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\section{\label{sec-ii}Top quark decays in the \sm\ and beyond}

\subsection{Born predictions}

\subsubsection{The total rate}

Based on the Standard Model, the lower limit of the top quark mass
$m_t&gt;91~GeV$ has been shifted to a level well above the real $W$ threshold
for the decay mode $t \to b$+$W^+$.  In the three-family SM the CKM matrix
element of the $tb$ coupling $V_{tb}$ is well known to be very close to unity.
As a result, the width of the top quark is dominated by decays into a $W$
plus a $b$
jet and can be precisely predicted in the SM \cite{acta,bigi2}~:
\begin{eqnarray}
\Gamma \left( t \to b+W \right) &amp;=&amp;
|V_{tb}|^2 \frac{G_F m^3_t}{8\sqrt 2\pi}
\frac{2P_W}{m_t}
  \left\{ \left[1-\left(\frac{m_b}{m_t}\right)^2\right]^2
+ \left[ 1 + \left( \frac{m_b}{m_t} \right)^2 \right]
\left( \frac{m_W}{m_t}\right)^2 \right. \nonumber\\
&amp;\ &amp;- 2 \left.\left( \frac{m_W}{m_t} \right)^4 \right\}
\ \approx\  \frac{G_Fm^3_t}{8\sqrt 2 \pi}
\left( 1 - \frac{m^2_W}{m^2_t}\right)^2
\left( 1 + 2 \frac{m^2_W}{m^2_t} \right)
\label{eq:1}
\end{eqnarray}
\par\noindent
The branching ratios BR($t \to q+W^+$) for q=b,s,d follow from
\cite{BurasFleischer} 

\begin{eqnarray}
0.9991\le |V_{tb}| \le 0.9993
  &amp; \Rightarrow &amp; BR(b) \approx 1\nonumber\\
0.0353\le |V_{ts}| \le 0.0429
 &amp; \Rightarrow &amp; BR(s) \approx0.2\%\nonumber\\
4.5\cdot 10^{-3} \le |V_{td}| \le 13.7\cdot 10^{-3}
 &amp; \Rightarrow &amp; BR(d) \approx 0.01\%
\label{eq:2}
\end{eqnarray}

While the search for $s$+$W^+$ final states, though difficult, is not
impossible, the $t$ production rates will be too small to look for $d$+$W^+$
decay products.

For $m_t$ around 175 GeV the width is already close to the asymptotic form
\begin{equation}
\Gamma \left(t \to b+W^+\right) \to 175 MeV \cdot \left(\frac{m_t}{
m_W}\right)^3
\label{eq:3}
\end{equation}
and the lifetime drops below $10^{-23}s$.  This is illustrated in Fig.
\ref{Fdecrat}.

\begin{figure}
 \vspace{9.5cm}
 \caption[]{\label{Fdecrat}{\it The width of the top quark in the \sm. 
The degree of longitudinal polarization is shown in the insert.}}
\end{figure}


The final state consists of a $b$-quark jet plus a lepton pair
or two other jets from the $W$ decay which, being a color singlet system,
fragments independent from the environment.

The decay rate can be split into the contribution from $W$'s with longitudinal
($h_W=0$) and transverse ($h_W=+1$) polarizations.
Neglecting b-quark mass effects, the ratio
\begin{equation}
\Gamma _L / \Gamma _T = m^2_t / 2m^2_W
\label{eq:4}
\end{equation}
amounts to 2.35 for  $m_t$ = 175~GeV.

\subsubsection{Spectra}

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 &amp; = &amp; \frac{m^2_t + m^2_W - m^2_b}{2m_t} \nonumber\\
E_h &amp; = &amp; \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
from the combined effects of perturbative QCD and from the finite width
of the $W$ will be treated in the next section.

The energy spectra of the leptons from the decay of the real $W$, and
correspondingly the invariant $(bl)$ mass
are strongly affected by the chirality of the ($tb$) charged weak current
for moderate top masses.  Assuming the charged lepton current to be of the
well-established $V\!-\!A$ form, the probability to observe
a charged lepton of maximal energy $E_l = m_t/2$ vanishes
for the conventional $V\!-\!A$ form of the $(tb)$ current
while it is nonzero for a speculative $V\!+\!A$ coupling.
\begin{equation}
\frac{dN}{dx} = f_{\pm}(x,y) \qquad
(V \pm A)_{tb} \quad \mbox{coupling}
\label{eq:6}
\end{equation}
with $\qquad y \leq x \leq 1$~,
where $x=2E_l /m_t$, $y=m^2_W /m^2_t$, $m_b =0$, and
\begin{eqnarray}
f_- (x,y) &amp; = &amp; 6x(1-x)/[(1-y)^2 (1+2y)] \nonumber\\
f_+ (x,y) &amp; = &amp; 6(x-y)(1+y-x)/[(1-y)^2 (1+2y)]
\label{eq:7}
\end{eqnarray}
For a mixture of $V\!-\!A$ and $V\!+\!A$ amplitudes of
relative weights $w_-$ and $w_+$ with $w^2_+ + w^2_- =1$
\begin{equation}
{\cal M} \sim \bar u _b \not{\epsilon}_W
\left( w_- (1-\gamma _5) + w_+
(1+ \gamma _5)\right) u_t
\label{eq:8}
\end{equation}
(assuming $m_b =0$) one obtains an incoherent sum of their
respective contributions
\begin{equation}
\frac{dN}{dx} = w^2_- f_- + w^2_+ f_+
\label{eq:9}
\end{equation}
A deviation from the $V$-$A$ structure would thus lead to
a stiffening of the charged lepton spectrum, and in particular
to a nonvanishing distribution at the upper end point.

Equivalently, the distribution of the invariant $(bl)$ mass
$\mu ^2 = m^2_{bl} /m^2_t$ is given by the same functions as above
\begin{equation}
\frac{dN}{d \mu ^2} = \ f_{\pm}(\mu^2+y,y) \qquad
(V \pm A) \quad \mbox{coupling}
\label{eq:10}
\end{equation}
with $0\leq\mu^2\leq 1-y$~.

\subsubsection{Angular distributions}

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 ) \nonumber\\
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 $\theta$.

For an arbitrary admixture of right- and lefthanded couplings
(and neglecting $m_b$)
\begin{eqnarray}
P_t \to \left( w^2_L - w^2_B \right) P_t
\label{eq:14}
\end{eqnarray}

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,jk2}
\begin{equation}
\frac{dN}{dx d \cos\theta} = f(x) (1 + P_t \cos\theta )/2
\label{eq:16}
\end{equation}

This factorization holds true for arbitrary $m_t$ and even including the
effect of the nonvanishing $b$-quark mass.  For righthanded $tb$ coupling,
however,
\begin{equation}
\frac{dN}{dx d \cos\theta} = f_+ (x) + g_+ (x) P_t \cos\theta
\label{eq:17}
\end{equation}
with $f_+$ as given in (\ref{eq:7}) and \cite{jk2}
\begin{eqnarray}
\frac{g_+}{f_+} &amp; = &amp; \frac{2y}{(1+y-x)x} - 1 \nonumber\\
 &amp; = &amp; \cases { 1 &amp; for $x = y$ or 1 \cr
 \frac{8y}{(1+y^2)} - 1 &amp; for $x = (1+y)/2$ \cr }
\label{eq:18}
\end{eqnarray}


Even after 
including QCD corrections \cite{jk2,czar2} the spectrum
of both charged lepton and neutrino can be cast into the form
\begin{equation}
  { {\rm d} N\over {\rm d}x{\rm d}\cos\theta}
= A(x) + B(x) \cos\theta
\end{equation}
The shape of the charged lepton spectrum is hardly different from the
lowest order result \cite{jk2} with main corrections towards the end
point.
 $B_e(x)\approx A_e(x)$ remains
valid to extremely high precision \cite{czar2}.  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 dependent
part of the 
neutrino spectrum $B_\nu$, however, is changed significantly 
(Fig.~\ref{fig-2}).  This observation \cite{jk239} 
could provide a useful tool in
the search for new couplings \cite{schmitt}.
\begin{figure}
\begin{center}
\leavevmode
\epsffile[70 320 550 500]{fig1.ps}
\caption{\it 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 
(from \protect\cite{jk239}).
\label{fig-1} }
\end{center}
\begin{center}
\leavevmode
\epsffile[70 320 550 500]{fig2.ps}
\caption{\it 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
(from \protect\cite{jk239}).
\label{fig-2}  }
\end{center}
\end{figure}


To summarize:  $V\!-\!A$ and $V\!+\!A$
currents lead to drastically different predictions.  However, a
small admixture of $V\!+\!A$ will be difficult to detect as a
consequence of the
incoherent addition of right- and lefthanded amplitudes.

\subsection{Radiative corrections}

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 \underline{decay rate} is usually written in the form
\begin{equation}
\Gamma = \Gamma _{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:
\begin{eqnarray}
f &amp; = &amp; {\cal F}_1 / {\cal F}_0 \nonumber \\
{\cal F}_0 &amp; = &amp; 2(1-y)^2 (1+2y) \nonumber \\
{\cal F}_1 &amp; = &amp; {\cal F}_0 \left[ \pi ^2 + 2 Li_2 (y) - 2Li_2
(1-y)\right] \nonumber \\ 
 &amp; &amp; + 4y(1-y-2y^2) \ln y + 2(1-y)^2 (5+4y) \ln(1-y) \nonumber \\
 &amp; &amp; - (1-y)(5+9y-6y^2)
\label{eq:21}
\end{eqnarray}
where $y=m^2_W / m^2_t$.
This formula was confirmed in \cite{den1}.  %,li,czar}.
  In the limit $m^2_t &gt;&gt; m^2_W$ $f$ approaches the
value $f(0) = 2/3 \pi ^2 - 5/2 \approx 4$.
For $m_t\approx 175$ 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.


The complete $\alpha_s^2$ 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.  Diagrams with
light fermion insertions into the gluon propagator have been
calculated numerically \cite{smith} and analytically \cite{ac95a} in
the limit $m_t\gg M_W$
\begin{eqnarray}
  \Gamma &amp;=&amp; \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\\
&amp;&amp;+
\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]
\end{eqnarray}
The BLM prescription \cite{BLM} suggests that the dominant
$\alpha_s^2$ coefficients can be estimated through the replacement

\begin{eqnarray}
-{2n_f\over 3} \to \left( 11 -{2n_f\over 3}\right)
\end{eqnarray}

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 $\mu = m_t$.  The applicability of the BLM recipe has been
verified for the $b\to c$ transitions at the points of maximal and of
minimal recoil \cite{czprl76,czmel}.



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} &lt; m_t - m_W
\label{eq:23}
\end{equation}

Including finite W-width effects and $m_b \not= 0$ the differential hadron
energy 
(and $W$ energy) distribution has been calculated in \cite{czar3}. 
The hadron energy
distribution is shown in Fig.\ref{fig:2.4} for $m_t = 180~GeV$.



\begin{figure}

\psfig{figure=fig28norm.ps,width=140mm,bbllx=-100pt,%
bblly=50pt,bburx=530pt,bbury=800pt}
\caption[]{\label{fig:2.4}{\it Normalized 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. 
 The shape is hardly different from the
lowest-order result \cite{jk2}.  

Results for the corrected angular distributions of charged leptons
and neutrinos
from the decay of polarized t~quarks are also available
in the literature \cite{czar2}.  It has been
shown that the deviations from the factorizable form (\ref{eq:16})
are below $10^{-3}$ such that
the Born approximation is well justified for this purpose.

\ifnum\wg=0%Lampe
\subsubsection{QCD corrections to invariant mass distributions}
The observation of two muons with large invariant mass from the decay
chain
$$t\rightarrow \mu^+ \nu b (\rightarrow \mu^-\bar\nu c)$$
may provide a clean signal for the presence of a top quark in hadronic
collisions and in electron positron annihilation. The invariant mass
distribution depends in a sensitive way on the mass of the top quark.
QCD corrections affect not only the normalisation but also the shape
of the distribution\cite{Lampe}.
The distribution of the invariant mass $a^2=(p_{\mu^+}+p_{\mu^-})^2$ and
of the angle between the directions of $\mu^+$ and $\mu^-$ are shown in
Fig.\ref{Flampe}.
%
\begin{figure}
\vspace{15cm}
\caption[]{\label{Flampe}Histograms for the distribution $\Delta\Gamma/\Delta a$ and
$\Delta\Gamma/\Delta \cos\theta$ for $m_t=150 GeV$ and $200 GeV$.
Dashed line: lowest order result. Solid line: With QCD corrections
(from \cite{Lampe}).}
\end{figure}
%
The distortion is more evident in Table\ref{Tlampe} which lists the
ratio between the corrected distribution and the distribution in Born
approximation. The shift towards lower values of the $\mu^+\mu^-$ mass
is clearly visible.
\begin{table}
\vspace{15cm}
\caption[]{\label{Tlampe}Ratio of QCD corrected result over Born term for the
distributions shown in Fig.\ref{Flampe}.}
\end{table}
\fi

\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
depicted above, provides the dominant contribution.
Defining the Born term by means of the Fermi coupling $G_F$,
one derives in this limit
\begin{eqnarray}
\Gamma &amp; = &amp; \Gamma(G_F)_{BORN}\left[ 1 + \delta_{EW} \right] \\
&amp; &amp; \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 we observe only a very weak
dependence of $\delta_{EW}$
on the top and the Higgs masses. 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.

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] &amp; $\alpha_s(m_t)$ &amp; $\Gamma^{\rm Born}_{\rm nar.\,w.}$ [GeV]&amp;
$\delta^{(0)}_\Gamma$ [\%] &amp; 
$\delta^{\rm nw}_{\rm QCD}$ [\%] &amp; 
$\delta_{\rm EW}$ [\%] &amp; 
$\Gamma$ [GeV] \\
\hline
170 &amp; .108 &amp; 1.41 &amp; -1.52 &amp; -8.34 &amp; 1.67 &amp; 1.29 \\
\hline
180 &amp; .107 &amp; 1.71 &amp; -1.45 &amp; -8.35 &amp; 1.70 &amp; 1.57 \\
\hline
190 &amp; .106 &amp; 2.06 &amp; -1.39 &amp; -8.36 &amp; 1.73 &amp; 1.89 \\
\hline
\end{tabular}
\caption{\it 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.

\subsection{Non--standard top decays}

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 chapters.
\ifnum\wg=0
They demonstrate the great
potential of top production in $e^+e^-$ collisions,
even if the novel decay processes are fairly rare.
\fi

\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,
\[
m(H^\pm) &gt; m(W^+) \mbox{ [mod. rad. corr.]}
\]
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) = \sqrt{\frac{G_F}{\sqrt{2}}} \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 &lt; \tan\beta = \frac{v_2}{v_1}  &lt; \frac{m_t}{m_b} \sim 50
\end{equation}
\noindent with $v = \sqrt{v_1^2 + v_2^2}$ 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)} {\it Branching ratios for the decays $t \ra b W^+$ and $t \ra b
 H^+$ in
two--doublet Higgs models.}}
\addtocounter{figure}{-1}

\vspace{10cm}

\caption[]{{\bf (b)} {\it Branching ratios for charged Higgs decays to $\tau$ leptons
 and
quarks.}}
\end{figure}
\addtocounter{figure}{-1}

\begin{figure}[hbt]

\vspace{9cm}

\caption[]{{\bf (c)} {\it Inclusive branching ratios for $t\ra l + X$ in two--doublet
 Higgs models
and breaking of the lepton universality. $BR_l(t) = 1/9$ in the \sm\  is
indicated by the broken line.}}
\end{figure}

The branching ratio of this novel Higgs decay mode is
compared with the $W$ decay mode in Fig.\ref{FHiggs}a.
In the medium $\tan\beta$ range  the $W$ decay mode is
dominant; the Higgs decay branching ratio is in general small, yet large
enough, in particular for small 
$\tan\beta \stackrel{&gt;}{\sim} 1$ 
and large
$\tan\beta \stackrel{&lt;}{\sim} m_t/m_b \sim 50$, 
 to be clearly observable \cite{PIK}.

The detection of this scalar decay channel is facilitated by the
characteristic decay pattern of the charged Higgs bosons
\[
H^+ \ra \tau^+ + \nu_\tau \quad \mbox{ and }\quad c + \overline{s}
\]
into \ssm particles. 
Since $H^\pm$ bosons couple preferentially to down--type fermions
\cite{ns38} for $\tan\beta &gt; 1$,
\begin{eqnarray}
\Gamma(H^+ \ra \tau^+ \nu_\tau) &amp; = &amp;    \frac{G_F m^2_\tau}{\sqrt{2}}
\frac{m_H}{4\pi} \tan^2\beta \\
\Gamma(H^+ \ra c \overline{s} ) &amp; = &amp;   \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, Fig.\ref{FHiggs}c.
Other possible decay modes of charged Higgs bosons, in 
particular chargino/neutralino decays for small $\tan \beta$, also lead 
to characteristic final states.

\subsubsection{Top decay to stop}

Another exciting decay mode in supersymmetry models is the decay of the top 
quark to
the SUSY scalar partner stop plus neutralinos, mixtures of neutral
gauginos and higgsinos
\cite{ns31,borz}. 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 (and eventually the bottom) system.

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}
\[
{\cal{M}}^2 = \left\| \begin{array}{cc}
          m_{\tilde{t}_L}^2 + m_t^2 + (\frac{1}{2} - \frac{2}{3} s^2_w) \cos 
2 \beta m^2_Z
&amp; \delta  \tilde{m}_{LR}^2 \\
          \delta \tilde{m}_{LR}^2 &amp; m_{\tilde{t}_R}^2 + m_t^2
+\frac{2}{3} s^2_w \cos 2 \beta m^2_Z
        \end{array}
\right\|
\]
Large Yukawa interactions  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 drop below the top
quark mass,
\[
m_{\tilde{t}_1} &lt; m_t \mbox{\ \ [ : possible ]}
\]
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 $\tilde{\gamma}$, for
example, the mass of which we neglect in this estimate,
\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} &amp; \ra &amp; b \tilde{W}, \tilde{W} \ra W \tilde{\gamma} \mbox{\
\ or\ \ }
l\tilde{\nu} \mbox{\ \ {\it etc.}\ \ } \nonumber \\
\tilde{t} &amp; \ra &amp; 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 \cite{PIK,Venturi}.


\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}:

\[
\hspace{5cm} \begin{array}{lclclcl}
              BR (t \ra cg) &amp; \sim &amp; 10^{-10} &amp; \quad &amp; BR(t \ra cZ) &amp; \sim &amp;
              10^{-12} \\
              BR(t \ra c\gamma) &amp; \sim &amp; 10^{-12} &amp; &amp; BR (t \ra cH) &amp; \sim &amp;
              10^{-7}
             \end{array}
\]

\vspace*{2cm}

At this level, the FCNC $t$ decays generated by the  \sm\  cannot 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}:
\[
 \begin{array}{lccccc}
   \mbox{\underline{vector quarks}: } &amp; \quad \cdots \quad &amp;
         \left[ \begin{array}{c} t \\ b \end{array} \right]_L &amp;
         \begin{array}{c} t_R \\ b_R \end{array} &amp; \qquad
         \left[ \begin{array}{c} U \\ D \end{array} \right]_L &amp;
         \left[ \begin{array}{c} U \\ D \end{array} \right]_R  \\
\\
   \mbox{\underline{mirror quarks}: } &amp; \quad \cdots \quad &amp;
         \left[ \begin{array}{c} t \\ b \end{array} \right]_L &amp;
         \begin{array}{c} t_R \\ b_R \end{array} &amp; \qquad
         \begin{array}{c} U_L \\ D_L \end{array} &amp;
         \left[ \begin{array}{c} U \\ D \end{array} \right]_R
 \end{array}
\]
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 can easily be  
measured at the LHC.

\vspace{1cm}

{\it Summa.}
For the illustrative examples outlined above, the \sm\  decay mode $t
\ra b + W^+$ is in general dominant so that $t$ quarks can be tagged easily.
Non--standard decays are correspondingly rare, yet frequent enough to be
 detected
in mixed \sm\  -- non--\sm\  $t\overline{t}$
pair decays at $e^+e^-$ colliders. Very rare decay modes, in particular
FCNC decays with clear signatures, can be investigated at the LHC.

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