\chapter{The Profile of the Top Quark}
Hadron collider experiments at the TEVATRON have firmly established
the existence of the top quark and already provide a fairly accurate
determination of its mass.  The couplings of the top quark to the
gauge bosons are uniquely fixed by the SM. Thus all its properties
--- its production cross section and its decay rate and distributions
--- can be predicted unambiguously.  

The study of real top quarks at high energy colliders, in particular
the observation of a peak in the invariant mass of its decay products,
is certainly the most impressive proof of existence.  Nevertheless, the
indirect evidence for a top quark and the determination of its mass is
not only of historical interest.  The arguments which anticipated the
existence of the top quark and its mass around 180 GeV illustrate the
rigid structure of the SM, its selfconsistency and beauty.  They will be
presented in section \ref{indir}. 

These theoretical arguments have inspired the experimental searches.
The upper limit on the top mass around 200 GeV
deduced already relatively early from electroweak
precision studies has provided encouragement  that energies of
present colliders were suited to complete this enterprise.  The
agreement between the most recent indirect mass determinations both
through radiative corrections and through direct observation
strengthens  the present belief into the quantum field aspect of the
theory.  It furthermore justifies the corresponding line of reasoning
concerning the search for the last remaining ingredient of the SM, the
Higgs boson.  Section \ref{indir} of this first chapter will, with this
motivation in mind, be devoted to a discussion of the indirect
information on the top
quark, its existence and its mass.  Top decays, including aspects of
radiative corrections, polarisation effects and decays induced by
physics beyond the SM will be covered in section \ref{decays}.

\section{Indirect information}
\label{indir}
\subsection{Indirect evidence for the top quark}
Several experimental results already prior to its discovery did
provide 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
\[ \!
   { \left[ {\;    u \;    } \atop d \right]_L }
     \quad { u_R \atop d_R } \qquad
   { \left[ {\, \, c \, \, } \atop s \right]_L }
     \quad { c_R \atop s_R } \qquad
   \,
   { \left[ {\, \, t \, \, } \atop b \right]_L }
     \quad { t_R \atop b_R }
\]
does include the top quark, imprinting the same multiplet structure on
the third family as the first two families. The evidence is
based on theoretical selfconsistency (absence of anomalies),  the
absence of flavour changing neutral currents (FCNC) and
measurements of the weak isospin of the $b$ quark which has 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 argument for the existence of top quarks follows
from a theoretical consistency requirement. 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*{5mm}
\psfig{figure=triangle.ps,width=50mm,bbllx=210pt,bblly=490pt,%
bburx=630pt,bbury=550pt}  
\hspace*{-5mm}
   & & \\
   \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
\[
   { \left[ c \atop s^{'} \right]_L } \quad { \atop b^{'}_L }
   \hspace{2cm} {
     \! s^{'}_L = \: s_L \cos \vartheta^{'} + b_L \sin \vartheta^{'} \atop
     \: b_L^{'} =   -s_L \sin \vartheta^{'} + b_L \cos \vartheta^{'} }
\]
generates non-diagonal neutral current couplings, i.e. the breaking of the
GIM mechanism



\begin{eqnarray*}
   <I_3> & = & + \frac{1}{2} \left( \bar c_L,c_L \right)
               - \frac{1}{2} \left( \bar s^{'}_L, s^{'}_L \right) \\
         & = & {1\over 2}\left( \bar c_L,c_L \right)-
-{1\over 2}\left( \bar s_L,s_L \right) \cos^2\vartheta^\prime
-{1\over 2}\left( \bar b_L,b_L \right) \sin^2\vartheta^\prime
\\
&& - \frac{1}{2} \sin \vartheta^{'} \cos \vartheta^{'}
              \left( \left( \bar s_L, b_L \right)
                   + \left( \bar b_L, s_L \right)\right).
\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.
The relative strenth of neutral versus charged current induced rate is
essentially given by
\be
{\Gamma_{\rm NC} \over \Gamma_{\rm CC} }\sim
{1\over 2} \left({M_W^2\over M_Z^2}\right)^2 
{(v_b^2+a_b^2)(v_e^2+a_e^2)\over (1+1) (1+1)}
\sim 0.06.
\ee
Taking the proper momentum dependence of the matrix element and the
phase space into account one finds \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  by
the UA1 Collaboration \cite{albaj3,PDG}
\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} }{0.103 \pm 0.005 }.
\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 $Z$ boson couples to quarks through vector and axial--vector
charges with the well--known strength
 
{\hangindent=4cm \hangafter=-5
%****************************************
\[
\hspace*{-20mm}
\psfig{figure=Zqq.ps,width=35mm,bbllx=210pt,bblly=410pt,%
bburx=630pt,bbury=550pt}  
=\sqrt{\frac{G_F m_Z^2}{2\sqrt{2}}}\gamma_\mu \left[v_q - a_q \gamma_5
\right].
\]
%****************************************
}
 
 
\noindent Depending on the isospin assignment of righthanded and
lefthanded quark fields these charges are defined as
%*****************************************
\be
v_q&=&2 (I_q^{3L}+I_q^{3R})-4e_q\swin \\
a_q&=&2(I_q^{3L} -I_q^{3R})
\ee
%*****************************************
 
For the present application the {\em Born approximation} in the
massless limit provides an adequate
representation of the partial $Z$ decay rate
%****************************************
\begin{equation}
\label{eqrate}
\Gamma_B(Z \rightarrow b \overline{b}) \approx
 \frac{G_F m_Z^3}{8\sqrt{2}\pi}\beta
\left( v_b^2 + a_b^2 \right).
\end{equation}
%****************************************
In the Standard Model $2I_q^{3R}=0$ and $2I_q^{3L}=\pm 1$ for up/down
quarks respectively. 


 
The ratio between the predictions in the context of a topless model
and the SM amounts to 
\be
{\Gamma^{\rm topless}\over \Gamma^{\rm SM}} 
= { (4Q_b \sin^2\theta_W)^2 \over (1+4Q_b \sin^2\theta_W)^2 + 1}
\approx 1/13
\ee
whereas theory and  LEP experiments are well consistent
\be
{ \Gamma(Z\to b\bar b) \over \Gamma(Z\to {\rm had})} =
\left\{
\begin{array}{lll}
0.2155\pm 0.0004 & {\rm theory} & \cite{HollikBrus} \\
0.2219\pm 0.0017 & {\rm experiment} & \cite{HollikBrus}
\end{array}
\right.
\ee
ruling out the $I=0$ assignement for the $b$-quark.
The forward-backward asymmetry at the $Z$ resonance 
\be
A^{\rm FB} = {3\over 4} A_eA_b
\ee
with 
\be 
A_f \equiv {2v_f a_f \over v_f^2 + a_f^2}
\ee
is sensitive toward the relative size of vector and axial $b$ quark
couplings. Up to a sign, the first of these factors, $A_e\approx 0.15$,
can be interpreted as the degree of longitudinal $Z$ polarisation,
$P_Z= - A_e$, which is induced by the electron coupling even for
unpolarized beams. For longitudinally polarized beams it can be
replaced by unity.  The second factor represents essentially the
analyzing power of $b$ quarks. With a predicted value of 0.93 it is
close to its maximum in the SM.
In fact, this high analysing power is the reason for the large
sensitivity of $A^{FB}$ toward $\sin^2\theta_W$ \cite{djoua1}.
  For a fictitious topless model $A_b$ is
zero. The most recent experimental results from LEP and SLC are
displayed in Fig.~\ref{fig:sudong}.  
\begin{figure}
\psfig{figure=sudong.ps,width=150mm,bbllx=-40pt,bblly=50pt,%
bburx=590pt,bbury=600pt}  
\caption{Experimental results for the asymmetry parameter $A_b$
\protect\cite{SuDong}.} 
\label{fig:sudong}
\end{figure}

A remaining sign ambiguity is finally resolved by the interference
between NC and electromagnetic amplitude.  It leads to a forward
backward asymmetry at low energies
\be
A_{\rm FB} = -{3G_F s\over 16\sqrt{2} \pi\alpha} {a_e a_b\over Q_eQ_b}
\ee
which has been studied in particular at PEP, PETRA, and most recently
with highest precision at Tristan at a cm energy of 58 GeV \cite{Trist}.   
 Using the data available
shortly after the turnon of LEP and  combining $\Gamma_{b\bar
b}$,  $A_b$, and $A_{\rm FB}$
\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}
has been obtained already some time ago \cite{SchailZ}.
As shown in Fig.~\ref{FSchailZ} all
measurements are nicely consistent with the predictions  of the 
SM\footnote{For a
discussion of the most recent results for $R_b$, 
however, see \protect\cite{HollikBrus}.}.
The isospin assignment of the
Standard Model is thus well confirmed.

\begin{figure}[hbt]

\vspace{11.5cm}

\caption[]{\label{FSchailZ}%
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}

\newpage

\subsection{Mass limits and indirect mass determinations}

\subsubsection{Theoretical constraints}

Present theoretical analyses of the \sm\ are based
almost exclusively on 
perturbation theory. If this method is assumed to apply also to the
top-quark sector, in particular when linked to the Higgs sector,
the top mass must be bounded as the strength of the Yukawa-Higgs-top
coupling is determined by this parameter. The following consistency
conditions must be met:

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

\nopagebreak
Defining the Yukawa coupling in the  \sm\  through
\be
 {\cal L}_Y = g_Y \left({v+H\over \sqrt{2}}\right)
\left( \bar t_L t_R+ \mbox{h.c.} \right)
\ee
the coupling constant $g_Y$
is related to the top mass by
\beq
   g_Y \left( ttH \right) = m_t \sqrt{2\sqrt{2} G_F}.
\eeq
Demanding the effective expansion parameter $g^2_Y/4 \pi$
to be smaller than 1, the top mass is bounded to
\beq
   m_t < \sqrt{ \frac{\sqrt{2}\pi}{ G_F} } \approx 620 \mbox{GeV}.
\eeq
For a top mass of 180 GeV the coupling $g^2_Y / 4 \pi \approx 0.085$ is
comfortably small so that perturbation theory can safely be applied in this
region.


\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 $\left| \mbox{Re} \, a_0 \right| \le 1/2$.
This condition translates to
\beq
   m_t < \sqrt{\frac{2 \sqrt{2} \pi}{3 G_F}} \approx 500 \, \mbox{GeV}.
\eeq
The bound improves by taking into account the running of the
Yukawa coupling \cite{cabib1}.
These arguments are equally applicable for any additional species of
chiral fermions with mass induced via spontaneous symmetry breaking. 

\newpage
\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 moderate values of the top mass,
$m_t \le 77$ GeV,
these radiative
corrections would have generated a lower bound on the Higgs mass
of 7 GeV.
With the present experimental lower limit $m_H> 65$ GeV and 
the top quark mass determined around 180 GeV this bound is of no practical
relevance any more.
At high energies the radiative corrections make
$\lambda$
rise 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
$\lambda$
\beq
   \frac{\partial \lambda}{\partial \log s} = \frac{3}{8 \pi^2} \left[
      \lambda^2 - 4g^4_{Y} + \mbox{gauge couplings} \right]
\eeq
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}. Depending on the cut-off scale $\Lambda$
where new physics may set in, 
\begin{figure}[hbt]
      \vspace{10cm}
      \caption[]{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}.}
\label{FLindner}
\end{figure}
the top mass is bounded to $m_t \le 200$ GeV
if $\Lambda$
exceeds the Planck scale but it rises up to 400 to 500 GeV if the
cut-off is reached at a level of 1 TeV and below. The estimates are similar
to the unitarity analysis in the preceeding
subsection. Lattice simulations of the Yukawa model have arrived at
qualitatively similar results (see e.g. \cite{Munster94}).

These theoretical analyses have shown that for the top
mass around 180 GeV
the \sm\ may be valid up to a cut-off at the Planck scale. [The hierarchy
problem, that is not touched in the present discussion,
may enforce nevertheless new additional physical phenomena already
in the TeV range.]

In the context of the SM the top Yukawa coupling is simply present as
a free parameter. In the minimal supersymmetric model, however, the
picture is changed completely.  A large Yukawa coupling may play the
role of a driving term for the spontaneous breaking of $SU(2)\times
U(1)$, as discussed in \cite{HallThis} and in fact the large mass of the
top quark had been predicted on the basis of these arguments prior to
its experimental discovery.

\subsubsection{Mass estimates from radiative corrections}
%
First indications of a 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, not 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 is much more advanced [2-7].
%\cite{bardin,HollikBrus,Swartz,Kniehl,Sirlin,degrassi}
Since Higgs mass effects are
weak as a result of the screening theorem \cite{veltm1},
the top mass is the dominant
unknown  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, from the forward-backward
asymmetry and  from
LR polarization measurements, the top quark mass 
has been
determined up to a residual uncertainty of less than 10 GeV
plus an additional uncertainty of  about 20 GeV
induced through the variation of $m_H$
\be
 m_t= 178\pm 8 ^{+17}_{-20}\; {\rm GeV}. 
\ee

The close agreement between direct and indirect top mass determination
can be considered a triumph of the Standard Model. Its predictions are
not only valid in Born approximation, as expected for any effective
theory, also quantum corrections play an important role, and are
indirectly confirmed.  Encouraged by this success and in view of the
improved accuracy of theory and experiment it is conceivable that the
same strategy can lead to a rough determination of the Higgs mass, or,
at least, to a phenomenologically relevant upper limit.


\subsection{The quadratic top mass dependence of $\delta\rho$}
The quadratic top mass dependence of $\delta\rho$ is a cornerstone of
the present 
precision measurements \cite{veltm2}.  
In view of its importance and the pedagogical
character of these lectures it is perhaps worthwhile to present a
fairly pedestrian derivation of this result.  

Let us first consider the definition of the weak mixing angle in Born
approximation.  It can be fixed through the relative strength of
charged vs.~neutral current couplings:
\be
\hspace*{10mm}
\psfig{figure=Wbb.ps,width=35mm,bbllx=210pt,bblly=410pt,%
bburx=630pt,bbury=550pt}  
& =&  {g\over 2\sqrt{2}} \gamma_\mu (1-\gamma_5)
\\[8mm]
\hspace*{10mm}
\psfig{figure=Ztb.ps,width=35mm,bbllx=210pt,bblly=410pt,%
bburx=630pt,bbury=550pt}  
& =&  {g\over 4\cos\bar \theta_W} 
\left[\left(2I_3^f-4Q_f\sin^2\bar\theta_W\right)\gamma_\mu 
+2I_3^f\gamma_\mu \gamma_5
\right]
\\[4mm]\nonumber
\ee
with the SU(2) coupling related to the electromagnetic coupling
through $g=e/\sin\bar \theta_W$.  Alternatively $\sin^2\theta_W$ is
defined through the mass ratio
\be
M_W^2/M_Z^2 = 1 - \sin^2\theta_W
\ee
These two definitions coincide 
in Born approximation 
\be
\theta_W = \left. \bar \theta_W \right|_{\rm Born}
\ee
However, the self energy diagrams depicted 
in Fig.~\ref{fig:self} lead to marked differences between the two
options, in particular if $m_t \gg M_{W,Z} \gg m_b$. 
\begin{figure}[htb]
\hspace*{60mm}
\begin{tabular}{c}
\psfig{figure=Wplus.ps,width=50mm,bbllx=210pt,bblly=410pt,%
bburx=630pt,bbury=550pt}
\end{tabular}\\[10mm]
\hspace*{35mm}
\begin{tabular}{cc}
\psfig{figure=W0t.ps,width=50mm,bbllx=210pt,bblly=410pt,%
bburx=630pt,bbury=550pt} 
& 
\psfig{figure=W0b.ps,width=50mm,bbllx=210pt,bblly=410pt,%
bburx=630pt,bbury=550pt}    
\end{tabular}\\[15mm]
\caption{Self energy diagrams which induce the mass splitting between
$W^\pm$ and $W^0$  and a $\rho$ parameter different from one.}
\label{fig:self}
\end{figure}


This difference can be traced to a difference in the mass shift for
the $W$ and the $Z$ boson.  For a simplified discussion consider, in a
{\em first step},  $\sin^2\theta_W=0$ and hence the $SU(2)$ part
of the theory only.  The neutral boson will be denoted by $W^0$. 
In the lowest order this implies
\be
M_- = M_0 \equiv M
\ee
and the couplings simplify to
\be
\psfig{figure=Wbb.ps,width=35mm,bbllx=210pt,bblly=410pt,%
bburx=630pt,bbury=550pt}  
& =&  {g\over 2\sqrt{2}} \gamma_\mu (1-\gamma_5)
\\[8mm]
\psfig{figure=Ztb.ps,width=35mm,bbllx=210pt,bblly=410pt,%
bburx=630pt,bbury=550pt}  
 & =&  {g\over 4}  2I_3^f \gamma_\mu (1-\gamma_5)
\\[8mm]\nonumber
\ee
In order $g^2$ the propagators of charged and neutral bosons  are
modified by the self energies
\be
W^+:\qquad\qquad 
{1\over M^2 - s} & \Rightarrow & {1 \over M^2 - s - \Pi_+(s)}
\nonumber \\
W^0: \qquad\qquad 
{1\over M^2 - s} & \Rightarrow & {1 \over M^2 - s - \Pi_0(s)}
\ee
The mass shifts individually are given by
\be
\delta M_{+,0}^2 = M_{+,0}^2 - M^2 = - Re\, \Pi_{+,0}(M^2).
\ee
They are most easily calculated through dispersion relations from
their respective imaginary parts. These can be interpreted as the
``decay rate'' of a fictitious virtual boson of mass $\kappa$:
\be
\kappa \Gamma(W_{+}^*\to t + \bar b) = Im\,\Pi_{+}(\kappa^2)
\nonumber\\
\delta M^2_{+} = {1\over \pi} \int {d\kappa^2
\over \kappa^2-M^2} Im\,\Pi_{+}(\kappa^2).
\label{stern}
\ee
The decay rates of the virtual bosons are easily calculated
($m\equiv m_{top}$)
\be
\kappa \Gamma(W_{+}^* \to t\bar b) = 
{3\over 12\pi} \left( g\over 2\sqrt{2}\right)^2
\kappa^2 2 \left(1-{m^2\over 2\kappa^2} -{m^4\over
2\kappa^4}\right) 
\left(1-{m^2\over \kappa^2}\right) 
\Theta(\kappa^2-m^2).
\label{eq:rates}
\ee
The factor 3 originates from color, the factor 2 from the identical
vector and axial contributions, the  squared
matrix element and the  phase space 
 are responsible for the second and third factors in
brackets respectively.

Similarly one finds
\be  
\lefteqn{
\kappa \Gamma(W_{0}^* \to b\bar b) +
\kappa \Gamma(W_{0}^* \to t\bar t) =
}
\nonumber \\
&&
{3\over 12\pi} \left( g\over 4\right)^2
\kappa^2
\left\{
2\Theta(\kappa^2) + 2
\left(1-{m^2\over \kappa^2}\right)
\left(1-{4m^2\over \kappa^2}\right)^{1/2}  
\Theta(\kappa^2-4m^2)\right\}
\nonumber\\
\lefteqn{
\delta M^2_0 =  {1\over \pi} \int {d\kappa^2
\over \kappa^2-M^2} Im\,\Pi_{0}(\kappa^2).
}
\label{eq:simil}
\ee

With the large $\kappa^2$ behaviour of ${\rm Im}\,\Pi$ given by ${\rm Im}\,\Pi\sim
\kappa^2$ the 
dispersive integral eq.~(\ref{stern}) is  evidently quadratically
divergent.  In the limit of large $\kappa$ the leading ($\sim
\kappa^2$) and next-to-leading ($\to$ const) terms of
eqs. (\ref{eq:rates}) and (\ref{eq:simil}) coincide.  The leading and
next to leading divergences can therefore be absorbed in  a $SU(2)$
invariant 
mass renormalization.  The relative mass shift, however, the only
quantity accessible to experiment, remains finite and is given by
\be
\delta M^2 &=& \delta M^2_+ - \delta M^2_0 
\nonumber\\
&=& {3\over 12 \pi} {g^2\over 8} {1\over \pi} 
  \int_0^\infty {d\kappa^2\over \kappa^2-M^2}\kappa^2
\nonumber\\
&& \times \left\{ 
\begin{array}{ll}
-2\left(1-{m^2\over \kappa^2}\right) 
   \left( 1 -{m^2\over 2\kappa^2} - {m^4\over 2 \kappa^4} \right)
&
\times \Theta(\kappa^2-m^2) \\
+1 & \times \Theta(\kappa^2) \\
\left(1-{m^2\over \kappa^2}\right) 
   \left( 1 -{4m^2\over \kappa^2} \right)^{1/2}
&
\times \Theta(\kappa^2-4m^2) 
\end{array}
\right.
\ee
We are only interested in the leading top mass dependence: $m^2 \gg
M^2$.  The leading term is obtained by simply setting $M^2\to 0$ in
the integrand.  Introducing a cutoff $\Lambda^2 \gg m^2$ the leading
contributions to the three integrals are given by 
\be
\left\{ \rule{-1mm}{5mm} \right.
&&-2\Lambda^2 + 3m^2 \ln{\Lambda^2\over m^2} + {3\over 2}m^2+ \ldots
\nonumber\\
&& +\Lambda^2
\nonumber\\
&& +\Lambda^2 -  3m^2 \ln{\Lambda^2\over m^2}
 \left.  \rule{-1mm}{5mm}
\right\}
\ee
and hence
\be
\delta M^2 = {3\over 12 \pi^2} {g^2\over 8} {3\over 2}m^2
\ee
Up to the proportionality constant this result could have been guessed
on dimensional grounds from the very beginning.

 It has become customary to express the $SU(2)$ coupling in terms of
$G_F$ and the $W$ mass
\be
{g^2\over 8} = {G_F\over \sqrt{2}} M_W^2
\ee
such that
\be
{\delta M^2 \over M^2} = {3\over 16 \pi^2} \sqrt{2}G_Fm^2.
\label{eq:deltaMM}
\ee

The ratio of neutral versus charged current induced amplitude at small
momentum transfers is thus corrected by a factor
\be
{M_+^2\over M_0^2}  \equiv 1 + \delta\rho 
=\left(1 + {\delta M^2\over M^2} \right)
\ee
with $\delta\rho$ given in eq. (\ref{eq:deltaMM}).


To discuss the phenomenological implications of this result it is now
necessary to reintroduce the weak mixing between the neutral $SU(2)$
and $U(1)$ gauge bosons.  The gauge boson masses are induced by the
squared covariant derivative acting on the Higgs field
\be
D_\mu \phi \to -i\left( g\vec W_\mu {\vec \tau\over 2} + g^\prime
BY\right) 
\left( \begin{array}{c} 0\\ v/\sqrt{2} \end{array} \right)
\ee
giving rise to the following mass terms in the Lagrangian
\be
{\cal L}_M &=& {v^2\over 8} (g^2+ g^{\prime 2} )
\left({g \over \sqrt{g^2 + g^{\prime 2} }} W_3
-{g^\prime \over \sqrt{g^2 + g^{\prime 2} }} B\right)^2 
\nonumber \\
&& + {v^2\over 8} g^2 (W_1^2 + W_2^2) 
(1+\delta\rho)
% + {v^2\over 8} g^2 \delta\rho (W_1^2 + W_2^2) 
\label{lagr}
\ee
The last term has been added to represent a contribution from a non
vanishing $\delta \rho$, induced e.g.~by the large top mass.  The
finite mass shift has been without loss of generality entirely
attributed to the charged $W$ boson.

The mass eigenstates are easily identified from eq.~(\ref{lagr})
\be
W^\pm &=& (W_1 \mp iW_2)/\sqrt{2}
\nonumber \\
Z &=& (\bar c W_3 - \bar s B)
\nonumber \\
A &=& (\bar c B + \bar s W_3)
\ee
with the weak mixing angle $\bar\theta_W$
\be
\bar c \equiv \cos \bar \theta_W 
       \equiv {g \over \sqrt{g^2 + g^{\prime 2} }} 
\nonumber \\
\bar s \equiv \sin \bar \theta_W
       \equiv {g^\prime \over \sqrt{g^2 + g^{\prime 2} }} 
\label{couplconst}
\ee
defined through the {\em couplings}.  This definition is, of
course, very convenient for measurements at LEP, where couplings are
determined most precisely.  The couplings of the photon and the $Z$
boson are thus also given in terms of $\bar s$.  The masses are read
off from eq.~(\ref{lagr})
\be
m_A^2 &=& 0
\nonumber\\
m_Z^2 &=& {v^2\over 4} (g^2 + g^{\prime2})
\nonumber\\
m_W^2 &=& {v^2\over 4} g^2(1+\delta\rho)
\ee
and 
\be
{m_W^2\over m_Z^2} = \bar c^2(1+\delta\rho)
\ee
which constitutes the standard definition of the $\rho$ parameter.
Alternatively one may define the mixing angle directly through the
mass ratio
\be
s^2 \equiv \sin^2\theta_W \equiv 1-M_W^2/M_Z^2
\label{massratio}
\ee
The two definitions coincide in the Born approximation; they differ,
however, for $\delta\rho \neq 0$:
\be
s^2 = 1-\bar c^2(1+\delta\rho) = \bar s^2 - \bar c^2\delta \rho
\ee
It is, of course, a matter of convention and convenience, which of the
two definitions (or their variants) are adopted.  The choice of input
parameters  and observables
will affect the sensitivity towards $\delta\rho$ --- and hence
towards $m_t$.  The observables which are measured with the highest
precision at present and in the forseable future are the
fine-structure constant $\alpha$, the muon life time which provides a
value for $G_F$ and the $Z$ boson mass.  To obtain the dependence of
$\sin^2\theta_W$ on $\delta\rho$ we predict $M_W^2$ from these
observables.  We start from
\be
{g^2\over 8} {1\over M_W^2} = {G_F\over \sqrt{2}}
\ee
and express $g^2$ through $G_F$ and $M_Z^2$ 
\be
g^2 = 4\sqrt{2} G_F M_Z^2 c^2
\ee
Alternatively $g$ can be related to the fine structure constant
\be
g^2 = e^2/\bar s^2
\ee
Note the appearance of $c^2$ in eq.~(\ref{massratio}) and of $\bar
s^2$ in eq.~(\ref{couplconst}).

One thus arrives at
\be
e^2 = 4\sqrt{2}  G_F^2 M_Z^2 c^2 (s^2 + c^2 \delta\rho)
\ee
or, equivalently, at
\be
{4\pi \alpha\over 4 \sqrt{2} G_F M_Z^2} {1\over 1+\cot^2\theta_W
\delta\rho} = s^2 c^2.
\ee
Solving for $\sin^2\theta_W$ (defined through the mass ratio) one
obtains on one hand
\be
\sin^2\theta_W \approx {1\over 2} \left[ 1 - \sqrt{1-{4\pi \alpha\over
\sqrt{2}G_F M_Z^2}}\; \right] -
{c^4\over c^2-s^2}\delta\rho
\label{eq:thet}
\ee
where the Born values for $c^2$ and $s^2$ can be taken in the
correction term. 
The definition of $\sin^2\bar \theta_W$ through the relative strength of
$SU(2)$ and $U(1)$ couplings leads on the other hand to
\be
\sin^2\bar\theta_W \approx {1\over 2} \left[ 1 - \sqrt{1-{4\pi \alpha\over
\sqrt{2}G_F M_Z^2}}\; \right] -
{s^2c^2\over c^2-s^2}\delta\rho
\label{eq:thetbar}
\ee

For the actual evaluation the running coupling $\alpha(M_Z)^{-1}
\approx 129$ must be employed \cite{Swartz}.  
Eq.~(\ref{eq:thet})  and eq.~(\ref{eq:thetbar})  exhibit rather
different sensitivity towards a 
variation of $\delta \rho$ and hence of $m_t$, with a ratio between the
two coefficients of $c^2/s^2 \approx 3.3$.  
 For a precise comparison between theory and
experiment subleading one-loop corrections must be included, and subtle
differences between variants of  $\bar\theta_W$ must be taken into
consideration, with  $\theta_{\overline{\rm MS}}$ and $\theta_{\rm eff}^{\rm
lept}$ as most frequently used options \cite{bardin}.  

With increasing experimental accuracy numerous improvements must be
and have been
incorporated into the theoretical predictions.

\begin{itemize}
\item
The full one loop corrections including all subleading terms are know
since long and are certainly indispensible (see
\cite{Swartz,bardin,avdeev,chetyr} and references therein).  
\item
 Two loop
purely weak corrections increase proportional $(G_Fm_t^2/16\pi^2)^2$.
A detailed discussion can be found in \cite{degrassi}.
\item
QCD corrections are available at the two- and even three-loop level 
\cite{Kniehl,avdeev,chetyr}.
\end{itemize}

With $\alpha$, $G_F$, and $M_Z$ fixed one may determine $m_t$ either
from $M_W$ or alternatively  from $\sin^2\bar\theta_W$ 
(corresponding to a measurement
of the left right asymmetry with polarised beams, the $\tau$
polarisation or forward backward asymmetries of unpolarised beams).
These measurements are in beautiful agreement with the determination
of $m_t$ in production experiments at the TEVATRON
(Fig.~\ref{fig:logo}).
\begin{figure}[htb]
\vspace*{10cm}
\caption{ }
\label{fig:logo}
\end{figure}


\section{Top Decays}
\label{decays}

Various aspects of top decays have been scrutinized in the literature.
The large top decay rate predicted in the SM governs top quark
physics.  Radiative correctons from QCD and electroweak interactions
have been calculated for the decay rate and for differential
distributions of the decay products.  Non-standard top decays are
predicted in SUSY extensions of the SM, with $t\to Hb$ and $t\to
\tilde t\tilde \gamma$ as most promising and characteristic
signatures.  Born predictions and radiative corrections (at least in
part) have been worked out also for these decay modes.  Beyond that a
number of even more exotic decay modes, in particular FCNC decays, have
been suggested.

\subsection{Qualitative aspects -- Born approximation}
\label{sect:qualit}
The decay of the top quark into $b+W$ is governed by the following
amplitude
\be
{\cal M}(t\to bW) = {ig\over \sqrt{2}} \bar b \not \varepsilon^W
{1-\gamma_5\over 2} t
\ee
Adopting the high energy limit ($m_t^2>M_W^2$) for the polarisation
vector $\epsilon_L$ of the longitudinal $W$ (corresponding to helicity
$h^W=0$)  
\be
\varepsilon^W_L = 
\left(
\begin{array}{c}
p_3^W\\
0\\
0\\
p_0^W
\end{array}
\right)
{1\over M_W}
=
\left(
\begin{array}{c}
p_0^W\\
0\\
0\\
p_3^W
\end{array}
\right)
{1\over M_W}
+
{\cal O}(M_W/m_t)
\ee
the amplitude is dominated by contribution from longitudinal $W$'s
\be
{\cal M}_L &=& {ig\over \sqrt{2}} \bar b \not \varepsilon_L^W
{1-\gamma_5\over 2} t 
\approx 
{ig\over \sqrt{2}} {m_t\over M_W} 
\bar b  {1+\gamma_5\over 2} t 
\nonumber \\
&=& i\sqrt{2} {m_t\over v} 
\bar b  (1+\gamma_5) t 
\ee
This  part is thus 
proportional to
the Yukawa coupling 
\be
g_Y = \sqrt{2} {m_t\over v} 
\ee
with a rate growing proportional $m_t^3$.  In contrast, 
the amplitude for the decay
into transverse $W$'s, is obtained with the polarisation vectors
\be
\varepsilon^\pm_T = 
{1\over \sqrt{2}}
\left(
\begin{array}{c}
0\\
1\\
\pm i\\
0\\
\end{array}
\right)
\ee
and remains constant in the high mass limit.
The rate is governed by the gauge coupling $g$ and increases only
linearly with $m_t$.
The longitudinal or transversal 
$W$ is produced in conjunction with a lefthanded $b$ quark.  The
production of $W$'s with helicity $h^W=+1$ is thus forbidden by angular
momentum conservation (see fig.~\ref{fig:TopDec}).  

\begin{figure}[h]
\begin{minipage}{16.cm}
\hspace*{1cm}
$
\mbox{
\hspace*{18mm}
\begin{tabular}{|cll|}
\hline
 & & $S^t = +1/2$  \\
\psfig{figure=TopDec1.ps,width=50mm,bbllx=0pt,bblly=300pt,%
bburx=630pt,bbury=420pt}
& $S_z^b = +1/2$ & $h^W = 0$
\\[2mm]
\hline
 & & $S^t = -1/2$  \\
\psfig{figure=TopDec2.ps,width=50mm,bbllx=0pt,bblly=300pt,%
bburx=630pt,bbury=420pt}
& $S_z^b = +1/2$ & $h^W = -1$
\\
\hline
 & & $S^t = \pm1/2$  \\
\psfig{figure=TopDec3.ps,width=50mm,bbllx=0pt,bblly=300pt,%
bburx=630pt,bbury=420pt}
& $S_z^b = +1/2$ & $h^W = +1$
\\
\hline
\end{tabular}
}
$
\end{minipage}
\caption{Top decays: angular momentum conservation}
\label{fig:TopDec}
\end{figure}
In total one finds 
\be
(h^W=-1) : (h^W=0) : (h^W=+1) = 1:{m_t^2\over 2 M_W^2}:0.
\ee
The implications for the angular distributions of the decay products
will be discussed below.  The decay rate
\be
\Gamma &=& {G_Fm_t^3\over 8 \sqrt{2}\pi} 
\left(1-{M_W^2\over m_t^2}\right)^2
\left(1+2{M_W^2\over m_t^2}\right)
\nonumber \\
&\approx& 175\, \mbox{MeV} 
\left({m_t\over M_W}\right)^3
\ee
increases with the third power of the quark mass and, for a
{\em realistic} top mass around 180 GeV amounts to more than 1.5 GeV,
exceeding significantly all hadronic scales.  Before we discuss the
implications of this fact let us briefly pursue the close similarity
between the coupling of the longitudinal $W$ to the $tb$ system and
the decay into a charged Higgs boson in a two Higgs doublet model
(THDM).  The decay rate is given by (see also section \ref{nonstan}) 
\be
\Gamma(t\to H^+b)= {G_Fm_t^3\over 8 \sqrt{2}\pi} 
\left(1-{m_H^2\over m_t^2}\right)^2
\left[
\left({m_b\over m_t}\right)^2\tan^2\beta + \cot^2\beta
\right]
\ee
The similarity between this rate and the rate for the decay into
longitudinal $W$'s is manifest  from the cubic top mass
dependence.  The minimal value of the term in brackets is assumed for
$\tan\beta = \sqrt{m_t/m_b}$.  Adopting $m_b(\mbox{running})\approx 3$
GeV, the minimal value of the last factor amounts to about $1/30$. On
the other hand, in any plausible THDM  the value of 
$\tan\beta$ should not exceed
$m_t/m_b$.  The $W$ decay mode will hence never be swamped by the
Higgs channel. (This fact is of course also implied by the actual
observation of the top quark at the TEVATRON.)  Up to this point we
have, tacitely, assumed the CKM matrix element $V_{tb}$ to be close to
one.  In fact, in the three generation SM one predicts (90\% CL)
\begin{eqnarray}
V_{tb} = 0.9990 \pm 0.0004 & \Rightarrow & BR(b) \approx 1\nonumber\\
V_{ts} = 0.044 \pm 0.010 & \Rightarrow & BR(s) \approx0.2\%\nonumber\\
V_{td} = 0.011 \pm 0.009 & \Rightarrow & BR(d) \approx 0.01\%
\label{eq:2}
\end{eqnarray}
on the basis of CKM unitarity.  In a four generation model, however,
sizeable mixing between third and fourth generation could arise.
Methods to determine the strength of $V_{tb}$ either through single
top production at a hadron collider or through a direct measurement of
$\Gamma_t$ in $e^+e^-$ colliders 
will be discussed below in sects.~\ref{singletop} and 3.2.

The large top decay rate provides a cutoff for the long distance QCD
dynamics.  The implications can be summarised in the statement: ``$t$
quarks are produced and decay like free quarks'' \cite{acta}. In
particular the angular distributions of their decay products follow
the spin 1/2 predictions.  This is in marked contrast to the situation
for $b$ quarks, with $B$ mesons decaying isotropically.  The arguments
for this  claim are either based on a comparison of energy scales, or,
alternatively, on a comparison of the relevant time scales.

  Let us
start with the first of these two equivalent viewpoints:
The mass difference between $B^{**}$ and $B$ mesons amounts to 450
MeV.  In the nonrelativistic quark model the $B^{**}$ is interpreted
as orbitally excited $b\bar q$ state.  With increasing mass of the
heavy quark this splitting remains approximately constant: it is
essentially governed by light quark dynamics. The hyperfine splitting
between $B^*$ and $B$, in contrast, is proportional to the color
magnetic moment and hence decreases $\sim 1/m_Q$.  Given a decay rate
of about 1.5 GeV it is clear that $T$-, $T^*$-, and $T^{**}$- mesons
merge and act coherently, rendering any distinction between individual
mesons meaningless.  In fact even individual toponium states cease to
exist.  From the perturbative QCD potential an energy difference
between $1S$ and $2S$ states around 1.2--1.5 GeV is predicted.  This
has to be contrasted with the toponium decay rate $\Gamma_{t\bar
t}\sim 2\Gamma_t \sim 3$ GeV.  All resonances merge and result in an
excitation curve which will be discussed in chapter 3.

A similar line of reasoning is based on the comparison between
different characteristic time scales:  The formation time of a hadron
from a locally produced $t$ quark is governed by its size which is
significantly larger than its lifetime
\be
\tau_{\mbox{\small Formation}} \approx \mbox{size} \approx 1/0.5 \mbox{GeV}
\gg \tau_{\mbox{\small Decay}} \approx 1/\Gamma_t
\ee
Top quarks decay before they have time to communicate hadronically
with light quarks and dilute their spin orientation.
For sufficiently rapid top quark decay even $t\bar t$ bound states
cease to exist.  The classical time of revolution $T_{\rm rev}$ 
for a Coulombic
bound state is given by $\left(\alpha_{\rm eff} = {4\over
3}\alpha_s\right)$
\be
T_{\rm rev} = \pi \alpha_{\rm eff} \sqrt{{m_{\rm red}\over 2|E|^3}}
\ee
With $E=-\alpha_{eff}^2 m_{\rm red}/2$ for the ground state
\be
T_{\rm rev} = \pi/|E| \gg 1/\Gamma_{t\bar t}
\ee
is obtained. The lifetime of the $t\bar t$ system is too small to
allow for the proper definition of a bound state with sharp binding
energy. 


\subsection{Radiative corrections to the rate}

Perturbative corrections to the lowest-order result affect the total decay
rate as well as differential distributions.
Their inclusion is a necessary
prerequisite for any analysis that attempts
a precision analysis of top decays.
Both QCD and electroweak corrections are well under control and will be
discussed in the following.

\subsubsection{QCD corrections}

The correction to the {\em decay rate} is usually written in the form
\begin{equation}
\Gamma = \Gamma _{\rm Born}\left(1+\delta _{QCD} \right)
 =\Gamma _{\rm Born} 
\left( 1 - \frac{2}{3} \frac{\alpha _s}{\pi} f \right)
\label{eq:20}
\end{equation}
The correction function $f$ has been calculated in \cite{jk3}
for nonvanishing and vanishing
$b$ mass.
 In the limit $m^2_b /
m^2_t \to
0$ the result simplifies considerably, but remains a valid approximation
(Fig.\ref{fig:2.3}):
\begin{eqnarray}
f & = & {\cal F}_1 / {\cal F}_0 \nonumber \\
{\cal F}_0 & = & 2(1-y)^2 (1+2y) \nonumber \\
{\cal F}_1 & = & {\cal F}_0 \left[ \pi ^2 + 2 Li_2 (y) - 2Li_2 (1-y)\right] \nonumber \\
 & & + 4y(1-y-2y^2) \ln y + 2(1-y)^2 (5+4y) \ln(1-y) \nonumber \\
 & & - (1-y)(5+9y-6y^2),
\label{eq:21}
\end{eqnarray}
where $y=m^2_W / m^2_t$.
  In the limit $m^2_t \gg m^2_W$ $f$ is well approximated by
$f(y) = 2/3 \pi ^2 - 5/2 -3y +9y^2/2-3y^2\ln y \approx 4$.
For $m_t \approx 180~GeV$
the QCD correction amounts to
\begin{equation}
\delta _{QCD} \approx -3.7 \alpha _s / \pi
\label{eq:22}
\end{equation}
and lowers the decay rate by about 10\%.  This has a non-negligible impact
on the height and width of a toponium resonance or its remnant.

\begin{figure}

\psfig{figure=fig27y2.ps,width=180mm,bbllx=40pt,bblly=230pt,%
bburx=670pt,bbury=580pt}

\caption{\label{fig:2.3}QCD correction function $f$ for the top quark
decay rate.  Solid line:  exact form; dashes:  $m_b = 0$, dotted:
approximate form for large $m_t/M_W$.}
\end{figure}



The $\alpha_s^2$ corrections are presently unknown, and the scale
$\mu$ in $\alpha_s(\mu^2)$ is uncertain.  Indications for a
surprisingly large correction of order $\alpha_s^2$, corresponding to
a rather small scale have been obtained recently.  Diagrams with
light fermion insertions into the gluon propagator have been
calculated numerically \cite{smith} and analytically \cite{ac95a} in
the limit $m_t\gg M_W$
\be
\Gamma &=& \Gamma_{\rm Born} \left[
1-{2\over 3} {\alpha_{\overline{\rm MS}}(m^2_t)\over \pi}
\left(4\zeta_2-{5\over 2}\right)
\right.
\nonumber\\
&&+
\left.
\left( {\alpha_{\overline{\rm MS}}\over \pi} \right)^2 
\left( -{2n_f\over 3} \right)
\left( {4\over 9} -{23\over 18}\zeta_2 - \zeta_3\right)
\right]
\ee
The BLM prescription \cite{BLM} suggests that the dominant
$\alpha_s^2$ coefficients can be estimated through the replacement
\be
-{2n_f\over 3} \to \left( 11 -{2n_f\over 3}\right)
\ee
and absorbed through a change in the scale.  For the problem at hand
this corresponds to a scale $\mu = 0.122m_t$ resulting in a fairly
large effective value of $\alpha_s$  of $0.15$  instead of
0.11 for $\mu = m_t$. 

% OLD TEXT
\subsubsection{Electroweak corrections}

Electroweak corrections to the top quark decay rate can be found in
\cite{den1,eil}. They involve a large number of diagrams.
For asymptotically  large top masses the Higgs exchange diagram
 provides the dominant contribution.
Defining the Born term by means of the Fermi coupling $G_F$,
one derives in this limit
\begin{eqnarray}
\Gamma & = & \Gamma(G_F)_{\rm Born}\left[ 1 + \delta_{EW} \right], \\
 \delta_{EW} &=& \frac{G_F m_t^2}{4\sqrt2\pi^2} \left[ \frac{17}{4} + \log
\frac{m_H^2}{m_t^2}  \right] + \mbox{subleading terms}. \nonumber
\end{eqnarray}
While the Higgs--top coupling is the origin of the strong quadratic dependence
 on the
top mass, the Higgs itself is logarithmically screened in this limit.
However, the detailed
analysis reveals that the subleading terms are as important as the leading
terms so that finally one observes only a very weak
dependence of $\delta_{EW}$
on the top and the Higgs masses, Fig.\ref{FDenner}. The numerical value
of the corrections turns out to be small, $\delta_{EW} \approx +2 \%$.
Electroweak corrections in the context of the two Higgs doublet model
can be found in \cite{DennerHoang} and are of comparable magnitude.
\begin{figure}[hbt]

\vspace{9cm}

\caption[]{\label{FDenner}QCD and electroweak radiative
corrections to the top decay width;
adapted from Ref.\cite{den1}.}
\end{figure}

The positive correction $\delta_{EW}$ is nearly cancelled by the
negative correction $\delta_\Gamma$ of -1.5\% from the nonvanishing
finite width of the $W$.  The complete prediction  taken from
\cite{JezK} is displayed in table~\ref{compilation} for the choice
$\alpha_s(\mu^2=m_t^2)$. 
\begin{table}[h]
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$m_t$ [GeV] & $\alpha_s(m_t)$ & $\Gamma^{\rm Born}_{\rm nar.\,w.}$ [GeV]&
$\delta^{(0)}_\Gamma$ [\%] & 
$\delta^{\rm nw}_{\rm QCD}$ [\%] & 
$\delta_{\rm EW}$ [\%] & 
$\Gamma$ [GeV] \\
\hline
170 & .108 & 1.41 & -1.52 & -8.34 & 1.67 & 1.29 \\
\hline
180 & .107 & 1.71 & -1.45 & -8.35 & 1.70 & 1.57 \\
\hline
190 & .106 & 2.06 & -1.39 & -8.36 & 1.73 & 1.89 \\
\hline
\end{tabular}
\caption{Top width as a function of top mass and the comparison of the
different approximations. }
\label{compilation}
\end{table}
For $\mu = 0.112 m_t$ the QCD correction amounts to -11.6 \% instead
of -8.3\%. This variation characterises the present theoretical
uncertainty, which could be removed by a full $\alpha_s^2$ calculation
only.  Additional uncertainties, e.g. from the input value of $\alpha_s$
($\sim 1\%$) or from the fundamental uncertainty in the relation
between the pole mass $m_t$ and the experimentally measured excitation
curve (assuming perfect data) of perhaps 0.5 GeV can be neglected in the
forseable future. 

Hence, it appears that the top quark width (and similarly the spectra
to be discussed below) are
well under theoretical control, including QCD and electroweak corrections.
The remaining uncertainties are clearly smaller than the experimental error
in $\Gamma_t$, which will amount to 5-10\% even at a linear collider
\cite{Fujii}. 

\subsection{Decay spectra and angular distributions}

\subsubsection{Born predictions}
Arising from a two body decay, the energy of the $W$ and of the hadronic
system ($\equiv b$ jet) are fixed to
\begin{eqnarray}
E_W & = & \frac{m^2_t + m^2_W - m^2_b}{2m_t} \nonumber\\
E_h & = & \frac{m^2_t + m^2_b - m^2_W}{2m_t}
\label{eq:5}
\end{eqnarray}
as long as gluon radiation is ignored.  The smearing of this $\delta$
spike 
by the combined effects of perturbative QCD and from the finite width
of the $W$ will be treated below.

Top quarks will in general be polarized through their electroweak
production mechanism.  For unpolarized beams and close to threshold
their polarization is given by the right/left asymmetry which would
be measured with longitudinally polarized beams \cite{stre}:
\begin{equation}
P_t = A_{RL}
\label{eq:11}
\end{equation}

For fully longitudinally polarized electron (and unpolarized positron)
beams the spin of both $t$ and $\overline{t}$ is aligned with the spin
of the $e^-$.  Quark polarization then leads to angular distributions of
the decay products which allow for various tests of the chirality of the
$tbW$ vertex.

The angular distribution of the longitudinal
and transverse W's is analogous
to those of $\rho$ mesons from $\tau$ decay $(m_{\tau} \to m_t,
m_{\rho} \to m_W)$
\begin{eqnarray}
\frac{dN_{T/L}}{d \cos\theta} = \frac{1}{2}
(1 \mp P_t \cos\theta )\quad  f\!or \quad
h_W = \left\{ {+1 \choose 0 } \right.
\label{eq:12}
\end{eqnarray}
and, after summation over the W polarizations
\begin{eqnarray}
\frac{dN}{d \cos\theta} = \frac{1}{2} \left( 1 +
\frac{m^2_t - 2m^2_W}{m^2_t + 2m^2_W} P_t \cos\theta \right)
\label{eq:13}
\end{eqnarray}
The angle between top quark spin and direction of the W
is denoted by $\theta$.
In the limit of $m_t \gg M_W$ the coefficient of the $P_t \cos\theta$
term rises to 1, for $ m_t= 180$ GeV, however, it amounts to 0.43
only. 

The angular distribution of leptons from the chain
$t \to b + W(\to \ell ^+\nu )$
will in general follow a complicated pattern with an energy
dependent angular distribution
\begin{equation}
\frac{dN}{dx d \cos\theta} = f(x) + g(x) P_t \cos\theta.
\label{eq:15}
\end{equation}

In the SM, however, a remarkable simplification arises.
Energy and angular distribution factorize \cite{stre,jk1}
\begin{equation}
\frac{dN}{dx d \cos\theta} = f(x) (1 + P_t \cos\theta )/2.
\label{eq:16}
\end{equation}
 
This factorisation holds true for arbitrary $m_t$ and even including
the effect of the nonvanishing $b$-quark mass \cite{JezK}.


%%%%%%%%%% C
\subsubsection{QCD corrections}
The $\delta$ spike in the energy distribution of the hadrons from the
decay $t \to b+W$ is smeared by quark fragmentation (not treated in this
context).

Hard gluon radiation leads to a slight shift and distortion of the energy
spectra with a tail extending from the lower limit given by two-body kinematics
upwards to $m_t - m_W$
\begin{equation}
\frac{m^2_t + m^2_b - m^2_W}{2m_t} \leq E_{had} < m_t - m_W
\label{eq:23}
\end{equation}

Including finite W-width effects and $m_b \not= 0$ the differential hadron
energy distribution has been calculated in \cite{cjk}. The hadron energy
distribution is shown in Fig.\ref{fig:2.4} for $m_t = 180$ GeV.

\begin{figure}

\psfig{figure=fig28.ps,width=120mm,bbllx=-100pt,bblly=50pt,bburx=530pt,bbury=800pt}
\caption[]{\label{fig:2.4}Distribution of
the W energy for $m_t = 180~GeV$
without (dashed) and with (solid curve) QCD corrections.}
\end{figure}

The lepton spectrum (as well as the neutrino spectrum)
receives its main
correction close to the end points where the counting rates are fairly
low. 
%%%%%%%%%%%%%%%%%% END of C

Including QCD corrections \cite{jk1,czajk91,cjkk94,cj94} the spectrum
of both charged lepton and neutrino can be cast into the form
\be
{ {\rm d} N\over {\rm d}x{\rm d}\cos\theta}
= A(x) + B(x) \cos\theta
\ee
The shape of the charged lepton spectrum is hardly different from the
lowest order result \cite{jk1} with main corrections towards the end
point.
 $B_e(x)\approx A_e(x)$ remains
valid to extremely high precision \cite{czajk91}.  The charged lepton
direction is thus a perfect analyser of the top spin, even after
inclusion of QCD corrections.  A small admixture of $V+A$ couplings
will affect spectrum and angular distributions of electron and
neutrino as well.  Assuming a $V+A$ admixture of relative rate
$\kappa^2 = 0.1$, the functions $A_e$, $B_e$ and $A_\nu$ are only marginally
modified (Fig.~\ref{fig-1} and \ref{fig-2}).  The angular dependence
part of the 
neutrino spectrum $B_\nu$, however, is changed significantly 
(Fig.~\ref{fig-2}).  This observation could provide a useful tool in
the search for new couplings.
\begin{figure}
\begin{center}
\leavevmode
\epsffile[70 320 550 500]{fig1.ps}
\caption{The coefficient functions a) ${\rm A}_\ell(x)$ and
b) ${\rm B}_\ell(x)$ defining the charged lepton angular-energy
distribution for $y=0.25$ and $\alpha_s(m_t)=0.11$ : \ \
$\kappa^2=0$ -- solid lines and $\kappa^2=0.1$ -- dashed lines.
\label{fig-1} }
\end{center}
\begin{center}
\leavevmode
\epsffile[70 320 550 500]{fig2.ps}
\caption{The coefficient functions a) ${\rm A}_\nu(x)$ and
b) ${\rm B}_\nu(x)$ defining the neutrino angular-energy
distribution for $y=0.25$ and $\alpha_s(m_t)=0.11$ : \ \
$\kappa^2=0$ -- solid lines and $\kappa^2=0.1$ -- dashed lines.
\label{fig-2}  }
\end{center}
\end{figure}




\subsection{Non--standard top decays}
\label{nonstan}
The theoretical study of non--standard top decays is motivated by
the large top quark mass which could allow for exciting novel
decay modes, even at the Born level.
A few illustrative, but characteristic examples will be discussed
in some detail in the following section.

\subsubsection{Charged Higgs decays}

Charged Higgs states $H^\pm$ appear in 2--doublet Higgs models in
which out of the eight degrees of freedom three Goldstone bosons build up the
longitudinal states of the vector bosons and three neutral and two charged
states correspond to real physical particles. A strong motivation for this
extended Higgs sector is provided by supersymmetry
which requires the \sm\
Higgs sector to be doubled in order to generate masses for the up and
down--type fermions. In the minimal version of that
model the masses of the charged Higgs particles
are predicted to be larger than the $W$ mass, mod.\ radiative corrections,
\[
m(H^\pm) > m(W^+) \mbox{ [mod. rad. corr.]}
\]
We shall adopt this specific model for the more detailed discussion in the
following paragraphs.

If the charged Higgs mass is lighter than the top mass, the top quark may
decay into $H^+$ plus a $b$ quark \cite{ns30},
\[
t \ra b + H^+
\]
The  coupling of the charged Higgs to the scalar $(t,b)$ current is defined by
the quark masses and the parameter $\tan \beta$,
\begin{equation}\label{gl10}
J(b,t) = {1\over \sqrt{2}v} \left[ \left( m_b \tan \beta + m_t \cot
 \beta
\right) - \gamma_5 \left( m_b \tan \beta - m_t \cot \beta  \right) \right].
\end{equation}
The parameter $\tan\beta$ is the ratio of the vacuum expectation values of
the
Higgs fields giving masses to up and down--type fermions, respectively. For
the sake of
consistency, related to grand unification, we shall assume
$\tan\beta$ to be bounded by
\begin{equation}\label{gl11}
1 < \tan\beta = \frac{v_2}{v_1}  < \frac{m_t}{m_b} \sim 60
\end{equation}
\noindent with $v = \sqrt{v_1^2 + v_2^2}=(\sqrt{2}G_F)^{-1/2}$ corresponding
to the ground state of the \sm\  Higgs field.
The width following from the coupling (\ref{gl10}) has a form quite
similar to the \sm\  decay mode [see e.g.\ \cite{bigi2}],
\begin{equation}
\Gamma(t\ra b + H^+) = \frac{G_F m_t^3}{8\sqrt{2}\pi} \left[ 1 -
\frac{m_H^2}{m_t^2} \right]^2 \left[ \left( \frac{m_b}{m_t} \right)^2
\tan^2\beta + \cot^2\beta \right]
\end{equation}

\begin{figure}[p]

\vspace{8.5cm}

\caption[]{{\label{FHiggs} \bf (a)} Branching ratios for the decays $t
\ra b W^+$ and $t \ra b  H^+$ in two--doublet Higgs models \cite{reya1}.}
\addtocounter{figure}{-1}

\vspace{10cm}

\caption[]{{\bf (b)} Branching ratios for charged Higgs decays to
$\tau$ leptons  and quarks \cite{reya1}.}
\end{figure}



The branching ratio of this novel Higgs decay mode is
compared with the $W$ decay mode in Fig.\ref{FHiggs}a
(The behaviour is qualitatively similar for $m_t=180$ GeV.)
In the parameter range  eq.~(\ref{gl11}) the $W$ decay mode is
dominant; the Higgs decay branching ratio is in general small, yet large
enough to be clearly observable \cite{PIK}.
The Higgs branching ratio is
minimal at $\tan \beta = \sqrt{m_t / m_b} \sim 6-8$. QCD corrections
to the $t\to Hb$ mode have been calculated in \cite{CzarDav} and
electroweak corrections in \cite{EWHiggs}.

The detection of this scalar decay channel is facilitated by the
characteristic decay pattern of the charged Higgs bosons
\[
H^+ \ra \tau^+ + \nu_\tau \quad \mbox{ and }\quad c + \overline{s}
\]
Since $H^\pm$ bosons couple preferentially to down--type fermions
\cite{ns38} for $\tan\beta > 1$,
\begin{eqnarray}
\Gamma(H^+ \ra \tau^+ \nu_\tau) & = &    \frac{G_F m^2_\tau}{\sqrt{2}}
\frac{m_H}{4\pi} \tan^2\beta \\
\Gamma(H^+ \ra c \overline{s} ) & = &   \frac{3G_Fm^2_c}{\sqrt{2}}
\frac{m_H}{4\pi} \left[ \left( \frac{m_s}{m_c} \right)^2 \tan^2 \beta
+ \cot^2 \beta \right]
\end{eqnarray}
\noindent the $\tau$ decay mode wins over the quark decay
mode [Fig.\ref{FHiggs}b], thus
providing a clear experimental signature. A first signal of top decays into
charged Higgs particles would therefore be the breakdown
of $\mu, e$ {\it vs.}\ $\tau$
universality in semileptonic top decays.

An interesting method for a determination of $\tan\beta$ is based on
an analysis of the angular distribution of Higgs bosons in the decay
of polarized top quarks
\be
{ {\rm d} N \over  {\rm d} \cos\theta}
\sim
m_t^2 \cot^2\beta (1+\cos\theta_H)
+m_b^2 \tan^2\beta (1-\cos\theta_H),
\ee
an immediate consequence of the couplings given in (\ref{gl10}).

\subsubsection{Top decay to stop}

Another exciting decay mode in supersymmetry models is the decay of
the top to 
the SUSY scalar partner stop plus neutralinos, mixtures of neutral
gauginos and higgsinos
\cite{ns31,ns39}. This possibility is intimately
related to the large top mass  which leads to novel phenomena induced
by the strong Yukawa interactions. These effects do not occur in light--quark
systems but are special to the top.

The mass matrix of the scalar SUSY partners $(\tilde{t}_L, \tilde{t}_R)$ to
the left-- and right--handed top--quark components $(t_L, t_R)$ is
built--up by 
the following elements \cite{ns40}
\[
{\cal{M}}^2 = \left\| \begin{array}{cc}
          m_{\tilde{t}_L}^2 + m_t^2 & \delta  \tilde{m}_{LR}^2 \\
          \delta \tilde{m}_{LR}^2 & m_{\tilde{t}_R}^2 + m_t^2
        \end{array}
\right\|
\]
Large Yukawa interactions lower the diagonal matrix elements $\sim
-m_t^2$ with respect to the common squark mass value in supergravity
models, and they mix the $\tilde{t}_L$ and $\tilde{t}_R$
states with the strength $\sim m_t$ to form the
mass eigenstates $\tilde{t}_1, \tilde{t}_2$. Unlike the five light quark
species, these Yukawa interactions of ${\cal{O}}(m_t)$
can be so large in the top sector that after
diagonalizing the mass matrix, the smaller eigenvalue may fall below the top
quark mass,
\[
m_{\tilde{t}_1} < m_t \mbox{\ \ [ : possible ]}.
\]
The decay modes
\[
t \ra \tilde{t} + \mbox{neutralinos}
\]
\noindent then compete with the ordinary $W$ decay mode. Identifying
the lightest SUSY particle with the photino $\tilde{\gamma}$ 
(the mass of which we neglect in this estimate) one finds
\begin{equation}
\frac{\Gamma(t \ra \tilde{t} \tilde{\gamma})}{\Gamma(t \ra bW)}  \approx
\frac{8
\sqrt{2}\pi \alpha}{9 G_F m_t^2}  \frac{[1 - m_{\tilde{t}}^2/m_t^2 ]^2}{[1 -
m_W^2 /m_t^2 ]^2 [1 + 2m_W^2/m_t^2] }
\end{equation}
\noindent This ratio is in general less than 10\%. The subsequent
$\tilde{t}$ decays
\begin{eqnarray}
\tilde{t} & \ra & b \tilde{W}, \tilde{W} \ra W \tilde{\gamma} \mbox{\
\ or\ \ }
l\tilde{\nu} \mbox{\ \ {\it etc.}\ \ } \nonumber \\
\tilde{t} & \ra & c \tilde{\gamma} \nonumber
\end{eqnarray}
\noindent lead to an overall softer charged lepton spectrum and,
as a result of the escaping photinos, to an increase of the  missing energy, the
characteristic signature for SUSY induced phenomena.

Depending on the SUSY parameters however, stop decays could even be
more enhanced if the top is heavy. Decays into strongly coupled,
fairly light higgsinos could thus occur frequently.

\subsubsection{FCNC decays}

Within the Standard Model, FCNC decays like $t\ra c\gamma$ are forbidden at
the tree level by the GIM mechanism. However, they do occur in
principle at the one--loop
level, though strongly suppressed. The suppression is particularly severe for
top decays since the quarks building up the loops, must be down--type quarks
with $m_b^2$ setting the scale of the decay amplitude, $\Gamma_{FCNC} \sim
\alpha G_F^2 m_b^4 m_t$. A sample of branching ratios is given below
\cite{ns41}:

\[
\begin{array}{c}
\psfig{figure=t2c.ps,width=60mm,bbllx=0pt,bblly=300pt,%
bburx=630pt,bbury=420pt}
\end{array}\\
\hspace*{-5mm} \begin{array}{lclclcl}
              BR (t \ra cg) & \sim & 10^{-10} & \quad & BR(t \ra cZ) & \sim &
              10^{-12} \\
              BR(t \ra c\gamma) & \sim & 10^{-12} & & BR (t \ra cH) & \sim &
              10^{-7}
             \end{array}
\]

\vspace*{1cm}

At this level, no \sm\  generated $t$ decays can be observed, even given
 millions
of top quarks in proton colliders. On the other hand, if these decay modes
were detected, they would be an undisputed signal of new physics beyond the
\sm. From such options we select one illustrative, though very speculative
example for brutal GIM breaking. It is tied to the large top mass and holds out
faint hopes to be observable even in low rate $e^+e^-$ colliders.

The GIM mechanism requires all $L$ and $R$ quark
components of the same electric charge
in different families to carry identical isospin quantum numbers,
respectively. This rule is
broken by adding quarks in LR symmetric vector
representations \cite{ns32} to the
``light'' chiral representations or mirror quarks \cite{ns33}:
\[
 \begin{array}{lccccc}
   \mbox{\underline{vector quarks}: } & \quad \cdots \quad &
         \left[ \begin{array}{c} t \\ b \end{array} \right]_L &
         \begin{array}{c} t_R \\ b_R \end{array} & \qquad
         \left[ \begin{array}{c} U \\ D \end{array} \right]_L &
         \left[ \begin{array}{c} U \\ D \end{array} \right]_R  \\
\\
   \mbox{\underline{mirror quarks}: } & \quad \cdots \quad &
         \left[ \begin{array}{c} t \\ b \end{array} \right]_L &
         \begin{array}{c} t_R \\ b_R \end{array} & \qquad
         \begin{array}{c} U_L \\ D_L \end{array} &
         \left[ \begin{array}{c} U \\ D \end{array} \right]_R
 \end{array}
\]
Low energy phenomenology requires the masses $M$ of the new $U,D$ quarks to be
larger than 300 GeV.

Depending on the specific form of the mass matrix,
mixing between the normal chiral states and the new states
may occur at the level $\sim\sqrt{m/M}$, so
that FCNC $(t,c)$ couplings of the order $\sim\sqrt{m_tm_c/M^2}$ can be induced.
FCNC decays of top quarks, for example,
\[
BR(t \ra cZ) \sim \mbox{fraction of }\%
\]
\noindent are therefore not excluded. Such branching ratios would be at the
lower edge of the range accessible at $e^+e^-$ colliders.


