
\vspace*{5cm}
 
The search for the fundamental building blocks of matter and their
interactions is the
central objective of high energy physics.
While the domain of light quark matter has well been explored in the
past decades, heavy flavor physics has advanced very rapidly in 
recent years, but  
still poses a large number of important problems which can 
 be solved in the years to come at existing and future colliders.
 
In the context of the Standard Model \cite{gsw1} three points are
outstanding.
 
\noindent
(i) The long awaited discovery of the top quark \cite{kobma1} which completes
the fermionic spectrum of the Standard Model [\ssm], had been expected with
certainty in the ongoing
experiments at the Tevatron or the future proton colliders
%\cite{reya1}
[see the preceding version of this report]. These theoretical predictions 
have been realized experimentally 
when the top quark was discovered at the Tevatron \cite{cdf1}.
This is a triumph of quantum field theory and the  high
precision measurements of electroweak radiative corrections at LEP and
in deep
inelastic lepton--nucleon scattering \cite{lefr} from which a value  
$m_t = 162 \pm 16 \pm 19$ GeV of the top quark mass could be 
derived before the particle was isolated in $p\overline{p}$ collisions.
Future $e^+e^-$  linear colliders \cite{eacco}
 in
the 350 GeV to 1 TeV c.m.\ energy range will provide a high precision
determination of the top quark mass within a fraction of one GeV
\cite{PIK}.
Anticipating the
development of a flavor theory in the future, the demand of
such  high accuracy appears
meaningful as the heavy top quark with a mass of the order of the electroweak
symmetry breaking scale will play a key r\^{o}le among the
fermions in such a theory. In the \ssm this is apparent from the large
Yukawa coupling between the Higgs boson
and the top quark, giving rise to sizeable indirect and direct effects in
$t\overline{t}$ production.
 
 
\noindent
(ii) A similarly fundamental problem
is the embedding of $\cal{CP}$ violation in
the Standard Model \cite{jarls1}.
Being a rationale for the existence of three
families \cite{kobma1}, $\cal{CP}$ violation enters through the quark mixing
in
the weak
interactions. While the information from the
neutral $K-\overline{K}$ complex is
nearly exhausted, the experimental investigation of $B^0 - \overline{B^0}$
mixing and 
of $\cal{CP}$ violation in
the $b$
quark sector allows for conclusive tests of the validity
of this parametrization.
Even though the Standard Model does not shed light on the physical origin of the
$\cal{CP}$
asymmetry [in the same way as it does not predict the mass spectrum of quarks
and leptons], it is nevertheless of utmost importance to prove experimentally
that $\cal{CP}$ violation
can be traced back to quark mixing. In particular, since in extended theories
like supersymmetry additional $\cal{CP}$ violating mechanisms are operative. 
 
\noindent
(iii) It has already been possible to
tag $b$ quarks with high efficiency  and  purity 
in jet final states of high energy processes. They may therefore serve as a
powerful instrument to discriminate quark jets from gluon jets, a necessary
step to refine the exploration of
QCD at short distances \cite{bethk1}. This method facilitates considerably the
isolation of the three--gluon vertex, the measurement of gluon--jet
properties and many other applications.
 
Heavy quarks are the fermionic particles closest to
the magic TeV scale which is likely to be the energy threshold to a more
profound theory,
encompassing the Standard Model and resolving some of its enigmatic patterns.
Mixing effects of order $m_i m_j / m_W^2$, too small to be observable in the
light quark sector, may manifest themselves through flavor--changing
neutral current processes between heavy quarks \cite{fritz1} and
through rare heavy--meson
decays \cite{buchm1}.
 
This report is divided into two parts. In the first chapter we will
summarize  the conclusions
 which can be drawn from precision measurements of $Z$ decays
into
$b$--quark pairs, the partial width and the forward--backward
asymmetry \cite{kuehn1}.
The important subject of $\cal CP$-violation
                                      is adressed separately in an extensive
report elsewhere in this book and it will  not be treated in this
article.
In the second chapter the properties of top
quarks in the Standard Model and beyond will be described, followed by a
 discussion of $t$
production in high energy proton--(anti)proton
\cite{reya1} and \epm\ colliders \cite{Zersaar,SLACschool}.
 
\newpage
 
\vspace{1cm}
 
\chapter{$b$ \hspace{2mm}QUARK}
 
A substantial fraction of all $Z$ decay modes are decays into bottom
quarks. The branching ratio was predicted in the Standard Model to be
close to
%****************************************
\[
BR(Z\rightarrow b\overline{b })  \approx  15\%
\]
%****************************************
so that approximately $3\,\times
\, 10^6$ bottom hadrons are produced in a sample of $10^7$ $Z$ decays. [A
similarly large
number of charm quarks has been produced, as expected from 
 a branching ratio  $ BR(Z\rightarrow
c\overline{c}) \approx 12\%$. However, for theoretical as well as 
experimental reasons  we will concentrate on $b$ quarks.]
 
Three different time steps can be distinguished in the evolution of the final
state. The decay process $Z \rightarrow b \overline{b}$ is completed after
%*********************************
\begin{figure}[hbt] \vspace{4cm} \end{figure}
%*********************************
a time of order $m_Z^{-1} \sim 10^{-3}$ fm, including the most important
genuine electroweak vertex corrections. Perturbative gluon radiation and
non--perturbative fragmentation proceeds up to a time scale of $\gamma
\Lambda_{QCD}^{-1} \sim 10$ fm in the lab frame. Given a lifetime of
\hbox{$\sim 1$ ps}, 
the $b$--flavored hadrons travel to macroscopic distances of order $\gamma
\tau_B  \sim 3$ mm before they decay weakly.
 
The total number of $b$ final states is sufficiently large
to carry out precision measurements in
the \lu{$b$ $quark\, sector$} \cite{kuehn1}. Fundamental aspects of the
Standard Model can
be tested in the heavy quark sector by determining the vector and axial
vector electroweak charges from cross sections and asymmetries. The
accuracy of $\swin$ which has been achieved by measuring the
forward--backward asymmetry of $b$ jets,
is  at the level of $\pm$ 0.001 and better, and thus competes
well with other methods in the lepton sector.
Moreover, vertex corrections due to the exchange of new particles 
predicted in theories 
beyond the Standard Model, affect the partial decay width of $Z$
bosons to $b$ quark pairs. Thus, these experiments allow for
a glimpse at the physics beyond the standard or, at least, lead to 
interesting constraints.
 
QCD inspired fragmentation models developed to describe the hadronization
of quarks and gluons in jets, can be scrutinized experimentally 
in $Z$ decays to heavy--quark jets \cite{kuehn1}.
Because
heavy quark pair creation in the jet evolution is strongly suppressed, the
fragmentation of heavy quarks provides the most powerful tool to tackle this
rather
involved problem. The tagging of heavy quark jets will allow us to isolate
clearly gluon jets in 3-- and multi--jet final states so that their profile and
their interactions can be explored.
 
The properties of \lu{$b$ $hadrons$} can be investigated
experimentally in great detail \cite{kuehn1,donog1}.
The high particle energies make feasible the measurement of individual
lifetimes in vertex detectors. The mixing parameters
in the $B_d$ system can well be
determined by analyzing the time evolution of the $B_d^0-\overline{B}_d^0$
states. The oscillations in the $B_s^0-\overline{B}_s^0$ system,
however, are so rapid that,
even though nearly complete mixing can easily be established by measuring
time--integrated observables, a quantitative measurement of the mixing
parameter appears very difficult, yet powerful lower
bounds could already be established.

 
\section{Electroweak properties of $b$ quarks}
 
\subsection{{Production of $b$ quarks}}
 
The $Z$ boson couples to quarks through vector and axial--vector
charges with the well--known strength
 
{\hangindent=4cm \hangafter=-5
%****************************************
\[
\mbox{}=\sqrt{\frac{G_F m_Z^2}{2\sqrt{2}}}\gamma_\mu \left[v_q - a_q \gamma_5
\right]
\]
%****************************************
}
 
 
\noindent In the Standard Model these charges are defined as
%*****************************************
\[
\left. \begin{array}{rcl}
v_q&=&2I_q^{3L}-4e_q\swin \\
a_q&=&2I_q^{3L} \end{array}\right\} \qquad 2I_q^{3L}=\pm \> 1 \qquad
\hbox{\ for up/down particles}
\]
%*****************************************
 
For many applications the \underline{Born approximation} provides an excellent
representation of the partial $Z$ decay rate
%****************************************
\begin{equation}
\label{eqrate}
\Gamma_B(Z \rightarrow b \overline{b}) = \frac{G_F m_Z^3}{8\sqrt{2}\pi}\beta
\left[ \frac{3-\beta^2}{2}v_b^2 + \beta^2 a_b^2 \right]
\end{equation}
%****************************************
The mass dependence which
enters through the velocity $\beta = \sqrt{1-\mu_b^2}\,\, [\mu_b^2=4m^2_b/s]$
is
${\cal O}(\mu_b^2)\, = {\cal O}(10^{-2})$ in the axial
term for $ b$ quarks; in the
vector term the correction is of order $\mu_b^4 \sim 10^{-4}$ and thus
of minor importance. Defining the strength of the $Z$ coupling to
$q\overline{q}$ by 
the
Fermi $\mu$ decay constant $G_F$ and the electroweak mixing angle
through $\swin = 1-m^2_W/m^2_Z$ the {\em bulk} of radiative
corrections to the width, i.\ e.\ the logarithmic corrections from the running
 of
$\alpha_{QED}$, are automatically incorporated.
 
\underline{QCD corrections} to the width
$\Gamma(Z\rightarrow q\overline{q}\ldots)$ are
known for non-zero (but small) quark masses up to
second order and for zero quark masses even up
to third order in $\alpha_s$ (for  a recent review see \cite{ckkpr}).
 They are
different for vector and axial--vector couplings not
only because masses break chiral invariance but also due to the large mass
splitting between bottom and top quarks. Decomposing the width into the vector
and the axial--vector part we expand both parts separately in the strong
coupling constant
%*****************************************f3
\begin{eqnarray}
\Gamma(Z\rightarrow q\overline{q}+\ldots)& = &\phantom{\mbox{} +
      \mbox{}}
      \Gamma_B^V\left[ 1+c_1\left(\frac{\alpha_s}{\pi}\right)+
      c_2\left(\frac{\alpha_s}{\pi}
      \right)^2 + \cdots \right] \nonumber \\
  &  & \mbox{} + \Gamma^A_B\left[1+d_1\left(
       \frac{\alpha_s}{\pi}\right) + d_2 \left(
\frac{\alpha_s}{\pi}\right)^2 + \cdots
\right]
\end{eqnarray}
%*****************************************
The {\em vector} coefficients $c_1,c_2,c_3$ have
been calculated for massless quarks.
In the ${\overline{MS}}$ scheme one obtains \cite{CKT,Kat}
\begin{eqnarray}
c_1 & = & 1 \nonumber \\
c_2 & = & 1.9857-0.1153N_f = 1.4092 \nonumber \\
c_3 & = & -6.6368-1.2001N_f-0.0052N_f^2  = -12.77
\end{eqnarray}
and
\begin{eqnarray}
\frac{\alpha_s(s)}{\pi}&=&\frac{4}{\beta_0L}
\Bigl[1-\frac{1}{\beta_0L}\frac{\beta_1
\ln L}{\beta_0} +\frac{1}{\beta_0^2L^2}
\Bigl(\frac{\beta_1^2}{\beta_0^2}(\ln ^2 L
-\ln L-1)+\frac{\beta_2}{\beta_0}\Bigr))\Bigr]  \\
\beta_0&=& 11 -\frac{2}{3}N_f,\qquad\beta_1=102-\frac{38}{3}N_f,
\qquad\beta_2=\frac{
2857}{2}-\frac{5033}{18}N_f+\frac{325}{54}N_f^2 \nonumber\\
\end{eqnarray}
 
 
In the total hadronic decay rate one must include a contribution to $c_3$
which arises from 3 gluon intermediate states and hence
corresponds to a flavor singlet configuration. This piece cannot  be
assigned easily to a specific flavor. For $\swin=0.23$ it amounts to
\begin{eqnarray}
c_3(singlet)=-1.2395\frac{(\sum v_f)^2}{\sum 3v_f^2}=-0.405
\end{eqnarray}
This small term, which affects the rate by less than $0.5\cdot 10^{-4}$,
indicates an inherent limit for flavor separation in Z decays. In fact,
already in $O(\alpha_s^2)$ the separation of $\Gamma_b$ requires same care:
b quarks may be produced from virtual gluons radiated off light quarks and
hence are present in the $Z\to u\overline{u}$ or $d\overline{d}$ channel.
However,
these quarks are soft and can, therefore, be separated kinematically.
 
\noindent
A similar discussion applies for the {\em axial} decay rate.
For massless quarks $d_1$ and $c_1$ coincide
$$d_1=1$$
However, the coefficient $d_2$ deviates from $c_2$ even for massless quarks
 \cite{knieh1}. This is a consequence of the cut triangular diagrams
exemplified below.
%****************************************
\begin{figure}[hbt]\vspace{2cm}\end{figure}
%****************************************
Whereas vector couplings do not give a contribution because of $\cal{C}$
invariance,
axial couplings do. This is due to the 4$b$ final state in the last diagram and
the mass difference between the top and the bottom quark in the first two
diagrams. The large mass splitting between top and bottom breaks the symmetry
in this weak isodoublet so that up and down contributions do not add up to
zero any longer. In the $\overline{MS}$ renormalization scheme one finds for
$N_F=5$ flavors
%*****************************************f4
\begin{equation}
 \begin{array}{rcl}
d_2&=&1.41\mp f_2(m_t) \qquad\qquad \mp \mbox{ for } q={\rm up/down}
\end{array}
\end{equation}
%*****************************************
$f_2(m_t)$ is well parametrized by
%*****************************************
\begin{equation}
f_2(m_t)=2\log\left(\frac{m_Z}{m_t}\right)-3.083+0.346
\left(\frac{m_Z}{2m_t}\right)^2 + 0.211
\left(\frac{m_Z}{2m_t}\right)^4
\end{equation}
%*****************************************
For a top quark mass of $175\pm 5$ GeV 
the function  $f_2(m_t)$ amounts to $-4.34\pm 0.06$. 
 
Also $d_3$ differs from $c_3$. 
The leading logarithms as well as the constant parts in 
the additional term have been evaluated in \cite{ChK308} and
\cite{CT327} respectively:
\begin{equation}
  f_3={23\over 12}\ln^2{m_Z\over m_t} +{67\over 18}\ln{m_Z\over
  m_t}-15.99
\approx -17.6.
\end{equation}
 
QCD corrections for arbitrary quark masses have been calculated up to first
order \cite{schwi1,jersa1,chety1} and recently even up to second order
\cite{hoa452,cks371}
for vector and axial vector contributions. For the present purpose an
expansion in $\mu_b=2m_b/m_Z$ is adequate. The leading terms are given
by
\begin{eqnarray}
\label{eqcd}
c_1 & = & 1+ 3\mu_b^2 + \dots \nonumber \\
d_1 & = & 1+3\mu_b^2\log (4/\mu_b^2) +\dots
\end{eqnarray}
The coefficient $d_1$ exceeds unity by $\sim 0.20$ for b quarks. The
large logarithm of $\mu_b$ that arises in $d_1$ (and in $d_2$,\dots as well as in
$c_2$,\dots) can be absorbed in a redefined running $b$ mass
\begin{eqnarray}
\overline{m}(s) & = & \overline{m}(\overline{m}^2)
\Bigl(\frac{\alpha_s(s)}{\alpha_s
(\overline{m}^2)}\Bigr)^{\gamma_0/\beta_0}
\Bigl\{ 1+\Bigl(\frac{\gamma_1}{\beta_0}-
\frac{\beta_1\gamma_0}{\beta_0^2}\Bigr)
\Bigl(\frac{\alpha_s(s)}{4\pi}-\frac{\alpha_s
(\overline{m}^2)}{4\pi}\Bigr)\nonumber \\
& & +\frac{1}{2}\Bigl[\Bigl( \frac{\gamma_1}{\beta_0}-
\frac{\beta_1
\gamma_0}{\beta_0^2}\Bigr) \Bigl( \frac{\alpha_s(s)}{4\pi}-
\frac{\alpha_s(\overline{m}^2)}
{4\pi}\Bigr)\Bigr]^2 \nonumber \\
& & +\frac{1}{2} \Bigl(\frac{\gamma_2}{\beta_0}+
\frac{\beta^2_1\gamma_0}
{\beta_0^3}-\frac{\beta_1\gamma_1}{\beta_0^2}-
\frac{\beta_2\gamma_0}{\beta_0^2}
\Bigr)
\Bigl( (\frac{\alpha_s(s)}{4\pi})^2-(\frac{\alpha_s(\overline{m}^2)}
{4\pi})^2\Bigr)\Bigr\}
\end{eqnarray}
%
\[ \gamma_0=4,\qquad\gamma_1=\frac{202}{3}-\frac{20}{9}N_f,
\qquad\gamma_2=1249-(
\frac{2216}{27}+\frac{160}{3}\zeta(3))N_f-
\frac{140}{81}N_f^2 \]
Recently even the four loop beta function $\beta_3$ \cite{beta3} 
and anomalous mass dimension $\gamma_3$ \cite{gamma3} have been
calculated. 
The ``on-shell mass'' $m$ which appears in (\ref{eqcd}) and
$\overline{m}(\overline{m}^2)$ are, for the $b$ quark, 
related through \cite{Schilcher}
\[ m=\overline{m}(\overline{m}^2)\Bigl(1+\frac{4}{3}
\frac{\alpha_s(\overline{m}^2)}{\pi} 
+9.6 (\frac{\alpha_s(\overline{m}^2)}{\pi})^2\Bigr)  \]
%
Evaluating higher order terms in $\alpha_s$, but keeping only corrections
of order $m_b^2/m_Z^2$ and summing logarithms of $m_b^2/m_Z^2$ one finds the
following correction for the partial width \cite{chety1,ckkpr}
%
\begin{eqnarray}
\Gamma(Z\to b\overline{b}+\dots)
&=&
\phantom{+} \Gamma^V_b
\biggr \vert_{m_b=0} \biggl[ 
1+\frac{\alpha_s}{\pi} +1.41(\frac{\alpha_s}{\pi})^2 
                       -12.77 (\frac{\alpha_s}{\pi})^3
+\dots
%\end{eqnarray}
%\begin{eqnarray}
%\quad\qquad\qquad
\nonumber\\
&&\qquad\qquad
+12\frac{\overline{m}^2(m^2_Z)}{m_Z^2}\Bigl(\frac{\alpha_s}{\pi}+
8.7(\frac{\alpha_s}
{\pi})^2 +45 (\frac{\alpha_s}{\pi})^3 +\dots\Bigr)\biggr]\nonumber \\
&&+\Gamma^A_b\biggr\vert_{m_b=0}
\biggl[1+\frac{\alpha_s}{\pi} +\Bigl(1.41+f(m_t)\Bigr)(\frac{\alpha_s}
{\pi})^2 -12.77 (\frac{\alpha_s}{\pi})^3 +\dots\nonumber \\
&&\qquad\qquad
-6\frac{\overline{m}^2(m^2_Z)}{m_Z^2}\Bigl(1+\frac{11}{3}
\frac{\alpha_s}{\pi}+
(11.3+\ln {m_Z^2\over m_t^2} )(\frac{\alpha_s}
{\pi})^2  +\dots\Bigr)\biggr]
\nonumber \\ && {}
\end{eqnarray}
%
The tiny QED corrections to the width $\delta\Gamma^{QED}/\Gamma^{QED}
=(1+\frac{3}{4}e_q^2\frac{\alpha}{\pi})$, being 0.017\% for down and
0.068\% for up quarks, are included in $\Gamma^V$ and $\Gamma^A$, as well
as the electroweak corrections.
 
A quantity of considerable practical interest is the ratio $\Gamma_b/
\Gamma_{had}$. The bulk of QCD corrections cancels in this ratio and
one finds \cite{ckkpr}
%
\begin{eqnarray}
\frac{\Gamma(Z\to b\overline{b}+\dots)}{\Gamma(Z\to had)}
&\approx& 
 \frac{\Gamma_b}{\sum_q\Gamma_q}\biggr\vert_{m_b=0}
\left\{
1-0.0031-0.005{\alpha_s\over \pi} 
-2.3 \left( {\alpha_s\over \pi}\right) ^2 
-6.9 \left( {\alpha_s\over \pi}\right) ^3
\right\}.\nonumber\\
&&
\end{eqnarray}
%
The error in the leading coefficient from the uncertainty in the
bottom mass amounts to $3\cdot 10^{-4}$ and is thus neglibible for all
practical purposes. For $m_b=4.7$ GeV and $\alpha_s=0.12$
corresponding to $\overline{m}_b(m_Z^2)=2.77$ GeV 
the overall correction factor yields
$(1-0.7\cdot 10^{-3})$.
At this point we should reiterate that b quarks can be produced also through
virtual gluons radiated off light quarks. These final states may be
interpreted as part of the $Z$ decay rate
into light quarks and treated in this
context. The $b$ energy spectrum is particularly  soft. These events
must be subtracted if one aims at an experimental determination of
$\Gamma_{b\overline{b}}$ as calculated in the electroweak theory.
 
The relative rate for this ``secondary'' $b$ and $c$ quark production
amounts to $\approx 2\cdot 10^{-3}$ and $(1-2)\cdot 10^{-2}$
\cite{HJKT}. 

Being proportional to the square of $\alpha_s$, the result is quite
sensitive to the choice of $\alpha_s$ and, furthermore, to the mass of
the bottom quark. These predictions are in fair agreement with the
experimental results $\overline{n}(g\to c\overline{c})=(2.27\pm
0.28\pm 0.41)\cdot 10^{-2}$ \cite{OPAL353} and 
 $\overline{n}(g\to b\overline{b})=(2.10\pm
1.1\pm 0.09)\cdot 10^{-3}$ \cite{DELPHI97}.

\lu{Genuine electroweak corrections} \cite{akhun1}--\cite{djoua1}, on the
other hand,
become sizeable for a large top mass $m_t \simeq 175$ GeV. They are also
sensitive to the value of the
Higgs mass. Two categories can be distinguished\footnote{The on--shell
renormalization scheme
with $\swin\equiv 1-m_W^2/m_Z^2$
will be adopted in the following discussion.}.
 
\noindent
(i) The universal electroweak corrections are built--up by loop corrections
to
the $W$,$Z$ propagators and $\gamma$--$Z$ mixing. They induce a universal
shift
of the Fermi coupling and the electroweak mixing angle
\begin{eqnarray}
\ltowidth{G_{F}}{\swin}  & \rightarrow &  G_{eff}=G_F
                                                 (1 + \Delta\rho) \\
\swin  & \rightarrow &  \sin^2\theta_{eff}=\swin [1 + \Delta \kappa_{SE}]
\nonumber
\end{eqnarray}
\noindent
The leading contributions to the Veltman parameter \cite{mvelt}
$\Delta\rho$ and to $\Delta\kappa_{SE}$ are due to $(b\overline{b}),
(t\overline{b})$ and $(t\overline{t})$ loops,
%******************************
\begin{eqnarray}
\Delta\rho_t & = & \frac{3\sqrt{2}G_F m_t^2}{16\pi^2} + \cdots \\
\Delta\kappa_{SE} & = & \vphantom{\frac{2^2}{3\pi^2}}
        \cot^2\theta_W \Delta\rho + \, \mbox{subleading $t$
         and $H$ contributions} \nonumber
\end{eqnarray}
%******************************
The electroweak mixing angle is related to the electric and weak coupling
constants in the usual way
\begin{equation}
m_Z^2 \cos^2\theta_W \swin = \frac{\pi \alpha(m_Z^2)}{\sqrt{2}G_F
(1+cot ^2 \theta_W\Delta \rho_t)}
\end{equation}
with the run\-ning e\-lec\-tric char\-ge $\alpha(m_Z^2)=1/128.896...$
\cite{seidel} and
subleading top and Higgs contributions again neglected.
 
\noindent
(ii) Of particular interest are the electroweak vertex corrections for
$b$--quark final states because they involve the exchange of virtual top
quarks.
These diagrams alter the vector and axial--vector charge of $b$ quarks by
the same amount
 
{\hangindent=7.9cm \hangafter=-5
\begin{eqnarray}
v_f & \rightarrow & v_{eff}=v_f + \delta_b \frac{2}{3} \Delta\rho\\
a_f & \rightarrow & a_{eff}=a_f + \delta_b \frac{2}{3} \Delta\rho \nonumber
\end{eqnarray}
}
 
\noindent
where subleading  t contributions have been suppressed again.

In view of the high precision reached in the  LEP experiments, even 
the calculation of mixed
QCD-electroweak corrections has become necessary. The top
contributions to the $\rho$ parameter and $\Delta r$ have been
calculated in order $G_F m_t^2 \alpha_s$ \cite{DVER,KKST} and  the
three--loop results of order $G_F m_t^2 \alpha_s^2$ are now available
\cite{AFMT,CKSt}. This leads to an overall shift in $m_t$ of roughly
10 GeV, which should be compared with the current experimental error
in precision measurements of 7 GeV. To interprete correctly the
precise determination of $R_b$ in the context of the SM (or its SUSY
extensions), the mixed QCD-electroweak corrections to the $Zb\bar b$
vertex must be included. The leading terms of order $G_F m_t^2
\alpha_s$ and $G_F \alpha_s \ln(m_t^2/m_Z^2)$ have been obtained
  in Ref.~\cite{FTJR}. Non-enhanced corrections of order $G_F \alpha_s
  m_Z^2$ are
  presently  available for light flavour final states only
  \cite{czk}. 
 
Utilizing effective couplings, 
$G_{eff}, \sin^2\theta_{eff}$ and $v_{eff}, a_{eff}$,  
 in the Born expression for the partial
width $\Gamma(Z \rightarrow b\overline{b})$ in eq.(1) and the
forward--backward asymmetry to be discussed later, the dominant electroweak
corrections are incorporated in a compact and elegant form.


%******************************
%***************************
%\begin{figure}[hbt]
%\vspace{9.5cm}
%\caption[dummi]{ \label{ratedecom} a) The dependence of 
%$\Gamma_b/\sum_q\Gamma_q$ on $m_t$
%($m_b=0$, no QCD corrections)
%b) The decomposition of $\Gamma_b$ into vector and axial vector parts.}
%\end{figure}
%******************************


\begin{table}[hbt]
\begin{center}
\begin{tabular}{|c||c|c||c|l|}
\hline
& & & & \\
 & TAG   &  $BR (Z \to b \overline{b})$  & TAG & $BR (Z \to c \overline{c})$ \\
& & & & \\
\hline
\hline
& & & & \\
Aleph   & multi &  0.2156 (9)(11)  & $D^{*\pm}$ & 0.1724 (98)(100)   \\
& & & lepton & 0.1649 (70)(106) \\
Delphi  & multi  & 0.2202 (14)(17)  & c$\_$count $\oplus D^{*\pm}$ & 
0.1692 (80)(83) \\
& & & & \\
 L3  & lifetime & 0.2185 (28)(32)  && \\
& & & & \\
Opal & multi   & 0.2175 (14)(15) & c$\_$count & 0.1670 (110)(120)  \\
& & & & \\
\hline
& & & & \\
SLD & multi    & 0.2149 (34)(16) & & \\       
& & & & \\
\hline
\hline
& & & & \\
Average & & $0.2177\pm 0.0011$ & & $0.1722\pm 0.0053$ \\
& & & & \\
\hline
\hline
& & & & \\
Standard Model & & $0.2158\pm 0.0003$ & & $0.1723\pm 0.0001$ \\
& & & & \\
\hline
\end{tabular}
\end{center}
\caption[table]{\label{tabbranch}
{\it Excerpts of experimental results; for a comprehensive summary see 
\cite{rclare}. Measurements are quoted for $bb$ only if the stat/syst
errors are less than (40)(40), for $cc$ if less than (120)(120).}}
\end{table}


The $b$ partial width of the $Z$ boson including all QCD, QED and genuine
electroweak corrections may then be summarized as 


\begin{eqnarray}
\Gamma(Z \rightarrow b\overline{b} + \cdots)\biggr\vert_{m_b=0} &
= & \phantom{\mbox{} + \mbox{}}
\Gamma_B^V(G_{eff},v_{eff}) \biggr\vert_{m_b=0}
\left[ 1 + \frac{3}{4} \frac{e^2_b \alpha}{\pi}
\right] \left[ 1 + c_1 \left(\frac{\alpha_s}{\pi}\right) + \cdots \right]
\nonumber \\
\nonumber\\
& & \mbox{} + \Gamma_B^A(G_{eff},a_{eff}) \biggr\vert_{m_b=0}
\left[ 1 + \frac{3}{4} \frac{e^2_b \alpha}{\pi}
\right] \left[ 1 + d_1 \left(\frac{\alpha_s}{\pi}\right) + \cdots \right]
\nonumber\end{eqnarray}
%***********************************************************

%%%%%%%%%%%%%%%%%%

\noindent
The partial widths including all these corrections are given for $b$ 
decays by 
\begin{eqnarray}\Gamma (Z \rightarrow b \overline{b} + \dots)
/\Gamma_{had} & = & 0.2157 \pm \ 0.0003 \nonumber \\
\end{eqnarray}

\noindent
and for $c$ decays by

\begin{eqnarray}
\Gamma (Z \rightarrow c \overline{c} + \dots)
 /\Gamma_{had} & = & 0.1724 \pm 0.0001  \nonumber \\
\end{eqnarray}
These values, derived within the Standard Model, are based on 
$\sin^2 \vartheta^{l, eff}_w = 0.2317 \pm 0.0003 $ and a top mass
of $175 \pm 6$ GeV in the higher--order corrections.

The experimental results for $Z$ decays to $b$ quarks and $c$ quarks
are collected in Table \ref{tabbranch} for the four LEP experiments
Aleph, Delphi, L3 and Opal, and the SLC experiment SLD. The world
average of the $b$ decay branching ratio agrees with the
theoretical prediction within less than two standard deviations.
While the Aleph results match the theoretical prediction
almost perfectly, the slight positive deviation in the overall picture
leaves room for small contributions from areas beyond the Standard Model,
{\it notabene} supersymmetric theories. However,  
supersymmetric theories do not necessarily 
lead to  deviations from the 
Standard Model which could be detected within the present experimental
accuracy.   

%\begin{figure}[p]
%\vspace{10cm}
%\caption[figur]{ The dependence of d, b and c forward-backward
%asymmetries on the top and Higgs masses.}
%\end{figure}
%***************************

 
 
The close agreement of the experimental data with the predictions of the
Standard Model allows us to draw interesting physical conclusions.

\newpage
 
\noindent (i) The \lu{weak isospin of the left-- and right--handed 
bottom quarks} are, respectively, 
\begin{eqnarray}
I_3(b_L)  & = & -1/2  \nonumber \\
I_3(b_R) & = & 0 \nonumber
\end{eqnarray}
A topless model with $I(b) = 0$ would have been in gross disagreement
with the $Z$
decay width to $b$ quarks
\begin{equation}
\frac{\Gamma(b\overline{b})^{topless}}{ \Gamma(b\overline{b})^{SM} } =
\frac{\left(4 e_b \swin \right)^2}{\left( 1 + 4 e_b \swin
\right)^2 + 1 } \approx \frac{1}{13}
\end{equation}

%
%\begin{figure}[hbt]\vspace{9cm}
%\caption[figur]{ \label{figT} T as a function of
%$m_2=m_3$ for $\alpha_s=0.12\pm 0.02$ with the experimental $1\sigma$ bound.
%The \ssm value is
%indicated by diamonds together with the uncertaintities in $\alpha_s$.}
%\end{figure}
%******************************
 
\noindent (ii) The universal electroweak corrections can be eliminated from
a combination of $Z$ decay observables and the specific \lu{$b$ vertex
corrections} \cite{girar1} can be isolated in this way:
\begin{eqnarray}
T & = & \frac{3}{59} \frac{\Gamma_{had}}{\Gamma_l} - \frac{30}{59}
\frac{9}{\alpha(m_Z^2)} \frac{\Gamma_l}{m_Z} \\
& \propto & 1 + \Delta_V^b \nonumber
\end{eqnarray}
%********************************
The \ssm top contribution to $ \Delta_V^b =
-\frac{20}{19} \frac{\alpha}{\pi} \frac{m_t^2}{m_Z^2} - \cdots $ is
negative for $m_t > m_Z$. And so are vertex corrections in the minimal
supersymmetric extension of the Standard Model. $Z$ mixing with new E(6) type
$Z'$ bosons, on the other hand, may either lower or raise the vertex
correction $\Delta_V^b$ relative to the Standard Model.
The same conclusion applies for vertex corrections in a general two--Higgs 
doublet model
with large $\tan \beta=v_2/v_1$, if the masses of scalar $(m_1,m_2)$ and
pseudoscalar $(m_3)$ Higgs bosons are not too large \cite{denne1}.
 
 
%\include{heaviref}
%\end{document}
\subsection{{Forward--backward asymmetry}}
 
 
For quarks tagged at an angle $\vartheta$ the differential production
cross section $e^+e^-\to b+\dots$
is a binomial in $\cos\vartheta$
\begin{equation}
{d\sigma\over d\cos\vartheta}=\frac{3}{8}(1+\cos^2\vartheta)\sigma_U +
 \frac{3}{4}\sin^2\vartheta\sigma_L + \frac{3}{4}\cos\vartheta\sigma_F
\end{equation}
$U$ and $L$ denote the contributions of unpolarized and longitudinally
polarized gauge bosons along the $\vartheta$-axis, and $F$ denotes the
difference between right and left polarizations. The total cross
section is the sum of $U$ and $L$,
\begin{equation}
\sigma = \sigma_U +\sigma_L
\end{equation}
The forward-backward asymmetry for a fixed angle $\vartheta$ vs.\
$\pi-\vartheta$ can be cast into the form
\begin{equation}
A^{FB}(\vartheta)={2\cos\vartheta\over 1+\alpha_Q\cos^2\vartheta}\beta_Q
\end{equation}
with the parameters
\begin{equation}
\alpha_Q={\sigma_U-2\sigma_L\over\sigma_U + 2\sigma_L} \qquad\qquad
\beta_Q={\sigma_F\over\sigma_U+2\sigma_L}
\end{equation}
If only a limited angular range $|\cos\vartheta|\leq x$ is covered,
the integrated asymmetry is modified to
\begin{equation}
A_{FB} (|\cos\vartheta|\leq x) = {3\over4}{x\sigma_F\over \sigma_U +
\sigma_L
-(1-x^2)(\sigma_U-2\sigma_L)/4}
\end{equation}
Including the full angular range this simplifies to
\begin{equation}
A^{FB}={3\over 4}{\sigma_F\over\sigma}
\end{equation}
 
The $\sigma^i$ can be
expressed in terms of the cross sections for the massless case ($\mu=0$);
in Born approximation,
\begin{eqnarray}
\sigma_B^U &=&\beta\sigma^{VV}+\beta^3\sigma^{AA}\nonumber \\
\sigma_B^L &=&{1\over 2}(1-\beta^2)\beta\sigma^{VV}\nonumber \\
\sigma_B^F &=&\beta^2\sigma^{VA}
\end{eqnarray}
with
\begin{eqnarray}
\sigma^{VV} &=&{4\pi\alpha^2(m^2_Z)e^2_ee^2_Q\over s} +
       {G_F\alpha(m^2_Z)\over\sqrt{2}}e_ee_Qv_ev_Q
       {m^2_Z(s-m^2_Z)\over(s-m^2_Z)^2+({s\over m_Z}\Gamma_Z)^2}\nonumber \\
&& +{G_F^2\over 32\pi}(v^2_e + a^2_e)v^2_Q
       {m^4_Zs\over(s-m^2_Z)^2+({s\over m_Z}\Gamma_Z)^2}\nonumber \\
\sigma^{AA} &=& {G_F^2\over 32\pi}(v^2_e+\alpha^2_e)a^2_Q
       {m^4_Z s\over(s-m^2_Z)^2+({s\over m_Z}\Gamma_Z)^2}\nonumber \\
\sigma^{VA}&=&{G_F\alpha(m^2_Z)\over\sqrt{2}}e_ee_Qa_ea_Q
  {m^2_Z(s-m^2_Z)\over (s-m^2_Z)^2+({s\over m_Z}\Gamma_Z)^2} \nonumber \\
  && +{G^2_F\over 8\pi}v_ev_Qa_ea_Q
  {m^4_Z s\over (s-m^2_Z)^2+({s\over m_z}\Gamma_Z)^2}
\end{eqnarray}
On top of the $Z$ resonance the asymmetry is dominated by the  $Z$
amplitude, so that to a very good approximation
\begin{eqnarray}
A^{FB}_B &=&{3\over 4}{2v_ea_e\over v^2_e+a^2_e}
  {2v_Qa_Q\beta\over v^2_Q(3-\beta^2)/2 + a^2_Q\beta^2}\nonumber \\
&\approx &
  {3\over 4}{2v_ea_e\over v^2_e + a^2_e}
  {2v_Qv_Q\over v^2_Q+a^2_Q}
\end{eqnarray}
As a consequence of an accidental cancellation among the couplings
$(v^2_b\approx a^2_b/2)$, the mass correction
$\simeq\mu^2(-v^2_Q+a^2_Q/2)/(v^2_Q+a^2_Q)$ is strongly
suppressed to ${\cal O}(10^{-4})$.
 
The first
factor,
$$P_{long} = - 2v_e a_e / (v_e^2 + a_e^2)$$
measures the degree of $Z$
polarization along the $e^-$ axis while the same factor for the $b$ quarks
describes their analysing power.
 
In the neighborhood of the $Z$ peak the variation of the asymmetry
is approximately linear in the energy
\begin{equation}
A_{FB}(\sqrt s) - A_{FB}(m_Z)=
      {8\pi\alpha\sqrt2\over G_\mu m_Z^2} A_{FB}(m_Z)
\biggl ({e_e e_Q\over v_e v_Q} -
                              {e_e e_Q\over a_e a_Q}{4\over 3} A_{FB}\biggr)
                               {\sqrt s - m_Z \over m_Z}
\end{equation}
\[
\approx 8.6 {|e_Q| \over v_Q^2 + 1 } {\sqrt s - m_Z \over m_Z}
     ={\sqrt s - m_Z \over m_Z} \left\{ \begin{array}{ll}
8.6 & \mbox{ for muons} \\
5.0 & \mbox{for charmed quarks} \\
1.9 & \mbox{for bottom quarks} \end{array} \right. \]
 
%In contrast to the asymmetry itself the slope is quite insensitive
%towards the exact value of $\sst$.
 
 
 
%******************************
\begin{equation}
A^{FB}(b) = \frac{3}{4} \frac{2 v_e a_e}{v_e^2 + a_e^2}
\frac{2 v_b a_b }{v_b^2 + a_b^2}
\end{equation}
%*****************************
 
As a result of the general discussion in the preceding section, the vector
charges $v_e$ and $v_b$ are defined by their respective effective mixing
angles $\swin^{eff}$ in eq. (12,15), thus incorporating the
leading $m_t$ dependent
electroweak corrections in the asymmetry.


%******************
%\begin{figure}[hbt]
%\vspace{9.5cm}
%\caption{\label{AFB}
%The dependence of $d, b$ and $c$ forward--backward asymmetries on the
%top and Higgs masses.}
%\end{figure}
%******************
 
 
 
Once QCD and QED corrections are included, a careful definition of the
scattering angle is necessary. Two different definitions appropriate
for heavy quark production have been used:
\goodbreak
(1) \underline{$b$ quark direction:} $e^+e^-\to b(\vartheta_b)+\ldots$~~~~
(2) \underline{thrust axis:} $e^+e^-\to j_b(\vartheta_T)+j$
\nobreak
\vspace{3cm}
\goodbreak
 
QCD corrections to the cross section and
$A_{FB}$, based on the first definition, have been
calculated in \cite{jersa1}.
For small $\mu$ the formulae simplify considerably.
On top of the $Z$, the corrections to the
various terms  can be cast into the form
\begin{equation}
\sigma_i=\sigma^0_i \biggl(1+c_i {\alpha_s\over\pi}\biggr)
\end{equation}
with \cite{djoua1, arbuz}
\begin{eqnarray}
c_F &=&{8\over 3}\mu + {\mu^2\over3} \biggl(7 + {\pi^2\over6} + \log
{4\over\mu^2} + {1\over4} \log^2 {4\over\mu^2} \biggr) + \dots \approx 0.367\nonumber \\
c_V &=& 1+3\mu^2 +\ldots\approx 1.033\nonumber \\
c_A &=& 1+3\mu^2\log\frac{4}{\mu^2}+\ldots\approx 1.196
\end{eqnarray}
(numerical values for $\mu = 2\times 4.8\  GeV/91\ GeV$).
After introducing  the running quark mass in the Born
and ${\cal O} (\alpha_s)$ expressions, the large logarithm
in $C_A$ is absorbed (see eq. 1.10). A similar study for $c_F$ has not been
performed so far. 
Note that $c_F$ vanishes in the zero-mass limit.
The correction to $A_{FB}$ on top of the resonance then reads
\begin{equation}
A^{FB}_0\to A^{FB}\approx
         A^{FB}_0\biggl(1-{\alpha_s\over\pi}\biggl(1-c_F+3\mu^2
          {v_Q^2 + a_Q^2\log\frac{4}{\mu^2}\over v_Q^2 + a_Q^2}\biggr)\Biggr)
\end{equation}
with the leading part of the correction due to the change of the normalization.
The dominant effect is thus a reduction of the asymmetry by about
4\% through a factor $(1-\frac{\alpha_s}{\pi})$  for $\mu^2 = 0$;
mass corrections lower the the coefficient of $\alpha_s\over\pi$
from $1$ to about $0.8$.
If $\vartheta_b$ is not summed over,
corrections are needed for the vector
and axial-vector parts of the U and L terms separately and can be found
in \cite{djoua1}.
               Analytical results for QCD corrections, including quark
mass effects and a cut on the gluon energy, can be deduced from
\cite{Leike}.
%****************************
\begin{table}[hbt]
\begin{minipage}[t]{10cm}
\caption[table]{\label{djouadi} {\it QCD corrections to forward--backward 
asymmetries of
$b$ quarks in 2--jet events \cite{djoua1} . }}
\end{minipage}\hspace{1.5cm}
\begin{tabular}[t]{|c|c|}
\hline
y    &   $k_A$ \\
\hline
0.02 & -0.05 \\
0.04 & -0.14 \\
0.08 & -0.34 \\
\hline
\end{tabular}
\end{table}
%****************************
 
Similar results are obtained if the thrust axis is chosen
as the reference axis.
If the jet masses are
bounded to values
below $m_j^2 \le y m_Z^2$, the QCD correction in 2--jet events
%******************************
\begin{equation}
A^{FB}(\mbox{2--jet}) = A^{FB}(b) \left[1 + k_A \left( \frac{\alpha_s}{\pi}
\right) \right]
\end{equation}
%******************************
remains small, Table \ref{djouadi}. This is intuitively expected as infrared and
collinear gluon bremsstrahlung leave the $b$--quark direction nearly
unaffected.
 
 
The impact of  QED corrections on $A_{FB}$ for $b \bar{b}$ final
states is small for loose cuts on the photon energy and angle.
 
Final state radiation is treated in complete analogy to gluon
radiation. Defining the scattering angle through the direction of the
$b$ quark (as adopted above) the asymmetry is reduced by a factor
$(1+{3\over 4}Q^2_b{\alpha\over\pi})^{-1}$ through the increase in the
overall normalization, an effect completely negligible.
 
Analytical results for the influence of the interference between
$\gamma$-emission from
the initial and final states can be found in \cite{Rieman}, [12]
for the $\mu$-pair final states.
Although two different definitions of the scattering angle are adopted,
for loose cuts the effect is found to be negligible, just as for the total
cross
section \cite{JKSW}.
Around the $Z$ peak the corrections are of relative
order $<{\alpha\over\pi}(\Gamma_Z/m_Z)$.
 
The influence of initial state radiation is strongly dependent on the energy.
It is modest below and on top of the $Z$ peak. A full treatment,
including the impact on the kinematics and the change of the
effective energy can only be performed in a Monte Carlo calculation.
Formulae have been developed for the $\mu$-pair asymmetry,
which are based on the approximation of collinear radiation; they
are accurate to better than
$0.3\times 10^{-2}$ and can easily be improved to include also
multiple photon emission. Without any
cut on the photon energy \cite{JW,Citoun} (see also \cite{Rieman})
\begin{equation}
\langle A_{FB}\rangle = \frac{3}{4}
\frac{\int^1_0 dz F(s,z)({1-z\over 1+z})^2
\sigma_F(sz)}{\int^1_0 dz F(s,z)\sigma (sz)}
\end{equation}
 
\noindent
To ${\cal O} (\alpha)$, the distribution $F(s,z)$ is given by the
Bonneau-Martin formula,
to ${\cal O} (\alpha^2)$ -- including exponentiation -- $F(s,z)$ is
given in \cite{Burgers}.
For $\sqrt{s}$ about 200 MeV above $m_Z$ the corrected
asymmetry coincides with $A_0^{FB}(m_Z)$.
A simple analytical calculation gives
$\delta W=\alpha(\ln(s/m_e^2) - 1)\Gamma_Z)/2=217\ MeV$ for this shift.
Conversely it is clear from eq. (28) that the asymmetry for $\sqrt s=m_Z$
is lowered by
\[
\delta A|_{QED}=8.6{|e_Q|\over v_Q^2 + 1}{\delta W\over m_Z} =
\left\{ \begin{array}{ll}
        0.46\times 10^{-2} & \mbox{for bottom  quarks} \\
               1.2 \times 10^{-2} & \mbox{for charmed quarks}
\end{array} \right. \]
 
 
 
 
{\lu{Mixing}}.
Forward--backward asymmetries of quarks are extracted by measuring
the asymmetries of mesons and baryons endowed with this particular quark
flavor. For $b$ quarks this is not a one--to--one correspondence due to
mixing in the neutral $B_d^0$ and $B_s^0$ systems \cite{bigi1}.
Because associate
production of $b\overline{b}$ pairs in the jet development, perturbative as
well as non--perturbative, can safely be neglected, the hadronic asymmetries
are related to the asymmetry of the primarily produced $b$ quarks in the
following way.
 
\noindent(i)~{\em b tagging through} $B_u^-$, $\Lambda_b$:
\nopagebreak
%****************************************
\begin{equation}
A^{FB}(B^-_u,\Lambda_b)=A^{FB}(b)
\end{equation}
%****************************************
 
\noindent(ii)~{\em b tagging through}
$\overline{B}_d^0$ or $\overline{B}_s^0$:
 
\nopagebreak
\vspace{-\parskip}
 
\noindent These particles mix with their partners $\overline{B}_d^0
\leftrightarrow B_d^0$
and $\overline{B}_s^0 \leftrightarrow B_s^0$ so that the $b$ flavor is changed
at a rate comparable or much more frequent than the decay rate for
$\overline{B}_d^0$ and $\overline{B}_s^0$, respectively. Denoting the
time--integrated
probability by $\chi_q$
for observing a $B_q^0$ particle in a beam of originally produced
$\overline{B}_q^0$ particles, these parameters have been determined in
 time--integrated and time--dependent measurements (see
Sect. \ref{sec:mix}),  
 for $B_d^0$: $\chi_d= 0.174\pm 0.006 $, or are bounded strongly
by the
Standard Model and experimentally very close to 1/2 for $B_s^0$.
 Mixing reduces the observed asymmetry so
that
%****************************************
\begin{equation}
A^{FB}(\overline{B }_q^0) = (1-2\chi_q)A^{FB}(b)
\end{equation}
%****************************************
While the impact of $B_d$ mixing is already quite noticeable $\sim 35\%$,
the asymmetry in the $B_s$ system is almost completely washed out.
 
\noindent(iii)~{\em b tagging through} $l^-$:
 
\vspace{-\parskip}
 
\nopagebreak
 
\noindent Denoting the probability to find a $\overline{b}$ quark in a $b$ beam
by
$\overline{\chi}$, a weighted mixture of $B_d$ and $B_s$ mixing parameters,
this
probability is directly measurable in the like--sign dilepton rate,
%****************************************
\begin{equation}
R_{ll}=\frac{l^\pm l^\pm }{\mbox{all }ll }= 2\overline{\chi
}(1-\overline{\chi })
\end{equation}
%****************************************
With $\overline{\chi}=0.126\pm 0.008$  \cite{Danilov} the ensuing
lepton \ass 
%****************************************
\begin{equation}
A^{FB}(l^-)=(1-2\overline{\chi })A^{FB}(b)
\end{equation}
%****************************************
is reduced to $\sim 75\%$ of the original $b$ quark \ass.

%****************************
%\begin{table}[hbt]
%\begin{center}
%\begin{tabular}[t]{|c|c|}
%\hline
%    &   $A_{FB}(l^-)$ \\
%\hline
%Aleph & $0.093\pm 0.021\pm 0.005$ \\
%Delphi &$0.115\pm 0.043\pm 0.014$ \\
%L3 &$0.084\pm 0.025\pm 0.010$\\
%Opal &$0.072\pm 0.042\pm 0.010$\\
%\hline
%LEP &$0.090\pm 0.015\pm 0.005$ \\
%\hline
%\end{tabular}
%\end{center}
%\caption[]{\label{tabafb} Recent experimental results on $A_{FB}(l^-)$,
%         not corrected for mixing (from\cite{Roudeau}).}
%\end{table}
%****************************
 
The forward--backward \ass of $b$--quarks is highly sensitive to the precise
value of the electroweak mixing angle $\swin$. This follows from the fact
that the term
%****************************
$$P_{long}(b) = -2v_b a_b/(v_b^2+a_b^2)\approx -0.93$$
%****************************
in the \ass is close to its maximum
and almost constant so that $A^{FB}(b) \approx -\frac{3}{4} P_{long}$ nearly
coincides with the degree of the $Z$ polarization and the
corresponding LR asymmetry. This asymmetry
is well known to be strongly dependent on $\swin$ because
$2v_e a_e/(v_e^2+a_e^2)\sim 2(1-4 \swin )$ falls
steeply near $\swin \sim 1/4$. Let us assume, for example, that $b$ quarks
are tagged through the semileptonic decays of $B$ mesons, then
%****************************************
\begin{equation}
A_{FB}(l^-)=A_{FB}(b)\biggl[1-2\overline{\chi }
\biggr]\biggl[1-\kappa\frac{\als }{\pi }  \biggr]
\end{equation}
%****************************************
For $\overline{\chi}=0.15$ and $\kappa\frac{\als}{\pi}= 0.03$ this leads to a
sensitivity of
%****************************************
\[
\delta\swin = \delta A_{FB}(l^-)/3.8 \quad .
\]
%****************************************
Because the error migrating from the mixing parameter $\overline{\chi}$ to
$A_{FB}$ is proportional to the \ass itself, its impact is small. An
error
of $\delta\sin^2\vartheta_W \sim 0.0004$ has been achieved so far, 
so that  
this method competes well with other measurements of
$\swin$ in the purely leptonic $\mu$ and $\tau$ sectors.  
The FB asymmetry of $b$ quarks measured at LEP \cite{rclare}, 
$A^b_{FB}  = 0.0979 \pm 0.0022$, 
is  smaller than the expectation in the Standard Model,
i.e. 01017. This gives rise to a slightly larger value of the 
electroweak mixing angle compared to the LEP + SLC average of 
$\sin^2 \vartheta^{lep}_{eff} = 0.23153 \pm 0.00023$.
 The SLC value of the 
LR/FB asymmetry ($A^b = 0.897 \pm 0.047$) coincides with the
\ssm prediction (= 0.934), albeit with larger error due to 
the reduced statistics. The evolution of the asymmetry from 
PETRA/PEP energies via TRISTAN to LEP/SLC is illustrated in 
Fig.~\ref{com}.
 
\begin{figure}[hbt]
\vspace{10cm}
\caption[]{\label{com} {\it Comparison of the Standard Model prediction for the
forward-backward asymmetry of $b$ quarks with data; full line:
$b$ asymmetry, broken line: observed $\ell^-$ asymmetry including
mixing effects. (Adopted from \cite{Abe}.)}}
\end{figure}
 
 
 















































































