\subsection{{Final State Polarization}}
Due to the parity violation of the electroweak interactions, quarks
are produced with non-zero polarization in \epem annihilation. For
``light'' heavy flavours ($c,b$), only the longitudinal component along
the flight direction is sufficiently large \cite{kuehn2} to be
accessible experimentally, while transverse and perpendicular
components are suppressed $\sim m_Q/m_Z$. For $\sqrt s= m_Z$ and
neglecting non-zero-mass corrections the variation of the
longitudinal polarization with the scattering angle $\vartheta$ is given by
\begin{equation}
P_L(\vartheta) = -{\lambda_Q(1+\cos^2\vartheta)
+\lambda_e\cdot 2\cos\vartheta
\over (1+\cos^2\vartheta)+\lambda_e\lambda_Q\cos\vartheta}
\end{equation}
where $\lambda_Q=2v_Qa_Q/(v^2_Q+a^2_Q)$ etc.
Because the $Z$ vector coupling to
electrons is small, the variation with $\vartheta$ is moderate and, in
contrast to the $\tau$ polarization, the average value
($\sin^2\theta_W=0.23$)
\[
P_L=-\lambda_Q=\left\{ \begin{array}{ll}
-0.68 &\mbox{for charm}\\
-0.94 &\mbox{for bottom} \end{array} \right. \]
provides an adequate approximation. Thus the longitudinal quark
polarizations are almost maximal.
The observable polarization is strongly affected by the
hadronization.
Assuming 100\% longitudinal quark polarization, one expects
{\it vector and pseudoscalar
mesons\/} to be produced with the relative proportion
50:25:0:25 for the different spin configurations
$[S,S_z]=$ $[1,1]:[1,0]:[1,-1]:[0,0]$.
Similar relations could be derived
for baryons. Experimentally little is known about the spin
transfer in hard reactions.
A variety of decay modes have been investigated to measure the
baryon polarization, in particular for $\Lambda_c$ and $\Lambda_b$. The
semileptonic $\Lambda_c\to \Lambda\ell^+\nu_\ell$ decay
\cite{Sehkozer} as well as the
nonleptonic decays \cite{Bjork} appear to be
promising candidates.
The polarization of $\Lambda_b$ can be studied in particular
through an analysis of the neutrino energy spectrum or its
moments \cite{czarn}.
Besides the parity-violating polarization itself, spin-spin
correlations between \linebreak
quarks and antiquarks have been studied. The formalism
is somewhat involved and we refer to the original papers
\cite{kuehn2,Dalitz} for details.
\section[Fragmentation]{Fragmentation}
The \fras of quarks into hadrons can qualitatively be deduced from QCD, yet a
rigorous quantitative understanding is still lacking due to its complex
nature. The intertwining of perturbative and nonperturbative mechanisms has
therefore been formulated in algorithmic models. The gross characteristics of
hadron (energy) distributions are adequately described in the independent jet
\fras model \cite{field1}--\cite{ali1}. This model is based on just one
general assumption:
that quark--gluon configurations in the femto--universe transform into jets,
bunches of hadrons with small relative transverse momenta, at large distances.
Not relying on unproven theoretical hypotheses, the model provided a solid
base for establishing experimentally the existence of gluon jets.
Picturing
the basic jet structures sufficiently well, the model is too crude though to
account properly for the small particle flow between the jets. Two different
approaches have been pursued which refine this picture. Non--perturbative
strong forces are the driving forces in the string picture \cite{ander1}, an
approximation to the color flux tube stretched between quarks in QCD.
Perturbative gluon bremsstrahlung, on the other hand, is considered to be the
main mechanism for building--up jets in parton shower models \cite{march1}.
Even though a mixture of both elements is responsible for the jet development
as a whole, it remains an important physical problem to explore the dynamical
driving mechanism of the jet formation. Heavy quark jets
are the simplest systems
to study the hadronization mechanism because spontaneous heavy quark--pair
production is suppressed in the jet evolution.
Since the non--perturbative binding of a heavy quark $Q$ into a hadron
$(Q\overline{q})$ is the final step after the perturbative radiation of gluons,
the fragmentation function
%****************************
\begin{equation}
D(z) = \int^1_z \frac{d\xi}{\xi} d_{NP}(z / \xi) d_{PT}(\xi)
\end{equation}
%****************************
is built--up by the convolution of the non--perturbative fragmentation function
$d_{NP}$ with the perturbative fragmentation function $d_{PT}$. $\xi$
is the fraction of energy [in units of the initial energy]
residing in the quark $Q$ after the early
perturbative
gluon radiation, $z$ the fraction of energy
in the $Q$ flavored hadron.
The average energy fraction $\langle z \rangle$ is the product of
the energy fraction $\langle z \rangle_{PT}$ after gluon radiation and the
fraction $\langle z \rangle_{NP}$ transferred from the quark to the hadron
non--perturbatively,
%****************************
\begin{equation}
\langle z \rangle = \langle z \rangle_{NP} \langle z \rangle_{PT}
\end{equation}
%****************************
Both mechanisms are described in the next two
subsections.
In contrast to light quarks, heavy quark fragmentation is hard because of the
inertia of heavy quarks. The energy transfer by soft non--perturbative
hadronization is small, and similarly for perturbative gluon radiation as long
as $\log E /\log m_Q \rtowidth{\gg}{} \ctowidth{/}{\gg} 1$.
\subsection{{Gluon bremsstrahlung}}
\oldsize=\hsize \skipsize=6cm \advance\oldsize by -\skipsize
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Perturbative radiation is described by the non--singlet Altarelli--Parisi
equation \cite{altar1}, the solution of which leads to an energy fraction
\begin{equation}
\langle z \rangle_{PT} = \left[ \frac{\als(E^2)}{\als(m_Q^2)}
\right]^{\frac{32}{3(33 - 2N_f)}}
\end{equation}
%************
left for the heavy quark after the perturbative gluon radiation
\cite{bigi2}.
For $E=m_Z/2$ in $Z$ decays, the $b$ quark looses about $20\%$ of its initial
energy to perturbative gluons. In next--to--leading order perturbation
theory, the perturbative fragmentation function $d_{PT}(z)$ has a sharp
maximum \cite{mele1}
just below 1 so that the fragmentation function vanishes at the
boundary as a result of repeated infrared radiation, Fig.\ref{shape}. \par
}
%******************************
%\begin{figure}[t]
%
%\vspace{11cm}
%\caption[figur]{\label{shape}
%Shape of the perturbative $b$ fragmentation
%function \cite{mele1}.}
%\end{figure}
%******************************
%****************************************
%\begin{figure}[b]\vspace{5cm}
%\caption[figur]{\label{Bj} Momentum distribution of the hadrons produced in \epem
%$\rightarrow \overline{Q} Q$, viewed in the \epm\ center--of--mass frame;
%from
% \cite{bjork1}.}
%\end{figure}
%****************************************
Perturbative gluon radiation alone is insufficient to account
for the spectrum of $b$ mesons as measured at LEP, $\langle z \rangle_B =
0.705\pm 0.011$ (to be compared with $\langle z\rangle _D=0.5088\pm 0.015$)
\cite{Roude}. Furthermore, with a probability of a few percent,
events without
radiation are predicted which require non--perturbative mechanisms to bridge the
color gap between quarks and antiquarks in the final state.
This is evident in particular for heavy
quarks where perturbative gluon radiation
\begin{equation}
dN_g = \frac{4\als}{3\pi}
\frac{\vartheta^2 d\vartheta^2}{\left[ \vartheta^2 + (m_Q/E)^2
\right]^2 }
\end{equation}
is concentrated on the surface of a cone about the $Q$ direction
with half--aperture $\vartheta_c = m_Q/E$ [$\sim 6^o$ for $b$ quarks at LEP]
while
the interior is depleted from gluons.
\subsection{{Non--perturbative fragmentation}}
In contrast to light--quark hadronization the \fras of heavy quarks is hard
\cite{bjork1}--\cite{peter1} because the inertia carried by the heavy quarks
is retained in the heavy--flavour
hadrons. This can most transparently be proven by analyzing the hadronization
$Q\,\rightarrow\, (Q\overline{q})\, + \, q$ in the rest frame of the heavy
quark
$Q$. In this frame the left--over \linebreak
%******************************
\begin{figure}[t]
\vspace{11cm}
\caption[figur]{\label{shape}
{\it Shape of the perturbative $b$ fragmentation
function \cite{mele1}.}}
\end{figure}
%******************************
%****************************************
\begin{figure}[b]\vspace{5cm}
\caption[figur]{\label{Bj} {\it Momentum distribution of the hadrons produced in \epem
$\rightarrow \overline{Q} Q$, viewed in the \epm\ center--of--mass frame;
from
\cite{bjork1}.}}
\end{figure}
%****************************************
\clearpage
%\subsection{{Non--perturbative fragmentation}}
%
%In contrast to light--quark hadronization the \fras of heavy quarks is hard
% \cite{bjork1}--\cite{peter1} because the inertia carried by the heavy quarks
% is retained in the heavy--flavour
%hadrons. This can most transparently be proven by analyzing the hadronization
%$Q\,\rightarrow\, (Q\overline{q})\, + \, q$ in the rest frame of the heavy
%quark
%$Q$. In this frame the left--over
\noindent
light--quark $q$ is endowed only with
little energy $\sim m_q^{eff}$ so that, when boosting the particles
back to the \epem~
c.m.\ frame with the $\gamma$ factor $\gamma=\sqrt{s}/2m_Q$, $q$ develops a
light quark jet of energy $E_q\sim \gamma m_q^{eff}\sim\sqrt{s}/2m_Q \times
\mbox{GeV} $. The average [scaled] energy to the heavy hadron
($Q\overline{q}$)
is therefore expected to be
%****************************************
\begin{equation}
\langle z\rangle_{NP}\sim 1-\Lambda_{\mbox{ QCD}}/m_Q
\end{equation}
%****************************************
The ensuing particle distribution is depicted in Fig.\ref{Bj}. Several
detailed forms have
been elaborated to parametrize the shape of the fragmentation functions.
%****************************************
\subsubsection[]{Peterson et al.~form \cite{peter1}}
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The transition amplitude for a fast moving quark $Q$ fragmenting into
$(Q\overline{q})+q$ is proportional to the inverse of the energy transfer
$\Delta E^{-1}$. Denoting by $z$ the fraction of energy ($\approx$ momentum)
retained in the heavy meson, it follows that
%****************************************
\begin{eqnarray*}
\Delta E & = &
\sqrt{m^2_Q+P^2}-\sqrt{m^2_H+(zP)^2} \\
& & \mbox{} - \sqrt{m_q^2+[(1-z)P]^2} \\
& \sim & 1 - \frac{1 }{z }-\frac{\varepsilon_Q }{1-z }
\end{eqnarray*}
%****************************************
\par
}
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\noindent The parameter $\epsilon_Q$ is the squared ratio of the
effective light to the heavy quark mass
%****************************************
$$
\begin{array}{lrcccl}
\ltowidth{\mbox{charm }}{\mbox{bottom }}:
\ & \varepsilon_c & \approx & \frac{m_q^2 }{m_c^2 } & \sim & 0.10 \\[2mm]
\mbox{bottom }:\ & \frac{\varepsilon_b }{\varepsilon_c } & \approx &
\frac{m_c^2 }{m_b^2 } & \sim & \frac{1 }{10 } \\
\end{array}
$$
%****************************************
Including the longitudinal phase space factor,
we derive the following form of the heavy--quark \fras function
%****************************************
\begin{equation}
d_{NP}(z) = \frac{N }{z\left[1-\frac{1 }{z }-\frac{\varepsilon_Q }{1-z }
\right]^2 }
\end{equation}
%****************************************
which is shown in Fig.1.4. The normalization factor $N$ is given by
$4\sqrt{\epsilon_Q}/\pi$ for small $\epsilon_Q$. The average scaled
energy of the heavy meson $=1-\sqrt{\epsilon_Q}$ approaches unity
at the order of $1/m_Q$.
\par}
\flowcap{0}{ {\bf Figure 1.4:} \em Peterson et al.~form of the heavy quark
fragmentation function for charm and bottom. }
\stepcounter{figure}
Even though data is presently insufficient to distinguish between
different forms of the heavy quark \fras functions, it is clearly
established \cite{Roude} that they are hard.
\subsubsection[]{String form \cite{artru1}--\cite{morri1}}
In string fragmentation two different approaches have been elaborated
to describe the non--perturbative heavy--flavor fragmentation. While
in one model the momentum--space aspects are emphasized \cite{ander2},
the other model, which will be discussed in more detail, is based on
the space--time properties of the string, leading to a cluster
picture, yet with sharply peaked mass distributions for heavy--flavor
particles \cite{bowle1,morri1}.
\vspace{\parskip}
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At large distances the force between heavy quark--antiquark pairs approaches
the constant value
%****************************
\begin{equation}\label{gl25}
\sigma = 1\, \mbox{GeV}/\mbox{fm} = (400 \,\mbox{MeV})^2
\end{equation}
%****************************
independent of the distance. This corresponds to a linearly rising potential
between the quarks, as nicely demonstrated in lattice QCD simulations
\cite{campb1}.
The associated flux tube may be approximated by a string \cite{artru1} with the
string tension $\sigma$ given by the value (\ref{gl25}). The mechanical
equation of motion
for the quark $Q$
\begin{equation}
\frac{d}{dt} \frac{m_Q v}{\sqrt{1 - v^2}} = - \sigma
\end{equation}
is solved by the position vector
\begin{equation}
x(t) = \frac{1}{\sigma}\left[ E_0 - \sqrt{(p_0-\sigma t)^2 + m_Q^2}\, \right]
\end{equation}
\par}
%\oldsize=\hsize \skipsize=7.5cm \advance\oldsize by -\skipsize
%\vbox{\parshape=12
%0cm\hsize 0cm\hsize 0cm\hsize
%0cm\hsize 0cm\hsize
%\skipsize\oldsize\skipsize\oldsize \skipsize\oldsize
%\skipsize\oldsize \skipsize\oldsize \skipsize\oldsize
%0cm\hsize
\noindent and the momentum vector
\begin{equation}
p(t) = p_0 - \sigma t
\end{equation}
\noindent
$E_0$, $p_0$ denote the initial values of energy and momentum.
The quarks move on
hyperbolae in the 4--dimensional world, spanning a string
between the two color
sources that sweeps the area between the $Q$ and $\overline{Q}$
branch of the hyperbolae, Fig.~\ref{hyp}.
%}
%******************************
\begin{figure}[t]
\vspace{6cm}
\caption[figur]{\label{hyp}
{\it Motion of the heavy quarks in the 4--dimensional world and the breaking
of the color string.}}
\end{figure}
%******************************
%\vspace{\parskip}
%\vbox{
%\hangindent=7.5cm \hangafter=-10
In analogy to point--particle decays an area law is postulated for the breaking
of the string by the
spontaneous creation of a light--quark pair at the position
$x$. The probability of string breaking after an area $A$ has been swept, is
taken to be
\begin{equation}
dP_{BREAK} = \lambda e^{-\lambda A} dA
\end{equation}
$A$ is defined as the area bounded by the world lines of the two quarks and a
light--like line along which the string break occurs randomly. Relating the
break coordinates $(x,t)$ to the energy and momentum of the daughter string
spanned between $x$ and $Q$
%}
\vspace{-\parskip}
%****************************
\begin{equation}
E = E_0 - \sigma x \qquad \mbox{and} \qquad p = p_0 - \sigma t
\end{equation}
%****************************
the invariant area is found to be
\goodbreak
%****************************
\begin{eqnarray}
A & = & \int^{\xi_-}_0 d\xi_- \left[ \xi_+(Q) - \xi_+(\overline{Q}) \right] \\
& = & \frac{m_Q^2}{2\sigma^2} \left[ \frac{M^2}{m_Q^2} \frac{1}{z} -1 -\log
\left( \frac{M^2}{m_Q^2} \frac{1}{z} \right) \right] \nonumber
\end{eqnarray}
%****************************
in the high--energy limit. $M$ is the mass of the string piece including $Q$.
$\xi_+ = (t \pm x) /\sqrt{2}$ are the light--cone
coordinates and $z$ is the scaled energy $E/E_0$. Elaborating
the mechanism for
finite energies and accounting for repeated
breakings of the daughter strings, the
energy spectrum of the final ($Q\overline{q}$) clusters develops a pronounced
peak close to $z < 1$, and the cluster masses accumulate at the minimum
heavy--flavor
particle mass. The numerical results \cite{morri1}
are shown in Fig.\ref{bow} for charm
and bottom \fras functions in $Z$ decays. A reasonable parametrization is
provided by the simple form
%****************************************
\begin{equation}
D_Q(z)\sim\frac{(1-z)^a }{z^{1+bm_Q^2} }\exp\left[-\frac{bm_Q^2 }{z }
\right]
\end{equation}
%****************************************
with $b=0.8\,\mbox{GeV}^{-2}$ and $a\approx 0.5$, the charm/bottom
quark masses
identified with the masses of the lowest--lying vector meson states.
\begin{figure}\vspace{10cm}
\caption{\label{bow} {\it Heavy quark fragmentation in the Bowler--Morris
string approach.}}
\end{figure}
This form of the string \fras function is surprisingly close to the
Peterson et al.\
form as evident by comparing Fig.1.8 and Fig.\ref{bow}. The spectrum is
considerably softer than the Lund form as a result of the
different factor in front which shifts the maximum of the
distribution to lower $z$ values. The average $z$ value predicted by this
space--time motivated string picture $\langle z\rangle\approx 1-1\mbox{GeV}
/m_Q$ coincides with
Bjorken's conjecture outlined at the beginning of this section. This is not
surprising because in both approaches the hadronic interaction energy between
the heavy and the light quark is provided by the energy of the
string piece or of
the flux tube, independent of the nature of the quarks as gluonic field
sources. The string picture of heavy quark \fras may therefore be considered
as an approximation to (non--perturbative) QCD that comes closest to
intuitive expectations.
The average $b$--hadron energy \cite{Roude} at LEP $\langle z_b
\rangle = 0.705
\pm 0.011 $ can adequately be understood by the convolution of perturbative
$(\langle z\rangle _{PT}=0.8)$
with non--perturbative $(\langle z\rangle _{NP}=0.9)$
fragmentation mechanisms.
\section{Tagging $b$ quarks: QCD tests}
Many of the QCD tests are refined considerably if gluon jets can be
discriminated from quark jets. Most obvious is the measurement of gluon jet
properties: hadron multiplicities, transverse momentum spectra etc. From a
variety of other points, two are particularly important, shedding new light on
the basic couplings in the QCD interaction density and the particle
distributions
in multi--jet events.
\subsection{{$ggg$ Coupling in 4--jet events}}
A key property of QCD is the self--interaction of gluons, a consequence of
the non--zero color charges of the gauge particles. The self--coupling is a
necessary ingredient to overcome the fermionic screening of color sources
and to render the quark--gluon interaction asymptotically free. A variety of
tests for the triple--gluon vertex have been discussed in the past,
comprising deep--inelastic lepton--nucleon scattering, high--transverse
momentum jets in hadron--hadron collisions and heavy quarkonium decays. A
qualitatively new test ground has opened with the huge number of high--energy
4--jet events in $Z$ decays. Sufficiently large invariant masses of all jet
pairs ensure that these jets truly reflect the distribution of quark and gluon
quanta in the femto--universe and thus reveal the basic couplings of the
QCD Lagrangian.
The triple--gluon coupling affects jet distributions only at the 4--parton
level in the Born approximation. Several methods have been proposed to isolate
the $ggg$ diagram \cite{koern1}--\cite{bengt1}.
%*****************************************
\begin{figure}[hbt]\vspace{4.5cm}\end{figure}
%*****************************************
For this purpose, QCD jet distributions are compared with an abelian gauge
theory in which only the two last diagrams are present. Endowing quarks
with three color degrees of freedom and adjusting the quark--gauge boson
coupling as $\alpha_A=\frac{4}{3}\alpha_{QCD}$, the total hadronic $e^+ e^-$
annihilation cross section, the three--jet cross section and all three--jet
distributions
are perfectly well described by such an effective theory. Large discrepancies
between QCD and this model however occur in 4--jet events. Whereas the
dominant
contribution in QCD is due to the triple gluon diagram and the 4--quark final
state is highly suppressed, the different color flow in the abelian model
raises the $q\overline{q}q\overline{q}$ contribution to the same level as the
$q\overline{q}g_A g_A$ bremsstrahl cross section. This induces characteristic
differences in the 4--jet distributions even if gluon jets are not tagged.
Two examples in which LEP data are compared with the predictions
derived from QCD and
the abelian theory, are displayed in Fig.\ref{gggex}.
%****************************
\begin{figure}[t]
\vspace{11cm}
\caption[figur]{\label{gggex} {\it Comparison of 4--jet angular distributions with prediction
from QCD and the abelian theory. Data are from the
L3 \cite{adeva2}
Collaboration at LEP.}}
\end{figure}
%****************************
The comparison proves that the angular distributions are sensitive to the
form of the 3--gluon vertex, its Lorentz and color structure. The strength of
the coupling has been measured by fitting the ratio of the Casimir invariants
\cite{bara}
%****************************
$$N_{C}/C_F = 2.20\pm0.09 \pm 0.13$$
%****************************
to be compared with the value $9/4$ in QCD and $0$ in the abelian toy model.
The direct discrimination of the gluon bremsstrahl diagram from the gluon
splitting diagram can most convincingly be achieved by tagging quark {\em vs.\
}gluon
jets. Besides AI techniques \cite{loenn1}, $b$ quark tagging provides the most
efficient method \cite{bethk1}.
\vspace{0.5cm}
\noindent (i) {\em Gluon alignment \cite{koern1}}
\vspace{2mm}
\vspace{-\parskip}
\nopagebreak
\noindent The alignment of the daughter
gluons in the splitting process
$g\rightarrow gg$ forces
the two planes, each formed by a high--energy jet together with a low--energy
jet, to orient preferentially parallel, with the low--energy jets pointing
into the same hemisphere. Unlike the abelian model where the planes tend to
be perpendicular to each other. This is demonstrated in Fig.\ref{gggth} where
non--zero jet resolutions are properly taken into account.
\vspace{0.5cm}
\noindent (ii) {\em NR polar angle \cite{nacht1}}
\vspace{2mm}
\vspace{-\parskip}
\nopagebreak
\noindent Virtual helicity--zero gluons of low invariant
mass that are radiated from
almost back--to--back high--energy quark--antiquark jets cannot decay
into gluon
jets at $90^o$ relative to the quark axis. Exploiting
the only zero
in the spin--one rotation matrix, the range
of large polar angles is depleted
from
low--energy jets in QCD and the jets
accumulate at small angles. This is just opposite
to virtual gluon decays into quark--antiquark
pairs which cannot be emitted along the
helicity--zero axis in the massless limit of vector
theories. Fig.\ref{gggth}.
\vspace{0.5cm}
\noindent (iii) {\em Azimuthal angle $\chi$ \cite{bengt1}}
\vspace{2mm}
\vspace{-\parskip}
\nopagebreak
\noindent Gluons radiated from quarks and antiquarks in $e^+ e^-
\rightarrow q\overline{q}g
$ are linearly polarized to a high degree {\em in} the $q\overline{q}g$ final
state plane. Denoting the cross sections for polarizations in and
perpendicular to this plane by $d\sigma_{\parallel}$ and $d\sigma_{\perp}$,
respectively, QCD predicts
%*****************************************
\begin{eqnarray}
P(x_g) & = & (d\sigma_{\|} - d\sigma_\bot)/(d\sigma_\| + d \sigma_\bot) \\
& = & 2(1-x_g)/(x_q^2 + x_{\overline{q}}^2){\rm {}} \nonumber
\end{eqnarray}
%*****************************************
where, as usual, $x_i=2E_i/\sqrt{s}$ are the scaled quark and gluon energies
in the laboratory frame. The fragmentation of a linearly polarized gluon into
daughter partons depends on the azimuthal angle $\chi$ between the final
state plane and the polarization vector. The asymmetric term is just opposite
in sign for $gg$ and $q\overline{q}$ decays,
\stepcounter{equation}
%*****************************************f10a 10b
$$
D_{g\rightarrow gg}(z,\chi) = \frac{6}{2\pi} \left\{\frac{(1-z+z^2)^2}
{z(1-z)} + z(1-z)\cos 2 \chi \right\} \eqno{(\arabic{equation}a)}
$$
$$
D_{g\rightarrow q\overline{q}}(z,\chi) = \frac{1}{2\pi} \left\{
{\frac{1}{2}}[z^2+(1-z)^2]-z(1-z)\cos2\chi \right\}
\eqno{(\arabic{equation}b)}
$$
%*****************************************
%****************************
\begin{figure}[p]
\vspace{19cm}
\caption[figure]{\label{gggth} {\it 4--jet angular
distributions in QCD and the abelian model if
quark and gluon jets can be distinguished; Ref.~\cite{bethk1}.}}
\end{figure}
%****************************
\noindent
Quark jets accumulate perpendicular to the polarization vector
with a
maximally pronounced angular distribution
$\sim$$[1-\cos 2\chi]$ for $z=\frac{1}{2}$. The
distribution of
gluon jets is more isotropic, $\sim[1+\frac{1}{9}\cos 2\chi]$ even for
$z=\frac{1}{2}$, so that the angular distribution in QCD is quite distinct
from abelian theories. In
4--jet events of $e^+ e^-$ annihilation we therefore expect the angle between
the plane formed by the two gluon jets
and the plane formed by the high--energy
primordial quark--antiquark jets to be distributed
nearly isotropically in QCD while these planes should be preferentially
perpendicular in abelian theories. This is borne out by a detailed 4--jet
analysis, Fig.\ref{gggth}.
\subsection{{Particle flow in 3--jet events}}
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In the string picture of quark--antiquark hadronization \cite{ander1}
the boost exerted
on the string by the emission of a high--energy gluon, depletes
the angular region between the
quark and antiquark from particles, while accumulating the
particles between the
quarks and the gluon.
This phenomenon \cite{barte1},
however, may also be explained, in an alternative way, as a
perturbative color coherence effect \cite{azimo1}. In QED,
soft $\gamma$ radiation
from a configuration of charged antennae as shown on the left,
is predicted
by Low's theorem:
%*****************************************
\begin{equation}
M_\gamma = M_{\mbox{\scriptsize BORN}} \times F_{\mbox{\scriptsize RAD}} \qquad
{\rm with} \qquad F_{\mbox{\scriptsize RAD}} = \Sigma
\frac{ e_i\vec{n}_i\vec{\varepsilon} }{ \vec{n}_i\vec{n} }
\end{equation}
%*****************************************
where $\vec{n}$ is the $\gamma$ flight direction. Apparently
$F_{RAD}(\vec{n})$ vanishes for $\vec{n}$ midway between $\vec{n}_1$ and
$\vec{n}_2$.
\par
\vspace{1mm}
}
This approach, translated to QCD, predicts many fine details of the particle
flow in multi--jet events. For a given $q\overline{q}g$ jet configuration,
kinematically fixed by tagged heavy quarks, the angular
distribution of the particle flow between the jets can be written as
\cite{azimo1}
%*****************************************f11
\begin{equation}\label{gl36}
\frac{dN^{q\overline{q}g}}{d\vec{n}} \propto [gq] + [g\overline{q}] -
\frac{1}{N_c^2}[q\overline{q}]
\end{equation}
%*****************************************
where
%*****************************************
\[
[ij] = \frac{a_{ij}}{a_ia_j}\qquad
\left\{\begin{array}{rclcrcl}
a_i&=&1-\cos\theta_i&\qquad&\theta_i&=&[\vec{n},\vec{n}_j] \\
a_{ij}&=&1-\cos\theta_{ij}&\qquad&\theta_{ij}&=&[\vec{n}_i,\vec{n}_j]
\end{array} \right.
\]
%*****************************************
The first two terms in (\ref{gl36}) agree with the
celebrated string effect while the last
negative term is the result of interference effects between the radiation
from the $q$, $\overline{q}$ and $g$ antennae, not accounted for by independent
string fragmentation. These interference effects enhance the asymmetry of
particle yields between quark--gluon and quark--antiquark jets, a consequence
of the repulsive force between $q$ and $\overline{q}$ in a colour octet state.
The result can best be illustrated for 3--fold symmetric
$q\overline{q}g$ events
$(\theta_{q\overline{q}}=\theta_{qg}=\theta_{g\overline{q}}=120^o)$.
The ratios of the
multiplicity flows projected onto the $q\overline{q}g$ plane in the $qg$ and
$q\overline{q}$ sectors, are
%*****************************************
\begin{equation}
\frac{N_{qg}}{N_{q\overline{q}}} \simeq \frac{5-1/N_c^2}{2-4/N_c^2} =
\left\{
\begin{array}{lcl}
3.14&\mbox{ }&\mbox{with interference} \\
2.5&\mbox{ }&\mbox{without }[N_c \rightarrow \infty]
\end{array} \right.
\end{equation}
%*****************************************
Thus with interference this ratio is markedly different from the prediction
of non--interacting strings. [In the case of untagged jets the ratio
is considerably reduced due to the smearing which is caused by the
mismatch of quark and gluon jets.]
\section{Inclusive production of $J/\psi$,
$\Upsilon$ and $B_c$}
In addition to the resonant production of spin 1 quarkonia
at low energies these states can also
be produced in $Z$ decays through a variety of processes.
The clean initial state configuration typical for $e^+e^-$ annihilation
is ideal to investigate the relative importance of different
quarkonium
production mechanisms. Two distinctly different situations have been
considered: high energy reactions like $Z$ decays with large event
rates available at LEP and alternatively the 10 GeV region that can be
explored at present at CESR or in the near future at the $B$-meson
factories. Three mechanisms have been identified at which contribute
in the high energy region with comparable rates.
The reaction \cite{keu}
\begin{equation}
Z\to \jpsi c\bar c+X
\end{equation}
requires the production of two $c\bar c$ pairs with a rate
proportional to $\alpha_s^2|R(0)|^2$. The second mechanism
\cite{cit1,keu2}
is the
splitting of a virtual gluon in a color octet $c\bar c$:
\begin{eqnarray}
Z\to q\bar q (c\bar c)_8
\end{eqnarray}
with the subsequent nonperturbative conversion of $(c\bar c)_8$ into
$\jpsi$. The rate for this mechanism is proportional to
$\alpha_s^2\langle{\cal O}^8\rangle$ where the second factor
characterizes the nonperturbative matrix element and must be
considered as an additional free parameter. Rates and distributions
calculated in this approach are fairly close to those expected in the
``color evaporation'' model \cite{branco}.
The third,
color singlet, contribution \cite{Stirling}
\begin{eqnarray}
Z\to q\bar q \jpsi gg
\end{eqnarray}
is strongly suppressed by the factor $\alpha_s^4|R(0)|^2$ and,
furthermore, by the small phase space. The branching ratios of the
three reactions are given by $0.8\cdot 10^{-4}$, $1.9\cdot 10^{-4}$,
$0.2\cdot 10^{-4}$, respectively. The total inclusive rate is
reasonably consistent with the observations by the OPAL collaboration
\cite{OPAL}
of $(1.9\pm 0.7 \pm 0.5 \pm 0.5)\cdot 10^{-4}$. However, a statement
about the dominance of any of these processes seems premature. The
analysis of $\jpsi$ energy and momentum distributions, however, could
help to settle this issue.
Also $B$ meson factories and CESR give rise to a large sample of
events with prompt $\jpsi$ production. Two mechanisms have been
proposed which might well describe complementary kinematical regions.
The leading process in the CSM
\begin{eqnarray}
e^+ e^- \to \jpsi +gg
\end{eqnarray}
is proportional to $\alpha_s^2|R(0)|^2$. It leads to a three body
final state and hence to a continuous energy distribution.
Predictions for the rate, the angular and the momentum distribution
and the polarization can be found in \cite{Driesen}. The alternative
approach \cite{bra} is based on ``color octet production'', $e^+e^-\to (c\bar
c)_8+g$. The rate is of order \as and multiplied by a
nonperturbative matrix
element. The $\jpsi$ energy is essentially fixed at
$E_{max}=(s+m^2_\psi)/(2\sqrt{s})$. The angular distribution is
proportional to $(1+\cos^2\theta)$. These features are identical to
the predictions of the ``color evaporation model'' \cite{Fritsch}
formulated a long time ago. An excess of $\jpsi$ at this special
kinematical point with the predicted angular distribution would be a
strong indication for this ``octet mechanism''. The angular
distribution of the $\jpsi$ in the CSM is of the form $1+\alpha(y)
\cos^2\theta$ where $\alpha(y)$ depends on $y\equiv E_\psi /E_{Beam}$
and approaches roughly $-0.8$ at the endpoint.
This difference will be crucial in disentangling the two mechanisms.
Charmed bottom mesons $B_c$ can be produced at an appreciable
rate only through $\left( \bar b c \right) \rightarrow B_c$
fusion in perturbatively generated $Z\to b\overline{b}, c\overline{c}$
final states. [Spontaneous nonperturbative $c\bar c$ pair
creation in a gluonic flux tube is suppressed at a level of
$10^{-11}$]. Folding the 4-quark production amplitude with the
$b\bar c$ quark-antiquark bound-state wave function, a branching
ratio for $Z \rightarrow B_c$
of order $10^{-4}$ to $10^{-5}$ has been estimated in
\cite{Amiri,Bra,Kis} so that
order 100 to 1,000 $B_c$ are expected in $10^7~Z$ decays. The final
state has a remarkable signature: an isolated $B_c$ meson plus
2 jets. The lifetime of the $B_c$ meson
is determined by the combination of charmed and bottom quark
lifetimes. Decay modes like $B_c\to J/\psi + \pi^-$ or $J/\psi+l\bar
\nu$ with branching ratio around $1\%$ \cite{Leike2} have allowed to
set nontrivial bounds on this channel \cite{DelNT}.
\newpage
\section{Lifetimes and mixing of $b$--flavor hadrons}
The measurement of lifetimes and mixing of hadrons built--up by $b$ quarks is
made possible by the large time dilation factor $\gamma$ in high energy
machines, leading to flight distances of macroscopic size.
\subsection{{Lifetimes of $b$--particles}}
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Hadrons with $b$ flavor are known to have a long lifetime
(see e.g. \cite{Shepherd,Ratoff,Richman}).
This is the consequence of the small Kobayashi--Maskawa coupling between
$b$ and $c$ quarks that neutralizes the large mass $m_b$ in the Fermi decay
of the $b$ quark. Because the \fras of the $b$ quarks to mesons and baryons
is hard, they have a large $\gamma$ factor $\leq$ 8 when they are produced in
$Z$ decays. The average flight distance is
about 2~mm, the average impact parameter
in the transverse plane is about
0.2 mm,
so that the particles can well be analyzed in high--resolution vertex
detectors. Three methods are employed to measure the lifetimes: (i)~Average
lifetimes can be determined from the impact parameter $\delta$ of the charged
lepton
in semileptonic decays. At LEP this parameter is
insensitive to the momentum of the decaying particle and directly
proportional to $\tau$ $[\delta\sim c\tau]$.
(ii) Exclusive $B$ lifetime
measurements can be carried out by reconstructing fully hadronic
decays. This method is ideal for hadron collider experiments with the
$J/\psi K$ or $J/\psi K^*$
channels as prominent examples. However, it suffers from
low statistics in $Z$ decays. (iii) Partial reconstruction of $B$
mesons, for example in semileptonic decays to $D$ or $D^*$, allows at
the same time for the separation between $B^0$ and $B^-$ and the
identification of $B$ meson decay vertices.
\par}
With the advent of sophisticated microvertex detectors the exponential
decay distribution of $B$ mesons has been nicely established
(Fig.~\ref{fig:DELPHI}).
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Unlike charm, the \underline{spectator model} is expected to provide a valid
approximation for $b$--particle decays.
In this model the lifetime is determined by the $b$--quark, irrespective of
the other quarks in the hadronic environment.
As a result, the lifetimes of $b$--particles [except $B_c$, see later] are
approximately universal. The model includes perturbative QCD corrections of
the $b$ Fermi decay and phase space effects due to non--zero quark masses
which -- because being not well defined in the present context -- introduce
some uncertainties.
These have been reduced during the past years through the evaluation
of higher order perturbative corrections and the machinery of Heavy
Quark Effective Theory. From the decay rate
the dominating {\em CKM} matrix element can be extracted.
Note that this matrix element, though coupling quarks of two neighboring
families, is of the order of the Cabibbo angle squared.
%\par}
\begin{figure}[htbp]
\begin{center}
\leavevmode
\vspace*{5cm}
\end{center}
\caption{\it Plot of excess decay times showing an exponential
distribution (from \protect\cite{Shepherd}).}
\label{fig:DELPHI}
\end{figure}
\lu{Non--spectator diagrams} render the lifetimes individually
different within a small margin. This is illustrated for the most prominent
examples \cite{bosma1,NeubertStech}:
\vspace{-0.7cm}
%****************************
\begin{center}
\renewcommand{\arraystretch}{1}
$\begin{array}{rclcl}
B^- & = & (b\overline{u})& : & \mbox{destructive interference between
$\overline{u}'s$,
caused by the Pauli}\\
& & & &
\mbox{principle in the spectator diagram, prolonging
the
lifetime;}\\
\overline{B}_d^0 & = & (\overline{b}d)& :& \mbox{W exchange diagram }
\overline{b}d\rightarrow\overline{c}u, \mbox{ shortening the lifetime;}\\
\overline{B}_s^0 & = & (\overline{b}s)& :& \mbox{W exchange } \overline{b}s
\rightarrow \overline{c}u, \mbox{ shortening the lifetime;}\\
\Lambda_b & = & (bud) & : & \mbox{W exchange }bc\rightarrow ud,
\mbox{ shortening the lifetime; but overwhelming}\\
& & & &
\mbox{$dd$ Pauli suppression;}
\end{array}$
\renewcommand{\arraystretch}{1.4}
\end{center}
\vspace{-0.3cm}
%****************************
\noindent The differences in the hierarchy
%****************************************
\begin{equation}
\tau(B^-) > \tau(\overline{B }^0_s) \ge \tau(\overline{B }^0_d) >
\tau(\Lambda_b)
\end{equation}
%****************************************
have conservatively been estimated to be at most at
a level of 10\%.
The most recent experimental results for $B$ mesons \cite{Shepherd,Richman}
\begin{eqnarray}
\tau(B^-) &=& 1.65\pm 0.04 \mbox{ ps}, \nonumber \\
\tau(B^0) &=& 1.55\pm 0.04 \mbox{ ps}, \nonumber \\
\tau(B_s) &=& 1.52\pm 0.07 \mbox{ ps}; \nonumber \\
\tau(B^-)/\tau(B^0) &=& 1.04\pm 0.04, \qquad
\tau(B_s)/\tau(B^0) = 0.98\pm 0.05,
\end{eqnarray}
are well consistent with these expectations.
The present measurement of $\tau(\Lambda_b)$ \cite{Ratoff},
however, with
\begin{eqnarray}
\tau(\Lambda_b)=1.21\pm 0.06 \mbox{ ps}, \qquad
\tau(\Lambda_b)/\tau(B^0) = 0.78\pm 0.04, \qquad
\end{eqnarray}
is difficult to accomodate in calculations based on HQET
\cite{NeubertStech}.
\subsection{{$B-\overline{B}$ mixing}}
\label{sec:mix}
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Oscillations between neutral $B_d^0$, $B_s^0$ particles and antiparticles are
induced by higher order weak diagrams as shown below. A $B^0$ beam at $t = 0$
develops at time $t$ a $\overline{B}^0$ component and vice versa
\cite{BurasFleischer},
%****************************************
\begin{eqnarray}
B^0 & \!\!\!\! \to \!\!\!\! & \left[
\cos\left( {\Delta m t\over 2} \right) B^0
+ {iq\over p} \sin\left( {\Delta m t\over 2} \right)
\overline{B}^0
\right]
\nonumber
\\
\overline{B }^0 & \!\!\!\! \to \!\!\!\! &
\left[
{ip\over q} \sin\left( {\Delta m t\over 2} \right) B^0
+
\cos\left( {\Delta m t\over 2} \right)
\overline{B}^0
\right]
\nonumber \\
&& {}
\end{eqnarray}
%****************************************
[$\Delta \Gamma\equiv \Gamma_H-\Gamma_L=0$ and hence $|p/q|=1$
has been assumed for the moment].}
Defining the average width
of the long-- and short--lived state by
$\Gamma=\frac{1}{2}(\Gamma_H+\Gamma_L)$, the oscillatory behavior is governed
by the ratio of the oscillation frequency to the decay rate,
%\par
%\vspace{-\parskip}
%****************************************
\begin{equation}
x=\frac{\Delta m}{\Gamma}
\end{equation}
%****************************************
with $\Delta m = m_H-m_L$ being the mass difference of the weak eigenstates.
The strength of the oscillation amplitude is maximal for vanishing
%****************************************
\[
y=\frac{\triangle \Gamma }{2\Gamma },
\]
%****************************************
a valid approximation for the $B_d^0$ system. The particle/antiparticle
components can be detected by recording, for example, the lepton charges in
semileptonic $B^0\rightarrow l^+$ and $\overline{B}^0\rightarrow
l^-$ decays. [This applies clearly to $B_d^0$; for $B_s^0$ the
assumption $\Delta \Gamma\ll \Gamma$ has been disputed \cite{Datta}
and
a more general analysis would then be required.]
%****************************************
\begin{eqnarray}
B^0\rightarrow
\left\{
\begin{array}{ll}
N[B^0\stackrel{t}{\rightarrow} l^+]
& \sim e^{-\Gamma t}[1+\cos\Delta m t] \\
N[\overline{B}^0\stackrel{t}{\rightarrow} l^-]
& \sim e^{-\Gamma t}[1-\cos\Delta mt]
\end{array}\right.
\end{eqnarray}
%****************************************
Integrated over all times, the mixing probability follows from
%****************************************
\begin{eqnarray}
\chi & = & \frac{P\left\{B^0\rightarrow \overline{B }^0 \right\} }{P
\left\{B^0\rightarrow B^0 \right\}+P\left\{B^0\rightarrow \overline{B }^0
\right\} }=\frac{1}{2}\frac{x^2+y^2 }{x^2+1} \\
& \approx &\frac{1}{2}\frac{x^2 }{x^2+1}\qquad
\mbox{for }B^0_d\mbox{ and presumably also }B^0_s. \nonumber
\end{eqnarray}
%****************************************
The oscillation frequency in the $B_d$ system,
%****************************************
\begin{equation}
\Delta m \sim |V_{td}|^2m^2_tf^2_{B_d} \eta_B
\end{equation}
%****************************************
is determined by three factors: (i) the {\em CKM}
coupling of $t$ quarks to $d$ quarks [the
{\em CKM} element $(tb)$ is very close to unity]; (ii) the
top quark mass which makes the $t$ exchange dominating over $u$, $c$ exchange
contributions (note that the CKM elements are of order $\lambda^3$ for
all three exchange amplitudes); (iii)
the quark wave function of the meson at the origin; and (iv)
perturbative QCD corrections.
The ratio
$y_d$ on the other hand is expected to be small since the dominant decay
channels are not self--conjugate.
The mixing parameter $x_s$ can be estimated in the Standard Model by
comparing the box diagrams for $B_s^0$ with $B_d^0$ transitions:
%****************************************
\begin{equation}
x_s\approx x_d\left|\frac{V_{ts} }{V_{td} } \right|^2
{\tau(B_d)\over \tau(B_c)}
{m(B_s)\over m(B_d)}\xi^2
\approx
x_d \left|\frac{V_{ts} }{V_{td} } \right|^2
\end{equation}
with the non-perturbative parameter $\xi$ which parametrizes the
deviation from SU(3) symmetry.
%****************************************
In the Wolfenstein expansion of the Cabibbo--Kobayashi--Maskawa matrix
%****************************************
\[
V=\left\|\begin{array}{ccc}
V_{ud} &V_{uc} &V_{ub} \\
V_{cd} &V_{cs} &V_{cb} \\
V_{td} &V_{ts} &V_{tb}
\end{array}\right\|
=\left\|
\begin{array}{ccc}
1-\frac{\lambda^2}{2} & \lambda & A\lambda^3[\rho-i\eta] \\
-\lambda & 1-\frac{\lambda^2}{2} & A\lambda^2 \\
A\lambda^3[1-\rho-i\eta] & -A\lambda^2 & 1
\end{array}
\right\|
+{\cal O}(\lambda^4)
\]
%****************************************
[with $\lambda \approx 0.22$ being the Cabibbo angle] the ratio
%****************************************
\begin{equation}
\left|\frac{V_{ts} }{V_{td} } \right|^2=\frac{1 }{\lambda^2
\left[(1-\rho)^2+\eta^2 \right] }
\end{equation}
%****************************************
is constrained from an analysis \cite{BurasFleischer}
of the unitarity triangle to the range
between 10 and 40. For SU(3)
symmetric quark wave functions, the Standard Model brackets thus
$x_s$ to
%****************************************
\[
7.39.2$ ps$^{-1}$ leads to interesting constrains on
quark mixing already now.
\vspace{2mm}
While the observation of $B_d^0 - \overline{B}_d^0$ mixing
by ARGUS gave the first indication of a very heavy top quark in
the past, the time dependence of $B_s-\overline{B}_s$ oscillations
will serve in the future to determine
the {\em CKM} matrix elements $V_{td}$ and $V_{ts}$. With the mass
parameter $m_t$ measured experimentally, two cases can be
realized. If the presently unknown decay constant $f_{B_d}$ could
be calculated reliably or measured in $B \rightarrow \tau
\nu_\tau$ decays, the mixing parameter $x_d$ determines the {\em CKM}
element $|V_{td}|$. If $f_{B}$ cannot be determined reliably,
this {\em CKM} element must be extracted from the ratio $x_s/x_d$ which
depends less strongly on the mesonic wave functions. Since $|V_{ts}|$
is fairly close to $|V_{cb}|$ and can therefore be measured in $B$
decays,
the {\em CKM}
matrix element $|V_{td}|$ can be deduced from the comparison of $B_d^0$ and
$B_s^0$ mixing data.