads to  an upper limit of 181~GeV (95\% CL).
The present data are summarized in Fig.\ref{FLEPData} 
in the $m_t$, $m_H$ plane. The predictions derived from the 
radiative corrections and the high--precision electroweak data
are compared with an overall fit including the direct 
measurement of the top mass at the Tevatron. 
These analyses, in particular the $\rho$ value, are quite stable when
possible extensions of the Standard Model are taken into account. 

With the enhanced precision for $M_W$  and $m_t$ expected from the
next generation of experiments, it may well be possible to determine
the mass of the Higgs boson by the same method with a precision around
20 \%. 

\begin{figure}[hbt]
   \vspace{13cm}
\caption[]{\label{FLEPData}{\it The 68\% confidence level contours in $m_t$
and $m_H$ for the fits to LEP data only (dashed curve) and to all data 
including the CDF/D0 $m_t$ measurement (solid curve); Ref.~\cite{rclare}.}}
\end{figure}

\vspace*{5mm}

Present theoretical analyses of the \sm\ are based
almost exclusively on perturbation theory. Since the $t$ mass is still
sufficiently small, this method can also be applied  the
top-quark sector.
 The following consistency conditions are fulfilled: 

\vspace*{3mm}
\noindent
\underline{Yukawa coupling $g_Y \left( ttH \right)$}
\nopagebreak
\vspace{-\parskip}
\vspace{2mm}

\nopagebreak
\noindent
In the \sm\ the Yukawa coupling is related to the top mass by
\beq
   g_Y \left( ttH \right) = m_t \sqrt{\sqrt{2} G_F}
\eeq
\noindent
For a top mass of 175 GeV the coupling $g^2_Y / 4 \pi \approx 0.04$ is
comfortably small so that the perturbation theory can safely be applied in 
the top--Higgs sector.
 
\vspace*{3mm}
\noindent
\underline{Unitarity bound} \\[0.2cm]
At asymptotic energies the amplitude of the zeroth partial wave for elastic
\ttb\ scattering in the color singlet same-helicity channel \cite{chano1}
\begin{eqnarray}
   a_0 \left(\ttb \ra \ttb \right) & = & - \frac{3 g^2_Y}{8 \pi} \\
                                   & = & - \frac{3 G_F m^2_t}{4 \sqrt{2} \pi}
                                         \nonumber
\end{eqnarray}
grows quadratically with the top mass. Unitarity, however,
demands this real
amplitude to be bounded by $\left| \mbox{Re} \, a_0 \right| \le 1/2$. With
$|a_0| \simeq 0.06$, this condition is met for a top mass of 175 GeV.
 
\newpage

%\vspace*{3mm}
\noindent
\underline{Stability of the Higgs system: top-Higgs bound} \\[0.2cm]
The quartic coupling
$\lambda$
in the effective Higgs potential
\[
V = \mu^2 \left| \phi \right|^2 + \frac{\lambda}{2} \left| \phi \right|^4
\]
depends on the scale at which the system is interacting. The running of
$\lambda$
is induced by higher-order loops built-up by the Higgs particles themselves,
the vector bosons and the fermions in the \sm\ \cite{cabib1,lindn1}.
For small values of the top mass,
$m_t \le 77$ GeV,
these radiative
corrections generate a lower bound on the Higgs mass
$\left[ < 7 \, \mbox{GeV} \right]$.
At high energies they make
$\lambda$ rise,  
\beq
   \frac{\partial \lambda}{\partial \log s} = \frac{3}{8 \pi^2} \left[
      \lambda^2 - 4g^4_{Y} + \mbox{gauge couplings} \right]
\eeq
\noindent
up to the Landau pole at the cut-off parameter
$\Lambda$
beyond which the \sm\ in the present formulation cannot be continued
[''triviality bound'', as this bound could formally be misinterpreted
as requiring the low energy
coupling to vanish ]. If for a fixed Higgs mass the top mass is increased,
the top--loop radiative corrections lead to negative values of the quartic
coupling $\lambda$.
Since the potential is unbounded from below in this case, the Higgs
system becomes instable. Thus the stability requirement defines an upper
value of the top mass
$m_t$
for a given Higgs mass
$m_H$
and a cut-off scale
$\Lambda$.
The result of such an analysis is presented
in Fig.\ref{FLindner}. If the fundamental  particles remain
weakly interacting up to the Planck scale, a top mass of 175 GeV restricts
the range of possible Higgs mass to a narrow gap between 130 and 180 GeV.


\begin{figure}[hbt]\label{FLindner}
      \vspace{10cm}
      \caption[]{\it Bounds on the Higgs and top masses
      following from triviality of the Higgs's quartic self-coupling and the
      stability of the Higgs system; from \cite{lindn1}.}
\end{figure}

















